ANALYSIS

I N SPACE

OF THE PARAMETERS OF THERMODYNAMIC

NUCLEAR POWER GENERATION PLANTS

D.D. Kalafti

CYCLES

UDC 629.194.6

The opt imt tm p a r a m e t e r s of t h e r m o d y n a m i c c y c l e s in s p a c e n u c l e a r p o w e r g e n e r a t i n g p l a n t s a r e aa~alyzedfor two b a s i c cond i t i ons : wi th cons t an t and v a r i a b l e t h e r m a l r e a c t o r p o w e r . So lu t ions a r e ob ta ined fo r each of the cond i t i ons by u s i n g two b a s i c c r i t e r i a : m i n i m u m s p e c i f i c r a d i a t o r s u r f a c e a r e a and m i n i m u m s p e c i f i c m a s s of the p lant . An a n a l y t i c a l p r o c e d u r e i s d e v e l o p e d fo r a n a l y z i n g the o p t i m u m p a r a m e t e r s . I t is shown that fo r e v e r y g iven r e a c t o r and s t e a m g e n e r a t o r t h e r e is a de f in i t e l i m i t i n g e l e c t r i c a l p o w e r c o r r e s p o n d i n g to m i n i m u m s p e c i f i c m a s s . F u r t h e r i n c r e a s e of e l e c t r i c a l p o w e r l e a d s to an i n c r e a s e of the s p e c i f i c s u r f a c e a r e a of the r a d i a t o r and, s i m u l t a n e o u s l y , to an i n c r e a s e of the s p e c i f i c m a s s of the p lan t .

Conditions f o r Minimum Surface A r e a o f the R a d i a t o r , a t C o n s t a n t T h e r m a l R e a c t o r P o w e r

F o r s p a c e n u c l e a r p o w e r g e n e r a t i n g p l an t s , the m i n i m u m s p e c i f i c s u r f a c e a r e a of the r a d i a t o r , f2( m 2 / k W ) , is u sed in a n u m b e r of r e p o r t s as the c r i t e r i o n fo r s e l e c t i n g the o p t i m u m hea t d i s s i p a t i o n t e m p e r a t u r e T 2.

The r e q u i r e d r a d i a t i n g s u r f a c e a r e a of an i s o t h e r m a l r a d i a t o r is

F2 = e C ~ ' (1 )

where C is the coefficient of radiation of a perfectly black body; e is the degree of blackness of the

radiator surface, and the heat dissipated for an ideal cycle at an average heat dissipation temperature

Tin v is

JV t--~qt 2u T2 . (2) Q2= t - -~ - - t = ~av_:r2

The s p e c i f i c s u r f a c e a r e a of the r a d i a t o r i s

F_._L2 ~ i f2~= Nt eC(Ttav--T2) T~, (3}

and the cond i t ion fo r m i n i m u m f2 f o r an i d e a l cyc le , c h o s e n with T tav = cons t [1, 2] is

T2=0.75T~ o r X t = T--~-~ =0.75 . (4) TI av

If we take into account the internal relative efficiency of the turbine ~0i =const, * the specific surface area of the radiator assumes the form

f _ i [ t l--no, 7 . 2e -- ~C~lo, - - (~av-~'2) T~ ~- T~ j , (5) T A B L E 1. V a l u e s of X e, ~t, and ~e C o r - r e s p o n d i n g to M i n i m u m f2e and the condi t ion fo r m i n i m u m f~e with T tav = cons t

t .0 0.7 0.5 0.0

o 7iaV

0,75 0.25 0.77 0.23 O, 78 0 22 O. 80 O. 9,0

0.25 0 , i6 (). i t 0.00

* If the hea t f r o m the l o s s in the e l e c t r i c a l g e n e r a t o r wi th e f f i c i e n c y 77 i s a l s o d i s s i p a t e d by the r a d i a t o r , g then V0i can be c o n s i d e r e d as the p r o d u c t V0iVg = cons t , and the e n e r g y g e n e r a t i o n e f f i c i e n c y V~ can be found i m m e d i a t e l y .

