2.1. Maturation pond (dispersed flow)

(1) For the hydraulic retention time (O), the methodology is iterative and the way to carried out is to propose a retention time, later are determined the fecal coliform concentration and the BOD in the output of the pond. The norm indicates that the pollutants must be equal or less than 1000 MPN/100 mL and 75 mg/L respectively

(a) For determining the volume of the pond we have:

(1)

(b) Area of the pond:

(2)

(c) Width-length ratio X = 3:

(3)

(4)

(d) For defining the superior width and length we have:

(5)

(6)

(e) Area of the superior water surface:

(7)

(f) Flow in the output of the lagoon:

(8)

(g) Fecal coliform decay with screens at 70 percent of the length:

(9)

(10)

(h) Bacterial decay coefficient:

(11)

(i) For "a" constant we have:

(12)

(j) Number of fecal coliform in the output of the pond:

(13)

(k) Number of fecal coliform corrected by evaporation:

(14)

(l) Kinetic coefficient:

(15)

(m) BOD concentration of the pond:

(16)

(n) BOD removal efficiency:

(17)

(o) BOD modified by evaporation:

(18)

Fig. 1 shows the operative route to perform the optimal design of a maturation pond.

Fig. 1 Flow chart for the optimal design of a maturation pond (). 

3.1 Objective function

The cost of land was considered to be $ 750.00 per square meter; perimeter wall: $ 1200.00 per meter; cost of floor slab: $1200.00 per square meter; cost of screens: $ 1200.00 per meter. With these data the expression (20) is determined. In the expression (20) is substituted the land area for the expressions (5) and (6) of the methodology, then the perimeters is substituted for: 2B _Sup + 2L _Sup. The floor slab area is substituted by the formulas (5) and (6); and the screen length can by represented by the multiplication of the number of screens and the length, which according to the methodology is the 70 percent of the lagoon length. The expression (21) shows the substitutions: The expression (21) can by simplified by using the length: width ratio of 3. The expression (22) shows the mentioned above:

(20)

In the expression (20) is substituted the land area for the expressions (5) and (6) of the methodology, then the perimeters is substituted for: 2B_Sup + 2L_Sup. The floor slab area is substituted by the formulas (5) and (6); and the screen length can by represented by the multiplication of the number of screens and the length, which according to the methodology is the 70 percent of the lagoon length. The expression (21) shows the substitutions:

(21)

The expression (21) can by simplified by using the length: width ratio of 3. The expression (22) shows the mentioned above:

(22)

In order to define the decision variables (O and N _Mamp ); must be cleared the volume in the expression (2) and replaced by the formula (1); as seen in formula (23):

(23)

Later the expression (23) is substituted in formula (3); in this case the slope of the perimeter wall is cero, so B_Sup is equal to B_Prom and is obtained:

(24)

Then the expression (24) is substituted in formula (22), as seen in expression (25):

(25)

Finally the formula (24) and (25) are substituted in the objective function (21), and is obtained:

(26)

The formula (26) can by simplify as shown in expression (27):

(27)

3.2. Model constraint

Following the criterion used by ^{Martínez, Cansino, García, Kalashnikov, and Rojas (2014)}, for determining the model's constraints of a facultative pond are taken the formulas (28) and (29): where

(28)

(29)

(30)

(31)

(32)

(33)

The superior width and length (formulas (24) and (25)) are substituted in the expressions (28) and (33), that is the way to link the constraints to the objective function (27); the model depends on the decision variables (O and N _Mamp ). The above process is shown below:

(34)

(35)

Finally the optimization model is presented complete (expression (36)), the fecal coliform and the BOD are restricted by the norm NOM-001-ECOL-96. Also, it is added a non-negativity condition for the decision variables (O _M, N _Mamp ≥ 0).

(36)

4.1. Results and discussion

With the data outputs from Table 1 is designed the maturation pond using the traditional methodology, without considering screens. Table 2 shows the results.

Fig. 3 shows the dimensions of the pond system determined with the traditional methodology, as already indicated there were no screens considered in the maturation pond. In accordance with Table 2, were obtained the fecal coliform and the BOD within the maximum pollutant allowed by norm.

