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Performance curve available for non-steady flow through Venturi-like reverse flow diverter
Flow Measurement and Instrumentation ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Short Communication
Performance curve available for non-steady flow through Venturi-like reverse flow diverter Cong Xu n Institute of Nuclear and New Energy Technology, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Tsinghua University, Chengfu Road, Haidian District, P.O. Box 1021, Postcode 102201, Beijing 100084, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 5 November 2014 Received in revised form 1 June 2015 Accepted 17 June 2015
A reverse flow diverter (RFD) consists of a driving nozzle, a diffuser, and a suction gap that separates the nozzle and diffuser. Thus, the RFD is a Venturi-like fluidic component with three ports. The jet flow emanating from the driving nozzle exit can entrain the ambient fluid and transport it to a high elevation. During this time, the flow through the RFD is non-steady, which makes it difficult to measure the flow depending on the pressure drop. In this study, a series of tests was carried out to evaluate this fluid flow with different contraction ratios, suction gap lengths, fluid properties, inlet flow rates, and inlet pressures. A performance curve was formulated that can be expressed as an exponential equation correlating the non-dimensional Euler number, pressure ratio, and suction factor. The performance curve is not affected by the driving nozzle exit diameter and suction length of the RFD. The performance curve makes it possible to measure the flow out of a RFD depending on the pressure drop. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Non-steady flow Reverse flow diverter Performance curve Flow measurement Venturi Entrainment
1. Introduction Reverse flow diverter (RFD) pumps are widely used in the nuclear, metallurgy, and chemical industries to transport hazardous liquids or liquid–solid mixtures that are radioactive, corrosive, or hot. Fig. 1 shows a schematic diagram of an RFD pump, and its detailed working principles and procedure are given in the literature [10,11,13–15]. RFD pumps are briefly introduced here. The core of the RFD pump is the RFD with three ports, and it directly determines the pumping capacity. As shown in Fig. 2, the RFD is a Venturi-like fluidic component consisting of a Venturi tube (i.e., driving nozzle), an expanded tube (i.e. diffuser), and a gap that separates the Venturi and expanded tubes (i.e., suction gap or port). The overall operating cycle of an RFD can be divided into two modes: forward flow and reverse flow. In the forward flow mode, the ambient liquid is sucked into the displacement vessel (DV) through the suction gap and then the driving nozzle. After the forward flow mode, the liquid filling the DV is forced through the driving nozzle to create a jet flow over the suction gap when the compressed air acts on the liquid in the DV. The reverse flow mode involves the flow from the driving nozzle to the diffuser. Owing to the Venturi phenomenon, a low-pressure area forms n
Fax: þ 86 10 62771740. E-mail address: [email protected]
around the jet core, and the ambient liquid is entrained into the diffuser, i.e., entrainment occurs. The entrained liquid and jet flow mix and are transported together to a high elevation through the diffuser and lifting pipe. Because of the entrainment, the RFD has a larger outlet flow than inlet flow. However, when the flow resistance of the lifting pipe is too great, the speed of the liquid emanating from the driving nozzle exit is not large enough to form a low-pressure area. The entrainment is destroyed, and portions of the liquid emanating from the driving nozzle exit turn into the surroundings through the suction gap. In this case, the outlet flow of the RFD is less than the inlet flow. Because more liquid can be transported, entrainment is expected in the reverse flow mode. Therefore, RFDs are generally operated in the entrainment regime in practice. The suction factor q ¼Qout/Qin represents the entrainment, where Qout and Qin are the inlet and outlet flows, respectively, of the RFD. Entrainment takes place if q 41. Therefore, the flow through a RFD is not a constant mass flow (non-steady flow). The flow and pressure drop through a RFD are two key parameters that represent the hydrodynamic characteristics. A performance curve formulated using the flow and pressure drop has potential for practical application. Venturi flowmeters have a RFDlike structure [1,9]. In contrast with a RFD, a Venturi flowmeter only has an inlet and outlet port, and the flow through it is a steady and constant mass flow. An import relationship between the flow and the pressure drop through a Venturi flowmeter can
