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Nuclear Engineering and Design 36 (1976) 5- 16 North-Ho lland Publishing Com pany

C O S T O P T I M I Z A T I O N O F V E R T I C A L N A T U R A L - C I R C U L A T I O N S T EA M G E N E R A T O R SA m i r N . N A H A V A N D I , M i ch a el A . V O R K A S a n d V i n c e n t J . D ' E M I D I ONew Jersey Institute o f Technology, Newark, New Jersey 0 7102, USAReceived 1 October 1975

A m athematical m odel for the cost optimization of U-tube vertical natural-circulation steam gene rators, used in Pressur-ized-water nuclear power plants, is developed. The to tal annual cost function is expressed as a function of the heat exchangearea and the pum ping power. Param etric studies indicate tha t the global minim um cost is on the line of the low est constantinside diameter and the lowest constant inside surface roughness. This mathem atical formulation is useful for incorporationin the cost optimization of the entire nuclear power plant.

1 . I n t r o d u c t i o n

O p t i m i z a t i o n o f h e a t e x c h a n g e r d e s i g n is o f i n t e re s ti n ma n y e n g i n e e r in g a p p l i c a ti o n s . A p o o r c h o i c e o f d e -s ig n p a r a me t e r s , s u c h as th e n u mb e r o f t u b e s a n d t h et u b e d i a m e t e r , c a n a d v e r s e ly a f f e c t th e c o s t o f t h e h e a te x c h a n g e r a n d t h e r e b y r e d u c e t h e e c o n o mi c e f f e c t i v e -n e s s o f t h e h e a t t r a n s f e r e q u i p m e n t [1 ] .

A c o m p u t e r i z e d d e s ig n o f t he m i n i m u m c o s t o f ac r o s s- f lo w g as c o o l e r w a s u n d e r t a k e n b y M o t t e t a l.[ 2 ] . T h e c o m p u t e r p r o g r a m d e v e l o p e d p r o v i d e s a me t h -o d f o r r a p i d l y e v a lu a t in g t h e e c o n o m i c p o t e n t i a l o fava i l ab le hea t t rans fer mat r i ces fo r a par t i cu la r hea te x c h a n g e r a p p l i c a t io n . A s i mi la r s t u d y w a s c o n d u c t e db y D e h n e [ 3 ] f o r t h e e c o n o m i c a l d es ig n o f a ir - c o o le dh e a t e x c h a n g e r s . H e e m p l o y s a n o p t i m i z a t i o n p r o g r a mo n ma n y t y p e s o f h e a t e x c h a n g e r s to a r r i ve a t t h e m o s te c o n o m i c a l s o l u ti o n f o r a g i v en h e a t t r a n s f e r p r o b l e m .

L a H a y e e t a l. [ 4 ] s h o w e d a n e w me t h o d o f p re -s e n ti n g t h e h e a t t r a n s f e r d a t a f o r t h e p r e d i c t i o n o fh e a t i n g su r f a c e p e r f o r ma n c e a n d h e a t e x c h a n g e r o p t i -miza t ion . T hei r ana lys i s l eads to a d imens ion les s per -f o r ma n c e p l o t b e t w e e n a ' h e a t t r a n s f e r p e r f o r ma n c ef a c t o r ' a n d a ' p u mp i n g p o w e r f a c t o r ' w i t h a n o n - d i -me n s i o n a l ' f l o w l e n g t h b e t w e e n ma j o r b o u n d a r y l a y e rd i s t u r b a n c e s ' a s a v a r y i n g p a r a m e t e r . T h i s p l o t p e r m i t st h e r a p i d a s s e s s me n t a n d c o mp a r i s o n o f h e a t t r a n s fe rgeom et r i es fo r a g iven app l i ca t ion an d i s va luab le in op-t imiz ing a des ign accoun t ing fo r space l imi t a t ion , eco-n o m i c r e s t r a in t s a n d s y s t e m c o n s i d e r a t io n s s u c h a sp u m p i n g p o w e r a n d e f f e c t iv e n e s s t r a d e - o f f s.

