BKF2422

Chapter 2

Principles of Steady-state Heat Transfer in

Conduction(Part 1)

Topic Outcomes

It is expected that students will be able to:

• Explain the mechanisms of conduction heat transfer

• Solve problems using steady-state conduction principles

Contents• 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• For steady state, the equation can be integrated,

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

2112

2

1

2

1

TTxx

k

A

q

dTkdxA

q T

T

x

x

x

Conduction

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

TTTT

x

kAq 2121

21

ConductionConduction Through Solids In Series

T1 A B C

q T2 T3

∆xA ∆xB ∆xC T4

• The heat flow, q must be the same in each layer;

C

RRR

TTTT

X

AkTT

X

AkTT

X

Akq

BAC

c

B

B

A

A

41433221

Ak

xR

A

AA

Ak

xR

B

BB

Ak

xR

C

CC

Try This

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 concrete. The wall surface temperature is 255.4 K inside the cold room and 297.1 K at the outside surface of concrete. The conductivites, k for pine: 0.151; cork board: 0.0433; and concrete: 0.762 W/m.K. Calculate

a)the heat loss in W for 1 m2 and b)the temperature at the interface between wood and cork board.

Answer: (-16.48 W, 256.79 K)

ConductionConduction Through Solids In Parallel

• The heat flow, q must be the sum of the heat flow through A and B,

q

A

B

T1T2

T3T4

4321 TTX

AkTT

X

Akqqq

B

B

A

ABA

T1=T3 and T2=T4 21

2121 11

//TT

RRXAk

TT

XAk

TTq

BABBAA

Try This

From figure above:a)Draw the electrical analog. b)Derive the heat flow equation.

A

B

C

D

E

F

G

T1 T2 T3 T4 T5

ConductionConduction Through A Hollow Cylinder T2 L

r2

q T1

r1

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

• The rate of heat transfer,

dr

dTk

A

q

Conduction Through A Hollow Cylinder

2

1

2

12

2r

r

T

TdTk

r

dr

L

q

rLAdr

dTk

A

q

12

12

12

12

22ln

22

ln

LrLr

LrLr

AA

AAAlm

12

21

rr

TTkAq lm

or

2112ln

2TT

rr

Lkq

Where:

kL

rrTT

q

2ln 12

21

Try This

A thick wall cylindrical tubing of hard rubber (k=1.15 W/m.K) having an 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. Calculate the heat flow.

(Answer: -15.2 W)

ConductionConduction Through a Multilayer Cylinders

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

T1T2T3

T4

r1r2

r3

r4q

A

B

C

Conduction• At steady-state, the heat-transfer rate q, be the

same for each layer.

• The rate of heat transfer,

or

Lkrr

TT

LkrrTT

Lkrr

TTq

CBA 2ln

2ln

2ln 34

43

23

32

12

21

CBA Lkrr

Lkrr

Lkrr

TTq

2ln

2ln

2ln

)(

342312

41

ClmCBlmBAlmA Akrr

Akrr

Akrr

TTq

342312

41 )(

Try This

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 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

a)the heat loss and b)the temperature at the interface between the metal and the insulation.

Answer: (331.7 W, 805.5 K)

ConductionConduction Through a Hollow Sphere T2 r2

q r1 T1

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

• The rate of heat transfer,

dr

dTk

A

q

Conduction

2

1

2

124

r

r

T

TdTk

r

drq

k

rr

TTTT

rr

kq

41111

4

21

2121

21

ConductionCombined Conduction and Convection and

Overall CoefficientTemperature Profile for Heat Transfer with convective

boundary: Plane Wall

Metal wall

Warm liquid ACold fluid B

T1

T2

T3

T4

q

hi

ho

Δx

A

ConductionCombined Convection and Conduction and Overall

Coefficients.• Heat flow with convective boundaries: plane wall

41

41

11TTUA

hkxh

TTAq

oi

AhkAxAh

TTq

oi 1141

4303221 TTAhTTx

kATTAhq i

Where: oi hkxh

U11

1

Overall

Coefficients

ConductionTemperature Profile for Heat Transfer with convective boundary: Cylindrical Wall

T1

T2

T3

T4

ro

ri

ho

hi

Aiq

Ao

Conduction

ooiioiii hAAkLrrAh

U

2ln1

1

Since the areas depends on inside diameter and outside diameter

Where,

and

ooioii AhkLrrAh

TTq

12/ln141

Overall Coefficients of inside area

Overall Coefficients

of outside area

4141 TTAUTTAUq ooii

oiooiioo hkLrrAhAA

U12ln

1

Try This

Water flows at 50C inside a 2.5 cm inside diameter tube such that hi=3500 W/m2.C. The 1 m tube has a wall thickness of 0.8 mm with a thermal conductivity of 16 W/m.C. The outside of tube heat loss by convection in air with ho= 7.6 W/m2.C. Calculate the overall heat transfer coefficient and heat loss to surrounding air at 20C.

(Answers; Uo=7.58 W/m2.C, q=19 W)