The momentum balance

Remember that all of this chapter is simply the manipulation and application of Eq. 7.1. 

The mass elements of a flowing fluid transfer momentum. This is understood to be the product of the mass and velocity. A mass element dM, which flows at a velocity of transports a momentum w dM = w gdV. The total momentum / transported in a fluid of volume V(/,) is therefore 

According to Newton s second law of mechanics the change in momentum of a body with time is equal to the resultant of all the forces acting on the body 

The forces F- attacking the body can be split into two classes in body forces which are proportional to the mass and in surface forces which are proportional to the surface area. 

The body forces have an effect on all the particles in the body. They are far ranging forces and are caused by force fields. An example is the earth s gravitational field. The acceleration due to gravity g acts on each molecule, so that the force of gravity on a fluid element of mass AM is 

It is proportional to the mass of the fluid element. The body force is defined by 

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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... 

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

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.). 

The momentum balance for homogeneous flow can be factored to a form which enables integration as ... 

HEM for Two-Phase Pipe Discharge With a pipe present, the backpressure experienced by the orifice is no longer qg, but rather an intermediate pressure ratio qi. Thus qi replaces T o iri ihe orifice solution for mass flux G. ri Eq. (26-95). Correspondingly, the momentum balance is integrated between qi and T o lo give the pipe flow solution for G,p. The solutions for orifice and pipe now must be solved simultaneously to make G. ri = G,p and to find qi and T o- This can be done explicitly for the simple case of incompressible single-phase (hquid) inclined or horizontal pipe flow The solution is implicit for compressible regimes. 

A staggered temporal mesh can also be constructed from the normal temporal mesh in a way similar to that described for the spatial temporal mesh, as shown in Fig. 9.7. The staggered temporal mesh points are at the midpoints of the mesh intervals. Some codes integrate the momentum balance equation, (9.3), on the staggered temporal mesh while the normal temporal mesh is used to integrate the other governing equations [18], [20], [21]. 

The energy conservation equation is not normally solved as given in (9.4). Instead, an evolution equation for internal energy is used [9]. First an evolution equation for the kinetic energy is derived by taking the dot product of the momentum balance equation with the velocity and integrating the resulting differential equation. The differential equation is... 

The momentum balance equation for the solid particles in the direction... 

Note that e is defined in terms of the components of the velocities, not the vector velocities, whereas the momentum balance is defined in terms of the vector velocities. To solve Equations 2-32 and 2-33 when all the velocities are not colinear, one writes the momentum balances along the principal axes and solves the resulting equations simultaneously. 

Equation 2.58 is the momentum balance for horizontal turbulent flow ... 

The terms which must be considered in the momentum balance for the X-direction are ... 

The momentum balance equation at the evaporation front has (neglecting the effect of viscous tension and changing surface tension along of meniscus) the following form ... 

The Chapman-Enskog theory of flow In a one-component fluid yields the following approximation to the momentum balance equation (Jil). 

For the steady, planar Couette flow to be examined In a later section, the momentum balance equation yields... 

Force and velocity are however both vector quantities and in applying the momentum balance equation, the balance should strictly sum all the effects in three dimensional space. This however is outside the scope of this text and the reader is referred to more standard works in fluid dynamics. 

In the fluid model the momentum balance is replaced by the drift-diffusion approximation, where the particle flux F consists of a diffusion term (caused by density gradients) and a drift term (caused by the electric field ) ... 

The fluid model is a description of the RF discharge in terms of averaged quantities [268, 269]. Balance equations for particle, momentum, and/or energy density are solved consistently with the Poisson equation for the electric field. Fluxes described by drift and diffusion terms may replace the momentum balance. In most cases, for the electrons both the particle density and the energy are incorporated, whereas for the ions only the densities are calculated. If the balance equation for the averaged electron energy is incorporated, the electron transport coefficients and the ionization, attachment, and excitation rates can be handled as functions of the electron temperature instead of the local electric field. 

A separated flow model for stratified flow was presented by Taitel and Dukler (1976a). They indicated analytically that the liquid holdup, R, and the dimensionless pressure drop, 4>G, can be calculated as unique f unctions of the Lockhart-Martinelli parameter, X (Lockhart and Martinelli, 1949). Considering equilibrium stratified flow (Fig. 3.37), the momentum balance equations for each phase are... 

