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Ind. Eng. Chem. Res. 1997, 36, 4670-4680
A Two-Compartment Convective-Diffusion Model for Slurry Bubble Column Reactors† S. Degaleesan and M. P. Dudukovic* Chemical Reaction Engineering Laboratory, Department of Chemical Engineering, Washington University, St. Louis, Missouri 63130
B. A. Toseland and B. L. Bhatt Air Products and Chemicals, Inc., Lehigh Valley, Pennsylvania 18002
Bubble columns and slurry bubble column reactors (SBCRs), due to their superior heat transfer characteristics, are the contactors of choice for conversion of syngas to fuels and chemicals. The multiphase fluid dynamics in these systems determines the mass and heat transfer and greatly affects reactor volumetric productivity and selectivity. Here we show how the recently collected data for liquid velocity, turbulence parameters, and holdup profiles can be used in guiding model development for liquid backmixing and in providing the needed model parameters. Predictions of the two-compartment convective-diffusion model for the residence time distribution of a nonvolatile tracer, which are based on measured liquid recirculation driven by the gas holdup profile and turbulence caused by the rising gas bubbles, are shown to be in good agreement with tracer data. The adaptation of model parameters needed to achieve predictability for industrial tracer data is discussed. The ability of the model to predict the liquid tracer responses at various locations in a SBCR during methanol synthesis is illustrated. Introduction Synthesis gas can be made from a variety of coal, natural gas, environmentally distressed materials such as petroleum coke and biomass. There are considerable reactor design and scale-up problems associated with synthesis gas conversion technologies which arise due to the special characteristics of these processes. Common to all of the reactions involved in these technologies is that they are highly exothermic. The excellent heat management capability of the slurry bubble column reactors (SBCRs) allows a higher conversion per pass than that in fixed bed reactors and avoids the hot spot problems. In addition, large gas throughputs must be handled necessitating large diameter reactors; high pressure is required for good mass-transfer rates and volumetric productivity; large reactor heights are needed to reach high conversion per pass. Bubble columns and slurry bubble column reactors (SBCRs) can meet the above requirements and have emerged as leading candidates for a variety of gas conversion processes (Krishna et al., 1997). Over the past 15 years, Air Products and Chemicals, using a series of contracts from the Department of Energy, has demonstrated on a pilot scale the feasibility of using SBCRs to produce a variety of synthetic fuels such as methyl tert-butyl ether, dimethyl ether, methanol, and Fischer Tropsch products which offer environmental advantages over petroleumderived fuels. Further progress in synfuels technologies involving SBCRs, their scale-up to very large scale, and commercialization is dependent on the improved understanding of multiphase fluid dynamics in these systems. Due to complex flow patterns in bubble columns and slurry bubble columns, the design and scale-up of these * To whom correspondence should be addressed. Phone: (314) 935-6021. Fax: (314) 935-4832. † Paper for presentation at the Banff Engineering Foundation Conference on Chemical Reaction Engineering VI: Chemical Reactor Engineering for Sustainable Processes and Products. S0888-5885(97)00200-5 CCC: $14.00
reactors pose a rather intriguing problem and have so far been based on empiricism and on ideal reactor models (e.g., plug flow of gas and complete backmixing of liquid) or their simple extensions (e.g., axial dispersion model for gas and liquid, tanks in series without/ with backflow for each phase, etc.). In recent years considerable effort has been made to obtain a fundamentally based description of multiphase fluid dynamics in bubble columns and use the emerging computational fluid dynamics (CFD) codes in solving the problem (Svendsen et al., 1992, 1996; Sokolichin and Eigenberger, 1994; Ranade, 1992). However, the uncertainty regarding the phase interaction terms and turbulence closure schemes, as well as problems associated with computation of large flow fields, have delayed the full implementation of these models in practice. The Department of Energy sponsored initiative for multiphase hydrodynamics attempts to fill this gap by providing the needed experimental data for the testing of multiphase fluid dynamic codes and for the development of the appropriate closure schemes. At the same time, as an interim measure, phenomenological models are developed for a better description of flow patterns and mixing in these columns. Here, we focus on the progress made in describing liquid mixing in bubble columns, which is important for the assessment of reactor volumetric productivity, selectivity, and heat removal rates. Our objective is to show that a phenomenological model based on experimentally observed reproducible phenomena and measured fluid dynamic quantities, such as recirculation liquid velocities and liquid eddy diffusivities, is capable of predicting (not just being fitted to) the liquid residence time distribution. Using physical arguments we discuss the steps needed in extrapolating the model parameters to industrial conditions and show how well the model can capture the key features of the system in predicting the local liquid tracer responses at various axial locations of the column. The motivation for this study was provided by the need for an improved © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4671
Figure 1. Liquid tracer impulse response in an air-water bubble column and model prediction (data of Myers et al., 1986). Column diameter ) 19 cm, gas-liquid height ) 2.44 m, Ul ) 1 cm/s, Ug ) 10 cm/s.
