Chemical Engineering Science 58 (2003) 3973–3983www.elsevier.com/locate/ces

A class of discontinuous dynamical systems IV.A laboratory air–water system

Kannan M. Moudgalyaa ;∗, Shivesh Kumar Singha, K. P. Madhavana, Gaurav Jainb

aDepartment of Chemical Engineering, Indian Institute of Technology, Powai, Mumbai 400 076, IndiabDepartment of Aerospace Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India

Received 14 August 2002; received in revised form 9 May 2003; accepted 21 May 2003

Abstract

An air–water experimental system consisting of two inlets and one outlet is constructed and characterised. It reaches the state of slidingmode, or equivalently, two phase slug 4ow. The linear hydraulic model proposed in the literature is adequate to describe it. Experimentaldata are used to tune this model. The resistance to the 4ow of air through the outlet valve during the two phase 4ow is much larger thanthat when air alone 4ows out. At the operating range, the resistance to water 4ow is not a6ected by the presence of air.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Discontinuous systems; Two phase 4ow; Slug 4ow; Sliding motion; Long time behaviour; Optimisation

1. Introduction

Recently Moudgalya and co-workers (Moudgalya &Ryali, 2001; Moudgalya & Jaguste, 2001; Moudgalya,2001) have studied a class of discontinuous dynamical sys-tems. They have demonstrated that when a gas and a liquidare sent into a reactor and withdrawn through one outlet,discontinuous behaviour will result and that the system willreach a state of sliding motion. An industrial High DensityPolyethylene (HDPE) reactor has been shown to belong tothis class.

The current work is concerned with a laboratory air–watersystem with two inlets and one outlet. The objectives ofundertaking this study are the following:

(1) To verify the predicted sliding behaviour in a labora-tory setting. Note that the HDPE reactor reported byMoudgalya and Ryali (2001), because of the industrialrequirements, had some limitations. For example, onecould not “see” the sliding behaviour as the reactor wasmade of stainless steel with no ports for observation.

The reactor studied in this work is made of Plexiglasand hence one can observe the contents of the system.

∗ Corresponding author. Tel.: +91-22-576-7213;fax: +91-22-572-6895.

E-mail address: kannan@che.iitb.ernet.in (K. M. Moudgalya).

(2) To characterise the performance of the system and toidentify suitable operating regimes.

(3) To understand the e6ects of a two phase out4ow in thefunctioning of the system. For example, when a phase(say gas) has to compete with another (say liquid) for4ow through a pipe, the 4ow rate could be lower thanthe maximum 4ow rate if it were to 4ow alone. This inturn could a6ect the in4ow to the system as well.

(4) To explore the possibility of using this system to studyair–water slug 4ow, a current topic of research (Triplett,Ghiaasiaan, Abdel-Khalik, LeMouel, & McCord, 1999;Yu, France, Wambsganss, & Hull, 2002).

(5) To explore the viability of running this as a “small”experimental system in a student curriculum so as toprovide the students with an insight into the dynamicsof a multivariable process with a novel arrangement forout4ow of gas and liquid.

This work details the result of a study undertaken with theabove aims. It is shown that the sliding behaviour or equiv-alently, slug 4ow, indeed results. Through several experi-mental studies, the predictions of Moudgalya and Ryali havebeen shown to be true. The required improvements in theirmodel are also identiIed and implemented to match the ob-served behaviour of the system.

This report is organised as follows. A detailed descriptionof the experimental system and the characterisation studies

0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0009-2509(03)00261-6

3974 K. M. Moudgalya et al. / Chemical Engineering Science 58 (2003) 3973–3983

are given Irst. Various attempts made to tune the model tomatch the experimental data are given next. The modiIca-tions required to the model of Moudgalya and Ryali (2001)are given next. The Inal sections are devoted to limitationsand the improvements required to the experimental deviceand conclusions.

2. An experimental air–water system

The experimental system is constructed on the lines ofthe ideal gas liquid system studied by Moudgalya and Ryali(2001), see Fig. 1. If the liquid level is above the outlet tube(known as dip tube) opening, the liquid comes out, else thegas comes out. The essential features of this system are

• The feed consists of both liquid and gas.• The gas and the liquid phases are both present in the tank.• There is only one outlet.

