**3. Selected mathematical models**

The characterization of the wastewater presented in this study is based on the application of the STAR ASA model.

#### **3.1. Water balance**

In the low load application Type I, the water balance allows for the determination of the hydraulic load that has to be applied to the soil to at least fulfil the crop's requirement, considering the part of water that percolates to the groundwater based on the permeability of the least permeable layer. Equation 1 shows the balance between the applied water (depth and precipitation) and use of water due to percolation and evapotranspiration [5, 9, 11]. Figure 1 shows a conceptual framework of this balance.

$$\mathbf{L}\_{\rm w} = \mathbf{E}\mathbf{t}\_{\rm c} - \mathbf{P}\mathbf{r} + \mathbf{P}\_{\rm w} \tag{1}$$

Lw= Wastewater hydraulic loading rate — based on soil permeability [mm/d]

Etc = Evapotranspiration rate [mm/d]

Pr = Precipitation rate [mm/d]

Pw =Design percolation rate [mm/d]

**Figure 1.** Water balance framework based on the soil permeability

Ln= Wastewater hydraulic loading rate — based on soil permeability [mm/d]

Etc = Evapotranspiration rate [mm/d]

Pr = Precipitation rate [mm/d]

The success in the application of effluents with a high organic load onto the soil will depend on the correct interpretation of the phenomena that occur in the soil and its relation with the plant. For this reason, the computer model STAR ASA has been developed. This model uses a set of mathematical equations taken from different authors, that, when applied within a process, allow the estimation of the conditions to be considered when the wastewaters are applied to the soil, making the decision process easier. The selected models were selected from those that obtained positive results and the most conservative results, in order to avoid the saturation of the soil and the contamination of groundwater. The application of this model will provide the wastewater volume that can be applied to the soil, the required land surface for the application, the minimum necessary time between applications, and the amount of

The characterization of the wastewater presented in this study is based on the application of

In the low load application Type I, the water balance allows for the determination of the hydraulic load that has to be applied to the soil to at least fulfil the crop's requirement, considering the part of water that percolates to the groundwater based on the permeability of the least permeable layer. Equation 1 shows the balance between the applied water (depth and precipitation) and use of water due to percolation and evapotranspiration [5, 9, 11]. Figure 1

Lw= Wastewater hydraulic loading rate — based on soil permeability [mm/d]

wc w L Et Pr P = -+ (1)

nutrients incorporated to the soil.

the STAR ASA model.

**3.1. Water balance**

64 Agroecology

**3. Selected mathematical models**

shows a conceptual framework of this balance.

**Figure 1.** Water balance framework based on the soil permeability

Etc = Evapotranspiration rate [mm/d]

Pw =Design percolation rate [mm/d]

Pr = Precipitation rate [mm/d]

Pw =Design percolation rate [mm/d]

In the application of low load Type II, the water balance focuses on the better use of the irrigation water than in a water treatment system. However, it is considered in this study since it allows the determination of the minimum hydraulic load rate required by the crop for its optimal development. This model includes in the calculation the leaching requirement and the irrigation efficiency, which is defined as the capacity of irrigating a determined volume of water into a uniform land surface. Equation 2 [6,9] shows this balance, which can also be seen in the conceptual framework of Figure 2.

$$\text{Lw} = \left(\text{Et}\_c - \text{Pr}\right) \* \frac{\left(1 + \text{LR}\right)}{\text{E}\_i} \tag{2}$$

Lw = Wastewater hydraulic loading [mm/d]

Etc = Crop evapotranspiration [mm/d]

Pr = Precipitation rate [mm/d]

LR = Leaching requirement (fraction)

Ei = Distribution system efficiency (fraction)

**Figure 2.** Water balance framework based on irrigation

The leaching requirement is used in the application of low load Type II in order to estimate the amount of water to be added in addition to the crop's requirement, avoiding the accumu‐ lation of salts in the root area of the plant, therefore preventing adverse effects for the crop development. Equation 3 [10, 14] represents this calculation, with help of the electric conduc‐ tivity of the applied wastewater and the electric conductivity that the soil can withstand without affecting its performance.

$$\text{LR} = \frac{\text{CE}\_{\text{w}}}{\text{5(CE}\_{\text{e}}) - \text{CE}\_{\text{w}}} \tag{3}$$

LR = Leaching requirement (fraction)

CEw = Average conductivity of irrigation water [dS/m]

CEe = Required conductivity in drainage water to protect de crop [dS/m]

Ln= Wastewater hydraulic loading rate — based on soil permeability [mm/d]

Etc = Evapotranspiration rate [mm/d]

Pr = Precipitation rate [mm/d]

LR = Leaching requirement (fraction)

#### **3.2. Oxygen balance**

As mentioned previously, the application of wastewater from the food and beverage industries implies the use of large concentrations of organic matter with a high BOD5. Consequently, it is important to estimate the amount of equivalent oxygen required for the oxidation of this organic matter by means of the microorganisms, maintaining aerobic conditions in the soil. It must be considered that the BOD5 concentration does not show the Total Oxygen Demand (TOD) existing in the wastewaters. Therefore, Equation 4 [12] estimates the TOD by means of the addition of the BOD5 and the concentration of the oxygen demand of the nitrogenous compounds (NOD):

$$\text{TOD} = \text{BOD} + \text{NOD} \tag{4}$$

TOD = Total oxygen demand [g/m3 ó mg/l]

BOD5 = Biochemical oxygen demand [g/m3 ó mg/l]

NOD = Nitrogenous oxygen demand [g/m3 ó mg/l]

$$\text{NOD} = 4.56^\circ\\\text{Nitrifiable Nitrogen}\tag{5}$$

The oxygen demand by the nitrogenized compounds is calculated from the wastewater nitrificable nitrogen concentration [mg N-NH3l] multiplied by an estequiometric coefficient equal to 4.56 mg of oxygen [12, 17].

#### **3.3. Analysis of the diffusive transportation in the soil**

In this study, the estimate of the soil oxygenation by means of the application of wastewaters with a load of organic matter is performed considering the diffusive transportation of the oxygen through the soil as the main source of oxygenation. To prove this supposition, it was necessary to use the analytic solution developed by Papendrick and Runkles [18], based on the second law of Fick (Equation 6). This solution allows the estimation of the concentration of oxygen in the soil, at a given and determined depth and time, considering a constant respiration (Equation 7). The solution proposed by these authors was proven by Kanwar [19] and Prasanta [20], demonstrating that the flux of oxygen (mass O2/area x time) is directly proportional to the gradient of oxygen concentration between the atmosphere and the soil, and with the microbial oxidation rate. Those researchers consider the microbial oxidation rate to be constant. However, in field conditions, the respiration rate increases with the fertility of the soil and consequently the flux of oxygen in the soil.

In this study, the estimate of the oxygenation of the land irrigated with wastewater with high loads of organic matter, considers the oxygen diffusive transport through the soil as the main source of oxygen, which is used by the model developed in [18] that calculates the concentra‐ tion of oxygen in depth and time.

To calculate the required oxygen flow for soil aeration, an equation is used that considers the elapsed time, the diffusivity of oxygen in soil and the oxygen concentration in the atmosphere and soil [12, 17], through a balance between the amount of equivalent oxygen that requires the organic matter from wastewater and the time that must elapse to reach the oxygenation of the soil through diffusive transport.

$$\frac{\partial \mathbf{C}}{\partial t} = \mathbf{D} \frac{\partial^2 \mathbf{C}}{\partial \mathbf{x}^2} - a \tag{6}$$

The initial and threshold conditions for the application of this model are:

$$\begin{aligned} \mathbf{C}(\mathbf{x},0) &= \mathbf{C}\mathbf{t} \\ \mathbf{C}(0,\mathbf{t}) &= \mathbf{C}\mathbf{o} \\ \delta \mathbf{c}(\mathbf{t})/\delta \mathbf{x} &= 0 \end{aligned}$$