http://dx.doi.org/10.1016/j.flowmeasinst.2015.06.019 0955-5986/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: C. Xu, Performance curve available for non-steady flow through Venturi-like reverse flow diverter, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.06.019i
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Nomenclature A A1, A2 C D d ds Eu F Ls ls m Pin Pout Pr ΔP ΔPf ΔPs Q Qin
throat area in Venturi flowmeter, m2 parameters, dimensionless flow coefficient, dimensionless pipe diameter, m2 driving nozzle exit (or diffuser entrance) diameter, m2 slipping interface diameter, m Euler's number, dimensionless function, dimensionless suction length, m slipping interface length, m entrained ratio, dimensionless inlet pressure, Pa outlet pressure, Pa pressure ratio, ¼Pin/Pout , dimensionless pressure drop, Pa pressure drop caused by the viscous friction, Pa pressure drop caused by the separated flow, Pa fluid flow, m3 s 1 inlet flow, m3 s 1
Qout q t1,t2 u0 u1 uin uout uslip,0 uslip,1 uslip,a X
outlet flow, m3 s 1 suction factor, dimensionless parameters, dimensionless velocity at the driving nozzle exit, m s 1 average velocity at the diffuser entrance, m s 1 inlet velocity, m s 1 outlet velocity, m s 1 slip velocity at the driving nozzle exit, m s 1 slip velocity at the diffuser entrance, m s 1 average slip velocity across the suction gap, m s 1 parameter, ¼Pr , dimensionless
Greek letters
β ε λ μ ξ ρ
contraction ratio, dimensionless expansion coefficient of fluid, dimensionless friction coefficient, dimensionless liquid viscosity, Pa s 1 resistance coefficient, dimensionless fluid density, kg m 3
is impossible because the steady flow (mass and energy conservation) is an important base for establishing Eq. (1). Although some performance equations have been proposed to describe the liquid flow through RFDs [16–18], some issues still exist; for example, these equations change according to the driving nozzle exit diameter and diffuser entrance diameter. In this study, a series of tests was carried out for different driving nozzle exit diameters, suction gap lengths, liquid properties, inlet flow rates, and inlet pressures to obtain a performance equation available for a nonsteady flow through RFDs. Based on the experimental data, a correlation was identified to represent the relationship between the pressure drop and flow.
2. Experimental setup and procedure
Fig. 1. Schematic diagram of RFD pump.
Fig. 2. Configuration and dimensions of RFD.
be expressed using Eq. (1) [6].
Q = ACε ΔP/[ρ(1 − β 4 )]
(1)
This equation applies to single-phase flows, and its modified versions have used to measure multi-phase flows [2,5,12]. Eq. (1) is deduced based on mass and energy conservation, in which a number of assumptions are required: steady state, incompressible flow, no external energy introduction, no friction, and internal energy and pressure uniform across the inlet and outlet sections [3]. A subsequent expectation is whether Eq. (1) can also express the performance of RFDs with non-steady flow. Unfortunately, this
As shown in Fig. 2, all of the tested RFDs were symmetrical (i.e., the diffuser had the same dimensions as the driving nozzle but was placed opposite to the driving nozzle). The symmetrical configuration is an optimized design for RFDs and has been widely adopted. The inner diameter D of the pipes connecting the driving nozzle inlet and diffuser outlet was 25 mm, and the taper angle of the driving nozzle and diffuser was 10°. The inlet section (section in–in, Fig. 2) was 316 mm apart from the driving nozzle exit (section 0–0), and the outlet section (section out–out, Fig. 2) was also 316 mm apart from the diffuser entrance (section 1–1). The pressures at the inlet and outlet sections were measured with pressure transducers (span: 500 kPa, accuracy: 70.5%). Fig. 3 shows a flowchart of the process used to evaluate the RFD performance. During the reverse flow mode, the liquid flow in an RFD can be considered quasisteady because the gas pressure in the DV is constant. A centrifugal pump was used to force liquid through the driving nozzle exit and create a jet across the suction gap to simulate a flow in the reverse flow mode. Valves V-1 and V-2 were used to adjust the pressure and flow rate at section in–in. To observe the effects of the liquid properties on the RFD performance, three aqueous glycerin solutions at 17–21 °C were used as the transported liquids. The viscosity and density were 1.08 10 3 Pa s and 998.7 kg m 3, respectively, for liquid I; 1.97 10 3 Pa s and 999.6 kg m 3, respectively, for liquid II; and 3.81 10 3 Pa s and 1125 kg m 3, respectively, for liquid III.
