*2.2. Balance of Energy*

Once flows and pressures are estimated along the time in the whole network, the energy equation (Reynolds Theorem) must be implemented to consider the energy balance in the system [35].

According to Figure 2, a generic irrigation network with all possible elements (reservoir, pumps, turbines, and compensation tanks) is presented. The conservation of energy equation is defined as:

$$\frac{d\mathbb{E}}{dt} = \frac{dQ}{dt} + \frac{\mathcal{W}\_{shaft}}{dt} = \frac{d}{dt} \left[ \iint\limits\_{CV} \rho \left(gz + u + \frac{v^2}{2}\right)dV + \iint\limits\_{CS} \left(gz + u + \frac{\mathbf{P}}{\rho} + \frac{v^2}{2}\right) \rho \left(\stackrel{\dashv}{v} \cdot d\vec{A}\right) \right. \tag{4}$$

where:

Ñ

*dE dt*= exchange of energy per unit time in the control system;

*dQ dt*= exchange of heat per unit of time (heat power);

*Wshaf t dt*= power transmitted directly to or from the fluid (e.g., pump);

dV = differential volume of control volume for integration;

Ñ *v* = velocity vector of fluid;

*d A* = differential area of control surface for integration;

ρ = fluid density;

*gz* = potential energy per unit mass;

*u* = internal energy per unit mass;

*v*2 2 = kinetic energy per unit mass;

*P ρ*= height of pressure per unit mass;

Within the control system, the following simplifications can be made:

The water density is constant.

Flow is uniform in each interval.

Exchange of heat between fluid and surroundings is negligible (adiabatic system).

The shaft work is the power transmitted directly to/from the fluid in the case that a pump or turbine exists in the network.

There is no compensation tank in the network, therefore, the time energy variation inside of the control volume as function of time is negligible.

**Figure 2.** Energy balance in the pressurized irrigation water network adapted from [36].

If an irrigation system operates by gravity (Figure 3), the equation of energy applied to any control system along a time interval is defined by Equation (5):

$$\gamma Q\_D H\_D \Delta t = \sum\_{i=1}^{n} \gamma Q\_{oi} H\_{oi} \Delta t + \rho \left( \sum\_{i=1}^{n} \left( Q\_{oi} \mu\_{oi} - Q\_{Di} \mu\_{Di} \right) \right) \Delta t \tag{5}$$

where:

Δ*t* = time interval (s);

*n* = total number of irrigation points;

*i* = individual irrigation points;

γ = specific weight of the fluid (N/m3);

*QD* = total flow demanded by the network (m3/s);

*HD* = piezometric head of the reservoir. For a pumped system, the value is the manometric height; *Qoi*= flow demanded by each irrigation point (m3/s);

*Hoi*= piezometric head of the consumption node (m);

γ*QDHD* = total energy (kW) supplied to the system. This term is equal to *ET*, which is later defined;

ř *ni*"1γ*QoiHoi* = energy consumed by all irrigation points (kW). This term will be defined as *ERI* plus *ETRI*;

ρ `ř *ni*"<sup>1</sup> p*Qoiuoi* ´ *QDiuDi*q˘ = Exchange of internal energy. In an adiabatic system, it is equal to friction losses. This term will be defined as *EFR*.

Leakages are not considered in this analysis because the drip irrigation network is still new (minimum leakages), the maintenance and repair plans are usually undertaken (which reduce possible losses), and finally, these networks are not as automated as drinking systems so unmeasured volumes and leakages are difficult to discern. If an energy audit were made, this volume should be considered or estimated [6,36]. Furthermore, the installation of hydraulic machines does not affect the water quality of the final use (*i.e.*, irrigation).

**Figure 3.** Scheme of irrigation network.

When a global energy balance of the network is established, it is possible to define different terms of energy. Such as lines, hydrants and irrigation points, as follow (Figure 4):


$$E\_{T\_i} \left( kWh \right) = \frac{9.81}{3600} Q\_i \left( z\_o - z\_i \right) \Delta t \tag{6}$$

where:

*Qi* is the flow circulating by a line that supplies to more unfavorable irrigation point (most disadvantageous consumption node in terms of need of the pressure) (m3/s);

*zi* is the geometry level above reference plane of the irrigation point. In this case, the reference is sea level (m);

*z*0 is the geometry level above reference plane of the free water surface of the reservoir. In this case, the reference is sea level (m);

Δ*t* is the time interval (s).


$$E\_{FR\_i}\left(k\mathcal{W}h\right) = 2.725 \cdot 10^{-3} Q\_i \left(z\_0 - \left(z\_i + P\_i\right)\right) \Delta t \tag{7}$$

where:

*Pi* is the service pressure in any point of the network when consumption exists. The units are meter water column (m w.c.).

