Sébastien Depraz

Marie-Yvonne Y Perrin

Anouar Soufiani
email: anouar.soufiani@em2c.ecp.fr

Infrared emission spectroscopy of CO 2 at high temperature. Part I: Experimental setup and source characterization

Keywords:

come

Infrared emission spectroscopy of CO2 at high temperature. Part I: Experimental setup and source characterization

temperature measurement, tube effects, CO 2 infrared radiation

Introduction

The knowledge of very high temperature absorption and emission spectra of CO 2 remains a challenge for radiative transfer in applications such as combustion and planetary atmospheric entry. Simulations for heated CO 2 -N 2 mixtures at equilibrium have shown for instance that the IR emission of the CO 2 molecule remains predominant at temperatures as high as 4000 K [START_REF] Babou | Spectroscopic data for the prediction of radiative transfer in CO 2 -N 2 plasmas[END_REF].

Several experimental studies have been devoted to this molecule during the last fifty years. We describe in the following high temperature measurements and try to classify them according to the confinement of the absorbing or emitting gas in tight cells, flows inside heated cells or the use of combustion products.

Tight cells enable accurate measurements with well controlled temperature, pressure and mixture composition. The achievable temperatures are however limited to about 1000 K due to the soldering of the optical windows with the cell or the thermal tolerance of O-rings. Low spectral resolution absorption measurements with tight cells and grating spectrometers were undertaken for instance by Levi Di Leon and Taine [START_REF] Di Leon | Infrared absorption by gas mixtures in the 300-850 K temperature range I-4.3 µm and 2.7 µm CO 2 spectra[END_REF] at temperatures up to 850 K. Diode laser absorption experiments enable very high spectral resolution but only cover limited spectral ranges. Rosenmann et al [START_REF] Rosenmann | Diode-laser measurements of CO 2 line intensities at high temperature in the 4.3 µm region[END_REF][START_REF] Rosenmann | Diode laser measurements of CO 2 hot band line intensities at high temperature near 4.3 µm[END_REF] have performed such measurements in the 4.3 µm region up to 800 K, and Mihalcea et al [START_REF] Mihalcea | Diode-laser absorption measurements of CO 2 near 2.0 µm at elevated temperatures[END_REF] investigated the 2 µm region in order to determine the most suitable absorption lines to be used as a diagnostic tool in combustion applications. André et al [START_REF] André | FTIR measurements of 12 C 16 O 2 line positions and intensities at high temperature in the 3700-3750 cm -1 spectral region[END_REF] have measured CO 2 absorption spectra in the range 3700-3750 cm -1 using high resolution Fourier transform infrared (FTIR) spectroscopy and optical paths of about 2 m up to 700 K. Lower spectral resolution FTIR measurements were performed by Parker et al [START_REF] Parker | High temperature absorption measurements and modeling of CO 2 for the 12 micron window region[END_REF] around 12 µm up to 800 K and by Medvecz et al [START_REF] Medvecz | Experimental determination of line strengths for selected carbon-monoxide and carbon-dioxyde absorption lines at temperatures between 295 and 1250[END_REF] who measured absorption spectra of CO/CO 2 /N 2 mixtures up to 1250 K in the 4.3 µm region. Phillips [START_REF] Phillips | Band-model parameters for 4.3 µm CO 2 band in the 300-1000 K temperature region[END_REF] has also performed FTIR measurements in the 4.3 µm region up to 1000 K with 0.06 cm -1 spectral resolution and has derived band model parameters from the experimental data.

Higher temperatures can be obtained using heated leaky cells inside furnaces and flowing gas mixtures. This technique enabled absorption measurements with grating spectrometers up to 1500 K in the 2.7 and 4.3 µm regions at different pressures and optical paths [START_REF] Burch | Infrared radiation emitted by hot gases and its transmission through synthetic atmospheres[END_REF][START_REF] Darell | Laboratory investigation of the absorption and emission of infrared radiation[END_REF]. Such absorption cells were also used in Refs. [START_REF] Clausen | FTIR transmission emission spectroscopy of gases at high temperatures: Experimental set-up and analytical procedures[END_REF][START_REF] Fleckl | Experimental verification of gas spectra calculated for high temperatures using the HITRAN/HITEMP database[END_REF] in association with moderate resolution FTIR spectroscopy up to 1273 and 1373 K respectively.

In more recent studies, Modest and Bharadwaj [START_REF] Modest | Medium resolution transmission measurements of CO 2 at high temperature[END_REF] and Bharadwaj and Modest [START_REF] Bharadwaj | Medium resolution transmission measurements of CO 2 at high temperature-an update[END_REF] developed a drop tube technique which enabled FTIR measurements at 4 cm -1 spectral resolution up to 1550 K. They studied the 2, 2.7 and 4.3 µm regions and compared their measurements with several spectroscopic data bases and statistical narrow-band data. The temperature gradients close to the tube end had apparently no significant effects.

Experiments through combustion products enable to reach higher temperatures but are also affected by uncertainties resulting from the partial knowledge of temperature and composition distributions, and boundary effects. Different authors have attempted to produce quasi two dimensional flames to reduce boundary layer effects [START_REF] Ludwig | Measurements of the curves-of-growth of hot water vapor[END_REF][START_REF] Soufiani | Sensitivity of temperature and concentration measurements in hot gases from FTIR emission spectroscopy[END_REF].

Pioneering studies were conducted by Ferriso and Ferriso et al [START_REF] Ferriso | High temperature spectral absorption of 4.3-micron CO 2 band[END_REF][START_REF] Ferriso | Spectral emissivity measurements of 4.3 µm CO 2 band between 2650 and 3000 degrees K[END_REF] who used a supersonic kerosene-oxygen burner at stoichiometric composition, and a Foelsch-type nozzle to avoid shock waves and to produce a quasi uniform cone at the exit of the divergent. The mixture temperature was measured by the emission/absorption technique and the composition was calculated at chemical equilibrium. Absorption spectra were recorded at 4 cm -1 spectral resolution in the 4.3 µm region for temperatures up to 3000 K. A similar study was conducted by Coppalle and Vervich [START_REF] Coppalle | Spectral emissivity of the 4.3-µm CO 2 band at high temperature[END_REF] who measured low spectral resolution absorption spectra of a methane-oxygen flame at temperatures up to 2900 K in the 4.3 µm. Diode laser absorption measurements of a C 2 H 4 -air flame have been carried out by Webber et al [START_REF] Webber | In situ combustion measurements of CO, CO 2 , H 2 O and temperature using diode laser absorption sensors[END_REF] in the 2 µm region. Selected spectral lines were used in this last study to measure the temperature and CO, CO 2 and H 2 O molar fractions.

The aim of the present study is to measure emission spectra from gases produced by a microwave discharge which enables to reach temperatures as high as 6000 K. Particular attention is given to CO 2 emission in the 2.7 and 4.3 µm regions at temperatures never reached before and at high to medium spectral resolution. The microwave discharge through a CO 2 flow at atmospheric pressure produces a mixture close to local thermodynamic equilibrium (LTE) containing mainly CO 2 , CO, O 2 , C 2 and atomic O species.

The emission of CO in the ground electronic state in the range 3800-4400 cm -1 allows accurate determination of the radial temperature distribution in the hot parts of the plasma. Local CO 2 emission coefficients or line of sight integrated emission intensities can thus be determined and compared to predictions from spectroscopic databases to study their validity at elevated temperatures. We present in this part the experimental setup (section 2) and analyze the effects of confinement tubes on plasma emission spectra (section 3). The plasma characterization using CO emission is then presented in section 4.

Experimental setup

[Figure 1 Two different MPT systems were used. The first one (3 kW maximum power) has been originally designed by LITMAS company and the second one (6 kW) by SAIREM company. For both systems, microwaves are generated by a magnetron at 2.45 GHz frequency, pass through an isolator, and then a wave-guide. A triple-head impedance tuner, positioned between the isolator and the plasma cavity, allows to match the impedance in order to decrease the amount of reflected power to less than 1% of the forwarded power.

Adjustment of this tuner is necessary from one experiment to the other as the impedance of the plasma cavity changes with the plasma operating conditions.

The plasma cavity, designed for the purpose of this study, is composed of a WR 340-type aluminum rectangular waveguide, hollowed across its axis to hold a vertical confinement tube of 45 mm outer diameter and 2.5 mm wall thickness. The position of the tube within the plasma cavity was calculated so that the electric field of the microwaves is maximum at the center of the tube without plasma. The tangential injection of the gas at the bottom of the cavity produces a swirled gas flow which stabilizes the plasma in the tube center and avoids tube overheating. The plasma produced at atmospheric pressure by this type of devices was analyzed in a previous work using visible and UV emission spectroscopy, and the medium was shown to be very close to thermal and chemical local equilibrium, at least in the central region of the plasma [START_REF] Babou | Spectroscopic study of microwave plasmas of CO 2 and CO 2 -N 2 mixtures at atmospheric pressure[END_REF][START_REF] Depraz | An experimental setup for IR gas radiative property measurements at temperatures up to 6000 K: application to CO 2 and CO 2 -N 2 mixtures[END_REF]. The first attempt to measure CO 2 emission in the 2.7 µm region is also described in Ref. [START_REF] Depraz | An experimental setup for IR gas radiative property measurements at temperatures up to 6000 K: application to CO 2 and CO 2 -N 2 mixtures[END_REF] where the optical path was not under vacuum.

The line of sight integrated emission spectra are recorded in the IR range using a high resolution Fourier transform infrared spectrometer (DA8, BOMEM company). The spectrometer is equipped with a CaF 2 beam splitter and a liquid nitrogen cooled InSb detector which allow measurements in the near and mid-infrared parts of the spectrum (above 1800 cm -1 ) at a resolution as high as 4.10 -3 cm -1 . The spectrometer is under vacuum to prevent absorption by CO 2 and H 2 O in ambient air, and the optical path between the MPT and the spectrometer is also under vacuum as shown in Fig. 1. A nitrogen purged cavity is placed around the confinement tube in order to avoid residual absorption outside the tube. This last cavity is made of a refractory material which can operate at high temperature and do not perturb microwave propagation. The spectrometer optical axis is at a height h above the exit of the waveguide; studying the plasma at different horizontal sections defined by the height h allows us to explore different tem-perature profiles and temperature levels since the plasma is cooled down by heat transfer phenomena as h increases.

The emission spectra are calibrated using the emission of a blackbody radiation source (Landcal company R1200P) at about 900 K with ± 3 K uncertainty. The spatial resolution in the radial direction is determined by the diameter of the pinhole placed in front of the InSb detector (0.5 mm) and from the CaF 2 collection lens magnification factor (0.25). This leads to a spatial resolution of about 2 mm. The emission spectra are recorded at different radial positions by moving the microwave plasma torch with a step of 0.5 mm.

