Reaeration in Open-Channel Flow
GEOLOGICAL SURVEY PROFESSIONAL PAPER 737 Reaeration in Open-Channel Flow By J. P. BENNETT and R. E. RATHBUN
GEOLOGICAL SURVEY PROFESSIONAL PAPER 737
An evaluation of procedures for measuring and predicting the reaeration coefficient of open-channel flows
UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1972 UNITED STATES DEPARTMENT OF THE INTERIOR
ROGERS C. B. MORTON, Secretary
GEOLOGICAL SURVEY
V. E. McKelvey, Director
Library of Congress catalog-card No. 72-600020
For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price $1 (Paper Cover)
Stock Number 2401-2146 CONTENTS
Page Page Abstract ...... 1 Experimental determination of the reaeration Introduction...... - .. . 1 coefficient Continued Theories for predicting the reaeration coefficient. . 7 Techniques for determining the reaeration Diffusion equation...... 7 coefficient Continued Conceptual models for the gas-liquid Disturbed-equilibrium technique 30 mass-transfer process...... 9 Tracer technique...... 31 Film model. . . .. 9 Analysis of errors in reaeration-coefficient Renewal models-..-..-...-.- 10 calculation techniques-.-- 33 Penetration model.... 10 Accuracy of measured dissolved-oxygen Film-penetration model...... 11 balance parameters.-.-.-...-... 33 Surface renewal-damped-eddy Dissolved-oxygen balance technique. .. 34 diffusivity model 13 Disturbed-equilibrium technique. .. 39 Kinetic theory..-.--.....--...... 14 Tracer technique .. 42 Large-eddy model...--...-.-..-.. . . 15 Laboratory measurements...... 43 Turbulent-diffusion model... . 16 Prediction of the reaeration coefficient - . 44 Semiempirical models ...... 17 Reaeration-coefficient data sets-.... 44 Krenkel model..--..-....-...... 18 Field data.-...... -..--....--.-...-.. 44 Thackston model.-.....-...... -...-..- . . 18 Flume data.-...... -.....-..-...... - 46 Empirical prediction equations.. .. 19 Evaluation of reaeration-coefficient prediction Streeter-Phelps equations.-. . 19 equations...... 47 Churchill-Elmore-Buckingham equations. 20 Development of a reaeration-coefficient Owens-Edwards-Gibbs equation. . 20 prediction equation ...... 48 Langbein-Durum equation...... 21 Regression analysis.-. , . 51 Isaacs-Gaudy equations.. .. . 21 Results of regression analysis. 52 Negulescu-Rojanski equations...... 21 Effect of detergents and wind on the reaeration Experimental determination of the reaeration coefficient...... 21 coefficient...... -...... 56 Measurement of dissolved oxygen in water. . 22 Conclusions and recommendations 58 Techniques for determining the reaeration Conclusions.--..-..-....-.. .. 58 coefficient-.....-...... - . 22 Recommendations.--...-.--...-.....-.. 58 Dissolved-oxygen balance technique. . 22 References....--...... 59 BOD test and KI...... 23 Appendix A: Effect of a corrosion reaction on Bottom deposits, KS, and m...... 24 reaeration-coefficient measurements in a Photosynthesis and respiration...... 26 laboratory flume--...-....-..-...---.. 67 Other parameters...... 28 Symbols used in appendix A...... 70 Calculation of k% in a recirculating flume. 29 Appendix B: Collected data...... 73
ILLUSTRATIONS
[All illustrations are graphs] Page FIGURE 1. Example of a DO deficit sag curve...... 3 2. Computation of reaeration coefficient from recirculating-flume data...... -. . 30 3-10. Dissolved-oxygen balance technique: 3. Sensitivity of reaeration coefficient to rate constant for BOD decay.-. 35 4. Sensitivity of reaeration coefficient to photosynthetic-oxygen production rate . 35 5. Sensitivity of reaeration coefficient to upstream DO deficit.. . 35 6. Sensitivity of reaeration coefficient to upstream BOD.-.-..-...-...... --.... - 36 7. Sensitivity of reaeration coefficient to traveltime ...... 36 8. Sensitivity of reaeration coefficient to downstream DO deficit ...... 37 9. Effect of a systematic error in DO measurement procedure on calculated reaeration coefficient--...... -.....---..-.-...-...... ---...-- . - 38 10. Effect of ignoring photosynthetic-oxygen production rate-...... --.-. - -- 38 in IV CONTENTS
FIGURES 11-16. Disturbed-equilibrium technique: Page 11. Sensitivity of reaeration coefficient to photosynthetic-oxygen production rate...... 39 12. Sensitivity of reaeration coefficient to upstream DO deficit...... 40 13. Sensitivity of reaeration coefficient to downstream DO deficit...... 40 14. Sensitivity of reaeration coefficient to traveltime...... 40 15. Effect on calculated reaeration coefficient of ignoring a respiration rate difference...... 40 16. Variation of calculated reaeration coefficient with respiration-rate ratio...... 41 17. Reaeration coefficient as a function of depth, from prediction equations of table 10 ...... 50 18. Synthetic flume data from the upstream sampling station...... 70 19. Determination of 61 and K2 from M values calculated at two different DO deficits...... 71
TABLES
Page TABLE 1. Basic parameter sets used in the sensitivity analyses of the DO balance equation...... 34 2. Assumed errors in measurements of DO balance parameter...... 37 3. Slopes of the sensitivity-analysis plots...... 37 4. Errors in the calculated reaeration coefficient resulting from errors in parameter measurement, DO balance technique...... 37 5. Basic parameter sets for the sensitivity analysis of the disturbed-equilibrium technique...... 39 6. Assumed parameter errors and slopes of sensitivity-analysis plots for disturbed-equilibrium technique...... 41 7. Errors in the calculated reaeration coefficient resulting from errors in parameter measurement, disturbed-equilibrium technique...... 41 8. Statistical data concerning measured gas-transfer coefficient...... 42 9. Ranges of experimental data from the literature-.--.-..-.--.--...... -.--..--.....-...... --.-..-....-...... 45 10. Performance of k2 prediction equations in terms of standard error of estimate...... 49 11. Results from regression analysis...... 52 12. Values of Student t for comparison of the exponents in equation 162 with those for the two corresponding field and flume equations of table 11...... 53 13. Dependence of turbulent intensity on the mean hydraulic parameters...... 54 14. Results of the Student t test of the exponents in the flume equations and in the field equations of table 13...... 54 15. Prediction equations for mean velocity, obtained from the data used in table 11...... 54 16. Results of the Student t test of exponents in the regression equations of this study and corresponding equations in the literature...... 55 17. Parameters used in calculating the synthetic flume data and the resulting calculated reaeration coefficient...... 69 18-26. Reaeration coefficients and hydraulic data from: 18. Churchill, Elmore, and Buckingham (1962)...... 73 19. Owens, Edwards, and Gibbs (1964)...... 73 20. Gameson, Truesdale, and Downing (1955)...... -...... 74 21. O'Connor and Dobbins (1958) (Streeter and Phelps, 1925)...... 74 22. O'Connor and Dobbins (1958) (various sources)...... _...... 74 23. Tsivoglou and others (1967, 1968)...... 74 24. Thackston (1966)...... 75 25. Krenkel (I960)...... 75 26. Negulescu and Rojanski (1969)...... 74
SYMBOLS
Symbol Definition a Parameter in the surface-renewal-damped-eddy diffusivity model (King, 1966). fli Constant in the equation for consumption of dissolved oxygen by bottom deposits. a-i Constant in the Edwards and Rolley (1965) equation for rate of consumption of dissolved oxygen by bottom deposits. flo Rate of addition of dissolved oxygen to the reach by drainage accrual, in milligrams per liter per day. A Temperature relation in the equation of Dobbins (1965); ^ = 9.68+0.054 (7-20°), where T is temperature, in degrees Celsius. A^A\.. .AN Coefficients in the linear-regression equation. CONTENTS V
Symbol Definition 4e Parameter in the large-eddy model (Fortescue and Pearson, 1967), in centimeters per second. ^o Rate of addition of dissolved oxygen to the reach by drainage accrual, in grams per square meter per hour. 4 S Water surface area, in square meters. Ax Cross-sectional area of the reach at longitudinal position jc, in square feet. b Proportion of available molecules in the surface layer that actually escape. b\ Exponent on dissolved-oxygen concentration in the Edwards and Rolley (1965) equation for rate of consumption of dissolved oxygen by bottom deposits. B Temperature relation in the equation of Dobbins (1965); 5 = 0.976 + 0.0137 (30°-7):!/2, where T is temperature, in degrees Celsius. BI Population regression coefficient. ^F Factor in the Dobbins (1965) equation, equal to 1.0 + F2, where F is the Froude number. BOD Biochemical oxygen demand, in milligrams per liter. BPS Basic parameter set; a set of values assumed for the parameters of the dissolved-oxygen balance equation or the disturbed-equilibrium technique equation, c Concentration fluctuation, or the difference between the instantaneous concentration and the time-averaged concentration, in milligrams per liter or in grams per cubic meter, ci, c2, c;i, c4, c-r, Constants in the equations of Dobbins (1964b, 1965). fi, c3', cs' Constants in the Metzger-Dobbins (1967) model. C Dissolved-oxygen concentration, in milligrams per liter or in grams per cubic meter. C _ Mean concentration of dissolved oxygen in the reach, in milligrams per liter. C, d Mean concentrations of dissolved oxygen in the reach for the two levels of dissolved-oxygen concentration, disturbed-equilibrium technique, in milligrams per liter. Ca, Ch Dissolved-oxygen concentrations at the upstream and downstream ends of the reach, respectively, at the first level of dissolved-oxygen concentration of the disturbed-equilibrium technique, in milligrams per liter. Ca, C'b Dissolved-oxygen concentration at the upstream and downstream ends of reach, respectively, at the second level of dissolved-oxygen concentration of the disturbed-equilibrium technique, in milligrams per liter. Ci Dissolved-oxygen concentration at the gas-liquid interface, in milligrams per liter. C/i, Co/ Initial and final concentrations, respectively, of dissolved oxygen in the dark bottle of the light- and dark-bottle test for photosynthesis, in milligrams per liter. CH, Cif Initial and final concentrations, respectively, of dissolved oxygen in the light bottle of the light- and dark-bottle test for photosynthesis, in milligrams per liter. Dissolved-oxygen concentration at saturation, in milligrams per liter or in grams per cubic meter. Saturation concentrations of dissolved oxygen for the two levels of dissolved-oxygen concentration of the disturbed- _ equilibrium technique, in milligrams per liter. Cs Mean saturation concentration of dissolved oxygen in the reach, in milligrams per liter. Ct Dissolved-oxygen concentration, in milligrams per liter; subscript t indicates variability with respect to time at a particular point in space. CS Concentration of the tracer gas in the water, in milligrams per liter. CL Dissolved-oxygen concentration in the bulk liquid phase, in milligrams per liter. Ct Concentration of the conservative liquid tracer in the water, in milligrams per liter. AC Difference in dissolved-oxygen concentration between upstream and downstream ends of the reach, in milligrams per liter. AC Difference in dissolved-oxygen concentration between upstream and downstream ends of the reach at sunrise, in milligrams per liter. d\, dt Molecular diameters of gases 1 and 2, in angstroms. do Rate of transfer of oxygen through the water surface, in milligrams per liter per day. dia Depth of bottom deposit, in centimeters. D Dissolved-oxygen deficit, or the difference between the saturation concentration and the concentration of dissolved oxygen, in milligrams per liter. ^a Dissolved-oxygen deficit at upstream end of reach, in milligrams per liter. D'a Dissolved-oxygen deficit at upstream end of reach at the second level of dissolved-oxygen concentration of the disturbed-equilibrium technique, in milligrams per liter. A> Dissolved-oxygen deficit at downstream end of reach, in milligrams per liter. £>b Dissolved-oxygen deficit at downstream end of reach at the second level of dissolved-oxygen concentration of the disturbed-equilibrium technique, in milligrams per liter. Di Combination of the molecular- and turbulent-diffusion coefficients at the surface, Krenkel and Orlob (1963) model, in square feet per day. Dm Molecular-diffusion coefficient, in square feet per day. Do Rate of transfer of oxygen through the water surface, in grams per square meter per hour. Ar Longitudinal dispersion coefficient, in square feet per day. DH Parameter that includes the effect of photosynthesis, plant respiration, and removal of dissolved oxygen by benthal layer; in milligrams per liter per day. DM, DK Average dissolved-oxygen deficit in the reach at that time in the morning and evening, respectively, when the production of oxygen by photosynthesis is zero, in milligrams per liter. VI CONTENTS
Symbol Definition DO Dissolved oxygen. E Rate of energy dissipation per unit mass of liquid, equal to USg for open-channel flow, in ergs per second per gram. E Combination of the activation energies for molecular diffusion and viscous flow, in degrees Kelvin. ~E Average kinetic energy of a roll cell, large-eddy model (Fortescue and Pearson, 1967), in square centimeters per second per second. Ea Activation energy for the absorption of oxygen by water, in calories per gram mole. EP Percent standard error of estimate for the reaeration coefficient, k2, in percent. Es Standard error of estimate for the reaeration coefficient, k2, in reciprocal days. ESL Standard error of estimate for the reaeration coefficient, £2, based on common logarithms. Ev Rate of energy dissipation per unit volume of liquid, in ergs per second per cubic centimeter. E( Rate of energy dissipation per unit mass of liquid at the surface, in ergs per second per gram. / Dissolved-oxygen exchange coefficient, in meters per hour. f(k2) Functional relation of the reaeration coefficient, common logarithm base; used in the sensitivity analyses of the dissolved-oxygen balance and disturbed-equilibrium techniques. f'(k2) First derivative of the functional relation of the reaeration coefficient, common logarithm base; used in the Newton- Raphson iterative method for solving the dissolved-oxygen balance and disturbed-equilibrium equations for the sensitivity analyses. fi(s) Rate-of-surface-renewal function, Thackston and Krenkel (1969a) model. f2(H) Depth-of-flow function, Thackston and Krenkel (1969a) model. F Volumetric water flow rate, in cubic meters per hour or cubic feet per second. Fc Source-sink term, in milligrams per liter per day. Tc Source-sink term averaged over the depth of flow, in milligrams per liter per day. F Froude number, equal to U/VgH. g Acceleration of gravity, in feet per second per second. G General functional relation between dependent and independent variables in linear regression analysis. h Henry's Law constant, in liters per milligram. H Mean depth of flow, in feet or meters. H' Depth above minimum low-water stage, Streeter and Phelps (1925) prediction equation for the reaeration coeffi cient, in feet. j Exponent on depth of flow in the general form of the empirical prediction equation for the reaeration coefficient. J Constant in the Thackston and Krenkel (1969a) model. kt Deoxygenation coefficient or rate constant for biochemical oxidation of carbonaceous material, common logarithm base, in reciprocal days. k2 Reaeration coefficient or rate constant for oxygen absorption from the atmosphere, common logarithm base, in reciprocal days, or in reciprocal minutes. (£2)0 Reaeration coefficient calculated from the results of an experimental test, common logarithm base, in reciprocal days. (k2)e Reaeration coefficient estimated from a linear regression analysis equation, common logarithm base, in reciprocal days. k3 Rate constant for the removal of BOD by sedimentation and adsorption, common logarithm base, in reciprocal days. kL Overall mass-transfer coefficient based on liquid phase, equivalent to k2H, in centimeters per second or in feet per day. Ki Deoxygenation coefficient or rate constant for biochemical oxidation of carbonaceous material, natural logarithm base, in reciprocal days. K2 Reaeration coefficient or rate constant for oxygen absorption from the atmosphere, natural logarithm base, in reciprocal days. K2(, Mass-transfer coefficient for the tracer gas, in reciprocal days. K2j, Reaeration coefficient at 7° Celsius, in reciprocal days. K2lt Reaeration coefficient at 20° Celsius, in reciprocal days. K3 Rate constant for the removal of BOD by sedimentation and adsorption, in reciprocal days. KG Overall mass-transfer coefficient based on the gas phase, in milligrams per square centimeter per second per atmosphere. KL Overall mass-transfer coefficient based on the liquid phase, in centimeters per second or in feet per day. KN Rate constant for the oxidation of nitrogenous material, in reciprocal days. L Biochemical oxygen demand of carbonaceous material, in milligrams per liter. La Biochemical oxygen demand of carbonaceous material at upstream end of reach, in milligrams per liter. Lb Biochemical oxygen demand of carbonaceous material at downstream end of the reach, in milligrams per liter. Lf Film thickness, in feet or meters. LI Average film thickness, in feet or meters. Lt Biochemical oxygen demand of carbonaceous material, in milligrams per liter; subscript / indicates variability with respect to time at a particular point in space. LK Rate of addition of BOD along the reach, in milligrams per liter per day. (Includes addition of BOD with runoff, the diffusion of partially degraded waste from the benthal layer into the water, and the scour and resuspensibn of bottom deposits.) CONTENTS VII
Symbol Definition m Rate of addition of BOD to water from bottom deposits, in milligrams per liter per day. M Molecular weight. Ms Modulus of compression at the surface, in grams per second per second. n An integer. ns Number of new surface layers exposed per unit of time, in reciprocal seconds. A7 Mass-transport rate of oxygen, in milligrams per square centimeter per second. N Average rate of mass transport of oxygen, in milligrams per square centimeter per second. A7 Number of data sets in linear regression analysis. A'jr Concentration of nitrogenous material at longitudinal position x, in milligrams per liter. P Rate of production of dissolved oxygen by photosynthesis, in milligrams per liter per day. P Rate of production of dissolved oxygen by photosynthesis at the second level of dissolved-oxygen concentration of the disturbed-equilibrium -technique, in milligrams per liter per day. p(f) Distribution function for surface-renewal time or for distance below the interface to which renewal occurs (/ is dummy variable for either t or yO, Harriott (1962) model. PI Partial pressure of the diffusing gas, oxygen, at the gas-liquid interface, in atmospheres. Pt Rate of production of dissolved oxygen by photosynthesis, in milligrams per liter per day; subscript / indicates variability with respect to time at a particular point in space. PJ-,I Rate of production of dissolved oxygen by photosynthesis at longitudinal position x at time /, in milligrams per liter per day. PH Barometric pressure, in atmospheres. PC Partial pressure of the diffusing gas, oxygen, in the gas phase, in atmospheres. P Rate of gross primary production of dissolved oxygen by photosynthesis, in grams per square meter per hour. PI Rate of production of dissolved oxygen by photosynthesis in the light bottle of the light- and dark-bottle test, in milligrams per liter per day. PH Gross rate of photosynthetic production of dissolved oxygen by attached plants, in grams per square meter per hour. PP Gross rate of photosynthetic production of dissolved oxygen by phytoplankton, in grams per square meter per hour. P Peclet number, equal to AAf/Dm. q Rate of change of dissolved-oxygen content in the reach, in milligrams per liter per day. q\ Rate of change of dissolved-oxygen content in the reach at sunrise, in milligrams per liter per day. q\t, qn Rate of change of dissolved-oxygen content in the reach at that time in the morning and evening, respectively, when the production of oxygen by photosynthesis is zero, in milligrams per liter per day. Q0 Rate of change of dissolved-oxygen content in the reach, in grams per square meter per hour. Qi, Qa Average rate of change of dissolved-oxygen concentration per unit area in the reach for the two levels of dissolved- oxygen concentration, disturbed-equilibrium technique, in grams per square meter per hour. r Rate of consumption of dissolved oxygen by plant respiration, in milligrams per liter per day. re Rate of entry of gas molecules into the surface layer, in grams per second. r0 Rate of loss of gas molecules from a surface layer, in grams per second. r, Rate of consumption of dissolved oxygen by plant respiration, in milligrams per liter per day; subscript t indicates variability with respect to time at a particular point in space. rf, t Rate of consumption of dissolved oxygen by plant respiration at longitudinal position x at time /, in milligrams per liter per day. R Rate of dissolved-oxygen utilization by all processes, in grams per square meter per hour, R Ratio of the oxygen consumed by respiration at the two levels of dissolved-oxygen concentration, disturbed- equilibrium technique. Rd, RI Rate of consumption of dissolved oxygen by respiration in the dark and light bottles, respectively, of the light- and dark-bottle test for photosynthesis, in milligrams per liter per day. Ri Equals the quantity (Ai B>)jSbi, used to test the significance of the regression coefficient A-,. RB Rate of consumption of dissolved oxygen by respiration by attached plants, in grams per square meter per hour. RBOD Rate of consumption of dissolved oxygen by the oxidation of organic wastes in the stream water, in grams per square meter per hour. Rc Ideal-gas constant, in calories per gram mole per degree Kelvin. RH Hydraulic radius, equal to the cross-sectional area of flow divided by the wetted perimeter, in feet. RM Rate of consumption of dissolved oxygen by bottom deposits, in grams per square meter per hour. RP Rate of consumption of dissolved oxygen by respiration of phytoplankton, in grams per square meter per hour. RMS Root-mean-square, or square root of the mean value of the square of the deviations of the measured parameter from the mean parameter. * Average rate of surface renewal, in reciprocal hours. 5 Water-surface or channel slope, in feet per foot. Sbt Standard error of the regression coefficient A-,. SEf, t Rate of removal of BOD by sedimentation at longitudinal position x at time /, in milligrams per liter per day. t Time, in days or hours. t' Flow time, or the time that must elapse between dissolved-oxygen measurements at the two sampling points in a recirculating flume, in seconds. VIII CONTENTS
Symbol Definition ti, tz The time at which sunrise occurred, and the time in the evening when the rate of change of dissolved-oxygen con tent in the reach returned to its sunrise value. ta, / Times at which the concentrations of dissolved oxygen are measured at the upstream and downstream ends, respectively, of the reach; t^-ta is the flow time, or the time required for a slug of water to flow through the reach, in hours. te Age of surface at which surface elements are replaced, in days or hours. T Temperature, in degrees Celsius. T Absolute temperature, in degrees Kelvin. M* Shear velocity, equal to Vg//5, in feet per second. 7T' Longitudinal turbulent intensity, in centimeters per second. ue Eddy velocity in the longitudinal direction, large-eddy model (Fortescue and Pearson, 1967), in centimeters per second. M, Velocity fluctuation in the xi direction (tensor notation), or the difference between the instantaneous velocity and __ the time-averaged velocity, in feet per day. ~i«c~ Product of the velocity fluctuation and the concentration fluctuation, in milligrams per liter times centimeters per second. (Overbar indicates average with respect to time.) U Mean flow velocity in the longitudinal direction, in feet per day. Ui Velocity in the jc, direction (tensor notation), in feet per day. ve Eddy velocity in the vertical direction, large-eddy model (Fortescue and Pearson, 1967), in centimeters per second. V Volume of the system, in cubic centimeters. w Exponent on velocity in the general form of the empirical prediction equation for the reaeration coefficient. W Width, in feet or meters. WG Weight of gaseous tracer (krypton) injected for a tracer-procedure determination of the reaeration coefficient, in pounds. WL Weight of conservative liquid tracer (tritiated water) injected for a tracer-procedure determination of the reaeration coefficient, in pounds. W0(i Weight of gaseous tracer (krypton) injected per unit of cross-sectional area of flow, equal to Wc,IAx, in pounds per square foot. WVL Weight of conservative liquid tracer (tritiated water) injected per unit of cross-sectional area of flow, equal to WLJAX, in pounds per square foot. x Longitudinal position or direction, in feet. x Longitudinal position in a coordinate system moving with the mean flow velocity, equal to x Ut. *a> *b Longitudinal position of the upstream and downstream ends of the reach, in feet. Xi Coordinate direction in tensor notation: Xi, longitudinal direction; x2, vertical direction; .v:i, lateral direction. A* Length of reach, or Xb-xa, in feet. X Distance between measurement points in a recirculating flume, in feet. Xt Amount of carbonaceous waste oxidized up to time t, in milligrams per liter. Xn, X-2\.. .Ajv, Independent variables in the linear regression equation. y Vertical position or direction, in feet. y\ Vertical distance below the interface to which surface renewal occurs, Harriott (1962) model, in feet. y Dependent variable in linear regression equation. Ye> Estimate of the dependent variable, Yi, from the linear regression equation. ^f Dependent variable / in linear regression equation. Z Constant in the Streeter and Phelps (1925) prediction equation for the reaeration coefficient. <* >$ Parameters in the distribution function for surface-renewal time or for distance below the interface to which sur face renewal occurs, Harriott (1962) model. ft Turbulence parameter for a flow layer in the Marchello and Toor (1963) model, equal to s(LfH-2)2, in square feet per hour. F Gamma function, T(a) = f^ e-«ya- l dy. 8 Mean relative increase in flow velocity per 5-foot increase in gage height, Streeter and Phelps (1925) prediction equation for the reaeration coefficient. d Indicates partial differentiation, e Eddy-diffusion coefficient for mass, equal to eT+Dm, in square feet per day (y and x subscripts indicate vertical and longitudinal directions, respectively; i and j subscripts are tensor notation). T Turbulent diffusion coefficient, in square feet per day (/' and j subscripts are tensor notation). Eddy-diffusion coefficient for momentum, in square feet per day. 17 Exponent on velocity in the Streeter and Phelps (1925) prediction equation for the reaeration coefficient. 6 Temperature coefficient, equal to (K2T/K220) T-'2(f K Slope parameter in logarithmic velocity-distribution law of von Karman. K(; Gas-film coefficient for mass transfer, in milligrams per square centimeter per second per atmosphere. KL Liquid-film coefficient for mass transfer, in liters per square centimeter per second. A Macroscale of turbulence, in centimeters. /i Absolute viscosity of the liquid, in grams per centimeter-second. CONTENTS IX
Symbol Definition /AO Velocity normal to the interface that a gas molecule must have to be able to leave the liquid, in centimeters per second. v Kinematic viscosity of the liquid, in square centimeters per second. IT The constant 3.1416. p Mass density of the liquid, in grams per cubic centimeter, cr Surface tension of the liquid, in dynes per centimeter. (/) Distribution function for surface ages. X Constant in the Isaacs and Gaudy (1968) equation. i/; Constant in the empirical prediction equation for the reaeration coefficient. o> Dissolved-oxygen consumption rate by bottom deposits, in grams per square meter per hour.
