Assessing and auditing water transport systems by applying the energy equations

E_e

electric energy consumed

E_hp

hydraulic energy delivered by pumps (E_hp=E_e.η_e.η_p)

E_N

natural energy supplied to the system

E_T

total energy supplied to the system (=E_e+E_N)

E_ou

output energy useful (supplied to users)

E_ol

output energy lost (in leaks)

E_o

output energy (=E_ou+E_ol)

E_oum

minimum energy required by users

E_ΔE

excess (Δ) of energy supplied to users (=E_ou−E_oum)

H_oi,m

minimum piezometric required

h_p

pumping head

I_e

energy intensity

real energy intensity

P_t

power supplied by compensation tanks

P_N

power gravitational (natural)

P_Sl

total power system losses

P_pl

power pumping station losses

P_T

total power supplied

P_eh

hydraulic power input (entrance) to the system

P_e

total electric power input

P_f

power dissipated by friction

P_oh

output hydraulic power

P_ohl

power lost in leaks

P_ohu

useful delivered power

P_ohum

useful (minimum required) delivered power

P_ml

power lost (other miscellaneous losses)

P_ΔP

power in excess (Δ) supplied

p_m

minimum pressure

Q_t

tank net flow

u

internal energy

z

elevation

W_s

hydraulic energy delivered by pumps (W_s=E_hp)

friction energy variation

ΔE_G

gravitational energy variation

thermal energy variation

ΔT

temperature variation

electric motor efficiency

pump efficiency

global efficiency

global efficiency considering the minimum standard pressure

The first question to address is why the extended Bernoulli equation is preferentially used in hydraulic flows whereas the energy integral equation is, surprisingly, rarely applied. The primary reason is that since most hydraulic flows are assumed to be incompressible and one-dimensional, the extended Bernoulli equation fits these problems perfectly. However, the energy integral equation (White 1973), which includes thermal and mechanical terms as well as various inputs and outputs, is more general and therefore not as immediately applicable. In fact, it can be used to analyze any problem because it performs a general energy balance over a control volume (CV) limited by a control surface (CS). Furthermore, as these two equations have historically different origins, they can only be compared after formulating restrictive hypotheses to simplify the energy integral equation to equate it to the extended Bernoulli equation, a generalization of the original equation formulated by Bernoulli in the 18th century (Rouse & Ince 1963).

This form was derived by separating the flow work through the CS (between the system and the external medium) and the shaft work, commonly associated with turbomachines. According to the sign convention for work (work done by the system is considered positive), pump work is negative and turbine work is positive. The other left-hand side term of the equation, the heat transfer rate, ⁠, represents the thermal energy exchange between the external medium and the system through the CS because of the existence of a temperature gradient (heat transfer to the system is considered positive). Other variables are, ρ, the fluid density; dV, the differential of volume inside the CV; p, the pressure and⁠, the differential flow through the CS. This formulation has few constraints and is widely used in thermal fluid mechanics.

Since the Bernoulli equation (Equation (3)) involves only mechanical energy, whereas the energy integral equation (Equation (1)) includes both mechanical and thermal energy, a crucial point in comparing the equations is related to the coupled nature of the physical problem, that is, with the possible conversion between thermal and mechanical energy, which is the clear boundary between thermal fluid mechanics and hydraulics.

In hydraulics, the special case of incompressible fluids, with both density and viscosity constants, is of considerable importance (Batchelor 1967; Panton 2013). The continuity and the momentum equations are simpler and, more importantly, decoupled from the energy equation (the momentum equation does not include the temperature). Consequently, the mass and momentum equations form a set of four scalar equations with four mechanical unknowns (⁠ and p). Later, when necessary, the obtained velocity field can be substituted into the energy equation to determine the temperature field. On the other hand, for an incompressible fluid, thermal and mechanical energies are only tied by friction.

