Che Eng Commu Flow Regime

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Flow Regime Transition in a Trickle Bed

ReactorAJAY BANSAL

a , R. K. WANCHOO

a & S. K. SHARMA

a

a Department of Chemical Engineering and Technology , Punjab

University , Chandigarh, India

Published online: 25 Jan 2005.

To cite this article: AJAY BANSAL , R. K. WANCHOO & S. K. SHARMA (2005) Flow Regime Transition

in a Trickle Bed Reactor, Chemical Engineering Communications, 192:8, 1046-1066, DOI:

10.1080/009864490522597

To link to this article: http://dx.doi.org/10.1080/009864490522597

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Flow Regime Transition in a TrickleBed Reactor

AJAY BANSAL, R. K. WANCHOO, ANDS. K. SHARMA

Department of Chemical Engineering and Technology,Punjab University, Chandigarh, India

Hydrodynamic behavior of a trickle bed reactor is dependent on the flow regime

that arises when gas and liquid flow concurrently downward in the reactor. Empiri-cal or semi-analytical models available in the literature do not predict to satisfactionthe transition between the trickle and pulse flow regimes in cocurrent down flow in aTBR.

In the present study, flow transition between trickle flow (low-interaction) and pulse flow (high-interaction) regimes was experimentally observed for differentnon-foaming Newtonian and non-Newtonian liquid phases under concurrent down

flow conditions with air in a packed bed using different bed configurations. Theeffect of bed geometry along with fluid phase properties has been considered tomodify the flow map coordinates, so as to collapse the data corresponding to differ-ent bed configurations and Newtonian liquid-phase properties onto a single curve(within 20%). An empirical correlation for the prediction of Newtonian liquid-

phase mass velocity at the transition between high interaction and low interactionregimes was developed, which assumes the form:

Lt

G

kn

S 2

S 21

1=4¼ 5:73 G

ke

4=3

The correlation was found to be in good agreement with the experimental data avail-able in the literature for different Newtonian liquid-phase systems including data onnonspherical particles. This correlation is found to predict the transition satisfac-torily when liquid-phase viscosity was replaced by apparent viscosity in the caseof viscoinelastic liquid-phase systems. The correlation was further extended to pre-dict the transition for a viscoelastic liquid-phase system with the introduction of theWeissenberg number as:

Lt;ve ¼ Lt;vi 1 þ ffiffiffi

2p We2

Keywords Trickle bed reactor; Two-phase flow; Flow regime transition;Newtonian=non-Newtonian systems

Introduction

Trickle bed reactors (TBR) are the most widely used type of reactors in the processindustry. The gas and liquid concurrently flow downward over a fixed bed of catalyst

Address correspondence to R. K. Wanchoo, Department of Chemical Engineering andTechnology, Punjab University, Chandigarh, India 160014. E-mail: [email protected]

Chem. Eng. Comm., 192:1046–1066, 2005Copyright # Taylor & Francis Inc.ISSN: 0098-6445 print/1563-5201 onlineDOI: 10.1080/009864490522597

1046


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particles. Trickle bed reactors are used in the petroleum, petrochemical, and chemi-cal industries, in wastewater treatment, and biochemical and electrochemical proces-sing. With ever-increasing demand for light oil and other products based on TBRtechnology, any improvement in the understanding of the complex phenomena tak-

ing place inside the TBR can lead to major technical and=or economical break-through. The various factors affecting design, operation, and performance of aTBR include hydrodynamic parameters, transfer coefficients, and reaction kinetics.But a priori to knowing all these parameters is the ‘‘flow regime’’ prevailing, corre-sponding to the operating variables and physical properties of the system of interest.In addition, the flow regime transition is strongly influenced by the foaming andnon-foaming character of the two fluids involved in a TBR. Knowledge of the flowregime is prerequisite because other parameters like pressure drop, holdup, transferrates, and dispersion coefficient depend on the prevailing regime.

