New Procedure for Optimal Design of Sequential Experiments in Kinetic Models

Transcription

1 62 Ind. Eng. Chem. Res. 1994,33,62-68 New Pocedue fo Optiml Design of Sequentil Expeiments in Kinetic Models Vincenzo G. Dovi,' Ande P. Revebei, nd Leond0 Acevedo-Dutet ISTIC-UnivesitB di Genov, Vi Ope Pi, 15, Genov, Itly The high coeltion between peexponentil fctos nd ctivtion enegies is sevee dwbck in the estimtion of kinetic pmetes. This difficulty is genelly ovecome using two diffeent techniques, Le., epmetiztion nd pmete seption. It hs been demonstted tht in sequentil expeimentl design the fome technique cn only occsionlly led to bette numeicl esults, with no theoeticl dvntge ove the non-epmetized model. In this wok we show tht the sepbility popety cn be convenient tool lso fo the optimiztion of expeiment plnning. A numeicl exmple is consideed, nd both cses of constnt bsolute nd eltive eos e exmined. Intoduction Kinetic te models cn be witten s = f(x,b) (1) whee is the vecto of mesued tes, X is the vecto of the opeting vibles, nd 0 e the unknown pmetes. Typiclly the kinetic te models cn be cst into the fom k = H+(E,cY,p) (5) whee is the vecto {j) nd H+ is the pseudoinvese of the vecto hi = e-e/rtc,~cz~... (6) the genel definition of the pseudoinvese of n m X n mtix A of nk being given by whee kjj indicte peexponentil fctos nd Eij ctivtion enegies. fij e suitble functions of concenttions c, nd the pmetes CY e elted to the odes of the ections. Thus the kinetics of single ievesible ection is descibed by = ke-e/rtc,c$... (2) The high coeltion between peexponentil fctos nd ctivtion enegies mkes it difficult to estimte kinetic pmetes ccutely. The influence of this coeltion on convegence in the itetive estimtion of the pmetes 0 cn be successfully educed if the epmetiztion suggested by Box (1965) is intoduced, nmely, k' 11,, = k. 41,e-Eii./RT whee!p is efeence tempetue. Since k' nd E tun out to be much less coelted thn the oiginl pmetes k nd E, the estimtion pocedue is considebly esie. Anothe sttegy bsed on the method of seption of vibles (Lwton nd Sylveste, 1971; Golub nd Peey, 1976) mkes it possible to cy out the whole estimtion pocedue in tems of the pmetes E, CY, nd p (which in the sequel will be efeed to s nonline pmetes, due to the fct tht they ente eltion 2 nonlinely), with k being evluted, in the cse of single ievesible ection, by the eltion of conditionl optimlity given by the expession (3) (4) whee R is nonsingul X mtix nd K nd H e two othogonl mtices given by the (not uniquely defined) othogonl decomposition of A A = HRKT Anothe ppoch, bsed on noml equtions nd diect substitution, hs been ecently poposed by Chen nd Ais (1992) fo simple cse. A typicl Guss-Newton step of the egession pocedue bsed on lest sques is povided by e(n+l) = g(n) - (DH*~)+H~~ (7) whee B is the vecto {E,cY,@, p e the esiduls (expeimentl vlues minus computed vlues of the ection tes), H* is defined s (1- H+H), nd DE* is the Fechet deivtive of the mtix HI (Golub, 19761, defined s At convegence, the stndd devitions of {E,cY,/~) cn be estimted employing the usul ppoximtion bed on the Hessin mtix of the likelihood function, whees the vince of k cn be evluted using the genel eltion * To whom coespondence should be ddessed. t Pesent ddess: Diptmento de Ingeniei Quimic, Univesidd Industil de Sntnde, Bucmng, Colombi. The im of optiml design of sequentil expeiments is to detemine vlues of the independent vibles T, c1, CZ,... t which to tke the new expeiment(s), so tht suitble 0S~S-5S85/94/ ~04.50/ Ameicn Chemicl Society

