steady heat conduction

ENT255 HEAT TRANSFERSchool fo Mechatronic EngineeringUniversiti Malaysia Perlis 1

STEADY HEAT CONDUCTION

AZIZUL BIN MOHAMAD

tTcq

zTk

zTk

rrTkr

rr p

2

11


Steady State 1-D Conduction Assumptions

– Temperature varies in only one coordinate directions

– Energy Balance still applies but time term goes to zero

Soln’s dependent on coordinate systems– Plane Wall

» Linear T» Constant heat flux and rate

– Cylinder» Logarithmic T» Constant heat rate

1,1,2, sss TLxTTxT

2,1,"

2,1,

ssx

ssx

TTLkq

TTALkq

1,2

21

1,2, )/ln(/ln s

ss TrrrrTT

xT

2,1,12 )ln(

2ssr TT

rrLkq


1-D HEAT CONDUCTION1-D heat transfer through a simple or

composite body exposed to convection from both sides to mediums at temperatures Ts,1 and Ts,2 can be expressed as

Where Rtotal is the total thermal resistance between the two mediums.

,1 ,2s s

total

T TQ

R


Thermal Resistance Will utilize circuit analysis methods

making analogy between heat flux and electric current.

Electricity

Thermal Conduction RateFlowPotentialRIVRIRV

/

kAL

LTTkATT

qTT

Rss

ss

x

sscondt

2,1,

2,1,2,1,,


Convection Resistance

Thermal Circuit Planar WallConstant q throughout the circuit

hATThATT

qTT

Rss

ss

convx

ssconvt

1

2,1,

2,1,

,

2,1,,

AhTT

kALTT

AhTT

q

qqq

ssssx

condconvx

2

2,2,2,1,

1

1,1,/1

)(/

)(/1

)(

totx R

TTAhkALAh

TTq

)(/1//1

)( 2,1,

21

2,1,


For a plane wall exposed to convection on both sides, the total resistance is expressed as:

This relation can be extended to plane walls that consist of two or more layers by adding an additional resistance for each additional layer.

,1 ,21 2

1 1total conv wall conv

LR R R Rh A kA h A


The elementary thermal resistance relations can be expressed as follows:

Conduction resistance (plane wall):

Conduction resistance (cylinder):

Conduction resistance (sphere):

wallLRkA

2 1ln( / )2cyl

r rRLk

2 1

1 24sphr rR

r r k


Convection resistance

Interface resistance

Radiation resistance

2 2

1

1

1

( )( )

conv

cinterface

c

radrad

rad s surr s surr

RhA

RR

h A A

Rh A

h T T T T

Rc = thermal contact resistance

hc = thermal contact conductance


Composite Wall Heat rate still constant

across. But Temperature gradient changes

Can use to find internal temperature distributions

AhAkLAkLAkLAhTT

RTT

q

ccBBAA

totx

41

4,1,

4,1,

/1////1)(

)(


Parallel Walls Uniform Temperature

at 1 and 2 Heat flux split between

sections Thermal Resistance

based on local areas Set up system of

equations based upon energy balance into nodes


A 3-m-high and 5-m-wide wall consists of long 16-cm x 22-cm cross section horizontal bricks (k = 0.72 W/m.°C) separated by 3-cm thick plaster layers (k = 0.22 W/m.°C).

There are also 2-cm- thick plaster layers on each side of the brick and a 3-cm-thick rigid foam (k = 0.026 W/m.°C) on the inner side of the wall.

The indoor and the outdoor temperatures are 20°C and -10°C, and the convection heat transfer coefficients on the inner and the outer sides are h1=10W/m2.°C and h2=25W/m2.°C, respectively.

Assuming one-dimensional heat transfer and disregarding radiation, determine the rate of heat transfer through the wall.

Ans: q = 4.38 W Qtotal = 263 W


Radial Steady State Conduction

01

rTkr

rr

1,2

21

1,2, )/ln(/ln s

ss TrrrrTT

xT

2,1,12 )ln(

2ssr TT

rrLkq


Derivation of Cylindrical Conduction

kLrr

R condt 2

ln1

2

,


Composite Radial Wall Similar concept

to the planar wall Apply thermal

circuit to each wall

Add the resistances that are in series


The radial heat rate is constant across the circuit

Can analyze in sections or across complete circuit depending on problem


Spherical Coordinates 2,1,

2111

4ssr TT

rr

kq

21

,11

41

rrkR condt


0

kq

xT

x

Conduction with generation

22

12

2,1,1,2,2

22ssss TT

LxTT

Lx

kLqxT


With equal wall temps Becomes parabolic

sTLx

kLqxT

2

221

2


Once the rate of heat transfer is available, the temperature drop across any layer can be determined from:

The thermal resistance concept can also be used to solve steady heat transfer problems involving parallel layers or combined series-parallel arrangement

T QR


Radial Generation

Typical in electrical heating elements (wires)

01

kq

rTr

rr

so

o Trr

krqxT

2

221

4

steady heat conduction

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