Suppose the liquid level is such that the outlet tube dipsinto the liquid inside the reactor. Then the liquid will comeout of the reactor through the tube. The incoming gas willgo to the gas space above the interface. The pressure in thetank will go up, with a consequent increase in the liquidout4ow from the tank. When the outlet liquid 4ow rate isgreater than that in the feed, the liquid level in the reactorwill drop and eventually the gas will come out. When thegas comes out of the tank, the pressure will start dropping.After a short duration of time, the liquid level will rise dueto the continued liquid feed, and will seal the tube opening,resulting in liquid 4ow once again. This process will berepeated.

At room temperature and pressure, air can be assumedto be ideal and insoluble in water; also the amount of wa-ter vapour can be assumed to be negligible. Moreover, han-dling of water and air is easy. Because of these reasons, anair–water system is constructed to verify the predictions ofMoudgalya and Ryali (2001), valid for an ideal gas–liquidsystem.

A schematic of the air water system is given in Fig. 2.It consists of a sealed Plexiglas tank with two inlets andone outlet, a compressor and a receiver, a pump and a wa-ter reservoir and associated instrumentation and control ele-ments. The chief component of the rig is a sealed Plexiglastank of volume 3:5 l with two inlets and one outlet. A wa-ter reservoir acts as a source of water for the system. Thewater 4ows from the reservoir through two 50 W monoblocpumps in series, into the sealed Plexiglas tank and back intothe reservoir through the outlet. A miniature air compressorwith a 7 l receiver is used as a source of air. Note that thevolume of the receiver is twice that of the Plexiglas chamber.

The inlet labelled I1 allows water to 4ow into the glasstank while the inlet I2 feeds compressed air to the sealedglass tank. The only outlet in the tank is the dip tube sus-pended midway into the tank. The inlet I2 is deliberately

PIC PIC

GasLiquid

Liquid

GasLiquid

Gas

Fig. 1. Schematic of the discontinuous system.

placed near the top of the tank to prevent accidental 4ow ofwater into it and through it, into the compressor. The waterlevel in the tank is not expected to rise high enough to reachthe end of I2 during the normal operation of the rig.

The control elements installed in the system are the threenormally closed proportional solenoid valves labelled CV1,CV2 and CV3. CV1 controls the 4ow rate of water intothe tank, CV2 controls the air 4ow rate and CV3 controlswhatever phase may be going out of the tank at any giveninstant of time. Each valve accepts an input voltage of 0–30 VDC. The valves have a dead band of around 5 V afterwhich they start opening and are fully open at 30 VDC. Theresponse time of the valves has been found experimentallyto be around one second.

The valves CV1, CV2 and CV3 are connected to endcontrol elements EC1, EC2 and EC3, which are essentiallyidentical versions of 6 × (gain = 6) voltage ampliIers withcurrent boost and isolation. The end control elements acceptan input of 0–5 VDC and output a proportionally highervoltage in the range of 0–30 VDC.

The end control elements are directly connected to theDAC channels of the Data Translation I/O card DT2811installed on the PC. DT2811 is a general purpose I/O cardwith 16 single ended or eight di6erential analog inputs andtwo analog outputs.

Flow transmitters FT1 and FT2 are turbine type 4ow sen-sors that give a 0–5 VDC output proportional to the 4owrate of water and air, respectively. Both 4ow sensors aredirectly connected to the I/O card DT2811.

Pressure transmitter PT1 incorporates a piezo resistiveelement on diaphragm type of sensor with inbuilt electronicsso that the output is an industry standard 4–20 mA signal.As the 4–20 mA signal is not suited for the ADC of the I/Ocard DT2811, a converter C1 is placed between the two. C1is a current to voltage converter which converts a 4–20 mAcurrent signal to a proportional voltage signal suitable forinput to the I/O card.

The PC shown in the Fig. 2 is an Intel Pentium II basedsystem running Redhat Linux 6.2. A home grown linuxCOMEDI (Schleef, 2002) based software is used for dataacquisition and screen display.

