Determine the modified Sherwood number (mathrm{Sh}_{text {gas,partial_pressure }}=frac{mathrm{k}_{mathrm{p}} mathrm{d}_{text {tube }}left(mathrm{p}_{mathrm{B}} ight)_{mathrm{lm}}}{D_{mathrm{AB}} mathrm{p}_{text {tot }}}) for gas-side-controlled

Determine the modified Sherwood number \(\mathrm{Sh}_{\text {gas,partial_pressure }}=\frac{\mathrm{k}_{\mathrm{p}} \mathrm{d}_{\text {tube }}\left(\mathrm{p}_{\mathrm{B}}\right)_{\mathrm{lm}}}{D_{\mathrm{AB}} \mathrm{p}_{\text {tot }}}\) for gas-side-controlled mass transfer for distillation of ethanol-water in a wetted wall column. Column diameter is \(10.0 \mathrm{~cm}\). Measurements are at very low ethanol concentrations where flowing liquid can be assumed to be pure water. Total pressure is 1.0 bar. Liquid water at its boiling temperature is flowing down a vertical column at a volumetric flow rate per meter of circumference of \(q=0.0000075 \mathrm{~m}^{2} / \mathrm{s}\). Ethanol is diffusing through an upward-flowing vapor (almost pure water) at 1.0 bar and its boiling temperature. The upward vapor velocity is \(0.81 \mathrm{~m} / \mathrm{s}\). Densities and viscosities of pure water liquid and pure water vapor are available in Perry's Handbook of Chemical Engineering. Use parameters in Table 15-2 to estimate diffusivity of ethanol-water in the vapor.

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TABLE 15-2. Lennard-Jones potential parameters and values of the collision integrals for ideal Fickian gas diffusivity calculation with the Chapman-Enskog equation (15-21b) Compound () /KB (K) KBT/ Air 3.711 78.6 0.30 2.662 Ammonia 2.900 558.3 0.50 2.066 0.9154 Benzene 5.349 412.3 0.75 1.667 0.9055 2-butane 5.278 330.1 1.00 1.439 0.9067 1-butane 4.687 531.4 1.50 1.198 0.9117 Carbon dioxide 3.941 195.2 2.00 1.075 0.9149 Ethane 4.443 215.7 2.5 0.9996 0.9145 Ethanol 4.530 362.6 5.0 0.8422 0.9086 Methanol 3.626 481.8 10 0.7424 0.9007 Propane 5.118 237.1 40 0.5960 0.8872 Water 2.641 809.1 100 0.5130 0.8722