theoretical model for the performance of liquid ring pump ... ·...

Research ArticleTheoretical Model for the Performance of Liquid Ring PumpBased on the Actual Operating Cycle

Si Huang, Jing He, XueqianWang, and Guangqi Qiu

School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510640, China

Correspondence should be addressed to Si Huang; [email protected]

Received 17 May 2017; Revised 9 July 2017; Accepted 27 July 2017; Published 28 August 2017

Academic Editor: Ryoichi Samuel Amano

Copyright © 2017 Si Huang et al.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Liquid ring pump is widely applied in many industry fields due to the advantages of isothermal compression process, simplestructure, and liquid-sealing. Based on the actual operating cycle of “suction-compression-discharge-expansion,” a universaltheoretical model for performance of liquid ring pump was established in this study, to solve the problem that the theoreticalmodels deviated from the actual performance in operating cycle. With the major geometric parameters and operating conditionsof a liquid ring pump, the performance parameters such as the actual capacity for suction and discharge, shaft power, and globalefficiency can be conveniently predicted by the proposed theoretical model, without the limitation of empiric range, performancedata, or the detailed 3D geometry of pumps. The proposed theoretical model was verified by experimental performances of liquidring pumps and could provide a feasible tool for the application of liquid ring pump.

1. Introduction

Liquid ring pump is a kind of rotary displacement machines,including liquid ring vacuum pump and liquid ring com-pressor. Owing to the advantages of isothermal compressionprocess, simple structure, and liquid-sealing, liquid ringpump is widely applied in petroleum, chemical, power,metallurgy, pharmaceutical, and light industry and otherfields, particularly suitable for the pumping of flammable,explosive, toxic, and harmful gases like hydrogen, oxygen,nitrogen, chlorine, ethylene, acetylene, propane, natural gas,and carbon dioxide. However, there are still some problemssuch as a higher power consumption, lower suction vacuumdegree, and unstable operation in application of liquid ringpump. The theoretical result considerably deviates from theactual performance due to the lack of the fundamentalstudy on the performance and restricts the improvement ofperformance and reliability of liquid ring pump. Therefore,it is of significant scientific and application values to carryout the fundamental study on the operating theory andperformance of the liquid ring pump.

The operating principle of the liquid ring pump isillustrated in Figure 1. With the revolution of the impeller,the operating liquid is ejected outwards due to centrifugal

forces and forms a liquid ring. Owing to the eccentricity ofthe impeller to the casing, a crescent-shaped chamber formsbetween the impeller hub and the liquid ring. The impellerblades divide the crescent-shaped chamber into several cellsof different volumes. As the impeller runs, the cells in therange of the suction port become larger and suck the gasthrough the suction port. In the part opposite to the suctionport, the gas is compressed in the reducing cells and emittedthrough the discharge port. Each revolution of the impellercorresponds to one operating cycle. Consequently, liquid ringpumps are the rotating machines with reciprocating pumpcharacteristics and the ring liquid takes on the function of areciprocating piston.

Since the first liquid ring pump was invented, manyscholars have carried out a variety of work on the design andindustrial application of liquid ring pumps [1–6]. Comparedwith the work on the other types of pumps and compressors,however, there were few reports of the fundamental studieson liquid ring pumps. Referring to ideal operating cycletheory of “suction-compression-discharge” in reciprocatingcompressor, Pfleiderer [7] derived an equation for theoreticalsuction capacity in liquid ring pump and developed a theo-retical model for the compression ratio and the liquid ring

HindawiInternational Journal of Rotating MachineryVolume 2017, Article ID 3617321, 9 pageshttps://doi.org/10.1155/2017/3617321

https://doi.org/10.1155/2017/3617321

2 International Journal of Rotating Machinery

Table 1: Calculation models of gas volume flow and shaft power.

