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Sensitivity of Process Design to Phase EquilibriumA New Perturbation Method Based Upon the Margules Equation Paul M. Mathias* Fluor Corporation, 3 Polaris Way, Aliso Viejo, California 92698, United States ABSTRACT: It is widely recognized by experts that the computer-based design of chemical processes depends strongly on the correlated thermodynamic and transport properties, and the effect of property uncertainties should be incorporated into the design. The most significant source of property uncertainties on process design is from the correlations of phase equilibrium. Many approaches have been proposed, but uncertainty analysis is not a routine component of today’s industrial practice, mainly because education and awareness is lacking, and the proposed methods are difficult to apply. The purpose of this paper is to report a new approach to uncertainty analysis that is intuitively appealing and easy to apply in process simulation. The proposed approach focuses on activity coefficients and is based upon the simplification that, for the purpose of perturbation, the liquid mixture can be treated as a set of pseudobinaries described by the Margules equation. The resulting perturbation in the activity coefficient of component i goes to zero when its mole fraction approaches unity, and also is relatively small when its estimated activity coefficient is close to unity (i.e., near-ideal systems are likely to be modeled more accurately than highly nonideal systems). In practice, the proposed uncertainty analysis is performed by varying a single parameter for each component in the mixture. The utility of the proposed uncertainty method is demonstrated by application to two problems: (1) a propylene−propane superfractionator for which small changes in correlated relative volatilities have a large effect on the design of the distillation column, and (2) a dehexanizer column that separates a mixture containing many close-boiling components. It is demonstrated that the proposed analysis provides quantitative insight into the effect of property uncertainties and helps to quantify the safety factors that need to be imposed upon the design. While the proposed method is applied to activity-coefficient models, the same idea is applicable to other models such as equations of state.

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are of “unequal importance,” and tend to be most important under low-temperature conditions. Mah10 has shown that in some cases rough approximations of physical properties are quite adequate. By “unequal importance,” Streich and Kistenmacher meant that in some cases there is extreme sensitivity of design to physical properties, and in other cases, the design is insensitive to property uncertainties; the latter is Mah’s observation. Several studies have illustrated the effects of the uncertainty in physicalproperty models on process design,11−14 however these publications do not provide an effective way to quantify the propagation of property uncertainty into design variability. Macchietto, Maduabeuke, and Szcepanski15 developed a method to connect exact derivatives of calculated properties to flowsheet simulations (e.g., derivative of ethylene purity in an ethylene/ethane superfractionator to the ethane−ethylene binary interaction parameter), and presented examples to demonstrate “that very valuable information can be obtained as to which properties matter in a

INTRODUCTION Quantifying the uncertainty in physical properties is clearly recognized as being of paramount importance by scientists and technologists in the chemical industry. In a forward to the Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results,1 then-director of NIST, Dr. John W. Lyons, wrote, “It is generally agreed that the usefulness of measurement results, and thus much of the information that we provide as an institution is, to a large extent, determined by the quality of the statements of uncertainty that accompany them.” Six key journals in the field of thermodynamics (Journal of Chemical and Engineering Data, Journal of Chemical Thermodynamics, Fluid phase Equilibria, Thermochimica Acta, and the International Journal of Thermophysics) have mandated reporting of combined uncertainties together with the experimental data tables.2−7 Kim et al.8 have highlighted the availability of online resources as an enabler to incorporate uncertainty analysis into chemical-engineering education. The design of chemical processes today invariably uses computer simulations, and many technologists have recognized the need to understand and evaluate the impact of uncertainties in property models on process design and plant operability. Streich and Kistenmacher9 pointed out that physical properties © XXXX American Chemical Society

Special Issue: In Honor of Grant Wilson Received: August 15, 2013 Accepted: September 25, 2013

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dx.doi.org/10.1021/je400748p | J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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γm i is calculated from the chosen activity-coefficient model, and is not constrained by the pseudobinary approximation.

process and where.” Whiting and co-workers have made extensive efforts to emphasize the importance of understanding uncertainties in physical properties as well to quantify their effect on process design and plant operation.16−21 For example, they demonstrated strategies for quantification of process uncertainties induced by property uncertainties via Monte Carlo simulations. While the importance of property uncertainty analysis is clearly recognized in the technical literature, it is hardly used in chemical-engineering practice; as Kim et al.8 observed, “Its practical implementation in a variety of scientific and engineering fields has typically seen less emphasis than it deserves.” There are several barriers to adopting quantitative uncertainty analysis in chemical-engineering industrial practice. Kim et al.8 identified education as a key barrier and proposed the use of online properties with clearly identified uncertainties as the means to incorporate uncertainty analysis into undergraduate and graduate courses. Another barrier is that the methods proposed by researchers such as Macchietto, Maduabeuke, and Szcepanski15 and Whiting and co-workers16−21 are not easily available in commercial process simulators. A key barrier in the experience of the present author is that simple, intuitive methods to perturb properties are not available, and the purpose of this paper is to propose such a method and to demonstrate its application through representative examples. This work focuses on phase equilibrium, which is considered to be the most important physical property,2 and develops a semiempirical method to perturb activity coefficients based upon the simplification that, for the purpose of perturbation, the liquid mixture can be treated as a set of pseudobinaries described by the Margules equation. The present approach assumes that the pure-component vapor pressures are described accurately and hence the effective uncertainty results from the modeling of the mixture nonideality or the residual Gibbs energy. The end result is an intuitive method where the activity coefficient (and consequently, the K-value) of any component in the mixture can be perturbed in a manner approximately conformal with thermodynamic consistency. This method is then used in flowsheet simulations to quantitatively relate uncertainties in K-values with the variability in process design.

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ln(γi p) = Ai (1 − xi)2

(2)

Equation 1 defines the perturbation to the activity coefficient (γpi ) that is added to the activity coefficient calculated from the model (γm i ); the perturbation can be applied to any activitycoefficient model. Equation 2 describes the perturbed activity coefficient. The pseudobinary approximation used here is that component i forms a binary with all the other components lumped as a single one. The base-model activity coefficient of component i can be used to get a rough idea of the value of Ai. Ai0 = ln(γim)/(1 − xi)2

(3)

For the purposes of perturbing the activity coefficient, a perturbation parameter δi is introduced, and Ai is defined as follows: Ai ≡ δi

|Ai0| 1 + |Ai0|

(4)

The rationale behind eq 4 is that for near-ideal systems (where A0i is small relative to unity or the calculated activity coefficient is close to unity), the model estimate of the activity coefficient (γm i ) is expected to be accurate even when xi is much smaller than unity. On the other hand, the full perturbation will be in effect when A0i is large compared to unity and xi is small relative to unity. The final expression for the perturbation to the activity coefficient is given by eq 5. ⎡ ⎤ |ln(γi m)| ⎥ ln(γi p) = δi(1 − xi)2 ⎢ ⎢⎣ (1 − xi)2 + |ln(γi m)| ⎥⎦

(5)

Equation 5 is phenomenological, but has desired characteristics. The perturbation in γi is small when γm i is close to unity, and also is small when the mole fraction of component i is high (i.e., close to unity). For components at very low concentration and with activity coefficients significantly different from unity, the relative change in γi is approximately equal to {exp(δi) − 1}.

MARGULES-BASED PERTURBATION SCHEME

The perturbation scheme proposed in this work focuses on mixture nonideality rather than pure-component properties. There are examples of property discrepancies due to erroneous pure-component properties (for example see the styrene− ethylbenzene example in Zeck14), but these can often be eliminated by good engineering practice. Kim et al.8 and Hajipour and Satyro22 have shown how to relate uncertainty in pure-component properties to the quality of the design. Mixture properties are more challenging because significant uncertainties resulting from mixture nonideality usually remain even with high-quality data and sound engineering practice. The proposed approach applies the simplification that for the perturbation to the activity coefficient, the mixture can be divided into a set of pseudobinaries, which are then described by the Margules equation. ln(γi) = ln(γi m) + ln(γi p)

Figure 1. Percentage change in γ∞ calculated by eq 5 as a function of γ∞ and at various values of δ.