Additional data on the instrumentation system is givenin Table 1. The other design and operational variables aregiven in Table 2.

K. M. Moudgalya et al. / Chemical Engineering Science 58 (2003) 3973–3983 3975

Fig. 2. Layout of the rig.

Table 1Instrument data

Item Make Range Accuracy Output

Air 4ow McMillan 2–10 l=min ±3% 0–5 V DCWater 4ow McMillan 0–2:5 l=min ±3% 0–5 V DCPressure transmitter Keller 0–5 bar ±0:5% 4–20 mAValve (air) Aalborg 0–100 l=minValve (water) Aalborg 0–2:873 l=min

Table 2Parameter values

Parameter Value

V 3:5 lVd 2:2 lT 300 KPout 1 atm

Here V is the volume of the chamber and Vd is the volumein the chamber below the dip tube.

3. Mathematical model

Let FG and FL be the constant molar feed 4ow rates ofair and water, respectively. Let G and L, respectively, be themolar air and water out4ow rates. The holdups in the system

are MG moles of air and ML moles of water. The followingmodels are suggested by Moudgalya and Ryali (2001).

If ML=�L¿Vd (liquid model)

dMGdt

= FG; (1)

dMLdt

= FL − L; (2)

MGRTP

+ML�L

= V; (3)

L= kLx(P − Pout); (4)

Else If ML=�L¡Vd (gas model)

dMGdt

= FG − G; (5)

dMLdt

= FL; (6)

3976 K. M. Moudgalya et al. / Chemical Engineering Science 58 (2003) 3973–3983

MGRTP

+ML�L

= V; (7)

G = kGx(P − Pout): (8)

Here �L is the density of the liquid in moles per unit vol-ume. The system pressure and temperature are denoted byP and T , respectively. The valve opening is denoted byx; 0¡x6 1. The valve coeOcients for water and air 4owsrespectively are kL and kG, which, need not be equal.

Moudgalya and Ryali (2001) have shown that this systemwill reach a state of sliding motion from all starting points.In this particular case, sliding approximately refers to thestate of both air and water leaving the system together.

4. Characterisation of the experimental set-up

For characterisation of the experimental set-up and iden-tiIcation of a useful operating range for the system, a seriesof experiments were conducted by varying the operatingvariables of the system. Water and air 4ow rates were variedup to 3.86 and 0:014 mol=s, respectively. Experiments werealso conducted for various valve positions. In each case,the experimental run covered dry startup to sliding modecondition. A typical proIle of the response of the system isshown in Fig. 3. The responses can be characterised by tworegimes—initial regime of response till the water levelreaches the dip tube and the Inal regime of response,characterised by increase in pressure and decrease in bothwater and air 4ow rates till a stable sliding mode is attained.The following patterns were observed:

(1) Liquid 4ow rate has a dominant e6ect on the pressureand the air 4ow rate at sliding mode condition of oper-ation. As liquid 4ow rate is increased, the air 4ow ratedrops drastically.

(2) At moderate to high levels of water 4ow rate, the cham-ber pressure and relative resistance to air 4ow rateincreases.

(3) With high 4ow rates of water, the e6ective operatingrange of the outlet valve becomes narrow a6ecting theoperability of the process.

(4) The chamber pressure is a6ected by the valve positionand the settings of the in4ow valves in air and wa-ter supply lines. For Ixed settings of these valves, thechamber pressure decreases with increase in outlet valveopening.

(5) Constraints on air 4ow rates:(a) High air 4ow rates: Fig. 4 shows the e6ect of high

4ow rates of air in the system. A dip in air 4owrate occurs when sliding starts, as the compressortakes 10–15 s to create adequate driving forces tomeet the higher pressure head demands of the sys-tem, to sustain such high 4ow rates. This dip hasbeen avoided by operating the system at or below0:012 mol=s.

Fig. 3. Typical proIles of pressure and 4ow rates of air and water.

