HIGH P E R F O R M A N C E, SHORT T I ME DURATION, MHD G E N E R A T OR S Y S T EM A. Sherman''

HIGH P E R F O R M A N C E , SHORT T I M E DURATION, MHD G E N E R A T O R S Y S T E M

A. Sherman''⁴

G e n e r a l E l e c t r i c S p a c e S c i e n c e s L a b o r a t o r y , V a l l e y F o r g e , P a . A b s t r a c t

T h e p o t e n t i a l p e r f o r m a n c e of a c h e m i c a l l y fueled MHD g e n e r a t o r o p e r a t i n g on an open c y c l e h a s b e e n c o n s i d e r e d . T h e e s s e n t i a l new f e a t u r e i s t h a t c h e m i c a l fuels o t h e r t h a n f o s s i l a r e c o n s i d e r e d . By a p p r o p r i a t e l y s e l e c t i n g u l t r a high t e m p e r a t u r e f u e l s , e x t r a o r d i n a r i l y high p o w e r d e n s i t i e s c a n b e a c h i e v e d . In the p r e s e n t p a p e r a t h e o r e t i c a l i n v i s c i d a n a l y s i s is c a r r i e d out of the c h a n n e l flow for s u c h a g e n e r - a t o r in o r d e r to d e t e r m i n e i t s s i z e , g e o m e t r y , a n d w e i g h t . C a l c u l a t i o n s show t h a t r e l a t i v e l y m o d e s t m a g n e t i c fields a r e p e r m i s s i b l e , so t h a t a p p r o p r i a t e c o n f i g u r a t i o n s of p e r m a n e n t m a g n e t s a r e c o n s i d e r e d for t h i s a p p l i c a t i o n . S o m e e s t i m a t e s a r e m a d e of t h e w e i g h t of s u c h a p e r m a n e n t m a g n e t s y s t e m . C o n s i d e r a t i o n is a l s o given to t h e h e a t t r a n s f e r c a u s e d by t h e e x t r a o r d i n a r i l y high gas t e m p e r a t u r e s . T e n t a t i v e c o n c l u - s i o n s a r e t h a t t h e h e a t l o s s in s u c h g e n e r a t o r s c a n , in all p r o b a b i l i t y , be h a n d l e d by a c o m b i n a t i o n of r e g e n e r a t i v e and r a d i a t i v e c o o l i n g . F r o m t h e s e t h e o r e t i c a l c a l c u l a t i o n s a n d e s t i m a t e s the fuel plus o x i d i z e r flow r a t e for s u c h an MHD g e n e r a t o r s e e m s to be in t h e r a n g e of 3 K g / k w - h r , w h e r e a s t h e i r s p e c i f i c w e i g h t s r a n g e f r o m 6 K g / k w down to 3 K g / k w for t h e l a r g e r s i z e s . A n u m b e r of p o t e n t i a l a p p l i c a t i o n s for s u c h an MHD g e n e r a t o r s y s t e m a r e d i s c u s s e d .

P r e s e n t e d at the ARS S p a c e P o w e r S y s t e m s C o n f e r e n c e , Santa M o n i c a , C a l i f o r n i a , S e p t e m b e r 2 5 - Z 8 , 1962.

^ C o n s u l t i n g E n g i n e e r , M a g n e t o h y d r o d y n a m i c P o w e r G e n e r a t i o n . T h i s r e s e a r c h h a s b e e n p e r f o r m e d u n d e r the

s p o n s o r s h i p of t h e A e r o n a u t i c a l S y s t e m s D i v i s i o n , A i r F o r c e S y s t e m C o m m a n d , W r i g h t - P a t t e r s o n A i r F o r c e B a s e , Dayton, Ohio, C o n t r a c t No. A F 3 3 ( 6 1 6 ) - 7 5 3 9 .

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

C h e m i c a l l y fueled MHD g e n e r a t o r s o p e r a t i n g on an open c y c l e h a v e b e e n c o n s i d e r e d for s e v e r a l y e a r s a s topping units for c e n t r a l s t a t i o n p o w e r p l a n t s . In t h e s e a p p l i c a t i o n s , g e n e r a l l y , f o s s i l fuels a r e c o n s i d e r e d , and t h e p e r f o r m a n c e of the MHD unit is a c c o r d i n g l y l i m i t e d by t h e c h o i c e of fuel.

M o r e r e c e n t l y t h e r e h a s b e e n i n t e r e s t in applying t h e s a m e type of MHD g e n e r a t o r to obtain l a r g e a m o u n t s of p o w e r for s h o r t p e r i o d s of t i m e . Since the l i m i t a t i o n to f o s s i l fuel d o e s not m a k e p o s s i b l e the full d e v e l o p m e n t of s u c h an MHD g e n e r a t o r ' s p o t e n t i a l , t h e p r e s e n t p a p e r c o n s i d e r s t h e u s e of the p r o d u c t s of c o m b u s t i o n of the h o t t e s t f l a m e known to m a n the c y a n o g e n - o x y g e n f l a m e w h i c h b u r n s a t a t e m p e r a t u r e of 5000°K at one a t m o s p h e r e p r e s s u r e . By s e e d i n g t h i s f l a m e w i t h a s m a l l p e r c e n t a g e of e a s i l y i o n i z a b l e s u b s t a n c e s u c h a s c e s i u m , e x t r e m e l y high e l e c t r i c a l c o n d u c t i v i t i e s a r e p o s s i - b l e due to the v e r y high t e m p e r a t u r e . T h e useful c o n s e q u e n c e of s u c h high e l e c t r i c a l c o n d u c t i v i t y is an e x t r e m e l y high

p o w e r d e n s i t y for t h e MHD g e n e r a t o r so t h a t i t s v o l u m e c a n be m i n i m i z e d . A s a c o n s e q u e n c e of s u c h a high p o w e r d e n - s i t y the h e a t t r a n s f e r l o s s is r e d u c e d and t h e m a g n e t r e - q u i r e m e n t s a r e put at a m i n i m u m .

In c h o o s i n g t h e p r o d u c t s of c o m b u s t i o n of c y a n o g e n and oxygen for our MHD g e n e r a t o r s t u d i e s a n u m b e r of t h i n g s h a v e to be c o n s i d e r e d . F i r s t of a l l w e o b s e r v e t h a t c y a n o g e n is p o i s o n o u s although no m o r e so than c a r b o n m o n o x i d e . In a d d i t i o n , it d o e s h a v e a pungent odor and is r e a d i l y d e t e c t e d . C y a n o g e n liquifies r e a d i l y at r o o m t e m p e r a t u r e and a c c o r d - ingly p r e s e n t s no s t o r a g e p r o b l e m when u s e d a s a fuel.

When c y a n o g e n and oxygen a r e b u r n e d s t o i c h i o m e t r i c a l l y t h e y yield c o m b u s t i o n p r o d u c t s w h i c h a r e a m i x t u r e of CO and N . In fact, the p r i n c i p a l r e a s o n for t h e v e r y high f l a m e t e m p e r a t u r e i s t h e fact t h a t both CO and N a r e not r e a d i l y d i s s o c i a t e d so t h a t l i t t l e of the c o m b u s t i o n e n e r g y goes into d i s s o c i a t i o n w h i l e e s s e n t i a l l y a l l of it goes into t r a n s l a t i o n a l e n e r g y o r t e m p e r a t u r e . A c a r e f u l c a l c u l a t i o n h a s b e e n m a d e of the p r o d u c t s of c o m b u s t i o n of c y a n o g e n and oxygen w h e n s e e d e d w i t h v a r y i n g p e r c e n t a g e s of c e s i u m . F l a m e t e m p e r - a t u r e s w e r e a l s o c a l c u l a t e d . T h e s e r e s u l t s h a v e s e r v e d a s the s t a r t i n g point of the p r e s e n t a n a l y s i s and h a v e s e r v e d to v e r i f y the a s s u m p t i o n t h a t the c o m b u s t i o n p r o d u c t s c a n be

assumed to consist of only CO and N . In using this cyano- gen flame for our power generator, one must recognize that the energy per pound is somewhat lower than with fossil fuels and considerably lower than for the hydrogen oxygen flame. Nonetheless, the very high electrical conductivity which can be obtained makes feasible an extremely efficient extraction of energy from the flowing s t r e a m . In the present study calculations were limited to a total p r e s s u r e of one atmosphere. As a result of this and as a r e s u l t of the desire to limit COT (GO = electron cyclotron frequency, T= mean elec- tron collision time) for practical r e a s o n s , it has been found that field strengths as low as 15, 000-20, 000 gauss yield high performance MHD g e n e r a t o r s . With such low fields it then becomes reasonable to consider the application of permanent magnets to such an MHD generator. Also as a result of the fact that the magnetic field d e c r e a s e s in magni- tude rapidly from its initial value at the entrance to the channel, a permanent magnet design which makes full use of the fringing magnetic field can be used. A permanent magnet of such a design appears to have considerable weight advan- tages, and the resulting MHD generator system is not at all too heavy. In addition, the permanent magnet consumes none of the electrical energy generated and accordingly does not reduce the s y s t e m ' s efficiency.

