A Mathematical Model for the
Trickling Filter
E. L. SWILLEY, USPHS Trainee
B. ATKINSON, Assistant Professor
Department of Chemical Engineering
Rice University
Houston, Texas
INTRODUCTION
The trickling filter for fixed bed bio-oxidation of organic materials was developed over half a century ago (1). Design criteria for this basic tool of the
environmental engineer have been developed purely on an evolutionary basis. No
existing theory or mathematical model adequately describes the situation and few
attempts have been made in the past to remedy this. In fact, as late as 1962,
McKinney (2) made the following statement:
"Thus far, the theoretical aspects of trickling filters have not progressed
to the point where the engineer can design a filter for a given waste and
predict in advance the results to be expected. The number of variables
and their interrelationships in trickling filter design have complicated
the development of a complete fundamental theory. The development
of a sound fundamental theory for trickling filters is one of the great uncharted areas of research."
Thispaperis part of a broad program to consider the analogy between the biochemical oxidation problem and mat ot a postulated surface reaction at the wall
of a film flow reactor model, the relevant equations being the same. Two limiting cases have been suggested: one based on "food" limitation, the other based
on oxygen limitation. It is the purpose of the program to differentiate between
the controlling mechanisms.
Once the important mechanisms have been described and their relative importance assayed , it will be possible to develop a mathematical model for the
system based on a realistic interpretation of the transport phenomena involved.
Such a model would then lead to a situation where the transport rates could be
optimized and attention concentrated on the basic criteria that require laboratory
evaluation. These would include reaction rate coefficients, diffusion coefficients,
and adsorption phenomena, as well as techniques for their determination. It should
be noted that no information is available on specific reaction rate coefficients
because all work has been conducted and reported in terms of the overall system,
and these overall coefficients are influenced by the geometry of the system
studied.
The first attempt to formulate a relevant theory describing the mechanisms
of the trickling filter was made by Velz (3), who stated: "The rate of extraction
of organic matter per interval of depth of a biological bed is proportional to the
remaining concentration of organic matter, measured in terms of its removability. "   This may be expressed in the following equation:
f = e-KlD       . (1)
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