The integral energy equation (Equation (1)) can be applied to both incompressible and compressible flows. One example is the analysis of a water hammer (elastic conduit with slightly compressible fluid) from an energy perspective (Karney 1990), where velocity and pressure changes are related to account for water compressibility (elastic effects), while the density is assumed only a function of the pressure (which is captured via the bulk modulus of elasticity). Thus, the flow is considered essentially isothermal (due to temperature, there is a negligible change in flow properties) and the thermal and mechanical variables are decoupled.

The above analysis confirms that, for an incompressible fluid flow, friction is the only link between thermal and mechanical energy. From a mechanical perspective, friction takes its toll converting useful mechanical energy into dissipated heat energy. If there is no heat transfer, the temperature of the fluid increases because of the friction, and, in the end, u₂ becomes greater than u₁. On the other hand, when the flow is not adiabatic (the system is not well insulated), due to the temperature gradient, heat will flow to the surroundings. Last, for an isothermal flow (both temperature and u, constants), the system has a loss of heat, q, at the same rate that friction converts mechanical into thermal energy.

In short, the same problem can be analyzed from different perspectives using both equations. With different origins (one being integral and thermal, the other differential and mechanical) they follow opposite paths: generalization, in the case of the Bernoulli equation, and simplification, for the integral energy equation, where an initial global power balance integrated over time becomes an energy balance. This last versatile equation can be employed for energy diagnosis and auditing but requires more information than the simplest Bernoulli equation. The latter is recommended to diagnose and audit simple systems and, when applied repeatedly, it can solve more complex systems (such as hydraulic networks), albeit in snapshots.

To show the particularities of each equation, this paper is organized as follows. The next section is a short reminder, through the energy intensity concept, of how to apply the Bernoulli equation to simple and complex systems. The rest of the study is devoted to the integral energy equation. After some previous remarks, two real examples are discussed. First, a closed and hydraulic system, that shows the complementarity of both equations. Later, a multi-topology system, with multiple inputs and outputs, proved the versatility of the global energy equation. A final summary collects all these recommendations.

For simple systems (one input and one output), these concepts have been well developed (Del Teso et al. 2023) and can be summarized as follows. On one side, the real energy intensity, I_er, is known from field data, including the energy consumed (billed kWh) and the measured pumped water volume (m³) at the delivering point. On the other side, the energy intensity objective, I_eo, is determined by setting target efficiency values. The difference I_er – I_eo indicates the margin of improvement.

This estimation has a weak point. Bernoulli is a stationary equation and during the period considered, the system's conditions can change (e.g. pumping groundwater systems, with water table level variations along the period of analysis). Therefore, the difference I_er – I_eo provides a snapshot assessment, that is strictly valid for a given instant. To some extent, this drawback can be mitigated by using average values over the whole considered period.

Bernoulli's equation can also be used to assess complex systems (many output nodes and, sometimes, more than one input node). A crucial aspect is to identify the final node of the equation because the first one is always the source. This final node, named the critical node, is the one among all the systems that need more energy to achieve the required pressure. Applying the equation between the source and the critical node provides the unitary energy that the system needs, and this maximum value is extrapolated to the whole system. But again, additional care must be taken with the Bernoulli equation, assessing systems with variable time conditions along the considered period of time. In fact, the critical node can depend on the scenario and, even being the same node, the energetic requirements to be covered between both nodes can be different. Again, the energy used to assess the system is the average energy required over the period.

Another fact to consider is that, in the rest of the output nodes, the pressure, although higher than required, is not known. This drawback can be addressed by applying successively the Bernoulli equation between the source and all these output nodes, as many times as necessary up to cover the whole network. However, this is most of the time an awkward strategy due to the high number of nodes. Therefore, a complete physical picture of how the system works can be calculated easily with Bernoulli's equation, although at the expense of simplicity (Cabrera et al. 2021).

In conclusion, Bernoulli is the appropriate energy equation to assess and audit simple systems. Being one-dimensional, both stages (assessment and audit) are, in practice, covered simultaneously. With these results, the energetic system's behavior can be easily labeled (Cabrera et al. 2023; del Teso et al. 2023).

