Powder Technology 211 (2011) 127–134

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Interfacial stress in non-Newtonian flow through packed bed

Suresh Kumar Patel, Subrata Kumar Majumder ⁎Department of Chemical Engineering, Indian Institute of Technology Guwahati, Guwahati-781039, Assam, India

⁎ Corresponding author. Tel.: +91 3612582265; fax:E-mail addresses: skmaju@iitg.ernet.in, rsmaju@yah

0032-5910/$ – see front matter © 2011 Elsevier B.V. Aldoi:10.1016/j.powtec.2011.04.010

a b s t r a c t

a r t i c l e i n f o

Article history:Received 7 February 2011Received in revised form 2 April 2011Accepted 13 April 2011Available online 20 April 2011

Keywords:Packed bedInterfacial areaNon-Newtonian liquidWettabilityShear stress

This study investigates the pressure drop characteristics, shear stress in packed bed with shear thinningpower law type non-Newtonian liquid. A mechanistic model has also been developed to analyze the pressuredrop and interfacial stress in packed bed with non-Newtonian liquid by considering the loss of energy due towettability. The Ergun's and Foscolo's equations were used for comparison with the experimental data. TheErgun equation was modified to account for the effect of flow behavior index of non-Newtonian fluid in thecolumn. The intensity factor of shear stress and the friction factor were analyzed based on energy loss due towettability effect of liquid on the solid surface.

+91 3612582291.oo.com (S.K. Majumder).

l rights reserved.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Packed bed reactors aremost commonly employed in the chemicalprocess industries among the several possible types of multiphasecatalytic reactors. Their popularity halts from their effectiveness interms of performance as well as low capital and operating costs. Thenon-Newtonian fluid flow through particulate bed system is impor-tant in a variety of chemical and biochemical processes [1]. Variousexamples of applications of the particulate system have beendescribed by many authors [1–4]. Studies on the flow of fluidsthrough porous media were restricted mostly to Newtonian fluids.Recently, the flow of non-Newtonian fluids through packed beds andporous media has received considerable attention because of itsimportance in various industrial applications. Considerable researchefforts have been expended in exploring and further understanding ofthe basic phenomena of momentum, heat andmass transfer processeswith and without chemical reactions in particulate system.

1.1. Previous work

Voluminous literature available on the flow of a variety of non-Newtonian materials through packed beds has been critically reviewedpreviously [5,6]. Wu and Pruess [6] described the non-Newtonian flowbehavior in packed bed including beds of uniform size and of multi-sizeparticles. Some other different studies related to Newtonian and non-

Newtonian flow behavior in packed bed has been thoroughly reportedin literature [7]. From the studies it is concluded that each study has itssome uniqueness in its morphology which is contributing in somemeasure to the complexity of the problem. Similarly, often inadequaterheological characterization alsoadds to the complexityof suchsystems.While it is usual for most non-Newtonian materials to exhibit shearthinning behavior, many other features including time-dependency,viscoelasticity, yield stress, etc., are also present but not oftenmeasured.Certainly, the major research effort has been directed at developingsimple and reliablemethods of predicting the frictional pressure loss forthe flow of non-Newtonianfluids through packed beds. Kozicki et al. [8]generalized the average velocity–pressure gradient relationship forarbitrary time-independent non-Newtonian fluids in porous media,based on the Blake–Kozeny capillary model. Mishra et al. [9] describedaverage shear stress–shear rate relationship topredict theflowbehaviorof power law as well as non-power law fluids. The wall factor is alsoanother phenomenon to affect theflowbehavior in porousmediawhichis a function of both diameter ratio and particle Reynolds number [10].For Reynolds number below 1.0, this dependency is nearly same forsettling in Newtonian and non-Newtonian liquids. In the range of1bRe∞b200, wall effect can be estimated for the non-Newtonian casefrom the relation applicable to settling in Newtonian liquids. Zhu andSatish [11] studied thedragphenomenawhichdecreasewith a decreasein flow behavior index and with an increase in the characteristic time.They found that both the normal stress difference and the bed voidagehave a great influence on the resistance of visco-elastic flow through apacked bed. Rao et al. [12] studied the pressure loss-throughputbehavior for the flow of inelastic power law fluids through randomlypacked spherical particles and over wide ranges of operating andphysical conditions. Sabiri et al. [13] investigation covers a large range ofReynolds number including creeping and inertial flow regimes. They

