Question

A membrane process is being designed to recover solute A by dialysis through a membrane.

The solute moves from a dilute feed solution (c1 =5 x 10-2 kg mol A/m3) to a fresh dialysate (permeate) solution (c2 = 0.3 x 10-2). The membrane thickness is 30 µm.

You also need to know that the distribution coefficient is K’ = 0.75, Ds = 3.5 x 10-11 m2/s in the membrane, the mass-transfer coefficient in the dilute solution is kc1 = 2.5 x 10-6 m/s, and kc2 = 1.9 x 10-5.

a. Calculate the individual resistances for each section of this system for mass transfer, and then calculate the total resistance for mass transfer.

b. Calculate the flux of the solute at steady state.

c. Find the total area needed (in m2) to achieve a rate of 0.01 kg mol solute/h

Answer 1

A) to find individual resistances

K_c1 = 2.5×10^-6 m /s

K_c2 = 1.9×10^-5

The resistance in dilute solution

= 1/K_c1 = 1/(2.5×10^-6) = 400000 s/m

K' = 0.75

The resistance in concentrated solution

= 1/K_c2 = 1/(1.9×10^-5) = 52631.5 s/m

To find overall resistance

1/K = (1/k_c1) +(0.75)(Kc₂)

1/K = 400000 + 0.75(52631.5) = 439473.625 s/m

K = 2.275 ×10^-6 m/s

B)

To find mass flux

N_A = D_s(C₁-C₂) /X

N_A = (3.5×10^-11) (0.05-0.003) /(30×10^-6)

N_A = 5.4833 ×10^-8 Kmol/m²s = 1.973×10^-4 Kmol/m²h

C)

Area needed to achieve 0.01 Kmol/h

0.01/(1.973×10^-4) = 50.684 m²

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