Optimal design of integrated agricultural water networks

Computers and Chemical Engineering 84 (2016) 63–82

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Computers and Chemical Engineering journal homepage: www.elsevier.com/locate/compchemeng

Optimal design of integrated agricultural water networks Eusiel Rubio-Castro a,∗ , José María Ponce-Ortega b , Maritza Elizabeth Cervantes-Gaxiola a , Oscar Martín Hernández-Calderón a , Jesús Raúl Ortiz-del-Castillo a , Jorge Milán-Carrillo a , José Francisco Hernández-Martínez a , José Antonio Meza-Contreras a a b

Chemical and Biological Sciences Department, Universidad Autónoma de Sinaloa, Culiacán, Sinaloa 80000, Mexico Chemical Engineering Department, Universidad Michocana de San Nicolás de Hidalgo, Morelia, Michoacán 58060, Mexico

a r t i c l e

i n f o

Article history: Received 16 November 2014 Received in revised form 7 August 2015 Accepted 11 August 2015 Available online 20 August 2015 Keywords: Water integration Fertilizer reuse Agricultural water Multi-period optimization

a b s t r a c t This paper presents a mathematical programming model for the optimal design of water networks in the agriculture. The proposed model is based on a new superstructure that includes all configurations in terms of use, reuse and regeneration of water in a field constituted by a number of croplands. The model also includes the allocation of pipelines, pumps and storage tanks in different irrigation periods. The objective function consists in maximizing the annual profit that is formed by the economic incomes owing to the crop sell minus the costs for fresh water, fertilizer, storage tanks, treatment units, piping and pumping. The proposed multi-period optimization problem is formulated as a mixed integer non-linear programming formulation, which was applied to a case study to demonstrate the economic, environmental and social benefits that can be obtained. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Agriculture is the activity with the highest water demand in the world. Besides it contributes to the water pollution due to the use of nutrients (fertilizers) and pesticides that are discharged as wastewater. Agriculture helps to the ecosystems biodiversity, provides elements to the flue gas capture, contributes to enhance the landscape, is an important factor to combat the world hunger and represents great economic profits (OECD, 2013). In this sense, agriculture is vital to the human being, therefore several technologies and methodologies have been developed to improve seeds, fertilizers, pesticides, process equipment, storage and preservation feeds as well as irrigation systems. The irrigation technologies have been focused on the reduction of fresh water and the application of this resource for increasing the yield of crops. Here, water reduction can be analyzed from two perspectives: (a) Inside the process: It consists in finding the best irrigation technique as well as the reuse of wastewater, (b) Outside the process: It is related to the reuse of wastewater coming from industrial activities. While a combination of the above scenarios produces a simultaneous scheme to consider the reuse of treated water, the irrigation technique as well as the water reuse. Nevertheless, the simultaneous scheme could be a good strategy to optimize the water use in agriculture, the reported works have

∗ Corresponding author. Tel.: +52 667 7137860x115. E-mail address: [email protected] (E. Rubio-Castro). http://dx.doi.org/10.1016/j.compchemeng.2015.08.006 0098-1354/© 2015 Elsevier Ltd. All rights reserved.

been focused on reducing the fresh water consumption. In this way, Wilson and von Broembsen (2010) studied the advantages and disadvantages to reuse wastewater in greenhouses. Anderson (2003) presented an analysis about the environmental benefits related to water reusing. Lazarova and Bahri (2004) proposed a strategy for planning water reusing. Furthermore, the water reusing has been studied from different points of view, including removing heavy metals (Petruzzelli, 1989; Wu et al., 1998), health risks (Chang et al., 1996; Shuval et al., 1997) and irrigation costs (Schleich et al., 1996; Schwarz and Mcconnell, 1993). Hussain et al. (2002) presented a review about the characteristics and international regulations of wastewater utilized in agriculture, and the positive as well as negative impacts owing to the use of treated water. Besides, the rainwater harvesting has been the most used strategy to meet with the water demand for the agriculture in several parts of the world. In this regard, several works dealing with the hydrological impact on watersheds due to the application of domestic and agricultural rainwater harvesting have been reported (Ghimire and Johnston, 2013). Other approaches have implemented economic analysis of different rainwater harvesting structures (Goel and Kumar, 2005). Furthermore, Hatibu et al. (2006) determined the economic aspects of harvesting rainwater, He et al. (2007) analyzed the factors that affect the rainwater harvesting, and Jiang et al. (2013) presented the benefits in the consumption of fresh water and energy associated to the rainwater harvesting. In addition, some reviews about rainwater harvesting have been

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Notation

Cfe

Parameters ˛ exponent to consider the economies of scale Ac crops area, ha Cf fixed cost of equipment and accessories, US$ irrigation criteria Cr Cusl unitary sell price of crops, US$/ton c op Cuw unitary cost of fresh water, US$/m3 op Cul unitary cost of fresh fertilizer, US$/kg op Cue unitary cost of electricity, US$/kWh op Cup unitary cost for pipe lines maintenance, US$/kg op unitary cost for storage tanks maintenance, US$/kg Cus op Cuu unitary cost for treatment units maintenance, US$/kg variable cost of pipelines, Cvpip US$/m3 /h Cvpu variable cost of pumps, US$/kg variable cost of storage tanks, US$/m3 Cvsw tu Cv variable cost of treatment units, US$/m3 distance between units, m D Dr root depth, cm Fwpc precipitated water in the crops, m3 Fwpst precipitated water in the storage tanks, m3 capillary rise of water table, m3 Fwrc g acceleration due the gravity, m2 s−1 gc conversion factor for the acceleration due the gravity, kg m N−1 s−2 He height between process components, m  efficiency KF factor used to annualize the inversion, year−1 m fractional interest rate per year, % number of years of operation, year n v velocity, m/s density, kg/m3   fc field capacity, cm3 of water/cm3 of soil permanent wilting point, cm3 of water/cm3 of soil pwp in Qwc volumetric flowrate, m3 /h max,pu ˝Fcc upper limit of the pump capacity to handle the flowrate between crops, kg/h max,pip Fcc upper limit of the pipeline capacity to handle the flowrate between crops, m3 /h max,pu ˝Fstc upper limit of the pump capacity to handle the flowrate from storage tanks to crops, kg/h max,pip ˝Fstc upper limit of the pipeline capacity to handle the flowrate from storage tanks to crops, m3 /h max,pu ˝Fcst upper limit of the pump capacity to handle the flowrate from the crops to storage tanks, kg/h max,pip ˝Fcst upper limit of the pipeline capacity to handle the flowrate from the crops to storage tanks, m3 /h max,pu upper limit of the pump capacity to handle the ˝Fctu flowrate from the crops to treatment units, kg/h max,pip ˝Fctu upper limit of the pipeline capacity to handle the flowrate from the crops to treatment units, m3 /h max ˝Cfe upper limit for the concentration of each fertilizer in l,t

max ˝Fst max ˝Ftu max ˝w,t

the environmental discharge, kg of fertilizer/total kg upper limit for the capacity of storage tanks, m3 upper limit for the capacity of treatment units, m3 limit for each type of fresh water in the periods, m3

Variables Capc capital cost, US$/year fertilizer concentration in the crops, kg of fertilCfc izer/total kg

fertilizer concentration in the environment discharge, kg of fertilizer/total kg Cfs fertilizer concentration in the storage tanks, kg of fertilizer/total kg fertilizer concentration in treatment unit, kg of ferCft tilizer/total kg Copc operating cost, US$/year total flowrate in the crops, kg/h Fctot Fcc flowrate between crops, kg/h Fcccapc,pu pump capacity to handle the flowrate between crops, kg/h Fcccapc,pip pipeline capacity to handle the flowrate between crops, m3 /h Fce flowrate from crops to environment discharge, kg/h flowrate from crops to the storage tanks, kg/h Fcst Fcstcapc,pu pump capacity to handle the flowrate from crops to the storage tanks, m3 Fcstcapc,pip pipeline capacity to handle the flowrate from crops to the storage tanks, m3 /h flowrate from crops to treatment units, kg/h Fctu Fctucapc,pu pump capacity to handle the flowrate from crops to treatment units, m3 capc,pip Fctu pipeline capacity to handle the flowrate from crops to treatment units, m3 /h Fe flowrate in the environment discharge, kg/h Ffc flowrate fertilizer in the crops, kg/h Ffcc fertilizer flowrate between crops, kg/h Ffctu fertilizer flowrate from crops to treatment units, kg/h Fff fresh fertilizer flowrate, kg/h Ffrc reused fertilizer flowrate, kg/h fertilizer flowrate from the storage tanks to crops, Ffrt kg/h fertilizer flowrate from the storage tanks to crops, Ffst kg/h Fftue fertilizer flowrate from treatment units to environment discharge, kg/h fertilizers cost, US$/year Flc Fstcapc storage tanks capacity, m3 Fsttot total flowrate in the storage tanks, kg/h Fstc flowrate from storage tanks to crops, kg/h Fstccapc,pu pump capacity to handle the flowrate from storage tanks and crops, kg/h Fstccapc,pip pipeline capacity to handle the flowrate from storage tanks and crops, m3 /h capc treatment units capacity, m3 Ftu Ftutot treatment units flowrate, kg/h flowrate from treatment units to the environment Ftue discharge, kg/h fresh water cost, US$/year Fwc Fwc abs absorbed water flowrate by the crops (soil and plants), m3 /h ev lost water flowrate by evapotranspiration in the Fwcc,t Fwc Fwcc Fwct Fwctu Fwf Fwfs Fwrt Fwst

