A PROPOSED MODIFICATION OF THE PHELPS-STREETER OXYGEN
SAG FORMULATION AND OF THE RELATION BETWEEN
K2 AND TEMPERATURE
Warren E, Howland, Professor Emeritus
School of Civil Engineering
Purdue University
West Lafayette, Indiana 47907
Robert H. L. Howe, Research Scientist
Eli Lilly and Company
Tippecanoe Laboratories
Lafayette, Indiana 47905
INTRODUCTION
Methods based upon the Phelps-Streeter formula for estimating the course of oxygen
depletion along a polluted stream have recently been published! 1 ].   They differ from the
original formulation by allowing for two (not one) ingredients of oxidizable matter in
the stream, each with a different rate of decomposition.   They also make allowance for
a delay in time of onset of decomposition of one of the ingredients.   The modifications
proposed in this paper result in a somewhat similar shape of sag curve but are based upon
different assumptions and do not involve the choice of an arbitrary time delay.   They also
include an allowance for the effect of sedimentation in a manner proposed by one of the
authors in a much earlier publication [2].   Of interest, one of the authors has found in
his laboratory investigation that K2, the reaeration coefficient, increases (and does not
decrease) as the temperature goes down.   On the other hand, K3 decreases as the temperature rises.
In essence, this study of the deoxygenation kinetics in the stream water is based upon
the hypothesis that the oxidation of the polluting matter introduced into a stream takes
place in two steps: (a) the oxidation of the primary compound present or introduced
into the stream at time zero; and (b) the oxidation of another compound resulting from
the breakdown of the primary compound which, presumably, takes place at a different
rate. It is quite possible that some resistant or recalcitrant organic compounds may have
a third oxidized derivative, etc.
The details of the hypothesis will now be expressed in mathematical form as the derivation of a general formula for the oxygen sag curve is presented.   Notation will be explained as needed and the corresponding information will be indexed in a special section
at the close of the paper.  The notation of earlier workers will be followed as closely as
is practicable.
ASSUMPTIONS
Assumption I
That the primary oxidizable matter whose concentration in a stream is L at time t
(L = L0 when t = 0) decreases as it proceeds downstream at a rate given by the equation:
dL/dt  =  — KjL (This equation defines KT) (1)
This is the total rate of reduction due to all causes!   On integration this becomes:
L  =   Lo  e-KT' (2)
Assumption II
That the portion of this total rate of reduction of the primary compound due to its
own oxidation is given by the equation:
252