Figure 1 and Figure 2 show the percentage change in the infinite-dilution activity coefficient as a function of the infinitedilution activity coefficient γm∞ and the perturbation parameter δ. Figure 1 indicates that the perturbation in γ approaches {exp(δ) − 1} when the base activity coefficient is large. Figure 2 gives an indication about the range of values of δ to be chosen

(1)

where γi is the effective activity coefficient after perturbation, γm i is the activity coefficient calculated by the model, and γpi represents the perturbation due to the uncertainty. Note that B



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perturbation scheme to quantify process uncertainties for this important industrial application. The property option chosen in Aspen Plus is NRTL-RK, which uses the NRTL26 model for activity coefficients and the Redlich−Kwong27 equation of state to describe the vapor phase. The vapor-pressure correlations for propane and propylene were taken from the correlation of Starling,28 which is the source used by Howat and Swift.25 The Starling correlation agrees with the NIST WebBook29 to about (0.1 to 0.2) % for vapor pressures above 100 kPa, which is the pressure range of interest to the present application, and hence we conclude that uncertainties in vapor pressure are a minor contribution to model uncertainty. All other pure-components parameters were taken from the Aspen Plus V7.3 database. The NRTL model has two parameters (αij and τij). αij is a symmetric binary parameter, and its value has been set to 0.4. τij is a temperature-dependent binary parameter,

Figure 2. Percentage change in γ∞ calculated by eq 5 as a function of δ and at various values of γ∞.

τij = aij + bij + eij ln(T ) + fij /T

for a particular case. As will be seen in Example 2 (Dehexanizer Column), the largest values of the infinite-dilution activity coefficients are in the range 1.5 to 2, and estimated uncertainties in the activity coefficients is about 5 %. Thus an appropriate range of δ values to be studied for this case is between + 0.2 and − 0.2. Equation 5 has been implemented into the Aspen Plus V7.3,23 and all calculations in this paper have been done using this version of the software. However, the present methodology can easily be applied to any software package. In the next section, two examples are presented that illustrate the wide range of analyses that may be performed with the proposed activity-coefficient perturbation scheme. The paper concludes by summarizing the benefits and discussing future work.

(6)

The fitted values of the αij and τij parameters are presented in Table 1. The “relative volatility,” α, which is defined as ratio of Table 1. NRTL Parameters for the C3H6(1)−C3H8(2) Binary

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EXAMPLE APPLICATIONS The two example applications used to illustrate the perturbation scheme are as follows: Example 1: Propane−Propylene Splitter. This application quantifies the extreme sensitivity to small mixture nonidealities in the separation of close-boiling mixtures. Example 2: Dehexanizer Column. The perturbation scheme is demonstrated to be of practical value in a case involving a large component slate. A point that should be emphasized is that the first step in the proposed analysis is qualitative understanding and quantification of the uncertainties in the mixture nonideality. For example, for a binary mixture of lose-boiling components, a small change in nonideality may cause an azeotrope and this will result in a significant qualitative change to the design. In the two chosen examples, both aspectsqualitative understanding and quantificationof the mixture nonideality have been given considerable attention. Example 1: Propane−Propylene Splitter. Propane− propylene splitters separate a close-boiling mixture, and are therefore highly sensitive to the phase-equilibrium correlation. Streich and Kistenmacher9 wrote, “If the process engineer must know the number of theoretical trays in the C2 or C3 splitters within 5 %, he will have to predict the K-values for ethane within 0.6 % and the K-values for propane within 0.3 %. We will define the accuracy level for a tray number within 5 % as the ‘desirable’ accuracy.” Hsu24 found that the Howat and Swift25 correlation is highly accurate for this binary mixture, and we have therefore used this correlation with the present

parameter

value

αij = αji a12 a21 b12 b21 e12 e21 f12 f 21

0.4 0.15911241 −0.3975538 −42.621622 131.685669 0 0 0 0

the two component K-values, is an important measure of the ease or difficulty of separation by distillation. Figure 3 compares

Figure 3. Relative volatility of propylene to propane at temperatures from (227.6 to 360.9) K. The points are the calculated results reported by Howat and Swift,25 and the lines are from the present model.

