Pitfalls in Parameter Estimation
For Oxygen Transfer Data
WILLIAM C. BOYLE, Professor
P.M. BERTHOUEX, Assistant Professor
Civil Engineering Department
University of Wisconsin
Madison, Wisconsin 53706
THOMAS C. ROONEY, Research Supervisor
Environmental Process Department
Rexnord, Inc.
Milwaukee, Wisconsin 53214
INTRODUCTION
The testing of aeration equipment has received extensive scrutiny over the last fifteen
years. During this period investigators have given attention to both physical and chemical
factors which may influence aerator test results. Given that the investigator handles all these
factors in the best possible way, he has left the problem of using the data obtained to
estimate parameters that describe the efficiency of the aeration system. It is the primary
objective of this paper to discuss some of the pitfalls of methods of parameter estimation
applicable to the analysis of oxygen transfer data.
Considerable research effort has been invested in developing and checking models for
oxygen absorption in water. A majority of these efforts have dealt with the effects of the
physical environment on the mass transfer coefficient. Such information is of great value
when designing aeration systems. Even so, engineers most often employ the overall mass
transfer coefficient Kl a to characterize a specified piece of equipment under a specified set
of physical conditions.
It will be assumed throughout this paper that the first order oxygen transfer model is
the correct model. We have, therefore, by decree satisfied the first two requirements for
fitting data to estimate the parameters in a model: 1) write the correct model, and 2) collect
data according to sound experimental procedures.
FIRST ORDER MASS TRANSFER MODEL
This model is so familiar that it is given only to clarify nomenclature and make clear
which mathematical function is being treated as the model.
The rate of oxgyen transfer is: dC/dt =K]^a(C* -C) (1)
where C* is the dissolved oxygen saturation concentration under the test conditions of
temperature, pressure, and salinity, C is the dissolved oxygen concentration at time t, and
Kj^a is the overall mass transfer coefficient. For simplicity Ki a is hereafter noted simply as
For an initial dissolved oxygen concentration of Co at time zero (tQ) the integrated
form of Equation 1 is:
*
ln    C*"-C°     =K(t'tD) (2)
or C = C*-(C*-C0) exp(K(t-t0)) (3)
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