Material control volume

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

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. 

In this chapter we will apply the conservation of mass principle to a number of different kinds of systems. While the systems are different, by the process of analysis they will each be reduced to their most common features and we will find that they are more the same than they are different. When we have completed this chapter, you will understand the concept of a control volume and the conservation of mass, and you will be able to write and solve total material balances for single-component systems. 

Therefore, the sum of the component balances is the total material balance while the net rate of change of any component s mass within the control volume is the sum of the rate of mass input of that component minus the rate of mass output these can occur by any process, including chemical reaction. This last part of the dictum is important because, as we will see in Chapter 6, chemical reactions within a control volume do not create or destroy mass, they merely redistribute it among the components. In a real sense, chemical reactions can be viewed from this vantage as merely relabeling of the mass. 

Consider a small control volume V = SxSySz (Fig. 4.27), where the inner heat generation is Q "(T) (heat production/volume) and the heat conductivity is A(T). The material is assumed to be homogeneous and isotropic, and the internal heat generation and thermal conductivity are functions of temperature. 

In reactor design, we are interested in chemical reactions that transform one kind of mass into another. A material balance can be written for each component however, since chemical reactions are possible, the rate of formation of the component within the control volume must now be considered. The component balance for some substance A is... 

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 ... 

Figure 1.3 Control volumes of finite (V) size (a) and of differential (dV) size (b) with material inlet and outlet streams and heat transfer (G. SQ)...

The input and output terms of equation 1.5-1 may each have more than one contribution. The input of a species may be by convective (bulk) flow, by diffusion of some kind across the entry point(s), and by formation by chemical reaction(s) within the control volume. The output of a species may include consumption by reaction(s) within the control volume. There are also corresponding terms in the energy balance (e.g., generation or consumption of enthalpy by reaction), and in addition there is heat transfer (2), which does not involve material flow. The accumulation term on the right side of equation 1.5-1 is the net result of the inputs and outputs for steady-state operation, it is zero, and for unsteady-state operation, it is nonzero. 

In further considering the implications and uses of these two points of view, we may find it useful to distinguish between the control volume as a region of space and the system of interest within that control volume. In doing this, we consider two ways of describing a system. The first way is with respect to flow of material ... 

FI) Continuous-flow system There is at least one input stream and one output stream of material the mass inside the control volume may vary. 

Consider a reaction represented by A +. . . - products taking place in a batch reactor, and focus on reactant A. The general balance equation, 1.51, may then be written as a material balance for A with reference to a specified control volume (in Figure 2.1, this is the volume of the liquid). 

To obtain an expression for tj, we first derive the continuity equation governing steady-state diffusion of A through the pores of the particle. This is based on a material balance for A across the control volume consisting of the thin strip of width dx shown in Figure 8.10(a). We then solve the resulting differential equation to obtain the concentration profile for A through the particle (shown in Figure 8.10(b)), and, finally, use this result to obtain an expression for tj in terms of particle, reaction, and diffusion characteristics. 

For a first-order reaction, A - products, and a spherical particle, the material-balance equation corresponding to equation 8.5-7, and obtained by using a thin-shell control volume of inside radius r, is... 

For continuous operation of a CSTR as a closed vessel, the general material balance equation for reactant A (in the reaction A vr C +. .. ), with a control volume... 

The material balance for a PFR is developed in a manner similar to that for a CSTR, except that the control volume is a differential volume (Figure 2.4), since properties change continuously in the axial direction. The material balance for a PFR developed in Section 2.4.2 is from the point of view of interpreting rate of reaction. Here, we turn the situation around to examine it from the point of view of the volume of reactor, V. Thus equation 2.4-4, for steady-state operation involving reaction represented by A+. ..- vcC +. . ., may be written as a differential equation for reactant A as follows ... 

The volume of the recycle PFR may be obtained by a material balance for A around the differential control volume dV. Equating molar flow input and output, we obtain... 

Consider a material balance for A around the differential control volume shown in... 

