Question 30.3: Trichloroethylene (TCE), a common industrial solvent, is oft......
Trichloroethylene (TCE), a common industrial solvent, is often found at low concentrations in industrial waste waters. Stripping is a common process for removing sparingly soluble, volatile organic solutes such as TCE from aqueous solution. A wetted-wall column is used to study the stripping of TCE from water to air at a constant temperature of 293 K and total system pressure of 1.0 atm. The column inner diameter is 4.0 cm, and the height is 2.0 m. In the present process, the volumetric air flow rate into the column is 500\ \mathrm{cm}^{3}/\mathrm{s}\left(5.0\times10^{-4}\,\mathrm{m}^{3}/\mathrm{s}\right), and the volumetric flow rate of water is 50\operatorname{cm}^{3}\!/\!\operatorname{s}\ (5.0\times10^{-5}\,\mathrm{m}^{3}\!/\!\mathbf{s}). At 293 K, the density of liquid water is 998.2\,{\mathrm{kg/m}}^{3}, and so the mass flow rate of water wetting the column is w = 0.05 kg/s. Estimate K_{L}, the overall liquid-phase mass-transfer coefficient for TCE across the liquid and gas film. Assume that water loss by evaporation is negligible.
Relevant physical property data are provided below. The process is very dilute so that the bulk gas has the properties of air and the bulk liquid has the properties of water. The equilibrium solubility of TCE in water is described by Henry’s law of the form
p_{A, i}=H x_{A, i}
where H is 550 atm at 293 K. The binary gas-phase diffusivity of TCE in air is 8.0\times10^{-6}\,{\mathrm{m}}^{2}/{\mathrm{s}} at 1.0 atm and 293 K, as determined by the Fuller–Shettler–Giddings correlation. The binary liquid-phase diffusivity of TCE in water at 293 K is 8.9\times10^{-10}\,{\mathrm{m}}^{2}/{\mathrm{s}}, as determined by the Hayduk–Laudie correlation.
With this physical property information in hand, our strategy is to estimate the gas–film coefficient k_{G}, the liquid film coefficient k_{L}, and then the overall mass-transfer coefficient K_{L}. First, the bulk velocity of the gas is
\displaystyle\mathrm{v}_{\infty }=\frac{4Q_{g}}{\pi D^{2}} = \frac{4\left(5.0\times10^{-4}\frac{{\mathrm{m}}^{3}}{\mathrm{s}}\right)}{\pi(0.04\,{\mathrm{m}})^{2}} = =0.40\,{\mathrm{m/s}}
The Reynolds number for air flow through the inside of the wetted-wall column is
\mathrm{Re}={\frac{\rho_{\mathrm{air}}\mathrm{v}_{\mathrm{\infty }}D}{\mu_{\mathrm{air}}}}={\frac{\left(1.19{\frac{\mathrm{kg}}{\mathrm{m}^{3}}}\right)\left(0.40{\frac{\mathrm{m}}{\mathrm{s}}}\right)(0.04\,\mathrm{m})}{\left(1.84\times10^{-5}{\frac{\mathrm{kg}}{\mathrm{m}\cdot\mathrm{s}}}\right)}}=1035
and the Schmidt number for TCE in air is
\mathrm{Sc}={\frac{\mu_{\mathrm{air}}}{\rho_{\mathrm{air}}D_{\mathrm{TCE-air}}}} = \frac{1.84\times10^{-5}\frac{\mathrm{k }}{\mathrm{m.s}}}{\left(1.19\frac{\mathrm{kg}}{\mathrm{m^{3}}}\right)\left(8.08\times10^{-6}\frac{\mathrm{m^{2}}}{\mathrm{s}}\right)} = 1.91
where the properties of air are found from Appendix I. As the gas flow is laminar ({\mathrm{Re}}\lt 2000), equation (30-21) for laminar flow inside a pipe is appropriate for estimation of k_{c}. Therefore,
