Overall Momentum Balance

Ocily n. - 1 of the n equations (4.1) are independent, since both sides vanish on suinming over r, so a further relation between the velocity vectors V is required. It is provided by the overall momentum balance for the mixture, and a well known result of dilute gas kinetic theory shows that this takes the form of the Navier-Stokes equation... 

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... 

The overall momentum balance of droplets discharging through liquid can be expressed as ... 

The reactor is modeled by three partial differential equations component balances on A and B [Eqs. (6.1) and (6.2)] and an energy balance [Eq. (6.3) for an adiabatic reactor or Eq. (6.4) for a cooled reactor]. The overall heat transfer coefficient U in the cooled reactor in Eq. (6.4) is calculated by Eq. (6.5) and is a function of Reynolds number Re, Eq. (6.6). Equation (6.7) is used for pressure drop in the reactor using the friction factor /given in Eq. (6.8). The dynamics of the momentum balance in the reactor are neglected because they are much faster than the composition and temperature dynamics. A constant... 

Recently, Agrawal et al. (2006) had modeled the LDPE reactor as an ideal plug flow reactor and presented all the model equations and parameters for use by researchers. The model equations include ordinary differential equations for overall and component mass balances, energy balance and momentum balance. The reactor model of Agrawal et al. (2006) is adopted, and cost expressions and economic objectives are... 

MACROSCOPIC MOMENTUM BALANCE. A momentum balance, similar to the overall mass balance, can be written for the control volume shown in Fig. 4.3, assuming that flow is steady and unidirectional in the x direction. The sum of all forces acting on the fluid in the x direction, by the momentum principle, equals the increase in the time rate of momentum of the flowing fluid. That is to say, the sum of forces acting in the x direction equals the difference between the momentum leaving with the fluid per unit time and that brought in per unit time by the fluid, or... 

Determining the local values of P and t for all the internal surface of even a relatively simple device like this nozzle is a formidable task. For a complicated shape, it is beyond our current ability. Nevertheless, we computed the overall force (which we were seeking) by the momentum balance fairly easily. 

Although the system as a whole is not moving in the y direction, some parts of it are, because of the internal fuel flows Thus, the overall system has some y-directed momentum, but since this presumably is not changing with time, d mVy)ldt =0. During the motor-starting period, this simplification is not correct, but for a rocket standing still it is always small compared with the other terms in the momentum balance. 

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... 

In this discussion overall or macroscopic mass balances were made because we wish to describe these balances from outside the enclosure. In this section on overall mass balances, some of the equations presented may have seemed quite obvious. However, the purpose was to develop the methods which should be helpful in the next sections. Overall balances will also be made on energy and momentum in the next sections. These overall balances do not tell us the details of what happens inside. However, in Section 2.9 a shell momentum balance will be made to obtain these details, which will give us the velocity distribution and pressure drop. To further study these details of the processes occurring inside the enclosure, differential balances rather than shell balances can be written and these are discussed in other later Sections 3.6 to 3.9 on differential equations of continuity and momentum transfer. Sections 5.6 and 5.7 on differential equations of energy change and boundary-layer flow, and Section 7.5B on differential equations of continuity for a binary mixture. 

A momentum balance can be written for the control volume shown in Fig. 2.6-3, which is somewhat similar to the overall mass-balance equation. Momentum, in contrast to mass and energy, is a vector quantity. The total linear momentum vector P of the total mass Af of a moving fluid having a velocity of v is... 

Substituting Equations (2.8-2), (2.8-6), and (2.8-7) into (2.8-3), the overall linear momentum balance for a control volume becomes... 

B Overall Momentum Balance in Flow System in One Direction... 

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... 

D Overall Momentum Balance for Free Jet Striking a Fixed Vane... 

In Sections 2.6,2.7, and 2.8 overall mass, energy, and momentum balances allowed us to solve many elementary problems on fluid flow. These balances were done on an arbitrary finite volume sometimes called a control volume. In these total energy, mechanical energy, and momentum balances, we only needed to know the state of the inlet and outlet streams and the exchanges with the surroundings. 

These overall balances were powerful tools in solving various flow problems because they did not require knowledge of what goes on inside the finite control volume. Also, in the simple shell momentum balances made in Section 2.9, expressions were obtained for... 

