Federico Bertoni
email: federico.bertoni4@unibo.it

Giovanna Citti
email: giovanna.citti@unibo.it

Alessandro Sarti
email: alessandro.sarti@ehess.fr

LGN-CNN: a biologically inspired CNN architecture

Keywords: CNN, LGN, Visual system, Retinex theory, Minimal functional symmetry properties

In this paper we introduce a biologically inspired Convolutional Neural Network (CNN) architecture called LGN-CNN that has a first convolutional layer composed by a single filter that mimics the role of the Lateral Geniculate Nucleus (LGN). The first layer of the neural network shows a rotational symmetric pattern justified by the structure of the

net itself that turns up to be an approximation of a Laplacian of Gaussian (LoG). The Each cortex receives information from other cortices, processes it through horizontal connectivity, forward it to higher areas and send feedback to previous ones. The structure is very complex and not totally ordered as physiologically described for example in [START_REF] Hubel | Indirect activation elicits strong correlations between light and electrical responses in on but not off retinal ganglion cells[END_REF]. Geometrical model of the first visual cortex, we refer to [START_REF] Ferraro | Lie transformation groups, integral transforms, and invariant pattern recognition[END_REF], [START_REF] Petitot | Neurogéométrie de la vision[END_REF], [START_REF] Citti | A Cortical Based Model of Perceptual Completion in the Roto-Translation Space[END_REF].

The first neural nets have been inspired by a simplification of this structure, and present a hierarchical structure, where each layer receives input from the previous one and provides output to the next one. Despite this simplification, they reached optimal performances in processes typical of the natural visual system, as for example object-detection [START_REF] Redmon | You only look once: Unified, real-time object detection[END_REF], [START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF] or image classification [START_REF] He | Deep residual learning for image recognition[END_REF], [START_REF] Simonyan | Very deep convolutional networks for largescale image recognition[END_REF].

More recently relations between CNNs and human visual system have been widely studied, with the ultimate scope of making the CNN even more efficient in specific tasks.

A model of the first cortical layers described as layers of a CNN has been studied in [START_REF] Serre | Robust object recognition with cortex-like mechanisms[END_REF]. In [START_REF] Yamins | Using goal-driven deep learning models to understand sensory cortex[END_REF] and in [START_REF] Yamins | Hierarchical modular optimization of convolutional networks achieves representations similar to macaque it and human ventral stream[END_REF]) the authors were able to study higher areas by focusing on the encoding and decoding ability of the visual system. Recurrent Neural networks have been introduced to implement the horizontal connectivity (as for example in [START_REF] Sherstinsky | Fundamentals of Recurrent Neural Network (RNN) and Long Short-Term Memory (LSTM) Network[END_REF])), or feedback terms (for example in [START_REF] Liang | Recurrent convolutional neural network for object recognition[END_REF]). A modification of these nets, more geometric and more similar to the structure of the brain, have been recently proposed in [START_REF] Montobbio | KerCNNs: biologically inspired lateral connections for classification of corrupted images[END_REF].

It is well known that both V1 RFPs and the first convolutional layer of a CNN are mainly composed by Gabor filters. We refer to [START_REF] Daugman | Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters[END_REF], [START_REF] Jones | An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex[END_REF], [START_REF] Lee | Image Representation Using 2D Gabor Wavelets[END_REF], [START_REF] Petitot | Neurogéométrie de la vision[END_REF]) for the visual system and to [START_REF] Yamins | Using goal-driven deep learning models to understand sensory cortex[END_REF], [START_REF] Yamins | Hierarchical modular optimization of convolutional networks achieves representations similar to macaque it and human ventral stream[END_REF], [START_REF] Girosi | Regularization Theory and Neural Networks Architectures[END_REF] for properties of CNNs.

Biological based models of V1 in terms of Gabor filters have been made in [START_REF] Zhang | Convolutional neural network models of V1 responses to complex patterns[END_REF]) and [START_REF] Serre | Robust object recognition with cortex-like mechanisms[END_REF]) and the statistic of the RFPs of a macaque's V1 was studied in [START_REF] Ringach | Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex[END_REF], but a comparison between these results and the statistics of learned filters is still missing.

Invariance properties in CNNs

Gabor invariance properties are mainly invariance with respect to translation and rotations. CNNs are translation equivariant since they are defined in terms of convolutional kernels (see [START_REF] Cohen | Group Equivariant Convolutional Networks[END_REF], [START_REF] Cohen | A General Theory of Equivariant CNNs on Ho-mogeneous Spaces[END_REF]). Rotation invariance properties can be imposed either obtaining the whole bank of filters from a learned one [START_REF] Marcos | Learning rotation invariant convolutional filters for texture classification[END_REF]) and [START_REF] Wu | Flip-rotatepooling convolution and split dropout on convolution neural networks for image classification[END_REF], or rotating any test image [START_REF] Fasel | Rotationinvariant neoperceptron[END_REF], [START_REF] Dieleman | Rotation-invariant convolutional neural networks for galaxy morphology prediction[END_REF], [START_REF] Dieleman | Exploiting cyclic symmetry in convolutional neural networks[END_REF]. A different kind of pooling or kernel procedure are used in [START_REF] Laptev | TI-pooling: transformationinvariant pooling for feature learning in convolutional neural networks[END_REF]) and [START_REF] Gens | Deep symmetry networks[END_REF], while (Barnard and Casasent 1.3 LGN, Retinex and contrast perception 1991) studied invariances with respect to other feature spaces.

LGN, Retinex and contrast perception

In the human visual system, the process in V1 operated by Gabor filters, is preceded by a preprocessing operated by radially symmetric families of cells, both in the retina and in the LGN (see [START_REF] Hubel | Indirect activation elicits strong correlations between light and electrical responses in on but not off retinal ganglion cells[END_REF]). The RFPs of cells can be approximated by a LoG which is rotational symmetric (for a review see for example [START_REF] Petitot | Neurogéométrie de la vision[END_REF] [START_REF] Land | The Retinex[END_REF] and further developed by E. H. Land and J.

McCann in [START_REF] Land | Lightness and retinex theory[END_REF]. Several developments are due to [START_REF] Brainard | Analysis of the retinex theory of color vision[END_REF], [START_REF] Provenzi | Mathematical definition and analysis of the retinex algorithm[END_REF] and [START_REF] Lei | An investigation of retinex algorithms for image enhancement[END_REF] [START_REF] Solomon | Suppressive surrounds and contrast gain in magnocellular-pathway retinal ganglion cells of macaque[END_REF]). Variational approaches have been proposed by [START_REF] Kimmel | A Variational Framework for Retinex[END_REF], [START_REF] Morel | A PDE formalization of Retinex theory[END_REF]) and [START_REF] Limare | Retinex Poisson equation: a model for color perception[END_REF].

A geometrical model which makes a first step in relating the architecture of the visual system and invariance of RFPs has been presented in [START_REF] Citti | A gauge field model of modal completion[END_REF]. The action of radially symmetric RFP is interpreted as a LoG, while the horizontal connectivity is modeled as an inverse of the Laplacian operator, and allows to recover the given image up to a contrast shift.

Our contribution

In the present paper, we introduce a new neural architecture inspired by the structure of visual cortices and study the properties of the filters of the first and second layers. The first layer contains a single filter and models the LGN. We show that it has the same radially symmetric shape, as LGN receptive profiles and is able to reproduce a Retinex effect. Then we show that the statistics of filters of the second layer much better fits the observed distributions, comparing with the esperimental results of [START_REF] Ringach | Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex[END_REF].

The paper is organized as following.

In Section 2 we recall the structure of

LGN and V1 and the RFPs of their cells. We described an interpretation of the Retinex model given by Morel [START_REF] Morel | A PDE formalization of Retinex theory[END_REF] and a statistical study on the RFPs of the distributions of V1 cells in a macaque by Ringach [START_REF] Ringach | Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex[END_REF].

In Section 3 we introduce our LGN-CNN

The visual system

The visual system is one of the most studied part of the brain. We describe here the aspects important for our study and refer the reader for a more general description to [START_REF] Sundsten | The human brain: an introduction to its functional anatomy[END_REF], [START_REF] Jessell | Central visual pathways. Principles of neural science[END_REF].

The retina is a light-sensitive layer of tissue which receives the visual stimulus and translates it into electrical impulses. These impulses first reach the LGN whose cells preprocess the visual stimulus. Then the impulse is processed by the cells of V1, whose output is taken in input to all the other layers of the visual system.

We are mainly interested in the cells of the LGN and in the simple cells of V1. Each cell receives the electrical impulse from a portion of the retina Ω called receptive field (RF). The RF of each cell is divided in excitatory and inhibitory areas which are activated by the light and that can be modeled as a function Ψ : Ω ⊂ R 2 → R called receptive field profile (RFP). Thus, if the excitatory areas are activated the firing rate of the cell increases whereas it decreases in case of inhibitory areas activation. Figure 1 shows the RFP of an LGN cell that can be modeled by a LoG.

LGN RFP: neural interpretation of

Retinex model

The Retinex algorithm, introduced in [START_REF] Land | Lightness and retinex theory[END_REF], mimics the contrast invariant process performed by our visual system and associates to an image I :

Ξ ⊂ R 2 → R the perceived image I.
In [START_REF] Morel | A PDE formalization of Retinex theory[END_REF]) and [START_REF] Limare | Retinex Poisson equation: a model for color perception[END_REF]

Out LGN (I) = ∆(G σ * I) ≈ ∆I
The horizontal connectivity in this layer is radially symmetric and modeled as the fundamental solution log( x 2 + y 2 ) whose associated operator is the inverse of the Laplacian ∆ -1 and allows to recover the function

I: I = ∆ -1 * Out LGN (I).
As a result

∆ I = Out LGN (I) ≈ ∆I,
is the Retinex equation. In general, I will not coincide with I, but will differ by a harmonic function.