C o m p u t e r o p t i m i z a t i o n o f d r y c o o l in g t o w e r h e a te x c h a n g e r s h a s b e e n u n d e r t a k e n b y ma n y a u t h o r s .Andeen and Gl i cksman [5 ] deve loped a d ig i t a l com-p u t e r p r o g r a m f o r t h e o p t i m i z a t i o n o f th e e n t i r e p o -wer p l an t cons ider ing a l l i n t e rac t ions be tween bo i l e r ,t u r b i n e a n d h e a t e x c h a n g e r . T h e y f o u n d t h e o p t i mu mp o i n t w h e r e t h e i n c r e me n t a l i n c r e as e i n c o s t o f p o w e rgenera t ion , as a resu l t o f the app l i ca t ion o f the d ryc o o l in g t o w e r s , is a m i n i m u m . A r m s t r o n g a n d Sc h e r -me r h o r n [ 6 ] s t u d ie d t h e e c o n o m i c e f f e c t s o f d r y c o o l -in g t o w e r s w h e n a p p l i e d t o a n u n f i r e d c o m b i n e d - c y c l ep lan t . They showed tha t fo r th i s par t i cu la r case , t hed r y t o w e r s o f f e r a m i n o r c o s t p e n a l t y a s c o mp a r e d t oo t h e r a l t e rn a t i v e c o o l in g me t h o d s .

O p t i mi z a t i o n o f h e a t t r a n s f e r d es ig n s b y g e o me t r i cp r o g r a m m i n g h a s b e en p e r f o r m e d b y O s w a ld a n d K o -chenberger [7 ] . App l i ca t ions a re shown fo r the se l ec-t i o n o f h e a t t r a n s f e r me d i a c o n s i d e r in g p o w e r r e q u i r e -me n t s , h e a t e x c h a n g e r c o s t , p i p e d i a me t e r , v e l o c i t y ,t e mp e r a t u r e a n d p h y s i c a l p r o p e r t i e s .

T h e p u r p o s e o f t h i s s t u d y i s t o e x a mi n e t h e c o s to p t i m i z a t i o n o f t h e U - t u b e n a t u ra l - c i rc u l a t io n s t e a mgenera to rs used in p res su r i zed-water nuc lear powers t a t ions . In par t i cu la r , t he ob jec t ive is t o show the e f -f e c t s o f t h e n u m b e r o f t u b e s , t h e i ns id e d i a m e t e r a n dthe roughness on the cos t o f the s t eam genera t ing un i t s .T h e ma t h e m a t i c a l f o r mu l a t i o n d e v e l o p e d i n t h is s t u d yis u s e f u l f o r in c o r p o r a t i o n i n t h e c o s t o p t i m i z a t i o n o ft h e e n t i r e n u c l e a r p o w e r p l a n t w h e n c o n s i de r i n g t h ein te rac t ion be tween the reac to r vesse l , s t eam genera-t o r s , ma i n c o o l a n t p u mp s a n d t h e p r e s s u r iz e r .


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6 A.N. Nahavandi et al. / Cost opti miza tion o f steam generators2 . M a t h e m a t ic a l f o r m u l a t i o n

Figure 1 is a schematic diagram of a vertical U-tubenatural-circulation steam generator. The primarycoolant flows through the inside of many hundredsof small stainless-steel tubes in the heat exchanger sec-tion of the steam generator. These heat exchangertubes are surrounded by the water of the secondarysystem, which is heated by the primary coolant. Wetsteam is formed which flows upward through the riserand enters the steam separator portion of the steamgenerator. Here the moisture is removed and returned(together with the cold water feed) to the heat ex-changer section through the downcomer. The'dry andsaturated steam leaves the top of the steam dryer andgoes to the steam turbine. The mathematical formula-

~ T G '& i~ i T ~ " r l IRR | NF

P A R A T O R

A T O R

tion is based on the following assumptions:(1) The primary and the secondary coolants are at

subcooled and two-phase flow condit ions, respectively.The secondary coolant temperature is treated as aconstant.(2) The steam generating unit is considered to becontinuously working at steady-state operating con-ditions.

The cost of a U-tube natural-circulation steam gen-erator C can be expressed as a function of the heat ex-change area A and the pumping power per u~ait area Erequired to pump the primary coolant through the U-tubes, orC = A C a R + A E O C p , (1)

where the heat exchange area A and the pumping po-wer per unit area E are calculated on the basis of thesteam generator thermal and hydraulic design con-siderations, respectively. The capital recovery factorR, cost per unit area C a, and cost per unit energy Cpare determined on the basis of economics, equipmentand power costs, respectively. The pump running time0 is established on the basis of the operat ing time, as-sumed to be continuous. The calculation of these pa-rameters is discussed in the following sections.

