Equation dimensionless forms

These reactions simply account for the number of CO and H2 molecules required to form hydrocarbon chains (expressed as CH2 units) and methane, respectively, but contain no mechanistic significance. At steady state, CO and H2 mass-balance equations (dimensionless form) within a catalyst pellet give... 

STEP 3 Hook up an analog computer according to the block diagram and display the outputs A, B, C, and D vs t on an oscilloscope or xy plotter. 2-16. Solve Prob. 2-15 by analytical integration of the differential equations (dimensionless form). 

Equation-of-state parameter (dimensions depend on equation) Dimensionless form of equation-of-state parameter a Equation-of-state parameter (m- /mol)... 

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... 

When such a function is estabUshed or assumed, it will still exist even after the variables are intermultiplied in any manner whatsoever. This means that each variable in the equation can be combined with other variables of the equation to form dimensionless products whose dimensional vectors are the 2ero vector. Equation 4 can then be transformed into the nondimensional form as (eq. 5) ... 

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. 

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... 

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... 

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. 

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 ... 

We note that equation 15 is an example of equation 26. It can be shown that all dimensionless parameters, arrived at in the conventional manner by writing equations of type 2k in completely dimensionless form, can be expressed as quotients of CT s. 

Solution On first inspection, the selectivity appears to depend on five parameters ao, bo, kj, kji, and ti,atch- However, the governing equations can be cast into dimensionless form as... 

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

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 ODEs are linear with constant coefficients. They can be converted to a single, second order ODE, much like Equation (11.22), if an analytical solution is desired. A numerical solution is easier and better illustrates what is necessary for anything but the simplest problem. Convert the independent variable to dimensionless form, = z/L. Then... 

Solution The problem requires solution of the convective diffusion equation for mass but not for energy. Rewriting Equation (8.71) in dimensionless form gives... 

A dimensionless form of the balance equation, can be obtained by substituting the following dimensionless variables... 

The model is written in both dimensional and dimensionless forms. This example provides experience in the use of dimensionless equations. 

The model equations are expressed in dimensionless form and defining new variables in terms of the fractional deviation from steady state. 

The model equations can then be recast into dimensionless form as... 

The tubular reactor, steady-state design equation is of interest here. The dimensional and dimensionless forms are compared for an nth-order reaction. 

The dimensionless form of these equations when combined with the total balance with constant F, apply both to the filling and full periods. These are... 

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