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PergamonComputers chem EngngVol. 20, No. 5, pp. 483-493, 1996

Copyright 1996 Elsevier Science LtdPrinted in Great Britain. All rights reserved

0 0 9 8 - 1 3 5 4 9 5 ) 0 0 2 1 2 - X 0098-1354196 15.00 + 0.00

U S E O F O RT H O G O N A L T R A N S F O R M AT I O N S I N D ATA

C L A S S I F I C AT I O N R E C O N C I L I AT I O N

MABEL S.~NCHEZ I and Jo s~ ROMAGNOLI 2 t

IPlanta Pi ioto de Ingenier ia Quimica , UN S-CO NIC ET, 12 de Octubre 1842, 8000) Bahia Blanca ,Argent ina ; 2IC I Labora tory of Process System Eng ineer ing, D epartment of Chemical Engineer ing,

Universi ty of Sydney, Sydney, NSW 2006, A ust ra l ia

Received 4 April 1994; inal revision received 18 May 1995)

Abstract In this paper, th e use of ortho gonal factorizations, mo re precisely theQ-R decomposi t ion,to analyze, decompose and solve the l inear and bilinear data reconciliation problem is furtherinvestigated. It is shown th at the decom position provides additional insight in identifying structuralsingularit ies in the system topology, allowing the problem to decompose into lower dimension sub-problem s. Energ y balances are explicit ly considered. Two examples of application are presen ted.

1. INTRODUCTION

I n t h e c o u r s e o f d a i l y o p e r a t i o n o f a c h e m i c a l p l a n t ,i t i s c o m m o n p r a c t i c e t o a d j u s t t h e m e a s u r e m e n t st a k e n f r o m t h e p r o c e s s , s o t h a t r a n d o m m e a s u r e -m e n t e r r o r s c a n b e c o m p e n s a t e d f o r . T h e a p p l i -c a t i o n o f t h e s e m e t h o d s t o l a rg e - s c a l e c o m p l e xc h e m i c a l p l a n t s c r e a t e s p r o b l e m s o f v e r y l a rg ed i m e n s i o n a l i t y w h i c h a r e d i f fi c u lt t o s o l v e . T h i s l a s tf e a t u r e m o t i v a t e d V ~ i c la v e k 1 9 6 9) t o a t t e m p t t or e d u c e t h e s i z e o f th e l e a s t - s q u a r e s p r o b l e m t h r o u g ha n e l e g a n t c l a s s i fi c a t io n o f t h e m e a s u r e d a n d u n m e a -

s u r e d p r o c e s s v a r i a b l e s f o r l i n e a r s y s t e m s . S u c hc l a s s if i c a ti o n a l lo w e d t h e s i ze r e d u c t i o n o f t h e i n i t ia lp r o b l e m a n d i t s e a s i e r s o l u t io n . I n a la t e r w o r kV ~ c l a v e k a n d L o u c k a 1 9 7 6 ) c o v e r e d a l s o t h e c a s eo f b i l i n e a r b a l a n c e s .

A s i m i la r a p p r o a c h w a s u n d e r t a k e n b y M a het al.1 9 7 6 ) in t h e i r a t t e m p t t o o rg a n i z e t h e a n a l y s i s o f

t h e p r o c e s s d a t a a n d t o s y s t e m a t i z e t h e e s t i m a t i o na n d m e a s u r e m e n t c o r r e c t io n p r o b l e m . A s i m p l eg r a p h - t h e o r e t i c p r o c e d u r e f o r s i n g l e c o m p o n e n tf lo w n e t w o r k s w a s d e v e l o p e d . T h e y l a t e r e x t e n d e dt h e t r e a t m e n t , f i rs t t o m u l t i c o m p o n e n t f lo wn e t w o r k s K r e t s o v a l i s a n d M a h , 1 98 7) a n d t h e n t ot h e g e n e r a l i z e d p r o c e s s n e t w o r k s i n c l u d i n g e n e rg yb a l a n c e s a n d c h e m i c a l r e a c t i o n s K r e t s o v a l i s a n dM a h , 1 9 8 8 a, b ) .

R o m a g n o l i a n d S t e p h a n o p o u l o s 1 9 80 ) p r o p o s e da n e q u a t i o n o r i e n t e d a p p r o a c h . S o l v a b i l i ty o f th en o d a l e q u a t i o n s w a s e x a m i n e d a n d a n o u t p u t s e ta s s i g n m e n t a l g o r i t h m S t a d t h e r ret al., 1974) wase m p l o y e d t o c la s s if y s i m u l t a n e o u s l y m e a s u r e d a n du n m e a s u r e d v a r i a b e s .

t To whom all correspo ndenc e should be addressed.

M o r e r e c e n t l y, a g e n e r a l t r e a t m e n t u s i n g p r o j e c -t i o n m a t r i c e s w a s p r o p o s e d b y C r o w eet al. 1983)f o r li n e a r sy s t e m s a n d e x t e n d e d l a t e r C r o w e , 1 98 6,1 9 89 ) f o r b i l i n e a r sy s t e m s . C r o w e s u g g e s t e d a u s e f u lm e t h o d t o d e c o u p l e t h e m e a s u r e d v a r i ab l e s f r o mt h e c o n s t r a i n t e q u a t i o n s , u s i n g a p r o j e c t i o n m a t r i xt o e l i m i n a t e t h e u n m e a s u r e d p r o c e s s v a r i a b l e s .O r t h o g o n a l f a c t o r i z a t i o n w e r e f i r st u s e d b y S w a r t z1989) i n t he con tex t o f success ive l i nea r i za t i on

t e c h n i q u e s t o e l i m i n a t e th e u n m e a s u r e d v a r i a b l e sf r o m t h e c o n s t r a i n t e q u a t i o n s .

I n t h i s p a p e r , t h e u s e o f o r t h o g o n a l f a c t o r i z a t i o n s ,m o r e p r e c i s e l y t h eQ - R d e c o m p o s i t i o n , t o a n a l y z e ,d e c o m p o s e a n d s o l v e t h e l i n e a r a n d b i l i n e a r d a t ar e c o n c i l i a t io n p r o b l e m i s f u r t h e r i n v e s t i g a t e d . As e q u e n c e o f s i m p l e e x p r e s s i o n s t o b e a p p l i e d i ni n s t r u m e n t a t i o n a n a l y s i s a n d d a t a r e c o n c i l i a t i o n i so u t l i n e d a n d t h e y a r e o b t a i n e d u s i n g s u b - p r o d u c t s o fQ-R f a c t o r i z a t io n s . F u r t h e r m o r e , t h e u s e o f t h i sm e t h o d , w h e n e n e rg y b a l a n c e s a r e i n c l u d e d i n t h ese t o f p rocess cons t ra in t s , i s a l so d i scussed . Resu l t so f t h e a p p l i c a t i o n f o r l i n e a r a n d b i l i n e a r s y s te m s a r ep r o v i d e d i n t e rm s o f tw o f l o w s h e e t i n g e x a m p l e s , o n eo f t h e m b e i n g a n e x i s t i n g o p e r a t i n g p l a n t .

2. LINEAR CASE

2.1. Problem statement

I n t h e a b s e n c e o f s y s t e m a t i c e r r o r s , w e c o n s i d e rt h e f o ll o w i ng m e a s u r e m e n t m o d e l

~ = x + e 1 )

w h e r e x i s a g x 1) v e c t o r o f m e a s u r e d v a r i a b l e s ,i s a g 1 ) vec to r o f me asur ed va lues a nd e is ag x 1) v e c t o r o f r a n d o m e r r o r s . T h e m e a s u r e m e n t

483

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484

e r r o r s a r e a s s u m e d t o b e n o r m a l l y d i s t r i b u t e d w i t hz e r o m e a n a n d k n o w n v a r i a n c e - c o v a r i a n c e m a t ri xt tt x. A se t o f m l i nea r ba l an ce equ a t ions fo r as t e a d y - s t a t e p r o c e s s c a n b e w r i t t e n a s :

A I x + A 2 u = 0 (2 )

w h e r e u i s a ( n x 1) v e c t o r o f u n m e a s u r e d v a r i a b l e sa n d A l m g ) , A2 m u) a r e m a t r i c e s o f k n o w nc o n s t a n t s .