Our aim is to replace the action of the RFP with the filter learned by the LGN-CNN. If it is a good approximation of the associated ∆G σ , then its inverse will allow to recover the perceived image I in problems of contrast perception. In Section 5 we will describe in detail the process.

Statistics of V1 RFPs

As first discovered by Daugmann, RFPs of the primary cortex V1 can be approximated by Gabor functions defined as follows:

h(x , y ) =Ae (-(x / √ 2σx) 2 -(y / √ 2σy) 2 ) cos(2πf x + φ) (2)
where (x , y ) is translated and rotated from the original coordinate system (x 0 , y 0 )

x =(x -x 0 ) cos θ + (y -y 0 ) sin θ y = -(x -x 0 ) sin θ + (y -y 0 ) cos θ.

as shown in Figure 2.

Recently, Ringach in [START_REF] Ringach | Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex[END_REF] has proved that RFPs are not uniformly distributed with respect to all the Gabor parameters, but they have a very particular statistic. Ringach defines two coefficients n x and n y which estimate the elongation in x and y directions respectively

(n x , n y ) = (σ x • f, σ y • f ).
In particular, if f = 0 the function h in (2) simplifies to a Gaussian since the cosine becomes a constant. Otherwise it is elongated:

-Fitting a Gabor function defined in equation (2) to the RFPs; In particular, if we consider a classical CNN we can add before the other convolutional layers, a layer 0 composed by only one filter Ψ 0 of size s 0 ×s 0 and a ReLU function. Note that after the first layer 0 we will not apply any pooling. In this way taking a classical CNN and adding 0 will not modify the structure of the neural network and the number of parameters will only increase by s 0 × s 0 . Furthermore, Ψ 0 will prefilter the input image without modifying its dimensions; this behavior mimics the behavior of the LGN which let the neural network to be closer to the visual system structure. Figure 3 shows a scheme of the first steps of the visual pathway (i.e., LGN and V1) in parallel with the first two layers 0 and 1 of the

-Comparing the results on (n x , n y ) = (σ x • f, σ y • f ) plane.
LGN-CNN architecture.

The theoretical idea behind this structure can be found in a simple result on rotational symmetric convex functionals. In particular, we recall that a rotational symmetric convex functional F has a unique minimum ω. Since F is rotational symmetric,

F (ω •g) = F (ω)
for a rotation g. Thus, since the minimum is unique, ω = ω • g, implying the rotational symmetry of ω. There are several results on symmetries of minimum for functionals as for example in [START_REF] Lopes | Radial symmetry of minimizers for some translation and rotation invariant functionals[END_REF], [START_REF] Gidas | Symmetry of positive solutions of nonlinear elliptic equations in RN[END_REF]. Our aim is to extend these results in the case of CNNs in particular on our architecture that we name as Lateral Geniculate Nucleus Convolutional Neural Network (LGN-CNN).

We will also show that the Gabor-like filters in the second convolution layer, reprojected in the (n x , n y ) plane introduced by Ringach and recalled above, satisfy the same properties of elongation which characterizes the RFPs of simple cells in V1. This analysis should enforce the link between our architecture and the visual system structure, at least as regards simple cells in V1.

Applications of LGN-CNN

Settings

In this Section we describe the settings for testing our architecture. We use Mat-Lab2019b for academic use.

We train our LGN-CNN architecture on a dataset of natural images called STL-10

(see [START_REF] Coates | An Analysis of Single Layer Networks in Unsupervised Feature Learning[END_REF])) that contains 5000 training images divided in 10 different classes.

We have modified the training set in the following way:

-Changing the images from RGB color to grayscale color using the built-in function rgb2gray of MatLab; that the retina performs as described in [START_REF] Carandini | Normalization as a canonical neural computation[END_REF]. Indeed, one of the main difference is the subtraction of the mean value µ performed by the batch normalization layer defined as follows

x i = x i -µ √ σ 2 + ,
where x i is the element to normalize, µ is the mean value of the batch, σ is the standard deviation of the batch and is a small value that prevents bad normalizations in case of really small standard deviations. However, since the input images have zero mean then the convolution with Ψ 0 have still zero mean. Thus, the batch normalization layer between 0 and 1 has similar characteristics as the biological one. On the other hand,

F (I) :=(σ • F C 3 • F C 2 • F C 1 • p 2 m • R • b • 4 • p 2 m • R• b • 3 • p 2 m • R • b • 2 • p 2 m • R • b • 1 • R • b • 0 )(I) (3)
A cross-entropy loss for softmax function defined as in equation ( 4) is applied to the functional (3) where z is the label selected by the neural network and y(I) is the true label.

L(F (I), y(I)) = log( z e (Fz(I)-F z (I)) )+ F z (I) -F y(I) (4)
We have trained the neural network for 30 epochs with an initial learning rate of 0.01, a learning rate drop factor of 0.97 and a piecewise learning rate schedule with a learning rate drop period of 1. The mini batch size is 128 with an every-epoch shuffle, the L2 regularization term is 0.02 and the momentum is 0.9.

In Table 1 2 shows that the rotational symmetry of Ψ 0 is stable with respect to variations of L2 regularization term and adding convolutional layers.

Furthermore, we have studied the properties of Ψ 0 varying the data augmentation (DA) applied to the dataset. Table 3 shows the correlation of Ψ 0 with the LoG and the correlation of Ψ 0 with Ψ 0 S with three different DA applied to the dataset. In the first one we have not applied any DA (No DA); in the second one we have randomly rotated the images of an angle of 0, π 2 , π, 3π 2 radiant (Mild DA); in the third one we have applied the DA described in Section 4.1 (Hard DA).

From Table 3 it emerges that the introduction of rotation invariances in the dataset by rotating the images lightly affects the correlation with the LoG but gives stability to Ψ 0 allowing it to be more rotational symmetric.

In conclusion, thanks to the analysis performed on the properties of Ψ 0 , we can argue that the structure of the architecture itself Table 3: Correlation between Ψ 0 and its LoG approximation and between Ψ 0 and Ψ 0 S varying the data augmentation (DA) applied to the dataset. influences the shape of the filters and that the introduction of 0 with a single filter Ψ 0 is a good model of the LGN.

The second layer of LGN-CNN

To enforce the link between our architecture and the structure of the visual system, we have studied the filters in the second layer comparing them with some real data obtained on monkey in [START_REF] Ringach | Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex[END_REF]. Therefore, we have trained two different CNNs, an

LGN-CNN defined by the functional ( 5)

F (I) :=(σ • F C 1 • R • 3 • p 4 a • R • 2 • p 4 m • R • 1 • R • 0 )(I) (5)
and a classical CNN defined by the functional ( 6) in which we have eliminated the first convolutional layer 0 and its following ReLU R, characteristic of our architecture.

F (I) :=(σ • F C 1 • R • 3 • p 4 a • R • 2 • p 4 m • R • 1 )(I) (6)
Let us note that in both architectures 1 contains filters with Gabor shapes after training. This is a well-known result on the filters of the first convolutional layer of CNNs as for example in [START_REF] Serre | Robust object recognition with cortex-like mechanisms[END_REF]), [START_REF] Yamins | Using goal-driven deep learning models to understand sensory cortex[END_REF]; however, the introduction of a first layer composed by a single filter does not change this behavior. Indeed, we have studied the statistical distribution We follow the same step as Ringach in [START_REF] Ringach | Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex[END_REF]) by plotting in the (n x , n y ) plane. In order to compare the plots, we looked for the distribution that best fits the Figure 10 shows the three plots. Let us note that introducing 0 modifies the elongation of Gabor filters in 1 . In particular, in classical CNN the filters are often more elongated in the x direction as we can see from the slope of the interpolating line in Figure 10a. In Figure 10b we can see that the slope changes greatly and that the filters become much more elongated in the y direction. This behavior is the same in the case of RFPs (Figure 10c) in which the distribution has a similar slope of LGN-CNN. This enforces more the link of LGN-CNN with the structure of the visual system motivating us to pursue in this direction.

Retinex algorithm via learned kernels

In this section we test the rotational symmetric filter on Retinex effect and contrastbased illusions. We described in Section 2 how the model of [START_REF] Morel | A PDE formalization of Retinex theory[END_REF]) have been neurally interpreted in [START_REF] Citti | A gauge field model of modal completion[END_REF] and applied to LoG. Our approach aims to find the reconstructed image I for a general operator

M : Ξ ⊂ R 2 → R, (7)
solving the following problem

M * I = M * I. ( 8
)
by finding the inverse operator of

M M : Ξ ⊂ R 2 → R. ( 9
)
By definition the inverse M of the operator M satisfies

M * M = δ. ( 10
)
Applying the steepest descent method, we obtain an iterative process, which can be formally expressed as

M t+1 = M t + dt • (M * M t -δ). ( 11
)
The algorithm will stop at time T when We start by comparing the inverse of the discrete Laplacian with respect to its ex-act inverse operator given by the function log( x 2 + y 2 ). Figure 11b shows the approximation of the inverse of the Laplacian and 11a shows the exact inverse in the interval [-5, 5]. We can note that the approximation is close to the exact one.

|| M T +1 -M T || L 1 dt < ,
In the second row of Figure 11 we have compared the inverse operators of a LoG and Ψ 0 where Figure 11c shows the LoG inverse operator and Figure 11d shows the Ψ 0 inverse operator. Since their inverse operators are close we expect similar Retinex effects in the next Section.

Application of the algorithm

In this Section, we aim to test our al-

Circles on a gradient background

We start with a simple grayscale image (see Figure 12a) in order to see how the different operators work. We are using the same image as in [START_REF] Morel | A PDE formalization of Retinex theory[END_REF]) and [START_REF] Limare | Retinex Poisson equation: a model for color perception[END_REF]) in which they obtain remarkable Retinex effects. It has a background that shifts from white (value 1) to black (value -1) maintaining the gradient constant from left to right. There are also two gray dots (same value 0), the left one with a brighter background, the right one with a darker one. Because of the different backgrounds, our visual system perceives the two dots differently: the left one is perceived as darker with respect to its true color; the right one is perceived as brighter.

