research article modeling of hydrodynamics in a 25mm...

Hindawi Publishing CorporationISRN Chemical EngineeringVolume 2013, Article ID 547489, 10 pageshttp://dx.doi.org/10.1155/2013/547489

Research ArticleModeling of Hydrodynamics in a 25 mm 𝜙 Pulsed Disk andDoughnut Column

Rajnish Kumar,1 D. Sivakumar,2 Shekhar Kumar,2 and U. Kamachi Mudali2

1 Atomic Energy Regulatory Board, Mumbai 400094, India2 Process Development and Equipment Section, Reprocessing R&D Division, Indira Gandhi Centre for Atomic Research,Kalpakkam 603102, India

Correspondence should be addressed to Shekhar Kumar; [email protected]

Received 30 June 2013; Accepted 25 July 2013

Academic Editors: G. D’Errico, J. A. A. Gonzalez, C.-T. Hsieh, F. Lefebvre, and E. A. O’Rear

Copyright © 2013 Rajnish Kumar et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The hydrodynamic parameters, namely, dispersed phase holdup and flooding throughput, have been investigated in 25mmdiameter pulsed disk and doughnut column (PDDC), in no mass transfer conditions. In this work, using existing correlations onplate pulsed columns, the dispersed phase holdup and the flooding throughput are empirically modelled well using the slip velocityconcept. A good agreement is observed between experimental values and predicted values obtained from empirical correlation.The experimental data for dispersed phase holdup and flooding throughput has been modelled using the Van Delden model todescribe the hydrodynamics characteristics of a PDDC and necessary adjustable parameters for drop size distribution and dispersedphase holdup are updated for 30% TBP-nitric acid system. The model parameters were estimated by minimizing the absoluteerror between experimental and theoretical values of flooding throughput and holdup data. It was found that the measured valuesand observed trends could be described accurately using this model after fitting holdup and flooding data. The error between theexperimental and theoretical values of flooding throughput and holdup was found to be less than 10%.

1. Introduction

Liquid-liquid extraction, a major unit operation in the chem-ical process industries, is also backbone of the aqueous routeof nuclear fuel reprocessing. The irradiated fuel from thenuclear reactor contains amixture of uranium and plutoniumalong with large number of fission products. For efficientreuse of uranium and plutonium, these are recovered fromspent nuclear fuels by PUREX process. The pulsed column,equipped with perforated plate internals and having 23%free area, has traditionally been used in the PUREX processfor almost the last six decades. Due to absence of movingparts, the pulsed columns have clear advantage over othermechanical contactors/extractors while processing corrosiveor radioactive solutions. A new type of pulsed columnwith internals, comprising an assembly of alternate disk anddoughnuts, has gained attention in recent times.However, theamount of literature available on these columns is severelylimited and briefly reviewed as follows.

Jahaya et al. [1] compared the performance of pulsed discand doughnut column with pulsed sieve plate column of75mm diameter and 23% free area for toluene-acetone-watersystem. While comparing these two designs of extractioncolumns, theymaintained the entire operating envelope iden-tical, for both of the columns. They observed that althoughtotal throughput per unit cross-sectional area through thePDDC was less, the mass transfer performance (and in turn,holdup of the PDDC) was better. Hong et al. [2] reportedexperimentally measured values of holdup and drop size ina 46.6mm diameter disc and doughnut column using theMIBK(d)-water(c) system. They also reported a functionalrelationship of the Sauter mean diameter as a function ofpulse intensity. Recently, Van Delden et al. [3, 4] reportedmodels to predict holdup, the Sauter mean drop diameter,flooding throughput, and mass transfer performance withrespect to operational conditions of PDDC.The models havebeen tested for extraction of caprolactam with toluene in40mmdiameter, 26.7% free area, and 4.24meter active height

2 ISRN Chemical Engineering

column. The interfacial area, available for mass transfer ina countercurrent differential extractor, depends upon themean droplet size and dispersed phase holdup. It is, therefore,important at the design stage to predict both these parametersfor any given system, column geometry, and set of operatingconditions.

The present study reports results of the dispersed phaseholdup measurement and correlation for organic phase dis-persion in PDDC, using the slip velocity and characteristicvelocity concept. Hydrodynamic model for pulse columnwas extended by Van Delden et al. [3, 4] for PDDC. Theapplicability of theVanDeldenmodel is discussed to describethe flooding characteristics, drop size, and dispersed phaseholdup, and this model was tested with 30% TBP-nitric acidsystem at no mass transfer conditions in 25mm diameter,23% free area, and 2m active height.The corresponding fittedparameters, different from conventional pulsed column, areinvestigated and compared with values reported by VanDelden for a similar PDDC.

