Design and Performance Data from EPA Pond Studies


Peterborough, New Hampshire

Kilmichael, Mississippi

Eudora, Kansas

Corinne, Utah

Organic Loading (kg BOD ha"1 d1)

State Design Standard



Primary cell.

Entire system.

Estimated from dye study.



Actual (1974-1975)



Source: Data from Middlebrooks et al. (1982) and USEPA (1983).

Theoretical Detention Time

State Design Actual

Standard Design (1974-1975)

None 57 107

None 79 214

None 47 231

Months Effluent BOD Exceeded 30 mg/L

October, February, March, April

November, July March, April, August


The Gloyna method was evaluated using the data referenced in Table 4.2. The equation giving the best fit of the data is shown below as Equation 4.2; despite the considerable scatter to the data, the relationship is statistically significant:


BOD = BOD5 in the system influent (mg/L). LIGHT = Solar radiation (langleys). V = Pond volume (m3). Q = Influent flow rate (m3/day). T = Pond temperature (°C).

4.2.3 Complete-Mix Model

The Marais and Shaw (1961) equation is based on a complete-mix model and first-order kinetics. The basic relationship is shown in Equation 4.3:


Cn = Effluent BOD5 concentration (mg/L). C0 = Influent BOD5 concentration (mg/L). kc = Complete-mix first-order reaction rate (d-1). tn = Hydraulic residence time in each cell (d). n = Number of equal-sized pond cells in series.

The proposed upper limit for the BOD5 concentration (Ce)max in the primary cells is 55 mg/L to avoid anaerobic conditions and odors. The permissible depth of the pond, d (in meters), is related to (Ce)max as follows:

where (Ce)max is the maximum effluent BOD (55 mg/L), and d is the design depth of the pond (in meters).

The influence of water temperature on the reaction rate is estimated using Equation 4.5:


Variation of the Plug-Flow Reaction Rate Constant with Organic Loading Rate

Organic Loading Rate


Variation of the Plug-Flow Reaction Rate Constant with Organic Loading Rate

Organic Loading Rate


kp (d-1)











Reaction rate constant at 20°C.

Source: Neel, J.K. et al., J. Water Pollut. Control Fed, 33, 6, 603-641, 1961. With permission.

Source: Neel, J.K. et al., J. Water Pollut. Control Fed, 33, 6, 603-641, 1961. With permission.

where kcT = Reaction rate at water temperature T (d-1).

T = Operating water temperature (°C).

4.2.4 Plug-Flow Model

The basic equation for the plug-flow model is:


Ce = Effluent BOD5 concentration (mg/L). C0 = Influent BOD5 concentration (mg/L). kp = Plug-flow first-order reaction rate (d-1). t = Hydraulic residence time (d).

The reaction rate (kp) was reported to vary with the BOD loading rate as shown in Table 4.3 (Neel et al., 1961). Theoretically, the reaction rate should not vary with loading rate; however, that is what was reported.

The influence of water temperature on the reaction rate constant can be determined with Equation 4.7:

where kpT = Reaction rate at temperature T (d-1).

T = Operating water temperature (°C).

4.2.5 Wehner-Wilhelm Equation

Thirumurthi (1974) found that the flow pattern in facultative ponds is somewhere between ideal plug flow and complete mix, and he recommended the use of the following chemical reactor equation developed by Wehner and Wilhelm (1956) for chemical reactor design:


Ce = Effluent BOD concentration (mg/L).

C0 = Influent BOD concentration (mg/L).

a = (1 + 4kfD)0S, where k is a first-order reaction rate constant (d-1), t is the hydraulic residence time (d), and D is a dimensionless dispersion number = H/vL = Ht/L2, where H is the axial dispersion coefficient (area per unit time), v is the fluid velocity (length per unit time), and L is the length of travel path of a typical particle.

e = Base of natural logarithms (2.7183).

