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Damelys Zabala and Aurélie Wender / Revista Ingeniería UC , Vol. 23, No. 3, Diciembre 2016, 237-246 241 10. Calculate the concentration profile for the reference component, “ ρ s (z) ” with equation (22). ∑nc [ ( J = ρi µi (ρ i ) − µ e )] i − (P (ρ) − P e ) i=1 Y i, j = ( ) 2 dρs = dz ∑ nc i=1 [ ( ) ( )] dρi dρ j c i j dρ s dρ s 2J ∑ nc (22) j=1 Y i, j If (dρ s /dz≥0), the concentration profile of the reference component (ρ s ) is monotonic as it was stated for doing the independent variable change according equation (3), the interface position can be calculated from step 11. On the contrary, other reference component has to be selected and the calculation process must be repeated from step 7. 11. The interface position is calculated with (dρ s /dz) = B. For example, for the first point z 1 from an arbitrary origin z 0 , using the equation (23). ∫ z1 ∫ ρs1 1 dz = z0 ρ s0 B dρ s (23) By the trapezoid method, the equation (24) is obtained: z1 − z0 = (ρ ( s1 − ρ s0 ) 1 + 1 ) (24) 2 B 0 B 1 The general expression for the interface position “z” at step “k” is in equation (25): z k = z k−1 + ∆ρ ( s 1 + 1 ) (25) 2 B k B k−1 12. Calculate the interfacial tension “σ”, with the equation (26) (if the concentration profile of the reference component from the step 10 is monotonic): ∑nc [ ( J = ρi µi (ρ i ) − µ e )] i − (P (ρ) − P e ) i=1 σ = ∫ ρ L s ρ V s Y i, j = [ ( ) ( )] dρi dρ j c i j dρ s dρ s √ 2J ∑ nc ∑ nc i=1 j=1 Y i, j dρ s (26) 3. Analysis and Discussion of Results Two binary and one ternary mixtures were studied using the Peng-Robinson (PR) equation of state. The pure components are n-butane (nC 4 ), n- decane (nC 10 ) and carbon dioxide (CO 2 ). Table 1: T c , P c , acentric factor of pure components. Component(∗) T c [K] P c [Pa] w CO 2 304.21 7.383 × 10 6 0.223621 nC 4 425.12 3.796 × 10 6 0.200164 nC 10 617.7 2.110 × 10 6 0.492328 (*): Experimental value from Diadem Pro Gold R 3.1. Binary mixtures The critical properties of pure components (critical temperature T c , critical pressure P c and acentric factor w) are shown in Table 1. The expressions of the saturation pressure P sat i (T) and saturation density ρ sat i (T) of pure components can be calculated with equations (27) and (28): ρ sat i (T) = P sat i B A (1+(1− T C) D) (27) (T) = exp [A + B ] T + ClnT + DT E (28) The parameters A, B, C, D and E are shown in Table 2 and 3. Table 2: Parameters of saturation liquid pressure of pure components. Saturation pressure CO 2 nC 4 nC 10 (Pa) A 47.0169 66.343 112.73 B -2839 -4363.2 -9749.6 C -3.86388 -7.046 -13.245 D 2.811 15 × 10 −16 0.0000094509 0.0000071266 E 6 2 2 Values from Diadem Pro Gold R The calculated influence parameters for the pure components used in the mixtures are shown in Table 4. Revista Ingeniería UC, ISSN: 1316–6832, Facultad de Ingeniería, Universidad de Carabobo.
242 Damelys Zabala and Aurélie Wender / Revista Ingeniería UC , Vol. 23, No. 3, Diciembre 2016, 237-246 Table 3: Parameters of saturation liquid density of pure components Saturation density CO 2 nC 4 nC 10 (kmol/m³) A 2.768 1.0677 0.41084 B 0.26212 0.27188 0.25175 C 304.21 425.12 617.7 D 0.2908 0.28688 0.28571 E 0 0 0 Values from Diadem Pro Gold R Table 4: Calculated influence parameter c i of pure components (PR). Figura 1: Interfacial tension vs pressure at 344.3K, (CO 2 /nC 10 mixture, PR). Experimental data [16]. Component Tref [K] σ pure [mN/m] (*) c i [Jm 5 /kmol 2 ] CO 2 216.58 (**) 16.6924 2.6147E-14 nC 4 272.65 14.90103 1.75337E-13 nC 10 447.305 10.26487 1.2415E-12 (*): Experimental value from Diadem Pro Gold R (**): TPT is used for CO 2 instead NBP of this cubic equation for the estimation of the liquid equilibrium values at high pressure. The monotonic trend of the density profile inside the interface, for the n-decane is observed in Figure 2. This behavior confirms the correct choice of the nC 10 as the reference component. Mixture CO 2 /nC 10 at T = 344.3K. For this mixture, the PR binary interaction parameter ki j=0.12 was used. The IFT was calculated at four different pressures. The results are shown in Table 5 and in Figures 1, 2 and 3. Table 5: Interfacial tension vs pressure at 344.3K for CO 2 /nC 10 mixture (PR). P(bar) IFT [mN/m] 69.4 7.24 103.4 2.77 117.3 1.37 125.5 0.75 Figure 1 shows that the IFT values obtained by the density gradient theory for the CO 2 /nC 10 mixture are very close to the experimental data with an absolute average deviation (AAD) below 10 %, when the experimental values are above 1 mN/m. For low experimental IFT values, the AAD increases abruptly over 60 %. This behavior was obtained also by other author [11] working DGT with RKS equation of state with a binary mixture methane + butane and it was stated the limitation Figura 2: Density profiles at 344.3K and different pressures, for reference component, nC 10 , (CO 2 /nC 10 mixture, PR). Mixture CO 2 /nC 4 at T = 344.3K. For this mixture, the PR binary interaction parameter kij=0.15 was used. The IFT was calculated at three different pressures. The results are shown in Table 6 and in Figures 4, 5 and 6. In Figure 4, the IFT values obtained by DGT for the CO 2 /nC 4 mixture are close to the experimental data (15 %), but there is a bigger deviation (24 %) in the interfacial tension value below 1mN/m, like in the previous case. The monotonic trend of the density profile for the n-butane inside the interface Revista Ingeniería UC, ISSN: 1316–6832, Facultad de Ingeniería, Universidad de Carabobo.
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