Local bubble‐size distribution in fluidized beds

Santana, D.; Macías-Machín, A.

doi:10.1002/aic.690460708

Cited by 16 publications

(14 citation statements)

References 13 publications

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“…Similarly, as the sample size increases the other higher-order moments approach to their respective experimental value. The convergence of these moments to the experimental values is initially very fast as the sample size increase, indicating that the sample size can be reduced drastically compared to the non-parametric approaches proposed by Liu et al (1996Liu et al ( , 1998 and Santana and Macías-Machín (2000).…”

Section: Resultsmentioning

confidence: 92%

“…Furthermore, Santana and Macías-Machín (2000) showed that, knowing the velocity vector, shape factor and the bubble diameter, the conditional probability density function of the chord-length measured by the probe can be given by…”

Section: Raw Moment Estimatorsmentioning

confidence: 99%

“…Notice that the procedure can be performed avoiding the use of the backward transformation previously proposed by Liu and Clark (1995) for bubbles with a purely vertical rise and revised later on by Santana and Macías-Machín (2000) for bubbles with a non-vertical rise.…”

Section: Maximum Entropy (Maxent) Estimatementioning

confidence: 99%

“…Note that for these approximations, a record of only 350 bubbles were used, whereas other non-parametric methods, i.e. Liu et al (1996Liu et al ( , 1998 and Santana and Macías-Machín (2000), require almost 5000 bubbles and need to use a backward transformation to obtain a reliable distribution. Consider that, if the bubble shape cannot be obtained experimentally, it must be inferred from the bubble velocity, for example, and, consequently, the error in the calculations of the raw sampled moments would increase due to the uncertainty in the bubble shape estimation.…”

Section: Experimental and Simulation Proceduresmentioning

confidence: 99%

“…Furthermore, Clark and Turton (1988) and Turton and Clark (1989) proposed a non-parametric numerical method, which was shown to be numerically unstable for small sampling records. Later on, Liu et al (1996Liu et al ( , 1998, using a Parzen window with ellipsoidal bubbles, proposed a non-parametric model that Santana and Macías-Machín (2000) extended to consider bubbles with non-zero angles of attack. Unfortunately, these methods incorporate the disadvantages that the bubble size distribution inferred is not in a closed form and the data must be divided in bins to take into account the bubble shape and/or velocity variation.…”

Section: Introductionmentioning

confidence: 99%

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Characteristic lengths and maximum entropy estimation from probe signals in the ellipsoidal bubble regime

Santana

Rodríguez-Rodríguez

Almendros-Ibáñez

et al. 2006

International Journal of Multiphase Flow

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The bubble size, surface and volume distributions in two and three phase flows are essential to determine energy and mass transfer processes. The traditional approaches commonly use a conditional probability density function of chord lengths to calculate the bubble size distribution, when the bubble size, shape and velocity are known. However, the approach used in this paper obtains the above distributions from statistical relations, requiring only the moments inferred from the measurements given by a sampling probe. Using image analysis of bubbles injected in a water tank, and placing an ideal probe on the image, a sample of bubble diameter, shape factor and velocity angle are obtained. The samples of the bubble chord length are synthetically generated from these variables. Thus, we propose a semi parametric approach based on the maximum entropy (MaxEnt) distribution estimation subjected to a number of moment constraints avoiding the use of the complex backward transformation. Therefore, the method allows us to obtain the distributions in close form. The probability density functions of the most important length scales (D, D 20 , D 30 , D 32 ), obtained applying the semi parametric approach proposed here in the ellipsoidal bubble regime, are compared with experimental measurements.

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Section: Resultsmentioning

confidence: 92%

Section: Raw Moment Estimatorsmentioning

confidence: 99%

Section: Maximum Entropy (Maxent) Estimatementioning

confidence: 99%