Chapter 4 Unsteady-State Conduction 4-1 INTRODUCTION Application of separation-of variables method in the determination of temperature distribution in.

Chapter 4 Unsteady-State Conduction

T

x

T 12

2

1

2

2

x

][0,20

][0,00

][20,01

cLxat

bxat

aLxatTTii

022

2

Xdx

Xd 02 Hd

dH

2

)sincos( 21 exCxC

4-1 INTRODUCTION

Application of separation-of variables method in the determination of temperature distribution in an infinite plate subjected to sudden cooling of surfaces.

)()(),( HxXx Assuming results in

02

1TT

......3,2,12

nL

n

1

]2/[

2sin

2

n

Lnn L

xneC

......3,2,14

2sin

1 2

0

nn

dxL

xn

LC

L

iin

1

]2/[

1

1 ......3,2,12

sin14 2

n

tLn

ii

nL

xne

nTT

TT

From boundary condition [b], C1=0

From boundary condition [c]

02sin L or

Final series form of the solution is

4-2 LUMPED-HEAT-CAPACITY SYSTEM

d

dTVcTThAq )(

0 atTT o

]/[ cVhA

o

eTT

TT

hA

Vc

Time constant ( 时间常数 )

Energy balance:

When the time equals to time constant,

8.36.0

TT

TT

o

Applicability of Lumped-Capacity Analysis

1.0)/(

k

AVh

BinumbrBiotk

hs _

Characteristic dimension: AVs /

4-3 TRANSIENT HEAT TRANSFER IN A SEMI-INFINITE SOLID

T

x

T 12

2

0),0(

)0,(

forTT

TxT

o

i

2

),( xerf

TT

TxT

oi

o

2/ 22

2

x

dex

erf The initial temperature of the semi-infinite solid is Ti, the surface is suddenly lowered to T0. Seek an expression for the T distribution in the solid as a function of time.

The problem is solved by Laplace-transform technique.

Gauss error function:

2/ 22),( x

oi

o deTT

TxT

Constant surface temperature

x

TkAqx

4/

4/

2

2

)2

(2

)(

xoi

xoi

eTT

x

xeTT

x

T

)( io

o

TTkAq

At surface the heat flow is

2/ 22),( x

oi

o deTT

TxT

Heat flow at any x position:

Constant Heat Flux on Semi-Infinite Solid

iTxT )0,(

00

forx

Tk

A

q

x

o

)2

1()4

exp(/2 2

x

erfkA

xqx

kA

qTT ooi

Energy Pulse at Surface

)4/exp(])(/[ 22/1 xcAQTT oi

as x allfor 0iTT

4-4 CONVECTION BOUNDARY CONDITIONS

00)(

xx x

TkATThA

)](1[)][exp(1 2

2

k

hXerf

k

h

k

hxerfX

TT

TT

i

i

)2/( xX

solid of re temperatuinitialTi uret temperatenvironmenT

T

x

T 12

2

For a semi-infinite solid with a convection boundary condition

The solution is:

k

h2

2

x

k

hx

)

k

h,

2

x(f

TT

TT

i

i

)k

h,

2

x(f

TT

TT

i

i

Heisler Charts

TT

TT

TrTorTxT

oo

ii

),(),(

TT

TT

T),x(T

oo

ii

k

hsBnumberBiot i

22o sFNumberFourier

cs

k

TT

TT

T),x(T

oo

ii

k

hsBnumberBiot i

22o sFNumberFourier

cs

k

TT

TT

T),r(T

oo

iik

hsBnumberBiot i

22o sFNumberFourier

cs

k

TT

TT

T),r(T

oo

ii

k

hsBnumberBiot i

22o sFNumberFourier

cs

k

TT

TT

T),r(T

oo

ii

k

hsBnumberBiot i

22o sFNumberFourier

cs

k

TT

TT

T),r(T

oo

ii

k

hsBnumberBiot i

22o sFNumberFourier

cs

k

TT

TT

T),x(T

oo

ii

oi

o

i

TT

TT

T),r(T

oo

ii

oi

o

i

oi

o

i

TT

TT

T),r(T

oo

ii

iio cVTTcVQ )(

iio cVTTcVQ )(

iio cVTTcVQ )(

The Biot and Fourier Numbers

k

hsBnumberBiot i

22o sFNumberFourier

cs

k

oiFBcs

k

k

hs

cs

h

cV

hA 2

Applicability of the Heisler Charts

2.02 s

Fo

In Lumped Heat Capacity analysis, characteristic dimension can be defined as

A/Vs The time constant becomes

4-5 MULTIDIMENSIONAL SYSTEMS

T

z

T

x

T 12

2

2

2

Governing eq.

Initial and boundary conditions:

iTzxT )0,,(

)),,0((0

TzThdx

dTk

x

)),0,((0

TxThdz

dTk

z

)),,2(( 12 1

TzLThdx

dTk

Lx

)),2,(( 22 2

TLxThdz

dTk

Lz

TT

TT

i

1

2

2

2

2

zx

h

dxd

kx

0

h

dxd

kLx

12

h

dzd

kz

0

h

dzd

kLz

22

1)0,,( zx

Definition:

Governing eq.

