Pressure drop and mixing behaviour of non-Newtonian fluids in a static mixing unit

Pressure Drop and Mixing Behaviour ofNon-Newtonian Fluids in a Static Mixing UnitGunjan Kumar and S. N. Upadhyay*

Department of Chemical Engineering and Technology, Centre of Advanced Study, Institute of Technology,Varanasi 221 005, India

Static or motionless mixers have received wide application in chemical and allied industries due to their low cost and high efficiency. Thepressure drop and mixing behaviour of such mixers have been widely studied. However, the available information for non-Newtonian fluidsis scanty. The results of pressure drop and mixing studies conducted with a locally made motionless mixer (MALAVIYA mixer) and fournon-Newtonian fluids—aq. CMC, PVA, and PEG solutions are reported in this article. The new mixer causes less pressure drop compared tosome of the commercial mixers. Mixing behaviour of the unit is more closer to plug flow and a two-parameter model correlates the dispersion data.

Les melangeurs statiques ou immobiles ont fait l’objet d’une large application dans l’industrie chimique et connexe en raison de leur faible cout etde leur haute efficacite. La perte de charge et le comportement de melange de tels melangeurs ont ete largement etudies. Neanmoins, l’informationdisponible pour les fluides non-Newtoniens est limitee. Les resultats d’etudes de perte de pression et de melange realisees avec un melangeurstatique fabrique localement (MALAVIYA) et quatre fluides non-Newtoniens, a savoir des solutions aqueuses de CMC, PVA et PEG, sont decritsdans cet article. Le nouveau melangeur cause moins de perte de charge comparativement a certains melangeurs commerciaux. Le comportementde melange de l’unite est plus proche d’un comportement piston et un modele a deux parametres correle les donnees de dispersion.

Keywords: static mixer, non-Newtonian fluids, pressure drop, mixing

INTRODUCTION

Static (or motionless) mixers have been applied as mixingdevices in liquid–liquid, gas–liquid, solid–liquid, andsolid–solid systems quite effectively since 1970s. Small over-

all space requirement, low cost, low power consumption, absenceof moving parts, short residence time, near plug flow behaviour,good mixing, high heat and mass transfer efficiencies, low shearrate, self-cleansing, and interchangeable or disposable nature arethe major advantages of these mixers over agitated vessels (Bor,1971; Baker, 1991; Thakur et al., 2003). In cooling processes usingstatic mixers, skinning due to boundary-layer solidification is alle-viated because of better radial mixing (Baker, 1991). Static mixersare also efficient in reducing fouling and coking and enhancingthe heat transfer during oil and tar residue treatments.

Commercial static mixers have a wide variety of basic geome-tries and in order to control their performance many adjustableparameters need to be optimized for specific applications (Byrdeet al., 1999). Thakur et al. (2003) presented an exhaustive reviewon the state-of-the-art of these mixers and highlighted the areasneeding more work particularly for systems involving viscousNewtonian and non-Newtonian fluids.

Static mixer systems are widely used with complex fluids inthe polymer and food processing industries, but measurement ofpressure drop for non-Newtonian fluids has been the subject ofonly a few studies. Limited reported data are available in the lit-erature for viscous and viscoelastic fluids (Shah and Kale, 1991,1992; Chandra and Kale, 1992, 1995). Shah and Kale (1991, 1992)compared the data for viscoelastic solutions of polyacrylamidewith inelastic solutions of carboxymethyl cellulose (CMC) andconcluded that elasticity always increased the friction factor. Thisis expected since elasticity is important in the entrance regionflow, and the sequential elements in a static mixer system createa sequence of entrance region flows.

Shah and Kale (1991, 1992) correlated their pressure drop datafor polymeric fluids using friction factor and the Metzner-Reed

∗Author to whom correspondence may be addressed.E-mail addresses: [email protected], [email protected]. J. Chem. Eng. 86:684–692, 2008© 2008 Canadian Society for Chemical EngineeringDOI 10.1002/cjce.20040

| 684 | THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING | | VOLUME 86, AUGUST 2008 |

generalized Reynolds number defined as:

f = �PsmDtε2

2L�u2(1)

and

Re = �u2−nDnt

K8n−1ε2−n(2)

The parameter K is the consistency index and n is the flowbehaviour index of fluid. Li et al. (1997) suggested another defini-tion of Reynolds number which is more general. This generalizedReynolds number is written as:

Reg = �uDt

�∗ ε(3)

Here, �∗ is the apparent viscosity corresponding to the shearrate at the wall.