466-751 O - 72 - 2
REAERATION IN OPEN-CHANNEL FLOW
By J. P. BENNETT and R. E. RATHBUN
ABSTRACT purification capacity of a stream therefore will Reaeration, the physical absorption of oxygen from the vary as the hydraulic conditions in the stream atmosphere, is the primary process by which a stream re change with time. places oxygen consumed in the biodegradation of organic Although the flow in a specific stream may have wastes. The reaeration process in a stream is characterized varied greatly from day to day and from year to by its surface reaeration coefficient. Hence, knowledge of the reaeration coefficient permits determination of the quantity year, the total annual runoff for the United States of waste that can be discharged into a stream without causing as a whole showed only a small, apparently cyclic, serious depletion of the dissolved-oxygen content of the stream. variation with time for the years 1895 to 1955 (Leo The three basic methods for measuring the reaeration co pold and Langbein, 1960). On the other hand, the efficient are the dissolved-oxygen balance, disturbed-equilib population and the number of industrial facilities rium, and tracer procedures. The dissolved-oxygen balance method consists of measuring the various sources and sinks have increased tremendously, and this growth has of dissolved oxygen and determining by difference the amount placed an ever increasing burden on the rivers and of reaeration needed to balance the equation. The disturbed- streams of the United States. In many instances, equilibrium method consists of artificially producing dissolved- the purification capacities of streams and rivers oxygen deficits by adding sodium sulfite to the stream and have been exceeded. subsequently measuring upstream and downstream concentra The fact that municipal and industrial wastes tions of dissolved oxygen at two different concentration levels. The tracer method consists of using an inert radioactive gas are discharged into a stream does not necessarily as a tracer for oxygen and correlating the rate of desorption mean that the stream is polluted, because the of the tracer gas with the rate of absorption of oxygen. stream does have the capacity for self-purifi Various theoretical models of the oxygen-absorption process cation. The definition of pollution has received exist; however, these models are generally not suited for pre considerable attention. The Colorado Supreme diction of the reaeration coefficient in streams because the model parameters have not been adequately related to bulk- Court in 1934 (Gindler, 1967, p. 5) defined pollution flow hydraulic variables. Semiempirical and empirical equa as follows: tions developed from experimental data adequately predict For the purpose of this case, the word "pollution" means an reaeration coefficients for streams of the type on which the impairment, with attendant injury, to the use of water that the equations were based, but large errors may occur when the plaintiffs are entitled to make. Unless the introduction of equations are applied to other types of streams or to condi extraneous matter so unfavorably affects such use, the condi tions outside the range of variables considered in the original tion is short of pollution. In reality, the thing forbidden is in correlation. jury. The quantity introduced is immaterial. Other definitions were discussed by Haney (1966), INTRODUCTION and he proposed (p. 110) that "pollution is the im pairment of water quality, with resultant sig Rivers and streams are used for the disposal of nificant interference with beneficial water use." municipal and industrial wastes, and Hull (1963) Apparently, from these definitions, the determina suggested that this use is one of the most impor tion of when pollution exists depends both on the tant factors contributing to the general health and intended use of the stream and on some water welfare of the people of the United States. Dis quality parameter that will insure that the water charge of wastes into a stream, however, does not is of a quality adequate for the intended purpose. simply dilute the wastes. Each stream and river Of the various indicators of water quality (such has a natural capacity for oxidizing biodegradable as the coliform bacteria count, the hydrogen ion wastes, thus purifying the waters. This purifica concentration, and the concentrations of phos tion capacity is dependent upon many factors, in phates, nitrates, synthetic organics, and industrial cluding the water discharge, the depth of flow, the chemicals), the one most often used as a measure velocity of flow, and the various sources and sinks of pollution by biodegradable organic wastes is of dissolved oxygen along the stream. The natural DO (dissolved oxygen) concentration (Wolman, REAERATION IN OPEN-CHANNEL FLOW 1960). This report uses DO concentration as the Streeter and Phelps (1925) assumed that the DO principal measure of water quality. Thus the con deficit was determined by these two independent dition of a stream is determined from a DO balance processes. Therefore, the DO balance was which includes all the sources and sinks of DO . M\. \ l-i l^ (4) along the reach of interest. The various types of dt DO balances for streams and rivers that have The solution of equation 1 is appeared in the literature are discussed in the following paragraphs. L=La exp(-/M f). (5) The purification capacity of a stream depends Substituting equation 5 in equation 4, the solution upon its DO resources and its ability to replace the of equation 4 is oxygen consumed in the oxidation of organic jy T T^ *^l L", [exp(- ATi 0~exp(- AT2 0] +Da exp(- wastes. One of the first studies of the DO balance A.2 A.I was the classical study of Streeter and Phelps (6) (1925). These authors assumed one sink and one where La is the BOD of the carbonaceous material source of DO. and Da is the DO deficit just upstream from the The oxygen sink was the biochemical oxidation point at which the waste is added to the stream. of carbonaceous wastes by bacteria in the stream. The time, /, is expressed in terms of flow time along It is generally assumed that biochemical oxidation the stream. If / is flow time, then dD/dt is equiva is a first-order process; that is, the rate of bio lent to U dD/dx, where U is the mean flow velocity, chemical oxidation is proportional to the amount in feet per day, and x is longitudinal distance, in of carbonaceous material remaining to be oxidized. feet. The factors D and L are usually expressed in This process may be described by parts per million or milligrams per liter, and K\ and K2 are usually expressed in reciprocal days. f K, L, (1) A plot of D as a function of t gives the classical sag where L is the BOD (biochemical oxygen demand) curve for DO deficit. An assumption inherent in of the carbonaceous material, / is time, and Ki is equation 6 is that the initial pollution load is added the deoxygenation coefficient or the rate constant at the upstream end of the reach and thereafter for the oxidation of carbonaceous wastes. Note is affected only by biochemical oxidation and not that the concentration of biodegradable waste is by the addition of either pollution or dilution flow expressed in terms of the oxygen demand. The con along the reach. centration of biodegradable waste in the stream is Streeter and Phelps (1925) were aware that the not directly measurable and is of importance only assumption of a slug injection of BOD was an over because it controls the demand for DO. The DO simplification of the actual situation; hence, they demand is determined by a BOD test, and standard also solved equation 4 for a BOD source uniformly procedures for the test were described by the distributed along the reach of interest. This BOD American Public Health Association (1965). If the distribution resulted in a much more complicated DO concentration is expressed in terms of the sat solution. Other assumptions regarding the distri uration deficit, D, or the difference between the bution of the BOD load are possible, and the com DO saturation concentration and the actual con plexity of the solution, equation 6, increases as centration at time /, then the complexity of the assumed distribution for ~dt = ~~di' (2) L increases. An example of an oxygen sag curve for an La of The oxygen source considered by Streeter and 9.0 mg/1 (milligrams per liter), a Da of 1.0 mg/1, a Phelps (1925) was reaeration, or the physical Ki of 0.1 days- 1 (reciprocal days), and a K2 of absorption of oxygen from the atmosphere by the 0.2 days" 1 is presented in figure 1. Just below flowing stream. Adeney and Becker (1919, 1920) the point of addition of the pollution load, the oxi showed that the absorption of oxygen by water is dation of the waste predominates, and the DO def a first-order process; that is, the rate of absorption icit increases. However, reaeration begins as is directly proportional to the saturation deficit. soon as a DO deficit develops, and the rate of re- This process may be described by aeration increases as the deficit increases. Finally, - K n at the minimum deficit of the oxygen sag curve, dt K* D> (3) the rate of supply of oxygen by reaeration equals where K2 is the reaeration coefficient or the rate the rate of consumption of oxygen by biochemical constant for the absorption of oxygen from the oxidation. Thereafter, the reaeration process pre atmosphere and D is the DO deficit. dominates, and the DO deficit is gradually reduced INTRODUCTION With these sources and sinks considered, the DO balance has the form
"^ Ti Urr"*-' _ Lfyrj 2 ~rA.2l,Csi If (f f^\C,J /Vi/-yIS I L>BIn (.&)(Q\ where C is the DO concentration, Cs is the satura tion DO concentration, x is longitudinal distance along the reach, U is the mean stream velocity, Do- is the longitudinal-dispersion coefficient, and DB includes the removal of DO by diffusion into the benthal layer and the effect of photosynthesis and respiration by plants. Therefore, the algebraic value of DB may be positive or negative, depending on the relative magnitudes of the three processes. 10 15 20 TIME, IN DAYS Dobbins (1964b) used the convention that DB is positive when the rate of removal of DO exceeds FIGURE 1. Example of a DO deficit sag curve. the rate of addition. A BOD balance analogous to the DO balance may be written as follows: to zero. Also shown in figure 1 is the increase in the DO deficit in the absence of reaeration (#2 = 0). The oxygen sag equation of Streeter and Phelps where K3 is the rate constant for the removal of (1925) has been widely used, and different forms BOD by sedimentation and adsorption, and LR is of the equation have appeared in the literature. the rate of addition of BOD along the reach, in Some authors have used common logarithms (base cluding the addition of BOD with runoff, the diffu 10), and the deoxygenation and reaeration coeffi sion of partially degraded waste from the benthal cients under these circumstances are generally layer into the water, and the scour and resuspen- represented by ki and A2, respectively, where sion of bottom deposits. Ki = 2.30*i (7) In general, steady-state and uniform-flow con and ditions are assumed in a DO balance study. With #2 = 2.30 #2. (8) these assumptions, the dC/dt and dL/dt terms are A mathematical analysis of the Streeter-Phelps zero, and the water discharge, the input BOD load, equations was presented by Fair (1939). the temperature, and the rate constants at each Processes other than biochemical oxidation and point in the reach are assumed independent of reaeration also may be important in the DO time. Other assumptions inherent in equations 9 balance of a stream. Although Streeter and Phelps and 10 are as follows: (1925) chose to include only these two fundamental 1. The DO and BOD are uniformly distributed processes in their oxygen balance, they were aware over each cross section so that equations 9 of the possible effects of inflow along the reach, and 10 can be written in the one-dimensional sedimentation and scouring in the reach, and form. photosynthesis (Streeter and Phelps, 1925, p. 10, 2. The processes described by the rate constants 22). Other authors, notably Dobbins (1964b) and Ki, K2, and K3 are first-order processes; that Camp (1963), have expanded the fundamental is, the rate of removal of BOD is proportional Streeter-Phelps equation to include in the oxygen to the amount of BOD remaining, and the balance these and other sources and sinks of DO. rate of reaeration is proportional to the DO Dobbins (1964b) included the removal of BOD by deficit. sedimentation or adsorption on bottom sediments, 3. The processes represented by LR and DK are uni the removal of DO from the water by diffusion into formly distributed along the reach. the benthal layer and by plankton and fixed-plant 4. Only the carbonaceous demand of the waste is respiration, the addition of BOD along the reach significant. If the nitrogenous demand is by runoff and by scour of bottom deposits, the important, an additional term must be added addition of DO by the photosynthetic process of to equations 9 and 10. plankton and fixed plants, and the continuous Each of the terms in equations 9 and 10 repre redistribution of the BOD and DO by the effects sents a time-rate of change of the BOD or DO of longitudinal dispersion. concentration, and the units used are generally REAERATION IN OPEN-CHANNEL FLOW parts per million per day or milligrams per liter the reach. Stream flow is also assumed to be con per day. stant and uniform over the reach of interest. Dobbins (1964b) solved equations 9 and 10 using The factor QQ is defined by these assumptions and the added assumption that the effect of longitudinal dispersion is negligible O = (C C ) (15) compared with the other terms for natural where F is the volumetric rate of flow of water, streams. For the BOD equation, the solution As is the water surface area of the reach, and Cb showed that the BOD may either decrease or in and Ca are the concentrations of DO at the down crease along the reach, depending on whether La stream and upstream ends, respectively, of the is greater or less than LnKKi + Ks), where La is the reach at times tb and ra, where tb ta is the time initial BOD load at the upstream end of the reach. required for a slug of water to flow through the Dobbins (1964b) suggested a procedure for the reach. evaluation of the various constants, and he also The factor Do is defined by discussed the DB term, which includes several different processes. D9=fi£.-C), (16) Camp (1963) presented DO and BOD balance where / is the exchange coefficient for the reach, equations similar in form to equations 9 and 10. and Cs and C are the mean saturation and the mean Primary differences are in the use of dC/dt and concentration of dissolved oxygen in the reach, dL/dt as opposed to UdC/dx and UdL/dx and in the respectively. treatment of the DB term. With the assumptions Each term in equations 14 to 16 has the dimen of steady-state and uniform-flow conditions, sions of grams per square meter per hour, or equivalent, and is a rate per unit of surface area. jrui- (11) To put these equations in dimensions comparable to the dimensions of the Streeter and Phelps (1925), Therefore, Camp's equations (1963, p. 295,297) may Camp (1963), and Dobbins (1964b) equations, each be written as term must be divided by the mean depth of flow. =/C2(Cs-C)-/CiL+/7 (12) Thus, the exchange coefficient, /, divided by the dt mean depth of flow is equivalent to the reaeration for the DO balance and coefficient, K2. The basic DO balance, equation 14, was modified as needed to meet the conditions in the stream for the BOD balance. In equation 12, p is the rate under study. For example, Owens and Edwards of production of oxygen by photosynthesis; and (1966) used in equation 13, m is the rate of addition of BOD to QO=PB+PP-RB-RP±DO, a?) the water from bottom deposits. For streams where the B subscript refers to bottom deposits where longitudinal mixing is important, Camp and attached plants, and the P subscript refers to (1963, p. 311) included a Dx(dz/dx2) term on the right- phytoplankton, which live unattached in the water. hand side of equations 12 and 13. In the River Lark, in England, just below a sewage Other forms of DO balances have been used. The works, Owens and Edwards (1963) found that the Water Pollution Research Laboratory group at consumption of oxygen by the bottom deposits or Stevenage, England (Edwards and Owens, 1962; mud was a very important factor. Thus the respira Edwards, 1962; Edwards and Owens, 1965; Owens, tion term, RB, for bottom deposits was divided into 1965), has used a DO balance which consists simply two terms, RB and RM, where the subscript B refers of an algebraic summation of the contributions or to attached bottom plants (benthos) and the sub debits of the various sources and sinks of DO. The script M refers to mud or bottom deposits. The equation is resultant equation, Qo=P±Do-R, (14) where P is the rate of production of oxygen by was discussed by Owens and Knowles (1968). If photosynthesis, R is the rate of oxygen utilization, significant organic pollution exists in the water Do is the rate of transfer of oxygen through the of the stream, then an additional oxygen sink, /?BOD> surface of the stream, and Qo is the rate of change must be added to the right-hand side of equation 18. of DO content in the reach of interest. Equation Bain (1968) used a modified form of the Streeter- 14 is an integral form of the DO balance in which Phelps equation for predicting the variation of the the various coefficients are assumed to be inde DO concentration with time at a point. His DO pendent of time and longitudinal position within balance was INTRODUCTION
dCt on the DO balance of streams receiving discharges =pt -rt-Kl (19) dt of municipal waste. where the t subscript indicates variability with O'Connor (1967) solved the general DO balance time at a particular point in space. Bain (1968) (eq 21) analytically using the assumptions that the wrote equation 19 for each of six 4-hour periods, photosynthetic rate varied sinusoidally with time giving a set of six simultaneous equations for each and that the maximum rate of photosynthesis and day of observation. These equations were solved the factors rx,, and SEX,, were independent of posi by a numerical technique using sequential sub tion and time. The solution was applicable to a sun stitution, and the result was a dynamic DO balance light period of 12 hours. O'Connor discussed in for a 24-hour period at a specific point in the stream. general terms two limiting forms of the solution. In O'Connell and Thomas (1965) considered the the first situation the stream is heavily polluted, is assumption of steady-state conditions as the shaded, and contains large amounts of suspended major fault of the Streeter-Phelps equation and materials and (or) small amounts of nutrient therefore used materials. Under these conditions, p*,, and rflt are small and are negligible with respect to the BOD (20) factors, and the assumption of steady-state condi tions is reasonably valid. The solution reduces to They solved equation 20 using the assumption the oxygen sag equation of Streeter and Phelps that the p and r factors were constant with respect (1925) with a term added for the oxygen demands to time and position. Hence, the solution is appli of the bottom deposits. In the second situation, the cable only at a point for a short interval of flow stream is in the latter stage of self-purification. time. Oxidation of organic material is essentially com Li (1962) presented a generalized solution of the plete, and bacterial respiration is negligible in Streeter-Phelps equation that was expanded to comparison with algae respiration. The assump include the effects of BOD and runoff additions tion of steady-state conditions is not valid, and the along the stream and of fluctuations of the BOD solution reduces to an equation that describes the load and DO concentration at the point of addition temporal distribution of DO as a result of photo- of the pollution to the stream. He assumed that the synthetic activity. variations in BOD load and DO concentration were The variety of DO balances that have appeared cyclic and were repeated cycle after cycle. in the literature demonstrates an important fact: O'Connor (1967) wrote a general DO balance for each stream and each reach of a stream must be a one-dimensional stream in which the oxygen de considered as a separate problem. Generalizations mand of the waste is divided into the requirements are possible, however. For example, McConnell for the carbonaceous material and those for the and Sigler (1959) found in a shallow, swift stream nitrogenous material. The result is that the effect of the stream metabolism on the DO content was negligible compared with the effect of reaeration. On the other hand, in the -KN Nx -rx,t -SEx,t, (21) River Ivel, in England, where the water velocity where F is the volumetric rate of flow, Ax is the seldom was greater than 6 centimeters per second, cross-sectional area of flow at longitudinal position reaeration was relatively unimportant (Owens, x, KM is the rate constant for the oxidation of nitrog 1965). Therefore, in an initial DO balance on a enous material, Nx is the concentration of nitrog stream, all possible sources and sinks of DO should enous material at x, SE is the rate of removal of be considered. The unimportant factors may then BOD by sedimentation, p is the rate of production be eliminated as they are shown to be negligible. of DO by photosynthesis, r is the rate of consump With the development of computers, the compu tion of dissolved oxygen by plant respiration, and tation of DO distributions from complex analytical the x,t subscript indicates the longitudinal posi solutions of DO balances was simplified greatly. tion x at time t. The oxygen demand of the nitrog Frankel (1965) developed a dynamic model of the enous material has not received as much attention sag equation for computer solution. His model was as the oxygen demand of the carbonaceous ma an extension of the model of Li (1962) to include a terial. However, work by Courchaine (1968), variable DO source or sink caused by the presence O'Connell and Thomas (1965), Wezernak and of algae. Gannon (1968), and O'Connor, St. John, and DiToro In addition to simplifying computations, com (1968) has shown the importance of nitrification puters made possible numerical solutions that REAERATION IN OPEN-CHANNEL FLOW previously would not have been feasible. Dresnack tration, Cs, as a result of temperature changes. and Dobbins (1968) presented numerical solutions Cs was assumed to vary linearly with distance of the BOD and DO equations where BOD and DO downstream. Three situations were considered: inputs varying with time and space prevented ana (1) the configuration of treatment plants that lytical solutions. These solutions were limited to would meet the DO standards with minimum cost; steady-state and uniform-flow conditions. Three (2) same as No. 1 except that the minimum DO examples were considered. These were: (1) The standard was reduced 0.1 mg/1; and (3) same as No. 1 effect of the longitudinal-dispersion coefficient on except the requirement was added that all plants the concentration profile of a conservative pollu provide the same percentage of BOD removal. tant introduced at the upstream end of the reach The general application of the concepts of linear in a periodically varying manner; (2) the effect of programing to the solution of the DO balance sinusoidal inputs of BOD and DO at the upstream equation and the determination of the degrees of end of the reach on the BOD and DO profiles; and treatment required for minimum basin cost were (3) the effects of sinusoidally varying inputs of discussed by ReVelle, Loucks, and Lynn (1967). The BOD and DO and a sinusoidal variation in the application of linear programing to the data from algae effects. Bella and Dobbins (1968) presented the Willamette River, Oreg., was discussed by a modeling procedure using finite-difference Revelle, Loucks, and Lynn (1968). The results were techniques for BOD and DO profiles in streams essentially the same as those obtained with and estuaries; this procedure considered temporal dynamic programing. and spatial variations in hydraulic conditions. Most DO balances give one value for the DO con The development of computers also has resulted centration at a downstream point. Similarly, most in the advancement of the concepts of systems DO quality standards do not distinguish as to the analysis, and these concepts in turn have been degree or duration of a pollution violation. This applied to the modeling of the DO resources of a consideration is important because a DO concen stream. A system may in general be defined as an tration that is sufficient on the average to assure entity which operates on or transforms an input healthy fish does not prevent fish kills. Thayer and to produce an output. Thus the system consists of Krutchkoff (1967) presented a stochastic model the input, the output, and the transformation be applicable to steady-state flow and input BOD and tween the two. Thomann (1963) presented a general DO conditions that gives the probability dis model for describing the variations of DO with tributions for the BOD and DO concentrations at time in a finite number of sections of a body of downstream points. With the confidence limits water and applied this model (1965) to the Dela determined from the distributions, the proportion ware River Estuary. He determined the transient of the time that the BOD will be greater than any response of the estuary to an instantaneous in specified concentration can be determined. The jection of waste, demonstrated the ease of extend model can be used with any flow condition, such as ing the systems-analysis model to two dimensions, the expected minimum flow condition, provided and determined the steady-state conditions for the the parameters ATi, K2, K3, and DB associated with one-dimensional model. He also applied the linear the flow condition are known. programing concept to the steady-state conditions Loucks and Lynn (1966) developed probabilistic to determine the most economical combination of models for predicting the probability distribution waste treatments for the various waste sources of minimum DO concentration at points down that would assure a minimum steady-state con stream from a waste-treatment plant. The models centration of DO of 2 mg/1. The latter problem was were based on different mean daily stream and solved for two values of the reaeration coefficient. waste flow conditions. Loucks and Lynn suggested Liebman and Lynn (1966) applied dynamic pro that the use of a maximum allowable probability graming in a simplified example of river-basin of the DO concentration falling below a specified planning based on data from the Willamette River, concentration for a given number of consecutive Oreg. The objective was to determine the amount days was a more realistic approach than the deter of BOD removal required of each waste discharger ministic DO standards generally used. so that the water-quality standards for DO would Kothandaraman and Ewing (1969) considered be met with minimum total cost to the basin. The the effect of variations in Ki and K2 on the down solution of the Streeter-Phelps equation was used stream DO distributions and developed procedures with Ki and K2 assumed known and constant in for taking the expected variations into account each reach. The equation was modified, however, in predicting the DO. A Monte Carlo simulation to account for changes in the saturation concen technique was used to analogize the variations in THEORIES FOR PREDICTING THE REAERATION COEFFICIENT
Ki and K2. The variations of Ki and K2 within their DIFFUSION EQUATION practical ranges were found to have significant effects in the prediction of the DO concentrations, A gas moves into and through a flowing liquid and the errors increased with temperature by the process of diffusion. This process may be increase. described by the general diffusion equation DiToro and O'Connor (1968) studied the effect of (Hinze, 1959) time-varying discharge and mean velocity on the -Z7+Uiac , rr ac =a [D / m ac- u i\,c\ + F~ c, /r»r»\(22) BOD and DO distributions. They found that the dt dxt dxt\ dXi I standard deviations of tlie resultant BOD and DO given here in tensor notation. The capital letters distributions were approximately equal to the co represent time-averaged quantities, the lowercase efficient of variation of the velocity multiplied by letters represent fluctuating components or the the BOD or DO distribution calculated using the instantaneous deviations from the time-averaged mean velocity. quantities, C is the concentration of the diffusing Despite the sophistication of the mathematical substance, LJi is the velocity in the xi direction, Dm techniques permitted by the development of com is the molecular diffusivity of the dissolved gas puters, the various parameters of the DO balance in the liquid, Fc is a source-sink term, and the over- still must be measured or estimated. Thomann bar on Hie indicates an averaging procedure with (1965) stated that his systems-analysis study of the respect to time. Equation 22 applies to a dispersant Delaware River Estuary pointed out "the impor which is in solution in a steady-state flow of an tance of good engineering evaluation of the system incompressible liquid. parameters of flow, diffusion, reaeration, and With the common assumptions that (1) turbulent decay coefficients." diffusion may be described by a turbulent-diffusion A critical analysis of previous theoretical and coefficient, er0, such that experimental work concerning the reaeration co d^fT it MI*-iTc' . \&tj)(9Vi efficient in open-channel flows is given on the following pages. The discussion covers theoretical (2) the total diffusion may be described by a diffu and empirical models for prediction of K2, tech sion tensor, ey, where the turbulent and molecular niques for experimental determination of K2, an diffusion are independent and therefore additive, analysis of previous experimental data, a sug or gested prediction equation, and suggestions for ey= ry +Dm ; (24) future research. From the introductory discus and (3) the coordinate axes coincide with the prin sion of the DO balance for open-channel flows, it cipal axes of the diffusion tensor, Sayre and Chang is apparent that if K2 is to be calculated using a (1968) reduced equation 22 to DO balance equation, all other parameters appear ?JL+ Ut |f=~(e, |f) + /V (25) ing in the equation must be known. The procedures for determining these parameters are therefore For uniform two-dimensional flow in a straight discussed also. channel of constant cross section, equation 25 may be expressed as ac ac= a2C ( dt dx~ x dx2 .. (26) THEORIES FOR PREDICTING THE REAERATION COEFFICIENT Equation 26 is expressed in the coordinate system usually used for open-channel flow situations. That is, the mean flow is in the x, or longitudinal, The gas-absorption process at the interface be direction, and U is the flow velocity in that direc tween a flowing liquid and a gas has received con tion; y is in the vertical direction, or the direction siderable attention in recent years. The process of the channel depth; and the mean flow in the y direction is zero. is very complicated and only superficially under 1 CH stood; so theoretical treatments have been only Defining averages of the type C=T7 Cdy, and partially successful. Still, considerable useful in "Jo formation concerning this phenomenon is given in noting that the mass transport across the lower the technical literature of chemical and sanitary flow boundary is zero UJ/-T- =0], one can aver- engineering. The purpose of this section is to de scribe the various theories for predicting the age equation 26 over the flow depth H and obtain l (* dC\\ reaeration coefficient or the gas-liquid mass- ^-+T-|, J7rc/C J=r-^l ley^ I transfer coefficient for open-channel flows. dt dx\ / dx2 H\ y dy/\y = H
