Flow Measurement and Instrumentation 22 (2011) 406–412

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Flow Measurement and Instrumentation

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Application of small size cavitating venturi as flow controller and flow meterHojat Ghassemi ∗, Hamidreza Farshi FasihSchool of Mechanical Engineering, Department of Aerospace Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran

a r t i c l e i n f o

Article history:Received 21 June 2010Received in revised form15 March 2011Accepted 9 May 2011

Keywords:Cavitating venturiMass flow rateFlow control systemFlowmeterPassive flow controller

a b s t r a c t

The cavitating venturi is using to provide constant mass flow rate of liquid which is passing througha passage, independent of downstream pressure changes. The flow rate is a function of the upstreampressure, the throat area, the density and saturation pressure of the liquid. An experimental setup withcapability of supplying water flow rate and constant upstream pressure was designed andmanufactured.Three cavitating venturis with throat diameter of 5, 2.5, and 1 mmwere designed and built to investigatethe effect of venturi size on its mass flow rate. Three different sets of experiments were conducted toinvestigate the performance of the venturis. In the experiments, themass flow rateswere examined underdifferent downstream and upstream pressure conditions and time varying downstream pressure. Theresults show for the ratio of downstream pressure to upstream pressure less than 0.8, the mass flow rateis constant and independent of the downstream pressure. Whenever the pressure ratio exceeds 0.8, theventuri acts like an orifice. This pressure ratio has been predicted analytically to highlight the affectingparameters, mainly the geometry of the venturi and viscous losses. It is found that the venturi size hasno effect on its expecting function to keep mass flow rate constant. Also, it is shown that by applying adischarge coefficient and using only upstream pressure, the cavitating venturi can be used as a flowmeterwith a high degree of accuracy in a wide range of mass flow rate.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In many applications it is necessary to deliver a very smallamount of liquid flow rate constantly. For example, in a labscale monopropellant or hybrid rocket motor a few grams ofliquid oxidizer is demanded per second. To provide such flowrate independent of variable chamber pressure, using a cavitatingventuri is a practical and economical solution. The mass flow rateof a cavitating venturi is proportional to its throat area. Therefore,a venturi with a small throat diameter can supply a very low massflow rate. Regarding the small size of the throat, some problemsmay arise from viscous phenomena and varying downstreampressure in the performance of the venturi. On the other hand, suchventuris have received very little attention in the literatures. Thisstudy is an attempt to clarify the performance of the small venturisunder dynamically variable downstream pressure.

The behavior of a cavitating venturi is investigated extensivelyunder a steady state condition, numerically and experimentally.The cavitating flow of liquid helium through a 1 mm throatventuri channel is numerically investigated to realize the furtherdevelopment and high performance of newmulti-phase superfluidcooling systems [1]. It is clear that the phase change effectively

∗ Corresponding author. Tel.: +98 21 77491228; fax: +98 21 77240488.E-mail addresses: h_ghassemi@iust.ac.ir (H. Ghassemi),

hr_farshifasih@mecheng.iust.ac.ir (H.F. Fasih).

0955-5986/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.flowmeasinst.2011.05.001

occurs in the downstream of the venturi’s throat section and thata cloud cavity which consists of concentrated small bubbles isformed in the wall surface vicinity of the throat section. Using anumerical method for different flow regimes, the identification ofbubbly cavitating depending on the value of the back pressure,in venturi is investigated [2]. When the back pressure is lowered,the flow becomes choked, and a steady bubbly shock wave formsin the diverging section of the nozzle. Also, cavitation and phasechange of water is modeled numerically using a homogeneousflow model and CFD techniques for venturis with different sizesof throat diameter ranging from 2.5 to 5.3 mm [3]. It is shown thatthe phase change begins from the throat of the flow passage whichcontains the throat and diverging parts. An experimental study isconducted in order to prove the cavitation of liquid helium in aventuri with 5mmdiameter [4]. High-speed video camera picturesindicate that the flow separation and cavitation happens in thethroat region of the venturi.