128 S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134

also studied a large range of Reynolds number including inertial effectsof pressure gradient through porousmedia in the case of purely viscousfluid flow. Vossoughi et al. [14] reported that the pressure drop of aporous media flow is only due to a small extent to the shear force. Theyalso studied effect of matrix polymer as an additive to pressure dropphenomena in the packed bed. Basu et al. [15] developed a model toelucidate thewall effect on pressuredropandmassflux. They found thatthe pressure drop increases with the increase in flow rate. The pressuredrop also increases with the increase in solute concentration for a givenflow rate. Chhabra et al. [16] critically analyzed the flow of complexfluids through unconsolidated fixed beds and fluidized beds. Theyparticularly focused on the prediction of macro-scale phenomena offlow regimes, pressure drop in fixed and fluidized beds. Gandhidasan etal. [17] studied the irrigatedpressure drop and found that the structuredpackinghas the lowerpressuredrop andhigher capacity comparedwithrandom packing. Nemec et al. [18] investigated the wall effect onpressure drop and concluded that the effect is negligible as long as thecolumn-to-particle diameter ratio is above 10. By phenomenologicaland empirical analyses they upgraded the original Ergun equation. Theyreported thatwith the proposed upgraded Ergun equation one is able topredict single-phase pressure drop in a packed bed of any shape ofparticles. Alopaeus et al. [19] developed a model to analyze thehydrodynamic parameters in packed-bed based on one-dimensionalmaterial and momentum balances for gas and liquid phases. Montilletet al. [20] studied the pressure drop through packed beds of spheres.They interpreted the behavior of dense packing which is characterizedby porosities in the range 0.36–0.39 (uniform spheres) or even less.Bendova et al. [21] observed the creeping flow behavior of fluids ofdifferent rheological behaviors through fixed beds of spherical and non-spherical particles. Yilmaz et al. [22] studied the Newtonian (distilledwater) and non-Newtonian (polyacrylamide solutions with concentra-tions 5 and 10 ppm) flow behaviors in porousmedium. They found thatthe permeability for the distilled water is almost constant. Thepermeability of the non-Newtonian visco-elastic fluid flow in porousmedium significantly depends on the pressure drop in the system. Fromthe literature it is found that voluminous work is available for non-Newtonian fluid flow through packed beds without considering theinterfacial stresses. Also there is a lack of studies on pressure dropcharacteristics based on the wetting effect of the solid–liquid surfaceduring the flow. The objective of the present study is to study thepressure drop characteristics, shear stress phenomena in packed bedwith shear thinning power-law type non-Newtonian liquid system anddevelopment of mechanistic model to analyze the pressure drop andinterfacial stress in packed bed based on the wetting effect.

2. Theoretical background

2.1. Macroscopic model for pressure drop

The fluid dynamic aspect of single phase through packed beds hasbeen described in this section using an internal flow model based onanalogy with flow through pipes. The dynamic interaction betweenliquid and solid wall is taken into account in modeling the flow byintroducing the rate of energy dissipation. The mechanical energybalance is used to calculate the pressure drop which can be lookedupon as either the force per unit area of cross section required toovercome frictional forces or the energy dissipation per unit volume.The model is presented with the following assumptions:

(i) the flow is steady and isothermal with the voidage and holdupbeing uniform.

(ii) acceleration effect is negligible due to absence inter-phasemass transfer.

(iii) frictional loss is considered as uniform throughout the columnfor a particular liquid flow rate.

The mechanical energy balance equation for the liquid phase isgiven by

ΔPlsAcVsl−gΔZAcρlVsl−El = 0: ð1Þ

In Eq. (1) the first term is “energy due to liquid–solid pressure(N.m/s)”, second term is potential energy (N.m/s) and the third termis energy dissipation per unit packed volume due to friction andwettability between liquid and solid. Eq. (1) can be represented as

ΔPls−ρlgΔZ = El = VslAc: ð2Þ

Theenergydissipation in solid–liquid surfaceoccursdue todrag forceexerted by thefluid on particle [23]. In Eq. (3), the amount ofmechanicalenergy (El) is irreversively converted to thermal energy due to friction.The total energy dissipation can be calculated from the product of theforce exerted by the fluid on a single particle, the fluid velocity and thetotal amount of particle present which can be represented as:

El = Fd Vsl = εð ÞNp = Cd14πd2p

12ρl Vsl =εð Þ2 Vsl = εð ÞNp ð3Þ

Np =ΔZ 1−εð ÞAc

1= 6ð Þπd3p: ð4Þ

Therefore Eq. (2) can be written as:

ΔPls−ρlgΔZ = 3 = 4ð ÞCDρlV2sl1−εε3

ΔZdp

: ð5Þ

From the experiment the total pressure drop can be obtained assummationof frictional pressure drop and thehydrostatic pressure dropas

ΔPls = ΔPfl + ρlgΔZ ð6Þ

where ΔPfl is the frictional pressure loss due to liquid flow. Therefore,Eqs. (5) and (6) give

ΔPfl = 3 = 4ð ÞCDρlV2sl1−εε3

ΔZdp

: ð7Þ

To determine the frictional losses due to liquid flow in the columna model can be formulated on the basis of the following assumptions:(i) the friction factor for liquid phase is a constantmultiple,α′ of that ifonly flow of single liquid phase without packing takes place in thecolumn. (ii) The area of contact of the liquid phase with wall is α″times to that of only flow of single phase without packing in thecolumn. From these assumptions a simple overall momentum balancefor liquid phase can be represented as [12]:

ΔPfl Cross sectionalareað ÞlsΔPfl0 Cross sectionalareað Þl0

=Wall shear stressð ÞlsWall shear stressð Þl0

×Areaof contactwithwallð ÞlsAreaof contactwithwallð Þl0

⇒ΔPflAcεΔPfl0Ac

=0:5fρlV

2sl

� �ls

0:5fρlV2sl

� �l0

×Areaof contactwithwallð ÞlsAreaof contactwithwallð Þl0

=flsfl0

:V2sl

� �ls

V2sl

� �l0

×Areaof contactwithwallð ÞlsAreaof contactwithwallð Þl0

= α′l:1ε2

:α″l =

αl

ε2

ð8Þ

⇒ΔPfl =αlΔPfl0

ε3ð9Þ

where, the subscripts “l0” refer to liquid single phase, “ls” refers toliquid–solid wall, “sl” refers to liquid superficial, (Vsl)ls=(Vsl)l0/ε,

129S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134

αl=α′lα″l and cross-sectional area for the flow of liquid is equal to εltimes the cross-sectional area of the column. α′l is defined as fls/fl0 andα″l is defined as (area of contact)ls/(area of contact)l0. The value of α″lis equal to the specific interfacial area in the packed bed. Theparameter αl is an intensity factor which signifies intensity ofinterfacial shear stress. ΔPfl0 is the frictional pressure drop due toliquid when only liquid phase flows through the column.

2.2. Energy loss due to wettability between liquid and solid wall

The total rate of energy loss due to wettability of liquid with thetotal surface of solid wall is the summation of the energy loss due towettability between liquid and surface of column wall and the energyloss due to wettability between liquid and surface of packing. This canbe represented as:

Ew =πdcVslσls

ε+

Vslσls

ahεð10Þ

where ah is the hydraulic or effective specific area of packing. Thehydraulic specific area of packing can be calculated from thecorrelations [24]:

aha

= ChRe0:5l Fr0:1l for Relb5 ð11Þ

aha

= 0:85ChRe0:25l Fr0:1l for Rel≥5 ð12Þ

where Rel=(Vslρl)/(aμeff), Frl=(Vsl2a)/g. Values of specific surface area

of packing (a) and Ch are characteristic of the particular type and sizeof packing. The value of Ch represents the shape factor of the particle.In the present system, for ceramic rashing ring of 10 mm size thevalues of a and Ch were taken as 440 m2/m3 and 0.791 respectively[24]. Wettability is the affinity of the solid matrix for the aqueousphases. It is normally quantified by the value of the contact angle. Thecontact angle θbπ/2 indicates that the solid is wetted by the liquid,and θNπ indicates non-wetting. The limits θ=0 and π define completewetting and complete non-wetting, respectively. The energy loss dueto wetting of liquid depends on the dynamic contact angle betweenliquid and solid wall [25]. The dynamic contact angle (θ) isapproximately equal to Ca1/3 [25], where Ca is the capillary numberwhich is defined as, Ca=μeffVsl/εσl [25]. The surface tension betweenliquid and solid column wall (σls) can be calculated from Young'sequation as

σl cosθ = σls ð13Þ

where σls, is the interfacial surface tension at the boundaries betweenliquid (l) and solid (s). Here, σ represents the force needed to stretchan interface by a unit distance. In the present study, it is found that thecontact angle is below 90°, which indicates that the solid surface iswetted by the liquid. The range of superficial liquid velocity used inthe present study is 0.004 to 0.05 m/s.

