Solucionario Fundamentos de transferencia de calor F. Incropera Edición 4
Heat Incropera
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Transcript of Heat Incropera
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Rev. cjc. 22.07.2014
Incropera [1]
4.4.- Finite Difference Equations
4.4.1.- The nodal network
4.4.2 Finite difference for of the heat e!"ation #$idien%ional ca%e&
'!"ation% #4.27& to #4.(2&
4.4.( The ener)* $alance ethod
'!"ation% #4.((& to #4.(+&
4.4.(.a ,pplication of the 'ner)* alance ethod
Finite difference e!"atiin for an internal corner of a %olid
with %"rface convection
'!"ation% #4.(/& to #4.44&
Ta$le 4.2. "ar* of nodal finite difference e!"ation
'!"ation% #4.(2& to #4.47&
'aple 4.2
%in) the ener)* $alance ethod3 derive the f inite-difference e!"ation
for the 3n nodal point located on a plane3 in%"lated %"rface of a
edi" with "nifor heat )eneration3
4..- Finite 5ifference ol"tion%
4..1.- The atri inver%ion ethod
'!"ation% #4.4+& to #4.1&
'aple 4.(
Two dien%ional teperat"re di%tri$"tion of a col"n %"pportin) a
f"rnace. ol"tion "%in) the atri inver%ion etho%.
4..2.- 6a"%%-eidel iteration
'!"ation% #4.2& to#4.(&
'aple 4.4
Two dien%ional teperat"re di%tri$"tion of a col"n %"pportin) a
f"rnace #%ae a% epl. 4.(&. ol"tion "%in) 6a"%%-eidel Iteration
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certain
conduc-
of simple
literature
involve
In thesedierence
speed
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4.4.- Finite-5ifference e!"ation% [1]3 pa)e 142
4.4.1.- The nodal network
x
TT
x
T
x
TT
x
T
nmnm
nm
nmnm
nm
=
=
+
+
,,1
,2
1
,1,
,2
1
5.4Figure
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4.4.2 Finite difference form of the heat equation
'!"ation% 4.27 to 4.(2
Thi% approiate finite-difference for of the heat
e!"ation a* $e applied to an* interior node that
i% e!"idi%tant fro it% fo"r nei)h$orin) node%.
Finite differences equations [1]3 pa)e 144
)32.4(04 ,,1,1,11, =+++ ++ nmnmnmnmnm TTTTT
y,n
x,m
m,n+1
m,n-1
m++1
m-1,n
m,n
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Rev. cjc. 22.07.2014
( )( )
( )
( )
( ) ( )
( ) ( )
)32.4(04
022
022
022
)31.4(2
)30.4(2
)27.4(
0
equationheatstatesteadyldimensionaTwo
,,1,1,11,
,1,1,,,1,1
2
,1,1,
2
,,1,1
2
,1,1,
2
,,1,1
,
2
2
,
2
2
2
,1,1,
,
2
2
2
,,1,1
,
2
2
2
,1,,,1
,
2
2
,1,
,
,,1
,
,,
,
2
2
2
2
2
2
2
1
2
1
2
1
2
1
=+++
=+++
=
++
+
=
=
++
+=+
+
+
=+
++
++
++
++
+
+
+
+
+
+
nmnmnmnmnm
nmnmnmnmnmnm
nmnmnmnmnmnm
nmnmnmnmnmnm
nmnm
nmnmnm
nm
nmnmnm
nm
nmnmnmnm
nm
nmnm
nm
nmnm
nm
nmnm
nm
TTTTT
TTTTTT
x
TTT
x
TTT
yx
con
yTTT
xTTT
yT
xT
y
TTT
y
T
x
TTT
x
T
x
TTTT
x
T
x
TT
x
T
x
TT
x
T
x
xT
xT
x
T
y
T
x
T
,n
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4.4.3 The energy alance method'!"ation% #4.((& to #4.(+&
4.4.( The ener)* $alance ethod [1]
The finite-difference e!"ation for a node a* al%o
$e o$tained $* appl*in) con%ervation ener)* to a
control vol"e a$o"t the nodal re)ion3 accordin)
e!"ation 1.10a
It i% a%%"ed that all the heat flow% into the node.
Th"%3 all heat flow i% con%idered in-flow.