In order to decrease frictional losses in the process of pressurereduction, the cavitating venturi is built in the shape of aconverging–diverging nozzle. The converging and diverging partsare connected by a cylindrical throat. While the upstream pressureis constant, this kind of venturi is able to keep mass flow rateconstant independent of downstream pressure. Ulas [5] for thepurpose of flow control in liquid rocket engines, kept upstreampressure constant by two methods and used an 11 mm cavitatingventuri to provide a constant mass flow rate, independent ofdownstream pressure.

H. Ghassemi, H.F. Fasih / Flow Measurement and Instrumentation 22 (2011) 406–412 407

By using the converging–diverging nozzle as a venturi, thecapability of the venturi to keep mass flow rate constant isinvestigated,while the downstreampressure to upstreampressureratio is less than 0.8 [3,6]. For the pressure ratio more than 0.8,the cavitating venturi acts as an orifice and does not control theflow rate. Liou and Chen [7] indicated that if the pressure ratio ofdownstream toupstream ismore than0.8, unchoking andoverflowphenomena will be observed in the venturi.

In this paper the performance of various cavitating venturiswith different sizes is examined. Some venturis are designed,manufactured and tested to pass a certain amount of water flowrate. Three venturis with different sizes have been used in theexperiment. The selected throat diameters of the venturis are 1,2.5, and 5 mm. The mass flow rates have been covered from30 g/s to 1500 g/s at an upstream pressure equal to 30 bar. Theexperiments are conducted to realize a cavitating performancerange for different downstream pressures. At first, for different(but constant) upstream pressure, the mass flow rates havebeen measured for different downstream pressures. Experimentalvalues have been compared with the theoretically predictedvalues. The results confirm that while the ratio of downstreampressure to upstream pressure is less than about 0.8, the cavitatingventuri will be choked.

Then, the experiments have been conducted to experiencethe behavior of the venturi and its response time when thedownstream pressure changes dynamically. In these tests, aconstant upstream pressure is applied, while a dynamicallyvariable pressure has been set downstream. Because of theexcellent agreement of measured to predicted mass flow rate,the cavitating venturi can be used as a flowmeter by using apressure transducer and applying a discharge coefficient. Thedetermined discharge coefficients are about 0.94 for all venturis inthis study. Also, a theoretical expression is presentedwhich revealsthe affecting parameters’ rules on the pressure ratio leading to achoking condition in the throat.

2. Design of cavitating venturi

2.1. Theory of cavitating venturi

In a pipeline, the cavitating venturi provides a converg-ing–diverging passage. When the static pressure in the throat ofthe venturi is lower than the vapor pressure of the liquid, a veryrapid partial transition from liquid to gas phase occurs. The waterin the upstreamof venturi can be assumed to be compressed liquid.Moreover, at normal condition, water vapor saturation pressure issmall (e.g. 3.169 kPa at 25°C [8]). The acceleration process in theconverging part of the venturi reduces the pressure to the satu-ration pressure of water. More pressure reduction causes a phasechange from liquid to vapor. So, the quality of saturated water in-creases from zero to a small value. The pressure drop in the venturican be assumed as a constant enthalpy process. The constant en-thalpy process for the two phase state accompanies a reduction inthe temperature. It is similar to the well known throttling processwith positive Joul–Tomson coefficient [8].

According to the thermodynamics tables of water, the quality ofsaturatedwater, x, at the throat has been shown to be very small. Inspite of the small value of the saturated water quality, the volumeof producing vapor is very large. In the following, the ratio of vaporvolume to liquid volume, α, is expressed.

α =x

1 − xρLiquid

ρVapor(1)

where ρLiquid and ρVapor are the density of liquid and vapor, withtypical values of 1000 kg/m3 and 0.0231 kg/m3, respectively.

Table 1Pressure loss coefficients of venturis.