2. Experimental Details

The PDDC pilot setup is schematically shown in Figure 1.The active column section is comprised of four glass sectionseach 50 cm in length and 25mm in internal diameter. Theactive section was closed on either ends by settlers (height of42 cm and an inner diameter of 50mm). The glass sectionswere connected using PTFE flanges having sample ports inthe form of needle valves. The column internals consistedof an alternating sequence of disc and doughnut plates thatwere held in place with three tie rods. The tie rods werehoused inside spacer sleeves and arranged in a triangularpitch as shown in Figure 2. The internals and spacer sleeveswere made of stainless steel (SS-304) material to preventwetting by the organic dispersed phase. The dimensionsof the column and the plates are shown in Table 1. Theexperimental conditions are shown in Table 2. The aqueousphase was introduced at the top and the organic phaseat the bottom. In the hydrodynamics experiments, bothphases leaving the column returned to their respective storagetanks for recycling. The organic stream was allowed to leavethe column via organic exit port. The interface level wascontrolled by an interface controller working on a jack-legprinciple. Pulsing limb was attached to the bottom side ofthe column. Pressure and vacuum were created in the pulselimb through an arrangement of digital timer, solenoid valves,and an ejector. This periodic reversal of pressurization anddepressurization in the pulse leg created periodic upwardsand downwards movement of liquid column in the pulse legwhich also created similar movement in the liquid columninside the PDDC. This periodic movement is known aspulsing, and this continuous reciprocating movement of theliquid column past the disc and doughnut internals causesturbulence with drop formation. The aqueous and organicphases were pumped by using two valveless digital meteringpumps (ISMATECMFPProcess drives withQ3 Pump heads)with maximum capacity of 2300mL/min each.

Table 1: Geometrical characteristics of the PDDCused in this study.

Parameter Dimensions of disc anddoughnut

Diameter of column 2.5 cmActive column height 200 cmHeight between disc and doughnut 1.0 cmFree area 23%Pulse leg diameter 1.0 cmDiameter of disc 2.195 cmInternal diameter of doughnut 1.2 cmSettlers diameter 5.0 cm

Spacer tubes SS304L tube 0.25OD ×

0.18 ID × 1.0 cm length

Tie rod position from centre 1.8 cm (triangularpitch, equispaced)

Table 2: Range of variables studied.

Variables RangeFlow rate (𝑈

𝑐and 𝑈

𝑑) 0.17–1.36 cm/s

Plate spacing,𝐻 1.0 cmFrequency (𝑓) 1-2HzAmplitude (𝐴) 1.04–4.6 cm

3. Experimental Procedure

3.1. Property Measurements. In all these experiments, thecontinuous phase was dilute aqueous nitric acid solution(0.5N HNO

3). 30%TBP (v/v) solution in normal paraffinic

hydrocarbon, an industrial substitute of n-dodecane, wasused as dispersed phase. Technical grade TBP (99%), ARgrade HNO

3(69–72% M/S RANKEM), commercial normal

paraffinic hydrocarbon (density ∼0.75 g/mL), and distilledwater were used to make the respective solutions. Den-sity of solutions was measured by an automatic vibrating-tube densitometer (Anton-Paar DMA-5000) with a built-in thermoelectric Peltier element module for maintainingtemperature of sample in a thermostated U-tube. The mea-surement accuracy was 10−6 g/cm3. Kinematic viscositieswere determined for the equilibrated phases at 298K, usingCannon Fenske glass viscometers. The measurement accu-racy was 0.01 cP. Interfacial tensions were determined forthe ternary TBP-HNO

3-water system at 298K with a Du

Nouy tensiometer using a standard Pt ring of 5.992 cm ascircumference, and results were corrected by the procedureof Harkinson and Brown [5]. The standard deviation of theinterfacial tension measurements was less than 0.3mN⋅m−1.The physical properties of the aqueous-organic pair used inthis study are listed in Table 3.

3.2. Experimental Runs with PDDC. As per the standardpractice, the column was first filled with the continuousphase (dilute aqueous nitric acid of 0.5N concentration).Thepulsationwas started after setting the frequency to the desired

ISRN Chemical Engineering 3

Organicfeed pump

Aqueousfeed pump

Organic feed inlet

Organic tank

Aqueous tank

Aqueousfeed inlet

Organicoutlet

Solenoidvalve 1

SV 2

To vent

SV 3

Surge tank

PressureregulatorCompressor

Aqueousoutlet

Aqueous out

Samplingpoint 4

Samplingpoint 5

Samplingpoint 3

Samplingpoint 2

Samplingpoint 1

Interface level

controller

Vent

Tomanometer

Tomanometer

Tomanometer

Tomanometer

Tomanometer

Figure 1: Schematic diagram of the experimental setup.

value with the help of a digital timer.The pulsation amplitudewas set to the desired values in the pulse leg by adjustingthe compressed air pressure with help of pressure regulator.Then, the organic phase (30% TBP/NPH) was pumped intothe column. A fixed ratio between the organic and aqueousflow rates was maintained. The interface was maintained at agiven height at the top of the columnwith help of an interfacecontroller.

