Computational methods in chemical engineering with Maple by Ralph E. White
By Ralph E. White
This e-book is helping chemical and different engineers increase their abilities for fixing mathematical versions utilizing Maple. those mathematical types can include platforms of algebraic, usual, and partial differential equations. Maple’s ‘dsolve’ is used to acquire options for lots of of those versions. Maple worksheets are supplied at the Springer site to be used by way of readers to resolve the instance difficulties during this e-book. simply click digital Supplementary fabric and insert the ISBN.
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First, the ‘limit’ command is applied with respect to the first parameter, k1 → 1 . 2. "); 36 2 Initial Value Problems Fig. 2 Concentrations of A, B, and C as functions of time We can solve for the time at which B is at its maximum by differentiating Cb with respect to t: > Eqmax:=diff(cb,t); Next, solve 'eqmax' to find the time at which Cb is at a maximum in terms of the rate constants k1 and k 2 : > tmax:=solve(Eqmax,t); Substitute this value for time into the concentration equation for B(cb(t)) to find the maximum value of Cb as a function of the rate constants k1 and k 2 .
2. 10)) > restart: > with(linalg): > with(plots): Specify the equations. > eq:=diff(Ca(t),t)=-k1*Ca(t)+k2*Cb(t); > eq:=diff(Cb(t),t)=k1*Ca(t)-k2*Cb(t)-k3*Cb(t)+k4*Cc(t); > eq:=diff(Cc(t),t)=k3*Cb(t)-k4*Cc(t); Specify the variables. 1 Linear Ordinary Differential Equations 39 Specify the right hand side of the equations. 3)]; Generate the coefficient matrix A. > A:=genmatrix(eqs,vars,A); Generate the exponential matrix of A. > mat:=exponential(A,t): Set the initial conditions vector: > Y0:=matrix(3,1,[1,0,0]); Obtain the solution.
The first row of the sol vector corresponds to ca(t), and the second row corresponds to cb(t). Note that the analytical solutions for ca and cb are obtained as functions of the parameters (rate constants) and the independent variable, t. Lower case letters are used for these variables with k1 and k 2 as parameters. 13. For a given set of rate constants, plot the concentration profiles. 1. 1. ' > eqs:=[rhs(eq),rhs(eq)]; Now use the 'genmatrix' Maple command to produce the coefficient matrix A.