Gas holdup and solid-liquid mass transfer in Newtonian and non-Newtonian fluids in bubble columns.

Ghosh, U.K. ; Upadhyay, S.N.

INTRODUCTION

Bubble columns, with or without circulation loops, are used as bio-reactors in biochemical processes as well as absorbers and strippers, and reactors in various chemical processes. The degrees of circulation of dispersed and continuous phases in bubble columns depend upon the size of equipment, the nature of phases involved, the velocities of various phases, internals, etc. Flow regime, bubble size and size-distribution, bubble rise velocity, bubble coalescence characteristics, and gas and liquid phase holdups, are the primary parameters, while mass and heat transfer coefficients, gas-liquid interfacial areas, gas-, liquid-, and thermal dispersion coefficients etc. are the secondary parameters needed for developing design correlations and also for performance evaluation of bubble columns. Accordingly, hydrodynamic and heat and mass transfer characteristics of these contactors have been studied by several workers. Some excellent reviews are available in the literature on the behaviour of bubble columns (Shah et al., 1982; Chhabra et al., 1996; Boyer et al., 2002; Kantarci et al., 2005)

Gas Holdup

Gas holdup is one of the important primary parameters required for establishing a suitable model for the design of bubble column contactors. Several workers have critically reviewed the available published information on gas holdup. Some workers have developed empirical correlations where as some have tried to develop CFD based models for predicting gas holdup.

Shah et al. (1982) critically reviewed the then available correlations for gas holdup. The divergent nature of these correlations was clearly apparent even for a simple system like air-water. Ruzicka et al. (2001) used hydrodynamic coupling between gas and liquid phases to develop a simple physical model for homogeneous--heterogeneous regime transition in bubble columns. Model predictions showed good agreement with experimental data. Boyer et al. (2002) presented a critical review on the various experimental techniques (intrusive and non-intrusive) used for investigating the behaviour of bubble and slurry bubble columns. Krishna and van Baten (2003) developed a CFD model to describe the hydrodynamic and gas-liquid mass transfer behaviour of a bubble column operating under homogeneous and heterogeneous flow regimes. They reported an excellent agreement between model results and available experimental data. Kantarci et al. (2005), in their review, focused on the description, design and operation, application areas, fluid dynamics, flow regimes encountered and parameters characterizing the operation of bubble columns. Mouza et al. (2005) studied the effect of liquid properties (surface tension and viscosity) on the performance of bubble columns. They also developed a correlation for gas holdup in the homogeneous regime. Behkish et al. (2006) conducted comprehensive literature survey on holdup in bubble and slurry bubble columns and developed separate correlations for the total gas holdup and that of large gas bubbles.

Solid-Liquid Heat and Mass Transfer

Heat transfer from contactor wall or surface of immersed heating rods and coils to the gas-liquid dispersion in bubble column contactors has been studied by several investigators. A summary of the published work on heat transfer in bubble columns and stirred vessel contactors, was presented by Steiff and Weinspach (1978) and Smith et al. (1987).

Deckwer (1980) assumed turbulent flow and suggested a method to predict heat transfer coefficient in bubble columns using surface renewal theory of heat/mass transfer coupled with the Kolmogoroff's theory of isotropic turbulence and developed a generalized correlation. The Deckwer's equation can be written as,

[St.sub.m] or St = 0.1 [{Re Fr [(Sc or Pr).sup.2]}.sup.-0.25] (1)

Patil and Sharma (1983) studied solid-liquid mass transfer in bubble columns (i.d. = 0.146, 0.38, 1.0 m) with and without downcomers by measuring the rate of dissolution of copper from the wall of the column in to acidic potassium dichromate solution. Both, average and local mass transfer coefficients were measured. The proposed correlation is basically the modification of equation proposed by Deckwer (1980) and Joshi et al. (1980). Patil and Sharma (1983) showed that Equation (1) predicted average [k.sub.c] values about 80-125% higher than their experimental data. The Kolmogoroff's theory assumed that all the energy was dissipated in causing the liquid circulation in bubble columns. Joshi and Sharma (1979) pointed out that complete dissipation of energy was not possible and only a part of the energy input was dissipated in this way. Patil and Sharma (1983), therefore, modified the coefficient 0.1 in Equation (1). From analysis of their own experimental data on mass transfer they showed that,

