Microsoft Word - 8 Andrei Simion UVAB.doc Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XI, Issue 4 – 2012 49 MODELLING OF THE THERMOPHYSICAL LACTIC ACID AQUEOUS SOLUTIONS. DENSITY AND VISCOSITY *Andrei I. SIMION1, Cristina G. GRIGORAŞ1, Loredana E. BARDAŞU1, Adriana DABIJA2 1Vasile Alecsandri University of Bacău, Department of Chemical and Food Engineering, Bacău, Romania, asimion@ub.ro, cristina.grigoras@ub.ro 2Ştefan cel Mare University of Suceava, Faculty of Food Engineering, Suceava, Romania, dadianadabija@yahoo.com *Corresponding author Received 10 November 2012, accepted 2 December 2012 Abstract: Lactic acid is an industrially important product with a large and rapidly expanding market due to its attractive and valuable multi-function properties. It is widely used in various fields such as food industry, in pharmaceuticals and cosmetics etc. and knowing and predicting the evolution of its thermophysical properties at any moment may be very useful. In this paper various mathematical relations between lactic acid concentrations and temperature with density and dynamic viscosity were established. The known data were fitted in different equations in order to assess and select a suitable mathematical model. Taking in consideration the level of precision and the simplicity of formulation several equations were generated for each thermophysical property. For density, two equations were formulated with average relative errors (absolute value) of 0.062% and respectively R2 = 0.9999 and average relative errors of 0.055% for intervals of temperature of 298.15 to 353.15 K and dry matter concentration range between 9.16 to 85.32%. For the dynamic viscosity an equation based on Arrhenius mathematical model with average relative errors of 1.11% and an equation with other mathematical formulation with R2 = 0.9997 were generated in the same range of temperature and citric acid concentration. The obtained equations can be uploaded in computer software for storing, organizing and manipulating data available both for industrial and academic users and so facilitating the sizing and optimization calculations of various technological processes and equipments. Keywords: lactic acid, thermo-physical properties, mathematical modelling 1. Introduction Lactic acid is an organic acid found in many products of natural origin. The first reports on isolation of lactic acid from milk can be found in as early as 1780 and the solidification by self-esterification some years later [1]. Lactic acid can be obtained via chemical synthesis [2-4] or carbohydrate fermentation [5-6] with the help of microorganisms such as Lactobacillus rhamosus [7], Lactococcus lactis [8] or Lactobacillus helveticus [9]. This last technique has a significant advantage in that by choosing a strain of microorganism able to produce only one enantiomer, an optically pure product can be obtained, whereas synthetic production results in a racemic mixture [10]. One of the most important steps of lactic acid production lies in the separation process which is needed to recover and purify the product from the fermentation broth. Different methods are available to this purpose. Among them, solvent extraction followed by centrifugal short path distillation [11], extraction with aliphatic amines [12] or liquid ions [13], nanofiltration [14-15], vapor-permeation assisted esterification [16], reverse osmosis downstream process [17], chromatography [18], electrodialysis [19-21], adsorption Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XI, Issue 4 – 2012 50 technology [22] conducted to satisfactory results. Lactic acid and its derivatives are widely used in the food [23-24], pharmaceutical leather, and textile industries [1]. Recently, there has been an increased interest in lactic acid as a raw material for production of polylactic acid [25-27], a polymer used as a specialty medical and environmental-friendly biodegradable plastic as an alternative for substituting conventional plastic produced from petroleum oil. This new material can be employed as film barrier on intra- abdominal adhesion formation [28], as colloidal drug delivery system [29] as membrane for periodontal guided tissue regeneration [30] etc. For proper and adequate process development it is necessary to use a lactic acid with high purity. Its production and further employment depend on some of its thermophysical properties. Among them density and viscosity are two of the most important ones. As for other types of products [31-32], the evolution of these lactic acid characteristics are related to parameters such temperature and solution concentrations. Different data are available in literature in this field but their use is rather difficult. As consequence, this work intended to establish mathematical relations between aqueous solutions of lactic acid density and viscosity and the above mentioned parameters. Relative error, ANOVA test and correlation coefficient were used in order to verify the similarity between the experimental data and that proposed by the obtained mathematical models. 2. Experimental Experimental data provided by the scientific publications (Tables 1 and 2) concerning the variation of aqueous lactic acid solutions density and dynamic viscosity with concentration and temperature were used as primary data for the regression analysis. Table 1. Variation of lactic acid aqueous solutions density with temperature and lactic acid content [1] Density, ρ [kg/m3] Temperature, T [K] Lactic acid concentration, C [% w/w] 293.15 298.15 303.15 313.15 323.15 333.15 343.15 353.15 9.16 1019.5 1018.1 1015.8 1011.3 1006.7 1000.7 995.04 988.99 24.35 1056.7 1054.4 1051.8 1047.1 1041.4 1035.1 1029.5 1022.6 45.48 1109.8 1105.3 1101.8 1094.2 1087.0 1079.2 1072.1 1063.9 64.89 1155.2 1151.8 1147.2 1139.8 1132.0 1123.5 1115.3 1099.6 75.33 1178.6 1174.8 1170.1 1161.3 1152.6 1142.5 1134.0 1125.1 85.32 1198.9 1194.8 1190.1 1181.3 1171.8 1163.1 1153.6 1144.3 Table 2. Variation of lactic acid aqueous solutions dynamic viscosity with temperature and lactic acid content [1] Dynamic viscosity, µ . 103 [Pa . s] Temperature, T [K] Lactic acid concentration, C [% w/w] 298.15 303.15 313.15 323.15 333.15 343.15 353.15 9.16 1.150 1.030 0.809 0.671 0.572 0.473 0.416 24.35 1.670 1.460 1.130 0.918 0.746 0.632 0.532 45.48 3.090 2.740 2.030 1.590 1.260 1.020 0.843 64.89 6.960 6.010 4.220 3.120 2.380 1.850 1.470 75.33 13.03 10.55 7.080 4.980 3.570 2.730 2.080 85.32 28.50 22.60 13.91 9.400 6.400 4.590 3.400 Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XI, Issue 4 – 2012 51 Microsoft Excel™ 2010 software was employed for typical data integration, graphical representations and ANOVA analysis. ANOVA analysis tool is able to compare the experimental and the calculated data generated by the established mathematical models. Complex and atypical data plotting in 2D (“vapour pressure” model, “heat capacity” model etc.) were performed with CurveExpert® software and 3D representations as a surface response were fitted and analyzed in TableCurve 3D® v.4 software. Thermo-physical property vs. Temperature, Thermo-physical property vs. Lactic acid concentration in aqueous solutions were plotted and different types of regression techniques, involving the method of least squares, relative error ε (Equation 1) and ANOVA were used to reveal the best-fit equation. [%]100   calculated calculatedalexperiment Data DataData  (1) 3. Results and discussions 3. 