2.3. Determination of model parameters

The single liquid phase (without packing) frictional pressure dropwas calculated using Fanning's equation:

ΔPfl0 = 2fl0 ρlV2slΔZ = dc ð14Þ

the friction factor is calculated as

fl0 =16Ren

for laminar flow ð15Þ

fl0 =0:079

n5 Renð Þ 2:6310:5n

for turbulent flow ð16Þ

where Ren is the Reynolds number for non-Newtonian fluid flowwhich is defined as [1]

Ren =dncV

2−nsl ρ

8n−1K4n

3n + 1

� �n

: ð17Þ

For transition flow, the friction factor f can be deduced from thegeneralized pressure loss equation as [26]:

fl0 = 0:125 nffiffin

p0:0112 + Re−0:3185

n

� �h ifor transition flow: ð18Þ

Substituting the Eq. (9) for ΔPfl into Eq. (6), one gets

ΔPls =αlΔPfl0

ε3+ ρlgΔZ: ð19Þ

From Eq. (19) αl can be expressed as

αl =ε3

δfl0δls−1½ � ð20Þ

where, δls=ΔPls/(ρlgΔz), δfl0=ΔPfl0/(ρlgΔz). Again from Eqs. (7) and(9), the drag coefficient can be expressed as

Cd =43

αlΔPfl0ρlV

2sl 1−εð Þ

dpΔZ

: ð21Þ

In Eq. (20), except αl, all parameters are known. Using theexperimental pressure drop, the corresponding values of αl can becalculated for different variables fromEq. (20). To estimate the values ofαl, the experimental data of ΔPls for different operating conditions ofnon-Newtonian flow in the packed bed were taken from theexperimental results. Once the value of αl will be calculated fromEq. (20), the drag coefficient can be calculated from Eq. (21). From thedefinition of αl, one can calculate the effective friction factor, fls in thepacked bed as:

fls =αl fl0α″

l:ð22Þ

The value of α″l is equivalent to the hydraulic specific interfacial areain the packed bedwhich is calculated by Eqs. (11) and (12) for differentranges of Reynolds number. The Blake–Kozeny's hydraulic modelassumes that the bed of solid particles consists of irregularly shapedchannels provided by the space between the particles in the bed. Thesechannels are considered as large numbers of capillary tubes runningparallel to the direction of flow. The tortuous path of the channel, Zc,traversed by the fluid elements is more than the bed length, Z. Theaverage liquid–solid surface interfacial shear stress is expressed as:

τi =12

flsρlV2sl

� �= ε2 = Rh

ΔPflZc

ð23Þ

where Rh is called hydraulic radius which is defined as:

Rh =dpϕs

6ε

1−ε

� �: ð24Þ

Shape factor is defined as:

ϕs =vpap

:6dp

: ð25Þ

130 S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134

Tortuosity factor,

β =ZcZ: ð26Þ

Thus,

fls =dpϕs

3βZε3

1−ε

!ΔPflρlV

2sl

=αlfl0α″

l:ð27Þ

Then the tortuosity factor can be expressed as

β =dpϕs

3Zε3

1−ε

!ΔPflρlV

2sl

α″l

αlfl0=

14Cdα″

lϕs

αlfl0: ð28Þ

If there is a slip at the interface, Eq. (23) can be expressed as

τi =12

flsρlV2sl

� �= ε2 = Rh

ΔPflβZ

= K3n + 1

4n

n 2 Vsl−Visð ÞεRh

n: ð29Þ

3. Experimental setup and procedure

The schematic diagram of the experimental setup is shown inFig. 1. A perspex column of inner diameter 0.050 m and of length0.49 m packed randomly with ceramic rashing rings of 10 mmdiameter and 10 mm height is used for the present experiment. Theshape factor of the packing is 0.791. The bottom of the packed bed isconnected to a pipe to drain liquid from the packed bed after theexperiment is over. There is a bypassing arrangement after the pumpwhich returns the liquid back to storage tank. The column is kept

