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. ...

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