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Feedback control optimization of a single fluid heat exchanger or nuclear reactor
NUCLEAR ENGINEERING AND DESIGN 7 (1968) 40-48. NORTH-HOLLAND PUBLISHING COMPANY, AMSTERDAM
FEEDBACK CONTROL O P T I M I Z A T I O N OF A SINGLE FLUID HEAT EXCHANGER OR NUCLEAR REACTOR R. G. W A T T S Department of Mechanical Engineering, Tulane University, New Orleans, La., USA and R. J . S C H O E N H A L S School of Mechanical Engineering, Purdue University, Lafayette, Ind., USA Received 21 D e c e m b e r 1967 The p e r f o r m a n c e of a single fluid heat exchanger is analyzed under conditions of proportional control of heat flux. The e r r o r is obtained from a single t e m p e r a t u r e probe whose axial location can be varied. The usual p r o c e d u r e is to place the probe at the outlet and to select the best proportional gain constant for this situation. However, r e s u l t s of this study show that simultaneous optimization of gain constant and axial probe location can produce l a r g e i m p r o v e m e n t s in s y s t e m p e r f o r m a n c e . F o r p u r p o s e s of i l l u s t r a tion, quantitative r e s u l t s a r e given for the case of v a r i a b l e inlet t e m p e r a t u r e whose s p e c t r a l power density is constant up to a high frequency limit, and is zero beyond this limiting value. A physical a r g u m e n t is suggested which r e l a t e s optimum probe location to a single p a r a m e t e r defined in t e r m s of s y s t e m time constants. Many nuclear r e a c t o r s y s t e m s can be r e g a r d e d as single fluid heat exchangers for p u r p o s e s of calculating outlet t e m p e r a t u r e dynamics, since the heat generation rate usually responds a g r e a t deal f a s t e r than the fluid t e m p e r a t u r e when a control rod position is changed. The general method outlined is also applicable to boiling water r e a c t o r s in which the outlet void fraction r a t h e r than the outlet t e m p e r a t u r e is the controlled variable.
1. I N T R O D U C T I O N T h e n u m b e r of p u b l i s h e d p a p e r s c o n c e r n i n g t h e d y n a m i c s of h e a t e x c h a n g e r s i s l a r g e a n d g r o w i n g . S i n c e t h e t h e s i s of P r o f o s [1], a n d e s pecially in the past ten years, transfer functions h a v e b e e n d e r i v e d f o r m a n y d i f f e r e n t t y p e s of h e a t e x c h a n g e r s . A r e v i e w of t h e l i t e r a t u r e o n h e a t - e x c h a n g e r d y n a m i c s b e f o r e 1961 h a s b e e n p u b l i s h e d b y W i l l i a m s a n d M o r r i s [2]. A c o m p i l a t i o n of t r a n s f e r f u n c t i o n s d e v e l o p e d b y m a n y a u t h o r s h a s b e e n g i v e n b y H s u a n d G i l b e r t [3]. Several papers have been published recently and a r e c o v e r e d i n r e f s . [2] a n d [3]. A m o n g t h e s e a r e t h e w o r k s of S t e r n m o l e a n d L a r s o n [4], Y a n g [5], a n d n e m p e l [6]. Transfer functions for heat exchanger syst e m s t y p i c a l of t h o s e e n c o u n t e r e d i n n u c l e a r r e a c t o r s h a v e b e e n g i v e n b y G r o s s m a n [7] a n d C h r i s t e n s e n [8]. C h r i s t e n s e n ' s w o r k i n d i c a t e s that the transfer function relating outlet fluid q u a l i t y to h e a t g e n e r a t i o n r a t e i n b o i l i n g w a t e r
reactors has essentially the same form as those derived in the references cited above. I n s p i t e of t h e g r e a t i n t e r e s t i n h e a t - e x c h a n g e r control which is demonstrated by the many pap e r s o n d y n a m i c s , c o m p a r a t i v e l y few p a p e r s e x ist in the literature which deal with the problem of h e a t - e x c h a n g e r c o n t r o l l e r d e s i g n [2]. M o s t of t h e p a p e r s o n t h i s s u b j e c t , h o w e v e r , s e e m to i n d i c a t e t h a t t h e p r o b l e m of d e s i g n i n g a c o n t r o l l e r for a given heat exchanger is not a particularly d i f f i c u l t o n e [2]. It i s , t h e r e f o r e , n o t t h e i n t e n t i o n of t h i s p a p e r t o f u r t h e r e x p l o r e t h e p r o b l e m of d e s i g n i n g a s t a b l e , r e s p o n s i v e c o n t r o l l e r f o r a g i v e n t y p e of h e a t e x c h a n g e r . I n s t e a d , a t t e n t i o n i s f o c u s e d o n t h e p r o b l e m of s u b s t a n t i a l l y improving the outlet-temperature c o n t r o l of a h e a t e x c h a n g e r b y u t i l i z i n g t h e u n i q u e n a t u r e of the exchanger, namely, its distribution in space. It h a s b e e n k n o w n f o r s o m e t i m e t h a t t h e o u t l e t - t e m p e r a t u r e c o n t r o l of o i l h e a t e r s c a n b e i m proved by locating the control-temperature probe s o m e d i s t a n c e a h e a d of t h e o u t l e t [9]. T o t h e
FEEDBACK CONTROL OPTIMIZATION
a u t h o r s ' knowledge t h e r e is no evidence in the l i t e r a t u r e of p r e v i o u s a t t e m p t s to p r e d i c t a c c u r a t e l y what this p o s i t i o n ideally should be. The p r o b l e m which is c o n s i d e r e d is that of p r e d i c t i n g the optimum p o s i t i o n of a m e a s u r e m e n t p r o b e for a p a r t i c u l a r type of heat exchanger. The extent of p e r f o r m a n c e i m p r o v e m e n t over that obt a i n e d by locating the probe in its u s u a l position at the exit is examined. E x a m p l e s of heat e x c h a n g e r s which can be t r e a t e d by the a n a l y q i s to be p r e s e n t e d below i n clude v a p o r - l i q u i d heat e x c h a n g e r s and n u c l e a r r e a c t o r s in which a meaningful m i x e d - m e a n outlet t e m p e r a t u r e can be m e a s u r e d , and whose heat g e n e r a t i o n r a t e r e s p o n d s much f a s t e r than does the fluid t e m p e r a t u r e . The method should be p a r t i c u l a r l y applicable to s u p e r h e a t e r t e m p e r a t u r e c o n t r o l in n u c l e a r and conventional power plants. It is also applicable to boiling w a t e r r e a c t o r s when outlet quality is to be controlled.
2. PROBLEM FORMULATION C o n s i d e r a s i m p l e t u b u l a r heat exchanger whose outlet t e m p e r a t u r e is c o n t r o l l e d by a u t o m a t i c a d j u s t m e n t of the outside t u b e - w a l l heat flux as shown in fig. 1. It is shown in Appendix 1 that the t r a n s f e r functions r e l a t i n g the t e m p e r a t u r e at a p o s i t i o n x, m e a s u r e d f r o m the inlet, to the heat flux 7/and i n l e t t e m p e r a t u r e ei a r e O(~S, x) e-h(s)x 0i(s~) =
(i)
and O(s, x) _ 1 - e - h ( s ) x ~s'
s (s g(s ) = -X
+ ~ + x) .
The d i m e n s i o n l e s s p a r a m e t e r s s , ~, and X a r e defined in t e r m s of the s y s t e m t i m e constants. Thus s = s't d , = td/t f , X = td/tw , where s ' = Laplace v a r i a b l e , t d = s y s t e m holdup t i m e ( L / V f ) , tf = fluid t i m e constant ( p f c f A f / h P ) , tw = wall t i m e constant ( P w c w A w / h P ) . The q u a n t i t i e s 0i(s), O(x,s), and ~?(s) a r e L a place t r a n s f o r m e d d i m e n s i o n l e s s t r a n s i e n t v a r i ables and a r e d i s c u s s e d in Appendix 1. Suppose t h e r e exists a s t a t i o n a r y r a n d o m fluctuation 0i in the inlet t e m p e r a t u r e , and that it i s d e s i r e d to c o n t r o l the outlet t e m p e r a t u r e 0L in such a m a n n e r that its m e a n - s q u a r e deviation from a d e s i r e d value is m i n i m i z e d . Since OL, as pointed out in the foregoing p a r a g r a p h , is a t i m e - v a r y i n g quantity an a p p r o p r i a t e ~ n t i t y to be m i n i m i z e d is 0L--L-2~(T). The value of 0LZ(7) can be w r i t t e n as 0L2(T) = f ~ {W(J~2)[ 2 ~(j~2)d~2 , o
(3)
w h e r e W(s) is the t r a n s f e r function r e l a t i n g 0L to 0i for the s y s t e m - c o n t r o l l e r c o m b i n a t i o n , and • ( j ~ ) is the p o w e r - d e n s i t y s p e c t r u m of the i n p u t - t e m p e r a t u r e d i s t u r b a n c e [10]. A s s u m i n g p r o p o r t i o n a l control, the c o r r e c tive heat flux in fig. 1 is given by ~(s) = -CO(x1, s) ,
(4)
where C is the p r o p o r t i o n a l gain constant and x 1 defines the p o s i t i o n of the m e a s u r i n g probe. The inlet-temperature disturbance causes a variation in the t e m p e r a t u r e at position x 1 as defined by eq. (1). The total change in t e m p e r a t u r e of a fluid p a r t i c l e at position Xl in the heat exchanger is given by the sum of those changes r e s u l t i n g f r o m the i n l e t - t e m p e r a t u r e fluctuations and the c o r r e c t i v e action of the c o n t r o l l e r . Thus
where h(s) = s + ~ -
and
(2)
g(s)
41
s+X
i
O(Xl, s) = Oi(s)
'
I
x
:K,l
(5)
Substituting eq. (5) into eq. (4) gives the e x p r e s sion for the heat flux if(s) i n t e r m s of Oi(s):
X'X,
Fig. 1. Single-fluid heat exchanger with feedback control of external wall-heat flux.
e-h(s)xl + ~(s) L1- e-h(S)Xl-] g(s) j.
= ~(s)
-Cg(s) e - h ( s ) x l g(s) + C(1 - e - h ( s ) x l ) 0i(s) .