crops, m3 /h water flowrate on each crop, m3 /h water flowrate between crops, m3 /h water flowrate from the storage tanks to crops, m3 /h water flowrate from crops to treatment units, m3 /h flowrate available fresh water, m3 /h fresh water flowrate on each crop, m3 /h reused water flowrate, m3 /h water flowrate from to crops to storage tanks, m3 /h

E. Rubio-Castro et al. / Computers and Chemical Engineering 84 (2016) 63–82

Fwtue HY Lic,t P P kw Pip Proffit Pump Swc Tuc Sr Ti

water flowrate from the treatment units to the environment discharge, m3 /h annual operating time, h/year depth water, m3 ha−1 crop yield, ton/ha power consumption for the pumping devices, kW piping cost, US$/year annual profit, US$/year pumping cost, US$/year storage water cost, US$/year treatment cost, US$/year sales, US$/year irrigation time, h

Binary variables z binary variable to determine the existence or not of accessories and equipment processes, 0 or 1 Subscript c crop l fertilizer st storage tank t period u treatment unit freshwater type w Superscript ab absorbed cap capital evapotranspiration ev in inlet ir irrigation max upper limit operating op out outlet pip piping pump pu rem removal sl sell tot total treatment unit tu Scalars NUMPT

number of periods

reported (Biazin et al., 2012; Moges et al., 2011; Oweis and Hachum, 2006). With respects to the optimal use of water, several methodologies have been proposed for improving the irrigation process through the optimization of irrigation scheduling (Garg and Dadhich, 2014; Naadimuthu et al., 1999; Pereira, 1999; Rao et al., 1988), the minimization of water losses during the irrigation process to increase the yield crop (Burke et al., 1999), the optimal geometric design of irrigation piping (Theocharis et al., 2006), the increment of the operating efficiency of reservoirs for irrigation (Moradi-Jalal et al., 2007), the evaluation of the crop yields in terms of the water use efficiency as well as the application of supplemental irrigation as function of soil water deficit (Xiao et al., 2007), the optimal allocation for the reuse of treated water (Bin et al., 2012), the multi-objective optimization of irrigation under uncertain conditions (Li and Guo, 2014), and the optimal design of irrigation processes through an evolutive strategy (Belaqziz et al., 2014).

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It is important to note that currently there is not a systematic strategy for determining the optimal water integration system in agriculture, which involves optimal water recycling, reusing and storing (such as in the industrial processes). For example, some works have addressed the optimal use of water in cities (Bishnu et al., 2014; Nápoles-Rivera et al., 2013; Rojas-Torres et al., 2014). In the industrial case, the wastewater treatment (Mussati et al., 2002a,b; Wang and Smith, 1994) and the optimal design of single water networks (Alva-Argaez et al., 1999; Karuppiah and Grossmann, 2008; Ponce-Ortega et al., 2010; Quaglia et al., 2013; Sueviriyapan et al., 2014; Takama et al., 1980) have represented significant reductions of fresh water consumption. This is summarized in the reviews by Bagajewicz (2000), Dunn and El-Halwagi (2003) and Foo (2009). Furthermore, additional benefits can be obtained when the interplant water integration is considered (Alnouri et al., 2014; Chen et al., 2010; Lowe, 1997; Rubio-Castro et al., 2011, 2012). Besides, recently López-Villarreal et al. (2014) have incorporated pollution treading in designing wastewater networks. Therefore, this paper proposes a multi-period mixed-integer non-linear programming formulation to design agricultural water networks to identify the optimal water distribution (use, reuse, recycling and storage) with the minimum production cost. The proposed model is based on a new superstructure that includes several potential configurations for water integration for a field that includes specific number of crops. Besides, the model considers the wastewater treatment to reuse it and to meet environmental constraints. Furthermore, treated water storing and dispatching is considered as an option to use it in the period when it is required. The objective function consists in maximizing the annual profit formed by the sale of crops minus the cost for fresh water, storage, treatment, piping and pumping. The proposed model was coded and solved in the software GAMS (Brooke et al., 2014). 2. Model formulation The proposed model is based on the superstructure shown in Fig. 1; this is a schematic representation that includes four crops, four storage tanks and one dam. There is considered the water exchange between crops, from crops to storage tanks, from storage tanks to crops, from crops to the environmental discharge, from crops to treatment units, from treatment units to the environmental discharge and the rainwater harvesting in the storage tanks as well as the rainwater capturing by the crops. It is important to mention that it is possible to reuse water from the fields because not all the inlet water to a field is used by the crops, and a lot of water is drained to channels. This way, the water balance in the fields is determined by inputs (irrigation, precipitation and groundwater charge), minus outputs (evapotranspiration, absorption and surface runoff). Where, the surface runoff represents the stream that can be reused directly (field to field) and/or indirectly (storage tank to field). Whereas the reused water between fields is determined accounting for the type of crops accounting fort the type and amount of needed fertilizers. This way, previous to the optimization stage, there are determined the lower and upper limits of the inlet fertilizers and this condition determines the allowed and forbidden matches between fields. For example, if the fertilizer used for the crop 1 is not allowed for the crop 2, then the outlet water of crop 1 cannot be reused in the crop 2. On the other hand, the available storage tanks are determined before to implement the optimization depending on the specific case study considered. In the superstructure, schematically there is indicated one tank per field, but this number depends on the specific geography for each case. Furthermore, treatment is considered as an option to treat the wastewater that is discharged to the channels to satisfy the environmental regulations for the used chemicals. There are also included pumps and pipelines to transport the water through

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Fig. 1. Proposed superstructure.

Table 1 Field capacity and permanent wilting point for different types of soil. Types of soil

Field capacity (cm3 of water/cm3 of soil)

Permanent wilting point (cm3 of water/cm3 of soil)

Franco Sandy Clay soil Clay loam Sandy loam Clayey sandy

0.22 0.10 0.45 0.35 0.20 0.20

0.10 0.05 0.31 0.23 0.12 0.10

for each type of soil like it is shown in Table 1 (Sifuentes-Ibarra and Macías-Cervantes, 2013). This way, the field capacity is transformed into water depth for each time period as follows: Fig. 2. Behavior of crop yield as function of humidity content.

Lic,t = fc Drc,t Crc , the different system components. In this sense, the mathematical representation for the above mass exchange process is presented below, and for this purpose the next subscripts are used: w is used to denote the type of fresh water, c denotes the crops, l the fertilizers, s the storage tanks, u the treatment units and t the time period. Here, the fresh water represents the water coming from the dam.

(1)

where Drc,t is the depth of roots on each time period and Cr is the in ) is irrigation criterion. The demanded water in each crop (Fwcc,t calculated as follows, in Fwcc,t =

2.1. Mass balance for the required water on crops The irrigation management has a great importance in the crop performance, in terms of quality and quantity, due to the water content of crops is around of 80%. For example, Fig. 2 presents the relationship between the crop performance and the moisture content. It should be noted that there is an ideal moisture content for each crop, which is called field capacity ( fc ), and when the moisture is greater than the field capacity the soil is saturated and the crops are stressed due to lack of oxygen in the roots. Otherwise, if the moisture content is lower than the field capacity, the crop performance is affected, and it is important that the moisture content does not reach the minimum level of available moisture, which is the permanent wilting point (pwp ). It is because in this point the crops cannot absorb water and die. Both  fc and pwp are specific

c ∈ C; t ∈ T

Lic,t Acc ir c

,

c ∈ C; t ∈ T

(2)

where Acc is the total area for crops and ir c is the irrigation efficiency.

