Comparison of Methods for Estimating Critical Properties of Alkyl

Comment/Reply pubs.acs.org/jced

Reply to Comment on “Comparison of Methods for Estimating Critical Properties of Alkyl Esters and Its Mixtures” Manuel García,* Juan-José Alba, Alberto Gonzalo, José Luis Sánchez, and Jesús Arauzo Aragón Institute of Engineering Research (I3A), Universidad de Zaragoza, Zaragoza, Spain

R

Table 1. Recalculated Normal Boiling Point and Critical Properties of Pure FAME with Package 1: CG and WJ

ecently, Wang et al. have performed a detailed analysis of the paper entitled “Comparison of Methods for Estimating Critical Properties of Alkyl Esters and Its Mixtures”,1 which is referred as the “Original paper” in this reply. Wang et al. have noted some mistakes we presumably did in our work, and they have summarized them in their document, which is referred as the “Comment” in this reply. Before starting with the answers for each comment it is important to point out that the detailed analysis of Wang et al. has confirmed the main idea of the Original paper by checking our results and obtaining similar results that prove the usability of the proposed estimations methods for critical properties of alkyl esters and its mixtures. In the following lines we have tried to answer all comments in a short form. Some of the comments/answers need to be clarified in order to find the reasons of different results. Comment 1. Recently, Garciá et al. quantitatively compared three different packages in predicting the critical properties (Pc, Tc, Vc) and the normal boiling points (Tnb) of pure methyl esters (FAMEs) and ethyl esters (FAEEs). The three packages were composed of the following methods: Constantinou and Gani (CG), Marrero and Pardillo (MP), Wilson and Jasperson (WJ), Ambrose (A), Joback (J), Lee-Kesler equations (LK), and the Yuan correlation (Y). However, the authors have perhaps accidently made several mistakes in the application of these methods, which leads to some erroneous results. The significant issues are detailed briefly in this comment. Answer. As Wang et al. pointed out in their Comment, some mistakes were done in the Original paper which are totally corrected in this answer. In this answer we also show that not all comments made by Wang et al. are correct. To clarify the mathematical procedure we describe in this document all relevant methods used in the Original paper. As a result, we present here the corrected results and conclusions. Comment 2. The CG method was used to estimate the Tnb, critical temperature (Tc) and critical volume (Vc) in package 1 in their study. We recalculated the Tnb, Tc, and Vc with the CG method. Our results are consistent with those obtained by the embedded CG method in the Aspen plus software.2 However, the values of Tnb and Tc in Table 3 and Table 4 are unexpectedly high while the values of Vc are unexpectedly low for all the compounds in their paper. Additionally, the CG method was used to compute the Tnb for ethyl esters in package 3. The values of Tnb for ethyl esters listed in Table 8 are also erroneous. Answer. We partially agree with this comment. Unfortunately we made one error using the CG method when splitting the molecules into functional groups, this mistake is corrected and the new results for critical properties are presented in Tables 1 and 2 for FAME and FAEE respectively. © 2013 American Chemical Society

Tnb

Tc

Pc

Vc

methyl ester

K

K

kPa

cm3·mol−1

ω

C8:0 C10:0 C12:0 C14:0 C15:0 C16:0 C17:0 C18:0 C20:0 C22:0 C24:0 C16:1 C17:1 C18:1 C18:1 OH C20:1 C22:1 C24:1 C18:2 C18:3

471.78 506.2 535.64 561.37 573.11 584.22 594.75 604.77 623.44 640.55 656.33 580.88 591.58 601.75 625.65 620.69 638.02 653.99 600.21 598.67

645.14 677.90 705.64 729.68 740.60 750.90 760.65 769.90 787.09 802.79 817.24 749.00 758.85 768.19 786.67 785.53 801.36 815.92 767.83 767.46

2376.42 1997.98 1704.37 1470.66 1370.98 1280.71 1198.62 1123.71 992.12 880.54 784.98 1303.58 1219.46 1142.77 1131.45 1008.21 894.25 796.77 1165.23 1188.39

568.70 680.30 791.90 903.50 959.30 1015.10 1070.90 1126.70 1238.30 1349.90 1461.50 1004.52 1060.32 1116.12 1123.05 1227.72 1339.32 1450.92 1102.73 1089.34