While flowing downward, gas and liquid rush towards the void space. Thisresults in different flow patterns. Based on the intensity of interaction involvedbetween the gas and liquid phase, these regimes are categorized as low interactionor high interaction. Although the high interaction regime is further subclassifiedas pulse, spray dispersed bubble, the transition from trickle to pulse (i.e., low to highinteraction) is of much more industrial importance as most of the reactors in indus-trial applications operate near the trickle to pulse transition (Satterfield, 1975).Information on regime transition available in the literature is briefly reviewed here.

Flow Regime Transition

A number of flow maps for regime transition are available in the literature.

Weekman and Myers (1964), Sato et al. (1973), and Satterfield (1975) used G versusL as the coordinates for the flow map, whereas Turpin and Huntington (1967) repre-sented the map as L=G versus G. Charpentier and Favier (1975) plotted the dimen-sionless group (L=G)kw as a function of G=k, where k and w were originallyproposed by Baker (1954) for two-phase flow with empty horizontal pipes. Differentinvestigators have reported the effects of physical properties of the fluid on regimetransition. Chou et al. (1977) observed that with increase in viscosity or decreasein surface tension the transition shifted to lower gas and liquid superficial velocities.These authors have further concluded that even the use of Baker coordinates doesnot cause the transition lines for various systems to coincide. In fact, the availabletransition plots are mostly based on an air-water system and do not predict very wellthe transition for other systems; for example, the transition for 93.9 wt% ethanoland for 20 ppm heptyl alcohol in water differ by a factor of three. Chou et al.(1977) and Sai and Varma (1988) observed that at a constant gas flow rate, the tran-sition from trickling to pulsing occurred at higher flow rates as the bed void fractionincreased, but the data of Sato et al. (1973) did not show any significant variationwith respect to changes in void fraction. Gianetto et al. (1978) deemed it necessaryto consider the effect of void fraction in the flow regime map coordinates and pro-posed G=ke as the modified abscissa, against G=k as proposed by Charpentier andFavier (1975). On the other hand, Sai and Varma (1988) selected (L=G)kw0.5 asthe ordinate, but the transitions, instead of collapsing to a single curve, showed a

family of curves corresponding to each system. These authors have reported dataon Newtonian and viscoinelastic non-Newtonian liquid phases using different pack-ings with aspect ratios (Dc=dp) in the range of 4.74 to 15.7. However, these authors

Flow Regime Transition in a Trickle Bed Reactor 1047


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The column was initially flooded with liquid alone so as to completely wetthe packing, and it was drained before the liquid was introduced into the columnat the desired rate. Air from the surge tank, through the saturator, was introduced

into the column at the desired rate. The column was allowed to attain the steady state.The flow pattern across the glass column was visually observed. For each run the gasflow was kept constant and the liquid flow rate was gradually increased in steps.

Figure 1. Experimental setup.



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The liquid flow rate at which transition was observed was noted. Corresponding tothe visual observation of the onset of pulsing the manometer fluid was found to indi-cate the fluctuations. In all systems, the onset of pulsing started at the bottom of thecolumn and with increased gas flow rates of liquid at a constant gas rate, the point of pulse inception moved towards the column top. To observe the fluctuations corre-sponding to transition precisely, manometers with different manometer fluids likewater, CCl4, and mercury were used. For systems resulting in small fluctuations atthe onset of transition, water was used as manometer fluid. The transition was alsoconfirmed from the~P=Z versus. L plots. It is reported in the literature that a sharp

rise in pressure drop is observed when the regime changes from low interaction tohigh interaction (Chou et al., 1977). A log-log plot was plotted in each case and thebest fit curve, for low and high interaction regimes were drawn separately to intersectat a point; the point of intersection was found to match fairly well with the visualobservations of the onset of pulsing. Sicardi et al. (2003) have recently confirmedthe reliability of non visual techniques for the detection of flow regime.

