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

Kumar, Gunjan ; Upadhyay, S.N.

INTRODUCTION

Static (or motionless) mixers have been applied as mixing devices in liquid-liquid, gas-liquid, solid-liquid, and solid-solid systems quite effectively since 1970s. Small overall space requirement, low cost, low power consumption, absence of moving parts, short residence time, near plug flow behaviour, good mixing, high heat and mass transfer efficiencies, low shear rate, self-cleansing, and interchangeable or disposable nature are the major advantages of these mixers over agitated vessels (Bor, 1971; Baker, 1991; Thakur et al., 2003). In cooling processes using static mixers, skinning due to boundary-layer solidification is alleviated because of better radial mixing (Baker, 1991). Static mixers are also efficient in reducing fouling and coking and enhancing the heat transfer during oil and tar residue treatments.

Commercial static mixers have a wide variety of basic geometries and in order to control their performance many adjustable parameters need to be optimized for specific applications (Byrde et al., 1999). Thakur et al. (2003) presented an exhaustive review on the state-of-the-art of these mixers and highlighted the areas needing more work particularly for systems involving viscous Newtonian and non-Newtonian fluids.

Static mixer systems are widely used with complex fluids in the polymer and food processing industries, but measurement of pressure drop for non-Newtonian fluids has been the subject of only a few studies. Limited reported data are available in the literature 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 polyacrylamide with inelastic solutions of carboxymethyl cellulose (CMC) and concluded that elasticity always increased the friction factor. This is expected since elasticity is important in the entrance region flow, and the sequential elements in a static mixer system create a sequence of entrance region flows.

Shah and Kale (1991, 1992) correlated their pressure drop data for polymeric fluids using friction factor and the Metzner-Reed generalized Reynolds number defined as:

f = [DELTA][P.sub.sm][D.sub.t][[epsilon].sub.2]/2L[rho][u.sup.2] (1)

and

Re = [rho][u.sup.2-n][D.sup.n.sub.t]/K[8.sup.n-1] [[epsilon].sub.2-n] (2)

The parameter K is the consistency index and n is the flow behaviour index of fluid. Li et al. (1997) suggested another definition of Reynolds number which is more general. This generalized Reynolds number is written as:

[Re.sub.g] = [rho]u[D.sub.t]/[[mu].sup.*] [epsilon] (3)

Here, [[mu].sup.*] is the apparent viscosity corresponding to the shear rate at the wall.

This article presents the results of pressure drop and residence time distribution experiments conducted with a new laboratory made static mixer and four non-Newtonian fluids--aqueous solutions of CMC, polyvinyl alcohol (PVA), and polyethylene glycol (PEG). The purpose has been to come out with the design of a new motionless mixer which could be easily fabricated and used in place of commercial mixers.

EXPERIMENTAL

Test Fluids

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

CMC (medium viscosity) was obtained from Cellulose Products Ltd. (Ahemadabad, India), CMC (high viscosity) from S. D. Fine Chemicals 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 dissolving the polymers in deionized water following the procedure used earlier (Lal, 1980).

The rheological properties of all solutions were determined from the flow-curves prepared using the flow and pressure drop data collected with the help of a capillary tube viscometer as described earlier (Lal, 1980).

Static Mixer

The static mixer (MALAVIYA mixer) elements were fabricated in laboratory using PMMA sheets and tubes. The diagram, dimensions, and photograph of static mixer used in this work are shown in Figures 1 and 2, respectively. The S-shaped curved portions were made by cutting out 1/3rd part of a 0.0257 m PMMA tube and the discs were made from a 6 mm thick PMMA sheet. Two curved parts were glued together to give S-shape elements as well as to the disk by placing the glued segments radially on either side of the disk. Such an attachment provided S-shaped protrusions aligned diametrically on either side of the disk. A hole was also drilled diametrically in the edge of the disk for ensuring proper alignment of elements in the tube.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Experimental Set-Up

Experimental set-up used is shown in Figure 3. The test column was 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 of copper wire of 1 mm diameter, which was rigidly fixed at the two ends of the test pipe with help of rubber corks. Open tube manometers were used for measuring the pressure drop and the liquid flow rate was controlled using a calibrated rotameter. Tank 1 contained the test fluid and Tank 2 was used for storing the tracer--the NaCl solution prepared in the respective test fluid. The samples of tracer bearing test fluid at the outlet end were collected from the sampling port. Test fluids were pumped to the column through a rotameter with the help of a centrifugal pump.

