From the species balance around the separator = + xFnF xRnR yPnP (xn)R (xn)F = (yn)P (9) Combine equations. (8) and (9) with equation (4) = we obtain d(xn) yp)dA, Q’O2(xP = yPnP Q’O2dA Separate the variables and integrate = (R F) yPnP Q’O2 = (R F)Am yPnP Q’O2 = (10) yPnP Am Q’O2lm where the log mean average lm is defined as lm = = (xP = (11) yp)lm Equation (10) expresses the molar flow rate of oxygen as a function of the yPnP permeance or mass transfer coefficient, area of membrane for mass transfer, Am Q’O2 and an average driving force lm across the membrane. Similarly, the molar flow rate of nitrogen in the permeate stream can be found (1yP)nP = [(1x)P (1y)p)]lmAm (12) Q’N2 The oxygen species balance, = + can be written in dimensionless xFnF xR( nF nP) yPnP, form using the definition of the cut = nP/nF, = 1 ) + (13) xF xR( yP Similarly, equations (7), (10), and (12) in dimensionless forms are = * (14) = yPnP Am Q’O2lm = (xr yP y)lm = (1 )*(xr (15) yPKR y)lm where (xr is defined by Eq. (11) y)lm (1yP) = (1 )[(1x)r (1y))]lm (16) KR where [(1x)r (1y))]lm is defined by Eq. (11) with and x’s y’s replaced by (1-x)’s and (1-y)’s
From the species balance around the separator xFnF = xRnR + yPnP (xn)R (xn)F = (yn)P (9) Combine equations. (8) and (9) with equation (4) d(xn) = Q’O2(xP yp)dA, we obtain yPnP = Q’O2dA Separate the variables and integrate yPnP = Q’O2 (R F) yPnP = Q’O2 (R F)Am yPnP = Q’O2lm Am (10) where the log mean average lm is defined as lm = = (xP yp)lm = (11) Equation (10) expresses the molar flow rate yPnP of oxygen as a function of the permeance Q’O2 or mass transfer coefficient, area of membrane Am for mass transfer, and an average driving force lm across the membrane. Similarly, the molar flow rate of nitrogen in the permeate stream can be found (1yP)nP = Q’N2 [(1x)P (1y)p)]lmAm (12) The oxygen species balance, xFnF = xR( nF nP) + yPnP, can be written in dimensionless form using the definition of the cut = nP/nF, xF = xR( 1 ) + yP (13) Similarly, equations (7), (10), and (12) in dimensionless forms are = * (14) yPnP = Q’O2lm Am yP = (xr y)lm yPKR = (1 )*(xr y)lm (15) where (xr y)lm is defined by Eq. (11) (1yP) KR = (1 )[(1x)r (1y))]lm (16) where [(1x)r (1y))]lm is defined by Eq. (11) with x’s and y’s replaced by (1-x)’s and (1-y)’s
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