Continuity equations, general form

The mass transfer, KL-a for a continuous stirred tank bioreactor can be correlated by power input per unit volume, bubble size, which reflects the interfacial area and superficial gas velocity.3 6 The general form of the correlations for evaluating KL-a is defined as a polynomial equation given by (3.6.1). 

No assumption has been made as to continuity, in general, but it will now be assumed that all functions have continuous derivatives of order n + 1. Then the t satisfy a linear ordinary differential equation of order n + 1, which can be written in the form... 

In the TFM, both the gas phase and the solid phase are described as fully interpenetrating continua using a generalized form of the Navier-Stokes equations for interacting fluids. The continuity and momentum equations for the gas phase are given by expressions identical to Eqs. (40) and (41), except for the gas solid interaction term ... 

There is a nonzero mass source in the continuity equation, Sm, arising from the summation of all species equations. A general form of this source term is given in Table 1. Use has been made of the assumption that summation of interspecies diffusion within the gas phase is equal to zero. Specifically, one has... 

Based on the spherical control volume shown in Fig 3.15, derive the mass-continuity equation. Begin with the general statement of the Reynolds transport theorm in integral form (Eq. 2.19)... 

Begin with the general vector form of the continuity equation as... 

The objective is to derive a system of equations in general vector form that describes the overall gas-phase mass continuity and the species continuity equations for A and all other species k in the mixture. Assume that there is convective and diffusive transport of the species, but no chemical reaction. 

Derive the general vector form of the overall mass-continuity equation, recognizing that the droplet evaporation represents a source of mass to the system. 

For an incompressible fluid, the continuity equation is written in general vector form as... 

The purpose of this appendix is to spell out explicitly the Navier-Stokes and mass-continuity equations in different coordinate systems. Although the equations can be expanded from the general vector forms, dealing with the stress tensor T usually makes the expansion tedious. Expansion of the scalar equations (e.g., species or energy) are much less trouble. 

When DJuL is found to be large and the tracer response curve is skewed, as in Fig. 2.23b, but without a significant delay, a continuous stirred-tanks in series model (Section 2.3.2), may be found to be more appropriate. The tracer response curve will then resemble one of those in Fig. 2.8 or Fig. 2.9. The variance a2 of such a curve with a mean of tc is related to the number of tanks / by the expression a2 = t2/i (which can be shown for example by the Laplace transform method 7 from the equations set out in Section 2.3.2). Calculations of the mean and variance of an experimental curve can be used to determine either a dispersion coefficient Dl or a number of tanks i. Thus each of the models can be described as a one parameter model , the parameter being DL in the one case and i in the other. It should be noted that the value of i calculated in this way will not necessarily be integral but this can be accommodated in the more mathematically general form of the tanks-in-series model as described by Nauman and Buffham 7 . 

General Considerations. The continuity equations for a column vector of species taking place in a reaction-diffusion wave phenomenon take the form... 

Figure 1-6. Diagram showing the dimensions and the flux densities that form the geometric basis for the continuity equation. The same general figure is used to discuss water flow in Chapter 2 (Section 2.4F) and solute flow in Chapter 3 (Section 3.3A).

The Navier-Stokes equations and the flow continuity equation together give the general flow model other cases associate various forms of the energy conservation equation to this model. 

It is convenient to continue considering operator equations in general form. Following the Uterature, we define the quantity... 

Fixed Coordinate Approaches. In the fixed coordinate approach to airshed modeling, the airshed is divided into a three-dimensional grid for the numerical solution of some form of (7), the specific form depending upon the simplifying assumptions made. We classify the general methods for solution of the continuity equations by conventional finite difference methods, particle in cell methods, and variational methods. Finite difference methods and particle in cell methods are discussed here. Variational methods involve assuming the form of the concentration distribution, usually in terms of an expansion of known functions, and evaluating coeflBcients in the expansion. There is currently active interest in the application of these techniques (23) however, they are not yet suflBciently well developed that they may be applied to the solution of three-dimensional time-dependent partial differential equations, such as (7). For this reason we will not discuss these methods here. 

This equation was first obtained by Clapeyron, a French engineer who continued the work of Carnot. He derived the equation for the evaporation of a liquid. In its general form it was established by Clausius, and is therefore called the Clausius-Clapeyron, or briefly the Clausius equation. By means of this equation we can calculate the change in pressure dp produced by an arbitrary change in temperature dT from the quantities L, 2, Vj, and T, which can all be determined by experiment. 

A set of points M is said to be a -dimensional manifold if each point of M has an open neighborhood, which has a continuous 1 1 map onto an open set of of R , the set of all w-tuples of real numbers. Consider an w-dimensional Riemannian manifold with metric G. In an arbitrary coordinate system x, .. . , x", the volume -form is generally given by u> = dx a a dx . Here, g is the determinant of the metric in this basis, and a denotes the wedge or antisymmetric tensor product. For a flow field on the manifold prescribed by x = x) with density f x, t), a continuity equation for f x, t) can be obtained by considering the number of ensemble members >T t) within a volume Q of phase space given by... 

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