Ind. Eng. Chem. Res. 2004, 43, 7445-7448
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KINETICS, CATALYSIS, AND REACTION ENGINEERING Modeling and Predictions of Solids Dynamic Holdup in Gas-Flowing Solids-Fixed Bed Contactors Aleksandar P. Dudukovic´ ,* Nikola M. Nikacˇ evic´ , and Z ˇ eljko V. Kuzeljevic´ Faculty of Technology and Metallurgy, University of Belgrade, 11000 Belgrade, Karnegijeva 4, Serbia & Montenegro
A model based on first principles and free of any semiempirical parameters is presented for trickling solids holdup in gas-flowing solids-fixed bed contactors. A small double-cone cavity is used to characterize the voids in packed beds. A comparison of model-predicted value holdup and numerous values reported in the literature is presented. Countercurrent gas-flowing solids-fixed bed contactors are a relatively new type of equipment gradually gaining more attention because of their favorable properties. In these contactors, gas is introduced at the bottom of the column and fine solid particles (flowing solids) are introduced at the top, and they flow countercurrently through the stationary bed of larger solid particles (second solids phase). The first recorded industrial use of such devices, contactors, occurred in 19651 in heat recovery, but the idea was patented as early as 1948.2 The potential of these contactors for efficient heat exchange, adsorption, purification of gases, etc., has been shown in a number of studies focused mainly on fluid dynamics of flowing solids systems,3-14 including heat- and mass-transfer investigations.8,15-17 Later studies were expanded to investigate the use of the flowing solids principle in chemical reactors.17-21 Reliable predictions of basic characteristics such as pressure drop and flowing solids holdup are needed for further applications and the design of this type of equipment. However, no satisfactory fundamentally based equations have been offered to date. Proposed models6,10,22,23 are all semiempirical in nature. Most of them6,10,22 are based exclusively upon experimental data taken in a single study conducted by the same investigators and are not tested on data of others. Useful empirical correlations for the prediction of pressure drop24 and solids dynamic holdup25 were proposed based on numerous experimental data from various sources, but they lack a fundamental basis. Modeling The objective of the present approach is to develop a phenomenological model for the flowing solids fluid dynamics and for the prediction of solids dynamic holdup without use of any empirical parameters. Our previous attempt23 started with the balance of forces on the flowing particle in the interstices (voids) between packed-bed elements. However, it was shown that the voids are too short for the particle terminal velocity to * To whom correspondence should be addressed. E-mail:
[email protected].
Figure 1. Sketch of a simplified bed void model.
be reached.23 Hence, the particles accelerate in each void before they collide with packed-bed particles and come to an almost complete stop at the end of the void. This process then repeats itself and was taken into account via an empirical factor based on the available data. In this approach, we attempt to overcome the need for any empirical contribution, taking into account the acceleration of the flowing solids particles. The governing equations for the flowing solids particle position and velocity along the vertical axis of a void are
dz/dt ) uS
(1)
3CD duS FS - F ) g(u + u′g)2 dt FS dSFS S
(2)
and
where uS + u′g is the relative velocity between the gas
10.1021/ie0401105 CCC: $27.50 © 2004 American Chemical Society Published on Web 10/07/2004
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Table 1. Studies of the Hydrodynamics of Gas-Flowing Solids-Fixed Bed Contactors
and solid particle, FS and F are flowing solids and gas densities, respectively, and CD is the drag coefficient, calculated from the Turton and Levenspiel equation:26
CD )
24 0.413 (1 + 0.173Rep0.6567) + Rep 1 + 16300Re
packing separately. For the sake of simplicity and generality, in this approach a simplified picture of an average void is used, requiring only the equivalent diameter of the voids, evaluated directly from the mean diameter of the packing elements, as shown below:
-1.09
p
(3)
dV )
2deq 3(1 - )
(5)
The gas velocity in the interstices is calculated as
u′g ) ug/( - β)
(4)
The vertical velocity of the solid particle as a function of the position in the void can be found by solving eqs 1-3 numerically, assuming uS ) 0 and z ) 0 at t ) 0. The record of the particle velocity as a function of the axial position in the void is kept for subsequent calculations. The average velocity in the column can be assumed to be equal to the average velocity in all of the voids. However, the shapes of the packed-bed particles could be of different geometry and, consequently, voids could be of very different shape and orientation, requiring a number of parameters to describe each type of
If we describe the shape of an average void by a double cone as shown in Figure 1, the vertical distance between the solid walls of the void is a function of the radial position as shown below:
z0(r) ) dV - 2r
(6)
The average particle velocity in a void at position r can be found as the mean value of the previously calculated local particle velocity obtained by solving eqs 1 and 2:
uS(r) )
1 z0(r)
∫0z (r)uS dz 0
(7)
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pressure drop and dynamic holdup, based exclusively on parameters known a priori. Acknowledgment The authors are indebited to the Ministry of Science, Technology and Development of Serbia for their financial support. Nomenclature
Figure 2. Comparison of predicted and experimental values for dynamic flowing solids holdup. Symbols are listed in Table 1.
Finally, the average particle velocity for the whole void is the weighted mean value:
uAV )
∫0d /2uS(r) r dr
2
V
(dV/2)2
(8)
After eqs 7 and 8 are solved numerically, the dynamic holdup can be calculated as
βdyn ) S/FSuAV
(9)
where S is the flowing solids mass flux. Results Predictions of solids dynamic holdup, using the model presented above, were compared with all of the data available in the literature (listed in Table 1). The comparison of predicted and experimental values is presented in Figure 2. The average error for 564 data points was 28.8%. Taking into account the different designs and scales of equipment, the wide range of experimental conditions used, the different shapes and properties of both flowing solids particles, and the packing elements employed, the agreement between experimental values and predictions is very good. Summary and Conclusion The proposed model for flowing solids holdup is based on first principles and does not contain any empirical parameters. Simplifications in geometry incorporated in the model made this model applicable to all types of packing elements in the column. For a great variety of dimensions, ranges of operating conditions, and properties of both solids phases, the proposed model gives good predictions of dynamic holdup for flowing solids. This model also eliminates the need for an empirical input to our pressure drop prediction in gas-flowing solidsfixed bed contactors23 because it provides the needed value of solids dynamic holdup. Both models taken together now have the capability of predicting the
a ) surface area of packing per unit bed volume, m2/m3 D ) diameter of column, m CD ) drag coefficient dS ) flowing solids particle diameter, m deq ) equivalent diameter of the packing particle [)6(1 )/(a + 4/D)], m dV ) equivalent diameter of voids, eq 5, m g ) gravity acceleration, m/s2 m ) mass of flowing solids in the bed, kg Rep ) particle Reynolds number r ) radial coordinate, m S ) mass flux of flowing solids, kg/(m2 s) t ) time, s ug ) superficial gas velocity, m/s u′g ) corrected superficial gas velocity, eq 4, m/s uS ) flowing solids velocity, m/s uAV ) average flowing solids velocity, m/s z ) axial coordinate, m β ) flowing solids holdup [)(m/FSV)] βdyn ) dynamic holdup ) fixed-bed void fraction F ) gas density, kg/m3 FS ) skeletal density of the flowing solids particles, kg/m3
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Received for review April 6, 2004 Revised manuscript received August 9, 2004 Accepted August 25, 2004 IE0401105
