13.6. CROSS-FLOW MODEL FOR GAS SEPARATION BY MEMBRANES

13.6A. Derivation of Basic Equations

A detailed flow diagram for the cross-flow model derived by Weller and Steiner (W3, W4) is shown in Fig. 13.6-1. In this case the longitudinal velocity of the high-pressure or reject stream is large enough that this gas stream is in plug flow and flows parallel to the membrane. On the low-pressure side the permeate stream is almost pulled into vacuum, so that the flow is essentially perpendicular to the membrane.

This model assumes no mixing in the permeate side as well as no mixing on the high-pressure side. Hence, the permeate composition at any point along the membrane is determined by the relative rates of permeation of the feed components at that point. This cross-flow pattern approximates that in an actual spiral-wound membrane separator (Fig. 13.3-1) with a high-flux asymmetric membrane resting on a porous felt support (P2, R1).

Referring to Fig. 13.6-1, the local permeation rate over a differential membrane area dA_m at any point in the stage is

Equation 13.6-1

Equation 13.6-2

where dL = dV and is the total flow rate permeating through the area dA_m. Dividing Eq. (13.6-1) by (13.6-2) gives

Equation 13.6-3

This equation relates the permeate composition y to the reject composition x at a point along the path. It is similar to Eq. (13.4-6) for complete mixing. Hwang and Kammermeyer (H1) give a computer program for the solution of the above system of differential equations by numerical methods.

Weller and Steiner (W3, W4) used some ingenious transformations and were able to obtain an analytical solution to the three equations as follows:

Equation 13.6-4

where

The term u_f is the value of u at i = i_f = x_f/(1 - x_f). The value of θ* is the fraction permeated up to the value of x in Fig. 13.6-1. At the outlet where x = x₀, the value of θ* is equal to θ, the total fraction permeated. The composition of the exit permeate stream is y_p and is calculated from the overall material balance, Eq. (13.4-9).

The total membrane area was obtained by Weller and Steiner (W3, W4) using some additional transformations of Eqs. (13.6-1)-(13.6-3) to give

Equation 13.6-5

where

Values of θ* in the integral can be obtained from Eq. (13.6-4). The integral can be calculated numerically. The term i_f is the value of i at the feed x_f, and i_o is the value of i at the outlet x_o. A shortcut approximation of the area without using a numerical integration, available from Weller and Steiner (W3), has a maximum error of about 20%.

13.6B. Procedure for Design of Cross-Flow Case

In the design for the complete-mixing model there are seven variables, and two of the most common cases were discussed in Section 13.4B. Similarly, for the cross-flow model these same common cases occur.

Case 1

The values of x_f, x_o, α*, and p_l/p_h are given and y_p, θ and A_m are to be determined. The value of θ* or θ can be calculated directly from Eq. (13.6-4) since all other values in this equation are known. Then y_p is calculated from Eq. (13.4-9). To calculate the area A_m, a series of values of x less than the feed x_f and greater than the reject outlet x_o are substituted into Eq. (13.6-4) to give a series of θ* values. These values are then used to numerically or graphically integrate Eq. (13.6-5) to obtain the area A_m.

Case 2

In this case the values of x_f, θ, α*, and p_l/p_h are given and y_p, x_o, and A_m are to be determined. This is trial and error, where values of x_o are substituted into Eq. (13.6-4) to solve the equation. The membrane area is calculated as in Case 1.

EXAMPLE 13.6-1. Design of a Membrane Unit Using Cross-Flow The same conditions for the separation of an air stream as given in Example 13.4-2 for complete mixing are to be used in this example. The process flow streams will be in cross-flow. The given values are xf = 0.209, θ = 0.20, a* = 10, ph = 190 cm Hg, pl = 19 cm Hg, Lf = 1 × 106 cm3 (STP)/s, = 500 × 10-10 cm3 (STP) · cm/(s · cm2 · cm Hg), and t = 2.54 × 10-3 cm. Do as follows: Calculate yp, xo, and Am. Compare the results with Example 13.4-2. Solution: Since this is the same as Case 2, a value of xo = 0.1642 will be used for the first trial for part (a). Substituting into Eq. (13.6-4), Solving, θ* = 0.0992. This value of 0.0992 does not agree with the given value of θ = 0.200. However, these values can be used later to solve Eq. (13.6-5). For the second iteration, a value of xo = 0.142 is assumed and is used again to solve for θ* in Eq. (13.6-4), which results in θ* = 0.1482. For the final iteration, xo = 0.1190 and θ* = θ = 0.2000. Several more values are calculated for later use and are: for xo = 0.187, θ* = 0.04876, and for xo = 0.209, θ* = 0. These values are tabulated in Table 13.6-1. Table 13.6-1. Calculated Values for Example 13.6-1 θ* x yp Fi 0 0.209 0.6550 0.6404 0.04876 0.1870 0.6383 0.7192 0.0992 0.1642 0.6158 0.8246 0.1482 0.1420 0.5940 0.9603 0.2000 0.1190 0.5690 1.1520 Using the material-balance equation (13.4-9) to calculate yp,

To calculate y_p at θ* = 0, Eqs. (13.6-3) and (13.4-17) must be used, giving y_p = 0.6550.

To solve for the area, Eq. (13.6-5) can be written as

Equation 13.6-6

where the function F_i is defined as above. Values of F_i will be calculated for different values of i in order to integrate the equation. For θ* = 0.200, x_o = 0.119, and from Eq. (13.6-4),

From Eq. (13.6-5),

Using the definition of F_i from Eq. (13.6-6),

Other values of F_i are calculated for the remaining values of θ* and are tabulated in Table 13.6-1. The integral of Eq. (13.6-6) is obtained by using the values from Table 13.6-1 and numerically integrating F_i versus i to give an area of 0.1082. Finally, substituting into Eq. (13.6-6),

For part (b), from Example 13.4-2, y_p = 0.5067 and A_m = 3.228 × 10⁸ cm². Hence, the cross-flow model yields a higher y_p of 0.5690, compared to 0.5067 for the complete-mixing model. Also, the area for the cross-flow model is 10% less than for the complete-mixing model.