Oil Characterization from Simulation of Experimental Distillation Data

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Oil Characterization from Simulation of Experimental Distillation Data Marco A. Satyro* and Harvey Yarranton Department of Chemical and Petroleum Engineering, the UniVersity of Calgary, Department of Chemical and Petroleum Engineering, Calgary, Alberta, Canada T2N 1N4 ReceiVed January 9, 2009. ReVised Manuscript ReceiVed June 1, 2009

The characterization of crude oil involves dividing the oil into pseudocomponents and allocating mole fractions, molar mass, specific gravity, average boiling point, and critical properties to each component. The characterization is typically based on distillation data reported in terms of true boiling points. Standard assay types such as the ASTM D86 or ASTM D1160 vacuum distillation do not provide well established saturated bubble temperatures and require empirical interconversion curves to convert the assay data into true boiling point (TBP) data. Recently developed assays such as the ASTM D5236 and Bruno’s new distillation assay methodology do provide well-defined saturated bubble temperatures that correspond to actual thermodynamic state points but lack an established interconversion method to a TBP, that is, a method to determine the TBP of the fluid based on the measured temperatures of the assay. In this work, a methodology is presented to determine pseudocomponent mole fractions that match the boiling point data from these new assays. The fluid is divided into pseudocomponents of different average boiling point, and the molar mass and other physical properties of each component are determined using established correlations. A simulation of the distillation is optimized to match the assay data by adjusting the mole fraction of each pseudocomponent. The characterization can also be constrained to match other data such as the bulk density and molar mass of the fluid. The proposed methodology is tested on naphtha and Alaska crude oil and then verified through three heavy oil case studies. The methodology is entirely general and can be applied to a compositional analysis from a distillation of any material.

Introduction The characterization of heavy oils and bitumen is a fundamental step in the design, simulation, and optimization of solvent extraction plants and distillation facilities. In refinery applications, the oil is typically characterized based on a distillation assay. This procedure is reasonably well-defined and is based on the representation of the mixture of actual components that boil within a boiling point interval by hypothetical components that boil at the average boiling temperature of the interval. These pseudocomponents are also constrained to match the average physical properties of the original bulk fluid, such as molecular weights and densities. The set of these hypothetical components with their compositions is then used as a facsimile for the actual fluid, since its actual composition is usually too difficult to obtain using analytical techniques. Once the normal boiling point, density, and molecular weight of the pseudocomponents are determined, important physical properties such as vapor pressures, critical constants, and ideal gas heat capacities can be estimated using established correlations.1-3 These properties are the inputs required to use an equation of state or other thermodynamic model to calculate the thermodynamic equilibrium of the mixture. * To whom correspondence should be addressed. E-mail: msatyro@ ucalgary.ca. (1) Whitson, C. H.; Brule, M. R. Phase Behavior, Monograph Vol. 20, SPE, Henry L. Doherty Series, Richardson, TX, 2000. (2) Riazi, M. R. Characterization and Properties of Petroleum Fractions, 1st ed.; ASTM Manual Series, American Society of Testing and Materials: Philadelphia, PA, 2005. (3) Pedersen, K. S.; Fredenslund, A.; Thomassen, P. Properties of Oils and Natural Gases; Gulf Publishing: Houston, 1989.

The technique hinges on our ability to develop a true boiling point curve (TBP) for the material of interest. The TBP corresponds to a distillation performed under high reflux and large number of theoretical in order to obtain as sharp a separation as possible between the actual components that make up the oil making it expensive to run. Commonly other distillation experiments are performed under low reflux as well as at vacuum, and empirical correlations are used to convert this data into TBP curves. The most common techniques are briefly discussed below; for a more extensive description, see Riazi.2 Traditionally, distillation assays were designed to provide the maximum amount of fractionation while performing a batch distillation.4 The reasoning behind this is very simple: we wish to have the maximum amount of separation between the different constituents that make up the oils. In theory, if a batch distillation apparatus was operated at very high reflux rates and a very large number of equivalent equilibrium stages, a good discrimination of the original hydrocarbon mixture could be obtained. This is the motivation for true boiling point (TBP) assays. This idea has found universal use in the refinery industry, and several different distillation methods are available and described in the technical literature.5 To mitigate the costs associated with time when running TBP experiments as well (4) Kaes, G. L. Refinery Process Modeling - A Practical Guide to SteadyState Modeling of Petroleum Processes, Athens Publishing, Atlanta, GA: 2000. (5) American Petroleum Institute. Technical Data Book - Petroleum Refining, 5th ed.; Refining Department, AIP: Washington, DC, May 1992.

10.1021/ef9000242 CCC: $40.75  2009 American Chemical Society Published on Web 07/09/2009