Venturi S.C G.C G.E S.E Keffective

1 0.34 0.02 0.65 0.67 1.732 0.36 0.02 0.65 0.71 1.793 0.39 0.02 0.65 0.90 2.01

According to Eq. (1) for a small value of x, α has a large value. Asan example, consider that the pressure at the throat of the venturidecreases from normal saturation pressure to 2 kPa. Assuming aconstant enthalpy process results in 17.5 °C, 0.013, and 863 fortemperature, x, and α, respectively. The large value of vapor voidfraction causes the vapor spread throughout the downstream ofthe venturi throat. Finally, this phase change causes the flow tochoke and prevents the downstream pressure change affecting theupstream.

The thermophysical properties of water are used during designcalculations of the cavitating venturi. Required design parametersare throat area, entrance and exit diameter of venturi, andconverging–diverging angles. Driving the Bernoulli’s equationbetween the free surface of the water in upstream and the throatof the cavitating venturi, the mass flow rate can be written as

m = Ath

2ρ(P1 − Pth) (2)

where P1 and Pth are the water pressure at stagnation upstreamand venturi’s throat, respectively. Ath is the throat area and ρ isthe liquid water density at the throat which can be assumed asa constant. In order to see the cavitation at the venturi throat,the throat pressure must be less than the water vapor saturationpressure. Finally, throat area and the venturi’s throat diameterwould be calculated by choosing the mass flow rate of water anddetermining the upstream pressure.

2.2. Non cavitating venturi theory

If the static pressure in the throat does not decrease as lowas the fluid saturation pressure, no phase change appears in thethroat. Therefore, all the fluid in the venturi is liquid and the flowis not choked. In this case, no cavitation occurs in the venturi andthe mass flow rate changes according to the variation of pressuredifference between the upstream and the downstream. The massflow rate of water can be calculated by using Bernoulli’s equationbetween the free surface of the water in the upstream and theventuri’s exit as

P1 +12ρV 2

1 = P3 +12ρV 2

3 + fLD12ρV 2

3 + (Keffective)12ρV 2

3 . (3)

Point 3 is located at the exit of the venturi. Keffective is the sumof pressure loss coefficients of the different elements in theflow path. The pipe interior loss (fL/D) and the tank interiorvelocity (1/2 ρV 2

1 ) are neglected due to their small values. Also,it is assumed the water passing area is the same in all sections.Therefore, the mass flow rate can be calculated as the following

m = A3

2ρ(P1 − P3)(1 + Keffective)

(4)

where A3 is the exit area of the venturi. The major parts in thewater flow path which cause the significant pressure loss are thefully open ball valve and the venturi. The pressure loss coefficientfor such a ball valve is 0.05. The pressure loss coefficient of theventuri is the sum of pressure loss coefficients of entrance ofventuri (Sudden Contraction), exit of venturi (Sudden Expansion),converging part (Gradual Contraction) and diverging part (GradualExpansion). These coefficients are tabulated in Table 1 [9,10].

For some downstream to upstream pressure ratios, the throatpressure does not reach its saturated value. The Bernoulli’s

408 H. Ghassemi, H.F. Fasih / Flow Measurement and Instrumentation 22 (2011) 406–412

Fig. 1. The breaking pressure ratio versus A3/Ath for different Keffective .

equation between the entrance and throat of the cavitating venturican be written as

P2 +12ρ2V 2

2 = Pth +12ρthV 2

th (5)

where V2 and P2 are the velocity and pressure at the entrance andVth is the velocity at the throat of venturi. Also, themass flow rate ofthe venturi is defined according to Eq. (4). Therefore, the continuityequation is derived as the following.

ρ2V2A2 = ρthVthAth = m (6)

whereA2 is the entrance area of the venturi. Combining Eqs. (4)–(6)and assuming ρ2 = ρth = ρ, the pressure in the throat of theventuri can be calculated as follows.

