MODULE 2: Worked-Out Problems

8/14/2019 MODULE 2: Worked-Out Problems

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MODULE 2: Worked-out Problems

Problem 1:The steady-state temperature distribution in a onedimensional wall of thermal conductivity

50W/m.K and thickness 50 mm is observed to be T(0

C)= a+bx2

, where a=2000

C, B=-20000

c/m2, and x in meters.

(a) What is the heat generation rate in the wall?

(b) Determine the heat fluxes at the two wall faces. In what manner are these heat fluxes

related to the heat generation rate?

Known: Temperature distribution in a one dimensional wall with prescribed thickness andthermal conductivity.

Find: (a) the heat generation rate, q in the wall, (b) heat fluxes at the wall faces and relationto q.

Schematic:

Assumptions: (1) steady-state conditions, (2) one dimensional heat flow, (3) constant

properties.

Analysis:(a) the appropriate form of heat equation for steady state, one dimensionalcondition with constant properties is

3520.

2

.

.

m/W100.2K.m/W50)m/CC2000(2q

bk2]bx2[dx

dk

)bxa(dx

d

dx

dkq

dx

dT

dx

dKq

==

==

+=

=

(b) The heat fluxes at the wall faces can be evaluated from Fouriers law,


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x

''x

dx

dTk)x(q =

Using the temperature distribution T(x) to evaluate the gradient, find

dx

dk)x(q

''x = [a+bx

2]= -2kbx.

The flux at the face, is then x=0

2x

''

20''x

x''

m/W000,10)L(q

m050.0)m/C2000(K.m/W502kbL2)l(q,LatX

0)0(q

=

===

=

Comments: from an overall energy balance on the wall, it follows that

0EEE g.

out

.

in

.

=+ 0Lq)L(q)0(q.

x''''

x =+

352"

x"x

.

m/W100.2m050.0

0m/w00,10

L

)0(q)L(qq =

=

=


3/22

Problem 2:A salt gradient solar pond is a shallow body of water that consists of three distinct fluid layers

and is used to collect solar energy. The upper- and lower most layers are well mixed and

serve to maintain the upper and lower surfaces of the central layer at uniform temperature T1

and T2, where T1>T2. Although there is bulk fluid motion in the mixed layers, there is no such

motion in the central layer. Consider conditions for which solar radiation absorption in thecentral layer provides non uniform heat generation of the form q=Ae-ax, and the temperature

distribution in the central layer is

cbxeka

A)x(T ax

2++=

The quantities A (W/m3), a (1/m), B (K/m) and C(K) are known constants having the

prescribed units, and k is the thermal conductivity, which is also constant.

(a)Obtain expressions for the rate at which heat is transferred per unit area from thelower mixed layer to the central layer and from central layer to the upper mixed layer.

(b)Determine whether conditions are steady or transient.(c)Obtain an expression for the rate at which thermal energy is generated in the entirecentral layer, per unit surface area.

Known: Temperature distribution and distribution of heat generation in central layer ofa solar pond.

Find: (a) heat fluxes at lower and upper surfaces of the central layer, (b) whetherconditions are steady or transient (c) rate of thermal energy generation for the entire

central layer.

Schematic:

Assumptions: (1) central layer is stagnant, (2) one-dimensional conduction, (3)constantproperties.

Analysis (1) the desired fluxes correspond to conduction fluxes in the central layer at the

lower and upper surfaces. A general form for the conduction flux is


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+==

+==

=

=

=

+

Bka

AkqqBe

ka

Akqq

Hence

Beka

Akq

"

)0x(cond

"

u

al"

0Lx(cond

"

l

ax"cond

(b) Conditions are steady ifT/t=0. Applying the heat equation,

t

T1

k

q

t

T.

2

2

=+

t

1e

k

Ae

k

A axax

=+

Hence conditions are steady since

tT

=0 (for all 0


5/22

Problem 3:The steady state temperatures distribution in a one-dimensional wall of thermal conductivity

and thickness L is of the form T=ax3+bx2+cx+d. derive expressions for the heat generation

rate per unit volume in the wall and heat fluxes at the two wall faces(x=0, L).

Known: steady-state temperature distribution in one-dimensional wall of thermalconductivity, T(x)=Ax3+Bx2+CX+d.

