Rodrigo Assar
email: rodrigo.assar@inria.fr

Alice Garcia
email: alice.garcia@inria.fr

David J Sherman
email: david.sherman@inria.fr

Modeling Stochastic Switched Systems with BioRica

Keywords: Hybrid Systems, Transition Systems, Switched Systems, Cell differentiation

Modeling physycal and biological dynamic systems needs to combine different types of models in a non-ambiguous way. We present an approach to integrate continuous, discrete, stochastic, deterministic and non-deterministic elements by using Transition Systems theory, reuse, composition of models, and the framework BioRica. The systems are described by interacting continuous and discrete models, and in addition continuous models are decomposed into two components: controlled and controller model. We define Stochastic Switched Systems whose continuous dynamics is modeled by differential equations and its discrete dynamics by transition systems, allowing stochastic and non-deterministic behaviours. We illustrated the use of our approach with examples of intrinsically and approximated hybrid systems. Our approach allows us to give a first step to integrate and to extend models of complex systems, such as cell differentiation.

Introduction

1 Physical phenomena often are described by combinations of different types of equations. These systems are called Hybrid systems ( [START_REF] Henzinger | The theory of hybrid automata[END_REF], [START_REF] Johansson | Hybrid control systems[END_REF]) due to the use of continuous and discrete features or Switched systems because they switch its equations over time ( [START_REF] Branicky | Stability of switched and hybrid systems. Decision and Control[END_REF], [START_REF] Shorten | Stability criteria for switched and hybrid systems[END_REF], [START_REF] Tavernini | Differential automata and their discrete simulators[END_REF]). Switched systems are a way to introduce discrete behavior into continuous models. They are Hybrid systems with discipline.

In dynamic models, one considers two groups of variables: dependent and factors. Models give the dynamics of the dependent variables, considering factors affecting it. One talks of continuous model if the variables change continuously over time and it is relevant to know the behaviour at any time. Continuous models use functional relations between the variables, being a common type the differential equations. On the other hand, if the variables make discrete changes at instantaneous points in time, the model is discrete. Hybrid models join both types of models: some variables have continuous dynamics while other ones have discrete dynamics. These models allow the interaction of diverse components of a system to contribute to complete descriptions of the behaviour. Switched systems are one kind of Hybrid system that restricts way discreteness is added. Discrete variables modify the behaviour of the continuous model by controlling its coefficients. Models try to accurately represent the reality, by using empirical observations and knowledge. With limited observations one wants to build models that are valid to explain the system in general conditions without testing it on all the conditions. As result, models are strongly dependent of the studied conditions. In order to get most valued out of existing models and to refine models to include more complex behaviours, it is necessary to define how to compose models. A system is built in a hierarchical way, composed of subsystems, where behaviours emerge from the association of components and its diversity ( [START_REF] Kitano | Computational systems biology[END_REF]). Complex biological processes can, in this way, be defined by interactions between basic functional entities called modules ( [START_REF] Callebaut | Modularity: Understanding the Development and Evolution of Natural Complex Systems[END_REF]) and, to explain its behaviour, to each module is associated a model that represents it.

The existence of different types of models to explain connected processes makes it necessary to define theory and tools to integrate them. Such theory must be able to unify processes with different timescales and whose models have different stiffness levels. Sometimes, to see the changes in the behaviour it is necessary to compare nearby times, but other times the changes happen in distant times. Equations with high stiffness require a superior level of discretization to obtain good approximations. Both characteristics, different timescales and stiffness affect computation times and accuracy. To compose models one must consider these characteristics, and be capable of giving an unique and non-ambiguous semantics to the composition. Modeling biological dynamic systems by composition requires a framework in which existing models of different types can be combined without need of rewriting them.

Many dynamic biological systems, such as physiological processes ( [START_REF] Tyson | Modeling the cell division cycle: cdc2 and cyclin interactions[END_REF]), are represented by ordinary differential equations ( [START_REF] Ch | BioModels database: An enhanced, curated and annotated resource for published quantitative kinetic models[END_REF]). Changes in the environmental conditions modify the development of the processes, switching coefficients in its continuous dynamics. In Gene Regulatory Networks, some biological facts such as a gene is activated by a transcription factor or regulator give power to the idea of using switched models.

We can group the different approaches of Hybrid Systems in two kinds: function and implementation oriented. Function oriented approaches favor human comprehension of models, while implementation orientation focuses in descriptions easy to interpret by machines. In the first orientation, the dynamics of systems are defined by functions. Continuous dynamics is commonly represented by differential equations, and the discrete dynamics affects it by switching its equations ( [START_REF] Branicky | Stability of switched and hybrid systems. Decision and Control[END_REF], [START_REF] Shorten | Stability criteria for switched and hybrid systems[END_REF], [START_REF] Tavernini | Differential automata and their discrete simulators[END_REF]). Systems are called Switched System. The models are easy to understand, but too restrictive with respect to the dynamics. Implementations oriented approaches present more general descriptions of Hybrid systems, by using an abstract representation to implement the model ( [START_REF] Henzinger | The theory of hybrid automata[END_REF], [START_REF] Johansson | Hybrid control systems[END_REF]). Models describe the rules of the dynamics allowing many types of continuous dynamics.

Here, we relax the concept of Switched Systems to allow possible stochastic or non-deterministic changes in the continuous dynamics. Such systems, called Stochastic Switched Systems, are described from function and implementation orientations using Transition Systems ( [START_REF] Arnold | Construction and analysis of transition systems with MEC[END_REF], [START_REF] Alfaro | Stochastic transition systems[END_REF]). We analize Gene Regulatory Networks ( [START_REF] Gebert | Modeling gene regulatory networks with piecewise linear differential equations[END_REF]), by approximating them by Switched Systems. Our representation of the recent osteo-chondro differentiation model ( [START_REF] Schittler | Cell differentiation modeled via a coupled two-switch regulatory networks[END_REF]), as Stochastic Switched System composed by two interacting components, allows us to improve the differentiation stimuli models separately and so improve the complete model. We suggest some experiments to model the effect of the Wnt pathway on the bone formation (osteogenic lineage) and include it as stimulus.

To simulate Hybrid models we use BioRica, available in BioRica. It is a high-level modeling framework that integrates discrete, continuous, stochastic, non-deterministic and timed behaviors in a non-ambiguous way allowing multi-scale dynamics, composition of models and hierarchical relations. The modeling language is an extension of the AltaRica Dataflow language ( [START_REF] Arnold | The AltaRica formalism for describing concurrent systems[END_REF]), allowing hybrid systems and stochastic behaviors.

Approach

Modeling

The dependent variables are called state variables (in analogy with Transition Systems), while continuous and discrete factors are considered controllers. These systems are described using a mixture of continuous, discrete dynamics and logical relations to allow multiple interacting components.

DISCRETE MODEL

CONTINUOUS MODEL

x mod

RESIDUAL MODEL

Hybrid approach

Separating the continuous model 1). We consider that the dynamics of state variables is modeled by ordinary differential equations (explicit representation, equation 1) including continuous and discrete variables.

ẋ(t) = F (x(t), u(t), mod(t)) (1)
The continuous dynamics is given by the changes in x over time, with F a function from

R n × R k × M to R n .
The discrete dynamics is given by the evolution of the mode variable denoted mod ∈ M , where M is a finite or enumerable set and we denote M = {1, . . . , M } by identification of its elements.

