PlantâModel Mismatch Analysis in Deethyleniser Simulation. 1

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SEPARATIONS Plant-Model Mismatch Analysis in Deethyleniser Simulation. 1. Methodology Anthony H. Kamphausen and John A. O’Donnell* Chemical Engineering, RMIT University, Melbourne, Australia 3001

Confidence in the accuracy of a simulation model and by implication the thermodynamic data model(s) used, particularly for separation equipment such as a deethyleniser, requires validation against actual plant operating data. A methodology has been developed that quantifies plantmodel mismatch with a Calculator module appended to a PRO/II (Simulation Sciences, Inc.) Column module to facilitate an assessment of the relative accuracy of alternative simulation model formulations. The methodology is based on a manual stepwise optimization procedure using a preferred formulation of the simulation model with the key adjustable modeling parameter defined as the effective average equilibrium tray efficiency ηeff i . The methodology seeks to minimize an Error Objective Function that calculates the sum of the weighted and variance-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. 1. Introduction Process simulation can be broadly defined as the use of mathematical models to formulate and solve steady state material and energy balances, equipment operating or design equations, and equations providing information on production, consumption of feedstocks, costs, and similar data for industrial processes. While providing valuable results in chemical process design, debottlenecking, retrofitting, and both on-line and off-line optimization, even the most advanced computer simulation model gives at best only a close approximation to a given separation problem and generally requires experimental validation to allow assessment of whether the simulation model is acceptable for the desired purpose. In the development and validation of a computer simulation model it is common industrial practice to select one or more key operating variables for the mismatch calculation depending on the number of fitting parameters. The difference between the measured operating data and the computer-calculated results for these key operating variables is then minimized by adjusting the parameters as, for example, the minimization of reflux rate error and reboiler duty error in the determination of independent rectifying section and stripping section fitting parameters. In some work where only one fitting parameter is to be determined, only one variable is assessed for mismatch and a simple difference is used as quantification of the mismatch. This procedure does not lead to a comprehensive single “Global” measure of plant-model mismatch to quantify the relative adequacy of the different thermodynamic data models, mixing rules, alpha formulations, * To whom correspondence should be addressed. Tel.: +61 3 9925 2079. Fax: +61 3 9925 3746. E-mail: [email protected].

and data model parameters that can be used in simulation models. In other words, the plant-model mismatch presented represents a “Local” or series of “Local” criteria whose values can change both in absolute and relative terms depending on the formulation of the simulation model and the thermodynamic data models used. The word “Global” is used here to imply the combined plant-model mismatch of steady state plant operating data consisting of a wide variety of measurement types and accuracy similar to the criteria for datamodel mismatch of vapor-liquid equilibrium data used in the rigorous optimization (or tuning) of thermodynamic data model parameters by Prausnitz et al.1, particularly the binary interaction parameters. It is the assumption and rationale for this work that the results of such data-model mismatch analysis in selecting and tuning thermodynamic data models are not necessarily the same as the selection and tuning results obtained from a comprehensive plant-model mismatch analysis since the former analysis is usually restricted to binary system information while the latter allows for the modifying presence of other components and the relatively restricted range of operating conditions that are often characteristic of the multicomponent systems to be separated. This lack of transferability of binary system tuning results to multicomponent systems has been observed by, among others, Knudsen et al.2 The simulation model fitting parameters are usually based on the concept of theoretical plates, and therefore by implication the overall column or section efficiency, although some authors have examined the use of alternative parameters such as the average Murphree tray efficiency. The starting point for deterministic simulations including feed rate and condition, column pressure and pressure drop, and product quality, etc., are usually the

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same as the corresponding reconciled measured operating data (e.g., Bevan3 and Urlic et al.4) and are therefore not part of the mismatch quantification. Also, little consideration is given to the fact that the same simulation could theoretically be initiated in a different way with, for example, the use of different Performance Specifications or the feed quantity necessary for a specified product rate being the required calculational outcome. The resulting plant-model mismatch and optimized simulation model parameters of different simulation model formulations could then appear quantitatively in a very different way. One of the major objectives of this work was to develop a methodology that quantifies plant-model mismatch for the simulation of a deethyleniser and by extension for other difficult separation processes in a way that is more comprehensive and accurate than previously reported methods and which can be used for evaluating the use of alternative data models in terms of minimizing such mismatch. The methodology was therefore required to provide a “Global” criterion of plant-model mismatch involving as many plant measurements as practical. This paper describes the development of such a methodology and is based on a manual stepwise optimization procedure with the key adjustable modeling parameter defined as the effective average equilibrium tray efficiency ηeff i . The methodology seeks to minimize an Error Objective Function (EOF) that calculates the sum of the weighted and variancenormalized least-squares errors around the deethyleniser taking into account the wide range of accuracy inherent in the various measurements. The methodology involves many assumptions that are examined as to their validity. A complete and exhaustive check of the consequences of all the assumptions made was not practical. The examination includes the optimization of alternative simulation model formulations with respect to the Performance Specification/Process Variable combinations defined in the input files using alternative optimization paths as well as spot simulation checks for selected assumptions. The assumptions are examined individually and in limited interaction with each other. The plant operating data available for model validation or plant-model mismatch analysis did not meet mass and energy balance constraints around the process plant being simulated even after the application of carefully determined flow meter calibration factors and operating conditions that were as close to steady state as possible. Rigorous statistical data reconciliation techniques were therefore applied to the deethyleniser operating data using the statistical analysis software tool DATACON5 before any plant-model mismatch analysis was undertaken on the reconciled results. 2. Process Plant Equipment, Experimental Procedure, and Data Analysis 2.1. Process Plant Description. The deethyleniser under investigation is part of an ethylene plant where ethylene and other cracked hydrocarbons are separated in a series of fractionation towers (Figure 1). After drying, the gas enters the “chilling train”, which condenses the gas in three stages to liquid, except for a hydrogen stream containing small amounts of methane, which is used as fuel gas. A vapor/liquid separation drum is provided after each heat exchanger and the liquid bottoms stream from each of the three flash

Figure 1. Schematic representation of ethylene plant (driers to deethyleniser).

separation drums flows to a demethaniser tower. The demethaniser function is to remove any hydrogen and methane not removed in the chilling train. The demethaniser bottoms at known bubble point conditions is throttled and passes through a preheater to be flashed as deethyleniser feed to the vapor stream above the liquid on the feed tray. These upstream unit operations allow a precise estimate of the vapor/liquid ratio (approximately 50% vapor and 50% liquid depending on operating conditions) of the feed stream entering the deethyleniser. The deethyleniser separates ethylene from the heavier components of the demethaniser bottoms. It produces a polymer-grade ethylene product of purity greater than 98%. The deethyleniser feed stream and an intermittently on-line ethane recycle stream produce an overhead ethylene vapor stream, ethylene liquid draw stream, and bottoms liquid stream. Attached to the deethyleniser is a mixed-phase condenser and a baffled thermosiphon reboiler. The downstream deethaniser obtains feed from the deethyleniser bottoms and separates ethane from propylene and heavier components. The ethane from the deethaniser overhead is recycled back to the furnaces for cracking, with a side stream sometimes recycled to the deethyleniser at the base of the column. The flow rate of the recycle side stream to the deethyleniser is zero on most days (i.e., the stream is off-line) as was the case in the particular test run on which this analysis is based. When there is a measurable recycle side stream to the deethyleniser it enters as dry vapor at the base of the column at about -4.7 °C, 2200 kPa(abs) and 95% ethane, 25% ethylene which is the approximate composition of the overhead from the downstream deethaniser source. A test run and the associated analysis involving a recycle side stream of approximately 1% by mass of total input to the column is presented in Kamphausen.6 2.2. Plant Data Acquisition and Manipulation. Process operating data from the test run were collected together with laboratory composition measurements and augmented by data obtained from proprietary inhouse computer programs. Preliminary plant data analysis on normal production runs revealed mass and

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component balance inconsistencies over the whole plant especially around the deethyleniser. These discrepancies were markedly reduced after the application of flowmeter calibration factors but material and component balance constraints were still not satisfied. The plant data consisted mainly of hourly averaged mass flow rates, temperatures, pressures, compositions, and tank levels obtained from the distributed control system (DCS). All hourly averages obtained on-line represent the average of readings taken at 6-min or 1-min intervals. Sufficient data were taken to enable material and component data reconciliation around an envelope encompassing the outlet of the driers and the top and bottom products of the deethyleniser. Three stream samples were each taken over a 6-h period for the effluent stream from the driers, demethaniser feed drum overhead, demethaniser off-gases, deethyleniser vapor product, deethyleniser bottoms liquid, and deethyleniser actual tray 84 vapor. The laboratory analysis was made available as a composition slate of 22 components as used throughout the plant. Some of the components were in fact not present in the sections of the plant being modeled and were assigned compositions of 0.00 mole fraction while others were amalgamated to simplify the calculations. These simplifications resulted in a twelve-component composition slate as used in data reconciliation calculations. A further simplification to nine categories was made for simulation purposes and to six categories for the purpose of calculating plant-model mismatch EOF values. The final six categories encompassed ethylene, ethane, propylene, propane, butanes, and pentanes or higher. All data were manipulated to provide 5-h averages, rolling averages, and rolling absolute and relative standard deviations to be used as data in the subsequent steady state validation, data reconciliation, and plantmodel mismatch calculations. The operating time period of 6 h selected for the calculation of these averages and standard deviations was to a large extent dependent on sample bomb stream sampling times and was seen as a reasonable compromise between the residence time of the plant and the data requirements for smoothing unsteady state effects such as varying deethyleniser tray holdup, reflux drum levels, bottoms ethylene concentration, etc. There was no rigorous steady state criterion implemented and the approach to steady state validation was based on an assessment of relative process standard deviations. Most measurements were reasonably constant for the rolling 5-h test data with the relative standard deviation for the deethyleniser reflux rate as low as 0.01. By far the most variable was the liquid draw rate with a relative standard deviation of 0.38. 2.3. Data Reconciliation Setup. Since material and component balance constraints were still not satisfied after the application of flow-meter calibration factors, the data reconciliation software tool DATACON5 was used to statistically manipulate the test-run plant operating data in an attempt to remove any remaining random error, bias, or gross error to provide consistent reconciled measured plant data for the analysis of plantmodel mismatch. DATACON performs plant data reconciliation based on an advanced sum of weighted least squares optimization technique, subject to a set of heat and material balance equality constraints. The program adjusts the measured data and gives estimates to unmeasured

variables where possible, in such a way that this set of measured as well as estimated data satisfies heat and material balance equations. In the data reconciliation calculations, DATACON uses weighting factors generated from the standard deviations assigned to the measurements. In practice, the weighting factors are an indicator of both the inherent random error of the instrument and the unsteadiness of the process. At the end of a run statistical tests were performed to indicate the presence of likely gross errors at a global level and also at an individual measurement level. The reliability of these tests is enhanced in the program by a serial elimination strategy. The measurements which are flagged as containing gross errors and serially eliminated from further reconciliation are critically dependent on the values of the standard deviations assigned to the measurements. Further details of the DATACON data reconciliation method as used in this work are described in the DATACON Keyword Input Manual.5 To achieve convergence in this particular case the data reconciliation proceeds in two distinct stages encompassing first the effluent stream from the driers to the demethaniser products as Stage 1 and second the deethyleniser only as Stage 2. The results from the output file of Stage 1 provide the deethyleniser feed stream data used in producing the deethyleniser-only data reconciliation results in the output file for Stage 2 that ultimately form the basis of the deethyleniser plant-model mismatch analysis. This first step was necessary as no direct measure of deethyleniser feed composition was available and had to be obtained indirectly by laboratory analysis of the effluent stream from the driers. Mass and component balance constraints were applied to all sections of the plant subject to data reconciliation. There were insufficient data available to apply energy balance constraints to the deethyleniser, but energy balance constraints as well as vapor-liquid equilibrium constraints were applied in upstream sections of the plant. The approach taken here to estimate standard deviations for DATACON was to use three times the average process standard deviations for the mass flow rates and the calculated standard deviation for feed and bottom product composition based on two or three measurements. The standard deviation for overhead product composition was assumed to be twice the standard deviation of the average measured process composition. The multiplying factors were necessary because of the approximate nature of the information used, particularly the use of hourly averaged information in determining the standard deviations for flow rates rather than the use of actual measurements taken at 6-min intervals. 2.4. Data Reconciliation Results. No gross errors were flagged for any measurements involved in the data reconciliation. The output file from Stage 1 reconciliation shows that the measurements passed both the DATACON Global Test and the DATACON Measurement Test at the 95% Confidence Level. The output file from the Stage 2 reconciliation shows that the measurements again passed both tests at the 95% Confidence Level. Both Global and Measurement tests are described in the DATACON Keyword Input Manual.5 The ratio of reconciled flow rates to measured values adjusted using calibration factors, ranged from 0.940 to 1.009 in bottoms and overhead. For compositions, the corresponding ratio for ethylene composition measurements


ranged from 1.007 to 1.000 in feed and bottoms, respectively, with an overall low of 0.50 for pentanes plus and an overall high of 1.05 for propane, both in the feed. The relatively large composition correction factors to adjust for the results of data reconciliation for most components other than ethylene or ethane reflect a combination of factors including unsteady-state conditions, low concentrations and subsequent analytical uncertainties, and the fact that butanes and in particular pentanes plus are a large number of components lumped together under the headings 1-butene and 1,3 cyclopentadiene, respectively. The fact that no gross errors in composition were flagged is due mostly to the large relative standard deviations that were initially assumed, ranging from 10% for methane to 30% for cyclopentadiene. The incorporation of two tank modules in a trial data reconciliation input file to represent the deethyleniser reflux drum and reboiler vessel and allowing the levels to vary between maximum and minimum levels to compensate for unsteady-state effects did not result in any significant change in the reconciled flow rates and compositions. No quantitative check could be made on the effect of variable deethyleniser tray hold-up in the column but any effect was assumed to be allowed for in the multiplication factor applied to the process flow rate standard deviations. A detailed data reconciliation analysis and discussion including a compilation of standard deviations used is available in Kamphausen.6 3. Plant-Model Mismatch Analysis Setup 3.1. Plant-Model Mismatch Analysis Overview. The development of a methodology that quantifies plantmodel mismatch for the purpose of evaluating alternative simulation model formulations involves the following subsidiary objectives: (a) the postulation and evaluation of three alternative PRO/II7 simulation model formulations that include simulation model parameters whose values are dependent on column configuration and process operating conditions; (b) the formulation of an appropriate EOF that quantifies the plant-model mismatch for the three alternative PRO/ II simulation models; (c) the development and evaluation of alternative optimization paths that seek to evaluate a minimum for the selected EOF and the corresponding optimized PRO/II simulation model parameters; and (d) the testing of the validity of some of the many assumptions that are required for development of the methodology in the one particular simulation model and optimization path selected from the postulated alternatives. The K-value data model that is used to develop and test the methodology is the PRO/II Soave-RedlichKwong (SRK) CEOS with default quadratic mixing rules, default alpha formulations as defined by the Simsci Data Bank in PRO/II7, and a tuned (optimized) ethylene/ethane binary interaction coefficient of kij ) 0.0152 as determined by Urlic et al.4 using the binary data of Barclay et al.8 The vapor and liquid enthalpy and vapor density data model is chosen as the PRO/II default Peng-Robinson (PR) CEOS, while liquid density is defined by the Rackett data model. The methodology that is developed is based on a manual stepwise optimization procedure and reflects a compromise between accuracy and time constraints. This methodology is an approximation that acknowledges the limitations of the sequential modular nature of the PRO/II simula-

tion program and makes numerous assumptions that are subsequently tested. 3.2. Computer Simulation Model Outline. The PRO/II computer simulation input file begins with a Stream Calculator module that defines, as close as possible, the expected bubble point conditions of the demethaniser bottoms liquid at the measured pressure and data-reconciled mass flow rate and composition. The product with calculated bubble point temperature at known bubble point pressure represents the demethaniser bottoms liquid stream. The demethaniser bottoms stream is preheated and throttled, with both actions amalgamated in a simple hypothetical Heater module to vaporize part of the demethaniser bottoms stream so as to simulate the stream in which the feed temperature probe is situated. A hypothetical Valve module adjusts the pressure at the feed temperature probe to the pressure at the deethyleniser feed tray stream, by allowing for the hydrostatic head due to the position of the temperature probe below the feed plate. The simulated feed temperature will therefore depend on the simulated column pressure. A Column module using the I/O algorithm simulates the deethyleniser by taking the feed stream and an ethane recycle stream to produce an overhead vapor product stream, liquid draw stream, and a bottoms liquid stream. The mixed condenser pressure is estimated from the measured reflux liquid temperature. The baffled thermosiphon reboiler produces a bottoms boil up return stream with a liquid fraction (LF) i.e., a liquid/vapor ratio assumed to be fixed at its design value of 65%. An appended Fortran-like Calculator module is used to calculate the deethyleniser reboiler duty from superheated propylene mass flow rate, temperature and pressure measurements, and temperature measurements of the sub-cooled propylene heating medium. The calculated duty is then used as one of the measurement terms in the evaluation of the plant-model mismatch EOF. A second Calculator module is used to evaluate the defined EOF by comparing measured operating conditions with the corresponding simulated conditions. 3.3. Preliminary EOF Analysis. A PRO/II computer simulation program as previously described was appended with Optimization and Case Study modules in an attempt to automate a procedure for obtaining “Global” minimum values of a total EOF that included all measured variables. The attainment of a minimum total EOF by variation of initial estimates of the simulation model parameters, together with initial estimates of the real value of the base case process variables by the use of Case Study modules, was found not to be practical, however, because of convergence difficulties. Consequently, the use of an “Automated” procedure that would accommodate an Error-in-Variables approach similar to that of Kim et al.,9 was abandoned, and a simplified “Manual” procedure requiring only the rectifying and stripping section simulation model parameters to be changed manually in independent increments of 0.01 tray efficiency units was developed. The 0.01 increments were chosen as being adequate after preliminary investigations showed that this degree of resolution was capable of clearly differentiating calculational outcomes for even minor changes in the definition of the K-value thermodynamic data models used in a simulation. The inability to automate the search for an optimum that would have allowed an Error-in-Variables approach to the total EOF

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Table 1. Performance Specification Combinations (PS1, PS2, and PS3) Ap PS1 PS2 PS3

ethane impurity concentration in overhead product (mole fraction) ethylene bottoms concentration (mole fraction) liquid draw rate (tonnes/hour)

Bp

Cp

reflux rate (tonnes/hour)

ethane impurity concentration in overhead product (mole fraction) reboiler duty (million kJ/hour)

ethylene bottoms concentration (mole fraction) liquid draw rate (tonnes/hour)

minimization, and the impracticality of doing this using a “Manual” optimization approach, necessarily reduced the “Global” nature of the EOF. Since the attainment of a “Global” optimum could not be automated, the validity of the assumptions inherent in the preferred manual optimization approach needed to be assessed and this was done by comparing the results with those obtained by using three alternative optimization paths with three alternative simulation models. It is expected that advanced versions of the simulation software tools that are currently under investigation will allow the successful development of an automated methodology based on the manual procedure outlined here. An automated procedure that includes additional and more optimally selected and weighted variables would allow a rigorous Error-in-Variables analysis and so provide a more “Global” criteria of plant-model mismatch. 3.4. Computer Simulation Model Definitions. Many alternative forms to the industry-preferred computer simulation model can be formulated for the purpose of comparing plant operating data with computer-simulated data. One of the key differences between these simulation models is in the Performance Specification/Process Variable combination chosen for the distillation algorithm. Either a 2*2 or a 3*3 combination can be chosen with the possible Process Variables being reboiler duty, condenser duty, and liquid draw rate. The choice of Performance Specifications can be from a wider range of options but must be chosen with care to be capable of providing a converged solution. Although the most common choice in the simulation of the deethyleniser is to specify the ethane impurity concentration in the overhead vapor product and the ethylene losses as ethylene concentration in the bottoms, other Performance Specifications can be defined such as reflux rate, reboiler duty, liquid draw rate, etc., provided only that in combination with the Process Variables chosen, a converged solution can be obtained. Since the Performance Specifications in this work are set to test operating data, then any EOF calculated from a converged solution will necessarily show a zero component error for these measurements. The Performance Specifications however, even if they have been previously data-reconciled, can still be far enough from their true value to significantly affect the best estimate of the minimum total EOF value. This, in turn, could affect the validity of comparing alternative thermodynamic data models used in simulations as well as the optimum value of the simulation model parameters inferred from these calculations. To test the assumption that the Performance Specifications used in the preferred simulation model formulations contain no significant bias two additional formulations are defined. All the simulation model formulations defined here can be described in terms of Xp where X defines the algorithm Performance Specification/Process Variable combination and p represents the number of simulation model parameters. Although the algorithm Performance Specification combinations in the simulation models used in the present work change, the Process Variable

liquid draw rate (tonnes/hour)

combinations of reboiler duty, condenser duty, and liquid draw rate are maintained. The designation for p is either 1 or 2, where 1 assumes a uniform average equilibrium tray efficiency throughout the column and 2 assumes one uniform value in the rectifying section and a second independent uniform value in the stripping section. The Performance Specifications for the three alternative simulation model formulations selected for assessment from the large list of models available are given in Table 1. The simulation model formulations that are used commercially and in Kamphausen,6 to evaluate alternative K-value data models are of type Ap: i.e., A1 and A2. Types Bp and Cp have been examined by performing sample calculations to assess the validity of any conclusions reached using type Ap. They are unlikely to be used in an industrial setting. The simulation models selected for examination all use as one of the Performance Specifications the liquid draw rate coupled with itself as a Process Variable. This was a technique employed to assist convergence in the distillation algorithm and to allow more control over the manual optimization procedure. As a consequence, the iterative procedure in this comparison between simulation models will not produce better estimates of the liquid draw rate value. 3.5. Mismatch Error Objective Function. To quantify the effect of changing the thermodynamic data model used on relative simulation model accuracy, it is necessary to define an EOF for plant-model mismatch whose value represents the difference between the measured (or data-reconciled) data and the computercalculated values for as many plant variables as practical. The difference is minimized by adjusting an appropriate simulation model parameter as well as estimates of the real value of the measured variables. Many studies select only a few key variables as a basis for comparison and do not allow for measurement errors, or for different degrees of accuracy of the various measurements. This work set out to consider all measured variables as part of a mismatch EOF by appropriately weighting and normalizing least squares errors and then minimizing the sum of these terms. The actual number of independent measurement variables considered was 31 as detailed in Kamphausen.6 The number of simulation variables for any given simulation model Xp and any given optimization path contributing to the minimum total EOF varied, but it was always considerably less than 31 since fixed simulation variables were set equal to the measured or data-reconciled values and therefore provided no contribution to the EOF. The EOF chosen, based on those used for parameter estimation in on-line optimization programs, can be stated as follows: N