Pth = P2 +

[A3A2

2−

A3Ath

2]

(1 + Keffective)(P2 − P3). (7)

Here, it is assumed the pipe friction losses from 1 to 2 arenegligible; therefore P2 ≈ P1. By dividing both sides of Eq. (7)by P2 and rearranging the equation, the downstream to upstreampressure ratio can be expressed by the Eq. (8).

P3P2

= 1 +(1 + Keffective)[A3A2

2−

A3Ath

2]

1 −PthP2

. (8)

Whenever the throat pressure closes to saturated pressure, it canbe assumed the Pth/P2 is negligible. Note that a typical value forPth is a few kilopascals and for P2 is 100 kPa. Therefore, the Eq. (8)reduces to the following.

P3P2

= 1 −(1 + Keffective)[A3Ath

2−

A3A2

2] . (9)

The above equation expresses the pressure ratio resulting incavitating flow in the throat. It depends strongly on the geometricalfeature of cavitating venturi, which is represented by the arearatios. Also, Keffective plays an essential role, which also depends onthe geometry of the cavitating venturi as well as the flow regime.This pressure ratio can be interpreted as a breaking pressureratio. For ratios greater than the break ratio, the flow remainsunchoked in the throat. For lesser values, the pressure at throat isthe saturationpressure of the liquid and themass flow rate remainsindependent of downstream pressure.

Eq. (9) shows the effects of two parameters on the breakingpressure ratio; flow losses and venturi’s geometry. Assuming theunit value for A3/A2, Eq. (9) is represented in Fig. 1. The figureshows the strong effect of Keffective on the breaking pressure ratio.It also shows, decreasing the throat area (increasing the A3/Ath),increases the pressure ratio. However, by decreasing the throat

Table 2Dimensions and mass flow rates of venturis at P1 = 30 bar.

Venturis m(kg/s)

Dth (mm) Dinlet (mm) Dexit (mm) Nominal Re

1 1.5 5 11 11 4300002 0.37 2.5 5 5 2120003 0.06 1 3 3 86000

area, pressure reduces and subsequently Keffective increases andresults in decreasing the pressure ratio. For small throat diameters,the total pressure loss is expected to increase and the breakingpressure ratio to decrease. Eq. (9) describes that in spite of smallthroat diameter and large value of losses, the breaking pressureratio remains nearly unchanged.

3. Experimental setup

3.1. Test equipment

A test rig has been designed, manufactured, and set up toexamine the behavior of the cavitating venturis. Fig. 2 shows theschematic of the current setup. It mainly contains the followingcomponents: Compressed air tanks, pressure regulator, 3-passvalve, water tank, manual ball valves, venturi, and plenum for thedischarging of water, compressed air capsule, and turbine typeflowmeter.

The water tank is pressurized using compressed air. Theregulator adjusts the pressure level in the water tank. The 3-passvalve connects the compressed air tanks to thewater tank. It is alsoused to relieve pressure from water tank when needed. The handvalve 1 is responsible for establishing and stoppingwater flow. Thecompressed air capsule is used to change the discharge plenumpressure via hand valve 2. Hand valve 3 discharges pressure forcollecting the water.

Three pressure transducers are used to record the pressure atthree different locationswhich are shown in Fig. 2. The transducersare selected fromTML companymodel PWF-PA. The instantaneousrate of flow is measured by using the turbine flowmeter type GLflow LX/NPT.

3.2. Venturis

Three cavitating venturis have been designed and built tostudy the effect of cavitating venturi size on its performance ina wide range of mass flow rates. A schematic drawing of theventuri is shown in Fig. 3. Three amounts of mass flow rates as1.5 kg/s, 0.37 kg/s and 0.06 kg/s are chosen at a water pressuretank (P1) equal to 30 bar. Throat diameters are calculated by usingEq. (2) and mentioned mass flow rates. The calculated diametersare 5 mm, 2.5 mm, and 1 mm, respectively. The entrance diameteris chosen to be equal to the exit diameter. The converging nozzleangle of 15° and diverging nozzle angle of 7° are selected forminimum pressure losses [5]. Finally in order to have steady flowat the entrance and exit of venturi, two pipes with a length of 5 cmare added to both sides of venturi. The diameter of each additionalpipe is selected to be the diameter of the venturi’s exit. These pipesare connected to the 1/2 inch pipeline.

