CEE Fall, 2007 CEE 5134 Deoxygenation – Reaeration and the The Streeter-Phelps Equation Thomas J. Grizzard 25 October, 2007

CEE Fall, 2007 Streeter-Phelps History Modeling of oxygen dynamics in flowing waters has become a sophisticated discipline in the 21 st century Modern models are built on the work of Streeter and Phelps on the Ohio River in the first quarter of the 20 th century (1914 – 1925) Developed a relationship to predict longitudinal oxygen profile in flowing waters as a function of: –Strength of degradable organic matter Consisting of mix of background BOD in stream and BOD introduced by a waste discharge –Rate of diffusion of oxygen into the water from the atmosphere

CEE Fall, 2007 Who Were Streeter and Phelps? Beginning in 1913, USPHS maintained a laboratory in Cincinnati dedicated to the study of “the manifold problems of stream sanitation” –Eventually morphed into the EPA Cincinnati Laboratory H.W. Streeter: Sanitary Engineer at the USPHS Cincinnati Lab E.B. Phelps: Professor of Sanitary Science at Institute of Public Health of the College of Physicians and Surgeons at Columbia University Studies on the oxygen dynamics of the Ohio River were commissioned in 1914 and 1915 by Surgeon General W.H. Frost –In 1925, Phelps dedicates his book, Stream Sanitation, to the memory of Frost

CEE Fall, 2007 Key Historical Publications Purdy, W.C., “A Study of the Pollution and Natural Purification of the Ohio River, Vol. I, The Plankton and Related Organisms,” Public Health Bulletin No. 133, U.S. Public Health Service, Washington, D.C., Streeter, H.W. and W.H. Frost, “A Study of the Pollution and Natural Purification of the Ohio River, Vol. II, Report on Surveys and Laboratory Studies,” Public Health Bulletin No. 143, U.S. Public Health Service, Washington, D.C., Streeter, H.W. and E. B. Phelps, “A Study of the Pollution and Natural Purification of the Ohio River, Vol. III, Factors Concerned in the Phenomena of Oxidation and Reaeration,” Public Health Bulletin No. 146, U.S. Public Health Service, Washington, D.C., 1925.

CEE Fall, 2007 Effect of Organic Wastes on Stream Ecosystems Streeter-Phelps Model: Dissolved Oxygen Sag Curve –The Streeter-Phelps Equation is integral to most of the widely used dissolved oxygen models in use today –Addition of degradable organic matter (BOD) to a flowing watercourse causes a slow decrease in O 2, caused by heterotrophic metabolism –Opposing “deoxygenation” is reaeration, which proceeds at a rate proportional to the concentration deficit (relative to the saturation concentration)

CEE Fall, 2007 Oxygen Sag Effects on Biological Communities

CEE Fall, 2007 Components of the Oxygen Sag Curve

CEE Fall, 2007 Definitions for the DO Sag Curve

CEE Fall, 2007 Streeter-Phelps Model Assumptions Stream behaves as an ideal plug flow reactor (PFR) Flow rate, stream cross section and longitudinal velocity are constant Physical, chemical, and biochemical reactions of interest are BOD exertion and O 2 transfer across air-water interface

CEE Fall, 2007 Streeter-Phelps Model Limitations Considers only one “sink” for DO – Degrading BOD –Missing: NOD, SOD, nonpoint sources, algal respiration, degradation of microbial products Considers only one “source” for DO – Atm. Reaeration –Missing: Algal photosynthesis Downstream movement is by advection only (ideal PFR). –Missing: Dispersion/Diffusion Velocity, depth, BOD exertion, and reaeration are invariant with distance

CEE Fall, st Order BOD Exertion Relationship It has been shown that, under experimental conditions approximating those prevailing in a stream containing reserve dissolved oxygen, this reaction an orderly and consistent one, proceeding at a measurable rate, and according to the following law: “The rate of biochemical oxidation of organic matter is proportional to the remaining concentration of unoxidized substance, measured in terms of oxidizability.” - Phelps, Earle B.

CEE Fall, 2007 Quantitatively Stating the BOD Exertion Relationship As long as oxygen is present, the rate of biochemical oxidation of organic matter is proportional to the amount of organic matter remaining….

CEE Fall, 2007 What does BOD exertion look like?