TABLE IIa. Properties of liquid phases: Newtonian fluids

Liquid phase

Density,

kg m

3

Viscosity,

Pa s Surface tension, mNm

1

Temperature,

C

Tap water 997.6 9.60 104 70 22.06 ppm surfactant

(sodium laurylsulfate)

999.5 11.13 104 59 16.0

12 ppm surfactant(sodium laurylsulfate)

999.5 11.13 104 55 16.0

60% glycerol(w=w)

1148.0 5.93 103 64 32.0

77% glycerol(w=w)

1192.7 27.04 103 63 32.0

Table I. Types of packing used

Type of packinga

Packing size,mm Dp, mm Porosity, e

Sphericity ,U

Glass beads I 3.337 3.337 0.376 1.0000Glass solid

cylinders, d h6.083 27.047 11.450 0.479 0.7163

Glass Raschig ringsid=od h

6.336=9.130 10.274

3.690 0.686 0.4225

Glass beads II 14.840 14.840 0.441 1.0000ZnO catalyst

pellets, d h4.522 7.677 5.302 0.406 0.8500

aAll packing materials exhibited complete wetting.

1050 A. Bansal et al.


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Results and Discussion

Newtonian Systems

Based on a wide historical database, Larachi et al. (1999) developed a neural network

correlation. The present observed flow transition data have been compared with thiscorrelation in Figure 2. The correlation predictions were in good agreement with theobserved transition for an air-water system on small spherical glass beads(dp ¼ 3.337 mm, e ¼ 0:404) with mean relative quadratic error (MRQE) Equation(9) of 0.0469. The model due to Grosser et al. (1988) was found to predict well thedata on the air-water system using glass beads I with MRQE ¼ 0.0783 (Figure 3).The effect of surface tension was also well predicted by the neural network corre-lation corresponding to 6 and 12 ppm surfactant (sodium lauryl sulfphate) solutionson 3.337 mm glass beads (MRQE 0.1379 & 0.0590 respectively). In such systems nofoaming was observed until flow transition was achieved. The typical effect of surfacetension of liquid phase on regime transition is shown in Figure 4(a). However, thecorrelation failed to predict the transition for non spherical particles (MRQE0.6111 and 1.1662). This model also failed to predict the flow transition for highlyviscous solutions of 60 and 77% (w=w) aqueous glycerol (MRQE 1.0120 and4.2357). The effect of liquid-phase viscosity is shown in Figure 4(b). However, itmay not be out of place to conclude that the model, although based on a large experi-mental database, is suitable for small-sized spherical packings and accounts for thevariation in surface tension but is not suitable for non spherical packings and forhighly viscous solutions.

The present data were also subjected to various other correlations available inthe literature (Table III). The correlations of Wang et al. (1994), Blok et al.

(1983), and Larachi et al. (1993) show good agreement for the air-water system onspheres with dp¼3.337 mm and e ¼ 0.404 (MRQE 0.1135, 0.3746, 0.0722 respect-ively). However, the correlation of Dudukovic and Mills (1986) under-predicted

Table IIb. Properties of liquid phases: Non-Newtonian (viscoinelastic) fluids

Liquid phase qL, kg m3 K, Pasn n rL, mNm1 T, C

CMC 0.15% 997.6 0.00252 1.0000 63 30.0

CMC 0.30% 997.8 0.01633 0.9157 59 30.0CMC 0.40% 998.5 0.34145 0.8909 57 30.0PEO 0.8% 998.1 0.00230 1.0000 57 29.0PEO 1.6% 1000.5 0.01360 1.0000 55 30.5

Molecular weight of CMC ¼ 1 105; molecular weight of PEO ¼ 3 105.

Table IIc. Properties of liquid phases: Non-Newtonian (viscoelastic) fluids

Liquid phase qL, kg m3 K, Pasn n K1, Pa sm m rL, N.m1 T, C

PAA 0.04% 1000.7 0.0340 0.673 0.0319 0.8265 68 18.0PAA 0.08% 1000.6 0.0640 0.644 0.0699 0.8131 67 17.0

Molecular weight of PAA ¼ 8.7 106.



