Dimensionless Variables and Numbers

The response of the system, flout, depends on at least five potentially manipulat-able parameters Q, R, L, k, fli ), and a plot showing the dependence of flout would need an impossible six dimensions. However, the solution to Equation 1.62 can be written as 

There are only two variables, flin/flout and kt, so an ordinary two-dimensional graph can be used. 

A dramatic reduction in dimensionality is often possible by converting a design equation from dimensioned to dimensionless form. Equation 1.62 contains the dependent variable a and the independent variable z. The process begins by selecting characteristic values for these variables. By characteristic value we mean some known parameter that has the same dimensions as the variable and that characterizes the system. Eor a PER, the variables are concentration and length. A characteristic value for concentration is flin and a characteristic value for length is L. These are used to define the dimensionless variables a = ala-m and zIL. The governing equation for a first-order reaction in an ideal PER becomes 

Find the equivalent to Equation 1.63 for second-order reactions. 

SOLUTION Begin with the case of one reactant, 2A — kC. The same variable transformations are used for a and z to give 

---