(b) Low air 4ow rates: Fig. 5 shows the e6ect of low4ow rates of air in the system. In this case, the com-pressor works for 5 s and then turns o6 for about15 s as the air requirement is small. The compressorcuts o6 when the pressure in the receiver reaches thepreset safety limit of 1:8 atm (gauge), and switcheson again when pressure in the receiver falls be-low the preset lower limit of 1:5 atm (gauge). Thiscauses the compressor cycle characteristics to besuperimposed on the normal characteristics of thesystem. At higher air-4ow rates, the compressor isalways on, and hence compressor cycles are absentin those data sets. Flow rates below 0:0025 mol=sare excluded from the operating region, to stay clearof the region of rapid compressor switching.

K. M. Moudgalya et al. / Chemical Engineering Science 58 (2003) 3973–3983 3977

Fig. 4. Air 4ow shows a dip at high 4ow rates.

Fig. 5. Compressor cycles at low air 4ow rates.

(6) Constraints on water 4ow rates:(a) High water 4ow rates: If high 4ow rates are used,

the water may rise and enter the compressor linethrough the air inlet pipe.

(b) Low water 4ow rates: Minimum water in4ow ratesof 0:93 mol=s have to be maintained to make surethat the pumps are not unduly stressed when thesystem pressure goes up, and the possibility ofreverse-4ow is ruled out. Water 4ow rates belowthis value cause the pumps to stop or stall duringthe experiment and are therefore avoided, to avoiddamage to the equipment.

Based on the observations made above, the operating rangeof the system was Ixed to cover a 4ow rate of water up to3 mol=s and air 4ow rate of up to 0:012 mol=s.

Manipulation of the outlet valve has been proposed as away to control the chamber pressure in this class of systems(Moudgalya & Ryali, 2001; Moudgalya & Jaguste, 2001).In the current experimental set-up, this objective was notachieved because of problems with the outlet control valve.While the valve worked well with continuous 4ow of eitherair or water, its performance on pulsating 4ow of air andwater during sliding mode was unsatisfactory.

Fig. 6. Experimental data for fully open outlet valve, Runs 1–3.

Several experiments were conducted with the outlet valveposition constant during the course of each run. In Table 3,the Irst three runs refer to the experiments in which the outletvalve was kept fully open. In these experiments, the initialair 4ow rate was about constant, while the water 4ow ratevaried. The proIles corresponding to these runs are given inFig. 6. In Run 1, at about 93 s, the pressure starts building upas water touches the dip tube with the consequent stoppageof air out4ow. After this, the water in4ow rate and air in4owreduce from their initial levels because of the rising pressureof the chamber against which the pump and the compressorhave to operate. This phenomenon can be observed in allthe runs.

Note that the initial air 4ow is constant in all three runs.The initial water 4ow, however, increases as the run number

3978 K. M. Moudgalya et al. / Chemical Engineering Science 58 (2003) 3973–3983

Table 3Pressure and 4ow rates, initial and during sliding motion, with constant outlet valve position

Run Valve FL FG FL FG Pno. position (initial) (initial) (Inal) (Inal) (atm)

(V) (mol/s) (mol/s) (mol/s) (mol/s)

1 30 1.29 0.0118 0.96 0.0086 2.02 30 1.43 0.0116 1.11 0.0077 2.23 30 1.54 0.0117 1.23 0.0068 2.4

4 26.4 1.88 0.012 1.15 0.0080 2.25 13.8 1.88 0.012 1.04 0.0067 2.46 12.6 1.85 0.012 0.98 0.0060 2.57 11.4 1.86 0.012 0.75 0.0050 2.8

increases. With an increase in water 4ow rate, the pressureincreases while sliding with a consequent drop in air 4owrate. For example, the air 4ow rate in Run 3 during slidingmotion is smaller than that in Run 1 even though the initialair 4ow rates are the same in both of them.

Runs 4–7 in Table 3 have been conducted with the valveclosed to di6erent extents. In these experiments, however,the in4ow rates of both air and water are constant. As thevalve gets closed more, the sliding mode pressure increaseswith a consequent decrease in the in4ow rates of both airand water. The proIles for Runs 4–6 have been shown inFig. 7 and those for Run 7 in Fig. 3. In all the runs (1–7),the experiments start with an empty chamber.

