Momentum balance expressions

Most authors who have studied the consohdation process of soflds in compression use the basic model of a porous medium having point contacts which yield a general equation of the mass-and-momentum balances. This must be supplemented by a model describing filtration and deformation properties. Probably the best model to date (ca 1996) uses two parameters to define characteristic behavior of suspensions (9). This model can be potentially appHed to sedimentation, thickening, cake filtration, and expression. 

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 procedure adopted here consists of taking a momentum balance on an element of fluid. The resulting Momentum Equation involves no assumptions concerning the nature of the flow. However, it includes an integral, the evaluation of which requires a knowledge of the velocity profile ux = f(y). At this stage assumptions must be made concerning the nature of the flow in order to obtain realistic expressions for the velocity profile. 

Momentum balance equations are of importance in problems involving the flow of fluids. Momentum is defined as the product of mass and velocity and as stated by Newton s second law of motion, force which is defined as mass times acceleration is also equal to the rate of change of momentum. The general balance equation for momentum transfer is expressed by... 

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... 

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... 

If the relative velocity is sufficiently low, the fluid streamlines can follow the contour of the body almost completely all the way around (this is called creeping flow). For this case, the microscopic momentum balance equations in spherical coordinates for the two-dimensional flow [vr(r, 0), v0(r, 0)] of a Newtonian fluid were solved by Stokes for the distribution of pressure and the local stress components. These equations can then be integrated over the surface of the sphere to determine the total drag acting on the sphere, two-thirds of which results from viscous drag and one-third from the non-uniform pressure distribution (refered to as form drag). The result can be expressed in dimensionless form as a theoretical expression for the drag coefficient ... 

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

Using the expressions for the shearing stresses in the momentum balance for the control volume and taking the limit of the resulting equations as the size of the control volume tends to zero then gives the following set of equations [6],[7],... 

This chapter has been concerned with flows in wb ch the buoyancy forces that arise due to the temperature difference have an influence on the flow and heat transfer values despite the presence of a forced velocity. In extemai flows it was shown that the deviation of the heat transfer rate from that which would exist in purely forced convection was dependent on the ratio of the Grashof number to the square of the Reynolds number. It was also shown that in such flows the Nusselt number can often be expressed in terms of the Nusselt numbers that would exist under the same conditions in purely forced and purely free convective flows. It was also shown that in turbulent flows, the buoyancy forces can affect the turbulence structure as well as the momentum balance and that in turbulent flows the heat transfer rate can be decreased by the buoyancy forces in assisting flows whereas in laminar flows the buoyancy forces essentially always increase the heat transfer rate in assisting flow. Some consideration was also given to the effect of buoyancy forces on internal flows. 

The resulting set of 10 equations, assuming toroidal symmetry and replacing the radial component of the ion momentum balance equation by an ad hoc diffusions ansatz (likewise the other radial transport coefficients are replaced by ad hoc anomalous expressions) is the basis for most current edge plasma simulation models. These anomalous ad-hoc coefficients are free model parameters. They, and their empirical scalings, can be determined by comparison with experimental plasma profile data, if one can be sure that all other terms in the equations, and in particular the source terms Sm resulting from atomic and molecular processes, are accurately known and implemented. 

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... 

By combining the force balance Eq. (66) with the spout momentum balance Eqs. (43) and (44) and the particle entrainment Eq. (59), Lefroy and Davidson were able to derive the following expression for spout diameter ... 

This is the relation for the momentum balance in the x-direction, and is known as the x-momentnm equation. Note that we would obtain the. same result if we used momentum flow rates for the left-hand side of this equation instead of mass limes acceleration. If there is a body force acting in the x-direction, it can be added to the right side of the equation provided that it is expressed per unit volume of the fluid. 

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. 

In [26] the velocity fields and thereby the power for stirrers with simple geometry (anchor stirrer and gate stirrer) have been calculated for the laminar case (highly viscous liquid with Newtonian or pseudoplastic flow behavior) by the help of the numerical solution of the continuity and momentum balance in connection with the rheological constitutive equation. In the case of Newtonian fluids the power characteristic in the laminar flow range could be calculated for all three stirrers with the help of the expression ... 

The alternative VOF models designed to describe stratified and dispersed flows with internal motions in all the phases are based on the whole domain formulation. The basis for this approach is a set of equations valid for the whole calculation domain, in which the governing mass and momentum balances are expressed as [132, 214, 32, 183, 164, 92, 222, 227] ... 

Substituting the expressions found for the two velocity components (7.23) and (7.24) into the continuity (7.20) and the angular momentum balance (7.19) gives an expression for the dependence of torque on fluid pumping ... 

The rate of transport of particles across a surface at a point, expressed as number per unit time per unit area, is culled the jinx at the point. Common dimensions for the flux are paTticle.s/cm. sec. Expressions for the diffusion flux and diffusion coefflcienl (hat apply to submicron particles are derived from first principles in this chapter. The presence of an external force Retd acting on the particles leads to an additional term in the flux. The transport of particles larger than about a micron is analyzed by solving a momentum balance that incorporates the external force fields. 

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... 

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. 

We consider first a cylindrical reactor of uniform diameter D. To derive an expression for the pressure drop, we write the steady-state momentum balance equation for a reactor element of length dL and cross-sectional area A ... 

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