For %tead*-%tate condition with )eneration !3 the
appropriate for of '!"ation 1.10 a i%
8on%ider appli*in) '!"ation 4.(( to a control vol"e
a$o"t the interior node 3n of Fi)"re 4.9
Ec. 4.32'!"ation% #4.27& to #4.(2&
)10.1( aEEEE stou tgin =+
)33.4(0=+ gin EE
m,n+1
m,n-1
m+1,nm-1,n m,n
x
x
y
Figure 4.6
equationheatstatesteadyldimensionaTwo
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( )( )
( )
( )
( ) ( )
( ) ( )
)32.4(04
022
022
022
)31.4(2
)30.4(2
)27.4(
0
,,1,1,11,
,1,1,,,1,1
2
,1,1,
2
,,1,1
2
,1,1,
2
,,1,1
,
2
2
,
2
2
2
,1,1,
,
2
2
2
,,1,1
,
2
2
2
,1,,,1
,
2
2
,1,
,
,,1
,
,,
,
2
2
22
2
1
2
1
2
1
2
1
=+++
=+++
=
++
+=
=
++
+=
+
+
+
=
+
++
++
++
++
+
+
+
++
+
nmnmnmnmnm
nmnmnmnmnmnm
nmnmnmnmnmnm
nmnmnmnmnmnm
nmnm
nmnmnm
nm
nmnmnm
nm
nmnmnmnm
nm
nmnm
nm
nmnm
nm
nmnm
nm
TTTTT
TTTTTTx
TTT
x
TTT
yx
con
y
TTT
x
TTT
y
T
x
T
y
TTT
y
T
x
TTT
x
T
x
TTTT
x
T
x
TT
x
T
x
TT
xT
x
x
T
x
T
x
T
yx
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y
( )
( )
( )
( )
( )
( ) ( ) ( )
( ) (
( ))38.4(04
111
)37.4(1
)36.4(1
)35.4(1
)34.4(1
01
)33.4(0)10.1(
2
,1,1,,1,1
,1,,1,,,1,,1
1,,,1,,1
,1,
),()1,(
,1,
),()1,(
,,1
),(),1(
,,1
),(),1(
4
1
),()(
=++++
++++=
+
+
=
=
=
=
=+
=+ =+
++
++
++
++
+
=
k
xqTTTTT
xxqTTTTTTTTk
yx
y
Txk
x
TTyk
x
TTyk
y
TTxkq
y
TTxkq
x
TT
ykq
x
TTykq
yxqq
EEaEEEE
nmnmnmnmnm
nmnmnmnmnmnmnmnm
nmnmnmnmnm
nmnm
nmnm
nmnm
nmnm
nmnm
nmnm
nmnm
nmnm
i
nmi
gin
stou tgin
( )04
2
,1,1,,1,1 =++++ ++k
xqTTTTT nmnmnmnmnm
)32.4(04 ,,1,1,11, =+++ ++ nmnmnmnmnm TTTTT
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Rev. cjc. 22.07.2014
( ) ( )
0
011 ,1,,
=
=+
+ yxqy
TTxk nmnmn
)38.4(
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!""lication of the Energy #alance $ethod [1]3 pa)e 149
Finite difference e!"atiin for an internal corner of a %olid
with %"rface convection
'!"ation% #4.(/& to #4.44&
tinf h
m,nm-1,n m+1,n
m,n+1
m,n-1
cond
cond
conv
cond
cond
conv
y
x
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( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( )
( )n1,mn1,m1nm,n1,-m
,
n1,mn1,m
,1nm,n1,-m
,
n1,mn1,m,
1nm,,n1,-m
,
n1,m,n1,m
1nm,,n1,-m
,
,n1,m,n1,m
,1nm,,n1,-m
,,
,n1,m,n1,m
,1nm,,n1,-m
,,,
con
,n1,m,1nm,
,n1,m,n1,m
,1nm,,1nm,
,n1,-m,n1,-m
cond
TT2
1TT
0!
"hT
!
"h.. .
.. .TT2
1.. .
.. .3TT
0!
"hT
!
"h.. .
.. .TT2
1-.. .
.. .T2T
0!
"hT
!
"h.. .
.. .T2T2
1.. .
.. .T2T
0T!
"h.. .
.. .TT2
1.. .
.. .TT
$%ea%%an$inandy"&ith
0T12
yhT1
2
"h.. .
.. .T
1
2
y!
