Monte Carlo simulation of the bubble size distribution in a fluidized bed with intrusive probes

International Journal of Multiphase Flow 44 (2012) 1–14

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International Journal of Multiphase Flow j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j m u l fl o w

Monte Carlo simulation of the bubble size distribution in a fluidized bed with intrusive probes Martin Rüdisüli a, Tilman J. Schildhauer a,⇑, Serge M.A. Biollaz a, J. Ruud van Ommen b a b

Paul Scherrer Institut (PSI), General Energy Research Department, CH-5232 Villigen PSI, Switzerland Delft University of Technology, Product and Process Engineering, Julianalaan 136, 2628 BL Delft, The Netherlands

a r t i c l e

i n f o

Article history: Received 30 September 2011 Received in revised form 10 February 2012 Accepted 16 March 2012 Available online 27 March 2012 Keywords: Monte Carlo simulation Intrusive probes Backward transforms Fluidized bed Maximum entropy method Bubble size distribution

a b s t r a c t Intrusive probes such as optical probes are commonly used to measure the bubble size distribution in a fluidized bed. However, usually only a chord length distribution is measured which is typically smaller than the actual centerline bubble size distribution of the pierced bubbles. Moreover, since small bubbles are less likely hit by the probe than large bubbles, the effective bubble size distribution in the entire bed is generally hidden to an intrusive probe. In order to elucidate the bubble size distribution in a fluidized bed measured by an intrusive probe, a Monte Carlo (MC) model is established. MC simulations are conducted with varying sample distributions (gamma and Rayleigh), varying probe positions, and varying spatial distributions of bubbles in the cross-section. Provided the bubble shape is ellipsoidal, it is shown that for all of these variations, the mean chord length can be taken as a representative measure of the mean bubble size in the bed. Furthermore, the applicability of statistical backward transforms (analytical, non-parametrical, and maximum entropy approach) to convert the chord length distribution to the overall bubble size distribution in the bed is assessed. None of these backward transforms outperforms the simple and straightforward approach to just take the mean chord length as the representative mean bubble size in the bed. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Bubbles in a fluidized bed are stable and particle-free voids rising up in the gas–solid emulsion of the bed (Kunii and Levenspiel, 1991; Yang, 2003). The influence of bubbles on the hydrodynamics in a fluidized bed is manifold and of utmost importance: Bubbles promote gas–solid mixing, heat and mass transfer as well as the bed expansion, solid entrainment, etc. Thus, bubbles can be named the ‘‘motor of fluidization’’ and their influence is vital to the efficiency, chemical conversion, and selectivity of a fluidized bed reactor. In order to understand the hydrodynamics in a fluidized bed, it is essential to characterize the bubble size distribution. A commonly used and simple method is to pierce rising bubbles by intrusive sensors such as optical (Glicksman et al., 1987; Guet et al., 2003; Xue et al., 2008; Yasui and Johanson, 1958) or capacitance probes (Werther, 1974a,b, 1978; Werther and Molerus, 1973a,b). However, with these intrusive methods, there is always the problem that the pierced bubble is most likely not pierced along its vertical centerline. Therefore, the measured chord length is typically smaller than the actual bubble size. Moreover, as large bubbles

⇑ Corresponding author. E-mail address: [email protected] (T.J. Schildhauer). 0301-9322/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmultiphaseflow.2012.03.009

are more likely pierced than smaller ones, the distribution of the pierced bubbles is biased and does not necessarily provide a representative distribution of bubble sizes in the entire bed. In literature, there are several methods to convert the measured chord length distribution to the overall bubble size distribution in the bed (Bai et al., 2005; Clark et al., 1996; Clark and Turton, 1988; Liu and Clark, 1995; Liu et al., 1998, 1996; Santana et al., 2006; Sobrino et al., 2009; Turton and Clark, 1989; Werther and Molerus, 1973a). These methods are based on the work of Werther and Molerus (1973a) and employ statistics and probability theory. However, all of these methods have their application boundaries and have often never been used with real fluidized beds. Moreover, according to Sobrino et al. (2009), most of these methods are numerically unstable and based on simplistic assumptions. Therefore, in order to investigate the applicability and accuracy of intrusive probes to determine the overall bubble size distribution in the bed from measured chord lengths, a Monte Carlo (MC) model is established. Already Liu and Clark (1995) and Clark et al. (1996) used Monte Carlo simulations to validate their backward transforms. However, their model only sampled bubbles around an imaginary probe. The entire bubble size distribution and geometric constraints in a planar cross-section of a fluidized bed have not been accounted for. The objective of this paper is to numerically investigate how the actual bubble size distribution in a fluidized bed is reflected by

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Nomenclature MC P (XjY) MaxEnt G(q; k) X Chord Bed Probe cdf NonPar PFM R r y a Q q n h Y dB H k i D

Monte Carlo probability of X given Y maximum entropy backward transform gamma distribution binary event variable ‘‘hit’’ pierced chord length overall bubble size in the bed pierced centerline bubble diameter cumulative distribution function non-parametrical backward transform pressure fluctuation measurement bubble radius (m) distance from bubble center (m) (pierced) chord length (m) proportionality constant (–) bubble wake factor (–) shape parameter (gamma) (–) number of data points (–) width of Parzen window (–) set of measured chord lengths (m) bubble diameter (m) Shannon entropy (bits) ordinal number (kth) (–) incremental number (–) column diameter (m)

intrusive probe measurements. In this respect, particularly the relationship between the measured chord length distribution of pierced bubbles and the overall bubble size distribution in the bed is elucidated by means of forward and backward transforms.

2. Theory From experiments with intrusive probes typically only the distribution of the pierced chord lengths is known. In order to obtain the more representative centerline bubble size distribution of bub-

Fig. 1. Intrusive probe intersecting a spherical bubble of radius R at a distance r from its centerline viewed (a) from the side and (b) from the top. The measured chord length y is typically smaller than the bubble size at its centerline.

L K z umf u0

a k

l / p b fit 0

mean std mf 0 ^ sv vol surf v –

sum of log-likelihoood (–) number of raw-moments (–) bed height (m) minimum fluidization velocity (m s1) superficial gas velocity (m s1) bubble shape factor (–) scale parameter (gamma) (–) shape parameter (Rayleigh) (–) bubble shape factor (–) probe bed fitted derivative arithmetic mean standard deviation minimum fluidization standardized, initial estimated surface–volume volume surface vertical mean value

bles pierced by the probe Pp(R) and the overall bubble size distribution in the bed Pb(R), a geometrical relationship between the chord length y and the centerline bubble radius R has to be known. This geometrical relationship is obtained by regarding a perfectly spherical bubble which intersects the probe tip at a distance r from its centerline (cf. Fig. 1). If bubbles are assumed to be uniformly distributed throughout a cross-section of the bed – or at least within a cylinder of radius R from the tip of the probe – the number of

Fig. 2. Bubble shape represented by (a) an ellipsoid and (b) a truncated ellipsoid. The ellipsoid is a function of the parameter a and the truncated ellipsoid is a function of parameters Q, a1, and a2, where usually a1 = a2 = a. For a perfectly spherical bubble, a and Q are equal to unity.

