Fig. 11. BatchColumn Simulator bang-bang approximation and variable reﬂux policies for p-xylene-n-hexane at total number of plates N = 5.
5. Conclusion
Nearly thirty years ago, literature works noticed with the help of simulations that the cyclic reﬂux policy (bang-bang type control) is more performant than variable reﬂux (singular type control) or constant reﬂux policy when the load amount of light product is low. However, the optimal control policy reported in the relevant and numerous literature, from the 60 s to 2000 was systematically an optimized variable reﬂux policy. This has prompted to revisit the well-known optimal control problem of a **binary** **non**-**ideal** zeotropic **mixture** distillation in an N-stage batch column under assumptions of total condensation and constant vapour ﬂow rate. The problem of maximizing the amount of distillate product with a desired purity in a ﬁxed time is investigated. Results are obtained by using the direct method based on a full discretization of the optimal control problem. A novel robust problem formulation is proposed based on full column dynamics and using the distillate ﬂow rate as control variable instead of the usual reﬂux and sev- eral assumptions made in literature’s optimal control formulation are released. The purity constraint is also handled as a new state variable, the purity deviation. For the ﬁrst time, the reﬂux policy is obtained as the optimal control when the separation is carried out using a plate number near its minimum for three **non**-**ideal** mixtures. The results are conﬁrmed by rigorous simulation of the batch distillation, showing that the cyclic policy improves by 13% the product recovery over the variable reﬂux policy. Inﬂuence of the relative volatility, vapour ﬂow rate, plate hold-up and initial load is discussed.

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3. Conclusions
Batch distillation inherent advantages has initiated recent search for process feasibility rules enabling the separation of azeotropic or difficult zeotropic **binary** mixtures thanks to the addition of an entrainer. A systematic procedure enabling to find a suitable **non** extractive batch distillation process and eventually a suitable entrainer for the separation of zeotropic or azeotropic **binary** **mixture** is described. It brings together into practical use batch distillation process feasibility rules, chemical affinity insight and thermodynamic data analysis available in the literature. The procedure has been implemented in a wizard computer tool and is illustrated on the separation of the water – acetonitrile **binary** homoazeotrope. Through this tool, all possible 224 feasibility rules and 326 batch distillation sequence processes are checked systematically for each entrainer. The graphical tools enables to compare efficiently the entrainer and select the candidates needing further investigation.

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Maximum distillate problem Direct method
a b s t r a c t
We revisit the maximum distillate optimal control problem of batch distillation of **non**-**ideal** **binary** zeotropic mixtures. The direct method with full discretization is used. The problem formulation is based on full column dynamics and the distillate ﬂow rate is used as control variable instead of the reﬂux. The purity constraint is handled as a new state variable, the purity deviation. Literature simulations showed that the cyclic reﬂux policy (bang-bang type control) performs better than variable reﬂux (singular type control) or constant reﬂux policy for small amount of light product in the load. For the ﬁrst time, a cyclic reﬂux policy is found as the optimal control solution. The results are conﬁrmed by rigorous simulation of the batch distillation, as the cyclic policy improves by 13% the product recovery over the variable reﬂux policy. Inﬂuence of the relative volatility, vapour ﬂow rate, plate hold-up and initial load is discussed.

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44 Chapter 5. Budget analysis
5.1 Introduction
Detailed sensitivity numerical studies have shown that the cell-size may have a drastic effect on the modelling of circulating fluidized beds. Typically the cell-size must be of the order of few particle diameters to predict accurately the dynamical behaviour of a fluidized bed. However, Euler-Euler numerical simulations of industrial processes are generally performed with grids too coarse to allow accurate prediction of local segregation effects. Those effects may be accounted for using the modelling using filtered approaches. A fil- tered approach is developed where the unknown terms, called sub-grid contributions, have to be modelled. Highly resolved simulations are used to develop the model. Those simula- tions based on Euler-Euler approach, are performed with grid refinement small enough to reach the mesh independence. Then spatial filters can be applied in order to quantify each sub-grid contribution appearing in the theoretical filtered approach. Such numerical simu- lations are very expensive and are restricted to very simple configurations. In the present study, highly resolved simulations are carried out in order to investigate the sub-grid contri- butions in case of **binary** particle mixtures in a periodic circulating gas-solid fluidized bed. A budget analysis is conducted to understand and model the effect of sub-grid contributions on the hydrodynamic of polydisperse gas-solid circulating fluidized bed.

