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Ind. Eng. Chem. Res. 2003, 42, 2530-2535
Solids Holdup and Pressure Drop in Gas-Flowing Solids-Fixed Bed Contactors Aleksandar P. Dudukovic´ * and Nikola M. Nikacˇ evic´ Faculty of Technology and Metallurgy, University of Belgrade, 11000 Belgrade, Karnegijeva 4, Yugoslavia
Dragan Lj. Petrovic´ † and Zlatica J. Predojevic´ Faculty of Technology, University of Novi Sad, 21000 Novi Sad, Bul. Cara Lazara 1, Yugoslavia
Fundamentally based model for pressure drop in gas-flowing solids-fixed bed contactors is presented, together with a phenomenological semiempirical model for prediction of dynamic holdup. These simple models do not require any parameters that need to be determined by measurements in the actual system of interest. The predictions are compared with all available data and give good agreement for a wide range of experimental conditions, different constructions, types, and dimensions of packing and for a variety of flowing solids properties. Countercurrent flow of gas and fine solids through packed beds was patented as an idea in 1948,1 and the first recorded industrial use occurred in 1965 (Compagnie de Saint Gobain)2 in heat-transfer applications. The fluid dynamics of such systems received considerable attention over the years,3-14 and this included heat- and mass-transfer studies.8,15-17 The interest in exploiting the unique features of the countercurrent gas-fine solids systems was enhanced by the studies of Westerterp and colleagues,18,19 who proposed the use of fine solids as a regenerative adsorbent flowing through the bed of catalyst for methanol synthesis. Additional reactor-oriented studies included catalytic oxidation of hydrogensulfide20 and regenerative desulfurization of flue gases.17,21 The application of this type of gas-flowing solidsfixed bed contactors would be enhanced if the fluid dynamics in these systems could be fully quantified, at least in the macroscopic engineering sense. Reliable prediction of pressure drop, flowing solids holdup, residence time distribution, back mixing, and so forth, are some of the quantities needed when assessing the applicability of the “flowing” or “trickling solids” systems in a variety of processes. Previous studies resulted in a semiquantitative description of the fluid dynamics of the system and empirical correlations for determination of some quantities of interest. Three flow regimes were observed, similar to gas-liquid systems: preloading, loading, and flooding. Gas-flowing solids interaction increases with the increase in gas superficial velocity. When the terminal velocity of gas relative to flowing particles is approached, a sudden increase in pressure drop and fine solids holdup occurs, together with accumulation of solids at the top of the bed and unstable operation, which is characteristic for flooding. The complexity of the fluid dynamics of these systems did not permit, so far, a unique pressure drop equation to emerge without empirical constants. The problem is * To whom correspondence should be addressed. E-mail:
[email protected]. † E-mail:
[email protected].
further complicated by the lack of consensus in accounting for the effects of particle shape, size, roughness, bed porosity distribution, and so forth. The models presented were often developed by fitting the data of a few studies and were not extensively tested, thus lacking in predicting ability. Moreover, the approach used was such that data on the system of interest were always needed to complete the correlation (i.e., one had to have data to predict them!).6,10 In our earlier study,22 we analyzed all the available pressure drop data and have developed correlations that fit all the data reasonably well, for both a preloading and loading regime. However, that, as well as other correlations, lacked a fundamental basis. The objective of this study is to develop a phenomenological model for the fluid dynamics of countercurrently flowing fine solids-gas systems in packed beds based on sound physical reasoning using appropriate approximations. The goal is to develop an approximate but physical model without any parameters that need to be determined by measurements in the actual system of interest. However, if parameters that cannot be predicted by theory appear in the model, the goal is to evaluate them from an independent set of experiments. Our final goal is a physical model-based correlation with a priori determined parameters that are capable of predicting well the dynamic fine solids holdup and pressure drop in these systems. The reasoning that led to such a model and correlations is outlined below. Dynamic Holdup Flowing solids holdup is the sum of dynamic and static holdup. Static holdup represents the fraction of fine solids that are at rest on the packing elements. When gas and solids flows are stopped, these particles will remain in the bed. Dynamic holdup represents the other part of flowing fine solids that are moving and trickling through the bed and that will flow out when the inlet of gas and flowing solids are closed. Static holdup of fine solids is usually assumed to be the dead (nonactive) portion of the flowing solids in the event when these serve as an adsorbent or catalyst.
