Chapter 3 Unidirectional transport - Chemical Engineering ...

10 CHAPTER 3. UNIDIRECTIONAL TRANSPORTz+ ∆ zzz=Lc*=0T*=0u*=0xzz=0c*=1T*=1u*=1xxFigure 3.2: Configuration for similarity solution for unidirectional transport.in the boundary conditions,c ∗ = 0at z = L at all tc ∗ = 1 at z = 0 at all t > 0c ∗ = 0 at t = 0 for all z > 0 (3.32)In this case, it is not possible to effect a reduction to a similarity form,because there is an additional length scale L in the problem, and so the zcoordinate can be scaled by L. A scaled z coordinate is defined as z ∗ = (z/L),and the diffusion equation in terms of this coordinate is∂c ∗∂t = D L 2 ∂ 2 c ∗∂z ∗2 (3.33)The above equation suggests that it is appropriate to define a scaled timecoordinate t ∗ = (Dt/L 2 ), and the conservation equation in terms of thisscaled time coordinate is∂c ∗∂t = ∂2 c ∗(3.34)∗ ∂z ∗2The boundary conditions, in terms of the scaled coordinates z ∗ and t ∗ , arec ∗ = 0at z ∗ = 1 at all t ∗