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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: