PlantâModel Mismatch Analysis in Deethyleniser Simulation. 2

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Plant-Model Mismatch Analysis in Deethyleniser Simulation. 2. Application to Data-Model Selection and Tuning Anthony H. Kamphausen and John A. O’Donnell* Chemical Engineering, RMIT University, Melbourne, Australia 3001

The focus of this investigation into difficulties associated with the simulation of a commercial deethyleniser is on the use of K-value data models based on equations of state as provided in PRO/II (Simulation Sciences, Inc.). For various K-value data-model definitions, the value of the temperature-independent form of the binary interaction coefficient, kij ) rij, of the dominant ethylene/ethane binary has been varied. All other thermodynamic data-model definitions were kept the same. The values of the optimized average equilibrium tray efficiency parameter, ηeff i , can vary markedly with the choice of the K-value data model and the kij values used and ranged from about 0.84 to about 1.01. The associated minimum total error objective function values are, however, remarkably similar, leading to the concept of “Equivalent” simulation models. For all three equation of state K-value data models examined, a quadratic relation was found to exist between a defined kij and the corresponding optimized average equilibrium tray efficiency. 1. Introduction One of the most critical factors in a distillation column simulation is often the physical and thermodynamic data models used.1-3 This is particularly so in the design, operation, and control of azeotropic distillation columns.4 The process of tuning appropriate equation of state (EOS) data models and using these EOSs in simulation studies in both their tuned and untuned form, together with classic quadratic mixing rules, has received considerable attention. Studies by Hernandez et al.5 on industrial superfractionators and Urlic et al.6 on the fractionation train of an industrial ethylene plant are among the most relevant to this investigation. In its widest sense, the concept of a thermodynamic data model is meant to encompass not only a particular EOS or liquid activity coefficient model but also the pure-component R formulation used for determining vapor pressure, the precise value of the pure-component and binary data-model parameters required for the particular data model, and the mixing rules applied to obtain mixture properties as a function of composition, temperature, and pressure. In distillation simulation, particularly for splitters, the most critical property in terms of the accuracy required is the vapor-liquid equilibrium (VLE) constant or K-value. Enthalpy is another important property but is less significant than the K-value in terms of the accuracy required for determining adequate simulation outcomes. Not only can the choice of a thermodynamic data model for evaluating K-values be significant in the simulation outcome, but it has been shown that for splitters the choice of critical binary interaction coefficient values, kij, can be more important than the choice of the actual data model.5 It has been suggested that in some separations the mixing rules applied to the EOS to determine K-values can also be more important than the EOS thermodynamic model to which it is applied. Consequently, difficulties encountered in producing an accurate computer simulation model of the commercial * To whom correspondence should be addressed. Tel.: +61 3 9925 2079. Fax: +61 3 9925 3746. E-mail: [email protected].

deethyleniser under investigation to meet operating needs were initially assumed to be at least partly related to the exact definition of the thermodynamic data model used. These needs were initially what-if studies associated with plans to reduce ethylene losses in the deethyleniser bottoms by increasing the maximum reboiler capacity, plans to increase maximum ethylene production levels by increasing column throughput, and the assessment of the effect of changes to equipment hardware and process operating conditions on the fractionation performance. The major objective in this work then was to apply the methodology developed in a previous paper by Kamphausen and O’Donnell7 for evaluating the relative accuracy of deethyleniser simulation models in terms of different definitions of the key K-value thermodynamic data model used. A comparative study was initiated where the most relevant commercially available PRO/II8 EOS data models for K-value generation were to be compared for applicability in deethyleniser simulation, as determined by the relative values of the minimum deethyleniser plant-model mismatch. The plant-model mismatch was evaluated in terms of an appropriate minimized error objective function (EOF) whose value represents the difference between the measured data and the computer calculated values. This difference is minimized by adjusting the best estimates of the real value of the measured variables as well as the value of an appropriate simulation model parameter defined in this work as the effective average equilibrium tray efficiency. The EOF chosen was defined in terms of the weighted and normalized least-squares errors around the deethyleniser involving as many plant measurements as practical, taking into account the wide range of accuracy inherent in the various measurements. The normalization term chosen was the best estimate of combined process and measurement variances. The EOS data models used were the Soave-RedlichKwong (SRK), Peng-Robinson (PR), and BenedictWebb-Rubin-Starling (BWRS) data models and were similar to those used in the studies of Hernandez et al.5

10.1021/ie049395b CCC: $27.50 © 2004 American Chemical Society Published on Web 10/30/2004

Ind. Eng. Chem. Res., Vol. 43, No. 24, 2004 7889 Table 1. Deethyleniser Operating Data (Averages)a process stream process variable

feed

bottoms liquid

liquid draw

vapor product

reflux

flow rate (TM/h) D/C(2) Inb flow rate (TM/h) D/C(2) outc flow rate (TM/h) PRO/II outd temperature (°C)d pressure [kPa (abs)]d

24.254 24.362 24.362e -17.86 2124

9.895 9.301 9.330 -2.91 2093

4.413 4.466 4.466 -27.29 2120

10.506 10.596 10.566 -27.30 2100

57.035 57.035 57.198 -27.30 2100

a D/C(2) ) DATACON Stage 2 reconciliation. b Values obtained from Stage 1 data reconciliation. c Values used for PRO/II input. Results based on the kij(ethylene/ethane) optimized SRK data model and optimized two-parameter simulation model. e Calculated as 52% liquid.

d

In addition to using the classic quadratic mixing rule as in previous studies, the Panagiotopoulos-Reid mixing rule was also examined as applied to both the SRK and PR cubic equations of state (CEOSs). The commercial EOS K-value data models were tuned by optimizing the associated ethylene/ethane binary interaction coefficient using the PRO/II8 associated software tool REGRESS9 as applied to the VLE data of Barclay et al.10 Manual optimization procedures were adopted after the REGRESS automatic optimization facilities, which make use of the weighted Orthogonal Distance Regression algorithm, had been shown to be inadequate for the task. The plant data used here were obtained from a controlled test run after a major revamp of the deethyleniser aimed at increasing throughput and efficiency. The test run involved the collection of over 40 independent measurements mostly as hourly averages of six minute data obtained from the distributed control system and laboratory-determined compositions taken over the period of the test run. Some of these measurements were combined as, for example, in the calculation of reboiler duty, and other measurements were used only for preliminary steady-state assessment as, for example, tank levels, leaving 31 independent variables to be part of the defined plant-model mismatch EOF. Measurement data include mass flow rates, compositions, temperatures, and pressures for the deethyleniser as well as upstream and downstream units. The nature of the thermodynamic data model used for ethylene and ethane K-value determination in the deethyleniser simulation model has been shown to be extremely important in the determination of the corresponding optimized simulation model parameter values. Although these values vary markedly with the choice of the K-value data model, the associated minimum total EOF values are very similar, leading to the concept of “Equivalent” simulation models as outlined in this work. Consequently, the original objective of unambiguously determining the relative accuracy of different K-value data models has not been achieved because on first inspection they would all seem to be approximately equally accurate although for widely different values of the associated tray efficiency simulation model parameters. It is, however, postulated that at least one of these “Equivalent” simulation models is, in fact, more accurate for use in extrapolation to other operating conditions. 2. Plant-Model Mismatch Analysis 2.1. Data Reconciliation. The plant operating data available did not meet mass and component balance constraints around the process plant to be simulated with mass balance inconsistencies around the deethyleniser averaging 11% over a period of 2 years. Even