In order to derive the basic equation for a laminar or turbulent boundary layer, a small control volume in the boundary layer on a flat plate is used as shown in Fig. 3.10-5. The depth in the z direction is b. Flow is only through the surfacesand dj and also from the top curved surface at 8. An overall integral momentum balance using Eq. (2.8-8) and overall integral mass balance using Eq. (2.6-6) are applied to the control volume inside the boundary layer at steady state and the final integral expression by von Karman is (B2, S3)... 

In Sections 3.6 and 3.7 we derived a differential equation of continuity and a differential equation of momentum transfer for a pure fluid. These equations were derived because overall mass, energy, and momentum balances made on a finite volume in the earlier parts of Chapter 2 did not tell us what goes on inside a control volume. In the overall balances performed, a new balance was made for each new system studied. However, it is often easier to start with the differential equations of continuity and momentum transfer in general form and then to simplify the equations by discarding unneeded terms for each specific problem. 

In the above conservations equations, expressions for the fluxes are required. This has already been accounted for in the energy and momentum balances (Eqs. 11 and 12, respectively) to provide second-order equations the reason being that they are highly coupled and remain general for many systems. However, understanding the fluxes and transport expressions for the material species including ions (Eq. 2) is critical in determining the resistances in the ionic and electronic phases and the overall response of the porous electrode thus, they are discussed in more detail. 

As the particle traverses a bend its velocity changes. Haag (1967) and Kovacs (1967) have analyzed the behavior of particles as they slow down in bends. The overall pressure loss in a bend may be made up of this deceleration of the particle in the bend plus the acceleration length term. Consider the horizontal bend shown in Fig. 5-4. The angle 0 varies from 0 to 90° for a right-angle turn in the horizontal plane. A single particle momentum balance on this system can be written as... 

The lack of an intrinsic angular momentum balance in the classical theory has seriously handicapped application of mixture theory to problems where constituent spin is a significant factor in the overall dynamics. This would include boundary control of constituent spin or particulate saltation [4], Thus the Twiss-Eringen theory appears to be an important development for flow situations in which strong spin of one or more constituents may occur. 

An important concern in FCC riser submodels is how to calculate the slip factor, ip, and the average voidage, s, of the riser. The slip factor is simply defined as the ratio between gas velocity and catalyst particle velocity. The slip factor plays an important part in determining the residence time of reactions, and thus, affects the overall conversion in the riser. Harriot describes a slip factor range of 1.2 to 4.0 for most FCC risers but also indicates that there is no reliable correlation available for prediction [44]. Previous authors have used a variety of approaches including constant shp factor [45], multiple slip factors [46] and correlations [47]. An alternative approach is to include additional momentum balance equations for the gas phase and catalyst phase [48]. This approach allows users to calculate velocity profiles for each phase and the overall pressure drop in the riser directly. 

To analyze this problem more completely, we write a mass balance on the solute, an overall mass balance on all species present, and a momentum balance to describe the flow. We then imagine small perturbations in the concentration or in the flow. If our balances indicate that these small perturbations get smaller with time, then the system is stable. If these perturbations grow with time, then the system is unstable, and free convection will occur. 

We now turn to the overall mass and momentum balances. Because the solute is sparingly soluble its dissolution has a negligible effect on the overall mass balance. Because the process is in steady state, the accumulation is zero. Thus the mass balance is... 

This overall mass balance can be supplemented by an x-momentum balance which in symbolic terms becomes... 

A great deal of use has been made of the overall momentum, mechanical-energy, and total-energy balances in various forms. Some of these applications are mentioned in the succeeding sections. 

As discussed in Chapters 2 and 3, in the integral method it is assumed that the boundary layer has a definite thickness and the overall or integrated momentum and thermal energy balances across the boundary layer are considered. In the case of flow over a body in a porous medium, if the Darcy assumptions are used, there is, as discussed before, no velocity boundary layer, the velocity parallel to the surface near the surface being essentially equal to the surface velocity given by the potential flow solution. For flow over a body in a porous medium, therefore, only the energy integral equation need be considered. This equation was shown in Chapter 2 to be ... 

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