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

American Journal of Computational Mathematics, 2012, 2, 331-335 http://dx.doi.org/10.4236/ajcm.2012.24045 Published Online December 2012 (http://www.SciRP.org/journal/ajcm)

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

Mohammedi Ferhat, Zergui Belgacem, Bensaada Said Laboratory Larhyss University, Biskra, Algeria Email: [email protected]

Received January 5, 2012; revised August 23, 2012; accepted September 5, 2012

ABSTRACT 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 perfor- mances of a continuous DF chemical laser, based on the exothermic reaction: FD2DFvjD , .

Keywords: Chemical Laser; Spectroscopy Vibrational; Rotational Processes; Lasers Cavity

1. Introduction The number of molecules in a vibrational state is given by: The rapid developments in new laser techniques and applications have extended the field of laser chemistry into N  Vhc12 N  exp (3) many other scientific fields, such as biology, medicine, V  QKTVV and environmental science, as well as into modern tech- nological processes. This “natural” invasion is a result of In any given vibrational level the rotational level the multidisciplinary character of modern laser chemistry. populations are given by: The success of simple models to simulate the kinetic be- Nn221 ggn1 (3a) haviour of gas lasers has amused a lot of interest in the numerical and theoretical study of such systems. Chemi- cal lasers find their origin in the study [1-3] of the radia- 2. Model Formulation tion emitted from chemical reactions. In many cases such In general the pumping chemical is using a very difficult radiation is of thermal origin. Energy levels and transi- and complexity; exigent primary cavity (chamber of re- tions; A laser chemical or otherwise, requires a mecha- active) and secondary cavity (the laser chamber for reac- nism to populate an excited state at a sufficiently fast rate tions). The mixing of flows is accompanied by the exo- such that at some time point there are more molecules in thermic reaction: an upper (higher energy) state than in a lower. Under FD2DF*D;  DF2DF*F these conditions the number of photons produced by stimulated emission cexceed those absorbed, and optical The reacting system can be presented in simplified amplification or gain will result (See Figures 1 and 2). forms as follows process us initiation, pumping reactions, Under equilibrium conditions the ratio [4,5] of the num- energy exchange reactions [6,7], (see Figure 3) termina- ber of molecules in an upper and lower state are given by tion and stimulated emission. The usual one dimensional the Boltzmann distribution: flow equations govern the jet, wherein the energy equa- Ng tion includes a loss term due to lasing. The rate equations uuexpEKT (1) for HF(v) have the form of radiative transport equation, Ng LL where ρ is density, u is flow velocity, and n(v)is mole-

This then leads to the concept of partial inversions cha- mass ratio of HF(v). The quantity γch(v) is a summation racterized by vibrational and rotational “temperatures”. over all chemical reactions, including energy transfer For a diatomic molecule, in the harmonic oscillator-rigid reactions, That affect HF(v): rotator approximation the energy levels are given by: dnv  uvv v1 (4) EvJ,12 v BJJ 1 (2) ch rad rad    er dx

the surfaces of caity in 3D Evoltuion chemicals reactions exothermic in cavity DF

0.5 1 0 z 0

-0.5 E(v,J -1 -1 -0.5 -1 2 2 -1 0 -0.5 1 1 0 0.5 0.5 1 1 x 0 0 y -1 -1 y (v) x( J ) -2 -2

the map cavity in 3D

1.5

0.5

-0.5

2 1 2 0 0 -1 Figure 1. Principle processes in laser chemistry [Telle & -2 -2 All].

The level curves of reactions E(v,J) = we(V+1/2).B.J(J+1) 40 0 0.8 .4 0.6 0.8 0 0 .4 .2 0.6 0 0 2 35 -0 . . .2 -0 .4 4 0.4 0 0 -0 - .2 30 0.2

- 0. 0 6 0 . 6 0 25 -

2 .

0 - - 0 4 y

. . 20 4 0 - 2 - 0 8 0. 0 . - 0 . 8 0 15 -

-0 .6 0.2 .6 10 -0 -0.2 0.4 0 0 .2 0 2 0. 2 0.6 5 - 0. 0.4 .4 0.6 0.8 0 0.8 5 10 15 20 25 30 35 40 x Figure 3. Reactions chemicals in primary and secondary Figure 2. The map surfaces of cavity and curves levels. reactions cavity.