after the application of carefully determined flowmeter calibration factors and operating conditions that were as close to steady state as possible, inconsistencies remained. Rigorous statistical data reconciliation techniques were therefore applied using the then latest available version of the engineering software tool DATACON11 to achieve mass and component balance closure. In addition, energy balance constraints and VLE constraints were selectively applied to the test run data before any plant-model mismatch analysis was undertaken on the reconciled results. Included in the data reconciliation results were average and standard deviation information on mass flow rates and compositions for deethyleniser feed and bottoms as well as for overhead vapor product and overhead liquid draw from the mixed-phase condenser. A 12-component composition slate as used in data reconciliation calculations was simplified to 9 categories for distillation simulation purposes and to 6 categories for the purpose of calculating plant-model mismatch EOF values for the deethyleniser. The final 6 categories encompassed ethylene, ethane, propylene, propane, butanes, and pentanes plus, where butanes and, in particular, pentanes plus are a large number of components lumped together under the headings 1-butene and 1,3-cyclopentadiene, respectively. A brief account of the two-stage data reconciliation procedure is given by Kamphausen and O’Donnell.7 Details of the data reconciliation procedure and results are available in work by Kamphausen.12 A summary of selected measurements for the deethyleniser is presented in Tables 1 and 2. The tables represent data reconciled values where available and for a comparison with simulated results include sample PRO/II output based on the concurrently kij(ethylene/ethane) optimized SRK data model and optimized two-parameter simulation model. Raw data and data adjustments to allow for calibration factors as well as operating data for other unit operations upstream are available in work by Kamphausen.12 2.2. Simulation Analysis. The methodology that is used is based on a manual stepwise optimization procedure labeled as the “Basic” optimization path as previously described by Kamphausen and O’Donnell.7 This methodology is an approximation that acknowledges the limitations of the sequential modular nature of the PRO/II8 simulation program and makes numerous assumptions that have previously been tested through either the use of alternative simulation models and alternative optimization paths or the use of spot simulation checks. The methodology adopted is based on a compromise between accuracy and time constraints. The starting point of the “Basic” stepwise manual optimization procedure is the selection of a computer simulation model Xp for the deethyleniser, where X defines a particular Performance Specification/Process

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Table 2. Deethyleniser Composition Data (Mole Fraction)a process stream stream component

feed D/C(2) inb

feed D/C(2) outc

bottoms D/C(2) ind

bottoms D/C(2) outc

bottoms PRO/II oute

ethylene ethane propyne propylene propane butadiene butenes butanes pentenes plus

0.6862 0.2595 0.0010 0.0159 0.0014 0.0030 0.0073 0.0065 0.0187

0.6912 0.2632 0.0011 0.0161 0.0015 0.0035 0.0075 0.0064 0.0093

0.1085 0.7596 0.0032 0.0468 0.0042 0.0103 0.0216 0.0183 0.0266

0.1085 0.7598 0.0032 0.0466 0.0042 0.0101 0.0216 0.0184 0.0269

0.1085 0.7604 0.0032 0.0467 0.0042 0.0101 0.0216 0.0184 0.0269

a D/C(2) ) DATACON Stage 2 reconciliation. The composition of the liquid draw, vapor product, and reflux is 99.93 mol % ethylene and 0.07 mol % ethane with trace impurities of other components at the ppm level. b Values obtained from Stage 1 data reconciliation. c Values used for PRO/II input. d Laboratory data. e Results based on the k (ethylene/ethane) optimized SRK data model and optimized ij two-parameter simulation model.

Variable combination and p defines the number of simulation model parameters. The feed to the deethlyeniser is defined by a known demethaniser bottoms stream at its bubble point that is passed through a valve and a heat exchanger before entering the deethyleniser. In this work, only deethyleniser models A1 and A2 are examined, where A is defined in terms of Process Specifications as the ethane impurity in the overhead product, ethylene in bottoms, and liquid draw rate together with the Process Variable combinations of reboiler duty, condenser duty, and liquid draw rate. The most likely value of the A1 average equilibrium tray efficiency, selected as the preferred simulation model parameter, is obtained by repeatedly evaluating the chosen EOF using the PRO/II Calculator module and plotting the results. The optimum A1 one-parameter simulation model is then expanded to the best A2 twoparameter model with independent parameter values in the rectifying and stripping sections, by repeatedly evaluating the EOF in a trial-and-error procedure. The EOF versus efficiency parameter relationships are presented graphically for the A1 simulation model, and these are used as a major means of comparing the performance of the various thermodynamic K-value data models in deethyleniser simulation. In addition, the major individual component errors of the EOF are extracted and compared for the optimized A1 and A2 simulation models to check for any differences in residual simulation errors (or discrepancies) due to the choice of data models. The simplified “Basic” optimization path approach adopted, using simulation models A1 and A2 only, resulted in a greatly reduced number of variables having values significantly greater than zero in the EOF because a large number had been set to and kept at their reconciled or measured values. 2.3. Data Models. These studies were undertaken to establish the relative suitability of the various K-value data models in accurately simulating a deethyleniser in terms of both A1 and A2 simulation models. The K-value data models that were examined are the PRO/II SRK and PR CEOSs with both quadratic and Panagiotopoulos-Reid mixing rules as well as the BWRS EOS. In order for such a comparative study to be meaningful, three distinct kij categories of the ethylene/ethane binary interaction coefficient were used for each K-value data model. The kij categories studied for all three EOSs were the kij ) 0.0000 (ideal mixing rule), kij PRO/II default, and kij-optimized data models using the ethylene/ethane binary data of Barclay et al.10 Only the

optimized ethylene/ethane binary interaction coefficients are used with the Panagiotopoulos-Reid mixing rule. The commercial data models examined all used the same PRO/II default PR enthalpy and vapor density together with Rackett liquid density data models, thus validating the comparison of their different K-value data models. 3. Thermodynamic Data Models 3.1. General Procedures. The SRK, PR, and BWRS EOSs used in this work are examples of commercial data models used for K-value generation. The quadratic mixing rule used in this work for calculating mixture terms am, bm, etc., is chosen on the basis of its widespread use in practical applications, its availability as an option for simulating distillation columns within the PRO/II software tool, and the ability of the associated software tool REGRESS to accurately generate optimized thermodynamic data-model parameters by the regression of experimental phase equilibrium data. The Panagiotopoulos-Reid mixing rule was examined as an alternative mixing rule and also on the basis of its availability as an option in both the PRO/II and associated REGRESS software tools and its ability to accurately generate optimized thermodynamic data-model parameters. For the EOS methods as applied to multicomponent systems, there are various pure-component R formulations, r(Tr) available. In this study, default pure-component R formulations for both the SRK and PR CEOSs are used in PRO/II and REGRESS throughout as defined by the Simsci Data Bank in PRO/II.8 The associated constant values are available from the internal database. Although the kij parameters are normally evaluated as constants, they are, however, essentially temperature-dependent and can be calculated using the software tool REGRESS. The REGRESS data regression software tool was applied to the ethylene/ethane VLE data of Barclay et al.,10 consisting of 77 data sets covering the complete composition range. Manual optimization procedures were adopted using the REGRESS program in its data validation mode, after the REGRESS automatic optimization facilities had been shown to be inadequate for the task of obtaining highly accurate and globally optimized binary interaction coefficients as required for systems of very low relative volatility. The reasons for this inadequacy for such binary systems can be shown to be due to the extreme sensitivity of the automatic optimized result to the initial estimate of the param-

