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and −dc∗c∗ ∂ζ+f(ξ)∂ζ+∂ζ+2c1 dc∗2c∗ ∂ζ=Perξ∂ξ ξ∂ξ +Pea∂ζ2 dn∗ ∂c∗ ∂c∗ ∂c∗ d2n∗ ∂c∗ 2 1 ∂ ∂c∗ 2 ∂2c∗ 212∗122 00 (4.15) Following Taylor,10 we assume that the time necessary for appreciable effects to appear due to convective transport is long compared with that needed for the radial variations to be reduced to a fraction of their initial value due to molecular diffusion, i.e., axial derivatives are independent of radial position. After integrating the first order equation radially and applying both radial boundary conditions, we obtain A solution for c∗1 is assumed of the form dn∗ ∂c∗1 ∂c∗1 2 ∂2c∗1 −dc∗c∗ ∂ζ+∂ζ=Pea∂ζ2 0 (4.16) (4.17) (4.18) Per dc∗0 ξ ξ′ ξ′′f(ξ′′)dξ′′ c∗1=g(ζ)+ 0 dξ′ 2 dζ 0 ξ′ Substituting equation 4.17 into equation 4.14 we obtain dn∗∂g∂g 2∂2g −dc∗c∗ ∂ζ+∂ζ=Pea∂ζ2 0 Integrating equation 4.18 by parts gives dn∗ 2 dg 1−dc∗ g=Pedζ (4.19) After substituting equation 4.12 into equation 4.19 and integrating, we obtain the c∗0 a solution for the axial contribution of the first order term, g (ζ), in the form dn∗ dc∗ 0 g(ζ)=B1exp 1− dc∗ (4.20) c∗0 c∗0 − n∗|c∗0 ′ dn∗ dc∗ Pedc∗ξ ξξ′′f(ξ′′)dξ′′ which when substituted in equation 4.17 yields ∗0r00′ c1 =B1exp 1− dc∗ + 2 dζ 0 ξ′ dξ (4.21) c∗0 c∗0 − n∗|c∗0 The method for evaluating the second order term is the same as the one out- lined above for the first order term. The example demonstrates that the radial terms 55PDF Image | TEMPERATURE SWING ADSORPTION COMPRESSION AND MEMBRANE SEPARATIONS
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