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Characterization of a centrifugal pump impeller under two-phase flow conditions
Journal of Petroleum Science and Engineering 63 (2008) 18–22
Contents lists available at ScienceDirect
Journal of Petroleum Science and Engineering j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p e t r o l
Research paper
Characterization of a centrifugal pump impeller under two-phase flow conditions Jose Caridad 1, Miguel Asuaje ⁎, Frank Kenyery 1, Andrés Tremante 1, Orlando Aguillón 1 Universidad Simón Bolívar, Laboratorio de Conversión de Energía Mecánica, Valle de Sartenejas, Venezuela, Caracas 1080-A, Venezuela
a r t i c l e
i n f o
Article history: Received 3 November 2004 Accepted 16 June 2008 Keywords: Electrical Submersible Pump (ESP) two-phase flow numerical simulation artificial lift CFD
a b s t r a c t Multiphase pumping is an area of primary interest, particularly for the petroleum industry, where fair amount of gas can be found in oil wells production. This study presents the results of numerical simulations carried out in a centrifugal pump impeller of an Electrical Submersible Pump (Ns = 2063) conveying an air– water mixture. The results include the impeller head and the relative flow angle at the outlet (β2) as a function of the liquid flow rate, as well as the phases distribution within the impeller. A sensibility analysis with regard to the Gas-Void Fraction (GVF) and the bubble diameter was also included. The deterioration of the head reported by other investigators (in the case of two-phase flow) is reproduced and substantiated by means of the forces acting on bubbles within centrifugal impellers. Finally, comparison with experimental data is excellent, which demonstrate that Computational Fluid Dynamics (CFD) is a useful tool in the analysis of turbomachinery. © 2008 Published by Elsevier B.V.
1. Introduction
2. Previous work
In the petroleum industry, the Electrical Submersible Pump (ESP) is the method of artificial lift commonly preferred in the case of high volume production wells. However, these systems perform most efficiently when handling single-phase flow. In fact, the presence of gas causes head degradation, low efficiency and higher operational costs for each barrel produced. Moreover, if the gas volumetric fraction exceeds certain value, the pump reaches the condition known as gas locking, where the pump is blocked by the gas and ceases to work. Despite the physical mechanism that governs these phenomena is not well understood, the segregation of phases and the difference in the velocities between gas and liquid (slippage) have been reported to be associated with the detriment of head experienced by ESP's systems under two-phase flow conditions. The objective of this work is to completely characterize the behavior of a centrifugal pump impeller in the case of two-phase flow. To accomplish this task, CFD calculations were implemented on a centrifugal impeller of known geometry in order to obtain parameters such as head, outlet relative flow angle (β2) and phases distribution as a functions of the liquid flow rate, the Gas-Void Fraction (GVF) and the bubble diameter.
2.1. Motion of bubbles through centrifugal impellers
⁎ Corresponding author. Tel.: +58 212 906 4134; fax: +58 212 906 4132. E-mail addresses: [email protected] (J. Caridad), [email protected] (M. Asuaje), [email protected] (F. Kenyery), [email protected] (A. Tremante), [email protected] (O. Aguillón). 1 Tel.: +58 212 906 4134; fax: +58 212 906 4132. 0920-4105/$ – see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.petrol.2008.06.005
The analysis of a bubble motion through a rotating field is of great importance for numerous engineering problems. Cavitation, boiling, heat and mass transfer applications are good examples. Besides, the nuclear and the petroleum industry are also interested in the subject. The former due to an eventual loss of coolant accidents (LOCA), which force the feeding pump to work under two-phase flow conditions, therefore, compromising the reactor safety. The latter due to the lower efficiency of ESP's applications in the presence of oil–gas mixtures. Nevertheless, the theoretical studies found in the literature make major assumptions in order to simplify the equations and permit their resolution with relatively low numerical effort. For instance, only single bubbles are considered, hence, overlooking the possible effect of coalescence or division of bubbles. Moreover, it is assumed that the bubble does not affect the liquid velocity or pressure fields. Schrage and Perkings (1972) performed an analytical and experimental study of the motion of an isothermal bubble through a liquid annulus rotating at angular velocities among 500 and 1500 rpm. Given that only single bubbles were considered, the equations of particle dynamics were implemented to estimate the bubble motion. It was assumed that the main forces acting on the bubbles were threefold: buoyancy, drag and virtual mass forces. The numerical results showed that the bubble describe an inward spiral which was corroborated for the experimental results. The authors recommended further studies to analyze the effect of bubble interactions.
