A New Methodology for Efficiency Scaling ofPumps-as-Turbines (PATs) and Its Application to two

South Africa 13kW and 100kW Micro HydropowerSystems

Ricardo N. Van Zellerricardo.van.zeller@tecnico.ulisboa.pt

Instituto Superior Tecnico, Universidade de Lisboa, Portugal

May 2020

Abstract

Hydropower is the most mature renewable energy resource in the world. Not only it represents a largeportion of the renewable energiy generated in the world, as it is estimated that most of its potential isyet to be explored, with the biggest focus on micro and small hydropower systems. However, the cost-effectiveness of these systems still needs to improve, with pump-as-turbines (PATs) playing a key roleon the reduction of the investment costs. The drawback of these types of turbomachines relies on thefact that pump manufacturers do not provide, nor test, the operating condition of a pump working as aturbine. In this thesis, it was developed a new methodology to predict the efficiency curves of a reverse-running centrifugal pump. This numerical method is based on a theory developed by Gulich to predict theefficiency curves of a centrifugal pump taking into account the influence of the roughness and Reynoldsnumber on the efficiency scaling. Later, the numerical method was implemented in Microsoft Excel inorder to create the Scaling of PAT Efficiency Curves program, or SPATEC. In addition, the numericalmodel was applied to an extensive number of cases that correlate different PATs found in literature. Asa consequence, the absolute and relative error ranges were successfully ascertained for both the flowrate and head efficiency curves. Finally, the same model was applied to two real-life cases of microhydropower systems and the precision of the efficiency prediction assessed.Keywords: Hydropower, energy saving, PAT, efficiency prediction, similarity laws

1. IntroductionTo combat the increasing demand of electricity,

researchers are seeking for improved techniquesto overcome some relevant issues associated withpower generation. One of the most suitable solu-tions is thought to be renewable energy, speciallyhydropower.[1] In many developing countries, spe-cially within the Sahara African region, small andmicro hydropower systems represent a valuablesolution to generate electricity. An energetically ef-ficient solution relies on the idea of pumps work-ing as turbines (also known as pump-as-turbinesor PATs). Whenever reverse-runned, pumps cangenerate and recover power in small and micro hy-dropower schemes with very satisfying efficiencies.Their simple layout, accessible purchase cost and“off-the-shelf” worldwide readiness are some ad-vantages of these PATs, which naturally promotestheir choice over hydraulic turbines. At an eco-nomic point of view, reverse-running pumps werefound to have a capital payback period lower than2 years, for capacities varying between 5 kW and

50 kW,[2, 3].However, there is still a significant drawback:

predicting the actual PAT performance for a spe-cific point of operation, specially the best efficiencypoint (BEP). This problem arises from the fact thatpump manufactures do not provide, nor test, anypump characteristics in turbine-mode.

In this thesis, it is presented an alternativemethodology to predict PAT performance curves.Based on turbine-mode operating points of severalpumps, the efficiency curves of a particularly in-tended PAT might be achieved (for a specific speedbetween 0.12 and 1.18). This work is based on amethodology developed by [7] to predict efficiencycurves for geometrically similar centrifugal pumps,taking into account the influence that the rough-ness and the Reynolds number have on the per-formance.

2. BackgroundA pump-as-turbine (PATs) is a pump that runs

with the impeller and the flow in an opposite direc-

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tion to that defined for the pump operation mode.On the contrary to conventional turbines, PATs

do not exhibit any type of regulation, which is a nat-ural consequence of running a pump in its oppositedirection. Additionally, PATs have a larger runner,with the blades’ curvature in an opposite directionas well. Despite the extra cost associated with alarger runner and the small efficiency loss inherentto the unregulated operation mode, PATs are stillcost-effective due to its mass production.

To avoid the decelerated flow issues normallyassociated with pumps, the pump’s impeller mustbe bigger (30% to 40% bigger) than the one ofa turbine, with long and gradually diverging chan-nels. In addition, reversed curvatures and smallangles at the tip of the blades are also some ge-ometrical characteristics that must be adopted toincrease the stability of the flow.

In the opinion of some researchers [13, 4], animmediately available PAT, that does not need tobe specially manufactured to suit particular specifi-cations, can be found with maximum power of 100kW and 250 kW. However, there are a few coun-tries, like Nepal and Pakistan that hardly own PATswith capacities superior to 15 kW, for relatively lowspecific speeds.

