气液两相流下离心泵性能分析的数据处理方法

Chinese Journal of Turbomachinery
Vol.62，2020,No.5
Data Reduction Procedure for Two-phase Flow Performance
Analysis in Centrifugal Pumps *
Ya-guang Heng 1,2
Kun-jian He 2
Wei-bin Zhang 2
G rard Bois 2,3
Qi-feng Jiang 2
Zheng-wei Wang 1
(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)
éAbstract：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 performances are degraded in case of liquid gas mixture at inlet section.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：Centrifugal Pump ，Data Reduction ，Two-phase Flow DOI :10.16492/j.fjjs.2020.05.0001
*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).
g h
H m N P Q R U V Greek symbols
αβηνσμ
ωφ
acceleration (m 2/s)enthalpy (m 2/s 2)
total head (m)mass flow rate (kg/s)rotational speed (rev/min)power (W)volume flow rate (m 3/s)radius (m)
impeller tip speed (m/s)
absolute velocity (m/s)
absolute flow angle (°),from tangential direction.relative flow angle (°),from tangential direction.global efficiency (%)kinematic viscosity (m 2/s)surface tension (N/m)slip coefficient,(-)μ=V U2/U 2
angular velocity (rad/s)flow coefficient (-)ψψth Ωs
Subscripts
t 1,1t 2,2df leak recirc mech hyd th tp r u g l t,tot
head coefficient (-)
theoretical head coefficient,(-)
ψth =ψ/ηspecific speed,
Ωs =ω
Q 0.5
(gH )0.75
impeller inlet section impeller outlet section due to disk friction due to leakage flow due to recirculation
due to mechanical losses hydraulic theoretical
related to two phase conditions related to the radial component related to the tangential component relate to gas phase relate to liquid phase total
Nomenclature
1Introduction
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,petrochemi-cals and oil extraction,the phenomenon of gas-liquid two-phase flow in pump occurs frequently.As the increases of in-let gas volume fractions,the performance of the centrifugal pump gradually deteriorates until the flow stops,which seri-ously 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 technolo-gies,the gas-liquid two-phase flow problem has gradually be-come a hot and difficult point for scholars domestic and over-seas[2].Consequently,more and more research teams are looking for better understanding and prediction on pump per-formances under two-phase flow conditions.However,open literature results are still not well documented and compara-tive results are not easy to handle because basic data reduc-tion procedure are not well-known or are not correctly ap-plied especially for the case of centrifugal pumps.About nu-merical 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 cen-trifugal pumps working under two-phase flows conditions.
2Test stand pump performance measure-ments
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 ob-tained the shaft power value.This power can be related to the impeller exit total enthalpy through the following power balancing equation:
P shaft=m(h t2-h t1)+P df+P leak+P recirc+P mech(1) Introducing the theoretical total pressure change inside the pump,assuming incompressible fluid,this equation can be also written as follow:
P shaft=Q.(P t2-P t1)th+P df+P lea+P recirc+P mech(2) =ΔP th,hyd/ηhyd+P df+P leak+P recirc+P mech
The hydraulic efficiencyηhyd can be then defined as:
ηhyd=Q*(P t2-P t1)/ΔP th,hyd(3) The quantity(P t2-P t1),corresponding to the pump pres-sure delivery,can be measured by means of wall static pres-sure 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 re-duction results.
Because of the presence of the parasitic power losses caused by disk friction P df and leakage P leak as well as flow re-circulation P recirc,that can have non-negligible values at very low flow rates.All these parasitic power losses need to be de-termined properly before Eq.can be used to calculate the im-peller exit total theoretical head H th=(P t2-P t1)/(ρ.g),also called Euler Head,from which the slip factor can be derived (see next section).The disk friction power loss can be esti-mated using formulations presented by Gülich[3],as well as the estimations of leakage and the recirculation loss.Appar-ently,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 in-compressible fluid,the disk friction power generally repre-sents3to5%of the measured shaft power.However,for low specific speed cases,it can reach20%.In case of two-phase inlet flow conditions,the shaft power generally decreases be-cause 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 in-creases compare with the theoretical hydraulic power.