CEE Fall, 2007 Summary of BOD Exertion As long as oxygen is present, the decline in BOD remaining (L) is exponential If the system is closed, such as in a BOD bottle, the DO supply is fixed, and no replenishment from the atmosphere can take place –This is the case for laboratory BOD measurements –Allows construction of a quantitative DO budget between start and the finish of the test What’s different about what happens in “the world?” –System is open –BOD is exerted, but DO depletion is opposed by continuous replenishment from the atmosphere

CEE Fall, 2007 Oxygen Replenishment by Atmospheric Reaeration Reaeration rate is 1 st order with oxygen deficit Rate coefficient is related to stream characteristics: –Velocity, Turbulence, Depth

CEE Fall, 2007 What Does Reaeration Look Like?

CEE Fall, 2007 Reaeration: O’Connor and Dobbins Formula Based on surface renewal theory –Model is that “parcels” of water are brought to air-water interface for some finite time period –Gas exchange takes place only while a water “parcel” is at the surface –After moving away from the surface, water “parcel” mixes with the liquid bulk

CEE Fall, 2007 Reaeration: Churchill et al. (1962) Formula Based on empirical studies of reaeration of under- saturated waters downstream of dams on the Tennessee River Correlated measured reaeration rates with velocity (U) and depth of flow (H)

CEE Fall, 2007 Reaeration: Owens and Gibbs (1964) Formula Conducted studies of British streams where oxygen was artificially depleted by sulfite additione Combined British and Tennessee River data –Correlated measured reaeration rates with velocity (U) and depth of flow (H):

CEE Fall, 2007 Atmospheric Reaeration Depth, (m) Depth, (ft) Method of Covar (1976) Uses formulae of: –O’Connor & Dobbins –Churchill –Owens-Gibbs Input stream velocity and depth of flow Select k r (d -1 ) at intersection of flow and depth coordinates

CEE Fall, 2007 Atmospheric Reaeration Method of Covar (1976) Uses formulae of: –O’Connor & Dobbins –Churchill –Owens-Gibbs Input stream velocity and depth of flow Select k r (d -1 ) at intersection of flow and depth coordinates

CEE Fall, 2007 Reaeration Coefficient Estimation from Stream Descriptions Water Body Descriptionk r (days 20 o C) Small ponds and backwaters Sluggish streams and large lakes Large streams of low velocity Large streams of normal velocity Swift streams Rapids and waterfalls> 1.15 Source: Peavy, Rowe and Tchobanoglous, 1985

CEE Fall, 2007 Temperature Corrections for Rate Coefficients Rule of Thumb is that biochemical reaction rates double with a 10 o C temperature increase (Van’t Hoff Rule) Arrhenius Equation may be used to more rigorously correct rate coefficients for differences in temperature : Arrhenius Van’t Hoff Source:

CEE Fall, 2007 Determine BOD ULT in Stream after Mixing with Discharge Construct mass balance on river flow and waste discharge to get BOD ULT of mixture: Waste Q w, Input L w River Flow Q r, L r Mixed Flow Q r +Q w, L 0

CEE Fall, 2007 Simplified Schematic Representation of Model Assume PF and define control volume as a unit rectangle Control volume moves downstream at constant velocity Determine the initial oxygen content after mixing (L 0 ) Compute DO at any time by solving differential equation for BOD exertion and atmospheric reaeration

CEE Fall, 2007 Differential Equation for Predicting Longitudinal DO Profile Identify a single fluid element in the stream –Constant volume –Constant velocity Write a mass balance for oxygen on the element as affected by BOD exertion and atmospheric rearation: –Accumulation = input – output ± reaction –Since the fluid element is “intact,” there is no flow in or out, and the mass balance becomes:

CEE Fall, 2007 Streeter-Phelps Development, continued

CEE Fall, 2007 Streeter-Phelps Development, continued

CEE Fall, 2007 Streeter-Phelps Development, continued

CEE Fall, 2007 Streeter-Phelps Development, continued

CEE Fall, 2007 Streeter-Phelps Development, continued

CEE Fall, 2007 Streeter-Phelps Development, continued

CEE Fall, 2007 Compute the Critical Deficit The critical deficit (D) occurs where the rate of change of D with time = 0 (dD/dt =0) May be computed from the original DE by setting the 1 st derivative equal to zero:

CEE Fall, 2007 Time to Reach the Critical Deficit (Lowest DO)

CEE Fall, 2007 Deoxygenation and Recovery in a Flowing Stream

CEE Fall, 2007 A Streeter-Phelps Model Example Problem Wastewater mixes with a river resulting in: –BOD = 10.9 mg/L –DO = 7.6 mg/L –Temperature = 20  C Deoxygenation constant = 0.2 day -1 Average flow = 0.3 m/s Average depth = 3.0 m What is the distance downstream of the maximum oxygen deficit? What is the minimum value of DO?