Solving Equations Numerically with Excel

Its square, if x, y, Z, Xz, yz, Zz), is the probability density of finding the first electron at point (xi, yi, Zi) and, simultaneously, the second electron at (xi, yi, Zz)-The Schrodinger equation is now more complicated, and an explicit solution for helium is not possible. Nevertheless, modern computers have enabled us to solve this equation numerically with high accuracy, and the predicted properties of helium are in excellent agreement with experiment. 

The advection—diffusion equation with a source term can be solved by CFD algorithms in general. Patankar provided an excellent introduction to numerical fluid flow and heat transfer. Oran and Boris discussed numerical solutions of diffusion—convection problems with chemical reactions. Since fuel cells feature an aspect ratio of the order of 100, 0(100), the upwind scheme for the flow-field solution is applicable and proves to be very effective. Unstructured meshes are commonly employed in commercial CFD codes. 

There are a variety of numerical methods available to solve the conservation equations. The most commonly used method in commercially available CFD software today is the finite volume method. Excellent descriptions of this method can be found in Refs. . With the finite volume method, an integral form of the conservation equations is solved by performing a mass and momentum balance over all faces of each computational cell. There are, however, many other methods available, such as the finite element method (where the equations are solved in differential form instead... 

It is well known that even for the simplest substances such as argon or krypton there is no satisfactory theory of the liquid state at the present time. The theories of Mayer (24), Kirkwood (25), and Born and Green (41) may for practical purposes be considered rigorous and would presumably give excellent agreement with experimentally observed thermodynamic properties of classical (i.e., nondegenerate) liquids with spherically or effectively spherically symmetrical molecules—but the equations which can be written down are so complicated that they cannot be solved for useful numerical results. The best that can be done along these lines at present is to use Kirkwood s superposition approximation. There are also a number of approximate theories of liquids, but none of these is really very adequate. 

Assumptions 1 and 2 have not been made in recent years in the models, however, since the numerical equations are actually easier to solve without these assumptions. Nonetheless, the spatial and temporal scales of mesoscale systems result in the two assuniptions being excellent approximations with respect to mesoscale-size systems. Assumption 3 is justified since the advection of (f) is much more significant in transferring

molecular motion on the mesoscale. 

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