k an--,j t1P7r F - KFUPM · 2013. 12. 31. · -re,vt(J tH'ah In!. D/5 I-r,' h t/f h'Dl "1'\ (i) 6tU...

$: k an--,j t1P7r F - KFUPM · 2013. 12. 31. · -re,vt(J tH'ah In!. D/5 I-r,' h t/f h'Dl "1'\ (i) 6tU f ,\k an--,j LtJfl/vi,' t1P7r F ~ . the ~¥Yr>y , IJ . YV . Y7 . 1 . lire . Y)$
-revt(J tHah In D5 I-r h tf hDl 1 (i) 6tU f k an--j LtJflvi t1P7r F ~

the ~yenYrgty

IJ YV Y7 1 lire Y) If)

o

Cartesian coordinates (x y z)

Cylindrical coordinates (r 8 z)

(B9-2)b

Spherical coordinates (r 8 cfJ)

(B9-3)1gt

15

Cartesian coordinates (x y z)

lt1gt = 2[(aV)2 + (aVy)2 + (aV)2] + [aVI + aVJ2 + [avz + aViJ2 + [av + avzJ2 _ ~ [avgt + avy avzJ2 (B7-1)ax ay az ax ay ay az az ax 3 ax ay + az

Cylindrical coordinates (r 0 z)

lt1gt = 2[(avr)2 + (1 avo + ~)2 + (avz)2] + [r ~ (vo) + 1aV1J2 + [1 avz + aVeJ2 [avr avzJ2 ar r ao r az ar r r a8 r a8 az + az + ar

(B7-2)

Spherical coordinates (r 8 cfJ)

lt1gt =2[(avr)2 + (1 avo + Vr)2 + (__ av + VI + V9 cot 8)2]ar r a8 r r sm 8 acfJ r

+ [r ~ (vo) + 1 avrJ2 + [~~(~) + _1 aveJ2 + [_1 aVr + ~ (V4raquoJ2ar r r a8 r ao sm 0 r sin 0 acfJ r sin 0 acfJ r ar r

2[la __ 1 a 1 aVrJgt] 2L - -3 2 ar (IV) + --ll a8 (vo sm 0) + ---- (B7-3)r r sm u r sm 0 acfJ

le~-f Ce771dt16)~~

I J ~Cift1 re

lt J7 vev

T I I Uniform heat I I production I I by electrical

II I heatingI

I I SeI

II II I I qr r +4shy

I I Heat in by I IHeat out by

II Iconduction I I conductionI II

L I~IIII ~I 11 I I IM-l~I I I I

II r R I I I I

II II I I J

_-l-t--=-_ --1-1-

lt--r--rJ I =

t7z---J h an L e

s J1e~13J be

I) S~ ~=vjV20 DM e oAm~s ~J ~u( J~ n r-of~foJ~

UYIfCgtTgt II rlc)traquo-lt i-cr-r- erJ ~ ooILC2-1b~ h ~ ~e Jo te CJ~ J-v

~l-lr

---

16Yl~ emiddotr-- 2- (em be 5gt Wlf )pol t3)

~ +tr-fJ) + ~ t

J shy

~-s

J vz ~V~

e jL

bull bull

Se r2 +- c Jvtr) -I- ~2-I - - -~ ~

gvvt~ul1 U77LJ) ~C

~c ( --~

gcz- =)

I-To

---

(f~To) ~re5~ f--~ ~f~vH YJjo8 n ~1 ~ hJ1 YIf oL1I1 e t-v J 69r it(7 middot

I ~ f e-~~vr 1Z~e ~ -

-==D

pc-- ~erlh~ ~ aspAerP j VI e +-v 11 tA ( LA t7 Y heuJ SDttyCe

Coolant

Sphere of fissionable

material

fa k e5 1tf C e (I +l1i7 is 5) 01 1e 5)pqltte ~ r=-O kgt Y lt= f2F l--ere j-s n-v ~~hv-r I VI l- N AI t( WI tj f))Jfl(j middot TA-OdH-r1 111

7UfYl)l-~ g1-3 IS SMr-eo irrshy

~~1 pound-1-0- k en e Yl B-vl-sl~d-tJJu~

FPY pound6 r e~ k 1 4( r L d 0) +

p r~ d~ ~J

Firrrflt r ( RC

j ) z J~)~ -h- -rr l( dJ() -== ~ - ~

ttl j~L GJ ~ ~ ~

-

evmM--Y)