M. Rüdisüli et al. / International Journal of Multiphase Flow 44 (2012) 1–14

bubble centers passing through a small annulus of width dr and radius r form the probe tip increases proportionally to r (Clark and Turton, 1988). Hence, the conditional probability P(rjR) for the distance between the center of the bubble and the probe tip is in direct proportion to r. The proportionality constant a is evaluated by integrating from 0 to R and stipulating that the integral must equal unity. Hence, for a spherical bubble a is calculated as

 PðrjRÞ ¼

ar

if 0 6 r 6 R

0

otherwise

!

Z

R

ar dr ¼ 1 ! a ¼ 2=R2

ð1Þ

0

The chord length y can now be calculated from trigonometric relationships

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi yðrÞ ¼ 2 R2  r 2 or rðyÞ ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y2 R2  4

ð2Þ

Thus, the likelihood of finding a chord length y for a given bubble radius R (i.e., P(yjR)) is obtained by using a geometrical probability approach and Eq. (2) to yield

   dr  y PðyjRÞ ¼ PðrjRÞ  ¼ 2 dy 2R

ð3Þ

Normally, the shape of a bubble is not spherical, but rather ellipsoidal with an indent at the bottom (cf. Fig. 9) (Kunii and Levenspiel, 1991). This indent constitutes the wake of the bubble. Therefore, more realistic bubble shape models are used: An ellipsoid (=oblate spheroid) with one additional parameter a (cf. Fig. 2a) and a truncated ellipsoid with two additional parameters a and Q. For the truncated ellipsoid, Sobrino et al. (2009) even distinguish between a1 and a2. However, these two parameters are usually equal (cf. Fig. 2b). According to these bubble shape models, the spherical bubble is a special case with a = 1 and Q = 1. In the following, only the ellipsoidal model is regarded and Eq. (3) can be rewritten as

PðyjR; aÞ ¼

y 2a2 R2

ð4Þ

From Bayes’ theorem on conditional probabilities and the law of total probability, it follows that the probability distribution of all possible chord lengths P(y) is

PðyÞ ¼

Z

1

PðyjRÞPp ðRÞ dR

ð5Þ

0

Typically, P(y) is known from measurements and P(yjR) is known from the geometrical relationship in Eq. (4), thus only the centerline bubble size distribution of pierced bubbles Pp(R) is unknown. Eq. (5) is known as a Fredholm integral equation of the first kind and usually ill-conditioned, i.e., small changes in the coefficient matrix of the linear system of equations results in a large change of the solution. For this purpose, numerical (Clark and Turton, 1988; Turton and Clark, 1989), analytical (Clark et al., 1996; Liu and Clark, 1995), non-parametrical (Liu et al., 1996; Santana and Macfas-Machfn, 2000), and maximum entropy (Santana et al., 2006; Sobrino et al., 2009) backward transforms are presented in the literature. All of these backward transforms have their advantages and disadvantages. Moreover, there are several assumptions which must be fulfilled, often in an oversimplified and unrealistic manner. Most notably, it is assumed that bubbles are uniformly distributed throughout the cross-section, i.e., although bubbles are considered heterogeneous in size, they are only considered homogeneous in space (Liu et al., 1998). For most fluidized bed reactors this assumption is oversimplified (Werther, 1974a) and an extension for non-uniformly bubbling systems needs to be made. Such a non-uniform approach is presented in Liu et al. (1998), however, also this approach is rather unstable and relies on a priori knowl-

3

edge of the bubble shape and the spatial distribution of bubbles in the cross-section (e.g., bivariate normal, etc.). The oldest, yet least stable approach is the numerical backward transform which is, moreover, cumbersome in its implementation and widely dependent on the discretization of y and the sample size (Sobrino et al., 2009). Therefore, this approach is regarded obsolete and not further discussed in this paper. For more information on the numerical approach refer to Clark and Turton (1988), Turton and Clark (1989). The analytical approach can be employed, if the type of the bubble size distribution in the bed is known or a priori fixed. Typical bubble size distributions are the gamma, Rayleigh, or log-normal distribution (Clark et al., 1996; Gheorghiu et al., 2003; Lim et al., 1990; Werther, 1974a). All of these distributions have in common that they are right-skewed. This right-skewness represents nonlinear bubble coalescence and splitting phenomena (Bai et al., 2005). The parameters of the distributions can be estimated by least square or maximum likelihood techniques. In this manner, Pfit(y) is fitted to the measured chord length distribution P(y). From the fitted distribution of the chord lengths, Liu and Clark (1995) derived the centerline bubble size distribution of bubbles pierced by the probe Pp(R) as

Pp ðRÞ ¼ aðPfit ð2aRÞ  2aR  P0fit ð2aRÞÞ

ð6Þ

where P 0fit is the first derivative with respect to y of the fitted distribution Pfit. In the work of Liu and Clark (1995), the gamma distribution with parameters q (shape parameter) and k (scale parameter)

Pðyjq; kÞ ¼

 x 1 xq1 exp  k k CðqÞ q

ð7Þ

and the Rayleigh distribution with parameter l

PðyjlÞ ¼

y

l2

  y2 exp  2 2l

ð8Þ

are used. Due to two adjustable parameters, the gamma distribution is more flexible and can be fitted more accurately. However, its numeric implementation in Eq. (6) is more demanding and less stable than the one-parametric Rayleigh distribution. Unfortunately, with experimental data, the type of the bubble size distribution is generally not known a priori. Therefore, a non-parametrical backward transform is presented by Liu et al. (1996). This non-parametrical approach is more versatile and powerful than the analytical one, since the nature of the underlying distribution does not have to be known a priori. In this manner, the measured chord length data set Y can directly be converted to the centerline bubble size distribution at the probe Pp(R). This is accomplished by employing the Parzen window function (Parzen, 1962) to smoothen the distribution of the chord length with a Gaussian kernel

" # n 1 P ðy  Y i Þ2 Pn ðyÞ ¼ pffiffiffiffiffiffiffi exp  2 nh 2p i¼1 2h

ð9Þ

where n is the number of samples in Y and h is the width of the Parzen window. Substituting Eq. (9) into Eq. (6) yields

" # n a P ð2aR  Y i Þ2 ffiffiffiffiffiffi ffi p Pp ðRÞ ¼ exp  2 nh 2p i¼1 2h " # n 2a2 R P ð2aR  Y i Þ2 þ 3 pffiffiffiffiffiffiffi ð2aR  Y i Þ exp  2 2h nh 2p i¼1

ð10Þ

A crucial point in the non-parametrical approach is the choice of the Parzen window width h. If h is too small, the estimated curve becomes spiky and irregular probability values arise. In turn, if h

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M. Rüdisüli et al. / International Journal of Multiphase Flow 44 (2012) 1–14

with ymean being the mean value and ystd the standard deviation of the sampled chord length data set Y. The latest method to calculate the bubble size distribution irrespective of an a priori bubble size distribution is the maximum entropy (MaxEnt) method presented by Santana et al. (2006) and Sobrino et al. (2009). The maximum entropy principle by Jaynes (1957) claims to establish a probability distribution on a minimum amount of a priori knowledge, and thus leads to the least biased estimate possible on the given information. In the case of finding the unknown centerline bubble size distribution at the probe Pp(R), the idea is to estimate a probability distribution subject to a given number of known raw-moments of the chord length distribution

is too large, the estimated curve becomes too flat and departs from the original distribution (Clark et al., 1996). In order to estimate h, an iterative maximum likelihood approach is used n max!P

L¼

logðP p ðRi ÞÞ

ð11Þ

i¼1

with Ri = yi/2a. An empirical equation to estimate h, which can also be used as the starting value in Eq. (11), is proposed by Clark et al. (1996) as

h

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ymean  ystd n1=5

ð12Þ

bubble center probe reactor wall

6

4

x [cm]

2

0

−2

−4

−6

−6

−4

−2

0

2

4

6

y [cm]

Fig. 3. 10,000 uniformly distributed bubble centers simulated by the Monte Carlo (MC) model described in Section 3. The horizontal black line in (b) reaching from the wall to the center of the cross-section (D/2) is the intrusive probe.

M. Rüdisüli et al. / International Journal of Multiphase Flow 44 (2012) 1–14

P(y). The sought function is the probability distribution that maximizes the Shannon entropy max!

H¼

Z

R

Pp ðRÞ lnðPp ðRÞÞ dR

ð13Þ

5

With Eq. (19) and the presented backward transforms, it is now possible to infer the unknown bubble size distribution in the bed Pb(R) from the experimentally accessible chord length distribution P(y).

0

while fulfilling the following k raw-moment constraints

Z

R

Rk Pp ðRÞ dR ¼ hRk i for k ¼ 1; . . . ; K

3. Simulations

ð14Þ

0

where hRki is the k-th raw-moment of R. An efficient and stable algorithm to solve this optimization problem is presented by Rockinger and Jondeau (2002) and discussed in van Erp and van Gelder (2008). It becomes obvious that the major challenge of the MaxEnt approach is to find the required raw-moments of the unknown distribution Pp(R) from the known raw-moments of the chord length distribution P(y). A procedure to find the respective raw-moments is described in Santana et al. (2006) as

hRk i ¼

 k n kþ2P Yi 2n i¼1 a

ð15Þ

Apparently, the mean (i.e., the first raw-moment) of the centerline bubble size distribution at the probe Pp(R) is a factor of 1.5 higher than the mean of the chord length distribution P(y). This is in agreement with experimental results obtained on nearly spherical bubbles by Kalkach-Navarro et al. (1993) and with a comparison of optical probe data and video recordings by Glicksman et al. (1987). For non-spherical bubbles, a factor of 1.744 is calculated by Karimipour and Pugsley (2011). The overall bubble size distribution in the bed Pb(R), which also includes bubbles which are not pierced by the probe, i.e., mainly small bubbles, can be related to the centerline bubble size distribution at the probe Pp(R) by means of Bayes’ theorem. The conditional probability that a bubble of radius R, which is uniformly distributed across the cross-section of a fluidized bed of diameter D, is hit (? binary event variable X) by a probe in the center of the bed (=D/2) is given by

( PðXjRÞ ¼

R2 ðD=2RÞ2

if R 6 D=4

1

otherwise

ð16Þ

which is an extension of

PðXjRÞ ¼

R2 R2max

if R 6 Rmax  D=2

ð17Þ

as proposed by Werther (1974a), Liu and Clark (1995). Eq. (17) is only valid if the largest horizontal bubble radius in the bed Rmax is much smaller than the bed diameter D and bubbles are only found in the center of the bed. This is reportedly the case in the study of Werther (1974a), where bubble sizes have been evaluated at 8 cm and 20 cm above the distributor in a 20 cm ID and 100 cm ID bed, respectively. In turn, if bubbles grow to sizes close to the bed diameter D and geometrical interactions with the wall can no longer be neglected, Eq. (16) must be applied. From Eq. (16) and Bayes’ theorem on conditional probability, it follows that

P b ðRÞPðXjRÞ ¼ Pp ðRÞ PðRjXÞ ¼ R D=2 Pb ðRÞPðXjRÞ dR 0

ð18Þ

where P (RjX) is indeed the same as Pp(R), i.e., the probability that a bubble has centerline size R given it is pierced by the probe (=X). Finally, the inverse, Pb(R) can be found as

Pb ðRÞ ¼ R D=2 0

P p ðRÞ=PðXjRÞ ½Pp ðRÞ=PðXjRÞ dR

ð19Þ

Monte Carlo (MC) simulations are used to find the probability distribution of a random variable by numerically solving a non-linear problem which is otherwise analytically not solvable or only solvable with great difficulty. The idea is to randomly pick sets of input parameters of the problem from given sample probability distributions and to approximate the solution by repeatedly solve the problem for a large number of times (typically more than 1000-times). In this paper, a MC model is established to simulate rising bubbles hitting an immersed probe in a circular fluidized bed crosssection of D = 145 mm diameter. By default, the MC model randomly samples 10,000 spherical bubbles (a = 1) with a diameter taken from a gamma distribution with q = 7 and k = 0.5 (?G(7; 0.5)). The parameters q and k of the gamma distribution are chosen such that the sampled bubble sizes are representative of a real bubble size distribution and such that the likelihood to sample a bubble size larger than the bed diameter D is minimal. Theoretically, for an infinite number of samples, every gamma distribution will produce bubbles larger than D, since the gamma distribution ranges from 0 to plus infinity. Bubble centers are uniformly distributed across the cross-section of the bed and have to comply with the geometrical boundary condition that bubbles must not cross the circumference of the circular bed. Moreover, all bubbles are assumed to rise in a perfectly vertical direction. As a further default, the radial distance of the probe tip from the center of the bed and the diameter of the probe are a priori fixed as 0 mm, i.e., the probe tip is in the center of the bed and infinitesimally small. As extensions, the MC model can simulate various other bed configurations: Ellipsoidally shaped bubbles, other sample distributions (e.g., Rayleigh), alternative probe positions, etc. After the simulation of all bubbles in the bed, the chord length and the centerline bubble diameter of each bubble pierced by the probe is evaluated. This forward transform is made for the default case described above and simulations with alternative sample bubble size distributions. Moreover, a non-uniform spatial bubble distribution with more realistic probe diameters and other radial probe positions are evaluated. The obtained chord length distribution P(y) is further analyzed with respect to the backward transforms described in Section 2, namely the analytical (Gamma and Rayleigh), the non-parametrical (NonPar), and the maximum entropy (MaxEnt) approach. These backward transforms are in particular assessed with respect to their practical applicability and numerical stability. 4. Results 4.1. Forward transforms 4.1.1. Default case Fig. 3a shows the scatter plot of 10,000 uniformly distributed bubble centers sampled from the MC model with the sample bubble size distribution G(7; 0.5). The corresponding 3-D histogram is given in Fig. 3b. Apparently, in the middle of the bed, the bubble centers are (nearly) uniformly distributed, whereas towards the walls of the cross-section the density of bubble centers decreases radially. This decrease can be explained by the fact that only small bubbles can reach the outer regions of the circular cross-section,