Please cite this article as: C. Xu, Performance curve available for non-steady flow through Venturi-like reverse flow diverter, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.06.019i
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uslip,1 = u0d2/[(1 − m)d2] − (q − 1)u0d2/(md2) = [1/(1 − m) − (q − 1)/m]u0 =0
(3)
uslip,1 is zero because the velocity field over the diffuser entrance is considered to be uniform. Consequently, the average slip velocity across the suction gap uslip,a is equal to (uslip,0 þuslip,1)/2.0, which is 0.5u0. Therefore, ΔPf can be expressed as follows: 2 2 ΔPf = 0.5λ(ls /ds )ρuslip , a = 0.25λ(ls /ds )ρu 0
(4)
λ is the friction coefficient, and ls and ds are the length and diameter, respectively, of the slipping interface. The pressure drop ΔPs can be represented by Eq. (5) ΔPs = 0.5ξρu12 = 0.5ξq2ρu02 Fig. 3. Schematic of experimental apparatus for RFD evaluation: (1) centrifugal pump, (2) RFD, (3) ultrasonic flowmeter, (4) storage tank, (5) pressure transmitter, (6) data acquisition system, (7) V1–V2 valves, and (8) gas–liquid separator.
The rate of the flow into and out of the RFD was measured using ultrasonic flowmeters (Siemens Ltd., FUS1010, accuracy: 70.5–1.0%) and corrected using flow calibration curves for the three liquids. Driving nozzle exit and diffuser entrance diameters d of 6, 8, and 10 mm and suction lengths Ls of 6, 8, 10, 15, and 20 mm were tested in this study. Generally, pressure and flow pulsations in a centrifugal pump are inevitable because of the wake flow from the impeller blade trailing edge, large-scale turbulence and vortices generated by flow separation, and flow recirculation at part load [4]. The wake flow is the strongest source of the pulsations. To reduce the influences of pressure and flow pulsations, all pressures and flows discussed in the following section were time-averaged over at least 3 minutes.
3. Analysis and results As expected, entrainment can result in more liquid being transported. The following analysis was only focused on scenarios where q4 1.0. The entrained liquid is known to exchange energy and momentum with the jet flow across the suction gap and diffuser; during this time, the velocity of the entrained liquid increases while a slip velocity develops between the entrained liquid and jet flow. In addition, when the liquid mixed by the jet flow and entrained liquid flow through the diffuser, the expansion of the flow area can cause a separated flow near the wall. One hypothesis suggests that the pressure drop through an RFD in the reverse flow mode with q 41.0 is mainly caused by the viscous friction between the entrained liquid and the jet flow over the suction gap and the local flow resistance from the separated flow in the diffuser. The pressure drops caused by the viscous friction and separated flow are expressed by ΔPf and ΔPs, respectively. An approximate analysis for ΔPf can be performed. The slip velocity is defined as the difference between the averaged velocities of the jet and entrained flows. At the driving nozzle exit, the velocity of the jet flow is u0, and the velocity of the entrained flow is zero. Thus, the slip velocity uslip,0 at the driving nozzle exit is u0. At the diffuser entrance, the average velocity of all liquids is u1 and the suction factor is q. The entrained ratio m is defined as follows:
m = (q − 1)u0 /u1 = (u1 − u0)/u1 = 1 − u0 /u1 = 1 − 1/q
(2)
At the diffuser entrance, the entrained liquid occupies a flow area of 0.25πmd2, while the jet flow occupies a flow area of 0.25π (1 m)d2. Thus, the slip velocity at the diffuser entrance uslip,1 can be expressed as follows:
(5)
The total pressure drop can subsequently be expressed by Eq. (6)
ΔP = ΔPf + ΔPs = 0.5ρu02[0.25λ(ls /ds ) + ξq2] = 0.5F (Pr , q)ρu02
(6)
The components in brackets to the right of the second equal sign are represented by the function F. To determine F, the dependent variables need to be determined first. The quantities λ, ls, ds, and ξ are all directly influenced by the inlet pressure Pin, inlet velocity uin (or u0), outlet pressure Pout, and outlet velocity uout (or u1) when the RFD dimensions are constant. Moreover, the effects of liquid properties such as the density and viscosity on the pressure drop – specifically on λ, ls, ds, and ξ – are also reflected implicitly in Pin, uin, Pout, and uout. Namely, the variables Pin, uin, Pout, and uout are the dependent variables of F. To avoid dimension mismatch for F, these variables need to be rearranged in dimensionless groups because F is dimensionless. The most simple and intuitive way to construct the dimensionless groups is to define two assembling parameters Pr and q. Here, q (¼ uout/uin ¼u1/ u0) is the suction factor as defined above, and Pr is the ratio of the inlet pressure Pin to the outlet pressure Pout (i.e., Pin/Pout). Then, F can be expressed as the function F(Pr, q) of Pr and q, which is only related to the pressures (i.e., pressure drop) and flows. When F(Pr, q) is determined, the correlation between the flows and the pressure drop through an RFD can also be determined. Eq. (6) can be rearranged as follows:
Eu =
ΔP = F (Pr , q) ρu02 /2
(7)
where Eu is a non-dimensional number, i.e., Euler's number. To obtain an expression for F(Pr, q), the experimental data were analyzed. First, all pressure drop data were divided by the corresponding values of ρu02/2 to determine the detailed values of Eu. The pressure drop data ΔP ( ¼Pin Pout) were calculated from Pin and Pout measured with pressure transducers. u0 was calculated by measuring uin with the ultrasonic flowmeter. During the analysis, the parameter X was defined as Prq, and all values of Eu were plotted against the corresponding X values. F(Pr, q) can be expressed as the following exponential function:
Eu =
ΔP = F (Pr , q) = A1exp( − X /t1) + A2 exp( − X /t2) + y ρu02 /2
(8)
This is a fitting formula for experimental data and was used to determine the parameters A1, A2, t1, t2, and y. Fig. 4 shows the results for RFDs with different d and Ls values using Liquid II as the working fluid. The figure shows the detailed values of the parameters in Eq. (8). The experimental data points clearly agree closely with the performance curve expressed by Eq. (8), and the performance curve is not influenced by the driving nozzle diameter (or diffuser entrance diameter) and suction gap length.
