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Gas-Liquid Mass Transfer in Benchscale Stirred TankssFluid Properties and Critical Impeller Speed for Gas Induction Martijn M. P. Zieverink, Michiel T. Kreutzer,* Freek Kapteijn, and Jacob A. Moulijn Delft UniVersity of Technology, Reactor and Catalysis Engineering, Julianalaan 136, 2628 BL Delft, The Netherlands
This work is concerned with the gas-liquid mass transfer in stirred tanks with gas-inducing impellers. Experiments were performed to determine the critical impeller speed for the onset of gas induction and the mass-transfer group kLa, using water, n-hexadecane, toluene, acetone, and sunflower oil at various temperatures as liquids and hydrogen and nitrogen as gases. The use of different liquids and gases allowed the influence of fluid properties on the mass transfer to be studied. Special emphasis is on reducing the experimental effort in downscaling, i.e., the use of benchscale autoclaves to mimic industrial conditions. In benchscale tanks, the capillary pressure is significant, with respect to the static liquid pressure at the impeller, and must be taken into account to describe the critical impeller speed for gas induction. Downscaling implies relatively low mass-transfer rates, which can be achieved by operating close to the critical impeller speed. The critical impeller speed must be included in models for the mass-transfer rate. Our data set of kLa values was best described by a dimensionless correlation, in which a Froude group (based on the stirrer speed N and the critical stirrer speed Ncr) was used to account for the gas-induction rate, together with the Reynolds number and Schmidt number, to account for turbulence intensity and fluid properties. 1. Introduction Stirred tanks are notoriously difficult to scale-up, and this is true even more so for multiphase stirred slurry reactors. Not only is it difficult to understand the scaling phenomena, stirred tanks are fundamentally poor mixers, and the bigger they get, the lower the mass-transfer rates that can be achieved in them.1 The reverse is also true: if an industrial-scale stirred-tank process is conducted in a benchscale stirred-tank autoclave, the mass-transfer limitations that were present on the industrial scale likely have disappeared on the small scale. Typical numbers are illustrative: in industrial units, kLa is often ∼0.1 s-1 and rarely exceeds 0.3 s-1, whereas in benchscale autoclaves operated at maximum stirrer speed, the value of kLa readily exceeds 1 s-1. For benchscale units, one must only consider the size of the stirrer engine, which is usually larger than the vessel for which it provides the stirring energy, to ensure that ample stirring energy is available. The subject of this paper is the gas-liquid mass transfer in such small benchscale units, with the application of downscaling in mind. In other words, we are interested in obtaining the relatively low mass-transfer rates of industrial units in laboratory autoclaves. On an industrial scale, the stirring power can be measured from the electric current to the stirrer, and then the power dissipation per unit volume can be used, together with the holdup, to estimate the mass-transfer rate using the theory of isotropic turbulence.2 On the bench scale, this is much more difficult: the sealing of the stirrer shaft in high pressure autoclaves generates too much friction to obtain reliable data, and most systems are not equipped with a torque meter. Most benchscale units are dead-end autoclaves with an internal gasinducing impeller. Therefore, neither power dissipation, nor gas flow rate, nor holdup are easily determined for benchscale units, and the popular correlations based on these quantities cannot be used. * To whom correspondence should be addressed. Tel.: +31 15 278 90 84. Fax: +31 15 278 50 06. E-mail:
[email protected].
