Pellets 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. 

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). 

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... 

F(c/c,) denotes the dimensionless form of an arbitrary rate expression./(x) is a nonuniform, normalized catalyst activity distribution inside the pellet. A(x) is an auxiliary function, subject to the following linear differential equation ... 

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... 

Consider diffusion with a first order isothermal reaction in a rectangular pellet.[ll] [8]. The governing equation and boundary conditions for concentration in dimensionless form are ... 

Consider mass transfer in a spherical pellet.[l] The governing equation in dimensionless form is... 

It is instructive to write Eq. (13-2) in dimensionless form by introducing the conversion x and dimensionless coordinates r and z based on the diameter of the catalyst pellet ... 

Thermal effects constitute a significant portion of the study devoted to catalysis. This is true of electrochemical reactions as well. In general the reaction rate constants, diffusion coefficients, and conductivities all exhibit Arrhenius-type dependence on temperature, and as a rule of the thumb, for every 10°C rise in temperature, most reaction rates are doubled. Hence, temperature effects must be incorporated into the parameter values. Fourier s law governs the distribution of temperature. For the example with the cylindrical catalyst pellet described in the previous section, the equation corresponding to the energy balance can be written in the dimensionless form as follows ... 

The pellet is thus described by the two dimensional system formed of equations (5.100) and (5.102). These equations can be put in a dimensionless form as follows ... 

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... 

A pellet may not be isothermal when heat effects are significant as in hydrocracking. While complications may arise due to the vaporization of liquid, it will be assumed in what follows that the pellet is completely wetted. Steady-state, one-dimensional balance equations for a slab-like pellet can be written in dimensionless form as ... 

The transfer coefficient can be correlated in the form of a dimensionless Sherwood number Sh(= h0dp/D). The particle diameter dp is often taken to be the diameter of the sphere having the same area as the (irregular shaped) pellet. Thaller and Thodos(38> correlated the mass transfer coefficient in terms of the gas velocity u and the Schmidt number Sc(= p/pD) ... 

As a typical example of this type of reaction, the transformation A) — products may be considered, where the kinetics are described by a simple power rate law of the order n. Since this reaction is completely characterized by specifying the conversion of reactant A. the above system of diflcntial equations (eqs 11-18) may be readily expressed in a convenient, nondimcnsional form. For this purpose, the reactant concentration and the temperature arc related to their corresponding values in the bulk fluid phase (eqs 24 and 25), and the radius coordinate r is divided by the pellet radius R to introduce a dimensionless coordinate (eq 26). 

For forced convection, the heat transfer coefficient is normally correlated in terms of tliree dimensionless groups the Nusselt number, Nu, the Reynolds number, Re, and the Prandtl number, Pr. For the single spherical pellets discussed here, Nu and Re take the following forms ... 

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