The next step is to define how the discrete variables mod evolve over time. By considering hybrid systems as switches between continuous systems, represented by sets of differential equations, one talks of Switched systems and Switched models ([3], [START_REF] Shorten | Stability criteria for switched and hybrid systems[END_REF], [START_REF] Tavernini | Differential automata and their discrete simulators[END_REF]). Independent of x, the value of the right hand side function F changes as a function of the value of discrete variable mod (equation 1) that affects the form of the equations. We will say that changes in discrete variables values carry switches between model configurations. An example of Switched System is the behaviour of a radiator that controls the temperature (T ) of a room (Fig. 2). A thermostat is activated when the temperature is detected to be low and it is regulated, if the temperature is high the system is turned off. This behaviour can be modeled in different ways: deterministic, non-deterministic or stochastic.

(A) (D) (C) (B) node RADIATOR

Composing Models

The act of building a model that is made of two or more modules is called Composition. The composition of two models is the model that explains the behaviour of both interacting modules. This allows us to the model to learn and to integrate knowledge of diverse types. The model can be extended and improved by introducing new modules that relate different functions. To be capable of describing the behaviour of a biological system over time, one needs to combine different types of models in a non-ambiguous way. A good implementation of this concept is essential to take advantage of the modularity of biological systems to build accurate and complete models. It must be sufficiently flexible to be capable of joining modules defined with different types of models, and reuse modules that have been a priori defined.

Our approach is based on the use of a global semantics to compose interacting modules. Flow conections and synchronizations allow to connect modules. Local clocks and solvers allow us to consider diverse types of dynamics.

The BioRica semantics is based on the automata semantics of AltaRica ( [START_REF] Arnold | Construction and analysis of transition systems with MEC[END_REF], [START_REF] Arnold | The AltaRica formalism for describing concurrent systems[END_REF]), and Stochastic Transition Systems ( [START_REF] Alfaro | Stochastic transition systems[END_REF]) that allow the inclusion of randomness and non-determinism. Given a BioRica node, one computes the probability of the state dynamics and considers non-deterministic decisions solved by random schedulers. The resulting semantics it is preserved with respect to flow relations and event syncronizations.

An important fact we approach here is that different processes have different timescales and stiffness levels. The use of modules solves in part this problem: each module has an specific timescale and discretization level. With this strategy, we improve the precision and the cumputation time. The processes with small timescales are observed at small time steps, while in case of long timescales we use longer time steps to reduce the number of simulations. So, if the equations solved by a module are stiff one uses small time intervals only at that module.

We implement models with these considerations in BioRica. A common specification of Systems Biology models is SBML ( [START_REF] Hucka | The Systems Biology Markup Language SBML: Language Specification for Level 3 Version 1 Core Release 1 Candidate[END_REF]), maybe the most popular abstraction for biochemical reactions models governed by temporal differential equations. Our framework includes a SBML parser that translates SBML models into BioRica models. So, it is possible to reuse and compose models previously specified in SBML to obtain more general models.

Stochastic Switched Systems

Our approach mixes both point of view of Hybrid Systems: function and implementation oriented. It is adapted to Hybrid models whose continuous dynamics is represented by differential equation systems, but we give more flexibility to the discrete dynamics allowing not only deterministic behaviours. We give function and implementation oriented descriptions of such systems.

The system dynamics is represented by transitions between different states. The interpretation of a model as dynamic entity, with possible changes on its form, turns it into a Transition Systems ( [START_REF] Arnold | Construction and analysis of transition systems with MEC[END_REF]) with two types of transitions: state transitions and mode transitions. The state transitions are internal and controlled by the continuous dynamics of the model. The mode transitions are transitions in the sense of Transition Systems theory and they can be deterministic, non-deterministic or stochastic. The transitions can be modeled with stochastic components or including non-determinism to allow different behaviours (Fig. 2).

In the general theory of Stochastic Transition Systems ( [START_REF] Alfaro | Stochastic transition systems[END_REF]) the transitions have possible stochastic behaviours. Given an action producing a transition, in this case a mode change, the next mode is randomly chosen according to transition probabilities if they exist. We will call Stochastic Switched Systems the extension of Switched systems that allows stochastic or non-deterministic transitions between modes. We consider two randomness sources: the moment (time) at which happens the action of changing mode, and the new mode that is chosen. The conditions that provoke the mode transitions are called guards. They are boolean formulas defined over state variables, external controllers and modes values. For each mode i, the mode transition arriving to i corresponds to an event (action in Transition Systems theory, [START_REF] Arnold | Construction and analysis of transition systems with MEC[END_REF] and [START_REF] Alfaro | Stochastic transition systems[END_REF]). In the radiator example (Fig. 2(B.1)), T ≤ 18 is a guard condition provoking that event turn on assigns the value 25 to the mode variable K.

We formally define a Stochastic Switched System as a hybrid system whose model is given by the equations 2-5. The first one defines the continuous dynamics and 3,4,5 the discrete dynamics at any time t. We denote P (ev|(x(t), u(t), mod(t)) the probability of choosing the event ev when the values of the state variables x are x(t), the values of the continuous control variables u are u(t) and the value of the mode variable mod is mod(t), time((x(t), u(t), mod(t)), ev) denotes the delay time of the event ev that is modeled to have distribution Dist ev {p event,1 , . . . , p event,m }.

ẋ(t) = F mod(t) (x(t), u(t)), (2)
P (ev|(x(t), u(t), mod(t)) = w ev e∈A(x(t),u(t),mod(t))

w e , (3)
time((x(t), u(t), mod(t)), ev) ∼ Dist ev {p event,1 , . . . , p event,m },

where w e is the probability weight assigned to the event e, and A(x(t), u(t), mod(t)) is the set of available actions when x takes the value x(t), u the values u(t) and the mode variable m the value mod(t)) given by the equation 5.

A(x(t), u(t), mod(t)) = {ev ∈ EV EN T S : G ev (x(t), u(t), mod(t)) = T RU E} , (5)
and EV EN T S is the set of events of the system (in Fig. 2 EV EN T S = {turn on, turn of f }).

Non-determinism appears with the presence of two or more available actions given a tuple (x(t), u(t), mod(t)) . We simulate non-deterministic systems by using random schedulers with weights given by the external directives choice, so the event turn on has probability 2 3 for the non-deterministic radiator (Fig. 2(B.3)). The time delays can be stochastic, with law < ev >: Dist ev{pev 1, . . . , pev m} denoting that the time delay of the event ev has distribution Dist ev {pev 1 , . . . , pev m } with parameters {pev 1 , . . . , pev m }. Between the possible distributions we include T (a deterministic time T ), the Gaussian distribution N ormal{µ, σ}, the exponential distribution Exponential{r} with r the rate and the uniform with parameters a, b (a < b) U nif orm{a, b}.

In the radiator example (Fig. 2), the simpler model is to consider a deterministic behaviour of the thermostat, where at any temperature at most one mode transition is observed: the radiator can be or active or not. For this model, it is obtained by considering the temperature of activation lower than the temperature where the radiator is turned off (18 and 20 respectively, Fig. 2(B.1)). With more ambiguous guard rules, given a temperature the radiator is accepted to be activated and turned off. This non-deterministic behaviour is obtained in case of activation if temperature is between 18 and 20 and deactivation if it is in the same interval (Fig. 2(B.2)).

BioRica Description of Stochastic Switched Systems

We represent and simulate Stochastic Switched Systems with BioRica. BioRica uses automata theory to represent Stochastic Switched Systems as a particular type of Hybrid Automaton ([1], [START_REF] Johansson | Hybrid control systems[END_REF]) where the continuous dynamics is given by differential equations. The continuous dynamics (equation 3) is described in eqdiff while the discrete one (equations 4 and 5) is described by transitions that change the form of the model. In Fig. 2 is defined the BioRica node RADIATOR, with state variable T (temperature) and mode K. The possible events are defined with the keyword event, turn on and turn of f , and its effect is described in trans: turn on provokes the assignment K := 25 when T ≤ 18, turn of f assigns K := 15 if T ≥ 20. With the keyword eqdiff one codes a set of differential equations, where dxi = f i(x1, . . . , xn, u1, ..., uk, mod) means that the rate of change of xi with respect to the time is equal to f i(x1, . . . , xn, u1, ..., uk, mod). In init, one defines the initial values of the variables. It is possible to define constant, const, and formula expressions, f ormula, to use in the equations.