Please cite this article as: C. Xu, Performance curve available for non-steady flow through Venturi-like reverse flow diverter, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.06.019i
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ports such as RFDs. The inlet pressure Pin, inlet velocity uin, outlet pressure Pout, and outlet velocity uout are all included in the relationship represented by Eq. (8). Pin and Pout can easily be measured with pressure transducers, and uin can be calculated through the energy equation of the flow from the DV to the RFD inlet. Consequently, the pumping capacity uout can also be easily determined using Eq. (8) regardless of the values of d and Ls. However, note that Eq. (8) is only available when q 41.0, i.e., the most common case for RFD operation.
Acknowledgments This work was supported by Program IRT13026 for Changjiang Scholars and Innovative Research Team in University in PR China.
References Fig. 4. Effects of d and Ls on performance curve.
Fig. 5. Effects of liquid properties on performance curve.
Fig. 5 shows the effects of the liquid properties on the performance curve. The performance curve is clearly affected by the liquid properties.
4. Conclusions In conclusion, a simple performance equation for the correlation between the flow and pressure drop through an RFD was obtained from experimental data. The performance equation is not affected by d and Ls but is affected by the liquid properties. The performance curve expressed by Eq. (8) is available for the nonsteady flow through Venturi-like fluidic components with three
[1] A. Beaulieu, E. Foucault, P. Braud, P. Micheau, P. Szeger, A flowmeter for unsteady liquid flow measurements, Flow Meas. Instrum. 22 (2) (2011) 131–137. [2] A. Collins, S. Clark, Evolution of wet gas Venturi metering and wet gas correction algorithms, Meas. Control 46 (1) (2013) 15–20. [3] R.W. Fox, A.T. McDonald, Introduction to Fluid Mechanics, John Wiley & Sons, New York, 1992. [4] J.F. Guelich, U. Bolleter, Pressure pulsations in centrifugal pumps, J. Vib. Acoust. 114 (1992) 272–279. [5] A.H.A.M. Hasan, G.P. Lucas, Experimental and theoretical study of the gas– water two phase flow through a conductance multiphase Venturi meter in vertical annular (wet gas) flow, Nucl. Eng. Des. 241 (6) (2011) 1998–2005. [6] W.J. Jiang, Y.Y. Dai, H.J. Gu, Chemical Fundamentals, Tsinghua University Press, Beijing, 1992. [7] W. Jitschin, M. Ronzheimer, S. Khodabakhshi, Gas flow measurement by means of orifices and Venturi tubes, Vacuum 53 (1–2) (1999) 181–185. [8] W. Jitschin, Gas flow measurement by the thin orifice and the classical Venturi tube, Vacuum 76 (1) (2004) 89–100. [9] A. Lupeau, B. Platet, P. Gajan, A. Strzelecki, J. Escande, J.P. Couput, Influence of the presence of an upstream annular liquid film on the wet gas flow measured by a Venturi in a downward vertical configuration, Flow Meas. Instrum. 18 (1) (2007) 1–11. [10] G.H. Priestman, J.R. Tippetts, Development and potential of power fluidics for process flow control, Chem. Eng. Res. Des. 62 (1984) 67–80. [11] G.H. Priestman, J.R. Tippetts, Characteristics of a double-acting fluidic pump with hot and cold water, J. Fluid Control 15 (4) (1985) 51–62. [12] R. Steven, A dimensional analysis of two phase flow through a horizontally installed Venturi flow meter, Flow Meas. Instrum. 19 (6) (2008) 342–349. [13] J.R. Tippetts, G.H. Priestman, D. Thompson, Developments in power fluidics for application in nuclear plant, J. Dyn. Syst. Meas. Control 103 (1981) 342–351. [14] C. Xu, Y. Huang, Experimental characteristics of pneumatic pulse jet pumping system with a Venturi-like reverse flow diverter, Int. J. Chem. React. Eng. 9 (2011) A34. [15] C. Xu, B.B. Liu, S. Jing, Compact pneumatic pulse-jet pump with Venturi-like reverse flow diverter, Chin. J. Chem. Eng. 19 (4) (2011) 626–635. [16] C. Xu, S. Jing, Dimensionless performance and design guidelines for reverse flow diverters during reverse flow mode, Chem. Eng. Commun. 199 (7) (2012) 912–924. [17] C. Xu, H. Yu, X.D. Meng, Dimensionless performance and characteristic numbers of reverse flow diverters during reverse flow mode, Asia-Pac. J. Chem. Eng. 8 (3) (2013) 396–404. [18] C. Xu, H. Yu, Prediction of the pumping capacity for reverse-flow diverter pumps, Chem. Eng. Res. Des. 92 (2014) 1219–1226.