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labeled C3-R2-n m 4-6It) and even with the **binary** 16-state
**binary** turbo code, which displays a lower decoding threshold than the **binary** 64-state code.
A comparison of the computational complexity was also carried out. First, Table III presents the values of the upper bound on the number of ACS for the proposed reduced- complexity decoding algorithm obtained according to (6) and that of the full Min-Log-MAP algorithm obtained with (5), as well as the percentage gain for the three simulated configurations.

disturb measurements.
In fact, an important issue with **binary** **mixture** is to know if the system is in a single phase. Multiple pressure sensors along the die would be necessary to check this assumption [27] . However, our system was set with only one sensor at the entrance of the die. Thus, an evaluation of the exact location of desorption in the die was made. A linear depressurisation was assumed from the entrance to the exit [28] . This linear depressurisation was conﬁrmed by modelling the die ﬂow with Comsol Multiphysics 1

Keywords: gas-solid flow, circulating fluidized bed, polydisperse **mixture**, filtered approach, subgrid drag force, particle-particle colli- sion
1. Introduction
Gas-solid reacting circulating fluidized beds are used in many industrial processes such as fluid catalytic cracking (FCC) in petroleum refineries or fossil combustion in power plants. Modelling of gas-particle flows, using the Two-Fluid Model (TFM) approach closed by the Kinetic Theory of Granular Flows (KTGF) is well established [3, 5, 11]. However, for A-type par- ticles, according to Geldart’s classification, the numerical sim- ulations with coarse grids fail to predict the behaviour of the solid phase due to the inaccurate prediction of the solid clus- ters [2, 13, 14]. For instance, Parmentier et al. [14] shows that neglecting the small structures in dense fluidized bed simulation leads to an overestimation of the bed height. According to Ozel et al. [13] and Andreux et al. [1] the insufficient mesh refinement leads to underestimate the vertical solid mixing and residence time in circulating fluidized bed. In addition, in bi-solid circu- lating fluidized bed, Batrak et al. [3] show that the "coarse mesh" simulation leads to underestimate the momentum coupling be- tween particle species with large density ratio and, consequently, the entrainment rate of the coarse particles by the small ones. The modelling strategy consists in splitting the local instanta- neous variables in resolved and subgrid contributions. The corre- sponding governing equations are obtained by filtering the parti- cle kinetic moment transport equations leading to unknown sub- grid terms which need to be modelled in terms of the computed resolved variables. The first step is the budget analysis of the fil- tered transport equation and the evaluation of the resolved and subgrid contributions of each terms. This step highlights which term(s) need(s) to be modelled, leading to the subgrid modelling step to account accurately for the effect of the small scale struc- tures in the filtered equations. For monodisperse simulations, the drag force is overestimated by the resolved contribution and may lead to wrong predictions of the global hydrodynamic behaviour [15]. Several sub-grid models have been proposed by Ozel et al. [13], Parmentier et al. [15] to take into account the effect of the unpredicted solid cluster on filtered fluid-particle drag force mod- elling. For polydisperse simulations, the particle-particle mo-

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Currently, cluster analysis is more and more concerned with large datasets where obser- vations are described by many variables. This large number of predictor variables could be beneficial to data clustering. Nevertheless, the useful information for clustering can be con- tained into only a variable subset and some of the variables can be useless or even harmful to choose a reasonable clustering structure. Several authors have suggested variable selec- tion methods for Gaussian **mixture** clustering which is the most widely used **mixture** model for clustering multivariate continuous datasets. These methods are called “wrapper” since they are included into the clustering process. Law et al. (2004) have introduced the feature saliency concept. Regardless of cluster membership, relevant variables are assumed to be independent of the irrelevant variables which are supposed to have the same distribution. Raftery and Dean (2006) recast variable selection for clustering into a global model selection problem. Irrelevant variables are explained by all the relevant clustering variables according to a linear regression. The comparison between two nested variable subsets is performed using Bayes factor. A variation of this method is proposed in Maugis et al. (2007) where irrelevant variables can only depend on a relevant clustering variable subset and variables can have different sizes (block variables). Since all these methods are based on a variable selection procedure included into the clustering process, they do not impose specific con- straints on Gaussian **mixture** forms. On the contrary, Bouveyron et al. (2007) consider a suitable Gaussian **mixture** family to take into account that data live in low-dimensional subspaces hidden in the original space. However, since this dimension reduction is based on principal components, it is difficult to deduce from this approach an interpretation of the variable roles.

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Computational experiments for the general case: Random instances: results depend on ratio m. n , adding 2-links helps[r]

§
Correspondence author. Fax: +33534322899, Email: akhouzam@imft.fr
ABSTRACT In this paper, a numerical and analytical analysis is performed in order to improve the
species separation process in a **binary** fluid **mixture** by decoupling the thermal gradient from the convective velocity. The configuration considered is a horizontal rectangular cavity, of large aspect ratio, filled with a **binary** fluid. A constant tangential velocity is applied to the upper horizontal wall. The two horizontal impermeable walls are maintained at different and uniform temperatures T 1 and T 2 with =T T 1 -T 2 . Species separation is governed by two control parameters, the temperature difference T and the velocity of the upper plate Ue x . The intensity of the thermodiffusion is controlled by the temperature, while the velocity Ue x controls the convective flow. This problem depends on six dimensionless parameters, namely, the separation ratio, , the Lewis number, Le, the Prandlt number Pr, the aspect ratio of the cell, A and two control parameters: the thermal Rayleigh number, Ra and the Péclet number Pe. In this study, the formulation of the separation (mass fraction difference between the two ends of the cell) as a function of the Péclet number and the Rayleigh number is obtained analytically. For a cell heated from below, the optimal

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Coupling sc-CO 2 and extrusion modifies the rheological properties of the polymer while flowing through the barrel of the extruder. The reduction of viscosity decreases the mechanical constraints and the operating temperature within the extruder. Thus, this may allow the use of fragile or thermolabile molecules, like pharmaceutical ones. Furthermore, the absence of residues in the final material is also an advantage for a pharmaceutical application. Using a new sc-CO 2 -assisted extrusion process that leads to the manufacturing of microcellular polymeric foams, we have elaborated microcellular foams of a biocompatible amorphous polymer [4, 5]. However, understanding and improving such a process requires the knowledge of physical properties, like the solubility of CO 2 into the polymer, the diffusion coefficient and the viscosity of the **mixture**. An extruder provides a high shear rate, particularly in the die. However, the viscosity of the **binary** system under process conditions is very difficult to reach in conventional rheometers. Viscosity under pressure can be measured in different ways [6]. One group of measurements is based on vibrating surface or falling ball rheometers [7]. These techniques require Newtonian or low viscosity polymers. The other main group of measurement techniques is based on capillary rheometry. This technique is widely used for viscous polymers and implementations exist to measure the rheology of mixtures of polymer and sc-CO 2 [8]. We have implemented this on-line technique on a single- screw extruder in order to estimate the viscosity of a bio-sourced polymer and of the **binary** system under process conditions.

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nodes, when the recursive updating of [10] is adopted.
3. Simplified decoding in the logarithm domain
The algorithms presented in the previous section are mathematically equivalent and present the same performance. The main restriction of all these is the limitation on the field order at a reasonable hardware complexity. We present in this section the EMS algorithm which represents an interesting simplified solution for reduced-complexity **non**-**binary** LDPC decoding. We focus on two implementations: the one presented in [4] and a more recent and simplified one [8].

de sa relation à ce grand frère. Puis il parle un peu de ses copains. Je demande s’il y a aussi des filles dans la bande. Il dit oui. Je demande si c’est différent. Il dit : **non**, on est tous pareil. Je m’exclame : oh, quel ennui ! Il me regarde en souriant, interloqué, intéressé. Un espace de parole me semble pouvoir s’ébaucher et Romain parle plus volontiers. Il m’explique que c’est difficile depuis deux ans. Sa mère est triste, ça se voit. Il aimerait connaître son histoire familiale. Nous parlons de ce qu’il en sait ou imagine. Il voudrait connaître des choses sur son père, comprendre pourquoi il est comme ça. Comme ça ? Je demande. C’est compliqué, il est plein de chose, il est paranoïaque… je n’aurai pas le temps d’expliquer là…. Cette ébauche d’échange ne durera pas. Avant la séance suivante, Madame m’appelle pour dire que Romain était en chemin puis a fait demi-tour. Il était trop mal pour venir, il a appris dans la journée qu’il risquait une exclusion. La séance suivante, Romain demeure silencieux tout en regardant longuement les livres sur la bibliothèque à côté du fauteuil. Au bout d’un moment je lui dis qu’il peut les prendre pour les regarder. Il me regarde, surpris, puis extirpe un livre dont le titre semble l’avoir retenu, lit la 4 ème