This article presents the results of pressure drop and residencetime distribution experiments conducted with a new laboratorymade static mixer and four non-Newtonian fluids—aqueous solu-tions of CMC, polyvinyl alcohol (PVA), and polyethylene glycol(PEG). The purpose has been to come out with the design of anew motionless mixer which could be easily fabricated and usedin place of commercial mixers.

EXPERIMENTAL

Test FluidsAqueous solutions of CMC, PVA, and PEG were used as test fluids.These solutions obey power-law model and do not suffer fromthixotropy or change on aging.

CMC (medium viscosity) was obtained from Cellulose ProductsLtd. (Ahemadabad, India), CMC (high viscosity) from S. D. FineChemicals Ltd. (Mumbai, India) and PVA (MW 10000) and PEG(MW 9000) from Lab-Chemical Industry (Mumbai, India).

Test solutions of known concentrations were prepared by dis-solving the polymers in deionized water following the procedureused earlier (Lal, 1980).

The rheological properties of all solutions were determinedfrom the flow-curves prepared using the flow and pressure dropdata collected with the help of a capillary tube viscometer asdescribed earlier (Lal, 1980).

Static MixerThe static mixer (MALAVIYA mixer) elements were fabricated inlaboratory using PMMA sheets and tubes. The diagram, dimen-sions, and photograph of static mixer used in this work are shownin Figures 1 and 2, respectively. The S-shaped curved portionswere made by cutting out 1/3rd part of a 0.0257 m PMMA tubeand the discs were made from a 6 mm thick PMMA sheet. Twocurved parts were glued together to give S-shape elements as wellas to the disk by placing the glued segments radially on either sideof the disk. Such an attachment provided S-shaped protrusionsaligned diametrically on either side of the disk. A hole was alsodrilled diametrically in the edge of the disk for ensuring properalignment of elements in the tube.

Figure 1. Static mixer (MALAVIYA mixer) element.

Figure 2. Photograph of static mixer (MALAVIYA mixer) element.

| VOLUME 86, AUGUST 2008 | | THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING | 685 |

Figure 3. Experimental set-up.

Experimental Set-UpExperimental set-up used is shown in Figure 3. The test columnwas made of a perspex tube of 0.042 m ID and 1.1 m length.Static elements were suspended in the column with the help ofcopper wire of 1 mm diameter, which was rigidly fixed at thetwo ends of the test pipe with help of rubber corks. Open tubemanometers were used for measuring the pressure drop and theliquid flow rate was controlled using a calibrated rotameter. Tank1 contained the test fluid and Tank 2 was used for storing thetracer—the NaCl solution prepared in the respective test fluid. Thesamples of tracer bearing test fluid at the outlet end were collectedfrom the sampling port. Test fluids were pumped to the columnthrough a rotameter with the help of a centrifugal pump.

Experimental Procedure

Pressure drop measurementDesired test fluid was taken in Tank 1 and pumped to thestatic mixer unit from the bottom of the column. Pressure drop(�Psm = h�g) at a given flow rate was calculated by measuringthe difference in liquid heights in the manometer tubes. Flowrates were calculated by measuring the time for collecting 1 L ofthe test fluid.

Friction factor f, was calculated from the pressure drop �Psm,diameter of tube Dt, fluid velocity u, porosity ε, density of fluid�, length of static elements Le, and number of static elements N.Reynolds number, Re was calculated using the superficial velocityand rheological constants. Void volume fraction, ε was calculatedby measuring the volume of water displaced due to static mixer.The average value of the void fraction, was found to be 0.815.

RTD measurementThe RTD data were generated by injecting the tracer solution(NaCl solution in the test fluid) into the test section at some time,t = 0 and then measuring the tracer concentration, C in the effluentstream as a function of time. These experiments were performedwith all the test fluids by giving step input and pulse input oftracer to the static mixer unit in separate runs. Concentrationswere calculated by measuring conductivity of the effluent stream

with the help of a conductivity meter and using the calibrationgraph. The calibration curves were straight line when plotted asconductivity versus

√concentration.