We first consider the discrete Laplacian operator since we expect to obtain the same Retinex effects as in [START_REF] Morel | A PDE formalization of Retinex theory[END_REF].

Figure 12c shows the result obtained with our method. It is clearly the same result of the experiments performed by Morel and the values obtained in the position of the dots are close to -0.9 and 0.9 (indeed really close to completely black and white dots) whereas the entire background is gray with 0 values. Thus, we have considered the exact inverse of the Laplacian operator log( x 2 + y 2 ).

From Figure 12b it is clear that the Retinex effects occur to the gray dots. Furthermore we can note that the contours of the two dots are more defined w.r.t. the discrete Laplacian. Also in this case the values in the position of the dots are close to -0.9 and 0.9.

Then, we have performed our experiment with the inverse of a LoG. We can note from Figure 12d that the Retinex effects occur also in this case (with values close to -0.9 and 0.9), similarly to the Retinex effects of the discrete Laplacian. Also in this case the contours of the dots are not completely clear.

Finally, we have tested the inverse of the convolutional operator Ψ 0 of an LGN-CNN introduced in Section 4. In Figure 12e we can see that this operator shows Retinex effects where the values of the dots are close to -0.9 and 0.9. In this case the contours of the dots are even clearer than the ones obtained with the LoG and the discrete Laplacian.

To summarize, we have shown that our method reproduces the same results of [START_REF] Morel | A PDE formalization of Retinex theory[END_REF] in the case of the Laplacian operator. Furthermore, we have tested it on other operators obtaining remarkable results.

It is particularly interested the case of the convolutional operator Ψ 0 since it is able to show Retinex effects even if it is a learned filter with no a-priori structure, enforcing the link between the LGN-CNN architecture and the visual system.

Adelson's checker

We have also tested our algorithm on the Adelson's checker shadow illusion as in [START_REF] Morel | A PDE formalization of Retinex theory[END_REF]) and [START_REF] Limare | Retinex Poisson equation: a model for color perception[END_REF]).

Since we are more interested in the Retinex effects of LoG and Ψ 0 operators we have analyzed their abilities on the grayscale image (see Figure 13a). It shows a checkerboard with light gray and dark gray square with a cylinder on it that shadows a part of the squares. In particular, the square labeled 'A' and the square labeled 'B' have the same grayscale value (in our case, since -1 is black and 1 is white, they have -0,4953 value). The illusion is built in such a way that, even if they have the same value, they are perceived in a completely different way. Indeed square 'A', which is outside the shadow and surrounded by light gray squares, is perceived as a dark gray square. On the other hand, square 'B', which is inside the shadow and is surrounded by dark gray squares, is perceived as lighter.

We expect that the two operators should

Information transmission efficiency

We are now interested to see if the layer 0 have similar properties as regards the information transmission efficiency of the LGN. Indeed, it is well established (see e.g. [START_REF] Reinagel | Temporal coding of visual information in the thalamus[END_REF], [START_REF] Zaghloul | Different circuits for on and off retinal ganglion cells cause different contrast sensitivities[END_REF], (Uglesich et al. 2009), (Im andFried 2015), [START_REF] Pregowska | Information processing REFERENCES in the LGN: a comparison of neural codes and cell types[END_REF])) that the average firing rate of the retinal neurons that drive information to the LGN is much bigger than that of the LGN. In particular, LGN is able to delete spikes preserving the more informative ones leading to a loss of information.

A Appendix

A.1 Rotation symmetry of Ψ 0

In this section we define the setting in which we study the rotation symmetry of Ψ 0 .

Let us consider the architecture of an LGN-CNN in which we can split the first convolutional layer composed by only one filter from the rest of the neural network which will be fixed. Thus, this first layer can be approximated by a function Ψ 0 : R 2 → R, assuming Ψ 0 ∈ L 1 loc (R 2 ). A general image can be defined as a function I : R 2 → R where we assume I ∈ L 1 loc (R 2 ). We can consider a subset Γ ⊂ L 1 loc (R 2 ) of all the images where for each image I is defined a labelling y : Γ → R, where y(I) is the corresponding label to image I.

We require some rotational invariant properties on this set Γ . In particular, if we consider a rotation R θ of an angle θ on R 2 plane around its center then the composition

I θ = R θ (I) = I(R -θ (x)
) is still an image and we can also assume that I θ ∈ Γ (i.e., that the subset Γ is close under rotation). Furthermore, the rotated image should maintain the same label, i.e., y(I θ ) = y(I).

Since the images we will consider in this problem are grayscale ones (thus with values between 0 and 1) we can also assume that the images are normalized with ||I|| L 1 loc ≤ 1. Summarizing all these properties we can assume that Γ is a compact set on L 1 loc , Γ = {I ∈ L 1 loc ; ||I|| L 1 loc ≤ 1} Thus, the rest of the neural network will be defined by a nonlinear functional

C : L 1 loc (R 2 ) → L 1 loc (R 2 ) C(f (x)) = max(0, f (x)) (12)
which is one of the more frequently nonlinear function used in CNN, called ReLU. And then we can define

F : L 1 loc (R 2 ) × L 1 loc (R 2 ) → R F (I, Ψ 0 ) := R 2 C (I * Ψ 0 )(z) dz. (13)
Then F (I, Ψ 0 ) is the label that our architecture associates to the image I and should eventually approximates y(I).

Thus, our aim is to find a function Ψ 0 in such a way that F approximates well the known functional y(I). In particular we would like to minimize the following functional min

Ψ 0 ∈L 1 loc (R 2 ) Γ |F (I, Ψ 0 ) -y(I)|dµ(I) (14)
where the integral done over the set Γ is a Bochner's integral (see e.g., Section 5 of chapter V of [START_REF] Yosida | Functional Analysis[END_REF] and [START_REF] Mikusinski | The Bochner Integral[END_REF]).

Our aim is to find a function Ψ 0 that attains the minimum of ( 14) where F is defined in (13) and C is defined in (12). We would like to find out if there exist some rotational invariant properties on the function Ψ 0 . Let us note that the functional defined in ( 14) is convex thanks to the convexity of the function C defined in (12) and continuous. Then the existence and uniqueness of a solution is guaranteed (see e.g., Section 1.4

of [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF].

Remark 1 Let us consider two function f, g ∈ L 1 (R 2 ) and a rotation R θ of an angle θ. Then Fig. 16: Comparison between the filter Ψ 0 and a Gaussian. We can see that Ψ 0 is really close to a discrete approximation of a Gaussian fitted to the data.

f * R θ (g)(x) = (R -θ (f ) * g)(R -θ (x)) Proof f * R θ (g)(x) = R 2 f (x -y)R θ (g(y))dy = R 2 f (x -y)g(R -θ (y))dy = then we substitute y = R -θ (y) whose Jacobian has determinant equal to 1 = R 2 f (x -R θ (y ))g(y )dy = R 2 f (R θ (R -θ (x)) -R θ (y ))g(y )dy = R 2 f (R θ (R -θ (x) -y ))g(y )dy = R 2 R -θ (f (R -θ (

  latter function is in turn a good approximation of the receptive field profiles (RFPs) of the cells in the LGN. The analogy with the visual system is established, emerging directly from the architecture of the neural network. A proof of rotation invariance of the first layer is given on a fixed LGN-CNN architecture and the computational results are shown. Thus, contrast invariance capability of the LGN-CNN is investigated and a comparison between the Retinex effects of the first layer of LGN-CNN and the Retinex effects of a LoG is provided on different images. A statistical study is done on the filters of the second convolutional layer with 1 Introduction 1.1 Architecture of the visual system and CNNs The visual system is composed by many cortices that elaborate the visual signal received from the retina via the optical nerve.

  among others. An analysis of retinal and cortical components of the Retinex theory has been performed by several research teams in the past years (for a review see e.g. (Yeonan-Kim and Bertalmío 2017), (Valberg and Seim 2013)) and it has been shown that the Magnocellular cells are the ones involved in contrast perception (see e.g. (Enroth-Cugell and Robson 1966),

  architecture and describe it in detail. The new architecture is implemented in Section 4. It provides classification performances comparable to the classical one, but it enforces the development of rotation symmetric filter during the training phase. As a result, the filter is a good approximation of RFP of LGN. In Section 5 we test our filter on contrast perception phenomena. Indeed, we use the learned kernel to repeat and justify the Retinex model. We test the model on a num-ber of classical Retinex illusion, comparing with the results of (Morel et al. 2010). Thus, in Section 4.3 we show that the filters in the second layer of the net mimics the shape of Gabor filters of V1 RFPs. We have already recalled that this happens also in standard CNN. We compare the statistics of the filters with the results obtained on the RFPs of a macaque's V1 from the work of Ringach (Ringach 2002). By construction both in our net and in the visual system the second layer do not exhibit LoG, which are collected in the first layer. Also, other statistical properties of the filters are present in our LGN-CNN. The proof of the symmetry of the filter of the first layer is collected in appendix A. It is provided in the simplified case of a neural network composed by only a layer composed by a single filter.

Fig. 1 :

 1 Fig. 1: On the left: RFP of an LGN cell where the excitatory area is in white and the inhibitory one is in gray. On the right: Its approximation by a LoG. From: (DeAngelis et al. 1995).

