5802
Ind. Eng. Chem. Res. 2004, 43, 5802-5809
Expansion of Gas Fluidized Beds Derek Geldart† Powder Research Ltd. and Department of Chemical Engineering, University of Bradford, Bradford, West Yorkshire BD7 1DP, U.K.
A knowledge of the degree of expansion of fluidized beds is needed by designers in order to calculate conversions in fluidized-bed chemical reactors and, in combustors or incinerators, the relative areas of heat-transfer surfaces immersed in the bed and the freeboard. Because differential and absolute pressures are relatively easy to measure in fluidized beds even at high temperatures, they can provide a means for measuring and controlling conditions in the bed. In this paper, protocols are presented for (a) predicting bed expansion and density and (b) using pressure measurements to deduce hydrodynamic conditions in the bed, notably bubble velocities, which determine the gas residence time. To test the validity of the theory presented, experiments have been done in two columns, one of which was 300 mm internal diameter using four sands at low gas velocities and the other in which a fluidized catalytic cracking catalyst and hydrated alumina were fluidized at velocities up to and beyond the velocity at which transition to turbulent fluidization occurs in a column of 290 mm internal diameter. The rise velocities deduced are alarmingly high compared with values predicted from correlations in current widespread use. 1. Introduction An understanding of the factors influencing the expansion of a fluidized bed is important for several reasons: In some fluidized-bed reactor systems, the designer needs to know the mass of solids per unit bed volume (the bed density FB) because this influences the chemical conversion. In others (e.g., fluidized-bed combustors), the heat-transfer surfaces are both in the bed and in the splash zones, and the overall rate of heat transfer depends on the relative surface areas in each region, which, in turn, depend on bed expansion. Experimental measurements of the bed density can be used to infer the hydrodynamic conditions (e.g., bubble velocity) inside the bed, and this is an essential parameter in many fluidized-bed reactor models. This paper is in two parts. In the first part, an overall theoretical approach to predicting the bed expansion and bed density is given, and in the second part, experimental measurements of the bed density were made in two fluidized beds of 300 and 290 mm internal diameter, and the latter had an internal cyclone to permit operation at high velocities. Measurements of pressure differentials are used to provide information on the hydrodynamic conditions in the bed, notably the rise velocity of the bulk of the gas.
The bubble holdup depends on the flow rate of gas through the bubble phase and on the bubble velocity, which, in turn, depends on the bubble size. A fluidized bed in which bubble growth is restricted through bubble splitting or by the careful choice of the fraction of fines in the size distribution will, therefore, have a higher average expansion (lower bed density) than a freely bubbling bed under otherwise identical conditions. 3. General Case (Figure 1) The following analogy, in which gas bubbles are imagined to be hollow balls, helps us to understand why a bubbling fluidized bed expands. Imagine a tank of water into which many spheres (e.g., ping-pong or table tennis balls) are suspended at various levels. The level of the water in the tank rises by an amount equal to the volume of the balls. Now imagine that we have some means of feeding balls into the bottom of the tank and skimming them off the surface. The amount by which the water level rises is
2. Part 1: Theoretical Approach The overall bed expansion is influenced by two factors: the degree of expansion of the dense phase, that is, by the amount that its voidage, D, increases beyond mf, and the bubble holdup. The expansion of the dense phase is generally believed to be very small in Geldart group B and D solids, so that D ) mf, though there are few published data. In groups A and A/C powders, D can be significantly larger than mf. †
Present address: 7 Westminster Gate, Burn Bridge, Harrogate, North Yorkshire HG3 1LU, U.K. Fax: +44 (0)1423 873375. E-mail:
[email protected].
Figure 1. Diagrammatic sketch of a freely bubbling bed of group B and D solids in which there is bubble growth with the distance above the distributor.
