Balance equations momentum

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 numerical solution of the energy balance and momentum balance equations can be combined with flow equations to describe heat transfer and chemical reactions in flow situations. The simulation results can be in various forms numerical, graphical, or pictorial. CFD codes are structured around the numerical algorithms and, to provide easy assess to their solving power, CFD commercial packages incorporate user interfaces to input parameters and observe the results. CFD... 

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

Consider the mass, thermal and momentum balance equations. The key assumption of the present analysis is that the Knudsen number of the flow in the capillary is sufficiently small. This allows one to use the continuum model for each phase. Due to the moderate flow velocity, the effects of compressibility of the phases, as well as mechanical energy, dissipation in the phases are negligible. Assuming that thermal conductivity and viscosity of vapor and liquid are independent of temperature and pressure, we arrive at the following equations ... 

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... 

Full mathematical models also include momentum balance equations, which have been omitted here. 

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... 

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... 

Evaluation of p and Km2 requires determination of the void fraction and the two-phase pressure drop. Crossflow is determined from the appropriate lateral momentum balance equation. The interchange due to mixing, represented by w is determined by the turbulent transverse fluctuating flow rate per foot of axial length (lb/hr ft), where... 

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. 

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 energy and momentum balance equations are drawn across planes at points 0, 1, and 2, as illustrated for a general case in Fig. 23-31. 

Moment of inertia, exponents of dimensions in absolute, gravitational, and engineering systems, 8 584t Momentum balance equation, 21 347-348 Momentum equation, 11 738, 739-743 Momentum flowmeters, 11 671 Monactin, chelating agent, 5 710 Monazite, 5 671 14 636 24 756-757 digestion of, 14 638 processing, 5 673 Monel, 14 14 Monel alloy, 9 595 Monel alloy 400, 17 100 Monel cathodes, 11 837 Monensin, 20 132, 133, 135, 136, 137, 139 Monensin A, 20 120... 

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. 

We now turn to the momentum and energy balance of the electromagnetic field. In analogy with conventional deductions, Eq. (1) is multiplied vectorially by and Eq. (2), by eoE. The sum of the resulting equations is then rearranged into the local momentum balance equation... 

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. 

The ash simulation model (Konstandopoulos et al., 2003 Rodriguez-Perez et al., 2004) consists of ash transport and ash layer evolution equations describing the interaction between ash deposition and re-entrainment in the channels along with the gas mass balance and momentum balance equations in the inlet and outlet channels of the DPF. Ash re-entrainment is initiated by flow... 

Fig. D.5 The mesh network to solve the momentum equation for the axial velocity distribution in a rectangular channel. As illustrated, the control volumes are square. However, the spreadsheet is programmed to permit different values for dx and dy. Because of the symmetry in this problem, only one quadrant of the system is modeled. The upper and left-hand boundary are the solid walls, where a zero-velocity boundary condition is imposed. The lower and right-hand boundaries are symmetry boundaries, where special momentum balance equations are developed to represent the symmetry. As illustrated, there is an 12 x 12 node network corresponding to a 10 x 10 interior system of control volumes (illustrated as shaded boxes). The velocity at the nodes represents the average value of the velocity in the surrounding control volume. There are half-size control volumes along the boundaries, with the corresponding velocities represented by the boundary values. There is a quarter-size control volume in the lower-left-hand corner.

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. 

Writing mass-, heat-, energy-, and/or momentum-balance equations to obtain the model equations that relate the system input and output to the state variables and the physico-chemical parameters. These mathematical equations describe the state variables with respect to time and/or space. 

---