T r a n s l a t e d f r o m A t o m n a y a ]~nergiya , Vol. 22, No. 6, pp. 432-439, June, 1967. O r i g i n a l a r t i c l e s u b m i t t e d A p r i l 1, 1965.

545

fo r a r e a l i s t i c c y c l e is

~ , 2 ~ - V ~ (6 ) X~ = 1 2 n 0 ~ - -

Hence, the t h e r m a l e f f i c i ency of a r e a l i s t i c c y c l e ~?t = 1 - X e and the e l e c t r i c a l e f f i c i ency of the s t e a m t u r b i n e uni t , c o r r e s p o n d i n g to the m i n i m u m f2e i s

~e ~lt~bi ~ 0.625-- 0.5 If l .5625 - - ~10~. (7)

The d e p e n d e n c e of Xe, the e f f i c i ency of the c y c l e and of the p l an t c o r r e s p o n d i n g to m i n i m u m f2e by Eq. (6) and (7) , on %i is shown in Tab le 1.

A c c o r d i n g to t h e s e da ta , the cond i t ion for m i n i m u m s p e c i f i c r a d i a t o r s u r f a c e a r e a wi th T la v = cons t c o r r e s p o n d s to a r e l a t i v e l y low e f f i c i ency : the t h e r m a l e f f i c i ency Vt = 20-25% and wi th %i = 0 . 5 - 0 . 7 the e l e c t r i c a l e f f i c i ency if of the o r d e r ~?e = 11-16%.

If we d e t e r m i n e the a v e r a g e t e m p e r a t u r e of suppIy of hea t to the c y c i e

:~q (8) T~ av=~ hs~

for SNAP-2 (USA) o p e r a t i n g on m e r c u r y vapo r , a c c o r d i n g to da ta f r o m [3] the r a t i o of the t e m p e r a t u r e s X e fo r the c y c l e of th is p lan t is 0.785, and fo r the c y c l e of SNAP-8 (F ig . 1), Xe=0 .766 , which is c l o s e to the cond i t ions fo r m i n i m u m f2e with T lav=COns t and fo r v a l u e s of % i = 0 . 5 - 0 . 7 (Tab le 4, co lumn 2).

However , the cond i t ion T , a V = c o n s t c o r r e s p o n d s only to a cho ice of T 2 fo r a s p e c i f i e d r e a c t o r wi th a c o n s t a n t t h e r m a l p o w e r Q1 = eons t , wi th the a s s u m e d s u r f a c e a r e a of the s t e a m g e n e r a t o r and coo lan t supp ly . Such a c a s e i s p o s s i b l e at h igh (about 2000 ~ K) l i m i t i n g t e m p e r a t u r e s of the n u c l e a r fuel of the UO 2 type, when an i n c r e a s e of T l a v of the c y c l e to the op t imum e f f i c i ency va lue is l i m i t e d by the d u r a b i l - i ty of the fuel e l e m e n t s s e a l s or by the t u rb ine b l a de s . Even if u r a n i u m with a low l i m i t i n g t e m p e r a - t u r e of the o r d e r of T o = 1000 ~ K is used , a c h a r a c t e r i s t i c f e a t u r e of des ign ing the i n s t a l l a t i o n is the f e a s i b i l i t y of r e p l a c i n g the a v e r a g e t e m p e r a t u r e s of the coo lan t and c y c l e with the c o r r e s p o n d i n g change of t h e r m a l p o w e r of the r e a c t o r .

A c c o r d i n g to e x p r e s s i o n (3), the s p e c i f i c s u r f a c e a r e a of the r a d i a t o r is u s u a l l y m o r e dependen t on T~a V than on T 2. Consequen t ly , wi th Q l = v a r i a b l e , op t imum t e m p e r a t u r e s of supp ly and r e m o v a l of hea t in the c y c l e m u s t be c h o s e n s i m u l t a n e o u s l y .