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eters to be optimized and the existence of literally hundreds of local optima along the different optimization paths that result from changes to initial estimates. Regardless of whether the default initial estimate generator was used or whether various more likely initial estimates were entered to override the default values prior to the automatic optimization, in the vast majority of cases, the automatic optimization results refer to an inadequate local optimum rather than the global optimum that is required. An automated optimization approach currently being developed by Hulley13 of the Applied Simulation Research Group at RMIT University depends on the location of the global optimum from a large number of local optima. This approach involves interfacing the REGRESS software with a spreadsheet serving as a front end both to sequentially submit a large number of initial estimates for the parameter(s) to be optimized by REGRESS and also to retrieve the corresponding large number of automatic local optimization results from which to determine the global optimum value(s). The initial estimates are defined in terms of a starting value(s), final value(s), and increment value(s) for the parameter(s) to be optimized. The automated optimization approach has been successfully applied by Hulley13 to VLE data for the propylene/propane binary system. The results show that provided the boundary values and increments are appropriately chosen, clearly defined globally optimized results are obtained for both temperature-independent and temperature-dependent parameters for the binary interaction coefficient either individually or as a multiparameter kij data set. These results can be visually depicted for each parameter as a set of seemingly randomly generated EOF values plotted against the calculated kij parameter value to generate a parabolic envelope surrounding an interior filled with hundreds of locally optimized results. The minimum of this parabolic EOF envelope defines a global optimum for each kij parameter defined by the data model. There appears to be little or no relationship between the initial estimate for a parameter and the local optimum generated by the automatic optimization routine in REGRESS and therefore no systematic way of determining an adequate initial estimate. The REGRESS program, in its data validation mode, was used to generate values of average absolute percent deviation (AAD %) in the K-value for both ethylene and ethane and also the average of these for the SRK, PR, and BWRS EOSs, with different values of the binary interaction coefficient kij ) Rij, over the range kij ) 0.0000 to Optimum Plus. The value of 0.0000 represents ideal mixing rule behavior, and the highest value represents a value significantly greater than the PRO/ II default value. The optimum value is then obtained graphically by minimizing the sum of the AAD % in the K-value for both ethylene and ethane as applied to the data of Barclay et al.10 3.2. Equations of State. A two-parameter CEOS can be described in terms of the general form

P ) [RT/(V - b)] - a(T)/(V2 + ubV + wb2)

(1)

where u and w are constants, typically integers. For the SRK CEOS, u ) 1 and w ) 0, while for the PR CEOS, u ) 2 and w ) -1. For all CEOSs, P is the system pressure, T is the system temperature, V is the system volume, R is the

gas constant, b is the covolume parameter, and a(T) is the temperature-dependent attractive energy parameter given in terms of the R formulation r(Tr) by

a(T) ) a(Tc) R(Tr)

(2)

where Tc is the critical temperature and Tr is the reduced temperature. Pure-component R formulations can take many forms and can vary from one component to another, e.g., in REGRESS, and in the PRO/II SIMSCI data bank, the default formulations used in the SRK and PR CEOS are

R(Tr)Ethylene ) Tr2(c2-1) exp[c1(1 - Tr2c2)] c3(c2-1)

R(Tr)Ethane ) Tr

and

exp[c1(1 - Trc2c3)] (3)

where c1, c2, and c3 are equation constants. The BWRS EOS as modified by Starling14 is an improvement on the empirical BWR equation in terms of accuracy and can be presented in the following form:

P ) FRT + {B0RT - A0C0/T 2 + D0/T 3 - E0/T 4}F2 + (bRT - a - d/T)F3 + R(a + d/T)F6 + cF3/T 2(1 + τF2) exp(-τF2) (4) The 11 parameters for pure components (B0, A0, etc.) are generalized as functions of component acentric factor, critical temperature, and critical density. The BWRS EOS can correlate pure-component properties for light hydrocarbons very accurately when experimental data covering entire ranges are available. 3.3. Mixing Rules. It has been concluded15 that, generally speaking, simple quadratic mixing rules lead to the most accurate prediction of VLE data. The accuracy of correlating VLE data using a CEOS can, however, sometimes be improved further16 by choosing an appropriate mixing rule for calculating am [)am(T)] and bm, the attractive energy parameter and the covolume parameter for mixtures. The two alternative mixing rules investigated in this work for the SRK and PR EOSs are the quadratic and Panagiotopoulos-Reid. The mixing rules for the 11 mixture parameters of the BWRS EOS are analogous to the mixing rules used for the BWR equation. (i) Quadratic Mixing Rule. Classic quadratic mixing rules are used by the SRK and PR commercial data models to give mixture parameters cm in terms of component mole fractions xi and xj and cross-coefficients cij and are of the general form

cm )

∑i ∑j xixjcij

(5)

which for the attractive energy parameter takes the form

am )

∑i ∑j xixjaij

where, for aij ) aji, aij ) (1 - kij)xaiaj (6)

A general form of the temperature-dependent binary interaction parameter kij is given by

kij ) Rij + βij/T + γij/T 2 + δij ln T

(7)

where rij, βij, γij, and δij are equation constants and T

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is the absolute temperature and all terms are supported by PRO/II except the logarithmic term. Only the temperature-independent form

kij ) Rij

(8)

is examined in this work. (ii) Panagiotopoulos-Reid Mixing Rule. Panagiotopoulos and Reid17 proposed an asymmetric mixing rule containing two parameters applicable to both the SRK and PR EOSs. The interaction parameter they proposed for the attractive energy parameter term am is defined by

am )

∑i ∑j xixjaij

where,

for aij ) aji, aij ) {(1 - kij) + (kij - kji)}xaiaj (9) The two adjustable interaction parameters are kij and kji. The asymmetric definition of the binary interaction parameters can significantly improve the accuracy in correlating binary data for polar and nonpolar systems. This mixing rule has been used to test several systems, including low-pressure nonideal systems, high-pressure systems, three-phase systems, and systems with supercritical fluids. The results in all cases reported are in good agreement with the experimental data. 3.4. Parametrization Strategy. In the deethyleniser under consideration, the feed reasonably approximates a binary mixture of ethylene (the light key) and ethane (the heavy key), which together comprise about 95 mol % of the total, with the remaining 5 mol % consisting of C3, C4, C5, and higher hydrocarbons. The column has a large number of stages, with the main objective being the production of ethylene in the overhead product at a strict distillate specification of 99.93% ethylene, with the remainder consisting essentially of an ethane impurity. A subsidiary objective is to minimize the ethylene losses in the bottoms. There is also a low relative volatility between the key components. These criteria, according to Urlic et al.,6 identify the deethyleniser as a process critical separation unit that requires fine-tuning of the simulation model fitting parameters. To fine-tune the deethyleniser, the strategy adopted involves concurrent tuning not only of the key thermodynamic data-model parameter(s) but also of the effective average equilibrium tray efficiencies determined independently for both the rectifying and stripping sections. This approach was deemed preferable to tuning data-model parameters to average process conditions for the column either as a whole or separately in the two column sections and using theoretical stages as the simulation model parameters. In addition to expected improvements in simulation accuracy as compared with using data-model and theoretical stage tuning, the resulting simulation models are expected to be more stable over a larger range of operating conditions. The rectifying section exhibits a pinch zone near the top of the column and, as the most sensitive zone in the column, clearly requires fine-tuning of the fitting parameters. The stripping section is not comparably sensitive, and no separate tuning of data-model parameters would seem to be indicated, but because of the concentration of heavy components from the feed, this section can be expected to exhibit a lower average tray efficiency than exists in the rectifying section. This