By simple techniques an estimate is made of the heat flux that may be anticipated in such an MHD generator unit. For the MHD generators considered here the heat loss may

reduce the overall generator efficiency on the order of 20%.

In addition this heat loss can in all probability be handled by a judicious combination of regenerative and radiation cooling.

It should be recognized, however, that there is one mode of operation of such a system in which the heat transfer will be no problem; that is, if a burst of a very high power is

required for only a few seconds, then the t h e r m a l lag of the channel walls will be sufficient to prevent over-heating of the m a t e r i a l s . Finally in relation to the heat transfer problem, a word should be said about m a t e r i a l s . Since the products of combustion considered in the present case a r e non-

oxidizing, it would s e e m feasible to use tungsten electrodes, while for the electrical insulator one of the nitrides such as boron nitride may not be attacked by the cesium. In this way it has been assumed that the electrodes can be operated up to

a t e m p e r a t u r e 3000°K, w h e r e a s t h e i n s u l a t o r s c a n p r o b a b l y be o p e r a t e d up to a t e m p e r a t u r e of 2500°K. It is t h e s e high m a t e r i a l o p e r a t i n g t e m p e r a t u r e s a s w e l l a s the v e r y high p o w e r g e n e r a t i n g d e n s i t y that m a k e t h e s t e a d y - s t a t e h e a t t r a n s f e r p r o b l e m a s o l v a b l e one. It should be r e m e m b e r e d , h o w e v e r , t h a t the f o r e g o i n g c o n s i d e r a t i o n s a r e b a s e d only on e s t i m a t e s and m a y h a v e to be r e v i s e d w h e n e i t h e r e x p e r i - m e n t a l or m o r e r e f i n e d t h e o r e t i c a l s t u d i e s a r e a v a i l a b l e .

F i n a l l y , in o r d e r fully to d e t e r m i n e t h e p e r f o r m a n c e of an MHD g e n e r a t o r o p e r a t i n g w i t h the c y a n o g e n - o x y g e n f l a m e an i n v i s c i d c h a n n e l flow a n a l y s i s h a s b e e n c a r r i e d out. Since it h a s b e e n found d e s i r a b l e in t h i s study to c o n s i d e r l a r g e p e r - c e n t a g e s of s e e d , a l l r e a l gas effects h a v e b e e n i n c l u d e d in t h e a n a l y s i s . T h e r e s u l t s of this a n a l y s i s should p r o p e r l y be i n t e r p r e t e d a s t h e u p p e r l i m i t of our s y s t e m s p e r f o r m a n c e .

G e n e r a t o r C o n f i g u r a t i o n

T h e p h y s i c a l c o n f i g u r a t i o n of t h e g e n e r a t o r to be c o n s i d - e r e d in t h e p r e s e n t study is shown in F i g . 1. G e n e r a l l y s p e a k i n g t h e c h a n n e l flow a r e a i n c r e a s e s in t h e d o w n s t r e a m d i r e c t i o n . In a d d i t i o n , t h e a p p l i e d m a g n e t i c field i s c o n c e n - t r a t e d in t h e e n t r a n c e r e g i o n and a l l o w e d to d e c r e a s e in m a g n i t u d e r a p i d l y w i t h a x i a l d i s t a n c e . T h e d i v e r g i n g a r e a of the c h a n n e l w a s found n e c e s s a r y in o r d e r to m a i n t a i n t h e Mach n u m b e r of the flow a p p r o x i m a t e l y c o n s t a n t t h r o u g h o u t t h e l e n g t h of the g e n e r a t o r . M o r e w i l l b e s a i d about the d e s i r a b i l i t y of m a i n t a i n i n g an a p p r o x i m a t e l y c o n s t a n t M a c h n u m b e r . S i n c e it w a s a l s o d e s i r a b l e t o k e e p t h e H a l l

p a r a m e t e r CUT w i t h i n r e a s o n a b l e b o u n d s , it w a s found n e c e s s a r y to d e c r e a s e the a p p l i e d m a g n e t i c field a s the p r e s s u r e d r o p p e d in t h e d o w n s t r e a m d i r e c t i o n . Such a field v a r i a t i o n can be o b t a i n e d n a t u r a l l y if one c o n c e n t r a t e s the m a g n e t i c field p r o d u c e d by an i r o n c o r e e l e c t r o m a g n e t or a p e r m a n e n t m a g n e t n e a r t h e c h a n n e l e n t r a n c e and m a k e s u s e of the n o r m a l fringing field. A m a g n e t of t h i s c o n f i g u r a t i o n is a l s o shown in F i g . 1. By e s t a b l i s h i n g a m a g n e t i c field in t h i s w a y it is found t h a t r e a s o n a b l e m a g n e t w e i g h t s c a n be a c h i e v e d . It is e s s e n t i a l to n o t e , h o w e v e r , t h a t s u c h a s c h e m e is only p o s s i b l e when an i r o n r e t u r n p a t h is a v a i l a b l e to

r e t u r n the m a g n e t i c flux in a r e g i o n a w a y f r o m the c h a n n e l . If one w i s h e d to c o n s i d e r t h e a p p l i c a t i o n of a s u p e r c o n d u c t i n g

a i r c o r e s o l e n o i d to such a g e n e r a t o r c o n f i g u r a t i o n one would h a v e to u s e e i t h e r a Saddle c o i l g e o m e t r y o r a H e l m h o l t z c o i l p a i r of the p r o p e r s p a c i n g a n d s h a p e . With t o d a y ' s v e r y r a p i d p r o g r e s s in s u p e r c o n d u c t i n g r e s e a r c h , it i s not a t a l l u n r e a s - a b l e to e x p e c t t h a t s u p e r c o n d u c t i n g c o i l s w i l l be a v a i l a b l e for MHD g e n e r a t o r a p p l i c a t i o n s ; h o w e v e r , in t h e p r e s e n t i n v e s t i - gation m a g n e t i c fields of i n t e n s i t y g r e a t e r than 20, 000 g a u s s w e r e not found to be n e c e s s a r y . If s t r o n g e r m a g n e t i c fields had b e e n u s e d , v e r y l a r g e v a l u e s of the H a l l p a r a m e t e r U)T w o u l d h a v e b e e n found to e x i s t . S i n c e a s i d e f r o m the fact t h a t t h e p r e s e n t a n a l y s i s is p r o b a b l y not v a l i d for s u c h e x t r e m e l y l a r g e v a l u e s of CĆT it is a l s o a n t i c i p a t e d t h a t t h e s e ooT v a l u e s w i l l r e s u l t in i n c r e a s e d n o n - u n i f o r m i t y in the c h a n n e l flow and c o n s e q u e n t l y a d d i t i o n a l l o s s e s . Of c o u r s e h i g h e r p r e s s u r e s c o u l d be c h o s e n for the c h a n n e l o p e r a t i o n t h a n w e r e c h o s e n in t h e p r e s e n t study; h o w e v e r , it w a s felt t h a t a p r e s s u r e of 1 a t m w a s a b o u t a l l t h a t c o u l d be p e r m i t t e d in o r d e r to p r e v e n t e x c e s s i v e h e a t flux and c o n s e q u e n t l y h e a t l o s s e s .

A l t h o u g h t h e m a g n e t s y s t e m a s shown in F i g . 1 i s of r e a s - o n a b l e w e i g h t a s w i l l be d i s c u s s e d l a t e r , one c o u l d e s s e n t i a l l y h a l v e i t s w e i g h t by the s i m p l e e x p e d i e n t of u s i n g the fringing m a g n e t i c field on the o t h e r s i d e of t h e m a g n e t gap. In o t h e r w o r d s , one would c o n s t r u c t the MHD g e n e r a t o r so a s to c o n s i s t of two o p p o s e d c h a n n e l s w i t h a c o m m o n c o m b u s t o r feeding into t h e m in the r e g i o n b e t w e e n the m a g n e t pole f a c e s . Such a d e s i g n would h a v e the a d d e d and not i n c o n s i d e r a b l e a d v a n t a g e of e l i m i n a t i n g the t h r u s t on the c h a n n e l due to t h e k i n e t i c e n e r g y of t h e e x h a u s t i n g c o m b u s t i o n p r o d u c t s . T h i s w o u l d be p a r t i c u l a r l y a d v a n t a g e o u s in a s p a c e p o w e r s y s t e m .