L I N E

O W N )

P R I M A R Y G O O LA P L N TbN

Fig. 1. Schematic diagram of natural-circulation U-tube steamgenerator.

2 . 1. H e a t e x c h a n g e a r eaApplying an energy balance on a small U-tube ele-

ment along the primary coolant flow givesa = d q / d A = W p Cp ( d T p / d A ) , (2a)Q = U ( T p - Ts). (2b)The inverse of the overall heat transfer coefficient U(from the primary to the secondary fluid) is given asthe sum of the primary film, tube wall, fouling factorand secondary film resistances1 / U =Rp +R w +R f +R s =R e +R s . (3)It should be noted that since the boiling film resistanceR s is dependent on the heat flux, the integration ofeqs. (2a) and (2b) is not straightforward.

The primary film resistance is given byR p = D o / D i h p , (4)where the primary film convective heat transfer coef-


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A . N . N a h a va n d i e t a l. / C o s t o p t im i za t io n o f s t ea m g en era to rs 7f ic ient is ca lcula ted f rom ref . [8]

k p - 0 " 0 2 3 \ A - - ~ ! \ - 'F p-p (5)The tube wall resis tance is calculated fromR w = [D o In ( D o / D i ) ] / Z k w . (6)The foul ing factor is cons idered con s tant . T he sec-ond ary f i lm resistance is calcula ted by co mb ining theRoh senow 's boi l ing heat t ransfe r corre la t ion [9]c f ( T w - T S ) h f g= c w s [ ( ~ [ \ g ( p f cO -g ) ] ] l / 2 1 1 / 3 j

(V ) , (7)o r s imp lyTw _ Ts = F ( P s ) Q l / 3 (8)w i th the s econdary hea t f lux r e la t ionQ = ( T w - T s ) l R s ( 9 )to ob ta inR s = F ( P s ) Q - 2 / 3 . (10 )Com bining eqs . (2b) , (3) and (10) y ie ldsT p - T s = R c Q + F ( P s ) Q 1 / 3 . (11)In tegrat ing eq . (2a) an d n ot ing th at T s is cons tan tg i v e s

2a f WpCp d(T p- p/O ] . ( 12 )1Subs t i tu t ing eq . (11) in to (12) a nd perform ing the in-t eg ra t ion , the h ea t exchange a r ea i s ob ta ined in t e rmsof the tube hea t f luxes a t the in le t and ou t l e t :

Q2A = WpCp f { d [ Re e + F ( P s ) O I / 3 ] / O }O lo rA = W p c p f R c l n ~ 2 + F ( P s ) ( Q 2 2 / 3 - Q 1 2 / 3 ) ] . ( 1 3 )

Em ploy ing eq . (11 ), the tube hea t f luxes a t the in le tand out le t (Q1 and Q2) are fou nd f rom eqs . (14) and( 1 5 ) b y t h e N e w t o n - R a p h s o n i te r at iv e m e t h o d :

T p l - T s = R c Q 1 + F ( P s ) Q l l / 3 ,Tp2 - T s = R cQ 2 + F ( P s ) Q t 2 / 3 .

(14)(15)

2 .2 . P u m p i n g p o w e r p e r u n i t a r eaThe pum ping pow er per uni t area is calcula ted in

terms o f the f r ic t ional power losses expe nded per uni ts u rf ace to pu mp the p r imary coo lan t th rough the hea texchanger tubesE = ( A i / A ) ( f O p [ 7 3 / 8 g c ) (16)The pr imary coolant veloci ty Vp is re la ted to o therpa ramete r s byW p = 7rD 2 U p p V p . (17~The f r i c t ion f ac to rs fo r s moo th and rough cond i t ionsare calcula ted f rom the fo l lowing corre la t ions , respec-t ively , which agree with the wel l -known Mo ody 'schar t :f = 0.184 (Wp D i / A p U p) - ' 2 (18 )and

e 106 Ap /ap~ "333f = 0 . 00 5 5 [ 1 + ( 2 1 0 4 ~ i i + W - --~ i ] _]. ( 1 9 )The tub e ins ide and ou ts ide d iam eters are re la ted bythe ASM E Power Boi ler Code form ula for wal l th ick-ness for high-pressure, high -tem pera ture piping [10]t = (D o - Di) /2 = P p D o / ( 2 S + 2 y P p ) . ( 2 0 )2 .3 . C a p i t a l r e c o v e r y f a c t o r