Thus, the energy integral equation differs from the Bernoulli equation in dimensionality (a CV versus a stream tube, Figures 1 and 2) and the capability to analyze transient flows. This general equation has been previously applied to diagnose (Cabrera et al. 2015) and audit (Cabrera et al. 2010) complex systems. In this study, it is extended to multi-topology systems.

The terms of Equation (7) are physically meaningful. Firstly, because the shaft work added (by pumps) or removed (by turbines) from the system balances the change with the time of the energy stored inside the CV, first integral (it is non-zero if there are tanks inside the CV, with level's variation), and second integral which is a flow term that represents a power global balance between the outgoing and incoming flows. Both integrals include the kinetic energy (negligible in our analysis), the internal energy (its change represents the power dissipated by friction, Equation (5)) and the gravitational head. The pressure head is the only difference between both integral terms.

Some remarks concerning this equation follow:

Systems can be open or closed (with or without flow throughout the CS, respectively). When dealing with pressurized water transport, most of the systems to analyze are open. In closed systems, because there are no input nor output flows of energy, it does not make sense to define efficiency.

There is a growing interest in modeling the water temperature evolution along urban water networks (Agudelo-Vera et al. 2020) because its value impacts the water quality. In fact, some countries have set a maximum temperature threshold of 25 °C at the customer's tap (Blokker & Pieterse-Quirijns 2013). In the context of climate change, with increasingly hot summers, this issue becomes more challenging. Obviously, for other uses (as irrigation), this question has no interest.

In any case, thermal and dynamic problems are decoupled because water is an incompressible fluid. Although the main focus of EPANET (Rossman 2000) is on water quality issues, its solver ignores any potential thermal exchange between soil and water. Even its recent update (Rossman et al. 2020) has ignored this question. Conversely, the models proposed to determine the water temperature evolution (Burch & Christensen 2007; Blokker & Pieterse-Quirijns 2013) are, as expected, decoupled of the hydraulic equations.

In summary, the adiabatic hypothesis assumed in this work may not apply in some cases, but this has no influence on the mechanical energy analysis being performed. The hydraulic variables calculated, and therefore the temperature variation ΔT due to friction, will be the same no matter whether the heat transfer soil–water effect is included or not. The only changed value should be the final water temperature, important from the water quality side, but irrelevant for the purpose of this analysis.

Gravitational energy, which depends on the potential elevation change of the considered mass, has no absolute value. Therefore, it is a crucial issue to set correctly the system's reference level. This is the lowest active node (consumption or source). Its elevation is set to zero. If real elevations (over the sea level) are kept, the gravitational energy is oversized although the final energy balance will be satisfied because inlet and outlet flows are equal (ρ=C) and the gravitational terms linear. Both inlet and outlet gravitational terms are equally incremented. However, a tank inside the CV breaks the energy balance, because the changes in the energy stored in the tank depend on the square elevation of the level while inlet and outlet flows are no longer equal. Therefore, node elevations must all refer to the lowest node, an invariable reference when it is a consumption node. Nevertheless, if the lowest node is a tank, its elevation varies with the surface level but can be kept as a reference. This will be further analyzed in the following section.

For the considered period, Equation (10) shows that in the closed system the loss of gravitational energy, ΔE_G, equalizes the positive internal energy variation (ΔE_T= ΔE_f). Assuming a quasi-static flow (Rao & Bree 1977), without changes of energy inside the pipe in the period t, water thermal changes came from the absorption of heat generated by friction between pipe and fluid.

The negative value confirms the loss of gravitational energy. The first term is the energy lost at the first tank, equal to the weight of the discharged volume multiplied by the change of elevation of the center of gravity (assuming a constant reference level). The second term is the energy gained by the lower tank during the filling with respect to the initial reference. Therefore, this term is the correction of the error introduced because of the variation of the reference level.