T h e r e i s a s e r i o u s p r o b l e m c o n c e r n i n g m a t e r i a l s w i t h w h i c h to build the c h a n n e l , s i n c e t h e t o t a l t e m p e r a t u r e of the c o m b u s t i o n p r o d u c t s in t h e c h a n n e l w i l l b e a p p r o x i m a t e l y 4500 K. It i s c l e a r t h a t the w a l l m u s t b e c o o l e d s i n c e t h e r e a r e not m a t e r i a l s t h a t w i l l w i t h s t a n d s u c h a t e m p e r a t u r e and not m e l t or s u b l i m e . S i n c e t h e c o m b u s t i o n p r o d u c t s , b e i n g c o m p o s e d of C O and N , should be n o n - o x i d i z i n g , it should be f e a s i b l e to u s e t u n g s t e n e l e c t r o d e s . With t h i s m a t e r i a l , t e m p e r a t u r e s of p e r h a p s 3 0 0 0 ° - 3 5 0 0 ° K should be p o s s i b l e for m o d e r a t e l e n g t h of t i m e . T h e p r o b l e m i s m o r e s e v e r e for the e l e c t r i c a l i n s u l a t o r s i n c e m o s t good e l e c t r i c a l i n s u l a t o r s

c o n s i s t of oxides w h i c h would be r e d u c e d by t h e CO in the c o m b u s t i o n p r o d u c t s . In addition, t h e r e is t h e unknown effect of c e s i u m a t t a c k . H o w e v e r , s o m e of the n i t r i d e s s u c h a s b o r o n n i t r i d e h a v e b e e n u s e d a s v e r y high t e m p e r a t u r e e l e c t r i c a l i n s u l a t o r s in e a r l i e r MHD g e n e r a t o r s t u d i e s and h a v e p e r f o r m e d r e a s o n a b l y w e l l . O t h e r p e r h a p s even h i g h e r t e m p e r a t u r e e l e c t r i c a l i n s u l a t i n g m a t e r i a l s c a n p e r h a p s be found t h a t w i l l not be r e d u c e d by t h e CO and w i l l not be a t t a c k e d by t h e c e s i u m . In any event, it d o e s not s e e m u n r e a s o n a b l e to a s s u m e t h a t s o m e e l e c t r i c a l i n s u l a t i n g m a t e r i a l c a n be found w h i c h c a n be o p e r a t e d a t a b o u t 2500°K for a r e a s o n a b l e l e n g t h of t i m e in the f o r e m e n t i o n e d a t m o s - p h e r e .

A n u m b e r of p o s s i b i l i t i e s e x i s t for t h e h a n d l i n g of t h e h e a t flux t h r o u g h t h e g e n e r a t o r w a l l s . A p r i m e c o n s i d e r a t i o n m u s t of c o u r s e be s i m p l i c i t y . It h a s b e e n a s s u m e d in t h e p r e s e n t i n v e s t i g a t i o n that liquid oxygen and liquid c y a n o g e n w i l l be c a r r i e d a s fuels and c a n be u s e d to r e g e n e r a t i v e l y cool at l e a s t t h e high flux a r e a s of t h e MHD g e n e r a t o r , w h i c h is t h e r e g i o n n e a r t h e m a g n e t gap. F o r t h e r e s t of t h e c h a n n e l , s i n c e it w i l l be r e a s o n a b l y far a w a y f r o m a n y confining s t r u c t u r e , it would s e e m t h a t r a d i a t i v e c o o l i n g could h a n d l e the r e m a i n i n g h e a t flux. Of c o u r s e t h e h e a t flux in the d o w n - s t r e a m r e g i o n of the c h a n n e l w i l l be m u c h l e s s than in the u p s t r e a m p o r t i o n due to t h e l o w e r p r e s s u r e and m u c h t h i c k e r b o u n d a r y l a y e r s .

C h o i c e of O p e r a t i n g C o n d i t i o n s

T h e n u m b e r of p o s s i b l e m o d e s of o p e r a t i o n of the MHD g e n e r a t o r a r e e x t r e m e l y l a r g e a s t h e r e i s an a l m o s t l i m i t l e s s s e l e c t i o n of p a r a m e t e r s p o s s i b l e . It would be d e s i r a b l e to c o n s i d e r s o m e of t h e s e p a r a m e t e r s , and to d i s c u s s t h e r e a s o n s for the c h o i c e s t h a t h a v e b e e n m a d e in t h e p r e s e n t c a s e . In t h e f i r s t i n s t a n c e one m u s t c o n s i d e r w h e t h e r o r not solid or

s e g m e n t e d e l e c t r o d e s should be u s e d . If s e g m e n t e d e l e c t r o d e s a r e c h o s e n , s o m e p r o b l e m s c a n be a n t i c i p a t e d . F i r s t , the s e g m e n t s should be v e r y s h o r t w i t h r e s p e c t to t h e d i s t a n c e a c r o s s the c h a n n e l . T h e n , s i n c e in t h e f i r s t m o d e of o p e r a t i o n of s u c h a g e n e r a t o r e a c h opposing s e t of e l e c t r o d e s w i l l be c o n n e c t e d to a s i n g l e s e p a r a t e e x t e r n a l load, t h i s w o u l d

a m o u n t to an e x t r e m e l y l a r g e n u m b e r of e x t e r n a l l o a d s . T h i s ,

of c o u r s e , would be u n d e s i r a b l e . A n o t h e r p o s s i b i l i t y e x i s t s w h i c h is t h e c o n n e c t i o n of e l e c t r o d e s on o p p o s i t e s i d e s of the c h a n n e l in a s k e w e d fashion s o t h a t only one p a i r of w i r e s n e e d be c a r r i e d out to the e x t e r n a l load. H o w e v e r , the d e s i g n of s u c h an e l e c t r o d e a r r a y i s e x t r e m e l y difficult and m a y be i m p r a c t i c a l . It m i g h t be n e c e s s a r y to h a v e in c e r t a i n r e g i o n s of the c h a n n e l long e l e c t r o d e s c o m p a r e d to the c h a n n e l w i d t h . T h e e f f e c t i v e n e s s of the s e g m e n t e d e l e c t r o d e s in p r e v e n t i n g H a l l c u r r e n t s w o u l d t h u s b e d e s t r o y e d . On the o t h e r hand, if the i d e a l i z e d s i t u a t i o n of s e g m e n t e d e l e c t r o d e s i s a s s u m e d to e x i s t in an MHD g e n e r a t o r c h a n n e l , it i s found t h a t t h e MHD g e n e r a t o r of the s a m e s i z e and c o n f i g u r a t i o n and m a g n e t i c field w i l l p r o d u c e a g r e a t e r a m o u n t of p o w e r a s c o m p a r e d to one w i t h solid e l e c t r o d e s . A c c o r d i n g l y , t h e c h o i c e b e t w e e n s o l i d or s e g m e n t e d e l e c t r o d e s i s : for a high p e r f o r m a n c e a n d h i g h p o w e r d e n s i t y g e n e r a t o r , one would h a v e to t r y to d e s i g n t o w a r d a s e g m e n t e l e c t r o d e s g e o m e t r y . If, h o w e v e r , s i m p l i c i t y and r e l i a b i l i t y w e r e i m p o r t a n t c o n s i d e r a t i o n s then a s o l i d e l e c t r o d e g e o m e t r y m i g h t be m o r e d e s i r a b l e . In s o m e c a s e s , s u c h a s t h e c a l c u l a t i o n s to be d e s c r i b e d for the v e r y l a r g e m u l t i - m e g a w a t t MHD g e n e r a t o r , it w i l l be found t h a t t h e a s s u m p t i o n of s e g m e n t e d e l e c t r o d e s i s not n e c e s s a r y in o r d e r to obtain a r e a s o n a b l e MHD g e n e r a t o r g e o m e t r y . In t h i s c a s e s o l i d e l e c t r o d e s a r e m u c h s i m p l e r and p r o v i d e t h e p e r f o r m a n c e d e s i r e d .

One of the m o s t i m p o r t a n t c o n s i d e r a t i o n s in t h e d e s i g n of an MHD g e n e r a t o r of t h e type c o n s i d e r e d h e r e is t h a t a m a x i m u m p o w e r d e n s i t y be o b t a i n e d in o r d e r t h a t the d e v i c e be p h y s i c a l l y s m a l l . Keeping t h e s i z e s m a l l i s d e s i r a b l e for s e v e r a l r e a s o n s , e. g. , t h e m a g n e t w e i g h t d e p e n d s v e r y s t r o n g l y on t h e c h a n n e l s i z e . In a d d i t i o n , the d e v i c e should be r e a s o n a b l y c o m p a c t if w e a r e to b o o s t it into s p a c e w i t h a r o c k e t of p r e s c r i b e d d i m e n s i o n s . T h e p o w e r d e n s i t y s h o u l d b e k e p t a t a m a x i m u m , but t h i s m a x i m u m should be c o n s i s t e n t w i t h s o m e p r a c t i c a l l i m i t on t h e v a l u e of озТ. T h e p o w e r p e r unit v o l u m e o r p o w e r d e n s i t y i s p r o p o r t i o n a l to cru²B², w h e r e or m u s t be r e p l a c e d by a / [1+ (a)T)³ ] if s o l i d e l e c t r o d e s a r e u s e d . In a d d i t i o n , COT i s p r o p o r t i o n a l to ( B / T ) f(P). T h e function f(P) i s a d e c r e a s i n g function. In o t h e r w o r d s , a s P i n c r e a s e s f r o m z e r o , f d e c r e a s e s . A c c o r d i n g l y a l a r g e r v a l u e of B c a n be o b t a i n e d for t h e s a m e v a l u e of COT if f b e c o m e s s m a l l e r . I n c r e a s i n g B c o r r e s p o n d i n g l y i m p r o v e s

t h e p o w e r d e n s i t y s i n c e it d e p e n d s on B^. S o m e c a l c u l a t i o n s h a v e b e e n m a d e b a s e d on the p r e s e n t s y s t e m to i l l u s t r a t e t h i s effect. The r e s u l t s a r e shown in F i g . 2. F o r a fixed v a l u e of p the v a l u e of B a t w h i c h OUT e q u a l s and b e g i n s to e x c e e d 1 is l a r g e r a s t h e p e r c e n t a g e s e e d is i n c r e a s e d . T h e obvious c o n c l u s i o n f r o m t h e s e c a l c u l a t i o n s is t h a t a s l a r g e a p e r c e n t a g e of s e e d a s i s p r a c t i c a l should be u s e d in o r d e r to a c h i e v e high p o w e r d e n s i t i e s and s t i l l r e t a i n s o m e p r a c t i c a l l i m i t on CUT. If w e h a v e s o l i d e l e c t r o d e s ooT should p r o b a b l y lie b e t w e e n 1 and 2. If w e h a v e s e g m e n t e d e l e c t r o d e s it i s p r o b a b l y p e r m i s s i b l e to allow GOT to be in t h e r a n g e 3 to 4.

