Solutions - University of North Texas

UCRL-52863 Rev. 1 Distribution Category UC-38

UCRL—52863-Rev.l DE87 012387

Conduction Heat Transfer Solutions

James H. VanSant

Manuscript date: August 1983

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees.^ makes, any warranty, express ox impUed, <w assumes any legal liability os responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, iecom-mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United Slates Government or any agency thereof.

LAWRENCE LIVERMORE NATIONAL LABORATORY University of California • Livermore, California • 94550 I, Available from: National Technical Information Service • U.S. Department of Commerce 5285 Port Roval Road • Springfield. VA 22161 • Ml.50 per copy • (Microfiche M.50 )

CONTENTS

Preface v

Nomenclature vxi

Introduction 1 Steady-State Solut ions

1. Plane Surface - Steady Sta te 1.1 Solids Bounded by Plane Surfaces 1-1 1.2 Solids Bounded by Plane Surfaces

—With Internal Heating 1-27 2. Cyl indr ica l Surface - Steady Sta te

2.1 Solids Bounded by Cyl indr ica l Surfaces —No Internal Heating 2 - i

2.2 Solids Bounded by Cyl indr ica l Surfaces —With Internal Heating 2-33

3. Spherical Surface - Steady S ta te 3.1 Solids Bounded by Spherical Surfaces

—No Internal Heating 3-1 3.2 Solids Bounded by Spherical Surfaces

—With Internal Heating 3-10 4. Traveling Heat Sources

4.1 Traveling Heat Sources 4-1 5. Extended Surface - Steady S ta te

5.1 Extended Surfaces—No In te rna l Heating 5-1 5.2 Extended Surfaces—With In te rna l Heating 5-31

Transient Solutions 6. I n f i n i t e Solids - Transient

6.1 I n f i n i t e Solids—No I n t e r n a l Heating 6-1 6.2 I n f i n i t e Solids—With In t e rna l Heating 6-22

7. Semi-Inf ini te Solids - Transient 7.1 Semi-Infinite Solids—No In te rna l Heating 7-1 7.2 - i m i - I n f i n i t e Solids—With In te rna l Heating . . . . 7-22

8. Plane Surface - Transient 8.1 Solids Bounded by Plane Surfaces

—No In te rna l Heating 8-1 8.2 Solids Bounded by Plane Surfaces

—With In ternal Heating 8-52

9. Cyl indr ica l Surface - Transient 9.1 Solids Bounded by Cyl indr ica l Surfaces

—No Internal Heating 9-1 9.2 Solids Bounded by Cyl indr ica l Surfaces

—With Internal Heating 9-24 10. Spherical Surface - Transient

10.1 Solids Bounded by Spherical Surfaces —No Internal Heating 10-1

10.2 Solids Bounded by Spherical Surfaces —With Internal Heating 10-19

11. Change of Phase 11.1 Change of Phase—Plane Interface 11-1 11.2 Change of Phase—Nonplanar Interface 11-13

12. Traveling Boundaries 12.1 Traveling Boundaries 12-1

Figures and Tables for Solutions F-l Miscellaneous Data

13. Mathematical Functions 13-1 14. Roots of Some Charac te r i s t i c Equations 14-1 15. Constants and Conversion Factors 15-1 16. Convection Coefficients 16-1 17. Contact Coefficients 17-1 18. Thermal Propert ies 18-1

References R-l

iv

PREFACE

This text is a collection of solutions to a variety of heat conduction problems found in numerous publications, such as textbooks, handbooks, journals, reports, etc. Its purpose is to assemble these solutions into one source that can facilitate the search for a particular problem solution. Generally, it is intended to be a handbook on the subject of heat conduction.

Engineers, scientists, technologists, and designers of all disciplines should find this material useful, particularly those who design thermal systems or estimate temperatures and heat transfer rates in structures. More than 500 problem solutions and relevant data are tabulated for easy retrieval. Having this kind of material available can save time and effort in reaching design decisions.

There are twelve sections of solutions which correspond with the class of problems found in each. Geometry, state, boundary conditions, and other categories are used to classify the problems. A case number is assigned to each problem for cross-referencing, and also for future reference. Each problem is concisely described by geometry and condition statements, and many times a descriptive sketch is also included. At least one source reference is given so that the user can review the methods used to derive the solutions. Problem solutions are given in the form of equations, graphs, and tables of data, all of which are also identified by problem case numbers and source references.

The introduction presents a synopsis on the theory, differential equations, and boundary conditions for conduction heat transfer. Some discussion is given on the use and interpretation of solutions. Also, some example problem solutions are included. This material may give the user a review, or even some insight, on the phenomenology of heat conduction and its applicability to specific problems.

Supplementary data such as mathematical functions, convection correlations, and thermal properties are included for aiding the user in computing numerical values from the solutions. Property data were taken from some of the latest publications relating to the particular properties listed. Only the international system of units (SI) is used.

v

Consistency in nomenclature and terminology is used throughout, making this text more readable than a collection of different references. Also, dimension-less parameters are frequently used to generalize the applicability of the solutions and to permit easier evaluation of the effects of problem conditions.

Even though some of the equational solutions are lengthy and include several different mathematical functions, this should not pose a formidable task for most users. Modern computers can make complicated calculations easy to perform. Even many electronic calculators can be used to compute complex functions. If, however, these tools are not available, one can resort to hand computing methods. The table of mathematical functions and constants would be useful in Chat case.

Heat conduction has been studied extensively, and the number of published solutions is large. In fact, there are many solutions that are not included in this text. For example, some solutions are found by a specific computational process that cannot be described briefly. Moreover, new solutions are constantly appearing in technical journals and reports. Nevertheless, this collection contains most of the published solutions.

The differential equations and boundary-condition equations for heat flow are identical in form to those for other phenomena such as electrical fields, fluid flow, and mass diffusion. This similarity gives additional utility to the heat conduction solutions. The user needs only identify equivalence of conditions and terms when selecting a proper solution. This practice is prescribed in many texts on applied mathematics, electrical theory, heat transfer, and mass transfer.

A search for particular solutions has frequently been a tedious and difficult task. Too often, countless hours have been spent in searching for a problem solution. Locating and obtaining a proper reference can require considerable effort. Also, it is frequently necessary to study a theoretical development in order to find the applicable solution. In so doing, thsre are sometimes misinterpretations which lead to erroneous results. This text should help alleviate some of these problems.

Science gives us information for reaching new frontiers in technology. It is, thus, appropriate to give something back. I hope this text is at least a small contribution.

James H. VanSant

vi

NOMENCLATURE

2 A = Area, m b = Time constant, s c = Specific heat, J/kg' C C = Circumference, m d, D = Diameter, depth, m

, 2 o h = Heat transfer coefficient, W/m * C k = Thermal conductivity, W/m- C

-1 m = V hC/kA, m I, L = Length, m

2 q = Heat flux rate, W/m q = Heat flux in x, y, z directions, W/m' 2

3 q"' = Volumetric heating rate, W/m' Q = Heat transfer rate, W r, R = Radius, m

o t, T = Temperature, C, K V = Velocity, m/s w = Width, m

x , y , z , = C a r t e s i a n c o o r d i n a t e s , m 2

a = Thermal d i f f u s i v i t y , k / p c , m / s

S = Temperature c o e f f i c i e n t , c

Y = Heat of e v a p o r a t i o n , J /kg

y = La ten t hea t of fus ion , J/kg

A = Di f f e r ence e = E m i s s i v i t y fo r thermal rad ia t ion

„ ,_. (actual heat transferred) H = Fin e f f e c t i v e n e s s , -*r—r—r 2 3—TTC—T~ci—T

(heat t rans ferred without f i n s ) (actual heat transferred <)> = F in e f f e c t i v e n e s s ,

3 p = D e n s i t y , kg/m

a = 2Vox, m —2 —4 a = Stefan-Boltzraann constant , W/m "K

T = Time, s

dT = Radiation configuration—emissivity factor

(he t transferred from infinite conductivity fins)

vii

DIMENSIONLESS GROUPS

B i j , Bi = B io t modulus = h8,/k

Bf „ , Bf = Bi Fo = hi/pcS, 2 Fo„ , Fo = F o u r i e r modulus = CIT/S.

Fo „ , Fo = Modif ied Four i e r modulus = 1/(2 V Fo) 3 2 Gr„ , Gr = Grashof number = gBAt£ /V

Ki„ , Ki = K i r p i c h e v number = qJl/kAt

Nu, , Nu = N u s s e l t number = hd/k d 2

Pd„, Pd = Predvoditelev modulus = hi. /a Po. , Po = Pomerantsev modulus = q ' ' ' J . /kAt Pr = Prandtl number = v/a R = Radius r a t i o = r / r

o Re,, Re = Reynolds number = vd/v X = Length ratio = x/l

Y = Width ratio = y/w Z = Length ratio = z/2. MATHEMATICAL FUNCTIONS

exp = Exponential function Ei = Exponential integral erf = Error function erfc = Complementary error function i erfc = Complementary error function integral I = Modified Bessel function of the first kind n J = Bessel function of the first kind n K = Modified Bessel function of the second kind n &n = Natural log Y = Bessel function of the second kind n P = Legendre polynomial of the first kind T = Gamma function

viii

INTRODUCTION

1. HEAT CONDUCTION

Energy in the form of heat has been used by man ever since he began walking on th i s ea r th . Moreover, the t ransfer of heat is e s s e n t i a l to our very ex is tence . Not only do our own physiological functions require some form of heat t ransfer , but so do most l i f e - sus ta in ing processes of nature and many man-contro l led a c t i v i t i e s . The importance of the thermal sciences in the t o t a l sphere of science can, thus , hardly be disputed.

Conduction i s one of the three p r inc ipa l heat transfer modes, the others being convection and rad ia t ion . I t is customarily dist inguished as being an energy diffusion process in materials which do not contain molecular convection. Kinet ic energy i s exchanged between molecules resul t ing in a net t ransfer between regions of d i f fe ren t energy l e v e l s , these energy l eve l s are commonly ca l led temperature. Pa r t i cu l a r ly , heat conduction in metals i s mainly a t t r i buted to the motion of free electrons and in so l id e l e c t r i c a l insu la to rs to the longi tudinal o sc i l l a t i ons of atoms. In f l u i d s , the e l a s t i c impact of molecules i s considered as the heat conduction process .

The process of heat transfer in mater ia l s has been studied for many cen tu r i e s . Even ear ly Greek philosophers, such as Lucretius (c. 98-55 B.C.), meditated on the subject and recorded the i r conclusions. Much l a t e r , the famous mathematical phys i c i s t , Joseph B. J . Fourier (1768-1830), developed a mathematical expression tha t became the bas i s of p rac t i ca l ly a l l heat conduct ion so lu t ions . He postulated that a loca l heat flux ra te in a mater ia l i s proport ional to the loca l temperature gradient in the d i rec t ion of heat flow:

where a i s the heat flow in the x-di rect ion per unit area as i l l u s t r a t e d in F ig . l a . Material p roper t ies are accounted for by including a propor t ional i ty constant :

V - k S ' ( 2 )

1

t ,

J 3t * q x

t

• 3x .. X

Q

(a)

*«-Ax-» x (b)

FIG. 1. Illustration of heat flow and temperature gradient.

where the constant k is called thermal conductivity. (The minus sign must be included to satisfy the second law of thermodynamics.) This equation is called Fourier's law for heterogeneous isotropic continua.

A simpler form of Fourier's law is for homogeneous isotropic continua. For example, consider a plate of this type material having isothermal surfaces and insulated edges as shown in Fig. 1. The plate has a width Ax, surface area A, and thermal conductivity k. The heat flow in the plate is expressed as

<t. -kA V

Ax (3)

This expression becomes Eq. (2) when Ax diminishes to zero.

fx lim At _ . a t A " \ " ~ K Ax+0 Ax " " K 3x

(4)

The heat flux q i s presumed to have both magnitude and d i r e c t i o n . Thus,

i t can be given as a vec tor , q which i s normal to an isothermal sur face . For

example, in Cartesian coordinates

q = V + V + q z k (5)

2

where i , j , k are unit vectors in une x- , y - r and z -d i rec t ions , respec t ive ly . Since Eq. (4) defines q x = -k3t /3x, and s imi la r ly q y = -k3t/3y, q z = -k3t /3z , we can s t a t e

q = -k (i3t/3x + j3 t /3y + k3t/3z) (6)

q = -kVt . (7)

In anisotropic continua the direction of the heat flux vector is not necessarily normal to an isothermal surface. Example materials are crystals, laminates, and oriented fiber composites. In such materials we may assume each component of the heat flux vector to be linearly dependent on all components of the temperature gradient at a point. The vector form of Fourier's law for heterogeneous anisotropic continua becomes

i = -K • Vt , (8)

where K is the conductivity tensor; the components of th i s tensor are called the conductivity coe f f i c i en t s .

In Cartesian form, Eq. (8) i s

"^11 3x + k ^ + k ^ K12 3y K13 3z,

S = "(k21 M + k22 U + k23 ft) ( 9 )

_ _/. at at 3 t \ q z ~ ^31 3x + k 32 3y + k 33 3z / '

To compute heat flow by Fourier's law, a thermal conductivity value i s needed. It can be estimated from theoretical predictions for some ideal materials, but mostly, i t i s determined by measurement and Fourier's law, Eq. (2) . As i l lustrated in Fig. 2, thermal conductivity can have a large range, which depends on materials and temperature. For example, copper at 20 K has a thermal conductivity of approximately 1000 W/m'K and diatomaceous earth at 200 K has a conductivity of 0.05 W/m*K. Consequently, heat flow in materials can have a very large range, depending on a combined e f f ec t of temperature gradient and material property.

Thermal properties of some selected materials are given in Section 18.

3

"i 1 r -] r

0.01' I i L 200 400

-Teflon

Copper

Lead Iron

. Fused quaru

• Diatomaceous earth

±

•

J i_ 600 800 Temperature — K

1000 1200 1400

FIG. 2. Thermal conductivity of some selected solids.

4

2. DIFFERENTIAL EQUATIONS OF HEAT CONDUCTION

Solutions to heat conduction problems are usually found by some mathematical technique which begins with a differential equation of the temperature field. The appropriate equation should include all energy sources and sinks pertinent to a particular problem. Also, the equation should be expressed in terms of a convenient coordinate system such as rectangular, cylindrical, or spherical. Then analytical or differencing methods can be used to solve for temperature or heat flow.

A common method for deriving the generalized differential equation for heat conduction is to apply the first law of thermodynamics (conservation of energy) to a volume element in a selected coordinate system. By accounting for all the thermal energy transferred through the element faces, the change of internal energy and thermal sources or sinks in the element, and by letting the element dimensions approach zero, the differential equation can be derived. This procedure is typified by the following heat energy accounting of the rectangular solid element shown in Fig. 3. The net heat flow though its six faces is

net Q . + Q *x+Ax *y Q . + Q *y+Ay z •^z+Az (10)

FIG. 3. Coordinate systems for heat conduction equations: (a) rectangular, (b) cylindrical, (c) spherical.

5

where typ ica l ly

Q = -AyAz x ML

Q =-AxAZ(k g ) ,

Q = -AxAy

and k , k , and k are d i r ec t i ona l conduct iv i t i es . x' y ' z An increase in in te rna l energy of the element is tepresented by

AI = AxAyAzpc | ^ , (11)

where t is the mean temperature of the element, p i s the mater ia l densi ty , and c i s i t s spec i f i c heat.

In te rna l energy sources can be expressed as

Q U I _ q i i i A x A y A z f (12)

where q " ' i s the unit volume source r a t e . Examples of i n t e rna l heating in mater ia ls are j o u l e , nuclear, or r ad ia t ion heating. Summing these energies in accordance with the energy conservation law yields

AI-W"'. ^ ^ ^ + ! ^y^ + ! ^ ^ + * ' " - C13,

For the l imi t s Ax, Ay, Az + 0, we obta in

at ,,, B qx ! i 3 q «

6

where

*x = " k x 8 i >

q y ~ " k y 3y '

q = -k •5— . ^Z Z dz

Using Bq. (14) as a general d i f fe ren t ia l equation, we can derive the following speci f ic equations.

2.1 Rectangular Coordinate System

For i so t ropic heterogeneous media

„ at a /.. at\ .3 /,. 3t\ . 3 A, at\ p c 3? " 3x" \ k 37) + 37 \ k 3y"j + 37 [k 37j + q" (15)

For i so t rop ic homogeneous media t h i s becomes

JLt _ k_ 3T ~ pc

3 2t + 32t + 3 2t 3x 2 3y2 3z 2

J 1 * * 2 + - 1 T — - aV t + pc pc (16)

When q " ' = 0 , Eq. (16) becomes Four ie r ' s equation. In s teady-s ta te conditions, 3t/3x = 0 and Eq. (16) becomes the Poisson

equation. When q " ' = 3t/3x • 0, Eq. (16) reduces to the Laplace equation. Nonisotropic mater ials , such as laminates, can have direct ional ly

sensi t ive p roper t i es . For such mater ia ls the conduction di f ferent ia l equation in two dimensions i s expressed in the following form:

pc | | = ( k c cos 2B + k n s i n 2 a) ^ - f + (k ? s in 2 B + k.) cos 2 B ) ~

2 (17)

7

FIG. 4. Coordinate system for a nonisotropic medium.

where k r and k are di rec t ional thermal conduc t iv i t i e s , and B is the angle of laminations as indicated in Fig. 4. When the geometrical axes of the nonisotropic material are oriented with the p r inc ipa l axes of the thermal conduc t iv i t i e s , then Eq. (17) s impl i f ies to the form of Eq. (14)

p c f i = k $ ! | + k & + q"> 3T X . . .2 y g y 2 ax'

(18)

2.2 Cyl indr ica l Coordinate System

Rectangular coordinates can be transformed in to cyl indr ica l coordinates by the r e l a t i ons x = r cos 9, y = r sin 9, and z = z. The p a r t i a l d i f f e r e n t i a l equations (15) and (16) transformed to cy l ind r i ca l coordinates are thus

at . i a_ frk at\ i s_ / at\ a_/ at\ , 3x r 3r \ I K 3r ) r 89 \* 38 ^ + 3z \ K Zz ) + q (19)

2 t „ a / l ! t i a t + j L 3 ! t + l ! t \ a i i i * W r 8 r r 2 3 9 2 3 z 2 / ' C

(20)

For nonisotropic materials with the conduccivity and geometry axes aligned as in Eq. (18) the d i f fe ren t ia l equation i s

3t r 3 / 3 t \ ^ k 8 3 t ^ , 3 t p c aT = T a7 r 17 + T 73 + k

z 77 + q

r 38 3z

(21)

8

2.3 Spherical Coordinate System

A transformation from rectangular to spherical coordinates can be accomplished by substituting the relations x = r sin \\> cos if), y = r sin I)J sin $ and z = cos \|) into Eqs. (15) and (16), which yield the partial differential equations for isotropic heterogeneous and homogeneous materials, respectively.

3t 1 3 / 2 . 3 t \ x 1 3 / . 3 t \

1 3 2 • i r s i n i|)

h ( k s i n * f t ) +<J"" <22>

at _„/i!t 2 3t i aft i it\ + al!A ( 2 3 )

\ 3 r r sin i|) 36 r tan \\i T /

The d i f f e r e n t i a l equation for nonisotropic mater ia ls with aligned conductivity and geometric axes is

„ 3t k r 3 2 t , *& 8 2 t , *S|) 3 , l h 3t . „ , , , , , . , p c a 7 = Trr + 2 : 2 , 7 7 + 2 . . 1+ 8 l n * i * + * • ( 2 4 )

3r r s in ip 3ij) r s in V

3 . SPECIAL DIFFERENTIAL EQUATIONS

Some defining equations can have implied assumptions and boundary condit ions. They are usually employed in special cases to simplify the method of solut ion. However, Four ie r ' s law i s the basis for der iving these special equations.

3.1 Combined Conduction-Convection

Thin ma te r i a l s having r e l a t i ve ly high thermal conduct ivi ty have very small l a t e r a l temperature gradients . If the surfaces are convectively heated or cooled, the convection condition becomes part of a heat accounting on a d i f f e ren t i a l element. Referring to Fig. 3 , l e t Az be the thickness b of a th in so l id . On the surface z and z + Az, the so l id has a convection boundary described by q = h (t - t ) , where t . i s the convection f l u i d temperature.

Applying the same pr inciples used to develop the general equations for rectangular coordinate systems should r e s u l t in

p c f f + 2 ^ t - t f , = q " - + f c ( k f i ) + f 7 ( k g ) . ,25,

If the geometry i s a thin rod of circumference C, the appropriate equation is

pof7 + i r (t " t £) - q ' " + t b ( k f f ) - (26) at j c 3T A

3.2 Moving heat sources

The general heat conduction Eq. (15) can also be used for moving heat sources, but a simpler quas i -s teady-s ta te equation can be derived by coordinate transformation. If the coordinates are r e l a t i v e to the t ravel ing source, the temperature d i s t r ibu t ions appear to be s t a t i ona ry . For example, if a point source of strength Q i s moving a t a veloci ty U p a r a l l e l to the x-axis , the transformation would be x = x ' + UT, where x' i s the x-direct ion distance from the source. By s u b s t i t u t i o n in Eq. (16) we can obtain

fc-("fe-)*fe(-e)*fe(»e)*«- £••>•••-• • The appl icable equation for a moving source in a th in rod that i s

convectively cooled i s

M k H O * * " ? ? + * • • • - * £ ( t - v • ( 2 8 )

The moving s o u r c e s t r e n g t h i s a c c o u n t e d for i n t h e boundary c o n d i t i o n s

for a p a r t i c u l a r problem s o l u t i o n .

4 . BOUNDARY CONDITIONS

S o l u t i o n s t o h e a t conduct ion p rob lems r e q u i r e s t a t e m e n t s of c o n d i t i o n s .

For g e n e r a l s o l u t i o n s t h e r e must be ; i v e n a t l e a s t a d e f i n i t i o n of t h e s o l u

t i o n r e g i o n , such a s i n f i n i t e , s e m i - i n f i n i t e , q u a r t e r - i n f i n i t e , f i n i t e , e t c .

A d d i t i o n a l l y , l i m i t s can be s p e c i f i e d f o r any of t h e s e r e g i o n s .

10

If specif ic solut ions are needed, then complete condit ions must be

defined. They could include, for example, i n i t i a l , i n t e r n a l , and surface

condit ions. Other conditions might include property d e f i n i t i o n s . Whether

deriving a solut ion or searching for ex i s t ing solu t ions , one must decide which

conditions are appl icable to the problem and how they can be sui tably

expressed.

4.1 Initial Condition

Unsteady-state problems must have an initial condition defined. Mostly, this implies a temperature distribution at T = 0 but, also, internal or surface conditions could have initial values. A problem solution for T > 0 depends, of course, on whatever is specified at T = 0.

4.2 Surface Conditions

The most commonly employed surface conditions in heat conduction problems are prescribed surface convection, temperature, heat flux, or radiation. It is even acceptable to prescribe two of these for the same surface, such as combined radiation and convection. Other surface conditions could include phase change, ablation, chemical reactions, or mass transfer from a porous solid.

4.2.1 Convection boundary

Conduction and convection heat transfer rates on a surface are equated to satisfy continuity of heat flux according to Newton's law:

3t G'H T) -k ^ — = h[t(xb, T) - t f] , (29)

3x

where t is the temperature of the convection fluid, and x. is the boundary location. The convection coefficient h must be determined from suitable sources that give predicted values satisfying the conditions of the fluid. (Some correlations of convection coefficients are given in Section 16.) The method for defining h can vary depending on the type of convection or the methods prescribed by those researchers who have supplied values for the coefficient. However, h is usually defined as

11

TABLE 1. Sample convection coefficient values .

Fluid Condition h, W/nT-K

Air Free convection on v e r t i c a l p la tes Air Forced convection on p la tes Air Forced flow in tubes Steam Forced flow in tubes Oil Forced flow in tubes Water Forced flow in tubes Water Nucleate boiling Liquid helium Nucleate boiling Steam Film condensation Liquid metal Forced flow in tubes Steam Dropwise condensation Water Forced convection boi l ing

10 100 200 300 500

2,000 5,000 8,000

10,000 20,000 50,000

100,000

h = t ( x b , x) - t f ' (30)

where t can be given as

t f = e[t ( V T, - t j + t m ( (31)

and where 3 £ 1, and t^ is the temperature outside the thermal boundary layer of the fluid. In this respect, one must take care to use the proper fluid temperature and convection coefficient.

Some typical order-of-magnitude values for the convection coefficient h are given in Table 1.

4.2.2 Surface temperature

Of all boundary conditions, this is probably the simplest in a mathematical sense. It can be variable or constant with respect to position and time. In the real sense, it is very difficult to achieve a prescribed surface temperature, but it can be closely approached by imposing a relatively high convection rate.

12

4.2.3 Heat flux

Four ier ' s law defines the flux on a boundary by

3t(x , T ) q = -k — ^ • (32)

An adiabat ic surface can be defined by e i the r se t t ing q in Eq. (32) or h in Eq. (29) to zero. Inversely, if the s o l i d ' s temperature d i s t r i b u t i o n has been solved, then Eq. (32) can be used to determine the surface heat f lux. Understandably, heat flux can be, in p a r t i c u l a r cases, time and posit ion dependent.

4.2.4 Thermal rad ia t ion

Heat t ransfer from an opaque surface by radiat ion can be expressed as

r , .-l 3 t ( X . , T)

a j r [ r \ v T) - T J = -k — ^ , (33)

where a is the Stefan-Boltzmann rad ia t ion constant , <#"is the combined configurat ion-emissivi ty factor for mui l t ip le -sur face r ad ia t ion exchange, and T is the sink or source temperature for r ad ia t ion . Because Eq. (33) i s a nonlinear expression, i t is frequently d i f f i c u l t to find exact solut ions to problems with t h i s condit ion.

A common method for dealing with rad ia t ion problems i s to t r e a t the radia t ion boundary as a convection boundary. According to Eq. (29) we can wri te

3t(x , T) ~ k — 3 * = h r [ t ( V * ~ fcs] • ( 3 4 >

where

h r = OJT[-T{xb, x) + T ] [ r 2 ( x b , T) + T^J

Using t h i s method means that T(x. , T) must f i r s t be estimated in order to compute a value of h . After a value for T(x. , T) has been computed from the problem so lu t ion , then the estimated value for h can be improved. This , of course, becomes an i t e r a t i v e process .

13

4.3 Interface Conditions

4 .3 .1 Contact

Two contacting solids, either similar of dissimilar, will almost always have some interface thermal resistance to heat flow between them. The magnitude of this resistance can depend greatly on the condition of the two contacting surfaces. Properties that can effect the surface condition include cleanliness, roughness, waviness, yield strength, contact pressure, and the thermal conductivities of the solids and interstitual fluid. Since there are so many influences on the contact thermal.resistance, it is difficult to theoretically predict its value. Consequently, experimental results are frequently used. Some representative values of the inverse thermal contact resistance, commonly referred to as the thermal contact coefficient, are given in Section 17.

The generally accepted definition of the contact coefficient is

hc=AE. ' i 3 5 )

where q i s the steady heat flux corresponding to a f i c t i t i o u s interface temperature drop of At. defined by ext rapola t ing the v i r t u a l l inear temperature gradient in each solid to the contact cen te r l i ne . This temperature drop, which is i l l u s t r a t e d in Fig. 5, would diminish to zero i f the interfaces were in perfect contac t .

4.3.2 Phase change

Other in terface conditions include those caused by endothermic reactions such as melting, s o l i d i f i c a t i o n , sublimation, vaporization, and chemical d i ssoc ia t ion .

A statement of an interface react ion condition defines the difference of heat flux across the in te r face . If the in te r face which separa tes two phases ot n material i s located a t x = x . , the heat balance for a phase change reaction is given in the form

k 2 3x " k l 3x " 2 dx * t 3 6 ]

14

FIG. 5. Illustration of interface contact between solids.

where y is the latent heat or chemical heat capacity, and subscripts 1 and 2 refer to the two phases.

5.0 Solutions

5.1 Extending solutions

A solution can be retrieved after identifying a problem by boundary conditions, geometry, and other pertinent data. Usually, a temperature solution is given, but heat flow can be derived from the temperature

15

d i s t r ibu t ion by using Four ie r ' s law, i . e . Eq. (2) . If cumulative heat flow i s required, a time and surface in tegrat ion of local heat flux i s necessary.

-/ .£*&*«- • where n i s the d i rec t ion normal to the surface s.

Steady-state solut ions can be considered as the in f in i t e - t ime condition for unsteady-state so lu t ions . That i s , problems which have a time-asymptotic solut ion exhibit s teady-s ta te solutions for x •> °°. Thus, s t eady-s ta te solutions can be derived from t rans ien t so lu t ions .

A steady surface temperature condition can be implied from a convection boundary condition. For h + °°, the surface temperature approaches the f luid temperature. Therefore, a solution which includes a convection boundary can be transformed into a constant temperature boundary solution by solving for the implied l imiting case.

5.2 Dimensionless parameters

Grouping pa r t i cu la r variables yie lds dimensionless numbers t ha t can be usefu l . Symbolically, they can shorten an equational expression. But, they can a l so give insight to the behavior of heat transf.er in a pa r t i cu l a r problem.

One very useful parameter is the Biot number, Bi = hj,/k, which r e su l t s from convection boundary condi t ions . This parameter i s proport ional to the r a t i o of the conduction res i s tance to the convection r e s i s t ance . Thus, we could say tha t for

Bi > 1, conduction is highest resistance to heat transfer, Bi < 1, convection is highest resistance to heat transfer, Bi « 1, the solid behaves like k = ™.

2 Another dimensionless parameter is the Fourier number, Fo = ai/l , which

is found in cransient solutions. This number is a dimensionless time value, but it is also considered an indicator of the degree of thermal penetration

2 into a solid. Since ax/SI = (kxAt/U)/(pc£At) , it is proportional to the ratio of conduction heat transferred to thermal capacity. Thus, an increasing Fo value implies approaching thermal equilibrium.

16

The product of Bi and Fo numbers yields the parameter Bf « hx/pdf. which occurs in transient problems having a convection boundary. This is also a dimensionless time parameter, but it is based on convection heat transfer instead of conduction ?s in the Fourier number.

Solutions to problems having an internal heat source q'" usually have a dimensionless heating parameter called the Pomerantsev modulus Po = q"''2,2/kAt. This number is a ratio of internal heating to heat conduction rates. Large values of Pr> imply large temperature differences will occur in the solid.

The parameter Fo = X,/2 rr is a form of the reciprocal of the Fourier number and occurs in many solutions for transient temperatures in semi-infinite solids.

When time dependent boundary conditions have a time constant, the solution will frequently include a dimensionless group called the Predvoditelev modulus, Pd = bi /d, where b is the inverse time constant. Small values of Pd imply a slow changing condition. It signifies the ratio of the change rate of the boundary condition to the change rate of the solid temperature.

5.3 Example Problems

5.3.1 Steady heat-transfer in a pipe wall

Hot water flows at 0.5 m/s in a 2.5 cm i.d., 2.66 cm o.d. smooth copper pipe. The pipe is horizontal in still air and covered with a 1-cm layer of polystyrene foam insulation. For a 65°C water temperature and 20°C air temperature, estimate the heat loss rate per unit length. The solution given in case 2.1.2 is

2n (t^ - t 2) q =

1 „ r2 . 1 . r3 ^ 1 _, 1 r— In — + 7— An — + — r - + — r -kl El k2 r 2 rl nl r3 h3

17

From t h e problem des c r i p t i o n

fcl fc4

o = 65 C

= 20°C

r 1 = 1.25 cm

r 2 = 1.33 cm

*-> = 2.33 cm

k = 400 W/m" C (from T a b l e 18.1)

k 2 = 0.038 W/m-°C (from Table 18.2)

h d / k = 0.0155 P r ° " 5 R e ° " 9 (from S e c t . 16 .1 )

h l = ( k w a t e r / 2 r l > ( ° - 0 1 5 5 ^ ° ' 5 ^ ° ' 9 ) k u = 0.659 W/m-°C (a t 65°C)

wa te r v

Pr = 2.73

Re = 2pvr l / ] i

p = 980 kg/m

v = 0 .5 m/s

U = 4 .3 x 10 kg/m"s

2 ( 9 8 0 ) ( 0 . 5 ) ( 0 . 0 1 2 5 ) Re = 4 .3 x 10 -4 = 28 488

( 0 . 6 5 9 / 0 . 0 2 5 ) ( 0 . 0 1 5 5 ) ( 2 . 7 3 ) ° " 5 ( 2 8 4 8 8 ) 0 , 9 = 6895 W/(m' 2 o, 'O h3 " ' W 2 ^ 1 C ( G r d P r > k a i r = Q " 0 2 5 w / m " ° c

Pr = 0 .71

Gr = g B ( t ,

(from S e c t . 16 .8 )

t 3 ) ( 2 r 3 ) 3 / \ ) 2

g - 9 .8 V s

B = 1 /T 4 = 1/293 K - 1

V = 16.55 x l O - 6 m 2 / s

= (9.8) (0.0466) (0.71) = ^ ( _ (

(293) (16.55 x 1 0 " b r 4

C = 1.14, m = 1/7 (from Table 16.3)

h 3 = ( 0 . 0 2 5 / 0 . 0 4 6 6 ) ( 1 . 1 4 ) ( 8 7 7 4 ) 1 / 7 = 2 .24 W/m 2 '°C

1°C)

q = 400 i n IS) 0.038 In

2ir (65-20

(0 .0125) (6895) ( 0 . 2 3 3 ) ( 2 . 2 4 )

2w(45)

1.55 x 10 + 14.76 + 0 .012 + 19.16 = 8 .33 W/m

18

2 i r r 3 h 3

8.33 271(0.0233) (2.24) = 25.4"C

Using t h i s new e s t i m a t e of ( t - t ) , we can r e c a l c u l a t e h .

h 3 = 2.24 ( 2 5 . 4 ) 1 / 7 = 3.56 W/m 2 ' °C,

q = 10.54 W/m.

Additional iterations on h would little improve this result. Note that the copper tube and water film have a small effect on the

results because they present little resistance to heat transfer by comparison to the insulation and air film.

5.3.2 Transient heat conduction in a slab

A billet of 304 stainless steel measuring 2 x 2 x 0.1 m thick and having a uniform temperature of 30 C is heated by sudden immersion into a 450 C molten salt bath. The mean convection coefficient is 350 W/m " C. Determine the time required for the center temperature of the billet to reach 400°C. The solution is found in Section 8.1 (case 8.1.7 and Fig. 8.4a):

400 450 t. l

30 450 0.119

From Table 18.1 k = 21 W/m*°C#

-6 2 , a = 7 x 10 m /s, k_ = 21 hi, (350) (0.05) 1.2 .

-h.t.

From Fig. 8.4a „2 QLT/i, 3.4,

3.4Jt2 _ (3.4H0.05) 2

7 x 10 -6 1214 s

19

5.3.3 Transient heat conduction in a semi-infinite plate

For the conditions given in 5.3.2, find the billet center temperature at 0.05 m from the edges and sides of the billet.

The solution is found in case 7.1.21 and Fig. 9.4a for a semi-infinite plate and expressed as T = (P)(Fo)'S(X). Values of P(Fo) are given in Fig. 8.4 and values of S(X) are given in Fig. 7.2.

hVax BiVfo

Fo

o\l -6. 350\(7 x 10 ) (1214) 21 1.54

V V 2Vafr

0^05

2V(7 x 10"6) (1214) 0.27

If (T = (t - t£)/(t. - t f) , S(X) = 1 - 0.45 = 0.55 (from Fig. 7.2).

The P(Fo) value is given in 5.3.2. Thus, the solution is

I

t ~ t £ P(Fo) S(X) = (0.119)(0.55) = 0.066

t = (0.066)(30 - 450) + 450 = 423°C

5.3.4 Extended surface steady-state heat transfer

A 160 °C uniform-temperature copper plate has a long rectangular rib brazed to it. All surfaces are convectively cooled by 30 C air having a convection coefficient of 53 W/(m • C). The rib is yellow brass extending 4 cm from the flat surface and 2 cm wide. Estimate the additional heat loss from the flat surface caused by the rib.

The solution for temperature distribution in the rib is given in Case 1.1.17 for f(x) = t., w = t and 1 = a.

V \ T

20

* - * £ ~ Bi cos ( X n X ) ! x n c o s h [ X n ( B - Y)l + Bi s i n h [ X n ( B - Y)l I

1 " tf £-\ cos (X ) ( B i 2 + X 2 + B i ) fx cosh (X B) + Bi sinh (X B)l n=l n n L n n n J

X tan (X ) = Bi ( c h a r a c t e r i s t i c equation)

B = b /a = 4 /1 = 4

X = x / a , Y = y /a

Bi = -r2- = ( 5 3 > ( Q - 0 1 ) = o.oo4 (k value from Table 18 .1) k 130

From Table 1 4 . 1

\ x = 0 . 0 6 3 2 , X 2 = 3 .1429, X 3 = 6 .2838 , X 4 = 9 .4252 , X 5 = 12 .5667 .

Heat l o s s form the convec t ive ly cooled surfaces i s determined by Eq. 32

applied to the boundary y = 0 .

_ , . f 3 t ( x , o ) q = 2k / — dx o 3y

°° Bi tan (X )[X sinh (X B) + Bi cosh (X B)] = 4k(t. - t f ) £ " " S «—

n=l (Bi + X'' + Bi)[X cosh (X B)+ Bi sinh (X B)] n n n n

= 666 W/m

Heat loss without the rib attached would be q = hA (t± - t f) = 53 (0.02)(160 - 30) = 138 W/m .

The additional heat loss is thus Aq = 666 - 138 = 528 W/m ,

5.3.5 Rectangular fin heat transfer

Use the straight rectangular fin solution to estimate heat loss from the rib described in 5.3.4.

The solution is found in Case 5.1.4 and Fig. 5.2.

/— Vka / 5 3

= 6 .385 m • 1 /— Vka V (130)(0 .01) = 6 .385 m

1 = c

0.1 54 + 0 .01 = 0.05 m

mAc = 0 3193

21

5.3.6 Semi-inf ini te plate heat t ransfer

Find an equation for the heat transfer ra te through the edge of the semi-in f in i t e p l a t e described in case 1.1.5 with f(x) = t .

Using the given temperature solut ion and Eqs. (2) and (37) we can find the heat t ransfer in the following manner:

S = ~k I? = " I ( t

2 " V S ' w h e r e T = T7=\

H | Y = 0 = -2 ]£ s in (mtX)[l - cos (nit)] n=l

00

Q Y|Y = 0 = l l q Y|Y = 0 ^ = 2 k ( t 2 ~ V / S S i n ^ ^ C 1 _ c o s ( n 1 , ) ] d x

n=l

GO 00

= 2 k ( t 2 - t x ) 2L El - cos(mr)3 2 = f * ( t 2 - t > 2 , i , n = 1, 3 , 5, n=l mr n=l

22

»

I f in » a.

i

>

REPRODUCED FROM BEST AVAILABLE COPY

Section 1.1. Solids Bounded by Plane Surfaces—No Internal Heating.

Case No. References Description Solution

1.1.1 19, Convectively heated and p. 3-103 cooled plate.

W Jyt 2

q = " i ( t i - V Bix + 1 + i\/^2)

t - t. B ^ + 1 fc2 " *1 ~ B i l + 1 + (W

^ 1.1.2 19, The composite plate, p. 3-103

( t 0 " V n

£ (W./ki + 1/h^ + l /h Q

k , h_4 h n-2 h n-1

V h H

l /»<«*^>v">^

k- U-\-*n

i - 1

Temp in the j th layer:

j - l S (S,./k i + 1 / IK) + (Xj/kj) + ( l / h 0 )

^ («,./k i + 1/h.) + ( l /h Q ) i= l

, J > 1



1.1.3 1, p . 138 Pla te with temperature dependent conduct ivi ty , k = k^ + S ( t - tx).

M

q = K fci - to 1 v 2 m J,

t - t 1 B \ 1 + 2 e k m ( t 2 - V f - X

k = (k. + k_)/2 in 1 2

i H . 4 2, p. 221 Porous plate with internal fluid flow, t = t x , x = 0. t = t_ , x = 6. P a Poros i ty .

u.t 0 ,k, ,p, .c

P = porosity

V P ~-*2

t ~ t, t

t r - ^ - = expl"- E 5 (1 - x/6)"l , 0 < x < 6

2 "O

t - t 0 exp(5 fx)

t 2 - t l exppp6 - 1]

Mean temp:

, -<*> < x < 0

^S—^ = ^ [1 - e*P< V ] ' ° 1 X i 6

c„ = p f u c

' 5- = p f u c

' p k p ( l - P) ' •'f k f (1 - P)

(See Pig . 1.1)



1.1.5 2, p. 122 Semi-Infinite plate. 9 , p . 164 t = t x , x = 0 , A, y >. 0 .

t - f ( x ) , 0 > x > A, y - 0. t - t = 2 2^ exp(-imY) s i n (rarX)

n=l

i x J l"f (X) - t i " | s i n (mrX) dX

For f (x ) = t 2 :

V t = f<x) - — = — / — exp(-mrY) s i n (mrX)fl - cos (nir)"] t a — t 1 IT fc-» n — -1

n=l

1 .1 .6 3 , p . 250 Rectangular s e m i - i n f i n i t e rod. t = t , x « 0. t = t_ on other s u r f a c e s . 2 "1 "2 t - 2 2

on co m+n (-1) exp • ( x n x / w ) 2 + a m x / f c ! 2

n*0 m=0 X X n m

x cos (X ) cos (X ) n m

X = (2n + \)\ , \ = (2m + l ) f n 2 m *•


Case No. References Description S o l u t i o n

1 . 1 . 7 2 , p . 130 Rectangular i n f i n i t e r o d . t = F ( x ) , 0 < x < I, y =

t = F ( x ) , 0 < x < I, y = w.

i

t m h + fcn + 'in + fciv

t = G ( x ) , x = I, 0 < y < w

c = r- \x.) , u •> x ^ *,, y = w. ^ t » G (x) , x = 0 , 0 < y < w. fcI = 2 Z

1 n=l

^ . sinh [ ^ ( l - Y)l

slS S/ff s i n < n " X L > f * t W s i n < n 7 r x > d x

t=G 1 W>-

t = F, (x)

1

t = F2(x)

s i n h (mr/L) s i n ( r f l , X L > / F 2 ( X > s i n ( m r X ) d X

n=l

-t = G2W) n=l

= 2 J^j2_LniLii_z_xLL s i n ( f f l r y ) , G < i ( y ) s i n ( n i r y ) d y "IV " ^ s i n h (niTL)

n=l

L = %/M

i i

) f G l (

J0

*-h ,f"4f(i--)1 . (mix) , 2 1 ncosh I

F x ( x ) = F 2 ( x ) = t 2 , G 2 ( y ) = G x ( y ) = t



1 .1 .8 2 , p . 147 Thin rectangular p l a t e . t = t Q , x = 0 , 0 < y < x .

t « t , 0 < x < 4 , y = 0.

t » t , 0 < x < %, y = w.

t - G(y) , x - A, 0 < y < w.

h i r t 0 a t z « 6 .

h 2 , t 0 a t z » 0.

y h v t 0 to

i m

t = G(y)

*o h 2 ' *o «•

t - t Q = 2 IS n=l

nh U + nW)*xJ s . n m

s inh |_(Bi + n IT L ) \ |

f [G<Y) - t Q ] sin (MTY) dY

Bi - (h + h 2 ) i / k 6 , L = l/v

1 . 1 . 9 4 , p . 41 I n f i n i t e rectangular rod in a s e m i - i n f i n i t e s o l i d .

/7777W77777777777777777777777

h- fco

n 1 „ i L r 3 . S ( d + K/h)1 R ~ k ( 5 . 7 + f _ ) ta[.0.25b0.75 J



1.1.10 4, p. 41 Infinitely long thin plate Vertical plate: in a semi-infinite solid. , ,. . .

k ( t i - V d

n t

Q * 7 - — T O T ' o . 5 < - 

1.1.11 4, p. 43 Thin rectangular plate on the kwn(t - t ) surface of a semi-infinite Q = solid. *(%)



1.1.12 4, p. 44 Thin rectangular plate in an 2irwk(t. - t_) infinite solid. Q «- —

In (1)

V 1.1.13 5, p. 54 Rectangular parallelpiped *"* with wall thickness of 6. Q t = [$(dw + db + wb) + 2.16{d + w + b) + 1.261 (t - t

= total heat flow through six walls



1.1.14 4, p. 37 Infinite hollow square rod.

Tl W I

q » 2TTk(t2 - tx)

J S _ + i n i^>8w + J k _ h l C 0 2 r 0 2 h 2 W

, — h 2 , t 2

1.1.15 9, p. 166 Rectangular infinite rod. t - f ( x ) , 0 < x < a, y = 0.

t - t = V A n s in (mrx) sinh [{1 - Y) ( ^ f ) 1 cosech 0 ^ \

n=l

1

A = 2 J [f(X) - O s in (miX)dX , L = Jt/w

For: f(X) = t 2"

t - t tprr = J 2 K s i n <nirx) s i n h [ ( 1 " Y) ft)] o o s e c h ft)' n=l n = 1. 3 , 5, 7



1.1.16 9, p. 167 Rectangular infinite rod. t = f ( x ) , 0 < x < H, y = 0. q = 0, 0 < x < H, y = w.

a = 0, x = 0, 0 < y < w.

~n - n , t f

t = f(x)

^ , Ma i 2 + X 2 J c o s {X X) cosh ["(1 - Y)WX 1

^ 2 n=l [ ( B i 2 + X n) + B i ] c o s h (V"

x / Tf(X) - t f ] cos (XnX)dX

X t an (X ) = Bi , Bi = hSL/k , W = w/X. n n

F o r : f ( x ) = t

fc- *f fcl" fcf

= 2 Bi 1 n=l

cos (XnX) cosh [ ( 1 - 5f)wXn]

K * >i) + B l c o s (X ) cosh (X W) n n



1.1.17 9, p. 168 Case 1.1.16 with q = 0, y = w, 0 < x < fc i s replaced by convection boundary h , t - .

J v ( B i 2 + X 2 ) cos (X X){X cosh Tx (W - y)~| + Bi sinh Tx (W - Y)1} . _ . - T > \ , " ' n n L n J *- n ^ _

f , i f B i 2 + X2 J+ Bil{X cosh (X W) + Bi sinh (X W)} n=l L\ n / J n n n

JL • • '0 a

I [f(X) - tf~j cos (XnX)<

X tan (X ) = Bi , Bi = hA/k , W = w/X. n n

For: f(x) = t , :

t - t_ J!. Bi cos (XX) {X cosh fX (W - y)] + Bi sinh [\ (W - Y)l} t ^ _ _ x n n *- n j _ *- n -* fcl ~ fcf , cos (X ) |( B i 2 + X2 J + BilrX cosh (X W) + Bi sinh (X W)l

n=l n LA n / j L n n n J



1.1.18 9, p. 168 Case 1.1.16 with t = t f, y = w, 0 < x < it.

J!, ( B i 2 + X 2 ) cos (X X) s inh fX (W - Y)"| /- 1

t - t . = 2 > ^ 1-7-^ ^ =i — ~ x I Tf (X) - t . 1 cos (X X)dX

' n=l L( B i + X n ) + B i ] S i n h <V> l l ^ X tan (X ) = Bi , Bi = hA/k, W = v/l n n

For: f (x ) = t :

i

M t - t . __ cos (X X) s i n h TX (W - Yf| t__ __. X n L n J : x. = 2 B i y n L n

fcl " fcf ^ Y B i 2 + \ 2 \ + B i l c o s (X ) s i n=l LV n / J n

nh (X W) n

1.1.19 9, p. 169 Case 1.1.16 with t = t x, y = 0, 0 < x < %.

t = t , y = w, 0 < x < A.

t - t i^k"»l cos (XnX) { s i n h [* n (W - Y)] - s inh (XnY) ( t 2 - t f ) / ( t x - t f ) }

j f B i + X 2 j + Bi cos (X ) s inh (X W 1 f . | [ Bi + X_ ) + Bi ] cos (X_) s inh (X_W)

X tan (X ) = Bi , W = w/SL



t - t„ 1.1.20 9, p. 178 Rectangular parallelpiped. t = t r x = 0 r 0 < y < w , " " U0 _ 1 6 V V [sinh (L - LX) + T sinh (LX)] s in (CTY) s in (nnrZ) . . ^ , t , - t„ = _2Z- Z , (nm) sinh (L)

n=l m=l n = 1, 3, 5,

0 < z < d "0 TT t = t 2 , x = i , 0 < y < w , 0 < z < d . Remaining surfaces at t n.

(run) sinh (L)

, m = 1, 3, 5,

L 2 = (rnrVw)2 + (imrVd)2 , z = z/d

i - 1

i

*S 1

c—-r M c—-r

0 / \ X

h e

T = ( t 2 - t 0 ) / ( t l - t 0 )



1.1.21 9, p. 179 Rectangular parallelpiped. t = t y , x = 0 , -w < y < +w , -d < z < +d . t = t 2 , x = i. ,

-w < y < +w f -d < z < +d . Remaining surfaces convection boundary with h,t,.

00 CO t - -fc. - , * — * - > Tsinh (L - X) + T sinh (LX)] cos (X Y) cos (B Z) — = 4 Bl D > > j--= 5 =y-z = z r

c l 'f , . c o s (X ) cos (B ) U + Bi + Bi[B + Bi D + Bi D l s inh (L) n=i m=± n m |_ n Jv in /

X tan (X ) = B i , B tan (B ) = Bi D, Bi = hw/k n n m m

L 2 = x V / w 2 + B m J l 2 / d 2 , D = d/w, X = x/SL, Y = y/w

Z = z / d , T = ( t - t ) / ( t - t )

Section 1.1. solids Bounded by Plane Surfaces—No Internal Heating.


1.1.22 9, p. 180 Case 1.1.21 except t = t , x = 0 , - w < y < + w , -c < z < +c . Remaining faces are correction boundaries with h,t_.

t - t _ „ [ABi sinh (L - LX) + L cosh (L - LXQ cos (X Y) cos (6 Z) t x - t f ~ B l D 2-> 2* ~ [A Bi sinh (L) + L cosh (L)] NM cos (X ) cos (f$m) n*l m=l

A = l/v , N = X 2 + Bi 2 + Bi , M " S 2 + Bi 2 C 2 + BiC. n m X , fi , L , D , X , Y , Z, and Bi are defined in n in case 1.1.21 .

1.1.23 2, p. 175 Infinite plate with Case 2.2.10 cylindrical heat source.



1.1.24 9, p. 428 Infinite strip with stepped temp boundary, t - t 0 . y « 0 . -oo < x < + 0 0 .

t = t l f y = w , x > 0 .

t = t 2 , y = w , x < 0 .

k " k 1 , 0 < y < w , x < 0

k » k _ , 0 < y < w , x > 0

s in *2

LXi V ^

fc- fco = y 2 [<VV - l]k2 f (-1)" * ( k l + V n=l n '

t _iiio_ T l 21>- <yy>i y (-n° t 2 - t x II ( k r + k 2 ) ^ n

s i n (mtY)exp(-mrX) , X > 0

s i n (mnf)exp(nirX) , X < 0

X = x/w , Y = y /w



1.1.25 9, p. 452 Heated planes on a semi - in f in i t e medium. t = t , x < - & , y = 0 . t = t 2 , x ^ + A , y = 0 . q = 0 , -i. < x < +1 ,

y - 0 .

Q = — cosh *Gk t 2) , -xx < x < -i.

xi • y l2

i +e

1.1.26 9, p. 453 Heated parallel planes in an infinite medium. t » t , x > _ 0 , y « s . t - t, , -<*• < x + « .

Y

Heat flow from bottom side of semi-infinite plane: Q = k Rx j / s ) + U A ) ] ( t L - t 2 ) , 0 < x < xx .

Heat flow from top s ide of semi - in f in i t e plant

Q = £ Jin[(xrxj/s) + l j i ^ - t 2 ) , 0 < x < x x .

H 1 h—



1.1.27 9, p. 454 Infinite right-angle corner. t « t j , x > 0 , y = 0 . t « t 1 , x = 0 , y > 0 . t = t 2 , x > v1 , y = w 2 . t • t 2 , x • wx , y > w .

Q = k w 2 W ; L TTW 2 \ « J T, \ 4 W l w 2 /

f ^ ' t a n _ 1 ( ^ ) j ( t i " V ' ° *x * x i ' ° K Y * y i X I * 1

I r

w 2 I * y

. w l 4 -

J 4

For wx = w 2 = w , x1 = yj = x :

i * 0 0 Q = k{2(I<x/w) - 1 ] + 0.559} ( t x - t 2 )

1.1.28 9, p. 462 A wedge with stepped surface temp. t - t x ± x ./*" sin X[i,n(r0/r)] cc

t 2 - t L

= I + it JQ A cosh (A6Q) cosh(X6)dX

Section 1.1. Solids Bounded by Plane Surfaces—No internal Heating.


1.1.29 15 Semi-infinite strip with convection boundary. t = t , x « 0 , w , y = Convection boundary at y = 0 .

y

exp(-mrY) sin (mrx) 2 2 rnr + (n it /Bi)

1, 3, 5, f 1 n=l

Heat transfer into strip at y - 0: CO

kit - t 1 = F " 2 ' n = 1 ' 3 ' 5 ' " * M r l V " n=l n + (nîr/Bi) See Tables 1.2a and 1.2b.

1.1.30 88 27

Periodic strip heated plate q^Wjy) = 0 , on 2b wide strips. t(w,y) •» t , on (2a - b) wide strips spaced 2a on centers.

tf,h-

See Fig. 1.6a and b for values of maximum differences on y = 0 surface, i.e. Ct(0,a) and heat transfer.

t<0,0)] = At m,



1.1.31 28 Case 1.1.30 except Q = surface heat flux on (2a - b) wide strips spaced 2a on centers.

tfx, y) - t Qa H - = |j- (1 + BiX) + 2

%_ T?

Bi sin (mrB) cos (mry) cosh (nirx) + — sinh (mrX)J

n=l .»[, sinh (mrw) + -^ cosh (mrW) B = b/a , Bi = ha/k , W = w/a , X = x/a , Y = y/a . See Fig. 1.7 for values of T = t(0 , a) - t(0 , 0).

1.1.32 29 Spot insulated infinite plate with constant temp on one face and convection boundary with an insulating spot on the other.

See Fig. 1.9 for values of 6(r/w,z/w) = Tt(r , z) - tfJ/rt(» , w) - tf"j at r = 0 , z = w .

M« MZLW/A I

\* T

"7" w


Case No. References Descr iption Solution

1.1.33 32 Infinite plate containing an insulating strip. t « t 1 , - a > < x < ° 0 , y = w .

t - t f l , -«" < x < «> , y - -w

0 f 0 < x < S , r y = 0 . sm

I O

t ( x ' y» - fto 1 1 - i —r r.— -1 + w c o s

{[p 2

+ G 2

+ l + V ( F 2 F

+ G 2

+ l , 2 - 4 F 2 j 4 f V

2yv p99WW9v9 • •

2 ( l + c o s h 1>(2X - L)H c o s r2TrY)]} 1 + cosh (TTL)

2 s inh [it(2X - L ) ] s i n (27TY) 1 + cosh (TIL)

F o r L •*• <*• :

F = 2 c o s (2mf)exp(-2TTX) - 1 , G = 2 s i n (2ltY)exp<-2ltX)-l ,

X • x/w , Y - y/w , L - «,/w

I n f i n i t e t h i n p l a t e wi th t - t^, K0(Br/<S) 1 . 1 . 3 4 1 9 , p . 3-110 heated c i r c u l a r h o l e .

t " fcl ' c " c l fc3 " «*» V B r l / 6 ) , r > r ,

" r ' l

cf B = V B i j + B i j , t r o = ( t x + H t 2 ) / ( 1 + H) , H = B i ^ B i j

h 2 , t 2

• h

Section 1.1 . Sol ids Bounded by Plane Surfaces—No Internal Heating.


1.1.35 19, p. 3-110

Case 1.1.34 with t replaced by a heat source of strength q.

k6(t - t j K 0(Br/6) q = 2Tr{Bri/(5) K^Brj/6) ' r > r

x

B and t defined in case 1.1.34 .

1.1.36 19, p. 3-111

Case 1.1.34 with h = 0. t - t. I„ (Br/6) t 3 - t x 1^8^/6) ' t > r 1

B given in case 1.1.34 .

1.1.37 60 Infinite plate with wall 19, cuts as shown. Heat flow p. 3-123 normal to cuts,

q

See Table 1.3 for conductance data K/K K ,. = ka/X. uncut Q = K t l - t 2

uncut



1.1.37.1 84 Infinite medium with single and multiple insulating cuts.

See Ref. 84 for temperature and heat flow solutions.

1.1.38 61 19. p. 3-124

Case 1.1.37, cut (c), with only one cut.

Conductance:

[<«, - d)/a] + (d/b) + (4/TI) Sin {sec (it/2) CI - <b/a)]f Q = K t.

1.1.39 62 19,

Infinite rib on an infinite plate with

p. 3-126 convection. t - t w , x - ±6/2 , y > i . t = t w , x > |6/2| , y - H . Convection boundary at y = 0

±

8 IA/WI

y

T vn

See Fig. 1.10 for temp a t x = 0 , y > 0

Section 1.1. Sol ids Bounded by Plane Surfaces—No In te rna l Heating.


1.1.40 79 Finite plate with centered hole. M t . V

TTd 2w + S,nl

2TT M

C4}_JZ igiT«

F 1 . 1 . 4 1 79 Tube centered in a

finite plate.

4 1 fed2r

| 1 t1 r

2itk(t. q = V

X-n (i) r < 10

w/d 1.00 0.1658 1.25 0.0793 1.50 0.0356 2.00 0.0075 2.50 0.0016 3.00 4.00

0.0003 1.4 x 10~

CD 0



1.1.42 79 Tube in a raulti-sided infinite solid.

2 1 1 , 5 ( t l " V l n ^ - C ( n ,

n = No. sides r l < r 2 y ' 1 0 f o r n = 3

n 3 4 5 6 7 8 9

10

C(n) 0.5696 0.2708 0.1606 0.1067 0.0761 0.0570 0.0442 0.0354

0

1.1.43 79 Infinite square pipe.

T w 1

2TTk(ti - t 2) 0.93 In (w/d) - 0.0502 2ttk(t.

, w/d > 1.4

V 0.785 An (w/d) , w/d < 1.4

1.1.44 87 Partially adiabatic rectangular rod with an isothermal hole.

q = Sk(t x - t Q ) See Fig. 1.11 for values of S.

/"'""'

*-J&. 2b

T 2a

1

I

" ^ r77777777 ~ 2 b



i m



Section 1.2. Solids Bounded by Plane Surfaces—With Internal Heating.


1.2.1 1, Infinite plate, p. 169 t = t , x = 0 .

t = t 2 , x = A . fc2~ fcl

X + Po X U - X)/2

h-H 1.2.2 4, p. 50 Infinite plate with

convection boundaries.

V h i — -h,.t,

tj_ - t 2 1 + Bi 2 + H + Bi 2 + 2 U * '

Bi^l + Po(l/Bi 2 + 1/2)1(1 - X) 1 + Bi, + H

H = h 2/h 1



1.2.3 3, p. 130

k

Infinite plate with temperature dependent conductivity. k = Jcf + 6 ( t - t f ) .

(t - t,)kc r • — T . ,

^ = f[-l+Vl + 2 B ( l - X 2 ) ] q " 1 ! .

t f .h_ k(t)

B = 6 q ' " L 2 / 2 ^

See Fig. 1.2.

-h,t.

1.2.4 p . 215

I n f i n i t e p l a t e with temperature dependent i n t e rna l hea t ing .

( t - t Q ) k

t = t Q , x ±9. . % •••I2 ^ B

cos CVPoQ X)

cos (VPO£)

q , , . „ q „ . + B ( t - t ( J ) Po e - 3*Vk

h— q'"(t)

k I

Mean temp:

< t m - y k

%"l2 Po! 372 t a n l V F 5 3 )



1.2.5 2, Infinite plate with

i to

p. 217 radiation heating. = 0 t = t Q , x = 0 ,

q ' " • qye g <* radiation energy flux. Y " mean radiation absorption coeff.

H«H

(t - t0)Jqr = 1 - yxe -Y*- _ e-Y*



1.2.6 2, p. 223

Porous plate with internal fluid flow. P = Porosity.

u.x0.kt.pf.c q'".kp,P

—I

t - t - = | - [_1 - e p J + X , 0 < _ X < 1

C I (See Fig 1.3)

fc " fc0 1 ( 1P) V c e,

Mean temp:

X < 0

t m - fc0 l / F + "V| + l

^p k p ( l - P) ' ^f K £ ( 1 - P) ' ^ " p f u c



1.2.7 2, p. 227 Porous plate with internal D X f luid flow and temperature S t + 1 _ e 1 ( 6 t l + 1 D 2 dependent internal heating. t = t 2 , x = I, q ' " = q j " < l + Bt) , P = Porosity.