The cap i t a l recovery f ac to rR , the s um o f the r e-turn (on the inves ted capi ta l ) and s inking fun d depre-c ia t ion com ponen t s , i s a func t ion o f the in te r est r a t e iand the s team genera tor li fe n [11] , i .e.R = i (1 + i )n / [ ( l + i ) n - 1] . (21)


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8

Table 1.Steam generator specif ica t ion.

A.N. Nahavandi e t a l . / Cos t op t imiza t ion o f s team genera tors

British units SI unit sHeat t ransfer loadPr imary f low, lopCoolan t in l e t tempera tu re , T p lCoolan t ou t l e t t em pera tu re , T p2Primary pressure, P pSecondary pressure, PsSeconda ry t empera tu re , TsFou l ing fac to r , RfRange of numbe r of U-tubes, NRange of U-tube i .d . , D iU-tube roughness, e

4.18 X 109 Btu/hr62.25 X 106 lb/h r598 . 5F5 4 7 . 8 F2100 ps ia770 psla513 . 8F0.0003 hr-f t2- F/B tu2 0 0 - 5 0 0 0 00 . 4 - 1 . 5 i n0 - 0 . 0 0 0 1 f t

1225 MW7.843 k g/sec314.7C286.6C14.48 X 106 Pa5.31 106 Pa267.7C5.283 X 106 m2-C/W2 0 0 - 5 0 0 0 01 0 . 1 6 - 3 8 . 1 0 m m0 - 0 . 0 3 m m

2 . 4 . U n i t c o s t sT h e s t e a m g e n e r a t o r c o s t p e r u n i t a r e a C a i s o b t a i n e d

f r o m t h e s o - c a l l e d ' s i x - t e n t h s r u l e ' , w h i c h h a s g i v e n g o o dr e s u lt s i n e s t im a t i n g c a p i t a l e q u i p m e n t c o s t s [ 7 ]C a / C a o = ( A / A o ) 0 " 6 . ( 2 2 )T h i s r u l e g iv e s t h e u n i t c o s t o f a n e w p i e c e o f e q u i p -m e n t i n t e r m s o f t h e u n i t c o s t o f a n o t h e r c a p a c i t y fo rw h i c h c o s t d a t a a r e a t h a n d . T h e c o s t p e r u n i t e n e r g yC p i s c o n s i d e r e d c o n s t a n t .

3 . P r e s e n t a t io n o f r e s u lt s an d c o n c l u s i o n s

T h e m a t h e m a t i c a l f o r m u l a t i o n d e r i v e d i n t h e p r e -v i o u s s e c t i o n is a p p l i e d t o t h e a n a l y s i s o f a t y p i c a l U -

Table 2.Inp ut para me ters used in the analys is.

t u b e v e r t ic a l s te a m g e n e r a t o r , u s e d i n p r e s s u r i z e d - w a t e rr e a c t o r s y s t e m s , s p e c i f i e d i n ta b l e 1 . T o f a c i l i t a t e t h i sa p p l i c a t i o n , t w o d i g i ta l p r o g r a m s w e r e d e v e l o p e d u s i n gt h e a b o v e f o r m u l a t i o n :

( 1 ) A p a r a m e t r i c s t u d y p r o g r a m w h i c h c o u l d v a r yo n e o r m o r e d e s ig n p a r a m e t e r s a t a t i m e a n d d e t e r m i n et h e c o s t i n t e r m s o f t h e s e p a r a m e t e r s . T h i s p r o g r a mp r o v e d t o b e u s e f u l i n g a i n i n g i n s i g h t i n t o t h e n a t u r eo f th e p r o b l e m .

( 2 ) A n o p t i m i z a t i o n s t u d y p r o g r a m u s i n g a n a v ai l-a b l e d i r e c t se a r c h p r o g r a m f o r a u t o m a t e d o p t i m a l d e -s i g n [ 1 2 ] .