Any v a l u e s h i g h e r than t h i s would p r o b a b l y l e a d to difficulties in the d e s i g n and p e r f o r m a n c e . It is a l s o t r u e t h a t i n c r e a s i n g the p r e s s u r e p p e r m i t s us to i n c r e a s e B w h i l e k e e p i n g CUT the s a m e . Again this l e a d s to a l a r g e r p o w e r d e n s i t y d e s p i t e t h e fact that the c o n d u c t i v i t y is r e d u c e d . T h i s is t r u e s i n c e c o n - ductivity v a r i e s i n v e r s e l y a s the s q u a r e r o o t of p . F o r e x a m p l e , i n c r e a s i n g B and p e a c h by f a c t o r s of two d o e s not c h a n g e COT, and the p o w e r d e n s i t y is i n c r e a s e d by a f a c t o r of (2)² / \]2 - 2 . 8 3 . A s h a s b e e n m e n t i o n e d , h o w e v e r , t h e h e a t t r a n s f e r p r o b l e m m a k e s it v e r y difficult to go to too high a p r e s s u r e . In t h e p r e s e n t c a l c u l a t i o n s p r e s s u r e s of 1/5 and 1 a t m w e r e both t r i e d . T h e p e r f o r m a n c e o b t a i n e d w i t h the h i g h e r p r e s s u r e w a s b e t t e r by a l m o s t a f a c t o r of 2. T h e r e - f o r e , a l l c a l c u l a t i o n s r e p o r t e d in t h e p r e s e n t p a p e r h a v e a s s u m e d an i n i t i a l t o t a l p r e s s u r e of 1 a t m . H i g h e r p r e s s u r e s m a y be p e r m i s s i b l e , but a m o r e c a r e f u l i n v e s t i g a t i o n of t h e h e a t flux and h e a t l o s s p r o b l e m would h a v e to be m a d e .

C h a n n e l F l o w A n a l y s i s

In the p r e s e n t s e c t i o n the i n v i s c i d n o n - h e a t c o n d u c t i n g flow of a r e a l gas p l a s m a t h r o u g h an MHD g e n e r a t o r c h a n n e l w i l l be c o n s i d e r e d . The p r o c e d u r e w i l l be to a s s u m e a q u a s i - o n e - d i m e n s i o n a l flow. B e c a u s e of the m o d e r a t e v e l o c i t i e s in t h e g e n e r a t o r the gas w i l l be a s s u m e d to be in t h e r m a l e q u i l i b r i u m . E x t r a - t h e r m a l i o n i z a t i o n ( e l e c t r o n s h o t t e r than gas) w i l l be n e g l e c t e d . P r o v i s i o n w i l l be m a d e to c a r r y out t h e c a l c u l a - t i o n s for infinitely finely s e g m e n t e d e l e c t r o d e s o r s o l i d e l e c t r o d e s . In o r d e r to i n c l u d e r e a l gas effects, in this a n a l y s i s , w i t h o u t an e x o r b i t a n t a m o u n t of w o r k the following g a s m o d e l w a s c h o s e n .

The products of combustion of the cyanogen and oxygen flame a r e assumed to be exclusively CO and N . A seed atom cesium is added to this mixture and is assumed not to interact with it. In addition to neglecting dissociation of the CO and N it will be assumed that both the CO and N a r e fully excited in vibration as well as rotation. Electronic excitation above the ground state will be neglected for all species including the cesium atom.

The three conservation equations can be written as follows Continuity:

puA = p u A = m (1)

*o o o

i B (2) У ^z

- i E (3) У У

Where the flow is in the x direction, the applied magnetic field is in the z direction (induced fields being neglected), and c u r r e n t to the electrodes flows in the y direction. Also

p = m a s s density u = flow velocity

A = c r o s s sectional flow a r e a iy = c u r r e n t density

B = applied magnetic field E = generated electric field

h = enthalpy p = p r e s s u r e

In addition to the foregoing equations others a r e needed such as: an equation of state relating p, p, and T; an equation giving h in t e r m s of p, and T; and Ohm's law relating i to E and Bz. The required relationships a r e given next.

Momentum:

pu du dx" ⁺ Energy:

pu r d h dx

dp dx

+ u

=

du dx

E q u a t i o n of S t a t e :

P = (1 + e) [ p R T /M] (4)

w h e r e

ç = d e g r e e of i o n i z a t i o n - r a t i o of e l e c t r o n n u m b e r d e n s i t y t o o r i g i n a l g a s n u m b e r d e n s i t y

T = g a s t e m p e r a t u r e R - u n i v e r s a l g a s c o n s t a n t

M = m e a n m o l e c u l a r w e i g h t of n o n - i o n i z e d g a s m i x t u r e - (1 - P) n + Рм

^ ; ' n n rps m o l e s of s e e d

P = p e r c e n t a g e by v o l u m e of s e e d g a s

t o t a l o r i g i n a l m o l e E n t h a l p y :

^[(^»Ч!« ¹ *')']

w h e r e

I = i o n i z a t i o n p o t e n t i a l of s e e d g a s O h m ' s Law:

i = ri r- Г Е - u B l (6)

y 1 + l (сот)2 ^L y ^J

w h e r e

0" = e l e c t r i c a l c o n d u c t i v i t y

(j} - e l e c t r o n c y c l o t r o n f r e q u e n c y 7- = e l e c t r o n m e a n f r e e t i m e

I - 1 w h e n E = 0, = 0 when ix = 0.

F i n a l l y , e q u a t i o n s a r e n e e d e d t o d e t e r m i n e a, ooT, € a s functions of the s t a t e of t h e g a s (i. e. , p and T ) . The Saha equation y i e l d s e in t e r m s of p and T . It i s

I 5 / ^{2 g}A \

*

p = - R T

z *

*

( Т ~ ) <

>

(2тгте)з/2 кв/2 + In ^з

p €*

(i + e)

( P - 0

w h e r e K P

g^ = i o n i c a t o m i c s t a t e d e g e n e r a c y

g = n e u t r a l s e e d a t o m a t o m i c s t a t e d e g e n e r a c y

T h e e q u a t i o n s for c a l c u l a t i n g 0" and COT a r e given in Ref. 2 a n d a r e r e p r o d u c e d h e r e for c o n v e n i e n c e :

0.85 n e 2 / m e e

a = ( 8 к Т / т г те) 1 / з [ nnQe n + ns /Qe s / +neQe i] (8) eB

a n d

COT =

w h e r e

( 8 к Т / т г те) 1 / г [ nnQe n + ns/ Qe s, + neQe i]

1/3

(9)

Q

ei

° -

( - k T - W

C^r) ]

n^ *nr

nn - ( 1 - P ) n0

ns, = ( P - € ) n0

a n d w h e r e

n = o r i g i n a l g a s n u m b e r d e n s i t y n = e l e c t r o n n u m b e r d e n s i t y n = n e u t r a l n u m b e r d e n s i t y

ng/ = u n i o n i z e d s e e d n u m b e r d e n s i t y T = deg K

P = a t m

Q e s '

e l e c t r o n n e u t r a l c r o s s s e c t i o n

e l e c t r o n u n i o n i z e d s e e d c r o s s s e c t i o n Q • = e l e c t r o n ion s e e d c r o s s s e c t i o n

If w e define, for c o n v e n i e n c e E = kuB

У

üeff 1 + (cor)2 t

and s u b s t i t u t e E q . (6) into E q s . (2) and (3) w e get

du dp

> u — + -7Г- dh du

0 Ud ^ ⁺d í ^ e f f ^ z ^{( k}-^{1 }}

^[S-Sj-eff^ ² ^^

(10)

(11) If h is e x p r e s s e d in t e r m s of p and T, and then p and p a r e eliminated by Eq. (1) and (4). Eqs. (10) and (11) a r e two s i m - ultaneous equations for u and T. They will be first o r d e r and linear and can be solved simply. Taking h from Eq. (5) and substituting into Eq. (11) we get

-[е ^{< 9} - 4 Р + 5 е , ^К! ^{а Т +} 9Й ^{+ и} £] _(I2 ,

= cr r r u 2 B² ( k - l ) k eti z

Thus we need an expression for de/dx. We find this from Eq. (7).