Fig. 1. Schematic diagram of the experimental setup.

approximately full-filled with packing for all trials. The inlet fluid wasprepared using cold distilled water by gradually adding the carboxymethyl cellulose (CMC) powder to the water with gentle stirring untilthe solution becomes homogeneous. The CMC solution is fed into thepacked bed by using the centrifugal pump. The volumetric flow rate ismeasured by rotameter. The CMC solution moved up through thepacked bed. The U-tube manometer connected to the packed bed isused to measure the pressure drop across the bed at different flowrates and CMC concentrations. Because of the time needed to preparethe concentrated CMC solution, first the testing was started at lowflow rates of CMC to obtain as many trial runs as possible. After thecolumn reached a steady state, the data for pressure drop (before andafter the addition of CMC) were recorded. The pressure tapconnections to each column were situated in the packing sections atlocations 5.0 cm from the top and from the bottom of the bed, to yielda direct pressure drop measurement without the necessity ofcorrection for end effects. Pressure differences were measured withdifferential carbon tetrachloride (sp. gr.=1.65) or mercury in glassmanometers depending on the pressure drop range. The void fractionof liquid or specific liquid holdup is 0.650. The readings were repeatedfour times to ensure the reproducibility.

3.1. Physical properties and effective viscosity of the liquid

The properties of non-Newtonian fluids cannot be described withNewton's law of viscosity as the viscosity of these fluids proves to bedependent on the rate of shear. Moreover, the viscosity of these fluidscan increase or decrease due to the changes of the rate of shear which,again, is subject to the nature of the fluid. The non-Newtonian fluids,unlike the Newtonian, are defined as materials which do not conformto a direct proportionality between shear stress and shear rate. In thepresent study the carboxy methyle cellulose (CMC) is used as a shear-thinning power-law type non-Newtonian liquid. The rheologicalparameters of the non-Newtonian liquid for its different concentra-tions are shown in Table 1. The effective viscosity of the liquid flowingthrough the porous media can be calculated as [27]:

μeff = 12n′−1K′ 150Krεð Þ 1−n′� �

=2: ð30Þ

The parameters n′and K′ are called Metzner and Reed parameter[28]. For power law fluids, the values of n′and K′ are equal to n and K[(1+3n)/4n]n respectively. Substitution of n′ and K′ in Eq. (30) gives

μeff =K12

3 + 9nn

� �n

150Krεð Þ 1−nð Þ=2: ð31Þ

The parameter Kr in Eqs. (30) and (31) is called permeability of theporousmediumwhich can be defined fromBlake–Kozeny equation [29]as

Kr =d2pε

3

150 1−εð Þ2 : ð32Þ

Table 1Physical properties of the non-Newtonian liquid.

CMC conc.(wt%)

Density(kg/m3)

Consistency(K, Pa.sn)

Flow indexproperties (n)

1.0 1000.96 0.00318 0.9481.5 1001.13 0.00419 0.9102.0 1001.37 0.0059 0.8712.5 1001.50 0.00692 0.850Water 998.50 – –

0.02

0.04

0.06

0.08

0.10

0.12

α l/α

l'', [-

]

Symbol CMC soln. 1.0 wt% 1.5 wt% 2.0 wt% 2.5 wt%

0

Ren, [-]

800700600500400300200100

Fig. 3. Variation of intensity factor with non-Newtonian Reynolds number.

131S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134

4. Results and discussion

4.1. Pressure drop characteristics for non-Newtonian fluid flow inpacked bed

In this present study the frictional pressure drop characteristic hasbeen analyzed by a mechanistic model based on the viscousinteraction and wetting of the non-Newtonian liquid. The intensityfactor (αl) of the liquid–solid surface interfacial shear stresscharacterizes the intensity of the pressure drop in the fluid flow.Variation of the frictional pressure drop (ΔPfl/Δz) against Reynoldsnumber based on non-Newtonian flow (Ren) for different CMCconcentrations is shown in Fig. 2. It is seen that the pressure dropincreases with the increase in liquid flow rate. This is due to the factthat the interfacial shear stress increases with increase in liquidvelocity. Also the viscous friction at the wall increases with theincrease in liquid velocity, which increases the pressure drop. Further,the pressure drop increases with the increase in CMC concentrationfor a given flow rate. The increase of CMC concentration leads to theincrease in effective viscosity which leads to increase the shear stressand increase of pressure drop in the packed bed. Basu [15] also got thesimilar trend of pressure drop in non-Newtonian fluid flow throughpacked bed but with different shapes of packing materials. Hereported that the increase in pressure drop with liquid velocity isdue to increase in consistency index of the non-Newtonian fluid. Thisis true because increase in consistency index enhances the effectiveviscosity of the fluid. A similar behavior was also observed by theother investigators [30–33].