T h e m a t h e m a t i c a l f o r m u l a t i o n p r e s e n t e d h e r e i n i ss u f f i c ie n t l y ge n e r a l t o o p t i m i z e t h e s y s t e m w i t h r e s p e c tt o a l l d e s ig n v a r ia b l e s. H o w e v e r , t h e o p t i m i z a t i o n p r o -c e d u r e i s p e r f o r m e d w i t h o n l y t h r e e d e s ig n v a r ia b l e s- n u m b e r o f U - t u b e s , t h e i r i n s id e d i a m e t e r a n d t h e i rr o u g h n e s s .

Bri t ish uni ts SI uni tsCos t pe r un i t a rea o f thereference s team gen erator , CaoHeat exchange area of thereference s team generato r , AoCost per uni t energy, CpInteres t ra te , iSteam generator l i fe , nAllowable stress in U-tubes, SCoef f i c i en t y o f the ASM E Boi le r CodeSteam gene ra to r ope ra t ing t ime , 0

60.00 $/f t 2 645.83 $/m 250 000 f t 2 4645.15 m 20.012 $/kWhr 3.33 X 10 -9 $/J0.08 0.084 0 y r 4 0 y r23 300 psia 160.6 X 106 Pa0.4 0.48760 hr 31.536 X 106 sec


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A.N . Nahava ndi et a l. / Cost op t im iza t ion o f s team genera tors 9

; ~ , u "

I1 .0U )q ll U

i .o " t ~ o . I S "

O. IS"

, , 0 .4 "

s x , o *

4 . J o ' 6 I J I i I~ o . o o o 2 o , o o o 3 o , o o oN U M B E R O F T U B I [ S ~ N

Fig. 2. He at exchange area versus num ber o f tubes w ith inside diameter as a parameter.T h e e f f e c t s o f th e n u m b e r o f t u b e s a n d t h e i r in s i de

d i a m e t e r a n d r o u g h n e s s o n t h e f o l l o w i n g d e si g n va r ia -b l e s w e r e e x a m i n e d :

( 1 ) h e a t e x c h a n g e a r e a a n d i t s c o n t r i b u t i o n t o t h et o t a l c o s t ( f ig s . 2 a n d 3 ) ;

( 2 ) r e a c t o r c o o l a n t p u m p i n g p o w e r a n d i t s c o n t r i b u -t i o n t o t h e t o t a l c o s t ( f i g s . 4 a n d 5 ) ;

( 3 ) t o t a l c o s t ( f ig . 6 ) ; a n d( 4 ) t h e e f f e c t o f t u b e r o u g h n e s s ( fi g s . 7 a n d 8 ) .F i g u r e 2 s h o w s t h e v a r i a ti o n o f t h e h e a t e x c h a n g e r

a r ea v e rs u s th e n u m b e r o f t u b e s , w i t h i n s i de d i a m e t e rc o n s i d e r e d a s a p a r a m e t e r . T h e h e a t e x c h a n g e a r e a r e-q u i r e d t o t r a n s f e r a c o n s t a n t t h e r m a l l o a d d e c r e a s e sw i t h a d ec r e a se i n t h e n u m b e r o f tu b e s a n d t h e i r i n-


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1 . 2 X I O s, I .4"

~ . o x , o 6

n..


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A.N. Nahavandi et at / Cost optimization of steam generators 11

%l,

tk l

,. Jn -I -a *U Ja .n .u J04 .

zO .Xo .

I 0

. 0 1

. 0 0 1

. 0 0 0 1 o ; . I .~ " I " O ; ; L ~ ' ~ , ~ m " ~ l . l " II 0, 0 0 0 2 0 , 0 0 0N U M B E R O F T U B E S , N

, 0 . 9 "

I

~ . o 5 "

~ . o . ~ '

I3 0 , 0 0 0

Fig. 4. Coolant pu mping pow er per unit area of the steam generator versus numb er of tube s with inside diameter as a parameter.