1 dT 1 dp

w h e r e

d<T= f r / 5 I N 1 dT j . dp dx L ^ 2 RT ) T dx " p dx

f (1 + €)(P-C)

2(l+€){P-c) + f (2C + 1-P) (13)

If we eliminate p and p, by replacing p from Eq. (4) using Eq. (1) to eliminate p , we obtain

d£

dx g(P-g) 2P-Ç

r /- 3 I v 1 dT 1 du 1 dA-i , , „,

[C 2

R ^ ) T ^

ï ï d ^

A dx"]

Combining E q s . (12) and (14) then gives one equation for T and u a s functions of x.

P - e I [ | _ ( 9 . 4 P + 5 0 + ( | R T + I ) ^

L u V 2 J 2 P - € u J dx

f(P-0 C - —N 2

r A — CT us

m eff

в

( к - 1 ) к - 1 ^ я т

Л ^ ^ 1

z д V ² У 2 P - e A dx J o r

dT du a — + a —

i dx 2 dx (15)

We c a n a l s o d e r i v e a s i m i l a r e q u a t i o n f r o m Eq. (10) u s i n g E q . (1) for t h e e x p r e s s i o n for d p / d x . We f i r s t find d p / d x in t e r m s of T a n d u.

dp ( l + Q R m dx I, uA

f ( P - O f T + _{( R T} ^I ₎

(1 + 0 (2P-C) + 1 dT

dx ( U f ) R T rh

Д U2A (1 + C)RT m

[<Й

L ( i +

0 ( 2 P - 0

«(P-O

0(2P-e) - ¹ ]

(16)

Next, Eq. (10) b e c o m e s , after s u b s t i t u t i o n of (16), (1 + €)RT 1 , f ( P - f )

1 + U

( i + Q R M ^u

u s

CÏÏ + C)(2P-0 " V J <bc

* ( P - 0 V 2 ⁺ RT У

(l + €)(2P-€) + 1 dT

dx (l + g)RT J_ r € ( P - Q

ц uA V (l + 0 ( 2 P - 0

v dA ) dx

o r

+ — a r r u B² ( k - l ) m eff z , du , dT

i dx 2 dx ³ (17)

E q u a t i o n s (15) and (17) c a n now be s o l v e d for d u / d x and d T / d x . T h e r e s u l t is s i m p l y

du „, m

- = G ( u . T . . . . )

(18) (19) w h e r e

_ a3 Ц a b

l l

a b

sK

- - - -

apb3

a b

2 P

a b 3 p 2 2

T h e i n i t i a l c o n d i t i o n s a r e t h a t w h e n x = 0 w e h a v e T = TQ a n d u = u . T h e v a r i o u s a and b a r e a p p a r e n t f r o m t h e e q u a t i o n s a n d n e e d n o t b e r e p e a t e d .

T h e final i t e m n e e d e d t o c o m p l e t e t h e c a l c u l a t i o n p r o c e d - u r e i s t h e s p e e d of sound in s u c h a m i x t u r e . We h a v e , for a d i a t o m i c i n e r t w i t h r o t a t i o n a n d v i b r a t i o n fully e x c i t e d

as = y p / p (20)

w h e r e 9-4P+5f 2c (P-c) r 5

l+€ 2 P - C + CP

1 J— ~f

2 ⁺ R T ] 7-4P+3C 2e(P-f) 7 - 4 P + 3 f

H e 2 P - r + <rP\ 2(1 + 0

I n²

2

B a s e d on t h e p r e c e d i n g a n a l y s i s a c o m p u t e r p r o g r a m h a s b e e n s e t up t o c a l c u l a t e t h e p e r f o r m a n c e of a n MHD g e n e r a t o r . In o r d e r to s t a r t t h e c a l c u l a t i o n i n i t i a l c o n d i t i o n s m u s t b e known a c c u r a t e l y . Such i n i t i a l c o n d i t i o n s a r e a v a i l a b l e f o r t h e o x i - c y a n o g e n f l a m e c o m b u s t i o n s e e d e d w i t h up t o 10% c e s i u m o v e r a l i m i t e d r a n g e of p r e s s u r e s . U s e h a s b e e n m a d e of t h e s e r e s u l t s in c a r r y i n g out t h e c a l c u l a t i o n s , a n d two c a l c u l a t i o n p r o c e d u r e s h a v e b e e n d e v e l o p e d . In t h e f i r s t t h e a r e a a n d m a g n e t i c field d i s t r i b u t i o n w i t h x a r e s p e c i f i e d a n d t h e v e l o c i t y , t h e r m o d y n a m i c v a r i a b l e s , a n d p o w e r g e n e r a t e d a r e c a l c u l a t e d a s functions of x. In t h e s e c o n d t h e M a c h n u m b e r i s kept a p p r o x i m a t e l y c o n s t a n t o v e r t h e e n t i r e c h a n n e l length.

T h i s i s done s i n c e M i s p r o p o r t i o n a l t o u^/T, a n d t h e object i s to e x t r a c t e l e c t r i c a l e n e r g y f r o m both t h e flow k i n e t i c e n e r g y u²/ 2 a n d t h e flow t h e r m a l e n e r g y T . In c o n t r a s t to t h e f i r s t c a l c u l a t i o n p r o c e d u r e t h e s e c o n d one i s c a r r i e d out s t e p by s t e p . A s e a c h s t e p of t h e c a l c u l a t i o n i s c o m p l e t e d t h e M a c h n u m b e r i s c h e c k e d . If i t i s r i s i n g o r d r o p p i n g t h e a r e a a n d / o i t h e m a g n e t i c field a r e a d j u s t e d to k e e p i t c l o s e to i t s i n i t i a l v a l u e .

In o r d e r t o d e s i g n a n MHD g e n e r a t o r duct both of t h e p r o -

cedures described a r e n e c e s s a r y . F i r s t , one would determine the channel shape neglecting viscous effects, by keeping the Mach number approximately constant, or by using some other similar constraint. Then once this flow and channel shape a r e known a boundary layer analysis will yield the distribution of boundary layer thickness. Knowing this one can a s s u m e the channel shape to be reduced accordingly and have a specified channel shape. The calculation can then be made according to the first procedure described. That is, the channel shape and magnetic field a r e assumed known and the properties calculated.

Since the immediate interest is in determining, approxi- mately, the maximum power that can be generated from given initial conditions it was decided to pursue the stepwise calcu- lation procedure keeping M approximately constant. This then gives the optimum generator for each set of initial conditions with only one calculation. A variety of initial conditions w e r e investigated to find an optimum generator. The principal quantities varied were k (generator coefficient), P(% seed), Mach number, initial p r e s s u r e and t e m p e r a t u r e , and AQ the initial a r e a . Without describing all the r e s u l t s it can be reported that the optimum percent seed was ten %, the highest for which initial conditions were available and the optimum initial total p r e s s u r e was 1 atm. Accordingly all calculations w e r e made with a 10% seed and for an initial p r e s s u r e of 1

atm. Calculations have been c a r r i e d out for two different size MHD generators, and the results a r e described.

Small Sizes

1) Solid electrodes:

M0 = 0.9 k = 0.5 BQ = 20, 000 gauss Br = 7500 gauss Ar = 5.84 cm2 Xr

f ⁶ f f

Power out = 5.34 kw 77 = 18%

2) Segmented electrodes:

M = 0.9 k = 0.5 B = 20, 000 gauss A = 1 cms

o o o B = 4500 gauss A = 63 cms X = 10 cm

AQ = 1 cm2 10 cm

P o w e r out = 10.87 kw 77 = 36%

3) S e g m e n t e d e l e c t r o d e s :

ML = 1 . 5 k = 0.5 B = 20, 000 g a u s s A = 1 c m s

o o o B = 2500 g a u s s A = 18 c m s X = 5 c m

P o w e r out = 7.5 kw r\ = 25%

4) S e g m e n t e d e l e c t r o d e s :

M = 1.5 k = 0.7 B = 2 0 , 0 0 0 g a u s s A „ = 1 c m s o o ^& o

Bf = 2500 g a u s s A = 14 cm2 X = 10 c m

P o w e r out = 10 kw T? = 33%

L a r g e Size

1) S e g m e n t e d e l e c t r o d e s :

M0 = 1 . 5 k = 0.9 B0 = 15, 000 g a u s s AQ = 2000 c m s B = 4000 g a u s s A = 12,700 cm^ X = 70 c m P o w e r out = I 9 . 5 mw 77 = 32%

2) Solid e l e c t r o d e s :

M = 1.5 k = 0.9 B = 1 5 , 0 0 0 g a u s s A = 2000 c m s o o ^& o B = 5000 g a u s s A = 9 , 4 5 0 cm2 X = 200 c m P o w e r out = 17 mw Г) = 28%

To i l l u s t r a t e the f o r m of the v a r i a t i o n of v a r i a b l e s a s w e l l a s the a c t u a l c h a n n e l g e o m e t r y , c a s e s 1 and 4 for t h e s m a l l

s i z e s a n d 2 for t h e l a r g e s i z e a r e p l o t t e d in F i g s . 3 t h r o u g h 8.