(6)
42
R . G. W A T T S a n d R. J . S C H O E N H A L S
Substitution of this e x p r e s s i o n into eq. (2) g i v e s the component of the outlet t e m p e r a t u r e which r e s u l t s f r o m the c o r r e c t i v e heat flux ~(s). This quantity is then added to the contribution owing to the fluctuating inlet t e m p e r a t u r e to give the va lu e of 0L(S), the t r a n s f o r m of the d i m e n s i o n l e s s t r a n s i e n t outlet t e m p e r a t u r e : 0L(S) = ~-i(s) e -h(s) -
C--e-h(s)xl(1-e-h(s)) g(s)+C(1 -e -h(s)X1)
W(s)
is t h e r e f o r e given by
W(s)
= 0-L(S)_ = e_h(s) 0i(s)
O. 16! %=0 5
2.O
0.12 0.71
. Q08
~i(s) •
C e-h(s)xl(1-e -h(s))
x=O.Ol
o.s 15
0.04
1.0
(7) 0 0
0 C
C
(8)
g(s)+C(1-e-h(s)xl) '
F o r a given p o w e r - d e n s i t y s p e c t r u m ~(j~2) eqs. (3) and (8) can be used to d e t e r m i n e the value of Xl which m i n i m i z e s ~L20-). F o r the p r e s e n t p r o b l e m , it is a s s u m e d that the inlet t e m p e r a t u r e c o n s i s t s of a s t a t i o n a r y r a n d o m signal containing equal components of a ll f r e q u e n c i e s b et wee n z e r o and s o m e m a x i m u m f r e q u e n c y e * . Thus, the value of (I,(fl2) is taken as unity for 0 ~< f~ -.< f~* and z e r o for a ll o th e r v a l u e s of f~. The equation for the m e a n - s q u a r e o u t l e t - t e m p e r a t u r e v a r i a t i o n can now be f o r m u lated as follows:
OL-~(z) = f 0
,='= 1.0 A= 0.01
C e-h(ja)xl(1-e-h(ja)) e-h(J Q) - g(jf~)+C(1-e -h(j~2)xl)
2
a~.
(9)
3. R ES ULTS
i)8
'
s j~,olo
'
'o.~
'
I.©
ODS
~/L/
~=5.0 2
004
~= 0.5 x= 3.0
O0
I
I
5
I0
I
I
15
20
25
O0
I
I
]
2
4-
6
I
8
i
12
C
C
Fig. 2. Mean-square outlet-temperature er r o r for various proportional gain constants and temperature probe locations. X'l = Xl, opt. A f a c t o r of i m p r o v e m e n t , fined as F -
F,
is d e -
0L2(T)Xl = 1.0 6L20-)Xl = Xl, opt
Eq. (9) was p r o g r a m m e d f o r an IBM 7090 d i g ital c o m p u t e r . The m e a n - s q u a r e o u t l e t - t e m p e r a t u r e e r r o r was m i n i m i z e d for given v a l u e s of ;t and ~ by v a r y i n g the v a l u e s of C and Xl until a m i n i m u m was located. The c o m p u t e r r e s u l t s in di cate that a v e r y definite o p ti m u m p r o b e p o s i t i o n e x i s t s , and that locating the p r o b e at this p o s i tion can r e d u c e the m e a n - s q u a r e o u t l e t - t e m p e r a t u r e deviation c o n s i d e r a b l y . Fig. 2 shows four c a s e s of OL2-~ plotted a g a i n s t the gain constant C with Xl as a p a r a m e t e r . The v a l u e of f~* u s e d to g e n e r a t e t h e s e c u r v e s was 15 r a d i a n s . C u r v e s s i m i l a r to those of fig. 2 w e r e g e n e r a t e d f o r t w e l v e c o m b i n a t i o n s of (; and X, and optimum gains and p r o b e p os it io n s w e r e picked f r o m t h e s e plots. T a b l e 1 shows s o m e of the p e r t i n e n t n u m e r i cal i n f o r m a t i o n a r r i v e d at in this study. The d i m e n s i o n l e s s gain v a l u e s l i s t e d a r e optimum gains for the p a r t i c u l a r locations Xl = 1.0 and
A f a c t o r of i m p r o v e m e n t of 2.0 i n d i c a t e s that the value of 0 L 2 ~ obtained by o p t i m i z i n g the cont r o l l e r gain with the p r o b e at the outlet is t w i ce as l a r g e as the o p t i m u m value with x 1 = x 1.opt. It is i n t e r e s t i n g to note that a c o m m e r c i a l fleat e x c h a n g e r t e s t e d by Hansen et al. [11] has (r and ~ v a l u e s of (~ = 1.65 and )t = 17. F o r a heat ex changer with ~ = 1.0 and ~ = 12 (values r e a s o n ably c l o s e to t h o s e of Hansen), the v al u e of 0L2(~-) can be r e d u c e d by a f a c t o r of about 30 by p r o p e r location of the co n t r o l p r o b e (see table 1). A simulated SPERT-IA moderator-coolant channel t e s t e d by Zivi and Wright [12] i n d i c a t e s v a l u e s of ~ and k of a p p r o x i m a t e l y ~ = 1 and ~t = 3.0. The i m p r o v e m e n t f a c t o r f o r this c a s e should l i e b e t w e e n 2 and 5 a c c o r d i n g to table 1. T h e s e r e s u l t s r e p r e s e n t i d e a l i z e d c a s e s , of c o u r s e , but the m ag n i t u d e of the i m p r o v e m e n t is in each c a s e n e v e r t h e l e s s quite s t r i k i n g and indic a t e s the value of the t ech n i q u e u n d e r d i scu ssi o n .
FEEDBACK CONTROL OPTIMIZATION
43
Table 1 Tabulation of some optimum results. (7
~
Xl, opt
Optimum value of C when x I = Xl. opt
Optimum value of C when x I = 1.0
0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 3.0 3.0 3.0
1.0 3.0 5.0 20.0 0.01 1.0 5.0 6.0 12.0 0.01 1.0 5.0
0.56 0.70 0.75 0.9 0.50 0.62 0.775 0.79 0.865 0.675 0.725 0.81
10.5 7.25 5.0 11.5 2.0 9.0 7.0 7.0 9.0 2.7 4.5 6.0
1.0 1.0 1.0 1.5 7.0 2.0 0.7 1.0 1.0 30.0 18.0 35.0
Fig. 2 shows some interesting trends which a r e c o r r o b o r a t e d by o t h e r c a s e s . W h e n cr i s s m a l l and ~ i s l a r g e , t h e m i n i m u m v a l u e of ~L2(T) is h i g h l y d e p e n d e n t on p r o b e p o s i t i o n , w i t h o p t i mum gain increasing as Xl,opt is approached. T h i s is t h e c o n d i t i o n of l a r g e tf and s m a l l t w. I n c r e a s i n g c ( d e c r e a s i n g tf) o r d e c r e a s i n g )t (inc r e a s i n g tw) g i v e s r i s e to a l e s s e r d e p e n d e n c e of the m i n i m u m 0L2(Ti on p r o b e p o s i t i o n . N o t i c e that for these conditions the optimum gain decreases s h a r p l y a s X l , o p t i s a p p r o a c h e d . T h i s , of c o u r s e , p o i n t s t o w a r d saving~s in e q u i p m e n t c o n t r o l l e r c o s t s . T h e s m a l l d e p e n d e n c e of t h e m i n i m u m 0L2(~-) on x 1 w i t h l a r g e c and s m a l l ~ r e s u l t s f r o m t h e f a c t that a t h i c k w a l l ( i n d i c a t e d by a l a r g e v a l u e of t w) h a s a t e n d e n c y to t r a n s f e r h e a t f r o m one p a r t i c l e to o t h e r s n e a r it. T h i s o c c u r s b e c a u s e a r e l a t i v e l y hot p a r t i c l e can c a u s e a t e m p e r a t u r e i n c r e a s e in t h e w a l l a r e a in c o n t a c t w i t h it. T h e a d d i t i o n a l e n e r g y is p a s s e d along to c o o l e r fluid p a r t i c l e s w h i c h flow p a s t t h i s a r e a l a t e r . T h i s effective]Ly s p r e a d s out any s h a r p temperature variations throughout the space ins i d e t h e h e a t e x c h a n g e r tube. In s y s t e m s w i t h t h i n w a l l s on the o t h e r hand, t e m p e r a t u r e v a r i a t i o n s in s p a c e a r e v e r y s h a r p l y d e f i n e d , and c o n t r o l l e r o p t i m i z a t i o n is m u c h m o r e d e p e n d e n t on Xl.
4. A S I M P L I F I E D A P P R O X I M A T E A P P R O A C H It is i n t e r e s t i n g and u s e f u l to a t t e m p t to p r e d i c t , on p u r e l y p h y s i c a l g r o u n d s , a p p r o x i m a t e l y w h a t the o p t i m u m po.'~ition X l , o p t s h o u l d be. W h e n a d e v i a t i o n f r o m t h e d e s i r e d v a l u e of fluid t e m p e r a t u r e i s d e t e c t e d at t h e o u t l e t of an e x c h a n g e r , a s i g n a l is s e n t f r o m t h e m e a s u r i n g d e v i c e to t h e c o n t r o l l e r w h i c h a l t e r s t h e h e a t flux in o r d e r to
Improvement factor F 1.53 2.74 3.70 45.5 1.01 1.61 5.43 8.85 30.3 1.06 1.84 4.36
o f f s e t the e x i s t i n g t e m p e r a t u r e e r r o r . In a h e a t e x c h a n g e r w i t h z e r o w a l l c a p a c i t y and z e r o fluid h e a t u p lag, the c o r r e c t i v e a c t i o n is i m m e d i a t e l y a p p l i e d d i r e c t l y to the fluid. When the w a l l h a s f i n i t e t h i c k n e s s and the fluid h e a t u p t i m e i s not z e r o , s o m e t i m e lag e x i s t s b e t w e e n the a p p l i c a t i o n of the c o r r e c t i v e h e a t flux to t h e o u t s i d e w a l l and i t s a p p l i c a t i o n to t h e fluid. T h e r e f o r e , by t h e t i m e the c o n t r o l l e r r e s p o n d s and a c o r r e c t i v e h e a t flux is a p p l i e d to t h e fluid, the fluid p a r t i c l e w h o s e t e m p e r a t u r e w a s in e r r o r m a y h a v e left the s y s t e m . If the m e a s u r e m e n t p r o b e is p l a c e d a c e r t a i n d i s t a n c e u p s t r e a m f r o m the e x i t , s u c h that the flow t i m e b e t w e e n t h i s point and t h e e x i t i s about the s a m e as the d e l a y in t h e a p p l i c a t i o n of t h e h e a t flux to t h e fluid, t h e n the a l t e r e d h e a t flux t e n d s to c o r r e c t f l u i d - p a r t i c l e temperatures just before these particles leave the s y s t e m . In A p p e n d i x 2 an a p p r o x i m a t e t r a n s f e r f u n c t i o n r e l a t i n g the o u t s i d e - w a l l h e a t flux ~(s) to the i n s i d e - w a l l h e a t flux ~(s) is shown to be
~(s) ~-~
s+(7+~
(10)
Suppose the inlet temperature to the heat exchanger fluctuates with some frequency ~. The error measured by the probe fluctuates at the same frequency, Assuming proportional control, the inside-wall heat flux ~(s) lags behind the error s i g n a l by p h a s e l a g = t a n -1 ( ~ + ~ )
ill)
a c c o r d i n g to eq. (10). S i n c e the s i g n a l s at p o s i t i o n s x = Xl (the p r o b e p o s i t i o n ) and x = 1.0 a r e e s s e n t i a l l y r e l a t e d to one a n o t h e r t h r o u g h a d i s t a n c e - v e l o c i t y lag, t h e p h a s e lag b e t w e e n OL and O(x1) i s
44
R.G. WATTS and R. J. SCHOENHALS phase lag =
~2(I
1.4
-Xl) •
(12)
I
I
I
]
I
I
;
I
O,
I
I
t
!
com~h~r solution of equation 19)
®
e~Jation ( t 41
--
I
I 2
I
, 4
I
i
I
G
t B
I
I I0
I
I 12
;
I ~4
I
d 16
I
18
L 20
s i s t e n t l y below the c u r v e , indicating that s o m e additional flow t i m e should be al l o w ed between the p o s i t i o n of the c o n t r o l - t e m p e r a t u r e p r o b e and the outlet. This is probably b e c a u s e eq. (14) a l lows only for delay b et w een the e x t e r n a l - w a l l heat flux and the i n s i d e - w a l l heat flux. Slightly m o r e t i m e is n e c e s s a r y f o r the fluid p a r t i c l e to a c t u a l l y change in t e m p e r a t u r e . This is the r e a son for the inequality of r e l a t i o n (13).
