A cross-flow molecular filtration device equipped with a mesoporous membrane is used to separate the enzyme lysozyme from a fermentation broth, as shown in the figure (right column). Water at 25°C flows over the top surface of the flat plate membrane at a velocity of 5.0 cm/s. The length of the membrane in the direction of flow is 10 cm, and the thickness of the membrane (e) is 0.10 cm. Uniform parallel pores of 30 nm diameter run down thickness of the membrane, and 30% of the surface area of the membrane is occupied by the pore openings. The dissolved enzyme diffuses through the pores of the membrane and across the boundary layer of the flowing fluid, so that there are two mass- transfer resistances in series. It is desired to understand how the fluid boundary layer affects the overall flux of enzyme across the membrane. At 25°C, the molecular diffusion coefficient of the lysozyme enzyme in water (D_AB)is 1.04 × 10^-6cm²/s^*, and the molecular weight (M_A) is 14,100 g/gmole. For diffusion of the 4.12-nm diameter lysozyme through the 30-nm diameter pores, the effective diffusion coefficient of lysozyme through the membrane (D_Ae) is 5.54 × 10^-7cm²/s, using the Renkin equation for hindered solute diffusion in solvent-filled pores, equation (24-62).

a. Show that the steady-state flux across the membrane and boundary layer is given by 

b. Estimate the average molar flux of the enzyme across the membrane and boundary layer, assuming the enzyme concentration at the bottom surface of the membrane (c_Ao) is maintained at 1.0 mmole/m³ and the bulk concentration of enzyme in the flowing liquid phase (c_A,ˆž) is maintained at 0.40 mmole/m³. Also, estimate the flux for two limiting cases, one where convection mass-transfer limits the process, and another where molecular diffusion through the membrane limits the process.
c. What is the Biot number (Bi) for the enzyme mass-transfer process? Reflect on the importance of convective mass transfer.
d. How does the maximum thickness of the hydrodynamic boundary layer compare to the thickness of the membrane?

Transcribed Image Text:

DAe = DA-H,0F1(44)F2(4a) = 1.04 x 10-6 Cm (0.744)(0.716) = 5.54 × 10- cm? (CA0-CA,0) NA (28-66) ke' DAe