S e v e r a l i n t e r e s t i n g c o n c l u s i o n s can be d r a w n f r o m the r e s u l t s p r e s e n t e d h e r e . F i r s t , it i s o b s e r v e d t h a t t h e p r e s - s u r e d r o p s c o n t i n u o u s l y and r a p i d l y f r o m i t s v a l u e at t h e e n t r a n c e to t h e c h a n n e l . T h e v e l o c i t y and t e m p e r a t u r e both a l s o d e c r e a s e a s the M a c h n u m b e r i s m a i n t a i n e d a p p r o x i - m a t e l y c o n s t a n t . F o r the s o l i d e l e c t r o d e c a s e s COT w a s m a i n - t a i n e d below 1 in one c a s e and 2 in the o t h e r c a s e . A l s o , in

a l l c a s e s c o n s i d e r e d , it w a s n e c e s s a r y to d e c r e a s e the a p p l i e d m a g n e t i c field B in o r d e r to k e e p coTfrom b e c o m i n g u n r e a s o n a b l y l a r g e . F o r t h e s m a l l e r g e n e r a t o r s it w a s a d v a n t a g e o u s to c o n s i d e r t h e s e g m e n t e d e l e c t r o d e g e o m e t r y . In fact, the p o w e r output w a s e s s e n t i a l l y doubled, a s w a s the g e n e r a t o r efficiency. It w a s a l s o found to w o r k b e t t e r a t s u p e r s o n i c v e l o c i t i e s M = 1.5 for the s m a l l s e g m e n t e d e l e c - t r o d e g e n e r a t o r , in o r d e r to k e e p the a r e a i n c r e a s e w i t h i n r e a s o n a b l e b o u n d s . F o r c a s e 2 t h e final a r e a w a s 63 t i m e s t h e i n i t i a l a r e a , w h e r e a s for c a s e 3 it w a s only 14 t i m e s a s l a r g e .

In the c a l c u l a t i o n s for the s m a l l - s i z e g e n e r a t o r s the

g e n e r a t o r coefficient k w a s c h o s e n to be 0.5. At t h i s v a l u e of k the p o w e r d e n s i t y in the s i m p l e t h e o r y is s u p p o s e d to be a m a x i m u m . It w a s i n d e e d found t h a t w h e n k w a s c h o s e n to be s o m e w h a t l a r g e r t h a n 0.5, t h e t o t a l p o w e r g e n e r a t e d for the s a m e s i z e c h a n n e l w a s c o n s i d e r a b l y l e s s . F o r t h e l a r g e r s i z e g e n e r a t o r s , a g e n e r a t o r coefficient k of 0.9 w a s c h o s e n . It w a s n e c e s s a r y to c h o o s e t h i s v a l u e to k e e p a r e a s o n a b l e g e n e r a t o r g e o m e t r y . F o r e x a m p l e , if t h e c h a n n e l h e i g h t i s i n i t i a l l y 45 c m , and t h e bulk of t h e p o w e r i s g e n e r a t e d in a d i s t a n c e of 3 0 - 4 5 c m , the c h a n n e l a s p e c t r a t i o , a t l e a s t a s far a s the e n t r a n c e is c o n c e r n e d , is e s s e n t i a l l y 1. It c a n be shown t h a t c o n s i d e r a b l e end l o s s e s c a n e x i s t w i t h a s p e c t r a t i o s a s s m a l l a s 1. One w o u l d n o r m a l l y d e s i r e an a s p e c t r a t i o of 5 or 10 to 1 in o r d e r to m i n i m i z e end l o s s e s . T h e r e - f o r e , for t h e l a r g e s i z e g e n e r a t o r s , a t t e m p t s w e r e m a d e to k e e p the g e n e r a t o r long c o m p a r e d to i t s e n t r a n c e h e i g h t in o r d e r to avoid, a s m u c h a s p o s s i b l e , end l o s s e s . To a c h i e v e t h i s , k w a s t a k e n to be 0.9 and t h e a p p l i e d m a g n e t i c field w a s r e d u c e d to 15, 000 g a u s s . F u r t h e r r e d u c t i o n s in B w i t h k ~ 0.5 should y i e l d s i m i l a r r e s u l t s a s w e l l a s a l i g h t e r m a g n e t .

F r o m F i g . 4, 6, and 8, it c a n be s e e n t h a t a g e o m e t r y of the g e n e r a t o r being c o n s i d e r e d in t h i s p a p e r i s i n d e e d a r a p i d l y d i v e r g i n g o n e . It is h o p e d t h a t t h i s r a p i d d i v e r g e n c e w i l l not i n v a l i d a t e t h e q u a s i - o n e - d i m e n s i o n a l a p p r o x i m a t i o n . We a l s o n o t e t h a t t h e i n i t i a l c r o s s - s e c t i o n a l a r e a of e a c h of the c h a n n e l s w a s c h o s e n to be s q u a r e . T h i s w a s done in o r d e r to m i n i m i z e the h e a t l o s s , s i n c e t h e h e a t t r a n s f e r s u r f a c e i s a m i n i m u m for a s q u a r e a s c o m p a r e d to a

r e c t a n g l e . When t h e d i s t a n c e b e t w e e n e l e c t r o d e s i s fixed at

the e n t r a n c e , s u c h a s w h e n the known i n i t i a l c r o s s s e c t i o n i s a s s u m e d to be s q u a r e , then the p o t e n t i a l b e t w e e n the e l e c - t r o d e s , if t h e y a r e c o n t i n u o u s , i s s p e c i f i e d . Then, s i n c e the e l e c t r i c field at e a c h p o s i t i o n along t h e c h a n n e l i s given by k u B , a l l t h r e e q u a n t i t i e s of w h i c h a r e known at e a c h point, the d i s t a n c e b e t w e e n e l e c t r o d e s i s c o m p l e t e l y s p e c i f i e d along the e n t i r e l e n g t h of the c h a n n e l . F o r s e g m e n t e d e l e c t r o d e s t h i s is not n e c e s s a r i l y the c a s e . F o r e x a m p l e , if it w a s c h o s e n to k e e p the c h a n n e l c r o s s - s e c t i o n a l a r e a s q u a r e o v e r t h e e n t i r e l e n g t h of the c h a n n e l , then t h e d i s t a n c e b e t w e e n e l e c - t r o d e s i s s p e c i f i e d . A s b e f o r e , the e l e c t r i c field E i s d e t e r - m i n e d s i n c e ku and B a r e known. T h e r e f o r e , the v o l t a g e b e t w e e n e l e c t r o d e p a i r s i s h i g h e s t at the c h a n n e l e n t r a n c e and a m i n i m u m at the c h a n n e l exit.

One of the m o s t i m p o r t a n t g a i n s f r o m the f o r e g o i n g a n a l y s i s i s an u n d e r s t a n d i n g of the a p p r o x i m a t e s c a l i n g r u l e s for s u c h MHD g e n e r a t o r s . F o r e x a m p l e , it i s s u p p o s e d t h a t a s m a l l MHD g e n e r a t o r i s d e s i g n e d to p r o d u c e r o u g h l y 10 kw a n d fix t h e i n l e t c o n d i t i o n s of p r e s s u r e and t e m p e r a t u r e , m a g n e t i c field, flow v e l o c i t y , and k. T h e n , if it i s w i s h e d to g e n e r a t e 20, 000 kw or 20 mw, o n e ' s f i r s t i n c l i n a t i o n m i g h t be to s c a l e out t h e i n i t i a l a r e a by the f a c t o r 20, 000 o v e r 10, or in o t h e r w o r d s , to m a k e the i n i t i a l a r e a 2, 000 c m s . In t h e s m a l l g e n e r a t o r the p o w e r d e n s i t y t h a t i s a c h i e v e d i s on t h e o r d e r of 1.5 k w / c m 3 . Now, if it i s w i s h e d to g e n e r a t e 20 mw in an MHD g e n e r a t o r w i t h e x a c t l y the s a m e c o n d i t i o n s and an i n i t i a l a r e a of 2, 000 c m 2 , one n e e d t r a v e l only 10 c m . A c t u a l l y , t h i s g i v e s us 20, 000 c m s and at 1 k w / с т з , a p p r o x i m a t e l y 20 m w . H o w e v e r , the g e o m e t r y of t h i s g e n e r a t o r i s a b s u r d . If it is a s s u m e d to h a v e a s q u a r e c r o s s s e c t i o n i n i t i a l l y , it i s 45 c m on a s i d e and only 10 c m long; in o t h e r w o r d s , it h a s an a s p e c t r a t i o of 1 0 / 4 5 o r r o u g h l y 1/5. T h e g e n e r a t o r of s u c h g e o m e t r y w i l l n e v e r p e r f o r m in any w a y t h a t m i g h t be e x p e c t e d due to t h e end l o s s e s . T h e obvious fact i s t h a t t h e e x t r e m e l y high p o w e r d e n s i t y t h a t w a s d e s i r e d and w a s found n e c e s s a r y in t h e s m a l l MHD g e n e r a t o r yielding 10 kw i s of a b s o l u t e l y no v a l u e for t h e l a r g e MHD g e n e r a t o r for w h i c h it i s d e s i r e d to p r o d u c e 20 mw. It w a s for t h i s r e a s o n t h a t k w a s l o w e r e d to 0.9 and the a p p l i e d m a g n e t i c field l o w e r e d to 15, 000 g a u s s , a n d it w a s s t i l l p o s s i b l e to c o n s i d e r a solid e l e c t r o d e g e n e r a t o r . In t h i s w a y , d e n s i t y w a s r e d u c e d to a p p r o x i m a t e l y 0.175 k w / c m s ; a r e d u c t i o n of a p p r o x i m a t e l y 10 in p o w e r d e n s i t y , and yet t h e

generator scales up geometrically. The singular conclusion is that a large megawatt or multi-megawatt MHD generator generally should be much easier to build than a small MHD generator. In particular, the requirements on electrical conductivity for such a large generator should not be nearly so severe as for smaller MHD units.