2 nom of the expected vincecovince mtix of the pmetes fte the new expeiments is lest. If ndom eos e nomlly distibuted with constnt vince u2, the vincecovince mtix cn be ppoximted by v = (F~F)-V (9) whee Fij (floj)(xi,oj) (Bd, 1974). A fequently used nom is the volume of the confidence egion, whose line ppoximtion with espect to 8 is given by (Agwl nd Bisk, 1985) (0-8)T(FTF)(0-8) = e whee 8 is the cuent vecto of pmetes nd e is constnt depending on the numbe of degees of feedom, the selected pobbility level, nd the eo vince. In fct it cn be shown tht the volume of the confidence egion is invesely popotionl to the deteminnt of V (Box nd Lucs, 1959). It hs been demonstted by Rimensbege nd Rippin (1986) tht the intoduction of epmetiztion does not chnge the sequence of optiml settings fo the independent vibles, even if mino numeicl impovements e poduced, s epoted by Agwl nd Bisk (1985). In fct the confidence egion volume is popotionl to IGI I(FTF)-')'/' whee G is the tnsfomtion mtix k', wl G = h', h', :j Since the elements of G do not depend on the inde- pendent vibles T, c1, nd CZ, thee is no diffeence in minimizing J(FTF)11/2 o )GI I(FTF)I-l/z with espect to the independent vibles fo the detemintion of optiml settings. On the othe hnd tnsfomtion (eqs 5 nd 6) does depend on the independent vibles Ti, Cli, nd Czi. This mens tht diffeences e to be expected between the tnsfomtion ppoch nd the seption of vibles ppoch in the genel optimiztion sttegy. In pticul the ltte ppoch will be shown to be moe menble to numeicl solution due to simplified nlyticl fom. Design Pocedues Using Sepble Models The exmple epoted by Agwl nd Bisk (1985) is used to illustte the method. It cn be genelized to multivite evesible models long the guidelines discussed in DovI et l. (1987). The kinetic model is given by = g = k4(e,,/31t,xl,x,) = ke-e/rtptpt with the limittions P1,min 5 PI 5 P1,m. ~2,min I PZ I P2,m Tmin I T I Tm. The ssumption of constnt pessue (mde by Agwl Ind. Eng. Chem. Res., Vol. 33, No. 1, nd Bisk) hs been dopped, so tht the intoduction of thid inet component could be voided. The kinetic tes i cn be consideed subject to nomlly distibuted ndom eos with vince u2 given by u2 = b which is the common wy of epesenting heteoscedsticity of dt (Reilly et l., 1977). Suppose the estimtes k, E,, nd hve been obtined fte the fist N expeiments nd let Ve be the vince-covince mtix of the estimtes of the nonline pmetes E,, nd p obtined using the minimiztion method, whose genel itetion is given by eltion 7. Let us suppose futhe tht the citeion used fo optiml design is given by the minimiztion of the-deteminnt of the expected vince-covince mtix V obtined fte the new expeiment. In this cse we hve (Bd, 1974) whee B is the vecto (DglDE, DglD, DglDp) computed t the new setting. DglDE, DglD, DglDp indicte totl deivtives of g, i.e., they tke into ccount chnges in k, ccoding to eltion 5, s the pmetes E,, nd pvy. We shll exmine fist the cse (b = 2) which coesponds to constnt eltive eos. Reltion 10 cn be witten s m whee Setting we obtin g = k4 k2ve,ij-1 uij T, 2 det[litj + uijl = mx (11) - - Due to the one-to-one coespondence between 1 -PI, nd [3 pz, the optimiztion cn be cied out in tems of the I fte suitble modifictions of the constints. On the othe hnd simple lgebic mnipultions chnge poblem 11 = E12(u,2u33 - u2:) + [2(u11u33 - ul:) + t:(u22ull - ul:) + 2f1E2(u13u23 - u12u33) + 2f1ES(u12u23 - u22u13) + 2&&3(ulZu13 - ~ 11~23) = mx (12) Sttiony points of this expession cn be detemined solving 3 X 3 system of line equtions. Howeve, since

