2. [15 points] The following differential equation describes the steady-state concentration of a substance that reacts with first-order kinetics in an axially-dispersed plug-flow reactor, as shown in Figure 1: d² c D dx² dc dx U kc = 0, where D = the dispersion coefficient (m2/h), c = concentration (mol/L), U = the velocity (m/h), and k = the reaction rate (1/h). The boundary conditions can be formulated as Ucin = Uc(x = 0) - D dc (x 0) dx dc (x = L) = 0, dx where Cin = the concentration in the inflow (mol/L), and L = the length of the reactor (m). Use the finite-difference method to solve for concentration as a function of distance, given the following parameters: D = 5000 m²/h, U = 100 m/h, k = 2 1/h, L = 100 m, and cin = 100 mol/L. Use centered finite-difference approximations with Ax = 10 m to obtain your solution. Compare your numerical results with the analytical solution: C= Ucin (U — DÀ1)À2e^2ª — (U — DÀ2)À¹³1 × (+20 +212 - 1+1+2) - 4kD ---/ (+1√17) - 10 (1√+420) U 1 = 2D 4kD U 12 = 1- U2 2D U2 x=0 X x= L Figure 1: Axially-dispersed plug-flow reactor

2. [15 points] The following differential equation describes the steady-state concentration of a substance that reacts with first-order kinetics in an axially-dispersed plug-flow reactor, as shown in Figure 1: d² c D dx² dc dx U kc = 0, where D = the dispersion coefficient (m2/h), c = concentration (mol/L), U = the velocity (m/h), and k = the reaction rate (1/h). The boundary conditions can be formulated as Ucin = Uc(x = 0) - D dc (x 0) dx dc (x = L) = 0, dx where Cin = the concentration in the inflow (mol/L), and L = the length of the reactor (m). Use the finite-difference method to solve for concentration as a function of distance, given the following parameters: D = 5000 m²/h, U = 100 m/h, k = 2 1/h, L = 100 m, and cin = 100 mol/L. Use centered finite-difference approximations with Ax = 10 m to obtain your solution. Compare your numerical results with the analytical solution: C= Ucin (U — DÀ1)À2e^2ª — (U — DÀ2)À¹³1 × (+20 +212 - 1+1+2) - 4kD ---/ (+1√17) - 10 (1√+420) U 1 = 2D 4kD U 12 = 1- U2 2D U2 x=0 X x= L Figure 1: Axially-dispersed plug-flow reactor

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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