C HA P TE R
2
FITTING MODELS TO DATA
2 . 1 2,2 2.3 2.4 2.5 2.6
Classification of Models I low to Build a Model
liitting l,'unctions to llmpirical Data 'I'he Method of Least Squares Factorial Experimental Designs
Iritting a Model to Data Subject to Constraints
36 4 l 43 50 57 60 62 63 References
Problems
I
34
l l l l l N t r l \ l t l l l l l \ l r t l r \ l \ l q
( ' o t t s l l r r n l s nt ollllnt/itltott lltolrlcttts irtts(' ltottt Pltvstt';tl lrottttrl'. ltl llll V H I ' i t t l r l e s . c n l l ) t t t ( ' : t l t t ' l l t l t o t t s , lllryslt:tl l:tWs, lttttl s() ()ll lls tttt'ttlloltt'tl ttt St't l'l T h c t t r r r t l l c r r l l r t i c i r l le l l r l i o t t s t l c s t ' t i b i l l F l l l c l ) l o t ' t ' s s : l l s ( ) ( ( ) l l l l l l l \ ( ' ( ( ) l l \ l l ; l l l l l "
H 0 w l . t c s t i n l i l t c t l t c v l r l r r e s o l t l r c c o e l l i t i c r t l s i t t l l t e t t t o t l t ' l s l t t t t l r l c v t ' l r t ; t ( ' l l l l l l l S H I t t . l r r l i o r r s u s i n g l l t c r r r t ' l l r o t l s ( ) l ( ) l ) l i n l i / l l l i o t t i s l l t e l o l t t t o l l l t t s t l t r t I l r t M H l l t c r t r i r t i c r r l lr o t l c l s i u c c n l l ) l ( ) y e r l r r r l r l l ; t l e l t s t t l s t ' i t ' t t t ' t ' , ( ' l l P , l l l ( ' ( ' l t t t ; ' , , ; t t t , l l r t t ' , t l l € t H l ( ) s ( ) l v o l)t()l)lcrils, rlcsigrr L:(luil)nrcrtl. ittlct'ptcl tlrtllt. ittltl ( ( ) l l l l l l l l l l l t , l l ( ' l l l f t t f t t t r t t i o r r . l l y k l r o l l ' ( 1 9 7 4 ) l l r s t l c l i r r c t l l t t t t l t l l t c t t t i t l i t ' r t l t t t o t l t ' l , r ' , ' i t f € p l ' c s c r r t i r t i o r r o l ' 1 h c c s s c l l l i i t l l s l ) c c l s o l ' i t t t c x i s t i t t g s y s l c l l l ( ( ) t : l \ \ ' \ l t ' l r t l l l r r
€ t t t l g l I l c t c ( l ) w l r i c h p l c s c n l s k t r o w l c t l g c o l ' t h t ( s y s l c t l t i t t r t t t s i t l r l t ' li r t t r r " l'ot lltr' p U t , ; t o s c o l ' o p l i n t i z l r t i o n . w c s h t r l l b c c r t t t c c r t t c t l w i l h t l c v c l r t ; ) i l l A ( l l i l l l l r l , r l r \ r ' ( F t n l l t c r r r i r t i c l r l ) c x p r c s s i o n s l i r r t l t c s y s l c n l t t t o t l c l s o l l t t t t w c r c : t l l t l \ ( ' l l t t ' t t t l r " , , r l f f i g l l t C t t u t t i C s i t t t d c t l t t t l t t t t c l c l t l c t l l i l t i ( ) l l s t t t c x l r l t c l t t s c l ' t t l c o t t c t ' p l s
M o c l c l i n g c i l n b c v : t l u i l b l c b c c i r t r s c i l i s i r r t l t l " r s ( r i t c t i o t t rt t t r l l t t ' l p ' , ; t t , ' t r l
; 3 p e l i l i v c c x p o r i n t c n t i l l i o n a r r d ( ) b s c r v i l t i ( ) t l s . l l o w c v c t ' , t l t c p o t c t r l r ; r l t , r ' , 1 , t t t r l t l m c r r r v i r r g s tt l l ' c r c d b y u s i r r g i l n r l l h c r n u t i c i t l n t o d c l t l l t t s t h c w e i g l t r ' t l ; t 1 1 , t t t t ' , 1 l l l t ' h C l t l r r r t l h c n r o c l o l ( ) n l y i n l i t l t c s r c i l l i t y i t n d d o c s t t o t i l t c o t l . t o t ' r t l t ' l t l l l t ' r t l t t r r " ' l l 1 f i 6 l r r r r l s y s l o n t b c i n g r " r . r o c l c l c c l . I l c n c c , t h c p r o c c s s ( ) l ' i t l ( o r c s t e ( ) t ) l i t l l l s l t r l o l l l l ' l l l o n r r o t r c i r t l i l y a v a i l u b l c ( ) r p c r h i r p s l t ( ) l c v c l t v l r l i c l i r t l l t c l t t o t l c l . l t t l l r t ' r l t ' t r ' l 6 p l l l c n t ( ) l ' i l t r . ! o d c l . tl r c u s c r m u s t d c c i c l c w h r t t l i r c t o r s i t r c t c l c v : t l l l i t l l ( l l l ( ) \ \ ' g g n t p l c x tlrc ntrlclcl slroukl be. For cxatnplc, us it trtodclcr yott sltottltl t'rtttrtrlt't l h e l i r l l o w i n g q t t o s l i o r l s :
l , S l t o t r k l t l r c p r o c c s s b c n - r o c l c l c c l ( ) n a nrilcr()scopic ot ttlielostrtpit lt'r't'l ,ttrtl w l t r r t l c v c l o l ' c l l i r r t w i l l b c r o q u i r c d l i r r c i t h c r i t p p r ( ) : r c l l ' ?
l , ( ' r t r r t l r c p r o c c s s b c c l c s c r i b o d t d c q u a t o l y u s i n g p r i t r c i p l e s o l ' c l t t ' t t t t s l t \ ' : t t t r l
p l r v s i c s ' l
t , W l r i r t i s l h c d c s i r c c l a c c u r a c y o l ' t h c m o c l o l a n d l t o w t l o c s i t s r t t ' t ' t t t r t t v t t t l l r r c r r c c it s u l t i r n a l c u s c ' ?
l , W l r i r l n l c i t s u r c n t 0 n t s a r c a v a i l a b l c a n d w h a t d a t a i l f c i l v i t i l t t l ) l c l i r t t t t o t l c l r c t t l l e r r t i o r r ' ?
t , l x l l r c l)r'()ccss i t c t u a l l y c o m p o s c d o f s m a l l c r , s i m p l c r s u b s y s l c t t r s w l t t t l t lilorT r.:irsilv iulillvzccl'?
