Effect of Systematic and Random Errors in Thermodynamic #0dc

7/27/2019 Effect of Systematic and Random Errors in Thermodynamic #0dc

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Effect of Systematic and Random Errors in Thermodynamic Models

on Chemical Process Design and Simulation: A Monte Carlo

Approach

V. R. Vasquez and W. B. Whiting*

Chemical Engineering Department, University of NevadasReno, Reno, N evada 89557-0136

A Monte Ca rlo method is presented to separa te a nd study the effects of systema tic and r a ndomerrors present in thermodynamic data on chemical process design and simulation. From analysisof t h e rm od yn am ic d a ta fou n d in t h e l i te ratu re , t h e re is c lear e vid en ce of t h e p re sen ce o fsyste m atic e rro rs, in p art icu lar , for l iqu id-l iqu id e qu il ib ria d ata . F o r syste m atic e rro rs, th ed ata are p e rtu rb e d syste m atical ly with a re ctan g u lar p ro b ab il i ty d istr ib u tio n , an d to an alyz erandom errors, the perturbation is carried out randomly with normal probability distributions.T h e rm o d yn am ic p aram e te rs are o b tain e d fro m ap p ro p riate re g re ssio n m e th o d s an d u se d tosimulate a given unit operation and to obtain cumulative frequency distributions, providing aqu a n ti ta t ive r isk asse ssm en t an d a b ett e r u n d e rstan d in g o f th e ro le of u n certa in ty in p rocessdesign and simulation. The results show that the proposed method can clearly distinguish whenone type of error is more significa nt. P otential a pplica tions a re sa fety fa ctors, process modeling,and experimental design.

1. Introduction

The quality of any computat ional method used forsimulat ion and predict ion is m easured in terms of theaccuracy and precision of the output results, which areaffected by all the uncertainty sources present in thecomputa tiona l procedure. These include err ors presentin the input da ta or variables, numerical a nd modelingerrors, and any assumptions made. The evaluation ofthe effects of these uncerta inty sources on t he qua lityof the computational model often cannot be carried outu s in g a n a l y t i ca l s t a t i st i ca l m et h od s b eca u s e of t h ecomplexity involved. From the chemical process simula-t ion standpoint , we can visualize the computer model

as a set of numerical steps to produce the evaluation ofa g ive n u n it op era t io n or p roce ss . F o r in s t a n ce , t h eperformance evaluation of a distillation column involvesa ba s ic s t e p s e q u e n ce de fin ed by t h e a c q u is i t ion ofexperimental data, thermodynamic modeling, modelparameters regression, and modeling of the distillationprocess. The beha vior of the error propaga tion in t hesek in ds of comp ut e r mode ls is h ig h ly n on lin e a r a n dcomplex, so Monte Carlo methods are preferred to studyt h e u n c e rt a in t y p ro p a g a t io n . W h e n u n c e rt a in t y a n dsensitivity analyses are performed, it is usually implic-it ly assumed that the thermodynamic and unit-opera-t ion models are reasonable representat ions of reality.Thus, these studies are mainly conducted based on the

experimental dat a errors a s the m ain source of uncer-ta inty. Even th ough th e idea of using the experimentaldata errors as the main source can be useful for manypractical situa tions, it ha s been common to include onlythe ra ndom experimental errors, neglecting th e effectof systematic or bias errors.

In t his work, a new Monte Ca rlo method is introducedt o e v a l ua t e t h e i m p a ct of s y st e m a t ic e r r or s a n d t o

differentiate their effects from those of ra ndom errorsin predicted process performance. The method consistsof the identificat ion of the systemat ic trends and ra ndomerrors present in the experimental data, definit ion ofappropriat e probability distributions for both types oferrors, and Monte Ca rlo simula tion using sam ples fromthese distributions. The method w as tested using botha synthetic example and case studies of l iquid -liquidextraction operat ions. The results show tha t t he methodis capable of distinguishing when one type of error hasa dominant effect over the other. For the liquid -liquide x t ra c t io n c a s e s s t u die d, i t is s h o wn t h a t s y s t e ma t icerrors can have a significant impact on the predicted

process performan ce, a nd therefore notoriously influencet h e u n c ert a in t y a n a ly s is .

Section 2 conta ins a brief description of the cha ra c-teristics of random and systematic errors. In section 3,the ba sic uncertainty propaga tion mechan ism throughcomputer models of the type used in chemical engineer-ing a pplicat ions is discussed, followed by a detaileddescription of the new Monte Carlo approach for theu n c e rt a in t y a n a ly s is o f t h e s e t wo t y p e s o f e rro rs incomplex models. Applications for the prediction of unitopera tions performa nce are presented in section 4 withtw o case st udies of liquid-liquid extra ction. In add ition,the method presented can be used to facilitate decision

ma k in g in f ie lds re la t e d t o s a f e t y f a c t ors s elect ion ,mode lin g , a n d e xp erimen t a l da t a me a s u reme n t a n ddesign.

An ot h e r import a n t p roblem n ot a ddre s se d in t h iswo rk is t h e e ff ect of mode lin g e rror in u n ce rt a in t ypropagation analysis, which is certaintly an open prob-lem f or f u rt h er re se a rch . M ode lin g e rrors ca n h a v esignificant effects on process design and simulation aspointed out by Vasquez and Whiting 1 u s in g t h e U N I -QUAC and NRTL models in the process performanceprediction of liq uid-liquid extraction operations.

* To whom correspondence should be addressed. E-mail :ww [email protected]. Tel.: 1-775-784-6360. Fa x: 1-775-784-4764.

3036 In d. E ng. Chem. Res. 1999, 38 , 3036-3045

10.1021/ie980748e CCC: $18.00 1999 American C hemical SocietyP ubl ish ed on Web 06/22/1999


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2. Random and Systematic Errors

I n g e n e r a l , r a n d o m a n d s y s t e m a t i c e r r o r s c a n b edefined as a measure of the state of i nform ati on a v a i l-able for a given set of experimental dat a or observablepara meters. The definit ion of the st at e of informa tionfor experimenta l measur ements is a controversial issueand usually is related to the concept of dat a space. Someauthors defined the data space as all the conceivableinstrumental responses,2 bu t t h is de fin it ion is v a g u efrom t he pract ical sta ndpoint because an experimenta lmeasurement is a function of many additional variablesexternal to the instrument itself . For instance, system-at ic externa l f luctua tions in t he w orking environmenta s we ll a s e xt e rn a l ra n do m f luc t u a t ion s s ign ifica n t lyaffect the instrument output in many situat ions. Theexperimentalist h as the ta sk of report ing either qua n-titat ively or qualitat ively these effects, but the toolsavailable are st ill quite basic. Addit ionally, the errorsources identificat ion process is not straightforward,ma king it even more difficult t o report estima tes for thetrue experimental condit ion. Tra dit ional pract icesinvolve only report ing the sample average (xj) w i t h avar iability mea sure based on the instrum ent (precision)

s t a t is t ics . H o we ve r , a t lea s t a q u a l i t a t iv e idea of s y s -tematic and other external errors should be reported.According t o ANS I/ASME P TC 19.1 St a nda rd, 3 a m e a -surement uncerta inty a na lysis should be able to identifymo re t h a n f o u r s o u rc e s o f s y s t e ma t ic o r bia s e rro rs(ot h er w i se , t h e r e i s t oo g r ea t a ch a n ce t h a t s om eimportant sources will go unrecognized), and authorss u c h a s H a y w a r d 4 in dica t e t h a t a g ood u n ce rt a in t yana lysis might reveal dozens of primary m easurementerrors.