466-751 O - 72 - 3 8 REAERATION IN OPEN-CHANNEL FLOW Over much of the flow depth of a turbulent open- For reasons of convenience and tradition, com channel flow, mass transport in the y direction as pletely aside from the fact that the turbulent- a result of vertical diffusion is very large compared transport process in open-channel flows is not with the mass transport through the surface. sufficiently understood to permit evaluation of Hence, the vertical gradient of C generally is very y near a free surface, the second term on the right small except near the surface and within the bulk in equation 28 is generally expressed using one of the flow the value of C should approach C. This of two mass-transfer coefficients, KL or K2. The has been verified by W. W. Sayre (written commun. more general coefficient KL, the liquid-film co 1968), who numerically solved the equation efficient, is defined as follows: = ( 2, ). He assumed ey to be equal to the mo di oy \ oyI ' (29) mentum transport coefficient of a flow having a where CL is the DO concentration in the bulk liquid logarithmic velocity distribution and solved the phase. Comparison of equations 9 and 28 shows equation for C as a function of time at 50 points in that the reaeration coefficient, K2, can be ex the vertical using values of \ y )\ corre- pressed in terms of the film coefficient as \ dy/\y = H K2 = KJH. (30) spending to the range of such values observed for streams in nature. Initially, the flow was assumed (Eq 9 is the same as eq 28, except the source-sink to be unsaturated except for the top increment, terms have been lumped into the term Fc in eq 28.) and he found that even for very short times, there Theories for the prediction of KL or K2 can be was no appreciable concentration gradient in the checked in certain idealized situations by mea vertical. In support of Sayre's work, Churchill, surements of CL, evaluation of the source-sink Elmore, and Buckingham (1962) found no vertical terms, and substitution into an integrated version concentration gradients in a series of DO measure of equation 28. ments in a deep, low-velocity stream, and Edwards The prediction of mass-transfer coefficients and Owens (1962) found that the DO concentration from solute and solvent properties and from the of a slow-flowing English stream was greater than parameters of the flow is a problem which has 98 percent of the surface concentration except in received much discussion in the fluid-mechanics a relatively small area near the bed. Also, Whipple, literature. At the present time (1971) fairly reliable Hunter, Davidson, Dittman, and Yu (1969, p. 116) theories are available for the prediction of heat- concluded that the centerline DO concentration at and mass-transfer coefficients for flows near the a depth of 1 foot was a good estimate of the mean solid boundaries in straight circular pipes (Hinze, cross-sectional concentration of DO for the slow- 1959). These fall into two general categories: the flowing Passaic River in New Jersey. When the eddy-diffusion or gradient-diffusion theories and concentration of the dissolved substance is con the renewal theories. stant in the vertical, equation 27 may be written Because of the gas-liquid interface, side-wall effects, and secondary currents, the mechanics of flow in straight open channels with uniform ^ B ^ + /V (28) dt cross sections are considerably more complex than the mechanics of flow in straight circular It is generally assumed that the diffusion equa pipes. Furthermore, mass-transfer coefficients tion in this form is applicable to computing the usually are desired not for straight channels with longitudinal distribution of DO in streams. uniform cross sections but for natural channels, Because of the experimental and theoretical which are neither straight nor of uniform cross evidence indicating no concentration gradient of section. Because of these complexities, reliable DO in the vertical direction of an open-channel theories comparable to the eddy-diffusivity flow, the use of the overbar on concentration in theories for pipe flows have not been developed for equation 28 to indicate an average with respect tq the prediction of mass-transfer coefficients from the depth of flow in unnecessary. Hence, this over- mean flow parameters in open-channel flows. The bar on concentration will not be used in the theories which have arisen are in general less remainder of the report. For consistency, the over- rigorous, in the fluid-mechanics sense, than are bar on the velocity will also be dropped, with the the eddy-diffusivity theories. They fall into three understanding, however, that U is the flow velocity general groups: (1) the conceptual theories, based averaged over the depth of flow or the mean veloc on assumed mathematical models of the mass- ity of flow in the longitudinal direction. transfer processes within the flow; (2) the semi- THEORIES FOR PREDICTING THE REAERATION COEFFICIENT empirical theories, qualitatively relating the oxygen-water system is mass-transfer coefficients to some measurable N\ T = KL(CI( 4~* 4~*CL) \ = KG /(p . G pi), \ /O(31) 1 \ flow property, the constants in the relation being where CL is the concentration of oxygen in the determined by multiple regression; and (3) water phase, d is the concentration of oxygen at empirical theories resulting from dimensional the interface, pc and p\ are, respectively, the analysis, again with the constants being deter partial pressures of oxygen in the gas phase and mined by multiple-regression techniques. The at the interface, and KL and KG are, respectively, remainder of this section is devoted to a discus the liquid-film and gas-film coefficients. Equilib sion of the models of these types which have rium conditions usually are assumed at the inter appeared in the literature. face, so that C, and /?, are related by Henry's Law, or pi=hd, where h is the Henry's Law constant. CONCEPTUAL MODELS FOR THE GAS-LIQUID The film coefficients are expected to be de MASS-TRANSFER PROCESS pendent on the properties of the liquid and the Conceptual models are derived from equations gas and on the flow parameters, but independent describing an interpretation of the physical con of the concentration of the dissolved gas in the ditions governing mass transfer at a gas-liquid liquid (Sherwood and Pigford, 1952). interface. Of these equations, the one generally Concentrations and partial pressures at the used is the differential equation describing non- interface are difficult to measure, however; so convective diffusion. The boundary conditions information on the individual film coefficients is chosen vary from model to model. In some models, difficult to obtain. Thus, equation 31 usually is the parameters arising from the solution of the written in terms of the overall liquid-film and gas- differential equation with the given boundary con film coefficients, or ditions are not clearly related to the mean flow N = KL(CS CL) = KG (PG PS), (32) parameters such as mean velocity, depth of flow, where ps is the partial pressure corresponding to or turbulent intensity. In such models, discovering the saturation concentration, Cs, and KL and KG the relation of the former parameters to the latter are, respectively, the overall liquid-film and gas- is a problem requiring considerable thought and film coefficients for mass transfer. experimental verification. The models may be To be of use in engineering calculations, the divided into the following general groups: (1) Film overall coefficients should be dependent on the models; (2) renewal, including penetration, film- properties of the liquid and of the gas and on the penetration, and surface-renewal-damped-eddy flow parameters, but be independent of the con diffusivity models; (3) kinetic-theory models; (4) centration of dissolved gas. From equations 31 large-eddy models; and (5) turbulent-diffusion and 32 and Henry's Law, models. The usefulness of a particular model ulti -L=1+J-. (33) mately must be determined by how well it predicts KL KL n KG reaeration coefficients for natural streams. Thus, for any gas-liquid pair that obeys Henry's Law, the overall liquid-film coefficient is inde FILM MODEL pendent of the gas concentration in the liquid. The earliest model for mass transfer at a gas- For a slightly soluble gas that does not react chem liquid interface was the two-film model of Lewis ically with the liquid, h is large, so that equation and Whitman (1924). They postulated that laminar 33 reduces to films of gas and liquid exist at the interface KL = KL. (34) between the two phases. Because mass transport Equation 34 is equivalent to saying that the gas- in a direction perpendicular to the mean-flow direc film resistance is negligible with respect to the tion in a laminar film is by molecular diffusion liquid-film resistance and that KL is independent alone, a process much slower than the turbulent of the concentration of dissolved gas for a slightly diffusion within the phases, the resistance to mass soluble gas that obeys Henry's Law. The oxygen- transfer is concentrated in the two films. The water system is such a gas-liquid pair. interface between the gas and the liquid films Because mass transfer in the laminar film is by seems to offer no resistance to mass transfer molecular diffusion, and if the concentration (Sherwood and Pigford, 1952). If the gas and the gradient across the film is assumed to be linear, liquid phases outside of the film region are then assumed to be completely mixed, then the mass N = Dm (Cs ~ CL\ (35) transport, N, of oxygen through the films for the 10 REAERATION IN OPEN-CHANNEL FLOW where L/ is the thickness of the film. By comparing elapsed between the formation of the film on the equations 32 and 35, it follows that surface and its replacement by fresh liquid from KL =2*. (36) the bulk flow. The difference between the Higbie Ar and the Danckwerts theories lies in the replace In equation 36, the solute and solvent properties ment process visualized by the authors. are accounted for by the molecular diffusivity, Dm, Molecular diffusion in a laminar layer is and the hydrodynamics of the flow are accounted governed by the differential equation for by the layer thickness, L/. For a truly laminar *x j. ^^ * ft -\ 2' ' (37) layer to exist on the surface of an open-channel dt dy flow, the flow must be very tranquil. Such flows For an infinitely deep layer the boundary condi are seldom found in field situations, however, tions are: so that equation 36 does not suffice to describe C = d, t=Q, the variation of the reaeration coefficient with flow parameters. The concept of diffusion into C=CS, (38) laminar patches of infinitesimal area existing for where CL is the concentration in the bulk-liquid short times on the water surface before being re phase, y is the distance from the interface, and t is placed by liquid from the bulk liquid has been used, the time elapsed since the formation of the inter however, in subsequent models describing the face. The solution to equations 37 and 38 is reaeration process. (Carslaw and Jaeger, 1959)
RENEWAL MODELS = C erfc (39) The renewal models are analogous to those for and the rate of transport of solute through the predicting mass-transfer coefficients for fluid surface is flows near solid boundaries. In the renewal models, diffusion into a surface layer is generally assumed (40) to be purely molecular. The surface layer is some dy times assumed to be of infinite thickness, and Higbie (1935) visualized the surface elements as sometimes to have a thickness, L/, which may or all being of the same age and as all being replaced may not be randomly distributed about an average simultaneously on reaching an age te. Alter thickness, L/. Below the layer, the diffusivity in natively, the ages of the differential elements of the liquid is assumed to be infinite, the concentra the laminar surface film can be visualized as being tion independent of depth. After diffusion into an distributed according to the distribution function such that element of the surface layer has been taking place for some time, the element is assumed to be replaced by liquid from the bulk flow. The fre quency of renewal is governed by an assumed (41) probability density function. The mass-transfer 0 t>te. coefficients predicted by these models are func The average rate of mass transport, N, through a tions of the molecular diffusivity, an average unit of surface area for either case is renewal frequency, and, generally, an average thickness of the surface layer. The renewal models N=-\<=ff'"«;.-, (C»-CL\ Dm dt Jt = ()(/->2(Cs-C r> L\) I1 *- (42) include (1) the penetration model, (2) the film- tejo TTt \\TTte penetration model, and (3) the surface-renewal- damped-eddy diffusivity model. Because N = KL (Cs -Ci), one has (43) PENETRATION MODEL Danckwerts (1951) assumed that the surface The penetration model was first postulated by consists of an infinite number of differential ele Higbie (1935) and later refined by Danckwerts ments of area which have their ages distributed (1951). Both authors visualized the surface of the according to liquid as being a laminar film, the thickness of 0(0=«?-*, (44) which is large with respect to the depth which where s is the average rate of surface renewal. can be penetrated by molecular diffusion during The average rate of mass transport through a unit the life of the film. The film lifetime is the time of surface area is then THEORIES FOR PREDICTING THE REAERATION COEFFICIENT 11 =/" N (C.-Ci) Jo >l/2 (50) so that For small renewal rates, equation 50 approaches (46) the film-model limit, and for large renewal rates it Neither author attempted to relate either l/te approaches one-half the penetration-model predic or s to the mechanics of flow in open channels, but tion for KL. it is clear that each of these variables must repre Another modification of the film-penetration sent a parameter related to the average residence model was presented by Harriott (1962). His theory time of the large flow eddies at the surface. was designed for application to mass transport at FILM-PENETRATION MODEL fluid-solid interfaces, but it could probably be The film-penetration model was developed by applied also to open-channel flows, especially in Dobbins (1956) and later by Toor and Marchello the presence of surfactants. In this theory it is (1958). Certain modifications have been discussed postulated that diffusion is molecular, according in the literature (see, for example, Marchello and to equation 37, and that renewal occurs not neces Toor, 1963, and Harriott, 1962), but essentially, the sarily all the way to the surface but to some dis film-penetration theory consists of assuming tance yi below the surface. In applying this theory, molecular diffusion into laminar surface elements one must solve equation 37 with the boundary conditions of differential area and thickness, L/, which exist C=CS, y=0, t^O, on top of a turbulent liquid which has uniform C=CL, j^°o, /i=0, solute concentration, CL- The ages of the surface C=f(y) O^y^ , f=0. (51) elements are assumed to be distributed according to Danckwerts' (1951) distribution function for The probability density functions used by Harriott surface age as given by equation 44. (1962) to describe the distributions of the renewal The differential equation governing the diffu period, /, and of the distance below the surface to sion process is equation 37. The boundary condi which renewal occurs, yi, were of the form tions for the film-penetration model are C=CL, /=0, 0 w _ I *^ I _ I ** \ / ff n\ (64) and where cs and c4 are constants, v is the kinematic (65) viscosity of the liquid, and Eg and E are, respec tively, the energy dissipated per unit mass of where c2 and c4 are the same constants as in the liquid at the surface, and the energy dissipated Dobbins theory and ci, c3, and cs are new constants, per unit mass of liquid by the entire flow. Dobbins and where c5 is independent of the flow dynamics. also postulated that the rate of expenditure of For distilled water, Metzger and Dobbins (1967) THEORIES FOR PREDICTING THE REAERATION COEFFICIENT 13 and lead to deviations of the predicted from the found -=1. , and Metzger (1968) found this Ms gram true mass-transfer coefficients in situations where same value in the presence of 0.5 mg/1 of alkyl ben the flow conditions differ appreciably from those zene sulfonate (a surfactant) in water. The under which the experimental model coefficients significance of having Ms rather than a- in the de were determined. For the Dobbins (1956) model, nominator of equation 65 is that Ms increases in the the O'Connor and Dobbins (1958) method for deter presence of surface-active agents while cr de mining s is perhaps overly simplified, and it creases. Thus equation 65 is in qualitative agree requires no experimental coefficients to be eval ment with the observation that surface-active uated. That this method predicts kL values for agents retard surface renewals. natural streams as well as it does is therefore a The localized-mixing model of Marchello and little surprising. The Dobbins (1964b) and Metzger Toor (1963) can be applied not only to the surface and Dobbins (1967) methods are more rational and layer of an open-channel flow but also to lower more sophisticated, and they require the evalua layers within this type of flow and to all layers in tion of two experimental coefficients. Equation 60 other types of flow, such as pipe flows. When this used by Dobbins (1964b) to predict c4 was obtained is done, for any layer: from a plot of field data that contains considerable scatter. This scatter is, however, apparently of minor importance, because if one checks the pre diction equation using the data from which equa (66) tion 60 was evaluated, there is only a small amount of scatter about the line of perfect agreement where ey is the eddy diffusivity for mass, and /8' is between predicted and observed k2 values. the turbulence parameter for the layer, such that \2 SURFACE-RENEWAL DAMPED-EDDY DIFFUSIVITY MODEL (67) Similarly, after applying the localized-mixing The surface-renewal-damped-eddy diffusivity model to momentum transfer, Marchello and Toor model was presented by King (1966), who postu obtained lated that the mass transport near the surface is due not only to Dm, but also to turbulent diffusion caused by small-scale eddies. The coefficient (68) characterizing the turbulent diffusion caused by the small-scale eddies is the eddy diffusivity for mass, y, which is assumed to be of the form where ev is the eddy diffusivity for momentum. y =ayn, where a and n are parameters independent Thus, if one can determine the momentum eddy of time, and y is the distance from the interface. diffusivity for a particular layer, one can solve The surface renewals are assumed to be due to the equation 68 for /8' and determine the eddy diffu large-scale eddies, and the surface ages are sivity for mass for the layer from equation 66. assumed to be distributed according to the distri The combination of the film concept of Lewis and bution function for uniform surface ages (eq 41). Whitman (1924) and the penetration concept of The diffusion equation applicable to the above Higbie (1935) and Danckwerts (1951) into one model model is has considerable merit in that the combined model «\ acl can accurately predict reaeration coefficients for TO' natural streams over a much wider range of con and the boundary conditions used by King (1966) ditions than can either of the models alone. The are film-penetration models, however, are idealized C = CL, t=Q, y>0, in that the diffusion process is visualized as taking \_^C ^-'LjCi *f>0 ^9 Jv »°° 9 place in two separate steps, rather than as being C=CS, r>0, y=0. (70) governed by the classical diffusion equation em Equation 69 along with the boundary conditions bodying both molecular and turbulent diffusion. 70 can be solved analytically only in special cases, The first step is molecular diffusion; the second such as for a = Q and all «; for all a and « = 0, 1, °°; step is turbulent diffusion, or nongradient diffu and for all a and all n with large t. For large t, the sion, embodied by a renewal or mixing process. solution of equations 69 and 70 (King, 1966) gives This idealization will, of course, lead to difficulties KL7-77-= n sin-,. TT (71) in relating model parameters to flow parameters, i-i/n -. n 7 14 REAERATION IN OPEN-CHANNEL FLOW where t may be interpreted as the surface age at must attain normal to the interface to be able to renewal. leave the liquid. If the surface-renewal-damped-eddy diffusivity If equation 73 is to account for the turbulence model is to be useful in predicting film coefficients, properties of the flow as well as for solute and the parameters a and n in equation 69 as well as an solvent properties, the critical parameter would average surface age, 7, must be estimated. From have to be /x0'- That ^ could account for the turbu equation 71 it is seen for large surface ages that lence properties of tHe flow is reasonable, because the exponent n can be estimated from the varia M(/A0')2/2 may be viewed as an activation energy tion of KL with molecular diffusivity, if the other term, and as pointed out by Bird, Stewart, and factors remain constant. Kozinski and King (1966) Lightfoot (1960, p. 27), the activation energy concluded that KL^Dm0-75 for gas-liquid transfer of liquids is influenced by local values of the shear at large surface ages; thus « = 4. From the work of stress, which can in turn be related to local turbu Calderbank and Moo-Young (1961), King (1966) con lence properties. Apparently, no studies, either cluded that theoretical or experimental, have been made in Eyp which such relations have been presented. a = 0.006 (72) 2 > Another kinetic-theory model was presented by where Ev is the energy-dissipation rate per unit of Tsivoglou (1967). It too arises from consideration liquid volume, p is the mass density of the liquid, of the difference between the rate of entry and the and fji is the absolute viscosity of the liquid. From rate of loss of gas (oxygen) molecules from a sur figure 3 of King (1966) and from equation 71 for face layer of liquid (water). The loss rate, r0, is n = 4, it is seen that if surface-renewal effects con r0 =b(LfCnsA s) (74) trol, the surface-renewal-damped-eddy dif where fusivity model yields film-coefficient predictions C=concentration of gas in the liquid, of the same order as those measured in large open Lf= thickness of surface layer from which gas channels and rivers. molecules can escape to the atmosphere, The surface-renewal-damped-eddy diffusivity ns = number of new surface layers exposed per model is physically more realistic than previous unit of time, models because the artifice of a surface layer is As = surf ace area, and avoided. Also, the eddy diffusivity is specifically b= the proportion of available molecules in the accounted for in the diffusion equation, rather surface layer that actually escape. than by being lumped into surface renewal, so that The entry rate, re, which is dependent on the gas gradient and nongradient diffusion are more concentration in the air above the water, is a con clearly separated. Practically, however, the model stant. It is equal to the loss rate at C = CS, so that is cumbersome in that three parameters must be evaluated, and equation 69 must be solved numer re = b(LfCs nsAs). (75) ically in most cases. King (1966) showed that if The time rate of change of the liquid concentration n = 4 and a is given by equation 72, s should be is then '£F//1 2/3 dC re r0 proportional to for open-channel flows. l(Cs -C\ (76) . P J dt This has not been checked experimentally. so that K2 = (77) KINETIC THEORY A 1 Because -rr=77» in which Kis the volume of the sys- The kinetic theory as developed by Miyamoto V n tern and H is its depth, (1932a, b) was presented by Krenkel and Orlob (1963). The model is based on the frequencies and KL =bLfns. (78) velocities of bombardment of the interface by Equation 78 is mostly of theoretical interest, solute molecules from both the liquid and the gas because at the present time none of the param phases. The resulting equation is eters on the right-hand side can be measured. i to 7"'\i/2 However, some of the results obtained by Tsivoglou , (73) (1967) by manipulation of the equations of this where RG is the ideal-gas constant, T' is the abso model are of interest. First, because the entry rate lute temperature, M is the molecular weight of the is a function only of the gas concentration in the gas, and /x0 is the velocity which a solute molecule layer above the fluid, Tsivoglou concluded that the THEORIES FOR PREDICTING THE REAERATION COEFFICIENT 15 factor b varies with temperature according to the lence conditions was studied for each pair of gases relation and each type of apparatus. On the basis of these measurements, Tsivoglou (1967) concluded that (b)fL.\ Tt = (b)/l_\ T, (r(Cs)r, , ', (79)fr7 Q\ (<-s)T2 the ratio of the gas-transfer coefficients was a con where the subscripts indicate two different tem stant within the limits of experimental error for peratures. Combining equation 79 with equation the range of temperature and turbulence condi 77 yields tions studied. This supports his model as given in equation 77. -^=-^Ii=e(T2- Ti\ (80) Tsivoglou (1967) suggested that his experimental measurements of the gas-transfer ratios for the in which 6 is a temperature-correction coefficient different pairs of gases together with molecular- for the mass-transfer coefficient. The tem diameter data from the literature support equa perature-correction coefficient will be discussed tion 83. However, a linear regression analysis of in greater detail later. At present, the value the logarithms of the transfer-coefficient ratios usually accepted for 0 is 1.0241, given by Elmore and of the molecular-diameter ratios yielded and West (1961). The values calculated by Tsivoglou :6 for B from equation 80 for oxygen and water for (84) temperatures from 0° to 30°C range from 1.018 to OQ2 where the 95-percent confidence limits on the 1.026, which lends credence to Tsivoglou's model. exponent are 1.09 and 1.44. Other experimental Tsivoglou (1967) presented further support for evidence suggests that equation 83 and the argu his model. Assuming the average molecular trans ment that Dm can be obtained from Einstein's law port within the body of a turbulent liquid to be due of diffusion are not strictly valid. Kozinski and to an average concentration gradient between King (1966) found that the exponent on Dm in the volume elements, and equating the mass trans relation kL « Dmn varied from about 0.50 to 0.75 for fer at the surface to the average transfer between the stirred-tank data of Dobbins (1964a); these the volume elements, Tsivoglou showed that for values are considerably different from the 1.26 the simultaneous transfer of two gases, indicated by equations 84 and 82. These apparent discrepancies suggest that Einstein's law of diffusion is not strictly appli When considering the ratio of the mass-transfer cable to this situation, but note that these discrep coefficients of two gases, equation 81 leads to the ancies do not in any way reflect on the validity of same conclusion as the film-model equation (36) the, model as represented by equation 81. The only namely that the ratio of the mass-transfer requirement of the model here is that the gas- coefficients of the two gases is directly propor transfer-coefficient ratio be a constant, as tional to the first power of the ratio of their molec was found in a laboratory study by Tsivoglou, ular diffusivities. O'Connell, Walter, Godsil, and Logsdon (1965) and In further developing his model, Tsivoglou (1967) Tsivoglou (1967). pointed out that if mass transport through the surface is governed by Einstein's law of diffusion, then LARGE-EDDY MODEL 4 (82) (D«)2 466-751 O - 72 - 4 16 REAERATION IN OPEN-CHANNEL FLOW The parameter Ae is a measure of E, the average much larger than the molecular diffusivity, so that kinetic energy of the roll cell, because 2 1 f A f A (90) :=^b (86) 41* JO JO The differential equation describing the gas con An example in which equation 90 can be said to centration distribution in the large-eddy model is govern was presented by Kishinevsky and Sere- C/O C/O j-k bryansky (1956). They reported studies in a rapidly W#>~T r Vt> ^/-/M dx dy stirred tank in which the measured film co with boundary conditions efficients for hydrogen, oxygen, and nitrogen were C=CS, y=Q, all roughly the same, despite the fact that the =0, molecular diffusivity of hydrogen is about twice dx x=n\, that of oxygen and nitrogen. They also reported, C=CL, y=\, O^Jc^A, (88) however, that at lower stirring speeds the molec where n is an integer. Equation 87 with boundary ular diffusivity influenced the measured film co conditions 88 must be solved numerically. efficient. The fact that in open-channel flows the Dimensional analysis of equation 87 along with measured values of kL are a function of tempera equation 85 shows that the characteristic param ture indicates that in this type of flow turbulence eter describing the large-eddy model is P, levels do not reach a magnitude sufficient to make the Peclet number, where P=A.AeIDm. Fortescue equation 90 applicable. and Pearson (1967) reported that for P values "over Note that the models discussed predict a range the range that was revelant," the numerical solu of dependence of kL on Dm varying from an ex tion of equations 85, 87, and 88 yields ponent of 1 for the film model to 0 for the turbulent- Dm u'\ 1/2 diffusion model. The film-penetration model kL = (89) predicts an exponent on Dm which varies from 0.5 A ) ' to 1.0, whereas the surface-renewal-damped-eddy where u' is the longitudinal turbulent intensity. diffusivity model can predict an exponent any Using the O'Connor and Dobbins (1958) relation for where from 0 to 1.0. Experimental data taken in u'l\ = UIH, Fortescue and Pearson reported that stirred tanks (Kozinski and King, 1966) indicated their model gives "rather better" predictions than that the exponent is in the 0.5 to 0.75 range. There the Higbie (1935) penetration model (eq 43) with fore, each model (with the possible exception of the llte=u'l\=U/H, for the river data used by O'Connor film model) seemingly is applicable to a limited and Dobbins (1958). range of turbulence conditions. However, the The large-eddy model is different in concept surface-renewal-damped-eddy diffusivity model from the other models presented in this section, is the only one which is applicable over the entire but the concept is as realistic as, for example, the range of turbulence conditions. film-penetration concept. An individual cell is a All the conceptual models for the gas-liquid considerably idealized turbulent eddy, and the sur mass-transfer process involve two physical prop face film is a film of eddies, rather than a stagnant erties of the oxygen-water system. These are the film. The model avoids the question of surface age molecular diffusivity and the saturation concen by allowing the age to vary in a set pattern across tration for oxygen in water. Each of these prop the top of the cell. It would be interesting to find erties has been studied in considerable detail, and the effect on the computed film coefficient of the following paragraphs summarize briefly the assumed distribution functions for varying cell results of these studies. sizes and kinetic energy. A more realistic relation The determination of the molecular diffusivities for surface values of w'/A might improve the pre of dissolved gases in liquids was reviewed by dictions made by the model. Himmelblau (1964), and the experimental deter minations of the molecular diffusivity of oxygen in water were discussed by Duda and Vrentas TURBULENT-DIFFUSION MODEL (1968). The experimental data for molecular diffusivity In discussing the turbulent-diffusion model, at 25°C fall into two groups, one group indicating Kishinevsky (1955) postulated for high turbulence a value of about 2.5xlO~5 cm2 per sec (square centi levels that the turbulent diffusivity, ey, is not meters per second) and the second group indicat damped out at the water surface. He further stated ing a value of about 2.0xlO~5 cm2 per sec. The that the turbulent diffusivity at the surface is larger values in general have been used (Isaacs THEORIES FOR PREDICTING THE REAERATION COEFFICIENT 17 and Gaudy, 1968; Metzger and Dobbins, 1967; For elevations above 3,000 feet and temperatures Kozinski and King, 1966; Dobbins, 1964b; O'Connor above 25°C, corrections for the water vapor pres and Dobbins, 1958). Himmelblau (1964) also believed sure should be made (Am. Public Health Assoc., that the smaller values at 25°C were probably in 1965, p. 409). correct, and Baird and Davidson (1962) believed Equation 91 demonstrates a fact perhaps not that the larger values obtained with the wetted generally recognized. For a temperature of 20°C sphere procedure were more accurate than the and a pressure of 760 mm Hg (millimeters of values obtained with the annular jet. On the other mercury), the saturation concentration is 9.02 hand, Kolthoff and Izutsu (1964) concluded that a mg/1 (Committee on Sanitary Engineering Re modified equation incorporating a second pertur search, 1960). If a concentration of 5.0 mg/1 is re bation term should be used in the calculation of quired for prevention of fish kills, then a range of oxygen diffusivities from polarographic measure- about 4 mg/1 of DO is available for a stream at an merits. This modification resulted in diffusivities elevation of about sea level for example, along of 1.94xlO-5 and 1.97xlO'5 cm2 per sec at 25°C as the east coast of the United States. However, for a compared with values of 2.56xlO~5 and 2.50 xlO~5 stream at an elevation of about 5,000 feet in the cm2 per sec obtained without the second perturba Rocky Mountain region, the saturation concen tion term. Duda and Vrentas (1968) concluded from tration is about 7.5 mg/1; hence, a range of only their consideration of the available experimental about 2.5 mg/1 of DO is available for use in oxida data that the evidence tended to support the tion of wastes. smaller diffusivity values, in agreement with their The saturation concentration decreases with in laminar-jet result. Controversy still exists concern creasing chloride concentration in the water. The ing the correct value for the molecular diffusivity effect ranges from a decrease of 0.017 mg/1 of DO of oxygen in water; so diffusivity data for the per 100 mg/1 of chloride at 0°C to a decrease of oxygen-water system should be used with this con 0.008 mg/1 of DO per 100 mg/1 of chloride at 30°C sideration in mind. (Am. Public Health Assoc., 1965, p. 409). Sewage in The saturation concentrations presented by the the water may affect the DO saturation value. American Public Health Association (1960, 1965) Moore (1938) found that the solubility of oxygen in and the Committee on Sanitary Engineering Re domestic sewage was about 95 percent of the search (1960) in general have been accepted and solubility in distilled water, and Rand (1959) con used as standards. Truesdale, Downing, and cluded that the solubility of oxygen in sewage was Lowden (1955) obtained values that were signifi not significantly different from the solubility in cantly different from the American Public Health pure water. Association values, but later work by Montgomery, Thorn, and Cockburn (1964) showed that the values were low because of losses of iodine vapor during SEMIEMPIRICAL MODELS analysis for DO. More recent studies of the satura tion concentration were discussed by Montgomery The semiempirical models result from a consid (1969). He concluded that the values from these eration and combination of the factors that should studies were in general slightly smaller than the affect mass transfer at a free surface. These values reported by the American Public Health models rely on regression-correlation analysis of Association (1960, 1965) and, for temperatures experimental data for establishment of a predic above 15°C, were slightly larger than values ob tion equation and for verification of the assumed tained by the Committee on Sanitary Engineering relations between the mass-transfer coefficient Research (1960). and the flow parameters. In general, the steps in The saturation concentration decreases with the derivation of these models are: (1) From intui increasing temperature and decreases with de tion, a knowledge of previous conceptual models, a creasing barometric pressure. The temperature knowledge of the results of previous empirical dependence of the saturation concentrations ob studies, or a combination of these, to postulate that tained by the Committee on Sanitary Engineering the film coefficient kL (or the reaeration coefficient, Research (1960) was expressed in terms of a cubic A:2) is a function of certain of the parameters of the equation. The dependence on barometric pressure, flow; (2) if necessary, to manipulate the assumed pB, may be expressed in terms of a simple ratio, or relation to obtain a relation for kL (or /c2) in terms of easily measured mean-flow parameters; and (3) to use regression-correlation analyses to fit i . (91) PB, experimental data to the assumed relation and to 18 REAERATION IN OPEN-CHANNEL FLOW check its validity. The determination of the use where U is in feet per minute, g is in feet per fulness of these models must be based on the reli minute per minute, and E is the energy dissipated ability of the data from which they were derived, per unit mass of flowing fluid. Alternatively, from on how well they predict mass-transfer coefficients a separate regression analysis of the same data, for natural streams, and on ease of use. The first Krenkel (1960) and Krenkel and Orlob (1963) re two points are discussed in a later section of this ported that kz at 20°C can be determined from paper; the last, as the models are presented. The (£2)20°= 1-141 xlO- 4 E°-408 //-°-66 (96) models discussed in this section are those pre where, again, kz is in reciprocal minutes. When sented by Krenkel (1960) and by Thackston (1966). equation 96 is to be used for flow at some tempera The limits on the ranges of data from which the ture other than 20°C, adjustment of kz must be semiempirical and empirical models were derived made using a suitable temperature correction are discussed in a later section. formula. (See eq 104.) Experimental differentiation between equations KRENKEL MODEL 92 and 93 may be very difficult if Dt = Dm because for a small temperature range (for example, ±10°C) The Krenkel model was presented by Krenkel -ocexpC E'/7'), (97) (1960) and Krenkel and Orlob (1963), who reasoned that where E' is a combination of the activation energies for molecular diffusion and viscous flow ^[^Ib^]' (92) and is independent of temperature. Thus, a plot where the first term in brackets is characteristic of log k2 versus 1/7' for constant e///2 will show a of the rate of absorption at the surface, and the linear variation, whether molecular diffusivity or second term is characteristic of the rate of surface the activation energy for oxygen absorption is the renewal. In equation 92, £>, is a combination of controlling factor. Most likely, Dm (or eq 92) con molecular and turbulent diffusivity at the surface, trols, because equation 93 in effect states that and e is an eddy-diffusion coefficient. However, interfacial resistance controls; but Davies and both Krenkel (1960) and Krenkel and Orlob (1963) Rideal (1963) and Sherwood and Pigford (1952) believed that the reaeration coefficient is better stated that interfacial resistance is negligible for represented by the absorption of slightly soluble gases. kz <* s exp [- EJRc T' ], (93) An important factor to be considered when using where s=e///2, Ea is the activation energy for equations such as 94 and 95 is that they were oxygen absorption, RG is the ideal-gas constant, developed for flows in straight channels with uni and T' is the absolute temperature. In using equa form cross section, whereas natural streams tions 92 and 93, the parameter most characteristic generally have nonuniform cross sections and con of the eddy-diffusion coefficient, e, is Dx, the longi siderable curvature in plan. In such channels, Dx tudinal dispersion coefficient of the flow. can be considerably greater than the £>x for an Regression analysis by Krenkel and Orlob (1963) equivalent straight channel with uniform cross of 58 laboratory measurements of k-2 made at section, so equations 94 and 95 might not apply. depths from 0.08 to 0.20 foot in a 1-foot-wide flume Equations of the type of 96 might, however, be yielded more valid in natural streams because the rate of /:2 = 1.418exp [ EalRcT'^-j^ 0.0069, (94) energy dissipation includes the effects of nonuni form cross sections and curvature. Finally, as where E« = 5,875 ^, *c =1.99 m<^oK, T' is the mentioned above, Dx is difficult to measure or