differential equations for the density of inversion of laser ch vri riL r (5) r populations with 3 or on 4 levels, the designations used in this diagram are: ω the probability of transitions sti- LK Cri K Cri (6) rr i r i mulated in the channel from oscillation (channel 2-1); ωp ii for the radiation of pumping in the channel (1-3); 1/τ, the 3. Analytical Model probability of spontaneous transitions in channels 1-3 and 2-3, ni(z,t) the density of population of the ith level Before you begin to format your paper, first write and and by n’ the total number of active centre per unit of save the content as a separate text file. We simulate the volume. The equations of the assessment for the popula-

tion’s n1, n2, n3 have the following equations form Ode’s (Ordinary differential equations) [8,9];

nn12n3 nn21p  nn 13 t   31

nn22n3  nn21    (7) t 32

nnn333  p ()n1 n3 t 32 31

With condition nt nt nt0 (8) 123 If shock waves relaxation-dissoc iation

nnn123 n (9) Then 1

nt2 nt1 (10) 0.5

3.1. ODE’s Solvers in MATLAB 0 A list of ODE solvers and of other routines that act on -0.5 functions is returned by help funfun, and a documenta- -1 tion window is opened by doc funfun. The two main rou- 5 tines of interest are ODE45, an explicit single-step inte- 0 5 0 grator, and ODE15s, an implicit multistep integrator that -5 -5 works well for stiff systems. We demonstrate the use of y x ODE45 and ODE15s for a simple batch reactor with the Figure 4. The cavity shock waves relaxation-dissociation. two elementary reactions

AB C and CB  D contribute to the population inversions observed; The following set of differential equations describes (v = 2→ 1 for HF and both 3 → 2 and 2 → 1 for DF). the change in concentration three species in a tank. The UF65 h  F UF reactions ABC occur within the tank. The con- stants k1, and k2 describe the reaction rate for AB F H2 HF F  H 32 K cal mol and B C respectively. The following ODE’s are ob- H UF65 HF UF  H 46 K cal mol tained: dCa KCa1 3.2. Simulation with MATLAB dt Systematic Approach: The following reaction data has dCb K12Ca K Cb (11) been obtained from a simple decay reaction; AB , dt Use MATLAB to plot the concentration of component A dCc  KCb2 in mol/L against the reaction time, t, in minutes. Title the dt plot, label the axes, and obtain elementary statistics for where k1 = 1 mn–1 and k2 = 2 mn–1 and at time t = 0, Ca the data. The idea of the simulation part was to form a = 5 mol and Cb = Cc = 0 mol. Solve the system of equa- basic setup for a future test stand for chemical cavities tions and plot the change in concentration of each species lasers within a numerical routine. The framework for the over time (Figure 4). Select an appropriate time interval numerical routine [10,11] has been MATLAB7, a nu- for the integration. We wish to use a general notation merical computing environment by “The MATH- system for IVPs, and so define a state vector, x, that WORKS”, (Figure 5) since it offers a large range of in- completely describe the state of the system at any time built functions and visualization tools. In principle, the sufficiently well to predict its future behavior; The very simulations could have been done with any other UF6-H2 chemical laser takes advantage of the low mo- sophisticated programming environment or language, but ment of inertia of HF and the effectiveness of UF6 as a the MATLAB language offered the quite comfortable photolytic fluorine atom source 15. Two reactions could advantage of vector based variables useful for this prob-

reactions vibrationals directions 12 Figure 5. Propagations shock waves in cylindrical lasers cavity. 10 lem. Cavity parameter variation: Thus, we simulated the 8 work of a continuous chemical H2 F2 laser with simultaneous lasing on the V; jv1 1 , j (j = 14 - 6 13) and vj , vj, 1 (j = 17 - 16) transitions. Sup- pression of the lasing on vibrational and rotational transi- 4 tions increased the energy extraction for purely rotational transitions Parameters of simulation cavity: Equations (7), 2 (8) and (11) governing physical phenomena in DF chemical laser cavity is to be discretized. There exist a lot of 0 numerical methods for solving this type of system. Time -2 stepping method, Boundary conditions and the resonator -2 0 2 4 6 8 10 12 is equipped with two flat mirrors of which separate dis- Figure 6. Oscillations shock waves lasers modes in cavity. tances (Fabry-Perrot).

The usual initial conditions for the translation tem- The concentration of atoms is given by the equation perature, pressure, density, and Mach number immedi- ately behind the shock are given the Rankine-Hugoniot dxat [12-14] relations (Figure 6), State-resolved vibrational dt kinetics are simulated by the master equation: 2  xx 21kf mol  k  1 at  dfxat at1 at  mat11 mol at   kiat  mfkmi at  ifm    dt  i at xx at21kf  mol kijmnffkmnijff,, ,, at mol 1  mol i j mol m n at1 mol ijn,, 2  xxmol 1 d mol xxat mol  fm  kmol 1   mol xtmol d 1 at mol 2 xx (14) at mol at kmat  fm k at  m  The last two terms in Equation (13), simulate the vi- atx mol at bration-dissociation coupling. Evib(T) and Evib(t) are vi-  2  brational energies: xat mol xmol kmmol  fm k mol  m   x  at mol mol EEfvxvvib  v v e 11e fv (14)

0,,,ijmn diss , i m where fv can either the equilibrium or a nonequilibrium (13) distribution function.

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