5. Optimisation of system parameters

The results obtained from the experimental runs describedin the earlier section can be used to obtain the best estimatesof the unknown parameters in the model. The maximumsource of uncertainty is in the out4ow model described bythe hydraulics when both liquid and gas 4ow under slidingmode conditions. Various 4ow models have been tried inan attempt to get the most appropriate model for the out-4ow section. In the models, kG and kL have been treated asunknown parameters. Runs 1–3 have been used to tune themodel parameters. The following objective function, whichis the sum of squares of di6erence between the model pre-diction and experimental data, has been minimised.

minkG;kL

f(P) =∑

(Px(ti) − P(ti))2; (9)

where Px denotes experimentally measured pressure and Pcorresponds to model prediction. Note that the sum is overall the discrete points ti at which the experimental data areavailable.

Two loop constructs have been used to solve this problem.The inner loop is concerned with the generation of the objec-tive function for a chosen set of parameters kG and kL. This isdone as follows. The mathematical model is Irst integratedto get the transient response P(ti), where ti are the discretetime points at which experimental data Px(ti) are available.

Fig. 7. Experimental data for partly open outlet valve, Runs 4–6.

K. M. Moudgalya et al. / Chemical Engineering Science 58 (2003) 3973–3983 3979

Select Model

Parameters

Can

till final

time?

yes

stop

yes

Hasoptimumreached? no

Getobjectivefunction

noLSODE integrate

Fig. 8. Flowchart of Box-Complex optimisation technique with LSODEgenerated objective function.

The state of the art ODE solver LSODE (Radhakrishnan &Hindmarsh, 1993) is used to solve this system of equations.If for whatever reason, the model cannot be integrated untilthe Inal time, the trial point (kG; kL) is considered as an in-valid point and rejected. If the integration is successful, theobjective function f(P), given in Eq. (9), is evaluated.

In the outer loop, the optimiser generates random points(kG; kL), passes them to the inner loop and receives the ob-jective function. The outer loop generates trial points in anintelligent manner using the Box-Complex code developedby Malik (1979). This procedure is continued until the op-timum is reached. The 4owchart in Fig. 8 summarises this

Fig. 9. Comparison of linear model generated pressure with experiment.

algorithm. This is the same as the methodology adopted byMoudgalya and Jaguste (2001).

The result of Box-Complex minimisation is given next.The tuned values are kL=1:114 and kG=0:161. The objectivefunction corresponding to these values is 13.14.

In Fig. 9, the model generated pressure proIle P for thisoptimal (kG; kL) set is compared with the experimental pro-Ile Px. This is the best It that can be obtained through thechosen model.

As the match is not good, a di6erent hydraulic model isattempted: a square root relation is now used for the out-4ow. That is, Eqs. (4) and (8) are replaced by the followingrelations:

L= kLx√P − Pout ; (10)

G = kGx√P − Pout : (11)

The tuned values now are kL = 1:25 and kG = 0:043. Theobjective function is 8.62. Although the objective functionis smaller than before, it is still large. The model predictedproIle is similar to the one obtained in Fig. 9.

Note that both of these simple linear and square root mod-els do not seem to work. In some of the simulations (notshown here), it was seen that the linear model Itted theearly part of the pressure response better than the squareroot model. On the other hand, as shown above, the objec-tive function of the square root model has come out to bebetter. As a result, initially a linear model is used and whenthe pressure starts building up, a power other than 1 is used.In other words, the following hydraulic model is attempted:

if t ¡ tp thenL= kLx(P − Pout)G = kGx(P − Pout)

elseL= kLx(P − Pout)a

G = kGx(P − Pout)a

end if

Note that tp is the time when pressure starts building upin the chamber, which in turn corresponds to the time when

3980 K. M. Moudgalya et al. / Chemical Engineering Science 58 (2003) 3973–3983

water seals the dip tube. This corresponds to 93:4 s in Run 1.Di6erent values in the range of 0.5–3 have been tried forthe exponent a. It has been found that for a = 2, best It isobtained. The tuned values in this case are kL = 1:11 andkG=0:192. The objective function is 5:3, which is also large.Indeed, in this case as well, the model predicted proIleshows a maximum in pressure proIle, as in Fig. 9.