10.1021/ie020541s CCC: $25.00 © 2003 American Chemical Society Published on Web 03/13/2003
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2531
Dynamic holdup could be found from
βdyn )
S F Su S′
(1)
but the average flowing solids velocity uS′ has to be determined. Simple balance of forces on a particle moving opposite the gas stream leads to a well-known equation for relative terminal velocity of particles:
uR )
x
4dSg(FS - F) 3FCD
(2)
The drag coefficient can be found from the correlation given by Turton and Levenspiel:23
CD )
24 0.413 (1 + 0.173Re0.6567 )+ p Rep 1 + 16300Re-1.09
(3)
p
To take into account that the particles are moving inside the voids between particle elements, one has to adopt the Richardson and Zaki 24 equation so that the corrected relative (slip) velocity is
uR′ ) uR × 10-dS/dV
(4)
with a correlation factor of 0.8. The values of ψ were in the range between 0.025 and 0.14, which illustrates how small the fraction of terminal velocity can be reached in the interstices between the particles in the packed bed. Pressure Drop The pressure drop is the basic hydrodynamic characteristic of gas-flowing solids-fixed bed contactors. It is well-known that even for a much simpler case, the flow of gas through the packed bed, there is no unique equation for satisfactory prediction of pressure drop. The Ergun equation is widely used, but for each type of packing a different set of constants is needed. For our case of three-phase gas-flowing solids-fixed bed contactors we start from the simple assumption that there are two major contributions for overall pressure drop: resistance of the packing and drag due to complex interactions between gas and flowing solids:
∆p ∆p ∆p + ) L L PB L FS
( )
2deq 3(1 - )
(5)
For countercurrent systems relative velocity is a sum of the velocity of particle (uS) and real gas velocity
ug′ )
ug ( - β)
(7)
(8)
where ψ is a correction factor. The rigorous analysis, which would lead to a correction factor, would be very complicated. However, for the sake of simplicity, one can find an empirical correlation for ψ taking into account that dimensionless parameters of major influence would be FS/F and dS/dV. To check this assumption, we used all the available data from the literature (Table 1). The correlation obtained on the basis of 452 data points was
ψ ) 0.4186
x
F dV ‚ + 0.0042 FS dS
)
PB
]
ug2F (1 - ′) A(1 - ′) +B ‚ ‚ Rep deq ′3
(11)
where ′ is the voidage of the bed corrected for the presence of flowing particles:
′ ) - β
is not a real velocity of flowing particles in the column because the terminal velocity from eq 2 cannot be reached. In fact, particles accelerate in the interstices, then collide with packing elements, and so continue to accelerate and decelerate through the bed. The real average velocity can be expressed as
uS′ ) ψ‚uS
( ) [ ∆p L
(12)
(6)
where ug is the superficial gas velocity, is the packed bed void fraction, and β is the total holdup of flowing solids in the column. The velocity of particles, found as
uS ) uR′ - ug′
(10)
The first term can be found from the Ergun equation with constants A and B determined (in two-phase systems) for each type of packing,
where dV is an equivalent diameter of voids:25
dV )
( )
(9)
The second term in eq 10, that is, the flowing solids contribution, is a result of the sum of drag forces exerted by the flowing solids particles. However, the effective drag coefficient to be used has to be corrected for two reasons. The first is the fact that the flowing solids particle interacts with suspension, not with gas alone. The second is the effect of blockage ratio, that is, the influence of the presence of walls of packing elements on drag force. The first effect is taken into account through the use of corrected Reynolds number (Rep′), based on density and viscosity of suspension. Suspension viscosity can be estimated by the equation25
µ′ 101.82(βdyn/′) ) µ (1 - βdyn/′)
(13)
For the effect of blockage ratio, the following was proposed:26
CD′ )
CD dS 1D
(
)
1.78
(14)
The ratio of diameters in this equation takes into account the ratio of cross-section area of object perpendicular to flow direction and free cross-section area available for flow. However, in our case in each cross
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Table 1. Studies of the Hydrodynamics of Gas-Flowing Solids-Fixed Bed Contactors
a
Dynamic holdup. b Pressure drop.