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the results with MRQE equal to 0.7614. The data corresponding to solid cylinders(dp ¼ 11.450 mm, e ¼ 0:479), Raschig rings (dp ¼ 3.690 mm, e ¼ 0:686), and largeglass spheres (dp ¼ 14.840 mm, e ¼ 0:441) were fairly well predicted by the corre-lation of Wang et al. (1994) and Larachi et al. (1993) with MRQE between 0.1390

Figure 3. Model predictions for air-water system (Grosser et al., 1988).

Figure 2. Parity between Lt,exp and Lt,neural at transition.



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and 0.3261. However, these correlations failed to account for the decrease in surfacetension or increase in viscosity. With decrease in surface tension or increase inviscosity the correlation predictions were much higher, with MRQE greater than 1in most of the cases. Except for the air-water system on 3.337 mm spheres, thepredictions of Blok et al. (1983) were also high (MRQE>1). The predictions byDudukovic and Mills (1986) were low and MRQE was between 0.3476 and

0.7614. From the above observations it is clear that various empirical correlationsare valid for specific conditions, and the variations in fluid properties and bed char-acteristics lead to large errors in correlation predictions.

Figure 4. Effect of liquid-phase properties on regime transition (low to high interaction): (a) effectof surface tension; (b) effect of viscosity; (c) effect of rheological parameters K and n (non-Newtonian viscoinelastic fluids); (d) effect of viscoelasticity (non-Newtonian viscoelastic fluids).



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Different kinds of flow maps are also available to predict the transition. As dis-cussed earlier, different investigators chose different coordinates to plot the flowmaps. The most reliable of these maps is the one proposed by Gianetto et al.(1978), the coordinates for this map being (L=G)kw versus G=ke. The data availablein the literature and the present data, corresponding to different bed geometries andliquid phase properties, are plotted using the axes proposed by Gianetto et al.

(Figure 5). It is observed that the data do not collapse to a single line and lie withinþ150% and 50%. The large scatter or inability to predict the transition by the flowmaps may be explained by the following observations:

. Sicardi and Hoffmann (1980) observed the influence of the shape of packing onpulsing inception, but the maps available in the literature do not include shape fac-tor as a parameter.

. As pointed out by Chou et al. (1977), there is not a theoretical basis to apply thechoice of k and w in trickle bed transitions as originally it was proposed for emptyhorizontal pipes by Baker (1954) and therefore may not properly take into accountthe properties of the liquid and gas phase.

. A flow map of gas velocity versus liquid superficial velocity for the air-water sys-tem does not provide an accurate prediction for other fluids or for beds with dif-ferent porosity or wetting characteristics (Chou et al., 1977).

TABLE III. Empirical correlations used for comparison

Reference Correlation(s)

Blok et al. (1983)

V 0;t

e ffiffiffi

d p p ¼ 0:045 V G

as

Wang et al. (1994) Ltkw ¼ 4:864 G k

0:337

Dudukovic and Mills(1986)

G =ke ¼ 1

Lt ¼ min 103G

kw ; 5:43

e

w

ke

G

0:22" #

Larachi et al. (1993) Ltkwu

G ¼ G

k

1:25

Larachi et al. (1999) Neural network correlationPresent study For Newtonian liquid phase:

Lt

G

kn

S 2

S 21

1=4¼ 5:73 G

ke

4=3For non-Newtonian (viscoinelastic) liquid phase:

Lt;vi

G

kn0

S 2

S 21

1=4¼ 5:73 G

ke

4=3For non-Newtonian (viscoelastic) liquid phase:


2p

We2



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It is therefore worthwhile to develop an empirical correlation that may take intoaccount the bed geometry including bed porosity and shape factor along with theliquid phase properties.