Figure 4 shows the variation of the coolant pumpingpower per unit area versus the number of tubes, withinside diameter considered as a parameter. The pumpingpower required to transfer a constant thermal load in-creases with a decrease in the number of tubes and thetube inside diameter, as expected. The reduction in the

number of tubes and their inside diameter, with a con-stant reactor coolant flow rate, increases the reactorcoolant velocity, leading to an increase in the pumpingpower required. The contribution of the coolant pump-ing power to the operating cost (the second term inthe right-hand side of eq. (1)) is shown in fig. 5. These


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12 A . N . N a h a v a n d i e t a l. / C o s t o p t i m i z a t i o n o f s t e a m g e n e r a t o r si 0 e

101'

n *-a i o 6

0, 4

I,-m00Z

IL0

t o 4 -

J J o - ' I t "I 0 , 0 0 0

D ; , O . 4 "

D I ' G S "

N U M B E Ra o , o o o

O F T U B E S , N

- - . O l , 0 . 6 .

~ . D i , G T "

~.~ O lmO. ", , ~ D L Om3 o ,o o o

Fig. 5. Annual operating cost of the steam generator versus number of tubes with inside diameter as a parameter.

curves exhibit a pattern similar to the pumping powerCHIVES.

Finally, the to tal annual cost of the steam generatoris presented in fig. 6. These curves show the variationof the cost versus number of tubes and their insidediameter. The lines of constant diameter have localminima which form a skewed curve oriented in the

'northwest' direction, tilted upward for smaller num-bers of tubes. The global minimum cost is located onthe line of lowest constant inside diameter. This pa-rametric study is confirmed by the optimization pro-gram using the direct search technique. It is apparentthat increasing the diameter increases the cost. For thisreason, characteristics above 1.5 in. i,d. are not shown.


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A.N . Na h a va n d i e t al . / Co s t o p t imi za t i o n o f s t ea m g en era to rs 13i 0 - -

EtaJ

0

0

J

Z

-1

i O s

, I . 5 ", I . 4 ", I . 3 ", I . 2 " I . I , I . O e, 0 . 9 "

, O . a " 0 . 7 "" 0 . 6 ", 0 . 5 ' O A "

I0 TM i0 4N U M B E R O F T U B E S , N

Fig. 6. To tal annual co st of the steam generator versus number o f tubes with inside diameter as a parameter.A l t h o u g h t h e m i n i m u m c o s t d e c r e a s e s w i t h d e c r e a s ! n gi n s i d e d i a m e t e r , t e c h n i c a l c o n s i d e r a t i o n s s u c h a s s c a l ef o r m a t i o n a n d m a i n t e n a n c e r e q u i r em e n t s p r o h i b it t h eu s e o f s m a l l d i a m e t e r t u b e s .

A p a r a m e t r i c s t u d y w a s c o n d u c t e d t o d e t e r m i n e t h ee f f e c t o f p ip e r o u g h n e s s o n t h e t o t a l a n n u a l c o s t . T h el o c i o f t h e m i n i m u m c o s t p o i n t s f o r th e c o n s t a n t d i a m -e t e r c o n t o u r s a r e s h o w n i n fi g. 7 at t w o d i f f e r e n t t u b er o u g h n e s s e s e = 0 . 0 0 0 1 a n d 0 . 0 0 0 0 0 5 f t . F ig . 8 s h o w sp l o t s o f t h e l o c al m i n i m a a t t w o d i f f e r e n t t u b e r o u g h -n e s se s : ( I ) f u l l y s m o o t h t u b e s u r f a c e a n d ( 2 ) t u b er o u g h n e s s e q u a l t o 0 . 0 0 0 0 0 5 f t. T h e s e c u r v es s h o wt h a t f o r ea c h t u b e d i a m e t e r , t h e n u m b e r o f t u b e s a n dt h e t o t a l a n n u a l c o s t r e q u i r ed t o t r a n s f e r t h e t h e r m a ll o a d a r e s m a ll e r f o r th e s m o o t h p i p e. T h i s c o n c l u s i o n

a g r e e s w i t h t h e a c c e p t e d e n g i n e e r i n g p r a c t i c e t h a ts m o o t h t u b i n g is t h e m o s t e f f i ci e n t m e a n s f o r h e a tt r a n s m i s s i o n [ 1 3 ] .