2.2. Irrigation time period The irrigation time period for each crop (Tic,t ) depends on the in ) and demanded water, volumetric flowrate (Qwcc,t

Tic,t =

in Fwcc,t in Qwcc,t

,

c ∈ C; t ∈ T

(3)

E. Rubio-Castro et al. / Computers and Chemical Engineering 84 (2016) 63–82

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Table 2 Correlations to calculate the crop yield. Crop Corn

Flood irrigation

Sprinkling irrigation

Y = 0.506 + 0.113·10−2 X − 0.119·10−6 X2 + 0.226·10−1 N − 0.382·10−4 N2 + 0.161·10−5 XN 2 Raj = 0.918

Y = 0.506 + 0.154·10−2 X − 0.141·10−6 X2 + 0.210·10−1 N − 0.361·10−4 N2 + 0.140·10−5 XN 2 Raj = 0.995

Y = 0.936 + 0.845·10−3 X − 0.217·10−6 X2 + 0.176·10−1 N − 0.439·10−4 N2 + 0.453·10−5 XN 2 Raj = 0.969

Y = 0.891 + 0.815·10−3 X − 0.196·10−6 X2 + 0.168·10−1 N − 0.499·10−4 N2 + 0.48823·10−5 XN 2 Raj = 0.982

Y = 0.871 + 0.951·10−3 X − 0.145·10−6 X2 + 0.172·10−1 N − 0.336·10−4 N2 + 0.249·10−5 XN 2 Raj = 0.971

Wheat

Barley

Y = 0.762 + 0.888·10−3 X − 0.114·10−6 X2 + 0.185·10−1 N − 0.538·10−4 N2 + 0.272·10−5 XN 2 Raj = 0.983

Y = 4.522 + 0.295·10−2 X − 0.196·10−6 X2 + 0.296·10−1 N − 0.254·10−3 N2 + 0.163·10−5 XN 2 Raj = 0.972

Alfalfa

Y = 4.266 + 0.291·10−2 X − 0.196·10−6 X2 + 0.279·10−1 N − 0.274·10−3 N2 + 0.282·10−5 XN 2 Raj = 0.976

Y = 0.325 + 0.573·10−3 X − 0.128·10−6 X2 + 0.200·10−1 N − 0.656·10−4 N2 + 0.127·10−5 XN 2 = 0.982 Raj

Sunflower

Y = 0.521 + 0.502·10−3 X − 0.920·10−7 X2 + 0.189·10−1 N − 0.649·10−4 N2 + 0.174·10−5 XN 2 Raj = 0.980

Y = 1.324 + 0.444·10−3 X − 0.210·10−7 X2 + 0.172·10−1 N − 0.930·10−4 N2 + 0.104·10−5 XN 2 = 0.975 Raj

Rice

Note: Y is the crop yield in ton/ha; X is the used water in m3 ha−1 ; N is the employed nitrogen in kg/ha.

2.3. Mass balance for fresh water The inlet fresh water in each field during each time period tot ) is equal to the sum of fresh water sent to each field (Fwfw,t (Fwfsw,c,t ), tot Fwfw,t =



Fwfsw,c,t ,

w ∈ W; t ∈ T

In the same way, the fertilizer flowrate is constituted by the fresh fertilizer (Fffl,c,t ), the reused fertilizer from other crops (Ffccl,c1,c,t ) and the recirculated fertilizer from the storage tanks (Ffstl,s,c,t ) in Ffcl,c,t = Fffl,c,t +



Ffccl,c1 ,c,t +

c1∈C

(4)



Ffstl,s,c,t ,

l ∈ L; c ∈ C, t ∈ T

s∈S

(9)

c∈C max ), There is an upper limit for each type of fresh water (˝w,t because this depends on the availability, tot max Fwfw,t ≤ ˝w,t ,

w ∈ W, t ∈ T

(5)

Therefore, the reused water and fertilizers on each period (Fwrtc,t , Ffrcl,c,t ) are calculated as follows:

It should be noted that in a given field there can be different types of freshwater.

Ffrcl,c,t =

2.4. Crop performance

Fwrtc,t =

The performance of each crop (Pc ) is a function of the fertilizer and water: Pc = f (water, fertilizer),

c∈C

(6)

Martínez et al. (2002) reported a set of correlations (see Table 2) to calculate the yield of different crops (corn, wheat, barley, alfalfa, sunflower and rice). 2.5. Mass balance in the inlet of each crop in , Ffc in ) deterThe inlet water and fertilizers to each field (Fwcc,t l,c,t

tot,in ) for each period t, mine the total inlet flowrate (Fcc,t tot,in in Fcc,t = Fwcc,t +



in Ffcl,c,t ,

c ∈ C, t ∈ T

(7)

l∈L

where  is the water density. Furthermore, the water on each period is formed by the freshwater (Fwfsw,c,t ), the precipitated water (Fwpcc,t ), the capillary rise from water table (Fwrcc,t ), the reused water from other crops (Fwccc1 ,c,t ) and the recirculated water from the storage tanks (Fwsts,c,t ), in Fwcc,t

=



Fwfsw,c,t + Fwpcc,t + Fwrcc,t +

w∈W

+

 s∈S



Fwccc1 ,c,t

c1∈C

Fwsts,c,t ,

c ∈ C, t ∈ T

(8)



Ffstl,s,c,t +

s∈S





Ffccl,c1 ,c,t ,

l ∈ L; c ∈ C, t ∈ T

(10)

Fwccc1 ,c,t ,

c ∈ C, t ∈ T

(11)

c1∈C

Fwsts,c,t +

s∈S

 c1∈C

Finally, the next equation is used for determining the inlet ferin ), tilizer concentration in each crops and period (Cfcl,c,t in Cfcl,c,t =

in Ffcl,c,t

, l ∈ L; c ∈ C; t ∈ T (12) tot,in Fcc,t in the above equations c1 is an alias of c used to model the flowrate between crops.

2.6. Mass balance in the outlet of each crop The outlet flowrate from the crops over each time period tot,out (Fcc,t ) is constituted by the water and fertilizers that were not out , Ffc out ), consumed in the crops (Fwcc,t l,c,t tot,out out Fcc,t = Fwcc,t +



out Ffcl,c,t ,

c ∈ C, t ∈ T

(13)

l∈L

where the outlet water is equal to the inlet water minus the water ab ) and the water lost by evapotranspiabsorbed by the crop (Fwcc,t ev ). Here, the water absorbed is referred to the filtrated ration (Fwcc,t water in the soil as well as the retained water in the plant capillary. out in ab ev Fwcc,t = Fwcc,t − Fwcc,t − Fwcc,t ,

c ∈ C; t ∈ T

(14)

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E. Rubio-Castro et al. / Computers and Chemical Engineering 84 (2016) 63–82

The absorbed and lost water are related to the efficiencies for cu storage and use (st c , c ): ab Fwcc,t = st c Lic,t Ac ,

c ∈ C; t ∈ T

(15)

ev Fwcc,t = cu c Lic,t Ac ,

c ∈ C; t ∈ T

(16)

where Lic,t Ac is the water depth for the total crop area. In addition, the outlet fertilizer is the difference between the inlet fertilizer and ab ), the one absorbed by the crops (Ffcl,c,t out in ab Ffcl,c,t = Ffcl,c,t − Ffcl,c,t ,

l ∈ L; c ∈ C; t ∈ T

l ∈ L; c ∈ C; t ∈ T

+





Fccc,c1 ,t +



c1 ∈C

Fctuc,u,t ,

It is needed to know the outlet fertilizer concentration from each out ): crop over each time period (Cfcl,c,t out Ffcl,c,t tot,out Fcc,t

,

l ∈ L; c ∈ C; t ∈ T

(20)

And the outlet fertilizer concentration is used to determine the fertilizer that is exchanged between crops, out Ffccl,c,c1 ,t = Cfcl,c,t Fccc,c1 ,t , lL; cC; c1 C; tT



Ffccl,c1 ,c,t ,



s ∈ S; t ∈ T ; l ∈ L

(25)

l∈L



tot Fsts,t−1

=

+

tot Cfsl,s,t Fsts,t

 