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M. Rüdisüli et al. / International Journal of Multiphase Flow 44 (2012) 1–14

9

0.35

40

0.2

MC simulation Mean (MC)

0.15

60

50

8

Shape parameter q

pdf [cm−1]

0.25

20

MC simulation Mean (MC) Theory (Eq. 5) → P(y) MC simulation Mean (MC) Theory (Eq. 18) → Ppr (dB)

0.3

10

B

30

G(7;0.5) → Pbed (d )

40

7

30

20

10 6

20

10

5

4

0.1

10

G(7;0.5) 3

0.05

0.3

0.4

0.5

0.6

0.7

Scale parameter λ 0 0

2

4

6

8

10

12

14

Chord length/Bubble diameter [cm]

G(7;0.5) G(3;0.25) G(3;0.75) G(9;0.25) G(9;0.75)

1

pdf [cm−1]

0.8

20

30

40

50

60

Fig. 6. Percentage of pierced bubbles for different sample gamma distributions.

9 G(7;0.5)

1.15

1.05

1.

1

8

Shape parameter q

Fig. 4. Outcome of the MC simulation from Fig. 3. The overall bubble size distribution in the bed (blue line, ), the chord length distribution (red line, o) and the (vertical) centerline bubble size distribution (black line, +) of bubbles pierced by the probe. Along with the distributions from the MC model also the corresponding theoretically calculated distributions are displayed. Vertical lines represent the mean values of the distributions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

10

1.05

7

1.1

5

1.0

1

6

1

1.05

0.95

1

5

0.95 1

0.9

0.95

0.9

4

0.85

0.9

0.6

0.85

0.85

3

0.3

0.4

0.8

0.5

0.6

0.7

Scale parameter λ

0.4 0.8

0.2

0.85

0.9

0.95

1

1.05

1.1

1.15

mean("Bed") / mean("Chord") 0 0

5

10

15

Bubble diameter [cm]

1.5

5

G(7;0.5)

4

3

6

8

4

3 6

2

Shape parameter q

2 5

7

4

3

5

2

4

3

1 3

0.4

0.5

0.6

1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5

2 0.3

mean("Bed") / mean("Chord")

1.4

9

0

0.7

Scale parameter λ

Fig. 5. (a) Sample gamma distributions of the bubble diameter in the bed and (b) their mean bubble diameter (in centimeters).

0.5

1

1.5

2

2.5

3

μ

Fig. 7. Ratio of the chord length mean (‘‘Chord’’) and the mean of the overall bubble size in the bed (‘‘Bed’’) for different sample (a) gamma distributions and (b) Rayleigh distributions.

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M. Rüdisüli et al. / International Journal of Multiphase Flow 44 (2012) 1–14

4.1.2. Influence of the sample distribution In order to see the influence of the sample distribution on the ratio of the chord length mean and the mean of the overall bubble size distribution in the bed, MC simulations are repeated by varying the parameters of the gamma distribution: 3 6 q 6 9 and 0.25 6 k 6 0.75. A graphical representation of the extremes of these additional sample (gamma) distributions is shown in Fig. 5a along with the default case G(7; 0.5). Fig. 5b shows the corresponding mean bubble size. Fig. 6 displays the percentage of pierced bubbles for all simulated sample (gamma) distributions. From Fig. 5b it follows that if the mean bubble size increases, also the percentage of pierced bubbles increases. Moreover, Fig. 6 shows that for most of the bubble size distributions in the bed, only a small portion of the bubbles is effectively pierced by the probe, thus most bubbles are missed by the probe. This fact endorses the use of the MC model to investigate the relations between the bubble size distribution in the bed Pb(dB) and the pierced chord length distribution of the probe P(y). Fig. 7a shows the ratio between the mean chord length of pierced bubbles and the mean bubble size of all bubbles in the bed simulated from the gamma distributions. Evidently, the influence of q (=broadness of the distribution) is larger than that of the scale parameter k. Generally, the ratio is close to unity for a large number of sample distributions. Thus, the finding from Fig. 4 that the mean chord length is a good approximation of the mean bubble size in the bed can be corroborated. In other words, if the mean chord length is taken as the representative bubble size in the bed, the relative error from Fig. 7a is generally less than 20% and no cumbersome backward transforms have to be applied (cf. Section 4.2). Also, if the Rayleigh distribution from Eq. (8) is taken as the sample distribution and its parameter l is varied between 0.25 and 3, the ratio between the mean chord length and the mean bubble size distribution in the bed is close to unity for a wide range of l (cf. Fig. 7b). Therefore, if the bubble size distribution is assumed to be wide enough, this finding is independent of the underlying distribution. The situation is different for narrow distributions such as if all bubbles have the same size, then an error of approximately 33% is made. However, for real fluidized beds, the bubble size distribution is typically wide and within the ranges of the investigated gamma or Rayleigh distributions Clark et al. (1996); Bai et al. (2005). 4.1.3. Influence of the bubble shape Since non-spherical bubbles are the normal case in real bubbling fluidized beds, in this subsection the influence of the bubble shape on the estimation of a representative bubble size is elucidated.

1

0.5

0.9 0.8

Bed height [m]

0.85

0.7 0.8

0.65

0.65

5

0.7

0.7

0.7

0.7

0.75

0.4

0.7

0.75

0.1

0

0.85

0.8

5

0.9

0.9

0.95

0.8

0.1

0.65

0.85

0.8

0.2

0.75

0.8

0.75

0.3

0

0.9 5 0.6

0.6

0.6

0.6 0.5

0.55

5

0.7

0.95

6

0.

0.5

0.65

while larger bubbles are geometrically constrained to the center of the bed. Fig. 4 shows the corresponding pierced chord length distribution P(y), the centerline bubble size distribution of pierced bubbles Pp(dB), and the overall bubble size distribution in the bed Pb(dB). Fig. 4 further shows the theoretically calculated Pp(dB) from Eq. (18) and P(y) from Eq. (5) as well as the sample distribution G(7; 0.5). The theoretically calculated distributions (dashed lines) match very well the corresponding distributions from the MC simulation (=symbols). Hence, the MC model and the theory from Section 2 match. Fig. 4 also graphs the mean values of the three simulated distributions (? thick vertical lines). Apparently, the mean of the chord length distribution P(y) and the mean of the overall bubble size distribution in the bed Pb(dB) are almost identical. This finding, however, must be corroborated with further simulations in the following subsections.