4.2. Intensity factor of interfacial shear stress

The intensity factor of interfacial shear stress is decreasing withincrease in non-Newtonian liquid Reynolds number (Ren) as shown inFig. 3. The intensity factor depends on the effective viscosity of theliquid. As the flow rate increases the effective viscosity decreaseswhich causes the decrease in viscous force. At the same time theinertial force also increases but the rate of increase in inertial force ishigher compared to the rate of decrease in viscous force. Henceincrease in Reynolds number leads to increase the intensity factor.This can also be interpreted in terms of hydraulic specific interfacialarea. The intensity factor increases with increase in hydraulic specificinterfacial area. The intensity factor by definition depends on the ratioof area of contact of liquid with the solid wall with packing to that ofwithout packing. The area of contact of liquid with the solid wall withpacking is measured by the hydraulic specific interfacial area which

00

50

100

150

200

250

300

350

400

ΔPfls

/ΔZ

, [P

a/m

]

Ren, [-]

Symbol CMC (wt%) 1.0 1.5 2.0 2.5

800700600500400300200100

Fig. 2. Frictional pressure drop over a packed bed as a function of Reynolds number.

depends on the Reynolds number as per Eqs. (11) and (12). Thereforeit is concluded that the intensity factor is a function of Reynoldsnumber. Also as the concentration of CMC solution increases, theeffective viscosity decreases at constant flow rate and results indecrease of intensity factor with increase in concentration of CMCsolution as shown in Fig. 3. The variation of shear stress intensityfactor with non-Newtonian liquid Reynolds number can be expressedby developing a correlation (Eq. (33)) as:

αl =3:484α″

l n10:98

Re0:814n2:557

n

: ð33Þ

The parity of goodness of fit of the predicted value of shear stressintensity factor with the value calculated by the equation is shown inFig. 4. The correlation coefficient of the correlation is 0.992 andstandard error 0.00546. The standard error was calculated as perEq. (34).

STEYX =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n−2ð Þ ∑ y−yð Þ2− ½ x−xð Þ y−yð �2∑ x−xð Þ2

sð34Þ

0.000.00

0.02

0.04

0.06

0.08

0.10

0.12

45o line Symbol CMC soln. 1.0 wt% 1.5 wt% 2.0 wt% 2.5 wt%

α l/α

'' l-pr

edic

ted,

[-]

αl/α''l-experimental, [-]

0.120.100.080.060.040.02

Fig. 4. Parity plot of experimental and predicted values for intensity factor of shearstress at different CMC solutions.

1

10

100

Symbol CMC soln. 1.0 wt% 1.5 wt% 2.0 wt% 2.5 wt%

f lsα'

' l, [-

]

0

Ren, [-]

800700600500400300200100

Fig. 5. Friction factor is a function of Reynolds number at different CMC solutions.

0.00.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

f lsα l

'', [-

]

Ewx104, [N.m/s]

Symbol CMC(wt%) 1.0 1.5 2.0 2.5

8.07.06.05.04.03.02.01.0

Fig. 7. Friction factor is a function of energy loss due to wettability at different CMCsolutions.

132 S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134

where, STEYX is the standard error which is a measure of the amount oferror in the prediction of y for an individual x. y's are thedependent datapoints (predicted values) and x's are the independent data points(experimental values). x and y are themeans of total datapoints of x andy respectively.