theoretical consideration, together with the on-site experience of plant engineers, was the subsequent justification for using the concurrent tuning approach implemented. Application of the criteria proposed in the work of Gros et al.,18 viz., a strict key component specification and low relative volatility, leads to a pinch zone in the rectifying section of the deethyleniser. The key thermodynamic data-model parameters for a high-purity ethylene column operating very much like a ethylene/ ethane splitter in the top of the column at least are, according to Gros et al.,18 those required for accurate prediction of the dominant separation variables. These are the equilibrium ratios (Ki ) yi/xi) of the key component being removed in each sensitive zone or column section. In the pinch zone of the rectifying section, the composition of the heavy key component being removed, ethane, changes rapidly in relative terms, while the global composition and temperature of the mixture remain almost constant. Therefore, the dominant data-model parameter in the top of the column is Ki(ethane). In the stripping section where the separation goal is to remove as much of the ethylene light key as possible, the dominant data-model parameter is Ki(ethylene). Therefore, the dominant thermodynamic separation variable in the deethyleniser column as a whole is the relative volatility between the key components ethylene and ethane. This is in agreement with the conclusion reached by Urlic et al.6 in their sensitivity analysis of distillation processes as applied to the modeling of a splitter. The binary interaction parameter required for finetuning the dominant separation variable Ki(ethane) in the pinch zone of the rectifying section is clearly kij(ethylene/ethane) because this is close to being a binary mixture. In the stripping section, the bottoms product is approximately 75 mol % ethane and 11 mol % ethylene, with the remaining 14% consisting of a mixture of higher MW hydrocarbons, of which propylene at less than 5 mol % is present in the greatest amount. Although there is no pinch zone in the stripping section, the binary interaction parameter required for finetuning the dominant thermodynamic variable Ki(ethylene) is clearly also kij(ethylene/ethane) because the composition of ethane in the bottoms is about 5 times the composition of all remaining components other than ethylene. Although the data-model parameters according to Gros et al.18 should ideally be fitted to each sensitive zone temperature and at low concentrations of the key component, the approach in this work has been to fit kij(ethylene/ethane) over the whole range of data available in work by Barclay et al.10 This value is then used in the simulation model tuning procedure using effective average equilibrium tray efficiencies as semiempirical fitting parameters that adjust for any inadequacies in the data-model tuning procedure. It is envisaged that further research could be carried out with the objective of determining whether independent kij(ethylene/ethane) tuning in both rectifying and stripping sections according to the criteria given by Gros et al.18 above, concurrent with the tuning of independent rectifying and stripping section efficiencies, is warranted. Studies by Kamphausen12 using the data of Barclay et al.10 of the effect of kij tuning over a successively narrower pressure range more in agreement with the


operating pressure range of the deethyleniser show that, although the optimized kij values do indeed change, so do the values of the concurrently optimized effective equilibrium tray efficiency parameters, with the result that the optimized simulation models give “Equivalent” simulation models of approximately the same accuracy. For both the SRK and PR CEOSs, the studies show a progressive decrease in the optimized kij values as the pressure range is narrowed. This is accompanied by a steady decrease in the optimized average tray efficiency values.

Table 3. SRK (1), PR(1), and BWRS(1) K-Value Resultsa param kij ) kji

ethylene

ethane

average

SRK(1) manual optimized SRK(1) PRO/II default SRK(1) ideal mixing rule PR(1) manual optimized PR(1) PRO/II default PR(1) ideal mixing rule BWRS(1) manual optimized BWRS(1) PRO/II default BWRS(1) ideal mixing rule

0.0157 0.0112 0.0000 0.0154 0.0119 0.0000 0.0055 0.0030 0.0000

0.38 0.51 1.44 0.30 0.47 1.52 0.51 0.80 1.49

0.56 0.95 2.33 0.42 0.71 2.22 0.75 1.56 2.66

0.470 0.730 1.885 0.360 0.590 1.870 0.630 1.180 2.075

a

4. K-Value Data-Model Tuning Results A plot of the K-value AAD % against kij indicates an optimum kij value of about 0.0157 for the SRK data model when default R formulations are used. The value of 0.0152 quoted by Urlic et al.6 is consistent with a value of 0.0153 obtained by Kamphausen,12 if the original acentric factor formulation of Soave19 is used for the R term. These values for kij are in contrast to the PRO/II default value of 0.0112 and a value of 0.0178 obtained from REGRESS in automatic regression mode. Again a plot of the K-value AAD % against kij indicates an optimum kij value of about 0.0154 for the PR data model and is in contrast to the PRO/II default value of 0.0119 and a value of 0.0228 in automatic regression mode. Similarly, an optimum kij value of about 0.0055 is obtained for the BWRS data model and is in contrast to the PRO/II default value of 0.0030 and a value of 0.0065 in automatic regression mode. The explanation for the discrepancies offered by the Applied Thermodynamics group of Simulation Sciences is that the PRO/ II default values were, in fact, not optimized by Simulation Sciences but obtained by combining and averaging the results of different published optimum kij data. These published data were originally obtained by regressing experimental data from various literature sources available prior to that of Barclay et al.10 with possibly different EOF formulations as criteria. The reason REGRESS does not give the best value of kij in automatic data regression mode is that the initial estimate generator is adequate only for generating approximate parameter values and that the regression algorithm is inadequate for the required accuracy. Changing the initial estimates does not lead to appreciably better automatic regression results. A stepwise manual iterative procedure of data verification for different kij parameter values distributed around the regressed value with interpolation of the results is therefore required to obtain the optimum parameter value. The evidence presented by Kamphausen12 strongly suggests that appropriately optimized kij parameter values for a given EOS can be considerably more important than the choice of EOS or the associated choice of the mixing rule in determining the accuracy of a K-value correlation for binary data. This evidence is presented in Tables 3 and 4, where, for example, with quadratic mixing rules the average % K-value mismatch between experimental and simulated values for the three kij-optimized commercial EOSs is about the same or smaller (viz., SRK/PR/BWRS ) 0.47/0.36/0.63, Maximum Optimized EOS Spread ) 0.27), than the % K-value mismatch EOF between the kij ideal mixing rule, the kij PRO/II default, and the corresponding kijoptimized data model for each of the three commercial EOS data models under investigation (viz., SRK )

K-value (AAD %)

K-value data model

EOS(1) denotes one-parameter kij values.

Table 4. SRKP(2) and PR(2) K-Value Resultsa K-value data model SRKP(2) manual optimized SRKP(2) PRO/II default PRP(2) manual optimized PRP(2) PRO/II default a

parameters kij kji

K-value (AAD %) ethylene ethane average

0.0176

0.0144

0.40

0.50

0.450

0.0112

0.0112

0.51

0.95

0.730

0.0162

0.0146

0.30

0.39

0.345

0.0158

0.0171

0.35

0.49

0.420

EOS(2) denotes two-parameter kij values.