NomenclatureA = t o t a l h e a t e x c h a n g e a r e a ( b a s e d o n t u b e o . d . )( m 2 )A i = t o t a l h e a t e x c h a n g e a r ea ( b a s e d o n t u b e i. d . )( m 2 )A p = p r i m a r y c o o l a n t f lo w c r o s s s e c t io n a l a r e a ( m 2 )A 0 = t o t a l h e a t e x c h a n g e a r e a ( b a s e d o n o . d . ) f o r a

s t e a m g e n e r a t o r w h o s e c o s t p e r u n i t a r ea C a0is k n o w n ( m 2 )


10/12

14 A .N . N a h a v a n d i e t al . / C o s t o p t i m i z a t i o n o f s te a m g e n e r a t o r s

C ~_.C a =Ca0 =

Cp =Cp =C f =Cws -~

O OD i =E -F ( e s ) =fgc =g ~-~hfg =h p =

,.~, 1.13:o_z

i v II..-0W_azWmI--

OJIOS

,sTUSEO U G H N E S S 0 . o 0 0 1 F T .~ 0,00005FT.I I I t I I I I

t O 4

N U M B E R O F T U B E S , N

Fig. 7. Loci of the mi nimum cost with tu be rou ghness as a parameter.t o t a l ye a r ly s t e a m ge ne r a to r c os t ( $ /y r )c os t o f s t e a m ge n e r a to r pe r un i t a r e a ( $ /m 2 )a kno w n r e f e r e nc e c os t o f a s t e a m ge ne r a to rw i th a t o t a l he a t e xc ha nge a r e a A 0 ( ba se d ontube o . d . )c os t pe r un i t e ne r gy ( $ /w s )p r im a r y c oo la n t spe c i f ic he a t ( J / ( k g K ) )spe c i fi c he a t o f s a tu r a t e d l i qu id f o r t he s e-c o n d a r y c o o l a n t ( J / ( k g K ) )c oe f f i c i e n t de pe nd ing on w a l l / f l u id c om bina -t i on f o r t he s e c onda r y c oo la n t ( sugge s t e d va lue0 . 013 , [ 9 ] )U - tube o . d . ( m )U - tube i . d . ( m )p u m p i n g p o w e r p e r u n i t a r ea ( W / m 2 )a f unc t ion o f p r e s su re de f ine d b y e qs . ( 7 ) a nd( 8 ) ( C / ( j1 /3 m 2 /3 ) )f r i c ti on f a c to r f o r p r im a r y c oo la n t f l ow32 . 2 l bm f t / l b f se c 2 ( e qua l t o d im e ns ion l e s sun i ty i n S I un i ts )32 . 2 f t / s e c 2 ( e qua l t o 9 . 81 m / se c 2 i n S I un i t s )l a t e n t he a t o f e va po r a t i on f o r t he s e c onda r yc o o l a n t ( J / k g )p r im a r y c oo la n t f i lm c onve c t ive he a t t r a ns f e r

c oe f f i c i e n t ( W / ( m 2 K ) )i = i n t e r e s t r a t e ( y r - 1 )k p = p r i m a r y c o o l a n t t h e r m a l c o n d u c t i v i t y ( W / ( m K ) )k f = t he r m a l c ond uc t iv i t y o f s a tu r a t e d l i qu id f o r t hes e c o n d a r y c o o l a n t ( W / ( m K ) )k w = U - tube w a l l t he r m a l c ondu c t iv i t y (W / ( m K ) )N = n u m b e r o f U - t u b e sn = s t e a m ge ne r a to r l i fe , y rP p = p r im a r y c oo la n t p r e s su r e ( P a )q = h e a t t r a n sf e r b e t w e e n p r i m a r y a n d s e c o n d a r y

c oo la n t s ( W )Q = h e a t f lu x b e t w e e n p r i m a r y a n d s e c o n d a r y

c oo la n t s ( Q = d q / d A ) , ( W / m 2 )Q 1 = he a t f l ux a t t he U - tube i n l e t ( W /m 2 )Q 2 = he a t f lux a t t he U - tube ou t l e t ( W /m 2 )R = c a p i t a l r e c ove r y f a c to r ( y r - 1 )R f = f ou l ing f a c to r ( ( m 2 K ) /W )R p = p r i m a r y c o o l a n t f i lm r e si s ta n c e ( ( m 2 K ) / W )R s = s e c ond a r y c oo la n t f i lm r e s is t a nc e ( ( m 2 K ) / W )R w = U - tube w a l l r e s i st a nc e ( ( m 2 K ) /W )R c = R f + R p + R wS = a l l ow a b le s t r es s due t o t he t ube i n t e r na l p r e s -

su r e a t t he ope r a t i ng t e m pe r a tu r e ( P a )Tp = p r im a r y c oo la n t t e m pe r a tu r e ( Tp~ = p r im a r y