In cases of non deterministic behaviours, BioRica decides what event select considering an aleatory decision between the possible events with weights given in the choice option of extern. In Fig. 2(B.3) the event turn on has weight 2, then at arriving to temperatures between 18 and 20 it selects the event turn on (activate the radiator) with probability 2 3 and turn off with probability 1 3 . The keyword extern allows the inclusion of external directives about distributions of event delays and priority between events.

To decompose the dynamics we use the ideas of Fig. 1. With the keyword flow, one includes inputs (outputs) from (to) other BioRica nodes. Continuous and discrete dynamics can be modeled separately, the node MAIN represents the complete system and the keyword sub is used to define instances of other node (Fig. 3).

Application: Approximating Regulatory Systems by Stochastic Switched Systems

The dynamics of the radiator (Fig. 2) is hybrid in its nature because the switches are directly associated to the value of the mode. So, the mode K of the radiator is a piecewise-constant function. In less restrictive cases, where one identifies different underlying behaviours, we can use Switched Systems too. One can choose a set of factors to consider as piecewise-order k and approximate them to obtain a Switched model. A particular case is given by the regulation models with reduced Hill functions.

Reducing Hill Functions

Hill functions [START_REF] Hill | The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves[END_REF] are sigmoidal curves used to measure the continuous influence of an element on a target, depending on the concentration of the affecting element x, an exponent m to control the curve steepness, and on the mean point of influence θ. We denoted

h + (x, θ, m) = x m
x m +θ m the positive influence and

h -(x, θ, m) = 1 -h + (x, θ, m) the negative influence.
Dynamics systems with interacting elements often generate differential equation systems with Hill functions. The solution of such differential equation systems, equations 7-9 in case of the osteo-chondro cell differentiation model, can be complicate and use high computation times. More influence relations more difficult to solve the system. To simplify them, we reduce them into switched systems choosing some influence functions to be represented with piecewise-dependent behaviours. Thus, the system dynamics is obtained from the interaction of continuous and discrete dynamics.

Here, we show two reductions of the Hill functions: Piecewise-constant and Piecewise-linear approximation. The first idea is considering Hill function as step functions. It is to say, piecewise-constant functions to be 0 when x is lower or equal than the threshold θ and 1 after this threshold. With this simplification, the model moves between different modes in function of how high or low are the concentrations x with respect to the thresholds θ. The thresholds divide the state variables space into cuboids, each one with an associated system of equations. Despite one obtains information of the system behaviour by looking the form of the equations in each cuboid of the state spaces, the observations are only qualitative.

The second reduction of Hill functions is the Piecewise-linear approximation, in which the transition between 0 and 1 is smoothed by a linear function. It is to say, we use the approximation of equation 6 below:

h + (x, θ, m) ≈ l + (x, θ 1 , θ 2 ) =        0 if x ≤ θ 1 x-θ 1 θ 2 -θ 1 if θ 1 < x < θ 2 1 if x ≥ θ 2 (6)
With this second alternative, the switched system transits between a big set of modes according to how high are the mRNA concentrations compared with the thresholds θ 1 and θ 2 of each influence function.

An Osteo-chondro Differentiation Model

An application of Gene Regulatory Networks is cell differentiation modeling. Each possible differentiation of a cell is associated to the mRNA concentration of an specific gene. Here we use the model of Schittler et al. ([13]) to differentiate progenitor cells into osteoblasts (bone cells) or chondrocytes (cartilage cells). They are considered two mutually inhibiting genes, so called the osteo-chondro switch, one associated to the osteogenic differentiation (Runx2) and another (Sox9) to the chondrogenic option. A third gene (Tweak) is associated with the progenitor maintenance role that inhibits both genes of the osteo-chondro switch. The mRNA concentration associated to the progenitor state is denoted x P , the mRNA concentration of the osteogenic state is denoted x O and the associated to the chondrogenic differentiation is denoted x C . To incorporate the external prodifferentiation, pro-osteogenic and pro-chondrogenic stimulus are included three inputs: z D , z O and z C with positive value. The increase of any differentiation stimulus provokes an increase of the expression of the const n=2;ap=0.2;bp=0.5;mp=10;cpp=0.1;kp=0.1;ao=0.1;ac=0.1;bo=1; const n=2;ap=0.2;bp=0.5;mp=10;cpp=0.1;kp=0. associated gene. The model is given by the equations 7-9 above.

ẋP (t) = a P •x n P +b P m P +z D +c P P •x n P -k P • x P , (7)
ẋO (t) = a O •x n O +b O +z O m O +c OO •x n O +c OC •x n C +c OP •x n P -k O • x O , (8)
ẋC (t) = a C •x n C +b C +z C m C +c CC •x n C +c CO •x n O +c CP •x n P -k C • x C , (9)
with n = 2, a P = 0.2, b P = 0.5, m P = 10, c P P = 0.1,

k P = 0.1, a O = a C = 0.1, b O = b C = 1, m O = m C = 1, c OO = c CC = c OC = c CO = 0.1, c OP = c CP = 0.5, k O = k C = 0.1 known parameters.
We obtain the same results for the scenarios analyzed in [START_REF] Schittler | Cell differentiation modeled via a coupled two-switch regulatory networks[END_REF], but the BioRica representation gives flexibility to the model. In Fig. 3 we considered another scenario, where pro-differentiation, pro-osteogenic and pro-chondrogenic stimulus happen with exponential probabilities over time (a Poisson process). The system corresponds to a strict Stochastic Switched System, in which delay times have random behaviours.

Since we separate between stimulus (node STIMULUS) and differentiation dynamics (DIFF), to specify each differentiation stimulus one needs only to modify such a node. The dynamics of STIMULUS controls the lineage decision by switching the values of the z coefficients, and depends on external factors. A factor that affects the lineage decision is the activation of the Wnt/β-catenin pathway. Since Runx2 is a Wnt target gene, the accumulation of β-catenin in the nucleus stimulates the expression of Runx2, and consequently favors the bone formation ( [START_REF] Krishnan | Regulation of bone mass by wnt signaling[END_REF]). One can include this effect, on the synamics of z 0 , by measuring the concentration of nuclear β-catenin over time and using LiCl to activate the pathway. We used the theory of Stochastic Transition Systems ( [START_REF] Arnold | Construction and analysis of transition systems with MEC[END_REF] and [START_REF] Alfaro | Stochastic transition systems[END_REF]) to define an special type of Hybrid System: Stochastic Switched System (section 2.3). The model is formed by the continuous dynamics of the state variables, given by differential equations, and the discrete dynamics of the modes that change over time to transform the differential equations. We allow stochastic and non deterministic behaviours, and implemented such systems using BioRica.

We defined the osteo-chondro cell differentiation model in [START_REF] Schittler | Cell differentiation modeled via a coupled two-switch regulatory networks[END_REF] as a Stochastic Switched System by composing STIMULUS and DIFF (differentiation) components giving more flexibility to extensions. As example, stimulis are considered with aleatory behaviour. By considering the activation of the Wnt/β-catenin pathway as factor of bone formation and measuring its effect, it is possible to improve the model.

We defined a non-ambiguous way to describe a complex system by decomposing it into different types of interacting models. Behaviour laws change over time, which is modeled by discrete changes of mode variables that transform the continuous dynamics, and complex processes are modeled by composing diverse models with flow connections and synchronization of events. Our approach allows us to reuse SBML specified models and exploits modular properties of systems, which can be separated into modules in function of the type of process and its timescale, and the complexity or type of model.