1.885/0.73/0.47, Maximum SRK Spread ) 1.415; PR ) 1.87/0.59/0.36, Maximum PR Spread ) 1.51; BWRS ) 2.075/1.18/0.63, Maximum BWRS Spread ) 1.445). It can also be noted that, for the strictly binary VLE information used in this analysis, there would appear to be only a marginal gain in using the kij-optimized Panagiotopoulos-Reid mixing rule as opposed to the kijoptimized quadratic mixing rule in terms of the average % K-value mismatch between experimental and simulated values (viz., SRK, 0.45/0.47; PR, 0.345/0.36). Any gain in simulation accuracy is not likely to justify the additional complexity required when one considers that the above kij-optimized results could be substantially modified in a multicomponent environment. 5. Plant-Model Mismatch Results 5.1. Total EOF Results. 5.1.1. Analysis by Equilibrium Tray Efficiencies ηeff i . For each of the three EOS categories using the quadratic mixing rule, the relative positions of the total EOF curves for simulation model A1 in Figures 1-3 for the three selected kij categories remain the same, for both the inferred A1 simulation model parameter values and the minimum total EOF value itself. For each EOS(kij), the kij ) ideal mixing rule or EOS(i) gives the lowest inferred (optimized) A1 simulation model parameter values of PR/ SRK/BWRS ) 0.90/0.88/0.84, followed by the kij ) PRO/ II default or EOS(d) values of PR/SRK/BWRS ) 0.98/ 0.95/0.88, with the manually optimized kij or EOS(o) curves giving the highest A1 simulation model parameter values of PR/SRK/BWRS ) 1.01/0.99/0.92. It can alternatively be stated that for each of the three kij categories the relative positions of the total EOF curves for simulation model A1 for the three selected EOS categories remain the same. For example, the BWRS(kij) EOS always results in the lowest inferred optimum A1 parameter value followed by the SRK(kij) EOS and the PR(kij) EOS always results in the highest parameter value. These relativities are again maintained for both the rectifying and stripping section parameters of the A2 simulation model. This constant

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Figure 1. SRK: simulation model A1.

Figure 2. PR: simulation model A1.

relativity is consistent with the previously made observation that the presumably optimized PRO/II default kij values were determined from a quite different VLE data set compared to the data of Barclay et al.10 used in this work. For all three EOSs, there is a power relationship, best described as quadratic, between the optimized equiliband the defined EOS(kij) rium tray efficiencies ηeff i value for both the overall A1 simulation model (see Figure 5) and the rectifying section of the A2 simulation model. There is no such discernible relationship evident for the A2 stripping section. For the SRK EOS, further studies by Kamphausen12 over a greater range of kij values show that this can perhaps be more accurately described as a highly correlated cubic relationship. There is again no such clear relationship evident for the A2 stripping section, and this can be largely explained by the high uncertainty that can be demonstrated to

Figure 3. BWRS: simulation model A1.

exist for the value determined for this parameter. Finally it can be observed, as shown in Table 5, that there is no simple relationship between the minimum data-model mismatch EOF and the minimum plantmodel mismatch EOF. This is further examined by Kamphausen.12 5.1.2. Analysis by Plant-Model Mismatch EOF. The above EOS relativities for the optimized tray efficiency parameters are basically the same for the minimum total EOF of both the A1 and A2 simulation models, implying that the BWRS(kij) EOS with the lowest minimum total EOF is the best available for both the A1 and A2 simulation models followed by the SRK(kij) and PR(kij) EOSs, with both of these approximately equal in accuracy. This is shown here for the EOS(o) category in Figure 4 with additional results available in Table 5 and in the work of Kamphausen.12 This conclusion would be contrary to the expectation, as judged from the data-model mismatch EOF, that the PR(kij) EOS is the best available. The minimum total EOF values are for EOS(i), PR/ SRK/BWRS ) 3.48/3.42/2.22, for EOS(d), PR/SRK/ BWRS ) 4.97/4.88/2.25, and for EOS(o), PR/SRK/BWRS ) 6.57/7.06/4.37. It is clear from these values, as depicted in Figures 1-3, that the A1 minimum total EOF for each EOS is obtained for EOS(i) followed by EOS(d), with EOS(o) always giving the highest value. This sequence is again opposite to that expected because the minimum data-model mismatch EOF is greatest in terms of the K-value AAD % for the EOS(i) category followed by the EOS(d) category and lowest for the EOS(o) category, as shown in Table 3. No particular significance can be attached to this reversal of expected sequence, however, because differences are extremely small and the one parameter A1 simulation model itself can be shown to be inadequate for precision applications. The A1 simulation model is, in fact, not used in industrial settings with a two-parameter A2-type simulation model required for careful analysis. The result of changing the mixing rule from quadratic to Panagiotopoulos-Reid for manually optimized kij values is shown in Figures 1 and 2 for the SRK and PR EOSs. It is clear that any improvement, if it can be quantified,

Ind. Eng. Chem. Res., Vol. 43, No. 24, 2004 7895 Table 5. Simulation Model Optimization Results EOS(kij) type

SRK(i)

SRK(d)

SRK(o)

SRKP(o)

PR(i)

PR(d)

PR(o)

PRP(o)

BWRS(i)

BWRS(d)

BWRS(o)

EOS kij EOS kji K-value EOF A1 parameter η minimum total EOF A2 parameter ηR A2 parameter ηS minimum total EOF

0.0000

0.0112

0.0157

0.0000

0.0119

0.0154

0.0030

0.0055

0.73 0.95 4.88 0.97 0.84 3.15

0.47 0.99 7.06 1.00 0.88 3.80

1.88 0.90 3.48 0.90 0.90 3.48

0.59 0.98 4.97 0.99 0.88 2.79

0.36 1.01 6.57 1.02 0.89 3.25

0.0164 0.0146 0.345 1.01 6.16 1.02 0.89 3.10

0.0000

1.89 0.88 3.42 0.89 0.87 3.08

0.0176 0.0144 0.45 0.99 6.7 1.00 0.87 3.70

2.075 0.84 2.22 0.84 0.84 2.22

1.18 0.88 2.25 0.89 0.81 1.53

0.63 0.92 4.37 0.94 0.80 1.69

is marginal and does not warrant the additional computational resources required for this system at these operating conditions. The corresponding optimized A2 simulation model results are presented in Table 5 and indicate the same relative positions for the rectifying section, ηR, and stripping section, ηS, tray efficiency parameter values for the same three EOSs and three kij categories as those for the A1 simulation model. In all cases the inferred stripping section efficiency was found to be lower than the rectifying section efficiency, and this was most pronounced for the optimized data models. This finding was qualitatively in general agreement with previous industrial experience. For all three EOSs, there is a reduction in the minimum plant-model mismatch total EOF in going from the A1 to the corresponding A2 simulation models, and this reduction increases markedly as a percentage of the minimum A1 value as kij values increase. The reduction is marginal for the kij ) 0.0000 category and increases to an approximately 50% reduction for the kij ) optimum category as detailed in Table 5. The final A2 simulation model results show a remarkable approach to convergence of minimum total EOF values for all EOS and for all kij values. This observation, together with the fact that uncertainties in the residual A2 minimum total EOF are very high, especially because of probable operating data bias in the stripping section, means it is not possible to conclude that any one EOS(kij) combination is, in fact, superior for the purpose of optimizing simulation model accuracy. Certainly, the apparent marginally superior results for the BWRS EOS