Where:

( )

CE LR

CEe = Required conductivity in drainage water to protect de crop [dS/m]

Ln= Wastewater hydraulic loading rate — based on soil permeability [mm/d]

LR = Leaching requirement (fraction)

Etc = Evapotranspiration rate [mm/d]

LR = Leaching requirement (fraction)

TOD = Total oxygen demand [g/m3 ó mg/l]

equal to 4.56 mg of oxygen [12, 17].

BOD5 = Biochemical oxygen demand [g/m3 ó mg/l] NOD = Nitrogenous oxygen demand [g/m3 ó mg/l]

**3.3. Analysis of the diffusive transportation in the soil**

Pr = Precipitation rate [mm/d]

**3.2. Oxygen balance**

66 Agroecology

compounds (NOD):

CEw = Average conductivity of irrigation water [dS/m]

w e w

As mentioned previously, the application of wastewater from the food and beverage industries implies the use of large concentrations of organic matter with a high BOD5. Consequently, it is important to estimate the amount of equivalent oxygen required for the oxidation of this organic matter by means of the microorganisms, maintaining aerobic conditions in the soil. It must be considered that the BOD5 concentration does not show the Total Oxygen Demand (TOD) existing in the wastewaters. Therefore, Equation 4 [12] estimates the TOD by means of the addition of the BOD5 and the concentration of the oxygen demand of the nitrogenous

The oxygen demand by the nitrogenized compounds is calculated from the wastewater nitrificable nitrogen concentration [mg N-NH3l] multiplied by an estequiometric coefficient

In this study, the estimate of the soil oxygenation by means of the application of wastewaters with a load of organic matter is performed considering the diffusive transportation of the oxygen through the soil as the main source of oxygenation. To prove this supposition, it was necessary to use the analytic solution developed by Papendrick and Runkles [18], based on the second law of Fick (Equation 6). This solution allows the estimation of the concentration

5 CE CE <sup>=</sup> - (3)

TOD BOD NOD = + (4)

NOD 4.56\*Nitrifiable Nitrogen = (5)

C(x, t) = Oxygen concentration at a given depth (x) and at a time (t) [cm3 O2/cm3 air]

Ct = Initial concentration of oxygen in the ground [cm3 O2 / cm3 air]

Co = Concentration of oxygen in the atmosphere [cm3 O2 / cm3 air]

t = Time [h]

D = Diffusivity of oxygen in the soil [cm2 / h]

x = Depth [cm]

α = Edaphic respiration rate [h-1]

The following solution is established:

$$\mathbf{C}\{\mathbf{x},\mathbf{t}\} = \mathbf{C}\_{\mathbf{t}} - \alpha t + \left(\mathbf{C}\_{o} - \mathbf{C}\_{\mathbf{t}}\right)^{\mathbf{\*}} \text{erfc}\frac{\mathbf{x}}{2\sqrt{\mathbf{D}^{\mathbf{\*}}\mathbf{t}}} + \alpha \left[ \left(\mathbf{t} + \frac{\mathbf{x}^{2}}{2\mathbf{D}}\right) \text{erfc}\frac{\mathbf{x}}{2\sqrt{\mathbf{D}^{\mathbf{\*}}\mathbf{t}}} - \mathbf{x} \sqrt{\left(\frac{\mathbf{t}}{\pi^{\mathbf{\*}}D}\right)^{\mathbf{\*}} \mathbf{e}^{\frac{-\left(\mathbf{x}^{2}\right)}{4D^{\mathbf{T}}}}} \right] \tag{7}$$

Where:

erfc = Complementary error function

In order to determine the oxygen flux in the soil, Equation 7 was applied with typical values of oxygen respiration and diffusivity (D). Table 4 shows three sets of values or tests used in the research.