Cumulative distribution or F(t) versus t curves were drawnfrom the RTD experimental data with step input of tracer for alltest fluids. Flow rate of test fluids was maintained constant at 1L/min and the tracer flow rates were maintained constant at 60mL/min. The conductivity values of solutions were converted intocorresponding concentrations from the standard curve. Then F(t)values were calculated and cumulative distribution curves weredrawn by plotting F(t) versus t.

E(t) curves were drawn from the pulse input data. In this case20 mL of NaCl solution was injected instantly into the static mixercolumn and then the effluent NaCl conductivities were measured.These E(t) versus t curves were used to compare the experimentalresults with the proposed RTD model.

All measurements were performed with mixers having differentorientations of static elements, that is, 0◦, 45◦, and 90◦ with respectto each other.

RESULTS AND DISCUSSION

Rheological PropertiesThe rheological properties of 1% aqueous CMChv, 1.5% aqueousCMCmv, 4% aqueous PVA, and 10% aqueous PEG solutions weredetermined with a capillary tube viscometer. The experimentswere carried out at a controlled temperature of 25◦C. The valuesof shear stress at the wall (�w = D�P/4L) and the correspond-ing pseudo-shear rate (8u/D) were calculated after correcting thepressure gradient in the capillary tube viscometer for entranceeffect at the average fluid velocity. The rheological constants weredetermined by fitting the power-law model to these plots.

The polymer solutions used were pseudoplastic in nature hav-ing flow behaviour indices less than unity, which was one of themain considerations for selecting these particular fluids for thepresent study. The values of both K and n at a given temperaturewere found to be constant in the range of shear stress studied.Rheological constants for different test fluids and their densityare given in Table 1.


Table 1. Rheological constants and density of test fluids

1% 1.5% 4% 10%CMChv CMCmv PVA PEG

n 0.465 0.710 0.774 0.859

K (Pa.sn) 1.037 0.136 0.045 0.01

� (kg/m3) 1007.6 1030.2 1040 1039.2

Pressure DropPressure drop estimation is the first parameter for selection of aproper static mixer system. The basic equation for pressure dropduring flow of a homogeneous, isothermal, incompressible fluidin a circular tube can be easily extended to that with static mixersand can be written as:

�Psm = 2f�u2

DtL = 2f�u2

DtNLe (4)

or

f = �PsmDt

2NL�u2or f = �PsmDe

2NLe�u2(5)

Due to reduction in the cross-sectional area for flow, the averagesuperficial velocity of fluid through the static mixer system wouldbe higher than that for the empty column used for the mixerassembly. Hence for static mixer some workers have replaced

diameter Dt with hydraulic mean diameter, De. Here N is thenumber of mixing element and Le is the effective length of one ele-ment. The friction factor, f is a function of Reynolds number andis determined experimentally or by CFD method for a particularmixer.

The same general concepts apply to flow in static mixers asthose for the open tube except that the transition values for Re arelower by a factor of about 2. Flow is generally laminar for Re < 50and turbulent for Re > 1000. The static inserts cause systematicdisturbances to the flow field so that complex but fairly repro-ducible flow behaviour can be expected in the intermediate range50 < Re < 1000. Actual range of transition, however, depends onthe design of the static elements including their aspect ratio. Forhelical and Kenics mixer elements, this region begins at Re around43 for Le/De ≤ 0.8, but is delayed to Re ≈ 55 when Le/De is 1(Jaffer and Wood, 1998). The influence of aspect ratio has beenconfirmed by Joshi et al. (1995), who also concluded that lowaspect ratios were better for heat transfer. However, the set-upused for generating most of the experimental data are for systemswith aspect ratio of 1.5 (Rauline et al., 1998, 2000). For SulzerSMX elements, Li et al. (1997) reported that the laminar regimeprevails up to Re = 15 while the turbulent regime begins whenRe = 1000.