description of liquid backmixing in SBCRs and the rare opportunity to get and interpret tracer data from an operating reactor for liquid phase methanol synthesis. Liquid Backmixing in Bubble Columns The usual assumption made by a design engineer is that the liquid in a slurry bubble column is perfectly mixed. Tracer runs confirm close to complete backmixing and a continuous flow stirred tank reactor (CSTR) model is usually employed to describe liquid mixing. However, even a small departure from complete backmixing, such as the one observed in Figure 1, can lead to significant differences in required reactor size. For example, for a second-order reaction and over 98% required conversion the size of a CSTR would be over 20 times larger than that of the actual reactor exhibiting the liquid residence time distribution (RTD) of Figure 1. Hence, the use of a conservative design may lead to prohibitive capital expenditures. As a usual remedy in arriving at more realistic reactor sizes the axial dispersion model (ADM) is used. While ADM can usually be fitted to the observed tracer response in a SBCR rather well, the calculated dispersion number ((Daxj)/(Ul l)), often termed as inverse Peclet number, is always very large, pointing to the lack of a physical basis for the ADM. Due to the broadness of the RTD, selectivity in nonlinear processes can vary greatly, depending on the micromixing at hand. Selectivity predictions are then subject to the micromixing model used. Since predictions of axial dispersion coefficients in bubble columns for scale-up has not been a successful art (Fan, 1989), and since the ADM does not provide information on the nature of the micromixing in the system, alternatives to it must be sought. It should be recognized that the widespread use of ADM for SBCRs owes its popularity not to a good physical basis for the model, but to its simplicity, as it is a one parameter model. Hence, correlations for the axial dispersion coefficient are abundant, such as those by Ohki and Inoue (1970), Deckwer et al. (1974), Baird
and Rice (1975), Joshi (1980), and Kantak et al. (1994), to name just a few. Unfortunately, there is wide scatter in their predictions (Fan, 1989), which is caused to a great extent by the unsuitability of ADM in describing liquid backmixing with a single parameter. Moreover, almost none of the correlations for the axial dispersion coefficient were based on data taken at high pressure, which is of industrial interest. We depart from the ADM and let experimental observations of the fluid dynamics in bubble columns guide us in the selection of a proper model. Such an approach was also used in the development of phenomenologically based models of Myers et al. (1986), Wilkinson et al. (1993), and Krishna et al. (1993). We have the advantage that additional experimental information has emerged in the last few years. Extensive studies utilizing computer aided radioactive particle Tracking (CARPT) (Devanathan, 1991; Degaleesan, 1997) have revealed that, at sufficiently high superficial gas velocities and in large aspect ratio columns, in a time-averaged sense a large scale liquid circulation cell occupies most of the column heightwise, with liquid ascending along the central core region and descending along the annular region between the core and the walls. While a single one-dimensional axial liquid velocity profile exists in this large recirculation cell, with negligible radial and azimuthal liquid velocities (Hills, 1974; Nottenkamper et al., 1983; Menzel et al., 1990; Devanathan et al., 1990; Dudukovic et al., 1991), two- and three-dimensional velocity profiles are evident in the distributor and free surface (disengagement) region. The entry and disengagement zones are approximately one column diameter each when the column is in churn turbulent flow (Devanathan, 1991), which occurs at high gas velocities that are of industrial interest. Liquid recirculation is driven by nonuniform radial gas holdup profiles (i.e., there is more gas in the center than at the walls) which, in the time-averaged sense, have been shown to approach a parabolic shape for churn turbulent flow (Hills, 1974; Yao et al., 1991; Nottenkamper et al., 1983; Kumar et al, 1995). Superimposed on this recirculation are turbulent fluctuations in the axial, radial, and azimuthal direction due to the eddies induced by the wakes of the rising gas bubbles. Mixing of the liquid phase is therefore primarily due to convective liquid recirculation, driven by the existing nonuniform radial gas holdup profile and turbulent dispersion due to bubble wake interactions and turbulent eddies. This experimental evidence of the flow pattern is used for the phenomenological modeling of liquid mixing in bubble columns.