Model 𝑞 𝑁Pfleiderer [7] 𝑞th = 12𝜇𝜔𝑏𝑟22 [(1 − 𝛼)2 − 𝜐2]

Prager [10]

𝑞𝑠 = 𝜆𝜆max 𝑞𝑠maxFor vacuum pumps:𝜆𝜆max = 0.147𝑋3 − 1.297𝑋2 + 0.150𝑋 + 1For compressors:𝜆𝜆max = 0.707𝑋3 − 1.527𝑋2 + 0.1782𝑋 + 1,

𝑋 = (𝑝𝑑/𝑝𝑠 − 1)[(𝑝𝑑/𝑝𝑠)max − 1]Powle [11] 𝑞𝑑 = 𝑞𝑑max𝑝𝑎 (𝑝𝑚 − 𝑝𝑚+1V /𝑝)(𝑝𝑚+1𝑎 − 𝑝𝑚+1V )Bodik [14] 𝐴𝑢32 + 𝐵𝑢2

Discharge Suction

Discharge portImpeller

Casing

Liquid ring

Direction of rotation

Suction port

Figure 1: Operating principle of a liquid ring pump.

surface at each circumferential angle in the pump. Schulz [8]and Segebrecht [9], respectively, deduced similar equationsfor the theoretical suction capacity. So far, the theoreticalmodel and the equations of theoretical suction capacity havebeenmainly used as reference for design of liquid ring pump.As shown in Table 1, the theoretical suction capacity, deter-mined by the impeller rotational speed and the geometricparameters of liquid ring pump, remains constant regardlessof the compression ratio of discharge pressure to suctionpressure. This result obviously deviates from the actual situa-tion, especially when the compression ratio is relatively high.To solve this problem, Prager [10] correlated an empiricalformula for the ratio of the actual suction capacity to themaximum actual suction capacity, based on experimentalperformance data of liquid ring pump, but the validity of thestatistical formula was restricted by the number of samplesand the unknown maximum actual suction capacity. On the

other hand, it was found from the actual applications that theminimum suction pressure of the liquid ring pump was closeto the saturated vapor pressure of the operating liquid. Thus,Powle [11] developed an equation for the ratio of the actualdischarge capacity to the maximum actual discharge capacityas a function of vapor pressure of the operating liquid andindex of expansion process in the pump, but the maximumactual discharge capacity should be given in advance.

Concerning the power of liquid ring pump, due to theabsorption and the removal of the heat by the operatingliquid, the gas compression process in the liquid ring pumpis generally regarded as isothermal one, so the formulafor the isothermal compression power of the ideal gas iscommonly used to calculate the effective power of the liquidring pump [12, 13]. Bodik [14] deduced an approximaterelation between the shaft power and rotational speed ofliquid ring pump (Table 1). Besides the effective power, theshaft power includes the power consumed by volumetriclosses, flow losses, and mechanical losses, but the coefficientsin Bodik’s power formula could not be determined. Recently,some investigators [15–18] used CFD numerical approach tosimulate 3D flows and predicted performance of liquid ringpumps. Although numerical approach plays an importantrole for understanding internal flow field of liquid ring pump,it is still hard for practical application, due to the restrictionof detailed 3D geometry of liquid ring pump, consumptionof computation time, and the accuracy of gas-liquid two-phase flow model. In this paper, therefore, a convenient anduniversal theoretical model is proposed for the performanceprediction of liquid ring pump based on the actual operatingcycle, to obtain the rules of operating cycle performanceand of power consumption in liquid ring pumps, withoutlimitation of statistical range or the detailed 3D geometry.

2. Theoretical Model

2.1. Actual Operating Cycle of Liquid Ring Pump. The tra-ditional theory [7] of liquid ring pump was based on theideal operating cycle of “suction-compression-discharge,”

International Journal of Rotating Machinery 3

0

p

q

1 2

34pd

ps

(a) Ideal

0

p

q

1

2

3

4pd

ps

q0

qe

(b) Actual

Figure 2: Diagram 𝑝-𝑞 of gas operating cycle in a liquid ring pump.

Compressionzone

Suctionzone

Expansionzone

A

B

C D

O

ea

R

Dischargezone

Liquid ringsurface

r2

r1

r

O

Figure 3: Geometric parameters in a liquid ring pump.