Please cite this article as: C. Xu, Performance curve available for non-steady flow through Venturi-like reverse flow diverter, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.06.019i
Contents lists available at ScienceDirect
Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
Short Communication
Performance curve available for non-steady flow through Venturi-like reverse flow diverter Cong Xu n Institute of Nuclear and New Energy Technology, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Tsinghua University, Chengfu Road, Haidian District, P.O. Box 1021, Postcode 102201, Beijing 100084, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 5 November 2014 Received in revised form 1 June 2015 Accepted 17 June 2015
A reverse flow diverter (RFD) consists of a driving nozzle, a diffuser, and a suction gap that separates the nozzle and diffuser. Thus, the RFD is a Venturi-like fluidic component with three ports. The jet flow emanating from the driving nozzle exit can entrain the ambient fluid and transport it to a high elevation. During this time, the flow through the RFD is non-steady, which makes it difficult to measure the flow depending on the pressure drop. In this study, a series of tests was carried out to evaluate this fluid flow with different contraction ratios, suction gap lengths, fluid properties, inlet flow rates, and inlet pressures. A performance curve was formulated that can be expressed as an exponential equation correlating the non-dimensional Euler number, pressure ratio, and suction factor. The performance curve is not affected by the driving nozzle exit diameter and suction length of the RFD. The performance curve makes it possible to measure the flow out of a RFD depending on the pressure drop. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Non-steady flow Reverse flow diverter Performance curve Flow measurement Venturi Entrainment
1. Introduction Reverse flow diverter (RFD) pumps are widely used in the nuclear, metallurgy, and chemical industries to transport hazardous liquids or liquid–solid mixtures that are radioactive, corrosive, or hot. Fig. 1 shows a schematic diagram of an RFD pump, and its detailed working principles and procedure are given in the literature [10,11,13–15]. RFD pumps are briefly introduced here. The core of the RFD pump is the RFD with three ports, and it directly determines the pumping capacity. As shown in Fig. 2, the RFD is a Venturi-like fluidic component consisting of a Venturi tube (i.e., driving nozzle), an expanded tube (i.e. diffuser), and a gap that separates the Venturi and expanded tubes (i.e., suction gap or port). The overall operating cycle of an RFD can be divided into two modes: forward flow and reverse flow. In the forward flow mode, the ambient liquid is sucked into the displacement vessel (DV) through the suction gap and then the driving nozzle. After the forward flow mode, the liquid filling the DV is forced through the driving nozzle to create a jet flow over the suction gap when the compressed air acts on the liquid in the DV. The reverse flow mode involves the flow from the driving nozzle to the diffuser. Owing to the Venturi phenomenon, a low-pressure area forms n
Fax: þ 86 10 62771740. E-mail address: [email protected]