(14)
idL(j~)/~(jf~)i
5. F U R T H E R DISCUSSION It was m en t i o n ed p r e v i o u s l y that f~* = 15 r a dians was u s e d to e v a l u a t e the r e s u l t s given in fig. 2 and in t ab l e 1. S i m i l a r computations w e r e c a r r i e d out for s e v e r a l d i f f er en t v a l u e s of ~*. A l l of the r e s u l t s obtained showed that v a l u e s of Xl,opt obtained f r o m solution of eq. (9) a r e e s I
T
I
[ I I I I I I I
×,=0.8
0.5
S
I
o'+~.
It has b e e n shown [2, 6, 13] that r e s o n a n c e o c c u r s in the Bode plot of in the vicinity of ~ =(2k+ 1)g w h e r e k is an i n t e g e r . It is , t h e r e f o r e , not u n r e a s o n a b l e to expect that the p a r t i c u l a r v al u e ~ = g ( c o r r e s p o n d i n g to k = 0) would a l low an a p p r o x i m a t e p r e d i c t i o n of Xl,opt to be made. The v a l u e s of Xl,opt d e t e r m i n e d f r o m the c o m p u t e r solution of eq. (9) c o r r e l a t e r e a s o n a b l y well with eq. (14) as shown in fig. 3. The plotted data points c o r r e s p o n d to the i n f o r m a t i o n given in table 1. The q u a l i t a t i v e a g r e e m e n t s e e m s quite good in view of the s i m p l i c i t y of the d e r i v a tion of eq. (14). The computed points fall con-
r.o
l
Fig. 3. Optimum probe location and comparison with approximate prediction.
?7
I
]
0.2
F o r f~ = ~, the p e r i o d is tw ic e the holdup t i m e of a fluid p a r t i c l e within the heat exchanger. If ~2 = ~T is i n s e r t e d into eq. (13), the p r e d i c t e d u p p e r l i m i t of Xl,opt b e c o m e s
I
I
0.6
(13)
I
I
0.8
w h e r e Xl,opt denotes the o p t im u m v a l u e of Xl. Solution for Xl,opt g i v e s
I
[
XI ,opt.
--~l~opt) ,
(Xl ,opt) u p p e r l i m i t = 1 - tan-l(~/(cr+~))
I
0
0.4
Xl,opt ~< 1 - tan-l(~2/(a+;~)) ~2
I
1.0
®
tan-1
:
J.2
Si nc e p r o p e r c o r r e c t i v e c o n t r o l a c t i o n should be a ppli ed to fluid p a r t i c l e s b e f o r e they l e a v e the heat e x c h a n g e r , the lag between e r r o r d e te c t io n and c o r r e c t i v e action on the fluid should be l e s s than the t r a n s p o r t a t i o n lag between x = Xl,opt and x = 1.0. Thus ~2
;
I0
15
DIMENSIONLESS FREQUENCY, ~IRADIANS) Fig. 4. Integrand of eq. (9) with b e s t gain f o r v a r i o u s probe locations (if= 0.5, X = 3.0).
FEEDBACK
CONTROL
s e n t i a l l y independent of ~* if ~* is l a r g e r than 6 or 7 radians. An explanation of this is s u g g e s t e d by e x a m i n a t i o n of pl~ts of i W(j~) i2 f o r v a r i o u s v a l u e s of Xl. Fig. 4 shows such a plot f o r the c a s e of ~ = 0.5 and ~ = 3.0. The v a l u e s of C u s e d for t h e s e c u r v e s a r e in all c a s e s the optimum v a l u e s f o r the p a r t i c u l a r v a l u e s of x 1 used. The high value cl o s e to ~ = ~ f o r the c a s e of x 1 = 1.0 is g r a d u a l l y r e d u c e d t o w a r d z e r o as x I a p p r o a c h e s the optimum value of Xl., o,nt -0.7 (see table 1). S i m u l t a n e o u s l y the peak of t h e s e c u r v e s i s n a r r o w e d and shifts to the right. Values of IW(j~2)i2 at high ~2 a r e h a r d ly a f f e c t e d at al l by v a r i a t i o n s in x l . A f t e r Xl is r e d u c e d below x].,opt the peak, which a l ways o c c u r s n e a r ~ : 1 = ~, b e g i n s to b e c o m e b r o a d cau s i n g the a r e a under the c u r v e to i n crease. It is noted that when Xl = 1.0, the l a r g e s t v a l u e s of I W(j~)]2 a r e contributed by f r e q u e n c i e s n e a r ~ = ~. A c a r e f u l study of fig. 4 r e v e a l s that t h e r e is actually a n o th e r s m a l l peak n e a r ~ = 3~ for the c a s e of xl = 1.0. It a p p e a r s that the v a l ue of 0L--L-~(T)can be f u r t h e r r e d u c e d if a second p r o b e is l o c a t e d at p o s i t i o n x 2 w h e r e x 2 = 1 - tan-l(3~/(~+~)) 3~
(15)
Of c o u r s e , the gains a s s o c i a t e d with the two p r o b e s r e q u i r e a p p r o p r i a t e adjustment. T h i s p r o b l e m has b e e n i n v e s t i g a t e d m o r e p r e c i s e l y in ref. [13] fo r five c o m b i n a t i o n s of ~ and ~. It was found that, for the c a s e s c o n s i d e r e d , the o p t imum p o s i t i o n for the second p r o b e is v e r y c l o s e ly a p p r o x i m a t e d by e q . ( 1 5 ) as shown in fig. 5. H o w e v e r , the value of ~ L 2 ~ is d e c r e a s e d in each c a s e by l e s s than 0.1%. It is concluded that,
'"q~_l
I
r
I
I
r
F
J
I
r
[
I
I
I
I
I
I
I
I
1.0-0.~ ~ -
"~'e'm~-~
0.6~_
PJ
computer sok~ion
--
equatlo. (15} m
--
m
m
o
f o r the type of i n l e t - t e m p e r a t u r e v a r i a t i o n s t r e a t e d h e r e , addition of a second p r o b e does not i m p r o v e the s y s t e m p e r f o r m a n c e v e r y m u c h as f a r as the m e a n - s q u a r e e r r o r is c o n c e r n e d . The f o r e g o i n g r e s u l t s of this study indicate that a su b st an t i al r e d u c t i o n in m e a n - s q u a r e outl e t - t e m p e r a t u r e e r r o r in the p r e s e n c e of a r a n domly fluctuating inlet t e m p e r a t u r e is p r o v i d e d by placing a s i n g l e t e m p e r a t u r e p r o b e at Xl,opt. In p r a c t i c a l s i t u a t i o n s it is often n e c e s s a r y to i n s u r e e l i m i n a t i o n of s t e a d y - s t a t e e r r o r s in outlet t e m p e r a t u r e which can r e s u l t f r o m p a r a m e t e r changes or f r o m d r i f t in the steady c o m p o nents of d i s t u r b a n c e s such as inlet t e m p e r a t u r e and heat l o s s e s to the e n v i r o n m e n t . The s t e a d y state oulet-temperature e r r o r resulting from a constant d ev i at i o n f r o m the d e s i r e d value of inlet t e m p e r a t u r e can be d e t e r m i n e d by using eq. (7) with ~-i(s) = ~i/s and applying the final value t h e o r e m to the t r a n s f o r m e d v a r i a b l e ~L(S). Thus I.
.
l i m ~L(t) = l i m S~L(S)=eil 1 t-* ~ s-~0 ~
.
.
I
C
l+Cxl"
i
(16)
T h e r e f o r e , u n l e s s C = 1 / ( 1 - Xl), a s t e a d y - s t a t e e r r o r can o c c u r . Optimum v a l u e s of Xl and C w i l l not g e n e r a l l y sat i sf y this relation. In ref. [13] it is shown that it is p o s s i b l e to e l i m i n a t e the s t e a d y - s t a t e e r r o r by l o c a t i n g a second p r o b e at the outlet and p r o v i d i n g it with i n t e g r a l action, while the p r o b e at Xl,opt p r o v i d es only p r o p o r t i o n a l control. A second t e m p e r a t u r e p r o b e does not a p p r e c i a b l y i m p r o v e the m e a n - s q u a r e o u t l e t - t e m p e r a t u r e e r r o r in the p r e s e n c e of the r a n d o m i n l e t - t e m p e r a t u r e f l u c tuations a s s u m e d for p u r p o s e s of i l l u s t r a t i o n in this paper. H o w e v e r , it does a p p e a r d e s i r a b l e to p r o v i d e a second p r o b e at the outlet of s y s t e m s r e q u i r e d to m a i n t a i n z e r o s t e a d y - s t a t e e r r o r .
6. SUMMARY
1.2--
0.4
45
r
/
X2,o~
OPTIMIZATION
i
I 2
I
I 4
III
I
6
8
I
I
I0
I
I
J2
f
I
14
I
I
16
I
I
18
J
Fig. 5. Optimum location for second probe and comparison with approximate prediction.