0.9 0.95

0.2

0.3

0.4

0.5 α [−]

0.6 0.55 0.5

Excess gas velocity u0 − umf [m/s] Fig. 8. Ellipsoidal bubble shape factor a as a function of the bed height z and the excess gas velocity u0  umf according to the model of Werther (1976). A perfectly spherical bubble has a = 1.

According to Werther (1974a), the shape of an ellipsoidal bubble at a certain bed height z can be expressed by the empirical ellipsoid shape factor a

aðzÞ ¼ f1  0:3 exp½8ðu0  umf Þg expð/zÞ

ð20Þ

with

/ ¼ 7:2ðu0  umf Þ exp½4:1ðu0  umf Þ

ð21Þ

Fig. 8 displays a from Eq. (20) as a function of the bed height z and the excess gas velocity u0  umf. Apparently, the bubble shape factor is usually in the range between 0.6 and 0.9. Almost spherical bubble can only be found in the very bottom of the bed and for relatively large excess gas velocities. The higher up in the bed and the lower the excess gas velocity, the more flattened the bubble turns. In the previous section, it has been shown that the mean pierced chord length is a good approximation of the mean bubble size in the bed, irrespective of the underlying bubble size distribution in the bed. If the assumption of Werther (1974a) that all rising bubbles within a local array of the bed have roughly the same bubble shape factor a (cf. Fig. 8) holds true, then the above made approximation also holds true for ellipsoidal bubbles. This can be

Fig. 9. Fit of a truncated ellipsoid to a X-ray photograph of a real fluidized bed bubble and conversion of the truncated ellipsoid to a ellipsoid having the same volume with bubble shape factor a (adapted from Kunii and Levenspiel (1991)).

8

M. Rüdisüli et al. / International Journal of Multiphase Flow 44 (2012) 1–14 Table 1 Mean chord length (‘‘Chord’’) and mean bubble size in the bed (‘‘Bed’’) determined from different dimensions k of the bubble according to Eq. (28). The underlying bubble size distribution is simulated by the default case MC model.

Factor for equivalent bubble diameter [−]

1.8 Volume equivalent Surface equivalent Surface−volume equivalent

1.7 1.6

Mean determined w.r.t. (cm)

Chord

Bed

D (%)

1.5

Diameter (k = 1) Surface (k = 2) Volume (k = 3)

3.350 3.737 4.090

3.510 3.747 3.981

4.6 0.3 2.7

1.4

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi! a2 1 þ 1  a2 p pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln  ¼ 0:5 þ 4 1  a2 1  1  a2

1.3 1.27 1.2

or the surface–volume equivalent diameter, which is the diameter of a sphere having the same external-surface-area-to-volume ratio as the non-spherical shape (Yang, 2003)

1.1 1 0.5

0.6

0.7

0.8

0.9

1

dB;sv ¼ 1  dB;v

Bubble shape factor α [−] Fig. 10. Factor to convert the mean (vertical) chord length to a representative equivalent bubble diameter. Example: A bubble with shape factor a = 0.7 and mean =2 cm, would have a volume equivalent diameter chord length y dB,vol=1.27  2 cm = 2.54 cm.

explained by the fact that with an ellipsoid all vertical dimensions are scaled proportional to the factor a (cf. Fig. 2a). Moreover, the above made finding is also valid for truncated ellipsoids (cf. Fig. 2b), since a truncated ellipsoid can be converted to a volumetrically identical ellipsoid with bubble shape factor a according to the formula (Sobrino et al., 2009).

a¼

a2 Q ð3  Q 2 Þ þ 2a1 4

ð22Þ

A graphical illustration of this numerical transformation of a truncated ellipsoid – which is visually fitted to an X-ray photography of Rowe et al. (1965) – is shown in Fig. 9. It can be seen that the corresponding bubble shape factor is 0.78, which is in a typical range of Fig. 8. This is an important finding, since it allows using the approximation of the mean bubble size in the bed by the mean chord length regardless of the bubble shape, as long as the bubble shape can be represented by either an ellipsoid or a truncated ellipsoid. It must be noted that the mean chord length is generally only an approximation of the vertical dimension of the bubble. Even though many literature correlations approximate the bubble size as the vertical bubble diameter (Karimipour and Pugsley, 2011), it is often desired to have a more generic and more representative measure of the bubble size which can be used if bubbles of different shapes have to be compared.  is taken as a good If a is known and if the mean chord length y representation of the mean (vertical) bubble size in the bed dB,v, i.e., the following approximation is true

 dB;v  y

ð23Þ

then the volume equivalent bubble diameter dB,vol (Santana et al., 2006) of a sphere having the same volume as the non-spherical bubble can be calculated with

dB;v ol ¼ a2=3  dB;v

ð24Þ

For chemical processes in a fluidized bed, rather the mass transfer between the bubble phase and the dense phase is of interest, hence the surface equivalent diameter (Santana et al., 2006)

dB;surf ¼ 1=2 a1  dB;v with

ð26Þ

ð25Þ

ð27Þ

is sought. Geometrical relationships and the derivations of Eqs. (24)–(27) can be found in Santana et al. (2006). Fig. 10 shows how the measured mean chord length can be converted to the various equivalent bubble diameters. In summary, the following procedure has to be applied if bubbles are measured with intrusive probes and a representative bubble size is sought: 1. Pierced chord lengths are measured and their mean is assumed to be the mean (vertical) bubble size in the bed. 2. The bubble shape factor a is approximated, e.g., from Fig. 8 or from an X-ray image. 3. Depending on the use of the bubble size, Eqs. (24)–(27) are used to obtain a representative equivalent bubble diameter. In this respect, it is important to know that with intrusive probes the mean bubble size is always determined from the linear dimension (=vertical length scale) of the bubble. However, if the mean bubble size is determined from other measurement methods (e.g., radiography (Rowe, 1976), pressure fluctuations measurements (van der Schaaf et al., 2002), etc.), which are based on other  2 physical dimensions of the bubble, i.e., the bubble surface  dB  3 or the bubble volume  dB , then the obtained bubble sizes cannot be compared. As an illustrative example of these relations, a set of n = 1000 spherical (a = 1) bubble diameters and the corresponding set of chord lengths Y are simulated from the default case G(7;0.5) MC model in Section 4.1.1. Subsequently, the corresponding means ) are calculated based on the bubble (e.g., the mean chord length y diameter (k = 1), the bubble surface (k = 2), and the bubble volume (k = 3) according to the formula

ðkÞ ¼ y



n 1P Yk n i¼1

1 k

ð28Þ

In Table 1, the obtained mean chord lengths and the mean bubble sizes are listed. Apparently, the mean chord length and the mean bubble size always correspond to each other, independent of k. Thus, no matter how the mean is determined (i.e., volume, surface, or diameter), the ratio between the mean chord length and the mean bubble size in the bed match within an error of ±5%. This coincidence occurs due to the two counteracting mechanisms: On the one hand, large bubbles are more likely hit by the probe and, on the other hand, pierced bubbles feature a chord length most likely smaller than their (vertical) centerline bubble diameter. Contrarily, the obtained means vary considerably if they are based on the bubble diameter (=3.510 cm) or the bubble volume (=4.090 cm), even though the same underlying bubble size