' l ' h t rtrrswcls lo tlrcsc qucstions dcpcnd on thc usitgo ol'tltc tttotlt:1. A s l l t t ' l r r o t l t ' l t t f l l r c p r o c c s s b c c o n r c s m o r o c o m p l o x , o p t i m i z a t i o t r i t l v ( ) l v i t l ! , t l t c t t t t t r l t ' l r r s t t r t l l y h F r ' n t l t c t r n o l c t l i l l i c t r l t .
l r r l l t i s e l l l p t c r w c w i l l d i s c u s s a n u m b c r o l l l r c I i r c t o t s t l t ; t l l t : t v t ' l o l x ' l I k e t t i r r l o t ' o r r s i t l c n r ( i o n i n c o n s t r u c t i n g a p r o c c s s n t o d c l . l r t r r t l t l i l i t t r t , r v c r v t l l
€ t g l t i n c l l r t . r r s c o l o p t i r n i z u t i r ' r r . r t o a s s i s t i n c l o v c l o l t i r r g t l l i t l l t r : t l l i t l l ( ' i t l t t t o r l t ' l s , 1 , F , , l p c s l i l t i r l c l l r c v i t l u o s ( ) l ' u l l k r ) ( ) w n c ( ) c l l i c i c n l s i t t t t t o t l c l s , y i e lr l i r r p l : t t o t t t ; t : t t l i l l t t l t r . r t s l l l r $ c 1 c l ' r 1 c s c r 1 t ; t t i ( ) n o l p l o r ' e s s t l i r t i r . A t l t l i t i o t t r t l i t t l i r t t t t : t l t o t t t ; t t t f t e l i r r r r r r l i r r l t . x l l r o o k s s p c r c i r r l i z i n g it t t t t : t l l t c r t t t i t ( i c t r l r r r o t l c l i r r g ( l t y k l r o l l . l ( ) / ' 1 . l l l t t r t r r c l l r l i r r r . l ( ) / O ; S e i r r l r ' k l : r n t l l ; r I t r l t t . , , l t ) / . 1 ; l l r x , l l t t t t l c : t ' . it t t t l l l r t t r l c t . l ( ) / X . l , t t y l r c r r , l ' ) / I , l ' 1 i 1 ' 1 l l y , l( ) / ) )
. f 6 t i l i l N ( r r \ t o t r t t , . to tr,\t.\
2 . 1 ( ' l , A s s i i l , ' t ( ' A ' t ' t ( ) N ( ) t , ' M()t)t,:t,li
' l w o
l . l c r r c r i r l e l t l c g ( ) r ' i c s o l r u o t l e l s c x i s l : l , ' l ' l r o s c l' r i r s c t l o n p h y s i c : r l t l r c o r y
l ' I l r o s e b i t s c : t l o t t s l r i c t l y c r n p i r i c i r l c l c s c l i P l i o r r s l s o , c l r l l c r l [ ' r l r r c k - b o x r n o c l c l s ) M i r l l r r ' r r u r l i t ' l r l n r o d c l s b a s e d on physical ancl clrcmical laws (c.g., mass and en- t ' t g v l r i t l i t t t t ' e s . t l r c r r n o d y n a m i c s , c h e m i c a l r c a c t i o n k i n e t i c s ) arc frcqucntly em- I k r v t ' r l r r r o g r t i n r i z : r l i o n a p p l i c a t i o n s ( s e e Chaps. l0 through l4). Thcsc models l i l ( ' ( ' o i l ( ' ( ' l ) l r r r r l l y i r l l r a c t i v e b e c a u s e a g e n e r a l m o d e l for any system size can be rlcvr'lrrpt'rl. t'vcn l'rclirrc the system is constructed. On the other hand, an empiri- t'rrl rrrorlcl cirn bc dcvised which simply correlates input-output data without any lrltvstt'ot'ltt:ltticlrl irttalysis of the process. The following examples illustrate differ- r r r l c i r l c g o r i c s o l ' m o d e l s that might be employed in association w i t h t h e o p t i m i z - l r l r o t r o l e lt c r r r i c i r l p r o c e s s e s .
M r x l c l s o l ' l R c a c t o r
A t t t s o l l l c t t t t i r l l r r b t r l a r re a c t o r operates with axial dispersion of concentration.
I r r o | r l c r ' l o o p l i r n i z c thc reactor operations, i t i s i m p o r t a n t to be able to predict f lrc r'rrrrct'nllirtiorr <lf a valuable component A at the reactor exit. Wen and Fan 1 l t ) 7 5 ) g i v c ir lnrnsicnt modcl for such a reactor, obtained by physical analysis of l l r c s y s l c r r r .
Rcuclrlr Modcl I
t ^ ,
( t ( ' d ' c A ( , . 1 = E,
l = , - il , . r * ' w l t c r c r' conccrrlrirlior.t o l ' ; 1
I l illrc
l , : t r x i i r l d i s p c r s i o n c o c l l i c i c n t : r l i s t : r r r c c l l o n g t h c r o i l c t o r t t l t v 0 r i t g c l l r r i d v c l o c i l y
l l r r ( c o l ' p l o d u c l i o n ol A : l'k ,)
Y o r r c r n s o l v c thc dillcrcntill cquation to obtilin t,(:, t) for appropriatc bounJ- i t t v t ' o t t t l i t i o t t s . A t l t t l l c r n a t c tnodcl cvolvcs front dividing thc rcactor vtrlume l t t l t r 1r 1'1;11111lctcly r n i x c d c ( ) l n p l r t r r r o r l s o l ' c c l L r a l s i r c . V l p . urrlngc{ in sorics, a s s l l o w l l il r l"ig. 2. l.'I'his ltrrirrlBcllrcnl i s k n o w n l r s ir n r i x i n g - c c l l n r o d e l . l l c c u u s c o f l l r c t l i s P c t s i o t t , i t b l r c k - l l ( ) w t c n l , /i', cxists ls wcll irs ir lirrwur-tl l l o w l i r r c a c h c ( ) l l l l ) i l l ' l l l l c l l t . ' l ' l t e tttittct'iltl b l t l l t r t c c c t l r r l r l i o n l i r l l h c r r l h c o r n p i r r t r l c p l is lk,lclor Mrxlcl 2
( l, )
',:,' tr'('l ,, ' r rr
,, | 'r ,, , ) | l 1 r ' , , r , r ) , ,, (;, )
l7
( l l / ) l ( l | /)r
F l l r r t r , l , l ( r)nrl)iulnr(:nls itt setics witlt lt;tek-llow tttotlcl (2 slitgcs).
