diagnostic procedures for establishing the quality of fluidization of gas-solid systems

Diagnostic Procedures for Establishing the Quality of Fluidization of Gas-Solid Systems

S. C. Saxena N. S. Rao V. N. Tanjore Department of Chemical Engineering, The University of Illinois at Chicago Chicago, Illinois

• Fluidization experiments were conducted on two dolomite beds of average diameters 1490 and 2010 /xm in a 0.153 m square fluidized bed at ambient conditions. In particular, the pressure drops across the distributor plate and the test bed were measured as a function of fluidization velocity. In addition, temperature and pressure histories were recorded at different fluidization velocities. The data are employed to compute minimum fluidization velocity, bed voidage, relative heat transfer coefficient, variances in temperature and pressure fluctuation data, probability density function, skewness, kurtosis, autocorrelation function, and power spectral density function. These parameters and functions were assessed as to their potential to characterize the quality of fluidization and thereby establish a methodology of diagnostic procedures for such a task based on the knowledge of those statistical functions that have proven successful in this endeavor.

Keywords: fluidized bed, quality of fluidization, temperature and pressure probes, probability density function, power spectral density function

I N T R O D U C T I O N

Gas-solid fluidized bed contactors and reactors are frequently employed in industry, and the success of the process involved depends to a large extent on the quality of gas-solid contact and solids mixing. These two characteristics are invariably referred to by the term quality of fluidization. Further, the extent to which these features will be achieved depends on the nature of the bed fluidization. For example, in a discrete bubbling bed, gas bypassing and solids mixing will be less pronounced, but they will increase as the fluidization regime shifts to bubble coalescing and vigorously bubbling states. In a turbulently fluidized bed, the gas-solids contact and solids mixing will improve in comparison to a bubbling bed. Similarly, in a slugging bed, both gas-solids contact and solids mixing are poor. Thus the nature of fuidization controls whether the process details that may be desired in a fluidized bed are achieved. As is well known, the nature of bed fluidization is influenced by the operating conditions and reactor geometry. The present work is prompted to develop techniques that could assist in identifying the bed fluidization condition or regime.

It will be very useful if diagnostic techniques can be developed to estimate quantitatively the quality of flu-

idization that is achieved, as different hydrodynamic regimes are established either by changing the operating parameters such as gas velocity, temperature, and pressure or by choosing different system parameters such as the physical properties of the fluidizing gas and solid particles, the geometry of the reactor, and the nature of internals present in the bed. Saxena and Rao [1] proposed the use of a heat transfer thermocouple probe immersed in the bed as a sensitive tool to respond to the changing environment on its surface, which is characteristic of the gross hydrodynamic activity in the bed. They measured relative values of the local heat transfer coefficient and temperature history records in an air-glass bead system. This was related to the bubble or void renewal frequency, fvr , which in turn characteristically represents the quality of bed fluidization. Saxena and Rao [2] also proposed to employ a pressure-measuring transducer probe to register pressure fuctuations in the bed over a time period and infer from such records the values of the void or bubble renewal frequencies, fve, which are uniquely representative of the bed quality fluidization. It may be pointed out that the temperature- and pressure-measuring probes respond only to the fluidization condition that prevails in the bed close to their physical location. In a practical bed, multiple probes will have to be installed to interpret the

Address correspondence to Professor S. C. Saxena, Department of Chemical Engineering, The University of Illinois, P.O. Box 4348, Chicago, IL 60680.

Experimental Thermal and Fluid Science 1993; 6:56-73 © 1993 by Elsevier Science Publishing Co., Inc., 655 Avenue of the Americas, New York, NY 10010 0894-1777/93/$5.00

56

Establishing the Quality of Huidization 57

fluidization characteristics in different bed regions and their variation from one bed location to another. Saxena and coworkers [3, 4] demonstrated the potential of such approaches by examining fluidized beds that used nonspherical particles of sand [3]; coal, dolomite, and their mixtures [4]; and spherical particles of glass beads [1, 2, 5].

The present work is motivated by the success of these efforts [1-5] and further extends the scope of such measurements, particularly in relation to the application of coal combustion in fluidized beds. Hence we have considered dolomite beds of two different particle sizes, average diameters of 1490 and 2010/xm. According to the Saxena and Ganzha [6] powder classification scheme, both of these beds belong to group liB, and hence abrupt changes of physical characteristics associated with changes in group number will not be encountered here. This, however, is not considered to be a drawback as the aim of this work is to establish the potential of the new diagnostic procedures being investigated and to outline the procedures for the computation of statistical quantities and the way in which this information can be used to predict the nature of bed fluidization. These dolomite beds of relatively close size particles will provide an opportunity for a crucial check because both beds belong to the same fluidization group. Alternatively, they will reveal the sensitivity of various parameters to such differences in particle diameter.

The new procedures are based on a more elaborate analysis of the temperature and pressure history records and the computation of additional statistical functions, which may become the basis for quantitatively specifying the quality of bed fluidization. Hence it was necessary to record the temperature and pressure fluctuation data over a longer period (92 s) than in the earlier efforts (10 s). This necessitated upgrading the data acquisition and analysis system. The results taken at ambient conditions are reported here.

In future efforts, particles belonging to different groups in powder classification schemes [6, 40] will be investigated with the aim to establish limits that discriminate different groups from each other and the way in which fluidization behavior changes from one group to another.

EXPERIMENTAL SETUP

The 153 mm square fluidized bed is described in earlier papers [1, 7]. Very briefly, it has a 0.52 m high plenum chamber, 0.61 m tall test section, and 1.51 m long free- board section. The plenum chamber and the fluidized bed air distributor plates are equipped with fiat-top bubble caps screwed on in a square arrangement with 2.8 cm center-to-center spacings. The fluidizing air is supplied by two 18.65 kW compressors and is dried and filtered before entering the fluidized bed metering system, which com- prises several precision rotameters. The air exits from the bed into the atmospheric environment through an offgas system composed of two cyclones and a fabric filter.

The design details of the temperature-registering heat transfer probe are given by Saxena and Rao [1]. The probe is a cylindrical 40 mm long probe made of a 12.7 mm diameter brass tube with a 200 W calrod heater that is energized by a dc power source with a regulation of 0.01%. The probe is held in a cylindrical Teflon cap, which is held in the fluidized bed wall with O-ring seals. Three copper-constantan thermocouples are cemented on the

probe surface in milled grooves at distances of 5, 15, and 20 mm from its tip and angularly staggered by 120 °. The thermocouple located on the lateral side of the probe, 120 ° angularly staggered from the thermocouple located on the downstream side, is used to record the temperature history for a period of 92 s. The thermocouple has a time constant (response time) of 0.42 s, and thus it can respond to a signal frequency of 2.4 Hz most efficiently by registering 63.2% of the temperature change. Faster signal changes will be recorded with reduced efficiency. This is not considered to be a serious limiting factor for the present effort, as our primary goal is to test the proposed diagnostic procedures for their general appropriateness to detect fluidization quality.

The pressure-measuring transducer probe is described by Saxena and Rao [2]. The corrosion-resistant rugged stainless steel transducer is mounted in an aluminum cylindrical plug, 60 mm in diameter and 65 mm long, and communicates with the fluidized bed through a 6 mm diameter channel. The channel terminates with a fine wire mesh that restricts the entry of solid fines into the transducer housing. The transducer assembly is mounted flush with the fluidized bed wall with two O-ring seals. The temperature and pressure probes are installed across each other at an elevation of 280 mm above the bubble cap air distributor plate. The time constant of the transducer is 5 ms, and hence it can respond to pressure changes taking place with a frequency up to as high as 200 Hz.

The electric signals from both probes are measured by the data acquisition unit HP3852A, voltmeter HP44071A, compiler HP98563, monitor HP98786A, dual disk drive HP9122D, keyboard HP46021A, printer HP2225A, and HP color pro-plotter. This measuring system can display, record, and plot the temperature and pressure history records of the fluidized bed. The signals are recorded with a speed of about 11 Hz for a total of 999 readings in 92 s.

DATA ANALYSIS PROCEDURE

Pressure-fluctuation data have been used from time to time to predict the quality of fluidization in gas-solid systems [2, 8-13] and in gas-liquid systems [14-16]. In a similar manner, temperature fluctuation records have been studied in such reactors [1, 17, 18] to predict the varying hydrodynamic regimes. A brief discussion of the statistical functions that we plan to employ for the analysis and interpretation of temperature and pressure fluctuation data is presented below.

Let us assume that N instantaneous values of a fluctu- ating signal have been recorded at equal time intervals over a time period t. The sampling frequency f is given by the relation

N = f t (1)

If the ith signal value is represented by x i, then the mean, /x, of N such signal values is given by

1 N ~L : --~ E Xi (2)

i=l

/x represents a measure of the central tendency of the data [19]. The scatter of data from this mean value is referred to as the dispersion of the data, and its quantita- tive measure is the standard deviation tr, or variance ~r 2,

58 S.C. Saxena et al

defined such that

0 .2- - 1 N

( N - 1) y'~ ( x i - tx)2 i=1

0-2 is also referred to as the second moment about the mean, m 2. Here, 0-p is used for pressure fluctuation data and 0-~ for temperature fluctuation data.

In fluidized beds the pressure (or temperature) fluctuations are caused by bubble formation, bubble propagation in the bed, and the eventual eruption of bubbles at the bed surface [20, 21]. Increasing gas velocity above the minimum fluidization velocity will generally increase the bubble size, and larger pressure fluctuations will result. This will cause the standard deviation or variance to increase. When the gas velocity is increased beyond a critical value at which the bubbles have attained a maximum stable size, they start breaking into smaller bubbles. The pressure fluctuations are then reduced in magnitude, and smaller values of standard deviations are observed. On the other hand, the temperature fluctuations of an immersed surface in the reactor and the variance will both decrease with the increase in gas velocity. This is because of the improvement in solids movement. In any case, the change in the value of variance signals a change in the fluidization quality and in the hydrodynamic regime. This property can therefore be employed for regime characterization. However, the knowledge of variance in itself is not enough to understand the detailed nature of the bed hydrodynamics and its structure. For this reason we must look into the detailed nature of the fluctuations of temperature a n d / o r pressure signals. This is done by computing the probability density function and investigating its departure from the normal or Gaussian distribution in terms of skewness and kurtosis. A brief discussion and defining relations of these functions follow.

The probability density function (pdf) of an ergodic process gives the probability that the data will have a value within a defined range at a given instant. If N x is the number of data points in the range dx around x, then the probability density function p ( x ) , is given by [22]

p ( x ) = N , : / ( N d x )

The estimation of pdf is not unique because it depends on the number of class intervals selected for analysis and the width of those intervals. The selection of class interval is therefore important, and Simpson and Kafka [23] have established five criteria for its selection. The application of the knowledge of pdf to physical data is done to establish a probabilistic description of the instantaneous values of the data. The existence of the periodic phenom- ena can be confirmed by the saddle shape about the mean of the pdf. A bell-shaped pdf with a sharp peak will signify only one most probable value of the temperature or pressure signal, implying uniform fluidization behavior with a single source causing the fluctuations in signal of a constant frequency. The shape of a pdf also reveals the extent of the interference of the smaller fluctuations superimposed on the major fluctuation. This feature is investigated by computing the skewness and kurtosis.

Skewness is a measure of the symmetry of the distribution about the mean. For a symmetrical distribution, the mean, median, and mode (most frequently occurring value)

are identical. The more the mean moves away from the mode, the larger the asymmetry of skewness. The distance between the mean and the mode is Karl Pearson's basis

(3) for measuring skewness [23, Chapter 13]. Skewness is most often measured in order to compare distributions. How- ever, the same amount of skewness has different meanings in distributions with small variations and in distributions with large variations, which must be taken into account if one is to make valid comparisons between the skewness in two or more distributions.

Pearson's coefficient of skewness is defined as

m e a n - m o d e

s k p = standard deviation (5)

Skewness is best measured by moment statistics, which is based on the exact value of each observation. Absolute skewness of a distribution is given by the third moment about the mean. The third moment (m 3) is defined as

1 U

m 3 = -N i~=l ( x i _ /.£)3 (6)

Relative skewness ( a 3) is defined as

m3 m3 a3 -- or 3 m 3 / 2 (7)

For a normal distribution, a 3 = 0. If the distribution is skewed to the left, a 3 is negative; and if the distribution is skewed to the right, a 3 is positive. This information is helpful in identifying the nature of the secondary signals associated with the primary signal and hence the sources generating them. The above expression for relative skewness can be simplified for the purpose of numeri- cal calculations as

N U ( X i _ /z) 3

a3 = ( N - 1 ) ( N - 2 ) 2.~ ~3- (8) i=1

Another property used for the description and comparison of distributions is the peakedness of the distribution. Peakedness is measured in terms of kurtosis. From the

(4) standpoint of kurtosis the normal curve is mesokurtic, having intermediate peakedness. Flat-topped curves are called platykurtic, while pronouncedly peaked curves are called leptokurtic [23]. The measure of kurtosis (a 4) is the fourth moment of the data points about their mean ( m 4) expressed in dimensionless form as

a 4 = m 4 / o -4 (9)

For a normal distribution, a 4 = 3. For this reason, the coefficient of kurtosis is sometimes defined by a 4 - - 3 , which is positive for a lepokurtic distribution, negative for a platykurtic distribution, and zero for the normal distribution. For computational purposes [24], the expression for the coefficient of kurtosis is

N ( N + 1) X ( X i __ /./,) 4

a 4 -- 3 = ( N - I ~ N - - 2 ~ U - 3) y'~ o.4 i=1

3 ( N - 1 ) ( N - 1) - 0 o )

( N - 2 ) ( N - 3 )

Establishing the Quality of Fluidization 59

The magnitude of kurtosis therefore signifies the range of variation in the probable signal value, and its sign specifies the relationship to the normal distribution.

A function that relates the influence of values at any time over values at a future time is the autocorrelation function (acf). Any deterministic data will have an autocorrelation function that persists over all time displacements as opposed to random data, which diminish to zero over large time displacements. There are two ways to compute an autocorrelation function [22]. The first is the direct method, which involves the computation of average products among the sample data values, that is,

1 fo~X(t)x( t + "c)dt (11)

The second indirect method of computing an acf involves calculating an autospectral density estimate using fast Fourier transform (FFT) procedures and then computing the inverse transform of the autospectrum. The first method is easier to program and represents the more logical approach from the viewpoint of basic definition. The second approach takes advantage of the dramatic computational efficiency of FFT algorithms and hence is much less expensive to execute in terms of computer time. The first method is adopted here, and the computational procedure is briefly outlined in the following.

Consider N data points x i, i = 1 . . . . . N, sampled at equally spaced time intervals, At, of a transformed record that is stationary with /z = 0. From the basic definition, an autocorrelation function can be estimated at various time delays (lags), T + rAt , by

]_ N - r

Rxx(rA t ) ( U - r)~r 2 ~ (xi - ]'£)(Xi+r - - ]db) (12) i = l

Here x is the random variable and r is referred to as the lag number. The number of possible products at each lag number r in the above equation is only N - r. Hence the division by N - r gives an unbiased estimate of the autocorrelation function. An autocorrelation measurement clearly provides a tool to detect the deterministic nature of the data, which might be masked in a random back- ground. The frequency of this periodic component of the signal is determined from the first major peak in the plot of acf versus T after ~- = 0. The reciprocal of this value of r at which the first major peak occurs gives the frequency for the periodic component [47], referred to as the major frequency (]m)' The autocorrelation function expresses the behavior of a signal in the time domain, and sometimes a better interpretation is possible by expressing the signal in the frequency domain, which is obtained through the power (or auto) spectral density function (psdf).

The power spectral density function, S~x(W), of x(t) is the Fourier transform of its autocorrelation function [25], that is,

1 Sxx(O~)=-f R~(~)e-J~d~ (13)

77" J - ~c

Our data were subjected to differencing to filter out any trend, [25, 26], and then this stationary data set was employed to obtain the psdf according to the procedure SPECTRA in the SAS package [24]. This is referred to as the first method. The major importance of the psdf for a

given data set is to establish the frequency composition of the data and its important relationship to the physical system involved. However, this method does not lead to the identification of the major frequency or frequencies in a unique fashion. Hence a second method is adopted [25], which is known as the univariate Box-Jenkins (UBJ) model and also as the autoregressive integrated moving average (ARIMA) model. In the ARIMA model, the random component of the signal is filtered out. In ARIMA model calculations, three stages are involved. In the first stage, the model and its tentative order are identified as autoregressive on the basis of the autocorrelation and partial autocorrelation functions. The second stage involves the computation of the coefficients of this model using the software of Scientific Computing Associates (SCA) [27]. In the third stage, the adequacy of the model is checked by calculating the autocorrelation function of the residuals for the existence of any probable structure in them. The coefficients are also checked for stationarity and invert- ibility. If needed, the order of the model is altered to a higher order and the entire calculation procedure is repeated until the residuals exhibit no structure. For the present data, the autoregressive model of order 6 and higher is found adequate. These model-based data, which are likely to be free of the random component of the raw data, are used for the computation of the psdf according to the following relationship [28]:

Sxx (to) = p( to)/trz 2 (14)

where

P(to)2tr22/[ 1 - 4'1 e x p ( - j t o ) - 4'2 e x p ( - j 2 t o )

and

. . . . . 4'p exp(- jpto)]2, for0 < to _< 7r (15)

~rz2 = ~a2/[ 1 - / 3 1 ( ] ) 1 - P 2 4 ' 2 . . . . . P p 4 ' p ] ( 1 6 )

4'1, tl)2 . . . . . 4'p are the autoregressive parameters of an AR(p) process, and Pl, P2 . . . . . pp are the theoretical autocorrelation functions, o-a 2 is the variance of the residuals. The defining relations for Pl, P2 . . . . , pp, are given by Box and Jenkins [28].

The most important characteristic of a psdf for a particular run is the presence of a sharp peak [45] as will be seen later. The concentration of signal power in a frequency band of width less than 1 Hz points to the pressure of a periodic component in the signal. Thus such plots also help in establishing the periodic nature of the temperature and pressure signals and in determining the frequency of this periodic component, which is referred to as the major frequency (fro)"

EXPERIMENTAL RESULTS

Experiments were conducted at ambient conditions in the square fluidized bed with two dolomite beds of the particle size distributions and average diameters given in Table 1. In this table the uncertainty reported in average particle diameter is computed on the basis of probable error in weighing the sieve sample. In Table 2 are listed some of the other characteristic properties of these particles and of their beds along with estimated uncertainties in each of them. The pressure drop values across the bubble cap air


Table 1. Size Distribution of Dolomite Particles

U.S. Sieve Average Mass Fraction of Number Size ( izm) Solids Retained

8-10 2180 0.4792 10-12 1850 0.5208 12-14 1550 0.7800 14-16 1290 0.2200

Average particle diameter(~m) 2010 ± 12 1490 ± 20

distributor plate, APD, as a function of superficial air velocity, U, yielded the relationship presented in Fig. 1. The uncertainty in AP u values is about 0.5%. The agreement between theAP o values for increasing and decreasing U values is excellent. This, however, is expected. A regression analysis of the data yielded the relationship

AP D = 51.73U 2'°7 (17)

This suggests that the gas flow through the distributor plate follows predominantly the characteristics of the flow through an orifice [29]. The combined pressure drop data taken across the bed of dolomite particles and the distributor plate as a function of decreasing and increasing air velocity in conjunction with Eq. (17) were employed to generate the bed pressure drop data. These are graphically displayed in Fig. 2 for the two beds. The well-known procedure of determining the minimum fluidization velocity values [30] from such data produced the values listed in Table 2.

The bed voidage is measured across two sections, lower and upper, of the bed. The lower section is a 170 mm tall section of the bed located 65 mm above the gas distributor plate, and the upper section is the adjoining 150 mm high portion. The bed voidage is computed from the pressure drop data taken across that section of the bed at a particular gas velocity on the basis of the relation

Ap = Lg(1 - E)( as - Pg) (18)

Here Ps and pg are the densities of solid and gas, respectively, g is the acceleration due to gravity, L is the length of the bed section whose mean bed voidage is e, and A p is the pressure drop across it. The bed voidages computed in this fashion for the lower (e L) and upper (e u) bed sections are presented in Fig. 3 for the two dolomite beds as a function of fluidization number. The uncertainty in e is about 2%. The latter is defined as the ratio of superficial air velocity (U) to minimum fluidization velocity (Urn f). The mean values of e read for U = Umf from these plots are represented as e,,f, and these are also recorded in Table 2. The computed values of the Archimedes number

Table 2. Properties of Dolomite Particles and of Their Beds

dp Umf (txm) (m / s) Emf Ps Remf Ar Group

2010 0.41 0.38 2463 57.1 ±12 ±0.01 10.008 125 ±1.9 (7.3 ±0.3) x 105 I I B 1490 0.37 0.41 2463 38.2 &20 i0.01 ±0.007 ±25 ±1.2 (2.96±0.1) x 105 I I B

200

&Pn=51.73U 2.o~ 150 o Increasing U /

I I

~1~~ 100 • DecreasingU / _

so ~ Tb = 303 K

0 ~ ' " ' - I , 0.0 0.5 1.0 1.5 2.0

U (m/s) Figure 1. Variations of gas distributor pressure drop values with U.