Apart from power, PATs might also be limited ac-cording to their specific speed, Ω. The specificspeed is a non-dimensional parameter of turboma-chines used to understand which flow properties(namely, flow rate Q and head H) are accepted tocirculate through the machine with a particular ro-tational speed. Low specific speeds are associ-ated with a high head potential and a low flow rate,whilst a low head and a high flow rate induce highspecific speeds.

In a general way, it is not possible to ascertaina particular range to define lower and upper spe-cific speed limits of applicability of PATs. When itcomes to lower specific speeds, some authors like[15] defend that PATs cannot compete with Peltonturbines for Ω ≤ 0.2, whilst others ([6, 8, 9, 14] statethat multistage PATs can still be a cost-effective so-lution in countries where the cost of manufacturingis competitive enough against impulse turbines.

On the other hand, high specific speed PATs (de-scribed by their axial flows) have disadvantageslike higher cost, lack of experimental informationwhen operating in turbine-mode [16], and bladeshape impairments on the stability of the flow onceit was designed to operate in pump-mode, [10]. Be-sides, they have a bias towards cavitation and lowefficiencies so that, high specific speed axial-flowor mixed-flow PATs might be the most suitable PATtypes to install in low head hydropower sites, thatrepresent most of the small hydro potential in theworld [11].

3. Dimensionless coefficients and similarity lawsIn a pump with incompressible flow there are dif-

ferent types of variables that characterize its opera-tion, such as control variables, fluid properties andgeometric variables. The control variables consistin parameters that are deliberately altered to regu-late the operation of the turbomachinery. As for theproperties of the fluid, it is easily noticed that prop-erties such as the specific mass and the viscosityconsiderably affect the operating characteristics ofthe turbomachine. Finally, when it comes to ge-ometric variables, changing the turbomachine im-plies a naturally different functioning of the system:its operation not only depends on the absolute di-mensions of the machine (for example, the outletimpeller diameter), but also on a large number ofangles and dimensionless relations α, β, etc.

The mathematical theorem, known as Bucking-ham’s theorem, along with the control variables,the fluid’s properties and the geometric variablesthat characterize a turbomachine, allow the deter-mination of the remaining variables of the system,which are dependent of the former parameters. Forsome of these variables it might be useful to disre-gard or assume certain dimensionless parametersas constant and also to disregard the so knownReynolds number.

Any dependent variable can be defined asa function of dimensionless parameters, whichmeans that they do not depend on the unit sys-tem considered, as long as it remains consistent.These dimensionless variables allow the charac-terization of a single curve for a family of geo-metrically similar turbomachines, instead of rep-resenting a curve for each turbomachine consid-ered. This happens because geometrically similarpumps imply similar dimensionless coefficients [5].

In addition, the exchange between dimension-alized and non-dimensionalized curves is ratherstraightforward: both the transformation from thedimensionless curve to the dimensional curve, andvice-versa, only requires the knowledge of twocharacteristics of the pump’s operation apart fromthe flow rate and head, for example, the diameterand the rotation speed. This might be useful foranyone that works with a unit system different fromthe one with which the pump operating curves aredrawn. The exchange between dimensional curvesis also possible without incurring to the dimension-less representation of the pump curve. The solu-tion simply consists in equalizing the dimension-less coefficients of both curves and is useful tomodify a certain pump characteristics.

Apart from changing between dimensionalcurves for a certain pump, non-dimensional coeffi-cients also allow the exchange of operating curvesbetween a family of geometrically similar pumps,

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even if they have distinct absolute dimensions. Theresulting equations are known as similarity or affin-ity laws. Equations 1 to 5 present the most usualnon-dimensional correlations.

ηB = ηA (1)ΩB = ΩA (2)

QBQA

=NBNA

(D2B

D2A

)3

(3)

HB

HA=

(NBNA

)2 (D2B

D2A

)2

(4)

PBPA

=ρBρA

(D2B

D2A

)(NBNA

)3

, (5)

with subscripts ”A” and ”B” representing two dis-tinct points of operation.

The fact that geometrically similar turboma-chines imply similar dimensionless operatingcurves, allows the estimation of the dimensionlesscurve of a reduced model to perceive the behav-ior of the respective prototype in the conditions ofthe system that is intended to design. However, itis important to realize that, due to the neglectionof factors like the Reynolds number and geometricscaling effects, the working curve of the prototypewill present different results from those verified inreality. The scale effects result from the naturalimpossibility of the prototype to obtain a geometryperfectly similar to that of the model. For exam-ple, the relative roughness and axial clearancesbetween the rotor and the stator are absolute di-mensions that change depending on the size of theturbomachine.