I
20
It has been shown that when the outlet t e m p e r a t u r e of a t u b u l ar heat e x c h a n g e r is c o n t r o l l e d in the p r e s e n c e of a fluctuating inlet t e m p e r a t u r e , location of the c o n t r o l - t e m p e r a t u r e p r o b e at a p o s i t i o n x 1,opt can su b st an t i al l y r e d u c e the m e a n - s q u a r e o u t l e t - t e m p e r a t u r e e r r o r . The opt i m u m p o s i t i o n Xl,opt has been c a l c u l a t e d for a wide r an g e of a and ~ values. To a good a p p r o x i mation, this position depends on a s i n g l e p a r a m e t e r (a+k) f o r the c a s e s studied h er e. A p h y s i c a l a r g u m e n t has s u g g e s t e d a s i m p l e r e l a t i o n which c o r r e l a t e s the data quite well. The p o s s i b i l i t y of i m p r o v i n g c o n t r o l by i n s e r t i n g a second t e m p e r a t u r e p r o b e has been in-
46
R.G. WATTS and R. J. SCHOENHALS
vestigated. Based on the p r e s e n c e of a r a n d o m l y fluctuating inlet t e m p e r a t u r e , the p r e s e n t study indicated that a second probe p r o v i d e s v e r y little reduction of the m e a n - s q u a r e o u t l e t - t e m p e r a t u r e e r r o r . It has been shown in ref. [13], however, that u n d e r c e r t a i n conditions s t e a d y - s t a t e e r r o r s in the outlet t e m p e r a t u r e can r e s u l t from changes i n h e a t - e x c h a n g e r p a r a m e t e r s or from d i s t u r b ance v a r i a b l e s . In o r d e r to m a i n t a i n zero s t e a d y state e r r o r in outlet t e m p e r a t u r e u n d e r all conditions it is n e c e s s a r y to locate a second probe at the h e a t - e x c h a n g e r exit.
and PwCwAw -aTw -=qe
" P e - h P ( T w - Tf)
at
(18)
Eqs. (17) and (18) can be n o n d i m e n s i o n a l i z e d by u s i n g the following d i m e n s i o n l e s s v a r i a b l e s : x
X'
L td = -vf - ,
= - -
L'
T=
Tf - Tfi o O-
T w - Two
TfLo _ Trio
,
Ow-
tf = pfcfAf hP '
ACKNOWLEDGEMENT Support of this r e s e a r c h by P u r d u e R e s e a r c h Foundation through XR G r a n t P R F 3015 is g r a t e fully acknowledged.
t ~-d ,
TfLo _ Trio
tw = PwcwAw hP '
= td,
x = tA
tf
tw
,
(19)
q eP eL H = p f c f A f V f ( T f L ° _ Trio ) .
APPENDIX 1 Thus An e l e m e n t a r y length of the t u b u l a r heat exchanger analyzed is shown in fig. 6. F l u i d is heated as it flows through the tube. The following a s s u m p t i o n s a r e made: 1) The fluid i n s i d e the tube is i n c o m p r e s s i b l e . 2) The velocity and t e m p e r a t u r e p r o f i l e s a r e flat with Vf and Tf denoting the fluid velocity and mixed m e a n t e m p e r a t u r e r e s p e c t i v e l y . 3) The velocity does not vary with time. 4) The o u t s i d e - w a l l heat flux is independent of position. 5) Axial conduction in the fluid and in the wall is negligible. 6) The tube t e m p e r a t u r e is independent of r a dial position. E n e r g y b a l a n c e s on the fluid and wall control s u r f a c e s yield aTf pfcfAf - - + p f c f A f V f at
aTf ax' = h P ( T w - T f )
(17)
ae ~@ [Two - Tfio] ~--~-+ ~-x = cr(Ow - @) + Cr\TfL° _ T f i j
(20)
and - -
( TwO -
=--
Tfi°~
aT@w X~H - ~ (Ow - O) - ~ \TfLo - Trio] "
(21)
Separation of the d i m e n s i o n l e s s heat flux and the t e m p e r a t u r e s into t i m e - v a r y i n g and s t e a d y state components, O = Oo(X ) + O(x,-r), 0 w = 0wo(X) + 0w(X, 1-), and H = 7/0 + n(1-), yields a0 aO o ~0 ~7+-~-+-~='~o(~w -
0o)+~(0w -
~"
/Tw°-Tfi°~
- ~J + % T - - i ~ o
= T--~o)
(22) and a0w X X / Two-Trio\ a--~--=~ no + ~ n-h(0wo - 0 o) -h(0w - 0 )-k( TfLo_ T f i J . (23)
,//////~//////J/ Vf
/ / , ~./~/ / / / / /~
!"
S u b t r a c t i o n of the steady state equations gives the d i f f e r e n t i a l equations for the d i m e n s i o n l e s s t r a n sients, namely aO
aO
+~-=
~(0 w - 0 )
(24)
a0 w = '" n - X ( e w - e ) . aT
(25)
and //////~//////////////fl/////// l
l
Fig. 6. Element of a single-fluid heat-exchanger.
Taking the Laplace t r a n s f o r m of these two equations gives
FEEDBACK CONTROL OPTIMIZATION
sO + ~d0 = a(0w - 0)
(26)
and
sOw =-~ ~-X(O-w - 0 ) .
(27)
S i m u l t a n e o u s s o l u t i o n of t h e s e two e q u a t i o n s y i e l d s t h e t r a n s f e r f u n c t i o n s r e l a t i n g 0(s, x) to i n l e t t e m p e r a t u r e and h e a t flux.
g(s, x) Ei(s ) = 0(s, x)
e x p [ - ( s + a - ~-++~X)x]
- ~~X )x]
1 - exp[-(s+a
g(s)
sx-l(s+~+X)
(28)
(29)
C F
= controller gain constant = f a c t o r of i m p r o v e m e n t
g(s) h(s)
= (s/x)(s+~+x) = s+ ~- ~V(s+ x)
h L P Pe q" q~ s' s t T V W(s)
= heat-transfer coefficient = l e n g t h of h e a t - e x c h a n g e r f l u i d - f l o w p a t h = inside tube perimeter = external tube perimeter = i n t e r n a l t u b e - w a U h e a t flux = external tube-wall heat flux = Laplace variable = dimensionless Laplace variable, s't d = time = temperature = velocity = s y s t e m ( c l o s e d loop) t r a n s f e r f u n c t i o n = distance measured from tube inlet = dimensionless distance, x'/L = d i m e n s i o n l e s s h e a t flux = d i m e n s i o n l e s s t r a n s i e n t h e a t f l u x at e x ternal tube surface = dimensionless temperature = dimensionless time-varying temperature = system parameter, td/t w = density = system parameter, td/tf = dimensionless time, t/t d = p o w e r - d e n s i t y s p e c t r u m of i n l e t t e m p e r a ture disturbance = d i m e n s i o n l e s s t r a n s i e n t h e a t flux at i n t e r nal tube surface. = d i m e n s i o n l e s s f r e q u e n c y in r a d i a n s
x H 7/
APPENDIX2 A good e s t i m a t e can b e o b t a i n e d f o r t h e t r a n s f e r f u n c t i o n r e l a t i n g ~the i n s i d e - w a l l h e a t flux to t h e o u t s i d e - w a l l h e a t flux by i g n o r i n g the a x i a l t e m p e r a t u r e g r a d i e n t in t h e l i q u i d a s a f i r s t a p p r o x i m a t i o m F o r t h i s c o n d i t i o n eqs. (26) and (27) become s~ = ~(0w-0)
(30)
s 0 w =-~ ~-~ (8-w-0).
(31)
(9 0 X p a T
and
T h e s e e q u a t i o n s a r e not p r e c i s e l y c o r r e c t , of c o u r s e . T h e i d e a h e r e is to o b t a i n a s i m p l e e x p r e s s i o n w h i c h a d e q u a t e l y d e s c r i b e s the d y n a m i c b e h a v i o r of t h e t u b e w a l l if a s m a l l d e g r e e of a c c u r a c y i s s a c r i f i e d . C o m p a r i s o n of t h e a p p r o x i m a t e and t h e m o r e e x a c t r e s u l t s in figs. 3 and 5 forms a partial justification for this procedure. The dimensionless time-varying inside-wall heat flux ~ i s s i m i l a r to ?? in its d e f i n i t i o n , but it is r e l a t e d to t h e t r a n s i e n t t u b e to fluid t e m p e r a t u r e difference. The expression for ~ is
= ~(0w-0)
•
~(s)
X
s+(r+x
NOTATION A c
= cross-section = specific heat
area
~b
Subscripts d f io Lo w wo 1 2
= = = = = = = =
d e a d t i m e o r holdup t i m e fluid tube inlet, steady state tube outlet, steady state wall wall, steady state indicates measuring probe indicates a second measuring probe
(32)
C o m b i n i n g e q s . (30), (31), and (32) y i e l d s t h e t r a n s f e r f u n c t i o n r e l a t i n g ~(s) and W(s). T h e r e sult is V(s) _
47
(33)
REFERENCES [1] P. Profos, Die Behandlung yon Regelproblemen vermittels des Frequenzganges des Regelkreises, PhD thesis, Technical University, ZUrich, Switzerland (1943). [2] T. J. Williams and H. J. Morris, A Survey of the Literature on Heat Exchanger Dynamics and Control, paper presented at the Joint Automatic Control Conference, AIChE, Cambridge, Mass., September 1960. [3] J . P . H s u and N. Gilbert, T r a n s f e r Functions of Heat Exchangers, AIChE J. 8 (1962) 593.
48
R. G. WATTS and R. J. SCHOENHALS
[4] F. J. Steramole and M. A. Larson, The Dynamics of Flow Forced Distributed P a r a m e t e r Heat Exchangers, AIChE J. 10 (1964) 688. [5] W . J . Y a n g , Transient Heat T r a n s f e r in a VaporHeated Exchanger with A r b i t r a r y T i m e w i s e - V a r i ant Flow Perturbation, J. Heat T r a n s f e r , Trans. ASME, Series C 86 (1964) 133. [6] A. Hempel, On the Dynamics of Steam Liquid Heat Exchangers, ASME P a p e r No. 60-WA-110 (1960). [7] S. R. Grossman, T r a n s f e r Functions of Heat Exchangers and Heat Exchanger P r o c e s s e s Applicable to Nuclear Reactors, AEC R e s e a r c h and Development Report IDO-16486 (1958). [~l H. Christensen, P o w e r - t o - V o i d T r a n s f e r Functions, Argonne National Laboratory, ANL-6385 (1961).
[9] B. T h a l - L a r s e n , Dynamics of Heat Exchangers and Their Models, ASME P a p e r No. 59-A-117 (1959). [10] G. C. Newton, L.A. Gould and J. F. K a i s e r , Analytical Design of Linear Feedback Controls (John Wiley and Sons, Inc., New York, 1957). [11] P. D. Hansen, S.H. Goodhue and A. R. Catheron, Control of Shell and Tube Heat Exchangers, ASME P a p e r No. 59-IRD-14 (1959). [12] S. Zivi and R.W. Wright, Power-Void T r a n s f e r Function M e a s u r e m e n t s in a Simulated SPERT-IA Moderator Coolant Channel, Argonne National Laboratory, ANL-6205 (1960). [13] R . G . W a t t s , Some Aspects of the Dynamics, Stability, and Control of Heat Exchangers, PhD t h e s i s , Purdue University, Lafayette, Ind. (1965).