The critical liquid flow rate, Lt, corresponding to the inception of pulsing,depends upon: gas flow rate, physical properties of liquid phase, porosity, particleshape factor or sphericity, and physical properties of the gas phase. This dependencecan be expressed as:

Lt ¼ f ðG ; qL; lL; rL; qG ; lG ; d p; e;/Þ ð1ÞIt is always desirable to club the different parameters so as to yield the dimen-

sionless groups. The dependence of gas and liquid phase properties is consideredto be represented by k and n respectively. The definition of k is taken from the litera-ture (Baker, 1954) as:

k ¼ qG qair

qLqW

1=2ð2Þ

Figure 5. Regime flow map (Gianetto et al. 1978).



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and the present data when subjected to multivariable regression yielded the form of nas:

n

¼ rW

rL

3:5lL

lW

0:5qW

qL

1=3

ð3Þ

The exponent of rw=rLð Þ seems to be rather high, which may be due to the smallrange of rL over which the present data has been generated. The 6 and 12 ppm sur-factant solutions in water showed little or no foaming in the low-interaction regimebut severe foaming was observed in the high-interaction regime. Data observedunder non-foaming conditions have been used to develop Equation (3). However,Equation (3) may not extend to foaming systems and should be used with cautionfor rLvalues beyond the range of data upon which this correlation is based.

Wanchoo et al. (in press), based on their studies on particles of different shapeand size, observed that the effect of bed configuration on hydrodynamics of a trickle

bed reactor can satisfactorily be represented in terms of two parameters, S1 and S2.These parameters, which have been so defined that they include particle effectivediameter, specific surface area, porosity of the bed, characteristic length of nonsphe-rical particles and sphericity, are:

S 1 ¼ asd pe

ð4Þ

and

S 2

¼

l

d ps

1/

ð5

ÞLt, which depends on the gas and liquid phase property groups k and n and bed

configuration parameters S1 and S2, may be expressed as:

Lt ¼ f ðG ; S 1; S 2; e; n and kÞ ð6Þ

Rearranging Equation (6):

Lt

G kn ¼ a G

ke b

S 1 c

S 2ð Þd ð7Þ

Using non linear regression technique (the Nelder and Mead method) the observeddata on transition were fitted to the above equation. The constants of the equationthus obtained are:

a ¼ 5:73 0:13b ¼ 4

3 0:05

c

¼1

2 0:06

d ¼ 14

0:08



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With these constants, Equation (7) assumes the form:

Lt

G

kn

S 2

S 21

1=4¼ 5:73 G

ke

4=3ð8Þ

R2¼ 0:98, variance ¼ 43:75, rmsd ¼ 0:77.The regime flow map based on the correlation equation is shown in Figure 6. It

is observed that the data collapses on a single curve within 20%. The correlationequation, when used to calculate the liquid phase transition mass velocity data, couldpredict the data of other investigators for different liquid phase properties and bedconfigurations to within 20%, as shown in Figure 7, whereas the present data,which had wide variations in the liquid phase properties and bed configurations,remained within 15% (Figure 8).

To verify the validity of the correlation Equation (8), the data from differentsources available in the literature was assigned to Equation (8). The mean relative

Figure 6. Proposed regime transition flow map.



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quadratic error (MRQE), defined below, was calculated:

MRQE ¼ ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiP e p

e 2n 1s ð9Þ

where ‘e’ is experimental and ‘p’ is predicted value of L at transition. MRQE valuesfor different systems are in Table IV.

Non-newtonian Systems

Viscoinelastic Systems

Different concentrations of carboxymethyl cellulose (CMC) and polyethylene oxide(PEO) have been considered to assess the effect of fluid rheology. Different concen-

trations of CMC exhibited pseudo-plastic behavior, however, for the differentconcentrations of low molecular weight PEO used in the present study, the flowbehavior index, n, remained as unity, and the solutions are expected to show a purely

Figure 7. Parity between present model Equation (8) and literature data.



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T a b l e

I V b .

S t a t i s t i c a l c o m p a r i s o n o

f p r e s e n t c o r r e l a t i o n E q u a t i o n ( 8 ) w i t h l i t e r a t u r e d a t a

R e f e r e n c e

C h o u

e t a l . ( 1 9 7 7 )

C h o u e t a l .

