Data Reduction Procedure for Two-phase Flow Performance Analysis in Centrifugal Pumps*

Wei-bin Zhang Gé rard Bois,3 Qi-feng Jiang Zheng-wei Wang

(1.Department of Energy and Power Engineering,Tsinghua University,2.Key Laboratory of Fluid and Power Machinery,Xihua University,3.Univ.Lille,CNRS,ONERA,Arts et Metiers Institute of Technology)

Abstract：This paper presents what should be best practice data reduction procedures for pump performance analysis with emphasis on two-phase inlet flow conditions.This becomes a mandatory step especially when pumps performancesare degradedin case of liquidgas mixtureat inletsection.Most of following recommendations are based on existing rules that must be recalled for researchers and end users performing centrifugal pump tests and more specifically when comparing the results between each other or/and with CFD approaches.

Keywords：CentrifugalPump，DataReduction，Two-phaseFlow

Nomenclature

gacceleration(m2/s)

henthalpy(m2/s2)

Htotal head(m)

mmass flow rate(kg/s)

Nrotational speed(rev/min)

Ppower(W)

Qvolume flow rate(m3/s)

Rradius(m)

Uimpeller tip speed(m/s)

Vabsolute velocity(m/s)

Greek symbols

αabsolute flow angle(°),from tangential direction.

βrelative flow angle(°),from tangential direction.

ηglobal efficiency(%)

νkinematic viscosity(m2/s)

σsurface tension(N/m)

μslip coefficient,(-)

ωangular velocity(rad/s)

φflow coefficient(-)

ψhead coefficient(-)

ψththeoretical head coefficient,(-)

Ωsspecific speed,

Subscripts

t1,1 impeller inlet section

t2,2 impeller outlet section

df due to disk friction

leak due to leakage flow

recirc due to recirculation

mech due to mechanical losses

hyd hydraulic

th theoretical

tp related to two phase conditions

r related to the radial component

u related to the tangential component

g relate to gas phase

l relate to liquid phase

t,tot total

*Fund Program:This research was funded by the project"Research on mechanism of internal energy conversion and hydraulic loss in unsteady flow within hydraulic ma⁃chines(51876099)",National Key R&D Program of China(2018YFB0905200),National Natural Science Foundation of China(51769035),and“Young Scholars”pro⁃gram of Xihua University(Z202042).

1 Introduction

Centrifugal pump is the heart of fluid delivery systems and is widely used in various sectors of the economy.In the high-end technical fields such as nuclear power,petrochemicals and oil extraction,the phenomenon of gas-liquid twophase flow in pump occurs frequently.As the increases of inlet gas volume fractions,the performance of the centrifugal pump gradually deteriorates until the flow stops,which seriously affects the safety and stability run of the systems[1].In recent years,due to the needs of engineering problems and the rapid development of new measurement technologies,the gas-liquid two-phase flow problem has gradually become a hot and difficult point for scholars domestic and overseas[2].Consequently,more and more research teams are looking for better understanding and prediction on pump performances under two-phase flow conditions.However,open literature results are still not well documented and comparative results are not easy to handle because basic data reduction procedure are not well-known or are not correctly applied especially for the case of centrifugal pumps.About numerical results,a lack of information often exists,not enable to compare numerical and experimental results in a proper way.The present paper intends to recall what should be good practices on experimental data reduction and numerical set up procedures for existing and future research works on centrifugal pumps working under two-phase flows conditions.

2 Test stand pump performance measurements

Most of pump test stand equipment use torque meters generally placed between the power supply and the rotating pump shaft.Knowing the rotational speed value,one can obtained the shaft power value.This power can be related to the impeller exit total enthalpy through the following power balancing equation:

Introducing the theoretical total pressure change inside the pump,assuming incompressible fluid,this equation can be also written as follow:

The hydraulic efficiency ηhydcan be then defined as:

The quantity(Pt2-Pt1),corresponding to the pump pressure delivery,can be measured by means of wall static pressure sensors placed at pump inlet and outlet sections and an evaluation of the dynamic pressure based on volume flow rate delivered by the pump.In general,both pressures should be measured separately.Inlet pressure measurement allows the evaluation of possible cavitation onset inside the impeller that must be checked to avoid misunderstanding on data reduction results.

Because of the presence of the parasitic power losses caused by disk frictionPdfand leakagePleakas well as flow recirculationPrecirc,that can have non-negligible values at very low flow rates.All these parasitic power losses need to be determined properly before Eq.can be used to calculate the impeller exit total theoretical headHth=(Pt2-Pt1)/(ρ.g),also called Euler Head,from which the slip factor can be derived(see next section).The disk friction power loss can be estimated using formulations presented by Gülich[3],as well as the estimations of leakage and the recirculation loss.Apparently,with all these empirical models,more uncertainties are introduced for the derived slip factor.However,without more direct velocity measurement of the flow field,this is an approximate procedure that can help us to estimate the slip factor for pumps from the test data.

