Momentum conservation equation derivation

The conservation equations of mass, momentum, and energy of a single-phase flow can be obtained by using the general conservation equation derived previously. 

Thus, in order to derive the disperse-phase mass- and momentum-conservation equations, it suffices to multiply the GPBE by m(i) and vm( ), respectively, and to integrate over the phase-space variables. 

However, as far back as 1829 Gauss provided an alternative derivation based on the inclusion of a surface energy in the energy conservation equation, corresponding to the inclusion of surface tension in the momentum conservation equation (Dussan V. 1979). A simple demonstration of the minimum energy derivation has been given by de Gennes (1985), and we repeat it here. 

Let us turn now to derivation of the momentum conservation equation. We will need a formula following from (5.7) and (5.12) ... 

The evolution of the height of the liquid inside the groove is derived from the continuity and the momentum conservation equation as... 

Equations to describe random pseudo-turbulent fluctuations have to be derived 1) from fluid mass and momentum conservation laws, and 2) from the Langevin equation for one particle. Taking the fluctuation parts of the mass and momentum conservation equations (the corresponding mean equations were formulated in Section 5) and multiplying the Langevin equation by the particle number concentration, we arrive at the following set of equations governing particle and fluid fluctuations ... 

Before studying multiple bubble interactions, we must understand the behavior of a single bubble in response to acceleration of the surrounding continuous liquid. Neglecting surface tension effects andp turbulent fluctuations smaller than the scale of the bubble, and assuming incompressible adiabatic flow, Stewart and Crowe [20] derived the averaged momentum conservation equations for dispersed bubbles and the surrounding continuous liquid [20]. These can be expressed, respectively, by ... 

The source term S here represents a general mass source or sink and therefore also allows for phase transition phenomena. The momentum conservation equation is derived from Newton s second law as a fixed control volume, whereby the... 

In general, pressure is defined as the net momentum flux through an imaginary plane that moves with the local average velocity in the system. Since momentum is a vector quantity and the direction of transfer is described by a vector, pressure is, in general, a tensor quantity. In the next chapter, we will show that the pressure tensor arises naturally in the derivation of the macroscopic momentum conservation equation from the Liouville equation. It is written as... 

However, the second moment leads to a force-dependent contribution to the nonequilibrium momentum flux, which can be derived as before, beginning with (76) and substituting the equilibrium expressions for r (79) and py (98). The time derivative of the momentum flux now generates terms involving uf + fu from the momentum conservation equation on the h time scale [cf. (99)] ... 

The analysis presented in Chapter 8 was solely in terms of the conservation equations for the particle phase of a fluidized suspension. However, the full one-dimensional description is in terms of the coupled mass and momentum conservation equations for both the particle and fluid phases eqns (8.21)-(8.24). These equations correspond to those derived in Chapter 7, except for the inclusion of the particle-phase elasticity term on the extreme right of eqn (8.22). 

As stated earlier, our goal is to derive general equations that relate these quantities to rheological variables such as shear stress, shear rate, and normal stress differences. Based on the direction of the imposed velocity, the cylindrical symmetry of the flow geometry, and the assumptions that the fluid is incompressible and flow occurs under isothermal conditions, the equations of continuity (Appendix 8.A) yield v = Vg(r,z)eg. Neglecting inertia, the differential linear momentum conservation equations (Appendix 8.B) can be simplified to give... 

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. 

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... 

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... 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

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 performance of a chemical reactor can be described, in general, with a system of conservation equations for mass, energy, and momentum. To solve this system we must have a model for the reaction on the basis of which we can derive the intrinsic rate equation on one side, and a model of the reactor in which we want to run the reaction on the other side. Both tasks are, of course, interconnected and difficult to solve without reduction of more general equations to a suitable limiting reactor type to be used for each particular reaction system [4,26],... 

A system of conservation equations, whose solution describes the velocity, temperature, and composition fields. These equations usually take the form of partial differential equations that are derived from physical laws governing the conservation of mass, momentum, and energy. 

Overall our objective is to cast the conservation equations in the form of partial differential equations in an Eulerian framework with the spatial coordinates and time as the independent variables. The approach combines the notions of conservation laws on systems with the behavior of control volumes fixed in space, through which fluid flows. For a system, meaning an identified mass of fluid, one can apply well-known conservation laws. Examples are conservation of mass, momentum (F = ma), and energy (first law of thermodynamics). As a practical matter, however, it is impossible to keep track of all the systems that represent the flow and interaction of countless packets of fluid. Fortunately, as discussed in Section 2.3, it is possible to use a construct called the substantial derivative that quantitatively relates conservation laws on systems to fixed control volumes. 

Deriving the compressible, transient form of the stagnation-flow equations follows a procudeure that is largely analogous to the steady-state or the constant-pressure situation. Beginning with the full axisymmetric conservation equations, it is conjectured that the solutions are functions of time t and the axial coordinate z in the following form axial velocity u = u(t, z), scaled radial velocity V(t, z) = v/r, temperature T = T(t, z), and mass fractions y = Yk(t,z). Boundary condition, which are applied at extremeties of the z domain, are radially independent. After some manipulation of the momentum equations, it can be shown that... 

Kadanoff and Swift have considered that the time evolution of the state is described by the Liouville equation. They also wrote down conservation equations for the number, momentum, and energy density similar to the ones given by Eqs. (1)—(3). The only difference was that in the treatment of Kadanoff and Swift the densities and currents are operators. The time derivative of the densities are replaced by the commutator of the respective density operator and the Liouville operator, L (as the Liouville operator governs the time evolution). The suffix op in the following equations stands for operator ... 

Due to the infinitesimal dimension of the pore sizes, Equation (3.3) is often inapplicable to porous media. Therefore, the momentum conservation for the fluid flow through the porous electrodes is often substituted by the phenomenologically derived constitutive equations, such as Darcy s law given by... 

The fundamental quantities which are conserved in a collision are the mass of a particle and each component of the total momentum of the colliding particles. Thus, conservation equations can be obtained by the use of the transport theorem. Although the energy equation can also be derived from the transport theorem, the total kinetic energy of... 

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