3.3. Holdup. Dispersed phase holdup is defined as the frac-tion of the active column section volume occupied by thedispersed phase. Before carrying out the experiments, bothphases were mutually saturated, and the pulse amplitudeand frequency were adjusted to the desired values. Thecontinuous-phase and the dispersed-phase flow rates werethen set to the required flow rate, and the system wasstabilized to allow steady state to be reached. Then, the inlet


Table 3: Physical properties of the equilibrated aqueous organicpair.

Property ValueDensity of continuous phase, that is,aqueous phase 0.5N HNO3 (𝜌𝑐)

1.0155 g/cm3

Density of dispersed phase, that is, organicphase (30% TBP/NPH) (𝜌

𝑑)

0.8085 g/cm3

Interfacial tension (𝛾) 9.95 dynes/cmDynamic viscosity of continuous phase (𝜂

𝑐) 1.05 cP

Dynamic viscosity of dispersed phase (𝜂𝑑) 2.09 cP

and outlet flows were stopped simultaneously.The dispersionwas then allowed to coalesce at the interface. The holdup wasthenmeasured either by determining the change of interfacialheight or by displacing the solvent layer into a measuringcylinder.

3.4. Flooding. Flooding throughput was measured keepingthe pulse velocity and the continuous phase flow rate constantand increasing the dispersed phase flow rate until the onsetof flooding. The measurement of pressure drop across thecolumn was a useful indicator for predicting the onset offlooding. At the onset of flooding, the pressure drop acrossthe column became unsteady. When the column approachedflooding, the interface position was unstable and kept onvarying in spite of adjusting the withdrawal rate of aqueousphase. The incoming and the exiting flow rates did notmatch at floodingwhich indicated buildup of dispersed phasewithin the column. When flooding occurred, all flows wereimmediately stopped and then the holdup was measured.

4. Results and Discussion

4.1. Equations for Estimating Hydrodynamic Model Parame-ters. According to the theory for pulsed sieve-plate columns,the column operation may be characterized by the Sautermean drop diameter (d

32) and the dispersed phase holdup, 𝜀

of which the latter is dependent on the operational regime inthe column.The operational window of the column is limitedby flooding that may arise due to either insufficient pulsationor phase inversion or excessive entrainment at high pulsation.

The parameters of the equations describing the columnhydrodynamics may be divided into (i) physical properties,(ii) operational variables, and (iii) geometric dimensions ofthe column and its internals. The physical properties includethe continuous and dispersed phase densities (𝜌

𝑐and 𝜌𝑑), the

corresponding density difference (Δ𝜌 = 𝜌𝑐−𝜌𝑑), the dynamic

viscosities of the phases (𝜂𝑐and 𝜂

𝑑), and the interfacial

tension, 𝛾. The operational parameters include the phasesuperficial velocities,𝑈

𝑐and𝑈

𝑑, and hence the corresponding

volumetric flux and flow ratio and the pulsation intensity𝐴𝑓. The model is developed for the purely hydrodynamicsituation and therefore assumes equilibrium conditions inorder to avoid the influence of mass transfer.

H

Figure 2: Schematic representation of the internals of the pulseddisk and doughnut column.

4.2. Dispersed Phase Holdup Using Slip Velocity Concept.The slip velocity is a useful concept that is widely usedin extraction literature for obtaining the relative velocitiesbetween phases that are either in cocurrent or countercurrentflow. Once an expression for slip velocity is available in termsof the geometric, operational parameters as well as the systemproperties, it could be used to predict holdup. Dispersedphase holdup was one of the important first parameters ofcolumn to be addressed by Gayler and Pratt in the early fifties[6, 7] relating to packed columns, followed by Thornton [8]in 1956. Hartland and Kumar did extensive work in this fieldand they reported correlations for pulsed perforated [9, 10],reciprocating [11], and general [12] columns.

4.3. Correlations Incorporating Slip Velocity for Organic PhaseDispersion. The basic idea of Gayler and Pratt [6, 7] was tocalculate the holdup in terms of slip velocity and characteris-tic velocity as follows:

𝑈slip =𝑈𝑑

𝜀

+

𝑈𝑐

(1 − 𝜀)

= 𝑉𝑜(1 − 𝜀) , (1)

where 𝑉𝑜is the characteristic velocity, which is the mean

droplet velocity relative to the continuous phase, when𝑈𝑐= 0

and 𝑈𝑑→ 0.

