A catalyic tubular reactor is shown in Figure 2. Catalytic region Dilute solution of Dilute solution of A and B in solvent S A in solvent S A - B 2R Fully developed flow z = 0 z= L Figure 2 Boundary conditions for a tubular reactor. A dilute solution of solute A in a solven S is in fully developed, laminar flow in the region z < 0. When it encounters the catalytic wall in the region 0 < z < L, solute A is instantanously and irreversibly rearranged to an isomer B. Assume that the flow is isothermal and neglect the presence of isomer B. Under these conditions, the governing equation for the diffusion of species A can be describe by Eq. 1, 1 a = DAS rr ("ar Vz Eq. (1) az Subject to the boundary conditions, .. 1: at z = 0 CA = CA0 Eq. (2) .. 2: at r = R CA = 0 Eq. (3) .. 3: at r = 0 CA = finite Eq. (4) An asymptotic so lution can be obtained for small values of z. The basic assumption underlying this solution is that curvature effects can be neglected, and the problem treated as though the wall were flat. Introducing the variable y = R r, this assumption implies that the fluid may be regarded as extending from the wall (y = 0) to y = 0. (a) Show that assuming the velocity profile is linear near the wall gives (Eq. 5), 2vmaxy v,(y) = R Eq. (5) (Po PL)R2/(4ul). Subsequently, substitute the linear velocity profile into Eq. 1 and change the geometry near the wall from cylindrical to cartesian (i.e., from r to y) where vmax = to obtain, a? CA = DAS ? 2umax Eq. (6) az (10 marks) (b) Show that the introduction of the non-dimensional variables, f= CA. CAO y (2vmaxR\3 9DASZ R leads to the governing partial differential Eq. 6 becoming an ordinary differential equation, azf af + 3n? an? = 0 Eq. (7)