FEEDBACK CONTROL O P T I M I Z A T I O N OF A SINGLE FLUID HEAT EXCHANGER OR NUCLEAR REACTOR R. G. W A T T S Department of Mechanical Engineering, Tulane University, New Orleans, La., USA and R. J . S C H O E N H A L S School of Mechanical Engineering, Purdue University, Lafayette, Ind., USA Received 21 D e c e m b e r 1967 The p e r f o r m a n c e of a single fluid heat exchanger is analyzed under conditions of proportional control of heat flux. The e r r o r is obtained from a single t e m p e r a t u r e probe whose axial location can be varied. The usual p r o c e d u r e is to place the probe at the outlet and to select the best proportional gain constant for this situation. However, r e s u l t s of this study show that simultaneous optimization of gain constant and axial probe location can produce l a r g e i m p r o v e m e n t s in s y s t e m p e r f o r m a n c e . F o r p u r p o s e s of i l l u s t r a tion, quantitative r e s u l t s a r e given for the case of v a r i a b l e inlet t e m p e r a t u r e whose s p e c t r a l power density is constant up to a high frequency limit, and is zero beyond this limiting value. A physical a r g u m e n t is suggested which r e l a t e s optimum probe location to a single p a r a m e t e r defined in t e r m s of s y s t e m time constants. Many nuclear r e a c t o r s y s t e m s can be r e g a r d e d as single fluid heat exchangers for p u r p o s e s of calculating outlet t e m p e r a t u r e dynamics, since the heat generation rate usually responds a g r e a t deal f a s t e r than the fluid t e m p e r a t u r e when a control rod position is changed. The general method outlined is also applicable to boiling water r e a c t o r s in which the outlet void fraction r a t h e r than the outlet t e m p e r a t u r e is the controlled variable.
1. I N T R O D U C T I O N T h e n u m b e r of p u b l i s h e d p a p e r s c o n c e r n i n g t h e d y n a m i c s of h e a t e x c h a n g e r s i s l a r g e a n d g r o w i n g . S i n c e t h e t h e s i s of P r o f o s [1], a n d e s pecially in the past ten years, transfer functions h a v e b e e n d e r i v e d f o r m a n y d i f f e r e n t t y p e s of h e a t e x c h a n g e r s . A r e v i e w of t h e l i t e r a t u r e o n h e a t - e x c h a n g e r d y n a m i c s b e f o r e 1961 h a s b e e n p u b l i s h e d b y W i l l i a m s a n d M o r r i s [2]. A c o m p i l a t i o n of t r a n s f e r f u n c t i o n s d e v e l o p e d b y m a n y a u t h o r s h a s b e e n g i v e n b y H s u a n d G i l b e r t [3]. Several papers have been published recently and a r e c o v e r e d i n r e f s . [2] a n d [3]. A m o n g t h e s e a r e t h e w o r k s of S t e r n m o l e a n d L a r s o n [4], Y a n g [5], a n d n e m p e l [6]. Transfer functions for heat exchanger syst e m s t y p i c a l of t h o s e e n c o u n t e r e d i n n u c l e a r r e a c t o r s h a v e b e e n g i v e n b y G r o s s m a n [7] a n d C h r i s t e n s e n [8]. C h r i s t e n s e n ' s w o r k i n d i c a t e s that the transfer function relating outlet fluid q u a l i t y to h e a t g e n e r a t i o n r a t e i n b o i l i n g w a t e r
reactors has essentially the same form as those derived in the references cited above. I n s p i t e of t h e g r e a t i n t e r e s t i n h e a t - e x c h a n g e r control which is demonstrated by the many pap e r s o n d y n a m i c s , c o m p a r a t i v e l y few p a p e r s e x ist in the literature which deal with the problem of h e a t - e x c h a n g e r c o n t r o l l e r d e s i g n [2]. M o s t of t h e p a p e r s o n t h i s s u b j e c t , h o w e v e r , s e e m to i n d i c a t e t h a t t h e p r o b l e m of d e s i g n i n g a c o n t r o l l e r for a given heat exchanger is not a particularly d i f f i c u l t o n e [2]. It i s , t h e r e f o r e , n o t t h e i n t e n t i o n of t h i s p a p e r t o f u r t h e r e x p l o r e t h e p r o b l e m of d e s i g n i n g a s t a b l e , r e s p o n s i v e c o n t r o l l e r f o r a g i v e n t y p e of h e a t e x c h a n g e r . I n s t e a d , a t t e n t i o n i s f o c u s e d o n t h e p r o b l e m of s u b s t a n t i a l l y improving the outlet-temperature c o n t r o l of a h e a t e x c h a n g e r b y u t i l i z i n g t h e u n i q u e n a t u r e of the exchanger, namely, its distribution in space. It h a s b e e n k n o w n f o r s o m e t i m e t h a t t h e o u t l e t - t e m p e r a t u r e c o n t r o l of o i l h e a t e r s c a n b e i m proved by locating the control-temperature probe s o m e d i s t a n c e a h e a d of t h e o u t l e t [9]. T o t h e
FEEDBACK CONTROL OPTIMIZATION
a u t h o r s ' knowledge t h e r e is no evidence in the l i t e r a t u r e of p r e v i o u s a t t e m p t s to p r e d i c t a c c u r a t e l y what this p o s i t i o n ideally should be. The p r o b l e m which is c o n s i d e r e d is that of p r e d i c t i n g the optimum p o s i t i o n of a m e a s u r e m e n t p r o b e for a p a r t i c u l a r type of heat exchanger. The extent of p e r f o r m a n c e i m p r o v e m e n t over that obt a i n e d by locating the probe in its u s u a l position at the exit is examined. E x a m p l e s of heat e x c h a n g e r s which can be t r e a t e d by the a n a l y q i s to be p r e s e n t e d below i n clude v a p o r - l i q u i d heat e x c h a n g e r s and n u c l e a r r e a c t o r s in which a meaningful m i x e d - m e a n outlet t e m p e r a t u r e can be m e a s u r e d , and whose heat g e n e r a t i o n r a t e r e s p o n d s much f a s t e r than does the fluid t e m p e r a t u r e . The method should be p a r t i c u l a r l y applicable to s u p e r h e a t e r t e m p e r a t u r e c o n t r o l in n u c l e a r and conventional power plants. It is also applicable to boiling w a t e r r e a c t o r s when outlet quality is to be controlled.
2. PROBLEM FORMULATION C o n s i d e r a s i m p l e t u b u l a r heat exchanger whose outlet t e m p e r a t u r e is c o n t r o l l e d by a u t o m a t i c a d j u s t m e n t of the outside t u b e - w a l l heat flux as shown in fig. 1. It is shown in Appendix 1 that the t r a n s f e r functions r e l a t i n g the t e m p e r a t u r e at a p o s i t i o n x, m e a s u r e d f r o m the inlet, to the heat flux 7/and i n l e t t e m p e r a t u r e ei a r e O(~S, x) e-h(s)x 0i(s~) =
(i)
and O(s, x) _ 1 - e - h ( s ) x ~s'
s (s g(s ) = -X
+ ~ + x) .
The d i m e n s i o n l e s s p a r a m e t e r s s , ~, and X a r e defined in t e r m s of the s y s t e m t i m e constants. Thus s = s't d , = td/t f , X = td/tw , where s ' = Laplace v a r i a b l e , t d = s y s t e m holdup t i m e ( L / V f ) , tf = fluid t i m e constant ( p f c f A f / h P ) , tw = wall t i m e constant ( P w c w A w / h P ) . The q u a n t i t i e s 0i(s), O(x,s), and ~?(s) a r e L a place t r a n s f o r m e d d i m e n s i o n l e s s t r a n s i e n t v a r i ables and a r e d i s c u s s e d in Appendix 1. Suppose t h e r e exists a s t a t i o n a r y r a n d o m fluctuation 0i in the inlet t e m p e r a t u r e , and that it i s d e s i r e d to c o n t r o l the outlet t e m p e r a t u r e 0L in such a m a n n e r that its m e a n - s q u a r e deviation from a d e s i r e d value is m i n i m i z e d . Since OL, as pointed out in the foregoing p a r a g r a p h , is a t i m e - v a r y i n g quantity an a p p r o p r i a t e ~ n t i t y to be m i n i m i z e d is 0L--L-2~(T). The value of 0LZ(7) can be w r i t t e n as 0L2(T) = f ~ {W(J~2)[ 2 ~(j~2)d~2 , o
(3)
w h e r e W(s) is the t r a n s f e r function r e l a t i n g 0L to 0i for the s y s t e m - c o n t r o l l e r c o m b i n a t i o n , and • ( j ~ ) is the p o w e r - d e n s i t y s p e c t r u m of the i n p u t - t e m p e r a t u r e d i s t u r b a n c e [10]. A s s u m i n g p r o p o r t i o n a l control, the c o r r e c tive heat flux in fig. 1 is given by ~(s) = -CO(x1, s) ,
(4)
where C is the p r o p o r t i o n a l gain constant and x 1 defines the p o s i t i o n of the m e a s u r i n g probe. The inlet-temperature disturbance causes a variation in the t e m p e r a t u r e at position x 1 as defined by eq. (1). The total change in t e m p e r a t u r e of a fluid p a r t i c l e at position Xl in the heat exchanger is given by the sum of those changes r e s u l t i n g f r o m the i n l e t - t e m p e r a t u r e fluctuations and the c o r r e c t i v e action of the c o n t r o l l e r . Thus
where h(s) = s + ~ -
and
(2)
g(s)
41
s+X
i
O(Xl, s) = Oi(s)
'
I
x
:K,l
(5)
Substituting eq. (5) into eq. (4) gives the e x p r e s sion for the heat flux if(s) i n t e r m s of Oi(s):
X'X,
Fig. 1. Single-fluid heat exchanger with feedback control of external wall-heat flux.
e-h(s)xl + ~(s) L1- e-h(S)Xl-] g(s) j.
= ~(s)
-Cg(s) e - h ( s ) x l g(s) + C(1 - e - h ( s ) x l ) 0i(s) .