which was composed of three processes. During the suctionprocess (Curves 1-2 shown in Figure 2(a)), the suctionpressure 𝑝𝑠 keeps constant and reaches the maximum suc-tion capacity 𝑞th. In the compression process (Curves 2-3),the gas volume reduces and the pressure increases to reachthe discharge pressure 𝑝𝑑. In the discharge process (Curves3-4), the discharge pressure 𝑝𝑑 keeps constant and all the gasis discharged from the pump. The corresponding zones in aliquid ring pump are divided into the suction zone (SectorDOA), the compression zone (SectorAOB), and the dischargezone (Sector BOC) as shown in Figure 3. Like the actualoperating principle of reciprocating compressor, however,some residual gas (capacity is 𝑞𝑜) is still available in the liquidring pump after gas discharge and returns to the suction zone;the residual gas changes to 𝑞𝑒 in capacity after a process ofexpansion (Curves 4-1 in Figure 2(b)) and goes to the nextoperating cycle. Thus, the actual operating cycle is composedof four processes: suction-compression-discharge-expansion,in which expansion process takes place at the expansion zone(Sector COD) shown in Figure 3. The actual suction capacity

in a liquid ring pump is 𝑞𝑠 = 𝑞th − 𝑞𝑒 (if 𝑝𝑠 ≤ 𝑝V, 𝑞𝑠 = 0).The residual gas and its expansion process were not taken intoaccount in the theoretical model for ideal operating cycle ofliquid ring pump, resulting in deviation between theoreticaland practical results.

Therefore, the theoretical model in the present work is tosupplement an expansion process on the basis of the idealoperating cycle model. For the integrity of the model, theformulas in each zone are given in detail below.

(1) Suction Zone. The definition of geometric parameters inthe liquid ring pump is shown in Figure 3. The distance fromany point on the liquid ring surface in the suction zone to thecenter of the impeller is expressed as

𝑟𝜑 = 𝑟2√2 (𝑅 − 𝜌)𝜇𝑟2 + 𝜐2, (1)where

𝜌 = 𝑟2√1 + ( 𝑒𝑟2)2 + 2 𝑒𝑟2 cos𝜑. (2)

Theoretical suction capacity 𝑞th is defined as the gas volumeflow through the OA section assuming that the liquid ringpump inhales fully:

𝑞th = 12𝜇𝜔𝑏𝑟22 [(1 − 𝛼)2 − 𝜐2] . (3)(2) Compression Zone. The distance from any point on theliquid ring surface in the compression zone to the center ofthe impeller is

𝑟𝜑 = 𝑟2√ (1 − 𝛼)2 + (𝜎 − 1) 𝜐2𝜎 . (4)The compression ratio 𝜎 of the pressure at the circumferen-tial angle 𝜑 to the suction pressure needs to solve an algebraicequation:

1𝜎2 = 𝐴2 (1 − 𝜎 − 1𝜀 ) , (5)where 𝐴 = 2(𝑅 − 𝜌)/𝜇𝑟2[(1 − 𝛼)2 − 𝜐2]; 𝜀 = 𝑔𝜔2𝑟22/2𝑝𝑠.


(3) Discharge Zone.The distance from any point on the liquidring surface in the discharge zone to the center of the impelleris expressed as

𝑟𝜑 = 𝑟2√𝜐2 + 2 (𝑅 − 𝜌)𝜇𝑟2 √13 (1 + 1𝜀 ). (6)

(4) Expansion Zone. The distance from any point on theliquid ring surface in the expansion zone to the center of theimpeller is also calculated by (6). Therefore, the full shape ofliquid ring surface in the liquid ring pump can be obtainedaccording to (1), (4), and (6).

The capacity of gas returning to the suction zone of liquidring pump is calculated by

𝑞0 = 𝜇∫𝑟𝜑=2𝜋𝑟1

𝜔𝑟𝑏 𝑑𝑟 = 12𝜇𝜔𝑏 (𝑟2𝜑=2𝜋 − 𝑟21) , (7)where 𝑟𝜑=2𝜋 is the distance from the liquid ring surface at thecircumferential angle 𝜑 = 2𝜋 to the center of the impeller.The gas capacity 𝑞0 changes to 𝑞𝑒 after a polytropic expansionprocess and then enters the next operating cycle:

𝑞𝑒 = 𝑞0 (𝑝𝑑𝑝𝑠 )1/𝑚 . (8)

Therefore, the actual suction capacity of liquid ring pump isexpressed as

𝑞𝑠= {{{{{{{

0 𝑝𝑠 ≤ 𝑝V12𝜇𝜔𝑏𝑟22 [(1 − 𝛼)2 − 𝜐2] − 𝑞0 (𝑝𝑑𝑝𝑠 )

1/𝑚 𝑝𝑠 > 𝑝V.(9)

Then the actual discharge capacity 𝑞𝑑 can be calculatedaccording to the isothermal compression process:

