COMMON ERRORS IN NUMERICAL PROBLEM SOLVING HOW CAN THEY BE DETECTED AND PREVENTED

Transcription

1 COMMON ERRORS IN NUMERICL PROBLEM SOLVING HOW CN THEY BE DETECTED ND PREVENTED Mordecha Shacham, Ben-Guron Unversty, Beer-Sheva, Israel Mchael B. Cutlp, Unversty of Connectcut, Storrs, CT 6269, US Mchael Elly, Intel Corporaton, Qryat Gat, Israel Introducton The ntroducton of mathematcal software packages such as Excel 1, MTLB 2 and Polymath 3 nto the feld of engneerng problem solvng has brought the benefts of hgher speed and precson n obtanng the results. However, numercal soluton technques have also ntroduced some new sources of errors. These errors may often pass undetected because of the lack of experence n the use of the new computatonal tools. few examples of errors orgnatng from numercal sources follow. 1. Regresson and nalyss of Data The most common errors n regresson of data orgnate from the use of regresson models wth too many or too few parameters [1], fttng polynomal models wthout proper scalng (standardzaton) of the temperature data, correlaton of data when the model equatons are mproperly lnearzed [1] or when expermental desgn for obtanng the data s not satsfactory [3]. 2. Systems of Nonlnear lgebrac Equatons (NLE) There are many examples n the lterature showng that dentfcaton of good ntal guesses and formulatng the problem correctly for enablng convergence to a soluton of an NLE system may represent a major challenge [4]. Even f the soluton s found, t may be nfeasble n the physcal sense (soluton wth negatve concentratons, for example) or a false soluton caused by mproper varable and/or functon scalng. 3. Ordnary Dfferental Equatons (ODE) Indscrmnate use of default error tolerances of the ODE solver tools s the most common source of errors n solvng ordnary dfferental equatons [5]. Falure to use the proper ntegraton algorthm (stff vs. non-stff), carelessly roundng numbers n the model equatons, usng the model outsde the doman of ts valdty and the use of nadequate resoluton n presentng the results have been documented [1],[2] as addtonal common sources of errors n solvng ODEs. Beng aware of potental causes to errors n numercal problem solvng s a prelmnary requrement to detecton and preventon of such errors. In ths paper three potental causes of errors are descrbed and demonstrated. Means to prevent these errors are proposed. No Consderaton Gven to Problem Parameter Uncertantes The mportance of the analyss of the senstvty of the problem soluton to small changes n the parameter values has been recognzed n the feld of optmzaton. Edgar et al. (21), for example, wrote "You should always be aware of the senstvty of the optmal answer Parameter values usually contan errors or uncertantes. Informaton concernng the senstvty of the optmum to changes or varatons n a parameter s therefore very mportant n optmal process desgn." However, senstvty analyss s rarely carred for other types of problems. In 1 Excel s a trademark of Mcrosoft Corporaton ( 2 MTLB s a trademark of The Math Works, Inc. ( 3 POLYMTH s a product of Polymath Software (