The procedure to extract local emission coefficients from line of sight integrated emission intensities, when the medium can be assumed to be optically thin, is based on the classical Abel inversion. The axisymmetry of the calibrated spectra has been checked and a spatial Butterworth-based filter is applied to the intensity curves for each wavenumber. Details on data processing are given in Appendix A. This leads to local absolute emission coefficients (in W.m -3 .sr -1 .(cm -1 ) -1 ) in the optically thin regions.

Refraction, reflection, absorption and emission by the tube

Plasma confinement inside a quartz or sapphire tube is necessary for the stability of the flow and to avoid diffusion and convection mixing with the outer gaseous medium. However, the tube affects direct measurement of the intensity emitted by the plasma through refraction, reflection, absorption and emission phenomena. The measured temperature on the external wall of the tube reaches 800 K and its own emission is not negligible. The aim of this section is to analyze these effects and to determine an accurate procedure for measurement correction when possible. We assume in the following that the plasma and the tube are perfectly axisymmetric and that tube interfaces are smooth so that refraction and reflection can be described by Fresnel's equations. The tube has an internal diameter R 1 =20 mm and an external one R 2 =22.5 mm.

Refraction effects

We denote by y the distance between the optical axis of the spectrometer and the line parallel to this axis and crossing the tube center (see Fig. 2), and by y ′ the distance between the tube axis (point O) and the optical path inside the tube that is conjugated with the spectrometer axis through the tube.

[Figure 2 about here.] Elementary geometrical calculations, using Fresnel's equations, are reported in Appendix B and lead simply to y ′ = y. This important result shows that refraction by quartz or sapphire tubes leads to a deviation of the optical axis but that the optical path associated to the spectrometer axis is not affected by this phenomenon as long as the system can be considered as axisymmetric. Thus, there is no correction to perform in Abel inversion due to refraction. The tube acts however as a slightly divergent lens but this effect, which could affect the spatial resolution of the measurements is neglected in the following.

Tube reflection, absorption and emission effects

Optical properties of quartz and sapphire

We have measured the transmissivity curves of half-tubes made of quartz and sapphire at room temperature, and the emission curves of the tubes in the plasma operating conditions, just after the extinction of the plasma after each experiment. The last measurements were done at 4 cm -1 spectral resolution to enable fast recording of the spectra before a significant decrease of tube temperature. Examples of these measurements are shown on Fig. 3 for transmission and on Fig. 4 for emission. On this last figure, plasma emission is also presented for comparison. Figure 3 shows also the theoretical normal transmittance, accounting for reflection, but not for absorption, as computed from

T σ = 1 -ρ σ (θ = 0) 1 + ρ σ (θ = 0) , and
ρ σ (θ = 0) = n σ -1 n σ + 1 2 , (1)
where ρ σ (θ = 0) is the interface normal relectivity, and from the values of the real part of the refractive index n σ , calculated from the Sellmeier-type dispersion relation

n 2 λ -1 = B 1 λ 2 λ 2 -C 1 + B 2 λ 2 λ 2 -C 2 + B 3 λ 2 λ 2 -C 3 , (2)
with the constants given in Table C.1.

Figure 3 shows first a good agreement between theoretical and measured transmittances in the transparency regions of quartz and sapphire. It shows also that absorption by the tubes occurs below 3700 cm -1 for quartz and 2500 cm -1 for sapphire. This is in agreement with the spectra of the absorption index that may be found in Refs. [START_REF] Dombrovsky | Use of Mie theory to analyze experimental data to identify infrared properties of fused quartz containing bubbles[END_REF] and [START_REF] Thomas | Infrared transmission properties of sapphire, spinel, yttria, and AlON as a function of temperature and frequency[END_REF] for instance, and with the present tube emission measurements shown on Fig. 4. Thus, the effects of tube reflection, absorption and emission must be taken into account to correct plasma emission measurements.

[Table 1 about here.]

[Figure 3 about here.]

[Figure 4 about here.] Moreover, it is well known that sapphire absorption coefficient increases with temperature (see e.g. Ref. [START_REF] Thomas | Infrared transmission properties of sapphire, spinel, yttria, and AlON as a function of temperature and frequency[END_REF]). We have thus measured the absorption spectrum of the whole sapphire tube just after plasma extinction, using a globar as emission source, and extracted from this measurement, and from the theoretical values of the refractive index, the transmissivity of a halftube. The result shown on Fig. 5 indicates clearly the strong variation of tube transmissivity with tube temperature. Half-tube transmissivity in the operating conditions will be considered for the analysis of the 4.3 µm CO 2 region.

[Figure 5 about here.]

Relation between measured intensity, tube emitted intensity, and plasma emitted intensity

In order to establish the relation between these different intensities, we assume a perfectly axisymmetric geometry, which yields, in association with Fresnel's equations, remarkable symmetry properties for multiple reflection phenomena. Calculation details are given in Appendix C and the general result, accounting for both components of radiation polarization, is

I tr = a I e p + b I e t , (3)
where I tr is the measured intensity, I e p and I e t are the intensities emitted by the plasma column and the tube column, and the constants a and b are given by:

a = τ t (1 -ρ t )(1 -ρ p ) 1 -τ p [ρ p + ρ t τ 2 t (1 -ρ p ) 2 ] , b = (1 -ρ t ) [1 -τ p ρ p + τ t τ p (1 -ρ p ) 2 ] 1 -τ p [ρ p + ρ t τ 2 t (1 -ρ p ) 2 ] , (4)
with the notations given in Appendix C.

Different scenarios can occur depending on the spectral region and the nature of the confinement tube. If the plasma is optically thin (τ p ≃ 1), which is the case for CO 2 emission in the 2.7 µm region and for CO overtone bands characterized by ∆v=2, and if the tube can be assumed to be transparent (τ t ≃ 1, I e t ≃ 0, ρ = 0), Eqs. 3 and 4 reduce to:

I tr = (1 -ρ t ) 1 -ρ t (1 -ρ p ) I e p . (5)
The measured intensity I tr is then equal, to the second order in the reflectivity ρ t or ρ p , to the intensity emitted by the plasma column. This is easily explained by the fact that the losses by reflection at the tube interfaces are compensated by the intensity emitted by plasma columns symmetrical to the considered column, and reflected by the tube towards the detector. Figure 6 shows that for τ p ≃ 1 and τ t ≃ 1, the factor a = 1 2 (a + a ⊥ ) remains very close to 1 and the difference is smaller than 0.005 for y < 19 mm. This is the reason why the calibration of the intensity with a blackbody source is done without tube.

For an optically thin plasma (τ p ≃ 1) but a partially absorbing tube (τ t = 1) as for the quartz tube near 3700 cm -1 , Figure 6 shows that the factor a is very close to τ t (differences limited to 2% for y < 18 mm) and a simple division of the measured intensity by the half-tube transmissivity yields an accurate correction. When the plasma can no longer be considered as optically thin (case of CO 2 in the 4.3 µm region), reflections by the tube are not compensated by other internal reflections and Fig. 6 shows that, for τ p =0.1 for instance, the error introduced by the simple division of I tr by τ t leads to an underestimation of the plasma intensity by about 13% at y=0. Measurements of CO 2 emission in the 4.3 µm region should then be corrected to account for self-absorption. But as the plasma spectral optical thickness is not determined experimentally, this phenomenon will introduce an important source of uncertainty in this spectral region.

[Figure 6 about here.] Finally, Fig. 4 and Eqs. 3 and 4 show that emission from the tube should be subtracted from the measured intensity to adjust the signal background. This simple subtraction is rigorous when the plasma is optically thin (τ p ≃ 1) since the measured tube emission is the same with and without plasma, as long as this measurement is carried out very quickly after plasma extinction.

A further correction is required again in the 4.3 µm region where the plasma is not optically thin. Figure 4 shows however that sapphire emission remains small in comparison with the maximum level of CO 2 emission.

To summarize the above analysis of tube effects, we have shown that refraction effects can be neglected in data processing and put forward the following corrections concerning reflection, emission and absorption:

• When the plasma is optically relatively thin (2.7 µm region and CO emission in the range 3800 < σ < 4400 cm -1 ), reflections of the plasma emitted intensity are compensated by other internal reflections and the intensity calibration must be carried out without tube.

• In the same spectral regions, no further corrections are required for sapphire tubes, while the measured intensity should be divided by the half-tube measured transmissivity τ t for quartz tubes.

• tube emission is systematically subtracted from the measured intensity.

The main source of uncertainty due to the confinement tubes results from reflection effects and high plasma optical thickness in the 4.3 µm region.

These phenomena may lead to an underestimation of the measured plasma intensity by about 15%.

Source characterization

At a given height h above the exit of the microwave cavity, the temperature profile T (r, h) was determined from CO emission in the 4000-4400 cm -1 region. As shown on Fig. 7, CO overtone bands corresponding to ∆v = v ′v" = 2 are well isolated and the emission signal obviously contains significant contributions from hot bands, up to v ′ =8 or 9, which insures a good sensitivity to the temperature. After calibration and Abel inversion, several techniques can be used to extract the temperature from the local emission spectrum (see e.g. [START_REF] Soufiani | Sensitivity of temperature and concentration measurements in hot gases from FTIR emission spectroscopy[END_REF]). The Boltzmann method, or the ratio of two selected line intensities is very sensitive to the background continuum, to line overlapping, and to line shapes and broadening parameters. We use here a least square adjustment between normalized theoretical spectra and experimental ones to determine the local temperature. In order to avoid the knowledge of CO partial pressure, the measured local emission spectrum, deduced from Abel inversion, is normalized by dividing it by its maximum value, and then compared to normalized theoretical spectra calculated at different temperatures. Comparisons of the absolute levels of the spectra are used afterwards to check the chemical equilibrium hypothesis.

[Figure 7 about here.] Theoretical line by line calculations of CO spectra, carried out with 0.01 cm -1 spectral resolution, are based on line intensities calculated by Chackerian and Tipping [START_REF] Chackerian | Vibrational-Rotational and Rotational Intensities for CO Isotopes[END_REF] and line positions deduced from the Dunham coefficients given in Ref. [START_REF] Guelachvili | Dunham coefficients for seven isotopic species of CO[END_REF]. Collisional line broadening coefficients are taken from Ref. [START_REF] Hartmann | Accurate calculated tabulations of CO line broadening by H 2 O, N 2 , O 2 and CO 2 in the 200-3000 K temperature range[END_REF] for different collision partners. The simple harmonic oscillator (with w=2143.27 cm -1 ) and rigid rotor model is used for the calculation of the internal partition sum of CO, and is found in very good agreement with direct summation calculations (see e.g. [START_REF] Babou | High-temperature and nonequilibrium partition function and thermodynamic data of diatomic molecules[END_REF][START_REF] Capitelli | Tables of internal partition functions and thermodynamic properties of high-temperature Mars-atmosphere species from 50K to 50000[END_REF]) with differences less than 3% for temperatures up to 7000 K. We have also checked that line positions and intensities agree well (typically within less than 0.003 cm -1 for positions and 2% for intensities) with the HITRAN data [START_REF] Rothman | The HITRAN 2008 molecular spectroscopic database[END_REF] for the cold lines belonging to this last data base, and with HITEMP data [START_REF] Rothman | the high-temperature molecular spectroscopic database[END_REF] with the same accuracy.