Bt . + 1 E \ B t , + 1

* °2 D l E = e - e

e - 1 D2X

U - V k f « r - C

q*"(t)

D i - V 2 + [ ( V 2 ) 2 " P O B] [' C„/2 - | ( ? „ / 2 ) 2 - POgl'5

B t x = ( D 1 - D2> ( B t 2 + 1 } - ^ p - E

* E - EC

* D D p uci. E = D l e - D2e , S p - k ( 1 . p ) . *° = q»'Bl-Ap

2

1.2.8 3, p. 220 Rectangular rod. I t - t - jK i 2

t = t n , x = ±b, -a < y < + a. ^ = r- (1 - O - 2 0 - I ' l ' a ^ '

t = t , -b < x < +b, y = ±a. q ' - ' a

V I b

\ q'" a • • r

X

• * —

X

^ (-1)" cosh |(2n + l ) j x | cos |(2n + 1 ) | Y |

n=0 |(2n + l ) j j cosh |(2n + 1 ) ? B 1

See Fig. 1.4, Table 1 .1 . Y = y /a , X = x/a , B = b/a



1.2.9 3 , p . 469 Inf ini te triangular bar. t = t Q , x = ±y. t - t 0 , x = «..

" - ° - - fe < * 2 - W * ) r< •>t2

y/l



1.2.10 2, p. 197 Infinite rectangular rod with temperature dependent internal heating. q'" = q^"(l + St). t = t , x = ±b, -a < y < +a t = t Q, -b < x < +b, y = ±a

V t b

oo oo n+m (t - t Q) %-* •%- (-1) cos U nX) cos (XmY)

4Po„ 2 , Z l/B + t n 6 n=l m=l X X [x 2A + (X 2/A\ - PoJ n m|_ n \ m / 3J

Nf

i u>

q"'(t)

X = 27T{2n - 1) , X = 2Tl(2ni -1 ) n in

X = x / b , Y = y / a , Po & = q^ • • ab6/k , A = a /b

See F ig 1 .5 .

a *• x

^ 1.2.11 9, p. 171 Infinite rectangular rod (t - t^Jk

with convection q

tangular rod (t - t )k , ^ ion boundary. ^— = ^~- + ±-(1 - x) - 4 £ ivity in x-dir. q ' " a / 1 n=l k = conductivity

k = conductivity in y-dir. s i n (X ) c o s (X X) cosh (X AY) n n n

X n [ 2 X n + sin (2Xn)] [ ( ^ ) sinh (AXJ + cosh (AX n)]

X n tan ( A [ ] ) = B i r B i x = h i a / k x , B i 2 = h 2 a / k x

A = r-t f ,h .

| ViT/iT", X = x / b , Y = y / a , B = b/a



1.2.12 9, p. 423 Infinite plate with point source. t = t Q , z = 0 , «,. Source of strength Q is located at r. , 9 , b.

(t - t 0 ) W Q

R

n=l sin (nirz) sin (rnrB)Kfl (nitR)

Ja[r2*'rJ -2^00. (8-8^]

i w

•H B = b/i, , Z = z/l

1.2.13 29 Spot insulated infinite plate. Insulated on one side and an insulating spot with convection boundary on the other.

M f

mkrm \ r J

////////////A'///////////// i Z

Solutions of |"t(r

at r » 0 , z = 0 , ' Z ) " tf] /[ t (°° ' W ) " fcf] = e ( r / w ' Z / W )

w are given in Fig. 1.8.

Section 1.2. Sol ids Bounded by Plane Surfaces—With In t e rna l Heating.

Case No. References Descr ipt ion Solut ion

i <J1

« « 1... — — » « - « - — « • * • — B""" a-


t J

t/5

I


Section 2.1. Solids Bounded by Cylindrical Surfaces—No Internal Heating.


2.1.1 4, p. 37 Infinite hollow cylinder. 2Trk(tQ - t±)

In ( r Q / r i ) + (1/fci^ + ( l /Bi Q )

t - t„ An ( r / r Q )

t . - t Q In ( r . / r 0 ) + ( l / B i Q + ( i / f e y

Bi . = h . r . / k , Bi Q = h Q r 0 / k

2.1.2 19, p . 3-107

The composite cylinder. 2TT{t - t . ) n 1__

n-1 . . n

Temp in the j t h l ayer :

^n-V j - 1

I i=l

k i \ r i / r i h i

n^l

k. I r . I r . h . 3 \ 3 / 3 3

-. J > 1

£*i \ ' i ; 4 ^

Section 2.1. Solids Bounded by Cylindrical Surfaces—No Internal Heating


2.1.3 3, p. 121 Insulated tubes. 2irk(t. V

teR In

See Fig. 2.1 Max heat loss occurs when r n = k/h.

w w 2.1.4 1, p. 138 Inf ini te cylinder with

temperature dependent . thermal conductivity, k = kfl + B(t - t 0 ) .

k = kQ at r Q .

k = k. at r . . l I

^ y t j - v In ( r ( J / r i )

( t ~ V g

= ^ , 2 B k m fctl<V^ k0 l n < V E i »

km = (kQ + k . , / 2



2.1.5 2, p. 148 Cylindrical surface in an infinite medium, t = f <<j» , r = r r

t = t„ , r - - .

t - t. m / \ n

2^ (-7) \ \ cos (n<}>) + B n sin (n<j))J f r > r

2TT ,n = M [f «t>) - t j cos (nij))d(t)

2n ni)))d$

2.1.6 3, p. 226 Infinite cylinder with specified surface temperature, t - f{<f>), r = r n .

t = W

t = fW), r = r0

n=l

"0 2TT f «t>)d<)>

00

t = a + y R n [ a cos (n<|>) + b s i n (mj>)"|

=1 21T

± r 2TT I

J o

2 a n = * /

2H b = i / f (<(i) s i n (ncf>}<a4> n IT I

-'o

' 0 2TT

f() c o s (w(>)d<|>

For f(4>) t Q , 0 < <)> < TT

( 0 , TT C (j> < 2Tt

n=l

Section 2,1. Solids Bounded by Cylindrical Surfaces—No Internal Heating.


2.1.7 3 , p . 228 I n f i n i t e cylinder with specif ied surface heat f lux , q = f (<f>) , r = r n .

q = f (0)

I

fc " fc0 = a 0 + 7_ R a cos (rap) + b s in (nd>) n = l Ln n J

2m

= i f e / f l * ) d * ' a n = TThd + n/Bi) /

27T

—1 f (1 + n/Bi) J

f(ip) cos (n(p)dip

b n = n h f(<p) s in (nip)dip

For q = q Q s in (cp) , 0 < tp < n 0 , u < (p < 2ir

( t - t 0 ) h 1 R s in (tt>) 2 V 2n cos (2mr) " * 2 C 1 + {1/Bi)1 ' * £ (4n2 - l)[l + ( 2 n / S i )]



2.1.8 2, p. 133 Infinite half-cylinder with specified surface temperature, t = t n , <ji = 0 and iu

ao -IT

- t = - A R n s in (n<|>) I ff m - t I s in (n<|>)d<t> 0 *• £ 1 Jo

t = f m , r - r 0" For f(<t>) f t ,

t '= mr fc - fc0 4 y i n t , - t„ TT * - n

sin (n(()) -1 "0 " n=l For f (40 = t x , s in (4>)

it * ^ n - — ? _ = R s i n <*) - 7 L tR"sin <"*> fcl " fc0 * n=l

2.1.9 3 , p . 230 Cyl indr ica l s h e l l sect ion with speci f ied surface heat flux and temperature. t = t . , r = r . . I I q = f(<j>) , r - r n .

q = f(0>

- t . - a 0 An (R) + £ a ^ j ^ " - R 0

n j cos (Xn*)

n=l

I \ -X \ 2 r n A

X n = rni/$0 , R = c/ri

Section 7..1. Solids Bounded by Cylindrical Surfaces—No Internal Heating.


2 . 1 . 1 0 2 , p . 148 I n f i n i t e q u a r t e r c y l i n d e r , t = t , <(> = 0 and i t / 2 .

t = f (<t>) , r = r n .

t = f<0)

<" Tl/2 t ~ fco = £ 2 R 2 n s i n ( Z n " / t f '" ~ 'oJ s i n (2n<1')<

n=l JQ For f(<ji) = t :

fc ' fc0 2 , - 1 — = — t a n

fcl " fc0 *

2R 2 s i n (2*)

1 - R *

2.1.11 2, p. 133 Finite cylinder with two surface temperatures, t = t x , z = 0. t - t t , c - c„. t = t 2 , z = A.

t - t, . . . „ sinh (X Z)J (X R) 1 = 9 \ n 0 n t, - t. L X sinh (XL)J (Xn) 2 1 , n n l n n=l

J o ( V = ° ' z = z / r o ' R = r / r o ' L = £ / r o For t 2 = f ( r )

s i n h (X Z)J (X R) , ^ s i n h (A Z ) j (A R) f

~ ^ " * * ~7t^Jt7 \ R^(R) " o 'V* r n . s i n h (XL) J , (X_) Jn

r„ , s i n h (X ^ , u , i„ , 0 n=l n I n



2.1.12 3, p. 253 Case 2.1.11 with t = t , r = O r r = Z .

t = t Q , r = r Q , n < <t> < 2n.

t - t„ -_~_-0_ _ 2 V W * s i n (V> t, - t„ ~ Ti Z nIn(A„Rn)

n=l „ _ T-^fcJ) s i n t X„ z) s i n W> . o_ x x m n n

2 Z Z mnl (XE. n=l m=l

A = nrr , R = r/JL , z = z/i n

n l n 0'

2.1.13 3, p. 234 Semi-infinite rod with variable end temperature, t = f (r) , z = 0. fc " fco ' r " V

t = f(r)

n=l

fc " fc0 = 2 Vo ( Xn R ) eXp (-V> =1

1

2 [apm -topo(XnR) SB.

A = n J i<V

Z = Z / r Q , J 0 ( X n ) = 0 , X n > 0

If f ( r ) = t ,

exp (-X n Z)J Q (X n R) fci - fcn Z X J (X )

1 0 __. n 1 n n = l



2.1.14 3, p. 234 Case 2.1.13 with t = t , 2 = 0 , and convection boundary h , t c

a t r = r„

t , - t f

" Bi J 0 f t n R ) exp <-XnZ)

n=l K + *i2> W X n W = B i W

2.1.15 3, p. 238 Infinite rod with a traveling boundary between two temperature zones, t = t L, r = r Q, z < 0.

to I CD

t = t 2, r V z " °" Velocity of boundary (z- = 0) = v.

t, - t„ x Z-n=l

[l + ( X n / P ) 2 ] 1/2 W> Vl'V

exp[(l + [ l + ( V P ) 2 ] 1 / 2 ) P * ] ' Z < °

3l fc-fc2 _ V t, - t, Z

n=l

1 + [1 + ( A / o , 2 ] 1 ^

x exp [ | l - [ l + ^ n / P ) 2 ] 1 / 2 ) p z ] , Z > 0

Z = z / V P = v r Q / a r J 0 ( X n ) = 0



2.1.16 3, p. 238 Finite rod with band heating. q = 0, z = + b. ^z — Convection boundary at r = r , -b < z < +b. Surface heati -a < z < +1 . Surface heating (q ) at r = r ,

-2b-

I h-2\Z-^{ rr n =izr i

Heating band

t - t

q,

s i n (X L ) I (X pR) cos (X Z)

r . / k Bi Z X J "B i I (X p) + (X p ) I (X p)"| 0 _ nl_ i) n n x n J n=l

X = rm, B i = h r r t / l c , L = l/b, Z = z / b n u

R = r / r Q , p = r Q / b

2 . 1 . 1 7 4 , p . 37 E c c e n t r i c hol low c y l i n d e r ,

t = t 2 , r = r 2 .

t = V r = r r

2TTk(t 2 - t x )

D 2 -In

R

V(R + l ) 2 - S 2 + V(R - D 2 - - s 2

In

R

_V(R + l ) 2 - S 2 - V(R -

= r2/r1> S = s/r.^

D 2 - - s 2

See Ref. 80 for other solutions to eccentric cylinder.



2.1.18 4, p. 39 Pipe in semi-infinite solid.

d > 4 r r

2 l T k l < t 2 - t l )

, xi \ r i / K L B i 2 j Bi

h a . t j

innii)ii>MHin>»>»»i)»»»i»>iiHi

i

h i r i h 2 d k 2 d B i l = T f ' B i 2 = ^ K = k7'D = r7

2.1.19 4, p. 40 Row of pipes in semi-in f in i t e so l id .

2TTk(t2 - tj_)

rr- + «-n .^«-H°^])] , for one p ipe ,

/Z>?zmz7zzpmz?zmzm7777??77 d

1 Jk „. V l „. h 2 d

n d B l i = T - ' B l 2 • — ' D " I

^;v viV vi-v <H



2.1.20 4, p. 40 Row of pipes in a wall.

yyT7777777T77777777Ty777777777

q = 4irk( t 2 - t x )

! l i L i V L B l :

• h l r l h 2 d d B 1 i = — • B 1 2 = - r - D = i

, for each p i p e ,

yyyyyyyyyyyyyyyyy/y/yyyyyyyyy/

2.1.21 4, p. 38 Two pipes in a semi-infinite solid. q l =

2 irk( t 1 - t Q ) z±

d' J k 1 r 2 / 6 1 C 0 ^ + ( d x - d 2 ) Z

/2d \ 2d, / s2 + (d + a . ) \ 2

For q_, interchange indices 1 and 2



2.1.22 4, p. 43 Circular disk on the q = 4 r 0 M t 2 - c^) 9, p. 215 surface of a semi-infinite

solid. t = t L , 2 t - t ,

t 2 - t j L 7T 2 . - if

= — sin !; i [ ( R - l , 2

+ Z

2 ] 1 / 2

+ [(R + l ) 2

+ Z

2 ] 1 / 2 .

'W/////MWW"' k z T.

Z = z/r„

2.1.23 4, p. 44 Circular disk in an infinite solid. t = t , z + ± <*>.

q = 8r QK(t 2 - 4 )

r2 I 1

l

* 1 1 k

2.1.24 4, p. 42 Circular ring in a semi-infinite solid.

M 0

t k i 1 -2r„ V

4Tf r 2 ( t

In (^j + Jin I " V " 4 r 2 "

4Tf r 2 ( t

In (^j + Jin A-4 , rx « d « r 2

Section 2.1. Solids Bounded by Cylindr' "\1 Surfaces—No Internal Heating.


2.1.25 4, p. 42 Vertical cylinder in a serai-infinite solid.

h,t n

q = 2D B l ,

\ln

//////////JWA

d

1 k

M"tf. n+ T K k | t i - V

k D = d/r„

<*> 2.1.26 9, p. 216 Two semi-infinite regions of different conductivities

t - t„ 2k „

connected by a c i r c u l a r d i s k . fcl ~ fc0 ^ l + k 2 }

t = t , z + +».

t = t 1 # z -»• -<=. q z = 0, r > cQ, z = 0.

x s in | [ ( R - D :

+ Z Z ] l / 2

+ [(R + l , 2

+ Z

2 ] 1 / 2 , Z < 0

'1 '///////nsm mm*//////

t - t„ fcl " fc0

2k, = 1 - TTl^ + k 2 )

x s i n 1 [ ( R - l ) 2

+ Z 2 ] 1 / 2

+ [(R + l ) 2

+ Z 2 ] 1 / 2

q = [ 4 r 0 k l V ( k l + k 2 ' ] ( t 0 - V ' * = 2 / r 0

, Z < 0



2.1.27 9, p . 218, F in i t e cylinder with 69 J n (RXJ sinh ([1 - ZjLXn) t = £(r), z = 0, 0 < r < r . - V r ° n

on- - r ~ c0 ~ <£ n sinh (LX ) t = t Q, z - a, 0 < r < r Q. ^ n Convection boundary at r " V ° with h, t. r = r Q , 0 < z < &,

2X 2 n n ( B i 2 + X 2 ) J 2 ' ( X ) D 0 n

T R[f(R) - t 0 ] j Q (RX n )dR Jo

I

L = J . A 0 , Bi = h r n A , X nJ- (Xn) + Bi J Q (X n ) = 0, X n > 0

For f ( r ) = t 1"

t - t J 0 (RX n ) sinh ([_ - Z]LXn)

t . ° - 2 B I y 7-2—1\ 1 ° n=l ( B i + VW s i n h < L V

2.1.28 9, p . 219 Case 2.1.27 with t = f ( r ) , z = 0, 0 < r < r Q . t - t Q - J V o ( 1 V Remaining surfaces convection boundaries h, t.. n=l

X cosh (£1 - z"]LX ) + Bi sinh (LI - Z3U ) n n n X. cosh (LX ) + Bi sinh (LX ) n n n

C , X , L, and Bi are defined in case 2 .1 .27 . n n

Section 2d. Solids Bounded by Cylindrical Surfaces—No Internal Heating.


2.1.29 9, p. 220 Case 2.1.27 with t = f (z). °° x ( m

r = r , 0 < z < H. t - t Q = 2 t = t Q, z = 0, I, 0 < r < r Q. n=l

_ I Q (mrR) r

2 USE) s i n <™ z ) / [f (z) " fco] s i n (n1,Z)dz

R = r/S.

2.1.30 9, p. 221 Case 2.1.27 with t = f(z), K C O S a n 2 ) * BH Si" ( X n Z ) I 0 ( X n R ) ] r = r Q, 0 < z < I. t - tQ = 2 ^ p- 2 2 , Remaining surfaces convection n=l (,BlA + X n + 2 B l l J I 0 { X n R 0 >

boundaries h, t . 1

/f(Z)["X cos (\nZ) + Bi^ sin (XnZ)] dz

•'o

tan (X ) = (2X Bi„)/(X2 + Bi 2 ), X > 0, R = i/% n n Jo n v, n



2 . 1 . 3 1 9 , p . 220 F i n i t e c y l i n d e r with s t r i p h e a t i n g and c o o l i n g .

q f = - q ^ r = r f l I 5, > z > «, - b .

q = 0 , r = r „ , S, - b > z > - £ + b . T: 0 q = +q , c = C , -8, + b > z > - 8 , .

q z = 0 , z = ±i., r = r Q .

t = t Q , z = 0 , 0 < r < r Q .

( t - t n ) k D ^ ( - l )" l 0 (miTR) V " _ 8_ V ,5- ~ 2 Z,

s i n (imrb/J,) s i n (nmZ) TT- n = Q m i ^ i r n i R ^

m = 2n + 1 , R = r /2« , , Z = z/2«. , B = b/2J,

q

t ///////////// //////// q

j I i ^ ro I 1 t 1 °\ hf/}///////) T X "

2 . 1 . 3 2 9 , p . 220 F i n i t e hol low c y l i n d e r w i t h : t - t Q = 2 ^ ^ s i n (nnz) / [ f (Z) - t Q ] s i n (mrZ) dZ

n=l

F Q = I0(nTTR) K 0 (mrR 0 ) - K 0(nirR} I 0 ( m t R 0 )

P l = I i ( n 1 T R i ) K

1 ( n l r R

0 ) " ^ ( m r R ^ I ^ m r R )

R = l/l



2.1.33 9, p. 221 Case 2.1.32 with q = f (z), <» „ r l t - t Q = - ||- V -—- sin (mrz) / f (Z) sin (nirZ)dZ r = r., 0 < z < l .

Remaining surfaces at t . n=l ^ "0

F defined in case 2.1.32

P x = i^n i rR^ K 0(wiR 0) + K 1(mrR i) i 0 (miR 0 )

For f (z) = q , w < z < (£ - w)

i = o , 0 < ! < w and (I - w) < z < l :

( t - t n ) k ^ F,

q V k v Fo

TS = > —5— cos (rnrw) s in (nuz) , n = 1,3,5, 1 ~ n F, n=l 1

W = w/Jl



2.1.34 9, p. 222 Case 2.1.32 with t = f(r), z = 0, i t < t < t Q .

Remaining surfaces at t z = 0 , r . < r < r Q . t - t Q

__.y ^ 'VW s i n h [ ( 1 - z ) U n ] 2 n=l [ J 0 ( V - J0 (Vo>] S i n h ( L V

1 x f [Rf (R) - t Q ] O 0(RX n)dR

J0

U„(X ) = 0 , X > 0 , R = r / r . , Z = _/«., L = l/i. O n n l l

r U 0 ( R V • J0 (RAn> V X n V ~ W V Y 0 ( R V i-" CO — — — — — — — — — — — — - — — - — - — — _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ — _ _ _ — _ _ ^ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

2.1.35 9, p. 222 Case 2.1.32 with t = f(z), ™ - -.I-/. , A COS (A Z) + Bl Sin (A Z)IG(R,n)

• • _• , |X 2 + Bi 0 + 2Bi„|G{r.,n) Remaining surfaces convection n=l |_ n x. S.J 0 boundary with h, ...

= r i , 0 < z < «.. t - t f = 2 Y n=]

x j [t (Z) - tfJj"X_ cos (XnZ) + B i £ s i n (AnZ)] -X)

tan (XJ = 2X^1^1 - B i J ) . X_ > 0

G(H,n> = V ^ n ' t W W - B 1 l K O f V o ' ]

dZ

+ I f l < t t n ) [ V l ( W + B i VW]



2.1.36 9, p. 223 Semi-infinite cylinder with: Jn^J

t = f ( z > , r = V z > 0. t - t Q = ^ — j - 5 - / [f(Z) - t Q ] exp ( -X^ + X^ ) t = t n , z = 0, 0 < r < r . n=l J0

i

exp (-X Z + X Z)dZ r v n n Jn(X ) = 0, X > 0, Z = z/r u n n u

2.1.37 9, p. 223 Case 2.1.36 with t = f ( z ) , r = r 0 , z > 0. Convection cooling at z =» 0 with h, t f .

t - exp (-X Z + X Z) n n „ J n R A J r _ ,

n=l 1 n J 0

J 0 ( X J = 0, X n > 0



2.1.38 9, p. 223 Finite cylinder with heated disks on (t „ - t_ .)k I, (mm..)

q z = +9 a , z = 0, Jl, 0 < r < r L < V IT2 n^dWH,,) L l u A

- K^mrRg) i^mrR^] , n = l, 3, 5, Remaining surfaces insulated

i i

^

-— T r i lro

i o £-

R = z/l t . = it m,0 t „ = mean temp, over region 0 < r < r.. , z = £• t . = mean temp, over region 0 < r < r,, z = 0 m, 0 1

2.1.39 9, p. 224 Case 2.1.32 with t = t ^

z = 0 and l, t. < r < r Q. t = f(z), r = r Q, 0 < z < A. q = 0, r = r.r 0 < z < l .

/-t=Hz) z = I

* i .

r1 >;/////;///;/;//;/;/ , r o t v//////////////////,

t „ ^ " ^ - t n

t - t 0 = 2 °°

2* ^ s i n (n'flZ) / f"f(Z) - t 0"l s i n (mrZ)dZ

n=l 2 -T)

F l = I i < n 7 r R i J K 0 ( n 1 T R ) + ^(fflTR^ IQ(n7TR)

F 2 = I 1 (mrR.) K0(n7TR0) + K^mrR^ I n(n7TR 0)

R = r/fc, Z = z / d



2.1.40 9, p. 426 Infinite cylinder in an infinite medium having a linear temp gradient. At large distances from cylinder, t = t Qz. Cylinder, conductivity = k . Medium conductivity = k .

z = *A„

2.1.41 9, p. 434 Sector of a cylinder, t = tx, 0 e = o, e n

t = t 1 # 0 < r < r Q ,

fc = V r • V 0 < 9 < 9„ .

fc-fcl — tan

sin (TT9/9 0)

Isinh K-©]



2.1.42 9, p. 451 Two infinite cylinders in an infinite medium.

airkC^ - t 2)

cosh ( * ^ )

R = r]/ r2' s = s / r2

w 2.1.43 9, p. 462 Spot heated semi-infinite solid. K q z = q 0, 0 < r < r Q f z = 0 .

q z = 0, r > r Q , z = 0 . t = t r r > 0 , z -* =° .

( t " U k - t j k r roqo i

exp (-\z)jQaR)J1a)^

%

H<H



2.1.44 73 Infinite cylinder with two internal holes. t m V r " r0 • t = t., hole surface . hole radius = A. .

q = Sk(tfl - t.)

S = shape factor See Fig. 2.6

2.1.45 79 Inf in i te cylinder with multiple internal holes.

2nk{t, V

®-i-® ' In

n = No. holes

r 2 < r x , n > 1



2.1.46 79 Ellipsoidal pipe. 2irk(t x - t 2 )

Jin ( ^ )

2.1.47 79 Ell ipsoid with a constant temperature s l o t .

2TTk{t. V Jin (*f*)


Case Ho. References Description Solution

2.1.48 79 Off-center tube row in an infinite plate. q =

% <5 © © ©~

1 -» |2rf«- «-2s-» ^ 7

2Ttk(t1 - t 2 ) / 6 2 ( a / 2 d ) \ ( , & ,

CO

9 (a/d) = 2 , exp iirs/n + | Y / 2 d cos [(2n + l)na/2d] n=l

e l ( o) = ire 2e 3e 0

OO

8 2(0) = ]£ exp kits (n + | ) 2 / 2 d n=l

9 = 1 + V e x P (itsn 2/2d) n=l

8 = 1 t 2 2 (-!•)" e x P (ittsn2/2d) n=l



2.1.49 79 Single off-center tube in an infinite plate. q =

2 * 1 ^ - t 2 )

H-P^D'-flS) T~T

t2_/ t , y

2.1.50 73 Tube centered in a f i n i t e p late . q =

2TTk{t1 - t 2 )

An

Y////////A (—)*¥• \Ttt / 2w

W////M



2.1.51 79 Heat flow between two pipes in an insulated infinite plate. q =

2nk(t 1 - t 2 )

w \nr /

2 . 1 . 5 2 79 i

Tube in a semi-infinite p late .

'zzm-zz*-WAW/MUW

q = 2 irk( t 1 - t 2 )

2a + an (-*) 2w \Tr)

2 . 1 . 5 3 79 Ell ipsoid in an in f in i t e medium.

Q -4TTVI - a 2 / b 2 k{tx - t 2 )

^ — T T T : arctanh ( \ l - a / b )


Case No. References <?cription S o l u t i o n

2 . 1 . 5 4 79

«o I

oo 2 . 1 . 5 5 79

E l l i p s o i d a l s h e l l . a irkf^ - t 2 )

arctanh ( c / b ) - arctanh ( c / b . )

c = yb^ - 3]^ = Vb 2 - a

Rod i n an i n f i n i t e medium, t = t 2 ; x , y , z + °° .

4 t f k < ^ - V r

An (28,/r) ' I T < 0 . 1

,-4 ± 28- -<T

2 . 1 . 5 6 79 Short c y l i n d e r i n an i n f i n i t e Q = ck ( t , - t 2 ) zned. jnu t = t 2 , - x , y , z + oo .

^ 1 2r

_ ^ £ 0 8 0 .25 10 .42 0 .5 12 .11 1.0 14 .97 2 .0 19 .87 4 .0 27 .84

-28-

Section 2.1. Solids Bounded by Cylindrical Surfaces—No Internal Beating.


2.1.57 79 Toroidal ring in an infinite a-n^k(t- - t- ) medium. 1 2 t = t 2. x,y,z-~. Q ~ In (Sr /r.,) ' rl / I :2 * Z U

2.1.58 79 Thin flat ring in an infinite medium, t = t 2. x,y,z-»».

4ir'k(t - t ) Q = to a6 t l/r 2) ' V r 2 > 1 0

2.1.59 79 Two parallel rods in an infinite medium.

I ^

26

Ty

47rk(t 1 - t 2 )

^ E Z H ] s > 5r a » r

For parallel strips of width 2w, use r = w/2



2.1.60 79

, ~

Two aligned rods in an in f in i t e medium.

T

28-

V,

4irk(t 1 - t 2 )

'[* ffWJ-H - , s - 21 > 5r

For aligned strips of width 2w, use r = w/2

2.1.61 79 Parallel disks in aii in f in i t e medium.

1 1 M-H

1 1 s

1

Q = iitktt^ - t 2 )

'.I- - arctan (r/s) , s > 5r

2.1.62 86 Inf in i te cylinder with symmetric isothermal caps.

kTTt^ - t 2 ) q ~ 2 Hn [ 2 ( 1 + cos 9 ) / s i n 8]



M



to i

Section 2.2. Solids Bounded by Cylindrical Surfaces—With Internal Heating.


2.2.1 4, 51 Infinite cylinder with convection boundary. tn,h.

( t - t Q ) k

§ i - J

2.2.2 4 , 51

I u> ui

Hollow i n f i n i t e c y l i n d e r w i t h ( t - t F ) k noxiow i n r i n i c e c y x i n a e r wicn (c - C _ ; K . i . r _-i _ _ i w i t h c o n v e c t i o n boundary on — = -r r v l - R, + 1 - R + 2R. Jin (R)l o u t a i d e s u r f a c e . q " ' r „ « I B 1 L J-J i I q r - 0 , r = i L

c/cit Bi = h r Q / k



2.2.3 4, 51 Hollow infinite cylinder with (t - tf) k .. , . p , -i „ , i convection cooled inside — = ~ | — RQ- 1 + 1 - R + 2R In (R)|

to I

surface. q r = 0. r = r 0

q-"r:

R = r/r , Bi = hr./k

2.2.4 2, p. 189 Infinite cylinder with temperature dependent heat

t = t Q, r = rQ

q'" = q^"d + 3t)

J (VPS: R) s H r i - § — • P oR - ten"*!'* B t 0 + X Jn(VPo7) B ° ° I

Jo l""S' Bt m + l Mean temp: -g 2 J

1

( V ? 5 i )

Max temp:

0 t ° + 1 ^ V ^ 0t + l max _ 1

6 t o + 1 '^(Gfy See Fig. 2.2



2.2.5 2, p. 189 Case 2.2.4 with:

to I

j (VPO" R) J (VPO~ R> t = t„, r - r , 0 < $ <ir. t'Vt = 2 ~ ~ * 7 > „ ZZT" '

q... = q «.(i + 6t) n = 1, 3, 5, ...

Pojj-B^'rjA

2.2.6 2, pg 191 Case 2.2.4 with convection boundary.

-O 8t i- l 0 t f + l

3 o ^ R > VPOT

' o 1 * 5 5 ^ - ~ i i J i ( V K i )

, Po g = B q ' - ' r ^ A

Mean temp: 2 J l(V5oT)

See Ref. 85 for nonuniform convection boundary solution.



2.2.7 2, p. 193 Infinite hollow cylinder with „ temperature dependent heat Q

source. fc " V r = r 0 • t •> t i # r = r i . q ' " - q n " ( l + St) .

t . + 1 \ (0t„ + 1)

V ^ ' ( B t . + i) '0 Y„ (VfSlRj.)

( 3 t 0 + l ) (Bt. + D V ^ B V " J o ( V ^ )

a0(VPoTR)

Y0(VPO^R)

M I

B = J 0 (R.VPO^) Y Q ( V P V " J o ( V 5 ° B ' Y o ( R i V 5 ° B )

p °B = B W k ' R = r / r o

2.2.8 2, p. 194 Infinite hollow cylinder with temperature dependent thermal conductivity, t - t 0, r - r.. fc = V r = V k = k Q [ l + B<t = t n ) ] .

( t - t Q ) k .„ 2 2PoD |Po B 6

(1 - R 2) (' - 0 In (1/R)

Zn (k) 1/2

Po - , » T 0 B A 0 , K - r / r Q



2.2.9 6, p. 272 Electrically heated wire with temperature dependent thermal and electrical conductivities. t " V r = ro • k t A t Q -.l + 3 t ( t - t Q) . ke / keO = 1 + B e(t - t n) .

B = ^ & , . - X - ^ 3 k t o L

I (Jj -J

k = thermal conductivity k = electrical conductivity e J

E = voltage drop over length L

2.2.10 2, p. 175 Cylindrical heat source in an infinite plate. t = t . r

—VASA c 1-V/ # ^v.

T L

2 , n 2

t ~ %> V™ 0

q'/kW 2n Bk^B]

See Fig. 2.3

B = r0^hx + h 2 )/kw

h ^ + h 2 t 2

^ " h 1 + h 2

q' = heating rate in cylindrical source



2 . 2 . U 9, p. 224 Fini te cylinder with steady surface temperature. t = t , z = 0 and SL, 0 < r < r . t » t Q f r = rQ, 0 < z < I.

l t ' V* z - z2 I„(nTrR)

q " ' * , 2

4 y *o *3 £i n 3V™V

s i n (nuZ) ,

i CO

2 N q"

n = 1, 3 , 5, . . .

Z - z/l, R = c/SL

2.2.12 9, p. 224 Fini te hollow cylinder. (t - t„)k „ „2 0 Z - Z q'"Si

i _ V s in (mrZ) . _1

n = 1, 3 , 5,

F = ijtniTR.jK (nirR) + I (mrR)K (mm.)

F 2 = ij^tnirRjJKjjtmrRjj) + I 0 (mrR 0 )X 1 (mrR i )

Z = z/fc, R = r/S,



2.2.13 9, p. 224 Case 2.2.11 with q'" = q^"[i + Pit - tQ)]

( t " V k _ c o s [(VFo72 - YPQ"ZJ _ 1_

q ' " H Po cos (VPo/2) Po

. r - . I . ( X R) s i n (nirz) 4_ X u n

"^nTl V W n X n , n = 1 , 3 , 5 ,

X2 = (2n + 1) 2TT 2 - Po , Po = q ' " B J ! . 2 A , Z = z/l, R = r / J l

2.2.14 9, p. 423 Finite cylinder with line heat

t = t Q, r = r Q, 0 < z < «.. t = tQ, 0 < r < r Q, z = 0, SL.

q'

unit length is located at r , 8 , Line source of strength g' per unit length i b < z — c o s (rairB) s i n (imrZ)

m=l

I (mTTR) ? [I (nnTR )K (nmK.) - K (mpR ) I {nfflR j] ~ i„ (nnrR„) L n ° n l n ° n 1 J n=0 V ^ V

x E n c o s [ n ( 9 - e i ) l , 0 < R < R

E = 1 i f n = 0 , E = 2 i f n > 0 , m = 2 n i - l n n

B = b/SL, R = r A , Z = z/l

For R < R < R i n t e r c h a n g e R f o r R. .

I



2.2.15 9, p. 423 Case 2.2.14 with a point . .. „ °° (t - tQ)KJ(, L _

source of strength Q located = — y sin (mirz) sin (mirz..) at r . , 9 . , z . . m=l

_ I (mnR) * Z E a H 5 i j [ I n < M B l l n ( l r t l l ' " * n < n i m ) ' n ( B , , V ]

n=0 n °

x cos["h(9 - 9 )~| i 0 < R < R

For R < R < R , interchange R for R n . R = r/SL, Z = z/SL, E = 1 if n = 0, E = 2 if n > 0 n n

2.2.16 9, p. 423 Infinite cylinder with ,. => ~ 0 0 1 V

point source of strength Q. g = 2? A c o s [n'® ~ ®i'] fc = fc . r = r . n~~°°

0 0 Source located at r l f 6 ^ ^ exp(-X| z | ) J n & n R) J

n^n

R{>

X>0

4%C-—f J n ( X n ) = 0 , Z = z / r 0 , R = r / r Q



2.2.17 16

I

Hollow cylinder with temperature dependent heating and thermal conductivity. t = V r = V q t = 0, r = r Q.

V r

q' k

r = 0. k Q + a(t 0 - t),

['" = 1'" + b(t. t),

Solution by numerical method . See Fig. 2.4 for values of t at r = r. .

l P i = I - (r./r Q), K = k 0/at 0, R = atybr^, G = g- • '/btQ

For a = b = 0: .

(t n - t,)kn — — - T(l - p.) + - n/3-^-J - -*o 0

(Eg. (5) in Fig. 2.4)



2.2.18 64 Triangular arrayed cylindrical cooling surfaces in an infinite so l id . t = t Q , r = r 0 .

<m'o,... at?

tt - t„)k o ' K V5 • i t 2 TT = ^ to <R)

2 £ (f) V (;f] - •<« See Table 2.1 for values of &. . See Fig. 2.5 for t values , 3 max S - s / V R = r/rQ

Equilateral triangular array



2.2.19 64 Case 2.2.18 with a square array of holes.

i

O Ox0 o mo

Square array

<fc - V k 2 ,,,2 = ? £ n ( R )

q'' "s

j=i See Table 2.2 for values of 6.. See Fig. 2.5 for t values. 3 max



2.2.20 67 Eccentric, hollow, infinite See Ref. 67 for equational solution, cylinder.

K> i

2.2.21 73 Cylinder cooled by ring of See Ref. 73 for max temperatures, internal holes.

Insu/ated surface



to t •> vn


Case No. Heferences Description Solution

<*>

REPRODUCED FROM S

BEST AVAILABLE COPY

Section 3.1. Solids Bounded by Spherical Surfaces—No Internal Heating.


3.1.1. 4, p. 37 Spherical shell. 4nr0k(t. - t Q) <r0/r.) - 1 + (r0/r.) (k/h.r.) + k / h ^

t - t„ (rQA) - 1 fci " fc0 ( r0 / ri» " * + ( r 0 / r i > ( k / r i h l » + k / r 0 h 0

i 3.1.2 92 Composite sphere. 4 i r ( t i " V

n-1

?. 'i ('i " rLl) + ?. r2h i = l i = l I

*i-*i i [^( ' i" r ^) + ^J + ^(^'") n-l n

i-1 1 V 4 1 + l / i - 1 h i r i

n 1 - J > 1

t. = local temperature in j layer

Section 3.1. Solids Bounded by Spherical Surfaces—Ho Internal Heating.


3.1.3 1, p. 139 Sphere with temperature-dependent thermal conductivity, t • t., r • r. . t - t Q, c - t Q . k - k 0 + B(t - t Q) .

4Trr.k (t. - t„) 0 m i 0 (r^r.)

*m " ( k0 + V / 2

k - k 0. t - tfl

k = k., t = t A

v 6 v

3.1.4 2, p. 137 Sphere with variable surface temperature, t » f(9), r - r n .

t • ^ a r nP (cos 8) n=0

f(9) a - * & an 2 £{9) P (cos 9) sin 9 d9 n



3.1.5 2, p. 137 Case 3.1.4 with: t 1 , 3 D D . „„ Q, 7 _,3 _ , „„ „, t = t 0, 0 < 8 < IT/2 . c 0 Z * * i t > J

t = Q, n/2 < 8 < it . . t i i R5 Pfcos 8) . . .

32 5

3.1.6 2, p. 137 Spherical shell with specified » n 2 n + 1 _ n 2 n + 1

surface temperatures. fc ~ fc0 = X (n + 2/—/"' 2n+l \ — (l) pn(c°s 9)

t « t„, r - r„ . n=0 \,R0 " 7 t = f(9), r = tL

f < I P (cos 8) [f(cos 8) - 1] sin 6 d6

R = r/r. I

t = W )



3.1.7 4, p. 41 Sphere in a semi-infinite 4irr k(t - t ) solid. Q = i 1 + 2[(d/r Q) + (l/Bi)"]

'/A\V////////////A B i = h r ° A

I

3.1.8 i, p. 43 Hemisphere in the surface Q = 2rrr k{t - t ) of a semi-infinite solid.

J2v



3.1.9 9, p. 423 Sphere in an infinite medium having linear temp gradient. For large distances from sphere t = t_z. Sphere conductivity is k, and medium conductivity is r— k ° K0 .

t 3 k Q

S + (R) (2k° + kJ)J cos 9, R > 1

3.1.10 59 19, p. 3-111

Irradiated spherical thin shell. See Fig. 3.1 for T/T . q = source heat flux . s T = sink temp . a = absorptivity . T =

a / E I / E 2 + 1/4V

1 \ G l / e 2 + X I

1/4

î.



3.1.11 79 Sphere in an infinite medium. Q = 4TT rQk(t., - t„) t = t 2, r + » .

3.1.12 79 Two spheres separated a large distance in an i n f i n i t e medium.

4irrk(t - t ) Q m 2(1 - r/s) ' s > 5 r

t.

2r

for s > 2r, error = 1%



3.1.13 79 Two spheres separated a small . ._ ... ,. .Hi L / , , / * 2 , . . . 3 , - , . ' e Q =2rrrk(t1 - t ? ) U + n/s + (r/s) + (r/s) + 2 (r/s) distance (see case 3.1.12) in an infinite medium.

+ 3 ( r / s ) + - ] , 2r < s < 5s

3.1.14 79 Two spheres of d i f f e ren t r a d i i i n an i n f i n i t e medium.

4 T r r 2 k ( t l - t 2 )

/

3© r l

\ ( V S ) 4

r l 1 - ( r 2 / s ) 2

- 2r / s , s > 5 r 2

2r,

Section 3.1. Solids Bounded by Spherical Surfaces—Ho Internal Heating.


i

• i <*„r£aces—No Internal Heating Section 3.1. Solids Bounded by Spherical Surfaces

Section- 3.2. Solids Bounded by Spherical Surfaces—With Internal Heating.


3.2.1 Spherical shell with specified inside surface heat flux. fc " V r = C0 • q r - q r r = r. .

<t - tQ)k q , ,'r i

qi ri **<

R = r/r.

2 < R - R0 ' 2 2

R* + R0 ' R

Ro - R

RR~

u 1

3.2.2 1, p. 190 Sphere with temperature dependent heating.

, 1 1 1 _ — I I I + 0(t - tn)

( t " V g _ 1 sin (KVPO) "*0

t - t,

sin (VPO) 1 , Po = 0rVk

max 0

t - t,

0 2 = 1 - R , Po = 0

0 sin (TTR) ,/-— max 0

( tmax- V f VPO sin (VPO) - 1

Section 3.2. Solids Bounded by Spherical Surfaces—With Internal Heating.


3.2.3 6, p. 274 Sphere with nuclear type heating. (t - t.Jk fc" V r = V q 1 " =q u"[i - 3< r/ r

0) 2]

^ • - * [ « - • ' - » « - ' ' ] *0 0

3.2.4 9, p. 232 Solid sphere with convection boundary.

(t - tf)k = i[L - R 2 + (2/Bi)]

See Ref. 85 for nonuniform convection boundary solution.



3 .2 .5 9, p. 232 Solid sphere in an in f in i te medium. k = k Q f 0 <. r < r Q . It = kj_, r > r Q . q ' " = q j " , 0 < r < r Q . q ' " = 0, r > r Q . h = contact coeff ic ient at t = f. t = t^, r > «. .

<fc " V Q 1 2 2— = pi - R + 2/Bi + 2 ^ / ^ ) , 0 < R < 1

o ' ' l t~ q 0 r 0

(t - t j k = 1/3R, r > r

% X0

Bi = hr 0 /k Q


Case No. References Descciptioii Solution



I

I


Section 4.1. Traveling Heat Sources.


4.1.1 2, p. 189 Traveling l ine heat source in 9, p. 267 an inf in i te so l id .

t = t r x •»• «°, y •*• ± ° ° .

•• V

» » v

(t - t„)k t 0 ) l t 1 / vx\„ /v V 2 _,_ 2 \ ' — = 2? e x P ( -2ETMto V x + y )

y = velocity of line source g' = heating rate in line per unit length

4.1.2 2, p. 291 Traveling line heat source in an infinite plate. t = t Q, x •* °°, y + ±°°.

to.h,

II y- "o-'M

£L .

t Q ' h 2 ^

(t - t 0)w 1

q'A 2n exp ( - t>. [A^*&)^ v = velocity of line source q' = heating rate of line



4.1.3 2, p. 286 Traveling plane heat source in an infinite medium, t = tQ, x + <*>.

(t - tQ)kv

(t - t0)kv

exp (-S)

1, x « 0

, x > 0

q"a q" = heating rate of the plane

4.1.4 1, p. 352 Traveling point heat source t - t A _ nee 5L — =— 9, p. 266 in an infinite solid.

t = tQ, r + ». q'/kr 4n " 4 " 2a y'j

q' = heating rate of the point



4.1.5 9, p. 268 Traveling plane source in an infinite rod. t = t Q , x + ±».

to.h-

X V

(t - t Q)kC q~i

= <*£ (vx - |x|u\

1/2 U = (v2 + 4a 2 hC/kA)

v = velocity of source q' = heating rate of source C = circumference of rod A = cross-sectional area

4.1.6 9, p. 268 Traveling point source on the surface of an infinite plate and no surface loss. t = t Q , i •*• =>.

n w/Mr* <% ^

CO ' 1/21

n=l

x cos (•?) exp(vx/2a) 2 2 ^ 2

r = x + y v = source velocity in x-direction q' = heating rate of point source



4.1.7 9, p. 268 Traveling line source on the surface of an infinite plate with no surface losses, t = t Q, x + ±«.

( t - t Q)vwk 1 + N — cos(rmz/w) (1 - N) exp(vx/2a)

n=l

/ j

I -J/<WA N = Cl + (2aim/w»> 2] 1 / 2

v = source velocity in the x-direction q' = heating rate of source

4.1.8 9, p. 269 Traveling infinite strip source on the surface of a semi-infinite solid with no surface losses, t = tfl, x •* ±».

( t • V * _ 2 ^ + B

j 'a ir / "'X-B

exp (X)K 0[{Z 2 + X 2 ) I / 2 ] d X

X = vx/2a, z = vz/2a, B = bv/2a See Fig. 4.1 v = strip velocity in the x-direction q' = heating rate of strip



4 .1 .9 30 Traveling i n f i n i t e s t r i p source on a s emi - in f in i t e so l id with convection boundary, t = t f , x + ±°°, and z.

"" /-Source = q s

(t(x,z) - t f)kirv X+L

•L OO

/

V

m V Z

2

+ n , 2

, 2 -2 . TTH exp (HZ) J T exp ( H ' T ~ ) e r fc (Z/2T + HT)

0

x {erf [(X + D/2T + t ] - erf [(X - L)/2r + i :}}*:

v = ve loc i ty of s t r i p H = 2ah/kv, L = vJl/2a, X = vx/2a, z = vz/2a

See F ig . 4 .2 .



4.1.10 30 Traveling band source on an infinite cylinder with convection cooling.

( t - t f)TTk

r 0 * s = | r + — y s i n (w}>) exp [-VJ(1 - R ) / V 5 ]

Bi VR ^ . n=l

( *

Source = q 5

c o s (w 2 Fo - n-t - CW(1 - R) /VTl - TI /4 } + Bi c o s Cw 2Fo - n«t> - W(l - R ) / V 2 ] \

n(W 2 + B i 2 + V3 W Bi) /

W tQywi/a

01 = r o t a t i o n speed of source 2d> = angular width of source See Fig . 4.3 a and b.


Case No. References D e s c r i p t i o n S o l u t i o n

4 . 1 . 1 1 34 I n f i n i t e c y l i n d e r wi th

moving r ing heat source

on i t s s u r f a c e .

(t

J

- t f ) 2 n r 0 k " X 2 J Q (X n R) 4 . 1 . 1 1 34 I n f i n i t e c y l i n d e r wi th

moving r ing heat source

on i t s s u r f a c e .

(t

J

Q0 " A , (Nu2/4) + X2 W n=l n

/h.t, ^ t ( z , r . t )

(t

J e x p ( f ±Vfce 2/16) +x;;C

^(p^/ie) + xj

Nu = 2 h r 0 / k , Pe = 2 v r Q / a , 5 = - * OT

Nu J n ( X ) = 2X j (X ) O n n 1 n

+ for 5 < 0 - for £ > 0

i/J-'^ *~-Ring source | t °0 \

(t

J e x p ( f ±Vfce 2/16) +x;;C

^(p^/ie) + xj

Nu = 2 h r 0 / k , Pe = 2 v r Q / a , 5 = - * OT

Nu J n ( X ) = 2X j (X ) O n n 1 n

k- •- r ) •

e x p ( f ±Vfce 2/16) +x;;C

^(p^/ie) + xj

Nu = 2 h r 0 / k , Pe = 2 v r Q / a , 5 = - * OT

Nu J n ( X ) = 2X j (X ) O n n 1 n

' R = r / r 0 ) •

e x p ( f ±Vfce 2/16) +x;;C

^(p^/ie) + xj

Nu = 2 h r 0 / k , Pe = 2 v r Q / a , 5 = - * OT

Nu J n ( X ) = 2X j (X ) O n n 1 n

See Figs. 4.4 a and b.



4.1.12 34

I

Inf in i te cylinder with a moving ring heat source on i t s inner surface.

At ,

(t - tf)2Tvrjk C - I s n=l

J n ( x R) W Y n ( ^ R) 0 l n*' Y,{X ) 0 ' n 1 n

Xn[WV W ~ W Wo>] = f N"[VXnV W - ^< VVXnV] .

C / 2

• n " ^ f(Pe /16) +\' exp I (Pe/4) ± \ (Pe 2 /16) + \ M ,

+ for? < 0 - for? > 0

[ww - ww] w Bn • "I { [ W o - W - W V W ] *

+ [WWV " WWo^n} -f[WW " WWT

R = r / r . , Nu = 2hr./k, Pe = 2vr . /a , £ l l l

See Figs. 4.5 a and b .

Z + vr



4.1.13 70 Infinite cylinder traveling Temperature solution given in source Ref. 70. through temperature zones.

^ , , y- h2-t2

1 r v ) C — z — ~ < i

Zone boundary

Traveling plane source in a Temperature solution given in source Ref. 71. thin rod with change of phase.

* if ' i Melt zone —' IQ = Source

4.1.15 71 Traveling plane source in an Temperature solution given in source Ref. 71. infinite medium with change of phase.

4.1.14 71 i



i O



<J\

!

|


Section 5.1. Extended Surfaces—No Internal Heating.


5 .1 .1 1, p. 209 Inf in i te rod. t « t Q , x = 0. t = t_ , x -»• «°.

l

I S*

" > =

3

exp (-nx)

5.1.2 7, p. 43 Finite rod, insulated end. t = t 0 , x = 0. g^ = 0, x - I.

y Z. h.t,

E? i

Total heat l o s s :

Q nCkA tan {ml) ( t Q - t J

t - t f_ cosh [m(ft - x)] cosh (ml) 0 f

See Fig. 5 . 1 .



5.1.3 7, p. 43 Finite rod with end loss, t = t Q , x = 0.

mkA Q =

fti2/mk)+ tan (m&f| ( t Q - t f ) 1 + (h /mk) tan (mJl)

h i - t f -h„t, V ^ ! " H

P t - t cosh QnU - x)] + (h /mk) sinh [m(£. - x)]

cosh (md) + (h /ink) sinh {ml)

iu = kA

M 5.1.4 7, p. 44 Straight rectangular fin.

< h.t,

°7Eu

Use m = •Wrr in case 5.1.3 .

tanh (ml ) .

—a- 1 - - *c =* + ! See Fig. 5.2

Recommended design criteria: 2k/hb > 5 .

Minimum weight criteria: 2X,/b = 1.419V2k/hb



5.1.5 7, p. 52 Straight triangular fin.

r ^ bf^-to

q = Y2hkb I± (2>p)

I Q (2-vfI)

2hH kb

t - t f I 0 ( 2 ^ x ) fc0 " fcf I (2V0I)

± I 1 (2TÎ) See Fig . 5 .2 ,

Optimum %/h r a t i o : V b = 0.655-j | j r .

5 .1.6 1, p . 228 S t r a igh t f in of minimum mass, t - t " ^ n ~ fcp'x

fc0" fcf i q'

q- = total heat loss. Profile:

y = 2k( X " <* • \ 2

y = 2k( X " h(t Q - v ) f q ' - h(t Q - V



5.1.7 10, p. 95 Straight fin of convex parabolic profile.

t - 1

t o - fcf W m = V2h/kb

q = kbm(t0 - t f )

wi_,„(f^V / 4) -1/3V

211

-1/3

(M (H

(H 1 I 2 / 3 ( ? m i l )

-1/3

, See F ig . 5.2

5.1.8 10, p. 175 Straight f i n of trapezoidal t - t g

parabolic pro f i l e . t - t , ~ \Jl/

f W » ' R - i + i 6 " 2 + 2 VI + 4 m'Jl 2„2

^ ( t , 9 - T" ( to I

3 + 1 '

V See F ig . 5.2



5 .1 .9 10, p. 175 Straight f in of trapezoidal

p r o f i l e . t - t . K1«WB> + W V B )

fc0 " fcf = WW + W W Kl((y W - W W h S c ( t 0 V

H 2 cbs G _ 1 e 0 c 1 e 0 c .

B K (BJ 1,(3J - 1,(0JK (0 ) c 1 e 1 c 1 e 1 c

2H2* W W + WW

0 = 2H

2H

\ x + b (1 - tan 9)

e _ 2 tan 0

= 2H M - tan 9)

2 tan 9

VI + b (1 - tan 9) e

2 tan 9 , H = Vh/k s i n 9 ,

SL = £ + — C 2


Case Mo. References Description Solution

5.1.10 7, p. 54 Cylindrical fin—rectangular profile.

I (x m)K (x m) - K (x ra) I (x m) _ _ „ Mrmffr ^ \ I D 1 e 1 D y - <«rxbDKmtt0 - t f j x m ) K ( x m ) + K ( x m ) T ( x m ) V e ' <T b l l e ' ( K b

«WuwJ ^-h,t,

m = Y2h/kb

t - t K. (x m)I <rm) + I {x m)K n(rm) t_ _ 1 e 0 1 e 0 t Q - t f - K l ( x e m ) l 0 ( x b m ) + I ^ B O K ^ X J B . )

f, , / ,21 I . (x,m) + BK„(x.i ' I ^ x ^ ) - BK^x m)

,m) , See Fig. 5.3

8 = I , ( x / x ) / K (x / x . ) 1 e o 1 e b

Minimum material: use Fig. 5 . 4 .

5.1.11 7, p. 58 Cylindrical fin—triangular 91 „. n

prof i l e . x bm[l - ( x e / x b ) 2 ]

- I - 2 / 3 ( 6 ) + g I 2 / 3 ( 5 )

i 1 / 3 ( 6 ) + Bl_ 1 / 3 ( (S)

m = V2h7kb, 6= - xbm, - I _ 2 / 3 fe) See Fig. 5 .5.

2/3 i ,3/2-



5.1.12 10, p. 106 Cylindrical fin— hyperbolic profile.

b x b Q 2 . W * , , , t 0

t f ) [ l _ a / 3 ( B 8 ) I _ l y 3 ( B b ) - I 2 / 3 ( B e ) I 1 / 3 ( B b ) ]

( t - t f ) ( t 0 " V

(x_\1/2 [ I 2 / 3 ( B e ) I l / 3 ( B ) " J , ^ b / [^'V1 1/3 <V

M ' V ' - l f l 1 " ] I - 2 / 3 ( B e ) I - l / 3 ( B b » ]

1 / 4 0 ( 1 - 0 ) \ 1 / 2 [ I 2 / 3 { B e > I - 2 / 3 " ' - 2 / 3 ( B e» *2/3 < V ] n \ d + P)^n (1/pJ P ^ ' V ^ ' V " '2/3 <Be> *l/3 ( V ]

M = Y2hAbx B = | M x 3 / 2 , B = f MX 3, 7 2, B = | M x V 2

b e 3 e b 3 b 3

,3 /2 D = <xe - *h) ' M/Vin l / p , p = x f a / x e



5.1.13 1, p . 234 Cyl indr ica l f in of minimum m a t e r i a l .

UI i

hux' Q = -j± ( x ^ + 2) ( x e / ^ - 1) ( t Q - t £ )

P ro f i l e :

y W " ^ ( x j " 2 x e

+ 6x

5.1.14 10, p . 114 Pin f i n s—cyl ind r i ca l type.

^ h . t ,

t o — - * - — t , — ^

Q = ~ kd2m tanh (mK.) ( t Q - t f )

<). = • t a n h J m a ) , "I = > W k d , See Fig. 5.1



5.1.15 10, p. 115 Pin fins—rectangular type. Depth of fin = b.

Same as for case 5.1.14 except I — d I is replaced by (bd) (H .1 C"

<,-/Fr^ 5.1.16 10, p . 117 Pin f ins—conical type. , . 2„ I„ (M)

U Bl I^M) l C 0 V

t - t f ft \ (M> A) t Q - t f \ x I l ( M)

, See Fig. 5.1

4I2(M)

Ml]!")" , M = 2V2 ml, m = V2h/kb



5 . 1 . 1 7 1 0 , p . 119 Pin f i n s — c o n c a v e p a r a b o l i c s = W^-3]<t 0 - V

t - t

^ f

_ i + Ww 2 2

<t> 1 + (I + | M ^ ) 1/2

, M = V4h/kb I , See P i g . 5 . 1

5 . 1 . 1 8 1 0 , p . 121 Pin f i n s — c o n v e x p a r a b o l i c

t y p e . Q^jAh^[t _ t ) u 16A I Q ( M ) ( t o c r

« . b r ' * \ , / 2

t - t f I 0 l lM(x/Jl)] 3 / 4

I 0 (M) ' M " 3 V kb

2 V M > M IQ(M) , See F i g . 5 .1



5.1.19 10, p. 134 Infinite fin heated by square See Fig. 5.6 arrayed round rods.

ui I

— Sn

5.1.20 10, p . 135 Infinite fin heated by equilateral-triangular arrayed round rods,

h.t.

E ~P k-«n

See Fig. 5.7



5.1.21 10, p. 185 Straight rectangular fin with nonuniform cooling. h(x) " + (iy-h = av coeff.

a

/ h(x).t,

I ft

t - t. i&M t Q - t f 2 l / ( Y + 2 >

1 / (Y+2) I Xil <V

1 i + i J + 2 , tu D) J i

* L (m* , 2 ^ + 1 > J

Y+2

V(Y+2) r i

Y+2 '

(U)

(Y+D/(Y+2) ( V 1 (Y+D/(Y+2) ( V

V+2

L Y+2.

2 V Y + T r - Y / 2 v ( Y + 2 ) / 2 Y + 2

U „ = ^ *• L ml, m = V2h /kb J, Y + 2 a

5.1.22 10, p. 191 Same conditions as case 5.1.21 with:

h(x) = h. ["(1 + a) (x/fc +

{(1 - = ) a + 1 " « cT a+l

n| I {0)1. „(U 0 ) - I „(U)I , ( 0 0 ) / u \ " n ' ^ l - n ' ^ ' * - n 1 " ' n - l 1 " ^

W I . W W V * I-B<D0 ) I»-l (D.

2(1 - n) T / , 2 ( l - n n "oL1 " ( W J

V l ' V W V " H-nô'Vi'V WWV "WVWV J

- - K^) 1/2 (x/& + c) 1/n

(1 + c ) a - n ) / n _ c ( l - n ) / n ml

u, x = 0; \iQ = u, x = l , n = — , m = V2h /kb 2a'



5.1.23 10, p. 187 Same conditions as case 5.1.22 2m£ _ / xc\ n = — ' U = n V 5 e X P ( - 2l) with: x '

[ , , ... _ i|) = nVa exp(c), 8 = nVa 1 - (a/c) p. - exp (-c)J J If n is an integer:

0 = . 3f ^1 ~h2 v 2m£(l - (a/c)Cl - exp(-c)]) A 3 - A 4

\ = [*„_!<*> - V l 1 * 1 ] [ Jn-1 ( B> " V l ( S > ] A2 " [ J n - 1 ^ " W " * ] [ V l ( B ) " W e > ] A3 = J n ( B ) [ V l ^ - V l ' * 1 ]

A 4 = Y m ( e , [ J n - l W " Jn +l '*>]

If n is not an integer:

. c . A l " A 2

2(raH)2(l - (a/c)Cl - exp(-c)]) A3 A4 A l = [" n J-n ( l | ; :! " ^ l - n W i r - " ^ ' 6 ' + 0 J n - l ( e ) ]

A2 = [-Jn(*> + *Jn_1(W][-nJ.n(B) - BJ^tB)]

A3 = J n ( e ) [ ~ n J - n ( , , , , " ^ l - n ^ ' l A4 = J - n

( 0 ) [ ~ n J n ( , t ' ) + ^ n - l W ]



5.1.24 10, p. 205 Straight fin of rectangular profile radiating to free space. t = t 0, x = 0. T = T , X = l . e Space temp = OK.

OK \>. r -

Z ^

4 20oeT_ kb B(0.3, 0.5) - B (0.3, 0.5)

B = complete beta function B = incomplete beta function

Q = 2 k b \ f | h (To " T e ) ' s e e F i g s- 5 , B a n d 5- 9

2 >l/z3 - L/z8 =

**' 20o«!/kb ' V e

2 3 Optimum length: b/i. = 2.486aeT A



5.1.25 10, p. 211 Straight fin of rectangular profile radiating to nonfree space. T = T , x = 0. T = T , x = I.

K = incident radiation absorbed from surroundings.

K 2

I

2 K „ T : _f lL. . l pL!g v r - [ kbT 0 \ 5kbZ

Ql/kb1Q

2ae!L2Tpkb

4 3 2 5 Z \ Z + Z + Z + Z + 1 — K 1 T 0

1/2

, K = 2ae , z = T / T 1 u e

See F i g s . 5 .10 and 5 .11

Optimum l e n g t h : b/SL2 = 2.365aeT:?/k



5.1.26 10, p. 218 Straight trapezoidal fin radiating to nonfree space. T = T , x = 0. K = incident radiation absorbed from surroundings.