From all dimensionless parameters, the mostfrequently used to correlate geometrically simi-lar pumps is the specific speed Ω. The specificspeed is also a characteristic parameter of turbo-machines: it defines the geometry with which amachine must be designed.

4. Power Balance and EfficienciesWhenever operating with a turbomachine, there

are losses associated with the process of machin-ing. According to [7], these losses are dissipatedinto heat and, after some rearrangement, can bedivided into three major groups: mechanical lossesPmec , volumetric losses Pvol , hydraulic lossesPhyd.

Inside a centrifugal pump, mechanical losses aredue to the friction forces exerted on the radial bear-ings, axial bearing and shaft seals. The volumetriclosses consist of leakages through several com-ponents of the pump that result in kinetic energylosses and, consequently, head and efficiency re-ductions. Any friction or vortex dissipation of en-ergy that takes place between the suction and dis-

charge nozzle is defined as a hydraulic loss. Skinfriction losses are a consequence from the shearstresses created between the fluid and the bound-ary layer of a body. These losses are highly de-pendent on the roughness ε, not so much on theReynolds number Re and, unlike vortex losses,they have more importance in thin boundary lay-ers and attached flows. As for vortex dissipations,also known as mixing losses, they constitute an-other important set of losses, within the hydrauliclosses - the mixing losses, that constitutes a majorpart of the total losses, particularly for high spe-cific speeds. These losses are caused by turbu-lent dissipation that occurs when the streamlinesexchange momentum between themselves.

5. Materials and Methods5.1. New methodology for PAT efficiency scaling

The methodology used to calculate the efficiencyand characteristics of a centrifugal pump workingas a turbine (PAT) takes as reference the efficiencyand characteristics of a geometrically similar pump(i.e. with the same specific speed). The pump,whose characteristics needed to be determined,was defined as the prototype (subscript ”p”), whilstthe reference pump is known as the model (sub-script ”M ”).

The implementation of such procedure providesan understanding of the impact that the Reynoldsnumber, roughness and specific speed have on thedifferent types of losses and, subsequently, on theefficiency of a PAT. Additionally, this method canalso be used in any type of flow conditions and sur-face roughnesses and still provide stable results.

Hydroelectric projects are one of the many caseswhere efficiency scaling is applied. Herein, the realsite flow conditions and the planned infrastructure(prototype) are reproduced in a small scale with thebest possible precision (model) and tested in a lab-oratory to understand and evaluate the real projectprior to its implementation. For that, it is mandatoryto know several model and prototype characteris-tics, see Table 1.

Table 1: Required model and prototype characteristic for theapplication of the numerical model.

From a list of 57 PATs available in the literature[12], two geometrically resembling pumps were se-lected based on the dimensionless specific speed.One of them represents the model, while the otherdescribes a pump with a geometry, not similar, butresembling to the prototype’s geometry, 1.

This resembling prototype ”rp” not only is con-

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Figure 1: Scheme of the numerical model

sidered as the reference to evaluate the dispersionof the estimated values, but also as the elementrequired to determine the prototype’s initial char-acteristics. In order to evaluate the proximity be-tween the prototype and the resembling prototype,some characteristics of the former must be stipu-lated as equal to the corresponding characteristicsof the latter. In this way, the optimum power androtational speed of the prototype are assumed asequal to those of the resembling prototype. Withboth the power and rotational speed of the proto-type, the remaining operating characteristics arecalculated by means of the model characteristicsand the similarity laws enunciated previously. Thesimilarity laws also state that the efficiency of themodel is equal to the efficiency of the prototype.

The presence of the Reynolds number androughness on a pump is frequently neglected.However, it can be of significant influence. Apartfrom the mechanical losses, which are assumedto be constant both for the model and prototype,the remaining losses are Reynolds and roughnessdependent. As such, there are some factors thatare used to correct the characteristics of the pro-totype that were once estimated through the simi-larity laws: efficiency correction factor fη; flow ratecorrection factor fQ; and head correction factor fH .These correction factors represent the influencethat the Reynolds number and roughness have onthe PAT operating characteristics. Detailed infor-mation regarding the calculation of these factors isfound in [7].