10.1021/ie040180b CCC: $27.50 © 2004 American Chemical Society Published on Web 07/28/2004
Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5803
again dependent on the volume of the balls beneath the surface (the holdup), but in order to calculate this, we need to know the volumetric rate at which the balls are being pumped in (in cubic meters per second) and how long they spend (in seconds) in the water. The latter, the residence time, depends on the velocity at which the balls rise. The actual situation is even more complicated than this because in a fluidized bed many small bubbles are entering the bottom through the gas distributor and, as they rise, they combine to produce fewer, faster, larger bubbles, and this is what we need to model. A first attempt to address this problem was made almost 30 years ago1 when much less was known about the fundamental behavior of fluidized beds and few correlations for bubble size were available in the literature. Consider a small element of bed of height dh (Figure 1) in which the bubble velocity is constant. The residence time of one bubble in the element is ∆h/UBsh, where UBsh is the average bubble rise velocity relative to the column wall (the absolute rise velocity) at height h above the distributor. If the “visible” bubble flow rate is QB, then the volume of bubbles ∆VB in the element is QB∆h/UBsh. In the limit
dVB ) QB
dh UBsh
(1)
and over the whole bed
VB )
∫0Hh QB UdhBsh
(2)
UBsh is a function of the bubble size, which, because of coalescence, grows with distance h above the distributor. Substitution of a full correlation for UBsh and its relationship to bubble size using, for example, eq 16 leads to a very complex equation. As outlined later, a calculation of bubble sizes at several levels and use of an average value for the entire bed are sufficient for engineering purposes. If we know the average bubble velocity UBs over the entire bed, then
H h VB ) QB UBS
VB HA
)
Y ) 1.64Ar0.2635
(8)
Upon substitution in eq 4 for QB/A from eq 7, the fraction of the bed occupied by bubbles
B )
Y(U - Umf) UBS H - HD H
B )
(9)
(10)
Substituting in eqs 4 and 9 and rearranging gives
R)
H ) Hmf U
BS
UBS - Y(U - UD)
(11)
This is the general equation for the bed expansion ratio caused by the bubbles. We shall now look at several practical situations. It should be remembered that with one exception all correlations for fluidized beds are approximations and are subject to error, so that one should not expect agreement with experimental measurements better than (30%. The one exception is that the pressure drop across a fully fluidized bed is equal to the mass of solids in the bed per unit area. 4. Bed Expansion When There Is Bubble Growth (a) Group B Powders. In these sandlike solids, there is the complication that the bubbles grow both with the distance above the gas distributor and with the excess gas velocity (U - Umf). However, the voidage of the dense phase (D), its height (HD), and the gas velocity in the dense phase (UD) are essentially identical with the values at minimum fluidization, so that eq 11 can be written as
(3) R)
and the fraction of the bed occupied by bubbles
B )
where Y ) 1 for slugging beds of all powders, while for bubbling beds of group A powders, Y ≈ 0.8. For group B powders, HD ∼ Hmf and Y is a function of the Archimedes number2
QB 1 A U
BS
(4)
BS
According to the two-phase theory
QB/A ) U - UD
(5)
where UD is the gas velocity in the dense phase, usually taken as the minimum fluidization velocity Umf for Geldart group B powders. In practice
QB/A < U - UD
(6)
because gas short circuits from bubble to bubble, and the true situation can be approximated by the expression
QB/A ) Y(U - Umf)
H ) Hmf U
(7)
UBS - Y(U - Umf)
(12)
To solve eq 12, we need an expression for the average, absolute bubble velocity, US. The local bubble velocity at any height above the distributor, h, can be obtained from several different correlations. The most commonly used is
UBsh ) UB + U - Umf
(13)
where UB is the rise velocity of a single isolated bubble, given by 0.71(gdBV)0.5. However, on the basis of many data taken from beds of different diameters, Werther3 proposed
UBsh ) φ(gdBV)0.5
(14)
φ increases with the bed diameter, is larger for group A powders, and is independent of the gas velocity. φ depends on the type of powder (group A or B) and the diameter of the column, D.