Model Formulation The data base for model development consists of our CARPT and CT (computed tomography) data for airwater systems obtained in columns of different diameters and at different gas superficial velocities. In a given column, at a chosen gas superficial velocity first γ-ray CT scans (Kumar et al., 1995) are obtained for the time-averaged gas holdup distribution at various heights above the distributor. This identifies the length of the column in which the holdup distribution is almost insensitive to axial distance, indicating the location of the fully developed recirculation cell. Azimuthal and axial averaging of the gas holdup distribution results in the radial gas holdup profile that drives liquid
4672 Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997
Figure 2. Schematic of the CARPT facility and data interpretation.
recirculation. This holdup profile is fitted by eq 1 (Kumar et al., 1995).
G(ξ) ) ˜ G
m+2 (1 - cξm) m
(1)
where ξ is the dimensionless radius and c and ˜ G are parameters available for fitting. The cross-sectional mean gas holdup is given by
jG ) ˜ G
m + 2(1 - c) m
(2)
Parameter m approaches the value of 2 in churn turbulent flow, while parameter c appears to be a function of column diameter and decreases with an increase in column diameter. Theoretically, at the column wall (ξ ) 1) gas holdup is zero, but this happens in a layer much thinner than the spatial resolution of our CT scanner, which is about 4 mm. Moreover, even in CFD modeling of larger columns there is a finite cell size by the wall which has some gas holdup that represents the average gas holdup in that cell. In that sense eq 1 allows for non-zero gas holdup at the wall. The tracing of the liquid (water) by a neutrally buoyant single radioactive particle (Sc-46) using the CARPT technique produces the data outlined in Figure 2. From the record of filtered particle positions in time, instantaneous Lagrangian velocities are obtained by time differencing. Performing the particle tracing experiments for a sufficiently long time at a fixed superficial gas velocity ensures that the particle will occur in each column “compartment”, to which velocities are assigned, numerous times. Several hundreds of occurrences are considered adequate for good statistics. Ensemble averaging then provides the average velocities for each compartment. The differences in instantaneous and average velocity can then be utilized to calculate root-mean-square (rms) velocities, Reynolds stresses, kinetic energy of turbulence, etc. CARPT results for the turbulence parameters have been shown to compare satisfactorily with hot wire anemometer measurements of Menzel et al. (1990) for the Reynolds shear stress, and with LDA measurements of Mudde et al. (1997) for axial normal stresses (Degaleesan, 1997).
Since with CARPT we monitor the motion of a single radioactive particle, using this technique we can measure the Lagrangian correlation coefficients from which various eddy diffusivity components are calculated (Devanathan, 1991). While our original approach utilized the isotropic theory in calculating eddy diffusivities from CARPT data, we have now accounted for the anisotropy and for the contribution to eddy diffusivity arising due to the presence of the mean velocity gradient (Degaleesan, 1997). Here we are interested in the radial and axial components of the eddy diffusivity which are obtained from the Lagrangian correlation coefficients as follows:
Drr ) Dzz )
1 d 2 y ) 2 dt r
1 d 2 y ) 2 dt z t ∂uz 0 ∂r
∫
{
|
yr(t′)
∫0tvr′(t)vr′(τ) dτ ) ∫0tRrr(τ) dτ
∫0t′vz′(t)vz′(τ) dτ) + vz′(t)vz(t′)
(
|
t∂uz ) 0 ∂r
∫
yr(t′)
(3)
}
dt
(4)
∫0t′Rrz (τ) dτ) dt′ + ∫0tRzz(t′) dt′
(
The eddy diffusivity in its general form is a second-order tensor, but we assume that the elements off the principal diagonal are small compared to the diagonal elements and CARPT data support this. We find that the eddy diffusivities are strong functions of superficial gas velocity and column diameter (Degaleesan, 1997). The axial eddy diffusivities are an order of magnitude larger than the radial ones, and both are only weakly dependent on axial position in the middle section of the column where a one-dimensional time-averaged velocity profile is established. For example, for an air-water system in a 19 cm diameter column with a L/D of 10 at a superficial gas velocity of Ug ) 10 cm/s and superficial liquid velocity of Ul ) 1 cm/s we obtain the liquid holdup, liquid time-
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Figure 4. Model schematic and equations for the end zones.