It is not too difficult to measure the mass-transfer behavior of a benchscale unit: most autoclaves are equipped with a frequently sampled pressure transducer, which has a sensor response time well below 1/(kLa). A pressure step experiment3-6 provides a simple and fast way to obtain mass-transfer rates from the decay of the headspace pressure in time. Therefore, because mass-transfer experiments are fast and straightforward, one might be tempted to suggest that a few such pressure step experiments suffice before embarking on a campaign of reactive experiments. The failure of this approach for fat hydrogenation was recently documented and elegantly explained by Fillion et al.7,8 Fillion et al. measured the value of kLa in untreated edible oil prior to hydrogenation and attempted to use these mass-transfer rates to account for mass-transfer limitations in the post-processing of their hydrogenation runs. This gave unsatisfactory results, because, during the reactive experiments, the fluid properties changed as the conversion progressed: edible oils become more viscous as they are hydrogenated or “hardened”. The higher viscosity, in turn, reduced the mass-transfer rate. This is perhaps a somewhat exotic example: most reactions are not accompanied by such dramatic changes in fluid properties. A more common problem is that taking liquid-phase samples in the course of a batch experiment changes the liquid level in the reactor, which increases the masstransfer rate, as will be discussed shortly. In short, one should not only do several pressure step experiments prior to reactive experiments, but also understand how the mass transfer changes as the reactive experiment progresses. The first aim of this paper is to determine, for one of our autoclaves, in detail, the mass-transfer characteristics at the low values of kLa that are needed to mimic an industrial process. In particular, we were interested in the influence of fluid properties and liquid level on kLa. The second aim of the paper is to reduce the amount of experimental work that would be needed to determine these characteristics in different autoclaves. We have limited ourselves to experiments that can be performed with existing equipment, foregoing the torque meters and forced-
10.1021/ie060092m CCC: $33.50 © 2006 American Chemical Society Published on Web 05/17/2006
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gas circulation that are not readily available on high-pressure autoclaves. We realize that the number of geometric parameters is large in stirred tanks, and we wish to avoid the hubris of suggesting correlations of universal validity. Rather, our aim is to provide enough understanding of the transport phenomena, so that, in different reactors, a minimal experimental effort is needed to obtain an estimate of the mass-transfer characteristics and to confidently incorporate mass-transfer limitations in the modeling of scaled-down processes. The paper is organized as follows. First, we briefly review theory and correlations from the open literature. Subsequently, we describe our own experiments. Finally, we discuss, for users of benchscale autoclaves, which experiments are most suited to quickly determine the mass-transfer behavior of their equipment. 2. Previous Work A gas-inducing impeller has a hollow shaft that provides a passage for the gas phase from the headspace above the liquid to holes in the impeller. When a gas-inducing impeller is rotating, the pressure difference between the holes in the top of the hollow shaft and holes on the impeller can be described using the inertial term and the hydrostatic term of the Bernoulli equation:
∆p ) R
( )
Futip2 - FgH 2
(1)
where R is a constant that describes the inducing efficiency, F the density of the liquid, utip the velocity of the stirrer tip, and H the submerged depth of the holes in the impeller. For ∆p > 0, the suction induced by the rotating impeller draws gas into the liquid. Setting ∆p ) 0 and substituting utip ) πNdst gives the critical impeller frequency (Ncr) at which gas is just drawn in. White and de Villiers9 used this analysis to arrive at a constant critical Froude number (Frcr) for different submerged depths:
Frcr )
Ncr2dst2 ) constant gH
(2)
in which dst is the impeller diameter and g is the gravitational constant. Equation 2 indicates that the critical stirrer speed is not a function of any liquid property and should be constant for a given reactor geometry. There is agreement over the limited impact of density, which, in the Froude number Fr, is cancelled out. Sawant and Joshi10 measured the critical impeller speed for viscosities up to 80 mPa s, and they found only a weak dependence on viscosity: Ncr2 ∝ (µ/µref)-0.11. This weak effect has been confirmed by others, although the exponential constant 0.11 is subject to debate and probably is dependent on the stirrer geometry. For instance, Poncin et al.11 found Ncr2 ∝ (µ/µref)-0.07. The critical impeller speed is highly relevant for gas-liquid mass transfer: for N < Ncr, only mass transfer at the top surface of the liquid occurs. At higher stirrer speeds (N > Ncr), bubbles are created at the impeller, and the mass-transfer rate increases rapidly as the stirrer speed increases. In fact, the critical stirrer speed may be estimated rather accurately from the change in slope of kLa versus N. The mass transfer due to surface aeration is much lower than the mass transfer at higher stirrer speeds (see, for instance, Figure 7 presented later in this work). Joshi and Sharma12 found that
