Modelling the Cavity of Continuous Chemical Lasers ( CCLs ) Using Matlab Applications

The subject that concerns us in this work is the numerical simulation and optimal control of equilibrium of the continuous chemical lasers (CCLs). Laser Chemistry: Spectroscopy, Dynamics and Applications are a carefully structured introduction to the basic theory and concepts of this subject. In this paper we present the design and discuss the performances of a continuous DF chemical laser, based on the exothermic reaction:   ,    F D2 DF v j D .


Introduction
The rapid developments in new laser techniques and applications have extended the field of laser chemistry into many other scientific fields, such as biology, medicine, and environmental science, as well as into modern technological processes.This "natural" invasion is a result of the multidisciplinary character of modern laser chemistry.The success of simple models to simulate the kinetic behaviour of gas lasers has amused a lot of interest in the numerical and theoretical study of such systems.Chemical lasers find their origin in the study [1][2][3] of the radiation emitted from chemical reactions.In many cases such radiation is of thermal origin.Energy levels and transitions; A laser chemical or otherwise, requires a mechanism to populate an excited state at a sufficiently fast rate such that at some time point there are more molecules in an upper (higher energy) state than in a lower.Under these conditions the number of photons produced by stimulated emission cexceed those absorbed, and optical amplification or gain will result (See Figures 1 and 2).Under equilibrium conditions the ratio [4,5] of the number of molecules in an upper and lower state are given by the Boltzmann distribution: This then leads to the concept of partial inversions characterized by vibrational and rotational "temperatures".For a diatomic molecule, in the harmonic oscillator-rigid rotator approximation the energy levels are given by: The number of molecules in a vibrational state is given by: In any given vibrational level the rotational level populations are given by:

Model Formulation
In general the pumping chemical is using a very difficult and complexity; exigent primary cavity (chamber of reactive) and secondary cavity (the laser chamber for reactions).The mixing of flows is accompanied by the exothermic reaction: The reacting system can be presented in simplified forms as follows process us initiation, pumping reactions, energy exchange reactions [6,7], (see Figure 3) termination and stimulated emission.The usual one dimensional flow equations govern the jet, wherein the energy equation includes a loss term due to lasing.The rate equations for HF(v) have the form of radiative transport equation, where ρ is density, u is flow velocity, and n(v)is molemass ratio of HF(v).The quantity γ ch (v) is a summation over all chemical reactions, including energy transfer reactions, That affect HF(v):  the surfaces of caity in 3D

Analytical Model
Before you begin to format your paper, first write and save the content as a separate text file.We simulate the differential equations for the density of inversion of laser populations with 3 or on 4 levels, the designations used in this diagram are: ω the probability of transitions stimulated in the channel from oscillation (channel 2-1); ω p for the radiation of pumping in the channel (1-3); 1/τ, the probability of spontaneous transitions in channels 1-3 and 2-3, n i (z,t) the density of population of the ith level and by n' the total number of active centre per unit of volume.The equations of the assessment for the popula-tion's n 1 , n 2 , n 3 have the following equations form Ode's (Ordinary differential equations) [8,9]; Then

ODE's Solvers in MATLAB
A list of ODE solvers and of other routines that act on functions is returned by help funfun, and a documentation window is opened by doc funfun.The two main routines of interest are ODE45, an explicit single-step integrator, and ODE15s, an implicit multistep integrator that works well for stiff systems.We demonstrate the use of ODE45 and ODE15s for a simple batch reactor with the two elementary reactions The following set of differential equations describes the change in concentration three species in a tank.The reactions occur within the tank.The constants k1, and k2 describe the reaction rate for and respectively.The following ODE's are obtained: where k1 = 1 mn -1 and k2 = 2 mn -1 and at time t = 0, Ca = 5 mol and Cb = Cc = 0 mol.Solve the system of equations and plot the change in concentration of each species over time (Figure 4).Select an appropriate time interval for the integration.We wish to use a general notation system for IVPs, and so define a state vector, x, that completely describe the state of the system at any time sufficiently well to predict its future behavior; The UF 6 -H 2 chemical laser takes advantage of the low moment of inertia of HF and the effectiveness of UF6 as a photolytic fluorine atom source 15.Two reactions could contribute to the population inversions observed; (v = 2→ 1 for HF and both 3 → 2 and 2 → 1 for DF).

Systematic Approach:
The following reaction data has been obtained from a simple decay reaction; , Use MATLAB to plot the concentration of component A in mol/L against the reaction time, t, in minutes.Title the plot, label the axes, and obtain elementary statistics for the data.The idea of the simulation part was to form a basic setup for a future test stand for chemical cavities lasers within a numerical routine.The framework for the numerical routine [10,11] has been MATLAB7, a numerical computing environment by "The MATH-WORKS", (Figure 5) since it offers a large range of inbuilt functions and visualization tools.In principle, the very simulations could have been done with any other sophisticated programming environment or language, but the MATLAB language offered the quite comfortable advantage of vector based variables useful for this prob- lem.Cavity parameter variation: Thus, we simulated the work of a continuous chemical laser with simultaneous lasing on the V;

A B 
, j (j = 14 -13) and (j = 17 -16) transitions.Suppression of the lasing on vibrational and rotational transitions increased the energy extraction for purely rotational transitions Parameters of simulation cavity: Equations ( 7), ( 8) and (11) governing physical phenomena in DF chemical laser cavity is to be discretized.There exist a lot of numerical methods for solving this type of system.Time stepping method, Boundary conditions and the resonator is equipped with two flat mirrors of which separate distances (Fabry-Perrot).
The usual initial conditions for the translation temperature, pressure, density, and Mach number immediately behind the shock are given the Rankine-Hugoniot [12][13][14] relations (Figure 6), State-resolved vibrational kinetics are simulated by the master equation:  The concentration of atoms is given by the equation The last two terms in Equation ( 13), simulate the vibration-dissociation coupling.E vib (T) and E vib (t) are vibrational energies: where f v can either the equilibrium or a nonequilibrium distribution function.

Conclusion
We are now ready to treat more complex problems of greater relevance to chemical engineering practice.We begin with the study of initial value problems (IVPs) of ordinary differential equations (ODEs), in which we compute the trajectory in time of a set of N variables x i (t) governed by the set of first-order ODEs.We start the simulation, usually at t 0 = 0, at the initial condition, x(t 0 ) = x[0].Such problems arise commonly in the study of chemical kinetics or process dynamics.While we have interpreted above the variable of integration to be time [15][16][17][18][19], it might be another variable such as a spatial coordinate.The following curves is produced upon execution, to be able to solve higher order ODE's in MAT-LAB, they must be written in terms of a system of first order ordinary differential equations.We have successfully designed, built and tested a in Military applications DF generator for a chemical H-F or DF laser A laser powered by an array of μSOG chips would be useful for a variety of industrial applications.The comparison of the results of numerical simulations performed with the use of our model with the results of experimental studies.

Figure 3 .
Figure 3. Reactions chemicals in primary and secondary reactions cavity.

Figure 6 .
Figure 6.Oscillations shock waves lasers modes in cavity. d