[St.sub.m] = 0.052 [([Re Fr Sc.sup.2]).sup.-0.25] (2)

Rai and Upadhyay (1986) used small stationary particles mounted at strategic locations in the external loop of a bubble column for obtaining mass transfer data. The close agreement between their results and those of others (Sano et al., 1974; Sanger and Deckwer, 1981) demonstrated the usefulness of this simple technique. Shah and Sharma (1987) recommended the use of Sanger and Deckwer (1981) equation for the calculation of solid-liquid mass transfer since it covered a wide range of experimental parameters. Prakash et al. (2001) conducted hydrodynamic studies and measured local heat transfer in a 0.28 m diameter slurry bubble column using suspension of yeast cells ([d.sub.p] = 8 [micro]m) as the liquid phase. Gas holdup exhibited a maximum value at [U.sub.G] = 0.10 m/s. Local heat transfer coefficients were measured at different axial and radial positions. Transfer rate in the foam section was significantly lower than in the main slurry section.

Khamadieva and Bohm (2006) investigated the effect of liquid viscosity on mass transfer at the wall of packed and un-packed bubble columns using Newtonian and non-Newtonian liquids. The results for both types of column were correlated using an expression of the form [St.sub.m] = f(Re, Fr, Sc).

From the literature review it is seen that sufficient experimental data is available on the gas holdup and heat and mass transfer in bubble columns involving Newtonian fluids, but the studies involving non-Newtonian fluids are relatively less (Chhabra et al., 1996). In order to fill in the existing gap, an attempt has been made in this work to study experimentally the average gas holdup and solid-liquid mass transfer in a bubble column using both Newtonian and non-Newtonian fluids as the continuous phase.

The results for both types of column were correlated using an expression of the form Stm = f(Re, Fr, Sc). From the literature review it is seen that sufficient experimental data is available on the gas holdup and heat and mass transfer in bubble columns involving Newtonian fluids, but the studies involving non-Newtonian fluids are relatively less (Chhabra et al., 1996). In order to fill in the existing gap, an attempt has been made in this work to study experimentally the average gas holdup and solid-liquid mass transfer in a bubble column using both Newtonian and non-Newtonian fluids as the continuous phase.

[FIGURE 1 OMITTED]

EXPERIMENTAL

Experimental Set-up

The schematic diagram of the experimental set-up used is shown in Figure 1. It essentially consisted of a test column, an air-supply unit, rotameter and pressure and temperature indicators. The test column was a vertical 'Perspex' column of diameter ([D.sub.C]) 0.145 m and of height ([H.sub.C]) 2.9 m. At the top of this column a 0.2 m diameter foam disengagement section was attached. Air to the test column was supplied at its bottom through an air distribution system. Three different single opening spargers of diameter ([d.sub.o]) 0.7, 1.0 and 1.3 mm were used. A laboratory air compressor was used as the source of air supply. Air flow rate was measured with the help of calibrated rotameter (Eureka, Pune, India). A pressure gauge and a thermistor probe were also used in the flow line prior to the test column to measure the air pressure and temperature, respectively. Ports were provided in the test column at appropriate locations (at the heights of 1.58 and 2.31 m) to connect graduated side tubes for holdup measurements.

For solid-liquid mass transfer measurements, three ports were provided in the column at 0.8, 1.55 and 2.05 m from the tip of the air sparger. The support wire holding the test specimen was attached to a split rubber cork through a connecting gland which in turn was mounted on the test column radially through a 'Perspex' gland in a manner to position the test specimen along the axis of the column.

Test Fluids

Demineralized water, 60% aqueous propylene glycol and 0.5 and 1.0% aqueous carboxymethyl cellulose (CMC) solutions were used as the test liquids. Demineralized water was obtained from an IAEC laboratory demineralizing plant. Chemically pure propylene glycol (s.d. Fine-Chem Pvt. Ltd., Boiser, India) was used for making aqueous propylene glycol solutions. The sodium carboxymethyl cellulose (CMC) used for making polymer solutions was obtained from Robert-Johnson (India). Aqueous polymer solutions were prepared by dissolving a known weight of powder in a known volume of demineralized water. Special care was taken to avoid the formation of tiny encapsulated lumps of polymer powder or 'cat's eyes'. The procedure followed was the same as used by Kumar and Upadhyay (1981).