( 1 9 7 7 )

C

h r i s t e n s e n

e t a l .

( 1 9 8 6 )

I l i u t a a n d t h y r i o n

( 1 9 9 7 )

I l i u t a e t a l .

( 1 9 9 6 )

T o s u n

( 1 9 8 4 )

S y s t e m

A i r - w a t e r ,

s p h e r e s ,

2 . 9 m m

A i r - 2 . 5 p p m

T e r g i t o l N P X ,

s p h e r e s ,

2 . 9 m m

A

i r - w a t e r ,

s p h e r e s ,

3 . 0 m m

A i r - w a t e r ,

s p h e r e s ,

3 . 3

3 m m

A i r - w a t e r ,

s p h e r e s ,

3 . 0 m m

A i r - w a t e r ,

s p h e r e s ,

1 . 9 m m

M R Q E

E q u a t i o n ( 9 )

0 . 1

1 7 8

0 . 2

4 8 7

0 . 0

6 3 2

0 . 1

1 8 8

0 . 2

6 2 5

0 . 1

5 9 6

R e f e r e n c e

T o s u n

( 1 9 8 4 )

S a i a n d

V a r m a

( 1 9 8 8 )

S a i a n d

V a r m a

( 1 9 8 8 )

S a i a n d

V a r m a

( 1 9 8 8 )

S a i a n d

V a r m a

( 1 9 8 8 )

S a i a n d

V a r m a

( 1 9 8 8 )

S y s t e m

A i r - 2 8 %

g l y c e r i n ,

s p h e r e s ,

1 . 9 m m

A i r - w a t e r ,

c y l i n d e r s ,

5 . 2

7 m m

A

i r - w a t e r ,

s p h e r e s ,

4 . 3

2 m m

A i r - w a t e r ,

s p h e r e s ,

2 . 3

9 m m

A i r - w a t e r ,

R a s c h i g r i n g s ,

1 . 5

9 m m

A i r - 2 0 %

g l y c e r i n ,

c y l i n d e r s ,

5 . 2

7 m m

M R Q E

E q u

a t i o n ( 9 )

0 . 0

9 2 6

0 . 2

4 6 5

0 . 2

5 1 0

0 . 1

7 3 5

0 . 8

6 9 4

0 . 3

7 3 2

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where K1 and m are model parameters. To evaluate N1 the method describedby Abdel-Khalik et al. (1974) has been used. The fluid characteristic time keff isevaluated from the shear dependent viscosity data (s and N 1 versus _cc) and isdefined as:

keff

¼

N 1

2_

ccs ð15

ÞThe shear rate _cc, evaluated at wall conditions of a capillary tube, is related to thepacked bed apparent shear rate, 8V 0=eDe, at the same wall shear stress by a shift fac-tor of k i 2 (Kozicki et al., 1967),

_ccw ¼3n þ 1

4n

8V 0eDe

ki2

ð16Þ

where k i is 4.17 using the Blake-Kozeny equation (Blake, 1922).Viscoelasticity of the liquid phase is accounted for in terms of the Weissenberg

number, We, defined as:

We ¼ keff :_ccw ð17Þ

Figure 9. Parity plot for viscoinelastic liquid phase using Equation (13).



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Correlation for Viscoelastic Systems

The effect of various parameters other than viscoelasticity has already been takeninto account while developing the correlation to evaluate L at transition correspond-ing to viscoinelastic non-Newtonian systems. For viscoelastic systems the depen-

dence of L at transition on liquid-phase elasticity may be expressed as:Lt;ve

Lt;vi ¼ 1 þ AWeB ð18Þ

A similar dependence has been previously used by Carlos et al. (1997) for flow of viscoelastic fluid through a packed bed.