' l ' l r c cllcct ol'rrsirrg tlrc conrplrtrncnt ntodcl irr lictr ol'tlrc tlillcrcttli:rl tttorlt'l l r t l r r r t lo l t r r o d c l 2 t l r c c o n c c n l r l t i o r t o l ' z l c x i s l s o t t l y u l d i s c r c t c 1 ' r o i t t l s ( w ' r t l r f € r l r t ( , 1 l o t l i s t u n c c ) . n r l h c l t l u r n c o n t i n u o u s l y ( a s a contitrurtus l i t n c l i o n o l . ' ) ; t s f t r f l h e i u l i t l y ( i c i t l s o l r r l i o r r o l ' t h c d i l l c r c n l i l l c q u : r l i o n . lI l h c r c i r c l ( ) r ' ( ) l x ' t i t l r ' s l t l l h e r t c r r t l y s t i t l c n l ( ) s l o l ' t h c l i r r c , A c l A t - Ac,,flt:0 ancl ntotlcl 2 t'ctlttt't's l o r t t € l o l ' p l i n c i r r u l g c b t ' u i c c q u a l i o t r s (i n p u n k n o w t l s ; ( ' 1 , ( ' r . . . . , r ' , , ) l l t i t t t r t t t l r t ' f t t l v e r l li r r t h c r c u c l o r ( ) u l l c l c ( ) n c c n t r a t i o n , t ' , , . Y o u c o u l d u s c s t t c l t : t t t r o t l c l : t s r t g t t l r t r n i l t in clrlculirling tlrc optimal volumc of thc roitcttlr. givcrr thc sclltrrp p f [ , t ' o l l h c p r o c l u c l z 1 lr r d t h c c o s t s o f l i r b r i c a t i n g a n c l o p c r i t t i t l g ll t c t e r t t l o t , Modcls ol' an l,llectrostatic Prccipitator
A c o u l e o n t b r r s l i r l t t p i l o t p l u r t t is u s o c l t o o b t l r i n c l l i c i c n c y d l r t l r 6 f ;rutlicrrlirtc r t t i r t t c r b y i r r t c l c c t r o s t a t i s p r c c i p i t a t o r ( l l S l ) ) . Fgtrecr lrirs bccrt vuriccl by clranging thc surlitcc arca o[ thc F l g r r r r 2 . 2 s l r o w s th c c l l i c i c t r c y d l ( a ( 4 ) a s a f u n c t i o n o f t h c t f € { ( . . 1 ) . t t t c r r s r r r c t l i t s ltlrrtc itrcit/volunlclric l l o w r a t c .
' l ' w o rrrodcls o l ' d i l l ' c r c r r l c o r n p l c x i t y h a v o b c c n p r o p o s c t l B 0 l l t T r l t t t i t :
l ) r c c i p i l r t t o r M o c l c l l : 4 - brA + bt
. . . 1 . , ' ' ( ' " ' I
l ) r ' c c i l . r i t r r t o r M o t l c l l : r t - l(X)l | "
I ; ' ' l ; ' , l l. l
M u t l e l I i s l i r r c l r l i n t h c e o c l l i c i c r r l s , a r t c l M o d c l 2 i s s c r n i c n r p i r i c l l . ' l ' l t c s o l i t l l i r r t ' I n l ' i I . l . l w l r s t l r i t w t t t r s i t t g b ' : 0 . 1 - 5 6 lr n c l b , - tt3.t3. Modcl I wrts tlt:tivt'rl l U l l r r g i n l o i r c c o r r r r t tl t c l"rlrysicll clt:rritclcrislics ol'llrr: plrlicrrlitlc tttrtllct, ittclttrl l l l g g r r r r ' l i r ' l t ' s i z t : i r r r t l c l c c l l i c l r l p l o p c r l i c s , l r c r t c c i t s c r l t t t p l c x l o r t t t . l ' l t t r l)lllttlllclct 1 r t t ' n n l r e o b l r r i r r c t l l i ' o r n c l u r r i r c l c r i z a t i o t t o l ' t l r c p r r r t i c l c s i z c t l i s ( r i b t r l i o r r ( l r l t r l W $ r r . ( l 1 i t l t o 0 . 0 . 1 li r r t l r c t l i r l t r s l t o w t t i r r l r i g . 2 . l ) . ' l ' l r c l ) i l r i l t l t c t c r s ; ' . , l r t t t l i ' l w c l ( ' i l l l t r r r r t t . r l t o l x ' 0 . 0 1 | l r r r t l 0 . ( X ) t l , r' e s p c c t i v c l y , lc l r d i r r g l o t l t c t l r r s l t c t l l i r t c i r r l ' i p l l l . A r t ' i r n b c s c c r r i r r l , i g . 2 . 2 , r n o d c l 2 p r o v i d c s i t b c l t c r l i l o l l h c t l r t l i t l l t : t t t f f i u r l c l I o v t , t l l t c l i t r r g c o l , 4 c o n s i d c r c r l . O n c c r t s r r i t i t l ' r l c p c t l i r t ' t t r : t t t t : c t t t o t l e l t s r c l t r e , l t . r l . r l t o r r l t l l r r : r r s c r l it s i r c o r r s t r i t i r r l i r r c l l c r r l i t t i t t g l l t t : o p l i r t r l t l e o s l o l l l t c G 0 l l c r ' l t o n s y \ l c n t .
A l l l r o t r S l r w e l t i r v c i l l u s l r l r t c r l s ( ) n r ( : l)r'()ccss t t t o r l c l s l t l r o v t ' . l l t e t r r ; r j o l t l r l l i g l l l l V llutl plt,rrr'rirllv t ' x i s l s i t r t l t ' v t ' 1 o P 1 1 1 1 ' s t t t ' l t t t t o t l c l s i s y o t t t : r l r i l i t v l t t t l t ' s t t t l r t ' F f o ( ' r \ r ( . \ l r V ( l u i u l l t l i t l r v t ' r r r ; r l l r t ' r r r ; r l t r ' i t l l c l i t l t o t t s . M o s l l)l()(css ('(ltlll)lll('lll l\ \() o t t t l t c e o l l t ' t l i o t t ' l ' h c lrSl'pctlirt
c o l l c c t i r r g p l ; r l r ' . ' s p c c i l i t t t t l l t ' t lt , r l t
t o l i l l l r c p c t l i r t
. r y r i l i l F . U M ( ' | ) l { l i r a r l t ^ l . A
?' percent collection emciencv
,b o /
ample, thermodynamic t'wrruwuJ'i1u'c or ctremicat tineti be answered. For ex- or cnemlcal kmetics data which are not available may be required in such a model, increasing,tr" he level of effort necessary to formulate thet"u"t of effort necesserv rn rnr-,,r-+^ +r^
ff n: ft :ff '::: iil 1" :l:i:'T q:y;6;:;' fi # r':",.' Hn:t*J:;
fr 3l:i: ]i: ;T Ll" "::.iT, j: :'."ff y.a3r " 1""G ;;il; #il,H;' f"11
ff,ff ':TH:i:llL"*'i:'?g' j":.':':'4;;;;;ffi:;##'jh:Ti:#:
ill5' ;,T "j,*#:l "T-'i: j: | "-'aJi -ii ; ;;? ; ;:TdffiT ;ft':#
f :'"'#r1flT";::',,::':3yi":::jT'^".'9,:,'ril";"";#"11H"""T1'"::?ffi:
J,il#f ::::# j::,: j:::irur"J[_i""."a"i,,""]i,ll'":,LT;,:*"""",il'i;
different operating conditions.
, l , l ( ' l , A s s l l f l ( ' A l t o l , , t t*,ttltltlt s 3 9
L I N E A R V S . N O N l , l N l : n n . L l n a u r m o d o l t l e x h i b i t t h e im p o r t a n t p r o p c r t y r t l ' ition; nonlincar ()1c!t {g ttot, lrquntions (and hence modcls) lrc lincttr il' dcpendent variablcs or thcir dcrivativcs appear only to thc lirst p()wcrl
ise they are nonlincar. In practice the ability to use a linear modcl for tt is of great significance, since the manipulation and solution of lincrtr which may be required during optimization calculations is an orclcr ol
easier than lor nonlinear ones.
