Momentum conservation equations

I) CFD equation set It consists of overall mass conservation equation, momentum conservation equation, and its closure equations. It aims to find the velocity distribution (velocity profile) and other flow parameters. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

Since angular momentum is conserved, equation (A3.11.192) may be rearranged to give the following implicit equation for the time dependence of r ... 

The equation of motion is based on the law of conservation of momentum (Newton s second law of motion). This equation is written as... 

The conservation of mass gives comparatively Httle useful information until it is combined with the results of the momentum and energy balances. Conservation of Momentum. The general equation for the conservation of momentum 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 ... 

Flow and Performance Calculations. Electro dynamic equations are usehil when local gas conditions (, a, B) are known. In order to describe the behavior of the dow as a whole, however, it is necessary to combine these equations with the appropriate dow conservation and state equations. These last are the mass, momentum, and energy conservation equations, an equation of state for the working duid, an expression for the electrical conductivity, and the generalized Ohm s law. 

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

Cauchy Momentum and Navier-Stokes Equations The differential equations for conservation of momentum are called the Cauchy momentum equations. These may be found in general form in most fliiid mechanics texts (e.g., Slatteiy [ibid.] Denu Whitaker and Schlichting). For the important special case of an incompressible Newtonian fluid with constant viscosity, substitution of Eqs. (6-22) and (6-24) lead to the Navier-Stokes equations, whose three Cartesian components are... 

Conservation equations Expressions that equate the mass, momentum, and energy across a steady wave or shock discontinuity ((2.1)-(2.3)). Also known as the jump conditions or the Rankine-Hugoniot relations. 

Lagranglan codes are characterized by moving the mesh with the material motion, u = y, in (9.1)-(9.4), [24]. The convection terms drop out of (9.1)-(9.4) simplifying all the equations. The convection terms are the first terms on the right-hand side of the conservation equations that give rise to fluxes between the elements. Equations (9.1)-(9.2) are satisfied automatically, since the computational mesh moves with the material and, hence, no volume or mass flux occurs across element boundaries. Momentum and energy still flow through the mesh and, therefore, (9.3)-(9.4) must be solved. 

Several discrete forms of the conservation of momentum equation, (9.3), can be derived, depending on the type of mesh and underlying assumptions. As an example, assume the equation will be solved on staggered spatial and temporal meshes, in two dimensions, in rectangular geometry, and with the velocities located at the nodes. Assume one quarter of the mass from each adjacent element is associated with the staggered element as shown in Fig. 9.11. 

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 conservation equations developed by Ericksen [37] for nematic liquid crystals (of mass, linear momentum, and angular momentum, respectively) are ... 

Thermal plumes above point (Fig. 7.60) and line (Fig. 7.61) sources have been studied for many years. Among the earliest publications are those from Zeldovich and Schmidt. Analytical equations to calculate velocities, temperatures, and airflow rates in thermal plumes over point and line heat sources with given heat loads were derived based on the momentum and energy conservation equations, assuming Gaussian velocity and excessive temperature distribution in... 

Considered are mass conservation of air and species (contaminants and humidity). Momentum equations are not considered on a global scale but have been used in some cases for the definition of the airflow-pressure relation of the individual links. Heat fluxes and thus energy conservation equations are not considered. 

Conservation is a general concept widely used in chemical engineering systems analysis. Normally it relates to accounting for flows of heat, mass or momentum (mainly fluid flow) through control volumes within vessels and pipes. This leads to the formation of conservation equations, which, when coupled with the appropriate rate process (for heat, mass or momentum flux respectively), enables equipment (such as heat exchangers, absorbers and pipes etc.) to be sized and its performance in operation predicted. In analysing crystallization and other particulate systems, however, a further conservation equation is... 

The Chapman-Jongnet (CJ) theory is a one-dimensional model that treats the detonation shock wave as a discontinnity with infinite reaction rate. The conservation equations for mass, momentum, and energy across the one-dimensional wave gives a unique solution for the detonation velocity (CJ velocity) and the state of combustion products immediately behind the detonation wave. Based on the CJ theory it is possible to calculate detonation velocity, detonation pressure, etc. if the gas mixtnre composition is known. The CJ theory does not require any information about the chemical reaction rate (i.e., chemical kinetics). 

Analytical methods relate the gas dynamics of the expansion flow field to an energy addition that is fully prescribed. A first step in this approach is to examine spherical geometry as the simplest in which a gas explosion manifests itself. The gas dynamics of a spherical flow field is described by the conservation equations for mass, momentum, and energy ... 