Later, Godfrey and Slater [13] in 1991 proposed thefollowing

𝑈slip = 𝑉𝑜(1 − 𝜀)𝑚

,

𝑈slip = 𝑉𝑜 (1 − 𝜀) exp (𝑏𝜀) .(2)

The holdup data is further correlated employing thefollowing slip velocity forms. Kumar andHartland [14] corre-lated the holdup in terms of slip velocity as a power function


0 1 2 3 40

0.5

1

1.5

2

2.5

3

3.5

4

4.5

+10%

−10%

Usli

p cal

(cm

/s)

Uslip expt(cm/s)

Figure 3: Comparisons of slip velocity measurement with thecorrelation using equation (3).

(1 − 𝜀). Since 𝑉𝑜depends on specific power dissipation, it is a

function of (𝐴𝑓). The final form for organic phase dispersionand the correlations of slip velocity in terms of 𝐴𝑓 and 𝜀 areas follows:

𝑈slip = 3.4855(𝐴𝑓)−0.4669

(1 − 𝜀)−0.7619

, (3)

𝑈slip = 3.3241(𝐴𝑓)−0.4713

(1 − 𝜀) 𝑒2.2780𝜀

. (4)

The calculated 𝑈slip values are plotted against experimental𝑈slip data in Figures 3 and 4 respectively. The data are alsocorrelated using holdup ratio as the correction factor forPratt’s equation and the resulting equation is as follows:

𝑈slip = 7.6812(𝑓𝐴)−0.4482

(1 − 𝜀) (

𝜀

1 − 𝜀

)

0.281

. (5)

The slip velocities calculated with equation (5) are plottedagainst the experimental slip velocities in Figure 5. Thoughequation (5) predicts the slip velocities well, this equationshould be strictly treated as empirical within the range ofvariables covered in the present study; as 𝜀 → 0, 𝑈slipbecomes zero instead of reducing to𝑉

𝑜, which is not expected.

The response of the dispersed phase holdup to the oper-ational variables in the present PDDC is thus qualitativelysimilar to the response of the liquid pulse and reciprocatingpulse columns to the flow rates of dispersed, continuousphases, and the pulsation intensity.

4.4. Correlation for Flooding Using Slip Velocity Concept. InPDDC, the flow regime might correspond to mixer settler,dispersion, and emulsion regimes although clear demar-cation was not possible based on the visual observations.In the mixer settler regime, pulsation amplitude less than1 cm/s typically leads to large globules of dispersed phase

0 1 2 3 40

0.5

1

1.5

2

2.5

3

3.5

4

4.5

+8%

−8%

Usli

p cal

(cm

/s)

Uslip expt(cm/s)


0 1 2 3 40

0.5

1

1.5

2

2.5

3

3.5

4

4.5

+10%

−10%

Usli

p cal

(cm

/s)

Uslip expt(cm/s)


flowing through the test section. In the dispersion regime,with pulsation amplitude of 1–4 cm/s, occasional big dropswere seen, while in the emulsion regime corresponding topulsation amplitude of 4 cm/s and above, the drops wereuniform. In this study, throughout the experiment, pulsationamplitude was maintained larger than 1 cm/s.

The flooding throughput is modelled in the present studyas a slip velocity concept:

𝑈𝑑𝑓

𝜀𝑓

+

𝑈𝑐𝑓

(1 − 𝜀𝑓)

= 6.22 exp (−0.10𝑓𝐴) (1 − 𝜀𝑓) . (6)


The constants in equation (6) are dependent upon thephysical properties of the system, the column geometry, andthe material of construction of the internals. Experimentaldata is compared with predicted values from equation (6) inFigure 6.

4.5. The Applicability of Unified Equation. Hartland andKumar correlated the holdup directly from the physicalproperty data, energy input, and flow rates, avoiding theproblematic concept of characteristic velocity. In a seriesof articles they reported correlations for sieve-plate pulsedcolumns [10], Karr column [11], and a unified correlationfor pulsed, packed, and reciprocating columns [12]. Unifiedcorrelation of Kumar andHartland [12] has been used by VanDelden et al. [3, 4] for PPDC as follows:

𝜀 = ΠΦΨΓ, (7a)

where

Π = 𝐶Π+ [

Ε

𝑔

⋅ (

𝜌𝑐

𝑔 ⋅ 𝛾

)

0.25

]

𝑛1

, (7b)

Φ = [𝑈𝑑⋅ (

𝜌𝑐

𝑔.𝛾

)

0.25

]

𝑛2

exp[𝑛3⋅ 𝑈𝑐⋅ (

𝜌𝑐

𝑔 ⋅ 𝛾

)

0.25

] , (7c)

Ψ = 𝐶Ψ⋅ (

Δ𝜌

𝜌𝑐

)

