The General Conservation Equations

The method we have described for setting up PDEs has a cumbersome feature attached to it. It must be repeated each time there is a change in geometrical configuration or in the process conditions. A switch from rectangular to cylindrical coordinates, for example, requires a new balance to be made. So does the inclusion of reaction terms. 

both del dot and del square tend to be scalar expressions. Del u, on the other hand, is a vector. 

Some thought will lead us to the conclusion that del dot terms will likely arise in flowing systems, whereas the Laplacian will most probably appear in the description of diffusion processes. This is indeed the case. 

The use of these operators in the formulahon of mass balances leads to the following generalized conservation equations. We have, for the component mass balance 

Two restrictions apply to these expressions. First, they are confined to incompressible flow, i.e., systems in which density changes can be neglected, such as liquid flow or gas flow involving low pressure drops. Second, the formulation requires continuity of the concentration within the flow field. Systems in turbulent flow in which xmdergoes an abrupt transition from linear gradient in the film to a constant value in the fluid core caimot be accommodated by these expressions. We must, in these cases, revert to the use of the classical shell balance. 

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The general conservation equation for any process system can be written as ... 

From the fluid continuity equation (8.3.13), the first term on the left-hand side cancels out with the last term on the right-hand side, giving the general conservation equation... 

Since the volume v is arbitrarily selected, the general conservation equation in a single phase of fluid can be written in the form... 

The conservation equations of mass, momentum, and energy of a single-phase flow can be obtained by using the general conservation equation derived previously. 

A conservation equation for Z may be derived from the general conservation equations. Under assumptions 1, 2, and 3 of Section 1.3, if the binary diffusion coefficients are equal then equation (1-37) is obtained, and from equation (1-4) it is readily found that... 

When the second order approximations to the pressure tensor and the heat flux vector are inserted into the general conservation equation, one obtains the set of PDEs for the density, velocity and temperature which are called the Burnett equations. In principle, these equations are regarded as valid for non-equilibrium flows. However, the use of these equations never led to any noticeable success (e.g., [28], pp. 150-151) [39], p. 464), merely due to the severe problem of providing additional boundary conditions for the higher order derivatives of the gas properties. Thus the second order approximation will not be considered in further details in this book. 

In conclusion, when these first order expressions for the pressure tensor and the heat flow vector are substituted into the general conservation equations (2.202), (2.207) and (2.213), we obtain the following set of partial differential equations ... 

The flow of an SCF under isothermic conditions is governed by the general conservation equation... 

The general conservation equations for the solid and reactant species for the volume reaction model in dimensionless form are as follows ... 

First, we will briefly review the general conservation equations for mass and momentum, for continuous media. Then we will use these equations to describe the transport of electrons and holes in a semiconductor. The results will correspond to those which are used in device modeling, such as in the SEDAN ( ) and MINIMOS (4) programs, and will demonstrate the role of momentum conservation. 

With increasing complexity of the models, one gets more accurate predictions but also the computation times become considerably larger. Keeping in mind that many of these evaporation models have been developed for use in CFD spray simulations, where hundreds of thousands of droplets have to be considered, computational costs become a primary issue. Therefore, the models discussed in this chapter are limited to the second and third category. The presentation starts with the general conservation equations for mass, species and energy, from which the simplified models are derived. 

Substitution of (19.7)-(19.10) into (19.6) leads to the general conservation equation in integral form... 

We will develop an appropriate mathematical model for the pilot scale reactor studied by Clough and Ramirez (1976). The mathematical model is derived by simplification of the general conservation equations. The reduction of the general equations is made possible by order-of-magnitude scaling agreements based upon reasonable values for system dependent and independent variables. An important approximation which is implied in the model development is that the flow through the packed bed reactor can be described by both axial and radial dispersion mechanisms. 

To aid the reader in the use of these equations, we have compiled a "dictionary" of the operator symbols, which provides a translation into scalar form for the three principal geometries (rectangular, cylindrical, and spherical). We can use this dictionary. Table 2.3, to extract several important subsidiary relations. For example, in the absence of flow and reaction, the general conservation equation becomes... 

Use the general conservation Equation 2.24a and Table 2.2 to derive the differential equation for uniform diffusion and reaction in a... 

In transient diffusion, the concentration of a species varies, as we have seen, with both time and distance. The underlying process of diffusion may take place in isolahon or it may be accompanied by a chemical reaction, by flow, or by both reaction and flow. These more complex cases are not taken up here, and we limit ourselves instead to the consideration of purely diffusive processes. Furthermore, with one or two exceptions, the treatment is confined to a single spatial coordinate represented by the Cartesian x- (or z-) axis or by the radial variable r. The last is used in formulating diffusion in a sphere, or in the radial direction of a cylinder. Pick s equation for these three cases can be deduced from the general conservation equation (Equation 2.24a) and Table 2.3. They are as follows ... 

This is the classical and much-studied expression known as Laplace s equation. It can be obtained from Pick s equation by omithng the transient term, or from the general conservation equation (Equahon 2.24a) by omitting transient, reaction, and flow terms. 

Note that the mass conservation (or continuity) equation (Equation [15.2]) is equivalent to the general conservation equation with 0 assigned to be unity. These equations are non-linear and cannot be solved analytically in the vast majority of practical cases (Patankar, 1980 Versteeg and Malalasekera, 1995). The equations must be linearised and solved over many small control volumes (the computational mesh). For determination of the flow-field, CFD simulations require input of geometry, boundary conditions and fluid properties. 

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