Solution of the Energy Equation

In the finite element solution of the energy equation it is sometimes necessary to impose heat transfer across a section of the domain wall as a boundary condition in the process model. This type of convection (Robins) boundary condition is given as... 

The inconsistent streamline upwind scheme described in the last section is fonuulated in an ad hoc manner and does not correspond to a weighted residual statement in a strict sense. In tins seetion we consider the development of weighted residual schemes for the finite element solution of the energy equation. Using vector notation for simplicity the energy equation is written as... 

Temperature variations are found by the solution of the energy equation. I he finite element scheme used in this example is based on the implicit 0 time-stepping/continuous penalty scheme described in detail in Chapter 4, Section 5. 

The above relation is obtained if the constant a obtained from the work of Zel dovich [7] is assumed to be a = Nu.U (which was not suggested in the work of Zel dovich [7]), where k is the thermal dif-fusivity and d is the characteristic length. However, Nu/Pe is a parameter in the solution of energy equation for flames with heat losses above certain value for which no solution of the energy equation exist (see, e.g., Ref. [8], p. 108)... 

Figure 3. Using the slow flow technique, which allows us to follow time scales larger than those required by an explicit solution of the energy equation, reduces the required computational time to three years...

The prediction of convective heat transfer rates will, however, always involve the solution of the energy equation. Therefore, because of its fundamental importance in the present work, a discussion of the way in which the energy equation is derived will be given here [2],[3],[5],[7]. For this purpose, attention will be restricted to two-dimensional, incompressible flow. 

Having established that similarity solutions for the velocity profile can be found for certain flows involving a varying ffeestream velocity, attention must now be turned to the solutions of the energy equation corresponding to these velocity solutions. The temperature is expressed in terms of the same nondimensional variable that was used in obtaining the flat plate solution, i.e., in terms of 8 = (Tw - T)f(Tw -Tt) and it is assumed that 0 is also a function of ij alone. Attention is restricted to flow over isothermal surfaces, i.e., with Tw a constant, and T, of course, is also constant. 

Since the variation of 8 with x has been derived by solving the momentum integral equation, Eqs. (3.153) and (3.156) together constitute the solution of the energy equation. The variation of A with Pr that they together give is shown in Fig. 3.14. 

Now the first term in this equation is zero by virtue of Eq. (3.247) while the second term is zero by virtue of Eq. (3.244). Hence, Eq. (3.261) is a solution of the energy equation. The constants Cj and Ci are found by applying the boundary conditions. These give ... 

Having established the form of the velocity profile, attention must now be turned to the solution of the energy equation (4.4). Since v was shown to be equal to 0 in fully developed flow, this equation reduces in this case to... 

A forward-marching implicit finite-difference solution of the energy equation will again be considered. In order to obtain this solution, a series of nodal lines running parallel to the x and y-axes are again introduced as shown in Fig. 10.15. 

Theoretical approaches to the prediction of H x,y,t) would involve the solution of the boundary layer equations for coupled energy and momentum transport or, more simply, the solution of the energy equations in conjunction with a constructed wind field. The application of such approaches to the prediction of inversion height has not yet been reported. Now, empirical models offer the only available means to estimate H. For those areas where it is necessary only to account for temporal variations in H, interpolation and extrapolation of measured mixing heights may be sufficient. When it is important to estimate // as a function of x,y, and t, a detailed knowledge of local meteorology is essential. 

Prandtl Number Equal to Unity. If Pr = 1, considerable simplification results. Equation 6.38 acquires a form identical with Eq. 6.37, / being analogous to A solution of the energy equation, therefore, is directly expressible in terms of the velocity distribution as... 

Similarly to the considerations for the solution of the energy equation, one can also note that 1// = ij/(6). This leads to the following general solution of Eq. 24 ... 

CONVECTION HEAT TRANSFER IN POLYMERIC SYSTEMS SOLUTIONS OF THE ENERGY EQUATION IN CIRCULAR CONDUITS... 

Other experimental studies for polymer solutions involved the work of Bassett and Welty [97], Popovska and Wilkinson [99], and Joshi and Bergles [103]. The first of these dealt with the thermal entry region and a constant wall flux. Popovska and Wilkinson used a numerical solution of the energy equation to test experimental data over Graetz numbers that ranged from 80 to 1600. Joshi and Bergles used a constant wall flux and correlated for power law fluids the effect of fluid property variation with temperature. 

Other work that considered parallel plates or slits were the papers of Lin and Hsu [105], Ybarra and Eckert [106], and Dinh and Armstrong [107]. All three studies involved theoretical solutions of the energy equation that included vis-... 

Several authors [22-24] have assumed uniform viscous heat dissipation in dealing with the solution of the energy equation for non-Newtonian systems. Why is this assumption generally invalid Are there any situations in which the assumption could be valid or partially valid ... 

Using first principles and dimensionless forms, we will derive the basic format describing the heat transfer coefficient. Next, we will use experimental data or combination of analytical solutions of the Energy Equation and experimental data to obtain equation for h. 

S-3.3.5 Numerical Diffusion. Numerical diffusion is a source of error that is always present in finite volume CFD, owing to the fact that approximations are made during the process of discretization of the equations. It is so named because it presents itself as equivalent to an increase in the diffusion coefficient. Thus, in the solution of the momentum equation, the fluid will appear more viscous in the solution of the energy equation, the solution will appear to have a higher conductivity in the solution of the species equation, it will appear that the species diffusion coefficient is larger than in actual fact. These errors are most noticeable when diffusion is small in the actual problem definition. 

The film and surface temperatures are obtained from the solutions of the energy equation applied to the lubricant film and contact surfaces, using a method proposed by Tevaarwerk [12],... 

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