The yearly waste water application rate (Lw(p)) needed in the design of a wastewater irrigation system is based on the amount NO3- -N loading to ground water allowed by regulatory agencies. The yearly water and nitrogen mass balance equations used to derive the design equation were outlined by   Metcalf and Eddy Inc (1990) and WCPF (1989). The nitrogen uptake by the crop is related to evapotranspiration  and  crop yield using the evapotranspiration production function. The derivation below is for metric units, english unit equations were also derived.

Lw(p) = ET – Pr + Wp    ………………  [Eq.1]

Where:

Lw(p) = Wastewater hydraulic loading rate (m/yr), the volume of wastewater applied per unit area of land per unit time.

ET = Design evapotranspiration rate (m/yr)

Pr = Design precipitation rate (m/yr)

Wp = Design percolation rate (m/yr)

The wastewater nitrogen loading to ground water based on the nitrogen mass balance equation is:

Ln = U + D + 10 Wp Cp  ………………………………………  [Eq. 2]

Where:

U = Crop nitrogen uptake (kg/ha/yr)

D = Denitrification (kg/ha/yr)

Wp = Percolating water (m/yr)

Cp = Percolate nitrogen concentration (mg/L)

Ln = 10 Lw Cn  ……………………………………….. [Eq.3]

Where

Lw = Wastewater applied (m/yr)

Cn = total nitrogen in applied wastewater (mg/L)

Solve for Wp in Eq. 2 yields:

Wp = (Ln – U- D)/ 10 Cp

Substitute Wp term in Eq. 1 yields:

Lw = ET – Pr + (Ln –U – D)/ 10 Cp      ……………………………….. [Eq. 4]

The fraction of applied nitrogen removed  by nitrification  and volatilization (F) can be expressed as:

F = D/Ln   ……………………………………………… [Eq. 5]

D= Denitrification (kg/ha/yr)

Solve for D in Eq. 5 and substitute in Eq. 4 yields:

D = F Ln

Lw = ET – Pr + (Ln –U – F  Ln)/ 10 Cp    ……………………… [Eq. 6]

Plug in Eq. 3 into Eq. 6 yields

Lw = ET – Pr + ( 10 Lw Cn –U – (10 F Lw Cn)/ 10 Cp    ……………. [Eq. 7]

Simplified Eq. 7:

Lw = (ET – Pr) + [ 10 Lw Cn (1 – F) – U ]/ 10 Cp

10 Lw Cp = 10 Cp (ET –Pr) + 10 Lw Cn (1-F) –U

10 Lw Cp – 10 Lw Cn (1-F) = 10 Cp (ET –Pr) –U

10 Lw (Cp - Cn (1-F) = 10 Cp (ET –Pr) –U

Lw (Cp - Cn (1-F) = Cp (ET –Pr) –U/10

Lw (Cp - Cn (1-F) = Cp (ET –Pr) –U/10

Lw = [Cp (ET – Pr) – U/10] / (Cp – Cn (1-F))

Lw = [Cp (ET – Pr) – U/10] / (Cp – Cn (1-F)) …………………………….. [Eq. 8]

Express the unit of Lw in mm/yr and multiply Eq. 8 by –1 yields

Lw = [Cp (Pr - ET) + 100 U] / (Cn (1-F)-Cp) ……………………………….. [Eq. 9]

The amount of nitrogen taken up by the tree (U) can be expressed in terms of the Evapotrnspiration production function and the nitrogen concentration in plant tissues [Cc] in (%):

U = (a+ b ET) Cc  …………………………….. [Eq. 10]

Plug in Eq. 10 into Eq. 9 yields:

Lw = [Cp (Pr - ET) + (100 (a+ b ET) Cc)] / (Cn (1-F)-Cp)  ……………….. [Eq. 11]

Where:

ET = design ET rate (mm/yr)

Pr  = design precipitation rate (mm/yr)

Cp = total nitrogen in percolating water (mg/L)

Cn = total nitrogen in applied wastewater (mg/L)

Cc = nitrogen concentration in plant tissues (%): See table

## a  = intercept of the Evapotanspiration production function (kg/ha)

b  = slope of the Evapotanspiration production function (kg/ha/mm)

## 100 = conversion factor, unitless

Irrigation efficiency for the design is  Et/Lw(p)

After the Lw(p) is determined, a water and nitrogen balance model is run on a daily time step to implement the design criteria. The daily operational procedure  must result in the same irrigation efficiency as calculated in the seasonal design equation.

Execution

The final design equation 11 has been programmed in an excel spreadsheet. Because deficit irrigation can occur, Et can vary from 0 to the maximum Et under non stress soil moisture conditions. Cp can vary between 0 and 10 mg/L based on limitation in the United States imposed by EPA and state regulator agencies. Cc depends on the crop grown and does not change for a given crop. The slop and intercept of the evapotranspiration production function can be acquired from an internet site at http://weather.nmsu.edu  under the water management of NM crop section. These water production functions are applicable for the Western United States and similar aired area in the world. They do not apply for area with humid conditions. The fraction of total nitrogen removed by dentrification is set for 0 2 only because information is lacking on the effect of irrigation management and high water tables on denitrification rates. A conservative number would be to assume that 0 denitrification occurs. A copy of the model can be downloaded in metric or english units. The model is set up to use Solver in Excel to solve the equation based on maximizing the irrigation efficiency. The limits for Et must be changed for the selected crop and for the minimum yield per acre and  associated Et that is economically viable. Use the evapotranspiration production function to determine the minimum Et setting that should be used when using Solver.

References

Metcalf and Eddy, Inc. 1991.  Wastewater engineering:treatment, disposal, and reuse.  McGraw Hill:New York, 815 pp.
W.C.P.F.  1990.  Natural systems for wastewater treatment.  Manual of Practice FD-16.  Water Pollution Control Federation, Alexanderia, VA. 270 p.