Experimental Procedure

Pressure drop measurement

Desired test fluid was taken in Tank 1 and pumped to the static mixer unit from the bottom of the column. Pressure drop ([DELTA][P.sub.sm]=h[rho]g) at a given flow rate was calculated by measuring the difference in liquid heights in the manometer tubes. Flow rates were calculated by measuring the time for collecting 1 L of the test fluid.

Friction factor f, was calculated from the pressure drop [DELTA][P.sub.sm], diameter of tube [D.sub.t], fluid velocity u, porosity [epsilon], density of fluid [rho], length of static elements [L.sub.e], and number of static elements N. Reynolds number, Re was calculated using the superficial velocity and rheological constants. Void volume fraction, [epsilon] was calculated by measuring the volume of water displaced due to static mixer. The average value of the void fraction, [euro] was found to be 0.815.

RTD measurement

The 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 effluent stream as a function of time. These experiments were performed with all the test fluids by giving step input and pulse input of tracer to the static mixer unit in separate runs. Concentrations were calculated by measuring conductivity of the effluent stream with the help of a conductivity meter and using the calibration graph. The calibration curves were straight line when plotted as conductivity versus [square root of (concentration)].

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

E(t) curves were drawn from the pulse input data. In this case 20 mL of NaCl solution was injected instantly into the static mixer column and then the effluent NaCl conductivities were measured. These E(t) versus t curves were used to compare the experimental results with the proposed RTD model.

All measurements were performed with mixers having different orientations of static elements, that is, 0[degrees], 45[degrees], and 90[degrees] with respect to each other.

RESULTS AND DISCUSSION

Rheological Properties

The rheological properties of 1% aqueous [CMC.sub.hv], 1.5% aqueous [CMC.sub.mv], 4% aqueous PVA, and 10% aqueous PEG solutions were determined with a capillary tube viscometer. The experiments were carried out at a controlled temperature of 25[degrees]C. The values of shear stress at the wall ([[tau].sub.w] = D[DELTA]P/4L) and the corresponding pseudo-shear rate (8u/D) were calculated after correcting the pressure gradient in the capillary tube viscometer for entrance effect at the average fluid velocity. The rheological constants were determined by fitting the power-law model to these plots.

The polymer solutions used were pseudoplastic in nature having flow behaviour indices less than unity, which was one of the main considerations for selecting these particular fluids for the present study. The values of both K and n at a given temperature were found to be constant in the range of shear stress studied. Rheological constants for different test fluids and their density are given in Table 1.

Pressure Drop

Pressure drop estimation is the first parameter for selection of a proper static mixer system. The basic equation for pressure drop during flow of a homogeneous, isothermal, incompressible fluid in a circular tube can be easily extended to that with static mixers and can be written as:

[DELTA][P.sub.sm] = 2f[rho][u.sup.2]/[D.sub.t] L = 2f[rho][u.sup.2]/[D.sub.t] N[L.sub.e] (4)

or

f = [DELTA][P.sub.sm][D.sub.t] / 2NL[rho][u.sup.2] or f = [DELTA][P.sub.sm][D.sub.e] / 2N[L.sub.e][rho][u.sup.2] (5)

Due to reduction in the cross-sectional area for flow, the average superficial velocity of fluid through the static mixer system would be higher than that for the empty column used for the mixer assembly. Hence for static mixer some workers have replaced diameter [D.sub.t] with hydraulic mean diameter, [D.sub.e]. Here N is the number of mixing element and [L.sub.e] is the effective length of one element. The friction factor, f is a function of Reynolds number and is determined experimentally or by CFD method for a particular mixer.