T1
2
y!.. .
.. .T
1"!T
1y!
so'e%o,equalemust%atesheatallosumThe
T12
yhT1
2
"hq
(4.43)ase"*%essede
mayq%ateheatconectiontotalThe
(4.42)T
12
y!q
(4.41)T
12
y!q
(4.40)T
1"!q
(4.3+)T
1y!q
ase"*%essede
mayq%ateheatconductionThe
+++
++
+
++
+
++
+
++
+
++
+
+
++
++
+++
=+
+++
++
=+
+++
++
=
+
+++
++
=+
+++
++=
=++
+
+
+
+
+
+
=
=
=
=
=
nm
nm
nm
nm
nm
nm
nm
nm
nm
nmnm
nmnm
nmnm
nmnm
nmnm
nmnmnm
nmnm
nmnm
nmnm
nmnm
T
T
T
T
T
T
T
T
T
TT
TT
TTx
T
x
T
y
T
x
T
TT
x
T
x
Ty
T
x
T
( )
( )
( ) T!
"hTT
2
1TT
0!
"h3...
..T!
"hTT
2
1TT
0!
"hT
!
"h...
...3TT
2
1TT
n1,mn1,m1nm,n1,-m
,,
n1,mn1,m1nm,n1,-m
,
,n1,mn1,m1nm,n1,-m
++++
=
++++
=+
++++
+++
+++
+++
nmnm
nm
nm
TT
T
T
( ) T!
"hTT
2
1TT n1,mn1,m1nm,n1,-m
++++ +++
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Rev. cjc. 22.07.2014
0!
"h3
)44.4(
.
, =
nmT
)44.4(0!
"h3 , =
nmT
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Talr 4.2. %ummary of nodal finite difference equation
2 : T-13n; T3n;1; T3n-1; 2:h:
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#4.(2&
#4.44&
#4.4&
#4.49&
0"
0
,
,
=
=
nm
n
T
0
m,n
m,n+1
m-1,n
m,n-1
0, =nm
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E&am"le 4.2 [1] pa)e 14/%in) the ener)* $alance ethod3 derive the f inite-difference e!"ation
for the 3n nodal point located on a plane3 in%"lated %"rface of a
edi" with "nifor heat )eneration3
q1
q2
q3
q4
m,nm-1,n
m,n+1
m,n-1
k,q.
x/2
x
y
Insulation
)33.4(0=+ gin EE
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( )
( )
( ) ( )
0!
"q4-TTT2
0!2
"q-T-T2T2
0!2
"qT
2
1T
2
1T
y"
withand
012
"q
T1
2
"!0.. .
.. .T
12
"!
T1y!
equationalanceene%$yin then$sustitutiand
T1
2
"!q
0q
T
12
"!q
T1y!q
whe%e
012
"qqqqq
q%ateat the
$ene%ationheatmet%icwith oluthat,ollows
itnode,nm,with theasociated12
"
%e$ionaout thesu%acecont%oltheto4.33,/q.
t%equi%emenonconse%atiene%$ythe**lyin$
2
,1nm,1-nm,n1,-m
2
,1nm,,1-nm,,n1,-m
2
,1nm,,1-nm,,n1,-m
,1nm,
,1-nm,,n1,-m
,1nm,4
3
,1-nm,2
,n1,-m1
4321
=
+++
=
+++
=
+++
=
=
+
++
+
+
=
=
=
=
=
++++
+
+
+
+
+
nm
nmnmnm
nmnmnm
nm
nmnm
nm
nm
nm
T
TTT
TTT
yy
T
y
T
x
T
y
T
y
T
x
T
y
y
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4..- Finite-5ifference ol"tion%
4.'.1.- The matri& in(ersion method [1]3 pa)e 11'!"ation% #4.4+& to #4.1&
8on%ider a %*%te of > finite-differece e!"ation% corre%pondin) to > "nknown te
%in) atri notation
[,] : [T] = [8]
where 8iefficient atri3 #> >& ol"tion vector
[,] = T =
a11
: T1; a
12: T
2; a
1(: T
(; ... ; a
1>: T
> = 8
1
a21
: T1; a
22: T
2; a
2(: T
(; ... ; a
2>: T
> = 8
2
a>1
: T1; a
>2: T
2; a
>(: T
(; ... ; a
>>: T
> = 8
>
a11
a12
a1(
... ; a1>
T1
a21
a22
a2(
... ; a2>
T2
a>1
a>2
a>(
... ; a>>
T>
$11
$12
a1(
... ; $1>
T1 =
[,]-1= $21 $22 $2( ... ; $2> T2 =
$>1
$>2
$>(
... ; $>>
T> =
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perat"re%
#4.4+&
The %ol"tion vector a* now $e epre%%ed a%
[T] = #4.0&
#4.4/&
8on%tant% vector
8 =
#4.1&
"ltipl*in) $oth %ide% $* the inver%e atri ,-1
[,]-1: [,] : [T] = [,]-1: [8]
[,]-1: [8]
81
82
8>
$11
: 81; $
12: 8
2;3?; $
1>: 8
>
$21
: 81; $
22: 8
2;3?; $
2>: 8
>
$>1
: 81; $
>2: 8
2;3?; $
>>: 8
>
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Finite differences) e&am"le 4.3 *1+) "age 1'2
$atri& in(ersiuon method
T% = 00 @
,ir with Tinf = (00 @
10:0.2 < 1 = 2. h = 10 A
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>ode% 13 ( and are interior point% for which the finite-difference e!"ation% a* $e
inferred fro '!"ation 4.(2.