3 64 Ind. Eng. Chem. Res., Vol. 33, No. 1, 1994 Tble 1. Flow Cht of the Algoithm fo Optiml Settings Computtion 1. Set k = 1 2. Set = 0 3. Set [k = [f' 4. Compute ukj [j#k = -[k Uhk 5. If [jmh I [j 5 [j,- Compute the objective function nd goto If k = 1 then set m = 2 else if k = 2thenset m = 3elseset m = 1 7. Set b = 0 9. If jmh I [j I [j,- Compute the objective function t :"', ;'& nd goto Compute the objective function t [:"',[;',[y' Compute the objective function t - [:"',[$',$' 11. If b = 0 then set b = 1 nd goto If = 0 then set = 1 nd got Set k = k + 1. If k > 3 then stop 14. Goto 2 thee e no line tems in, the solution cnnot be locted on inteio point of the fesible domin. Thus the solution of this mximiztion poblem cn be obtined by combintoilly setting some of the ti to thei uppe (lowe) limits nd solving simplified system of line equtions with espect to the emining vibles. The ovell combintoil scheme nd the coesponding optiml vlues of tj e illustted in Tble 1, whee the nottion ti(o) = ti,min 5;') = hs been used. On the othe hnd the stong coeltion eduction between nonline pmetes fte the seption pocedue cn be tken dvntge of. In fct if off digonl elements of Ve cn be neglected, V0-l will lso be nely digonl. Thus if Uij N U the mximiztion poblem educes to Tble 2. Initil Expeimentl Conditions expt no. T, "C PI PZ Tble 3. Initil Estimtes. 10% noise 40% noise A & BC this wok A & BC this wok k (103) > > E, kcl/mol * > Tue vlues: k = 4.62, E = * Repoted vlue is infeed fom gph. Agwl nd Bisk (1985). Tble 4. Estimtes of Rection Odes tue vlue fte 6 ex& fte 15 ex& (Y * & B subject to In ode to estimte the degee of ccucy of the ppoximtion descibed, let us neglect the qudtic tems in the off-digonl vibles uij contined in eltion 12 (in fct if the coeltion coefficients Uij/(UijUjj)'/' e lge, the ppoximtion could not be used nywy). We obtin * = 512u22u33 + 5?u1u u22U11-2'!1E2U12U33 - nd consequently 2t153U13u U23Ull whee 42 = UllU22 mx This poblem cn be esily solved by septe optimiztion of p-+=mx de R (i.e., T = Tm, o?' = Tmi; ccoding to the cuent mgnitudes of k nd kle nd the sign of dklde) nd whee the symbols 51(O), 52(O), 5 3(O), nd *O efe to the ppoximte solution nd A* is the chnge of the objective function bought bout by nonzeo vlues of u12, u13, nd ~23. Thus fo ny pedetemined vlue of the tio A*/ 'ko, it is possible to decide if the ppoximtion descibed cn be employed. Let us conside now the cse of constnt bsolute eo. We hve to mximize det[m + Vi'] = I det V;'[ which coesponds to the mximiztion of whee + BVJ3Tl 2E 2 2P Y ' = k Y 2 = k Y 3 = F + k hp{ = mx subject to suitble bounds on the t. Sttiony points of this poblem e given by

4 i.e., i Ye + =O Ind. Eng. Chem. Res., Vol. 33, No. 1, Tble 5. Coeltion Fctos between Pmetes (40% Noise) Repmetized Model In k hwr) B SeDted Model I Reltions 15 nd 16 mke it possible to expess 51 nd 52 s line functions of 53. Substituting in 14 gives n eqution in the single vible 53. Thus by scnning suitble nge of (3, we cn locte ll the sttiony points of ([1,[2,[3). This mkes it possible to locte sttiony points inside the fesible domin. In this cse too, mxim on the domin boundy cn be detemined by combintoilly setting some of the ti to thei uppe (lowe) limits nd epeting the pocedue descibed bove fo the emining vibles. Vlues of the heteoscedsticity pmete b othe thn 0 o 2 genelly led to objective functions moe difficult to optimize due to the nonstightfowd detemintion of ll unconstined mxim. Howeve, knowing optimum settings fo the cses b = 0 nd b = 2 cn povide guidnce concening the optimum vlues fo intemedite cses. Thus genel outline of the method hs been completely descibed. Benefits Resulting fom the Use of Sepble