The main measurement error sources, as classifiedby ANSI /ASME ,3 a re in (a) calibra tion, (b) data a cquisi-t ion, and (c) dat a reduction procedures. On one hand,calibrat ion errors come from the assumption that the

in s t ru me n t c a l ibra t io n is do n e in s u c h a wa y t h a t i t sre sp on s e is iden t ica l t o s o me k n own s t a n da rd in t h eworking environment. On the other hand, data acquisi-t ion errors are related to error sources in the processsuch as excitation voltages, signal conditioning, probee rrors , a n d s o f or t h . F in a l ly, da t a re duct ion e rrorsinvolve errors in calibrat ion curve fit t ing and compu-tational resolution. All these error sources are subjectto random and systematic variat ions, which should beq u a n t i f ied. D e comp os in g t h e t o t a l e rror f or a g ive nexperiment a s defined by ANSI /ASME ,3

where is a fixed bias error and is a random precisionerror. The bias error is assumed constant for a givenexperiment. The methodology suggested to estimate is based on bia s limits defined as follows:

wh e re S represents an average of the elemental biaserrors, j includes t he categories involved (i.e., (a ), (b),and (c) error types), and i defines the sources within agiven category. Similarly, the same approach is used tod ef in e t h e t ot a l r a n d om e rr or b a s ed on i n di vi d ua l

standard deviat ion est imates:

A similar approach for including both random andbias errors is presented by Dietrich,5 with minor varia-tions. The ma in difference lies in the use of a G a ussia ntolerance probability mu lt ip ly in g a q u a dra t u re s u m

of both types of errors:

where is the value used to define uncerta inty interva lsf o r me a n s o f la rg e s a mp le s o f G a u s s ia n p o p u la t io n sdefined as ( . Addit ional formulas are presentedwhen dealing with small sam ples, replacing the Ga uss-ian probability term by a toleran ce probability fromthe t student distribution, which is equivalently usedt o d ef in e u n ce rt a i n t y i n t er v a l s f or m ea n s of s m a l ls a m pl es a s xj ( t s. Also, formulat ions for effect ivedegrees of freedom a nd t effective values are given whenboth types of errors are present. On one hand, problem

with this formulat ion as pointed out by D ietrich is tha tit is very difficult to estimate the effects on the outputresults due to systema tic errors. On the other ha nd, theANS I/ASM E P TC 19.13 e s t a bl is h e s t h a t t h e ra n do mu n ce rt a in t y is e st ima t e d by t h e s t a t is t ica l a n a ly s is o fexperimental da ta , while the systematic uncertaint y ise st ima t e d by n on s t a t is t ica l me t h ods , s u g g es t in g t h a tone should keep the random uncertainty separate fromthe systema tic one. The Int ernat ional Organiza t ion forSt a n da rdiza t ion (I SO ) in t h e docu me n t G u i d e t o t h e Expr ession of U ncert ain ty i n M easur ement6 also con-c e p t u a l ly de f in e s t h e ro le o f ra n do m a n d s y s t e ma t ice rror on t h e me a s u reme n t p roce s s, a n d t h e u s e o faddit ive correction factorsis suggested to decrease t heeffect of systema tic errors wh en th e source is identified.A s n o t e d by I SO , t h e s y s t e ma t ic e rro r a n d i t s c a u s ecannot be completely known; therefore, the compensa-t ion cannot be complete. Thus, ISO does not presents pe ci fi c m e t h od s a n d t e ch n iq u es t o d ea l w i t h t h i sproblem. The combination of random and systematicuncertaint ies for stat ist ical analysis has became evenmore controversial in recent years, as pointed out byDietrich,5 due to the difficulties found in linking randomand systematic probability distributions when dealingwith highly complex models.

3. Propagation of Uncertainty in ComputerModels

A computer model, in a broad sense, can be defineda s a n y comp u t er code t h a t impleme n t s ma t h e ma t ica lmode ls f or a g ive n p h en ome n on , wh ich ca n n o t bean alyzed directly or a na lytically beca use of its complex-ity. Any unit operation simulation process, involving theu s e o f e x p e rime n t a l da t a a s a n in i t ia l s t e p t o o bt a inpara meters, easily falls into this cat egory. The ra ndomand systemat ic error propaga tion in this kind of modelbecomes very complex from the ana lyt ical sta ndpoint ,part icular ly the considera tion of bias errors. Most of thework regar ding error propaga tion theory is defined interms of Taylors series expansion techniques 3,7 of themode l in v olv ed f or in dep en de nt ra n do m e rrors a n dma inly considers only the first tw o terms in th e series.F or in s t a n ce , f or a f u n ct ion wit h t wo in dep en de n t

) + (1)

S ) [j)1

3

i)1

K

S , i j2]1/2 (2)

SR ) [j)1

3

i)1

K

SS , i j2]

1/2(3)

U ) [SR2

+ S2]

1/2(4)

Ind. Eng . C hem. Res. , Vol. 38, No. 8, 1999 3037


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varia bles, the expansion is expressed a s

wh e re x a n d y a r e t h e e rr or s a s s oci a t ed w i t h t h evariables x a n d y. R2 is the remainder after two terms,which is generally assumed not significan t. On the ba sisof eq 5, the ANSI /ASME P TC 19.1 Sta nda rd 3 presentsformulas for the equivalent variance term propagatedfor several functions based on the va ria nce terms of theindependent variables. For example, for a function r )f(x,y) the expressions a re of the type