(Ar) and the Reynolds number at minimum fluidization velocity (Rein f) on the basis of the following relations are also listed in Table 2.

A r = p, - (19)

W/A

3p Ii

.o [ J ', Tb= 3 0 7 + 8 K

11"; ', . I inoU

0 ~ u.0 0.5 1.0 1.5 2.0

U (m/s)

W/A ....

T ; 3 0 6 + 8 K

u.0 0.5 1.0 1.5 2.0

U (m/s) Figure 2. Variations of AP b values with U for dolomite beds of dp 1.49 mm (A) and 2.01 mm (B).

o.48 + u ,it." + " "..+, 1 • • 1.01mm

2 • e 1 . , , m m j ~ , , , ' ~

0.44 2 • •

_ . y % 2 ' ..i o.,o

A . . . j , . ' ~

I

0"360 1 2 3 4 5

U/U

and

Figure 3. Variations of bed voidage (E v and E L) for the two dolomite beds as a function of fluidization number (U/Umf).

250

Remf = dpUmfPg/tZg (20)

The relative values of the heat transfer coefficient (h w) are also measured on the basis of the heat transfer probe [1] as a function of the fluidization velocity for the two dolomite beds. These data are graphically shown in Fig. 4, and the variations are typical of a fluidized bed system [31, 32] and are consistent with the general trend [33-36] of a bubbling fluidized bed. h w increases rapidly as the solids movement in the bed increases with increasing U, attains a maximum value, and thereafter decreases owing to increased bed voidage or reduced solids concentration in the region around the immersed surface in the bed. The vertical bars on the data points in Fig. 4 are the estimates of the errors in individual measurements, about + 5%.

In Figs. 5 and 6 are presented the pressure history curves for the two dolomite beds at five representative air velocities characterized by U/Umf values. These records are sampled at the recording speed of about 11 Hz for a

23O

210

170

150 1

A = 2.01 mm o ,. 1.49 mm

2 3 4 5

U/U


Figure 4. Variations of the relative heat transfer coefficient (hw) for the two dolomite beds as a function of fluidization number (U/Umf ).

period of 92 s. Similarly, the temperature fluctuation data for these two beds for the same time period and recording speed are shown in Fig. 7. The computed standard deviations for these data, crp and (r r, are given in Figs. 8 and 9. Similarly, the computed probability density functions for these data of two beds at several fluidization numbers are sketched in Figs. 10 and 11, with the corresponding skewness and kurtosis values in Figs. 12 and 13. Representative acf plots for the bed of 1.49 mm average particle diameter from pressure and temperature fluctuation data are given in Figs. 14 and 15 respectively. In the latter case, because of the temperature drift, a differencing method is used [26]. Computed psdf plots for the same bed by the two methods from the pressure and temperature fluctuation data are given in Figs. 16-19. All of these results are discussed in the following section.

DISCUSSION OF RESULTS

According to the Ergun correlation [37], Urn/ is proportional to d . z in the laminar flow regime (Re m. < 20) and is proportional to dp in the turbulent flow regJime (Remf > 1000). For the two dolomite beds, the minimum fluidization velocity ratio is 1.11 and the mean particle diameter ratio is 1.35. The dependence of Umr on dp is reproduced rather inadequately by the Erguncorrelat ion, which is known to be quite successful for beds of spherical particles. We attribute this disagreement to the uncertainty in establishing an appropriate particle size for nonspherical particles. Dolomite particles are known to be quite nonspherical in shape [30], and this fact must be kept in consideration while discussing the hydrodynamic properties of such beds or beds of nonspherical particles in general. However, for our present purpose, to assess the appropriateness of statistical functions to characterize the bed fluidization quality, this is not a limiting factor.

In an attempt to delineate the different fluidization regimes--particulate, bubbling, bubble coalescing, slugging, transitional, and turbulent--several qualitative criteria involving characteristic numbers have been proposed from time to time. These criteria have been reviewed by Kunii and Levenspiel [37] and more recently in greater detail by Saxena [38]. Of all the schemes, the classification proposal of Geldart [39] is the most comprehensive and the most widely used. However, as discussed by Saxena and Ganzha [6, 38], the scheme of Geldart [39] is inca- pable of simultaneously characterizing the hydrodynamic and heat transfer properties of fluidized beds. They [6, 40] have therefore proposed a new scheme and have shown [41-44] that it does not suffer from such a deficiency.

Saxena-Ganzha [6] group liB particle beds are confined between the limits of Archimedes number 1.3 × 105 to 16.0 x 105. However, because of the nonspherical shapes of these particles it is likely that a dolomite bed of 1.49 mm average diameter may have properties characteristic of group IIA particles. Thus for nonspherical particle beds, regime characterization becomes difficult and somewhat ambiguous. This is also reflected in the relative qualitative variation of ~mf with Remf or Ar. This, of course, is of no major concern because the related uncertainty in computing Re,, , /and experimental uncertainty in ~:mf a r e large enough to justify the observed variations and small changes in the values of these quantities. On the


t~

12

0.6

0.2

0.8

0.2

0.6

0.2

0.6

0.2

0.4

0.0 0 10 20 30 40 50 60 70 80 92

Time (S) Figure 5. Variations of pressure fluctuation with time for the dolomite bed of 1.49 mm average particle diameter at ambient conditions.

contrary it must be emphasized that the use of the newly proposed statistical functions for regime del ineat ion will be more appropr ia te and accurate. To this extent, these two dolomite beds provide a more appropr ia te basis for checking the consistency and uniqueness of the statistical functions and their variations with operat ing parameters . This will reflect the appropriateness and the extent to which a statistical function can be used to characterize bed fluidization behavior.