In order to avoid important differences between theoretical and experimental spectra resulting from different line shapes and from FTIR apparatus function, both spectra are convolved with a triangular function of 0.8 cm -1 spectral resolution at the triangle base before the least square adjustment.

Examples of measured and theoretical CO overtone spectra, calculated at the temperature yielding the least square difference, are shown in Fig. 8 for a height above the plasma cavity h=6 mm and for different radii. The plasma is confined in a quartz tube in this example. The agreement is very good in the central regions with differences limited to a few percents of the maximum signal. At the center of the plasma, the measured temperature is about 5475 K and band heads up to v ′ =8 are clearly identified. Higher differences are observed in the outer regions of the plasma where CO emission becomes weak. However, the difference between calculated and measured normalized emission coefficients do not present a spectral structure, correlated with the emission coefficient itself, which indicates that the medium is very close to local thermodynamic equilibrium with a single temperature Boltzmann distribution of vibrational and rotational CO levels.

[Figure 8 about here.]

In order to check chemical equilibrium, we compare in the following the absolute levels of the measured and calculated emission coefficients. The latter require however the knowledge of CO molar fraction. Assuming local chemical and thermodynamic equilibrium at the measured temperature, the local composition of the medium is computed by solving the law of mass action for dissociation (Guldberg-Waage equation) and for ionization (Saha equation) reactions, together with electrical neutrality, the perfect gas law and the total conservation of the nuclei [START_REF] Babou | Spectroscopic data for the prediction of radiative transfer in CO 2 -N 2 plasmas[END_REF][START_REF] Babou | High-temperature and nonequilibrium partition function and thermodynamic data of diatomic molecules[END_REF]. The retained chemical species for CO 2 plasmas are CO 2 , CO, O 2 , C 2 , C, O, C + , O + , CO + and e -. Figure 9 shows the computed composition in the temperature range of interest. CO molar fraction becomes less than 0.01 for T <1600 K and CO emission is expected to be very low on the periphery of the plasma.

[Figure 9 about here.] Comparisons between absolute levels of local emission coefficients are given in Fig. 10 for the same experiment as in Fig. 8. Here again, a good agreement is observed in the central region of the plasma indicating the reliability of temperature measurement and of chemical equilibrium assumption. For r=10 and r=15 mm, the calculated emission level is smaller than the experimental one with typical differences of about 15% and 30% respectively. As displayed in Fig. 9, CO molar fraction is very sensitive to the temperature in the 2000-2500 K range. If we assume chemical equilibrium and use the absolute emission spectrum to adjust the local temperature, we find for instance 2567 K instead of 2525 K for r=10 mm and 2150 K instead of 2075 K for r=15 mm. The spectral structure of the residual in this last adjustment becomes however more pronounced. So, a plausible explanation of the differences observed on the borders of the plasma is a departure from chemical equilibrium. CO concentration is however very small in these regions and does not affect significantly CO 2 concentration. Indeed, the experimental CO emission spectrum is about one hundred times smaller at r=15 mm than at the plasma center.

[Figure 10 about here.] Figure 11 displays typical temperature profiles determined from the normalized and from the absolute CO emission spectra with a confinement tube in quartz and at the height h=6 mm. At this height, the very high temperature narrow central region shown in the figure is confirmed visually by the high luminosity of this region in the visible range. As CO concentration becomes too small on the boundaries (r >15 mm), its emission signal becomes very weak and unusable to determine the temperature. The temperature profile is thus extrapolated in these regions by using the last reliable measurement point using CO spectra and wall temperature. The latter has been measured at the external tube wall by a contact K-type thermocouple and was found in the range 800±50 K for the heights h considered here. Two types of extrapolations where assumed, a simple linear one, and a smooth one preserving the temperature derivative at the last measurement points. These extrapolations are also shown on Fig. 11. The uncertainty on the temperature profile, especially in the borders of the plasma, will constitute the most important source of uncertainties in this experimental study and will be accounted for when comparing experimental and theoretical CO 2 spectra.

[Figure 11 CO concentration vanishes at the periphery. A good agreement is obtained between these intensities with, as may be expected, a better agreement for the external chords when the temperature profile deduced from absolute levels of CO emission coefficient is used.

[Figure 12 about here.]

Summary

An original setup was developed to measure IR emission spectra of gas mixtures at temperatures up to 6000 K. It includes a microwave discharge in a flowing gas, CO 2 in the present study, and a high resolution FTIR spectrometer under vacuum.

The optical effects due to the confinement of the gas mixture inside quartz or sapphire tubes were discussed in detail. The analysis and tube transmittance measurements showed that refraction effects may be neglected and that reflection on smooth tube surfaces and absorption by the tube can be accurately taken into account in the spectral regions where the gas mixture is optically thin. Significant uncertainties remain however when the medium is optically thick like for CO 2 in the 4.3 µm region.

For CO 2 plasma, the temperature radial distribution was determined from CO emission in the 4000-4400 cm -1 spectral range. A least square adjustment between normalized calculated and Abel inverted experimental emission coefficients yielded accurate determination in the central regions of the mixture. In the same regions, the absolute levels of CO local emission confirmed that the atmospheric plasma is at local chemical and thermodynamic equilibrium. However, due to very low CO concentration in the relatively cold outer regions, the temperature profile had to be extrapolated in theses regions. The uncertainties resulting from this extrapolation will be accounted for in the comparisons between theoretical and experimental CO 2 emission spectra presented in the companion paper. and then the angle θ 2 from

R 2 2 = R 2 1 + AB 2 + 2R 1 × AB × cosθ 2 . (B.3)
Finally, the distance y ′ is given by

y ′ R 1 = sinθ 4 = n sinθ 2 . (B.4)
Using the trigonometric relation sin 2 θ + cos 2 θ = 1 for the angles θ 2 et θ ′ 2 , we find respectively

y ′ nR 1 2 + R 2 2 -R 2 1 -AB 2 2R 1 AB 2 = 1, (B.5) y nR 2 2 + R 2 2 -R 2 1 + AB 2 2R 2 AB 2 = 1. (B.6)
The combination of these two relations leads to y ′ = y.

Appendix C. Relation between the measured intensity and the intensities emitted by the plasma and by the tube

We adopt the following notations (see also Figure 2) and omit subscripts to indicate parallel and perpendicular polarizations which are understood throughout the following notations and relations:

-I tr : Intensity transmitted to the detector (measured quantity),

-I e p : Intensity emitted by a plasma column at the distance y from the center (quantity to be compared to theoretical data), -I e t : Intensity emitted by a tube column. As the tube is relatively optically thin, this intensity is assumed to be the same for tube emission in the inward and outward directions, -I i int : Incident intensity on the internal tube interface, -I i ext : Incident intensity on the external tube interface -I l int : Intensity leaving the internal tube interface, -I l ext : Intensity leaving the external tube interface, ρ p : Specular reflectivity at tube interface for radiation propagating from the plasma to the tube, ρ t : Specular reflectivity at tube interface for radiation propagating from the tube to the outer medium, τ t : Transmissivity of the tube (not accounting for reflections), τ p : Transmissivity of the plasma column.

The two reflectivities ρ p and ρ t are calculated, for each polarization, from the general expressions related to radiation travelling from medium i with real refractive index n i to medium t with index n t , and neglecting the imaginary part of the index in comparison with its real part:

ρ = n i cosθ t -n t cosθ i n i cosθ t + n t cosθ i 2 , ρ ⊥ = n i cosθ i -n t cos θ t n i cosθ i + n t cosθ t 2 . (C.1)
The different angles used in this expression are calculated from the expressions given in Appendix B for each distance y to the plasma axis. The axisymmetry of the problem and Eq. C.1 show that ρ p , for instance, is also equal to the specular reflectivity at tube interface for radiation propagating from the tube to the plasma, which is required to express I l int . With these notations, one can write the expressions of each intensity and for each polarization:

I tr = (1 -ρ t )I i ext (C.
I l int = (1 -ρ p )τ t I l ext + (1 -ρ p )I e t + ρ p I i int (C.5) I l ext = ρ t I i ext . (C.6)
Equation C.2 means that the measured intensity I tr is simply equal to the incident intensity on the outer tube interface multiplied by the interface transmission factor (1ρ t ). In Eq. C.3, the incident intensity on the outer tube interface I i ext is written as the sum of the incident intensity on the inner tube interface, multiplied by a transmission factor, and the intensity emitted by the tube. Equation C.4 reflects the fact that the incident intensity on the inner tube interface is equal to the intensity leaving the inner interface, multiplied by plasma column transmissivity, plus the intensity emitted by this plasma column. In Eq. C.5, the intensity leaving the inner interface is expressed as the sum of three terms, a directly reflected intensity on this interface, a transmitted intensity coming from tube emission, and the transmitted part of the intensity leaving the outer interface. Finally, since no significant radiation comes from outside, the intensity leaving the outer interface is simply expressed in Eq. C.6 as the reflected part of the incident intensity on the same interface.

The solution of this system leads to the expression of I tr :

I tr = a I e p + b I e t , (C.7) with a = τ t (1 -ρ t )(1 -ρ p ) 1 -τ p [ρ p + ρ t τ 2 t (1 -ρ p ) 2 ] , b = (1 -ρ t ) [1 -τ p ρ p + τ t τ p (1 -ρ p ) 2 ] 1 -τ p [ρ p + ρ t τ 2 t (1 -ρ p ) 2 ]
.

(C.8) The data for quartz and sapphire are from Ref. [START_REF] Malitson | Interspecimen comparison of the refractive index of fused silica[END_REF] and [START_REF] Malitson | Refraction and dispersion of synthetic sapphire[END_REF] respectively and are widely adopted in the literature (see e.g. [START_REF] Dombrovsky | Thermal radiation in disperse systems: An engineering approach[END_REF]).