(~hY~It- of tj

ctSS 11 r1 1~ C ~J V(cti1 T~~~VP

--------

c -==-C) gel I

gez _ Silo [e~ +b k G

F

I

f c~ =

Ij bull

---

-----

rz ~o ~F

-GILI-

6-t~

_ gt11 poundr ~

( I + ~V~~ - ~) +Tu Ie ~ -3t-c shy

F~ u ~ h oL)~cvhUn ) bewer(Wl CoYICBrlfrc cy(JMi

Outer cylinder moves with angu2a ~e~oity n

If - 11 5 rf) t e h1 L-rt [tJYL 5 ))0r en e y 7J I(f1Jk

be1-lJeeil l D _ -XJ em c~MC ~1 rtelf-n J vf e f-

1 5 (O Lg J ~ ~JuY) FY~-t tAl e t-~~vt Me

Hn ~tp oe+vfe-vt ~Irtje-rs cltvCt-~e fzrr -f fv1l h EYYI (ty f- t~ rJ IO ble~ Ca VI

b tZ su ked J V1 C lc-t(fEsllC h (ODrcyt~ rII r t dn dt WI or 1- Yj t yenI amp1 J-c J

w JA ~ c LY1r 1--r -gt -~ I W f (-DJ ~

~_ s_u~f~ce moves with velocity vb = RQ II amiddot bull

x

Stationary surface

rb ] --I

gc 5 X -= 0 Vc --0

J 11

X=-O

----

----

eLLJi l2- ( (oj +Cz ~ -reg~

Jv -- - -Ie -2

+ - ~~J rZJJ

- ~d ff + c h +2J -- 2shyv pound -Y

-v +b T -To

-1 tl2 A-f [ - fJ + f (u -~ C-

-C IIiIcc b 101 f JJ vi gt beuro

J -T~ ~ ~ gr - [1-1J-+-Tk -To---shy

GYr1 -vtpJe if fI - L f 1- F_ _ ~ ~ L-~t1LY 0amp ~ 1

C7 g-cJ-trf +he h~ c Egt7f1Ahc~~ roUe vY1

-tJ71t d Ft1middot J h rv~ 5 ~ heJ )(09

le7tAVtC- Je) (lVl~ I-he 2-J(ech~ OI~ tA sen e +- M e S It3 b J -r~ ~ffwr ~ J 0 if -V1 r ~ ( e gt oR V1 t h ( Jgt) Y lt2ej1 Yv1

]-v ~h bJ ~r ~ 1 RecJ epv4

g q ~ vrYl apoundJ-ev- Vamp~0-11Vgt J1f pound ~ s~ 877 e cJM ~s~J Cpoundhvl~ tjy

bull

In YM5 COgteuro [Ale Ave cztl~~ Cffl e~rCf l-erWJ

A 11 fJI(1 (J2IfS of -Jd elwhJy whct--

s h~ Of5 fJr [Inl V-aoJIVIEshy

A - J2 L vV V -- L W 2g

dLT __ h IT- -T ==- 0 -J22- L g ~

I-----L-shy

Heat out by transfer to air stream

x

VValltemperature known to be T 14

2shy

d ZtR _ J1 L tS2 ==0

d~2~ klS

JCtJ7A1~ 19 6J

J2tf1 ~~ h amp = 0 ~12-

tg

- = T tV C-gt tJ) == Ii ~ J

_- IT- dtr1=0 -~=-~~Q ~ = I

Atf~fn~C _ If ot ~gt cf2 C cos~ JfSjf 1= -+ fiJ C- S~I ~ if ~ ~

gel -

5 h[~ 1)J + o = ~

C2- CD~j~ Zi fl~(

-) C2- =- - +-- h Jeuroi ~-)

--0 tJ7 g~ ~- J=~ g~5J-=-- C85

~~ ~ z]

poundX01rl1pJe 5 rree Con veamp)l~ ffe I-Me er Iwo VtwhW flOfi~ bull

~

Temperature tl 1T distribution - f

Tlty)

