heat incropera

7/25/2019 Heat Incropera

1/43

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


f"rnace #%ae a% epl. 4.(&. ol"tion "%in) 6a"%%-eidel Iteration


2/43


3/43

certain

conduc-

of simple

literature

involve

In thesedierence

speed


4/43

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


5/43

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


6/43

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


7/43

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


8/43

( )( )

( )

( )

( ) ( )

( ) ( )

)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


9/43

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


10/43


11/43

Rev. cjc. 22.07.2014

( ) ( )

0

011 ,1,,

=

=+

+ yxqy

TTxk nmnmn

)38.4(


12/43


13/43

!""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


14/43

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( )

( )

( )

( )

( )

( )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

++++ +++


15/43

Rev. cjc. 22.07.2014

0!

"h3

)44.4(

.

, =

nmT

)44.4(0!

"h3 , =

nmT


16/43

Talr 4.2. %ummary of nodal finite difference equation

2 : T-13n; T3n;1; T3n-1; 2:h:


17/43

#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


18/43

Rev. cjc. 22.07.2014


19/43

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


20/43

( )

( )

( ) ( )

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


21/43

Rev. cjc. 22.07.2014


22/43

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> =


23/43

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

>


24/43

Rev. cjc. 22.07.2014


25/43

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


26/43

>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:


27/43

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


28/43

1 &

'


29/43

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


30/43

>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 ! "


31/43

-100 ((/.1

4

! = 2 : h : [ 5


32/43


33/43

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

>>

>>

>>


34/43

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(


35/43

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(


36/43

(/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


37/43

Rev. cjc. 22.07.2014

for the teperat"re

)

)52.4()1


38/43

(/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+


39/43

E&am"le 4.4


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


40/43

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


41/43

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+


42/43

[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


43/43

d%. Incropera

heat incropera

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