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2. Overview
The study of linear stability of compressible boundary layers on a flat plate is not new. Motivated by applications related to the civil or military aeronautical sector, different works have very quickly sought to understand the influence of real gas effects on the stability of a boundary layer, including vibrational excitation, dissociation and recombination of gas species, ionization, radiation and surface ablation (for example, Malik & Anderson 1991 ) and thermochemical **non**-equilibrium reactions (for example, Stuckert & Reed 1994 ). However, these studies do not deal with certain aspects of **non**-**ideal** gases, such as a strong thermodynamic stratification or the modification of transport properties (speed of sound, viscosity, conductivity, etc.), that occur when the fluid evolves close to the critical point. **Non**-**ideal** fluids represent a research field of great importance for a wide range of applications; the work of Ren et al. ( 2019b ) aims to fill this gap.

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Abstract. Accurate observational data are available for the eclipsing double-lined spectroscopic **binary** system RS Cha, composed of two stars in the pre-main sequence stage of evolution: masses, radii, luminosities and effective temperatures of each component and metallicity of the system. This allows to build pre-main sequence stellar models representing the components of RS Cha and to constrain them in terms of physical ingredients, initial chemical composition and age.

This study investigated the use of a **mixture** of n-hexane and methanol as a **binary** solvent for the direct oil extraction and resin separation from Calophyllum seeds, in a single step. Optimal oil and resin yields and phy- sicochemical properties were determined by identifying the best extraction conditions. The solvent **mixture** tested extracted oil and resin eﬀectively from Calophyllum seeds, and separated resin from oil. Extraction con- ditions aﬀected oil and resin yields and their physicochemical properties, with the n-hexane-to-methanol ratio being the most critical factor. Oil yield improved as n-hexane-to-methanol ratio increased from 0.5:1 to 2:1, and resin yield increased as methanol-to-n-hexane ratio increased from 0.5:1 to 2:1. Physicochemical properties of oil and resin, particularly for acid value and impurity content, improved as the n-hexane-to-methanol ratio de- creased from 2:1 to 0.5:1. The best oil (51% with more than 95% triglycerides) and resin (18% with more than 5% polyphenols) yields were obtained with n-hexane-to-methanol ratios of 2:1 and 0.5:1, respectively, at a temperature of 50 °C, with an extraction time of 5 h. The best values for physicochemical property of oil were a density of 0.885 g/cm 3 , a viscosity of 26.0 mPa.s, an acid value of 13 mg KOH/g, an iodine value of 127 g/100 g, an unsaponiﬁable content of 1.5%, a moisture content of 0.8% and an ash content of 0.04%.

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To cite this version : Özel, Ali and Fede, Pascal and Simonin, Olivier. Effect of
unresolved structures on the Euler-Euler simulation of 3D periodic circulating fluidized of **binary** **mixture**. (2013) In: 8th International Conference on Multiphase
Flow - ICMF 2013, 26 May 2013 - 31 May 2013 (Jeju, Korea, Republic Of).

C0 ð1 C0 Þð T h T c Þ (13)
To make this measurement, it is necessary that the walls be set in motion with the optimal velocity U Popt . The determination of the optimal
velocity requiresthe mass diffusivity of the **binary** **mixture** to be known. In what follows, we show that it is also possible to access the mea- surement of the mass diffusion coefficient D using Equation (7) in the case of f -1.

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A B S T R A C T
In the presence of a gravity field or under microgravity, pure thermo-diffusion leads to very weak species sep- aration in **binary** mixtures. To increase the species separation in the presence of gravity, many authors use thermo-gravitational diffusion in vertical columns (TGC). For a given **binary** **mixture**, the species separation between the top and the bottom of these columns depends on the temperature difference, ΔT, imposed between the two vertical walls facing each other, and the thickness, H, between these two walls (annular or paral- lelepipedic column). These studies show that, for a fixed temperature difference, the species separation is optimal for a thickness, H opt , much smaller than one millimetre. The species separation decreases sharply when the thickness H decreases with respect to this optimum value. It decreases progressively as H increases with respect to H opt . In addition, for mixtures with a negative thermo-diffusion coefficient, the heaviest component migrates towards the upper part of the column and the lightest one towards the lower part. The loss of stability of the configuration thus obtained leads to a brutal homogeneity of the **binary** solution.

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