Figure 4. kij ) Optim; simulation model A1.

are not consistent with the associated K-value datamodel deviations where, for example, the optimized BWRS average K-value data-model mismatch is 0.63% compared with 0.36% for the optimized PR data model. They are also not consistent with the extremely improbable optimum simulation model parameter value of 0.92 using the manually optimized BWRS kij value of 0.0055 (see Table 5). More probable values as estimated by industry are tray efficiency parameter values closer to 1.0. The calculated variables having a significant effect on the value of the total EOF were the reflux rate, reboiler duty, actual tray 84 vapor ethylene mole fraction, actual tray 80 temperature, bottoms temperature, and reboiler return temperature. As deviations from optimum kij values increased, errors associated with the demethaniser bottoms temperature and deethyleniser feed temperature also increased. This was not unexpected because these measurements had been given a small initial bias so as to minimize their contribution to the total EOF for the specific case of the kij-optimized SRK K-value data model. Of all of the variables in the “Basic” optimization path approach using Ap simulation models, the calculated reflux rate was found to be by far the most susceptible to error, and the minimization of this component error was critical to the total EOF minimization. The component EOF results can be organized by EOS and kij categories, as was done previously for total EOF

Figure 5. Binary interaction coefficient vs model A1 optimized tray efficiency.

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Figure 6. SRK: models A1/A2. The acronym EOSp,(kij) is used for the above bar charts, where for EOS S ) SRK, P ) PR, and B ) BWRS, for p 1 ) model A1, 2 ) model A2, and for kij i ) ideal mixing rule, d ) PRO/II default, and o ) manually optimized.

Figure 7. kij ) Optim: models A1/A2. The acronym EOSp,(kij) is used for the above bar charts, where for EOS S ) SRK, P ) PR, and B ) BWRS, for p 1 ) model A1, 2 ) model A2, and for kij i ) ideal mixing rule, d ) PRO/II default, and o ) manually optimized.

results. The SRK EOS results are presented as a function of kij for both A1 and A2 simulation models in Figure 6, and the optimized kij results are presented as a function of EOS for both A1 and A2 simulation models in Figure 7. Very similar results can be presented for the PR and BWRS EOSs and the ideal mixing rule, where kij ) 0.0000 and PRO/II default kij are values as shown by Kamphausen.12 5.1.3. Analysis by EOS Category. Within each EOS category, the main reason for the paradoxical increase

in the minimum total EOF for the A1 simulation model as the ethylene/ethane kij value increases from kij ) 0.0000 for the ideal mixing rule to kij ) PRO/II default value to kij ) optimized value is an increase in the component EOF of the actual tray 84 mole fraction of ethylene with other changes close to zero. This contribution to the minimum total EOF virtually disappears as an additional independent simulation model parameter is introduced for the A2 simulation model. The result is essentially three “Equivalent” A2 simulation models within each EOS category, with only slight variations in the component contributions to the total EOF. For the SRK EOS, for example, the minimum total EOF for the three kij data models ranges from 3.08 to 3.80, i.e., a maximum difference of 0.72 EOF units. It is very similar for the PR data model with a range of 2.79-3.48 (0.69 EOF units) and for the BWRS data model with a range of 1.53-2.22 (0.69 EOF units). For an assumed 10 significant variables that contribute to the minimum total EOF, this gives an average spread of approximately 0.07 EOF units for each significant variable across the three kij categories for each EOS. To gain some perspective as to what the EOF values indicate, it should be remembered that each component EOF value is equal to the {(Simulated Value - Measured Value)/Standard Deviation}2.7 Therefore, a spread of 0.07 units per variable is equal to a spread of x(total EOF/10) ) x(0.07) ) 0.25 standard deviations for each significant variable. This is hardly sufficient to differentiate the relative accuracy of the kij values used in optimized A2 simulation models. Furthermore, if all optimized A2 simulation models are taken together for all EOSs and both mixing rules, the minimum total EOF ranges from 1.53 to 3.80, i.e., a maximum spread of 2.27 EOF units or x(total EOF/ 10) ) x(0.27) ) 0.51 standard deviations for each significant variable across all EOSs and all kij values. This is again hardly sufficient to clearly differentiate the relative accuracy of alternative K-value data models and is the quantitative basis for the concept of “Equivalent” simulation models, as tentatively defined in section 5.2. 5.1.4. Analysis by kij Category. For the kij ) ideal mixing rule category, there are significant error contributions from the demethaniser bottoms temperature, and the deethyleniser feed temperature, that are not present in the other two categories. This is because these temperatures had been initially adjusted by an energy reconciliation procedure using the optimized SRK data model that was consistent with the deethyleniser feed composition and temperature and the demethaniser bottoms temperature and pressure to leave zero residual error when this particular data model was used. Because the deethyleniser feed temperature in the PRO/II flowsheet is simulated by first a bubble-point routine and then constant heat transfer to the defined demethaniser bottoms stream, a change in the K-value data model is going to introduce an error component to the total EOF that would increase further as the data model deviated from optimized kij values. Arguably, a better analysis would have kept the upstream K-value data model as the optimized kij SRK data model throughout the investigation, with only the downstream deethyleniser K-value data model varied. The only other significant error contributions are due to the deethyleniser bottoms and the reboiler return temperatures. The minimized total EOF results for all


EOSs and for both the A1 and A2 simulation models are remarkably similar although the associated tray efficiency parameters are up to 0.06 units lower for the BWRS data model, as shown in Table 5. For the kij ) PRO/II default category, the inferred spread between optimized A2 rectifying and stripping section parameters is much larger than that for kij ) 0.0000 and increases considerably from the BWRS at 0.89/0.81 to 0.97/0.84 for the SRK and 0.99/0.88 for the PR EOS. The major difference between A1 and A2 component EOF values is the almost complete elimination of those stripping section errors due to actual tray 80 temperature and actual tray 84 ethylene composition. These errors were not even inferred in the ideal mixing rule category. Residual A2 component errors are, as before, due to the deethyleniser bottoms and the reboiler return temperatures. There are also very small residual errors due to demethaniser bottoms and deethyleniser feed temperatures for the PR and BWRS data models. For the kij ) manually optimized category, where for all three EOSs kij is larger than kij for the corresponding PRO/II default value, there is a further increase in the spread between optimized A2 rectifying and stripping section parameters to 0.94/0.80 for BWRS, 1.00/0.88 for SRK, and 1.02/0.89 for PR. Differences between A1 and A2 component EOF values are almost the same as those for the previous PRO/II default kij values except that the A1 models have significantly higher actual tray 80 temperature and actual tray 84 ethylene composition errors before the use of the second A2 modeling parameter virtually eliminates these errors. It can be noted that for all A1 and A2 simulation models the error contribution from the deethyleniser bottoms temperature and the reboiler return temperature remains roughly constant. 5.2. “Equivalent” Simulation Models. This concept was initially conceived for the A1 simulation model when it was noticed that, although the optimized tray efficiency parameter could vary between 0.84 and 1.01 for the various data models, no matter how wrong the assumed kij ethylene/ethane binary interaction coefficients were, the minimum total EOF values rarely varied by more than a factor of 2. Although this may seem large, in the context of the increase in the total EOF value by factors of 7 or more within 0.05 parameter units of the inferred optimum for the same EOS(kij) data model, this does not seem excessive. When the various A2 simulation models were optimized, it was found that, for the three EOSs and two mixing rules over the range of kij values examined (see Table 5), the highest ratio for the minimum total EOF values was 2.48. Because the subjectively determined uncertainty in the minimum EOF was also of this order or greater, it was decided that for the purpose of this work the definition of “Equivalent” simulation models would be as follows: For a particular column configuration and operating conditions, the ratio of the highest to the lowest minimum EOF in a set of “Equivalent” simulation models using different K-value data models should not exceed 2.5. It is this arbitrary definition that applies when the terminology is used here. This, of course, means that the absolute value of the minimum EOF values is not relevant to the definition, and these values may indeed change substantially from one set of operating conditions to another particularly as the standard deviation