averaged velocity profile, and axial and radial eddy diffusivities as shown in Figure 3. The above experimental evidence for the fluid dynamics of the bubble columns forms the physical basis for the development of the present liquid mixing model, which is schematically represented in Figure 4. The model equations for the upflow and downflow sections are
∂C1 ∂2C1 Dri ∂ + u j 1C1 ) D1 (C - C2) ∂t ∂z a2 1 ∂z2
(5)
∂C2 ∂2C2 Dri ∂ + u j C ) D2 (C - C2) ∂t ∂z 2 2 a2 1 ∂z2
(6)
Axial eddy diffusivities D1 and D2 represent the crosssectional averages of Dzz for cell 1 and 2, respectively, mean liquid velocities u j 1 and u j 2 are also cross-sectional averages of uz for cell 1 and 2, respectively. Dri is the radial eddy diffusivity at the inversion point, i.e., obtained by averaging Drr in the vicinity of the inversion point. The above model equations have been derived using a finite volume discretization approach to simplify the original fundamental two-dimensional convectiondiffusion equation for the liquid tracer species balance in bubble columns (Degaleesan, 1997). The area factors a1 and a2, having units of length square, arise from this discretization and are given by
() ( )( ) 1 r* R i r
(7a)
2 R2 - r* R i 4r*
(7b)
a1 ) a2 )
Figure 3. CARPT-CT determined fluid dynamic parameters (19 cm diameter column, air-water, Ug ) 10 cm/s, Ul ) 1 cm/s). (a) Liquid holdup profile; (b) axial liquid velocity profile; (c) axial and radial eddy diffusivities.
where r* is the radial position of flow inversion (between cell 1 and cell 2) and i is the liquid holdup at the inversion point. From eqs 5 and 6 we recognize the recirculation and cross flow with dispersion (RCFD) model (Degaleesan et al., 1996) which was derived on
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the basis of heuristic arguments and utilized a cross flow per unit length, K, as a fitting parameter, instead of Dri. Slight variations in area factors are noted in the two formulations. The model of course is a variation on the Hochman and McCord (1970) recycle with crossflow approach. In our previous publication (Degaleesan et al., 1996) we have shown that the upflow and downflow sections need to be connected to a common cell at each end which is assumed well-mixed. Danckwerts type of boundary conditions are to be applied at the entrance and exit of the upflow and downflow section while CSTR mass balance is used for each of the stirred cells. The additional equations are shown in Figure 4 along with the schematic of the model. We have shown that changing the volume, i.e., capacitance of the two CSTRs at the ends does not affect the results much. The model is now applicable to both steady flow of liquid (Fo finite) or batch liquid (Fo ) 0).