Reynolds number for venturis based on its throat diameter andcavitation number can be defined as

Re =ρVthDth

µ=

4mπDthµ

(10)

k =P1 − PthP1 − P3

(11)

where Dth is throat diameter and µ is the viscosity of water. Thenecessary information about venturis is presented in Table 2. Thesmallest venturi has a 1 mm diameter. Such a small size besides

H. Ghassemi, H.F. Fasih / Flow Measurement and Instrumentation 22 (2011) 406–412 409

Table 3Specification of different experimental conditions for venturi 1.

P1 = 6 bar P1 = 11 bar P1 = 16 bar P1 = 20 barP3 (bar) k P3 (bar) k P3 (bar) k P3 (bar) k

0.765 1.140 0.814 1.077 0.800 1.051 0.774 1.0393.501 2.388 0.816 1.077 0.805 1.051 0.775 1.0393.541 2.427 3.861 1.536 6.331 1.651 9.501 1.9024.801 4.978 4.481 1.683 6.351 1.655 10.041 2.0054.950 5.684 6.881 2.663 9.771 2.564 13.821 3.232

7.441 3.082 9.911 2.622 15.112 4.0858.891 5.201 12.691 4.826 16.730 6.1079.050 5.625 12.831 5.039 17.130 6.958

14.270 9.230 19.680 62.40114.310 9.449

Fig. 2. Schematic of the experimental setup.

Fig. 3. Schematic of venturi.

two larger diameters allows us to study the effect of cavitatingventuri size on its performance. These sizes cover a wide rangeof Reynolds numbers, from 43000 corresponding to the minimummass flow rate of the smallest venturi to 460000 for themaximummass flow rate of the largest one.

4. Experimental method

To investigate the behavior of venturis with different sizes,three sets of experiments have been conducted. In the first set,the mass flow rates have been measured at different upstreampressures, while the downstream pressure (pressure of plenumin Fig. 2) was fixed. In this set, Eq. (2) has been examined. Inthe second set, for a different but constant upstream pressure,the flow rates have been measured for different downstreampressures. Using this type of experiment, the dependency offlow rate to downstream pressure has been investigated. In thethird set, providing a constant upstream pressure, a dynamicallyvariable pressure has been applied to the downstream. This kindof experiment clarifies the performance of a venturi as a passiveflowcontrol device. Dynamic variable pressure for the downstreamhelps to examine the response time of the venturi with respect topressure fluctuations.

4.1. Different upstream pressures versus constant downstream pres-sure

In the first set of experiments, the pressure of plenum P3was kept constant at 1 bar. The pressure of water tank P1 orupstream pressure, was set to different values in different tests.

The experiment was done for all three venturis. At first, plenum ispressurized by a compressed air capsule and hand valve 2. Duringa certain time, by using hand valve 1, the water flows into the pipeand passes through the flowmeter and venturi, and discharges intothe plenum. The time durations of each experiment for venturis1 to 3 are 10 s, 11 s, and 22 s, respectively. The longer durationfor venturi 3 is needed due to its low mass flow rate. Finally, theplenum is discharged and thewater inside is collected. The averagemass flow rate is calculated simply by dividing water mass by testduration.

4.2. Different downstream pressures versus constant upstream pres-sure

In the second set of experiments, upstream pressures weredifferent but constant and downstream pressures were differentfor each individual test. The experiment was done in the samemanner as stated in Section 4.1. The time durations of experimentsfor venturis 1 and 2 are 11 s and 14 s, respectively.