J. Caridad et al. / Journal of Petroleum Science and Engineering 63 (2008) 18–22
Later on, Minemura and Murakami (1980) carried out an outstanding work with regard to the movement of air bubbles in a centrifugal pump impeller. The equation of motion for bubbles is stated, including the effects of five different forces, namely: • The drag force due to the velocity differences between air and water (slippage). • Force caused for the pressure gradient, which exist around the bubble. • Force due to the difference in densities between the phases. • Force associated with the virtual mass effect. • Force related to the history of acceleration (Basset term). The authors demonstrated that the forces that govern the bubble motion within centrifugal impellers are those corresponding to the slippage (drag force) and the force due to the pressure gradient around the bubble. Besides, the bubbles tend to deviate from the streamlines of liquid. This tendency enhances as the bubble diameter increases. All the same, experimental results, though limited, seem to agree with the numerical results. On the other hand, Sterrett et al. (1996) applied a method similar to that of Schrage and Perkings (1972) to describe the kinematics of a single bubble passing through a centrifugal pump impeller. The study considered two different geometries: a logarithmic spiral impeller and radial vane impeller. The authors pointed out that the dominant forces in the analysis are drag force and the buoyancy force (due to the pressure gradient). It was found that the bubble diameter have a significant influence in the kinematics of the bubble and, for a given condition, there is a maximum bubble size for which the bubble is able to exit the impeller. Despite their simplicity, these studies have contributed to a better understanding of the key forces that influence the movement of a single bubble in a rotating motion, which allows to explain its behavior in centrifugal pump impellers. 2.2. Centrifugal pumps handling air–water mixtures Quite a few investigators have reported a deterioration in the performance of centrifugal pumps when handling two-phase mixtures. As it was mentioned before, the nuclear and the petroleum industry have to cope with this problem and, also, have conducted intensive research in the area. In the frame of experimental works, biphasic flow causes a drop in the pump head. In fact, for a constant liquid flow rate, this drop increases at larger values of the GVF. Lea and Bearden (1982) collected data to define the performance of three different centrifugal pumps under various flow and pressure conditions. Air–water and diesel–CO2 were the work fluids. The authors proved that the flow rate, the suction pressure and the percent of free gas are variables of preponderant importance to characterize these phenomena. More recently, Cirilo (1998) analyze the performance of three pumps (two of mixed flow and the other of radial flow) handling air– water mixtures. As Lea and Bearden, Cirilo demonstrated that the suction pressure, the flow rate and the Gas-Void Fraction (GVF) strongly influence the performance of this kind of a device in the case of two-phase flow. The author attributes the head degradation of the pump to gas accumulation within the impeller. Pessoa and Prado (1999) conducted an experimental study in a 22 stages pump using a mixture of air–water as working fluid. The main contribution of this work is that the pressure changes were measured stage-by stage. The authors concluded that the average performance of the pump is quite different from that of each stage.
19
As for theoretical approaches, several attempts have been made for the purpose of predicting the biphasic performance of centrifugal pumps, especially in the nuclear industry. However, limited success has been achieved. The principal shortcomings are that these models: • Overlook the effect of suction pressure, which, according to the experimental results, has proved to be an important factor to describe the problem. • Consider low GVF values (b10%). Given that industrial applications fall beyond this value, the range of the models needs to be increased. • Consider that the slip factor for two-phase flow is the same as for single-phase flow. In other cases, they simply do not consider the slip factor. This assumption can lead to important errors in the prediction of the performance of centrifugal pumps, as Caridad and Kenyery (2005) pointed out. • The physical mechanism that causes the head degradation is not well understood. 3. Numerical simulations To completely characterize a centrifugal impeller when handling two-phase mixtures, numerical simulations were carried out on an Electrical Submersible Pump impeller of known geometry. The methodology can be summarized as below. 3.1. Geometry A seven bladed centrifugal impeller (Ns = 1960) was selected as the geometry of interest given that experimental results were available from the work of Añez et al. (2001). Geometric and operation characteristics of this impeller are summarized in Table 1. With regard to the computational domain, it was possible to take advantage of the axisymmetry of the flow. Because of this, just 1/7 of the geometry was simulated, hence, diminishing the computational effort. On the other hand, as in any CFD problem, a sensibility analysis was performed to guarantee the independence of the results with respect to the grid. The final grid comprises 27,552 elements and can be seen in Fig. 1. Further details at this respect can be found in the work of Caridad and Kenyery (2004). 3.2. Extent of the domain In the modeling of turbomachinery through CFD, it is necessary to solve some distance upstream and downstream of the blade passage (extent of domain): The boundaries should be placed far enough the blade in order to smooth any inconsistency regarding turbulence quantities, usually not well known. The extent of domain also gives space to capture secondary flows and recirculations (especially in off-design operation cases) and allows room for elliptic influence of the flow. To address the issue of how far upstream and downstream the boundaries need to be placed, knowledge of the flow physics and experience are the best allies.
Table 1 Characteristics of the Impeller of Interest Characteristic
Value
HN (m) QN × 103 (m3/s) N (rpm) D2/D1 NS Z Exit blade angle (°)
8.5 4.6 3000 2.22 1960 7 30
20
J. Caridad et al. / Journal of Petroleum Science and Engineering 63 (2008) 18–22
Fig. 3. Comparison with experimental results (whole stage).
3.4. Boundary conditions Fig. 1. Domain of analysis and final grid.
In any case, it is a trade off situation. If the boundaries are placed too close to the blade, the problems mentioned above will arise. On the other hand, if the boundaries are placed too far, tangential velocity becomes much larger than meridional velocity, which is particularly critical at the outlet of the impeller, enhancing numerical errors. Moreover, additional grid is required and the simulation procedure is less efficient. In this work, an extent of domain of 0.5 (D2 − D1) was chosen as a criterion to cope with the problems identified above. 3.3. Models The Navier–Stokes equations were solved coupled with the continuity equation by means of a Computational Fluid Dynamics (CFD) commercial software. The Eulerian approach was chosen for the purpose of obtaining phases distribution and its influence on the pressure and velocity fields. As for multiphase model, the option regarded as two-fluid model (non-homogeneous) is the more appropriate because it allows separate velocity fields for each phase (the pressure field is shared by all fluids). The two-phase κ−ε model was chosen as turbulence model. The homogeneous option, which solves a common turbulence field, was implemented according to the recommendations stated in the software manual (AEA Technology, 1997).
