ARTICLE pubs.acs.org/IECR
Experimental Analysis and Modeling of the Separation of Triacylglycerol and Free Fatty Acid Mixtures Using Molecular Distillation Guillermo Arzate-Martínez,†,§ Arturo Jimenez-Gutierrez,‡ and Hugo S. García*,† †
UNIDA, Instituto Tecnologico de Veracruz, M.A. de Quevedo 2779, Col. Formando Hogar, Veracruz, Ver. 91897 Mexico Departamento de Ingeniería Quimica, Instituto Tecnologico de Celaya, Avenida Tecnologico y A. García Cubas S/N, Col. Alfredo B. Bonfil, Celaya, Gto. 38010 Mexico § Departamento de Ingeniería Agroindustrial, Universidad Politecnica de Guanajuato, Avenida Universidad Norte S/N, Loc. Juan Alonso, Cortazar, Gto. 38483 Mexico ‡
ABSTRACT: Experimental results for the separation of triacylglycerols and free fatty acids mixtures by molecular distillation are reported. The experiments were carried out so as to evaluate the effect of relevant operating conditions on the performance of the separation. Mixtures with different compositions were distilled at pressures of 1 and 4 mbar, with temperatures ranging from 40 to 80 °C, mass flows from 0.95 to 7.65 g/min, and wiper speeds from 0 to 400 rpm. Distillate and retentate products were weighed and evaporated fractions were calculated. The results were compared with those predicted by the use of the Langmuir Knudsen equation together with a functional group contribution method to estimate the vapor pressure of the components. The approach provided a good basis to model the experimental results for the separation by molecular distillation of binary mixtures of free fatty acids and triacylglycerols, as well as for mixtures of several free fatty acids.
’ INTRODUCTION Molecular distillation, or short path distillation, is a separation technique suitable for the purification of heat-sensitive substances. It is typically carried out at low pressures, which reduces the boiling point of the mixture to help the separation of compounds with high boiling points. Other characteristics are the very short distance between the condenser and the evaporator and the thin liquid film formed that ensures a greater surface area and improves mass transfer. Both conditions help minimize the residence time, and therefore the thermal decomposition of the distilled material.1 Molecular distillation has been used to separate glycerides as early as in the middle of the past century, as reported by Kuhrt et al.,2 who obtained high-purity monoacylglycerols mixtures (97%). It is also a suitable method to purify lipids, because lipid thermal decomposition can be avoided or greatly reduced.3,4 Other laboratory and industrial applications of molecular distillation include the preparation of cholesterol-reduced materials; Liang and Hwang5 obtained cholesterol-free concentrated mixtures of DHA and EPA (ω-3 fatty acids), whereas Lanzani et al.6 reduced the cholesterol content in butter and lard. This operation is also useful to concentrate vitamins such as A, D, and E, to eliminate undesirable byproducts formed in lipase-catalyzed reactions, and even to deodorize oils in the refining process.3 Brandt and Jensen7 investigated the extent of recovery of volatile substances such as ethylacetate, 2-butanone, diacetyl, and acetic acid from edible oils. Molecular distillation has also been used as a step in the isolation of substances with vitamin activity from vegetable oils such as carotenoids from palm oil8 and tocopherols form soybean oil.9,10 Esters from different materials have been purified, such as ethyl esters from squid visceral oil5 and steryl r 2011 American Chemical Society
esters from soybean oil deodorized distillates.11 Campos et al.12 modified the milk fat melting profile by concentrating long chain triacylglycerols. Technologies have been patented in Malaysia that involve the use of molecular distillation for deodorization and deacidification in the processing of red palm oil.13 This unit operation has also been a useful method to reduce the concentration of undesirable free fatty acids resulting from lipase or chemically catalyzed esterification reactions,1416 or to concentrate other substances such as monoglycerides.17 Xu et al.18 prepared structured triacylglycerols rich in medium-chain fatty acids via enzymatic acidolisis of capric acid and rapeseed oil. Triacyglycerols were purified using two molecular distillation steps; in the first one, they evaporated mainly capric acid and in the second one long-chain free fatty acids (two steps were necessary to avoid the condensation