Chapter 3 Unidirectional transport - Chemical Engineering ...

8 CHAPTER 3. UNIDIRECTIONAL TRANSPORTexpress the conservation equation 3.16 in terms of the variable ξ alone. Whenz and t are expressed in terms of ξ, the concentration equation becomes(−z2D 1/2 t 3/2 ) ∂c∗∂ξ = D Dt∂ 2 c ∗∂ξ 2 (3.26)After multiplying throughout by t, the equation for the concentration fieldreduces toξ ∂c ∗2 ∂ξ + ∂2 c ∗∂ξ = 0 (3.27)2Equation 3.27 validates the inference that the non-dimensionalised concentrationfield is only a function of ξ. It is also necessary to transform theboundary conditions 3.9 into conditions for the ξ coordinate. The transformedboundary conditions arec ∗ = 0 as z → ∞ at all t → as ξ → ∞c ∗ = 1 at z = 0 at all t > 0 → at ξ = 0c ∗ = 0 at t = 0 for all z > 0 → as ξ → ∞ (3.28)It is useful to note that the original conservation equation, 3.16, is a secondorder differential equation in z and a first order differential equation in t, andso this requires two boundary conditions in the z coordinate and one initialcondition. The conservation equation expressed in terms of ξ is a secondorder differential equation, which requires just two boundary conditions forξ. Equation 3.28 shows that one of the boundary conditions for z → ∞ andthe initial condition t = 0 turn out to be identical conditions for ξ → ∞,and therefore the transformation form (z, t) produces no inconsistency in theboundary and initial conditions.Equation 3.27 can be easily solved to obtain∫ ( )∞c ∗ (ξ) = C 1 + C 2 dξ ′ exp − ξ2ξ2(3.29)The constants C 1 and C 2 are determined from the conditions c ∗ = 1 at ξ = 0,and c ∗ = 0 for ξ → ∞, to obtainc ∗ (ξ) =√2π∫ ∞(z/ √ Dt)dξ ′ exp(− ξ22)(3.30)