A convenient way of representing pressure drop data is todirectly correlate the friction factor with the Reynolds number. Inlaminar flow, the classic relationship between f and Re is usuallyobtained as:

f = C1

Re(6)

Table 2. Constants for f–Re correlationsS. no. Reference Mixer element C1 C2 m a

1. Grace (1971) Helical 77.76 10.88 0.5

2. Sir and Lecjaks (1982), Lecjaks et al. (1984) Helical 85.5 0.34 0.0

3. Cybulski and Werner (1986) Helical (for Re ≤ 50) 115.2 0.5Helical (for Re ≤ 100, Re ≤ 1000) 6.592 0.5Sulzer SMX 160–1600Sulzer SMV 1040–4800Inliner Lightnin 144–118.4 11.2 0.0N form 240–272Komax 400Hi-Toray 608ISG 4000–4800LPD 5407.5 (D/L)

4. Shah and Kale (1991) Kenics 64.06n3.68n4

(n + 1)1.36n

(n + 3)

Sulzer 350n10.26n

(n + 1)2.32n

(n + 3)

5. Chandra and Kale (1992) Komax 176.5n7.56n

(n + 1)1.72n

(n + 1)6. Li et al. (1997) Sulzer SMX

Re ≤ 15 39815 ≤ Re ≤ 1000 220 0.8 0.8Re ≥ 1000 12 0.25

7. Rauline et al. (1998) Sulzer SMX LPD 160–960139.5146.08 (D/L)

8. Cavatora et al. (1999) Sulzer SMX (20 ≤ Re ≤ 1000) 78.4 0.148 0.0

9. Present work Modified disk mixer (0◦ orientation) 70 0.75 0.1390◦ Orientation 72.2 0.24 0.1345◦ Orientation 88 0.24 0.13


where C1 is a constant greater than 16. For laminar and transitionrange it is:

f = C1

Re+ C2

Rem(7)

The second term is intended to reflect the effect of radial flowcaused by the mixing elements. The values of C1, C2, and mreported by earlier workers for various mixing elements are listedin Table 2.

Owing to the limited use of static mixers in turbulent flow,fewer correlations of pressure drop in this regime are avail-able (Bourne et al., 1992). Pahl and Muschelknautz (1982) andCybulski and Werner (1986) presented correlations for the fric-tion factor for two ranges of Reynolds number, 1200 < Re < 7000and 7000 < Re < 30 000. A typical correlation used for turbulentflow is of the form:

f = C3

Rem′ (8)

where C3 is a constant. The exponent m′ itself has been found tobe a function of Reynolds number, typically decreasing at highervalues of Re. Cybulski and Werner (1986) presented results forthe Kenics, LPD and Komax mixers. At higher Reynolds numbers,m′ approaches 0 and f becomes constant. A similar behaviour isobserved in empty pipes with fempty → 0.02 as Re → ∞. Limitingf values for Kenics, Hi-Toray, SMX, and SMV mixers are 3, 11, 12,and 6–12, respectively (Pahl and Muschelknautz, 1982).

Figure 4 shows typical variation of friction factor, f withReynolds number, Re on logarithmic coordinates for all the fluidsstudied and the three orientations used between two consecu-tive elements. The line representing f–Re relation for empty tubeis also shown in these figures for comparison. It is seen that thepressure drop in static mixer is about 3.5–5 times higher than thatfor empty tube. From these plots it is also seen that for Re < 35, thelog f varies linearly with log Re. For Re > 35, upward deviation off values from the linear relation is observed indicating transitionfrom the creeping flow. The additional pressure losses are due tocontributions from the creeping flow as well as the boundary layerflow with in the transition region. The secondary flows created inthe direction perpendicular to the main flow rather than the turbu-lence are attributed to be responsible for the improved convectivetransfer in some cases of static mixers (Morris and Proctor, 1977).Thus, the skin friction and to some extent the form drag aroundthe elements may contribute more to the increased pressure drop(Shah and Kale, 1991).

A relation similar to Equation (7) can be used to correlate thefriction factor data. The constants C1, C2, and m for various sit-uations are reported in Table 2. It is revealed from Table 2, thatthe new static mixer offers less pressure drop as compared to Ken-ics and Sulzer static mixers. The correlation constants C1 and C2

both are smaller compared to those for Kenics and Sulzer staticmixers and are nearly similar to those for the helical mixers. Itis also seen that the value of C1 is higher for 45◦ orientation ofstatic elements. At 45◦ orientation of elements, fluid streams splitinto four parts at the entrance and number of flow paths increasein number as they proceed upward in static mixer, but at 0◦ and90◦ orientation of static element, the fluid streams split only intofour parts at the entrance and remain in the same situation in thefollowing sections.

Figure 4. f versus Re plot for static mixer assembly with (a) 0◦, (b) 45◦,and (c) 90◦ orientation of elements.