1. Density Using Microsoft Excel™ 2010 spreadsheets and CurveExpert® software, 8 quadratic correlations (taking in consideration the best fit and simplicity in formulation) between lactic acid concentrations C [% w/w] and density ρ [kg/m3], at constant temperature T, [K] have been established: 2 321 CaCaa  (2) The a1 and a2 values are presented in Table 3 and the regression coefficients R2 are greater than 0.99, thus indicating a good correlation of variables. In order to correlate a1, a2 and a3 coefficients with temperature T, [K], several models were uploaded in CurveExpert® software (1st, 2nd and 3rd Table 3. Coefficients for equation no. 2 Equation 2 coefficients Temperature, T [K] a1 a2 a3 R2 293.15 994.6371 2.6771 -0.0032 0.999 298.15 994.1297 2.5713 -0.0024 0.999 303.15 992.4598 2.5202 -0.0023 0.999 313.15 989.0405 2.4335 -0.0020 0.999 323.15 984.9243 2.3689 -0.0020 0.999 333.15 980.2156 2.2684 -0.0014 0.999 343.15 975.2492 2.2243 -0.0016 0.999 353.15 969.7177 2.1592 -0.0013 0.999 degree polynomial equations, “vapor pressure” model, “heat capacity” model etc.). The best fit model is the quadratic equation with good regression coefficients (Table 4). 2 321 TbTbbtCoefficien  (3) Table 4. Coefficients for equation no. 3 Equation 3 coefficients Equation 2 coefficients b1 b2 b3 R2 a1 - 0.0574246013 0.0003189759 -4.54E-07 0.9044 a2 11.8306041781 - 0.0506144371 5.8356E-05 0.9912 a3 813.3102959923 1.4872913211 -0.0029584854 0.9993 Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XI, Issue 4 – 2012 52 Combining the equations 2 and 3 and, the final form of proposed equation model (Equation 4) is: 22 131211 2 232221 2 333231 )()()( CTbTbbCTbTbbTbTbb aaaaaaaaa  (4) Using the relative error equation the calculated data given by the density mathematical model and the existing experimental data were compared (Table 5) obtaining a final average of 0.062%. Table 5. For densities of citric acid aqueous solutions the absolute value of relative errors for calculated data versus tabular data Density, ρ [kg/m3] Temperature, T [K] 293.15 298.15 303.15 313.15 L ac tic a ci d co nc en tr at io n, C [% w /w ] ED* CD* ε, % ED CD ε, % ED CD ε, % ED CD ε, % 9.16 1019.5 1019.1 0.044 1018.1 1017.3 0.082 1015.9 1015.3 0.051 1011.4 1011.1 0.029 24.35 1056.7 1057.8 0.103 1054.5 1055.3 0.079 1051.8 1052.6 0.076 1047.2 1047.1 0.009 45.48 1109.8 1109.5 0.022 1105.4 1106.1 0.068 1101.8 1102.6 0.073 1094.3 1095.5 0.113 64.89 1155.2 1154.7 0.046 1151.8 1150.7 0.098 1147.2 1146.6 0.053 1139.9 1138.4 0.130 75.33 1178.6 1178.1 0.042 1174.8 1173.8 0.085 1170.1 1169.5 0.052 1161.3 1160.8 0.042 85.32 1198.9 1199.8 0.081 1194.8 1195.4 0.052 1190.1 1190.9 0.070 1181.3 1181.9 0.047 Average ε, % 0.057 Average ε, % 0.077 Average ε, % 0.062 Average ε, % 0.061 Temperature, T [K] 323.15 333.15 343.15 353.15 ED CD ε, % ED CD ε, % ED CD ε, % ED CD ε, % 9.16 1006.7 1006.4 0.038 1000.8 1001.2 0.040 995.0 995.5 0.043 989.0 989.3 0.032 24.35 1041.5 1041.2 0.029 1035.1 1034.9 0.019 1029.6 1028.4 0.116 1022.6 1021.5 0.106 45.48 1087.0 