𝑞𝑑 = 𝑞𝑠𝑝𝑠𝑝𝑑 . (10)2.2. Shaft Power Model of Liquid Ring Pump. The effectivepower of the liquid ring pump is generally calculated accord-ing to the isothermal compression power. However, accurateprediction of shaft power 𝑁 is essential for estimating globalefficiency and studying how to reduce power consumptionin a liquid ring pump. The shaft power is mainly composedof two parts: the total power consumption 𝑁𝑔 of the gas andthe friction loss 𝑁𝑙 of the operating liquid. The total powerconsumption of the gas is the adiabatic compression work:

𝑁𝑔 = 𝜅𝜅 − 1𝑝𝑠𝑞th [(𝑝𝑑𝑝𝑠 )(𝜅−1)/𝜅 − 1] . (11)

The empirical formula of the liquid friction loss 𝑁𝑙 in liquidring pump for the turbulent flow was given by Prager [19]:

𝑁𝑠 = 0.708𝜌2𝜔3𝑟52Re−0.1732, (12)Re = 𝜔𝑟22

]. (13)

It is found that axial width 𝑏 of the impeller is usually big inliquid ring pumps but not yet considered in Prager’s equation(12). By referring to the formula of disk friction power incentrifugal pump [20], the equation (12) of liquid frictionloss 𝑁𝑙 can be corrected as

𝑁𝑙 = 0.708𝜌2𝜔3𝑟52Re−0.1732 (1 + 𝑓 𝑏𝑟2) . (14)Therefore, the shaft power of liquid ring pump is written as

𝑁 = 𝑁𝑔 + 𝑁𝑙= 𝜅𝜅 − 1𝑝𝑠𝑞th [(𝑝𝑑𝑝𝑠 )

(𝜅−1)/𝜅 − 1]+ 0.708𝜌2𝜔3𝑟52Re−0.1732 (1 + 𝑓 𝑏𝑟2) .

(15)

The global efficiency 𝜂 is finally estimated by the followingexpression [13]:

𝜂 = 𝑝𝑠𝑞𝑠ln (𝑝𝑑/𝑝𝑠)𝑁 . (16)3. Performance Tests of Liquid Ring Pumps

To validate the feasibility of the proposed theoretical modelson performance of liquid ring pumps, the performancetests of the liquid ring pumps were performed. Three typesof single-stage double-acting liquid ring vacuum pumps(2BE1103, 2BE1253, and 2BE1353) were selected for perfor-mance tests. The design and operating parameters of theliquid ring vacuum pumps are listed in Table 2. Waterand air were the operating liquid and gas in the tests, androom temperature was maintained at 28∘C. The tests wereperformed under different rotational speed of the pumps byfrequency control. A multiparameter data acquisition andprocessing systemwere used in the performance tests and thedetailed measurement method was introduced in [21]. Theexperimental facility and the layout of measurement deviceare shown in Figure 4.

4. Comparisons of Performance betweenTest and Theoretical Models

4.1. Suction Capacity of Liquid Ring Pump. Figure 5 shows thecomparisons between the tested suction capacity 𝑞𝑠 valuesand theoretical ones from (3), Prager’s equation, Powle’sequation, and (9) (𝑚 = 1.4 for adiabatic expansion)under different rotational speeds, in which 𝑞𝑠max in Prager’sequation and 𝑞𝑑max in Powle’s equation are provided by (9).From the performance curves, it can be seen that the resultsof (9) under the adiabatic expansion process are close to thetest results. The predicted suction capacities of the liquidring vacuum pumps are within 15% relative error for allrotational speeds. The average relative errors of predictedvalues by different models are within 10% for three types ofpumps at various rotational speeds as shown in Table 3. This


(a) Experimental apparatus

Motor

Liquid ring pump

Suction

Suction

Discharge

Liquiddischarge

12

3

4

5

67

89

10

11

12

13

14

(b) Layout of monitoring points

Figure 4: Experimental facility. 1—suction temperature; 2—discharge temperature; 3—liquid suction temperature; 4—liquid dischargetemperature; 5—drive bearing temperature; 6—nondriven bearing temperature; 7—suction pressure; 8—discharge pressure; 9—air capacity;10—liquid capacity; 11—power; 12—rotational speed; 13—X/Y/Z-direction vibration; 14—substructure vibration.