Fig. 2 :

 2 Fig. 2: First row: RFPs of two simple cells of V1 where the excitatory area is in white and the inhibitory one is in black. Second row: their approximations by Gabor functions. From: (Sarti and Citti 2011).

  the authors have formalized the Retinex theory as the solution of the following discrete Poisson PDE -∆ d I = M (I) (1) where ∆ d is the classical discrete Laplacian and M is a modified version of the discrete Laplacian. In (Citti and Sarti 2015) a neural interpretation of the model has been introduced. The RFP of the LGN cell which takes in input the visual signal acting by convolution on it, modeled as a modified discrete LoG, M (G σ ) ≈ ∆G σ , where G σ is a Gaussian bell Hence the action on an input image is the following:

Figure

  Figure 10c shows the statistical distribution of RFPs of V1 cells in monkeys in (n x , n y ) plane obtained by Ringach in (Ringach 2002). In (Barbieri et al. 2014) the authors have studied the same statistical distribution with respect to the Uncertainty Principle associated to the task of detection of position and orientation.

Fig. 3 :

 3 Fig. 3: Scheme of the LGN and V1 in parallel with the first two layers 0 and 1 of the LGN-CNN architecture.

--

  Applying a circular mask to each image, leaving unchanged the image in the circle and putting the external value to zero; Rotating each image by 5 random angles augmenting the dataset to 25000 images; thanks to the previous step no boundary effects are present; -Cropping the 64×64 centered square that does not contain the black boundaries; -Subtraction of the mean value in order to have zero mean input images. Thus, after these steps we have obtained a rotation invariant training set composed by 25000 64 × 64 images. We have applied the same steps to the test set but we have rotated each image to just one random angle. Since the images are 64 × 64 we have decided to use quite large filters in the first and second layer (7 × 7 and 11 × 11 respectively) in order to obtain more information about their shapes.

Figure 4

 4 Figure 4 shows the architecture of our CNN. Let us note that between each convolutional layer and its ReLU function there is a batch normalization layer b with the same size of the number of filters of the corresponding convolutional layer. The input is an image of size 64 × 64 × 1. Then there is the first convolutional layer 0 composed by only Ψ 0 of size 7 × 7 followed by a ReLU R. After 0 the batch normalization layer performs a normalization similar to the one

Fig. 4 :

 4 Fig. 4: Architecture of LGN-CNN.

  tures to obtain better results on the classification task without losing the properties of the first layers of the LGN-CNN. Since our focus was to analyze the structures that arises in the first two layers, we have not further investigated the performance of the CNN.4.2 The first layer of LGN-CNNAfter the training phase we analyze the neural network focusing on the first layer in this Section. Figure5shows the filter Ψ 0 and 5b shows its approximation with minus the LoG.The two have a high correlation of 95.21% computed using the built-in function of MatLab corr2.Moreover, Figure6ashows the 2D image of Ψ 0 obtained after the training phase.Then, in Figures6b and 6cwe plot a 2D approximation of Ψ 0 as a 280 × 280 filter and minus the LoG in which the rotational symmetric pattern is clearer.We would like to point out that in theLGN there exists both cells with on-center/offsurround as well as off-center/on-surround RFPs. In order to model these kinds of cells we have modified the current architecture by adding a second filter in the layer 0 .Since the STL10 dataset contains natural images with many contours we should expect to have oriented filters. This is not the case and Figure7shows the on-center/offsurround and off-center/on-surround obtained after the training phase. For simplicity, we have considered the model with a single filter. In order to quantify the rotational properties of the filter, we compute the correlation between the filter obtained in the first layer of the CNN and a new one, obtained via a rotation invariance symmetrization. We will test the behavior of the first layer of LGN-CNN for different values of the L2 regularization term and adding more convolutional layers. In particular, we added to the previous architecture described in Section 4.1 two convolutional layers composed by 32 filters of size 3 × 3 × 32 (each one followed by a batch normalization layer b and ReLU R ) after the third layer 3 . We have added other two convolutional layers with the same characteristics after the layer 4 . The normalized rotational invariant filter is obtained by the following procedure: -Using the function imresize with scale 3 and bilinear method to enlarge the filter; -Rotating the filter with imrotate by 360 discrete angles between 1 and 360 degrees and summing them up; -Applying again the function imresize with scale 1/3 and nearest method to recover a filter with the same size of Ψ 0 ;(a) Filter Ψ 0 of first layer of LGN-CNN. (b) Minus the Laplacian of Gaussian (LoG).

Fig. 5 :

 5 Fig. 5: Comparison between the filter Ψ 0 and minus the LoG. The two have a high correlation of 95.21% computed using the built-in function of MatLab corr2.

( a )

 a Filter Ψ 0 of first layer of LGN-CNN. (b) Approximation of the filter Ψ 0 of first layer of LGN-CNN. (c) Minus the Laplacian of Gaussian (LoG).

Fig. 6 :

 6 Fig. 6: The first figure shows the 7 × 7 filter Ψ 0 of the neural network. To better visualize the filter Ψ 0 , we provide an approximating 280 × 280 filter and minus the LoG.

Fig. 7 :

 7 Fig. 7: On-center/off-surround and offcenter/on-surround filters of 0 with 2 filters.

Fig. 8 :

 8 Fig. 8: On the left: filters from LGN-CNN. On the right: their approximation with the function (2).

Fig. 9 :

 9 Fig. 9: On the left: filters from classical CNN. On the right: their approximation with the function (2).

( a )

 a Distribution of filters in 1 of classical CNN defined by functional (6). (b) Distribution of filters in 1 of LGN-CNN defined by functional (5).

( c )

 c Distribution of RFPs of simple cells from (Ringach 2002).

Fig. 10 :

 10 Fig. 10: Comparison between the statistical distribution on (n x , n y ) plane of filters of a classical CNN, of our architecture and of RFPs of real data.

  for a fixed error > 0. Thus, ||M * M T -δ|| L 1 < and indeed M T would be a good approximation of M . Finally, we can compare the image I with the reconstructed one I and see if any Retinex effects occur. Let us note that one difference between our approach and the one proposed by Morel is that he imposed the Neumann boundary conditions to the Poisson equation whereas, in our model, the Neumann boundary conditions imposed to the filter are inherited from the inverse operator itself. Furthermore, we have faced the problem for a Laplacian operator to see if the results are similar to the Morel ones. Indeed, in our Retinex algorithm we convolve a given visual stimulus I with a fundamental solution M and we reconstruct the perceived image I using the inverse operator M .5.1 Study of the inverse operatorsFirstly, we show the inverse operators obtained through the algorithm described by equation (11). We start from the classical discrete Laplacian operator and then we will move to convolutional operators, in particular a discrete LoG and the filter Ψ 0 of an LGN-CNN. In order to compare the inverse operators, we will show the 2D plots obtained by selecting a slice of the 3D inverse operator itself.

  gorithm and our different operators to see if Retinex effects occur. Let us note that Retinex is an algorithm that mimics our color perception and does not try to improve the image quality. Indeed, the grayscale values should modify towards our color perception intensity values.

( a )

 a Exact inverse of a Laplacian log( x 2 + y 2 ) in the interval [-5, 5]. (b) Inverse of discrete Laplacian. (c) Inverse of LoG. (d) Inverse of the first layer of an LGN-CNN.

Fig. 11 :

 11 Fig. 11: Comparison between inverse operators. First row: inverse of the discrete Laplacian and its exact inverse operator. Second row: inverse operators of a discrete LoG and the first filter Ψ 0 of an LGN-CNN.

( a )

 a Starting image with two gray dots on a gradient background. (b) Retinex effects of exact inverse of Laplacian. (c) Retinex effects of discrete Laplacian. (d) Retinex effects of LoG.(e) Retinex effects of Ψ 0 of LGN-CNN.

Fig. 12 :

 12 Fig. 12: Retinex effects of some inverse operators on starting image 12a.

  Figures 13d, 13d and 13f highlight the two squares 'A' and 'B' in the starting image and in the recovered images using the LoG and Ψ 0 . In this way it is clearer that the Retinex effects occur in both cases. To summarize, we have shown that the filter Ψ 0 considered as a convolution operator shows Retinex effects really closed to Retinex effects of LoG. This enforces again the link between the structure of our architecture and the structure of LGN.

  Thus, we have studied the information loss on the 8000 images of STL10 test set by convolving each image with Ψ 0 and computing the entropy from the histogram of gray scale values via the built-in MatLab function entropy. It turns out that on average the entropy decreases from 7.04 to 5.97 with a loss of 15.27 % of the information. Thus, we have reconstructed the images using the Retinex algorithm described in Section 5. The average entropy increases to 6.92 leading to a loss of 1.83% of the information w.r.t. the original dataset. This suggests that almost the entire information contained in the visual stimulus can be reconstructed via some feedback or horizontal connections, where the reconstructed stimulus becomes invariant w.r.t. lightness constancy. Figure 14 shows an example of the convolution and the reconstruction performed on an image of the dataset with the corresponding gray values histogram with respect to the entropy is calculated.

  Grayscale Adelson's checker shadow illusion: the two squares. (c) Retinex effects of LoG. (d) Retinex effects of LoG: the two squares. (e) Retinex effects of Ψ 0 of LGN-CNN. (f) Retinex effects of Ψ 0 of LGN-CNN: the two squares.

Fig. 13 :

 13 Fig. 13: Comparison between the Retinex effects of some operators on grayscale Adelson's checker shadow illusion.

Fig. 14 :

 14 Fig. 14: In order on the left: a grayscale image I, the convolved image I and the reconstructed image I via eq. (8). On the right: the corresponding histograms of the grayscale values.

2 RCC.

 2 Fig. 15: Filter Ψ 0 of LGN-CNN obtains after training.

  (a) Filter Ψ 0 of first layer of LGN-CNN. (b) Gaussian function.

  ). It does not seem that an analogous layer is present in classical CNN, but it is known that it is cru-

cial for human contrast perception. It has been investigated in Retinex theory formulated in 1964 by E. H. Land in

Table 2 :

 2 Correlation between Ψ 0 and Ψ 0

	S

let Ψ 0 to be the closest rotational symmetric filter is 0.02. This is the filter we have shown in Figures

5 and 6

and we have used through the paper. The Table

* This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sk lodowska-Curie grant agreement No 754362. * * This project has received funding from GHAIA -Marie Sk lodowska-Curie grant agreement No 777822.