T o = ;060 r, ~ ~ tzr =705~

gO0 t t r ~ g ~ f ~ 7 o c j~i~ =577? 800 / P,=lg atm ]

700 , re~5#oK [ 500

500

400

300

200

100

P2~1,3 aim

02 4 0.6 08 1,0 1,2 1.4 $, kcaI/kg~I<

F ig .1 . T h e r m o d y n a m i c c y c l e of the S N A P - 8 p o w e r g e n e r - a t ing p lan t .

C o n d i t i o n s f o r M i n i m u m R a d i a t o r S u r f a c e A r e a f o r a F i x e d E l e c t r i c a l a n d V a r i a b l e T h e r m a l R e a c t o r P o w e r

Le t us c o n s i d e r the condi t ion fo r m i n i m u m r a d i a t o r s u r f a c e a r e a for a v a r i a b l e t h e r m a l r e a c t o r power , Q ~ = v a r i a b l e (as in the p r e v i o u s ca se , for a g iven r e a c t o r and s t e a m g e n e r a t o r wi th a chosen coo lan t s u p p l y ) . Wi th a f ixed l i m i t i n g n u c l e a r fuel t e m p e r a t u r e T O = const , the t h e r m a l p o w e r of the r e a c t o r depends l i n e a r l y on the a v e r a g e coo lan t and c y c l e t e m p e r a t u r e s [4]:

()~ :_ q, (To__ Tlay (9)

w h e r e q~= cons t is the t h e r m a l output in d e g r e e of the t e m p e r a t u r e d rop ( T 0 - T l a v ) . The e l e c t r i c a l p o w e r fo r an idea l cyc le , a s a funct ion of t e m p e r a t u r e , is

= q~n~ = ql (To - - Z'I~,) ( ~ .... I', N~ (10)

Since, in the i d e a l c y c l e

Q2 :=Q~ r~ (1L) TI av '

according to Eqs. (i) and (9), the surface area of the radiator is

qt (To-- Ttav) (12) [2-- eCTIavT a

546

ForQi=variable, the condition for minimum F 2 at two alternative temperatures of the cycle, Tta v and T2, assoeiatedwith the supplementary condition N t = const for T O = eonst, gives the optimum relation between the ratios of the temperatures in the ideal cycle and in the plant in the form:

For a real is t ic plant, a r e a is

1'2 = 0,75 r, av x~ 0.75Y ~ (13) T, av ~ or

taking account of the turbine efficiency for ~0i = const, the radiator surface

qt (To--1", av (1 -- ~10i) ] , (14) F2e= ecT~ ~[ ~T-~2 ~1~ + ~1 aV

and the optimum relat ion between the temperature ratios is

where the coefficient ~opt (Fig. 2) is --e

opt op opt X e - - ~ ~*~ , (15)

Olo~ --0.25) ~ePt-}- (I -- ~loi) ( 16 )

Using the nomenclature f rom Eq. (10), we obtain

( i - x ~ ) ( i - Y~) = Nr qiTo~loi '

and, assuming that Ye = X e / ~ e, we find

(17)

_ _

Xe(1-xr ( i _ X e ) Ne ( 1 8 )

q~To~loi

~e can be eliminated by equating Eqs. (18) and (18) ; however, in this case we obtain an equation of the third degree in X e and therefore for fixed values of Ne, ql, To and V0i the value X e is found more simply by tr ial and e r r o r until coincidence of ~e in Eqs. (16) and (18). After this we find Ye, and then the extreme values

T e x t v ~ ot~. - - ex t ~.opt_ i a v = ~ e *o a n a l ' 2 - ~ ) t e Tiav. (19)

When N e =0, it follows from Eq. (8) that ~ e = X e and formula (16) t rans forms to (6), i .e . , in the limit, the optimum rat io of the temperatures of the real is t ic cycle with Q~ =var iable tends to an optimum distribution of tempera tures with QI= const. K N e > 0, then from Eq. (18) X e < ~e, i .e . , for V0i= the

.0 , \ \

aN , 0.5 ~d

Fig.2. Optimum relation ~opt between temperature ratios of the cycle T 2 /Tiav and the plant Tlav/T 0 for real- istic cycles at different values of Xe ~

and ~0i.

same and with the optimum value of the tempera tures , the thermal efficiency of the cycle v t = l - X e with Ql=cons t is

opt always higher than with Ql=var iab le or Vt > 20-25%.