**Table 3.** Typical values of oxygen respiration and diffusivity.

In each test, an equation with seven depth values (x) and six different aeration times (t) was applied. The relative concentrations of oxygen obtained in each case are shown in Table 4.


Characterization of Industrial Highly Organic Wastewater to Evaluate Its Potential Use as Fertilizer in Irrigation… http://dx.doi.org/10.5772/59999 69


**Table 4.** Oxygen concentrations with values of α and D.

( ) ( )

Where:

68 Agroecology

the research.

of air in the soil)

Respiration (α)

(g of O2/m3 of air in the soil\*d)

a

t ot

erfc = Complementary error function

Initial concentration of oxygen (Ct) (cm3 of O2/cm3

**Table 3.** Typical values of oxygen respiration and diffusivity.

**Relative Concentration of Oxygen (C/C0)**

x xx t C x,t C C C \*erfc t erfc x \*e 2 D\*t 2D 2 D\*t \* *<sup>t</sup>*

æ ö æ ö ê ú =-+ - + + ç ÷ - ç ÷ ê ú

In order to determine the oxygen flux in the soil, Equation 7 was applied with typical values of oxygen respiration and diffusivity (D). Table 4 shows three sets of values or tests used in

In each test, an equation with seven depth values (x) and six different aeration times (t) was applied. The relative concentrations of oxygen obtained in each case are shown in Table 4.

X (cm) t = 0 t = 4 t = 8 t = 12 t = 16 t = 20

 1 0.983 0.974 0.9673 0.20997 0.9566 1 0.972 0.955 0.9421 0.20995 0.9210 1 0.966 0.942 0.9231 0.20994 0.8922 1 0.962 0.933 0.9090 0.20994 0.8691 1 0.961 0.927 0.8990 0.20993 0.8510 1 0.960 0.924 0.8919 0.20993 0.8369

15 1 0.970 0.953 0.9398 0.92853 0.9186 30 1 0.958 0.927 0.9021 0.88051 0.8613 45 1 0.954 0.914 0.8800 0.84983 0.8225

P1 0 1 1 1 1 1 1

P2 0 1 1 1 1 1 1

Diffusivity of oxygen in the soil (D) (cm2 of soil / h) 259.2 89.1 89.1

 a ( ) <sup>2</sup> <sup>x</sup> <sup>2</sup>

é ù -

è ø è ø ë û

**1st Test (P1) 2ª Test (P2) 3ª Test (P3)**

0.21 0.21 0.21

0.002125 0.0025 0.0125

4D\*t

(7)

*D*

p

Once the relative concentrations of oxygen are calculated, the curves were drawn for each test, obtaining the polynomial equations based on the depth, as shown in Figures 3, 4, and 5.

**Figure 3.** Relative concentrations vs. depth, Test 1 (P1).

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

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

Using the previous diagrams, the polynomial equations that describe the profile of the oxygen concentration in the soil for each time step were obtained. The derivation of the polynomial, evaluated at x=0 (soil surface), presents the flux for each time. For a constant degradation, in a given period, the area under the curve flux vs. time will calculate the amount of consumed oxygen. In this way, the mass of oxygen represents the load of BOD that could be accepted per day. Table 6 shows the oxygen flux in the soil for each one of the performed tests.


**Table 5.** Oxygen flux estimated using typical values of respiration and diffusivity.

With these results, the flux of oxygen that the soil can receive, using typical values of respiration and diffusivity, was determined to be between 102.5 y 386.5 g/m2 . On the other hand, it was observed that the flux of oxygen in the soil is directly proportional to the increase of the edaphic respiration and depends on the coefficient of diffusivity of the soil. These two factors determine the increase of the gradient of the oxygen concentration of the soil, which increases the flux of oxygen.

Considering the obtained results, this research established the application of wastewater with oxygen requirements lower than 100g/m2 as a maximum, avoiding soil anaerobic conditions.