Comparison of Model Predictions and Experimental Results A. Air-Water System. Here we consider the data of Myers et al. (1986). In a 19 cm diameter column at Ug ) 10 cm/s and Ul ) 1 cm/s the response to a pulse injection of dye at the bottom of the column was monitored in the column overflow, at the top, by spectrophotometry. The results are shown in Figure 1. Model predictions of the tracer outlet response are based on CARPT-CT data for the fluid dynamic parameters shown in Figure 3. From CARPT-CT data the following model parameters were obtained:
D1 ) 285 cm2/s, D2 ) 440 cm2/s, Dri ) 34 cm2/s, u j1 ) 12.5 cm/s, u j 2 ) 7.7 cm/s Comparison of the model prediction and data is shown in Figure 1. Clearly the model predicts the tracer impulse response well. A good model prediction of the observed tracer response is encouraging but not surprising since all the model parameters were obtained in the same system on which the tracer study was performed. Now we will attempt to extend the model in predicting the results of a tracer study in an industrial liquid phase methanol synthesis bubble column reactor. B.1. Liquid Tracer Experiments in the Methanol Synthesis Reactor. These tracer experiments were performed in a high-pressure slurry bubble column reactor (the Alternate Fuels Development Unit (AFDU) in LaPorte, Texas), during methanol synthesis in order to assess the extent of slurry backmixing. Powered methanol catalyst (≈45 wt % loading) suspended in an inert hydrocarbon oil forms the batch slurry phase. Synthesis gas is bubbled through a sparger placed at the bottom of the reactor. The gas disengages from the oil in the freeboard section of the reactor, and the unreacted feed gas is recycled back to the reactor. Tracer studies were performed at 5.2 MPa at gas superficial velocities of 14 and 25 cm/s and at 3.6 MPa at a gas superficial velocity of 36 cm/s. We will focus our attention here to the run at 5.2 MPa at Ug ) 25 cm/s, which exhibited an average gas holdup of 0.39. A schematic of the AFDU as equipped for tracer studies is shown in Figure 5. The reactor has an internal diameter of 46 cm and a height of 1524.0 cm,
Figure 5. Schematic of the tracer experiments in the methanol synthesis SBCR.
with the liquid-solid-gas dispersion level maintained at 1325.0 cm (L/D ) 28.8). Radioactive manganese (Mn56) particles (50 µm in diameter) mixed in oil were used for the slurry phase tracing. It is assumed that the liquid-solid suspension behaves as a pseudohomogeneous medium since there was no evidence of catalyst settling. Four impulse injections were made at each given process rate: (1) lower nozzle N2, at 11.4 cm from the wall, (2) nozzle N2, at the wall, (3) upper nozzle N1, at 11.4 cm from the wall, and (4) nozzle N1, at the wall. The axial levels of these injection points N1 and N2 are shown in Figure 5. The injections made at 11.4 cm from the wall will be referred to as “center injections” as they are made into the core part of the column (r < r*) where the liquid is known to flow upward in the time-averaged sense. Radiation measurements of tracer responses were made using 5 cm by 5 cm NaI scintillation detectors positioned outside the column at various axial levels as shown in Figure 5. Sets of four detectors were placed at 90° angles at seven heights. All detectors were shielded on the sides. The measurements from the four detectors at a given axial level are averaged to obtain the response to be compared to the model predictions and are shown in Figure 6. The radiation intensity received at each detector is a complex function of geometry and distribution of the radioactive tracer, among other factors. For the conditions of the present experiments the intensity detected at a given axial level can be shown to be proportional to the liquid tracer concentration at that level (Degaleesan et al., 1996). The radioactive tracer experiments outlined above require a three-dimensional model for complete description. However, our objective was to assess whether the two-compartment convective-diffusion model introduced here can capture the key features of the system and its
Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4675
Figure 6. Detector responses at different axial locations to a tracer pulse injection at N1-wall.