Fig. 2. Numerical results for biphasic impeller head.
The equations describing the fluid flow through a specific domain need to be numerically closed stipulating the so-called boundary conditions. At the inlet of the domain, a total pressure condition was set. This condition is the most accurate due to the inflow energy is defined and the simulator is allowed to obtain gradients in velocity and pressure. At the outlet the mass flow was specified. Each flow rate of interest corresponds to a different mass flow. As it was mentioned before, the computational effort demanded for the solution of the problem was reduced solving 1/7 of the impeller (a single inter-blade channel). Thus, periodic boundaries were placed to numerically represent the symmetry of the problem. The non-slip condition (relative velocity equal to zero) was specified in every wall of the domain. Finally, in addition to single-phase results, two different Gas-Void Fractions (10 and 15%) and three different bubble diameters (0.1, 0.3 and 0.5 mm) were analyzed. 4. Results Based on the methodology described above, the numerical results including the head of the impeller, phases segregation and relative flow angle (β2) are shown in Figs. 2–7. The simulation conditions are detailed in Table 2.
Fig. 4. Phases distribution in the inter-blade channel (QL = 3.1E − 03 m3/s, GVF = 10%, d = 0.5 mm).
J. Caridad et al. / Journal of Petroleum Science and Engineering 63 (2008) 18–22
21
Fig. 7. Influence of the bubble diameter on the impeller head (GVF = 15%). Fig. 5. Forces acting on a single bubble within a centrifugal impeller.
4.1. Head of the impeller The head delivered for the isolated impeller as a function of the liquid flow rate is presented in Fig. 2 (single-phase and GVF = 15%). The experimental curves for the whole stage (impeller + diffuser) are also included for comparison purposes (taken from Añez et al., 2001). The impeller head was estimated through the following expression, where X is defined as the quality of the mixture in mass. "
2 2 # V −V ðP2 −P1 Þ H ¼ ð1−xÞ þ 2 1 γ 2g
"
LIQ
2 2 # V −V ðP2 −P1 Þ þ 2 1 þx γ 2g
GAS
The curves obtained from numerical simulation follow the tendency of those corresponding to the experimental results. However, the head predicted by means of CFD is greater for all cases. For instance, in the case of single-phase flow and nominal flow rate, the difference is close to 20%. This discrepancy is due to the losses of the diffuser, which were not taken into account. At this respect, Bastardo (2003) used the results of this study as an input to characterize the diffuser of the ESP tested for Añez et al. (2001) for two-phase flow. As a result, including the diffuser losses, Fig. 3 was generated for a bubble diameter of 0.1 mm. In the case of single-phase flow and GVF= 10% the results are excellent (see Fig. 3). For GVF = 15% the predicted results tend to overestimate the head of the stage, which suggest that the bubble diameter should be increased in the simulations, as will be explained later. 4.2. Phases distribution Fig. 4 exhibits the phases distribution within the inter-blade channel for GVF = 10% and d = 0.5 mm (for visualization purposes).
A gas pocket is obtained in the pressure side of the blade, as reported by other investigators (Murakami and Minemura, 1974; Kouidri et al., 2001). The detriment of the head experimented by centrifugal pumps conveying biphasic mixtures has been attributed in the past (Cirilo, 1998) to this segregation of phases and its effect on the hydraulic losses, which, for a constant liquid flow rate, increases for larger values of the GVF due to the increase of the difference between the velocities of the liquid and the gas (Caridad and Kenyery, 2005). The accumulation of gas on the pressure side of the impeller can be substantiated by means of the forces acting on a single bubble. As it was mentioned before, these forces are the drag force (FD) and the force due to the pressure gradient (F▽P). Both forces are showed schematically in Fig. 5. FD tends to push the bubble to the outlet because of the slippage (the liquid velocity is larger than the bubble velocity). On the other hand, F▽P tends to slow down the bubble (the pressure gradient is adverse). Moreover, the resultant force acting on the bubble (FR in Fig. 5) acts towards the pressure side of the blade, where it accumulates due to the low relative velocity of the liquid (hence, low FD). 4.3. Relative flow angle In addition to an increase in the hydraulic losses, the segregation of phases causes a diminishment in the outlet relative flow angle (β2). In fact, the gas pocket can be visualized as a change in the inter-blade geometry, modifying the streamlines of the liquid. The outlet relative flow angle as a function of the liquid flow rate is presented in Fig. 6. For a constant QL, β2 decreases for larger Gas-Void Fractions (larger pocket size). The effect of this result can be understood through the concept known as the theoretical head for a finite number of blades (HT), given by: HT ¼ U2 ðU2 −VM Ctg ðβ2 ÞÞ: HT represents the head that a given impeller can deliver (without taking into account the hydraulic losses). Thus, as the GVF increases, HT decreases. Table 2 Simulations condition
Fig. 6. Outlet relative flow angle as a function of the liquid flow rate (d = 0.1 mm).
