Generic balance equation

Rather than considering the individual transport processes separately governing the balance of mass, momentum, species, etc., we focus now on a single, abstract, generic conservation law, known under the name general balance law, governing the transport of all extensive physical properties, in continuous three-dimensional media, and then in discontinuous media. Ultimately, the generic balance equations will be applied in later chapters to specific physical circumstances. 

What does a generic balance equation describe ... 

Note that the rate of formation of A is rA, as defined in section 1.4 for a reactant, this is a negative quantity. The rate of disappearance of A is (-rA), a positive quantity. It is this quantity that is used subsequently in balance equations and rate laws for a reactant. For a product, the rate of formation, a positive quantity, is used. The symbol rA may be used generically in the text to stand for rate of reaction of A where the sign is irrelevant and correspondingly for any other substance, whether reactant or product. 

By a natural extension of the concepts developed in the previous chapter (structural estimability), if the generic rank of the composite matrix (Aj A2) is not less than n (n number of unmeasured variables), then the system does not include structural singularities. Furthermore, if all the unmeasured nodes are determinable, then there are no isolated variables, which cannot be computed from the balance equations. 

Equation (23) represents the total energy balance or first law of thermodynamics. It includes all forms of energy transport. An independent energy equation, which does not represent a generic balance relation, is obtained by performing the operation v (equations of motion) and is... 

The generic balance relations and the derived relations presented in the preceding section contain various diffusion flux tensors. Although the equation of continuity as presented does not contain a diffusion flux vector, were it to have been written for a multicomponent mixture, there would have been such a diffusion flux vector. Before any of these equations can be solved for the various field quantities, the diffusion fluxes must be related to gradients in the field potentials . 

We denote by 07 = Hi/HijS the dimensionless variables corresponding to the energy flow rates Hiy i = 1,..., N (the subscript s denotes steady-state values). Appending a generic representation of the overall and component material-balance equations, with xtfc IRm being the material-balance variables, the overall mathematical model of the process in Figure 6.1 becomes... 

To solve the mass balance equations, we considered the extractor being divided in N layers, being obtained a system discretisation in relation to the bed height. The generic layer n is described by the following two equations ... 

To establish the integral form of the basic conservation laws for mass, momentum and energy, the fundamental approach is to start out from a system analysis and then transform the balance equations into a control volume analysis by use of the transport theorem. However, to achieve a more compact presentation of this theory it is customary to start out from a generic Eulerian form of the governing equations. That is, the material control volume analysis is disregarded. 

Consider the generic transport equation for the property -0 and assume that the velocity held and the fluid properties are known. The starting point for the FVM is the integral form of the balance equation ... 

After the given approximations of the terms have been substituted into the generic equation, and dividing all the terms by At, the balance equation yields ... 

Figure 2 Simplified example of a generic phototrophic organism for setting up balance equations.

The relations (10.146) and (10.147) constitute the generic, volumetric, and surface balance equations for continuous three-dimensional media at each point of the continuum with a surface of discontinuity. 

Wynn (1992,1993) have presented short-cut studies of solids process flowsheeting. Custom-written software to solve the model equations has been used by a number of authors. It has also been proposed that symbohc manipulation packages, like Mathematical, can be used to solve the population balance equations as a stand-alone modelling environment or as a set of modules to be linked to a generic process simulation package such as SPEEDUP (Hounslow, 1989 Sheikh and Jones, 1996). 

Given these general equations, one approach to mathematical modeling is to apply appropriately and simphly these equations for the particular problem. This involves a clear statement of assumptions and an estimation of the relative importance of various terms (e.g. using dimensional analysis see Chapter 4). In Example 3.3, the energy balance equation has been simplifed from its generic form to an equation describing the heat conduction in one dimension (Example 3.1). 

The mole balances for gas-phase reactions are given in Table 6-2 in terms of the number of moles (batch) or molar flow races for the generic rate law for the generic reaction, Equation (2-1). The molar flow rates for each species Fj are obtained from a mole balance on each species, as given in Table 6-2. For example, for a plug-flow reactor... 

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