Edoardo Provenzi
email: edoardo.provenzi@telecom-paristech.fr

Achromatic induction: A variational interpretation of Rudd-Zemach's edge integration model

Keywords: Achromatic induction, edge integration models, variational principles, phenomenology of visual perception

Induction is the term used to describe the fact that visual perception depends not only on the absolute luminance of a patch in a visual scene, but also on the luminance of the surrounding patches. Induction can be achromatic or chromatic, depending on the presence of only shades of gray, or color, respectively. Achromatic induction can be measured through quite simple psychophysical matching experiments, whose data are well fitted by the so-called edge integration models. In this paper we will examine Rudd-Zemach's edge integration model and we will show that it can be interpreted in terms of a variational framework that shares many features with a phenomenologically-based visual perception theory recently proposed. The relevance of this result is twofold: firstly, it allows interpreting induction in terms of local contrast enhancement, secondly, it provides a validation of the variational framework in terms of quantitative measures, opening the possibility to determine important parameters of this framework.

I. INTRODUCTION

Human perception of the color of a patch in a visual scene is not determined only by luminance of the patch itself, but also by that of the surrounding patches. This phenomenon is called chromatic induction, to stress the fact that color perception of a patch is induced (and altered) by its surround. What just stated is also true for the so-called achromatic colors, i.e. perceived shades of gray, in which case one talks about achromatic induction.

Following a common nomenclature, we will call lightness the perceived luminance reflected by a non selfluminous patch, while brightness will refer to the perceived luminance emitted by a source of light. Thus, the terms lightness and brightness already incorporate the effect of induction in their definition.

The most elementary example of induction is the simultaneous contrast phenomenon, depicted in Fig. 1: the inner squares have exactly the same luminance, however we perceive them very differently because we are strongly influences by their distinct surrounds.

Induction can be measured through psychophysical experiments. The first quantitative measure of achromatic induction has been performed by H. Wallach in [START_REF] Wallach | Brightness constancy and the nature of achromatic colors[END_REF]. In his classical experiment, see Fig. 2 (courtesy of [START_REF] Rudd | Quantitive properties of achromatic color induction: An edge integration analysis[END_REF]), Wallach considered two disks, T and M for Target and Match, embedded in black background B, with luminance D T and D M , surrounded by two rings of luminance R T and R M , respectively. He showed this configuration to a set of observers adapted to the light condition of a dimly illuminated room keeping D T and R M fixed, using R T as an independent variable that he could fix in every experiment, and D M as a dependent variable that the users could adjust in order to achieve a perceptual match between the two disks T and M . Notice that the luminance values selected fall in the scotopic (nearly mesopic) range, to assure that only achromatic stimuli are generated. If the luminance of the surrounding rings failed to influence the perception of the achromatic color of the disks, then the match among the two disks would be simply the photometric match, i.e. D M = D T , instead Wallach found that a fairly good achromatic color match among the two disks was obtained when the ratios between the disk and the ring luminances were identical on the two sides of the display, i.e.

D M R M = D T R T , (1)
a formula called Wallach's Ratio Rule. By taking the Briggs (decimal) logarithms at both sides and solving for D M we find:

LogD M = LogD T + LogR M -LogR T , (2)
thus, according to Wallach's Ratio Rule, the plot of the perceptual match in the plane of coordinates (x, y) = (LogR T , LogD M ) should be a straight line with slope -1, against the slope 0 that a photometric match would measure.

More recent measures on the classical Wallach's experiment show that this slope is actually between -1 and 0, as it can be seen in Fig. 3 (adapted from [START_REF] Rudd | Quantitive properties of achromatic color induction: An edge integration analysis[END_REF]).

Figure 3. Quantitative measures of Wallach's achromatic color induction experiment for four observers performed in [START_REF] Rudd | Quantitive properties of achromatic color induction: An edge integration analysis[END_REF]. The best-fit regression line slopes and associated 95% confidence limits reported in [START_REF] Rudd | Quantitive properties of achromatic color induction: An edge integration analysis[END_REF] are the following: -0.639±0.033 (LT), -0.791±0.034 (IKZ), -0.723±0.047 (JL), and -0.657±0.042 (AD).

To account for these new psychophysical data, M.E. Rudd and I.K. Zemach have proposed a more sophisticated model than Wallach's. Rudd-Zemach's model will be detailed in Section II. In Section III a generalization of Rudd-Zemach's model to arbitrary spatial configurations and its variational interpretation will be discussed, followed Section IV, in which the relationship with the perceptuallyinspired variational framework of [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF] will be analyzed. The interest of this result is twofold: from one side, the variational framework allows understanding induction in terms of local contrast enhancement, on the other side, the psychophysical measures on which Rudd-Zemach's model rely give the first quantitative validation of the variational framework introduced in [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF] and can be used to determine important parameters within this framework.

II. RUDD-ZEMACH'S EDGE INTEGRATION MODEL FOR ACHROMATIC INDUCTION

Rudd and Zemach in [2] repeated Wallach's experiment adding a non-black background B. The experimental setting is described in detail in [START_REF] Rudd | Quantitive properties of achromatic color induction: An edge integration analysis[END_REF]. As in Wallach's experiment, D T and R M are fixed and the observer's task is to adjust D M to achieve an achromatic color match to the test disk as a function of R T . R T is varied from trial to trial by random sampling from a set of six luminance values spaced equally in units from 2.54 to 6.31 cd/m 2 (0.405 to 0.800 Log cd/m 2 ).

Rudd and Zemach claim that the inspiration for their model of achromatic induction is the Retinex theory of color perception of Land and McCann [START_REF] Land | Lightness and Retinex theory[END_REF], however, they use a version of Retinex without the so-called threshold and reset mechanisms, see [START_REF] Provenzi | Mathematical definition and analysis of the retinex algorithm[END_REF] for more details.

Let L i and L j be the luminance of two points i and j in an image, the ratio Lj Li can be decomposed as a chain multiplication of the local luminance ratios at borders encountered along a path connecting j and i:

L j L i = j-1 ∏ k=i L k+1 L k , ( 3
)
by taking the logarithm at both sides we get

Log L j L i = j-1 ∑ k=i Log L k+1 L k . ( 4
)
Rudd and Zemach introduced induction strength weights w k , in order to take into account locality of vision, i.e. the different influence of nearby vs. far away luminance, and defined the logarithmic brightness of i as follows:

Φ i ≡ j-1 ∑ k=i w k-i+1 Log L k+1 L k . ( 5
)
Notice that, if the luminances L k and L k+1 are equal, then their ratio does not give any contribution to Φ i . A meaningful contribution to Φ i is given only by the luminances of points lying at the border of an edge. So, Φ i represents the summed influence of all the edges present within the spatial surround on the target point, suitably weighted. The index of the weights is such that small values of the index label patches close to i and large values of the index label patches far away from i. With this convention, and knowing that induction strength declines with distance [START_REF] Wallach | The perception of neutral colors[END_REF], we have that w 1 > w 2 > . . ., i.e. w2 w1 < 1, and so on.

Rudd and Zemach call their model of achromatic induction 'Weighted Log Luminance Ratio', or WLLR for short. If WLLR were the true model of achromatic induction, then an observer would match the brightness (perceived luminance) of the disks D M and D T of Fig. 2 when this equation holds

w 1 Log D M R M + w 2 Log R M B = w 1 Log D T R T + w 2 Log R T B , (6)
solving this equation w.r.t. LogD M we have

LogD M = LogD T + ( 1 - w 2 w 1 ) LogR M - ( 1 - w 2 w 1 ) LogR T . ( 7
)
The WLLR model thus predicts that if we plot the Log of observer's matching disk settings versus the Log of the test ring luminance, the data should fall on a straight line having a slope β = -

( 1 -w2 w1
) ∈ (-1, 0), which is coherent with Rudd and Zemach's observations. The estimation of the ratio w2 w1 from their interpolated data for the four observers is: 0.361 (LT), 0.209 (IKZ), 0.277 (JL), and 0.343 (AD).

If we set

c 1 ≡ LogD T + ( 1 -w2 w1 ) log R M and c 2 ≡ w2 w1 -1, then LogD M = c 1 + c 2 LogR T = c 1 + LogR c2 T , so D M = 10 c1+LogR c 2 T , i.e. D M = kR T c2 , ( 8
)
where k = 10 c1 . This means that the brightness of D M is proportional to a power function of R T with exponent given by w2 w1 -1, which, in the experiments of Rudd and Zemach, ranged from -0.639 to -0.791.

A key assumption of edge integration models, as WLLR, is that the total achromatic color induction produced by a complex background is the sum of the individual induction effects produced by the luminance borders comprising that background. Rudd and Zemach performed experiments to directly test this assumption by predicting and then measuring the magnitude of the total induction effect produced by combining three circular edges located at different distances from the test disk.

This was done after first measuring the magnitudes of the induction effects produced by the individual edges. The combination of three inducing edges was produced by surrounding the test disk with two concentric rings as in Fig. 4 (courtesy of [START_REF] Rudd | Quantitive properties of achromatic color induction: An edge integration analysis[END_REF]). The observers in the two-ring experiment adjusted D M as a function of R2 T to achieve an achromatic color match between the test and matching disks. R1 T was fixed at the value 6.31 cd/m 2 (0.800 Log cd/m 2 ) and R2 T was randomly varied from 2.54 to 6.31 cd/m 2 (0.405 to 0.800 Log cd/m 2 ) in six steps of equal units. D M and R M were fixed at the values 1.02 cd/m 2 (0.009 Log cd/m 2 ) and 3.96 cd/m 2 (0.597 Log cd/m 2 ) respectively. The width of the matching ring was 0.35 deg (same as in the one-ring experiment).