tracer responses at various locations. From Figure 6 (and similar ones for tracer injections at different locationssnot shown) several conclusions can be reached. Since overshoots appear first at the detector above the injection point for “center” injection and below the injection point for wall injection, this supports the model presented here. Unfortunately, when the detectors were calibrated prior to an experiment, the radioactive tracer was not removed to sufficient distance from the detectors, thus resulting in nonuniform normalization of the detectors. This causes the response of various detectors at steady state to settle at different levels. Therefore an absolute match between model predictions and experimental responses cannot be attempted. By normalizing both model predictions and detector responses with respect to the maximum value for each, the general shape of the curve and the position of the predicted peak and the actual peak (if one is exhibited) can be compared. B.2. Extrapolation of Data. In order to use the present model to predict the tracer distribution in the SBCR during methanol synthesis, it is first necessary to evaluate the input model parameters under the existing operating conditions. However, the existing CARPT-CT data base for the various fluid dynamic parameters is limited to air-water systems at atmospheric pressure, in certain column sizes, and up to certain gas velocities. Therefore, we need a methodology for extrapolating the data to the methanol synthesis reactor operating under different conditions (high gas velocity, high pressure, and presence of heat exchanger tubes). To evaluate the model parameters under these conditions, using physical arguments, we have proposed and tested the following procedure with reasonable success (see Table 1). To begin with, we use the existing air-water data to study the effects of gas velocity and column diameter
on the fluid dynamic parameters in the churn-turbulent flow regime. First, using experimental data for the overall gas holdup, from the present work and from the literature (Reith et al., 1968; Nottenkamper et al., 1983; Reilly et al., 1986; Guy et al., 1986; Myers et al., 1986) for air-water systems in large diameter columns, we arrive at the following equation for jG as a function of gas superficial velocity, Ug (cm/s) and column diameter, Dc (cm):
jG ) 0.07Ug0.474-0.00626Dc;
Dc > 10 cm
(8)
The above equation accounts for the effects of gas velocity and column diameter on the average gas holdup. The second effect is negligible at lower gas velocities (Hammer et al., 1984; Wilkinson, 1992) but becomes significant at higher gas velocities, Ug g 20 cm/s (Krishna et al., 1997). The mean liquid recirculation velocity, defined as the average upflow velocity in section 1 (Figure 4), is correlated with gas superficial velocity and column diameter as
u j 1(cm/s) ) 2.2(UgDc)0.4
(9)
The experimental data base considered in developing the above correlation includes the data generated in our laboratory (CREL) along with data for the liquid velocity and holdup from the literature for air-water systems (Menzel et al., 1990; Nottenkamper et al., 1983) up to gas velocities of 80 cm/s and column diameters of 60 cm. In a similar manner, the cross-sectional average eddy h rr, are correlated for air-water diffusivities, D h zz and D
4676 Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 Table 1. Schematic for Parameter Estimation in Systems Other Than Air-Water
systems, in the churn-turbulent regime, using the CARPT data base as
D h zz(cm2/s) ) -
2325 + 106.6(DcUg)0.3; 0.8 Dc Ug > 5 cm/s (10) h zzP4(ξ) Dzz(ξ) ) D
D h rr(cm2/s) ) -
350 + 13.0(DcUg)0.3; Dc0.8 h rrP2(ξ) Drr(ξ) ) D
(11) Ug > 5 cm/s (12) (13)
Equations 11 and 13 are used to calculate the radial profiles of the axial and radial turbulent diffusivities, respectively. The polynomials appearing in these equations are obtained from CARPT data at lower gas velocities and are reported by Degaleesan (1997). Due to the currently limited database from CARPT, the above equations have yet to be verified with experimental data at higher gas velocities and using liquids of different physical properties. The above developed correlations are strictly applicable only to air-water systems in churn-turbulent flow at atmospheric pressure. It is well-known that tracer impurities and distributor changes, etc., can affect two-phase flow considerably. It seems, however, that such effects are much more pronounced for bubbly flows and tend to diminish in the churn-turbulent regime, with increase in gas velocity. There is evidence, for example, that in the churn-turbulent flow regime, at sufficiently high gas velocity, the holdup profile is
Table 2. Estimated Fluid Dynamic Parameters for the Methanol Synthesis Reactor Operating at Ug ) 25 cm/s, P ) 5.2 MPa, and T ) 250 °C u j 1 (cm/s) u j 2 (cm/s) j1 j2
47.7 37.8 0.464 0.752
D1 (cm2/s) D2 (cm2/s) Dri (cm2/s)
868.6 1042.4 46.3
most often parabolic (Kumar, 1994; Menzel et al., 1990; Nottenkamper et al., 1983). Therefore, we assume that for air-water systems in the churn-turbulent regime, at sufficiently high gas velocities, the fluid dynamic parameters are predominantly the function of superficial gas velocity and column size (diameter). A change in system properties (physical properties of the fluid, presence of solids, etc.) and operating conditions (pressure and temperature) directly affects bubble sizes and distribution, and thereby the global gas holdup and holdup profiles. This in turn influences the liquid recirculation and turbulence characteristics. For example, an increase in the system pressure tends to reduce the bubble sizes which delays transition to turbulent flow and therefore results in the increase in gas holdup compared to values expected at atmospheric conditions. Experimental results in the literature also suggest that at high gas velocities, that are of industrial interest, well in the churn-turbulent regime, the bubble characteristics of the larger bubbles (Krishna and Ellenberger, 1996) are unaffected by system properties and pressure. Independent measurements of the local holdup profile in high-pressure bubble columns, at high gas velocities (Adkins et al., 1996), indicate that the holdup profile is parabolic in shape. This implies that in the churn-turbulent flow regime at high gas velocities, similar bubble size distribution characteristics are present in the system, irrespective of the system proper-
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Figure 7. Comparison of model predictions and experimentally determined responses for tracer injection at N1-wall.