pre dict, whereas equations of the type of 96 use readily absolute temperature in degrees Kelvin, and the measurable mean-flow parameters. Thus, from units of k2 are reciprocal minutes. Because the pro this point of view, use of equation 96 would be rec cedure for measuring Dx is difficult and time con ommended over the use of equation 94. suming, this parameter is not one of the commonly available stream-flow parameters. However, for THACKSTON MODEL uniform open-channel flows in the depth range given above, Krenkel (1960) showed by a regression The Thackston model was presented by Thack analysis that Dx can be related to the commonly ston (1966) and Thackston and Krenkel (1969a), measured mean flow parameters by who postulated that Dx = 6.425 E0-296 //1-243, (95) '- --^l(5) (98) where Dx is in square feet per minute, E-USg, THEORIES FOR PREDICTING THE REAERATION COEFFICIENT 19 where s is the average surface renewal rate and H EMPIRICAL PREDICTION EQUATIONS is the flow depth. They further postulated that fi(s)cc y, where e^ is the vertical eddy diffusivity The empirical prediction equations result from at the water surface, and that / >(//) a://2. They least-squares multiple-regression analyses of the assumed that variables considered pertinent by the investigator u or from dimensional analysis of the mean-flow and , CC y =-//«,, (99) gas-liquid parameters. For water and air in the temperature range encountered in the field, it is where ey is the average value of the vertical eddy generally assumed that the temperature depen diffusivity, «* is the shear velocity, \/gHS, and K is dency of k-2 resulting from the temperature depen von Karman's constant. The resulting prediction dence of the molecular diffusivity, kinematic equation is of the form viscosity, surface tension, and other physical -/"*./ , (100) properties can be embodied in a single function of the temperature. This dependency is generally where J is a constant to be determined from regres removed before the least-squares analysis is sion analysis. begun. The empirical temperature-correction Based on 40 values of A2 measured by Thackston equation (the form of which can be derived from (1966) in a 2-foot-wide flume at flow depths ranging rate process considerations) is of the form from 0.05 to 0.23 foot, a regression analysis yielded K2T =K2w e(T~2(f\ (104) where T is in degrees Celsius and the 0 generally = 0.000215 (101) H used is the 1.0241 suggested by Elmore and West (1961). where A2 is in reciprocal seconds, «* is in feet per The empirical prediction equations are generally second, and H is in feet. More importantly, equa of the form tion 101 fits rather well to the field data used by TJW O'Connor and Dobbins (1958) and the data collected 20°^7r, (105) by Churchill, Elmore, and Buckingham (1962). where i// is a constant obtained from a regression Furthermore, the fit is improved if a Froude- analysis of experimental data, and w and j may be number relation is included in the regression anal chosen in this fashion; or w and./ may be chosen for ysis. A regression analysis which included these dimensional homogeneity of the prediction equa field data plus the flume data of Thackston (1966) tion. Judgment of a particular prediction equa yielded tion must be based on the quality of the data from T, (102) which it is derived, and on how well it predicts H field reaeration coefficients. The equations dis where F is the Froude number. cussed in this section are based generally on field- Some question exists as to the universal applica scale data and utilize readily measurable bility of an equation of the type of 102, because mean-flow parameters for prediction of reaeration when the data of Krenkel (1960) were analyzed in coefficients. Presented in the following sections the same fashion as those yielding equation 101, are equations by Streeter and Phelps (1925), the prediction equation Churchill, Elmore, and Buckingham (1962), Owens, Edwards, and Gibbs (1964), Langbein and Durum (103) (1967), Isaacs and Gaudy (1968), and Negulescu and Rojanski (1969). was obtained. Thackston and Krenkel (1969a) ex plained the difference between equations 101 and 103 by stating that because Krenkel's 1-foot-flume STREETER-PHELPS EQUATIONS data do not fit field data, whereas Thackston's 2- foot-flume data do, the conditions in the 1-foot- The pioneering study of reaeration coefficients flume were not really characteristic of field of large-scale natural streams was presented by conditions. More likely, however, this discrepancy Streeter and Phelps (1925). In this study, the clas indicates that prediction equations of the type of sical oxygen sag equation 101 and 102 are not general enough and that some (m is t is T\ ft (\a\ additional parameter such as the width-depth j-=KiL-K2 D UOo) ratio, WIH, should be included in the prediction was developed and was used in its integrated form equation. to calculate reaeration coefficients from measured 20 EEAERATION IN OPEN-CHANNEL FLOW values of the time of travel, upstream and down minations were made for 30 different mean-flow stream values of the DO deficit and BOD, and the situations. The discharge, mean depth, and mean BOD reaction rate constant, Ki. velocity ranged, respectively, from 952 to 17,270 The prediction equations were developed from cfs, from 2.12 to 11.41 feet, and from 1.85 to 5.00 fps. data taken in 1914 and 1915 on the Ohio River Churchill, Elmore, and Buckingham (1962) per between Pittsburgh, Pa., and Louisyille, Ky. The formed many multiple-regression analyses on the study reach was 611 river miles longj; it was sepa pertinent stream flow, gas, and liquid parameters rated into 11 sub-reaches for experimental mea corresponding to the kz measurements. None of the surements. During the study period at the various prediction equations was found to be statistically points in the study reach, the mean monthly values any more significant than any of the others, so of discharge, depth, and velocity varied, respec that the simplest one was suggested by these tively, from 1,670 to 309,000 cfs (cubic feet per authors for general usage. This equation, which second), from 16 to 48 feet, and from 0.1 to 4.6 fps has a coefficient of multiple correlation of 0.82, is (feet per second). Eighty-six average values of kz #2 = 5.026 £/°-969 //-1-673 (1.0241)r- 20°, (108) were reported as corresponding to the monthly where kz is in reciprocal days, V is in feet per averages of the flow parameters existing over the second, H is in feet, and T is in degrees Celsius. period of measurement on the various sub-reaches. Equation 108 is much more convenient to use The prediction equation suggested by Streeter than equation 107 because it utilizes only the and Phelps (1925) was readily available parameters U, H, and 7. (*2V=pp (107) OWENS-EDWARDS GIBBS EQUATION where //' is the depth above minimum low water stage and kz is in reciprocal days. The coefficient Z Owens, Edwards, and Gibbs (1964) reported is a function of what Streeter and Phelps called reaeration-coefficient measurements for several an irregularity factor, which is defined as the English streams. These coefficients were deter number of changes in channel bottom slope pro mined using the disturbed-equilibrium method of ducing a change in elevation greater than 1 foot, deaeration with sodium sulfite and cobalt catalyst. expressed on a per-mile basis. For irregularity Use of this procedure eliminates the necessity of factors of 2.0 to 3.0, Z=[0.39(101 -16)5 + 17], and for considering parameters of the DO balance that irregularity factors of 3.0 to 5.0, Z = 1.1S2-3, where 5 are independent of the DO concentration. is the channel water-surface slope, in feet per mile. The study by Owens, Edwards, and Gibbs (1964) The coefficient 17 is a function of 8, or T7=(8-0.17)/ included measurements on 21 different reaches of (8 1.17), where 8 is the mean relative increase in six streams. They listed 32 separate observed flow velocity per 5-foot increase in gage height. values of kz for stream discharges, depths, and Because of the need for computing the irregu mean velocities ranging, respectively, from 1.50 larity factor and the coefficient 8, equation 107 is to 36.2 cfs, from 0.34 to 2.44 feet, and from 0.13 to difficult to use. Also, because //' is not a true 1.83 fps. They also listed six values of A2 from streamflow parameter, a prediction equation in Gameson, Truesdale, and Downing (1955) for two terms of this parameter probably is not completely reaches of a different stream for discharges, general. Other authors (for example, O'Connor and depths, and mean velocities within the above Dobbins, 1958) have, however, reanalyzed some of ranges. the Streeter-Phelps data to yield prediction equa Multiple-regression analysis of the data by tions with a little more generality. Owens, Edwards, and Gibbs (1964) yielded A:2 = 10.90 £/°-73 //- 1 -75 (1.0241)r- 20°. (109) CHURCHILL-ELMORE-BUCKINGHAM EQUATIONS When Owens, Edwards, and Gibbs (1964) included the data of Gameson, Truesdale, and Downing Probably the best field study of the reaeration (1955) and that of Churchill, Elmore, and Bucking coefficient has been that by Churchill, Elmore, ham (1962) in the analysis, the resultant equation and Buckingham (1962). Their measurements were was made in stretches below dams, where the water £2 = 9.41 £/« «* #-»-M (1.0241)r-20P. (110) released was deficient in DO because of prolonged In these equations, kz is in reciprocal days, U is in storage under conditions of thermal stratification, feet per second, H is in feet, and T is in degrees and where generally the BOD was negligible. Celsius. Equation 110 was obtained from 68 kz The study covered 16 different reaches on five values, which had a 300-fold range from about 0.25 rivers. Five hundred and nine individual kz deter to 75 days"1. Again, equations 109 and 110 are very EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 21 convenient to use because they utilize only the Equations of the type of 112 through 114 are con readily available parameters U, H, and T. venient to use, and if the constant is expressed as in equation 115, they are dimensionally homo LANGBEIN-DURUM EQUATION geneous. There is some question as to whether the data from which equation 112 is derived truly Langbein and Durum (1967) combined the river represent streamflow conditions (Thackston and data of O'Connor and Dobbins (1958) and of Church Krenkel, 1969b; Bella, 1969), so that use of this ill, Elmore, and Buckingham (1962) with the equation for prediction of stream reaeration co laboratory data of Krenkel and Orlob (1963) and of efficients is questionable. However, equation 113 Streeter, Wright, and Kerr (1936) and obtained the should be applicable to streamflows, at least for the equation range of flow parameters over which the measure = 3.3- U (111) ments were taken, because it is based on field data. The disparity between the constants in equations where fe is in reciprocal days, U is in feet per 113 and 114 probably indicates the effects of various second, and H is in feet. They also considered in a nonhomogeneities in the data collection and general way the effect of downstream and regional modeling procedures and also suggests that all the variations of stream discharge on the reaeration pertinent variables have not been included in the coefficient computed from equation 111. They prediction equations. found that the computed reaeration coefficients decreased in the downstream direction with about NEGULESCU-ROJANSKI EQUATIONS the square root of the discharge. The equations given by Negulescu and Rojanski ISAACS GAUDY EQUATIONS (1969) were derived by regression analysis from a series of 18 fe and Dx measurements in a 66-foot- Isaacs and Gaudy (1968) made a study of reaera long by 0.66-foot-wide recirculating flume. Not all tion in a circular tank with moving walls intended the data are given, but for those available, the to simulate streamflow conditions. They reported depths range from 0.164 to 0.492 foot, and the 52 individual fe observations for 20 different velocities range from 0.656 to 1.903 fps. The equa simulated streamflow conditions. The simulated tions obtained by Negulescu and Rojanski are depth ranged from 0.50 to 1.50 feet, and the simu (116) lated mean velocity, from 0.55 to 1.63 fps. A multiple-regression analysis of the data by and /fAl.63 Isaacs and Gaudy (1968) yielded exponents of fe = 1.423 D. (117) 1.0027 and -1.4859 on mean velocity and depth, \H) ' respectively; so a second analysis was performed where fe is in reciprocal days, U/H is in reciprocal in which these exponents were held to 1.0 and seconds, and Dx is in square feet per day. Compar -1.5. The results of this analysis were ing values of fe predicted by equations 116 and 117 2«°. (112) and values from a set of river measurements (which are not reported), Negulescu and Rojanski Using 29 of the 30 data points of Churchill, Elmore, (1969) concluded that equation 117 is the more and Buckingham (1962), Isaacs and Gaudy obtained accurate. 20P, (113) Again, from the standpoint of convenience in obtaining measured parameters for prediction of but using Krenkel's (1960) flume data they fe in field streams, equation 116 is the desirable one obtained to use. (114) EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT In equations 112, 113, and 114, fe is in reciprocal days, U is in feet per second, and H is in feet. All the experimental techniques for determining Isaacs and Gaudy (1968) pointed out that the the reaeration coefficient in open-channel flow, multiplicative constants in equations 112, 113, and except the tracer technique of Tsivoglou (1967) and 114 can be interpreted as Tsivoglou and others (1965, 1968), use some inte / .>' \ grated form of a DO balance equation for (115) 8 example, an integrated form of equation 12. The where the constant X is dimensionless. determination of K> requires, therefore, that all 22 REAERATION IN OPEN-CHANNEL FLOW other parameters in the equation be measured. dispersive conservation of mass for a gas dissolved These parameters are determined by using a in a liquid (eq 28 with ex negligibly small). The third variety of techniques, all of which depend on the technique utilizes an integrated form of this equa measurement of the amount of oxygen dissolved tion for an instantaneous input of tracer material in a volume of water. which is convected and dispersed longitudinally The purpose of this section is to discuss (1) the by the flow. The basis and procedures of these basic DO measurement techniques, (2) the various techniques are discussed in the following techniques for determining K2, and (3) an analysis paragraphs. of the errors expected from the different tech niques for determining the reaeration coefficient. DISSOLVED-OXYGEN BALANCE TECHNIQUE MEASUREMENT OF DISSOLVED OXYGEN IN WATER Studies utilizing the DO balance technique are of two types: those in recirculating flumes, and The basic method for the determination of DO in those in natural streams or in nonrecirculating water is the Winkler procedure, which was first flumes. Studies of the first type utilize the non- described in the literature in 1888. (See Am. Public steady-state nondispersive form of the conserva Health Assoc., 1965, p. 414, for reference.) Since tion of mass equation, equation 28 with e.r=0. Those that time, the procedure has been improved by of the second type utilize the steady-state nondis advances in equipment, technique, and instrumen persive form of this equation that is, equation 28 tation, and it is still the basis for most analytical methods for the determination of DO in water. with the x and the dC/dt terms equal to zero. We Oxidizing or reducing substances in the sample know of no techniques which have been applied to the calculation of K2 in nonsteady dispersive con interfere with the procedure, however, and various modifications of the basic Winkler method have ditions; however, the numerical techniques of been developed. A common interference in waters Thomann (1963, 1965) and Dresnack and Dobbins receiving sewage and effluents from sewage treat (1968) could be modified to compute K2 from mea ment plants is nitrate nitrogen, and the Alsterberg sured variations of DO with time. modification is used under these conditions. The general DO balance technique for natural Details of the procedure were given by the Amer streams is based on a more sophisticated version ican Public Health Association (1965) and by of the equations originally presented by Streeter Rainwater and Thatcher (1960). The Alsterberg and Phelps (1925) for example, equations 12 and modification is the procedure generally used for 13, as presented by Camp (1963). These one- routine analysis of natural waters. Details of other dimensional steady-state equations are applicable modifications of the basic Winkler procedure were to conditions which meet the assumptions previ also given by the American Public Health Associa ously given for equations 9 and 10. In addition, tion (1965), and several of these modifications were longitudinal dispersion is generally assumed to compared by Montgomery, Thorn, and Cockburn have a negligible effect on the longitudinal DO (1964). profiles in natural streams. Various instrumental methods for the deter With these assumptions, the solutions to equa mination of DO in water have also been developed. tions 12 and 13 yield the BOD and the DO deficit at These include, for example, gas-exchange, gas- a downstream station, Jt=jcb, in terms of the BOD chromatographic, radiometric, voltammetric, and and DO deficit at an upstream station, x=xa. The membrane-electrode methods. The procedures solutions are used in these methods and the advantages and dis advantages of these methods and of titrimetric (118) methods were discussed by Mancy and Jaffe (1966). and TECHNIQUES FOR DETERMINING THE REAERATION COEFFICIENT Three general techniques are used for deter io-**+o.io-'.«. (119) mining the reaeration coefficient of open-channel flow. These are the dissolved-oxygen balance where the flow time is f=*b «? the rate constants technique, the disturbed-equilibrium technique, and the tracer technique. The first two techniques are to the base 10, and equation 12 has been solved utilize an integrated form of the equation of non- in deficit form. EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 23 Equation 119, in its most general form, must be Westfield (1965). The sample is placed in a sealed solved for k2 by trial and error. Before this can be cell equipped with a DO measurement probe, and done, however, the other parameters appearing in the DO concentration is measured continuously. equations 118 and 119 must be evaluated. The When the DO concentration decreases to 2 mg/1, techniques used in evaluating these parameters the cell is aerated with compressed air. This process are discussed briefly in the following paragraphs. is continued for as long as desired. The respirom eter has the same advantages as the reaeration BOD TEST AND K^ technique in that no dilution and seeding are necessary, and it has the additional advantage that Of the various parameters in a DO balance, the the original sample is kept intact. The literature rate constant, Ki, for the oxidation of organic on other types of improved techniques for BOD wastes probably has been studied to the greatest determinations was reviewed by Gannon, Pelton, extent. The oxidation process is generally con and Westfield (1965), and a complete review of sidered to be a first-order process as represented respirometric methods was presented by Mont by equation 1. The two parameters in equation 1, gomery (1967). KI and L, may be determined from a BOD test. The The results of a BOD test are usually in the form earliest recorded use of a BOD test was in 1870 of either BOD or amount of waste oxidized at time (Theriault, 1926). Since that time, numerous t as a function of t. The applicable equation is studies have been made of the BOD test. The lit ^ (120) erature prior to 1927 on the BOD test was sum marized by Theriault (1927), and a bibliography of where Xt is either the BOD or the amount of waste the literature on the BOD test since 1927 was com oxidized up to time t, and L is the total first stage, piled by Hull (1959). or carbonaceous, BOD of the sample. Because both The BOD test consists basically of obtaining two L and k\ are unknown, several different techniques samples of the water containing organic wastes, have been developed for determining L and k\ from determining the DO concentration in one of the the results of BOD tests. Among these techniques samples at the beginning of the test, incubating are the least-squares procedure of Reed and the second sample in darkness at 20°C for a period Theriault (1931), the log-difference method of of time, usually 5 days, and determining the DO Fair (1936), the slope method of Thomas (1937), the concentration in the second sample at the end of moments method of Moore, Thomas, and Snow the incubation period. The difference in DO con (1950), the graphical procedure of Lee (1951), and centrations is the 5-day BOD of the sample. the ratio methods of Williams (1951) and Sheehy In practice, however, the procedure is not so (1960). The Reed-Theriault procedure is the most simple. To prevent complete depletion of the DO accurate, but it also requires the most time for in the test sample, the sample must be properly the calculations because it is a trial-and-error pro diluted before the start of the test. The dilution cedure. Of the other procedures, the moments water must be properly prepared and seeded. The method gives the most accurate results for the time type and amount of seed to be used in the dilution consumed (Ludzack and others, 1953; Schroepfer water for the particular waste of interest must be and others, 1960; Gannon, 1966; Zanoni, 1967). determined by experience. If the sample is alkaline The extensive series of BOD measurements by or acidic or contains residual chlorine or other Streeter and Phelps (1925) suggested that ki has a toxic substances, then it also must be treated value of 0.1 day-1 at 20°C for all types of sewage before the start of the test. The complete details and organic wastes. Later work, however, has of the standard 5-day, 20°C BOD test were given shown that ki at 20°C may range from about 0.01 by the American Public Health Association (1965). day- 1 for some industrial wastes and well-oxidized The reaeration technique of Elmore (1955) for treatment-plant effluents to about 0.30 day- 1 for BOD determinations has the advantage that dilu readily oxidizable wastes (Camp, 1963, p. 247). tion water and seeding are not required. This pro Schroepfer, Robins, and Susag (1960) found that cedure, however, has the disadvantages that the the ki values for raw wastewaters and primary 5-day BOD must be less than 25 mg/1, large volumes settled effluents from treatment plants in the of sample are required, and considerable attention Minneapolis-St. Paul area varied from 0.14 day- 1 on the part of the technician is necessary during on Sundays and holidays to 0.24 day-1 on weekdays. the initial stages of the test. They concluded that k\ depended greatly on the A respirometer assembly for BOD determina degree of stabilization of the effluent. Kothan- tions was described by Gannon, Pelton, and daraman and Ewing (1969) found that k\ values for 466-751 O - 72 - 5 24 REAERATION IN OPEN-CHANNEL FLOW the Ohio River downstream of Cincinnati, Ohio, advantage because the conditions in the streams were normally distributed and had a mean value may change considerably while the k\ value is of 0.075 day- 1 and a standard deviation of 0.028 being determined. Camp (1965) found that the day- 1. Camp (1965) also noted considerable daily water temperatures varied widely during the day variation of ki values for several stations on the light hours on the Merrimack River in Massa Merrimack River in Massachusetts. Thus, ki chusetts and concluded that it was pointless to apparently cannot be considered a constant, but determine the BOD in the laboratory at a constant ki values must be determined for the waste, the temperature. Hull (1966, 1969) and Camp (1965) reach, and the time of interest. recommended using a dark bottle suspended in The ki values determined at 20°C in the labora the stream at the point at which the sample was tory are converted to the appropriate river tem obtained. This procedure eliminates delay and the perature by means of the equation need for a temperature correction, and it gives a (if \ tif\ Q(T-20P) n 9~n BOD value for the period of sample incubation. To \H-lJrp \*Vl/0|\o" , V-*-^-1 -*-/ determine ki, however, requires BOD values for where 6 is the temperature coefficient, which has different times or continuous recording of the DO a value of 1.047 for temperatures from 10° to 37°C concentration in the submerged bottle, and one is (Camp, 1963, p. 247). Zanoni (1967) found the same again faced with the problem of changing condi value for 0 for a somewhat smaller temperature tions in the stream. range (15° to 32°C) and 6 values of 0.985 for tem peratures from 32° to 40°C and 1.126 for tempera tures from 2° to 15°C. He also found that L increased BOTTOM DEPOSITS, Ks, AND m with temperature. Widespread use of the laboratory 5-day 20°C In addition to the biochemical oxidation of BOD test has made apparent several aspects of organic wastes in the flowing stream, wastes may the standard BOD bottle test, as well as the afore also be removed from the stream by settling to the mentioned problems with dilution and seeding, bottom. Bottom deposits form in three general that require additional study. These aspects are ways (Velz, 1958): (1) deposition of settleable solids concerned with how well the diluted and seeded from the wastes; (2) deposition resulting from the sample incubated in darkness under quiescent flocculation or coagulation of solids within the conditions in a constant temperature bath approxi stream; and (3) biological extraction and accumu mates the behavior of a slug of water of similar lation of wastes by growths attached to the size under dynamic streamflow conditions. For bottom of the stream. example, the turbulence conditions are obviously The first type of deposit is the result strictly of different, and increased values of ki have been a sedimentation process. The second type may be found when the BOD bottles were stirred or rotated considered as resulting from a process analogous (Owens and Edwards, 1966; Gannon, 1966). The in to an activated sludge system in which the growth creased turbulence prevents deposition of sus is dispersed throughout the flowing stream. Under pended material and facilitates the transport of certain conditions, the growths flocculate, and the the wastes to the organisms. The large surface-to- floes increase in size until sedimentation occurs. volume ratio of the bottle may result in increased The third type of deposit may be considered as re growth of organisms on the inside of the bottle sulting from a process analogous to that occurring and, hence, faster DO consumption than would oc in a trickling filter. Biological extraction and cur in an unconfined slug of water of the same size accumulation of wastes were studied by Velz and (Symons, 1966; Hull, 1969). The bottle test does not Gannon (1963). The rate of sedimentation is gen allow for the possible decrease in the respiration erally assumed to be proportional to the amount rate of organisms with decreasing DO concentra of BOD, or tion in the bottle (Owens and Edwards, 1966). When rate of sedimentation =K3 L. (122) algae are present, the bottle test conducted in darkness could give a high value for the BOD of A velocity of 0.6 fps was considered by Velz (1958) the sample because of respiration demands of the to be the critical velocity at which organic wastes algae (Owens and Edwards, 1966). In the stream, begin to deposit. however, the algae replenish the oxygen consumed Even though the formation of a bottom deposit by their respiration by the process of photo may result in a reduction of the BOD of the water synthesis during daylight hours. The 5-day period from which the waste was removed, the deposit required for the standard bottle test is also a dis- may be a sink for DO to other masses of water EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 25 passing over the deposit. Several processes may be which K3 can be determined if Ki has been deter involved: mined from laboratory measurements. The term 1. BOD additions to the overlying water as a result DB, which includes the addition of BOD to the of resuspension of the bottom deposits. A water from bottom deposits, in general must be velocity of 1.0 to 1.5 fps is considered necessary determined by difference from the DO profile for resuspension, the actual velocity necessary equation. This procedure, however, requires an being dependent upon the degree of decompo independent estimate of the reaeration coefficient, sition and compaction of the deposit (Velz, K-Z. Camp (1965) recommended that m, the rate of (1958). addition of BOD to the overlying water, be deter 2. Diffusion of the anaerobic decomposition prod mined from in situ measurements by the technique ucts of the deposit into the overlying water. of Fair, Moore, and Thomas (1941) and Camp (1963, 3. Diffusion of DO from the overlying water into p. 310), and that Ks be determined from the BOD the aerobic zone of the deposit. profile equation after K\ has been determined from laboratory BOD tests. Fair, Moore, and Thomas (1941) concluded that the In addition to the studies of the overall effect oxygen demand of sludge deposits was controlled of bottom deposits on the DO and BOD balances, by process 2 and that process 3 was relatively the consumption of DO by muds has been studied unimportant. The relative importance of processes in some detail with somewhat contradictory re 1 and 2 may be seen qualitatively from an experi sults. For example, Baity (1938) and Fair, Moore, ment of Edwards and Rolley (1965) in which the and Thomas (1941) found that the DO consumption DO consumption rate of a mud sample increased rate, w, was dependent on the depth, C/M, of the approximately sixfold when the sample began to deposit according to scour. Similarly Brown and Clark (1968) reported large decreases in the DO concentrations during a) = ai UM0-485, (123) dredging operations in a tidal estuary. They also in which ai is a constant. On the other hand, pointed out that resuspension of bottom deposits Edwards and Rolley (1965) found that the DO con decreases light penetration and, consequently, the sumption rate was independent of the deposit additions of DO by photosynthesis. depth for depths greater than 2 cm, and McDonnell The importance of bottom deposits on DO bal and Hall (1969) concluded that the consumption ances has been demonstrated by several field rate was essentially independent of the sample studies, including those by Velz (1958), Owens and depth. Edwards (1963), and Camp (1965). Velz (1958) dis With respect to the dependence of the DO con cussed the effect of sludge deposits on DO profiles sumption rate on the DO concentration of the for several situations. Owens and Edwards (1963) overlying water, Baity (1938) found that the con found on the River Lark in England that they sumption rate was independent of the DO concen obtained very poor agreement between measured tration for the range of 2 to 5 mg/1 for deposits 0.5 and observed DO profiles until the DO demands of cm thick. Edwards and Rolley (1965), however, the mud were included in the DO balance. Edwards found that the DO consumption rate was depen and Owens (1965, fig. 1) presented the percentage dent on the DO concentration in the overlying of the total DO consumed by mud as a function of water according to the water depth for BOD and mud consumption (124) values typical of English streams. Their graph shows that for depths of flow less than 1.25 meters, where bi is 0.45 for the muds studied. This relation the mud consumed more oxygen than was con was valid for DO concentrations in the range from sumed by the biochemical oxidation of the wastes 2 to 8 mg/1 but was not satisfactory for muds con in the water. Camp (1965) found on the Merrimack taining large populations of midge larvae. Similar River in Massachusetts that 50 to 80 percent of the results were obtained by McDonnell and Hall (1969). BOD removal was by sedimentation. He also noted An attempt by Rolley and Owens (1967) to corre that most of the BOD removed by sedimentation late the DO consumption rates with various chemi consisted of solids formed by bioflocculation. cal properties of the muds was unsuccessful. They The effects of removal of BOD by sedimentation attempted to correlate five chemical properties and addition of BOD to the overlying water by with DO consumption rates for both winter and bottom deposits have been included in the DO summer samples from 12 English rivers but found balances of Dobbins (1964b) and Camp (1963). no correlation, although a strong correlation was Dobbins (1964b) described three situations for found among all the chemical properties. 