0.66 0.74 0.81 0.80 0.84 0.87 0.91 0.95 1.02 1.09 1.26 0.85 0.89 0.92 1.05 1.00 1.07 1.14 0.91 0.89

Wang et al. recalculated the normal boiling point (Tnb), critical properties (Tc, Pc, and Vc) and acentric factor for pure FAME and FAEE using Package 1. They pointed out that their results are consistent with those obtained with the Aspen Plus2 simulation software. Actually, application of the CG method (or any other methods) should lead to the same results independently of the used software. Considering the comment of Wang et al. we perform the same calculations and found several deviations with the “corrected” results from Wang et al. To clarify the mathematical procedure, we present here two examples of calculations for saturated and unsaturated methyl and ethyl esters. The method of Constantinou and Gani (CG)3 uses the UNIFAC groups to describe the molecules, by this way an organic compound can be described as a combination of functional groups. Table 3 shows the application of the CG method to FAME molecules, saturated and unsaturated methyl esters, for the estimation of the Tnb. Received: July 31, 2013 Accepted: August 22, 2013 Published: September 3, 2013 2689

dx.doi.org/10.1021/je400697n | J. Chem. Eng. Data 2013, 58, 2689−2694

Journal of Chemical & Engineering Data

Comment/Reply

What’s worse, the authors mistook the molecular weights of methyl esters for those of ethyl esters. Additionally, the group CH2 and COO[] should be used for the application of the MP method for ethyl esters, but the authors still used the group CH3 and COO[], which is a fatal error. Answer. As recommended by Wang et al. we have performed a revision of our calculation and the new results are shown in Tables 4 and 5. They are identical to those reported by Wang et al. in the Supporting Information.

Table 2. Recalculated Normal Boiling Point and Critical Properties of Pure FAEE with Package 1: CG and WJ Tnb

Tc

Pc

Vc

ethyl ester

K

K

kPa

cm3·mol−1

ω


484.55 517.04 545.06 569.70 580.99 591.68 601.85 611.53 629.62 646.24 661.61 588.46 598.78 608.61 631.76 626.95 643.78 659.32 607.12 605.63

653.57 684.96 711.71 735.01 745.63 755.65 765.16 774.18 790.99 806.37 820.55 753.80 763.39 772.51 790.58 789.46 804.97 819.25 772.16 771.80

2151.60 1826.81 1570.10 1362.82 1273.63 1192.43 1118.28 1050.33 930.33 827.94 739.81 1213.03 1137.13 1067.64 1057.06 945.04 840.56 750.70 1087.98 1108.91

643.70 755.30 866.90 978.50 1034.30 1090.10 1145.90 1201.70 1313.30 1424.90 1536.50 1079.52 1135.32 1191.12 1166.82 1302.72 1414.32 1525.92 1177.73 1164.34

0.68 0.71 0.73 0.73 0.73 0.73 0.72 0.71 0.68 0.65 0.61 0.71 0.71 0.70 0.82 0.67 0.64 0.61 0.69 0.69

Table 4. Recalculated Normal Boiling Point and Critical Properties of Pure FAME with Package 2: MP and WJ

A contribution factor is assigned to each group and the factors are related to the physical property by means of a correlation. In the case of Tnb, the equation is Tnb(K ) = 204.359 ln[∑ Nk · (tb1k) + W · ∑ Mj · (tb2j)] k

j

Using this methodology we have recalculated our critical properties after solving the mistakes kindly pointed out by Wang et al. Comment 3. The MP method was used to estimate the Tnb, Tc, and Vc in package 2. The application of the MP method in predicting the Tnb for most methyl esters is correct, but errors are found for a few methyl esters and all the ethyl esters. The authors have misused the molecular weight of methyl ester C14:0 to calculate the Tnb for methyl esters C8:0, C10:0, and C12:0, which leads to erroneous Tnb. The values of Tnb are also erroneous for methyl esters C24:0, C17:1, C18:1 OH, C22:1, and C24:1, but the reason for their miscalculations is unclear. The erroneous Tnb leads to erroneous values for the corresponding Tc by the MP method and the critical pressure (Pc) by the WJ method. Additionally, the predicted critical volumes of methyl esters C18:1 OH and C24:1 are unexpectedly wrong, although those of the remainder methyl esters are correct. As for the ethyl esters, the mistakes in the calculation of the properties of methyl esters are inherited.

