Iranian Journal of Science & Technology, Transaction B, Engineering, Vol. 29, No. B5 Printed in The Islamic Republic of Iran, 25 Shiraz University NE ND SIMPLE MODEL FOR SURFCE TENSION PREDICTION OF TER ND ORGNIC LIQUID MIXTURES R. THERY ND H. MODRRESS Dept. of Chemical Engineering, mir Kabir University of Technology, P.O. Box: 5875-443, Tehran, I. R. of Iran Email: hmodares@aku.ac.ir bstract In this paper a new simple equation for surface tension prediction of binary aqueousorganic solutions over the entire range of compositions is presented and used for the calculation of surface tension of 5 binary aqueous-organic solutions. The overall average absolute deviation is.53%. The parameters of the new equation are calculated for the binary systems under study and it is shown that in the temperature range used for surface tension calculations they are independent of temperature and can be expressed in terms of the normal boiling point of organic compounds. Keywords Surface tension, binary aqueous-organic solutions, surface tension coefficient. INTRODUCTION Thermodynamic and transport properties are fundamental in the design of process units that involve fluid flow and interface heat and mass transfer. Surface tension is one of the most interesting thermophysical properties due to its manifestation in many of the naturally occurring phenomena, as well as in many industrial applications. Surface effects have great importance in dealing with heterogeneous catalysis applications. Moreover, processes like lubrication, corrosion and reactions in electrochemical cells are also related to surface effects. In fact, liquid-vapor interfaces are critical to the chemical engineering separation processes such as gas absorption, condensation and distillation. In the past years, some equations have been derived in various degrees of generality for the prediction of binary aqueous-organic solutions. Schuchowitzky [] and Belton and Evans [2] developed semi-empirical equations for ideal or nearly ideal solutions in the bulk phase. However, these equations are empirical, and the constant terms have no apparent relation to the properties of the pure components, therefore the calculation of surface tension is complicated. Guggenheim [3] used a quasicrystalline model to derive an equation for the surface tension of regular solutions as a function of the heat of mixing and simplified it for ideal solutions. Hoar and Melford [4] developed a theoretical model for nonpolar solutions with the assumption that the bulk liquid and surface layer behave like regular solutions and that the molecules approximate hard spheres, Sprow and Prausnitz [5], by using the regular solution theory, derived an equation for predicting surface tension of binary solutions. This equation applies only to solutions having zero excess volume and entropy. In the same trend of investigation, Semenchenko [6] and Feinerman [7] developed equations for predicting the surface tension of binary solutions. Goldsack and Sarvas [8] assumed ideal solution behavior in both bulk and surface phases and obtained an equation for predicting surface tension of binary solutions. Connors and right [9] derived an equation for calculating the surface tension of binary aqueous organic solutions. lvarez et al. [-5] used Connors and right's [9] equation for modeling the surface tension of binary aqueous solutions and computed model parameters for each solution. Lee et al. [6] used Received by the editors pril 2, 24; final revised form ugust 6, 25. Corresponding author
52 R. Tahery / H. Modarress Connors and right's [9] equation for modeling the surface tension of,3-propandiol + ater. ccording to their calculations, model parameters are temperature dependent. Zihao et al. [7] calculated the surface tension of binary aqueous organic solutions with an equation that was derived on the basis of the ilson equation. Zhibao and Benjamin [8] derived an equation based on the Davis theory [9] and calculated the surface tension of some binary aqueous-organic solutions. In this work a new equation is proposed and used for calculating the surface tension of binary aqueous-organic solutions. The results show that the calculated surface tensions by this equation are in better agreement with experimental data compared with similar equations. a) Background 2. THEORY Connors and