The Momentum Balance Equation

The momentum of a body is the product of its mass and velocity. Since velocity is a vector, momentum is also a vector. The momentum balance equation describes the conservation of momentum it is also referred to as the equation of motion. 

Momentum can be transported by convection and conduction. Convection of momentum is due to the bulk flow of the fluid across the surface associated with it is a momentum flux. Conduction of momentum is due to intermolecular forces on each side of the surface. The momentum flux associated with conductive momentum transport is the stress tensor. The general momentum balance equation is also referred to as Cauchy s equation. The Navier-Stokes equations are a special case of the general equation of motion for which the density and viscosity are constant. The well-known Euler equation is again a special case of the general equation of motion it applies to flow systems in which the viscous effects are negligible. 

In polymer flow systems, the inertia and body forces are generally negligible. For these systems, the momentum balance equation in Cartesian coordinates can be written as  

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. 

An example of the use of momentum balance in pipe flow of a Newtonian fluid is 

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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... 

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. 

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... 

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. 

The CRE approach for modeling chemical reactors is based on mole and energy balances, chemical rate laws, and idealized flow models.2 The latter are usually constructed (Wen and Fan 1975) using some combination of plug-flow reactors (PFRs) and continuous-stirred-tank reactors (CSTRs). (We review both types of reactors below.) The CRE approach thus avoids solving a detailed flow model based on the momentum balance equation. However, this simplification comes at the cost of introducing unknown model parameters to describe the flow rates between various sub-regions inside the reactor. The choice of a particular model is far from unique,3 but can result in very different predictions for product yields with complex chemistry. 

For the EMS mode the momentum balance equation includes the additional forces F, and Fm. Because of the result E C = 0 the energy balance equation (15) of a plane EMS wave will on the other hand be the same as for the EM wave. 

When writing the boundary conditions for the above pair of simultaneous equations the heat transferred to the surroundings from the reactor may be accounted for by ensuring that the tube wall temperature correctly reflects the total heat flux through the reactor wall. If the reaction rate is a function of pressure then the momentum balance equation must also be invoked, but if the rate is insensitive or independent of total pressure then it may be neglected. 

The system of equations with initial and boundary conditions formulated above allows us to find the velocity distributions and pressure drop for the filled part of the mold. In order to incorporate effects related to the movement of the stream front and the fountain effect, it is possible to use the velocity distribution obtained285 for isothermal flow of a Newtonian liquid in a semi-infinite plane channel, when the flow is initiated by a piston moving along the channel with velocity uo (it is evident that uo equals the average velocity of the liquid in the channel). An approximate quasi-stationary solution can be found. Introduction of the function v /, transforms the momentum balance equation into a biharmonic equation. Then, after some approximations, the following solution for the function jt was obtained 285... 

There are only a few exact or analytical solutions of the momentum balance equations, and most of those are for situations in which the flow is unidirectional that is, the flow has only one nonzero velocity component. Some of these are illustrated below. We end the section with a presentation of the, which today is widely accepted to model the flows that occur during mold filling processes. 

The momentum balance equations can be written in a form that is valid for the Navier-Stokes equations as well as low Reynolds number non-Newtonian flow equations ... 

The internal energy balance equation for the fluid is based on the momentum balance equation. The assumption of local thermodynamic equilibrium will enable us to introduce the thermodynamic relationships linking intensive quantities in the state of equilibrium and to derive the internal energy balance equation on the basis of equilibrium partial quantities. By assuming that the diffusion is a slow phenomenon, 1" J/p pv2, the change of the total energy of all components per unit volume becomes... 

All of these rheological expressions (equations 13.16, 13.17, and 13.18) can be used to analyze the flow under the doctor blade in tape casting. Using the momentum balance equation 13.14 and one of the preceding equations for the shear stress, the differential equation which governs the velocity V,., can be determined. For Newtonian fluids, the solution is given by... 

Newtonian (and non-Newtonian) flow into a die is a complicated process due to the free surface of the fluid where the boimdary condition of the shear stress, being 0, is defined and this fi surface position changes with time. This boundary condition is used with the momentum balance equation to determine the velocity profile in the mold at any... 

Problem 1.2 (Worked Example) Compute the shear-rate profile in a cone-and-plate geometry. For a cone angle a of 0.1 radians, what percentage increase occurs in shear rate as one migrates from the plate to the cone Hint Look at the component of the momentum balance equation in spherical coordinates.)... 

We work in spherical coordinates and define the polar angle as 0 = nil — ot (see Fig. Al-2). Now we go to the momentum-balance equations in spherical coordinates in the Appendix. By symmetry, derivatives with respect to r and 0 are zero and Eq. (A-9) reduces to... 

In addition to the momentum balance equation (6), one generally needs an equation that expresses conservation of mass, but no other balance laws are required for so-called purely mechanical theories, in which temperature plays no role (as mentioned, balance of angular momentum has already been included in the definition of stress). If thermal effects are included, one also needs an equation for the balance of energy (that expresses the first law of thermodynamics energy is conserved) and an entropy inequality (that follows from the second law of thermodynamics the entropy of a closed system cannot decrease). The entropy inequality is, strictly speaking, not a balance law but rather imposes restrictions on the material models. 

The state of the art procedure for design of cyclic PSA or TSA processes using activated carbon adsorbents is to simultaneously solve the partial differential equations describing the mass, the heat, and the momentum balance equations for each step of the process using the appropriate initial and boundary conditions. These numerical calculations are carried out over many cycles for the process until a cyclic steady-state performance solution is achieved. Many different numerical integration algorithms are available for this purpose. The core input variables for the solution are multicomponent gas adsorption equilibria, heats, and kinetics for the system of interest [37]. 

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. 

The first term on the right indicates the pressure drop due to friction, and the second indicates the pressure drop due to change in velocity (kinetic energy). In many apphcations, the seeond term in Eq. 7.5.2 is small in comparison to flie first, and noting that u — v/A, the momentum balance equation reduces to... 

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