Dimensionless form

The differential material balances contain a large number of physical parameters describing the structure of the porous medium, the physical properties of the gaseous mixture diffusing through it, the kinetics of the chemical reaction and the composition and pressure of the reactant mixture outside the pellet. In such circumstances it Is always valuable to assemble the physical parameters into a smaller number of Independent dimensionless groups, and this Is best done by writing the balance equations themselves in dimensionless form. The relevant equations are (11.20), (11.21), (11.22), (11.23), (11.16) and the expression (11.27) for the effectiveness factor. 

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

As In the case of the material balance equations, the enthalpy balance can be written in dimensionless form, and this introduces new dimensionless parameters in addition to those listed in Table 11.1. We shall defer consideration of these until Chapter 12, where we shall construct the unsteady state enthalpy and material balances, and reduce them to dimensionless form. 

In section 11.4 Che steady state material balance equations were cast in dimensionless form, therary itancifying a set of independent dimensionless groups which determine ice steady state behavior of the pellet. The same procedure can be applied to the dynamical equations and we will illustrate it by considering the case t f the reaction A - nB at the limit of bulk diffusion control and high permeability, as described by equations (12.29)-(12.31). 

The resulting equations (12.32)-(12.34) have a corresponding dimensionless form, which we will not bother to write out since it is an obvious simplification of equations (12.40)-(12.42). Finally, approximations based on the large size of led to equations (12.33)-(12.37) and, since... 

For the same reaction in a pellet of finely porous structure, where Knudsen diffusion controls, the appropriate dynamical equations sre (12.20) and (12.21) if we once more adopt approximations which are a consequence of Che large size of K. These again have a dimensionless form, which may be written... 

In the absence of body force, the dimensionless form of the governing model equations for two-dimensional steady-state incompressible creeping flow of a viscoelastic fluid are written as... 

Values of the mass-transfer coefficient k have been obtained for single drops rising (or falling) through a continuous immiscible Hquid phase. Extensive Hterature data have been summarized (40,42). The mass-transfer coefficient is often expressed in dimensionless form as the Sherwood number ... 

Asymptotic Solution Rate equations for the various mass-transfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, Np, for axial dispersion. For pore and sohd diffusion, q = / // p is a dimensionless radial coordinate, where / p is the radius of the particle, if a particle is bidisperse, then / p can be replaced by the radius of a suoparticle. For prehminary calculations. Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. 

In dimensionless form, a reduced HETP, h HETP/dp, analogous to the reduced HTU (cf Fig. 16-13), is obtained as a function of the dimensionless velocity ReSc ... 

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

The flow coefficient is the capacity of the flow rate expressed in dimensionless form... 

Tabic 9-1 is a handy little chart to visualize a vertical, single-cylinder compressor and the basic functions. The functions are normalized to keep them in a dimensionless form [4]. With the following set of equations, the X and y components of the inertia forces for a single cylinder can be calculated. For the derivation, the reader is referred to references [4,. 5], Figure 9-4 depicts the generalized stage to aid in the definitioii of terms... 

This expression can be represented graphically in dimensionless form to simplify its use. It is generally expressed as the so-called filtration number , defined as follows E, = /iR, / 2APT3 jr x . The filtration number, E, is dimensionless and varies from zero at Rf = 0 to a large value when there is an increase in the viscosity of the sludge and Rf or a decrease in pressure drop, auxiliary time, specific cake resistance and the ratio of cake volume to filtrate volume. It may be assumed in practice that F(, = 0 to 10. If washing and drying times are constant and independent of filtration time, they may be added directly to the auxiliary time. In... 

Equation 46 is a general expression that may be applied to the treatment of experimental data to evaluate exponent a. This, however, is a cumbersome approach that can be avoided by rewriting the equation in dimensionless form. Equation 42 shows that there are n = 5 dimensional values, and the number of values with independent measures is m = 3 (m, kg, sec.). Hence, the number of dimensionless groups according to the ir-theorem is tc = 5 - 3 = 2. As the particle moves through the fluid, one of the dimensionless complexes is obviously the Reynolds number Re = w Upl/i. Thus, we may write ... 

Treating the other terms in a similar manner, the linear momentum equation in a dimensionless form is obtained ... 

The dimensionless form of the continuity equation (4.278) ( , = 0) in Uv o-dimensional boundary layer flow is... 

As I mentioned above, it is conventional in many engineering applications to seek to rewrite basic equations in dimensionless form. This also applies in quantum-mechanical applications. For example, consider the time-independent electronic Schrodinger equation for a hydrogen atom... 

The physical quantities h, e and all tend to get in the way, so the first task is to write the Hamiltonian in dimensionless form (each variable is now the true variable divided by the appropriate atomic unit). I showed you how to do this in Chapter 0. The electronic Hamiltonian... 

Before discussing the on.set, and nature, of fluid turbulence, it is convenient to first recast the Navier-Stokes equations into a dimensionless form, a trick first used by Reynolds in his pioneering experimental work in the 1880 s. In this form, the Navier-Stokes equations depend on a single dimensionless number called Reynolds number, and fluid behavior from smooth, or laminar, flow to chaos, or turbulence,... 

For both of these cases, Eqs. (13)—(15) constitute a system of two linear ordinary differential equations of second order with constant coefficients. The boundary conditions are similar to those used by Miyauchi and Vermeulen, which are identical to those proposed by Danckwerts (Dl). The equations may be transformed to a dimensionless form and solved analytically. The solutions may be recorded in dimensionless diagrams similar to those constructed by Miyauchi and Vermeulen. The analytical solutions in the present case are, however, considerably more involved algebraically. 

Clark and Vermeulen (C8) later reported an extensive experimental study of power requirements in agitated gas-liquid systems. They correlated their data in dimensionless form as a function of fractional gas holdup, Weber number, and a geometrical factor. Their correlation is shown in Fig. 5. 

Recalling that 0i=Q/Cjiraax and defining =z/L, where L is the thickness of the catalyst film, one can write equations (11.16) to (11.18) in the following dimensionless form ... 

Among the dimensional variables of the problem, five parameters have independent dimensions and Eq. (6.41) may be written in dimensionless form. Choosing parameters Pl, CpL, U, ATs, and taking into account r-theorem (Sedov 1993), Eq. 

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