According to the WLLR model, an achromatic color match should be achieved for this stimulus when this equation holds

w 1 Log D M R M + w 3 Log R M B = w 1 Log D T R1 T + w 2 Log R1 T R2 T + w 3 Log R2 T B , (9)
where

R1
LogD M = LogD T + ( 1 - w 3 w 1 ) LogR M - ( 1 - w 2 w 1 ) LogR1 T + ( w 3 w 1 - w 2 w 1 ) LogR2 T . ( 10
)
This equation predicts that a Log-Log plot of the observer's matching disk settings versus outer ring luminance should form a straight line having the slope w3 w1 -w2 w1 . To test this prediction, they first measured the induction strength ratios w 3 /w 1 and w 2 /w 1 using eq. ( 7) in independent experiments in which the test disk was surrounded by a single ring only. Thus, accordingly to the WLLR model, the slopes of the lines depicting Log D M in function of Log R2 T should be given by the difference between the ratios w 3 /w 1 and w 2 /w 1 previously measured. The results are in accordance with the predictions.

III. GENERALIZATION OF RUDD-ZEMACH'S MODEL TO ARBITRARY SPATIAL CONFIGURATIONS AND ITS VARIATIONAL INTERPRETATION

It is natural to search for a generalization of Rudd-Zemach's model valid for arbitrary spatial configurations and not only the very special one discussed in their experiments. To do that, let us introduce some convenient notation. We denote the visual spectrum as Λ = (380, 780)nm and the 2-D projection of the spatial domain of the scene as Ω ⊂ Z 2 . x = (x 1 , x 2 ) and y = (y 1 , y 2 ) will be the coordinates of two arbitrary points in Ω.

The spectral luminance function ℓ : Ω × Λ → (0, +∞) can be decomposed in three luminance functions relative to the retinal cones, performing a weighted integration over Λ with weights given by the spectral sensitivity functions r(λ), ḡ(λ), b(λ) of the cones:

     L r (x) = ∫ Λ ℓ(x, λ)r(λ) dλ, L ḡ (x) = ∫ Λ ℓ(x, λ)ḡ(λ) dλ, Lb(x) = ∫ Λ ℓ(x, λ) b(λ) dλ.
(11)

Achromatic induction can be studied on the scalar components of ⃗ L(x) = (L r (x), L ḡ (x), Lb(x)), that will be written as L(x) without the subindex for a clearer notation.

With this notation in mind, the achromatic logarithmic brightness of an arbitrary point x in Rudd-Zemach's model can be written as follows

Φ(x) = ∫ Ω w(x, y)Log L(y) L(x) dy (12)
or, equivalently,

Φ(x) = ∫ Ω w(x, y) [Log L(y) -Log L(x)] dy. ( 13
)
The following proposition provides a variational interpretation of this formula.

Proposition 1: Given the functional

E w (LogL) = 1 2 ∫ Ω ( Log L 0 (x) L(x) ) 2 dx - 1 4 ∫ ∫ Ω 2 w(x, y) ( Log L(y) L(x) ) 2 dxdy, ( 14
)
where L 0 (x) is the photometric luminance of x and must be interpreted as a constant, the Rudd-Zemach achromatic logarithmic brightness Φ(x) of an arbitrary point x ∈ Ω satisfied this equation

Φ(x) = argmin Log L E w (Log L) + Log L 0 (x). ( 15
)
The proof of this proposition is provided in the Appendix. Here, we would like to stress its interpretation: it is convenient to write the functional terms appearing in the energy as follows, D(LogL) ≡

1 2 ∫ Ω ( Log L0(x) L(x)
) 2 dx and call it quadratic dispersion term around the original logarithmic luminance, and

C w (LogL) ≡ 1 4 ∫∫ Ω 2 w(x, y) ( Log L(y) L(x)
) 2 dxdy and call it local quadratic contrast term in the logarithmic domain. The minimization of E w (LogL) = D(LogL)-C w (LogL) is reached through the simultaneous minimization of D(LogL) and maximization of C w (LogL) (because of the minus sign in front of the contrast term). In particular, the constant term Log L 0 (x) is generated by the minimization of the dispersion term and it does not appear in Rudd-Zemach's edge integration model, this is why it must be added in order to get Φ(x).

Thus, we can say that Rudd-Zemach's logarithmic brightness Φ(x) corresponds to the maximization of the local quadratic contrast in the logarithmic domain, where locality is given by the weight function w(x, y).

IV. RELATIONSHIP BETWEEN RUDD-ZEMACH'S MODEL AND VARIATIONAL THEORIES OF PERCEPTUAL COLOR CORRECTION

The result just obtained has many similarities with the theory of perceptually-inspired color correction developed in the papers [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF], [START_REF] Bertalmío | Perceptual color correction through variational techniques[END_REF], [START_REF] Bertalmío | Issues about the retinex theory and contrast enhancement[END_REF]. In fact, also the functionals considered in these papers balance dispersion and local contrast to perform perceptual color correction, i.e. to modify the intensities I(x) of pixels in digital images to approach the lightness and brightness of the points in the visual scene that pixels represent.

In [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF], it has been proven that the most general perceptually-inspired energy functional E φ w,α,β (I) = C φ w (I) + D α,β (I) that fulfills the basic HVS properties (local contrast enhancement, adjustment around the average value, color constancy and Weber-Fechner's law of contrast perception) is given by the sum these terms

C φ w (I) ≡ 1 4 ∫ ∫ I w(x, y) φ ( min(I(x), I(y)) max(I(x), I(y))
) dxdy,

(16) the functional parameter φ being a strictly increasing positive function, and

D α,β (I) ≡ α ∫ I ( µ log µ I(x) -(µ -I(x))
) dx

+ β ∫ I ( I 0 (x) log I 0 (x) I(x) -(I 0 (x) -I(x)) ) dx. (17)
Let us discuss the meaning of C φ w (I) and D α,β (I), starting with the contrast term. Notice that the ratio defined by min(I(x), I(y))/ max(I(x), I(y)) is minimized when the minimum decreases and the maximum increases, which of course corresponds to a contrast stretching. Thus, minimizing an increasing function of that ratio will produce a contrast enhancement. Moreover, recalling that a global illuminant change can be represented as the transformation I(x) → λI(x), λ > 0, the contrast enhancement term will be unaffected by such a change, coherently with the color constancy property. Finally, it can be proven [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF] that this definition of contrast is coherent with Weber-Fechner's law.

The dispersion term is based on the relative entropy distance [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] between I and µ (first term) and between I 0 and I (second term). Given the statistical interpretation of entropy, we can say that minimizing D α,β (I) amounts to minimizing the disorder of intensity levels around the average µ and around the original data I 0 (x). Thus, D α,β (I) serves as a global adaptation term.

By minimizing the energy E φ w,α,β , e.g. through a gradient descent technique, we have the explicit algorithm implementation of this model:

∂ t log I = -δE φ w,α,β (I), (18)
t being the evolution parameter and δ the first variation of E. For practical implementations this scheme must be discretized: choosing a finite evolution step ∆t > 0 and setting I k (x) = I k∆t (x), k ∈ N, with I 0 (x) being the original image, we have

I k+1 (x) = I k (x) + ∆t ( αµ + βI 0 (x) + 1 2 R φ w,I k (x)
)

1 + ∆t(α + β) .

(19) As can be seen, α and β represent the strength of the attachment to 1/2 and to I 0 , respectively, while R φ w,I k represents the contrast enhancement. When φ ≡id, the identity function, then

R id I k (x) := ∫ Ω w(x, y) I k (y) I k (x) H(I k (x) -I k (y)) dy - ∫ Ω w(x, y) I k (x) I k (y) H(I k (y) -I k (x)) dy, ( 20
)
I k is the image at the k-th step, I 0 is the original image, H is the Heaviside step function, i.e. H(t) = 1 when t > 0, H(t) = 0 when t ≤ 0.

When α = β = 0, i.e. the adjustment coefficients are set to 0, both equation ( 19) and ( 13) express the brightness computation as a generalized edge integration. The main difference is given by the fact that in (13) there appear logarithmic luminance values, while (19) is written in terms of pixel intensities. This difference is expected, since the transformation from real-world luminance to digital image intensity is a concave function, as it is the logarithm.

It would be interesting to perform psychophysical experiments to test if the attachment to the original luminance and to the average value µ play a significant role in improving the ability of Rudd-Zemach's model to measure achromatic induction.

V. CONCLUSION

We have discussed a variational interpretation of Rudd-Zemach's model of achromatic induction, showing that it shares many similarities with recently built variational models for perceptually-inspired color correction.

We have proven that the edge integration formula on which Rudd-Zemach's is based can be interpreted as the maximization of local quadratic contrast of logarithmic luminances, where locality is given by induction weights quantitatively measured by Rudd and Zemach in [START_REF] Rudd | Quantitive properties of achromatic color induction: An edge integration analysis[END_REF].

This shows with even greater strength the exigence to extend Rudd-Zemach's experiments in many directions: to color stimuli, to a wider number of users, and to a larger luminance range, in order to have more reliable results that would allow building a proper metrology of chromatic and achromatic induction.

This would be highly beneficial, for example, for the quantitative determination of the weight function used in the color correction models [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF], [START_REF] Bertalmío | Perceptual color correction through variational techniques[END_REF], [START_REF] Bertalmío | Issues about the retinex theory and contrast enhancement[END_REF]. Up to now, these functions are Gaussians whose standard deviation can be regulated by users to following their personal tastes and needs in the correction of images. However, in order to be as close as possible to human perception, a quantitative measure of induction weight, as that started by Rudd and Zemach, can set the parameter function w(x, y) in the (25) Thus, the logarithmic brightness Φ(x) can be written as Φ(x) = argmin LogL E w (LogL) + LogL 0 (x) and so Proposition 1 is proven.

2

Figure 1 .

 1 Figure 1. Top: Chromatic simultaneous contrast. Bottom: Achromatic simultaneous contrast.

Figure 2 .

 2 Figure 2. Wallach's classical experiment.

Figure 4 .