Figure 1 .

 1 Figure 1.Modeling schema of Complex Biological Systems by Hybrid models. First it is identified the discrete and continuous interacting dynamics, then the continuous dynamics is separated into two interacting models: the X MODEL that describes the dynamics of X, and the residual model. Let x = (x 1 , . . . , x n ) ∈ R n be the state variables of the model. The variables u = (u 1 , . . . , u k ) ∈ R k are the continuous control variables, and mod are the discrete controllers (Figure Fig.1). We consider that the

Figure 2 .

 2 Figure 2. The radiator. (A)Schema of the Switched system. (B)Three models. (B.1) deterministic: it is turned on if T ≤ 18 • C and turned off if T ≥ 20 • C, (B.2) non-deterministic: both events can happen if 18 ≤ T ≤ 20, and (B.3) stochastic: turn on with probability 2 3 . (C)Temperature dynamics for (B.1), (D) For (B.3).

Figure 3 .

 3 Figure 3. BioRica code and simulation of an osteo-chondro differentiation model ([13]). The pro-differentiation stimulus happens at time exponential with rate 0.01 (expected value E(t) = 100), the pro-osteogenic stimulus happens with rate 0.002 (E(t) = 500) and the pro-chondrogenic with E(t) = 1000.

Project-team (EPC) MAGNOME common to INRIA, CNRS, and U. Bordeaux 1, an EMR of UMR 5800 LaBRI

Acknowledgements

RA supported by an INRIA CORDI-S scolarship.


A Lebouvier

F Fresnet

F Fabry

V Boch

V Rohani

F Cauneau

L Fulcheri

Experimental and theoretical study of exhaust gas fuel reforming of Diesel fuel by a non-thermal arc discharge for syngas production

Keywords: Plasma reformer, syngas, diesel fuel reforming, NO x trap

An experimental set-up has been developed to study two typical operating points of Diesel powered vehicle, corresponding to high load and low load points. A sensibility study over O/C ratio, injected electric current and mass flow rate have been carried out. The plasma reformer performances have been evaluated in terms of energy efficiency and conversion rate. At low engine load, an energy efficiency of 40% and a conversion rate of 95% have been reached which correspond to a syngas dry molar fraction of 25%. For the most favorable case, only 12 s are needed to regenerate the NOx trap catalyst. The 1D multistage kinetic model developed has shown good trend correlation with experimental results. It has been demonstrated that the oxygen from CO 2 and H 2 O almost does not intervene in the exhaust gas Diesel fuel reforming. At the contrary, CO 2 and H 2 O decrease temperatures, the kinetic reaction speed and the energy efficiency compared to PO x reaction. To higher the temperature, more oxygen is needed but local combustion can happen and promote H 2 O and CO 2 production.

Introduction

In Europe, Euro stage VI regulation for Diesel engines will come into force in September 2014. This new regulation sets a 56 % reduction of NOx emissions compared to Euro stage V and forces car manufacturers to develop new efficient solutions. Three-Way Catalysts (TWC) used for gasoline engines after-treatment can efficiently decrease NOx emissions which operate under close to stoichiometric condition. However, TWC technologies become ineffective in Diesel engines since, in this case, exhaust gases have a high oxygen content.

For Diesel engines after-treatment, technologies based on NO x trap catalysts are one of the technological solutions under development to meet the further emission regulations. The NO x trap is based on a cyclic operating mode. In storage mode, the NO x trap stores NO x emitted from the engine on a catalyst material. Once full, the classical way to regenerate NO x trap catalysts consists in operating the engine under rich combustion conditions for a short while in order to produce reducing species in the exhaust gas that will convert NO x into N 2 . This method is not totally satisfying since it suffers from a significant drawback known as oil dilution problem. To overcome oil dilution problem, it is possible to produce reducing species such as H 2 and CO by Diesel fuel reforming whose species will be used for NO x trap regeneration. In this case, a part of the Diesel engine exhaust gas is by-passed to the reformer and is mixed with a small amount of Diesel fuel, which provides the necessary species to regenerate the NO x trap catalyst. We present in this paper an alternative to catalytic reforming method consisting in using a non-thermal plasma torch. The plasma torches dedicated to reforming have been reviewed in [1]. Contrary to catalysts, plasma processes are non-sensitive to sulfur, light and compact device, and have short-transient time.

Experimental setup

The plasma reactor is composed of two consecutive zones: a plasma zone and a postdischarge zone. The plasma zone is the part where the arc plasma really takes place. The postdischarge zone is a passive zone, located downstream of the plasma zone where most of the reforming reactions ignited in the plasma zone continue to take place depending on their kinetic speed. The power supply is a resonant converter controlled in current [2]. The current can be precisely tuned in the range of 0.22-0.66 A. A high voltage is applied to the tip electrode (anode as we work in inverse polarity), and the cylinder electrode (cathode) is grounded. The cathode is a stainless steel cylinder with an 8 mm inner diameter and 75 mm long.

Both operating conditions studied are given in Tab. 1. They correspond to the exhaust gas composition of a diesel engine for two fuel/air equivalence ratios (Φ) of 0.66 (high load) and 0.32 (low load) for a Renault 2.0 L 16v dCi turbocharged engine.

Plasma reformer performances are analyzed in terms of energy efficiency and conversion rate. The energy efficiency is based on the LHV of syngas produced over the LHV of diesel fuel and plasma power. The conversion rate accounts for the fuel transformation and is an indicator of the mass balance on carbon atoms.

Results and Discussion

Influence of O/C ratio

In Fig. 1, for condition 2, the diesel fuel decomposition is higher than 80% for O/C greater than 1.3. The second operating condition (Φ = 0.32) reaches an energy efficiency of 40% against only 15% for the first one (Φ = 0.66). The strong difference in energy efficiencies is essentially due to the oxygen rate in the gas mixture, which is twice higher at low engine load, and CO 2 and H 2 O, which are twice lower. In both cases, the temperature grows linearly with the O/C ratio. Oxygen has the role to bring energy to the system and allows reaching higher temperatures for low engine load.

In addition, at high load, an important part of calories are absorbed by CO 2 and H 2 O present in high concentrations as in an EGR system. At low load, the higher the temperatures, the better the decomposition of diesel fuel, the faster the kinetic reactions, and, therefore, the better the energy efficiency.

The peak efficiencies are reached at an O/C greater than the PO x stoichiometry reaction (respectively, for O/C equals 1.3 and 1.5 instead of O/C = 1). First, temperature is one of the most important parameters.

As mentioned above, reforming reactions need a lot of energy to set quickly. More oxygen is needed to activate reforming reactions and thus a higher O/C ratio, but a part of this additional oxygen forms CO 2 (cf. Fig. 2). The nonhomogeneity of the plasma reformer can lead to a local combustion reaction that raises the temperature and leads to better performances, but higher CO 2 and H 2 O production.

For each O/C ratio, Fig. 2 also shows that the plasma reformer promotes CO 2 production. The dry reforming reaction does not take place because a decrease of the CO 2 molar fraction compared to the initial composition is not observed. Concerning steam reforming, it maybe takes place when O/C is lower than 1. Indeed, the H 2 and CO molar fractions are quite high (4 and 8%, respectively) and the CO 2 fraction increases. For O/C lower than 1, the high excess of fuel induces a high production of CH 4 and a very low conversion rate, which can be associated with methanation. The CH 4 mole fraction must be decreased at maximum as it is a regulated pollutant.

For O/C higher than 2, conditions are getting close to the combustion reaction and high temperature, and high CO 2 and low CH 4 production are observed. The H 2 /C ratio of diesel fuel is equal to 0.92, and thus, generally more CO than H 2 is produced.