of volatiles in the pumps). Molecular distillation has several advantages over other methods for oil refining, such as fewer triacylglycerol losses than those found in neutralization, and more efficiency in the separation of glycerides than that achieved by conventional distillation.3 In this paper, we report the experimental results from the use of a molecular distillation unit for the separation of mixtures of triacylglycerols and/or free fatty acids. This problem arises very often after the reaction step to produce structured lipids. The reaction products (consisting mainly of mixtures of free fatty acids and triacylglycerols) require the correct selection of the operating conditions in order to achieve the desired separation; Received: January 15, 2011 Accepted: August 19, 2011 Revised: August 10, 2011 Published: August 19, 2011 11237
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Table 1. Fitting Constants for the Ceriani and Meirelles Vapor Pressure Model component
f0
f1
s0
S1
0.00444
0.4476
0.0751
esters
0.2773
acylglycerols
0
0
0
fatty acids
0.001
0
0
0
alcohols
0.7522
0
0
group
A1k
B1k
D1k 103
C1k
A2k 103
B2k
C2k 103 1.06
7232.3
22.7939
3.38
63.3963
CH2
8.4816
10987.8
1.4067
1.67
0.91
6.7157
COOH CH=cis
8.0734 2.4317
20478.3 1410.3
0.0359 0.7868
2.07 4
3.99 0
63.9929 0
CH=trans
1.843
526.5
COO
7.116
49152.6
CH3
OH CH2CHCH2
117.5
0.0203
28.4723 688.3
16694 349293
36.1
0.041
D2k 105 1.5 0.126
1.32 0
1 0
0
0
0.6584
3.68
0
2.337
8.48
2.79
3.257
0
4.85
0
0
0
1.45
0
0
0
181.4
122.5
for example, in the work by Xu et al.18 mentioned above, the reaction product consisted of a mixture of medium- and longchain free fatty acids and triacylglycerols. The effect of several operating variables such as process temperature, pressure, mass flow, glyceride chain length, species initial weight ratio and wiper speed on the extent of separation is reported. The experimental data are compared with those obtained with the CerianiMeirelles model19 and LangmuirKnudsen equations.
’ MATHEMATICAL MODELING OF MOLECULAR DISTILLATION Previous efforts on the modeling of molecular distillation include the works by Moraes and co-workers,20,21 who used a model with momentum, energy, and mass balance equations for the thin, falling film, and the Langmuir equation to estimate the evaporation rate. A rigorous modeling of the vapor phase in terms of molecular motions and collisions has been presented by Batistella et al.22 Typical modeling approaches such as those described above require information on physical properties of the molecules to be distilled that are not always available in the literature, such as thermal conductivity, mass diffusivity, evaporation enthalpy, or molecular diameter. For this reason, one of the objectives of this work was to develop a simpler approach for the use of molecular distillation on the separation of triacylglycerols and free fatty acids. The modeling approach was based on the use of the Ceriani and Meirelles model for vapor pressure and the LangmuirKnudsen equation for the evaporation rate. The model was then compared with the experimental data, which allowed for the model to be complemented with suitable empirical correction factors. The modeling procedure is described below. ’ BASIC EQUATIONS Surface Evaporation Equation. The Clausius equation is used for the calculation of the mean free path
1 λ ¼ pffiffiffi 2 2πσ N
0
ð1Þ
where λ is the mean free path (the mean path that a molecule can move before having a collision with another molecule), σ is the molecule diameter, and N is the number of molecules in one
0 10.0396
0.34
0.295
cubic centimeter. Alternative forms to represent this equation include kB T RT λ ¼ pffiffiffi 2 ¼ pffiffiffi 2 2πσ p 2πσ NA P
ð2Þ
where kB is the Boltzmann constant, T is the temperature, NA is Avogadro’s constant, and P is the pressure. Assuming ideality of the gas phase, one can calculate the number of evaporated molecules from p0i ffi ni ¼ Aτpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2πMi RT
ð3Þ