{wi(Vci - Vei )2/σ2i } ∑ i)1

(1)

where for any steady-state operating condition, N is the


number of independently measured and preferably reconciled plant measurements Vei such as flow rates, composition, temperature, and pressure, and Vci are the corresponding computed values. Also σ2i , the normalizing term, is the estimated variance of the particular measurement, and wi is a weighting term that simultaneously allows a measurement to be excluded if the value is set to zero. To gain some perspective as to what the EOF values indicate, it should be remembered that for a weighting of 1.0, each individual EOF value is equal to the {(Simulated Value - Measured Value)/Standard Deviation}2. Although the absolute value of total EOF on its own has no real physical significance since it is dependent on the number of terms in the summation, the square root of the average value can have a statistical interpretation. On the basis of the 12 dependent simulated variables which have been found to be significant if all three Xp simulation models are considered, the average Model Xp discrepancies (or simulation errors) is x(total EOF/12) standard deviations for each significant variable. It is clear that all average standard deviation multiples would be considerably reduced if all 31 variables including fixed simulation variables were taken into account. 3.6. Simulation Model Parameter. Since tray efficiencies change at different flooding percentages (Anderson et al.10), most significantly near weeping or flooding conditions, it is common practice to use some measure of it as an adjustable modeling parameter. On the basis of recommendations in the PRO/II Input Manual7 for fitting plant data to a simulation model, the average equilibrium tray efficiency was chosen as the adjustable simulation model parameter from a range of options including Murphree tray efficiency, vaporization tray efficiency, individual component efficiencies, or overall column efficiency as used by Bevan.3 The definition of equilibrium tray efficiency, ηeqlb , i for component i is eqlb ηeqlb ) (Keff - 1) i i - 1)/(Ki

(2)

where Keff i is the effective K-value for component i and is the equilibrium K-value for component i at the Keqlb i temperature and pressure on the tray. In the context of this work it should be understood that the equilibrium tray efficiency parameters referred to as optimized or inferred simulation model parameters are not the “True” , but rather ηeff equilibrium tray efficiencies, ηeqlb i i , the “Effective” equilibrium tray efficiencies, which take into account unknown and variable “Data Model Correction Factors” that compensate for the inaccuracies inherent in the assumed thermodynamic data model. These correction factors could theoretically vary from tray to tray or be aggregated into section correction factors in the same way as the tray efficiencies themselves. 4. Plant-Model Mismatch Optimization Path 4.1. Optimization Path Overview. The basis of the EOF evaluation procedure for all three optimization paths involves, most importantly, average and standard deviation estimates of the various measurements. Where simulated values equal measured values and are not allowed to vary in the PRO/II input file, the contribution to the total EOF is necessarily zero and the corresponding standard deviation estimate is irrelevant. Where the

simulated value is allowed to vary, the standard deviation estimate of a measurement can become very significant since the contribution of the normalized individual EOF of this variable to the total EOF varies inversely as the square of the standard deviation. The individually determined EOF values determined from batch submissions of at least 10 input files were plotted to obtain the optimized simulation model parameters at the minimum value of the total EOF. The above procedure assumes that the data reconciliation previously outlined would provide feed and composition data, as well as light and heavy key concentrations in the bottoms and overhead that was already optimized with respect to the mismatch calculations, and that other assumptions would have a negligible effect on the results, effectively leaving only the simulation model parameters as variables to be manipulated. Alternative optimization paths labeled “Simple” and “Complex”, and alternative simulation models Bp and Cp, were assessed to validate some of the “Basic” optimization path assumptions. Additional checks were subsequently applied to test other key assumptions. The major assumptions tested were that the Performance Specifications provided no contribution to the total EOF and that the effect of possibly inaccurate reflux rate and pressure measurements and of assuming the liquid/ vapor fraction in the reboiler return was equal to its design value would not change the essential nature of any results. The “Basic” optimization path approach proved very successful in comparing the performance of alternative thermodynamic data models with a comprehensive application of the methodology to evaluating the use of alternative K-value data models described in Kamphausen.6 4.2. Optimization Path Definitions. Optimization path definitions are arbitrary and have been chosen on the basis of usefulness and time constraints. The main (“Basic”) optimization path to be used in the intended application of this work and the two alternative (“Simple” and “Complex”) optimization paths used for checking some of the assumptions inherent in the “Basic” optimization path are summarized below in terms of the required manually executed steps. Various “Basic” path assumptions have been checked by the use of alternative optimization paths that vary sequentially, in different order and iteratively, variables which had previously been fixed. All of the three Xp computer simulation models previously defined have been used for all of the three defined optimization paths. Optimization Path Steps. Various manually executed steps are defined as Step Nx where N (equals 1 to 8) refers to the sequence of a particular Step and x refers to the optimization path which is either b for “Basic”, s for “Simple”, or c for “Complex”, or remains as x if any of the optimization path definitions are applicable. “Basic” Non-Iterative Optimization Path. The “Basic” optimization path is a four-step procedure for determining the minimum total EOF and associated simulation model parameters. The single parameter value in the X1 models is varied until a minimum is achieved (Step 1b) and then in a three-step procedure both parameters are varied independently in the X2 models (Steps 2b to 4b) until a new lower minimum is achieved after each Step. Step 2b involves continuously decreasing the stripping section efficiency in increments of 0.01 units. Step 3b consists of increasing the rectify-

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Figure 2. “Simple” non-iterative optimization.

Figure 3. “Complex” iterative optimization path.

ing section efficiency by 0.01, while Step 4b again involves continuously decreasing the stripping section efficiency in increments of 0.01 units. Further increases in rectifying section efficiency were not required. The “Basic” optimization path involves no check of any of the assumptions made and is intended for the evaluation of alternative thermodynamic data models in its Ap form only. The maximum of four steps, viz., Steps 1b to 4b, is equivalent to Steps 1s to 4s only, of the “Simple” optimization path for all Xp simulation models. “Simple” Non-Iterative Optimization Path. The “Simple” Non-Iterative Optimization Path is a six-step procedure and extends the previous result by testing the assumptions of no error in the assumed deethyleniser column pressure and the design value of 65% for the liquid fraction reboiler return being accurate. The overall results are plotted as in Figure 2. After the twoparameter minimum total EOF solution is reached, Steps 1s to 4s, as in the “Basic” optimization path, the tray efficiency parameters are held constant and the simulated column pressure is varied until a new minimum is obtained (Step 5s). Finally the reboiler return liquid fraction is changed to further minimize the EOF (Step 6s). “Complex” Iterative Optimization Path. The “Complex” Iterative Optimization Path is an eight-step procedure developed to test the important Performance Specification assumptions in addition to the assumptions of accurate column pressure and accurate reboiler return liquid fraction. The procedure involves obtaining a solution to the one-parameter models X1 (Step 1c), averaging the original (OR) Performance Specification values obtained from the three solutions, and re-running the input files with these new values to obtain amended (AM) one parameter solutions (Step 2c). These new solutions provide twice amended Performance Specification values by a second averaging procedure. The twiceamended one-parameter models are then further sequentially optimized by varying the simulated column pressure (Step 3c) in increments of 10 kPa followed by optimization to the two-parameter X2 models (Step 4c