In short, for a variable reference elevation, the compensation term of Equation (16) restores the balance and therefore can be kept as a reference level. If it is a reservoir, the lowest level is constant and the correction is null (⁠⁠). In a reverse flow situation (pumping line), the lower tank is emptied while the upper one is filled. In that case, Equation (14) is the same, but the signs of the terms are interchanged.

Data of the system depicted in Figure 3 are: D₁ = 40 m; D₂ = 20 m;₁ = 200 m;₂ = 100 m; ξ₁₀ = 6 m; ξ₂₀ = 1 m; D_p = 100 mm and L = 5,000 m. The volume transferred between both tanks is 1,256.64 m³. Two scenarios are considered: (a) valve completely open (no local losses; k ≈ 0) and (b) valve almost closed (local losses are relevant; k = 500).

This result is equal to the loss of gravitational energy, ΔE_G, 350.99 kWh. This value has been obtained from Equation (14) with h_o = 105 m and h_f = 104 m. From this value, the water temperature variation due to friction is ΔT = 0.24 °C, Equation (17), a minuscule value, because of the high specific heat of water. In fact, the energy intensity to heat water by 1 °C is 1.16 kWh/m³ (Equation (5)), which is equivalent to elevating 1 m³ of water by 426 m. Thus, it is reasonable to ignore thermal effects due to hydraulic friction.

If the valve is open (K = 0), only the pipe dissipates energy but, if partially closed, the energy is lost at both elements (valve and pipe), although the integral energy equation is unable to break down their respective contribution. For this purpose, assuming a quasi-stationary flow (a valid assumption because ⁠), it is necessary to model hydraulically, with EPANET, both elements. Results are:

• (a) Valve open: Time required to transfer 1,256.64 m³, 32 h 40 min; initial velocity, 1.38 m/s; final velocity, 1.34 m/s; pipe average losses 102.5 m.

• (b) Valve partially closed: Time required to transfer 1,256.64 m³, 40 h 00 m; initial velocity, 1.13 m/s; final velocity, 1.10 m/s; pipe losses, at t = 0 h, 72.35 m and at t = 40 h 00 m, 68.90 m; average pipe friction losses, 70.63 m. Valve losses, at t = 0 h, 32.67 m and at t = 40 h 00 m, 31.06 m, average valve's losses, 31.87 m. Therefore, the valve is responsible for 31.09% of the total losses, because it is practically closed (k = 500).

From this simple but conceptually strong case study, it can be concluded that the initial level of the lowest tank, even being variable, can be set as the reference elevation node. Furthermore, it evidences that to complete the audit, it is necessary to solve the hydraulic system's behavior, showing that the discussed equations are also complementary to each other.

A final concept must be considered: the minimum energy to be supplied (H_om,i), defined through the minimum pressure, because if the pressure exceeds the standard (minimum) value, p_m, then H_o,i>H_om,i and the system will deliver more energy than needed. That is, to some extent, an inefficiency that should be avoided.

The assessment of a specific topology within a multi-topology system without energy storage (no tanks inside the CV) can be conducted by comparing output with input powers, disregarding what occurs inside the CV. Within this approach for the diagnosis phase, the system can be assimilated to a black box, defined as ‘a fictitious system representing a set of concrete systems onto which stimuli S impinge and out of which reactions R emerge. The constitution and structure of the box are altogether irrelevant to the approach under consideration, which is purely external or phenomenological’ (Bunge 1963).

If the CV includes internal tanks, the flow Q_t, calculated from a differential quasi-static continuity relationship (Equation (20)), has to be known in order to perform the assessment. The additional inputs/outputs of these tanks (as water levels, z), although internal to the CV, are similar to the external inputs/outputs of the system.

Therefore, at the diagnostic stage, the system can be associated with a black box, with or without energy storage. Nevertheless, at the next stage, the audit, the whole inefficiency, ⁠, must be decoupled in (Equation (25), second term). To do that, each system's element must be hydraulically characterized to assess individually its work, complementing the integral energy equation. So, each possible topology must be diagnosed and audited. Nevertheless, it is convenient to carry out the analysis of each scenario to identify the most inefficient ones and to improve them.