In condition 2 at O/C = 1.3, 15% and 8% dry molar fractions of CO and H 2 , respectively, for a deposited power of 720 W have been reached. In these conditions, the dry molar fraction of CH 4 is only 1.6%.

The Euro V passenger car engines have to emit less than 180 mg/km of NO x , and typically their real NO x emission is close to 150 mg/km of NO x . The Euro VI regulation imposes a NO x emission lower than 80 mg/kg. Consequently, 70 mg/ kg of NO x has to be treated by the NO x trap. The homologation of European vehicles is based on NEDC (New European Driving Cycle), which is 11 km long. Assuming one regeneration during the cycle and that the NO x emitted is essentially NO 2 , the NO x trap has to store 16.7 × 10 -3 mol of NO x . 5 mol of syngas are needed to reduce 1 mol of NO x . Without scaling up the mass flow rate, in the second operating condition, 7 × 10 -3 mol/s of syngas is produced and leads to a NO x trap regeneration duration of 12 s, which is a very promising. On the contrary, the first condition leads to a NO x trap regeneration duration of 45 s and is not competitive compared to catalytic processes.

Influence of the current

The monitoring of the input current directly affects the input power injected in the system. Fig. 3 shows the performances of the reformer as a function of input current. In the input current range [0.25-0.6 A], the deposited power varies quasi-linearly with the current as long as we stay in the glidarc zone. In NO x trap regeneration conditions, the quasicontinuous regime, which gave the best results with ethanol, E85, and gasoline [3,4] cannot be reached anymore.

The performances at high load are quite low, and even a current of 0.6 A (P = 940 W) cannot reach an adequate temperature to quicken the PO x reaction. At low load, the energy efficiency grows quasilinearly with the input current and hence with the deposited power until 0.4 A. The energy efficiency and the conversion rate reached 31% and 58%, respectively, for I = 0.4 A and 35% and 60%, respectively, for I = 0.6 A while the deposited power rises from 730 to 1180 W.

Influence of the exhaust gas flow rate

The influence of the exhaust gas mass flow rate has been studied in a range of 2.5-5% of the total exhaust gas mass flow rate. One can observe in Fig. 4 that the higher the exhaust gas flow rate, the lower the volume power injected and the lower the performances. The syngas molar flow rate is quasiconstant at 1.9 × 10 -3 mol/s for the first engine condition. For the second condition, a better syngas molar flow rate is attained (4.8 × 10 -3 mol/s) between 30 × 10 -3 and 40 × 10 -3 mol/s exhaust gas flow rate. This exhaust gas molar flow rate corresponds to a 3.5-4.5% range of total exhaust gas emitted by the engine.

Comparison with a 1D multi-stage kinetic model

The 1D multistage model, presented in Fig. 5, has been detailed in [5]. This model is based on the following assumptions: (i) The medium is adiabatic.

(ii) Only a fraction of reactants' inlet flow passes through the arc discharge. (iii) These two fractions, that is, cold and hot streams, respectively, are perfectly and instantaneously mixed at the reactor exit. The arc is modeled by a perfectly stirred reactor (PSR) where a homogeneous input power is applied. Finally, the postdischarge is modeled by means of plug flow reactor (PFR). The model uses the Chemkin II package [6]. The diesel fuel surrogate molecule used is n-heptane. The kinetic mechanism is composed of 160 species and 1540 reactions [7,8].

A shift in energy efficiency between experiments and the 1D model can be in Fig. 6. First, the energy efficiency discrepancy mainly comes from thermal losses. The model assumes the adiabaticity of the medium. Second, the perfect and instantaneous mix at the torch exit is far removed from the experimental torch exit, which is highly nonhomogeneous. Local nonhomogeneities could also appear in the reactor, leading to H 2 O and CO 2 production instead of H 2 and CO. However, the model trends are similar to the experimental trends.

For O/C higher than 1.5, the experiments and the 1D model are very close to thermodynamics results. For the most favorable case, only 12 s are needed to regenerate the NO x trap catalyst. In the other case, a hybrid catalyst-plasma solution is envisaged.

Conclusions and Perspectives

Figure 1 .

 1 Figure 1. Experimental performances as a function of O/C for both operating conditions (left, condition 1; right, condition 2).

Figure 2 .

 2 Figure 2. Dry molar fraction as a function of O/C (left, condition 1; right, condition 2)

Figure 3 .

 3 Figure 3. Experimental performances as a function of input current (left, condition 1; right, condition 2).

Figure 4 .

 4 Figure 4. Experimental performances as a function of exhaust gas flow rate(left, condition 1; right, condition 2).

Figure 5 .

 5 Figure 5. Diagram of the 1D multistage kinetic model describing the low current plasma reformer.

Figure 6 .

 6 Figure 6. Comparison between experimental data, 1D multistage model and thermodynamics for condition 2.

Table 1 .

 1 Operating conditions.

	Conditions	1 High load	2 Low load
	Φ	0.66	0.32
	O 2 (% mol )	6.8	13.9
	N 2 (% mol )	75.8	77.4
	CO 2 (% mol )	9.1	4.5
	H 2 O (% mol )	8.3	4.1

  An experimental set-up has been developed to study two typical operating points of Diesel powered vehicle, corresponding to high load and low load points. A sensibility study over O/C ratio, injected electric current and mass flow rate have been carried out. The plasma reformer performances have been evaluated in terms of energy efficiency and conversion rate. At low engine load, an energy efficiency of 40% and a conversion rate of 95% have been reached which correspond to a syngas dry molar fraction of 25%. The 1D multistage kinetic model developed has shown good trend correlation with experimental results.It has been demonstrated that the oxygen from CO 2 and H 2 O almost does not intervene in the exhaust gas Diesel fuel reforming. At the contrary, CO 2 and H 2 O decrease temperatures, the kinetic reaction speed and the energy efficiency compared to PO x reaction. To higher the temperature, more oxygen is needed but local combustion can happen and promote H 2 O and CO 2 production.


A Lebouvier

C Delalondre

F Fresnet

F Fabry

V Boch

V Rohani

F Cauneau

L Fulcheri

3D MHD unsteady state modeling of a high voltage -low current plasma torch operating with air

Keywords: MHD modeling, low current, plasma torch

A non-transferred high voltagelow current DC plasma torch operating with air has been simulated using Code_Saturne CFD software for current lower than 1 A. The 3D MHD model considers two injection configurations: axial and vortex. The current and the air flow rate have been tuned in the range 0.3 to 0.6 A and 0.16 to 0.5 g/s, respectively. The influence of these parameters on the arc behavior and the arc characteristics has been studied in terms of temperature, velocity, electrical potential and Joule heating.

Introduction

Thermal plasma models become of interest because they are more and more sophisticated and get close to realistic conditions. Numerous MHD fluid models have been reported in the literature mostly on direct current (DC) arc discharge modeling operating at high current, typically I > 100 A [1,2].

Unfortunately, the literature is very poor concerning MHD arc discharge modeling for low currenthigh voltage plasma torches. One of the explanation for the lack of studies dedicated to this purpose is probably that below 1 A, the self-magnetic field becomes negligible and the convection effects induce a highly irregular motion of the arc column which imply very unstable physical phenomena. Thus, one of the most challenging issues in the field of plasma modeling is to be able to simulate very low-current plasma (typically lower than 1 A) as a result of many numerical instabilities deriving from physical instabilities.

A non-transferred high voltagelow current DC plasma torch operating with air has been simulated using Code_Saturne CFD software for current lower than 1 A and air flow rate up to 0.5 g/s which conditions correspond to typical experimental parameters. The study has been carried out with two injection configurations: axial and vortex. The current and the air flow rate have been tuned in the range 0.3 to 0.6 A and 0.16 to 0.5 g/s respectively.

