Chapter_2 Conduction Heat Transfer

BKF2422 HEAT TRANSFERChapter 2

Principles of steady-state heat transfer in conduction

1

It is expected that students will be able to: Solve problems using steady-state conduction

principles for one-dimensional solid conduction heat transfer in parallel and series

Calculate overall heat transfer coefficient to solve problems related to combined conduction and convection heat transfer mechanism.

Solve the problem related to internal heat generation and determine the critical thickness and of insulation for a cylinder

Apply shape factor to estimate the multidimensional heat transfer

TOPIC OUTCOMES

2

One Dimensional Conduction Heat Transfer◦ Conduction Through a Plane Wall ◦ Conduction Through Solids In Series◦ Conduction Through Solids In Parallel◦ Conduction Through a Hollow Cylinder◦ Conduction Through a Multilayer Cylinders◦ Conduction Through a Hollow Sphere

Combined Conduction and Convection and Overall Heat Transfer Coefficient

Conduction with Internal Heat Generation Critical Thickness of Insulation for a Cylinder Multidimensional Conduction Heat Transfer -

Shape Factor

CONTENTS

3

For steady state, the equation can be integrated,

This equation is basically a matter of putting in values to solve.

HEAT TRANSFER – CONDUCTION

2112

2

1

2

1

TTxx

k

A

q

dTkdxA

q

x

T

T

x

x

x

4

Conduction Through a Plane Wall.

The temperature various linearly with distance.


0 ΔxΔx Distance,x (m)

T1

T2

Temperature, (K)

T2

T1q

R

TT

kAx

TTq 2121

5

Calculate the heat loss per m2 of surface area for an insulating wall composed of 25.4 mm thick fiber insulating board, where the inside temperature is 352.7 K and the outside temperature is 297.1K.(Thermal conductivity for fiber insulating board is 0.048 W/m.K.)

Answer: 105.1 W/m2

EXERCISE 1

6

Conduction Through Solids In Series.

T1 A B C q T2 T3

DxA DxB DxC T4

The rate of heat transfer,

where, C

RRR

TT

R

TT

R

TT

R

TTq

BACBA

41433221

Ak

xR

A

AA

Ak

xR

B

BB

Ak

xR

C

CC


7

A cold storage room is constructed of an inner layer of 12.7 mm of pine, a middle layer of 101.6 mm of cork board and an outer layer of 76.2 mm of concrete. The wall surface temperature is 255.4 K inside the cold room and 297.1 K at the outside surface of concrete. The conductivities for pine, 0.151; cork board, 0.0433; and concrete, 0.762 W/m.K. Calculate the heat loss in W for 1 m2 of the wall and the temperature at the interface between wood and cork board.

Answer: (-16.48 W, 256.79 K)

EXERCISE 2

8

Conduction Through Solids In Parallel


A

B

C

D

E

F

G

1 2 2 3 2 3 2 3( ) ( ) ( ) ( ) .......

T A B C D E F G

C CA A B B D DA

A B C D

q q q q q q q q

k Ak A k A k Aq T T T T T T T T

x x x x

9

Conduction Through A Hollow Cylinder. T2 L

r2

q r1

The cross-sectional area normal to the heat flow is, A =2prL.



dr

dTk

A

q

10

CONDUCTION THROUGH A HOLLOW CYLINDER

2

1

2

12

r

r

T

TdTk

r

dr

L

qdr

dTk

A

q

12

12

12

12

12

12

ln

)(2

22ln

22

ln

rr

rrL

LrLr

LrLr

AA

AAAlm

12

21

rr

TTkAq lm

or

2112ln

2TT

rr

Lkq

Where:

kL

rr

kA

rrR

R

TT

kArr

TTq

lm

lm

2

ln 1212

21

12

21

11

A thick wall cylindrical tubing of hard rubber having and inside radius of 5 mm and an outside radius of 20 mm is being used as a temporary cooling coil in a bath. Ice water is flowing rapidly inside, and the inside wall temperature is 274.9 K. The outside temperature is at 297.1 K. A total of 14.65 W must be removed from the bath by the cooling coil. How many m tubing are needed?(k=1.15 W/m.K)

Answer: 0.964 m

EXERCISE 3

12

Conduction Through a Multilayer Cylinders.

