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0 0 1.3E−02 399.85 0.98 0.021 0.0009 0 0 2.3E−02 402.25 0.98 0.022 0.0009 0 0 2.4E−02 404.05 0.98 0.015 0.0006 0 0 1.6E−02 409.15 0.97 0.03 0.0013 0.0001 0 3.3E−02

      The coexisting liquid compositions are given in Table 2.4 for each temperature reported by Vu [31].

Comparison of the two data sets is informative, but the Sanz et al. data should be corrected for the missing oligomers. Only approximate analysis can be performed from the published data. The oligomerization model equations can be used to convert the Sanz et al. data for comparison. The relation suggests a method to determine p, but the p values found using the Sanz et al. data with the relation range from 0.23 to 0.29 when the p values should be increasing with lactic acid concentration as in Figure 2.6. Values determined using the Sanz et al. data with the alternative relation are reasonable and range from 0.007 to 0.64 when the value K = 0.2023 is used. Thus, for the Sanz et al. data, the apparent composition can be calculated from the reported x _W and using Equation 2.10 with K = 0.2023 to determine p, and then using Equations 2.14 and 2.17 to find the apparent moles of lactic acid and water, from which the apparent mole fractions are calculated and plotted in Figure 2.7.

A similar analysis can be performed on the data of Vu. Using Equation 2.10, the p values from the data range from 0.05 to 0.22. Analysis using and comparison with shows a reasonable parity as shown in Figure 2.6, indicating that the data are much more consistent with equilibria. A comparison of the data of Vu with the manipulated data of Sanz et al. is shown in Figure 2.7. The Vu data at high concentrations are more scattered indicating some control issues with the recirculating apparatus. The data of Sanz et al. vary smoothly to higher concentrations indicating better circulation control. Though both data sets have some shortcomings, the agreement is remarkable for the complex experimental system. The data provide reasonable approximations for practitioners.

FIGURE 2.6 Parity plot of p values determined by two methods as explained in the text for data of Vu.

FIGURE 2.7 VLE data for lactic acid + water at 101.33 kPa as reported by Vu compared to the manipulated data of Sanz et al. Data of Vu are filled symbols with triangles for the vapor phase and squares for the liquid phase.

2.7 DENSITY OF AQUEOUS SOLUTIONS

Holten [28] fitted density data from Troupe et al. [32] using a third‐order polynomial with each wt% fitted separately. The individual fits represent the experimental data within 0.2%. To provide practitioners a more rapid calculation, we have refitted the data of Troupe et al. using Equation 2.26 where coefficients A and B are linear functions of the apparent wt% lactic acid, w. The calculated densities, ρ , match experimental data within 0.3% as shown in Figure 2.8.

(2.26)

      (2.27)

(2.28)

2.8 VISCOSITY OF AQUEOUS SOLUTIONS

Troupe et al. [32] observed that the viscosity increased over time as did the titratable acidity when a 75 apparent wt% lactic acid sample was freshly prepared from 85 wt%. At room temperature, it took more the 20 days to reach equilibrium. Equilibration of solutions is critical to obtain reliable viscosity data, and equilibration is slow at 20°C. Suggested equilibration times are provided by Vu et al. [27]. Troupe et al. fitted their viscosity data with Equation 2.29 where each wt% was fitted separately. The individual fits represent the experiments within 5%. For practitioners, we provide a rapid way to interpolate the viscosity. We have regressed the coefficients A and B for the common log of viscosity

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