To test for the linearity versus nonlinearity of a model, examinc thc cqttrt- S) that represents the process. If any one term in the equations is nonlincttr,
the model itself is nonlinear. By implication, the process is nonlinear, Examine models I and2 for the electrostatic precipitator. Is modcl I lincitr /? Model 2? The superposition test in each case is: Does
o
6o7/6
J(ax, * bxr) : al(xt) + bJ(xr)
J(kx): kJ(x)
J is any operator contained in the model such as square, diffcrcntiutiolt.
fo on, /c is a constant, and x, and x, are variables. ESP model I is linclr in zl J ( b / + b ) : b J ( A ) + b z
ESP model 2 is nonlinear because
( 2 , 1 u )
( 2 . t h ' )
7
--- n: toof
' - =o',01" o o"' L o.o4 + o.oortJI
F,3ure 2.2 ,," "o,'"",llll; ":,:"ol:,1"'on "."u
complicated in operation that application of conservati^' vFvrolrv' r'4r alrplrca'on
or conservation laws leads to in_
numerable questions in modeling,
-urry of which cannot be answererr F^- --
For model I of the reactor, the differential equation is linear if r is a lincur of c because differentiation is a linear operation. For examplc, a sccond ive is linear
(,,' { ;:;:,;) . (ffi) . (5;",7
jF;, tor, + bcr) :
" # c, * b # r,
In addition to the crassification of models as theoretically based versus empirical, we can generally group modets accoraing to the following types:
0l0t and 0l0z are linear operators. However, if the term r is nonlincitr f, then thc model is also nonlinear. The same comments apply to thc mixittg
lpproximation, model 2.
ITICADY STATII VS. UNSTEADY STATE. Other synonyms for steady statc tlmo-invtriunt, static, or stationary. These terms refer to a proccss in which VglUes of thc dcpendent variables remain constant with rcspect to timc. tJtt- ntate processcs are also called nonsteady state, transicnt, or dyntmic, und t thc situation in which the process-dependcnt variablcs changc with A typicul exttmplc of an unstcady state proccss is thc opcration of a batch
l o n c o l u m n . w h i c h w o u l d c x h i b i t a t i m e - v a r y i n g p r o d u c t c o m p o s i t i o n , l n ttletor crumple given nbovc wc ntentioncd that a transicnt mudcl reduccs to
rtutc modcl *1116 r1/r)t * 0, Virtually all optimizltion problcms trcnted
Linear vs. nonlinear
Steady state vs. unsteady statc Lumpcd parameter vs. distribuicd purtrrrclcr ('orrtinuous vs. <Jiscrcto vurjlblei
O / o
Specific collection area ,4 (s/m)
thh book thut involve nrorlclr uro harcd on rtoady rlute mtxlels. Optimizution
ltI
ilI
4 0 r ' l I n N ( i M ( lr ) r ' r s r ( ) l A I A
p r o b l e m s in v o l v i n g d y n a n r i c rn o d c l s u s u i t l l y p c r l l i n t o " o p l i n t a l c o n t r ( ) 1 " p r o b - lems (see Sec. ti.10).
3 DISTRIBUTED VS. LUMPED PARAMETERS. Briefly, a lumped-parameter representation means that spatial variations are ignored, and that the various properties and the state of the system can be considered homogeneous through- out the entire volume. A distributed-parameter representation, on the other hand, takes into account detailed variations in behavior lrom point to point throughout the system. The differential equation for the reactor (model 1) above illustrated a distributed parameter model in which c was dependent on z. In contrast, in the mixing cell approximation (model 2) each compartment repre- sented a lumped-parameter section of the reactor model because the concentra- tion in the nth mixing cell is "lumped" into an average concentration cn (even though in the process cn will vary axially across the distance represented by the cell). All real systems are, of course, distributed in that there are some variations of states throughout them. Because the spatial variations often are relatively small, thcy may be ignored, leading to a lumped approximation. If both spatial irrrd trnnsicnl chartctcristics arc to be included in a model, a partial differential etprirliorr or ir scrics of slagcs is required to describe the process behavior.
l l i s r r o l c i r s y lo t l c l c r n r i n c w h c t h c r lu m p i n g i n a s i n g l e c o m p a r t m e n t in a pt'o('crtfr rrrorltl ir vrrlitl l'rrr rcprcscrtling thc proccss. A good rule of thumb is that i l ' l l t c . t' r . ; l l r t t x c o l ' l h c lr'ocuis is csscrrtiirlly t h c s a m c a t a l l p o i n t s i n t h e p r o c e s s , l l t r , r t ll t r , p l n f r r l r r r o t l c l t ' r r r r b c l r r r r r p c t l i r s l s i r r g l c u n i t . l f t h e r e s p o n s e s h o w s r i g r r i l l c r r r r l i n n l n n l n n c o r r s r l i l l ' c r c r r c c s i r r i r n y d i r c c t i r l r r a l o n g t h e v e s s e l , t h e n i t r l t o r r l r l h e l t e r r l c r l rn i i r r p r r r r r r p p t o p l i i r l r : t l i l l ' c r c r r t i l l c c p r a t i o n o r s e r i e s o f c o m - p u t l n t c n l B , l n r r t t o p l i t t r i r r r l i o r r p r o b l c t t t it i s r l c s i r l b l c t o s i m p l i f y a d i s t r i b u t e d r r r o r l r l lt y r r r i n g r r r r e r p r i v i r l c r r l l r r n r l ' r c t l - p i r r i u l l c l c r s y s t c m , a l t h o u g h y o u m u s t b e e r t r c l i t l lo i t v o i d r r r i r s k i t t g t l t c s i r l i c r t t l c l l u r c s o l t h c d i s t r i b u t e d e l e m e n t (h e n c e b u i l t l i n g l r r r i n l r d c q u i r l c r n o d c l ) . In t l r i s l c x t . w 0 w i l l m a i n l y c o n s i d e r o p t i m i z a - t i o r r t c c h n i q u c s a p p l i c d t o l u m p o d s y s t c m s .
4 CONTINUOUS VS. DISCRETE VARIABLES. Continuous means that a vari- able can assume any value within an interval; discrete means the variable can take on only distinct values in the interval. An example of a discrete variable would be one that assumes integer values only. Often in chemical engineering discrete variables and continuous variables occur simultaneously in a problem.
If you wish to optimize a compressor system, for example, you must sclect thc number of compressor stages (an integer) in addition to the suction and produc- tion pressure of each stage (positive continuous variables). Optimization prob- lems without discrete variables are far easier to solve than thoso with cvon ()no discrete variable.
