Page 566 - Modelling in Transport Phenomena A Conceptual Approach

546 APPENDIX B. SOLUTIONS OF DIFFERENTIAL EQUATIONS

with a rate constant of k = 2h-'. If the initial number of moles of species A is
1.5mo1, determine the variation in the number of moles of A during the first one
hour of the reaction. Compare your results with the analytical solution.

Solution
The inventory rate equation based on the moles of species A is

Analytical solution

Equation (2) is a separable equation with the solution
nA = nA, exp( - kt) (3)

in which nA, aS the initial number of moles of species A.
Numerical solution

In terms of the notation of the Runge-Kutta method, Eq. (2) is expressed as

dY - =-2y
dt (4)
with an initial condition of
y(0) = 1.5 (5)
Therefore,

f(t,Y) = -2Y
yo = 1.5
Integration of Eq. (4) from t = 0 to t = 1 by using fourth-order Runge-Kutta
method urith a time step of h = 0.1 is given cbs follows:
Calculation of y at t = 0.1 hour
First, it is necessary to determine kl, kz, k3, and k4:
kl = hf(Y0)
= (O.l)(- 2)(1.5) = - 0.3000
k2 = hf (Yo + ik,)

( Of)
= (O.l)(- 2) 1.5 - - = - 0.2700