(9 10) Transient Heat Conduction

8/13/2019 (9 10) Transient Heat Conduction

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TRANSIENT HEAT CONDUCTION

Associate Professor

IIT Delhi

E-mail: [email protected]


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1.Lumped System Analysis

. ,

long cylinders, and spheres with spatial effects

3. Transient heat conduction in semi-infinite solids


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Lumped System

Cooling of a hot metal forging

P.Talukdar/Mech-IITD


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Energy balanceHeat transfer into

the body during dt

The increase in

the energy of the

body during dt

=

dTmCdt)TT(A.h ps =Vm =

)TT(ddT =

ttanconsT =

dtVC

A.h

TT

)TT(d

p

s

=

tVC.

TTtln

p

s

i =

btT)t(T A.h 1


i TT s

VCp=


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)s(

A.hb 1s =

b has the unit of 1/s.

p

t

i

eTT

=

This equation enables us to determine thetemperature T(t) of a body at time t, or

alternatively, the time t required for the

empera ure o reac a spec e va ue .



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The temperature of a body approaches the )s(A.h

b 1s =ambient temperature Texponentially.

The temperature of the body changes rapidly at

the beginning, but rather slowly later on.

p

A large value of b indicates that the body will

approach the environment temperature in a short

time.e arger e va ue o e exponen , e

higher the rate of decay in temperature.

Note that b is proportional to the surface area,

specific heat of the body. This is not surprising

since it takes longer to heat or cool a larger

mass es eciall when it has a lar e s ecific

heat.



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Heat TransferRate of convection heat transfer between the body and its environmentat that time

The total amount of heat transfer between the body and the surrounding medium

=

= T)t(ThA)t(Q s Watt

body:

[ ]ip T)t(TmCQ = kJ

[ ]ipmax TTmCQ = kJ



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LAL

LLthicknesswallPlanec

==2

)2(

Criterion for lumped

s

cA

VL =

k

hL

Bic

=

When the convectioncoefficient h is high and k is

low, large temperature


inner and outer regions of a

large solid.


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Transienttemperature

distribution



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Fourier Number()



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The specified surface temperature corresponds

o e case o convec on o an env ronmen awith a convection coefficient h that is infinite



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Transient heat conduction in large plane

walls, long cylinders, and spheres with

spat a e ects



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2

?tx2 =

trr

rr =

-



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Plane WallGoverning equation

T1T2

2 =

Transient temperature profiles in a plane wallexposed to convection from its surfaces for Ti > T

Governing equation along with boundary conditions

shows that T is a function of x, t, k, , h, Ti and T



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- .

=

TT

TT

Governing equation (Non-dimensional form)

=

2

2

X

Initial condition Boundary conditions

, =

0=

=

.Bi

The non-dimensionalization enables us to present the

temperature in terms of three parameters only:X, Bi, and P.Talukdar/Mech-IITD


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Solution

=

2

2

Solution takes the form of an

The solution of 1D transient heat conduction

involves infinite series, which are difficult to dealwith. However, the terms in the solutions

infinite series ,

> 0.2, keeping the first term and neglecting all

the remaining terms in the series results in an

error under 2 percent. We are usually interested)X(cos)exp(A n2

nn =

in the solution for times with > 0.2, and thus itis very convenient to express the solution using

an one term approximation.

1n=

nn

nn

2sin2

sin4A

+

=

See next slide for this expressionsBitan nn =



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Transient mid-plane temperature chart for a

plane wall of thickness 2L initially at a uniform


empera ure i su ec e o convec on rom o

sides to an environment at temperature T with a

convection coefficient of h.


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Transient heat transfer

chart for a plane wall of thickness2L initially at a uniform

temperature Ti subjected toTTVCTTC.m == kJ


environment at temperature T

with a convection coefficient of h.


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Transient Conduction in Semi-Infinite Solid

Consider a semi-infinite solid (extends to infinity in all

but one direction with a single identifiable surface) ( ) iTxT =Case 1

T(x,0) = TiT(0,t) = Ts

Case 2

T(x,0) = Ti

Case 3

T(x,0) = Ti

T&

0x0xx =

==

( )t,0TTh

xk

0x

=

=

T Q ,

x x x

Ts x, Tt

tt

Ti

Ti

Ti


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( )

=

x

erf

Tt,xT s ( ) ( )TTk

tQ iss

=&

Case 1 Constant Surface Temperature: T(0,t) = Ts

and

t2TT si

xxxtq2 2o2

1

&

Case 2 Constant Surface Heat Flux: Qs = Qo

=t2

er ckt4

expk

Tt,xT i

T ,x 0x

=

=

2

Case 3 Surface Convection:

+ + = kt2erfckkexpt2erfcTT,

2i

The Gaussian error function, erf w , is a standard 2mathematical function (see Table B.2 for values)

Complementary Error Function, erfc(w) = 1 erf(w)

( )

=0

dvewerf


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BA

i,BBi,AAs

mm

TmTmT

++=Ts between 2 solids



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(9 10) Transient Heat Conduction

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