F ro m a p ra ct ica l s t a n dp oin t , i t is o bv iou s t h a t t h ist y p e o f a p p roa c h ca n n o t be impleme n t ed e ve n f ormoderately complex computer models because of theinability to develop the expressions for the Tay lorsseries expansion. Extremely poor resolution of the erroris obtained wh en th e complexity of th e model increases.T o g e t a ro u n d t h is p ro ble m, t h e u s e o f M o n t e Ca rlomethods are very popular. Ima n a nd Conover 8 a n d I ma na n d H e lt on 9 p ion e ere d t h e u n ce rt a in t y a n a ly sis of

com pu t er m od el s t h r ou g h t h e u s e of M on t e C a r l omethods. Ba sica lly, their approach consists of assigningprobability distributions to the model parameters in-v olv ed, a n d f rom t h e re p e rform s a mp lin g ov er t h edistributions to obtain pa ra meter sets, which are usedto study the variat ion or imprecision of the output ordependent variables of the model from the collect ivevaria tion of the independent para meters. The para metersets a re obtained via ra ndom-sam pling procedures, withmodifica t ion s t o a c ce le ra t e t h e con v erg e n ce of t h ecumulative frequency distributions for the output vari-ables. The most common modificat ion is t he use ofs t ra t i f ied s a mp lin g , in p a rt icu la r , L a t in h y p ercu besampling (LHS ),10 where the sample space is dividedin t o in t e rva ls of e q u a l p ro ba bil it y a n d on e s a mp le istaken at random from within each interval.

All of these uncertainty analyses methods based onMonte Carlo simulation using probability distributionsfor the model parameters have the disadvantages thatthey cannot incorporate the effect of systematic errorsa n d t h e p a r a m e t e r c o r r e l a t i o n h a s t o b e t a k e n i n t oa c cou n t t o obt a in re lia ble re s u lt s . D e t e rmin in g t h eparameter correlat ion is a diff icult task when dealingwith nonlinear models, as shown by B at es and Wat ts 11,12

an d Cook and Witmer.13 Ima n a nd Conover14 suggesteda p a ir in g p roce dure t o a p prox ima t e t h e p a ra me t e rscorrelation w hen sa mpling using LH S. H owever, Vasq u-e z e t a l .15 s h owe d t h a t t h e re solu t ion of t h is p a ir in gprocedure for highly nonlinear models can be very poor.

The latter developed a new sampling technique callede q u a l p ro ba bil i t y s a mp lin g ( E P S) t h a t a u t o ma t ic a l lyta kes into account the pa ra meter correlat ion, increasingt h e re solu t ion of t h e p a ra me t e r s pa c e s a mp lin g s u b-s t a n t ia l ly .

Most of the emphasis given in the literature for thea n a l y s is of s y st e m a t ic e rr or s i s r el a t ed t o s pe ci fi cp h y s ic a l s i t u a t io n s o r c a s e s s u c h a s p a ra me t e r bia selimination for log-transformed data 16 or the report ofexperimental dat a ra t ios to reduce bias effects.17

4. The Monte Carlo Approach

The two previous sect ions clearly demonstrate thenecessity for further development of methods for incor-

p ora t in g t h e e ff ect of ra n do m, s y s t e ma t ic, a n d e ve nmodeling errors in uncertainty and sensitivity analysesof computer models. Because of the obvious complexityof error propagation in a computer model, which canin v o lv e h u n dre ds o f e q u a t io n s a n d v a ria ble s , M o n t eCa rlo me t h o ds s e e m t o be t h e mo s t p ra c t ic a l wa y o fincorpora t ing the effect of these types of errors in theuncertainty analysis.

4.1. Method. I t is re a s o n a ble t o in corp ora t e t h e

systemat ic and ra ndom error effects in the uncerta intyana lysis at the sta ge of the computer model where theya r e i n i t i a l l y d e t e c t e d . I n o t h e r w o r d s , i f w e h a v e acomp u t er mode l t o s imu la t e t h e p erf orma n c e of aliquid-liquid extraction operation involving fluid-phaseequilibria predictions (assuming no modeling error), thep re se nce of s y st e ma t i c a n d r a n d om e rr or s w i ll b ed et e ct e d m a i n ly i n t h e e xp er i me nt a l d a t a u s ed t oregress any parameter of the computer model. Thus, themain idea behind the approach proposed is to performperturbations in the experimental data according to thetype of errors present and their probability distribu-tions, an d then t o obta in the necessary m odel para metersets to perform Monte C ar lo simulat ions of th e systemunder t he uncerta in conditions detected. This a pproa ch

h a s t h e a dv a n t a g e o f n o t e x p lic i t ly de a l in g wit h t h ee s t ima t io n o f e x p e rime n t a l e rro r p ro p a g a t io n t o t h ep a ra me t e r e s t ima t e s , wh ic h is n e e de d t o de f in e a p -propriate probability distributions of the para meters ifthe uncertainty analysis is carried out starting from thep a ra me t e r s p a c e a s u s u a l ly don e .8,9 Addit ionally, theis su e o f a n a ly zin g s y s t e ma t ic t re n ds p res en t in t h eexperimental dat a becomes easier.

As a f irs t s t e p, t h e a p p roa c h con s ist s of de fin in gprobability distributions for the random errors basedon experimental evidence. This includes informationa bo u t in s t ru me n t s t a t is t ic s a n d a n y o t h e r s o u rc e o finformation available. Second, one defines the n a t u r e of any systema tic error present in the experimental d at a

by c o mp a ris o n wit h s t a n da rd v a lu e s , wi t h l i t e ra t u resources, a nd/or from the experimenta l conditions usedt o o bt a in t h e da t a . W it h t h is in f o rma t io n , o n e e s t a b-lishes a ny bia s limits a nd dyn am ic dependencies of thebias errors for the model varia bles to define appropriat eprobability distributions. Once the sta t ist ical informa-tion for the errors involved ha s been generated, t herear e tw o possibilit ies to set up t he uncerta inty a nalysis:(a) separately study the random effects and the system-a tic ones or (b) ana lyze th e combined effect of both ty pesof errors. In this w ork, we a re interested in qua ntifyingt h e in f lu e n c e o f t h e bia s a n d ra n do m e rro rs o n t h euncertainty analysis as well as the combined effect ofboth of them.

For each experimental datum, using its probabilitydist r ibu t ion de fin ed f rom t h e e xp erimen t a l ra n do minformation [in practice, often the probability distribu-tions will be assumed normal N(xj,sx2)], one generates ns a mp les a c cordin g t o t h e p recis ion de sired f or t h eest imated cumulat ive distributions. These values arethen collated to produce n pseudo-experimental datasets. Then, one regresses n sets of model parametersu s in g t h e n pseudo-experimenta l da ta sets a s input .Eva luat ions of the computer m odel a re performed usingt h e n para meter sets obtained from t he previous step,an d th e cumulative frequency distr ibutions (cfd) for t heoutput va ria bles ar e constr ucted. These cfds representt h e p roba bili t y dis t r ibu t ion s of t h e ou t p ut v a r ia bleunder the influence of random errors. The stat ist ical

f(x+x,y+y) f(x,y) +f

xx +

f

yy + R2 (5)

Sr2

) (rxSx)2

+ (rySy)2

(6)

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characterizat ion of the cfd quantifies the effect of therandom errors on the computer model.