Models in the Design Pocedue Any pocedue fo the design of sequentil expeiments consists of two steps: (1) Estimtion of pmetes by mens of egession clcultions using the expeimentl dt vilble. (2) Detemintion of optimum vible settings fo the next expeiment(s). If pmetes e highly coelted, the fist step is genelly difficult nd cn be delt with only using specil techniques, such s epmetiztion nd seption of vibles, the ltte technique possessing the dditionl dvntge of educing the numbe of independent pmetes. The second step is difficult tsk, becuse it implies nonconvex, nonline mximiztion. It is genelly tckled by setting up gid ove the pemitted expeimentl egion, ech point of the gid being used s stting point fo locl convex mximiztion. By educing the dimensions of the gid meshes (i.e., incesing the numbe of gid points), we cn be confident tht the globl mximum hs been locted. Howeve, if the numbe of vibles exceeds 2, this pocedue cn be computtionlly pohibitive, even if the selected gid is comptively cose. Thus fequent compomise is to limit the selection of new expeimentl settings to the points of the gid, t ech of which the objective function is evluted (Rimensbege nd Rippin, 1986). As hs been pointed out in the intoduction, the use of epmetiztion does not llevite this difficulty, whees the method of seption of pmetes leds, s demonstted in the pevious pgph, to simple combintoil lgoithm, cpble of locting the globl minimum with modest numbe of evlutions of the objective function. Thus the benefit esulting fom the use of sepble models is two-fold () Afte ech expeiment the infomtion vilble is used effectively fo the estimtion of pmetes. This cn pevent dditionl unnecessy expeiments. (b) Globlly optimum settings e detemined fte ech egession step, so tht ech new expeiment povides the lgest mount of infomtion. Tditionl techniques e less likely to detemine the sme sequence of pmetes nd settings, due to the numeicl difficulties descibed bove. This cn led to gete numbe of expeiments thn necessy. A Numeicl Exmple To test the vlidity of the pocedue descibed we hve simulted n expeimentl cmpign, using the model nd the initil six expeimentl dt points poposed by Agwl nd Bisk (19851, nd epot the esults in Tble 2. Although ptil pessues insted of mol fctions wee used, we wee ble to use the sme numeicl vlues with ptil pessue being expessed in tmosphees nd the peexponentil fcto possessing suitble dimensions. To epoduce the test we dded ndom 10 % nd 40% eltive nd bsolute eos to the exct dt in the initil set, s well s in the te dt obtined by simulting the ection t the optimum settings detemined. The esults obtined e epoted fo the fou cses, but compison with those povided by Bisk nd Agwl is possible only fo the two constnt eltive eo cses. In both the low (10%) nd high (40%) noise level cses the initil estimtes of E,, nd /3 wee diffeent fom those poposed by Agwl nd Bisk (1985), with lowe objective function thn tht obtined using thei poposed vlues, which wee the stting vlues used in ou minimiztion pocedue (see Tble 3). In othe wods seption of pmetes detemines the globl minimum, whees the non-epmetized model s well s the epmetized model loctes locl optimum, which seems to coincide with the esults obtined by Rimensbege nd Rippin (1986). The vlues of nd /3 e estimted vey pecisely nd ccutely even using only the fist 6 dt points, s comped with 38.9% devition fte 15 expeiments if the epmetized model is used (see Tble 4). Similly the initil coeltion coefficients between the nonline pmetes e considebly lowe thn those between pmetes in epmetized nd non-epmetized models (see Tble 5).