Domain
1 inter-blade channel (periodic condition)
Grid Inflow B.C. Outflow B.C. Turbulence model Multiphase model Gas-Void Fraction (%) Bubble diameter (mm)
27,552 elements Total pressure (100,000 Pa) Mass flow (variable) κ–ε Two-fluid 0, 10, 15 0.1, 0.3, 0.5
22
J. Caridad et al. / Journal of Petroleum Science and Engineering 63 (2008) 18–22
It is worth mentioning that the diminution of β2 does not represent a loss but a decrease in the ability of the impeller to change the kinetics momentum of the working fluid. 4.4. Influence of the bubble diameter Previous researchers (Minemura and Murakami, 1980) have reported d = 0.3 mm as commonly found in centrifugal impellers. In order to investigate the effect of this variable on the phenomena, three different bubble diameters were set in the simulations, namely, 0.1, 0.3 and 0.5 mm. As can be seen from Fig. 7, the head degradation is more precipitous for larger bubble diameters. Another important detail is that, for d = 0.3 and 0.5 mm, the characteristic unstable zone for low liquid flow rates, in the case of two-phase flow, is obtained. This behavior is explained again by means of the forces acting on a single bubble. While FD is proportional to the surface area of the bubble (~ d2), F▽P is proportional to the volume (~d3). In this way, for a larger bubble diameter, the force due to the adverse pressure gradient increases more significantly than the drag force and, as a consequence, the size of the gas pocket increases. Therefore, the hydraulic losses enhance and the outlet relative flow angle (β2) diminishes. Experimental and analytical studies (Lea and Bearden, 1982; Cirilo, 1998) have proven that the suction pressure is a key factor in the performance of centrifugal pumps conveying two-phase flow mixtures. The authors believe that the suction pressure and the bubble diameter are variables strongly connected. Additional research is in order. 5. Conclusions The biphasic characterization of a centrifugal impeller pump was carried out by means of CDF tools. The distinctive detriment of head reported by experimental studies is obtained. This detriment has been linked to an accumulation of the gas phase on the pressure side of the impeller. Indeed, the gas pocket increases the hydraulic losses and diminishes the capacity of the impeller to change the kinetic momentum of the mixture. As a consequence the head delivered by the pump lessens. Furthermore, this segregation of phases is substantiated with the use of the forces acting on a single bubble passing through a centrifugal impeller, namely, the drag force (FD) and the force due to the pressure gradient (F▽P). A sensibility analysis with respect to the bubble diameter was carried out. It was found that the larger the bubble diameter, the larger the detriment in head experimented by the impeller. Again, this behavior was explained by means of FD and F▽P. Additionally, the unstable zone observed in experimental studies at low liquid flow rates was evidenced in the case of d = 0.3 and 0.5 mm. The numerical results (including the diffuser losses) show excellent agreement with the experiments, which prove that CFD is a reliable and economic resource in the prediction of the performance of centrifugal pumps. Nomenclature B.C. Boundary condition [−] CFD Computational Fluid Dynamics [−]
d D ESP F G GVF H N Ns P Q U V X Z γ β
Bubble diameter [mm] Diameter [mm] Electrical Submersible Pump [−] Force [N] Acceleration due to the force of gravity [m/s2] Gas-Void Fraction [adim] Head [m] Rotational speed [rpm] pffiffiffi h pffiffiffiffiffiffiffiffii N Q GPM Pump specific speed Ns ¼ 0:75 ¼ rpm 0:75 ðH Þ ðftÞ Pressure [Pa] Flow rate [m3/s] Blade velocity [m/s] Absolute velocity [m/s] Quality of the mixture in mass [adim] Number of blades [adim] Specific weight [N/m3] Relative flow angle [°]
Subscripts Gas Gas Liq Liquid 1 Impeller inlet 2 Impeller outlet D Drag L Liquid M Meridional N Nominal R Resultant T Theoretical U Tangential ▽P Due to the pressure gradient References AEA Technology, 1997. “CFX 4.3 Solver” (User Manual for the Computational Fluid Dynamics Software CFX-4.3). Añez, D., Kenyery, F., Escalante, S.I., Teran, V.M., 2001. ESP's performance with twophase and viscous flow. Proceedings of ETCE 2001, Petroleum Production Technology Symposium, Houston, USA. Bastardo, R., 2003. Simulación Numérica del Flujo Bifásico Líquido-Gas en el Difusor de una Bomba Centrífuga Multietapa. M.S. Thesis, (in Spanish), Universidad Simón Bolívar, Venezuela. Caridad, J., Kenyery, F., 2004. CFD analysis of Electric Submersible Pumps (ESP) handling two-phase mixtures. ASME J. Energy Resour. Technol. 126, 99–104 JUNE. Caridad, J., Kenyery, F., 2005. Slip factor for centrifugal impellers under single and two phase flow condition. ASME J. Fluid Eng. 127, 317–321 March. Cirilo, R., 1998. Air–Water Flow Through Electric Submersible Pumps, M.S. Thesis, The University of Tulsa, USA. Kouidri, S., Belamri, T., Bakir, F., Rey, R., 2001. On a general method of unsteady potential calculation applied to the compression stages of a turbomachine: part II— experimental comparison. ASME J. Fluids Eng. (JFE) 123, 787–792. Lea, J.F., Bearden, J.L., 1982. Effect of gaseous fluids on submersible pump performance. J. Pet. Technol. 2922–2930 December. Minemura, K., Murakami, M., 1980. A theoretical study on air bubble motion in a centrifugal pump impeller. ASME J. Fluids Eng. (JFE) 102, 446–455. Murakami, M., Minemura, K., 1974. Effects of entrained air on the performance of a centrifugal pump (1st report, performance and flow conditions). Bull. JSME 17 (110), 1047–1055. Pessoa, R., Prado, M., 1999. Experimental Investigation of Two-Phase Flow Performance of Electrical Submersible Pump Stages. SPE, p. 71552. Schrage, D.L., Perkings Jr, H.C., 1972. Isothermal bubble motion through a rotating liquid. J. Basic Eng. 187–192 March. Sterret, J.D., Knight, R.W., Reece, J.W., 1996. Fluids engineering division conference. FED 236, 365–372.