The same general concepts apply to flow in static mixers as those for the open tube except that the transition values for Re are lower by a factor of about 2. Flow is generally laminar for Re < 50 and turbulent for Re > 1000. The static inserts cause systematic disturbances to the flow field so that complex but fairly reproducible flow behaviour can be expected in the intermediate range 50 < Re < 1000. Actual range of transition, however, depends on the design of the static elements including their aspect ratio. For helical and Kenics mixer elements, this region begins at Re around 43 for [L.sub.e]/[D.sub.e] [less than or equal to] 0.8, but is delayed to Re [approximately equal to] 55 when [L.sub.e]/[D.sub.e] is 1 (Jaffer and Wood, 1998). The influence of aspect ratio has been confirmed by Joshi et al. (1995), who also concluded that low aspect ratios were better for heat transfer. However, the set-up used for generating most of the experimental data are for systems with aspect ratio of 1.5 (Rauline et al., 1998, 2000). For Sulzer SMX elements, Li et al. (1997) reported that the laminar regime prevails up to Re = 15 while the turbulent regime begins when Re = 1000.

A convenient way of representing pressure drop data is to directly correlate the friction factor with the Reynolds number. In laminar flow, the classic relationship between f and Re is usually obtained as:

f = [C.sub.1]/Re (6)

where [C.sub.1] is a constant greater than 16. For laminar and transition range it is:

f = [C.sub.1] / Re + [C.sub.2] / [Re.sup.m] (7)

The second term is intended to reflect the effect of radial flow caused by the mixing elements. The values of [C.sub.1], [C.sub.2], and m reported by earlier workers for various mixing elements are listed in Table 2.

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

f = [C.sub.3] / [Re.sup.m'] (8)

where [C.sub.3] is a constant. The exponent m' itself has been found to be a function of Reynolds number, typically decreasing at higher values of Re. Cybulski and Werner (1986) presented results for the Kenics, LPD and Komax mixers. At higher Reynolds numbers, m' approaches 0 and f becomes constant. A similar behaviour is observed in empty pipes with [f.sub.empty] [right arrow] 0.02 as Re [right arrow] [infinity]. Limiting f 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 with Reynolds number, Re on logarithmic coordinates for all the fluids studied and the three orientations used between two consecutive elements. The line representing f-Re relation for empty tube is also shown in these figures for comparison. It is seen that the pressure drop in static mixer is about 3.5-5 times higher than that for empty tube. From these plots it is also seen that for Re < 35, the log f varies linearly with log Re. For Re > 35, upward deviation of f values from the linear relation is observed indicating transition from the creeping flow. The additional pressure losses are due to contributions from the creeping flow as well as the boundary layer flow with in the transition region. The secondary flows created in the direction perpendicular to the main flow rather than the turbulence are attributed to be responsible for the improved convective transfer in some cases of static mixers (Morris and Proctor, 1977). Thus, the skin friction and to some extent the form drag around the elements may contribute more to the increased pressure drop (Shah and Kale, 1991).

A relation similar to Equation (7) can be used to correlate the friction factor data. The constants [C.sub.1], [C.sub.2], and m for various situations are reported in Table 2. It is revealed from Table 2, that the new static mixer offers less pressure drop as compared to Kenics and Sulzer static mixers. The correlation constants [C.sub.1] and [C.sub.2] both are smaller compared to those for Kenics and Sulzer static mixers and are nearly similar to those for the helical mixers. It is also seen that the value of [C.sub.1] is higher for 45[degrees] orientation of static elements. At 45[degrees] orientation of elements, fluid streams split into four parts at the entrance and number of flow paths increase in number as they proceed upward in static mixer, but at 0[degrees] and 90[degrees] orientation of static element, the fluid streams split only into four parts at the entrance and remain in the same situation in the following sections.

[FIGURE 4 OMITTED]

Residence Time Distribution

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

The RTD was determined experimentally by injecting NaCl tracer at time t = 0 in the static mixer filled pipe and then measuring the tracer concentration C, in the effluent stream as a function of time. These experiments were performed for step input of tracer with all the test fluids. Cumulative residence time distribution function, F(t), obtained from a sudden step input of an inert tracer is a convenient parameter for judging the mixing effectiveness of a system. The residence time distribution data obtained are converted to F(t) values. Figures 4a,b shows the cumulative distribution F(t) versus [bar.t] data for static elements with 45[degrees] and 90[degrees] orientations as typical example. Data for all test fluids with static elements at a particular orientation are plotted on the same graph. Smoothed curves are also drawn in these figures. The first appearance time, [[bar.t].sub.first] of tracer is not exactly clear, however, it is slightly more than 0.5 in each case. Figures 5a,b show that the first appearance time ([[bar.t].sub.first]) for 45[degrees] orientation of static elements is slightly higher as compared to 90[degrees] orientation. This is due to more complicated flow path provided by 45[degrees] orientation of static elements as compared to 90[degrees] orientation.

