APPLICATION OF INFORMATION ENTROPY AND FRACTIONAL CALCULUS IN EMULSIFICATION PROCESSES

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1 14 th European Conference on Mixing Warszawa, September 2012 APPLICATION OF INFORMATION ENTROPY AND FRACTIONAL CALCULUS IN EMULSIFICATION PROCESSES Barbara Tal-Figiel Cracow University of Technology, ul. Warszawska 24, Kraków, Poland; Abstract. A new concept of investigation on emulsification process using mechanical mixing and ultrasonication is presented. A description of this process has been proposed based on Rényi s residual entropy and fractional entropy. This method is very useful since it can be applied to measure tail heaviness even for the distributions for which the kurtosis measure does not exist. Rényi generalized Shannon entropy. The cumulative residual entropy (CRE) has been found to be a new measure of information that paralells Shannon entropy. The folded normal, lognormal and folded Cauchy, distributions which are commonly used in reliability modeling, have been characterized using this measure. The analysis of experiments was carried out employing the probability droplet distribution variation with the time of emulsification process. Keywords: Emulsification, Rényi entropy, residual Rényi entropy, shape measure, stochastic order, tail heaviness. 1. INTRODUCTION In most emulsification processes the droplet size distribution is regarded as main criterion for product quality. Unfortunately, basic statistical distribution models utilized frequently, to describe typical liquid-liquid emulsions, fail in the case of contemporary cosmetic and pharmaceutical emulsions, which are usually heavily loaded with surfactants and rheology modifiers therefore their droplet size distributions often can be regarded as heavy tailed. Since many chemical engineering processes, including mixing, can be modeled using the probability terms, information entropy offers a good chance to treat such phenomena [1]. Shannon entropy, used in information theory, does not guarantee the best generalization. Split criteria based on generalized entropies offer different compromise between purity of nodes and overall information gain. Rényi generalizes the Shannon entropy [2,3] as:, ln d, 1, 0. (1) It should be noted, that as β 1, the Rényi entropy tends to Shannon entropy, which can be regarded as the negative expected loglikelihood. Considering Rényi entropy as a function of β, H(X,β) may be called the spectrum of Rényi information. It has been observed that the gradient of the spectrum at β = 1 is the negative half of the variance of the loglikelihood. The variance of the loglikelihood can be used to measure the intrinsic shape of the distribution. Nadarajah and Zografos [4] have derived expressions of L f for different univariate distributions. L f (t) is called residual Rényi information measure or shape measure for used item. These expressions of L f (t) are derived for 3 univariate distributions, that are heavy tailed or for which kurtosis measure does not exist but are useful in reliability analysis. Based on 461

2 Rényi entropy datasets of emulsification process have been tested. This measure has a lot of applications in describing nonlinear dynamical and chaotic systems. Analysis of experiments was carried out employing the probability droplet distribution variation with the time of emulsification process. 2. RÉNYI INFORMATION MEASURES FOR DIFFERENT DISTRIBUTIONS Levy flights are Markovian stochastic processes whose individual jumps have lengths that are distributed with the probability density function (PDF) λ(x) decaying at large x as with 0<α<2. Due to the divergence of their variance,, extremely long jumps may occur, and typical trajectories are self-similar, on all scales showing clusters of shorter jumps intersparsed by long excursions. In fact, the trajectory of a Levy flight has fractal dimension d f = α. Similar to the emergence of the Gaussian as limit distribution of independent identically distributed (IID) random variables with finite variance due to the central limit theorem, Levy stable distributions represent the limit distributions of iid random variables with diverging variance [1 3]. In that sense, the Gaussian distribution represents the limiting case of the basin of attraction of the so-called generalized central limit theorem for α = 2. For sums of independent, IID random variables with proper normalization to the sample size, the generalized central limit theorem guarantees the convergence of the associated PDF to a Levy stable (LS) PDF even though the variance of these random variables diverges. In general, an LS PDF is defined through its characteristic function of the probability density, that is, its Fourier transform:, ;, F, ;, d, ;, where expiµk σ k 1 iβ ωk, a, (2) if α 1, 0 2, ωk, a ln k if α 1. (3) tan Thus, one can see that, in general, the characteristic function and, respectively, the LS PDF are determined by the four real parameters: α, β, μ, and σ. The exponent α [0, 2] is the index of stability, or the Levy index, β [ 1, 1] is the skewness parameter, μ is the shift parameter, and σ > 0 is a scale parameter. The index α and the skewness parameter β play a major role in our considerations, since the former defines the asymptotic decay of the PDF, whereas the latter defines the asymmetry of the distribution. The shift and scale parameters play a lesser role in the sense that they can be eliminated by proper scale and shift transformations,, ;,, ; 0,1. (4) Due to this property, in what follows we set μ = 0, σ = 1, and denote the Levy stable PDF, ; 0,1 by,. We note the important symmetry property of the PDF, namely,,. For instance, the asymptotics of the PDF,, x 0 (x>0) or x have the same behavior as,, x 0 (x<0) or x. One easily recognizes that a stable distribution, is symmetric if and only if β = 0. Stable distributions with skewness parameter β = ±1 are called extremal. One can prove that all extremal stable distributions with 0 < α < 1 are one sided, the support being the positive semiaxis if β=+1, and the negative semiaxis if β= 1. For instance,, is only defined for x