26 REAERATION IN OPEN-CHANNEL FLOW PHOTOSYNTHESIS AND RESPIRATION found in a summer study on the Merrimack River in Massachusetts that about 67 percent of the DO was furnished by photosynthesis and that reaera The presence of phytoplankton, benthic algae, tion was relatively unimportant, and he concluded and rooted aquatic plants in a stream or river re "that reliance must be placed on the photosynthetic sults in the production of oxygen through the pro production of DO." Merritt, McDonald, and Paul- cess of photosynthesis. The plants combine carbon son (1968) found on the Iowa River that photo dioxide and water in the presence of sunlight to synthesis was a contributing factor in the DO produce carbohydrates and oxygen. At the same balance, and they concluded that "whether or not time, however, the plants require oxygen in their the photosynthetic effect can be relied upon respiration process. In general, the rate of produc quantitatively, it remains an important factor of tion of oxygen exceeds the rate of consumption the oxygen balance in streams." Similar photo- during daylight hours whereas consumption is synthetic effects have been observed in European predominant during nighttime hours. This may rivers, and this literature has been summarized result in a pronounced diurnal variation of the by Edwards and Owens (1965). Leffel (1968) found DO concentration in the stream, the maximum DO on the Chao Phraya River in Thailand that photo- concentration occurring in the afternoon and the synthetic production of DO was small in compar minimum DO concentration occurring just before ison with reaeration. Bain (1968) stated that in a dawn. deep turbid stream the photosynthetic production In the past, the photosynthesis and respiration of DO_ often is not sufficient to offset the respira processes have been of interest only to biologists, tion demands of the plants, even if the stream is who in their studies attempted in a very general well mixed. O'Connell and Thomas (1965) pointed way to allow for additions of DO by reaeration. out that if the photosynthetic production of DO Engineers, on the other hand, with the exception is by benthic algae or rooted aquatic plants rather of a few such as Hull (1960), ignored photosyn than phytoplankton, then photosynthesis is thesis and concentrated their efforts on the a liability because the oxygen produced during reaeration process. When photosynthesis is ig daylight hours by the attached plants is not in the nored, grossly incorrect values of the reaeration same mass of water from which oxygen is con coefficient, K2, may be obtained if the usual com sumed by nighttime respiration. An example of putation procedure is used. For example, Gunner- this cited by O'Connell and Thomas is the shallow son and Bailey (1963) found negative values of K2 rapidly-flowing Truckee River below the outfall of for two sampling periods on the Sacramento River, the sewage treatment plant of Reno, Nev. In a deep Calif. Lynch (1965) also found negative K2 values slowly-flowing stream, the response of the stream for nocturnal sampling periods on the Tioughnioga to a nutrient source is the development of phyto River in New York, and he believed that the com plankton which move with the stream. Hence, the puted K2 values for daytime sampling periods production and consumption of oxygen occur at would be larger than the true values if a separate different points in space but in essentially the factor for photosynthesis were not included in the same mass of water. In streams with attached DO balance. Similarly, Camp (1965) concluded that plants, the production and consumption of oxygen the Kz values for streams of the type of the Merri- occur at the same point in space but in different mack River were considerably smaller than had masses of water. Edwards (1962) suggested that been previously thought because of including sometimes plant growths may be beneficial even photosynthetic effects with reaeration. when there is no net gain of DO because the plants Opinions as to whether the DO produced by produce oxygen during the summer conditions of photosynthesis is an asset or a liability to a stream low flows and high temperatures and remove oxy vary widely. Camp (1963) stated that the DO pro gen during the decay process in winter when high duced by photosynthesis should not be relied upon flows and low temperatures exist. Gunnerson because there is no assurance that this DO will be (1966) suggested that photosynthesis may be ne available during the critical low-flow periods of glected if the time period over which the data are summer when it is needed most. Similarly, Imhoff averaged is sufficiently long-for example, longer and Fair (1956) suggested that the DO produced than 1 year. by photosynthesis should not be relied upon in engineering calculations because it is available Obviously, then, considerable controversy exists only during daylight hours and the warmer sea concerning photosynthesis and respiration in a sons of the year. On the other hand, Camp (1965) natural stream. Hull (1966) pointed out, however, EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 27 that the controversy generally is whether the oxy excluded, and the third part is analyzed for the DO gen production by photosynthesis is significant concentration. The light and dark bottles are sus and not whether a term for photosynthesis should pended in the stream at the sampling point for a be included in the DO balance. Thus Camp (1963), period of time, usually 24 hours or a multiple Dobbins (1964b), the Water Pollution Research thereof. At the end of the desired period, the light Laboratory group (Edwards and Owens, 1962,1965; and dark bottles are analyzed for their DO concen Edwards, 1962; Owens, 1965), O'Connor (1967), and trations. Equations for the light-bottle and dark- others have included terms for the photosynthetic bottle DO concentration are, respectively, production of oxygen in their DO balances. Clf-Cn=Pi-Ri (125) The production of oxygen by plant photosyn Cd,-CV=/?d, (126) thesis depends on several factors, including light where the / and d subscripts refer to light and dark intensity at the water surface, optical density of bottles, respectively; / and / are subscripts refer the water, and weight and distribution of plants ring to initial and final DO concentrations; R is the in the stream (Owens and Edwards, 1964). The gross respiration, or BOD; C is DO concentration; depth of the stream affects the photosynthetic and P is the photosynthetic oxygen production. For production rate in that the extent of light penetra reasonably short periods of time, Ri = Rd, and be tion varies inversely with depth. Photosynthesis is cause Cu = CM, it follows that limited to the euphotic zone, which is generally G/-Cd/ =P*. (127) assumed to extend to the depth at which the light Hull (1963) believed that the light- and dark-bottle intensity is 1 percent of the light intensity at the technique probably is more accurate than the surface (Owens and Edwards, 1966). The optical standard BOD test and that the technique in gen density of the water depends on the concentra eral gives conservatively low estimates of the tion, type, and size of suspended particles and on oxygen production by photosynthesis. the color of the water produced by dissolved Symons (1966) discussed several problems with materials. An example of a situation where the the classical light- and dark-bottle technique, in environmental conditions were suitable for a high cluding unnaturally large surface-to-volume ratios level of photosynthetic oxygen production but in the bottles, the relatively long time required for where actual production was limited to a very low the test, the lack of mixing in the bottles, and the level because of the turbidity of the water is the necessity for removal of samples from the stream, Chao Phya River in Thailand (Pescod, 1969). The which may result in harm to the algae. He dis weight and the distribution of the plants affect the cussed a three-probe chamber device designed to net DO production mainly by affecting the total eliminate these problems. O'Connell and Thomas respiration rate. Thus, as the plant density (1965) also used a chamber device but did not increases, the photosynthetic production rate re measure DO continuously. mains approximately constant because of self- The upstream-downstream method uses mea shading by the plants, although the respiration surements of the DO concentrations at the ends of rate continues to increase (Edwards, 1968). the reach of interest for a 24-hour period. The basis Several procedures have been developed for the of the procedure is a DO balance of the form measurement of photosynthetic and respiration (Odum, 1956) effects in natural streams. These include the QO=P-R+DO+AO, (128) upstream-downstream method (Odum, 1956; where Q0 is the rate of change of DO content, P Owens, 1965), the single-curve procedure (Odum, is the gross primary production, R is the respira 1956), which is a simplification of the upstream- tion rate, D0 is the rate of reaeration, and A 0 is the downstream procedure; and the light- and dark- rate of addition of DO to the reach by drainage bottle technique. accrual. Each term of equation 128 is on an area The light- and dark-bottle technique requires basis for example, in units of grams per square collection of water samples at a number of verticals meter per hour. in the cross section and at a number of points at If each term in equation 128 is divided by H, the each vertical so that the number of samples is mean depth of flow, the result is sufficient to yield an accurate value of photosyn q =p- r+ d0 + Si(h (129) thesis and respiration for the cross section. Each where the terms are now on a concentration basis sample is divided into three parts; one part is for example, in units of milligrams per liter per placed in a transparent glass bottle, another part day. is placed in an opaque bottle so that all light is The gross primary production, P, is defined as 28 REAERATION IN OPEN-CHANNEL FLOW the sum of the net DO plant production and com time, as usually occurs after sunset, then at that munity respiration during the daylight hours. instant, because the reaeration is zero, the respi Therefore, it follows that if r and do are indepen ration rate is equal to the rate of change of DO in dent of time and a0 is negligible or independent of the reach, assuming AQ is negligible. time (Odum, 1956), The single-curve procedure (Odum, 1956) is a simplification of the upstream-downstream pro P=H (130) cedure in which the diurnal variation of DO con where F is the volumetric water flow rate, As is centration at a single station is used in the the surface area of the reach, AC is the DO con calculation of P and R. This procedure assumes centration difference between the upstream and that the entire reach experiences uniform changes downstream stations, and the subscripts 1 and 2 in the DO concentration, and hence the curves at refer, respectively, to the time at which sunrise the upstream and downstream stations would be occurred and the time in the evening at which q identical. Owens (1965) compared the two calcula or AC returned to its sunrise value. When equation tion procedures for two stations on the River Ivel 130 is used to determine P, equation 128 must have in England which had nearly identical diurnal DO units of weight per unit of area per day. curves and found that the calculated P and R In general, the oxygen supplied by reaeration, values were different for the two procedures and do, is not independent of time because the rate of also different for the two stations. Edwards and reaeration depends on the DO deficit. If a reaera Owens (1962) also found differences between the tion coefficient is available for the reach, then a two procedures. correction can be applied for each hour in the cal In addition to their direct effects on the DO culation of the q curve from the DO measurements. balance, plants also exert indirect effects on the The reaeration coefficient can be calculated from DO balance. For example, heavy plant growths the diurnal DO measurements if r and ao are can increase the depth of flow and reduce the assumed independent of time. From measure water velocity, which in turn reduces the addition ments before dawn and in the evening, when the of oxygen by reaeration. At the same time, the p values are zero, it can be shown (Odum, 1956) that reduced velocity increases the deposition of solids. If the plants extend through the surface, then tur M E bulence at the air-water interface may be in where D is now the average DO deficit in the reach, creased, thereby facilitating the absorption of and the M and E subscripts refer to the morning oxygen by the water. Finally, the presence of the and evening times, respectively. The reaeration plants increases the surface area on which the coefficient calculated from equation 131 is the Kz bacteria can live and oxidize wastes present in for the average deficit of the reach as compared the stream. with an integral K2 obtained by integrating the deficit over the DO change. Both equations 130 and 131 assume that r is OTHER PARAMETERS independent of time and, hence, independent of the DO concentration level. Opinions differ as to Other parameters that are necessary in a DO the validity of this assumption. Odum (1956) con balance study include the DO concentrations at cluded on the basis of the available information the upstream and downstream ends of the study that it was simplest to assume r constant. On the reach and the flow time through the reach. other hand, Edwards, Owens, and Gibbs (1961) The flow time can be determined from mean concluded that the respiration rate of plants velocity measurements and the distance between depended on the DO concentration level. There the ends of the study reach, or fore, corrections for the dependence of r on DO A,=-.A (132) concentration may be necessary. The respiration rate can be determined from the Distances may be obtained from topographic maps, results of an upstream-downstream experiment in and mean velocities from current meters or from the following ways (Odum, 1956). First, if the rate stage-discharge ratings and cross-section measure of reaeration is known and A0 is negligible, then ments. Long study reaches are desirable; however, for the hours of darkness the respiration rate can practical considerations such as variations in the be calculated from the observed rate of change of channel cross section, location of sewage outfalls, DO in the reach. Second, if during the hours of and personnel requirements generally limit the darkness the water is saturated with DO at some length of reach that can be studied. EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 29 The flow time can also be determined from tracer 119 reduces to the classical Streeter-Phelps sag measurements, using either radioactive or fluores equation, equation 6. Examples of reaches in which cent tracers. The procedure consists of instanta p, m, and k3 were not negligible have been given neously injecting a quantity of the tracer far previously. In the reaches chosen specifically for enough upstream from the reach of interest so that the study of reaeration by Churchill, Elmore, and the tracer can be uniformly mixed in both the Buckingham (1962), the BOD, k3, m, and p param lateral and vertical directions before it enters the eters were all negligible, so that equation 119 reach. Procedures for estimating the distances reduced to required for complete mixing have been presented Db =Da 10-fc2<. (133) by Sayre (1965) and Fischer (1967). The flow time Thus, kz could be calculated simply from DO mea is the time required for the tracer cloud to move surements on samples obtained at two points in through the study reach. In some instances, the the reach separated by the flow time, t. In general, peaks of the tracer concentration versus time however, equation 119 or a similar equation should curves have been used in flow-time determina be the starting point in a DO balance calculation tions; however, it is more correct to use the cen- of k2 for a natural stream, and parameters should troids of the curves. The flow time is the difference be eliminated only after they have been shown to between the times at which the centroid of the dye be negligible for the reach and hydraulic condi cloud passes the downstream and upstream ends tions of interest. of the reach. The upstream and downstream DO concentra CALCULATION OF k2 IN A RECIRCULATING FLUME tions are determined from samples obtained at the ends of the study reach. One of the procedures In a laboratory flume the sources and sinks of previously described is used to determine the DO DO can be controlled so that the BOD, k3, m, and p concentrations of the samples. The samples at the parameters will in general be zero. With these downstream point usually are obtained at a time conditions, the reaeration coefficient can be cal lag of A*/Lf after the samples at the upstream culated from equation 133. Flume lengths, point. This is done because DO balance equations however, are limited by space and economical con generally are written in a coordinate system mov siderations so that the available flow times and ing with the mean flow velocity. the consequent change in DO deficit generally will The number of samples obtained at each cross be very small. Thus the calculated reaeration section must be sufficient to define a mean DO coefficients will be subject to large errors. This concentration for the cross section. As mentioned problem can be alleviated by recirculating the previously, gradients of DO in the vertical direc water through the flume. tion in a flowing stream are generally negligible. In recirculating flumes, the water is partly de- The lateral DO distribution depends on many fac aerated before the start of an experiment, usually tors, including distance from sewage outfalls and using sodium sulfite in the presence of a cobalt stream tributaries, cross-section shape and tor catalyst. As the water circulates through the tuosity of the channel, and size, shape, and spac flume, it reaerates at a rate which is dependent on ing of bed roughness elements. Churchill, Elmore, the hydraulic conditions in the flume and on the and Buckingham (1962) observed consistent characteristics of the flume return-flow system. U-shaped DO distributions across the stream For steady-state hydraulic conditions, the form of channel, and they attributed these to (1) incom equation 9 generally assumed to be applicable is plete mixing of tributary inflows, (2) increased <*Q+U <*P-=-2.3k2 D, (134) reaeration near the banks because of the longer traveltime, and (3) unequal effects across the where the DO concentrations have been expressed stream width of photosynthesis and respiration by as deficits. Making the substitutions x'=x Ut and attached aquatics. Thus the possibility of lateral t' = t into equation 134, one obtains variations in the DO concentration should be / 9^Lr £-dK 2 LJ)n \J-OO}n^i considered. To determine k2 in the most general field situa and integrating equation 135 with D = Da at f'=0 tions, all the parameters in equation 119 must be gives evaluated using the methods previously described. D=Da .10- fc2<, (136) In specific reaches, however, some of the factors or in equation 119 may be of negligible importance. 1 (137) For example, if p, ni, and k3 are negligible, equation 30 REAERATION IN OPEN-CHANNEL FLOW Equation 137 is identical in form with equation variations in DO concentration caused by photo 133; however, in equation 137, because conditions synthesis may be used. If P, /?, K2, U, and Cs are are not steady state, the flow time t' is a time lag constant for the reach during the period of the which must elapse between the DO measurements experiment and therefore independent of the at the two sampling stations, whereas in equation DO concentration, then by manipulation of equa 133, because conditions are steady state, the DO tion 14: measurements at the two stations may be made at K2 =-ln^a "^a , (138) any time. In a recirculating flume, it is common experimental practice to obtain an average log where Ca and Ca are the upstream DO concentra deficit difference as compared with the single tions at the two levels of concentration, Cb and Cb' measurement of the difference obtained in a are the downstream DO concentrations measured natural stream or a nonrecirculating flume. The at a time t after the upstream concentrations, and average log deficit is determined by the procedure t is the flow time for the reach of interest. illustrated in figure 2. The parallel lines show the Although the respiration rates of bacteria are variation with time of the logarithm of the deficit independent of the DO concentration for concen at two points in the flume a distance X apart. Be trations larger than 1 mg/1 (Edwards and others, cause the water is recirculated, several points can 1961), the respiration rates of plants (Edwards and be obtained on each line, and thus the posi others, 1961), and of bottom deposits (McDonnell tions and slopes of the lines can be accurately and Hall, 1969; Edwards and Rolley, 1965) are not. determined. Furthermore, photosynthetic DO production can cause large variations in DO concentration with time on a specific day and from day to day. There fore, when plants are present, Edwards, Owens, and Gibbs (1961) suggested that the experiments be conducted at night, when the photosynthetic contributions are negligible. To account for the dependence of the plant respiration on DO concen tration, they showed that K2 = _J^-^Qi ^_ t (139) where Ci and C2 are the average DO concentra tions for the study reach for the two levels of DO concentration, Q\ and Q2 are the average rates of change per unit area of the DO concentration in the reach at each of the two levels of DO concen tration, CS2 and Ctl are the DO saturation concen trations at the times of the measurements for the DO concentrations at the two levels, and R is the TIME ratio of the oxygen consumed by respiration at FIGURE 2. Computation of reaeration coefficient, K%, from DO concentration C2 to that consumed at DO con recirculating-flume data. centration Ci. The reaeration coefficient obtained from equation 139 is the K2 for the average DO DISTURBED-EQUILIBRIUM TECHNIQUE deficit in the study reach. The present authors derived from equation 119 The disturbed-equilibrium technique developed an equation that will give an integral reaeration by the Water Pollution Research Laboratory coefficient rather than one based on the average (Gameson and Truesdale, 1959) is based on a DO DO deficit for the reach. If equation 119 is written balance described by equation 14. The procedure for upstream deficits of D a and D a and the consists of measuring the DO concentrations at difference taken, the result is the upstream and downstream ends of the reach of interest at two different levels of DO concen (140) tration. The second DO concentration level is obtained usually by adding sodium sulfite and a cobalt catalyst to the stream, although the diurnal where p and p include the respiration rates of both EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 31 plants and bottom deposits. In equation 119, the mination of a true reaeration coefficient in the parameter p is the net production of oxygen by presence of pollution and photosynthetic effects, photosynthesis. However, the production of DO by provided the value of the ratio of gas-transfer photosynthesis is assumed to be constant or zero coefficients is known for the inert gas and oxygen. during the period of the measurement, so that the Thus, the tracer technique is unique in that it does p parameter becomes strictly a respiration term. not require the measurement of any of the param The derivation of equation 140 also assumes that eters in a DO balance. The technique was first Ki, Ks, La and m are constant during the period of described by Tsivoglou, O'Connell, Walter, Godsil, the measurements. Because K2 appears on both and Logsdon (1965), a field application was sides of equation 140, a trial-and-error solution is described by Tsivoglou, Cohen, Shearer, and Godsil necessary. (1968), and a complete discussion of the technique The disturbed-equilibrium technique requires was presented by Tsivoglou (1967). measuring the upstream and downstream DO con The technique uses an instantaneous injection centrations and the respiration rate in the reach of three tracers at a point upstream from the reach at two different values of the upstream DO deficit. over which the reaeration coefficient is to be mea The different values of the upstream deficit may sured. The tracers are (1) a fluorescent dye, the be produced by diurnal variations in photosyn purpose of which is to enable field personnel to thesis and respiration or in BOD inflow, or by the follow the movement of the tracers; (2) tritiated addition of sodium sulfite along with a cobalt water, which is used as a conservative dispersion catalyst. In general, the technique is very con tracer; and (3) the radioactive tracer gas krypton- venient to use, especially if the DO variations are 85, which is used to measure the gas transfer capac produced by natural conditions in the stream. In ity of the flow. Samples of stream water are this case, however, accurate measurements of the removed from the flow as the dye peak passes the respiration rates p and/?' are necessary, quantities upstream end and again as it passes the down which are often not easily obtained. If the deficit stream end of the reach. The transfer capacity of is produced by sulfite dosing, where p and p' are the flow for the tracer gas is obtained from the small, there is less uncertainty in the measure relative concentrations of the gas and tritiated ments of p and p'; however, their variation may water tracers at the two ends of the reach. still be significant. The amount of sulfite required Tsivoglou (1967) showed on the basis of an argu places limitations on the size of stream that can ment involving numbers of krypton and tritium be dosed and on the length of time that the dose atoms and an exponential expression for gas can be applied. Furthermore, it must be assured exchange that that the sulfite is completely mixed with the flow and that the reaction has proceeded to completion (141) before the flow enters the study reach. Finally, in (Co] either method, the experimenter must assure him \C£/a self that the effect of the time variations in DO where C* is concentration; t is flow time, or and BOD do not cause serious errors in the com where A* is the distance between the downstream putation of the reaeration coefficient. and upstream stations indicated respectively by the subscripts "b" and "a"; the term K2c is the gas- TRACER TECHNIQUE transfer coefficient of the tracer gas; and the G and L subscripts refer respectively to the krypton Equations 81 and 83 resulting from the kinetic- and tritium tracers. theory model of Tsivoglou (1967) are the basis of the The tracer-gas transfer coefficient, KzG, can be tracer technique for determination of the reaera converted to a reaeration coefficient, K2, by means tion coefficient. As discussed previously, Tsivoglou of the gas-transfer ratio determined by Tsivoglou (1967) concluded on the basis of laboratory mea (1967). He found that the average ratio of the rate surements that the gas-transfer-coefficient ratio of krypton desorption to the rate of oxygen absorp in equations 81 and 83 was a constant within the tion was 0.83. Hence, limits of experimental error for the range of tem perature and turbulence conditions studied. K'=m- (142) Because the ratio is constant, an inert gas that The crucial assumption in the tracer technique does not participate in the various biochemical is that this ratio is a constant that is, indepen and biological processes in a stream can be used dent of the flow conditions, temperature, and as a tracer for oxygen. This permits the deter pollutants present. 466-751 O - 72 - 6 32 REAERATION IN OPEN-CHANNEL FLOW We developed a theoretical basis for the tech point. To obtain a K2 representative of the entire nique from solutions of the conservation-of-mass cross section of a reach, the tracers must be in equation for the two tracers. The one-dimensional jected sufficiently far upstream so that they will be diffusion equation in a steady-uniform flow for a uniformly distributed in the cross section before conservative tracer is entering the reach. Under these conditions, the one-dimensional diffusion equation is applicable, ^r+u^r=D*^' (143) and as shown previously, a theoretical basis exists where C\ is the tracer concentration. The cor for the tracer-technique equation obtained from responding equation for the gaseous tracer is intuitive arguments by Tsivoglou (1967). ,c* aaa ^ ^ The assumptions on which the tracer technique dt ~ dx dx2 "'<> CG' (144) is based were listed by Tsivoglou, Cohen, Shearer, where C*G is the tracer gas concentration. Implicit and Godsil (1968, p. 302) as follows: in the use of the one-dimensional diffusion equa (1) The tritiated water undergoes dispersion only in the tion is the assumption that the tracers are uni stream, and is not lost from the stream water in any significant formly distributed in the cross section of the amount; (2) the dissolved krypton-85 undergoes the same dis persion as the tritiated water, in addition is lost to the atmo stream. Therefore, the upstream sampling point sphere, and is not otherwise lost in any significant amount; must be far enough downstreanrfrom the injec (3) the ratio (0.83) of gas transfer coefficients for dissolved tion point so that complete mixing in both the krypton-85 and oxygen is not affected significantly by tempera lateral and vertical directions has been achieved. ture, turbulence, or the presence of usual water pollutants, in The solutions to equations 143 and 144 for instan the range of interest***. taneous inputs of weight WL of the liquid and WG The tritium and krypton tracers were chosen with of the gaseous tracer at ;c=0 and t=0 were pre the first two assumptions in mind, and therefore sented by Camp (1963). For equation 143, the these assumptions should be valid. The third solution is assumption was verified experimentally, as dis cussed previously. The tracer technique has several advantages (145) over the DO balance and disturbed-equilibrium and for equation 144, techniques for determination of reaeration co efficients. The most obvious advantage is that only the concentrations of two radioactive sub (146) stances need be determined as compared with all the various parameters that must be measured in in which WoL=WLIAx and W*G =WG\AX, where Ax is the other two techniques. In addition, the nuclear the flow area. For any longitudinal position x at counting techniques are very accurate, and be time t, the ratio of the gas concentration to the cause only ratios of the counts are used, determi liquid concentration is nation of the counting efficiency is not necessary, provided it is the same for the counting of all e-'V . (147) samples. The possibility exists also of determining with the tracers the discharge and longitudinal- If one now takes the ratio of equation 147 for the dispersion characteristics of the stream at the downstream station at a time t=bx!U to equation same time that the reaeration coefficient is mea 147 for the upstream station at t=Q, the result is sured. This would require additional effort, identical with equation 141 given previously by however, because the complete concentration- Tsivoglou (1967). Using equations 141 and 142 and versus-time curve at each cross section is nec the four measured tracer concentrations, the essary whereas only the peak concentrations reaeration coefficient K2 can be computed. are needed for the reaeration-coefficient deter The arguments used by Tsivoglou (1967) in mination. developing equation 141 require no assumptions There are also disadvantages to the tracer tech regarding vertical and lateral mixing of the nique. Radiation exposure received by the per tracers. However, the reaeration coefficient ob sonnel conducting the study and the public along tained from the tracer technique using a point- the stream may be significant. Tsivoglou (1967) source injection and samples from the centerline reported that radiation exposure could be elimi of the channel is representative only of the fila nated with a minimum of trouble for discharges ment of the stream traversed by the tracers in less than 200 cfs. However, for larger flows, more moving from the injection point to the sampling stringent precautions are required because of the EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 33 larger amounts of tracer necessary. Another dis For example, as has already been mentioned, advantage of using radioactive tracers is that the not only is accurate measurement of BOD in process for obtaining a license to use radioactive bottles difficult due to dilution and seeding prob material in a public waterway is quite involved. lems, but also there is considerable question as to Finally, the nuclear counting equipment neces how accurately the behavior of a sample of water sary for tritium and krypton is expensive and not incubated under quiescent conditions in a constant field-type equipment; hence, samples must be temperature bath simulates the behavior of a returned to a laboratory for analysis. similar slug of water under dynamic streamflow conditions and varying temperature. There are ANALYSIS OF ERRORS IN REAERATION-COEFFICIENT similar problems with the techniques for deter CALCULATION TECHNIQUES mining the rate of oxygen consumption by the The determination of the reaeration coefficient benthos and for determining the rates of photo by any of the techniques described previously is synthesis and respiration of suspended and subject to many sources of errors. The major attached plants. sources of error are (1) differences between the Completely aside from the problem of simulating dynamic stream condition and the quiescent con field conditions with measurement techniques is dition in the various bottle techniques developed the problem of obtaining representative samples for measuring the DO balance parameters, (2) the in the field. The uniformity of DO and BOD distri difficulty and expense of obtaining a sufficient bution in a cross section will depend on a multiplic number of representative samples to describe ity of factors, among which the most important accurately the stream conditions at the cross are distance from sewage outfalls and stream trib section of interest, (3) deviations from the assump utaries, shape of cross section, tortuosity of tions inherent in the derivation of the DO balance channel, and bed roughness size, shape, and spac equation, and (4) the accuracy of the DO measure ing. For example, Churchill, Elmore, and Bucking ment procedure used in the various techniques ham (1962, figs. 3-9) found a consistent U-shaped for determining the DO balance parameters. DO distribution across the stream channel and A general statistical analysis of the effect of attributed this distribution to (1) incomplete mix these various sources of error on the calculation ing into the main stream of flow from upstream of the reaeration coefficient is beyond the scope of tributaries, (2) increased reaeration near the banks this report and probably is not feasible with the due to a longer traveltime, and (3) unequal effects present state of our knowledge. However, it is across the stream of photosynthesis and respira instructive and important to consider the range tion by attached aquatics. Considering the large of possible errors in the measured DO balance cross-sectional DO variation and the small longi parameters and the effect of these errors on the tudinal variation in average DO illustrated in DO balance equation used to calculate the reaera these figures, one would expect considerable error tion coefficient. in the calculated k2 if DO were sampled at only one The purpose of this section is to present a dis point in the cross section. Another example of the cussion of (1) the sensitivity of the DO balance difficulty of obtaining representative samples for technique to errors in the measured parameters; the computation of the DO balance parameters (2) the sensitivity of the disturbed-equilibrium was given by Owens and Edwards (1961), who took technique to errors in the measured parameters; extensive samples of attached aquatic plants from (3) possible errors in the tracer technique; and several English streams for analysis of the photo (4) reproducibility of reaeration coefficients mea synthesis-respiration parameter. In their studies, sured in flumes. the standard deviation of the amount of the attached aquatics per unit area averaged 32 per ACCURACY OF MEASURED DISSOLVED-OXYGEN BALANCE cent of the mean. The size of this standard devia PARAMETERS tion in terms of the mean indicates the difficulty of obtaining a reliable estimate of photosynthesis The accuracy of the measured value of a DO or respiration without sampling the plants on a balance parameter is dependent not only on the large area of the stream bed. accuracy of the DO measurements but also on the extent to which the measurement conditions under Because of the difficulty of reproducing the which the measurements are made simulate the stream environment in the measurement tech conditions in the flow, and on how representative niques for DO balance parameters, and because of the samples are of the total flow. the difficulty and expense of obtaining representa- 34 REAERATION IN OPEN-CHANNEL FLOW tive samples, it is apparent that if all parameters and calculations were stopped when the absolute in the DO balance equation are significant, then value of the ratio on the right-hand side of equa the two conditions discussed above, and not DO tion 149 was less than 0.001 times the true value measurements themselves, limit the accuracy of k2. which can be obtained in the calculation of k2. The sensitivity analysis consisted of the follow Because of the present state of uncertainty about ing steps: these two situations, the accuracy obtainable from DO balance calculations is largely a matter of con 1. Assume a typical set of parameters for use in jecture. The error analysis techniques used in this the DO balance equation, equation 119. This section have been designed with this in mind. Rea set of parameters is called a BPS (basic sonable average values for the various DO balance parameter set). parameters and what the authors consider to be 2. Substitute the BPS into equation 119, and cal reasonable values of errors are.given in this culate the corresponding BPS value of £>b. section. 3. Use equations 148 and 149 to compute the sen sitivity of equation 119 to each of the param eters in turn. DISSOLVED OXYGEN BALANCE TECHNIQUE In some calculations, the classical sensitivity Because of the complexity of the general DO analysis procedure was changed in that several balance equation, the number of parameters in of the parameters were varied simultaneously. volved, and the difficulty of reproducing field The BPS values are given in table 1. Typical sen conditions for replications of k2 determinations, a sitivity analysis plots are given in figures 3 rigorous statistical analysis of the effect of errors on the calculated k2 cannot be performed at the TABLE 1. Basic parameter sets used in the sensitivity analy ses of the DO balance equation present time. Alternatively, however, one can perform for k2 a sensitivity analysis, similar to that BPS ki *2 fa of Kothandaraman and Ewing (1969), on the down No. (days-') (days-') (days-1) pe^daV) (mgE/1) 900 500 PHOTOSYNTHETIC PRODUCTION RATE, IN MILLIGRAMS PER LITER PER DAY FIGURE 4. Sensitivity of reaeration coefficient to photo- synthetic-oxygen production rate, dissolved-oxygen balance technique. 