Conduction finite volume

The focus of the remainder of this chapter is on interstitial flow simulation by finite volume or finite element methods. These allow simulations at higher flow rates through turbulence models, and the inclusion of chemical reactions and heat transfer. In particular, the conjugate heat transfer problem of conduction inside the catalyst particles can be addressed with this method. 

Similar equations apply to cylindrical and spherical coordinate systems. Finite difference, finite volume, or finite element methods are generally necessary to solve (5-15). Useful introductions to these numerical techniques are given in the General References and Sec. 3. Simple forms of (5-15) (constant k, uniform S) can be solved analytically. See Arpaci, Conduction Heat Transfer, Addison-Wesley, 1966, p. 180, and Carslaw and Jaeger, Conduction of Heat in Solids, Oxford University Press, 1959. For problems involving heat flow between two surfaces, each isothermal, with all other surfaces being adiabatic, the shape factor approach is useful (Mills, Heat Transfer, 2d ed., Prentice-Hall, 1999, p. 164). 

The radiative source term is a discretized formulation of the net radiant absorption for each volume zone which may be incorporated as a source term into numerical approximations for the generalized energy equation. As such, it permits formulation of energy balances on each zone that may include conductive and convective heat transfer. For K—> 0, GS —> 0, and GG —> 0 leading to S —> On. When K 0 and S = 0N, the gas is said to be in a state of radiative equilibrium. In the notation usually associated with the discrete ordinate (DO) and finite volume (FV) methods, see Modest (op. cit., Chap. 16), one would write S /V, = K[G - 4- g] = Here H. = G/4 is the average flux... 

This process was first recognized by Ostwald and is known as Ostwald ripening. The mathematical details were worked out independently by Lifshitz and Slyozov and by Wagner ° and is known as the LSW theory. However, this theory is based on a mean field approximation and is restricted to low volume fraction systems. Voorhees and coworkers extended the LSW theory to finite volume fraction systems and conducted a series of flight experiments designed to test this and similar theories. ... 

To design a cooling system for the fuel cell stack the thermal parameters of the stack and its components had to be determined accurately. For this reason a 3D FV (finite volume) thermal model was created. Special emphasis was placed on the thermal conduction of the various layers responsible for heat flow in all directions of the fuel cell stack. The basic parameters are given in Table 8-1. 

In this code, a 1-dimensional electrochemical element is defined, which represents a finite volume of active unit cell. This 1-D sub-model can be validated with appropriate single-cell data and established 1-D codes. This 1-D element is then used in FLUENT, a commercially available product, to carry out 3-D similations of realistic fuel cell geometries. One configuration studied was a single tubular solid oxide fuel cell (TSOFC) including a support tube on the cathode side of the cell. Six chemical species were tracked in the simulation H2, CO2, CO, O2, H2O, and N2. Fluid dynamics, heat transfer, electrochemistry, and the potential field in electrode and interconnect regions were all simulated. Voltage losses due to chemical kinetics, ohmic conduction, and diffusion were accounted for in the model. Because of a lack of accurate and detailed in situ characterization of the SOFC modeled, a direct validation of the model results was not possible. However, the results are consistent with input-output observations on experimental cells of this type. 

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. 

TE Module Modeling and Evaluation Procedure. The light condensed by the water lens had a long rectangular shape, and a TE module was set at this long focus, as shown in Fig. 1. Table 1 lists the temperature dependencies of the material properties and the sizes of the TE elements, electrode, and insulator in reference to a commercial one. The power generation simulations were conducted numerically based on the finite-volume method adding TE phenomena on the commercial software... 

Basara B, Alajbegovic A, Beader D (2004) Simulation of single- and two-phase flows on sliding unstructured meshes using finite volume method. Int J Numer Meth 45 1137-1159 Batchelor GK (1959) SmaU-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity. J Ruid Mech... 

Kakimoto and Liu [10] developed a partly 3D global model that takes into account feasible 3D global modeling with moderate requirements of computer memory and computation time. AH convective and conductive heat transfers, radiative heat exchanges between diffuse surfaces and the Navier-Stokes equations for the melt are all coupled and solved simultaneously by a finite-volume method in a 3D configuration. 

The cell is represented by a single compartment of finite volume II in contact across a living membrane with an external medium I of fixed composition and pH and of unlimited dimension. The whole layer which lies between the vacuole of an aquatic plant cell and the aqueous solution in which it lives (Figure 1). is treated globally as a single transversally homogeneous membrane. In principle it is assumed that an increased sophistication of the model, e.g., distinction between tonoplast and plasmalemma membranes, sets of ionic conductance of separate specific channels should not affect the overall validity of the phenomenological relations. Thus, the results are also valid for the plasmalemma alone. 

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