Equations (27) and (29) provide, in power terms, the instantaneous efficiency (snapshots) for different topologies L_j. The whole system assessment over a period t (month or a year) to calculate the average efficiency, η, can be obtained by integrating Equation (27) (or Equation (29)) over time along all the scenarios that the system evolves through. However, it is easier and more direct to calculate η through an energy balance, from:

• Average piezometric heads (at the inlets/outlets of the system along t), based on the recorded pressures along t, and the volumes measured at the inlets/outlets of the system for the same period t.

• The electric energy supplied to the system, E_ei, the sum of the electricity bills during period t.

• Over a long period of time, the contribution of the internal tanks is negligible (therefore, the third addend of the denominator, is zero).

Once the system's energy losses have been determined with the assessment, it proceeds, when necessary (a poor performance result) to audit the system. This involves breaking down the inefficiencies. The audit will quantify all the power lost through friction,⁠, power embedded in leaks,⁠, excess of delivered power due to overpressure, and power inefficiency at the pumping station, ⁠. This breakdown is essential to identify what must be improved in the system.

This balance shows that the total power supplied, P_T,,j, is equal to the hydraulic input power ⁠, plus the pumping station power losses, ⁠, and equalizes the sum of the minimum useful output power, ⁠, the excess of the delivered power, ⁠, and the powers required to overcome internal inefficiencies (friction, and leaks, ⁠), plus losses at the pumping station, ⁠. After the audit, the terms of Equation (39) are known.

Formulating a global audit in terms of power in multi-topology systems is neither useful nor physically meaningful, but it is in terms of global energy as proposed in the assessment. It is better to analyze each topology with its inefficiencies, examine their causes, and amend them. The feasible solution is to improve, one by one, the efficiency of each system's topology, paying more attention to the most energy-consuming one.

There are different energy sources (pumps or tanks) and different critical nodes for each topology. To assess each system with the Bernoulli equation is impractical because the initial and final nodes change permanently. However, the use of the energy integral equation is straightforward. Table 2 details the results of the diagnostic calculated with Equation (28). These results have been obtained from the inlet/outlet shown in Figure 6.

The water comes from the wells, which are the lowest nodes of the system. Therefore, only when both good pumps are ON (topology 1), is there a gravitational (or natural) input power contribution from well 1 (well 2 is the lowest). The total input power is the sum of the pump's power minus the power invested to fill the tanks, which only supply power to the system when they are emptying. This occurs in topologies 4 and 5. Summing up, for all topologies (except for L4 and L2.1), the pumping power exceeds the total input power because the internal tanks are being filled.

Efficiency is the ratio between the power output and power input (P_ohu/P_T) while their difference (P_T – P_ohu) gives the power lost, P_Tl, for each topology. The two most inefficient ones are L1 and L3 when Pumps 2 and 3 are simultaneously working.

The mathematical model is used to perform the audit (Equations (33) to (39)), calculating in which parts of the system the power is lost. Table 3 shows the results for topology (T1).

Similar tables for the other topologies will show their respective weak points. In this topology 1, major inefficiencies are located at the pumping stations.

A possible solution to enhance efficiency could be to remove tank 1 and pump 3. This would entail a direct pumping from well 2 to tank 2. Consequently, P_ml would disappear and less head would be needed in pump 2. Additionally, feeding the tanks from the bottom should be considered as an improvement, as it requires less head.

A final global assessment (in energy terms) provides relevant information. This value can be obtained from the different topologies' efficiencies, with an appropriated weighted average. However, it is often simpler to determine this value from annual inputs and outputs data. Table 4 presents volumes (elevated and delivered to users), annual energy consumption by the pumps, average pressure at the system outlets, elevation nodes, and minimum standard pressure.

