#### 2.1. Brief Summary of the Granados System

Granados System is a pipe-sizing method indicated for the design of branched water distribution systems. It is a gradient based methodology. It is, in fact, based on the ‘change gradient’ concept: The change gradient is defined as the cost of reducing one meter of head loss by increasing the pipe diameter from

${\varnothing}_{\mathrm{i}}$ to the next bigger one

${\varnothing}_{\mathrm{j}}=\text{}{\varnothing}_{\mathrm{i}+1}$:

where P

_{i} and P

_{j} are prices of pipes of length L and diameters

${\varnothing}_{\mathrm{i}}$ and

${\varnothing}_{\mathrm{j}}$, respectively; ∆hi

^{q} and ∆hj

^{q} are the head losses of a pipe of length L and diameters

${\varnothing}_{\mathrm{i}}$ and

${\varnothing}_{\mathrm{j}}$, respectively, when a flow rate q is circulating. When the head losses are calculated using Manning’s formulation, the change gradient’s expression is:

with n being Manning’s friction coefficient.

The definition of the change gradient also needs to include some other associated costs such as excavation expenditure or reinforcements against chemically aggressive environment or hydraulic transients. During the optimization stage, hydraulic transients are usually taken into account in a simplified way, for instance, extra thickness of the pipes to withstand water hammer transient pressure. Once the optimum design is achieved, proper assessment of water hammer should be carried out to verify the initial hypothesis. These associated cost factors increase the pipe price, and therefore, they have to be accounted in the design through P_{i} and P_{j}.

Increasing the pipe’s diameter means a reduction in the head loss along the pipeline, but it has the extra cost of the wider and more expensive tube. Since the change gradient is the cost of reducing the head loss by one meter, it needs to be compared to the cost of the energy required for pumping the water at one meter height throughout the entire life of the facility, C

_{E1}, which reads:

where Ca

_{E}_{1} is the annual energy cost per meter, f

_{A} is the discount factor, n

_{c} is the duration of the construction period, n

_{u} is the useful life of the installation, i is the discount rate, V is the annual volume of water to pump, μ

_{B} and μ

_{M} are the pump and engine efficiency, and lastly p

_{E} is the unit price of energy.

As it is justified in Granados’ work, with the exception of the pump performance μ

_{B}, the other variables in the previous equation are relatively known data: V is the total demanded volume to pump, p

_{E} is the unit price of the energy that has been hired, and f

_{A} is calculated from the discount rate i, the service life n

_{U} of the water pipeline which depends on the chosen material, and the construction period n

_{C}. Regarding the engine efficiency μ

_{M}, although it varies theoretically depending on the model chosen and the operating point of the pump, the variations in engine performance are so small that it can be considered constant across different models and manufacturers. Therefore, the above equation can be simplified to:

As previously said, Granados System consists of comparing the cost of building a wider pipe (change gradient) and reducing the head loss by a meter, to the cost of pumping one meter of head loss (C_{E1}). The reasoning is the following:

If ${\mathrm{GC}}_{{\varnothing}_{\mathrm{i}}\text{}\to \text{}{\varnothing}_{\mathrm{i}+1}}^{\mathrm{q}}$ < C_{E1} ⇒Diameter ${\varnothing}_{\mathrm{i}+1}$ is preferable to ${\varnothing}_{\mathrm{i}}$.

If ${\mathrm{GC}}_{{\varnothing}_{\mathrm{i}}\text{}\to \text{}{\varnothing}_{\mathrm{i}+1}}^{\mathrm{q}}$ > C_{E1} ⇒Diameter ${\varnothing}_{\mathrm{i}}$ is preferable to ${\varnothing}_{\mathrm{i}+1}$.

But since C_{E1} depends on the pump efficiency µ_{B}, the procedure changes to:

If ${\mathrm{GC}}_{{\varnothing}_{\mathrm{i}}\text{}\to \text{}{\varnothing}_{\mathrm{i}+1}}^{\mathrm{q}}{\mathrm{C}}_{\mathrm{E}1}\Rightarrow {\text{}\mathrm{GC}}_{{\varnothing}_{\mathrm{i}}\text{}\to \text{}{\varnothing}_{\mathrm{i}+1}}^{\mathrm{q}}{\mathrm{KC}}_{\mathrm{E}1}\text{}\frac{1}{{\mathsf{\mu}}_{\mathrm{B}}}\Rightarrow {\text{}\mathsf{\mu}}_{\mathrm{B}}\frac{{\mathrm{KC}}_{\mathrm{E}1}}{{\mathrm{GC}}_{{\varnothing}_{\mathrm{i}}\text{}\to \text{}{\varnothing}_{\mathrm{i}+1}}^{\mathrm{q}}}\text{}\Rightarrow \text{}\mathrm{Move}\text{}\mathrm{to}\text{}{\varnothing}_{\mathrm{i}+1}$.