right [9] considered binary solutions as a distinct two-phase system: the bulk phase and surface phase. In their equation the surface tension of the solutions is written as an average of the surface tensions of the pure components weighted by the total surface concentrations of the components. By using Langmuir adsorption isotherm Connors and right derived a two parameter equation for the surface tension of binary solutions. Zihao et al. [7] proposed a surface tension equation based on the thermodynamic definition of surface tension and the expression of Gibbs free energy. By using the ilson equation to represent the excess Gibbs free energy, a two parameter surface tension equation was derived for liquid mixtures. Zhibao and Benjamin [8] developed a semi theoretical model for calculating surface tension of pure liquids and binary mixtures based on the Davis theory [9]. In this work a new model is derived for binary aqueous-organic solutions. The essence of the theory is described in the following section. b) Derivation of the new model For an ideal solution of organic compound () and water (), the surface tension is presented as = x + x () Ideal where and the mole fraction. lso, for an ideal solution a reduced surface tension can be defined as are the surface tensions of pure organic component and water respectively, and x is Ideal = Ideal (2) By using Eqs. () and (2) it is shown that Ideal = (3) Similarly, for a real solution the reduced surface tension, Real = Real Real, is defined as (4) s for an ideal solution Ideal =, it is convenient to present Real as = Φ x (5) Real Iranian Journal of Science & Technology, Volume 29, Number B5 October 25
new and a simple model for surface tension prediction of 53 whereφ is a new solution property defined as the surface tension coefficient and for an ideal solution it is equal to unity. Therefore the excess reduced surface tension, which is the difference between reduced surface tensions of real and ideal solutions, can be presented in the following form: nd therefore, by using Eqs. (3), (5) and (6) we obtain Real Ideal = (6) = ( Φ ) (7) For the new property introduced as the surface tension coefficient, Φ, we propose the following simple functionality: Φ = + ax bx (8) where a and b are the parameters of this new surface tension equation for binary aqueous-organic solutions. ccording to Eq. (8), as x the surface tension coefficient should tend to unity ( Φ ) and also as x, the limiting value of surface tension coefficient is Φ (+ a b)/( b). By combining Eq. (7) and Eq. (8) we obtain a more applicable form of the surface tension equation proposed in this work as = Φ = ax Equation (9) shows that variation of /(Φ ) versus / x is linear and the parameters a and b can be evaluated from the slope and intercept of this plot. Figure shows, as examples, the linear variation of /(Φ ) versus / x for binary aqueous solutions of formic acid, ethanol, methanol and monoethanolamine (ME). The values of a and b for each binary solution evaluated at 5 C intervals from slopes and intercepts of this plot are presented in Table. s it is seen from the results in Table, the parameters a and b are independent of temperature. The same independency of temperature for the parameters a and b of all binary aqueous solutions studied in this work is observed. The calculated parameters, a and b, as average values for the studied temperature range are reported in Table 2. lso in this table, the average absolute deviation percent (D %) in calculating surface tension of solutions by the new equation and the (D %)s by the equations proposed by Connors and right [9], Zihao et al. [7], and Zhibao and Benjamin [8] are reported. The lower D% values of surface tension calculated by the new equation compared with those calculated by Zihao et al. [7] and Zhibao and Benjamin's [8] equations indicate the accuracy of the new equation. However as it is seen from the results in the Table 2, the (D %)s calculated by the Connors and right [9] equation are less than those calculated by the new equation, but the superiority of the new equation is inherent in the fact that parameters a and b are independent of temperature. It is worth noting that the parameters in the Connors and right [9] equation vary with temperature, and therefore for each experimental data the specific parameters should be calculated, thus reducing the predictability of Connors and right is [9] equation. It is usual to correlate the surface tension of pure fluids in terms of characteristic substance properties such as critical temperature, pressure and normal boiling point by utilizing the principle of correspondence states. In this respect, the proposed correlation of Hakim et al. [2] for polar fluids and Brock and Bird [2] for a number of substances ranging from noble gases to diatomic and simple inorganic molecules can be mentioned. b a (9) October 25 Iranian Journal of Science & Technology, Volume 29, Number B5