4.3. Analysis of friction factor and shear stress

The friction factor can be calculated from Eq. (22) by knowing thevalue of the intensity factor from Eq. (20). Fig. 5 presents some of thetypical friction factor–Reynolds number plots for the differentconcentrations of non-Newtonian liquid (CMC). The friction factordepends on the intensity factor of shear stress and flow behaviorindex of non-Newtonian liquid. The dependency of friction factor canalso be correlated as:

fls =900:7n4:844

α″lRe

1:094n0:832n :

ð35Þ

The value of α″l is the value of specific interfacial area of the packedbed. The above correlation is made in the range of 0.1bRenb730 with

00

2

4

6

8

10

12

14

16

18

20

45o line

Symbol CMC soln. 1.0 wt% 1.5 wt% 2.0 wt% 2.5 wt%

flsα''l-experimental, [-]

f lsα'

' l-pr

edic

ted,

[-]

2018161412108642

Fig. 6. Parity plot of experimental and predicted values for the friction factor at differentCMC solutions.

visco-elastic non-Newtonian fluid. The correlation coefficient and thestandard error of the correlation are 0.9943 and 0.62. The parity ofgoodness of fit is shown in Fig. 6. The present study is beyond therange for Renb0.1. The further study will be done in the creepingflow of non-Newtonian fluid in packed bed forRenb0.1. The frictionfactor can also be represented by the rate of energy loss due towettability. The friction factor decreases with the increase inwettability. The energy loss due to wettability increases with theincrease in surface tension and decrease in porosity. This is because ofcapillary effect of flowing of liquid through porous media. The surfacetension increases with the increase in concentration of CMC whichleads to increase in loss of energy due to wettability. So as the surfacetension increases the friction factor increases and hence increase infrictional losses. The variation of friction factor with the loss of energydue to wettability is shown in Fig. 7. A correlation has been made tointerpret the fact as a function of energy loss due to wettability effecton friction factor as:

fls =1:69 × 10−4n−5:64

E1:09n−0:15

w

: ð36Þ

0.00.0

4.0

8.0

12.0

16.0

20.0

K[(3n+1)/4n] n[2(Vsl-Vis)/(εRh)]nx104

Symbol CMC(wt%) 1.0 1.5 2.0 2.5

12.010.08.06.04.02.0

τ ix10

2 , [N

/m2 ]

Fig. 8. Variation of shear stress with effective shear rate at different CMC solutions.

1001011

10

100

Cd-

pred

icte

d, [-

]

Cd-experimental, [-]

Symbol CMC(wt%) 1.0 1.5 2.0 2.5

Fig. 10. Parity plot of experimental and predicted values for the drag coefficient atdifferent CMC solutions.

133S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134

The shear stress is a function of the non-Newtonian flow behaviorindex and the dynamic variables like flow velocity and the wall slip ifany. The data for different concentrations of CMC solution for shearstress obtained are shown in Fig. 8. At higher flow rates it deviatesfrom the laminar flow curve showing that inertial effects arebeginning or dominating the viscous effects. In the present experi-mental investigation, slip effects are assumed to be negligible.

4.4. Analysis of drag coefficient

The drag coefficient includes both hydrodynamic drag forces onparticles and the column wall. The drag coefficient was calculatedfrom Eq. (21) by using the values of experimental pressure drop dataat above the Reynolds number (Rep) greater than 0.1. The dragcoefficient from the present study has been compared with Ergun'sequation (Eq. (37)) [34] and Foscolo's equation (Eq. (38)) [35] of dragcoefficient which is represented as

Cd =43

150Rep 1−εð Þ + 1:75

!ðErgunÞ ð37Þ

Cd =43

17:3Rep

+ 0:336

!ε−1:8 ðFoscoloÞ: ð38Þ

From the graphical representation (Fig. 9), it is seen that the dragthat is obtained by the present work is smaller than predicted byErgun's equation and greater than Foscolo's equation. It is found thatthe present result is well approximate to the Ergun's result. Thedeviationmay be the non-Newtonian characteristics of the fluid. SinceCd is a function of frictional pressure drop a pressure drop relationsuch as the Ergun's equation was adapted to develop the model topredict the drag coefficient for the present study with non-Newtonianliquid within the stipulated range of the experimental variables. Inthis case two parameters k1 and k2 were introduced to signify theeffect of flow behavior index of the non-Newtonian liquid. Eq. (42)describes the drag coefficient in the present stipulated range ofexperimental variables of: 0.004buslb0.05 m/s, CMC concentration1.0 to 2.5% (wt) which is represented as:

Cd = k143

150Rep 1−εð Þ + 1:75

!" #k2ð39Þ

0

5

10

15

20

25

30

35

40

Cd,

[-]

Rep, [-]

Symbol CMC (wt%) n

1.0 0.948 1.5 0.910 2.0 0.871 2.5 0.850 As per Ergun eqn. As per Foscolo eqn.

