Internal partition function calculated with where N is the particle density (cm 3 ) Influence of Electronically Excited States on Thermodynamic Properties of LTE Hydrogen Plasma

Ratio of inner enthalpy to total enthalpy at different pressure. The total enthalpy of all the system is calculated as follows: while the inner part of enthalpy is equal to:

the total specific heat is diveded in two parts: the frozen specific heat the reactive specific heat the inner specific heat is : Ratio of inner Cp to total Cp at different pressure Ratio of inner Cp to Cp frozen at different pressure

Ratio of Cp, using Debye-Huckel theory, to Cp calculated with cut-off, at 10 8 Pa. Ratio of Cp, using Debye-Huckel theory, to Cp calculated with cut-off, at 10 5 Pa. Internal specific heat and his component at 10 8 Pa.Internal specific heat and his component at 10 5 Pa.

The transport properties of a partially ionized thermal hydrogen plasma has been calculated by taking into account electronically excited states with their “abnormal” transport cross sections. The results show a strong dependence of these transport properties on electronically excited states specially at high pressure. Translational thermal conductivity Viscosity Heavy particles transport propertiesElectron transport properties Translational thermal conductivity Electrical conductivity Reactive thermal conductivity “Usual” Collision integrals“Abnormal” Collision integrals Complete set of data (see text) Influence of Electronically Excited States on Transport Properties of LTE Hydrogen Plasma

Model Species We have considered an hydrogen plasma constituted by molecular hydrogen, atomic hydrogen (12 atomic levels), H + ions and electrons: H 2 H(n=1,12) H + e - Reactions we consider the dissociation process and ionization reactions starting from each electronic states of hydrogen atom.

Equilibrium Composition The equilibrium composition is obtained by using Saha and Boltzmann laws. First we calculate the equilibrium composition by considering only four species and two reactions which take into account the total atomic hydrogen without distinction among the electronic states. Then we use the Boltzmann distribution for calculating the distribution of the electronic states of atomic hydrogen.

Collision Integrals I: General Aspects Transport cross sections can be calculated as a function of gas temperature according to the equations where Considered interactions: neutral-neutral (H 2 -H 2, H 2 -H, H-H) ion-neutral (H + -H 2, H + -H) electron-neutral (e-H, e-H 2 ) charged-charged (H + -e, H + -H +, e-e) For the present calculation we need the collision integrals of different orders depending on the different approximations used in the Chapman-Enskog method. To this end we have used a recursive formula Note that we have used the reduced collision integrals i.e. the collision integrals normalized to the rigid sphere model  ij *

Collision Integrals II: e-H(n) Diffusion type collision integrals for the interactions e-H(n) are calculated by integrating momentum transfer cross sections of Ignjatovic * et al.. Viscosity type collision integrals have been considered equal to diffusion type ones * L.J. Ignjatovic, A.A. Mihajlov, Contribution to Plasma Physics 37 (1997) 309. T=10 4 K T= K

Collision Integrals III: H + -H(n) Diffusion type and viscosity type collision integrals have been calculated by Capitelli et al. * and fitted according to the following expressions. with * M. Capitelli, U.T. Lamanna, J. Plasma Phys.12, 71 (1974). The elastic contribution to  (1,1)* has been evaluated with a polarizability model

Collision Integrals IV: H(n)-H(n) Viscosity type collision integrals for the interactions H(n)-H(n) up to n=5 have been calculated by Celiberto et al. * By using potential energy curves obtained by CI (configuration interaction) method. The data have been interpolated at different temperatures with the equation where * R. Celiberto, U.T. Lamanna, M.Capitelli, Phys.Rev A 58, 2106 (1998). T=10 4 K T= K

Collision Integrals V: H(n)-H(m) M.Capitelli, P.Celiberto, C.Gorse, A.Laricchiuta, P.Minelli, D.Pagano, Phys. Rev. E 66,016403/1 (2002)

Transport Coefficients I: General Aspects Transport coefficients have been calculated by using the third approximation of the Chapman-Enskog method for the electron component and the first non-vanishing approximation for heavy components. In general we have considered 12 electronically excited states; at high pressure we have reduced the number of excited states to 7 to take into account the decrease of the number of the electronically excited states with increasing pressure. Cut-off criterium We include in the electronic partition function all the elctronic states with radius less than the average distance between particles where

Transport Coefficients II: Translational Thermal Conductivity Heavy particlesElectrons Chapman-Enskog method Second order approximationThird order approximation

Transport Coefficients III: Reactive Thermal Conductivity For a gas constituted by chemical species and  independent reactions the reactive thermal conductivity can be calculated by means of Butler and Brokaw theory where

Transport Coefficients IV: Viscosity Viscosity has been calculated by means of the first approximation of the Chapman-Enskog method The H ij are expressed as a function of temperature, collision integrals and molecular weight of the species, while  i represents the molar fraction of the i-th component