kLa - kLa0 ∝ (N - Ncr)1.2
(3)
Poncin et al.11 measured mass transfer for different liquid levels and stirrer speeds, and they proposed a dimensionless correlation that was based on the Froude number Fr:
k La ∝
(Fr - Frcr)1.1 1 + 0.132(Fr - Frcr)1.1
(4)
This correlation agrees with eq 3, in the sense that below Ncr, the mass transfer is not dependent on the stirrer speed anymore (and, in fact, it becomes insignificant). However, eq 4 was obtained in air-water systems and, most likely, does not extrapolate to different fluid properties: note that no fluid properties appear in eq 4. Dietrich et al.4 proposed a correlation from experiments with water, ethanol, and partially hydrogenated adiponitrile, in which the mass-transfer group was made dimensionless, using the stirrer diameter as the characteristic length (using a Sherwood number, which was defined as Sh ) kLadst2/D):
Sh ∝ Re1.45Sc0.5We0.5
(5)
In this correlation, the critical stirrer speed is not included, and Dietrich et al. noted that the proportionality constant for eq 5 was a function of the liquid level H. (Here, Re is the Reynolds number, Sc is the Schmidt number and We is the Weber number.) Even if the liquid height is taken into consideration in the proportionality constant, eq 5 still does not predict a change in slope at Ncr. This was probably not a problem in the experiments at high stirrer speeds (N . Ncr) upon which this correlation is based, but it obviously will be a problem if such an equation is to be used in the proximity of Ncr. In sparged agitated tank reactors, gas-to-liquid mass transfer is often correlated as
k La ∝
()
Pg n m VL G
(6)
where (Pg/VL) is the power input per unit liquid volume in the presence of gas and G is the gas holdup.13-16 Poncin et al.11 demonstrated that such correlations based on power input nicely predicted mass-transfer rates for gas-inducing impellers, using measurements of gassed power input and holdup. In summary, a reasonable understanding of the mass-transfer behavior for gas-inducing agitated tanks is available. If the power input and (internal) gas holdup can be readily determined, the mass-transfer group may be predicted from eq 6. If only the stirrer speed, liquid level, and fluid properties can be used, then the stirrer speed should be taken into account, using Re, Fr, or We or a combination of these dimensionless groups. It is still unclear which of these dimensionless numbers, or combination thereof, should be used. Most correlations in the open literature base Re, Fr, and We on the stirrer speed N, whereas most of the experimental data suggests that (N - Ncr) is the relevant stirrer speed. For a more extensive overview of literature on gas-inducing impellers in the open literature, the interested reader is referred to Patwardhan and Joshi17 and Lemoine et al.18 3. Experimental Section Equipment. A 400-mL high-pressure autoclave (flat bottom, diameter of 0.064 m), with four symmetrically placed baffles
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Figure 1. Schematic diagram of the stirrer used. Table 1. Properties and Composition of Sunflower Oil fatty acid
content (wt %)
stearic (saturated) oleic (mono-unsaturated) linoleic, linolenic (poly-unsaturated) others (C16 and C20 fatty acids)
11 22 59 8
(W/dtank ) 0.1), was used. The reactor was equipped with a six-bladed gas-inducing Rushton stirrer (dst ) 31.8 mm; see Figure 1), with 2.5-mm holes in the hollow shaft. The holes in the stirrer were 22.5 mm above the bottom of the reactor. The liquid temperature was controlled using an external heating jacket. Both pressure and temperature in the reactor were recorded by a computer. For visual observation of the onset of gas induction, the high-pressure vessel was replaced by a transparent plastic tank of the same dimensions. Commercially available sunflower oil and technical-grade toluene, acetone, and n-hexadecane were used as liquids. The composition of the sunflower oil is given in Table 1. Physical Properties. The diffusivity of N2 and H2 in acetone, toluene, and hexadecane was calculated using the Wilke and Chang equation.19 All relevant fluid properties for these liquids are given in Table 2. The diffusivity of hydrogen and nitrogen in vegetable oil was estimated using the experimental data and correlations that were compiled by Fillion and Morsi.20 The viscosity of the sunflower oil used was determined using a rotating drum viscometer, and the measured values agreed with the correlation given by Veldsink et al.21 The density of sunflower oil was estimated using data from Bailey’s Industrial Oil and Fat Products.22 The surface tension of the sunflower oil was calculated using the relation given by Bartsch.23 (See also Supporting Information.) Critical Stirrer Speed. The critical stirrer speed was measured by slowly increasing the stirrer speed until the first bubbles appeared. Figure 2 shows four photographs of the stirrer in a transparent tank of the same size as the high-pressure autoclave vessel, for increasing stirrer speeds. Bubbles appear at a stirrer speed of >12.3 Hz, indicating that the critical stirrer speed in this case must be between 12.3 Hz and 14.5 Hz. In the measurements of the critical stirrer speeds, small increments in stirrer speed were used, and the critical stirrer speed could be measured with an accuracy of 0.1 Hz.