Find: expressions for the heat generation rate in the wall and the heat fluxes at the two wallfaces(x=0, L).

Assumptions: (1) steady state conditions, (2) one-dimensional heat flow, (3) homogeneousmedium.

Analysis:the appropriate form of the heat diffusion equation for these conditions is

0kq

dxTd

.

2

2

=+ Or2

2.

dxTdkq =

Hence, the generation rate is

]B2Ax6[kq

]0CBx2Ax3[dx

dk

dx

dT

dx

dq

.

2.

+=

+++=

=

which is linear with the coordinate x. The heat fluxes at the wall faces can be evaluated fromFouriers law,

]CBx2Ax3[kdx

dTkq 2"x ++==

Using the expression for the temperature gradient derived above. Hence, the heat fluxes are:

Surface x=0; (0)=-kC"xq

Surface x=L;"xq (L) = -K [3AL

2+2BL+C]

COMMENTS: (1) from an over all energy balance on the wall, find

BkL2AkL3E

0E]CBL2AL3)[K()kC()L(q)0(q

0EEE

2''

g

.

g

.2

x"

x"

g

.

out

.

in

.

=

=+++=

=+

From integration of the volumetric heat rate, we can also find

BkL2AkL3E

]Bx2Ax3[kdx]B2Ax6[kdx)x(qE

2g

''.

L

0

L0

2.

L0

''

g

.

=

+=+==


6/22

Problem 4:The one dimensional system of mass M with constant properties and no internal heat

generation shown in fig are initially at a uniform temperature Ti. The electrical heater is

suddenly energized providing a uniform heat flux qo at the surface x=0. the boundaries at

x=L and else where are perfectly insulated.

(a)Write the differential equation and identify the boundary and initial conditions thatcould be used to determine the temperature as a function of position and time in the

system.

(b)On T-x coordinates, sketch the temperature distributions for the initial condition(t


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Boundary conditions: x=0 0t)x/Tkq 0"o >=

x=L 0)x/T L = Insulated

(b) The temperature distributions are as follows:

No steady-state condition will be reached since is constant.in..

outin

.

EandEE

(c) The heat flux as a function of time for positions x=0, L/2 and L appears as:

( d) If the heater is energized until t=to and then switched off, the system will eventually reach

a uniform temperature , Tf. Perform an energy balance on the system, for an interval of time

t=te,

.

stin

.

EE = es"os

"0

t0inin tAqdtAqQE

e === )TT(McE ifst =

It follows that = OR es"o tAq )TT(Mc if += if TT

Mc

tAq es"o


8/22

Problem 5:

A 1m-long steel plate (k=50W/m.K) is well insulated on its sides, while the

top surface is at 100

0

C and the bottom surface is convectively cooled by a fluidat 200C. Under steady state conditions with no generation, a thermocouple at the

midpoint of the plate reveals a temperature of 850C. What is the value of the

convection heat transfer coefficient at the bottom surface?

Known: length, surfacethermal conditions, and thermal conductivity of aPlate. Plate midpoint temperature.

Find:surface convection coefficient

Schematic:

Assumptions: (1) one-dimensional, steady conduction with no generation, (2)Constant properties

Analysis: for prescribed conditions, is constant. Hence,

2/L

TTq 21"cond

= = 20

m/W1500k.m/W50/m5.0

C15=

K.m/W30h

m/W1500

W/K.m)h/102.0(

C30

)h/1()k/L(

TT"q

2

2

2

01

=

=+

=+

=

Comments: The contributions of conduction and convection to the thermal

resistance are

W/K.m033.0

h

1R

W/K.m02.0K

LR

2cond,t

"

2cond,t

"

==

==


9/22

Problem 6:

The wall of a building is a composite consisting of a 100-mm layer of common

brick, a 100-mm layer of glass fiber(paper faced. 28kg/m

2

), a 10-mm layer ofgypsum plaster (vermiculite), and a 6-mm layer of pine panel. If the inside

convection coefficient is 10W/m2.K and the outside convection coefficient is

70W/m2.K, what are the total resistance and the overall coefficient for heat

transfer?

Known: Material thickness in a composite wall consisting of brick, glass fiber,

and vermiculite and pine panel. Inner and outer convection coefficients.

Find: Total thermal resistance and overall heat transfer coefficient.