I

  about here.] The experimental setup shown on Fig. 1 is aimed at producing and analyzing stable plasma flows at high temperature, atmospheric pressure and local thermodynamic equilibrium for different gas mixtures. It is composed of a microwave plasma torch (MPT) which is coupled to an IR Fourier transform spectrometer.

  about here.] Finally, we have compared the measured CO intensities (in absolute levels and without Abel inversion) to the calculated intensities obtained by solving the radiative transfer equation along a ray at different distances y from the plasma axis. The results are plotted in Fig. 12 using both temperature profiles deduced from normalized and absolute CO emission coefficient. The intensities are plotted on Fig. 12 for the central chord (y=0) and for y=5 and y=10 mm. They are not sensitive to temperature extrapolation since

  2)I i ext = (1ρ p )τ t I i int +

  tr appears then to be a linear combination of the intensities emitted by the plasma and by the confinement tube with the constants a and b depending on the different reflectivities and transmissivities. Equations C.7 and C.8 are in fact written for each polarization: I tr = a I e p + b I e t , (C.9)I tr ⊥ = a ⊥ I e ⊥ p + b ⊥ I e ⊥ t ,(C.10)where the coefficients a , b , a ⊥ and b ⊥ are calculated from Eqs. C.8 with reflectivities ρ p and ρ t depending on the polarization state according to Eqs.C.1. As the emission by the plasma and the tube are not polarized (I e p = I e ⊥ p =1 2 I e p and I e t = I e ⊥ t = 1 2 I e t ), the total transmitted intensity is given by:I tr = I tr + I tr ⊥ b ⊥ )I e t . (C.11) 12 Comparison between absolute levels of chord integrated intensities for the ∆v = 2 CO bands for h=6 mm and different distances to the axis. The temperature profile deduced from normalized CO emission coefficient is used in the left graphs and the one deduced from absolute emission coefficients is used in the right graphs. . . . . . . . . . . . . . . . . . . . . . . . . 44

Figure 1 :Figure 2 :

 12 Figure 1: Experimental setup.

Figure 3 :

 3 Figure 3: Measured and calculated transmissivities of quartz and sapphire half-tubes (2.5 mm width, room temperature). The calculations only take into account reflections at the interfaces while the measurements include also absorption effects.

Figure 4 :

 4 Figure 4: Calibrated spectra of total emission and emission by the tube alone ((a) sapphire and (b) quartz) recorded just after plasma extinction at y=0 and h=6 mm. Only the spectral region above 2800 cm -1 is processed for quartz tubes.

Figure 5 :

 5 Figure 5: Transmissivity (ignoring reflection at the interfaces) of a half-tube of sapphire (2.5 mm width) at room temperature and in the plasma operating conditions.

Figure 6 :

 6 Figure 6: Multiplicative factor a in the relation I tr = aI e p + bI e t for optically thin (left) and thick (right) plasma.

Figure 7 :

 7 Figure 7: Calculated absorption coefficient of pure CO at atmospheric pressure and 3500 K. The 8 first graphs show the contributions of the vibrational bands 2-0 to 9-7 ; the 9th graph shows total absorption spectrum and the 10th graph an example of measured emission spectrum.

Figure 8 :Figure 9 :

 89 Figure 8: Comparison between normalized experimental and adjusted theoretical CO overtone emission coefficients for h=6 mm and different radii.

Figure 10 :

 10 Figure 10: Comparison in absolute levels of local emission coefficients of ∆v = 2 CO bands for h=6 mm and different radii.

Figure 12 :

 12 Figure 12: Comparison between absolute levels of chord integrated intensities for the ∆v = 2 CO bands for h=6 mm and different distances to the axis. The temperature profile deduced from normalized CO emission coefficient is used in the left graphs and the one deduced from absolute emission coefficients is used in the right graphs.

B

  Table C.1: Constants for the dispersion relation, Eq. 2 where the wavelength λ is in µm.

		1	B 2	B 3	C 1	C 2	C 3
	Quartz 0.69617 0.40794 0.89748 0.0046792 0.013512 97.934
	Saphir	1.024	1.058	5.281	0.003776 0.01225 321.4

Acknowledgments

We acknowledge the French National Research Agency (ANR) for partial financial support through the project Rayhen and the funding from the European Community Seventh Framework Program under grant agreement 242311.

Appendix A. Data processing and Abel inversion

We briefly describe in this appendix the data processing algorithm that enables to transform line of sight integrated intensities of CO overtone emission to local emission coefficients through Abel inversion. This procedure was first developed in Ref. [START_REF] Deron | Rayonnement thermique des plasmas d'air et d'argon : Modélisation des propriétés radiative et étude expérimentale[END_REF] where more details may be found .

If the plasma can be considered as axisymmetric and optically thin for a given wavenumber σ, the measured intensity I σ (y) at a distance y from the plasma center is given by:

where J σ (r) is the local emission coefficient (in W.m -3 .sr -1 .(cm -1 ) -1 ) at radius r and R p is the plasma total radius. The emission coefficient is retrieved by the Abel inversion formula:

In practice, measurements are carried out with a spatial step ∆y = 0.5 cm on both sides of the plasma center. The signal is then made symmetrical by finding the local extremum corresponding to the plasma center and by averaging the values at equal distances from this center. A Butterworth filter is then applied to eliminate experimental noise, especially at the plasma borders. The transfer function of this filter is given by:

where the cutoff frequency ω c was taken equal to 0.4 and the filter order n was fixed to 2. The axisymmetric and filtered signal was then Abel-inverted 20 for each wavenumber using a cubic spline method. A local third order polynamial is used to extrapolate the signal between each couple of adjacent measurement points using the continuity of the polynomials and of their derivatives. Abel inversion reduces then to the calculations of integrals of the form:

These integrals have the analytical expressions:

Appendix B. Refraction effects

The different angles θ 1 to θ 4 , involved in the following are defined in Fig. 2.

If we assume that the indices of refraction of the plasma and of the outer medium are equal to 1, and denote by n the refraction index of the tube we have

The distance AB between the points of the optical path intersecting the tube interfaces can be deduced from

List of Figures

List of Tables

C. [START_REF] Babou | Spectroscopic data for the prediction of radiative transfer in CO 2 -N 2 plasmas[END_REF] Constants for the dispersion relation, Eq. 2 where the wavelength λ is in µm. The data for quartz and sapphire are from Ref. [START_REF] Malitson | Interspecimen comparison of the refractive index of fused silica[END_REF] and [START_REF] Malitson | Refraction and dispersion of synthetic sapphire[END_REF] respectively and are widely adopted in the literature (see e.g. [START_REF] Dombrovsky | Thermal radiation in disperse systems: An engineering approach[END_REF]). . . . . . . . . . . . . . . . . . . . . . 46


S Depraz

M Y Perrin

Ph Rivière

A Soufiani
email: anouar.soufiani@em2c.ecp.fr

Infrared emission spectroscopy of CO 2 at high temperature. Part II: Experimental results and comparisons with spectroscopic databases

Keywords: FTIR emission spectroscopy, microwave plasma source, spectroscopic databases, high temperature CO 2 infrared radiation

Measurements of CO 2 emission spectra at high temperature in the 2.7 µm and 4.3 µm regions are presented and compared to predictions from the spectroscopic databases, CDSD-4000 (Tashkun SA, Perevalov VI, JQSRT 2011; 112:1403-10 ) and HITELOR (Scutaru D et al, JQSRT 1994; 52:765-81). The measurements near 2.7 µm show that CDSD-4000 provides better agreement with experimental data, especially in the band wing corresponding to very hot band emission. In the 4.3 µm region, experimental intensities are generally lower than theoretical predictions but the measurements are more affected by the use of sapphire confinement tubes. The two spectroscopic databases provide closer results than in the 2.7 µm region although CDSD-4000 yields here also better agreement with experimental data in band wing. Analysis of the two spectroscopic databases is carried out in terms of vibrational and rotational energy cutoff and total band emissivities.

Introduction

As discussed in the literature review presented in Part I [START_REF] Depraz | Infrared emission spectroscopy of CO 2 at high temperature. Part I: Experimental setup and source characterization[END_REF], there is a lack of very high temperature measurements of CO 2 IR radiative properties. These measurements are useful to study the accuracy of spectroscopic databases when the latter are used in the extrapolation range of energy levels. We present in this second part FTIR emission measurements using a microwave post-discharge in CO 2 flow as emission source. The measurements are compared to predictions from spectroscopic databases including the recently published CDSD-4000 database [START_REF] Tashkun | CDSD-4000: High-resolution, hightemperature carbon dioxide spectroscopic databank[END_REF] and the older HITELOR database [START_REF] Scutaru | Approximate intensities of CO 2 hot bands at 2.7, 4.3 and 12 µm for high temperature and medium resolution applications[END_REF].

Among the attempts to build approximate high temperature CO 2 spectroscopic databases, the HITELOR database has been developed for applications at medium to low spectral resolution and temperatures up to about 2500 K, corresponding to combustion applications. The HITRAN-1992 database was extended in the 2.7 µm, 4.3 µm and 12µm regions for the two major isotopologues 12 C 16 O 2 and 13 C 16 O 2 . The intensities of the most important hot bands were calculated by pragmatic extrapolation of band intensities of the same type identified in HITRAN-1992. The extrapolations concerned the ν 3 transitions in the 4.3 µm region, the ν 1 + ν 3 transitions near 2.7 µm, and the ν 3 -ν 1 transitions in the 12 µm region. Vibrational energies were calculated by diagonalizing the vibrational Hamiltonian for each Fermi polyad with the spectroscopic constants of Ref. [START_REF] Chedin | Carbon-dioxide molecule -potential, spectroscopic, and molecular constants from its infrared spectrum[END_REF]. Hot lines were generated for lower energy levels characterized by v 1 ≤ 10, v 2 ≤ 10 and v 3 ≤ 5 and up to a rotational quantum number J=200, and were added to HITRAN-1992 lines.

In the Carbon Dioxide Spectroscopic Databank (CDSD), line positions and intensities are calculated from effective Hamiltonian and effective dipole moment operators. The parameters of these operators were adjusted to fit experimentally observed line positions [START_REF] Tashkun | Global fitting of 12 C 16 O 2 vibrational-rotational line positions using the effective Hamiltonian approach[END_REF] and intensities [START_REF] Tashkun | Global fit of 12 C 16 O 2 vibrational-rotational line intensities using the effective operator approach[END_REF]. Three versions, CDSD-296 [START_REF] Tashkun | CDSD-296, the carbon dioxide spectroscopic databank: version for atmospheric applications[END_REF], CDSD-1000 [START_REF] Tashkun | CDSD-1000, the high-temperaure carbon dioxide spectroscopic databank[END_REF] and CDSD-4000 [START_REF] Tashkun | CDSD-4000: High-resolution, hightemperature carbon dioxide spectroscopic databank[END_REF] were successively developed and correspond to the temperature at which a cutoff in intensities of 1.0 10 -27 cm molecule -1 was employed. The latest version CDSD-4000 contains about 628 million lines among which 574 million lines are for the main isotopologue. The maximum energy considered for this isotopologue is 43522 cm -1 and is close to the dissociation limit of the ground electronic state of CO 2 .