section a number (n) of particles are present. The number n can be estimated from
dS2π D2π ‚βdyn ) n‚ 4 4
(15)
Substituting n from eq 15 in the ratio of cross-section area of particles and free cross-section area available for flow gives
n‚dS2π/4 2
‚ D π/4
)
βdyn
(16)
and the drag coefficient corrected for the blockage ratio is
CD′ )
CD (1 - xβdyn/)
(17)
Rigorously, from the value of the static holdup should be subtracted. However, the values of static holdup are usually very small, the data in the literature are extremely scarce, and there is no available correlation to predict them. So for the sake of simplicity, static holdup was neglected in this approach.
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2533 Table 2. Values of Constants in the Ergun Equation for Different Packings (Values Are Either Reported by the Authors or Calculated from Their Two-Phase Experiments)
Figure 1. Comparison of predicted and experimental values for dynamic holdup. Symbols are listed in Table 1.
Pressure drop in the gas is the result of drag force on all of the flowing particles in the bed,
FD,overall ∆p ) Afree
a
Calculated from two-phase experimental results.
(18)
where
dS2π FuR′2 ‚ 4 2
FD,overall ) N‚CD′‚
(19)
and Afree is the free cross-section area available for flow:
Afree )
D2π ( - βdyn) 4
(20)
The number of particles in the bed (N) can be found from 3
dS π D2π Lβdyn ) N 4 6
(21)
From eqs 19-21, we obtain 2 3 CD′βdynFuR′′ ∆p ) ‚ L FS 4 ( - βdyn)dS
( )
(22)
For βdyn the predicted values from eq 1 are to be used. CD′ can be found from eq 17 and uR′′ is the corrected relative velocity
uR′′ ) ug′ + uS′
(23)
where ug′ and uS′ can be found from eqs 6 and 8, respectively. Results Predictions for dynamic holdup, using eqs 1-9 are compared with all available data from the literature (listed in Table 1), and the comparison is presented in Figure 1. For a wide range of experimental conditions, different dimensions of equipment, flowing particles, and packing elements of different shapes, the predictions give good agreement with experimental results with an average error of 20.8%, for 452 data points.
Figure 2. Comparison of predicted and experimental values for pressure drop. Symbols are listed in Table 1.
Pressure drop was calculated using eqs 10, 11, 22, and 23, together with values of constants for different types of packing (Table 2). The average contribution of the first term in eq 10 was 75%, but it varied from system to system in the range from 25% to 85%. Results are compared with all available literature data (sources in Table 1) in Figure 2. The agreement is reasonably good, with an average error of 40.1% for 435 data points. However, it can be concluded from Figure 2 that the data are somewhat underpredicted. This is probably the consequence of neglecting the static holdup contribution. But, as stated before, data on static holdup are extremely scarce; there is no available way to predict it and its value is strongly dependent on the type of packing. So the error caused by neglecting the static holdup contribution is the price we have to pay for the sake of simplicity.
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Conclusion
Literature Cited
Simple phenomenological model for pressure drop in gas-flowing solids-fixed bed contactors is presented, together with a fundamentally based, but semiempirical, model for prediction of dynamic holdup. Taking into account the wide range of operating conditions, and great variety of packing elements and flowing solids types and dimensions, the results are in good agreement with all available experimental results. The agreement is good especially when one considers the available predictions for a much simpler case of two-phase systems. Proposed models are based exclusively on parameters known a priori and do not require any experimental data from an actual system.