54 R. Tahery / H. Modarress 7 Methanol 6 /(Φ ) 5 4 3 Formic cid ME Ethanol.8 2.6.4.2 /x Fig.. The variation of /(Φ ) versus / x for indicated aqueous solutions at 25 C ( ); 3 C ( ); 35 C ( ); 4 C ( ); 45 C ( ); 5 C ( ). The points are the experimental data and the lines are the calculated values of/(φ ) Table. The surface tension coefficient parameters ( a,b ) as expressed by Eq. (8) at 5 C intervals Temp ME Methanol Ethanol Formic acid ( C) a b a b a b a b 2.8497.8624.9279.9552.76.842 25.6264.9459.8484.8632.929.9547.7573.8428 3.628.9459.8468.8637.9276.9552.769.843 35.6282.9453.844.8645.9277.955.759.842 4.6252.9462.846.8669.9264.9557.7565.8433 45.6255.9457.8382.868.9259.9564.7559.844 5.6239.946.8346.8698.9267.956.7557.8436 Surface tension experimental data at 2 C for ater + ME (Monoethanolamine) solution is not available. In this work, for calculated parameters a and b obtained from the slope (/a ) and the intercept ( b/ a) of the plot of /(Φ ) versus/ x, a simple correlation in terms of reduced boiling point T br = T b / T c can be proposed, where T b and T c are the normal boiling point and critical temperature of the organic compound, respectively. This correlation can be represented by the following equation.5.5 T br / a =.9253( Γ / a).69 () The correlation factor for linearity of Eq. () is.998 for the 5 binary aqueous solutions named in Table 2 and the characteristic parameter Γ in Eq. () can be expressed as: lso, the average ratio of 2 3 4 5 6 7 Γ =.33( T br).28 ().5 ( b / a.5 ) for parameters a and b of the studied solutions is.5.5 b / a =.9823 (2).5 The average standard deviation of ( b / a.5 ) values for each binary solution from the average value.5.5 b / a is.28. nd as it is generally convenient to represent the deviation of surface tension of a solution from ideal behavior in terms of E, the excess surface tension, in the following form [-5]: Iranian Journal of Science & Technology, Volume 29, Number B5 October 25..2.3.4
By using Eqs. (2), (4), (8) and (9) we can write new and a simple model for surface tension prediction of 55 = ( x + x ) (3) E ax E = Dividing both sides of Eq. (4) by ( ) following form x ( ) bx (4), we obtain a reduced surface tension deviation in the E ax = x bx (5) Compound Table 2. Surface tension parameters ( a,b ) for the new equation and the average absolute deviations percent (D %) calculated by various surface tension equations for 5 binary aqueous-organic solutions Temp. ( C) a b (D%) (D%) 2 (D%) 3 (D%) 4 n Monoethanolamine (ME) 25-5.6264.9459.4.3.36 n.a. 3 2 Diethanolamine (DE) 25-5.6676.9557.2 n.a. n.a. n.a. Triethanolamine (TE) 25-5.957.9582.3 n.a. n.a. n.a. N-Methyldiethanolamine (MDE) Data source 25-5.88.9653.46.4.5 n.a. 4 4 -mino-2-propanol (MIP) 25-5.6943.9638.59 n.a. n.a. n.a. 2 8 3-mino--propanol (P) 25-5.687.952.36 n.a. n.a. n.a. 2 8 2-mino-2-methyl-- propanol (MP) 25-5.8332.9794.7.9.72 n.a. 4 2 Formic acid 2-5.7557.8436.3.24.39.84 4 3 cetic acid 2-5.796.956.22.2.79.9 4 3 Propionic acid 2-5.99.9855.3.22.95 2.53 4 3 Methanol 2-5.8346.8698.42 n.a..72.7 4 Ethanol 2-5.9267.956.57 n.a..9.68 4 -Propanol 2-5.9777.9933.6 n.a..94 3.5 4 2-Propanol 2-5.97.9832.45 n.a. 2.75.97 4,3-Propandiol 25-5.785.8952.45 n.a. 6.55 n.a. 7 5 Overall average D%.53%.29%.6%.77% (D%), (D%) 2, (D%) 3 and (D%) 4 refer to the average absolute deviations percent based on Eq. (9), and Connors and right [9], Zihao et al. [7] and Zhibao and Benjamin [8] equations. n: Number of experimental data n.a.: Not available In the next section the proposed equations, Eqs. (4), (5) and (8), are applied to 5 binary aqueousorganic solutions and the results are compared with those obtained by application of similar equations proposed by Connors and right [9], Zihao et al. [7], and Zhibao and Benjamin [8]. 3. RESULTS ND DISCUSSION The calculated parameters a and b of Eq. (9) for 5 binary aqueous-organic solutions are reported in Table 2. In Fig., the values of /(Φ ) versus / x have been plotted for aqueous solutions of methanol, ethanol, formic acid and monoethanolamine (ME) at different temperatures. The points indicate the experimental values of/(φ ) obtained by inserting the experimental data in Eqs. (4) and (5). The lines October 25 Iranian Journal of Science & Technology, Volume 29, Number B5