5804 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004
φB ) 0.64
for D e 0.1 m
) 1.6D0.4 ) 1.6 φA ) 1
for 0.1 < D < 1 m for D g 1 m
for D < 0.1 m
) 2.5D0.4 ) 2.5
for 0.1 < D < 1 m for D g 1 m
(15)
These coefficients should be used with caution, but qualitatively they do reflect observed trends. There are many equations for the diameter of bubbles as a function of the level in the bed and gas velocities, and we shall use that of Darton et al.:4
dBV )
0.54 (U - Umf)0.4(h + 4N-0.5)0.8 g0.2
(16)
We are now in a position to formulate a procedure to estimate the bed expansion as follows. Step 1. Calculate dBV at h ) 0.4Hmf. If dBV > D/3, the bed is slugging so go to section 5a. Step 2. If dBV < D/3, calculate dBV at h ) 0 and Hmf using eq 16. Note that N is the number of holes per square meter in the distributor. Step 3. Calculate the corresponding values of UBsh from eq 13 or eq 14, and take the arithmetic average, UBS. Step 4. Calculate Y from eq 8. Step 5. Calculate H from eq 12, and make it equal to H′, the first estimate of the expanded bed height. Step 6. Recalculate dBV and UBS when h ) H′ as in steps 1 and 2, and then obtain the second estimate of the bed height, H′′, from eq 12. Usually two iterations are sufficient to obtain H, so that the expansion ratio can be calculated from
R ) H/Hmf
(17a)
Step 7. If required, calculate the bed density from
FB ) FBmfHmf/H
(17b)
For a bed containing many horizontal tubes, Xavier et al.5 found that their data (using group B and D solids) could be expressed by eq 12 but with Y ) 1 and UBS ) 1 + U - Umf. Their data, like those of all other researchers, show considerable scatter, and only moderate accuracy should be expected. (b) Group D Solids. For moderate excess gas velocities (U - Umf < 0.5 m/s), the same approach to that described in section 4a can be used but using the bubble growth equation of Cranfield and Geldart,6 based on coarse solids equation 18. This is based on measure-
dBV ) 2.25(U - Umf)1.11h0.81
(18)
ments made in beds 3 the following equation predicts the transition velocity:
Rec ) 0.371Ar0.742
(38)
These are similar conditions to those used in our experiments, and when the physical properties for our FCC and hydrated alumina are inserted into this equation, the velocities of transition to turbulence are calculated as 0.41 and 1 m/s, respectively. For the catalyst, the effective bubble rise velocities at excess gas velocities above 0.4 m/s are in the range of 2-3 m/s, somewhat higher than the average void velocities on the order of 2 m/s measured directly by Ellis et al.14 using a cross-correlation technique based on capacitance measurements. In the alumina, surprisingly high void velocities on the order of 6-8 m/s were deduced from the differential pressure measurements.
8. Conclusions A procedure for calculating the expansion ratio and bed density of fluidized beds of powders is proposed and shown to agree moderately well with experimental data gathered using sands in a column of 300 mm diameter at velocities up to 0.08 m/s. A second set of experiments were carried out using two fine powders, FCC catalyst and hydrated alumina, in another column of 290 mm diameter fitted with an internal cyclone and fluidized at velocities up to and beyond the transition to turbulence. Differential pressure measurements made on the bed axis were used to calculate the effective rise velocity of bubbles/voids and found to have values up to 8 m/s when the superficial gas velocity was in the turbulent regime.