Tnb

Tc

Pc

Vc

methyl ester

K

K

kPa

cm3·mol−1

ω


468.41 507.75 544.37 578.83 595.38 611.52 627.28 642.70 672.60 701.38 729.17 613.27 629.01 644.40 691.53 674.24 702.97 730.72 646.11 647.83

641.73 676.87 709.77 741.72 757.63 773.65 789.85 806.32 840.41 876.53 915.32 780.99 796.27 811.80 864.19 843.93 877.98 914.52 818.44 826.28

2345 1990 1722 1512 1424 1345 1274 1209 1096 1003 923 1385 1309 1240 1279 1120 1021 937 1274 1310

569.10 682.30 795.50 908.70 965.30 1021.90 1078.50 1135.10 1248.30 1361.50 1474.70 1001.30 1057.90 1114.50 1127.30 1227.70 1340.90 1454.10 1093.90 1073.30

0.62 0.72 0.80 0.87 0.89 0.90 0.91 0.90 0.86 0.79 0.68 0.86 0.87 0.88 0.99 0.86 0.81 0.72 0.85 0.80

Comment 4. The authors stated that the Yuan correlation (eq 7 in their paper) was used to estimate the Tnb for methyl esters in package 3. We recalculated the Tnb for methyl esters with eq 7, but the results are inconsistent with those listed in Table . We found that the authors actually have applied the following equation3 to calculate the Tnb for most methyl esters rather than eq 7. Tnb = 206.9 ln CN + 27.0

(1)

For methyl esters C16:0 and C18:1, the authors took the values reported by Graboski and McCormick4 instead of those calculated by the above equation. Answer. We partially agree with this comment. The equation used in the Original paper to compute the Tnb of pure FAME is the equation stated by Wang et al. and not the

Table 3. Factors for Tnb and Molecular Groups for the CG Method Applied to FAME Molecules first order group (tb1k)

second order group (tb2j)

Nk

Tnb (K)

Mj

factor

0.8894

0.9225

3.636

1.8433

−0.1406

groups

CH3 (1)

CH2 (2)

CH3COO (1)

CHCH (2)

CH2−CHmCHn (m = 1, n = 1)

C8:0 C10:0 C22:1

1 1 1

6 8 18

1 1 1

0 0 1

0 0 2

2690

471.78 506.20 638.02



Comment/Reply

Table 5. Recalculated Normal Boiling Point and Critical Properties of Pure FAEE with Package 2: MP and WJ

Table 6. Recalculated Normal Boiling Point and Critical Properties of Pure FAME with Package 3: A and J

Tnb

Tc

Pc

Vc

Tnb

Tc

Pc

Vc

ethyl ester

K

K

kPa

cm3·mol−1

ω

methyl ester

K

K

kPa

cm3·mol−1

ω


484.48 522.61 558.29 591.98 608.20 624.03 639.51 654.67 684.09 712.46 739.89 625.74 641.20 656.33 702.66 685.71 714.03 741.41 658.00 659.69

653.78 688.13 720.84 753.04 769.21 785.56 802.18 819.14 854.41 892.00 932.57 791.89 807.57 823.57 877.59 856.84 892.28 930.51 829.15 835.91

2135 1833 1600 1416 1338 1267 1203 1145 1043 957 885 1302 1234 1172 1211 1064 973 897 1201 1233

634.30 747.50 860.70 973.90 1030.50 1087.10 1143.70 1200.30 1313.50 1426.70 1539.90 1066.50 1123.10 1179.70 1192.50 1292.90 1406.10 1519.30 1159.10 1138.50

0.67 0.76 0.84 0.89 0.90 0.90 0.90 0.89 0.83 0.74 0.62 0.87 0.88 0.88 0.95 0.84 0.77 0.66 0.85 0.82


447.40 496.16 535.99 569.67 584.75 598.85 612.10 624.58 647.60 668.43 687.44 595.85 609.10 621.58 621.58 644.60 665.43 684.44 639.20 639.20

614.60 664.23 702.04 732.25 745.31 757.28 768.32 778.57 797.10 813.49 828.19 755.60 766.57 776.75 776.75 795.14 811.42 826.03 800.78 802.85

2311 1978 1729 1536 1454 1381 1315 1255 1150 1061 985 1412 1343 1280 1350 1171 1079 1001 1307 1334

565.5 677.5 789.5 901.5 957.5 1013.5 1069.5 1125.5 1237.5 1349.5 1461.5 993.5 1049.5 1105.5 1118.5 1217.5 1329.5 1441.5 1085.5 1065.5