R Jalain

J Bonnety

G Legros

A Matynia

Doping rich ethylene premixed flames: influence of C 3 -C 5 alcohols on the structure of the steady one-dimensional laminar flame

Keywords: Flame structure, alcohol addition, alcohol isomer, soot precursors

    Doping rich ethylene premixed flames: Influence of C 3 C 5 alcohols on the structure of the steady one-dimensional laminar flame

Introduction

Among the main features of the oxygenated fuels are their ability to reduce the emissions of particulate matter [1,2,3,4,5] together with their propensity to form toxic byproducts such as aldehydes or ketones [START_REF] Mcenally | Proceedings of the Combustion Institute[END_REF][START_REF] Oßwald | [END_REF]8]. Heavy alcohols, with their isomeric properties, can mitigate both effects, by choosing the correct isomer or mixture. Moreover, they exhibit higher energy content, lower corrosivity, and are less hygroscopic than ethanol, which is widely used in gasoline engines. Interestingly, propanol, butanol, and pentanol can all be produced by microbial processes ([9, 10, 11, 12]) and they are compatible with an utilization as additives to both gasoline and diesel.

While numerous studies have been dedicated to propanol isomers, i.e. 1-and 2-propanol [13,14,15,16,17,18,19], and all the isomers of butanol (namely, 1-butanol, 2-butanol, i-butanol/2-methyl-1propanol, t-butanol/ 2-methyl-2-propanol) [START_REF] Mcenally | Proceedings of the Combustion Institute[END_REF]20,21,[START_REF] Oßwald | [END_REF]22,23,24,25], the literature on pentanol isomers is relatively sparse. Over the four primary alcohol isomers, three, i.e. 1-pentanol, 2-methylbutanol, and i-pentanol/3-methyl-1-butanol, seem to be the most investigated ones ( [26,[START_REF] Togbé | Proceedings of the Combustion Institute[END_REF][START_REF] Heufer | [END_REF]29,30,31]), followed by the secondary alcohols, 2-pentanol and 3-pentanol [32,8]. In contrast, the only tertiary alcohol isomer, i.e. t-pentanol/2-methyl-2-butanol [31], the secondary 3-methyl-2-butanol and the primary 2,2dimethyl-propanol suffer from an apparent lack of data. For the latter, its solid state below 325 K (at atmospheric pressure) makes it tricky to blend at a controled content to a gaseous stream.

As a result, kinetic mechanisms have been developed for propanol and butanol isomers, as well as some pentanol isomers. Among others, the CRECK modeling group developed a free-access kinetic mechanism including propanol and butanol [16,15]. Recently, Li et al. [18] published a mechanism on propanol isomers. Sarathy et al. [4] also delivered a mechanism for the propanol and butanol isomers. Several kinetic mechanisms have been developed for 1-pentanol [START_REF] Heufer | [END_REF][START_REF] Togbé | Proceedings of the Combustion Institute[END_REF], 2methyl-1-butanol [33,30], and i-pentanol [26,4]. Finally, Kölher et al. built a mechanism specifically developed for the three linear pentanol isomers [8].

Throughout the extensive literature on butanol, which includes the three types of alcohols, some characteristics have reached a consensus. For instance, the unstretched laminar flame speed seems to decrease while adding methyl groups, i.e. 1-butanol and t-butanol exhibit the highest and lowest speed, respectively [21,23]. A similar conclusion was drawn for pentanols isomers in Ref. [31]. On the opposite, conclusions about sooting tendencies are more blurry. Most of the studies state that tertiary alcohols generate more hydrocarbon intermediates leading to more soot, while primary and secondary alcohols produce more oxygenated intermediates, preferentially forming aldehydes or ketones [START_REF] Mcenally | Proceedings of the Combustion Institute[END_REF][START_REF] Oßwald | [END_REF]. That being said, a recent study unveiled an opposite trend, showing that among the butanol isomers in the specific conditions of the study the highest rate in particulate matter reduction was achieved by t-butanol [25].

Within the scope of the current study, the investigations will focus on alkene production by alcohol dopants in rich flames of a well-studied sooting fuel, i.e. C 2 H 4 , which might the soot production pathways. As detailed in Sarathy's review Alcohol combustion chemistry [4], two major ways of alkene production can be identified.

Firstly, at high temperatures (above 1500 K) and in fuel-rich conditions, the consumption of alcohols is dominated by unimolecular decomposition. Such reactions are important for correctly simulating soot emissions, and non-premixed diffusion flame structures. In all alcohols, these reactions consist of simple C-C and C-H bond scission reactions. Unimolecular water elimination (i.e. dehydration) reactions are also important for alcohols. The four-center reactions involving the β hydrogen atoms are the most thermodynamically favored and result in the formation of an alkene and water (R1).

C x H y OH C x H y-1 + H 2 O (R1)
The reaction rate coefficients of four-center dehydration reactions vary with the extent of substitution be-cause this affects the number of abstractable β hydrogen atoms and their bond dissociation energies (i.e., primary, secondary, or tertiary).

Secondly, fuel radicals can undergo unimolecular decomposition via β-scission at relatively high temperatures (e.g., 900 K). In these reactions, the bonds located in the β position of the radical carbon are weakened and break to form an unsaturated species and another radical. β-scission of C-C and C-H bonds leads to the formation of unsaturated alcohols or alkenes (i.e. C=C bonds, reaction R2), whereas β-scission of the O-H bonds results in the formation of a carbonyl group (i.e. C=O bond).

• C x H y-1 OH C x H y-1 + • OH (R2)
The present work intends to characterize the intermediate hydrocarbons production attributed to each of the C 3 to C 5 alcohols (cf. Tab. 1) when it seeds the unburnt C 2 H 4 /air stream of an atmospheric laminar flat premixed flame. The set of experiments especially leads to the measurements of the mole fraction profiles of C 1 to C 5 hydrocarbons, with isomeric separation of butene and pentene.

OH OH OH OH 2-methyl-1-butanol 3-pentanol 3-methyl-2-butanol 2,2-dimethyl-1-propanol

Experimental setup

Flames and burner

A rich flat flame (φ=1.7) of ethylene/air established over a McKenna burner at atmospheric pressure is seeded by a single additive, selected among the different isomers of propanol, butanol, and pentanol (cf. Fig. 1). The experimental parameters are reported in Tab. 2. A base flame of pure ethylene with a flowrate of 0.984 nl/min is also investigated.

As extensively outlined in Ref. [34], the liquid additive is to be prevaporized, then blended at a controlled content with the flow of ethylene which is here the carrier gas. The liquid flow rate is adjusted by a Bronkhorst Cori-Flow mass flow controller while a Bronkhorst EL-Flow one regulates the ethylene flow rate. The liquid additive is evaporated and mixed with ethylene in a CEM (Controlled Evaporator Mixer). The liquid fuel flow is set to get 3.5 % (in mole) of vapor X vap in the carrier gas. Connected to the outlet of the CEM, a heated line (referred to as HL in the following) feeds the burner with the unburnt gaseous mixture. A second heated line supplies the burner with air, the flow of which is regulated by a second Bronkhorst EL-Flow mass flow controller. The total mass flow rate is set to 12 g/min. A nitrogen shield flows around the burner to prevent any perturbation from the environment. A water circuit inside the burner allows to regulate the temperature of the porous section. The regulation is ensured by a heating cartridge as hot source and fans as cold source.

To further improve the flame stability, especially for rich flames, a disk of non coated cellular monolith ceramic is held at 1 cm above the burner output (see Fig. 1 c)). The ceramic disk is meshed by 600 cells per square inch to allow the circulation of burnt gases and limit the upstream propagation of the local perturbations.

To compare the behaviour of different fuels, experimenters have to choose which content to be held constant. There are several options such as exit velocity [35], volume fraction [36], molar fraction [37,34,[START_REF] Abboud | Impact des suies issues de biocarburants sur le filtre à particules[END_REF] or total carbon mass [START_REF] Yang | [END_REF]. Each one has side effects, (i) constant exit velocity and constant volume fraction might be unadvised if the range of fuel densities is important, (ii) molar fraction might be unadvised if the range of fuels number of carbon is important, (iii) total mass carbon might be unadvised for molecules containing atoms different from carbon and hydrogen. For instance by keeping the total carbon mass constant for propanol and pentanol would lead to add more oxygen in the case of propanol than pentanol, thus improving the oxydative potential of the mixture, which could, for instance, lead to shift the balance regarding sooting tendency of the dopants. Usually in the case of dopant, the amount staying low, the base flame must not be too affected, therefore the choice might not impact the tendencies too much. Moreover if the experimental result are supported by numerical simulations the impact will be taken into account.

In the current paper, the liquid fuel flow has been set to 3.5 % (in mole) of vapor X vap . This amount is expected to not change significantly the ethylene flame properties (such as temperature and thickness) and allows to control the alcohol flow properly and to measure significantly the discrepancies among the isomers. Constant molar fraction has been chosen to keep the same quantity of matter into the combustion. Propanol, butanol and pentanol having different molecular weights, mass flow had to be modified to keep the molecular amount constant.

Sampling and analysis

Gaseous sampling was performed along the vertical axis z at different heights above the burner (HAB) with a quartz probe, connected to a vacuum pump, and stored in spherical glass flasks (volume V=1 l). The quartz probe orifice is about 60 µm diameter. Flasks are vacuumed before each experiment and then filled by the sampled gas up to 4 kPa. A sampling can be done by step of 100 µm along the vertical axis z of the flame by moving the burner downwards with a stepper motor. The samples are then injected into a Gas Chromatography device (GC) equipped with a Flame Ionization Detector (FID). To this end, the gas to be analyzed is transferred in a compression vessel connected to a sample loop of 200 µl and a piston is used to adjust the pressure to 20 kPa before the injection into the Varian 4000 GC/MS.