Nomenclature A ) cross-sectional area of the column (m2) Ar ) Archimedes number {Fg(Fp - Fg)gd3}/σ2 dp ) mean size of the powder calculated from eq 21 D ) diameter of the column (m) dsv ) mean surface volume diameter of the powder (m) dBV equivalent volume diameter of a bubble (m) dBeq ) equilibrium volume diameter of bubbles in a bed of group A powder (m) F45 ) mass fraction of the size distribution smaller than 45 µm g ) gravitational constant, 9.81 m/s2 H ) average height of the fluidized bed of the powder at superficial velocity U (m) HD ) height of the dense phase (m) Hmf ) height of the fluidized bed at minimum fluidization velocity Umf (m) h ) distance above the distributor (m) Q ) total volumetric flow rate of gas into the bed (m3/s) QB volumetric flow rate of gas appearing as bubbles (m3/s) R ) bed expansion ratio H/Hmf U ) superficial gas velocity Q/A (m/s) UD ) superficial velocity of gas in the nonbubbling or dense phase (m/s) Umf ) minimum fluidization velocity (m/s) UB ) rise velocity of a single isolated bubble (m/s) UBS ) rise velocity of a bubble in a freely bubbling bed (m/ s) UBsh ) rise velocity of bubbles in a freely bubbling bed at a height h above the gas distributor (m/s) B ) volume fraction of the bed occupied by bubbles or voids mf ) voidage of the dense phase at minimum fluidization conditions FB ) bed density (kg/m3) FF ) apparent density of the particles (kg/m3) FBmf ) bed density at the minimum fluidization velocity (kg/m3) σ ) gas viscosity (Ns/m2)
Literature Cited (1) Geldart, D. Predicting the expansion of gas fluidized beds. In Fluidization Technology; Keairns, D. L., Ed.; McGraw-Hill: New York, 1975; Vol. 1, p 237. (2) Baeyens, J.; Geldart, D. Solids Mixing. In Gas Fluidization Technology; Geldart, D., Ed.; John Wiley & Sons: Chichester, U.K., 1986; Chapter 3, p 105. (3) Werther, J. Hydrodynamics and mass transfer between the bubble and emulsion phases in gas fluidized beds of sand and cracking catalyst. In Fluidization IV; Kunii, D., Toei, R., Eds.; Engineering Foundation: New York, 1983; p 93. (4) Darton, R. C.; LaNauze, R. D.; Davidson, J. F.; Harrison, D. Bubble growth due to coalescence in fluidized beds. Trans. Inst. Chem. Eng. (London) 1977, 55, 274. (5) Xavier, A. M.; Lewis, D. A.; Davidson, J. F. The expansion of bubbling fluidized beds. Trans. Inst. Chem. Eng. (London) 1978, 56, 274. (6) Cranfield, R. R.; Geldart, D. Large particle fluidization. Chem. Eng. Sci. 1974, 29, 935. (7) Canada, G. S.; McLaughlin, M. H.; Staub, F. W. Flow regimes and void fraction distribution in gas fluidization of large particles in beds without tube banks. In Fluidization: Application to Coal Conversion Processes; Wen, C. Y., Ed.; AIChE Symposium Series 176; AIChE: New York, 1978; Vol. 74, p 14. (8) Matsen, J. M.; Hovmand, S.; Davidson, J. F. Expansion of Fluidized Beds in Slug Flow. Chem. Eng. Sci. 1969, 24, 1743. (9) Baker, C. G. J.; Geldart, D. An investigation into the slugging characteristics of large particles. Powder Technol. 1978, 19, 177. (10) Abrahamsen, A. R.; Geldart, D. Behaviour of gas fluidized beds of fine powders Part 1 homogeneous expansion. Powder Technol. 1980, 26, 35. (11) King, D. F. Estimation of dense phase voidage in fast and slow fluidized beds of FCC catalyst. In Fluidization VI; Grace, J.
Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5809 R., Shemilt, L. W., Bergougnou, M. A., Eds.; Engineering Foundation: New York, 1989; p 1. (12) Geldart, D. The Behaviour of freely bubbling fluidized beds. PhD. Dissertation, University of Bradford, Bradford, U.K., 1971. (13) Xue, Y. Flow regimes, Bed expansion and entrainment in fluidized beds of fine particles. In Coarse and fine powder fluidization; Electric Power Research Institute Project 8006-16, Final Report TR102428; Geldart, D., Avontuur, P. P. C., Xue, Y., Eds.; University of Bradford: Bradford, U.K., Dec 1993; Section 2.
(14) Ellis, N.; Bi, H. T.; Lim, C. J.; Grace, J. R. Hydrodynamics of turbulent fluidized beds of different diameters. Powder Technol. 2004, 141, 124.
Received for review June 18, 2004 Revised manuscript received July 1, 2004 Accepted July 2, 2004 IE040180B