Figure 2. Visualization of critical stirrer speed. The liquid in these photographs was water.
Mass Transfer. To determine the gas-liquid mass transfer, a physical adsorption method was used, as described by Dietrich et al.4 This method, which is also known as the dynamic pressure-step method, was used as follows: (1) The liquid is degassed under agitation until an equilibrium pressure P0 is reached (with P0 ) Patm). (2) Without agitation, the autoclave is pressurized to a pressure Pm. (3) At t ) 0, the stirrer is started, and the pressure is recorded as a function of time until a new equilibrium pressure Pf is reached. Integration between t ) 0 (P ) Pm) and t(P) gives the mass balance between the gas and liquid phases:
ln
(
) (
)
P m - Pf P m - P0 ) k a(t - t0) P - Pf Pf - P0 L
(7)
By plotting the left-hand side of eq 7 versus time, kLa can be determined from the slope. An example of this graphical method is given in Figure 3. 4. Results Critical Stirrer Speed. From the measured Ncr value, the critical Froude number Frcr was first calculated using eq 2. The resulting Frcr value (Figure 4) was not independent of liquid height. In fact, the pressure term due to bubble formation in
Table 2. Fluid Properties (at Room Temperature, unless Indicated Otherwise) property
oil (at 295 K)
oil (at 438 K)
toluene
acetone
n-hexadecane
viscosity, µL (mPa s) density, FL (kg/m3) surface tension, γ (N/m) diffusion coefficient (× 10-9 m2/s) N2 H2
58.8 915 0.036
2.73 822 0.029
0.586 868 0.0285
0.422 791 0.0237
3.45 773 0.0276
1.17 1.98
10.9 19.3
4.25 7.24
6.63 11.3
1.13 1.93
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Figure 5. Critical Froude numbers, in which the capillary pressure is added to the static head.
Figure 3. Graphical determination of the gas-liquid mass transfer.
Figure 4. Critical Froude numbers for onset of gas induction, without correcting for the capillary head.
the stirrer, 2γ/rhole, is significant, with respect to the static head; therefore,
FgH +
2γ ≈ F(Ncrdst)2 rhole
(8)
and, hence, a better criterion for gas induction is given by
F(Ncrdst)2 ) constant FgH + 2γ/rhole
(9)
This equation may be rewritten to give a correction term for Frcr:
Frcr )
(
)
(Ncrdst)2 2γ/rhole ) constant × 1 + gH FgH
(10)
This corrected value is plotted for different liquid levels in Figure 5, and this Frcr value was constant within experimental error for a given liquid. The critical stirrer speed was measured using acetone, soybean oil, n-hexadecane, and water at 296 K to estimate the effect of viscosity. For water, Frcr ) 0.155 in the limit of high liquid levels, and, arbitrarily, the viscosity of water was used as the reference viscosity to determine a powerlaw correction for Frcr. Frcr is plotted versus the liquid viscosity (µ) in Figure 6, and the following power-law dependence was determined:
Frcr ) 0.155
( ) ( µ µref
0.09
1+
)
2γ/rhole FgH
(11)
Figure 6. Dependence of the critical Froude number, at high liquid level, on the liquid viscosity. Table 3. Experimental Campaign for the Mass-Transfer Measurements set
liquid
gas
T (K)
H (m)
N (Hz)
A B C D E F
sunflower oil sunflower oil acetone toluene toluene n-hexadecane
H2 N2 H2 H2 H2 N2
295-440 315-390 295 295 295 295
0.055 0.055 0.055 0.055 0.055-0.063 0.039-0.063
17-22 13-22 10-20 10-22 17.5 12-22
The effect of viscosity is somewhat less than that predicted by Sawant and Joshi,10 who found Fr ∝ µ0.11, but is somewhat greater than that determined by Poncin et al.;11 thus, the weak viscosity dependence seems to be related to the shapesand, perhaps, sizesof the stirrer. Mass Transfer. Pressure-step experiments were performed using four liquids and two gases. In the experiments using sunflower oil, the temperature was varied from 295 K to 440 K, which decreased the viscosity from 0.06 Pa s at room temperature to 0.003 Pa s at 440 K. High viscosity results in a lower diffusion coefficient, and it reduces the intensity of turbulence. Table 3 gives an overview