−



 tot Cfsl,s,t−1 Fsts,t−1

l∈L

Fcstc,s,t −



c∈C

 

 out Cfcl,c,t Fcstc,s,t

c∈C l∈L

Fstcs,c,t −



 Cfsl,s,t Fstcs,c,t

c∈C l∈L

s ∈ S; t ∈ T

(26)

c ∈ C; c1 ∈ C; t ∈ T

In Eqs. (25) and (26), the term of the left hand side represents the fertilizer and water in the storage tanks over the time period t, the first term of the right hand side represents the species in the period t − 1, the second term of the right hand side is the fertilizer (Ffstl,s,c,t ) and water (Fwstc,s,t ) that are sent from the crops to the storage tank over the time period t and the last one represents the fertilizer (Ffrtl,s,c,t ) and water (Fwcts,c,t ) that are recirculated from to the storage tank to the crops over the time period t. Hence, the last variables are calculated as follows: Ffstl,s,c,t = Cfsl,s,t Fcsts,c,t ,

(21)

With the fertilizer flowrate exchanged between crops is possible to determine the water on this stream as follows: Fccc,c1 ,t = Fwccc,c1 ,t +

Cfsl,s,t Fstcs,c,t ,

(19)

u∈U

out Cfcl,c,t =

tot Fsts,t −

+ Fwpsts,t ,

c ∈ C, t ∈ T

c∈C

c∈C

s∈S

out Cfcl,c,t Fcstc,s,t

Cfsl,s,t is the concentration of fertilizers in the storage tanks over each time period. Next relationship is required to determine the water in the storage tanks over each time period:

−

Fcstc,s,t





c∈C

(18)

The outlet flowrate from each crop can be sent to the environment (Fcec,t ), to other crops (Fccc,c1 ,t ), to the treatment units to meet the environmental constraints (Fctuc,u,t ) and/or to the storage tanks (Fcstc,s,t ), tot,out Fcc,t = Fcec,t +

−

(17)

Here, the absorbed fertilizer is determined by the efficiency to absorb fertilizers of each crop (abs ), l,c ab in Ffcl,c,t = abs Ffcl,c,t , l,c

tot tot Cfsl,s,t Fsts,t = Cfsl,s,t−1 Fsts,t−1 +

(22)

c1 ∈C

Fwsts,c,t = Fcsts,c,t −



l ∈ L; s ∈ S; c ∈ C; t ∈ T Cfsl,s,t Fcsts,c,t , s ∈ S; c ∈ C; t ∈ T

(27) (28)

l∈L out Ffrtl,c,s,t = Cfcl,c,t Fstcc,s,t ,

Fwctc,s,t = Fstcc,s,t −

l ∈ L; c ∈ C; s ∈ S; t ∈ T



out Cfcl,c,t Fstcc,s,t ,

c ∈ C; s ∈ S; t ∈ T

(29) (30)

l∈L

2.7. Mass balance in the storage tanks The mass balance for each storage tank is stated as follows: tot tot Fsts,t = Fsts,t−1 +

 c∈C

+ Fwpsts,t ,

Fcstc,s,t −



Fstcs,c,t

Fstcs,c,t = Fwsts,c,t +

c∈C

s ∈ S; t ∈ T

In this regard, the total recirculated flowrate from the storage tank to crops (Fstcs,c,t ) and the total flowrate sent from the crop to the storage tank (Fcstc,s,t ) are determined as follows:

(23)

s ∈ S; c ∈ C; t ∈ T

(31)

is the total inlet flowrate to each storage tank over the tot is the total inlet flowrate of each storage tank in period t. Fsts,t−1 the period t − 1. Fcstc,s,t is the flowrate sent from each crop to each storage tank in the period t, Fstcs,c,t is the flowrate recirculated from the storage tanks to each crop in the period t and Fwpsts,t is the precipitated water in the storage tanks in the period t. The stored water before to the first period of time is set as zero: (24)

It should be noted that the storage tanks are composed by water and fertilizer. In this regard, the next expression allows calculating the fertilizer in each storage tank for each time period:



Ffrtl,c,s,t ,

c ∈ C; s ∈ S, t ∈ T

(32)

l∈L

tot Fsts,t

s∈S

Ffstl,s,c,t ,

l∈L

Fcstc,s,t = Fwctc,s,t +

tot Fsts,t=0 = 0,



2.8. Mass balance in the treatment units There is needed to install treatment units to meet the environmental constraints (see Fig. 1), the inlet flowrate to each treatment unit over each time period (Ftutot u,t ) is equal to the segregated flowrates from the crops (Fctuc,u,t ) and the outlet flowrate sent to the environmental discharge (Ftueu,t ). Therefore, the mass balance on each treatment unit is represented as follows: Ftutot u,t =

 c∈C

Fctuc,u,t − Ftueu,t ,

u ∈ U; t ∈ T

(33)

E. Rubio-Castro et al. / Computers and Chemical Engineering 84 (2016) 63–82

And the component balance for the fertilizer is represented as follows: in Cftl,u,t Ftuu,t =



out out Cfcl,c,t Fctuc,u,t − Cftl,u,t Ftueu,t ,

c∈C

l ∈ L; u ∈ U; t ∈ T

(34)

2.9. Mass balance in the environmental discharge The environmental discharge over each time period (Fet ) is constituted by the flowrate segregated from the crops to the environmental discharge (Fcec,t ) and the flowrate coming from the treatment units (Ftueu,t ): Fet =

in Ftuu,t − Cftl,u,t Ftuu,t

=

 

Fctuc,u,t −

c∈C





Fcec,t +



Cfel,t Fet =

c∈C



out Cfcl,c,t Fcec,t +

l ∈ L; u ∈ U; t ∈ T

(35)

Similarly to Eqs. (25) and (26), from the terms of Eqs. (34) and (35) is possible to determine the fertilizer and water flowrates (Ffctul,c,u,t , Fwctuc,u,t ) that are sent from the crops to each treatment unit, and the fertilizer and water in the segregated flowrates from the treatment units to the environmental discharge (Fftuel,u,t , Fwtueu,t ): out Ffctul,c,u,t = Cfcl,c,t Fctuc,u,t ,



(43)

l ∈ L; c ∈ C; u ∈ U; t ∈ T out Cfcl,c,t Fctuc,u,t ,



out Cftl,u,t Ftueu,t ,

l ∈ L; t ∈ T

(44)

u∈U

c∈C

out − (Ftueu,t − Cftl,u,t Ftueu,t ),

t∈T

The corresponding component balance is expressed as follows:

out Cfcl,c,t Fctuc,u,t

Fwctuc,u,t = Fctuc,u,t −

Ftueu,t ,

u∈U

c∈C



69

And the following relationship is needed to meet the environmental constraints related to the fertilizer: max Cfel,t ≤ ˝Cfe ,

l ∈ L; t ∈ T

l,t

(45)

max is an upper limit for the concentration of each fertilizer where ˝Cfe l,t

in the environmental discharge. 2.10. Objective function

(36)

c ∈ C; u ∈ U; t ∈ T

l∈L

(37)

The objective function is to maximize the annual profit (Proffit), which is constituted by the sales (Sr) minus the operating (Copc) and capital (Capc) costs: Proffit = Sr − Copc − Capc

(46)

where the operating and capital costs are calculated as follows: out Fftuel,u,t = Cftl,u,t Ftueu,t ,

Fwtueu,t = Ftueu,t −



l ∈ L; u ∈ U; t ∈ T out Cftl,u,t Ftueu,t ,

(38)

u ∈ U; t ∈ T

(39)

l∈L

Therefore, the total flowrate that is sent from the crops to the treatment units (Fctuc,u,t ) and the total flowrate segregated from the treatment units to the environment discharge (Ftueu,t ) are calculated from the next mathematical relationships: Fctuc,u,t =



Ffctul,c,u,t + Fwctuc,u,t ,

c ∈ C; u ∈ U; t ∈ T

(40)

l∈L

Ftueu,t =



Fftuel,u,t + Fwtueu,t ,

u ∈ U; t ∈ T

(41)

Copc = Fwc + Flc + Pumpop + Pipop + Swcop + Tucop cap

Capc = Pump

+ Pip

cap

+ Swc

cap

+ Tuc

cap

(47) (48)

In the above equations, Fwc is the fresh water cost, Flc is the fertilizer cost, Pumpop is the operating cost for pumps, Pipop is the operating piping cost (for example the piping maintenance), Swcop is the storing water cost, Tucop is the operating treatment cost, Pumpcap is the capital cost for pumps, Pipcap is the capital cost for pipes, Swccap is the capital cost for storage devises, Tuccap is the capital cost for treatment units. Then, the sales are determined as follows: Sr =



l∈L

Cusl c Pc

(49)

c∈C

The outlet fertilizer concentration in the treatment units is determined using the efficiency of each treatment unit to remove each fertilizer (rem ): l,u

where Cusl c is the unit selling price of crops. While the mathematical expressions used to calculate Fwc, Flc, Pumpop , Pipop , Swcop and Tucop are given as follows:

out in Cftl,u,t = Cftl,u,t (1 − rem ), l,u

Fwc =

l ∈ L; u ∈ U; t ∈ T

(42)