If ${\mathrm{GC}}_{{\varnothing}_{\mathrm{i}}\text{}\to \text{}{\varnothing}_{\mathrm{i}+1}}^{\mathrm{q}}{\mathrm{C}}_{\mathrm{E}1}\Rightarrow {\text{}\mathrm{GC}}_{{\varnothing}_{\mathrm{i}}\text{}\to \text{}{\varnothing}_{\mathrm{i}+1}}^{\mathrm{q}}{\mathrm{KC}}_{\mathrm{E}1}\text{}\frac{1}{{\mathsf{\mu}}_{\mathrm{B}}}\Rightarrow {\text{}\mathsf{\mu}}_{\mathrm{B}}\text{}\frac{{\mathrm{KC}}_{\mathrm{E}1}}{{\mathrm{GC}}_{{\varnothing}_{\mathrm{i}}\text{}\to \text{}{\varnothing}_{\mathrm{i}+1}}^{\mathrm{q}}}\text{}\Rightarrow \text{}\mathrm{Keep}\text{}{\varnothing}_{\mathrm{i}}\text{}$.

This means that, whenever there is a pump on the market whose efficiency can be greater than the calculated µB, the optimum diameter will be ${\varnothing}_{\mathrm{i}}$. In the event that no commercial pump can reach that performance because it is very high, it will be necessary to move to the next diameter ${\varnothing}_{\mathrm{i}+1}$. Therefore, to apply this method it is necessary to know the maximum efficiency that pumps can reach. This is an uncertainty of the Granados System, and up to now, typically, pump efficiency values around 80% are already indicative.

#### 2.2. Methodology

Among all variables affecting Granados System, the pump performance is the key factor for calculating the diameter of the hydraulic conduction. Nevertheless the results may vary significantly depending on the pump efficiency; indeed, µ_{B} presents a wide range of possible values that typically goes from 70% up to 90%. This means a variability of almost 20% in the estimation of the cost, which is a substantial difference. Therefore the aim of this paper is to define the relationship between the pump’s performance and the other parameters involved in the procedure, such as the flow rate, the pumping head or the required power. For this to be done, we select 400 commercial pumps from the catalogues of several manufactures. We discard custom-made pumps since the cost of these pumps is much higher than the ordinary ones listed and the offering in commercial catalogues is already very wide. The selected pumps vary from each other in their type, impeller diameter, number of stages, rotation speed (electrical current frequency, number of poles), brand, etc. For all these models, we study the pump efficiency; in particular, we register the optimum value together with the correspondent flow rate, head and power consumed. Nevertheless, some pumps are ruled out of the sample and presentation of the results because they either were similar to those of other manufacturers, or because they were very specific for some industrial or sanitary engineering uses. In the end, the sample consists of 226 hydraulic pumps.

For the detailed study of the pump performance, this research has focused on the most common type of pump for the applications in civil engineering (supply, irrigation, sanitation, etc.)—centrifugal pumps. Within centrifugal pumps, both horizontal and vertical axes are selected, mostly with a radial flow configuration, with the exception of submersible pumps, for which the axial arrangement is more common. The sample includes regular horizontal and vertical pumps, split case pumps, multistage and submersible pumps. The manufacturers used for this analysis were IDEAL, WILO, ESPA and HASA. More specifically, the commercial models were:

Split case pumps: CP/CPI/CPR series.

Horizontal pumps (normalized in the European Union): RNI/RN series.

Multistage horizontal pumps: APM series.

Vertical pumps: VS/VG series.

Submersible vertical pumps: SVA/SVH series.

Multistage pumps perform with the same efficiency for a specific flow rate and different heights (which is the number of stages multiplied for the unitary head). To avoid this dispersion that could make it difficult to draw conclusions, it was decided to only use the optimum operating point correspondent to a single stage.

Using the data collected for the optimum operating points for all the 226 hydraulic pumps previously mentioned (pump’s optimum efficiency and associated flow rate, head, speed, power, frequency, diameter, etc.) we carried out an analysis to establish relationships among the design variables of a water drive.