the present correlation to the calculated results of Howat and Swift25 for the propylene−propane relative volatility, and Figure 4 focuses on the propylene-rich portion of the chart. In Figure 4, ± 0.5 % error bars are shown for the 327.6 K isotherm. The agreement between the present model and the Howat and Swift correlation is better than 0.5 %, and is estimated to be ± 0.3 %. C



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propane and propylene that is at its bubble-point temperature at 963 kPa. The feed stage is 75 numbered from the top. The column specifications are 0.99 mol fraction propylene as distillate and 0.97 mol fraction propane as the bottoms product. The calculated condenser and reboiler duties for the base case are −125.7 GJ/h and 123.4 GJ/h, respectively. Sensitivity studies have been performed to evaluate how the calculated reboiler duty changes as the product purity specifications are changed, and as the two component activity coefficients are perturbed. Figure 6 shows how the calculated

Figure 4. Relative volatility of propylene to propane at temperatures from (227.6 to 360.9) K. This figure is a subset of Figure 3 with a focus on propylene-rich compositions. The error bars for the 327.6 isotherm show a ± 0.5 % deviation from the Howat and Swift25 model.

Figure 6. Effect of the propane product purity on the calculated reboiler duty (base case).

reboiler duty increases as the propylene concentration in the propane product is reduced from a mole fraction of 0.03 to 0.005 while the propylene purity is kept fixed at a mole fraction of 0.99, and Figure 7 presents the analogous change in reboiler Figure 5. Relative volatility of propylene to propane in the purepropylene limit. Comparison among various correlations: Howat and Swift,25 Manley and Swift,30 Harmens,31 Funk and Prausnitz,32 and the present model. The error bars show ± 1 % deviation from the Howat and Swift correlation.

Figure 5 attempts to estimate the uncertainty of the calculated relative volatility by comparing model predictions from various correlations at the propylene-rich end of the concentration scale. The models compared are those from Howat and Swift,25 Manley and Swift, 30 Harmens, 31 Funk and Prausnitz, 32 and the present correlation. The present correlation agrees well with the model of Howat and Swift, and this, of course, is because it was fit to this model. There is reasonably good agreement (< 1 %) between the Howat and Swift model and the correlation presented by Funk and Prausnitz32 for the propylene-propane relative volatility. The correlations of Manley and Swift,30 and Harmens31 show significant departures from the Howat and Swift model (> 1 %) at low temperatures, say less than 275 K. The uncertainty studies performed in this work are at temperatures between (290 and 300) K, and here we assume that the relative volatilities from the present model are probably within ± 0.5 %, and most likely within 1 %. The purpose of the present study is primarily to demonstrate the relationship between phaseequilibrium uncertainty and variability in process design. The property perturbation scheme has been implemented in a representative C3-splitter distillation column. The column has a total of 116 stages, including a reboiler and a total condenser. The column is assumed to operate at 125 psig (963 kPa), and the feed rate of 1000 kmol/h is for an equimolar mixture of

Figure 7. Effect of the propylene product purity on the reboiler duty (base case).

duty due to a decrease in the propane concentration in the propylene product from a mole fraction of 0.01 to 0.002 while the propane purity is kept at its base value. It is clear that there is a larger relative increase in reboiler duty due to an increase in the propylene purity than in the propane purity, and this is because the relative volatility in much larger at the propane-rich end than at the propylene-rich end (Figure 3). Figure 8 presents the results of perturbing the propylene activity coefficient. This was done by varying δC3H6 (eq 5) and relating this value to the percentage change in the infinitedilution relative volatility for propane-rich mixtures. The effect of perturbing the propylene activity coefficient is quite modest. At the base purity of 0.03 mol fraction propylene in propane a 1 % reduction in the relative volatility causes 1 % increase in the reboiler duty, and if the purity is modified to 0.01 mol fraction propylene in propane, the increase in reboiler duty is 2.7 %. D