F o r t h e ca s e f or s y s t ema t ic e rrors , t h e a p p roa c hconsists of randomly generating pseudo-experimentalda t a s e t s f ro m in s ide t h e bia s l imit s by s h i f t in g t h eoriginal data set according to a representat ive rectan-gular probability distribution. Notice that the system-atic or bias limits can be dynamic with the independentvar iables, so the shifting procedure has t o take this int oa ccount. The meth odologies suggest ed for the est ima tiona n d a n a ly s is o f s y s t e ma t ic e rro rs by t h e a u t h o rs a n dorganiza tions mentioned in section 2 do not include thepossibility of any dynamic behavior of the systematicerrors with the variables involved in the experiment ormeasuring process. With the n pseudo-experimentald a t a s et s s y st e ma t i ca l l y ob t a i n ed , a g a i n t h e m od elp a ra me t e rs a re re g res s ed a n d e va lu a t io n s o f t h e m inthe computer model are d one to obtain the cumulat ivefrequency curves qua ntifying t he influence of the sys-temat ic errors on the output var iables.

Having the cumulat ive frequency distributions pro-duced by considering the systematic and random errorsseparately, now it is possible to separately evaluate theeffect of both types and to determine which has a more

pronounced influence on the output varia bles, t husimp ro v in g t h e s t a t e o f in f o rma t io n o f t h e mo de l f o rdecision-ma king purposes. Addit iona lly, as mentionearlier, it is of interest to evaluate the combined effecto f bo t h t y p e s o f e rro rs o n a n o u t p u t v a r ia ble o f t h ecomputer model. This can be done by systematicallyshift ing the experimental data set as explained abovefor the systema tic case a nd then shifting each individua ldatum by a random normal deviate using the methodol-ogy suggested for the analysis of random errors. Thisprocedure is repeated n t imes to generat e the pseudo-e xp er i m en t a l d a t a s et s n ece ss a r y t o p er f or m t h eparameter regressions, and the computer model evalu-at ions or simula tions. The cumulat ive frequency dist ri-bution produced by this approach quantifies the com-

bin e d e ff ect of bot h t y p es of e rrors . Th e a p p roa c hproposed is demonstrated below with a synthetic ex-am ple ta ken from th e literat ure and then w ith specificthermodynamic cases.

4.2. A Basic Example. A numerical exa mple involv-ing an exponential function presented by Dolby andLipton 18 i s u s e d t o d e m o n s t r a t e t h e u s e o f t h e n e w a p p ro a c h f o r a n a ly zin g t h e e f f e c t s o f s y s t e ma t ic a n drandom errors using Monte Carlo methods. The modelis of the form

wh e re x is the independent varia ble and R, , a n d a re

p a ra me t e rs . F or t h e c a s e o f a n a ly s is o f ra n do m e rroreffects, the experimental da ta were simulat ed by addingrandom normal deviates to a hypothetical set of truevalues defined by points sat isfying the r elat ionship

Th e h y p ot h e t ica l t ru e v a lu es we re obt a in e d byevaluating xk+1 ) k in eq 8, for k ) 0.0, ..., 6.0 with unitincrements. The normal distributions used for each oft h e da t a p oin t s a re N(y(xi),y) a n d N(x,x) for i ) 1, ...,7, an d y is computed as 10%of y(xi) and x is 10%of xi.

T h e bia s e rro r e f f e c t s a re s imu la t e d by ra n do mlychoosing values from a uniform distribution given byU[4.0,8.0]. Thus, eq 7 with R ) 1.0, ) 8.0, and )

0.55 a n d wit h R ) 1.0, ) 4.0, a nd ) 0.55 definedthe synt hetic bias upper a nd lower limits for y. Figure1 shows t hese two limits and h ypothetical experimenta ldata showing the behavior of the random error used.

One hundred pseudo-experimental data with randomdeviat es were used to regress the pa ram eters R, , a n d of eq 7 using a direct search optimizat ion method(Powells method 19). With the parameter sets obtained,eq 8 wa s evalua ted at x ) 1.0. The result s a re present edin Figure 2b, which is the cumulative frequency distri-bu t ion . I t s h ows t h e in flu en ce o f t h e ra n do m e rrorssimulat ed on tha t specific value of the model. Similarly,100 para meter sets (R, , ) w ere regressed from pseudo-experimental dat a sets using systemat ic deviates for yas explained before. Figure 2a presents the cumulativef re q u e n c y dis t r ibu t io n o f e q 8 e v a lu a t e d a t x ) 1.0.Co mpa rin g F ig u re 2, p a rt a a n d b, i t is clea r t h a t t h eeffect of the systematic error is more significant thant h e on e p rod u ce d b y t h e r a n d om e rr or s . Th i s w a sexpected in this case, showing that the methodology isable to distinguish random and systematic effects whenthey a re significan tly different. Another importa nt issueis t h a t t h e me a n of t h e c f d f or t h e ra n do m e rrors is ag ood e st i m a t e f or t h e u n kn ow n t r ue v a l u e of t h eoutput variable from the probabilistic standpoint. How-ever, this is not the case for the cfd obtained for the

y(x) ) R + x (0 < < 1) (7)

y(x) ) 1.0 + 6.0(0.55)x (8)

Figure 1. U p p e r a n d low e r s y n t h e t ic s y s t e m a t ic e r r or l im it sdefined for the model y(x) ) R + ()x. The points shown alsoinclude a 10%random normal variat ion on hypothetical experi-m e nt a l d a t a .

Figure 2. Comparison of the systematic and random effects onthe cumulat ive frequency curve est imation for the function valuea t x ) 1.0 in t he equat ion y(x) ) R + ()x.



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systematic effects; any value on that distribution canbe t h e u n k n o wn t ru e v a lu e . T h is is be c a u s e e v e ryvalue is equa lly likely in t his case, but t he knowledgeof the cfd broadness becomes very importa nt for decisionm a k in g , e v en m or e i m por t a n t t h a n f or t h e ca s e ofrandom error effects. The cfd broadness is a measureof how uncertain a nd sensit ive ar e the output va riablesto systema tic effects. Also notice tha t, because th is is asimple model, the effect of systematic input deviatesfrom a uniform distribution produces a uniform effecto n t h e dis t r ibu t io n o f t h e o u t p u t v a r ia ble . F o r mo recomplex models, a s shown lat er, this effect is no longerobserved because of nonlinear effects on the uncertaintypropagation.