Oil Characterization

as address the lack of standardized specifications for TBP apparatuses, the American Society for Testing of Materials (ASTM) has defined a series of useful experiments for the characterization of conventional hydrocarbon mixtures such as the D866 and D28877 assays. The D86 assay is one of the oldest and simplest methods for the measurement of boiling points of petroleum fractions and is used mostly for naphthas, gasolines, kerosenes, gas oils, and fuel oils. The method cannot be used for mixtures containing very heavy materials that cannot be vaporized and should be used with caution when temperatures are higher than 250 °C because of possible thermal cracking of material in the still. The D2887 assay is based on gas chromatography (GC) and results in a simulated distillation curve (SD). It is now a common way to represent distillation data. Since the analysis is done via chromatographic analysis, the results are presented in %mass distilled coordinates instead of the most common %volume distilled of other assays. Results from SD are similar to TBP results, but interconversion methods are required2 and it is debated if conversion methods should be used at all and how to handle heavily aromatic fluids.4 Other assays, such as the D1160 assay,8 are designed to handle high boiling materials and the need to conduct distillation under vacuum to avoid thermal cracking that would occur if the assay was conducted at higher pressures. The American Petroleum Institute provides well-defined procedures for the conversion of distillation data collected using these standard methods to TBP data. From this point onward the data is converted into pseudocomponents that are used to model the thermodynamic properties of the mixture, which are then used for reservoir and process simulation as well as the design and simulation of separation equipment.4,9 There are two problems associated with the use of these standard methods. First, the data must be converted into an equivalent TBP curve. Although API took great care in the development of conversion procedures, inconsistencies are sometimes unavoidable because of the empirical nature of the methods such as the problems reported by Daubert10 and Satyro.11 Second, the measured values from these procedures are not easily associated with well-defined thermodynamic state points such as bubble point temperatures. The procedures are usually defined based on the performance of a related batch distillation. It is necessary to model the equipment used for the measurements very accurately if one were to associate the measurements from a normal distillation assay with true thermodynamic state points. Two recent methods, Bruno12 and D5236,13 offer a solution to these problems. Bruno12 developed a thermodynamically consistent assay procedure in which the measured distillation temperature corresponds to a true thermodynamic state point. This was (6) ASTM D86 - 07b Standard Test Method for Distillation of Petroleum Products at Atmospheric Pressure; http://www.astm.org/Standards/D86.htm (accessed Nov 30, 2008). (7) ASTM D2887 - 06a Standard Test Method for Boiling Range Distribution of Petroleum Fractions by Gas Chromatography; http:// www.astm.org/Standards/D2887.htm (accessed Nov 30 2008). (8) ASTM D1160 - 06 Standard Test Method for Distillation of Petroleum Products at Reduced Pressure; http://www.astm.org/Standards/ D1160.htm (accessed Nov 30 2008). (9) VMGSim User’s Manual, Version 4.0, Virtual Materials Group, Inc., Calgary, Alberta, 2009. (10) Daubert, T. E. Hydrocarbon Process. 1994, 73 (9), 75–78. (11) Satyro, M. A.; Satyro, M. A. Life, Data and Everything. Pure Appl. Chem. 2007, 79 (8), 1403–1417. (12) Bruno, T. J. Ind. Eng. Chem. Res. 2006, 45, 4371–4380. (13) ASTM D5236 - 03(2007) Standard Test Method for Distillation of Heavy Hydrocarbon Mixtures (Vacuum Potstill Method); http://www.astm. org/Standards/D5236.htm (accessed Nov 30 2008).

Energy & Fuels, Vol. 23, 2009 3961

accomplished by carefully positioning the thermometer in the boiling fluid, thus ensuring the measurement of instantaneous bubble temperatures. Bruno’s procedure also allows for the measurement of compositions, although this feature is not necessary for the procedure proposed in this paper. Bruno and co-workers have used this method to analyze a wide variety of different mixtures.14-16 They also propose the use of the method in association with a comprehensive equation of state package17 for the development of surrogate mixtures (mixtures of known compositions of pure components designed to represent more complex mixtures). This approach has proven successful and has been used in the development of consistent thermodynamic models for biodiesel.18 In Canada, more and heavier hydrocarbon fluid characterizations are being performed using the D5236 ASTM method.13 The method was developed to assist in the characterization of high boiling point materials, and the equipment was designed in such a way as to allow distillation under high vacuum. This method is similar to Bruno’s method in one very important feature: the measured temperatures are also the instantaneous bubble temperature. Hence, for both the Bruno method and the D5236, the measured temperature has a precise thermodynamic meaning and can be calculated from a fluid characterization directly without a detailed knowledge of the equipment actually used for the measurements. Empirical methods of interconversion and detailed models of the apparatus are not required. However, neither the Bruno nor the D5236 method has a conversion procedure to TBP, thus limiting their use. The objective of this work is to develop an interconversion method for these assays. We propose a methodology based on Eckert and Vanek’s19 detailed mathematical procedure for the modeling of D86 experiments. Their work was not based on raw hydrocarbon fluids but rather synthetic mixtures made of hydrocarbons ranging from 4 to 10 carbon numbers including paraffins, naphthenes, and aromatics. They used the ideal gas model for the gas phase and the NRTL model for the liquid phase, and their predictions compared reasonably well with the measured data. For most distillation assays, the complexity of the mathematical model necessary for the solution of the nonsteady-state material and energy balance equations makes it difficult to use. In addition, the model must still be validated against the actual distillation campaign and idiosyncrasies related to the actual equipment. Fortunately, a simpler modeling is possible for the Bruno and D5236 assays because the temperatures correspond to instantaneous values of the mixture boiling point and can be determined directly from the fluid characterization. The gist of the proposed method is to determine what initial composition of the pseudocomponents defining the fluid corresponds to the same trajectory of measured temperature versus volume of material distilled. In this paper, the methodology is developed, its internal consistency is tested on two data sets (one a mixture of well-defined pure components the other a (14) Bruno, T. J.; Smith, B. L. Ind. Eng. Chem. Res. 2006, 45, 4381– 4388. (15) Smith, B. L.; Bruno, T. J. Ind. Eng. Chem. Res. 2006, 46, 310– 320. (16) Smith, B. L.; Bruno, T. J. Energy Fuels 2007, 21 (5), 2853–2862. (17) Huber, M. L.; Lemmon, E. W.; Diky, V.; Smith, B. L.; Bruno, T. J. Energy Fuels 2008, 22 (5), 3249–3257. (18) Ott, L. S.; Bruno, T. J. Energy Fuels 2008, 22 (4), 2861–2868. (19) Eckert, E.; Vanek, T. Chem. Pap. 2008, 62 (1), 26–33.

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trajectory of the bubble temperature as a function of material distilled from the batch apparatus. The fundamental equation is the non-steady-state component material balance,24 eq 1: dxj ) jx - jy dξ

Figure 1. Batch distillation without refluxing: residue curve map generation.

mixture of pseudocomponents), and then it is applied to three case studies. Modeling The unifying concept for the understanding and modeling of distillation assays is the residue curve. This concept was probably put forth first by Schreinenmakers in 190120-22 and since then has been used with great success as a tool for the conceptual design of distillation systems.23 The residue curve is the profile of saturation composition versus boiling temperature as a distillation progresses. If the original composition of a mixture is known and the pressure is specified then, for a batch apparatus, the profile can be determined based on rigorous thermodynamics and material balance equations. It is important to notice that the technique is a close representation of the measurement of a distillation assay only if the measured temperature in the assay corresponds to the actual bubble temperature of the material inside the batch apparatus and that no rectification happens during the experiment. These are the two experimental characteristics that make Bruno’s and D5236 distillation assays ideal for mathematical representation using residue curves. The trajectory calculation and a procedure to find the pseudocomponent compositions that fit a given trajectory are outlined below. Trajectory Calculation. The mathematics for modeling the temperature trajectory of a batch distillation were developed in detail by Doherty and Perkins24 and shown in Figure 1. This technique is usually referred to as residue curVe map and is the representation of the trajectory of all saturation conditions between an initial feed and a final state where the batch distillation process is stopped. Usually residue curve analysis is used to identify the topology of the bubble temperature space, such as minimum and maximum azeotropes and related distillation boundaries. In our case, we are concerned with another piece of information encoded in the residue curve map: the (20) Schreinenmakers, F. A. H. Z. Phys. Chem. 1901, 36, 257–289. (21) Schreinenmakers, F. A. H. Z. Phys. Chem. 1901, 36, 413–499. (22) Schreinenmakers, F. A. H. Z. Phys. Chem. 1903, 43, 671. (23) Doherty, M. F. Malone, M. F. Conceptual Design of Distillation Systems; McGraw-Hill: New York, 2001. (24) Doherty, M. F.; Perkins, J. D. Chem. Eng. Sci. 1978, 33 (3), 281– 301.