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propane K-value of 0.3 %, which is in broad agreement with the result we have obtained. Further, Fair33 concluded that for a relative volatility of 1.1 a 10 % error in K-values leads to a 100 % error in equipment cost. Similar examples have been given by Nelson, Olson and Sandler,34 and by Peridis, Magoulas, and Tassios.35 The extra value that the present perturbation scheme provides is that one can quickly and easily make a quantitative estimate of the effect of phase equilibrium on process design for any specific design case. Example 2: Dehexanizer Column. Dehexanizer columns remove C5 and C6 paraffins from a heavier slate of components that are mostly aromatic. The components in a typical dehexanizer column are presented in the Appendix (Table A1). Here we need to deal with 61 components ranging from lower aliphatics to C10 aromatics. The first column of Table A1 contains the component IDs while the second column provides the chemical name. For the purposes of identification of the compounds the chemical formulas, molecular weights, and normal boiling points are also provided. The feed mole fraction is given in column 6. The column has 30 theoretical stages, with a reboiler and total condenser, and the feed tray is #23, numbered from the top. The condenser pressure is 160 kPa with a pressure drop of 2.9 kPa in the condenser. The pressure drop is equal over the remaining stages, and the reboiler pressure is 227 kPa. The column has two specifications: the total mole fraction of 22DMP and higher in the distillate is 0.02, and the total mole fraction of BENZENE and lower in the bottoms is 0.02 (note that the component names used here match those in the first column of the Appendix). These specifications are achieved by varying the distillate rate and the reboiler duty. The property option chosen in Aspen Plus is again NRTLRK. The pure-component properties have been taken from the Aspen Plus V7.3 database, and these are expected to be

Figure 8. Effect of changing the infinite-dilution relative volatility on the calculated reboiler duty. The calculations have been done at various values of the propane purities. The infinite-dilution relative volatility (α) is representative of mixtures with very low concentrations of propylene in propane.

Figure 9. Effect of changing the infinite-dilution relative volatility on the calculated reboiler duty. The calculations have been done at various values of the propylene purities. The infinite-dilution relative volatility (α) is representative of mixtures with very low concentrations of propane in propylene.

Figure 9 presents analogous results of perturbing the relative volatility at the propylene-rich end, and the results for this case are dramatically different. Here a 1 % reduction in the relative volatility for the base case of 0.01 mol fraction propane in the propylene product causes a 13 % increase in reboiler duty, and the increase in reboiler duty rises to 28 % if the propylene purity specification is fixed at a mole fraction of 0.998. These results are, of course, qualitatively expected. As noted above, Streich and Kistenmacher9 wrote that reducing the uncertainty in the number of stages to 5 % requires an uncertainty in the

Figure 10. Activity coefficients of key components in the dehexanizer column.

Table 2. Ordered Mole Fractions of Components in the Overhead (Distillate), and Infinite-Dilution Activity Coefficient of BENZENE in These Components at 313.15 K component

OHD (x)

cum (x)

order

MW

γ∞

NRTL fit?

NC5 IC5 2MP NC6 3MP BENZENE 22DMB 23DMB 24DMP MCP

0.306716 0.204997 0.123511 0.118615 0.086975 0.055396 0.053790 0.026106 0.011325 8.06 × 10−03

0.307 0.512 0.635 0.754 0.841 0.896 0.950 0.976 0.987 0.995

1 2 3 4 5 6 7 8 9 10

72.15 72.15 86.18 86.18 86.18 78.11 86.18 86.18 100.20 84.16

1.70 1.63 1.64 1.50 1.55 1.00 1.80 1.86 1.47 1.39

Y N Y Y N

E

Y Y Y Y



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Table 3. Ordered Mole Fractions of Components in the Bottoms and Infinite-Dilution Activity Coefficient of Key Light Components (NC6, MCP, CH, 22DMP, 24DMP, and 223TMP) in These Components at 363.15 K. The Cells Shaded Green Indicate Those Binaries for Which NRTL Parameters Are Available

contribute to the design uncertainty. The main goal of our analysis is to quantitatively understand the uncertainties in the model activity coefficients and to use the proposed perturbation scheme to relate these uncertainties to variability in the column design and operation. Figure 10 presents the calculated activity coefficients of some key components. There are mainly two families of species in the dehexanizer column: the saturated aliphatic compounds and the aromatic compounds. It is generally expected that mixtures containing a single family of compounds will exhibit near-ideal behavior (activity coefficients close to unity), while components

sufficiently accurate and thus not a primary source of phaseequilibrium uncertainty. The NRTL parameters were taken from the Aspen Plus V7.3 databases where available, and these were supplemented with fits using binary VLE data from NISTTDE.36 Finally, the UNIFAC predictive method37 was used to fill any remaining gaps in the NRTL binary parameters. The Appendix presents component splits for the base case in the last two columns of the table. Many components split strongly to either the distillate of the bottoms; it is the components that split somewhat evenly between the two column products that are of the most interest, as these F