(6)
42
R . G. W A T T S a n d R. J . S C H O E N H A L S
Substitution of this e x p r e s s i o n into eq. (2) g i v e s the component of the outlet t e m p e r a t u r e which r e s u l t s f r o m the c o r r e c t i v e heat flux ~(s). This quantity is then added to the contribution owing to the fluctuating inlet t e m p e r a t u r e to give the va lu e of 0L(S), the t r a n s f o r m of the d i m e n s i o n l e s s t r a n s i e n t outlet t e m p e r a t u r e : 0L(S) = ~-i(s) e -h(s) -
C--e-h(s)xl(1-e-h(s)) g(s)+C(1 -e -h(s)X1)
W(s)
is t h e r e f o r e given by
W(s)
= 0-L(S)_ = e_h(s) 0i(s)
O. 16! %=0 5
2.O
0.12 0.71
. Q08
~i(s) •
C e-h(s)xl(1-e -h(s))
x=O.Ol
o.s 15
0.04
1.0
(7) 0 0
0 C
C
(8)
g(s)+C(1-e-h(s)xl) '
F o r a given p o w e r - d e n s i t y s p e c t r u m ~(j~2) eqs. (3) and (8) can be used to d e t e r m i n e the value of Xl which m i n i m i z e s ~L20-). F o r the p r e s e n t p r o b l e m , it is a s s u m e d that the inlet t e m p e r a t u r e c o n s i s t s of a s t a t i o n a r y r a n d o m signal containing equal components of a ll f r e q u e n c i e s b et wee n z e r o and s o m e m a x i m u m f r e q u e n c y e * . Thus, the value of (I,(fl2) is taken as unity for 0 ~< f~ -.< f~* and z e r o for a ll o th e r v a l u e s of f~. The equation for the m e a n - s q u a r e o u t l e t - t e m p e r a t u r e v a r i a t i o n can now be f o r m u lated as follows:
OL-~(z) = f 0
,='= 1.0 A= 0.01
C e-h(ja)xl(1-e-h(ja)) e-h(J Q) - g(jf~)+C(1-e -h(j~2)xl)
2
a~.
(9)
3. R ES ULTS
i)8
'
s j~,olo
'
'o.~
'
I.©
ODS
~/L/
~=5.0 2
004
~= 0.5 x= 3.0
O0
I
I
5
I0
I
I
15
20
25
O0
I
I
]
2
4-
6
I
8
i
12
C
C
Fig. 2. Mean-square outlet-temperature er r o r for various proportional gain constants and temperature probe locations. X'l = Xl, opt. A f a c t o r of i m p r o v e m e n t , fined as F -
F,
is d e -
0L2(T)Xl = 1.0 6L20-)Xl = Xl, opt
Eq. (9) was p r o g r a m m e d f o r an IBM 7090 d i g ital c o m p u t e r . The m e a n - s q u a r e o u t l e t - t e m p e r a t u r e e r r o r was m i n i m i z e d for given v a l u e s of ;t and ~ by v a r y i n g the v a l u e s of C and Xl until a m i n i m u m was located. The c o m p u t e r r e s u l t s in di cate that a v e r y definite o p ti m u m p r o b e p o s i t i o n e x i s t s , and that locating the p r o b e at this p o s i tion can r e d u c e the m e a n - s q u a r e o u t l e t - t e m p e r a t u r e deviation c o n s i d e r a b l y . Fig. 2 shows four c a s e s of OL2-~ plotted a g a i n s t the gain constant C with Xl as a p a r a m e t e r . The v a l u e of f~* u s e d to g e n e r a t e t h e s e c u r v e s was 15 r a d i a n s . C u r v e s s i m i l a r to those of fig. 2 w e r e g e n e r a t e d f o r t w e l v e c o m b i n a t i o n s of (; and X, and optimum gains and p r o b e p os it io n s w e r e picked f r o m t h e s e plots. T a b l e 1 shows s o m e of the p e r t i n e n t n u m e r i cal i n f o r m a t i o n a r r i v e d at in this study. The d i m e n s i o n l e s s gain v a l u e s l i s t e d a r e optimum gains for the p a r t i c u l a r locations Xl = 1.0 and
A f a c t o r of i m p r o v e m e n t of 2.0 i n d i c a t e s that the value of 0 L 2 ~ obtained by o p t i m i z i n g the cont r o l l e r gain with the p r o b e at the outlet is t w i ce as l a r g e as the o p t i m u m value with x 1 = x 1.opt. It is i n t e r e s t i n g to note that a c o m m e r c i a l fleat e x c h a n g e r t e s t e d by Hansen et al. [11] has (r and ~ v a l u e s of (~ = 1.65 and )t = 17. F o r a heat ex changer with ~ = 1.0 and ~ = 12 (values r e a s o n ably c l o s e to t h o s e of Hansen), the v al u e of 0L2(~-) can be r e d u c e d by a f a c t o r of about 30 by p r o p e r location of the co n t r o l p r o b e (see table 1). A simulated SPERT-IA moderator-coolant channel t e s t e d by Zivi and Wright [12] i n d i c a t e s v a l u e s of ~ and k of a p p r o x i m a t e l y ~ = 1 and ~t = 3.0. The i m p r o v e m e n t f a c t o r f o r this c a s e should l i e b e t w e e n 2 and 5 a c c o r d i n g to table 1. T h e s e r e s u l t s r e p r e s e n t i d e a l i z e d c a s e s , of c o u r s e , but the m ag n i t u d e of the i m p r o v e m e n t is in each c a s e n e v e r t h e l e s s quite s t r i k i n g and indic a t e s the value of the t ech n i q u e u n d e r d i scu ssi o n .
FEEDBACK CONTROL OPTIMIZATION
43
Table 1 Tabulation of some optimum results. (7
~
Xl, opt
Optimum value of C when x I = Xl. opt
Optimum value of C when x I = 1.0
0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 3.0 3.0 3.0
1.0 3.0 5.0 20.0 0.01 1.0 5.0 6.0 12.0 0.01 1.0 5.0
0.56 0.70 0.75 0.9 0.50 0.62 0.775 0.79 0.865 0.675 0.725 0.81
10.5 7.25 5.0 11.5 2.0 9.0 7.0 7.0 9.0 2.7 4.5 6.0
1.0 1.0 1.0 1.5 7.0 2.0 0.7 1.0 1.0 30.0 18.0 35.0
Fig. 2 shows some interesting trends which a r e c o r r o b o r a t e d by o t h e r c a s e s . W h e n cr i s s m a l l and ~ i s l a r g e , t h e m i n i m u m v a l u e of ~L2(T) is h i g h l y d e p e n d e n t on p r o b e p o s i t i o n , w i t h o p t i mum gain increasing as Xl,opt is approached. T h i s is t h e c o n d i t i o n of l a r g e tf and s m a l l t w. I n c r e a s i n g c ( d e c r e a s i n g tf) o r d e c r e a s i n g )t (inc r e a s i n g tw) g i v e s r i s e to a l e s s e r d e p e n d e n c e of the m i n i m u m 0L2(Ti on p r o b e p o s i t i o n . N o t i c e that for these conditions the optimum gain decreases s h a r p l y a s X l , o p t i s a p p r o a c h e d . T h i s , of c o u r s e , p o i n t s t o w a r d saving~s in e q u i p m e n t c o n t r o l l e r c o s t s . T h e s m a l l d e p e n d e n c e of t h e m i n i m u m 0L2(~-) on x 1 w i t h l a r g e c and s m a l l ~ r e s u l t s f r o m t h e f a c t that a t h i c k w a l l ( i n d i c a t e d by a l a r g e v a l u e of t w) h a s a t e n d e n c y to t r a n s f e r h e a t f r o m one p a r t i c l e to o t h e r s n e a r it. T h i s o c c u r s b e c a u s e a r e l a t i v e l y hot p a r t i c l e can c a u s e a t e m p e r a t u r e i n c r e a s e in t h e w a l l a r e a in c o n t a c t w i t h it. T h e a d d i t i o n a l e n e r g y is p a s s e d along to c o o l e r fluid p a r t i c l e s w h i c h flow p a s t t h i s a r e a l a t e r . T h i s effective]Ly s p r e a d s out any s h a r p temperature variations throughout the space ins i d e t h e h e a t e x c h a n g e r tube. In s y s t e m s w i t h t h i n w a l l s on the o t h e r hand, t e m p e r a t u r e v a r i a t i o n s in s p a c e a r e v e r y s h a r p l y d e f i n e d , and c o n t r o l l e r o p t i m i z a t i o n is m u c h m o r e d e p e n d e n t on Xl.
4. A S I M P L I F I E D A P P R O X I M A T E A P P R O A C H It is i n t e r e s t i n g and u s e f u l to a t t e m p t to p r e d i c t , on p u r e l y p h y s i c a l g r o u n d s , a p p r o x i m a t e l y w h a t the o p t i m u m po.'~ition X l , o p t s h o u l d be. W h e n a d e v i a t i o n f r o m t h e d e s i r e d v a l u e of fluid t e m p e r a t u r e i s d e t e c t e d at t h e o u t l e t of an e x c h a n g e r , a s i g n a l is s e n t f r o m t h e m e a s u r i n g d e v i c e to t h e c o n t r o l l e r w h i c h a l t e r s t h e h e a t flux in o r d e r to
Improvement factor F 1.53 2.74 3.70 45.5 1.01 1.61 5.43 8.85 30.3 1.06 1.84 4.36
o f f s e t the e x i s t i n g t e m p e r a t u r e e r r o r . In a h e a t e x c h a n g e r w i t h z e r o w a l l c a p a c i t y and z e r o fluid h e a t u p lag, the c o r r e c t i v e a c t i o n is i m m e d i a t e l y a p p l i e d d i r e c t l y to the fluid. When the w a l l h a s f i n i t e t h i c k n e s s and the fluid h e a t u p t i m e i s not z e r o , s o m e t i m e lag e x i s t s b e t w e e n the a p p l i c a t i o n of the c o r r e c t i v e h e a t flux to t h e o u t s i d e w a l l and i t s a p p l i c a t i o n to t h e fluid. T h e r e f o r e , by t h e t i m e the c o n t r o l l e r r e s p o n d s and a c o r r e c t i v e h e a t flux is a p p l i e d to t h e fluid, the fluid p a r t i c l e w h o s e t e m p e r a t u r e w a s in e r r o r m a y h a v e left the s y s t e m . If the m e a s u r e m e n t p r o b e is p l a c e d a c e r t a i n d i s t a n c e u p s t r e a m f r o m the e x i t , s u c h that the flow t i m e b e t w e e n t h i s point and t h e e x i t i s about the s a m e as the d e l a y in t h e a p p l i c a t i o n of t h e h e a t flux to t h e fluid, t h e n the a l t e r e d h e a t flux t e n d s to c o r r e c t f l u i d - p a r t i c l e temperatures just before these particles leave the s y s t e m . In A p p e n d i x 2 an a p p r o x i m a t e t r a n s f e r f u n c t i o n r e l a t i n g the o u t s i d e - w a l l h e a t flux ~(s) to the i n s i d e - w a l l h e a t flux ~(s) is shown to be
~(s) ~-~
s+(7+~
(10)
Suppose the inlet temperature to the heat exchanger fluctuates with some frequency ~. The error measured by the probe fluctuates at the same frequency, Assuming proportional control, the inside-wall heat flux ~(s) lags behind the error s i g n a l by p h a s e l a g = t a n -1 ( ~ + ~ )
ill)
a c c o r d i n g to eq. (10). S i n c e the s i g n a l s at p o s i t i o n s x = Xl (the p r o b e p o s i t i o n ) and x = 1.0 a r e e s s e n t i a l l y r e l a t e d to one a n o t h e r t h r o u g h a d i s t a n c e - v e l o c i t y lag, t h e p h a s e lag b e t w e e n OL and O(x1) i s
44
R.G. WATTS and R. J. SCHOENHALS phase lag =
~2(I
1.4
-Xl) •
(12)
I
I
I
]
I
I
;
I
O,
I
I
t
!
com~h~r solution of equation 19)
®
e~Jation ( t 41
--
I
I 2
I
, 4
I
i
I
G
t B
I
I I0
I
I 12
;
I ~4
I
d 16
I
18
L 20
s i s t e n t l y below the c u r v e , indicating that s o m e additional flow t i m e should be al l o w ed between the p o s i t i o n of the c o n t r o l - t e m p e r a t u r e p r o b e and the outlet. This is probably b e c a u s e eq. (14) a l lows only for delay b et w een the e x t e r n a l - w a l l heat flux and the i n s i d e - w a l l heat flux. Slightly m o r e t i m e is n e c e s s a r y f o r the fluid p a r t i c l e to a c t u a l l y change in t e m p e r a t u r e . This is the r e a son for the inequality of r e l a t i o n (13).