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M. Rüdisüli et al. / International Journal of Multiphase Flow 44 (2012) 1–14

bubble center probe

6

reactor wall 4

x [cm]

2

0

−2

−4

−6

−6

−4

−2

0

2

4

6

y [cm]

Fig. 11. 10,000 Bubble centers simulated with the Monte Carlo model. The bubble size is sample from a gamma distribution G (7; 0.5) and the location of the bubble center is non-uniformly distributed in the cross-section, i.e., more bubbles are sampled from the center of the bed. In contrast to Fig. 3, the (optical) probe has a more realistic diameter of 3.6 mm.

distribution is used. Therefore, it should be kept in mind that bubble sizes obtained from different measurement techniques cannot directly be compared. 4.1.4. Influence of the radial probe position If bubbles are uniformly distributed across the bed cross section, except for the wall region, the radial positioning of the probe tip in the bed does not have a large influence on the number and

size of pierced bubbles (cf. Fig. 3b). However, from literature it is known that bubbles are often not uniformly distributed across the cross-section of a fluidized bed (Liu et al., 1998; Werther, 1974a). This is explained by the fact that coalescence statistically drives bubbles towards the center of the bed as they rise (Lim et al., 2007). Therefore, the MC model is re-run with bubble centers more often simulated in the center of the bed (cf. Fig. 11). Moreover, the diameter of the probe is set to a more realistic 3.6 mm.

10

M. Rüdisüli et al. / International Journal of Multiphase Flow 44 (2012) 1–14 0.35

0.35 G(7;0.5) → Pbed (dB)

pdf [cm−1]

0.25

MC simulation Mean (MC)

0.2

0.15

0.3

0.25

pdf [cm−1]

MC simulation Mean (MC) Theory (Eq. 5) → P(y) MC simulation Mean (MC) Theory (Eq. 18) → Ppr (dB)

0.3

0.2

0.15

0.1

0.1

0.05

0.05

0

0 0

2

4

6

8

10

12

0

14

2

0.35

0.35

0.3

0.3

0.25

0.25

pdf [cm−1]

pdf [cm−1]

4

6

8

10

12

14

12

14

Chord length/Bubble diameter [cm]

Chord length/Bubble diameter [cm]

0.2

0.15

0.2

0.15

0.1

0.1

0.05

0.05

0

0 0

2

4

6

8

10

12

14

Chord length/Bubble diameter [cm]

0

2

4

6

8

10

Chord length/Bubble diameter [cm]

Fig. 12. Outcome analogs to Fig. 4 of MC simulations with a non-uniform spatial bubble distribution (cf. Fig. 11) for different positions of the probe tip relative to the bed diameter D.

This implies that only bubbles which intersect the full circumference of the probe are allowed for evaluation. Eventually, the probe tip is positioned at D/2, D/3, D/4, and D/5, measured from the central axis of the column. According to Table 2, the number of pierced bubbles reduces drastically, as the probe tip is moved more towards the wall: From 31% in the center of the bed (=D/2) to merely 6% at D/5. By looking at Fig. 12, it seems that the match of the sampled and the theoretically calculated distributions of the chord length and the centerline bubble size distribution at the probe, is even better at D/3 Table 2 Fraction of bubbles pierced by the probe depending on the position of the probe tip for the MC simulation in Fig. 11. Probe tip

Pierced bubbles

%

D/2 D/3 D/4 D/5

3091/10,000 1449/10,000 856/10,000 582/10,000

31 14 8.6 5.8

compared to the default position of D/2. This has the additional advantage that the hydrodynamics in the bed are less disturbed. Therefore, a probe position at D/3 seems to be advisable. Also, for this non-uniform spatial bubble distribution in the bed, the mean of the chord length and the mean of the overall bubble size in the bed match considerably well. Therefore, the approximation of the mean bubble size by the mean chord length does neither depend on the employed sample distribution, nor on the bubble shape as long as it can be approximated by a (truncated) ellipsoid, nor on the spatial bubble distribution, nor on the exact positioning of the probe tip. As a comparison to this simple and straightforward approximation of the bubble size, the backward transforms from Section 2 are evaluated in the next section.

4.2. Backward transforms 4.2.1. Analytical approach The analytical backward transform, as described in Section 2, is typically done by fitting the measured chord length distribution to

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M. Rüdisüli et al. / International Journal of Multiphase Flow 44 (2012) 1–14 0.35

(a)

0.3

MC simulation → P(y) Rayleigh fit Mean (Rayleigh) Gamma fit Mean (Gamma)

pdf [cm−1]

0.25 0.2

Mean (MC) → P(y) Mean (MC) → Ppr (dB)

0.15 0.1

Mean (MC) → Pbed (dB)

0.05 0

(b)

0.3

MC simulation → Ppr (dB) Gamma fit Mean (Gamma) Rayleigh fit Mean (Rayleigh)

pdf [cm−1]

0.25 0.2 0.15 0.1 0.05 0 −0.05

(c)

0.3

MC simulation → Pbed (dB) Rayleigh fit Mean (Rayleigh)

pdf [cm−1]

0.25 0.2

gamma). For the underlying Rayleigh distribution, the calculated centerline bubble size distribution is less peaked (=platykurtic) than the sampled one. On the other hand, for the underlying gamma distribution, the fit is again slightly too right-skewed and features unrealistic negative values for small bubble diameters. These negative values result from numerical instabilities, since for small bubble sizes the second term in Eq. (6) (=first derivative) becomes larger than the first one. If again only the mean of the calculated distributions (i.e., Rayleigh and gamma) is regarded, both match the mean of the MC sampled distribution almost perfectly. Due to the negative values for small bubble sizes in the gamma distribution, Eq. (19) cannot be used to calculate the overall bubble size distribution in the bed (?Pbed(dB)). Therefore, the analytical backward transform with a gamma distribution is not suitable for practical applications. In turn, the Rayleigh distribution does not show these negative values and can be used to estimate Pb(dB), such as depicted in Fig. 13c. However, the overall bubble size distribution in the bed obtained from the underlying Rayleigh fit does not match the sampled distribution G(7; 0.5) too well. Also their mean values are off by about 20%. Therefore, the analytical backward transform with a Rayleigh distribution, albeit numerically stable, only gives a rough approximation of the overall bubble size distribution in the bed. However, this can also be achieved by just taking the mean of the sampled chord length distribution as the representative bubble size in the bed (cf. Section 4.1.1).