(1)

where jx represents the saturated liquid mole fraction vector, jy represents the saturated vapor mole fraction vector, and ξ represents the “warped time”. The warped time is a nondimensional time used to describe the trajectory of the composition inside a vessel undergoing batch distillation without a reflux. At any given warped time, the temperature and vapor-phase composition can be determined from the liquid-phase composition and a bubble point calculation. Doherty and Perkins also show how to convert the warped time information into hold-up information, thus allowing the distillation trajectory to be traced as a function of material distilled and the associated bubble temperature. The holdup in the batch is given by eq 2: H ) H° exp(-ξ)

(2)

where H is the liquid holdup (remaining moles of liquid in distillation flask) at a given warped time and H° is the initial holdup. The combination of eqs 1 and 2 allow for rapid determination of the temperature trajectory as a function of amount of material distilled (in mass, mole, or volume basis). If the actual distillation campaign was to be modeled, real time and real duties associated to the experiment would be required. In our case, time is immaterial; we are just interested in the bubble temperature trajectory. Given an initial state, eq 1 can be readily integrated using a variety of methods. A simple Euler first-order method was used here as follows: ∆xjm ) (xjm-1 - jym-1)∆ξ

(3)

jxm ) jxm-1 + ∆xjm

(4)

Tsat,m ) f(xjm, P)

(5)

j mjxmT jym ) K

(6)

ξm ) ξm-1 + ∆ξ

(7)

Hm ) Ho exp(ξm)

(8)

where m and m - 1 are integration step counters, ∆ξ is the integration step in warped time, ∆xj and ∆yj are the change in liquid- and vapor-phase composition, respectively, Tsat is the saturation pressure determined from a bubble point calculation at the liquid-phase composition and distillation pressure, P, and j is the vector of equilibrium ratios determined during the K saturation temperature calculation. An integration step of 0.01 was deemed adequate after verifying the trajectories calculated using several different integration steps. More sophisticated integration algorithms will make the calculations faster, but this feature was not explored in this study. Saturation temperature calculations were performed using the commercial software VMGThermo25 and the ideal solution model. Usually, hydrocarbon mixtures are assumed to behave ideally at low pressures where the ideal gas and Raoult’s law approximately apply, making calculations very fast. An ideal (25) VMGThermo User’s Manual, Version 5.0, Virtual Materials Group, Inc., Calgary, Alberta, 2008.



model is not required by the characterization method and any thermodynamic model could be used for the calculation of bubble temperatures, such as a cubic equation of state or reference equations of state as used by Huber et al.17 Reference equations of state have been successfully used to model systems where a better understanding of the chemical makeup of the mixture is available. Initial Mixture Definition. In the absence of any additional information, we have to guess the type and initial composition of the pseudocomponents. At this point the problem is too undefined, and some assumptions must be made. We assume that the pseudocomponents are allocated as boiling cuts between an initial boiling point and a final boiling point. These values are necessarily empirical, but reasonable values can be easily set depending on the mixture of interest. In this case, the fluids of interest are heavy oils and bitumen and an initial boiling point of 50 °C and a final boiling point of 1200 °C are adequate guidelines. Pseudocomponents with average boiling temperature intervals between 10 and 30 °C are usually adequate. This gives us an initial set of pseudocomponents used to define the mixture. Any additional information about the mixture can be provided, such as average gravity, average molecular mass, or property curves such as gravity versus fraction distilled. At its barest minimum, all physical properties of interest can be estimated based on the average boiling point, although this is not recommended. Usually at least bulk gravity is available and the pseudocomponent physical properties can be estimated from the average boiling point and gravity.1,2 The composition of the mixture is still unknown, and we assume as an initial guess that all components are present in identical proportions. The proposed characterization procedure is then a mathematical recipe designed to determine the initial composition fed to the distillation apparatus based on the initial pseudocomponent slate defined above. Characterization. The following characterization procedure is proposed: 1. Define the component slate as suggested in the “Initial Mixture Definition” section. 2. Set the initial number of moles vector equal to the mole fraction vector as shown in eq 9: j o ) jxo N

(9)

Now define the vector of optimization variables Vj, eq 10: j o) Vjo ) ln(N

(10)

The use of logarithmic coordinates to represent the composition avoids the possibility of negative mole numbers or mole fractions and considerably reduces the mathematical complexity of the optimization algorithm. 3. Set the pressure observed during the experiment. Note that saturation temperatures measured under vacuum or equivalent atmospheric results can be used. 4. Choose a thermodynamic model for the mixture. We assumed ideal solutions in this study. 5. Integrate eq 1 (eqs 3-8), collecting the fraction distilled, the saturation temperature, and the current holdup. Usually, for heavy oils and bitumen, only a fraction of the material will distill,26 and the integration can stop when we reach the same point as defined by the lab data. (26) Nji, G. N.; Svrcek, W. Y.; Yarranton, H. W.; Satyro, M. A. Energy Fuels 2008, 22 (5), 3559.