𝑛4

⋅ (

𝜂𝑑

𝜂𝑐

)

𝑛5

, (7d)

Γ = 𝐶Γ⋅ 𝑒𝑛6⋅ [𝐻 ⋅ (

𝜌𝑐⋅ 𝑔

𝛾

)

0.5

]

𝑛7

, (7e)

𝐸 = (

Π2

2

)(

1 − 𝑒2

𝑒2𝐶2

𝑜𝐻

)(𝐴 ⋅ 𝑓)3

. (7f)

Equation (7a) is a semiempirical relation valid in thedispersion and emulsion operational regime. The group Πallows for the influence of the energy input per unit mass,Φ for the effect of phase velocities, Ψ for the influence ofthe physical properties, and Γ for the dimensions of theinternals. The value for 𝐶

Ψwas taken unity since the system

operated without mass transfer and 𝑛6was set to 0 as all the

pulsed columns studied had the same value of the fractionalplate free area (0.23). Van Delden suggested that the energyinput would be different for a PDDC compared to a pulsedsieve-plate or the Karr column. Therefore, the parameters𝐶Π, 𝐶Γ, and 𝑛

1might differ for a PDDC compared to the

literature values. Same reasoning applies for 𝑛5when the

viscosity influence in equation (7a) gets changed comparedto the literature. The deviation in 𝑛

5is probably not large as

long as the continuous phase viscosity is close to the value ofwater.The parameter values are presented in Table 4 togetherwith original values derived for pulsed sieve-plate column byKumar and Hartland and Van Delden fitted limited numberof parameter for PDDC.

4.6. Dispersed Phase Holdup Comparison between Experimen-tal Data ofThisWork with Van Delden Correlation. Thefittedparameters, suggested by Van Delden, were based on limited

0 1 2 3 4 50

1

2

3

4

5

+5%

−5%

Usli

p cal

Uslip expt

Figure 6: Comparisons of slip velocitymeasurement with the corre-lation using equation (6).

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

Measured 𝜀

Pred

icte

d 𝜀

Figure 7: Comparison of holdupmeasurements with the Van Deld-en model.

number of experimental data for caprolactam extraction inPDDC having column diameter 40mm and plate spacing ofabout 9mmwith 26.3% free area. Figure 7 shows comparisonof experimental data with the original Van Delden model.As column diameter is different in this study and aqueous-organic pair is also different, the parameters reported in thiswork (𝐶

Πand 𝑛

1) differ from the values reported by Van

Delden. New fitted parameters are 𝐶Π= 3.1054 and 𝑛

1=

0.6768 for 95% confidence interval. The calculated holdup


Table 4: Adjustable parameters of correlations for estimating holdup.

𝜀 𝐶Π

𝐶Γ

𝑛1

𝑛2

𝑛3

𝑛4

𝑛5

𝑛7

Kumar and Hartland 0.27 6.87 0.78 0.87 3.34 −0.58 0.18 −0.39Van Delden 2.39 0.45 0.34 — — — −0.08 −0.12Present work 3.1054 — 0.6768 — — — — —

Table 5: Adjustable parameters of correlations for estimating dropsize.

(a)

𝑑32equation (8) 𝐶

1𝐶2

𝐶3

𝑛1

𝑛2

𝑛3

𝑛4

Kumar and Hartland 1.38 0.16 1.25 0.3 0.18 0.14 0.06Delden 2.84 — −2.59 — — — —Present work 3.7557 — −0.7517 — — — —

(b)

𝑑32equation (8) 𝐶

Ψ𝐶Ω

𝐶Π

𝑛 𝑛1

𝑛2

Kumar and Hartland 1∗ 1.55 0.42 0.32 −0.35 1.15∗No mass transfer.

profile using the new fitted parameters is shown in Figure 8together with experimental data. From Figure 8, it may beobserved that the experimental data are describedwell within±10%.

4.7. Mathematical Model for Estimation of the Sauter MeanDropDiameter. Knowledge of the drop size is of fundamentalimportance in the design of liquid-liquid extraction columns.It affects the dispersed-phase holdup, the residence time ofthe dispersed phase, and the allowable throughputs. Kumarand Hartland [14, 15] made major study of the drop size,and based on wide range of experimental conditions, theydeveloped unified correlations for drop size, as a function ofsystem parameters for a wide variety of columns.The correctdescription of influence of energy input on drop size has notyet been found.

In 1986, Kumar and Hartland developed the followingmodel for the pulsed sieve-plate column [14]:

𝑑32

√𝛾/Δ𝜌𝑔

= 𝐶1𝑒𝑛1(𝐻 ⋅ √

𝜌∗𝑔

𝜎∗

)

𝑛2

(

𝜂𝑑𝑔0.25

𝜎0.75

∗𝜌0.25

∗

)

𝑛3

(

𝛾

𝜎∗

)

𝑛4

[𝐶2+ exp(

𝐶3(𝐴𝑓)

𝑒

(

𝜌∗

𝜎∗𝑔

)

(1/4)

)] .