Contents lists available at ScienceDirect
Journal of Petroleum Science and Engineering j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p e t r o l
Research paper
Characterization of a centrifugal pump impeller under two-phase flow conditions Jose Caridad 1, Miguel Asuaje ⁎, Frank Kenyery 1, Andrés Tremante 1, Orlando Aguillón 1 Universidad Simón Bolívar, Laboratorio de Conversión de Energía Mecánica, Valle de Sartenejas, Venezuela, Caracas 1080-A, Venezuela
a r t i c l e
i n f o
Article history: Received 3 November 2004 Accepted 16 June 2008 Keywords: Electrical Submersible Pump (ESP) two-phase flow numerical simulation artificial lift CFD
a b s t r a c t Multiphase pumping is an area of primary interest, particularly for the petroleum industry, where fair amount of gas can be found in oil wells production. This study presents the results of numerical simulations carried out in a centrifugal pump impeller of an Electrical Submersible Pump (Ns = 2063) conveying an air– water mixture. The results include the impeller head and the relative flow angle at the outlet (β2) as a function of the liquid flow rate, as well as the phases distribution within the impeller. A sensibility analysis with regard to the Gas-Void Fraction (GVF) and the bubble diameter was also included. The deterioration of the head reported by other investigators (in the case of two-phase flow) is reproduced and substantiated by means of the forces acting on bubbles within centrifugal impellers. Finally, comparison with experimental data is excellent, which demonstrate that Computational Fluid Dynamics (CFD) is a useful tool in the analysis of turbomachinery. © 2008 Published by Elsevier B.V.
1. Introduction
2. Previous work
In the petroleum industry, the Electrical Submersible Pump (ESP) is the method of artificial lift commonly preferred in the case of high volume production wells. However, these systems perform most efficiently when handling single-phase flow. In fact, the presence of gas causes head degradation, low efficiency and higher operational costs for each barrel produced. Moreover, if the gas volumetric fraction exceeds certain value, the pump reaches the condition known as gas locking, where the pump is blocked by the gas and ceases to work. Despite the physical mechanism that governs these phenomena is not well understood, the segregation of phases and the difference in the velocities between gas and liquid (slippage) have been reported to be associated with the detriment of head experienced by ESP's systems under two-phase flow conditions. The objective of this work is to completely characterize the behavior of a centrifugal pump impeller in the case of two-phase flow. To accomplish this task, CFD calculations were implemented on a centrifugal impeller of known geometry in order to obtain parameters such as head, outlet relative flow angle (β2) and phases distribution as a functions of the liquid flow rate, the Gas-Void Fraction (GVF) and the bubble diameter.
2.1. Motion of bubbles through centrifugal impellers
⁎ Corresponding author. Tel.: +58 212 906 4134; fax: +58 212 906 4132. E-mail addresses: [email protected] (J. Caridad), [email protected] (M. Asuaje), [email protected] (F. Kenyery), [email protected] (A. Tremante), [email protected] (O. Aguillón). 1 Tel.: +58 212 906 4134; fax: +58 212 906 4132. 0920-4105/$ – see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.petrol.2008.06.005
The analysis of a bubble motion through a rotating field is of great importance for numerous engineering problems. Cavitation, boiling, heat and mass transfer applications are good examples. Besides, the nuclear and the petroleum industry are also interested in the subject. The former due to an eventual loss of coolant accidents (LOCA), which force the feeding pump to work under two-phase flow conditions, therefore, compromising the reactor safety. The latter due to the lower efficiency of ESP's applications in the presence of oil–gas mixtures. Nevertheless, the theoretical studies found in the literature make major assumptions in order to simplify the equations and permit their resolution with relatively low numerical effort. For instance, only single bubbles are considered, hence, overlooking the possible effect of coalescence or division of bubbles. Moreover, it is assumed that the bubble does not affect the liquid velocity or pressure fields. Schrage and Perkings (1972) performed an analytical and experimental study of the motion of an isothermal bubble through a liquid annulus rotating at angular velocities among 500 and 1500 rpm. Given that only single bubbles were considered, the equations of particle dynamics were implemented to estimate the bubble motion. It was assumed that the main forces acting on the bubbles were threefold: buoyancy, drag and virtual mass forces. The numerical results showed that the bubble describe an inward spiral which was corroborated for the experimental results. The authors recommended further studies to analyze the effect of bubble interactions.