Heat Transfer

Obviously, at a total t e m p e r a t u r e of 4500°K, the heat flux and heat loss to the walls of an MHD generator operating continuously will be a serious problem. If s h o r t - t i m e opera- tion is feasible this should no longer be a problem. An immediate question is whether or not regenerative cooling would be feasible in order to remove the heat flux through the walls of such an MHD unit. Therefore, it is of interest to

estimate the heat that could be absorbed in heating and vapor- izing the oxygen and cyanogen liquids that would be c a r r i e d as fuels. In heating liquid oxygen from its boiling point to 100°F, it is found that 179 Btu/lb can be picked up with a m a s s flow for the smaller MHD generators considered earlier of 6.6 g / s e c . The total heat that could be picked up would be approximately 1 kw. The heat absorbed in bringing the cyanogen liquid up to 100°F from roughly 0°F, again for a total m a s s flow of 6.6 l b / s e c . , is found to be one third of a kilowatt. Using both liquids for regenerative cooling approx- imately 1-1/3 kw can be absorbed.

Whether or not the heat loss is of the same order as or smaller than, the foregoing figure can be determined only when the actual heat flux for the particular device being considered has been estimated. In order to estimate the heat flux accurately, one would have to c a r r y out a boundary layer analysis based on the channel flow calculations already

described. No such detailed analysis has been c a r r i e d out.

However, a crude estimate of what the heat flux should be has been made. It is assumed that the heat flux can be r e p r e s e n t - ed as

k Ah q - —

where k is the t h e r m a l conductivity of the plasma, C its

average specific heat, h its enthalpy, and 5 the boundary layer thickness, oh. represents the change in enthalpy between the free stream and the channel wall. The enthalpy of the free stream depends on the degree of ionization, of course. The enthalpy at the wall is essentially independent of ionization, since the temperature there is considerably lower than the free stream. By estimating Ah in this way in the heat flux is included the diffusion of chemical species to the wall where they recombine. For nitrogen at approximately 7, 000°K, its thermal conductivity is approximately 10⁴ ergs/cm» sec • K.

Since the gas is partially ionized (e ^ .028), k should be some- what higher. Therefore, k is assumed to be approximately

5 x 10 . An expression for C in our mixture can be derived by taking (dh/òT)p. The result is:

9-4P

P 2 ^f \2 ^{l +}RJ|_TVRT 2 Д2Р-е(1-Р)Ш

where <r = 0.028 P = 0 . 1 0 T = 4000°K Č - 5 x 10⁸ erg/°K g mole It is known that

9 - 4 P4 n m ^ 5

h = C^) ^{R T +} Cf ^{RT +} 0

at the wall c = 0, and it is assumed that the wall temperature is 2800°K so that

h ^ 10i^Q erg/g mole CO ° °

In the free stream the calculation gives hro - 1.5 x IO12

so that Ah = 5 x IO-11. Thus 5 x 10-3

q^ kw/cm2

when б i^s i-ⁿ centimeters. It should be noted that the 2800°K wall temperature was a compromise between the anticipated

electrode operating temperature of 3000°K and the electrical insulator operating temperature of 2500°K.

The next problem is to obtain an estimate of the boundary layer thickness 6. First, one must estimate the Reynolds

n u m b e r , w h i c h i s Re = o u L / д . F o r M ^ 1:

p ^ 1.77-6 m / i n s u ^ 3000 fps #

A l s o , for N a t the t e m p e r a t u r e of i n t e r e s t , t h e v i s c o s i t y i s

? #

fi = 0.28 m / f t - h r

B a s e d on t h e s e v a l u e s of the p r o p e r t i e s , the R e y n o l d s n u m b e r b a s e d on t h e d i s t a n c e x, w h e r e x i s in c e n t i m e t e r s , i s

a p p r o x i m a t e l y 3 700 x. A c o n s e r v a t i v e e s t i m a t e of 6 c a n be o b t a i n e d f r o m t h e w e l l - k n o w n r e s u l t of t h e B l a s i u s b o u n d a r y l a y e r :

6 0

F o r a 1 c m length, 6 i s a p p r o x i m a t e l y 1/60 c m s o t h a t

^ ^ 0.3 k w / c m²

If i t i s a s s u m e d t h a t t h e s u r f a c e a r e a in t h e s m a l l MHD g e n e r a t o r w h i c h w i l l be e x p o s e d to t h e m o s t s e v e r e h e a t flux w i l l be the f i r s t 2 c m , the t o t a l h e a t t r a n s f e r s u r f a c e w i l l be a p p r o x i m a t e l y 8 c m². With a h e a t flux of 0.3 k w / c m², t h i s a m o u n t s to ~ 2 kw h e a t l o s s . T h e r e i s a r e a l p o s s i b i l i t y t h a t r e g e n e r a t i v e c o o l i n g c o u l d h a n d l e m o s t of t h i s h e a t l o s s . T h e r e m a i n d e r could c o n c e i v a b l y be r e m o v e d s i m p l y by r a d i a t i o n into s p a c e f r o m the d o w n s t r e a m s u r f a c e of the MHD unit.

B e f o r e d i s c u s s i n g t h e influence of g e n e r a t o r s i z e on the h e a t l o s s , it w o u l d be of v a l u e to c o m p a r e t h e p r e v i o u s l y c a l c u l a t e d r e s u l t s w i t h s o m e e x p e r i m e n t a l h e a t flux m e a s u r e - m e n t s m a d e by Way, who o p e r a t e d an MHD g e n e r a t o r using c o m b u s t i o n p r o d u c t s at p ° ~ 1 a t m , T ~ 3000°K, and M ~ 1.

Way found t h a t t h e h e a t flux t h r o u g h t h e w a t e r - c o o l e d c e r a m i c w a l l s w a s a p p r o x i m a t e l y 0.01b k w / c m², w h e r e a s t h e h e a t flux t h r o u g h w a t e r c o o l e d c o p p e r w a s a p p r o x i m a t e l y 0.408 k w / c m². It is i n t e r e s t i n g and s o m e w h a t s u r p r i s i n g t h a t the p r e s e n t a u t h o r ' s a p p r o x i m a t e c a l c u l a t i o n falls b e t w e e n t h e s e two v a l u e s .

In g e n e r a l , s i n c e t h e l a r g e r MHD unit w i l l be e a s i e r to d e s i g n , it w i l l , in m o s t i n s t a n c e s , be s h o r t e r t h a n t h e MHD g e n e r a t o r for s m a l l p o w e r s - a t l e a s t r e l a t i v e to i t s i n i t i a l c r o s s - s e c t i o n a l a r e a . To t h e e x t e n t t h a t t h i s i s the c a s e , the

h e a t t r a n s f e r p r o b l e m should be l e s s s e v e r e in a l a r g e g e n e r - a t o r s i m p l y b e c a u s e of i t s s m a l l e r s u r f a c e a r e a . H o w e v e r , a n o t h e r f a c t o r c o m p l i c a t e s t h e s i t u a t i o n : in t h e l a r g e r MHD g e n e r a t o r s i z e s t h e R e y n o l d s n u m b e r i s l a r g e r , and w h a t p r e v i o u s l y w a s a s s u m e d to be a l a m i n a r b o u n d a r y l a y e r m i g h t i n d e e d t u r n out to be a t u r b u l e n t one. If so, t h e h e a t l o s s would be c o n s i d e r a b l y h i g h e r t h a n had b e e n e s t i m a t e d . T h i s

m i g h t be p r e v e n t e d b e c a u s e the b o u n d a r y l a y e r s in t h i s g e n e r a t o r a r e being v e r y s t r o n g l y c o o l e d , and it h a s b e e n shown by n u m e r o u s i n v e s t i g a t o r s t h a t a highly c o o l e d b o u n d a r y l a y e r i s m u c h m o r e s t a b l e and w i l l r e m a i n l a m i n a r to the v e r y m u c h h i g h e r R e y n o l d s n u m b e r s . Only c a r e f u l b o u n d a r y l a y e r a n a l y s e s would be a b l e to r e v e a l w h e t h e r l a m i n a r b o u n d a r y l a y e r s m a y be e x p e c t e d along t h e w a l l s of our MHD g e n e r a t o r . It should be noted, h o w e v e r , t h a t it i s l e s s i m p o r t a n t for the b o u n d a r y l a y e r to r e m a i n l a m i n a r o v e r the d o w n s t r e a m p o r t i o n of the g e n e r a t o r than o v e r the i n i t i a l p o r t i o n of the g e n e r a t o r w h e r e the h e a t flux is g r e a t e s t .