The upstream absolute pressures, downstream absolute pres-sures, cavitation number and Reynolds number are tabularized inTables 3 and 4, for venturis 1 and 2, respectively. Each set of P3, kand Re in a row is a separate experiment.

4.3. Dynamically variable downstream pressure

In the third set of experiments, a constant upstream pressureand a dynamically variable downstream pressure are applied. Thevariable downstream pressure is provided by using hand valve 2and a compressed air capsule. Then themass flow rate of water hasbeen measured by a flowmeter. Also, the instantaneous flow ratesare recorded on a personal computer for processing purposes.

5. Results and discussion

In order to effectively describe the experimental results, first,variation of mass flow rate versus upstream pressure has beenconsidered. Then, dependency of mass flow rate to downstream

410 H. Ghassemi, H.F. Fasih / Flow Measurement and Instrumentation 22 (2011) 406–412

Table 4Specification of different experimental conditions for venturi 2.

P1 = 10 bar P1 = 15 bar P1 = 19 bar P1 = 22 barP3 (bar) k P3 (bar) k P3 (bar) k P3 (bar) k

0.913 1.097 0.894 1.061 1.047 1.057 0.940 1.0433.984 1.657 5.511 1.577 5.005 1.355 5.325 1.3174.203 1.720 8.126 2.178 9.069 1.910 8.855 1.6716.911 3.227 11.270 4.013 13.348 3.356 12.171 2.2356.931 3.248 12.980 7.410 15.500 5.420 14.845 3.070

18.700 6.657

Fig. 4. Comparison of mass flow rates for different venturis.

Fig. 5. Mass flow rate versus downstream pressure for venturi 1.

pressure has been investigated. Finally, performance of a venturias a passive flow control device and the response time of a venturiin supplying the system has been studied.

5.1. Different upstream pressure

According to the different upstream pressures, the mass flowrates of three venturis are shown in Fig. 4. In this figure, foreach venturi, there are three curves. The first one belongs to themeasured mass flow rate. The second curve shows the mass flowrate based on the theoretical values for the cavitating venturiaccording to Eq. (2). The comparison of these two curves revealsno significant difference. It means the venturi acts as a cavitatingventuri. Otherwise, the flow rate should follow Eq. (4), which isalso depicted as the third curve in Fig. 3. In spite of the wide rangeof flow rates for three venturis, the behavior and performance ofall venturis are similar. It confirms that a venturi with a throatdiameter as small as 1 mm follows the cavitating theory precisely.Here, the cavitating venturi with a diameter smaller than 1mmhasnot been examined. But Fig. 4 illustrates that such venturis performin the expected manner.

5.2. Different downstream pressure

The measured mass flow rates of venturis versus downstreampressure for different upstream pressures are shown in Figs. 5

Fig. 6. Mass flow rate versus downstream pressure for venturi 2.

Fig. 7. History of mass flow rate at dynamically variable downstream pressure forventuri 2.

and 6. The test conditions are indicated in Tables 3 and 4. Inthese figures, results of the experiments with a cavitation numberless than 5 are shown. The highlighted values are discussed inSection 5.4. The performances of venturis indicate their capabilityof passing a constant mass flow rate even though downstreampressure changes. This constant flow rate is determined byupstream pressure.

5.3. Variable downstream pressure

In former experiments, a constant pressure has been imposedon the plenum in each test. The pressure depends on desired testcondition. It is desirable to use a cavitating venturi as a passiveflow controller to keep the mass flow rate constant. Therefore, itis necessary to investigate its behavior under variable downstreampressure. A sample of such pressure is shown in Fig. 7 for venturi 2.The variable downstream pressure has no specific pattern. As isshown in Fig. 7, the mass flow rate and upstream pressure areconstant. This confirms the venturi plays its controlling role inthe case of dynamically variable downstream pressure. In thisexperiment, the downstream pressure P3 is always much less thanupstream pressure P1.