0.2 0.4 0.6 0.8 1-0 7f> BIOCHEMICAL OXYGEN DEMAND DECAY RATE CONSTANT, IN RECIPROCAL DAYS FIGURE 3. Sensitivity of reaeration coefficient to rate con stant for BOD decay, dissolved-oxygen balance technique. The sedimentation rate constant, k3, and m, the rate of addition of BOD from bottom deposits, were not considered in figures 3 through 8 because the sensitivity analyses showed that the dependence of k2 on these parameters was very small. For k3, the most sensitive example was for BPS 5, where kz changed only 23 percent of its BPS value for a change of k3 from 0 to 0.8. For m, the most sensitive example was for BPS 10, where k2 changed only 13 percent as m varied from 0 to 200 mg/1 per day. The results shown in figures 3 through 8 can be UPSTREAM DISSOLVED-OXYGEN DEFICIT, clarified using an approximate form of equation IN MILLIGRAMS PER LITER 119. Because the exponents in this equation are FIGURE 5. Sensitivity of reaeration coefficient to upstream much less than 1, the exponential terms may be DO deficit, dissolved-oxygen balance technique. 36 REAERATION IN OPEN-CHANNEL FLOW -J1200 BPS 16 tnioooUJ 800 600 400 BPS 6 BPS 5. 0.05 0.10 0.15 0.20 0.25 100 150 200 TRAVELTIME, IN DAYS UPSTREAM BIOCHEMICAL OXYGEN DEMAND, IN MILLIGRAMS PER LITER FIGURE 7. Sensitivity of reaeration coefficient to traveltime, FIGURE 6. Sensitivity of reaeration coefficient to upstream dissolved-oxygen balance technique. BOD, dissolved-oxygen balance technique. approximated by the first two terms in their coefficient calculation resulting from an error in Taylor-series expansions. In this case, equation a measured parameter requires information on 119 reduces to the magnitudes of the errors to be expected in the measured parameters. However, as discussed previously, the different techniques developed for The first term in the brackets of equation 150 measuring the DO balance parameters are subject explains why the sensitivity of kz to variations in to various types of errors, and the determination flow time, t, decreases as Db approaches Da (fig. 7). of such errors is difficult. Therefore, for the pur The second and third terms in the brackets pose of an error analysis we have made what we explain the linear variation of kz in terms of ki, p, consider to be reasonable assumptions of the RMS and La (figs. 3, 4, and 6). The second term in the (root-mean-square) and maximum errors to be brackets explains the changes in sensitivity of kz expected in the various measured parameters. to ki with changes in La. Finally, the absence of k3 These estimates are presented in table 2. The RMS and m from equation 150 explains the lack of sen error is the square root of the mean value of the sitivity of the calculated kz to these parameters. square of the deviations of the measured param The solution of equation 150 for kz using BPS 1 is eter from the true value of the parameter. 0.94, and that using BPS 5 is 1.37; thus, the The errors in a calculated kz corresponding to approximations in the equation are too great for the estimated parameter errors given in table 2 it to be used for calculation of kz directly, but it is are given by the product of the estimated error valuable for showing quickly the relative serious and the slope of the sensitivity-analysis plot. It is ness of the errors in the calculated kz which can apparent from figures 3 through 8 that the slopes result from errors in the other parameters. of the sensitivity-analysis plots depend on the The extension of the sensitivity analysis to particular BPS considered. Therefore, the mini the determination of the error in a reaeration- mum, average, and maximum slopes were deter- EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 37 1200 tion coefficient. These errors are presented in table 4. It is apparent that, with the exception of m, errors in all the parameters may cause consider able error in the calculated kz value. As the value of k2 decreases, the percentage of error increases. hiooo - TABLE 3. Slopes of the sensitivity-analysis plots Slope Minimum Average Maximum 800 - *, 1.6 10 80 *3 0.01 0.05 0.50 Da 1.4 (mg/1 per day)" 1 2.0 (mg/1 per day)" 1 3.0 (mg/1 per day)" 1 0b 1.6 (mg/1 per day)" 1 2.5 (mg/1 per day)" 1 6.9 (mg/1 per day)- 1 J-a 0.02 (mg/1 per day)" 1 0.035 (mg/1 per day)" 1 0.08 (mg/1 per day)- 1 0.11 (mg/1)-' 0.20 (mg/1)" 1 0.30 (mg/1)- 1 P 1x10-* (mg/1)- 1 5x10"' (mg/1)" 1 2x10"" (mg/1)- 1 1 5.0 days"2 40 days"2 150 days-2 600 - TABLE 4. Errors in the calculated reaeration coefficient resulting from errors in parameter measurement, DO bal ance technique 400 - Error in calculated klt in days-1 Minimum slope Average slope Maximum slope RMS Maximum RMS Maximum RMS Maximum *, 0.08 0.48 0.50 3.0 4.0 24 *3 .001 .005 .005 .025 .05 .25 oa .14 1.4 .20 2.0 .30 3.0 200 0b .16 1.6 .25 2.5 .69 6.9 i-a .04 .20 .07 .35 .16 .80 P .22 1.1 .40 2.0 .60 3.0 m 5xlO"4 2xlO"3 2.5 xlO"3 .01 .01 .04 t .01 .025 .08 .20 .30 .75 Figures 5 and 8 show the sensitivity of the cal culated kz to the upstream and downstream DO DOWNSTREAM DISSOLVED-OXYGEN DEFICIT, deficits, respectively. Because the lines slope in IN MILLIGRAMS PER LITER opposite directions, the effect of systematic errors FIGURE 8. Sensitivity of reaeration coefficient to downstream in the procedure used to determine the DO should DO deficit, dissolved-oxygen balance technique. tend to cancel. This was verified with calculations based on BPS 14. The reaeration coefficient was calculated from equations 148 and 149 using a TABLE 2. Assumed errors in measurements of DO balance parameters systematic error in the upstream and downstream DO deficits, and the results are presented in figure Assumed error Parameter Root-mean-square Maximum 9. The abscissa is in terms of the systematic error fcl 0.05 days-1 0.30 days-1 in DO and is expressed as a percentage of the up ks .10 days"1 .50 days"1 Da .10 mg/1 1.0 mg/1 stream deficit or a percentage of the deficit range. Db .10 mg/1 1.0 mg/1 It is apparent from comparison of figures 9 and 5 La 2.0 mg/1 10 mg/1 P 2.0 mg/1 per day 10 mg/1 per day or figures 9 and 8 that the effect of the systematic m 5.0 mg/1 per day 20 mg/1 per day error is much less severe than the effect of a t .002 day .005 day random error of the same magnitude in one of the DO deficits. The use of an incorrect value of the saturation concentration, Cs, in the computation mined for each parameter so that the entire range of the DO deficits will have an effect identical with of conditions covered by the 19 different BPS a systematic error in the DO measurement would be considered. The slopes are presented in procedure. table 3. The minimum, average, and maximum The effect on the calculated k2 of ignoring photo- slopes were combined, each in turn, with the RMS synthetic production is shown in figure 10. The and maximum error estimates from table 2 to downstream DO deficits for BPS 16,18, and 19 were give the expected errors in the calculated reaera calculated with the photosynthetic production 38 REAERATION IN OPEN-CHANNEL FLOW LJ DISSOLVED-OXYGEN ERROR, IN PERCENTAGE term in the equations. Then the k2 was calculated D OF DISSOLVED-OXYGEN DEFICIT RANGE -400 -300 -200 -100 0 +100 +200 +300 +400 for the same conditions except that photosynthesis 2 160 1 1 1 1 1 1 1 1 was ignored. The effect is very definite; for t C/5 BPS 14 example, for a photosynthetic production rate of £150 -\ 2 mg/1 per day, the calculated kz is 130 percent of t the BPS value of 1.0 days- 1. Thus, if a term for a 140 - \ _ photosynthetic production is not included in the £L \ DO balance equation and this production is signif O \ icant, the calculated k-2 will be significantly larger 00 \ than the true value. The effect of photosynthesis LL. O was discussed previously by Camp (1965) and - LJ 120 - Lynch (1965). 0 \ The effect on the calculated reaeration coeffi PERCEN' o - - cient of simultaneous errors in two or more of the \ parameters was investigated in this study to a limited extent, and it was found that the overall - 100 - > - H errors were in general not equal to the sums of Z LJ x^ the individual errors. Thus each individual error - ^ g 90 - set must be investigated as an entity; however, LL. LJ this is beyond the scope of the present report. O X O z 80 From the results of the sensitivity analysis pre o sented in figures 3 through 10 and in table 4, several general conclusions are possible. Because o: 7n 1 1 1 1 1 1 1 I -40 -30 -20 -10 0 +10 +20 +30 +40 of the limited scope of the sensitivity analysis, DISSOLVED-OXYGEN ERROR, IN PERCENTAGE estimates of the errors will be qualitative only. It OF INITIAL DISSOLVED-OXYGEN DEFICIT is expected in the calculation of the reaeration FIGURE 9. Effect of a systematic error in DO measurement coefficient from equation 119 that: procedure on calculated reaeration coefficient, dissolved- 1. If the BOD levels are small, and if attached or oxygen balance technique. floating plants are present in significant quan tities, then neglect of the photosynthetic produc tion is probably the largest source of error in the 500 calculation of the reaeration coefficient. 2. If the BOD levels are small, and if errors in the upstream and downstream DO deficit are inde pendent, then these errors are probably the second largest source of error in the calculation of the reaeration coefficient. 3. For small to moderate upstream BOD levels, the errors in the calculated £2 resulting from errors in kt and La are generally small but signif icant for £2 ^1.0 days-1 . For an upstream BOD of greater than about 50 mg/1, the errors in the cal culation of k2 resulting from errors in /ct are especially significant. 4. The other parameters should cause no signif icant errors in the calculated k2 unless ta^O.l days-1. 5. By summing the squares of the RMS errors for the average slope column of table 4 and taking the square root, an expected relative RMS error 5 10 15 20 of 65 percent of a true k2 value of 1.0 days-1 is PHOTOSYNTHETIC PRODUCTION RATE, obtained. IN MILLIGRAMS PER LITER PER DAY 6. If errors in the upstream and downstream DO FIGURE 10. Effect of ignoring photosynthetic-oxygen pro deficits are the result of a systematic error in the duction rate, dissolved-oxygen balance technique. DO measurement procedure or of the use of an EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 39 incorrect DO saturation value, then these errors The results of the sensitivity analyses of equa compensate each other to some extent so that the tion 151 are presented in figures 11 through 14, in error in the calculated reaeration coefficient is not which the ordinates are in terms of percentage of as great as it would be if the error had occurred the EPS value of the reaeration coefficient, and independently in Da or £>b. the abscissas are in the dimensions of the variable 7. The effects of simultaneous errors in two or being investigated. In general, the sensitivity in more parameters are not linearly additive, so that creases as the deficit range decreases. the results of this study cannot be used to deter mine the effects of more than one parameter error 250 on the calculated k2 values. Future studies should be directed to determining the sensitivity of the DO balance equation to errors in two or more param eters. The most ideal study would be to use Monte Carlo techniques, which would give the distribu tion function of the reaeration coefficient cal culated from the DO balance equation for known distributions of the other parameters. DISTURBED-EQUILIBRIUM TECHNIQUE A rigorous statistical analysis of the disturbed- equilibrium technique is presently impractical for the same reasons that such an analysis is imprac tical for the DO balance technique. However, a sensitivity analysis was performed, using the same approach as for the DO balance technique. The results of the sensitivity analysis and a compari son of the results of calculating reaeration coeffi cients from equations 139 and 140 are discussed in the following paragraphs. The equations used in the sensitivity analysis were equation 140 written in the form -8 -6 -4 -2 0 PHOTOSYNTHETIC PRODUCTION RATE, l ~ Da+ 2JUj' IN MILLIGRAMS PER LITER PER DAY (151) FIGURE 11. Sensitivity of reaeration coefficient to photosyn- and a Newton-Raphson iterative solution as given thetic-oxygen production rate, disturbed-equilibrium tech by equation 149. The EPS values studied are listed nique. in table 5. These were chosen to correspond to the EPS used in the sensitivity analysis of the DO balance technique. With the exception of the traveltime, two mea surements of each parameter are necessary in the disturbed-equilibrium technique. Because of the additive nature of the variances, we have multi TABLE 5. Basic parameter sets for the sensitivity analysis of the disturbed-equilibrium technique plied by 1.5 the estimated parameter errors pre sented previously, in table 2, and used in the EPS ft, / Da Da P P. Db Db sensitivity analysis of the DO balance technique. No. (days-') (days) (mg/1) (mg/1) p^m|;'y) p^ay) (m*ll) lm«M These estimates of the parameter errors and the 1 1 0.05 3 5 0 0 2.73 4.51 2 1 .05 3 5 -1 0 2.73 4.46 slopes of sensitivity-analyses plots for the 3 1 .05 3 8 0 0 2.73 7.18 4 1 .05 3 8 -1 0 2.73 7.14 disturbed-equilibrium technique (figs. 11 through 5 .2 .05 3 5 0 0 2.99 4.94 6 .2 .05 3 5 -1 0 2.99 4.89 14) are presented in table 6. The possible errors in 7 10 .05 3 5 0 0 .98 1.62 8 10 .05 3 5 -1 0 .98 1.59 the calculated reaeration coefficient obtained from 9 1 .05 3 5 -1 0 2.73 4.46 10 1 .05 3 5 -5 0 2.73 4.27 the different combinations of RMS and maxi 11 1 .05 3 5 -10 0 2.73 4.04 12 1 .05 3 8 -5 0 2.73 6.95 mum errors and minimum, average, and maximum 13 1 .05 3 8 -10 0 2.73 6.71 slopes are presented in table 7. 40 REAERATION IN OPEN-CHANNEL FLOW 1400 0.10 0.15 0.20 0.25 0 1234 UPSTREAM DISSOLVED-OXYGEN DEFICIT, TRAVELTIME, IN DAYS IN MILLIGRAMS PER LITER FIGURE 12. Sensitivity of reaeration coefficient to upstream FIGURE 14. Sensitivity of reaeration coefficient to travel- DO deficit, disturbed-equilibrium technique. time, disturbed-equilibrium technique. 1200 RESPIRATION RATE DIFFERENCE, IN 123456 MILLIGRAMS PER LITER PER DAY DOWNSTREAM DISSOLVED-OXYGEN DEFICIT, IN MILLIGRAMS PER LITER FIGURE 15. Effect on calculated reaeration coefficient of FIGURE 13.- - Sensitivity of reaeration coefficient to down- ignoring a respiration rate difference, disturbed-equilibrium stream DO deficit, disturbed equilibrium technique. technique. EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 41 TABLE 6. Assumed parameter errors and slopes of sensitivity-analysis plots for disturbed-equilibrium technique Assumed error Slope Parameter - Root-mean-square Maximum Minimum Average Maximum p 3.0 mg/1 per day 15 mg/1 per day 0.09 0.20 0.40 (mg/1)-1 Da .15 mg/1 1.5 mg/1 .1.5 (mg/1 per day)"1 2.5 (mg/1 per day)"1 4.0 (mg/1 per day) -1 Db .15 mg/1 1.5 mg/1 2.0 (mg/1 per day)"1 6.0 (mg/1 per day)"1 14 (mg/1 per day)"1 t .002 day .005 day 2 days-2 20 days"2 200 days"2 TABLE 7. Errors in the calculated reaeration coefficient re 700 sulting from errors in parameter measurement, disturbed- equilibrium technique Error in calculated kt, in days" 1 BPS 2 Minimum slope Average slope Maximum slope RMS Maximum RMS Maximum RMS Maximum 600 p 0.27 1.35 0.60 3.0 1.2 6.0 £> .22 2.25 .38 3.8 .60 6.0 Db .30 3.0 .90 9.0 2.1 21 l .004 .01 .04 .10 .40 1.0 500 Figure 15 shows the effect of ignoring a differ ence in the respiration rates for the two levels of DO concentration of the disturbed-equilibrium technique. Reaeration coefficients were calcu lated from equation 151 for various respiration rate 400 differences, and then the calculations were re peated with no respiration rate difference. It is apparent from figure 15 that the effect of ignoring the difference between respiration rates is defi 300 nitely significant. For example, neglecting a res piration rate difference of 1.0 mg/1 per day causes about 12 percent error in the calculated reaeration coefficient, and the error increases rapidly as the difference in respiration rate increases. Either equation 139 or 140 can be used to cal culate k2 from measurements by the disturbed- equilibrium technique when the respiration rates are different for the two levels of DO concentra tion. Equation 140 gives an integral reaeration coefficient, whereas equation 139 gives a reaera tion coefficient based on the average DO deficit for the reach. Figure 16 shows the variation with respiration-rate ratio of /c2 as calculated from equa tion 139 for EPS 2, 4, and 13. A comparison of 0.2 0.4 0.6 0.8 figures 15 and 16 shows that the sensitivity of the RATIO OF RESPIRATION RATE reaeration coefficient calculated from equation 140 to the difference in respiration rates is con FIGURE 16. Variation of calculated reaeration coefficient siderably less than the sensitivity of the reaera with respiration-rate ratio, disturbed-equilibrium technique. tion coefficient calculated from equation 139 to the respiration-rate ratio. a change from 2 to 10 mg/1. From figure 16, how In general, the respiration-rate ratio for a ever, it is apparent that equation 139 cannot give disturbed-equilibrium-technique experiment the correct reaeration coefficient of BPS 2, 4, and should not deviate too much from a value of 1.0. 13 if fl>0.64. Therefore, because the BPS values From figure 7 of Edwards, Owens, and Gibbs (1961), were chosen as representative of conditions in it is estimated that R' = 0.85 for a DO concentra natural streams, the use of equation 140 is tion change from 2 to 4 mg/1, and that R' = 0.65 for preferred. 42 REAERATION IN OPEN-CHANNEL FLOW .From the results of the sensitivity analysis pre not significantly different at the 95-percent level sented in figures 11 through 16 and in table 7, of significance, and weighting the means in inverse several general conclusions are possible. These proportion to their variances yielded a mean ratio conclusions, as follows, are qualitative only, of 0.83. The 95-percent confidence limits for the because of the limited scope of the sensitivity weighted mean were ±0.034, and the most probable analysis. error was 0.011, or about 1.3 percent of the mean 1. Errors in all parameters of the disturbed- ratio. Thus, the value of 0.83 may be regarded as equilibrium technique are significant in the cal an accurate estimate of the mass-transfer coeffi culation of the reaeration coefficient. The most cient ratio for the krypton-oxygen gas pair. significant is the error in the downstream DO deficit, followed by errors in the net photosyn- TABLE 8i. Statistical data concerning measured gas-transfer thetic production parameter, upstream DO deficit, coefficient and traveltime. 2. Neglecting a respiration-rate difference of 1.0 Transfer Range of K2 (days-') coefficient Mean Minimum Maximum Median mg/1 per day causes about 12 percent error in the ratio Minimum Maximum calculated reaeration coefficient; therefore, ~i<7 neglecting the effect of DO concentration on the 0.82 0.76 0.91 0.81 0.04 1.5 12.5 respiration rate may be significant. .846 .789 .909 .846 .045 4.4 68.6 3. The use of equation 140 for disturbed- equilib fTsivoglou, O'Connell, Walter, Godsil, and Logsdon (1965). rium-technique calculations is preferred. JTsivoglou (1967). 4. The square root of the sum of the squares of the RMS errors for the average-slope column of The effect of different conditions on the ratio was investigated extensively by Tsivoglou, table 7 yields an expected relative RMS error of O'Connell, Walter, Godsil, and Logsdon (1965) and 115 percent for a reaeration coefficient of Tsivoglou (1967). They measured mass-transfer 1.0 days-1. coefficient ratios in two different experimental systems for seven different gases for transfer TRACER TECHNIQUE both into and out of the water. They also used a Two major sources of error must be considered range of temperature and turbulence conditions in an error analysis of the tracer technique. The that produced reaeration coefficients comparable first is the accuracy of the value of the ratio of the to the upper range of reaeration coefficients mass-transfer coefficient of the tracer gas to that observed in natural streams. They concluded that of oxygen. The second is the accuracy of the pro the mass-transfer coefficient ratio was not in cedures involved in measuring the mass-transfer fluenced significantly by (1) temperature in the coefficient of the tracer gas in an actual reaera- range from 10° to 30°C, (2) the degree of turbulent tion-coefficient determination. mixing over a wide range of mixing conditions, The value of the mass-transfer coefficient ratio (3) the presence or absence of a broken water sur can cause errors in the calculated reaeration coef face, and (4) the direction of gas transfer. ficient in two ways. First, there are errors inherent In addition to this experimental evidence, some in the ratio because of errors in the laboratory of the conceptual models discussed previously and procedures used to determine the ratio. Second, a theoretical argument presented by Tsivoglou the conditions of turbulence and temperature and (1967) suggest that the transfer coefficient ratio the presence of surfactants and pollutants in the is equal to the ratio of molecular diffusivities. measurement situation may be different from the Thus, this observation also suggests that the conditions under which the ratio was determined, transfer coefficient ratio should be a constant, if and this may result in a different ratio value. differences in the temperature dependences of the The mass-transfer coefficient ratio for the diffusivities are neglected. krypton-oxygen gas pair was determined under a The accuracy of the procedure used to measure variety of laboratory conditions by Tsivoglou, the mass-transfer coefficient of the tracer gas in O'Connell, Walter, Godsil, and Logsdon (1965) and an actual reaeration-coefficient determination by Tsivoglou (1967). The results of these tests are depends on the accuracy of the nuclear counting summarized in table 8. Also shown in table 8 are techniques, the measurement of the flow time, and the maximum and minimum reaeration coeffi sampling errors. The counting techniques used by cients observed in the two series of tests. A statis Tsivoglou (1967) involved splitting each sample tical analysis showed that the mean values were into three parts and determining the number of EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 43 counts of each sample for at least two 20-minute measurements on three different gases at constant periods. Tsivoglou (1967) reported, turbulence conditions in a mixing tank; in two This procedure, involving considerable replication and long cases, four repetitions were made, and the result total counts, provided excellent counting statistics, and a high ing variances were 3.6 percent and 8.6 percent of level of accuracy for river samples whose count rates might be the mean. In one case, involving seven repetitions, in the neighborhood of only twice the background count rate. the variance was 8.5 percent of the mean. Isaacs Flow time in the tracer technique is determined and Gaudy (1968) reported measurements in a from measurements of the concentrations of the "simulated" stream which in essence was a mixing water-soluble fluorescent dye and, hence, should tank with moving walls, used in an attempt to dup be very accurate. Furthermore, the flow time de licate velocity conditions in streams. They reported termined by this procedure should be identical two situations where five repetitions were made in with the flow time for the conservative liquid and which the variance was 6.3 and 6.9 percent of the gas tracers. Errors resulting from sampling should mean. In five situations where three repetitions be small because the only quantity determined were used, the variance was from 1.6 to 4.1 percent from the samples is the ratio of the concentrations of the calculated k2 value. Dobbins' k2 determina of the gas and liquid tracers. Care must be exer tions were based on measurements of the volume cised, however, to insure that the gas tracer is not of gas absorbed, whereas Isaacs and Gaudy's were lost from the samples during handling and made using the azide modification of the Winkler counting. test. If one compares the average variance of about The overall accuracy of the tracer technique is 5 percent of Isaacs and Gaudy's measurements difficult to estimate. A qualitative indication of with the 0.1 percent of Cs reproducibility expected the reproducibility of k2 values determined from in the DO measurements with the Winkler method, the tracer technique can be obtained from the work the comparison indicates that as the complexity of of Tsivoglou and others (1967, 1968). This work the measurement procedure increases, the error included two or three repetitions of k2 measure increases considerably, so that the variance of ments for 14 river reaches. The average deviation flume measurements would be expected to be even from the mean for nine reaches was less than ±10 larger than 5 percent. Furthermore, the compari percent; the average deviation was less than ±6 son suggests that the accuracy of the DO measure percent for six of the reaches; and the maximum ment procedure should not be considered as deviation for any reach was±18.2 percent. Because indicative of the accuracy of the complete deter two days generally elapsed between measurements mination procedure for the reaeration coefficient. on a specific reach, some of the variation could The difference between the reproducibility of have been the result of actual changes in the the Winkler method for DO determination and the reaeration coefficient. Thus, on the basis of these expected reproducibility of flume measurements limited data, errors in reaeration coefficients shows that other factors are important in the determined by the tracer technique may be of the flume measurement procedures. These factors order of ±15 percent of the mean. may be errors in reproducing or measuring the hydraulic conditions describing the flow. In addi LABORATORY MEASUREMENTS tion, another factor not previously considered by researchers is the possibility of a corrosion reac In the laboratory, most of the sources and sinks tion in the flume system. in the DO balance can be eliminated, so that the Many laboratory flumes are constructed from complete DO balance, equation 119, reduces to the metals, and because metals corrode or react with simple equation 133 for a laboratory flume. Hence, oxygen to varying degrees, a DO sink may be laboratory flume measurements of the reaeration present that the researcher has not considered. coefficient should provide a limit on the repro The reaeration coefficient calculated by the pro ducibility that can be expected from the DO cedure illustrated in figure 2 may be incorrect balance technique in general. Unfortunately, the when a corrosion reaction is occurring in the literature does not contain much information con channel. Furthermore, the magnitude of the error cerning reproducibility of reaeration-coefficient has been shown to depend on the relative magni measurements in flumes. It does contain, however, tudes of the reaeration coefficient and the rate some information for mixing-tank studies, an constant for the corrosion reaction and also on the experimental situation considerably simpler than initial DO deficit. (See appendix A.) the flume situation because only one plot of log D Therefore, the first step of any laboratory study versus time is involved. Dobbins (1964a) reported of the reaeration coefficient should probably be 44 REAERATION IN OPEN-CHANNEL FLOW the determination of the significance of possible are well enough developed or have been suffi corrosion reactions together with a determination ciently verified so that they can be used to of the reproducibility of the experimental measure accurately predict reaeration coefficients from the ments. The reproducibility of the measurement mean hydraulic parameters. Because of this, one procedure can be determined by replication of the must rely for the prediction of reaeration coeffi experimental runs, and the significance of possible cients on empirical regression equations based on corrosion reactions can be determined by the pro experimental data. The mean parameters com cedures outlined in appendix A. monly available for use as the independent vari The presence of an oxygen-consuming reaction ables in such equations are width, depth, slope, in the laboratory equipment shows one way in and velocity. Another parameter which has been which systematic errors can be introduced into used as an independent variable in &2 prediction flume measurements of the reaeration coefficient. equations is the longitudinal-dispersion coeffi This might explain one of the puzzling facts cient, Dx, but because of a scarcity of experimental present in the literature on reaeration in flumes. Dx data, this parameter has not been used herein. This puzzle is the inconsistency between predic The purpose of this section is to (1) present a tion equations obtained from data taken in two summary of the known experimental data con different but supposedly hydraulically equivalent cerning the reaeration coefficient in open-channel flumes. The work of Krenkel (1960) and Thackston flows, (2) evaluate the presently available equa (1966) illustrates this inconsistency. The flumes tions for prediction of reaeration coefficients, and used by these investigators were not significantly (3) present what we consider to be the most rea different hydraulically, except that Krenkel's was sonable prediction equations that can be obtained 1.0 foot wide and Thackston's was 2.0 feet wide. by regression analysis from the available data. The results, however, obtained from their studies over similar ranges of hydraulic variables may be REAERATION-COEFFICIENT DATA SETS given as (Thackston, 1966): The reaeration-coefficient data sets available (Krenkel) &2 =0.000469^ (152) H in the literature can be judged against two basic and criteria. The first of these is the accuracy with which the basic data were measured and the care (Thackston) k2 = 0.000215^-. (153) H fulness with which the coefficients were calcu The constants in the equations differ by a factor lated. The measurement techniques used must be of 2.2. considered, and if the DO balance technique or The reasons for the discrepancy between these the disturbed-equilibrium technique is used, the equations is not known. However, it might be due effects of all neglected parameters must be to the systematic error introduced by an oxygen- evaluated. consuming reaction proceeding in the channel of The second criterion is the completeness of the one or the other of the experimental flumes. data in the set. Specifically, this concerns the Another possibility for the discrepancy is the number of independent variables reported and the failure to include some pertinent hydraulic vari ranges of the independent and dependent able in the equation for example, the width-depth variables. ratio to account for the different widths. However, From the points of view of scale and of data- Thackston and Krenkel (1969a) wrote: "It is possi collection methods, the reaeration-coefficient data ble that some sort of wall effect is involved, inas fall into two general classes. These classes are data much as the values of the width-to-depth ratio from laboratory flumes and data from natural were essentially the same in both studies***." streams. The flume data were in general collected Additional experimental work will be necessary to at flow depths one to two orders of magnitude lower than the field-data flow depths, and at slopes determine the effect of the width-depth ratio on the reaeration coefficient. one to two orders of magnitude higher. In terms of scaling, the data of Owens, Edwards, and Gibbs (1964) to some extent bridge the gap between the PREDICTION OF THE REAERATION rest of the field data and the flume data. COEFFICIENT FIELD DATA None of the presently available models of the The best available data were presented by oxygen-absorption process in open-channel flows Churchill, Elmore, and Buckingham (1962). The PREDICTION OF THE REAERATION COEFFICIENT 45 DO balance technique was used, for the most part variation exceeded the variation in DO between under such idealized conditions, and with such the ends of the test reach. This necessitated taking attention to detail that a high degree of accuracy DO samples at several points in the cross section is expected in the reaeration coefficients. These and using the average of the measurements as the measurements were made in river reaches down cross section DO concentration. For most cross stream from deep storage impoundments. A DO sections, seven samples were used. deficit was present because the water released was For all but seven of the 30 reaches studied, low in DO concentration as a result of prolonged Churchill, Elmore, and Buckingham (1962) believed storage under thermally stratified conditions. For that except for k2, all of the parameters in the DO the reservoirs studied, the BOD of the water was balance equation were negligible. For the seven satisfied during storage, so that it was zero at reaches they felt that correction was necessary release time. for the photosynthesis and respiration of attached Over the study period on each reach, the stream plants. The corrections were made using laboratory discharge was held constant by keeping the dis measurements of the photosynthesis and respira charge constant at the upstream hydroelectric tion rates of plants attached to rocks removed plant. The discharge was determined at local from the river bottom. For reasons discussed pre stream gaging stations, and the mean velocity was viously, and as pointed out by Hull and DeFilippi determined by dividing the discharge by the aver (1963), this type of correction is dubious, so that age cross-sectional area. The flow time was deter these seven values are less reliable than the other mined by dividing the length of the test reach by values reported. However, the problem was recog the mean velocity. Because the test reaches were nized, and an attempt was made to compensate for carefully selected for uniformity, and because it. In addition, the corrections were for less than cross sections were measured at intervals roughly 25 percent of the calculated coefficients, which, as equivalent to the stream width, the accuracy of will be shown later, is of the order of magnitude this method of determining the flow time is of the scatter around the best-fit regression equa probably consistent with the accuracy of the other tions obtainable from the field data. steps in the DO balance procedure used for the The data obtained by Churchill, Elmore, and calculation of k2. Buckingham (1962) are listed in table 18 of appen The Winkler method, with a special end-point dix B. In terms of the commonly measured mean determination procedure which produced a stan hydraulic parameters, this data set is considered dard deviation of only 0.007 mg/1, was used to complete, because all the variables U, H, S, and W determine the DO concentrations. The measured are reported. The ranges of the measured vari DO concentrations showed no variation in the ver ables are listed in table 9. These data are repre tical, but for many cross sections, the lateral DO sentative of moderate-size streams. TABLE 9. Ranges of experimental data from the literature *2 H U W 5x10" (days- 1) Reference (feet) (fps) (feet) (ft per ft) Maximum Minimum Maximum Minimum Maximum Minimum Maximum Minimum Maximum Minimum Churchill and others (1962)...... 5.558 0.225 11.41 2.12 5.00 1.85 667 90 23.514 1.2571 Owens and others (1964)...... 