An engineer typically strives to treat discrcte vnriablcs as conlirrrrous orlcs e v e n a t t h e c o s t o f a c h i e v i n g a s u b o p t i m a l s o l u t i o n w h c n t h c c o r r t i r r r t r l r r s v i r r i - a b l e i s r o u n d e d o f f C o n s i d e r th c v a r i a t i o n o f l l r c c o s l o l ' i r r s r r l l r t i o l r w i t l r t l r i c k - n e s s s h o w n i n F i g . I i l . l . A l t h o u g h i r t s u l i r t i o n is o r r l v rrvrrilrrhlc i r r ( 1 . 5 - i r r c l r
ll
h o | o n t e r t l * . i t i s l r c l p l i r l i n t t n o p t i n t i z i t l i o t t p t ' t l b l c n r t ( ) tr s c i l c t t l t l i t t t t . t t s l l p l ) l . r l
; 1 i 6 , f t r r t5 c t l r i c i r r c s s . s i l l u i r r i r t c a l i r r t l r c r . r i ' l i x i t t t t ' l c J . l ' , , 1 l l o w ' l ' o ltl'Ill,l) A Mol)lll'
M O f l O l - h r r i [ l i t t g c i r t l t r c { i v i { c t l l i r r c t l t t v c n i c t l c c r l l ' p r c s c n t i t t i g t r i t l l p l i r r t r p l t t t r t c t r t t o b l o m d c l i n i t i o n i t t t d l l t r t n t t l i t t i o n , p r c l i n r i n i t r y i t n d d c t i r i l c d i t t t r t l v s i n ' c v t t l t t t t
, n n d i n t c r p r c t i r t r o n i a p f l l c i l t i o n . y o u shtlulcl kccp in rnind llrut rrrorlcl l r r r r l r l l r u n i t c r n t i v c p r , , c c . l u r c ' F i g u r o 2 ' 3 s u m m a r i z c s t h c . i t c t i v i t i c s
l t ' l T l l t l : " , : l l h 0 t u r c d i s c u s s c d b c l o w . i h c c o n t c n t o f t h i s s c c t i o n i s t l r r i l c l i t r t i l c r l t t t il trtrr" rrctuillly cnrbirrking ()n il c()nlprchcnsivc rnodcl tlcvcl,t,'ttrl:.,],,.11:::
. ' . ' i l ( ) w t o [ t t t t t t A M t ) l r t ' l 4 l
y , r u * f i u . , f . f c o ' * u l t . t h c r f c x t b o o k s , s u c h a s t h o s c n r c n t i . t t c t l it t l l t c t t t l t r t I o t h i s c h a P t c r .
p n o B l , l : M l ) l c l ' l N l ' l ' l ( ) N A N t ) l ' o R M L J l , A ' l ' l ( ) N l' l l A s l ' l ' l t t t l r i s lrltttnt' v o t t l i d o l t n . t l r c p r o b l c m t o b c s o l v c d a n d t . i d c n t i f y t h c i t t t p . r l i r t t l c l t ' t t t r t t l r
lMr""sfi;l l_.'biMj
I ' R ( ) l l l l r M l ) l r l . l N l l l ( , N
I ' l I A S I '
I ) l i s l ( i N I ' l l A s l '
t i v A l t l A l ' l ( ) N l ' l l A s l r
l , l M r t l o t r t r ' l l v l t l n r l t t t t t r r r l r ' l l r r r l l r l t t l p l r l l r r l l l l l l l r y l l l ' $ l l o l l Sclcct kcY variablcs,
physicul principlcs to hc applicd' tcst Plun to bc uscd
l)cvcloP modcl
l i s t i m a t c plr rittnctcrs
l i v n l u o t c o n d vcrify ntodcl
"fie
4 2 u i l r N ( i v r ( ) r r r r s r r ) r) A r A
t h a t p c r t a i n to thc problctrl itn(l its solution. 'l'lrc tlcgrr:c ol'irccunrcy 1cc1c6 in t h e m o d c l a n d thc pcltcntial uscs ol'tlrc rrroclol r r r u s ( b c c l c t c r m i n c d . y o u must also evaluate the structurc and complcxity ol' thc rrroclcl; asccrtain
l. the number of independent variables to be includcd in thc m.dcl
2. the number of independent equations required to describe the systcrn (some- times called the "order" of the model)
3. the number of unknown parameters in the model
In the previous section we have addressed some of these issues in the con- text of physical vs. empirical models. These issues are also intertwined with the question of model verification; what kinds of data are available for determining that the model is a valid description of the process? Model-building is an itera- tive process, as shown by the recycle of information in Fig. 2.3.
Before carrying'out the actual modeling work, it is important to evaluate the economic justification for (and benefits of) the modeling effort, and the capa- bility of support staff for carrying out such a project. primarily you should determine that a successfully developed model will indeed help solve the optimi- zation problem.
2 DESIGN PHASE. Activities in the design phase include specification of the information content, general description of the programming logic and algo- rithms necessary to develop and employ a useful model, formulation of the mathematical description of such a model, and simulation of the model. First, define the input and output variables and determine what are the ..system" and the "environment." Also, select the specific mathematical representation(s) to be uscd in the model, as well as the assumptions and limitations of the model rcsulting from its translation into actual computer code. Computer implementa- tion of thc modcl rcquires that you verify the availability and adequacy of com- putcr hurdwarc and software, specify computer input/output media, develop progrilm logic and llowsheets, and define program modules and their structural relationships. Usc of existing subroutines and data bases saves your time but can add complcxity to an optimization problem for the reasons explained in C h a p . 1 4 .
3 EVALUATION PHASE. This phase is intended as a final check of the model as a whole. Testing of individual model elements should be conducted during earlier phases. Evaluation of the model is carried out according to thc cvalua- tion criteria and test plan established in the problem definition phtsc. Ncxt, carry out sensitivity testing of the model inputs and parametcrs 1n{ {otcrmino if the apparent relationships are physically meaningful. Use actual dltl irr thc model when possible. This step is also referred to as <Jiagnoslic clrcckirrg, lrrcl may entail statistical analysis of the fitted paramctcrs (llirnnrclblrrrr, 1970: lkrx.
Hunter, and Hunter, 1978).
F i n g e r a n d Naylor (1961) havc stltccl that nrodcl virlitlirliorr c o r r s i s l s o l ' t h r e e p a r t s : v a l i d a t i o n ol'logic. vitlidltiorr ol'rrodcl itssrrrrrlrliorrs, i r r r r l v i r l i r I r l i o r r t l l m o c l c l h c l t l r v i o l ' . ' l ' l t c s t ' l r t s k s it t v o l v c conrptrlison w i l l r l r i s l o r i c i r l r r r l r r r t o u l p l l
2.3 TO
. ! | I lillN(l t r N ( i l ( , N s t { r t , M n l t l t A l l r A l A . l . t
d a t a , o r d a l a i n l h c l i t e r i r t r r r c . u o r r r p r r r i s o r r w i t h p i l o t p l l n l p c r l i r t t t t i t t t c c , t t t t t l s i m u l a t i o n . In g c n c r a l , c l a l l u s c d ir r l i r r r r r u l l t i n g a m o c l c l s h o u l d t t o t b c t t s c t l lo v a l i d a t c it i f a t a l l p o s s i b l c . l l c c i t u s e n r o c l c l c v a l u a t i o n in v o l v c s tt t u l t i l ' r l c c t ' i l c r t r t , i t i s h c l p f u l t o o b t a i n e x p c r t o p i n i o n i n t h c v e r i f i c a t i o n o l ' n r o d c l s , i. c . , w l t i t l n thc imprcssion of the model when reviewed by peoplc who know tlte: l)t()('('h:.
being modeled'?
No single validation procedure is appropriate for all moclols. Ncvctlltck'r',, it is appropriate to ask the question: What would you really likc tltc rrorlt'l lo do? In the best of all possible worlds, you would like the moclcl to plctlit'l llrr' desired features of the process performance with suitablc accurlcy, brrl lltis tr often an elusive goal.