'c. 4.(2
3n;1 >ode% 13 ( *
>ode 1 T2 ; T( ; 00 ; 00 - 4:T1 0-13n 3n ;13n >ode ( T1 ; T4 ; T ; 00 - 4:T( = 0
>ode T( ; T9 ; T7 ; 00 -4:T = 0
3n-1
'!"ation% for point% 23 4 and 9 a* $e o$tained in a like anner3 or %ince
the* lie on a %*ert* adia$at3 $* "%in) '!"ation 4.4 #Ta$le 4.2& with h = 0
#4.4&
#with h = 0&
>odo T1 T2 T( T4 T T9 T7
1 -4 1 1 0 0 0 0
2 2 -4 0 1 0 0 0
( 1 0 -4 1 1 0 0
at, = 4 0 1 2 -4 0 1 0
0 0 1 0 -4 1 1
9 0 0 0 1 2 -4 0
7 0 0 0 0 2 0 -/
+ 0 0 0 0 0 2 2
T3n;1
;T3n-1
; T;13n
;T-13n
- 4:T3n =
0
2 : T-13n
; T3n;1
; T3n-1
; 2:h:
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at, : atT = at8
Invat, : at, : atT = Invat,:at8
atT = Invat,:at8atT = DT#Invat,3at8&
-0.((1(/ -0.0/+/ -0.12797(+ -0.094(9 -0.009 -0.0(0+1 -0.0094 -0.0041
-0.1/7+7 -0.((14 -0.12+711 -0.12797 -0.0919( -0.009 -0.00+( -0.0094
-0.12797 -0.0944 -0.(+1/+(7 -0.12/7 -0.14079 -0.07299 -0.017++ -0.01009
Invat, = -0.12+71 -0.1277 -0.2/4/+ -0.(+1/+ -0.14(2 -0.14079 -0.02012 -0.017++
-0.009 -0.0(0+ -0.1407929 -0.07299 -0.(9714 -0.11/0 -0.044+4 -0.01+21
-0.0919( -0.009 -0.14(12 -0.14079 -0.2(+11 -0.(9714 -0.0(942 -0.044+4
-0.01(0/ -0.00+( -0.0(71( -0.02012 -0.0+/9+ -0.0(942 -0.1209 -0.017/4
-0.0199 -0.01(1 -0.0402(7 -0.0(7 -0.072+4 -0.0+/9+ -0.0(+/ -0.1209
,eat flow rate from the column
= 0.2
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1 &
'
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Rev. cjc. 22.07.2014
E&am"le 4.3 1Two dien%ional teperat"re di%tri$"tion of a col"n %"pportin) a
f"rnace. ol"tion "%in) the atri inver%ion ethod.
, lar)e ind"%trial f"rnace i% %"pported on a lon) col"n of firecla* $rickwhich i% 1 $* 1 on a %ide. 5"rin) %tead* %tate operation3 in%tallation
i% %"ch that three %"rface% of the col"n are antained at %"rface
teperat"re T%3 while the reainin) %"rface i% epo%ed to an air a$ient
%trea with a teperat"re Tinf and and a convection coefficient h.
deterine the two dien%ional teperat"re di%tri$"tion in the col"n
and the heat rate to the air %trea per "nit len)th of col"n.