In parallel to the development and improvement of the atmospheric HI-TRAN databank, a high temperature analogous databank, called HITEMP, has been developed for fewer molecules (H 2 O, CO 2 , CO, NO and OH) but encompassing many more bands and transitions than HITRAN. The 1995 version of the HITEMP databank [START_REF] Rothman | HI-TRAN, HAWKS and HITEMP high-temperature molecular data-base[END_REF] was based for CO 2 on a mix between HITRAN lines and DND calculations [START_REF] Wattson | Direct numerical diagonalization: Wave of the future[END_REF]. HITEMP-1995 predictions of CO 2 radiative properties were accurate in the 2 µm, 2.7 µm and 4.3 µm regions for temperatures up to about 1000 K but it was shown that they overestimate absorption at higher temperatures (see e.g. [START_REF] Bharadwaj | Medium resolution transmission measurements of CO 2 at high temperature-an update[END_REF]). The latest version, HITEMP-2010 [START_REF] Rothman | the high-temperature molecular spectroscopic database[END_REF] is mainly constituted for CO 2 of an updated version of CDSD-1000 databank including an improvement of 13 CO 2 data and lines of three less abundant isotopologues.

For the very high temperatures considered in this study, we will compare the experimental results with the predictions from HITELOR and CDSD-4000 databanks. In addition to line positions and intensities provided in these databanks, one needs the internal partition sum (discussed in Sec. 4) and line shapes to perform line by line calculations to be compared to the measurements. We use here Voigt line shapes with the collisional broadening parameters of Ref. [START_REF] Rosenmann | Accurate calculated tabulations of IR and Raman CO 2 line broadening by CO 2 , H 2 O, N 2 , O 2 in the 300-2400-K Temperature range[END_REF] for HITELOR and the semi-classical calculations of Ref. [START_REF] Bykov | Semi-empiric approach to the calculation of H 2 O and CO 2 line broadening and shifting[END_REF], also provided in CDSD-4000, when this databank is considered.

A line wing cutoff of 50 cm -1 was found sufficient and was used in all the present calculations.

The paper is organized as follows. A discussion on experimental uncertainties is provided in Sec.2 and, then, the experimental results and the comparisons with theoretical predictions are presented for the 2.7 µm and 4.3 µm regions in Sec. 3. An intrinsic comparison between the two spectroscopic databases is finally performed and discussed in terms of energy cutoff and total band emissivities.

Experimental uncertainties

The first type of uncertainties is directly related to the measurement of CO 2 emission intensities and include confinement tube effects discussed in Part I [START_REF] Depraz | Infrared emission spectroscopy of CO 2 at high temperature. Part I: Experimental setup and source characterization[END_REF], measurement calibration with a blackbody source, and other causes due to the experimental setup. But as one of the goals of this study is to compare the measured spectra to calculated ones, there is a second type of uncertainties resulting from the temperature distribution and CO and CO 2 molar fractions to be used in the calculations. As discussed in the first part, CO emission bands ∆v=2 are used to determine the local temperature and CO molar fraction, but this emission becomes very weak on the periphery of the plasma and the temperature field has to be extrapolated in these regions.

Concerning the experimental uncertainties, we have analyzed the following points:

• the spatial resolution of the measurements is about 2 mm and spatial convolution may affect the results. A numerical simulation of this effect has been undertaken. The temperature profile measured with sapphire tubes at the height h=6 mm was used as input to compute CO line of sight integrated emission spectra. These spectra were then spatially convolved and Abel inverted. The temperature profile obtained from these convolved spectra was then compared to the initial one. This numerical simulation showed that temperature differences as high as 250 K are observed at the plasma center but, for the region where CO 2 emission is significant (T <5000 K and r >1.5 mm), the spatial convolution induces temperature changes smaller than about 30 K.

• The uncertainty on the temperature of the blackbody source used to calibrate the emission signals is estimated to be about 5 K, although the instrument specifications indicate less than 3 K. A temperature difference of 5 K around 900 K induces variations of the Planck's function of about 1.6% at 2000 cm -1 and 3.3% at 3700 cm -1 . Measurements of blackbody emission at different temperatures showed also that the ratio of Planck's functions was recovered within ±1.5%.

• The use of a fluoride window, normal to the optical path, at the end of the tube connecting the plasma cavity to the spectrometer, induced some Fabry-Pérot interferences with the highest amplitude of ±2.2% at the lowest wavenumbers (4.3 µm region). This configuration was nevertheless preferred to an inclined window since it allowed to minimize residual absorption by ambient air.

• The analysis of confinement tube effects shows that accurate corrections may be carried out except for the measurements in the 4.3 µm region where the plasma is optically relatively thick. Due to reflections by the sapphire tube interfaces, the measured intensity may be underestimated by up to 15%.

For the uncertainties holding on the temperature and concentration profiles, there are also several issues that have been analyzed:

• In the central part of the plasma, the good agreement between the temperature deduced from the normalized and absolute CO spectra indicates accurate determination of both the temperature and CO molar fraction. Carbon atom conservation in the mixture leads then to an accurate determination of CO 2 concentration as well. The major uncertainty comes from temperature extrapolation on the plasma periphery.

Although it does not significantly affect CO 2 concentration which is practically equal to 1 below 2000 K, this extrapolation will induce important uncertainties on the comparisons, especially for the peripheral chords, and in the spectral regions where the cold boundaries absorb significantly the radiation emitted by the central regions.

• The reliability of the CO spectroscopic data used in the least square adjustment of the temperature was checked by comparing different databases (see Part I). The uncertainties on CO line intensities were estimated to about ±2% for temperatures up to 7000 K.

• CO data analysis, including symmetrization, spatial filtering and Abel inversion introduces also uncertainties on temperature and concentration determination. We have checked by numerical simulation that uncertainties resulting from data processing remain of the same order than the uncertainties on the initial emission signals.

It appears from the above discussion that, except from the problem of sapphire tube effects in the 4.3 µm region, the major source of uncertainties is due to temperature and CO 2 concentration profiles, and particularly on the periphery of the plasma. Although this uncertainty concerns the calculated emission spectra, it is transformed, for figure legibility reasons, into error bars holding on the measured intensities. The temperature profile used in the calculations is the one deduced from the adjustment of the normalized CO spectra with linear extrapolation in the peripheral regions, and the height of the error bars is taken equal to the maximum difference between the spectra calculated using the four different temperature profiles, i.e. based on the normalized or absolute CO emission and using a linear or smooth extrapolation, with HITELOR database.

Experimental results and comparison with theoretical data

Measurements were carried out for several heights h above the plasma cavity and several displacements y from the central chord with a step δy=0.5 mm.

We present in the following the results obtained for h=6 and h=20 mm. The experimental operating conditions for the experiments using quartz and sapphire confinement tubes are given in Table 1.

[Table 1

about here.]

The temperature profiles, as determined from CO normalized spectra and linear interpolations on the boundaries, are shown on Fig. 1 for the experiments with both quartz and sapphire confinement tubes. CO 2 and CO molar fractions deduced from local chemical equilibrium calculations are also shown on this figure. A very high temperature central region appears for h=6 mm and, for h=20 mm, diffusion, convection, and probably radiation processes lead to smoother temperature profiles. Due to the very low CO 2 concentration in the central region r < 2 mm at h=6 mm, CO 2 emission from this region corresponding to temperatures greater than typically 4500 K will not significantly contribute to the emission signal. The temperatures are slightly higher with the sapphire confinement tube due to the higher microwave injected power.

[Figure 1 about here.]

The 2.7 µm region

Figures 2 presents the experimental absolute measured and calculated

intensities at h=6 and 20 mm, and for the chords y=0 (central chord), y=5, 10 and 15 mm with quartz confinement tubes in the 2.7 µm region. Figure 3 shows the results for the same positions obtained with a sapphire confinement tube. For legibility reasons, the experimental and calculated intensities are presented at low spectral resolution. A simple convolution with a 10 cm -1 wide rectangular function was used. The general trends and conclusions from quartz and sapphire experiments are similar.

It is first observed that the error bars are greater for the peripheral chords corresponding to low temperature levels and to the highest uncertainties on the temperature. For the central chords, the error bars are also more important in the spectral regions corresponding to "cold" lines, e.g. 3400-3750 cm -1 typically.

Though the error bars are very important for y=15 mm for instance, a good agreement between experimental and theoretical results is observed. This shows that the linear extrapolation of the temperature profile is a reasonable approximation and that both spectroscopic databases are reliable in the 2.7 µm regions for temperatures below typically 2000 K. Indeed, the results from HITELOR and CDSD-4000 are very close for y=10 and y=15 mm.

The results for central chords are more interesting since error bars are smaller and the contribution of high temperature regions to emission becomes predominant. The agreement between experimental data and theoretical predictions is obviously better with CDSD-4000, especially in the band wing (3000-3400 cm -1 ) and for h=6 mm corresponding to higher temperature levels. Results from HITELOR underestimate clearly the emission in this spectral region, indicating that many "hot" lines are missing in this database.

Note that for some chords, even CDSD-4000 results seem to be slightly outside the experimental error bars (Fig. 2, h=20 mm, y=10 mm and Fig. 3, h=6 mm, y=10 mm). In fact the error bars should be slightly larger since they only take into account the uncertainties on the temperature profile and not the purely experimental uncertainties.

[Figure 2 about here.]

[Figure 3 about here.]

The 4.3 µm region

The measurements in the 4.3 µm region were carried out using sapphire confinement tubes since quartz absorbs and emits strongly in this region.

The comparisons between experimental data and theoretical ones are shown on Fig. 4 for h=6 and 20 mm, and for the chords y=0 and y=5 mm. The errors bars are more important than those obtained in the 2.7 µm region, especially in the 2150-2400 cm -1 spectral range. This is due to the high optical thickness of the medium in this spectral range and to the important absorption by the cold layers of radiation emitted in hotter central regions.

It should be noted that the difference between the results from the two spectroscopic databases are less pronounced than those observed in the 2.7 µm region.