The losses discussed in the previous section4 are important to consider the efficiency of thePAT. All of them can be calculated through moreor less extensive equations, see [7] detailed infor-mation. With each specific loss identified, the effi-ciency correction factor can be determined. As aconsequence, the performance correction factorsthat represent the influence of the roughness andReynolds number are finally characterized. Thesefactors, when multiplied by the PAT correspond-ing characteristics, result in the new performancevalues that take into account the roughness and

Reynolds number variants for a scaled version ofthe model, Eqs. 6 to 9 .

ηB = ηA · fη (6)

QBQA

=NBNA

(D2B

D2A

)3

· fQ (7)

HB

HA=

(NBNA

)2 (D2B

D2A

)2

· fH (8)

PBPA

=ρBρA

(D2B

D2A

)(NBNA

)3

· fη (9)

One of the initial considerations stipulated uponthe implementation of this procedure states thefollowing: the model and prototype are undoubt-edly geometrically similar. Recalling geometricalsimilarity, two pumps with similar geometry havethe same dimensionless characteristics and so,the same specific speed. Therefore, the specificspeed of the prototype must be equal to the spe-cific speed of the model. However, the new charac-teristic values of the PAT comprehend an error as-sociated with the dimesionless specific speed: thisparameter, when estimated with the new charac-teristics of the PAT, differs from the specific speedof the model. Such result leads to the conclusionthat some assumption is misleading the calculationof the performance correction factors.

The presence of the efficiency correction factorin the power affinity law (Eq. 9, states that oneof its terms must change in order to maintain theprototype’s power.

At the beginning of this numerical method, thepower and rotational speed of the prototype weredefined as constraint variables, meaning, the resultof the calculations that follow them depends on thevalue that these parameters assume. They cannotbe modified in the process as they are consideredas certain. In addition, any parameter that charac-terizes the model cannot vary as well, the modeland prototype densities must be equal and theirnumber of stages must be the same so that the ge-ometric similarity condition holds true. Therefore,the only variable that can compensate the variationimposed by the efficiency factor is the impeller di-ameter of the prototype. This means that, after theacquisition of the PAT operating characteristics, acorrection of the impeller diameter still has to betaken into consideration. Afterwards, it is manda-tory to evaluate the difference between the impellerdiameter estimated initially and the corrected im-peller diameter in order to verify if the geometricsimilarity condition remains valid.

5.2. SPATEC – Efficiency Curves Scaling ProgramThe Scaling of PAT Efficiency Curves program

or SPATEC is the Microsoft Excel implementation

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of the numerical methodology presented in chapter5. This program has a database of 57 PATs, fromwhich 35 are reverse-running centrifugal pumps.

The numerical methodology developed in thiswork relies on a PAT model to calculate the sys-tem characteristics of a geometrically scaled pro-totype. Subsequently, it uses a third PAT (previ-ously defined as resembling prototype) that has ageometry quite similar to the model and prototypeto compare the estimated results.

To understand which PATs have the most simi-lar geometries, the dimensions of both machinesmust be scaled with similar ratios. Such relationbetween the prototype and the resembling proto-type cannot be verified entirely once literature lacksinformation regarding pump geometries. There-fore, it is assumed in this work as a first approxi-mation that, PATs with close specific speeds implyresembling geometries. Naturally, this condition isnot necessarily true. However, at the end of the nu-merical model the deviation between both the pro-totype and resembling prototype diameters is cal-culated and its modulus evaluated.

As a result of this geometry assumption, the 57PATs were re-organized by specific speed to under-stand which PATs come closer to each other geo-metrically. Afterwards, each PAT was consideredas the model and possible resembling prototypeswere selected according to the following condition:the specific speed deviation between the prototypeand the resembling prototype should be lower orequal to 0.05, i.e ∆Ω < 0.05. From this criterion itwas obtained 95 cases that relate different PATs.

The specific speed deviation condition is neces-sary in the screening process but not sufficient. Inaddition to this restriction, the cases must also beanalyzed with respect to their geometric category,which is provided in each PAT formulae. If one ofthe previous cases combines two PATs with distinctgeometries, then it shall be removed from the testdata.

The prediction of the prototype’s efficiency curvedepends on the diameter of the model. Wheneverthis parameter is unknown for both PATs, the esti-mation of the efficiency curve is not possible andthe respective case must be excluded as well.