The relatively small diameter of the probe orifice allows minimizing the sampling volume (and therefore the potential shift between the burner and the sampling position) together with the flow perturbations. A pressure level of 20 kPa has been set after having characterized a linear response between the injection pressure and the FID response, to maximize the FID response while keeping a reasonable time to achieve the samples. In addition, a level of 4 kPa has been selected to have an appropriate quantity of gas to be injected into the GC/MS at the required pressure. It is worth mentioning that filling a flask requires between 2.25 and 3.5 minutes depending on the position into the flame.

C 1 to C 5 hydrocarbons were analyzed by GC/MS with a Restek Rt-Alumina BOND/MAPD separation column. At the exit of the column, a mass spectrometer (MS) enables the identification of unknown peaks and quantitative measurements are provided by FID. This column does not separate alcohol species. Typical chromatograms, the identified species and their order of elution in the Restek Rt-Alumina BOND/MAPD Column are available in the supplementary material.

Molar fraction profiles

Two calibration methods have been used to calibrate the GC/MS measurements. First, a standard bottle was used to link the peak areas to the molar fractions of 8 species (cf. Table 3) by the relation :

X i = A i A std i X std i (1)
with X i the sample molar fraction, A i the sample peak area, A std i the standard peak area and X std i the standard molar fraction of the species i.

Concerning the species absent from the standard bottle (propadiene, butene isomers and pentene isomers), FID Response Factors (referred as to RF) were used. A. Tipler, from PerkinElmer, published in an Application Note [START_REF] Tipler | PerkinElmer Application Note[END_REF] a chromatogram (obtained with a FID) including all the species presented in the current paper. From this chromatogram, where species molar fractions were known, it was possible to compute absolute response factor RF abs :

RF abs i = A i X i (2)
with A i the area of the ith species (V • s) and X i the mole fraction (%). Then, the relative response factors were defined between a reference species re f (here 1,3butadiene), and each of the species i to be calibrated:

RF rel i = RF abs i RF abs re f (3)
Finally, the molar fraction were computed from the relation:

X i = 1 RF rel i        A i A std re f X std re f        (4)
These calculations are presented on Tab. 4. The reference species 1,3-butadiene has been selected because (i) its FID response was very stable and (ii) most of the species absent from the standard bottle has a RF rel rather close to 1. To evaluate the validity of the method, the molar fraction of the standard (X std ) were computed by this method. The last column (X std ) shows the molar fraction of the species present in the standard bottle calculated from the RF method. The maximal differences between the reference values (see Tab. 3) and the calculated values are 12 % (propene and n-butane). By these two methods, 17 chemical compounds have been identified and quantified. Profiles were realized between a HAB of 0.2 mm and 2 mm. The distance between these two points will be referred as to Z (i.e. 1.8 mm) and will be used in the next sections.

To ensure the robustness of the flame analysis, each profile is the combination of at least two flame profiles (same flame generated twice), realized in two consecutive days (except the 2,2-dimethyl-propanol due to the complexity of the experiment). Points spaced by only 100 µm are always from two different profiles, i.e. two different sets of measurements. Also, realizing a whole profile requires around one hour and during this time the content of alcohol dopant is stable and the vaporization complete.

Modeling

Simulation parameters

Experimental data were compared to simulations realized with CANTERA [START_REF] Goodwin | Cantera: An Object-Oriented Software Toolkit For Chemical Kinetics, Thermodynamics[END_REF] software and the burner flame configuration with mixture-averaged molecular diffusion and the resolution of the energy equation.

Reaction mechanisms used in the present study are summarized in Tab. 5 and Tab. 6 presents the input conditions used for all the simulations. Because there is no temperature measurement, this latter has been computed by solving the energy equation. As the concentration of additive remains quite low, there are small differences in the temperature profiles among all the flames. The downstream temperature (z = 5 cm) for each simulation ranges from 1723 to 1743 K.It is worth mentioning that the burner flame model, when the mass flow term is provided, estimates the heat losses to solve the energy equation. By comparison, the mass flow rate and flame temperature obtained for simulations of perfectly adiabatic flames (free flame model) are 0.342 kg/s/m 2 and 2068 K, respectively. Besides, the flames studied in the present work do not emit yellow/orange light, therefore soot radiative heat losses are likely to be negligible. That being said, it is important to verify how much a wrong estimation of the heat losses may change the analyses and the conclusions established in this paper.

In order to investigate the impact of temperature profile on species concentration profiles, complementary simulations were conducted with imposed temperature profiles (i.e. without resolution of the energy equation). The four alternative profiles tested are shown in Fig. 2. For two of them, the flame thickness is preserved while the maximum temperature reached at the end of combustion is increased by 10% (≈ 170 K) to assess a bad evaluation of the temperature or decreased by 20% to take into account probable heat losses (≈ 340 K, see profiles A and B, respectively). The two other profiles have more/less pronounced temperature gradients and the flame thickness is therefore decreased/increased (see profiles C and D, respectively). In spite of these important modifications, the results showed that the maximum molar fraction of studied species never exceeds a factor 1.5 with the reference temperature profile and the observed qualitative trends never change.

Propanol and butanol

Flames doped by propanol isomers were computed by CRECK [15,16], Li et al. [18] and Sarathy et al. [4] mechanisms. Flames doped by butanol isomers were also computed by CRECK and Sarathy et al. mechanisms.

CRECK mechanism has been developed to simulate the combustion blends of toluene primary reference fuel and alcohols. The mechanism contains 254 species and 7568 reactions. Concerning the propanol isomers, the validation of the kinetic mechanism relies, among others, on structures of counterflow non-premixed flames and previously reported data over a wide range of configurations and conditions. The model has also been validated by comparing simulations made using this kinetic model with previous and new experimental data on counterflow non-premixed flames of n-butanol and i-butanol. The structures of these flames were measured by gas sampling from the flame and analyzed by gas chromatography.

Li et al. mechanism contains 156 species and 1413 reactions. The kinetic model of propanol isomers was developed and validated against the experimental data of the paper (flow reactor pyrolysis and laminar flame propagation), as well as other experimental data in literature covering wide ranges of temperatures, pressures and equivalence ratios.

Sarathy et al. mechanism contains 354 species and 2625 reactions and includes detailed high-temperature and low-temperature reaction pathways with reaction rates assigned to describe the unique oxidation features of linear and branched alcohols. Experimental validation targets for the model include low pressure premixed flat flame structure, fundamental laminar flame speeds, rapid compression machine and shock tube ignition de-lay, and jet-stirred reactor species profiles. The agreement with these various data sets spanning a wide range of temperatures and pressures is reasonably good.

The idea behind confronting several models is to find out where there is room for improvement and where it might have consensus.

Pentanol

To the authors' knowledge there is no mechanism including all isomers, therefore a single mechanism including the three linear ones has been used. Indeed, comparing numerical and experimental results for 8 isomers and adding more models could make the paper reading too tedious.

Simulation of pentanol isomers is realized thanks to the Köhler et al. mechanism (225 species, 24 526 reactions) [8], developed for the combustion of the three linear isomers of pentanol. This kinetic mechanism was built thanks to the open-source software RMG [42], based on the C 1 -C 4 USC mechanism [43] ("seed mechanism" given on RMG) and tested on low pressure H 2 /O 2 /Ar flames doped by pentanol isomers.

Results

Overall results

In the present study, all flames are composed of 96.5 % of ethylene and 3.5 % of alcohol isomer. Therefore, temperature and main species molar fractions are likely driven by ethylene combustion rather than the dopants. To support this statement the molar fraction profiles of four species, i.e. C 2 H 4 (the main fuel), CH 4 , C 2 H 6 , and C 3 H 8 , are presented in Fig. 3. For the sake of clarity, the error range, estimated at ±20 %, is displayed at z=1 mm for the minimum value in gray and for the maximum value in black. By this way the maximum error of the whole set of experiment is displayed, one can see that for the species presented (considered unaffected by the dopants) , there is always a common part between both error ranges.

With 17 species measured for 14 dopants, the experimental data consists in 238 profiles (17 × 14). To simplify the comparisons, the average molar fractions along Z have been computed by the following equation:

i = 1 Z 2 0.2 X i (z)dz (5)
This method allows mostly to obtain one single value by profile and eventually to encompass noisy data and larger peaks.

Every average molar fraction X i for every dopants are displayed on Fig. 5. The logarithmic y-axis scale allows to point out the similarities and discrepancies among the species. For instance, as already mentioned, C 2 H 4 , CH 4 , C 2 H 6 , C 3 H 8 but also C 2 H 2 and C 4 H 10 levels are not influenced by the different dopants. This figure will be referred to numerous times in the following sections. All profiles are available in the supplementary material.

Propanol

Propanol is the smallest alcohol presenting an isomeric structure. The two isomers are the primary alcohol 1-propanol and secondary alcohol 2-propanol. Flame structure analysis revealed that among all the intermediate species measured, only C 3 H 6 molar fraction profile significantly differs when 1-propanol or 2propanol are added to ethylene flames.

Figure 4 presents the profiles measured for each of the two flames, as well as, the X (eq. 5), representing the average level of the species along z. It can be noted that the molar fractions of C 3 H 6 are almost twice as high for the flames doped with 2-propanol. Similar trends were obtained in the study by Li et al. [13] who investigated rich (φ=1.8) propanol/O 2 /Ar premixed flames at 4 kPa. Peak molar fraction measured were then 250 and 390 ppm for 1-propanol and 2-propanol, respectively. A second important aspect of the curve, in the case of 2-propanol, is that the maximum molar fraction is obtained for a much lower z, suggesting production reactions starting earlier.

Finally, contrary to the 2-propanol flame, the molar fraction of propene measured in the 1-propanol-doped flame is similar to the ones measured in the base flame and the butanol and pentanol flames (maximum value and location), suggesting that the origin of propene for this flame is mainly driven by the chemical decomposition of ethylene.