(14)
idL(j~)/~(jf~)i
5. F U R T H E R DISCUSSION It was m en t i o n ed p r e v i o u s l y that f~* = 15 r a dians was u s e d to e v a l u a t e the r e s u l t s given in fig. 2 and in t ab l e 1. S i m i l a r computations w e r e c a r r i e d out for s e v e r a l d i f f er en t v a l u e s of ~*. A l l of the r e s u l t s obtained showed that v a l u e s of Xl,opt obtained f r o m solution of eq. (9) a r e e s I
T
I
[ I I I I I I I
×,=0.8
0.5
S
I
o'+~.
It has b e e n shown [2, 6, 13] that r e s o n a n c e o c c u r s in the Bode plot of in the vicinity of ~ =(2k+ 1)g w h e r e k is an i n t e g e r . It is , t h e r e f o r e , not u n r e a s o n a b l e to expect that the p a r t i c u l a r v al u e ~ = g ( c o r r e s p o n d i n g to k = 0) would a l low an a p p r o x i m a t e p r e d i c t i o n of Xl,opt to be made. The v a l u e s of Xl,opt d e t e r m i n e d f r o m the c o m p u t e r solution of eq. (9) c o r r e l a t e r e a s o n a b l y well with eq. (14) as shown in fig. 3. The plotted data points c o r r e s p o n d to the i n f o r m a t i o n given in table 1. The q u a l i t a t i v e a g r e e m e n t s e e m s quite good in view of the s i m p l i c i t y of the d e r i v a tion of eq. (14). The computed points fall con-
r.o
l
Fig. 3. Optimum probe location and comparison with approximate prediction.
?7
I
]
0.2
F o r f~ = ~, the p e r i o d is tw ic e the holdup t i m e of a fluid p a r t i c l e within the heat exchanger. If ~2 = ~T is i n s e r t e d into eq. (13), the p r e d i c t e d u p p e r l i m i t of Xl,opt b e c o m e s
I
I
0.6
(13)
I
I
0.8
w h e r e Xl,opt denotes the o p t im u m v a l u e of Xl. Solution for Xl,opt g i v e s
I
[
XI ,opt.
--~l~opt) ,
(Xl ,opt) u p p e r l i m i t = 1 - tan-l(~/(cr+~))
I
0
0.4
Xl,opt ~< 1 - tan-l(~2/(a+;~)) ~2
I
1.0
®
tan-1
:
J.2
Si nc e p r o p e r c o r r e c t i v e c o n t r o l a c t i o n should be a ppli ed to fluid p a r t i c l e s b e f o r e they l e a v e the heat e x c h a n g e r , the lag between e r r o r d e te c t io n and c o r r e c t i v e action on the fluid should be l e s s than the t r a n s p o r t a t i o n lag between x = Xl,opt and x = 1.0. Thus ~2
;
I0
15
DIMENSIONLESS FREQUENCY, ~IRADIANS) Fig. 4. Integrand of eq. (9) with b e s t gain f o r v a r i o u s probe locations (if= 0.5, X = 3.0).
FEEDBACK
CONTROL
s e n t i a l l y independent of ~* if ~* is l a r g e r than 6 or 7 radians. An explanation of this is s u g g e s t e d by e x a m i n a t i o n of pl~ts of i W(j~) i2 f o r v a r i o u s v a l u e s of Xl. Fig. 4 shows such a plot f o r the c a s e of ~ = 0.5 and ~ = 3.0. The v a l u e s of C u s e d for t h e s e c u r v e s a r e in all c a s e s the optimum v a l u e s f o r the p a r t i c u l a r v a l u e s of x 1 used. The high value cl o s e to ~ = ~ f o r the c a s e of x 1 = 1.0 is g r a d u a l l y r e d u c e d t o w a r d z e r o as x I a p p r o a c h e s the optimum value of Xl., o,nt -0.7 (see table 1). S i m u l t a n e o u s l y the peak of t h e s e c u r v e s i s n a r r o w e d and shifts to the right. Values of IW(j~2)i2 at high ~2 a r e h a r d ly a f f e c t e d at al l by v a r i a t i o n s in x l . A f t e r Xl is r e d u c e d below x].,opt the peak, which a l ways o c c u r s n e a r ~ : 1 = ~, b e g i n s to b e c o m e b r o a d cau s i n g the a r e a under the c u r v e to i n crease. It is noted that when Xl = 1.0, the l a r g e s t v a l u e s of I W(j~)]2 a r e contributed by f r e q u e n c i e s n e a r ~ = ~. A c a r e f u l study of fig. 4 r e v e a l s that t h e r e is actually a n o th e r s m a l l peak n e a r ~ = 3~ for the c a s e of xl = 1.0. It a p p e a r s that the v a l ue of 0L--L-~(T)can be f u r t h e r r e d u c e d if a second p r o b e is l o c a t e d at p o s i t i o n x 2 w h e r e x 2 = 1 - tan-l(3~/(~+~)) 3~
(15)
Of c o u r s e , the gains a s s o c i a t e d with the two p r o b e s r e q u i r e a p p r o p r i a t e adjustment. T h i s p r o b l e m has b e e n i n v e s t i g a t e d m o r e p r e c i s e l y in ref. [13] fo r five c o m b i n a t i o n s of ~ and ~. It was found that, for the c a s e s c o n s i d e r e d , the o p t imum p o s i t i o n for the second p r o b e is v e r y c l o s e ly a p p r o x i m a t e d by e q . ( 1 5 ) as shown in fig. 5. H o w e v e r , the value of ~ L 2 ~ is d e c r e a s e d in each c a s e by l e s s than 0.1%. It is concluded that,
'"q~_l
I
r
I
I
r
F
J
I
r
[
I
I
I
I
I
I
I
I
1.0-0.~ ~ -
"~'e'm~-~
0.6~_
PJ
computer sok~ion
--
equatlo. (15} m
--
m
m
o
f o r the type of i n l e t - t e m p e r a t u r e v a r i a t i o n s t r e a t e d h e r e , addition of a second p r o b e does not i m p r o v e the s y s t e m p e r f o r m a n c e v e r y m u c h as f a r as the m e a n - s q u a r e e r r o r is c o n c e r n e d . The f o r e g o i n g r e s u l t s of this study indicate that a su b st an t i al r e d u c t i o n in m e a n - s q u a r e outl e t - t e m p e r a t u r e e r r o r in the p r e s e n c e of a r a n domly fluctuating inlet t e m p e r a t u r e is p r o v i d e d by placing a s i n g l e t e m p e r a t u r e p r o b e at Xl,opt. In p r a c t i c a l s i t u a t i o n s it is often n e c e s s a r y to i n s u r e e l i m i n a t i o n of s t e a d y - s t a t e e r r o r s in outlet t e m p e r a t u r e which can r e s u l t f r o m p a r a m e t e r changes or f r o m d r i f t in the steady c o m p o nents of d i s t u r b a n c e s such as inlet t e m p e r a t u r e and heat l o s s e s to the e n v i r o n m e n t . The s t e a d y state oulet-temperature e r r o r resulting from a constant d ev i at i o n f r o m the d e s i r e d value of inlet t e m p e r a t u r e can be d e t e r m i n e d by using eq. (7) with ~-i(s) = ~i/s and applying the final value t h e o r e m to the t r a n s f o r m e d v a r i a b l e ~L(S). Thus I.
.
l i m ~L(t) = l i m S~L(S)=eil 1 t-* ~ s-~0 ~
.
.
I
C
l+Cxl"
i
(16)
T h e r e f o r e , u n l e s s C = 1 / ( 1 - Xl), a s t e a d y - s t a t e e r r o r can o c c u r . Optimum v a l u e s of Xl and C w i l l not g e n e r a l l y sat i sf y this relation. In ref. [13] it is shown that it is p o s s i b l e to e l i m i n a t e the s t e a d y - s t a t e e r r o r by l o c a t i n g a second p r o b e at the outlet and p r o v i d i n g it with i n t e g r a l action, while the p r o b e at Xl,opt p r o v i d es only p r o p o r t i o n a l control. A second t e m p e r a t u r e p r o b e does not a p p r e c i a b l y i m p r o v e the m e a n - s q u a r e o u t l e t - t e m p e r a t u r e e r r o r in the p r e s e n c e of the r a n d o m i n l e t - t e m p e r a t u r e f l u c tuations a s s u m e d for p u r p o s e s of i l l u s t r a t i o n in this paper. H o w e v e r , it does a p p e a r d e s i r a b l e to p r o v i d e a second p r o b e at the outlet of s y s t e m s r e q u i r e d to m a i n t a i n z e r o s t e a d y - s t a t e e r r o r .
6. SUMMARY
1.2--
0.4
45
r
/
X2,o~
OPTIMIZATION
i
I 2
I
I 4
III
I
6
8
I
I
I0
I
I
J2
f
I
14
I
I
16
I
I
18
J
Fig. 5. Optimum location for second probe and comparison with approximate prediction.
