Analytic Solution of the Linear Case

Here we assume that Da = 1.4 and that the reaction is first-order. The solution of the simulation problem is as follows. 

Solution of the second-order differential equation (5.24) for the first-order reaction 

This is a linear second-order ordinary differential equation that can be solved explicitly by using the characteristic equation 

From the theory of linear differential equations, the solution of the second-order linear differential equation (5.27) has the general form 

Now we study the design problem where Da is unknown, the conversion xa is to be equal to 0.75, and therefore the dimensionless concentration at the exit is x( ) = 0.25 (= 1 — xa)- We follow the same procedure above until we reach the expression (5.29) for Ai and A2. This expression now contains an unknown Da and the variable Pe. 

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