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from one family at low concentration in a mixture dominated by the other family will have activity coefficients greater than unity. Figure 10 has this expected behavior since the activity coefficient of BENZENE is greater than unity at the top of the column, while the activity coefficients of the saturated aliphatic compounds are greater than unity at the bottom of the column (mainly an aromatic mixture). The results of the NRTL correlation are thus at least semiquantitatively correct. The main component that has activity coefficients different from unity at the top of the column is BENZENE. Table 2 presents the components in the overhead in order of decreasing mole fraction. Table 2 also presents the infinite-dilution activity coefficient of BENZENE in these components and whether NRTL parameters are either available from the Aspen Plus database or fitted using experimental data (last column of Table 2). The activity coefficients are expected to be accurate because in most cases the NRTL binary parameters are available and, when they are missing, the UNIFAC prediction is expected to be reliable; for example, IC5 is similar to NC5, and 3MP is similar to NC6. The next issue that needs to be studied is the activity coefficients of the key light components in the lower part of the column. Table 3 presents the components in the bottoms in order of decreasing mole fraction. Table 3 also shows the infinite-dilution activity coefficient of key light components (NC6, MCP, CH, 22DMP, 24DMP, and 223TMP) in these high-concentration species at 363.15 K (representative bottoms temperature). The shaded cells indicate those binaries for which NRTL parameters are available. It is clear that many key NRTL binary parameters are missing, but it is reassuring that the activity coefficients are as expected. For example, the paraffin−paraffin activity coefficients are close to unity, while the paraffin−aromatic activity coefficients are higher than unity. We thus expect that the NRTL model will at least give reasonable results.

Table 5. Effect of Activity-Coefficient Perturbation on Dehexanizer Column Design

no. data points

temp range (K)

38

10

348 to 374

39

26

345 to 368

40 41

10 10

342 to 349 349 to 357

components NC6−BENZENE− TOLUENE NC6−BENZENE− TOLUENE NC6−MCP−BENZENE NC6−MCP−CH− BENZENE−TOLUENE

δ-heavies

δ-lights

base ↑↑ ↓↓ ↓↑ ↑↓

0 0.20 −0.20 −0.20 0.20

0 0.20 −0.20 0.20 −0.20

3.139 3.125 3.171 2.855 3.505

boilup ratio

% change in reboiler duty

% change in condenser duty

0.847 0.837 0.859 0.805 0.901

−0.8 1.2 −5.0 6.5

−0.4 0.8 −6.9 8.9

while a down arrow means that the component δ values have been decreased. The first arrow refers to the heavies and the second arrow refers to the lights. For example, ↑↓ means that the δ values of the heavies have been increased and the δ values of the lights have been decreased. Examination of the results in Table 5 provides the following insights: 1. Changing the activity coefficients of the heavies and lights in the same direction (cases ↑↑ and ↓↓) has only a small effect on the column design. This is to be expected since ↑↑ improves the stripping section and hurts the rectifying section, while ↓↓ has the opposite effect, but it is reassuring that the model gives this expected result. 2. Lowering the activity coefficients of the heavies and raising the activity coefficients of the lights (case ↓↑) reduces the reboiler and condenser duties resulting in an optimistic design. Again, this is an expected result (both the stripping and rectifying sections are improved), but obtaining expected results increases confidence in the approach used here. 3. Finally, raising the activity coefficients of the heavies and lowering the activity coefficients of the lights results in a conservative design. Perturbing the activity coefficients by about 5 % results in increases in the reboiler and condenser duties by 7 % and 9 %, respectively. Hence the approach used here provides a quantitative relationship between the activity coefficient uncertainty and the column design.