0.59 0.68 0.77 0.85 0.89 0.93 0.96 1.00 1.07 1.14 1.20 0.91 0.95 0.99 1.03 1.06 1.13 1.19 0.98 0.97

equation we mentioned in the Original paper (eq 7). In fact, the equation we used is also a correlation developed by Yuan et al.4 based on the experimental data from Graboski and McCormick, which can be used to estimate the Tnb of saturated FAME with carbon number from 8 to 24 with an R2 of 0.986. We apologize for this misunderstanding. To correct the presented results we have calculated the Tnb of FAME using the following equation as stated in our paper:

and Pc with the Ambrose method. Our results are consistent with those obtained by the embedded Ambrose method in the Aspen plus software.2 Answer. We agree with this comment and sincerely apologize for this mathematical mistake. As recommended by Wang et al. we have performed a revision of our calculation and the new results are shown in Tables 6 and 7, and are obviously identical to those reported by Wang et al. in the Supporting Information with the exception of Tc of C18:1(OH) where we

Tnb = 218.9 ln CN − 6.933

The results are shown in Table 6 of this document and are completely consistent with those reported by Wang et al. Comment 5. Additionally, the method for the prediction of Tnb for C18:1 OH in Table 7 is unclear. Obviously, it should be neither the CG method nor the MP method, because the Tnb for C18:1 OH in Table 7 is inconsistent with that in Table 3 or Table 5 in their paper. Answer. We totally agree with Wang et al., the value used in the Original paper for C18:1(OH) (Table 7 in our paper) is 686.15 K, which actually is the experimental normal boiling point of methyl ricinoleate. We used this value instead of a prediction for error. It was also an error not to mention it in the paper. We apologize for this mistake. To give more consistency to our model, the value we use in this answer and further calculations is the direct application of Yuan equation. Results are shown in Table 6. Comment 6. The Ambrose method was used to estimate the Tc and Pc in package 3, but it was not applied correctly by the authors. They took the number of carbon atoms in the molecule as the number of carbon atoms in alkyl groups for all the compounds except for the methyl and ethyl ester C18:1 OH. In fact, the number of carbon atoms in alkyl groups should be the number of carbon atoms in the molecule excluding the carbon atom of the aliphatic functional group CO−O. Moreover, one double bond is missing when it was applied by the authors to calculate the Tc and Pc of methyl linolenate (C18:3) and ethyl linolenate (C18:3). We recalculated the Tc

Table 7. Recalculated Normal Boiling Point and Critical Properties of Pure FAEE with Package 3: A and J Tnb

2691

Tc

Pc

Vc

ethyl ester

K

K

kPa

cm ·mol−1

ω


484.55 517.04 545.06 569.70 580.99 591.68 601.85 611.53 629.62 646.24 661.61 588.46 598.78 608.61 631.76 626.95 643.78 659.32 607.12 605.63

656.74 684.37 706.99 726.13 734.69 742.70 750.24 757.34 770.47 782.41 793.35 740.60 748.25 755.48 784.21 768.79 780.87 791.93 755.42 755.39

2131.28 1845.04 1626.56 1454.34 1381.21 1315.08 1255.00 1200.16 1103.71 1021.60 950.86 1343.04 1280.42 1223.39 1286.33 1123.31 1038.36 965.36 1247.42 1272.32

621.50 733.50 845.50 957.50 1013.50 1069.50 1125.50 1181.50 1293.50 1405.50 1517.50 1049.50 1105.50 1161.50 1174.50 1273.50 1385.50 1497.50 1141.50 1121.50

0.64 0.72 0.81 0.89 0.93 0.96 1.00 1.04 1.10 1.17 1.23 0.95 0.99 1.03 1.07 1.10 1.16 1.22 1.02 1.00

3



Comment/Reply

Table 8. Experimental and Recalculated Pure FAME Densities with All Packages exp. ρ

package 1

package 2

package 3

FAME

kg·m−3

ρ/kg·m−3

error (%)

ρ/kg·m−3

error (%)

ρ/kg·m−3

error (%)

C8:0 C10:0 C12:0 C14:0 C15:0 C16:0 C17:0 C18:0 C20:0 C16:1 C18:1 C22:1 C18:2 C18:3 ARD (%)