The influence of these parameters on the arc behavior and the arc characteristics has been studied in terms of temperature, velocity, electrical potential and Joule heating. When possible, the model results have been compared to experimental data.

Mathematical model

The three-dimensional model studied is based on the following main assumptions: -The plasma is considered as a single continuous fluid (Air). -The plasma is optically thin and at Local Thermodynamic Equilibrium (LTE). -The gas is treated as incompressible.

-The gas flow is laminar and time-dependent.

-Gravitational effects are taken into account. Let us point out that whereas high current arcs or high power-density plasmas [3] are commonly assumed to be at LTE. For low current or low power density arcs, this hypothesis is generally not correct and this assumption can imply great differences in the arc behavior. However, in this study, due to highly challenging numerical issues linked with the low current, the LTE assumption has been made with the objective of getting preliminary information on transport phenomena including mass, momentum, and energy as well as on the gas flow and temperature fields.

The model is defined by the set of fluid dynamics Navier-Stokes equations and Maxwell electromagnetic equations respectively. The resistive MHD equations have been solved using Code_Saturne v. 1.3, developed by EDF R&D [4]. The Code_Saturne electric arc module has been used. Fig. 1 presents the schematic of the plasma torch. The mesh is composed of 348 800 hexa-cells. As we work in inversed polarity, we apply to the anode a positive voltage and the cathode is grounded.

The time step is set to 10 µs. At the first time step, the anode potential is 1000 V, and we create a hot channel between both electrodes to ignite the plasma. For 50 µs, low inflow is injected to stabilize the arc. Then, an inflow velocity ramp is applied. The results presented have been performed with a current of 400 mA and an air flow rate of 0.32 g/s. Fig. 2 shows the evolution of the temperature versus time. We can see that the arc root moves linearly in the same section plane as for this first preliminary model, the injection is axial. The arc column seems very stable. The arc root is attached at the cathode tip, a few millimeters above the outlet. This phenomenon was also observed experimentally (see Fig. 3). The displacement of the arc root inside the nozzle, once the velocity ramp has reached its maximum value, is 15.2 m/s, the injection velocity being 12.3 m/s. This difference is due to thermal expansion of the gas which increases the arc root velocity. The temperature of the arc core reaches 5500 K.

Results and Discussion

In Fig. 4, between 1.3 and 6 ms, we observe that the voltage undergoes numerical instabilities with a 20 kHz frequency due to sudden stretch of the plasma arc. The voltage increases almost linearly with the arc length. From 10 ms, the arc root reaches the cathode tip, and a voltage fluctuation around 2.7 kV is observed. This fluctuation comes from a coming and going movement of the arc plume with a 1.85 kHz frequency. Experimentally, we observe a voltage fluctuation of the same order of magnitude, between 0.5 and 3 kHz, but with a voltage of 1.9 kV in these conditions. The overestimation of the voltage is probably due to the LTE assumption, as demonstrated by Trelles et al. [5].

The magnetic field is very low for a current of 400 mA (< 10 µT). Consequently, the Lorentz forces are also very low (< 120 N/m 3 ). Convective effects become predominant over the magnetic effects and lead to the instabilities observed. Fig. 3 shows also that the experimental cathodic arc root emits more radiation than the rest of the arc column. As the model does not include the metal vapor emitted by the electrodes, the model underestimates the radiative emission of the arc root (cf. Fig. 5). The radius of the experimental arc is estimated to 1 mm by naked eye. According to the model results, the radiation losses are the largest within a radius of 0.83 mm, which is very close.

Influence of the current

The current was varied over a range of 300 mA to 600 mA, corresponding to the range of experiments. The air mass flow rate is set to 0.32 g/s.

Fig. 6 shows that, the higher the current, the lower the voltage drop and the higher the deposited power (1000, 1100 and 1200 W for I = 300, 400 and 600 mA, respectively), a phenomenon observed experimentally. We can also notice that the higher the current, the higher the instabilities magnitude before and after 10 ms. The oscillation are in the same frequency range, and thus do not depend on the current.

The lower the current, the higher the difference between simulated and experimental mean voltage. This phenomenon is probably due to the nonequilibrium effects which increase when decreasing the current. Fig. 7 shows that the higher the current, the warmer the arc core, and the wider the temperature profile. Consequently, the arc root velocity in the nozzle increases (14.3, 15.2 and 16 m/s for I = 300, 400 and 600 mA respectively).

Influence of the air mass flow rate

The air flow rate has been varied over a range of 0.16 to 0.5 g/s, the current being 400 mA.

Figs. 8 and9 show the influence of the air flow rate on the voltage and the temperature profile in the middle of the cathode. The higher the air flow rate, the longer the arc, the higher the voltage and the higher the instabilities magnitude. For a flow rate of 0.5 g/s, the arc is very disturbed by the flow instabilities. The arc must increase its voltage to self-sustain. These instabilities are probably due to Kelvin-Helmholtz instabilities but need further investigations to confirm this hypothesis.

The higher the flow rate, the narrower the temperature profile because the energy released is carried by the gas flow. The maximum temperatures reached for the three cases are very close (~ 5200 K).

Vortex injection results.

The vorticity of the flow can be tuned. Fig. 10 shows a helical shape of the arc column. The shape has been confirmed by a high-speed camera video. If the vorticity is too high, the arc cannot be no longer sustained. The mean voltage is 2.17 kV and thus closer to experimental results.

Conclusions and Perspectives

A 3D MHD unsteady model of a low current arc has been successfully implemented. The model allows obtaining significant information on low current arc which are sorely experimentally measurable.

The results have been compared to experimental results and show good correlations and good trends. Indeed, the arc root attachment and the arc shape are very similar to the experimental observations. The model overestimates the voltage drop probably because of the LTE assumption. The nonequilibrium effects are significant and result in a discrepancy between experimental and numerical voltage values. The vortex injection model shows a helical shape of the arc and a better voltage estimation.

This preliminary work is likely to open the way towards a better understanding of low current arc discharges which technologies are currently encountering an important development in many application fields.

Figure 1 .

 1 Figure 1. Schematic of the plasma torch.

3. 1 .

 1 I = 400 mA. Q air = 0.32 g/s. Axial injection.

Figure. 2 .

 2 Figure. 2. Evolution of the temperature for three different times (t = 2, 5 and 20 ms). I = 400 mA. Q air = 0.32 g/s.

Figure 3 .

 3 Figure 3. Photograph of the experimental plasma arc.

Figure 4 .

 4 Figure 4. Numerical voltage versus time. I = 400 mA. Q air = 0.32 g/s.

Figure 5 .

 5 Figure 5. Numerical radiation losses. I = 400 mA. Q air = 0.32 g/s.

Figure 6 .

 6 Figure 6. Numerical voltage versus time for three different currents. Q air = 0.32 g/s.

Figure 7 .

 7 Figure 7. Numerical temperature profile over radial distance for three different currents in the middle of the cathode. Q air = 0.32 g/s.

Figure 8 .

 8 Figure 8. Numerical voltage versus time for three different air flow rates. I = 400 mA.

Figure 9 .

 9 Figure 9. Numerical temperature profile vs. radial distance for three different air flow rates. I = 400 mA.

Figure 10 .