Fig. 8 shows another sample of pressure-flow rate in a testwith venturi 3. As is depicted in this figure, sometimes severalvariations happen in mass flow rate. The upstream pressure is

H. Ghassemi, H.F. Fasih / Flow Measurement and Instrumentation 22 (2011) 406–412 411

Fig. 8. History of mass flow rate at dynamically variable downstream pressure forventuri 3.

Fig. 9. Nondimensional parameters in a histogram for venturi 3.

about 21 bar. The mass flow rate of the venturi is constant whilethe downstream pressure is less than about 16 bar. When thedownstream pressure increases to further values, the mass flowrate falls severely. While the downstream pressure decreases toless than 10 bar, the mass flow rate increases to reach to itsprevious constant level. It shows that the mass flow rate of theventuri is not completely independent of the downstream. Asa matter of fact, when the downstream pressure is sufficientlyclose to upstream pressure, the mass flow rate of the venturi willdecrease. The ratio of downstream pressure to upstream pressuredetermines the state of the venturi whether or not the flow ischoked. This experiment clarifies the performance of the venturiencountering a variable downstream pressure which causes thepressure ratio to exceed 0.8.

5.4. Cavitating venturi performance range

Fig. 8 shows that the venturi does not pass a constant massflow rate for the whole range of downstream pressure at a fixedupstream pressure. To investigate the range of P3, the flow ratethat can be fixed by the venturi, Fig. 8 is represented usingnon-dimensional variables as Fig. 9. The mass flow rate becomesdimensionless by dividing it to the theoretical mass flow rateobtained from Eq. (2). Also, the downstream pressure, P3 is turnedto dimensionless form by the upstream pressure, P1. Fig. 9 showsthe mass flow rate ratio is constant, while the pressure ratio P3/P1is less than about 0.8.

As is depicted in this figure, mass flow rate ratio decreaseinitiates at t1 = 7 s, when the pressure ratio P3/P1 goes beyond 0.8.The second important point, t2 = 12.5 s, indicates the time whenthe pressure ratio falls down from 0.8. The flow behavior betweent1 and t2 is not at cavitating condition. Therefore, the venturi actslike a simple non-cavitating venturi, or an orifice. Different flowregimes may happen under this variable condition. About 1 s aftert2, the mass flow rate ratio reaches to its previous constant rate.

Fig. 10. Performance of venturi 1.

Fig. 11. Performance of venturi 2.

This time elapses to compensate the upstreamwater supply delay.This delay time depends on upstream component characteristics.Similar conditions are observed for venturis 1 and 2.

Referring to Tables 3 and 4, the experiments had beenconducted under different P3 and P1. Some test conditions inthese tables are highlighted. These conditions indicate the non-cavitating flow in the venturis. To show this fact, all data obtainedfrom the experiments for each venturi are scattered as pairs of non-dimensional mass flow rate and non-dimensional downstreampressure, in Figs. 10 and 11. These figures obviously show thatwhile the pressure ratio is less than 0.8 the mass flow rate ratiomexp/mthe remains constant. Also, when the pressure ratio is about0.8 there is a break in the mass flow rate ratio. For a ratio of morethan about 0.8, pressure in the throat does not reach saturationpressure and no cavitation occurs.

The pressure ratios that the mass flow rate break happens canbe predicted using Eq. (9). To examine this equation for venturi1 and 2, the pressure ratios are 0.87 and 0.81, respectively. Forcalculating these values, the geometrical properties of venturis inTable 2 are used. The breaking pressure ratio of about 0.8 has beeninvestigated by Dale and Hermann [6]. It is shown that the use of acavitating venturi provides a substantially stable liquid flow rate atReynolds numbers of about 60000 or less and a pressure recoveryof at least 80%.