Both of these dolomite beds belong to Geldar t [39] group D powders. For such beds the bubble coalescence is not predominant , and as a result the gas velocity in the dense phase is high and bed expansion is appreciable and increases with increasing gas velocity. This is evident from Fig. 3 inasmuch as for both beds e U and e L increase with increasing values of U/Umf. Further, e U is always greater than e L at a given U/Umu. This trend, which has been observed in beds of spherical glass beads [1, 5] also, is indicative of a very important characteristic of fluidization. The coalesced bubbles in the lower bed section break up and degenerate into smaller bubbles in the upper bed section so that eL~ is greater than e L. Further , smaller

bubbles are more efficient in solids mixing and mass transfer propert ies than the larger bubbles. Thus, the upper bed section will have more favorable heat and mass transfer characteristics than the lower section. This bubble coalescence and breakup physical picture is also at least partially supported by the varying bed pressure that a bubble experiences as it rises in the bed. However, such a fluidization characteristic is also shown by powders of different groups and hence cannot be employed for regime delineation. The relative heat transfer coefficient values presented in Fig. 4 refer to the upper bed section, where, as ment ioned above, a bubbling bed condition prevails and hence we get a typical variation of h w corresponding to such a condition [29-35]. Thus, the results of Figs. 3 and 4 support and supplement each other. It is clear that procedures of predicting bed quality fluidization based on e and h w are to some extent generally inadequate, because as employed here these represent gross behavior averaged over an appreciable region of the bed. Local measurements will be more appropr ia te to characterize that region of the bed. To that extent, tempera ture and pressure probes are more adequate and will be discussed below. To


0.8

0.4

0.6

0.2

0.4

C., 0.2

0.2

0.2

0.0 0 ao 20 30 40 50 60 70 80 92

Time (s) Figure 6. Variations of pressure fluctuation with time for the dolomite bed of 2.01 mm average particle diameter at ambient conditions.

determine the overall bed quality, multiple probes will be needed.

The standard deviation plots of ~rp and err in Figs. 8 and 9 exhibit a clear monotonic increase in their values with increase in U/Umf. The scatter in the values of er r is relatively more than in o- v values. This is due to the relative locations of these probes in the bed. The pressure probe is located on the bed wall, whereas the heat transfer temperature probe occupies more of the central bed region. The pressure probe thus registers only somewhat damped pressure fluctuations that reach the bed wall, whereas the temperature probe is exposed to more vigor- ous and violent movements of bubbles and dense phase and therefore exhibits large temperature fluctuations. However, both probes unmistakably exhibit a clear and unambiguous increase in standard deviation values with increase in air velocity. This is because as the air velocity increases the residence time of the bubble or dense phase on the heat transfer surface increases. The monotonic variation of err and ~p with increasing U/Umf is due to the maintenance of the same hydrodynamic regime over the entire U/U,,,¢ range. An abrupt change in the err and

erp values will occur when the fluidization regime undergoes a transition from one regime to the other as in our work with spherical glass beads [5].

Computed values of the probability density functions for the two beds at several fluidization numbers from the pressure and temperature fluctuation data are plotted in Figs. 10 and 11, respectively. Many of these pdf plots clearly represent two peaks. The first peak is identified as referring to the bubble phase, and the second peak as referring to the emulsion phase [46]. In the temperature fluctuation data plots, the two peaks are more evident than in the pressure fluctuation data plots. This is obvi- ously due to the relative location of the two probes. The temperature probe is located in the dense bed and hence registers the two phases directly, while the pressure probe located at the bed wall is affected only by the descending solids while bubbles are drifting away from the wall to rise in the central bed region. The corresponding skewness and kurtosis plots for the two beds are presented in Figs. 12 and 13 as a function of fluidization number. For both beds, the distributions are not normal, and as a result the values of relative skewness and kurtosis are finite, as seen


79.0

78.5

76.0

.u v

7~.5

6 7 . 0

65.0

72.5

71.5

0.20

,I 80.0

U~ U,., - Sigl) "

=U/Urn, - 3:01~

= " ' ~ ' 1 1

U/Umf" - 2.s4

U/Umf - 1.1gl

%(" !

0 20 T i m e (st

0 20

0.15

0.10

0.05

0.00

A = 2.01 mm a = 1.49 mm a "~

i i

3 4 U / U ~

Figure 8. Variations of standard deviation in pressure fluctuation data with fluidization number for the two dolomite beds.

4O 6O 8O

Time (s)

Figure 7. Temperature history records at different fluidization numbers for beds of dolomite, (A) 1.49 mm and (B) 2.01 mm average particle diameters.

0 . 6

A = 2.01 m m o = 1.49 mm

0.4 a

~ A A

~ 0.2

&

i i i 0.0 2 3 4 5

U / U ~

Figure 9. Variations of standard deviation in temperature fluctuation data with fluidization number for the two dolomite beds.


12 12 U/U.~ = 4.52 u ~ ~ = 4.21 I i I

6 6

0

2 u~, . f . ,~+.

~ 075.0 79J

u~.~.~.~,

~.f-,~,

/-V\

. !i! io o . o + t 75-~ 76.5 77.5 4 U/IJaf= 3.94

6 ~ 6 U / U ~ = 3.0i I I "+ i i' i " 12 U/U~ = 3.78 1 u/U.a = 3.23 71.0 72.0 .0 079.0 ' 80.0 '~ 81.0

6 ~ ~ 4 iJ/U'~'2"54 4U/U'=3"201 ,1

. .~o ~ - ~ o ' \ k 12

6

u~J-,. 3.ts ~ zs

o I~"\ ,, j 12 U/U'B/' = 2.J3 ' 25 U/I.JBI = ~..48

I

6 ~ , ~ to !

o \ / 0 0.2 0.4 0,6 0,S 1.0 0.2 0.4 0.6 0.8 1.0

Figure I0. Variations of probability density function at different fluidization numbers from pressure fluctuation data for dolomite beds of average particle diameters 1.49 mm (A) and 2.01 mm (B).

in Figs. 12 and 13. The distributions of Fig. 10 for both beds are skewed to the right at larger gas velocities. At smaller gas velocities, the distribution functions for the bed of larger dolomite particles are skewed to the left, and for the bed of smaller particles, they are skewed to the right.

The important point to note in Fig. 12 is that both skewness and kurtosis have systematic variations with increasing U/Um[ that are monotonic. This is because both of these dolomite beds belong to the same hydrodynamic regime characterized by group liB of the Saxena and Ganzha powder classification scheme. When a group change occurs, implying a change in the fluidization regime, abrupt changes in skewness and kurtosis take place. The skewness is mostly negative in Fig. 13A, implying that the distributions of Fig. 11, at least at larger gas velocities, are not normal and are skewed to the left. Similarly, the kurtosis plots of Fig. 13B suggest that the distributions of Fig. 11 are platykurtic. Again the smooth monotonic variations of the two properties with U/U,,,f indicate that no change in the hydrodynamic regime has taken place.

Computed values of autocorrelation functions from Eq. (12) for the pressure and temperature fluctuation data at

o(~.0 67.0 4 U/U~ = I . U

o 64.0 65.0 66.0

4 lu~'f" 'A'P l

(A)

..£'-'~ Q/~.0 74.0 8 U/U~ = 2.83

i ' 4 h / o / 69.0 70.0 8 Jrdj = 2.59

71.0

o /~ ~- 63.0 64.0 65.0

.

Tempe~tme (*C) (B)

Figure 11. Variations of probability density function at different fluidization numbers from temperature fluctuation data for the two dolomite beds of average particle diameters 1.49 mm (A) and 2.01 mm (B).

different fluidization numbers are presented in Figs. 14 and 15, respectively. Unlike the records of Figs. 5-7, the plots of Figs. 14 and 15 clearly establish that the temperature and pressure signals are composed of deterministic and random components. A maximum is observed at zero time lag (r = 0), and the acf behaves like a sinusoidal function with a fairly constant period as the time lag is increased [45]. It is this deterministic component in which we are interested, and for this purpose the signal is analyzed by computing the psdf as a function of frequency. As mentioned before, there are two ways to obtain the psdf. In the first method, the raw data are employed, with the results reported in Figs. 16 and 17. It is clear that the identification of the major frequency or frequencies is not uniquely possible. This brings out the importance of the second method, in which the random component is filtered out from the signal and only the deterministic component is employed. These results are presented in Figs. 18 and 19, where the major frequencies are also listed.

