Momentum balance applications

Material balances, often an energy balance, and occasionally a momentum balance are needed to describe an adsorption process. These are written in various forms depending on the specific application and desire for simplicity or rigor. Reasonably general material balances for various processes are given below. An energy balance is developed for a fixea bed for gas-phase application and simphfied for liquid-phase application. Momentum balances for pressure drop in packed beds are given in Sec. 6. 

A momentum balance for the flow of a two-phase fluid through a horizontal pipe and an energy balance may be written in an expanded form of that applicable to single-phase fluid flow. These equations for two-phase flow cannot be used in practice since the individual phase velocities and local densities are not known. Some simplification is possible if it... 

Consider the vertical column of fluid illustrated in Fig. 4-1, but now imagine it to be on an elevator that is accelerating upward with an acceleration of az, as illustrated in Fig. 4-3. Application of the momentum balance to the slice of fluid, as before, gives... 

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. 

Our discussion of multiphase CFD models has thus far focused on describing the mass and momentum balances for each phase. In applications to chemical reactors, we will frequently need to include chemical species and enthalpy balances. As mentioned previously, the multifluid models do not resolve the interfaces between phases and models based on correlations will be needed to close the interphase mass- and heat-transfer terms. To keep the notation simple, we will consider only a two-phase gas-solid system with ag + as = 1. If we denote the mass fractions of Nsp chemical species in each phase by Yga and Ysa, respectively, we can write the species balance equations as... 

An application of an internal momentum balance to determine the pressure drop in a sudden expansion is given in Section 2.4. 

The advanced process control strategies that are most applicable to the optimization of the distillation process are usually based on white-box modeling, where the theoretical dynamic models are derived on the basis of the mass, energy, and momentum balances of this well-understood process. Although the optimization techniques described here can improve productivity and profitability by 25%, this goal will only be achieved if the distillation process is treated as a single and integrated unit operation and the variables, such as flows, levels, pressures, etc., become only constraints, and the controlled and optimized variables are productivity and profitability. 

In the 1960s, the start of application of computers to the practice of marine research gave a pulse to the development of numerical diagnostic hydrodynamic models [33]. In them, the SLE (or the integral stream function) field is calculated from the three-dimensional density field in the equation of potential vorticity balance over the entire water column from the surface to the bottom. The iterative computational procedure is repeated until a stationary condition of the SLE (or the integral stream function) is reached at the specified fixed density field. Then, from equations of momentum balance, horizontal components of the current vector are obtained, while the continuity equation provides the calculations of the vertical component. The advantage of this approach is related to the absence of the problem of the choice of the zero surface and to the account for the coupled effect of the baroclinicity of... 

In practical applications considering fluid particles one normally neglects all the interface terms in (3.91), except the interfacial tension terms. Hence, the jump momentum balance is given by ... 

In most reaction operations, it is not necessary to use the momentum balance equation. For gas-phase reaction, when the pressure of the reacting fluid varies substantially and it affects the reaction rates, we apply the momentum balance equation to express the pressure variation. This occurs in rare applications (e.g., long tubular reactor with high velocity). The last section of Chapter 7 covers the application of the momentum balance equation for plug-flow reactors. 

Equation 7.5.20 provides a relation for the change in the pressure along a packed-bed reactor. In most applications, the gas velocity in packed-bed reactors is much lower than the sound velocity. Hence, to formulate the reactor design, we should solve design equations (Eq. 7.1.1), energy balance equation (Eq. 7.1.16), and momentum balance equation (Eq. 7.5.20) simultaneously, subject to specified initial conditions. The solutions provide Z, 0, and P/Pq as functions of dimensionless space time, t. 

To illustrate the application of the momentum balance, we consider first two very simple examples not involving fluids. 

SOME STEADY-FLOW APPLICATIONS OF THE MOMENTUM BALANCE... 

The application of this steady-flow one-dimensionai momentum balance will be illustrated by several examples. 

In this section we first discuss the two types of fluid flow that can occur laminar and turbulent flow. Also, the Reynolds number used to characterize the regimes of flow is considered. Then in Sections 2.6, 2.7, and 2.8 the overall mass balance, energy balance, and momentum balance are covered together with a number of applications. Finally, a... 

A quite common application of the overall momentum-balance equation is the case of a section of a conduit with its axis in the x direction. The fluid will be assumed to be flowing at steady state in the control volume shown in Fig. 2.6-3 and also shown in Fig. 2.8-1. Equation (2.8-13) for the x direction becomes as follows since u = v. ... 

Another application of the overall momentum balance is shown in Fig. 2.8-2 for a flow system with fluid entering a conduit at point 1 inclined at an angle of a, relative to the... 

Two-fluid models presented earlier are applicable for two-phase flow of gas/Newtonian liquid systems. It has been successfully determined that the momentum balance equations for Newtonian fluids also apply to non-Newtonian fluids [52]. 

The analysis of many flow problems can be simplifled by considering only one component of the equation of motion, the one in the direction of flow. Further simplifying assumptions are often necessary in order to solve the problem. In the analysis of isothermal processes, only two balance equations are needed, the mass and momentum balance. In order to solve the problem, additional information is required. This is information on how the fluid deforms under application of various stresses. This information is described by the constitutive equation of the fluid see also Section 6.2 on melt flow properties. 

Both the mechanical energy equation, (1.6.12), and momentum balance equation, (17.7), are used for the analysis of isothermal flow problems. Most applications fall into one of three classes ... 

The momentum balance can be obtained by application of Newton s second law on a moving fluid element. Over a volume element of a chemical reactor, the balance of momentum in direction / can be written as ... 

Newton s law is applied in order to derive the linear momentum balance equation. Newton s law states that the sum of all forces equals the rate of change of linear momentum. The derivation/application of this law ultimately provides information critical to any meaningful analysis of most chemical reactors. Included in this information are such widely divergent topics as flow mechanisms, the Reynolds number, velocity profiles, two-phase flow, prime movers such as fans, pumps and compressors, pressure drop, flow measurement, valves and fittings, particle dynamics, flow through porous media and packed beds, fluidization— particularly as it applies to fluid-bed and fixed bed reactors, etc. Although much of this subject matter is beyond the scope of this text, all of these topics are treated in extensive detail by Abulencia and Theodore. ... 

A one-dimensional form of the momentum balance equation for engineering applications in flow networks and engineered equipment is (Todreas and Kazimi, 1990)... 

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