Control volume defined

Consider again a reaction represented by A +. .. - products taking place in a single-stage CSTR (Figure 2.3(a)). The general balance equation, 1.5-1, written for A with a control volume defined by the volume of fluid in the reactor, becomes... 

Consider the control volume defined by the dashed lines in C. Employ Equation (3.23) ... 

Since all substrate is consumed in the thin layer, we can assume that the layer has a flat geometry. Consider a control volume defined by the element at r with thickness dr, as shown in Figure 3.7. 

A key concept in describing environmental chemical fate is the principle of mass preservation. A chemical in a par-ticirlar location at a specific time can remain at that location, can be transported elsewhere, or can be transformed into another chemical. A mass balance can be formulated for a specific subsection of the environment, called the system or control volume. Defining a system boundary involves a decision of what is considered a part of the system and what is part of its srrnoundings. The mass balance then accormts for how much chemical crosses the system botmdary and how much chemical is generated and lost within the system dtuing a particular time interval (Fig. 5) ... 

Elementary single-component systems are those that have just one chemical species or material involved in the process. Filling of a vessel is an example of this kind. The component can be a solid liquid or gas. Regardless of the phase of the component, the time dependence of the process is captured by the same statement of the conservation of mass within a well-defined region of space that we will refer to as the control volume. 

I FIGURE 11.4 Aspect ratio and biasing for control volumes. Aspect ratio is defined as Ax, Ay for cell (. Biasing in the x direction is defined as Ax, /Ax,. 

The basic conservation laws, as well as the transport models, are applied to a system (sometimes called a control volume ). The system is not actually the volume itself but the material within a defined region. For flow problems, there may be one or more streams entering and/or leaving the system, each of which carries the conserved quantity (e.g., Q ) into and out of the system at a defined rate (Fig. 1-2). Q may also be transported into or out of the system through the system boundaries by other means in addition to being carried by the in and out streams. Thus, the conservation law for a flow problem with respect to any conserved quantity Q can be written as follows ... 

As discussed in Chapter 1, the basic principles that apply to the analysis and solution of flow problems include the conservation of mass, energy, and momentum in addition to appropriate transport relations for these conserved quantities. For flow problems, these conservation laws are applied to a system, which is defined as any clearly specified region or volume of fluid with either macroscopic or microscopic dimensions (this is also sometimes referred to as a control volume ), as illustrated in Fig. 5-1. The general conservation law is... 

In the IBM, the presence of the solid boundary (fixed or moving) in the fluid can be represented by a virtual body force field -rp( ) applied on the computational grid at the vicinity of solid-flow interface. Considering the stability and efficiency in a 3-D simulation, the direct forcing scheme is adopted in this model. Details of this scheme are introduced in Section II.B. In this study, a new velocity interpolation method is developed based on the particle level-set function (p), which is shown in Fig. 20. At each time step of the simulation, the fluid-particle boundary condition (no-slip or free-slip) is imposed on the computational cells located in a small band across the particle surface. The thickness of this band can be chosen to be equal to 3A, where A is the mesh size (assuming a uniform mesh is used). If a grid point (like p and q in Fig. 20), where the velocity components of the control volume are defined, falls into this band, that is... 

A control volume is a volume specified in transacting the solution to a problem typically involving the transfer of matter across the volume s surface. In the study of thermodynamics it is often referred to as an open system, and is essential to the solution of problems in fluid mechanics. Since the conservation laws of physics are defined for (fixed mass) systems, we need a way to transform these expressions to the domain of the control volume. A system has a fixed mass whereas the mass within a control volume can change with time. 

The Reynolds transport theorem is a general expression that provides the mathematical transformation from a system to a control volume. It is a mathematical expression that generally holds for continuous and integrable functions. We seek to examine how a function fix, y, z, t), defined in space over x, y, z and in time t, and integrated over a volume, V, can vary over time. Specifically, we wish to examine... 

Let us see how to represent changes in properties for a system volume to property changes for a control volume. Select a control volume (CV) to be identical to volume V t) at time t, but to have a different velocity on its surface. Call this velocity, w. Hence, the volume will move to a different location from the system volume at a later time. For example, for fluid flow in a pipe, the control volume can be selected as stationary (w = 0) between locations 1 and 2 (shown in Figure 3.4, but the system moves to a new location later in time. Let us apply the Reynolds transport theorem, Equation (3.9), twice once to a system volume, V(t), and second to a control volume, CV, where CV and V are identical at time t. Since Equation (3.9) holds for any well-defined volume and surface velocity distribution, we can write for the system... 

This is the relationship we seek. It says that the rate of change of a property / defined over a system volume is equal to the rate of change for the control volume plus a correction for matter that carries / in or out. This follows since v w is the relative velocity of matter on the boundary of the control volume. If v — w = 0, no matter crosses the boundary. As we proceed in applying Equation (3.12) to the conservation laws and in identifying a specific property for/, we will bring more meaning to the process. We will consider the system to be composed of a fluid, but we need not be so restrictive, since our analysis will apply to all forms of matter. 

