Powder Technology, 59 (1989)
117  123
117
Interpreting Probe Signals from Fluidized Beds R. TURTON
Department of Chemical Engineering, Center for Fluidization Research, West Virginia University, Morgantown, WV 26506 (U.S.A.) and N. N. CLARK
Department of Mechanical and Aerospace Engineering, Center for Fluidization Research, West Virginia University, Morgantown, WV 26506 (U.S.A.) (Received September 6,1988;
in revised form April l&1989)
SUMMARY
Bubble size distribution in fluidized beds may have a profound effect on reactor mass transfer and hydrodynamics. Although several different types of probes are available to monitor bubble events in fluidized beds, the probe signals must be processed carefully to yield bubble size information, because a probe will not always intersect a bubble centrally and because bubble rise velocity will depend on bubble size (usually velocity increases as the square root of linear size). Using a truncated ellipsoid as a model of bubble shape, the relationship between probe signals and bubble size has been explored using probability density functions. A backtransform technique has been presented to yield the bubble size distribution for a given distribution of time intervals when the probe ‘cuts’s bubble. This backtransform has been demonstrated for two different bubble size distributions. When too few bubble measurements are taken or too many intervals in the discrete bubble size distribution are used, instabilities in the backtransform may result.
INTRODUCTION
Fluidized beds are widely used in the engineering industries for such processes as drying, coating, combustion, reaction and agglomeration. Many of these processes are conducted in bubbling beds, where bubbles of gas rise through a dense phase of fluidized particles. The size distribution of these bubbles has a profound effect on the performance of the fluidized bed, by influencing 00325910/89/$3.50
solids movement and circulation, gas residence time and reaction in the bed. However, it is very difficult to determine the size distribution of bubbles present in a fluidized bed, even in the research mode. Thin, transparent, twodimensional beds permit direct observation of the bubbles but are not generally representative of the hydrodynamics in larger threedimensional beds. This paper discusses the use of probe signals to infer the bubble size distribution, with emphasis on backtransforming a distribution of measured bubble intersection times into the true size distribution. Accurate bubble size measurement will permit future improvements in bed design, particularly the configuration of the grid or distributor plate.
BACKGROUND AND LITERATURE
Presence of a bubble at a point in a fluidized bed can be determined using optical, capacitance or pressure probes. General reviews of fluidized bed probes are given by Fitzgerald [l], Cheremisinoff [Z] and Atkinson and Clark [3]. The most detailed research on capacitance probes has been presented by Werther [4,5] and Werther and Molerus [6,7]. Their miniature probe was able to detect the passage of a void and produce a pulse, the duration of which was representative of the time interval for which the probe was immersed in the bubble. However, as Werther observes, the bubble will not necessarily cut the probe along its central axis so that a distribution of time pulses cannot be converted directly to a bubble size @ Elsevier Sequoia/Printed in The Netherlands
118
\
1
i_RQ=aR”
It kR
Fig. 2. Ellipsoidal bubble used to approximate bubble shape in a fluidized bed. 01
I
0
\I 1
2 Time
3
4
5
, sets
Fig. 1. Typical pressure signal data from differential pressure probes.