If, by choosing T o =1600~ for the SNAP-8 unit and with a constant e lect r ical power N e =60 kW, it becomes possible to change the average tempera tures of the coolant and cycle to the extreme values given in Table 4 (column 4), then in place of the value assumed for the surface area of the radiator F 2 = 54 m 2, we find the required minimum value of F ~ m = 4 6 . 1 m2

Thus, for a given reac to r and steam generator , the extreme tempera tures of the cycle Tin v and T 2 can be found for each specified e lectr ical power N e =eonst and, also, the corresponding minimm:~ surface a rea of the radiator can be found. In this case, the g rea te r is the stated electr ical power N e for a given reactor , the g rea te r also the c o r , responding minimum surface area of the radiator, as shown in Fig. 3 where, the line of minimum f~e is drawn through the

547

1,2

10

03 ~D

v.m

~7

0,5

0,4

800

~ . N ~ . Ne =0 . . . .

850 g00 ,q50 1000 T1av~

Fig.3. Dependence of the specif ic r ad ia - to r sur face a rea on the e lec t r i ca l power of the plant and of the t e m p e r a t u r e d is - t r ibut ion T 2 and Tlav: 1) f~e m for Tla v =const ; 2) f ~ m for T iny=va r i ab l e ; 3) f2e for rome m with Tlav =const .

minimum of the specific surface areas of the radiator for

different electrical power but with N e = const; this increases

monotonically with increase of N e.

On the same graph, the envelope is drawn below the

region of possible values of fPe- It is obvious that this

envelope corresponds to the minima of the radiator specific

surface area for Tin v = const, which we considered above,

with a larger specific surface area f2e for the same electric

power. Only at the point N e = 0 do the lines of minimum f2

with Tin v = variable and T I = constant merge.

On the other hand, with a given radiator surface area

F2=const, the criterion for selecting the optimum para-

meters of the cycle is maximum electrical power. All

practicable units must satisfy this condition.

Since the value found in this case will be themaximum

electrical power according to Eq. (i0) with the supplemen-

tary condition F2=const , determined by Eq. (12) or (14), in comparison with the previous criterion only the initial

condition is changed. Consequently, for the optimum re-

lation between the ratios of the cycle and unit temperatures,

we obtain the same result. Thus, the line of minimum

values of f2e in Fig. 3, with Tlav=Variable for Ne=const , is simultaneously the line of maximtu~n values of electrical

power for F2=const. With the stated extreme ratio of

temperature for a given value of Fpe , the extreme temper-

atures Tia v and T 2 can be found also from Eq. (14), which,

for a r e a c t o r of the SNAP-8 type with Q1 =var iab le , co r responds to another power N e and are given in Table 4 (column 5).

For a given r e a c t o r and s team genera tor with F2= eonst, the condition for max imum e lec t r ica l eff ic iency coincides s imul taneously with the condition of minimum specif ic m a s s of the genera tor .

Conditions of Minimum Specific Mass of Unit

with Variable Thermal Reactor Power

The minimum value of the specific mass of the generator per unit of electrical power is the second

most important criterion for selecting the parameters.

We shall use this criterion as before with a given reactor and steam generator and also with the

assumed coolant supply and limiting temperature of the nuclear fuel T O = const. The thermal power of the

reactor depends on the average temperatures of the coolant and of the cycle Tla v according to Eq. (9).

Taking into account the slow change of vapor pressure of liquid metals as a function of temperature, we shall assume the mass of the reactor and steam generator, with constant surface area of heat exchange,

to be invariable. We shall neglect the change in inass of the turbine* and we shall assume the mass of

the isothermal radiator to be proportional to the radiating surface.