25020015010050

Fig. 9. Drag coefficient as a function of particle Reynolds number at different CMCsolutions.

where

k1 = 0:7087n−0:1042 ð40Þ

and

k2 = 2:3817−1:2738n: ð41Þ

It is seen that the present experimental data is well fitted for dragcoefficient with predicted values by the correlation (Eq. (39)). Thestandard error of the predicted values with the experimental data ascalculated by Eq. (34) is 1.10. The parity of experimental data withpredicted data of drag coefficient is shown in Fig. 10.

5. Conclusions

In this present study the frictional pressure drop characteristic hasbeen analyzed by a mechanistic model based on the viscousinteraction and wetting of the non-Newtonian liquid. The increaseof CMC concentration leads to the increase in effective viscosity whichleads to increase the shear stress and increase of pressure drop in thepacked bed. The intensity factor is a function of Reynolds number.Also as the concentration of CMC solution increases, the effectiveviscosity decreases at constant flow rate and results in decrease ofintensity factor with increase in concentration of CMC solution. Theintensity factor (αl) of the liquid–solid surface interfacial shear stresscharacterizes the intensity of the pressure drop in the fluid flow. Thefriction factor decreases with the increase in wettability. The energyloss due to wettability increases with the increase in surface tensionand decrease in porosity. This is because of capillary effect of flowingof liquid through porousmedia. The surface tension increaseswith theincrease in concentration of CMC which leads to increase in loss ofenergy due to wettability. The drag that is obtained by the presentwork is smaller than predicted by Ergun's equation and greater thanFoscolo's equation. It is found that the present result is wellapproximate to the Ergun's result. The deviation may be the non-Newtonian characteristics of the fluid. The present study may beuseful for further understanding and modeling of specific multiphasereactor in industrial applications.

Nomenclaturea Specific interfacial area [1/m]Ac Column cross-sectional area [m2]ah Hydraulic specific interfacial area of packing [1/m]

134 S.K. Patel, S.K. Majumder / Powder Technology 211 (2011) 127–134

ap Surface area of single particle [m2]Cd Drag coefficient [−]Ch Constant, defined in Eq. (21) [−]dc Column diameter [m]dp Particle diameter [m]El Energy dispersion per unit packed column [N.m/s]Fd Drag force [N]Ew Energy loss due to wettability [N.m/s]fls Friction factor of liquid–solid system [−]flo Friction factor of single liquid system [−]g Gravitational acceleration [m/s2]Kr Permeability of porous medium [m2]K Consistency of fluid [Pa.sn]n Flow behavior index [−]Np Number of particles [−]Pls Liquid–solid pressure [N/m2]Pfls Frictional pressure due to liquid flow [N/m2]Pflo Frictional pressure due to single liquid-phase [N/m2]Rh Hydraulic radius [m]STEYX Standard error defined in Eq. (34) [−]Vis Interfacial slip velocity [m/s]vp Volume of single particle [m3]Vsl Superficial liquid velocity [m/s]ΔZ Height of packing [m]

Dimensionless groupsCa Capillary number (=(μeffVsl)/(εσl)) [−]Frl Liquid Froude No. (=Vsl

2/gdp) [−]Rel Liquid Reynolds No. (=(Vslρl)/(aμeff)) [−]Ren Non-Newtonian liquid Reynolds number ( = dnc V

2−nsl ρ

8n−1K4n3n + 1

� �n

) [−]

Rep Particle Reynolds number (=(ρlVsldp)/μeff) [−]

Greek lettersαl Parameter defined in Eq. (9) [−]β Tortuosity factor [−]δflo Ratio of frictional pressure to hydrostatic pressure [−]δls Ratio of total pressure to hydrostatic pressure [−]ε Porosity [−]μeff Effective dynamic viscosity [kg/m.s]Φs Sphericity [−]ρl Density of liquid [kg/m3]σl Liquid surface tension [N/m]σls Surface tension between liquid and solid [N/m]τ i Interfacial shear stress [N/m2]θ Liquid–solid contact angle [radian]

Subscriptsc columnf frictionali interfaciall liquid phasels liquid–solid0 Singlep particles superficial, solidw wettability

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