Residence Time DistributionMixing in static mixers is affected by scores of parameters. Grosz-Roll (1980) tabulated more than 50. These parameters are notalways clearly defined and are also not easy to compare with eachother. There is no single criterion suitable for all applications, andeach has its advantages and disadvantages.

The RTD was determined experimentally by injecting NaCltracer at time t = 0 in the static mixer filled pipe and then measur-ing the tracer concentration C, in the effluent stream as a functionof time. These experiments were performed for step input of tracerwith all the test fluids. Cumulative residence time distributionfunction, F(t), obtained from a sudden step input of an inert tracer


Figure 5. Cumulative distribution curves for static mixer assembly with(a) 90◦ and (b) 45◦ orientation of elements.

is a convenient parameter for judging the mixing effectivenessof a system. The residence time distribution data obtained areconverted to F(t) values. Figures 4a,b shows the cumulative dis-tribution F(t) versus t data for static elements with 45◦ and 90◦

orientations as typical example. Data for all test fluids with staticelements at a particular orientation are plotted on the same graph.Smoothed curves are also drawn in these figures. The first appear-ance time, tfirst of tracer is not exactly clear, however, it is slightlymore than 0.5 in each case. Figures 5a,b show that the first appear-ance time (tfirst) for 45◦ orientation of static elements is slightlyhigher as compared to 90◦ orientation. This is due to more com-plicated flow path provided by 45◦ orientation of static elementsas compared to 90◦ orientation.

F(t) curves show that the flow behaviour is a combinationof that for plug flow reactor (PFR) and stirred tank reactor(CSTR). The exact combination of PFR and CSTR, which showsan equivalent behaviour of static mixer, is determined by mod-elling. Dimensionless variance of the residence time distributionis another common indicator of degree of mixing. It can be calcu-

Figure 6. The proposed RTD model.

lated using:

�2 = 2 ∫∞0 1 − F(t)t dt

(t)2− 1 (9)

The dimensionless variance is zero for plug flow. It is theo-retically infinite for laminar flow without diffusion, but becomesfinite in all real systems due to molecular diffusion (Nauman,1982; Nauman and Buffham, 1983). From the experimental dataof pulse input it is seen that the dimensionless variance forstatic mixer is in the range of 0 ≤ �2� ≤ 1. So one can formu-late a two-parameter model based on the geometry of eachelement (Nauman and Buffham, 1983; Li et al., 1996). It isessentially a sub-system consisting of a stirred tank and a PFRcoupled in parallel. This sub-system behaves as a single staticelement. There are N number of static elements present so thewhole system is arranged in the form of a series of sub-systems(Figure 6).

The composite density function for a system in parallel is theweighted sum of the component density functions in both Laplaceand time domain (Nauman and Buffham, 1983). Thus for theabove system of static element (for N = 1):

fs(s) = �fc(s) + (1 − �)fp(s) (10)

where fc(s) is the Laplace transform for stirred tank reactor whichis given by Equation (11):

fc(s) = 11 + � ′s

(11)

and fp(s) is the Laplace transform for PFR which is given byEquation (12):

fp(s) = e−�′s (12)

where

� ′ = Vε

NQ= space time for subsystem (13)

The generalized formulae for N subsystems in series is givenby Equation (14):

fs(s) = f1(s) × f2(s) × · · · × fN(s) (14)


Figure 7. E(t) versus t curve for (a) 1.0% CMChv and (b) 0.5% CMCmvsolutions.

But f1(s) = f2(s) = · · · = fN(s), so the generalized formulae interms of Laplace transform for proposed system is given by:

G(s) = [�fc(s) + (1 − �)fp(s)]N (15)

or

G(s) =(

�

1 + (ˇ/�)� ′s+ (1 − �) exp

(−(1 − ˇ)� ′s(1 − �)

))N

(16)

Equation (16) was converted to time domain, and simulationof the above model for pulse input was carried out. The mainadvantage of this model is that it can be applied for any type ofstatic mixer in practice.