The theral cond"ctivit* of the firecla* $rick i% k #froTa$le ,.(3 at a
teperat"re of a$o"t 47+ @3 [1] &
T% = 00 @
Tinf = (00 @h = 10 A
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>odo 7
2:T ; T+ ; 00 ; 2:2.:(00 - 2:#2.;2&:T7 = 0
2:T ; T+ ; 00 ; 100 - /:T7 = 0
2:T ; T+ ; 2000 - /:T7 = 0
>odo +
2:T9 ; T7 ; ;T7; 2:2.:(00 - #2.;2&:T+ = 0
2:T9 ; 2:T7 ; :(00 - /:T+ = 0
2:T9 ; 2:T7 ; 100 - /:T+ = 0
>odo% 23 4 * 9
>odo 2 2:T1 ;00 ; T4 - 4:T2 = 0>odo 4 2:T( ;T2; T9 - 4:T4 = 0
>odo 9 2:T ;T4 ; T+ - 4:T9 = 0
(
T+
0
0 -1000 4+/.(
0 -00 4+.2
0 -00 472.1
0 at8 = 0 atT = 492.0
1 -00 DT#Invat,3at8& 4(9./
1 0 41+.7
-/ -2000 (7.0
m,n ! "
m,n+1 ! #
m-1,n ! $
m,n-1 ! $%%
m,n ! #
m,n+1 ! "
m-1,n ! 6
m,n-1 ! "
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-100 ((/.1
4
! = 2 : h : [ 5
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4.'.2.- auss-%eidel teration [1] pa)e 1
'!"ation% #4.2& to #4.(&
,pplication of the 6a"%%-eidel iteration to %olve the %*%te repre%ent
$* '!"ation 4.4+.
1.- '!"ation% %ho"ld $e reordered to provide dia)onal eleent% who%ea)nit"d are lar)er than tho%e of other eleent% in the %ae row.
That it i% de%ira$le to %e!"ence the e!"ation% %"ch that
a11
: T1; a
12: T
2; a
1(: T
(; ... ; a
1>: T
> = 8
1
a21
: T1; a
22: T
2; a
2(: T
(; ... ; a
2>: T
> = 8
2
a>1
: T1; a
>2: T
2; a
>(: T
(; ... ; a
>>: T
> = 8
>
(.- ,n initial #k = 0& val"e i% a%%"ed for each teperat"re Ti.
3321
23232221
13131211
,...,
,...,
,...,
NNNN aaaa
aaaa
aaaa
>>
>>
>>
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k
0 4+0 470 440 4(0 400
k0 4+0 470 440 4(0 400
1 477. 471.( 41./ 441.( 42+.0
0.2 : F1
0.2 : '1 ; 0. : F1
0.2 :51 ; 0.2 : 60 ; 0.2 : C
0. : 51 ; 0.2 : 60 ; 12
0.2 : '0 ; 0.2 : F0; 20
.- %in) e!. #4.2& the iteration i% contin"ed
9.- The iteration i% terinated when
T1 T
2T
(T
4T
4.- >ew val"e% of Tiare then calc"lated
T1
T2
T(
T4
T
T1#k&= 0.2 : T
2#k-1& ; 0.2 : T
(#k-1&; 20
T2#k&= 0. : T
1#k& ; 0.2 : T
(#k-1&; 12
T(#k&= 0.2 : T
1#k& ; 0.2 : T
4#k-1&; 0.2 : T
#k-1&; 12
T4#k&= 0.2 : T
2#k&; 0. : T
(#k&; 0.2 : T
9#k-1&
T#k&= 0.2 : T
(#k& ; 0.2 : T
9#k-1& ; 0.2 : T
7#k-1& ; 12
T9#k&= 0.2 : T
4#k&; 0. :T
#k&; 0.2 : T
+#k-1&
T7#k&= 0.222 : T
#k&; 0.111 : T+#k-1&; 222.2
T+#k&= 0.222 :T
9#k&;0.222 :T
7#k& ; 199.97
)52.4()1(
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2.- 'ach of the > e!"ation% %ho"ld $e written in eplicit ford a%%ociated with the dia)onal eleent.
