announcement
DESCRIPTION
Announcement. Next class is in ECJ 302 (computer lab) No class on October 17 Conference Make up will be an out of class time final project presentation (during the final week). Lecture Objectives:. Analyze the unsteady-state heat transfer numerical calculation methods. Example:. - PowerPoint PPT PresentationTRANSCRIPT
Announcement
• Next class is in ECJ 302 (computer lab)
• No class on October 17 – Conference– Make up will be an out of class time final
project presentation (during the final week)
Lecture Objectives:
• Analyze the unsteady-state heat transfer numerical calculation methods
Unsteady-state heat transfer(Explicit – Implicit methods)
QT
mcp
• Example:
Ti To
Tw
Ao=Ai
To - known and changes in timeTw - unknownTi - unknownAi=Ao=6 m2
(mcp)i=648 J/K(mcp)w=9720 J/K
Initial conditions: To = Tw = Ti = 20oCBoundary conditions:
hi=ho=1.5 W/m2
Time [h] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
To 20 30 35 32 20 10 15 10
Time step =0.1 hour = 360 s
boundariesatp QT
mc _
Conservation of energy:
Explicit – Implicit methods example
wiwoww
wp TThATThATT
mc
Conservation of energy equations:
Wall:
iwii
ip TThATT
mc
Air:
wioww TTTTT 2)(3 Wall:
iwii TTTT )(3.0 Air:
After substitution: For which time step to solve:+ or ?
+ Implicit method Explicit method
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
=360 sec
2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
T [C
]
time
To Tw Ti
UNSTABILE
There is NO system of equations!
Tim
e
Explicit method
Problems with stability !!!
Often requires very small time steps
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
Explicit methods information progressing during the calculation
QT
mcp Ti To
Tw
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
T
p
Discretization 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
IIII
II
p TTxkTTxkTT
xc II
)(/)(/ 11
IIII
II
p TTxkTTxkTT
xc
xx For uniform grid
Physical approach(finite volume method)