5 66 Ind. Eng. Chem. Res., Vol. 33, No. 1, 1994 Tble 6. Numeicl Results Using 6 Initil Dt Points exp. noise, exp. noise, no. T pl p2 % eo k E, kcl/mol no. T pl p2 % eo k E, kcl/mol f 4.8% f 6.7% f 4.5% f 5.3% f 17.8% f 8.1% f 17.8% f 8.4% f 4.6% f 5.0% f 4.3% f 4.6% f 16.2% f 8.1% f 16.4% f 9.9% f 4.0% f 5.1% f 3.4% f 4.2% f 15.3% f 8.3% f 15.0% f 7.7% f 3.6% f 4.2% f 3.6% f 5.5% f 16.3% f 7.8% f 12.7% f 6.7% f 3.2% f 4.9% f 3.7% f 4.4% f 15.3% f 8.1% f 14.5% f 6.8% f 3.4% f 4.3% f 3.2% f 3.9% f 11.2% f 7.1% f 11.7% f 6.9% f 3.0% f 3.2% f 3.0% f 3.6% f 11.1% f 7.3% f 13.1% f 6.1% f 3.2% f 3.2% f 3.5% f 3.6% f 11.4% f 6.2% f 12.4% f 6.0% f 3.1 % f 3.7 % f 2.3% f 3.6% f 11.1% f 5.7% f 10.8% f 6.0% f 2.2% f 3.3% f 3.2% f 3.3% f 11.9% f 5.7% f 10.6% f 5.5% f 2.3% f 2.8% f 2.7% i 3.8% f 10.0% f 5.5% f 12.1% f 4.8% f 2.5% f 2.8% f 2.4% f 3.1% f 11.7% f 4.6% f 9.6% f 5.0% n f 2.4% f 2.7% f 2.4% f 2.6% f 8.6% f 5.1% f 9.5% f 4.6% f 2.2% f 2.8% f 2.3% f 2.8% f 7.0% k 4.7% f 8.2% f 4.3% f 2.4% f 2.5% f 2.6% f 2.2% f 9.2% f 4.5% f 9.2% f 4.6% f 1.9% f 2.7% f 2.3% f 2.8% f 8.6% f 4.4% f 7.3% f 4.1% f 1.8% f 2.4% f 1.9% f 2.4% f 8.9% f 4.1% f 9.1% f 4.4% f 2.0% f 2.5% f 2.1% f 2.6% f 6.9% f 4.2% f 6.8% f 4.4% f 1.6% f 2.3% f 2.1% f 2.0% f 7.6% f 3.5% f 6.9% f 4.3% f 1.9% f 2.4% f 1.8% f 2.4% f 7.3% f 3.7% f 7.7% f 3.5% f 2.1% f 2.3% f 2.0% f 2.5% f 6.2% f 4.0% f 7.4% f 3.1% f 1.9% f 2.3% f 1.8% f 2.5% f 6.0% f 3.2% f 6.3% f 3.2% f 1.7% f 2.0% f 1.6% f 2.3% i 7.4% f 3.3% f 5.7% f 3.4% f 1.6% f 2.1% f 1.8% f 2.0% f 7.0% f 3.3% f 6.8% f 3.6% As fo the two pmetes k nd E, it ws not possible to cy out complete compison, becuse the vey stting vlues wee diffeent. Afte the initil estimte the sequence of optiml ltente expeimentl settings detemintion nd egession clcultions ws cied out using the methods descibed in this ppe (i.e., the lgoithms defined in Tble 1 o by eqs fo settings detemintion nd the lgoithm defined by eq 7 fo the estimtion of pmetes fte ech new expeiment). The esults obtined fom this pocedue (optimum settings, pmete estimtes, nd stndd devitions of pmetes fte ech dditionl expeiment) e shown in Tble 6. They cn be summized s follows: () Thee is hdly ny influence of noise level nd type of eo on optimum settings, nd thei influence on the estimtes obtined fte ech new expeiment is not vey stong eithe, if comped with the esults epoted by Agwl nd Bisk (19851, which seems to indicte dmping influence of the seption of pmetes on the ccucy of the estimtes. (b) In no cse did the optiml estimtes of k nd E tend to the tue vlues. (c) The sme (uppe) vlue of tempetue ws clled fo t ech stge, with 12 expeiments out of 24 being exctly the sme. This nomlous behvio ws ttibuted to n insufficient numbe of initil dt points (six dt points fo fou pmetes). In fct eltion 10, computed t the cuent vlues of the pmetes, is vlid only if the ltte e sufficiently close to the tue vlues. Othewise stongly bised estimtes my cll fo the sme expeiment mny times sequentilly. On the othe hnd the sme expeimentl settings my fil to povide enough infomtion fo bis coection nd eventully this cn led to wong symptotic estimtes. To test this possibility, we incesed the numbe of initil dt points to 12 nd epeted the sme pocedue incesing the numbe of sequentil expeiments coespondingly. The esults e epoted in Tble 7.

