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X j Ŵ ∂φ ∂x j dV. 6) The diffusion term takes into account the transport of φ by diffusion. 6) can be treated in a similar way to the convective term, Eq. 3). s. ∂φ n d A. 7) Using the same notation as in the convective case, Eq. 7) can be evaluated to give a similar expression: − AŴ ∂φ ∂x − w AŴ ∂φ ∂x + AŴ e ∂φ ∂y − AŴ s ∂φ ∂y + AŴ n ∂φ ∂z AŴ t ∂φ ∂z . 3 The source term The last term in the general transport equation is the source term, Sφ dV . v. The source term takes into account any generation or dissipation of φ.
Some examples of simplifications in the solved Example 1 are the following. r The problem could be treated as 1D due to symmetries. A problem in 2D or 3D would, of course, generate more cells; in this case a 3D treatment would give 1000 cells instead of 10, assuming that the grid density was kept constant and the computational domain had a cubic geometry. The cells were placed with constant spacing, generating a so-called equidistant grid. r Further, the presence of a constant velocity made the solution process easier.
For non-ideal gases many choices can be found in the literature, the most common of which are the law of corresponding states and the cubic equations of state. 33) RT where Z is a function of the reduced temperature and pressure. 34) P= − 2 V −b V + ubV + wb2 Z= where a, b, u and w are parameters. Depending on the parameters, they form van der Waals, Redlich–Kwong, Soave and Peng–Robinson equations of state. For liquids the pressure dependence can often be neglected and a simple polynomial can describe the temperature dependence: ρ = A + BT + C T 2 + DT 3 + · · · .
Computational fluid dynamics for engineers by Bengt Andersson; et al