J. Caridad et al. / Journal of Petroleum Science and Engineering 63 (2008) 18–22
Later on, Minemura and Murakami (1980) carried out an outstanding work with regard to the movement of air bubbles in a centrifugal pump impeller. The equation of motion for bubbles is stated, including the effects of five different forces, namely: • The drag force due to the velocity differences between air and water (slippage). • Force caused for the pressure gradient, which exist around the bubble. • Force due to the difference in densities between the phases. • Force associated with the virtual mass effect. • Force related to the history of acceleration (Basset term). The authors demonstrated that the forces that govern the bubble motion within centrifugal impellers are those corresponding to the slippage (drag force) and the force due to the pressure gradient around the bubble. Besides, the bubbles tend to deviate from the streamlines of liquid. This tendency enhances as the bubble diameter increases. All the same, experimental results, though limited, seem to agree with the numerical results. On the other hand, Sterrett et al. (1996) applied a method similar to that of Schrage and Perkings (1972) to describe the kinematics of a single bubble passing through a centrifugal pump impeller. The study considered two different geometries: a logarithmic spiral impeller and radial vane impeller. The authors pointed out that the dominant forces in the analysis are drag force and the buoyancy force (due to the pressure gradient). It was found that the bubble diameter have a significant influence in the kinematics of the bubble and, for a given condition, there is a maximum bubble size for which the bubble is able to exit the impeller. Despite their simplicity, these studies have contributed to a better understanding of the key forces that influence the movement of a single bubble in a rotating motion, which allows to explain its behavior in centrifugal pump impellers. 2.2. Centrifugal pumps handling air–water mixtures Quite a few investigators have reported a deterioration in the performance of centrifugal pumps when handling two-phase mixtures. As it was mentioned before, the nuclear and the petroleum industry have to cope with this problem and, also, have conducted intensive research in the area. In the frame of experimental works, biphasic flow causes a drop in the pump head. In fact, for a constant liquid flow rate, this drop increases at larger values of the GVF. Lea and Bearden (1982) collected data to define the performance of three different centrifugal pumps under various flow and pressure conditions. Air–water and diesel–CO2 were the work fluids. The authors proved that the flow rate, the suction pressure and the percent of free gas are variables of preponderant importance to characterize these phenomena. More recently, Cirilo (1998) analyze the performance of three pumps (two of mixed flow and the other of radial flow) handling air– water mixtures. As Lea and Bearden, Cirilo demonstrated that the suction pressure, the flow rate and the Gas-Void Fraction (GVF) strongly influence the performance of this kind of a device in the case of two-phase flow. The author attributes the head degradation of the pump to gas accumulation within the impeller. Pessoa and Prado (1999) conducted an experimental study in a 22 stages pump using a mixture of air–water as working fluid. The main contribution of this work is that the pressure changes were measured stage-by stage. The authors concluded that the average performance of the pump is quite different from that of each stage.
19
As for theoretical approaches, several attempts have been made for the purpose of predicting the biphasic performance of centrifugal pumps, especially in the nuclear industry. However, limited success has been achieved. The principal shortcomings are that these models: • Overlook the effect of suction pressure, which, according to the experimental results, has proved to be an important factor to describe the problem. • Consider low GVF values (b10%). Given that industrial applications fall beyond this value, the range of the models needs to be increased. • Consider that the slip factor for two-phase flow is the same as for single-phase flow. In other cases, they simply do not consider the slip factor. This assumption can lead to important errors in the prediction of the performance of centrifugal pumps, as Caridad and Kenyery (2005) pointed out. • The physical mechanism that causes the head degradation is not well understood. 3. Numerical simulations To completely characterize a centrifugal impeller when handling two-phase mixtures, numerical simulations were carried out on an Electrical Submersible Pump impeller of known geometry. The methodology can be summarized as below. 3.1. Geometry A seven bladed centrifugal impeller (Ns = 1960) was selected as the geometry of interest given that experimental results were available from the work of Añez et al. (2001). Geometric and operation characteristics of this impeller are summarized in Table 1. With regard to the computational domain, it was possible to take advantage of the axisymmetry of the flow. Because of this, just 1/7 of the geometry was simulated, hence, diminishing the computational effort. On the other hand, as in any CFD problem, a sensibility analysis was performed to guarantee the independence of the results with respect to the grid. The final grid comprises 27,552 elements and can be seen in Fig. 1. Further details at this respect can be found in the work of Caridad and Kenyery (2004). 3.2. Extent of the domain In the modeling of turbomachinery through CFD, it is necessary to solve some distance upstream and downstream of the blade passage (extent of domain): The boundaries should be placed far enough the blade in order to smooth any inconsistency regarding turbulence quantities, usually not well known. The extent of domain also gives space to capture secondary flows and recirculations (especially in off-design operation cases) and allows room for elliptic influence of the flow. To address the issue of how far upstream and downstream the boundaries need to be placed, knowledge of the flow physics and experience are the best allies.
Table 1 Characteristics of the Impeller of Interest Characteristic
Value
HN (m) QN × 103 (m3/s) N (rpm) D2/D1 NS Z Exit blade angle (°)
8.5 4.6 3000 2.22 1960 7 30
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Fig. 3. Comparison with experimental results (whole stage).
3.4. Boundary conditions Fig. 1. Domain of analysis and final grid.
In any case, it is a trade off situation. If the boundaries are placed too close to the blade, the problems mentioned above will arise. On the other hand, if the boundaries are placed too far, tangential velocity becomes much larger than meridional velocity, which is particularly critical at the outlet of the impeller, enhancing numerical errors. Moreover, additional grid is required and the simulation procedure is less efficient. In this work, an extent of domain of 0.5 (D2 − D1) was chosen as a criterion to cope with the problems identified above. 3.3. Models The Navier–Stokes equations were solved coupled with the continuity equation by means of a Computational Fluid Dynamics (CFD) commercial software. The Eulerian approach was chosen for the purpose of obtaining phases distribution and its influence on the pressure and velocity fields. As for multiphase model, the option regarded as two-fluid model (non-homogeneous) is the more appropriate because it allows separate velocity fields for each phase (the pressure field is shared by all fluids). The two-phase κ−ε model was chosen as turbulence model. The homogeneous option, which solves a common turbulence field, was implemented according to the recommendations stated in the software manual (AEA Technology, 1997).
Fig. 2. Numerical results for biphasic impeller head.