Schematic:

Kb Kgl

Brick Gypsumglass

Pine panel,Kp

100mm 10mm 6mm

hi=10W/m

2.

K

h0=70W/m2.K

0h

1

b

b

K

L

gl

lg

k

L

gy

gy

k

L

p

p

K

L

ih

1

Assumptions: (1) one dimensional conduction, (2) constant properties, (3)

negligible contact resistance.

Properties: T= 300K: Brick, kb=1.3 W/m.K: Glass fiber (28kg/m3), kg1=

0.038W/m.K: gypsum, kgy=0.17W/m.K: pine panel, kp=0.12W/m.K.

Analysis: considering a unit surface Area, the total thermal resistance


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W/K..m93.2R

W/K.m)1.00500.00588.06316.20769.00143.0(R

W

K.m

10

1

12.0

006.0

17.0

01.0

038.0

1.0

3.1

1.0

70

1R

h

1

K

L

k

L

k

L

K

L

h

1R

2"tot

2"tot

2"tot

ip

p

gy

gy

1g

1g

B

B

0

"tot

=

+++++=

+++++=

+++++=

The overall heat transfer coefficient is

.K.m/W341.0U

)W/K.m93.2(R

1

AR

1U

2

12

tot"

tot

=

===

Comments: an anticipated, the dominant contribution to the total resistance is

made by the insulation.


11/22

Problem 7:

The composite wall of an oven consists of three materials, two of which are

known thermal conductivity, kA=20W/m.K and kC=50W/m.K, and known

thickness, LA=0.30m and LC=0.15m. The third material, B, which is sandwiched between materials A and C, is of known thickness, LB=0.15m, but unknown

thermal conductivity k

B

B. Under steady-state operating conditions, measurements

reveal an outer surface temperature of Ts,0=200C, an inner surface temperature

of Ts,i=600 C and an oven air temperature of T =800 C. The inside convection

coefficient h is known to be 25W/m .K. What is the value of k

0 0

2BB?

Known: Thickness of three material which form a composite wall and thermal

conductivities of two of the materials. Inner and outer surface temperatures of

the composites; also, temperature and convection coefficient associated with

adjoining gas.

Find: value of unknown thermal conductivity, kB.

Schematic:

KA KB

LA

T=8000C

h= 25W/m2.K

LB LC

KC

LA=0.3mLB=LC=0.15mkA=20W/m.KkC=50W/m.K

Ts,i=6000C

Ts,0=200C

Assumptions: (1) steady state conditions, (2) one-dimensional conduction, (3)

constant properties, (4) negligible contact resistance, (5) negligible radiation

effects.

Analysis: Referring to the thermal circuit, the heat flux may be expressed as


12/22

2

B

B

0

C

C

B

B

A

A

0,si,s"

m/WK/15.0018.0

580

K.m/W50

m15.0

K

m15.0

018.0

m3.0

C)20600(

K

L

K

L

K

L

TTq

+=

++

=

++

=

The heat flux can be obtained from

2''

02i,s

"

m/W5000q

C)600800(K.m/W25)TT(hq

=

==

Substituting for heat flux,

.K.m/W53.1K

098.0018.05000

580018.0

q

580

K

15.0

B

"B

=

===

Comments:radiation effects are likely to have a significant influence on the

net heat flux at the inner surface of the oven.


13/22

Problem 8:

A steam pipe of 0.12 m outside diameter is insulated with a 20-mm-thick layer

of calcium silicate. If the inner and outer surfaces of the insulation are at

temperatures of Ts,1=800K and Ts,2=490 K, respectively, what is the heat lossper unit length of the pipe?

Known: Thickness and surface temperature of calcium silicate insulation on

a steam pipe.

Find: heat loss per unit pipe length.

Schematic:

D2=0.16m

D1=0.12m

Steam

Ts,1=800K

Ts,2=490K

Calcium silicate insulation

Assumptions: (steady state conditions, (2) one-dimensional conduction, (3)constant properties.

Properties: calcium silicate (T=645K): k=0.089W/m.K

Analysis: The heat per unit length is

m/W603q

)m12.0/m16.0ln(

K)490800)(K.m/W089.0(2q

)D/Dln(

)TT(K2

q

qq

'r

'r

12

2,s1,s

L

r'r

=

=

==

Comments: heat transferred to the outer surface is dissipated to the surroundings

by convection and radiation.


