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A study of the convective flow as a function of external parameters in high-pressure mercury lamps

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A study of the convective flow as a function of external parameters in high-pressure mercury lamps
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  J. Phys. D: Appl. Phys.  29  (1996) 753–760. Printed in the UK A study of the convective flow as afunction of external parameters inhigh-pressure mercury lamps K Charrada † , G Zissis †  and M Stambouli ‡ †  Centre de physique des Plasmas et de leurs Applications de Toulouse,118 route de Narbonne, F-31062 Toulouse, France ‡  Ecole Normal Sup´erieur d’Enseignement Technique, 5 Avenue Taha Hussein,1008 Tunis, TunisiaReceived 26 June 1995, in final form 15 October 1995 Abstract.  This paper deals with the modelling of the convection processes inhigh-pressure mercury arcs. Temperature and velocity fields have been calculatedby using a 2D semi-implicit finite-element scheme for the solution of conservationequations relative to mass, momentum and energy. After validation, this model wasapplied to the study of the influence of arc external parameters such as mercurycharge and tube diameter on the convective processes. It was found that stablelaminar mono-cellular convection flow occurs at low values of tube diameter for acylindrical burner and/or a low mercury charge. Finally, we considered in detail theregion behind the electrodes, where an accumulation of mercury is observed. Theevaluation of this amount of mercury ‘trapped’ in these regions is of primeimportance for a good description of the distribution of mercury in the burner andfor a correct evaluation of the total discharge pressure. 1. Introduction Transport phenomena in electric arcs are responsible foreffects of considerable importance to stable operation of the device. If the convection flow is unstable, undesirablephenomena appear and several discharge characteristics areaffected. Elenbaas (1951) and Kenty (1938) interpretedsuch instabilities observed at sufficiently high pressure as atransition from laminar to turbulent convective flow at highconvective velocities. However, Zollweg (1977) believedthat the instability of mercury arcs can be explained as ahelical instability resulting from the self-magnetization of the arc.Modelling of the convection processes in high-pressuremercury arcs has been discussed in numerous papersincluding those of Zollweg (1978), Lowke (1979) andChang  et al  (1990). However, to the author’s knowledge,there is no precise quantitative description of the effectof mass transport on the high-pressure mercury lamp,neither in these papers nor in other references. Moreover,except for a short paper of Weber (1986) dealing withan experimental mapping of Hg and Xe densities in high-pressure lamps, we looked in vain, in the literature, for anaccurate verification of the amount of mercury distributedin the three different discharge zones: the regions aboveand behind respectively the lower and higher electrodes,and the inter-electrode volume. Obviously, the dischargeoccurs practically in the region between the electrodes. So,only the amount of mercury in this zone is consideredto be the active part of the total amount of mercuryintroduced into the burner. Mercury accumulated in theother regions can be considered as a mercury loss. Anyway,we cannot eliminate this loss. However, a good evaluationof the amount of mercury ‘trapped’ in these regions is of prime importance for a good description of the mercurydistribution in the burner and for a correct determination of the total discharge pressure.In the case of an atmospheric pressure discharge theflow is highly dependent on the total amount of mercury andthe current as well as on the arc tube dimensions. In fact, inhigh-pressure lamps these parameters are linked to practicalconstraints related to wall loading and arc tube voltage.However, in this paper we neglect these constraints and weconsider these quantities to be completely independent of each other. This assumption will allow us to study a largerange of mercury loadings and arc tube radii in order toevaluate their relative importance. Furthermore, we focusmuch attention on the region behind the electrodes, wherean accumulation of mercury is observed. This simulation of arc properties is performed with a 2D code. Mass, energyand momentum continuity equations are solved in order tocalculate transport flows. Numerical resolution is achievedby using a finite-element semi-implicit scheme. In order toconfirm the validity of the model adopted for this study, acomparison is made with measurements and calculations of other authors. 