(1) De Directie Van De Staatsmijnen in Limburg, Procede pour augmenter la concentration des particules solides dans un courant de milieu gazeuz. French Patent 978287, 1948. (2) Compagnie de Saint-Gobain, Produit intermediaire pout la fabrication du verre et autres silicates, et procede et appareillages pour sa fabrication. French Patent 1469109, 1965. (3) Kaveckii, G. D.; Planovskii, A. N. Flow Study of Solids in Up Flowing Gas in Packed Columns (Issledovanie techeniya tverdogo zernistogo materiala v voshodyaschem potoke gaza v kolonah s nasadkoi). Khim. Tekhnol. Topl. Masel. 1962, No. 11, 8. (4) Kaveckii, G. D.; Planovskii, A. N.; Akopyan, L. A. Calculation of Axial Mixing of Gas and Solids in Packed Column (Ob uchete prodolnogo peremishivaniya gaza i tverdogo zernistogo materiala v kolone s nasadkoi). Khim. Prom. 1963, No. 6, 449. (5) Claus, G.; Vergnes, F.; Le Goff, P. Hydrodynamic Study of Gas and Solid Flow Through a Screen-Packing. Can. J. Chem. Eng. 1976, 54, 143. (6) Roes, A. W. M.; van Swaaij, W. P. M. Hydrodynamic Behavior of a Gas-Solid Counter-current Packed Column at Trickle Flow. Chem. Eng. J. 1979, 17, 81. (7) Roes, A. W. M.; van Swaaij, W. P. M. Axial Dispersion of Gas and Solid Phases in a Gas-Solid Packed Column at Trickle Flow. J. Chem. Eng. 1979, 18, 13. (8) Large, J. F.; Naud, M.; Guigon, P. Hydrodynamics of the Raining Packed-Bed Gas-Solids Heat Exchanger. J. Chem. Eng. 1981, 22, 95. (9) Verver, A. B.; van Swaaij, W. P. M. The Hydrodynamic Behavior of a Gas-Solid Trickle Flow over a Regularly Stacked Packing. Powder Technol. 1986, 45, 119. (10) Westerterp, K. R.; Kuczynski, M. Gas Solid Trickle Flow Hydrodynamics in a Packed Column. Chem. Eng. Sci. 1987, 42, 1539. (11) Predojevic´, Z. J.; Petrovic´, D. Lj.; Dudukovic´, A. Flowing Solids Dynamic Holdup in the Countercurrent Gas-Flowing Solids-Fixed Bed Reactor. Chem. Eng. Commun. 1997, 162, 1. (12) Predojevic´, Z. J. Fluid Dynamics of Countercurrent GasSolid-Packed Bed Contactor. Ph.D. Thesis, Faculty of Technology, Novi Sad, Yugoslavia, 1997. (13) Predojevic´, Z. J.; Petrovic´, D. Lj.; Martinenko, V.; Dudukovic´, A. Pressure Drop in a Gas-Flowing Solids-Fixed Bed Contactor. J. Serb. Chem. Soc. 1998, 63, 85. (14) Stanimirovic´, O. P. Prediction of Pressure Drop and Dynamic Holdup of Solid Phase in a Gas-Solid-Packed Bed Reactor. B.S. Thesis, Faculty of Technology, Novi Sad, Yugoslavia, 1998. (15) Roes, A. W. M.; van Swaaij, W. P. M. Mass Transfer in a Gas-Solid Packed Column at Trickle Flow. Chem. Eng. J. 1979, 18, 29. (16) Saatdjian, E.; Large, J., F. Heat Transfer in Raining Packed Bed Exchanger. Chem. Eng. Sci. 1985, 40, 693. (17) Kiel, J. H. A.; Prins, W.; van Swaaij, W. P. M. Modelling of Non-Catalytic Reactions in a Gas-Solid Trickle Flow Reactor: Dry, Regenerative Flue Gas Desulphurisation Using a SilicaSupported Copper Oxide Sorbent. Chem. Eng. Sci. 1992, 47, 4271. (18) Westerterp, K. R.; Kuczynski, M. A Model for a Countercurrent Gas-Solid-Solid Trickle Flow Reactor for Equilibrium Reactions. The Methanol Synthesis. Chem. Eng. Sci. 1987, 42, 1871. (19) Kuczynski, M.; Oyevaar, M. H.; Pieters, R. T.; Westerterp, K. R. Methanol Synthesis in a Countercurrent Gas-Solid-Solid Trickle Flow Reactor. An Experimental Study. Chem. Eng. Sci. 1987, 42, 1887. (20) Verver, A. B.; van Swaaij, W. P. M. The Gas Solid Trickle Flow Reactor for the Catalytic Oxidation of Hydrogen Sulphide: A Trickle-Phase Model. Chem. Eng. Sci. 1987, 42, 435. (21) Kiel, J. H. A.; Prins, W.; van Swaaij, W. P. M. Mass Transfer Between Gas and Particles in Gas-Solid Trickle Flow Reactor. Chem. Eng. Sci. 1993, 48, 117. (22) Predojevic´, Z. J.; Petrovic´, D. Lj.; Dudukovic´, A. P. Pressure Drop in a Countercurrent Gas-Flowing Solids-Packed Bed Contactor. Ind. Eng. Chem. Res. 2001, 40, 6039. (23) Turton, R.; Levenspiel, O. A Short Note on the Drag Correlation for Spheres. Powder Technol. 1986, 47, 83. (24) Richardson, J. F.; Zaki, W. N. Sedimentation and Fluidization. Trans. Inst. Chem. Eng. 1954, 32, 35.