 4 Figure 4. Rudd-Wallach's experimental setting to test the additivity hypothesis underlying the WLLR model with two rings.

  most adequate way in relation to achromatic and chromatic color perception.ACKNOWLEDGMENTThe author acknowledges FUI project CEDCA.APPENDIX: PROOF OF PROPOSITION 1Let us compute the variation of the functional E w (LogL) = D(LogL) -C w (LogL) with respect to LogL. To do this, it is convenient to write Log L(x) ≡ L(x). The variation of the first term of the functional is trivial to compute and gives: δD( L, J) being a generic functional perturbation of L(x). The variation of the second term instead can be written as follows δC w ( L, J) the role of the 'mute' variables x and y in the second integral and using the symmetry of the induction weight, we can write δC w ( L, J) δE( L, J) = δD( L, J) -δC w ( L, J), we have that the Euler-Lagrange equation δE( L, J) = 0 for all perturbations J, is satisfied if and only if the argmin L(x) satisfies L0 (x) -L(x) -) -L(x) ] dy -L0 (x).

  T is the luminance of the inner ring surrounding the test disk, R2 T is the luminance of the outer ring, and the subscripts on the induction weights indicate the distance between the border with which they are associated and the test disk perimeter, so w 1 stands for w(0 deg), w 2 stands for w(0.18 deg), and w 3 stands for w(2.48 deg).Solving this equation w.r.t. LogD M we have


Laszlo Hegely

Vincent Gerbaud

Peter Lang

General Model for Studying the Feasibility of Heterogeneous Extractive Batch Distillation

By extending our former model for batch heteroazeotropic distillation, a general model is developed for the batch heterogeneous extractive distillation. An entrainer-rich and an entrainer-lean phase are present in the decanter, where the holdup of both phases can be reduced, kept constant, or increased, thanks to batch operation mode. Any fraction of both phases can be refluxed or withdrawn as distillate. The entrainer is fed continuously either onto one of the plates of the column, or into the decanter. The still path equation is derived for both cases. A discussion is given on the literature occurrence and practical applicability of the 16 possible operational policies, and the effect of continuous entrainer feeding on the still path direction of all operational policies is determined. The still path directions are validated by rigorous simulations for multiple operational policies for the mixture of ethanol, water, and n-butanol.

INTRODUCTION

Batch distillation is a widely applied separation technology in the pharmaceutical and specialty chemical industries. [START_REF] Mujtaba | Batch Distillation: Design and Operation[END_REF] Azeotropic and close-boiling mixtures are often encountered, and the separation of these mixtures requires the application of a special distillation method, such as pressure swing, extractive, and heteroazeotropic distillation. The addition of a new component, the entrainer (E), might alter the relative volatilities of the original components favorably. In the case of homoazeotropic distillation, E is added to the charge in the still at the start of the process, and no heteroazeotrope is present.

If E is added to the charge, and a heteroazeotrope is present in the system, formed either by the entrainer and one of the original components or by two original components, the separation method is called batch heteroazeotropic distillation (BHAD). The original azeotropic composition can be crossed due to the separation of the two liquid phases by decantation. The BHAD is used in the industry only in batch rectifiers equipped with a decanter. In this configuration, the heteroazeotrope must be the unstable node of the residue curve map, so that the top vapor composition lies in the twoliquid-phase region.

Two main types of BHAD can be distinguished, on the basis of whether distillation and the decantation of the two liquid phases occur sequentially (Mode I) or simultaneously (Mode II). [START_REF] Skouras | Feasible separations and entrainer selection rules for heteroazeotropic batch distillation[END_REF][START_REF] Skouras | Analysis and control of heteroazeotropic batch distillation[END_REF] By Mode I, the composition of the reflux and distillate is identical and the final distillate product is heterogeneous. By Mode II, since distillation and phase separation occurs simultaneously, the reflux usually has a different composition than that of the condensate, unlike batch homoazeotropic distillation. Skouras et al. [START_REF] Skouras | Feasible separations and entrainer selection rules for heteroazeotropic batch distillation[END_REF][START_REF] Skouras | Analysis and control of heteroazeotropic batch distillation[END_REF] distinguished two strategies for Mode II. The entrainer-rich phase is totally refluxed by Strategy A, and only partially refluxed by Strategy B. Rodriguez-Donis et al. [START_REF] Rodriguez-Donis | Feasibility of heterogeneous batch distillation processes[END_REF] investigated the BHAD with variable E-rich phase decanter holdup. The E-lean phase had constant decanter holdup, and distillate was withdrawn only from this phase. The partial reflux of the E-lean phase is also allowed, as it can increase the recovery by maintaining the phase split longer. In the model of Lang and Modla, [START_REF] Lang | Generalised method for the determination of heterogeneous batch distillation regions[END_REF] any fraction of either liquid phase can be refluxed (and withdrawn as distillate); however, the holdup of both phases is constant.

Hegely et al. [START_REF] Hegely | Generalised model for heteroazeotropic batch distillation with variable decanter hold-up[END_REF] proposed a general model of BHAD, where both liquid phases can be refluxed or withdrawn as distillate. Moreover, the holdup of both liquid phases in the decanter also can be increased, decreased, or kept constant. The still path equation was derived, and the possible operational policies were identified. Denes et al. [START_REF] Denes | New double column system for heteroazeotropic batch distillation[END_REF][START_REF] Denes | Generalised closed double-column system for batch heteroazeotropic distillation[END_REF] suggested new, closed double-column configurations for BHAD. The charge is divided between the two reboilers, where the two original components are accumulated at the end of the process. The reflux strategy was Mode II, Strategy A.

In batch extractive distillation (BED), the entrainer is fed continuously to the column. The components do not form heteroazeotropes, or even if a heteroazeotrope is present, its existence is not exploited. The entrainer is conventionally chosen in industrial practice as a heavy component, [START_REF] Luyben | Design and Control of Distillation Systems for Separating Azeotropes[END_REF][START_REF] Seader | Separation Process Principles[END_REF][START_REF] Lang | Batch extractive distillation under constant reflux ratio[END_REF][START_REF] Rodriguez-Donis | Thermodynamic Insights on the Feasibility of Homogeneous Batch Extractive Distillation, 1. Azeotropic Mixtures with a Heavy Entrainer[END_REF] and the separation is performed in a rectifier column. However, it is also possible to use a light or intermediate boiling component. [START_REF] Rodriguez-Donis | Thermodynamic Insights on the Feasibility of Homogeneous Batch Extractive Distillation. 3. Azeotropic Mixtures with Light Entrainer[END_REF][START_REF] Rodriguez-Donis | Thermodynamic Insights on the Feasibility of Homogeneous. Batch Extractive Distillation. 4. Azeotropic Mixtures with Intermediate Boiling Entrainer[END_REF] Lelkes et al. [START_REF] Lelkes | Feasibility of extractive distillation in a batch rectifier[END_REF] developed a method to study the feasibility of BED, by calculating feasible profiles of different column sections. Rodriguez-Donis et al. [START_REF] Rodriguez-Donis | Thermodynamic Insights on the Feasibility of Homogeneous Batch Extractive Distillation, 1. Azeotropic Mixtures with a Heavy Entrainer[END_REF] published a general feasibility criterion for BED.

Based on industrial experiences of BED in a rectifier, where a very long start-up phase was observed, Lang et al. [START_REF] Lang | Industrial application of a new batch extractive distillation operational policy[END_REF] suggested a new operational policy, where entrainer feeding is started already during startup. Nonconventional variants of BED was also studied, e.g., BED in a middle-vessel column. [START_REF] Safrit | Extending continuous conventional and extractive distillation feasibility insights to batch distillation[END_REF][START_REF] Warter | Operation of a batch distillation column with a middle vessel: Experimental results for the separation of zeotropic and azeotropic mixtures[END_REF] Batch heterogeneous extractive distillation (BHED) can be regarded as a combination of batch extractive and heteroazeotropic distillation, as the entrainer, which is fed continuously into the column and forms a heteroazeotrope with one of the original components, whose existence makes the separation feasible.

Koehler et al. [START_REF] Koehler | Zur Diskontinuierlichen Rektifikation Azeotroper Gemische mit Hilfsstoffeinsatz[END_REF] was the first to mention BHED. The authors described the industrial application of the process; however, they did not present any theoretical analysis. Modla et al. [START_REF] Modla | Batch heteroazeotropic rectification of a low relative volatility mixture under continuous entrainer feeding: Feasibility studies[END_REF][START_REF] Modla | Batch heteroazeotropic rectification of a low relative volatility mixture[END_REF] extended the feasibility of Lelkes et al. [START_REF] Lelkes | Feasibility of extractive distillation in a batch rectifier[END_REF] to heterogeneous extractive distillation, and also performed rigorous simulations for the separation of dichloromethane and acetone using water as entrainer. The authors stated that the mixed addition (mixing to the charge and continuous feeding) of E provided the best results. Rodriguez-Donis et al. [START_REF] Rodriguez-Donis | Heterogeneous batch-extractive distillation of minimum boiling azeotropic mixtures[END_REF] stated that BHED is particularly useful for mixtures where E forms a saddle azeotrope with one of the original components. By analyzing the isovolatility curves and the volatility order diagram, it is possible to predict the distillate, which can either be an original component (as in the case of BED), or a heterogeneous saddle azeotrope of the entrainer and an original component. In the latter case, a rectification section is not even required for the feasibility if the extractive stable node (and, thus, the overhead composition) lies in the VLLE region. Petlyuk [START_REF] Petlyuk | Distillation Theory and Its Application to Optimal Design of Separation Units[END_REF] called the (continuous) heterogeneous extractive process as heteroextractive distillation. One of the phases in the decanter plays the role of the entrainer, but to allow the column section above the feed to function as an extractive one, an additional amount of entrainer, fed externally, is required. Van Kaam et al. [START_REF] Van Kaam | Heterogeneous extractive batch distillation of chloroform-methanol-water: Feasibility and experiments[END_REF] presented the experimental validation of the BHED process, where the saddle binary heteroazeotrope was withdrawn as the distillate. Barreto et al. [START_REF] Barreto | Optimization of Heterogeneous Batch Extractive Distillation[END_REF] performed the optimization of BHED using a genetic algorithm.