The frequencies, fm, are also computed for the dolomite bed of 1.49 mm average particle diameter using the auto-


2

- - - 1 ,2.'

io

-2 2

2

, - . 1

o 0 I:

- 1

A

B

5

[] = 1.49 mm

, I * I i

3 4 U/Umf(-) ' Dolomite

i ! i I i

3 4

2

0

-1

3

2 I

v

0

-1

-22 5 -2

Figure 12. Variations of (A) skewness and (B) kurtosis of the probability density function for pressure fluctuation data with fluidization number.

correlation plots of Figs. 14 and 15. These values are listed in Table 3 along with the corresponding values obtained from psdf plots of Figs. 18 and 19. The two sets of values are in satisfactory agreement at all gas velocities and for both temperature and pressure signals. This lends support to the statistical methods employed in computing acf and psdf values from the temperature and pressure signals.

These major frequencies for the temperature and pressure fluctuation signals are plotted in Fig. 20 as a function of fluidization number for the two beds of dolomite particles. It is clear that for both beds the variations of fm with U/U,,f are in good agreement with each other. On the other hand, fm values obtained from temperature fluctuation data are greater than those obtained from the pressure fluctuation data over the entire fluidization velocity range. This is in general agreement with the understanding of fluidization quality of a bubbling bed- - the bubbles drift toward the central region of the fluidized bed, and bubble frequency is greater in the central part of the bed than in the region close to the wall. The temperature-measuring thermocouple probe is located more in the central region of the bed, while the pressure- measuring transducer probe is located at the bed wall. The particle size has no influence in the bed quality

A

a

i i i

Dolomite a ,, 1.4g mm & = 2.01 mm

A A

" ' ~ . , t ~ - - ' ~ ' ~ a

B

, I i I , I ,

2 3 4

Dolomite A ; 2 . 0 1 m m ~ . a : 1 . 4 9 m m

o

°,o []

i I , I i I i

2 3 4 U/Umf(-)

5

5

Figure 13. Variations of (A) skewness and (B) kurtosis of the probability density for the temperature fluctuation data with fluidization number.

fluidization, and again this is not surprising as both beds belong to group liB of the Saxena and Ganzha particle classification scheme. The change in the statistical properties reported in Figs. 10-13 and in Fig. 20 around a U/Umf value of 3 is suggestive of a change in the structure of the bed quality fluidization. However, a detailed discussion of this nature involving the changes in fluidization regime is postponed pending more detailed investiga- tions with particles of controlled shape over a wide size range. Such experiments are being planned with beds of spherical glass beads over a wide size range so that the fluidization regime delineation criteria can be established unambiguously for different fluidization groups [6, 39] as well as the trends and values of characteristic numbers distinguishing one group from the other.

SIGNIFICANCE OF DIFFERENT DIAGNOSTIC PROCEDURES

In this section an assessment is presented of the fluidization quality detection capabilities of the two probes together with the recording computer system and their impact on the nature of the information generated. Fur-


+0.5

0.0

--- -0.5 ,2,'

+ 0 . 5 0 o

o 0.0

o u o

"~ -0 .5 <

+0.5

LAA

I I

UIU~ = 3.3s

v VVVVV~ v

I I U/Um~ = 3As

L/ AAA ~AA]^ AJ,,AA Vvvvvvvvv vv~vvv

I I

tA 2 4 6 8

Lag (s)

-0 .5 0 10 0

AA

i I U/U~ = 3.7e

V- v~ vV~ w v

rVV

I I

U / U n d = 4.03

A IAAA -JvAv V V v jVU

1 I U / U m f = 4.52

/

Vv ,"v

2 4 6 8 10 Lag(s)

Figure 14. Variations of autocorrelation function at different fluidization numbers from pressure fluctuation data for the dolomite bed of 1.49 mm average particle diameter.

ther, the appropriateness and relative significance of the various statistical functions in detecting the fluidization nature of the bed are also pointed out. It may be mentioned at the outset that the aim of this research effort was to examine and discriminate among the various properties and statistical function-based procedures that have been used to establish the quality of bed fluidization. The results of this investigation will be useful to those who would like to establish criteria for delineating various fluidization regimes.

The thermocouple temperature probe used here has a relatively slow thermal response, and the sampling rate could have been faster; indeed, thin metal film thermocouples with fast response have been employed to measure heat transfer coefficient and temperature histories [47]. In the hostile environment of a fluidized bed coal combustor that employs nonspherical dolomite particles, only a sturdy probe such as the one used here will have a good chance of lasting service. Further, even if sources of fast and weaker temperature signals in the bed are missed, it is of no major consequence because we are presently more interested in establishing the procedures of the

analysis of temperature-t ime records than in making sure that all sources of fluctuation have been registered.

To further substantiate the above inference, an experi- ment was conducted in which the recording thermocouple was replaced by a faster response copper-constantan thermocouple of 10-20 ms (100-50 Hz). The temperature history records indicated almost no change in the frequency of the variations in the temperature signal, which was 1.1-1.2 Hz at the fluidizing velocity of 1.75 m / s for spherical glass bead particles of average diameter 2093 ~m. This also indicated that the recording speed of about 11 Hz was appropriate. Large bubbles moving with a fast speed will not be detected if their residence time is less than 0.4 s. At the highest gas velocity (1.73 m/s) , bubbles of average diameter 4.5 cm will have a residence time of about 180 ms in the bed. Such large bubbles will not be detected by the thermocouple probe but will be detected by the pressure probe. In our experiments such large bubbles were observed erupting from the bed surface only.

Diagnostic procedures based on bed voidage variations with changes in superficial gas velocity are of only limited


+0.5

0.0

I I

U/Um~ = 2.o6

;' v'vv,,y " ' " v v ' v v v " ' " " v " w

,2' -0.5

~0 +0.5

< +0.5

0.0

I I

U/Um~ = 1.~

I~,~¢;.,, -,..~,A ,~ 'A ^ ,[ ~A, , .v-~:v,,A,

i I

U / U m f = 1.58

/,~_A ,.... ~.~AA A ,~ ~^ ^~, " "V v ~-v ~, --'v v

-0.5 0 2 4 6 8 10 0

L a g (S)

i !