3Euler Head Coefficient
Based on the ideal conservation law of angular momen-tum in centrifugal pumps,the theoretical head H E(or Euler head)can be expressed,in a one-dimensional form and un-der simplified assumptions,as:
H
E
=
ω2R22
g
-
Qω
2πgb2tanβ2-Qω
2πgb1tanα1(4) All assumptions and developments can be found in sev-eral turbomachinery publications such as Gülich[3]or Zhu and Zhang[4].This expression is obtained assuming one-di-mensional inlet and outlet mean flow conditions.
Introducing non-dimensional theoretical head coeffi-cientψE and flow coefficientφ,one can obtained the follow-ing relation:
ψ
E
=gH
E
/(U2)2=1-ϕtanβ2-(R1R2)2V1u U1(5) WhereψE=gH E/(U2)2andφ=Q/(2π·R2·b2·U2).
4Example of theoretical head coefficient evaluation discrepancies form torque mea-surements
Figure2presents 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 proce-dure.Figure3presents the global efficiency directly derived from the torque meter indication and the hydraulic efficiency (with disk friction correction).What can be seen is that hy-draulic efficiencies values are higher compared with global ones and that the consequences on corresponding Euler curves,that are given in figure4,give more consistent re-
Chinese Journal of Turbomachinery
Vol.62，2020,No.5
sults related to the application of the moment of momentum
equation.A single straight black line representing simplified theoretical head coefficient (corresponding to Eq.5with 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 leakag-es 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 accord-ing to the pressure that have to be measured.
5Two-phase flow performance measure-ments 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 rotat-ing channels is added,based on flow visualization tech-niques.
Another part presents a dimensional analysis that ex-plains what are the most relevant parameters that influence two-phase performances in pumps and the corresponding co-efficients that must appear for any publication related to this topic.
5.1Two-phase classification in non-rotating straight tubes
Figure 5illustrates the classification of flow mixture fig-ures proposed by Baker [6].Different flow mixture configu-ration in horizontal cylindrical tubes are shown which are de-pending on liquid fraction coefficient on x axis and gas frac-tion 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 cir-cuit is too resistive,one can go through plug flow to strati-fied flow conditions with the same inlet gas fraction amount.
x :volumetric fraction of gas σ:water surface tension tension in air at 20°C =73.10-3N/m
a x =(σeau /σL )(νL ρeau /νeau ρL )1/3a x =(σeau /σL )=1,when gas is air and fluid is water
a y =(ρair ρeau /ρG ρL )0.5a y =1,when gas is air and fluid is water
G =mass flow rate of each phase (x axis for liquid ,y axis for gas)/tube section area.
5.2Two-phase classification in rotating impeller passages
Due to the additional forces that are acting on gas
bub-
Fig.2Experimental head characteristics [5]
Fig.3
Experimental efficiencies:with and without disk friction
correction [5]
Fig.4
Head coefficient evolution versus flow coefficient:with
and without disk friction correction
[5]
Fig.5Two-phase mixture patterns in horizontal pipe
configuration.Extracted from Baker [6]
bles with rotation (see next section for basic physical analy-sis development),local void fraction distribution and value is strongly modified.This can be seen from visualization tech-niques,and a schematic representation is given in figure 6in-
side an impeller and in figure 7for 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.7Gas-liquid flow structures in impeller and diffuser flow channels
[8]
(a)Bubbly flow
(b)agglomerated bubbly flow (c)Gas Pockets (d)Annular flow
5.3Two-phase flow test results and performance parameters
Dimensional analysis
For a given design,global pump head performances de-pends on several parameters that are listed below.