[FIGURE 5 OMITTED]

F(t) curves show that the flow behaviour is a combination of that for plug flow reactor (PFR) and stirred tank reactor (CSTR). The exact combination of PFR and CSTR, which shows an equivalent behaviour of static mixer, is determined by modelling. Dimensionless variance of the residence time distribution is another common indicator of degree of mixing. It can be calculated using:

[[sigma].sup.2] = 2 [[integral].sup.[infinity].sub.0] 1 - F(t)t dt / [([bar.t]).sup.2] - 1 (9)

[FIGURE 6 OMITTED]

The dimensionless variance is zero for plug flow. It is theoretically infinite for laminar flow without diffusion, but becomes finite in all real systems due to molecular diffusion (Nauman, 1982; Nauman and Buffham, 1983). From the experimental data of pulse input it is seen that the dimensionless variance for static mixer is in the range of 0 [less than or equal to] [[sigma].sup.2] [tau] [less than or equal to] 1. So one can formulate a two-parameter model based on the geometry of each element (Nauman and Buffham, 1983; Li et al., 1996). It is essentially a sub-system consisting of a stirred tank and a PFR coupled in parallel. This sub-system behaves as a single static element. There are N number of static elements present so the whole system is arranged in the form of a series of sub-systems (Figure 6).

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

[[bar.f].sub.s](s) = [phi][[bar.f].sub.c](s) + (1 - [phi]) [[bar.f].sub.p](s) (10)

where [[bar.f].sub.c](s) is the Laplace transform for stirred tank reactor which is given by Equation (11):

[[bar.f].sub.c](s) = 1 / 1 + [tau]'s (11)

and [[bar.f].sub.p](s) is the Laplace transform for PFR which is given by Equation (12):

[[bar.f].sub.p](s) = [e.sup.-[tau]'s] (12)

where

[tau]' = V[epsilon]/NQ = space time for subsystem (13)

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

[[bar.f].sub.s](s) = [[bar.f].sub.1](s) x [[bar.f].sub.2](s) x ... [[bar.f].sub.N](s)) (14)

[FIGURE 7 OMITTED] But [[bar.f].sub.1](s) = [[bar.f].sub.2](s) = ... = [[bar.f].sub.N](s), so the generalized formulae in terms of Laplace transform for proposed system is given by:

G(s) = [[[phi][[bar.f].sub.c](s) + (1 - [phi]) [[bar.f].sub.p](s)].sup.N] (15)

or

G(s) = [([phi] / 1 + ([beta]/[phi])[tau]'s + (1 - [phi]) exp (-(1 - [beta])[tau]'s / (1 - [phi]))].sup.N] (16)

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

The simulation results of the model are compared with the experimental results of pulse input of tracer (NaCl) to the static mixer system with 45[degrees] orientation of static elements. The measured effluent concentrations were converted to dimensionless form E(t), [t.sub.m], and [[sigma].sub.t]. The results are shown in Figures 7a through Figure 8b, respectively. The RTD model agrees satisfactorily with the experimental results only upto certain value of t. It does not conform to the experimental results in which a long tail of concentration is found with time. The values of parameters which give a good fit are:

[phi] = 0.69 + 9.5 x [10.sup.-3] log(Re) (17)

[beta] = 0.47 (18)

CONCLUSIONS

The pressure drop results show that new static mixer gives lower pressure drop as compared to some commercial mixers. The pressure drop is more when the elements are oriented at 45[degrees] to each other. This is due to increase in the number of flow paths in the axial direction due to splitting of the primary flow. The radial flow created by new static mixer is nearly independent of orientation of mixer elements with respect to each other.

The cumulative distribution curves show that static mixer (MALAVIYA mixer) with 45[degrees] orientation of static elements is more closer to plug flow behaviour compared to 90[degrees] orientation. This is due to more number of flow paths or more splitting of streamlines offered by 45[degrees] orientation of static elements.

The results of RTD model follow the experimental results only upto certain range of time (t = 1.5[bar.t]), this is due to taking parameter [beta] (fraction of total volume corresponding to CSTR) as independent of Reynolds number and number of static elements.