of the experimental conditions. All experimental data are listed in the Supporting Information. The mass-transfer rates measured varied from 0.005 s-1 to 0.5 s-1, depending on the fluid properties, liquid height, and stirrer speed. The stirrer speed was varied from below Ncr to ∼21 Hz. A representative plot of kLa, as a function of the stirrer speed, is given in Figure 7 for two different liquids. For all liquids, a sharp increase in mass-transfer rate was observed as the stirrer speed increased above Ncr. In this work, the masstransfer rates below Ncr were so low, compared to the rates at higher stirrer speeds, that only a small error is introduced by
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Figure 7. Mass-transfer rate kLa versus stirrer speed. The data are taken from experimental campaigns C and D of this work. (See also Table 3). Ncr ≈ 10 Hz for all experiments.
assuming that kLa ) 0 at Ncr. For modeling purposes, this means that all models based on kLa ∝ Rea or Fra must fail to describe the data near the critical stirrer speed. Rather, data should be described by kLa ∝ (N - Ncr)a, or, in dimensionless form,
Sh ∝ (Re - Recr)a
(12a)
Sh ∝ (Fr - Frcr)a
(12b)
or
so that kLa ) 0 at Ncr is ensured. Dimensional analysis reveals that Sh, Re, Fr, We, and Sc are the relevant dimensionless groups. We have fitted our data to six dimensionless relations, which are summarized in Table 4. In all relations, we have made the mass-transfer rate dimensionlesssquite arbitrarily, but in agreement with Dietrich et al.4s using the stirrer diameter as a characteristic length:
Sh )
kLadst2 D
(13)
The liquids that we used in this study were very similar in surface tension and density, which makes it difficult to include the Weber number in a meaningful way as a separate group in the fitting procedure. In our parameter estimations, we have tried several equations that included the Weber number. When the Weber number was included in any of the models of Table 4, then the uncertainty in the fitted exponent for We was approximately equal to the absolute value of the fitted parameter, which renders that parameter meaningless. All data were fitted to the model equations using the nonlinear regression capabilities of Athena Visual Studio (Stewart and Associates, Madison, WI) with weighted least squares. The fitted parameters and their 95% confidence intervals are listed in Table 4. If penetration theory holds around the bubbles, then the mass transfer coefficient kL scales with the root of the diffusion coefficient, or, in dimensionless form:
Sh ∝ xSc
(14)
In Figure 8, we have used penetration theory and plotted Sh/Sc0.5. Figures 8a, b, c, and d respectively show how well relations I, IV, III, and V of Table 4 fit the experimental data. 5. Discussion The large number of experiments (30 for determining Ncr, 75 to investigate kLa) is sufficient to determine a good
correlation for the gas-inducing stirrer used in our experiments. In the following paragraphs, we discuss the results, to determine what limited set of experiments would suffice to find a correlation for laboratory autoclaves of different dimensions. Critical Impeller Speed. The theory for the critical impeller speed is well-established, and, provided the capillary pressure is taken into account, only the constant in eq 10 and the exponent for the influence of viscosity must be determined. To determine the critical impeller speed Ncr, only four experiments would be needed: two for a low-viscosity liquid and two for a highviscosity liquid. For both liquids, one experiment at a high liquid level and one at a low liquid level would be required. Preferably, these experiments are performed in a transparent tank of the same dimensions, because visual observation gives high accuracy. Alternatively, if such a tank is unavailable or cumbersome experimentally, one could also determine the critical impeller speed from a curve such as that presented in Figure 