I~f Velocity--shy

distributionQl I vzlty) t ~ S Q

Q

1tl

2os QI ur ~

If 11

ti f-B---shy

z y 1

Cr71poundJo-er f-N C~ 4- amp( (11 cerJfYJd

bvl-vv e 8-r1 f-wu V-erJ-cJ fl ~J-u 5epay fet

by jJs~tt(e JLf One faJmiddot~ s h~()Vjftv(

~h) I--I~ oJ---0Y ts Cao)~ ThIS YJdr1eJ

nl FlnV I1JcdeJl-D () VI 0 yt1 Y] (1 ~ (r

It II~h ~ ovos uf =nJ cvlelHJ)fY

fjDC S )) vJh

FrSd J-h euro ve Dc~ ~ f 71) Ie Rec~ eztah

~ G CltA af~ $] -f1jfcoJ(TV -

o = - -1of T d dZy~z- Fe dcfL -f ~ ltJ-z middot

~ZVt _ JfoIj2- - -Jpound T S~

heY( vsltDsi 7 J~5 OltsectV1tne~ c~~- b-

-he d enci l 15 n ~ If hi l~ vve t~

g 0 111~ J tn isJ CX-fTD J( WI hM etYf)(gtf1J

g ltLj Iv c- A~ CJ f- I 1 Tcy ~ 5 ampT- J q ~fOJYl ~ ~~ c-Vr D vtd ofV~

r~f~ T= ~ --i2~

~ 91 q 91 I--==-a =- _ + O I- I + shyT-= T T=7

~ -rOJv (e hoe cDerhc~middotvJ ~fh~ ~ft0YlSJCJ-gt

= b ~J ~ -~ ~ =~ J =sect - sect~ amp--1j

n~c6lsoI-eV -1 tL lter1~ ~~ 1-1

A f~ So WI fIt ()eV~

-efT =-0

~~

___) J = T- ~2-J~d = -e Tc Tz-L--~G)

lj -- g T= T(

(i) V J--v CD

~~ J1~ ~~~ T~~ ~ Jif tT l

in -erA~ ~ dLfI~ ~C s -

=--gt V 9 0 f ~ ~ Id ts tdJt JJl ~~) +

ri-_ 1 +~drt1J- ] - jJ Q~ V LJ8J ~ I

~

be 0 bk I t1 (J j ~ WI ~s ccrnJ gtampr- v-vAg

nef h1~ 9vJ y-cJ-e 01] 2-drecil-

b-~D 0(

f5

[ 5) v~ 11 = a

-~

1 h ~s Wlf es

c gar~poundlt

f~JAshy

le~-f Ce771dt16)~~

I J ~Cift1 re

lt J7 vev

T I I Uniform heat I I production I I by electrical

II I heatingI

I I SeI

II II I I qr r +4shy

I I Heat in by I IHeat out by

II Iconduction I I conductionI II

L I~IIII ~I 11 I I IM-l~I I I I

II r R I I I I

II II I I J

_-l-t--=-_ --1-1-

lt--r--rJ I =

t7z---J h an L e

s J1e~13J be

I) S~ ~=vjV20 DM e oAm~s ~J ~u( J~ n r-of~foJ~

UYIfCgtTgt II rlc)traquo-lt i-cr-r- erJ ~ ooILC2-1b~ h ~ ~e Jo te CJ~ J-v

~l-lr

---


~ +tr-fJ) + ~ t

J shy

~-s

J vz ~V~

e jL

bull bull

Se r2 +- c Jvtr) -I- ~2-I - - -~ ~

gvvt~ul1 U77LJ) ~C

~c ( --~

gcz- =)

I-To

---


I ~ f e-~~vr 1Z~e ~ -

-==D


Coolant


material





p r~ d~ ~J

Firrrflt r ( RC

j ) z J~)~ -h- -rr l( dJ() -== ~ - ~

ttl j~L GJ ~ ~ ~

-

evmM--Y)