estimates change. Although the definition can be separately used for both A1 and A2 simulation model categories, it is perhaps justified here, in terms of data uncertainties, only for the A2 models, and this is the main context in which it is used. The “Equivalence” of the A2 simulation models can alternatively be expressed by the highest EOF ratios of 1.23 for the four SRK, 1.25 for the four PR, and 1.45 for the three BWRS data models. It can also be expressed by the highest ratios of 1.57 for the three ideal mixing rule, 1.82 for the three PRO/II default, and 2.25 for the five manually optimized data models. From this, it would appear that the EOS category is more important than the kij category in determining the closeness of approach to “Equivalence” within a category. This is in no way inconsistent with the idea that the best kij value for any given EOS can provide better simulation results than the best EOS using inaccurate kij values as implied by Hernandez et al.5 Because, as far as the minimum total EOF for the deethyleniser is concerned, the various simulation models can be tuned to be roughly “Equivalent” within the bounds of experimental uncertainty, it would seem at first that the whole objective of the methodology developed in part 1 of this series of papers might be brought into question. This objective was, of course, based on the hypothesis that some K-value data models are very much better than others in the optimization of simulation model accuracy and that the best available data model could be identified as the one that gave the lowest minimum total EOF in any assessment of alternative optimized simulation models. There are, however, several reasons why the methodology is still expected to be very useful after further refinements are applied. For any reasonable K-value data model as defined by EOS, pure-component R formulation, kij value, and associated mixing rules, the methodology allows the quick and accurate determination of the corresponding optimized simulation model parameters for the given operating conditions. For some separations other than for the deethyleniser under investigation, it is quite conceivable that one or more of the optimized simulation models will, in fact, not be “Equivalent” as far as accuracy is concerned and that there will be a K-value data model that clearly provides the greatest simulation accuracy potential as identified by the relative values of the minimum total EOF. The approximate “Equivalence” of simulation models for the deethyleniser holds only for the specified operating conditions used in the EOF minimization procedure. This “Equivalence” does not necessarily hold when extrapolations are made to other operating conditions from identical base-case operating conditions. This is shown for the deethyleniser at least, in work by Kamphausen,12 where a number of “Equivalent” simulation models at identical base-case operating conditions are used to predict widely diverging simulation outcomes. It is, therefore, necessary to assume that, at least for all but one K-value data model, there will be a change in the optimized simulation parameter values in going from one set of operating conditions to another. Consequently, the methodology could be used to identify the best data model as the one that gives the minimum ratio of optimized simulation model parameter values over the likely operating range of the process concerned.

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The ratio(s) would, in fact, be 1.0 for the best possible data model because this would imply that the one simulation model can be confidently used without having to periodically redetermine the optimized simulation model parameter values over the operating range considered. There is, of course, the alternate possibility of keeping the optimized simulation model parameter values for one set of operating conditions the same and varying the K-value data-model parameters. The best possible simulation model could in this case be identified as the one where the ratio of optimum K-value datamodel parameter values was closest to 1.0. Using either approach, the methodology should allow the determination of the most accurate and most stable simulation model over a predefined range of operating conditions. 5.3. Advanced Applications of the Methodology. In principle, the methodology developed for optimizing simulation model accuracy as defined by the use of various features of the PRO/II simulation environment and applied to the deethyleniser should be readily transferable to other complex separation processes. Such processes include azeoptropic distillation, reactive distillation, and liquid-liquid extraction. In these processes, liquid activity coefficients are additional or alternative categories of data-model parameters that may require fine-tuning as, for example, in the separation of isobutanol from acetic acid using water to enhance the separation. Also the number of theoretical stages may be alternative simulation model parameters that require fine-tuning instead of tray efficiencies as, for example, in liquid-liquid extraction, where there is usually some form of continuous medium in which mass transfer takes place rather than on individual trays. Moreover, it is envisaged that the methodology as defined by a PRO/II Unit Operations module used in conjunction with the PRO/II Calculator and Optimizer modules is applicable, in principle at least, to any other chemical process such as a chemical reactor involving concurrent data-model tuning and simulation model tuning. At least some of these processes are likely to present more difficulties in the choice of key tuning parameters for each sensitive zone than the deethyleniser example presented here. There is every likelihood that multiple tuning parameters will be required for a simulation and some of these may require a temperature dependence to be built into them. Furthermore, a particular datamodel parameter may need to be tuned more than once to different sections of the process because of significant changes in process conditions and a particularly high sensitivity to these changes. Such conditions are likely to exist, for example, in a column where two phases exist in the actual column but three phases appear in the decanter or in the low- and high-pressure sections of a distillation train such as in the cryogenic separation of air, as illustrated in a PRO/II Casebook.20 This would make the development of an automated procedure for accurately minimizing an appropriate EOF for the process essential because the manual procedures described in this paper for concurrently tuning both datamodel parameters and simulation model parameters would be far too time-consuming. Such an automated procedure is currently under development and has been successfully applied by Hulley13 to fine-tuning a PRO/II simulation model for a propylene/propane splitter. The automated procedure involves interfacing a spreadsheet with the PRO/II simulation program for interactive pre- and postpro-

cessing of hundreds or more case studies in batch submissions similar to the automated procedure previously described for REGRESS. Each case study solves an optimization routine in PRO/II that results in a local minimum of an appropriate EOF, as determined by a Calculator module attached to the Column module in PRO/II. A batch of case studies is initiated in the spreadsheet environment by the specification of basecase values, increment sizes, and final values for the initial estimates of each of the fitting parameters. Postprocessing involves retrieving local optima from the simulation environment back into the spreadsheet environment for all case studies and searching for the global optimum. Finally, it is envisaged that the methodology is transferable in whole or in part to other simulation environments such as Aspen Plus and Hysys provided only that the appropriate regression and simulation modules are available and that the simulation environment can be interfaced with a spreadsheet environment to facilitate pre- and postprocessing of large numbers of simulation runs, each varying only by the values of the initial estimates of the parameters to be optimized. 6. Conclusions Of the commercial data models examined, the PR EOS provides the greatest accuracy in correlating the basic binary VLE data used in this work and would seem to offer the greatest potential as an effective commercial K-value data model in deethyleniser simulation and optimization. The results for the SRK EOS are very similar to those for the PR EOS, with the BWRS EOS showing the least promise as far as basic K-value data-model accuracy is concerned. The Panagiotopoulos-Reid mixing rule would appear to offer only a marginal advantage to the classic quadratic mixing rule in terms of correlation accuracy for a binary system but at the expense of greater complexity and computational time required. These conclusions are not necessarily transferable to multicomponent systems as they exist in the dethyleniser. The optimized ethylene/ethane kij parameters as determined in this work are significantly greater than the corresponding PRO/II default values for all of the EOSs considered in this work. There are two major sets of adjustable parameters used to minimize deethyleniser plant-model mismatch. They are the K-value data-model parameters defined in terms of the ethylene/ethane kij binary interaction coefficient and the simulation model parameters defined in terms of the effective average equilibrium tray efficiency, ηeff i . These parameters cannot, however, be independently tuned, and the choice of K-value datamodel parameters determines the tuning results for the simulation model parameters or conceivably vice versa. Careful tuning of the effective average equilibrium tray efficiency parameters, ηeff i , at defined ethylene/ethane kij parameters values has been shown to be vital in minimizing plant-model mismatch. Such tuned (optimized) simulation model parameters can be confidently used only for the particular K-value data-model and operating conditions that were used in the tuning procedure. For the one-parameter simulation models, the value of the optimized simulation model parameter, ηeff i , has been shown to increase almost linearly with the value of the assumed ethylene/ethane kij parameter for all three EOSs.