Note that this model, although better than the previousones, is still not adequate. In all these models, the simulatedpressure reaches a maximum value unlike in the actual re-sponse. This indicates that although our guess that the 4owbehaviour changes just around the time sliding starts is qual-itatively correct, our modelling of it is not.

The correlations that are available to predict the pressuredrop in a two phase 4ow are complicated and they involveseveral parameters (Triplett et al., 1999; Yu et al., 2002):Voidage and frictional losses have to be calculated Irstbefore the pressure drop can be estimated. Further, entrancee6ects to the two phase section cause distinctive 4ow phe-nomena. It is not clear whether the existing correlationscould be applied to this case. In view of this, a simple modelthat lumps these e6ects on valve coeOcients is attempted.

It is felt that the modelling of air 4ow process may haveto be viewed in a di6erent light. It is necessary to identifytwo regimes of operation for the air 4ow: when the liquidlevel is below the dip tube and during the sliding motion.In the case of water, the initial rate is not governed by theoutlet valve equation. The valve 4ow equation comes intopicture when water Irst touches the initial period duringwhich only water 4ows, after which both air and water start4owing. Thus there may be two regimes for water 4ow aswell. As a result, one can think of two valve coeOcientsfor each phase—one during single phase 4ow and the otherfor sliding mode. Indeed, as the resistance to 4ow duringsliding motion is higher than that during individual 4ow, thek values during the two phase 4ow should be lower than thatduring the single phase 4ow. We get the following algorithmas a result:

if t ¡ t0 thenkG = kG1

kL = kL1elsekG = kG2

kL = kL2end if

where, t0; kG1; kL1; kG2; kL2 are the Ive parameters to betuned. Linear dependence on pressure is assumed for out-4ow, both for water and air. This model is based on the ob-servation that the resistance to 4ow out of the chamber intwo phase 4ow could be di6erent from that in a single phase4ow.

We expected this switching time t0 to be around the timewhen sliding begins. Hence for Run 1, this would be atabout 100 s. As a measure of abundant caution, however,we allowed t0 to be in the range of 80–120 s while tuning.

Fig. 10. Comparison of experimental proIle with changing k modelgenerated proIle, Run 1.

Note that this range envelops both the time when pressurestarts building up as well as the time when sliding begins.Box-Complex minimisation is attempted once again withthese Ive tuning parameters. The best It obtained from thedata of Run 1 is given in Fig. 10, in which proIles of pressureand volume of water in the chamber are given as functionsof time. The tuned values are found to be kL1 = 3:48; kG1 =0:27; kL2 = 3:05; kG2 = 0:012. The value of the objectivefunction is 0.22. The switch happens at t0 = 94:51 s. Notethat this time of switch is more or less the time water levelreaches the dip-tube = 93:4 s. As no water would have leftthe system prior to this time, kL1 is used for just about 1 s.This indicates that a single kL value can be used both duringthe two phase 4ow as well as during the time only water4ows. This can be represented by the following algorithm:

if t ¡ t0 thenkG = kG1

elsekG = kG2

end if

The valve coeOcient for liquid takes the constant valueof kL.

With this logic, the following tuned values are obtained:kG1 = 0:27; kL = 3:05; kG2 = 0:012 and t0 = 93:12 s. Thevalue of the objective function is still 0.22. We get Iguresidentical to the one shown in Fig. 10.

K. M. Moudgalya et al. / Chemical Engineering Science 58 (2003) 3973–3983 3981

Table 4Comparison of tuned values for data from Runs 1–3

Parameter Run 1 Run 2 Run 3

kL 3.05 3.36 2.85kG1 0.26 0.24 0.25kG2 0.012 0.0084 0.0067t0 93.12 80.23 71.25Objective function 0.22 0.56 0.98

Fig. 11. Comparison of experimental proIle with changing k modelgenerated proIle, Run 2.

In other words, the presence of air does not seem to a6ectthe water 4ow. This is not surprising because the water 4owis a lot more powerful compared to air and hence it muzzlesits way through.