I n t h e p r e s e n c e o f m e a s u r e m e n t e r r o r s, t h eb a l a n c e e q u a t i o n s a r e n o t s a t i s f i ed e x a c t l y. To c o m -p e n s a t e f o r r a n d o m m e a s u r e m e n t s e r r o r s, a g e n e r a ld a t a r e c o n c i l i a t i o n p r o c e d u r e m u s t s o l v e t h e f o l l o w -i n g l e a s t - s q u a r e s p r o b l e m :

m i n ( x - i ) r q J x l ( X - i ) ( 3)

s . t . A~x +A 2u = 0 .

M. SANCHEZand J. ROMAGNOLI

o f m a t r i x A 2 i s e a s i l y a c c o m p l i s h e d . F r o m o n e c o d ei n s t r u c t i o n , m a t r i c e s Q . , R . a n d t h e p e r m u t a t i o nm a t r i x l I . a r e o b t a i n e d , s u c h t h a t :

A21-I. = Q .R . (7)

w h e r e Q . a n d R . c a n b e d i v i d e d in t o :

a s i s i n d i c a t e d i n t h e A p p e n d i x , w i t h r u = r a n k ( A 2 ) =r a n k ( R . , ) .

I n t h e s a m e w a y, t h e u n m e a s u r e d p r o c e s s v a r i -a b l e s c a n b e p a r t i t i o n e d i n t o t w o s u b s e t s:

[ - I .L U n r . J

n o t a t i o n , t h e b a l a n c e e q u a t i o n s ( 2 )i t h t h i sb e c o m e :.2. Solution usingQ - R factorizat ions

S e v e r a l t e c h n i q u e s w e r e p r o p o s e d t o r e d u c e t h es i z e o f t h e r e c o n c i l i a t i o n p r o b l e m b y e l i m i n a t i n gu n m e a s u r e d v a r i a b l e s ( V ~ ic l av e k , 19 69 ; M a het al.,1 97 6; R o m a g n o l i a n d S t e p h a n o p o u i o s , 1 9 8 0) . A ne l e g a n t a n d u s e f u l w a y o f o b t a i n i n g t h i s d e c o m p o -s i t i o n w a s d u e t o C r o w eet al. ( 1 9 8 3 ) . T h e m e t h o dw a s b a s e d o n t h e u s e o f a p r o j e c t i o n m a t r i x P. I t w a sd e f i n e d s u c h t h a t p r e - m u l t i p l y i n g m a t r i x A 2 w i th Py ie lds :

PA2 = 0 (4)

w h e r e t h e r o w s o f P s p a n t h e n u l l s p a c e o f A 2 , a n d

t h u s t h e u n m e a s u r e d v a r i a b l e s a r e e l i m i n a t e d . T h ec o n s t r a i n e d l e a s t - s q u a r e s p r o b l e m f o r t h e o v e r a l lp l a n t ( 3 ) c a n b e r e p l a c e d n o w b y t h e e q u i v a l e n t t w o -p r o b l e m f o r m u l a t io n .

( i ) Least-squares estimation ofx:

m in (x - ~)TqJx I(X -- ~) 5 )

s . t . G x = 0 .

T h e s o l u t i o n o f th i s p r o b l e m i s g i v e n b y :

~ = i - q J , G r G W x G r )- lG i (6)

w h e r e G =P*AI.( i i ) Estimation of u using ~ and the balance equa-

tionsB o t h t h e a p p l i c a t i o n o f C r o w e s m a t r i x p r o j e c t io n

m e t h o d a n d t h e s o l u t i o n o f t h e r e d u c e d l e a s t -s q u a r e s p r o b l e m c a n b e s i m p l i f i e d b y u s i n gQ - Ro r t h o g o n a l t r a n s f o r m a t i o n s . A b r i e f d e s c r i p t i o n o fQ - R t r a n s f o r m a t i o n i s i n c l u d e d i n t h e A p p e n d i x .T h e a p p l i c a ti o n o fQ - R f a c t o r i z a t i o n s t o d i f f e r e n ts t a g e s o f t h e d a t a r e c o n c i l i a t i o n p r o b l e m ( 3 ) fo l l o w s .

2.2.1. El iminat ion o f unmeasured variables.B y a p p l y i n g s o f tw a r e p a c k a g e s f o r m a t r i x c o m p u -

t a t io n , s uc h as M AT L A B , t h eQ - R d e c o m p o s i t i o n

The Q,~ ma t r ix i s such tha t i t s rows span the nu l lspace o f A~ . Tha t i s :

TQ . 2 A 2 - O (11)

s o Q ,~ w o r k s a s t h e p r o j e c t i o n m a t r i x P p r o p o s e d b yC r o w e . I t d i ff e r s , h o w e v e r , f r o m P i n t h a t t h en u m e r i c a l v a l u e s h a v e n o p h y s i c a l s ig n i f ic a n c e .P r e - m u l t i p l y i n g th e s y s t e m o f e q u a t i o n s ( 2 ) b y Q rt h e u n m e a s u r e d v a r i a b l e s a r e e l i m i n a t e d .

R e m a r k 1 . I n M A T L A B , t h eQ - R d e c o m p o s i t i o nc a n b e e a s i l y a c c o m p l i s h e d t h r o u g h o n e c o d ei n s t r u c t io n . F u r t h e r m o r e , Q . r i s o b t a i n e d a s s u b -p r o d u c t o fQ - R A 2 )w i t h o u t e x t r a c o m p u t i n g e f f o r tfo r t he use r.

2.2.2. Least-squares estimation o f x.T h e f o l l o w -i n g p r o b l e m m u s t b e s o l ve d :

m i n ( x - i ) r ~ x l ( X - i ) ( 12 )

s . t . G x x = 0

w h e r e

a x T= Q, :A, . (13)

Z e r o c o l u m n s o f G x c o r r e s p o n d t o n o n - r e d u n d a n tm e a s u r e m e n t s ; t h e o t h e r s b e l o n g t o r e d u n d a n tm e a s u r e m e n t s a s i n d ic a t e d b y C r o w eet al. (1983)a n d C r o w e ( 1 98 9 ).

We c a n s o l v e t h i s c o n s t r a i n e d p r o b l e m b y t h eL a g r a n g i a n a p p r o a c h ; h o w e v e r , u s i n gQ - R f ac to r i -z a t i o n s , it c a n b e t r a n s f o r m e d i n t o a n u n c o n s t r a i n e dp r o b l e m . T h e c o n s t r a i n t s e q u a t i o n s m a y b e u s e d t os o l v e , f u n c t i o n a l l y, f o r a s m a n y v a r i a b l e s a s t h e r ea r e c o n s t ra i n t s . T h e n a n u n c o n s t r a i n e d l e a s t - s q u a r e s

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Orthogonal transformations in data classification-reconciliation

p r o b l e m i s s o l v e d t o e s t i m a t e t h e r e m a i n i n g v a r i -a b l e s . T h e p r o c e d u r e i s a s f o ll o w s .i ) C o m p u t a t i o n o f t h e g e n e r a l so l u t i o n o f t h e u n d e -

t e r m i n e d s y s t e m G , x = 0 ) :A Q - R o r t h o g o n a l f a c t o r i z a t i o n o f G . g i v e s

Q~, R~, H~ and a l l ows one to ob t a inQ~, Q w R~, R~:, x ,~ , Xg . . . such tha t

G.I-Ix=Q~R~ 14)

Q~=[Q~ Q~I] R = J R , ,R. ~] 15 )

1-ITx [ xgX~rx] 16 )

w i t h r ~ = r a n k R ~ l ) = r a n k G 0 . T h e g e n e r a l s o l u t i o no f t h is p r o b l e m i s :

Xr,= - - R~ lR~2 x~_ x 1 7 )

wh ere xg_~. i s a rb i t r a ry.i i) F o r m u l a t i o n o f th e u n c o n s t r a i n e d p r o b l e m :

U s i n g th e p r e v io u s re s u l ts , t h e v e c t o r x - i ) f r o mt h e o b j e c t i v e f u n c t io n i s m o d i f i e d , a s e q u a t i o n 1 8 )i n d i c a t e s :

- - ix - i ) = [ I ~ , I ~ ] xg ,

= ( I . 2 - 1 ~ , R ~ ' R ~ 2 ) X g _ , - i 18)

w h e r e

l l - I ~ = [ I . , I , : ] I = I .~ - I ~R ~ , ' R~ 2 . 19)I i s a g x g ) i den t i t y ma t r ix an d [ i s a [g x g - r 0 ]m a t r i x w i th i n d e p e n d e n t c o l u m n s .i i i) E s t i m a t i o n o f x :

T h e s o l u t i o n o f t h e u n c o n s t r a i n e d p r o b l e m i s :

_ ~ = ( F ~ ; ~ i ) - ~ F ~ ; ~ 20)

wi th t he va lue o f i g_r, , one can so lve fo r x r. us ing17) .