Finally, for the last restriction it is stipulated thatthe rotational speed of the resembling prototypemust be constant for the points of operation used inthe analysis. These points must have the same ro-tational speed so that, when compared to the pro-totype points of operation, both efficiency curvesfall over the same coordinate plane. This restric-tion could be avoided by using similarity laws totransform the different points of operation into thesame plane of rotational speed. However, in thisthesis, it was decided not to apply this procedure

so that the experimental results (resembling proto-type points of operation) would not be subject toadditional errors.

The numerical model used to predict PAT effi-ciency curves was developed with the aid of Mi-crosoft Excel. The program is divided into foursections: Reynolds and roughness correction; spe-cific speed correction; diameter correction - itera-tive process; and PAT performance.

At the first stage, the program requires the inputof the best efficiency point operating characteris-tics of both the model and resembling prototype.With such data, the numerical calculations resultin the initial correction factors. At the subsequentstage, the former factors and the impeller diame-ter are corrected as a consequence of the specificspeed correction.

The parameters determined in the two first sec-tions are used in the third stage to evaluate the de-viation between the estimated and the correcteddiameter ∆D2p and, subsequently, the worthinessof the next iteration: if ∆D2p ≤1 mm, then the it-erative process stops; otherwise, the initially esti-mated diameter shall be replaced by the correcteddiameter. In this way, the calculations associatedwith the first and second stages are repeated, andthe resulting correction factors are attained with ahigher precision. Three iterations are applied foreach case. The PAT performance stage repro-duces the prototype operating points. Such pointsare achieved through the application of the correc-tion factors and the affinity laws to the various op-erating points of the model.

At the end of the program, the model, proto-type and resembling prototype efficiency curvesare represented graphically as a function of theflow rate and, subsequently, as a function of thePAT’s head. These graphics allow to understandthe variation between the model and the prototypeand to evaluate the closeness of the prototype andresembling prototype curves, as shown in Fig. 2

6. Analysis and Discussion of ResultsFrom the 40 cases that were selected to test

the numerical model, some predicted quite suc-cessfully the prototype efficiency curves. However,there were other cases that fall short on what wasverified experimentally with their respective resem-bling prototypes. A lower, an intermediate and anupper specific speed group were assumed as thebest way to divide all cases, see fig. 3 to perceivethe group division delimited by the black intermit-tent lines.

Whilst these case groups have specific speedsvarying between 0.12 and 0.89 it does not meanthat it is not possible to apply the numerical modelto a PAT with a higher or lower specific speed. Oneof the problems faced when developing this theory

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Figure 2: Efficiency curves as a function of the flow rate andthe head

is that it is impossible to know whether resemblingprototypes have the same geometry as the proto-types to which they were correlated. Due to thelack of information regarding the geometry of thePATs and, in order to proceed with the efficiencyprediction study, it was assumed that equal spe-cific speeds imply equal geometries. However, asexplained above, it is evident that if the prototypediameter differs considerably from the resemblingprototype diameter, then the PATs have differentgeometries and, therefore, the similar geometry isinfringed and there is a larger error associated withthe efficiency prediction.

Viewed as one of the most important variablesin this work, the specific speed allows not only torepresent the geometry of a PAT family, but also toevaluate the accuracy of the PAT efficiency predic-tion. This accuracy naturally depends on the devi-ation between the specific speed of the prototypeand resembling prototype.

One of the outputs of the numerical model char-acterizes the prototype diameter. This parameter,together with the resembling prototype diameterrepresents a crucial feature of the case to be an-alyzed. Through these two variables it is possibleto draw conclusions about the geometric proximity,assumed to be similar for the comparison of effi-ciency curves. Therefore, to understand how di-ameters differ across the 40 cases it is necessaryto estimate the respective absolute ∆Dabs and rel-ative ∆Drel errors. The first represents the met-ric difference between the prototype and the re-

sembling prototype diameters. The second char-acterizes the percentage difference that the pro-totype diameter presents compared to the resem-bling prototype diameter.

If the relative error is greater than 10%, then thesimilar geometry condition is automatically refuted,and the error associated with the efficiency predic-tion higher. Otherwise (i.e. 610%), the geometricrelationship between the prototype and the resem-bling prototype is maintained as similar and the er-ror associated with efficiency prediction lower.