To consider the control volume form of the conservation of mass for a species in a reacting mixture volume, we apply Equation (2.14) for the system and make the conversion from Equation (3.12). Here we select/ = pt, the species density. In applying Equation (3.13), v must be the velocity of the species. However, in a mixture, species can move by the process of diffusion even though the bulk of the mixture might be at rest. This requires a more careful distinction between the velocity of the bulk mixture and its individual components. Indeed, the velocity v given in Equation (3.13) is for the bulk mixture. Diffusion velocities, Vi, are defined as relative to this bulk mixture velocity v. Then, the absolute velocity of species i is given as... 

The conservation of momentum or Newton s second law applies to a particle or fixed set of particles, namely a system. The velocity used must always be defined relative to a fixed or inertial reference plane. The Earth is a sufficient inertial reference. Therefore, any control volume associated with accelerating aircraft or rockets must account for any differences associated with how the velocities are measured or described. We will not dwell on these differences, since we will not consider such noninertial applications. 

It is necessary first to define the region or control volume for which the momentum equation is to be written. In this example, it is convenient to select the fluid within the nozzle as that control volume. The control volume is defined by drawing a control surface over the inner surface of the nozzle and across the flow section at the nozzle inlet and the outlet. In this way, the nozzle itself is excluded from the control volume and external forces acting on the body of the nozzle, such as atmospheric pressure, are not involved in the momentum equation. This interior control surface is shown in Figure 1.9(a). 

For most numerically solved models, a control-volume approach is used. This approach is based on dividing the modeling domain into a mesh. Between mesh points, there are finite elements or boxes. Using Taylor series expansions, the governing equations are cast in finite-difference form. Next, the equations for the two half-boxes on either side of a mesh point are set equal to each other hence, mass is rigorously conserved. This approach requires that all vectors be defined at half-mesh points, all scalars at full-mesh points, and all reaction rates at quarter-mesh points. The exact details of the numerical methods can be found elsewhere (for example, see ref 273) and are not the purview of this review article. The above approach is essentially the same as that used in CFD packages (e.g.. Fluent) or discussed in Appendix C of ref 139 and is related to other numerical methods applied to fuel-cell modeling. ... 

Let us consider a water volume defined by two river cross sections at x and x + dx, respectively. As a starting point, we assume that the mean total (dissolved plus sorbed) concentration of a chemical i in the water of this slice of the river, Ct(x), is controlled by only two types of processes (1) in-situ production or consumption of the chemical, R and (2) fluxes at the water surface or at the sediment-water interface, T ... 

Consider the system and control volume as illustrated in Fig. 2.2. The Eulerian control volume is fixed in an inertial reference frame, described by three independent, orthogonal, coordinates, say z,r, and 9. At some initial time to, the system is defined to contain all the mass in the control volume. A flow field, described by the velocity vector (t, z,r, 9), carries the system mass out of the control volume. As it flows, the shape of the system is distorted from the original shape of the control volume. In the limit of a vanishingly small At, the relationship between the system and the control volume is known as the Reynolds transport theorem. 

The expression V ndA is the scalar product (dot product) between the velocity vector and the outward-pointing normal unit vector that describes the control surface. Since n is defined as an outward-normal unit vector, a positive value of fcs r)p ndA indicates that. V leaves the control volume. By definition, however, N remains in the system. 

The shear-stress convention is a bit more complicated to explain. In a differential control volume, the shear stresses act as a couple that produces a torque on the volume. The sign of the torques defines the positive directions of the shear stresses. Assume a right-handed coordinate system, here defined by (z, r, 9). The shear-stress sign convention is related to ordering of the coordinate indexes as follows a positive shear xzr produces a torque in the direction, a positive xrg produces a torque in the z direction, and a positive x z produces a torque in the r direction. Note also, for example, that a positive xrz produces a torque in the negative 6 direction. 

The equation for the value of the velocity at each node is based on a momentum balance for each control volume. In the interior of the domain, the control volume has a momentum flux crossing each of the four sides. The flux depends on the sign of the velocity gradient and the outward-normal unit vector that defines the face orientation. In discrete, integral form, the two-dimensional difference equation emerges as... 

Figure 12. Control volume of surface reaction zone. The component molar balances are defined by equations 27 and 28.

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. 

The construction of a mass balance model follows the general outline of this chapter. First, one defines the spatial and temporal scales to be considered and establishes the environmental compartments or control volumes. Second, the source emissions are identified and quantified. Third, the mathematical expressions for advective and diffusive transport processes are written. And last, chemical transformation processes are quantified. This model-building process is illustrated in Figure 27.4. In this example we simply equate the change in chemical inventory (total mass in the system) with the difference between chemical inputs and outputs to the system. The inputs could include numerous point and nonpoint sources or could be a single estimate of total chemical load to the system. The outputs include all of the loss mechanisms transport... 

The actual distributions of microbubble diameters in sea water, measured by Weitendorf in the optically defined control volume, ranged between 20 and 117 pm. Within this diameter range, the usual number of measured microbubbles per cm3 was of the order of 10 to 100, yielding an approximate microbubble concentration of 104-105/liter for these ocean experiments (ref. 

The scope definition is similar to the definition of the control volume in the thermodynamic analysis or the battery limits in process design, and for the LCA in terms of space and time (e.g., we follow the use of product X in the process from the raw materials to the time it is disposed by the consumer. Throughout the lifetime of the product, we analyze the environmental burden). The reasons for the study are also clearly defined (e.g., is the study necessary to make a decision about a process ), as well as an answer must be given as to who is performing the study and for whom. Consider the following hypothetical example ... 

---