Deep bed filtration: A new look at the basic equations

Mathematical models of deep bed granular filters are usually based on two linked partial differential equations. The first is an equation of mass balance; the second expresses the kinetics of the process. The combination of the precise mass balance with the simple kinetic equation most commonly used leads to an inconsistency which has hitherto been resolved either by attempting to express the equations in “filter” rather than “absolute” time, or by assuming the offending terms are negligible. The inconsistency does not arise when a new, rational kinetic equation is combined with the more complicated, precise form of the mass balance equation. The new equations are, however, awkward to solve, and there remains an incentive to use simplified approximations which offer the possibility of exact solution. Comparison of series solutions for the simple and precise equations indicates that there is no loss of accuracy in using the approximations. It is also concluded that the concept of filter time does not simplify the mathematics.