0022-3727/96/030753+08$19.50 c   1996 IOP Publishing Ltd  753  K Charrada  et al  2. Description of the model 2.1. Basic assumptions We assume an axially symmetric arc in local thermody-namic equilibrium (LTE) and at steady state. Under suchassumptions, all discharge properties can be deduced fromthe temperature distribution. Thus, the required materialfunctions, namely thermal and electric conductivity, spe-cific heat of arc plasma and net emission coefficient, areassumed to be fully described in terms of the local tem-perature only. It is also assumed that the plasma flow islaminar and that the electrodes are of cylindrical shape. Inthis work all phenomena at the electrode surface and elec-trode regions are omitted. Thus our model results can beconsidered to be valid a few mean free paths distant fromthe electrodes. We also neglect the viscous energy dissipa-tion and we suppose that the electric field is purely axial.We assume that the contribution of magnetic force is neg-ligible. 2.2. The governing equations Under these conditions, the positive column plasmais governed by usual balance equations concerningmass, radial momentum, axial momentum and energyconservation. These equations in cylindrical coordinates( r  and  z ) are as follows.For mass conservation1 r∂∂r(rρv r ) + ∂∂z(ρv z ) = 0 .  (1)For radial momentum conservation v r ∂v r ∂r + v z ∂v r ∂z =− 1 ρ∂p∂r + 1 ρ  1 r∂∂r  rη∂v r ∂r  + ∂∂z  η∂v r ∂z  − 2 ηv r r 2  .  (2)For axial momentum conservation v r ∂v z ∂r + v z ∂v z ∂z =− 1 ρ∂p∂z − g + 1 ρ  1 r∂∂r  rη∂v z ∂r  + ∂∂z  η∂v z ∂z  .  (3)For energy conservation ρc p  v r ∂T ∂r + v z ∂T ∂z  = σE 2 − U  +  1 r∂∂r  rκ∂T ∂r  + ∂∂z  κ∂T ∂z  .  (4)One also has the ideal gas law p = RM ρT   (5)and Ohm’s law I   = 2 πEq    R w 0 rσ   d r.  (6)The basic variables defined by these equationsare density  ρ , axial velocity  v z , radial velocity  v r ,temperature  T   and pressure  p . The plasma material Figure 1.  Boundary conditions. functions are viscosity  η , specific heat  c p , net radiativeemission  U  , electrical conductivity  σ   and thermalconductivity  κ . All these material functions are supposedto be functions of temperature. Other quantities of theseequations are electric field  E , gravity  g , tube radius  R w ,electric current  I  , elementary charge  q , atomic mass  M  and the ideal gas constant  R .We note that the contribution of the energy transferdue to the velocities of electron flow was neglected in theenergy balance equation (4). The effect of this term in thecentral region is small, because in this region temperaturegradients are relatively small. It does have an effect inthe region of the arc near the cathode, where temperaturegradients are relatively large. According to Lowke  et al (1992), because this term gives a cooling effect for thisregion of the arc where they expect a heating effect due tothe electron temperature being higher than the neutral gastemperature, it is better to omit the term.Electric conductivity, thermal conductivity and viscos-ity included in this model are calculated by using the firstapproximation of the gas kinetic theory as developed byHirchfelder  et al  (1954) assuming a Maxwellian shape forthe electron energy distribution function and the Lennard-Jones interatomic potential. The value corresponding to amonatomic ideal gas is used for  c p  (Chase  et al  1986). Fi-nally, the net emission coefficient is calculated accordingto Stormberg and Sch¨afer (1983). This coefficient includesUV and visible lines as well as continuum emission fromthe plasma. It is established for a parabolic radial tem-perature profile. In this work we used an interpolation todetermine the net emission coefficient for a non-parabolicradial profile. 2.3. Boundary conditions The boundary conditions are taken to be similar to thoseof Lowke (1979). They are summarized with reference tofigure 1, in table 1.The experimental value,  T  w , corresponding to a 400 Wcommercial mercury discharge is taken for the walltemperature (equal to 1000 K). We also suppose that thisvalue does not vary much with power supply. This is 754  Convective flow in high-pressure mercury lamps Table 1.  Boundary conditions.AH CDEF FG and BC HG and AB v  r   v  r   = 0  v  r   = 0  v  r   = 0  v  r   = 0 v  z  ∂ v  z  ∂ r   = 0  v  z   = 0  v  z   = 0  v  z   = 0 T   ∂ T  ∂ r   = 0  T   =  T  w   T   =  T  ( z  )  T   =  T  elc  Table 2.  Discharge characteristics used by Zollweg andKenty.Parameter Zollweg KentyArc tube length (mm) 90  ≃ 170Inter-electrode length (mm) 70 155Internal diameter (mm) 18 33.3Electrode length (mm) 10  ≃ 7 . 