Table 2. Significant Individual EOF Variables descriptor RF RATE TP RATE BT RATE TP MF Ethan BT MF Ethyl BT MF Ethan TR85 MF Ethyl DUTY 2 TR81 TEMP RB TEMP BT TEMP PTOP

measurement variable reflux rate overhead ethylene vapor rate bottoms rate overhead ethane composition bottoms ethylene composition bottoms ethane composition actual tray 84 ethylene composition reboiler duty actual tray 80 temperature reboiler temperature bottoms temperature column top pressure

units TM/h TM/h TM/h mole fraction mole fraction mole fraction mole fraction MMkJ/h °C °C °C kPa(abs)

to 6c) similar to Steps 2b to 4b of the “Basic” path. The column pressures in the two-parameter models are again changed until the lowest possible EOF value for this procedure is reached (Step 7c), and finally the reboiler return liquid fraction is changed (Step 8c) as in the “Simple” path to provide a limiting minimum total EOF for the two-parameter models. The overall results are plotted as in Figure 3. 4.3. Optimization Path Process Variables. Thirtyone variables have been defined as contributing to the total EOF to be minimized, although most of the variables are fixed at their measured or data-reconciled values and therefore provide zero contribution to the EOF. These variables include all compositions in feed and products. The variables that are fixed and the variables that actively contribute to the total EOF are dependent on the simulation model formulation and the optimization path. Table 2 is a list of the direct or derived measurements that are most significant in determining the final calculated EOF result. 5. Plant-Model Mismatch Results 5.1. Simulation Models X1 Total EOF. The simulation models A1, B1, and C1 are manually optimized


Figure 4. Total EOF curves after iteration 1 (OR).

Figure 5. Total EOF curves after iteration 2 (AM).

in Step 1x for all three optimization paths as shown in Figure 4. Data-reconciled values are used for all the Performance Specifications except reboiler duty which is calculated from a combination of raw measurements. For the “Complex” optimization path the associated values of the Performance Specifications from each of the three optimized solutions are extracted and averaged and then used to replace the previous raw or datareconciled values. The input files are run again in Step 2c to produce new curves as in Figure 5. This approach increases the “Global” nature of the minimum total EOF by allowing two fixed simulation variables to become dependent simulation variables. All three simulation models infer approximately the same optimized efficiency parameter value of 0.99 after Iteration 1 (Step 1x) by pinpointing the minimum in very steep quadratic type curves of EOF versus average

equilibrium tray efficiency. The optimization path for model C1 shows a much steeper gradient in total EOF versus assumed single tray efficiency parameter values, than the gradient for the other two models. When the resultant curves are smoothed as in Figure 4 it is apparent that there is a maximum variation in optimum tray efficiency of 0.006 units with the corresponding minimum total EOF for model C1 almost twice that for models A1 and B1. The slight discrepancy in tray efficiency disappears after Iteration 2 as shown in Figure 5 with the minimum total EOF values obtained for the three X1 simulation models also more in agreement with each other. Although the precise minimum value in the total EOF for model A1 increases by about 15% after iteratively combining the initial results for all X1 simulation models, this EOF change is not significant when the uncertainties in the data and the very large increase in EOF with a small change in the assumed parameter value are taken into account. The change in EOF, however, does emphasize the sensitivity of the EOF value to the assumed tray efficiency parameter value. This is most noticeable for model C1. The results further suggest that simulation model C1 is the most susceptible to inadequacies in the optimization path assumptions. This is to be expected due to the use of the more approximate value of the derived and unreconciled reboiler duty as a fixed Performance Specification. The approximate nature of the reboiler duty is allowed for in models A1 and B1 by letting the simulation value vary and by assigning a relative standard deviation of almost 10% to the measurement. Approximate calculations of heat transfer across column and reboiler cladding and walls indicate that, at a maximum of 8.5 kW, this was so low as to be an unlikely source of simulated reboiler duty error. The results obtained by perturbing the values assigned to simulation model Performance Specifications led to the conclusion that the assumption of negligible effective errors in the Performance Specifications for model A1 and B1 are substantially correct. In other words, the assumptions of negligible error in the data reconciled values of overhead ethane impurity and bottoms ethylene composition each with relative standard deviations of approximately 5% and the measured value of reflux rate with a relative standard deviation of 0.5% do not significantly affect the value of the inferred efficiency parameter within the limits imposed by the assumed efficiency increments of 0.01 units. This observation also indirectly supports the assumption that errors in other data reconciled measurements such as feed rate and composition are not significant with relative standard deviations after data reconciliation of approximately 1.5% for feed rate and less than 1% and 2% for the critical ethylene and ethane feed compositions and thus helps to justify the setting of simulated values equal to the corresponding data reconciled values. Conversely, it confirms that the accuracy of the unreconciled reboiler duty with an assumed relative standard deviation of 10% is inappropriate to serve as a Performance Specification. The fact that perturbations of the reboiler duty Performance Specifications would be required for model Cp to obtain results consistent with the other two models Ap and Bp emphasizes the need for highly accurate fixed Performance Specifications in the simulation model analysis. A threshold accuracy of at least 5% relative standard deviation after

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Table 3. Individual EOF Contributions “Simple” Patha model:

A1

B1

C1

A2

B2

C2

A2

B2

C2

(ηR/ηS) PTOP kPa LF %

(0.99/0.99) 2043 65

(0.99/0.99) 2043 65

(0.99/0.99) 2043 65

(1.00/0.86) 2043 65

(1.00/0.86) 2043 65

(1.01/0.85) 2043 65

(1.00/0.86) 2063 95

(1.00/0.86) 2063 95

(1.01/0.85) 2053 99

step no.:

step 1x

step 1x

step 1x

step 4b/s

step 4b/s

step 4b/s

step 6s

step 6s

step 6s

TOTAL EOF RF RATE TP RATE BT RATE TP MF Ethan BT MF Ethyl BT MF Ethan TR85 MF Ethyl DUTY 2 TR81 TEMP RB TEMP BT TEMP PTOP “Other”

6.95 0.23 0.00 0.02 0.00 0.00 0.01 2.35 0.02 1.19 1.70 1.42 0.00 0.01

7.62 0.00 0.00 0.02 0.73 0.00 0.02 2.44 0.05 1.23 1.70 1.43 0.00 0.00

13.58 0.81 0.02 0.24 0.00 3.00 1.46 2.63 0.00 1.21 2.18 1.86 0.00 0.17

3.46 0.00 0.00 0.02 0.00 0.00 0.01 0.01 0.05 0.23 1.70 1.42 0.00 0.02

3.49 0.00 0.00 0.02 0.02 0.00 0.02 0.01 0.05 0.24 1.70 1.42 0.00 0.01

4.46 1.07 0.00 0.01 0.00 0.01 0.04 0.09 0.00 0.15 1.67 1.40 0.00 0.02

1.50 0.16 0.00 0.02 0.00 0.00 0.01 0.01 0.11 0.02 0.00 0.79 0.36 0.02

1.58 0.00 0.00 0.02 0.26 0.00 0.01 0.02 0.06 0.05 0.00 0.98 0.16 0.02

2.76 1.03 0.00 0.05 0.00 0.16 0.03 0.08 0.00 0.07 0.00 1.28 0.04 0.02

a Items in italics represent “Local” EOF giving same optimized Tray Efficiency as “Global” EOF. See Table 2 for Variable Definitions. ηR is the Rectifying section average equilibrium tray efficiency, and ηS is the Stripping section average equilibrium tray efficiency. “Other” refers to a combination of fixed feed stream simulation variables as well as marginally relevant variables such as reflux temperature and compositions other than ethylene and ethane.