With these specif ied conditions, the specif ic m a s s of the unit is

M~-~ bF2 (20)

where M i = const is the mass of the entire installation except the radiator, in kg; b is the mass per I m 2

of the radiator in kg/m 2.

* If the mass of the turbine changes proportionally as the power, this mass generally has no effect on

the parameter of the specific mass minimum. In future, therefore, the symbol M I will denote the mass

only of the primary loop and shielding.

548

TABLE 2. Extreme Trends of Cycle Temperatures with Different Criteria

Starting criterion

Minimum specific surface area of radiator, fz

Minimum specific mass, m t Minimum radiator surface area Fz,

with Nt: const Minimum specific mass of unit,

mt

Constant temper ature

T l a V ~ c o n s t

T l a v : c o n s t

/ T O := const

)

E x t r e m e t reuds

of absolute temperature

in the assumed nomenclature.

TT-•_z = 0 . 7 5 i a v

Formula (24)

TT_•2 =0.75 T~av ~ v t o -

Xt~0.75

X t ~ 0 . 7 5 Yt

n~ x:

0,25 < 0.25

| /0 .75T 2 1 - - u To

< 0.25)

T A B L E 3. S u c c e s s i v e A p p r o x i m a t i o n for Ca lcu la t ing m ra in

t6 47

Taking intoaeeount F 2 from Eq. (12) and N t from Eq. (i0), the specific mass of the unit as a function of

temperature for an ideal cycle is

bqt (To-- T~av) M~-- eCT~avT ~

m t = qi (T 0_T~ av) (1 -- T2/Ttav) " (21)

The required condition fo r minimum m t is that two of its partial derivatives with respect to the in-

dependent variables Tia v and T 2 should be simulta-

neously equal to zero. If we equate them to zero by the

usual method, we obtain two equations which cannot be so lved in expl ic i t f o rm and t he r e fo r e we use a speei.al method for so lv ing them, n a m e l y : we subs t i t u t e the s t a r t i n g funct ion i t s e l f at the point of m i n i m u m m~ n m for the r a t i o s of the n u m e r a t o r to the d e n o m i - h a t e r which occur in the e x p r e s s i o n for the d e r i v a t i v e s of the f r a c t i o n a l funct ion.

After simplifying, we obtain from the partial derivatives

bTo min 2 (22) eCr~ rat (Tiav -- TOT2) 0;

3b __rntrain_~ 0. (23) eCr~

we divide Eq. (22) by Eq. (23), we e l i m i n a t e m~ n i n and we find the op t imum r e l a t i o n between the If

r a t i o s of the t e m p e r a t u r e s for an ideal cyc le with condi t ions of m i n i m u m spec i f i c m a s s of the uni t , c o m p l e t e l y co inc id ing with r e l a t i o n s h i p (73), obta ined p r e v i o u s l y with condi t ions of m i n i m u m r a d i a t o r s u r f a c e a r e a F 2 for Q1 = va r i ab l e .

Tab le 2 c o m p a r e s the e x t r e m e t r ends be tween the r a t i o s of the t e m p e r a t u r e s , and the e f f i c i enc ies a r e g iven for d i f fe ren t c r i t e r i a and condi t ions for ideal cyc le s of the uni t .

F r o m e x p r e s s i o n (23), the t e m p e r a t u r e of heat r e m o v a l in an i s o t h e r m a l r a d i a t o r , fo r an ideal cyc le and c o r r e s p o n d i n g to the m i n i m u m spec i f ic m a s s of the uni t is

Text ~/ 3b 2t = V ~ (24) eCm~ rn

ext i . e . , for a known degree of b l a c k n e s s c, the t e m p e r a t u r e of heat r e m o v a l T2t in the ideal cycle of the un i t is d e t e r m i n e d only by the r a t io of the m a s s pe r m 2 of r a d i a t o r su r f a c e a r e a to the spec i f ic m a s s of the e n t i r e uni t b / m t m ext ext at i ts m i n i m u m point . We obtain Tla v with the va lue of T 2 found f rom Eq. (13).