56 R. Tahery / H. Modarress indicate the calculated /(Φ ) by inserting in Eq. (9) the fixed values of the parameters a and b as reported in Table 2. For all 5 binary aqueous-organic solutions studied in this work similar plots can be presented. Figure 2 shows the surface tension coefficient (Φ ) versus x for aqueous solutions of formic acid, acetic acid, propionic acid, methanol, ethanol, -propanol, 2-propanol, monoethanolamine, diethanolamine, and triethanolamine at 25 C. The points indicate the surface tension coefficient Φ is obtained directly by inserting the experimental data in Eqs. (4) and (5). The lines indicate the calculated surface tension coefficient Φ by using the fixed values of the parameters a and b as reported in Table 2 in Eq. (8). s it can be seen from this figure, the surface tension coefficient for the studied solutions increases nonlinearly with the mole fraction of water. This is the expected trend since according to Eq. (8) as x the surface tension coefficient should tend to unity (Φ ). lso according to Eq. (8) for x, the limiting value of surface tension coefficient is Φ (+ a b)/( b). By using the values of parameters a and b in Table 2, the values of Φ at x are equal to 46.93, 63.69 and 23.74 for -propanol, propionic acid and triethanolamine (TE) respectively. In Fig. 2 a large deviation from linearity is observed for x. This indicates that, even the presence of a very small amount of organic compound causes a large deviation from ideal behavior due to adsorption of the organic compound on the liquid-water surface. lso from Figure 2, it is obvious that surface tension coefficient (Φ ) for the ternary amine is larger than that of secondary and primary amines and the deviation from ideal behavior depends on the molecular structure of the amines. The deviation from ideal behavior increases from methanol to -propanol and from formic acid to propionic acid. This is seen from values of the surface tension coefficients (Φ ) for alcohols and organic acids presented in Fig.2. By plotting the excess surface tension ( E ) versus x in Fig.3, the deviation from the ideal behavior for aqueous solutions of methanol, ethanol, -propanol and 2-propanol at 2 C is also shown. The points indicate the excess surface tension E is obtained directly by using experimental data in Eq. (3). The lines indicate the calculated E by inserting the fixed values of parameters a and b as reported in Table 2 in Eq. (4). 2 Φ 6 2 8 ME DE TE Formic cid cetic cid Propionic cid Methanol Ethanol -Propanol 2-Propanol 4.2.4.6.8 x Fig. 2. The variation of Φ versus x for indicated aqueous solutions at 25 C. The points are the experimental data and the lines are the calculated values of Φ Iranian Journal of Science & Technology, Volume 29, Number B5 October 25
new and a simple model for surface tension prediction of 57.2.4.6.8 - E (mn/m) -2-3 -4-5 -Propanol 2-Propanol Ethanol Methanol x Fig. 3. The variation of E versus x for indicated aqueous solutions 2 C. The points are the experimental data and the lines are the calculated values of Figure 4 shows the variations of both excess reduced surface tension and reduced surface tension, versus for aqueous solution of formic acid at 2 and 5 C. The points are obtained by using the experimental data in Eqs. (5) and (7). The lines indicate the calculated and, by inserting the fixed values of the parameters a and b as reported in Table 2 in Eqs. (5) and (7). Figure 5 shows the same variations as Fig. 4, which, and versus for aqueous solution 2-amino-2-methyl-- propanol (MP) at 2 and 5 C. E Formic cid + ater. MP + ater.8 Ideal Solution.8 Ideal Solution,.6.4 ()real 2 C () 2 C ()real 5 C () 5 C,.6.4 ()real 2 C () 2 C ()real 5 C () 5 C.2.2.2.4.6.8 Fig. 4. The variations of and versus x aqueous solution of formic acid. The points are the experimental data and the lines are calculated values of and for..2.4.6.8 Fig. 5. The variations of and versus x for aqueous solution of 2-mino-2-methyl--propanol (MP). The points are the experimental data and the lines are the calculated values of and Figure 6 illustrates the agreement between the experimental values of the surface tension for the aqueous solution of monoethanolamine (ME) at 25, 35, 45 C and the calculated values. The points are experimental data and the lines are obtained by inserting the fixed values of the parameters a and b as October 25 Iranian Journal of Science & Technology, Volume 29, Number B5