Tab. 7 presents the experimental and numerical mean molar fraction of propene. The agreement between experimental and simulated profiles of C 3 H 6 is acceptable for 1-propanol doped flames (differences of mean molar fractions of about 30 %) given the measurement uncertainties. However, the highest molar fraction of propene observed in 2-propanol doped flames is not reproduced by the simulation, indicating that a pathway toward propene production might be missing or underestimated. Note that the "early" peak location of the propene is not reproduced either.

Propene being a small molecule, there are numerous ways to obtain it. Thus trying to explain this discrepancy seems difficult. However, 2-propanol has two carbons in β position from OH, whereas 1-propanol only one. Consequently dehydration (R1) and β-scission (R2) reactions are more likely to happen. McEnally et al. [START_REF] Mcenally | Proceedings of the Combustion Institute[END_REF], for instance, attributed the consumption of 1butanol by dehydration (R1) at 1% against 28% for 2butanol in CH 4 diffusion flame doped with 3500 ppm of butanol.

Butanol

Butanol is the smallest alcohol presenting a tertiary alcohol (solid below 300 K), and two primary alcohols. Flame structure analysis of butanol doped flames revealed no significant differences on the molar fraction profiles of C 1 and C 2 species, as well as propane (cf. Fig. 5). On the contrary, propene, propadiene (C 3 H 4 ), butadiene (C 4 H 6 ) and butene isomers molar fractions vary depending on the butanol isomer. The maximum molar fraction of propene is observed in i-butanol doped flames. The minimum molar fraction is observed, at equivalent levels, for the 2-butanol and t-butanol doped flames. On the other hand, propadiene molar fraction is maximum for t-butanol then quite similar for the three other isomers (level differences being close to the detection limit). Butadiene molar fraction is maximum for 2-butanol followed by 1-butanol and then the other two isomers. Finally, concerning the butene, the experimental setup allows distinguishing the three isomers. Maximum peak locations are located in the flame, for all cases, earlier than other intermediates (such as the propene profile of the 2-propanol flame).

Because of this early reactivity, reactions like R1 and R2, providing the most direct reaction paths to get butene from butanol, may play an important role. Fig. 6 presents these reactions applied to butanol isomers. In the case of β-scission, the first step (H-abstraction on β carbon) is not represented. Butene isomer molar fraction profiles and mean molar fractions are presented in Fig. 7. Observed trends, which can be linked with the aforementioned reaction types, are the following:

1. the major isomer produced in 1-butanol flame is 1-butene and the butene production is much lower than to other blends, 2. i-butanol mostly produces i-butene, 3. butene isomers produced in 2-butanol doped flames are mostly 1-and 2-butene. Molar fraction levels are quite similar (156 and 207 ppm on average, respectively), 4. i-butene is produced almost exclusively in tbutanol doped flames and in a high amount compared to other isomers in butanol doped flames.

Interestingly, Oßwald et al. [START_REF] Oßwald | [END_REF] measured mole fraction profiles of butene (without distinctions between isomers) in low-pressure rich (φ=1.7) premixed flames of butanol/O 2 /Ar and also found a much higher molar fraction in t-butanol flames. McEnally et al., in Ref. [START_REF] Mcenally | Proceedings of the Combustion Institute[END_REF], attributed the consumption of t-butanol by dehydration (R1) at 87%.

Tab. 8 provides a comparison between experimental measurements and numerical calculations of averaged butene molar fractions. For ease of reading, the molar fractions of butene isomers are summed. Both models compute higher molar fractions of butene in t-butanol-doped flame. However, no other similarities are found between experimental and numerical results. Models underestimate butene molar fractions in all cases. Moreover, contrary to experimental observations, there are no significant differences in butene molar fractions between 1-butanol and 2-butanol doped flames. As observed for propanol flames, it seems that the mechanisms do not correctly reproduce the reactivity of alkenes.

Butene isomers production by kinetic mechanisms

This subsection reports which reactions account the most in the production of butene isomers by butanol isomers, and then compare both mechanisms. This aside on specific reactions of kinetic mechanisms focuses on butanol for several reasons: (i) 2 kinetic mechanisms are available, (ii) butanol isomers include the 3 alcohol types, and (iii) there are 3 butene isomers which remains a reasonable number of species to study.

The rate of progress variable q i for the ith reaction is given by the difference of the forward and reverse rates as: where [X n ] is the concentration of the nth species and k f i and k r i are the forward and reverse rate constants of the ith reaction (given by the mechanism).

q i = k f i N n=1 [X n ] ν n,i -k r i N n=1 [X n ] ν n,i (6)
Then, the production rate ωn of the nth species can be written as a summation of the rate-of-progress variables for all reactions involving the nth species:

ωn = I i=1 ν n,i q i ( 7
)
where

ν n,i = ν n,i -ν n,i (8)
with ν n,i the reverse stoichiometric coefficients and ν n,i the forward stoichiometric coefficients of the nth species in the ith reaction and I the number of reaction involving the nth species.

Hence, computing ν n,i q i (referred as to ωn,i ) for the I reactions involving the nth species allows to evaluate the contribution of the ith reaction on the production rate. A positive value of ωn,i corresponds to the production of the nth species and negative value to its consumption.

To illustrate the methodology, Fig. 8 displays the partial production rates of t-butene profile for t-butanol flame modeled by CRECK mechanism. Three reactions are responsible of most of the i-butene production: dehydration (R1), β-scission (R2) and H-addition on butenyl. For the sake of readability, only the 10 most contributive reactions are displayed.

The determination of the contribution of ith reaction on the production of the nth species requires three steps.

Firstly, the reaction rate ωn,i is integrated over z when ωn,i > 0 :

A prod i = Z 0 ωn,i (z)dz (9)
Secondly, the overall production of this species is calculated by summing the terms for each reaction: A prod

0.6 0.
I = I i=1 A prod i (10)
Finally, the contribution of the ith on the production of the species n is deduced from:

c prod i = A prod i A prod I (11)
In the previous example, dehydration reaction (R1) accounts for 59%, H-addition for 28% and β-scission (R2) for 11%. These reaction contributions are useful to understand the evolution of alkene levels of the different kinetic mechanisms.

Tab. 9 presents the production of butene isomers computed in butanol-doped flames. Only the cases where reactions R1 and R2 take place are reported (2butene for 2-butanol, 1-butene for 1-,2-butanol and ibutene for i-, t-butanol). Both mechanisms can be quite different concerning the butene isomers. For instance, dehydration reaction never accounts for more than 15% with Sarathy mechanism whereas it reaches 59% for CRECK mechanism in the case of t-butanol flame. On the other hand, βscission reactions are the main pathways in many cases, with both mechanisms. Moreover, it worths noticing that there is no case where both mechanisms have similar pathways. It can be explained easily by the fact that for these reactions Arrhenius parameters (i.e. A and Ea) cover a rather large range of values. While the value of the activation energy in the case of dehydration of tbutanol seems to always be given around 65 kcal/mol, the other R1 and R2 reactions and associated Arrhenius constants vary more in the literature ( [START_REF] Mcenally | Proceedings of the Combustion Institute[END_REF]16,44,22]. This underline the need of supplementary works to define the reaction rate of such reactions with a better accuracy.

Pentanol

There are 8 pentanol isomers, 4 primary alcohols, 3 secondary alcohols and 1 tertiary alcohol. This paper presents, to the author's knowledge, the first flame structure of a flame doped by 2,2-dimethyl-propanol, which is solid below 325 K. Important distinctions on average molar fractions can be noticed on C 3 H 6 , C 3 H 4 , C 4 H 6 , butene and pentene isomers (cf. Fig. 5). There is almost a factor 3 on propene level among the isomers, 3-methyl-2-butanol being maximum with an average of 270 ppm and 1-and t-pentanol minimum with 100 ppm.

Propadiene reaches its highest level with 2,2dimethyl-propanol and t-pentanol (average of 25 ppm) then no major distinction can be given to the other isomers. Interestingly, similar trends are observed by butanol isomers: the t-butanol produced the most of propadiene, followed by the i-butanol (the branched primary alcohol).

Butadiene production varies also significantly, there is a factor 4 between 1-pentanol flame (minimum) and 3-pentanol flame (maximum).

Concerning the butene isomers, 1-butene is mostly produced by 3-pentanol (average of 56 ppm) and 2butene by 3-methyl-2-butanol and 2-methyl-1-butanol. Finally, i-butene is produced at a significant level by 2,2-dimethyl-propanol flame (average of 200 ppm), then by t-pentanol, and almost not produced by linear isomers (less than 1 ppm). i-butene seems the most natural product from 2,2-dimethyl-propanol, since reaction R1 and R2 cannot happen, and the position of the maximum i-butene is indeed farther in the flame.

Pentene production by pentanol is represented in Fig. 9. Similarly to propanol and butanol isomers, for all cases (except 2,2-dimethyl-propanol), a link can be drawn between observed molar fractions and reactions R1 and R2.

1. 1-pentanol produces mainly 1-pentene, 2. 2-methyl-1-butanol produces mostly 2-methyl-1butene (2 times higher than the other pentene isomers), 3. i-pentanol produces mostly 3-methyl-1-butene, 4. 2,2-dimethyl-propanol produces mostly 2-methyl-2-butene, and has the highest global level of pentenes among the primary alcohols, 5. 2-pentanol produces mostly 1-and 2-pentene, 6. 3-pentanol produces mostly 2-pentene (R1 and R2 forming exclusively 2-pentene by symmetry of the molecule), 7. 3-methyl-2-butanol produces mostly 3-methyl-1butene and 2-methyl-2-butene but interestingly 2methyl-1-butene as well, and the global level of pentenes is the highest among the secondary alcohols, The case of 2,2-dimethyl-propanol is interesting. Indeed, neither R1 or R2 can occur, because of the configuration of the molecule (double bonding impossible on the β carbon). However, the most produced pentene isomers remain the ones with a methyl in position 2 (2methyl-1-butene and 2-methyl-2-butene) and the position of the maximum is similar to the other pentanol cases.