14/22

Problem 9:

A long cylindrical rod of 10 cm consists of a nuclear reacting material

(k=0.0W/m.K) generating 24,000W/m3

uniformly throughout its volume. This

rod is encapsulated within another cylinder having an outer radius of 20 cm anda thermal conductivity of 4W/m.K. the outer surface is surrounded by a fluid at

1000C, and the convection coefficient between the surface and the fluid is

20W/m2.K. Find the temperatures at the interface between the two cylinders and

at the outer surface.

Known: A cylindrical rod with heat generation is cladded with another cylinder

whose outer surface is subjected to a convection process.

Find: the temperature at the inner surfaces, T1, and at the outer surface, Tc.

Schematic:

Assumptions: (1) steady-state conditions, (2) one-dimensional radialconduction, (3), negligible contact resistance between the cylinders.

Analysis: The thermal circuit for the outer cylinder subjected to the

convection process is


15/22

o

'2

2

1o'1

r2h

1R

k2

r/rlnR

=

=

Using the energy conservation requirement, on the inner cylinder,

g

.

out

.

EE =

Find that

21

.

1'

rqq =

The heat rate equation has the form hence,R/Tq '.

=

'''

2'1

'i R/Tandq)RR(qTT ==

Numerical values:

m/W0.754m)1.0(m/W000,24q

W/m.K0398.0m20.02K.m/W20/1R

W/m.K0276.0K.m/W42/1.0/2.0lnR

223'

2'2

'1

==

==

==

Hence

C13030100W/m.K0398.0m/W0.754C100T

C8.1508.50100W/m.K)cccc0276.0(m/W0.754C100T

00C

00i

=+=+=

=+=++=

Comments: knowledge of inner cylinder thermal conductivity is not

needed.


16/22

Problem 10:

An electrical current of 700 A flows through a stainless steel cable having a

diameter of 5mm and an electrical resistance of 6*10-4

/m (i.e. perimeter of

cable length). The cable is in an environment having temperature of 300C, andthe total coefficient associated with convection and radiation between the cable

and the environment is approximately 25W/m2.K.

(a) If the cable is bar, what is its surface temperature?

(b) If a very thin coating of electrical insulation is applied to the cable, with a

contact resistance of 0.02m2K/W, what are the insulation and cable surface

temperatures?

(c) There is some concern about the ability of the insulation to withstand

elevated temperatures. What thickness of this insulation (k=0.5W/m.K) will

yields the lowest value of the maximum insulation temperature? What is the

value of the maximum temperature when the thickness is used?

Known: electric current flow, resistance, diameter and environmental

conditions associated with a cable.

Find: (a) surface temperature of bare cable, (b) cable surface and insulation

temperatures for a thin coating of insulation, (c) insulation thickness which provides the lowest value of the maximum insulation temperature.

Corresponding value of this temperature.

Schematic:

T

Ts

Eg

q

Assumptions: (1) steady-state conditions, (2) one-dimensional conduction in r,(3) constant properties.


17/22

Analysis: (a) the rate at which heat is transferred to the surroundings is fixed by

the rate of heat generation in the cable. Performing an energy balance for a

control surface about the cable, it follows that or, for the bare cable,qEg.

=

.m/W294)m/106()A700(RIwithq).TT)(LD(hLRI 42'e2'

si'e

2 ====

It follows that

C7.778T

)m005.0()K.m/W25(

m/W294C30

Dh

qTT

0s

2

0

i

'

s

=

+=

+=

(b) With thin coating of insulation, there exists contact and convection

resistances to heat transfer from the cable. The heat transfer rate is

determined by heating within the cable, however, and therefore remains the

same,

h1R

)TT(Dq

LDh

1

LD

R

TT

LDh

1R

TTq

c,t

si'

ii

c,t

s

i

c,t

s

+

=

+

=

+

=

And solving for the surface temperature, find

C1153T

C30W

K.m04.0

W

K.m02.0

)m005.0(

m/W294T

h

1R

D

qT

0s

022

c,t

i

'

s

=

+

+

=+

+

=

The insulation temperature is then obtained from

e,t

s

R

TTq

=

Or


18/22

C7.778T

)m005.0(

W

K.m02.0

m

W294

C1153LD

RqC1153qRTT

0i

2

0

i

c,t"