6 Tble 7. Numeicl Results Using 12 Initil Dt Points Ind. Eng. Chem. Res., Vol. 33, No. 1, exp. noise, exp. noise, no. T pl p2 % eo k E,kcl/mol no. T pl p2 % eo k E, kcl/mol f 3.8% f 4.5% f 1.9% f 2.5% f 3.6% f 4.4% f 2.2% & 2.9% f 13.9% f 7.6% f 8.5% f 3.8% f 16.3% f 5.9% f 7.9% f 3.8% f 3.2% f 4.0% f 1.7% f 2.2% f 3.6% f 4.6% f 1.9% f 2.4% f 14.5% f 8.4% f 8.0% f 3.5% f 14.5% f 7.4% f 7.0% f 4.4% f 3.4% f 4.1% f 2.0% f 1.9% f 3.1% f 3.5% f 2.0% f 2.5% f 10.9% f 7.2% f 7.9% f 4.4% f 13.6% f 6.7% f 7.8% f 3.6% f 2.7% f 3.6% f 2.0% f 1.9% f 3.3% f 3.3% f 1.7% f 2.1% f 13.6% f 5.9% f 6.2% f 4.1% f 12.8% f 6.7% f 7.3% f 3.7% f 3.2% f 3.7% f 1.5% f 2.3% f 2.8% f 3.6% f 1.8% f 2.2% f 11.2% f 6.0% f 7.8% f 3.2% f 11.7% f 5.1% f 7.6% f 3.7% f 2.3% f 3.7% f 1.9% f 1.7% f 3.0% f 2.8% f 1.8% f 2.1% f 11.8% f 6.2% f 6.7% f 3.6% f 11.7% f 5.5% f 6.8% f 2.9% f 2.3% f 3.5% f 1.7% f 1.9% f 2.6% f 3.1% f 1.7% f 1.6% f 9.1% f 5.0% f 5.0% f 2.9% f 9.5% f 5.1% f 6.3% f 3.8% f 2.6% f 2.8% f 1.5% f 2.1% f 2.7% f 3.5% f 1.7% f 1.7% f 10.6% f 5.4% f 6.7% f 2.6% f 9.9% 11.1OOO f 4.2% & 6.1% f 3.7% f 2.4% f 2.4% f 1.6% f 2.1% f 2.5% f 3.1% f 1.6% f 2.0% f 11.1% f 4.8% f 4.9% f 2.8% f 10.5% f 4.1% f 6.5% f 2.6% f 2.0% f 2.5% f 1.6% f 1.5% f 2.5% f 2.7% f 1.5% f 1.8% f 10.9% f 4.3% f 6.7% f 2.9% f 8.4% f 4.9% f 6.5% f 3.1% f 1.8% f 2.6% f 1.6% f 1.4% f 2.2% f 2.6% f 1.1% f 1.9% f 9.7% f 4.6% f 6.6% f 2.8% f 8.3% f 4.6% f 5.5% f 3.0% f 2.1 % f 2.7 % f 1.5% f 1.4% f 2.2% f 2.7% f 1.4% f 1.7% f 8.3% f 3.8% f 5.3% f 3.3% f 8.3% f 4.2% f 5.6% f 3.3% As cn esily be seen both k nd E e now estimted without bis. Similly the new optiml settings e distibuted in fily unifom wy in the fesible egion. These two popeties cn incese, to consideble extent, the ovell efficiency of sequentil expeiment plnning. Conclusions Seption of peexponentil (line) pmetes fom the nonline ones (ctivtion enegies, odes of ection, etc.) shows two emkble dvntges. () The coeltion between the independent (nonline) pmetes is f less thn tht between line nd nonline pmetes in nonsepted models. Since coeltion is genelly sevee hindnce to pecise estimtion nd intepettion of the individul pmetes, its eduction mkes it possible to educe the numbe of expeiments, due to bette use of the infomtion vilble. (b) The objective function fo the detemintion of optimum settings tuns