3 Only in three particular cases can the PDF, be expressed in terms of elementary functions: (i) Gaussian distribution, α = 2, p x exp. (5) In the Gaussian case β is irrelevant since tan π = 0, and the variance is equal to 2. In this case, the generalized central limit theorem coincides with the traditional central limit theorem. (ii) Cauchy distribution, α = 1, β = 0, p, x. 6 Entropy functions based on fractional calculus are very interesting. This new entropy has the same properties as the Shannon entropy except additivity. This entropy function satisfies the Lesche and thermodynamic stability criteria. The Rényi entropy has similar properties as the Shannon entropy: it is additive, it has maximum ln(n)for p i =1/n. The residual Rényi entropy is defined as HX, β; t ln F dx, β 1, β 0, 7 where X has probability density function f and survival function. It is to be noted that, as t -, H(X, β; t) tends to Rényi entropy. Looking into the importance of Rényi entropy and corresponding information measure (or measure of tail heaviness, or shape measure), this section is devoted to the measures of shape for different univariate distributions. Here we have calculated Rényi entropy for a used item whose lifetime follows some univariate distribution. The corresponding measure and the Shannon entropy H(X;t) for used item are also reported here. (iii) Folded normal distribution Let the probability density function be given by fx µ exp, x 0. (8) The residual Rényi entropy is given by HX, β; t 1β ln βln21φµ ln 21Φµ ln β, 9 where Φ(.) is the cumulative distribution function of N(0;1). Then the residual Shannon entropy is given by where HX; t lnft ln ; µ F, (10) Ft 21Φ µ, t. (11) The Rényi entropy is then obtained as HX, β ln ln β, (12) which gives Shannon entropy as ln. (13) The corresponding shape measure for the life distribution of a used item is given by t AA" A, (14) A 463

4 where A1 21Φ µ, (15) A 1 µ µ exp, (16) and A " 1 µ exp µ 1µ. (17) The shape measure for the life distribution of a new item is ½. (iv) Folded Cauchy distribution The Rényi entropy is obtained as HX, β ln ln B,β βlnπ, (18) The Shannon entropy is thus ln2. (19) The corresponding shape measure for the life distribution of a used item is given by t ψ ;t ψ 1; t. 20 The shape measure for the life distribution of a new item is 3. (21) (v) Lognormal distribution Let the probability density function be given by f x exp, x 0, 0, 0. (22) The residual Rényi entropy is given by HX, β; t ln 1 σ Φ lnρt 1 σ β ln 1 Φ ln 1, (23) where Φ is the cumulative distribution function of the standard normal variate. The residual Shannon entropy is obtained as HX; t lnd t ln2πeσ lnρ, (24) Where 1Φ, 1 and is the standard normal density function. The Rényi entropy is given by HX, β ln 1 σ. (25) The Shannon entropy is given by HX ln2πeσ lnρ. (26) 3. EXPERIMENTAL An experimental investigation of cosmetic and pharmaceutical emulsions of various types obtained with mechanical mixing with high stirrer speed and ultrasonication has been carried out. Emulsifiers used in experiments were commercial products. SRO and Span 80 are liquids miscible with oils, while sodium lauryl glutamate and Tween 80 are water soluble surfactants. For the mechanical mixing turbine impeller or homogenisator was used. Ultrasonication was 464

5 carried out with generator with operating frequency of 22.5 khz with sonotrode diameter d s =0.02 m and ultrasound field intensity W/m 2. Sample size ranged from ml, depending on the required power density value. Cosmetical type O/W and W/O emulsions were made by stirring 1, 3, 5, 10 and 20 min at rpm. For pharmaceutical emulsions two step emulsification process was used. In the first stage emulsion was prepared by intensive homogenization ( rpm), and in the second stage emulsion was formed by ultrasonication, where irradiation times varied from s. The experimental droplet size distributions were obtained with Malvern Mastersizer 2000 device. The samples of emulsions were used to describe the droplet size distribution at different sampling moments. Figure 1. Wax emulsion droplet and their volumetric size distribution Volume (%) Particle Size Distribution Particle Size (µm) The frequency histograms obtained could be analyzed by means of various distributions. According to experimental results, the droplet size distribution at different sampling moments of investigated emulsions can be described by Folded normal distribution Lognormal distribution Folded Cauchy distribution 4. CONCLUSIONS This paper reviewed two important mathematical tools, namely the fractional calculus and the entropy; these concepts allow to improve the analysis of system dynamics of emulsification processes. The paper analysed multi-droplet systems with integer and fractional order behavior and demonstrated that the above concepts are simple and straightforward to apply. The generalized residual information measure based on Rényi entropy for a continuous random variable is very useful in practice, and its connection to the likelihood is established. An intrinsic loglikelihood-based residual life distribution measure is developed without finiteness of moments and/or smoothness assumptions. The proposed L f (t) measure exists for almost all life distributions even if the traditional kurtosis measure does not exists. This measure can be used in studying residual distributional shapes especially for heavy-tailed distributions. The forms of residual Rényi entropy, residual Shannon entropy, shape measure for the life distribution of used item have been derived for 6 different univariable distributions. Rényi entropy, Shannon entropy and shape measure for the life distribution of a new item have been obtained as particular cases. This, as well as applications of non-standard entropies to the text classification data, where small classes are very important, remain to be explored. The results presented here can serve as a reference for scientists and engineers in many areas. 465

6 5. REFERENCES [1] Ogawa K., Chemical Engineering. A New Perspective, Elsevier, Amsterdam. [2] Rényi A., On Measures of Entropy and Information, Proceedings of the Fourth Berkeley Symp. on Mathematics, Statistics and Probability, University of California Press, Vol [3] Życzkowski K.,2003. Renyi Extrapolation of Shannon Entropy, Open Sys. & Information Dyn. 10, [4] Nadarajah S., Zografos K., Formulas for Rényi Information and Related Measures for Univariate Distributions, Inform. Sciences, 155, [5] Masiuk St., Rakoczy R., Kordas M., Informational Analysis of the Grinding Process of Granular Material Using a Multi-ribbon Blender, Chem. Papers, 63, [6] Ubriaco M.R., Entropies based on fractional calculus, Physics Letters A, 373,