The comparison between numerical and experimental results is more restrained because the kinetic mechanism used included only the three linear isomers of pentanol, i.e. 1-,2-and 3-pentanol. In this mechanism, 2 pentene isomers are included (1-and 2-pentene). As previously for propanol and butanol, the Köhler mechanism does predict the expected isomers of pentene for each isomer of pentanol but the level is much lower than experimental ones, especially for secondary alcohols.

Similar trends among all studied alcohols

As already mentioned, the four kinetic mechanisms used do not predict the level of alkenes produced by the different dopants. Experimentally, a clear hierarchy of alkene production level as a function of the type of alcohol has been established: primary < secondary < tertiary.

To illustrate this statement, Fig. 10 presents the average molar fraction of propene for propanol flame, the sum of butene isomers average molar fractions for butanol-flames and the sum of pentene isomers average molar fractions for pentanol-flames as function of the type of alcohol. A factor 2.2 is observed between the lowest mean molar fraction from secondary alcohol and the highest level from primary alcohol, as well as a factor 1.7 between tertiary and secondary alcohols. A second feature can be extracted from Fig. 10: the branched fuels produce a higher level of alkenes than the linear ones, such as i-butanol compared to 1-butanol, or i-pentanol and 2-methyl-1-butanol compared to 1pentanol.

Experimental results show that among all the intermediates only the alkene with the same number of carbons as the dopant has its maximum molar fraction at a very low z. This trend suggests a direct pathway from the dopant to the alkene. Moreover, the results obtained by measuring distinctly the 3 butene isomers and the 5 pentene isomers highlights that the products of R1 and R2 are always the major isomers (for the 13 dopants). For instance, 2-butanol produces effectively 1-and 2-butene, and 3-pentanol produces mostly 2pentene. These reasons suggest the importance of these two types of reactions in the first steps of the alcohol decomposition. These reactions might occur for lower temperature than expected, i.e. below 900 K [4]. The results suggest also that, in alcohol-doped premixed rich flames of C 2 H 4 (φ=1.7), the alcohol could be degraded similarly than in diffusion flame. Indeed, the results are quite similar to Ref. [START_REF] Mcenally | Proceedings of the Combustion Institute[END_REF] (nearly 75% of the tertiary alco- On the other hand, numerical results do not reproduce (i) the peak location, (ii) the hierarchy among alcohol types, (iii) the hierarchy among linear and branched fuels, and mostly (iv) the amount of alkene produced (with a factor up to 20 for the case of butene produced by t-butanol in the simulation with Sarathy mechanism). With the lack of consensus among kinetic mechanisms regarding R1 and R2, the experimental database developed in the framework of this work shall be valuable for the validation of improved kinetic mechanisms. These improvements will allow a better prediction of alcohol conversion and, ultimately, could also lead to a better prediction of soot precursors.

Summary and conclusions

Fuel-rich, premixed atmospheric flames of ethylene doped by all the C 3 to C 5 alcohols were investigated by Gas Chromatography. 19 species have been identified and quantified, including the 3 butene isomers and the 5 pentene isomers. All flames exhibit similar macroscopic characteristics, such as major species profiles. However intermediate species pool measured for the isomeric fuels differs strongly. Firstly, the alcohol type seems to play a role in the production of alkenes with the same number of carbon as the fuels. Primary alcohols produce the least alkenes while tertiary alcohols generate the most. With a lower order of magnitude, branched fuels seem to produce more alkene than linear ones.

Two types of reactions, R1 and R2 are suggested to explain most of the conversion from alcohol to alkene. Indeed for every fuel, products of these reactions are the alkenes presenting the highest molar fractions. It suggests that these reactions might occur under temperatures lower than currently predicted by the four kinetic mechanisms. These differences could play a role in soot formation and need to be confirmed on other configurations.

Considering the probable important role of R1 and R2, these experimental results could help to the development of kinetic mechanisms including alcohol fuels up to C 5 , or eventually higher, since all C 3 -C 5 alcohols seem to have similar behavior regarding alkene of the same number of carbons. These improvements could lead to a better prediction of alcohol combustion inter-mediates and soot precursors, which ultimately would help to the modelization of soot emission.

Future works plan to study the same dopants in diffusion flames associated with optical diagnostics to measure soot characteristics such as volume fraction. The different alkene levels might lead to important discrepancies in sooting propensity among the different dopants.Then investigations should address the impact of pressure on the flames. Among others, Griffin et al. [45] showed that the sooting propensity of C 2 -C 4 alkenes seems to increase significantly with pressure, this trend being more important for ethylene than propene or butene.

Figure 1 :

 1 Figure 1: Experimental setup: a) Burner alimentation; b) 3D view of the burner with sampling probe and ceramic mesh disc; c) Picture of the flame on the McKenna burner.

Figure 2 :

 2 Figure 2: Temperature, t-butanol and i-butene profiles used to investigate the impact on simulation results

Figure 3 :

 3 Figure 3: Ethylene, methane, ethane and propane molar fractions profiles.

X

Figure 4 :

 4 Figure 4: C 3 H 6 profiles and X for 1-and 2-propanol flames.

Figure 5 :

 5 Figure 5: Overall experimental results: X of each dopant and each measured species.

OH+Figure 6 :

 6 Figure 6: Butanol to butene potential pathways in high-temperature rich-flame.

Figure 7 :

 7 Figure 7: Left : Butene isomers profiles ; Right : X of butene isomers.

Figure 9 :

 9 Figure 9: X of pentene isomers for pentanol flames.

Figure 10 :

 10 Figure 10: Comparison of alkenes production by type of alcohol (respectively propene, butene and pentene isomers for propanol, butanol and pentanol isomers).

Table 1 :

 1 Nomenclature of studied alcohols.

		OH
	OH
	1-propanol	2-propanol
		OH		OH
	OH		OH
	1-butanol	2-butanol	i-butanol	t-butanol
	OH	OH		OH
			OH
	1-pentanol	2-pentanol	i-pentanol	t-pentanol

Table 2 :

 2 Premixed flame parameter values.

	Parameter		Value
	CEM (K)			353 ±2
	HL (K)			323 ±5
	Burner (K)			323 ±2
	φ			1.70 ± 0.04
	X vap			0.035 ±4.10 -4
	N 2 (nl/min)			9.00 ±0.12
	Air (nl/min)			8.30 ±0.11
		prop.	but.	pent.
	C 2 H 4 (nl/min) 0.931 0.915 0.900 ±0.011
	Alcohol (g/h)	5.47	6.64	7.76	±0.04

Table 3 :

 3 Standard molar fractions provided by AirLiquide company.

Table 4 :

 4 FID Response Factor. The last column (X std ) shows the molar fraction of the species present in the standard bottle (Tab. 3) calculated from the RF method.

		Extracted for Ref. [40]	Eq. 2	Eq. 3 Eq. 4
	Species	X (%)	A (µV.s)	RF abs (µV.s/%) RF rel	X std
	Ethylene 2.0025	187178	93472	0.51	1055
	Ethane 4.0021	398308	99525	0.54	1003
	Propane 6.0185	877739	145840	0.80	216
	Propene 3.0038	421959	140475	0.77	224
	Propadiene	0.997	127910	128295	0.70	-
	n-Butane 3.9993	750601	187683	1.03	112
	1-Butene 1.9998	368216	184126	1.01	-
	i-Butene 1.0025	175064	174627	0.96	-
	trans-2-Butene 3.0061	548315	182401	1.00	-
	cis-2-Butene 1.9996	358177	179124	0.98	-
	1,3-Butadiene* 3.0107	550241	182762	1.00	100
	1-Pentene 0.4007	82292	205371	1.12	-
	trans-2-Pentene 0.1996	41556	208246	1.14	-
	cis-2-Pentene 0.4001	82919	207246	1.13	-
	2-m-2-Butene 0.1998	39758	198989	1.09	-
	3-m-1-Butene** Absent	-	198989	1.09	-
	2-m-1-Butene** Absent	-	198989	1.09	-

*Reference species **Response factor assumed to be equivalent to 2-m-2-Butene

Table 5 :

 5 Utilization of the different kinetic mechanisms

		Propanol	Butanol	Pentanol
	Mechanism	1-, 2-	1-, 2-, i-, t-	1-, 2-, 3-
	Li et al. [18]	X
	CRECK [15, 16]	X	X
	Sarathy et al. [4]	X	X
	Köhler et al. [8]			X

Table 6 :

 6 Premixed flame parameter values.

	Input		Value	Unit
	Φ			1.70 -
	P			1.0 bar
	T			323 K
	ṁ			0.07 kg/m 2
		prop.	but.	pent.
	X alcohol 0.367 0.361 0.356 %
	X C 2 H 4	10.1	9.96	9.81 %
	X N 2	70.7	70.8	70.9 %
	X O 2	18.8	18.8	18.8 %

Table 7 :

 7 Experimental and numerical level of X C 3 H 6 .

	Isomer	Exp.	Li	CRECK Sarathy
	1-propanol 136	95	105	114
	2-propanol 327 108	95	125

Table 8 :

 8 Experimental and numerical level of X C 4 H 8 .

	Isomer	Exp. Sarathy CRECK
	1-butanol	71	23	16
	i-butanol	148	27	34
	2-butanol 364	31	14
	t-butanol 1065	52	75

Table 9 :

 9 Reaction contribution (%) for the production of butene isomers by butanol isomers. Left columns : Sarathy; Right columns : CRECK.

		1-ol		i-ol		2-ol	t-ol
	Reaction	1-ene	i-ene	1-ene	2-ene	i-ene
	dehyd.	2	8	2 17 9 27 7 19 15 59
	β-scis.	49 32 77 53 15 4 69 33 73 11
	H-add.	13 0 11 28 23 0 23 17 11 28
	allyl + methyl 29 55 0	0 43 62 0	0	0	0