See Figs. 5.12a, 5.12b, 5.12c, and 5.12d. \ = b /b. = 0.75, 0.50, 0.25, and 0 (triangular fin) e 0 K = 2 ere.

5.1.27 10, p. 228 Straight fin of least material Profile: radiating to free space. T = 0, x = 0.

f(x) = _JSL. 2kaeT;

7/2 ( ! ) ' • SL = 3Q 4 2aeT„

OK fc-(fi

1/2



5.1.28 10, p. 230 Straight fin with constant Profile: temperature gradient radiating to nonfree space

T - T dT 0 e dx = SL

10kZ3(Z - 1 ) Z < L *• J KjT* *-*• Jl

K S,T4(Z5 - 1) Q = 5(Z - 1) K2l

Z = T Q/T e

K = 2 ere

z 5-i 5Z 5(Z - 1) KjT*

K_ = incident radiation absorbed

T = (K,/K,) 1 / 4

e,rain 2 1 t See Figs. 5.13 and 5.14

Section 5.1. Extended Surfaces—No Internal Hea<-i


5.1.29 10, p. 247

Radial fin of trapezoidal profile radiating to nonfree space. T = T„, r = r„. 0 0

Fin efficiency: (See Figs. 5.15a-5.15£) X = b /b„ = 0, 0.5, 0.75, 1.0 e o K 2 / K 1 T 0 = ° ' ° " 2 ' ° - 4

Profile No. = K^rjj/kbjj

i

K. = oz, K_ = incident radiation absorbed



5.1.30 10, p. 275

The capped cylinder, convection boundary, t = t Q, x = I. Insulated inside surfaces.

Mf-i 5 r„

-HI—S t.

1

Side: t - t. cosh (mx) + sinh (mx) fl, (mr )/l0(mr )~\ £_ LI e 0 e -1 t„ - t ~ cosh (mi) + sinh (mS,) r i , (mr )/I„(mr )1 O r L l e Q e J

sinh (rnS.) + [cosh (mS.) - l ] [ l (mr ) / I (mr >] ^ = mS,|cosh (mX.) + sinh (mS.) [ l 1 ( m r e ) / I ( ) { m r e ) ] }

Top: t - t . I 0 ( m r ) / I 0 ( m r e ) t . - t , cosh (m£) + sinh (m£) fl., (mr )/I„(mr )"l O r L l e O e J

2 I 1 ( m r e ) / I 0 ( m r e ) mr jcosh (mS.) + s inh (mJl) [ I (rar ) / I (rar )1 I

m = "Vh/kfi, See F i g s . 5.16a, and 5.16b

Section 5.1. Extended SucCaces—No Internal Heating.


5.1.31 10, Doubly heated straight, p. 407 rectangular fin.

t = t Q, x = 0. t = t &, x = I.

fc- "f t o - t f

= exp(mx) - 2B sinh (mx)

q Q = kbm(2B - 1) (tQ - t f)

i i-o o

r'' i 0

q f c = kbmJexp{mJl) - 2B cosh (mJl) ( t - t f ) ]

exp (mi) - ( t p - t ) / ( t - t ) 1 Trrr- 2 - m=V2h7kb 2 sinh (m£)

See Pig. 5.17

Section 5.1. Extended Surfaces—No Internal Heating-


5.1.32 10, p. 394

i M

Single T, straight, rectangular fin. t = t Q, x = b. All surfaces convectively cooled by h, t f .

%L I

n1

b id It

• ^

Vertical section:

t - t^

t> = — x mxb

— = FJcosh (m x) + "V2u/v tanh (m a) sinh (mxx)]

| s inh {m b) + V2u/v tanh (m a) £cosh (m b) - l j )

Horizontal section:

t - t , cosh (m-Ty) = F £' t Q - t f " cosh (ma)

F tanh (m a) m a

y

, See F igs . 5.18a, 5.18b

cosh (m b) + V2u/v tanh (ma) s inh (m^b)

m = VhTiw, m = Yh/ku x y



5.1.33 10, p. 413

i to

Double T, straight, rectangular fin. t ** V x i = b i -All surfaces convectively cooled by h, t f.

'/A yA X 2 b 2

H

m \ u i X, b, U, E

I

Vert ica l s ec t ions : t - t f

- — ^ - j - = F^Tcosh (nijjjXj ) + ^ sinh ( n > x l x 1 ) ] , 0 < x1 < b1

t - t^ I — - = ^ [ c o s h (m x 2 x 2 ) + V 2 V V 2 t a n h ^ V

x s inh (m x 2 x 2 ) ] . 0 < *2

 + K l ( c o s h ( m x l b l ) " D] . 0 < x x x2 " ^ [ S l n h ( m x 2 b 2 ' * V 2 V V 2 t a n h ( r ay2 a2>

x (cosh (m x 2 b 2 ) - 1)] , 0 < x 2 < b 2 .

Horizontal sec t ion :

t - t f cosh (m YX) t Q - t f

= F l cosh 0 " ^ ) ' ° < Y l < a l

t - t. cosh (m„ 2y 2) =— = F F y , 0 < v < a

t Q - t f 1 2 cosh ( m y 2 a 2 i ' *2 2

V = i v ^ t a n h W ' 0 < y i < a i

f y 2 = ^ a 7 t a n h ' m y 2 a

2 ' ' 0 < y 2

< a 2



5.1.33 continued F = 1 cosh (in _b ) + B sinh (m b )

2 cosh (rajc,t)2) + 1' 2 u2' / v2 t a n h ^ mv2 a2' S i n h { mx2 b2'

B = F2 " V v 2 / v i [ s i n n ( m x 2 b 2 ) + ' V 2 u 2 / v 2 t a n h ( my2 a2'

x cosh (m 2b_)l + 2Vu /v tanh (ra b )

Ol i mxl = W**!' \2 = V 5 E 7 1^2

m x = V2h/ku1, m 2 = V2h/ku 2



5.1.34 2, Insulated wires adjoined on t - t m 1

i

p . 179 a convect ively cooled p l a t e t - t^ 1 +(fe Kj(r H)/K (r Hj] ( e . g . , thermocouple wi res ) , t = t , r - co.

Wire therm, cond. = k . Insu la t ion therm, cond. = k. P l a t e therm, cond. = k . P

tS U T h 2 , l f 2 .

B = Wire radius = r . ~ VET + Y k T w wl w2

Wire in su la t ion thickness = 6.

kpWd^ + h 2 ) ( l / h i + S / k ^ l 1 7 2

r w

w' H = "V2(h + h 2 ) /kw



5.1.35 9, p. 157

i to

Composite finite rod. t = t Q, x = 0. t = t 2, x = x 2. A 1, A_ - areas. C. , C_ = circumferences.

Lh, , t f ph 2,t f

cosh (A) sinh (B) + H sinh (A) cosh (B) + (T /T ) sinh (Dx/x )

T ~ sinh (D) cosh (A) + H cosh (D) sinh (A)

0 < x < x .

A = m 2 (x 2 - X l ) , B = m 1 ( x 1 - x) , D = m ^ , T = t - t f

E " V W l V W z V »1 = V h l C l / k l A l ' m2 =V h2 C2 A2 A2

h—xi—^ U x. F—*x

2 J"l cosh (A) sinh (B) + H sinh (A) cosh (B) + (T /T ) sinh (m b - rax) sinh (D) cosh (A) + H cosh (D) sinh (A)

\ < X < X 2

A = m 1 x 1 , B = m 2 (x 2 - x ) , D = m2 (x 2 - x ) , T = t - t f



5.1.36 9, Case 5.1.35 with: t - t_ cosh (mlx. - IILX) p. 157 q = 0, x = x . t - t cosh (m x ) cosh (m,x2 - "• x ) + H sinh (m x ) sinh (m x - m x )

0 < x < x .

(t - t f) cosh (m 2x 2 - ra2x) t n - t- H cosh (m.x ) cosh (m x - m x ) + sinh (m x ) sinh (m_x - HLX ) '

X . < X < X . .

H = V 5 ^ V W A ' m i = V h l C l / k l A l ' m2 =V h2 C2 / k2 A2 oi - >"2"2 2""2' " 1 1 "1 1' 1 ^ - j r i ' l ~ r 2 y ~2 2 ' "2 2 to



5.1.37 31 Concave parabolic fin See Fig. 5.19 for values at <b(N ,N ). c s radiating to non-free space.

T = T , x = 0. K_ = incident absorbed

N c = (2eoF!.2T^)/kb0, N S = K 2/2eaFT*. radiating to non-free space. T = T , x = 0. K_ = incident absorbed F = view factor.

radiation. . Profile: f (x) = -|(1 - x/l) .

5.1.38 31 Convex parabolic fin radiating to non-free space. Conditions same as for case 5.1.37 except profile is:

f(x) = -|Vl - x/l.

See Fig. 5.20 for values of <j>(N , N ), N and N given in case 5.1.37. c s

5.1.39 81 Straight rectangular fin of variable conductivity. k = k0[l + 3(t-t 0,].

See Ref. 81 for approximate temperature and efficiency solutions.



5.1.40 83 Sheet fin for square array tubes with convection boundary, h f t f .

*

f

o

o o o o

- t — 1 - —

I J I

o o o o

JJ I

+x> +x>

fc- fcf

fco- fcf

2 , Z K

0 l V 5 I O x - mA-2 + < Y - ^ ) n=-»» m=-°°

jca jut

K, RUST) + i n {R>ET) ^ ^ KQ[AVBT(m2 + n 2 ) J

n=- c° m=-°° n , m 7s 0 .

21T

K>§l[(A/R) 2 - IT]

K (KVST) - I ^ R Y B I ) V V K [AVIi(ni2 + n 2j] , 1 / 2

=-m m=-03

K 0 ( K V 5 T ) + I 0(HVBT) ^ ^ K 0[AVBl(ni 2 + n 2 ) ] 1/2

n=-°° 111=-"

n, m ? 0 .

X = x / b , Y = y / b , R = r / b , Bi = hb/k, A = a/b



m i to



Section 5.2. Extended Surfaces—With Internal Heating.


5.2.1 10, Straight, rectangular fin. (t - t ) p. 194 q = q Q , x = 0.

q x = 0, x = b.

1 ,-h.t,

i Ul

I

f . i_ cosh (mx) . . , . , g'' 'b — mkb = -—-• I „. - sinh (mx) + a

tanh (m&) q0m

Optimum profile:

sinh (2mA) i - g 0 / h b ( t 0 - t f )

2mil 1/3 - q0/hb(t0 - tf) ' ° i V h b ( t 0 " V < 1 / 3

m = -V2h/kb, t = t Q , x = 0

5.2.2 1, p. 246 Straight in f in i t e rod. t = t Q , x = 0.

h.t, k q'" F 7

t - 1 , fco- fcf

= 1 + 2 1 -km (t f l - t f ) Miti



U1 i



in I

f\

I 1 I


Section 6.1. Infinite Solias—No Inte \i Heating.


6.1,1 9, p. 54 Infinite plate source. t = t Q , - b < x < b , x = 0. t = t . b < x < - b , T = 0.

i-rr - ![«* K - K)+ e r f K + F 0 ] * •°° < X < <"

See Fig . 6.1

x = "b x = b x

Section 6.1. Infinite Solids—No Internal Heating.


6 .1 .1 .1 74, Case 6 .1 .1 with a plate of p. 424 different properties.

i

© 0 ©

l^k^-TTil (-H rfc n=l

(2n - 1) - X 2VFO. b l

+ erfc <2n - 1) + X ZW%1

J , - 1 < X < +1

t - t. — = K e r £ c / X " 1 \ - K ( 1 + H ' V d H ) 1 1 - 1

\ D2/ n=l

/X - 1 + 2nVa^/ot \ x erfc [ -—=• , 1 < X < - 1

V " ^ /

P 2 C 2 k 2 1 + K

Interface temp

t - t 7 - = V . K ? > (-H) erfc/Fo ) , X = 1 fc» X + K (1 + K) 2 £ V b 1 '

Por Fo.„ small : DZ

fc2 ~ fc» _K . / X - 1 e r f c o °° ziss-y a + K)-

2K , / X - 1 + 2Vl\ erfcl— 2VPo^



6.1.1.2 74, Case 6.1.1 with a pulse source (t p. 427 of heat of strength Q J/m" at

X = 0, T = 0. t = t Q, -«> < x < °°, T = 0.

V P l C l b 1 / ( X 2 \ V n-1 c -^H^/\? ( _ H )

{ (2n - X)' + exp

n=l

(2n + X)' 4 Fo, - 1 < X < 1

«t ^ VP^c^b

I X (-H) n - l

x exp [ fx. - 1 + (2n - D V ^ T T ] 2 I j_L_l _ _ 2_^J_f, K X < - 1

K = Wi n l - K P 2 c 2 k 2 ' 1 + K

6 . 1 . 2 9 , p . 54 Case 6 . 1 . 1 w i t h : °?_ _ 1 -_ X t = t Q ( b - x ) / b , -b < x < b, t - t a

T = 0 .

t = t m , b < x < - b , T = o.

* K - K)+ H^ «*K+ K) - * ««K) j . ; ' a T ,

b TT ( L a

- (x + b) ' exp

• (x - b) '

a2 J 2 exp



6 .1 .3 9, p . 56 I n f i n i t e rod source. t = t Q , | x | < d, | y | d, | y | > b , T = o.

t - t I-TT; = i [ e r f K - F ox) + e r fK + K)]

x [erf (FO^ - Fo*) + erf (j Fo, + Fo b y )]

• + •

6.1.4 74, p . 432 I n f i n i t e cylinder source,

t = t Q r r < r 0 , T = 0. fc = fc=o' r > V T = °"

exp(-X Fo) C 0 " ^ " Jo

\ \ = K f exp(-X 2Fo) c 0 » IT SO

j 0 a R ) j 2 ( X ) J 1(X)Y Q(X) - J Q(X)Y 1(X)

J 0 (XR)J 1 (X)

f"j 1 (X)Y 0 (X)-J 0 (X)Y 1 (X)' j 2

a , » > •

dX , R < 1

See Fig . 6.2


Case No. References Description Solut ion

6 .1 .5 9, p . 55 Spherical source,

t = t 0 , r b , T = 0

t - t

t = t 0 , r ; • K 0 ) 2 ] ) 2 Fo

See F ig . 6.3



6.1.5.1 74, Case 6.1.5 with a sphere of p. 429 different thermal properties.

x erfcA-t-^ _ p VFO7\ - e c f \2VFO; 1 X 2VF07/I

fc0 ' R ( zVio J K2 " ! ^ V F O ^

-Vlj- exp(pjto2 - P2R - l)erfc/j-~^ - P.VFO^

t - t

^ 2 _ + K ^ (K2 - 1) (Kn

• ^ - e r f c / - 1 + 2 ^ " A + — K ^ ^ " 2 - 1 \ 2 V ^ ; ( K 2 - l ) ( K i + l )

x e x p f p 2 F o 2 - P 2 R - 1 + 2 'Va 2 / a 1 êr fc / R - 1 + 2y^/\

V ^

P 2 V F O ^ \ \ , R > 1

P l ° l k l k l K 2 " l K2 ~ X

K l = V ^ ' * 2 = V ? 1 = K 2 + V K 1 ' ? 2 = ^ ^ f (+R) = f (-R) - f(+R)



6.1.6 9, p. 6x Adjoined semi-infinite regions, t = t , x > 0, T = 0. t = t 2, x < 0, T = 0.

PVIKO-],-t - 1 , OQ < j; < 00

6.1.7 7, p . 89 Specified temperature d i s t r i b u t i o n , t = f (x) , - » < x < °°, T

CO

vl / > + X0)exp(-X )dX

Section 6.1. Infinite S o l i d s — N o Internal Heating.


en i CO

6.1.8 9, p. 88 Adjoined semi-infinite regions with contact resistance, t = t , x > 0, T =0. t = t_, x < 0, T = 0.

«2

t - t

t - t,

t^\m r h I1 + A [ e r f K ) + e x p ( v + H f 2 / 4 ) x erfc(Fo + HO /2Yj 1 , x > 0

x erfc( |Fo* | + H 2a 2/2YJ , x < 0

= hyP H = h ( 1 + A > H = - & < 1 + A) k ^ C t ' H l k ,A ' H 2 k ( 1 A )

1 2

6.1.9 9, p. 88 Adjoined semi-infinite regions—no contact resistance. t t

t , X > 0 , T = 0 .

t 2 , X < 0 , T = 0 .

__JL . _ i_[ x + A e r f (P c0j ,

t~rr2 - TÂ e r f (K 0' x < 0

A = ^ k l "2

x > 0



6.1.10 9, p. 89 Plane source of heating between adjoined semi-infinite regions. t = t . , < X < <=, T = 0. \ = q0r x = o, x > o

= heat source strength.

l X »i

( t - t >k , , . J- 1 = T - ^ T ierfclFo ),

nO. 1 + A \ x/' x > 0 "01

( t - V k2 q 0 a 2

= r T T i e r f c ( l F o x l ) ' x < 0

k * a 1 2

6 . 1 . 1 1 IF p . 341 Line source of h e a t .

9 , p . 261 t = t , 0 < r < rQ, T = 0 .

ill

t - t

t„ - t It Fo 0 °°

exp (- K 2) • r > V r0 * °

I f l i n e i s a s t e a d y source of s t r e n g t h q :

( t - t )k

"i/Vf



6.1.12 1, p . 342 Cyl indr ica l s h e l l source. t " V r i K r < V T = t = V r > r Q , T = 0.

t - t z~ = (Fo* - Fo 2 ] exp[-(Fo* 2 + Fo* 2 \ J „ / 2 i Fo Fo \

t o " f c - V r o V L V r r i /J °V r r J -v=i

at I

6.1.13 9, p . 247 Spherical surface a t steady temperature, t = t . f r > t , T = 0. t = t 0 , r = r f l , T > 0.

t ' fci 1 / * - — - = - e r f c (Fo r - Fo

0 l \ ; )



6.1.14 9, p. 247 Spherical surface with convection Ipoundary. t = t . , r > r Q , T = 0.

B i t - t. i _ _ t t - t . (Bi + 1)R f l

erfc fPo - Fo | exp[(Bi + 1){R - 1)

+ (Bi + l )Fo]erfc Fo* - Fo* + (Bi + 1)YFO I . r o J |

I P 6.1.15 9, p . 248 Spherical surface with

steady heat f lux, t = t i f r = r Q , T = 0. q r = V r - V

(t - t )k x r , * ~ = =r er fc (Fo - Fo

%r0 ; )

- exp(R - 1 + Fo)erfc(Fo^ - Fo^ + V F O | | \ r n

Section 6 . 1 . I n f i n i t e Solids—No I n t e r n a l Heating.

Case Ho. References Descr ipt ion Solution

6.1.16 1, p . 336 Point source. t - t. " °° C 0 / „ *2\ — = - exp - Fo , i " fc°° A C U V r / t = t . , o < r < v x = o. TT^X, = 7 T ^ e x p l" F°r"i ' r > V ro * °

t = t m , r > r , T = 0. X °° 6Vira T

If point is a steady source of strength Qfl:

( t S t ) k r f l

V r 0 " 47r 3 / 2Va 1 ? V T

— / e x p l - X Fo JdX ff A/= V r '

F o r T •+ <*>:

(t - t j k r 1 Q 0

= 41T



6.1.17.1 1, Spherical shell source. p . 340 t = t . , r. < r < r

1 1 T = 0 .

0 '

t - fc~ R 0 - 1 [ ( R - l ) :

t . - t_ 6VTTRVFO e x p L 4Fo exp (R + 1)

4Fo ' R > V R0 * °

R = r/r.

If the shel l i s a steady source of strength Q :

(t - O k VoT V r i 47t 3 / 2

For T •+ °°:

(t - t J k r

OO

exp

, 2 , 2 (r - r ^ X

4a - exp (r + r ^ 2 * 2

4a d)

_ 1 4ir

6 . 1 . 1 7 . 2 1 , p . 342 Plane s o u r c e .

t = t t , -a < t - o |x| > i, x = o.

9, p. 263 t = t., -A < x < +J-, T = 0.

/MAA

rtA(^y

I I

t - t_

1

If the plane is a steady source of strength q

r r ^ f - = ik exp(~ F o x 2 ) ' ixi * l>*+ °

(t - t j k

1/Vx A

2 2 exp(- x X /4a)dX

= VFO" o exp I (-«0-Jf ««.(»;„ i)



6.1.18 9, p. 335 Cylinder with steady

surface temp,

t = t i f r > r Q , t = 0. t = v r - v T y °-

CO

o i Jn

JQ(XR)Y 0(X) - YQ(XR)J0(X) Fo) —

See Fig. 6.4 <J r

n

Jg{A) + Yg(X)

1/2 k ( t „ - t . ) 0 l

For Fo < 0 .02:

1 ( t - tt) ( t 0 - t . ) " V5

ax_ x

. „ « , - ^ . i - l ( P ) - , i f c . . . . . - !

32R

6.1.19 9, p. 337 Cylindrical surface with convection

t = t i f r > r Q r T = 0.

' ""*: ^ f , Bi f boundary. - — = -2 — I t . - t_ if I exp(- X'Fo)

J 0 ( X R ) [ X Y 1 ( X ) + Bi YQ(X)] - YQ(XR)[XJ1(X) » Bi J0(X>] ^

[X^tX) + Bi J

0 ( ^ ) ] 2 + [X^lX) f Bi YQ{X)J



6.1.20 9, p. 338 Cylindrical surface with steady heat flux, t = t i f r > r Q, T = 0. qr = V r "" V T > °-

I

( t ~ V k 2 f r 2 1 J0 < X R ) Y 1 ( X ) " Tf0(XR)J (X)dX

— — — = - - / L 1 - e i cP(^ F°>J r , 5 — T ! ^ 0 * JQ ^jJ(X) + Y*(X)JX2

For Fo < 0.02:

(t - t.)k Vo

For Fo » 1

(t - t.)k

i'K <lFo . , /R - 1\ 3R + 1 ^=- .2 . /R - 1\ ^ i - = 2 V - ierfc(-^=) - - j j - VPS" I e r f c ^ - ^ )

= An/^f^ - 0.57722 vo ~*"U2>

Section 6.1. Infinite S^ Lds—No Internal Heating.


6 .1 .20 .1 74, Case 6.1.20 with two dif-p . 434 e ren t thermal m a t e r i a l s ,

t = t . , r > 0, T = 0. g c = q n , r = r 0 , T > 0.

en l

(t. - t . ) k J„(A R)J.t t)dA

Vo / J L v ^ X4<<!> +

( t - t 4 ) k

X R i * • v o \ ^ a X R ' *

^h=^ /"[l-expôj ~^f-

* " K J i ( X , 7 o W ? x ) ->§• V x > J i W r x

(<j) + ill')

J , (A)

, R < 1

(* + r )

dX , R > 1

"4 K 2



i

6.1.21 9, p . 342 I n f i n i t e conduct iv i ty

cyl inder in an i n f i n i t e

medium.

t = t Q , r > r Q , T = 0.

t = t , r < r Q , T = 0.

CO

fc " t 0 40 f , , 2 _ %dA R < 1

B = 2 p 0 c 0 / p l C l

A = [XJ„(X) - B J ^ A ) ] 2 + [XY0(X) - BY^X)] 1

See Fig . 6.5

6.1.22 9, p. 342 Infinite conductivity cylinder with steady heat source in an infinite medium. t = t , r > 0 , T = 0 . Heat rate = Q per unit length.

(t - tQ)k CO

1 - exp(-X2 Fo)]^-, R < 1 X 3A

B and A defined in case 6.1.21.

See Fig. 6.6.



6.1.23 9, p. 346 Cylinder with properties fc » j(XR)J<X)dX different from surrounding — = —fr I exp(-A Fo, ) —_r _ r — , R < 1 medium. t = t L, r < r Q, T = 0.

t - t r , N J (XR)J (X)d) rounding = -£ / exp(-A Fo ) — - p - ^T

fc- * 0 2Y f ( ,2, \ J l ( X ) [ J 0 ( e X R ) " ' - Y 0 ( 6 X R ) ' t ' ] d X

= —*- I exp -X Fo, I = -0

i|) * Y^cXj j j j jBX) - B J 0 ( X ) J 1 ( B A )

<j> = y J ^ X j y ^ B A ) - BJ o (A)Y 1 (0X)

R > 1 .

A X y B - V 5 ^ , Y = k A 00

See c a s e 6 . 1 . 4 f o r e q u a l m a t e r i a l p r o p e r t i e s .

6 . 1 . 2 4 9 , Po int source o f rec tangular ( t - too)4Trrk

p . 402 p e r i o d i c wave h e a t i n g . Q~ " e r c * "'**' T ^ + 1

CO

(ST) - T i

exp(-TX"")j e x p f - X 2 + T X 2 j - e x p ( - X 2 ) J sin(Fo*X)dX

x[l - exp(-X2)]

Q(T) = 0 , x < 0. „ ^ 2 j

Q ( T ) = Q Q , nT Q < T < nTQ

+ T.f n = 0 , 1 , . . .

Q(T) = 0 , n t f l + T < T < (n + 1 ) T 0 . 0 < T < T . n = Number of c y c l e s .

t = t ^ , r + co. T x = T 1 / T 0 , T = T / T Q , Fo* = r/V5f^

>* r

Section 8.2 . Solids Bounded by Plarie Surfaces—With In t e rna l Heating.

Case No. References Descr ipt ion Solution

8.2.3 9 , p . 131 I n f i n i t e p l a t e with time dependent i n t e r n a l heat ing, t = t . , -Jt. < x < ±SL,

T = 0. t = t , X = ±1, T > 0. q " ' = f ( T ) .

q"'

h-H U*|

fu i. \ *» V ( -1)" r(2n + l ) 1 (fc " t i ) = iiT Z Un + 1) C O S L 2 — " X J

n=0

x J f (T) exp -ir (2n Jo

+ l) 2a(T - x ' ) / i 2 Jd t

8.2.4 9, p. 132 Case 8 .2 .3 with f t t ) = q £ " e " b T .

(t - t j k

I n * ' *

i 1 i fcos (yy^d) .-] m , . ,

2 ^ (-1)" cos (X^X) exp (-X* Fo)

n=0

X = (2n + 1 ) TI/2 n

m

Oi B

•rt JJ 10 0> £ H fl c M 0) +J c c -H 0

-w 0 4J

f u & O •0 m •A 41 M a 0 0 )

4» t> •rf <a M4 s U-l c C 01 W u 0)

<M

, S r-( Pi M>

» c 0 o z •"* 1 .. 4-1 1 » o 2 d> 1 w to 1 O

6-2Q



i

Section 6.2. Infinite Solids—With internal Heating.


6 .2 .1 9, p . 265 Cylindrical source of heating, q ' " = q j " , 0 < r < r 1 #

-SL < z < +1, T > 0 .

q " ' = 0 , r > i t , | z | > J., T > 0. t = t, . , r > 0, | z | > 0, T = 0.

q"' = 0

•1 — 8,

5— = ^ r / 1 - exp(-r / 4au ) erf( X 0 * ' r l r i J0 V™

( t - t . ) k du, r = 0, Z = 0

For S, + » :

( t - t.)(c

7^T- = * £ ' exp(~ ^ ) ] " « Ei " ™ ) ' r = °' z = o r q 0 1

Po = OT/r

6.2.2 9, p. 347 Case 6.1.23 with a cylinder and internal hea t ing . t = t Q f r > 0, T = 0. — i i i — n t i i

to - — ( t ~ fc0)k0 _4_ f 11 - expi-\2Fo1)\jQ(\R)Jia)a\

q " 1 , 0 < r < r n . T > 0 . ( t " t n ) k - V kQ 2 f [ l - exp(-X FOjfl [J 0 (BXR)(|) - Y0(SXR)4)ldX

x 3 [ * 2 • * 2 ] R > 1,

1, <|)(and ii defined in case 6.1.23 .

Section 6.2. Infinite Solids—With Internal Heating.


6 . 2 . 3 9 , p . 348 Sphere wi th i n t e r n a l h e a t i n g ( t - t n ) k _ *2

i n an i n f i n i t e medium,

t = t Q , r > 0 , T = 0 .

- V " Fcflr L"rl ' 4 L

1 - 2R i e r f c

*0 0

' " = q j " , 0 < r < r Q , q T > o

0 , r > r .

T > 0 .

2 Fo .3 , •=—-— i e r f c

0 < R < 1 .

( t - t . ) k *2 ,

*0 0

0 ' FO f. 2 . /R - 1 \ x .2 /R + 1 \ * .3 c /R - 1 \

+ Fo i e r f c , R > 1 ,

Fo = 2-vSr/r



6.2.4 9, p. 349 Infinite conductivity sphere r • *• -u *• ' ^ u ( t " 0* 1 1 + Bi 6 2 _ .2 i n an i n f i n i t e medium with -— = ———. — K Bi , , . 2 3 B l TT

contact r e s i s t a n c e . ^0 0 L t = t , r > 0, T = 0. » 2

q ' " - q j " , 0 < r < r Q , T > 0. X ^ ^ ^ 2 " f^ V „ » . - 0 ° r > 0. r > 0° K [ u 2 d + Bi, - K B i ] 2

+ [ u 3 - K B i „ ] '

0 < R < 1 .

f \ o-.c- / For small values of T:

< V ' > r c r k l ( t " V " _ K Fo - 3 [1 - (K Bi Fo)/2 + . . . J a ' ' ' r q 0 0

For l a rge values of T :

a ' " r 2 q 0 0

lpL + Bi 1_ 2 + Bi(2 - K) 1 3L Bi " * F o " 2 B . K 7 i A / ^ J - J

3P c K = " T T ^

P 0 C 0



l to

6.2.5 74, p . 426 I n f i n i t e p l a t e with i n t e rna l heating in an i n f i n i t e medium, t = t_ , -I < x < I, T = 0. t = t m , «. < x < - £ , T = o.

i U "

fc J . i .

= 1 + PO Fo, erfc (2n - 1J_ + X 2VFo7

+ 4 Po Fo,

t - t„

tfi " ^

1 1 + K L, * ' n=l

- r i r [-* (1*4) * * - "2 <2«- (§*$

l < x < l .

n=l

'2riVa / a + X - i e r fc! i W S T -

2 / 2nV5q7c^ + X - 1 + 4 Po F o 2 i e r fc ^ ^== , 1 < X < - 1

K = 1 V , H = f — f , f (+X) = f (-X) - f(+X) P 2 C 2 k 2 1 + K

Section 6.2. Infinite Solids—With internal heating.


i to 01



I

7. Semi-Infinite Solids — Transient |

o o

D 111 O ID D O o: i l

ea < < > <

UJ W

CQ

Section 7 .1 . Semi-Infinite Solids—Ho Internal Heating.


7.1.1 9, p. 61 Constant surface temperature, t - t / *\ bx t = t Q + bx, x > 0, T = 0. t = t f x = 0 , T > 0 .

s

0 s

For b = 0:

V _2 k ( t s - t Q) -Vir

0 s

See Fig. 7.1

i

7.1.2 7, p. 89 Variable initial temperature, t = f (x), x > 0, T = 0. t = t , x = 0 , T > 0 .

s

t - t s ~ cryir J, / «,L[-(H*)}--.[- (^)1 dB f (B) = £W

Section 7.1. Semi-Infinite Solids—No Internal Heating.


7.1.3 5, p. 80 Convection boundary 74, p. 206

t = t i f x > 0, x = 0.

t,.h

t - t . 1 _

fcfi-fcl

h ( t f - 1 . )

1 - erf (FO^)

~ expK + B lx ^ i 1 ' e r f K + B i x * ^ l l

= exp(crrh 2 /k 2 )erfc(Vox hA) See Fig. 7.2

T 7.1.4 to

2, p. 275 Ramp surface temperature. t - t. t = t., x > 0, T = 0. t = t. + bT, x = 0, T >_ 0.

t = t. + br.

bT (l + 2 Fo^2) erfc (Fo;) - § F o ; exp(-Fo;2)

7.1.5 1, p. 258 Steady surface heat flux, t = t., x _> 0, T = 0. g^ = q Q, x = 0, T > 0.

(t - t.)k i^5f- " "* F o x «fc(Fo*) + exp(Fo^)

(t - t )k - " d - x = o q0V5r" VW See Fig. 7.7

Section 7 . 1 . Semi-Inf ini te Solids—No I n t e r n a l Heating.

Cases No. References Descript ion Solution

7.1.6 9, p. 62 Time dependent surface temperature. t = t Q, x = 0, 0 < T < T 0. t = t^, x = 0, T > T Q. t = t , x > 0, T = 0.

t - t. erfc K).

t - t

0 < T < T„

t. " t, ±- = erfc(F0;) + ^ - ^ erf c(^-2—-y) . T > T„

7.1.7 9, p. 63 Time dependent surface temperature, t = t., x > 0, x = 0. t = bT, x = 0 , T > 0.

bT-

^ i = (l + 2 F o ^ e r f c ^ ) - |= Fo* exp (-Fo^2)

7.1.8- 9, p. 63 Time dependent surface temperature. t = t . , x > 0, T = 0.

l t = bVx, x = 0, T > 0.

t - t . bt

- = expl-2 Fo "J - Vix Fo erfcfFo J

by/r-



/.1.9 9, p. 63 Time dependent surface temperature.

t - t. ^53* - A (f • i)i"-*.K)

t = t., X > 0, T = 0-i t = t t + br"^2, x = 0, T > 0. See Fig. 7.6

br' .n/2.

I 7.1.10 9, p. 64 Time dependent surface temperature, t = t., x > 0, T = 0. t = e , x = 0 , T > 0.

t - t ^ - i = | exp(- xVb/ty erfc /FO* - Vbr\

+ exp(xYb/ct)erfc(Fo + VbTJ



7.1.11 9, p. 74 Convection boundary with t - t. step fluid temperature t = t , x > 0, T = 0. t f = V 0 < T < T r t _ t

t f = t 2 , T > V

t f .h-

I ui

;— = erfc (FO*) - exp[Bi x + (H 2 /4}Jerfc |Fo* + (H/2)J

i - = erfc^Fo*) - exp[Bi x + (H 2/4)jerfc|Fo* + (H/2)J

F h a ( T - T 0 ) - | exp[Bi x + jj j

, 0 < T < T,

+ •— ^ I e r f c ^ " ' i 2Vct(T - TQ)

x e r fc 2Va(T - x Q )

+ k | ' T > T o

H = 2hV5r


Case No. References Descript ion Solut ion

7.1.12 9, p . 76 Step surface heat f lux . ( t - t . ) k t = t . , x > 0, T = 0. IQ*^*

= 2 i e r f c v x / T \ « & < T - x0)J

^ = q Q , X = 0 , 0 < T < T Q .

^ - 0 , X - 0 , T > V T > T

0

See case 7.1.5 for 0 < T < T. .

_j ' 7 .1.13 9, p . 76 Time dependent surface t - t . = 1, x = 0, 0 < T < T 0* heat f lux.

t = t l f x > 0, T « 0. t - t i = | s in _ 1 (VT 0 /T ) r x = 0, T > T 0

q = k/Vcrrir, x = 0 ,

° * T * V 2 / V ^ V expf- Fo*2,l + u 2 , l q x = 0, x - 0 , t > T Q . t - t i = ? y L

x ' u 2 ^ d u , T > T 0



7.1.14 9, p. 77 Time dependent surface heat flux. t = t . , x > 0, T = 0. q^ = l ^ 2 - x = 0, T > 0,

n = - 1 , 0, 1, 2 . . .

( t - t ^ H I + n/2) _ / o - /•> _ \ ' X U f I ' U ^ ^ 2 " r(| • f)

( t " X l , - 2 ( 4 ) n / 2 r ( l + n / 2 ) i n + 1 e r f c q0VOT T n/2 ;

See Fig . 7.8 for n = 2

K)

V 7.1.15 3 , p . 402

p l a t e and semi - in f in i t e so l id composite with constant surface temperature. t = t . , - b < x < <*>, x = 0. t = t , x = - b , T > 0.

n=0 '

er fc (2n + l ) b - x , -b < x < 0,

fc2 " fci _ _ 2 V n _ f u n + l ) b + 6x'

n=0

6 - V a i / a 2 , X = k 2 6 A r x B = p ^ y

, x > 0



7.1.15.1 74, Case 7.1.15 with: p. 409 t = t ,0 < x < b,

T = 0. t = t_, x > b, T = 0. t = t , x = 0, T> 0. o

t - t t~=\ = e r f° (F°x) * H 2 H n _ 1 e t f c ( 2 n F o b * F°x)

n=l

t, - t (t \ ) (1 + K) S ^ e r f 0 [ n=l

(2n - 1) Fo. + Fo, :]• O < X < 1 .

t - t 1 |_ = JUL y H""1 erfc TFO* - Fo* + Fo, t - t. 1 + 1 L L x x I ° n=l

.. + (2n - 1)B Fo. b b. a (fcl - V K

(tQ - t x) (1 + K) erfc Fo - Fo. + x b 2 K

(1 + K)'{tQ - tx)

y H n - 1 erfc |Fo* - Fo* + 2nB Fo* , X > 1 . n=l

Plclkl _ 1 - K _ W 2

Section 7 . 1 . Semi- Inf in i te Solids—No I n t e r n a l Heat ing.

Case No. References Descr ip t ion Solution

7.1.16 7, p. 99 Periodic surface temperature, t = (t„- t J cos I — — \ x = 0, T > 0. t - t , x > 0 , t = 0 .

0

l T o / ' i ^ = exp ( - V J x) cos (u,x - V J x ) , See Fig 7.3 t - t„ V" "o

T = maximum surface temperature m T 0 = per iod of pe r iod ic temperature a) = 2irn/T n

Maximum temp:

fcmax ~ fc0 _ o m ( yljlx\

Time for t t o reach xs max

St _ m X fc0 n 2 "Vamrx^

, m = Q, 1, 2,

Heat t r ans f e r r ed i n t o s o l i d , x < x < x : q V a A 0 1 T I %\ i ir\1

-— r-r- = cos l u x . - - I - cos (a)X_ - 7 )

Section 7.1. Semi-InLinite Solids—No Internal Heating.


7.1.17 7, p. 106 Periodic fluid temperature, t. - £ =(tfM " t 0) cos &2L\ T > 0. \ ° / t = t . JC > 0, T = 0. 0

t - to t Cw - t fM o

exp (-yiT* U)T 25 x - t a n

Cl + (2/H) + (2/H 2)] 1 / 2 !Mi)]

t f.h-

H = — 2 = - , ID = 2mr/x imlc °

t = max fluid temperature T. = period of periodic fluid temperature

Maximum temp:

fcM- fco exp ( - # * ) ( tfM " V [l + (2/H) + (2/ H 2 ) ] 1 / 2

7.1.18 9, p. 68 Square wave surface temperature.

0. *• + V X

2mT. < T < (2m + 1)T , 0 0

m = 0, 1, 2, . . . . t = t - t , x = ra D (2m + 1)T < T <

0 (2m + 2)T .

0,

" T> nTo X ° ' L ° _ x /t2n +1)1 X V 2 a r 0 J

irr

T = period of periodic temp 0 t = mean surface temp m r

L = deviation of surface temp from t

7.1.19 2, p. 248 Quarter-infinite solid. 7.1.20 2, p. 248 Eighth-infinite solid. 7.1.21 2, p. 248 Semi-infinite plate.

Dimensionless temperatures equal product of solution for semi-infinite solid (case 7.1.1 or 7.1.3) and solution for .infinite plate (case 8.1.6 or B.1.7). See Figs. 9.4a and 9.4b.



7.1.22 4, p. 84 Infinite cylinder in a semi-infinite solid. Cylinder assumed raassless.

Time for soil to reach steady state:

B|= C(d/D + 1/Bi I 1 " 4 4 , D

I

2 d

C = 4.6 for const, cylinder temperature. C = 6.0 for const, heat flux from cylinder.

The heat transferred during the time given above: -2SL

KD (t c- t f) 12(d/D + l/BiD) 1.25

Temperature of cylinder = t . Initial temperature of semi-infinite solid = t f. t > t,. c r

The time for the cylinder to re tu rn to temp t a f t e r steady s t a t e i s achieved and heat ing i s stopped:

^ = (d/D + VBi/"3 f (^f-D \ c f

See F ig . 7.4 for values of f,



7.1.23 9, p. 264 Semi-infinite surface heating, t = t., 0 < x < <*>,

- C D < r < +ot>, T = 0 .

% - v x = °--oo < y < 0 . q = 0 , x = 0 ,

0 < y < +°°.

( t - t . , f c 1

q(T/ax Vn Po

erfc K)+S*Ei W 2). , x = 0 .

7.1.24 9, p. 264 Infinite strip heated surface. t = t., 0 < x < o°, -=o < y < +=o/ 1 = 0 . q x = q Q, x = 0, -a < y < +a. g^ = 0, x = 0, y > a, y < -a.

: = -n=- 1 erf / 1 + erf / V -« {*" lâ ) + e r f( 2^J (zVKîr)

Ei 1 + Y. 2VFo\

i - y \2VFO" \ 0

Ei 1 - Y 2Vfo"

x = 0 .

See Fig. 7.5



7.1.25 9, p. 264 Heating through a circular surface.

% ' V ° < r " V X = 0 , T > 0. c r = 0 , r > r , x = 0 , T > 0 .

t = t . , 0 < r < ° ° ,

0 < x < °°, T = 0 .

( t - t . ) k

V c ~ = 2 > f \ r e r f c K ) ^xV 1 + ( r c / x ) 2 ] - i erfc , r = 0


Case Ho. References Description Solut ion

7.1.26 9, p. 265 Heating through a rectangular surface. t = t . , x > 0 , y > 0 , Z >. 0, T = 0 . % - V -w < y < +w, -% < z < +&, x = 0 ,

T > 0. g^ = 0, x = 0 | y | > w, |z j > I,

T > 0.

Maximum temp:

('max ~ fc i)k q0w

Average temp:

m l )k q0w l

| s i n h - 1 (L) + L s i n h - 1 QS\, -W < y < +w,

-£ < z < +S,, x = 0 .

2 - 1 — jsinh (L) + L sinh •HiHH + i

- ^ ^ ] | . - » < y < +w, -S. < z + £ , x = 0 .

L = l /w



7.1.27 9, p . 401 Rectangular wave surface heat f lux . t = t , x •*•<*>, i > 0.

CO

q = q n ( T ) , x = 0 , T > 0 .

q n(x) = o r x > o. 9 0(T) = q „ , n x 0 < x < nx Q + V

n = 0 r 1, . . .

V T ) " °' n T0 +

A T , < T < (n + l ) T n . ui x u

n = No. of cyc les .

At x = 0:

(t - t_)k

V a T o

( t " t I k

2 T _ ^ Vr + - — [(1 - T x) V^ - I [Tv T)/ViF],

0 < T < T,

<*> 2 ~- 2 T r 1/2 ~\ _ _ _ = __ ^ vr + - — ^ _ T i , V5-_ ( T . T i , _ x { T ] / T ) / V n J ,

T < T < 1 ,

CO

I ( T 1 # T) = /

q„(r)-

exp (- T X2) 1(1 - T ) exp (-X2) - exp (-X2 + X 2 ^ ) + T 1

X Z[l - exp (-X2)]

dX

T = T/TQ , Tx = T l A 0



7.1.28 9, p . 419 Semi- inf in i te cyl inder See case 9.1.20 with convection boundary.

t = t Q , 0 < cQ< c.., z > 0 , t = 0 .

t = t , , 0 < r < r Q , z = Q, T > 0.

Convection boundary of h, t . at r « t . i z > 0, T > 0.

7.1.29 9, p. 419 Case 7.1.28 with: See case 9 .1 .21 t - t x , r = r 0 , z > 0, T > 0. t = t n , 0 < r < r , z = 0 , T > 0 .



7.1.30 19, Plate and semi-infi-p. 3-83 nite solid composite

with constant surface heat flux, t = tQ, 0 < x 1< 6, 0 < X < <*>r T = 0 . k •*• ».

q = q", x = 0, T > 0.

k 2 ( t r V _ i 2 q" Vo^r 2 m VFo 2 L

exp (m Fo J erfc (m VFo ) -1

+ — , 0 < x < 6 VF 1

Heat flux at interface:

q /q" = 1 - exp (m Fo j erfc (m VFo ) , x = 6 ,

• = (P2 c 2/p x cx)

See Fig. 7.9

7.1.31 74, Semi-infinite rod with p. 211 convection boundary.

t = t . , x > 0 , x = 0 .

t = t _ , x = 0 , T > 0 .

— - ± - = 1 exp (VBTd X) e r f c ( j ^ - B i d F o d

+ exp ( B i , X) e r f c d n "—— + VBTTFOT

2 V F o , d d

T" t,. n

. cross section area d = : r , X = x/d perimeter Heat flux at x = 0:

V k(t f - t. = = exp (-Bi Fo J + V B T , erfc (VBi Fo,) IT VFoT d d d d d


Case No. References Description

7.1.32 74, Semi-infinite plate p. 465 with fixed surface

temperatures.

i

v 1

Solution

r } - = x + I 2, *=i*~ sir> (A x> exp (-X2 Fc>) fco " fci n n=l n n \ n /

CO

+ ^ 2, "^n1" s i n < x X) [exp (AnY) erfc

x rPo* + X n VfoJ + exp {-XnTC) erfc (FO* - X Vfo)

("Xn F O ) e r f c ( F Oy)] ' A n = nir.

- 2 exp

Section 7.1. Semi-Infinite Solids—No Internal Beating.


7.1.33 74, p. 466

Semi-infinite cylinder with fixed surface temperatures. t = t i f 0 < r < r Q, o < y < °°, T = o .

t = t ± , 0 < r < r Q ,

y = 0, T > 0.

fc • V r = r 0 ' ° < y

t - t . t - t . ,— J . (A R) r / o \ / * \

n=l

+ exp (X Z) erfc (Fo + X V F 5 )

+ exp (-X Z) e r fc ^Fo - X V F O \ " | .

< =», T > 0 . w =o

K7 H

7.1.34 74,

p . 466

Case 7.1.33 with: convection boundary h , t . a t r = r , and

t = t Q , 0 < r < r Q ,

Z = 0 , T > 0 .

t - t. 2 Bi J 0 (X n R) exp (-XnZ)

£L J 0 ^ ( B i 2 + X n )

Bi J n (X„R) ^ - Bz J [AR] r / * \ + / 7—= =\ exp (X Z) e r fc (X VFO + Fo }

~ J„(X ) f B i 2 + X 2 U n V n * ' n=l 0 n \ n /

exp {-X Z) e r fc f\ VFo - Fo*)

W " B i

Section 7.1. Semi-infinite Solids—No Internal Heating.


Section 7.1. Semi-infinite Solids—No Internal Heating.

Case No. References Description ' Solution

i

Section 7.2. Semi-infinite Solids—With Internal Heating.


7.2.1 9, p. 79 Steady surface temperature and initial temperature distribution, t « T. + bx, x >. 0, T - 0. t « tQ, x * 0, T > 0.

t - t, 0 * * * 2 / * \ 2 * * — = (1 + Po + Po Fo ) erf (Fo I + jpr Po Fo exp (" »D

bx _ * _ *2 + Po Fo . T. x

PO * a' • ' q'" at

7.2.2 9, p. 79 Case 7.2.1—variable heating, t • T, + bx, x >_ 0, T = 0.

t u

q ' " - q,

tQ, x = o, T > o. - q ^ " e _ B x .

(t - t o)k0" bx

T, Po - b " 1 ) * -Sx erf (Fo ) + 1 - e p

i • X - B * er fc (VX - Fo*) - \ e

A + p x e r fc (V* + F o * , ,

q ' 1 • Po*= — ~ , X = <*32T



7.2.3 9, p. 79 Case 7.2.1 with band heating, t = T., + bx, x >_ 0, T = 0.

t •> t , X • 0 , T > 0 .

g ' " - q j " , 0 < x < d. q ' " = 0, x > d.

i to

t - t, — - = Po* l l - 4 i erfc / F O j + 2 i 2 e r f c IFo* (1 + X) 1

i 2 e r f c [ F O * ( 1 - X ) ] j + § * + erf ( F G * ) . 1, 0 < X < l

t - t — - = 2 Po* J i 2 e r f c fo*(X - 1)1 + i 2 e r f c |Po*(X + 1)1

i 2 e r f c ( F o * ) ) + ^ + e r f ( F O * ) , 2 i

* q • • > ax Po _ JO

kT,

I , X > 1 .



7.2.4 9, p. 80 Insulated boundary and band heating. t » t . , i > 0, T = 0 .

X = 0 , T > 0 .

(t - t . )k

g^ = 0,

q ' " = q ^ " , 0 < x < d,

T > 0.

q ' " = 0, x > d.

, . , . * T = 1 - 2 i 2 e r f c [FO*{1 - X)l - 2 i 2 e r f c [FO*(1 + X)l ,

0 < X < 1 .

. . . a T " 2 j i 2 e r f c [ F o

d

( X " 1 } ] " i 2 e r f ° [ F ° d ( X + 1 } ] ) ' X > 1

I HJ

7.2.5 9, p. 80 Insulated boundary and variable heating, t = t Q + bx, x > 0,

T = 0 .

a = 0, x = 0, T > 0. q . . . = q j . , e - 6 x t

x > 0 , T > 0 .

r r n = — * " exp (-Bx) + — ^ M 0 Po \Fo Po

X \ X X

+ 2X1 erfc K) + j exp (X2 - Bx) erfc f\ - Fo*} + j exp (X2 + Bx) erfc (x + Fo*)

Po * l i " «

kbx , X = B Var

Section 7.2. Semi-infinite Solids—With Intecnal Heating.


7.2 .6 9, p. 323 Plate and semi- inf i - . . . n i te so l id composite. ' 1~ (T 1 t = t 0 , -b < x < °°f

T = 0. t = t , x = - b , T > 0.

*0 ' -b < x < 0, x > 0.

g- ' -ar = 1 - 4

e i erfc

0 , X '< 0 , T > 0 . X .2 „

i erfc 1 + X

n=0

f(2n + 1) - X~| X .2 . / 2 n + X\ L -2Vfo- J + 1 + X * e r f c \Z^ }

("Yvfe * ' ) ] - * < - < ° • ( V t « ) k " V k l 4 r . n f . 2 _ / 2 n + j x \

^ — = (in) z B { 1 e r f c VTvfs-J n=0

. f(2n + 2) + 6x1 , • 2 „ f(2n + 1) + 6x1 I v -, n f c L 2 v i s — J - 2 1 e r f c [ 2VP5—J J* x > °-

+ i 2 e r f

6 = VSp5T2, x = k 26/k l f 3 = f = - i FO = ¥-, x = * b



7 .2 .7 9, p. 307 Time dependent heating and convection boundary. t = t , x > 0, T = 0. q " 1 - q ^ " T S / 2 , s = - 1 , 0, 1, . . .

V h -

(t - t | J ) k - i ., rm + (s/2)] , , , „ T l + ( s / 2 ) [1 + (a/2fl F o ( s / 2 ) + l , _ n 4 ,s+2 q^'OT — J Fo^~' (-Bi x)

exp Fo + Bi V P o

X X I X , (si + B i 2 Fo \ erfc (\

s+2 / n . in

<* 7.2.8 9, p. 308 Constant temperature surface and band beating, t = t - , x > 0, x >» 0. t = t Q , x = 0, x > 0. q " ' = 0, 0 < x < x , x > x„.

q " " = q , ; " . X]_ < x < x 2 .

Surface heat f lux:

%'• at = 2 i e r fc (Fo \ - i er fc |Fo 1 , x =


Case No.. References Description Solution

7.2.9 54 19, p. 384

i to

Exponential heating, t = t Q / x > 0, T = 0. q " ' •= q ' " exp (-ux).

k(

2~q,

fc" V / *\ e x p ( _ 2 M F o x ) r 2i ^k= l e r f c K) — i i — - L1 - «* »n

_ HEjaa! [ e x p (. 2 .„ F c g e r f c ( F o * _ H )

- exp [ 2 H F o j er fc (Po + M^ .

Surface temp:

M t w " V = 0 5 6 4 2 _ 1 - e x p (M 2 ) e r f c (M) 2 q ( J 1 , V a r r 2M , x 0

M = U V o f See Fig . 7.10

7.2.10 74, p . 383 Planar heat pulse and convection boundary. Instantaneous heat pulse occurs a t T = 0 , x = x , with s t r eng th

•V m

t = t , X > 0 , T = 0 .

( t - t£)kx1

Qa = 2 V ^ ° | e x p

- Bi exp Bi (X -

(X - 1 ) ' 4 Fo + exp

(X -

1) + Bi Fo er fc X + 1 2 Fo + Bi VFn ]l


i to

Case No. Refe rences

7 . 2 . 9 54 1 9 , p . 384

D e s c r i p t i o n S o l u t i o n

E x p o n e n t i a l h e a t i n g . t = t Q, X > 0, T = 0. q i i i = gîi e x p J.^J

7.2.10 74, p. 383 Planar heat pulse and convection boundary. Instantaneous heat pulse occurs at T = 0, x = x, with strength Q J

m t = t

2

r X > 0 , T = 0 .

k ( t - t > o- /_ *\ e x p (" 2 M F o

. . . ,— = i e r f c (Fo 2 qA M VccF V x K) -*

X M ^ -

exp (M) i exp (M)

4M exp (- 2 M F Ox) Fo ) erfc (FO XJ \ X K - M)

- exp ( 2 » * > » ) ('

erfc [Fo + -)i Sur face temp: k ( V t o )

2 q J ' W = 0.5642 1 - exp (M ) e r f c (M)

2M x = 0

M = UYOfT

See F i g . 7 .10

( t - t f ) k x x

Qot = 2-VTTFO exp (x - i r

4 Fo + exp (X - 1) 4 Fo

- Bi exp .2 (X - 1) + Bi Fo e r f c X + 1

2 Fo + Bi YFn





i •o

OB

SB 9 «

H

s




8 .1 .1 9, p. 94 In f in i t e plate with steady surface temperature. t » t ( x ) , 0 < x < & , T = 0. t = t , x = 0, I, x £ 0.

K>V1

t - t Q = 2 > A s in (rnrx) exp (-nV Fo) n=l

*„ = / A = / f (X) s in (mrx) dx

See case 8.1.25 for convection cooling at x = i..

8 .1 .2 ° , p. 96 Inf in i te plate with steady surface temperature and l inear i n i t i a l temperature. t = t Q + bx, 0 < x<JL, T = 0. t = t , x = o, a, T >_ o.

t - t b SL

L = 2 V Szll n-1 2 2 s in (mrX) exp {-n IT FO)

n=l

7


Case (to. References Description Solution

8.1.3 9, p . 97 Infinite plate with steady surface temperature and bilinear i n i t i a l temperature. t = ( t c - t0){l - | x | ) / 0, + t Q ) , -i. < x < +SL, T •» 0. t = t , X = tl, T > 0.

CD I

CO fc ~ t Q = -i V i c o s [(2n + 1) 1 ["_ (2n +1) 2 ^ F o ] ' e ^ O ir2 \ <2n + l , 2 |_ * " M ^ L 4 J

n=0 l

= 1 - |X| - 2 V F O " J ( - l ) n f i e r f c ( 2 n + l X ' )

["(2n + 2) - |x|~|l L 2VIS Jl

N A ^ A ^ I

ierfc

— *0

>w\M See Fig. 8.1



8.1 .4 9, p. 98 Inf in i te plate with steady surface temperature and b i -parabolic i n i t i a l temperature. fc - <V V <*2" * 2 ) /

z2 + t , -a < x < +1, c

T = 0 .

t = t Q , x = ±1, T > 0 (see case 8 . 1 . 3 ) .

t ~ fc0 32 XT t - t n - 2 2

±UJ (2n + 1) TTX exp n O ( 2 n + 1 ) '

£_ iln^lii ^ p o

See Fig. 8.2

i 8 .1.5 9, p. 100 Inf in i te plate with steady surface temperature and cosine i n i t i a l temperature. / v t ~ fc0 - ( tc" V <*" ( i f) + t , -I < x < +& c T = 0 .

t = t n , x = ±1, T >_ 0 .

t„

V fco ( f x ) e x p ( - ^ F o )

UsA/ /V/



8.1.6 9, p. 97 Infinite plate with 74, p. 113 steady temperature and

constant initial temperature, t = t±, -I < x < I,

T t 0' ±1. T > 0.

NV»'V*>

^ +^/\s*

— * < >

oo

T=T: =2 E ^ « V> *** {< FO) i 0 . n x ' n=0 Mean temp:

££-»£ *-(•*» n X n=0 n

*«" fc0 /Fo ^ — ^ = 1 - 2 ^/- , FO < 0.1

Beat loss:

Q , V S i n ^ X n i1 * e xP (-Xn F o)] 2£pc (t.- t.) Z Z, X |"X + sin (X ) cos (X )' *^ i 0 , n I n n n n=l •- J

X = (2n + l)n/2, See Fig. 8.3



8.1.7.1 74, Infinite plate with p. 220 convection boundary

(general). t = f (x), -SL < x < +1, T = 0.

-h. t.

t - t f = 2 X cos (X X) exp (-X Fo0 ) f 1

— " n . V ",, ' I [f W - t-1 cos (X X) dX sin (X ) cos (X ) I L fj n

n n Jn

„ A cos L x + n=l

cot (X ) = X /Bi

X given in Table 14.1 n

8.1.7.2 3, p. 294 Infinite plate with 74, p. 223 convection boundary—

constant initial temp. t = t . , -I < x < +£,

l T = 0. ( see case 8 . 1 . 7 . 1 )

t - t„ ~ s i n (X ) cos (X X) exp l-\ Fo )

t . - t - Z , X + s i n {X ) cos (\ ) n=l

c o t (X ) = X / B i n n

t . - t f

t - t

1 - cos ( V B I X) exp (-Bi Fo) , Bi < 0 . 1

r~ = e r f Z 1 " X ) - exp | B i ( l - X) + B i 2 F o ] e r f c f1 ~ X + Bi "VFo) i f \ 2 V F O / L J \2 VFO /

fc l1 + X 1 - exp U3i(l + X) + B i 2 F o | e r f c ( •*• * X + Bi VFo") , \ 2 VFO"/ L J \2 VF^ /

+ erf + Bi VFO"1 VFO

Fo < 0 . 1 . See F i g s . 8 . 4 a , b , c , d and e and 9 . I d .



8.1.8 1, p. 268 Infinite plate with time varying surface temperature. t = t . , -I < x < S.,

l

T = 0. t = t . + bt , x - ± I, T > 0.

I t (T) -

' I I J

- t (T)

(t - t±) a = Fo +

[(2n + 1) | xj

»* " » ' n r 0 (2n + 1)

x cos I(2n + 1)

See Table 8 . 1 , Fig. 8.16 and F ig . 8.25.

8.1.9 9, p. 104 Inf in i te plate with time varying surface temperature (general), t = t . , -i. < x < l.

T = 0. t = X = ± I, T > 0.

t i + f{T) ,

, 2 2 - V (2n + 1) (-1)" _ f (2n + 1) "Tf „ - t . = Tt y J t J—*— Fo exp — ! j—* Fo n=0

x cos [• t t t 7 J a '] /"»[" , ^ V "1 ['!""•>] dT'



8.1.10 9, p. 104 Infinite plate with exponentially varying i _ 1 -bx cos (X"^9) _ 4_ ST (-1) surface temperature. T - t. e „/Trr, it ^ ,n ,•.,!"-, ,-> . i> 2 „2/;lr,,2n t = t., -I < x < I, * cos (VP3) n = Q (2n + 1) |_1 - (2n + 1) u /4Pd J

I T = 0. t = t. + T (1 - e l X = ±1, T > 0.

_bX)i x e x p [_ iaaîiV F oj c o s [ i2n_^ i i J L XJ (

b ft (2n + l ) 2 ir 2a/4A 2

^ 8.1.11 9, p. 105 Inf in i te plate with °° i. exponentially varying i _ ^bx cosh (XT^d) _ 4 X" (-1)

IT ^->

t = t . , -i, < X < I, surface temperature. I - t . ^ ^ TT ^ ( 2 n + 1 } r + 4 M / „ 2 ( 2 n + 1 ) 2 - j

T (2n +. l ) 2 i i 2 „ 1 f~(2n + l)ir „1 exp - J j - 1 Fo cos P -—<— XJ T = 0. bT

t = t . + Te i

x = ±JL, T > 0.



8.1.12 1, p. 299 Infinite plate with periodic surface temperature. fc - "ta - W

( ? ) • " * = ±1

rax mn 5 cos [(2TTTA 0 )+ *]

t = max surface temp mx t = mean surface temp mn

cycle time

tW-

M ^ « V b « l

t !

-tlr)

5 =

r 2 2 f-^Xa) + f2(XO)

f^(o) + t\{a)

1/2

4> = tan

See Table 8.2

£xia) f 2 (xa ) - f 2 ( a ) f ^ x a )

f x ( a ) ^ ( x a ) + f 2 ( a ) f 2 (xa )

f x ( a ) cos (a) cosh ( a ) , f»(a) = s in (a) sinh (a)

a = VirPd = JSVir/aT.