883.04 876.31 873.28 868.18 869.79 864.19 868.11 867.55 866.28 882.39 877.46 873.69 893.18 901.83

953.56 934.47 908.21 861.57 848.90 833.55 819.64 805.07 773.01 839.71 809.87 746.35 818.78 826.06

7.99 6.64 4.00 0.76 2.40 3.55 5.58 7.20 10.77 4.84 7.70 14.57 8.33 8.40 6.62

931.01 925.97 911.07 890.44 876.67 861.05 845.81 826.78 785.88 865.57 833.98 753.03 839.46 839.62

5.43 5.67 4.33 2.56 0.79 0.36 2.57 4.70 9.28 1.91 4.95 13.81 6.01 6.90 4.95

935.38 921.24 913.44 908.44 906.20 903.92 901.50 898.85 892.72 916.77 911.05 896.56 899.97 908.96

5.93 5.13 4.60 4.64 4.19 4.60 3.85 3.61 3.05 3.90 3.83 2.62 0.76 0.79 3.68

performed a revision of our calculation. Our new results have a slight deviation with those reported by Wang et al., the reason for this deviation is unclear, but the influence in the calculations is barely remarkable not having any influence either in the results or in the conclusions. Comment 9. The aforementioned errors have led to unreasonable predictions. For example, the unexpectedly low acentric factors of C18:0 OH in Table 5 and Table 6 calculated with package 2 in their paper. Answer. We agree with this comment. Please see Tables 4 and 5 for recalculated acentric factors of C18:1(OH). Comment 10. Misleading conclusions were also drawn. For example, the authors stated that “the error of the methods increase slightly in the case of low molecular weight methyl esters (C8:0 and C10:0)”1 as shown in Figure 2 in their paper. However, from Figure 1 in this comment, in which the recalculated normal boiling points of some esters by the CG method and the MP method are plotted with experimental data taken from the literature. It is evident that both the CG method and the MP method can give accurate predictions on normal boiling points for low molecular weight methyl esters. The average relative deviation (ARD %) defined the same as eq 8 in their paper is also recalculated and shown in Figure 1. It should be pointed out that the experimental normal boiling points of methyl esters are from reference 6 (reference 4 in this comment) rather than reference 7 indicated by Garciá et al. Answer. We totally agree with this comment and would like to apologize for the wrong conclusion. In addition, we would like to thank Wang et al. for the comment which has served us to correct the results. The new ARD (%) for pure FAME normal boiling point are 2.66 % and 1.74 % for CG and MP methods, respectively. These results confirm that both methods can be used to estimate the normal boiling point of pure FAME with a very high accuracy. Comment 11. On the basis of the recalculated properties, we further calculated the densities of methyl esters, ethyl esters, and biodiesels. It is found that several conclusions drawn by Garciá et al.1 should be corrected. First, it is package 3 (ARD = 3.68 %), rather than package 2 (ARD = 4.90 %) concluded by the authors, that showed the highest accuracy in predicting the densities of 14 pure methyl esters. Answer. We agree with this comment. According to Tables 8 and 9 in this reply, Package 3 is able to predict the densities of

have found a very small deviation (3 kPa) with the results presented by Wang et al. Comment 7. The Joback method was used to estimate the Vc in package 3. The application of the Joback method is correct for methyl esters, but incorrect for ethyl esters. For a certain methyl ester, one more >CH2 group should be added for the corresponding ethyl ester. However, the authors added one more CH3 group instead, which leads to the erroneous critical volumes for all the ethyl esters. Answer. We agree with this comment and sincerely apologize for this mathematical mistake. As recommended by Wang et al. we have performed a revision of our calculation and the new results are shown in Table 7, and are obviously identical to those reported by Wang et al. in the Supporting Information. Comment 8. (6) The Lee−Kesler equations were used to compute the acentric factor, but the authors failed to use the method properly. First, the constant 0.160347 in eq 4 in their paper should be 0.169347.5 Therefore, eq 4 should be written as α = −ln Pci − 5.92714 + 6.09648θ −1 + 1.28862 ln θ − 0.169347θ 6

(2)

Second, the critical pressure, Pci in the above equation is actually in the unit of atmosphere.5 To enable Pci using the SI unit, for example kPa, the above equation should be rearranged as α = −ln(Pci/101.325) − 5.92714 + 6.09648θ −1 + 1.28862 ln θ − 0.169347θ 6