6. An objective function can be constructed and is given by eq 11: np

OF )

∑ (T

c i

- Tei )2

(11)

i)1

where OF stands for the objective function value, np is the number of data points, T is the temperature, i is the data point number, and the subscripts c and e stand for calculated and experimental respectively. 7. The next step is to change the initial composition and physical properties in order to search for a new and hopefully better jx. The Nelder-Mead algorithm27 was found to be effective as long as logarithmic coordinates were used. The optimization algorithm will provide a new guess for the variable vector Vj1. With this new variable vector, a new mole number vector is calculated, eq 12: j 1 ) exp(ν¯ 1) N

(12)

The new mole fractions are calculated based on the new number of moles, eq 13: xj )

N1,j

(13)

nc

∑N

1,j

j)1

where j stands for the pseudocomponent index and 1 is just the optimization iteration counter value. j k-1 is smaller than a set tolerance, jk - N 8. When the norm of N the optimization is terminated. The composition and physical properties of the pseudocomponents are defined, and the characterization is complete. In this study a tolerance of 10-6 was found to be adequate. 9. If the optimization is not complete, calculate the new composition using eq 13 and return to Step 2. Methodology Tests The best possible test for the methodology would be the direct comparison of a TBP experiment with a Bruno or D5236 experiment; however, an extensive literature review did not provide a useable comparison. Nevertheless, there are other types of data that allow for a reasonable test of the methodology. For example, if a detailed analysis of a fluid is available, the theoretical TBP can be determined simply by ordering the chemical components in boiling point order and corresponding amounts. The residue curve material balance, eq 1, can also be integrated (eqs 3-8) to determine exactly what a theoretical Bruno or D5236 curve would be, as long as a reliable thermodynamic model is available for the mixture of interest. For hydrocarbons at low pressures, such reliable thermodynamic models exist.2,28,29 In this case, an experimental Bruno or D5236 assay is not required because the predicted instantaneous boiling points correspond exactly to what is measured. Test 1: Complete Analytical Data. Cady and co-workers30 measured distillation data for well-defined mixtures of actual (27) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T., Flannery, B. P. Numerical Recipes: The Art of Scientific Computing, 3rd ed.; Cambridge University Press: New York, 2007. (28) VMGSim for Natural Gas Processing, Virtual Materials Group, Inc., Calgary, Alberta, Canada, 2004. (29) Pedersen, K. S. Christensen, P. L. Phase BehaVior of Petroleum ReserVoir Fluids; CRC Press: Boca Raton, FL, 2007. (30) Cady, W. E.; Marshner, R. F.; Cropper, W. P. Ind. Eng. Chem. 1952, 1859–1863.

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Table 1. Virgin Naphtha Compositional Analysis Based on Data from Cady et al.30 a

Table 2. TBP Calculated Based on Pure Component Boiling Points and Composition from Cady et al.30

component

liquid volume fraction

vol% distilled

TBP (°C)

2-methylpropane n-butane 2-methylbutane n-pentane cyclopentane 2,2-dimethylbutane 2,3-dimethylbutane 2-methylpentane 3-methylpentane n-hexane methylcyclopentane benzene cyclohexane 2,2-dimethylpentane 2,4-dimethylpentane 1,1-dimethylcyclopentane 2,3-dimethylpentane 2-methylhexane trans-1,3-dimethylcyclopentane trans-1,2-dimethylcyclopentane 3-methylhexane n-heptane methylcyclohexane ethylcyclopentane toluene trimethylcyclopentane dimethylcyclohexane trans-1,2-dimethylcyclohexane ethylcyclohexane aromatics (modeled using ethylbenzene) paraffins (modeled using n-nonane) cycloparaffins (modeled using cyclononane) aromatics (modeled using n-proylbenzene)

0.24 2.49 5.10 6.42 0.59 0.12 0.12 4.16 2.48 9.15 4.40 0.24 2.98 0.35 0.13 0.25 0.25 3.20 1.30 4.16 4.41 8.31 7.01 1.17 1.16 2.14 4.69 1.61 3.09 1.16 8.91 7.01 1.18

0 1 2 3 4 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 96 97 98 99 100

260.41 271.27 281.42 291.51 299.38 303.49 311.44 333.50 338.15 342.39 344.38 356.59 364.68 364.83 366.62 371.67 373.25 374.62 387.43 393.87 414.37 430.58 440.91 451.75 453.92 456.08 458.25 460.42 462.58

a Note, a small fraction of the material (3%) had carbon numbers equal to or higher than 10 and was not included in the analysis.

components for which reliable physical properties are available. The distillations were performed at low pressure, and polar interactions in the mixture are negligible; hence, ideal solution behavior can be assumed without any significant loss of accuracy,31,32 although more complex models could be used. The compositional analyses of several naphthas (virgin, thermally cracked and catalytically cracked) were reported. For our purpose, any of the data is usable and we simply chose the analysis for virgin naphtha, reproduced in Table 1. The TBP curve was calculated for the analysis given in Table 1 based on the experimental boiling point of each component and the reported volume fraction, Table 2 and Figure 2. It is interesting to observe that the measured TBP and the recalculated TBP based on chemical analysis are virtually identical up to approximately 80% distilled. The measured TBP was not available as tabular data but rather as a graph and digitized. It is believed that the digitized reading errors are well below 1 K. Unfortunately, the paper does not give enough information to further discuss the discrepancies we see at the end of the curve. The compositional analysis from Table 1 is next used to predict a Bruno distillation assay by solving the residue curve eq 1. Ideal solution behavior was assumed and pure component data was obtained from the VMGSim process simulator.9 As mentioned previously, it was not necessary to simulate the actual apparatus; it was sufficient to determined instantaneous boiling temperatures. Briefly, the boiling temperature and equilibrium liquid and vapor compositions were determined for the initial (31) Walas, S. M. Phase Equilibria in Chemical Engineering; Butterworth: London, 1985. (32) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw Hill: New York, 1987.

condition. The incremental change in the liquid composition for a given incremental warped time was determined from eqs 3-8. The liquid composition was updated, and the vapor composition and instantaneous boiling temperature were determined from a bubble point calculation. This procedure is repeated until the complete distillation curve is calculated. The results are summarized in Table 3 and Figure 3. With a TBP and a Bruno distillation we can now test the new method. Using the Bruno distillation and the reported bulk gravity of 0.72430 and molecular weight of 97.0 g/gmol as the inputs, the fluid was characterized using the new procedure assuming an ideal solution. The specific gravity and molecular weight were used as constraints in the optimization; that is, the gravity and molecular weight of individual pseudocomponents were rescaled every time the composition was recalculated by the optimizer. Given the relatively small boiling point range for this fluid, the minimum and maximum temperatures were set to 300 and 600 K, respectively. Note, only the Bruno

Figure 2. Virgin naphtha TBP from Cady et al.30 data (diamonds) and the TBP determined from the pure component data (solid line) compared with TBP determined from Bruno data using the proposed methodology (dotted line).