Fig. 3 Treatment system redesign with the traditional method. 

4.2. Mathematical model application

In order to solve the minimization model, it is used the interior point algorithm. The model (expression (36)) must be written in code as shown below:

The model was solved by Matlab's Fmincon function, with the interior point algorithm. The model determines the variables, in this case the total cost, retention time and number of screens. The results are shown in Table 3.

Following the criteria of Fig. 1, the number of screens turned out to be a decimal number, so it is rounded to the next integer number, then N _Mamp = 6. With this data are calculated the rest of the results with the traditional methodology, as seen in Table 4.

According to Tables 2 and ⁴, the hydraulic retention time was reduced by 8.65 days, which represents the 48.18 percent. According to ^{CNA and IMTA (2007a}, ^2007b) the above reduction affects the area of land required. It is important to mention that unlike the traditional design, which is iterative, the interior point algorithm determined the optimum retention time and number of screens, and this would be very complicated to achieve with the traditional method because it has more than one independent variable.

With the Matlab software were inferred that 6 screens would be the best, about this, ^{Shilton and Mara (2005)}, ^{Abbas et al. (2006)}, ^{Shilton and Harrison (2003a)}, ^{Muttamara and Puetpaiboon (1997)}, ^{Bracho et al. (2006)}, ^{Winfrey et al. (2010)}, ^{Kilani and Ogunrombi (1984)}, ^{Muttamara and Puetpaiboon (1996)}, ^{Von Sperling, Chernicharo, Soares, and Zerbini (2002)}, ^{Shilton and Harrison (2003b)} carried out studies in stabilization ponds were different numbers of screens were considered, and They concluded that the uses of screens increases the elimination of the pathogens, also the hydraulic flow gets improved within the pond. The present paper confirms the conclusions made by these authors.

Tables 2 and 4 shows that the fecal coliform and the BOD are below the maximum contaminant allowed by norm NOM-001-ECOL-96 ^{DOF (1996)}. The area in the optimized design was reduced by 1 805.98 square meters: which represents the 48.16 percent. The difference in the area is an important saving in land requirement. According to ^{CNA and IMTA (2007a}, ^2007b), the major disadvantage of the ponds systems is the large area required.

The cost saving with the optimized lagoon, regard the traditional method, was 42.24 percent which represents $3,231,865.65. About this, ^{Olukanni and Ducoste (2011)} mentioned the feasibility to diminish the construction cost of a pond system as long as the constraints are defined correctly; i.e., when the mathematical model is applied all the pollutants are within the norm.

Fig. 4 shows the dimensions of the maturation pond and the number of screens for the optimization model.

Fig. 4 Treatment system redesign with the optimization model. 

4.3. Sensitivity analysis

According to ^{Anderson, Sweeney, and Williams (1999)}, sensitivity analysis can be done using a tornado diagram. The aforementioned study consists of modifying the values of the main variables in order to observe how the optimum solution is affected.

The tornado diagram uses bars to define sensitivity; i.e., the widest bar indicates the most sensitive parameter on which the constraints depend on. For sensitivity analysis of fecal coliform and BOD, we established the cells on which they depend, and then their values are changed ±10 percent (^{Muramatsu, 2011}).

Fig. 5 shows the sensitivity analysis for fecal coliform. The wide bar is the hydraulic retention time, it is interpreted that the concentration of fecal coliform increases when there is lower retention time. The second parameter that influences the fecal coliform removal is temperature: lower temperatures mean higher concentration of the indicator organism, in order of importance the tornado chart continues with the number of partitions.

Fig. 5 Sensitivity analysis of fecal coliform in the pond system. 

Regarding sensitivity analysis for biochemical oxygen demand (Fig. 6), BOD concentration in the influent represents the most sensitive parameter. Next, in order of importance, the temperature: the lower the temperature, the pollutant removal efficiency decreases and increases the concentration of organic matter in the effluent. Finally, the hydraulic retention time is interpreted that a shorter retention time increases the concentration of organic matter in the effluent.

Fig. 6 BOD sensitivity analysis in the pond system.