The CMC solutions were assumed to be pseudoplastic and were characterized by the Ostwald-de-Waele power law model,

[tau] = K[[gamma].sup.n] (3)

where [tau], [gamma] K and n are shear stress, average shear rate, power law fluid consistency index and flow behaviour index, respectively. The effective viscosity was defined as,

[[mu].sub.eff] = K[[gamma].sup.n-1] (4)

The K and n values of the CMC solutions and viscosity of the other liquids ([[mu].sub.L]) were determined with the help of a Brookfield cone plate viscometer (Model RVTDV-ICP) from the respective flow curves.

In order to calculate the effective viscosity of a non-Newtonian fluid in the bubble column it is necessary to know the effective average shear rate in the bubble column. Several workers (Godbole et al., 1984; Haque et al., 1986a, 1986b; Nakanoh and Yoshida, 1980; Schumpe and Deckwar, 1982) used the relation for the shear rate proposed by Nishikawa et al. (1977) for bubble columns for this purpose. This relation was used in the present study also. According to Nishikawa et al. (1977),

[gamma] = 5000[U.sub.G] (5)

where [gamma] and [U.sub.G] are in [s.sup.-1] and m/s, respectively.

PROCEDURE

Gas Holdup

For making the gas holdup measurements, the test fluid was filled in the column up to a desired height ([H.sub.C]). The air flow, after metering through a calibrated rotameter was allowed to bubble into the liquid through the sparger fitted at the bottom of the column. After allowing the flow to continue for some time at a steady rate, measurements of air flow rate, air pressure, air temperature, liquid temperature and liquid height in the column were made. The fractional gas holdup ([[member of].sub.G]) was calculated by measuring the change in liquid height in the side tube at various flow rates. Similar observations were repeated at different flow rates and with different fluids. In each case effects of initial liquid height and sparger were studied.

Solid-Liquid Mass Transfer

Solid-liquid mass transfer coefficients were measured by following the dissolution rate of a low soluble solute. Benzoic acid was chosen as the solute on the basis of its low solubility and amenability to pelletization without a binding agent. It was obtained from Sarabhai Chemicals, Baroda, India. Benzoic acid granules were compressed in the form of pellets in a single punch pelleting machine as described by Kumar et al. (1987). Four different sizes of pellets were prepared. The particle surface area and volume were calculated from the average dimensions of pellets measured with the help of a micrometer screw gauge. The support wires used for mounting the pellets in the test column were stainless steel wires, 1500 ?m in diameter and 0.08 m long. One end of the wire was heated just above the melting point of the acid and embedded radially in the pellet. Any molten acid sticking to the surface of the wire was removed carefully with the help of a fine file followed by polishing with tissue paper. All such test specimens were washed with water and dried in a desiccator before use in the actual runs. The equilibrium solubility and diffusivity values of the benzoic acid in the test liquids were taken from the literature (Kumar et al., 1987; Steinberger and Treybal, 1960; Lal et al., 1988).

A test particle, weighed to the nearest 0.05 mg, was placed in the column through the sample mounting port located at 1.55 m height from the sparger. The bubble column was filled with the test liquid and air flow at a known rate was diverted through the sparger. After a fixed interval of time, the test liquid was drained out of the column up to a level just below the test particle. The test particle was then taken out, dried in a desiccator for 36 h and reweighed. The loss in the weight thus obtained was used to calculate mass transfer coefficients. For runs made with 60% aqueous propylene glycol and aqueous CMC solutions, the test particles were washed with a saturated solution of benzoic acid in water before drying in order to remove the test liquid sticking to the pellet surface. Effects of test particle position, sparger diameter, and liquid height in column were studied in separate runs by making similar measurements by changing the operating condition in the column appropriately.

[FIGURE 2 OMITTED]

In a separate set of blank runs, the loss in weight of the pellets during mounting into and removal from the test column and washing, was also determined and was used for correction purpose.