The flow field in front of an expanding piston is characterized by a leading gas-dynamic discontinuity, namely, a shock followed by a monotonic increase in gas-dynamic variables toward the piston. If both shock and piston are regarded as boundary conditions, the intermediate flow field may be treated as isentropic. Therefore, the gas dynamics can be described by only two dependent variables. Moreover, the assumption of similarity reduces the number of independent variables to one, which makes it possible to recast the conservation equations for mass and momentum into a set of two simultaneous ordinary differential equations ... 

Strehlow (1975) achieved a solution by conducting a mass balance over the flow field. Such a balance can be drawn up under the assumptions of similarity and a constant density between shock and flame. The assumption of constant density violates the momentum-conservation equation, and is a drastic simplification. The maximum overpressure is, therefore, substantially underestimated over the entire flame speed range. An additional drawback is that the relationship of overpressure to flame speed is not produced in the form of a tractable analytical expression, but must be found by trial and error. 

The major mechanism of a vapor cloud explosion, the feedback in the interaction of combustion, flow, and turbulence, can be readily found in this mathematical model. The combustion rate, which is primarily determined by the turbulence properties, is a source term in the conservation equation for the fuel-mass fraction. The attendant energy release results in a distribution of internal energy which is described by the equation for conservation of energy. This internal energy distribution is translated into a pressure field which drives the flow field through momentum equations. The flow field acts as source term in the turbulence model, which results in a turbulent-flow structure. Finally, the turbulence properties, together with the composition, determine the rate of combustion. This completes the circle, the feedback in the process of turbulent, premixed combustion in gas explosions. The set of equations has been solved with various numerical methods e.g., SIMPLE (Patankar 1980) SOLA-ICE (Cloutman et al. 1976). 

Euler s equation (equation 9.7) may be recovered from Boltzman s equation as a consequence of the conservation of momentum, but only in the zeroth-order approximation to the full distribution function. Setting k — mvi in equation 9.52 gives, in component form. 

Since /S Tj0) = , its integral over the collision term is zero (conservation of momentum in a collision). Thus the result of multiplying the Boltzmann equation by and integrating is ... 

As developed from the equation for conservation of momentum (Ref 27), the thrust, F, on a rocket motor is... 

A detailed study of the influence of viscous heating on the temperature field in micro-channels of different geometries (rectangular, trapezoidal, double-trapezoidal) has been performed by Morini (2005). The momentum and energy conservation equations for flow of an incompressible Newtonian fluid were used to estimate... 

The quasi-one-dimensional model of flow in a heated micro-channel makes it possible to describe the fundamental features of two-phase capillary flow due to the heating and evaporation of the liquid. The approach developed allows one to estimate the effects of capillary, inertia, frictional and gravity forces on the shape of the interface surface, as well as the on velocity and temperature distributions. The results of the numerical solution of the system of one-dimensional mass, momentum, and energy conservation equations, and a detailed analysis of the hydrodynamic and thermal characteristic of the flow in heated capillary with evaporative interface surface have been carried out. 

The angular momentum conservation equation couples the viscous and the elastic effects. The angular profiles of the director and the effective viscosity data are computed for one set of material parameters based on published data in literature. The velocity profiles are also attained from the same dataset. The results show that the alignment of molecules has a strong influence on the lubrication properties. 

Equation 5.1.13 shows how heat release acts a volume source. Assuming that the combustion takes place in a uniform medium at rest (Mach < 0), and writing for small perturbations, a = a + a a = p, p, v), the linearized conservation equations for mass and momentum can be used to eliminate the density in 5.1.13 to obtain a wave equation for the pressure in the presence of local heat release ... 

By demanding that the new velocity w field fulfils both the momentum and the mass conservation equation, the following equations for the velocity and pressure correction are derived ... 

While we laud the virtue of dynamic modeling, we will not duphcate the introduction of basic conservation equations. It is important to recognize that all of the processes that we want to control, e.g. bioieactor, distillation column, flow rate in a pipe, a drag delivery system, etc., are what we have learned in other engineering classes. The so-called model equations are conservation equations in heat, mass, and momentum. We need force balance in mechanical devices, and in electrical engineering, we consider circuits analysis. The difference between what we now use in control and what we are more accustomed to is that control problems are transient in nature. Accordingly, we include the time derivative (also called accumulation) term in our balance (model) equations. 

The intrinsic constitutive laws (equations of state) are those of each phase. The external constitutive laws are four transfer laws at the walls (friction and mass transfer for each phase) and three interfacial transfer laws (mass, momentum, energy). The set of six conservation equations in the complete model can be written in equivalent form ... 

Mixture conservation of mass equation Mixture conservation of momentum equation Mixture conservation of energy equation Slip equation (concerning the difference in velocity)... 

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