Heat stored during half cycle: 2X.pc (t

mx t ) F(a)/0 mil/

F(a) = df^ (a) + df 2 (a) 1/2

See Fig. 8.5

dfx(a) cos (a) sinh (a) - sin (a) cosh (a)

df _ (0) = sin (0) cosh (o) + cos (0) sinh (0)



8.1.13 9, p. 105 Infinite plate with t - t. periodic surface ——_ = A sin (w + 9 + $) temperature. m i t - t t, -» < x < I, T = 0 .

CD I

ID

t<T)-

t = t , + (t - t . ) i m i

s i n (on: + 9 ) , x = ±Sb, T > 0. t = maximum surface m temperature.

^ \ » » * ^ . ^ *

\~H

•t(T)

H

+ 4ir Y ( - l ) n ( 2 n + 1) [4 Pd cos (9) - (2n + 1 ) 2 TT2sin (9)} / , 2 4 4 ~ 16 Pd + {2n + 1) IT n=u

[- (2n + I ) 2 * 2 Fo/4] cos [i2E_±_l)JL x ] .

cosh (X V2 Pd + cos (X -V2 Pd)

x exp

A = cosh {V2 Pd) + cos (Y2 Pd)

cosh ( x ^ + X ^ f i ) '

1/2 See F ig . 8.6a .

arg

. c o s h W ^ + V 2 l) See F ig . 8.6b .

Pd = 2. to/a .



8.1.14 9, p. 105 I n f i n i t e p l a t e with p e r i o d i c surface temperature on one s ide / constant temperature on the o ther , t = t . , 0 < x < I, T = 0 . * = fci + < f c . V s in (ut+ 9 ) , x = %, T > 0. t = t . , x = 0, T > 0. t = maximum surface m temperature.

-t(r)

t - t . — = A s i n (wr + 6 + <|>)

m I

CO

V n ( - l ) n [n2TT2 s in (9) - cos (9) PdJ £•> A A 7 A ?

n TI + urtr/ar n = l

x exp ( - n V F o ) s in (n IT X)

cosh (XV2 Pd) - cos (xVz~Pd) cosh (V2~Pd) - cos (V2~"Pd)

1/2

$ = arg sinh (x>pf + X-y/ l i)

, Pd = I a/a

hH

Section 8 . 1 . Sol ids Bounded by Plane Surfaces—No In t e rna l Heating.


8.1.15 9, p . 108 I n f i n i t e p l a t e with « X (T - T " ) X { T - T ' ) steady per iod ic fc 0 2 V ( - 1 ) n . e n - e n

surface temperature t , - t „ " rr A , n s l n ( m r X } X T . , . 1 0 n=l , n

on one s i d e , constant 1 - e temperature on the o the r . T* = x - nff, m » 1 . t = t Q , X = 0, T > 0. fc - X n ( T - T 1 - T ' - ) X n ( T - T " ) t - t 2 ^ ( H n

fc = V* =*- ir^r = * 2. -H—sin {mTX) s fi mT < T < IDT + T , X ° n = 1 1 - e n

m = 0, 1, 2, . . . . » t - t 0 , x - i . mr T " = T - (mT + T l ) f m » 1 . p : T < T - i , ( v 1 )

1

T - * = a n

2 , v T = per iod of cyc le . n



8.1.16 9, p. 109 Infinite plate with steady surface temperature on one side, insulated on the other, q = 0, x = 0, T > 0. t = t 1 # x = I,

mT < T < mT + T ,

m = 0 , 1 , 2 , . . . .

00

to

t = t , x = SL, mT

+ T x < x < (m + 1 ) T .

T = p e r i o d o f c y c l e .

fcl " fc0 * n 4 *> + 1 ° ° S L 2 X J

X (T..--TM X T - T ' ) n v 1 ' n v

e - e

1 - e X T n

,T ' = T - mT, m » 1

00

_ ! _ ! ? _ . i S (-1)" cos f(2n * l)it 1 t - t, t

X (T-T - T") X (T-T") n 1 n e - e

1 - e X T

n , T" • = t - (mT + T ) , m » 1

X = o (J2n + 1) 2 T T 2 / 4 A 2



00 I

8.1.17 4, p. 82 Infinite plate with 48 steady surface heat

flux and convection boundaries. t = t , 0 < x < I, T = 0. %f q0r X = «., T > 0.

t f ,h -

( t - V h Bi fio" 1 xp (X - l ) 2

4 Fo q 0 " 2 V , |-xp (X - l ) 2

4 Fo

- Bi Po exp (Bi + Bi + 4 Bi Fo

See Fig. 8 . 1 8 .

For Bi + » :

<fc ' V k VF5 f r (x - i ) 2 i v - 2vr b 4 Fo J ' tt|

+ exp (X + 1)' 4 Fo

>—(JvK*"* 5)]'

(X + 1)' 4 Fo

See case 8.1.20 for special solutions.

h-'-H



8.1.18 14 47 74, p. 174 flux and variable

Infinite plate with „ r „ °° steady surface heat q 0 1 ,_ , \ ,„ i, . * . ,, 2 V 1 r„,v , Cl

— £ - . J (X - 1) + (X - 1) + y + Fo - — £ — cos \m ( X - 1)J thermal conduct iv i ty , t = t Q , -I < x < +1,

0 0 L IT n=l n

2_2 X = 0. k = k ( t ) . a i s x exp (-n ir Fo) cons tan t . q x = V X = l' "lm

Wo k (T) dT

k(t)

-»-x

T = ( t - t 0 ) / t 0 , kQ= k ( t Q )

For k( t ) = kQ [ l + 3 ( t - t Q ) ]

tf k(T) dT = t - fcn S t n / t - t

' 0 0

For f$ = 0, see F ig . 8 .7 .

£(^) For 0 = 0 and q- terminated a t T = D, see F ig . 8 . 1 " .

Section 8.1. Solids Bounded by Plane Surfaces—No Internal Beating.


8.1.18.1 3, p. 279 Infinite plate with steady surface heat flux. t = t t , -I < x < +SL,

T = 0. q ^ q„, X = ±1, T > 0.

( t " V k „ X -TJL- = F° + 2 6

X = nn n

n=l X v ' cos (X X) n

M- -H



8.1.19 3, p. 350 Infinite composite plates with convection boundary and infinite thermal conductivity. t = t2,0 < x < i.2, T > 0. fc = V l2 < * < A2 + V T > 0. t = t, T - 0. <^- 0, x - 0, h_ is contact coefficient between 1 and 2.

.Q, 0 < x < *j+ i . r

-X„T -\,T t - t X,e _ X.e fco- fcf x x - x 2

/ -X„T -xn-r 4 - ^ " l ^ f . f e 2 -e 1

t„ - t c " t„ - t* 3 \ X, - X,

X l + X 2 - b l + b 2 + b 3

X l X 2 * bl b 3

bl - h2A C2 V b2 - V P1 Cl V b3 - V P1 Cl S.

V*\ '// V.-+<~\ k-»~ 1 • h r t f



8.1.20 3, p. 476 Infinite plate with steady surface heat flux and convection boundary, t = t f, 0 < x < %, T = 0. c^ = q Q , x = 0, T > 0.

(t - tf)k

(t - tfn

z \ V5Fo7 , 0 £ T < T r

40 - i p - t f • & • * « - - A ]

-h.t.

K-«H

x \1 - exp [ l +(Bi/3)]Bi Fo [1 - (T 0 A)]

1 + 2(Bi/3)+ 2(Bi 2 /15) T > T„

T Q = penetration time to X = 1

= 12/5CL

See case 8.1.17 for general solution.

8.1.21 3, p. 349 Infinite plate with unsteady surface heat flux; convection boundary and infinite thermal conductivity, t = t , 0 < x < £,, T = 0. a = q.cos (cot) , x

(t - t f)h COS (WT - b) -

Vm + to 2 ^ 2

exp (-mx)

m = h/pct, b = tan (to/ra)

k->«>| -h.t,

k-f—I



8.1.22 2, p. 248 Semi-infinite plate, quartet-infinite plate, infinite rectangular bar, semi- infinite rectangular bar, rectangular parallelepiped.

Dimensionless temperatures equal the product of solution for semi-infinite solid (case 7.1.1 or 7.1.3) and solution for infinite plate (case B.1.6 or 8.1.7). See Figs. 9.4a and 9.4b.

8.1.23 9, p. 113 Infinite plate with unsteady surface heat flux.

( t - t , ) k

V T 4 r - 2 m + 1 r (m/2 + 1 , VFO- y i - i . r t c r< 2 n + i> - x i * 2 n=0 L 2 VFO" J

. .m+1 , + I e r f c (2n + 1) + X"| 2 V?o J

k-i- l

Section 8 . 1 . Solids Bounded by Plane Surfaces—No Internal Heating.


8.1.24 9, p. 114 Case 8.1.23 with t = t . , x » 0, T > 0.

Solution

(t - t . )k

q 0 *T

1'"' = 2 n H " 1 T (m/2 + 1) VF5 J i 1 .m+1 m/2 n=0

fe f(2n -f- 1) - X l e t t c L — n S — J

.m+1 . I erfc f(2n + 1 ) + x ] L 2VFO" J

8.1 .25.1 9, p. 120 Inf in i te plate with steady surface temperature, convec-

i tion boundary and to variable i n i t i a l

temperature, t = f (x) , 0 < x < I, x t = t Q , x - 0, x < 0.

•"1*0

(t

co / 2 2\ V (Bi + \ ) s in i\ X)

- V = 2 y exp (-X F o V °> " n=l x ' Bi + X + Bi

n ,-1

* / [ t 0 - f (X)] s in (Xn X) dx -Jo

Bi = hVkf * cot (A } + Bi = 0 n n

See case 8 .1 .1 for t = t , x = I, x > 0

•h-«H



8.1.25.2 74, p. 239 Inf in i te plate steady surface and convection boundaries.

" i t h t - t n v fai2 + * 2 ) ["i - « * <* >1 / , \

t = t , 0 < x < SL,

T - 0 .

t - t f l , X = C, T > 0 .

I—I—I

X cot n

See Fig. 8.24

n=l [ Bi + Bi + \

+ Bi = 0

* 0 " - M ,



8.1.26 74, p. 236 Infinite plate with parabolic initial temperature and convection boundary.

o < x < a, T = o. t = t , x = 0 , T = 0 . c t = t , x = ±1, T = 0,

* - fcf 1 + 4 t s _ t f n = 1 K X n/

s in (X ) cos (X X) n n X + s in (X ) cos (X ) n n n

exp «»o X n tan (Xn) = *it

H" -H CD I

-h.t.

Section 8.1, Solids Bounded by Plane Surfaces—No in. .. ,u. Heating.


8.1.27 9, p. 125 Infinite plate with steady surface temp and convection boundary, t = t , 0 < x < H, T = 0. t • t 0, x = 0, t > 0.

-h.t,

Bi X 1 + Bi - 2 B

^ sin (X X) exp f -X Fo 1

n=l (Bi + B i 2 + X 2 J sin (X )

V cot (X \ n V V + Bi = 0

h-*—I 8.1.28 9, p. 126 Inf in i te plate with a

steady surface temp. Convection boundary and l inear i n i t i a l temperature, t = ( t j - tQ)x/l + t Q ,

0 < x < l, T » 0. t = t Q , x = 0, T > 0.

•h,t.

fc- fc0 X + [Bi ( t f - trf/^ - t Q ) - Bi - 1

Bi + 1

2 JBi [ ( t f - t 0 ) / ( t l - t 0 )]

^ s in (X X) exp ( -X 2 Pol

n=l (Bi2 + Bi + X 2 } s in (Xn) - Bi - 1

X cot (X ) + Bi = 0 n n

h-i-H

Section 8.1. Solids Bounded by Plane Surfaces—No Internal Beating.


8.1.29.1 9, p. 126 Infinite plate with two convection temperatures, t = t , -I < x < +1, T = 0.

fc" fco _ B J X _ y t - t - 2 (Bi + 1) B 1 _4-

s in ({5 X) exp « » ) n=l ( B I 2 + Bi + e 2 ) s i n (B )

t n , h-

os I to f-i- [-i-*-l

- h . t f

i »•' f cos (XnX) exp (-X^ Fo) 2 ~ 4- 7 ^ 2~\

n=l (Bi" + Bi + X 1 cos (X )

X tan (X ) = Bi , B cot (6 ) + Bi = 0 n n n n

8.1.29.2 74, p. 239 Inf in i te plate with two convection coe f f i c i ent s , t = t Q , 0 < x < +1,

T = 0 .

W - h 2 , t,

t - fco l + B i £ l X

fcf - to~ 1 + B i s , i + ( B W B i i 2 )

oo

-Xv n=l

B i J>l 1 / 2 cos (X XJ+ -r s i n (X X) exp [-X Fc n X n \ n

(r . .-, s in (X ) cos (X ) + X A n = | [ 1 + K l / B i 4 " z s i n C X ^ "

+ " X — s i n ( Xn> n

HM B i i l l = h ^ / k , B i ^ = h 2 V k

cot (X ) n B i U X n / V BilZI



8.1.30 9, p. 127 Infinite plate with convection boundaries and time dependent fluid temperature. t = t . , -I < x < +8,, x = 0. t f = 4> (T) .

-h.t,

as l

°° , 2

t - t . = 2 Bi I A cos (X X) exp i-\ Fo) n n r \ n / I2 n=l (&i + X 2 + B i 2 J cos (Xn)

<.( exp ( a x V A 2 ) * (T 1) dT' Ja

X tan (X ) = B i . n n



8.1.30.1 74, p. 316 Case 8.1.30 with t - t± cos (VPS X) exp (-Pd Fo) t, = t_ - (t„ - t.) - D T . — ;— = 1 - : — — f f m »» * • t f m - c. c o s ( V ? 3 ) _ J , ^ s i n { V M )

A cos (X X) exp (-X2 Fo) n n \ n / n=l 1 - X2/Pd n

( •

,,n+l _ „. A .2 ,2\l/2 -1) 2 Bi 1 Bl + X 1 ' n X n ^Bi 2 + Bi + X 2)

Mean Temp: t - t, m i exp (-Pd Fo) ^ m " fci Vpd" fcot {VPd) - ;rr VPd"l

» 2 I exp 2 Bi"

« » ) »-l i* (Bi2 * Bi * 1*) [l -(X=/M)]



8.1.31 9, p. 127 Case 8.1.30 with Case 8.1.30 with ^ _ °° / , 2 „ \

t _fv"». iik..x,,Mi - v > • * K F ° ) .• < T < x , t"~ -T r0 i n=l(Bi + X^ + Bi) — "-'

[t l r T > T 0. V n ) cos (Xn)

fc- fcl , , " f [T + ( 1 - T > e x p (Xn F o o ) ] - _ . - l - 2 Bi ^ T - ^ r -c l c i n=l ( B i + X^ + B i ) cos {X )

x c o s (X X) exp (-X Pol

<j> T - ( t - t ) / ( t - t . ) , F o . = OCT / i 2 , X tan (X ) = Bi i O i l i O O n n


Ref-Case No. erences Description S o l u t i o n

8 . 1 . 3 2 9 , Case 8 . 1 . 3 0 wi th p . 127 t f = ( t Q - t j ) s i n

I*'325 0.

oo I

t - t . Bi M ~ = ~M~2 S i n ( W T + E + B 0 " B l ' fco- fci

+ 2 Bi

iS„

2 Fo *2 cos (e) - s i n (e)

v- X' cos (X X) exp (-X FoA n ^ n /

„ = ! ( * + 4 F ° M A n ) ( x * + B i 2

+ B i ) cos <Xn)

~n * * * * * M e = cosh (X Fo ) cos (X Fo ) + i s i n h (X Fo ) (X Fo ) s i n (X Fo )

1 it it * * * * 1t M e = (Fo ) sinh (Fo ) cos (Fo ) - Fo cosh (Fo ) sin (Fo ) + Bi cosh (Fo )

* r * * * * * * x cos (Fo ) + i I Fo sinh (Fo ) cos (Fo ) + Fo cosh (Fo ) sin (Fo )

+ Bi sinh (L) sin (L)J

Fo = I V(»)/2a , X tan (X ) = Bi n n



8.1.33 9, p. 127 Case 8.1.30 with 89 t = tA+ bT, x > 0.

( t - V ° „ B i X 2 - B i - 2 = Fo + — • t>r 2 Bi

„ . V cos (X X) exp (-X Fo 1 + 2 Bi \ n \ n I n=l X 2 (Bi2 + X 2 + Bi\ cos (X ) n \ n / n

X tan (X ) = Bi, See Fig. 8.26. n n (See Fig. 8.1.25 for Bi = ™.)

perfect contact with an infinite plate of infinite conductivity, t = t., 0 < x < Jl + SL , T = 0 . q = 0 , x = 0 , T > 0 .

- q x = q Q , X = l 1 + l 2 .

T > 0 .

2

I—/,-*K~l

( t - t . ) k ' V _M_ L + i- 3 + M 1 ~£~ ' 1 + M L 1 2 " 6{1 + M)J

V c o s ^ x j e x p ^ F o J

n=l X2 (x 2 + M2 + M) cos (X ) n \ n / n

X cot (X ) = -M n n

« - ^ q . E o i = V / S v x = x / * i



8.1.35 9, p. 129 Case 8.1.34 with q = 0, x = I + I , T > 0. t = t 2, i.x< x < «,2, T = 0. t = t., 0 < x 0. (t - t.)k

= X - 2M Y sin (XnX) exp (-X2 FOjJ

n=l X |X 2 + M 2 + M ) cos (X ) n \ n / n

CD i

X tan (X \= M n \ n/

8.1.37 9, p. 128 Case 8.1.34 with t = t., x = 0, T > 0. t = t 2, & 1< x < %2, T = 0. t = t. , 0 < x 0.

( t - v ( t 2 - t . ) = 2M 2 sin (X X) exp « *•,)

n=l ( X 2 + M 2 + M) sin (X ) \ n / n

X tan i\ ) = M n \ nl

8.1.38 9, p. 129 Case 8.1.34 with b Q, x = 0, X 0, x = I, + &,, T > 0.

t = t Q, x = 0, X > 0. ( t ~ V . , f (Xn+"2) s i n(v) e xp(- Xn F ol) (t„ - t.) " _4, ^ ^2 + M 2 + ^ n=l

X tan (X ) = M n n



8.1.39 9, p. 129 Case 8.1.34 with 0^= 0, X = &J+ %2, T > 0.

V q0' X = °' T > °' (t- t )k(l +M) M , 2 M(3 + M)

— i ^ - j - — = M FO + 1 - x + 5 ( i - X ) - 6 ; 1 + M ; *0~1

- 2 (i + M) y (M 2 + A^J cos (XnX) exp (-A* FQ.J

n=l Xn (Xn + " 2 + M ) X cot (X ) + M = 0 n n

8.1.40 9, p. 129 Case 8.1.34 with convection boundary h , t f at x <= A x+ l 2 . t = t 2 , ^ < x < *.,, X = 0.

(t - t 2 ) _ = 2 V ( B V X n / M ) TOS ( X n X ) S X P (~Xn F ° l ) ( t i " t 2 ) n=l f A i , - \2/M\2 + X 2 + X2/M + B i l cos (X )

L\ 1 n / n n xj n X.ntan (An) = B ^ - A*/M, Bi = Wt.,/1^

(See Ref. 82 for solution of material 1 of low conductivity and material 2 of high conductivity.)



8.1.41 9, p. 129 Case 8.1.40 with <* .2 ,, „, / ,2 „ \ — X cos X X) exp l-X Fo, I n n V n 1/ -q =f(T),x = 0, (t - t ) _ 2a_ Y T > 0. kJT n=l I^Bi - X 2 / M V + (1 + 1/M)' X 2 + Bil cos (X )

x f f(T') exp (\âz'/lZ) dt'. Jo

X tan (X ) = Bi, - X 2/M n n 1 n

OD 8 . 1 . 4 2 9 , p . 129 Case 8 . 1 . 3 4 wi th a ( t - t . ) (M + l ) k 3 B i + 6 + M B i ii> c o n t a c t r e s i s t a n c e h r - r = Fo n + — (1 - X) - ——— ————

q-Jl.M 1 2 6 Bi (1 + M) between t h e two c

p l a t e s .

+ 2 3 i „ . „ V- X 2 - M Bi c

( X 2 - M Bi ) tan (X } = Bi X . V n c / n e n

P = X + X ^Bi + B i - 2 M Bi ) + M B i 2 (1 + M) X 2 . n n n \ c c c / c n

B i c - h c V k l •

Section 8.1. Solids Bounded by Plane Surfaces—No internal Heating.


8.1.43.19, p. 324 Composite plates of Region 1, -&1< x < 0: different materials. fc = V "V x < V (t - t„) x = 0. t - v

x = -X^, T < 0. t = t n, x. = l 2 , T > 0.

( t " t p ) = <!•-»> - 2 y exp -X 2 FO,

cos (A X) sin (BU ) - a sin (A X) cos (3LA ) ri li n n A Td + K32L) sin (A ) sin (PlA ) - B (K + L) cos (A ) cos (BLA )l n L n n n n J

Region 2, 0 < x < l^'

__r^-! (ti ~ V ~ L + K £1 exp -A Fo 1

sin (0LA - 0A X) n n A [(X + K32L) sin (A J sin (SLA J - 3 (K + L) cos (A) cos (SLA )]

cot (A ) + a cot (3U ) = 0 n n

B = V V ^ ' K = k 2 A l ' F ° l = V / ll' L = W X = X / £ l



8.1.43.2 74, p. 441 Composite plates of different materials with insulated surface and convect ion boundary, t = t . , 0 < x < I. +

-h,t,

rrf - x ~ 2 1 Hp c o s ( x n x / v s ) e x p ( - x n L 2 p o i / A ) ' ° < x K

f l n=l ir v ' v '

L .

t - t . T^T = x " 2 A xlJ ( c o s K ( x " L>1 c o s ^n 1 -/^)

f i n=l n ' L J

- K s in l"x (X - L)l s in (XnL/VE) } exp (-xh^Fo^h) , L < X < 1 .

• " [(X * *>** S P ) S i n ( X n > + X„ (TP) ( l + KL/VA) ra (Xj] cos ^XnL/VS^ + [^ i + KL/VA + ^ g p ) cos (X )

- \ (SP) (* + i & ) s i n <V] K s i n <x,y*> • LX X —"- ( 1 + L) t a n (X L / V A ) = 1 - -TT ( 1 + L) t a n (X )

- K tan (X ) tan (X L/ih) . n n

Bi = oyK.,, L = V V F O l = *f/%\. A - a ^ , K = ^ J



i

8.1.44.1 9, p. 416 Rectangular parallelpiped. t=t., 0 < x < w , 0 < y 0. t = t . , a l l

other faces , T > 0.

t - t . , , v V" s in (imrY) s in (mrZ) sinh | ( 1 - X)-v/X I i_ _ X6_ y y L \ m f n,o J t n - t . ~ 2 ^ , ^-t . . ,, ,

0 i IT — " — " - - . 1. M i

as co

m n s inh (X ) m,n,o

OO 00

m=0 n=0

i. s i n (nnrY) s in (mrZ) s in (JITTX) exp (-X . Fo ) ID f n t Xf w

0 m n X_ m,n,Jl

X „ = A2TT2 + (ntirw/b2) + (mrw/d) 2

IQ,n,x<

m = 2m + 1, n = 2n + 1, X = x/w, Y = y / b , Z = z/d



8.1.44.2 74, p. 287 Rectangular parallele- » » m piped with convection 0 _ . V y *\ boundary. t = t±, -i.x < x < +£ r

-l2 < y < +A 2, -Jl3 < z < +A 3, T = o.

n=l m=l s=l Anl Am2 As3 c o s <*„1 X ) « » <\n2Y>

OB I

• t l . t .

x cos ( x ^ z ) exp (\2

nl ?ox + X^2 P o 2 + x j 3 F o 3 )

( n , . , s ) + l / 2 + x2 \l/2 A = J . l i i n ,m,s, i j

n ' m ' s ' i X . ( W + B i . + X 2 . ) n , m , s f i \ I i n , m , s , i /

cot (X .) = ( r T - J X . , Fo. = ax/I2. v n , m , s , i ' \ B i . / n , m , s , i l l

Bii = WI../IC, i = 1, 2, 3 , X = x A r Y = y / £ 2 , Z = z A 3

<1 A



8.1.45 9, p. 417 Case 8.1.44.1 with t = t Q sin (UK + e), x = 0, 0 < y 0.

00 00 t - t . V ^ " s i l > ivm?) s i n (mrz) M s i n (urr + c + d> ) M n m,n * - fci _ i6 y t - t 2 " — ^0 i n m=0 n=

QO 00 00 ^ — O

„- V"* V* V" & s i n (nffiY) s i n (mrz) s i n (JITTX) (W COS E -X n s i n E) exp(-Xar/w )

- f Z 2 2 7-i—- 2—x-- mf"f* m=0 n=0 H=Q ( W + \ _ 1 m n

\ m,n,x./

m,n »P »m,n s i n h { [ ( S r w / b )

2

+ ( ^ w / d ) 2 + i w ] 1 / 2 }

2 *~ "~ W = w 0)/a, See case 8 .1 .44.1 for X ,.,, X, Y, Z, m, n m f n f .<£



8.1.46 9, p. 419 Wedge with steady surface temperature. i 6 L 2 V , ,.nr . , „ / n , t - t . . o < e < e n , t n - t . - 57 + e" L <-« [ S l n < 2 n + D^/BJ

i 0 0 l 0 0 n=l T = 0 . t = t , e = o, T > o. t = t . , e = e 0 , T > o.

/ exp (-<rruVr 2> a ( u ) d u

J n u s

09 I

s = (2n + 1) TT/8.

For t = t Q , 9 = 0 , 8 0 , T > 0:

t _ t = 1 " | ~ ^ s i n < s 9 > J e x P ( - o r u 2 / r 2 ) — J (u)du

0 n=0



8.1.47 33 Composite slabs with ramp surface temperature. t = t , 0 x < 5LX+ %2,

T = 0. t = fci + ( t o " V fcrc' x = 0, 0 < t < 1/b.

t = t , x = 0 f T > " l / b .

t . - t . = 2

OS

k-»oo

i

Y Pd (x^ + M2) exp f-A^ Fo ) [ l - exp / x 2 / P d ) j s i n (X X)

n=l \^ (\^ + M2 + K)

\ tan {An) = M, M = A

1 P 1

C

1 / J ! ' 2 P 2 C 2 ' X = x / X l

See cases 8.1.34 to 8.1.42 for other condi t ions .

h-it-H*-je2-H

Section 11.1. Change of Phase—Plane Interface.


1 1 . 1 . 9 9 , p . 292 Convect ion boundary a t

o r i g i n a l s o l i d - l i q u i d

boundary.

t = t 1 # x > 0 , T = 0 , t x > t m . t = m e l t i n g temp.

t - t F B i 2

_§ _S = B i T-^{1 + 2 ! Po ) t - t , x 2 x '

m £

P Bi

IT'1 + 3 1 F °x> + . . . , 0 < x < w .

V \ . ,

F 2 B i 3

F B i x F O x . _ _ x ( 1 + P ) F o 2 + . .

F = k s ( t m - V / a s P Y ' F o x = V / x ' B i x = h x A s

(solid) (liquid)

Section B.l. Solids Bounded by Plane Surfaces—No Internal Heating.


8.1.49 19, p. 3-30 Infinite plate of infinite conductivity with convection boundary and harmonic fluid temperature. fcf - W (tmx -• t ) mn COS (UK ) . t = mean temp, mn t = max temp, mx

I

V"—i • h . t ,

t - t mx mn Ve* COS (TU) - \p)

+ 1 ~ exp (-' + 1 V pel J

9 = upeVh, \|i = tan (9)

Instantaneous surface heat f lux:

J I _ . e h {t - t ) mx mn

See Fig . 8.9

mx mn" ^|ZT?~ s i n (Tto - i|))

(«-2i-*|

Section 8.1. . Solids Bounded by Plane Surfaces—Ho Internal Beating.

C, .a No. References Description Solution

8.1.50 40 Infinite plate of infinite conductivity with surface reradia-tion and circular pulse heating, t = t Q, T = 0,

c r ^ T 4 .

See Pig . 8.12 for maximum temperature values :

/3naj£

D =' heat ing durat ion

Sc = 'W111 l r r / D >

I k->»

|— 28—|



8.1.51 19, p. 3-44 Infinite composite 42 43 66

plates in perfect contact, convection boundary on one side, insulated on other side, t = t Q, 0 < x < &1 + S2, T = 0. o = 0, x = 6 + 6 , T > 0.

See Fig. 8.14 Pl6l = f V P i k i D / ° i

D = heating duration

fc2inax " fc0 t„ - t„

|k,^H tf.h *\ J

HH^-H



8.1.52 19, p. 3-49 Infinite plate with 44 radiation heating 68 or cooling.

t • t Q, 0 < x < S,

9 r - ^ ( T s - T x = o ) -

See Figs. 8.15a-e for heating.

See Figs. 8.15f-l for cooling,

t - t„ t •* t = f (Fo)

source temp.



8.1.53 19, Infinite plate with p. 3-68 time-linear surface 49 heat flux.

t = t 0, 0 < x < 6, T = 0. gx " W ^ ' T > °-

k ( t -

«0« Jfii Fo |

" F o D ^ k 2 x 3

12

% = ctD/6 2 , X - x/6

¥ - - f e ) .*<*>< F°D ' F ° D > 1 .

For Sc = q n , a x ( 1 - T / D ) ;

MH

k (t - t n) x 2 ±

= Fo + —- - X + — + (solution for V 3 V q x = qnu T/D ) •

D «• duration of heating.

TT < Fo < Fo , Fo„ > 1 2 D D —

8 . 1 . 5 4 1 9 , Case 8 . 1 . 5 3 wi th p . 3-68 a = a s i n wT/t>. 150 ^ ™ * X

See F i g s . 8 .19a & b

&S* ( t - t Q ) = f ( 2 i r t / D ) , QQ= ^ D



8.1.55 19, p. 3-73 Porous infinite 5-l plate with steady

surface heat flux and fluid flow, t = t Q, 0 < x < S, x = o. tfl = fluid source

09 I - V k f c ,

k-«H

See F igs . 8.20a & b y i t - t n)

F = ,Gf C f5/k e

k e = Pk f + (1 - P)k s

G = fluid mass flux P = plate porosity

Steady state solution:

^T7l = p exp (-Fx/6)



8.1.56 19, Two i n f i n i t e p l a t e s p . 3-75 in per fec t contact 52 with a s teady surface

heat f lux . t = t 0 , 0 < x x < fiir

0 < x 2 < 6 2 , t = 0. ^ = q 0 , x x - 0 , T > 0.

See Pigs . 8.21 a-d

( t ~ V k i V i

f (Fo >

CO I I 3 1

H i * K H 8.1.57 19,

p . 3-75 48

Case 9.1.56 with k 2 + » .

See Fig . 8.22

q 6 - = f < F ° l > V i

8.1.58 19, p . 3-75 53

Case 8.1.56 with k_ + ™ and

% * Snax S l n 2 ffT/D-

See F igs . 8.23 a, b , and c t ~ fc0

t h - t„ = £ U 1 T T / D )

eqb 0

D = durat ion of pulse



8.1.59 74, p. 245 Thin plate with two convection conditions, t = t , -A < x < +1, T = 0.

CO

I

V h i

t , .h.Vh-i—

h,.t« fcr^" \H

t - t_ cosh K z 0 tvE

fcf - *o cosh ( B i ^ 2 L ) • ^ sinh ( B i ^ ) L

- 2 X n s i n X n C O S ( X n X ) e x p [" ( X n * B i l w L 2 ) F o w J

( X n + B i l w L 2 ) [ X „ + s i n ( M C O S ( X

n ) ] Xn tan(X n)= B i 2 w , B i l w = h l W A , B i 2 w = h^ /k

X = x/w, L = Vw

Mean temp:

t_ - t_ tanh ( B i * ' 2 L ) Bi , t " " t ° " » ^ - ^ - 2 " - K 2 -)

X ' °n «» f- ('n * " l . L ' ) ">.] n=l

X 2 + B i . L 2

n lw

Bi B_ = lw

\2 fai2 +Bi, + X 2) n \ lw lw n /



8.1.60.1 74, p. 320 Infinite plate with evaporation and convection boundary. t = t . , -S, < x < +S,,

T = 0 . -bx m = m. e , x = ±SL.

m = evaporat ion r a t e .

H- -H

-h.t«

t - t . M cos (VPdY) exp (-Pd Fo) tf ~ fci c o s (VPd) - ^ r VPd s i n (Ypd)

= 1

-2 n=l

1 - M 1 - Pd/X

B i

A c o s (X X) exp -X Fo n n n

2 s i i n ( X n ) . M = YÔ fcf - fcwb

n X + s i n / X \ c o s (X \' h ( t . - t . ) t c - t . n ( n | I n | f i ' f 1

c o t / X ) = X / B i , t . = wet bulb temp.



8.1.60.2 74, p. 321 Case 8.1.60.1 with t - t asymmetrical boundary i

s

tf,h

h X + sin/X )cos/X \' "m X_ - cos/X_\sinfX

cot/X ^= X / B i , tan/X \= -X /Bi

Mean temp:

" l t - t . m l i - « - )

- 2 — - 2 l { X - 2 ) X 2 / . 2 + B . + x 2 ) n=l n \ n /



i o


Case No. References Description Solution i _____—_ —



8.2.1 3, p. 276 Infinite plate with uniform internal heating. t = t., -SL < x < SL, i T = 0. t = t ± , X = ±SL. T > 0,

CD I KJ

( t - V k 1 2 ~ = f (1 - XZ) - 2

X = (2n + 1J1T/2 n See Fig. 8.8

y 1=11: n ^

n=0 n

exp ^-X2 Fo\ cos (X X)

k*. -i,-»j



8 .2 .2 .1 74, p . 363 Inf in i te plate with steady surface heat flux and variable internal heating. t = t . , -a < x < +i, x = o . q " ' = q,J" (1 - * / * ) • \ = q„» x = ±a, x > o.

CD I

in U)

%" -<1o

i i " V k Fo * i " * 2

™ cos f(2n - l)7rxl cos (X X) _ ^ ( 2 n . 1}4 ~

n=l

°.n !

1 - exp f-xjj Fo) + *

« n=l \ n i

t - W - f )cos ,XnX) n=l W

x exp X = mr n

8.2.2.2 74, p. 363 Case 8.2.2.1 with (t - t,)k .1 I I — — • • • = q>" (1 - X-). — i - - f Fo - J ("I)" cos (XnX, [l - exp (-X* Fo)] + *

0 n=l n X and * given in case 8.2.2.1 n r

Section 8.2. Solids Boi id by Plane Surfaces—With Internal Heating.


8.2 .2 .3 74, p. 364 Case 8 . 2 . 2 . 1 with q > " = g j " exp (-bX). irjr • H t" - '-b!] * I t1" -1'" - H 7W77)

0 n=l n \ n /

(X X) j l - exp (-A 2 Fo) + $ X COS

X and $ given in case 8.2.2.1

8.2.2.4 74, p. 364 Case 8.2.2.1 with (t - t )k ± — = Fo (1 + Pd Fo) + $

T 1 ' " " l A " * 1 + * > • q-" 9? 4 given in case 8.2.2.1



6.1.23 9, p. 346 Cylinder with properties fc » j(XR)J<X)dX different from surrounding — = —fr I exp(-A Fo, ) —_r _ r — , R < 1 medium. t = t L, r < r Q, T = 0.

t - t r , N J (XR)J (X)d) rounding = -£ / exp(-A Fo ) — - p - ^T

fc- * 0 2Y f ( ,2, \ J l ( X ) [ J 0 ( e X R ) " ' - Y 0 ( 6 X R ) ' t ' ] d X

= —*- I exp -X Fo, I = -0

i|) * Y^cXj j j j jBX) - B J 0 ( X ) J 1 ( B A )

<j> = y J ^ X j y ^ B A ) - BJ o (A)Y 1 (0X)

R > 1 .

A X y B - V 5 ^ , Y = k A 00

See c a s e 6 . 1 . 4 f o r e q u a l m a t e r i a l p r o p e r t i e s .

6 . 1 . 2 4 9 , Po int source o f rec tangular ( t - too)4Trrk

p . 402 p e r i o d i c wave h e a t i n g . Q~ " e r c * "'**' T ^ + 1

CO

(ST) - T i

exp(-TX"")j e x p f - X 2 + T X 2 j - e x p ( - X 2 ) J sin(Fo*X)dX

x[l - exp(-X2)]

Q(T) = 0 , x < 0. „ ^ 2 j

Q ( T ) = Q Q , nT Q < T < nTQ

+ T.f n = 0 , 1 , . . .

Q(T) = 0 , n t f l + T < T < (n + 1 ) T 0 . 0 < T < T . n = Number of c y c l e s .

t = t ^ , r + co. T x = T 1 / T 0 , T = T / T Q , Fo* = r/V5f^

>* r

Section 8.2 . Sol ids Bounded by Pleê Surfaces—With I n t e r n a l Heating.

Case No. References Descr ipt ion Solution

8.2.3 9, p. 131 Infinite plate with time dependent internal heating. t = t . , -a < x < ±SL,

T = 0. t = t , X = ±1, T > 0. q ' " * f (T ) .

c -« J- s i •&$» - P ^ -] n=0

x J f fr) exp -IT (2n Jo

+ l) 2a(x - T ' ) /X 2 J* : '

8.2.4 9, p. 132 Case 8 .2 .3 with f(r) = q j " e " b T .

( t - t j k V * l fcos (xVgd) ."1 o % m , . .

2 V t-l>" cos U nX) exp (-X 2 Fo) + ™ \ X ( l - X 2 /Pd)

n=0 n \ n / X = (2n + 1 ) IT/2 n



8.2.5 9, p. 132 Case 8.2.1 with q' «• • fix). fr"J-5:2H«-Pi} tr k , n n=l

1 - exp (-=¥-)] x I £ (X) cos ( ~ x)

J-X dX

CD I m



8 . 2 . 6 9 , p . 132 I n f i n i t e p l a t e wi th uniform 41 i n t e r n a l hea t ing and

c o n v e c t i o n boundary, t = t f , -l < x < +1, T = 0 .

t „ h -

(*-!(,—|*-X,-»|

-h,t,

(t - t j k Bi „. . 2 f T . B l B i X

^ = 1 + T'~2~ q - ' T

cos (X X) 2 B i '

~ X 2 f \ 2 + B i 2 + B i n= l n V n J cos (Xn) exp K -)

X t a n (X ) » Bi , See P i g . 8 .13 n n

8 . 2 . 7 9 , p . 132 I n f i n i t e p l a t e wi th uniform i n t e r n a l h e a t i n g and c o n v e c t i o n boundary on one s i d e , t = t , 0 < x < «., T = 0 .

t = t - , x = 0, T > 0.

- M f

{ t " t f ' k (1 + B i /2?X _ x f

l ' " i 2 1 + Bi

+ 4 B i ^ s i n {X X ) f l - cos (X )~1

, X 2 (\2 + B i 2 + Bi\ s i n (2X ) n = l n \ n / n exp « *>)

X c o t (X ) + Bi = 0 n n

H-<M



8.2 .8 9, p. 311 Inf in i t e p late with time-dependent heating, t - t , 0 < x < I, T = 0. t = t n , x = 0, I, x > 0.

q " - = l * 8 / 2 , S - - 1 , 0, 1, 2, . . . .

I U1

I — A — I

(t - t 0 ) k

Bar l+s /2 (1 + s/2) 1 - T(2 + s /2)2 s+2

^•['"-(^•'"-(l^)]}

8.2.9 9, p. 404 Infinite plate with variable internal heating. (t - t„)k t = t n , 0 < x < l, x = 0. „ i . i p 2 B ] c o s ( V l )

u 9 0 t = t Q , x = «., T > 0.

0' _ 1 c o s ( V l x ) _

. t i t _ - H I q j " + B t t - t Q ) . a = 0, x = 0, T >. 0.

16 V 1-1)" exp {L-t2n -v \)2T\2 + tej go/4J cos L(2n + l ) n x / 2 j * ~ f4B - (2n + l ) 2 i i 2 ] (2n + 1)

x

:

n=0

B = &8T A



8.2.10 9, p. 405 Case 8.2.9 with convection boundary h, t at x = I.

(t- t f l)k Bi cos (VEX) 1_ , , ,.2 ~ B[Bi cos (VTf) - V T sin (V"B) B

90*"*

+ 2 Bi ^ cos {X X) exp (B - X 2) Fo

\=o( B- Xn)[ Xn + B i ] COS <V

CD I

X tan (X ) = Bi, B = 04 /k n n

8.2.11 19, p. 3-29

Infinite plate of infinite G Hn (l - (Q/G)\ + 9 = LH/C conductivity, variable specific heat, convection : boundaries and steady heating. (1 + q^" £)/hL(t - t fJ, C = (1 - c f)/c Q, H = hT/pc£ t = t , 0 < x < 28,, T = 0. c = c Q + S(t - t Q ) . a• * > = a l ' ' .

%

- * • x

H evaluated at fc„, c, evaluated at t_. 0 E f L = H x (surface area/volume) = 1 9 = (t - t 0)/(t f - t Q)

-h,t f

I—2S-H



8.2.12 19, Case 8.2.11 for infinite p. 3-29 square cod and cube.

See case 8.2.11, set: L = 2 for square L = 3 for cube

rod

8.2.13 19, p. 3-32

Infinite plate of infinite conductivity with surface radiation and steady heating, t = t , 0 < x < 2£, T = 0. q t = adr(i* - T 4 ) . T = source temp.

4" =%"-

In + 2 tan - 1(8/N) = 2 tan_1(l/N) + 4 MN 3 (6 + N) (1 - H)"| (6 + N) (1 + N)J

/ 4 4 \ 1 / 4 3 6 = T/TQ, N = [q^'SL/a&T0 + 8*J , M = a^T^/pcl

<*r - ^ r

H-2JH



8.2.14 19, p . 3-33

09 I

Case 8.2.13 with s imultaneous surface convection and r ad i a t i on . ^ = h ( t f - t) + a-jr^T4 - T 4 ) , x = 0, 211, x > 0.

M = Ai *n ( F T ^ ) + A Jin 9 - R \ A

wnrj + r^ (9 - n ) 2 + d> 2 l

2 2 ( i - n.) + <t> :

A4 r - i /e - n\ .. - i A - n\"l

^ = JBVU - n 2 [n Vv - n 2 + (v/2 + n2)]J

A 2 = jsVv - n 2 [nVv - n 2 - (v/z + n 2 ) ] |

A 3 = n/(v 2 + 8n4)

A, = (V/2 - V 2 ) / ( V 2 + 8I1 4) 4

•\f 2 \ ( 2 R1 = - n - > v - r i , R 2 = - n + ' y - n * = Vn + B/4, n = ^672

\ V 3

*-(££+&^F + (£-JF^r v = - JN 4 + 32/4

s = h/a#-T„

N = I q " ^ / ^ ^ + 9* + s9 fl 1/4



8.2.15 19, Infinite plate of infinite p. 3-34 conductivity with prescribed

heating rates. t = t f l , T = 0.

a. s tep input; _TTT = 1 ' q?"p = 1 Tnax Tnax

g i l l Qf I I b. l inear input; ^nr = T , «,,-,' = 0.5

Tnax Tnax Q I I • Q I I I

c. l inear input; _rrr =» (1 - T) , * , , , p = 0 . 5 Tnax Tna~ nax

r l l l

sax ) • 11 di n"' 2 — 0 ' "

«* d. circular pulse; g , , , = s in irr , • , , , D

Toax Tnax 0.5

b. ^ e = 7 2

PC«5 Q">

peg

= T ( 2 - T )

sin27TT

e. power pulse; JL1L = ( 3 T ) - 3 e ~ ( l A - 3 ) f aLlL- = 0 . 5473 e. &*. Tnax Tnax

0.06767 f -(I* 1 ) "(l /T-3)

f. power pulse, J l l = ( 5 T ) - 5

e - ( V c - 5 ) f QLLL_ m 0 < 2 7 9 5

Tnax Tnax

pC6 Q "

- 6 = 0.006869/'-^r- + ~Z7 + Z + x J \ 6 T 2T T /

• ( l / T - 5 )

exponential pulse; _rrr = < 1 0 T ) e 1 _ 1 ° T , £7775 = 0.2717 g. &~ 9 = 1-001 [ l - (10T + 1) e 1 0 T J Tnax <TnaxD

9 = t - t_, D = heating duration T = x/D, Q''' = total heat input per unit volume during time D See Pig. 8.10



8.2.16 19, Infinite plate of infinite See Fig. 8.11 p. 3-35 conductivity with surface

reradiation, steady surface heating, and steady internal heating. t = t Q / T = 0. ^ = q " + O&^l - T 4 ) , x = 0, 26. T = source temp.

k-q"

I — - ' 5 — |



8 . 2 . 1 7 7 4 , p . 386 I n f i n i t e p l a t e wi th two symmetrical planar heat p u l s e s and c o n v e c t i o n boundar ie s , t = t f , -ft < x < +&, T = 0 .

Ins tantaneous p u l s e occurs a t T = 0 , x = ± x , w i t h

2 s t r e n g t h Q(J/m ) .

CD i

UI

t f ,h -

/

• x 1

X

-h,t f

(t - t f ) U = 2 I

n=l

, ; : T;—; 7,—r c o s (X X, ) cos (X X) X + s i n (X ) c o s (X ) n 1' n

x exp {<")• X tan (X ) = Bi . n n

For t = f (x) , -H < x < +«,, T = 0 : m

X ( t - t f ) k & 03 = 2 / T : : 7\—; 7\—r cos (X X) exp (-X Fo)

<L >. + s i n (X ) c o s (X ) n r \ n / n=l

1

x / [ f ( X ) - t f ] c o s (X X)dX .

Mean temp:

( t m - t f)kX, ^ . B i 2 c o s (Xn\) exp ( - X 2 Foj

& ~ s i n (X )X ( B i 2 + B i + X 2 V n=l n n \ n /

For Bi = °°:

( t - t f )kX.

Q5 = 2 Y ( - l ) n + 1 cos J"(2n - 1)TTX "I cos [(2n - l ) i r x ]

n=l

exp [ - (2n - 1) IT 2 F o / « ] .



8.2.18 74, p. 387 Infinite plate with a planar heat pulse, t = t , 0 < x < I, T = 0. Instantaneous pulse occurs at x = x., T = 0 with

2 strength Q(J/m ) .

rri

(t - t Q )k£ V 2 2 = 2 > s in (mrx ) s i n (mrX) exp (-n v Fo)

£j i n=l

L CO I

h-M





1° n

I i




9.1.1 2, p. 240 Infinite cylinder with 74, p. 131 steady surface temp.

t = t i , 0 < r < r Q , T = 0.

t = t 0 , r = r Q , T > 0.

i

t ~ t, c ~ 0 „ V 1 ( ,2 _ \ V n

n=l

J (A ) = 0, see Fig. 9.2

Cumulative heating:

J, (X R) s--»sM--k-)â n=l n

e 0 = 7 I r u p o ( t o - t i )

0x"n

9.1.2 X, p. 269 Infinite cylinder with time 9, p. 328 dependent surface temp.

t.f 0 < r < r Q, T 0. t. + cr, r = r Q, T > 0.

t - t

cr2/a i = Po + J (R2 - 1) + 2 V exp (-A2 Fo) J ^ J L L

n 1 n n=l

W = ° See Table 9.1 and Fig. 9.8



9.1.3.1 74, p. 270 Infinite cylinder with convection boundary and variable initial temp.

n=l

W > e XP l'Xn Fol

f n=l J 0 < V + J l ( X n >

Bi w - V i ( V

R)dR

w 9 .1 .3 .2 3, p. 298 Inf in i te cylinder with ^ convection cooling.

t = t t , 0 < r < r Q , T =

h,t.

fc- fcf . V B i V \ R ) °*P ( - X 2 F o ) fci - fcf " Z , (X 2

+ B i 2 ) J n ( X ) n=l \ n / 0 V n

X J,(X ) + Bi J.(X ) = 0 n 1 n u n

See Figs . 9.1a-c

Cumulative heating:

J!. B i 2 | l - exp (-X 2 F o ) | ,

n=l n \ n /

For Fo < 0.02, R » 0:

( t " t i ) , n . /Fo" . . A - R\ _,_ , „. Fo

(t„ -1.) • 2 B i v r i e r f c vr^re) + 4 B l vs



9.1.4 9, p. 199 Infinite cylinder with initial temp varying with radius. t = t . - b r 2 ,

0 < r < r , T = 0 .

t = t 0 . r = r 0 , x > 0 .

t - t„

br! I exp (-X2

n Fo) J Q ( R V

n=l

J 0 ( X n ) = 0

n 1 n

l j J (X ) + 2 J (X ) I 1 n 2 n

9 . 1 . 5 9 , p . 201 I n f i n i t e c y l i n d e r wi th p e r i o d i c s u r f a c e temperature , t =» t . , 0 < r < r_ , x = 0.

t = ( t - t . ) s i n (wr + e ) , ni i

r n , T > 0 .

t = max f l u i d temp, m

'(t - t.)sin(wT + e) m i

t - t .

t - t . m I

r e a l I n ( R Vi Pd) 1

. . . . . . . . , i i n ( v* pa) e x p C i ( w r + e ) l

+ 2 2 n=l

exp (-Xn F ° ) Xn [ P d 2

cos (e) - X s i n (e) n

( x 4 + P d 2 ) J (X ) \ n / J- n

J 0 ( R V

J o ( V = °' p d = r o t J / a

Section 9,1. Solids Bounded by Cylindrical Surfaces—No Internal Heating.


9 . 1 . 6 . 1 9, p . 202 I n f i n i t e cyl inder with 89 convection cooling and

l i nea r time-dependent f lu id temp, t = t t , 0 < r < r Q , t = 0.

I l b

( t - t . ) a exp

b r o ' - 4 V - V

n l n ' B i J 0 ( X n >

Mean temp: t - t .

m i • i / w L l

Kii J»(*_ R)

~ X2 ( B i 2 + \ 2 ) j n ( \ ) n=l n \ n) 0 n

«•>) bx

n=l n \ n /

See F ig . 9.9



9.1.6.2 74, p. 317 Case 9.1.6.1 with fc yjg F ( j )

-or x _ , 0 t_ = t , - (t.. - t . ) e T r~ = l , f f m £ m l ' fcfm

_ti V p a j n { V p d ) - ^ P d J , ( Vfd) 0 B l X

2Bi V W * e X P ( ' Xn F °) n=l W (Xn + B i 2 ) [ l - ( X n / M ) ]

Mean temp : t - t . 2J (VPd) exp (-Pd Fo)

£ fcfm " fci VPd J Q ( VPd) - i j - Pd jj_ VPd

exp - 4 B i ' » 2 2 S »5 W * «*) L1 - (»MJ



9.1.6.3 74, p. 320 Infinite cylinder with fc _ MJ n( V P T R ) exp ,-Pd Fo) convection and evaporation i_ _ 0 boundary. t f - t. J Q ( VPd) - ( VPd/Bi) ( VP<3)

1 - T T rr=r r A cos (A R) exp (-A Fo ]

t = tj, 0 < r < r Q, T = 0. m = m ne = evaporation rate.

0 - 2 h-h n=l

~{\j S " < W (4••'*)' " = h "< A = h(tc - t.)'

0 n l n n



9.1.7 9, p . 202 I n f i n i t e cyl inder with per i od i c convection boundary, t = t . , 0 < r < r Q , T = 0. t . = ( t - t . ) s i n (<DT + e) 1 m i t = wax f l u id temp, m

J h,t.

t - t . f Bi I (R Vi pd) exp (ion + ie) = r e a l TT

t - t . m I

i[VT"Pd I 1 ( YlHPd) + Bi I Q ( VFPd)] + 2 Bi

V / 2 \ X n C P d o o s ( e l " s i n ( e ) l J o ( X n R )

A exp ^ P o ) (V + rd

2Wx2

+ B r W > n=l Vn / V n / 0 l n

V i { V = B i J o ( V ' P d = r o u / a

9.1 .8 9, p. 203 Inf in i te cylinder with steady surface heat f lux, t = t . , 0 < t < rQ, T = 0.

q r = V r = r 0 ' T > °-

( t _ V k 2 V / 2 \ — *— = 2 Fo + (R /2) - (1/4) - 2 y exp /-A* Foj V o n = l

\£ J, (X ) = 0, See Fig . 9.3 I n ' '

0 For Fo < 0.02, R » 0:

, /Fo" . , / l - R\ ^ Fo(l + 3R) . 2 . (\ - R\ (t - t t ) k

V o 2R



t - t„ 9.1.9 9, p. 207 Infinite hollow cylinder

with steady surface temp. c ~ ^0 V / -,2 „ \ t - t „ r / < r < r n , x = 0. 1 " ^ = * 2 «-P K F ° )

t » t Q , r = c i f r Q , t > 0.

ID i

n=l

J0(Vn> [ J Q ( X n R ) W ~ W W * ] V W + W

VWVV " V W W = ° R = r / r Q , Fo = oer/r0



9.1.10.1 9, p. 207 Infinite hollow cylinder with different surface temp, t = t.f r. < r < r Q r T = 0. t = t Q , r = V T > 0. t = t , r = r i , x > 0.

f- = TT £ exp (-X* Fo) n=l

W V R V Jo ( a„> + W T.

1

C W - < W W ] J0(an)UQ(RXn)

n=l J^(a ) - J?(b ) O n 0 l n

exp « »•)

T0 [ ( T 1 / T 0 ) l n l 1 / R ) + &n(R/R i)] Tt Jlnd/Rj)

an = V n ' bn = V °0 ( R Xn' = W ' W ~ W V Xn R>

W i ' W ~ W V V W i ' " °

R = r/r0, Fo = ax/r Q

See Fig. 9.7a for t, = t., and Fig. 9.7b for t„ = t., 1 l 0 l



9.1.10.2 74, p. 282 Infinite hollow cylinder with °° two convection boundaries and _ y / 2 \ variable initial temperature- f Z* n ^ \ n ^ l n' ' * general case. n=l t = f (R), r.< r < r , T = 0 .

V^

W(XnR) = - [ B i . Y 0 ( X n ) + X n Y l ( X n ) ] a Q (X n R)

+ [ W V + B i i W ] W > . 2 T T 2 , 2 En " T K K V W " W W ] '

R 0 x [ R [£(R, - t f ] W(Xn,R)dR j(xj + BiJ) [Bi. J ^ ,

+ W V ] 2 - ( xn + Bil) CBii V W " W W ] 2 ) " . 1

CBii W + V l M CBi0 V W " W W ] " LBi0 J 0 ( W " W W ] P 1 ! V V " W V ] = °'

B i i - W * Bi0 " V i / k ' R = r / r 0



9.1.11 3, p. 409 Infinite hollow cylinder with steady inside surface temperature.

i , _ x i \2 _ \ 1 n 0 n 1 n o n

v ^ T = " 4 e x p ( n Fo) Vo 'v-Wi ' B i ' n=l

W = W V W - V W W Al(R.) = V X n ) J l ( X nV - V V V W W W i > - J i f t „ | Y o ( Vi ' - ° Po = ocr / r 0 , R = r / r Q

9.1.12 74, p . 200 I n f i n i t e hollow cylinder with ( t - t . ) k steady surface heat f lux . t = t i ? c± < r < r Q , x = 0.

- q r - q 0 / * = r 0 , x > o. q r = 0, r = r i f T > 0.

V o 1 - R' i h - i ' r - " ! "» fe * rh») x An (R 1 ) + -

j . a R . ) j , a > + i | U Y ir u l l A n * i , u l * V 4JJ nTl^l'W " W

[ J 0 <*„ R >VW - ^ V l ^ f t ^ ) ] exp (-X* Fo)

Po = Qtr / r 0 , R = r / r Q

j i u . ] ! . a ) = Y a p..?J. (Xj l n x l n l n i i n



9.1.13 9, p . 210 I n f i n i t e cylinder with v a r i able i n i t i a l temperature and constant surface temp, t = f ( r , 9 ) , 0 < r < r Q , x = 0.

t = t Q f r = r Q , X > 0.

CO 0 0

t - fc0 = 2 I [V» c o s < n 9 ) + Bn,m s i n ( n 6 ) ] m=l n=0

x J (XmR) exp K-!)

ID I

- 1 -If

\ = r / / R|f(K,9) - t n"| J n (X R)dR dB °'m * [ w ] 2 I 4 [ o J ° m

1 IT

\ = - / | R p E ^ e ) - t . l cos (n9)J (X r)dR dB n'm *[W] ^ I L ° J ' 0 -IT

1 IT

n,m * OW]* Jo 4, JI / R|"f(R,9) - t J s i n (n9) J n(X r aR) dR

n •i.tr

de

J (X ) = 0 n 4 m'



9.1.14 9, p. 211 Infinite cylinder with vari- °° °° able initial temp and convec- . . V V r» / n, , ,-. , o>n T , I T,, / I 2 r, \

. . , t - t = ) > A cos (n9) + C s in (n9) J (X R) exp (-X Fo) tion boundary-general case. f Z_, £-, \_ n,m n,ra 'J n l n V n / t = f ( r , 9 ) , 0 < r < r , x = 0. ra=l n=0

x 2 - 1 -* A = —j-z 5? I I Rff (R,9) - t,~| J . (X R)dR d9

° ' m n (x 2 + B i 2 ^ f j i j t ! l 2 l J L fj 0 m

2X2

m

""•(^" !- !)[W] !

1 IT

x / / R[~f{R,9) - t 1 cos (n9)J (XR)dR d9

2X2

m n ' m " TT(X2 + B i 2 - n 2 ) [ J (X )1 : s i - n ; ivvj 2

1 .It

x / / R[f(R,9) - t j s in |f(R,9) - t , 1 s in (n9)J (X R) dR d9 1 * • n m ' 0 - i r

K?'JXJ + B i J n < X J = ° m n m n m

See case 9.1.3.2 for f(r,9) = t. or

case 9.1.3.1 for f(r,9) = f(r)



9.1.15 9, p. 212 Infinite cylindrical sector with steady surface temperature. 0 8 V sin(j8) \* /,2 _\ j m f _ T ,_, .._ t = t i f 0 < r < rQ, 0 <P9 < 9Q, T — r - 7 1 2^Y 1 «* K ^T^TP j j'

n=0 m=l D m "'O T = 0 t = t Q, r = r Q, 9 = 0, x > 0. j = (2n + l)ir/8 , J.(X ) = 0

— 0

9.1.16 2, p. 248 Semi-infinite cylinder. Dimensionless temperature equals product of solution for semi-infinite solid (case 7.1.1 or 7.1.3) and solution for infinite cylinder (case 9.1.1 or 9.1.3). See Figs. 9.4a and 9.4b.

9.1.17 2, p. 248 Finite cylinder. Dimensionless temperature equals product of solution for infinite plate (case 8.1.6 or 8.1.7) and solution for infinite cylinder (case 9.1.1 or 9.1.3). See Figs. 9.4a and 9.4b.

9.1.18 Infinite cylinder in a semi-infinite solid.

See case 7.1.22

Section 9.1. Solids Bounded by Cylindrical Surfaces—No internal Heating.


9 . 1 . 1 9 9 , p . 418 F i n i t e c y l i n d e r . t = t„, 0 < r < V

o < z < a , T = o . t = t 1 # 0 < r < r Q ,

z = 0, T > 0 . t = v 0 < r < r , z = I, T > 0.