Table 4. Individual EOF Contributions “Complex”’ Patha model:

A1

B1

C1

A2

B2

C2

A2

B2

C2

(ηR/ηS) PTOP kPa LF %

(0.99/0.99) 2093 65

(0.99/0.99) 2073 65

(0.99/0.99) 2043 65

(1.00/0.89) 2093 65

(1.00/0.90) 2073 65

(1.01/0.82) 2043 65

(1.00/0.89) 2083 95

(1.00/0.90) 2073 95

(1.01/0.82) 2053 99

step no.:

step 3c

step 3c

step 3c

step 6c

step 6c

step 6c

step 8c

step 8c

step 8c

TOTAL EOF RF RATE TP RATE BT RATE TP MF Ethan BT MF Ethyl BT MF Ethan TR85 MF Ethyl DUTY 2 TR81 TEMP RB TEMP BT TEMP PTOP “Other”

5.43 0.01 0.01 0.05 0.06 0.22 0.06 2.32 0.09 0.26 0.71 0.54 1.00 0.10

6.33 0.21 0.01 0.06 0.38 0.33 0.10 2.32 0.02 0.53 1.12 0.89 0.36 0.00

7.50 0.20 0.00 0.05 0.06 0.17 0.04 2.44 0.02 1.20 1.80 1.52 0.00 0.00

2.93 0.09 0.01 0.05 0.06 0.22 0.06 0.09 0.09 0.00 0.72 0.54 1.00 0.00

3.51 0.21 0.01 0.06 0.12 0.32 0.10 0.18 0.02 0.10 1.12 0.89 0.36 0.02

4.02 0.23 0.00 0.02 0.06 0.00 0.02 0.50 0.02 0.05 1.69 1.42 0.00 0.01

1.96 0.02 0.01 0.05 0.06 0.22 0.06 0.09 0.07 0.03 0.00 0.70 0.64 0.01

2.38 0.21 0.01 0.06 0.12 0.32 0.10 0.18 0.02 0.10 0.00 0.89 0.36 0.01

2.42 0.21 0.01 0.05 0.06 0.22 0.05 0.45 0.02 0.02 0.00 1.29 0.04 0.00

a Items in italics represent “Local” EOF giving same optimized Tray Efficiency as “Global” EOF. See Table 2 for Variable Definitions. ηR is the Rectifying section average equilibrium tray efficiency, and ηS is the Stripping section average equilibrium tray efficiency. “Other” refers to a combination of fixed feed stream simulation variables as well as marginally relevant variables such as reflux temperature and compositions other than ethylene and ethane.

data reconciliation would therefore seem to be indicated as a requirement for Performance Specifications. It was accepted that one iteration using model Ap (or model Bp) would be sufficient to obtain X1 simulation model parameter values and minimum total EOF values adequate for the stated ultimate purpose of evaluating the use of alternative thermodynamic data models at the conditions of the operating data used for the simulation. Model Ap was selected as the preferred simulation model to be used for comparative EOF analysis despite the very low relative standard deviation of the reflux rate in model Bp because it is the industry preferred form and also because the reflux rate had not been subjected to data reconciliation therefore possibly incorporating an undetected bias. 5.2. Simulation Models X2 Total EOF. The optimized simulation models A1, B1, and C1 are further manipulated to provide optimized A2, B2, and C2 simulation models for the previously defined optimiza-

tion paths. In general it is apparent from Tables 3 and 4 that the values assigned to the tray efficiency parameters are extremely important in minimizing the total EOF for each simulation model Xp. It is also clear from Figures 2 and 3 that a two-parameter model X2 is capable of approximately halving the minimum total EOF for all X1 simulation models considered and this is consistent with the industry practice of using A2 simulation models. The C2 simulation model, however, still has a minimum total EOF significantly greater than those of either the A2 and B2 models, which are nearly identical, and this is again consistent with the clearly inaccurate assumption of the simulated reboiler duty being equal to the derived measurement value. For simulation models X2 the minimum total EOF for the “Basic” path is arrived at in Step 4s which may or may not be the same result as for Step 3s depending on how quickly the minimum is obtained. This minimum for model A2 which represents the lowest point


Figure 6. 3-D EOF surface for simulation model A2.

of a 3-D spoon-like surface can be extracted from the three-dimensional plot in Figure 6 which shows a very strong dependence of total EOF on the assumed rectifying section tray efficiency. A narrow steep surface along the X-axis represents the rectifying section parameter and a long shallow surface along the Y-axis represents the stripping section parameter. In all cases the inferred stripping section efficiency was found to be lower than the rectifying section efficiency. This finding was qualitatively in general agreement with previous industrial experience. The stripping section efficiency data over the range 0.84-0.88 for model A2 at the optimum rectifying section efficiency of 1.00 gives a change in EOF from 3.68 to 3.58 EOF units with an optimum of 3.46 EOF units at the optimum stripping section efficiency of 0.86. This emphasizes the weak dependence of the total EOF on the assumed stripping section tray efficiency. All simulation models and optimization paths provide mostly similar trends of decreasing total EOF as depicted in Figures 2 and 3. Simulation model A2 provides the lowest values of the total EOF at every step for all optimization paths. This is not unexpected as it is the only two-parameter simulation model in which two Performance Specifications have been subjected to prior data reconciliation. The corresponding EOF values for model B2 are only slightly greater. Corresponding values of the rectifying section parameters are the same at every step while stripping section parameters are the same within 0.03 units. For the “Simple” and therefore also the “Basic” optimization path the maximum difference in inferred optimum simulation model parameter values for all X2 models is 0.01 tray efficiency units. The results for models A2 and B2 are identical at 1.01 for the rectifying section and 0.86 for the stripping section. The greatest anomaly occurs for simulation model C2 in Step 1x of the optimization path as previously indicated. A further anomaly for model C2 is the fact that the parameter value for the stripping section is up to 0.08 units lower than those for the A2 and B2 models

even though the total EOF values are very similar. The corresponding C2 rectifying section parameter value is, however, only 0.01 units higher than the corresponding A2 and B2 values. The final minimum total EOF for either the “Simple” and “Complex” optimization path that tests several of the assumptions made in the “Basic” optimization path range from 1.50 to 2.76. The corresponding lowest value for the A2 model using the “Basic” path is 3.46. These results are consistent with the hypothesis that the Performance Specifications in A2 or B2 cannot be error free and that other effects must also distort the results for C2. Some of these effects are examined further but the main conclusion of the importance of a two-parameter model X2 in greatly reducing plant-model mismatch by approximately 50% compared with a one-parameter model X1, is still valid and consistent for all simulation models and all optimization paths. It is expected, however, that the percentage reduction in minimum total EOF could vary considerably depending on the makeup of the individual measurements that make up the total EOF. The results suggest, importantly, that, depending on the nature of the separation process, a two-parameter model might be required to rate, on a relative basis, alternative thermodynamic data models for simulation model accuracy. This approach would be expected to be necessary, for example, in distillation processes similar to the cryogenic separation of air where, in some simulation applications as illustrated in a PRO/II Casebook,11 even the data model parameter values for the critical binary interaction coefficients of the otherwise same thermodynamic data model, are defined differently for highpressure and low-pressure sections of the process. Overall it can be said that the value of the inferred optimum rectifying section parameter does not change significantly even with the greater accuracy and increased complexity of the alternative optimization paths and an increase in the “Global” nature of the EOF. The corresponding value of the stripping section parameter is, however, more uncertain with a wide range of values giving similar minimum total EOF values for the