ACKNOWLEDGEMENTS

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

NOMENCLATURE

A constant
b' mixing constant
[[bar.c].sub.1] average volumetric concentration of
 species i (mol/L)
[c.sub.ij] volumetric concentration of species i at
 sampling point j (mol/L)
[C.sub.1], [C.sub.2],
[C.sub.3] constants
[C.sub.i] initial concentration of tracer (mol/L)
[C.sub.f] final concentration of fluid (mol/L)
C(t), [C.sub.out](t) concentration of sample at time t at inlet
ends (mol/L) and outlet
C([infinity]) concentration at time t = [infinity] (mol/L)
D diameter of capillary tube (m)
[D.sub.c] diameter of static element (m)
[D.sub.e] equivalent diameter (m)
[D.sub.t] diameter of tube filled with static mixer (m)
E(t) = [C.sub.out](t)/[[integral].sup.
 [infinity].sub.0] [C.sub.out] (t)dt
f Fanning friction factor with static mixing
 inserts
[f.sub.empty] Fanning friction factor in an empty pipe
F(t) (C(t) - [C.sub.i])/(C([infinity]) - [C.sub.i])
K consistency index (Pa [s.sup.n])
L height of curved portion of static element
 (m)
L length (m)
[L.sub.e] length of one static element (m)
m, m' constant
n flow behaviour index
N number of static elements
P pressure (N/[m.sup.2])
[DELTA][P.sub.sm] pressure drop (N/[m.sup.2])
[DELTA]P pressure difference in capillary tube
 (N/[m.sup.2])
Q volumetric flow rate (L/s)
Re Reynolds number
[Re.sub.g] generalized Reynolds number
t residence time (s) [= (v / a)] (s)
[t.sub.m] mean residence time (= (V / Q)] (s)
[bar.t] dimensionless time (= t / [tau]) (s)
u superficial velocity (m/s)
v fluid velocity in capillary tube (m/s)
V volume of static mixer system ([m.sup.3])
[w.sub.d] width of disc of static element (m)
[w.sub.c] width of curved (s-shaped) part of static
 element (mm)

Greek Symbols

[[??].sub.w] rate of shear at wall ([s.sup.-1])
[epsilon] static mixer system void fraction
[phi] RTD model parameter
[phi] RTD model parameter
[beta] RTD model parameter
[mu] viscosity (Pa s)
[[mu].sup.*] viscosity based on [[??].sub.w] for non-
 Newtonian fluids (Pa s)
[rho] density of fluid (kg/[m.sup.3])
[[sigma].sup.2] variance ([s.sup.2])
[[sigma].sub.2 dimensionless variance [= ([[sigma].sup.2]
.sub.[tau]] / [t.sup.2.sub.m])]
[[tau].sub.w] shear stress (Pa)
[[tau].sub.s] space time (s)

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

[FIGURE 8 OMITTED]

REFERENCES

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

Cybulski A. and K. Werner, "Static Mixers-Criteria for Applications and Selection," Int. Chem. Eng. 26, 171-180 (1986).

Grace, C. D., "Static Mixing and Heat Transfer," Chem. Process Eng. 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 Mixing in 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 Static Mixer: New Data and Proposed Correlations," 59, 265-271 (1995).

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Li, H. Z., Ch. Fasol and L. Choplin, "Hydrodynamics and Heat Transfer of Rheologically Complex Fluids in a Sulzer SMX Static Mixer," Chem. Eng. Sci. 51, 1947-1955 (1996).

Li, H. Z., Ch. Fasol and L. Choplin, "Pressure Drop of Newtonian and 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 on Forced Convection in the Kenics Static Mixer," Ind. Chem. Eng. Process Des. Dev. 16, 406-412 (1977).

Nauman, E. B., "Reactions and Residence Time Distributions in Motionless Mixers," Can J. Chem. Eng. 60, 136-140 (1982).

Nauman E. B. and B. A. Buffham, "Mixing in Continuous Flow Systems," Wiley, New York (1983).

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Rauline, D., P. A. Tanguy, J. Leblevec and J. Bousquet, "Numerical Investigation of the Performance of Several Static Mixers," Can. J. Chem. Eng. 76, 527-535 (1998).

