Energy Assessment of Pressurized Water Systems

An ideal system is one where there are no friction head losses or leaks (while the kinetic energy is disregarded, a common practice in network analysis). There is no excess pressure because at the critical point, $p_{ci}$, is equal to the required service pressure, $p_0$. In this case, the critical point pressure is also the required pressure (Fig. 1), and then $p_{ci}=p_{hi}=p_0$. But it must be underlined that in real cases this may not be the case.

The application of the continuity equation to the system shown in Fig. 1 (and for a given steady flow time period) is straightforward. The total injected volume at the pumping station ($V$) is equal (in the absence of water losses) to the sum of the demands of all the consumption nodes ($V=\sum v_j$). The supplied energy in that period iswhere $H_{hi}$ is the piezometric head at the highest node (ideally equal to all nodes). Additionallybeing $\gamma$ the specific weight of water, $E_{uo}$ the minimum energy required by the users, and $E_{ti}$ the topographic energy (elevation potential energy) of the system. This latter term depends on the terrain’s irregularities, thus its suggested name. In Fig. 1, the shaded area is proportional to $E_{ti}$ (the area between the top horizontal line and the dashed line). In a flat network, $E_{ti}$ would be zero. This energy also represents the amount of energy that, in theory, could be recovered by installing PATs at every consumption node. The energy to be recovered in $\mathrm{\Delta }t$ at any demand node by the corresponding PAT is $v_j$ (the volume consumed at node $j$ in the considered time period) times the available topographic pressure, $p_{jt,i}$. In reality, recovering all this energy is impossible. In many systems, this excess (when considered from the user’s perspective) energy is dissipated with PRVs (see “Dissipation and Recovery of Energy (with PRV or PATs) in a PWTS”).

When the pressure at the critical point is as required (as detailed in Fig. 1), the supplied energy is the same as the minimum required energy or base energy $E_{bi}$. Therefore

In an ideal frictionless system and for any period of time, the hydraulic grade line (HGL), sum of the natural (or gravitational) head, $h_{ni}$, and the pump head, $h_{pi}$, (or total dynamic head, TDH) is constant and equal to $H_{hi}=H_{ji}=H_{li}$. However, the total supplied energy is proportional to the corresponding demand of each period.

In a real system, friction losses depend on flow and leakage rates. As a result, the supplied head (usually the pump head) must be adjusted to meet the steady demand of each period. This implies that a control system [such as a variable frequency drive (VFD)] must be present at the pumping station. Energy balance calculations are mathematically static and then only valid during brief periods of time ($\mathrm{\Delta }t$). Therefore, for longer time periods (h, day, month or year), the total supplied energy must be calculated by integration (Cabrera et al. 2010)—meaning an extended period simulation (Rossman 2000). Alternatively, this energy can be assessed by adding the wire energy (electricity bill) and gravitational energy, the latter being $\gamma h_{ni}(V+\mathrm{\Delta }V)$, assuming a constant suction water level ($z_n$).

Finally, it is emphasized that, in this ideal case, the type of supplied energy ($h_{pi}$ or $h_{ni}$) is irrelevant for assessing an energy balance. After all, every single kWh is the same from a physical perspective, regardless of its origin.

The only difference between the systems in Figs. 1 and 2 is that in the second case the supplied energy (pressure) is above the minimum necessary value. Fig. 2 shows a pressure value at the critical point higher than the required, $p_o$ ($p_{ei}$ represents excess pressure). To distinguish this case from the previous case, the pressure and head terms are represented with an asterisk.

The higher pressure ($p_{hi}^{*}>p_o$) is attributable to excess energy entering the system

In this case, the system’s input energy is

$E_{ti}$ is obviously the same as in the previous case (and proportional to the shaded area), whereas the excess energy, $E_{ei}$, (avoidable most of the time) must be corrected with operational measures.

The energy to be supplied, $E_{uo}$, remains constant (Fig. 3). However, the topographic energy of the system, $E_{tr}$, (shaded area) changes because it is linked to the hydraulic head line. Energy losses, which are referred to as global reducible energy, $E_{rg}$, are now different from zero, and include:

In summary, the real energy supplied to the system is

In a real system, however, the minimum required energy or base energy depends on its hydraulic operation and, therefore, defining a term equivalent to $E_{bi}$ makes no sense. However, $E_{bi}$ will still represent the lower limit of this base energy, which corresponds to the case of a real system in which losses tend to zero. It is also noted that the hydraulic head is no longer a straight line because head losses are not uniform.

Eq. (6) should include the surplus energy if the pressure at the critical point (the point with the lowest pressure in the network) exceeds the required pressure. Thenwith $\mathrm{\Delta }V$ being the total water losses in the system, and $E_{er}$ the surplus energy (which can be significant because the whole input volume to the system is subject to this excess pressure).

The minimum required energy (both in real and ideal systems) is $E_{uo}$, whereas the supplied energy in an ideal system is $E_{si}$. If a part of the topographic energy included in $E_{si}$ is recovered, the energy efficiency of the PWTS will improve. Therefore, it seems reasonable to define two different ratios depending on this fact, ${\eta }_{wi}$ (with) and ${\eta }_{ai}$ (without recovery).