where ni is the number of evaporated moles, A is the surface area, τ is the time, p0i is the vapor pressure, Mi is the molar mass, R is the gas constant, and T is the temperature. On the basis of these principles, Langmuir and Knudsen obtained the equation that defines the molar evaporation rate per squared unit and per time unit (ji) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 ji ¼ pi ð4Þ 2πMi RT where pi0 is the vapor pressure of component i. As reported by Nguyen and Le Goffic,23 this equation can also be used for multicomponent systems rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 ji ¼ xi pi ð5Þ 2πMi RT where xi represents the molar fraction of component i. Glycerides Vapor Pressure Model. Ceriani and Meirelles19 (2003) proposed a functional group contribution method based on 1220 experimental data points to estimate the vapor pressure for major components occurring in vegetable oils, such as free fatty acids, esters, alcohols, and acylglycerols (tri-, di-, and mono-). The reported model is given by B1k ln p0i ðPaÞ ¼ Nk A1k þ 1:5 C1k ln T D1k T T k " # B2k þ Mi Nk A2k þ 1:5 C2k ln T D2k T þ Q T k
∑
∑
ð6Þ 11238
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where Nk is the number of the k-type functional groups present in the i component, A1k, B1k, C1k, D1k, A2k, B2k, C2k, and D2k are parameters estimated by correlation with experimental data (with values given in Table 1), T is the temperature, Mi is the molar mass of the i component, and Q is a correction term, given by Q ¼ ξ1 q þ ξ2
8 Recalculate the molar fraction of the i component for each (k+1) step as follows: xi, kþ1 ¼
β γln T δT T 1:5
ξ1 ¼ fo þ Nc f1
ð8Þ
Dk ¼ Dk1 þ ΔDk
ð10Þ
L ¼ F D204
where α, β, γ, and δ are fitted parameters with values reported also in Table 1; f0, f1, s0, and s1 are parameters dependent on the type of compound (Table 1), Nc is the number of carbon atoms in the molecule, and Ncs is the number of carbon atoms in the alcoholic fraction of the molecule.
’ MODELING PROCEDURE We used a finite differences method to calculate the extent of separation achieved by molecular distillation. The effect of several process variables on the amount of both distillate and retentate products and their compositions was analyzed. The model calculations were done according to the following procedure (which was implemented in Lahey Fujitsu FORTRAN 95) 1 Set initial values for the variables (temperature T, pressure P, feed molar flow F, and feed composition), and set z = 0 (z represents the distance in meters traveled by the feed in the distiller). 2 Calculate of the vapor pressure of each component (p0i ) using the model of Ceriani & Meirelles.18 3 Set k = 0 and Δz = 0.001 m. 4 Use Jik, to account for the cumulative evaporated amount for the i component per time unit until the kth step. 5 For the kth step, calculate the evaporating rate of each species according to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 2πrΔz ð11Þ ΔJi, k ¼ xi, k pi 2πMi RT where ΔJik is the evaporated rate of the i component as the feed travels through length Δz in the distillator, xi,k is the molar fraction of the i component in the mixture as it advances within the unit (for each kth step), p0i is the i component vapor pressure, Mi is the molar mass of the i component, and r is the distillator radius (in this case, r = 0.0254 m). 6 Calculate the total evaporated rate for the kth step (ΔDk) ΔDk ¼
n
∑ ΔJi, k i¼1
ð12Þ
where n is the number of components in the mixture 7 Calculate the cumulative evaporated rate for the i component. Ji, k ¼ Ji, k1 þ ΔJi, k where Ji,1 = 0
ð13Þ
ð15Þ
where D1 = 0 9 Make k = k + 1 and z = z + Δz and repeat steps 58 until k = 204 (equivalent to z = 0.204 m). 11 Calculate the nonevaporated rate (retentate) as
ð9Þ
and ξ2 ¼ so þ Ncs s1
ð14Þ
Where xi,k+1 and xi,0 are the molar fraction of the i component in the (k+1)th step and in the feed respectively; F is the molar feed rate, and Dk is the cumulative evaporated rate up to the kth step, calculated as follows
ð7Þ
where q¼α þ
xi, 0 F Ji, k F Dk
ð16Þ
12 Calculate vapor molar fractions (yi) and liquid molar fractions (xi) of each component, yi, 204 ¼
Ji, 204 D204
ð17Þ
xi, 204 ¼
xi, 0 F Ji, 204 F D204
ð18Þ
The resulting molar fractions for the retentate and distillate (xi,204 yi,204) were converted to mass fractions for comparison with experimental data. The evaporated mass fraction ψ was calculated as Ψ¼
distillate mass rate feed mass rate
ð19Þ
From the set of calculations, mass fractions of the components in the retentate and distillate and ψ are the variables that will be used to compare the results of the model with the experimental values. To evaluate the agreement between the modeled and experimental data we calculated the root mean squared error (RMSE) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi un u ^ i Þ2 u ðXi X t1 RMSE ¼ ν
∑
^ i the predicted value, ν are where Xi is the experimental value, X the degrees of freedom calculated as ν = (number of experimental data 1).