around the jet core, and the ambient liquid is entrained into the diffuser, i.e., entrainment occurs. The entrained liquid and jet flow mix and are transported together to a high elevation through the diffuser and lifting pipe. Because of the entrainment, the RFD has a larger outlet flow than inlet flow. However, when the flow resistance of the lifting pipe is too great, the speed of the liquid emanating from the driving nozzle exit is not large enough to form a low-pressure area. The entrainment is destroyed, and portions of the liquid emanating from the driving nozzle exit turn into the surroundings through the suction gap. In this case, the outlet flow of the RFD is less than the inlet flow. Because more liquid can be transported, entrainment is expected in the reverse flow mode. Therefore, RFDs are generally operated in the entrainment regime in practice. The suction factor q ¼Qout/Qin represents the entrainment, where Qout and Qin are the inlet and outlet flows, respectively, of the RFD. Entrainment takes place if q 41. Therefore, the flow through a RFD is not a constant mass flow (non-steady flow). The flow and pressure drop through a RFD are two key parameters that represent the hydrodynamic characteristics. A performance curve formulated using the flow and pressure drop has potential for practical application. Venturi flowmeters have a RFDlike structure [1,9]. In contrast with a RFD, a Venturi flowmeter only has an inlet and outlet port, and the flow through it is a steady and constant mass flow. An import relationship between the flow and the pressure drop through a Venturi flowmeter can
http://dx.doi.org/10.1016/j.flowmeasinst.2015.06.019 0955-5986/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: C. Xu, Performance curve available for non-steady flow through Venturi-like reverse flow diverter, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.06.019i
C. Xu / Flow Measurement and Instrumentation ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
Nomenclature A A1, A2 C D d ds Eu F Ls ls m Pin Pout Pr ΔP ΔPf ΔPs Q Qin
throat area in Venturi flowmeter, m2 parameters, dimensionless flow coefficient, dimensionless pipe diameter, m2 driving nozzle exit (or diffuser entrance) diameter, m2 slipping interface diameter, m Euler's number, dimensionless function, dimensionless suction length, m slipping interface length, m entrained ratio, dimensionless inlet pressure, Pa outlet pressure, Pa pressure ratio, ¼Pin/Pout , dimensionless pressure drop, Pa pressure drop caused by the viscous friction, Pa pressure drop caused by the separated flow, Pa fluid flow, m3 s 1 inlet flow, m3 s 1
Qout q t1,t2 u0 u1 uin uout uslip,0 uslip,1 uslip,a X
outlet flow, m3 s 1 suction factor, dimensionless parameters, dimensionless velocity at the driving nozzle exit, m s 1 average velocity at the diffuser entrance, m s 1 inlet velocity, m s 1 outlet velocity, m s 1 slip velocity at the driving nozzle exit, m s 1 slip velocity at the diffuser entrance, m s 1 average slip velocity across the suction gap, m s 1 parameter, ¼Pr , dimensionless
Greek letters
β ε λ μ ξ ρ
contraction ratio, dimensionless expansion coefficient of fluid, dimensionless friction coefficient, dimensionless liquid viscosity, Pa s 1 resistance coefficient, dimensionless fluid density, kg m 3
is impossible because the steady flow (mass and energy conservation) is an important base for establishing Eq. (1). Although some performance equations have been proposed to describe the liquid flow through RFDs [16–18], some issues still exist; for example, these equations change according to the driving nozzle exit diameter and diffuser entrance diameter. In this study, a series of tests was carried out for different driving nozzle exit diameters, suction gap lengths, liquid properties, inlet flow rates, and inlet pressures to obtain a performance equation available for a nonsteady flow through RFDs. Based on the experimental data, a correlation was identified to represent the relationship between the pressure drop and flow.
2. Experimental setup and procedure
Fig. 1. Schematic diagram of RFD pump.
Fig. 2. Configuration and dimensions of RFD.
be expressed using Eq. (1) [6].