For centrifugal flow pump working in single phase incompressible fluid,the disk friction power generally represents 3 to 5% of the measured shaft power.However,for low specific speed cases,it can reach 20%.In case of two-phase inlet flow conditions,the shaft power generally decreases because of the strong decrease of the liquid flow rate.Since the disk friction power is independent of the liquid flow rate,the relative importance of disk friction power comparatively increases compare with the theoretical hydraulic power.

3 Euler Head Coefficient

Based on the ideal conservation law of angular momentum in centrifugal pumps,the theoretical headHE(or Euler head)can be expressed,in a one-dimensional form and under simplified assumptions,as:

All assumptions and developments can be found in several turbomachinery publications such as Gülich[3]or Zhu and Zhang[4].This expression is obtained assuming one-dimensional inlet and outlet mean flow conditions.

Introducing non-dimensional theoretical head coefficientψEand flow coefficientφ,one can obtained the following relation:

WhereψE=gHE/(U2)2and φ =Q/(2π·R2·b2·U2).

4 Example of theoretical head coefficient evaluation discrepancies form torque measurements

Figure 2 presents the total head coefficient values from experiments in a centrifugal special design presented by Heng et al.[5]using only water as working fluid.For this pump,head coefficients do not fulfilled the usual similarity laws due to the corresponding special design,however,this test case has been chosen to illustrate what is obtained on the theoretical head curves depending on data reduction procedure.Figure 3 presents the global efficiency directly derived from the torque meter indication and the hydraulic efficiency(with disk friction correction).What can be seen is that hydraulic efficiencies values are higher compared with global ones and that the consequences on corresponding Euler curves,that are given in figure 4,give more consistent re-sults related to the application of the moment of momentum equation.A single straight black line representing simplified theoretical head coefficient(corresponding to Eq.5 with no pre-swilling flow)versus flow coefficient can be drawn as shown on figure 4.Note that one approximation remains,since the real flow rate,going through the impeller,is higher than the delivered flow rate given by the pump due to leakages that are present in the fluid domain.

As a conclusion,data reduction procedure must include a disk friction evaluation when establishing theoretical head curves versus pump flow coefficient.Important care must be taken,when choosing the pressure transducer range according to the pressure that have to be measured.

Fig.2 Experimental head characteristics[5]

Fig.3 Experimental efficiencies:with and without disk friction correction[5]

5 Two-phase flow performance measurements requirements

This section presents what should be prepared when two phase performance are performed in addition with single phase cases that have been discussed in the previous section.A quick reminder is presented first on the usual classification in simple configuration such as for straight horizontal tubes,then the extended classification for the case of impeller rotating channels is added,based on flow visualization techniques.

Another part presents a dimensional analysis that explains what are the most relevant parameters that influence two-phase performances in pumps and the corresponding coefficients that must appear for any publication related to this topic.

5.1 Two-phase classification in non-rotating straight tubes

Figure 5 illustrates the classification of flow mixture figures proposed by Baker[6].Different flow mixture configuration in horizontal cylindrical tubes are shown which are depending on liquid fraction coefficient on x axis and gas fraction coefficient on y axis.Bubbly flows only exist for high liquid flow rates comparatively to gas flow rates.

When applied to pump inlet pipes,one has to check in which mixture category the two-phase flow may beyond to.When pump rotational speed is reduced,or if the pump circuit is too resistive,one can go through plug flow to stratified flow conditions with the same inlet gas fraction amount.

Fig.5 Two-phase mixture patterns in horizontal pipe configuration.Extracted from Baker[6]

x:volumetric fraction of gasσ:water surface tension tension in air at 20°C=73.10-3N/m

ax=(σeau/σL)(νLρeau/νeauρL)1/3ax=(σeau/σL)=1,when gas is air and fluid is water

ay=(ρairρeau/ρGρL)0.5ay=1,when gas is air and fluid is water

G=mass flow rate of each phase(xaxis for liquid,yaxis for gas)/tube section area.

5.2 Two-phase classification in rotating impeller passages

Due to the additional forces that are acting on gas bub-bles with rotation(see next section for basic physical analysis development),local void fraction distribution and value is strongly modified.This can be seen from visualization techniques,and a schematic representation is given in figure 6 inside an impeller and in figure 7 for the case of an impeller and vaned diffuser configuration.

Fig.6 Schematic representation of flow patterns inside a rotating impeller pump.From left to right figures,flow pattern corresponds to:bubble flow,agglomerate bubble flow,gas-pocket flow and segregated flow.[7]

Fig.7 Gas-liquid flow structures in impeller and diffuser flow channels[8]

5.3 Two-phase flow test results and performance parameters

Dimensional analysis

For a given design,global pump head performances depends on several parameters that are listed below.

Compared with single phase working fluid,additional parameters such as second phase density and viscosity and surface tension between the two involved fluids are added.