ties except, perhaps, when viscosity is very high. Therefore, one can apply the correlations for the liquid recirculation and turbulent diffusivities developed for
air-water systems, to any system of interest in the turbulent flow regime if one knows a priori the gas holdup in the system of interest. For the present
4678 Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997
methanol synthesis reactor experiments, global holdup measurements were made using differential pressure (DP) transducers. Using the measured holdup in the system of interest, for a given column diameter, along with eq 8, we can evaluate an equivalent gas superficial velocity, Uge, that would exist at atmospheric pressure in such a column in an air-water system. For the conditions of the methanol synthesis reactor considered here, i.e., Ug ) 25 cm/s and jG ) 0.39, Uge ) 47 cm/s, which is significantly different from the original gas velocity, indicating that increasing the pressure of the system results in an increase in the gas holdup and will thereby influence the fluid dynamics in the column. Then using eqs 9-13 we can calculate the parameters of our model at this equivalent gas superficial velocity, Uge. D1 and D2 are calculated by cross-sectionally averaging eq 11 in the upflow and downflow regions. The holdup profile is calculated using results for the mean gas holdup from DP measurements, along with the line average holdup measured using nuclear gauge densitometry (NDG), and extracting the radial holdup profile (Degaleesan, 1997). This can be used to estimate the average holdups in the upflow and downflow sections, and thereby estimate u j 2. The input model parameters evaluated in the above manner still do not take into account the effects of the heat exchanger tubes in the column. There are 24 tubes, 2.54 cm in outer diameter, arranged in an annular ring, which extend over the entire length of the gas-liquid-solid dispersion and occupy about 7.5% of the cross-sectional area of the reactor. The presence of these tubes is accounted for only in the radial eddy diffusivity, since the arrangement of the tubes will physically restrict the radial length scales of turbulence in this region, thereby reducing the radial eddy diffusivities. The characteristic distance between the tubes, which is about 5.7 cm, is used as an effective diameter to estimate the radial diffusivity in the region between the tubes (eq 12). The modified profile for the radial eddy diffusivity is then used to calculate Dri, which is considerably lower than the value that would be obtained on the basis of the column diameter. The above procedure, based on physical arguments, thus enables us to evaluate the input fluid dynamic model parameters under the existing reactor conditions, using the air-water database. B.3. Comparison of Model Predictions and Data. The model parameters evaluated by the above procedure are listed in Table 2. Comparison of model predictions and tracer responses (both scaled with respect to the maximum value) is shown in Figure 7. Clearly, the model does a fair to good job in predicting tracer responses at all locations. Most importantly the position of the peak, when it exists, is captured well indicating that the recirculation rate is predicted well. This also substantiates the method proposed in Table 1 for evaluating the model parameters in the system of interest, using air-water data. Exact comparison of the tails of the curves is impossible due to alreadymentioned problems with detector calibration. In contrast to the present model, the axial dispersion model does not do well. Its best predictions of data trends are at measurement points which are the furthest from the injection. However, even there (e.g., axial level 1) the ADM prediction based on the popular correlation of Baird and Rice (1975), for the axial dispersion coefficient, underpredicts the dispersion as evident from Figure 8. Other established correlations
Figure 8. Comparison of present model and axial dispersion model predicted response with experimental data, injection at N1wall, and measurement at level 1.