(~hY~It- of tj


--------

c -==-C) gel I


F

I

f c~ =

Ij bull

---

-----

rz ~o ~F

-GILI-

6-t~

_ gt11 poundr ~

( I + ~V~~ - ~) +Tu Ie ~ -3t-c shy










x

Stationary surface

rb ] --I

gc 5 X -= 0 Vc --0

J 11

X=-O

----

----


Jv -- - -Ie -2

+ - ~~J rZJJ


-v +b T -To

-1 tl2 A-f [ - fJ + f (u -~ C-


J -T~ ~ ~ gr - [1-1J-+-Tk -To---shy







bull




A - J2 L vV V -- L W 2g

dLT __ h IT- -T ==- 0 -J22- L g ~

I-----L-shy


x


2shy


d~2~ klS

JCtJ7A1~ 19 6J

J2tf1 ~~ h amp = 0 ~12-

tg


_- IT- dtr1=0 -~=-~~Q ~ = I


gel -

5 h[~ 1)J + o = ~

C2- CD~j~ Zi fl~(

-) C2- =- - +-- h Jeuroi ~-)

--0 tJ7 g~ ~- J=~ g~5J-=-- C85

~~ ~ z]


~


Tlty)

I~f Velocity--shy


Q

1tl

2os QI ur ~

If 11

ti f-B---shy

z y 1







fjDC S )) vJh









r~f~ T= ~ --i2~

~ 91 q 91 I--==-a =- _ + O I- I + shyT-= T T=7





-efT =-0

~~

___) J = T- ~2-J~d = -e Tc Tz-L--~G)

lj -- g T= T(

(i) V J--v CD

~~ J1~ ~~~ T~~ ~ Jif tT l



ri-_ 1 +~drt1J- ] - jJ Q~ V LJ8J ~ I

~



b-~D 0(

f5

[ 5) v~ 11 = a

-~

1 h ~s Wlf es

c gar~poundlt

f~JAshy

---


~ +tr-fJ) + ~ t

J shy

~-s

J vz ~V~

e jL

bull bull

Se r2 +- c Jvtr) -I- ~2-I - - -~ ~

gvvt~ul1 U77LJ) ~C

~c ( --~

gcz- =)

I-To

---


I ~ f e-~~vr 1Z~e ~ -

-==D


Coolant


material





p r~ d~ ~J

Firrrflt r ( RC

j ) z J~)~ -h- -rr l( dJ() -== ~ - ~

ttl j~L GJ ~ ~ ~

-

evmM--Y)

(~hY~It- of tj


--------

c -==-C) gel I


F

I

f c~ =

Ij bull

---

-----

rz ~o ~F

-GILI-

6-t~

_ gt11 poundr ~

( I + ~V~~ - ~) +Tu Ie ~ -3t-c shy










x

Stationary surface

rb ] --I

gc 5 X -= 0 Vc --0

J 11

X=-O

----

----


Jv -- - -Ie -2

+ - ~~J rZJJ


-v +b T -To

-1 tl2 A-f [ - fJ + f (u -~ C-


J -T~ ~ ~ gr - [1-1J-+-Tk -To---shy







bull




A - J2 L vV V -- L W 2g

dLT __ h IT- -T ==- 0 -J22- L g ~

I-----L-shy


x


2shy


d~2~ klS

JCtJ7A1~ 19 6J

J2tf1 ~~ h amp = 0 ~12-

tg


_- IT- dtr1=0 -~=-~~Q ~ = I


gel -

5 h[~ 1)J + o = ~

C2- CD~j~ Zi fl~(

-) C2- =- - +-- h Jeuroi ~-)

--0 tJ7 g~ ~- J=~ g~5J-=-- C85

~~ ~ z]


~


Tlty)

I~f Velocity--shy


Q

1tl

2os QI ur ~

If 11

ti f-B---shy

z y 1







fjDC S )) vJh









r~f~ T= ~ --i2~

~ 91 q 91 I--==-a =- _ + O I- I + shyT-= T T=7





-efT =-0

~~

___) J = T- ~2-J~d = -e Tc Tz-L--~G)

lj -- g T= T(

(i) V J--v CD

~~ J1~ ~~~ T~~ ~ Jif tT l



ri-_ 1 +~drt1J- ] - jJ Q~ V LJ8J ~ I

~



b-~D 0(

f5

[ 5) v~ 11 = a

-~

1 h ~s Wlf es

c gar~poundlt

f~JAshy

---


I ~ f e-~~vr 1Z~e ~ -

-==D


Coolant


material





p r~ d~ ~J

Firrrflt r ( RC

j ) z J~)~ -h- -rr l( dJ() -== ~ - ~

ttl j~L GJ ~ ~ ~

-

evmM--Y)