U/Umf = 2.54

V,.Vv,.',yy .... .v..v'-v,.,,vv.vv.vv,,,

1 I

U/Umf = 3.01

IwvvvvIvv'vvw vvvvvv rVVVV'

v I U/Umr = 4.s3

V,"V . . . . , v ' v " ' - ' w , ' v W F v " " ' e

2 4 6 8 10

L a g (s)

Figure 15. Variations of autocorrelation function at different fluidization numbers from temperature fluctuation data for the dolomite bed of 1.49 mm average particle diameter.

usefulness. The greatest limitation comes from the fact that it refers to the gross bed behavior and responds rather poorly to local variations that may be occurring in nearby regions of the bed. For an overall description it will be a good property and has been very commonly used because of its simplicity. In all our fluidization measurements [1, 5], it has been seen that bed voidage increases with bed height. This property will not help in delineating the fluidization regimes, though it will help in explaining heat and mass transfer properties if it is known how this • variation changes with fluidization quality.

The heat transfer coefficient method is similar to the temperature history-based method inasmuch as both occupy a certain portion of the bed and we need to know how h , variation depends on fluidization quality. At best, this can be achieved only in qualitative terms, and hence one is tempted to take the temperature history-based approach because such probes can be miniaturized and sophisticated statistical procedures can be adopted to quantitatively understand the dependence of h w on fluidization quality. This is discussed below. Similar comments apply for the pressure fluctuation history records, and hence both are discussed together.

Standard deviation variation with fluidization number can help in certain special conditions. In general, as in Figs. 8 and 9, we see that bed violence is related to fluidization increases with U/U, , , f . Such a plot will be more useful if transition from one regime to another regime occurs when these curves undergo an abrupt change. However, for the same fluidization regime such variations are also useful to know but provide only limited information as is the case here. For example, bubble coalescence will increase the magnitude of tTp as U increases.

Plots of pdf are quite indicative of the quality of fluidization in the bed region where such data have been recorded. In an actual bed, multiple probes will have to be installed. For example, Fan et al [8] used three pressure transducer probes at different vertical locations above the distributor plate to interpret the quality of fluidization. Here the temperature probe records of Fig. 11 suggest, in general, two-peak distribution curves for both sizes of particles at different air velocities. The first peak is due to the bubble phase, and the second peak is due to the emulsion phase. In Fig. 18 are seen the frequencies corresponding to the pdf plots of Figure 11A. The bubble phase


0 . 0 6 , 0 . 0 6 , ,

°° ,1 °° hL

"~ , I , O.O6

° II [I , m 0 . 0 0 "

~O 0.06 , 0.06 , , ~-~ UIUm! = 2.53 Ull~m ! = 4.52

! I.

0 . 0 3 0 . 0 3 111 . ,

, | , , JJl,

0"030 1 2 3 4 5 1 2 3 4 5

F r e q u e n c y ( I / z ) F r e q u e n c y ( H z )

Figure 16. Variations of power spectral density function at different fluidization numbers from pressure fluctuation data for the dolomite bed of 1.49 mm average particle diameter. The first method based on Eqs. (12) and (13).

frequency is about 2, but at higher air velocities this is not seen because of the insensitivity of the detecting thermocouple probe. The emulsion phase frequency of about 4 is relatively clearer at all air velocities and is recorded because of the relatively slower motion of the emulsion phase. This frequency is taken as the measure of the bed quality fluidization. In Fig. 20, this major frequency, fro, is potted for both particles as a function of U/U,,/. It is the contention of this endeavor that fm may be different for different regimes and hence can be employed for regime delineation and characterization. This, though, needs to be demonstrated.

In Fig. 10, the pdf plots of pressure fluctuation at the bed wall for both particle sizes are presented with the corresponding psdf plots reproduced in Fig. 18. It is to be noted that fluidization behavior at the wall is much different than in the central region of the bed at the same height. This follows from a comparison of Figs. 10 and 11

and Figs. 18 and 19. Only one major frequency is observed at the bed wall, and this physically represents the frequency of the descending emulsion phase. As stated earlier, this is a consequence of the fact that bubbles drift away from the wall to the central bed region. In Fig. 20, these major frequencies are plotted for both sizes of particles as a function of U/Umf and can be used to characterize the fluidization quality of a particular regime. It is interesting to take such data in different fluidization regimes to discover whether such fm values can be taken as a measure of fluidization quality at the wall. Further, such data will establish the nature of bed fluidization at the wall in different hydrodynamic regimes.

When the pdf plots are not normal and are asymmetri- cal on either side of the peak, the knowledge of functions as given in Figs. 12 and 13 is useful in establishing the detailed structure of fluidization behavior. For example, Fig. 12, which shows skewness values for 1.49 mm average


"7"

O ¢d

O

0.010

0.005

0.000

0.004

0.002

L * L , | ~ 0.000

0.004

0.002

0.000

u/0.,. .o6 I

,ill

UlOm,- i.sa

ulo,,,- i.sa I

0.02

0.01

0.00 -~-~

u/0.,. I

I

b

0 1 2 3 4 5

Frequency (Hz)

0.02

0.01

0.00

0.02

0.01

0.~

0, o, ri0, II ll]lai

1

U/0mf - ,i.,~

2 3 4 5

Frequency (Hz)

Figure 17. Variations of power spectral density function at different fluidization numbers from temperature fluctuation data for the dolomite bed of 1.49 mm average particle diameter. The first method based on Eqs. (12) and (13).

particle bed as positive for a range of U//Umf values, implies that secondary signals probably occur at greater pressure values than those at which the most probable primary signal occurs. This follows from the records of Fig. 12A. A comparison of Figs. 12A and 12B highlights the differences in the distribution of probable pressure amplitude for the two particles as the gas velocity is varied. Figure 12 also indicates that the flatness of the peak associated with this most frequently occurring signal lessens as the gas velocity increases. This is explicitly indicated in Fig. 12B, where the kurtosis is negative at lower gas velocities and becomes less and less negative as the velocity is increased.

CONCLUSIONS

From the extensive experimentation presented in this work, involving the measurement of bed voidage, local

heat transfer coefficient, and temperature and pressure fluctuations as a function of gas velocity at ambient temperature and for beds of two dolomite particles, several conclusions have been drawn. The underlying idea is to develop procedures for establishing the quality and nature of fluidization and see if the parameters obtained from such data can be employed to achieve this goal.

It is clear that gross measurements of bed voidage can lead to only limited details of fluidization, and multiple temperature and pressure probes are good tools to infer the statistical analysis of these temperature and pressure fluctuation signals through computation of the acf, which has revealed that these contain a deterministic periodic component that is used here to characterize the nature of fluidization. Plots of pdf and psdf provide the basic information in the amplitude and frequency domains, respectively, for both of these signals. Probability density function plots reveal the existence of bubble and emulsion

20,

15'

10

5 $ ~ Q

m o

~ t5 0

UIUrM=3.36

1 2 3 4

U/Uml = 3.15

, ' i ' i ' i ' i '

U/UmI = 2.53

oo

"1 A U/Uy = 32-8

-O 1 2 3 4 1.2~

0.8

0.4

0.0~ t

1

U/Urn/= 4.0B

1 2 3 4 5

U/Uml = 4.51

t 2 3 4 5 i ~ ~ 4 5 Frequency (Hz) Frequency (Hz)

Figure 18. Variations of power spectral density function at different fluidization numbers from pressure fluctuation data for the dolomite bed of 1.49 mm average particle diameter. The second method based on Eqs. (14) and (15).

phases, and psdf plots give the frequencies of these phases. It is shown that these major frequencies vary with gas velocity and are uniquely dependent on the quality of fluidization. It also appears that these frequencies are related to the nature of bed fluidization, and it will be worthwhile to launch an e labora te exper imental program based on the promise of the work presented here. This will involve exper imentat ion with beds of different part icle sizes and belonging to different fluidization regimes. The detai led nature of fluidization behavior involving the physical dimensions of the two phases may be inferred from the plots of such statistical functions as skewness and kurtosis. It will be important to establish whether the magnitudes of ~r r are significantly different when the fluidization regime undergoes a change and whether such changes are strong and abrupt enough to provide a viable diagnostic basis. The present work demonst ra tes that these statistical functions are capable of revealing fluidization behavior, and their use for regime del ineat ion remains to be established by further experimental work.