49.17 .31 2.44 .39 1.83 .13 39.8 11.0 106.0 1.56 Gameson and 57.7 34.3 .42 .34 .73 1.12 5.7 O'Connor and .92 .14 24.2 4.0 4.20 .19 1.97 .68 Tsivoglou and 3.373 .646 3.11 1.16 .636 .286 110 58 6.5' 52.50 5.78 .232 .052 2.320 .365 2.0 201.2 115.3 10.63 .200 .080 2.140 .243 1.0 239.9 7.5 Negulescu and 18.70 8.64 .492 .164 1.902 .656 .66 tStreeter and Phelps (1925) data only. The reaeration-coefficient study presented that either considerable weed growth or mild by Owens, Edwards, and Gibbs (1964) was per organic pollution was present. However, they did formed under conditions much less ideal than not state whether phytoplankton were present. those prevailing for the study by Churchill, The presence of plants is an important factor, Elmore, and Buckingham (1962). In 11 of the 32 because the disturbed-equilibrium technique was cases presented, Owens, Edwards, and Gibbs noted used to calculate the reaeration coefficients, and 46 REAERATION IN OPEN-CHANNEL FLOW different DO consumption rates of the plants at in that width values are not given. The Streeter- the two DO levels used in this test could have Phelps data typify moderate-size rivers on gentle significantly influenced the calculated reaeration slopes. Table 22 of appendix B lists data collected coefficients. by O'Connor and Dobbins from sources other than In the study by Owens, Edwards, and Gibbs Streeter and Phelps. Stream width is not included (1964), the discharge was determined by a dilution in this data set, nor in three cases is stream slope. technique. The flow time was determined from the Because the sources of the individual data points time of passage of the centroid of the tracer pulse vary, the ranges of the variables in this data set through the reach. The reliability of the reported have not been included in table 9. depths cannot be determined because the fre The only reaeration-coefficient data for streams quency of cross-sectioning was not reported. DO that have been determined using the tracer tech measurements were made using the Alsterberg nique are those given by Tsivoglou and others modification of the Winkler method. (1967, 1968). The data are rather limited in scope because the prime purpose of the study was not k-2 The paper by Owens, Edwards, and Gibbs (1964) determination, but verification of the measure contains two sets of data. The first set consists of ment technique itself. the 32 kz values determined by these workers. In the study by Tsivoglou and others (1967,1968), Eleven of the k2 values were corrected for oxygen the traveltime was determined from the time of consumption by the bottom mud or for oxygen con passage of the dye peak between the sampling sumption by attached plants. This data set does stations. The stream was cross-sectioned every not include the stream width, but this parameter 1000 feet, but as Tsivoglou stated (1967, p. 69), can be calculated. The second data set, consisting "The stream was characterized by a series of riffles of six kz values, was taken from Gameson, Trues- and pools and an associated considerable range of dale, and Downing (1955). In this data set, neither velocities and depths," so the average depth mea the width nor slope values are given, but width surements are questionable. The nuclear counting values can be calculated. The two data sets are techniques used were highly reproducible, and given in tables 19 and 20 in appendix B, and the sufficient replications were used so that even at ranges of the variables are given in table 9. The low krypton concentrations the tracer-gas to hydraulic parameters would be considered typical liquid-concentration ratio should have been very for steep, shallow streams. accurate. The tracers were introduced at the A summary of the pre-1958 field data was pre centerline of the injection cross-section and sented by O'Connor and Dobbins (1958). The data sampled from the flow at the centerline of each consist for the most part of those of Streeter and sampling cross section. In light of the results of Phelps (1925), screened by O'Connor and Dobbins the study by Churchill, Elmore, and Buckingham (1958) to minimize the effects of algae and sludge (1962), this sampling procedure would seem to be deposits and modified so that the reported depth questionable if the k? values obtained are to be values approximate the true stream depth rather representative of the entire cross section. Despite than the distance above mean low water. the questionable sampling technique, the repro- The modified Streeter and Phelps (1925) data ducibility of calculated k-2 values was good, with a must be viewed with some caution, because they maximum deviation from the mean of 18 percent were obtained from a river containing organic and with nine of 14 values having an average pollution and probably phytoplankton, and kz deviation from the mean of less than 10 percent. values were computed from the Streeter-Phelps The experimental data obtained by Tsivoglou form of the DO balance equation, which ignores all and others (1967, 1968) are listed in table 23, but the suspended and dissolved BOD oxidation appendix B. The channel slope is not given, and the and the surface reaeration. In many of the test width value was obtained by the present authors reaches the other terms in the DO balance equa by dividing the depth-velocity product into 82.5 tion probably would have influenced the calcu cf s, the average discharge for the time period. The lated &2 values, had they been available for use in ranges of the individual variables are listed in a more general form of the DO balance equation. table 9. The data of Streeter and Phelps (1925) as modi FLUME DATA fied by O'Connor and Dobbins (1958) are reproduced in table 21, appendix B. The ranges of the vari In all the laboratory studies reported herein, ables are given in table 9. The data set is incomplete the flumes used were the recirculating type, and PREDICTION OF THE REAERATION COEFFICIENT 47 the water in the flume was deaerated with sodium EVALUATION OF REAERATION-COEFFICIENT PREDICTION EQUATIONS sulfite before the start of the experimental measurements. The conceptual models discussed earlier have, in Thackston (1966) used a flume 60 feet long and 2 general, not been carried to the point where they feet wide, and three different bottom roughnesses. can be used by themselves to predict reaeration All DO measurements were made using the Azide coefficients from the mean stream parameters of Winkler method, and the traveltime measurements width, depth, slope, and velocity. The most useful were made using dissolved sodium chloride as a reaeration coefficient prediction equations are, tracer. The sampling stations in the flume were 42 therefore, those obtained from the semiempirical feet apart; at each station, 10 samples were taken models. These equations have coefficients derived at 2-minute intervals. With at least two of the in some fashion from field data, or derived entirely bottom roughnesses used, an oxygen-consuming by least-squares regression analysis from field reaction may have influenced the calculated data. Of course, such prediction equations should reaeration coefficients. See appendix A for a dis be used with caution in situations for which the cussion of this problem. mean stream parameters are outside the range of data from which the prediction equations were The data of Thackston (1966), also reported by derived. Thackston and Krenkel (1969a), are given in table In this section, we will, for the prediction equa 24, appendix B. The data set is complete, in that all tions available in the literature, evaluate their the mean hydraulic parameters commonly mea performance over the entire range of available sured are available. The ranges in the values of data. We will also assess on a common basis the the variables are listed in table 9. performance of the various equations when The flume used by Krenkel (1960) was 60 feet applied to the data sets from which the equations long and 1 foot wide, and the bottom roughness were derived. consisted of expanded metal mesh. In terms of The basis chosen for comparison of the perfor experimental methods and techniques, and in mance of the various prediction equations is the terms of quality of measurements, the study was standard error of estimate. For a prediction equa comparable to that by Thackston (1966). Again, tion yielding an estimated (k^)e for which the cal oxidation reactions within the flume channel may culated reaeration coefficient is (k2\, the standard have influenced the calculated reaeration coeffi error of estimate, Es, is cients. See appendix A. The data of Krenkel (1960), which were also used ± n '-, (154) by Krenkel and Orlob (1963), are given in table 25, where n is the number of kz values used. Another appendix B. Again, all the commonly measured definition of the standard error of estimate which mean hydraulic parameters are listed, so the data is convenient to use in conjunction with least- set is considered to be complete. The ranges in squares prediction equations derived from the values of the variables are reported in table 9. logarithms of k-2 is The flume used by Negulescu and Rojanski (1969) was 66 feet long and 0.66 foot wide and had a 40-foot test section. The bed roughness consisted ~ n ' <155> of 1-mm sand grains cemented to the flume floor. The DO measurements were made using the The term ESL can be expressed as a percent error, Winkler method. A 20-minute measuring period EP, defined as was used, and about 10 samples were collected at each sampling station. where EP is the percent standard error of estimate in terms of (A:2)e. The term EP will always be between Negulescu and Rojanski (1969) tabulated only 0 and 100 percent. When compared with Es as a per eight of the 18 values they determined. These are cent, it is a conservative estimate of the percent listed in table 26, appendix B. Values for slope and error of the prediction equation. It must be empha temperature were not included in the original sized that EP is not equal to the percent standard reference, so a temperature of 20°C was assumed. error of estimate that would be calculated from The ranges in values of the variables are listed the regression of k2 (rather than the transformed in table 9. data log kz). 48 REAERATION IN OPEN-CHANNEL FLOW In table 10 are reported the quantities Es when it is realized that none of the values of Es and EP for 13 of the equations discussed previously. listed in column 10 are less than 16 per day, a con For each of the equations, Es and EP have been siderable magnitude when compared with the kz computed first, for the data sets used by the values of less than 5 per day usually determined original authors, and second, for all of the data for natural streams. sets discussed in the previous section which are The reason for the large Es values given in the sufficiently complete for use in the particular 10th column under "Total applicable data" in table equation being evaluated. Table 10 also contains, 10 is further emphasized by reference to figure 17. for both groups of data, the number of data points In this figure, all the equations of table 10 have and the ranges of the individual variables. been plotted for an assumed slope of 0.0001 feet On the basis of the percent standard error of per foot and a mean velocity of 1.0 fps. The square estimate, the equations which seem to fit best ruled on the graph delimits the range of experi their original data sets are equation 96, with a mental data used. For the smaller depths at the standard error of estimate of 15 percent, and equa extreme left edge of the depth range, -there is tion 101, with a standard error of estimate of 25 sometimes more than two orders of magnitude percent. These equations were presented, respec difference between a given pair of prediction equa tively, by Krenkel (1960) and by Thackston (1966). tions. At the smaller depth values, the experimen The relatively good fit of these equations is not tal data fall into a range representative of the surprising, because they were obtained from bottom six prediction equations shown in the laboratory studies in which the interfering fac figure, so that large standard errors of estimate tors and the hydraulic variables could be much would be expected for the prediction equations better controlled than under the field conditions which yield large k-z values at shallow depths. represented by most of the other equations. In general, two different sets of equations are Again on the basis of the percent standard error apparent in figure 17: those derived from flume of estimate, the equations which best fit the entire data, which have slopes on the order of -1/2, and range of available data are equation 61, with a those derived from field data, which have slopes standard error of estimate of 43 percent; equations on the order of -3/2. It becomes apparent that 101 and 102, with standard errors of estimate of 45 there is a basic difference between the data sets percent, and equation 54, with a standard error of obtained in the two different situations. Because estimate of 46 percent. Equation 61, given by of this, then the use of equations derived from Dobbins (1964b), was derived from a consideration flume data for the prediction of reaeration coeffi of two sets of field data and one set of flume data; cients in natural streams is dubious. This matter equation 101, presented by Thackston (1966), was is discussed further in the following sections. derived from flume data; equation 102, again from Thackston (1966), was derived from flume data, DEVELOPMENT OF A REAERATION-COEFFICIENT along with two sets of field data; equation 54, pre PREDICTION EQUATION sented by O'Connor and Dobbins (1958), was derived without resorting to any experimental data. As might be expected, two of the best-fit As discussed previously, none of the available equations were derived from data which covered prediction equations can be considered applicable essentially the same range as the data in to the entire range of available data. Assuming the appendix B. data to be bias-free, there are two possible reasons In the 10th column under "Total applicable data" for this. First, none of the available equations were in table 10, it is seen that five of the equations, derived from a wide enough data range, and when applied to the entire available data range, second, and more likely, some basic difference have standard errors of estimate greater than 100 exists between the two data sets which precludes per day. With the exception of equations 56 and 89, writing an equation which is applicable to the these five equations were derived from river data entire range of data. In this section, these alterna and give much too high k2 predictions at the shallow tives are investigated using regression-analysis depths used in the flume studies. This fact illus procedures. The data used are those discussed trates the caution which must be used when apply previously. The criterion for judging the best equa ing kz prediction equations to situations in which tion to use is the standard error of estimate of the the mean flow parameters are well outside the prediction equation as obtained from the 62 field range of the data from which the prediction equa and (or) 110 flume data sets which contain all the tions were obtained. Further caution is indicated variables (/, //, W, and 5. PREDICTION OF THE REAERATION COEFFICIENT 49 ?D i I CO CD i I 1C if- co ^r ic co ic T-I HNI4I t- oo oo to to to i-8 >- . « r t- 0 0 IMTfitOOCOOOOSCO 00 IM CO 1C T-H 1C tO O O O OS ** 1C O T-I .2 as £ "S £ 'S LI) >> CO T-I IM O> T-I IM IM ^ T-I t- m co IM 05"«» »S .* CO O O O O O O c X X > XXX o o o o o s i g t- e- t- t- t> O> IM IM IM IM IM SO 5"8J O O O O O O 08 ^_- X T-4 T-4 T-* T-4 T-4 TH +J 09 XX > 09 < XXX T-I O C5 O O O T3 ^ O ** *: 0 T3 IM IM O) IM IM IM 3 CU 09 _C 0 3 O O O O O O C p p p p p p "a. g o M S Q. *,» a> o o oooooooo o o M § X tO IM IM OIMIMIMOOOO O O "3 09 09 S CO IM IM «IMC<|IMCOCOCOCO CO CO "S £ fi CO.* ^CO^^^COCOCOCOCOCO Q « s o «+-* ^^ x t- o c2 O O O C o o o c o o 09 co p c5 ppOC o o o c o o S 1-H 1C If3 icicinicicicicic ic m c ^4 ^» Tr CM ** * -^ (N IM IM IN IM IM 4 p T-4 rt T- pope O O "M i o * x X O> CO CO COCOCOCOWCOCOCO CO CO 3 09 IM 1C U 1C 10 1C U- 00 T-I T-2 S S ^ "" T-4 T-4 |«.*.. 3 co to o coooococococo co co J-slt g 3 T-I IM !M!M!M I1NI4I ^t -^t CO t-T-IIMCOIMCOCOCO IM (M -,- r ^« csi rr t-cotoocgtooic ^r co CO <* t oo in os c > 1C ** O If5 * tO 5 a £ "o to ^i tjj jj o t (M t- OS If5 1C 1C O T^ C/2 T3 a) * C ^ o o o o o o o C X X < X X > X £ S o o o o o o to SH OJ IM Cv) t- tO 0-1 1-1 c o a. 7 7 777 7 O O i s o o c5 O 09 ij X ^ X X X X X >< X 1 o o o o ** ** ic 2 S) ** to ** <* o o co 3 g T-4 CO G^ IM IM Oj C*J "8 O. 01 2 O O 00 OOO'f^flMO^f^ 1 00 to « c ^c O OS O OSOOOT-4COCOT- ; p T-< s>. g + » % OJ g T-4 IM « -i a. 3 5tJ O T3 o cS O O CO C5 T-I Tf T-I T-I O T-I C) IM CVI O C S O CO CO 00 t> tO O ^ S IM -^ Ift ^f in c*j o 1 1C OS O If5 m oo f 1 T-40000 -*J4in T-4T-4 OJ OS M G cS j^ C CO IM C^0 GO 00 O U3 O IM 00 C5 00 CO C T3-51) Si'S'-S T-I IM evj CO 1C ^* C5 CO CO tO O3 1C IM I-S 8.3-1*8 -4 T-H v c|& g * s -£ S T * >> N £* r 7 7 * ft ^j|» 0 Ur ^ & o a Sp" ? a|5 I ? ? 7 => t3 X O QP 1 S + 1 » * ? j 0 $15- a; <~!^ 5j oo : - ^ji J^* ,s ' C Ij ' 0 0 » C 0. M S || |l i oi "x "| -= p : S s 0) rt " ^ TJ to ^ Ci ^ ^ 00 "^ O> 0 C C Cl- ft 03 T-4 ^ ^^ S ffi OS O> ?j ^ s = 3=S- § 1 «£ II * S 1 g S I i X § ll 0) § * w i» 0:3 f^ aJ= Sw « « St^ _g-g ^ o os O euB5 bII ;! 1Q fcUll w H 3 o M'+* T, tO -4 OS to T-I c^ 00 OS O CO Tti to 1C 1C tO OOOSOOOO 1000 100 10 ^ i o.i o.oi - Mean velocity =1.0 feet per second Slope = 0.0001 o.ooi 0.01 100 DEPTH, IN FEET FIGURE 17. Reaeration coefficient as a function of depth, from prediction equations in table 10. PREDICTION OF THE REAERATION COEFFICIENT 51 REGRESSION ANALYSIS obtained which are entirely satisfactory within the range of data used in deriving them. These equa In general, regression analysis consists of com tions could, however, yield highly erroneous pre puting the coefficients for an assumed equation dictions if applied outside the data range. Second, form so that the sum of the squares of the de even though the model is correct, the prediction partures of the experimental data points from the equation will be incorrect if the data are incorrect curve given by the equation is a minimum. For or incomplete. Finally, in a regression analysis two example, if for the data set Yi = G (Xu, X2i, ..., Xni), types of significance must be considered: physical the linear model and statistical. For example, a variable may be YI ei Sloislis\.iiTSi2-A-2ii''*iStn-A-niA 4- A. Y -4- A Y -4- 4- A Y \±O(~\ x.H\ I) highly significant in the physical process yet have is used to obtain estimates Ye of the dependent little significance in the regression equation be variable Y, then the ,4,, /=0,... n, are chosen so that cause the range of this particular variable was not the quantity wide enough in the data group chosen. Further A" more, it is possible to have a high statistical sig j\N' L-iSF<2 ^wT L[Y * ft Y-1* ll 2 CITS'1V-L'-'Oy nificance in the regression equation and yet to i = 1 have assumed the wrong physical relation for that is minimized. In equation 158, N' is the number of particular variable in the model. For these reasons, data sets used. it is best before beginning a regression analysis The best measure of the goodness of fit of a to have a reliable physical model available. Un regression equation to a particular data group is fortunately, as will be emphasized later, there are the standard error of estimate Es, as defined by no such models presently available for the reaera- equation 154. A parameter which can be used to tion coefficient in terms of the mean values of the determine the relative significance of a particular flow parameters which will accurately predict variable to a given regression equation is the reaeration coefficients for the entire range of partial correlation coefficient. The partial correla available data. tion coefficient is the correlation coefficient In this report, most of the regression analyses between the dependent variable and the chosen were performed using a general linear model and independent variable, all other independent vari the logarithms of individual variables. The general ables being held constant. Its value ranges model assumed was between 1 and +1, and the larger its magnitude, the more significant is the chosen variable in the (160) regression equation. The standard error of esti which is more commonly expressed as mate and the partial correlation coefficient are (161) discussed at much greater length in standard statistical-methods texts such as that by Snedecor In several cases, one or more of the At were and Cochran (1967). assumed to be zero, and the effect on the standard The significance of a particular regression coef error of estimate of the prediction equation was ficient, At, may be tested using the fact that the investigated. The coefficients were computed distribution of the quantity using the regression program BMD03R of the Uni p _(Aj DJ) versity of California, Los Angeles, Health Sciences K i~ ^^ (159) obi Computing Facility (Dixon, 1965). is Student-r distribution with N'-n degrees of The results of a linear least-squares regression freedom. In equation 159, fi, is the population analysis on an equation of the form of 160 are regression coefficient, and 5&i is the standard error generally slightly different from the results of a of the regression coefficient At. (See Snedecor and least-squares regression analysis of an equation Cochran (1967) for more detail.) of the form of equation 161, because the distri In conducting a regression analysis, the limita bution of the variance of the logarithms of kz from tions of the method must be remembered. First, equation 160 will be different from the distribution the model to which the data are fit is preselected of the variance of kz from equation 161. If the so that there is freedom only to adjust the coeffi analysis is conducted on an equation of the form cients in the model. If the model is incomplete, a of 160 and the result is converted to the form of particular variable may appear to be of negligible equation 161, the average width of the scatter band significance, when in reality it may be of great about the prediction equation will increase directly significance. On the other hand, even though the in proportion with k-2, whereas if the analysis is model is incomplete, prediction equations might be conducted directly using equation 161, the scatter 52 REAERATION IN OPEN-CHANNEL FLOW band width will be a constant, independent of k2. In this study, it was decided to use an equation of £ S3 2 the type of 160 and to accept deviation from the pure least-squares regression analysis procedures because of the relative simplicity of the analytical rt rt O procedures involved in this type of analysis as com pared with a least-squares analysis based on equation 161. IH IH oo A prediction equation of the type of equation 161 was chosen for comparison primarily because this type of k2 prediction equation has been used frequently in the past. Furthermore, most of the conceptual models available can be reduced, at least for natural streams, to this type of equation. A number of equations of different forms from equation 161 were tried with negligible success in reducing the standard error of estimate. RESULTS OF REGRESSION ANALYSIS The available experimental data may be divided into two general groups: first, those from natural streams, and second, those from laboratory flumes. Basic data groups used in the analysis were the data sets for which all of the variables k2, H, U, W, and S were available. These consisted of the 62 I I 1 5>s s, field data sets from Churchill, Elmore, and Buck O O ingham (1962) and Owens, Edwards, and Gibbs TABLE 13. Dependence of turbulent intensity on the mean hydraulic parameters Partial correlation coefficients Number Equation ESL of data Type of data H U W S points 21 jpo.43» 50.399 O.Q85 0.302 0.560 0.115 20 Flume, rigid, //0. 396 50.50:1 .779 .700 .117 20 Do. f /0.143 IfO 798 eo.544 1 nnn <-> " j -.390 .121 .527 .590 .146 27 Flume, alluvial. 0.890 //O 158 [70.422 50.313 .219 .315 .381 .168 27 Do. 23.750 //°-°» [/°-764 ff0.75751.183 .234 .718 .490 .516 .087 43 Field. 0.121 //0.042 [71.050 50.090 .176 .871 .271 .098 43 Do. tests comparing the exponents on the variables partial correlation coefficients further indicate in the two flume situations with those of the field that U in the flume is less dependent on depth than situation of table 13 are listed in table 14. These in the field. The difference between the two situa Student-? values show that at least one exponent tions is, however, best illustrated by the Student-? in each flume equation differs from the corre test. Comparing the exponents of the third equa sponding exponent in the field equation at signifi tion with those of the first in column one of table cance levels less than 0.005. It must be concluded 15, the Student-? values for W. RH, and S are 1.65, that the flume regression equations for the tur -5.02, and -9.94, whereas in the second and fourth bulent intensity at one-tenth the flow depth below equations, for RH and 5, the Student-? values are the water surface are significantly different from -8.05 and -10.58. This indicates that the predic the field equation. tion equations for the two situations are different at some significance level less than 0.005. TABLE 14. Results of the Student-t test of the exponents in Possible reasons for the difference between the the flume equations and in the field equations of table 13 regression equations for flume and field reaeration Student-t values Equation H V W s coefficient data are (1) differences in the basic Flume, rigid...... 0.30 1.75 -0.89 5.16 physical processes, or misinterpretation of them, Do...... 79 2.33 1.54 Flume, alluvial...... 2.13 2.50 .15 4.02 (2) statistical quirks, and (3) deficiencies or bias Do...... 79 2.33 1.54 in the available data. Possible physical differences between the two As a final example of the difference between systems are easy to visualize. For example, the regression equations for flume and field data, con straight, uniform flume flow may have entirely sider equations for predicting mean velocity from different turbulent transfer properties than the W, //, and S obtained from the same data as the tortuous, nonuniform field stream flows. R. S. k2 prediction equations of table 11. These equations McQuivey (written commun., 1970) indicated that along with the partial correlation coefficients and the turbulence profile in a 200-foot-long by 8-foot- the standard errors of estimate are listed in wide flume at depths of about 1 foot does not fully table 15. develop at any point in the flume. This indicates some question about the similarity between the TABLE 15. Prediction equations for mean velocity, obtained turbulent transfer properties near the surface of from the data used in table 11 field and flume flows. In addition, the high partial Partial correlation coefficient between /c2 and W in the Equation correlation ESL Ep Type [/ = coefficients (percent) of flume as compared with u and U indicates a W RH S possible secondary circulation effect on k2 which is 37.55 ffo.187^.168 50.742 0.256 0.676 0.861 0.166 31.8 Field. not present in the other phenomena. If such an 98.80 /? > "" 5°- 4 .937 .891 .170 32.4 Do. effect is present, simple WjH scaling may not 29.82 fp0.33Sfl^.608 50 .423 .340 .468 .755 .138 27.3 Flume. account for it. As a further example, the comments 25.20 R^> 54050.400 .409 .721 .146 28.6 Do. of Haddock (1969) and Colby (1964) quoted pre viously could be applied equally as well to reaera The partial correlation coefficients and the small tion as to sediment transport. That is, the different change in standard error of estimate on omitting constraints on the two physical systems might pre W from the equations indicate considerably less vent similar mass-transfer regimes from develop dependency of U than kz on this parameter. The ing at the water surface. Finally, the difference PREDICTION OF THE REAERATION COEFFICIENT 55 between the two sets of equations may be due to a because in none of the studies reported, except basic misunderstanding of the turbulent-mass- possibly that of Churchill, Elmore, and Bucking transfer process near the surface that is, it may ham (1962), have sedimentation, bottom deposits, be due to a lack of knowledge of the pertinent and photosynthesis been accounted for adequately. scaling parameters, or to the fact that the correct Second, in the two sets of flume data, there is a parameters have not been measured. possibility that one or both of the data sets were in Certain quirks are inherent in regression anal fluenced by an oxygen-consuming reaction in the ysis which might cause field and flume regression flume channel, and this would have a tendency to equations for &2 to differ. In the first place, the bias the resulting data. (See appendix A.) differences between the two systems cause the In addition to the differences pointed out above experimental variables to be selected differently between regression equations derived from dif that is, in the flume, slope and discharge vary and ferent data sets, it can also be shown that the determine velocity, whereas in the field, depth and regression equations (listed in table 11) for the field discharge vary to determine velocity. This differ data are significantly different from most ence in selection of "independent" variables for comparable equations found in the literature. analysis can be reflected in the regression equa The Student-r test was used to compare expo tions, as can the differences in the adjustments of nents on various variables of the equations in the so-called independent variables to each other. table 11 with the corresponding exponents from In the second place, if a wide enough range of the equations in the literature, and the results are data is not available for each of the independent presented in table 16. The Student-f values show variables, the significance of the variable may be that all but the two equations of Owens, Edwards, missed in the regression analysis, or the variable and Gibbs (1964) have at least one exponent which may be represented incorrectly in the regression differs from the exponent in the corresponding equation. In this sense, dependency of the flume field equation at a level less than the 0.005 signifi equations on W is particularly questionable be cance level. Because the second equation of Owens, cause only two values were available in the Edwards, and Gibbs (1964) was derived, with the literature. exception of the six data sets of Gameson, Trues- Secondly, if the form of the regression equation dale, and Downing (1955), from the same data as chosen is incorrect, the equations for the two sys the corresponding field regression equation, it is tems of different scale might be expected to differ not surprising that these equations do not differ radically. This is, in a different sense, a statement significantly. of the scaling problem mentioned above. The previous discussion suggests at least two Finally, no regression equation is better than the reasons for pursuing the search for a new physical data used in its derivation, because if the data con model describing reaeration. The first is the dif tain errors or bias, this will be reflected in the ference observed between the regression equations prediction equations. In the field data, especially, for the field and flume situations. It would seem the accuracy of the data can be questioned. First, that equations describing the physical process of TABLE 16. Results of the Student-t test of exponents in the regression equations of this study and corresponding equations in the literature Regression equations Comparable equations Student-fo.oos (this study) in the literature 5 (percent) Source // t/ /yO.413 CO. 273 24.55(t/S£)°-*>8 -3.89 0.04 (Chezy C>17) -5.60 2.98 op K O'Connor and " ' Dobbins (1958). 5.026 {/»-«»//- '673 (7-=20°C) Churchill and -2.95 -5.04 " " others(1962). n in v" 6" 10.90 L"> -73//-1.75 Owens and 9 - 59 //1.665 " others (1964). 9.41 (/°«//- 185 (7"=20°C) -.23 -.07 ...... Do. f A0.85 Negulescu and -15.58 -3.01 (^) ' Rojanski (1969). (Chezy C< 17) -6.00 ...... K qc qq Q O'Connor and Dobbins (1958). o 18.58 Vff(j^)"2 -3.64 ...... 1.92 ...... Thackston (1966). {Equations by Fortescue and Pearson (1967) and Isaacs and Gaudy (1968) are also of this form. 56 REAERATION IN OPEN-CHANNEL FLOW reaeration should be applicable no matter what the widest range of field data presently available, the scale of the system. The second is the fact that this equation is probably the best available for none of the available theoretical models can be prediction of reaeration coefficients for natural related to the regression equation for field data. streams. This is because either the theoretical model is not EFFECT OF DETERGENTS AND WIND ON THE expressed in terms of the mean hydraulic param REAERATION COEFFICIENT eters, or because for the ones which are expressed in this fashion, it has been shown that the Up to this point, the reaeration process has been exponents in the field regression equation differ considered as occurring under ideal conditions significantly from those in the theoretical models. that is, in relatively pure water with no wind Furthermore, the commonly measured mean hy effects. The literature, however, provides evidence draulic parameters U, H, W, and S may not be that the presence of sewage and surfactants in the sufficient to model the phenomenon. Perhaps some water tends to reduce KL, the liquid-film coefficient other parameters must also be measured, such as for gas transfer, compared with the pure-water those characteristic of the near-surface tur value, whereas winds tend to increase 'KL with bulence, or of the velocity in the air above the flow. respect to the value existing for quiescent air con Using all the available data and using the stan ditions. Recall that KL = K2H, where K* is the dard error of estimate of logarithms as the basis reaeration coefficient and H is the mean depth of for judgment, one can see from table 11 that the flow. The results of some of these studies are sum best prediction equation for reaeration coefficients marized in the following paragraphs. of natural streams is Mancy and Okun (1965) discussed two theories r/0.413 C0.273 concerning the effect of surfactants on the rate *2 = 46.05 ^Jog . (163) of oxygen absorption. On the one hand, they cited However, considering the percentage variation four references suggesting that surfactants (table 11), the equation reduce the absorption rate only by decreasing the r/0.674 hydrodynamic activity at the interface. On the fe = 9-&977T56 <164> other hand, they cited six references suggesting has a percentage variation only 1.1 percent greater that surfactants reduce the absorption rate both than that for equation 163. Considering the diffi by increasing the diffusional resistance at the culty and expense of measuring accurately the interface and by decreasing the hydrodynamic slope of a field stream (Yao, 1969), an equation that activity at the interface. does not contain the slope is preferred for routine Eckenfelder, Raymond, and Lauria (1956) hy use as a prediction equation for k2 in natural pothesized that surfactants would reduce KL in streams. two ways: first, by increasing the diffusional In appendix B, in addition to the field data used resistance through a reduction in the surface ten in deriving equation 164, there are other field data sion, and second, by decreasing the surface which report only U and //. From the 121 sets of renewal rate through an increase in viscosity. field data which contain U and //, the least-squares Metzger (1968) studied the absorption of helium regression equation for k2 is and nitrogen under identical mixing conditions in r/0.607 a stirred tank containing 0.5 mg/1 of ABS (alkyl ^ re^ISSr (165) benzene sulfonate), a surfactant. He found that Equation 165 has a standard error of estimate of the film thickness and surface-renewal param logs of 0.203 (37.5 percent). The Student-/ test was eters of the Dobbins (1956) film-penetration model used to compare the exponents on U and //; com were larger and smaller, respectively, than the paring equation 165 with equation 164 (119 degrees values obtained for the same mixing conditions of freedom) for U and //, the Student-/ values are with distilled water. Downing, Melbourne, and -1.42 and 4.22, whereas comparing equation 164 Bruce (1957) studied the effects of household and 165 (60 degrees of freedom) gives Student-/ detergents and sewage on oxygen absorption in a values of 1.14 and -2.70. Thus equations 164 and stirred tank and in a wave tank. They found that 165 have one exponent which differs from its mixed household detergents reduced KL from the counterpart at a significance level less than 0.005. clean-water value by various amounts depending The standard error of estimate of equation 165 on the concentration of detergent, the initial KL has increased markedly over that of 164. This is value, and the way in which the water was probably due to a decrease in overall quality of the agitated. The effects of sewage effluents and data. However, because equation 165 is based on settled sewage containing detergents were similar EFFECT OF DETERGENTS AND WIND ON THE REAERATION COEFFICIENT 57 to the effects observed for mixed detergents, but Downing and Truesdale (1955) studied the effect the effects of sewage effluents and settled sewage of oil films on oxygen absorption in a stirred tank. were less when the detergents were not present. They found no significant effect until the film Gameson, Truesdale, and Varley (1956) studied thickness exceeded 10~ 4 cm. Thus, they concluded the effect of about 1 mg/1 ABS on oxygen absorp that oil films were not a problem because films tion by flowing water in a laboratory open channel. thicker than 10~ 4 cm were not likely to persist very They found that the percentage reduction in KL long in natural streams. depended on the degree of turbulence, expressed Poon and Campbell (1967) found that the in terms of KL. The maximum reduction occurred presence of small concentrations of biologically for intermediate KL values, equivalent to those of inert suspended particles increased the rate of lowland English streams. Lower and higher tur oxygen absorption in an aerator. This finding is in bulence conditions resulted in smaller percentage agreement with the observation of R. S. McQuivey reductions. Similarly, Metzger (1968) and Mancy (oral commun., 1970) that suspended sediment in and Okun (1965) found in stirred-tank studies that creases the intensity of turbulence over that the largest reduction in KL occurred for interme observed in a clear water flow. diate mixing intensities. Whereas surfactants and sewage generally re Gameson, Truesdale, and Downing (1955) added duce KL, the presence of winds and surface distur detergents to a small English stream and found bances produced by the winds generally increase that there was no apparent relation between the KL. Imhoff and Fair (1956, p. 291) discussed wind reduction in KL and the detergent concentration, effects in general terms. They suggested that in the range from 0.5 to 2 mg/1. The reduction in winds passing over the water surface may double KL decreased with increasing KL for values ranging the absorption rate, that wind-induced waves may from about 30 to 120 cm/hr (centimeters per hour). increase the absorption rate 10 times, and that No maximum was observed in the natural streams, whitecaps may, by droplet aeration, increase it apparently because KL values were all larger than 100 times. 