7, which has been obtained from several mass-transfer experiments. Mass Transfer. For the mass-transfer group kLa, there is much less agreement in the literature over the functional form, and the amount of experiments with different fluids is limited. The correlations found in the literature may be represented, in dimensionless form, using relations I, III, and IV. The three other relationssII, V, and VIswere added to obtain a complete and systematic set to investigate which combination of Re, Fr, and Ncr gives the best results. The first two model relations, I and II, use Re and Fr, respectively, based on the stirrer speed N, and represent models that are obtained when excluding the critical stirrer speed Ncr from analysis. The second pair of equations, labeled III and IV, subtract the critical value from Re and Fr, respectively, to ensure zero mass-transfer rate at the onset of gas induction. In the third pair of equations, labeled V and VI, we have used a combination of Re and Fr, with one of the two factors corrected for the onset of gas induction. If only the stirrer speed is varied (i.e., the fluid properties, stirrer speed, and liquid height remain constant), it is impossible to distinguish whether Re or Fr is the more suitable group. Figure 8a shows how the mass transfer correlates with Re (see model I). Clearly, the correlation for the low-viscosity liquids toluene and acetone is different than that for the high-viscosity compounds sunflower oil and n-hexadecane. Varying the exponent of the Schmidt number within the range of 0.2-1.0 did not bring these groups closer together. Figure 8b shows that, for the low-viscosity liquids (acetone and toluene), a reasonable correlation is obtained for the Froude group (Fr - Frcr) and the mass-transfer rate (model IV). Again, the data for the high-viscosity liquids do not agree. The data for n-hexadecane were measured at one temperature, and the points fall onto a single curve. For the sunflower oil at various temperatures, the fluid properties varied from very high viscosity at low temperature to low viscosity at high temperature. For these experimental points, no correlation was observed with the Froude group. Furthermore, for all the liquids, the correlation with the Froude group is different, so the Froude number alone is not sufficient to correlate the data. In a variation on the Froude group used by Poncin et al.,11 we have used a similar group based on the Reynolds number in Figure 8c. This group, (Re - Recr), gives a fairly good correlation for all the liquids and gases used. If only a single group is used to predict Sh/Sc0.5, then this group is the best choice. If we use the product of two groups to predict the masstransfer rate Sh/Sc0.5, then only one of the groups requires that the value at Ncr be subtracted. Poncin et al.11 demonstrated that
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4579 Table 4. Model Correlations and Estimated Parameters for Mass Transfer in Gas-Inducing Stirrersa
a
model
equation
a
b
I II III IV V VI
Sh ) Sh ) 10aFrbScm Sh ) 10a(Re - Recr)bScm Sh ) 10a(Fr - Frcr)bScm Sh ) 10aReb(Fr - Frcr)cScm Sh ) 10a(Re - Recr)bFrcScm
-4.41 (( 4.30) 4.67 (( 0.56) -4.01 (( 0.50) 4.67 (( 0.53) -3.35 (( 0.77) -3.14 (( 0.68)
1.66 (( 0.83) 2.12 (( 0.57) 1.60 (( 0.10) 1.09 (( 0.31) 1.50 (( 0.14) 1.48 (( 0.12)
10aRebScm
c
m
1.12 (( 0.11) 0.44 (( 0.26)
1.08 (( 0.42) -0.16 (( 0.16) 0.95 (( 0.07) -0.16 (( 0.15) 0.81 (( 0.09) 0.83 (( 0.09)
The values within brackets indicate the 95% confidence interval.
Figure 8. Mass-transfer group Sh/Sc0.5 (Sh/Sc0.81 in panel d) versus various dimensionless groups for toluene, acetone, n-hexadecane, and sunflower oil at various temperatures: (a) Model I, the Reynolds number Re; (b) Model IV, the Froude group (Fr - Frcr), as proposed by Poncin et al.;11 (c) Model III, a similar group (Re - Recr) based on Re; and (d) Model V, the best fit from the nonlinear regression, based on Re and (Fr - Frcr).