2 ths secton, the mportance of the senstvty analyss n solvng systems of nonlnear algebrac equatons wll be demonstrated. The example consdered was orgnally publshed by Froment and Bshoff (199), and ts soluton usng Mathcad was presented by Parulekar (26). The reacton consdered s the catalytc hydrogenaton of olefns (component ) n an sothermal CSTR. Dfferental mass balance on component yelds dc dt C = C τ r (1) where C s the nlet concentraton of (mol/l), τ = V / v s the space tme (s), V s the reactor volume (L), v s the nlet flow rate (L/s) and r s the reacton rate per unt reactor volume (mol/l s): r C = (2) ( 1 C ) 2 The concentraton of at steady-state s sought for the operatng condtons: C = 13 mol/l; V = 1 L; and v =.2 L/s. Fndng the steady-state soluton requres solvng the nonlnear algebrac equaton f(c ) = where f C C ( C ) = r (3) τ In Fgure 1, f(c ) s plotted versus C for the regon C 1. Observe that n ths regon there are three real solutons. The solutons as obtaned by the Polymath 6.1 software package are: C * = [ , , and ]. Polymath dsplays the results showng 7 sgnfcant dgts. Mathcad dsplays the same results wth 2 sgnfcant dgts (Parulecar, 26). The dscrepancy between the number of sgnfcant dgts n the CSTR parameters (one or two) and the number of those dgts reported n the soluton (up to twenty) cannot be mssed. The determnaton of the number of sgnfcant dgts to be shown n the documentaton of the problem soluton should be based, n ths partcular case, on the uncertanty assocated wth the parameter values. In Table 1, the values of C at the soluton are presented for varous levels of uncertanty n the flow rate v. ssumng an uncertanty of 1-4 L/s (thus flow rate of.21 L/s nstead of.2 L/s) yelds soluton values whch match the base soluton only n the frst two dgts. For uncertanty of 1-3 L/s there s only one correct dgt n the frst two roots and two correct dgts n the largest root. For an uncertanty of.1 L/s there s not even a sngle correct dgt n the frst two roots and one n the largest root. Thus, any realstc estmate for the uncertanty n the value of the flow rate leads to the concluson that the solutons for C should be rounded up to two sgnfcant dgts as all the addtonal dgts are uncertan. It should be emphaszed that ths analyss apples only to physcal quanttes whch obtaned as the fnal results of the computaton. There are other cases where roundng the numbers may lead to sgnfcant errors. Ths wll be demonstrated n one of the followng examples. s the value of the flow rate s specfed by one sgnfcant dgt, only one possble assumpton regardng the uncertanty could be that the flow rate was measured (or controlled)

3 wth such an accuracy, thus ts value could vary n the range of.15 L/s v.24 L/s. In the last two rows of Table 1 the solutons for the two extreme values of v are shown. For v =.24 L/s there s only one real root at C = mol/l whle for v =.15 L/s there s one real root at C =.3419 mol/l. Thus, n these cases there s only one root nstead of the three roots for the base value of v =.2 L/s, and the roots are completely dfferent. Use of Inapproprate Statstcs and Graphcal Representatons n nalyss of Regresson Models The use of napproprate statstcs and graphcal representaton may often lead to the acceptance unsatsfactory regresson models. Ths phenomenon wll be demonstrated by fttng the Clausus-Clapeyron equaton to vapor pressure (P V ) data of ethane. The vapor pressure data, contanng N = 17 ponts, are avalable [8] n the temperature range of T = 92 K (P V = 1.7 Pa) through T = 34 K (P V = 4.738*1 6 Pa). Ths range covers essentally the lqud phase regon between the trple pont temperature (= K) and the crtcal temperature (T C = K) of ethane. The Clausus-Clapeyron equaton s consdered napproprate for modelng vapor pressure for such a wde range of temperatures and we want to examne what knd of statstcs and/or graphcal representaton can reveal ths napproprateness. The Clausus-Clapeyron equaton s gven by B ln ( P V ) = + (4) T where and B are coeffcents calculated by regresson of expermental data. In Fgure 2 ln(p V ) exp s plotted versus 1/T. The straght lne ftted to the data s also shown. The lnear equaton obtaned s yˆ = x (5) where N 2 = 1 = N = 1 x = 1/ T and y ˆ = ln( P ) V calc. The correlaton coeffcent R2 s defned ( yˆ y) 2 R (6) ( ) 2 y y where y s the sample mean of the dependent varable. The value of R s bounded: R 1. If R s close to 1, there s a strong correlaton between the varables, whereas a value close to zero ndcates a weak or no correlaton. The R 2 value n ths case (as shown n Fgure 2) s.999, very close to 1. Thus, the plot of ln(p V ) exp versus 1/T and the statstcal R 2 both ndcate that the Clausus-Clapeyron equaton represents the vapor pressure data adequately over the entre range from the trple pont to T C. nother wdely used graphcal representaton for checkng adequacy of a correlaton s the plot of the calculated value of the dependent varable ŷ versus the expermental value y. In case of a good ft a straght lne should be obtaned wth the slope ~1.. In Fgure 3, ln(p V ) calc (as calculated by the Clausus-Clapeyron equaton) s plotted versus ln(p V ) exp. Observe that the data ponts are algned along a straght lne wth the slope of.9999 wth only small devatons. Thus, ths type of representaton ndcates also a good ft between the Clausus-Clapeyron equaton and the vapor pressure data over the entre range. However, comparng ln(p V ) calc and ln(p V ) exp at T = 92 K, for example, yelds ln(p V ) exp =.53; ln(p V ) calc =.95, whch s an ~ 8 %