Table 4

Global system assessment

.	Input .		Output .
	Well 1 .	Well 2 + Pump 3 .	Industrial .	Town .
Volume [m3]	641,879	524,774	9,853	1,100,473
Electric energy consumed (billed) [kWh]	403,085	338,885	–	–
Average output pressure [m]	–	–	50	38
Minimum output pressure [m]	–	–	30	30
Electric energy consumed, Ee	741,970
Natural energy supplied, EN	3,498
Total energy injected, ET	745,468
Energy supplied to users, Eou			357,401
Minimum energy required by users, Eoum			332,874
Average efficiency,	0.48
Average efficiency (minimum pressure),	0.45

.	Input .		Output .
	Well 1 .	Well 2 + Pump 3 .	Industrial .	Town .
Volume [m3]	641,879	524,774	9,853	1,100,473
Electric energy consumed (billed) [kWh]	403,085	338,885	–	–
Average output pressure [m]	–	–	50	38
Minimum output pressure [m]	–	–	30	30
Electric energy consumed, Ee	741,970
Natural energy supplied, EN	3,498
Total energy injected, ET	745,468
Energy supplied to users, Eou			357,401
Minimum energy required by users, Eoum			332,874
Average efficiency,	0.48
Average efficiency (minimum pressure),	0.45

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As expected, the average efficiency calculated over a one-year period (0.45), is an intermediate value between the maximum efficiency (0.92 in T2.1) and the minimum value (0.22 in T1 and T3).

Pressurized water transport systems can be divided into simple, complex (current networks) and multi-topology systems. All of them must be assessed in terms of their energy efficiency and, when it proceeds, audited. Previous papers have presented assessments and audits for simple (Cabrera et al. 2021; Cabrera et al. 2023) and complex systems (Cabrera et al. 2010; Cabrera et al. 2015) using either the Bernoulli equation or the integral energy equation.

The main objective of this study is to discuss which equation is most convenient to apply to a particular system for analyzing it from an energy perspective. Particular attention has been paid to multi-topology systems because they have not been analyzed previously. Each topology creates a different system that requires, at the assessment stage, an independent analysis.

Table 5 summarizes the recommended equation to be used in each analysis. Integral Energy (P) refers to the equation expressed in power terms, its original form, while an (E) indicates that the equation has been integrated over the analysis period to calculate the energy required or delivered.

Regardless of the energy equation used, the hydraulic behavior of the system (currently through its mathematic model) is needed to perform the audit. Otherwise, it is impossible to break down the losses.

The transport of pressurized water consumes a large amount of energy that should be reduced within the current context of climate change and growing energy costs. Therefore, the first step is to diagnose the system understanding its operational mechanisms and potential areas for energy savings. If there is an important margin for improvement the system should be audited to identify where and how much energy is lost. Both steps can be performed using either the energy integral equation or the Bernoulli equation. While these equations differ in their origins, once adapted, both are suitable to perform these analyses. However, as each equation has its advantages and disadvantages (spatiality and simplicity, respectively), it is advisable to discern the cases for which each equation is best suited. Additionally, any water audit must complement the energy equation with the hydraulic behavior of the system, currently its mathematical model.

This study has placed particular emphasis on the integral energy equation, rarely used to solve hydraulic problems. In particular, it has been applied to assess and audit multi-topologic water transport systems. In addition, the application to a closed simple system has evidenced that the reference elevation level must be the lowest active node (source of consumption) even when the elevation of the lowest node is a tank with a variable level.

The type of system dictates the applicability of the equations. For simple and stationary systems, the Bernoulli equation is accurate and easy to use, making it the recommended equation. Both equations are useful for complex systems, as long as the topology does not change over time. The energy integral equation requires more data but provides more information than the Bernoulli equation. The use of the Bernoulli equation corresponds to a more direct and intuitive approach, but only provides a snapshot of the system, at the moment the equation is applied. This study has established recommendations on when to apply each equation depending on the system under analysis and the phase of the analysis.

The authors acknowledge the staff of the water company FACSA for providing financial assistance, helpful advice, and the real case study presented in this work.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.