R e m a r k 2 . T h e p r e v i o u s a p p r o a c h h a s t w o a d v a n -t a g e s : i t a v o i d s t h e d i r e c t u s e o f t h e c o n s t r a i n t s i n tot h e l e a s t - s q u a r e s e s t i m a t i o n a n d r e d u c e s t h e n u m b e r

o f v a r i a b l e s t o b e s i m u l t a n e o u s l y e s t i m a t e d . T h ep r o c e s s o f e l i m i n a t i n g p a r t o f t h e v a r i a b l e u s i n g t h ec o n s t r i a n t s i s e a s i l y a c c o m p l i s h e d b y m e a n s o fQ - Rf a c t o r i z a t i o n s .

2.2.3. Estim ation of unmeasured variables.M a t r i x R . i n t h eQ - R f a c t o r i z a t i o n o f t h e A z m a t r i xc o n t a i n s t h e t o p o l o g i c a l i n f o r m a t i o n a b o u t t h es y s t e m i n t e r m s o f t h e a v a i l a b l e m e a s u r e m e n t s .

i ) i f r a n k R o ) = r. = n , a l l u n m e a s u r e d p r o c e s sv a r i a b le s a r e d e t e r m i n a b l e f r o m t h e a v a i l a b l ei n f o r m a t i o n .

485

i i ) i f rank Ru ) = ru < n, the n a t least n - ru + 1v a r i a b l e s c a n n o t b e c a l c u l a t e d f r o m t h e a v a i l-a b l e i n f o r m a t i o n .

F o r c a s e i i ) , t h e e s t i m a b i l i t y c o n d i t i o n o f u n m e a -s u r e d v a r i a b l e s c a n b e e x p r e s s e d i n t e r m s o fQ - Rd e c o m p o s i t i o n r e s u l t s o f A2 m a t r i x . T h e p e r m u -

t a t i o n m a t r i x F In a l l o w s t h e d i v i s i o n o f t h e u n m e a -sured p rocess va r i ab l e s i n to subse t s u r. and U. - ru .T h e s u b s e t u . _ r. c o r r e s p o n d s t o n - r u) i n d e t e r m i n -a b l e u n m e a s u r e d p r o c e s s v ar i a bl e s . R e g a r d i n g t h es u b s e t u ~ ., s o m e v a r i a b l e s c a n b e c a l c u l a t e d u s i n go n l y t h e r e c o n c il e d m e a s u r e m e n t v a l u es a n d s o m ed e p e n d a l s o o n t h e a s s u m p t i o n o f t h e u . _~ u v a r i a b l e s .T h i s r e s u l t is o b t a i n e d b y p r e - m u l t i p l y i n g t h e s y s t e me q u a t i o n s 1 0 ) b y Q r a n d r e o r d e r i n g t h e fi rs t r.e q u a t i o n s o f s y s te m 2 1 ) a s i n d i c a t e d b e l o w :

l r x lu,A~ Ru~r 2 u~, = 0 21)

Q,2A1 0I u

1 Tu~ = - R u , Qu lAlx - R~XR u 2 u . . . . 22)

To c l a ss i fy va r i ab l e s i n subse t u~ ., i t i s neces sa ry t ol o o k a t t h e r o w s o f t h e m a t r i x :

R w = R u l I R u 23)

T h e f o l lo w i n g c a n b e s t a t e d :

i ) A va r i ab l e i n subse t u r. i s sa id t o be de t e rm in-a b l e i f t h e c o r r e s p o n d i n g r o w i n t h eR wmat r ix i s z e ro .

i i) A v a r i a b l e i n s u b s e t u ~ i s s a i d i n d e t e r m i n a b l eo t h e r w i s e .

R e m a r k 3 . T h e c l a s s if i c a ti o n m a t r i x a n a l y s i s h a sb e e n d o n e b y C r o w e 1 9 8 9) in te r m s o f t h e p r o j e c -t i o n m a t r i x P. H e r e , s i m i l a r r e s u l ts a r e o b t a i n e d i n ac l e a r e r w a y u s i n gQ - R f a c t o r i z a t i o n . I n M AT L A B ,i n s p e c t i o n o f m a t r i x R m c a n b e e a s i l y a c c o m p l i s h e di n a n a u t o m a t i c w a y.

3 . N O N - L I N E A R C S E

3.1. Pro blem statement

L e t u s n o w c o n s i d e r a p r o c e s s c o n t a i n i n g K u n i t sd e n o t e d b y k = 1 . . . , K , a n d J o r i e n t e d s t r e a m sj = l , . . . J , w i th C c o m p o n e n ts c = 1 , . . . C . P l a ntt o p o l o g y i s r e p r e s e t n e d b y t h e i n c i d e n c e m a t r i x L ,w i t h r o w s c o r r e s p o n d i n g t o u n i t s a n d c o l u m n s t os t r e a m s . T h e nlkj= f s t r e a m j e n t e r s n o d ek, lkj = - 1i f s t r ea m j l e av es mo de k , l k j = 0 o the rw i se .

T h e g e n e r a l p r o c e s s c o n s t r a i n t s a r e a s f o l l o w s .To t a l m o l a r b a l a n c e s :

E l ~ + E E S k , r c ~ k ,rc=O. 24)j r c

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Co m po nen t mo la r ba l anc es : Tab le 1 . Ca tegori esof com ponent and energy flowrates

Category F m/ T

E l ~ j F j m / c E Sk r c~ kr =0 (25) M M2 U M

j r 3 M U U

E n e r g y b a l a n c es :

E IkjFjhj+EHk'r+qk=O 2 6 )I r

N o r m a l i z a t i o n e q u a t i o n s:

E ~rnjc - ~ = 0 (27)c

wh ere ~ i s t he t o t a l m ola r f l ow ra t e o f s t r eam j , Sk, rci s t h e c o e f f i c i e n t o f t h e s t o i c h i o m e t r i c m a t r i x ( C r o w eet a l . , 1 9 83 ) o f c o m p o n e n t c f o r r e a c t i o n r i n u n i t k ,

~k , i s t he ex t en t o f r ea c t ion r i n un i t k , m j . c i s t hem o l a r f r a c t i o n o f c o m p o n e n t c in s t r e a m j , h j r e p r e -s e n t s t h e s p e c if i c e n t h a l p y o f s t r e a mj , Hk . r i s t het o t a l h e a t o f r e a c t i o n r i n u n i t k a n d d e p e n d s o nSg, r,a n d ~ . , q k i s t h e v e c t o r o f p u r e e n e rg y f l o w s o f u n i tk .

To c o m p e n s a t e f o r r a n d o m m e a s u r e m e n t s e r r o r s ,t h e d a t a r e c o n c i l i a t i o n p r o c e d u r e m u s t s o l v e t h ef o l l o w i n g l e a s t - s q u a r e s p r o b l e m :

m i n ( y - y ) r U j y ~ ( y - y ) ( 28 )

s . t. W ( y, z ) = 0

w h e r e W ( y , y ) r e p r e s e n t s a s u b s e t o f b a l a n c e s a n dn o r m a l i z a t i o n e q u a t i o n s ; y a n d z a r e t h e v e c t o r s o fm e a s u r e d a n d u n m e a s u r e d v a r i a b le s fo r b i l in e a rp r o b l e m s .