1088.2 0.108 1079.3 1080.7 0.137 1072.2 1073.1 0.082 1064.0 1065.2 0.117 64.89 1132.1 1130.0 0.177 1123.6 1121.6 0.088 1115.3 1113.0 0.029 1099.6 1104.3 0.064 75.33 1152.6 1152.0 0.053 1142.5 1143.1 0.050 1134.1 1134.0 0.004 1125.1 1124.9 0.022 85.32 1171.8 1172.7 0.073 1163.1 1163.3 0.019 1153.6 1153.9 0.023 1144.3 1144.3 0.003 Average ε, % 0.079 Average ε, % 0.058 Average ε, % 0.049 Average ε, % 0.057 * ED – experimental data, CD – calculated data The ANOVA analysis was used to compare the values of density experimental and calculated data at 6 different concentrations in 8 temperatures variation. The results presented in Table 6 showed that the sample P-value is 0.991263 greater than the targeted alpha 0.05 and the F crit value is larger than the F-test value and as consequence the null hypothesis is not rejected indicating that is not a statistical difference between tabular and calculated data. By plotting experimental data for aqueous lactic acid solutions in TableCurve 3D® v.4 software (Figure 1) an equation for the response function was generated, chosen due to the accuracy and simplicity of formulation. The Equation 5 is a simple equation, Rank 33, Eqn. 1033 in TableCurve 3D® v.4 library with a precision of R2 = 0.999967394, FitSdErr = Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XI, Issue 4 – 2012 53 1.2166463, Fstat. = 20950.495. The coefficients values are presented in Table 7. 3 7 2 654 321 1 ln TaTaTaCa TaCaa    (5) Table 6. The ANOVA test summary Lactic acid concentration, C [% w/w] SUMMARY 9.16 24.35 45.48 64.89 75.33 85.32 Total Experimental data Count 8 8 8 8 8 8 48 Sum 8056.42 8338.99 8713.71 9064.73 9239.16 9397.9 52810.91 Average 1007.053 1042.374 1089.214 1133.091 1154.895 1174.738 1100.227 Variance 126.4038 153.4055 271.2956 373.6544 387.4041 397.5284 3960.208 Calculated data Count 8 8 8 8 8 8 48 Sum 8055.094 8338.821 8721.028 9059.337 9236.239 9402.199 52812.72 Average 1006.887 1042.353 1090.128 1132.417 1154.53 1175.275 1100.265 Variance 116.92 174.5367 260.073 337.1056 375.7415 409.9365 3956.772 ANOVA Source of Variation SS* df* MS* F P-value* F crit Sample 0.034015 1 0.034015 0.000121 0.991263 3.954568 Columns 348403.1 5 69680.62 247.094 1.13E-48 2.323126 Interaction 6.930834 5 1.386167 0.004915 0.999995 2.323126 Within 23688.04 84 282.0004 * SS – sum of squares, df – degrees of freedom, MS – mean square, P-value – level of significance Table 7. Coefficients for equation no. 5 Coefficient Value Coefficient Value a1 1004.2716 a5 0.0032418557 a2 3.6820806 a6 -8.8833524E-07 a3 2.8476715 a7 3.4556043E-08 a4 0.000950369 R2 0.9999 0 10 20 30 40 50 60 70 80Temperature, T [K] 360 350 340 330 320 310 300 Lactic acid c oncen tration , C [% , w/w] 950 1000 1050 1100 1150 1200 D en si ty ,  [k g/ m 3 ] Figure 1. Lactic acid aqueous solutions density values plotted in TableCurve 3D and fitted with simple equation type (Equation 5) with residuals 2. Dynamic viscosity A common mathematical model, for fitting the viscosity values is based on the equation of Arrhenius because it creates a good correlation between experimental and calculated values. TR Ea e    0 (6) where: µ – dynamic viscosity [Pa.s,], µ0 – water dynamic viscosity [Pa.s,], Ea – activation energy [kcal/mol], R – universal gas constant [1.987×10-3 kcal/mol], Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XI, Issue 4 – 2012 54 T – absolute temperature [K]. Taking logs of equation (6), it gets (7): TR Ea 1 303.2 loglog 0    (7) and:   T R Ea   loglog303.2 0 (8) By plotting in TableCurve 3D® v.4 software the results obtained from Equation 8, Figure 2, which make correlations between experimental data from Table 1 and water viscosity an equation for the response surface was generated (Equation 9). The Equation 9 is a polynomial equation, Rank 29, Eqn. 1049 in TableCurve 3D® v.4 library with a precision of R2 = 0.9998, FitSdErr = 4.6371739, Fstat. = 36032. 