Mass balance fluid mechanics

In practice, the loss term AF is usually not deterrnined by detailed examination of the flow field. Instead, the momentum and mass balances are employed to determine the pressure and velocity changes these are substituted into the mechanical energy equation and AFis deterrnined by difference. Eor the sudden expansion of a turbulent fluid depicted in Eigure 21b, which deflvers no work to the surroundings, appHcation of equations 49, 60, and 68 yields... 

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)... 

Fick s second law of diffusion can be derived from Fick s first law by using a mass balance approach. Consider the differential fluid element shown in Figure 4. This differential fluid element is simply a small cube of liquid or gas, with volume Ax Ay Az, and will be defined as the system for the mass balance. Assume now that component A enters the cube at position x by diffusion and exits the cube at x + Ax by the same mechanism. For the moment, assume that no diffusion occurs in the y or z directions and that the faces of the cube that are perpendicular to the y and z axes thus are impermeable to the diffusion of A. Under these conditions, the component mass balance for A in this system is... 

For the student, this is a basic text for a first-level course in process engineering fluid mechanics, which emphasizes the systematic application of fundamental principles (e.g., macroscopic mass, energy, and momentum balances and economics) to the analysis of a variety of fluid problems of a practical nature. Methods of analysis of many of these operations have been taken from the recent technical literature, and have not previously been available in textbooks. This book includes numerous problems that illustrate these applications at the end of each chapter. 

We win develop mass balances in terms of mixing in the reactor. In one limit the reactor is stirred sufficiently to mix the fluid completely, and in the other limit the fluid is completely unmtxed. In any other situation the fluid is partially mixed, and one cannot specify the composition without a detailed description of the fluid mechanics. We wiU consider these nonideal reactors in Chapter 8, but until then all reactors wiU be assumed to be either completely mixed or completely unmixed. 

As before, we consider only two ideal continuous reactors the PFTR and the CSTR, because any other reactor involves detailed consideration of the fluid mechanics. For a phase a the mass balance if the fluid is unmixed is... 

For any more complex flow pattern we must solve the fluid mechanics to describe the fluid flow in each phase, along with the mass balances. The cases where we can still attempt to find descriptions are the nonideal reactor models considered previously in Chapter 8, where laminar flow, a series of CSTRs, a recycle TR, and dispersion in a TR allow us to modify the ideal mass-balance equations. 

The themiodynamics of flow is based on mass, energy, and entropy balances, which have been developed in Chaps. 2 and 5. The application of tliese balances to specific processes is considered in this chapter. The discipline imderlying tlie study of flow is fluid mechanics, which encompasses not only the balances of thermodynamics but also the linear-momentum principle (Newton s second law). This makes fluid mechanics a broader field of study. The distinction between thermodynamics problems and fluid-mechanics problems depends on whether this principle is required for solution. Those problems whose solutions depend only on mass conservation and on the laws of thermodynamics are commonly set apart from the study of fluid mechanics and are treated in courses on thermodynamics. Fluid mechanics then deals with the broad spectmm of problems which require application of the momentum principle. This division is arbitrary, but it is traditional and convenient. 

The steady-state fluid mechanics problem is solved using the Fluent Euler-Euler multiphase model in the fluid domains. Mass, momentum and energy balances, the general forms of which are given by eqn. (4), (5), and (6), are solved for both the liquid and the gas phases. In solid zones the energy equation reduces to the simple heat conduction problem with heat source. By convention, / =1 designates the H2S04 continuous liquid phase whereas H2 bubbles constitute the dispersed phase 0 =2). 

You will be solving for Ca ), but you cannot evaluate the rate of reaction in Eq. (8.45) because you do not know Ca.s- You need a mass balance relating the rate of mass transfer to the catalyst to the rate of reaction. One form of that is Eq. (8.46), where is a mass transfer coefficient in units of m/s, determined from correlations derived in fluid mechanics and mass transfer courses, a is the surface area exposed per volume of the reactor (m /m ), and k is a rate of reaction rate constant (here rn /kmol s). Other formulations are possible, too ... 

The principles of physics most useful in the applications of fluid mechanics are the mass-balance, or continuity, equations the linear- and angular-raomentum-balance equations and the mechanical-energy balance. They may be written in differential form, showing conditions at a point within a volume element of fluid, or in integrated form applicable to a finite volume or mass of fluid. Only the integral equations are discussed in this chapter. 

Notice again that solving the rate-of-change form of the mass balance requires more inform.ation (here the rate of reaction) and more effort than solving the difference form of the mass balance. However, we also get more information—the amount of each species present as a function of time. In Sec. 2.4, which is optional and more difficult, we consider another, even deeper level of description, where not only is time allowed to vary, but the system is.not spatially homogeneous that is, the composition in the reactor varies from point to point. However, this section is not for the faint-hearted and is best considered after a course iri fluid mechanics. B... 

The Microscopic Mass Balance Equations in Thermodynamics and Fluid Mechanics 43... 

THE MICROSCOPIC MASS BALANCE EQUATIONS IN THERMODYNAMICS AND FLUID MECHANICS ... 

In this chapter we continue the quantitative development of thermodynamics by deriving the energy balance, the second of the three balance equations that will be used in the thermodynamic description of physical, chemical, and (later) biochemical processes. The mass and energy balance equations (and the third balance equation, to be developed in the following chapter), together with experimental data and information about the process, will then be used to relate the change in a system s properties to a change in its thermodynamic state. In this context, physics, fluid mechanics, thermodynamics, and other physical sciences are all similar, in that the tools of each are the same a set of balance equations, a collection of experimental observ ations (equation-of-state data in thermodynamics, viscosity data in fluid mechanics, etc.), and the initial and boundary conditions for each problem. The real distinction between these different subject areas is the class of problems, and in some cases the portion of a particular problem, that each deals with. 

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