However, the experimental results obviously underestimate the theoretical ones, except for the band wing where a good agreement is observed between experiments and CDSD-4000 calculations. The underestimation is more important at h=6 mm than at h=20 mm. It may be partly explained be sapphire tube effects discussed in detail in the first part [START_REF] Depraz | Infrared emission spectroscopy of CO 2 at high temperature. Part I: Experimental setup and source characterization[END_REF]. For the high optical thicknesses of the 4.3 µm band, reflection by tube interfaces of the radiation propagating towards the spectrometer is not compensated by other symmetrical internal reflections. The resulting underestimation was estimated to about 13% for τ p =0.1. We recall that, in the optically thin band wing, a good agreement between CDSD4000 and experimental data is achieved. A rigorous treatment of tube effects requires the knowledge of plasma transmissivity at high spectral resolution, but absorption measurements in a wide spectral range are not an easy task at the very high temperatures considered in this study. Another solution would be the use of CO 2 diluted with a transparent species such as argon but this would result in a reduction of CO concentration and thus of the emission signal used to determine the temperature profile.

A pragmatic correction was attempted here. The experimental spectrum was simply multiplied by a function f (σ) equal to 1.13 in the 2200-2400 cm -1 range and varying linearly between 1 and 1.13 in the range 1800-2200 cm -1 .

The result of this pragmatic correction, given in Fig. 5, shows that experimental and calculation predictions are much closer though some discrepancies remain at the height h=6 mm.

[Figure 4 In order to check the contributions of different regions in the plasma to the measured intensity on the central chord, partial intensities emanating from a given depth r (with -20< r <20 mm) were computed using CDSD-4000. Figure 6 shows the emission of the column (r, R) of the central chord with R=20 mm being the abscissa at the exit of the emitting column. The temperature profiles measured with a sapphire tube, shown on Fig. 1, were used for the simulation. It appears that the contribution of the region 8< r <12 mm, corresponding to temperatures between 2500 K and 3000 K, is predominant in band wings (below 2100 cm -1 in the 4.3 µm region and 3300 cm -1 in the 2.7 µm region). The contribution of the region 4< r <8 mm, corresponding to temperatures typically between 3000 and 4000 K remains however significant in these band wings. In contrast, the contribution of the central region 0< r <4 mm is very weak, though visible for h=20 mm, due to the very small CO 2 concentrations. We may then conclude that the present experimental conditions allow us to check the validity of the spectroscopic databases up to roughly 4000 K.

[Figure 6 about here.]

High resolution comparisons

The experimental emission intensities presented above were measured with a Full-Width at Half-Maximum (FWHM) spectral resolution of about 0.10 cm -1 . They were then convolved at 10 cm -1 and compared to numerical predictions at the same resolution. Measurements with 0.016 cm -1 FWHM resolution were also carried out for the central chord after each experimental series. They allow us to study the quality of the spectroscopic databases in terms of extrapolated line positions. An estimation of the average FWHM CO 2 Voigt line profile at 3000 K leads to 0.03 cm -1 in the 4.3 µm region and 0.034 cm -1 in the 2.7 µm region. The high resolution spectral measurements are thus sufficient to resolve line profiles although there remains a small convolution effect with the apparatus function.

The comparisons between experimental spectra and CDSD-4000 predictions are shown on Fig. 7 for selected spectral regions and for the experiment at h=20 cm with the sapphire tube. This experiment was selected because it provided a good agreement with CDSD-4000 at low spectral resolution for both 4.3 and 2.7 µm spectral regions. Four narrow spectral ranges were selected in band heads (around 2385 and 3761 cm -1 ) and band wings (around 2118 and 3280 cm -1 ) and, in the 4.3 µm region, attention was paid to select a region free from CO emission lines. To do this, calculations with and without CO contribution were compared. The only significant CO line in the selected narrow range of the 4.3 µm band wing is shown on the figure. It appears from Fig. 7 that a very good agreement between experimental and theoretical spectra is obtained in band heads. Due to the slightly different spectral resolutions, some differences are observed for the absolute signal levels but line positions are predicted very accurately. In contrast, in the band wings where emission arises from very hot lines, the agreement is less good and the spectra are not well correlated. This disagreement shows that the positions of very hot lines are not well predicted by CDSD-4000. However, the important density and overlapping of lines are so high that this issue should have no incidence in practical applications.

[Figure 7 about here.]

Comparisons between the spectroscopic databases

The aim of this section is to provide intrinsic comparisons between HITELOR and CDSD-4000 databases and to analyze the comparisons with experimental data given above. We discuss in a first step the vibrational and rotational energy cutoff in the two databases, then the difference between the internal partition functions used in association with each database, and conclude by comparing predicted band total emissivities at different temperatures and optical paths.

Figure 8 shows the absorption coefficients calculated from the two databases at different temperatures and in the two spectral regions of interest. The high resolution calculations are convolved at 10 cm -1 spectral resolution.

The agreement between the two predictions is good at 1000 K, satisfactory at 2000 K, and deteriorates gradually as the temperature increases. In order to fully understand this behavior, we have computed the cumulated distributions of line intensities of the main isotopologue 12 C 16 O 2 as a function of vibrational, rotational and total energy of the line lower level. These distribution functions are defined by:

[Figure 8 about here.]

F(E ′′ r ) = line i/ E ′′ r i <E ′′ r S i (T ), (1)
F(E ′′ v ) = line i/ E ′′ v i <E ′′ v S i (T ), (2)
F(E ′′ tot ) = line i/ E ′′ i <E ′′ tot S i (T ), (3)
where S i (T ) is the intensity (in cm -2 atm -1 ) of the line i at temperature T and E ′′ r , E ′′ v , E ′′ tot designate respectively the rotational, vibrational and total energy of the lower level. The cumulated distribution functions are shown on Fig. 9 in two spectral regions defined precisely by the 1800-2450 cm -1 and 2800-4400 cm -1 spectral ranges. As the vibrational levels in CDSD-4000 are denoted by a polyad index defined by P = 2v 1 + v 2 + 3v 3 , we have roughly estimated E ′′ v by P ω 2 where ω 2 ≃ 672 cm -1 is the vibrational energy of the mode ν 2 . This leads to the small discontinuities shown on Fig. 9 for CDSD-4000 even for high vibrational levels. The vibrational energy for HITELOR was taken equal to the energy of the level having the smallest rotational quantum number within a given vibrational configuration.

At 2000 K, we observe reasonable agreement between the distribution functions predicted from the two databases in the two spectral regions. The total sum of line intensities predicted from HITELOR is however 15% smaller than the one predicted from CDSD-4000 in the 2.7 µm region. At higher temperatures, HITELOR leads to a higher total sum of line intensities in the 4.3 µm region, but to a significantly smaller one in the 2.7 µm region, when compared to CDSD-4000. A discontinuity in the derivative of the function F(E ′′ v ) appears for the HITELOR database around E ′′ =12700 cm -1 for the 4.3 µm region and E ′′ =11000 cm -1 for the 2.7 µm region. This discontinuity is characteristic of a premature vibrational energy cutoff (see the description given in the introduction) and appears earlier in the 2.7 µm region, which explains the better global comparisons with experimental data in the 4.3 µm region. Although the rotational cutoff criteria correspond to J=200 in HITELOR and to J=300 in CDSD-4000, no discontinuity appears in the derivative of the function F(E ′′ r ), which means that the rotational cutoff was less critical than the vibrational one.

The different internal partition functions used in association with the two spectroscopic databases constitute an additional explanation of the different band shapes observed on Fig. 8. Several high temperature partition functions of CO 2 have been reported in the literature [START_REF] Gamache | Total internal partition sums in the temperature range 70-3000 K: Atmospheric linear molecules[END_REF][START_REF] Gamache | Total internal partition sums for molecules in the terrestrial atmosphere[END_REF][START_REF] Osipov | Partition sums and dissociation energy for 12 C 16 O 2 at high temperatures[END_REF]. They are compared on Fig. 10 for the main isotopologue 12 C 16 O 2 . The partition function used in HITELOR is the one tabulated by Gamache et al [START_REF] Gamache | Total internal partition sums in the temperature range 70-3000 K: Atmospheric linear molecules[END_REF] and also used in HITRAN-1992. For CDSD-4000, Tashkun and Perevalov [START_REF] Tashkun | CDSD-4000: High-resolution, hightemperature carbon dioxide spectroscopic databank[END_REF] use direct summation with the energy levels tabulated in their database. As the partition sum is not provided with a fine temperature step in Ref. [START_REF] Tashkun | CDSD-4000: High-resolution, hightemperature carbon dioxide spectroscopic databank[END_REF], we have used in the present calculations with CDSD-4000 the simple uncoupled harmonic oscillators and rigid rotor function with a rotational constant B=0.3916 cm -1 and the vibrational wavenumbers ω 1 =1353 cm -1 , ω 2 =672 cm -1 and ω 3 =2396 cm -1 . Figure 10 shows that this simple approximation leads to an excellent agreement with the partition function of Refs. [START_REF] Tashkun | CDSD-4000: High-resolution, hightemperature carbon dioxide spectroscopic databank[END_REF][START_REF] Osipov | Partition sums and dissociation energy for 12 C 16 O 2 at high temperatures[END_REF] up to 4000K. It shows also that the partition function used in HITELOR significantly underestimates the other calculations (5.8% at 2000 K, 25.6% at 3000 K and 43.5% at 4000 K in comparison with the direct sum used in CDSD-4000). This difference explains the band shapes observed at high temperature on Fig. 8 The difference in the predicted band shapes may have important effects in practical radiation heat transfer calculations. We have computed for instance the total band emissivity defined by

ǫ t (T, l) = σ 2 σ 1 (1 -exp(-κ σ l))I 0 σ (T ) dσ σ 2 σ 1 I 0 σ (T ) dσ , ( 4
)
where l is the column length, I 0 σ (T ) is the Planck's function, and σ 1 =1800 cm -1 , σ 2 =2450 cm -1 for the 4.3 µm region, σ 1 =2800 cm -1 , σ 2 =4400 cm -1 for the 2.7 µm region. The results from HITELOR and CDSD-4000 are shown on Fig. 11 for different path lengths. As the optical thicknesses are much smaller in the 2.7 µm region than near 4.3 µm, a logarithmic scale is used in the first region. For small optical paths, good agreement between the two spectroscopic databases is observed in the 4.3 µm region whatever is the temperature. In contrast, important discrepancies are observed for high optical paths in the same spectral region due to emission saturation in band center and to the different band shapes. In the 2.7 µm region, HITELOR results systematically underestimate CDSD predictions at very high temperatures.

[Figure 11 about here.]