Convection boundary h, tQ

a t r = r Q , x > 0.

t - t "" ~ "0 V" B i J p t ^ n ) s i n h [(1 - Z)X nL] fcl " fc0 ~ ( B i 2 + X 2 ^ J„(X ) s i nh (X L) n=l \ n / 0 n n

"I I

- 4ir Bi

mJ (RX ) s i n (nmZ) exp j - [ x 2 + (imr/L) 2 jFoJ J0 " w V

n = l m=l K L ) 2 + lm)2] [Bi* + Xn] V V Vi'V = B i W Bi = h r Q A , R = r / r Q , Z = z/X,, Fo = a r / r 2 , L = V r 0

9 . 1 . 2 0 9 , p . 418 Case 9 . 1 . 1 8 w i t h convect i o n boundary h, t

a t z = I, T > 0. h~ ko = 2 Bi

~ J,,(RX ) I X cosh ["X L ( l - Z)"| + Bi s i n h 1V(1 - 4

- 4 B

n-1 ( B i 2 + Xn) W [Xn c o s h <LV + B i s i n h <LV] " y » . ( B i 2 L 2 4 6 ; ) J o ( R X n ) s i n ( ze m ) exp [ - f a + ^ / L

2 ) F Q ]

Z

n

Z , fBi2

+ X 2 ) ( B i 2 L 2

+ B 2 + Bi L ) ( L 2 X 2

+ B 2 ) J n ( X n = l m=l V n / V ra / \ n m / 0 n

B cotB = - B i L, L = H/t„ m m u

See case 9 . 1 . 1 9 for X , B i , R, Z, and Fo n i i i



9.1.21 9, p. 419 Semi-infinite cylinder with convection boundary. t = t Q, 0 < r < r Q, z > 0, T = 0 .

t = t ± , 0 < r < r Q , z = 0, T > 0. Convection boundary h, t . a t r = r Q , z > 0, T > 0.

t - t, J 0 ( R V h-k^" ^ ?Bi2

+ X 2 ) J n ( X ) n=l \ n / 0 n 2 exp (-X Z) + exp (X Z) n n

x e r f c \ 2 Po*

+ Z Fo 1 - exp (-X Z) e r fc n 2 Fo Z FO

X J (X ) = Bi J (X ) n 1 n O n

Bi = h r Q / k , R = r / r Q , Z = z / r Q , Fo = r Q / 2 yjaJF

9.1.22 9, p . 419 Semi- inf in i te cyl inder with steady surface temperature, t = t Q , 0 < r < r Q , z > 0 , T = 0

t fcl' r = r 0 ' Z > ° ' T > 0.

t = t ( J , 0 < r < r Q ,

Z = 0 , T > 0 .

fc- fco J « < » * >

n=l

V O n ' I 2 \ * Z X J (X ) 2 e x p \T Xn F o j e c f ( Z F o > + e x p ( X n Z )

(z Fo* + — M + exp (-ZX ) e r fc (z Fo* " - _ ) V 2Fo / n ^ 2 Fo '

x e r f c i z Fo +

See case 9.1.20 for X , Fo , R,and z



9.1.23 9, p. 420 Cone with steady surface temperature. t = t . , o < e < e Q , T = o. t = t Q f e = e 0 , T > o.

= I + ~ {2n + 1)T [<n + l ) / 2 ] Pn(X)

^ 2 . nf (n/2) [dPn(X)/dn] x = x

n=0 0

•f exp ( - o r u / r * ) J n + 1 / 2 ( u ) d u

..3/2

P (X„) = 0 (Legendre polynomial) n 0

T 9.1.24 33 Infinite composite cylinder, t = t., r 3 < r < r , T = 0. t = t Q , r = r 3 , T > 0.

q r = 0, r = r ± .

k = °°

1 - ^ 2 - = TT J exp (-X 2 Fo) FG [ x ; J 0 ( R 2 X n ) - N X n J l ( R 2 X n ) ]

n=l

F = vw

[Wall "['n - N ( N " 2 / R2> Xn] " K J 0 < R

2 V " " V l ' V n ^ G ° J0(RV KWn' " "Wn'] " V^n'

^KvVi, 1 - H VI ( R 2V] {R - D 2 P , C

R = r / r i ' N = ^ — ^ T ~ ' J ° ( R 3 X n > = ° (R3 - 1) P 2 c 2



9.1.25 33

to I

boundary and insulated end. -~ fci V B i J 0 ( X n R ) c o a h [ Xn L ( 1 ~ X>] t = t,, 0 < r < rrt, 0 < x < I, t n - t. - Z A2 + B i 2 \ c M h

n=l \ n / O n n

Finite cylinder with convection al, r0

T = 0. t = tQ, 0 < r < r Q, X = 0, T > 0. q =0, 0 < r < r Q, x = Si, T > 0. Convection boundary h, h 2 , t Q : at r = r 0 .

h,t.

I

V — | x = A

- 271

2 1 (2m + l)BiJ 0 (X n R) cos [(2m-^~ -"J (1 - X)nJ

«=0 „-l L ' K + ( 2 m + X> V / 4 L 2 ] ("n + B i 2 ) W L [~,2 A (2m + 1) 2 TT 2 "11 x exp | -Fo |_Xn + J — - - J J |

X J . (XJ = Bi J (A,,) n l n O n

Bi = h r Q / k , L = J! /r 0 , R = r / r Q , X = x /£ , Fo = orr/r 2



t - t. *- Bi, Bi„ J n(X R) cosh fX L(l - X)l X £. U n i- n £_

9.1.26 33 Case 9.1.25 with convection V t. at r = rQ. ^ L . = 2 , - = Convection boundary 0 i , (X + Bi. ) J. (X ) X sinh (X L) + Bi_ cosh (X L)

n=l \ n 1 / O n L n n 2 n J h 2 , t 0 : a t x = 0.

4 in

" " , X m B i l B i 2 ^ + B i 2 J 0 (X n R) COS [X m L(l - X)] Z , ^ (X 2L 2 + X 2 ) (X2

+ B i 2 ) J n (X ) [(X2

+ B i 2 ) + Bi , l 11=1 n=l V n m/ \ n 1/ (T n |_\ m 2/ 2J

* e x p [-FO (xv + xy] V i ( V = B i i J o ( V ' x » t a n ( V = B i 2 Bi = h r A , Bi = h l A , L = V r . , H = r / r , X = x/ i



9.1.27 33

I

Finite cylinder with a linear time dependent temperature on one end. t = t.,, 0 < r <c r Q, 0 < x < %, T = 0. t = t t + b(t Q - tjjx, x = 0, 0 < T < b. t = t Q, x = 0, T > b. c^ = 0, 0 < r < r Q, x = l ,

T > 0. Convection boundary h/ t . : a t r = r Q .

t = f{r)-

^ x

h,t r

I X =

L t * 1

Bi J Q (3 n R) cosh [e n L(l - x)]

o " t i £, (ef + Bi 2 )j . (3„) cosh te D n=i \ n / o n n

B Bi j n ( 6 R) cosh f3 L(l - xjl in 0 n L m J

t„ -4_ Pd

CO CO

m=0 n=l . f r 2 + *m) ( B n + B i 2) W S i n «*»>

1 " « * [fa (6n + B > 2 ) ] « • ["Fo (Bn + e > 2 ) 0 m = (2m + UTT, U ( B ) = Bi j n ( 3 „ ) in n 1 n O n

Fo = oct/rjj, Bi = h r Q / k , L = l/a, R = r / r Q , X = x/B.,

Pd = rQb/a

9.1.28 33 Case 9.1.27 with convection boundary h_, t- at x = 0.

t £ = t. + b ( t 0

0 < T < 1/b. t = t Q, T > 1/b.

Solution identical to that given in case 4.1.25 except multiply the second summation series by

Pd {exp l^rr (X: + X~/L")| - 1} [k « • >>2)] Convection boundary h . , t . :

a t r = [ . .



9.1.29 65

UP l

Infinite rod attached to an infinite plate, t = t Q, 0 £ x <.<D , 0 < r < » , T = 0. g^ = q Q, x = 0, X > 0.

© b — ^

tf?mmzz;

te5S5S555!?£!

3 • s » ^ exp(-n2TT2 Fo) ,

X = 1 , R + » •

(fc - V k l H* - V k l l / k 2 P 2 C 2 \ 1 / 2

q0b " L g 0

b J ^ 1 V k l P l c l / R+oo

x J Fo j 4 i2erfc (^=) + 4 i2erfc (^=^) 4 i'erfc V « 4) 1 / 21

2 VFo J

W 2 / T ( 2 n + 2 ) 2 + R o i m l ( „ /n \ , f /n + l\

- erfc p * 4) 1/21 , (nfîiliir) . 2~?Fo J " e t f C \ L i~F5 J /

, X = 1 , R = 0

Fo = o^T/b", X = x / b , RQ = r Q / b





I to



9.2.1 9, p. 204 Infinite cylinder with steady » internal heating and constant 0' 1 ... _2 - V

4 (1 - R ) - 2 2, J 0(RX n) exp (-X2 FO)

surface temperature. r 2a"* 0*0 n=l i n n J„(X ) = 0

See Fig. 9.5

9.2.2 9, p. 205 Infinite cylinder with 90 steady internal heating

and convection boundary, t = t £, 0 < r < r Q, T = 0.

<t tf)k

r„q„ • * »

" exp (-A Po) J n(RA n) R 2 + 2/Bi) - 2 Bi J •" 7- "

*i x 2 (BI 2 + x 2) jn(x ) n=l n \ n/ 0 n

Bi j (AJ = X J (X ), O n n l n See Fig. 9.10.

Section 9 .2 . Solids Bounded by Cylindrical Surfaces—With Internal Heating.


9 .2 .2 .1 74, p . 372 Case 9 .2 .2 with q'' • = qj- 'e" 1 * 1 t ~ t

i

t = t . r 0 < r < r , T = 0. t f _ t i P d L J 0 (VPd,-4pJ l ( vPd,J CO

1 ( 1 - ^ ) * n W » » (-XnP0)

exp (-Pd Fo)

A = 2 Bi J„(A ) A. 0 n n ^"v V (x; + Bi»)'W"

9.2.3 9, p. 204 Inf in i te cylinder with exponential tine-dependent internal heating. t = t Q , 0 < r < r Q , T = 0. fc - V r - V T * °"

<* - V exp t-bt) (V R >y> Pd J n (VT3)

W

. I I I _ — • I I - « « " • -br

2_ V exP ( " ^ K ^ n 1

P d Z , X f x 2 / P d - l ) J,(X ) n=l n V n / I n



9.2.4 3, p. 363 Infinite pipe with steady Pipe temp: fluid flow and sudden heat generation in the wall. (t - t.)h P. / I \ 1 T + F r * Flu id : t = t . , T = 0. E l ^ - =AT + + ( r ^ T ) d " e > " e ( ^ " D

a ' " R (X - l l / i 2 \A - 1 / u. Pipe: t * t . , T = 0 . q 0 V 1 ) M

Fluid ve loc i ty = V.

\o

x * ( E , T * ) - X<f>(£,T*> + ^ - i - j . ) e _ X T A ( ? , T \ A ) ] , T + > (X - 1)C.

No a x i a l conduction. Fluid temp:

Wall heat ing: g• " = q ' " , T > 0 . u

( t . - t . ) h P. _,_ . + r

0 Insulated £ 1 i - = X T + - (1 - e~* T ) - e"^ J, outer wall q ( J " A p ( X - 1 ) / X 2

JM(M(/«l<lti«/<(tt(tllltM<«q _ „ . * * ~ X ^ * ~ x [ < V r + X " 1 ) , I ' ^ ' T ) _ A * ( u . T * 5 - e A ( C , T , X ) ] , T + > (X - l )C v-<^

6e~ 6 d6 h/MJJMM/MM/MMMd ± £ r n fT

I — ' * *(E,T ) = > - S — r / nfo <"'>2 J

F, » b -x /v , X = 1 + b f / b . T* = b„(x - x / v ) , 1 r P P a> T

b f - h Pi/PjCjAj. b p = h Pi/PpCpAp,, T

+ - b p x + ( 5 f T * , . £ j £ _ J fin+le-fl d f i

A, = flow area , A = pipe sec t ion a rea , n = 0 °

P. = pipe ins ide per imeter . °° r /. -, n .-(X-l)T 1 A«,T*,») = y M * - i ) ] f « n- 6 *

See F i g s . 9 .6a, 9.6b, and 9 .6c . *-* fnl) J n=0 * ' 0



9.2.5 9, p. 405 Infinite cylinder with linear temperature-dependent internal heating, t = t 0, 0 < r < r Q, T = 0.

V r = r Q, T > 0.

V' T > 0.

(t - t0)k x J Q ( R V P 5 ) X ^ exp [(-X n + Po)Fo] J Q(RX n) 4 2 z q-'-r*. ~ B JQ ( V P o ) P O

'0 0 ~ X (Po - X2)j1(X ) n=l nV n' 1 n

W = °' P° = B V k ' R = r/r0

2 If 6 > 0, steady state solution exists for Po < A..'.

9.2.6 19, Infinite cylinder of p. 3-29 infinite conductivity,

variable specific heat, convection boundary, and uniform heating.

See case 8.2.11, set h = 2, radius

9.2.7 19, Case 9.2.6 with a short p. 3-29 cylinder of length 2&.

See case 8.2.11, set L = 3, radius = i.



9 .2 .8 63 Hollow i n f i n i t e cyl inder with exponent ia l ly space varying heat generat ion, t = t . , r . < r < r Q . x = 0. t = t^Cr) , r = r 1 # T > 0. t = t 2 ( T ) , r - r 2 , T > 0.

q " 1 • im exp[-B(r - c , ) ] .

I CD

{t - t . )kB a • 2 J ?

n=l "O^n* "0""n"2 ' 2 = * I 2 ? W e XP (~Xn F°)

l/t J n< x J

FO

T^T) exp W-) d(Fo)

^ f (Xn) ( S ^ ) 2 / mx exp ( \ 2 Foj d(Fo)J

( t - t . ) kS a R = r / r i r Fo = t r r / r l f T ( T ) =

f*2 :(Xn) = / exp [-[Jr^R - 1)] RB0(XnR)dR

B 0(X nR, = J 0 (X n R)Y 0 (X n , - V X n R , J 0 ( V

WVVV = WVW



9.2 .9 74, p. 375 Inf in i te cylinder with a (t - t . )k steady surface heat flux — and steady internal heating. 1 n " r o t = t . , 0 < r < r„, T = 0. l 0 q

Po +

• " = *o" 0< r < r Q , T > 0. ^ _ g L FQ _ i ( 1 _ 2R2) _2 y z " V C = r 0 ' T " °" L ntl

J0(XnR) exp ( A ) -

I CO

a ' txj - o

9.2.10 74, p. 375 Case 9 .2 .9 with • I I I = C T * e • q ' " ( l - R ) . " - y - i „

J - = J Fo + * a ' r" y 0 0

[ 2 J W - W ] X n J 0<V

Ll2 x J Q (X n R) [ l - exp (-X' Fo)] + *

$ given in case 9 .2 .9



9.2.11 74, p. 375 Case 9.2.9 with q' = q£"(l + bT>. - - Fo ( 1 + ~ Pd Fol + <j>

a " ' r2

q 0 r 0 given in case 9.2.9

9.2.X2 74, p. 376 Case 9.2.9 with .•it . „ I M q'" eap (-br).

(t - t.)k -...,2 Pd

given in case 9.2.9 V ro

<=> 9.2.13 74, p. 376 Case 9.2.9 with qui * q'" cos (br).

(t - t )k 5 — = £r sin (Pd Fo) +

q""r*

*> given in case 9.2.9

9.2.14 74, p. 376 Case 9.2.9 with q... = q . . . b r n -

<* - v * (Pd Fo>" q.,.2 » + l

given in case 9.2.9


i


9.2.15 74, p. 393 Infinite cylinder with ._ . 2 » 2 „ „ , , , , „ . _ / ,2 _\ , . .7 (t - t , ) p c r „ , _~ A J„ (A R,) J . (X R) exp (-A Fo) pulse heating on a J f H 0 _ 1. V n 0* n V 0 V n ' ^ V n / c y l i n d r i c a l surface Q 1 ~ TT Z , L/L + ^ 2 \ j 2 . ^ and convection boundary n=l \ n/ 0 n h, t f , 0 < r < r 0 , T = 0. „ 0 / R =

Instantaneous pulse heating of strength Q. occurs at For Bi •+ »: 1 " * ' T " "• » - v°«o i £ ' . " . " ' . ' W -g (-'S*>)

Mean temp:

( t - t j p c r 2 , Z X J (X E,)B exp (-\2 Fo) 1 m f'K 0 l V n O n l ' n r V n / JS Q l * ^ J 2 r A ) 1 n-1 J l ( V

4 B i 2

1 1 A 2 (A 2

+ B i 2 )





Til •9

!

Hi I REPRODUCED FROM

BEST AVAILABLE COPY



10.1.1 2, p. 239 Sphere with steady •= 74, p. 126 surface temperature. 0 2 V (-1)n , 2 2 „ , . , „„.

t = tA, 0 < r < rfl, T = 0. t^ - t Q TT £, n n=l

' - V r • V x > °

- * - i £ [— ( 3 an^ s) - — ("ST^]- •»«i-n=l

Mean temp: 00

fcm " fc0 6 V 1 , K2 « ,

? I ^ y * ° * n=i n

Cumulative heat rate:

0 * n=l n

See Fig . 10.1



10.1.2.1 74, Sphere with convection p. 254 boundary-general case

o l

so

t = f ( r ) , 0 < r < r Q , t ' fcf = 2

T = 0.

h.t.

V X n S l n ( X n R ) exp K F o ) f „ r F, . ,. "1 Z T x - s in (X ) cos (X )1 R / R L f ( r ) " fcfJ

n=l L n n n J "0 x s i n (X R) dR n

t a n ( A n > - (1 - Bi) See Table 14.2 of roots

10.1.2.2 3, p. 298

Sphere with convection boundary. t = t £, 0 < r < r Q, T = 0. t, - t, 2 Z

Bi sin (X ) exp (-X Pol sin (X n \ n / n X - s i n (X ) cos (X )X R n n n n

R)

n=l X cos (X ) = (1 - Bi) s i n (X ) n n n

Cumulative heat rate:

" [s in (Xn) - Xn cos tXnQ [l - exp {-\\ Fo)] Q " 6 Z 7i

n=l X 3 fX - sin {X ) cos (X )1 n •- n n n ~>

See Figs. 10.2a, 10.2b, and 10.2c



10.1.3 9, p. 233 Sphere with initial temp f(r) and surface temp 4>(T)J general case. - 1 2

n=l

•If

2 2 exp{-n TT Fo) sin (rnrR)

Rf (R) sin (nirR)dR - nir( •l)nFoj exp(n 2 i r 2 Fo X)* (X)dX

o 10.1.4 9, p . 235 Sphere with surface temp <ii p ropor t ional to time.

t = t . , . 0 < r < r Q , T = 0.

t Q = br + t i # r = r Q , T > 0.

fc " fci / . , R 2 - l \ 2 V _tlll , K 0 i \ / i F o R , n

n= l

Fo) s in (nlTR)

See Table 10.1 and Fig . 10.8



10.1.5 9, p. 235 Sphere with variable initial temp. t - t„ 8 fc = ( t c " to* < r0 " r , / ( r 0 + fc0> fcc " fc0 " n 3R % (2n + 1 0 < r < r Q , T = 0. n ~ fc " V r B V T > °-

Y ~ j exp[-Tl 2{2n + l ) 2 Fo] s in [(2n + 1) TTR]

?

10.1.6 9, p. 236 Sphere with variable initial temp (parabolic). fc = (tc " V ('I " ^Ho 2 + fco)' 0 < r < r , T = 0 . t = t Q, r = r Q, T > 0.

t - t„

' c - fc0 12 7T3R I ^ n-1 2 2 sin (ffitR) exp(-n li Fo)

n=l



10.1.7 9, p. 237 Sphere with variable initial temp (trigonometric). ~ _0_ 1 ._„. , Jl „ . ._ .. .,,_,, i , i . 7 I"~ = n Sin (TTR) exp(-TT FO) t = t Q + (t^r) sin (irr/r0), t c - t Q R r i

o U1

10.1.8 9, p. 237 Sphere with variable initial temp (exponential). .. 0 2TT fc = fc0 + ( t c " V e x p L b ( r " roU' fcc ~ fc0 " R

0. 0 < r < r Q , x t = t ( ) , r = r Q , T > 0.

n=l (nV + B 2 ) 2

2 2 exp(-n IT Fo)

x s in (nTTR) [ ( - l ) n + 1 ( B 2 - 2B + n2TT2) - 2B exp (-B)] B = br„



10.1.9 9, p. 236 Sphere with two initial temperatures.

J

t - 1 0 2 Y" r s i n '•ri'nR

1'> i

tr^x = m Z L—m— - R i s i n < n i t R i > J "1 0 n=l 2_2 x sin (mrR) exp (-n n Fo)

10.1.10 9, p. 237 Sphere with polynominal initial temperature. t - t,

t - t x - t Q + b x r + b 2 r 2 + b 3 r 3 h - 0

+ . . . . , 0 < r < r , T = 0. t = t Q , c = r Q t T > 0.

S- =2 | « i . («m ( i H ) I H 1 * - ^ [ ( A 2 - 2, n=l

x ( - l ) n + 1 - 2 ] + - ^ ( n V _ enTT) ( - l ) n + 1

+ 4„4 n it 5„5 n TT

x [24 - < n V - 1 2 n V + 2 4 > ( - l ) n + . . . . ] ! e x p ( - n 2 n 2 Fo)

bX " V o / R ' b2 = b 2 r 0 / R ' b 3 = V o ' * '



10.1.11 9, p. 238 Sphere with convection boundary 89 and changing fluid temperature.

t = t., 0 < r < r , T = 0. t f = bT + t A , T > 0 .

o I

( t - V " „ R 2 Bi - 2 + Bi 2 B i

—i— =Fo + —m + i r -b r :

s i n (RX ) e x p (-Fo X )

, X 2 |~X2 + B i ( B i - 1)1 s i n (X ) n= l n L n -I n I

X cot{\ ) + n n Bi = 1 Mean temp: ( tm-

tn . = Fo - 1 15

( ' •

See Fig. 10, .9

V 6 X P (~Xn F ° ) ^ X 4 (X 2

+ B i 2 - B i ) n= l n V n '

S e c t i o n 1 0 . 1 . S o l i d s Bounded by S p h e r i c a l Surfaces—No I n t e r n a l Heat ing .

Case No. References D e s c r i p t i o n S o l u t i o n

1 0 . 1 . 1 1 . 1 7 4 , Case 1 0 . 1 . 1 1 wi th t - t . „ „ . . , v ^ T ^ > , „ J r, > , , . , _ K T i _ _ , R Bi s i n ( VPd R) exp (-Pd Fo)

- 1) s i n ( V P H ) + V"PcT cos

A R s i n (A R) exp (-A 2 Foj

P " 3 1 7 t , = t„ - ( t„ - t . ) e " b T . t . - t . "" x (Bi - 1) s i n (VTa") + V M cos ( V P 3 ) £ £m rm l £m I

o I

y ~ (l - A 2 / W ) A i = l \ n / n

( -D n + 1 [^ + (Bi - I ) 2 ]

- 2 B i

n=l

,21 1/2 A

( A 2 + B i 2 - B i ]

Mean temp:

fcm " fci _ 3 Bi Ctan ( V?d) - YPTLI exp (-Pd Fo) t f - t ± Pd [(Bi " l ) tan ( -VPd) + VPdJ

00 / 2 \ _ ^ exp (~A Fo)

- 6 B i 2 ^ V n ' , A 2 ( A 2 + B i 2 - B i ) ( l - A 2 / P d ) n=l n \ n / \ ti '



10.1.12 9, p. 238 Sphere with convection boundary and changing fluid temp. t = t., 0 < r < r„, T = 0. i 0 t = (t - t.) sin (OTT + G), T > 0. r m i t - max fluid temp.

m r

o

t " t i 2 Bi t - t . Pd R m I

X rX

Pd Lpd s i n ( e ) " c o s ( e l l ( B i " 1 ) s i n ^V \£x CMH* Xl (Xn + B i 2 " B i ) «» <V

x exp f-Fo \ J + — - s i n (urr + e + $ + <t>2)>

Pd = <»)rn/a

X c o t (X ) + Bi = 1 n n

A e = s i nh (u ) cos (u ) + I cosh (u> ) s i n (u> )

i < l > 2 A e = a) cosh (to ) + (Bi - 1) s i nh (ui )

co = R "Vpd/2, ii) = (1 + i ) VPd/2



10.1.13 9, p. 240 Sphere enclosed in a finite medium of infinite conductivity. t = t . , 0 < r < r , x = 0 .

t = t Q , rx < r < r 2 , x = 0 .

q r = 0 , r = r 2 , T > 0 .

a

o

Temp of t h e s p h e r e :

fc- fco 0 _ 1 2M V" n t . - t „ M + 1 " 3R Z w 2

M 2X 4 + 3(2M + 3 ) X 2 + 9 - n

MX + 9|M + l ) r n = l n n

s i n (RX )

x s i n (X ) exp (-X Fo J , 0 < R < 1 .

Temp of the surrounding medium: oo

t - t

3 1 C 1

See F i g . 1 0 . 3

t j - t „ M + 1 ^ M 2 A 2 + V n 1/ n=l n '

H 2 U 2 / 3 \ " ' M = Kc7 ( R 2 - V ' R = r / r l ' t a n <V

1 < R < R_

3X

3 + MX n



10.1.14 9, p. 241 Case 10.1.13, except the sur- Temp of surrounding medium rounding medium transfers ^ heat by convection to a fluid at temp t . t = t., 0 < r < r , T = 0. fc - V r > V T " °-Exterior surface area = A.

t - t i

- i L 0 t , - t„ Z , (3 Bi R2MX=) exp (-A* P o J

. M2X4 + 3X^{3 + 3M - 2M Bi R_) - 9 Bi R , ( l - Bi R_) n=l n n 2 Z 2

P,c_ y2 2 / 3 \ Bi = h r 2 A . , H = — (R2 - l ) , R = t / r L

tan(X ) 3X

3(1 - Bi R_) + MX 2 n

\



10.1.15 9, p. 242 Sphere with steady surface heat flux. <fc - V k

3 p o + 5R 2- 3 2 V sin (R2Xn) exp (-XR FO) t = t 0 < r < r , T = 0. q nr 0 " 10 " R L f s i n ( X )

n=l n n ^r = V r = V T > °'

i

tan(X ) = X v n n See Pig. 10.4

10.1.16 9, p. 246. Hollow sphere with two R. (T-/T. - R )R T /T . cos (mi) -R. surface temperatures. T_ = _i t 0 i i 2_ V 0 i i t = f ( r ) , c . < r < r„ , T = 0. T. ° R R Rir Z . n t = t . , r = r , , T > 0.

t = t 0 , r = r 0 , x > 0.

n=l

2(1 - R.) * 2 2 " v " i V x s in (nnR ) exp(-n n Fo) + ) s in

n=l (nTiR )

2 2 x exp(-n ir Fo) / \ , + R)f (R ) s in (nlTR ) dR

R* = (r - r , ) / ( r n - r , ) , R = r / r . , Fo = a T / ( r . - r . ) 2

• i " v 0 i' "0 i '



10.1.17 9, p. 247 Hollow sphere with steady surface flux, t = t t , r A < r < r Q , T = 0.

t = t i t r = r i r x > 0.

^ r = V r = V T > °'

?

(t - t^k (RQ - R)

V i RQR + 1 V (1+XD1/2 S i n ( X n R Q - X n R )

\=1 K K + <R0 " 1 ) X n J ,2 x exp (-X Fo)

tan (X R„ - X ) + X = 0 n 0 n n

R = r/r . , Fo = orr/r . I I

10.1.18 9, p. 350 Sphere of infinite conductivity enclosed by shell of f inite conductivity.

t - t,

fci- fc0

0 _ iJK R

'I sin [X (1 - R)] exp (-Xn Fo)

^ 2K(1 - R.)Xn + 4RtXn s i n 2 fr^l - R^] - K sin [ 2 \ n ( l - R.)]

R. < R < 1 1

R,K X cos fX (1 - R.)l = (x2R? - K) sin [\ (1 - R.)l i n L n i ' J v n x ' u nv i ' J

2 K = 3 P 2

C 2 / p i c l ' R = r / , , : 0 ' F o = c c r / r Q



10.1.19 9, p. 351 Composite sphere.

o i

fc " V r = V T > ° t - V 0 < r < V -r - 0. __±_ = _ 0 ^ ^ s i n ( R V s i n ( V s i n [ A < v 1 } X J

l u , n n=l

x exp (-X 2 Fo n) , 0 < R < 1 .

CO

fc - fcQ 2 R Q

t- ~ t . ~ R , . . 1 ° n=l n

x exp (-X n F o 2 ) , 1 < R < RQ .

(|)(X ) = K X s i n 2 TA X (R„ - 1)1+ A X (R„ - 1) s i n 2 (X ) n n ' - n u - ' n O n

+ [ (1 - KA)/A X n ] s i n 2 (Xn) s i n 2 [A X Q (RQ - 1)] .

X cotfA X (R„ - 1)"| + 1/A + K [X cot (X ) - l ] = 0 . n L n O ' J ' L n n J

Fo = CTC/r*, A = V\te7. K = \ / * 2 , R = r / r . , A(RQ - 1) i irrat ional

Section 10.1. Solids Bounded by Spherical Surfaces—Wo Internal Heating.


10.1.20 45 19, P. 3-57

Sphere with radiation cooling t = t Q, 0 < r < t x , T = 0.

q = a #"(T4 - T 4 ) , r = r,. ^r \ r=r x s/ 1 T = sink temp.

See Fig. 10.6

o I

10.1.21 72 Prolate spheriod with steady surface temp t = t., throughout solid, T = 0. t T > 0

t , on surface of solid.

See Figs. 10.7a & b for t - t.

1 t -w

t. x

vs Fo



10.1.22 74, p. 320 A sphere with convection and evaporation boundary. t = t., 0 < r < r„r T = 0. I 0 m = m e = evaporation rate.

t - t. = 1 - R M Bi s in ( VPd" R) exp (-Pd Fo)

(Bi - 1) s in (VPd) + VPa" cos (VPd)

H L i -(pdA 2 y n v n ;

n=l

A = n ( - l ) n + 1 B i [A 2

+ (Bi - I , 2 ] 1 / 2

2 2 X + Bi - Bi n , K = Ym Q /h( t f - tt)

A cot(A ) = 1 - Bi n n

Section 10,1. Solids Bounded by Spherical Surfaces—No Internal Heating.


o



o l }-• CD



10.2.1 1, p. 307 Sphere with steady surface temp and internal heating, t = t.f 0 < r < r Q, x = 0. t = v r - vT" °-q"' = q i " , 0 > r > r n, T > 0.

(t - tQ)k

*0 '0 i'-«2'*±Ife-^)-"" IT R ~;; "n n=l

2_2 x sin (nirR) exp (-n ir Fo). See Fig. 10.5 for condition t. = t ,



10.2.2 9, p. 243 Sphere with linear internal heating distribution, t = t Q, 0 < r < r n, T = 0. t - t Q , r = r Q , T > 0.

q - - q J " ( E 0 - r > / r 0 , 0 < r < r Q , T > 0.

( t - t Q ) k x

J 0 0

f j U - 2RZ + R 3) _ J3__ V* _ _ _ 1

^ n=0 ( 2 n + 1 }

x exp [-(2n + 1) TT Po] s in [(2n + 1) TTR]

> fO a

10 .2 .2 .1 74, p. 366

Case 10 .2 .2 with

«'" = %"^°n> b •» 1, 0, 1, 2, . . . t - t,,, 0 < r < r 0 , T = 0 t = t Q , r = r Q , T ? 0,

t^rr ' 2 R [ e r f c PYvfe" V ~ e r £ c ( 2 Yv*5 + R )J

U - r ( 2 + | )

n=l

Po Po 1 + (b/2)

h+2

V 1 T.b+2 . /2n - 1 - R 2. R L1 " ^ V 2VF5 j

b+2 c /2n - 1 - R't ,b+2 . (^f^)]) n=l

P o = q - ' T b / Z r J / b ( t 0 - t l



10.2.3 9, p. 243 Case 10.2.2, with q - - < J " ( r ; - r 2 ) / r ; f ^ ^ - fe (1 - R2, (V - 3R2, 0 < r < rn, T > 0. H0 0 n=l

(t - tn)k , 9 •> 12 V (-l)""1

"0' 2 2 x exp(-n li Fo) sin (nltR)

10.2.4 9, p. 244 Case 10.2.2, with ( t - V k 1 r 2 - , q'-'r. 5 — = -5- [l - exp{-TT Fo)J sin (TTR)

- J — a s l n d r r A 0 ) . q ' " r 2 ^ R

0 < r < r , T > 0 .

10.2.5 9, p . 244 Case 10.2.2 with ( t - t . ) k

q ' - ' r 2 B~

uase io.-i . i! witn i t - t JK , 1 q » . = q ' - - e x p [ b ( r - r 0 ) ] . — _ 2 _ . i j 11 - f - ( l - ^ ) exp(BR - B) - | ( | - l ) e B )

sin (mrR) , 2 2 „ %

j j 2 exp(-n TT Fo) 2_ V s in (mrR) Q n m , ^ 2 TTR Z ,

n = 1 n(n TI + B )

[ ( - l ) n ( 2 B - n2Tf2 - B 2) - 2Be" B ] B = r Q b .


Case Mo. References Description Solution

10.2.6 9, p. 245 Case 10.2.2 with q,., . q - ' e ^ .

i_ e"br [sin (RVTdL _ I _2_ Pd e LR sin(Vpd) "J 7 r3 R

CD

V (-1) s in (mm) Z . _ , 2_2 „ ,

2 2 exp(-n fl Fo) ,

n=l

Pd = rjjb/ct

n(n TT - Pd)

i to

10.2.7 9, p . 245 Case 10.2 .2 with q " ' = q j - ' e "* 1 * , rx<c< r 0 . q " > = 0, 0 < r < r ,

2 a " ' ' r q 0 0

_ e j f s i n (RV"Pd) _~| s in (VPd - RVpd) I " R Pd l [ s i n (V~Pd~) J s in (VPd) J

x [ V c o s (RjVPd) - ^ L g . s in (RjVPdjJ

00

+ h I ,z\ M 1 [- '- 1 '"+ T S F s i n """V n h It — Prll l-n(n TT - Pd)

R cos (mtR, . ) s in 2 2 (nuR) exp(-n TI Po) .

Pd = rjjb/a .



10.2.8 9, p. 245 Case 10.2.2 with ,,. . ,,, . i n - *.-. n - - - _ * c ~ zn>K

T > 0

q'" - f(r), 0 < r < r Q, ^ _ = _^_ y ±_^ exp(-n'ir' Fo) sin (nirR) / R'f(R") sin (mrR')dR' ~ fc0 2 V 1 2 2 / -= = -f" > ^ exp(-n IT Fo) sin (nTfR) / R'f(R') E* TTZR ~ n 2 J

n=l 0 jj I R , 2f(R') dR' + I R'f(R') dR" - I R'2f(R') dR'

10.2.9 9, p. 245 Sphere with steady internal ,. . .. •» • ,i „, / , 2 „ \ S 90 heating and convection ( t ~ t f ) l { 1 2 „ .„., „ Bi V s l n ( V > e xP H n

F°) i boundary. l t l 2 = ? ( 1 ~ R + 2 / B x ] ~ 2 T 2

- = t f , 0 < r < r Q

boundary. „ m , 2 6 R ^- ^2 ( \ 2 ^ r>-2 „-\ , i > t = t *0 < r < r n , T = 0. *0 r 0 „- l X J X n + B l " B l J s l n <V

q . . . = q ^ . . # o < r < r Q , X n c o t (Xn> = 1 - B i T > 0 .

See F i g . 10 .10

h.t f

Section 10.2 . Sol ids Bounded by Spherical Surfaces—With I n t e r n a l Heating.

Case No. References Descr ipt ion Solut ion

10 .2 .9 .1 74, p . 366 Case 8.2.9 with

q " ' = qJ"e"" b T -Pd L

x exp(-Pd Fo)

Bi s in (VPCTR) "1 R E(Bi - 1) s in fy Pd)+ V Pd cos

.A s in (X R) exp (-X Fo). Po \ n n \ n /

n=l n RX

2 Tsin (X ) - X cos (X )~| L n n n J

X - s in (X ) cos (X ) ' t a n ( X n } ~ 1 - Bi

10.2.10 9, p. 246 Sphere with pa rabol ic i n i t i a l o I

IO

(t - t , )k temp and convection boundary. 1* 1 2 |"l_ 1 1 I" 2 Bi

0 < r < r , T = 0 (temp d i s t . for steady state with const, surface temp = t 1 ) . . i n _ — 111 q'", 0 < r < r n, T > 0.

^. sin (XnR) exp(-Xn Foj -i

*£*_ (x* + Bi2 - Bi) sin (X ) -I n=l v n / n 2

X n cot (Xn, = 1 - Bi, Po = ° J°h)



10 2.11 9, p. 350 Sphere of infinite con- (t - t )k ductivity enclosed by a 0__ _ 1 /i 2^.\ _ M shell of finite al"r 2 3 ^ R V R

" sin [X n(R Q - 1)] sin [XJRp - R)] exp (-X* Fo)

^ X n J2K(R0 - l)Xn + 4Xn sin2 [ X ^ - 1)] - K sin [2Xn(R0 - 1)])

KXn cos [Xn(Rc - 1)] = (x* - K) sin [Xn(RQ - 1)] , 1 < R < % .

= 3p 2c 2/P 1c 1, R = r/r., Fo = a2T/r.

snexi or rinite q'"^ conductivity. ° 0 t = t Q, 0 < r < r Q, T = 0. q... = g^.., o < r < r Q f

T > 0. q'" = 0, t. < r < r Q,

T > 0.

o to Ol

10.2.12 19, pp. 3-29 Sphere of infinite con- See case 8.2.11, set L = 3, radius = & ductivity, variable specific heat, convective boundary, and steady heating.



10.2.13 74, p. 370 Sphere with a steady internal heating and steady surface flux, t = t., 0 < r < r Q, T = 0. -q r - q 0, r = r Q, x > o.

*o"> 0 < C < r 0 > T > 0.

I to

(t - ti)k Fo + $

4> = Po 3 Fo - j£ (3 - 5R2) - 2 sin (X R) exp (-X Poj

tana ) = X n n n n=l X R sin (X ) n n

10.2.13.1 74, Case 10.2.13 with p. 370 q"' = q'"(l - r/rn). (t - t.)k

• t i r2 4 ^

2 - (2 + \ 2 ) cos (X ) sin (X R) V n/ n n

V ro n=l 5 2 X R sin (X ) n n

x [l - exp(-X2 Fo)] + $

$ given in case 10.2.13.



10.2.13.2 74, Case 10.2.13 with P-370 q l I I = q , . . a _ R 2 h

( t - V 2 Fo a' "t q0 r0

- 2 ^ sin (X R) [l - exp (-X2 Fo)] z — ~ — = ~—+ * n=l X R sin (X ) m m


10.2.13.3 74, p. 370

o I

Case 10.2.13 with .111 — n i i l = qA" exp(-bR).

( t " fci)k 3 Fo [2 - exp(-b) (b2 + 2b + 2) q' • 'r ^0 r0

2X + 2b

n=l

- exp(-b) (2 + b + 2b + X ) sin (X ) sin (X R) n \ v\j_ n n X 2 sin2 (X ) (X2 + b 2 ) 2 R n n \ n '

x J"l - exp(-X2 Fo) 1 + * $ given in case 10.2.13.

10.2.13.4 74, p. 370

Case 10.2.13 with .111 — n 111 q'"(l + br),

( t ~ V k / 1 x 5 — = Fo (l + ± Pd Fo) + $ q0 r0


10.2.13.5 74, p. 371

Case 10.2.13 with q"' = q^-'expf-br),

<t - t.)k T~ = ^ L1 " exp(-Pd Fo)J + 0

q'"r 2 H0 0

Pd




10.2.13.6 74, p. 371

Case 10.2.13 with q'" = q'"cos(bT).

(t - t >k x

,,,2 " id S i n ( P d P o ) + *

9 given in case 10.2.13.

10.2.13.7 74, Case 10.2.13 with p. 371 T " ' = < - . • ' v*

( t - V k (Pd Fo) n r .

,,, 2 " n + 1 F o + *

o * given in case 10.2.13.



10.2.14 74, p. 390 Sphere with pulse heating on a spherical surface and a convection boundary, t = t f, 0 < r < r Q, T = 0. Instantaneous pulse of strength Q occurs at r = r x , T = 0 .

, 1 / 2 ( t - t f ) p c r ^ x ^ . [(Bi - l ) 2 + X 2 ] A n s i n ( X ^ ) s i n (XnR)

{£ ~ 4TF Z , Bi R. R l

A = (-1) n n+1

n = 1 x exp (-X^ Fo) .

2 B i [ V * ( B i - l ) 2 ] 1 / 2

X 2 + B i 2 - Bi

o I

t a n « x

n > - r ^ - R = r / r o For Bi •* »s

( t - t f ) p c r g ^ - / r^- s i n (mrR.) s i n (nTO) exp (-n TT Fo) 21T l _ j *î

For r. = 0: n=l

3 °° , 2 ( t - t f ) p ^ _ j ^ V X n S i n ( X n R ) e x p ( - X 2 ^ o )

21TR Z , X - s i n (X^JcosfX^) n=l

Mean temp:

( t - t . ) p c r ^ , ^ - r , o - r / 2 ! • Z 0 . I \ [ (Bi - l ) 2 + X 2 1 X B s i n (X R )

Q, 4irR, Bi AJ L nJ n n n 1 n=l

x exp (-X Fo)

6 Bi n , 2 / , 2 , „..2 X 2 (X 2 + B i 2 - B i ) n v n '

Section 10.2. Solids Bounded by Spherical SurCaces—With Internal Heating.


i u o



o u

o sr

REPRODUCED FROM j§ BEST AVAILABLE COPY "

e *a 3 -P ft



11.1.1 9, p. 285 Initial surface maintained at Solid temp: constant temp: freezing liquid. t = t n , J t = u , T > 0 . t = t , x > 0, T = 0. t = melting temp, m

°Y •w i «

(solid)

VV 1 "! (liquid)

' f c s - fc0 fcm" fc0

erf K) / e r f ( X ) , 0 < x < w.

Liquid temp;

t„ - t. a I / * \ i— —— t _ t = e r f c l F o ^ ) / e r t c ( X V c r / a A ) , x > w . m ~ 1 ^ '

XVVTT C

S

( t m " V exp(-X 2) , ( t m - V h e X P fo V * & )

erf(X) ( t m - t Q ) k s

2 a erfc~(X)

w = 2Xy a x s

11.1.2 9, p . 287 Solid region maintained a t melt ing temp: freezing l i q u i d . t = t , x < w, T > 0 . m t = t „ , x > 0, x = 0. t , < t , t = melting temp. 1 m m

Liquid temp:

= XVTer fc /po^VexpC-X ) , x > w .

w = 2X-ya.T

X exp(X 2) erfc(X) = ( t m - t ) C J / Y V T


_ase No. References Description Solution

11.1.3 9, p. 287 Melting of a solid in contact Solid temp: with another solid. t = t Q , x > 0, T = 0. t = t, > t m, x = 0, T > 0. t = melt ing temp. m

'NU,

i to

vv k. (solid)

I W k ! ! I (liquid) (solid)

fc

S" fc0 = erfc ( F o x s ) / e r f c (XVc^/cT) , x > w.

Liquid temp

= erf ( F o ^ l / e r f (X) , 0 < x < w .

exp( -X 2 ) + "s^Ô - V " P ^ V v _ XYV? e r f { X ) k. V5~(t , - t ) e r f c (XVa. /a ) c X , ( t l " V £ s 1 m SL s

w = 2X V a - t



11.1.4 9, p. 288 Liquid freezing on solid-properties of solidified liquid different from original solid. t = t „ , x < 0 , T = 0 . t

t 0 , X < 0, T

t , , x > 0, X = 0. t = melting temp, m

Original so l i d :

VA" fcsi - fco t - t„ VS" + K erf (X) m 0

[ l + erf (Fo* s l ) ] , x < 0 .

So l id i f i ed l i q u i d :

fcs2 - fc0 V * + K e r £ ( F °xs2> m 0 VA + K erf (X) , 0 < x < w

Liquid:

>1' i l 1 \Tai2-kn2 (solid) I (solid) (liquid)

t„ - t , t _ = e r fc ( F o ^ J / e r f c (XV a ^ / c ^ ) , x > w m i.

X Y V T K e xp( -X 2 ) ] " f c ^ ^ m - V 6 X P ( - X a B2 / a ft> c s 2 ( t m " t 0 ) VA"+ K erf (X) k , V X ( t - t . ) e r fc (XVa , / a . )

s2 X, m 0 s2 X,

w = 2X V a T r , K = k , / k . , A = a , / a 0 s2 s i s2 s i s2



11.1.5 9, p . 289 Melting of a solid in contact with another solid—properties of liquid different than original solid. t = t, > t , x < 0, T = 0.

1 m t = t Q , x > 0, T = 0.

w = 2XV^t

XYVTT c A ( t l V

k^Vaêxp (-Xz) H^7l +* s l Vêrf(X)

+ k * ^ < t x - y erfc (XVapoT^)

See case.11.1.4 for temperatures t = melting temperature , m

11.1.6 9, p . 289 Solid melts over temp range For case 11.1.1, change equation for X to:

>2, o f tml to W t - t ^ , x > 0, T = 0.

H * °H + Y / ( t m 2 " W '

exp [ t a s - H)VVaJ erfc ( X V a ^ ) ( t m 2 - t m l ) k £ V ^ erf (X) (t. ml V s * 5 *

For case 11.1.4, change equation for X to:

K exp[(a 5 2 - y X 2 / ^ ] erfc a ^ ^ \ ) _ ( t m 2 - t ^ ^ V T J V3T+ K erf (X) (t. ml t„> V^7

Use c." for c« in the above equations and solutions given in cases 11.1.1 and 11.1.4.



11.1.7 9, p. 290 Change of volume during t. - t ( t X(p s - P„) /a — -= = e r f c I Fo „ + n

s o l i d i f i c a t i o n . t_ - t , I xl P m t = t Q , X = 0 r T > 0.

t = t 1 # x > 0, T = 0.

P s > P r

t = melting temperature.

I 2 ) ( t l - V h ^ e * p ( - * 2 P s V P & a l ) X Y V ? erf (X) ~ (t_ - t n ) k „ V o 7 e r f c (Xp„VcT/pDVa;) c^ ( t m - tn) exp(-X )

*m " O ' ^ s 7 ^

W = 2 X V CX T s

J s T "B'WI'

I I " x

w P.-'WM

(solid)

t a . a s , k e , p s

(liquid)

11 .1 .8 9, p . 291 Constant heat f lux a t o r ig ina l s o l i d - l i q u i d boundary. t - t , x > 0, T = 0 . m q j £ = q Q , x = 0, T > 0.

t " so l i d temp, t < t . s r s m

( t s - V k s a s V

Qx 2x

_ - 2 ^ ( 1 + 2 F o J + - ^ - T d + 1 2 F o - + 1 2 P o »> x- 4 X 12x

. . . , 0 < x < w .

w = QT - \ Q 3 T 2 + | Q 5 T 3

Q = q 0 / a s P s Y • F ° x = < V / X * ' T = V

t = melting temp, m



1 1 . 1 . 9 9 , p . 292 Convect ion boundary a t

o r i g i n a l s o l i d - l i q u i d

boundary.

t = t 1 # x > 0 , T = 0 , t x > t m . t = m e l t i n g temp.

t - t F B i 2

_§ _S = B i T-^{1 + 2 ! Po ) t - t , x 2 x '

m £

P Bi

IT'1 + 3 1 F °x> + . . . , 0 < x < w .

V \ . ,

F 2 B i 3

F B i x F O x . _ _ x ( 1 + P ) F o 2 + . .

F = k s ( t m - V / a s P Y ' F o x = V / x ' B i x = h x A s

(solid) (liquid)

Section 11.1,, Change of Phase—Plane Interface.


11.1.10 55 19r p. 3-87

Solidification on plane semi-infinite solid with convection coefficient between solid-solid and liquid-solid, t = t , x < 0, T = 0.

c = 0. t = melting temperature.

h 2 ( t s m V

pyk

N =

= i - o r N - 1 I Bi M 2 LN(Bi + 1 ) - l J " N

V f c0 " V h (t s m V

, Bi = h w

s

See Fig. 11.1

(Solid) (Solid)

P.c

(Liquid)

-•Vô

11.1.11 56 19, p . 3-88

Case 11.1.10 with s o l i d i f i e d

layer of f i n i t e heat capacity

and heat flow at x = 0 given

as q = bt , where t i s temp w w at x = 0.

See Figs . 11.2a and b



11, . 1 . 12 1 9 , 77 p . 3. -88

1 1 . 1 . ,13 1 9 , p . 3--88

Case 11.1.10 with heat flow 4

at x = 0 given as q = b t M . See F igs . 11.3a and b

i CD

Decomposition of a semi- Char layer: i n f i n i t e s o l i d . t = t Q , x >_ 0, T = 0. t = t , x = 0, T > 0 . . t . = decomposition temp. £ - bVT.

Char Virgin

©

I — i - 1

(D

t - 1 ,

w d

erf KJ erf {b/2Va^> , 0 < x <_ I

Virgin layer:

t - t , erfc KJ t w - t d " erfc" (b/2 V a p , x > I

e x p ( - b 2 / 4 a 1 ) / P ^ 7 / t d - t A exp f b 2 / 4 a 2 ) _ ^ i r p / c ^ yb erf (b/2Va£) \ P-f^ \*v " t j « f c ( b / 2 V ^ ) 2 { t w - t f l )

See Figs . 11.4a and b



11.1.14 57 19, p. 3-93

i

Ablation of a semi-infinite solid by convection heating, t = t , x >_ 0, T = 0. t, = decomposition temp, d

See Figs. 11.5a, b and c

2 fcd" fc0 fc _ ° = 1 - exp [(hA) OT d] erfc [(h/k) VaTj]

T. = Time for free surface to reach temp t, d a Steady state ablation velocity and temp distribution:

v = h< f ca " V

"ss p (t d - tQ) + If]

^ = e x p [ - > ( * - * , ] H = heat of ablation



11.1.15 58 Case 11.1.15 with heating by See Fig. 11.6 19, a constant surface flux q". Preablation time: p. 3-95

T * r ^ - ^y Td= 4 p c k \ T 5 r r ~ 7

Ablation ve loc i ty:

' ss p [ c ( t d - t Q ) + H]

Section 11.1. Change of Phase-Plane interface.

Solution Case No. References Description

V

Section 11.1. Change of Phase—Plane intecface.


i to

Section 11.2. Change of Phase—Nonplanar Interface.


11.2.1 9, p. 29 Liquid freezing on a cylinder. t = t , 0 < r < R . m t = t , r -»• «, t < t x i n t = melting temp.

.£) ,

Liquid temp:

hî= Ei ( _ F O * 0 m (-x2)' r > r

s-X 2 expCX2) Ei (-X 2) + ^ ( ^ - t x ) / Y = 0 . r = 2X VaTr s I

(Solid) (Liquid)

11.2.2 9, p. 295 Liquid freezing on a sphere. t = t , 0 < r < R . t = t", r * » , t x < t m .

(Solid) (Liquid)

Liquid temp:

H ~ h = 2X ^n ~ fcl exp(-X 2) X V 7 erfc (X) (2 Fo*

exp (-3) rS,

- - erfc r > r_ K)) • X 2 exp(X 2) jexp(-X 2 ) - XTT erfc (X)J = ^ ( t m - tj

r s = 2X V v

Section 11.2. Change of phase—Nonplanar Interface.


11.2.3 9, p. 296

Steady line heat sink t = t , r > 0, x = 0, t > t . sink rate = q. t = melting temp.

Solid temp: ( t s " V k s 4ir Ei (-2) Eif.-X") , d < r < r s .

r = 2X VCTT s s

Liquid temp: ( t , - M

r > r m 1

xp(-X 2)/4ir + - ^ ft - t ) e x p ( - X 2 a s / a t ) / E i (-\\zaj = X 2 a s pY/q 0 .

11.2.4 9 , p . 296 Freezing on steady temp cylinder. t " V c " V T > 0 r

fc0 < V * = V r ' r o ' T = °* t = melting temp, m

Solid temp:

^ 4 " T - *n(r/r ) / t a ( V r ) , r Q < r < r s cm c 0

2 r s * » « V r 0 » " r s + ^ = 4 k s ( t m - t 0 , T / ^

(Solid) (Liquid)

Section 11.2 . Change oii Phase—Nonplanar Interface.

Case No. References 1 . -jription Solution

11.2.5 19, Freezing inside a tube, 2 p. 3-87 tube resistance neglected, ( m " a' _ Bi T(X_ l \ 2 _2 „ /l\"l

convection cooling at r = r Q . pyk " 2 LUi + 2 j ( 1 ~ R ' " R . n UAJ ' R - 1 -t = t m , r = r 1 . Bi = h6/k, R = ci/rQ

t = melting temp.

Liquid

Solid

11.2.6 19, Case 11.2.5 with freezing on 2 p. 3-87 outside of tube and convec- ' m a' _ Bi*" |/1_ 1\ ,Jl .. , _2

tion cooling on inside. pyk ^ lv °' i n ' v" """' Bi 171 1 \ ,Jl .. s „2 , R > 1

11.2.7 19, Case 11.2.5 with freezing ji n Is- ~ tJ p. 3-88 inside a sphere. - 1 f e r a i = ^ f [ ( i i " 0 <* " H3) + f d - R2>] . R < 1



11.2.8 19, p. 3- 88

Case 11.2.6 with freezing on outside of sphere. n n, " V ro a Bi2 [&-) 3 3 2 (RJ - 1) - f (R - 1) R > 1

I

Section 11 .2 . Change of Phase—Nonplanar interface .


i



4. CD

a o B a ex W " l M a on


Section 12.1. Traveling Boundaries.


12.1.1 9, p. 389 Semi-infinite solid with t - t linear initial temp, time - — ^ - - D{1 - 0) + G + - erfc [x(l - U)J dependent surface temp and internal heating. t = t + dx, x > 0, T =» 0. t « t + br, x » 0, T > 0. Sol id increasing at ve loc i ty u.

i

+ i exp(4UX2) erfc [ x ( l + U)] + | (§ + D + § )

x j (1 + U) exp(4DX2) erfc [ x ( l + U)] - (1 - U) erfc [x( l - U)]J.

B = b T / ( t x - t Q ) , D = d x / ( t x - t Q ) r G = q- ' -crr /M^ - t Q )

O = UT/X, X = K/2V0CT

12.1.2 9, p. 389 Semi-infinite so l id with convection boundary, t = t Q , x > 0, x = 0. Solid increasing at ve loc i ty u.

t . ° = \ (erfc [X(l - 0)] + tJZfi 0 exp(U/Fo) erfc [ x ( l + D)]

^ 0 exp[Bi( l - U + Fo Bi)] erfc [x(2 Fo Bi + 1 - 0) ] . + 2 Fo Bi - U Fo B

Bi = hx/k, Fo = crc/x , X = x/2 Var, 0 = ur/x



12.1.3 9 , p . 389 Semi-infinite so l i d with steady periodic temp, t = ( t f l - t ^cos tor r + 9 ) , x = 0, X > 0. Sol id increasing at ve loc i ty u.

t - t _ "* = exp[ux/2a - xVb cos (<j>/2)] cos [UT - xVb s in (<j>/2) + e]

^ + f = b exp(i<t» 4a

K> i

12.1.4 9, p. 389 Cylindrical boundary traveling in an infinite solid. t « tj, r > tfl, t > 0. fc = V r = ro- T > °" Cylinder traveling at velocity u.

J3-

t - 1 „ - - ..„ _ e I (O)K (UR) cos (n6) 0_ X n n n t 0 - t . " 2 K_,

n=0 ,(U)

+ I y exp (-a2 Fo) e cos (n6) i (U) IT {_, n n n=0 C° exp(-Fo X 2) fj (XR)Y (X) - Y (XR)J (X)~l XdX I L n n ii n J

1 0 (X 2+u 2) [a2(X) + Y 2 ( X ) ]

e Q = 1, e n = 2 if n > 1, Fo = ar/r 2, R = r/rQ

U = urQ/2a



12.1.5 9, p. 391 Thin rod with fixed end temps and convection cooling, t = t , 0 < x < %, T = 0. t = t , x - I, T > 0. t = t Q, x = 0, T > 0. The end boundaries traveling at velocity u. p = rod perimeter. A = cod sect ion area.

inh L\U 2X 2 + Bi X2J sinh t, - t„

X + Bi X J exp(DX - D) sinh [ V o + B i J

+ 2ir exp(DX - 0) £ ( - ^ " y i n imx)

n=l Bi + 0" + n it

[" x exp l-Fo (Bi + U + n n )J

Bi = hpi 2 /kA, Fo = ore A 2 , U = uA/2a, X = x/SL

I u •4 T

•-T



i



to i m

5" a era to s a. H s:


x/6

FIG. 1.1. Temperature d i s t r i bu t i on and mean temperature in a porous p la te (case 1.1.4, source: Ref. 2, p . 221, Fig . 9 . 2 ) .

0 0.2 0.4 0.6 0.8 1.0 x/B

FIG. 1.2. Temperature distribution in an infinite plate with internal heating and temperature dependent conductivity (case 1.2.3, source: Ref. 3, p. 132, Fig. 3.23). F_!

FIG. 1.3. Temperature d i s t r i b u t i o n in a heat generating porous p l a t e (case 1 .2 .6 , source: Ref. 2, p . 223, Fig. 9 . 3 ) .

0 1 2 3 4 5 6 7 8 9 10 11 12 -+ b/a (or a/b)

FIG. 1.4. Ratio of mean and maximum temperature excesses in an electrical coil of rectangular cross section (case 1.2.8, source: Ref. ?., p. 180, Fig. 10.5).

F-2

FIG. 1.5. First-term approximation to the maximum temperature in a solid rectangular rod with internal heating, ^ = -Vq 0" >/f$k 2b, \|)2 = "V<3o'''/3k 2a (case 1.2.10, source: Ref. 2, p. 198, Fig. 8.16).

F-3

•±" 1 0 " 3

FIG. 1.6a. Maximum temperature var ia t ions on the cooled surface of a f l a t p l a t e having equally spaced ad iaba t i c and constant temperature s t r i p s on the opposi te surface (case 1.1.30, source: Ref. 27) .

F-4

0.8 1.0

FIG. 1.6b. Heat flux through slabs held at a uniform temperature on one surface and having equally spaced constant temperature strips on the other (case 1.1.30, source: Ref. 88).

F-5

o ra

ha/k

FIG. : . . 7 . Maximum temperature va r i a t ion of the cooled surface of a f l a t p i havi-(> a l t e r n a t i n g adiabat ic and constant heat- f lux s t r i p s on the opposi te surface (case 1.1.31, source: Ref. 28).

P-6

10J

10

o

101

10L

"I 1 I I ! II I I

Insulatin

10 ,-2

hw/k

PIG. 1.8. Temperatures at the spot center of a spo t - insu la ted p la t e having a uniform in te rna l heat source (case 1.2.13, source: Ref. 29) .

• )0 J c 1—i—r i M I I I 1—i—i i i i i 11 1—i—I-T i i i i I 1—i—i i I I 11_

Insulating spot-

h, t „

o i n

' i i 1 1 ' i i i i i i i 1 1 i i i i i 1 1 1

10' hw/k

10J

FIG. 1.9. Temperatures on the cooled surface and at the spot center of a spot-insulated plate having a constant temperature heat source on one face (case 1.1.32, source: Ref. 29).

F-7

0.2 0.4 0.6 0.8 (y/B)/(1/Bi + 1)

1.2 1.4

PIG. 1.10. Hotspot temperatures along plate/rib centerline (Bi = hS,/k) (case 1.1.39, source: Ref. 19, p. 3-126, Fig. 77).

0.5 1.0 1.5 2.0 2.5 3., Ratio b/a

0.5 1.0 1.5 2.0 2.5 3.0 Ratio b/a

PIG. 1.11. Conductive shape factors for a rectangular sec t ion containing a constant temperature tube (case 1.1.44, source: Ref. 87) .

F-8

H" 1.0

rcA

FIG. 2 .1 . Heat l o s s from Insulated tubes (case 2 . 1 . 3 , source : Ref. 3, p . 123, F ig . 3 . 18 ) .

F-9

b

4

I 1 ' 1 1 T 12.25 1 \

1

1

t m ^ l \ -^ 3 1 \ + 1 \ £ Vb- r0 = 2.00 1 \

1 \ -

+ 2 \ 1 \ §

1

" '1.75 ^ 1 \

\— 1 ~ l.bU.

^ 1 \ \—

CSvV 0 i — 1 — o . o o — 1

+ o

* E

0 0.2 0.4 0.6 0.8 R

1.0

FIG. 2.2 . Temperature distribution and mean temperature in a cylinder with temperature dependent heat source (case?2.2 .4 , source: Ref. 2, p. 189, Fig. 8 .11) .

J

FIG. 2.3. Temperature distribution in an inf in i te plate with a cyl indrical heat Bource, B = V ( h x + h 2)/kw r g , t 0 = t at r 0 , t«, = t at r = °° (case 2 .2 .1 source: Ref. 2, p . 175, Fig. 8 . 2 ) .

F-10

0.1

0.01

0.001

= o0.00 0 (1

0.001

0.01

0.001 0.05

K = 10, R = 100 K= 10, R = 1 K = 10, R= 10-

K = 1, R = 100-

0.4 0.5

FIG. 2.4. The i n s ide surface temperature of an i n f i n i t e tube with temperature dependent conduct iv i ty and heating (case 2 .2 .17 , source: Ref. 16, Fig. 1 ) . Equation (5) given in case so lu t ion .

F - l l

s/r0

FIG. 2.5. Maximum (hot spot) temperatures in the cross sect ion of a heat-generating s o l i d (cases 2.2.18 and 2.2.19, source: Ref. 64, Fig. 3),

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

1/P = b/r0

FIG. 2.6. Shape factor for a cylinder with two longitudinal holes (case 2 .1 .44 , source: Ref. 73, Fig. 6 ) .

F-12

0.01 0.1 1 10 100

FIG. 3.1. Steady temperature of thin, nonrotating spherical shell in uniform radiation field (case 3.1.10, source: Ref. 19, pp. 3-112, Fig. 68).