0c,tsi

=

=

==

(c) The maximum insulation temperature could be reduced by reducing the

resistance to heat transfer from the outer surface of the insulation. Such a

reduction is possible Di Di =0.005m. To minimize the maximum temperature, which

exists at the inner surface of the insulation, add insulation in the amount.

mt

mDDDDt icri

0175.0

2

)005.004.0(

22

0

=

=

=

=

The cable surface temperature may then be obtained from

( )

C318.2T

(0.005m)

W

.Km0.02

m

W294

C692.5LD

R

qTqRTT

/RTTq,thatgrecognizin

C692.5T

2.25m.K/W

C30T

0.32)m.K/W0.66(1.27

C30T

m

W294

hence,

(0.04m).Km

W25

1

)2(0.5W/.

005)ln(0.04/0.

(0.005m)

.K/W0.02m

C30T

h

1

2

)/Dln(D

D

R

TTq

0

i

2

0

i

"

ct,

sc,t,si

c,t,is

0

s

0

s

0

s

2

2

0

s

rc,

irc,

i

"

ct,

s'`

=

===

=

=

=

++

=

++

=

++

=

Comments: use of the critical insulation in lieu of a thin coating has the

effect of reducing the maximum insulation temperature from 778.70C to

318.20C. Use of the critical insulation thickness also reduces the cable

surface temperatures to 692.50C from 778.7

0C with no insulation or fro

11530C with a thin coating.


19/22

Problem 11:

The steady state temperature distribution in a complete plane wall of three

different methods, each of constant thermal conductivity, is shown below.

(a)On the relative magnitudes of and of ."3"2andqq

"4

"3andqq

(b)Comment on the relative magnitudes of kA and kB and ok kB BB and kC.(c)Plot the heat flux as a function of x.

Known: Temperature distribution in a composite wall.

Find: (a) relative magnitudes of interfacial heat fluxes, (b) relative magnitudes

of thermal conductivities, and (c) heat fluxes as a function of distance x.

Schematic:

1 2 3 4A B C

Assumptions: (1) steady-state conditions, (2) one-dimensional conduction, (3)constant properties.

Analysis: (a) for the prescribed conditions (one-dimensional, steady state,

constant k), the parabolic temperature distribution in C implies the existence of

heat generation. Hence, since dT/dx increases with decreasing x, the heat flux inC increases with decreasing. Hence,

"

4

"

3 qq >

However, the linear temperatures distributions in A and B indicate no

generation, in which case

"

3

"

2 qq =


20/22

(b) Since conservation of energy requires that and dT/dx/",3"

,3 CB qq = B

Similarly, since thatfollowsit,dT/dx)dT/dx)andqq BA"

B2,

"

A2, >=

AK < KB.

(d)It follows that the flux distribution appears as shown below.

"

xq

x2 x3O x

"

2q

Comments: Note that, with dT/dx)4,C=0, the interface at 4 is adiabatic.


21/22

Problem 12:

When passing an electrical current I, a copper bus bar of rectangular cross

section (6mm*150mm) experiences uniform heat generation at a rate

. If the bar is in ambient air with h=5W/m.

222 A.m/W015.0where,alq == 2. K andits maximum temperature must not exceed that of the air by more than 300C,

what is the allowable current capacity for the bar?

Known: Energy generation, q (I), in a rectangular bus bar.

Find:maximum permissible current.

Schematic:

Assumptions: (1) one-dimensional conduction in x (W>>L), (2) steady-state conditions, (3)constant properties, (4) negligible radiation effects.

Properties: copper: k 400W/m. K

Analysis: the maximum mid plane temperature is

sT

K

qLT +=

2

.2

0

Or substituting the energy balance results,


22/22

,/.

hLqTTs +=

AI

KmWKmW

mmAmW

CI

hkLL

TTI

hence

hk

LLI

hk

LLqTT

o

o

1826

./5

1

./800

003.0003.0)./(015.0

30

)/12/(015.0

,

.1

2015.0

1

2

max

2

1

2

23

0

max

2

1

2.

=

+

=

+

=

+=

+=

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MODULE 2: Worked-Out Problems