Minor losses (pressure loss in particular network components like tees, valves, and similar) are considered as a percentage of friction losses. Associated with this term, the Energy Footprint of Water (EFW) can be calculated. Energy Footprint of Water is defined as the ration between energy dissipated due to friction losses *(EFRi)* over the distributed volume on the network (kWh/m3).


$$E\_{TN\_i}(k\mathcal{W}h) = 2.725 \cdot 10^{-3} Q\_i P\_{min\_i} \Delta t \tag{8}$$

where:

*Pmini* is the minimum pressure of service of a line or hydrant to ensure the minimum pressure in the most disadvantageous consumption node. The units are meter water column (m w.c.).


$$E\_{\rm RL\_i}(k\mathcal{W}h) = 2.725 \cdot 10^{-3} Q\_i P\_{\rm minI\_i} \Delta t \tag{9}$$

where:

*PminIi* is the minimum pressure of service of an irrigation point required to ensure the irrigation water evenly. The units are meter water column (m w.c.).


$$E\_{TA\_i} \, (kWh) = 2.725 \cdot 10^{-3} Q\_i \left( P\_i - P\_{min\_i} \right) \, \Delta t \tag{10}$$


$$E\_{TR\_i}(kWh) = 2.725 \cdot 10^{-3} Q\_i \left( P\_i - \max\left( P\_{\min\_i i}, P\_{\min\_i i} \right) \right) \Delta t = 2.725 \cdot 10^{-3} Q\_i H\_i \Delta t \tag{11}$$

where

> *Hi* is the value of head in irrigation point, hydrant or line (m w.c.), obtained as:

$$H\_l = P\_l - \max\left(P\_{\min\_i}; P\_{\min I\_i}\right) \tag{12}$$


$$E\_{NTR\_i} = E\_{TA\_i} - E\_{TR\_i} \tag{13}$$


*CRTi* " *ETRi ETAi* (14)

**Figure 4.** Scheme of hydraulic energies grade line.

*Qh* (Figure 4) is the flow circulating in a line or consumed by a hydrant (m3/s), *zh* is the geometry level above reference plane of the line or hydrant. *H*0 is the piezometric height of the reservoir that supplies the network. The units are meter water column (m w.c.). If reservoir is open, *H*0 is equal to *z*0.

When Equation (5) is applied in a point of the network, it is defined by Equation (15):

$$E\_{T\bar{i}} = E\_{FR\bar{i}} + E\_{R\bar{i}i} + E\_{TR\bar{i}} \tag{15}$$

When all irrigation points are considered (*zi* in Figure 4), the annual balance of energy is defined by Equation (16):

$$\sum\_{i=1}^{n} E\_{Ti} = \sum\_{i=1}^{n} \left( E\_{FRi} + E\_{Rli} + E\_{TRi} \right) \tag{16}$$

In the case of lines and hydrants (*zh* in Figure 4), the annual balance of energy is defined by Equation (17):

$$E\_T = E\_{FRh} + E\_{TA} + E\_{TN} = E\_{FRh} + E\_{RI} + E\_{TR} + E\_{NTR} \tag{17}$$

The energy footprint of water and the theoretical recoverable energy are crucial for the energy balance. The energy footprint on the network distribution can be obtained along the year and compared with average values analyzed in other distribution systems. Some of these values are: 0.31 kWh/m<sup>3</sup> in injected irrigation network [36], 0.18–0.32 kWh/m<sup>3</sup> according to California Energy Commission [37], 0.081 kWh/m<sup>3</sup> in Bangkok, 0.5 kWh/m<sup>3</sup> in Delhi and 0.13 kWh/m<sup>3</sup> in Tokyo [38].

Regarding the theoretical recoverable energy in a network, it mainly depends on the orography of the irrigation area. The networks with larger gradients between the supply and the consumption points have greater possibility to recover energy, if the appropriate machine is selected. The energy recovery can be analyzed in different parts of the network:


The presented methodology helps managers to estimate the theoretical recoverable energy in irrigation points, hydrants and branches (pipelines). According to Spadaro *et al.* [39], this recovery can contribute with a theoretical average reduction of greenhouse gases emission between 582 and 877 gCO2/kWh when compared to non-renewable energy solutions (e.g., coal and gas) and 1150 gCO2/kWh when compared to emissions of fossil fuel [40]. However, this reduction will depend on the water source (groundwater, superficial or residual water) and distribution (gravity or pumped) [41]. Future research should try to integrate all applications together (supply, irrigation and wastewater for better water management) in a strategy to improve system efficiency, thus reducing greenhouse gases [42].

The viability of these installations is subject to economic evaluation (incomes *vs.* costs). Zema *et al.* [43] proposed a simple method to evaluate the economic feasibility of micro-hydropower plants in irrigations systems. In preliminary studies, these methods are good indicators for taking decisions to develop more detailed projects. These decisions are focused on the selection of machine efficiency, temporal distribution of energy produced and investment analyses. Similarly, Castro [44] proposed the period simple return (PSR) and energy index (EI) as indicators of investment viability. These sorts of installations are viable if the PSR is less than six years and the energy index smaller than 0.6 €/kWh. PSR and EI are defined by the following equations:

$$PSR = \frac{IC}{I - \mathcal{C}} \tag{18}$$

$$I = P\_E E \eta \tag{19}$$

$$C = C\_0 E \eta \tag{20}$$

$$EI = \frac{IC}{E} \tag{21}$$

where: *IC* is the investment cost (€); *C* is the annual operating cost (€/year); *C*0 is the unit operating cost (€/kWh); *I* is the annual income (€/year); *PE* is the energy price (€/kWh); *E* is the theoretical energy recovery by the turbine (kWh/year) and η is the machine efficiency.

The investment cost using PAT is 50% lower than the cost of traditional turbines [45]. Carravetta *et al.* [20] estimated *IC* of 545 €/kW if the machine is electrically regulated. Incomes depend on generated energy, which depends on the price of energy, recovery energy and the efficiency of the machine. Based on the specific speed, expert literature indicates that efficiency varies between 50% and 60% [20,46]. Castro [44] established a sales price (PE) of 0.0842 €/kWh and C0 of 0.0145 €/kW. These solutions, due to the smaller cost of PATs, present a lower payback period.