Rauline, D., J. M. Leblevec, J. Bousquet and P. A. Tanguy, "A Comparative Assessment of the Performance of the Kenics and SMX Static Mixers," Chem. Eng. Res. Des. 78, 389-396 (2000).

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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).

Gunjan Kumar and S. N. Upadhyay * Department of Chemical Engineering and Technology, Centre of Advanced Study, Institute of Technology, Varanasi 221 005, India

* Author to whom correspondence may be addressed. E-mail addresses: [email protected], [email protected]

Table 1. Rheological constants and density of test fluids

 1% 1.5%
 [CMC.sub.hv] [CMC.sub.mv]

n 0.465 0.710
K (Pa.[s.sup.n]) 1.037 0.136
[rho] (kg/[m.sup.3]) 1007.6 1030.2


 4% 10%
 PVA PEC

n 0.774 0.859
K (Pa.[s.sup.n]) 0.045 0.01
[rho] (kg/[m.sup.3]) 1040 1039.2

Table 2. Constants for f-Re correlations

S. no. Reference Mixer element

1. Grace (1971) Helical

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

3. Cybulski and Helical (for Re [less
 Werner (1986) than or equal to] 50)

 Helical (for Re [less
 than or equal to] 100,
 Re [less than or
 equal to] 1000)

 Sulzer SMX

 Sulzer SMV

 Inliner Lightnin

 N form

 Komax

 Hi-Toray

 ISG

 LPD

4. Shah and Kale (1991) Kenics

 Sulzer

5. Chandra and Kale (1992) Komax

6. Li et al. (1997) Sulzer SMX
 Re [less than or
 equal to] 15

 15 [less than or
 equal to] Re [less
 than or equal
 to] 1000

 Re [greater than
 or equal to] 1000

7. Rauline et al. (1998) Sulzer SMX LPD

8. Cavatora et al. (1999) Sulzer S MX (20 [less
 than or equal to] Re
 [less than or equal
 to] 1000)

9. Present work Modified disk mixer
 (0[degrees]
 orientation)

 90[degrees] Orientation

 45[degrees] Orientation

Mixer element [C.sub.1] [C.sub.2]

Helical 77.76 10.88

Helical 85.50 0.34

Helical (for Re [less 115.2 0.5
than or equal to] 50)

Helical (for Re [less 6.592
than or equal to] 100,
Re [less than or
equal to] 1000)

Sulzer SMX 160-1600

Sulzer SMV 1040-4800

Inliner Lightnin 144-118.4 11.2

N form 240-272

Komax 400

Hi-Toray 608

ISG 4000-4800

LPD 5407.5 (D/L)

Kenics 64.06n 3.68[n.sup.4]/(n + 1)

Sulzer 350n 10.26n/(n + 1)

Komax 176.5n 7.56n/(n + 1)

Sulzer SMX
 Re [less than or 398
 equal to] 15

 15 [less than or 220
 equal to] Re [less
 than or equal
 to] 1000

 Re [greater than 12
 or equal to] 1000

Sulzer SMX LPD 160-960
 139.5
 146.08 (D/L)

Sulzer S MX (20 [less 78.4 0.148
than or equal to] Re
[less than or equal
to] 1000)

Modified disk mixer 70 0.75
(0[degrees]
orientation)

90[degrees] Orientation 72.2 0.24

45[degrees] Orientation 88 0.24

Mixer element m a

Helical 0.5

Helical 0.0

Helical (for Re [less
than or equal to] 50)

Helical (for Re [less 0.5
than or equal to] 100,
Re [less than or
equal to] 1000)

Sulzer SMX

Sulzer SMV

Inliner Lightnin 0.0

N form

Komax

Hi-Toray

ISG

LPD

Kenics 1.36n/(n + 3)

Sulzer 2.32n/(n + 3)

Komax 1.72n/(n + 1)

Sulzer SMX
 Re [less than or
 equal to] 15

 15 [less than or 0.8 0.8
 equal to] Re [less
 than or equal
 to] 1000

 Re [greater than 0.25
 or equal to] 1000

Sulzer SMX LPD

Sulzer S MX (20 [less 0.0
than or equal to] Re
[less than or equal
to] 1000)

Modified disk mixer 0.13
(0[degrees]
orientation)

90[degrees] Orientation 0.13

45[degrees] Orientation 0.13