Mathematical Methods in Nuclear Reactor Dynamics by Ziya Akcasu

By Ziya Akcasu

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By Ziya Akcasu

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W e do not consider n e u t r o n s escaping t h e reactor in t h e positive z direction a n d r e t u r n i n g back u n d e r t h e influence of gravity. 1). , (ijjn | 0 m > ^+ Snm . H o w e v e r , by c o m p a r i n g (21a) to (3a), we can guess that t h e set ^n(z)y defined as 2 12 i/j +(z) n = (2/a) I e+»*/ sin[(« + l>rs/5] (22) is o r t h o n o r m a l to i/*n(z) in t h e sense that + + < = K m (23) W e shall refer to ^n(z) as t h e " a d j o i n t " set. T h e y satisfy t h e following equation: 2 2 2 [(d /dz ) - Mldz) + Bn] +n+(z) = 0 (24) w h i c h differs from (20) in t h e sign of t h e first derivative.

Any vector e x p a n d e d into {| n>} as l<£> = | y can be £ a \n> n n=0 (14a) + where a n = <<^ | c/>y (14b) 4. T h e eigenvalues of a self-adjoint operator are real a n d positive, a n d its eigenfunctions form a c o m p l e t e o r t h o n o r m a l set. 5. T h e c o m p o n e n t s of | n> satisfy (closure p r o p e r t y ) 8(r - r') S(v - v ' ) 8„ = £ ^ w( ri' , V) ^ ( r , v ) (15) ( P r o b l e m 3). 2. S t a t i o n a r y R e a c t o r a n d t h e M u l t i p l i c a t i o n F a c t o r T h e p u r p o s e of this section is to i n t r o d u c e t h e concept of criticality a n d n e u t r o n cycle, a n d to define t h e multiplication factor in a stationary noncritical reactor.

Mathematical Preliminaries + w h e r e s is t h e outer surface of t h e reactor. Since satisfies (5), t h e surface integral vanishes because either + or 4> will always be zero on t h e surface in integrating over SI. T h e r e m a i n i n g t e r m can b e recognized as < — v • V+ | >, w h i c h proves t h e assertion. Example 2. Integral operator. T h e adjoint of J d*v' |V278(v' - > v, r)] (6a) j d*v'[vZB(v^v',r)] (6b) is which is obtained by interchanging t h e v and v ' in t h e integrand.

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