Table 2: Parameters of liquid ring vacuum pump.

Category Parameter Unit 2BE1103 2BE1253 2BE1353Casing Radius 𝑅 m 0.1365 0.3145 0.407

Impeller

Rotational speed 𝑛 r/min 1170, 1300, 1450 590, 740, 920 372, 490, 590Axial width 𝑏 m 0.215 0.5 0.708

Angle of blade 𝛽 Degree 40 42 45Impeller radius 𝑟2 m 0.105 0.25 0.36Hub radius 𝑟1 m 0.0535 0.125 0.177Eccentricity 𝑒 m 0.0136 0.0316 0.04

Blade thickness 𝛿 m 0.008 0.01 0.012Blade number 𝑍 12 15 18

indicates that the expansion process of the residue gas is nearthe adiabatic one due to a shorter expansion time and path.In addition, Prager’s and Powle’s equations can also providereasonable predictions of 𝑞𝑠, if 𝑞𝑠max and 𝑞𝑑max are calculatedby (9).

4.2. Shaft Power of Liquid Ring Pump. Figure 6 shows thecomparisons between the tested shaft power 𝑁 values andtheoretical ones by (15) in liquid ring pump under different

rotational speeds. FromFigure 6, it can be seen that the trendsof the shaft power curves calculated by (15) agree well withthe ones of the test results. The shaft power predictions of theliquid ring vacuum pumps are within 13% relative error forall rotational speeds. The average relative errors of predictedvalues by the proposed model are within 10% for three typesof pumps at various rotational speeds as shown in Table 3.This means the effect of axial width 𝑏 of the impeller shouldbe taken into account for liquid friction loss in liquid ring


0.00.00

0.02

0.04

0.06

0.08

0.10

Prager’s modelPowle’s modelProposed model (m = 1.4)Experimental

×105

qs

(Ｇ3/s

)

qＮＢ

ps (Pa)0.2 0.4 0.6 0.8 1.0

(a) 𝑛 = 1170 r/min, type: 2BE1103

0.00

0.02

0.04

0.06

0.08

0.10

0.12

×105

qs

(Ｇ3/s

)

0.0


qＮＢ

ps (Pa)0.2 0.4 0.6 0.8 1.0

(b) 𝑛 = 1300 r/min, type: 2BE1103

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

qs

(Ｇ3/s

)

0.0


qＮＢ

ps (Pa)0.2 0.4 0.6 0.8

×1051.0

(c) 𝑛 = 1450 r/min, type: 2BE1103

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

×105

qs

(Ｇ3/s

)

0.0


qＮＢ

ps (Pa)0.2 0.4 0.6 0.8 1.0

(d) 𝑛 = 590 r/min, type: 2BE1253

0.00.10.20.30.40.50.60.70.80.9

×105

qs

(Ｇ3/s

)

0.0


qＮＢ

ps (Pa)0.2 0.4 0.6 0.8 1.0

(e) 𝑛 = 740 r/min, type: 2BE1253

0.0

0.2

0.4

0.6

0.8

1.0

1.2

×105

qs

(Ｇ3/s

)

0.0


qＮＢ

ps (Pa)0.2 0.4 0.6 0.8 1.0

(f) 𝑛 = 920 r/min, type: 2BE1253

Figure 5: Continued.


0.0

0.2

0.4

0.6

0.8

1.0

×1050.0


qＮＢ

ps (Pa)0.2 0.4 0.6 0.8 1.0

qs

(Ｇ3/s

)

(g) 𝑛 = 372 r/min, type: 2BE1353

0.0

0.2

0.4

0.6

0.8

1.0

1.2

×105

qs

(Ｇ3/s

)

0.0


qＮＢ

ps (Pa)0.2 0.4 0.6 0.8 1.0

(h) 𝑛 = 490 r/min, type: 2BE1353

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

qs

(Ｇ3/s

)

0.0


qＮＢ

ps (Pa)0.2 0.4 0.6 0.8 1.0

×105

(i) 𝑛 = 590 r/min, type: 2BE1353

Figure 5: Comparisons of 𝑞𝑠-𝑝𝑠 curves between the test and theoretical models.

Table 3: Average relative errors of predicted models.