RESULTS AND DISCUSSION

Gas Holdup

Depending upon the superficial gas velocity, fluid viscosity, column diameter and sparger type, different regimes of gas-liquid flows persist in bubble columns. Visual observations during experimental runs indicated that primarily bubbly and churn-turbulent flow regimes persisted for all liquid heights and sparger diameters used within the range of superficial gas velocity covered. In a few cases slug flow was also observed. Figure 2 shows a simple flow regime chart for transition from churn turbulent to slug flow regime based on visual observations for both Newtonian and non-Newtonian fluids presented by Haque et al. (1986a). According to this chart churn turbulent flow prevails in the upper part of the graph i.e., above the line bounded by the continuous curve. Below this curve slug flow is encountered at higher viscosities. As the column diameter increases the bubble size necessary for slug flow becomes larger and is encountered at higher viscosities. Thus, Haque et al. (1986a) reported the flow transition to be occurring at a critical viscosity that increased with increasing column diameter. For larger diameter columns ([D.sub.c] > 0.38-1.0 m), they did not observe any slug formation within the range of the effective viscosity ([[mu].sub.eff] < 0.17 Pa.s) of fluids used by them. Godbole et al. (1984) reported the existence of churn turbulent flow in a 0.35 m diameter column and up to a viscosity of 8.1 Pa.s. Schumpe and Deckwer (1982) reported the formation of slugs at [[mu].sub.eff] [approximately equal to] 0.02 to 0.03 Pa.s, for a column of 0.14 m diameter. Patwari (1983) reported slug flow regime at viscosity as low as 0.018 Pa.s. for a column of 0.06 m diameter. The viscosity--diameter data for the two non-Newtonian fluids used in this work are also shown in Figure 2. It is clear that for CMC solution, the column operated in the slug flow regime. A further analysis of the present data in light of the inviscid theory also confirms this.

Slip velocity between the bubble and the liquid must be known for using the theoretical relations for predicting the gas holdup and other related parameters. This in turn requires a knowledge of the bubble rise velocity ([U.sub.br]). A relation can be obtained for calculating [U.sub.br] by applying the inviscid theory to rising bubbles. Whalley and Davidson (1974) and Vishwanathan and Rao (1983; 1984) analyzed this problem. Vishwanathan and Rao (1983; 1984) developed a simple non-numerical predictive model to calculate the flow strength and velocity profiles in the liquid circulation zone which led to the equation,

[U.sub.G.sup.2] = 4 [U.sub.br] [(g [D.sub.C]).sup.0.5] [[member of].sub.G.sup.2.5] (6)

It follows from Equation (6) that for a given column, [U.sub.br] can be obtained from the slope of the plot of [U.sub.G.sup.2] versus [[member of].sub.G.sup.2.5]. The experimental data obtained in this work were plotted as [U.sub.G.sup.2] versus [[member of].sub.G.sup.2.5] plots for various systems investigated. It was seen that the data for water and 60% aq. propylene glycol solution fell on fairly good straight lines, whereas those for 0.5 and 1.0% aq. CMC solutions remained non-linear. The fairly good linear relation in the case of water and aq. propylene glycol confirms the validity of inviscid circulation model for Newtonian fluids. The slope of these lines and, therefore, the [U.sub.br] values remain more or less constant with increasing ([H.sub.C]/[D.sub.C]). The sparger openings also do not appear to have much influence on the bubble rise velocity. Haque (1986) reported the validity of inviscid theory for non-Newtonian fluids on the basis of their data for several aq. CMC solutions obtained at ([H.sub.C]/[D.sub.C]) ranging from 1 to 5. The present data did not agree with this probably due to large ([H.sub.C]/[D.sub.C]) ratio (11 to 16) used in this work. The [U.sub.br] values estimated from the slopes of the approximate linearized plots of [U.sub.G.sup.2] versus [[member of].sub.G.sup.2.5] were used to estimate average bubble diameters using the procedure recommended by Haque (1986). For water and 60% aq. propylene glycol solution the average bubble diameters were estimated to be less than 0.035 m and those for 0.5% and 1.0% aq. CMC solutions were found to be of the order of 0.08 to 0.1 m and upto 0.26 m, respectively. This indicates that the flow regime for 1.0% aq. CMC is slug flow and that for other liquids is mostly churn-turbulent.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Gas holdup studies were performed with all the four test liquids. Three different orifice type air spargers of diameters 700, 1000 and 1300 [micro]m were used. Working liquid height was varied from 1.58 to 2.31 m. Typical plots of effect of gas holdup versus superficial gas velocity are shown in Figures 3 and 4. It is seen that the gas holdup increases with increasing gas velocity ([U.sub.G]). This increase, however, tends to become slightly slower at higher [U.sub.G] values, particularly for viscous liquids.