FITTING FUNCTIONS EMPIRICAL DATA
A model relates the output, i.e., the dependent variablc(s), to llrc il1111:pgrrrlt'trl variable(s). Each equation in the model usually includes onc or rnorc cocllir'tcttlr.
that are presumed constant. The term parameter as uscd hcrc will ntcltn t'ot'lli c i c n t a n d p o s s i b l y i n p u t o r i n i t i a l c o n d i t i o n . W i t h t h e h c l p o f c x p c l i n r r . : r r l i r l r l r r l r r , w c c a n d e t e r m i n e th e . / b r m o f t h e m o d e l , a n d s u b s e q u c n t l y ( o r s i n r r r l l i r t t c ( ) u s l y l
€Etimate the value of some or all of thc parameters in thc motlcl.
2.3.1 How to Determine the Form of a Model
Modcls can bc writtcn in a varicty of mathcmatical forms. liigrrre: l..l sltowr it l o w o f t h e p o s s i b i l i t i c s , s o m c o f w h i c h h a v e a l r c a d y b c c n i l l r . r s t l i r t c t l i n S c c , l. l , T h i s s c c t i o n fo c u s c s o n t h c s i m p l c s t c a s c , n a m c l y m o d c l s c o m p r i s o t l o l i t l g t h t r t t r ' O q u o t i o n s . E m p h a s i s h c r c w i l l b c o n c s t i m a t i n g t h c c o c f f i c i c n l s i t t s i r r t p k r r r o r l c l s
& i d n o t o n t h c c o m p l c x i t y o f t h e m o d c l .
S c l e c t i o n o l t h c f r r r m o [ a n c m p i r i c a l m o d c l r c q u i r c s . j u d g n t c t t t i r s w r ' l l r t s rcmc skill in rccogrrizing how rcsponsc pattcrns match possiblc llg,cbt'itic: lttttr' t l o n s . O p t i r n i z a t i o n n r c l h o c l s c a n h c l p i n t h c s o l c c t i o n o f t l r c m o d c l s l t u c l r t t t ' r t s W e l l u s i n t h c c s t i n r l t i o n c l l ' t h c u n k n o w r r c o c l l i c i o n t s . l f y o u c i r r t s p c c i l y ir ( p t i r l t . t l t a t i v e c r i l c r i o n w l r i c h d c l i r r c s w h a t i s " b c s t " i n r c p r c s c n t i n g t l r c d i r l l r , th r : t t ll t c F t o d c l r c p r c s c n l a t i o n c a n h c i m p r o v c d b y l c l . l u s t m c n t < l l ' l h c l i r r n r o l l l r c r r r o t l c l t o i m p r o v c t h c v l l u c o l ' t h c c r i t c r i o n . ' l ' h o b c s l r n o d c l p r c s u t n i r b l y c x h i h i t s t l t t : l g e r i c r r o r h e t w e c r r i r c l r r l l t l l t i r a n c l th c p r c c l i c t c d r c s p ( ) l ' l s c i t ' t s o t t t c s c n s c . ' l ' y g r i
€ € l r o l n t i o n s l i r r e t r t p i r i c i r l t t t o c l c l s m i g h t b c
l i r r c i r r i r t t l r c v r r r i i r b l c s i r t t t l e o c l l i c i e t t l s l i r r c i r r i n t h c c o c l l i c i c r r t s . r r o r r l i r r c i r r i r r t h c v i r r i i r l ' r l c s ( r,, r,,)
r r o r r l i r r c n l i r r r r l l Il r c t' o r r l l i r r i r r r r l s n n t t l i t t r n t tt r ll r r . t o r l l i c r c r r l / r
;t*+.
. f -(lo Irl ,r' Itt2.t; |"' . l ' - ' , l o I l r l \ i l , l , r . r ' , \ ' ; | " '
tJ(r) -
N t r
I
a r i l | . l t , \ | l r . ! , t
r r ( ll t ' l t '
4 4 | t t t t N r M ( ) t l ' r s r r ) r r l r A
A l g c b r a i c e q u a t i o n s (steady state, lumped parameter)
I rrtcgrtl cqultions (contrnu()us)
l ) i l l c r c r r t r n l l ) r l l c r c n e c c q u l t i l ( ) i l s c q u l t l t ( ) i l s ( c o n t i n u o u s ) ( t l i s c o n t i n u o u s )
Partial differential equatlons
Ordinary differential equatlons
Steady state (distributed parameter)
Unsteady state (distrifuted parameter)
Steady state (one distributed parameter)
Steady state (multidimensional connection of lumped parameter systems, e.9., srages)
Figure 2.4 Typical mathematical forms of models.
Unsteady state (one-dimensional)
When the model is linear in the coefficients, the coefficients can be estimated by a procedure called linear regression. If the coefficients appear in the function in a nonlinear fashion, the estimation of the coefficients is referred to as nonlinear regression. We will describe the linear regression problem in more detail in the next section (Sec. 2.4).
Graphical methods for determining the form of the function of a single variable (or two variables) can save considerable time. The response (y) versus the independent variable (x) can be plotted and the resulting form of thc modcl evaluated visually. Figure 2.5 shows experimental heat transfer data which havc been plotted on log-log coordinates. Since the plot appears to bc approximatcly linear over most of the range of Re, then a selcction of thc modcl lrl rcprcscnt Nu vs. Re is much easier than if the relation bctwccn Lhc Nu aud tlrc llc wcrc e x t r e m e l y n o n l i n e a r . A s t r a i g h t li n e i n F i g . 2 . 5 w o u l d c o r r c s p o n d to h r g N r r l o g a t b l o g R e o r N u : a ( R c ) ' . O b s c r v c t h c s c a l t c r o f c x 1 ' r c l i r r r c r r l l r l t l i r l i r i r r Fig. 2.5, espccially f<rr llrgc valrrcs o[ thc llc.
I f t w o i n d c p c n t l e n l v r l i i r h l c s lt r c i n v o l v c t l in l l r c r r r o r l c l . i r llhrt srrclt irs l,'ig.
2 . 6 c : t n b c o l ' l t s s i s l i u r c c i i n l l t i s c i r s c ll r c s c e o r r t l i n t l c p c r r t l t ' t t l v r r r i t r b l c b c c o t r r c s r l Unsteady
state (lumped parameter)
gI
,j
i l N ( l t l
a l
I ' t
@
't
@
rt
i l'{
, l l
O
,
r t i
' ; i
',1t l
l :
. ' J
l l t l
l l
'.,
r j o
&
tt
l l
2 &
i Hr l r l . ! ,
ll
&
v,t :
1 i
tt,
' t
$ t -
4t
{
-s ll
z
6 rrr'r"t'rNo M()r)u,s r'o t)Ar'A
Figure 2.6 Predicted Nusselt numbers for turbulent flow with constant well heat flux (adaptedfrom Bird, et al., 1964).
parameter that is held constant at various levels. Figure 2.7 shows a variety of nonlinear functions and their associated plots (Himmelblau, 1970). These plots can assist in selecting relations for nonlinear functions of y versus x.
Now let us examine an application of analyzing data with a model.