(8)

Later in 1996, Kumar and Hartland [15] presented a unifiedcorrelation for many types of columns. Its form for pulsedcolumns is as follows:𝑑32

𝐻

=

𝐶𝜓𝑒𝑛

(𝐶Ω(𝛾/Δ𝜌𝑔𝐻

2)0.5

)

−1

+(𝐶Π[(𝐸/𝑔) (Δ𝜌/𝑔𝛾)

0.25

]

𝑛1

[𝐻(Δ𝜌𝑔/𝛾)0.5

]

𝑛2

)

−1.

(9)These equations are valid in all operating regimes, take energyinput into account via pulsation intensity (𝐴𝑓), and are based

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

𝜀expt

𝜀 cal

+10%

−10%

Figure 8: Parity plot of holdup experimental values with predictedvalues based on new fitted parameters.

on the physical properties and geometrical properties of thesystem. Both equations are similar. However, Equation (8)takes into account the influence of dispersed phase viscosity.The parameters 𝜂∗ and 𝜎∗ represent reference density andsurface tension values of water at 298K.

However, the characteristic internal sizes, 𝑒 and 𝐻, aredifferent for a PDDC compared to sieve-plate internals, andthe resulting energy input is thus different as well. Therefore,the parameters 𝐶

1, 𝐶2, 𝐶3, and 𝑛

2in (8) and 𝐶

Ω, 𝐶Π,

𝑛1, and 𝑛

2in (9) might differ from the original values. In

the experiments conducted by Van Delden, however, thecharacteristics of the internals, 𝐻 and 𝑒, were almost thesame for the forward extraction and back extraction. So theparameters 𝑛

1and 𝑛

2could not be fitted, and the original

values were thus applied. 𝐶1and 𝐶

3of Equation (8) were

fitted, and the results were listed in Table 5.

4.8. Mathematical Model for Estimation of Flooding Through-put. This section is based on the concept of balances of forces,originally proposed by Baird and Lane [16]. Under steady sateconditions, the net weight of the dispersed phase droplets willbe balanced by the drag force acting on them. This may beexpressed in a simple quantitative way if it is assumed that thedroplets are rigid spheres of a uniform size 𝑑. Let us consider


the superficial velocity 𝑉𝑠of the continuous phase relative to

the droplets of dispersed phase as follows:

𝑉𝑠= 𝑈slip (1 − 𝜀) = (

1 − 𝜀

𝜀

)𝑈𝑑+ 𝑈𝑐. (10)

The frictional pressure gradient may be estimated from theErgun packed bed equation

(−

𝑑𝑝

𝑑𝑧

)

𝑑(1 − 𝜀)3

𝜌𝑐𝑉2

𝑠𝜀

=

150𝜀

Re+ 1.75, (11)

where Re = 𝑉𝑠𝑑𝜌𝑐/𝜇𝑐.

The net weight of the dispersed phase balances thefrictional force as follows:

(−

𝑑𝑝

𝑑𝑧

) = 𝜀𝑔 (Δ𝜌) . (12)

Substitution of Equation (12) into (11) provides

𝑑(1 − 𝜀)3

𝑔 (Δ𝜌)

𝜌𝑐𝑉𝑠

2=

150𝜀

Re+ 1.75. (13)

The substitution of Equation (10) into Equation (13) will yield

𝑑32(1 − 𝜀) 𝑔 (Δ𝜌)

𝜌𝑐𝑉2

slip=

150𝜀

Reslip (1 − 𝜀)+ 1.75. (14)

The equations proposed by Kumar and Hartland [15], how-ever, are of comparable form as shown in

𝑑32⋅ 𝑔 ⋅ Δ𝜌

𝜌𝑐𝑉2

slip⋅

4/3 ⋅ (1 − 𝜀)

1 + 4.56 ⋅ 𝜀0.73

=

24

Reslip+ 0.53. (15)

At flooding, maximum value of holdup is reached. Thesuperficial velocities 𝑈

𝑐and 𝑈

𝑑do not increase beyond their

value at flooding. Therefore,

(

𝜕 (𝑈𝑑)

𝜕𝜀

)

𝑓

= (

𝜕 (𝑈𝑐)

𝜕𝜀

)

𝑓

= 0. (16)

Kumar and Hartland derived the correlation for pulsedcolumn at flooding by applying the above criterion given byEquation (16). This correlation is listed in Equation (17a) asfollows:

[𝜀𝑓+ 𝑅 ⋅ (1 − 𝜀

𝑓)]

