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

Abstract

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.

Maximum entropy estimation of the bubble size distribution in fluidized beds

“…It has been shown that the backward transformation is an unstable problem for all the aforementioned methods. Recently, Santana et al (2006) reported a different method that avoids the use of the backward transformation. They found a semiparametric solution of the problem by applying the maximum entropy method and applied it successfully to gas-liquid systems assuming an elliptical bubble shape.…”

“…Moreover the sample size needed to obtain a reliable distribution is smaller than in the other methods. Santana et al (2006) successfully applied this method to gas-liquid systems assuming an elliptical bubble shape.…”

Voidage distribution around bubbles in a fluidized bed: Influence on throughflow

“…The bubble eccentricity was obtained as the eccentricity of the ellipse that has the same second moments as the bubble and the bubble orientation was defined as the angle (in degrees) between the horizontal axis and the major axis of such ellipse. The Probability Density Functions (PDF) were obtained as the distributions p(x) that maximize the Shannon's entropy ∫ − p(x) ln (p(x))dx, restricted to a series of constrains that the distribution p(x) has to satisfy, which are the raw moments of the distribution 〈x i 〉 [27,28]: max∫−pðxÞ ln ðpðxÞÞdx ð5Þ…”

“…The mathematical procedure followed to obtain p(x) was similar to the one described in detail in [27]. Fig.…”

“…Fig. 6(b) gives the PDF of the data of ε bc for d p = 350 µm (dashed line), d p = 600 µm (dotted line) and both together (solid line), obtained applying again the maximum entropy method [27,28]. The three group of data follow a normal distribution with the data distributed around the mean value µ with a standard deviation σ.…”