EXAMPLE 2.1 ANALYSIS OF THE HEAT TRANSFER COEFFICIENT Suppose the overall heat transfer coefficient of a shell and tube heat exchanger is monitored daily as a function of the flow rates in both the shell and tube sides (w"
and w,, respectively). U has the units of Btu/(h)('F)(ft2) and w" and w, are in lb/h.
Figures E2.la and E2.lb illustrate measured data patterns. Determine the form of a model based on physical analysis.
Solution. You could elect to simply fit U as a function of w" and w,; there appears to be very little effect of w" on U while U appears to vary linearly with w, (cxcept at the upper range of w, where it begins to level off). A single line could bc drawn through the data using visual evaluation, or a more quantitative approtch cun bc used based on a physical analysis of the exchanger. Firsl determine why w, hun no effect on U. This rcsult can be explained by thc formulu for thc ovcrsll hcnt transfer coeflicicrtt
t t t l
r + l
Ll h, h, h,
( l ) ' - u + f xI
v
A , l = -o,l +0.3x
v
B , l : o . l + 0 . 3 x
t
c , l = - 0 . 5 + 0 . 3 t v
D . | =0.5+0.3x
l Z l v = r + f , A ' v= -o.l +ffi E ' Y'2 *o:
c , /-4+oi3 D , y - 6 + o r 3
x - a + f x T
r - - 0 . 1 + 0 . 3 x
v
r - 0 , t + 0 , 3 x
v
r - - 0 , 4 + 0 . 3 x
v
r - r l + 0 , 3 r
v
t2 t 0 I
0 - 2 - 4 - 6 - 8
l ! l , ' t ' r T l N o l , u N ( " l l ( ) N s r ( ) l l M P t R l ( ' A l , l t A tA 4 7
Equation (2) l 0
1 0 7 l0s [06
104 8
v 6
4
2
A, f,
€, D,
Equation ( I )
Equation (l)
( a )
la'ililllrntedl
4 t l r r r i l N ' t \ , r r ) l ) r r s r ' t r ^ l A
( 4 ) t' rr/' t l . .t' 4r"' l l . .y 4v"' ( " ) : 4 . 1 r ) r D ' ):4x o 5
l ( r l 4 l 2 l 0 y 8
6 4 2
I I l r t l r r r r t r o n ( 4 )
( s ) v: aF' A. y :2(0.2)"
B. y :2(0.3)' C. y : 2(0.8)"
D. y :2(0.95)' E' Y :2(1.02\' F . Y :2(t.04) G . y : 2 ( r . 3 ) '
) . 2
3.0 2.8
2.4 2.2 2.0 1 . 8
0.8 0.6 0.4 0.2 0
x
f.lgurc 2.7 (continued) Functions of a singlc variablo (x) and thcir corrcsponding trajcct()rics.
in which h. anJ h, are thc shcll and tubc siclc hclt lransl'cr cocllicicnts, rcspcctively, itntl ft, is thc fouling cocflicicnt. ll'fi, is srnall and lr. is largc, [/ is tkrnrin:rtcd by h,, hcncc changcs in n'* huvc littlc cll'cct. Next cxanlinc thc tlutu I'rrr t/ vs. ru, in thc c ( ) n t c x t o l ' I r i g . 2 . 7 . li o r a r c a s o r u r b l c n r n g c o l ' u ' , t h c p : r t t c r n is s i n r i l l r l o c u r v c / ) i n cilsc :1. whcrc
\
, , a l / l v ( r )
w h i c : l t c u r r lr l s o b c w r i l l c n : r s
Equation (5)
l , r I ll
w . ( x l 0 r )
F l g u r c t r ) 2 . 1 a V l r i a t i o n o [ o v o r a l l h r : a t lrlnsl'cr cocllicicnt with shcll-sidc llow nrtc (w,) frrr w, ,- l{(XX).
I I i l t i l N | t | N ( l I ) N S r ) t , t ! | l , l l i l t A l l r A l A 4 "
U 60 40
20
_ 1 l l 80 l(x) | lo
w , ( x l 0 t )
l ' i g u r c t r ) 2 , 1 f V l t r i i t l i o t t o l o v c t t t l l l r r t l t r a n s f o r c o c l l i c i r : n t w i t h l u l x ' r t r h ' l I r t r ratc (w,) for w, 4(XX).
. . t 3 '
N o t c t h c s i m i l a r i t y o f ( < ' ) t o E q . ( a ) , w h e r e x : h t a n d . v : U . l i r o n r r t s l r t t r t l i t t t l l t t ' i t l lrlnsfcr cocllicicnt correlation (Bird ct al., 1964), you could lind thrrt ,rr itlsr) vrtrr('\
o c c o r d i n g to K , w j ) B ( K , i s a c o e l l i c i e n t t h a t d e p e n d s o n t h c l l u i t l p h y s i c l l p r r r l x ' r tics and exchangcr gcttmctry). Thus, the data in hig. E2.lh coultl bc ittk:tplclcrl ttt tcrms of a scmithcoretical model. Equation (a) could bc uscd as lhc tnrxlcl lirr lrcttl cxchirnger pcrformance, estimating thc unknowns K,and h,. (lr. corrltl bc igttott'tl ot
"lurnpcd" into ft1) from plant data.
I n t i t t i n g o i t h c r a n c m p i r i c a l o r t h e o r c t i c a l l y b a s c d m o ( l c l . k c c g t tt t t t t t t t r l l h e t t h c n u m b c r o f d a t a s e t s m u s t b c e q u a l t o o r g r e a t e r lh u n l l t c t t t t t t t b c t o l 1 ; 6 o l l i c i s n t s i n t h c m o d c l , t h c v a l u s s o f w h i c h a r c t o b c c s t i m u t 0 ( 1 . l ' o t c x r t t t t p l t ' . W l t h lh r c c d l t a p o i n t s ( ) f / vcrsus.{, y ( ) u c a n o s t i m a t c a t m ( ) s t tl t c v l t l t t c s o l l l t t r ' c 0 o o l l i c i c n t s . I i x a r n i n c l . ' i g . 2 . t t . A s t r a i g h t - l i n c c o r r c l a t i o n f t r r l l r c ( l t r c c p o t t t l s f i l g l r t h c i l d c q u a t c , b u t t h c d a l a c a n b c l i t t o c l o x i l c t l y v i a a q u u c l r i t l i c t u o r l c l
. V : ( ' o t ' r ' 1 . r * r ' 2 . x 1 ( : l) l l e c l t t l r r t t r p o i n l . a p i t i r ( . y , . r ' ) , w o u l c l c ( ) r r c s p o n d t ( ) ( ) r ' l c o q u u t i ( ) l t o l ' t ' ( r ) w i l l t l h r o c r n r k r r o w n c ( ) c l l i c i o r r t s . ' l ' h c s c l o l ' l h r c c d l l i t p o i n l s l l t c r c l i r t ' c y i c l t l s t l t t c e l l t t o n r c q u n l i o n s il t l l r r c c u r r k r r o w n s ( t h c cocllicicnls).'l'ltc s o l t t l i t l t t o l ' l l t c s c c t l t r i r l l o t t n c r r r r h c o h l i r i n c d b y u s i r t g ( i i t t t s s i i t n c l i r n i t t i t l i o t t ( s c c A p p . l | ) .