(~hY~It- of tj


--------

c -==-C) gel I


F

I

f c~ =

Ij bull

---

-----

rz ~o ~F

-GILI-

6-t~

_ gt11 poundr ~

( I + ~V~~ - ~) +Tu Ie ~ -3t-c shy










x

Stationary surface

rb ] --I

gc 5 X -= 0 Vc --0

J 11

X=-O

----

----


Jv -- - -Ie -2

+ - ~~J rZJJ


-v +b T -To

-1 tl2 A-f [ - fJ + f (u -~ C-


J -T~ ~ ~ gr - [1-1J-+-Tk -To---shy







bull




A - J2 L vV V -- L W 2g

dLT __ h IT- -T ==- 0 -J22- L g ~

I-----L-shy


x


2shy


d~2~ klS

JCtJ7A1~ 19 6J

J2tf1 ~~ h amp = 0 ~12-

tg


_- IT- dtr1=0 -~=-~~Q ~ = I


gel -

5 h[~ 1)J + o = ~

C2- CD~j~ Zi fl~(

-) C2- =- - +-- h Jeuroi ~-)

--0 tJ7 g~ ~- J=~ g~5J-=-- C85

~~ ~ z]


~


Tlty)

I~f Velocity--shy


Q

1tl

2os QI ur ~

If 11

ti f-B---shy

z y 1







fjDC S )) vJh









r~f~ T= ~ --i2~

~ 91 q 91 I--==-a =- _ + O I- I + shyT-= T T=7





-efT =-0

~~

___) J = T- ~2-J~d = -e Tc Tz-L--~G)

lj -- g T= T(

(i) V J--v CD

~~ J1~ ~~~ T~~ ~ Jif tT l



ri-_ 1 +~drt1J- ] - jJ Q~ V LJ8J ~ I

~



b-~D 0(

f5

[ 5) v~ 11 = a

-~

1 h ~s Wlf es

c gar~poundlt

f~JAshy


Coolant


material





p r~ d~ ~J

Firrrflt r ( RC

j ) z J~)~ -h- -rr l( dJ() -== ~ - ~

ttl j~L GJ ~ ~ ~

-

evmM--Y)

(~hY~It- of tj


--------

c -==-C) gel I


F

I

f c~ =

Ij bull

---

-----

rz ~o ~F

-GILI-

6-t~

_ gt11 poundr ~

( I + ~V~~ - ~) +Tu Ie ~ -3t-c shy










x

Stationary surface

rb ] --I

gc 5 X -= 0 Vc --0

J 11

X=-O

----

----


Jv -- - -Ie -2

+ - ~~J rZJJ


-v +b T -To

-1 tl2 A-f [ - fJ + f (u -~ C-


J -T~ ~ ~ gr - [1-1J-+-Tk -To---shy







bull




A - J2 L vV V -- L W 2g

dLT __ h IT- -T ==- 0 -J22- L g ~

I-----L-shy


x


2shy


d~2~ klS

JCtJ7A1~ 19 6J

J2tf1 ~~ h amp = 0 ~12-

tg


_- IT- dtr1=0 -~=-~~Q ~ = I


gel -

5 h[~ 1)J + o = ~

C2- CD~j~ Zi fl~(

-) C2- =- - +-- h Jeuroi ~-)

--0 tJ7 g~ ~- J=~ g~5J-=-- C85

~~ ~ z]


~


Tlty)

I~f Velocity--shy


Q

1tl

2os QI ur ~

If 11

ti f-B---shy

z y 1







fjDC S )) vJh









r~f~ T= ~ --i2~

~ 91 q 91 I--==-a =- _ + O I- I + shyT-= T T=7





-efT =-0

~~

___) J = T- ~2-J~d = -e Tc Tz-L--~G)

lj -- g T= T(

(i) V J--v CD

~~ J1~ ~~~ T~~ ~ Jif tT l



ri-_ 1 +~drt1J- ] - jJ Q~ V LJ8J ~ I

~



b-~D 0(

f5

[ 5) v~ 11 = a

-~

1 h ~s Wlf es

c gar~poundlt

f~JAshy

ttl j~L GJ ~ ~ ~

-

evmM--Y)