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different simulation models and different optimization paths. This is supported by the shallow curve obtained for model A2 in plotting total EOF against stripping section efficiency at a given rectifying section efficiency as shown in Figure 6. For the “Complex” optimization path the rectifying section tray efficiency results are identical to the corresponding results for the “Basic” and “Simple” optimization path for the three simulation models. The stripping section tray efficiency results from the “Complex” optimization path are a maximum of 0.04 units different from the corresponding results for the “Basic” and “Simple” optimization path for the three simulation models. The results for the preferred “Basic” optimization path for model A2 imply a large standard deviation for the inferred stripping section efficiency parameter of perhaps 0.04 units as opposed to a very small standard deviation for the rectifying section efficiency parameter of perhaps 0.01 units. It should be emphasized that since the rectifying section occupies 72 actual trays out of a total of 96 actual trays a much greater significance should be attributed to the more accurate rectifying section parameter value. The version of the simulation software used (PRO/II version 4.18) and the techniques employed did not allow further quantification, although the application of Monte Carlo simulation techniques should make possible the evaluation of more accurate statistical information. 5.3. Simulation Model X1: Individual EOF. Because of the various definitions of the Xp simulation models, the individual contributions to the total EOF at any given simulation model parameter value vary significantly, and those variables providing zero or near zero contributions also vary. For model A1 the most significant contributor to the total EOF away from the one parameter X1 optimum is the reflux rate, for model B1 it is ethane mole fraction in the ethylene vapor product, while for model C1 it is ethylene mole fraction in the bottoms followed closely by ethane mole fraction in the bottoms. Reboiler duty for model A1 as well as reflux rate, product rates, and bottoms temperature for model C1 change significantly with the value of the X1 simulation model parameter. Bottom liquid compositions are also free to vary for model C1 as are the bottoms and overhead vapor flow rates but these effects are relatively small. Other model X1 contributors are relatively very minor except as the X1 minimum total EOF is approached as depicted in Figures 7, 8, and 9. The major contributors to the total EOF increase as a power index of the distance between the assumed X1 simulation model parameter and optimized X1 simulation model parameter and mirror the increase in total EOF itself. Other contributors to the total EOF such as actual tray 80 temperature discrepancies for all X1 models change relatively slowly away from X1 simulation parameter optimum or not at all such as reflux temperature. Most individual EOF contributions remain essentially zero throughout the X1 tuning procedure by definition of the simulation model or by virtue of being linked to those variables that are fixed. Variables that contributed to the total EOF in at least one X1 model for at least one optimization path, but whose contributions were too small to be included in a detailed analysis, included reflux temperature and feed temperature, as well as certain product compositions. All X1 models optimize to the same average tray efficiency of 0.99 using either the same defined “Global”

Figure 7. A1 Optimization path step 1x.

Figure 8. B1 Optimization path step 1x.

EOF or the different selected “Local” EOF definitions. These “Local” criteria are reflux rate for model A1, ethane composition in vapor product for model B1, and ethylene composition in bottoms for model C1, and are selected according to the greatest rate of change for that simulation variable with a change in tray efficiency parameter away from the optimum. Although all X1 models optimize to the same average tray efficiency parameter value, the relative values of the minimum “Global” EOF are somewhat different from the corresponding “Local” EOF ratios. This may or may not be


Figure 9. C1 Optimization path step 1x.

Figure 10. Comparison of optimized simulation models.

important in assessing the relative adequacy of alternative K-value data models. Other contributions to the X1 minimum total EOF after Step 1x for all optimization paths are a function of simulation model formulation as shown in the first three column results of Table 3. It is apparent that the discrepancies in reflux rate, which is the most sensitive variable in model A1, and product ethane impurity, which is the most sensitive variable in model B1, have been virtually reduced to zero. Model C1, however, contains small residual discrepancies for the two most sensitive variables, viz ethylene and ethane composition in the bottoms. All models indicate that the significant residual discrepancies are in the stripping section with the actual tray 84 ethylene composition and stripping section temperatures contributing between 60% for model C1 and 95% for model A1. These temperature discrepancies include actual tray 80, bottoms temperature, and reboiler return temperature. The consistent reduction in individual EOF values in the stripping section for all Xp simulation models when a second adjustable parameter is introduced strongly supports the need for a two-parameter simulation model with independent rectifying and stripping section tray efficiency parameters to minimize plant-model mismatch. 5.4. Simulation Model X2: Individual EOF. Optimization path results that include final column pressure and liquid/vapor fraction reboiler return adjustments for different simulation models along the “Simple” path are compared in Figure 10 and Table 3. X1x is the optimized one-parameter model after Step 1x of all optimization paths, X2s is the “Basic” path optimized two-parameter model, and X2f is the final adjusted twoparameter model of the “Simple” optimization path. Although the individual errors of the total EOF vary along each optimization path for all the three simulation models, the final individual EOF results are similar, especially for models Ap and Bp. It is worth noting that

the first three columns of the bar graph shown in Figure 10 represent the lowest total EOF points at identical tray efficiencies of 0.99 of each of the three graphs depicted in Figures 7-9. A significant reduction in minimum total EOF for all models occurs in going from the “Basic” path oneparameter X1 models to the “Basic” path two-parameter X2 models but the reason for this reduction is greatly dependent on the simulation model definition. For model A2 and B2 the most significant reductions occur in the stripping section variables of actual tray 80 temperature and actual tray 84 ethylene composition. For model C2 there are, in addition to the above, significant reductions in residual bottoms ethylene and ethane mole fraction discrepancies. The most significant residual X2 contributions are bottoms temperature and reboiler return temperature with a lesser contribution from actual tray 80 temperature. The residual mismatch in actual tray 84 ethylene vapor composition could be partly explained by a badly calibrated instrument that had in the past been found to measure this composition about 5% absolute too low. Residual contributions to the total EOF from the reflux rate as well as actual tray 84 ethylene composition persist for model C2. Spot simulation checks indicate most of these residual mismatch contributions could be readily explained by suspected measurement biases particularly for temperatures, use of sub-optimal enthalpy data models, composition approximations that amalgamated C6 and higher hydrocarbons in the C5 definition of 1,3 cyclopentadiene, or some combination of these factors. A suspected bias in the temperature measurements is only partly allowed for by the assumption of normalizing standard deviations significantly greater than the measured process standard deviations. A conceivable bias of 2.0 °C could reduce the temperature discrepancies to zero. As there is a large composition standard deviation for pentanes and higher as inferred by the data reconciliation results and as

