By Jon Fridjonsson
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The local equilibrium assumption, used in the majority of the cases dealt with in this text and in the remainder of this chapter, is discussed more below. Table 2-2 summarizes the overall balance equations needed for a complete description of isothermal, multicomponent, multiphase flow in permeable media. Column 1 in Table 2-2 gives the differential form of the equation named in column 2. Column 3 gives the number of scalar equations represented by the equation in column 1. Columns 4 and 5 give the identity and number of independent variables added to the formulation by the equation in column 1.
Here we consider only the rate of work done against a pressure field WPV , although other types of work could be included (see exercise 2P). In this derivation, there is no compression or expansion work done on volume V since it is assumed to be fixed. 28 Returning to Fig. 2-4(b), consider an element in the multiphase, multicomponent flow field crossing ΔA. Since work is the product of force times a distance, the rate of work is force times a velocity. The element crossing ΔA is, therefore, doing work ΔWPV , where NP ΔWPV = −∑ Pj ΔAn iu j .
A careful examination of the possible configurations shows further that a state with one or two molecules on each side is more likely to exist than the initial state, which has a probability of only 2-4 or 1/16, the initial state being just one of 16 possible states. A true equilibrium state with two molecules on either side of the original partition (no gradients) is also more likely to exist (probability of 6/16 or 3/8) than the initial state. 35 Figure 2-6: Illustration of configurations for four molecules in a closed composite system (after Smith et al.