Pressure and temperature are consider to have negligible effects since cavitation is excluded for the present approach.TakenD,Nandρlas basic variables,it is possible to built 8 different non-dimensional groups of variables that depend on these 3 basic ones.This results to the follwing expression,

-ND2/νlrepresents the liquid phase Reynolds number

-ND2/νgrepresents the gas phase Reynolds number

-ρlN2D3/σrepresents the Weber number,the value of which is generally close to 3×106.Surface tension terms are consider to be negligible because very weak compared to all other terms

-g/DN2represents the gravitaional forces(also related to the Froude number)that is much lower than the centrifugal forces

-ρg/ρlis the density ratio between gas and liquid and so,is close to 10-3.This means that the inertia effects on the flow field are more associated with water than gas.In case of water and gas mixture,this ratio can be neglected,however,this assumption must be verified for other kind of mixtures like water and oil for example

Considering the previous statements,the flowing relation can be obtained

Introducing the variableβdefined as:

One can obtained the expression of the gas volumetric fractiona,that is expressed as:

Forαvalues lower than 10%,it can also be approximated as:

This implies the following new expression:

For most of gas liquid mixtures,the ratioρg/ρlis small and one can define a density mixtureρtpsuch as:

Using the pressure coefficientψtot,dpand flow coefficientφtp,one has

One can also use the void fraction and write

Finally,the pump void fraction mainly depends on the pump geometry,the rotational speed,the flow rate and the inlet void fraction αi,this last parameter can be measured as well as all others when performing experimental pump tests.The final relation can be written as follow:

5.4 Pump performance degradation coefficients

In order to quantify the performance pump modification under two-phase conditions,one can use the following coefficientψ*,defined as the ratio between real head(or pressure)in two-phase conditions and the head(or pressure)for liquid single phase:

Another coefficient can also been established as follow:

Which corresponds to the difference between theoretical head and real head obtained in two-phase conditions divided by the same difference obtained only for single phase conditions.

These coefficients are also used for the development of semi-empirical or two-dimensional models for two-phase performance predictions.The second one,that uses theoretical head is a interesting one because it introduces the pump geometry variables and more specifically to the blade outlet relative angle as a variable.

5.5 Example of two-phase pump experimental results(From Si et al.[9])

From the results presented by Si et al[9],it has been shown that head and efficiency curves have the same kind of evolution in both case of single and two-phase conditions with lower values of the head in case of two-phase inlet conditions,up to a limiting value of inlet void fraction for which head drops dramatically.Theoretical head can be obtained from the two-phase performance measurements using the same procedure expained in section 3.The corresponding simplified Euler curve given by Si et al[9],which is reproduced in figure 8,also presented a straight line.This means that a curve can be found,which is equivalent to the one given in Eq.(5)that represents the evolution of two-phase simplified Euler curve written below;

Fig.8 Theoretical head coefficient for different rotational speeds(extracted from Si et al.[9])

This leads to an important result:for inlet void fraction below 8%,and for a given rotational speed,the theoretical head coefficient and the simplified Euler curves are not modified when two-phase flows are present inside a pump,up to a certain value of inlet void fraction that generally corresponds to homogeneous and bubbly flow regimes.For these cases,one has:

5.6 Additional curves

Since both flow rate and head are strongly modified for two phase flow condition,the results presented in previous figures must be complete by what is called“two-phase mapping”figure like the one presented by Estevam and Barrios[10],as shown in figure 9.It allows to define,for a given pump geometry,several zones for which the two-phase patterns can be related to pump head modification in relation with liquid volume flow rate and not only pump head coefficient.

Fig.9 Example of two-phase mapping proposed by Estevam et al.[10]

6 Conclusions

In this study,explanation of basic phenomena and best practice consideration concerning experimental procedure have been presented,and the use of significant parameters and coefficients to describe quantitatively the performance degradation of pumps working under two-phase conditions have been developed.The best practice experimental and data reduction procedure are listed below,partial have also been mentioned in[11]:

1)Start with single phase measurements for several flow rates and rotational speeds

2)Check similarity laws validation with Reynolds number effects

3)For two-phase inlet flow conditions and for a given rotational speed:

i.Start experimental procedure with maximum flow rate,then set a value of air flow rate and perform all measurements for decreasing flow rate up to pump shut down.

ii.Restart same procedure for increasing inlet air flow rate up to maximum limiting value(Pump surge may happen for a small modification of air flow rate-Maximum air flow rate should depend on water flow rate and rotational speed)

4)Perform the same measurements with a defined constant inlet void fraction by changing both regulation flow rate valve and air flow rate.

5)Repeat the procedure with another rotational speed

6)After proper data reduction,calculate all pump parameters and always plot figures using non-dimensional coefficients such as:

i.Head coefficient versus Flow coefficient

ii.Theoretical head coefficient versus Flow coefficient

iii.Efficiency versus Flow coefficient.(Disk friction losses must be evaluated and cancelled)

iv.Normalized water volume flow rate versus normalized inlet air volume flow rate or inlet void fraction.

7)Cavitation onset must be also checked in case of high flow rates and/or high rotational speeds.