out to be moe esily menble to globl optimiztion thn tditionl nonsepted models. Nomencltue = fist pmete in the heteoscedsticity expession of eo b = second pmete in the heteoscedsticity expession of eo B = vecto (Dg/DE,Dg/D,Dg/DB) ci = concenttion of species i (mol/cms) D = Fechet opeto of mtix deivtive E = ctivtion enegy (kcl/mol) f = ection te s function of opeting vibles nd pmetes Fij = (f/@j)(xi,oj) g = ection te (mol/cm3.s) gi = g/ei G = pmete tnsfomtion mtix

7 68 Ind. Eng. Chem. Res., Vol. 33, No. 1, 1994 hi = see eq 6 H = pseudoinvese of hj k = peexponentil fcto pi = ptil pessue (tm) = ection te (mol/cm3-s) T = tempetue (K) uij = k:v,ij-l V = vnnce-covince mtix x1 = mole fction of component 1 x2 = mole fction of component 2 X = vecto of independent vibles Geek Symbols = ode of ection j9 = ode of ection yj = 28i/k 1 = k/e - k/rt t2 = k/ + k ~n p1 t3 = k/s + k In pz 8 = pmetes u2 = vince 4 = ection te divided by the peexponentil fcto Supescipts * = efeence tempetue + = pseudoinvese I = 1 - H+H Litetue Cited Agwl, A. K.; Bisk, M. L. Sequentil Expeimentl Design fo Pecise Pmete Estimtion. 1. Useof Repmetiztion. Ind. Eng. Chem. Pocess Des. Dev. 1986,24, Bd, Y. Nonline Pmete Estimtion; Acdemic Pess: New Yok, Box, G. E. P.; Hunte, W. G. The Expeimentl Study of Physicl Mechnisms. Technometics 1965, 7, Chen, N. H.; Ai, R. Detemintion of Ahenius Constnts by Line nd Nonline Fitting. AIChE J. 1992, 38 (4), Dovl, V. G.; Ab, E.; Mg, L. A moe genel fomultion of sepble lest sques. Mth. Model. 1987,8, Golub, E. H.; Peey, V. The diffeentition of pseudo-inveses nd nonline lest sque whose vibles septe. SIAM J. Nume. Anl. 1973, 10 (2), Golub, E. H.; Peey, V. Genelized Inveses nd Applictions; Acdemic Pess: New Yok, Lwton, W. H.; Sylveste, E. A. Elimintion of line pmetes in nonline egession. Technometics 1971, 13, Reilly, P. M.; Bjmovic, R.; Blu, G. E.; Bnson, D. R.; Suehoff, M. W. Guidelines fo the Optiml Design of Expeiments to EetimtePmetesinFist Ode KineticModels. Cn. J. Chem. Eng. 1977,55, Rimensbege, T.; Rippin, D. W. T. Comments on "Sequentil Expeimentl Design fo Pecise Pmete Estimtion. 1. Use of Repmetiztion". Ind. Eng. Chem. Pocess Des. Dev. 1986, 25, Received fo eview My 25, 1993 Revised mnuscipt eceived Septembe 13, 1993 Accepted Septembe 28, Abstct published in Advnce ACS Abstcts, Decembe 1, 1993.