distribution, even if the bubble velocity is known. Probes of the differential pressure type can also detect bubbles successfully [8,9, lo], because there is a vertical pressure gradient in the dense phase, but no significant gradient within the bubble [8], see Fig. 1. Differential pressure probes which resemble an ‘over and under’ shotgun [ll] are likely to be most suited to this role, requiring the simplest data interpretation. As with capacitance probes, both optical and pressure probes can produce data which may be transformed into a binary signal, denoting the presence or absence of a bubble at each moment in time. Werther [4] has argued that bubble velocities are erratic and difficult to characterize and has used a single value of bubble rise velocity to transform the distribution of time intervals during which the bubble surrounds the probe to a distribution of ‘pierced lengths’ representing the cutting of the bubble by the probe. It is from this distribution of pierced lengths that Werther [4,5] deduced the bubble size distribution, by assuming that the bubble shape could be approximated by an ellipsoid. This paper will show that such a transformation is also possible using a more accurate approximation of fluidized bed bubble shape, viz. the top greater section of an ellipsoid or sphere (see Fig. 2). In addition, it is possible to account for the dependence of bubble rise velocity on bubble size, rather than assume a constant rise velocity. THEORY
Consider that we have at our disposal a distribution of pierced lengths y, given by
true
the probability density function P(y). We wish to transform this into a bubble size distribution, given by P(R) where 2R = D is the largest horizontal dimension of a bubble, as shown in Fig. 2. This figure also shows that a truncated ellipsoid is a good representation of the spherical cap bubble which is most common in fluidized beds [ 41. Note also that at any radius P from the bubble center, pierced length y is uniquely defined. If, within the region of the probe, no bubble channeling occurs and bubbles rise with an even density through the region, the conditional distribution of the radii at which bubbles of a given size D intersect the probe is given by
P(rlR) = 5 =
2r
(1)
R2
Since y is known for each value of r and R, one can derive the conditional probability density function for finding a pierced length y from a bubble of size R. This analysis has been conducted previously by Clark and Turton [ 121, who showed that forO
=
&
@a)
and for 2aQR < y < oR(1 + Q) P(YIR)
= &(Y
(2b)
aQ)
else P(y(R) = 0. Using eqn. (2) for a given distribution of bubble sizes, which are pierced by the probe, P(R), the distribution of pierced lengths is readily found by R(Y) =
~JWWW 0
dR
(3)
119
WI = Cl, PW,) AR W2 = C2. PWl)
= C,, wrn
AR
+ Cz,,W32)
AR
,P(R,) AR + C,,,P(R,)
+c m, ,R(R,) yi+
where C,,j =
(4)
AR 1
P(Y I&)
s
AR + . . .
dy
Yi Chord Length y , cm
Fig. 3. Distribution of pierced lengths obtained for a uniform distribution of bubble sizes touching the probe.
Figure 3 shows the distribution of pierced lengths which can be expected for a uniform distribution of bubble sizes touching the probe. However, it is the backtransform, finding P(R) from P(y), that is usually of interest. Moreover, P(R) would be the distribution of bubble sizes touching the probe, and we must transform this further to the distribution of bubble sizes in the bed, P,(R), using a weighting factor of R* which accounts for the higher likelihood of larger bubbles touching the probe, assuming that the bubbles are uniformly distributed across the crosssection of the bed. Since the form of P(R) is unknown, Clark and Turton [12] have solved for P(R) using a numerical approach, similar to that of Werther [4]. Consider a set of data consisting of n observations of chord lengths y. Let us divide the chord lengths into ~lt equal length partitions such that Yi = Ymax (i++)
Ay
R max Y max = m a(1 + Q)m
withAR=
Note that Ci,j is zero for i < j, essentially because there is an upper limit to the pierced length that can be yielded by a bubble of a particular size. A numerical example of this approach is given below. Using the bubble size distribution given in Fig. 3 and shape given in Fig. 2, 5000 pierced lengths were synthesized using a Monte Carlo simulation. The backtransform matrix was applied, using 10 intervals of bubble size to yield a bubble size distribution in good agreement with that used to produce the chord lengths (see Fig. 4). However, the backtransform can become unstable if too many size divisions or too few pierced length observations are used. Figures 5(a) and 5(b) show the results of the backtransform when only 1000 and 100 pierced lengths are used. In extreme cases, even negative probabilities may result.
E
OGi
O
WldRj=R,,,jAR
2
0.35
a 0.30 Predicted
where Ay = ymax/m. Then, an approximation to the probability of finding a chord length y between yi and yi+ 1 is defined as w(Yi =
< Y G Yi+ 1) Number of chord lengths between yi and yi+ 1
solution
has the following
1
2 BUBBLE
Total number of chord lengths, n
The matrix gular form:
0
trian
4
3 RADIUS
R
, cm
Fig. 4. Comparison of backtransformed bubble distribution with the true distribution from which the chord length data were obtained. (5000 pierced lengths and 10 size intervals.)
5
120 ‘i
0.40 .l
6 P z
0.35 
l
dr R(tlR) = R(rlR) z
II
Prydicted
0.30 *
mqm
= P(Y
(a)
BUBBLE RADIUS R
(7)
For all the examples given below, the value of c was taken to be 0.71. Combining this with eqn. (2), we find that for 0 < [email protected] < 2aQR, that is,
, cm
2aQR P5 x
0.4.
8
0.3
& I
0.2  
C r
0.1
0” iih gi
Predicted /

JYltlR) = s
=
o.o.0.1
.