The effect of sparger diameter was studied with all the test fluids using spargers of diameters 700, 1000 and 1300 [micro]m keeping the liquid height same. The nature of the gas holdup--velocity curves remained practically unaffected by the sparger diameter, [d.sub.o], in the flow regime studied. Similar observations were also reported by Schumpe and Deckwer (1982) and Haque et al. (1986a, b).

[FIGURE 5 OMITTED]

Hydrodynamics of a bubble column is likely to be affected by the liquid height to column diameter ratio ([H.sub.C]/[D.sub.C]). Figures 3 and 4 show the effect of ([H.sub.C]/[D.sub.C]) ratio on fractional gas holdup as typical examples. In the present investigation the ([H.sub.C]/[D.sub.C]) ratio was varied from 11 to 16. No significant effect of ([H.sub.C]/[D.sub.C]) is observed on the gas holdup. Haque et al. (1986a, b), however, reported that for ([H.sub.C]/[D.sub.C]) [less than or equal to] 3, the gas holdup decreased with increasing ([H.sub.C]/[D.sub.C]). At low liquid heights, the equilibrium bubble size is not attained and hence the bubble rise and liquid circulation velocities are also not at their equilibrium values. For large ([H.sub.C]/[D.sub.C]), the bubble breakup due to shear sets in leading to equilibrium bubble size, hence a constant gas holdup.

The bubble column hydrodynamics is also expected to depend upon the column diameter. This effect is likely to be more pronounced in the churn-turbulent flow than in the bubbly flow regime. Godbole et al. (1984), reported a strong influence of column diameter on the gas holdup. Haque et al. (1986 a, b) reported that the holdup decreased as the column diameter increased. Haque et al. (1986a) quantified the effect of column diameter by plotting the mean bubble rise velocity ([U.sub.G]/[[member of].sub.G]) with respect to the column diameter and observed that the mean bubble rise velocity for larger diameter columns were higher and increased with increasing viscosity. Godbole et al. (1984) also reported a similar behaviour. As only one column has been used in the present work, no direct evaluation of this effect has been possible.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

In order to study the effect of liquid phase viscosity, [[mu].sub.L] or [[mu].sub.eff], on gas holdup, measurements were made with all the four test fluids using the same sparger and same liquid height in the column. From Figure 4, it is seen that, for a given air flow rate the gas holdup is lower for fluids of higher viscosity. This change is more pronounced for non-Newtonian fluids than for Newtonian, which can be attributed to the reduced surface tension. A typical cross plot of [[member of].sub.G] versus viscosity ([[mu].sub.L] or [[mu].sub.eff]) at various superficial gas velocities is shown in Figure 5. It is seen that the holdup varies as [([[mu.sub.L] or [[mu].sub.eff]).sup.-0.13]. Godbole et al. (1984), Schumpe and Deckwer (1982) and Haque et al. (1986a) also reported similar effect.

The Zuber-Findlay model (1965) type relation was used to correlate the experimental gas holdup data obtained in this work together with those of Haque (1986) and Abraham (1990). Haque (1986) obtained holdup data with aq. CMC solutions. Test columns of 0.2 to 1.0 m diameter and different types of sparger were used. Abraham (1990) also used aq. CMC solutions and a number of spargers. The ranges of various parameters covered in the three investigations are given in Table 1. Regression analysis of the data as plotted in Figure 6 showed that equation,