Rodriguez-Donis et al. [START_REF] Rodriguez-Donis | Column Configurations of Continuous Heterogeneous Extractive Distillation[END_REF] investigated the feasibility of continuous heterogeneous extractive distillation with several feed point strategies for the entrainer stream and the main azeotropic feed. A stripping section is always present, and, depending on the feeding locations, the column might contain a rectifying or an extractive section, both, or none. In a batch rectifier column, the azeotropic feedand, thus, the stripping sectionis missing, but different feed-point strategies for the entrainer stream can be adopted for the batch process.

The objective of this paper is to propose a general model of BHED, by extending our former model, which is valid for BHAD, [START_REF] Hegely | Generalised model for heteroazeotropic batch distillation with variable decanter hold-up[END_REF] and to study the effects of the entrainer feeding on the operational policies and still path directions. The entrainer can be fed onto any plate of the column, or mixed to the condensate stream, which is equivalent to feeding it into the decanter.

The paper is organized as follows. First, the general model and the equations describing the still path (the trajectory of the still composition) for two different entrainer feeding locations are presented. After that, the different operational policies of batch heterogeneous extractive distillation and the effect of entrainer feeding on the still path direction are discussed. Finally, rigorous simulation results obtained with the professional dynamic simulator of CHEMCAD for the separation of the mixture of water and ethanol with n-butanol by different operational policies are presented in order to validate the model.

THE EXTENDED MODEL FOR FEASIBILITY STUDIES

The generalized model of the batch heterogeneous extractive distillation (BHED; see Figure 1) is an extension of the model of batch heteroazeotrope distillation presented by Hegely et al. [START_REF] Hegely | Generalised model for heteroazeotropic batch distillation with variable decanter hold-up[END_REF] In this way, both liquid phases can be refluxed or withdrawn as distillate, and the holdup of both phases can be increased, decreased or kept constant in the decanter. Throughout the paper, we use the same notation as that used by Hegely et al. [START_REF] Hegely | Generalised model for heteroazeotropic batch distillation with variable decanter hold-up[END_REF] In the feasibility analysis, the following simplifying assumptions are applied:

(1) The composition of the condensate is constant and equals to that of the heteroazeotrope, (2) The overall liquid composition in the decanter equals that of the incoming stream (which might be the condensate, or the condensate mixed with the entrainer stream), (3) The vapor and liquid hold-ups of the column and of the condenser are negligible, (4) The entrainer is boiling point liquid, (5) Constant molar overflow.

It must be noted that the third assumption can even be omitted. In practice, during product withdrawal operation, the tray and condenser holdups are approximately constant. The material balances can be written for the still and the column together, and the results of the feasibility analysis will be valid for the average (still + column) hold-up composition, instead of the still composition.

Rodriguez-Donis et al. [START_REF] Rodriguez-Donis | Column Configurations of Continuous Heterogeneous Extractive Distillation[END_REF] presented four different possibilities for the entrainer feeding location for continuous heterogeneous extractive distillation. The entrainer could be (1) mixed with the azeotropic feed, (2) fed to the column at a point between the azeotropic feed and the reflux stream, (3) mixed with the refluxed stream, or (4) mixed with the condensate stream. In a batch process, the first case is not applicable, because there is no continuous azeotropic feeding. If the entrainer is fed to the column at a lower position than the top, a rectifying section exists above the feeding location and an extractive section below it. Feeding the entrainer at the top plate is equivalent to mixing it with the reflux stream. In this case, the column is only composed of an extractive section. During the feasibility analysis, as the column holdup is neglected, and constant molar overflow is assumed, the two latter configurations are treated together, as Case 1, since the existence of a rectifying section has no influence on our feasibility results. In Case 2, the entrainer stream is mixed with the condensate or, in a practical realization, fed directly into the decanter.

The top vapor of the column has a molar flow rate of V and molar composition of y 2 . Its condensate, a liquid stream with a flow rate of L 0 = V, and composition x 0 = y 2 , is usually a twophase liquid; however, in Case 2, a homogeneous condensate can also be feasible. For the following, the notations with an prime/accent mark will denote variables related to Case 2. For example, in Case 2, after the mixing of the condensate and the entrainer stream, this mixed stream entering the decanter (decanter inlet stream) has a flow rate of L 0 ′ and a composition of x 0 ′. Compositions x 0 (for Case 1) and x 0 ′ (for Case 2) lie in the heterogeneous region; that is, the condensate (Case 1) or the decanter inlet stream (Case 2) is a heterogeneous liquid. L 0,R and L 0,W are the flow rates of the E-rich and E-lean (Eweak) phase in the condensate. The phase split ratio of the condensate (η R ) is defined as the ratio of the flow rate of the Erich phase and the total flow rate:

η = L L R 0,R 0 (1)
Taking into account the first two simplifying assumptions, the phase ratio (η R ) and the composition of the phases leaving the decanter (x 1,R and x 1,W ) are also constant, although they may change during a real operation. [START_REF] Pommier | Heterogeneous batch distillation processes: Real system optimization[END_REF] Hegely et al. [START_REF] Hegely | Generalised model for heteroazeotropic batch distillation with variable decanter hold-up[END_REF] introduced the operational parameters r R (respectively r W ), defined as the ratio of the E-rich (respectively E-lean) phase entering the decanter and refluxed to the column:

= r L L R 1,R 0,R (2)
= r L L W 1,W 0,W (3)
where L 1,R and L 1,W are the flow rates of E-rich and E-lean phases refluxed, respectively. In Case 1, the liquid entering the decanter is the condensate; in Case 2, it is a mixture of the condensate and the entrainer stream. If the reflux ratio of the Erich phase (R R ) is defined as the ratio of the flow rates refluxed and nonrefluxed,

= - R L L L R 1,R 0,R 1,R (4)
r R has the following relation with R R :

= + r R R 1 R R R (5)
r R is always non-negative, and if R R tends to infinity, r R approaches unity. If the flow rate of the E-rich phase refluxed is higher than the flow rate of this phase in the stream entering the decanter (which means that the holdup of the E-rich phase in the decanter is decreasing), R R is a negative number and r R is greater than unity. Hegely et al. [START_REF] Hegely | Generalised model for heteroazeotropic batch distillation with variable decanter hold-up[END_REF] gives the differential total and component material balances of the still. H S denotes the holdup and x S represents the composition of the still. The total material balance of the still is given as

η η = - + - - H t r r V d d [ ( 1) ( 1)(1 )]
S R R W R (6)
The component material balance of the still is given as

η η = - + - - H x t r x r x V d( ) d [ ( 1) ( 1)(1 ) ]
S S R R 1,R W R 1,W (7)
From the above equations, the still path equation can be derived, which is a differential equation describing the evolution of the still composition in time:

η η = - - + - - - x t V H r x x r x x d d [(1 ) ( ) (1 )(1 )( )] S S R R S 1 , R W R S 1 , w (8)
For the BHED process, the entrainer feeding must be included into the material balances. The equations have different forms, depending on the entrainer feeding location, so Cases 1 and 2 are treated separately.

2.1. Case 1. In Case 1, the entrainer is fed to the column either at an intermediate or at the top tray, or, alternatively, it is mixed to reflux stream. The flow rate of the entrainer stream is F E , its composition is x E . The variables and equations concerning the decanter are unchanged, compared to Hegely et al., [START_REF] Hegely | Generalised model for heteroazeotropic batch distillation with variable decanter hold-up[END_REF] as the decanter is not directly influenced by the entrainer feeding; however, the differential material balances of the still must be modified.

The total material balance of the still is given as

η η = - + - - + H t r r V F d d [( 1) ( 1)(1 )] S R R W R E (9)
The component material balance is given as

η η = - + - - + H x t r x r x V Fx d( ) d [( 1) ( 1)(1 ) ] S S R R 1,R W R 1,W E E (10)
The equation of the still path by applying the product rule of differentiation is given as

= - H x t H x t x H t d d d( ) d d d s s ss s s ( 11
)
η η η η = - + - - + - - + - - - H x t r x r x V Fx r x r x V Fx d d [ ( 1) ( 1)(1 ) ]
[

S S R R 1,R W R 1,w E E R R S W R S ES ( 1) ( 1)(1 ) ]
η η = - - + - - - - - x t V H r x x r x x F H x x d d [(1 ) ( ) (1 ) (12)
S R R S 1 , R W R S 1 , w E S S E (1 )( )] ( ) S
In eq 13, besides the first two terms, which represent the removal of the E-rich and the E-lean phase from the column, respectively, a new term appeared, compared to eq 8, which describes the effect of the continuous feeding of the entrainer (that is, a change in the still composition toward the entrainer composition). As stated by Hegely et al., [START_REF] Hegely | Generalised model for heteroazeotropic batch distillation with variable decanter hold-up[END_REF] the removal of either liquid phase from the column can be realized by either withdrawing it as distillate, or accumulating it in the decanter. In the case of total reflux (no distillate, constant decanter holdup) of a liquid phase (r R = 1 or r W = 1), the corresponding term disappears from eq 13. Nevertheless, entrainer feeding also can be continued during total reflux operation, in which case the still composition is not constant, but rather approaches the entrainer composition.