I
20
It has been shown that when the outlet t e m p e r a t u r e of a t u b u l ar heat e x c h a n g e r is c o n t r o l l e d in the p r e s e n c e of a fluctuating inlet t e m p e r a t u r e , location of the c o n t r o l - t e m p e r a t u r e p r o b e at a p o s i t i o n x 1,opt can su b st an t i al l y r e d u c e the m e a n - s q u a r e o u t l e t - t e m p e r a t u r e e r r o r . The opt i m u m p o s i t i o n Xl,opt has been c a l c u l a t e d for a wide r an g e of a and ~ values. To a good a p p r o x i mation, this position depends on a s i n g l e p a r a m e t e r (a+k) f o r the c a s e s studied h er e. A p h y s i c a l a r g u m e n t has s u g g e s t e d a s i m p l e r e l a t i o n which c o r r e l a t e s the data quite well. The p o s s i b i l i t y of i m p r o v i n g c o n t r o l by i n s e r t i n g a second t e m p e r a t u r e p r o b e has been in-
46
R.G. WATTS and R. J. SCHOENHALS
vestigated. Based on the p r e s e n c e of a r a n d o m l y fluctuating inlet t e m p e r a t u r e , the p r e s e n t study indicated that a second probe p r o v i d e s v e r y little reduction of the m e a n - s q u a r e o u t l e t - t e m p e r a t u r e e r r o r . It has been shown in ref. [13], however, that u n d e r c e r t a i n conditions s t e a d y - s t a t e e r r o r s in the outlet t e m p e r a t u r e can r e s u l t from changes i n h e a t - e x c h a n g e r p a r a m e t e r s or from d i s t u r b ance v a r i a b l e s . In o r d e r to m a i n t a i n zero s t e a d y state e r r o r in outlet t e m p e r a t u r e u n d e r all conditions it is n e c e s s a r y to locate a second probe at the h e a t - e x c h a n g e r exit.
and PwCwAw -aTw -=qe
" P e - h P ( T w - Tf)
at
(18)
Eqs. (17) and (18) can be n o n d i m e n s i o n a l i z e d by u s i n g the following d i m e n s i o n l e s s v a r i a b l e s : x
X'
L td = -vf - ,
= - -
L'
T=
Tf - Tfi o O-
T w - Two
TfLo _ Trio
,
Ow-
tf = pfcfAf hP '
ACKNOWLEDGEMENT Support of this r e s e a r c h by P u r d u e R e s e a r c h Foundation through XR G r a n t P R F 3015 is g r a t e fully acknowledged.
t ~-d ,
TfLo _ Trio
tw = PwcwAw hP '
= td,
x = tA
tf
tw
,
(19)
q eP eL H = p f c f A f V f ( T f L ° _ Trio ) .
APPENDIX 1 Thus An e l e m e n t a r y length of the t u b u l a r heat exchanger analyzed is shown in fig. 6. F l u i d is heated as it flows through the tube. The following a s s u m p t i o n s a r e made: 1) The fluid i n s i d e the tube is i n c o m p r e s s i b l e . 2) The velocity and t e m p e r a t u r e p r o f i l e s a r e flat with Vf and Tf denoting the fluid velocity and mixed m e a n t e m p e r a t u r e r e s p e c t i v e l y . 3) The velocity does not vary with time. 4) The o u t s i d e - w a l l heat flux is independent of position. 5) Axial conduction in the fluid and in the wall is negligible. 6) The tube t e m p e r a t u r e is independent of r a dial position. E n e r g y b a l a n c e s on the fluid and wall control s u r f a c e s yield aTf pfcfAf - - + p f c f A f V f at
aTf ax' = h P ( T w - T f )
(17)
ae ~@ [Two - Tfio] ~--~-+ ~-x = cr(Ow - @) + Cr\TfL° _ T f i j
(20)
and - -
( TwO -
=--
Tfi°~
aT@w X~H - ~ (Ow - O) - ~ \TfLo - Trio] "
(21)
Separation of the d i m e n s i o n l e s s heat flux and the t e m p e r a t u r e s into t i m e - v a r y i n g and s t e a d y state components, O = Oo(X ) + O(x,-r), 0 w = 0wo(X) + 0w(X, 1-), and H = 7/0 + n(1-), yields a0 aO o ~0 ~7+-~-+-~='~o(~w -
0o)+~(0w -
~"
/Tw°-Tfi°~
- ~J + % T - - i ~ o
= T--~o)
(22) and a0w X X / Two-Trio\ a--~--=~ no + ~ n-h(0wo - 0 o) -h(0w - 0 )-k( TfLo_ T f i J . (23)
,//////~//////J/ Vf
/ / , ~./~/ / / / / /~
!"
S u b t r a c t i o n of the steady state equations gives the d i f f e r e n t i a l equations for the d i m e n s i o n l e s s t r a n sients, namely aO
aO
+~-=
~(0 w - 0 )
(24)
a0 w = '" n - X ( e w - e ) . aT
(25)
and //////~//////////////fl/////// l
l
Fig. 6. Element of a single-fluid heat-exchanger.
Taking the Laplace t r a n s f o r m of these two equations gives
FEEDBACK CONTROL OPTIMIZATION
sO + ~d0 = a(0w - 0)
(26)
and
sOw =-~ ~-X(O-w - 0 ) .
(27)
S i m u l t a n e o u s s o l u t i o n of t h e s e two e q u a t i o n s y i e l d s t h e t r a n s f e r f u n c t i o n s r e l a t i n g 0(s, x) to i n l e t t e m p e r a t u r e and h e a t flux.
g(s, x) Ei(s ) = 0(s, x)
e x p [ - ( s + a - ~-++~X)x]
- ~~X )x]
1 - exp[-(s+a
g(s)
sx-l(s+~+X)
(28)
(29)
C F
= controller gain constant = f a c t o r of i m p r o v e m e n t
g(s) h(s)
= (s/x)(s+~+x) = s+ ~- ~V(s+ x)
h L P Pe q" q~ s' s t T V W(s)
= heat-transfer coefficient = l e n g t h of h e a t - e x c h a n g e r f l u i d - f l o w p a t h = inside tube perimeter = external tube perimeter = i n t e r n a l t u b e - w a U h e a t flux = external tube-wall heat flux = Laplace variable = dimensionless Laplace variable, s't d = time = temperature = velocity = s y s t e m ( c l o s e d loop) t r a n s f e r f u n c t i o n = distance measured from tube inlet = dimensionless distance, x'/L = d i m e n s i o n l e s s h e a t flux = d i m e n s i o n l e s s t r a n s i e n t h e a t f l u x at e x ternal tube surface = dimensionless temperature = dimensionless time-varying temperature = system parameter, td/t w = density = system parameter, td/tf = dimensionless time, t/t d = p o w e r - d e n s i t y s p e c t r u m of i n l e t t e m p e r a ture disturbance = d i m e n s i o n l e s s t r a n s i e n t h e a t flux at i n t e r nal tube surface. = d i m e n s i o n l e s s f r e q u e n c y in r a d i a n s
x H 7/
APPENDIX2 A good e s t i m a t e can b e o b t a i n e d f o r t h e t r a n s f e r f u n c t i o n r e l a t i n g ~the i n s i d e - w a l l h e a t flux to t h e o u t s i d e - w a l l h e a t flux by i g n o r i n g the a x i a l t e m p e r a t u r e g r a d i e n t in t h e l i q u i d a s a f i r s t a p p r o x i m a t i o m F o r t h i s c o n d i t i o n eqs. (26) and (27) become s~ = ~(0w-0)
(30)
s 0 w =-~ ~-~ (8-w-0).
(31)
(9 0 X p a T
and
T h e s e e q u a t i o n s a r e not p r e c i s e l y c o r r e c t , of c o u r s e . T h e i d e a h e r e is to o b t a i n a s i m p l e e x p r e s s i o n w h i c h a d e q u a t e l y d e s c r i b e s the d y n a m i c b e h a v i o r of t h e t u b e w a l l if a s m a l l d e g r e e of a c c u r a c y i s s a c r i f i e d . C o m p a r i s o n of t h e a p p r o x i m a t e and t h e m o r e e x a c t r e s u l t s in figs. 3 and 5 forms a partial justification for this procedure. The dimensionless time-varying inside-wall heat flux ~ i s s i m i l a r to ?? in its d e f i n i t i o n , but it is r e l a t e d to t h e t r a n s i e n t t u b e to fluid t e m p e r a t u r e difference. The expression for ~ is
= ~(0w-0)
•
~(s)
X
s+(r+x
NOTATION A c
= cross-section = specific heat
area
~b
Subscripts d f io Lo w wo 1 2
= = = = = = = =
d e a d t i m e o r holdup t i m e fluid tube inlet, steady state tube outlet, steady state wall wall, steady state indicates measuring probe indicates a second measuring probe
(32)
C o m b i n i n g e q s . (30), (31), and (32) y i e l d s t h e t r a n s f e r f u n c t i o n r e l a t i n g ~(s) and W(s). T h e r e sult is V(s) _
47
(33)
REFERENCES [1] P. Profos, Die Behandlung yon Regelproblemen vermittels des Frequenzganges des Regelkreises, PhD thesis, Technical University, ZUrich, Switzerland (1943). [2] T. J. Williams and H. J. Morris, A Survey of the Literature on Heat Exchanger Dynamics and Control, paper presented at the Joint Automatic Control Conference, AIChE, Cambridge, Mass., September 1960. [3] J . P . H s u and N. Gilbert, T r a n s f e r Functions of Heat Exchangers, AIChE J. 8 (1962) 593.
48
R. G. WATTS and R. J. SCHOENHALS
[4] F. J. Steramole and M. A. Larson, The Dynamics of Flow Forced Distributed P a r a m e t e r Heat Exchangers, AIChE J. 10 (1964) 688. [5] W . J . Y a n g , Transient Heat T r a n s f e r in a VaporHeated Exchanger with A r b i t r a r y T i m e w i s e - V a r i ant Flow Perturbation, J. Heat T r a n s f e r , Trans. ASME, Series C 86 (1964) 133. [6] A. Hempel, On the Dynamics of Steam Liquid Heat Exchangers, ASME P a p e r No. 60-WA-110 (1960). [7] S. R. Grossman, T r a n s f e r Functions of Heat Exchangers and Heat Exchanger P r o c e s s e s Applicable to Nuclear Reactors, AEC R e s e a r c h and Development Report IDO-16486 (1958). [~l H. Christensen, P o w e r - t o - V o i d T r a n s f e r Functions, Argonne National Laboratory, ANL-6385 (1961).
[9] B. T h a l - L a r s e n , Dynamics of Heat Exchangers and Their Models, ASME P a p e r No. 59-A-117 (1959). [10] G. C. Newton, L.A. Gould and J. F. K a i s e r , Analytical Design of Linear Feedback Controls (John Wiley and Sons, Inc., New York, 1957). [11] P. D. Hansen, S.H. Goodhue and A. R. Catheron, Control of Shell and Tube Heat Exchangers, ASME P a p e r No. 59-IRD-14 (1959). [12] S. Zivi and R.W. Wright, Power-Void T r a n s f e r Function M e a s u r e m e n t s in a Simulated SPERT-IA Moderator Coolant Channel, Argonne National Laboratory, ANL-6205 (1960). [13] R . G . W a t t s , Some Aspects of the Dynamics, Stability, and Control of Heat Exchangers, PhD t h e s i s , Purdue University, Lafayette, Ind. (1965).