In an ideal system, all the topographic energy is recovered, and the total useful energy will be the sum of $E_{uo}$ and $E_{ti}$, whereas an ideal performance will be equal to

In this case, the only possible inefficiency can be the result of an excessive energy supply [$E_{ei}$, Eq. (4)]. If $E_{ei}=0$, performance would be one (${\eta }_{wi}=1$), a utopian value obtained from the ideal assumptions made.

In reality, $E_{ti}$ is (partially or totally) lost because energy recovery only makes economic sense in a few occasions. The most common scenario is one where no recovery exists. In this case, the system efficiency, ${\eta }_{ai}$, is equal to

If no excess energy is supplied, then $E_{ei}=0$, ${\eta }_{ai}$ and ${\theta }_{ti}$ are complementary with a sum equal to 1. The topographic parameter represents the fraction of the supplied energy that is lost (assuming that no energy is recovered) attributable to the system excess pressure (topographic pressure, $p_{jt,i}$, Fig. 1). If ${\theta }_{ti}$ approaches its upper limit (a system with an irregular topography), then installing PRVs to reduce pressure levels should be considered to minimize leaks and reduce stress in the pipes. In such cases, however, the possibility of introducing physical changes in the system to reduce $E_{ti}$ (dividing the system into different pressure areas) should have been considered previously. In ideal flat networks, ${\theta }_{ti}$ is zero, and then ${\eta }_{wi}$ is irrelevant.

Overall, these performances confirm an obvious truth: when energy losses are zero, supplying the minimum energy ($E_{bi}$) and recovering all the topographic energy ($E_{ti}$) results in a performace value of 1 [Eq. (8)]. Without topographic energy recovery and with $E_{ei}=0$, the maximum performace is one minus the fraction of topographic energy in the system [Eq. (9)].

Analyzing ideal systems enables establishing maximum values for system performances. Real systems share with ideal systems the numerator of the energy efficiency ($E_{uo}$), whereas the denominator changes to include energy losses. Then Eq. (7) becomeswhere energies and reducible losses are expressed on a per unit basis ($\theta$ for the energies and $\lambda$ for reducible losses). When these losses are broken into different terms, the contribution of each to the system inefficiencies can be clearly seen with the expression

Analyzing the efficiency of a system should start by calculating ${\eta }_{ar}$. The numerator ($E_{uo}$) is known, and the denominator ($E_{sr}$) is calculated, as said before, from the energy consumed by the pump (e.g., value from the electricity bill) plus the gravity (natural) supplied energy that is dependent on the characteristics of the system. The global value ($E_{sr}$) is then known, but not its components [second part of Eq. (11)]. Determining their values requires a system audit. The best strategies to improve performance can be selected once the contribution of each term is known.

This reduction requires recovering the total volume of reasonable leaks ($\mathrm{\Delta }{V}_o$), the sum of the losses ($\mathrm{\Delta }{v}_{jo}$) at each node for a given time period. The energy loss embedded in leaks would then be

This approximation implies concentrating all the leaks in a median pressure node. Its value can be estimated assuming an ideal behavior of the system ($p_{jt,i}/\gamma =z_h-z_j$, Fig. 1). This approximation is valid for losses of approximately 10–15%, but for higher ratios it could introduce a bias in the calculations. For this paper’s purposes, this is not a problem, and the level of losses must be low to set a target. In any case, the exact calculation of the energy embedded in the leaks requires an energy audit (Cabrera et al. 2010), because Eq. (17) does not consider the additional friction losses in pipes attributable to higher circulating flow rates as a result of leakage. Eq. (17) is suitable for the estimation of $E_{rl,o}$ (to calculate later ${\eta }_{ar,o}$).

This is the most uncertain estimation, especially in large systems with loops, redundancies, and parallel flows. It requires assigning an average energy loss, related to an average path for the total mobilized volume of water. This path, dependent on the demand and leak distributions, is very variable. Consumptions which are very close to supply points barely have losses. On the other hand, those further away have significant losses. An average path $L_{pm}$ must be estimated between the pumping station and the consumption nodes, and an average unit head loss ($J_m$) ($\mathrm{m}/\mathrm{km}$) for all pipes. Again, this is not an easy estimation. $J_m$ is a function of the square value of the flow rate (variable in time with demand and leakage). The preceding factors represent a significant risk of making a biased assessment of this factor.

Additionally, local head losses ($\mathrm{\Delta }{p}_{rf,p}$) must be added owing to the presence of filters, valves, and manifolds. With these assumptions, the following relationship results:

Friction head losses in PRVs will be considered later (see following section).

Pumping stations are usually responsible for a high percentage of energy losses. However, estimating a reasonable value is not difficult (using the average combined efficiency (${\eta }_{po}$) of the variable frequency drive/motor/pump and the corresponding pump head). When the pump’s head-flow curve is available, these losses are easy to calculate. Otherwise, they can be estimated using Eq. (19), in which $L_{pc}$ is the distance between the pumping station and the critical point, whereas $z_n$ represents the pump suction level

Further losses can be found in PWTS. Break pressure tanks (chambers to avoid depressions in high points are required in pipes with an irregular profile) can be responsible for important energy losses that should be avoided—similar to an excess of energy ($E_{er}$) injected in the system. Break pressure tanks (e.g., domestic tanks) are equally inconvenient; in such cases the value of the wasted energy ($E_{ro}$) can be calculated by multiplying the depressurized volume times the network pressure at the delivery node. In the absence of inefficiencies

These estimations enable calculating the performance reference value (${\eta }_{ar,o}$) [Eqs. (15) and (16)].