Δp tp =F (D ,N ,Q l ,Q g ,ρl ,ρg ,νl ,νg ,g ,σ)(6)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 negligi-ble effects since cavitation is excluded for the present ap-proach.Taken D ,N and ρl as basic variables,it is possible to built 8different non-dimensional groups of variables that depend on these 3basic ones.This results to the follwing ex-pression,
Δp tp /(ρl (ND )2)=F (Q l /(ND 3),Qg /(ND 3),ρg /ρl ,(ND 2)/νl ,
(ND 2)/νg ,(N 2D )/g ,ρl N 2D 3/σ)(7)
-ND 2
/νl represents the liquid phase Reynolds number -ND 2/νg represents the gas phase Reynolds number
-ρl N 2D 3/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 /DN 2represents the gravitaional forces (also related to the Froude number)that is much lower than the centrifugal forces
-ρg /ρl is 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 wa-ter 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 rela-tion can be obtained
Δp tot,tp /(ρl (ND )2)=F 1(Q l /(ND 3),ρl /ρg )(8)Introducing the variable βdefined as:β=(1+Q l /Q g )-1or(1-β)=(1+Q g /Q l )-1(9)One can obtained the expression of the gas volumetric fraction a ,that is expressed as:
a =β/(1-β)
For αvalues lower than 10%,it can also be approximat-ed as:
α=Q g /(Q g +Q l )≈Q g /Q l (10)
This implies the following new expression:Δp tot,tp /((1-β)ρl (ND )2)=F 2((1+Q g /Q l )*Q l /(ND 3),β)(11)For most of gas liquid mixtures,the ratio ρg /ρl is small and one can define a density mixture ρtp such as:
ρtp =(1-β)ρl (12)
Using the pressure coefficient ψtot,dp and flow coefficient φtp ,one has
ψtot,tp =Δp tot,tp /(ρtp (ND )2)=G (φtp ,β)(13)
One can also use the void fraction and write
ψtot,tp =Δp tot,tp /(ρtp (ND )2)=G 1(φtp ,α)(14)
Finally,the pump void fraction mainly depends on the pump geometry,the rotational speed,the flow rate and the in-let void fraction αi ,this last parameter can be measured as
Chinese Journal of Turbomachinery
Vol.62，2020,No.5
well as all others when performing experimental pump tests.
The final relation can be written as follow :
ψtot,tp =Δp tot,tp /(ρtp (ND )2)=G 2(φtp ,αi )(15)5.4Pump performance degradation coefficients In order to quantify the performance pump modification
under two-phase conditions,one can use the following coeffi-cient ψ*,defined as the ratio between real head (or pressure)in two-phase conditions and the head (or pressure)for liquid single phase:
ψ*=ψtp /ψ(16)
Another coefficient can also been established as follow :
Δψtp *=(ψth,tp -ψtp )/(ψth -ψ)(17)
Which corresponds to the difference between theoreti-cal head and real head obtained in two-phase conditions di-vided 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 theoreti-cal head is a interesting one because it introduces the pump geometry variables and more specifically to the blade outlet relative angle as a variable.
5.5Example of two-phase pump experimental re-sults (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 con-ditions,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 repro-duced in figure 8,also presented a straight line.This means that a curve can be found,which is equivalent to the one giv-en in Eq.(5)that represents the evolution of two-phase sim-plified Euler curve written below ;
ψth,tp =1-f tp ·φtp /tan β2(18)
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 modi-fied 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 :
ψth,tp =ψth =1-φtp /tan β2,with f tp =1(19)
5.6Additional 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 map-ping ”figure like the one presented by Estevam and Barri-os [10],as shown in figure 9.It allows to define,for a given pump geometry,several zones for which the two-phase pat-terns can be related to pump head modification in relation with liquid volume flow rate and not only pump head coeffi-cient.
6Conclusions
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 da-ta 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 num-ber 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 measure-ments 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 con-stant inlet void fraction by changing both regulation flow rate valve and air flow rate.
5)Repeat the procedure with another rotational speed
Fig.8Theoretical head coefficient for different rotational speeds
(extracted from Si et al.
[9])
Fig.9
Example of two-phase mapping proposed by
Estevam et al.[10]
6)After proper data reduction,calculate all pump pa-rameters and always plot figures using non-dimensional coef-ficients 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 normal-ized 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.
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