Table 3. Calculated Bruno Distillation Assay Using Cady et al. Experimental Dataa

a

% volume

T (K)

2.44 7.23 11.83 18.45 26.79 34.41 41.34 50.69 61.62 70.47 80.88 91.26 97.60

345.58 349.09 352.36 357.98 362.99 367.49 371.61 378.90 386.00 393.21 407.71 421.32 431.52

Table 5. Alaska North Slope Data33 a cut

AETb (°F)

yield (mass%)

SG

1 2 3 4 5 6 7 8 9 10 11 12 13 14

165 320 450 500 550 600 650 750 850 950 1050 1150 1250

3 9.6 9.6 4.7 5.4 4.4 5.4 10.5 11.1 6.5 4 5.3 5.7 14.8

0.6518 0.7555 0.8100 0.8468 0.8633 0.8697 0.8871 0.9053 0.9285 0.9548 0.9603 0.9656 0.9812 1.0029

a Average bulk gravity equal to 0.8966. equivalent temperature.

Ideal solution was assumed for the calculations.

b

AET ) atmospheric

Table 6. Summary of Methods Used for Pseudocomponent Physical Property Estimation

Figure 3. Bruno distillation assay simulated with proposed methodology (line) compared with synthetic assay generated using Cady et al. experimental data (symbols). Ideal solution was assumed for the calculations. Table 4. Pseudocomponents Determined Based on Bruno Distillation for Naphtha name

mole fraction

NBP (K)

liquid density (kg/m3)

molecular weight (g/mol)

cut[1] cut[2] cut[3] cut[4] cut[5] cut[6]

0.1409 0.0314 0.4438 0.0978 0.2799 0.0060

302.00 342.00 354.00 382.00 418.00 466.00

640.14 695.37 709.13 737.10 766.10 796.10

65.52 83.87 89.55 103.36 122.67 151.92

distillation curve was used, and all of the pure component data was excluded. The pseudocomponents and their compositions are shown in Table 4. Five pseudocomponents were sufficient to fit the Bruno distillation data, Figure 3, and to predict the original TBP curve, Figure 2. The TBP is predicted with an average error of 5.7 K using only six pseudocomponents. Test 2: Pseudocomponent Data. The next test is based on pseudocomponents instead of pure components since detailed analyses, such as the naphtha composition used in the previous example, are rare. The API recently made available detailed experimental TBP data for eight crude oils33 which provide an opportunity for an indirect test. The TBP data was sufficient to directly construct a detailed fluid characterization which was then used to simulate a Bruno assay. The proposed characterization methodology was applied to this synthetic Bruno assay as if it were an experimental assay. The resulting characterization was compared with the original characterization and also used (33) Sturm, G. P. Shay, J. Y. Comprehensive Report of API Crude Oil Characterization Measurements - Downstream Segment; API Technical Report 997, 1st ed., August 2000.

physical property

method

reference

liquid density molecular weight critical temperature critical pressure acentric factor vapor pressure

Katz-Firoozabadi Riazi-Daubert Lee-Kesler Lee-Kesler Lee-Kesler Lee-Kesler

Whitson23 Riazi20 Pedersen et al.24 Pedersen et al.24 Pedersen et al.24 Reid et al.25

to simulate the original TBP assay in the same way as in the previous naphtha example. An Alaska North Slope (ANS) oil was selected from the API data set since it has a sizable nondistillable residue (14.8 wt %) with an equivalent atmospheric boiling point above 1250 °F and a specific gravity of 1.0029. It is a heavy oil and significantly different from the naphtha used in the previous test. The relevant data for this oil is summarized in Table 5. Note, only fractions up to 950 °F were determined under favorable distillation conditions (packed column, timed reflux, and vacuum of approximately 15 mmHg) and are TBP measurements while the remaining fractions were distilled at 0.6 mmHg and no column packing nor reflux, thus being poorer approximations to TBP conditions. Our intent is to verify the self-consistency of the procedure, and although the final cut data is not ideal, it does not affect the test. The above data was entered in the process simulator VMGSim and characterized using the above data as TPB data together with the gravity curve and a bulk oil gravity of 0.8966. In the absence of any other experimental data, the pseudocomponent properties were calculated using the methods listed in Table 6. The relevant results from the characterization are summarized in Table 7. Thirty nine cuts were required to represent the oil using the boiling point range definitions recommended by Kaes.4 This characterization was then used to simulate a Bruno distillation experiment using the Advanced Peng-Robinson equation of state,25 Figure 4. The simulation of the experiment was performed as described in the naphtha example except that pseudocomponents were used instead of pure components. Again, since the residue curve of a Bruno test is simply the instantaneous boiling temperatures which are calculated using a flash routine and fugacity coefficients, the only source of error must be related to the thermodynamic model. This “synthetic” Bruno type distillation together with a bulk oil density equal to 896.6 kg/m3 is now processed using the proposed characterization procedure. Note that the characterization procedure has no knowledge of the thermodynamic model used to calculate the Bruno distillation curve. The only information provided is the distillation curve and the bulk oil

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Table 7. Physical Properties Calculated Based on Alaska North Slope Data name LD60 MW NBP Pc Tc Vc acentric mole vol mass name LD60 MW NBP Pc Tc Vc acentric mole vol mass name LD60 MW NBP Pc Tc Vc acentric mole vol mass

units

cut 1

(kg/m3)

635.45 92.13 (K) 333.49 (kPa) 2999.22 (K) 491.38 (m3/kmol) 0.36 factor 0.33 fraction 0.02412 fraction 0.01017 fraction 0.00721 units

cut 14

(kg/m3)