 

op

Cuw,t Fwfsw,c,t

(50)

w∈W c∈C t∈T

Flc = Pumpop = HY

Pipop = HY



kw Cue,t Pc,s,t + HY op





op

Cul,t Fffl,c,t

l∈L c∈C t∈T kw Cue,t Ps,c,t + HY op



kw + HY Cue,t Pc,c 1 ,t op

(51)



c∈C

s∈S

c∈C

c∈C

s∈S

c∈C

c1 ∈ C

u∈U

t∈T

t∈T

t∈T

t∈T



op

Cup,t Fcstc,s,t + HY



c∈C

s∈S

s∈S

c∈C

t∈T

t∈T

op

Cup,t Fstcs,c,t + HY



op

kw Cue,t Pc,u,t

Cup,t Fccc,c1 ,t + HY

op



c, c1 ∈ C

c∈C

t∈T

u∈U t∈T

op

Cup,t Fctuc,u,t

(52)

(53)

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E. Rubio-Castro et al. / Computers and Chemical Engineering 84 (2016) 63–82

Swcop = HY



tot Cus,t Fsts,t op

⎡

(54)

s∈S Pip

t∈T Tucop = HY



Cuu,t Ftutot u,t op

⎢ ⎢ capc,pip 1  pip Fcstc,s Dc,s ⎢ pip pip zc,s Cfc,s + C vc,s = KF ⎢ 3600vc,s ⎢ ⎣c ∈ C c∈C

cap

(55)

s∈S

u∈U t∈T HY is the annual operating time, Cu is the unit cost for different raw materials (freshwater, fertilizer and electricity) and maintenance (pipe lines, storage tanks and treatment units), Pkw is the power consumption for the pump used to transport water between the process components (crops, storage tanks and treatment units). kw , P kw , P kw kw HY , Pc,s,t s,c,t c,c1 ,t as well as Pc,u,t are determined by the following relationships: HY =



Tic,t NUMPT

−3

g gc

= 1 × 10

kw Pc,c = 1 × 10−3 1 ,t

kw Pc,u,t = 1 × 10−3



(Fcstc,s,t /3600Tic,t ) (Hec,s + 3.048)

 (57)

pu



(Fstcs,c,t /3600Tic,t ) (Hes,c + 3.048)

(58)



s,c (Fccc,c1 ,t /3600Tic,t ) (Hec,c1 + 3.048) pu



c,c1 (Fctuc,u,t /3600Tic,t ) (Hec,u + 3.048) pu

c,u

s∈S

+

+

c∈C

s∈S pip

capc,pip 2 Ds,c pip Fstcs,c

C vs,c



pip

zc,c1 Cfc,c1 +

c∈C

c∈C

c1 ∈ C

c1 ∈ C

pu pu zs,c Cfs,c

+

s∈S

c∈C

c∈C

s∈S



pu pu zc,c1 Cfc,c1

 (60)

3600vs,c

capc,pip

pip

C vc,c1

Fccc,c1

+

c∈C

c∈C

c1 ∈ C

c1 ∈ C



pu pu zc,u Cfc,u

+



c∈C

c∈C

u∈U

u∈U

u∈U

u∈U



zssw Cfssw

+



s∈S



3600vc,c1

⎤ capc,pip

pip Fctuc,u

C vc,u

+



⎥ ⎥ ⎥ ⎥ ⎦

4 Dc,u ⎥

3600vc,u

(62)

 capc ˛ C vsw ) s (Fsts

s∈S

zutu Cfutu

3 Dc,c 1

capc ˛ C vtu ) u (Ftuu

(63)

 (64)

u∈U

where Cf is the fixed cost of units and accessories (pump, storage tank, treatment unit and pipe lines), Cv is the variable cost of units and accessories, D is the distance between units, z is a binary variable used to model the existence of units and accessories, v is the velocity, ˛ is an exponent for the capital costs function used to consider the economies of scale, and KF is the factor used to annualize the inversion. The last one depends on the fractional interest rate per year (m) and the number of years of operation (n), m(1 + m)n

KF =

(65)

(1 + m)n − 1

Finally, to model the existence or not of storage tanks, the following logical relationships are needed (Vecchietti et al., 2003): tot Fsts,t

 capc

Fsts

pu capc,pu ˛ C vs,c (Fstcs,c )



c∈C

= KF

= KF

pip

zc,u Cfc,u +

u∈U

s∈S



Tuc

cap

pip



c∈C

(59)

⎢ ⎢  pu ⎢ pu pu capc,pu ˛ zc,s Cfc,s + C vc,s (Fcstc,s = KF ⎢ ) ⎢ ⎣c ∈ C c∈C 

Swc

cap



⎡

+

c∈C



pu

g gc

g gc

c,s

NUMPT is a scalar used to represent the number of time periods, pu is the pump efficiency installed to send water between the process components, He is the equivalent height between the process components, g is the acceleration due to gravity and gc is a conversion factor associated to the acceleration due to gravity. Furthermore, there is needed to calculate the terms Pumpcap , Pipcap , Swccap and Tuccap as follows,

Pumpcap

s∈S



+ g gc



pip

(56)

t∈T kw Pc,s,t = 1 × 10−3

pip

zs,c Cfs,c +



+

c∈C

kw Ps,c,t



+

s∈S

capc

≤ Fsts

,

s ∈ S; t ∈ T

max ≤ zssw ˝Fst ,

(66)

s∈S

(67)

In the same way, the existence of the remaining process units as well as pipelines are determined as follows: - Treatment units Ftutot u,t

pu capc,pu ˛ C vc,c1 (Fccc,c1 )

pu capc,pu ˛ ) C vc,u (Fctuc,u

 capc

Ftuu

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

capc

≤ Ftuu

,

u ∈ U; t ∈ T

max ≤ zutu ˝Ftu ,

u∈U

(68) (69)

- Water pumps and pipelines used to send flow rates from crops to other crops capc,pu

(61)

Fccc,c1 ,t ≤ Fccc,c1 capc,pu

Fccc,c1

pu

,

c, c1 ∈ C; t ∈ T

max,pu

≤ zc,c1 ˝Fcc

Fccc,c1 ,t capc,pip ≤ Fccc,c1 , Tic,t

,

c, c1 ∈ C

c, c1 ∈ C; t ∈ T

(70) (71) (72)

E. Rubio-Castro et al. / Computers and Chemical Engineering 84 (2016) 63–82 capc,pip

Fccc,c1 pu

pip

max,pip

≤ zc,c1 ˝Fcc

pip

zc,c1 = zc,c1 ,

,

c, c1 ∈ C

c, c1 ∈ C

(73) (74)

- Water pumps and pipelines used to send flow rates from storage tanks to crops capc,pu

Fstcc,s,t ≤ Fstcs,c capc,pu

Fstcs,c

pu

,

s ∈ S; c ∈ C; t ∈ T

max,pu

≤ zs,c ˝Fstc

Fstcc,s,t capc,pip ≤ Fstcs,c , Tic,t capc,pip

pip

Fstcs,c pu

,

s ∈ S; c ∈ C

(76)

s ∈ S; c ∈ C; t ∈ T

(77)

max,pip

≤ zs,c ˝Fstc

pip

zs,c = zs,c ,

(75)

,

s ∈ S; c ∈ C

s ∈ S; c ∈ C

(78) (79)

- Water pumps and pipelines used to send flow rates from crops to storage tanks capc,pu

Fcstc,s,t ≤ Fcstc,s capc,pu

Fcstc,s

pu

,

c ∈ C; s ∈ S; t ∈ T

max,pu

≤ zc,s ˝Fcst

Fcstc,s,t capc,pip ≤ Fcstc,s , Tic,t capc,pip

Fcstc,s pu

pip

,

c ∈ C; s ∈ S

(81)

c ∈ C; s ∈ S; t ∈ T

(82)

max,pip

≤ zc,s ˝Fcst

pip

zc,s = zc,s ,

(80)

,

c ∈ C; s ∈ S

c ∈ C; s ∈ S

(83) (84)

- Water pumps and pipelines used to send flow rates from crops to treatment units capc,pu