N o l c l l r i r l y ( ) l t c i u t p o s l u l a l c il w i ( l o r i l l l l l c ( ) l ' l i r t t c l i o r t i r l l i r r r r r s l i r r t(r), Itlch lls
( ' r r f l o ( . \ ) | r',11 1 ( . r ) | r ' r l / . , ( r ) ( l . t ) ( f o t exrttttplc. 1 y , , l : i l r k r g r ; r l ,
l h f e c u t t k l r o w n s w o u l ( l s t i l l l c s r r l t .
l / r ) . l r r i r l l c i r s c s ll t r c c l i r t c i t r ( : ( l t t i t l i ( ) t t s i t l
Y o r r r l r o u l t l u s ( : p l r y s i c l l i r r s i g l r t i t t t t l c : t t t t t t t t r r r s c n s c ir r l i t l i r r g tl i r l i t . lr r l ' i 9 , . l , i , n t p r r r r l l r r l i e l r r r r r ' l i o r r l i l t t : t l l o t l r c l l t r c e p o i t t l s c x l t i b i l s i t t l t i t x i t t t t t t r t l t t l t t t l l t t l t , c r 1 x . r ' r c l l ( ' r . ( r t plrysit'rrl l c l s o n i n p . ( ' n n in(lierlc il'it ntttxinrttttt ( ( t t tttitlirrrttrlr)
S 0 | r n N ( r M r ) t r r ' r s r { r r ) A r A
,r/\- t.in.o,
Figure 2.8 Linear vs. quadratic fit for three data points.
should occur, which may validate the model structure. In the absence of such information, the simplest adequate model (with the fewest number of coeffi- cients) should be used.
If p is the number of data sets and n is the number of undetermined coeffi- cients in the model, then you should collect enough data sets such that p > n rather than p: n. Although you now have an inconsistent set of equations, optimization can be used to obtain the best solution of the p equations, i.e., the
"best" values of the unknown coefficients, according to some selected criterion.
We describe a method in the next section called least squares that minimizes the sum of the squares of the errors between the predicted and the experimental values of the dependent variable y for each data point x. A quadratic objective function is minimized with respect to the unknown coefficients; for models in which the coefficients appear linearly, this procedure leads to a set of linear equations that can be solved uniquely.
2.4 THE METHOD OF LEAST SQUARES
Various criteria might be used to estimate the coefficients in a modcl lirrnr cx- perimental data. For each of p data points, we can dclinc thc 0rr()r r;i as thc d i f f e r e n c e b e t w e e n th e o b s e r v a t i o n Y ; , .i :1,2,..., p , a n d t h c p r c d i c t c d tn o d c l response yj(x)
Y i .r1: t1 . i : 1 , . . . , p lt,,
s t s , t r ' , r r r t s 5 l T h c i n d c p c n c l c n l v l r r i r r b l c s i r r t l t c v o c t o r x c l l l b c d i l l ' c r c r r t v n l i i r h l c s o r d i l l i ' t c t t l f u n c t i o n s o f t h c s a r n c v l r l i r r h l e , s u c h a s . x , . x 2 , . r 1 , c t c . ' l ' l t c i r t d c p c n t l c n l v i r t i i r h l t ' s e r e n s s u m c d t o b c k n o w n c x l c t l y , o r a t l c a s t th s c r r o r i n v o l v c d it t c n e l t c l c n r t ' t r l o f x i s s u b s t a n t i a l l y l e s s t h a n t h a t i n v o l v c d i n X Y o u m i g h t t h i n k t l r i r t t l r r ' o v e r l l l s u m o f t h e e r r o r s c o u l d b e o f u t i l i t y a s a n o b j c c t i v c fu r t c t i o n : lt o w t ' v r ' r . this iclca is not appropriate because such an objcctivc l'utrclitln itlkrws l)osrlr\'(' Bnd ncgativc crrors to cancel. A second criterion wcluld bc [o sr.rlr tltc itbsoltrlt' valucs o[ the errors
D
7 , : I lr;,r
i = |
Anothcr would be to minimize the absolute value of the maximunr clr'or Of these latter criteria can be used via library computer codcs.
H o w e v e r , th e c l a s s i c a l e n r o r c r i t c r i o n i s t h c q u a d r a t i c c r r o r s u n r r i r l r o r l
P
r ' - S , ' 2
. r 2 - L , ' j
Griterion .f, is different from criterion /', in that it wcights llrgc crrols nrrrr'lr l l l o r e t h a n e x t r e m e l y s m a l l o n e s i n e s t i m a t i n g th c c o c f f i c i c n t s . l r r i r d t l i l i o r r /r, W h e n m i n i m i z e d , l e a d s to a n a n a l y t i c a l s o l u t i o n fo r t h c u n k n o w n c o c l l i c i c r r t s . l r r lOmc cases, if the error in r;, is known or can be estimatcd, it is a1'lllrolrlirrlc to U t c w c i g h t i n g f a c t o r s f o r t h c r ; , t h a t a r c i n v c r s c l y p r o p o r t i o n a l ( o l l t c k r r o w r r llror, By this procedurc you rcducc the conlidence limits f<rr thc csliruirlctl rk.- p o n d c n t v a r i a b l c ( H a l d , 1 9 5 2 ) , b u t n o t c t h a t t h c c o n t i d c n c c li r r r i t s le r r l l v rrre begcd on the dcgrcc of thc cxpcrimcntal crror as dcscribcd in rrrost lcxls orl I t B t i s t i c s . l f w c i g h t s a r c i n c l u d c d in t h c s u m m a t i o n , y o u w o u l d u s c
.l'r: I w,fi
. i . l
L c t u s u s c t h c l i n c a r m o d c l y : f o + /i,.x to illustratc thc principirl leirlrrrcs
€ f t h c l c a s t s q u a r c s m c t h o d t o e s t i m a t c th c m o d c l c o c l l i c i o n t s w i t h u , , l . ' l ' l r c gbjcctivc function is
I , : L (Yi
i l
T h c r c l r e l w o u t t k r t o w t t c o c l l i c i c t t t s , / , , a n c l / 1 , , a n d 2 k n o w n p i r i l s o l ' c x p c l i - n e n t t l l v i t l u c s o l ' Y , l t t d . r . , . W c w a n t l o m i n i r r i z c . / , w i t h r c s p c c l tt l /1,, i r r r t l /1,, l g e t t l l l i o l n c i t l c u l t t s t h i t t y o t t t i t k c t h c l i r s t p a r t i r r l d c r i v t r l i v c s r l l ' / , i r r r t l c ( p l i t l ( : t h g m t o z c l o , l o g c t t l r c l l c c o s s i l r y c o r r d i l i o n s l o r l r r r r i r r i r n r r r n ( l h c nrlionirlc is d e r e r i h c d i r t n r o r c t l c t i r i l ir r S c c . 4 . . 5 )
t).1 :
,) ll,, )
l l I
.r,i)t : I (Yi fiu fi,r )2
I
( . 1 l )
l i r t l t
( .) .ll
( l . t )
( l.(r I
( ) , 7 u l
1 2 , 7 h l
( l I r v , llu
l l
L r t ;
/ , r \ i x l )
/ , r \ / X \ r )
r1t ,
rt;, , :