’ REAGENTS The feed composition consisted of 99% purity tributyrin (triester of butanoic acid (C:4) with glycerol), caproic acid (C:6) and caprylic acid (C:8) (all purchased from SIGMA). ’ EXPERIMENTAL PROCEDURE Samples of 100 g of mixtures of free fatty acids and/or triacylglycerols were distilled under different conditions in a molecular distillation apparatus (Pope Scientific Inc., Saukville, WI), consisting of a cylindrical externally heated glass body (evaporating wall or evaporator) and an internal coldfinger 11239
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Figure 1. Molecular distillation still. (a) Upper view, (b) side view, (c) wipers and thin film flow (Pope Scientific).
Figure 2. Wiper speed effect on the separation of tributyrin-caproic acid at 50 and 80 °C.
cooled down by recirculating water at 5 °C. The internal diameter of the evaporator was 2 in. and its length 20.4 cm (0.204 m), so the heating area was 0.0325 m2. The interior of the system was maintained under vacuum conditions. Three plastic wipers were mounted inside the still over an axis so they could rotate to make the feed a thin, evenly distributed film on the heated surface. The heat necessary for the evaporation was provided by a heating mantle surrounding the still. A rotary vane pump (RV3 BocEdwards) was used to achieve the desired
Figure 3. Correction factor used to correct the estimated data according to the observed for different pressures.
vacuum. A scheme of the apparatus is shown in Figure 1. At the end of each run, the distillate and the retentate products were collected, weighed, and analyzed. Samples were taken from the distillate and the retentate and analyzed to obtain species concentrations. Methyl esters of the total free fatty acids were prepared with a modified version of the method used by Ortega et al.24 Each sample (100 μL) was mixed with 1400 μL of chloroform:methanol (volume ratio 2:1); an aliquot part of this 11240
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Figure 4. Effect of the process temperature and pressure on the (a) % of retentate, and % of distillate, (b) retentate composition, and (c) distillate composition.
solution (400 μL) was then mixed with 1 mL of methanolic HCl 0.2 M and heated at 60 °C for 4 h, after which 200 μL of water were added and the resulting methyl esters were extracted with 2 mL of hexane. The extract was dried with sodium sulfate and centrifuged for 5 min at 4000 X g. Gas chromatography was used to determine the concentration of the fatty acid methyl esters, for which 1 μL of the methyl ester extract was injected into a Hewlett-Packard gas chromatograph (model 6890). The column used for the separation was a Supelco SP2560 100 mX0.25 mm; the split/splitless injector and FID detector temperatures were 200 and 230 °C, respectively. The temperature program for the column was as follows: 60 °C for 14 min, 60100 at 10 °C/min, 100 °C for 10 min, 100190 at 10 °C/min, and 190 °C for 22.5 min. Several sets of experiments were carried out to investigate the effect of process variables on the separation process, which included wiper speed, processing temperature, pressure, volumetric flow, and initial concentrations.
’ RESULTS AND DISCUSSION Wiper Speed Effect. The wiper is a rotating element that helps to incorporate the feed as a thin film inside the distillator. We investigated the extent at which the wiper speed helps the separation of the triacylglycerols and free fatty acids. A 60/40% weight caproic acid/tributirin mixture was distilled under a mass
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Figure 5. Effect of the mass flow on the (a) % of retentate, and % of distillate, (b) retentate composition, and (c) distillate composition.
flow of 2.3 g/min, a pressure of 1 mbar, and process temperatures of 50 and 80 °C; the wiper speed tested ranged from 0 to 400 rpm. Figure 2 shows the results from this test. It can be noted that as the process temperature decreased, the critical wiper speed needed to achieve total separation increased. At 50 °C, maximum separation was observed at about 25 rpm, whereas at 80 °C, separation occurred at about 10 rpm. Although this behavior is clearly exhibited, we conclude that the critical value for the wiper speed is very low to be considered as a relevant variable for this separation process. Effect of Process Temperature and Pressure. Another objective of this work was to investigate the applicability of the Ceriani and Meirelles equation for the estimation of vapor pressures for acylglycerols, together with the Langmuir-Knudsen equation to estimate acylglycerols separation. Using a mass flow of 2.3 g/min, wipers speed of 300 rpm, evaporator temperatures ranging from 50 to 80 °C, pressures of 1 and 4 mbar, and a condenser temperature maintained at 5 °C by using cold water provided by a lab refrigerating circulator, we distilled mixtures of tributirin/caproic acid 40%/60% (w/w). After comparing the experimental and model data, we observed that the model overpredicted the experimental values. This happened because the mass transfer model used here (the Langmuir-Knudsen equation) predicts the highest theoretical evaporation rate, which cannot be reached in practice because of the re-evaporation of condensed molecules and the pressure of residual air molecules. So, as expected, we observed that as the operation pressure increased, the evaporation rate decreased. 11241
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Figure 6. Effect of the initial tributyrin concentration (in a binary system with caproic acid) on the (a) % of retentate and % of distillate, (b) retentate composition, and (c) distillate composition.