×

[[[

[

(𝛽3− 𝛽1) ⋅ (1 − 2𝜀

𝑓) −

2 ⋅ 𝛽2⋅ 𝜀𝑓⋅ (1 − 𝜀

𝑓)

𝛽3⋅ (1 + 4.56 ⋅ 𝜀

0.73

𝑓)

2

(1 + 4.56 ⋅ 𝜀0.73

𝑓+ 3.33 ⋅ 𝜀

−1.27

𝑓⋅ (1 − 𝜀

𝑓))

]]]

]

+ (𝛽3− 𝛽1) ⋅ 𝜀𝑓⋅ (1 − 𝜀

𝑓) (𝑅 − 1) = 0,

(17a)

where

𝛽1=

24 ⋅ 𝜂𝑐

0.53 ⋅ 𝑑32⋅ 𝜌𝑐

; 𝛽2=

4 ⋅ 𝑑32⋅ 𝑔 ⋅ Δ𝜌

1.59 ⋅ 𝜌𝑐

;

𝛽3= (𝛽

2

1+

4 ⋅ 𝛽2(1 − 𝜀

𝑓)

(1 + 4.56 ⋅ 𝜀0.73

𝑓)

)

0.5

.

(17b)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

Total throughput expt

Tota

l thr

ough

put ca

lFigure 9: Parity plot of total throughput experimental values withpredicted values based on the Van Delden parameters.

Using Equation (17a) the holdup at flooding conditions can becalculated fromphysical properties and the Sautermean dropdiameter, which is described via Equation (8) or Equation (9).Since authors did not perform experiments for measurementof the Sauter mean drop diameter, they did reverse modelingto estimate adjustable parameters of Equation (8) or Equation(9) for drop size to minimize error between the experimentalflooding throughputs and theoretically estimated floodingthroughput via Equation (17a). The error minimization wasperformed in MATLAB.

The throughput values predicted by themodelwere foundto be in agreement with those obtained experimentally asshown in Figures 9 and 10. Figure 11 shows variation of theestimated Sauter mean drop diameter with pulsation velocity.

5. Conclusions

The dispersed phase holdup and flooding throughput weremodelled in the present study following the slip velocityconcept. The characteristic velocity was correlated to thepulsation intensity for organic phase dispersions. Usingexisting correlations on plate pulsed columns, the newparameters applicable for PDDCwere estimated. It was foundthat the measured values and observed trends could bedescribed accurately using this model after fitting holdupand flooding experimental data. A very good agreement isobserved between experimental values and predicted valuesobtained from empirical correlation.Using existing theory onpulsed columns, unified equations, and equations derived forpulsed sieve-plate, andKarr columns, a theoreticalmodel wasdeveloped by Van Delden to describe the operational char-acteristics of a PDDC. The observed trends and determineddata were described according to a previously developed


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

+12%

+12%

Total throughput expt

Tota

l thr

ough

put ca

l

Figure 10: Parity plot of total throughput experimental values withpredicted values based on new fitted parameters.

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

Af (cm/s)

This work using (8)Van delden work

d32(m

m)

Figure 11: Variation of the Sauter mean drop diameter withpulsation velocity of PDDC column for organic dispersion.

model, covering the operational window, drop size, andholdup. It was found that the measured values and observedtrends could be described accurately using this model afterfitting drop diameter, holdup, and flooding data.

Nomenclature

𝑑32, 𝑑max: The Sauter drop diameter and maximum

drop diameter, respectively, cm𝐷𝑐: Internal column diameter, cm

𝐷𝑑: Disc diameter, cm

𝐷𝑟: Diameter of ring aperture in doughnut, cm

𝑒: Fractional free cross-sectional area, 𝑒 = 1−𝐷2

𝑑/𝐷2

𝑐

𝑓: Frequency, s−1𝐴𝑓: Pulsation velocity, cm⋅s−1Flux: Total throughput, Flux = 𝑈

𝑑+ 𝑈𝑐, m⋅h−1

𝑔: Gravitational constant, 9.81 g⋅m−2⋅s−1𝐻: Disc to doughnut spacing, cm𝑅: Flow ratio, 𝑅 = 𝑈

𝑑/𝑈𝑐

Re: Reynolds number, Reslip = 𝑑32 ⋅ 𝑈slip ⋅ 𝜌𝑐/𝜇𝑐

𝑈slip, 𝑉𝑜: Slip velocity and characteristic velocity,respectively, cm⋅s−1.