2.2. Case 2. In Case 2, the entrainer is mixed to the condensate or fed to the decanter. In this way, the decanter is directly influenced by the entrainer feeding. The flow rate of the stream entering decanter is now given as

′ = + = + L L F V F 0 0 E E (14)
By mixing the entrainer with the condensate, the global composition (x 0 ′) of the decanter no longer equals to that of the heteroazeotrope, but, assuming that the entrainer stream is the pure component, x 0 ′ still lies on the same edge of the composition triangle as the heteroazeotrope. Moreover, unless the entrainer flow rate does not exceed a critical value, the global composition is in the heterogeneous region and, thus, the composition of the phases is not changed:

′ = x x 1,R 1,R (15)
′ = x x 1,W 1,W (16)
However, the phase split ratio in the decanter has a different value:

η′ = ′ ′ L L R 0,R 0 (17)
L 0,R ′ can be calculated from the following equation, which describes the decanter inlet flow as the sum of its E-rich and Elean phases:

′ ′ = ′ ′ + ′ ′ = ′ + ′ L x L x L x L x L x 0 0 0,R 0,R 0,W 0,w 0,R 1,R 0,W 1,W (18)
′ = ′ ′ -′ -′ L x L x L L x ( ) 0,R 1,R 0 0 0 0,R 1,W (19)
′ - = ′ ′ - L x x L x x ( ) ( ) 0,R 1,R 1,W 0 0 1,W (20)
′ = ′ ′ - - L L x x x x 0,R 0 0 1 , W 1,R 1,W (21)
This means that the phase split ratio in the decanter can be written as

η′ = ′ - - x x x x R 0 1 , W 1,R 1,W (22)
A similar equation can be derived for the phase split ratio of the condensate:

η = - - x x x x R 0 1 , W 1,R 1,W (23)
By subtracting eq 23 from eq 22, a relationship can be obtained between the two phase split ratios:

η η ′ - = ′ - - - - = ′ - - x x x x x x x x x x ( ) R R 0 1 , W 0 1 , W 1,R 1,W 0 0 1,R 1,W (24)
From the component material balance of the mixing of the condensate and the entrainer, x 0 ′ can be expressed:

′ ′ = + L x L x F x 0 0 0 0 E E (25)
′ = ′ + ′ = + + + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ x L L x F L x L L F x F L F x 0 0 0 0 E 0 E 0 0 E 0 E 0 E E (26)
By substituting eq 26 into eq 24, one obtains

η η ′ - = + - - + + + + ( ) ( ) ( ) x x x x x L L F F L F L F L F R R 0 E 0 1,R 1,W 0 0 E E 0 E 0 E 0 E (27)
By rearranging eq 27, the decanter split ratio can be expressed as a function of the condensate split ratio:

η η ′ = + + - - ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ F L F x x x x R R E 0 E E 0 1,R 1,W (28)
It can be seen that the phase split ratio of the decanter is always higher than that of the condensate, and increases with the entrainer flow rate. At a critical entrainer flow rate, η′ R reaches one.

In Case 2, according to their definition, the two operating parameters r R and r W have the following values:

= ′ r L L R 1,R 0,R (29)
= ′ r L L W 1,W 0,W (30)
The total material balance of the still is

= - H t L V d d S 1 (31)
The reflux flow rate L 1 can be written as

= ′ + ′ L r L r L 1 R 0,R W 0,W (32)
η η = ′ ′ + -′ ′ L r L r L (1 ) 1 R R 0 W R 0 (33)
In this way, eq 29 has the following form:

η η = ′ + -′ ′ - H t r r L V d d [ ( 1 ) ]
S R R W R 0 (34)
The component material balance of the still is given as

= - H x t L x Vx d( ) d S S 1 1 0 (35)
The component flow rate of the reflux stream is given as

η η = + = ′ ′ + -′ ′ L x L x L x r L x r L x (1 ) 1 1 1,R 1,R 1,W 1,W R R 0 1,R W R 0 1,W (36)
Meanwhile, the component flow rate of the top vapor can be expressed from the component material balance of the mixing of the condensate and the entrainer (eq 25), by taking advantage of the fact that V = L 0 :

= ′ ′ - Vx L x F x 0 0 0 E E (37)
If we substitute L 0 ′x 0 ′ from eq 18, we obtain

= ′ + ′ - Vx L x L x F x 0 0,R 1,R 0,W 1,W E E (38) η η = ′ ′ + ′ -′ - Vx L x L x F x (1 ) 0 0 R 1,R 0 R 1,W E E (39)
Using eqs 36 and 39, the component material balance of the still is given as

η η η η = ′ ′ + -′ ′ -′ ′ -′ -′ + H x t r L x r L x L x L x Fx d( ) d (1 ) (1 )
S S R R 0 1,R W R 0 1,W 0 R 1,R 0 R 1,W E E (40)
η η = - ′ ′ + - -′ ′ + H x t r Lx r Lx Fx d( ) d ( 1) ( 1)(1 )
S S R R 0 1,R W R 0 1,W E E (41)
The equation of the still path is obtained by applying the product rule of differentiation (eq 11):

η η η η = - ′ ′ + - -′ ′ + - ′ + -′ ′ + H x t r Lx r F x r r L x Vx d d ( 1) ( 1)(1 )
[

S S R R 0 1,R W R 0 1,W E E R R W R 0 S S ( 1 ) ]
Using eq 14 to substitute V, and rearranging the equation, yields

η η η η = - ′ ′ + - -′ ′ + - - ′ + - -′ ′ - H x t r Lx r Lx Fx r r Lx Fx d d ( 1) ( 1)(1 )
[

S S R R 0 1,R W R 0 1,W E E R R W R 0 S E S ( 1) ( 1)(1 )]
The final form of the still path equation for Case 2 is

η η = + - ′ - + - -′ - - - x t V F H r x x r x x F H x x d d [(1 ) ( ) (1 )(1 )( )] ( ) S E S R R S 1 , R W R S 1 , W E S S E (44)
The comparison of the still path equation of Case 2 to that of Case 1 (eq 13) reveals that while, the term describing the continuous entrainer feeding is the same, two differences occur:

(1) The condensate phase split ratio is replaced by the decanter phase split ratio, which always has a higher value. (2) The influence of the first two terms (removal of either the E-rich or the E-lean phase) is increased, as V is replaced by V + F E . In the case of total reflux operation, the first two terms disappear, and the still path equation for Case 1 and Case 2 become identical.

THE EFFECT OF THE CONTINUOUS ENTRAINER FEEDING

In this section, the effect of continuous entrainer feeding on the possible operational policies and on the still path directions is discussed.

3.1. The Effect of Continuous Entrainer Feeding on the Operational Policies. The model has four independent operational parameters: r R , r W , V, and F E . In BHAD, r R and r W determine the still path direction, and V only influences the speed of distillation; 6 in BHED, the still path direction is also determined by the relative magnitude of F E and V. A higher F E / V ratio obviously diverts the still path in a more prominent way toward the entrainer composition. Hegely et al. [START_REF] Hegely | Generalised model for heteroazeotropic batch distillation with variable decanter hold-up[END_REF] distinguished 16 possible operational policies, depending on the values of r R and r W , categorized into four distinct intervals: zero (0), between zero and one (<1), one (1), and higher than one (>1). The continuous entrainer feeding does not change the classification of the operational policies, as the value of the entrainer flow rate cannot be divided into separate categories (assuming that it is higher than the minimum value necessary for process feasibility, and not so high as to render the decanter holdup homogeneous). The operational policies are listed in Table 1, along with literature examples for the different policies. The majority of the literature suggests either Policy 9 or Policy 10. Using Policy 9, the E-rich phase is totally refluxed, while the E-lean phase is not refluxed at all. The E-lean phase could be accumulated in the decanter or withdrawn as distillate; the latter one is applied in the literature examples. Using Policy 9, the reflux ratio is completely determined by phase split ratio in the decanter. If this reflux ratio is insufficient for a feasible separation, or if the liquid-liquid split must be maintained longer than is possible via Policy 9, Policy 10 can be applied. In this case, beside the total reflux of the E-rich phase, the E-lean phase also is partially refluxed. Policy 10 is practically realized in the following way. A portion of the vapor is condensed and refluxed, while the remainder is led to the decanter, from which the total amount of the E-rich phase is refluxed at a constant Erich phase holdup. The E-lean phase is withdrawn as distillate. Van Kaam et al. [START_REF] Van Kaam | Heterogeneous extractive batch distillation of chloroform-methanol-water: Feasibility and experiments[END_REF] performed a BHED experiment, which was originally intended to be a realization of Policy 10. The top vapor was condensed and partially refluxed. The nonrefluxed part of the condensate was led to the decanter. Instead of refluxing the E-rich phase, the continuous entrainer feeding was increased to compensate for the loss of entrainer in the decanter. In this way, Policy 6 was realized, as both liquid phases were partially refluxed.

3.2. The Possible Still Path Directions. The still path direction is determined by the sum of the same three vectors (Figure 2) both for Case 1 (eq 13) and Case 2 (eq 44), which originate from x S , the actual still composition and have a direction of (x Sx 1,R ), (x Sx 1,W ) and (x Ex S ), respectively. The first two vectors, whose magnitude depends on r R and r W , were already present in the BHAD. These vectors point away from (toward) the E-rich phase composition x 1,R and the E-lean phase composition x 1,W , if the value of r R and r W is less (higher) than one, respectively. However, a new vector appears that always points toward the composition of entrainer composition, and whose direction is proportional to the entrainer flow rate. The direction of the resultant vector of the first two vectors (which are related to the decanter) depends on the operational parameters r R and r W , that is, on the operation policy. This direction is modified by the addition of the vector of the entrainer feeding to an extent that was dependent on the relative magnitude of the vectors, that is, on all the four operational parameters (r R , r W , V, and F E ) and also on the split ratio of the condensate or in the decanter.

The results are illustrated for a mixture of A, B, and E. A and B can be a homoazeotropic (minimum or maximum) or a close boiling mixture, while A and E form a binary heteroazeotrope, which is a saddle point of the system. With an azeotrope A-B, a distillation boundary may exist, but this does not affect the feasibility of the separation. [START_REF] Rodriguez-Donis | Feasibility of heterogeneous batch distillation processes[END_REF][START_REF] Lang | Generalised method for the determination of heterogeneous batch distillation regions[END_REF] The possible directions of the still path of BHAD can be classified into eight still path zones (see Figure 3a. [START_REF] Hegely | Generalised model for heteroazeotropic batch distillation with variable decanter hold-up[END_REF] ), covering in all possible directions. At least one operational policy belongs to each zone; that is, the still composition can be directed in any desired direction. Policy 11 (total reflux operation) is an exceptional case not belonging to any zone, as the still composition remains constant. Some zones, which point toward or away from the composition of one of the phases, only have one direction (Zones I, III, V, and VII), while the others span a set of directions.