The difference between theoretically calculated and experi-mentally obtained mass flow rates might come from several re-sources. The effect of the boundary layer on the decrease of thepractical throat diameter causes a decrease in actual mass flowrate. Also, to calculate the theoretical mass flow rate, the nominaldiameters of the venturi’s throat are used. The actual diametermaydiffer slightly, which significantly appears via its square in Eq. (2).

6. Application of a cavitating venturi as a flowmeter

Fig. 4 shows that the experimental and the theoretical massflow rate for all cavitating venturis are very close and conforming.

412 H. Ghassemi, H.F. Fasih / Flow Measurement and Instrumentation 22 (2011) 406–412

Also, Figs. 10 and 11 clearly show that in a wide range of pressureratios, the non-dimensional mass flow is constant. Therefore, thecavitating venturi can be used as a flow meter. Usually twopressure transducers are used for measuring the mass flow rate bya venturi. When cavitation occurs in a venturi, the flow rate can bemeasured by using only one pressure gauge.

The mass flow rate of the cavitating venturi is expressed byEq. (2). For a specific venturi, the throat area is constant. Also,the vapor saturation pressure is a function of the temperatureof flowing liquid. Its value may be negligible in comparison withupstream pressure. Also, whenever the ratio of downstream toupstreampressures is less than 0.8, the venturi provides a constantmass flow rate. Under such a condition, Eq. (2) can be written as

m ∼= βP1 (12)

where β = Ath√2ρ. Deviation of the experimentally obtained

mass flow rate from theoretically predicted values can be shownby a discharge coefficient. The discharge coefficient is defined asthe ratio of the experimental mass flow rate to a theoretical massflow rate. Therefore, the mass flow rate of a cavitating venturi canbe expressed as follows

m = CDβP1 (13)

where CD is the discharge coefficient. Generally, the dischargecoefficient is a function of flow conditions that are determinedby working pressure and the venturi’s geometry. Therefore, it isa function of venturi design andmanufacturing specifications suchas its dimension, converging and diverging angles, surface quality,etc. The roles of these parameters are embedded in the pressureloss coefficient. This pressure loss is used in Eqs. (4) and (5). Thedischarge coefficients for three venturis are specified by Figs. 9–11.These venturis have almost the same discharge coefficients,approximately 0.94. For simplicity, it is recommended to use asuitable upstream pressure instead of P1 in Eq. (13). The mostsuitable pressure is P2, which is the pressure of a point just beforethe venturi in Fig. 2.

Therefore, using only one pressure transducer for a simplecavitating venturi, the mass flow rate can be calculated precisely,applying Eq. (13). The cavitating venturi not only helps to measurethe mass flow rate, but also controls it.

7. Conclusion

Three venturis have been designed and built with a throatdiameter of 1–5 mm. These venturis have been examined in a

test rig with constant upstream pressure and variable downstreampressure to investigate the behavior of cavitating venturis withdifferent sizes. The results emphasize on the following concludingremarks:

1. The range of downstream pressure ratio resulting to cavitationflow is predictable by geometrical features and the pressureloss coefficient of venturi. An expression is derived to calculatethe pressure break (P3/P2 or P3/P1) for starting cavitationoccurrence.

2. In spite of very small throat diameter (1 mm), the cavitatingventuri acts in the theoretically expected manner. It isconcluded that the diameter of the throat has no significanteffect on the performance of the venturi. Therefore, any amountof liquid flow rate can be controlled by using cavitating venturi.

3. It is shown experimentally when the downstream pressurechanges severely with time; the cavitating venturi keeps theflow rate constant after some delay.

4. The excellent agreement between the theoretically calculatedand experimentally obtained mass flow rates introducesthe cavitating venturi as a suitable instrument for flowmeasurement. Using a discharge coefficient, it is possible to findthe mass flow rate in terms of the upstream pressure.

Acknowledgments

This work was fully supported by Engineering ResearchInstitute. The authors wish to thank Mr. Akbari Baseri and hiscolleagues.

References

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