Conclusion

CO 2 IR emission spectra have been measured at very high temperature in the 2.7 and 4.3 µm regions using a microwave discharge in CO 2 flows and a high resolution FTIR spectrometer. Compared to tight cell measurements, further uncertainties resulted from the bad knowledge of the temperature distribution in the peripheral regions and from the use of a confinement tube, especially in the 4.3 µm region with sapphire tubes. Nevertheless, these measurements allowed us to study the validity and completeness of two spectroscopic databases, HITELOR and CDSD-4000. HITELOR database appears to be sufficient for classical combustion applications (for which it was intended during its construction), typically below 2500 K, and at low spectral resolution. At higher temperatures as those encountered for instance in atmospheric entries, it is found to be insufficient due to the low energy cutoff of vibrational levels. CDSD-4000 yields generally very good agreement with experimental data, especially in band wings, indicating the reliability and the accuracy of this spectroscopic database for very high temperature applications. High spectral resolution comparisons revealed that very hot line positions are not accurately predicted in CDSD-4000. However, this fact should have no effects in practical applications due to the very high degree of line overlapping.

) Experimental intensity HITELOR CDSD4000 h=6mm, y=0 mm 2800 3000 3200 3400 3600 3800 σ (cm -1 ) 0

) Experimental intensity HITELOR CDSD4000 h=20 mm, y=0 mm 2800 3000 3200 3400 3600 3800 σ (cm -1 ) 0 .sr -

) HITELOR CDSD4000 Experimental h=20 mm, y=5 mm

) Experimental -corrected HITELOR CDSD4000 h=20 mm, y=5 mm .(

Experimental CDSD-4000 CO

  about here.] [Figure 5 about here.]

in the 4 . 3

 43 µm region: HITELOR predictions overestimate the absorption coefficient in band center due to the underestimation of the partition function, but they underestimate absorption in band wing due to the vibrational energy cutoff. [Figure 9 about here.] [Figure 10 about here.]

Figure 1 :

 1 Figure 1: Temperature profiles deduced from CO normalized emission spectra with quartz and sapphire tubes at two heights h above the plasma cavity. CO and CO 2 molar fractions are deduced from calculations at local chemical equilibrium.

Figure 3 :

 3 Figure 3: Experimental and theoretical line of sight integrated intensities at h=6 mm (left) and h=20 mm (right) and different distances y from plasma center; sapphire tube; 2.7 µm region.

Figure 4 :

 4 Figure 4: Experimental and theoretical line of sight integrated intensities at h=6 mm (left) and h=20 mm (right) and for y=0 (central chord) and y=5 mm ; sapphire tube ; 4.3 µm region.

Figure 5 :

 5 Figure 5: Experimental (corrected) and theoretical line of sight integrated intensities at h=6 mm (left) and h=20 mm (right) and for y=0 (central chord) and y=5 mm ; sapphire tube ; 4.3 µm region.

Figure 6 :

 6 Figure 6: Partial intensities on the central chord computed with CDSD-4000 and the temperature profile obtained with a sapphire tube at h=6 mm and h=20 mm.

Figure 7 :Figure 8 :

 78 Figure 7: Comparison between high spectral resolution experimental intensities, obtained with a sapphire tube on the central chord at h=20 mm, and CDSD-4000 predictions in four selected spectral regions.

Figure 9 :

 9 Figure 9: Cumulated distribution function of 12 C 16 O 2 line intensities vs lower level energy in the two spectral regions and for different temperatures.

Acknowledgments

We acknowledge the French National Research Agency (ANR) for partial financial support through the project Rayhen and the funding from the European Community Seventh Framework Program under grant agreement 242311. We are grateful to Prof. S.A. Tashkun for providing us with CDSD-4000 databank prior to its publication.

List of Figures

Temperature and profiles deduced from CO normalized emission spectra with quartz and sapphire tubes at two heights h above the plasma cavity. CO List of Tables


O J Boxma
email: boxma@win.tue.nl

B J Prabhu
email: bjprabhu@laas.fr

Analysis of an M/G/1 queue with customer impatience and adaptive arrival process

We study an M/G/1 queue with impatience and an adaptive arrival process. The rate of the arrival process changes according to whether an incoming customer is accepted or rejected. We analyse two different models for impatience : (i) based on workload, and (ii) based on queue-length. For the workload-based model, we obtain the Laplace-Stieltjes Transform of the joint stationary workload and arrival rate process, and that of the waiting time. For the queue-length based model we obtain the analogous z-transform. These queueing models might also be useful for capturing the interaction between congestion control algorithms and queue management schemes in the Internet.

I. INTRODUCTION

Data traffic in the Internet is regulated by distributed algorithms which require each flow to adapt its sending rate according to the level of congestion in the network. This information is conveyed to a flow by the links in the network using a binary feedback instructing the flow to either increase or decrease its sending rate. The link generates these feedback signals as a function of the occupancy of its input buffer: the higher the occupancy, the higher is the level of congestion, leading to a larger number of decrease signals.

A frequently employed binary feedback signal is packet admission/rejection : admission signals an increase and rejection signals a decrease. An easy to implement example of a packet admission control policy is to reject an incoming packet if the buffer is full and to accept it otherwise. However, this policy leads to synchronization among concurrent flows and hence to inefficient utilization of the link bandwidth, cf. [START_REF] Floyd | Random early detection gateways for congestion avoidance[END_REF]. To overcome such effects, various alternative policies, including probabilistic admission control, have been proposed, cf. [START_REF] Floyd | Random early detection gateways for congestion avoidance[END_REF] and [START_REF] Pan | CHOKe: A stateless active queue management scheme for approximating fair bandwidth allocation[END_REF].

The aim of this paper is to model and analyze the above described interaction between a data source and a link that has a probabilistic packet admission control policy. The packet arrival process is modelled as a time inhomogeneous Poisson process whose intensity varies depending on whether an incoming packet is accepted or rejected. The packet sizes are assumed to be independent and identically distributed, and the link is modelled as an infinite buffer served at a fixed rate. An incoming packet is accepted to the queue with a probability that is a function of the current workload in the queue. We obtain the Laplace-Stieltjes Transform (LST) of the joint stationary workload and arrival intensity process, which then leads to the LST of the waiting time of accepted arrivals.

In addition, we also study the model in which incoming packets are accepted depending upon the number of packets in the system. For this model we give the z-transform of the joint stationary distribution of the queue length and the arrival intensity process, which then leads to the z-transform of the number in the system seen by accepted arrivals.

A. Related work

Performance analysis of congestion control and admission control policies has been a subject of several studies, though most of them have neglected the interaction between the two (cf. [START_REF] Boxma | Analysis of an M/G/1 queue with customer impatience and an adaptive arrival process[END_REF] for a detailed discussion). Due to its complexity, the analytical study of this interaction has been more or less restricted to studying the dynamics of the expected values using deterministic differential equations (with or without feedback delay), cf. [START_REF] Mishra | Fluid based analysis of a network of AQM routers supporting TCP Flows with an application to RED[END_REF] and [START_REF] Kuusela | Modeling RED with idealised TCP sources[END_REF]. The model considered in the present paper attempts to capture this interaction within a stochastic framework for a certain class of probabilistic admission control policies.

A related work in which this interaction has been studied is that of [START_REF] Van Foreest | Analysis of a feedback fluid model for heterogeneous TCP sources[END_REF] in which the authors model a TCP (for Transmission Control Protocol, the predominant congestion control protocol in the Internet) source as a fluid source whose rate varies depending upon whether the finite buffer is full or not. As opposed to their fluid model, we model the TCP source as one which emits packets at distinct epochs thereby causing jumps in the buffer content process which no longer has a continuous sample path. Another important difference is in the feedback model itself. In [START_REF] Van Foreest | Analysis of a feedback fluid model for heterogeneous TCP sources[END_REF] there is a positive feedback if and only if the buffer is not full. It is thus a model for a Drop-Tail policy, whereas we study a probabilistic feedback policy -the feedback is positive with a probability that decreases as the buffer level increases.

The present model is also strongly related to some of the existing models on impatience investigated in queueing theory, and in the following we describe this connection.

B. Connection with queueing theory

The connection between queueing theory and performance analysis of congestion control algorithms has been known since long, cf. [START_REF] Wang | Performance analysis of a feedback congestion control policy under non-negligible propagation delay[END_REF]. In the context of queueing theory, several rejection rules have been studied for the M/G/1 queue, see, e.g., [START_REF] Perry | Rejection rules in the M/G/1 queue[END_REF], but the arrival rate typically does not change with the decision to accept or reject a customer.

The model under investigation in this paper can be seen as a generalization of the M AP/G/1 queue with impatient customers which was studied by Combé in [START_REF] Combé | Impatient customers in the MAP/G/1 queue[END_REF]. In that model, the arrival process changes states at each packet arrival instant. However, the dynamics of the arrival process do not depend on whether a packet is accepted or rejected, which makes our model a generalization of the one studied in [START_REF] Combé | Impatient customers in the MAP/G/1 queue[END_REF]. Our method of analysis is similar to the one in [START_REF] Combé | Impatient customers in the MAP/G/1 queue[END_REF] in that we obtain a system of Volterra integral equations of the second kind for the joint stationary distribution of the workload and arrival rate process, from which we obtain a system of recursive equations for the LST of the joint stationary process.

A part of the present work containing the LST of the joint stationary workload and arrival rate process appeared in [START_REF] Boxma | Analysis of an M/G/1 queue with impatience and an adaptive arrival process[END_REF].

C. Organization of the paper

The rest of the paper is organized as follows. In Section II, we describe the system model and state the assumptions. In Section III, we present the analysis of the model with workload-based impatience leading to the computation of the LST of the joint stationary workload and arrival rate process. Based upon this LST, we give the LST of the waiting time of the accepted arrivals. In Section IV, we analyse the queue-length-based model and obtain the z-transform of joint stationary queue-length and arrival rate process. Based upon this z-transform, we give the z-transform of the number of customers seen in the system by accepted arrivals.

Most of the results in this paper are presented without proof. The proofs are provided in the research report [START_REF] Boxma | Analysis of an M/G/1 queue with customer impatience and an adaptive arrival process[END_REF], which also contains a numerical example.

II. MODEL DESCRIPTION

Consider a variable data rate source which generates packets at Poisson intensity λ(t) ∈ L, where L is a finite set of cardinality N . The packet sizes are assumed to be i.i.d. with distribution function B i (•), mean µ -1 i , and LST B i (•), when the data source is in state i. These packets arrive at a queue, say a router in the Internet, which admits the packets based on the following admission control policy. An incoming packet which sees a workload level of x is admitted to the queue with probability exp(-νx), and rejected otherwise. We do not model the possibility of a rejected packet re-entering the queue at a later instant. The function exp(-νx) can also be thought of as an impatience function associated with customers arriving to a server. If an incoming customer sees a higher waiting time, then it is less likely to join the queue.