It is clear that, from the 40 cases to which thenumerical model has been applied, there is a highnumber of cases that do not illustrate the diameterdifference, as observed in Fig. 3.

Figure 3: (a) Diameter absolute error, (b) Diameter relative er-ror

In an overall point of view 72% of the cases (ex-cluding the invalid cases) exhibit approximately animpeller diameter difference lower or equal to 10%,which is assumed as a satisfying result.

To compare the efficiency curves between theprototype and the resembling prototype it is neces-sary to determine the efficiencies of the prototypethat correspond to the diverse flow rates and headsof the resembling prototype. This alignment allowsthe calculation of the absolute and relative errorsassociated with each operating point. A linear re-gression is then the most appropriate method toestimate the efficiencies.

By replacing the flow rates and heads of the re-sembling prototype on the prototype trendline, itis determined the absolute and relative error as-

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sociated with the numerical method developed inthis work. After some restriction rules are estab-lished, the absolute and relative errors associatedwith the prototype efficiency prediction are evalu-ated, as seen in Fig. 4. If any resembling prototypepoint of operation is out of the operating range ofthe prototype, the error calculation should be esti-mated via an extrapolation method.

Figure 4: (a) Flow rate efficiency curve errors, (b) Head effi-ciency curve errors

In order to understand the impact that each er-ror has, it is selected for all cases the minimumand maximum errors verified amongst all operatingpoints and its values plotted in the same graphic.Taking into account that there are absolute and rel-ative errors for both the head and flow rate effi-ciency curves, four graphics must be plotted to rep-resent the different characteristic errors, Fig. 5. Toanalyze the obtained results, the 40 cases are di-vided into distinct groups according to their specificspeed.. For each case, it is assumed that both therange length and its minimum (or initial) value canhave high or low values Table 2.

Table 2: Qualifier description according to range length andminimum error value

Depending on the type of error being analyzed(i.e. absolute or relative error), each qualifier has aparticular meaning. Without going into a much de-tailed explanation, it is important to acknowledge

that, in a general point of view, absolute errors varyapproximately between 5% and 20%. As a conse-quence, it is expected that both the flow rate andhead prototype efficiency curves exhibit differentshapes when compared to the resembling proto-type curves, but at least present an area wherethe efficiency prediction is carried out with success.As for the relative errors, although their averagerange differs considerably (8%-50% compared to8%-140% for flow rate and head curves, respec-tively), they represent the same meaning in theprediction of efficiencies. Their Large-Small quali-fier implies that the prediction uncertainty might below or high compared to the real efficiency value.

In most cases, the behavior of relative error asa function of flow is expressed by the Large-Smallqualifier, meaning, the numerical model estimatesthe prototype efficiencies with both low and highuncertainties.

The errors calculated demonstrate a numericalmodel that is associated not only with a high rela-tive error, but also an absolute. Such errors haveoften been justified on the premise that operatingpoints in the linear zone induce quite significant er-rors. In order to show that these operating pointsactually have a high impact on the overall errorranges, it was analyzed the different errors for flowrates and heads that are 10% greater or smallerthan the BEP operating characteristics. AlthoughFig.6 clearly imparts that the length of each er-ror range diminished significantly, it is essential tounderstand carefully what happened to each errorrange so that a general outcome of the results canbe formulated.

Table 3: Relative frequency of range variation

By analyzing Table 3, it is verified that in mostcases the total error range either reduces its max-imum error and withholds its minimum error, or in-creases its minimum error and reduces its maxi-mum error. It is only a few percentage of the casesthat increase the minimum error and keep the max-imum error constant. Apart from the 27th case ofthe flow rate efficiency curve, that remains with thesame error range length, the range restriction ap-plied to the total operating range truly increases thecertainty of the error associated with the efficiencycurves.

In addition to the analysis of the errors associ-ated with the efficiency prediction, as to the differ-ences between the specific speeds and diametersof the prototype and resembling prototype, it is alsointeresting to study the sensitivity of the numeri-cal model when certain parameters of its system

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Figure 5: Absolute and relative error ranges for the total operating range of the prototype. (a) flow rate absolute error range, (b)flow rate relative error range, (c) head absolute error range, (d) head relative error range.

Figure 6: Absolute and relative error ranges for ±10%BEP. (a) flow rate absolute error range, (b) flow rate relative error range, (c)head absolute error range, (d) head relative error range

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are subject to variations that are considered signif-icant.