To correct the model, we divided the evaporated rate of each species by a factor proportional to the operating pressure, and this empirical factor was used to match the experimental results for each pressure. Figure 3 shows the correlation for the empirical factor as a function of the pressure. A linear relationship (40P + 80) was obtained. Therefore, the evaporation rate was estimated as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 0 ð20Þ 2πrΔz ΔJi, k ¼ xi, k pi 2πMi RT 40P þ 80 where P is the pressure in mbar. Figure 4 compares the experimental data with those given by the corrected model in terms of the evaporated percentage, retentate and distillate compositions for several temperatures and pressures of 1 and 4 mbar. One can see that at low temperatures (50 °C) there is a higher effect of the pressure on the evaporation rate. At a pressure of 1 mbar, 53.31% of the feed was evaporated, whereas at 4 mbar 33.13% was evaporated. At higher temperatures (80 °C), the difference between the evaporation percentages at 1 mbar and 4 mbar was minimal (65.53 and 63.09, respectively). This can be explained because at lower temperatures, re-evaporation of molecules is not as high as that achieved at higher temperatures. Mass Flow Effect. Vaporliquid equilibrium is not reached during normal operating conditions because the vapor phase produced is not saturated as the vapor molecules that travel
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Figure 7. Molecular distillation for the 40%/60% w/w mixture of tributyrin/caprylic acid (a) % of retentate and % of distillate, (b) retentate composition, and (c) distillate composition.
across the short path condense immediately. Mass flow is a critical factor because the separation depends on the liquid residence time on the surface of the distiller. For the mixture of 40%/60% tributyrin/caproic acid, we tested different mass flow conditions ranging from 0.95 to 7.65 g/min at wiper speed of 300 rpm, 50 °C, and 1 mbar. Results from these experiments are shown in Figure 5. It can be noted that there is an important effect of the mass flow on the separation. Larger evaporated fractions were observed as the mass flow was reduced; for example, at 0.95 g/min, 58.69% of the feed was evaporated with a 100% w/w of tributyrin in the retentate and 96.78% w/w of caproic acid in the distillate, whereas at 7.65 g/min the process was 8 times faster and the separation was lower with only 35.40% of the feed being evaporated, providing a low purity in the tributyrin retentate (about 70% w/w). Another adjustment was necessary in order to fit the model to the experimental data. The proposed model was rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ð21Þ 2πrΔz ΔJi, k ¼ xi, k p0i 2πMi RT 40P þ ε=F where ε is the adjusted parameter, and F is the feed rate in g/min. After least-squares fitting, the best value for ε was 184. So, the proposed equation is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 0 ð22Þ 2πrΔz ΔJi, k ¼ xi, k pi 2πMi RT 40P þ 184=F 11242
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100 °C; this is because of the significant difference between vapor pressures of the species (at 100 °C, p0 =14.45, 3.04, and 2.56 103 mbar for caproic, caprylic, and oleic acid, respectively). For the separation of mixtures with close values of vapor pressures, a multistage distillation arrangement would be needed.
Figure 8. Molecular distillation for the 30/30/40% w/w caproic/ caprylic/oleic acid mixture: (a) % of retentate and % of distillate, (b) retentate composition, and (c) distillate composition.