A modified form of the chart prepared by Thirumurthi (1974) is shown in Figure 4.1 to facilitate the use of Equation 4.8. The dimensionless term kt is plotted vs. the percentage of BOD remaining for dispersion numbers ranging from zero for an ideal plug flow unit to infinity for a completely mixed unit. Dispersion numbers measured in wastewater ponds range from 0.1 to 2.0, with most values being less than 1.0. The selection of a value for D can dramatically affect the detention time required to produce a given quality effluent. The selection of a design value for k can have an equal effect. If the chart in Figure 4.1 is not used, Equation 4.8 can be solved on a trial-and-error basis as shown in Example 4.1 (see below). Middlebrooks (2000) has developed a spreadsheet that calculates the dimensions for a pond system using the Wehner-Wilhelm equation with options to use a wide range of variables. The spreadsheet takes the tedium out of the design procedure and eliminates the need to read an imprecise table. A copy can be obtained by contacting the author.

FIGURE 4.1 Wehner-Wilhelm equation. (Adapted from Thirumurthi, D., J. Water Pollut. Control Fed., 46, 2094-2106, 1974.)

To improve on the selection of a D value for use in Equation 4.8, Polprasert and Bhattarai (1985) developed Equation 4.9 based on data from pilot and full-scale pond systems:


D = Dimensionless dispersion number.

t = Hydraulic residence time (d).

v = Kinematic viscosity (m2/d).

The hydraulic residence times used to derive Equation 4.9 were determined by tracer studies; therefore, it is still difficult to estimate the value of D to use in Equation 4.8. A good approximation is to assume that the actual hydraulic residence time is half that of the theoretical hydraulic residence time.

The variation of the reaction rate constant k in Equation 4.8 with the water temperature is determined with Equation 4.10:

where kT = Reaction rate at water temperature T (d-1). k20 = Reaction rate at 20°C = 0.15 d1. T = Operating water temperature (°C).

Example 4.1

Determine the design detention time in a facultative pond by solving Equation 4.8 on a trial-and-error basis. Assume Ce = 30 mg/L, C0 = 200 mg/L, k20 = 0.15, D = 0.1, d = 1.5 m, Q = 3785 m3/day, and the water temperature is 0.5°C.


1. Calculate kT using Equation 4.10:

2. Assume for the first iteration that t = 50 d, and solve for a:

a = (1 + 4 kTDt)0.5 a = (1 + 4 x 0.028 x 0.1 x 50)0.5 a = 1.25

3. Solve Equation 4.8 and see if the two sides are equal:

C0 200 (1 +1.25)2e125/(2x01) - (1 -1.25)2e-125/(2x01)


Because 0.15 does not equal 0.283, repeat the calculation. 4. The final iteration assumes that t = 80 d:

The agreement is adequate, so use a design time of 80 days. 5. Using Equation 4.9, determine the length-to-width ratio that will yield a D of approximately the assumed value of 0.1: v = 0.1521 m2/d.

Volume = 80 d x 3785 m3/day = 302,800 m3. Divide the flow into two streams. Volume in one half of the system = 151,400 m3. Divide the half system into four equal-volume ponds. Volume in one pond = 37,850 m3. Surface area of one pond = 37,850/1.5 = 25,233 m2. Theoretical hydraulic detention time in each pond = 80/4 = 20 d. Assume length-to-width ratio is 4:1.

Equation 4.9 was developed using the hydraulic detention time determined by dye studies; therefore, it is reasonable to assume that the theoretical hydraulic detention time is not the correct value to use in Equation 4.9. A good approximation of the measured hydraulic detention time is to use a value of one half that of the theoretical value:

D _ 0.184[10 x 0.1521(79.4 + 2 x 1.5)f489(79.4)LS11

10 9720.8

To illustrate the effect of using the theoretical hydraulic detention time, a O value is calculated using the theoretical value, and both values of 0 are used in Equation 4.8 to calculate the effluent BOD5 concentration. The theoretical hydraulic detention time is used in Equation 4.8 because it was developed based on the theoretical value. The total detention time is used because the equation represents the entire system and not a component of the system:

The latter part of the denominator in Equation 4.8 was omitted because it is insignificant in this and most situations. For O = 0.149, Ce = 32.5 mg/L, and for O = 0.209, Ce = 35.4 mg/L. As shown by these calculations, small changes in O can have a significant influence on the effluent quality.

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