4 dfference. Ths defntely cannot be defned as adequate representaton of the expermental data. The proper graphcal means to test the goodness of ft between a regresson model and the data s the "resdual" plot. The resdual εˆ s defned as ˆ ε = y yˆ (7) random dstrbuton of the resduals around zero ndcates that the regresson model represents the data correctly. defnte trend or pattern n the resdual plot may ndcate ether lack of ft of the model or that the assumed error dstrbuton for the data (.e. random error dstrbuton n y) s ncorrect. In Fgure 4 the resdual: ln(p V ) exp - ln(p V ) calc s plotted versus ln(p V ) exp. Observe that there s defnte trend (curvature) n the resduals, ndcatng that the Clausus-Clapeyron equaton cannot represent the data adequately. Careless Roundng of Model Parameters. In the DIPPR database [9], the Redel equaton s recommended for modelng vapor pressure data. The Redel equaton can be wrtten B E ln ( P V) = + + C lnt + DT (8) T where,b,c,d and E are the parameters of the regresson model. Usually the value of E s set at 2 to enable calculaton of the rest of the parameters by lnear regresson. Usng the same vapor pressure data that were dscussed n the prevous secton, the DIPPR staff obtaned the followng parameter values: = E+1; B = E+3; C = E+; and D = E-5. In Fgure 5, the resdual plot of ln(p V ) calculated by the Redel equaton usng the DIPPR parameter values s shown. Observe that the resduals n ths case are smaller by at least one order of magntude than the resduals obtaned usng the Clausus-Clapeyron equaton. There are two separate regons. In the frst one (up to ln(p V ) ~ 12) the resduals are n the range of -.4 to.2, not showng any partcular trend. In the second regon (ln(p V ) > 12) there s an order of magntude reducton n the resdual values (all smaller n absolute value than.3), however there s some trend n the resdual values n ths regon. To obtan perfectly random dstrbuton of the resduals, a dfferent correlaton should be used (the Wagner equaton, for example). Improvement of the correlaton by use of a dfferent type of equaton, however, s outsde of the scope of the present paper. The "copy and paste" faclty s a very convenent and tme-savng opton when transferrng numercal and other data between varous data-bases and software packages. For example, the Redel equaton constants for varous compounds can be coped nto an Excel spreadsheet. fter pastng the parameter values nto Excel, they are stored wth the exact same value as they appear n the DIPPR database, however they are dsplayed rounded to two three decmal dgts: = 5.19E+1; B = -2.6E+3; C = -5.13E+; and D = 1.49E-5 when the default number dsplay format s used. Copyng these values from Excel and pastng t nto a document or a dfferent software package wll results n a loss of the addtonal dgts whch were ncluded n the DIPPR publshed values. Calculatng the vapor pressure usng the rounded values of the parameters leads to sgnfcant errors n the calculated pressure values. The resdual plots of P V calculated by the Redel equaton usng exact and rounded parameter values are shown n Fgure 6. Observe that the resdual (error) n the hghest pressure range s of the order of 2.5% when rounded

