13.5. COMPLETE-MIXING MODEL FOR MULTICOMPONENT MIXTURES

13.5A. Derivation of Equations

When multicomponent mixtures are present, the iteration method of Stern et al. (S1) is quite useful. This method will be derived for a ternary mixture of components A, B, and C. The process flow diagram is the same as Fig. 13.4-1, where the feed composition x_f is x_fA, x_fB, and x_fC. The known values are

The unknown values to be determined are

These eight unknowns can be obtained by solving a set of eight simultaneous equations using an iteration method. Three rate-of-permeation equations similar to Eq. (13.4-3) are as follows for components A, B, and C:

Equation 13.5-1

Equation 13.5-2

Equation 13.5-3

The three material-balance equations similar to Eq. (13.4-9) are written for components A, B, and C:

Equation 13.5-4

Equation 13.5-5

Equation 13.5-6

Also, two final equations can be written as

Equation 13.5-7

Equation 13.5-8

Substituting x_oA from Eq. (13.5-4) into Eq. (13.5-1) and solving for A_m,

Equation 13.5-9

For component B, Eq. (13.5-5) is substituted into Eq. (13.5-2), giving

Equation 13.5-10

Rearranging Eq. (13.5-10) and solving for y_pB,

Equation 13.5-11

In a similar manner Eq. (13.5-12) is derived for y_pC:

Equation 13.5-12

13.5B. Iteration Solution Procedure for Multicomponent Mixtures

The following iteration or trial-and-error procedure can be used to solve the equations above:

1.	A value of ypA is assumed where ypA > xfA.
2.	Using Eq. (13.4-2) and the known value of θ, Vp is calculated.
3.	The membrane area is calculated from Eq. (13.5-9).
4.	Values of ypB and ypC are calculated from Eqs. (13.5-11) and (13.5-12).
5.	The sum Σn ypn is calculated from Eq. (13.5-7). If this sum is not equal to 1.0. steps 1 through 5 are repeated until the sum is 1.0.
6.	Finally, xoA, xoB, and xoC are calculated from Eqs. (13.5-4), (13.5-5), and (13.5-6).

EXAMPLE 13.5-1. Design of Membrane Unit for Multicomponent Mixture A multicomponent gaseous mixture having a composition of xfA = 0.25, xfB = 0.55, and xfC = 0.20 is to be separated by a membrane with a thickness of 2.54 X 10-3 cm using the complete-mixing model. The feed flow rate is 1.0 × 104 cm3 (STP)/s and the permeabilities are = 200 × 10-10 cm3 (STP) · cm/(s · cm2 · cm Hg), = 50 × 10-10, and = 25 × 10-10. The pressure on the feed side is 300 cm Hg and 30 cm Hg on the permeate side. The fraction permeated will be 0.25. Calculate the permeate composition, reject composition, and membrane area using the complete-mixing model. Solution: Following the iteration procedure, a value of ypA = 0.50 is assumed. Substituting into Eq. (13.4-2) for step 2, Using Eq. (13.5-9), the membrane area for step (3) is Following step 4, the values ypB and ypC are calculated using Eqs. (13.5-11) and (13.5-12): [View full size image] [View full size image] Substituting into Eq. (13.5-7), For the second iteration, assuming that ypA = 0.45, the following values are calculated: The final iteration values are Am = 3.536 × 106 cm2; ypA = 0.4555, ypB = 0.4502, and ypC = 0.0943. Substituting into Eqs. (13.5-4), (13.5-5), and (13.5-6),