3 .2 . So lu t ion us ingQ - R f a c t o r i z a t i o n s

A s c h e m e f o r t h e s o l u t i o n o f t h e b i l i n e a r r e c o n c il -i a t i o n p r o b l e m i s p r o p o s e d b y C r o w e ( 1 98 6 ). I n t h i sw o r k , t h e a p p l i c a t i o n o fQ - R f a c t o r i z a t i o n s w i t h i nt h i s s c h e m e i s a n a l y z e d . F u r t h e r m o r e , t o t a l f l o w -r a t es a r e s e p a r a t e l y c o n s i d e r e d fr o m c o m p o n e n t a n de n t h a l p y f l o w r a t e s a n d e n e rg y b a l a n c e s a r e e x p l i c i tl yt a k e n i n t o a c c o u n t .

F o l l o w i n g C r o w e s p r o c e d u r e , t h e s o l u ti o n t op r o b l e m ( 2 8 ) c a n b e a c c o m p l i s h e d in f o u r s t e p s .

3 .2 .1 . M odif ica t ion of b i l inea r cons t r a in ts .T h el i n e ar t e r m s i n W ( y , z ) = 0 r e m a i n u n ch a n g e d .B i l i e n a r t e r m s a r e r e w r i t t e n u s i n g t h e c l a s s i f ic a t i o no f c o m p o n e n t a n d e n t h a l p y f lo w r a te s . C o m p o n e n tf l o w r a t e s a r e d i v i d e d i n t o t h r e e c a t e g o r i e s d e p e n d -i n g o n t h e c o m b i n a t i o n o f t o t a l f l o w r a t e s a n dc o n c e n t r a t i o n m e a s u r e m e n t s i n t h e s t r e a m a s s h o w n

i n Ta b l e 1 , w h e r e M a n d U in d i c a t e m e a s u r e d a n du n m e a s u r e d v a r i a b l e s r e s p e c t i v e l y.

F o r e n e rg y b a l a n c e s , a n e x p r e s s i o n o f s p e c i f i ce n t h a l p y a s fu n c t io n o f t e m p e r a t u r e ( T ) i s o b t a i n e db y u s i n g a t h e r m o d y n a m i c p a c k a g e , f o r a s t r e a mw i t h c o n s t a n t s t e a d y - s t a t e s i m u l a t e d v a l u e s o f p r e s -s u r e a n d c o m p o s i t i o n . Ta b l e 1 a l s o r e p r e s e n t s t h ec a t e g o r i z a t i o n o f e n e rg y f l o w r a t e s w h e n t h i sa p p r o a c h i s a p p l i e d .

C o m p o n e n t / e n e r g y b a l a n c e s :

B l f+ B2 Vd + n3v = 0 . (29)

N o r m a l i z a t i o n e q u a t i o n s :

E I f E 2 V d E 3 v E 4 F M E s F v = O (30)

w h e r e f i s t h e v e c t o r o f c o m p o n e n t o r e n t h a l p yf l o w r a t e s o f C a t e g o r y 1 ; d i s t h e v e c t o r o f m e a s u r e dc o n c e n t r a t i o n s a n d c a l c u l a t e d s p e c if i c e n t h a l p y f o rc o m p o n e n t o r e n t h a l p y f l o w r a t e s o f C a t e g o r y 2 ; v i st h e v e c t o r o f c o m p o n e n t o r e n t h a l p y f l o w r a t e s o fC a t e g o r y 3 , e x t e n t o f r e a c t i o n , u n k n o w n p u r ee n e rg y f l ow s , e t c ; FM s t a n d s f o r m e a s u r e d t o t a lf l o w r a t e s a n dF v f o r t h e u n m e a s u r e d o n e s ; V r e p r e -s e n t s th e d i a g o n a l m a t r i x o f u n m e a s u r e d t o t a l f l o wr a t e s f o r c o m p o n e n t a n d e n t h a l p y f l o w r a t e s o fC a t e g o r y 2 . T h e n u m b e r o f e n t r i e s f o r a s tr e a m i n Vi s e q u a l t o t h e n u m b e r o f e l e m e n t s o f d c o r r e s p o n d -ing to t h i s s t r eam.

T h e m e a s u r e d v a r i a b l e d i s r e p l a c e d b y a c o n s i s t -e n t m e a s u r e d v a l u e p l u s t h e c o r r e c t i o n t e r m6 d :

d = 0] + 6d) (31)

a n d a n e w v a r i a b l e 0 i s d e f i n e d a s :

0 = V6d. (32)

T h e t e r m s t h a t c o n t a i n v a r i a b l e d i n e q u a t i o n s( 2 9) a n d ( 3 0 ) a r e r e p l a c e d b y :

BE V d = BE 0 B2 V d

E 2 V d = E 2 0 E 2 V d . (33)

I n o r d e r t o d i s p l a y u n m e a s u r e d t o t a l f lo w r a t e s fo rs p e c if i c f l o w r a t e s o f C a t e g o r y 2 ( F u z) f r o m e q u a t i o n s(33) , B4 and E6 ma t r i ce s a re de f ined a s :

B4(d)Fu2 = B2 Vd

E6(d)Fu2 = E2 Vt]. (34)

E a c h c o l u m n o f B 4 a n d E 6 c o n t a i n s t h e s u m o f t h ec o l u m n s o f B 2 a n d E 2 f o r t h e s t r e a m m u l t i p l i e d b y

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t h e c o r r e s p o n d i n g c o n s i s t en t c o n c e n t r a t i o n o r s p e c i -f ic en tha lpy. T o gro up a ll unm easured to ta l f low-ra te s , ze ro co lu mns a re added to B4 and F6 i f it i snecessa ry. New B5 and E7 a re ob ta ined such tha t :

Bs(d )Fu = B2 V~I

ET(d )Fu = E2 Vd. (35)Diffe re n t l inea r ly independe nt se t s o f p rocess con-s t ra i n ts c a n b e f o r m u l a t e d . O n e o f t h e m m a y i n c lu d et o t a l m a s s b a l a n ce s , C - 1 c o m p o n e n t b a la n c e s,ene rg y ba lances and norma l iza t ion equa t ions .A n o t h e r o n e m a y c o n t a i n al l c o m p o n e n t a n d e n e rg yba lances and norma l iza t ion equa t ions . Us ing pre -v ious express ions , the la s t se t can be w r i t ten a s :

= 0 3 6 )E4 EI E2 E8 E3

whereE8 = E7 + Es . I f w e cons ide r the ad jus tme ntso f t o t a l f lo w r a t e s 8 F a n d t h e c o m p o n e n t a n dentha lp y f lows dr, the gene ra l r econc i l ia t ion prob lemcan be s ta ted a s :

m i n r 1 r6 FMUklFM O FM -{- ~ T I~I l o T -~- o Ttt l o I O )

s . t . [Bn B B33] = _ O1 BIE4 El

w h e r e

SFM] Ox B1 B2

g r u , g t , g 0 a n d g , a r e t h e v a r i a n ce - c o v a r ia n c ematr ice s fo r FM, f , 0 and d . g 0 i s de f ined a s :

g 0 = v g , V. ( 3 8)

Re m a r k 4 . By s e p a r a ti n g t o t a l f l ow r a t e s f ro m c o m -ponent and en tha lpy f lowra te s , c lea re r express ionsfor ins t rumenta t ion ana lys is and da ta r econc i l ia t ionca lcu la t ions may be ob ta ined .

3.2.2. Elimination of unm eas ure d variables.T h e s e v a r i a b l e s a r e e l i m i n a t e d f r o m t h e m o d i f i e dc o n s t ra i n t s u si n g Q - R o r t h o g o n a l t r a n s f o r m a t i o n sas follows.