#4.4+&
where i = 13 23?.3>
k i% the level of the iteration
Fro eaple 4.4
T1 = 0.2 : T2 ; 0.2 : T( ; 20
T2 = 0. : T1 ; 0.2 : T4 ; 12
T( = 0.2 : T1 ; 0.2 : T4 ; 0.2 : T ; 12
T4 = 0.2 : T2 ; 0. : T( ; 0.2 : T9
T = 0.2 : T( ; 0.2 : T9 ; 0.2 : T7 ; 12
T9 = 0.2 : T4 ; 0. :T ; 0.2 : T+
T7 = 0.222 : T ; 0.111 : T+ ; 222.2
T+ = 0.222 :T9 ;0.222 :T7 ; 199.97
Re!"ireent
52.4()1(
1
)(1
1
)(
+=
=
= kjN
ij ii
ijkj
i
j ii
ij
ii
iki T
a
aT
a
a
a
CT
()(1
)(
=N
ijki
ijiki T
aT
aCT
)52.4()1(
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(/0 (70 (0
(/0 (70 (0
411.+ (9.2 ((7.(
0.222222 :I1;0.222222 :E1; 199.97
0.222222 : C1 ; 0.111111 :@0; 222.2
0.2 : 61 ; 0. :C1 ; 0.2 : @0
0.2 : I0; 0.2 : E0 ; 12
0.2 : I0
0 ; 12
k
0 4+0 470 440 4(0 4001 477. 471.( 41./ 441.( 42+.0
2 4+0.+ 47.7 492. 4(.1 4(2.9
( 4+4.9 4+0.9 497.9 47.4 4(4.(
4 4+7.0 4+2./ 49/.7 4/.9 4(.
4++.1 4+4.0 470.+ 490.7 4(9.1
9 4++.7 4+4. 471.4 491.( 4(9.
7 4+/.0 4+4.+ 471.7 491.9 4(9.7
+ 4+/.1 4+.0 471./ 491.+ 4(9.+
/ 4+/.2 4+.1 472.0 491./ 4(9./
10 4+/.2 4+.10 472.00 491./4 4(9./0
11 4+/.( 4+.1 472.0 492.0 4(9./
12 4+/.( 4+.1 472.0 492.0 4(9./
1( 4+/.( 4+.1 472.1 492.0 4(9./
14.00 4+/.(0 4+.1 472.09 492.00 4(9./4
1.00 4+/.(0 4+.1 472.09 492.00 4(9./4
T9
T7
T+
T9
T7
T+
T1 T
2T
(T
4T
11 +== ij iij iiii aaa
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Rev. cjc. 22.07.2014
for the teperat"re
)
)52.4()1
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(/0 (70 (0411.+ (9.2 ((7.(
41(./ (.+ ((7.7
41./ (9.2 ((+.(
417.2 (9.9 ((+.9
417./ (9.7 ((+.+
41+.( (9.+ ((+./
41+. (9./ ((/.0
41+.9 (9./ ((/.0
41+.7 (7.0 ((/.0
41+.9/ (9./9 ((/.04
41+.7 (7.0 ((/.0
41+.7 (7.0 ((/.0
41+.7 (7.0 ((/.0
41+.7( (9./7 ((/.0
41+.7( (9./7 ((/.0
T9
T7
T+
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E&am"le 4.4
Two dien%ional teperat"re di%tri$"tion of a col"n %"pportin) a
f"rnace #%ae a% epl. 4.(&. ol"tion "%in) 6a"%%-eidel Iteration
>odo T1 T2 T( T4 T T9 T7
1 -4 1 1 0 0 0 0
2 2 -4 0 1 0 0 0
( 1 0 -4 1 1 0 0
4 0 1 2 -4 0 1 0
0 0 1 0 -4 1 1
9 0 0 0 1 2 -4 0
7 0 0 0 0 2 0 -/
+ 0 0 0 0 0 2 2
>odo 1 T2 ; T( ; 00 ; 00 - 4:T1 = 0 4 : T1 = T2 ; T( ; 1000
>odo 2 2:T1 ; T4 ; 00 - 4:T2 = 0 4 : T2 = 2:T1 ; T4 ; 00
>odo ( T1 ; T4 ; T ; 00 - 4:T( = 0 4 : T( = T1 ; T4 ; T ; 00