We are grateful to Dr. K. K. Ho for his cooperation and his support of this project. This work has been sponsored by the Illinois Depart- ment of Energy and Natural Resources through its Coal Develop- ment Board and its Center for Research on Sulfur in Coal and by the U.S. Department of Energy. This work is also supported in part from the Office of Solid Waste Research at the University of Illinois at Urbana-Champaign under grant number OSW-04-002. We also

"3" v

O

Q;

~d


O

U/Un¢=2~ rx

"ZJt 1 2 3 4 5

U/UmI = 1.88 /

1 2 3 4

U/Un~ = 1.58 fm= 1.79 Hz

UIUmf = 234

fm = 4.02 Hz

Z 1

1 2 3 4

UIUmf =3.01 A

4 2 I ~ ~ 6 f==a.59Hz

00 1 2 3 4 5

U/Un~ = 4.53

3

2

1

0 1 2 3 4 5 l 2 3 4 5 Frequency (Hz) Frequency (Hz)

Figure 19. Variations of power spectral density function at different fluidization numbers from temperature fluctuation data for the dolomite bed of 1.49 mm average particle diameter. The second method based on Eqs. (14) and (15).

appreciate the helpful comments of the reviewers, which improved the focus of this work.

NOMENCLATURE

Ar Archimedes number, dimensionless a 3 relative skewness, dimensionless a 4 kurtosis, dimensionless dp mean particle diameter , m d x small increment in the signal value, Pa

or K f frequency, Hz

fm major frequency, Hz

Table 3. Comparison of fm from psdf and acf

Pressure Temperature Signals Signals

U / Umf acf psdf U / Um/ aef psdf

2.53 2.25 2.06 1.58 1.57 1.79 3.15 1.84 1.79 1.88 5.11 5.00 3.36 1.72 1.68 2.06 4.08 4.13 3.78 1.75 1.74 2.54 3.83 4.02 4.03 1.67 1.63 3.01 3.63 3.59 4.51 1.70 1.68 4.53 3.45 3.48


6

, ~ 4 N

V

2

0

, ~ , 1,4dP (mm) a b 2.01 " &

A a - Temperature-fluctuation data b - Pressure-fluctuation data

I I = I = I I

2 3 4 U/Umf

5

Figure 20. Variations of major frequency with fluidization number for the temperature and pressure fuctuation data.

g

hw L

m2, m3, m4

N

P p(oJ)

p(x) Remf

r

Rxx(r) Sx.(~o)

skp

T

T0 t

U u,.i x(t)

X i

f v e bubble frequency determined from pressure-fluctuation data, Hz

f v r bubble frequency determined from temperature-fluctuation data, Hz accelerat ion due to gravity, m / s 2 heat transfer coefficient, W / ( m 2 • K)

length of a bed section, m second, third, and fourth moments, respectively, of the data points about their mean, Pa 4 or K 4

number of data points, dimensionless number of data points in the range dr, dimensionless pressure, Pa power spectrum of an autoregressive process, P a 2 o r K 2

probabil i ty density function, Pa 1 or K - 1 Reynolds number at minimum fluidization condition, dimensionless lag number, dimensionless autocorrelat ion function, dimensionless power spectral density function, dimensionless Pearson's coefficient of skewness given by Eq. (5), dimensionless total time, s bed temperature , K time period, s superficial air velocity, m / s value of U at minimum fluidization, m / s signal value at t, Pa or K ith signal value, Pa or K

G r e e k S y m b o l s A P pressure drop, Pa

A P b pressure drop across the bed, Pa A P D pressure drop across the distr ibutor plate,

Pa

tx

&

Pl, P2 . . . . . Pp

2 O" %2 ~z 2

T

4,. 62 . . . . . % O)

acf A R I M A

F F T pdf

psdf SAS SCA UBJ

A P r total pressure drop across the distr ibutor plate and the bed, Pa

At time interval, s e bed voidage, dimensionless

e L bed voidage of the lower bed section, dimensionless

emf bed voidage at Urn f, dimensionless e U bed voidage of the upper bed section,

dimensionless mean of a set of data points, Pa or K gas viscosity, k g / m . s gas density, k g / m 3 solid density, k g / m 3 theoretical autocorrelat ion functions, dimensionless standard deviation, Pa or K

trp standard deviation in pressure fluctuation signals, Pa

or w standard deviation in tempera ture fluctuation signals, K variance, Pa 2 or K 2 variance of residuals, Pa 2 or K 2 defined according to Eq. (16), Pa 2 or K e lag, s autoregressive parameters , dimensionless angular frequency, radian

A b b r e v i a t i o n s

autocorrelat ion function auto regressive integrated moving averages fast Four ier transform probabili ty density function power spectral density function statistical analysis system scientific computing associates Univariate Box-Jenkins

REFERENCES

1. Saxena, S. C., and Rao, N. S., Fluidization Characteristics of Gas-Fluidized Beds: Air and Glass Bead System, Energy, 14, 811-826, 1989.

2. Saxena, S. C., and Rao, N. S., Pressure Fluctuations in a Gas Fluidized Bed and Fluidization Quality, Energy, 15, 489-497, 1990.

3. Saxen, S. C., Rao, N. S., and Zhou, S. J., Fluidization Regime Delineation in Gas-Fluidized Beds, AIChE Syrup. Set., 86(276), 95-103, 1990.

4. Saxena, S. C., Rao, N. S., and Zhou, S. J., Temperature and Pressure Fluctuation Histories in a Gas-Solid Fluidized Bed for Regime Delineation, Sixth Annual Int. Pittsburgh Coal Conf., 2, 959-969, Sept. 25-29, 1989.

5. Saxena, S. C., and Rao, N. S., Determination of Fluidization Quality of Beds of Spherical Particles, Energy, 16, 1199-1206, 1991.

6. Saxena, S. C., and Ganzha, V. L., Heat Transfer to Immersed Surfaces in a Gas-Fluidized Bed and Particle Characterization, Powder Technol., 39, 199-208, 1984.

7. Saxena, S. C., and Vadivel, R., Heat Transfer and Hydrodynamic Studies in Gas Fluidized Beds, Energy, 14, 353-362, 1989.

diagnostic procedures for establishing the quality of fluidization of gas-solid systems

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