The simulation results of the model are compared with theexperimental results of pulse input of tracer (NaCl) to the staticmixer system with 45◦ orientation of static elements. The mea-sured effluent concentrations were converted to dimensionlessform E(t), tm, and �t. The results are shown in Figures 7a throughFigure 8b, respectively. The RTD model agrees satisfactorily withthe experimental results only upto certain value of t. It does notconform to the experimental results in which a long tail of con-

Figure 8. E(t) versus t curve for (a) 4% PVA and (b) 10% PEG solutions.

centration is found with time. The values of parameters whichgive a good fit are:

� = 0.69 + 9.5 × 10−3 log(Re) (17)

ˇ = 0.47 (18)

CONCLUSIONSThe pressure drop results show that new static mixer gives lowerpressure drop as compared to some commercial mixers. The pres-sure drop is more when the elements are oriented at 45◦ to eachother. This is due to increase in the number of flow paths in theaxial direction due to splitting of the primary flow. The radial flowcreated by new static mixer is nearly independent of orientationof mixer elements with respect to each other.

The cumulative distribution curves show that static mixer(MALAVIYA mixer) with 45◦ orientation of static elements is morecloser to plug flow behaviour compared to 90◦ orientation. This isdue to more number of flow paths or more splitting of streamlinesoffered by 45◦ orientation of static elements.

The results of RTD model follow the experimental resultsonly upto certain range of time (t = 1.5t), this is due to taking


parameter ˇ (fraction of total volume corresponding to CSTR) asindependent of Reynolds number and number of static elements.

ACKNOWLEDGEMENTSThe results reported in this article are based on the M.Tech.(Chem. Eng.) dissertation of Mr Gunjan Kumar. Authors are grate-ful to Prof. G. Djelveh, LGCB, Universite of Blaise Pascal, AubiereLedex, France, who encouraged them to initiate work in this area.

NOMENCLATUREA constantb′ mixing constantc1 average volumetric concentration of species i

(mol/L)cij volumetric concentration of species i at sampling

point j (mol/L)C1, C2, C3 constantsCi initial concentration of tracer (mol/L)Cf final concentration of fluid (mol/L)C(t), Cout(t) concentration of sample at time t at inlet and outlet

ends (mol/L)C(∞) concentration at time t = ∞ (mol/L)D diameter of capillary tube (m)Dc diameter of static element (m)De equivalent diameter (m)Dt diameter of tube filled with static mixer (m)E(t) = Cout(t)

∫∞0 Cout(t)dt

f Fanning friction factor with static mixing insertsfempty Fanning friction factor in an empty pipeF(t) (C(t) − Ci)/(C(∞) − Ci)K consistency index (Pa sn)l height of curved portion of static element (m)L length (m)Le length of one static element (m)m, m′ constantn flow behaviour indexN number of static elementsP pressure (N/m2)�Psm pressure drop (N/m2)�P pressure difference in capillary tube (N/m2)Q volumetric flow rate (L/s)Re Reynolds numberReg generalized Reynolds numbert residence time (s)

[= (va

)](s)

tm mean residence time[= (

VQ

)](s)

t dimensionless time(= t

�

)(s)

u superficial velocity (m/s)v fluid velocity in capillary tube (m/s)V volume of static mixer system (m3)wd width of disc of static element (m)wc width of curved (s—shaped) part of static element

(mm)

Greek Symbols�w rate of shear at wall (s−1)ε static mixer system void fraction� RTD model parameter� RTD model parameterˇ RTD model parameter� viscosity (Pa s)

�* viscosity based on �w for non-Newtonian fluids (Pa s)� density of fluid (kg/m3)�2 variance (s2)

�2� dimensionless variance

[=

(�2

t2m

)]�w shear stress (Pa)�s space time (s)

REFERENCESBaker, J. R., “Motionless Mixers Stir Up New Uses,” Chem. Eng.

Progress 87, 32–38 (1991).Bor, T., “The Static Mixers as a Chemical Reactor,” Br. Chem.

Eng. 610–612 (1971).Bourne, J. R., J. Lenzner and S. Petrozzi, “Micromixing in Static

Mixers: An Experimental Study,” Ind. Eng. Chem. Res. 31,1216–1222 (1992).

Byrde O. and M. L. Sawley, “Optimization of a Kenics StaticMixer for Non-Creeping Flow Conditions,” Chem. Eng. J. 72,163–169 (1999).

Cavatora, O. N., U. Bohm and A. M. C. Giorgio, “Fluid Dynamicand Mass Transfer Behavior of Static Mixers and RegularPacking,” AIChE J. 45, 938–948 (1999).

