Use of Dimensionless Variables

The use of dimensionless variables will be illustrated using Equation (8.12) but with an added term for axial diffusion  

When expressed in the scaled variables, the ajd and ajd terms have the same magnitude, but the d ajd term is multiplied by a factor of that will not be larger than 0.01. Thus, this term, which corresponds to axial diffusion, may be neglected, consistent with the conclusion in Section 8.2. 

The velocity profile is scaled by the mean velocity, m, giving the dimensionless profile z(- ) = Ez(r)/ . To complete the conversion to dimensionless variables, the dependent variable, a, is divided by its nonzero inlet concentration. The dimensionless version of Equation (8.12) is 

The stability criterion. Equation (8.29), can be converted to dimensionless form. The result is 

Example 8.6 Generalize Example 8.5 to determine the fraction unreacted for a first-order reaction in a laminar flow reactor as a function of the dimensionless groups and kt. Treat the case of a parabolic velocity profile. 

Simultaneous solution of the convective diffusion equations for mass and heat must be done numerically in all but trivial cases. The solutions can be based on dimensioned variables like z and T, and this has the advantage of keeping the physics of the problem close at hand. However, the solutions are then quite specific and must be repeated whenever a design or operating variable is changed. Somewhat more general solutions, while still numerical, can be obtained through the judicious use of dimensionless variables, dimensionless parameters, and dimensionless functions. Table 8.1 defines a number of such variables. 

The dimensionless equations for the convective diffusion of mass and heat are 

Dimensioned Variable Dimensionless Variable Type of Variable 

The usual way of defining a dimensionless variable is to divide the dimensioned variable by a quantity with the same dimensions that characterize the system. There are sometimes several possibilities. Reasonable choices for Tref include Tin, Twaii, 

Adiabatic A useful variable for heat transfer calculations when the heat of reaction is small is (T - r )/(7 waii - Tm). The dimensionless heat of reaction for this case becomes [AHum] j [pCp T x - Tin)]. 

---

Inequality constraints. Implicit constraints exist because of the use of dimensionless variables... 

The theoretical solution of a heat transfer problem is structured more clearly when dimensionless variables are used. It is therefore recommended that dimensionless variables should be introduced at the beginning of the problem solving process. The evaluation and representation of the solution will also be simplified by keeping the number of independent variables as small as possible through the use of dimensionless variables and groups. 

The need to use specific numerical values for the rate constants and initial conditions is a weakness of numerical solutions. If the specific values change, then the numerical solution must be repeated. Analytical solutions usually apply to all values of the input parameters, but special cases are sometimes needed. Recall the special case needed for Uo = bo in Equation 1.32. Numerical solution techniques do not have this problem, and the problem of specificity with respect to numerical values can be minimized or overcome through the judicious use of dimensionless variables, as... 

Solving differential equations in terms of dimensionless parameters generally does yield solutions that characterize whole families of specific experimental situations. This is a magnificent asset, especially where numerical solutions are required consequently, the use of dimensionless variables has become standard practice. 

Computational analysts often prefer the use of dimensionless variables for numerical efficiency... 

---