(Fr - Frcr) predicted the gas holdup when the liquid level in the tank was varied. We have measured the gas holdup from the increase in liquid level for several stirrer speeds and liquid levels when the critical stirrer speed was determined in the transparent tank. The results were inaccurate due to the large error in measuring the small increase in liquid level by visual observation. Nevertheless, the results plotted in Figure 9 agree, within experimental error, with the findings of Poncin et al.11 If (Fr - Frcr) is a good predictor for the holdup, which vanishes below the onset of gas induction, then the turbulence intensity in the tank may be estimated from the Reynolds number Re. This leads to model V, which gave the best fit for our data. Model VI is a slight variation, based on (Re - Recr) and Fr. This model did not result in a significant estimate for the exponent of Fr and thus reduces to model III, based on (Re - Recr) alone.
The result of the fitting procedure for model V, in Figure 8d, are based on experiments with four different liquids and two different gases, and the experiments were conducted for a sufficient range of stirrer speeds and liquid heights. This scope of the experiments gives confidence to the proposed use of model V for correlating mass-transfer data for gas-inducing impellers. With a small number of experiments, the four parameters of model V can be estimated for other autoclaves than ours. It should be stressed that the parameters that were estimated in the fitting procedure are valid only for our setup, and should not be applied to reactors of different geometry. The choice of experiments is also clear: two liquids are required, including a high-viscosity liquid for low Sh/Sc0.5 and a low-viscosity liquid for higher values of Sh/Sc0.5. The conditions of the experiments should be such that Re and Fr are varied independently. In other
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Figure 9. Gas holdup, as a function of the Froude group (Fr - Frcr), for acetone and n-hexadecane.
Pm ) starting pressure (bar) Re ) Reynolds number; Re ) FNdst2/µ Recr ) critical Reynolds number; Recr ) FNcrdst2/µ Sc ) Schmidt number; Sc ) µ/(FD) Sh ) Sherwood number; Sh ) kLadst2/D T ) temperature (K) utip ) stirrer tip velocity (m/s) V ) molar volume (m3/kmol) VL ) liquid volume (m3) W ) baffle width (m) We ) Weber number; We ) FN2dst3/γ g ) gas holdup (m3/m3) µ ) dynamic viscosity (Pa s) F ) density (kg/m3) γ ) surface tension (N/m) Literature Cited
words, it is important to vary the liquid height, as well as the stirrer speed, significantly. 6. Conclusions The gas-liquid mass-transfer characteristics of a small laboratory autoclave that is equipped with a gas-inducing stirrer were studied using different liquids. Special emphasis was on the influence of fluid properties. (1) In benchscale units, the capillary pressure should be taken into account when determining the critical impeller speed. (2) Mass transfer in gas-inducing autoclaves is best described using relation V of Table 4, i.e., Sh ∝ Rea(Fr - Frcr)bScc. This correlation fitted our experimental data best, and it agrees with the physical intuition that the mass transfer is dependent on the intensity of turbulence (Re) and on the holdup (the group Fr Frcr, where Frcr is the value of the Froude number at the onset of gas induction). The findings presented in this paper help to reduce the amount of experiments that must be performed, for a given autoclave setup, to determine the mass-transfer characteristics. Once these characteristics are obtained, one can reliably estimate the significance of the change in mass-transfer rate that is due to the withdrawal of liquid samples from the reactor or the change in fluid properties due to chemical reaction. Supporting Information Available: Description of the physical properties of sunflower oil, as a function of temperature, and raw experimental data for the mass-transfer experiments. This material is available free of charge via the Internet at http:// pubs.acs.org. Nomenclature D ) diffusivity (m2/s) dst ) stirrer diameter (m) Fr ) Froude number; Fr ) N2dst2/(gH) Frcr ) critical Froude number; Frcr ) Ncr2dst2/(gH) g ) gravitational constant (m/s2) H ) liquid height above impeller (m) kLa ) volumetric gas-liquid mass-transfer coefficient (s-1) M ) molecular weight (g/mol) N ) stirrer frequency (s-1) Ncr ) critical stirrer frequency (s-1) P ) pressure (bar) P0 ) atmospheric pressure (bar) Pf ) equilibrium pressure (bar) Pg ) power consumption in the presence of gas (W)
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ReceiVed for reView January 20, 2006 ReVised manuscript receiVed March 29, 2006 Accepted April 19, 2006 IE060092M