11/12

A.N. Naharandi e t al. / Cost optimiza tion o f s team generators 15

r e

0,,J$.J

:E-,):E

9 X l O s

8 X I 0 5

7 X 0 8

6 X l O m

5 X I010

SMOOTH

ROUGHNESS E ,LO.O00005 FT.

I i II0,000 20,000NUMBER OF TU BES t N

I30,000

F ig . 8 . T h e e f f e c t o f t u b e r o u g h n e s s o n c o s t.

coolant at the U-tube inlet, Tp2 = primarycoolant at the U-tube outlet) (K)

T s = secondary coolant temperature (K)t = U-tube wall thickness (m)U = primary to secondary coolant overall heat

transfer coefficient (W/(m 2 K))Vp = primary coolant veloci ty (m/sec)

Wp = primary coolant mass flow rate (kg/sec)y = a coefficient given by the ASME Boiler Code

G r e e k s y m b o l se = absolute roughness for the interior surface of


12/12

16 A.N. Nahavandi e t a l . / Cos t op t imiza t ion o f s team genera torsthe U-tubes (m)

0 = steam generato r operating time (sec/yr)/ap = primary coo lant viscosity (Pa sec)pf = density of saturated liquid for the secondar y

coolant (kg/m 3)pg = density of saturated vapor for the secondar ycoolant (kg/m 3)

o = surface tension of liq uid -va por interface forthe secondary coolant (N/m).

References[ 1 ] R. Rajasekaran and P.A. Lytle, A compact design of shelland tube heat exchangers, ASME Publication 73-HT-17(1973) 1-8.[2] J.E. Mott, J.T. Pearson and W.R. Brock, Computerizeddesign of minimum cost heat exchanger, AICHE-ASMEHeat Transfer Conference, Denver, Colorado, August1972; ASME Publication 72-HT-26 (1972) 1-8.[3] M.F. Dehne, Economical design of air-cooled heat ex-changers, Cooling Towers, Chem. Eng. Progr. AICHE(1972) 13-21.[41 P.G. LaHaye, F.J. Neugebauer and R.K. Sakhuja, A gen-eralized prediction of heat transfer surface performance

and exchanger optimization, ASME Publication 72-WA/HT-55 (1972) 1-7.[5] B.R. Andeen and L.R. Glicksman, Computer optimiza-tion of dry cooling tower heat exchangers, ASME Pub-lication 72-WA/Pwr-8(1972) 1-11.[6] C.H. Armstrong and R.S. Schermerhorn, Economics ofdry cooling towers applied to combined-cycle powerplants, ASME Publication 73-WA/Pwr-5 (1973) 1-5.[7] P.F. Oswald,and G.A. Kochenberger, Optimization ofheat transfer designs by geometric programming, ASMEPublication 72-WA/HT-15 (1972) 1- 8.[8] W.H. McAdams, Heat Transmission, 3rd edn, McGraw-Hill, New York (1954).[9] W.M. Rohsenow, A method of correlating heat-transferdata for surface boiling of liquids, Trans. ASME 74,Aug. (1952) 969-976.[10] W.R. Burrows, R. Michel and A.W. Rankin, A wallthickness formula for high-pressure high-temperature

piping, Trans. ASME, Apr. (1954) 427-444.[11] K.A. Gulbrand and P. Leung, Power system economics,A sensitivity analysis of annual f'txed charges, ASME Pub-lication 74-WA/Pwr-4 (1974) 1-8.[12] M. Pappas and C.L. Amba-Rao, A direct search algorithmfor automated optimum structural design, AIAA J. 9 (3)(1971) 387-393.[13] W.F. Cope, The friction and heat transmission coefficientsof rough pipes, Proceedings of the Instution of MechanicalEngineers, London, 1941, Vol. 145, 99-105.

special problem ref

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