F o r r e a I i s t i c s t e a m tu rb ine un i t s with rT0i =eons t , taking into aceount Eq. (14), the spec i f ic m a s s of the unit is

bqt (To-- T~ av) ~CT~ ( T--~av ~~ i--~~ )

m e q~ (To- r~av~ (i -- Tz/Tta v) ~1o~ (25)

549

From the partial der ivat ives of m e with re spec t to Tin v and T2, after s impl i f icat ion and subst i tu- tion of the ratio of the numerator and denominator by the value of the starting function at the point of min imum, we obtain for a rea l i s t i c unit:

5 ~Crl'lo~ [~loiT2To + (i - - tloi ) T~av] - - mrgm (T~av - T2To) ~- 0; (26)

b eCT~qoi ~ 3~1a+4(1 ~ \ r~av O. - - ,,ou --~-2 ; - - m ~ i n = (27)

If w e d i v i d e E q . (-26) b y (27) w e o b t a i n , a s b e f o r e w i t h f~nin, E q s . (15) a n d (16 ) .

W e f i n d f r o m E q . (27) t h e e x t r e m e t e m p e r a t u r e of h e a t r e m o v a l , c o r r e s p o n d i n g to t h e m i n i m u m of

t h e s p e c i f i c m a s s of a r e a l i s t i c u n i t

== , i -~ 0 75~10~xmm/ ' K,

rain (kg/W) depends on the type and power of the unit where the specific mass at the point of minimum m e (C = 5 . 6 7 x 10 -8 W / m 2 O K 4 ) .

S i n c e t h e r a t i o of t h e c y c l e t e m p e r a t u r e s X~ x t o c c u r s in Eq . ( 2 8 ) , a n d i s n o t k n o w n a p r i o r i , we u s e

s u c c e s s i v e a p p r o x i m a t i o n f o r r e a l i s t i c c y c l e s . If, f o r t h i s , w e a s s u m e t h a t

X Y - - r2 _ X2 P o �9 ,

T A B L E 4. C o m p a r i s o n of t h e C h a r a c t e r i s t i c s of S p a c e N u c l e a r P o w e r G e n e r a t o r s w i t h a

S N A P - 8 T y p e of R e a c t o r , f o r D i f f e r e n t C r i t e r i a in S e l e c t i n g the C y c l e P a r a m e t e r s

Parameters

Electrical power N e, kW Thermal power of reactor

Q:, kW Heat dissipated in radiator

Q2, kW Average temperature of sup-

ply of heat Tta v, ~K Temperature of dissipation

of heat T 2, OK Ratio of cycle temperatures

xe Ratio of temperatures in

unit Ye

Thermal efficiency of cycl~ ~t, %

Electrical efficiency of

unit ~1, e, % Surface area of radiator

F2 e , m2

Specific surface area of ra- diator f~e, m2/i kw

Mass of unit M, kg Specific mass of unit m e ,

kg/k W Fraction of radiator mass 6

SNAP-8 [3]

for T l a v = C O n s t , T ~ v a r for TO = const, T l a v = v a r Te = v a r

mernin min ramin 12e

2 3

56,5 81.2

480.7 480.7

424,2 399.5

840 840

654,6 573

0,779 0.682

0.792 0.792

22,t 3t.8

1t,8 t6.9

50.8 8t.7

0.898 1.006 t054 t363

t8.7 t6,77 0.48 0.60

i 6

min Nernax Y2e for for N e ~ COnSt F2 ~ eOllSt

4 5

60 65.6

348 372.5

288 306.9

901 890

609 595

0,676 0,669

0.850 0,840

32,4 33.t

17~2 t7,6

46,1 540

0.77 0.82 t007 1086

16.8 t6.55 0,46 0.50

60

480,7

420.7

84O

643

0.766

0,792

23.5

12.5

54.0

0.90 t086

t8.1 0.50

72.6

401.8

329.3

876

578

0-660

0.826

34,0

18.1

64.6

0.89 t192

t6.42 0.54

Note: In generai, the following values are taken for all variants: To= !060~ Mt=546 kg; q1=2.t85 kW/~ b = 1 0 k g / m ~; ~0i =0-532;e=0"8,

