Control volume approach

Let us use a control volume approach for the fluid in the boundary layer, and recognize Newton s law of viscosity. Where gradients or derivative relationships might apply, only the dimensional form is employed to form a relationship. Moreover, the precise formulation of the control volume momentum equation is not sought, but only its approximate functional form. From Equation (3.34), we write (with the symbol implying a dimensional equality) for a unit depth in the z direction... 

For most numerically solved models, a control-volume approach is used. This approach is based on dividing the modeling domain into a mesh. Between mesh points, there are finite elements or boxes. Using Taylor series expansions, the governing equations are cast in finite-difference form. Next, the equations for the two half-boxes on either side of a mesh point are set equal to each other hence, mass is rigorously conserved. This approach requires that all vectors be defined at half-mesh points, all scalars at full-mesh points, and all reaction rates at quarter-mesh points. The exact details of the numerical methods can be found elsewhere (for example, see ref 273) and are not the purview of this review article. The above approach is essentially the same as that used in CFD packages (e.g.. Fluent) or discussed in Appendix C of ref 139 and is related to other numerical methods applied to fuel-cell modeling. ... 

MOLD FILLING SIMULATIONS USING THE CONTROL VOLUME APPROACH 493... 

Based on the control volume approach and using the three-dimensional finite element formulations for heat conduction with convection and momentum balance for non-Newtonian fluids presented earlier, Turng and Kim [10] and [17] developed a three-dimensional mold filling simulation using 4-noded tetrahedral elements. The nodal control volumes are defined by surfaces that connect element centroids and sides as schematically depicted in Fig. 9.33. 

Equation E15.2-21 can also he derived by a controlled volume approach. Consider the a element confining node m in Fig. 15.6(b) (shaded area). For an incompressible fluid and under the same assumptions as earlier we can make the following flow rate balance... 

X. L. Luo, A Control Volume Approach for Integral Viscoelastic Models and Its Application to Contraction Flow of Polymer Melts, J. Non-Newt. Fluid Mech., 64, 173-189 (1996). 

An issue of fundamental importance is that the physical laws determining the fluid behavior are stated in terms of fluid systems, not control volumes. To formulate the governing laws in a control volume approach we must re-phrase the laws in an appropriate manner. In the integral formulation the Leibnitz-or Re3molds transport theorem provides the relationship between the time rate of change of an extensive property for a system and that for a control volume. For differential equations a similar interrelation between the system and control volume approaches is expressed through the substantial - or material derivative operator. Hence it follows that by use of these mathematical tools we may convert a system analysis to a control volume analysis. 

In this section, an analysis based on the Boltzmann equation will be given. Before we proceed it is essential to recall that the translational terms on the LHS of the Boltzmann equation can be derived adopting two slightly different frameworks, i.e., considering either a fixed control volume (i.e., in which r and c are fixed and independent of time t) or a control volume that is allowed to move following a trajectory in phase space (i.e., in which r(t) and c t) are dependent of time t) both, of course, in accordance with the Liouville theorem. The pertinent moment equations can be derived based on any of these two frameworks, but we adopt the fixed control volume approach since it is normally simplest mathematically and most commonly used. The alternative derivation based on the moving control volume framework is described by de Groot and Mazur [22] (pp. 167-170). 

S. G. Kandlikar and B. J. Stumm, A Control Volume Approach for Investigating Forces on a Departing Bubble under Subcooled Flow Boiling, ]. Heat Transfer (117) 990-997,1995. 

B. Dual-control Volume Approach for Gradient-driven Diffusion Monte Carlo versus Molecular Dynamics... 

Bruschke, M. V. and Advani, S. G., A finite element/control volume approach to mold filling in anisotropic porous media , Polymer Composites, 11(6), 398 05, 1990. 

The effect of compressibility on flow in microsystems has been discussed in this section. The term in equation (3.52) is composed of two parts that is, + p VP, where the first part is a temporal contribution to the change in pressure and the second part is a convective contribution to the change in pressure. Let us discuss the second contribution from a typical example of flow between parallel plates (Figure 3.15) using the control volume approach. The conservation of mass between the inlet and outlet can be written as... 

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