0.15 0.35

0.1

(a)

0.3

0.05

MC simulation → P(y) MaxEnt fit Mean (MaxEnt)

4

6

8

10

12

14

Chord length / Bubble diameter [cm] Fig. 13. Analytical backward transform with Rayleigh and gamma distribution: (a) Pierced chord lengths simulated by the MC model (o) and the fitted Rayleigh (full line) and gamma distribution (dash dotted line) including their mean values (vertical lines). Mean values from the MC simulation (? mean (MC)) are included in all subfigures. (b) Centerline diameter of bubbles pierced by the probe from the MC simulation (+) and calculated from the fitted Rayleigh (full line) and gamma distribution (dash dotted line). (c) Overall bubble size distribution in the bed from the MC simulation () and the fitted Rayleigh distribution (full line).

a Rayleigh or gamma distribution and analytically calculate the overall bubble size distribution in the bed. Fig. 13 shows the whole procedure of the analytical backward transform with a Rayleigh distribution (full line) and a gamma distribution (dash dotted line) to convert the MC sampled chord length distribution (Fig. 13a) via the centerline bubble size distribution at the probe (Fig. 13b), to the overall bubble size distribution in the bed (Fig. 13c). The maximum likelihood fit of the Rayleigh and the gamma distribution to the MC simulated chord length distribution (?P(y)) in ^ ¼ 2:64 0:07 (Rayleigh) as well as Fig. 13a yields parameters of l ^ ¼ 3:41 0:27 and ^ q k ¼ 0:97 0:08 (gamma) with 95% confidence interval, respectively. Surprisingly, the Rayleigh distribution fits the sampled data somewhat better than the gamma distribution. This is even more surprising since the data is sampled from a gamma distribution and the Rayleigh distribution features only one adjustable parameter l. The fitted gamma distribution is in particular slightly more right-skewed than the sampled and the Rayleigh one. However, the right skewness is a typical feature of the gamma distribution and in general also the gamma distribution yields a satisfactory fit with the sampled chord length distribution. Moreover, the mean values of both Rayleigh and gamma distribution perfectly match the mean of the sampled chord length distribution. The agreement of the calculated (Eq. (6)) and the sampled centerline bubble size distribution (?Ppr(dB)) in Fig. 13b is fairly good for both underlying chord length distributions (Rayleigh and

Mean (MC) → P(y) Mean (MC) → Ppr (dB)

0.2

Mean (MC) → Pbed (dB)

0.15 0.1 0.05 0

(b)

0.3

MC simulation → Ppr (dB) MaxEnt fit Mean (MaxEnt) NonPar fit Mean (NonPar)

0.25

pdf [cm−1]

2

0.2 0.15 0.1 0.05 0

(c)

0.3

MC simulation → Pbed (dB) MaxEnt fit Mean (MaxEnt)

0.25

pdf [cm−1]

0

pdf [cm−1]

0.25

0

0.2 0.15 0.1 0.05 0 0

2

4

6

8

10

12

14

Chord length / Bubble diameter [cm] Fig. 14. Backward transform with the non-parametrical (NonPar) and maximum entropy (MaxEnt) approach: (a) Pierced chord lengths simulated by the MC model (o) and the fitted MaxEnt distribution (dash dotted line) including their mean values (vertical lines). Mean values from the MC simulation (? mean (MC)) are included in all subfigures. (b) Centerline diameter of bubbles pierced by the probe from the MC simulation (+) and calculated from NonPar (full line) and MaxEnt (dash dotted line). (c) Overall bubble size distribution in the bed from the MC simulation () and the fitted MaxEnt distribution (dash dotted line).

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M. Rüdisüli et al. / International Journal of Multiphase Flow 44 (2012) 1–14 9

1.

Shape parameter q

5

2

2

1.

9

2

2

G(7;0.5)

G(7;0.5) 3

3 0.3

0.4

0.5

0.6

0.3

0.7

0.4

Scale parameter λ

1.4

1.4

1.5

2

8

2.

3

2.8 2.6

0.3

0.7

0.4

0.5

4

3

3.2

2.

2.8

6

2

3

3

2.

Shape parameter q

8

1.

2

2.4

1.6

2.2

G(7;0.5)

2.4

2

2.1

2

3

Scale parameter λ

1.8

2.

6

5

G(7;0.5)

0.6

2

1.8

3

3 0.5

2

2.4 2.6

2.

6

4

4

1.6

1.9

2.4

3

8

2.2

2.

4

7

2.

2

5

1.4

1.8

3

6

0.4

1.7

2.2

6 8

1.

2

2.2

4

0.3

1.6

2.6

1.

7 2.

0.7

3

8

8

3

1.5

3

1.8

2

2.4

1.45

9

2.2

9

1.35

0.6

2.

1.3

0.5

Scale parameter λ

2.8

1.25

Shape parameter q

6

2

4

1.5

8

1.

Shape parameter q

5

1 .5

1.5

1.5

1. 6

1.8

1.5

2

7

4

2

7

1.

1.

5 1 .4

1.5

1.

35

2

7 1.

1.

1.5

5

4

3

9 1.

6

45 1.

1.5

1 .4

7

1.

4

1.5

8

1.6

1.9

5

8

1.8

2

1 .3

1.5

5 1.4

9

0.6

0.7

Scale parameter λ

2.4

2.6

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

Fig. 15. Ratio of the first four raw-moments of the chord length and the centerline bubble size pierced by the probe for 100,000 simulated bubbles from different sample gamma distributions. In order to apply the maximum entropy backward transfrom, the ratios should be 1.5, 2, 2.5, and 3, respectively (Santana et al., 2006).

4.2.2. Non-parametrical and maximum entropy approach As opposed to the analytical backward transform, the non-parametrical (NonPar) and the maximum entropy (MaxEnt) backward transforms do not rely on an a priori probability distribution. Analog to the analytical approach in Fig. 13, Fig. 14 shows the backward transforms with the non-parametrical (full line) and the maximum entropy (dash dotted line) approach of the same MC sampled distributions of (a) the chord length, (b) the centerline bubble diameter at the probe, and (c) the overall bubble size in the bed. Since with the NonPar approach, the centerline bubble size distribution at the probe (?Ppr(dB)) is directly estimated from the chord length data set, no fitting of the chord length distribution is needed (cf. Section 2). Therefore, only the MaxEnt fit of the chord length distribution is shown in Fig. 14a. Due to the versatility of the MaxEnt approach, the match of the sampled and the MaxEnt fitted chord length distribution is very good. Only at their peaks, the MaxEnt fitted chord length distribution is more peaked than the sampled one. This misfit at the peak, however, could be improved if more than the default number of four raw-moment constraints would be used. Also, the mean of the sampled and the fitted distribution match perfectly.