The continuous E-feeding has the following effects on the still path zones. Comparing with the BHAD (Figure 3a), the still path zones are modified in the following way (see Figure 3b):

• The zones with only one direction are opening into the direction of the entrainer composition and every zone widens toward the entrainer feeding vector.

• As every zone has the entrainer feeding vector in common, every one of them are overlapping other zones. For example, the new Zone I partially overlaps with the new zones (Zones III, IV, and VIII). • There are zones that disappear (in this case, Zones II and VII), because they have become identical to other zones (here, Zones I and VI, respectively). • The operation under total reflux (Policy 11) appears as a new zone (Zone IX), pointing toward the entrainer composition. This is the only new zone with only one direction. Because the new still path cover zones in all possible directions, similar to the BHAD, the still composition can be moved into any desired direction by varying the value of the parameters r R and r W .

It can be still mentioned that, despite the continuous Efeeding, the still composition can remain constant using Policies 4 and 8 (Zone VIII), if the direction of the resultant of vectors related to the operational parameters r R and r W is just opposite to the vector of E-feeding and their lengths are equal.

Similar to BHAD, it is possible to direct the still path to achieve a desired final still composition by applying an appropriate operational policy or a combination of multiple policies. Reaching the A-B edge, except where the purity requirement for the product B is satisfied, is unwanted as, in this way, the original mixture to be separated is obtained once again. Using BHED, this problem is less likely to be encountered, because of the continuous feeding of the entrainer. Another objective could be to recover one of the original components (in this case, B) in the still, rendering a further separation step unnecessary. However, the concentration of E is increasing in the still, compared to that observed with BHAD, which can prevent recovering the original component.

3.3. The Practical Significance of the Operational Policies. Not all of the possible operational policies are of practical interest. The most conceptually straightforward one is Policy 9, the total reflux of the E-rich phase. If necessary, the reflux ratio can be increased by using Policy 10, that is, partially refluxing the E-lean phase besides the total reflux of the E-rich one. Policy 11, the total reflux of both phases, is applied in the start-up period, in order to obtain the heteroazeotrope as condensate. A desirable goal could be to reach the B-E edge, relatively far from the E vertex, so that B could be easily recovered in a second separation step. In order to reach this goal, the still path should be directed toward the B-E edge; that is, it should lie in one of the following zones (Figure 3b):

• Zone V (operation via Policy 9 or 10). These are the traditional operational policies. However, if the influence of the vector of E-feeding is large, they may result in a still path moving toward rather the E vertex than the B-E edge.

• Zone VI (operation via Policy 1, 2, 5, or 6). The two phases are not (or only partially) refluxed. If neither of the phases is refluxed, and the entrainer is fed into the column (Case 1), the column is operated according to the hybrid process. Policy 6 was already applied experimentally by Van Kaam et al., [START_REF] Van Kaam | Heterogeneous extractive batch distillation of chloroform-methanol-water: Feasibility and experiments[END_REF] even if due to practical limitations of the experimental setup. By these policies, the still path can move directly toward the B-E edge, and can even move away from the E vertex. In order to compensate the influence of the continuous entrainer feeding, the vectors pointing away from the E-rich and E-lean phase composition, respectively, should be as long as possible (r R and r W as low as possible). A low r R value moves the still composition toward lower E concentrations, while a low r W value moves it toward the B-E edge. Since a certain amount of reflux is usually necessary for the feasibility of the separation, these operational parameters might have minimum values.

• Zone VIII (operation via Policy 4 or 8). In this still path zone, the holdup of the E-lean phase in the decanter is reduced. This results in a vector pointing toward the E-lean phase composition (away from the B-E edge); therefore, although the still path can still move toward the B-E edge using these policies, these policies are not recommended.

• Zone IV (operation via Policy 13 or 14). Because of the reduction of the E-rich phase holdup in the decanter, the still path moves even faster toward higher entrainer concentrations. Similar to Zone VIII, although the still path can theoretically move in the right direction, these policies are not recommended.

The reduction of the E-rich phase holdup (Policies 13-16) can be replaced with the continuous entrainer feeding, which is more efficient, since this latter is not limited by the original holdup of E-rich phase in the decanter. However, the composition of the E-rich phase and that of the entrainer feeding are different to an extent depending on the liquidliquid equilibrium conditions. The smaller the difference between the composition of the E-rich phase and the entrainer, and the farther the still path is located from these points, the more similar are the effects of the continuous entrainer feeding and of the reduction of the E-rich phase hold-up.

As seen above, although the variation of the holdup of the phases in the decanter is possible, its effect is smaller than that via BHAD, because of the usually strong influence of the continuous entrainer feeding.

RIGOROUS SIMULATION

The results of the feasibility analysis are validated by rigorous simulation calculations performed with the dynamic module (CC-DCOLUMN) of the professional flow sheet simulator CHEMCAD. Our goal is to study the applicability of new BHED operational policies.

For the rigorous simulation, simplifying assumptions of the feasibility analysis are replaced with the following, more realistic ones:

• theoretical trays,

• constant volumetric liquid holdup on the trays and in the decanter, • negligible vapor holdup.

The calculations were performed with the CHEMCAD model (Figure 4), which comprises of the following parts. The column was modeled with a SCDS column equipped with a separate total condenser. One of the four Dynamic Vessels was the still, where a constant heat duty was applied; two other vessels were product tanks for the two liquid phases, while the fourth one served as the decanter. The two RAMP modules modified the liquid levels in the decanter with time, according to a piecewise linear function specified in the module.

The mixture studied is ethanol (A)-water (B) + n-butanol (E), which exhibit a minimum boiling point A-B homoazeotrope, while A and E form a saddle heteroazeotrope. The calculated residue curve map of the mixture along with the binodal curve at T = 25 °C is presented in Figure 5. The vapor-liquid-liquid equilibria were described with the NRTL model. The NRTL parameters used can be found in the Appendix.

The objective is the dehydration of the mixture by removing water from the decanter in the aqueous phase. First, Policy 9 is applied (total reflux of the E-rich phase only) after the startup. The composition of the charge (x F ) is 50 mol % A and 50 mol % B. The amount of charge is 100 mol (3.8 dm 3 at 25 °C) in the still pot, and 6.52 mol (0.25 dm 3 at 25 °C, which equals the maximum decanter holdup) in the decanter. The volume of decanter holdup has to be low, compared to that of the charge. The column has 50 theoretical trays; the heat duty applied is 2 kW. The holdup of the column is 0.01 dm 3 /plate. The column is operated for 82 min under total reflux; at the end of this period, the average (still + column) hold-up composition is x S,0 . The E-feeding (120 mol/h, 25 °C) is already started at t = 8 min into the column, that is, Case 1 of the entrainer feeding is applied. The feeding location is the second plate of the column (from the top), as a rectifying section of minimum one plate is required, if the decanter has been filled up before the start of the operation, in order to reach the two-liquid-phase region in the decanter. In the following, four examples are presented, which differ from each other in the policy applied after the start-up period. Table 2 gives an overview of the examples with the operation policy applied, containing the material balances and the recoveries.

Example 1.

After the startup under R = ∞, the entire amount of aqueous (E-lean) phase is withdrawn as distillate (Policy 9). The holdups of the two phases are constant and equal (125 cm 3 ) in the decanter. In the simulations, the assumption of negligible column holdup is omitted, and results of the feasibility analysis are valid for the average (still + column) hold-up composition instead of the still composition. [START_REF] Hegely | Generalised model for heteroazeotropic batch distillation with variable decanter hold-up[END_REF] The hold-up path calculated is shown in Figure 6, along with the influencing vectors at two different times, at the end of the startup (x S,0 ) and at 182 min (x S,1 ). The vapor and liquid profiles are also presented, at the end of the startup and at the end of the operation (t = 282 min). The liquid profiles start on the liquid-liquid envelope (composition of the reflux stream), and end at the still composition. The vapor profiles must always start in the two-liquid region, in order to have a liquid-liquid split in the decanter.

The hold-up path moves into the direction predicted by the feasibility analysis, that is, between the vector pointing away from the E-lean phase composition (x 1,W ) and that of pointing toward E. It must be noted that the E-lean phase composition is slightly changing during the operation, and always contains some ethanol. At the end of the operation, the still composition hardly contains water (0.53%). Since a large amount of E is needed for the separation, its concentration is high (91.55%) in the residue (see Table 2).

4.2. Example 2. The only difference between this and the previous examples is that Policy 5 (only partial reflux of E-rich phase) is applied after the startup instead of Policy 9. Ten percent (10%) of the E-rich phase is withdrawn as a distillate.

The hold-up path (Figure 7) is influenced not only by the two previous vectors, but also by a new vector pointing away from the point of E-rich phase (x 1,R ), whose direction is toward the B-E edge in this case. Consequently, the still composition approaches better the B-E edge (0.39 mol % water in the residue (at 282 min)). However, the ethanol recovery (Table 2) is reduced compared to that of Example 1, as the E-lean phase distillate has to be considered a waste, since it contains a high amount of water.

4.3. Example 3. The only difference between this and Example 1 is that Policy 6 (partial reflux of both phases) is applied after the startup: 10% of the E-rich phase and 60% of the E-lean phase is withdrawn as distillate.

The still path is influenced by • the vector of continuous entrainer feeding, which is unchanged, • the vector of the E-lean phase, which gets shorter than that observed in Example 1, and • the vector of the E-rich phase (similarly to Example 2).

(see Figure 8).

The resulting vector is located in Zone VI, and points less prominently in the direction of E; however, because of the large influence of the continuous entrainer feeding, the difference is not significant. Because of the fact that the vector of the E-rich phase also points toward the B-E edge, the final residue contains less water (0.42%) at the same time than in Example 1. The recoveries are slightly higher than those in Example 2 (see Table 2).