F-13

PIG. 4 .1 . Surface temperature of a serai-infinite sol id with heating on the surface over width 2b, which moves at ve loc i ty u (case 4 .1 .8 , source: Ref. 9, p. 270, Fig. 34).

3 -

2 -

1 -

• Heat source width - I

' I 1 1 1 1 1 — r '""

A<"°~" ,\\ sH = 0 L = 1

/ / H = 0 . lS \ w y^ H=I.O ^ -

1 1

L e a d i n g e d g e — ^

» < • 1 •~~ " v | ^ ^

-

x/8

PIG. 4.2. Surface temperature of a oonvectively cooled semi-infinite solid with a traveling strip heat source (case 4.1.9, source: Ref. 30).

F-14

o • ' ' ' • • — •

-0.10 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0 - rad

FIG. 4.3a. Surface "temperature of an i n f i n i t e cylinder with a r o t a t i n g surface band source, <f> =» 0.01 (case 4 .1 .10 , source: Ref. 30) .

n i T -0.10-0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08

fl - rad

FIG. 4.3b. (Same conditions as for Fig . 4.3a),

F-15

i

-1.5

(a)

fl(R.r) (4/Pe) 2 0 _|_

Nu = 2.0 Pe = 20.0 Pe = 2.0

-1.0 -0.5

Nu = 0.2 Pe = 20.0

-1.5 -1.0 -0.5 0 0.5 1.0 1.5

(b)

FIG. 4.4. Temperature distribution in a cylinder with a ring heat source (case 4.1.11, source: Ref. 34), 9 - (t - tf)2nrQk/Q0.

e(R.f) Nu = 0.2 R 0=1.2B Pe = 20.0

- Pe = 2.0

SIR,?)

Nu = 2.0 R 0 = 1.25 Pe = 20.0 Pe = 2.0

PIG. 4.5. Tempecature distributions in a hollow cylinder with an inside ring heat source (case 4.1.12, source: Ref. 34). 9 defined in Pig. 4.4.

n—i—i—i—i—i—r " i — i — i — r "1—I—i—i—r- 1 ' '2 ' n = c o ; y = y b (x /C) 2 ;0 =

Vb

U-n = -1;y = V b(x/B|;0 =

2s/T . u b = — - - C V h T k y ;

9 b

u b l , (u b )

n = O;y = y b ( x / C ) 1 / 2 ; 0 =

V 2 " . "b =—7—-KVh/ky b

2 !l("b) ub l 0(u b)

^ ? * &

tanh u h

n = 1/2;y = yb;tf>=-

u b =v^-f iVh7ky b " ub

J I I I I I I I L J I I L 1 J I I L J I I L 1.0 2.0 3.0

xVhTkVb

4.0 5.0

FIG. 5 .1 . Performance of pin f ins (cases 5 .1 .2 , 5 .1 .14 , and 5.1.16 through 5 .1 .18 , source: Ref. 8 and Ref. 7, p. 55, Fig. 3 .14 .

F-18

J 1_

i= 1/2;v = ; u h = » V h / k v b

tanh u b

= 1/3; y = V b (x/C) 1"; u b=—| ^Jhjk^

' 2/3< ub> 3

3 u b L, / 3<'Jb) = 0;y = y b (x /B) ;u b =28 V 'h /ky b

2 ' i K '

i = - 1 ; V = Vb(x/B) 3 / 2; u b =4C-./h/kyb

"b Mub) _2 , = ± oo; y = y b(x, 'C) 2;u b =Cx/h7kyb

2

1.0 2.0 3.0 4.0 5.0

ev^RVb"

FIG. 5.2. Performance of straight f ins (cases 5 .1 .4 , 5 . 1 . 5 , 5 .1 .7 , 5 .1 .8 , source: Ref. 8 and Ref. 7, p. 56, Fig. 3.15).

F-19

0.5 -

0.4

1.0 2.0 3.0 4.0 BVh7kVb

5.0

FIG. 5.3. Performance of circumferential fins of rectangular cross section (case 5.1.10, source: Ref. 8 and Ref. 7, p. 57, Fig. 3.16).

F-20

I G. 5.4. Curve for calculating dimensions of circular fin of rectangular rofile requiring least material (case 5.1.10, source: Ref. 1, p. 234,

Pig. 11.11). a - Q/nhx£(t0 - t f ) .

F-21

1.0 2.0 3.0

FIG. 5.5. Performance of cyl indrical f ins of triangular prof i le (case 5 .1 .11 , source: Ref. 8 and Ref. 7, p. 58, Fig. 3 .17) .

F-22

(re*-ro),/2h7kr0

FIG. 5,6. Efficiency of an infinite fin heated by square arrayed round rods (case 5.1.19, source: Ref. 11 and Ref. 10, p. 135, Pig. 2.22). t% = (2/W)s.

F-23

(re--r0)V5R7E5;

FIG. 5.7. Efficiency of an in f in i t e f in heated by equi lateral triangular arrayed round rods (case 5.1.20, source: Ref. 11 and Ref. 10, p. 136, Fig. 2 .23) . r* * (2V37n) 1 / 2 s .

F-24

to Ln

Z = T 0A" e

FIG. 5 .8 . Fin parameter as a function of Z for a s traight f in radiating to free space (case 5 .1 .24 , source: Ref. 10, p. 208, Fig. 4 . 3 ) .

Z = V Te FIG. 5.9. Fin efficiency for a straight fin radiation to free space (case 5.1.24, source: Ref. 10, p. 209, Fig. 4.4).

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Profile number C(K 1 T^ /2kb) 1 / 2

FIG. 5.10. Heat flow relationship for a straight fin of rectangular profile radiating to nonfree space (case 5.1.25, source: Ref. 10, p. 216, Fig. 4.8).

0.4 0.8 1.2

Profile number « ( K ^ T j ^ k b ) 1 7 2

FIG. 5 .11. Efficiency of a s t r a i g h t f i n of rectangular p r o f i l e rad ia t ing to nonfree space (case 5 .1 .25 , source: Ref. 10, p . 216, Fig . 4 . 9 ) .

F-26

X = — = 0.75

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Profile number K.,Tê2/kbn

FIG. 5.12a. Fin e f f ic iency for the long i tud ina l radia t ing f in of t rapezoidal prof i l e with a taper rat io of 0.75 (case 5 .1 .26 , source: Ref. 10, p. 223, Fig . 4 .12 ) .

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Profile number K^Tj^/kbrj

FIG. 5.12b. Fin e f f ic iency for the l ong i tud ina l radia t ing f in of t rapezoidal prof i l e with a taper ra t io of 0.50 (case 5 .1 .26 , source: Ref. 10, p . 224, Fig . 4 .13) . F _ 2 7

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Profile number K,T3(!2/kb0

FIG. 5.12c. Fin ef f ic iency for the long i tud ina l rad ia t ing f in of t rapezoidal p r o f i l e with a taper r a t i o of 0.25 (case 5 .1 .26 , source: Ref. 10, p . 224, F i g . 4 .14) .

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

\ Profile number K^rg^/kbn

FIG. 5.12d. Fin ef f ic iency for the longitudinal radiating f i n of t r iangular prof i l e (case 5 .1 .26 , source: Ref. 10, p. 225, Fig. 4.15).

F-28

1.2 1.3 1.4 1.5 1.6

Temperature ratio Z = TQ/T

1.7

PIG. 5.13. Profi le number of constant-temperature-gradient longitudinal radiating fin as a function of base-to-t ip temperature ratio (case 5 .1 .28, source: Ref. 10, p. 233, Fig. 4 .18) .

F-29

K 2 / K 1 T 0

0.4 0.8 1.2

Profile number £2l<1T^/kb

1.6

FIG. 5.14. Efficiency of constant-temperature-gradient longitudinal radiating fin as a function of profile number (case 5.1.28, source: Ref. 10, p. 232, Fig. 4.17).

0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 Radius ratio p = rQ/re

FIG. 5.15a. Radiation cin efficiency of radial fin of rectangular profile. Taper ratio, X = 1.00; environmental factor, K /K T Q = 0.00 (case 5.1.29, source: Ref. 10, p. 250, Fig. 4.24).

F-30

-©. > • u c a> \u

M-4 -01 c

0.25 0.35 0.45 0.55 0.65

Radius ratio p = rQ/r e

0.75

FIG. 5,15b. Radiation fin efficiency of radial fin of rectangular profile. Taper ratio, X = 1.00; environmental factor, K /K T = 0.20 (case 5.1.29, source: Ref. 10, p. 251, Fig. 4.25).

0.25 0.35 0.45 0.55

Radius ratio p = r n / r e

0.65 0.75

FIG., 5.15c. Radiation fin efficiency of radial fin of rectangular profile. Taper ratio, X = 1.00; environmental factor, K /K T n = 0.40 (case 5.1.29, source: Ref. 10, p. 251, Fig. 4.26).

F-31

0.25 0.35 0.45 0.55 0.65 0.75

Radius ratio p = rQ/re

FIG. 5.15d. Radiation f in eff ic iency of radia l f in of t rapezoidal p r o f i l e . Taper r a t i o , X = 0.75; environmental factor , K 2 / K , T ! ! = 0 . 0 0 (case 5 .1 .29, source: Ref. 10, p . 252, Fig. 4 .27 ) .

Radius ratio p = rQ/r e

FIG. 5.15e. Radiation fin efficiency of radial fin of trapezoidal profile. Taper ratio, X = 0.75; environmental factor, K./K.T. =0.20 (case 5.1.29, source: Ref. 10, p. 252, Fig. 4.28).

F-32

0.25 0.35 0.45 0.55 0.65

Radius ratio, p = rQ/re

0.75

FIG. 5.15f. Radiation f in eff iciency of radia l fin of t rapezoida l p r o f i l e . Taper rat io , \ = 0.75; environmental f ac to r , R / K T J = 0 . 4 0 (case 5.1.29, source: Ref. 10, p. 253, Fig. 4 .29) .

> u c

c c o

•o

0.25 0.35 0.45 0.55 0.65


0.75

FIG. 5.15g. Radiation fin efficiency of radial fin of trapezoidal profile. Taper ratio, X = 0.50; environmental factor, K /K.T^ source: Ref. 10, p. 253, Fig. 4.30).

0.00 (case 5.1.29,

F-33

0.25 0.35 0.45 0.55 0.65 0.75 Radius ratio, p = rQ/re

FIG. 5.15h. Radiation fin efficiency of radial fin of trapezoidal profile. 4 Taper ratio, X = 0.50; environmental factor, K

2 / K i T n = ° - 2 0 < c a s e 5.1.29, source: Ref. 10, p. 254, Fig. 4.31).

0.25 0.35 0.45 0.55 0.65 0.75

Radius ratio, p = rn/re

FIG. 5.15i. Radiation fin efficiency of radial fin of trapezoidal profile. Taper ratio, X = 0.50; environmental factor, K./K.K = 0.40 (case 5.1.29, source: Ref. 10, p. 254, Fig. 4.32).

F-34

0.25 0.35 0.45 0.55 0.65 0.75


PIG. 5.15J. Radiation fin efficiency of radial fin of triangular profile. Taper ratio, X •» 0.00; environmental factor, K /K T« = 0.00 (case 5.1.29, source: Ref. 10, p. 255, Fig. 4.33).

0.25 0.35 0.45 0.55 0.65 0.75

Radius ratio, p = rn/re

PIG. 5.15k. Radiation fin efficiency of radial fin of triangular profile. Taper ratio, X = 0.00; environmental factor, K /K T = 0.20 (case 5.1.29, source: Ref. 10, p. 255, Fig. 4.34).

F-35

-Q-> u c

0.6

0.5

0.4

0.3

0.2

0.1

"Z^^^^ —

—

^5^$^ ^r?^ — Profile number

I I I I I I I 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80

Radius rato, p = rQ/re

PIG. 5.158,. Radiation f in eff iciency of rad ia l fin of t r i angu la r p r o f i l e . Taper r a t i o , X = 0 .00; environmental f ac to r , KjAiTn = 0 . 4 0 (case 5.1.29, source: Ref. 10, p . . 256 , Pig. 4 .35) .

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Base paramater mR,

FIG. 5.16a. Efficiency of the s ide of a capped cyl inder f in (case 5.1.30, source: Ref. 10, p . 276, Fig. 5 .6 ) .

F-36

0.4 0.8 1.2

Base parameter mB

1.6

FIG. 5.16b. Efficiency of the top of a capped cylinder fin (case 5.1.30, source: Ref. 10, p. 277, Fig. 5.7).

F-37

0.4 1.6 2.0 2.4

Parameter ml

2.8 3.2 3.4 3.6

FIG. 5.17. Heat flow ratio q^/qg as a function of mb and temperature excess rat io (t^ - t f ) / ( t g - f f ) for the doubly heated rectangular f in (case 5 .1 .32 , source: Ref. 10, p. 410, Fig. 8 .10) .

F-38

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 m x 1 b l

FIG. 5.18a. Eff ic iency of the v e r t i c a l sec t ion of a s t r a i g h t , s i ng l e Tee f in for u » v (case 5 .1 .32 , source: Ref. 10, p . 398, Fig. 8 .4 ) .

1.0

0.9

0.8

0.7 •e?

| 0.6

% 0.5 i !

0.4

0.3

0.2

0.1

IK ' i i i i i i

-

\ \ /S . \ / S . y— Values of m .a.

-

- -

i 1 1 1 1 1 1

» « • « . -

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Parameter rn x 1 b 1

FIG. 5.18b. Efficiency of the hor izonta l sect ion of a s t r a i g h t , s ing le Tee f in for u • v (case 5 .1 .32 , source: Ref. 10, p . 399, Fig. 8 . 5 ) .

F-39

* 0.5

FIG. 5.19. Effectiveness of the concave parabolic f in radiating to non-free space (case 5 .1 .37 , source: Ref. 31) .

tf> 0.5 - J

FIG. 5.20. Effectiveness of the convex parabolic f in radiating to non-free space (case 5 .1 .38 , source: Ref. 31).

F-40

as 100 90 80 70 60 50 40 30 20 10 0

V I I l I l I

~ \ -- TO^ -_ y\ —

Y*. A/2R -

\ ^—1.5 ^ - 3 . 0 ^ - 5 . 0 ^—10.0

-

i i i 3?===—_ -

i i i I I— 1 ' i

FIG. 5.21. Effectiveness of sheet fin with square array tubes (case 5.1.40, source: Ref. 83, p. 294, Fig. 2).

J 0.5

< J \0.05M 1

.02\NAVi -

-

^î^fc 1,

1 ÔCs^

^î^fc 1,

FIG. 6 .1 . Temperatures i n an i n f i n i t e region of which the region | x | sfl!b: \ 0.01 = F.u ' 1

oTN -

O^** -

0.5 -

~1 "~

2^=> 2 ..! ,., I ^ S 2^=>

R

FIG. 6.2. Temperatures in an inf ini te region of which the region r < r Q i s i n i t i a l l y at temperature t 0 (case 6 .1 .4 , source: Ref. 9, p. 55, Fig. 4b).

i i

1.0 0 0 5 ^

O.ION

W>2\ \0 .0 l '= Fo ' ' 1

0.5 0 . 2 0 s

0.50

1.00

1 I ^ ^ ^ S :

FIG. 6.3. Temperatures in an infinite region of which the region r < r n is initially at temperature t 0 (case 6.1.5, source: Ref. 9, p. 55, Fig. 4c).

F-42

— 0.4

PIG. 6 .4 . Temperatures in an inf in i te region with steady temperature t 0 on the surface r = r 0 (case 6 .1 .18 , source: Ref. 9, p. 337, Fig. 41) .

log10(Fo)

FIG. 6.5. Temperature in a cylinder of infinite conductivity, initially at temperature t l r in an Infinite medium inditially at t 0. Numbers on the curves are values of 2PQCQ/P1C! (case 6.1.21, source: Ref. 9, p. 342, Fig. 45).

F-43

o

J, 0.2

FIG. 6.6. Temperature in a cylinder of i n f i n i t e conduct iv i ty , i n i t i a l l y a t temperature tg , in an i n f i n i t e medium. Numbers on the curves are values of 2Pnco/Pici (case 6 .1 .22 , source: ref . 9, p . 343, Fig. 46) .

0.2 1 2 10 20 100 200 1000 2000 10,000 Fo-»

PIG. 6.7. Temperature response of s o l i d sphere 0 < r < r 1 ( with kj_ + M and i n i t i a l l y at t 1 g, embedded in an i n f i n i t e so l id (r > r^) i n i t i a l l y a t t 2 g (case 6 .1 .26 , source: Ref. 19, pp. 3-64, Fig. 37).

F-44

0.001 0.005 0.010 0.050 0.100 0.500 1.000 5.000 10.000 F o -

PIG. 6.7. Temperature response of an infinite solid with a constant spherical surface temperature of t Q (case 6.1.13, source: Ref 74, p. 430, Fig. 10.9).

0.6

(III) 0.5

0.4 t

~° 0.3

0.2

0.1

0 0.02 0.04 0.06 0.08(1)0.1 I - *J I I

- • I _

l l l \ - II

III > II

I I I I S N

1.0

(II)

0.9

0.8

0.7

0.6

10.0

(I) 0.99

0.98

0.97

0.5 0 0.2 0.4 0.6 0.8(11)1.0 0 40 80 120 160(111)200

20 40 60 80 100 10 1 1 1 1 0.2b

0.08 - 0.20

0.06 0.15

0.04 — - 0.10

0.02 —

I 1 1 1

0.05

0 0 200 400

Fo

600 800 10 00

PIG. 7.1. Temperature distribution in the semi-infinite solid having a steady surface temperature (case 7.1.1 b = 0, source: Ref. 74, p. 95, Figs. 4.4 and 4- 5)> p-45

.3 - «3" I I

11 0.1

~ I J -

0.01

FIG. 7.2a. Temperature distribution in the semi-inf inite so l id with convection boundary condition (case 7 .1 .3 , source: Ref. 5, p. 82, Fig. 4 .6 ) .

F-46

0.01 . 0.05 0.10 0.50 1.00 5.00 10.00 50.00

FIG. 7.2b. The dimensionless excess temperature vs the number BijjfFOjj)1/2 and various Fourier numbers for semi-infinite solid with a convection boundary (case 7.1.3, source: Ref. 74, p. 207, Fig. 6.2).

1.0' 0.8 0.6 0.4 0.2

-0.2

-0.4

-0.6

-0.8 -1.0

0 0.2 0.4 0.6 0.8 1.0 x

FIG. 7 . 3 . Temperature d i s t r i b u t i o n in a semi- inf in i te region when the surface temperature i s harmonic (case 7 .1 .16 , source: Ref. 7, p . 99, Fig. 4-19) .

V-» 1 i i 1 \* \° — _ - v \

* \ -* V O — • ^

— CPV

7^° 2^s». 7^° 2^s». — JO"

~h¥ '<? — / * / / & 4 y/.& _ 7 A/ 7 A/ f i i 1 1 1

F-47

100 80 60 40

o 20 o

* „ 10

6 4

2

0 0.2 0.4 0.6 0.8 1.0 1.2

* & )

FIG. 7.4. Function f T(t-tf)/(tc-tf)] (case 7.1.22, source: Ref. 4, p. 84, Fig. 4.15).

- 2 - 1 0 1 2 x/a

FIG. 7.5. Temperature distribution across a heated strip of width 2a on the surface of a semi-infinite solid (case 7.1.24, source: Ref. 9, p. 265, Fig. 33).

F-48

•i- 0.6 -

"2- 0.4-

1.4 1.6

FIG. 7.6. Temperature response of semi-infinite solid (x >_ 0) with surface temperature t g suddenly increasing as power function of time, t s = t: + br n, (case 7.1.9, source: Ref. 19, pp. 3-66, Fig. 39).

1.0

1/ /— 0.6 1/V7T —

0.4

0.2

0

V \ I I I I I Semi-infinite solid

\ \ q 0 = const —

\ \ i k i 3 t ~~ J TIT la —

39

k 3t

" " " q 0 3x

'klt-t,) N ^

I t-2 q Q \ ^ r I t-0.4 0.8 1.2 1.6 2.0 2.4

FIG. 7.7. Temperature response1, temperature gradient, and heating rate in a semi-infinite s o l i d after exposure to a constant surface heat flux q 0

(case 7.15, source: Ref. 19, pp. 3-81, Fig. 49).

F-49

FIG. 7 .8 . Temperature response of a semi- in f in i t e so l id a f t e r sudden exposure to a l i nea r ly increas ing surface heat flux for a duration D, Q = D q m a x / 2 , q 0 = q m a x T/D (case 7 .1 .14, source: Ref. 19, pp. 3-82, F ig . 50) .

rri\/Fch

8 10 12 14 16

FIG. 7 .9 . Temperature response of an i n f i n i t e conduct ivi ty p la t e and semi-i n f i n i t e so l i d composite with a constant heat flux a t x^ = 0 (case 7 .1 .30, source: Ref. 19, pp. 3-84, Fig . 51) .

F-50

My/it-0.6

Fo* -+ x

FIG. 7.10. Temperature distributions in a semi- inf ini te so l id with exponential heating (case 7 . 2 . 9 , source: Ref. 19, pp. 3-85, Fig. 52).

1.0

0 . 8 -

i 0.6 U

J2 0.4

0 . 2 -

\ 0=Fo 1

_ o . o r H ^

6 ! O J * ^ « S S N \ ^

— oTio ,. ^ ^ -

0.4iQ

V

1.00

T r - - ^ 1 1 T r - - ^ 0.2 0.4 0.6

X 0.8 1.0

FIG. 8.1. Temperatures in a slab with a linear initial temperature distribution (case 8.1.3, source: Ref. 9, p. 97, Fig. 10).

F-51

1 .U o . o p " ^ OuT"" <

I I ^ ^ 0 = F o

I

0.8 ouT""* . -

'„ 0.6 — —

4-<

^ 0.4 ^0.40 \ \

0.2

1.0

N \ -

I I

N \ -

0.2 0.4 0.6 0.8 X

1.0

FIG. 8.2. Temperatures in a slab with a parabolic initial temperature distribution (case 8.1.4, source: Ref. 9, p. 98, Fig. 10).

FIG. 8.3. Temperatures in the infinite plate with a constant intial temperature and steady surface temperature (case 8.1.6, source: Ref. 9, p. 101, Fig. 11).

F-52

1.000

"3 I in

0.100 -

0.010 -

0.001 0 1 2 3 4 10 30 50 70 90 110 130 150 300 500 700

Fo

FIG. 8.4a. Hidplane temperature of an infinite plate of thickness 21 and convectively cooled. (case 8.1.7, source: Ref. 12, p. 227 and source: Ref. 5, p. 83, Fig. 4-7).

0.0001 0.0005 0.0010 0.0050 0.0100 0.0500 0.1000 0.5000 1.000 Fo

FIG. 8.4b. Surface temperature of an Infinite plate of thickness 21 and correction boundary (case 8.1.7.2, source: Ref. 74 p. 226, Fig. 6.6c).

i

0.0001 0.0005 0.0010 0.0050 0.0100 0.0500 0.1000 0.5000 1.0000 Fo

FIG. 8.4c. Surface temperature of an infinite plate of thickness 21 and convection boundary (case 8.1.7.2, source: Ref. 74, p. 226, Fig. 6.6d).

F-54

1.0

0.9

0.8

0.7 > ' 0.6 J? > 0-5

i 0.4

0.3

0.2

0.1

0 0.

' ' I ' ' I ' j ^ 5 ? ^ X - 0.2 ^ ^ y j ^ 5 ? ^

— —D-4""""" / /A f

/ /1/ -

— — °-6-* ^ /// -

JU& / -

/ -- ^ / -

A.O i l l . i 1 i i . i

01 0.1 1.0 1/Bi

10.0 100.0

PIG. 8.4d. Temperature as a function of midpiane temperature for an i n f i n i t e plate of thickness 2i and convectively cooled (case 8 .1 .7 , source: Ref. 5, p . 86 , Fig . 4-10).

1.0

0.8

a 0.6 9 6" 6' •=" fi, <o ST , _

O O' o' O' O' O' O' O' ©' •-' <V *l>" £" S '

Bi 2 Fo

FIG. 8.4e, Relative heat loss from an infinite plate of thickness 2% and convectively cooled (case 8.1.7, source: Ref. 5, p. 90, Fig. 4-14 and Ref. 13).

F-55

FIG. 8.5. The function F(O) (case 8.1.12, source: Ref. 1. p. 301, Fig. 14-3),

FIG. 8.6a. Variation of amplitude of the steady oscillation of temperature in an infinite plate caused by harmonic surface temperature (case 8.1.13, source: Ref. 9, p. 106, Fig. 13).

F-56

-600

-500 -

-400-

-300

-200-

FIG. 8.6b. Variation in phase of the steady oscillation of temperature in an infinite place caused by harmonic surface temperature (case 8.1.13, source: Ref. 9, p. 107, Fij. 14).

F-57

I in m

Fo

FIG. 8.7. Temperature distribution in an infinite plate with no heat flow at x = 0 and constant heat flux q 0 at x = I (case 8.1.18, source: Ref. 9, p. 113, Fig. 15).

1.0

0.8

so 0.6

S °- 4

0.2

1.0 I I I

- 0.8

olT~ -

=—~ 0.4

0.3

.

-

0.2

0.1

— 0.06

— 0.02 -Fo

I I I 1 — *

0.2 0.4 0.6 X

0.8 1.0

FIG. 8 . 8 . Temperatures in an i n f i n i t e p l a t e with constant i n t e r n a l heating q " ' and surface temperature t^ (case 8 . 2 . 1 , source: Ref. 9, p . 131, F ig . 20) .

FIG. 8 .9 . Quasi-steady s t a t e temperature of an i n f i n i t e conduct iv i ty p la t e with a harmonic f lu id temperature (case 8 .1 .49 , source: Ref. 19, Fig . 17b, P. 3 -31) .

F-59

FIG. 8.10. Plate temperature response for prescribed heat inputs (case 8.2.15, source: Ref. 19, p. 3-35, Fig. 18).

F-60

iooo.o~l " ~ 800.0 finnn (q" + q" 'S+x7t * ) /o7 t2 600.0

400.0 \ 200.0

1.0 0.0001

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

1.0000 o.ftg/pCS)

-0.90 i 1 r i r -0.80

-0.70 -0.60

0.50 jr- (q" + q'"« + a.?\^lo&\\

-0.40 -0.34

-0.30 -0.24

•0.20

—0.14

0.10

-0.07 -0.05

Thin plate cooling

-0.04 -0.02 -0.02 -0.01

J L 10 14 18 22 16 30

(b) (o.ftppC5)

FIG. 8.11. Temperature response of thin generating plate suddenly exposed to constant heat input with surface reradiation: (a) heating, (b) cooling (case 8.2.16, source: Ref. 19, p. 3-36, Fig. 19).

F-61

Thin plate

20 40 4 10

q m a x

D / P C 6 t 0

(a)

FIG. 8.12. Maximum temperature of t h i n insula ted p l a t e suddenly exposed to c i r cu la r heat pulse with surface r e r a d i a t i o n : (case 8 .1 .50 , source: Ref. 19, p . 3-38, Fig. 20) .

Fo

0.20 0.16 0.12 0.08 0.04 0 0.20

- 0.16 +•*

I

~ 0.12

-? 0.08

0.04-

l\ I I 1 A / 0 . 5 /

/ / 1.0 X. 1 A

/ 0 . 5 / / / 1.0

Thick plate ^ / 2 /

^^V /^S-' 3 0^

&^\ i

^ ^ ^ 3 ^ 0 . 0 -

I I \ 0 0.04 0.08 0.12 0.16 0.20

Fo (a)

FIG. 8.13. Temperature response of an infinite plate with uniform internal heating, I = half thickness (case 8.2.6, source: Ref. 19, p. 3-43, Fig. 24).

F-62

0.2 J _ -i. 0.1 0.2 0.3

1*2. max " W V 0.4 0.5

FIG. 8.14a. I n su la t ion weight for subs t ruc ture protect ion to t 2 m a x in heating duration D, Wx = Pi^ i (case 8 . 1 . 5 1 , source: Ref. 19 p . 5-48, Fig. 27) .

F-63

(a)

/i = n +(n + D/Bi,- , n = / 3 2 C 2 f i 2 / p 1 C l 5 l

1.0

5 10 20

Bi, =h6,/k 1

Fo: i

(b) FIG. 8.14b. Temperature response of thick plate (0 <_ x <_ 6 ) convectively heated at x. » 0 and in perfect contact at its rear face x = 6 with a thin plate (0 < x 2 < 5 2 ) insulated at x 2 = 6 2 : (a) x.,/6, = 0. (b) x . / ^ " 1 (case 8.1.51, source: Ref. 19, p. 3-46, Fig. 26).

F-64

y = t s 1.0

0.8

0.6

0.4

0.2

o&tiS/k-^ 1

i wn '/TT^ <•• 0.5 / / /

/ 0.33 / / / / / — / ' / 0.2 / iff / / a i / ' / / 0.05

/ / ^ - Th in plate

/ / / 0 - 0 3 3 /

P^ZZ- | |

^ • 0 2 t s = const

Heating: tg/t,. = 0 —

x/5 = 0

1 0.1 0.5 1 (a) 5 10 50 100 50 100

50 100

FIG. 8.15. Temperature response of thick plate (0 <_ >: < <5) with insulated rear face x = 6 after sudden exposure to uniform radiative environment t s at x = ("•: ""(a) heating, tn/t s = 0, x/6 = 0, (b) heating, to/t s

= °' x/& = 1' <c> heating, t 0/t s = 1/4, x/6 = ;, (d) heating, t 0/t s = 1/4, x/5 = 1 (case 8.1.52, source: Ref. 19, pp. 3-50, Fig. 29).

0.1 0.5 1 5 10 50 100 0.1 6.5 1 5 10 50 100 500 (f) Fo (h)

FIG. 8.15 (Cont.) . Temperature response of thick plate (0 <_ x <_ &) with insulated rear face x = 6 after sudden exposure to uniform radiative environment t s at x = 0: (e) heating, t Q/t s = 1/2, x/6 = 0, (f) heating, t 0/t s = 1/2, x/6 = 1 , (g) cooling, t 0/t s = 6, x/6 = 0 , (h) cooling, t 0/t s = 6, x/ = 1 .

500

500

FIG. 8.15 (Cont.). Temperature response of thick plate (0 <_ x £ 6) with insulated rear face x = 6 after sudden exposure to uniform radiative environment t 8 at x = 0: (k) cooling, t 0/t a = 2, x/6 = 0, (1) cooling, t 0/t s = 2, x/S = 1.

F-67

500

Thin plate.

Cooling: tn/ts = 2 x/8 = 1

50 100 500

FIG. 8.15 (Cont.). Temperature response of thick plate (0 <_ x <_ 6) with insulated rear face x = 6 after sudden exposure to uniform radiative environment t_ at x = 0: (i) cooling, tg/t a • 4, x/6 = 0, (j) cooling, t 0/t s - 4, x/5 - 1.

F-68

PIG. 8.16. Temperature response of thick plate (0 £ x £ 6) with surface temperature t„ at x = 0, increasing linearly with time and rear face x = 6 insulated or exposed to uniform convective environment t Q (case 8.1.8, source: Ref. 19, pp. 3-60, Fig. 33).

FIG. 8.17. Surface temperature response of a plate of thickness 21 exposed to a steady heat flux <jn on both sides for a time duration D (case 8.1.18, source: Ref. 19, pp. 3-67, Fig. 40).

F-69

•P 0,

J?

5 10 50 100 (a)

50 100 Fo

(b) FIG. 8.18. Temperature response of a plate with steady heatlnq qg at x = % and convection boundary at x - 0 to tg (a) x/£ - 1, (b) x/fc = 0 (case 8.1.17, source: Ref. 19, pp. 3-69, Fig. 41).

F-70

18

16

14

12

10

ef o

Q D / 2 J I 5 2

Thick plate = q m a x s in 2 nf l /D

x/5 = 0

3 -

2 -

I Thick plate

1 1 aD/27tfi2 -

1 1 x/6 = 1 0.1—v _ — a — —

0 . 6 - \ \

°- 4-\\ 400% <<S§ô.i °- 3 — \ \4 tf/y/\/ 0.08

~~ 0 . 2 - A * 0.16-M2

0 .14-2cW 0 12^225cOt£

•JX//0.06-

— Thin plate-/ \ S ' ^S 0.03" ' ^ V ^ - O ^ 0.025 i i ^ i - r - o . 0 2

!,**« S = ^ q _ - n n i d 0 1 2 3 4 5 6 7

2TTT/D

FIG. 8 .15. Temperature response of an i n f i n i t e p l a t e exposed to a c i r cu l a r heat pulse q^ a j t s i n 2 (TTT/D) at x = 0 and i n su la ted a t x = 6 (a) x/5 = 0, (b) x/6 = 1 (case 8 .1 .54 , source: Ref. 19, pp. 3-70, Fig. 42) .

F-71

0.05 0.1 0.5 1 5 10 OLeTlf>1

(b)

FIG. 8.20. Temperature response of an infinite porous plate after sudden exposure to a constant heat flux at x = 0 and cooled by steady flow through plate from x = 6 of a fluid initially at t 0 (a) x/6 = 0 , (b) x/5 = 1 (case 8.1.55, source: Ref. 19, pp. 3-74, Fig. 44).

F-72

1 I '• Two-layer plate q const lk2/k,)âx/a2 = 1/2

FIG. 8.21. Temperature response of an infinite plate with a constant heat flux q« at X i = 0, in contact with a plate at x-± = 6^ and insulated on exposed surface of second plate x 2 = 5 2 (case 8.1.56, source: Ref. 19, pp. 3-76, Fig. 45).

F-73

Two-layer plate y i ' i T=^~I—r

(k,/k1)VVs = 2 >°, *2'*1' V",'"2 x , = S r l\. 1/4

( S ^ l V S ^ .1/2

4 °°< " 4 oo^

£ J I L J . I ' l 1 0 1

i (0

0.2

0.4

0.6

0.8

1.0 2 3 4 5 6 7 E S

Fo, J 1

v -i i i i i - Two-layer plate

fc2/k,)^/u2

1 —

= 4. A i ' i ' i '

- \ vs 1 / 4 ^-—'

\

: \ "~ 1/2—1 " - 1 / 4 - ~ _ _ _ _ _ ^

r;, , 1 , 1 J i , n _ r;, , 1 , 1 J i , 1 . 1 , 1 ,

0

- 0.2

- 0.4

- 0.6

0.8

0 1 2 3 4 5 6 7 8 (d> Fo,

1.0

FIG. 8 .21. (continued).

P-74

FIG. 8.22. Temperature response of an infinite plate exposed to a constant heat flux q" at xi = 0 and in perfect contact at xi = 61 with a plate of thickness 62 insulated at x 2 = S 2 (case 8.1.57, source: Ref. 19, pp. 3-78, Fig. 46).

F-75

Two-layer plate

n = p 2 C 2 6 2 / p 1 C l S l =0.1 W t 0 - S i « W W n + 1 > k i

FO;/2TT = 3 , 0 / 2 ^ 2

FIG. 8.23. Temperature response of an infinite plate exposed to a circular heat pulse at ip = 0 and in perfect contact with an infinite conductivity plate of thickness &2 (case 8.1.58, source: Ref. 19, pp. 3-78, Pig. 47).

F-76

1000 I I Two-layer plate

_ (b)

0.01

1000 Two-layer plate

p 2 C 2 S 2 / p 1 C,6 1 = 10

100 -

- (0

0.01

FIG. 8 . 2 3 . ( cont inued) .

F-77

I

FIG. 8.24. Temperature response of an infinite plate surface having steady temperature and convection boundaries (case 8.1.25.2, source: Ref. 74, p. 240, Fig. 6.12).

FIG. 8.25. Temperature of a flat plate having a ramp surface temperature equal to t 8 = t + br (case 8.1.33, source: Ref. 74, p. 305, Fig. 7.1).

F-78

2Fo 1 + 2/Bi

2Fo 1 + 2/Bi

FIG. 8.26. Transient surface temperature of a slab convectlvely coupled to a linearly changing environment temperature equal to tg - t + bt (case 8.1.33, source: Ref. 89).

F-79

i

0.001 10 16 24 30 60 90 120 150 350 Fo

FIG. 9.1a. Axis temperature for an infinite cylinder of radius r Q (case 9.1.3, source: Ref. 5, p. 84, Fig. 4-8 and Ref. 12).

1.0

0.8

0.6

1 0.4

0.2

1 I 1 1 L_-| •

- r / r 0 = 0 . 2 - ^ ^ ^ >

— °ji_—-"^'^ / //t ' -

/ III -— Q&y^ /ll -

/1 1 -— / /1

0.8 . / / /

^***^ / I -— Q-\S / -

1 . 0 , / r ^ 1 i i 1 , I , ,

0.01 0.1 1.0

1/Bi

10 100

FIG. 9.1b. Temperature as a function of axis temperature in an i n f i n i t e cylinder of radius r Q (case 9 .1 .3 , source: Ref. 5, p . 87 and Ref. 12).

1.0 0.9 0.8

0. i 0.

°|cP o. 0. 0. 0. 0.

7 6 5 .4 ,3 .2 .1 0 10" 5

^ . f ^ i i j * . . t i t § / # #

a S e o o S ^ fy <o ' , / < o < a f i ,o o o o o o o <a o o !^ V* y ^ <v <o

FIG. 9.1c. Dimensionless heat IOSB Q/Q0 of an i n f i n i t e cyl inder of radius r 0 with time (case 9 . 1 . 3 , source: Ref. 5, p . 90, Fig. 4-15 and Ref. 13) .

F-81

I.U 0.9 0.8 0.7 0.6

V ' ~ r I I I I . I . _ Plate: m = 1 Cylinder: m = 2 Sphere: m = 3 —

i Range of these lines: \ F o B > 0 . 2

0.5 \ B i c <0 .01 -

•3-1 0.4 \ *-* "3-

i 0.3 \ _

0.25 - \

0.2 - \

0.15

0.1 I , I i I , I , I , \

U. I U 0.08 \ '

I I

0.06 0.05 0.04

-

0.03 — 0.02 -

0.01 0.008

v -

0.006 0.005 0.004

\ -

0.003 — \ -0.002 - \ -

0.001 I I I I \

mBifFog FIG. 9.Id. Center temperatures for plates, cylinders, and spheres for small values of h (case 9.1.3, 8.1.7 and 10.1., source: Ref. 5, p. 89, Fig. 4-13 and Ref. 12), C is half thickness of radius.

FIG. 9.2. Temperature distribution in an infinite cylinder with initial temperature t and steady surface temperature to (case 9.1.1, source: Ref. 9, p. 200, Fig. 24).

F-82

PIG. 9 .3 . Temperature in an in f in i te cylinder with constant heat f lux at the surface (case 9 .1 .8 , source: Ref. 9, p. 203, Fig. 25).

F-83

T = S(X)

Semi-inf ini te solid

T = S ( X , ) S < X 2 )

Quarter-infinite solid

T = S(X 1)S(X 2)S(X 3)

Eighth-inf inite solid

T = P(Fo) T = P(Fo)S(X) T = P(Fo)S(X 1)S(X 2)

Inf in i te plate

*—28

Semi-infinite plate Quarter-inf inite plate

T = P(Fo,)P(Fo2)

V

u~ | — Z * 1

In f in i te rectangular bar

.28,

T = P(Fo,)P(Fo2)S(X) T = P(Fo,)P(Fo 2)P(Fo 3)

r 28,

K - 2 8 ,

Semi-inf inite rectangular bar

A 28, T h + «3

±\. - \ Rectangular parallelepiped

• 26,

, T = C(Fo,)

i

A.

In f in i te cylinder

T = C(Fo1)S(X)

\S

T = C ( F 0 l ) P ( F o 2 )

Y- 2r, cb

-2r,

Semi-infinite cy l inder Short cyl inder

S = semi-infinite sol id P = plate C = cyl inder

FOT = G T / 6 2 = a r / r 2

X , = x 1 / 2 v / 5 f = F o J

T = ( t - t w ) / ( t 0 - t w )

FIG. 9.4a. Product solutions for internal and central temperatures in solids with step change in surface temperature. S(X) given in Fig. 7.2, P(Fo) given in Fig. 8.4, C(Fo) given in Fig. 9.1 (case 7.1.19, 7.1.20, 8.1.22, 9.1.16, and 9.1.17, source: Ref. 74, pp. 3-65, Fig. 38).

F-84

0.01 -

0.001

Fo, or/5?

FIG. 9.4.b. Central temperatures in an infinite plate, an infinite rod, an infinite cylinder, a cube, a sphere, and a finite cylinder of length equal to its diameter, with all surfaces at temperature tg (case 7.1.29, 7.1.20, 9.1.16, 9.1.17, and 8.1.22, source: Ref. 2, p. 248, Fig. 10-8).

V°

1.0

0.8

0.6

i 0.4 -

0.2

^ Q T 6 ^ - 4 = O I I I - 0 4 ^ ^ v

-o • 3 - _ ^ S s \ N s v -

- 0 - 2 — . — . ^ \ \ ^ v

— 0 -15 . N . N,

= 0 • 1 — _ _ _ _ _ _ _ ^ ^ * s N^vk -- 0 • 08 _ _ ^ ~ " - - « ^ _

—n • n f i - — ^"~"""

0 • 01

— 0 • 0° ~ Fo I I l ^p-^ 0.2 0.4 0.6 0.8 1.0

FIG. 9 . 5 . Temperature distr ibution in an i n f i n i t e cylinder with s teady internal heating and a surface temperature t 0 (case 9 . 2 . 1 , source: Ref. 9, p . 205, F ig . 26) . F 8 5

100.0

s 10.0 h

Juy>

• 9 -

AA?

•3-

0.4 0.8 2.0 3.0 4.0 fore?

FIG. 9 . 6 a . ( case 9 . 2 . 4 , s o u r c e : Ref. 3 , p . 356 , F i g . 7 - 5 ) .

FIG. 9 . 6 b . (case 9 . 2 . 4 , s o u r c e : Ref. 3 , p . 364, F i g . 7 - 8 ) .

0.1 1 10 50 S/IX-1)

PIG. S.6c. (case 9 .2 .4 , source: Ref. 3, p. 365, Fig. 7-9)

F-87

0.001 0.005 0.010 0.050 0.100 0.500 1.000 5.000 10.000

Fo, OT/TQ

FIG. 9.7a. Temperature response of a hollow cylinder with a steady ins ide surface temperature equal to the i n i t i a l temperature, t^ (case 9 .1 .10, source: Ref. 74, p . 156, Fig. 4 .24 ) .

0.001 0.005 0.010 0.050 0.100 0.500 1.000 5.000 10.000

Fo, ar/rg

FIG. 9.7b. Temperature response of a hollow cylinder with a steady outside surface equal to the initial temperature, t (case 9.1.10, source: Ref. 74, p. 157, Fig. 4.25).

F-88

T — i - r - i i i i

0 . 8 0 " " " " " " " ~ _ ^ - —

I I I

- /

- r i

0.66"" " ^ ^ - - " " ^

^ 0.50 " ^ \ ^ ^ ^ ^

I I I I

t 5 = t, + bT

I I I

I I

I 0.8

0.6 a 0.4

0.2

0 0.1 0.2 0.3 0 4 0.5 0.6 0.7 0.8 0.9 1.0 Fo

FIG. 9 .8 . Temperature of an i n t i n i t * cylinder having a ramp surface temperature equal to t s = t± + bx (case 9 .1 .2 , source: Ref. 74, p . 312, Fig. 7 .3 ) .

4Fo 1 + 2/Bi

4Fo 1 + 2/Bi

FIG. 9 .9 . Transient surface temperature of a cylinder convectively coupled t o a l inearly changing environment temperature equal to tf • t j + br (case 9 . 1 .6 , source: Ref. 89) .

F-89

100 i i i i i 1 1 i i i i t i i i

J i I • ' • • • I I I I i i • • ' 10 Fo

100 1000

FIG. 9.10. Central temperature of an in f in i t e cylinder with uniform internal heating and convection boundary (case 9 . 2 . 2 , source: Ref. 90) .

F-90

I ID

0.400

0.8 -

0.6

0.4 -

0.2 -

1 R - 1

^0 .95 ^

o.9o / y 0.85 /

/ O.BoX / ' /

—/ ' / 0.75 /

' / 0.66

— / ' / 0.50 •^-0.25

/ >^0.33X 'o

1 0 0.2 0.4 0.6 0.8 1.0

(1-R) 0.004 0.010 0.4 0.100

FIG. 10.1. Temperature distribution in a sphere of radius r Q with initial temperature t± and surface temperature t 0 (case 10.1.1, source: Ref. 74, p. 127, Figs. 4.15 and 4.16).

1.000

I

-i. 0.050 +•«

'._ 0.030

0.001 3.0 5.0 7.0 9.0 15.0 25.0 35.0 45.0 70.0110.0 170.0 250.0

Fo PIG. 10.2a. Center temperature of a convectively cooled sphere (case 10.1.2, source: Ref. 5, p. 85, Fig. 4-9 and source: Ref. 12).

1 c V ^ n

0.02 0.1 0.5 2.00 5.00 20.00 100.00

1/Bi

FIG. 10.2b. Temperature as a function of center temperature of a convectlvely cooled sphere (case 1 0 . 1 . 2 , source: Ref. 5, p . 88, Fig. 4-12, and Ref. 12).

aid P 0.5 -

Bi2Fo

FIG. 10.2c. Dlmenslonless heat loss Q/QQ of a convectively cooled sphere (case 10 .1 .2 , source: Ref. 5, p . 91 , Fig . 4-16, and Ref. 13) .

F-93

FIG. 10.3. Temperature of an infinite conductivity medium surrounding a spherical solid (case 10.1.13, source: Ref. 9, p. 241, Fig. 30).

0.2

s o

0 -

7- -0.1-

-0.2

-0.3

-

I I I

A 0

i ±1

f

2.8

2.4

2.0

1.6

1.2

0.8

0.4

-

I I I

A 0

i ±1

f

2.8

2.4

2.0

1.6

1.2

0.8

0.4 I

I I I

f A

I I I

\

i

0

i ±1

f

2.8

2.4

2.0

1.6

1.2

0.8

0.4 I

I I I

f A

I I I

\

i

0

i ±1

f

2.8

2.4

2.0

1.6

1.2

0.8

0.4 I

I I I

f A

I I I

\ „«I ^~*-~**5w

i

0

i ±1

f

2.8

2.4

2.0

1.6

1.2

0.8

0.4 I

I I I

f A

I I I

\

—

i

0

i ±1

f

2.8

2.4

2.0

1.6

1.2

0.8

0.4 I

I I I

f A

I I I

\

—

__ 0 . 0 8 ( X y /

i

0

i ±1

f

2.8

2.4

2.0

1.6

1.2

0.8

0.4 I

I I I

f A

I I I

\

0.2 0.4 0.6 0.8 R

1.0 0 0.2 0.4 0.6 0.8 Fo

FIG. 10.4. Temperature in a sphere caused by a steady surface heat flux (case 10.1.15, source: Ref. 9, p. 242, Fig. 31).

F-94

1 - 0

1 -0

FIG. 10.5. Temperature distribution in a sphere with steady internal heating (case 10 .2 .1 , source: Ref. 9, p. 244, Fig. 32) .

F-95

0.010 0.100 1.000 10.000 Fo

10.00

10.6. Temperature response of solid sphere (0 <_ r £ r- ) cooling by radiation to a sink temperature of (a) r/r x = 0, (b) r/c1 = 1/2, (c) r/rx = 1 (case 10.1.20, source: Ref. 19, pp. 3-57, Fig. 30).

T - T , 0.6

T w - T i

0.02 0.06 0.10 0.40 0.80

F o , —

FIG. 10.7a. Transient temperature d i s t r i bu t ion a t the center point (x = y = z = 0) for various p ro l a t e spheroids (case 10 .1 ,21 , source: Raf. 72, Fig. 1),

1.0

0.8

T - T i 0.6

T w - T i

0.01 0.04 0.08 0.20 0.601.00 _ a r

FIG. 10.7b. Transient temperature distribution at the focal point (x = y = 0, z - L2) for various prolate solids (case 10.1.21, source: Ref„ 72, Fig. 2).

F-97

FIG. 10.8. Temperature of a sphere having a ramp surface tep.perature equal to t s = t£ + br (case 10.1.4, source: Ref. 74, p. 310, Fig. 7.P1 .

0.5

6Fo 1 + 2/Bi

FIG. 10 .9 . Transient sur face temperature of a sphere convectively coupled to a l i n e a r l y changing environment temperature equal to tf = tj_ + bt (case 10 .1 .11 , source : Ref. 89).

F-98

100.0

10.0

I

CNO

r-[—i i i 11 I I 1—I I ! 111 1 1—i i i i i I J 1—i i i i I I

R = 0 80.0 —

J I I I I I I 100.0 1000.0

FIG. 10.10. Central temperature of a sphere with uniform internal heating and convection boundary (case 10.2.9, source: Ref. 90).

(h sVk)(t m-t a)r

FIG. 11.1. Solidification depth in semi-infinite liquid for which solidified .phase has negligible thermal capacity and is exposed to convective environments t a at free surface x = 0 and t 0 at interface x = w (case 11.1.10, source: Ref. 19, pp. 3-88, Fig. 54).

F-99

0.01 0.05 0.1 0.50 1

(b2/pck)T-

50 100 500 1000

10

5

0.5

0.1

0.05 -

0.01 0.001 0.005 0.01

(b2/pck)T

100

5 10 50 100

50 100 500 1000

FIG. 11.2. Temperature response and solidification depth in semi-infinite liquid initially at t^ (a) surface temperature for convective cooling q = bt w

at x = 0, (b) solidification depth (case 11.1.11, source: Ref. 19, pp. 3-89, Fig. 55).

F-100

0.01 0.001

0.05 0.1

(b2tJL/pck)r

0.006 0.01

50 100 500 1000

5 10

50 100

50 100 500 1000

(b 2 t 6m/pck)

FIG. 11.3 . Temperature response and so l i d i f i ca t i on depth in semi - in f in i t e l i qu id i n i t i a l l y a t t^ with r a d i a t i v e cooling q = b t w a t x = 0 (a) surface temperature a t x = 0, (b) s o l i d i f i c a t i o n depth (case 11 .1 .12 , source: Ref. 19, pp. 3-90, F ig . 55) .

F-101

- 0.4 -

Semi-infinite solid t = const w 0<x=£l

0 0.0001 0.0005 0.001

b/2V^7

0.005 0.01 FO:

0.05 0.1 0.5 1

xl

0.01 Fo' x2

PIG. 11.4. Temperature response of decomposing semi- inf ini te sol id after sudden change in surface temperature from tg to t,,: (a) 0 £ x £ SL, (b) x £ SL (case 11.1.13, source: Ref. 19, pp. 3-91, Fig. 56).

F-102

10

5

0.01.

(a)

0.1

• W ^ d - V

Semi-infinite solid t = const

(h/pH)tt a-t dh/^7J=1/2

0.5 1

r / r d - 1

10

5

i i i /

1

0.5 10-50-

1.0-0 .5 -0.1 -

dv

0.1

\ / , y\ - V ^ ' V o 1

0.05

n n i

V i

Semi-infinite solid

(h /pHM^- t^^ /T^ I 1 1

(b)

0.1 0.5 1

r/T d - 1

10

0.05

0.01 0.1

vwv

Semi-infinite solid

(h/pH)(ta-td)v/5d7d"=2

(c)

0.5 1

T/r, - 1

10

FIG. 11.5. Ablation depth of semi- inf inite sol id (x >_ 0) after sudden exposure to convective environment t-(> t^): (a) (h/pH) ( t a - t d ) ^td/a = 1/2, (b) (h/pH) ( t a - t d ) STga = l , (c) (h/pfl) ( t a - t d ) /Ta/a •» 2 (case 11 .1 .14 , source: Ref. 19, pp. 3-94, Fig. 58) .

F-103

10

5

1

0.5

0.1

0.05

0.011—<•

i—i—n—i | i i i I I 1—i—i—i—i I i i 11

Semi-infinite q " = const

0.01

- V?C(t d-t 0)/2Hv

0.05 0.1 5 10 50 100

<r/T d) - 1

FIG. 11.6. Ablation depth of semi- inf ini te sol id (x >_ 0) after sudden exposure to constant surface heat input q" (case 11.1 .15, source: Ref. 19, pp. 3-96, Fig. 59) .

F-104

TABLE 1.1. Mean and maximum temperature excesses and their ratio for electrical coils of rectangular and circular cross sections, 9 = t - t_,

m mean 0 6„ = t - t„ (case 1.2.8, source: Ref. 3, p. 220, Table 10-3). 0 max u

Cross Section, Rectangular Circular, b/a » 10 5 2.5 1.5 1 r = 2a//i

0.66 0.625 0.58 0.28 0.32

1.00 1.00 1.00 0.59 0.64

0.66 0.625 0.58 0.52 0.485 0.475 0.50

TABLE 1.2a. Values of (t - t ) / ( t f - t±) on the

surface y = 0 of a s emi - in f in i t e s t r i p

(case 1.-1.29, source: Ref. 15, Table 2) .

x/w hw/k hw/k hw/k hw/k hw/k = 100 = 10 = 1 » 0.10 = 0.01

0.05 0.889 0.448 0.094 0.011 0.001 0.10 0.931 0.594 0.146 0.018 0.002 0.20 0.964 0.725 0.212 0.027 0.003 0.30 0.973 0.781 0.250 0.032 0.003 0.40 0.977 0.807 0.270 0.035 0.004 0.50 0.978 0.814 0.277 0.036 0.004

TABLE 1.2b. Heat transfer rates through the surface y = 0 of a semi- inf ini te

s t r i p (case 1.1.29, source: Ref. 15, Table 1) 9 = (t - t ) .

hw/k 1000 200 100 20 10 1.0 0.1 0.01 0.001

Q/k6 8.603 6.827 5.984 3.971 3.107 0.792 0.097 0.01 0.001 % Error 1.8 0.5 0.3 0.08 0.05 0.02 0.02 0.02 0.02

U k / q ' - ' a ^ e

; 2 k / q " ' a 2 ) 9 n

F-105

TABLE 1.3. Conductance data for heat flow normal to wall cuts in an i n f i n i t e

p l a t e (case 1.1.37, source: Ref. 19, p . 3-124). These groups of data are for

the four kinds of wall p l a t e s shown in case 1.1.37.

(a) K/K uncut

(b) K/K uncut

c / a b /a = 1/2 1/4 1/8 c / a b/a = 1/2 1/4 1/8

4 0.902 0 .760 0.646 3 /2 0.747 0.520 0 .381

2 0.818 0 .618 0.480 3 /4 0.575 0.315 0.217

1 0.704 0 .465 0.339 1/2 0.430 0.209 0 .146

1 /2 0.610 0 .363 0.235 1/3 0.296 0.113 0 .080

1/4 0.564 0 .303 0.182

(c) K/K uncut

(d)

c / a d /a b /a = 1 /2 1/4 1/8 c / a d/a

1/8 0 .822 0.634 0.478 0.402

3 3 /8 0 .767 0.546 0.374 3 / 2 0.469

5 /8 0 .721 0.480 0.311 0.502

12/8 0 .599 0 .339 0.190 0.866

1/16 0 .740 0.486 0.342 1.010

3 / 2 3/16 0 .698 0.430 0.284 1.083

5/16 0 . 6 6 1 0.390 0.242 1 /2 1/2 12/16 0 .570 0.299 0.164

0 0 .627 0 .403 0.270

1 /2 1/16 0 .561 0.312 0.186

3/16 0 .527 0.276 0.152

K/K uncut

0.507

0.479

0.329

0.362

0.338

1/16

0.310

0.254

0.213 0.141

P-106

I

TABLE 2 . 1 . Values of A. for a t r i angu la r cooling a r ray of cy l inders (case 2 . 2 . 1 8 , source: Re— 64, Table 2 ) .

8 / r o Al A2 *3 A4 A5 A6 *7

4.0 -5.05072(10-2) -8.0891(10~4) -1.262(10~5) -1.97(10~7) -3.08(10~9) -4.8(10 - 1 1) -7(10- 1 3, 2.0 -5.04988(10-2) -8.0542(10~4) -1.217(10~5) -1.64(10 - 7) -1.33(10~9) 2.8(10 - 1 1) 2(10~ 1 2) 1.5 -5.02447(10~2) -6.9920(10~4) -1.374(10-6) 8.34(10~7) 5.23(10~7) 2.4(10 - 9) 8(10 - 1 1) 1.3 -4.90792(10~2) -2.1203(10~4) 6,354(10~5) 5.42(10~6) 2.99(10~7) 1.3(10~8) 4(10- 1 0) 1.2 -4.69398 (10 - 2) 6.7812(10~4) 1.749(10~4) 1.31(10~5) 6.27(10~7) 1.7(10~8) -K10- 1 0) 1.175 -4.60073 (10 - 2) 1.0619(10~3) 2.205(10~4) +1.56(10~5) 6.32(10~7) 9.8(10~10) -2(10~9) 1.15 -4.48345(10-2) 1.5384 (10~3) 2.736(10-4) 1.76(10~5) 4.21(10~7) -5 (10~8) -7(10"9) 1.10 -4.15694(10-2) 2.8029(10~3) 3.782(10~4) 1.08(10~5) -2.40(10-6) -5 (10~7) -5(10"9) 1.05 -3.68059(10"2) 4.3358(10~3) 3.169(10~4) 5.67(10~5) -1.51(10~5) -1.7(10 - 6) -9(10"8)

TABLE 2 . 2 . Values of 6 for a square cooling a r ray of cy l inders (case 2 .2 .19 , source : Ref. 64, Table 1)

3 / r o 61 S2 63 64 55 66 67

4.0 -1.25382 (10 _ 1) -1.0583(10-2) -6.120(10~4) -3.898(10~5) -2.42(10~6) -1.5(10~7) -8 do"9) 2.0 -1.25098 (10 - 1) -1.0428(10"2) -5.713(10-4) -3.177(10-5) -1.39(10~6) -2 (10~8) 3 (10~9) 1.5 -1.22597 (10 _ 1) -9.060 (10~3) -2.08 (10~4) 3.33 (10~5) 8.1 (10~6) 1 dO" 6) 1 (10 - 7) 1.3 -1.17022 (10 _ 1) -5.995 (10~3) 6.13 (10~4) 1.83 (10~4) 3.1 (10~5) 4 (10~6) 4 (10 - 7) 1.2 -1.10421 (10 - 1) -2.387 (10~3) 1.566(10~3) 3.50 (10~4) 5.3 (10~5) 6 (10 - 6) 5 (10"7) 1.15 -1.05352 (10 - 1) 3.172 (10 - 4) 2.233(10~3) 4.44 (10~4) 5.8 (10~5) 4.3(10"6) 1 (10_8> 1.1 -9.8721 (10~2) 3.6596(10-3) 2.91K10" 3) 4.655(10~4) 2.44(10~5) -1.K10" 5) -5. 3(10 - 6) 1.05 -9.0358 (10~2) 7.310 (10~3) 3.210U0 - 3) 2.06 (10~4) -1.20(10~4) -5.B(10~5) -1. 6(10"5)

TABLE 8.1. Values of (t - t,)/(t - t.) for an infinitely wide plate 1 s 1 whose surface temperature increases propor t iona l ly to t ime, t

(case 8 .1 .8 , source: Ref. 1, p . 268, Table 13 .5) . t . + bx

i

Fo

X 0.08 0.16 0.20 0.32 0.40 0.80 1.60 2.00 4.00

0 0.004 0.045 0.074 0.170 0.231 0.464 0.694 0.752 0.875 0.33 0.030 0.104 0.142 0.245 0.305 0.522 0.728 0.779 0.889 0 .5 0.082 0.194 0.238 0.346 0.402 0.594 0.770 0.814 0.906 0.66 0.214 0.353 0.398 0.498 0.545 0.698 0.830 0.862 0.930 0 . 8 0.418 0.548 0.586 0.664 0.699 0.802 0.889 0.911 0.955

r-l 1 1 1 1 1 1 1 1

•-1

TABLE 8.2. Values of the function X, (case 8.1.12, source: Ref. 1, p . 299).

X

a 0 1/8 2/8 3 /8 4 / 8 5/8 6 / 8 7/8 1

0 1 1 1 1 1 1 1 1 1

0 . 5 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.99 1

1.0 0.77 0.77 0.77 0.78 0.79 0.81 0.85 0.91 1

1.5 0.47 0.47 0.47 0.48 0.52 0.58 0.68 0.83 1

2 .0 0.27 0.27 0.28 0.30 0.36 0.45 0.58 0.77 1

4 . 0 0.04 0.04 o.or. 0.08 0.13 0.22 0.37 0.64 1

8 .0 0.00 0.00 0.01 0.01 0.02 0.05 0.14 0.36 1 00 0 0 0 0 0 0 0 0

F-108

TABLE 9 . 1 . Values of (t - t . ) / ( t - t . ) 1 s 1

for an i n f i n i t e l y long cylinder whose surface increases propor t iona l ly to time (t - t . = cT) (case 9 .1 .2 , source: Ref. 1, p . 269).

R Fo

0.08 0.16 0 .32 0.80

0 0 .016 0 .123 0.354 0 .691

0 .33 0.054 0 .191 0.420 0.725

0 .5 0 .122 0.287 0 .505 0.768

0 .66 0 . ' , 6 8 0.443 0 .628 0.828

0 . 8 0.470 0 .621 0 .755 0.888

1 1 1 1 1

TABLE 1 0 . 1 . Temperatures for a sphere whose surface temperature increases propor t ional ly to time ( t Q = t . + bt) (case 10 .1 .4 , source: Ref. 1, p . 269, Table 13-7).

R Po

0 .016 0.08 0 .16 0.32 0 .80

0 0 .00 0.054 0 .219 0.506 0 .792

0 .33 0 .00 0.090 0 .290 0.560 0 .725

0 . 5 0 .00 0.162 0 .385 0.626 0 .844

0 .66 0 . 0 2 0.312 0 .529 0.722 0 .884

0 .8 0 .14 0.516 0 .686 0.819 0 .925

1 1 1 1 1 1

F-109

2 35 n

a to


SECTION 13. MATHEMATICAL FUNCTIONS

13.1. The Error Function and Related Functions.

2 f 2 erf(x) = y== I exp(-z ) dz

H w n-0 (2n+l)nl

- 1 - exp£x ) V X 2 x 3

erf(0) = 0 , erf(~) = 1

erf (-x) • - erf(x)

erfc(x) • 1 - erf (x)

i n er fc (x ) = r !• l _ 1 e r f c ( z ) d

2n+l , for small values of x

i erfc(x) « erfc(x) 1 2 ierfc(x) » -& exp(-x ) - x erfc(x)

i2erfc{x) = j [erfc(x) - 2x ierfc(x)]

2n inerfc(x) » i n _ 2erfc(x) - 2x i n _ 1erfc(x)

lêrftx) - ^ e x p ( - x 2 )

d 2 .. . 4x , 2 . —_ erf (x) - - T E exp(-x ) dx z ^

for large

* values of x

13-1

Z-ZT

>

•L zz •; •;•:•: z z s E •; •; s; s;: z z z z ?

s s g s c s g s g s •;: 11z i l i i l l l i i l i i l i i i l i i l i l l i l l l i l l l l i l l

o o o o o o O a a o O a a o o o o © o o

S S 3 , emu V-i o O W o u o» w II

M> IB V) 09 f. 3 M a 2 d en ui A. u M

hi a at o en

O O O O O O O O O O O O O O O H

I 1 1 i O O O

o o a o M M Ul Jh *• in Ut 91 91 U» P P * •J IS -J s i W ~J N O IO K) W en m - J to - j u»

' - • • oi o u> o ui m ui ui o

o o o o o o c - o o O O O O O Q O O

_ 10 <A o « _ tC U3 10 10 O lO 10

10 ID U? U3 10 U) 40 10 ^ U) to 10 -- IS >0 10 10 CO C3 - - - - - - - » 5 » R »j o> oi wi m » O O* W U N M

10 10 10 " ! N » « « ? > F ' t = ° R ° S o i 3 K - « 5 f o w w a j i o f c o m o f

? ? ! ' W O J-" O Ul CD *

O Ul O N Ul h) Q CO 10 H « r > * - * U J W i W O N > K l M S 5 S S

o o o o o o o o o o a a o O O O O O O O ) - '

i i i S S o o o o a H> H M w m -j o w ui CD w

IS Ol M ID o c n u u i s j u i i o i a - j w u i o « a ?• s « S q - j g o w y . » > i y > = » * o » = S * » 10 UJ C f v u i U l < * e n W C " > M

Ul U 10 03 U l U l M C F i U I I A t O M

O O O O O O O O O O O O O O O O O O O O *-• !-•

I g g g g g g g p o !S H "! H a t- >o en in M ui

o o c a o o o o o

h - M M M W U l U l W * _ t n C D O * - J t - ' U I U ) *

«J » CO K> U) O * H O* 10 V> C\ CT1 *-l 00 "O

U -J K> M IO

O O O O

u n a m -a ui w ui

© o o o o

l l i l l l

o o o o

o o a a o o o w u m en * B B . i E S m W O N *

G O O D O O © O O O O O O

IIIIIIIIII o o o o o o a o

o o o r~i o o o o o a o o o o o o a o o o o o o o o — — o o o o o o o o o o o o o o o i- 1 \0 o\ *• pCk - J

o o a o o O P O O O O o o D o a o

o o o o o O O O O O O O h-n) o ro ui en ui CD en

o o o o o o o o

a o o o o

o o o o o D O O O O O o o o o

o a o o o o O O Q O O D O O O O o o o ** m en m o

w w w

H" Ul W

*» !

oo

1

I

ft 13-3

» 13.2. Exponential and Hypetbolic Functions.

e = 2.71828 . . . u exp(u) = e

u -u sinh (u) = - — i ~ ^ — , sinh (0) = 0 , sinh (=>) = 2

u e + -u

6 2

2 e

u - 1 tanh (u) = — , tanh (0) = 0 , tanh («) 2u , e + 1

TABLE 13 .2 . Exponential functions (source: Ref. 2, pp. 367-9).

0.00 1 .000 1.000

0 .02 1.020 0 .980

0.04 1 .041 0 .961

0.06 1 .062 0.942

0.08 1 .083 0.923

0.10 1 .105 0.905

0.12 1.128 0.887

0.14 1.150 0.869

0 .16 1.174 0 .852

0 .18 1.197 0.835

0.20 1 .221 0 .819

0 .22 1.246 0 .802

0.24 1 .271 0.787

0 .26 1.297 0 .771

0 .28 1 .323 0.756

0.30 1.350 0 .741

0 .32 1.377 0 .726

0.34 1 .405 0.712

0 .36 1.433 0.698

0.38 1 .462 0.684

0.40 1 .492 0.670

0 .42 1 .522 0.657

0 .44 1 .553 0 .644

0 .46 1 .584 0 .631

0.48 1 .616 0 .619

0 .50 1 .649 0 .606

0 .52 1 .682 0.594

0.54 1.716 0 .583

0 .56 1 .751 0 .571

0 .58 1 .786 0.560

e " u

0.60 1 .822 0.549

0 .62 1.859 0.538

0 .64 1.896 0.527

0 .66 1.935 0.517

0.68 1.974 0.507

0 .70 2 .014 0.497

0 .72 2 .054 0.487

0.74 2 .096 0.477

0 .76 2 .138 0.468

0.78 2 .182 0.458

0 .80 2 .226 0.449

0 .82 2 .270 0.440

0.84 2 .316 0.432

0 .86 2 .363 0.423

0.88 2 . 4 1 1 0.415

0.90 2 .460 0.407

0 .92 2 .509 0.398

0.94 2 .560 0 .391

0 .96 2 .612 0.383

0.98 2 .664 0.375

1.00 2 .718 0.368

1.02 2 .773 0 .361

1.04 2 .829 9.353

1.06 2 .886 0.346

1.08 2 .945 0.340

1.10 3 .004 0.333

1.12 3 .065 0.326

1.14 3 .127 0.320

1.16 3 .190 0.313

1.18 3 .254 0.307

e u

1.20 3.320 0 . 3 0 1

1 .22 3.387 0 .295

1 .24 3.456 0 .289 1 .26 3.525 0 .284

1 .28 3.597 0 .278

1.30 3.669 0 .272

1 .32 3.743 0 .267

1 .34 3 .819 0 .262

1 .36 3.896 0 .257

1.38 3.975 0 . 2 5 2

1 .40 4.055 0 .247

1 .42 4.137 0 . 2 4 2

1 .44 4 .221 0 .237

1.46 4.306 2 .232

1 .48 4.393 0 .228

1 .50 4 .482 0 .223

1 .52 4 .572 0 .219

1 .54 4 .665 0 .214

1 .56 4 .759 0 .210

1.58 4 .855 0 .206

1 .60 4.953 0 .202

1 .62 5 .053 0 .198

1 .64 5.155 0 .194

1.66 5.259 0 .190

1.68 5.366 0 .186

1 .70 5.474 0 .183

1 .72 5.584 0 .179

1 .74 5.697 0 .176

1 .76 5 .812 0 .172

1 .78 5.930 0 .169

13-4

TABLE 1 3 . 2 . (Continued).

u e u e " u u e u e " u u e u e " u

1.80 6.050 0 .165 2.40 11 .023 0.091 3 .00 20 .08 0 .050

1 .82 6 .172 0 . 1 6 2 2.42 11 .246 0.089 3 .10 22 .20 0 .045

1.84 6.296 0 .159 2.44 1 1 . 4 7 3 0.087 3 .20 24 .53 0 .041

1.86 6 .424 0 .156 2.46 11 .705 0.085 3 .30 27 .11 0 .037

1.88 6.554 0 .153 2.48 1 1 . 9 4 1 0.084 3 .40 29 .96 0 .033

1.90 6.686 0 .150 2.50 1 2 . 1 8 0.082 3 .50 33 .11 0.030

1.92 6 .821 0 .147 2.52 1 2 . 4 3 0.080 3 .60 36 .60 0 .027

1.94 6.959 0 .144 2.54 1 2 . 6 8 0.079 3 .70 40 .45 0 .025

1.96 7.U?9 0 .141 2.56 1 2 . 9 4 0.077 3 .80 44 .70 0 .022

1.98 7 .243 0 .138 2.58 1 3 . 2 0 0.076 3 .90 49 .40 0 .020

2 .00 7 .389 0 .135 2.60 1 3 . 4 6 0.074 4 .00 54 .60 0 .018

2 .02 7 .538 0 .133 2.62 1 3 . 7 4 0.073 4 .20 66 .69 0 .015

2 .04 7 .691 0 .130 2.64 1 4 . 0 1 0 .071 4 .40 8 1 . 4 5 0 .012

2 .06 7 .846 0 .127 2.66 1 4 . 3 0 0.070 4 .60 99 .48 0 .010

2 .08 8 .004 0 . 1 2 5 2.68 1 4 . 5 8 0.069 4 .80 1 2 1 . 5 1 0 .008

2 ,10 8 .166 0 .122 2.70 1 4 . 8 8 0.067 5 .00 1 4 8 . 4 0 .007

2 .12 8 .331 0 .120 2.72 1 5 . 1 8 0.066 5 .20 1 8 1 . 3 0 .006

2 .14 8 .499 0 .118 2.74 1 5 . 4 9 0.065 5 .40 221 .4 0 .004

2.16 8 .671 0 .115 2.76 1 5 . 8 0 0.063 5 .60 2 7 0 . 4 0 .004

2 .18 8 .846 0 .113 2.78 1 6 . 1 2 0.062 5 .80 330 .3 0 .003

2 .20 9 .025 0 .111 2.80 1 6 . 4 4 0 .061 6 .00 4 0 3 . 4 0 .002

2 .22 9 .207 0 .109 2.82 1 6 . 7 8 0.060 6 .50 6 6 5 . 1 0 .002

2 .24 9 .393 0 .106 2.84 1 7 . 1 2 0.058 7 .00 1096 .6 0 .001

2.26 9 .583 0 .104 2.86 1 7 . 4 6 0.057 7 .50 1808 .0 0 .001

2 .28 9 .777 0 .102 2.88 1 7 . 8 1 0.056 8 .00 2981 .0 0 .000

2 .30 9 .974 0 .100 2.90 18 .17 0 .055 8 .50 4914 .8 0 .000

2 .32 10 .176 0 .098 2.92 1 8 . 5 4 0.054 9 .00 8 1 0 3 . 1 0 .000

2 .34 1 0 . 3 8 1 0 .096 2.94 1 8 . 9 2 0.053 9 .50 13360 0 .000

2 .36 10 .591 0 .094 2.96 1 9 . 3 0 0.052 1 0 . 0 0 22026 0 .000

2 .38 10 .805 0 .093 2.98 1 9 . 6 9 0 .051

13-5

13.3 . The Gamma Function.

co

I 'M = / u*""1 e " u du, x > 0

Hx) = x x e - x ^ | L 12x „„„ 2

139 571

288x" 5184x3 2 4 8 8 3 2 0 X 4 + ...

T(x + 1) = xr<x) for large positive values of x

TABI£ 13.3. The gamma function (source: Ref. 17, p. 136)

X T(x) X T(x> X T(x) X T(x) X T(x)

1.01 0.99 433 1.21 0.91 558 1.41 0.88 676 1.61 0.89 468 1.81 0.93 408 1.02 0.98 884 1.22 0.91 311 1.42 0.88 636 1.62 0.89 592 1.82 0.93 685 1.03 0.98 355 1.23 0.91 075 1.43 0.88 604 1.63 0.89 724 1.83 0.93 969 1.04 0.97 844 1.24 0.90 852 1.44 0.88 581 1.64 0.89 864 1.84 0.94 261 1.05 0.97 350 1.25 0.90 640 1.45 0.88 566 1.65 0.90 012 1.85 0.94 561

1.06 0.97 874 1.26 0.90 440 1.46 0.88 560 1.66 0.90 167 1.86 0.94 869 1.07 0.96 415 1.27 0.90 250 1.47 0.88 563 1.67 0.90 330 1.87 0.95 184 1.08 0.95 973 1.28 0.90 072 1.48 0.88 575 1.68 0.90 500 1.88 0.95 507 1.09 0.95 546 1.29 0.89 904 1.49 0.88 595 1.69 0.90 678 1.89 0.95 838 1.10 0.95 135 1.30 0.89 747 1.50 0.88 623 1.70 0.90 864 1.90 0.96 177

1.11 0.94 740 1.31 0.89 600 1.51 0.88 659 1.71 0.91 057 1.91 0.96 523 1.12 0.94 359 1.32 0.89 464 1.52 0.88 704 1.72 0.91 258 1.92 0.96 877 1.13 0.93 993 1.33 0.89 338 1.53 0.88 757 1.73 0.91 467 1.93 0.97 240 1.14 0.93 642 1.34 0.89 222 1.54 0.88 818 1.74 0.91 683 1.94 0.97 610 1.15 0.93 304 1.35 0.89 115 1.55 0.88 887 1.75 0.91 906 1.95 0.97 988

1.16 0.92 980 1.36 0.89 018 1.56 0.88 964 1.76 0.92 137 1.96 0.98 374 1.17 0.92 670 1.37 0.88 931 1.57 0.89 049 1.77 0.92 376 1.97 0.98 768 1.18 0.92 373 1.38 0.88 854 1.58 0.89 142 1.78 0.92 623 1.98 0.99 171 1.19 0.92 089 1.39 0.88 785 1.59 0.89 243 1.79 0.92 877 1.99 0.99 581 1.20 0.91 817 1.40 0.88 726 1.60 0.89 352 1.80 0.93 138 2.00 1.00 000

13-6

the pos i t ive sign used if -n/2 < arg u < 31T/2 and the negat ive sign used if -3TT/2 < arg u < n / 2 .

I (u) - I (u) _ , . _ 7J_ —n n n w " 2 s i n (mr)

For n any positive integer:

n+2j Kn(u, = < - i > n + 1 | m (u/2) + Y | i n , „ , + ^- I iag__ j=0

n+3

2 m"1 + S ra_1

.m=l m=l j=0

V - 1 where for ]=0 r y m =

m=l

For large values of u:

v«->/H»^»'^ffi-••'*••»•'» 4 - J (u) = J . (u) - s J (u) = - J (u) - J _,, (u) du n n-1 u n u n n+1

| - y ( U) = y ( U ) - £ Y (u) = T; Y (u) - Y «u) au n n— 1 u n u n n+1

4 - I (u) = I . (u) - - I (u) = I _,, (u) + - I (u) du n ' ' n - l x ' u n v ' n+1 u n '

jr K (u) = -K (u) - £ K (u) = £ K (u) - K (u) au n n-1 u n u n n+1

13-7

13.4. Bessel Funct ions .

3 , „ / ^ + 2 j a ( u ) V ( - l ) J ( u / 2 ) " ^ , n i J

n

( u ) L j i r { n + j + i) i s r ea l and u may be complex

j=0

J (u) = (-1) J (u) , if n i s an integer n -n

J_ (u) cos (nir) - J__(u) V (u) = — : —;

n s in (nit)

J ,„/•,* n + 2 3 TIY

B (u) = 2 ( in (u/2) + Y | J n ( u , - I ^ i ^ f ^ ' j=0

j+n j n-1 ^ m"1 + ^ m" 1 - S ( u / 2 ) " n + 2 j (n -_J - 1)1

_m=l m=l J j=0

where y = 0 .5772 . . . i s Euler ' s cons tant , and n i s any pos i t i ve integer

j YQ(u) = {In (u/2) + y}3Q{M) + ( u / 2 ) 2 - ( l + j •& Z1L (21)

•(••WW--V d i ' ( 31T

V u ) Z , j i r t n + j + J-0

For large values o£ u:

I

n

( u ) "^rora .^ i * !^^!^) )

± = = e x p [ - u ± (n + | ) * i ] | l + 0 ( u ' 1 ) ) , V2TTU

13-8

TABLE 13.4.1. Zero and first-order Bessel functions of the first kind (source: Ref 2, pp. 369-71).

u J0(u) Jj_(U) u J 0(u) •^(u)

0.0 1.0000 0.0000 3.0 -0.2600 0.3391 0.1 0.9975 0.0499 3.1 -0.2921 0.3009 0.2 0.9900 0.0995 3.2 -0.3202 0.2613 0.3 0.9776 0.1483 3.3 -0.3443 0.2207 0.4 0.9604 0.1960 3.4 -0.3643 0.1792

0.5 0.9385 0.2423 3.5 -0.3801 0.1374 0.6 0.9120 0.2867 3.6 -0.3918 0.0955 0.7 0.8812 0.3290 3.7 -0.3992 0.0538 0.8 0.8463 0.3688 3.8 -0.4026 0.0128 0.9 0.8075 0.4059 3.9 -0.4018 -0.0272

1.0 0.7652 0.4400 4.0 -0.3971 -0.0660 1.1 0.7196 0.4709 4.1 -0.3887 -0.1033 1.2 0.6711 0.4983 4.2 -0.3766 -0.1385 1.3 0.6201 0.5220 4.3 -0.3610 -0.1719 1.4 0.5669 0.5419 4.4 -0.3423 -0.2028

1.5 0.5118 0.5579 4.5 -0.3205 -0.2311 1.6 0.4554 0.5699 4.6 -0.2961 -0.2566 1.7 0.3980 0.5778 4.7 -0.2693 -0.2791 1.8 0.3400 0.5815 4.8 -0.2404 -0.2985 1.9 0.2818 0.5812 4.9 -0.2097 -0.3147

2.0 0.2239 0,5767 5.0 -0.1776 -0.3276 2.1 0.1666 0.5683 5.1 -0.1443 -0.3371 2.2 0.1104 0.5560 5.2 -0.1103 -0.3432 2.3 0.0555 0.5399 5.3 -0.0758 -0.3460 2.4 0.0025 0.5202 5.4 -0.0412 -0.3453

2.5 -0.0484 0.4971 5.5 -0.0068 -0.3414 2.6 -0.0968 0.4708 5.6 0.0270 -0.3343 2.7 -0.1424 0.4415 5.7 0.0599 -0.3241 2.8 -0.1850 0.4097 5.8 0.0917 -0.3110 2.9 -0.2243 0.3754 5.9 0.1220 -0.2951

13-9

TABLE 13 .4 .1 . (Continued.)

u J 0(u) Jû) u J 0(u) ^(u) 6.0 0.1506 -0.2767 9.0 -0.0903 0.2453 6.1 0.1773 -0.2559 9.1 -0.1142 0.2324 6.2 0.2017 -0.2329 9.2 -0.1368 0.2174 6.3 0.2238 -0.2081 9.3 -0.1577 0.2004 6.4 0.2433 -0.1816 9.4 -0.1768 0.1816

6.5 0.2601 -0.1538 9.5 -0.1939 0.1613 6.6 0.2740 -0.1250 9.6 -0.2090 0.1395 6.7 0.2851 -0.0953 9.7 -0.2218 0.1166 6.8 0.2931 -0.0652 9.8 -0.2323 0.0928 6.9 0.2981 -0.0349 9.9 -0.2403 0.0634

7.0 0.3001 -0.0047 10.0 -0.2459 0.0435 7.1 0.2991 0.0252 10.1 -0.2490 0.0184 7.2 0.2951 0.0543 10.2 -0.2496 -0.0066 7.3 0.2882 0.0826 10.3 -0.2477 -0.0313 7.4 0.2786 0.1096 10.4 -0.2434 -0.0555

7.5 0.2663 0.1352 10.5 -0.2366 -0.0788 7.6 0.2516 0.1592 10.6 -0.2276 -0.1012 7.7 0.2346 0.1813 10.7 -0.2164 -0.1224 7.8 0.2154 0.2014 10.8 -0.2032 -0.1422 7.9 0.1944 0.2192 10.9 -0.1881 -0.1604

8.0 0.1716 0.2346 11.0 -0.1712 -0.1768 8.1 0.1475 0.2476 11.1 -0.1528 -0.1913 8.2 0.1222 0.2580 11.2 -0.1330 -0.2028 8.3 0.0960 0.2657 11.3 -0.1121 -0.2143 8.4 0.0692 0.2708 11.4 -0.0902 -0.2224

8.5 0.0419 0.2731 11.5 -0.0677 -0.2284 8.6 0.0146 0.2728 11.6 -0.0446 -0.2320 8.7 -0.0125 0.2697 11.7 -0.0213 -0.2333 8.8 -0.0392 0.2641 11.8 0.0020 -0.2323 8.9 -0.0652 0.2559 11.9 0.0250 -0.2290

13-10

TABLE 1 3 . 4 . 1 . (Continued.)

u J 0(u) Jû) u J 0(u) Jx(u)

12.0 0.0477 -0.2234 13.6 0.2101 0.0590 12.1 0.0697 -0.2158 13.7 0.2032 0.0791 12.2 0.0908 -0.2060 13.8 0.1943 0.0984 12.3 0.1108 -0.1943 13.9 0.1836 0.1165 12.4 0.1296 -0.1807 14.0 0.1711 0.1334

12.5 0.1469 -0.1655 14.1 0.1570 0.1488 12.6 0.1626 -0.1487 14.2 0.1414 0.1626 12.7 0.1766 -0.1307 14.3 0.1245 0.1747 12.8 0.1887 -0.1114 14.4 0.1065 0.1850 12.9 0.1988 -0.0912 14.5 0.0875 0.1934

13.0 0.2069 -0.0703 14.6 0.0679 0.1998 13.1 0.2129 -0.0488 14.7 0.0476 0.2043 13.2 0.2167 -0.0271 14.8 0.0271 0.2066 13.3 0.2183 -0.0052 14.9 0.0064 0.2069 13.4 0.2177 0.0166 15.0 0.0142 0.2051

13.5 0.2150 0.0380

13-11

TABLE 13.4.2. Zero and first-order Bessel functions of the second kind (source: Ref. 2 p. 371-3).

u Y0(u> Yx(u) u V u ) ^(u)

0.0 -co -CO 3.0 0.3768 0.3247 0.1 -1.5342 -6.4590 3.1 0.3431 0.3496 0.2 -1.0811 -3.3238 3.2 0.3070 0.3707 0.3 0.8073 -2.2931 3.3 0.2691 0.3878 0.4 -0.6060 -1.7809 3.4 0.2296 0.4010

0.5 -0.4445 -1.4715 3.5 0.1890 0.4102 0.6 -0.3085 -1.2604 3.6 0.1477 0.4154 0.7 -0.1907 -0.1032 3.7 0.1061 0.4167 0.8 -0.0868 -0.9781 3.8 0.0645 0.4141 0.9 0.0056 -0.8731 3.9 0.0234 0.4078

1.0 0.0883 -0.7812 4.0 -0.0169 0.3979 1,1 0.1622 -0.6981 4.1 -0.0561 0.3846 1.2 0.2281 -0.6211 4.2 -0.0938 0.3680 1.3 0.2865 -0.5485 4.3 -0.1296 0.3484 1.4 0.3379 -0.4791 4.4 -0.1633 0.3260

1.5 0.3824 -0.4123 4.5 -0.1947 0.3010 1.6 0.4204 -0.3476 4.6 -0.2235 0.2737 1.7 0.4520 -0.2847 4.7 -0.2494 0.2445 1.8 0.4774 -0.2237 4.8 -0.2723 0.2136 1.9 0.4968 -0.1644 4.9 -0.2921 0.1812

2.0 0.5104 -0.1070 5.0 -0.3085 0.1479 2.1 0.5183 -0.0517 5.1 -0.3216 0.1137 2.2 0.5208 0.0015 5.2 -0.3312 0.0792 2.3 0.5181 0.0523 5.3 -0.3374 0.0445 2.4 0.5104 0.1005 5.4 -0.3402 0.0101

2.5 0.4981 0.1459 5.5 -0.3395 -0.0238 2.6 0.4813 0.1884 5.6 -0.3354 -0.0568 2.7 0.4605 0.2276 5.7 -0.3282 -0.0887 2.8 0.4359 0.2635 5.8 -0.3177 -0.1192 2.9 0.4079 0.2959 5.9 -0.3044 -0.1481

13-12

TABLE 13.4.2. (Continued.)

u VQ(u) V x(u) u Y 0(u) ^(u)

6.0 -0.2882 -0.1750 9.0 0.2499 0.1043 6.1 -0.2694 -0.1998 9.1 0.2383 0.1275 6.2 -0.2483 -0.2223 9.2 0.2245 0.1491 6.3 -0.2251 -0.2422 9.3 0.2086 0.1691 6.4 -0.2000 -0.2596 9.4 0.1907 0.1871

6.5 -0.1732 -0.2741 9.5 0.1712 0.2032 6.6 -0.1452 -0.2857 9.6 0.1502 0.2171 6.7 -0.1162 -0.2945 9.7 0.1279 0.2287 6.8 -0.0864 -0.3002 9.8 0.1045 0.2379 6.9 -0.0562 -0.3029 9.9 0.0804 0.2447

7.0 -0.0260 -0.3027 10.0 0.0557 0.2490 7.1 0.0042 -0.2995 10.1 0.0307 0.2508 7.2 0.0338 -0.2934 10.2 0.0056 0.2502 7.3 0.0628 -0.2846 10.3 -0.0193 0.2471 7.4 0.0907 -0.2731 10.4 -0.0437 0.2416

7.5 0.1173 -0.2591 10.5 -0.0675 0.2337 7.6 0.1424 -0.2428 10.6 -0.0904 0.2236 7.7 0.1658 -0.2243 10.7 -0.1122 0.2114 7.8 0.1872 -0.2039 10.8 -0.1326 0.1973 7.9 0.2065 -0.1817 10.9 -0.1516 0.1813

8.0 0.2235 -0.1581 11.0 -0.1688 0.1637 8.1 0.2381 -0.1332 11.1 -0.1843 0.1446 8.2 0.2501 -0.1072 11.2 -0.1977 0.1243 8.3 0.2595 -0.0806 11.3 -0.2091 0.1029 8.4 0.2662 -0.0535 11.4 -0.2183 0.0807

8.5 0.2702 -0.0262 11.5 -0.2252 0.0579 8.6 0.2715 0.0011 11.6 -0.2299 0.0348 8.7 0.2700 0.0280 11.7 -0.2322 0.0114 8.8 0.2659 0.0544 11.8 -0.2322 -0.0118 8.9 0.2592 0.0799 11.9 -0.2298 -0.0347

13-13

TABLE 13.4.2. (Continued.)

_u * 0(u) Yx(u)

12.0 -0.2252 -0.0571 12.1 -0.2184 -0.0787 12.2 -0.2095 -0.0994 12.3 -0.1986 -0.1190 12.4 -0.1858 -0.1371

12.5 -0.1712 -0.1538 12.6 -0.1551 -0.1689 12.7 -0.1375 -0.1821 12.8 -0.1187 -0.1935 12.9 -0.0989 -0.2028

13.0 -0.0782 -0.2101 13.1 -0.0569 -0.2152 13.2 -0.0352 -0.2182 13.3 -0.0134 -0.2190 13.4 0.0085 -0.2176

u Y 0(u) Y x(u)

13.5 0.0301 -0.2140 13.6 0.0512 -0.2084 13.7 0.0717 -0.2007 13.8 0.0913 -0.1912 13.9 0.1099 -0.1798

14.0 0.1272 -0.1666 14.1 0.1431 -0.1520 14.2 0.1575 -0.1359 14.3 0.1703 -0.1186 14.4 0.1812 -0.1003

14.5 0.1903 -0.0810 14.6 0.1974 -0.0612 14.7 0.2025 -0.0408 14.8 0.2056 -0.0202 14.9 0.2066 0.0005

13-14

TABLE 1 3 . 4 . 3 . Zero and f i r s t order modified Bessel functions of the f i r s t kind

(source: Ref. 2, p . 373-4) .

u I 0 ( u ) I ^ u ) u I 0 ( u ) 3^(11)

0 . 0 l.ÔOO 0 .0000 3 . 0 4.881 3.953

0 . 1 1.0025 0 .0501 3 . 1 5.294 4.326

0 .2 1.0100 0 .1005 3 . 2 5.747 4.734

0 .3 1.0226 0 .1517 3 .3 6.243 5 . 1 8 1

0.4 1.0404 0 .2040 3 .4 6.785 5.670

0 .5 1.0635 0 .2579 3 .5 7.378 6.206

0 .6 1.0920 0.3137 3 .6 8.028 6 .793

0 .7 1.1263 0 .3719 3.7 8.739 7.436 0 . 8 1.1665 0 .4329 3 . 8 9.517 8.140

0 .9 1.2130 0 .4971 3 . 9 10.369 8.913

1.0 1.2661 0 .5652 4 .0 11.30 9 .76

1 .1 1.3262 0 .6375 4 . 1 12.32 10 .69

1.2 1.3937 0 .7147 4 .2 13.44 11 .71

1.3 1.4693 0 .7973 4 .3 14.67 12 .82

1.4 1.5534 0 .8861 4 .4 16 .01 14 .05

1.5 1.6467 0 .9817 4 .5 17 .48 15 .39

1.6 1.7500 1.0848 4 . 6 19.09 16 .86

1.7 0.8640 1.1963 4 .7 20.86 1 8 . 4 8

1 .8 1.9806 1 .3172 4 . 8 22.79 20 .25

1.9 2.1277 1 .4482 4 . 9 24 .91 22 .20

2 .0 2.280 1 .591 5 .0 27 .24 24 .34

2 . 1 2.446 1 .746 5 . 1 29.79 26 .68

2 . 2 2.629 1 .914 5 . 2 32.58 29 .25

2 .3 2 .830 2 .098 5 .3 35.65 32 .08

2 . 4 3 .049 2 .298 5 . 4 39 .01 35 .18

2 .5 3 .290 2 .517 5 . 5 42.70 38 .59

2 .6 3.553 2 .755 5 .6 46.74 42 .33

2.7 3 .842 3 .016 5 .7 51.17 46 .44

2 .8 4.157 3 . 3 0 1 5 . 8 56.04 50.95

2 .9 4.503 3 .613 5 . 9 61.38 55 .90

13-15

TABLE 13 .4 .4 . Zero and f i r s t order modified Bessel functions of the second kind (source: Ref. 2, p . 374-5) .

u iVu> K ( u > u ¥oM K ( u )

0 . 0 CO CO 2 . 0 0.072 0.089 0 . 1 1.545 6.270 2 . 1 0.064 0.078 0 .2 1.116 3.040 2 . 2 0.057 0.069 0.3 0.874 1.946 2 .3 0.050 0.060 0 . 4 0.710 1.391 2 . 4 0.045 0.053

0 . 5 0.588 1.054 2 . 5 0.040 0.047 0 . 6 0.495 0.829 2 . 6 0.035 0.042 0.7 0.420- 0.669 2 .7 0.031 0.037 0 . 8 0.360 0.549 2 . 8 0.028 0.032 0 . 9 0.310 0.456 2 . 9 0.025 0.029

1.0 0.268 0.383 3 . 0 0.022 0.026 1 .1 0.233 0.324 3 . 1 0.020 0.023 1.2 0.203 0.277 3 . 2 0.018 0.020 1.3 0.177 0.237 3 . 3 0.016 0.018 1.4 0.155 0.204 3 . 4 0.014 0.016

1.5 0.136 0.177 3 . 5 0.012 0.014 1 .6 0.120 0.153 3 . 6 0.011 0.013 1.7 0.105 0.133 3 . 7 0.010 0.011 1.8 0.093 0.116 3 . 8 0.009 0.010 1 . 9 0.082 0.102 3 . 9 0.008 0.009

13-16

13.5. Legendre Polynomials

The Legendre polynomial of degree n, of the first kind:

P (u) = ( - i ) n / 2 l • 3 ' 5 — < n - X ) Ti - HUU-1L U 2 n* ' l ; 2 • 4 • 6 ••• n I 21

n(n - 2)(n + 1} (n + 3) 4 , "l „ „ , + — i '•' ,. , ' ' - u + , , " , n = 2, 4, 6, .... 41 -1

n , , , l t(n-l)/2 1 » 3 • 5 •" n r (n - 1) (n + 2) 3 V U ) = »- X ) 2 • 4 • 6 — (n - 1) L u " 31 u

+ {n-l){n-3nn + 2){n + A) u 5 + . . . . " J # n „ ^ ^ ? | u ] < 1

P Q(u) = 1, Px(u) = u, P 2(u) = (3u2 - l)/2 ,

P (u) = (5u3 - 3u)/2r P (u) = P5u 4 - 30u 2 + 3)/8 ,

P 5<u) = (63u5 - 70u 3 + 15u)/8, |u|<l

The Legendre polynomial of degree n, of the second kind:

Qn(U, = (-i, < n + i> / 2 ^ . v . 7 . i ? - n ^ - [i - afcji-y- u2

+ n(n - 2) ( n ^ 1) ( , + 3) fl4 + .„.-] , n = 3 , 5 , 7 , . . . ,

O ( u , = ( _ i ) n / 2 ^ • i • 6 — a r _ <n - l)<n + 2) 3 Unvuj \ i.) l • 3 • 5 ••• (n - 1) L u 31 u

a. (n - 1) (n - 3) (n + 2) (n + 4) 5 . 1 . , ,

13-17

0 0 W) - i Q±W = QQ(u) Pû) - 1,

Q2(u) = QQ(u) P 2 (u) - 3u/2

Q3(u) » QQ(u) P 3 (u) - 5u 2 /2 + § ,

Q4('J) = QQ(u) P 4 (u) - 35u 3/8 + 55u 2/24 ,

Qn(u) = Q0(U) Pn(u, - - f - r - i l p n _ i ( u ) _ 122^ |L P n _ 3 ( u )

13-18

TABLE 1J .5 . The f i r s t five Legendre polynomials of the f i r s t kind (source: Ref. 2, p . 375-7).

u ^(u) P2(u) P3(u) P 4(u) P5(u)

0.00 0.0000 -0.5000 0.0000 0.3750 0.0000 0.01 0.0100 -0.4998 -0.0150 0.3746 0.0187 0.02 0.0200 -0.4994 -0.0300 0.3735 0,0374 0.03 0.0300 -0.4986 -0.0449 0.3716 0.0560 0.04 0.0400 -0.4976 -0.0598 0.3690 0.0744

0.05 0.0500 -0.4962 -0.0747 0.3657 0.0927 0.06 0.0600 -0.4946 -0.0895 0.3616 0.1106 0.07 0.0700 -0.4926 -0.1041 0.3567 0.1283 0.08 0.0800 -0.4904 -0.1187 0.3512 0.1455 0.09 0.0900 -0.4878 -0.1332 0.3449 0.1624

0.10 0.1000 -0.4850 -0.1475 0.3379 0.1788 0.11 0.1100 -0.4818 -0.1617 0.3303 0.1947 0.12 0.1200 -0.4784 -0.1757 0.3219 0.2101 0.13 0.1300 -0.4746 -0.1895 0.3129 0.2248 0.14 0.1400 -0.4906 -0.2031 0.3032 0.2389

0.15 0.1500 -0.4662 -0.2166 0.2928 0.2523 0.16 0.1600 -0.4616 -0.2:.98 0.2819 0.2650 0.17 0.1700 -0.4566 -0.2427 0.2703 0.2769 0.18 0.1800 -0.4514 -0.2554 0.2581 0.2880 0.19 0.1900 -0.4458 -0.2679 0.2453 0.2982

0.20 0.2000 -0.4400 -0.2800 0.2320 0.3075 0.21 0.2100 -0.4338 -0.2918 0.2181 0.3159 0.22 0.2300 -0.4274 -0.3034 0.2037 0.3234 0.23 0.2300 -0.4206 -0.3146 0.1889 0.3299 0.24 0.2400 -0.4136 -0.3254 0.1735 0.3353

0.25 0.2500 -0.4062 -0.3359 0.1577 0.3397 0.26 0.2600 -0.3986 -0.3461 0.1415 0.3431 0.27 0.2700 -0.3906 -0.3558 0.1249 0.3453 0.28 0.2800 -0.3824 -0.3651 0.1079 0.3465 0.29 0.2900 -0.3738 -0.3740 0.0906 0.3465

13-19

TABLE 13.5 . (Continued.)

u ^(u) P2(u) P3(u) P4(") P 5(u)

0.30 0.3000 -0.3650 -0.3825 0.0729 0.3454 0.31 0.3100 -0.3558 -0.3905 0.0550 0.3431 0.32 0.3200 -0.3464 -0.3981 0.0369 0.3397 0.33 0.3300 -0.3366 -0.4052 0.0185 0.3351 0.34 0.3400 -0.3266 -0.4117 0.0000 0.3294

0.35 0.3500 -0.3162 -0.4178 -0.0187 0.3225 0.36 0.3600 -0.3056 -0.4234 -0.0375 0.3144 0.37 0.3700 -0.2946 -0.4284 -0.0564 0.3051 0.38 0.3800 -0.2834 -0.4328 -0.075? 0.2948 0.39 0.3900 -0.2718 -0.4367 -0.0942 0.2833

0.40 0.4000 -0.2600 -0.4400 -0.1130 0.2706 0.41 0.4100 -0.2478 -0.4427 -0.1317 0.2569 0.42 0.4200 -0.2354 -0.4448 -0.1504 0.2421 0.43 0.4300 -0.2226 -0.4462 -0.1688 0.2263 0.44 0.4400 -0.2096 -0.4470 -0.1870 0.2095

0.45 0.4500 -0.1962 -0.4472 -0.2050 0.1917 0.46 0.4600 -0.1826 -0.4467 -0.2226 0.1730 0.47 0.4700 -0.1686 -0.4454 -0.2399 0.1534 0.48 0.4800 -0.1544 -0.4435 -0.2568 0.1330 0.49 0.4900 -0.1398 -0.4409 -0.2732 0.1118

0.50 0.5000 -0.1250 -0.4375 -0.2891 0.0898 0.51 0.5100 -0.1098 -0.4334 -0.3044 0.0673 0.52 0.5200 -0.0944 -0.4285 -0.3191 0.0441 0.53 0.5300 -0.0786 -0.4228 -0.3332 0.0204 0.54 0.5400 -0.0626 -0.4163 -0.3465 -0.0037

0.55 0.5500 -0.0462 -0.4091 -0.3590 -0.0282 0.56 0.5600 -0.0296 -0.4010 -0.3707 -0.0529 0.57 0.5700 -0.0126 -0.3920 -0.3815 -0.0779 0.58 0.5800 0.0046 -0.3822 -0.3914 -0.1028 0.59 0.5900 0.0222 -0.3716 -0.4002 -0.1278

13-20

TABLE 13.5. (Continued.)

u P-û) P2(u)

0.60 0.6000 0.0400 0.61 0.6100 0.0582 0.62 0.6200 0.0766 0.63 0.6300 0.0954 0.64 0.6400 0.1144

0.65 0.6500 0.1338 0.66 0.6600 0.1534 0.67 0.6700 0.1734 0.6B 0.6800 0.1936 0.69 0.6900 0.2142

0.70 0.7000 0.2350 0.71 0.7100 0.2562 0.72 0.7200 0.277 6 0.73 0.7300 0.2994 0.74 0.7400 0.3214

0.75 0.7500 0.3438 0.76 0.7600 0.3664 0.77 0.7700 0.3894 0.78 0.7800 0.4126 0.79 0.7900 0.4362

0.80 0.8000 0.4600 0.81 0.8100 0.4842 0.82 0.8200 0.5086 0.83 0.8300 0.5334 0.84 0.8400 0.5584

0.85 0.8500 0.5838 0.86 0.8600 0.6094 0.87 0.8700 0.6354 0.88 0.8800 0.6616 0.89 0.8900 0.6882

P 3(u) P 4(u) P5(u)

-0.3600 -0.4080 -0.1526 -0.3475 -0.4146 -0.1772 -0.3342 -0.4200 -0.2014 -0.3199 -0.4242 -0.2251 -0.3046 -0.4270 -0.2482

-0.2884 -0.4284 -0.2705 -0.2713 -0.4284 -0.2919 -0.2531 -0.4268 -0.3122 -0.2339 -0.4236 -0.3313 -0.2137 -0.4187 -0.3490

-0.1925 -0.4121 -0.3652 -0.1702 -0.4036 -0.3796 -0.1469 -0.3933 -0.3922 -0.1225 -0.3810 -0.4026 -0.0969 -0.3666 -0.4107

-0.0703 -0.3501 -0.4164 -0.0426 -0.3314 -0.4193 -0.0137 -0.3104 -0.4193 0.0164 -0.2871 -0.4162 0.0476 -0.2613 -0.4097

0.0800 -0.2330 -0.3995 0.1136 -0.2021 -0.3855 0.1484 -0.1685 -0.3674 0.1845 -0.1321 -0.3449 0.2218 -0.0928 -0.3177

0.2603 -0.0506 -0.2857 0.3001 -0.0053 -0.2484 0.3413 0.0431 -0.2056 0.3837 0.0947 -0.1570 0.4274 0.1496 -0.1023

13-21


u P x(u) P2(u)

0.90 0.9000 0.7150 0.91 0.9100 0.7422 0.92 0.9200 0.7696 0.93 0.9300 0.7974 0.94 0.9400 0.8254

0.95 0.9500 0.8538 0.96 0.9600 0.8B24 0.97 0.9700 0.9114 0.98 0.9800 0.9406 0.99 0.9900 0.9702 1.00 1.0000 1.0000

P 3(u) P 4(u) P 5(u)

0.4725 0.2079 -0.0411 0.5189 0.2698 0.0268 0.5667 0.3352 0.1017 0.6159 0.4044 0.1842 0.6665 0.4773 0.2744

0.7184 0.5541 0.3727 0.7718 0.6349 0.4796 0.8267 0.7198 0.5954 0.8830 0.8089 0.7204 0.9407 0.9022 0.8552 1.0000 1.0000 1.0000

13-22

13.6. Sine, Cosine, and Exponential Integrals. - X

Sine i n t e g ra l : Si (x)

Si(<=°) = IT/2

/ "¥**••

Cosine integral: Ci(x) = - / c o s ^ du =i n(yx) - j 1

0

An Y = 0.577215... (Euler's constant)

Exponential i n t e g r a l : Ei(x) = - / —— du, <0, °° > x > 0 x

CO

/ ^ du, <0,

*i(x) = J Logarithmic i n t e g r a l : Si (x) = / . " •• = E i ( i i x)

0

«,i(e x) = Ei(x)

FOE small values of x:

Si(x) = x,

JU(x) = -x /£n( l /x)

Ci(x) = Ei(-x) = Ei(x) = Jlnd/pc) = Y + An(x) - x + ( x 2 / 4 ) +

For l a r g e values of x:

Si(x) = TT/2 - cos (x) /x

Ci(x) = s in (x)/x

I i ( x ) = (e X /x) (1 + 11/x + 2 i /x + 3!/x + '•'), | x | » 1

13-23

TABLE 1 2 . 6 . Values of the s i n e , c o s i n e , l o g a r i t h m i c , and e x p o n e n t i a l i n t e g r a l s .

( source : Ref. 18 , p . 6 - 9 . )

S i ( x ) Ci(x) Ei(x) E i ( - x )

0.00 +0.000000 -CO —CO

0.01 +0.010000 -4.0280 -4.0179 0.02 +0.019999 -3.3349 -3.3147 0.03 +0.029998 -2.9296 -2.8991 0.04 +0.039996 -2.6421 -2.6013

0.05 +0.04999 -2.4191 -2.3679 0.06 +0.05999 -2.2371 -2.1753 0.07 +0.06998 -2.0833 -2.0108 0.08 +0.07997 -1.9501 -1.8669 0.09 +0.08996 -1.8328 -1.7387

0.10 +0.09994 -1.7279 -1.6228 0.11 +0.10993 -1.6331 -1.5170 0.12 +0.11990 -1.5466 -1.4193 0.13 +0.12988 -1.4672 -1.3287 0.14 +0.13985 -1.3938 -1.2438

0.15 +0.14981 -1.3255 -1.1641 0.16 +0.15977 -1.2618 -1.0887 0.17 +0.16973 -1.2020 -1.0172 0.18 +0.1797 -1.1457 -0.9491 0.19 +0.1896 -1.0925 -0.8841

0.20 +0.1996 -1.0422 -0.8218 0.21 +0.2095 -0.9944 -0.7619 0.22 +0.2194 -0.9490 -0.7042 0.23 +0.2293 -0.9057 -0.6485 0.24 +0.2392 -0.8643 -0.5947

0.25 +0.2491 -0.8247 -0.5425 0.26 +0.2590 -0.7867 -0.4919 0.27 +0.2689 -0.7503 -0.4427 0.28 +0.2788 0.7153 -0.3949 0.29 +0.2886 -0.6816 -0.3482

- C O

-4 .0379

-3 .3547

- 2 . 9 5 9 1

-2 .6813

- 2 . 4 6 7 9

-2 .2953

- 2 . 1 5 0 8

-2 .0269

-1 .9187

- 1 . 8 2 2 9

- 1 . 7 3 7 1

- 1 . 6 5 9 5

-1 .5889

- 1 . 5 2 4 1

- 1 . 4 6 4 5

- 1 . 4 0 9 2

- 1 . 3 5 7 8

- 1 . 3 0 9 8

- 1 . 2 6 4 9

- 1 . 2 2 2 7

-1 .1829

- 1 . 1 4 5 4

-1 .1099

- 1 . 0 7 6 2

-1 .0443

- 1 . 0 1 3 9

-0 .9849

-0 .9573

-0 .9309

13-24


X Si(x) Ci(x) Ei(x) Ei(-x)

0.30 +0.2985 -0.6492 -0.3027 -0.9057 0.31 +0.3083 -0.6179 -0.2582 -0.8815 0.32 +0.3182 -0.5877 -0.2147 -0.8583 0.33 +0.3280 -0.5585 -0.17210 -0.8361 0.34 +0.3378 -0.5304 -0.13036 -0.8147

0.35 +0.3476 -0.5031 -0.08943 -0.7942 0.3S +0.3574 -0.4767 -0.04926 -0.7745 0.37 +0.3672 -0.4511 -0.00979 -0.7554 0.33 +0.3770 -Q.4263 +0.02901 -0.7371 0.39 +0.3867 -0.4022 +0.06718 -0.7194

0.40 +0.3965 -0.3788 +0.10477 -0.7024 0.41 +0.4062 -0.3561 +0.14179 -0.6859 0.42 +0.4159 -0.3341 +0.17828 -0.6700 0.43 +0.4256 -0.3126 +0.2143 -0.6546 0.44 +0.4353 -0.2918 +0.2498 -0.6397

0.45 +0.4450 -0.2715 +0.2849 -0.6253 0.46 +0.4546 -0.2517 +0.3195 -0.6114 0.47 +0.4643 -0.2325 +0.3537 -0.5979 0.48 +0.4739 -0.2138 +0.3876 -0.5848 0.49 +0.4835 -0.1956 +0.4211 -0.5721

0.50 +0.4931 -0.17778 +0.4542 -0.5598 0.51 +0.5027 -0.16045 +0.4870 -0.5478 0.52 +0.5123 -0.14355 +0.5195 -0.5362 0.53 +0.5218 -0.12707 +0.5517 -0.5250 0.54 +0.5313 -0.11099 +0.5836 -0.5140

0.55 +0.5408 -0.09530 +0.6153 -0.5034 0.56 +0.5503 -0.07999 +0.6467 -0.4930 0.57 +0.5598 -0.06504 +0.6778 -0.4830 0.58 +0.5693 -0.05044 +0.7087 -0.4732 0.59 +0.5787 -0.03619 +0.7394 -0.4636

13-25


Si(x) Ci(x) Ei(x) Ei(-x)

0.60 +0.5881 -0.02227 +0.7699 0.61 +0.5975 -0.008675 +0.8002 0.62 +0.6069 +0.004606 +0.8302 0.63 +0.6163 +0.01758 +0.8601 0.64 +0.6256 r0.03026 +0.8898

0.65 +0.6349 +0.04265 +0.9194 0.66 +0.6442 +0.05476 +0.9488 0.67 +0.6535 +0.06659 +0.9780 0.68 +0.6628 +0.07816 +1.0071 0.69 +0.6720 +0.08946 +1.0361

0.70 +0.6812 +0.10051 +1.0649 0.71 +0.6904 +0.11132 +1.0936 0.72 +0.6996 +0.12188 +1.1222 0.73 +0.7087 +0.13220 +1.1507 0.74 +0.7179 +0.14230 +1.1791

0.75 +0.7270 +0.15216 +1.2073 0.76 +0.7360 +0.16181 +1.2355 0.77 +0.7451 +0.17124 +1.2636 0.78 +0.7541 +0.1805 +1.2916 0.79 +0.7631 +0.1895 +1.3195

0.80 +0.7721 +0.1983 +1.3474 0.81 +0.7811 +0.2069 +1.3752 0.82 +0.7900 +0.2153 +1.4029 0.83 +0.7989 +0.2235 +1.4306 0.84 +0.8078 +0.2316 +1.4582

0.85 +0.8166 +0.2394 +1.4857 0.86 +0.8254 +0.2471 +1.5132 0.87 +0.8342 +0.2546 +1.5407 0.88 +0.8430 +0.2619 +1.5681 0.89 +0.8518 +0.2691 +1.5955

-0.4544

-0.4454

-0.4366

-0.4280

-0.4197

-0.4115

-0.4036

-0.3959

-0.3883

-0.3810

-0.3738

-0.3668

-0.3599

-0.3532

-0.3467

-0.3403

-0.3341

-0.3280

-0.3221

-0.3163

-0.3106

-0.3050

-0.2996

-0.2943

-0.2891

-0.2840

-0.2790

-0.2742

-0.2694

-0.2647

13-26


X Si(x) Ci(Jt) Ei(x) Ei(-x)

0.90 +0.8605 +0.2761 +1.6228 -0.2602 0.91 +0.8692 +0.2829 +1.6501 -0.2557 0.92 +0.8778 +0.2896 +1.6774 -0.2513 0.93 +0.8865 +0.2961 +1.7047 -0.2470 0.94 +0.8951 +0.3024 +1.7319 -0.2429

0.95 +0.9036 +0.3086 +1.7591 -0.2387 0.96 +0.9122 +0.3147 +1.7864 -0.2347 0.97 +0.9207 +0.3205 +1.8136 -0.2308 0.98 +0.9292 +0.3263 +1.8407 -0.2269 0.99 +0.9377 +0.3319 +1.8679 -0.2231 1.00 +0.9461 +0.3374 +1.8951 -0.2194

1.0 +0.9461 +0.3374 +1.8951 -0.2194 1.1 +1.0287 +0.3849 +2.1674 -0.1860 1.2 +1.1080 +0.4205 +2.4421 -0.1584 1.3 +1.1840 +0.4457 +2.7214 -0.1355 1.4 +1.2562 +0.4620 +3.0072 -0.1162 1.5 +1.3247 +0.4704 +3.3013 -0.1000

1.6 +1.3892 +0.4717 +3.6053 -0.08631 1.7 +1.4496 +0.4670 +3.9210 -0.07465 1.8 +1.5058 +0.4568 +4.2499 -0.06471 1.9 +1.5578 +0.4419 +4.5937 -0.05620 2.0 +1.6054 +0.4230 +4.9542 -0.04890

2.1 +1.6487 +0.4005 +5.3332 -0.04261 2.2 +1.6876 +0.3751 +5.7326 -0.03719 2.3 +1.7222 +0.3472 +6.1544 -0.03250 2.4 +1.7525 +0.3173 +6.6007 -0.02844 2.5 +1.7785 +0.2859 +7.0738 -0.02491

2.6 +1.8004 +0.2533 +7.5761 -0.02185 2.7 +1.8182 +0.2201 +8.1103 -0.01918 2.8 +1.8321 +0.1865 +8.6793 -0.01686 2.9 +1.8422 +0.1529 +9.2860 -0.01482 3.0 +1.8487 +0.1196 +9.9338 -0.01304

13-27


x Si(x) Ci(x) IT(x) Ei (-x)

3.1 +1.8517 +0.08699 +10.6263 -0.01149 3.2 +1.8514 +0.05526 +11.3673 -0.01013 3.3 +1.8481 +0.02468 +12.1610 -0.028939 3.4 +1.8419 +0.004518 +13.0121 -0.027890 3.5 +1.8331 -0.03213 +13.9254 -0.026970

3,6 +1.8219 -0.05797 +14.9063 -0.026160 3.7 +1.8086 -0.08190 +15.9606 -0.025448 3.8 +1.7934 -0.1038 +17.0948 -0.024820 3.9 +1.7765 -0.1235 +18.3157 -0.024267 4.0 +1.7582 -0.1410 +19.6309 -0.023779

4.1 +1.738T -0.1562 +21.0485 -0.023349 4.2 +1.7184 -0.1690 +22.5774 -0.022969 4.3 +1.6973 -0.1795 +24.2274 -0.022633 4.4 +1.6758 -0.1877 +26.0090 -0.022336 4.5 +1.6541 -0.1935 +27.9337 -0.022073

4.6 +1.6325 -0.1970 +30.0141 -0.021841 4.7 +1.6110 -0.1984 +32.2639 -0.021635 4.8 +1.5900 -0.1976 +34.6979 -0.021453 4.9 +1.5696 -0.1948 +37.3325 -0.021291 5.0 +1.5499 -0.1900 +40.1853 -0.021148

6 +1.4247 -0.06806 +85.9898 -0.033601 7 +1.4546 +0.07670 +191.505 -0.031155 8 +1.5742 +0.1224 +440.380 -0.043767 9 +1.6650 +0.05535 +1037.88 -0.041245 10 +1.6583 -0.04546 +2492.23 -0.054157

11 +1.5783 -0.08956 +6071.41 -0.051400 12 +1.5050 -0.04978 +14959.5 -0.064751 13 +1.4994 +0.02676 +37197.7 -0.061622 14 +1.5562 +0.06940 +93192.5 -0.075566 15 +1.6182 +0.0462B +234 956 -0.071918

13-28

TABLE 1 3 . 6 . (Continued.)

X Si(x) Ci(x) X Si(x) Ci(x)

20 +1.5482 +0.04442 140 +1.5722 +0.007011 25 +1.5315 -0.00685 150 +1.5662 -0.004800 30 +1.5668 -0.03303 160 +1.5769 +0.001409 35 + 1.5969 -0.01148 170 +1.5653 +0.002010 40 +1.5870 +0.01902 180 +1.5741 -0.004432

45 +1.5587 +0.01863 190 +1.5704 +0.005250 50 +1.5516 -0.00563 200 +1.5684 -0.004378 55 +1.5707 -0.01817 300 +1.5709 -0.003332 60 +1.5867 -0.00481 400 +1.5721 -0.002124 65 +1.5792 +0.01285 500 +1.5726 -0.0009320

70 +1.5616 +0.01092 600 +1.5725 +0.0000764 75 +1.5586 -0,00533 700 +1.5720 +0.0007788 80 +1.5723 -0.01240 800 +1.5714 +0.001118 85 +1.5824 -0.001935 900 +1.5707 +0.001109 90 +1.5757 +0.009986 10 2 +1.5702 +0.000826

95 +1.5630 +0.007110 10 4 +1.5709 -0.0000306 100 +1.5622 -0.005149 10 5 +1.5708 +0.0000004 110 +1.5799 -0.000320 106 +1.5708 -0.0000004 120 +1.5640 +0.004781 10 7 +1.5708 +0.0 130 +1.5737 -0.007132 00 1/2TT 0.0

13-29

SECTION 14. ROOTS OF SOME CHARACTERISTIC EQUATIOHS

TABLE 14.1. First six roots of X tan X = C. (Source: Ref. 74, p. 217.)

c Xl X2 X3 X4 X5 X6 0 0.0000 3.1416 6.2832 9.4248 12.5664 15.7080 0.001 0.0316 3.1419 6.2833 9.4249 12.5665 15.7080 0.002 0.0447 3.1422 6.2835 9.4250 12.5665 15.7081 0.004 0.0632 3.1429 6.2838 9.4252 12.5667 15.7082 0.006 0.0774 3.1435 6.2841 9.4254 12.5668 15.7083 0.008 0.0893 3.1441 6.2845 9.4256 12.5670 15.7085 0.01 0.0998 3.1448 6.2848 9.4258 12.5672 15.7096 0.02 0.1410 3.1479 6.2864 9.4269 12.5680 15.7092 0.04 0.1987 3.1543 6.2895 9.4290 12.5696 15.7105 0.06 0.2425 3.1606 6.2927 9.4311 12.5711 15.7118 0.08 0.2791 3.1668 6.2959 9.4333 12.5727 15.7131 0.1 0.0311 3.1731 6.2991 9.4354 12.5743 15.7143 0.2 0.4328 3.2039 6.3148 9.4459 12.5823 15.7207 0,3 0.5218 3.2341 6.3305 9.4565 12.5902 15.7270 0.4 0.5932 3.2636 6.3461 9.4670 12.5981 15.7334 0.5 0.6533 3.2923 6.3616 9.4775 12.6060 15.7397 0.6 0.7051 3.3204 6.3770 9.4879 12.6139 15.7460 0.7 0.7506 3.3477 6.3923 9.4983 12.6218 15.7524 0.8 0.7910 3.3744 6.4074 9.5087 12.6296 15.7587 0.9 0.8274 3.4003 6.4224 9.5190 12.6375 15.7650 1.0 0.8603 3.4256 6.4373 9.5293 12.6453 15.7713 1.5 0.9882 3.5422 6.5097 9.5801 12.6841 15.8026 2.0 1.0769 3.6436 6.5783 9.6296 12.7223 15.8336 3.0 1.1925 3.8088 6.7040 9.7240 12.7966 15.8945 4.0 1.2646 3.9352 6.8140 9.8119 12.8678 15.9536 5.0 1.3138 4.0336 6.9096 9.8928 12.9352 16.0107 6.0 1.3496 4.1116 6.9924 9.9667 12.9988 16.0654 7.0 1.3766 4.1746 7.0640 10.0339 13.0584 16.1177 8.0 1.3978 4.2264 7.1263 10.0949 13.1141 16.1675 9.0 1.4149 4.2694 7.1806 10.1502 13.1660 16.2147 10.0 1.4289 4.3058 7.2281

14-1

10.2003 13.2142 16.2594

TABLE 14 .1 . (Contii)ued.)

