Forms of Governing Equations

The governing equations, (9.1)-(9.4), are approximated with discrete equations on the computational mesh. The discrete equations can be derived 

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. 

The topology of the grid may change by using slide surfaces and contact 

A slide surface is a surface where the tangential velocity can be discontinuous as shown in Fig. 9.9. Separate velocities are calculated for each side. Slide lines are useful for modeling phenomena such as sliding friction or flow through pipes. 

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. 

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Various forms of governing equations have been used in PEFC modeling, although all fall under the single-phase assumption. To clarify important subtleties with theoretical rigor, in this subsection we summarize a set of conservation equations and provide detailed comments of various terms that should be used. 

As already mentioned, the system ofEqs. (8.1-8.5) is supplemented by the Clausius-Clapeyron equation, as well as by the correlation that determines the dependence of enthalpy on temperature and describes the thermohydrodynamical characteristics of flow in a heated capillary. It is advantageous to analyze parameters of such flow to transform the system of governing equations to the form that is convenient for significant simplification of the problem. 

Derive a dimensionless form of the equations and thus obtain the important dimensionless groups governing the dynamic behaviour of the heat exchanger. 

Partitioning is governed mainly by free energy change. The net free energy describes the overall tendency of the system to make a specific change. The concept is in accord with the laws of thermodynamics and assumes that it is the natural tendency of a system to seek spontaneously a condition of minimum energy and maximum disorder [65,192-194]. The most common form of the equation is... 

These three equations in the three variables o>, and T are the set of equations that must be simultaneously solved subject to the correct boundary conditions to obtain the heat transfer rate. The original set of Equations from which they were deduced, i.e., Eqs. (2.70), (2.71), (2.72). and (2.77), contained four unknowns u, v, p, and T and this reduction in the number of variables and, therefore, the number of governing equation in itself constitutes a considerable simplification. The governing equations in terms of u, v, p, and T are often said to be expressed in primitive variable form. 

The limiting forms of the equations that result from the application of these conservation principles to this control volume as dx -+ 0 give a set of equations governing the average conditions across the boundary layer. These resultant equations are termed the boundary layer momentum integral and energy integral equations. 

ELSIM [90,92,99] is freely available from the author [12]. It has been updated since its earliest version around 1992, but is still DOS-based, that is, there is not a Windows version as yet. It accepts input in the form of reaction equations, in which case the program itself generates the governing equations or the user can enter the governing equations directly. ELSIM is not limited to a discrete number of mechanisms or experiments, these being determined by what the user enters. Even the method used for simulation can be chosen (within some limits, indicated by the program when necessary). It is written in C++ and the source code is available and can be modified by the user. 

Following Tribollet and Newman, the dimensionless form of the equation governing the contribution of mass transfer to the impedance response of the disk electrode is developed here in terms of dimensionless position... 

The model formulation used with moving meshes is of the arbitrary-Lagrangian-Eulerian (ALE) type. The integral form of the equations governing the incompressible Newtonian fluid in a time-varying control-volume V(t) is written as ... 

If we review the derivation of the Reynolds equation, (5 79), starting with the governing equations and boundary conditions, (5—69)—(5--72), we see that the present problem differs in that dh/dt = <)d/<)t = 0, but there is still a normal velocity at z = 0 that is due to blowing of air through the porous tabletop. Hence we can see from (5—75)—(5—77) that the appropriate form of Reynolds equation for the present problem should be... 

Transient problems of syneresis are of great interest. For example, the transient syneresis in a stagnant foam layer (U = 0) under the action of constant mass forces is governed by a complex nonlinear parabolic equation. Some self-similar solutions and traveling wave type solutions were found in [152] for some special forms of this equation. For one-dimensional barosyneresis (g = 0), Eq. (7.4.4) has the form... 

Equations 12.45, 12.46, 12.47, and 12.48 form a complete set of governing equations which are strongly coupled to each other. Therefore, these equations can be solved by nonlinear iterative procedures [133, 134, 198] and efficient second-order algorithms [1, 71,72,132]. 

The use of (6) and (8) for the prediction of the performance of a given system requires a brief explanation. It will be observed that each of these results contains the pressurizing gas density as a parameter. This, however, is a function of the gas temperature which is predicted by the equation itself. The form of the equations, as given, has assumed the gas to have a constant mean density. This was done in order to maintain linearity in (1), the governing differential equation. Hence, the use of (6) and (8) requires an iterative process. This is not difficult and the process is rapidly converging. 

It is obvious that the system of governing equations in the multi-tempierature approach is considerably simpler than the corresponding system in the state-to-state approach, since it contains much fewer equations. In the zero-order approximation of the Chapman-Enskog method, the system of governing equations takes the form typical for inviscid non-conductive flows. In this case equations (85), (86) read ... 

Equation (7.37) through equation (7.43) form the set of governing equations for the transport phenomenon taking place in the catalyst layer. Before proceeding on to discuss the parameters needed to close this set of equations, it should be pointed out that the transport of electrons and protons respond almost instantaneously to the change in electrical potential compared to the slow process in the transport of gas species. Thus, in equation (7.39) and equation (7.40), the transient terms are neglected. This behavior has been studied by Wu et al. [80]. 

The velocity profile is dictated by a boundary layer that moves out from the wall as illustrated in figure 6. The form of the equation governing this velocity-profile development means that just below turbulent flow at a Reynolds number of say 2300, a length of 138 diameters is needed to achieve fully-developed flow. (However for turbulent flow—see next section—lengths are much shorter because the relationship is given by L/d = 4.4 so that just above the transition to turbulence, we only need 16 diameters to obtain fuUy developed flow [7].)... 

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