550

19 - \

.e. ~o

b

\ 17

840' 1%00

/ /

/

| 1 I I I

~lr . ; \ ~ . ' 1

k w /

I~ NCas I

r II =16"42 876% ~'l t i J

850 900 gso

Fig .4 . Specif ic m a s s of an e n e r g y g e n - e r a t i on p lant as a funct ion of a s s u m e d e l e c t r i c a l power and d i s t r i b u t i o n of

cyc le t . empera tu re s T 2 and Troy: 1) m e ~ , r a i n for f2 m m with Tla v = const ; z) m e with

Tttav=Var; 3) m~e in with T ,a v = eons t .

we find the value

= :%ext

which we shall use together with Eq. (28).

If b, ~/0i, ~ and T~ are given and also the minimum r a i n specific mass of the unit m e is given as a function of the

type and power of the unit when designing, by assuming X e (a priori) as a function of ~?0i in accordance with Fig.2, we find from Eq. (28) the values of T2e and r by Eq. (]6), for which we determine X e by Eq. (29). If X e diverges from the value assumed a priori, we repeat the calculation with the va lue found for X e. The cyc le t e m p e r a t u r e s Which c o r r e s p o n d to m i n i m u m spec i f ic m a s s of the r e a l i s t i c un i t can a lso be found in the ca se when m e is known for other t e m p e r a t u r e s . F o r example , for a v a r i a n t of the SNAP-8 uni t with the p a r a m e t e r s shown in Fig. 1, we a s s u m e : b = 1 0 k g / m 2 ; q1=2,185 k W / ~ C; ~?0i =0.532; e =0.8 and the spec i f ic m a s s m e = 18.1 k g / k W (without tak ing sh ie ld ing into accoun t ) .

The sequence of the ca lcu la t ion is shown in Tab le 3, f i r s t of all fo r the s ta ted va lue of m e and then for the va lue found to a f i r s t app rox ima t ion ; the p a r a m e t e r s ob ta ined with minimum specific mass are given in Table 4 (column 6).

Since the optimum ratio of the temperatures according to Eq. (15) has been obtained generally, the line of minima of specific surface area for different values of N e = eonst in Fig. 3 is simultaneously the line of minima of specific mass for Troy=variable. We can plot similar criteria for different values of N e =eonst in the coordinates me-T1a v (Fig. 4). In this case, the line of minima of specific mass passes through the minimum of these curves. With increase of the electrical power of the plant, the line of specific

mass minima decreases initially, passes through the minimum minimorum m e at a definite value of , .m in i~ e and then increases again. Consequently, for every given reactor there is a definite limiting electrical power Nle im corresponding to the minimum minimorum of the specific mass. At a high

electrical power it increases as the minimum specific mass and also as the specific surface area of the

radiator and therefore, in selecting the parameters, only the lower limit should be taken for the electri- cal power of the plant.

For an ideal cycle, we obtain from Eqs. (3) to (23) the fraction of the radiator mass in the mini- mum mass of the unit

Q e x t e x I: e x 5t bf~t 2 T.~ t -- ~l~

m~ ~m 3N~X t 2/~-ext ~ext~ 3~},~xt ' o ~ - - l a V - - ~ 2 j

(30)

ext Similarly, for a realistic cycle we find 5 e from Eqs. (5) to (27)~ Since 5 < i, according to Eq. (30)

*)t is always > 0.25 for an ideal cycle with T~a v =variable, for the condition of minimum specific mass.

Condition of Minimum Specific Generator Mass

at Constant Thermal Reactor Power

For nuclear fuel with a high limiting temperature of the order of T O = 2000 ~ K, the temperature of the radiator T 2 is determined at constant thermal reactor power, i.e., Tic v = const.