Type of pump Average relative errors (%) Prager’s model Powle’s model Proposed model

2BE1103𝑛 (r/min) 1170 1300 1450 1170 1300 1450 1170 1300 1450Δ𝑞𝑠 5.23 4.25 5.41 4.68 5.36 10.06 1.77 1.83 5.32Δ𝑁 — — — — — — 6.43 4.87 6.69

2BE1253𝑛 (r/min) 590 740 920 590 740 920 590 740 920Δ𝑞𝑠 4.17 7.13 9.38 4.27 5.02 5.80 3.34 2.99 4.94Δ𝑁 — — — — — — 6.40 10.53 7.62

2BE1353𝑛 (r/min) 372 490 590 372 490 590 372 490 590Δ𝑞𝑠 8.85 7.38 7.07 12.95 4.01 4.52 8.66 1.88 1.40Δ𝑁 — — — — — — 5.02 4.66 4.46


0

2

4

6

8

10

n = 1170 r/min

n = 1300 r/min

N (k

W)

Proposed modelExperimental

n = 1450 r/min

0.0×105ps (Pa)

0.2 0.4 0.6 0.8

(a) Type: 2BE1103

n = 590 r/min

n = 740 r/min

n = 920 r/min

0

20

40

60

80

100


0.0×105ps (Pa)

0.2 0.4 0.6 0.8

N (k

W)

(b) Type: 2BE1253

0

20

40

60

80

100

120

140


0.0×105ps (Pa)

0.2 0.4 0.6 0.8

n = 372 r/min

n = 490 r/min

n = 590 r/min

N (k

W)

(c) Type: 2BE1353

Figure 6: Comparisons of 𝑁-𝑝𝑠 curves between the test and theoretical model.

pump. More sophisticated models for shaft power of liquidring pump are necessary to consider for higher predictiveaccuracy in further work.

5. Conclusions

Based on the actual operating cycle of “suction-compression-discharge-expansion,” a universal theoretical model for per-formance of liquid ring pump was established and verifiedby performance test in this study. Some conclusions aresummarized as follows:

(1) Considering expansion process of the residual gas inliquid ring pumps is an effective approach to solve theproblem that the theoretical models deviate from theactual performance in operating cycle.The expansionprocess is likely near the adiabatic one due to a shorterexpansion time and path.

(2) With the major geometric parameters and operatingconditions of a liquid ring pump, the performanceparameters such as the actual capacity for suction and

discharge, shaft power, and global efficiency can beconveniently predicted using the proposed theoreticalmodel, without the limitation of empiric range, per-formance data, or the detailed 3D geometry of pumps.

(3) The effect of axial width of the impeller should betaken into account for liquid friction loss in liquidring pump model.

Nomenclature

𝐴: Coefficient𝐵: Coefficient𝑁: Shaft power, W𝑅: Inner radius of casing, mRe: Reynolds number𝑎: Immersion depth of the blades, m𝑏: Impeller width, m𝑒: Impeller eccentricity, 𝑒 < 𝑅 − 𝑟2, m𝑓: Coefficient in friction power considering

effect of impeller width, 𝑓 = 0.15𝑚: Gas expansion index, 1 < 𝑚 < 𝜅


𝑛: Impeller rotational speed, r/min𝑝: Pressure, Pa𝑞: Volumetric capacity, m3/s𝑟: Radius, m𝑢: Circumferential speed of impeller, m/s𝑍: Blade number.Greek Letters

𝛼: Relative immersion depth of the blades𝛽: Angle of blade, degree𝛿: Blade thickness, m𝜀: Coefficient𝜂: Global efficiency of liquid ring pump𝜑: Circumferential angle, degree𝜅: Adiabatic index; for air, 𝜅 = 1.4𝜆: Flow coefficient𝜇: Coefficient considering the effect of bladethickness, 𝜇 = 0.68∼0.85

]: Kinematic viscosity, m2/s𝜌: Length, m; density, kg/m3𝜎: Compression ratio𝜔: Angular speed of impeller, 𝜔 = 2𝜋𝑛/60,rad/s𝜐: Ratio of hub radius to impeller radius,𝜐 = 𝑟1/𝑟2.

Subscripts

𝑎: Atmospheric𝑑: Discharge𝑒: Expansion𝑔: Gas𝑙: Operating liquid𝑠: SuctionV: Saturated vaporth: Theoretical0: Residual gas1: Impeller hub2: Impeller.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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