Table 4. Comparison between NRTL-RK K-Value Predictions and Four Sets of Multicomponent Data. All Measured Data Are at a Pressure of 101.325 kPa data ref

case

reflux ratio

average sum-squared % error in K-value

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5.3, 3.3, 6.0

CONCLUSIONS The purpose of this work has been to develop a simple, intuitive, and easily applicable method to perturb liquid-phase nonideality (in particular, the activity coefficient) in process simulations, and to demonstrate its utility by applying it to two representative cases. The first task that must be completed in any uncertainty analysis is to quantitatively understand the correlation uncertainties, and considerable effort has been devoted to explaining the details of this effort. The two examples reveal different benefits from the proposed perturbation approach. The propane−propylene splitter example demonstrates that the effect of property uncertainties for a close-boiling system can be quantified for the specific design of interest. The dehexanizer column example illustrates that even for a system containing many components, the approach clarifies the key uncertainties and enables a quantitative relationship between the property uncertainties and the variability in the design. The uncertainties in the dehexanizer column are about an order of magnitude higher than for the propane−propylene splitter, but the design uncertainties are much larger for the propane−propylene splitter. Qualitatively, this is an expected result; however, the value of the present

3.1, 1.0, 3.1 1.7, 6.5, 2.2 1.8, 5.1, 2.0, 1.6, 2.9

Table 4 presents comparisons between NRTL-RK K-value predictions and four sets of multicomponent data (Arai et al.,38 Saito,39 Rowan and Weber,40 and Weatherford and Van Winkle41). The last column in Table 4 presents the average sum-squared percentage differences between the NRTL-RK model predictions and the data. These sum-squared differences (errors) may be taken as an indication of the model uncertainty. As a conservative estimate, the uncertainty of the K-values is taken to be ± 5 %, and hence an appropriate range of values of δi is between − 0.2 and + 0.2 (Figure 2). These values of δi have been used in the perturbation approach and the results are presented in Table 5. The base case is the one where no perturbation has been done. Other cases are identified by up and down arrows. An up arrow means that the component δ values, and hence the activity coefficients, have been increased, G



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approach is that the design uncertainties can be quantified with modest effort. Future work is planned to apply the present approach to additional cases, and to extend the method to other models, for example equations of state.

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APPENDIX

The components of a typical dehexanizer column are presented in Table A1, along with the chemical formulas, molecular weights, normal boiling points, and feed mole fraction of each component.

Table A1 component split component ID

component name

formula

MW

Tb (K)

feed mole fraction

distillate

bottoms

C3 IC4 NC4 IC5 NC5 C5-OLFIN 22DMB 23DMB 2MP 3MP NC6 C6-OLIFN MCP CH BENZENE 22DMP 24DMP 223TMB 33DMP 23DMP 2M-HEX 3M-HEX 3E-PENT NC7 C7-OLEFN DM-CP MCH ECP TOLUENE IC8 NC8 C8-OLEFN PRPL-CP E-CH E-BENZ PXYLENE MXYLENE OXYLENE IC9 NC9 C9+OLFN BUTYL-CP PROPL-CH PROPL-BZ 1M-3E-BZ 1M-4E-BZ 135TM-BZ 1M-2E-BZ 124TM-BZ 123TM-BZ INDAN C10-PRFN BYTYL-CH

propane isobutane n-butane 2-methyl-butane n-pentane trans-2-pentene 2,2-dimethyl-butane 2,3-dimethyl-butane 2-methyl-pentane 3-methyl-pentane n-hexane trans-2-hexene methylcyclopentane cyclohexane benzene 2,2-dimethylpentane 2,4-dimethylpentane 2,2,3-trimethylbutane 3,3-dimethylpentane 2,3-dimethylpentane 2-methylhexane 3-methylhexane 3-ethylpentane n-heptane cis-2-heptene trans-1,3-dimethylcyclopentane methylcyclohexane ethylcyclopentane toluene 2-methylheptane n-octane trans-2-octene n-propylcyclopentane ethylcyclohexane ethylbenzene p-xylene m-xylene o-xylene 2-methyloctane n-nonane 1-nonene n-butylcyclopentane n-propylcyclohexane n-propylbenzene 1-methyl-3-ethylbenzene 1-methyl-4-ethylbenzene 1,3,5-trimethylbenzene 1-methyl-2-ethylbenzene 1,2,4-trimethylbenzene 1,2,3-trimethylbenzene indane 3-methylnonane n-butylcyclohexane