\mathrm{Sh}=1.86\left(\frac{\mathrm{v_{\infty }}D^{2}}{L D_{A B}}\right)^{1/3}=1.86\left(\frac{D{\mathrm{v_{\infty }}}}{L}\frac{D\nu}{\nu D_{A B}}\right)^{1/3}=1.86\left(\frac{D}{L}\mathrm{Re}\mathrm{~Sc}\right)^{1/3} (30-21)
k_{c}={\frac{D_{A B}}{D}}1.86\left({\frac{D}{L}}\mathrm{Re}\ \mathrm{Sc}\right)^{1/3}={\frac{8.0\times10^{-6}\ {\mathrm{m}}^{2}/s}{0.04\,\mathrm{m}}}1.86 \left(\frac{0.04\;\mathrm{m}}{2.0\,\mathrm{m}}(1035)(1.91)\right)^{1/3}=1.27\times10^{-3}\;\mathrm{m}/s
The conversion to k_{G} is
k_{G}={\frac{k_{c}}{R T}}={\frac{1.27\times10^{-3}{\frac{\mathrm{m}}{\mathrm{s}}}}{\left(0.08206{\frac{\mathrm{m}^{3}\cdot\mathrm{atm}}{\mathrm{kgmole}\cdot\mathrm{K}}}\right)(293\,\mathrm{K})}}=5.28\times10^{-5}{\frac{\mathrm{kgmole}}{\mathrm{m}^{2}\cdot\mathrm{s}\cdot\mathrm{atm}}}
The liquid–film mass-transfer coefficient is now estimated. The Reynolds number for the falling liquid film is
\mathrm{Re}_{L}={\frac{4w}{\pi D\mu_{L}}}={\frac{4(0.05\,\mathrm{kg/s})}{\pi(0.04\,\mathrm{m})(9.93\times10^{-4}\,\mathrm{kg/m}\cdot s)}}=1600
and the Schmidt number is
\mathrm{Sc}={\frac{\mu_{L}}{\rho_{L}D_{\mathrm{TCE-H_{2}O}}}}={\frac{\left(9.93\times10^{-4}{\frac{\mathrm{kg}}{\mathrm{m}\times s}}\right)}{\left(998.2{\frac{\mathrm{kg}}{\mathrm{m^{3}}}}\right)\left(8.90\times10^{-10}{\frac{\mathrm{m^{2}}}{s}}\right)}}=1118
where the properties of liquid water at 293 K are found in Appendix I. Equation (30-22) is appropriate for estimation of k_{L} for the falling liquid film inside the wetted-wall column:
\mathrm{Sh}={\frac{k_{L}z}{D_{A B}}}=0.433(\mathrm{Sc})^{1/2}\left({\frac{\rho_{L}^{2}\,g\,z^{3}}{\mu_{L}^{2}}}\right)^{1/6}(\mathrm{Re}_{L})^{0.4} (30-22)
k_{L}=\frac{D_{A B}}{z}0.433(\mathrm{Sc})^{1/2}\left(\frac{\rho_{L}^{2}g\,{z}^{3}}{\mu_{L}^{2}}\right)^{1/6}(\mathrm{Re}_{L})^{0.4}
=\frac{8.9\times10^{-10}{\frac{{\mathrm{m}}^{2}}{\mathrm{s}}}}{2.0\ \mathrm{m}}0.433(1118)^{1/2}\left(\frac{\left(998.2\frac{\mathrm{kg}}{\mathrm{m}^{3}}\right)^{2}\left(9.8\frac{\mathrm{m}}{\mathrm{s}^{2}}\right)(2.0\,\mathrm{m})^{3}}{\left(9.93\times10^{-4}\frac{\mathrm{kg}}{\mathrm{m}\cdot\mathrm{s}}\right)^{2}}\right)^{1/6}(1600)^{0.4}
=2.55\times10^{-5}\,{\mathrm{m}}/{\mathrm{s}}
Since the process is dilute, the Henry’s law constant in units consistent with k_{L} and k_{G} is
H=550\operatorname{atm}{\frac{\operatorname{M}_{H_{2}O}}{\rho_{L,H_{2}O}}}=\left(550\operatorname{atm}\right)\left({\frac{18~k g/\ g m o l e}{998.2\,\operatorname{kg/m^{3}}}}\right)=9.92{\frac{{\mathfrak{m}}^{3}\cdot{\mathrm{atm}}}{\mathrm{kgmole}}}
The overall liquid-phase mass-transfer coefficient, K_{L}, is estimated by equation (29-22):
\frac{1}{K_{L}}=\frac{1}{H\cdot k_{G}}+\frac{1}{k_{L}} (29-22)
{\frac{1}{K_{L}}}={\frac{1}{k_{L}}}+{\frac{1}{H k_{G}}}={\frac{1}{2.55\times10^{-5}{\frac{\mathrm{m}}{s}}}}+{\frac{1}{\left(9.92{\frac{\mathrm{atm}\cdot{\mathrm{m}}^{3}}{\mathrm{kgmole}}}\right)\left(5.28\times10^{-5}{\frac{\mathrm{kgmole}}{{\mathrm{m}}^{2}\cdot{\mathrm{s}}\cdot{\mathrm{atm}}}}\right)}}
or K_{L}=2.43\times10^{-5}\ \mathrm{m/s}, Since K_{L}\longrightarrow k_{L}, the process is liquid-phase mass-transfer controlling, which is characteristic of interphase mass-transfer processes involving a large value for Henry’s law constant.