Greek Letters

𝛾: Interfacial tension, N⋅m−1𝜀: Dispersed phase holdup defined as volume

fraction of the dispersed phase𝜂: Dynamic viscosity, kg⋅m−1⋅s−1𝜂∗: Reference dynamic viscosity of water at

298K, 𝜂∗ = 0.001 kg⋅m−1⋅s−1𝜌: Density (kg⋅m−3)𝜌∗: Reference density of water at 298K, 𝜌∗ =

997.0 kg⋅m−3Δ𝜌: Density difference, kg⋅m−3𝜎∗: Reference surface tension of water at

298K, 𝜎∗ = 0.0728N⋅m−1.

Subscript

aq: Aqueous organic phaseorg: Organic phase, respectively𝑐: Continuous phase𝑑: Dispersed phase𝑓: At flooding condition.

Acknowledgment

The authors sincerely acknowledge the help provided by Mr.S. Sundarmurthy of RRDD in conducting the experiments.

References

[1] A. B. Jahya, H. R. C. Pratt, and G. W. Stevens, “Comparison ofthe performance of a pulsed disc and doughnut column with apulsed sieve plate liquid extraction column,” Solvent Extractionand Ion Exchange, vol. 23, no. 3, pp. 307–317, 2005.

[2] K. Hong, P. Bang-sam, P. Hum-hwee, and R. Bo-sung, “Dropsize distribution and hold up in pulsed disk and doughnutextraction column,” Journal of Korean Institute of ChemicalEngineering, vol. 42, no. 1, pp. 35–42, 1984.

[3] M. L. van Delden, N. J. M. Kuipers, and A. B. de Haan,“Extraction of caprolactam with toluene in a pulsed disc anddoughnut column-part I: recommendation of a model forhydraulic characteristics,” Solvent Extraction and Ion Exchange,vol. 24, no. 4, pp. 499–517, 2006.


[4] M. L. van Delden, G. S. Vos, N. J. M. Kuipers, and A. B. deHaan, “Extraction of caprolactam with toluene in a pulsed discand doughnut column-part II: experimental evaluation of thehydraulic characteristics,” Solvent Extraction and Ion Exchange,vol. 24, no. 4, pp. 519–538, 2006.

[5] W. D. Harkins and F. E. Brown, “The determination of surfacetension (free surface energy), and the weight of falling drops:the surface tension of water and benzene by the capillary heightmethod,”The Journal of the American Chemical Society, vol. 41,no. 4, pp. 499–524, 1919.

[6] R. Gayler and H. R. C. Pratt, “Hold-up and pressure drop inpacked columns,” Transactions of IChemE, vol. 29, pp. 110–125,1951.

[7] R. Gayler, N. W. Roberts, and H. R. C. Pratt, “Liquid-liquidextraction studies: part IV. A further study of hold-up in packedcolumns,” Transactions of the Institution of Chemical Engineers,vol. 31, pp. 57–68, 1953.

[8] J. D. Thornton, “Spray liquid extraction columns: predictionof limiting hold-up and flooding rates,” Chemical EngineeringScience, vol. 5, no. 5, pp. 201–208, 1956.

[9] A. Kumar and S. Hartland, “Independent prediction of slipvelocity and hold-up in liquid/liquid extraction columns,” TheCanadian Journal of Chemical Engineering, vol. 67, no. 1, pp. 17–25, 1989.

[10] A. Kumar and S. Hartland, “Prediction of dispersed phase hold-up in pulsed perforated-plate extraction columns,” ChemicalEngineering and Processing, vol. 23, no. 1, pp. 41–59, 1988.

[11] A. Kumar and S. Hartland, “Prediction of dispersed-phaseholdup and flooding velocities in Karr reciprocating-plateextraction columns,” Industrial and Engineering ChemistryResearch, vol. 27, no. 1, pp. 131–138, 1988.

[12] A. Kumar and S. Hartland, “A unified correlation for the pre-diction of dispersed-phase hold-up in liquid-liquid extractioncolumns,” Industrial and Engineering Chemistry Research, vol.34, no. 11, pp. 3925–3940, 1995.

[13] J. C. Godfrey and M. J. Slater, “Slip velocity relationshipsfor liquid-liquid extraction columns,” Chemical EngineeringResearch and Design, vol. 69, no. 2, pp. 130–141, 1991.

[14] A. Kumar and S. Hartland, “Prediction of drop size in pulsedperforated-plate extraction columns,” Chemical EngineeringCommunications, vol. 44, no. 1–6, pp. 163–182, 1986.

[15] A. Kumar and S. Hartland, “Unified correlations for the predic-tion of drop size in liquid-liquid extraction columns,” Industrialand Engineering Chemistry Research, vol. 35, no. 8, pp. 2682–2695, 1996.

[16] M. H. I. Baird and S. J. Lane, “Drop size and holdup in areciprocating plate extraction column,” Chemical EngineeringScience, vol. 28, no. 3, pp. 947–957, 1977.

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


research article modeling of hydrodynamics in a 25mm...

Documents