487

(i) A Q - Rdecom pos i t ion of ( rob nb) ma t r ixB isaccompl ished , then :

B33II, = [Q B][ RB ]

=[QB, QB2][ RB 7 2 ] (39)

w h e r e , r , = r a n k ( RB1 ) a n dQB~is such that i ts rowsspan the nu l l space of B(3 so:

Q BrB33 = 0. (40)

( i i) Equ a t ion (37) is mul t ip l ied byQB~ so theunmeasured va r iab le s v a re e l imina ted and the p ro-cess constra ints are def ined as:

QB ~B nt+ QBrB22Fu = QBre. (41)

( ii i) A new (maxha) matr ix D i s de f ined and equa -t ion (41) is rewrit ten as:

QBrB nt+ DFu= QBre . (42)

(iv) A Q - R o r t h o g o n a l t r n s f o r m a t i o n i s p e r f o r m e do n m a t r i x D :

D I-IF,= [ QD ][ RD ]

4 3[QD 1 QD2] 0

w h e r e r f = r a n k ( RD t ) a n dQD r is such that i ts rowsspan the nu l l space of D r, then th e p rocess con-s t r a in ts can be r educed to :

QD ~QB~Bllt= QD ~Q Bre. (44)

3.2.3. Estimation of measured variables andunmeasured total flow rates.Afte r e l imina t ing unmeasured va r iab le s , the

reconc i l ia t ion of measure d va r iab le s and the e s t ima-t ion of unm easured to ta l f low ra tes a re accom -pl i shed by an i te ra t ive p rocedure :

Step 1:

U s i n g a n e s t i m a t i o n o f u n m e a s u r e d t o t a l f lo wrates, t lJ0 is evaluat ed. Th e follow ing l inear reconcil-ia t ion prob lem needs to be so lved :

m i n t r g t l t ( 45 )

s . t . G t t = b

w h e r e

Gt= QD rQB~B , ,

b = QD ~QB re (46)

wi th so lu t ion g iven by

i = Wt Gtr(GtWt G t )-~b. (47)

W h e n t h e Q - R d e c o m p o s i t i o n o f t is applied toes t ima te [ , the ca lcu la t ion has the same advantagesas ind ica ted for the L inea r Case .

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Step 2:

T h e e s t i m a t i o n o f u n m e a s u r e d t o t a l f l o w r a t e s i sd o n e b y u s i n g t h eQ - R o r t h o g o n a l d e c o m p o s i t i o n o fm a t r i x D . E q u a t i o n ( 4 2) c a n b e w r i t te n a s :

Q B f B u i + [ Q D I Q D 2 ]

0 0 kFuu,~_.,tj

w h e r e

Fv,: qI = I I ~ .F o . 49)

[ Fv,,~, i J

T h e s u b s e tFv.d ,:c o r r e s p o n d s t o t h e i n d e t e r m i n a b l et o t a l f l o w r a t e s . R e g a r d i n g t h e s u b s e tFv, :n o t h i n gc a n b e s a i d , s i n ce s o m e o f t h e s e v a r i a b l e s c a n b ec a l u l a te d d i r ec t l y f r o m t h e m e a s u r e m e n t s a n d s o m ed e p e n d s o n Fu,d_,. F u r t h e r i n f o r m a t i o n t o c l a s s i f yv a r i a b l e s i n s u b s e t F c a n b e o b t a i n e d p r e -m u l t i p l y i n g e q u a t i o n ( 4 8 ) b yQ D r a n d w r i t i n g t h ev e c t o r Fv,:i n te r m s o f t h e o t h e r v a r i a b l e s :

F v, = R D f l Q D (Q B ~ e - R D f l Q D ( Q B f B u t

- RD { IRD2 Fu ~ ~f. (50)

N o t i c e t h a t i f t h e l a s t t e r m i n t h e R H S o f e q u a t i o n(50) i s z e ro , a l l t heFu,:c a n b e c a l c u l a t e d f r o m t h ea v a i l a b l e i n f o r m a t i o n . I n o r d e r t o c l a s s i fy t h e v a r i -ab l e s i n Fv,:a m a t r i x RI i s de f ined a s :

R~F= R D ~ I R D 2 (51)

a n d t h e f o l l o w i n g c a n b e s t a t e d :

( i ) a v a r i a b l e in s u b s e t F % i s d e t e r m i n a b l e i f t h ec o r r e s p o n d i n g r o w i nRFi s z e ro ;

( i i ) a va r i ab l e i n subse tFur: i s i n d e t e r m i n a b l eo t h e r w i s e .

A t t h i s p o i n t t h e v e c t o r F v c a n b e d i v i d e d i n t o :

F u = LFu J

w h e r e F vd is t h e f e -d i m e n s i o n a l v e c t o r o f d e t e r m i n -a b l e t o t a l f l o w r a t e s ( fe

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a r e d e t e r m i n a b l e i f t h e t o t a l f l o w r a t e o f t h e s t r e a m i sm e a s u r e d o r d e t e r m i n a b l e . O t h e r w i s e , t h e y a r ei n d e t e r m i n a b l e . T h e i n c l u s io n o f i n t e n s i i v e p r o c e s sc o n s t r a i n t s c a n c h a n g e t h e c l a s s if i c a ti o n b u t i t i s n o to n t h e s c o p e o f t h i s p a p e r.

3.3. Fur ther discussion on energy balances

T h r o u g h t h e a b o v e d i s c u s s i o n , s im p l i f i e d e x p r e s -s i o n s o f s t r e a m s p e c i f i c e n t h a p y a s f u n c t io n o f t e m -p e r a t u r e a r e u s e d . T h e y h a v e t o b e u p d a t e d d u r i n gp r o c e s s o p e r a t i o n t o c o n s i d e r c h a n g e s i n s t e a d y -s t a t e c o m p o s i t i o n s .

T h e a p p l i c a t i o n o f a m o r e p r e c i s e e x p r e s s i o n f o re n t h a l p y , a t l e a st , a s a f u n c t io n o f t e m p e r a t u r e a n dc o m p o s i t i o n r e q u i r e s a n e w c a t e g o r i z a t i o n o fe n t h a l p y f l o w r a t e s . T h e y c a n b e d i v i d e d i n t o t h r e ec a t e g o r ie s d e p e n d i n g o n t h e c o m b i n a t i o n o f t o t a lf l o w r at e s , c o m p o s i t i o n a n d t e m p e r a t u r e m e a s u r e -m e n t s , a s i n d i c a t e d in Ta b l e 2 .

T h e p r o b l e m a r is e s fo r t h e l as t m e a s u r e m e n t c o m -b i n a t i o n . I t i s d u e t o t h e d i f fi c u l ty o f a d j u s t i n gt e m p e r a t u e m e a s u r e m e n t v a l u e s f or s t re a m s w h i c hc o m p o s i t i o n s a r e u n m e a s u r e d o r p a r t i a l ly m e a s u r e d .I n t h is c o n t e x t , t h e t e m p e r a t u r e o f a s t re a m j m a y b ea d j u s t e d o n l y f o r t h e f o l l o w i n g c o n d i t io n s :

- - a l l c o m p o n e n t m o l a r f r a c ti o n s a re u n m e a s u r e d ;

- - ru l e o f mix ing : h j = m~chjc;

- - h j c i s a p p r o x i m a t e d a s a l i n e a r f u n c t i o n o ft e m p e r a t u r e f o r t h e s t e a d y - s ta t e o p e r a t i o n

r a n g e .T h e f o l l o w i n g s o l u t io n s c h e m e c a n b e i m p l e m e n t e d :

i ) e s t i m a t i o n o f u n m e a s u r e d t o t a l f l o w r a t e s a n du n m e a s u r e d s p e c i e s f l o w r a t e s f o r s t r e a m sw i t h u n m e a s u r e d t e m p e r a t u r e s ;

i i ) s i m u l t a n e o u s e l i m i n a t i o n o f u n m e a s u r e d v a r -i a b le s . A t w o - s ta g e p r o c e d u r e o f d e c o m p o -s i t i o n d o e s n o t g i v e a d v a n t a g e s b e c a u s em e a s u r e m e n t s a r e i n v o l v e d i n C a t e g o r y 3f l o w r a t e s ;

i ii ) L e a s t - s q u a r e s a d j u s t m e n t o f m e a s u r e m e n t s ;i v ) e s t i m a t i o n o f u n m e a s u r e d v a r i a b l e s ;v ) i t e r a t i o n u n t i l c o n v e rg e n c e i s a c h i e v e d .

H e n c e , f a c t o r iz a t i o n m e t h o d s c a n b e o n l y a p p l ie dt o s o l v e p a r t i c l u a r c a s e s o f d a t a r e c o n c i l i a t i o n w h e n

Ta b l e 2 C a t e g o r i e s o f e n t h a l p y f lo w r a t e s

C a t e g o r y F T m

M M M

2 U M M3 M U U M

3 M U M U

489

e n e r g y b a l a n c e s a r e c o n s i d e r e d . O t h e r e q u a t i o no r i e n t e d t e c h n iq u e s , su c h a s P L A D AT S ~ n c he zetal. 1 9 9 2 ) , c a n b e u s e d t o t a c k l e t h e g e n e r a l p r o b -l e m .

