**3.4. Estimation of oxygen flux due to diffusive transportation**

**Figure 4.** Relative concentrations vs. depth, Test 2 (P2).

70 Agroecology

**Figure 5.** Relative concentrations vs. depth, Test 3 (P3).

Once it is established that the aeration of the soil is produced mainly by diffusive transportation phenomena, the estimation of the oxygen flux using Equation 7 will require information of the edaphic respiration, which is only obtained by field experimentation; this constitutes a limitation for the application of this model.

For this reason, the estimation of the oxygen flux in this research will be performed considering a simplified diffusive model that will require no information related to the edaphic respiration but will allow the adequate estimation of a minimum level of oxygen flux in the soil.

$$\frac{\partial \mathbf{C}}{\partial t} = \mathbf{D} \frac{\partial^2 \mathbf{C}}{\partial \mathbf{x}^2} \tag{8}$$

In order to determine the oxygen flux in a simplified manner, this research selected the analytic solution of the Law of Fick (Equation 8), proposed in [17] — the determination of the oxygen flux considers the elapsed time, the oxygen diffusivity of the soil, and the concentration of oxygen in the atmosphere and air in the soil, but it does not consider the edaphic respiration, as shown in Equation 9. This model establishes a minimum concentration of oxygen in the root area of the soil in order to avoid negative effects in the crop development.

$$\mathbf{N}\_{\mathrm{O2}} = 2 \left( \mathbf{C}\_{\mathrm{O2}} - \mathbf{C}\_{\mathrm{P}} \right) \* \sqrt{\mathrm{D} \mathbf{P}^{\star} \mathbf{t}\_{\circ}} \sqrt{\pi} \tag{9}$$

NO2 = Flux of oxygen crossing the soil surface [g/m2]

CO2 = Vapour phase O2 concentration above the soil surface [g/m3 o mg/l]

Cp = Vapour phase O2 concentration required in soil to prevent adverse yields or root growth [g/m3 o mg/l]

Dp = Effective diffusion coefficient [m2/d]

ta = Aeration time

$$\text{Dp} = 0.66^\* \,\, \epsilon^\* S^\* \text{Do2} \tag{10}$$

Tortuosity = 0.66 (fraction)

є= Porosity (fraction)

S = Fraction of air filled soil pore volume at field capacity (fraction)

DO2 = Oxygen diffusivity in air [m2/d]

**Figure 6.** Outline of oxygen balance

The adequate flux of soil oxygen is achieved by means of a balance between the amount of equivalent oxygen required by the organic matter present in the wastewater (Equation 4) and the time period needed in order to achieve the soil oxygenation by diffusive transportation, expressed in Equation 9. This balance can be observed in the framework shown in Figure 6.

#### **3.5. Time between applications**

oxygen in the atmosphere and air in the soil, but it does not consider the edaphic respiration, as shown in Equation 9. This model establishes a minimum concentration of oxygen in the root

Cp = Vapour phase O2 concentration required in soil to prevent adverse yields or root growth

Dp 0.66\* \* \* 2 = ò

The adequate flux of soil oxygen is achieved by means of a balance between the amount of equivalent oxygen required by the organic matter present in the wastewater (Equation 4) and the time period needed in order to achieve the soil oxygenation by diffusive transportation, expressed in Equation 9. This balance can be observed in the framework shown in Figure 6.

<sup>π</sup> = - (9)

o mg/l]

*S Do* (10)

( ) <sup>a</sup> O2 O2 P Dp\*t N 2C C \*

area of the soil in order to avoid negative effects in the crop development.

NO2 = Flux of oxygen crossing the soil surface [g/m2]

Dp = Effective diffusion coefficient [m2/d]

[g/m3

72 Agroecology

o mg/l]

ta = Aeration time

Tortuosity = 0.66 (fraction)

**Figure 6.** Outline of oxygen balance

DO2 = Oxygen diffusivity in air [m2/d]

є= Porosity (fraction)

CO2 = Vapour phase O2 concentration above the soil surface [g/m3

S = Fraction of air filled soil pore volume at field capacity (fraction)

In order to determine the time that is needed between applications of wastewater, Equation 10 is used [12]. This equation considers, in addition to the aeration time, the time that it takes to infiltrate the water in the soil (Equation 11) and the duration of irrigation.

$$\mathbf{t}\_{\rm ap} = \mathbf{t}\_{\rm air} + \mathbf{t}\_{\rm in} + \mathbf{t}\_{\rm r} \tag{11}$$

tap = Time between applications[d]

tair = Time of aeration [d]

tin = Time of infiltration [d]

tr = Time of irrigation [d]

$$\mathbf{t}\_{\rm ln} = \frac{\mathbf{L}\mathbf{n}}{\mathbf{k}}\tag{12}$$

Ln = Hydraulic load rate (depth) [mm]

k = Infiltration or permeability in the saturated soil [mm/d]

#### **3.6. Nitrogen balance**

This balance is established through the determination of the hydraulic rate of the wastewater to be applied based on its nitrogen content; the following process differs from the design of the hydraulic load developed in Equations 1 and 2. In this case, the required hydraulic load is estimated based on the amount of nitrogen that can be removed by a crop as part of its nutritional requirements, in a determined land area and time. Equation 13 determines the nitrogen load that is to be incorporated into the soil for the crop to remove it adequately. This equation also considers the factor of nitrogen loss due to denitrification and volatilization. The result of this equation, divided by the nitrogen concentration in the wastewater, determines the hydraulic load rate as shown in Equation 14 [12].

$$L\_n = \frac{\mathcal{U}}{(1-f)}\tag{13}$$

Ln = Nitrogen loading [kg/(ha\*year)]

U = Estimated crop uptake as a function of yield [kg/(ha\*year)]

f = Nitrogen loss factor (0.5–0.8) (fraction)

$$\mathbf{L} \mathbf{n}\_{\text{nitrogen}} = \frac{\mathbf{L}\_{\text{nitrogen}}}{\mathbf{C}\_{\text{nitrogen}}} \mathbf{\*} \mathbf{F} \tag{14}$$

Ln nitrogen = Hydraulic nitrogen load based on the nitrogen load [mm/d] Cnitrogen = Nitrogen concentration in the waste water [mg/l or g/m3]

#### F = Conversion factor

#### **3.7. Application area**

The determination of the required area for the application of wastewater in the soil is obtained from the relation between the volume of wastewater discharge and the hydraulic load rate to be applied in the soil, as shown in Equation 14 [6, 9, 12].

$$\mathbf{A} = \frac{\mathbf{Q}}{\mathbf{L}\mathbf{n}\_d} \mathbf{^\*\mathbf{F}} \tag{15}$$

A = Area required for the wastewater application [ha]

Q = Wastewater discharge volume [m3/d]

Lnd = Design hydraulic load rate [mm/d]

F = Conversion factor