90 80 70 60 50 40 30 20 10 Lac tic a cid con cen trat ion, C [ %, w /w] 360 350 340 330 320 310 300 Temperature, T [K]] -1100 -1000 -900 -800 -700 -600 -500 -400 -300 -200 -100 0 E a/ R Figure 2. Ea/R values plotted in TableCurve 3D and fitted with polynomial equation type (Equation 9) with residuals TaCa TaCaCaa R Ea    65 4 2 321 1 (9) Table 8. Coefficients for equation no. 9 Coefficient Value Coefficient Value a1 -230.01384 a4 0.67832811 a2 -9.7502949 a5 -0.01242596 a3 0.044803796 a6 0.0019299232 Combining the equations 6 and 9 and replacing the coefficients with numeric values, the final form of proposed equation model (Equation 10). TTaCa TaCaCaa e )1(0 65 4 2 321      (10) Appling the relative error equation the calculated data generated with the dynamic viscosity final equation and the existing tabular data were compared (Table 9) obtaining a final average of 1.11%. Table 9. For dynamic viscosities of citric acid aqueous solutions the absolute value of relative errors for calculated data versus tabular data Dynamic viscosity, µ . 103 [Pa . s] Lactic acid concentration, C [% w/w] 9.16 24.35 45.48 T em pe ra tu re , T [K ] ED* CD* ε, % ED CD ε, % ED CD ε, % 298.15 1.15 1.16 0.83 1.67 1.68 0.39 3.09 3.14 1.66 303.15 1.03 1.02 0.62 1.46 1.47 0.35 2.74 2.69 1.77 313.15 0.81 0.82 1.45 1.13 1.15 1.98 2.03 2.04 0.53 323.15 0.67 0.67 0.32 0.92 0.92 0.51 1.59 1.58 0.64 333.15 0.57 0.56 2.45 0.75 0.76 1.51 1.26 1.26 0.24 Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XI, Issue 4 – 2012 55 Dynamic viscosity, µ . 103 [Pa . s] Lactic acid concentration, C [% w/w] 9.16 24.35 45.48 T em p. , T [K ] ED* CD* ε, % ED CD ε, % ED CD ε, % 343.15 0.47 0.47 0.31 0.63 0.63 0.35 1.02 1.02 0.31 353.15 0.42 0.41 1.83 0.53 0.54 1.17 0.84 0.85 0.34 Average ε, % 1.11 Average ε, % 0.89 Average ε, % 0.78 Lactic acid concentration, C [% w/w] 64.89 75.33 85.32 ED CD ε, % ED CD ε, % ED CD ε, % 298.15 6.96 7.00 0.51 13.03 12.84 1.46 28.50 28.80 1.04 303.15 6.01 5.82 3.21 10.55 10.39 1.47 22.60 22.37 1.01 313.15 4.22 4.17 1.13 7.08 7.10 0.26 13.91 14.18 1.95 323.15 3.12 3.07 1.54 4.98 5.00 0.46 9.40 9.36 0.41 333.15 2.38 2.34 1.88 3.57 3.66 2.45 6.40 6.46 0.96 343.15 1.85 1.82 1.42 2.73 2.76 1.02 4.59 4.63 0.89 353.15 1.47 1.45 1.24 2.08 2.13 2.27 3.40 3.41 0.43 Average ε, % 1.56 Average ε, % 1.34 Average ε, % 0.95 * ED – experimental data, CD – calculated data The ANOVA analysis was used to compare the values of dynamic viscosity tabular and calculated data at 10 different concentrations in 4 temperatures variation. The results presented in Table 10 showed that the sample P-value is 0.9997 greater than the targeted alpha 0.05 and the F crit value is larger than the F-test value and as consequence the null hypothesis is not rejected indicating that is not a statistical difference between tabular and calculated data. Table 10. The ANOVA test summary (for dynamic viscosity, µ [Pa . s]) Lactic acid concentration, C [% w/w] SUMMARY 9.16 24.35 45.48 64.89 75.33 85.32 Total Experimental data Count 7 7 7 7 7 7 42.00 Sum 0.005121 0.007088 0.012573 0.02601 0.04402 0.0888 