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

Where:

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

ET = Design evapotranspiration rate (ft/yr)

Pr = Design precipitation rate (ft/yr)

Wp = Design percolation rate (ft/yr)

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

Ln = U + D + 2.7 Wp Cp  ………………………………………  [Eq. 2]

Where:

U = Crop nitrogen uptake (lb/acre/yr)

D = Denitrification (lb/acre/yr)

Wp = Percolating water (ft/yr)

Cp = Percolate nitrogen concentration (mg/L)

Ln = 2.7 Lw Cn  ……………………………………….. [Eq.3]

Where

Lw = Wastewater applied (ft/yr)

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

Solve for Wp in Eq. 2 yields:

Wp = (Ln – U- D)/ 2.7 Cp

Substitute Wp term in Eq. 1 yields:

Lw = ET – Pr + (Ln –U – D)/2.7 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 (lb/acre/yr)

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

D = F Ln

Lw = ET – Pr + (Ln –U – F  Ln)/2.7 Cp    ……………………… [Eq. 6]

Plug in Eq. 3 into Eq. 6 yields

Lw = ET – Pr + (2.7 Lw Cn –U – (2.7 F Lw Cn)/2.7 Cp    ……………. [Eq. 7]

Simplified Eq. 7:

Lw = (ET – Pr) + [2.7 Lw Cn (1 – F) – U ]/2.7 Cp

2.7 Lw Cp = 2.7 Cp (ET –Pr) + 2.7 Lw Cn (1-F) –U

2.7 Lw Cp - 2.7 Lw Cn (1-F) = 2.7 Cp (ET –Pr) –U

2.7 Lw (Cp - Cn (1-F) = 2.7 Cp (ET –Pr) –U

Lw (Cp - Cn (1-F) = Cp (ET –Pr) –U/2.7

Lw (Cp - Cn (1-F) = Cp (ET –Pr) –U/2.7

Lw = [Cp (ET – Pr) – U/2.7] / (Cp – Cn (1-F))

Lw = [Cp (ET – Pr) – U/2.7] / (Cp – Cn (1-F)) …………………………….. [Eq. 8]

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

Lw = [Cp (Pr - ET) + 4.4 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) + (4.4 (a+ b ET) Cc)] / (Cn (1-F)-Cp)  ……………….. [Eq. 11]

Where:

ET = design ET rate (inch/yr)

Pr  = design precipitation rate (inch/yr)

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

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

## a  = intercept of the Evapotanspiration production function (lb/acre)

b  = slope of the Evapotanspiration production function (lb/acre/inch)

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