Dimensionless form for

To find a dimensionless form for equation (11.23) we Introduce the dimensionless reaction rate r, defined by... 

Transformation of the independent variables to dimensionless form uses = r/R and jz = z/L. In most reactor design calculations, it is preferable to retain the dimensions on the dependent variable, temperature, to avoid confusion when calculating the Arrhenius temperature dependence and other temperature-dependent properties. The following set of marching-ahead equations are functionally equivalent to Equations (8.25)-(8.27) but are written in dimensionless form for a circular tube with temperature (still dimensioned) as the dependent variable. For the centerline. 

The forward traces may therefore be computed in dimensionless form for any values of the two parameters p and 0C. Of particular usefulness is the estimation of the effect of ohmic drop and double-layer charging on the peak characteristics. The exact values of the peak, peak potential, and peak width may be found in Table 1 of reference 19. As an example, the shifts undergone by the dimensionless peak potential are shown in Figure 1.9. 

The process of obtaining suitable dimensionless forms for the present model is very similar to those followed in the previous chapters, 2.8 and 3.4 we need first to choose a reference concehtration and a reference timescale. For the first of these, the concentration, there is a natural choice, as we have seen already in eqn (6.6), i.e. the inflow concentration of the reactant a0. Again, using Greek letters for dimensionless quantities, we have... 

The pressure profile can now be solved for and is presented in dimensionless form for various power law indices in Fig. 6.41. 

Dynamic Model A two-dimensional heterogeneous dynamic model was developed, which describes the mass and energy balances in both phases. In dimensionless form, for the nfc component and the temperature in the gas phase,... 

It is convenient to write this equation in dimensionless form. For this purpose the following dimensionless variables are introduced ... 

Another case of interest with LSV is the catalytic system, described above in the section on potential steps. The equations are the same except that the potential here is not constant and very negative, following (2.29) in its dimensionless form. For small and intermediate rates of the homogeneous chemical reaction (dimensionless constant K), the same procedure as mentioned above, that is, convergence simulations, must be used. For large K, however, the LSV curves become sigmoid, with a plateau equal to the current for the potential step, G = y/K. This can be used to test methods. Figure 2.7 shows LSV curves for some A -values, where this effect is seen. 

Now we can transform the model relations into dimensionless forms. For this purpose, we use the dimensionless temperature as a measure of a local excess with respect to the adjacent medium Tp = (t —tj)/ the dimensionless time... 

Dimensionless forms for the momentum equation and the species-conservation equation... 

The governing equations in dimensionless form for the three regions have been derived as follows [Harold et al., 1989]. 

Corresponding States (CS) The principle of CS applies to conformal fluids [Lehmd, T. L., Jr., and P. S. Chappelear, Ind. Eng. Chem., 60 (1968) 15]. Two fluids are conformal if their intermolecu-lar interactions are equivalent when scaled in dimensionless form. For example, the Lennard-Jones (LJ) intermolecular pair potential energy U can be written in dimensionless form as... 

In applying these factors, a useful device which avoids the possibility of error in the less simple cases, is to convert them into a dimensionless form. For example, as seen above, 1 liter-atm. is equivalent to 24.218 cal. hence, 24.218 cal./liter-atm., i.e., 24.218 cal. liter atm. , is equal to unity, without dimensions. It is then permissible to multiply one side of an expression by unity and the other side by 24.218 cal. liter" atm.". The subsequent cancellation of identical units with opposite exponents makes conversion from one set of units to another a relatively simple matter. The application of the foregoing ideas will be illustrated in subsequSit portions of the book. ... 

In this case, Laplace s equation still represents a good description of the potential distribution in the electrolyte, and all results discussed in Section III.l remain valid. Furthermore, the change of concentration caused by migration currents of the reacting species can be neglected, and the appropriate equations (first derived by Flatgen and Krischer ) on which the calculations are based, read in dimensionless form for the potential ... 

These two terms must be equal at steady state. Hence, introducing the dimensionless forms for r and 9, we can express this condition in the form... 

The new choice of characteristic length and time scales leads to a modified dimensionless form for (3-223), namely,... 

Another way to represent the controller gain is the proportional band PB), which is an approach that was in more common use 10 to 15 years ago. Proportional band can be expressed (as a percent) in terms of when is in dimensionless form. For example, the controller output and the error from setpoint can be scaled 0 to 100%, yielding a dimensionless K ... 

As indicated by equation (15-12), the simplified homogeneous mass transfer model for diffusion and one chemical reaction within the internal pores of an isolated catalytic pellet is written in dimensionless form for reactant A as... 

When the kinetics are first-order and irreversible in catalytic pellets with spherical symmetry, the mass transfer/chemical reaction model that focuses on intrapeUet diffusion is written in dimensionless form for carbon monoxide as... 

The next objective is to identify a time constant for each important mass transfer rate process and solve the system of equations for the two-phase CSTR in terms of these time constants. This approach allows one to develop generic solutions in dimensionless form. For example, six time constants can be defined for (1) convection in the liquid phase (r), (2) chemical reaction in the liquid phase (X), and (3-6) interphase mass transfer for each component (0y, j = B,C1,M,H). Obviously, these six time constants produce five dimensionless ratios. Remember that time constants represent order-of-magnitude estimates of the time scales of mass transfer rate processes. The time constant for convective mass transfer in the liquid phase is equivalent to the liquid s residence time ... 

Liquid-phase CSTR algebraic equations written in dimensionless form for each component ... 

With the help of these characteristic numbers, mass and heat balances can now be derived in dimensionless form for the ideal reactors considered. 

The governing equations in the dimensionless form for the system are 9Cbc... 

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