Fctuc,u,t ≤ Fctuc,u capc,pu

Fctuc,u

pu

,

c ∈ C; u ∈ U; t ∈ T

max,pu

≤ zc,u ˝Fctu

,

Fctuc,u,t capc,pip ≤ Fctuc,u , Tic,t capc,pip

Fctuc,u pu

pip

zc,u = zc,u ,

pip

c ∈ C; u ∈ U

(86)

c ∈ C; u ∈ U; t ∈ T

(87)

max,pip

≤ zc,u ˝Fctu

c ∈ C; u ∈ U

(85)

,

c ∈ C; u ∈ U

(88) (89)

where ˝max is an upper limit for the capacity of different process units and pipelines, the superscript capc means the capacity for the different process units and pipelines, and z is a binary variable. The last term is equal to one when its associated process unit or pipeline is selected for the optimal configuration. Otherwise, this is equal to zero. Remarks (a) The novelty of this manuscript consists in proposing a new optimization formulation for solving a very important problem. The used of water in the agriculture is a worldwide problem, because this activity represents the highest water consumer. Therefore, developing and implementing process integration techniques for solving this problem represent an attractive research problem. Furthermore, the proposed optimization formulation can be used for decision makers for the proper use of water in the agriculture. It is noteworthy that, to best of the authors’ knowledge, currently there is not reported any similar contribution. (b) The developed model considers simultaneously the use, reuse and recycling of water and fertilizers in the agriculture. The crop yield is given by correlations considering the consumed water and used fertilizers.

71

(c) With the proposed model is possible to represent and solve different scenarios; for example, water scarcity, unlimited available water, with or without integration and direct or indirect integration. (d) The existence or not of treatment units is modeled simultaneously to the crop irrigation. This way, the environmental constraints can be taken into account. (e) The pressure drops in the pipelines are included. (f) Different types of soils and crops can be considered simultaneously. (g) The capital cost associated to the equipment and accessories of the system includes fixed and variable costs. (h) There are several differences between industrial and agricultural water networks. First the pollutants considered are significant different, then, the way to recycle wastewater is totally different, furthermore, the periods involved in the different activities are different. In addition, the options for obtaining additional fresh water are different. Therefore, there is needed the proposed optimization formulation for designing water networks in the agriculture, because the industrial water networks cannot be used for solving this problem.

3. Results and discussion One case study under different scenarios and conditions is presented to demonstrate the application of the proposed optimization formulation. In this case, four croplands for maize cultivation are considered, whose distribution is shown in Fig. 3a and for which five irrigation periods are considered. The correlations reported by Martínez et al. (2002) for flood irrigation systems are used (see Table 2). Also, Table 3 shows the data for the economic and operating parameters used in the solution of this case study, where 150 tons of maize are considered as the minimum production for this lot of croplands to meet the market demands. Furthermore, the precipitated water and the capillary rise of water table are not considered. While the water lost by evapotranspiration is considered as 2.5% of the water depth. This example is solved considering that there is unlimited fresh water for irrigation (Case A) and considering that there are limitations in the availability of fresh water as shown in Table 4 (Case B). Each one of the above cases is solved under the following scenarios: Scenario 1 (without integration): The example is solved without considering water integration, it means under the traditional irrigation scheme; where only fresh water and fertilizer are used. This is shown in Fig. 3b, in which any interaction between different fields is included. Therefore, there is needed to fix all binary variables associated to the mass interchange flowrates equal to zero pu pu pu pu (zc,c1 , zs,c , zc,s and zc,u ). Scenario 2 (direct gravity integration): The example is solved by considering only the direct integration between crops and without additional pumping equipment. In other words, the storage tanks, treatment units and pumping equipment are not considered in the optimization formulation. This implies that water interchange is reduced in those cases where the gravity allows the mentioned exchange. Specifically, in this case study, the possible exchanges are from the crop 1 to crop 2 and 4, and similarly from crop 3 to crop 2 and 4 (see Fig. 3c). Therefore, all binary variables associated pu to the existence of flowrates from storage tanks to the crops (zs,c ), pu from the crops to the storage tanks (zc,s ), from the crops to the pu pu treatment units (zc,u ), and from the crops to other crops (zc,c1 ), are pu

pu

pu

pu

zero. With exception of z1,2 , z1,4 , z3,2 and z3,4 , which can be zero or one depending on the optimization.

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E. Rubio-Castro et al. / Computers and Chemical Engineering 84 (2016) 63–82

Fig. 3. Crop distribution (a) and Scenarios (b–d) for Example.

Table 3 Parameters for the Example. Parameter

Value

Parameter

Value

Parameter

Value

Parameter

Value

in Qwcc,t (m3 /h) Ac (ha) st c cu c abs l,c rem l,u Cusl c (US$/ton) op Cuw,t (US$/m3 ) op Cul,t (US$/kg) op Cue,t (US$/kWh) op Cup,t (US$/kg) op Cus,t (US$/kg) op Cuu,t (US$/kg) pu Hec,s (m) Hes,c (m) Hec,u (m) He1,1 (m) He1,2 (m) He1,3 (m) He1,4 (m) He2,1 (m) He3,2 (m)

108 5 1 0 0.65 0.95 400 0.015 0.3113 0.076 0 0 0 0.60 10 10 10 10 0 10 0 10 0

He3,3 (m) He3,4 (m) He4,1 (m) He4,2 (m) He4,3 (m) He4,4 (m) Cfpu (US$) Cvpu (US$/m3 ) ˛ Cfpip (US$) Cvpip (US$/m3 /h) sw Cfs (US$) 3 Cvsw s (US$/m ) n (number of years) m (fractional) max ˝Fst (m3 ) max ˝Ftu (m3 ) ˝max,pu (kg) ˝max,pip (m3 /h) 1 D1,1 (m) 1 D1,2 (m) 1 D1,3 (m) 1 D1,4 (m)

10 0 10 10 10 10 1712 0 0.8 250 7200 613.1 63.3 5 0.05 20,000 20,000 15 × 103 300 100 400 400 400

1 D2,1 1 D2,2 1 D2,3 1 D2,4 1 D3,1 1 D3,2 1 D3,3 1 D3,4 1 D4,1 1 D4,2 1 D4,3 1 D4,4 2 D1,1 2 D1,2 2 D1,3 2 D1,4 2 D2,1 2 D2,2 2 D2,3 2 D2,4 2 D3,1 2 D3,2

400 100 400 400 400 400 100 400 400 400 400 100 350 200 450 350 650 350 800 450 450 350

2 D3,3 2 D3,4 2 D4,1 2 D4,2 2 D4,3 2 D4,4 3 D1,1 3 D1,2 3 D1,3 3 D1,4 3 D2,1 3 D2,2 3 D2,3 3 D2,4 3 D3,1 3 D3,2 3 D3,3 3 D3,4 3 D4,1 3 D4,2 3 D4,3 3 D4,4

350 200 800 450 650 350 250 50 250 100 550 250 600 250 250 100 250 50 600 250 550 250

Cfpip , Cvpip and ˝max,pip were taken from Chew et al. (2008). Cfpu , Cvpu and ˝max,pu were given by Agroterra Company. ˛ was taken from El-Halwagi (2006). sw max max Cfs , Cvsw s , ˝Fst and ˝Ftu were taken from Tanks Direct Company. The values of remaining parameters were given by engineers of CONAGUA (2013).

(m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m)

(m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m)

E. Rubio-Castro et al. / Computers and Chemical Engineering 84 (2016) 63–82

73

Table 4 Availability of water for Cases A and B. Period

Available fresh water (m3 ) Case A

1 2 3 4 5

Enough Enough Enough Enough Enough

Case B 60,000 22,000 20,000 20,000 20,000

Scenario 3 (Direct integration with pumping): The example is solved without including the storage tanks and treatment units. However, the pumping devices are options in the optimization formulation; therefore, the water exchange among all crops is allowed, no matter their location. In this sense, the possible configurations in this scenario are shown in Fig. 3d. Then, to model this scenario, the binary variables associated to the existence of storage tanks (zutu ) as well as treatment units (zutu ) must be fixed as zero. Scenario 4 (Indirect integration): The example includes all configurations presented in the proposed superstructure (Fig. 1), except for the treatment units. In other words, this scenario provides the existence of water exchange among all crops, and between the storage tanks and crops. Because the treatment units are not considered, then, it is necessary to fix the binary variables associated to the treatment units equal to zero (zutu ). Finally, Fig. 4 presents the situations considered for the solutions of example 1. Also, the results for Case A are presented in Table 5. Notice that from Table 5 the discussed scenarios can be compared in terms of: (a) economic (profit and total annual cost), (b) environmental (demand and discharge of water and fertilizer), (c) social (maize production) and (d) computational issues (number of equations and computational time for execution). In this sense, the mentioned aspects are analyzed as follows.