The equations describing the fluid flow through a specific domain need to be numerically closed stipulating the so-called boundary conditions. At the inlet of the domain, a total pressure condition was set. This condition is the most accurate due to the inflow energy is defined and the simulator is allowed to obtain gradients in velocity and pressure. At the outlet the mass flow was specified. Each flow rate of interest corresponds to a different mass flow. As it was mentioned before, the computational effort demanded for the solution of the problem was reduced solving 1/7 of the impeller (a single inter-blade channel). Thus, periodic boundaries were placed to numerically represent the symmetry of the problem. The non-slip condition (relative velocity equal to zero) was specified in every wall of the domain. Finally, in addition to single-phase results, two different Gas-Void Fractions (10 and 15%) and three different bubble diameters (0.1, 0.3 and 0.5 mm) were analyzed. 4. Results Based on the methodology described above, the numerical results including the head of the impeller, phases segregation and relative flow angle (β2) are shown in Figs. 2–7. The simulation conditions are detailed in Table 2.
Fig. 4. Phases distribution in the inter-blade channel (QL = 3.1E − 03 m3/s, GVF = 10%, d = 0.5 mm).
J. Caridad et al. / Journal of Petroleum Science and Engineering 63 (2008) 18–22
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Fig. 7. Influence of the bubble diameter on the impeller head (GVF = 15%). Fig. 5. Forces acting on a single bubble within a centrifugal impeller.
4.1. Head of the impeller The head delivered for the isolated impeller as a function of the liquid flow rate is presented in Fig. 2 (single-phase and GVF = 15%). The experimental curves for the whole stage (impeller + diffuser) are also included for comparison purposes (taken from Añez et al., 2001). The impeller head was estimated through the following expression, where X is defined as the quality of the mixture in mass. "
2 2 # V −V ðP2 −P1 Þ H ¼ ð1−xÞ þ 2 1 γ 2g
"
LIQ
2 2 # V −V ðP2 −P1 Þ þ 2 1 þx γ 2g
GAS
The curves obtained from numerical simulation follow the tendency of those corresponding to the experimental results. However, the head predicted by means of CFD is greater for all cases. For instance, in the case of single-phase flow and nominal flow rate, the difference is close to 20%. This discrepancy is due to the losses of the diffuser, which were not taken into account. At this respect, Bastardo (2003) used the results of this study as an input to characterize the diffuser of the ESP tested for Añez et al. (2001) for two-phase flow. As a result, including the diffuser losses, Fig. 3 was generated for a bubble diameter of 0.1 mm. In the case of single-phase flow and GVF= 10% the results are excellent (see Fig. 3). For GVF = 15% the predicted results tend to overestimate the head of the stage, which suggest that the bubble diameter should be increased in the simulations, as will be explained later. 4.2. Phases distribution Fig. 4 exhibits the phases distribution within the inter-blade channel for GVF = 10% and d = 0.5 mm (for visualization purposes).
A gas pocket is obtained in the pressure side of the blade, as reported by other investigators (Murakami and Minemura, 1974; Kouidri et al., 2001). The detriment of the head experimented by centrifugal pumps conveying biphasic mixtures has been attributed in the past (Cirilo, 1998) to this segregation of phases and its effect on the hydraulic losses, which, for a constant liquid flow rate, increases for larger values of the GVF due to the increase of the difference between the velocities of the liquid and the gas (Caridad and Kenyery, 2005). The accumulation of gas on the pressure side of the impeller can be substantiated by means of the forces acting on a single bubble. As it was mentioned before, these forces are the drag force (FD) and the force due to the pressure gradient (F▽P). Both forces are showed schematically in Fig. 5. FD tends to push the bubble to the outlet because of the slippage (the liquid velocity is larger than the bubble velocity). On the other hand, F▽P tends to slow down the bubble (the pressure gradient is adverse). Moreover, the resultant force acting on the bubble (FR in Fig. 5) acts towards the pressure side of the blade, where it accumulates due to the low relative velocity of the liquid (hence, low FD). 4.3. Relative flow angle In addition to an increase in the hydraulic losses, the segregation of phases causes a diminishment in the outlet relative flow angle (β2). In fact, the gas pocket can be visualized as a change in the inter-blade geometry, modifying the streamlines of the liquid. The outlet relative flow angle as a function of the liquid flow rate is presented in Fig. 6. For a constant QL, β2 decreases for larger Gas-Void Fractions (larger pocket size). The effect of this result can be understood through the concept known as the theoretical head for a finite number of blades (HT), given by: HT ¼ U2 ðU2 −VM Ctg ðβ2 ÞÞ: HT represents the head that a given impeller can deliver (without taking into account the hydraulic losses). Thus, as the GVF increases, HT decreases. Table 2 Simulations condition
Fig. 6. Outlet relative flow angle as a function of the liquid flow rate (d = 0.1 mm).