821.15 191.86 (K) 514.89 (kPa) 1971.20 (K) 702.70 (m3/kmol) 0.72 factor 0.53 fraction 0.05063 fraction 0.03440 fraction 0.03150 units (kg/m3)

cut 27

914.26 369.09 (K) 694.97 (kPa) 1287.62 (K) 872.64 3 (m /kmol) 1.20 factor 0.91 fraction 0.02415 fraction 0.02835 fraction 0.02890

cut 2

cut 3

cut 4

cut 5

cut 6

cut 7

cut 8

cut 9

cut 10

cut 11

cut 12

cut 13

652.19 98.06 347.39 2887.23 508.56 0.38 0.34 0.02326 0.01017 0.00740

668.92 104.14 361.29 2782.29 525.77 0.41 0.35 0.02246 0.01017 0.00759

685.66 110.37 375.19 2684.96 542.96 0.43 0.36 0.02173 0.01017 0.00778

702.40 116.77 389.09 2596.04 560.11 0.46 0.37 0.02104 0.01017 0.00797

719.14 123.35 402.99 2516.23 577.22 0.49 0.38 0.02039 0.01017 0.00816

735.74 130.09 416.85 2445.69 594.19 0.51 0.39 0.01950 0.01002 0.00822

752.46 137.53 431.17 2380.42 611.55 0.54 0.41 0.02215 0.01177 0.00988

766.46 145.04 444.89 2314.27 627.52 0.57 0.42 0.02734 0.01505 0.01286

777.55 153.51 458.72 2236.12 642.60 0.59 0.44 0.02939 0.01687 0.01463

787.16 162.69 472.62 2154.16 657.19 0.63 0.46 0.03130 0.01881 0.01651

795.99 172.40 486.51 2073.30 671.43 0.66 0.49 0.03314 0.02088 0.01853

805.84 182.26 500.43 2004.15 685.95 0.69 0.51 0.03480 0.02289 0.02057

cut 15

cut 16

cut 17

cut 18

cut 19

cut 20

cut 21

cut 22

cut 23

cut 24

cut 25

cut 26

839.87 198.29 527.55 1984.81 719.20 0.73 0.54 0.05511 0.03784 0.03544

853.40 208.50 541.79 1954.01 735.18 0.75 0.56 0.03610 0.02565 0.02441

861.08 220.43 555.95 1887.25 748.99 0.78 0.59 0.03253 0.02421 0.02325

865.58 233.23 569.78 1805.39 761.32 0.82 0.62 0.03136 0.02457 0.02372

868.09 247.38 583.72 1713.91 772.89 0.87 0.65 0.02956 0.02450 0.02372

874.94 260.56 597.61 1656.30 785.95 0.91 0.68 0.02942 0.02548 0.02486

884.26 272.88 611.29 1619.78 799.75 0.94 0.70 0.02568 0.02304 0.02272

890.44 287.29 625.30 1563.63 812.55 0.98 0.74 0.02114 0.01983 0.01969

895.04 302.81 639.29 1500.69 824.65 1.02 0.77 0.01917 0.01886 0.01883

899.22 319.04 653.28 1438.61 836.51 1.07 0.80 0.01878 0.01938 0.01943

903.62 335.81 667.31 1380.97 848.43 1.12 0.84 0.02022 0.02185 0.02202

908.74 352.36 681.16 1331.67 860.48 1.16 0.87 0.02283 0.02574 0.02608

cut 28

cut 29

cut 30

cut 31

cut 32

cut 33

cut 34

cut 35

cut 36

cut 37

cut 38

cut 39

922.90 393.86 715.30 1229.98 890.74 1.26 0.96 0.04354 0.05404 0.05562

936.47 428.71 743.23 1167.57 916.35 1.34 1.02 0.03579 0.04765 0.04976

950.26 463.19 770.63 1115.47 941.75 1.40 1.06 0.02764 0.03918 0.04152

957.31 504.76 798.29 1032.22 964.24 1.50 1.12 0.01795 0.02753 0.02939

959.36 555.42 826.75 928.54 984.82 1.64 1.19 0.01442 0.02428 0.02597

961.47 606.60 854.31 837.80 1004.69 1.77 1.26 0.01251 0.02295 0.02460

963.92 662.88 882.65 754.14 1025.20 1.91 1.32 0.01188 0.02376 0.02554

969.97 713.59 910.02 695.49 1046.96 2.02 1.37 0.01281 0.02740 0.02964

981.65 789.53 951.62 624.35 1081.62 2.17 1.44 0.02350 0.05497 0.06018

997.25 892.04 1007.15 542.25 1128.60 2.37 1.53 0.02113 0.05497 0.06114

1012.84 993.01 1062.67 488.80 1179.41 2.54 1.58 0.01928 0.05497 0.06209

1025.92 1074.07 1109.27 458.22 1238.63 2.67 1.54 0.01225 0.03728 0.04266

density. The distillation curve was successfully simulated using 15 pseudocomponents with physical property data of Table 8. Figure 4 shows the comparison between the calculated and simulated Bruno curve, and Figure 5 shows the most important result of this exercise, the comparison between the original TBP curve and the TBP calculated from the characterization procedure developed in this work. The fluid model developed based on the Bruno type distillation captures all the basic features of the original TBP, even though the number of required pseudocomponents for the simulation of the Bruno curve was just 15. The molecular weight for the fluid estimated by the characterization procedure, 319 g/gmol, compares well with the value from the original detailed characterization, 308 g/gmol.

Figure 4. Calculated Bruno distillation for Alaska North Slope oil33 based on synthetic experimental data (symbols) and simulated Bruno distillation using 15 pseudocomponents (line).

Table 8. Pseudocomponent Physical Properties Used To Model Alaska North Slope Crude Using Calculated Bruno’s Distillation and Bulk Gravity component

mole fraction

boiling point, K

density, kg/m3

molecular weight, g/mol

cut[1] cut[2] cut[3] cut[4] cut[5] cut[6] cut[7] cut[8] cut[9] cut[10] cut[11] cut[12] cut[13] cut[14] cut[15]

0.02027 0.14187 0.14593 0.20342 0.14018 0.10045 0.09538 0.05185 0.02301 0.04581 0.02266 0.000077 0.001008 0.008082 0.000002

333.3 400.0 466.7 533.3 600.0 666.7 733.3 800.0 866.7 933.3 1000.0 1066.7 1133.3 1200.0 1266.7

694.8 762.5 806.5 838.8 867.2 895.1 922.5 940.3 954.2 965.4 974.5 982.2 988.7 994.3 999.1

82.1 114.5 153.6 202.1 261.9 335.1 424.8 539.0 682.2 861.0 1083.7 1360.1 1702.6 2126.0 2648.3

Case Studies. Three case studies are presented: one for a Bruno type distillation and two for a D5236 assay. The first two assays are both for bitumens from Western Canada. A limited comparison between the two characterizations stemming from different assays is made to verify the consistency of the simulations. The third case study is for a heavy hydrocarbon and includes some property and mixture data that are used to test the predictive capability of the characterization. Case 1: Bruno’s Distillation of a Bitumen. A Bruno type distillation was provided by Shell Canada for a bitumen sample. This sample had been previously distilled using ASTM D1160 to remove emulsified water. The kettle temperature was 275 °C with a vapor temperature of 100 °C and any light ends lost



Figure 5. Estimated TBP for Alaska North Slope oil based on Bruno’s distillation curve and bulk gravity information (symbols) compared with experimental TBP33 (symbols). The average error in temperature is 24.5 K or 3.9%.