Fig. 4. Scenarios to solve the Example 1.

3.1. Analysis of Case A for Example (a) Economy: Regarding the annual profit due to the sale of products, Scenarios 3 and 4 with a profit of $60,249 US represent the highest profit, while, Scenarios 1 and 2 show profits of $60,154 US and $58,230 US, respectively. In this sense, comparing Scenario 4 with respect to Scenarios 1 and 2, there is observed an increment in the profit of $2037 US and $113 US, respectively. That increase in Scenario 2 is due to the decrement of 32% in the cost of fresh water and 21% in the cost of fertilizer. Also, regarding to the Scenarios 1 and 2, the savings in Scenario 4 related to the demand of fresh water and fertilizer are 50% and 35%, respectively. However, to achieve these savings, an annual investment of $1037 US for the installation of pipes and pumps is required. Based on the above discussion, it is possible to say that for this example the improvement in economic terms achieved from the mass integration (water and fertilizer) is not substantial. However, the fresh water and fertilizer reduction is substantial due to the mass integration. (b) Environment: The environmental aspect is discussed in terms of the consumed fresh water and fertilizer as well as the

Table 5 Results for the Case A. Case A: Enough fresh water

Profit (US$/year) Sale of crops (US$/year) Total annual cost (US$/year) Capital cost (US$/year) Operating cost (US$/year) Fresh water cost (US$/year) Fertilization cost (US$/year) Pumping cost (US$/year) Pumping operating cost (US$/year) Pumping capital cost (US$/year) Piping cost (US$/year) Piping operating cost (US$/year) Piping capital cost (US$/year) Storage water cost (US$/year) Treatment cost (US$/year) Fresh water demanded (m3 ) Power consumption (kW) Fresh fertilizer demanded (kg) Environment water discharge (m3 ) Environment fertilizer discharge (kg) Number of crops Number of pumps Numbers of pipelines Yield per hectare (ton/ha) Overall yield (ton) Number of equations Continuous variables Binary variables CPU (s)

Scenario 1

Scenario 2

Scenarios 3 and 4

58,230 68,521 10,292 0 10,292 3524 6767 – – – – – – – – 234,945 0 10,000 114,536 3500 4 0 0 8.6 171 319 308 0 3

60,154 68,521 8367 119 8248 2665 5583 0 – – 119 0 119 0 0 177,677 0 8250 57,268 1750 4 0 2 8.6 171 828 677 16 23

60,249 68,521 8272 1036 7236 1806 4399 1822 1031 791 246 0 246 0 0 120,409 31 6500 0 0 4 2 4 8.6 171 850 759 16 66.5

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E. Rubio-Castro et al. / Computers and Chemical Engineering 84 (2016) 63–82

environmental discharge. In this sense, as can be seen in Table 5, Scenario 4 allows a saving of 50% of fresh water with respect to Scenario 1 and 35% with respect to Scenario 2. Moreover, the demand of fertilizer in Scenario 4 is 33% lower than in Scenario 1 and 19% lower than in Scenario 2. In addition to the demand of fresh resources, it is important to mention that from the configurations obtained for the Scenarios 3 and 4, the discharge of water and fertilizer to the environment is reduced in 100%, which were observed in the Scenarios 1 and 2. Therefore, it is evident that mass integration allows finding lower adverse impacts to the environmental, since the integrated scenarios demand less fresh water, which means less exploitation

of natural reservoirs (dams, rivers, lakes and groundwater) and coupled with the maximum reuse and recycle of water and fertilizer, as a consequence the discharges to the environment are reduced and therefore it has less deterioration of natural receptors (rivers, lakes, groundwater and oceans). (c) Social: The social benefits generated by the water integration in the agriculture and specifically in the example under study lies in the fact that the reduction of demand of water to the aforementioned economic activity allows a greater availability of water for human consumption; as the scarcity of drinking water to human consumption is an ongoing problem of great concern to both global governments and organizations. Added to this,

Fig. 5. Optimal configuration for Scenario 1 of Case A for Example (periods 1–5).

E. Rubio-Castro et al. / Computers and Chemical Engineering 84 (2016) 63–82

the fact that pollution in natural water reservoirs decreases, and as consequence the risk of health problems associated to the consumption or use of contaminated water by agriculture is reduced. Particularly, notice that Scenario 4 allows a reduction of 114,536 m3 and 57,268 m3 of wastewater compared to Scenarios 1 and 2, respectively. Also, in the same order, reductions of 3500 and 1750 kg of fertilizer (nitrogen) are reached. Finally, another aspect and probably the most important problem associated to the agriculture is the fact that through this activity a minimum amount of food should be produced, which in this example is reached in all scenarios; since the

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minimum required production is 150 tons and in each scenario 171 tons were produced. In this sense, agriculture also represents a source of jobs and the most important income for some families. In the example under study, when the water is available, is possible to cultivate maize in the four croplands, which means economic incomes for the families related to these croplands. (d) Computational effort: Although the mathematical representation of the proposed superstructure (Fig. 1), and its application represents significant savings of fresh water and fertilizer as well as a minor environmental discharge, also it is needed

Fig. 6. Optimal configuration for Scenario 2 of Case A for Example (periods 1–5).

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to mention that due to the multiple possible configurations available to choose the optimal one, the computational effort increases significantly. For example, from Table 5 is possible to identify that the Scenario 4 has 2120 equations, 1801 continuous variables and 52 binary variables, whereas the Scenario 1 (without integration) has 319 equations, 308 continuous variables and 0 binary variables. The solution of the Scenario 1 demands 3 s, whereas Scenario 4 requires 122.3 s.

Finally, the obtained configurations for each scenario in every period of time are presented in Figs. 5 and 6, and they are analyzed as follow.

(a) Scenario 1 (Case A): The configuration for this scenario is presented in Fig. 5, it should be noted that the mass integration allows reducing significantly the needed water and fertilizer (Fig. 5a–e). In this sense, 14,000 m3 of water required for each crop in the first period come from a central storage unit, and 500 kg of nitrogen needed for each crop in the same period are fresh nitrogen. Besides, it is worth to note that in each period 500 kg of nitrogen are fed to each crop, which indicates that the consumption of nitrogen per hectare in each crop for each period of time is 100 kg of nitrogen. This equitable distribution of the total optimal nitrogen was set during the optimization process; since the yield correlation used as a function of water and nitrogen (Martínez et al., 2002) depends on the total values and does not depend on its distribution along the development

Fig. 7. Optimal configuration for Scenarios 3 and 4 of Case A for Example 1 (periods 1–5).

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Table 6 Description of total annual cost of Case A. Case A: Enough water

Total annual cost (US$/year) Capital cost (US$/year) Operating cost (US$/year) Fresh water cost (US$/year) Fertilization cost (US$/year) Pumping cost (US$/year) Piping cost (US$/year) Storage water cost (US$/year) Treatment cost (US$/year)

Scenario 1

Scenario 2

Scenario 3

Scenario 4

10,292 0 100 34 66 – – – –

8367 1 99 32 67 0 1 0 0

8272 13 87 22 53 22 3 0 0

8272 13 87 22 53 22 3 0 0

of the plant. This situation does not occur for the consumption of water, as it is observed in Fig. 5, whose demand varies in each period of time, since it depends on the root growth which was modeled by Eq. (1), with exception of the first period where the amount of water was set from reviews by producers and the irrigation module manager. Also, Fig. 5 shows that the not consumed water and fertilizer are discharged to the environment. (b) Scenario 2 (Case A): In this scenario only was considered the water exchange by gravity and the configurations obtained in each period of time are shown in Fig. 6. Here, it is important to note that the savings of fresh water and fertilizer with respect to Scenario 1 in each period are 24–25% and 17–18%, respectively. The mentioned savings are obtained because in all periods the outlet flowrates from crops 1 and 3 are reused in the crop 4, whose values can be seen in Fig. 6. The reuse of water and fertilizer implies an annual investment of $113 US for the required pipes. As it can be seen in Table 5, the mentioned saving is reflected with an increment of 3.3% in the profit for Scenario 2. (c) Scenarios 3 and 4 (Case A): The optimal network for the water integration under the conditions of these scenarios (see Fig. 4) is presented in Fig. 7. In the first period, the waste flowrates

from crops 1 and 3 are reused in the crops 2 and 4. In addition, it presents a recirculation between the crops 2 and 4, which requires the installation of two pumps and four pipes. This requires an additional annual investment of $2067 US used to buy and operate the pumps and pipes utilized in this optimal configuration. And this configuration is kept in the later periods but with different flows. In summary, the reuse of water and fertilizer in the Scenarios 3 and 4 allows savings of 49% of water and 35% of fertilizer with respect to Scenario 1. In this sense, the amount of water and fertilizer required in Scenarios 3 and 4 are 32% and 21% lower than the one of Scenario 2, respectively. Owing to the Scenarios 3 and 4 have the same optimal configuration; then, if the available fresh water is such that is possible to satisfy the needs of the crops only with this type of water, the optimal configuration only includes the reuse of water between crops without water storage. On the other hand, Table 6 reports the description of the total annual cost for the analyzed configurations (Figs. 5–7). Here, it is important to note that the cost of fertilization has the biggest impact on the total annual cost (66% in Scenario 1, 67% in Scenario 2 and 53% in Scenarios 3 and 4).

