Mass Balance for Reactant

In general, the kinetic rate constant for nth-order kinetics with units of (volume/mole) Vtime is related to the heterogeneous forward rate constant kf with units of moles per area per time, and details of the porous pellet as follows  

The simplified homogeneous mass transfer model for diffusion and Langmuir- Hin-shelwood chemical kinetics within the internal pores of an isolated catalytic pellet is written in dimensionless form for reactant A or A2 (i.e., ua = —1)  

One-dimensional diffusion is expressed in terms of the Laplacian of molar density in rectangular coordinates as 

The basic information for molar density 4 a( ) is obtained by solving the dimensionless mass balance which includes the appropriate Hougen-Watson model  

This second-order ODE for 4 a(9) with split boundary conditions, given by equations (19-11) and (19-12), cannot be solved numerically until one invokes stoichiometry and the mass balance with diffusion and chemical reaction to relate the molar densities of aU gas-phase species within the pores of the catalytic pellet 

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Now consider the mass balance for reactant A over the control volume S8l. 

We note that to determine the effectiveness factor r/, we have to solve the mass balance for reactant Ai inside the porous pellet, which for the above chosen example ( i = 1, n2 = 0) takes the following form ... 

When a fixed bed of catalyst is operated isothermally as an id al piston-flow reactor without radial concentration gradients and without change in the gas density, the differential mass balance for reactant along the bed is given by... 

The model equation is a mass balance for reactant species inside a volume element of the slab catalyst in terms of dimensionless concennation f=c/cs and dimensionless ace coordinate x=z/t we... 

It is a simple matter to demonstrate that it is the interaction of reaction, geometric, and heat-transfer requirements that limits the value of scale-up. Consider a fixed-bed catalytic reactor. Suppose that the pressure drop has no effect on the rate and that plug flow exists. The mass balance for reactant is given by Eq. (12-2) as... 

For steady-state conditions and the assumptions of plug flow in the gas phase and complete mixing in the liquid phase, one can write the mass balance for reactant at any point in the reactor as... 

The primary objective of this section is to add a mass balance for reactant B to the set of coupled ODEs required to simulate the performance of this doublepipe reactor. The simplest approach to accomplish this task is to adopt the mass balance at high Peclet numbers for reactant A within the inner pipe, replace subscript A with subscript B, and replace subscript Rx for the reactive fluid in the inner pipe with subscript cool for the reactive cooling fluid in the annular region. Hence, modification of (4-76) yields ... 

Mass balance for reactant B, onto- tube Csoif-------= f Bflcooi, Xb)... 

The homogeneous diffusion model in spherical coordinates accounts for the fact that the surface area across which radial diffusion occurs increases qnadratically with dimensionless coordinate rj as one moves radially outward from the center of a spherically shaped catalyst. Once again, basic information for I a = /(i ) is obtained by integrating the dimensionless mass balance for reactant A with radial diffusion and chemical reaction... 

Solve the dimensionless mass transfer equation (i.e., the mass balance for reactant A) with homogeneous one-dimensional diffusion and zeroth-order irreversible chemical reaction to obtain an expression for 4molar density of reactant A. 

The dimensionless scaling factor in the mass transfer equation for reactant A with diffusion and chemical reaction is written with subscript j for the jth chemical reaction in a multiple reaction sequence. Hence, A corresponds to the Damkohler number for reaction j. The only distinguishing factor between all of these Damkohler numbers for multiple reactions is that the nth-order kinetic rate constant in the 7th reaction (i.e., kj) changes from one reaction to another. The characteristic length, the molar density of key-limiting reactant A on the external surface of the catalyst, and the effective diffusion coefficient of reactant A are the same in all the Damkohler numbers that appear in the dimensionless mass balance for reactant A. In other words. 

Calculation of the conversion of the reactant A (first-order irreversible reaction, Ta = kcj ) by means of the dispersion model is based on the mass balance for reactant A ... 

A mass balance for reactant A in a batch process is analogous to that for a first order chemical reaction. Assuming the apparent rate constant is constant in time, we find then ... 

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