Initial and boundary conditions:

1

21

2 1x

Initial and boundary conditions:

10

1 h

dxd

kx

22

2

2

h

dzd

kLz

1)0,(1 x

20

2 h

dzd

kz

12

1

1

h

dxd

kLx

1)0,(2 x

z

z1

22

2

For plate 1 with thickness 2L1

Initial and boundary conditions:

For plate 2 with thickness 2L2

),(),(),,( 21 zxzx

TTTxT

i

),(1

TTTzT

i

),(2

To be shown that

),(11 x ),(22 z

),(),(),,( 21 zxzx

21

2

22

2

xx

22

2

12

2

zz

1

22

1

21

2

222

2

1 xz

)(1

21

2

222

2

122

2

121

2

2 xzzx

1

21

2 1x

2

22

2 1z

Dimensionless temperature distribution can be expressed as a product of the solutions for the two plate problems

10

1 h

dxd

kx

h

dz

dk

z

0

h

dzd

kLz

22

111)0,()0,()0,,( 21 zxzx

210

12 h

dx

dk

x

h

dxd

kx

0

h

dxd

kLx

12

In a similar manner,

Conclusion: ),(),(),,( 21 zxzx

Heat Transfer in Multidimensional Systems

121

1ooototalo Q

Q

Q

Q

Q

Q

Q

Q

213121

111oooooototalo Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

4-6 TRANSIENT NUMERICAL METHOD

T

cy

T

x

Tk )( 2

2

2

2

)T2TT(x

1

x

Tn,m

pn,1m

pn,1m

p22

2

)T2TT(y

1

y

Tn,m

p1n,m

p1n,m

p22

2

p

nmpnm TTT ,1

,

pnm

pnmnm

pnm

pnm

pnm

pnm

pnm

p TT

y

TTT

x

TTT ,1

,2

,1,1,

2

,,1,1 1

)(

2

)(

)2(

yx

pnm

pnm

pnm

pnm

pnm

pnm T

xTTTT

xT ,21,1,,1,12

1, ]

)(

41[)(

)(

4)( 2

x

pm

pm

pm

pm T

xTT

xT ]

)(

21[)(

)( 21121

2)( 2

x

2)( xM

systems ldimensiona- two4

systems ldimensiona-one 2

)( 2

M

MxM

)(4

11,1,,1,1

1,

pnm

pnm

pnm

pnm

pnm TTTTT

For one-dimensional problem:

)(

TThAx

TkA w

wall

)()( 11 TTyhTTx

yk mmm

kxh

TkxhTT mm /1

)/(1

pnm

pnmp

nm

pnm

pnm

pnm

pnm

pnm

pnm

TTy

xcTTyh

y

TTxk

y

TTxk

x

TTyk

,1

,,

,1,,1,,,1

2)(

22

Boundary conditions

}42)(

22{()(

,

2

1,1,,121

,

pnm

pnm

pnm

pnm

pnm

Tk

xhx

TTTTk

xh

xT

}22)(

22{()(

2

121

,pm

pm

pnm T

k

xhxTT

k

xh

xT

case ldimensiona- twofor the )2(2

case ldimensiona-one for the )1(2)( 2

k

xhk

xhx

case ldimensiona- twofor the )2(2

case ldimensiona-one for the )1(2)( 2

k

xhk

xhx

if yx

Convergence condition:

pnm

pnm

pnm

pnm

pnm

pnm

pnm

pnm TT

x

TTT

x

TTT ,1

,2

1,

11,

11,

2

1,

1,1

1,1 1

)(

2

)(

2

1,2

11,

11,

1,1

1,12, ]

)(

41[)(

)(

pnm

pnm

pnm

pnm

pnm

pnm T

xTTTT

xT

k

xhBi

2)( xFo

Forward and Backward Differences

Forward difference and explicit formulation

pnm

pnm

pnm

pnm

pnm

pnm T

xTTTT

xT ,21,1,,1,12

1, ]

)(

41[)(

)(

Backward difference and implicit formulation

向前差分：将时间步长末时节点的温度用时间步长起点时周围节点的温度表示的差分方法。

向后差分：空间微分用当前时刻温度表示的差分方法。

4-7 THERMAL RESISTANCE AND CAPACITY FORMULATION

p

ip

i TTVc

E 1

iiii VcC

p

ip

ii

j ij

pi

pj

i

TTC

R

TTq

1

Forward difference:

j j

pi

ijiiij

pj

ipj T

RCCR

TqT )

11()(1

j iji RC

01

1

stabilityfor /1

min

jij

i

R

C

j

pi

ij

pi

pj

ii

pi T

R

TTq

CT )(1

Stability requirement:

Consideration on round-off error

Vqq ii

iradii Aqq '',

Heat source term:

For radiation input to the node,

'',radiq =net radiant energy input to the node per unit area

p

ip

ii

j ij

pi

pj

i

TTC

R

TTq

111

i iij

ip

iiijpjip

i CR

TCRTqT

/)/1(

)/()/( 11

Backward difference: