Thermal Conductivity: 11, Development of a Thermal Conductivity Expression for the Special Case

of Prolate Spheroids by MILTON ADAMS and A. 1. LOEB

A straightforward derivation of the heat-flow equation for a prolate spheroid is given. In its simplest form the expression is thermal conduc- tivity equals body factor (determined for the prolate spheroidal vessel) times power divided by temperature drop. The geometry of the pro- late spheroid is reflected in a detailed expression

for the body factor.

1. Introduction N the envelope type of thermal conductivity tests only

the sphere and the spheroid readily permit ,exact mathe- I matical derivations of heat-flow equations. The equa- tion for the case of the sphere is well known. For the special case of the prolate spheroid the derivation is more complex; however, an exact thermal conductivity expression can be developed from the fundamental equation of heat conduction,

The apparatus contains a prolate spheroidal core so con- structed that heat is generated uniformly per unit length along the axis of rotation as required for uniform heat flow. The ceramic specimen, whose thermal conductivity is to be de- termined, is designed and shaped so that its inner and outer surfaces are confocal with the surface of the core.

II. Table of Symbols The principal symbols employed in this paper are as fol-

lows; the corresponding c.g.s. units are also given.

Q = heat crossing surface per unit area per unit time (cal. per

k = thermal conductivity (cal. per sec. per “C. per sq. cm.

V T = temperature gradient. A T = temperature drop ( “C.) . T = temperature (“C.). P = power (watts). e = point of higher temperature. f = point of lower temperature. I = radial coordinate (cm.). e = axial coordinate (cm.). a = semifocal length of spheroidal coordinates (em.). A = area (sq. cm.). dil = a surface element of an isotherm. d r = an element of a line of flow directed tangent to the line of

$ = integration over a closed surface. t = constant: equation of a prolate spheroid. 7 = constant: equation of a hyperboloid of revolution of two

B = body factor.

sec. per sq. cm.).

through a cm.).

flow.

sheets.

111. Derivation It has been shown1 that heat flows out of the core along

hyperbolas normal to the spheroidal surfaces. The surfaces of equal temperature are called isotherms, and the hyperbolas normal to them, lines of flow. The temperature gradient or the rate of change of temperature with respect to the length of the path along the lines of flow is written V T (T = tempera- ture), If the gradient is taken in the direction of heat flow, it is negative.

A. L. Loeb, “Theory of Envelope Type of Thermal Conduc- tivity Tests,” J . Appl . Phys., 22,282-85 (1951).

The fundamental equation of heat conduction states that the amount of heat flowing per unit time across a unit area of the isotherm is proportional a t any point to the temperature gradient at that point.

4

Q = - k V T (1) -+ Q = amount of heat per unit time per unit area flowing across

k = thermal conductivity of the material, V T = temperature gradient. T = temperature.

Q is a vector quantity tangent to the liiles of flow; where lines of flow originate or terminate the vector quantity is said to have respectively a positive or negative divergence ; the diver-

gence of Q is indicated as V .Q. Since there is no source of heat flow in the ceramic material,

and since no heat is absorbed by it, the divergence of Q is zero.

an isotherm.

-+

4 +

+

(2)

Two quantities are measured in the determination of ther- mal conductivity: the power used in maintaining the core a t constant temperature, and the temperature drop between two measured points in the specimen. The power is the total amount of heat flowing in a unit time across any closed sur- face about the core. If any isothermal surface is considered,

then the power is the integral of Q over an isotherm.

v . Q = 0

-+.

P = $QdA (3) $ = integration over a closed surface. Q = magnitude of Q. dA = a surface element of an isotherm.

+

As the gradient represents a rate of change with distance, the drop in temperature between two points on a line of flow is given by

--t dr = an element of a line of flow directed tangent toline of flow. e = point of higher temperature. f = point of lower temperature.