Table 7 Description of total annual cost of Case B. Case B: Limited water

Profit (US$/year) Sale of crops (US$/year) Total annual cost (US$/year) Capital cost (US$/year) Operating cost (US$/year) Fresh water cost (US$/year) Fertilization cost (US$/year) Pumping cost (US$/year) Pumping operating cost (US$/year) Pumping capital cost (US$/year) Piping cost (US$/year) Storage water cost (US$/year) Treatment cost (US$/year) Fresh water demanded (m3 ) Fresh fertilizer demanded (kg) Power consumption (kW) Environment water discharge (m3 ) Environment fertilizer discharge (kg) Number of crops Number of pumps Number of pipelines Number of storage tanks Yield per hectare (ton/ha) Overall yield (ton) Number of equations Continuous variables Binary variables

Scenario 1

Scenario 2

Scenario 3

Scenario 4

14,557 17,130 2573 0 2573 881 1692 0 0 0 0 0 0 58,736 2500 0 28,634 875 1 0 0 0 8.6 43 319 308 0

30,078 34,261 4183 59 4124 1333 2792 0 0 0 59 0 0 88,839 4125 0 28,634 875 2 0 0 0 8.6 86 828 677 16

45,765 51,391 5626 586 5040 1355 3299 782 386.50 395 190 0 0 90,307 4875 15.59 0 0 3 1 1 0 8.6 128 850 759 16

39,809 68,521 28,712 20,919 7793 1806 4399 3170 1589 1582 379 18,958 0 120,409 6500 48.04 0 0 4 5 1 4 8.6 171 2120 1801 52

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Fig. 8. Optimal configuration for Scenario 1 of Case B for Example (periods 1–5).

Fig. 9. Optimal configuration for Scenario 2 of Case B for Example (periods 1–5).

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Fig. 10. Optimal configuration for Scenario 3 of Case B for Example (periods 1–5).

As was previously analyzed, when there is enough water to satisfy the irrigation needs of crops in each period, the justification for the mass integration (water and fertilizer) is based on environmental aspects (less use of natural resources and less adverse impact on natural reservoirs and groundwater) and social aspects (improved availability of water for human consumption and other economic activities, as well as less risk of damage to health), instead of economic aspects. However, if the amount of water available for agriculture is limited, then this availability determines the surface to cultivate and the yield for crops, for that reason, Example 1 is solved considering a limited amount of fresh water in each period of time as shown in Table 4 (Case B). Also, in the same way that in Case A, the Scenarios 1, 2, 3 and 4 are considered to solve the Case B (see Fig. 4), whose discussion is addressed in the next section.

3.2. Analysis of Case B for Example The results for Case B for each scenario are presented in Table 7, in which is possible to observe that in Scenario 1 the amount of available water only allows to seed one cropland with a total production of 43 tons (29% of the minimum overall production). This yields an annual profit of $14,557 US, and the associated configuration is presented in Fig. 8, this figure shows the total demands of fresh water, nitrogen and the environment discharge. On the other hand, in Scenario 2 the total production is 86 tons (57% of the minimum overall production) of maize obtained with the seeding of two cropland and the annual profit is $30,078 US, and the corresponding configuration is given in Fig. 9, in which the used water is sent from the crop 1 to crop 2 in the five periods. Besides, Table 7 presents the results for the Case B considering the Scenario 3; and

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Fig. 11. Optimal configuration for Scenario 4 of Case B for Example (periods 1–5).

in this scenario the available water in each period allows to seed three croplands for a total production of 128 tons (86% of the minimum overall production), which let an annual profit of $45,765 US and the optimal configuration is shown in Fig. 10. It is possible to observe that the waste stream from the crops 1 and 3 are sent to the crop 2 in each period, and the waste stream for the crop 2 is recirculated to the crop 1. Finally, in Scenario 4 is possible to seed the four crops to meet with the total production of 171 tons (which meets the minimum overall production) and an annual profit of $39,809 US (see Table 7). Notice that to seed the four crops there is needed to store the waste stream from the crop 2 (see Fig. 11), besides there is a direct reuse of water between some crops. As can be seen from the above analysis, the configuration with the highest profit is the corresponding to Scenario 3 as it is 15%

greater than Scenario 4, 52% higher than Scenario 2 and 214% higher than Scenario 1. However, the total production for the three crops in this scenario is less than the minimum needed to meet the market demands, and also under this scenario there is a producer without incomes. In this regard, Scenario 4 meets the minimum demand production (150 tons), since 171 tons in the four crops are produced. This is very important from a social point of view since the configuration associated to Scenario 4 allows meeting the minimum amount of food and also it represents an equitable distribution of monetary incomes; this way, the government incentives are justified. It should be noted that the most significant difference between Scenarios 3 and 4 in economic terms is the capital cost associated to the storage, pumping and piping.

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In addition, the flowrates involved in Figs. 5–11 are given in electronic Supplementary Material. From the above analysis it is important to highlight that the proposed methodology allows identifying the best economic, environmental and social scenario for the agriculture activity. Therefore, this strategy represents an excellent tool for irrigation districts and producer groups to determine the optimal water distribution in order to meet with the market demands, to ensure incomes to several producers and increase the water availability. 4. Conclusions This paper has presented a mixed integer non-linear programming model for the optimal design of agricultural water networks using an analogy of the traditional industrial water networks. The proposed model is based on a new superstructure that includes all the configurations of interest where can be found the optimal agricultural water networks involving several croplands in a field. These configurations include the optimal reusing, recycling and storing of water and fertilizers between croplands in different time periods. Also, the proposed model includes the selection of the optimal treatment of wastewater to meet crops and environmental constraints. The objective function accounts for the maximization for the overall profit and diverse constraints associated to the food demands and water availability. A case study for an irrigation district of Mexico for cultivating maize was considered to show the advantages of the proposed model. The results show that important economic benefits can be achieved for the implementation of the proposed methodology satisfying the specific demands for food, fresh water availability and environmental constraints for the wastewater discharged. Furthermore, the proposed model can be applied to different conditions, where cases where water scarcity were considered, also cases for different limitation in the initial investment were incorporated, and the results show attractive solutions that can be considered for the decision makers (i.e., producers or governments). Acknowledgements The authors acknowledge the financial support by the Universidad Autónoma de Sinaloa (PROFAPI 2013/176) and SEP-CONACYT (2013–2014). Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.compchemeng. 2015.08.006. References Agroterra Company. Sales catalog; 2014, Available at: http://www.agroterra.com/ p/hidraulicos-inox-8-de-12-a-120-m3-h-con-una-altura-manometrica-de211-mts/3088693 Alnouri SY, Linke P, El-Halwagi M. Optimal interplant water networks for industrial zones: addressing interconnectivity options through pipeline merging. AlChE J 2014;60(8):2853–74. Alva-Argaez A, Vallianatos A, Kokossis A. A multi-contaminant transhipment model for mass exchange networks and wastewater minimisation problems. Comput Chem Eng 1999;23(10):1439–53. Anderson J. The environmental benefits of water recycling and reuse. Water Supply 2003;3(4):1–10. Bagajewicz M. A review of recent design procedures for water networks in refineries and process plants. Comput Chem Eng 2000;24(9):2093–113. Belaqziz S, Mangiarotti S, Le Page M, Khabba S, Er-Raki S, Agouti T, et al. Irrigation scheduling of a classical gravity network based on the Covariance Matrix Adaptation – Evolutionary Strategy algorithm. Comput Electron Agric 2014;102: 64–72.

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