lead to similar or worse underpredictions. This is partly because these correlations have been developed at atmospheric pressure, in the absence of heat exchanger tubes. A larger value of the dispersion coefficient would be able to fit the tracer curve in Figure 8. On the basis of the present analysis, it is clear that an increase in system pressure and the presence of internals affects (increases) axial liquid mixing in the column. The recent correlation for the liquid axial dispersion coefficient proposed by Berg et al. (1995) that incorporates the effects of heat exchanger tubes on liquid mixing overpredicts the value of Dax as shown by comparison with data in Figure 8. Summary and Conclusions The present two-compartment convection-diffusion model has been formulated by considering a finite volume discretization of the two-dimensional, axisymmetric convection-diffusion model developed for bubble column flows. The model, which is solved by directly using experimental measurements from CARPT and CT for the various fluid dynamic parameters, predicts well the tracer response in an air-water system. Scale-up equations based on the experimental data (from present work and literature data) in air-water systems for the mean liquid recirculating velocity and turbulent eddy diffusivities are suggested for evaluation of these quantities in other systems and larger columns. The strategy proposed for evaluating the model parameters in an industrial slurry bubble column reactor (AFDU) during methanol synthesis results in a fairly accurate prediction of the characteristic mixing times within the column as measured by the radiation detectors at various axial locations. This substantiates the proposed methodology of using the gas holdup in churnturbulent flows at sufficiently high-gas superficial velocities for characterizing different systems of interest, and thereby the use of the results obtained for airwater systems to evaluate the fluid dynamic parameters of the model for an industrial reactor. The only experimental information required by the model in the system of interest is the gas holdup and its radial distribution. For the present case, this was obtained from measure-
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ments of the holdup using DP and NDG. While comparison of model prediction with experimental data has been shown here only for a single case, similar comparisons are observed for all the AFDU experiments, at the different operating conditions (gas superficial velocity and pressure) as well. Additional CARPT/CT experiments at selected conditions (higher gas velocities, high pressure, and presence of internals) are planned as future work, to substantiate the correlations for the turbulent eddy diffusivities and the proposed methodology of characterization of bubble columns in churn-turbulent flows. Once verified, this will serve as a tool by which data from a limited database can be utilized to model liquid mixing in different systems in the churn-turbulent flow regime. Acknowledgment The authors are grateful for the support from the Department of Energy (DOE-FC 22-95 PC 95051) which has made this study possible. S. Degaleesan and M. P. Dudukovic also acknowledge the support of industrial sponsors of CREL. Notation a ) area factor, cm2 c ) fitting parameter in eq 1 C ) tracer concentration, mol/cm3 Co ) inlet tracer concentration, mol/cm3 Dc ) column diameter, cm D1, D2 ) average axial eddy diffusivities in the upflow and downflow sections, respectively, cm2/s Dax ) effective axial dispersion coefficient for the ADM, cm2/s Drr ) radial turbulent eddy diffusivity, cm2/s Dri ) radial eddy diffusivity at inversion point, cm2/s Dzz ) axial turbulent eddy diffusivity, cm2/s F ) liquid volumetric flowrate, cm3/s H(t) ) Heavisides step function for the step input of tracer K ) crossflow coefficient, cm2/s l ) length of reactor, cm L ) length of the recirculating region, cm m ) void fraction exponent r ) radial position, cm r* ) radial position corresponding to zero axial liquid velocity, cm R ) Lagrangian correlation coefficient, cm2/s2 t ) time, s u j ) liquid interstitial velocity, cm/s uz ) liquid axial velocity as a function of radial position, cm/s Ug ) superficial gas velocity, cm/s Uge ) equivalent superficial gas velocity, cm/s Ul ) superficial liquid velocity, cm/s v′ ) Lagrangian fluctuating velocity, cm/s V ) volume, cm3 y ) displacement due to v′, cm z ) axial coordinate, cm Greek Letters ) liquid holdup j ) average liquid holdup jG ) mean gas holdup ξ ) nondimensional radial position Subscripts 1 2 a b
) ) ) )
upflow region, section 1 downflow region, section 2 inlet well-mixed region A exit well-mixed region B
i ) inversion point l ) liquid phase g,G ) gas phase r ) radial direction z ) axial direction
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Received for review March 10, 1997 Revised manuscript received May 12, 1997 Accepted May 21, 1997X IE970200S
X Abstract published in Advance ACS Abstracts, September 15, 1997.
