Applied Mathematics for Science and Engineering

6Numerical Solution of Ordinary Differential Equations

An Illustrative Example

In engineering and the applied sciences, transient mass and energy balances arise frequently, often leading to ordinary differential equations (ODEs). Suppose, for example, we have a jacketed process vessel in which an exothermic chemical reaction may occur. The entering (feed) stream has a temperature, T_in; the well-mixed contents have temperature, T; and the steam used to heat the vessel has temperature, T_s. A verbal statement of the appropriate energy balance might appear:

And written out symbolically, we would expect something like this:

(6.1)

The rate at which the reactant species, A, is consumed is r_A. We will let the mass flow rates in and out be the same, and we set the reference temperature equal to the inlet (feed) temperature and divide by :

(6.2)

Please note that every term in the equation has the dimension of temperature. The characteristic time, τ, that appears on the right-hand side is the total mass in the vessel divided by the mass flow rate, and it is the time constant for this system. ...