[[member of].sub.G] [([[mu].sub.L] or [[mu].sub.eff]).sup.0.13] [[D.sub.C].sup.0.15] [([H.sub.C]/[D.sub.C]).sup.0.2] = [U.sub.G]/(0.65 + 2.5 [U.sub.G.sup.0.666]) (7)

correlates all the Newtonian and non-Newtonian data used with an average deviation of [+ or -] 11.5% for a large column diameter ([D.sub.C] = 0.145 to 0.38m) and operating parameters ([U.sub.G] = 5 x [10.sup.-4] to 2 x [10.sup.-1]m/s and ([H.sub.C]/[D.sub.C]) = 2 to 16) range.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

Solid-Liquid Mass Transfer

Figure 7 shows the variation of mass transfer coefficient, [k.sub.c], with superficial gas velocity, [U.sub.G], for all the four test fluids and spargers of 700 and 1000 [micro]m diameter for test particles mounted at the same location inside the column. It is seen from these plots that [k.sub.c] versus [U.sub.G] plots have nearly similar shape. The data appear to follow the relation,

[k.sub.c] [infinity] [U.sub.G.sup.a] (8)

where, average value of a as observed in this work is 0.25 to 0.3 and it is very close to the value (0.29 to 0.33) reported by Patil and Sharma (1983) and others (Deckwer, 1980) for bubble columns and by Rai and Upadhyay (1986) for external loop airlift bubble columns.

In order to examine the effect of position of the mass transfer surface from the sparger tip on the mass transfer coefficient, three separate test particles were mounted at fixed axial locations measured from the tip of the air sparger. Typical results are shown in Figure 8. No apparent effect of height on the mass transfer coefficient is evident and the [k.sub.c] values are found to be nearly the same at a given flow rate. This is also in agreement with observations reported by earlier workers (Patil and Sharma, 1983; Deckwer, 1980; Rai and Upadhyay, 1986). For CMC solutions, [k.sub.c] values for test particles mounted at 2.05 m from the sparger are some what lower due to foam formation in this part of the column.

Figure 9 shows the plots of mass transfer coefficients versus UG for the spargers of 700 and 1000 [micro]m diameters. It is clear from these figures that the sparger diameter has no effect on the mass transfer coefficient.

Figure 10 shows a comparison of Equations (1) and (2) with the present experimental data as well as those of Patil and Sharma (1983). Present mass transfer results show the same form of functional relationship between [St.sub.m] and ([ReFrSc.sup.2]) as observed by Deckwer (1980) for heat transfer and Patil and Sharma (1983) for mass transfer. The data points, however, lie in between the two equations. The Deckwer's (1980) relation {Equation (1)} correlated present data with an average deviation of [+ or -] 21.7%. Efforts to reduce the degree of scatter by including other parameters like gas holdup, liquid height to column diameter ratio etc. in the correlation were not very successful.

CONCLUSIONS

Effects of several operating parameters on gas holdup and wall to liquid mass transfer rates have been investigated. The gas holdup increases with superficial gas velocity for all sparger diameters used. The liquid height-column diameter ratio and sparger diameter have no influence on the bulk gas holdup in the range covered in the study. For a given geometry and operating conditions, the gas holdup decreases with increasing effective liquid phase viscosity. The inviscid theory is inadequate for predicting the average bubble size for large liquid height to column diameter ratios (>5). Equation (7) predicts the present and published gas holdup data for both Newtonian and non-Newtonian fluids over a wide range of column diameter ([D.sub.C] = 0.145 to 0.38 m), liquid height to column diameter ratio ([H.sub.C]/[D.sub.C]) = 2 to 16) and superficial gas velocity ([U.sub.G] = 5 x [10.sup.-4] to 2 x [10.sup.-1] m/s).

Orifice diameter and the location of the mass transfer surface (beyond the sparger region) have no effect on the mass transfer rate. Equation (1) correlates the present data together with those of others with an average deviation of [+ or -] 21.7%.

ACKNOWLEDGEMENT

This paper is based on the Ph.D. thesis of Dr. U. K. Ghosh submitted to the Banaras Hindu University, Varanasi, India.