Example, heat is being transferred through the walls of an insulated pipe.


T1T2T3

T4

r1r2

r3

r4q

A

B

C

13

At steady-state, the heat-transfer rate q, is the same for each layer.


where,


CBACBA RRR

TT

R

TT

R

TT

R

TTq

41433221

Lk

rrR

AA 2

ln 12

Lk

rrR

BB 2

ln 23

Lk

rrR

CC 2

ln 34

14

A thick walled tube of stainless steel (A) having a k = 21.63 W/m.K with dimensions of 0.0254 m ID and 0.0508 m OD is covered with a 0.0254 m thick layer of insulation (B), k = 0.2423 W/m.K. The inside wall temperature of the pipe is 811 K and the outside is at 310.8 K. For a 0.305 m length pipe, calculate the heat loss and also the temperature at the interface between the metal and the insulation.

Answer: (331.7 W, 805.5 K)

EXERCISE 4

15

Conduction Through a Hollow Sphere T2 r2

q r1 T1

The cross-sectional area normal to the heat flow is, A = 4r2.



dr

dTk

A

q

16

R

TT

krr

TTTT

rr

kq

dTkr

drq r

r

T

T

21

21

2121

21

2

41111

44

2

1

2

1


17

EXERCISE 5 Consider a spherical container of inner

radius = 8 cm, outer radius = 10 cm, and thermal conductivity k = 45 W/m.K. The inner and outer surfaces of the container are maintained at constant temperatures of T1 = 200oC and T2 = 80oC, respectively, as a result of some chemical reactions occurring inside. Determine the rate of heat loss from the container.

Answer: 27.1 kW

18

Temperature Profile for Heat Transfer By Convection From One Fluid To Another.


film filmMetal wall

Warm liquid A

Cold fluid B

T6

T5

T4

T3

T2 T1

q

19

Region,T1 – T2 : turbulent fluid flow. Mainly convective heat transfer.T2 – T3 : velocity gradient very steep. No turbulent flow,

(i.e. only laminar). Mainly conductive heat transfer.T3 – T4 : conductive heat transfer.T4 – T5 : no turbulent in film, mainly conductive heat transferT5 – T6 : turbulent flow, convective heat transfer.T1 – T2 and T5 – T6 : different are small.

Convective coefficient for heat transfer through a fluid:q = hA(T – Tw)

where,h = convective heat transfer coefficient.T = average temperature in fluid.Tw = temperature of wall in contact.


20

Combined Convection and Conduction and Overall Coefficients.

Heat flow with convective boundaries: plane wall


oAi

oi

oAAi

oi

RRR

TT

AhAkxAh

TTq

11

oi

oAAi

oi TTUAhkxh

TTAq

11

TiT1

T2

To

q

hi

ho

DxA

21

Heat flow with convective boundaries: cylindrical wall with insulation.


T1

T2

T3

T4

rO

ri

r1ho

hi

A

B

22

oAAoioiioo

ooiAAiioii

ooii

ooAAioii

hAkArrhAAU

hAAAkArrhU

R

TTTTAUTTAUq

R

TT

AhAkrrAh

TTq

lm

lm

lm

1

1

1

1

11

414141

4141

Similarly,

Overall heat transfer coefficient,

where,

and

23

Other way we can used,

where,

HEAT TRANSFER

414141 TTAUTTAU

RRRR

TTq ooii

oBAi

Lk

rrR

A

iA 2

ln 1

Lk

rrR

B

oB 2

ln 1

iiiii AhhLr

R1

2

1

ooooo AhhLr

R1

2

1

24

Water flows at 50oC inside a 2.5 cm inside diameter tube such that hi = 3500 W/m2.oC. The tube has a wall thickness of 0.8 mm with a thermal conductivity of 16 W/m.oC. The outside of the tube loses heat by free convection with ho = 7.6 W/m2.oC. Calculate the overall heat transfer coefficient and heat loss to surrounding air at 20oC. Assume unit length.

Answer: 7.577 W/m2.oC; 19 W

EXERCISE 6

25

If outer radius < rcr: adding more insulation will increase heat transfer rate

If outer radius > rcr: adding more insulation will decrease heat transfer rate

CRITICAL THICKNESS OF INSULATION FOR A CYLINDER

h

kr cr )( 2

26

Calculate the critical radius of insulation for asbestos (k = 0.17 W/m.oC) surrounding a pipe and exposed to room air at 20oC with h = 3.0 W/m2.oC. Calculate the heat loss from a 200oC, 5 cm diameter pipe when covered with the critical radius of insulation and without insulation.

