HW2 Due date Next Tuesday (October 14). Lecture Objectives: Unsteady-state heat transfer -...
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Transcript of HW2 Due date Next Tuesday (October 14). Lecture Objectives: Unsteady-state heat transfer -...
Lecture Objectives:
• Unsteady-state heat transfer - conduction
• Solve unsteady state heat transfer equation for a wall
Implicit methods - example
wioww TTTTT 2)(3
iwii TTTT )(2.0
woiw TTTT 3)23(
iiw TTT )12.0(
400 800 1200 1600 2000 24000
10
20
30
40
50
60
70
80
T[C
]
time
To Tw Ti
=0 To Tw Ti
=36 system of equation Tw Ti
=72 system of equation Tw Ti
After rearranging:
2 Equations with 2 unknowns!
Explicit methods - example
wioww TTTTT 2)(3
iwii TTTT )(2.0
3
)23( owi
w
TTTT
2.0
)12.0(
iw
i
TTT
=0 To Tw Ti
=360 To Tw Ti
=720 To Tw Ti
=36 sec
2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
T [C
]
time
To Tw Ti
UNSTABILITY
There is NO system of equations!
Tim
e
Explicit methods - example
30
)230( owi
w
TTTT
2
)12(
iw
i
TTT
=0 To Tw Ti
=36 To Tw Ti
=72 To Tw Ti =36 sec
400 800 1200 1600 2000 24000
10
20
30
40
50
60
70
80
T[C
]
time
To Tw Ti
Stable solution obtainedby time step reduction
10 times smaller time step
Tim
e
Unsteady-state conduction - Wall
sourcep
qx
T
c
T
2
2
q
Ts
0
T
-L / 2 L /2
h
h
h
To
T
h omogenous wa ll
L = 0.2 mk = 0 . 5 W/ m Kc = 9 20 J/kgK
= 120 0 k g/mp
2
Nodes for numerical calculation
x
Discretization of a non-homogeneous wall structure
Fa
cad
e s
lab
Insu
latio
n
Gyp
sum
Section considered in the following discussion
Discretization in space
2
2
x
T
c
kT
pDiscretization in time
Internal node Finite volume method
2/
2/
2/
2/2
2I
I
I
I
XI
XI
XI
XI
pII dxdx
Tkdxd
Tc
2
2
x
Tk
Tcp
For node “I” - integration through the control volume
( x) I- 1 ( x)I
x I
I-1 I I+1q I -1 to I q I to I+1
Boundaries of control volume
Surface node
2/
2/
I
I
XI
XI
III TTxdxdT
1
111
2/2/
2/
2/
2/
2/2
2
I
III
I
iII
XIXI
XI
XI
XI
XI x
TTk
x
TTk
dx
dTk
dx
dTk
x
Tk
xx
Tk
II
I
I
I
I
Left side of equation for node “I”
Right side of equation for node “I”
dx
TTk
x
TTkdxd
x
Tk
I
III
I
IIIXI
XI
I
I
1
1112/
2/2
2
Mathematical approach(finite volume method)
- Discretization in Time
- Discretization in Space
Mathematical approach(finite volume method)
xx
dx
TTk
x
TTkdxd
x
Tk
I
III
I
III
XI
XI
I
I
1
111
2/
2/2
2
I
III
I
IIIIII x
TTk
x
TTkTTx 111
pc
Explicit method
and for uniform grid
Implicit method
I
III
I
IIIIII x
TTk
x
TTkTTx 111
pc
By Substituting left and right side terms of equation
2/
2/
2/
2/2
2I
I
I
I
XI
XI
XI
XI
pII dxdx
Tkdxd
Tc
we get the following equation for
Using
xdx
][1
111
I
III
I
III
x
TTk
x
TTk
Physical approach(finite volume method)( x) I- 1 ( x)I
x I
I-1 I I+1q I -1 to I q I to I+1
Boundaries of control volume
2
2
x
Tk
Tcp
Change of energy in x
Change of heat flux along x
xxT
kTcp
)(
qx
=
Change of energy in x
Sum of energy that goes in and out of control volume x
=
or
) (1
1 I toI I to1-I
qqx
Tcp
For finite volume x:
1 I toI I to1-I
qqT
xcp
( x) I- 1 ( x)I
x I
I-1 I I+1q I -1 to I q I to I+1
1 I toI I to1-I
qqT
xcp
1I toI I to1-I
qqT
xcp
)(/)(/ 111
IIIIII
p TTxkTTxkTT
xc II
)(/)(/ 11
IIIIII
p TTxkTTxkTT
xc
xx For uniform grid
Physical approach(finite volume method)
Internal node finite volume method
Explicit method
Implicit method
By defining time step on the right side we get
I
III
I
IIIII
IIpI
x
TTk
x
TTkTT
xc
111
I
II
I
IIIII
IIpI
x
TTk
x
TTkTT
xc
11
Internal node finite volume method
I
II
I
IIIII
IIpI
x
TTk
x
TTkTT
xc
11
Explicit method
Implicit method
I
III
I
IIIII
IIpI
x
TTk
x
TTkTT
xc
111
FTCTBTA III
11
),,( 11
IIII TTTfTRearranging:
Rearranging:
Implicit method(internal node)
2
112
1
I
III
I
IIIII
IpI
x
TTk
x
TTkTT
c
IIpI
II
II
I
IIpII
I
I Tc
Tx
kT
x
kcT
x
k
12212)()2()(
AI BI CIFI
Internal nodes
1 2 3 4 5
B1 C1
A2 B2 C2
A3 B3 C3
A4 B4 C4
A5 B5
x =
F1
F2
F3
F4
F5
T1
T2
T3
T4
T5
kI-1=kI+1=kI
Implicit method(surface nodes)
B1 C1
A2 B2 C2
A3 B3 C3
A4 B4 C4
A5 B5
x =
F1
F2
F3
F4
F5
Surface nodes
1 2 3 4 5external internal0 6
T O Air T I Air
F0
F6
B0 C0
A1
C5
B6A6
T1
T2
T3
T4
T5
T0
T6
Surface node: 0
x
For surface nodes: flux coming in = flux going out
2/
100_ x
TTkTTh AirOexternal
Surface node: 6
AirIernal TThx
TTk_6int
65
2/
Calculate B0 and C0
Calculate A6 and B6