4.4. Example 4. The difference between this and Example 1 is that the hold-up reduction is applied by the combination of Policy 9 (total reflux of the E-rich phase) and Policy 8 (partial reflux of E-rich phase, reduction of the holdup of the E-lean phase). Policy 9 is applied after the startup until the butanol content of the still exceeds 80 mol % (see Figure 9), then the operation is continued by Policy 8. By Policy 8, 10% of the Erich phase is withdrawn as distillate, and the E-lean phase holdup in the decanter is reduced by 75 cm 3 (by 60%) during a period of 10 min.

During the application of Policy 8 (between x S,1 and x S,2 ), the direction of the hold-up path is in Zone VIII, and it clearly deviates from trajectory of Example 1. Because of the large influence of the continuous entrainer feeding, however, it moves rather toward the A-E edge than the B-E one, which is not advantageous. The water content of the residue (at 282 min) is 0.62 mol %, although the E-content (91.49%) is slightly less than that observed in Example 1. The ethanol recovery is slightly lower than in Example 1, while that of water is higher, because of the reduction of the E-lean phase holdup in the decanter (see Table 2). We can conclude that the highest ethanol recovery was obtained by using Policy 9 (Example 1), while the recovery of the water was the highest by using a combination of Policies 9 and 8 (Example 4). The water content of the still residue was the lowest by using Policy 5 (Example 2).

The results of the rigorous simulations indicate that the still path directions are in accordance with the results of the feasibility analysis, and that batch heterogeneous extractive distillation can also be performed by new operational policies (e.g., Policy 5).

CONCLUSIONS

The model of Hegely et al. [START_REF] Hegely | Generalised model for heteroazeotropic batch distillation with variable decanter hold-up[END_REF] was extended to batch heterogeneous extractive distillation (BHED), by taking into consideration continuous entrainer feeding. The possibility of refluxing or withdrawing both liquid phases as distillate was maintained, as well as the variability of the hold-up of the phases in the decanter. Two different entrainer feeding locations are distinguished: in Case 1, the entrainer was fed into the column, whereas in Case 2, it was added to the decanter. The equation describing the evolution of the still composition was derived for both cases. Compared to batch heteroazeotropic distillation (BHAD), a new term, which is related to the continuous entrainer feeding, appeared, and by Case 2, the influence of the existing terms, which are related to the operation of the decanter, was increased.

Sixteen (16) operational policies can be distinguished on the basis of the two operational parameters r R and r W , which are the ratio of the flow rates of the E-rich and E-lean phase refluxed and condensed, respectively. The occurrence of the operational policies in the literature and potential practical applicability were discussed. The effect of the continuous entrainer feeding on the still path direction, which is now influenced by three vectors with magnitudes depending on the operational policy and the entrainer flow rate, was studied. The eight original still path zones of BHAD are modified: some of them disappear, and the remaining zones overlap each other. Beside the entrainer feeding, the still path zones still cover all possible directions (that is, by using an appropriate operational policy, it is possible to direct the still composition into any direction). In practice, however, the influence of the entrainer feeding is large, and it is difficult to move the still path away from the entrainer composition. For the same reason, the variation of the holdup of the phases in the decanter only has a small effect on the still path.

The still path directions were validated by rigorous simulation of the dehydration of the water-ethanol mixture, using n-butanol as an entrainer. Four examples were presented: operation by Policy 9 (total reflux of entrainer-rich phase only), Policy 5 (partial reflux of E-rich phase only), Policy 6 (partial reflux of both phases), and a combination of Policies 9 and 8 (partial reflux of E-rich phase, reduction of the E-lean phase hold-up in the decanter). The average (still and column) holdup composition moved in accordance with the feasibility results. By using the new Policy 5, it was possible to slightly reduce the water content of the still residue that mainly contained ethanol and butanol.

■ APPENDIX: VALUE OF THE PARAMETERS USED FOR THE PHASE EQUILIBRIA CALCULATIONS The NRTL parameters for the mixture A-B-E (where A is ethanol, B is water, and E isn-butanol), are given in Table A1.

Figure 1 .

 1 Figure 1. Batch heterogeneous extractive distillation (BHED) column model.

Figure 2 .

 2 Figure 2. Possible directions of the vectors influencing the still path.

Figure 3 .

 3 Figure 3. Direction of the still path for the different operational policies: (a) batch heteroazeotropic distillation (BHAD) and (b) batch heterogeneous extractive distillation (BHED).

Figure 4 .

 4 Figure 4. CHEMCAD model of the batch heterogeneous extractive column with variable liquid holdup.

Figure 5 .

 5 Figure 5. Residue curve map of the water (A)-ethanol (B)-n-butanol (E) mixture.

Figure 6 .

 6 Figure 6. Hold-up path calculated for Policy 9.

Figure 7 .

 7 Figure 7. Hold-up path calculated for Policy 5.

Figure 8 .

 8 Figure 8. Comparison of the hold-up paths calculated for Policy 6 and Policy 9.Figure 9. Hold-up path calculated for Example 4 (Policy 9 then Policy 8).

Figure 9 .

 9 Figure 8. Comparison of the hold-up paths calculated for Policy 6 and Policy 9.Figure 9. Hold-up path calculated for Example 4 (Policy 9 then Policy 8).

Table 1 .

 1 Literature Examples for the 16 Possible Operational Policies of BHED

		r R	r W	reflux a	example in the literature
	1	0	0	n or e flux
	2	0	<1	E-lean phase (partial)
	3	0	1	E-lean phase (total)
	4	0	>1	E-lean phase (reintr.)
	5	<1	0	E-rich phase (partial)
	6	<1	<1	E-rich (partial) and E-lean phase (partial) Van Kaam et al.: 24 BHED experiment
	7	<1	1	E-rich (partial) and E-lean phase (total)
	8	<1	>1	E-rich (partial) and E-lean phase (reintr.)
	9	1	0	E-rich phase (total)	Koehler et al., 19 Rodriguez-Donis et al.: 22 mixture ethyl-acetate-ethanol-water
	10	1	<1	E-rich (total) and E-lean phase (partial)	Modla et al., 20,21 Rodriguez-Donis et al.: 22 mixture acetonitrile-water-butyl-acetate, Van Kaam et al., 24 Barreto et al. 25
	11	1	1	E-rich (total) and E-lean phase (total)	total reflux operation with entrainer feeding
	12	1	>1	E-rich (total) and E-lean phase (reintr.)
	13	>1	0	E-rich phase (reintr.)
	14	>1	<1	E-rich (reintr.) and E-lean phase (partial)
	15	>1	1	E-rich (reintr.) and E-lean phase (total)
	16	>1	>1	E-rich (reintr.) and E-lean phase (reintr.)

a The abbreviation "reintr." denotes reintroduction, a greater amount of liquid is refluxed than what leaves the condenser.

Table 2 .

 2 Main Data and Results of the Examples

		Example 1 Example 2 Example 3 Example 4
	operation policy	Policy 9	Policy 5	Policy 6 Policies 9
					and 8
	charge [mol]	106.52	106.52	106.52	106.52
	entrainer [mol]	548	548	548	548
	final residue [mol]	592.34	579.78	580.78	592.11
	final decanter holdup	8.91	7.51	7.69	7.40
	[mol]
	organic phase distillate	0	31.24	29.17	1.10
	[mol]
	aqueous phase	44.86	27.77	28.57	45.50
	distillate [mol]
	ethanol recovery [%]	93.9	81.3	83.3	93.5
	water recovery [%]	83.6	51.6	53.8	85.20

■ AUTHOR INFORMATION

Corresponding Author *Tel.: +3614631707. E-mail: lang@mail.bme.hu.

Notes

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS


Christophe Boudry
email: boudry@enc.sorbonne.fr

Web 2.0 Applications In Medicine : Trends And Topics In The Literature

Background and objectives

Over the past two decades, the world wide web has changed researchers' habits. These changes were further extended when "web 2.0" appeared in 2005, bringing tools and platforms facilitating user collaboration, user-generated content and data sharing. These tools have gradually influenced the world of research, especially in medicine. Bibliometrics is a helpful and widely used tool for describing patterns of publication and interpreting temporal evolutions and the geographical distribution of research in a given field. Few studies employing bibliometrics, however, have been carried out on the correlative nature of scientific literature and web 2.0. The aim of this bibliometric analysis was to provide an overview of web 2.0 implications in the field of Medicine. The objectives were to assess growth rate of literature, key journals, authors and country contributions, and to evaluate whether the various web 2.0 applications are expressed within this biomedical literature, and, if so, how.

Methods

A specific query with keywords chosen to be representative of web 2.0 applications has been built for PubMed database. Articles dealing with web 2.0 were downloaded in XML and were processed through developed PHP scripts, then were imported to Microsoft Excel 2010 (Microsoft) for data processing. Journals 614 journals were identified during this analysis. Table 1 presents the top 10 most productive journals over the period studied producing articles related to web 2.0 .

Journal

Number of papers

Percentage of articles

Studies

Geographical repartition of authors (country contributions)

United States was by far the predominant country, in terms of author country. Europe came globally second whereas Africa and South America were very poorly represented.

Core population of authors related to web 2.0

4209 authors were found for the 1347 articles retained. Distribution of number of articles per author shows that the great majority of authors (n=3444; 91.54 %) wrote only one article, whereas 73 (1.9 %) wrote three or more. Thus, the core population of researchers working on web 2.0 in the biomedical field can be estimated to approximately 75.

Words frequency analysis

274 words in the field of web 2.0 were found after manual sorting of the 15878 words belonging to title and abstract fields of articles. Word frequency analysis reveals the word "blog" as the most recurrent, followed by "wiki", "web 2.0", "social media", "Facebook", "social networks", "blogger", "cloud computing", "Twitter" and "blogging". All categories of web 2.0 applications were found, indicating the successful integration of web 2.0 into the biomedical field. Blog was the predominant category of any web 2.0 application, followed by social networks and wiki (respectively 1279, 1199 and 824 occurrences). Micro-blogging, cloud computing, social bookmarking/document sharing and syndication were much less represented with respectively 332, 227, 183 and 175 occurrences. Remarkably, apart from wikis, applications specifically developed for science, biology or medicine were rare (e.g. PatientLikeMe,) or not represented (e.g. researchblogging.org for blogs, Researchgate and Academia for social networks).