Dividing equation (3) by equation (4) produces equation (5) :

Substituting for VT the expression given by equation (1) gives

-t

dr = magnitude of dr .

(7)

73

74 Journal of The American Ceramic Society-Adants v01. 37, No, 2 d€

a . L f r l p There remains the problem of solving equation (2) and

As indicated,' prolate spheroidal coordinates are conveniently used. These coordinates are defined relative to cylindrical coordinates as follows :

(15 ) substituting the value of Q thus obtained in equation (7). k = - - 2~a2$dq AT

q varies from - 1 to + 1 over an entire isotherm, hence, J f d q = 2 (16)

5 = constant: q = constant:

equation of a prolate spheroid. equation of a hyperboloid of revolution of two

sheets.

y = a(52 - 1 ) 1 / 2 (1 - ,2)%

e = a[q

The numerator of equation (15) should be integrated betmeen the two points where the temperature is measured. (')

[ and q are dimensionless quantities somewhat analogous to angles; n represents+he semifocal length of the isotherms and the flow surfaces, The isotherms are described in this system by the equation 5 = constant. The lines of flow lie in the surfaces described by the equation q = constant.

Equation (2) is a first-degree differential equation; its

The measurements Of temperature are Usually made on the minor axis of the Prolate spheroid; here 9 = 0, hence from equation (10)

r = ~ ( € 2 - 1)1 /2

:. 5 =

(19)

(20) (a2 + P ) % solution involves a single integration, hence one arbitrary

U constant. Its solution in prolate spheroidal coordinates is

constant (12) Substituting equation (20) into equation (18) gives Q = (p - I)% (€2 - q 2 ) ' / 2

(a2 + rj2)*/2 - a (a2 + re2)'/2 + u In Along an isotherm = constant, Q is seen t o depend on q . Although the temperature is constant in an isotherm, the amount of heat per unit area flowing across an isotherm varies along the isotherm. Isotherms crowd together where they are strongly curved; here the temperature gradient is largest, and the amount of heat flowing per unit area is also greatest.

The element of area dA of an isotherm is given in prolate spheroidal coordinates by

= - [ (a2 + r/2)1/2 -+ a (a2 + re2)1/a - u] - P STU AT

(21) r, = distance of point c from major axis of prolate spheroid. y r = distance of Pointf from major axis of Prolate spheroid.

1 (a2 + rf2)'/2 - a (a2 + r,2) '11 + a (a2 + f / 2 ) 1 ' Z + a . (a2 + Y,2) ' /? - a is the

87ra In [ The term

dA = 2*a2(.$2 - 1)'/2((2 - q 2 ) ' / 2 dq (I3) body factor for a prolate spheroidal apparatus. Equation

(21) can be expressed as follows: Thermal conductivity equals body factor times power divided by temperature drop.

( 2 2 ) P (14) k = - B -

AT Substituting equations (12), (13), and (14) into equation (7) gives B = body factor.

A line element of the lines of flow is given by

dr = a ( 5 2 - $2)1/2

( 5 2 - 1 ) V z d [

Thermal Conductivity: Ill, Prolate Spheroidal Envelope Method

Data for A1203, BeO, MgO, Th02, and ZrO2

by MILTON ADAMS

An absolute method of determining the thermal conductivity of refractory materials is described. This investigation confines itself to the steady state, i.e., where the temperature at any given point in the material is independent of time. The variation of conductivity with temperature is shown for alumina, magnesia, zirconia, beryllia, thoria, and insulating iirebrick B & W K-28. The results of previous studies are compared with

those of this investigation.

1. Introduction SIMPLE expression for the flow of heat by conduction is rate equals driving force divided by resistance. I n A this fundamental expression the rate is, of course, the

rate of heat flow; the driving force is the temperature drop across the material, since an inequality of temperature is necessary for heat flow; and the resistance term embraces the factors of surface area and heat-flow path length, and also includes the proportionality factor K , which denotes the ther- mal conductivity.