(~hY~It- of tj


--------

c -==-C) gel I


F

I

f c~ =

Ij bull

---

-----

rz ~o ~F

-GILI-

6-t~

_ gt11 poundr ~

( I + ~V~~ - ~) +Tu Ie ~ -3t-c shy










x

Stationary surface

rb ] --I

gc 5 X -= 0 Vc --0

J 11

X=-O

----

----


Jv -- - -Ie -2

+ - ~~J rZJJ


-v +b T -To

-1 tl2 A-f [ - fJ + f (u -~ C-


J -T~ ~ ~ gr - [1-1J-+-Tk -To---shy







bull




A - J2 L vV V -- L W 2g

dLT __ h IT- -T ==- 0 -J22- L g ~

I-----L-shy


x


2shy


d~2~ klS

JCtJ7A1~ 19 6J

J2tf1 ~~ h amp = 0 ~12-

tg


_- IT- dtr1=0 -~=-~~Q ~ = I


gel -

5 h[~ 1)J + o = ~

C2- CD~j~ Zi fl~(

-) C2- =- - +-- h Jeuroi ~-)

--0 tJ7 g~ ~- J=~ g~5J-=-- C85

~~ ~ z]


~


Tlty)

I~f Velocity--shy


Q

1tl

2os QI ur ~

If 11

ti f-B---shy

z y 1







fjDC S )) vJh









r~f~ T= ~ --i2~

~ 91 q 91 I--==-a =- _ + O I- I + shyT-= T T=7





-efT =-0

~~

___) J = T- ~2-J~d = -e Tc Tz-L--~G)

lj -- g T= T(

(i) V J--v CD

~~ J1~ ~~~ T~~ ~ Jif tT l



ri-_ 1 +~drt1J- ] - jJ Q~ V LJ8J ~ I

~



b-~D 0(

f5

[ 5) v~ 11 = a

-~

1 h ~s Wlf es

c gar~poundlt

f~JAshy

--------

c -==-C) gel I


F

I

f c~ =

Ij bull

---

-----

rz ~o ~F

-GILI-

6-t~

_ gt11 poundr ~

( I + ~V~~ - ~) +Tu Ie ~ -3t-c shy










x

Stationary surface

rb ] --I

gc 5 X -= 0 Vc --0

J 11

X=-O

----

----


Jv -- - -Ie -2

+ - ~~J rZJJ


-v +b T -To

-1 tl2 A-f [ - fJ + f (u -~ C-


J -T~ ~ ~ gr - [1-1J-+-Tk -To---shy







bull




A - J2 L vV V -- L W 2g

dLT __ h IT- -T ==- 0 -J22- L g ~

I-----L-shy


x


2shy


d~2~ klS

JCtJ7A1~ 19 6J

J2tf1 ~~ h amp = 0 ~12-

tg


_- IT- dtr1=0 -~=-~~Q ~ = I


gel -

5 h[~ 1)J + o = ~

C2- CD~j~ Zi fl~(

-) C2- =- - +-- h Jeuroi ~-)

--0 tJ7 g~ ~- J=~ g~5J-=-- C85

~~ ~ z]


~


Tlty)

I~f Velocity--shy


Q

1tl

2os QI ur ~

If 11

ti f-B---shy

z y 1







fjDC S )) vJh









r~f~ T= ~ --i2~

~ 91 q 91 I--==-a =- _ + O I- I + shyT-= T T=7





-efT =-0

~~

___) J = T- ~2-J~d = -e Tc Tz-L--~G)

lj -- g T= T(

(i) V J--v CD

~~ J1~ ~~~ T~~ ~ Jif tT l



ri-_ 1 +~drt1J- ] - jJ Q~ V LJ8J ~ I

~



b-~D 0(

f5

[ 5) v~ 11 = a

-~

1 h ~s Wlf es

c gar~poundlt

f~JAshy

---

-----

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k an--,j t1P7r F - KFUPM · 2013. 12. 31. · -re,vt(J tH'ah In!. D/5 I-r,' h t/f h'Dl "1'\ (i) 6tU...

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Transcript of k an--,j t1P7r F - KFUPM · 2013. 12. 31. · -re,vt(J tH'ah In!. D/5 I-r,' h t/f h'Dl "1'\ (i) 6tU...