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... 

Note that because momentum is a vector, this equation represents three component equations, one for each direction in three-dimensional space. If there is only one entering and one leaving stream, then = m0 = m. If the system is also at steady state, the momentum balance becomes... 

We will apply the steady state momentum balance to a fluid in plug flow in a tube, as illustrated in Fig. 5-6. (The stream tube may be bounded by either solid or imaginary boundaries the only condition is that no fluid crosses the boundaries other that through the inlet and outlet planes.) The shape of the cross section does not have to be circular it can be any shape. The fluid element in the slice of thickness dx is our system, and the momentum balance equation on this system is... 

Here, rw is the stress exerted by the fluid on the wall (the reaction to the stress exerted on the fluid by the wall), and Wp is the perimeter of the wall in the cross section that is wetted by the fluid (the wetted perimeter ). After substituting the expressions for the forces from Eq. (5-43) into the momentum balance equation, Eq. (5-42), and dividing the result by — pA, where A = Ax, the result is... 

It should be noted that in evaluating the forces acting on the system, the effect of the external pressure transmitted through the boundaries to the system from the surrounding atmosphere was not included. Although this pressure does result in forces that act on the system, these forces all cancel out, so the pressure that appears in the momentum balance equation is the net pressure in excess of atmospheric, e.g., gage pressure. 

This result can also be derived by equating the shear stress for a Newtonian fluid, Eq. (6-9), to the expression obtained from the momentum balance for tube flow, Eq. (6-4), and integrating to obtain the velocity profile ... 

The total friction loss in an orifice meter, after all pressure recovery has occurred, can be expressed in terms of a loss coefficient, ATr, as follows. With reference to Fig. 10-12, the total friction loss is P — P3. By taking the system to be the fluid in the region from a point just upstream of the orifice plate (Pj) to a downstream position where the stream has filled the pipe (P3), the momentum balance becomes... 

Many engineering operations involve the separation of solid particles from fluids, in which the motion of the particles is a result of a gravitational (or other potential) force. To illustrate this, consider a spherical solid particle with diameter d and density ps, surrounded by a fluid of density p and viscosity /z, which is released and begins to fall (in the x = — z direction) under the influence of gravity. A momentum balance on the particle is simply T,FX = max, where the forces include gravity acting on the solid (T g), the buoyant force due to the fluid (Fb), and the drag exerted by the fluid (FD). The inertial term involves the product of the acceleration (ax = dVx/dt) and the mass (m). The mass that is accelerated includes that of the solid (ms) as well as the virtual mass (m() of the fluid that is displaced by the body as it accelerates. It can be shown that the latter is equal to one-half of the total mass of the displaced fluid, i.e., mf = jms(p/ps). Thus the momentum balance becomes... 

The stress terms F at the front of the plug and B at the back of the plug, dependent on the pressure drop and powder properties, can be developed from a momentum balance but often times they are set equal to each other. Using the momentum balance... 

The source terms on the right-hand sides of Eqs. (25)-(29) are defined as follows. In the momentum balance, g represents gravity and p is the modified pressure. The latter is found by forcing the mean velocity field to be solenoidal (V (U) = 0). In the turbulent-kinetic-energy equation (Eq. 26), Pk is the source term due to mean shear and the final term is dissipation. In the dissipation equation (Eq. 27), the source terms are closures developed on the basis of the form of the turbulent energy spectrum (Pope, 2000). Finally, the source terms... 

While the mass balances given above are relatively straightforward (assuming that a suitable closure can be derived for the mass-transfer terms), the momentum balances are significantly more complicated. In their simplest forms, they can be written as follows ... 

When a fluid flows through a bed of particles, interactions between fluid and particles lead to a frictional pressure drop, (- AP). Calculation of (- AP) enables determination of both L and D, for a given W (or F). This calculation is done by means of the momentum balance, which results in the pressure gradient given by... 

Momentum Balance in Dimensionless Variables For pipe flow, it is necessary to solve the momentum balance. The momentum balance is simplified by using the following dimensionless variables ... 

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