551

~,~0-

0,8-

-20 rain i -Ig ~ f ~ ~f2e with Tlav=COnSt -/8 m~ rain m,~ \ 1 /

i I

___] 0/ i

550 573 500 6/5 655 T2,~

Fig.5. Change of specific mass and specific sur- face area of the radiator as a function of the

temperature T 2 with Tin v = eonst = 840~ in the region of the optimum.

rain

In this case, formula (24) for an ideal cycle and formula (28) for a realistic cycle remains valid and only the minimum specific mass in these equations according tO Eqs.(21) and (25) respec- tively should be determined at the give tempera,

ture Ttav=eonst. We find the quantity Xe=T 2 /Tin v in Eq. (28) through the previously-used temperature T 2 by successive approximation until the value coincides with that found by this formula.

At the bottom in Fig. 4 is drawn the line enveloping the values of the specific mass minima corresponding to their values at Tla v =const. The parameters with minimum specific mass of the generator for SNAP-8 with Tla v = 840~ are shown in Table 4 (column 3).

Thus, for Tiav=COnSt , there are always two values of the extreme temperature T~: a larger value for the min imum specif ic sur face a r ea of the rad ia to r f~e in and a lower value corresponding to the min imum specif ic m a s s of the genera to r

(Fig. 5). We can find the relation between these extreme, temperatures for an ideal cycle from the me .~min ra t io of fo rmula (24) with m t = m 2 e to its express ion with m~ nm according to Eq. (21):

m 4 /~-~mln T2 2t (31)

A s i m i l a r re la t ion for a r ea l i s t i c cycle can be obtained f rom Eq. (28).

However, in p rac t i ce both the c r i t e r i a considered are s imul taneously important . Therefore , we note that the der iva t ive d f J d m f rom the min imum f2 to the min imum m e va r i e s f rom 0 to - o0, i .e . , a reduction of the t e m p e r a t u r e T 2 f rom the min imum f2 gives at f i r s t a s m a l l e r inc rease of specif ic su r face a r ea of the r ad ia to r re la t ive to the reduction in the specif ic m a s s of the genera tor , until the der iva t ive r eaehes the f rac t ion of the m a s s of the rad ia to r at point A in Fig. 5 for:

b d]2dm =bfz=6",n (32)

Further reduction of T 2 should lead to a more rapid increase of f2 relative to the reduction of specific mass than is obtained in the generator and therefore it is unjustified. The derivative df2/dm is determined from the ratio of the derivatives dF2/dT 2 and dm/dT 2 respectively by Eqs. (3) and (21) for an ideal cycle or Eqs. (5) and (25) for a realistic cycle.

Consequently, the minimum of the specific radiator surface area and condition (32) are the limits of a region within which occurs the optimum radiator temperature T~ pt with Tin v = const.

Of course, in thefinal selection of parameters and working substance of the thermodynamic cycle of a space power generator it is necessary also to take into accourit the initial and final steam pressures, the permissible final moisture content, the non-isothermality of the radiator, the possibility of some change of mass of the primary circuit or shielding as a function of Tia v or T 2 etc, which falls outside the scope of the theoretical analysis carried out with the assumed model of the effect.

1.

2. 3.

4.

LITERATURE CITED

Blyu and Ingolad, Raketnaya tekhnika i kosmonavtika, No. 5, 168 (1963). Bernatovi ts , Voprosy raketnoi tekhniki, No. 1, 17 (1964). Vspomogate l 'nye yadernye ~nerget icheskie s i s t emy dlya kosmicheskikh poletov (Supplementary nuclear energy generat ing s y s t em s for space f l ights) , Atomnaya tekhnika za rybezhom (Foreign nuclear technology) [Russian translation], No. 7, 10 (1963). D.D. Kalafati , Thermodynamic Cycles of Nuclear Power Stations [in Russian], Atomnaya t~nergiya, Gos~nergoizdat , Moscow, No. 8, 5 (1960).

552