Equation of energy

The numerical solution of these equations is not trivial, since for reasonably low viscosities the flow becomes turbulent. A popular method of treating these equations (together with the equations of energy and mass conservation) is the MAC method [156,157]. For the case of immiscible fluids or moving internal interface a phase-field-type approach seems to be successful [78,158,159]. Because of the enormous requirements of computing ressources the development in this field is still relatively slow. We expect, however, an impact from the more widespread availability of massively parallel computers in the near future. 

This is the equation of energy balance, which states that the internal energy of a fluid changes because of the flow of energy into it, because of the pressure-volume work done on the surroundings by the fluid element, and because of loss of energy by viscous dissipation.6 The energy-balance equation may be written in an alternate form in terms of the temperature ... 

In general the flow of a pure fluid is described by the equation of continuity, the three equations of motion, and the equation of energy balance. In addition, one has to specify boundary and initial conditions and also the dependence of p on p and T (the thermal equation of state) and the dependence of Cv or U on p and T (the caloric equation of state). 

The equation of motion and the equation of energy balance can also be time averaged according to the procedure indicated above (SI, pp. 336 et seq. G7, pp. 191 et seq. pp. 646 et seq.). In this averaging process there arises in the equation of motion an additional component to the stress tensor t(,) which may be written formally in terms of a turbulent (eddy) coefficient of viscosity m(I) and in the equation of energy balance there appears an additional contribution to the energy flux q(1), which may be written formally in terms of the turbulent (eddy) coefficient of thermal conductivity Hence for an incompressible fluid, the x components of the fluxes may be written... 

Fluid flow rarely follows ihe commonly accepted idea of streamlines, since the velocities necessary for viscous flow of this nature are almost always lower than those found expedient to employ, Most flows are turbulent in nature. They become turbulent at a definite velocity, the value of which was studied by Reynolds and this value is incorporated in the well-known Reynolds Number. A general thermodynamic equation of energy of a fluid under flow conditions would be as follows ... 

It should be noted, that the constitutive assumptions (10) and the corresponding rates, respectively, as well as the relations (5) and (11) have already been included in the balance equation of energy. Furthermore, for the phases which are involved with mass exchange processes (tp1, partial density and the Jacobian of the corresponding phase have been considered as process variables. Due to the mass exchange, the partial density of [Pg.333]

Abstract In this contribution, the coupled flow of liquids and gases in capillary thermoelastic porous materials is investigated by using a continuum mechanical model based on the Theory of Porous Media. The movement of the phases is influenced by the capillarity forces, the relative permeability, the temperature and the given boundary conditions. In the examined porous body, the capillary effect is caused by the intermolecular forces of cohesion and adhesion of the constituents involved. The treatment of the capillary problem, based on thermomechanical investigations, yields the result that the capillarity force is a volume interaction force. Moreover, the friction interaction forces caused by the motion of the constituents are included in the mechanical model. The relative permeability depends on the saturation of the porous body which is considered in the mechanical model. In order to describe the thermo-elastic behaviour, the balance equation of energy for the mixture must be taken into account. The aim of this investigation is to provide with a numerical simulation of the behavior of liquid and gas phases in a thermo-elastic porous body. 

The volume-averaged equation of energy conservation in terms of internal energy is given by averaging Eq. (5.18) as... 

TABLE 2.5 The Equation of Energy in Terms of Energy and Momentum Fluxes in Several Coordinate Systems... 

The last term on the right-hand side can be obtained by solving the temperature profile in the solid bed. Consider a small, x-direction portion of the film and solid [Fig. 5.12(b)], We assume the solid occupies the region y > 5 (where 3 is the local film thickness) and moves into the interface with constant velocity vs.v. The problem thus reduces to a onedimensional steady heat-conduction problem with convection. In the solid, a steady, exponentially dropping temperature profile develops. The problem is similar to that in Section 5.4. The equation of energy reduces to... 

Finally the equation of energy using the definition of <5o becomes... 

The equation of energy, which for this case can be solved independently, reduces to... 

The equations for convection are the continuity or conservation of mass equation, the momentum equations and the energy equation. From the dimensionless equation of energy, useful dimensionless numbers are obtained. 

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