lecture objectives: analyze the unsteady-state heat transfer conduction introduce numerical...
TRANSCRIPT
Lecture Objectives:
• Analyze the unsteady-state heat transfer • Conduction
• Introduce numerical calculation methods • Explicit – Implicit methods
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/m2Time [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 )(2.0 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
NON-STABILE
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
2.0
)12.0(
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
kT
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/
2/2
2 I
I
I
I
I
I
XI
XI
XI
XI
XI
XI
pII dxdqdxdx
Tkdxd
Tc
sourcep qx
Tk
Tc
2
2
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
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
Internal node finite volume method
- Discretization in Time
- Discretization in Space
Internal node finite volume method
xx
dx
TTk
x
TTkdxd
x
Tk
I
III
I
IIIXI
XI
I
I
1
1112/
2/2
2
I
III
I
III
x
TTk
x
TTk 111
Explicit method
For uniform grid
Implicit method
I
III
I
III
x
TTk
x
TTk 111
Internal node finite volume method
Explicit method
Implicit method
Substituting left and right sides:
qx
TTk
x
TTkTT
xc
I
III
I
IIIII
III 111
qx
TTk
x
TTkTT
xc
I
II
I
IIIII
III
11
Internal node finite volume method
qx
TTk
x
TTkTT
xc
I
II
I
IIIII
III
11
Explicit method
Implicit method
qx
TTk
x
TTkTT
xc
I
III
I
IIIII
III 111
FTCTBTA III
11
),,( 11
IIII TTTfTRearranging:
Rearranging:
Energy balance for element’s surface node
III
ISIII
II
cxT
x
kT
x
kcxT
x
kT
2
21
Implicit equation for node I (node with thermal mass):
or
n
iiRadiatSiSConvASSI qTThTT
x
kTT
1.,. )( )(
2)(
Implicit equation for node S (node without thermal mass):
or
n
iiRadiatSConvA
n
iiRadiatConvSI qThThh
x
kT
x
kT
1.,.
1.,. )(
22After formatting:
Energy balance for element’s surface node
General form for each internal surface node:
After rearranging the elements for implicit equation for surface equations:
FTSTRTBTAn
iSiAII
11
FTBTA SI
General form for each external surface node:
Unsteady-state conductionImplicit method
1 2 3 4 5 6
Matrix equation
M × T = F
for each time step
Air Air
b1T1 + +c1T2
+=f(Tair,T1,T2
)
a2T1 + b2T2
+ +c2T3+=f(T1
,T2, T3
)
a3T2 + b3T3
+ +c3T4+=f(T2
,T3 , T4
)
a6T5 + b6T6
+ =f(T5 ,T6
, Tair)
………………………………..
M × T = F
Stability of numerical scheme
)2/()( 2 kxcp
Explicit method- simple for calculation- unstable
Implicit method- complex –system of equations (matrix) - Unconditionally stabile
What about accuracy ?
Unsteady-state conductionHomogeneous Wall
0 1 2 3 4 5 6 7 8 9 100.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Analytical solution Numerical -3 nodes, =60 min Numerical -7 nodes, =60 min Numerical -7 nodes, =12 min
(T-T
s)/(
To
-Ts)
hour
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
System of equation for more than one element
air
Left wall
Roof
Right wall
Floor
Elements are connected by:1) Convection – air node2) Radiation – surface nodes