Results

Conclusions

This study shows that the biomedical community is engaged in the use of web 2.0 and confirms its high level of interest for these tools. Therefore, changes in the ways researchers use information, generated by the arrival of the world wide web and web 2.0, seem far from over.

Malaga. Spain. October 2014

Growth of literature (annual number and cumulative number)

Growth of literature

  1347 articles were included in this study. The number of articles dealing with web 2.0 has been increasing since 2002 (average annual growth rate was 106.30 % with a maximum of 333 % in 2005)

	Country		Number	Percentage
				of articles	of articles
	United States		514	54
	United Kingdom		87	9.1
	Australia			44	4.6
	Canada			41	4.3
	China			40	4.2
	Germany			34	3.6
	Spain			26	2.7
	Netherlands		21	2.2
	France, Italy		14	1.5
	Greece, Japan		11	1.2
	New Zeland, Switzerland	10	1.1
	India, Israel			9	1
	Norway, Sweden		8	0.8
	Portugal			6	0.6
	Belgium, Ireland		4	0.4
	Austria,	Brazil,	Bulgaria,	3	0.3
	Egypt, Romania, Turkey
	Luxembourg, South Africa	2	0.2
	Argentina, Czech Republic,	1	0.1
	Poland, Singapore, Slovakia


Serge Vladimir

Emteu Tchagou

Alexandre Termier
email: alexandre.termier@irisa.fr

Jean-Franois Méhaut
email: jean-francois.mehaut@imag.fr

Brice Videau
email: brice.videau@imag.fr

Miguel Santana Stmicroelectronics

Ren Quiniou
email: rene.quiniou@inria.fr

Reducing trace size in multimedia applications endurance tests

Proper testing of applications over embedded systems such as set-top boxes requires endurance tests, i.e. running applications for extended periods of times, typically several days. In order to understand bugs or poor performances, execution traces have to be analyzed, however current trace analysis methods are not designed to handle several days of execution traces due to the huge quantity of data generated. Our proposal, designed for regular applications such as multimedia decoding/encoding, is to monitor execution by analyzing trace on the fly in order to record trace only in time periods where a suspicious activity is detected. Our experiments show a significant reduction in the trace size compared to recording the whole trace.

I. INTRODUCTION

Nowadays, there is a huge market for electronics consumer products such as smart-phones, tablets or set-top boxes. These products are powered by MultiProcessor System-on-Chip (MP-SoC), which are highly integrated chips having on a single die several generalist computation cores, specialized accelerators, memories and I/O components. Demanding applications such as 4K video decoding require to exploit all the processing power of MPSoC through a parallel approach. However parallel programming is notoriously difficult and even small timing errors between components may lead to low QoS of the applications. In such cases, traditional debuggers cannot be used as they are extremely intrusive: they will change the timings of components execution and hide potential errors. The solution used in industry is to continuously collect traces of the application execution through dedicated, low-intrusive tracing hardware. These traces are analyzed post-mortem in order to understand bugs or reasons for poor performance.

Industrial validation of the applications requires "endurance tests" (typically several hours/days of continuous execution) in order to verify that bugs do not occur after long period of time. Tracing such tests would generate hundreds of GigaBytes of traces, which are difficult to analyze with state of the art approaches based on visualization [START_REF] Cornelissen | Execution trace analysis through massive sequence and circular bundle views[END_REF] or data-mining [START_REF] Lagraa | Scalability bottlenecks discovery in mpsoc platforms using data mining on simulation traces[END_REF]. Hence currently these tests do not benefit from tracing, whereas they exhibit some of the most complex bugs to detect.

Based on the observation that most applications tested are multimedia applications, which exhibit regular behaviors, our proposal (Section II) consists in capturing only parts of the trace from long running tests that contain problematic execution periods (if any) and that can be analyzed by existing tools. For this purpose, we use machine learning to model correct behavior during the monitoring of the application and record the trace only when the detected behavior departs significantly from the correct behavior. Using such approach on real execution traces, our experiments (Section III) show that trace size can be significantly reduced.

II. METHODOLOGY

In this section, we describe our main contribution: an online approach to analyze the trace stream of a multimedia application, and record only suspicious portions of the trace. This approach is based on anomaly detection techniques from the data mining field.

Such techniques require two steps: first a learning step in order to model the correct behavior from a reference execution trace. Then a monitoring step where incoming trace data is compared to that model. If the difference is too large, the corresponding trace data is recorded to a storage device.

Data representation:

In order to perform these steps, an adequate trace representation has to be found. A raw trace is a sequence of timestamped events. In a streaming context, trace data is not provided event by event by the tracing hardware, but by windows of N consecutive events. The value of N is usually correlated to the size of the buffers of the tracing hardware. Such window will be our elementary processing unit.

Each window is transformed as a probability mass function (pmf ), i.e. a vector giving for each event type the number of occurrences of that event type in the window. Such vectors are easy to manipulate and give a good abstraction of the trace in a window.

Learning: Our approach exploits the Local Outlier Factor (LOF) [START_REF] Breunig | Lof: identifying density-based local outliers[END_REF] anomaly detection method, which is designed to handle such pmfs. The learning step of LOF is simple and requires only a reference trace for a correct execution of the application. This trace is divided in windows, and the pmf of each window is computed. This gives a set of points in an high dimensional space, which constitutes the model of the reference trace. Depending on objectives of the endurance tests planned, a reference trace can be simply the trace of the first few minutes of application execution, during which the developer noticed no QoS errors. A curated database of reference traces can be constituted in order to skip the learning step.

Online anomaly detection: Once the model is constructed, the anomaly detection step can take place. As trace windows arrive, their pmf representation is computed. The algorithm evaluates the similarity between N pmf , the pmf vector of the current window, and P pmf , the pmf vector associated with past windows. We use Kullback-Leibler distance [START_REF] Kullback | On information and sufficiency[END_REF] because it is well suited to compare pmf vectors. If N pmf and P pmf are similar, we consider that no significant change occurred, no anomaly detection tests are performed. In this case N pmf is merged with P pmf and P pmf is updated: this helps to detect slow changes of behavior.

On the contrary, if P pmf and N pmf are dissimilar, then a LOF computation is performed on N pmf in order to determine if the new window is an anomaly. The idea of the LOF computation is to place N pmf in the space of high dimensional points found during the learning set, and to compare the density of points around N pmf with the density of points around the K nearest neighbors of N pmf . If N pmf has the same density than its neighbors (LOF = 1), then it is embedded in a cluster of "regular" points and does not represent an anomaly. If N pmf has a lower density than its neighbors (LOF ≥ α > 1), then it is not in a cluster of "regular" points and is likely to represent an anomaly. α is a user given parameter of the approach.

If an anomaly is detected, the corresponding window of trace is recorded to a storage device.

III. EXPERIMENTS

In this section, we demonstrate the applicability of our approach on a real application by monitoring its trace in a controlled setting. In this preliminary experiment, the application monitored is GStreamer, a well known video decoding framework. GStreamer is run on a standard laptop with an Intel core i7 processor X 920 @ 2.00GHz and is limited to using one core. It is tasked to decode a video of 6h17m. Its trace output is monitored by our prototype. Other parameters are: windows size of 40ms, K = 20.

The first 300s of decoding are used as a reference in order to learn a trace model for the correct behavior. Then every 3 minutes, a "perturbation" is made during 20s through an heavy processing application. The goal of the experiment is to verify that our monitoring approach records trace mostly for windows impacted by the perturbation. Note that due to the Gstreamer buffering mechanism, the perturbation impact on the video is delayed (∆ s ) from start of the perturbation and the end of the perturbation impact is delayed (∆ e ) from the return to the normal video playback. The status of playback can be deduced from error messages sent by GStreamer. We computed a mean value ∆ avg s and ∆ avg e on a two minutes part of the video in order to fix these experimental parameters. During monitoring, each trace window W monitored is labeled as: The quality of anomaly detection is evaluated with the well known metrics of precision = T P/(T P + F P ) and recall = T P/(T P + F N ). precision gives the ratio of windows that are correctly reported as anomalous, the higher its value the higher the reduction rate of the trace. recall gives the ratio of actually anomalous windows found by our approach, a low value means that some potentially buggy parts of the trace were missed by our approach.

The values of precision and recall according to the LOF threshold α are given in Figure 1. With α = 1.2, the precision is 78.9% and the recall is 76.6%: these values indicate that most anomalies were correctly detected. In this setting, the recorded trace weights 418M B, instead of 5.9GB if recorded completely: this represents a 14-fold reduction in volume. This confirms the interest of our approach.

IV. CONCLUSION

We have presented an approach for online monitoring of multimedia applications in endurance tests, in order to reduce the volume of trace recorded and ease further debugging. Experiments show a reduction of one order of magnitude in the recorded trace size, with a good coverage of the anomalies happening in execution. We plan to apply our approach on larger scale endurance tests of actual embedded systems. We are also interested in further reducing the recorded trace size by exploiting the periodic behavior of the application.

Fig. 1 .

 1 Fig. 1. Precision and recall of anomaly detection

  True Positive (T P ) if it is in [P erturbation start + ∆ avg s , P erturbation end + ∆ avg e ] and GStreamer reports an error and LOF (W ) ≥ α, False Negative (F N ) if it is in [P erturbation start + ∆ avg s , P erturbation end + ∆ avg e ] and GStreamer reports an error and LOF (W ) < α, False Positive (F P ) if LOF (W ) ≥ α and either GStreamer does not report an error or the window is not in [P erturbation start + ∆ avg s , P erturbation end + ∆ avg e ], True Negative (T N ) in the other cases.