Q = ACε ΔP/[ρ(1 − β 4 )]
(1)
This equation applies to single-phase flows, and its modified versions have used to measure multi-phase flows [2,5,12]. Eq. (1) is deduced based on mass and energy conservation, in which a number of assumptions are required: steady state, incompressible flow, no external energy introduction, no friction, and internal energy and pressure uniform across the inlet and outlet sections [3]. A subsequent expectation is whether Eq. (1) can also express the performance of RFDs with non-steady flow. Unfortunately, this
As shown in Fig. 2, all of the tested RFDs were symmetrical (i.e., the diffuser had the same dimensions as the driving nozzle but was placed opposite to the driving nozzle). The symmetrical configuration is an optimized design for RFDs and has been widely adopted. The inner diameter D of the pipes connecting the driving nozzle inlet and diffuser outlet was 25 mm, and the taper angle of the driving nozzle and diffuser was 10°. The inlet section (section in–in, Fig. 2) was 316 mm apart from the driving nozzle exit (section 0–0), and the outlet section (section out–out, Fig. 2) was also 316 mm apart from the diffuser entrance (section 1–1). The pressures at the inlet and outlet sections were measured with pressure transducers (span: 500 kPa, accuracy: 70.5%). Fig. 3 shows a flowchart of the process used to evaluate the RFD performance. During the reverse flow mode, the liquid flow in an RFD can be considered quasisteady because the gas pressure in the DV is constant. A centrifugal pump was used to force liquid through the driving nozzle exit and create a jet across the suction gap to simulate a flow in the reverse flow mode. Valves V-1 and V-2 were used to adjust the pressure and flow rate at section in–in. To observe the effects of the liquid properties on the RFD performance, three aqueous glycerin solutions at 17–21 °C were used as the transported liquids. The viscosity and density were 1.08 10 3 Pa s and 998.7 kg m 3, respectively, for liquid I; 1.97 10 3 Pa s and 999.6 kg m 3, respectively, for liquid II; and 3.81 10 3 Pa s and 1125 kg m 3, respectively, for liquid III.
Please cite this article as: C. Xu, Performance curve available for non-steady flow through Venturi-like reverse flow diverter, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.06.019i
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uslip,1 = u0d2/[(1 − m)d2] − (q − 1)u0d2/(md2) = [1/(1 − m) − (q − 1)/m]u0 =0
(3)
uslip,1 is zero because the velocity field over the diffuser entrance is considered to be uniform. Consequently, the average slip velocity across the suction gap uslip,a is equal to (uslip,0 þuslip,1)/2.0, which is 0.5u0. Therefore, ΔPf can be expressed as follows: 2 2 ΔPf = 0.5λ(ls /ds )ρuslip , a = 0.25λ(ls /ds )ρu 0
(4)
λ is the friction coefficient, and ls and ds are the length and diameter, respectively, of the slipping interface. The pressure drop ΔPs can be represented by Eq. (5) ΔPs = 0.5ξρu12 = 0.5ξq2ρu02 Fig. 3. Schematic of experimental apparatus for RFD evaluation: (1) centrifugal pump, (2) RFD, (3) ultrasonic flowmeter, (4) storage tank, (5) pressure transmitter, (6) data acquisition system, (7) V1–V2 valves, and (8) gas–liquid separator.
The rate of the flow into and out of the RFD was measured using ultrasonic flowmeters (Siemens Ltd., FUS1010, accuracy: 70.5–1.0%) and corrected using flow calibration curves for the three liquids. Driving nozzle exit and diffuser entrance diameters d of 6, 8, and 10 mm and suction lengths Ls of 6, 8, 10, 15, and 20 mm were tested in this study. Generally, pressure and flow pulsations in a centrifugal pump are inevitable because of the wake flow from the impeller blade trailing edge, large-scale turbulence and vortices generated by flow separation, and flow recirculation at part load [4]. The wake flow is the strongest source of the pulsations. To reduce the influences of pressure and flow pulsations, all pressures and flows discussed in the following section were time-averaged over at least 3 minutes.