Using a nonlinear regression technique (the Nelder and Mead method), theobserved flow transition data on viscoelastic fluids were fitted to Equation (18),yielding the following correlation constants:

A

¼ ffiffiffi2

p

0:123

B ¼ 2R2 ¼ 0.94, variance ¼ 0.0979, rmsd ¼ 0.0757.

Figure10. Parity plot for viscoelastic liquid phase using Equation (19).



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Thus the flow transition correlation for viscoelastic fluids assumes the form:


2p

We2

ð19Þ

where Lt,ve is the liquid phase transition velocity for viscoelastic fluids and Lt,vi is theliquid phase transition velocity for viscoinelastic fluids predicted by Equation (13).The parity plot between the experimental and predicted values for Lt,ve shows a goodagreement, and the data are found to remain within 10% (Figure 10).

Conclusion

Different flow maps and empirical correlations are available in the literature but noneof them seems to be applicable to the systems having different bed configurations anddifferent liquid phase properties. Based on the experimental observations the effect of

particle shape and bed configuration has been taken into consideration, and thedependence of the physical properties of the liquid phase are reconsidered. Whiledeveloping the correlation equation the effect of bed configuration has been foundto be well represented by parameters S1 and S2. The correlation developed couldpredict the data of different investigators with reasonable accuracy of 20%(Figure 6) on the flow map as against the coordinates of Gianetto et al., where itwas within þ150% and –50%. Further, the present correlation predicts the datavery well for viscoinelastic systems when liquid phase viscosity is replaced byapparent viscosity. The present correlation has been extended for viscoelastic liquidsby introducing the Weissenberg number and predicts the flow transition up to 10%.

Experimental results obtained on large-sized packing materials with aspect

ratios less than 12 could be under influence of ‘‘wall flow,’’ and care must be takento use this data to predict the flow transition for such systems.

Acknowledgment

Financial support from All India Council for Technical Education, Government of India, New Delhi under grant number ‘F.NO. 8017=RDII=R&D - 617=1999–2000’ isgratefully acknowledged.

Nomenclature

A, B constants in Equation (18)a, b, c, d constants in Equation (7)as specific surface area, m

2 m3

CMC carboxymethyl cellulosedp effective particle diameter ¼ equivalent spherical particle diameter-

sphericity of the particle ¼ dps/ , mdps equivalent spherical diameter, i.e., the diameter of the sphere having the

same volume as that of the particle, m

De d pe=1:5 1e

ð Þ þd p

Dc, equivalent diameter, m

Dc column diameter, me experimental value , Equation (9)



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G superficial gas mass velocity, kg m2 s1

K flow consistency index, Pa sK1 parameter, Equation (14)ki constant, Equation (16)

L liquid-phase mass velocity, kg m2 s1Lt,neural Lt predicted by neural network correlation

(Larachi et al., 1999), kg m2 s1

l characteristic length of the nonspherical particle ordiameter of the spherical particle, m

MRQE mean relative quadratic error, Equation (9)m parameter, Equation (14 )n flow behavior indexN1 first normal difference, Equation (14)PAA polyacrylamide

PEO polyethylene oxidep predicted value, Equation (9)S1 asd p=re

S2 l =d ps 1

/

V0 superficial liquid-phase velocity, m s1

VG superficial gas-phase velocity, m s1

We Weissenberg number, Equation (17)

Greek Letters

_cc shear rate, s1

_

ccw shear rate at wall, Equation (16), s

1

e bed porosityk a parameter, Equation (2)keff fluid characteristic time, Equation (15), sla

viscosity of a-phase, Pa sla apparent viscosity of liquid phase, Pa sn, n0 parameters as defined by Equations (3) and (12) respectivelyqa

density of a-phase , kg m3

ra

surface tension of a-phase, Nm1

s shear stress, N m2

/ particle sphericity

w rW rL

lLlW

qW qL

2

1=3u correlation parameter, Larachi et al. (1993)

Subscripts

exp experimental valueG gas phaseL liquid phase

pred predicted valuet at transitionve viscoelastic liquid phasevi viscoinelastic liquid phasew water



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References

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