4 E X A M P L E S

4 .1. Exam ple I - -L inea r Case : app l ica t ion to asec tion o f an e thy lene p lan t

T h i s s e c t o r o f t h e O i e f i n p l a n t i n c l u d e s t h e e t h -y l e n e r e f r i g e r a t i o n a n d c o m p r e s s i o n t o C 2 s p l i t t e rs e c t io n s . A s i m p l i f ie d n o d e d i a g r a m o f t h e p r o c e s s i sg i v e n in F i g . 1 . C r a c k e d g a s e s c o m i n g o u t f r o m t h eg a s c o m p r e s s o r e n t e r t h e p r e - c o o l i n g a n d d r y i n gs e c t i o n s . T h e c o o l e d , c r a c k e d s t r e a m s e n t e r t h e d e -e t h a n i s e r c o l u m s n o d e s 3 a n d 4 ) w h e r e C 3 a n dh i g h e r h y d r o c a rb o n s a r e s e p a r a t e d a s b o t t o m p r o d -u c t . T h e t o p p r o d u c t o f u n i t 3 , c o n s i s t in g o f C2 a n dlowe r hydro ca rb ons C2H6, C2H4, C2H2, CH4, H2,e t c . ) e n t e r s t h e a c e t y l e n e h y d r o g e n a t i o n r e a c t o rw h e r e a c e t y l e n e i s h y d r o g e n a t e d t o e t h y l e n e . T h eh y d r o g e n a t e d g a s e o u s s t r e a m e n t e r s t h e c o l ds e c t i o n , w h e r e i t i s p a s s e d t h r o u g h a n u m b e r o f h e a te x c h a n g e r s a n d s e p a r a t o r s . A p o r t i o n o f l i q u i ds t r e a m f r o m u n i t 1 0 is u s e d a s t h e r e c y c l e s t re a m .H y d r o g e n i s s e p a r a t e d a s a g a s e o u s s t r e a m i n u n i t1 2. T h e l i q u i d s t r e a m s f r o m s e p a r a t o r s 7 - 11 e n t e rt h e d e - m e t h a n i s e r c o l u m n u n i t 1 3) . T h e t o p p r o d -u c t o f t h is c o l u m n i s m e t h a n e , w h i c h i s s e n t t o f u e lg a s s t r e a m v i a c o l d s e c t i o n a n d d r y i n g / p r e - c o o l i n g

s e c t i o n . T h e b o t t o m p r o d u c t e n t e r s t h e C a s p l i t t e rc o l u m n u n i t 1 5) as a f e e d . T h e t o p p r o d u c t o f th i sc o l u m n i s c o o l e d a n d c o m p r e s s e d a n d s u b s e q u e n t l ys t o r e d a s e t h y l e n e p r od u c t . T h e b o t t o m p r o d u c t o ft h e C 2 s p l it t e r c o l u m n i s e t h a n e , w h i c h i s s e n t b a c kt o t h e c r a c k i n g f u r n a c e a s f e e d s t o c k t h r o u g h t h ep r e - c o o l i n g s e c t i o n .

T h e Q - R f a c t o r i z a t i o n i s a p p l i e d t o a d j u s t m e a -s u r e d f l o w r a t e s o f t h e w h o l e s e c t o r , s u c h t h a t m a s sba l ances a re sa t i s f i ed . The sec t ion has 31 un i t s and6 3 p r o c e s s v a r i a b l e s o f w h i c h o n l y 2 9 a r e m e a s u r e d .F r o m t h e a n a l y s i s a r i s e s th a t :

a ) t h e r e a r e e ig h t r e d u n d a n t e q u a t i o n s c o n t a i n -i n g a ll t h e 2 9 m e a s u r e d v a r i a b l e s ;

b ) f ir s t t h e u n m e a s u r e d f l o w r a t e s c a n b e d i v i d e din to :

u = [ 3 5 10 24 34 35 36 38 39 40 4142 44 45 48 49 50 51 54 55 56 59 61]

un_ ,=[4 6 11 43 47 52 57 58 60 62 63] .

F u r t h e r m o r e , f r o m t h e i n s p e ct i o n o f th eRtum a t r i x , t h e f l o w r a t e s o f s tr e a m s 2 4 , 3 4 - 3 6 a n d 4 9 a r ed e t e r m i n a b l e .

C C E 2 0 : 5 8

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O r th o g o n a l t r an s fo rm a t io n s i n d a t a c l a s s i f i ca t io n - r eco n c i l i a ti o n

Table 3. D ata and results for Example 1

Measured ReconciledStream Variances values values

1 10.87 70.49 70.1052 0.2030 7.103 7.0967 2.624 13.04 11.9808 0.3970 35.38 35.5409 5.76 53.21 53.641

12 0.922 23.90 23.56013 0.608 0.00 - 0.02414 5.76 0.0765 - 0.15015 0.23 54.59 53.81616 1.44 12.78 11.93417 0.7060 23.42 23.00518 0.017 0.2378 0.22919 0.13 8.657 8.61820 0.09 5.087 5.41321 0.014 1.74 1.78722 0.0002 0.0255 0.02623 0.018 3.113 3.17825 0.09 5.407 5.35426 0.014 2.898 2.88927 0.36 11.83 13.23928 0.563 8.197 8.90729 0.023 1.364 1.39330 1.103 20.94 19.87231 0.008 1.051 1.06932 0.397 12.58 13.46533 0.152 4.999 5.33837 0.09 5.73 5.96946 0.13 4.25 4.59553 1.232 16.34 19.608

6

i l l

9 _ _ ~ ~ ] 1 0 _ .

Fig . 2 . S impl i f ied am onia p lan t .

M e a s u r e m e n t v a l u e s , v a r ia n c e s a n d re c o n c i l ed

v a l u e s f o r th e m e a s u r e d f l o w r a te s a r e i n c l u d e d i n

Ta b l e 3 .

4.2. Exa mp le 2--N onlin ear Case: a simplif iedammonia p l an t

T h e p r o c e s s f l o w s h e e t i s s h o w n i n F ig . 2 .M e a s u r e d v a r i a b l e s f or t h is p r o c e ss a r e p r e s e n t e d i n

Ta b l e 4 . S i m u l a t i o n v a l u e s f o r p r o c e s s v a r i a b l e s a r e

Table 4. Measured variables for Exam ple 2

Measurements Stream numbers

Total flowrate 3 5 6Composition of N2 1 2 3 6 7 8Composition of H2 4Composition of Ar 4Composition of NH3 2 3 5Temperature 1 3 4 5 6 7 8 9

491

o b t a i n ed f ro m S E P S I M M a n u a l ( A n d e r s e net al.1 9 9 1) . I t is c o n s i d e r e d t h a t a l l c o m p o n e n t s ( N 2, H E,

A r a n d N H 3 ) a r e p r e s e n t i n a ll s t r e a m s e x c e p t t h e

f e e d .

T h e s e t o f p r o c e s s c o n s t r a i n t s i n c l u d e s : c o m p o -

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

t io n s . E q u a l i t y o f c o n c e n t r a t i o n s a n d t e m p e r a t u r e s

o f t h e s p l i t te r s t r e a m s a r e n o t t a k e n i n t o a c c o u n t i n

t h i s p a r t i c u l a r e x a m p l e . E x p r e s s i o n s o f s t r e a m

e n t h a l p y a s f u nc t i o n o f t e m p e r a t u r e a r e o b t a i n e d

u s i n g t h e r m o d y n a m i c p a c k a g e s f o r s i m u l a t e d v a l u e s

o f p r e s io n a n d c o m p o s i t i o n . T h e s e e x p r e s s io n s h a v e

t o b e u p d a t e d t o c o n s i d e r c h a n g e s i n s t e a d y s t a t e .