Figure 2c shows the cumulat ive frequency obtainedfrom including both types of errors combined in theu n c e rt a in t y a n a ly s is . I t is o bs e rv e d t h a t , in t h is p a r-ticular case, the broadness of the curve is mainly definedby the systema tic effects. In other w ords, the individua luncertainty effects of the systematic and random errorsare not a ddit ive when both a ct together. This suggeststhat the main individual effect or error type controlsthe uncertainty propaga tion through the model.

Un der accepted experimenta l ana lysis and mea sure-me n t me t h ods , i t is e x pe ct e d t h a t t h e e xp erimen t a lvalue reported corresponds to the mean of a series ofexperimenta l measu rements ca lled replicates. Thus, inprinciple, the stat ist ics associated with the reportedvalues should also be included. Unfortuna tely, it is verycommon to f ind tha t t he lat t er is not done, and usua llyonly a vague percentage value for the random uncer-tainty present in the data is reported, with no mention-ing of any possible systematic error. This situation canaffect t he resolution of the pr oposed m ethod because ofthe inherent uncertainty present w hen choosing th e biaslimits an d a lso because one is using the reported m eanvalues t o incorpora te the ra ndom error effects. Theconsequences of this w ill be reflected a s a n overestima -tion of the uncerta inty effects. To eva luat e the resolutionan d/or sensitivity of the proposed meth od, uncerta intylevels were assigned to the definition of the bias limitsand to the precision of the estimation of the hypotheticalexperimental mean values.

As explained before, in Figure 2 (par ts a and c) thebia s l imit s we re a rbi t ra r i ly de fin ed u s in g values of4.0 a nd 8.0 for t he lower a nd upper limits, r espectively.For practical purposes it is convenient to define the biasl imit s in s u ch a wa y t h a t t h e y c a n re p res en t e xt re melimits or the worst case scenario. To see the effect ofusing uncertain bias limits for the systematic errors,we determined the cumulat ive frequency distribution(cfd) of the function y(x) ev a lu a t e d a t x ) 1.0 using a

n orma l dist r ibu t ion wit h me a n ) 6. 0 a n d s t a n da rddeviation ) 1.0 to calculate the lower and upper biaslimits of the para meter in eq 7. One hundred randompairs of values were drawn from this distribution todefine bias limits, and then the approach proposed wasu s ed t o g e n er a t e a cf d f or e a ch p a ir of b ia s l im i t sob t a i n ed d u r in g t h e s a m p li n g. Al l t h e cu m u la t i v efrequency distribution curves genera ted a re plotted inFigure 3. The main points to observe are the generalbroadness of all the cfd curves together and the generalshape of the distributions. Comparing these chara cter-is t ics w it h t h e on e s of F ig u re 2a , we ca n s e e t h a t t h elatt er has similar broadness and sha pe. This means tha tt h e u s e o f a p pro xima t e d e xt re me bia s l imit s (in t h is

e xa mp le a h y p ot h e t ica l wo rs t ca s e s ce n a rio) cov erre a s on a bly we ll t h e bia s l imit s n orma l dist r ibu t ionassumed.

Th e ca s e of e rror in t h e e xp erimen t a l ly re port e dmean-value random error est imate (SR) is studied byassum ing a n order-of-ma gnitude less relat ive error t ha nthe one present in single-experimental measurements.It is important to mention that when using a reasonablesam ple size, the varia nce of the mea n estima tion shouldbe v e ry s ma ll wh e n c o mp a re d t o t h e u n c e rt a in t y o fs in gu la r e xp erimen t s . I n t h e p res e nt e xa mp le, t h e

pseudo-experimental true values were at first perturbedby draw ing values from a normal distribution with t het r u e v a lu es a s me a n s a n d 1% s t a n da rd de v ia t io n , a n dthen the methodology proposed for analysis of randomerrors explained before w as used with each of the dat asets generated by the first ra ndom perturbat ion process.The results ar e presented in F igure 4, wh ich consistsof 100 cfd curves. Notice that the uncertainty of themean estima tion is not very significant in th is ca se, an dthe cfd curves ar e similar to th e ones in Figure 2b. Thissuggests that when the uncertainty of the mean estima-tion is small, it is reasonable to expect a good resolutionfrom the approach proposed.

The combined effect of uncerta inty on the bia s limitsdefinition a nd experimental mean estimat ion w as simu-lated by combining t he t wo cases explained a bove. Atf irs t , t h e bia s l imit s u n ce rt a in t y wa s in t rodu ce d, f ol-lowed by the uncertainty on the mean est imation. Theresults a re presented in F igure 5. As pointed out for theca s e w it h ou t u n c ert a in t y on t h e bia s l imit s a n d me a ne st ima t ion (s ee discu s s ion f or F ig u re 2), t h e e rrorpropagation is mainly defined by systematic errors inthis case, and the broadness of the cfd shown in Figure2c is similar to the general broadness of the cfd curvespresented in Figure 5.

In genera l, i f t here is evidence of significant uncer-t a in t y on t h e bia s l imit s de f ini t ion a n d e xp erimen t a lmean estimation, the methodology described above canb e i n cor por a t e d a s a n a d d i t ion a l s t ep t o s t u d y t h e

Figure 3. Systema tic error effects on th e cumulat ive frequency

curve est imat ion for th e function va lue at x ) 1.0 in t he equat iony(x) ) R +()x for 100 bias limits simula ting th e uncertainty effecton the bia s limits definit ion.



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impact of these uncertaint ies on the cumulat ive fre-quency distributions est imation, and in tha t w ay evalu-a t e i f mo re p recis ion in t h e in pu t da t a is re q u ired t oimprove the uncertainty analysis.

5. Thermodynamic Applications

To illustra te the a pplica tion of the uncertaint y a na ly-sis approach presented in t his w ork, tw o liquid-liquidextraction cases using the UNIQUAC activity coefficientmodel are studied, involving liquid -liquid equilibriapredictions for the ternary systems diisopropyl ether +acetic acid + wa ter a nd chloroform + acetone + w a t e r .

5.1. The UNIQUAC model. The UNIQUAC model( A bra ms a n d P ra u s n it z20) is wide ly u s e d f o r v a p o r -l iq u id a n d l iq u id-l iq u id e q u il ibr ia . I n t h is wo rk , t h ep a ra me t e rs re g re ss ed f or t h is mode l a re bi j a n d bjiaccording to the following equation and assumptions:20-23

wh e re

The objective function definit ion is based on t heminimization of the distan ces between experimental andestimated mole fra ct ions, using a inside-varia nce est i-mation method proposed by Vasquez and Whiting24 a sthe regression a pproach, which aut omatically reweightsth e objective function. The optimiza tion procedure usedwa s t h e mu lt idimen s ion a l P o we ll me t h od, dis ca rdin gthe direction of largest decrease to avoid the problemsof the quadratically convergent method (for details, seeP re s s e t a l .19).