0.183612 Average 0.000732 0.001013 0.001796 0.003716 0.006289 0.012686 0.004371714 Variance 7.76E-08 1.84E-07 7.43E-07 4.45E-06 1.73E-05 9.2E-05 3.46139E-05 Calculated data Count 7 7 7 7 7 7 42 Sum 0.005114 0.007146 0.012579 0.025668 0.043878 0.089219 0.18360409 Average 0.000731 0.001021 0.001797 0.003667 0.006268 0.012746 0.004371526 Variance 8.02E-08 1.84E-07 7.51E-07 4.4E-06 1.65E-05 9.27E-05 3.47515E-05 ANOVA Source of Variation SS* df* MS* F P-value* F crit Sample 7.45E-13 1 7.45E-13 3.9E-08 0.999843 3.973897 Columns 0.001468 5 0.000294 15.3644 2.93E-10 2.341828 Interaction 2.26E-08 5 4.51E-09 0.000236 1 2.341828 Within 0.001376 72 1.91E-05 * SS – sum of squares, df – degrees of freedom, MS – mean square, P-value – level of significance Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XI, Issue 4 – 2012 56 By plotting directly the tabular data for the dynamic viscosity in TableCurve 3D® v.4 software an equation for the response surface was generated (Figure 3). The Equation 11 is a linear equation, Rank 3, Eqn. 1071 in TableCurve 3D® v.4 library with a precision of R2 = 0.99973191, FitSdErr = 0.00010903979, Fstat. = 13258.82 and the coefficients are presented in table 11. TbCbCbCb TbTbTbCbCbb    10 3 9 2 87 3 6 2 54 2 321 1  (11) Table 11. Coefficients for equation no. 11 Coefficient Value Coefficient Value b1 0.1095492 b6 -2.4739564E-09 b2 -4.7563422E-06 b7 -0.027799358 b3 1.9479985E-08 b8 0.00022424967 b4 -0.00092119128 b9 -6.1981778E-07 b5 2.6073585E-06 b10 0.00051576641 01 02 03 04 05 06 070 80Lactic acid concentration, C [%, w/w] 360 350 340 330 320 310 300 Tem per atur e, T [K] 0 0.005 0.01 0.015 0.02 0.025 0.03 D yn am ic v is co si ty , µ [P a. s] Figure 3. Lactic acid aqueous solutions dynamic viscosity values plotted in TableCurve 3D and fitted with linear equation type (Equation 11) with residuals Combining the models developed for the calculation of dynamic viscosity and density of citric acid aqueous solutions, the kinematic viscosity (ν) can be calculated using equation 12:     [m2/s] (12) 4. Conclusion For density two equations were formulated with average relative errors (absolute value) of 0.062% (Equation 4) and respectively R2 = 0.9999 and average relative errors of 0.055% (Equation 5) for intervals of temperature of 298.15 to 353.15 K and dry matter concentration range between 9.16 to 85.32%. For the dynamic viscosity an equation based on Arrhenius mathematical model with average relative errors of 1.11% (Equation 10) and for a direct fitting of the experimental data in TableCurve 3D an equation with R2 = 0.9997 were generated in the same range of temperature and citric acid concentration. The proposed mathematical models can be loaded in the widespread PC software for storing, organizing and manipulating data and for targeted concentrations and temperature more precise values of the studied thermophysical properties can be found easier than using the existing experimental data in tabular form or graphic form. 5. References [1]. REN, J., Lactic Acid, in Biodegradable Poly(Lactic Acid): Synthesis, Modification, Processing and Applications, Springer Berlin Heidelberg, (2011). [2]. DISSELKAMP, R.S., HARRIS, B.D., HART, T.R., Hydroxy acetone and lactic acid synthesis from aqueous propylene glycol/hydrogen peroxide catalysis on Pd- black, Catalysis Communications, 9. 2250- 2252, (2008). Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XI, Issue 4 – 2012 57 [3]. 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