The air 4ow, on the other hand, comes down substantiallywhen water starts competing for 4ow. The valve coeOcientfor gas during two phase 4ow is only about 4% of that whenit can 4ow uninhibited. This can easily be explained in termsof increased resistance to 4ow of air when water also 4owsout.

This minimisation strategy is applied to the data obtainedin Runs 2 and 3 as well. Table 4 compares the tuned valuesfor these cases. Notice that the kL values are more or lessthe same, while kG1 values are almost identical. The valueof kG2, on the other hand, decreases as the air 4owrate decreases. The lower the air 4ow, the larger the resis-tance it faces at the outlet valve because of the competingwater 4ow.

A comparison of pressure proIles for Runs 2 and 3 areshown respectively in Figs. 11 and 12.

An attempt was made to check if a square root hydraulicmodel would be a better candidate, with the above presentedmodel that changes both kG and kL or kG only. It was foundthat these perform worse than the linear pressure dependencemodel presented above. Thus it can be concluded that themodel presented above is the best amongst the class of suchsimple models.

Fig. 12. Comparison of experimental proIle with changing k modelgenerated proIle, Run 3.

6. Limitations/improvements in the system

(1) The out4ow valve is required to operate at low 4owrates and should be rugged to withstand the pulsating4ow at the outlet of the air–water system. Indeed, asthe 4ows occur in slugs, the load on the valve is high.Typically, heavy duty valves are available only for high4ow rates. For problems of this type, one needs heavyduty valves capable of low 4ow rates.

(2) It will be nice to have a level sensor as well in theapparatus. Unfortunately, however, the demands onthe level sensor also are high. One would like to mea-sure the levels accurate at least to 1 mm. Indeed, abetter accuracy may be required if one has to under-stand the 4ow phenomenon at the entrance to the diptube. In the current experiments, the water level to bemeasured is of the order of about 20 cm. As the levelsensors generally have a large noise problem, it may bediOcult to make them work in the range of 1–20 cmand yet measure levels of fractions of a millimeteraccurately. One possible solution is to restrict the totalwater level to a small value, say of the order of a fewcentimetre. This in turn will reduce the time availablefor initial dynamics or restrict the water 4ow range thatcan be studied. Indeed, in the current set-up, there is aninitial gush of water as soon as the pump is turned on.

(3) Higher capacity for compressor and receiver will resultin a more stable compressed air source. This will helpincrease the range available for experiments.

(4) The valves used in the rig allow bi-directional 4ow.Unidirectional valves are preferable.

(5) Volume of tubing, sensors and actuators will play amore signiIcant role if the experiments have to be con-ducted under a small volume.

7. Conclusions

The experimental study has helped understand the be-haviour of the air–water system under various conditions ofoperation. The study has shown the impact of the design on

3982 K. M. Moudgalya et al. / Chemical Engineering Science 58 (2003) 3973–3983

the operational characteristics. The single outlet dischargesystem for both water and air places constraints on the 4owrates that can be handled by the system. The experimentaland modelling results show that the water 4ow rate has adominant role in 4ow processes when both liquid and gas4ows are handled under sliding mode conditions. This willplace constraints on G=L ratios that can be attained by thesystem. Depending upon the operating conditions, the sys-tem could move from a strongly water 4ow rate dominatedprocess to one where both air and water 4ow rates have in-4uence on the pressure proIles.

The 4ow processes in the outlet section when both airand water 4ow out are complex. Visual inspection showsslugs of 4ow of water with air. Normal valve 4ow equationsare not applicable to such pulsating 4ows. Despite theselimitations, a simple model with a linear pressure drop termand two coeOcients for air 4ow, appropriate to the tworegions of operation, is seen to It the experimental modelwell. Hence for modelling purposes, these approximationscould be acceptable.

The study has also conIrmed that the model structureproposed in the earlier work is applicable to the experimentalair–water system.

The following are the contributions of this work:

(1) There are limited experimental studies in chemical en-gineering dealing with dynamic systems exhibiting slid-ing motion. This provides a comprehensive study of onesuch system.