0.2 4 0
@I
1
2 BUBBLE
3 RADIUS
4
2aQR that is, for 
Q aR(1 + Q),
cWl+Q) cm
c&s
P(tlR)
It was assumed above that the pierced length was known from the time interval during piercing, t. This is simply determined if one assumes a constant bubble rise velocity as a first approximation [ 41 i This will also be valid if the bubbles are themselves rising in a fastmoving fluidized stream. On the other hand one may use pairs of probes vertically above one another with crosscorrelation of the signals to measure bubble rise velocity u.
= &
(5)
If we choose to adopt a inodel to predict bubble rise velocity as a function of bubble size, such as U=C’@
dR
P(t) = ji(tlR)Z’(R)
(2)
0
Once again we are more interested in the backtransform. Following the logic and notation used previously in this paper, with Wi= W(ti< t
1
P(tlRj)
dt
(10)
ti
the matrix for backtransformation is Wi = El, P(R,) AR W2
(6)
 aRQ)[email protected]
(ct
else P( t( R) = 0. For a given size distribution of bubbles touching the probe, P(t), the distribution of time intervals would be
Q+
then
@a)
2cw2R
R, cm
ACCOUNTING FOR BUBBLE RISE VELOCITY
=c&s
2gt
and for 2aQR < [email protected] c 5
Fig. 5. Comparison of backtransformed bubble distribution with the true distribution. (a), 1000 pierced lengths; (b), 100 pierced lengths. Severe instabilities are evident in (b).
y = ut
cm
= E2,
,PW
1) AR
+ E2,2P(R2)
AR
W, = IL,, P(R,) AR + E,, 2P(R2) AR +E m, JYR,)
AR
...
(11)
121 TESTING THE APPROACH
The matrix presented above was tested using computer simulations. For example, 5000 bubbles with a known size distribution and all of the same shape were simulated to rise from a surface past a point probe. The velocity of each bubble was in proportion to the square root of its diameter. This simulation yielded a distribution of time intervals during which the probe intersected the bubble as shown in Fig. 6(a). A computer program employing the matrix was then used to backtransform the distribution of time intervals into a discrete distribution of bubble sizes using 10 size intervals. The broken line in Fig. 6(b) shows that the agreement between the original and calculated distributions is good. However, as pointed out previously, if too few data points or too many size intervals
are used in the backtransform, severe inaccuracy may result and the backtransform may become quite unstable. Two examples of these instabilities are discussed below. Figure 7(a) shows the result of using only 1000, rather than 5000, bubbles in the simulation, while Fig. 7(b) shows the result of using 20 intervals rather than 10 intervals with 1000 bubbles. It is not possible to develop a firm prescription for the minimum number of bubbles or maximum number of size intervals that may be used, since the true nature of the bubble size distribution is not known a priori. It is possible, noting the mechanisms governing bubble generation and coalescence in fluidized beds, to have bimodal or multimodal bubble size distributions so that one cannot dismiss such calculated distributions as computational artifacts.
i
o.30 1
E
mr 0.00
0.01
E
”
B Y
.
0.03
0.04
0.05
0.06
PROBE INTERSECTION
(4 i
0.02
0.07
TIME
0.08
0.09
K P
0.10
t ,s
0.5 1
0.24
I
t
0.45
E
1
2 BUBBLE RADIUS
(a)
0.5ot
0
t
3
R,
0.7 1
c 5
4 cm
t
0.40 
p Y 2 c m
0.30 
8 5 i P 0
(b)
1
2
3
4
5
BUBBLE RADIUS R , cm
Fig. 6. Triangular distribution of bubble sizes used in the study of the backtransformation method. (a), Distribution of time intervals obtained by Monte Carlo simulation from the distribution of bubble sizes shown by the broken line in (b) (5000 bubbles were used); (b), comparison of the backtransformed distribution of bubble sizes with the true distribution.
(b)
BUBBLE RADIUS R , cm
Fig. 7. Comparison of the backtransformed bubble size distribution from the distribution of time intervals with the true bubble size distribution. (a), 1000 bubbles and 10 intervals; (b), 1000 bubbles and 20 intervals. Instabilities demonstrate that there are too many intervals used for the number of measurements taken.
122 DISCUSSION
be modified to take this into account discussed in detail elsewhere [ 151.