 10 Figure 10. Iso-temperature (3200 K) for the vortex injection. t = 80 ms. I = 0.4 A. Q air = 0.32 g/s.


Thierry Delort
email: tdelort@yahoo.fr

DERIVATION OF THE TULLY-FISHER's LAW FROM THE THEORY OF ETHER

Keywords:

 a result raising questions concerning the existence of dark matter and the validity of General Relativity, conclusion of his observations of 47 galaxies. He established the validity of Tully-Fisher's law for all those galaxies, which is unexplained and in contradiction with classical Cosmology based on General Relativity. Moreover, we remind that this classical Cosmology did not solve the enigma of the nature of dark matter, nor the enigma of its invisibility, nor the enigma of the constancy of the curve of velocities of stars in a galaxy. Consequently it appears that a new Cosmology, alternative to the classical Cosmology, is necessary, at least in the field of dark matter. In fact it exists already a new Cosmology, alternative to classical Cosmology and compatible with the Relativity Theory, which is the Cosmology of the Theory of Ether relative to dark matter. This new Cosmology gives the nature of dark matter, the origin of its invisibility and permits to obtain the constancy of the curve of velocities of stars in a galaxy. In this article, after having recalled the theoretical elements of this new Cosmology, we will show that it also permits to obtain theoretically Tully-Fisher's law whose Mc Gaugh observations confirmed the validity. We will note that all the theoretical results used and obtained in this article are compatible with the Special and General Relativity Principles.

1.INTRODUCTION

The theory exposed in this article is based on a new Cosmology that is the Cosmology of the Theory of Ether relative to dark matter. Despite of the fact that this new Cosmology has been elaborated inside the frame of a very general Theory of Ether, we will note that it is compatible with the Special and General Relativity Principles.

So in this article, we suppose the validity of the modern Theory of Ether exposed in the book [START_REF] Delort | Théorie moderne de l'Ether, from the book « Théories d'or 2 ième édition[END_REF] (That contains rectified and reactualized version of the articles [START_REF] Delort | Applications of Theory of ether[END_REF](5)(6)(7)(8)(9) [START_REF] Delort | Complements of Cosmology in the Theory of Ether[END_REF] that need to be improved). But we will use only theoretical elements compatible with the Relativity Theory that we will recall at the beginning of the article.

In 3 articles [START_REF] Delort | Applications of Theory of ether[END_REF] (2) [START_REF] Delort | Elements of Cosmology in the Theory of Ether[END_REF] , we exposed how this new Cosmology based on a Theory of Ether interprets the nature of dark matter. In this interpretation, dark matter is a substance, called Ether-substance, which constitutes what is usually called "The vacuum".

Besides, an astrophysicist, Stacy Mc Gaugh, established experimentally in the article [START_REF] Mc | A Novel Test of Modified Newtonian Dynamics with Gaz rich Galaxies[END_REF] the validity of Tully-Fisher's law, showing that the baryonic mass of a galaxy is proportional to the 4 th power of the velocity of the stars in this galaxy. This fact appears to be in contradiction with the existence of dark matter, because it implies that the velocity of a star depends only on the baryonic mass of the galaxy, and consequently with General Relativity that needs dark matter to explain that the stars are retained inside galaxies.

In reality, in the classical present Cosmology, we have 2 first enigmas: The nature of dark matter and the origin of its invisibility. The interpretation by the new Cosmology of the nature of dark matter, which is according to it a substance constituting what is usually called "The vacuum", explaining its invisibility, permits consequently to solve the 2 preceding enigmas and is consequently extremely interesting. But we obtain another very amazing result: Modeling the Ether-substance as an ideal gas, we obtained in each article [START_REF] Delort | Applications of Theory of ether[END_REF] (2) [START_REF] Delort | Elements of Cosmology in the Theory of Ether[END_REF] the constancy of the curve of stars in galaxies. The constancy of this curve is also unexplained and constitutes a 3 rd enigma in the present Classical Cosmology.

We are going to see in this article how our model of the Ether-substance as an ideal gas permits to solve a 4 th fundamental enigma that is the validity of Tully-Fisher's law.

It is clear that the fact that what we call "the vacuum" be constituted of a substance (called Ether-substance) owning a mass is in agreement with Special and General Relativity Principles. We will see that it is also the case for the whole of the new Cosmology used in this article in order to solve the 4 fundamental enigmas concerning dark matter.

RECALL

We remind that the Tully-Fisher's law is the following: Tully and Fisher realized some observations on spiral galaxies. They obtain that the luminosity L of a spiral galaxy is proportional to the 4 th power of the velocity v of stars in this galaxy. So we have the Tully-Fisher's law for spiral galaxies, K 1 being a constant:

L=K 1 v 4 (1)
But the baryonic mass M of a spiral galaxy is proportional to its luminosity. So we have also the law for a spiral galaxy, K 2 being a constant:

M=K 2 v 4 (2)
This 2 nd form of Tully-Fisher's law is known as the baryonic Tully-Fisher's law.

We remind that the Tully-Fisher's law (1) is not verified in general for galaxies that are not spiral galaxies. But the observations of Mc Gaugh [START_REF] Mc | A Novel Test of Modified Newtonian Dynamics with Gaz rich Galaxies[END_REF] show that the baryonic Tully-Fisher's law (2) seems to be true for all galaxies. This constitutes a new major enigma for the classical Cosmology, but we are going to see how we can derivate this law from the new Cosmology.

Let us recall how in the Chapter 2.2 Black mass of each article [START_REF] Delort | Applications of Theory of ether[END_REF] and [START_REF] Delort | Applications de la Théorie de l'Ether[END_REF] we solved the enigma of the constancy of the curves of velocities of stars a galaxy: In order to obtain this result, we first modeled the Ether-substance as an ideal gas: An element of Ether-substance with a mass m, a volume V, a pressure P and a temperature T verifies the law, k 0 being a constant:

PV=k 0 mT (3)
Which means, setting k 1 =k 0 T :

PV=k 1 m (4a)
Or equivalently, ρ being the density of the element:

P=k 1 ρ (4b)
We then emitted the hypothesis that a galaxy could be modeled as a concentration of Ethersubstance presenting a spherical symmetry, at a constant and homogeneous temperature T.

We then considered the sphere S(r) (resp.the sphere S(r+dr)) that is the sphere inside the concentration of Ether-substance with a radius r (resp. r+dr) and whose the center is the center O of the galaxy.

We then considered the following equation (5a) of equilibrium of forces on an element of Ether-substance with a surface dS, a width dr, situated between the 2 spheres S(r) and S(r+dr):

∫ = - + + r r dSP dx x x dSdr r r G dr r dSP 0 2 2 0 ) ( ) 4 ) ( )( ) ( ( ) ( π ρ ρ (5a)
We then verified that the density of the ether-substance ρ(r) satisfying the preceding equation of equilibrium was:

2 2 4 ) ( r k r π ρ = (5b)
The constant k 2 being given by, G being the Universal attraction gravitational constant:

G T k G k k 0 1 2 2 2 = = (6)
Using the preceding equation (5b), we obtain that the mass M(r) of the sphere S(r) constituted of Ether-substance is given by the equation:

r k dx x x r M r 2 0 2 ) ( 4 ) ( = = ∫ ρ π (7)
We then obtain, neglecting the mass of stars in the galaxy, that the velocity v(r) of a star of a galaxy situated at a distance r from the center O of the galaxy is given by v(r) [START_REF] Delort | Applications de la Théorie de l'Ether[END_REF] /r=GM(r)/r 2 and consequently :

v(r) 2 =Gk 2 =2k 1 =2k 0 T (8)
So we obtain in the previous equation ( 8) that the velocity of a star in a galaxy is independent of its distance to the center O of the galaxy, solving the 3 rd enigma concerning dark matter. We justify in details the equations (5b) and ( 6) in the Chapter 2.2 Black mass of any of the 2 articles [START_REF] Delort | Applications of Theory of ether[END_REF] and [START_REF] Delort | Applications de la Théorie de l'Ether[END_REF] , it is easy to verify them using the equation (4b). We note that the theoretical elements of the new Cosmology permitting to obtain the equations (5b)(6)(7)(8) are compatible with Special and General Relativity Principles.