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components higher than pentanes have been aggregated under the component name 1,3-cyclopentadiene, it is to be expected that there could be a significant residual mismatch for temperature measurements in the stripping section of the column. The residual individual mismatch could be expected to increase down the column as the concentration of high MW hydrocarbons increases, and this in fact tends to occur. The decrease in the “Basic” path stripping section discrepancies from X1 to X2 models is achieved by reducing the stripping section parameter by up to 0.14 units while the rectifying section parameter increases marginally by up to 0.02 units. The relatively large difference between optimized rectifying and stripping section parameters can probably best be understood in terms of the steep change in composition profile at the feed plate. Above the feed plate there is effectively only a binary mixture of ethylene and ethane, while at the feed plate and below there is a step change in composition to include significant quantities of C3, C4, C5, and traces of heavier hydrocarbons. The resultant increase in liquid viscosity in the stripping section would therefore be expected to result in a significantly lower average tray efficiency. In addition the multicomponent composition environment may indicate the need for a modified optimum ethylene/ethane binary interaction coefficient in the stripping section. For all X2 simulation models an examination of two of the “Basic” path assumptions leads to the “Simple” optimization path where an increase in the specified simulated pressure of up to 30 kPa, i.e., approximately 1.5% of the actual measurement, gives a net decrease of up to 30% in total EOF despite there now being an error contribution from this adjusted pressure. This discrepancy can be compared with an instrument accuracy of 2.5% assessed by plant engineers. The reduction is, as before, primarily due to further decreases in all stripping section discrepancies and strongly suggests that the measured pressure was too low. There is no significant change in the relative values of the EOF and the simulation model parameters are the same as before, therefore the assumption of no error in the measured pressure should not affect the results from any application to the rating of alternative thermodynamic data models. The effect of a possibly underestimated liquid fraction in the reboiler return indicates a probable value in excess of the assumed design value of 65% with the lowest total EOF occurring at about 95% L/V fraction for models A2 and B2 and 99% for model C2. The steady reduction in total EOF with increasing liquid/vapor fraction from 65% to either 95% or 99% is solely due to the reduction in reboiler return temperature discrepancy to zero. The optimum values for the liquid/vapor fraction are clearly unrealistic but the exercise suggests it is higher than 65%. The use of the real value in the simulation models would, however, again not affect the relative values of the minimum total EOF or the optimized values of the simulation model parameters. Therefore, the use of the design value of 65% is validated for the intended use of the methodology in this instant. Since the “Complex” optimization path consists of additional simulation variables that are allowed to vary along the optimization path, the number of nonzero components in the minimum EOF is increased but the final results as selectively presented in Table 4 are very similar to those for the “Simple” optimization path discussed above.

Spot simulation checks on the assumed accuracy of the measured but unreconciled reflux rate indicated that a bias of plus or minus 2% could lead to errors of plus or minus 0.01 and 0.02 in the optimized model X2 rectifying and stripping section tray efficiencies, respectively. This emphasizes the need for data reconciliation to include reflux rate if the optimized simulation model parameters are to be used for process optimization as ultimately intended, as distinct from the current stated purpose of evaluating alternative thermodynamic data models. For the “Basic” path all X2 models optimize to the same average tray efficiency parameters using the best single “Local” EOF definitions as with the corresponding “Global” EOF. However as for the X1 models this occurs at somewhat different relative values of the “Local” minimum EOF. Again, this may or may not become important depending on the ultimate application if “Local” criteria only are used in rating thermodynamic data models. Based on 12 significant variables, the average Model A2 discrepancies (or simulation errors) are 0.54 standard deviations for the “Basic” path, 0.35 for the “Simple”’ path, and 0.40 for the “Complex” path. The highest average standard deviation multiple at 0.61 is obtained for the “Basic” path Model C2. Average multiples would be reduced if all 31 variables are taken into account, with for example, the “Basic” path average value for Model A2 reduced to 0.33, although these values have less significance since so many simulated variables are fixed at the measured values to give zero individual EOF values. 6. Conclusions A methodology using the Calculator module of the commercial simulation tool PRO/II to quantify plantmodel mismatch has been developed that evaluates and clearly differentiates the minimum of an appropriately defined mismatch EOF at an optimum average value of the simulation model parameter(s) that is accurate to within 0.01 (or 1%) effective tray efficiency units for the A1 simulation model. Although the absolute value of any minimum total EOF has no real significance, the ratio of such values obtained for alternative simulation model formulations with respect to the K-value data models used, is available to serve as a guide to choosing the most suitable data model for the ultimate intended purpose of on-line process optimization. This concept can be extended for application to the optimization of simulation model accuracy for any given separation process. In the development of this methodology the following has been shown for the deethyleniser: (a) a normalized least squares EOF that includes a normalizing term estimated from the best available measurement variance over the selected “steady state” time span, is suitable for quantifying plant-model mismatch; (b) the preferred Ap simulation model, using effective average equilibrium tray efficiencies as simulation model parameters, is the most suitable of the three postulated simulation models for the evaluation of plant-model mismatch; (c) the preferred “Basic” manual optimization path is the most time-effective optimization path while still allowing the differentiation of a relatively sharp minimum in the total EOF at inferred optimum values of the simulation model parameter(s); and (d) many assumptions made in the development of the “Basic” optimization path for the preferred Ap simulation model have been justified by a comparison


of the results with results obtained from the use of alternative simulation model formulations and alternative optimization paths, as well as by spot simulation checks. In general it is apparent that the value assigned to the tray efficiency parameters, particularly in the rectifying section, is extremely important in minimizing the total EOF for each simulation model Xp. It is further clear that a two-parameter model X2 is capable of more than halving the one-parameter model X1 total EOF for the simulation models considered. This reduction is almost entirely due to discrepancy reductions in stripping section variables. Given the uncertainties in input data and the relatively large increase in total EOF values for rectifying section parameters only slightly removed from the optimum, it is expected that the “Basic” optimization path for model A2 is adequate for the purpose of assessing the relative effect of using alternative K-value data models on the value of the minimum total EOF and the corresponding inferred optimum simulation model parameters. This has been demonstrated by Kamphausen,6 where different K-value data models infer significantly different optimized simulation model parameters for the same process data. The use of sufficiently sophisticated data reconciliation techniques is critical to accurate minimization of plant-model mismatch and successful simulation model parameter estimation. The most likely source of erroneous results is the use of unreconciled operating data. It is expected that further development of the methodology using more advanced versions of commercially available data reconciliation and simulation tools will provide a more accurate and fully automated optimization procedure.

Literature Cited (1) Prausnitz, J. M.; Anderson, T. F.; Grens, E. A.; Eckert, C. A.; Hsieh, R.; O’Connell, J. P. Computer Calculations for Multicomponent Vapor-Liquid and Liquid-Liquid Equilibria; PrenticeHall: Englewood Cliffs, NJ, 1980. (2) Knudsen, K.; Stenby, E. H.; Fredenslund, A. A Comprehensive Comparison of Mixing Rules for Calculation of Phase Equilibria in Complex Systems. Fluid Phase Equilib. 1993, 82, 361368. (3) Bevan, D. J. Analysis of Distillation Column Performance using Computer Simulation. Proc. Chemeca ‘88, Sydney; 1988; pp 233-247. (4) 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. (5) DATACON Keyword Input Manual, Version 2.0; Simulation Sciences Inc.: Brea, CA, 1994. (6) Kamphausen, A. H. Selection and Tuning of Thermodynamic Data Models in Deethyleniser Simulation and Optimisation; Ph.D. Thesis, RMIT University, Melbourne, Australia, 1999. (7) PRO/II Keyword Input Manual, Version 4.0; Simulation Sciences Inc.: Brea, CA, 1994. (8) 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. (9) Kim, I. W.; Liebman, M. J.; Edgar, T. F. A Sequential Errorin-Variables Method for Non Linear Dynamic Systems. Comput. Chem. Eng. 1991, 15, 663-670. (10) Anderson, R. H.; Garrett, G.; Van Winkle, M. Efficiency Comparison of Valve and Sieve Trays in Distillation Columns. Ind. Eng. Chem. Process Des. Dev. 1976, 15, 96-100. (11) PRO/II Casebook No 5: Air Separation Plant; Simulation Sciences Inc.: Brea, CA, 1993.

Received for review November 25, 2003 Revised manuscript received July 1, 2004 Accepted July 12, 2004 IE034270C

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PlantâModel Mismatch Analysis in Deethyleniser Simulation. 1