1852
Ind. Eng. Chem. Res. 1994,33, 1852-1857
Bubble-Chord Length and Pressure Fluctuations in Three-phase Fluidized Beds Hyeuk W. Kwon, Yong Kang,t Sang D. Kim,’ Mutsuo Yashima3 and Liang T. Fan$ Department of Chemical Engineering, Korea Advanced Institute of Science and Technology, Taejon, 305-701, Korea
The bubble-chord lengths, ZV’S, and pressure fluctuations were measured in three-phase fluidized beds by focusing on the effects of gas flow rate, liquid flow rate, and particle size on them. The pressure fluctuations have been analyzed by resorting to fractal analysis. Specifically, the rescaled range analysis has been applied to time series of the pressure fluctuations, thus giving rise to the Hurst exponent, H. It has been revealed that as the gas flow rate in a bed increases, ZV and the standard deviation of its distribution increase but H decreases, and that the opposite is true as the liquid flow rate increases. The Hurst exponent, H, has been correlated in terms of the operational variables and the geometrical features of the beds, and 2~ has been correlated as a function of H and the radial position in the bed.
Introduction Three-phase fluidized beds are found in a number of industrial processes because these fluidized beds have various advantages, such as high rates of heat and mass transfer and chemical reaction due to effective contact among the phases, and ease of continuous operation (Epstein, 1981; Shah et al., 1982; Pandit and Josh, 1986; Fan, 1989). A three-phase fluidized bed generally comprises the fluidized-gas bubbles, fluidized-solid particles, and fluidizing liquid; the first and second exist as dispersed phases and the third exists as a continuous phase. The motion of the flow behavior of the dispersed phases have been identified as the influential factors affecting the performance of the bed as a processing device. Numerous investigators examined the bubbling phenomenon and flow behavior of three-phase fluidized beds (Kim et al., 1977; Morooka et al., 1982; Lasa et al., 1984; Matsuura and Fan, 1984; Kim and Kim, 1987; Yu and Kim, 1988;Han and Kim, 1990). Their results have amply demonstrated that the bubbling phenomenon in these fluidized beds is extremely complex; therefore, it is difficult to understand the effects of this bubbling phenomenon on the performance or behavior of the beds, which, in turn, renders any systematic analyses by conventional approaches extremely cumbersome. Analysis and characterization of the bubbling phenomenon in a three-phase fluidized bed in terms of fluctuations of its state variables could yield information useful for its fault diagnosis and control. Moreover, the bubbling phenomenon is probably most closely related to pressure fluctuations in the bed. Hence, the analysis of pressure fluctuations should play a crucial role for developing diagnostic tools for the bubbling phenomenon. The pressure fluctuations in multiphase flow systems, which are apparently highly stochastic and random, have been successfully treated by means of the rescaled range analysis based on the concept of fractional Brownian motion (Feder, 1988;Fan et al., 1990,1991,1993;Kang et al., 1992). This analysis is applied to three-phase fluidized beds in the present study. The results, specifically the values of the Hurst exponent, H,are correlated with the
* To whom correspondence should be addressed.
relative sizes of particles and beds, the sizes of rising bubbles, and the flow rates of the fluidizing media. Naturally, the bubble size is one of the most significant parameters governing the bubbling phenomenon and, hence, pressure fluctuations. Moreover, the bubble size can be most readily characterized by the bubble-chord length, Iv. Efforts are made, therefore, to correlate the bubble-chord length measured on-line with the Hurst exponent.
Theoretical Section To determine H for a given time series of pressure fluctuations, the sample and rescaled ranges are defined and constructed as follows (Feder, 1988; Fan et al., 1990, 1991, 1993; Kang et al., 1992). For the recorded times series of pressure fluctuations, X ( t ) ,evenly spaced in time from t = 1to t = T, their mean value within the subrecord from time (t + 1)to time (t 7 ) can be written as
+
( 1 / 7 ) [ X * ( t + T )- X*(t)l = ( X ( t ) ) ,
(1)
where X * ( t ) is defined as 1
X*(t) = C X ( u )
(2)
u-1
Let C(t,u) denote the cumulative departure of X ( t + u ) from the mean, ( X ( t ) ) , ,for the subrecord; then, C(t,u) = [X*(t+u)- X*(t)l - ( u / T ) [ X * ( ~ +-TX*(t)l )
(3) The sample sequential range, R(t,T),is defined as
R ( ~ , T=) max O