3.THEORY OF QUANTIFIED LOSS OF CALORIFIC ENERGY (BY BARYONS)

We saw in the previous equation ( 8) that according to the new Cosmology, the square of the velocity of stars in a galaxy is proportional to the temperature of the concentration of Ether-substance constituting this galaxy. So if we determine this temperature T, we then obtain the squared velocity of the stars in this galaxy. So we need to try to determine T: -A first possible idea is that the temperature T is the so called "Temperature of the fossil radiation". But this is impossible because it would imply that all stars of all galaxies are driven with the same velocity and we know that it is not the case.

-A second possible idea is that the temperature T is due to the absorption by the concentration of Ether-substance constituting the galaxy of a fraction of the photons emitted by the stars of this galaxy. But if it was the case, the temperature and consequently the velocity of the stars of the galaxy would only depend on the luminosity of the galaxy, and we should have a law analogous to the Law of Tully-Fisher (1) and we know that it is not the case.

-A third possible idea is that in any galaxy, each baryon interacts with the Ether-substance constituting the galaxy, and consequently it occurs for each baryon a loss of calorific energy communicated to the Ether-substance. A priori we could expect that this loss of calorific energy for each baryon (transmitted to the Ether-substance) depend on the temperature of this baryon, but if it was the case, the total calorific loss for all baryons would be extremely difficult to calculate and moreover we would not obtain that the total calorific loss depend on the baryonic mass of the galaxy.

The final idea is that indeed it occurs a calorific loss for each baryon (transmitted to the Ether-substance), but that this loss is quantified, depending only on the number of the nucleons of the baryon. This loss should be very low, but the calorific capacity of the Ethersubstance being also very low, it can involve an appreciable temperature of the concentration of Ether-substance constituting the galaxy. So we make the following hypothesis:

HYPOTHESIS OF QUANTIFIED CALORIFIC LOSS (OF BARYONS)

-Each baryon of a galaxy is submitted to a loss of calorific energy, transmitted to the concentration of Ether-substance constituting the galaxy.

-This loss of calorific energy depends only on the number of nucleons constituting the baryon (It is independent of its temperature). So if p is the power corresponding to the loss of calorific energy for a baryon with n nucleons, it exists a constant p 0 (loss of calorific energy per nucleon) such that: p=np 0 (9) According to the equation ( 9), the total power corresponding to the loss of calorific energy by all the baryons of a galaxy is proportional to the number of nucleons of the whole of those baryons, and consequently to the baryonic mass of this galaxy. So if m 0 is the mass of one nucleon, M being the baryonic mass of the galaxy, we obtain according to the equation ( 9) that the total power P r corresponding to the calorific energy received by the concentration of Ether-substance constituting the galaxy from all the baryons is given by the following equation, K 3 being the constant p 0 /m 0 :

P r =(M/m 0 )p 0 =K 3 M ( 10
)
Concerning the preceding Hypothesis of loss of calorific energy, it is important to remark: -The loss of calorific energy of a baryon transmitted to the Ether-substance is a quantum phenomenon, consequently it is not surprising that the power corresponding to the loss of calorific energy of a baryon be quantified.

-In physics of thermal transfer, the calorific loss of one or several other particles usually depend on their temperature. But it is always only thermal transfers from baryons to other baryons that are considered, and consequently it is not compulsory that it be also the case for transfers between baryons and Ether-substance.

-It is possible that this hypothesis be true only for baryons whose temperature be superior to a given temperature T S . Moreover, their temperature must be superior to the local temperature of the Ether-substance.

-The great simplicity of this hypothesis permits to obtain very easily the total power corresponding to calorific energy received by the concentration of Ether-substance (Equation ( 10)). If the loss of energy of a baryon depended on its temperature, then it would be incomparably more complicated, and maybe impossible, to obtain a simple expression giving this total power.

-This hypothesis is a priori compatible with the Special and General Relativity Principles, and also with classical Quantum Physics.

DEDUCTION OF THE TULLY-FISHER's LAW

In agreement with the previous model of galaxy, we model a galaxy as a concentration of Ether-substance presenting a spherical symmetry (and consequently being itself a sphere), at a temperature T and immerged inside a medium constituted of Ether-substance at a temperature T 0 (T 0 being the so called "Temperature of the fossil radiation), and with a density ρ 0 .

In order to obtain the radius R of the concentration of Ether-substance constituting the galaxy, it is logical to make the hypothesis of the continuity of ρ(r): R is the radius for which the density ρ(r) of the concentration of Ether-substance is equal to ρ 0 . So we have the equation:

ρ(R)=ρ 0 (11)
Consequently we have according to the equations (5b) and ( 6):

0 2 2 4 ρ π = R k (12) 0 2 0 4 1 2 ρ π = × R G T k (13)
So we obtain that the radius R of the concentration of Ether-substance constituting the galaxy is given approximately by the equation:

2 / 1 4 2 / 1 0 0 ) 4 2 ( T K G T k R = = ρ π (14)
The constant K 4 being given by : We can then consider that the sphere with a radius R of Ether-substance constituting the galaxy is in thermal interaction with the medium at a temperature T 0 in which it is immerged. We model this thermal interaction as a convection phenomenon. If φ is the thermal flow of energy on the borders of the sphere, the power P l lost by the sphere of Ether-substance constituting the galaxy is given by the equation:

P l =4πR 2 φ (16)
But we know that for a convection phenomenon between a medium at a temperature T and a medium at a temperature T 0 the flow φ between the 2 media is classically given by the expression, h being a constant depending only on ρ 0 :

φ=h(T-T 0 ) (17)
Consequently the total power lost by the concentration of Eher-substance is:

P l =4πR 2 h(T-T 0 ) (18)
We can consider that at the equilibrium, the thermal power P r received by the concentration of Ether-substance constituting the galaxy is equal to the thermal power P l lost by this concentration. Consequently according to the equations ( 10) and ( 18), M being the baryonic mass of the galaxy, we have:

K 3 M=4πR 2 h(T-T 0 ) (19)
Using then the equation ( 14) :

K 3 M=4πK 4 2 hT(T-T 0 ) (20)
Making the approximation T 0 <<T (We remind that T 0 is of the order 3°K) : Consequently we obtain the expression of T, defining the constant K 5 :

2 / 1 5 2 / 1 2 / 1 2 4 3 ) 4 ( M K M h K K T = = π (22)
And then according to the equation ( 8) :

v 2 =2k 0 T=2k 0 K 5 M 1/2 (23) So : 4 2 5 0 ) 2 1 ( v K k M = (24)
So we finally obtain :

M=K 6 v 4 (25)
The constant K 6 being defined by: So we obtain the baryonic Tully-Fisher's law (2), with K 2 =K 6 .

4.CONCLUSION

So we solved the 4 th enigma concerning dark matter, meaning that we justified theoretically the validity of the Tully-Fisher's law, using a new Cosmology that is the Cosmology of the Theory of Ether relative to dark matter. We note that the theoretical elements of this new Cosmology in order to solve this 4 th enigma are, as all the theoretical elements used in order to solve the 3 other enigmas, compatible with the Special and General Relativity Principles. In order to obtain the baryonic Tully-Fisher's law, we deepened our model of ideal gas for the Ether-substance, we introduced the Hypothesis of quantified loss of calorific energy (for baryons), and we used a simple thermal model for a galaxy. So we see that the presented here new Cosmology, being compatible with Special and Relativity Principles, appears to be much more powerful than the classical Cosmology in the field of dark matter. We remind that despite of its compatibility with the Relativity Theory, it has been elaborated in the frame of a very general Theory of Ether (4)(5)(6)(7)(8)(9) [START_REF] Delort | Complements of Cosmology in the Theory of Ether[END_REF] .