(3)

In fact, the authors may have directly applied eq 2 with Pci in bar, which leads to the erroneous acentric factors despite the correct predictions on critical properties and normal boiling points. Answer. We partially agree with this comment. Unfortunately the eq 4 in our paper was written with a typographical mistake, but after checking our calculations we have found that the right equation was used. Regarding the second comment, we agree with it and sincerely apologize for this mathematical mistake; the very small deviations found by Wang et al. are due to this mistake. As recommended by Wang et al. we have 2692



Comment/Reply

Table 9. Experimental and Recalculated Pure FAEE Densities with All Packages exp. ρ

package 1

package 2

package 3

FAEE

kg·m−3

ρ/kg·m−3

error (%)

ρ/kg·m−3

error (%)

ρ/kg·m−3

error (%)

C8:0 C10:0 C12:0 C14:0 C16:0 C18:1 C18:2 C18:3 ARD (%)

871.42 873.34 867.96 867.65 864.39 877.18 883.18 899.18

936.00 903.42 864.94 823.39 780.32 742.69 751.38 760.31

7.41 3.44 0.35 5.10 9.73 15.33 14.92 15.44 8.97

926.86 917.17 900.82 875.86 842.98 816.07 820.44 824.94

6.36 5.02 3.79 0.95 2.48 6.97 7.10 8.26 5.11

911.46 916.74 921.45 924.54 925.54 936.63 947.50 958.65

4.59 4.97 6.16 6.56 7.07 6.78 7.28 6.61 6.25

Table 10. Experimental and Recalculated FAME Biodiesel Densities with All Packages exp. ρ kg·m algae babassu beef tallow borage camelina oil canola oil castor choice white grease coconut 1 coconut 2 coconut 3 Coffee corn evening primrose hemp hepar. high IV hepar. low IV Jatropha linseed Moringa oleifera neem palm perilla seed poultry fat rice bran soybean sunflower used cooking oil yellow grease ARD (%)

−3

878.0 876.0 874.0 886.5 888.0 882.0 899.0 877.0 807.3 874.8 874.7 881.5 885.0 888.5 888.5 875.5 875.5 879.5 892.5 877.0 884.5 876.0 899.0 880.5 885.5 884.0 880.0 855.5 882.5

package 1 ρ/kg·m

−3

862.30 874.90 879.10 872.35 884.22 863.34 877.15 864.80 877.67 875.53 879.03 866.72 869.69 867.18 872.26 873.48 865.66 864.19 869.45 864.99 861.35 864.36 870.69 865.15 863.89 865.95 862.42 865.43 864.30

package 2

error (%) 1.79 0.13 0.58 1.60 0.43 2.12 2.43 1.39 8.72 0.08 0.50 1.68 1.73 2.40 1.83 0.23 1.12 1.74 2.58 1.37 2.62 1.33 3.15 1.74 2.44 2.04 2.00 1.16 2.06 1.83

ρ/kg·m

−3

879.55 875.62 889.04 898.97 914.33 884.41 904.34 877.95 876.95 876.88 880.55 883.33 890.56 892.85 898.98 887.42 876.13 882.76 898.98 883.57 876.56 872.84 901.26 880.77 882.63 888.70 882.24 886.84 882.48

package 3

error (%) 0.18 0.04 1.72 1.41 2.96 0.27 0.59 0.11 8.63 0.24 0.67 0.21 0.63 0.49 1.18 1.36 0.07 0.37 0.73 0.75 0.90 0.36 0.25 0.03 0.32 0.53 0.25 3.66 0.00 0.99

ρ/kg·m

−3

860.58 878.90 876.54 878.03 890.58 863.73 887.66 862.96 881.54 878.91 882.50 865.26 873.48 874.15 879.31 872.63 862.91 865.55 876.86 861.50 859.19 861.70 879.25 865.39 864.77 870.34 862.21 868.84 865.42

error (%) 1.98 0.33 0.29 0.96 0.29 2.07 1.26 1.60 9.20 0.47 0.89 1.84 1.30 1.62 1.03 0.33 1.44 1.59 1.75 1.77 2.86 1.63 2.20 1.72 2.34 1.55 2.02 1.56 1.94 1.72

critical properties of pure FAME and FAEE, these properties are used to compute the so-called “pseudo-component critical properties” following the procedure mentioned in the Original paper. The Lee−Kesler mixing rules are applied carefully to the mixture obtaining the critical properties of the mixture. By applying the Rackett-Soave eq [5] the density is calculated. Using the L−K mixing rules is not a trivial issue when 20 components are considered. After the comments by Wang et al. we have checked our calculation very carefully and realize that no errors or mistakes were done during the application of the mixing rules. Unexpectedly, the density values of FAME biodiesel we obtain using the Wang et al. recalculated critical properties are not consistent with those density values they report in the Comment.