Figure 6. Effect of number of pseudocomponents on the quality of fit for Bruno-type distillation data at 82.3 kPa. Note that the experimental values go only up to 75% volume distilled and the remaining of the distillation curves are extrapolated based on the optimal composition determined during the regression procedure.

Table 9. Experimental Distillation Data Determined Using Bruno’s Distillation Apparatusa

Table 10. Effect of Number of Pseudocomponents on Final Objective Function Value (eq 11)

a

volume %

temperature, K

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

596.75 619.15 640.75 654.55 669.05 676.15 683.55 689.65 694.85 700.15 703.95 710.05 715.05 719.75 726.75

The test pressure was 82.3 kPa.

were not replaced. The fluid density at 15 °C is approximately 986 kg/m3. The experimental data is summarized in Table 9, and the pseudocomponent properties were again calculated using the methods listed in Table 6. We stress that any additional information related to the fluid could be used in the characterization method. If a gravity or molecular weight curve was measured, these additional data could be entered to help define the pseudocomponents. In the same way, if additional knowledge related to the fluid under study in the form of specific estimation methods for pseudocomponent physical properties was available, a different set of estimation methods could have been selected. The initial and final boiling points were set to 300 and 1300 K, respectively. The optimum number of pseudocomponents to satisfactorily fit the data was determined by trial and error. Figure 6 shows that 30 pseudocomponents (each representing a 33.3 K average boiling point) were adequate, corresponding to an average error in temperature of just 0.22 K. The optimization results are summarized in Table 10. Note that the true boiling points calculated for volumes distilled above or below the experimental volume distilled are just estimates based on the optimization procedure. Hence, the boiling temperature trajectories above 70 vol% distilled have little meaning since there are no data to constrain them. If additional data, such as viscosity, were available, a more meaningful extrapolation could be made.

number of pseudocomponents

optimal of (K2)

5 10 20 30 50

17566 424 279 11.0 9.63

Table 11. Compositions and Pseudocomponent Physical Properties Used To Simulate the Bruno-Type Distillation component

mole fraction

boiling point, K

density, kg/m3


cut[1] cut[2] cut[3] cut[4] cut[5] cut[6] cut[7] cut[8] cut[9] cut[10] cut[11] cut[12] cut[13] cut[14] cut[15] cut[16] cut[17] cut[18] cut[19] cut[20] cut[21] cut[22] cut[23] cut[24] cut[25] cut[26] cut[27] cut[28] cut[29] cut[30]

0.000000 0.000000 0.000000 0.000000 0.000000 0.060587 0.000000 0.122751 0.000000 0.154310 0.000000 0.209197 0.368759 0.000000 0.000000 0.000000 0.084396 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

316.67 350.00 383.33 416.67 450.00 483.33 516.67 550.00 583.33 616.67 650.00 683.33 716.67 750.00 783.33 816.67 850.00 883.33 916.67 950.00 983.33 1016.67 1050.00 1083.33 1116.67 1150.00 1183.33 1216.67 1250.00 1283.33

754.27 796.73 830.36 857.20 879.02 897.33 913.37 928.09 942.21 956.16 970.10 983.93 997.28 1009.35 1018.22 1026.04 1032.98 1039.19 1044.76 1049.81 1054.39 1058.57 1062.40 1065.92 1069.17 1072.18 1074.97 1077.56 1079.99 1082.25

73.76 86.43 100.20 115.54 132.80 152.25 174.08 198.47 225.59 255.63 288.85 325.66 366.64 412.72 465.79 525.61 592.96 668.72 753.84 849.40 956.61 1076.78 1211.41 1362.14 1530.81 1719.43 1930.27 2165.83 2428.88 2722.52

The complete set of pseudocomponents for the 30 component case is shown in Table 11. Note that only six pseudocomponents are necessary in this case since during the optimization procedure compositions of several pseudocomponents were determined to be less than 0.000001 mol fraction. Limited experimentation with starting compositions demonstrated that the solution was not sensitive to the initial guess of the

3968



Table 12. Experimental D5236 Distillation Dataa volume %

temperature, K

1 4.57 11.08 18.93 22.86 30.32 35.66

596.09 658.34 684.47 720.63 745.13 755.70 766.90

a The equivalent pressure is 101.325 kPa (values were corrected to atmospheric using API procedure). Reported temperatures are measured in the kettle.

Table 13. Compositions and Pseudocomponent Physical Properties Used To Simulate the D5236 Distillationa component

mole fraction

boiling point, K

density kg/m3


cut[1] cut[2] cut[3] cut[4] cut[5] cut[6] cut[7] cut[8] cut[9] cut[10] cut[11] cut[12] cut[13] cut[14] cut[15] cut[16] cut[17] cut[18] cut[19] cut[20] cut[21] cut[22] cut[23] cut[24] cut[25] cut[26] cut[27] cut[28] cut[29] cut[30]

0.002648 0.000000 0.000000 0.000000 0.000000 0.000000 0.057470 0.000000 0.000000 0.223426 0.000000 0.000000 0.045198 0.000000 0.671258 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

316.67 350.00 383.33 416.67 450.00 483.33 516.67 550.00 583.33 616.67 650.00 683.33 716.67 750.00 783.33 816.67 850.00 883.33 916.67 950.00 983.33 1016.67 1050.00 1083.33 1116.67 1150.00 1183.33 1216.67 1250.00 1283.33

729.43 771.89 805.52 832.36 854.18 872.49 888.52 903.25 917.37 931.32 945.26 959.09 972.44 984.51 993.38 1001.20 1008.14 1014.34 1019.92 1024.97 1029.55 1033.73 1037.56 1041.08 1044.33 1047.33 1050.12 1052.72 1055.15 1057.41

74.53 87.97 102.47 118.52 136.50 156.68 179.28 204.50 232.52 263.54 297.85 335.85 378.10 425.54 479.95 541.17 609.97 687.21 773.85 870.95 979.70 1101.39 1237.49 1389.62 1559.55 1749.27 1960.98 2197.11 2460.36 2753.72

a

Figure 7. D5236 assay corrected to 101 kPa and model fit. Values above 35.66% cumulative volume are estimated based on the pseudocomponent slate and the allowed initial and final boiling points. The average error in distillation temperature is 2.0 K.