There are several factors that contribute to theefficiency differences between two PATs, most no-tably the roughness and the Reynolds Number. Tounderstand the impact that these factors have onthe efficiency prediction, in each of the 40 cases,it is studied separately the outcome resultant fromthe variation of ±10% of roughness or kinematicviscosity. To avoid a further extensive analysis ofthe final results it is feasible to directly comparethe correction factors obtained at the end of thenumerical model rather than to calculate the newprototype operating points and then compare theefficiency curves, Table 4.

Table 4: Percentage variation of the correction factors and im-peller diameter

From an overall point of view, an average vari-ation of approximately 0.05% for each correctivefactors is intuitively insignificant. As for the proto-type geometry, it is noticeable that the variation ofthese parameters does not affect the final diameterof the prototype at all.

7. Application of the numerical model to real-life mi-cro hydropower plantsOne of the main objectives of this work is to en-

able the a priori study of the economic viability ofa micro hydropower system based on the use of apump working as a turbine for power generation.To do so, it is essential to understand how the nu-merical model behaves when applied to such sys-tems. In this thesis, it was analyzed the applicationof the SPATEC program to two systems: Waterval’smicro hydropower system with 13kW located inWestern Cape, South Africa; and Teutberg’s micro-hydropower system of 117kW located on the SouthCoast of South Africa in the Roman Bay Sea Farm.Then, it was compared the theoretical and experi-mental results of the PATs efficiencies.

For the first case, the prediction presents an ex-tremely high level of success once the absoluteand relative errors for both the flow rate and headefficiency curves are satisfyingly below the errorranges established for PAT efficiency prediction. Asfor the prototype diameter, it was also obtained areasonable result. For the second case, the effi-ciency prediction does not exhibit the same quality

as the first, which was expectable since the specificspeed deviation between the prototype and the re-sembling prototype overcomes the acceptable limitof 0.05. Although it was verified a significant ab-solute error, its importance loses value when as-sociated with a moderately satisfactory relative er-ror. The estimated prototype diameter has a rela-tively high absolute deviation, which indicates thatthe geometry of the resembling prototype does notmatch successfully Teuteberg’s PAT geometry.

8. ConclusionsIn order to reflect about any problems that may

exist in the application of the numerical model, itshould be noted the possibility of the geometricsimilarity condition being invalid and what implica-tions this has in the prediction of efficiencies. Re-call the assumption that pumps with equal dimen-sionless coefficients have similar geometries.

In this work, there were cases with similar anddissimilar diameters. Taking into account that thelatest are associated with higher errors, it is up tothe user to decide whether or not these errors areacceptable, for example, in the preliminary eco-nomic study of an hydropower system.

When it comes to the extrapolation methodologydeveloped in this work, it is brought to a conclusionthat, despite having positive influence in the er-ror ranges estimation, the payback fall short whencompared to the amount of work invested in theextrapolating process.

In addition to the evaluation of the precision ofthe numerical model along both efficiency curves,it was also possible to draw conclusions regard-ing their behavior within a specific operating range,more precisely ±10% BEP. It is now known that tothe reduction of the operation range correspondsan almost certain reduction of the maximum error(95%). Likewise, it is concluded that the minimumerror is likely to increase or remain constant (55%and 41% respectively). Such results allow not onlyto realize that the error associated with the effi-ciency prediction improves, but also that the cer-tainty inherent to the error increases and thus, theresults naturally become more reliable.

Although the sample of real cases is reduced, itcan be concluded that the prediction of the BEP ef-ficiency was successfully carried out. In one case,the efficiency was predicted with a very satisfyingprecision, only diverging about 1.5% in both effi-ciency curves. In the other case, the efficiency de-viated a little further from the experimental value(11% in both efficiency curves as well), but stillmanaged to reach a satisfying result.

In addition, as it was intended to prove, it is nowpossible to conclude that the specific speed has asignificant impact on the efficiency curves of a PAT,as it can be seen in fig. 7.

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Figure 7: Specific speed influence on the PAT efficiency (a)PAT efficiency as a function of the flow rate coefficient Φ, (b)PAT efficiency as a function of the head coefficient Ψ.

That said, it is concluded that the SPATEC pro-gram satisfactorily forecasts the flow rate and headefficiency curves for pumps operating as turbines.

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on Hydropower Technologies, P. E. Agency,and C. H. Association. Hydropower and theworld’s energy future, November 2000.

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