Transport Coefficients V: Electrical Conductivity The electrical conductivity has been calculated by using the third approximation of the Chapman-Enskog method where The presence of electronically excited states can affect  e through the collisions e-H(n)

Results I: Diagonal Approximation (Viscosity) The small relative error indicate a sort of compensation between diagonal and off- diagonal terms. The differences calculated with the two sets of collision integrals are very higher. Including off-diagonal terms Neglecting off-diagonal terms This point can indicate the importance of using higher Chapman - Enskog approximations for the calculation of the viscosity in the presence of excited states. “usual” collision integrals: solid line “abnormal” collision integrals: dashed line

Results II: Heavy Particles Translational Thermal Conductivity The ratio between the translational thermal conductivity values calculated with the “abnormal” cross sections ( h a ) and the corresponding results calculated with the “usual” cross sections ( h u ) is reported as a function of temperature for different pressures. The small effect observed at 1 atm is due to the compensation effect between diagonal and off- diagonal terms in the whole representation of the translational thermal conductivity of the heavy components. This compensation disappears at high pressure as a result of the shifting of the ionization equilibrium

Results III: Electron Translational Thermal Conductivity In this case the presence of excited states affects only the interactions of electrons with H(n). The figure reports the ratio e a / e u calculated with the two sets of collision integrals as a function of temperature at different pressures. Again we observe that the excited states increase their influence with increasing the pressure. At high pressure the deviation decreases when considering only 7 excited states.

Results IV: Reactive Thermal Conductivity This contribution has been extensively analyzed in a previous paper *. The main conclusions follow the trend illustrated for  h and  e in this work. * M.Capitelli, P.Celiberto, C.Gorse, A.Laricchiuta, P.Minelli, D.Pagano, Phys. Rev. E 66,016403/1 (2002)

Results V: Viscosity The results for viscosity are in line with those discussed for the heavy particles translational contribution to the total thermal conductivity. The viscosity values calculated with the “abnormal” cross sections (  (a) ) are less than the corresponding results calculated with the “usual” cross sections (  (u) ). The relative error decreases when, at high pressure, seven excited states are included in the calculation.

Results VI: Electrical Conductivity The trend of the electrical conductivity follows that one described for the contribution of electrons to the total thermal conductivity.

Results VII: Number of levels in partition function Figures show the ratio between transport coefficients calculated by using “abnormal” and “usual” collision integrals at pressure of 1000 atm, as a function of temperature and for different number of atomic levels.

At high pressure the number of excited states decrease. However increasing pressure, the ionization equilibrium is shifted to higher temperatures so that the concentration of low lying excited states can be sufficient to affect the transport properties We can see that in this case already the first excited state (n=2) affects the results.

Conclusions The results reported indicate a strong dependence of the transport properties of LTE H 2 plasmas on the presence of electronically excited states. This conclusion is reached when comparing the transport coefficients calculated with the two sets of collisison integrals. Our results emphasize the importance of these states in affecting the transport coefficients specially at high pressure. But at high pressure a question is open concerning the number of excited states to be included in the calculation of the partition function. Another point to be discussed is the accuracy of the present calculations with respect to the Chapman-Enskog approximation used in the present work. These approximations are very accurate when neglecting the presence of excited states. In the presence of excited states with their “abnormal” transport cross sections these approximations could not be sufficient.

1.Excitation and de-excitation by electron impact 2.Ionization by electron impact and three body recombination 3.Spontaneous emission and absorption 4.Radiative recombination Collisional-Radiative Model for Atomic Plasma

Rate Equations Quasi-Stationary Solution(QSS) Stationary solution Time-dependent solution

QSS approximation The ground state density changes like the density of the charged particles and the excited states are in an instantaneous ionization-recombination equilibrium with the free electrons differential equation for the ground state system of linear equation for excited levels The system of equations is linear in X 1 X j (j>1) can be calculated when X 1, n e, T e are given

CR for Atomic Nitrogen Plasma: Energy-level Model

CR for Atomic Nitrogen Plasma with QSS X i vs level energy T e =5800 KT e =11600 K T e =17400 K

Time-dependent solution CR rate equations Boltzmann equation Rate coefficients for electron impact processes f(  ) electron energy distribution function  (  ) cross section v(  ) electron velocity level population plasma composition f(  ) rate coefficients

Atomic Hydrogen Plasma P=100 Torr, T g =30000 K, T e (t=0)=1000 K  H + =  e - =10 -8,  H =1,  H (1)=1,  H (i)=0 i>1 density (cm -3 ) vs time(s) X i = n i /n i SB vs time(s) T e vs time(s)

 H (i)/g(i) vs E i eedf(eV -3/2 ) vs E

reservoir exit throat Non-Equilibrium Kinetics in High Enthalpy Nozzle Flows

coupling state-to-state kinetics with fluid dynamic models - Numerical aspects - Coupling with kinetics - Chemical kinetics - Vibrational kinetics - Metastable state kinetics - Boltzmann equation - Coupling with chemical kinetics - EM fields contribution