5.2. Diisopropyl Ether(1) + Acetic Acid(2) +Water(3). The experimenta l da ta for t his system w eretaken from Treybal,25 Othmer a nd White,26 a n d H la v a t ya n d L in ek .27 Th e la s t t wo da t a s et s a re a ls o rep ort e dby Srensen and Artl. 28 Figure 6 shows the experimen-tal tie lines for the different data sets mentioned. Clearsystemat ic trends are observed in the left phase of thesystem, as well as systemat ic var iat ions on the t ie-line

Figure 4. Random effects on the cumulat ive frequency curve

est imation for the function va lue at x)

1.0 in t he equat ion y(x))

R + ()x for 100 pseudo-experimental data sets s imulat ing theeffect of having uncertainty on the experimental mean estimation.

Figure 5. Combined uncerta inty effect on the bia s limits defini-tion and pseudo-experimental mean estimation on the cumulativefrequency curve est imation for the function value at x ) 1.0 in

the equat ion y(x)) R +

()x

.

Figure 6. Systematic error limits defined for the left phase inthe liquid-liquid equilibria of the diisopropyl ether(1) + aceticacid(2) + water(3) ternary system at 25 C based on evidence ofbias errors found using experimental data from different sources.

ln i )i

xi

+z

2qi ln

i

i

- qi ln ti - qij

ji j

tj

+

li + qi -i

xi

j

xjlj (9)

i )qixi

qT

; qT )k

qkxk; i )rixi

rT

; rT )k

rkxk;

li )z

2(ri - qi) + 1 - ri; i j ) exp(bi j/T);

ti )k

kki; z ) 10 (9)



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slopes. According to the original references a nd ad-dit ional litera ture related to experimenta l dat a report-ing (see, for example, Higashiuchi and Sakuragi 29), itis u s u a l ly cla imed t h a t t h e s e k in d of e xp erimen t a lmeasurements a re a ccura te w ithin 1% of the experi-me n t a l v a lu e , wi t h e rro rs n o rma lly dis t r ibu t e d. T h eda t a s e t re port e d by H la v a t y a n d L in ek27 wa s u s e d t ointroduce the systema tic and ra ndom deviates a ccordingto the evidence presented (see Figure 6), to apply theapproach presented in this w ork. To simulat e the effecto f t h e ra n do m e rro rs , e a c h e x p e rime n t a l da t u m wa sindependently perturbed using ra ndom numbers gener-at ed from a normal distribution w ith the experimenta lvalue as t he mean a nd a sta ndar d deviat ion of 0.3236%of the mean, w hich defines limits of 1%as t he ma ximum

f or t h e e xp er i me nt a l d a t a . O n e h u n dr ed p se ud o-experimental da ta sets were generat ed in this wa y andused to regress 100 binary interact ion parameter setsfor the UNI QUAC equa tion. Then, 100 simulat ions ofa liquid-liquid extraction operation were performed tostudy the effect of the ra ndom errors on the predictedperformance. The unit operat ion used is an exampleselected from Treybal,25 w hich consis ts of 8000 kg/h ofa n a c et ic a c id-wa ter solution, conta ining 30% acid(ma ss), wh ich is to be counter-currently extra cted wit hdiisopropyl ether to reduce the acid concentration in thesolvent-free raffinate product. The column has 8 equi-librium st a ges, a nd t he solvent feed is 12500 kg/h. Thecolumn operates at 23.5 C. The output variable is thep erce n t a g e of a c et ic a cid e xt ra c t e d a t s t e a dy -s t a t econditions in the column or extractor. The extractionwa s s imu la t e d u s in g t h e A SP E N P lu s ( ASP E N P lu s isa trademark of Aspen Technology Inc., Cambridge, MA)process simulator.

Figure 7b shows the cumulat ive frequency distr ibu-tion obta ined for t his case, where th e varia tion observedin the percenta ge of acetic acid extra cted is between 73.3a n d 74.6%. F o r t h e a n a ly s is of t h e s y s t ema t ic e rroreffects on th e predicted performa nce of th is unit opera-t io n , t h e bia s l imit s de f in e d in F ig u re 6 we re u s e dt o g e t h e r wit h P ro c e du re 1 t o g e n e ra t e t h e re q u ire dpseudo-experimental data sets to regress the param-e t e rs o f t h e U N I Q U A C mo de l . T a ble 1 p re s e n t s t h especific limits used for water (component 3) for the data

s e t o f H l a v a t y a n d L i n e k .27 F ig u re 7a p res e nt s t h ecumulat ive frequency distribution obta ined under theinfluence of systematic errors. I t is observed that theeffect of this type of error is larger than the effect causedby random errors on the uncertainty of the predictedperforma nce. Additiona lly, th e effect of combining boths y st e ma t i c a n d r a n d om e rr or s w a s s t u di ed i n t h isexample. The pseudo-experimenta l da ta sets a re gener-a t e d by f irs t a p p ly in g P ro c e du re 1 t o in t ro du c e t h esystematic deviates in the data and second perturbing

a g a in t h e da t a g en e ra t e d f rom t h e f irs t s t e p by in t ro-ducing random deviates for each datum as explainedbefore. Figure 7c presents the cumulat ive frequencydistribution for the combined effect of both types oferror. It is observed that the distribution chara cteristicsar e very similar to the ones obtained from t he system-atic error effects, suggesting that the error type withlarger effects m ainly defines the uncerta inty propaga-tion. Another important observation is the shape of thecumulative distribution for the systematic or combinedeffects. Notice that its shape cannot be predicted orinferred from t he sta tistical properties of the systema ticde via t e s (i .e ., u n iform dis t r ibu t ion ) beca u s e of t h enonlinearities and complexity of the computer model.This is not the case when the model is relatively simple,

as presented before w ith t he basic example of Dolby andLipton18 (see eq).

Procedure 1. Method for pseudo-experimenta l da tageneration ba sed on systema tic limits defined for liquid -l iquid equilibria of terna ry syst ems.

1. I de n t i f y t h e p h a s e s wh e re s y s t e ma t ic e rro rs a represent.

2. Define approximate systematic limits on the ex-perimenta l-phas e envelope for th e phases a ffected. U sea binary plot to represent the phase equilibria (see, forexample, Figu re 6).