Chandra G. K. and D. D. Kale, “Pressure Drop for Laminar Flowof Viscoelastic Fluids in Static Mixers,” Chem. Eng. Sci. 47,2097–2100 (1992).

Chandra G. K. and D. D. Kale, “Pressure Drop for Two-PhaseAir/Non-Newtonian Liquid Flow in Static Mixers,” Chem.Eng. J. 56, 277–280 (1995).

Cybulski A. and K. Werner, “Static Mixers-Criteria forApplications and Selection,” Int. Chem. Eng. 26, 171–180(1986).

Grace, C. D., “Static Mixing and Heat Transfer,” Chem. ProcessEng. 57–59 (1971).

Grosz-Roll, F., “Assessing Homogeneity in Motionless Mixers,”Int. Chem. Eng. 20, 542–549 (1980).

Jaffer S. A. and P. E. Wood, “Quantification of Laminar Mixingin the Kenics Static Mixer: An Experimental Study,” Can. J.Chem. Eng. 76, 516–524 (1998).

Joshi, P., K. D. P. Nigam and E. B. Nauman, “The Kenics StaticMixer: New Data and Proposed Correlations,” 59, 265–271(1995).

Lal, P., “Dissolution of Solid Particles in Agitated Newtonian andNon-Newtonian Fluids,” Ph. D. Thesis, Banaras HinduUniversity, Varanasi, India. (1980).

Lecjaks, Z., I. Machac and J. Sir, “Pressure Drop in LiquidFlowing Through a Pipe With Built-In Helical Elements,”Chem. Eng. Process. 18, 67–72 (1984).

Li, H. Z., Ch. Fasol and L. Choplin, “Hydrodynamics and HeatTransfer of Rheologically Complex Fluids in a Sulzer SMXStatic Mixer,” Chem. Eng. Sci. 51, 1947–1955 (1996).

Li, H. Z., Ch. Fasol and L. Choplin, “Pressure Drop of Newtonianand Non-Newtonian Fluids Across a Sulzer SMX Mixer,”Chem. Eng. Res. Des. 75, 792–796 (1997).

Morris W. D., and R. Proctor, “The Effect of Twist Ratio onForced Convection in the Kenics Static Mixer,” Ind. Chem.Eng. Process Des. Dev. 16, 406–412 (1977).

Nauman, E. B., “Reactions and Residence Time Distributions inMotionless Mixers,” Can J. Chem. Eng. 60, 136–140 (1982).

Nauman E. B. and B. A. Buffham, “Mixing in Continuous FlowSystems,” Wiley, New York (1983).

Pahl M. H. and E. Muschelknautz, “Static Mixers and TheirApplications,” Int. Chem. Eng. 22, 197–204 (1982).


Rauline, D., P. A. Tanguy, J. Leblevec and J. Bousquet,“Numerical Investigation of the Performance of Several StaticMixers,” Can. J. Chem. Eng. 76, 527–535 (1998).

Rauline, D., J. M. Leblevec, J. Bousquet and P. A. Tanguy, “AComparative Assessment of the Performance of the Kenicsand SMX Static Mixers,” Chem. Eng. Res. Des. 78, 389–396(2000).

Shah N. F. and D. D. Kale, “Pressure Drop for Laminar Flow ofNon-Newtonian Fluids in Static Mixers,” Chem. Eng. Sci. 46,2159–2161 (1991).

Shah N. F., and D. D. Kale, “Two-Phase, Gas-Liquid Flows inStatic Mixers,” AIChE J. 38, 308–310 (1992).

Sir J. and Z. Lecjaks, “Pressure Drop and HomogenizationEfficiency of a Motionless Mixer,” Chem. Eng. Commun. 16,325–334 (1982).

Thakur, R. K., Ch Vital, K. D. P. Nigam, E. B. Nauman and G.Djelveh, “Static Mixers in the Process Industries—A Review,”Chem. Eng. Res. Des. 81, 787–826 (2003).

Manuscript received April 18, 2007; revised manuscriptreceived January 14, 2008; accepted for publication January 14,2008.


Pressure drop and mixing behaviour of non-Newtonian fluids in a static mixing unit

Documents

Transcript of Pressure drop and mixing behaviour of non-Newtonian fluids in a static mixing unit