5 parameter values are used n comparson to.3 % wth exact parameter values. Thus, careless roundng of parameter values causes the predcton error, n ths example, to ncrease by almost one order of magntude. Conclusons complete defnton of a problem should specfy the parameter values and the uncertanty levels for these parameters. Specfcaton of the uncertanty levels of the parameters enables a determnaton of the uncertanty of the soluton. Ths provdes justfcaton to number of sgnfcant dgts to be used n documentaton of the results and sometmes (as n the example presented) t may reveal that the proposed process cannot be mplemented n practce because of levels of uncertanty that are too hgh. The correlaton coeffcent ( R 2 ) and a smple plot of calculated values versus expermental values can be msleadng n determnng the goodness of the ft between the data and the regresson model. The resdual plot s the most relable means for examnng the goodness of the ft. transfer of numercal data between varous databases, documents and programs by electronc "copy and paste" may result n a loss of sgnfcant dgts of mportant parameter values. Beware of unntended roundng durng such transfer. Proper attenton to the potental causes of errors that are presented here and n prevous publcatons [1],[2],[3],[4],[5] can consderably mprove the relablty of the results n numercal problem solvng and n the correlaton of data by regresson. References 1. Brauner, N., M. Shacham and M. B. Cutlp, Computatonal Results: How Relable re They? - Systematc pproach to Modal Valdaton'', Chem. Engr. Educaton, 3(1), 2-25 (1996). 2. Shacham, M., N. Brauner and M. Pozn, "Potental Ptfalls n Usng General Purpose Software for Interactve Soluton of Ordnary Dfferental Equatons", cta Chmca Slovenca, 42(1), (1995). 3. Shacham, M. and N. Brauner, "Correlaton and Over-correlaton of Heterogeneous Reacton Rate Data", Chem. Engr. Educaton, 29(1) 22-25, 45 (1995). 4. Shacham, M., N. Brauner and M. B. Cutlp, Web-based Lbrary for Testng Performance of Numercal Software for Solvng Nonlnear lgebrac Equatons, Comput. Chem. Eng. 26(4-5), (22). 5. Shacham, M., N. Brauner, W. R. shurst, and M. B. Cutlp, "Can I Trust ths Software Package? n Exercse n Valdaton of Computatonal Results ", Chem. Engr. Educaton, 42(1), (28). 6. Edgar, T. F., Hmmelblau, D. M., and Lasdon, L. S., Optmzaton of Chemcal Processes, 2nd Ed., McGraw-Hll, New York, Froment, G.F., and Bshoff, K. B., Chemcal Reactor nalyss and Desgn, 2 nd Ed., Wley, New York, Parulekar, S. J., Numercal Problem Solvng Usng Mathcad n Undergraduate Reacton Engneerng, Chem. Engr. Educaton, 4 (1), 14 (26). 8. Ingham, H.; Frend, D.G.; Ely, J.F.; "Thermophyscal Propertes of Ethane"; J. Phys. Ref. Data, 2, 275(1991).

6 9. Rowley, R.L.; Wldng, W.V.; Oscarson, J.L.; Yang, Y.; Zundel, N.. DIPPR Data Complaton of Pure Chemcal Propertes Desgn Insttute for Physcal Propertes, (http//dppr.byu.edu), Brgham Young Unversty Provo Utah, 26. Table 1. Isothermal CSTR solutons for varous levels of uncertanty n the flow rate v. Feed flow rate (v ) L/s C, 1st Root C, 2nd Root C, 3rd Root Correct dgts are shown n bold f(c ) C Fgure 1. Plot of f(c ) versus C for v =.2 L/s 16 ln(pv )exp 12 8 y = x R 2 = /T (K -1 )

7 Fgure 2. Plot of ln(p V ) exp versus 1/T for ethane between the trple pont and T C ln(pv )Calc 8 y =.9999x ln(p V ) exp Fgure 3. Plot of ln(p V ) calc, calculated by the Clausus-Clapeyron equaton, versus ln(p V ) exp for ethane between the trple pont and T C.2.1 Resdual ln(p V ) exp Fgure 4. Resdual plot for ln(p V ) calculated by the Clausus - Clapeyron equaton for ethane between the trple pont T C Resdual ln(p V ) exp Fgure 5. Resdual plot for ln(p V ) calculated by the Redel equaton for ethane between the trple pont and T C

8 Exact Rounded 4.E+4.E+ Resdual -4.E+4-8.E+4-1.2E+5-1.6E+5.E+ 1.E+6 2.E+6 3.E+6 4.E+6 5.E+6 P Vexp (Pa) Fgure 6. Resdual plot for P V, calculated by the Redel equaton, for ethane between the trple pont and the T C