5 to 10 cm/hr, the level at which the maximum re Eloubaidy (1969) studied reaeration both with duction occurred in the laboratory studies. and without wind waves in a laboratory wind-water Ogden, Gibbs, and Gameson (1959) studied the tunnel. He concluded that the absorption rate was effect of salinity and the combined effect of significantly increased when waves developed on salinity, detergents, and sewage on the reaera- the water surface, and this increase was consider tion rate of flowing water in a flume. They found ably more than could be accounted for by the in that the percentage reduction in KL was equal to crease in surface area. He presented a prediction six times the square root of the salinity expressed equation containing the shear velocity, the as grams of salt per 1,000 grams of mixture, but kinematic viscosity, and the wind velocity. This that the reductions caused by detergents and equation, however, was verified only with a very sewage did not appear to depend on the presence limited amount of data. of salts in the water. Kehr (1938) studied the effect Downing and Truesdale (1955) studied the effect of sewage on reaeration in a laboratory flume and of surface wind on oxygen absorption in a labora found that the percentage reduction increased tory ripple tank. They found that there was little with the percentage of sewage and the flow veloc change in KL until a wind velocity between 3.0 and ity; however, the maximum velocity considered 3.5 meters per second was reached. For larger was only about 0.5 fps. velocities, KL increased rapidly. Similar results Lynch and Sawyer (1954) studied the effect on were obtained in tank experiments by Kanwishe; oxygen absorption of 11 synthetic detergents avail (1963), who found little change in the gas-exchange able on .the market in 1954. They found with one coefficient up to a velocity of about 3 meters per exception that all reduced the oxygen-transfer second but a coefficient that increased approxi coefficient, and they pointed out that because mately as the square of the velocity for wind these detergents were not biodegradable, they velocities between 3 and 10 meters per second. eventually entered the streams and rivers, where Downing and Truesdale (1955) also studied the they could significantly reduce the reaeration effect of wave height and frequency on KL. They coefficients. However, since 1954, biodegradable found that the rate of absorption of oxygen in detergents have been developed, and these should creased almost linearly with increase in wave reduce the effect that detergents have on reaera height, increasing approximately fourfold for a tion coefficients. fourfold increase in height. For irregular choppy 58 REAERATION IN OPEN-CHANNEL FLOW waves of the same frequency, KL increased less 6. The expected root-mean-square error of 15 rapidly. They also found that KL increased with an percent for the tracer technique is much less than increase in the frequency of the waves, although the 65-percent error expected from the DO balance there was more scatter in the results. Finally, technique and the 115-percent error expected from Juliano (1969) found that absorption-rate constants the disturbed-equilibrium technique. Thus, it measured in the Sacramento-San Joaquin estuary would seem well worthwhile to expend further time system varied directly with wind velocity. There and effort on developing the tracer technique for was considerable scatter in the relation; however, use in streams of all sizes. this scatter was attributed to the fact that wind 7. The best available reaeration-coefficient data effects vary greatly from point to point in the for natural streams is that of Churchill, Elmore, reach of interest. and Buckingham (1962). With these data and those of Owens, Edwards, and Gibbs (1964), 62 field data CONCLUSIONS AND RECOMMENDATIONS sets can be considered complete; that is, they con tain all the mean hydraulic parameters commonly CONCLUSIONS measured: velocity, depth, width, and slope. 8. Analysis of available reaeration-coefficient From the review of reaeration in open-channel data from natural streams and from laboratory flow presented in previous sections, we have flumes indicates that there is a significant dif arrived at several general conclusions. ference between the k2 regression equations ob 1. The approximations that can be made in the tained from the two groups of data. The most likely general DO balance equation (see pages 22-29, reasons for this are (a) an inability to scale cor 33-39) vary radically from situation to situation. rectly the turbulent mass-transfer process at the Therefore, careful consideration of all the cir water surface, (b) statistical quirks involved in cumstances must be made before any terms of the regression analysis using the complexly inter equation are disregarded. dependent hydraulic parameters of mean velocity, 2. The most general and realistic conceptual depth, width, and slope, and (c) experimental bias model for the reaeration process of open-channel in one or more of the individual groups of data. flows is the surface-renewal-damped-eddy diffu- 9. For the prediction of k2 in natural streams, sivity model of King (1966). This model merits regression analysis indicates that the equation further development in relating its parameters to with the smallest standard error of estimate con the mean flow parameters that are easily sistent with the ease of obtaining the mean flow measured. f/0.607 3. In theory, the tracer technique is far superior parameters is £2 = 8.76 -,1 689, where k-2 is in recipro to the other two techniques commonly used for determining the reaeration coefficient of open- cal days. channel flows. In practice, however, the use of RECOMMENDATIONS radioactive tracers involves problems from the A more detailed and reliable theoretical model standpoint of obtaining permission for use in than is presently available for the transfer of natural streams, and from the standpoint of pos oxygen through the surface of an open-channel sible hazard to the personnel involved in the study. flow is needed. The model should account for the 4. Sensitivity analysis of the DO balance tech effect of impurities in the flow such as sewage, nique shows that if photosynthetic-oxygen pro sediment, dissolved solids, and surfactants. It duction is present, its neglect is probably the most should consider the effect of wind blowing over serious source of error in the calculation of the the water surface, and it should be related directly reaeration coefficient. Following this in decreas to hydraulic parameters such as mean velocity, ing order of effect are errors in the upstream and depth, width, slope, and possibly near-surface ver downstream DO deficit, ki, and La. tical and (or) longitudinal turbulence. The model 5. Sensitivity analysis of the disturbed-equilib should, furthermore, scale the interface mass- rium technique shows that the most serious transfer phenomenon so that field and flume-scale sources of error in the calculation of the reaeration data fit one general equation. This would make coefficient are, in decreasing order of effect on the possible (with adequate independent verification calculated k2, errors in the downstream DO deficit, of the equation) the use of experimental results net photosynthetic production, upstream DO from laboratory flumes to predict field-scale deficit, and time of travel. reaeration coefficients. REFERENCES 59 For verification of the model in large-scale field Baird, M. H. I., and Davidson, J. F., 1962, Gas absorption, streams in which some degree of pollution, photo part II of Annular jets: Chem. Eng. Sci., v. 17, p. 473-480. synthesis, and other interferences may be present, Baity, H. G., 1938, Some factors affecting the aerobic decom position of sewage sludge deposits: Sewage Works Jour., or for obtaining more reliable k2 values than are v. 10, no. 3, p. 539-568. presently available, the tracer technique should Bella, D. 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SA-5, ment of the unimolecular expression Y=L(1 e~Kt), part p. 1-30. II of The statistical treatment of reaction-velocity data: _____1965, Recent results from a mathematical model of Jour. Phys. Chemistry, v. 35, pt. 2, p, 950-971. water pollution control in the Delaware estuary: Water ReVelle, C. S., Loucks, D. P., and Lynn, W. R., 1967, A Resources Research, v. 1, no. 3, p. 349-359. management model for water quality control: Water Thomas, H. A., Jr., 1937, The "slope" method of evaluating Pollution Control Federation Jour., v. 39, no. 7, p. 1164- the constants of the first-stage biochemical oxygen-demand 1183. curve: Sewage Works Jour., v. 9, no. 3, p. 425-430. 1968, Linear programming applied to water quality Toor, H. L., and Marchello, J. M., 1958, Film penetration management: Water Resources Research, v. 4, no. 1, model for mass and heat transfer: Am. Inst. Chem. Engi p. 1-9. neers Jour., v. 4, no. 1, p. 97-102. Rolley, H. L. 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V., Davidson, B., Dittman, F., Tsivoglou, E. C., O'Connell, R. L., Walter, C. M., Godsil, P. J., and Yu, S., 1969, Instream aeration of polluted rivers: Water Resources Research Inst., Rutgers Univ., New and Logsdon, G. S., 1965, Laboratory studies, part I of Brunswick, N. J., 196 p. Tracer measurements of atmospheric reaeration: Water Williams, W. E., 1951, Relation of BOD to stream deoxygena- Pollution Control Federation Jour., v. 37, no. 10, p. 1343- tion: Sewage and Indus. Wastes, v. 23, no. 9, p. 1161-1175. 1362. Wolman, Nathaniel, 1960, U.S. Senate Committee Pre-print Velz, C. J., 1958, Significance of organic sludge deposits: The 32, 86th Congress, 2d sess., Aug. Robert A. Taft Sanitary Eng. Center Technology Rept. Yao, K. M., 1969, Discussion of reaeration prediction in W58-2, p. 47-57. natural streams: Am. Soc. Civil Engineers Jour., v. 95, Velz, C. J., and Gannon, J. J., 1963, Biological extraction and no. SA-6, p. 1189-1190. accumulation in stream self-purification: Internat. Jour. Zanoni, A. E., 1967, Waste water deoxygenation at different Air and Water Pollution, v. 7, p. 587-595. temperatures: Water Research, v. 1, no. 8/9, p. 543-566. APPENDIXES APPENDIX A: EFFECT OF A CORROSION REACTION ON REAERATION-COEFFICIENT MEASUREMENTS IN A LABORATORY FLUME [Symbols used in appendix A are given in a list at the end of the appendix] A recirculating-flume system generally consists and of a headbox, a section of open channel, a tailbox, and a closed return-flow conduit. The system may D2 (jt2,0 = Cs [l-e "2 ] be conceptualized as a flow loop consisting of an b 9 open channel and a pipe in series. If first-order + (D0 -Dx)e- at e oxidation reactions proceed in the channel and in , (A-6) the return pipe simultaneously with reaeration in where from the channel, the characteristic differential equa (A-7) tions are: and D,(L,,0)=D2 (0,0): (A-8) (channel) (A-l) b2 L.2 b, _ and + . e (pipe) dt - dx2 " " (A-2) (A-9) where b is the first-order rate constant for the oxidation reaction, D is the DO deficit, C is the DO and concentration, K2 is the reaeration coefficient, U is the mean water velocity, / is time, x is longi tudinal position, and the 1 and 2 subscripts refer in which V is the volume of the water in the system, to the channel and pipe, respectively. These equa Q is the discharge of the system, Cs is the satura tions assume that the only source of oxygen is tion concentration, and Li is the length of the reaeration, the only sink is the oxidation reaction, channel. Because of the presence of the headbox, and longitudinal dispersion effects are negligible. tailbox, and pump, the dimensions L\ and L2 in a The oxidation reaction is assumed to follow the real flume are equivalent dimensions. basic law of chemical kinetics for a first-order Equation A-9 gives the deficit at the station heterogeneous reaction; that is, the rate of *i=0, x2 L2 as /-*oo. This non-zero value of the reaction is proportional to the concentration of the deficit is the value at which the oxygen consumed reactant, the dissolved oxygen. by the reactions is just replaced by that supplied Note that D\=D\ (x\, t) and £>2 = Z)2 fe, t) and also by surface reaeration. Equation A-5 shows that if that Jti = 0, JC2 = L2 designates the beginning of the there is an oxygen-consuming reaction in the channel, and XI = LI, ;t2 = 0 the end of the channel. channel or return pipe of a flume, then because of The initial and final conditions in the system are: the first term on the right of that equation and be Di(0,0)=D2 (L2,0) = D0 (A-3) cause of the non-zero D^ value, the logarithm D versus / relation at a station jct will not be a straight where L2 is the length of the pipe. The solutions to line as shown in figure 2. Thus the system does not equations A-l and A-2 (derived by the authors) follow the first-order model usually assumed valid under these conditions are: for flume systems. Also, if b\ and (or) b-> are not equal to zero, the curves of logarithm D versus t at two different stations in the channel will not be D, l-e parallel, as they would be were b\ and 62 = 0. Furthermore, it is expected, even discounting the bl -a)^l curvature of the lines and their deviation from parallel, that the quantity M= , as determined x. \ (j (A-5) by the method of figure 2 will be some weighted 67 68 REAERATION IN OPEN-CHANNEL FLOW average of Kz and b\ and will not equal the reaera- Equation A-18 states that, assuming one has tion coefficient, K2. Examples presented later show available DO-deficit measurements of two samples, that if bi=0, the error due to the curvature and one from the upstream station, and one from the deviation from the parallel is negligible. If bi^O, downstream station at a lag time r=Ajc/f/ later, equations A-5, A-9, and A-10 can be used to obtain then the K2 value as determined from A/ = Aln D/r an estimate of the true values of K2 and b\ from differs from the true value only if bi^Q. The values of M measured by the procedure of figure 2. amount of the deviation of M from K2 increases If as the deficit at the upstream station decreases. If the method of figure 2 is used, that is, if M is ul + D' e determined not from two instantaneous point samples but from experimental results which were forced to fit two parallel straight lines, the result (A-ll) ing curve-fitting errors will be added to those and resulting; from the non-zero b\ value. )fl2. To determine Kz and bi from experimental data, "l a plot of In D'i versus t at two different stations, as in figure 2, should be made without forcing the +D x e data to fit two parallel straight lines. For two sufficiently different values of D'n, one determines Ul M, as in figure 2, and computes bi and K2 from two + (D'0 -D'x)e e , - applications of equation A-18. If the data are where in the double subscripts the first subscript, forced to fit two parallel straight lines, M will be 1, indicates the channel and the second, 1 or 2, the same for all D'n, and b\ and K2 will be impos indicates one of two longitudinal locations in the sible to determine. Other methods for determining channel, Af=*12 Xll =-rr, and the deficits have been Kz, bi, and b2 may be derived from equations A-9, A-10, and A-18. These methods require only a nondimensionalized by dividing by Cs, which is in single determination of M, but they involve con dicated by the primed notation, then from equa ducting the experiment until f ><» to determine tions A-ll and A-12, experimentally Dx. These methods, in addition, Ax. require measurements of Li and L2, indeterminate quantities in real flume systems. Thus the pro cedure outlined at the start of this paragraph Equation A-13 may be written appears to be the best one for determining K2 in a flume system in which a first-order oxygen- Y h where 17= - (K2 + bi) and S= * Because rj is consuming reaction proceeds in the channel in conjunction with reaeration. small, e~v in the first term on the right of equa Equation A-18 may be rewritten as tion A-14 is adequately represented by the first ^M = |L(1 ~ Dn)> (A-19) two terms in its Taylor series expansion, and equa A.2 A.2 Lf 11 tion A-14 becomes from which it is seen that if b\ and D'n are con stants, the relative error in the measured Kz in Taking the natural logarithm of equation A-15, creases as Kz decreases. Thus if b\ is a constant in expressing the right-hand side in its power series a set of flume experiments, the resulting errors form, and neglecting all powers of 8 rj greater will be more severe at the lower end of the experi than 1, it may be shown that mental Kz range, with the possibility of severely (A- 16) biasing any resulting least-squares prediction equations. This could be a reason for the widely Because e~^ = different values of the multiplicative constants of the regression equations obtained by Thackston and Krenkel (1969a) from the analysis of their data (A-17) (Thackston, 1966; Krenkel, 1960). Ay- , The discussion above suggests a procedure for / conducting a series of flume experiments for (A-18) measuring K2 which would determine if oxygen- Lf\ consuming reactions are influencing the K2 mea- APPENDIX A 69 surements. The procedure involves running a In the examples, DO deficits at two sampling special experiment prior to the start of a test stations were calculated using equation A-5 and series. In this experiment, test conditions are the parameters listed in table 17, where a and Dx selected which are expected to yield a K2 value that were calculated from equations A-9 and A-10. would rank among the lowest expected in the test series. The experiment is conducted under con ditions typical for the entire set, but over a larger TABLE 17. Parameters used in calculating the synthetic DO-deficit range than normal-that is, for a long flume data and the resulting calculated reaeration coefficient time. If there is no oxygen-consuming reaction in process in the system, the curves of logarithm D Ki b, b., a M _L 1 1 1 1 Example D,, Q versus t observed at the two measurement stations No. sec sec sec sec sec will be straight and parallel, and Kz may be com xlO4 xlO5 xlO5 xlO4 xlO4 1 8.00 0 8.00 3.50 0.800 0.144 8.00 puted by the method of figure 2. If a reaction is in 2 8.00 0 8.00 3.50 .400 .144 8.00 3 8.00 4.00 8.00 3.65 .800 .179 7.81 progress in the system, this will be indicated, as t 4 8.00 4.00 8.00 3.65 .400 .179 7.27 becomes large, by curvature and nonparallelism 5 4.00 4.00 4.00 1.90 .400 .211 3.34 NOTE.- Parameters which were the same for all examples are: of the curves of logarithm D versus t at the two K=16 ft3, 0 = 0.3 cfs, t/,= 1.8 fps, t/2 =1.2 fps, tt = 36 ft, t2 = 40 ft, AT,i = 8 ft at upstream, and x, 2 = 28 ft at downstream measuring stations. If the reaction is in the return sampling stations. conduit, b\ as computed by the method suggested above will be zero, and as will be shown in the examples to follow, the analysis method of From the synthetic flume data, M was calculated figure 2 will be accurate enough. If &i^0, K2 must for each of the examples by the method of figure be calculated by the method given above. The trial 2. Least-squares straight lines were fit to 20- experiment described above will point the way for minute increments of the calculated values of the conducting the other experiments in the series. If logarithm deficit and t for each of the two sampling the reaction in the channel is insignificant in the stations. The 20-minute sampling period was test case, it will most likely also be insignificant in chosen because this was the period used by both the others. If it were significant, its effects should Thackston (1966) and Krenkel (1960). The slopes of be considered in the remainder of the experiments. the two lines were averaged, and new intercepts were computed for parallel least-squares lines EXAMPLES through the two sets of points. The quantity M was then computed from the new intercepts and the The usefulness of the K2 determination pro average slope. These calculated M values are listed cedure described above is best illustrated by in table 17. In this procedure, M is not a function example. The examples which follow utilize, with of time as it is if the lines are not parallel. the exception of the reaction-rate constants b\ Figure 18 shows plots of logarithm D versus t and b2, values of flume and flow parameters which for the upstream sampling stations in examples 3, are typical for small laboratory flumes. The values 4, and 5; the lines are the least-squares fits to the of hi and b2 must necessarily be highly speculative, synthetic data points. The curvature due to the because available flume data are not sufficient to non-zero Dx value is virtually unnoticeable over allow determination of typical values for these the 20-minute time period. In fact, if experimental parameters. For a mixing tank, however, Tsivog- data with its scatter were used, the curvature lou, O'Connell, Walter, Godsil, and Logsdon (1965) would probably be impossible to detect. Thus, over reported data which indicate that the reaction- a 20-minute experimental time, it is virtually rate constants in a largely metal system can be on impossible to detect a non-zero Dx , and it is the order of 2xlO~ 5 per second. The reaction is impossible to conclude whether or not an oxygen most likely due to corrosion of the walls of the sys consuming reaction is proceeding in the system. tem and would be expected to vary widely from As can be seen from table 17, for examples 3, 4, one system to another; consequently, slightly and 5, the deviation of M, the calculated K>, from larger values for b\ and b2 have been chosen for the true value is significant. Thus it is possible use in the examples described. The relative error for oxygen-consuming reactions to cause signifi in a measured K2 due to a particular value of bi is cant errors in calculated K2 values without their dependent on the ratio bi/K2, so that for the values presence being noticed in the experimental results. of b\ used in the examples, the relative errors will The values of M for examples 1 and 2 are not be more or less severe for respectively larger or significantly different from 8.0xlO~* seconds-'. smaller K2 values. This shows that, at least for short time periods, if 70 REAERATION IN OPEN-CHANNEL FLOW -.20 values obtained in the two ways indicates that despite the approximations made in its derivation, equation A-18 provides an accurate means for obtaining K-2 and b\ from two measured values of M. The procedure for determining K2 and b\ from experimental data is illustrated in figure 19. Synthetic data parameters shown in figure 19 are the same as those for example 4 in table 17. Deficits at the upstream and downstream sampling stations are plotted for two periods: from 0 to 10 minutes and from 50 to 60 minutes. The procedure consists of plotting the natural logarithm of deficit versus time, determining M graphically (as in fig. 2) for each of the 10 minute periods, and solving for bi and K2 using equation A-18 and the resulting two sets of D'n and M values. For the example given in figure 18, D'n and M for the initial 10- minute period are, respectively, 0.376 and 7.20xlO~4 -1.20 0 6 8 10 12 14 16 18 20 seconds"1, whereas for the period from 50 to 60 TIME, IN MINUTES minutes, they are 0.245 and 6.56 xlO"4 seconds" 1. Using equation A-18, the calculated bi and K> FIGURE 18. Synthetic flume data from the upstream sam values are, respectively, 4.51xlO"4 seconds"1 and pling station. 7.95xlO" 4 seconds" 1 . The agreement of these values with the bi and K2 values used in generat ing the synthetic data is very good, considering b\ is zero, the deviation of M from the true K2, due that the two M values were determined graphically. to the curvature induced by a non-zero Dx , is If in a flume study there is an error in the value negligible. Thus, when Cs is known, the K-2 cal of Cs being used, there will be a corresponding culation procedure illustrated in figure 2 will con error in all the computed deficits. In such a case, tain significant errors only if b\ is not zero. the true deficit, DT is related to the plotted deficit The upstream deficit values at f=0 minutes for by D=DT +d. The resulting equation for the plotted examples 3, 4, and 5, respectively, are 0.676, 0.356, deficit, assuming no oxygen-consuming reactions and 0.379. The corresponding values of M calcu in either the channel or the return conduit, is lated from equation A-18 are 7.82, 7.28, and _ _ X 3.34xlO~4 seconds" 1. These are very nearly equal -c* e 2 ulf (A-21) to the M values calculated by the procedure of figure 2 listed in column 8 of table 17. Even better Equation A-21 is similar to equation A-5, with correspondence between the measured M and the bi=Q. It has been shown that in the case of bi = Q, M calculated from equation A-18 is obtained if the error in the K-2 computed using the method of instantaneous values of the upstream and down figure 2 is negligible. Thus in a flume study, a stream deficits are used to calculate M. For negligible error is introduced by using a value of instance, in example 5 at t = lQ minutes the up Cs which is incorrect by a small amount Ad. This is stream deficit is 0.368, and the M value calculated in contrast to the nature of errors in analysis of from the deficits is 3.31xlO~4 seconds"1, whereas mixing-tank data where K2 is determined from the the M calculated from equation A-18 is 3.31 x 10~4 slope of a single line (Isaacs and Gaudy, 1968; seconds^1 . The close correspondence of the M Tsivoglou and others, 1965). SYMBOLS USED IN APPENDIX A Symbol Definition bi Rate constant for the first-order oxidation reaction in the channel of the flume, in reciprocal seconds. b-2 Rate constant for the first-order oxidation reaction in the return pipe of the flume, in reciprocal seconds. Ci Dissolved-oxygen concentration in the water in the channel of the flume, in milligrams per liter. C2 Dissolved-oxygen concentration in the water in the return pipe of the flume, in milligrams per liter. Cs Dissolved-oxygen concentration at saturation, in milligrams per liter. D Computed deficit of dissolved oxygen, in milligrams per liter. APPENDIX A 71 -0.91 -1.43 - -1.45 0 100 200 300 400 500 600 3000 3100 3200 3300 3400 3500 3600 TIME, IN SECONDS FIGURE 19. Determination of 61 and K? from M values calculated at two different DO deficits. 72 REAERATION IN OPEN-CHANNEL FLOW Symbol Definition Di Equivalent to Di(jci,/)f the dissolved-oxygen deficit at longitudinal position x in the channel of the flume at time t, in milligrams per liter. (Subscript 1 indicates the channel of the flume.) D2 Equivalent to D2 (jt2,0, the dissolved-oxygen deficit at longitudinal position x in the return pipe of the flume at time t, in milligrams per liter. (Subscript 2 indicates the return pipe of the flume.) Do Initial dissolved-oxygen deficit in the system at time /=0, in milligrams per liter. D* Final dissolved-oxygen deficit in the system at time / = °°, in milligrams per liter. Do Nondimensional deficit of dissolved oxygen in the system at time (=0, equal to D0/CS. D'» Nondimensional deficit of dissolved oxygen in the system at time ?=°°, equal to DX/CS. Dr True deficit of dissolved oxygen, equal to the computed deficit minus an error, or D-d, in milligrams per liter. Dr» Initial true deficit of dissolved oxygen, equal to the initial computed deficit minus an error, or D0 -d, in milligrams per liter. D'n Equivalent to D\(xn,t), the nondimensional deficit of dissolved oxygen in the channel at longitudinal position xt at time t. (Deficit nondimensionalized by dividing by Cs; first subscript in the double subscript (1) indicates the channel of the flume, and the second subscript (1) indicates upstream measurement point in the channel.) D'i2 Equivalent to D'i(jti2,/ + A/), the nondimensional deficit of dissolved oxygen in the channel at longitudinal position x» at time t+&t. (Deficit nondimensionalized by dividing by Cs; first subscript in the double subscript (1) indicates the channel of the flume, and the second subscript (2) indicates downstream measurement point in the channel; A? is the flow time or (xiz x\\)IU\, where Ui is the mean water velocity in the channel.) d Error in the saturation concentration of dissolved oxygen, in milligrams per liter. Kz Reaeration coefficient or rate constant for oxygen absorption from the atmosphere, natural logarithm base, in reciprocal days. Li Length of the flume channel, in feet. (Equivalent dimension because of the presence of the headbox and tailbox.) Lz Length of the return pipe, in feet. (Equivalent dimension because of the presence of the headbox, tailbox, and pump.) X AJC M Equal to A In Dj (nomenclature of fig. 2 and the main part of the report) or A In D'/ (nomenclature of appendix A); U U\ A" = Ajt=the distance, in feet, between the upstream and downstream measurement points in the flume channel; U=Ui = the mean water velocity, in feet per second, in the flume channel; A In D = A In D =the difference in the natural logarithms of the dissolved-oxygen deficits at the upstream and downstream measurement points with the downstream measurement lagging the upstream by the flow time, AJC/(/I. See figure 2. Q Water discharge, in cubic feet per second. t Time, in seconds. (/i Mean water velocity in the channel of the flume, in feet per second. Uz Mean water velocity in the return pipe of the flume, in feet per second. Xi Longitudinal position in the channel of the flume, in feet. xn Upstream measurement point in the channel of the flume, in feet. Xiz Downstream measurement point in the channel of the flume, in feet. Equal to -77 -77-?+(K2 +61)77- » in reciprocal seconds. v I Uz c/jj 8 Equal to bi/(Kz + bi). Equal to (Kz + bi). T Lag time or flow time, equal to AJC/(/I, in seconds. APPENDIX B: COLLECTED DATA TABLE 18. Reaeration coefficients and hydraulic data from TABLE 19. Reaeration coefficients and hydraulic data from Churchill, Elmore, and Buckingham (1962) Owens, Edwards, and Gibbs (1964) fcst H V S w tot H u s w (day-') (feet) (fps) (X10 1 ) (feet) (day-1 ) (feet) (fps) (X10<) (feet) 2.272 3.27 3.07 10.8502 229 2.35 1.21 0.14 3.94 28.7 1.440 5.09 3.69 8.6426 242 2.06 1.15 .14 3.94 25.8 .981 4.42 2.10 2.7058 344 3.20 1.09 .13 3.94 27.3 .496 6.14 2.68 2.9014 358 2.37 1.50 .37 3.94 27.7 .743 5.66 2.78 3.6970 376 4.57 1.08 .16 4.24 28.1 1.129 7.17 2.64 1.6448 313 2.09 .98 .15 4.24 28.2 .281 11.41 2.92 1.7994 312 1.18 1.12 .13 4.24 26.6 3.361 2.12 2.47 14.2342 617 3.18 1.46 .38 4.24 27.8 2.794 2.93 3.44 14.5225 635 6.18 1.31 .47 11.65 25.0 1.568 4.54 4.65 13.9099 667 .90 2.44 .23 8.95 17.9 .455 9.50 2.94 1.6893 374 1.66 2.03 .22 4.08 22.5 .389 6.29 2.51 2.9038 414 .75 1.74 .28 2.69 22.5 .270 7.52 3.15 3.0443 443 2.12 1.47 .37 8.72 20.1 .550 7.07 3.30 3.7775 450 1.41 1.82 .50 1.56 39.8 .544 5.44 3.11 5.9961 330 .31 2.41 .43 1.56 34.9 .604 8.06 4.28 5.7667 346 31.80 .72 1.37 106.00 21.9 1.251 3.88 2.73 6.0706 90 24.53 .87 1.19 84.00 20.9 .273 9.28 2.41 1.2571 537 24.57 .85 1.07 61.00 23.7 .225 10.19 3.06 1.7789 549 25.59 .40 .44 92.00 15.3 1.881 3.29 2.40 15.5378 520 28.34 .40 .56 61.50 12.1 .842 4.74 3.46 13.5458 535 22.80 .39 .63 60.50 11.0 .883 5.72 4.02 11.7032 542 49.17 .60 1.83 92.00 16.1 .915 6.95 4.52 10.1594 550 30.77 .69 1.81 61.50 14.2 .995 4.29 1.85 3.9709 517 18.46 1.00 1.54 60.50 11.5 .547 6.01 2.75 3.7523 531 21.05 .82 1.07 101.60 21.8 .881 7.16 3.23 3.6794 539 16.06 .78 1.27 83.00 19.3 .252 8.49 3.71 3.6430 548 12.04 .64 .46 61.20 17.3 5.558 3.42 5.00 23.5142 182 20.32 .48 .60 65.70 17.7 1.712 3.02 3.05 17.4603 124 18.90 .72 1.16 61.20 20.7 3.222 2.83 3.51 18.8692 115 20.25 .66 1.31 65.70 20.0 17.09 .67 1.30 65.70 19.9 tAdjusted to (fc2)20% USiing equation 104. 19.16 .69 1.25 65.70 20.0 tAdjusted to (fcs)20°, using equation 104. 73 74 REAERATION IN OPEN-CHANNEL FLOW TABLE 20. Reaeration coefficients and hydraulic data from TABLE 22. Reaeration coefficients and hydraulic data from Gameson, Truesdale, and Downing (1955) O'Connor and Dobbins (1958) (various sources) fczt H U W fc:t H u s (day : (feet) (fps) (feet) (day-1 ) (feet) (fps) (X101 ) 57.7 0.41 1.04 5.7 5.80 0.9 0.97 36 48.7 .41 1.12 5.3 3.07 1.9 .55 14 55.6 .42 .99 5.6 1.22 4.0 .73 4.9 46.2 .42 1.02 5.4 .92 3.1 .65 6.9 34.3 .42 .73 6.0 .23 9.2 1.37 .27 35.2 .34 .74 6.0 .28 9.0 1.57 .27 .30 8.9 1.63 .27 t Ad justed to (ki)io°, using equation 104. .048 12 .32 .018 32 .50 .026 37 .88 TABLE 21. Reaeration coefficients and hydraulic data from t Adjusted to (£2)20°, using equation 104. O'Connor and Dobbins (1958) (Streeter and Phelps, 1925) fc2t H u s (day-1 ) (feet) (fps) (X104 ) TABLE 23. Reaeration coefficients and hydraulic data from 0.45 9.2 2.32 0.68 Tsivoglou and others (1967,1968) .23 8.8 1.98 .68 .15 9.3 2.18 .68 fc2t H u W .34 6.5 1.41 .68 (day-1 ) (feet) (fps) (feet) .17 7.0 .59 1.97 1.273 1.76 0.636 73.7 1.142 1.94 .419 101.5 .26 5.8 .32 1.97 .977 2.09 .359 110.0 .20 5.3 .22 1.97 .646 3.11 .458 57.9 .14 4.8 .19 1.97 1.67 .19 6.4 .43 1.23 1.653 .570 86.7 .37 4.0 .23 1.23 .872 2.36 .346 101.0 1.251 1.99 .335 123.8 .22 8.6 .40 .95 1.170 2.26 .436 83.72 .71 7.6 .40 .95 1.261 .15 7.1 .22 .95 2.83 .286 101.9 .20 5.5 .51 1.67 3.373 2.30 .563 63.7 .26 13.8 2.50 1.97 1.163 1.95 .529 80 1.491 2.42 .441 77.3 .22 15.0 4.20 1.23 2.13 .31 8.6 1.36 1.23 2.344 .539 71.9 .21 7.2 .86 1.23 1.420 2.06 .425 94.2 .23 6.6 .53 1.23 .977 1.66 .457 108.8 .86 14.3 3.68 .95 t Adjusted to (fa) 20% using equation 104. .92 9.3 1.01 .95 .45 9.0 .62 .95 .19 18.7 3.72 1.67 TABLE 26. Reaeration coefficients and hydraulic data from .24 8.8 1.48 1.67 Negulescu and Rojanski (1969) .17 8.2 1.18 1.67 fc2t H u W .19 7.2 .91 1.67 (day-1 ) (feet) (fps) (feet) .16 7.3 .96 1.67 15.90 0.492 1.902 0.66 .14 24.2 3.90 .68 11.50 .492 1.640 .66 t Adjusted to (fc2)20°, using equation 104. 17.25 .164 .951 .66 18.70 .164 .853 .66 8.64 .492 1.050 .66 11.50 .492 .656 .66 8.64 .492 .886 .66 14.45 .361 1.082 .66 tAdjusted to (£2)20°, using equation 104. APPENDIX B 75 TABLE 24. Reaeration coefficients and hydraulic data from TABLE 25. Reaeration coefficients and hydraulic data from Thackston (1966) Krenkel (1960) tot H U S w feit H U s w (day-1 ) (feet) (fps) (X1 U. S. GOVERNMENT PRINTING OFFICE : 1972 O - 466-751