X 2

15.0 1.4729 4.4255 7.3959 10.3898 13.4078 16.4474 20.0 1.4961 4.4915 7.4954 10.5117 13.5420 16.5864 30.0 1.5202 4.5615 7.6057 10.6543 13.7085 16.7691 40.0 1.5325 4.5979 7.6647 10.7334 13.8048 16.8794 50.0 1.5400 4.6202 7.7012 1C.7832 13.8666 16.9519 60.0 1.5451 4.6353 7.7259 10.8172 13.9094 17.0026 80.0 1.5514 4.6543 7.7573 10.8606 13.9644 17.0686 100.0 1.5552 4.6658 7.7764 10.8871 13.9981 17.1093

CO 1.5708 4.7124 7.8540 10.9956 14.1372 17.2788

14-2

TABLE 14.2 . F i r s t five roo t s of 1 -• X cot X = C. ( s o u r c e : Ref. 20 p . 442.)

c Xl \ 2 X3 X4 X5

0.000 0.0000 4.4934 7.7253 10.9041 14.0662 0.005 0.1224 4.4945 7.7259 10.9046 14.0666 0.010 0.1730 4.4956 7.7265 10.9050 14.0669 0.020 0.2445 4.4979 7.7278 10.9060 14.0676 0.030 0.2991 4.5001 7.7291 10.9069 14.G683 0.040 0.3450 4.5023 7.7304 10.9078 14.0690 0.050 0.3854 4.5045 7.7317 10.9087 14.0697 0.060 0.4217 4.5068 7.7330 10.9096 14.0705 0.070 0.4551 4.5090 7.7343 10.9105 14.0712 0.080 0.4860 4.5112 7.7356 10.9115 14.0719 0.090 0.5150 4.5134 7.7369 10.9124 14.0726 0.100 0.5423 4.5157 7.7382 10.9133 14.0733 0.200 0.7593 4.5379 7.7511 10.9225 14.0804 0.300 0.9208 4.5601 7.7641 10.9316 14.0875 0.400 1.0528 4.5822 7.7770 10.9408 14.0946 0.500 1.1656 4.6042 7.7899 10.9499 14.1017 0.600 1.2644 4.6261 7.8028 10.9591 14.1088 0.700 1.3525 4.6479 7.8156 10.9682 14.1159 0.800 1.4320 4.6696 7.8284 10.9774 14.1230 0.900 1.5044 4.6911 7.8412 10.9865 14.1301 1.000 1.5708 4.7124 7.8540 10.9956 14.1372 1.500 1.8366 4.8158 7.9171 11.0409 14.1724 2.000 2.0288 4.9132 7.9787 11.0856 14.2075 3.000 2.2889 5.0870 8.0962 11.1727 14.2764 4.000 2.4557 5.2329 8.2045 11.2560 14.3434 5.000 2.5704 5.3540 8.3029 11.3349 14.4080 6.000 2.6537 5.4544 8.3914 11.4086 14.4699 7.000 2.7165 5.5378 8.4703 11.4773 14.5288 8.000 2.7654 5.6078 8.5406 11.5408 14.5847 9.000 2.8044 5.6669 8.6031 11.5994 14.6374 10.000 2.8363 5.7172 8.6587 11.6532 14.6870 11.000 2.8628 5.7606 8.7083 11.7027 14.7335 16.000 2.9476 5.9080 8.8898 11.8959 14.9251 21.000 2.9930 5.9921 9.0019 12.0250 15.0625

14-3

TABLE 14.2 . (Continued.)

c Xl X2 X3 X4 X5 31.000 3.0406 6.0831 9.1294 12.1807 15.2380 41.000 3.0651 6.1311 9.1987 12.2688 15.3417 51.000 3.0801 . 6.1606 9.2420 12.3247 15.4090 101.000 3.1105 6.2211 9.3317 12.4426 15.5537

TABLE 14 .3 . F i r s t five roots of J (X ) m n

(Source: Ref. 20, p. 443.)

2.4048 3.8317 5.1356 6.3802 7.5883

5.5201 7.0156 8.4172 9.7610 11.0647

8.6537 10.1735 11.6198 13.0152 14.3725

11.7915 13.3237 14.7960 16.2235 17.6160

14.9309 16.4706 17.9598 19.4094 20.8269

14-4

TABLE 14.4. First six roots of X J 1(X n) - CJQ(X ) = 0. (Source: Ref. 74. p. 217)

X6 0 0 3.8317 7.0156 10.1735 13.3237 16.4706 0.01 0.1412 3.8343 7.0170 10.1745 13.3244 16.4712 0.02 0.1995 3.8369 7.0184 10.1754 13.3252 16.4718 0.04 0.2814 3.8421 7.0213 10.1774 13.3267 16.4731 n 06 0.3438 3.8473 7.0241 10.1794 13.3282 16.4743 0.08 0.3960 3.8525 7.0270 10.1813 13.3297 16.4755 0.1 0.4417 3.8577 7.0298 10.1833 13.3312 16.4767 0.15 0.5376 3.8706 7.0369 10.1882 13.3349 16.4797 0.2 0.6170 3.8835 7.0440 10.1931 13.3387 16.4828 0.3 0.7465 3.9091 7.0582 10.2029 13.3462 16.4888 0.4 0.8516 3.9344 7.0723 10.2127 13.3537 16.4949 0.5 0.9408 3.9594 7.0864 10.2225 13.3611 16.5010 0.6 1.0184 3.9841 7.1004 10.2322 13.3686 16.5070 0.7 1.0873 4.0085 7.1143 10.2419 13.3761 16.5131 0.8 1.1490 4.0325 7.1282 10.2516 13.3835 16.5191 0.9 1.2048 4.0562 7.1421 10.2613 13.3910 16.5251 1.0 1.2558 4.0795 7.1558 10.2710 13.3984 16.5312 1.5 1.4569 4.1902 7.2233 10.3188 13.4353 16.5612 2.0 1.5994 4.2910 7.2884 10.3658 13.4719 16.5910 3.0 1.7887 4.4634 7.4103 10.4566 13.5434 16.6499 4.0 1.9081 4.6018 7.5201 10.5423 13.6125 16.7073 5.0 1.9898 4.7131 7.6177 10.6223 13.6786 16.7630 6.0 2.0490 4.8033 7.7039 10.6964 13.7414 16.8168 7.0 2.0937 4.8772 7.7797 10.7646 13.8008 16.8684 8.0 2.1286 4.9384 7.8464 10.8271 13.8566 16.9179 9.0 2.1566 4.9897 7.9051 10.8842 13.9090 16.9650 10.0 2.1795 5.0332 7.9569 10.9363 13.9580 17.0099 15.0 2.2509 5.1773 8.1422 11.1367 14.1576 17.2008 20.0 2.2880 5.2568 8.2534 11.2677 14.2983 17.3442 30.0 2.3261 5.3410 8.3771 11.4221 14.4748 17.5348 40.0 2.3455 5.3846 8.4432 11.5081 14.5774 17.6508 50.0 2.3572 5.4112 8.4840 11.5621 14.6433 17.7272 60.0 2.3651 5.4291 8.5116 11.5990 14.6889 17.7807 80.0 2.3750 5.4516 8.5466 11.6461 14.7475 17.8502

100.0 2.3809 5.4652 8.5678 11.6747 14.7834 17.8931 00 2.4048 5.5201 8.6537 11.7915 14.9309 18.0711

14-5

TABLE 14.5. First five roots of J 0(X n)Y 0(CX n) = Y Q(X n) J 0(CX n). (Source: Ref. 9, p. 493.)

c Xl X2 X3 X4 X5 1.2 15.7014 31.4126 47.1217 62.8302 78.5385 1.5 6.2702 12.5598 18.8451 25.1294 31.4133 2.0 3.1230 6.2734 9.4182 12.5614 15.7040 2.5 2.0732 4.1773 6.2754 8.3717 10.4672 3.0 1.5485 3.1291 4.7038 6.2767 7.8487 3.5 1.2339 2.5002 3.7608 5.0196 6.2776 4.0 1.0244 2.0809 3.1322 4.1816 5.2301

i 14-6

TABLE 1 4 . 6 . F i r s t s i x r o o t s of tan(X ) = -X / C . (Source : Ref. 7 4 , p . 322.)

c \ X2 X3 X4 X5 X6 0 1.5708 4.7124 7.8540 10.9956 14.1372 17.2738 0.1 1.6320 4.7335 7.8667 11.0047 14.1443 17.2845 0.2 1.6887 4.7544 7.8794 11.0137 14.1513 17.2903 0.3 1.7414 4.7751 7.8920 11.0228 14.1584 17.2961 0.4 1.7906 4.7956 7.9046 11.0318 14.1654 17.3019 0.5 1.8366 4.8158 7.9171 11.0409 14.1724 17.3076 0.6 1.8798 4.8358 7.9295 11.0498 14.1795 17.3134 0.7 1.9203 4.8556 7.9419 11.0588 14.1865 17.3192 0.8 1.9586 4.8751 7.9542 11.0677 14.1935 17.3249 0.9 1.9947 4.8943 7.9665 11.0767 14.2005 17.3306 1.0 2.0288 4.9132 7.9787 11.0856 14.2075 17.3364 1.5 2.1746 5.0037 8.0385 11.1296 14.2421 17.3649 2.0 2.2889 5.0870 8.0962 11.1727 14.2764 17.3932 3.0 2.4557 5.2329 8.2045 11.2560 14.3434 17.4490 4.0 2.5704 5.3540 8.3029 11.3349 14.4080 17.5034 5.0 2.6537 5.4544 8.3914 11.4086 14.4699 17.5562 6.0 2.7165 5.5373 8.4703 11.4773 14.5288 17.6072 7.0 2.7654 5.6078 8.5406 11.5408 14.5847 17.6562 8.0 2.8044 5.6669 8.6031 11.5994 14.6374 17.7032 9.0 2.8363 5.7172 8.6587 11.6532 14.6870 17.7481 10.0 2.8628 5.7606 8.7083 11.7027 14.7335 17.7908 15.0 2.9476 5.9080 8.8898 11.8959 14.9251 17.9742 20.0 2.9930 5.9921 9.0019 12.0250 15.0625 18.1136 30.0 3.0406 6.0831 9.1294 12.1807 15.2380 18.3018 40.0 3.0651 6.1311 9.1986 12.2688 15.3417 18.4180 50.0 3.0801 6.1606 9.2420 12.3247 15.4090 18.4953 60.0 3.0901 6.1805 9.2715 12.3632 15.4559 18.5497 80.0 3.1028 6.2058 9.3089 12.4124 15.5164 18.6209 100.0 3.1105 6.2211 9.3317 12.4426 15.5537 18.6650

GO 3.1416 6.2832 9.4248 12.5664 15.7080 18.8496

14-7

SECTION 1 5 . CONSTANTS AND CONVERSION FACTORS.

1 5 . 1 . M a t h e m a t i c a l C o n s t a n t s .

e

Hn 10

TI

2 .7182818 . . . .

2 . 3 0 2 5 8 5 1 . . .

3 . 1 4 1 5 9 2 6 . . .

Y = 0 . 5 7 7 2 1 5 6 . . . ( E u l e r ' s c o n s t a n t )

1 5 . 2 . P h y s i c a l C o n s t a n t s .

Standard a c c e l e r a t i o n

of g r a v i t y

J o u l e ' s c o n s t a n t

9 .80665 m/s

32.1742 f t / s 2

J = 1.0 N'm/J c = 778 .16 f f l b f / B t u

S te£ an-Boltzmann

c o n s t a n t a = 1.355 x 1 0 ~ 1 2 c a l / s ' c m 2 - K 4

= 1712 x 1 0 ~ 1 2 B t u / h r « f t 2 - R 4

= 5.673 x 1 0 " 8 W/m 2'K 4

U n i v e r s a l gas

cons tant R = 8 .3143 J/mol«K

» 0.08205 il'ativ'iiiol'K

» 1.9859 Btu/lbm*mol'R

« 1545 .33 f f l b f / l b m ' m o l ' R

" 6 . 4 1 x 10 j/kg*mol'K

1 5 - 1

15.3. Conversion Factors.

TABLE 15.1. Conversion factors for length.

Vim in. ft yd 1 m 1 cm 1 lint 1 A 1 in. 1 ft 1 yd

¥ to

= 1 100 10" 1 0 i u 39.37 3.2B0 1.0936 = 0.01 1 10 4 10 8 0.3937 0.0328 0.0109 = 10" 6 lO" 4 1 10 4 0.3937 x 1 0 ~ 4 0.0328 x 1 0 - 4 0.0109 x 10 - ID" 1 0 lO" 8 io- 4 1 0.3937 x 1 0 - 8 0.0328 x 1 0 ~ 4 0.0109 x 10 - 0.0254 2.540 25.4 x 10 4 2.540 x 10 8 1 0.0833 0.0277 - 0.3048 30.48 30.48 x 1 0 4 30.43 x 10 8 12 1 0.3333 = 0.9144 91.440 91.440 x 1 0 4 91.440 x 1 0 8 36 3 1

r 4

-8

TABLE 15.2. Conversion factors for area.

cm in. ft yd

1 cm l m 2

1 in. 2 = 1 f t 2 = 1 y d 2 =

10' 6.4516 929.034 8361.307

10 " 1 6.4516 x 10" 4

929.034 x io" 8361.307 x 10 -4

0.1550 1.07639 1.1960 x 10 1550 10.7639 1.1960 1 6.9444 7.7160 x 10 144 1 0.11111 1296 9 1

-4

TABLE 15.3. Conversion factors for volume.

Ul i

™ 3 cm . 3 in. ft3 ml liter gal 3

1 cm « 1 610.23 x 10~ 4 35.3145 x 1 0 - 4 999.972 x 1 0 - 3 999.972 x 10" -6 264.170 x lO" 6

1 in.3 = 16.3872 1 5.7870 x 10" 4 16.3867 16.3867 x 10" -3 432.900 x lO"5

1 f t 3 = 283.170 x 10 2 1728 1 28.3162 x 10 3 28.3162 7.4805 1 ml = 1.000028 610.251 x 10" 4 353.154 x 10~ 7 1 0.001 264.178 x 10" b

1 liter = 1000.028 61.0251 353.154 x 1 0 - 4 1000 . 1 264.178 x 10" J

1 gal => 3785.434 231 133.680 x 10" 3 3785.329 3.785329 1

TABLE 15.4. Conversion factors for mass.

lb slugs 9 kg ton 1 lb = 1 0.03108 453.59 0.45359 0.0005 1 slug = 32.174 1 1.4594 x 10 4 14.594 0.016087 1 g = 2.2046 x 10' -3 6.8521 x 10" -5 1 ID" 3 1.1023 x 1 0 - 6

1 kg = 2.2046 6.8521 x 10' -3 103 1 1.1023 x lo - 3

1 ton = 2000 62.162 9.0718 x 10 5 907.18 1

TABLE IS .5 . Conversion factors for density.

lbm/ft3 slug/in. lbm/in.3 lbm/gal . 3 g/cm

1 lbm/ft3 = 1 0.03108 5.787 x 10 - 4 0.13386 0.01602 1 slug/ft3 = 32.174 1 0.01862 4.3010 0.51543

3 1 lbm/in. = 1728 53.706 1 231 27.680 1 lbm/gal = 7.4805 0.2325 4.329 x 10" 3 1 0.11983 1 g/cm3 62.428 1.9403 0.03613 8.345 •-I

I

15-4

TABI£ 1 5 . 6 . C o n v e r s i o n f a c t o r s f o r p r e s s u r e .

l b f / i n / dyne/cm kgf/cm i n . Hg mm Hg i n . H O atm ba r

1 l b f / i n . 2 = 1

1 dyne /em 2 = 145 .0383

x 1 0 ~ 7

1 k g f / c m 2 = 14 .2234

1 i n . Hg = 0 .4912

1 ram Hy = 0 .01934

1 i n . H O = 0 .03609

1 atm

1 bar

14 .6960

= 14 .5038

689 .473

x 1 0 2

1

980 .665 3

x 1 0 J

338 .64

x 1 0 2

1333 .223

24.883

x 1 0 2

101 .325 4

x 10

1 0 6

0 .07031

101 .972

x l O - 8

1

0.03453

1.3595

x 1 0 ~ 3

2 .537

x 1 0 ~ 3

1.03323

2.0360

295.299 x 10~ 7

28.959

1

0.03937

0.0735

29.9212

1.01972 29.5299

51.715

750.062 x 10" 6

735.559

25.40

1

1.8665

760

750.0617

27.71

4.0188 -4

x 10 394.0918

13.608

0.5358

460.80

401.969

0.06805

986.923 x 10" 9

967.841 x 10 3

0.03342

1.315 x 1 0 - 3

2.458 x 10~ 3

1

986.923 x 10" 3

0.06895

io- 6

980.665 x 10~ 3

0.03386

1.333 x 1 0 - 3

2.488 x 10 1.01325

TABLE 15.7. Conversion factors for energy.

f f l b f abs joule int joule c a l I .T. cal

1 f f l b f 1 1.35582 1.355597 0.32405 0.32384 1 abs joule = 0.73756 1 0.999835 0.23885 0.238849 1 int joule = 0.737682 1.000165 1 0.239045 0.238889 1 cal 3.08596 4.18401 4.1833 1 0.99934 1 I .T. cal = 3.08799 4.18676 4.18605 1.000657 1

1 Btu 778.16 1055.045 1054.366 252.161 251.996 1 int kW'hr = 265.567 x 1 0 4 360.0612 x 1 0 4 360.000 x 1 0 4 860.565 x 1 0 3 860.000 x 1 0 3

1 hp'hr = 198.0000 x 10* 268.4525 x 1 0 4 268.082 x 1 0 4 641.615 x 1 0 3 641.194 x 10 3

1 l i ter-atm = 74.7354 101.3278 101.3111 24.2179 24.2020

Btu int kW»] hr hp»hr l i t er 'a tm

1 f f l b f 1 abs joule 1 int joule 1 cal 1 I .T. cal 1 Btu 1 int kWhr 1 hp'hr 1 l i t er 'a tm

128.5083 x 10 947.827 x 10" 6

0.947988 x 10 396.572 x 10' 396.832 x 10" 1 3412.76 2544.46 0.09604

-3

376.553 x 10 277.731 x 10~ 9

2.777778 x 10~ 7

116.2028 x 10~ 8

116.2791 x io~fi

293.018 x 10~ 6

1 0.74558 281.718 x 10~ 7

505.051 x 10 ' 372.505 x io - S

3.7256 x lo" 7

155.8566 x io" 155.9590 x io" 293.010 x 10~ 6

1.3412 1 377.452 x 10~ 7

133.8054 x 10 986.896 x 10 9.87058 x 10 412.918 x lo" 413.189 x lo" 10.4122 255.343 x 10 2

264.935 x 10 2

1

-3

Table 15.8. Conversion factors for specific energy.

abs joule/h ca l /g I .T. cal/gm Btu/lb ft-lb£/lbm in t . kw hr/g hp hr/ lb , . 2 . 2 t t / s

abs joule/g - 1 0.2390 0.2388 0.4299 334.53 2.777 x 10" 7 1.690 x 10" 4 10763 cal /g - 4.184 1 0.9993 1.7988 1399.75 1.162 x 1Q~6 7.069 x 10~ 4 4 .504 x 10 4

I.T. cal /g - 4.186 1.0007 1 1.8 1400.69 1.163 x 10"6 7.074 x 10"4 4 .506 x 10 4

Btu/lb . 2.326 0.5559 0.5556 1 778.16 6.460 x l O - 7 3.930 x in" 4 25,037 f f lb f / lbm - 2.989 x 10~ 3 7.144 x 10" -4 7.139 x 10~ 4 1 .285 x lo"" 1 8.302 x 1 0 ~ 1 0 5.051 x lo" 7 32.174 int . RWhr/g - 3.610 x 10 6 860,565 86C,000 \ .548 x 10 6 1.2046 x 10 9 1 608.4 3, ,876 x I D 1 0

hp-hr/lb • 5919 1414.5 1413.6 2545 1.980 x 10 6 0.001644 1 6. ,370 x 10 7

f t W - 9.291 X 10" 5 2.220 x 10" •5 2.219 x 10" 5 3, .994 x 10" 5 0.03108 2.580 x l o " 1 1 1.567 x 1 0 - 8 1

Table 15.9. Conversion factors for specific energy per degree.

a b s j ou le /g • >K Cal/g-K I .T. c a l / g ' •K Btu / lb 'R Ws/kg-K

abs joule/g*K = 1 0.2390 0.2388 0.2388 1 0 3

cal/g-K = 4.184 1 0.9993 0.9993 4184

I . T . cal/g-K = 4.186 1.0007 1 1 4186

btu/ lb-R

Ws/kg-K

4.186

l O " 3

1.0007

2.390 x 10" -4 2.

1

,388 x 1Q~ •4

1

2.388 x I d - 4

4186

1

Table 15.10. Conversion factors for thermal conductivity.

cal/s«cm-°C Btu/hr«ff°F Btu/hr-ft ' "F/ in . W/ m-°C

1 cal/s'cm* C = 1 241.9 2903 418.6 1 Btu/hr'ft- F = 4.13 x 10~3 1 12 1.73 1 Btu/hr'ft2- F/in. = 3.45 x 10~4 0.0833 1 1.44 x 10"1

1 W/m C = 23.89 5780 69350 1

Table 15.11. Conversion fa for thermal ''and momentum diffusivities

ft2/hr stokes m2/hr 2/ m /s

ft2/hr = 1 0.25806 0.092903 2.58 x 10~5

stokes = m2/hr =

3.885 10.764

1 2.778

0.36 1

io-4

1.778 x 10~ 4

m2/s = 38,750 10 4 3600 1

TABLE 15.12. Conversion factors for heat flux.

Btu W k c a l c a l

f t 2 « h r 2

m . 2 hr«m 2 S"cm

B t u / f t 2 « h r = 1 3.154 2 . 7 1 3 7.536 x 1 0 ~ 5

W/m2 = 3.170 x 1 0 7 1 8 .600 x 1 0 7 2389 2

k c a l / h r - m = 0.3687 1.163 1 36000 2 c a l / s ' c m = 13277 41868. 2 .778 x 10~ 5 1

TABLE 15.13. Conversion factors for heat transfer coefficient.

Btu W c a l k c a l

n r . f t

2 . ° F m 2 ' ° C 2 o„

s»cm • C h r - m 2 ' ° C

B t u / h r - f t 2 « ° F = 1 5 .678 1.356 x 1 0 " 4 4 .883

W/m 2 ' °C 1.761 x 1 0 7 1 2391 8.600 x 1 0 7

c a l / s e c - c m - ° C = 7376 4 .186 x 1 0 4 1 36000

k c a l / h r - m 2 - ° C = 0.2049 1.163 2 .778 x 1 0 " 5 1

15-9

SECTION 16. CONVECTION COEFFICIENTS

16.1. Forced Flow in Smooth Tubes.

16.1.1 Fully Developed Laminar Flaw (Source: Ref. 21)

— = 4.364 (constant heat rate) k

— = 3.658 (constant surface temp)

16.1.2 Fully Developed Turbulent Flow (Soui'ce: Ref. 21)

££ = 6.3 + 0.003 (Re Pr), Pr < 0.1 (constant heat rate)

— = 4.8 + 0.003 (Re Pr), Pr < 0.1 (constant surface temp)

jp = 0.022 Pr * Re " , 0.5 < Pr < 1.0 (constant heat rate)

— = 0.021 Pr " Re , 0.5 < Pr < 1.0 (constant surface temp)

jp- = 0.0155 Pr°' 5Re°* 9, 1.0 < Pr < 20

f- = 0.011 Pr°' 3Re°- 9, Pr > 20 k

Pr = Prandtl number, Re = Reynolds number

16.2. Forced Flow Between Smooth Infinite Parallel Plates.

16.2.1. Fully developed laminar flow (Source: Ref. 21): he — = 4.118 (constant heat rate on both sides)

hs — = 2.693 (constant heat rate on one side, other side insulated)

16-1

hs — = 3.77 (constant surface temperature on both sides) hs r— = 2.43 (constant surface temperature on one s ide , o ther side insulated)

s = spacing of p la tes

16 .3 . Forced Flow P a r a l l e l to Smooth Semi-Inf ini te F la t P l a t e s Laminar Flow. 1 6 . 3 . 1 . Laminar flow: hx k

temperature)(Source: Ref.7) = 0.332 Pr 1 / 3Re 1 / 2 [l - (XQ/X) 3 / / 4 ] ~ 1 / 3 , (constant surface

X = unheated starting length X = distance from leading edge

JP = 0.453 P r 1 / 3 R e x1 / 2 [l - (XQ/X) 3 / 4 ] " 1 / 3 , (constant heat rate)

(Source: Ref. 21) 16.3 .2 . Turbulent flow (Source: Ref. 21):

f = 0.0295 P r ° - 6 R e x

0 ' 8 [ l - ( X 0 / X ) 9 / 1 0 ] " 1 / 9 , (constant surface temperature)

hx 0.6 0.8 T 9/101-1/9 — = 0.0307 Pr Re x |_1 - (XQ/X) ' J ' , (constant heat rate)

16.4. Fully Developed Flow in Smooth Tube Annul!.

16.4.1. Laminar flow (Source: Ref. 21):

h i < a Q - d i > N U i i k " 1 " <qo/qX

h (d - d.) Nu o o i oo

i - VqX (Subscripts i and o re fe r to inner and outer surfaces, q i s surface heat f lux , Nu.. and Hu are inner and outer surface Nussel t numbers when only one surface i s heated and 0 i s an influence coeff ic ient given in Table 16.1.)

16-2

TABLE 16.1. Tube-annulus solutions for constant heat rate in fully developed laminar flow and temperature profiles.

r.r Nu.. Nu 0. 0 l o 11 oo l o 0 °° 4.364 °° 0 0.2 8.499 4.883 0.905 0.1041 0.4 6.583 4.979 0.603 0.1823 0.6 5.912 5.099 0.473 0.2455 0.8 5.580 5.240 0.401 0.299 1.0 5.385 5.385 0.346 0.346

16.4.2. Turbulent flow (Source: Ref. 22):

7 (d - d.) - 0.023 Re°' 8 Pr°" 4 (d /d.)°' 4 5. Re., 10 4, A d = d - d. k o I . . Ad o i Ad o I

16.5. Forced Flow Normal to Circular Cylinders.

16.5.1. Local Coefficients (Source: Ref. 22):

& = 1.14 Pr°- 4 Re°' 5 [l - (G/90)3] , 0 < 6 < 80°

0 = cylinder angle from stagnation point

16.5.2. Average Coefficients (Source: Ref. 7):

M = 0.43 + C Re"? Pr°' 3 1

k d

TABLE 16.2.

R S d 1-4,000

4,000-40,000 40,000-400,000

0.533 0.500 0.193 0.618 0.0265 0.805

16-3

16.6. Forced Flow Normal to Spheres.


^ = 0.37 Re, - 6 P r 0 " 3 3 , 20 < Re < 150,000

ifl = 2 + 0.37 Re°- 6 Pr 0- 3 3, Re., < 20 k d a

16.7. Free Convection on Vertical Plates and Cylinders.

16.7.1. Local Coefficients (Source: Ref. 7):

jp = 0.508(Pr)1/2(0.952 + Pr)~ 1 /' 4 ( G r x ) 1 / 4f GrPr < 10 4

|* = 0.0295 ( P r ) 7 / 5 [l + 0.494 ( P r ) 2 / 3 ] ~ 2 / 5 ( G r ) 2 / 5 , GrPr > 10 4

16.8. Free Convection on Horizontal Cylinders. >


f = C (Grd Pr)1"

TABLE 16 .3.

Gr dPr C m 0-10"5 0. 40 0 10" 5-10 - 1 0. 97 1/16 10 - 1-10~ 4 1. 14 1/7 10 4-10 9 0. 53 1/4 10 9-10 1 2 0 13 1/3

16-4

16.9. Free Convection From Horizontal Square Plates.

16.9.1. Average Coefficients (Source: Ref. 23);

•-1 = C (GrT Pr) m, L = plate dimension k L

TABLE 16.4.

Condition Gr Pr L Upper surface heated or lower surface cooled Lower surface heated or upper surface cooled Upper surface heated or lower surface cooled

10 5 - 2 x 10 7 0.54 1/4

3 x 10 5 - 3 x 1 0 1 0 0.27 1/4

2 x 10 7 - 3 x 1 0 1 0 0.14 1.3

16.10. Free Convection from Spheres.

16.10.1. Average Coefficients (Source: (Ref. 5):

^ = ?. + 0.43 (Grd P r ) 1 / 4 , 10° < Gr d Pr < 10 5

16.11. Free Convection in Enclosed Spaces.

16.11.1. Enclosed Vertical Air Spaces (Source: Ref. 1);

Gr B < 2,000

0.18 {Gr ) 1 / 4 ( s / L ) 1 / 9 , 2,000 < Gr < 20,000 s s

I 0.065 (Gr ) 1 / 3 ( s / L ) 1 / ' 9 , 20,000 < Gr < 10 7

8 S

16-5

k e = ef fec t ive thermal conductivi ty s = width of a i r space L = length of a i r space

16 .11 .2 . Enclosed Horizontal Air Spaces (Source: Ref. 1) i

0.195 (Gr ) 1 / 4 , 10 4 < Gr < 4 x 10 5

s s 0.068 (Gr ) 1 / 3 , 4 x 10 5 < Gr s s

16.12. Film Condensation.

1 6 . 1 2 . 1 . Ver t i ca l Plates—Average Coef f ic ien ts (Source: Ref. 5) :

P A ( P a - P v )g

1/3 = 1.47 Re„ -1/3 Re^ < 1800

& Hâ " p v ) g

1/3 = 0.007 R e j j 0 , 4 , *•% > 1800

R% ' < 4 h A t v„>/W* At = temp difference between sa tura ted vapor and wall i j = heat of evaporat ion Ug,ko,p,, = v i s c o s i t y , thermal conductivi ty and densi ty of l iquid

a t s a t u r a t i o n vapor temp

16 .12 .2 . Horizontal Tubes—Average Coeff ic ients (Source: Ref. 5):

j £ = 0.725 i(H ~ PI ) 1 9 Hvd

W * A t v w

1/4

16-6

16.13. Pool Boiling,

r -| 1/2 r CjAt -i

At = temp difference between wall and saturated vapor wv c„,u«,Pj,Pro = specific heat, viscosity, density and Prandtl

number at saturated liquid temperature a = surface tension at liquid vapor interface

i« = heat of evaporation C , = surface coefficient (see Table 16.5)

TABLE 16.5. Values of C and n (source: Ref. 19).

Surface-fluid combination C8f n

Water-nickel 0.006 1.0 Water-platinum 0.013 1.0 Water-copper 0.013 1.0 Water-brass 0.006 1.0 CCl.-copper 0.013 1.7 Benzene-chromium 0.101 1.7 ri-Pentane-chromium 0.015 1.7 Ethyl alcohol-chromium 0.0027 1.7 Isopropyl alcohol-copper 0.0025 1.7 35% K2C03-copper 0.0054 1.7 50% K.CO.-copper 0.0027 1.7 r»-Butyl alcohol-copper 0.0030 1.7

16-7

SECTION 17. CONTACT COEFFICIENTS

An empirical correlation developed by Shevts and Dyban (Ref. 24) gives estimated thermal contact coefficients for many common ferrous and non-ferrous metals in contact, including dissimilar metals.

hr/k = (TT/4) [l + 85(P/S) 0 - 8] k = thermal conductivity of the gas phase P = contact pressure r = height of a micro-element of roughness plus the '.leight of the wave

for one surface S = permissible rupture stress

A theoretical approximation of contact coefficients developed by French and Rohsenow (Ref. 25) can be found by using Fig. 17.1 and the following properties:

C = constriction number, P/M P = contact pressure M => Meyer hardness of the softer contact material B = gap number = 0.335 C™1

m = 0.315(V"A7r) 0" 1 3 7

A = interface area (one side) I = effective gap thickness, 3.56 (JL + I ) if (i, + I ) < 280

uin. (smooth contacts), or 0.46 (&, + I ) > 280 uin. (rough contacts)

H.,%. • mean (or rms) depths of surface roughness k, = equivalent conductivity of interstitial fluid, for liquids use

k. = k evaluated at t = (t., + t,)/2, for gases:

k ô Kf 1 + 8 Y (v/v) + a 2 - a x - a 2)/Pr(y + UJU^ + a 2

iole e t3

+ £1 + £ 2 " e i £ 2

17-1

k = fluid conductivity at zero contact pressure Pr = Prandtl number 'v = mean molecular velocity Y = ratio of specific heats v = kinematic viscosity evaluated at t a = accommodation coefficient e = surface emissivity a = Stefan-Boltzmann constant K = conductivity number, k (k + k )/2k k, k. = conductivity of first solid evaluated at

t =[\ + ( k ^ + ^tjj/tkj + k 2) J /2 k = conductivity of second solid evaluated at

t = [ f c 2 + ( k ^ + k j V / t k j , * ^ ) ] ^

Some thermal contact' data assembled by P. J. Schneider from various sources (Ref. 19) is given in Fig. 17.2. Also, some additional data from Ref. 26 showing the effects of machining processes and surface matching is given in Fig. 17.3.

17-2

TABLE 17.1. Interface conditions for contact data given in Fig. 17.2.

RMS surface Mean contact Curve Material pair finish (Uin.) Gap material temp (°F)

1 aluminum (2024-T3) 48-65 -4 vacuum (10 mm Hg) 110 2 aluminum (2024-T3) 8-18 —4 vacuum (10 ram Hg) 110 3 aluminum (2024-T3 6-0 (not flat) —4 vacuum (10 mm Hg) 110 4 aluminum (75S-T6) 120 . air 200 5 aluminum (75S-T6) 65 air 200 6 aluminum (75S-T6) 10 air 200 7 aluminum (2024-T3) 6-8 (not flat) lead foil (0.008 in.) 110 8 aluminum (75S-T6) 120 brass foil (0.001 in.) 200 9 stainless (304) 42-60 —4 vacuum (10 mm Hg) 85 10 stainless (304) 10-15 —4 vacuum (10 mm Hg) 85 11 stainless (416) 100 air 200 12 stainless (416) 100 brass foil (0.001 in.) 200 13 magnesium (AZ-31B) 50-60 (oxidized) —4 vacuum (10 mm Hg) 85 14 magnesium (AZ-31B) 8-16 (oxidized) —4 vacuum (10 mm Hg) 85 15 copper (OFHC) 7-9 —4 vacuum (10 mm Hg) 115 16 stainless/aluminum 30-65 air 200 17 iron/aluminum - air 80 IB tungsten/graphite — air 270

TABLE 17.2. Interface conditions for contact data given in Fig. 17.3.

Roughness RMS (Uin.)

Block Fluid in Temp

Curve Material Finish 1 2 Gap (°F) Condition

a Cold rolled steel Shaped 1000-1000 Air 200 Parallel cuts, rusted b Cold rolled steel Shaped 1000-1000 Air 200 Parallel cuts, clean c Cold rolled steel Shaped 1000-1000 Air 200 Perpendicular cuts, clean d Cold rolled steel Hilled 125-125 Air 200 Parallel cuts, rusted e Cold rolled steel Milled 125-125 Air 200 Parallel cuts, clean f Cold rolled steel Shaped 63-63 Air 200 Perpendicular cuts, clean g Cold rolled steel Shaped 63-63 Air 200 Parallel cuts, clean h Cold rolled steel Lapped 4-4 Air 200 Clean i 416 Stainless Ground 100-100 Air 200 J 416 Stainless Ground 100-100 Air 400 k 416 Stainless Ground 30-30 Air 200 X 416 Stainless Ground 30-30 Air 400 m Stainless Milled 195-195 Air Clean

* * ^ 6 X 1 0 " ' i-i 1-1

5X10" 4X10 ' 3X10" 1

2X10" 1

10"1 , 8 X 10-2

6X 10"2

4 X 10-2 2X10-2 'o- 2 , 8X10-3 6X10-3 4X10-3 2X 10"3

10~ 3 . 8 X 1 0 " 4

6 X 1 0 i 4 X 10" 4

2 X 10- 4

8 X 10-f 6 X 10"? 4 X 1 0 " 5

2 X 10" s

10~5

FIG. 17.1. Thermal contact coefficient from theory (source: Ref. 25),

17-5

J — I I I . . . I 5 10 50 100 200 500 1000

p - lb/in.2

FIG. 17.2. Thermal contact coefficient data (source: Ref. 19), 1

17-6

0.016

u

c o o

100 200 300 400

Contact pressure — lb/in. 2

500

Parallel cuts

Perpendicular cuts

PIG. 17 .3 . Thermal contact coeff ic ient for bare s t e e l surfaces, (source:

Ref. 26, sec t ion G502.5, p . 5 ) .

17-7

Table 18.1 Thermal properties of selected metals (source: Refs. 35 to 39).

Properties at 300 K Thermal conductivity (k), (W/m-K) x 10' ^T

(kg/m3) x 10* 3 (J/kg-K) x 10" 3 (W/m-K) x 10~ 2 (m?/s) x 10* 50 100 200 400 600 flOO 1000 1200

CO I

Aluminum: Pure 2.70 0.88 2.4 1.01 235. 117. 12.3 3.0 2.4 2.4 2.3 2.2 2028 (4.5* Cu, 1.5* Kg) 2.77 0.84 1.3 0.56 0.08 0.17 0.40 0.65 1.0 1.4 5086 (4.1* rig, 0.5* Mn) 2.65 0.90 1.3 0.55 0.08 0.17 0.40 0.63 0.85 6063 (0.7* Hg, 0.1* Si) 2.70 0.92 2.1 0.85 0.86 1.7 2.8 2.1 2.0

Beryllium: Pure 1.85 1.84 2.0 0.59 ia.o' 34.8 40.0 9.9 3.01 1.61 1.26 1.07 0.89 0.73 Chromium: Pure 7.19 4.56 0.90 0.03 3.9 6.0 3.2 1.6 1.1 0.B7 0.B1 0.71 0.65 0.62 Copper: Pure 8.96 0.38 4.0 1.17 196. 105. 12.2 4.8 4.1 3.9 3.8 3.7 3.6 3.4

Commercial bronze (10* Zn) 8.80 0.41 1.8 0.50 0.20 0.42 0.81 1.2 Yellow brass (35* Zn) 8.47 0.41 1.2 0.35 0.59 0.94 1.3 1.5 Naval brass (38* Zn, O.B* 5n) 8.41 0.38 0.75 0.23 0.85 1.01 German silver (22* Zn, 15* Hi) 8.55 0.40 0.25 0.07 0.028 0.07 0.15 0.17 0.20 Cupronickel (20* Ni, 2* Zn) 8.94 0.38 1.5 0.44 0.11 0.20 0.38 0.92 Constantan (40* Ni) B.90 0.42 0.23 0.06 0.08 0.15 0;17 0.19 Manganin (13* Hn, 4* Ni) 8.19 0.40 0.22 0.07 0.015 0.032 0.082 0.13 0.17 0.28

Gold: Pure 19.32 0.13 3.2 1.27 28.2 15.0 4.2 3.5 3.3 3.1 3.0 2.9 2.B 2.6 Iron: Pure 7.87 0.45 0.80 0.23 7.1 10.0 3.7 1.3 0.94 0.69 0.55 0.43 0.33 0.2H

Wrought iron (<0.S* C) 7.87 0.46 0.60 0.17 0.5B 0.49 0.39 Gray cast iron (3.0* C, 0.6* Si) 7.15 0.42 0.29 0.10 0.31 0.34 0.29 0.22 0.20 5AE 1095 steel (0.9* C, 0.3* Hn) 7.83 0.47 0.44 0.12 0.08 0.22 0.34 0.43 AISI 4340 steel (1.9* Ni, 0.7* Cr) 7.84 0.46 0.36 0.10 0.26 0.31 0.38 0.38 0.34 Nickel steel (9* Ni, 0.8* Hn) 7.96 0.46 0.29 0.08 0.18 0.24 0.33 0.34 0.33 invar (31* Ni, 5* Co) 8.00 0.46 0.14 0.08 0.12 0.16 0.19 0.22 SAE 4130 steel (1* Cr, 0.5* Mn) 7.86 0.46 0.41 0.11 0.06 0.16 0.17 0.37 AISI 304 stainless (19* Cr, 10* N1) 7.90 0.38 0.15 0.05 0.008 0.021 0.06 0.09 0.13 D.17 0.20 0.22 0.25 0.28 AISI 316 stainless (17* Cr, 12* Ni) 8.00 0.46 0.14 0.40 0.008 0.021 0.06 0.09 0.12 0.15 0.18 0.21 0.23 0.25

Lead: Pure 11.34 0.13 0.35 0.24 1.78 0.59 0.44 0.40 0.37 0.34 0.31 Solder (80* Pb, 20* Sn) 10.20 0.37 Solder (50* Pb, 50* Sn) 8.B9 0.18 0.46 0.29

Lithium: Pure 0.53 3.97 0.77 0.37 6.1 7.2 2.3 1.1 0.88 0.72 Magnesium: Pure 1.74 1.00 1.56 0.90 11.7 13.9 3.8 1.7 1.6 1.5 1.5 1.5

(6* Al, 2* Si) 1.84 1.00 0.70 0.38 0.50 0.62 9.BO (12* Al, 2* Si)

Molybednum: Pure 1.81

10.22 0.99 0.25

0.56 1.4

0.31 0.55 1.45 2.77 3.0

0.30 1.8

044 1.4

0.68 1 j 1.3 1.2 1.1 1.0

Table 1B.1 (Continued).

Properties at 300 K Thermal conductivity (It), (W/m-K) x 10~ ?

P, c, k, a, fetal (kg/m3) x 10~ 3 (J/kg-K) x 10" 3 (W/m-K) x 10" 2 (m?/s) x 10* 10 K 20 50 100 ?0D 400 600 800 1000 12O0

Ni ckel: Pure Duranickel (4.5* Al)

8.90 8.26

0.45 0.54

0.91 0.19

0.23 0.04

6.0 8.6 3.4 1.6 1.1 0.B0 0.66 0.67 0.72 0.76

Monel (30* Cu, 1.4* Fe) 8.84 0.43 0.23 0.05 0.08 0.14 0.17 0.20 Inconel X-750 (16% Cr, 8* Fe) 8.51 0.46 0.12 0.03 0.010 0.025 0.06! i 0.08! i 0.10 0.13 0.17 0.21 0.25 0.29 Nichrome (20* Cr) B.40 0.45 0.13 0.03 0.14 0.17 0.21 0.25 0.28 Nichrome V (24* Fe, 16* Cr) 8.25 0.44 0.10 0.03 0.12 0.15 0.19 0.22 0.26

Niobium: Pure 8.57 0.27 0.54 0.23 2.2 2.3 0.76 0.55 0.53 0.55 0.5B 0.61 0.64 0.68 Platinum: Pure 21.45 0.13 0.71 0.25 12.3 4.9 1.1 0.78 0.72 0.72 0.73 0.76 0.79 0.83 Plutonium: Pure 19.B4 0.15 0.67 0.23 0.03 0.48 0.90 Rhenium: Pure 21.04 0.14 0.48 0.16 14.0 8.4 0.96 0.60 0.51 0.46 0.44 0.44 0.45 0.46 Silver: Pure

Sterling (7.5* Cu) Eutectic (28* Cu)

10.49 10.38 10. U6

0.24 4.3 3.5

• 3.3

1.71 168. 51. 7.0 4.5 4.3 4.2 4.1 3.9 3.7 3.6

00 Tin: Pure 7.30 0.22 0.67 0.42 19.3 3.2 1.2 0.85 0.73 ' 0.62 1 S3

Solder (40* Pb) 8.44 0.19 0.51 0.32 0.43 0.56 0.52 0.54 0.52 Tantalum: Pure 16.6 0.14 0.58 0.25 1.08 1.5 0.72 0.59 0.58 0.58 0.59 0.59 0.60 0.61 Tungsten: Pure 19.3 0.13 1.8 0.72 82. 32. 4.7 2.4 2.0 1.6 1.4 1.3 l.Z 1.1 Titanium: Pure 4.51 0.50 0.22 0.10 0.14 0.28 0.40 0.31 0.25 0.20 0.10 0.20 0.71 0.2?

. A-110 AT (5* Al, 2.5* Sn) 4.46 0.53 0.075 0.03 0.088 0.11 0.14 Uranium: Pure 19.07 0.12 0.28 0.12 0.098 0.16 0.19 0.22 0.25 0.30 0.34 0.39 0.44 0.40 Zinc: Pure 7.13 0.39 1.2 0.43 43. 10.7 2.1 1.2 1.2 1.2 1.1

TMLE I B , 2 . Thernal p r o p e r t i e s of misce l l aneous a o l l d s {source: R e f s . 35 t o 3 8 ] .

P r o p e r t l e g a t 3Q0 K Thermal c o n d u c t i v i t y ( k ) , (W/m'K) > 10

(kg/aT) x 10" J (J/kg-lt) k.

( m / a ) * 10 10 K 1D0 200 400 600 1000 1200

Ceramics* ( F o l y c r y a t a l l i n e , 99 .5 ' pu r i t y ,98% solid}

A 1 2 0 3

B«0 MgO

S i 0 2 [high p u r i t y fused)

T W 2

TiO

sco 2

G l a s s e s t B o r o s i l i c a t e (Pyrex) Soda l i n e (751 SiO ) V i t r e o u s s i l i c a (1001 SiOj] Zinc crown (65% SiO )

I n s u l a t i o n s ; (Per high temps.) Aluqtina, fused (901 A1 ,0 3 ) AsbBStos paper [ laminated)

Dia tonaceous ear th . s i l i c a (powder) Pirebrick (58% SiO , 37% A1 20 3) Magnesite (65% MgO) Micro quart* fiber (blanket) Rockwool (loose)' Zirconia (grain)

Insulations: (For low temps.) Expanded glass (Foanglas) Fiberglass (board} Fiberglass (blanket) Silica aerogel (powdei) polystyrene foaa (latmabs) polystyrene foam (10~ atmabs)

3.84 0.79 36.0 0.12 2.97 1.00 272.0 0.92

3 . 2 1 0.92 48 .0 0.16

2 .21 0.75 1.4 0.008

9.58 0 .23 13.0 0.06

3 . 9 1 0 .71 8 .0 0.03

5.2B 0.46 1.6 0.007

2 .21 0 . 7 1 1.1 0.007 2.52 0.66 0 .9 0.005 2 .21 1.00 1.4 0.006 2 .60 0.67 1.1 0.006

2.70 0.83 5.2 0.02

0.35 0.84 0.048 0.002

0 .81 0.92 0.19 1.13 0.05 0.84 0.16 1.81 D.50

0.17

0 .18

0.05

0 .08

0 .08

0.08

0.34 0.005

0.050 0.002 0.036 0.009

0.046 0.18 0.002

0.062

0.040

0.038 0.022

0.036

0.017

10. 550 . 260 . 160 . 100. BO.

4240. 1960. 1110. 700. 470.

750 . 350. 220 . 140. 90 . 6 .9 1 1 . 15. 18 . 22 . 29.

ISO. 100. 70. 50. 40. 1D0. 70 . 50 . 4 0 . 30 .

1 7 . 18 . 19 . 19.

6. 9. a.

12. 1 1 .

15 . 19 .

0.31 0.12

0.07 0.12

0.51 0.24

0.17 0.24 0.12

0.73

0.79

0.61 0.48 0.66 1.9

1.0

5.2

0.89 1.0 2.1

1.3

6.6

1.3 1.3 2.2

TABU: IB .2 . (Continued.I

CO I

P r o p e r t i e s a t 300 K Thermal c o n d u c t i v i t y ( k ) , Wmfl) x 10~ 2

P, c , k . a . M a t e r i a l <kg/m 3) x 1 0 - 3 ( JAg-K) x 10 ~3 W/m-K (ra 2 /s) * 10* 10 K 20 50 100 200 400 600 800 1000 1200

P l a s t i c s : A c r y l i c , PHKA ( P l e x i g l a s ) l . i a 1.46 0.16 0.001 0.6 0 .7 1.2

Nylon 6 1.16 1.59 0 .25 0 .001 0 . 3 0 .9

P o l y v i n y l c h l o r i d e ( r i g i d ) 1.40 1.00 0.15 0.001 1.5 1.6 1.5

T e f l o n , PTFE 2.18 1.05 0.40 0.002 1.0 1.5 2 .0 2 .5 4 .5 5.5 P o l y e t h y l e n e , high d e n s i t y 0.95 2.30 0.50 0.002

Rocks:

G r a n i t e 2 .60 0.79 3.4 0.017 30. 2 " .

Marble 2.50 0.88 l .B 0.00B 1 7 . 1 1 . Sandstone 2 .20 0 .92 S.3 0.026 44. 30. s h a i a 2.60 0 .71 1.8 0.010 15 . 14.

Hoods: ( a c r o s s g r a i n , oven dry)

Ba l sa 0.16 0.059

Douglas f i r 0.46 2.72 a. i i 0.001

Oak, r e d 0 .6? 2.38 0.17 0.001

P i n e , w h i t e 0.40 2.80 0.10 0.001

Redwood 0.42 2 .90 0 .11 0.001

M i s c e l l a n e o u s : Carbon b l ack (powder] 0 .19 0.84 0 .021 0.001

Carbon (pet roleum coke) 2 .10 0.B4 1.9 0.011 2 1 . 25 . 26. 30 . G r a p h i t e (ATJ)

(U t o g r a i n s ) ' 1.73 0.B4 125.0 0.B9 S. 25 . 170. 5B0. 1200. 1180. 950. 770. 640. ( i t o g r a i n s ) 1.73 0.84 98 .0 0 .67 4 . 19 . 130. 420. B60. 900. 730. 590. 490.

G r a p h i t e ( p y r o l y t i c )

( | t o g r a i n s ] 2.20 0.84 2000.0 10. B BID. 4200. 23000. 50000. 32000. 14600. 9300. 6800. 5300. (1 t o g r a i n s ) 2.20 0.84 9.0 0.05 270. 1100. 1000. 390. 150 . 70. 40. 30. 20.

Conc re t e 2.28 0.67 l . B 0.012

Mica 1.96 0 .88 0.43 0.002

Rubber (hard) 1.19 1.88 0.16 0.001

Gypsun board 0.S2 0 .11

Sand (dry) 1.52 0.80 0.33 0.003

TABLE 1 8 . 3 . Thermal p r o p e r t i e s of some metals a t 10 K above t h e i r melt ing

point (source: Refs. 35 and 38) .

mp, k, P. c, Y Metal K W/m'k kg/m3 kJ/kg 'k kJ/kg

Aluminum 933.2 91 2390 1.09 395.4 Copper 1356. 167 7940 0.49 205. Iron 1810. 41 7020 0.82 281.6 Lead 600.6 16 10700 0.21 24.7 Lithium 453.7 43 520 4.25 663.2 Mercury 234.3 7 13650 1.34 11.3 Potassium 336.8 54 820 0.84 61.5 Sodium 371.0 87 930 1.34 114.6 NaK (eutectic) 262. 13 850 1.00 ~ Tin 505.1 31 6980 0.23 60.2 Zinc 692.7 50 6640 0.50 102.1

18-5

TABLE 18.4. Room temperature total emissivities (source: Ref. 37).

Silver (highly polished) 0.02 Brass (polished) 0.60 Platinum (highly polished) 0.05 Oxidized copper 0.60 Zinc (highly polished) 0.05 Oxidized steel 0.70 Aluminum (highly polished) 0.08 Bronze paint 0.80 Monei metal (polished) 0.09 Black gloss paint 0.90 Nickel (polished) 0.12 White lacquer 0.95 Copper (polj :..)ied) 0.15 White vitreous enamel 0.95 Stellite (polisned) o.ia Asbestos paper 0.95 Cast iron (polished) 0.25 Green paint 0.95 Monel metal (oxidized) 0.43 Gray paint 0.95 Aluminum paint 0.55 Lamp black 0.95

18-6

TABLE 18.5 . Total e m i s a i v i t i e s of miscellaneous m a t e r i a l s (source: Ref. 37) .

Temp, Emis- Temp, Emis-

Hater i a l °c s i v i t y Mate r i a l °c s i v i t y

Alloys I ron , rus t ed 25 0.65 20Ni-25Cr-55Fe, oxidized 200 0.90 wrought, d u l l 25 0.94

500 0.97 oxid ized 350 0.94 60Ni-12Cr-2BFe, oxidized 270 0.89 Lead, unoxidized 100 0.05

560 0.62 oxid ized 200 0.63 BONi-20Cr, oxidized 100 0.87 Mercury, unoxidized 25 0.10

600 0.87 100 0.12 1300 0.89 Molybdenum, unoxidized 1000 0.13

Aluminum, unoxidized 25 0.022 1500 0.19 100 0.028 2000 0.24 500 0.60 Monel meta l , oxidized 200 0.43

oxidized 200 0.11 600 0.43 600 0.19 Nickel , unoxidized 25 0.045

Bismuth, unoxidized 25 0.048 100 0.06 100 0.061 500 0.12

B r a s s , oxidized 200 0.61 laao 0.19 600 0.59 oxidized 200 0.37

unoxidized 25 0.035 1200 0.85 100 0.035 Platinum, unoxidized 25 0.037

Carbon, unoxidized 25 0.81 100 0.047 100 0.81 500 0.096 500 0.81 1000 0.152

Chromium, unoxidized 100 0.08 1500 0.191 Coba l t , unoxidized 500 0.13 S i l i c a br ick 1000 O.80

1000 0.23 1100 0.85 Columbiura, unoxidized 1500 0.19 S i l v e r , unoxidized 100 0.02

2000 0.24 500 0.035 Copper, unoxidized 100 0.02 S t e e l , unoxidized 100 o.oe

l i q u i d 0.15 l i q u i d 0.28 oxidized 200 0.6 oxid ized 25 0.80

1000 0.6 200 0.79 calor ized 100 0.26 600 0.79

500 0.26 S t ee l p l a t e , cough 40 0.94 ca lo r i zed , oxidized 200 0.18 400 0.97

600 0.19 c a l o r i z e d , oxidized 200 0.52

F i r e br*c* 1000 0.75 600 0.57 Gold, unoxidized 100 0.02 Tantalum, unoxidized 1500 0.21

500 0.03 2000 0.26 Gold enamel 100 0.37 Tin, unoxidized 25 0.043

Icon , unoxidized 100 0.05 100 0.05 oxidized 100 0.74 Tungsten, unaxidized 25 0.024

500 0.84 100 0.032 1200 0.89 500 0.071

c a s t , unoxidized 100 0.21 1000 0.15 l i q u i d 0.29 1500 0.23

c a s t , oxidized 200 0.64 2000 0.2S 600 0.78 Zinc, unoxidized 300 0.05

c a s t , s t rongly oxidized 40 250

0.95 0.95

18-7

TABLE IS.6. Electrical resistivity of some common metals (source: Ref. 37).

R e s i s t i v i t y , Temp. S p e c i f i c M e l t i n g

llfi*cm c o e f f i c i e n t g r a v i t y . p o i n t .

Metal 20°C 20°C g/cra °c Advance. See c o n s t a n t a n

Aluminum 2.824 0 . 0 0 3 9 2.70 659

Antimony 41.7 .0036 6 .6 630

A r s e n i c 33 .3 . 0 0 4 2 5 .73 — Bismuth 120 . 0 0 4 9 .8 2 7 1

Brass 7 . 002 8 .6 900

Cadmium 7 .6 .0038 8.6 321

C a l i d o . See nichrome

Climax 87 .0007 e . i 1250

Coba l t 9 .8 . 0033 B.71 1480

Constantan 49 . 00001 8 .9 1190

Copperi annealed 1.7241 .00393 8 .89 1083

hard-drawn 1.771 .00382 8.89 — Eureka. See c o n s t a n t a n

E x c e l l o 92 .00016 8 .9 1500

Gas carbon 5000 - . 0 0 0 5 — 3500

German s i l v e r , 18% Hi 33 .0004 B.4 1100

Gold 2 .44 .0034 1 9 . 3 1063

I d e a l . See c o n s t a n t a n

i r o n , 99 .984 pure 10 . 005 7 .8 1530

Lead 22 . 0 0 3 9 11 .4 327

Magnesium 4 .6 . 004 1.74 651

Manganin 44 . 0 0 0 0 1 8 . 4 910

Mercury 95.783 .00089 13 .546 - 3 8 . 9

Molybdenum, drawn 5 .7 . 0 0 4 9 . 0 2500

Monel metal 42 .0020 8 .9 1300

Nichrome 3 100 .0004 8 . 2 1500

N i c k e l 7 . 8 .006 8 .9 1452

Pal lad ium 11 . 0033 12 .2 1550

Phosphor bronze 7 . 8 . 0018 8 .9 750

Plat inum 10 .003 21 .4 1755

S i l v e r 1.59 . 0038 1 0 . 5 960

S t e e l , E. B. B. 1 0 . 4 . 005 7 .7 1510

S t e e l , B. B. 1 1 . 9 . 004 7 .7 1510

S t e e l , Siemens-Martin 18 . 0 0 3 7 .7 1510

S t e e l , manganese 70 . 0 0 1 7 . 5 1260

Tantalum 1 5 . 5 . 0 0 3 1 1 6 . 6 2850

T h e r l o a 47 . 0 0 0 0 1 8 . 2 — T i n 1 1 . E . 0 0 4 2 7 . 3 232

Tungsten , drawn 5 . 6 .0045 19 3400

Zinc 5 . 8 .0037 7 . 1 419

"Trade

18-8

TABIB IB.7. Thermal conductivity in tegra ls of miscellaneous mater ials {source: Ref. 76] .

Q - 7 !*_) , k_ - J t d t , W/n x 10~ 2

L i X -ir

C o p p e r s S t e e l M i • c. A l l o y s Temp, Hi-purity A l u m i n u m Stain- Inconel G l a s s a n d p l a s t i c s Soft

K O.F.H.C. annealed Brass (Pb) 2024-T4 6063-T5 SAE 1020 less (annealed) glass nylon perspex Constantan Ag solder solder

6 6. .1 166 0.053 0.080 0. .850 0.088 0.0063 0.0133 2. .11 X 10 " J 0. .32 x 10 ' ' 1.18 x 10"' ' 0.024 0.059 0.425 8 14. 5 382 0.129 0.197 2. .05 0.231 0.0159 0.0348 4. .43 0. .60 2.38 0.066 0.148 1.05 10 25. 2 636 0.229 0.347 3. .60 0.431 0.0293 0.0653 t. .81 1. .48 3.59 0.128 0.268 1.B3 15 61. 4 1270 0.594 0.872 9. .00 1.17 0.0816 0.182 13. .1 4. .10 G.69 0.375 0.688 4.16 20 110 1790 1.12 1.60 16. .5 2.22 0.163 0.356 20, .0 8. .23 10.1 0.753 1.25 6.86

25 169 2160 1.81 2.51 25. .B 3.52 0.277 0.592 27. ,9 13. .9 14.4 1.24 1.92 9.66 30 228 2410 2.65 3.61 36. .5 5.02 0.424 0.882 36. .8 20. B 19.6 1.B1 2.67 12.5 35 285 2580 3.63 4.91 48. .8 6.74 0.607 1.22 47. .1 29. 0 25.9 2.44 3.52 15.3 40 338 2700 4.76 6.41 62. 0 8.67 0.824 1.60 58. .6 3B. 5 33.0 3.12 4.47 1B.1 50 426 2860 7.36 9.91 89. 5 13.1 1.35 2.47 84. 6 60. 4 49.5 4.57 6.62 23.4

60 496 2960 10.4 14.1 117 18.1 1.98 3.45 115 B5. 9 6B.3 6.12 9.12 2B.5 70 554 3030 13.9 18.9 143 23.6 2.70 4.52 151 113 B8.5 7.75 12.0 33.6 76 586 3070 16.2 22.0 158 27.1 3.17 5.19 175 131 101. 8.75 13.9 36.7 SO 606 3090 17.7 24.2 167 29.5 3.49 5.66 194 142 110. 9.43 15.2 15.B 90 654 "140 22.0 30.1 190 35.5 4.36 5.85 240 173 132. 11.1 IB.7 44.1

100 700 3180 26.5 36.3 211 41.7 5.28 8.06 292 204 155. 12.8 22.6 49.4 120 788 3270 36.5 50.1 253 54.5 7.26 10.6 408 269 200. 16.2 31.1 60.3 140 874 3360 47.8 65.4 293 67.5 5.39 13.1 542 336 247. 19.7 40.6 71.4 160 956 3440 60.3 83.1 333 80.5 11.7 15.7 694 405 294. 23.2 51.0 82.6 ISO 1040 3520 ' 73.8 100 373 93.5 14.1 IB.3 85B 475 342. 26.9 62.2 93.8

200 1120 3600 B8.3 119 413 107 16.6 21.0 1030 545 390. 30.6 74.0 105 250 1320 3800 128 171 513 139 23.4 26.0 1500 720 510. 40.6 105 133 300 1520 4000 172 229 613 172 30.6 35.4 1990 895 630. 51.6 138 162

Table 18.8 Specific heat of miscellaneous solids (source: Rets. 35 to 38).

Specific heat (c), J/ kg-K Material 10 20 50 100 200 400 600 800 1000 1200

Aluminum; Pure 1.4 8.9 142 481 790 161 1026 1092 1054 1096 2024 418 515 737 920 1029 1318

Beryl H u m 0.389 1.61 19.2 199 1113 2134 1046 2804 3022 3223 Copper: Pure 0.86 7.7 99 254 360 423 427 434 439 443

Brass 30% Zn) 0.65 10.5 120 255 360 401 42B 448 469 Gold 2.2 15.9 72 108 122 134 135 138 140 142 Iron: Pure 1.24 4.5 55 216 389 485 582 677 774 665

AISI 316 stainless 0.80 13.1 110 265 420 510 560 580 590 600 : SAE 1010 steel 1.88 6.69 75 251 418 628 669 690 711 724

Lead 13.7 53 103 118 121 141 121 10B 103 96 Nickel 1.62 5.8 68 232 385 502 580 539 540 540 Platinum 1.12 7.4 55 100 121 133 142 146 150 155 Silver 1.8 15.5 108 187 226 243 251 259 267 280 Tin 4.2 40 130 189 200 242 Tungsten 0.234 1.89 33 88 122 133 138 142 146 130 Zinc 2.5 26 171 293 352 401 472 489 460 434

Carbon: Graphite 1.38 71 41.8 138 418 1004 1339 1590 1716 1799 Ceramic: A 12°3 0.08 0.84 14.2 125 502 920 1088 1172 1213 1255 Glasses; Borosi Licate

(pyrexi 4.2 25.9 125 272 544 837 1088 Quartz 16.7 16.7 96 255 502 879 1046 1172 1213 1255

Plastics Acryl ic, PMMA (plexiglas) 22 79 250 540 360 1640 NEMA G10 18 89 240 360 630 1110 1600 Teflon PTFE 20 75 200 380 700 1300

18-10

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