Domain
1 inter-blade channel (periodic condition)
Grid Inflow B.C. Outflow B.C. Turbulence model Multiphase model Gas-Void Fraction (%) Bubble diameter (mm)
27,552 elements Total pressure (100,000 Pa) Mass flow (variable) κ–ε Two-fluid 0, 10, 15 0.1, 0.3, 0.5
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It is worth mentioning that the diminution of β2 does not represent a loss but a decrease in the ability of the impeller to change the kinetics momentum of the working fluid. 4.4. Influence of the bubble diameter Previous researchers (Minemura and Murakami, 1980) have reported d = 0.3 mm as commonly found in centrifugal impellers. In order to investigate the effect of this variable on the phenomena, three different bubble diameters were set in the simulations, namely, 0.1, 0.3 and 0.5 mm. As can be seen from Fig. 7, the head degradation is more precipitous for larger bubble diameters. Another important detail is that, for d = 0.3 and 0.5 mm, the characteristic unstable zone for low liquid flow rates, in the case of two-phase flow, is obtained. This behavior is explained again by means of the forces acting on a single bubble. While FD is proportional to the surface area of the bubble (~ d2), F▽P is proportional to the volume (~d3). In this way, for a larger bubble diameter, the force due to the adverse pressure gradient increases more significantly than the drag force and, as a consequence, the size of the gas pocket increases. Therefore, the hydraulic losses enhance and the outlet relative flow angle (β2) diminishes. Experimental and analytical studies (Lea and Bearden, 1982; Cirilo, 1998) have proven that the suction pressure is a key factor in the performance of centrifugal pumps conveying two-phase flow mixtures. The authors believe that the suction pressure and the bubble diameter are variables strongly connected. Additional research is in order. 5. Conclusions The biphasic characterization of a centrifugal impeller pump was carried out by means of CDF tools. The distinctive detriment of head reported by experimental studies is obtained. This detriment has been linked to an accumulation of the gas phase on the pressure side of the impeller. Indeed, the gas pocket increases the hydraulic losses and diminishes the capacity of the impeller to change the kinetic momentum of the mixture. As a consequence the head delivered by the pump lessens. Furthermore, this segregation of phases is substantiated with the use of the forces acting on a single bubble passing through a centrifugal impeller, namely, the drag force (FD) and the force due to the pressure gradient (F▽P). A sensibility analysis with respect to the bubble diameter was carried out. It was found that the larger the bubble diameter, the larger the detriment in head experimented by the impeller. Again, this behavior was explained by means of FD and F▽P. Additionally, the unstable zone observed in experimental studies at low liquid flow rates was evidenced in the case of d = 0.3 and 0.5 mm. The numerical results (including the diffuser losses) show excellent agreement with the experiments, which prove that CFD is a reliable and economic resource in the prediction of the performance of centrifugal pumps. Nomenclature B.C. Boundary condition [−] CFD Computational Fluid Dynamics [−]
d D ESP F G GVF H N Ns P Q U V X Z γ β
Bubble diameter [mm] Diameter [mm] Electrical Submersible Pump [−] Force [N] Acceleration due to the force of gravity [m/s2] Gas-Void Fraction [adim] Head [m] Rotational speed [rpm] pffiffiffi h pffiffiffiffiffiffiffiffii N Q GPM Pump specific speed Ns ¼ 0:75 ¼ rpm 0:75 ðH Þ ðftÞ Pressure [Pa] Flow rate [m3/s] Blade velocity [m/s] Absolute velocity [m/s] Quality of the mixture in mass [adim] Number of blades [adim] Specific weight [N/m3] Relative flow angle [°]
Subscripts Gas Gas Liq Liquid 1 Impeller inlet 2 Impeller outlet D Drag L Liquid M Meridional N Nominal R Resultant T Theoretical U Tangential ▽P Due to the pressure gradient References AEA Technology, 1997. “CFX 4.3 Solver” (User Manual for the Computational Fluid Dynamics Software CFX-4.3). Añez, D., Kenyery, F., Escalante, S.I., Teran, V.M., 2001. ESP's performance with twophase and viscous flow. Proceedings of ETCE 2001, Petroleum Production Technology Symposium, Houston, USA. Bastardo, R., 2003. Simulación Numérica del Flujo Bifásico Líquido-Gas en el Difusor de una Bomba Centrífuga Multietapa. M.S. Thesis, (in Spanish), Universidad Simón Bolívar, Venezuela. Caridad, J., Kenyery, F., 2004. CFD analysis of Electric Submersible Pumps (ESP) handling two-phase mixtures. ASME J. Energy Resour. Technol. 126, 99–104 JUNE. Caridad, J., Kenyery, F., 2005. Slip factor for centrifugal impellers under single and two phase flow condition. ASME J. Fluid Eng. 127, 317–321 March. Cirilo, R., 1998. Air–Water Flow Through Electric Submersible Pumps, M.S. Thesis, The University of Tulsa, USA. Kouidri, S., Belamri, T., Bakir, F., Rey, R., 2001. On a general method of unsteady potential calculation applied to the compression stages of a turbomachine: part II— experimental comparison. ASME J. Fluids Eng. (JFE) 123, 787–792. Lea, J.F., Bearden, J.L., 1982. Effect of gaseous fluids on submersible pump performance. J. Pet. Technol. 2922–2930 December. Minemura, K., Murakami, M., 1980. A theoretical study on air bubble motion in a centrifugal pump impeller. ASME J. Fluids Eng. (JFE) 102, 446–455. Murakami, M., Minemura, K., 1974. Effects of entrained air on the performance of a centrifugal pump (1st report, performance and flow conditions). Bull. JSME 17 (110), 1047–1055. Pessoa, R., Prado, M., 1999. Experimental Investigation of Two-Phase Flow Performance of Electrical Submersible Pump Stages. SPE, p. 71552. Schrage, D.L., Perkings Jr, H.C., 1972. Isothermal bubble motion through a rotating liquid. J. Basic Eng. 187–192 March. Sterret, J.D., Knight, R.W., Reece, J.W., 1996. Fluids engineering division conference. FED 236, 365–372.