3. Choose an experimental data set representative ofthe pha se envelope. Give empha sis to experimenta l setsthat cover the regions with systematic trends.

4. Define the minimum an d ma ximum values for th ecomposition of the component in the x a x is , u s in g t h esystematic limits defined in the phase envelope (see, forexample, Table 1).

5. Represent the experimental t ie lines by straightlines of the follow ing form: y ) ax + b, where a a n d brepresent vectors containing the slopes and interceptsfor the t ie lines involved, respectively, a nd x i s t h ecomposition of the component in the x a x is , a n d s imi-larly for y.

6. G e n e ra t e f rom a u n iform dis t r ibu t ion U[0,1] afactor f1 for ra ndomly shift ing the pha se or pha ses withsystemat ic errors.

7. For a given phase, change the composit ion of the

component on the x axis by xine w

) ximi n

+ f1(xima x

- ximi n),

Figure 7. Uncerta inty of th e percentage of a cet ic acid extractedin t h e l iq uid-liquid extractor under different sources of uncer-taint y: systemat ic, random, and a combinat ion of both.

Table 1. Upper and Lower L imits Used To Define theSystematic Trends Found in the Li quid-Liquid TernarySystem Diisopropyl Ether(1) + Acetic Acid(2) + Water(3)a

t ie lin e low er (m ol %) upper (m ol %)

1 0.00 2.002 3.50 8.803 5.60 10.004 12.00 19.205 17.80 26.206 20.50 30.00

a The values presented correspond t o the minimum and maxi-mum composition of wa ter (component 3) used in P rocedure 1 forthe left phase only of the data set reported by Hla vat y and L inek.27



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wh e re i ) 1, ..., M, a n d M is the n umber of t ie lines inthe experimental data set .

8. Generate another factor f2 from U[0,1]. Addition-ally, randomly assign the sign (+ or -) to the factor f2.

9 . E s t ima t e a t y p ica l f ra c t ion a l v a r ia t ion gs for thetie-line slopes by ana lysis of the experiment al da ta setsavailable.

10. Let aine w

) ai + f2gsai.11. Recompute t he vector b using th e new vector ane w

and the data of the phase not being changed.12. R ecompute t he composition of the component on

t he y axis by yine w

) aine w

xine w

+ bine w.

13. R e pe a t t h e p roce dure t o g en e ra t e n pseudo-e xp erimen t a l da t a s et s , w h e re n i s t h e n u m b er ofsimulations to be performed.

5.3.Chloroform(1) + Acetone(2) + Water(3). Thisis a very w ell-known system, w ith several equilibriumdat a sets ava ilable in the litera ture. The experimentald a t a u sed i n t h is ex a m pl e a r e f rom B a n cr of t a n dH u ba rd,30 B r a n c k e r a n d H u n t e r ,31 R e i n d e r s a n d D eMinjer,32 a n d R u iz a n d P ra t s ,33 all of th em a lso reportedby Srensen and Artl.28 Figure 8 shows t he experimen-ta l t ie lines of these four da ta sets. I t can be seen tha tt h e re is a clea r s y s t ema t ic t re n d f or t h e le f t p h a s e o fthe syst em. On t he ba sis of original r eferences for thee xp erimen t a l da t a s et s , t h e a v e ra g e e x pe rime n t a l orra ndom err or reported is a round 1% for the composi-t ion s . As p oin t ed ou t in t h e p rev iou s e xa mp le, n oreference is made to the presence of systematic trends.O n o ne h a n d, t h e d a t a s et f rom R ei nd er s a n d D eMinjer 32 was chosen to study the effect of the randomand systematic errors on the predicted process perfor-mance of a liquid-liquid extraction operation. Any of theother da ta sets can be used for th is purpose, but speciala t t e n t ion h a s t o be t a k en in t o a c cou n t t o e n s u re t h a tthe systema tic errors a re well represented. On th e otherhand, for this part icular system, the larger systematicerrors according t o Figure 8 seem t o be on t he left phaseat relat ive low compositions of wa ter a nd a cetone in theternar y system. This means th at the regions w ith mores y s t ema t ic e rror a re n ot s t ron g ly t a k en in t o a c cou n twhen predicting process performance because one ex-pects a reasonably high separat ion in the equilibriumstages during the extract ion. To obtain a low acetone

composit ion in t he extract , a large a mount of wa ter int h i s c a s e h a s t o b e u s ed , m a k in g t h e p r oces s n otpractical from the engineering sta ndpoint. H owever, theuncertainty on the predicted process performance dueto this type of error is st ill significant and larger thant h e u n ce rt a i n t y p rop a g a t ed f r om r a n d om e rr or s a sshown later.

Using the m ethodology described a bove to study thera ndom error effects with 1% of the composit ions aserror limits, 100 pseudo-experimental dat a sets weregenerated and used to regress 100 binary interact ionparameter sets for the UNIQUAC model. An exampletaken from Smith 34 wa s used to simulate a un it opera-t ion wh e re w a t e r is u s e d t o s e p a ra t e a c hlorof orm -

a cetone mixture in a simple counter-current extra ctioncolumn with two equilibrium stages. The feed containsequal amounts of chloroform and acetone on a weightb a s i s . T h e c o l u m n o p e r a t e s a t 2 5 C a n d 1 a t m . Asolvent/feed ma ss ra tio of 1.565 is used. The outputv a r ia b l e f or t h i s ca s e i s t h e p er ce nt a g e of a c et on ee xt r a c t ed . Th e op er a t i on w a s s im u la t e d u s in g t h eASP EN P lus process simulat or.

The cumulative frequency distribution obtained fort h e p erce n t a g e of a c et o n e e x t ra c t ed is p res en t e d inFigure 10b, which shows a small effect of the experi-menta l ra ndom error on the predicted performa nce. Forthe systema tic error case, the bia s limits w ere definedusing t he left-phase experimenta l envelope a s r eferenceand they are indicated by dashed lines in Figure 9. Thenumerical values of the limits used for w at er (compo-nent 3) a re presented in Ta ble 2. One hund red pseudo-experimental da ta sets were obtained using Procedure1, a n d t h e n 100 bin a ry in t e ra c t io n p a ra me t e rs we reregressed and used to evaluate the predicted perfor-ma nce of the liquid-liquid extraction.

Figure 10a presents the cumulative frequency distri-bution for the effects caused by the systematic trendde t e c t e d in t h e e x p e rime n t a l da t a . I t s bro a dn e s s iss ig n if ic a n t ly la rg e r t h a n t h a t f o r t h e c a s e o f ra n do me rr or s (F i gu r e 1 0b ), s h ow i n g t h a t t h e l a t t er h a s as ma ller e ff ect t h a n t h e s y s t ema t ic e rro rs o n t h e p re -dicted performance, even though the total variation forthe percentage extra cted is n ot very large. In general,

Figure 8. Liquid -liquid equilibria experimental tie lines for thechloroform(1) + acetone(2) + w a t e r ( 3 ) t e r n a r y s y s t e m a t 2 5 Cfrom different experimental data sources.