The use in this work of effective average equilibrium tray efficiency parameters, compared with the usual industrial choice of using the number of theoretical trays required as simulation model parameters, together with the use of a statistically sound EOF definition, has expedited the procedure for the optimization of simulation model accuracy. For all of the K-value data models tested, a two-parameter simulation model has been shown to reduce the minimized total EOF values for a simulation model by up to a factor of approximately 2 compared to the corresponding one-parameter model. Although the optimized simulation model parameter values can vary markedly with the choice of the K-value data model in terms of both the EOS and kij values used, the associated residual minimum total EOF values for the deethyleniser are similar, leading to the concept of “Equivalent” simulation models, as outlined in this work. This concept is unlikely to have significant practical application because future investigations are likely to prove the superiority of one of these “Equivalent” simulation models in applications involving extrapolations to other specified operating conditions. For the deethyleniser at least, such a superiority in the accuracy of extrapolations should be most evident where changes in column concentration profiles occur, due to either changing product purity specifications, changing ethylene/ethane ratios in the deethyleniser feed, or both because the critical kij values for the ethylene/ethane binary are expected to vary at least slightly with composition, pressure, and temperature. Any such superiority could be assessed in the sense that any adjustments required to the simulation model parameter values to again minimize the total EOF at the new operating conditions are minimized. Improvements obtained in simulation model accuracy and expected stability over a range of process operating conditions, as compared with previous work in the field, depend on properly reconciled global plant simulation results as well as the accurate concurrent tuning of thermodynamic data-model parameters, defined here as binary interaction coefficients, and simulation model parameters, defined here as average equilibrium tray efficiencies. The methodology used in optimizing simulation model accuracy in the PRO/II simulation environment can be expected to be readily applicable to other difficult separation processes such as splitters, cryogenic, azeotropic and reactive distillation, liquid-liquid extraction, etc., and is particularly advantageous where the lack of highly accurate thermodynamic data-model parameters and simulation model parameters can lead to significant inaccuracies in the economic process optimization of an existing plant. Improvements in the methodology currently under development involving interactive spreadsheet pre- and postprocessing as currently successfully applied to a propylene/propane splitter are envisaged as enabling the automated concurrent tuning of the data-model and simulation model parameters to other complex separation processes and possibly other unit operations that need to be defined by both data-model parameters and simulation model parameters as fitting parameters for laboratory or industrial operating data.

It is envisaged that the automated methodology involving total EOF minimization in a PRO/II simulation environment can also be transferred in whole or in part to other simulation environments, provided only that appropriate data models, simulation modules, and software interface capabilities are available. Literature Cited (1) Zudkevitch, D. Imprecise Data Impacts Plant Design and Operation. Hydrocarbon Process. 1975, March, 97-103. (2) Sandler, S. I. Thermodynamic Model and Process Simulation. In Foundations of Computer-Aided Process Design; Mah, R. S. H., Seider, W. D., Eds.; Engineering Foundation: New York, 1980; Vol. II, pp 83-111. (3) Zeck, S. The Effects of Thermodynamic Data on the Design and Operation of Distillation Columns. Int. Chem. Eng. 1993, 33, 184-196. (4) Cairns, B. P.; Furzer, I. A. Azeotropic Distillations Simulation and Experiment. Proceedings of Chemeca ‘87, Melbourne, Australia, 1987; pp 57.1-57.6. (5) Hernandez, M. R.; Gani, R.; Romagnoli, J. A.; Brignole, E. A. The Prediction of Properties and its Influence on the Design and Modeling of Superfractionators. Proceedings of the International Conference on the Foundation of Computer-Aided Process Design, Snowmass, CO, 1983; pp 709-740. (6) Urlic, L.; Bottini, S.; Brignole, E. A.; Romagnoli, J. A. Thermodynamic Tuning in Separation Process Simulation and Design. Comput. Chem. Eng. 1991, 15, 471-479. (7) Kamphausen, A. H.; O’Donnell, J. A. Plant-Model Mismatch Analysis in Deethyleniser Simulation. 1. Methodology. Ind. Eng. Chem. Res. 2004, 43, 6680-6693. (8) PRO/II Keyword Input Manual, Version 4.0; Simulation Sciences Inc.: Brea, CA, 1994. (9) REGRESS Users Guide, Version 2.01; Simulation Sciences Inc.: Brea, CA, 1991. (10) Barclay, D. A.; Flebbe, J. L.; Manley, D. B. Relative Volatilities of the Ethane-Ethylene System from Total Pressure Measurements. J. Chem. Eng. Data 1982, 27, 135-142. (11) DATACON Keyword Input Manual, Version 2.0; Simulation Sciences Inc.: Brea, CA, 1994. (12) Kamphausen, A. H. Selection and Tuning of Thermodynamic Data Models in Deethyleniser Simulation and Optimisation. Ph.D. Thesis, RMIT University, Melbourne, Australia, 1999. (13) Hulley, R. Private Communication. Applied Simulation Research Group, RMIT University, Melbourne, Australia, 2004. (14) Starling, K. E. Fluid Thermodynamic Properties for Light Petroleum Systems; Gulf Publishing Co.: Houston, TX, 1973. (15) Shibata, S. K.; Sandler, S. I. Critical evaluation of equation of state mixing rules for the prediction of high-pressure phase equilibria. Ind. Eng. Chem. 1989, 28, 1893-1898. (16) Ashour, I.; Aly, G. Effect of Computation Techniques for Equation of State Binary Interaction Parameters on the Prediction of Binary VLE Data. Comput. Chem. Eng. 1996, 20, 79-91. (17) Panagiotopoulos, A. Z.; Reid, R. C. A New Mixing Rule for Cubic Equations of State for Highly Polar, Asymmetric Systems. ACS Symp. Ser. 1986, 300, 571-582. (18) Gros, H. P.; Urlic, L. E.; Brignole, E. A. Phase Equilibrium Dominant Parameter Matrix of Multicomponent Distillation Trains. Comput. Chem. Eng. 1997, 21, S733-S738. (19) Soave, G. Equilibrium Constants from a Modified RedlichKwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197-1203. (20) PRO/II Casebook No. 5: Air Separation Plant; Simulation Sciences Inc.: Brea, CA, 1993.

Received for review July 9, 2004 Revised manuscript received August 30, 2004 Accepted September 13, 2004 IE049395B

PlantâModel Mismatch Analysis in Deethyleniser Simulation. 2

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