NOMENCLATURE

a exponent
[C.sub.pL] heat capacity of liquid, [L.sup.2][t.sup.-2]
 [T.sup.-1]
[D.sub.c] column diameter, L
[D.sub.m] molecular diffusivity, [L.sub.2] [t.sup.-1]
[d.sub.o] orifice diameter, L
Fr Froude number, = [U.sub.G.sup.2]/[gL.sub.c]
g acceleration due to gravity, [Lt.sup.-2]
[k.sub.L] thermal conductivity of liquid, ML[t.sup.-3]
 [T.sup.-1]
H height column L
[H.sub.c] clear liquid height, L
[h.sub.w] wall heat transfer coefficient, ML[t.sup.-3]
 [T.sup.-1]
[k.sub.c] solid--liquid mass transfer coefficient,
 [Lt.sup.-1]
K power law fluid consistency index, [ML.sup.-1]
 [t.sup.n-2]
[L.sub.c] characteristic dimension, L
m exponent
n power law fluid flow behaviour index
Pr Prandtl number, = ([mu] [C.sub.pL]/[k.sub.L])
Re Reynolds number, = ([L.sub.c] [U.sub.G] [[rho]
 .sub.L]/([[mu].sub.L] or [[mu].sub.eff])
Sc Schmidt number, = ([[mu].sub.L]/[[rho].sub.L]
 [D.sub.m]) or ([[mu].sub.eff]/[[rho].sub.L]
 [D.sub.m])
St Stanton number for heat transfer, = ([h.sub.w]/
 [U.sub.G] [[rho].sub.L] [C.sub.pL])
[St.sub.m] Stanton number for mass transfer, = ([k.sub.c]/
 [U.sub.G])
[U.sub.G] superficial gas velocity, [Lt.sup.-1]
[U.sub.br] terminal bubble rise velocity, [Lt.sup.-1]

Greek Symbols

[[member of].sub.G] fractional gas holdup
g average shear rate, [t.sup.-1]
[[mu].sub.eff] effective viscosity of liquid, [ML.sup.-1]
 [t.sup.-1]
[[mu].sub.L] viscosity of liquid, [ML.sup.-1] [t.sup.-1]
[[rho].sub.L] density of liquid, [ML.sup.-3]
[tau] shear stress, [ML.sup.-1] [t.sup.-2]

Manuscript received May 31, 2006; revised manuscript received March 20, 2007; accepted for publication March 21, 2007.

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U. K. Ghosh ([dagger]) and S. N. Upadhyay *

Department of Chemical Engineering & Technology, Centre of Advance Study, Institute of Technology, Banaras Hindu University, Varanasi--221005, INDIA

* Author to whom correspondence may be addressed. E-mail address: upadhyaysnu@rediffmail.com

([dagger]) Presently at the Department of Paper Technology, IIT Roorkee, Saharanpur Campus, Saharanpur--247 001, India

Table 1. Range of parameters covered in holdup data used

Source [U.sub.G], m/s Sparger

Haque (1986) 8x[10.sup.-3] - Spargers of various
 2.1 x [10.sup.-1] geometries
 (single or multiple holes)
 [d.sub.o] = 2-19 mm

Abraham (1990) 0.02-0.14 Single orifice
 [d.sub.o] = 10-25 mm

Present Work 4.5 x [10.sup.-4] - Single orifice
 1.57 x [10.sup.-2]
 [d.sub.o] = 0.7-1.3 mm

Source Fluid

Haque (1986) Aq. CMC solutions
 n = 0.8-0.61
 K = 0.012-0.35 [Pa.s.sup.n]

Abraham (1990) Water
 0.25-1% aq. CMC
 solutions
 n = 0.77-0.62
 K = 0.147-2.01 [Pa.s.sup.n]

Present Work Water
 [[mu].sub.L] = 8.6 x [10.sup.-4] Pa.s at
 28.6[degrees]C
 60% aq. propylene glycol solution
 [[mu].sub.L] = 8.7 x [10.sup.-3] Pa.s at
 21.5[degrees]C
 0.5% aq. CMC solution
 n = 0.96
 K = 0.044 [Pa.s.sup.n] at 23.1[degrees]C
 1.0% aq. CMC solution
 n = 0.92
 K = 0.126 [Pa.s.sup.n] at 19.3[degrees]C

Source [H.sub.C]/[D.sub.C] [D.sub.C], m

Haque (1986) 2-5 0.2-1.0

Abraham (1990) 4-8 0.38

Present Work 11-16 0.145