A f t e r m o d i f i c a t i o n o f b i l i n e a r t e r m s , v e c t o r s F M , f,

F v, 0 a n d v a r e :

r T = IF F5 F6]

F ~ r = [ e , F2F4FTF8F9F o ] . . .

f r= I f3 ,f3.4fh3fh5f6. fh6fs 4]

o T = [ 0 1 , 1 0 h l 0 2 , 1 0 4 , 2 0 4 , 3

Oh407. 1 Oh7 08.1 08 Oh9 024 ]

~ ' = [ '1 1 ,2 / 2 1 .3 I ) 2 , 2 1 ' 2 . 3vh2v3.2113,3V 4 , 1

1 4 ,4 11 6 ,2 11 6 ,3 ~ ' 7 ,2 11 7 ,3 1 / 8 ,2 11 8 ,3V h lo

vhu ~v s,, v5, 2 vs,3 116.4 118, 117,4 H , ]

w h e r e f / , n a n dfhj a r e C a t e g o r y 1 c o m p o n e n t a n de n t h a l p y f l o w r a t e s , 0 r. . a n dOhj s t a n d f o r a d j u s t -m e n t s o f C a t e g o r y 2 c o m p o n e n t a n d e n t h a l p y f lo w -

r a t e s , v j , , a n d vh a r e C a t e g o r y 3 c o m p o n e n t a n de n t h a l p y f l o w r a t e s .

F o r v a r i a b l e c l a s s if i c a ti o n a n d d a t a r e c o n c i l i a t i o n

t h e f o l l o w i n g p r o c e d u r e i s a p p l i e d :

( a ) M a t r i c e sQB~ QB2 RBh RB2l -I , a n d v e c t o r sVr, a n d V, b -r , a r e o b t a i n e d b y t h eQ - Rd e c o m p o s i t i o n o f m a t r i x B 3 3.

( b ) A f t e r c a l c u l a ti n g m a t r i x D , aQ - R o r t h o g o n a ld e c o m p o s i t i o n o f D g iv e sQD~ QD2 RD~RD2 FIdm a t r i c e s a n d FUrd FUnd-rdv e c t o r s .R ~F i n s p e c t i o n a l l o w s t o c l a s s i fy u n m e a s u r e d

t o t a l f l o w r a t e s i n :

F T d = [ F , F2 F F7 Fs ]

F ~ = [ F 9 F 0 ] .

(C) Bzi Rtva n d RtF a r e o b t a i n e d T h e i n s p e c t i o no f t h e l a s t t w o m a t r i c e s i s u s e d t o c l a s s i f yu n m e a s u r e d v a r i a b l e s i n v :

V d = [ ~ 11 2 ,3 V 4 , I V 8 , 4 11 3 , 2 1 ) 4 , 4 V 3 , 3 q l l ' lJ 2 , 2

115.1116,4 v h 2v 7 4 H r ]

v i = [ ~. ,2116,3118,2111,3117,3118.3115,211h10111.2].

( d ) M a t r i x G t is c a l c u la t e d a n d m e a s u r e m e n t c l a s-s i f ic a t io n i s a c c o m p l i s h e d . N o n r e d u n d a n t

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492 M. SANCHEa and J . ROMAGNOLI

Table 5 . Measu red and reconciled values for Exam ple 2F= [mol kg/h) - T= [K])

Variable Mes. value Rec. value Variable Mes. value Rec. value

F3 174.616 172.720 TI 70 1.4 701.4F5 25.52 25.1423 c (2,1) 0.2419 0.2389F6 147.263 147.577 c(4 , 2) 0.5821 0.5726c(3, 1) 0.1880 0.1957 c(4, 3) 0.0172 0.01749c(3, 4) 0.2126 0.2144 T4 270.5 269.16T3 700.1 698.3 c(7 , 1) 0.2308 0.2306

T5 272.7 270.7 T 267.1 266.7c(6, 1) 0.2299 0.2305 c(8, 1) 0.2265 0.2304T6 267.1 266.6 T8 272.5 272 .0c(5, 4) 0.9853 0.9856 T9 282.0 282.0c( l, 1) 0.2504 0.2504 c(2, 4) 0.0359 0.0368

Table 6 . Est im ated values of u nmeasu red determinab le var iables for Exam ple 2 F= [mol kg/h];o = [Mo l kg/h];ohj,q, Q = [MJ/h]

Variable Est imation Variable Est imation Variable Est imation

/'1 101.608 v(4 , 1) 33.801 0(5, 1) 0.0002F2 202.491 v (8, 4) 7.267 v (6, 4) 12.2569F4 172.720 v (3, 2) 98.910 vh2 2853.141F7 46.694 v (4, 4) 37.039 v (7, 4) 4.989Fg 100.882 v (3, 3) 2.969 Hr 915.220

14.885 qll 2989.72v (2, 3) 2.969 v (2, 2) 143.566

( e )

( f )

m e a s u r e m e n t s a r e T 3, c ( 5 , 4 ) , c ( 1 , 1 ) , 7 1 a n d

Tg. T h e r e m a i n i n g m e a s u r e m e n t s a r e re d u n -

d a n t .

A n i t er a t iv e p r o c e d u r e is p e r f o r m e d t o a d j u s t

m e a s u r e m e n t s a n d t o e s t i m a te u n m e a s u r e d

d e t e r m i n a b l e t o t a l f l o w ra t e s . M e a s u r e d a n d

r e c o n c i l e d v a l u e s f o r th i s e x a m p l e a r e d i s -

p l a y e d i n Ta b l e 5 .

T h e n t h e e s t i m a t i o n o f d e t e r m i n a b l e v a r i -

a b l e s i n v i s d o n e . I n Ta b l e 6 t h e e s t i m a t e dv a l u e s o f u n m e a s u r e d d e t e r m i n a b l e v a r i a b le s

a r e p r e s e n t e d .

5 . CONCLUSIONS

I n th i s w o r k , t h e a p p l i c a t i o n o fQ - R f a c t o r i z a t i o nt o a n a ly z e , d e c o m p o s e a n d s o l v e t h e li n e ar a n d

b i l i n e a r r e c o n c i l i a t i o n p r o b l e m i s d i s c u s s e d i n t h e

c o n t e x t o f C r o w e ' s p r o j e c t i o n s c h e m e .

T h e p r o p o s e d a p p r o a c h h a s s e v e ra l c o m p u t a -

t io n a l a d v a n t a g e s w h e n c o m p a r e d w i t h r e s p e c t t o

t h e c o n v e n t i o n a l a p p r o a c h e s . F u r t h e r m o r e , i t a l lo w s

s t r a ig h t f o r w a r d i m p l e m e n t a t i o n w i t h i n th eM AT L A B e n v i ro n m e n t .

T h e s e p a r a t i o n o f to t a l f l o w r a te s f ro m c o m p o n e n t

a n d e n t h a l p y f l o w r at e s h as t w o i m p o r t a n t a d v a n -

t a g e s :

( a ) T h e r e i s a n o t a t i o n a l c o n v e n i e n c e . I t a l l o w s t o

o b t a i n m o r e c l e ar e x p r e s s i o n s fo r i n s t ru m e n -

t a t i o n a n a l y s i s a n d d a t a r e c o n c i l i a t i o n w h i c h

a r e n o t e x p l i c i t ly i n c l u d e d i n p r e v i o u s w o r k s .

T h e e x p r e s s i o n s a r e w r i t t e n i n t e rm s o f su b -

p r o d u c t s o fQ - R f a c t o r i z a t i o n s .

( b ) T h e u s e o f a s s u m p t i o n s f o r t o t a l f l o w r a t e

a d j u s t m e n t is a v o i d e d .

R e s u l t s o f t h e a p p l i c a t i o n f o r l i n e ar a n d b i l i n e a r

s y s t e m s w e r e p r o v i d e d i n t e rm s o f tw o f l o w s h e e t i n g

e x a m p l e s , o n e o f t h e m b e i n g a n e x i s ti n g o p e r a t i n g

p l a n t .