Acknowledgment We thank professor Levenspiel for inspiring this work as part of the joint US Yugoslav program for studies of countercurrent gas-flowing solids-fixed beds reactors, funded by the National Science Foundation. Unfortunately, the funding from this grant was prematurely terminated in the 1990s due to the events in Yugoslavia. We are indebted to the Ministry of Science of Serbia for resuming partial funding of this work, which enabled us to contribute these results to this special issue in Professor Levenspiel’s honor. Nomenclature A ) cross-section area, m2 A, B ) constants in Ergun equation (11) a ) surface area of packing per unit bed volume, m2/m3 D ) diameter of column, m CD ) drag coefficient CD′ ) corrected drag coefficient, eq 17 dS ) flowing solids particle diameter, m deq ) equivalent diameter of packing particle ()6(1 - )/ (a + 4/D)), m dV ) equivalent diameter of voids, eq 5, m FD ) drag force, N g ) gravity acceleration, m/s2 L ) fixed bed height, m m ) mass of flowing solids in the bed, kg N ) number of flowing solids particles in the bed, eq 21 n ) number of flowing solids particles in the cross-section element, eq 15 p ) pressure, Pa Rep ) particle Reynolds number () uRdSF/µ) Rep′ ) corrected particle Reynolds number ()uR′′dSFj/µ′) RePB ) packed bed Reynolds number ()ugdeqF/µ) S ) mass flux of flowing solids, kg/(m2 s) ug ) superficial gas velocity, m/s ug′ ) corrected superficial gas velocity, eq 6, m/s uR ) relative velocity between gas and flowing solids, m/s uR′ ) relative velocity, eq 4, m/s uR′′ ) corrected relative velocity, eq 23, m/s uS ) velocity of particles, eq 7, m/s uS' ) real average solids velocity, eq 8, m/s V ) volume of the fixed bed column, m3 β ) flowing solids holdup ()m/(FSV)) βdyn ) dynamic holdup ) fixed bed void fraction ′ ) corrected void fraction, eq 12 µ ) gas dynamic viscosity, kg/(m s) µ′ ) suspension viscosity, eq 13, kg/(m s) F ) gas density, kg/m3 FS ) skeletal density of the flowing solids particles, kg/m3 Fj ) suspension density ()(FSβdyn + F′)/′, kg/m3 ψ ) correction factor, eq 8
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2535 (25) Foust, A. S.; Wenzel, L. A.; Clump, C. W.; Maus, L.; Andersen, L. B. Principles of Unit Operations; John Wiley & Sons: New York, 1962; pp 452-453. (26) Dudukovic´, A.; Koncar-Djurdjevi_, S. K. The Effect of Tube Walls on Drag Coefficients of Coaxially Placed Objects. AIChE J. 1981, 27, 839. (27) Kiel, J. H. A. Removal of Sulphur Oxides and Nitrogen Oxides from Flue Gas in a Gas-Solid Trickle Flow Reactor. Ph.D. Thesis, University of Twente, Enschede, The Netherlands, 1990.
(28) Pjanovic´, R. The Basic Characteristics of Three-Phase Gas-Solid-Solid Contactor. M.Sc. Thesis, Faculty of Technology and Metallurgy, Belgrade, Yugoslavia, 1998.
Received for review July 23, 2002 Revised manuscript received January 9, 2003 Accepted January 13, 2003 IE020541S