Figure 9. Systematic error limits defined for the left phase inthe liquid-liquid equilibria of the chloroform(1) + acetone(2) +wa ter(3) ternar y system a t 25 C based on evidence of bias errorsfound using experimental data from different sources.



9/10

if the safety factors used in the design do not cover thev a ria t io n o bs e rv e d in t h e p e rc e n t a g e o f a c e t o n e e x -tra cted because of systema tic errors, it would be recom-mended to obtain more accurate experimental data toavoid under- or over-design problems because everypredicted performance value in Figure 10a is equallylikely. Notice t ha t this type of conclusion cannot beo bt a in wit h o u t a n u n c e rt a in t y a n a ly s is o f t h is k in d,which shows t he importa nce and va lue of these methodsin process design and simulation. Analyzing Figure 10c,which shows the cumulative frequency distribution forthe combined effect of systematic and random errors,one observes (a s in the previous example) tha t the err ort y p e wit h a la rg e r e f f e c t o n t h e u n c e rt a in t y a n a ly s isma inly defines the final error propaga tion.

6. Concluding Remarks

A n e w a p p roa c h ba s e d o n M o n t e Ca rlo me t h ods t os t u dy t h e e f f e c t o f s y s t e ma t ic a n d ra n do m e rro rs o n

uncerta inty ana lyses wa s developed. The results showthat the proposed method clearly dist inguishes whenone type of error has a larger effect t han the other oneon th e predicted process performa nce of specific studycases, and tha t systema tic errors can have a significan trole in t h e u n c ert a in t y a n a ly s is . A ddit ion a l ly , i t wa sfound that the error type with the larger influence ist h e o n e t h a t m a i n ly d ef in es t h e b r oa d n e ss on t h ecumulat ive frequency distributions in the uncertaint yana lysis. Also, t he sha pe of the cumulat ive frequencydistribution cannot be inferred from the probabilitydistributions used to introduce the systemat ic deviateswh e n t h e comp u t er mode l in volve d is s ign ifica n t lycomplex. For th e case of ra ndom normally distributederrors, the cfd for the output generally tends to havenormal characterist ics. In general, experimental dataart icles in the literature (part icularly thermodynamicdat a) do not a dequat ely assess and report the presenceof possible systematic error. I t wa s shown th at system-at ic errors may be a very importa nt a nd common sourceof uncertainty. In addit ion, the method presented canbe used to facilitate decision making in safety factorselect ion, modeling, and experimental data measure-ment and design. As pointed out in the Introduction

section, modeling errors can ha ve a significant impacton uncertainty propagation through compupeter models.I t is well-known in the field of thermodynamics thatmost of the molecular-based models have limitat ionsand these effects ha ve to be ana lyzed when performingu n ce rt a in t y a n d s e n si t ivi t y a n a ly s is . Th is is a n o pe nproblem for further r esearch. The goal is to be a ble toquantify separately the contributions of random, sys-tematic, and modeling errors and in that way improvedecision ma king a nd process design. Resear ch effortsa re being conducted to incorpora te t he effect of modelingerrors in the qua ntificat ion a nd a na lysis of uncerta intypropaga tion in thermodynamic models.

Acknowledgment

T h is wo rk wa s s u p p o rt e d, in p a rt , by t h e N a t io n a lScience Foundation Grant CTS-96-96192.

Literature Cited

(1) Vasquez, V. R.; Whiting, W. B. Uncertainty and SensitivityAn a ly s is of Th e r m od y n a m ic M od e ls U s in g Eq u a l P r ob a b i li t ySampling (EPS). Comput. Chem. Eng. 1999, in press.

(2) Tarantola, A. In verse Problem T heory: M ethods for D ataFi t t in g a n d Mo d e l Pa r a me t e r Es t ima t io n ; Elsevier Science Pub-lishers: New York, 1987.

(3) American Society of Mechanical Engineers. A N S I / A S M E PTC 19.1s1985: M easurement Uncert aint y Part 1; AS M E: N e w York, 1985.

(4) Hayward, A. T. J . Repeatabil ity and Accur acy; Mechanical

Engineering Publicat ion Ltd. : New York, 1977.(5) Dietrich, C. F. Uncertainty, Calibration and Probabil ity, 2nded.; Adam H ilger: New York, 1991.

(6) International Organization for Standardization (ISO). Gu id e to Th e Expression of U ncertai nty i n M easurement, I S O : S w it z e r -land, 1993.

(7) Taylor, J . R. A n I n t r o d u ct i o n t o E r r or A n a l y si s , 2nd ed. ;Science Books: Sa usa lito, CA, 1997.

(8 ) I m a n , R . L . ; C on ov e r , W. J . S m a ll S a m p le S e n s it iv it yAnalysis Techniques for Computer Models , w ith an Applicat ionto Risk Assessment. Commun . Stat.-Theor. M ethods1980, A 9 (17),1749-1874.

(9) Ima n, R. L. ; Helton, J . C. An Invest igat ion of Uncertaint yand Sensit ivity Analysis Techniques for Computer Models . RiskA n a l . 1988, 8 (1), 71-90.

(10) Ima n, R. L.; Shortencarier, M. J . A FORT RAN 7 7 Pr o gr a ma n d Us er s Gu id e f o r t h e Gen er a t i on o f L a t i n H y p er c u b e an d

Figure 10. Uncertainty of the percentage of acetone extractedin t h e l iq uid-liquid extractor under different sources of uncer-taint y: systemat ic, random, and a combinat ion of both.

Table 2. Upper and Lower L imits Used to Define theSystematic Trends Found in the Liquid-Li quid TernarySystem Chloroform(1) + Acetone(2) + Water(3)a

t ie lin e low er (m ol %) upper (m ol %)

1 2.00 6.002 2.90 6.253 4.00 7.004 5.75 8.005 8.75 11.006 12.00 14.257 12.50 14.758 14.75 16.509 16.25 18.00

10 19.90 19.9911 22.45 22.5012 25.16 25.1813 26.59 26.6014 28.86 28.8615 33.13 33.1316 34.12 34.1217 49.92 49.92

a The values presented correspond t o the minimum and maxi-mum composition of w at er (component 3) used in P rocedure 1 forthe left phase only of the data set reported by Reinders and DeMinjer.32



10/10

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Received for review November 30, 1998Revised manuscript received April 7, 1999

Accepted April 14, 1999

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