58 R. Tahery / H. Modarress reported in Table 2 in Eqs. (5) and (8). Figure 7 shows the same variation as Figure 6, that is, versus x for aqueous solution triethanolamine (TE) at 25, 35, 45 C. 75 ME+ater 75 TE+ater (mn/m) 65 55 25 C 35 C 45 C (mn/m) 65 55 25 C 35 C 45 C 45 45..2.4.6.8. 35..2.4.6.8. Fig. 6. The variations of versus x for aqueous solution of monoethanolamine at 25, 35, 45 C. The points are experimental data and the lines are calculated values Fig. 7. The variations of versus x for aqueous solution of triethanolamine at 25, 35, 45 C. The points are experimental data and the lines are calculated values 4. CONCLUSIONS new surface tension equation is proposed for binary aqueous-organic solutions, which is different from the existing equations proposed by the other authors. The parameters of this equation have been calculated for 5 binary aqueous-organic solutions. new property named surface tension coefficient is defined and calculated for the studied solutions and is used as an indication of deviations from ideal behavior. The overall D% for the correlation of surface tensions by the new equation is obtained as.53%, which is lower than the D% obtained by two similar equations proposed by other workers (Zihao et al. [7] and Zhibao and Benjamin [8]). lthough comparing with the D% obtained by one of the equations (Connors and right [9]), the new equation has a slightly higher D%, it is superior due to the fact that its parameters are independent of temperature, and therefore it has higher predictability. D verage absolute deviation a and b Coefficients of Eq. (9) n Number of experimental data T x Temperature (K) Mole fraction Greek symbols Φ Surface tension coefficient defined by Eq. (5) Γ Characteristic parameter defined by Eq. () NOMENCLTURE Iranian Journal of Science & Technology, Volume 29, Number B5 October 25
new and a simple model for surface tension prediction of 59 Surface tension (mn/m) Excess reduced quantity Superscripts Reduced quantity Subscripts Organic component Br Reduced quantity at boiling point C Thermodynamic properties at critical point E Excess property ater REFERENCES. Schuchowitzky,. (944). cta physicochim. URSS, 9, 76, 58. 2. Belton, J.. & Evans, M. G. (945). Trans. Faraday Soc., 4,. 3. Guggenheim, E.. (945). Trans. Faraday Soc., 4, 5. 4. Hoar, T. P. & Melford, D.. (957). Trans. Faraday Soc., 53, 35. 5. Sprow, F. B. & Prausnitz, J. M. (966). Trans. Faraday Soc., 62, 5. 6. Semenchenko, V. K. (973). Russ. J. Phys. Chem., 47, 63. 7. Feinerman,. E. (974). Colloid Polym. Sci., 252, 582. 8. Goldsack, D. E. & Sarvas, C. D. (98). Can. J. Chem. Eng., 59, 2968. 9. Connors, K.. & right, J. L. (989). nal. Chem., 6, 94.. lvarez, E., Vazquez, G. & Navaza, J. M. (995). J. Chem. Eng. Data, 4, 6.. lvarez, E., Vazquez, G., Rendo, R., Romero, E. & Navaza, J. M. (996). J. Chem. Eng. Data, 4, 86. 2. lvarez, E., Vazquez, G., Navaza, J. M., Rendo, R. & Romero, E. (997). J. Chem. Eng. Data, 42, 57. 3. lvarez, E., Vazquez, G., Sanchez-Vilas, M., Sanjurjo, B. & Navaza, J. M. (997). J. Chem. Eng. Data, 42, 957. 4. lvarez, E., Rendo, R., Sanjurio, B., Sanchez-Vilas, M. & Navaza, J. M. (998). J. Chem. Eng, Data, 43, 27. 5. lvarez, E., Cancela,., Maceiras, R., Navaza J. M. & Taboas, R. (23). J. Chem. Eng. Data, 48, 32. 6. Lee, J.., Park, S. B. & Lee, H. (2). J. Chem. Eng. Data, 45, 66. 7. Zihao,., enchuan,. & Chunxi, L. (2). Fluid Phase Equil., 75, 85. 8. Zhibao, L., Benjamin, C. & Lu, Y. (2). Can. J. Chem. Eng., 79, 42. 9. Davis, H. T. (975). J. Chem. Phys., 62, 342. 2. Hakim, D. I., Steinberg, D. & Stiel, L. I. (97). Ind. Eng. Chem. Fundam.,,74. 2. Brock, J. R. & Bird, B. (955). IChE J.,, 74. October 25 Iranian Journal of Science & Technology, Volume 29, Number B5