Answer: 5.67 cm; 105.7 W/m (w insulation); 84.8 W/m (w/o insulation)

EXERSICE 7

27

Conduction with Internal Heat Generation. Heat generated inside the conducting medium. Heat conducted only in the x direction. The other walls

are assumed to be insulated.


Plane wall with internal heat generation at steady state.

L LTo

Tw Tw

T

0

x -x

q’(generation rate)

28

The temperature profile is,

The center temperature is,

The total heat generated, q’T,

where, A = cross-sectional area.


LAqq

Tk

LqT

Txk

qT

T

wo

o

2

2

2

..

2.

2

.

29

A plane wall of thickness 0.1 m and thermal conductivity 25 W/m.K having uniform volumetric heat generation of 0.3 MW/m3 is insulated on one side, while the other side is exposed to a fluid at 92oC. The convection heat transfer coefficient between the wall and the fluid is 500 W/m2.K. Determine the temperature of the insulated surface. Assume unit area.

Answer: 212oC30

EXERCISE 8

Heat generation in cylinder,

The temperature profile is,

The center temperature is,

The total heat generated,


k

LTW TW

R R

q’ q’

To

wTrRk

qT 22

.

4

wo Tk

RqT

4

2.

LRqqT2

..

31

A long homogeneous resistance wire of radius ro = 0.5 cm and thermal conductivity k = 13.5 W/m.oC is being used to boil water at atmospheric pressure by the passage of electric current. Heat is generated in the wire uniformly as a result of resistance heating at a rate of = 4.3 × 107 W/m3. If the outer surface temperature of the wire is measured to be Ts = 108oC, determine the temperature at the centerline of the wire when steady operating conditions are reached.

Answer: 128oC32

EXERCISE 9

q

GRAPHICAL ANALYSIS◦This method is used to predict the 2 dimensional heat

flow.◦Accuracy depends on the method of sketching

MULTIDIMENSIONAL - SHAPE FACTOR

T1

T2

33

q

q

y

x

T

Curvilinear section

The heat flow across the curvilinear section follows the Fourier’s law:

Heat flow is the same within heat-flow lane. ∆x ≈ ∆y; so that the heat flow is proportional with ∆T

across the element.

N is the no. of T increment between inner and outer surface.

CONT’

(1)T

q k xy

overallTT

N

34

CONT’

The total heat transfer

M is the heat flow lanes, where M=8.2, and N=4.

The ratio of M/N is called the conduction shape factor (S).

2 1( )overall

M Mq k T k T T

N N

35

Determine the total heat transfer through the walls of the flue in figure of slide 33, if T1=600K, T2=400 K, k=0.90 W/m.K and L=5 m.

EXAMPLE 1

1 2

8.24 ( ) 4 0.90 5 (600 400)

4

Mq kL T T x x x

N

7380W

36

Scan table 3.1 pg 84 (Holman JP)

37

A horizontal pipe 15 cm in diameter and 4 m long is buried in the earth at a depth of 20 cm. The pipe wall temperature is 75oC, and the earth surface temperature is 5oC. Assuming that the thermal conductivity of the earth is 0.8W/m.oC, calculate the heat lost by the pipe.

The heat flow is

EXAMPLE 2

1 1

2 2 (4)

cosh ( / ) cosh (20 / 7.5)

LS

D r

15.35m

(0.8)(15.35)(75 5) 859.6q kS T W

38

For a 3D wall, separate shape factors are used:

3-DIMENSIONAL GEOMETRY

0.54

0.15

wall

edge

corner

AS

LS D

S L

A= Area of wall

L= Wall thickness

D= Length of edge

39

A small cubical furnace 50 by 50 by 50 cm on the inside is constructed of fireclay brick (k = 1.04 W/m.oC) with a wall thickness of 10 cm. The inside of the furnace is maintained at 500oC, and the outside is maintained at 50oC. Calculate the heat lost through the walls.

Answer: 8.592 kW

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EXERCISE 10

End of Chapter 2

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Chapter_2 Conduction Heat Transfer