Throughout the development given above it has been assumed that the rise velocity of a bubble is given by the rise velocity of a single bubble in an infinite medium, i.e., [email protected] For both liquidgas systems and solidgas fluidized systems when there are clusters of bubbles present, such a simple expression for bubble velocity may not be appropriate due to the motion of the dense phase surrounding the bubbles. A variety of drift flux models have been proposed for gasliquid systems [ 131 and gassolid fluidized systems [14] which take into account the relative motion of the continuous and discontinuous phases. The analysis presented here could be carried out for the case where a drift flux model is used to describe the bubble velocity. For example, we could assume a general form of the bubble velocity as [email protected]
P(tlR)
= P(YIE)(A
+ Em)
and then eqn. (8) becomes qt,R)
=
(A +BaQ2 t
(134
2a2R2 2cuQR OGt< P(t,R)
=
CONCLUSION
A technique has been presented for the translation of a signal from a probe into a bubble size distribution. Knowledge is required of the bubble shape and the relationship between bubble size and rise velocity. The signal must be translated into a number of time intervals, corresponding to bubbleprobe intersection, and these intervals are then transformed into a bubble size distribution. The theory presented above is for a point probe. For finitesized probes, some simple modification of the theory will be required.
ACKNOWLEDGEMENT
WI
The analysis would follow as given above, eqns. (6)  (11). For the model given in eqn. (2), we find that
[email protected]
and is
This work was supported by the Department of Energy, Morgantown Energy Technology Center, under Cooperative Agreement DEFC2187MC24027. The authors are grateful to Thomas W. Keech, Jr., of DOEMETC, for helpful discussion. Richard Turton would also like to acknowledge funding through NSF grant CBT 8657548. The views and opinions of the authors expressed herein do not necessarily reflect those of the U.S. Government or any agency thereof.
264+ [email protected] CY2R2
LIST OF SYMBOLS
x [(A + B&$)t
 crR&l
2aQR
W+Q)R
[email protected]
Wb)
‘[email protected]
Equations 13(a) and 13(b) are thus somewhat more cumbersome to use but since a numerical inversion technique is used to backcalculate the bubble size distribution it presents little difficulty. Another of the assumptions used to derive this technique was that all the bubbles were geometrically similar. According to Taitel et al. [13], this may be a poor assumption when there is a wide distribution of bubble sizes present. The transform technique can
A B c, c’ c D E g i i p”
constant in drift flux model, eqn. (12) constant in drift flux model, eqn. (12) constants relating bubble velocity to size coefficient in backtransform matrix (see eqn. (4)) horizontal bubble diameter coefficient in backtransform matrix (see eqn. (10)) acceleration due to gravity denotes chord length or time interval subdivision denotes bubble size subdivision number of chord length subdivisions denotes probability density
123
Q R t U
W Y
height of bubble at circle of truncation = 2aRQ bubble radius (= D/2) time bubble rise velocity denotes discrete probability chord length cut by probe through bubble
Greek symbol a! aspect ratio of ellipsoid
REFERENCES T. J. Fitzgerald, Review of Instrumentation for Fluidized Beds, Workshop Rensselaer Polytechnic Institute, Troy, NY, Oct. 1979. N. P. Cheremisinoff, Z & EC Proc. Des. Dev., 25 (1986) 329. C. M. Atkinson and N. N. Clark, Proc. 1 lth Powder and Bulk Solids Conf., Rosemont, Illinois, 1988, pp. 437  443.
4 J. Werther, Trans. Inst. Chem. Engrs., 52 (1974) 149. 5 J. Werther, Trans. Inst. Chem. Engrs., 52 (1974) 160. 6 J. Werther and 0. Molerus, Znt. J. Multiphase Flow, 1 (1973) 103. 7 J. Werther and 0. Molerus, Znt. J. Multiphase Flow, 1 (1973) 123. 8 0. Sitnai, Chem. Engng. Sci., 37 (1982) 1059. 9 C. M. Atkinson and N. N. Clark, Powder Technol., 54 (1988) 59. 10 N. N. Clark and C. M. Atkinson, Chem. Engng. Sci., 43 (1988) 1547. 11 Y. Oka, M.S. Thesis, Illinois Institute of Technology (1983). 12 N. N. Clark and R. Turton, Znt. J. Multiphase Flow, 14 (1988) 143. 13 Y. Taitel, B. Barnea and A. E. Duckler, AZChE J., 26 (1980) 345. 14 J. F. Davidson and D. Harrison, Fluidized Particles, Cambridge University Press, Cambridge, 1963. 15 R. Seiss, N. N. Clark and R. Turton, Inferring Bubble Size Distributions from Resistance Probe Measurements in GasLiquid, GasSlung and ThreePhase Systems, paper presented at the 3rd Int. Symp. on LiquidSolid Flows, ASME Winter Meeting, Nov. 27  Dec. 2,1988, Chicago, IL.