Figure 5 shows a comparison between the fitted and the observed data. The RMSE for the evaporated fraction was 2.18, whereas that for the phase composition was 2.79. Effect of Initial Species Concentration. The proposed model was tested by distilling mixtures of varying initial concentrations. The concentrations used for the tributyrin-caproic acid mixtures (w/w) were 30/70%, 40/60%, 50/50%, 60/40%, and 70/30%. The temperature was set at 50 °C, the pressure at 1 mbar, the mass flow at 2.3 g/min, and the wiper speed at 300 rpm. Figure 6 shows the results of this experiment. The RMSE was 1.83 for the evaporated fraction and 2.88 for the phase composition. Distillation of a Mixture of a Triacylglycerol and Caprylic Acid. The proposed model was tested for the distillation of a mixture of tributyrin and caprylic acid (C:8) 40%/60% (w/w), at 1 mbar, 4080 °C, with a feed of 2.3 g/min and wiper speed of 300 rpm. The results are shown in Figure 7. A good agreement was observed between the experimental and simulated data (RMSE was 1.97 for the evaporated fraction, and 3.67 for phase composition). From the data shown in Figure 7b we can observe that at 80 °C caprylic acid is completely eliminated from the retentate, whereas from Figure 7b it can be seen that caproic acid is completely eliminated from the retentate at 60 °C. This result is a consequence of the mixture properties; as the vapor pressure of the acid to be separated is lower, the temperature of the still needs to be higher in order to eliminate it from the retentate stream. Separation of a Free Fatty Acid Mixture. The model was also tested for the separation of mixtures of free fatty acids. A mixture of caproic, caprylic and oleic acids 30/30/40% w/w was distilled at 4 mbar, 60100 °C, feed rate of 2.3 g/min, and wiper speed of 300 rpm. The results of this experiment are depicted in Figure 8. Although the RMSE obtained is higher for this mixture (6.28 for the evaporated fraction, and 6.22 for the phase composition), the model follows the trends of the experimental data. It is important to note that separation of the two medium chain free fatty acids from the oleic acid occurred almost completely at
’ CONCLUSIONS An experimental analysis on the use of molecular distillation for the separation of mixtures of triacylglycerol and free fatty acids has been presented. Results include the effect of relevant operating variables on the separation system such as the still pressure, temperature, feed flow rate and wiper speed. The results showed that the most important variables were the temperature and the feed flow; this result is consistent with those reported in another work for monoacylglycerols purification.25 Although other factors are also important, their effects depend strongly on other operating conditions; for instance, pressure has an important effect for separation of tributyrin and caproic acid at low temperatures. For the conditions tested, the wiper speed did not influence the separation significantly for values over 25 rpm. Although some elaborated models have been reported for the design and analysis of molecular distillation, the experimental results obtained in this work can be used as a basis to develop fairly simple models for this separation system. We have shown how after some empirical factors were added, the models reported here provided a good description of the separation system. This modeling approach could also be considered for the separation of other mixtures by molecular distillation. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected] and
[email protected].
’ NOTATION A1k, B1k, C1k, D1k, A2k, B2k, C2k, D2k, q, Q, α, β, γ, δ, f0, f1, s0, s1, ξ1, ξ2 the Ceriani and Meirelles constants for the determination of vapor pressure λ mean free path (m) σ molecule diameter (m) N number of evaporated molecules in one cubic centimeter (m3) kB Boltzmann constant (J k1) T temperature (K) p0i vapor pressure for the i species (Pa) NA Avogadro’s Constant (mol1) P pressure (Pa), (mbar) for the fitting factor R ideal gas constant (J mol1 K1) A surface area (m2) Mi molar mass for the i component (kg/kmol) τ time (s) ni evaporated moles (mol) z, Δz distance traveled by the feed in the distiller, distance traveled by the feed for each kth step (m) ji molar evaporation rate per squared unit and per time unit for the i component (kmol m2 s1) Ji,k ΔJik cumulative evaporated rate for the i species until the kth step, evaporated rate for the i component as the feed travels through length Δz in the distillator (in the kth step) (kmol s1) xi, xi,k, xi,204 molar fraction for the i component in the liquid 11243
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Industrial & Engineering Chemistry Research phase, molar fraction for the i component in the liquid phase in the kth step, molar fraction for the i component in the retentate (dimensionless) r distillator internal radius (m) Dk, ΔDk, D204 cumulative evaporated rate for all the components until the kth step, evaporated rate for all the components in the kth step, distillate rate (kmol/s) k current step for calculation L retentate rate (kmol/s) n number of components in the mixture (dimensionless) ψ evaporated fraction (dimensionless) yi,204 molar fraction for the i species in the distillate (dimensionless) F feed rate (kmol/s), (g/min) for the fitting factor ε adjusted parameter for flow correction (g/min) Xi predicted data (for RMSE calculation) ^ i experimental data (for RMSE calculation) X
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