Figure 8. Comparison of calculated true boiling point curves from characterizations obtained from Bruno-type and D5236 assays.

Note that only five pseudocomponents are necessary.

composition and that an equimolar composition was a good starting point. Randomized starting points did not show any improvement over the simpler equimolar starting point. Also, restarting from the previous solution showed negligible changes when compared to the previously converged solution. Case 2: D5236 Distillation of a Bitumen. The D5236 distillation was provided by Shell Canada for another bitumen sample from the same source reservoir as the previous sample, Table 12. This sample had also been distilled to remove emulsified water but at a temperature less than 100 °C. Also, in this case, a Simdist was performed prior to the distillation, and light components were added back in after the distillation to recreate the original Simdist composition. The fluid density at 15 °C is approximately 980 kg/m3. In this example, the same physical properties and number of pseudocomponents as the previous example were used for the characterization. The calculated composition is given in Table 13, and the fit of the distillation data is shown in Figure 7. Note, only five pseudocomponents were required to fit the data. Comparison of Bruno and D5236 Results. The true boiling point curves and molecular weight distributions calculated from the Bruno and D5236 characterizations are shown in Figures 8

Figure 9. Comparison of molecular weight distributions from characterizations obtained from Bruno-type and D5236 assays.

and 9, respectively. The results are reasonably comparable for bitumen samples with different histories. Note that the D5236 composition includes a small fraction of low molecular weight material not present in the Bruno test, as expected from the sample preparation. Interestingly, given the similar density of the samples, the D5236 analysis indicates a somewhat higher molecular weight and higher boiling sample. Such property variations are not unusual in heavy oil reservoirs.34 (34) Larter, S.; Adams, J.; Gates, I. D.; Bennett, B.; Huang, H. J. Can. Petrol. Technol. 2008, 47 (1), 52–61.


Figure 10. D5236 data (symbols) and simulated D5236 curve (line) using procedure described in this paper. The error in the beginning of the distillation curve comes from solvent contamination in the heavy hydrocarbon fed to still.


Figure 12. D5236 data (symbols) and simulated D5236 curve (line) for mixture of heavy hydrocarbon and light hydrocarbon solvent. The simulation was based on pseudocomponents based on D5236 assay shown in Figure 10 and is entirely predicted.

based models, thus providing an expeditious and inexpensive way to incorporate VLE data while performing oil characterization distillations. Conclusions

Figure 11. Experimental and predicted liquid heat capacity data as a function of temperature for a heavy hydrocarbon fluid.

Case 3: D5236 of a Heavy Hydrocarbon. A D5236 assay was obtained for a heavy oil. The distillation data was collected at vacuum and converted into equivalent atmospheric boiling points by the lab. The data are proprietary and only unscaled plots are used to illustrate the results. Figure 10 shows the measured and simulated D5236 distillation assay based on the procedure described in this paper. It is interesting to note the discrepancy between the simulation and experiment for the very first measured point. Upon further analysis it was found that the sample was contaminated with solvent. In other words, the characterization was sufficiently rigorous to identify problems with the data itself. The characterization can also be used to provide accurate predictions of physical properties such as density and heat capacity. In this case density was constrained, but Figure 11 shows that the predicted liquid heat capacities compare well with the measured liquid heat capacities for the heavy hydrocarbon feedstock. No adjustments of any kind were needed. Figure 12 shows the measured D5236 distillation assay for a mixture of the same heavy hydrocarbon feedstock and light solvent. The distillation curve was then predicted from the characterization derived from the D5236 assay using the residue curve map technique described previously (solid line in Figure 12). Unfortunately these results are based on proprietary data, and details outside the sketches from Figures 10, 11, and 12 cannot be provided. Note, that data of the type shown in Figure 12 can be used for the determination of interaction parameters between pseudocomponents and defined components for equation of state

A simple characterization procedure that is easy to implement was proposed for D-5236 and Bruno’s distillation curves. The procedure allocates pseudocomponents and determines their mole fractions based on the distillation data from these assays. The procedure is completely general and is not based on a specific thermodynamic model for the mixture. Any equation of state or activity model can be used as the basis for the computation of the residue curve. The procedure was used to successfully model bitumen type hydrocarbon mixtures. These thermodynamic models can then be used as the basis for the development of reliable models for process simulation. This method allows the use of available distillation data and can be combined with the procedure developed by Nji et al.26 for the estimation of entire distillation curves for heavy hydrocarbons. Possible improvements include the use of a more sophisticated optimization algorithm for the determination of compositions and to reduce the number of residue curve calculations. The residue curve calculations can also be optimized by using a more sophisticated integration algorithm. Variable width average boiling points for pseudocomponents could also be used. The method stresses the need for close collaboration between experimentalists and modelers interested in this class of mixtures since additional information collected in the lab can be easily and efficiently added to the optimization procedure without any need for fundamental changes in the calculations outside of simple modifications to the objective function. Extra data will make the information encoded in the pseudocomponents more relevant and will help make mathematical models more relevant to process design and process simulation. Last but not least, the procedure allows for checking experimental data by identifying inconsistencies in the data. Acknowledgment. The authors are grateful to Shell Canada for providing Bruno’s and D5236 distillation data and Virtual Materials Group’s permission to publish this paper and use of its software. Mr. Gerd Nji’s comments on the paper are also acknowledged.

Appendix Nomenclature

f ) bubble temperature function

3970



H ) molar holdup j ) equilibrium ratio vector, y/x K j ) number of moles vector N nc ) number of pseudocomponents np ) number of data points OF ) objective function P ) pressure, kPa T ) absolute temperature, K Vj ) optimization variable vector jx ) mole fraction vector jy ) mole fraction vector

j ) pseudocomponent index k ) optimization step counter m ) warped time integration step counter sat ) saturation

Subscripts

ξ ) warped time

i ) measured data point index

EF9000242

Superscripts

c ) calculated e ) experimental 0 ) initial value Greek Letters

Oil Characterization from Simulation of Experimental Distillation Data

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