3. Analysis and results As expected, entrainment can result in more liquid being transported. The following analysis was only focused on scenarios where q4 1.0. The entrained liquid is known to exchange energy and momentum with the jet flow across the suction gap and diffuser; during this time, the velocity of the entrained liquid increases while a slip velocity develops between the entrained liquid and jet flow. In addition, when the liquid mixed by the jet flow and entrained liquid flow through the diffuser, the expansion of the flow area can cause a separated flow near the wall. One hypothesis suggests that the pressure drop through an RFD in the reverse flow mode with q 41.0 is mainly caused by the viscous friction between the entrained liquid and the jet flow over the suction gap and the local flow resistance from the separated flow in the diffuser. The pressure drops caused by the viscous friction and separated flow are expressed by ΔPf and ΔPs, respectively. An approximate analysis for ΔPf can be performed. The slip velocity is defined as the difference between the averaged velocities of the jet and entrained flows. At the driving nozzle exit, the velocity of the jet flow is u0, and the velocity of the entrained flow is zero. Thus, the slip velocity uslip,0 at the driving nozzle exit is u0. At the diffuser entrance, the average velocity of all liquids is u1 and the suction factor is q. The entrained ratio m is defined as follows:
m = (q − 1)u0 /u1 = (u1 − u0)/u1 = 1 − u0 /u1 = 1 − 1/q
(2)
At the diffuser entrance, the entrained liquid occupies a flow area of 0.25πmd2, while the jet flow occupies a flow area of 0.25π (1 m)d2. Thus, the slip velocity at the diffuser entrance uslip,1 can be expressed as follows:
(5)
The total pressure drop can subsequently be expressed by Eq. (6)
ΔP = ΔPf + ΔPs = 0.5ρu02[0.25λ(ls /ds ) + ξq2] = 0.5F (Pr , q)ρu02
(6)
The components in brackets to the right of the second equal sign are represented by the function F. To determine F, the dependent variables need to be determined first. The quantities λ, ls, ds, and ξ are all directly influenced by the inlet pressure Pin, inlet velocity uin (or u0), outlet pressure Pout, and outlet velocity uout (or u1) when the RFD dimensions are constant. Moreover, the effects of liquid properties such as the density and viscosity on the pressure drop – specifically on λ, ls, ds, and ξ – are also reflected implicitly in Pin, uin, Pout, and uout. Namely, the variables Pin, uin, Pout, and uout are the dependent variables of F. To avoid dimension mismatch for F, these variables need to be rearranged in dimensionless groups because F is dimensionless. The most simple and intuitive way to construct the dimensionless groups is to define two assembling parameters Pr and q. Here, q (¼ uout/uin ¼u1/ u0) is the suction factor as defined above, and Pr is the ratio of the inlet pressure Pin to the outlet pressure Pout (i.e., Pin/Pout). Then, F can be expressed as the function F(Pr, q) of Pr and q, which is only related to the pressures (i.e., pressure drop) and flows. When F(Pr, q) is determined, the correlation between the flows and the pressure drop through an RFD can also be determined. Eq. (6) can be rearranged as follows:
Eu =
ΔP = F (Pr , q) ρu02 /2
(7)
where Eu is a non-dimensional number, i.e., Euler's number. To obtain an expression for F(Pr, q), the experimental data were analyzed. First, all pressure drop data were divided by the corresponding values of ρu02/2 to determine the detailed values of Eu. The pressure drop data ΔP ( ¼Pin Pout) were calculated from Pin and Pout measured with pressure transducers. u0 was calculated by measuring uin with the ultrasonic flowmeter. During the analysis, the parameter X was defined as Prq, and all values of Eu were plotted against the corresponding X values. F(Pr, q) can be expressed as the following exponential function:
Eu =
ΔP = F (Pr , q) = A1exp( − X /t1) + A2 exp( − X /t2) + y ρu02 /2
(8)
This is a fitting formula for experimental data and was used to determine the parameters A1, A2, t1, t2, and y. Fig. 4 shows the results for RFDs with different d and Ls values using Liquid II as the working fluid. The figure shows the detailed values of the parameters in Eq. (8). The experimental data points clearly agree closely with the performance curve expressed by Eq. (8), and the performance curve is not influenced by the driving nozzle diameter (or diffuser entrance diameter) and suction gap length.
Please cite this article as: C. Xu, Performance curve available for non-steady flow through Venturi-like reverse flow diverter, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.06.019i
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4
ports such as RFDs. The inlet pressure Pin, inlet velocity uin, outlet pressure Pout, and outlet velocity uout are all included in the relationship represented by Eq. (8). Pin and Pout can easily be measured with pressure transducers, and uin can be calculated through the energy equation of the flow from the DV to the RFD inlet. Consequently, the pumping capacity uout can also be easily determined using Eq. (8) regardless of the values of d and Ls. However, note that Eq. (8) is only available when q 41.0, i.e., the most common case for RFD operation.
Acknowledgments This work was supported by Program IRT13026 for Changjiang Scholars and Innovative Research Team in University in PR China.
References Fig. 4. Effects of d and Ls on performance curve.
Fig. 5. Effects of liquid properties on performance curve.
Fig. 5 shows the effects of the liquid properties on the performance curve. The performance curve is clearly affected by the liquid properties.
4. Conclusions In conclusion, a simple performance equation for the correlation between the flow and pressure drop through an RFD was obtained from experimental data. The performance equation is not affected by d and Ls but is affected by the liquid properties. The performance curve expressed by Eq. (8) is available for the nonsteady flow through Venturi-like fluidic components with three
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Please cite this article as: C. Xu, Performance curve available for non-steady flow through Venturi-like reverse flow diverter, Flow Measurement and Instrumentation (2015), http://dx.doi.org/10.1016/j.flowmeasinst.2015.06.019i