NOMENCLATURE

Linear Casex = Ve c t o r o f m e a s u r e d v a ri a b le s ( g 1)

A l = M a t r i x f o r m e a s u r e d v a r i a b l es ( m x g )P = P r o j e c t i o n m a t r ix

G x = M a t r i x e q u a t i o n ( 1 3 ) [ ( m - r. ) x g ][Qu, Ru, l-I.] =QR A2)

r u = Ra nk (R.1 )u ~ u n _ , . = Pa r t i t i o n s o f u

lx~, Ix : = Pa r t i t i on s o f I for x ,x , xg_,.l = [ M a t r i x e q u a t i o n ( 1 9 ) [ g x ( g - r ~) ]u = Ve c t o r o f u n m e a s u r e d v a r i a b le s ( n x 1 )

A 2 = M a t r ix f o r u n m e a s u r e d v a r i a b le s ( m x n )G= PA1

1 = I d e n t i t y m a t r i x[Q~, Rx,n d QR Gx)

rx = R a n k ( R ~ )x ,~ , x g _ , = Pa r t i t i o n s o f x

Rw = M a t r i x e q u a t i o n ( 2 3 ) [ r , x ( n - r , ) ]

Non-linear CaseK = N u m b e r o f u n i tsk = U n i t i n d e xJ = N u m b e r o f s t r e a m s] = S t r e a m i n d e xF = M o l a r t o t a l f lo w r a t eS = S t o i c h i o m e t r i x m a t r i xr = I n d e x o f r e a c t i o n

m = M o l a r f r a c t i o n sh = Sp e c i f i c e n t h a l p y

B , = M a t r i c e s f o r c o m p . / e n t h a p y b a l a n c e sf = Sp e c i f i c f l o w r a t e s in C a t e g o r y 1

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O r t h o g o n a l t r a n s f o r m a t i o n s i n d a t a c l a s s i f ic a t i o n -r e c o n c i l ia t i o n

d = M m o la r f r ac t i o n s an d sp . en th a lp yv = S p ec i f i c f lo w ra t e s i n C a teg o ry 3V = D iag o n a l m a t r ix o f F v 2E i = M at r i ces fo r n o rm a l i za t i o n eq u a t io n s

Fvrf,F u d = P a r t i t i o n s o f F uB= QB~BzzC = N u m b e r o f c o m p o n e n t sc = C o m p . i n d ex

L = In c id en ce m a t r ixl = I n c i d e n t m a t r i x i n d e x

H = To ta l h ea t o f r eac t io nq = P u re en e rg y f l o wy = M easu red v a r i ab l e sz = U n m e a s u r e d v a r i a b le sT = Te m p e r a t u r e

t , w, e = Ve c to r s e q u a t io n (3 7 )B . = M a t r i ces eq u a t io n (3 7 )

i i = 1 , 3O = Z e r o m a t r ix

[QB, R B, Ho ] = O R(B33)[QD, RD,Hd] = O R (D)

Gt, b = E q u a t io n (4 6 )v, , , v,b , , = Pa r t i t ion s o f v

R tF, R ~ v, R /F i = In sp ec t io n m a t r i ces

Greek letterse = Ve c t o r o f r a n d o m e r r o r s

Wi = V a r i a n c e - c o v a r i a n c e m a t r i x o f i= Ve c t o r o f e x t e n t s o f r e a c ti o n

d ii = C orre ct io n o f i0 = Vec t o r eq u a t io n (3 2 )

Superscripts= Wi t h m e a s u r e d v a l u e s

/~ = Wi th r eco n c i l ed v a lu es

SubscriptsM , U = M e a s u r e d o r u n m e a s u r e d v a r i ab l e

d , i = D e t e r m i n a b l e o r i n d e t e r m i n a b l e v a r i a b le

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A n d e r s e n P. , F. G e n o v e s e a n d J. P e r r e g a a r d ,Manual forSteady State Simulator SEPSIM. In s t i t u t fo rK em i t ek n ik , 8 2 -1 0 4 (1 9 9 1 ) .

C r o w e C . M . , R e c o n c i l i a t i o n o f p r o c e s s f lo w r a te s b ym a t r i x p r o j e c t i o n P a r t I I : N o n l i n e a r C a s e .A. I . Ch .E .Jl 32 , 616-623 (1986) .

C r o w e C . M . O b s e r v a b i l it y a n d r e d u n d a n c y o f p r o c e s sd a t a fo r s t ead y s t a t e r eco n c i l i a t i o n .Chem. Engng Sci.44 , 2909-2917 (1989) .

C r o w e C M . , Y. A . G a r c i a C a m p o s a n d A . H r y m a k ,R eco n c i l i a t i o n o f p ro cess f l o w ra t e s b y m a t r ix p ro j ec t i o nP ar t I : L in ea r C ase .A.I .Ch .E . J l29 , 881-888 (1983) .

K r e t s o v a l is A . a n d R . S . H . M a h , O b s e r v a b i l it y a n dr e d u n d a n c y c l a s s i f i c a t i o n i n m u l t i c o m p o n e n t p r o c e s sn e t w o r k s .A.I .Ch .E . J l33 , 70 -82 (1987) .

K r e t s o v a l is A . a n d R . S . H . M a h , O b s e r v a b i l t y a n d re d u n -d an cy c l a s s i f i ca t i o n in g en e ra l i zed p ro cess n e tw o rk s I :t h e o r e m s .Computers chem. Engng1 2 , 6 7 1 -6 8 7 (1 9 8 8 a) .

K r e t s o v a l is A . a n d R . S . H . M a h , O b s e r v a b i l ty a n d r e d u n -d an cy c l a s s i f i ca t i o n in g en e ra l i zed p ro cess n e tw o rk s I I :a lg o r i t h m s . Computers chem. Engng12 , 689-703(1988b) .

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493

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P P E N D I X

L et A b e a g iv enm n m at r ix w i thm > ~ man d n l i n ea r lyi n d e p e n d e n t c o l u m n s . T h e n t h e r e e x i s ts a m m u n i t a rym at r ix Q an d a m n m a t r ix R , su ch th a t A =QR, w h e r e :

R = [ R ~ ] an d R 1 i s an u p p e r t r i an g lu l a r m a t r ix .

I f A i s r an k -d e f i c i en t , t h en a t l ea s t o n e d i ag o n a l en t ry i nRm is ze ro.

L e t u s e x a m i n e w h y t h eQ - R f ac to r i za t i o n ap p ro ach canf ia i i n th e c a s e w h e n r a n t ( A ) = r < n .

T h e m i s s i on o f a n y o r t h o g o n a l i z a ti o n m e t h o d i s t o c o m -p u t e a n o r t h o n o r m a l b a s i s f o r t h e r a n g e o fA , R ( A ) .I n d e e d , i f R ( A ) = R ( Q 1 ) w h e r e Q l = [ q l . . . . . q r ] h a so r t h o n o r m a l c o l u m n s t h e n A =Q~N f o r s o m e Ne3t n.U n f o r t u n a t e l y, i fr < n , t h e n t h e Q - R f ac to r i za t i o n d o es

n o t n e c e s s a r il y p r o d u c e a n o r t h o n o m r a l b a s i s fo rR ( A ) .H o w e v e r , t h eQ - R d e c o m p o s i t i o n c a n b e m o d i f ie d i n as i m p l e w a y s o a s t o p r o d u c e a n o r t h o n o r m a l b a s i s fo r A ' sr an g e . T h e m o d i f i ed a lg o r i t h m co m p u tes t h e f ac to r i za t i o n :

A I I = [Q ~ Q 2 ] [ R~ ~ 2 ]

w h e re Q 1 , Q 2, g l l an d R Iz a r e m a t r i ces o f d im en s io n(m r ) , [m (m - r ) ] , ( r r ) an d [ r (n - r ) ] respec t ively,H i s a p e rm u ta t io n m a t r ix an d R ~ i s u p p e r t r i n ag u la r. I f :

A H = [ a q . . . . . ac ] a n d Q = [ q l . . . . . qm]

t h e n f o r k = l , . . . , n w e h a v e :

min{r k}

ack = Z rikqiEspan{q1 . . . . . q ,} .i=1

A lso , i t fo l l o w s th a t fo r an y v e c to r s a t i s fy in gAx = b ,t h en :

T _ CF l r x = [ : ] a n d Q b - [ d ]

w h ere y an d e a r e r -d im en s io n a l v ec to r s , z i s an(n - r ) -d im en s io n a l v ec to r an d d i s an (m - r ) -d im en s io n a lv ec to r.