>odo 4 T2 ; 2:T( ; T9 - 4:T4 = 0 4 : T4 = T2 ; 2:T( ; T9
>odo T( ; T9 ; T7 ; 00 -4:T = 0 4 : T = T( ; T9 ; T7 ; 00
>odo 9 T4 ; 2:T ; T+ - 4:T9 = 0 4 : T9 = T4 ; 2:T ; T+
>odo 7 2:T ; T+ ; 00 ; 100 - /: T7 0 / : T7 = 2:T ; T+ ; 2000
>odo + 2:T9 ; 2:T7 ; :(00 - /:T+ = 0 / : T+ = 2:T9 ; 2:T7 ; 100
T1#k&= 0.2 : T
2#k-1& ; 0.2 : T
(#k-1&; 20
T2#k&= 0. : T1#k& ; 0.2 : T(#k-1&; 12T
(#k&= 0.2 : T
1#k& ; 0.2 : T
4#k-1&; 0.2 : T
#k-1&; 12
T4#k&= 0.2 : T
2#k&; 0. : T
(#k&; 0.2 : T
9#k-1&
T#k&= 0.2 : T
(#k& ; 0.2 : T
9#k-1& ; 0.2 : T
7#k-1& ; 12
T9#k&= 0.2 : T
4#k&; 0. :T
#k&; 0.2 : T
+#k-1&
T7#k&= 0.222 : T
#k&; 0.111 : T
+#k-1&; 222.2
T+#k&= 0.222 :T
9#k&;0.222 :T
7#k& ; 199.97
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T+ k
0 1000 0 4+0 470
0 00 1 477. 471.(
0 00 2 4+0.+ 47.7
0 0 ( 4+4.9 4+0.9
0 00 4 4+7.0 4+2./
1 0 4++.1 4+4.0
1 2000 9 4++.7 4+4.
-/ 100 7 4+/.0 4+4.+
+ 4+/.1 4+.0T1 = 0.2 : T2 ; 0.2 : T( ; 20 / 4+/.2 4+.1
T2 = 0. : T1 ; 0.2 : T4 ; 12 10 4+/.2 4+.10
T( = 0.2 : T1 ; 0.2 : T4 ; 0.2 : T ; 12 11 4+/.( 4+.1
T4 = 0.2 : T2 ; 0. : T( ; 0.2 : T9 12 4+/.( 4+.1
T = 0.2 : T( ; 0.2 : T9 ; 0.2 : T7 ; 12 1( 4+/.( 4+.1
T9 = 0.2 : T4 ; 0. :T ; 0.2 : T+ 14.00 4+/.(0 4+.1
T7 = 0.222 : T ; 0.111 : T+ ; 222.2 1.00 4+/.(0 4+.1
T+ = 0.222 :T9 ;0.222 :T7 ; 199.97
T1 T
2
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Rev. cjc. 22.07.2014
440 4(0 400 (/0 (70 (0
41./ 441.( 42+.0 411.+ (9.2 ((7.(
492. 4(.1 4(2.9 41(./ (.+ ((7.7
497.9 47.4 4(4.( 41./ (9.2 ((+.(
49/.7 4/.9 4(. 417.2 (9.9 ((+.9
470.+ 490.7 4(9.1 417./ (9.7 ((+.+
471.4 491.( 4(9. 41+.( (9.+ ((+./
471.7 491.9 4(9.7 41+. (9./ ((/.0
471./ 491.+ 4(9.+ 41+.9 (9./ ((/.0472.0 491./ 4(9./ 41+.7 (7.0 ((/.0
472.00 491./4 4(9./0 41+.9/ (9./9 ((/.04
472.0 492.0 4(9./ 41+.7 (7.0 ((/.0
472.0 492.0 4(9./ 41+.7 (7.0 ((/.0
472.1 492.0 4(9./ 41+.7 (7.0 ((/.0
472.09 492.00 4(9./4 41+.7( (9./7 ((/.0
472.09 492.00 4(9./4 41+.7( (9./7 ((/.0
T(
T4
T
T9
T7
T+
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[1] F"ndaental% of heat and a%% tran%fer
Frank G. Incropera and 5avid G. 5e Aitt
chool of echanical 'n)ineerin)3 G"rd"e niver%it*
Eohn Aile* H on%3 1/+
Ceat tran%fer. Finite difference% for a %tead* %tate %*%te. Iplicite and eplicite etho
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d%. Incropera