Balances control volume

Figure 16 Pump energy balance control volume...

For a first approach the energy balance equation has to be solved for a simplified stack model. The stack is assumed to be ideally insulated and the stack is in a steady state operation. The energy balance control volume is given in Fig. 20.4. 

To illustrate the use of the momentum balance, consider the situation shown in Figure 21c in which the control volume is bounded by the pipe wall and the cross sections 1 and 2. The forces acting on the fluid in the x-direction are the pressure forces acting on cross sections 1 and 2, the shear forces acting along the walls, and the body force arising from gravity. The overall momentum balance is... 

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. 

Mass balance Apphed to the control volume, the principle of consei vation of mass may be written as (Whitaker, Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981)... 

Momentum Balance Since momentum is a vector quantity, the momentum balance is a vector equation. Where gravity is the only body force acting on the fluid, the hnear momentum principle, apphed to the arbitraiy control volume of Fig. 6-3, results in the following expression (Whitaker, ibid.). 

A balanced equation for every extensive property in each control volume may be written as ... 

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. 

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. 

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. 

For the generalized control volume shown in Figure 2-33, and entropy balance can be stated as follows ... 

FIGURE 1.1 Control volume for total mass balance. 

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... 

While true, this result is not helpful. The derivation of Equation (1.6) used the entire reactor as the control volume and produced a result containing the average reaction rate, In piston flow, a varies with z so that the local reaction rate also varies with z, and there is no simple way of calculating a-Equation (1.6) is an overall balance applicable to the entire system. It is also called an integral balance. It just states that if more of a component leaves the reactor than entered it, then the difference had to have been formed inside the reactor. 

A differential balance written for a vanishingly small control volume, within which t A is approximately constant, is needed to analyze a piston flow reactor. See Figure 1.4. The differential volume element has volume AV, cross-sectional area A and length Az. The general component balance now gives... 

Export processes are often more complicated than the expression given in Equation 7, for many chemicals can escape across the air/water interface (volatilize) or, in rapidly depositing environments, be buried for indeterminate periods in deep sediment beds. Still, the majority of environmental models are simply variations on the mass-balance theme expressed by Equation 7. Some codes solve Equation 7 directly for relatively large control volumes, that is, they operate on "compartment" or "box" models of the environment. Models of aquatic systems can also be phrased in terms of continuous space, as opposed to the "compartment" approach of discrete spatial zones. In this case, the partial differential equations (which arise, for example, by taking the limit of Equation 7 as the control volume goes to zero) can be solved by finite difference or finite element numerical integration techniques. 

The population follows the change in the granule size distribution as granules are bom, die, grow, or enter or leave a control volume, as illustrated in Fig. 34. The number of particles between volume v and v - dv is n(v)dv, where n(v) is the number frequency size distribution, or the number density. The population balance for granulation is then given by... 

Figure 34. Control volume for performing a number balance on granules. (From Litster and Ennis, 1994.)...

CFD may be loosely thought of as computational methods applied to the study of quantities that flow. This would include both methods that solve differential equations and finite automata methods that simulate the motion of fluid particles. We shall include both of these in our discussions of the applications of CFD to packed-tube simulation in Sections III and IV. For our purposes in the present section, we consider CFD to imply the numerical solution of the Navier-Stokes momentum equations and the energy and species balances. The differential forms of these balances are solved over a large number of control volumes. These small control volumes when properly combined form the entire flow geometry. The size and number of control volumes (mesh density) are user determined and together with the chosen discretization will influence the accuracy of the solutions. After boundary conditions have been implemented, the flow and energy balances are solved numerically an iteration process decreases the error in the solution until a satisfactory result has been reached. 

The balance is made with respect to a control volume which may be of finite (V) or of differential (dV) size, as illustrated in Figure 1.3(a) and (b). The control volume is bounded by a control surface. In Figure 1.3, m, F, and q are mass (kg), molar (mol), and volumetric (m3) rates of flow, respectively, across specified parts of the control surface,6 and Q is the rate of heat transfer to or from the control volume. In (a), the control volume could be the contents of a tank, and in (b), it could be a thin slice of a cylindrical tube. 

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. 

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... 

Batch and continuous processes may also be compared by examining their governing mass-balance relations. As an elaboration of equation 1.5-1, a general mass balance may be written with respect to a control volume as ... 

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... 

For a continuous-flow reactor, such as a CSTR, the energy balance is an enthalpy (H) balance, if we neglect any differences in kinetic and potential energy of the flowing stream, and any shaft work between inlet and outlet. However, in comparison with a BR, the balance must include the input and output of H by the flowing stream, in addition to any heat transfer to or from the control volume, and generation or loss of enthalpy by reaction within the control volume. Then the energy (enthalpy) equation in words is... 

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 ... 

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