(2) It experimentally validates the theoretical results pre-dicted earlier. The validated 4ow rate vs. pressure drophas a very simple structure involving only the valve co-eOcient. Thus this may be useful in some cases whereone does not want to use the existing correlations fortwo phase 4ow that require elaborate calculations in-volving several parameters.

(3) The research in the area of two phase 4ow and in par-ticular, slug 4ow, is not mature enough to address manyof the two-phase 4ow situations. Our work makes thefollowing contributions in this area:(a) Many of the two phase 4ow study and correlations

correspond to 4ows in pipes, packed columns, bub-ble columns, etc. This set-up has an entirely di6er-ent conIguration with its own distinctive features.This sets in certain periodicity in the input 4ows tothe two-phase section. There is very little theoreti-cal or experimental study on the 4ow behaviour ofsuch a system.

(b) A quantitative value for the valve coeOcient forsingle phase and two phase 4ows have been ar-rived at. While going from single phase 4ow to twophase 4ow, the coeOcient for air comes down byan order of magnitude; for water, this value remainsconstant.

(c) The literature discusses the great lengths re-searchers have to go in devising experimental

set-ups that can produce slug 4ow. Our apparatusis capable of producing slug 4ow at ALL 4owrates—the only requirement is that both the phasesshould be 4owing. Moreover, the time average ofindividual 4ows can be measured as the in4ow isequal to the out4ow and the in4ow can be mea-sured. Perhaps it is because of this fact that wehave got excellent match between model predic-tion and experimental results. It is well known thatthere is usually quite a bit of measurement error inexperiments involving slug 4ow.

Thus this apparatus could be a valuable tool forresearchers working in the area of slug 4ow.

Notation

FG gas feed rate, mol/sFL liquid feed rate, mol/sG gas out4ow rate, mol/s and subscript to

indicate gaskL valve coeOcient for gas 4owkG valve coeOcient for liquid 4owL liquid out4ow rate, mol/s and subscript to

indicate liquidMG gas holdup in the system, molML liquid holdup in the system, molP system pressure, atmPout outlet pressure, atmPx pressure measured in experiment, atmR gas constant = 0:082 l atm=(mol K)T temperature, Kt time, stp time when exponent on pressure drop is

changed, st0 time when valve constants change, sV volume of the reactor, lVd volume of the reactor below the dip tube, lx valve opening�L density of liquid in mol/l

Acknowledgements

The authors gratefully acknowledge the Inancial supportgiven by the Ministry of Human Resource Development(No. F. 28-1/99 TS 1, 31 March 2000) to carry out this work.

References

Malik, R. K. (1979). Optimal design of 2exible chemical processes.Ph.D. thesis, University of Wisconsin, Madison.

Moudgalya, K. M. (2001). A class of discontinuous dynamical systems III.Degrees of freedom analysis. Chemical Engineering Science, 56(18),5443–5447.

Moudgalya, K. M., & Jaguste, S. (2001). A class of discontinuousdynamical systems II. An industrial slurry high density polyethylenereactor. Chemical Engineering Science, 56, 3611–3621.

K. M. Moudgalya et al. / Chemical Engineering Science 58 (2003) 3973–3983 3983

Moudgalya, K. M., & Ryali, V. (2001). A class of discontinuousdynamical systems I. An ideal gas–liquid system. ChemicalEngineering Science, 56, 3595–3609.

Radhakrishnan, K., & Hindmarsh, A. (1993). Description and use ofLSODE, the Livermore solver for ordinary di6erential equations.Reference Publication NASA RP-1327, NASA.

Schleef, D. (2002). Control and measurement device interface.http://www.llp.fu-berlin.de/pool/software/comedi/.

Triplett, K. A., Ghiaasiaan, S. M., Abdel-Khalik, S. I., LeMouel, A., &McCord, B. N. (1999). Gas–liquid two-phase 4ow in microchannels—Part ii: Void fraction and pressure drop. International Journal ofMultiphase Flow, 25, 395–410.

Yu, W., France, D. M., Wambsganss, M. W., & Hull, J. R. (2002).Two-phase pressure drop, boiling heat transfer, and critical heat 4uxto water in a small-diameter horizontal tube. International Journal ofMultiphase Flow, 28, 927–941.