C3H8 C4H10 C4H10 C5H12 C5H12 C5H10 C6H14 C6H14 C6H14 C6H14 C6H14 C6H12 C6H12 C6H12 C6H6 C7H16 C7H16 C7H16 C7H16 C7H16 C7H16 C7H16 C7H16 C7H16 C7H14 C7H14 C7H14 C7H14 C7H8 C8H18 C8H18 C8H16 C8H16 C8H16 C8H10 C8H10 C8H10 C8H10 C9H20 C9H20 C9H18 C9H18 C9H18 C9H12 C9H12 C9H12 C9H12 C9H12 C9H12 C9H12 C9H10 C10H22 C10H20

44.1 58.1 58.1 72.2 72.2 70.1 86.2 86.2 86.2 86.2 86.2 84.2 84.2 84.2 78.1 100.2 100.2 100.2 100.2 100.2 100.2 100.2 100.2 100.2 98.2 98.2 98.2 98.2 92.1 114.2 114.2 112.2 112.2 112.2 106.2 106.2 106.2 106.2 128.3 128.3 126.2 126.2 126.2 120.2 120.2 120.2 120.2 120.2 120.2 120.2 118.2 142.3 140.3

231.11 261.43 272.65 300.99 309.22 309.49 322.88 331.13 333.41 336.42 341.88 341.02 344.96 353.87 353.24 352.34 353.64 354.03 359.21 362.93 363.20 365.00 366.62 371.58 371.56 364.88 374.08 376.62 383.78 390.80 398.83 398.15 404.11 404.95 409.35 411.51 412.27 417.58 416.45 423.97 420.02 429.75 429.90 432.39 434.48 435.16 437.89 438.33 442.53 449.27 451.12 440.95 454.13

trace trace trace 0.033 0.050 trace 0.009 0.004 0.021 0.015 0.024 Trace 0.002 0.000 0.019 0.002 0.011 0.002 0.003 0.002 0.034 0.023 0.004 0.028 trace 0.004 0.001 trace 0.318 0.017 0.005 trace trace trace trace 0.066 0.071 0.149 0.002 trace trace trace trace 0.005 0.012 0.005 0.006 0.005 0.021 0.005 trace trace trace

1.0000 1.0000 1.0000 0.9993 0.9983 0.9980 0.9896 0.9597 0.9531 0.9269 0.8330 0.8013 0.6664 0.1427 0.4828 0.2352 0.1689 0.1051 0.0060 0.0003 0.0005 0.0002 0.0001 0.0000 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0007 0.0017 0.0020 0.0104 0.0403 0.0469 0.0731 0.1670 0.1987 0.3336 0.8573 0.5172 0.7648 0.8311 0.8949 0.9940 0.9997 0.9995 0.9998 0.9999 1.0000 1.0000 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

H



Article

Table A1. continued component split component ID

component name

formula

MW

Tb (K)

feed mole fraction

distillate

bottoms

C10-AROM BUTL-BZ 14DE-BZ 1245TMBZ C1+AROM 23HX 1M1E 1T2H

1,2-dimethyl-3-ethylbenzene isobutylbenzene 1,4-diethylbenzene 1,2,4,5-tetramethylbenzene n-pentylbenzene 2,3-dimethylhexane 1-methyl-1-ethylcyclopentane trans-1,2-dimethylcyclohexane

C10H14 C10H14 C10H14 C10H14 C11H16 C8H18 C8H16 C8H16

134.2 134.2 134.2 134.2 148.2 114.2 112.2 112.2

467.11 445.94 456.94 469.99 478.61 388.76 394.67 396.58

0.013 trace trace trace trace 0.007 0.001 0.000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

DEDICATION This paper is dedicated to the memory of Dr. Grant Wilson who showed us by his vast body of work how to measure quality data with defined uncertainties, to develop useful correlations, and to apply the data and correlations to design and improve chemical process technology. Grant Wilson’s career exemplified the highest excellence in the practice of chemical process technology.

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