Question

A multicomponent gaseous mixture having a composition of XfA=0.30, XfB = 0.45, Xfc = 0.25 is to be separated by a membrane with a thickness of 2.0×103 cm using the complete-mixing model. The feed flow rate is 2.0×104 cm³ (STP)/s and the permeabilities are PA' = 200×10-10 cm³ (STP) cm/(s-cm².cmHg), PB' = 50×10-¹0 cm³ (STP) cm/(s cm² cmHg), and Pc' = 25×10-10 cm³ (STP) cm/(s cm² cmHg). The pressure on the feed side is 300 cmHg and 30 cmHg on the permeate side. The fraction permeated will be 0.50. Calculate the permeate compositions, reject compositions, and membrane area using the complete-mixing model. Start with an initial guess of ypA = 0.45.

Answer 1

To solve the problem, we'll use the complete-mixing model for gas separation. The key equations are:

For the permeate flow rate (Qp):

Qp = Ap * Pp / (RT)

For the reject flow rate (Qr):

Qr = Ar * Pr / (RT)

For the total flow rate (Qt):

Qt = Qp + Qr

For the permeate compositions (XpA, XpB, XpC):

XpA = (Qp / Qt) * XfA

XpB = (Qp / Qt) * XfB

XpC = (Qp / Qt) * XfC

For the reject compositions (XrA, XrB, XrC):

XrA = XfA - XpA

XrB = XfB - XpB

XrC = XfC - XpC

For the permeability-corrected permeate flow rate (Qp_prime):

Qp_prime = Qp / (PA' * (Pf - Pp))

For the membrane area (A):

A = Qp_prime / (YP - XpA)

Now let's calculate the values step by step:

Given:

XfA = 0.30

XfB = 0.45

XfC = 0.25

Pf = 300 cmHg

Pp = 30 cmHg

YP (initial guess) = 0.45

Ap = Ar (unknown)

Pr = Pf - Pp = 270 cmHg

R = 82.06 cm³ cmHg / (mol K)

T = 273 K

Calculate the total flow rate (Qt):

Qt = 2.0×104 cm³ (STP)/s

Calculate the permeate flow rate (Qp):

Qp = Qt * 0.50 = 1.0×104 cm³ (STP)/s

Calculate the reject flow rate (Qr):

Qr = Qt - Qp = 1.0×104 cm³ (STP)/s

Calculate the permeate compositions (XpA, XpB, XpC):

XpA = (Qp / Qt) * XfA

XpB = (Qp / Qt) * XfB

XpC = (Qp / Qt) * XfC

XpA = (1.0×104 cm³ (STP)/s / 2.0×104 cm³ (STP)/s) * 0.30 = 0.15

XpB = (1.0×104 cm³ (STP)/s / 2.0×104 cm³ (STP)/s) * 0.45 = 0.225

XpC = (1.0×104 cm³ (STP)/s / 2.0×104 cm³ (STP)/s) * 0.25 = 0.125

Calculate the reject compositions (XrA, XrB, XrC):

XrA = XfA - XpA

XrB = XfB - XpB

XrC = XfC - XpC

XrA = 0.30 - 0.15 = 0.15

XrB = 0.45 - 0.225 = 0.225

XrC = 0.25 - 0.125