Control volume description

Until this point we have limited our thermodynamic description to simple (closed) systems. We now extend our analysis considering an open system. In this case the material control volume framework might not be a convenient choice for the fluid dynamic model formulation because of the computational effort required to localize the control volume surface. The Eulerian control volume description is often a better choice for this purpose. 

In accordance with the standard approach in continuum fluid mechanics, the abstract system or material control volume description is converted into a combined Eulerian framework by use of a generalization of the conventional Reynolds theorem. For general vector spaces the Re3molds theorem is written as [95, 96, 93, 94] ... 

In Chapter 2 we developed models based on analyses of systems that had simple inputs. The right-hand side was either a constant or it was simple function of time. In those systems we did not consider the cause of the mass flow—that was literally external to both the control volume and the problem. The case of the flow was left implicit. The pump or driving device was upstream from the control volume, and all we needed to know were the magnitude of the flow the device caused and its time dependence. Given that information we could replace the right-hand side of the balance equation and integrate to the functional description of the system. 

Pick s Law. Pick s law is a physically meaningful mathematical description of diffusion that is based on the analogy to heat conduction (Pick, 1855). Let us consider one side of our control volume, normal to the x-axis, with an area 4r, shown in Figure 2.3. Pick s law describes the diffusive flux rate as... 

The terminology of computational techniques is descriptive, but one needs to know what is being described. Table 7.1 lists some common terms with a definition relative to mass transport. Most computational techniques in fluid transport are described with control volume elements, wherein the important process to be computed is the transport across the interfaces of small control volumes. The common control volumes are cubes, cylindrical shells, triangular prisms, and trapezoidal prisms, although any shape can be used. We will present the control volume technique. 

In this form one sees an analogy in the vorticity equation to the other transport equations— a substantial-derivative description of advective transport, a Laplacian describing the diffusive transport, and possibly a source term. It is interesting to observe that the vorticity equation does not involve the pressure. Since pressure always exerts a normal force that acts through the center of mass of a fluid packet (control volume), it cannot alter the rotation rate of the fluid. That is, pressure variations cannot cause a change in the vorticity of a flow field. 

The discussion above provides a brief qualitative introduction to the transport and fate of chemicals in the environment. The goal of most fate chemists and engineers is to translate this qualitative picture into a conceptual model and ultimately into a quantitative description that can be used to predict or reconstruct the fate of a chemical in the environment (Figure 27.1). This quantitative description usually takes the form of a mass balance model. The idea is to compartmentalize the environment into defined units (control volumes) and to write a mathematical expression for the mass balance within the compartment. As with pharmacokinetic models, transfer between compartments can be included as the complexity of the model increases. There is a great deal of subjectivity to assembling a mass balance model. However, each decision to include or exclude a process or compartment is based on one or more assumptions—most of which can be tested at some level. Over time the applicability of various assumptions for particular chemicals and environmental conditions become known and model standardization becomes possible. 

Usually, the intrinsic average reflects the real physical property or quantity such as density and velocity, while the phase average gives a pseudoproperty or quantity based on the selection of control volume. Phase averages are used to construct the continuum of each phase to which Eulerian description can be applied. 

The mathematical description of simultaneous heat and mass transfer and chemical reaction is based on the general conservation laws valid for the mass of each species involved in the reacting system and the enthalpy effects related to the chemical transformation. The basic equations may be derived by balancing the amount of mass or heat transported per unit of time into and out of a given differential volume element (the control volume) together with the generation or consumption of the respective quantity within the control volume over the same period of time. The sum of these terms is equivalent to the rate of accumulation within the control volume ... 

Rapid developments have taken place in the fleld of laser anemome-try, and this technique has been applied successfully in a number of studies on measurements in gaseous flames. In these studies, the gas flow was seeded with micron or submicron particles, and the velocity of these particles was taken to be representative of the velocity of the local gas flow. For the study reported here, a laser anemometer was adapted for the special problem of measurements in a spray flame which initially contains a polydisperse cloud of droplets up to 300 /un in diameter. Droplets and carbon particles are present, and seeded particles are added to the annular air flow. For the particles larger than 1 /un, significant differences exist between velocities of particles and surrounding gas. A complete description of the velocity field requires simultaneous measurement of velocity and size of individual particles. This has not yet been achieved, and, for this study, the velocity of all particles passing through the measurement control volume of the laser anemometer are reported. 

In the finite volume method, discretized equations are obtained by integrating the governing transport equations over a finite control volume (CV). In this section, general aspects of the method are briefly discussed using a generic conservation equation for quantity, 0. Patankar (1980), Versteeg and Malalasekara (1995) and Ferziger and Peric (1995) may be referred to for a more detailed description. 

To convert the system balance description into an Eulerian control volume formulation the transport theorem is employed (i.e., expressed in terms of the... 

The next task in our model derivation is to transform the system description (3.55) into an Eulerian control volume formulation by use of an extended form of the generalized transport theorem (see App A). For phase k the generalized Leibnitz theorem is written ... 

It follows from the continuum assumption that the integrands in (3.415) are continuous and differentiable functions, so the integral theorems of Leibnitz and Gauss (see app. A) can be applied transforming the system description into an Eulerian control volume formulation. The governing mixture... 

Analysis of rate-limited processes usually begins with a differential mass balance. One description of this balance for our stationary control volume for a component A is ... 

In the first description of mass conservation for our system, we consider on arbitrarily chosen volume element (here called a control volume) of fixed position and shape as illustrated in Fig. 2-2. Thus, at each point on its surface, there is a mass flux of fluid pu n through the surface. With n chosen as the outer unit normal to the surface, this mass flux will be negative at points where fluid enters the volume element and positive where it exits. 

Figure 0 illustrates the control volume over which the balance equations are written. A detailed description of the basic equations have been already presented in Leclerc and Schweich. Thus the equations will be written only in ffieir final form. 

The following discussion represents a detailed description of the mass balance for any species in a reactive mixture. In general, there are four mass transfer rate processes that must be considered accumulation, convection, diffusion, and sources or sinks due to chemical reactions. The units of each term in the integral form of the mass transfer equation are moles of component i per time. In differential form, the units of each term are moles of component i per volnme per time. This is achieved when the mass balance is divided by the finite control volume, which shrinks to a point within the region of interest in the limit when aU dimensions of the control volume become infinitesimally small. In this development, the size of the control volume V (t) is time dependent because, at each point on the surface of this volume element, the control volnme moves with velocity surface, which could be different from the local fluid velocity of component i, V,. Since there are several choices for this control volume within the region of interest, it is appropriate to consider an arbitrary volume element with the characteristics described above. For specific problems, it is advantageous to use a control volume that matches the symmetry of the macroscopic boundaries. This is illustrated in subsequent chapters for catalysts with rectangular, cylindrical, and spherical symmetry. 

The description of diffusion and chemical reaction at the boundaries of the control volume is rather complex. 

Figure 5,3b Mathematical description for material balance in control volume the convective flux.

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