A p p l i c a t i o n s

A q u e s t i o n a r i s e s why s u c h a d e v i c e i s m o r e d e s i r a b l e than o t h e r s s u c h a s the t u r b o - a l t e r n a t o r , b a t t e r y , o r fuel c e l l . S i m p l y , s u c h a d e v i c e should be e x t r e m e l y r e l i a b l e

s i n c e t h e r e a r e no m o v i n g p a r t s . T h i s , of c o u r s e , a s s u m e s t h a t the h e a t t r a n s f e r p r o b l e m c a n be s o l v e d . In a d d i t i o n to b e i n g a s t a t i c d e v i c e it does not r e q u i r e e x t r e m e l y p r e c i s e

s p a c i n g b e t w e e n e l e c t r o d e s , so t h a t s m a l l a m o u n t s of a b l a t i o n o r c h a n n e l e x p a n s i o n should only h a v e a m i n o r effect on g e n e r a t o r p e r f o r m a n c e . T h i s type of g e n e r a t o r i s a l s o e x t r e m e l y c o m p a c t and t a k e s up l i t t l e v o l u m e in a s p a c e v e h i c l e , and a c c o r d i n g l y r e q u i r e s l i t t l e s u p p o r t i n g s t r u c t u r e , w h i c h i t s e l f c a n b e c o m e h e a v y . A n o t h e r a d v a n t a g e of t h e p r e s e n t a p p r o a c h o v e r o t h e r t y p e s of c h e m i c a l l y f i r e d MHD g e n e r a t o r s i s that it i s f e a s i b l e to u s e a p e r m a n e n t m a g n e t to obtain t h e m a g n e t i c field. In o t h e r d e s i g n s it i s g e n e r a l l y r e c o g n i z e d t h a t a c r y o g e n i c a l l y c o o l e d o r s u p e r - c o n d u c t i n g m a g n e t i s e s s e n t i a l . T h i s i s b e c a u s e t h e p o w e r d e n s i t y for l o w e r t e m p e r a t u r e c o m b u s t i o n s y s t e m s i s too low and m u s t be i n c r e a s e d by i n c r e a s i n g t h e m a g n e t i c field. In the p r e s e n t d e v i c e the t e m p e r a t u r e s a r e so high t h a t m o r e than a d e q u a t e c o n d u c t i v i t y i s obtained, and, t h e r e f o r e , p o w e r d e n s i t y i s sufficiently high to p e r m i t t h e u s e of w e a k e r m a g n e t i c f i e l d s .

By the use of a permanent magnet the difficult r e s e a r c h and development p r o g r a m s associated with superconducting mag- nets and their associated cryogenic equipment can be avoided.

A number of different applications can be imagined for the MHD generator considered h e r e . F i r s t , it must be r e c o g - nized that it is inherently a s h o r t - t i m e power generator. If power is desired for long periods of time, in the order of a year, it is obvious that a nuclear r a t h e r than chemical fuel should be used. However, for s h o r t - t i m e operations a num- ber of possibilities exist. F o r example, for n e a r - e a r t h space missions it might be desirable to c a r r y along a power si^pply on a space vehicle which is, perhaps, circling the moon.

When the space vehicle is behind the moon, it can take pic- t u r e s , and when the vehicle c r o s s e s to the front of the moon

such a generator might provide the n e c e s s a r y power for t r a n s m i s s i o n . F o r land-based applications it might be

desirable to have an extremely large generator available on a standby basis for relatively modest expense. Then if power w e r e needed for several minutes such an MHD generator

could be operated, since it could be left unattended for long periods of time and called into service almost immediately.

There is nothing in the system that could deteriorate with time or t e m p e r a t u r e .

Due to the high efficiencies that have been calculated for the present concept, it should be competitive with the fuel cell for which fuel r a t e s of approximately 2 Kg/kw-hr have been suggested, and batteries for which fuel r a t e s of

10 kg/kw-hr have been proposed. The p r e s e n t calculations have actually yielded fuel r a t e s of approximately 3 kg/kw-hr for an MHD generator.

In addition to the question of efficiency of a power gener- ating device, one must also consider the question of dead weight, especially if future space applications a r e anticipated.

In the present system the dead weight will be essentially magnet weight and an estimate of this would be valuable.

Calculations have been made of pole shapes for several permanent magnet designs to be used with a s y s t e m such as the present one. In general, the volume of magnetic material which is, of course, proportional to its m a s s , can be calcula- ted from a formula such as

f F B 2 V V = S_&.

m B H m m

H e r e t h e f a c t o r s f and F c o r r e s p o n d to fringing, B i s the m a g n e t i c field in the gap, V the gap v o l u m e , and BmH t h e m a g n e t i c e n e r g y p r o d u c t of the p e r m a n e n t m a g n e t m a t e r i a l . F o r a m a g n e t i c field gap 1-1/2 c m high w i t h a d i s t a n c e of 2 c m b e t w e e n the pole f a c e s and 15, 000 g a u s s in t h e c e n t e r of t h e gap, the m a g n e t w e i g h t is e s t i m a t e d to be a p p r o x i m a t e l y 130 lb. A s r e q u i r e d , the field d r o p s f r o m about 15, 000 g a u s s b e t w e e n t h e pole f a c e s to about 2000 g a u s s at t h e end of the c h a n n e l . T h e p o l e f a c e s c a n be s h a p e d p r e c i s e l y to give any v a r i a t i o n of m a g n e t i c field d e s i r e d . F o r the w e i g h t c a l c u l a t e d h e r e and a 10 kw p o w e r output f r o m t h e MHD g e n e r a t o r , t h i s a m o u n t s to 1 3 l b / k w , or if r e s o r t i s m a d e to the d e v i c e of c o n s t r u c t i n g two o p p o s e d MHD c h a n n e l s so t h a t t h e f r i n g e field on the o t h e r s i d e of the m a g n e t c a n be u s e d , t h i s a m o u n t s to a f i g u r e of r o u g h l y 7 l b / k w . A c t u a l l y , s u c h low f i g u r e s for l b / k w i s s u r p r i s i n g for a d e v i c e w h i c h u s e s a p e r m a n e n t m a g n e t to obtain the m a g n e t i c field. F o r t h e l a r g e r s i z e MHD g e n e r a t o r c a l c u l a t e d e a r l i e r , t h e 20 mw d e v i c e , a m a g n e t gap height of 65 c m a n d a d i s t a n c e b e t w e e n t h e pole f a c e s of 50 c m should b e a d e q u a t e to p e r m i t t h e r e - q u i r e d h e a t t r a n s f e r . In t h i s c a s e , t h e m a g n e t m a s s should be a p p r o x i m a t e l y 120, 000 lb w h i c h then y i e l d s a specific w e i g h t of 6 l b / k w for t h e s i n g l e d e v i c e or 3 l b / k w for t h e t a n d e m d e v i c e . T h e s e f i g u r e s a r e even b e t t e r t h a n t h e o n e s for t h e s m a l l e r unit s i n c e a c o r r e s p o n d i n g l y s m a l l e r p o r t i o n of t h e gap i s r e q u i r e d for c o o l i n g p u r p o s e s . F i n a l l y , it should be n o t e d t h a t a l a r g e p e r m a n e n t m a g n e t i s n o r m a l l y not thought of a s being p r a c t i c a l , a l t h o u g h w e a r e v e r y m u c h f a m i l i a r w i t h the s m a l l p e r m a n e n t m a g n e t s . T h e r e a r e , h o w e v e r , a p p l i c a t i o n s to c o s m i c r a y p h y s i c s in w h i c h v e r y l a r g e p e r m a n e n t m a g n e t s a r e r e q u i r e d and a r e , in fact, built.

F o r e x a m p l e , it h a s b e e n r e p o r t e d by C o u s i n s and N a s h ^ t h a t a l a r g e n u m b e r of p e r m a n e n t l a r g e m a g n e t s h a v e b e e n built a n d t h a t t h e d e s i g n c r i t e r i a a r e r e a s o n a b l y w e l l u n d e r s t o o d . One p a r t i c u l a r m a g n e t t h e y d e s c r i b e d h a s a pole face a r e a of 824 c m i and a 3.18 c m gap. T h e m a g n e t i c field w i t h i n the gap is 9000 g a u s s a n d the e n t i r e p e r m a n e n t m a g n e t w e i g h s only 1300 lb; t h u s , l a r g e s c a l e and h e a v y p e r m a n e n t m a g n e t s a r e i n d e e d a r e a l i t y and t h e s p e c i a l type of m a g n e t r e q u i r e d