FAME with the best accuracy compared with Packages 1 and 2. It is important to point out that the ARD (%) of the Packages did not change significantly after the correction as shown in the corresponding tables. This fact confirms the main conclusion that the models and methods proposed can be used to estimate the densities of pure FAME and FAEE. Comment 12. Second, package 1 is not able to predict methanol-based biodiesel density with ARD lower than 2.1 % because it has an ARD of 5.55 % according to our recalculation. Finally, since package 1 has the highest ARD (4.53 %) in predicting the ethanol-based biodiesel density, it is not a good option to estimate the properties of ethanol-based biodiesel. Answer. The estimation of the biodiesel density is obviously the main point of the Original paper. After computing the 2693



Comment/Reply

Table 11. Experimental and Recalculated FAEE Biodiesel Densities with All Packages exp. ρ palm soybean canola corn ricebran ARD (%)

package 1

package 2

package 3

kg·m−3

ρ/kg·m−3

error (%)

ρ/kg·m−3

error (%)

ρ/kg·m−3

error (%)

873.8 882.7 881.1 881.6 881.7

865.77 877.57 873.84 878.40 875.02

0.92 6.12 0.82 0.36 0.76 1.80

866.07 869.27 868.05 869.45 868.58

0.88 5.11 1.48 1.38 1.49 2.07

845.92 848.09 843.59 848.13 847.18

3.19 2.55 4.26 3.80 3.92 3.54

(4) Yuan, W.; Hansen, A. C.; Zhang, Q. Vapor pressure and normal boiling point predictions for pure methyl esters and biodiesel fuels. Fuel 2005, 84, 943−950. (5) Spencer, C. F.; Danner, R. P. Improved equation for prediction of saturated liquid density. J. Chem. Eng. Data 1972, 17, 236−241.

Tables 10 and 11 show the recalculated biodiesel densities for FAME and FAEE biodiesels. As can be observed, the FAME’s biodiesel ARD (%) for the Package 2 and 3 are very close to those reported by Wang et al. but not identical. In the case of Package 1 this difference is significant. Additionally, the current results are consistent with those reported in the Original paper showing that the selected methods are accurate enough to predict the biodiesel densities. As can be also observed in Table 11, the corrected results perfectly fit the original conclusions, Packages 1 and 2 being the “...the best option in case of ethanol based biodiesels.” as we concluded in our Original paper.

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GENERAL CONCLUSIONS AND COMMENTS We really want to thank Wang et al. for their detailed work reviewing our Original paper. As we answered in this document, some of the mistakes they appointed have been corrected and the results have been improved. With the new results, which actually show an improvement in the models performance, some conclusions of the Original paper were corrected and other conclusions were confirmed. The estimation method CG and MP were used to compute the Tnb, after the corrections indicated by Wang et al. the new ARD (%) for pure FAME normal boiling point are 2.66 % and 1.74 % for CG and MP methods, respectively. In case of FAME biodiesels the correction of the mistakes benefits the precision of the models indicating that all three can be used to estimate the FAME biodiesel density with good precision, Package 2, with an ARD of 0.99 %, being the more accurate. Although more experimental data for ethanol based biodiesel are needed, the corrected results shown in this answer meet those reported in the Original paper. Moreover, we want to apologize for the mathematical mistakes we made in our Original paper and want to thank again Wang et al. for helping us to correct and improve our calculations.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) García, M.; Alba, J.-J.; Gonzalo, A.; Sánchez, J. L.; Arauzo, J. Comparison of Methods for Estimating Critical Properties of Alkyl Esters and Its Mixtures. J. Chem. Eng. Data 2011, 57, 208−218. (2) Aspen Physical Property System: Physical Property Models; Aspen Technology Inc.: Burlington, MA, 2008. (3) O’Connel, J. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 5th ed.; McGraw-Hill: New York, 2001. 2694


Comparison of Methods for Estimating Critical Properties of Alkyl

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