For a router in the Internet the buffer occupancy in bits (the workload) is generally known to the router, and hence a workload-based impatience model is better suited for the analysis of such systems. However, in some cases the impatience probability may depend on the number of customers in the system rather than the total workload which may be unknown. For such a queue-length-based model, we shall assume that if an incoming packet sees n packets in the system, then it is admitted with probability p n , 0 < p < 1.

We shall assume that the variable data rate source is informed immediately whether a packet was admitted or rejected. In practice, there is a delay after which the source receives this information. The source reacts to the admission control policy by adapting its data rate in the following way. With state i of the source we associate a Poisson intensity λ i . The state of the source jumps from i to j with probability p ij if a packet is rejected, and with probability p * ij if a packet is accepted. Thus, the intensity of the arrival process potentially changes with each arrival to the queue. In a protocol like TCP, the state of the source will jump to a state j ≤ i if a packet is rejected and to a state j ≥ i if a packet is accepted. However, we shall not assume any particular structure for the matrices P = [p ij ] and

P * = [p * ij ].

III. IMPATIENCE BASED ON WORKLOAD

Let V i (t, x) denote the joint probability that at time t the workload is less than or equal to x and the input process is in state i. The server is assumed to work at unit rate. There are three events that can happen in a small interval [t, t + δt) : (i) there are no arrivals, in which case the workload is drained by an amount δt; (ii) an arrival occurs and is rejected, in which case the input process changes state; and (iii) an arrival occurs and is accepted, in which case the input process changes state and there is a jump in the workload process. The three terms on the RHS in the following equation correspond to the above three possible events.

V i (t + δt, x) = (1 -λ i δt)V i (t, x + δt) + j p ji λ j δt x 0 - (1 -exp(-νy))dV j (t, y) + j p * ji λ j δt x 0 - B j (x -y) exp(-νy)dV j (t, y), for x > 0, 1 ≤ i ≤ N .
From the above dynamics, we can derive the following integral equation for i = 1, . . . , N :

∂V i (t, x) ∂t = ∂V i (t, x) ∂x -λ i V i (t, x) + j p ji λ j x 0 -
(1 -exp(-νy))dV j (t, y)

+ j p * ji λ j x 0 - B j (x -y) exp(-νy)dV j (t, y). (1)
Let us now discuss the issue whether the joint steady-state distribution of workload and arrival rate process exists.

Proposition 1 (Stability). If 1) ρ max := sup i λ i µ -1 i
is finite, and 2) lim x→∞ f (x) = 0, then the joint workload and arrival rate process is stable.

In the sequel we assume that the two conditions of the above proposition hold.

Let Φ i (s) = ∞ 0 -exp(-sx)dV i (x) denote the LST of the joint distribution function, and let Φ(s) := [Φ i (s)] denote the row vector of the LST of the joint stationary distribution. Also, let V i (0) be the stationary joint probability that the workload is zero and the arrival intensity is λ i , and V (0) := [V i (0)] be the row vector of these joint probabilities. The following result relates Φ(s) to V (0).

Lemma 1. The LST of the joint distribution function Φ(s) is

given by the following infinite sum:

Φ(s) = V (0) ∞ i=0 D(s + iν)A -1 (s + iν)   i-1 j=0 C(s + jν)A -1 (s + jν)     , (2)
where the empty product is assumed to be unity, and

A(s) = sI-Λ(I-P), D(s) = sI, C(s) = Λ(P-B(s)P * ),
Λ is a diagonal matrix with λ i as its ith diagonal entry, B(s) is a diagonal matrix with B i (s) as its ith diagonal entry.

Remark 1. Instead of rejecting packets based on workload upon arrival, the router could reject packets based on the sum of the workload upon arrival and the service time of the arriving packet. For this rejection policy, the LST of the joint distribution function turns out to have the same form as (2)

where A(s) = sI -Λ(I -P), D(s) = sI, C(s) = Λ(B(ν)P -B(s + ν)P * ).

The only difference with respect to rejection based on workload upon arrival is in the matrix C.

We next proceed to determine the constants V i (0), i = 1, 2, ..., N , which will then completely characterize Φ(s).

Let γ i , i = 1, 2, ..., N , denote the ith eigenvalue of Λ(I -P), such that γ i ≤ γ j for i < j, and α i denote the corresponding right eigenvector. Since P is a stochastic matrix, the first eigenvector, α 1 , equals [1 1 . . . 1] T with eigenvalue γ 1 = 0. For the location of the other N -1 eigenvalues of Λ(I -P), we have the following result.

Lemma 2. The eigenvalues of the matrix Λ(I -P) have positive real parts.

Since A(s) = sI -Λ(I -P), A(s) is singular at the eigenvalues of Λ(I -P), i.e., det(A(s)) = 0 at s = γ i , i = 1, 2, ..., N . However, Φ(s) is analytic in the half-plane Re(s) ≥ 0, and hence the constants V i (0), i = 1, 2, ..., N , are such that the RHS of ( 2) is finite at s = γ i , i = 1, 2, ..., N .

In order to compute V (0) we shall make use of the above fact and the following representation

Φ(s)A(s) = V (0)M(s), (3)
where

M(s) =   ∞ i=0 D(s + iν)   i-1 j=0 A -1 (s + (j + 1)ν)C(s + jν)     .
Before stating the main result, we first make an assumption under which the result holds.

Assumption 1. For j ≥ 1, A(s + jν) is invertible at s = γ i , which is equivalent to the condition that γ i = γ k + jν for i = k and for every j, i.e., no two eigenvalues differ by an integer multiple of ν.

The above assumption ensures that A(s + jν) is invertible in the right-half plane for j ≥ 1. We shall later observe using numerical computations that when two eigenvalues differ by an integer multiple of ν, we can obtain the constants V i (0) by perturbing the entries of the matrix Λ(I -P).

Theorem 1. The joint probability vector V (0) is the unique solution of the following set of N linear equations:

V (0) =      (I + M(ν)A -1 (ν)Λµ -1 )α 1 M(γ 2 )α 2 . . . M(γ N )α N      -1      1 0 . . . 0      . (4)
Proof: Since A(s) = sI -Λ(I -P), every right eigenvector of Λ(I -P), α i , is also a right eigenvector of A(s) with eigenvalue (sγ i ). For i = 2, 3, ..., N , we right multiply (3) by α i and set s = γ i to get the following N -1 equations 0 = V (0)M(γ i )α i , i = 2, 3, ..., N.

(

For the final equation, we first note that M(s) is singular at s = 0. To see this, rewrite M(s) as

M(s) = D(s) + M(s + ν)A -1 (s + ν)C(s), (6)
and right multiply by α 1 . Using (1), we see that M(s)α 1 = sα 1 + M(s + ν)A -1 (s + ν)Λ(I -B(s))α 1 is equal to zero at s = 0, and that

lim s→0 M(s)α 1 s = (I + M(ν)A -1 (ν)Λµ -1 )α 1 , (7)
where µ is a diagonal matrix with µ i as its ith diagonal entry. We right multiply (3) by α 1 and use the normalization equation V(0)α 1 = 1 to obtain

1 = V (0)(I + M(ν)A -1 (ν)Λµ -1 )α 1 . (8)
Combining ( 8) and ( 5), we obtain the system of equations (4).

Let W be the waiting time of the accepted customers in steady state. The following result relates the LST of W to Φ i , which is the LST of the the joint steady-state probability that the workload is less than or equal to x and the input process is in state i. Combé [9] derived a similar result for the special case when the dynamics of the input process do not depend on whether a packet is accepted or rejected. Proposition 2.

E[exp(-sW

)] = i λ i Φ i (s + ν) i λ i Φ i (ν)
.

IV. IMPATIENCE BASED ON NUMBER OF CUSTOMERS

In this section we shall consider a discrete state-space model to study the joint behaviour of a variable data rate source and the queue-length at the buffer. As in the previous section, we assume that the variable rate source generates packets according to a Poisson process of rate λ i when it is in state i, i = 1, . . . , N . The packets arrive at a single server queue which admits the packets based on the following admission control policy. If an incoming packet sees n packets in the system, then it is admitted to the queue with probability p n , with 0 < p < 1. The background state now changes to j w.p. p * i,j . It is rejected with probability 1p n , and then the background state changes to j w.p. p i,j . Unlike in the previous section however, we shall restrict ourselves to the case when packet sizes are i.i.d. and exponentially distributed with rate µ i when the background state is i (i.e., the service speed may depend on the background state).

As in Proposition 1, we may conclude that the system is stable when p < 1. Let {q n,i }, n = 0, 1, . . . , and i = 1, . . . , N , be the steady-state probability that the system contains n customers while the background state is i. In the following, we shall obtain the z-transform of {q n,i } defined as

Q i (z) := ∞ n=0 z n q n,i , i = 1, . . . , N, (9)
and

Q(z) := [Q 1 (z) Q 2 (z) . . . Q N (z)]. (10)
The steps to obtain Q(z) closely follow the steps for obtaining Φ(s) in Section III. Let µ (resp. Λ) be a N × N diagonal matrix with µ i (resp. λ i ) as its ith diagonal element, and let q 0 := Q(0). Then, Lemma 3. The joint transform vector

Q(z) = q 0   ∞ i=0 D(p i z)A -1 (p i z)   i-1 j=0 C(p j z)A -1 (p j z)     , (11)
where the empty product is assumed to be unity, and A(z) := (z -1)µ + zΛ(I -P), C(z) := -zΛP + z 2 ΛP * , D(z) := (z -1)µ.

(12)

In order to determine q 0 we shall couple the fact that Q(z) is analytic in the unit disk {z : |z| ≤ 1}, i.e. it has no poles in the unit disk, with the fact that det(A -1 (z)) has N poles in the unit disk, and deduce that q 0 is such that the RHS of (11) should remain analytic at these N singularities. (13)

In order to determine q 0 , we make the following assumption on the eigenvalues γ i .

Assumption 2. For any pair i 1 and i 2 such that i 1 = i 2 , γ i 1 = p j γ i 2 , ∀j ∈ Z.

We now have the following result.

Theorem 2. The probabilities q 0 are the unique solution to

q 0 =      M(1)A -1 (1)α 1 M(γ 2 )α 2 . . . M(γ N )α N      -1      1 0 . . . 0      . ( 14
)
Let L be the number of customers in the system as seen by an accepted arrival in steady state. We can derive the ztransform of L by using arguments similar to those used for deriving the LST of the waiting time in the previous section. Proposition 3.

E[z

L ] = i λ i Q i (zp) i λ i Q i (p)
.

Lemma 4 .

 4 The N zeros of the polynomial det(A(z)) lie in the disk {z : |z| ≤ 1}.Let α i , i = 1, 2, ..., N denote the right eigenvectors of I + µ -1 Λ(I -P) corresponding to the eigenvalue γ i , and let j z)A -1 (p j+1 z)   D(p i z).