The derivation of the material-balance or continuity equation for reactant A is similar to that of equations 19.4-48 and -49 for nonreacting tracer A, except that steady state replaces unsteady state (cA at a point is not a function of t), and a reaction term must be added. Thus, using the control volume in Figure 19.15, we obtain the equivalent of equation 19.4-48 as ... 

Consider a material balance for reactant A around the control volume dV in Figure 20.1. The fraction of fluid entering the vessel between t and t + dt is E(t) dt that is, if the actual flow through the vessel is q0,... 

Figure 23.7 Schematic representation of control volume for material balance for bubbling-bed reactor model...

However, by the thermally thin approximation, T(x,t) T(t) only. A control volume surrounding the thin material with the conservation of energy applied, Equation (3.45), gives (for a solid at equal pressure with its surroundings)... 

The approach is to formulate the entire burning problem using conservation laws for a control volume. The condensed phase will use control volumes that move with the vaporization front. This front is the surface of a regressing liquid or solid without char, or it is the char front as it extends into the virgin material. The original thickness, l, does not change. While the condensed phase is unsteady, the gas phase, because of its lower density, is steady or quasi-steady in that its steady solution adjusts to the instantaneous input of the condensed phase. 

In all cases the weight of all material within the control volume must be included in the force-momentum balance, although in many cases it will be a small force. Gravity is an external agency and it may be considered to act across the control surface. The momentum flows and all forces crossing the control surface must be included in the balance in the same way that material flows are included in a material balance. 

Obviously, control decisions cannot be made on the basis of volume alone, but certainly the higher-volume materials deserve early scrutiny and consideration. 

The term on the left side of the equation is the accumulation term, which accounts for the change in the total amount of species iheld in phase /c within a differential control volume. This term is assumed to be zero for all of the sandwich models discussed in this section because they are at steady state. The first term on the right side of the equation keeps track of the material that enters or leaves the control volume by mass transport. The remaining three terms account for material that is gained or lost due to chemical reactions. The first summation includes all electron-transfer reactions that occur at the interface between phase k and the electronically conducting phase (denoted as phase 1). The second summation accounts for all other interfacial reactions that do not include electron transfer, and the final term accounts for homogeneous reactions in phase k. 

The fundamental physical laws governing motion of and transfer to particles immersed in fluids are Newton s second law, the principle of conservation of mass, and the first law of thermodynamics. Application of these laws to an infinitesimal element of material or to an infinitesimal control volume leads to the Navier-Stokes, continuity, and energy equations. Exact analytical solutions to these equations have been derived only under restricted conditions. More usually, it is necessary to solve the equations numerically or to resort to approximate techniques where certain terms are omitted or modified in favor of those which are known to be more important. In other cases, the governing equations can do no more than suggest relevant dimensionless groups with which to correlate experimental data. Boundary conditions must also be specified carefully to solve the equations and these conditions are discussed below together with the equations themselves. 

Model Equations. If the material in the bottle is taken as the control volume (Figure 1), application of the law of conservation of mass yields... 

All evaporating material is intercepted by the hemisphere, because the molecular beam exists only at angles less than 90° from the center line. Therefore, the mass flow from the source must equal the flux of the molecular beam leaving the control volume across the surface of the hemisphere. Under these conditions, the mass flow from the source is given by... 

Model Equations to Describe Component Balances. The design of PVD reacting systems requires a set of model equations describing the component balances for the reacting species and an overall mass balance within the control volume of the surface reaction zone. Constitutive equations that describe the rate processes can then be used to obtain solutions to the model equations. Material-specific parameters may be estimated or obtained from the literature, collateral experiments, or numerical fits to experimental data. In any event, design-oriented solutions to the model equations can be obtained without recourse to equipment-specific fitting parameters. Thus translation of scale from laboratory apparatus to production-scale equipment is possible. 

The rate at which the material leaves the control volume is governed by two mechanisms reflection and emission of the adsorbed species into the... 

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