The General Balance Equations for Distributed Systems

Consider a conserved quantity in a three-dimensional region A, with boundary 3A. x, y, and z are the space coordinates and t is the time coordinate. Let h = h(x, y, z, t) be its concentration, that is, the amount per unit volume, so that the total amount in a subregion a of A is 

Similarly, g(x, y, z, t) is the rate of generation per unit volume, so that 

Of course, we have to express f, g, and h in terms of a common variable, u for example, by means of a constitutive relation for the material under study often u = h. 

At a surface of discontinuity in a three-dimensional space (or a line in a two-dimensional space, or a point in a one-dimensional space), the flux normal to the surface must be continuous, for the surface has no capacity to hold anything or volume to generate anything. Because there can be no accumulation in the surface, the flux up to it from one side must equal the flux away from it on the other. Thus, 

When any quantity a varies over the cross-section, we can define an average across the tube in the usual way as 

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The differential operator in Eq. (102) will be recognized as the Laplacian and, taking the isothermal case with v = °° for simplicity, a balance of the same sort as was treated in the section entitled The General Balance Equations for Distributed Systems (3/3t = 0, f = —D grad c, g = /(c)) gives... 

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