Regarding the centerline bubble size distribution in Fig. 14b (?Ppr(dB)), a fairly good and an excellent agreement is found for the NonPar and the MaxEnt approach, respectively. Moreover, the mean of both approaches match the sampled mean almost perfectly. However, for small bubble sizes close to zero the NonPar approach shows an inconsistency. This inconsistency is due to a lower limit of validity of the NonPar approach. Below this lower limit, Eq. (10) becomes inappropriate to estimate the distribution of bubble sizes. For instance, if y = 0, Eq. (10) gives an infinite value, while the theoretical value is still finite (Clark et al., 1996). These numerical instabilities for small bubble sizes do not allow calculating the overall bubble size distribution in the bed (?Pbed(dB)). Moreover, the NonPar approach needs a large amount of data and large computation capacities to reliably obtain the bubble size distribution (Sobrino et al., 2009), thus also in this light the NonPar approach seems to be inappropriate. Unlike the NonPar approach, the MaxEnt approach yields a result for the overall bubble size distribution in the bed in Fig. 14c. Regarding only the mean, the match of the sampled and the MaxEnt fitted mean match perfectly. Therefore, the MaxEnt approach seems to be the most powerful and most accurate of all backward transforms employed in this study. However, also the MaxEnt

M. Rüdisüli et al. / International Journal of Multiphase Flow 44 (2012) 1–14

approach is prone to numerical instabilities in the conversion of Pp(dB) to Pb(dB) by means of Eq. (19). This numerical instabilities can be seen in Fig. 14c for small bubble diameters close to zero, where the calculated bubble size distribution surges to infinity. This may occur if the denominator in Eq. (16) becomes larger than the numerator. Since the MaxEnt approach relies on the raw-moments transformation given in Eq. (15), the ratio of the first four raw-moments of the sampled P(y) and Pp(dB) can readily be simulated with the MC model for alternative sample gamma distributions (cf. Section 4.1.2). From all of these sample gamma distributions, 100,000 bubbles are sampled and evaluated with respect to their raw-moment ratios. The corresponding results can be found in Fig. 15. As a comparison, the theoretical values from Eq. (15) should be 1.5, 2, 2.5, and 3 for the first, second, third, and fourth raw-moment, respectively. With most of the sample distributions, the ratio of the first rawmoment (=mean) agrees well with the theoretical value of 1.5. This has also been reported by Kalkach-Navarro et al. (1993), Glicksman et al. (1987). Also, the ratio of the second raw-moment (variance) is within an acceptable range of the theoretical value of 2 for all sample distributions. For the third and fourth raw-moment, the discrepancy with theory is more significant. While for several sample distributions the discrepancy is negligible, there is a significant discrepancy of 11%(=2.23) and 17%(=2.48), respectively, for the default case G(7; 0.5). Therefore, the MaxEnt approach yields good results, if only a small number of raw-moments (e.g., mean and variance) is considered. However, to retrieve information on higher order moments some caution has to be applied on the accurate raw-moments transformation. Moreover, due to the numerical instabilities and since the MaxEnt distribution does not have a closed-form analytical solution such as the Rayleigh or the gamma distribution, its application to retrieve the overall bubble size distribution in the bed may often not be straightforward.

5. Discussion From the analysis of the different backward transforms, only the analytical backward transform with a Rayleigh distribution yields a numerically stable result. Moreover, neither of the presented backward transforms yield indisputably good results. Encountered problems and simplistic assumptions of the backward transforms are

numerical instabilities,

lack of goodness-of-fit due to scarcity of adjustable model parameters,

necessary a priori knowledge of distributions (Rayleigh, gamma) and bubble shape,

assumed homogeneous bubble shape based on an ellipsoid model,

assumed uniform spatial bubble distribution across the bed cross-section. Due to the good agreement of the mean chord length and the mean bubble size in a bed with ellipsoidal bubbles (cf. Fig. 7) and due to the listed drawbacks of the backward transforms, the measured mean chord length of an intrusive probe can – for the sake of simplicity and parsimony – be taken as an appropriate approximation of the bubble size in the bed. This agreement is based on two counteracting mechanisms which allow for the fact that large bubbles are more likely hit by the probe while the measured chord length of the pierced bubble is most likely smaller than its (vertical) centerline diameter.

13

To a similar conclusion came Liu et al. (2010), who employed the median of the cumulative distribution function (cdf) of pierced bubble chord lengths as the representative bubble size in the bed. Their argumentation not to use any backward transform was that the shape of a bubble typically depends on interactions with other bubbles or with solids surfaces. Therefore, the assumption of bubbles having a single (ellipsoidal) shape – as presupposed by all backward transforms – does not account for the complex and irregular bubble shapes encountered in a real fluidized bed. Liu et al. (2010) also reasoned that the measured chord length typically overestimate the actual bubble size in the bed, since small bubbles frequently skirt the probes. However, this error would decrease with increased bed height and gas velocity, due to an increasing number of large bubbles in the bed. Although, according to Werther (1974a), the mean chord length does neither yield direct information on the overall bubble size distribution in the bed, nor on the mean bubble size, the MC simulations in this paper show that within an acceptable range of error, the mean chord length can indeed be taken as a representative measure of the mean bubble size in the bed. It must be noted that this simple approach imposes some degree of error on the interpretation of intrusive probe results. However, since none of the presented backward transforms clearly outperforms this simple approach, the interpretation of the mean chord length as the representative mean bubble size allows for a certain degree of consistency between different experimental settings. This also holds for the slugging regime and experiments with internals and elevated pressure, where the presented backward transforms are not applicable at all. 6. Conclusion In this paper, a Monte Carlo (MC) model is established to simulate how the actual bubble size distribution in the cross-section of a fluidized bed is reflected by intrusive probe measurements. Apparently, for ellipsoidal bubbles, the mean of the chord length is a decent approximation of the mean of the overall bubble size distribution in the bed. Surprisingly, under the assumption of a wide enough bubble size distribution, this finding does not depend on the underlying bubble size distribution. Moreover, this finding also holds for ellipsoidal bubble shapes, non-uniform spatial bubble distributions in the bed cross-section, and if the probe tip is located slightly off the central axis of the bed. Thus, a probe positioning at D/3 is generally more advantageous due to reduced interactions of the probe with the bed hydrodynamics. In the literature, there are several backward transforms based on conditional probability theory to convert a measured chord length distribution to the overall bubble size distribution in the bed. These backward transforms can be classified as analytical, non-parametrical, and maximum entropy methods. However, by means of the MC model, it is shown that all of these backward transforms are limited in their applicability: Due to strong simplifications and approximations of the underlying bubble characteristics (i.e., bubble shape, bubble rise pathways, etc.) and due to severe numerical instabilities, none of these backward transform has clearly outperformed the simple approach of just taking the mean chord length as a representative measure of the overall bubble size in the bed. Moreover, due to more consistency in the estimated bubble size, this approach can more readily be applied to compare bubble sizes from different fluidized beds conditions such as with internals and different pressures. Acknowledgments The authors thank Celia Sobrino from Carlos III University of Madrid, School of Engineering, for her support on the implementation

14

M. Rüdisüli et al. / International Journal of Multiphase Flow 44 (2012) 1–14

of the maximum entropy approach. For the funding of this project, ‘‘Verband Schweizerische Gasindustrie’’ (VSG/ASIG) is gratefully acknowledged.

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