5Electrode diameter (mm) 2  ≃ 5Pressure (atm) 2.89 1.2Current (A) 3.0 2.9Hg loading (mg cm − 1 ) 5.72 11.5 confirmed by using an optical pyrometer and measuring thewall temperature of a large number of lamps (it is foundthat wall temperature is almost constant within a limit of  T  w  = ± 200 K. Moreover, the wall temperature is notconstant along the wall; it is higher at the top than at thebottom. An accurate calculation of this temperature needsa resolution of the energy balance equation near the tubewall. Note that, in spite of the simple appearance of thisequation, a rigorous evaluation of this term may be verydifficult. For electrode temperature ( T  elc ) we have taken avalue of 2000 K. 2.4. The numerical procedure The partial differential equations (1)–(4) are solved by usinga finite-element scheme based on rectangular structuredelements and a variable step grid. The code has beenchecked for numerical diffusion effects; in all casestested no significant systematic error was detected. Thecalculation procedure, described in more detail elsewhere(Charrada 1995), is outlined as follows:(i) rectangular grid generation,(ii) initial arbitrary values of   T,v r ,v z  and  p  are selectedthroughout the whole region,(iii) all material functions are evaluated for each node,(iv) for a given current the electric field value iscalculated from Ohm’s law (equation (6)),(v) radial and axial components of convective velocityand pressure are obtained from equations (1)–(3),(vi) new values for temperature are obtained by usingequation (4) and(vii) the procedure is repeated from step (iii) untilconvergence. Figure 2.  ( a  ) A comparison between calculated (full line)and measured (symbols) temperature radial profiles.( b  ) The calculated axial velocity distribution on the arc axis;symbols denote results from Zollweg’s calculations. 3. Results and discussion 3.1. Model validation In order to confirm the validity of the model, wecompared our calculations with measurements given byother authors. Here we only show the comparisonbetween our calculations and the results of Zollweg (1978)and Kenty (1938) in the case of high-pressure mercurydischarges (characteristics given in table 2).Figure 2( a ) shows the experimental data and ourcalculation results concerning the temperature profile forthe middle cross section of the positive column. We notethe good agreement between our calculation results andexperimental results. Figure 2( b ) gives our results for axial 755  K Charrada  et al  Figure 3.  The calculated temperature field for two differentamounts of mercury: ( a  )  m  Hg   = 10 mg and( b  )  m  Hg   = 200 mg. ( c  ) The calculated map of the velocityvectors in the discharge vessel. Thick arrows illustratequalitatively the convection pattern. Note that the arrowgrey level is in relation to the gas temperature (the blackerthe arrow the hotter the gas). velocity distribution on the axis. Zollweg’s calculations(Zollweg 1978) are also included in the same plot. Theagreement between these two sets of results is found tobe satisfactory. The small difference between them canbe explained in terms of differences between numericalmethods and different approaches used for the evaluationof the plasma material functions.As reported by Kenty, the upward convective velocityin the centre of the middle plane of the arc tube is of the order of 40 cm s − 1 ; the calculations of Chang andDakin (1991) confirms this value. In the case of a similardischarge we obtain from our numerical code a value of 38 cm s − 1 , which is in acceptable agreement with the abovevalues. This result is also in good agreement with theempirical relation  v z  (cm s − 1 ) =  5 m 0 . 85 (mg cm − 1 ) givenby Fohl (1975). 3.2. Study of the influence of the total amount of mercury In this study the DC current value is maintained at 3.2 A.The burner has an electrode spacing of 7.2 cm with a totallength of 9.2 cm. The arc tube diameter is 1.85 cm. The1 cm long electrodes have a radius of 0.1 cm. Thesecharacteristics of the electrodes have been taken the samefor all cases studied in this paper. The amount of mercuryin the burner is taken in the range 10–200 mg, varying insteps of 10 mg.The calculated temperature distributions for the twoextreme amount of mercury (10 and 200 mg) are shown infigures 3( a ) and ( b ) respectively. Constriction at the lowerelectrode increases when the amount of mercury rises. Wealso remark that increasing the total mercury mass in thedischarge leads to the heating of the zone behind the higherelectrode. This is explained by the fact that the gas on thetube axis is heated because it is at the core of the arc wherethe gas near the wall is cooled by conduction. The densitydifference between the gas at the core and that at the arcperiphery establishes an upwind flow in the centre and adownwind flow adjacent to the wall. Thus, the hot gascoming out of the arc channel heats the region behind theupper electrode. Figure 3( c ) illustrates this convective flowin the discharge vessel. The arc constriction in the lowerelectrode region is caused by the cold gas from the arcperiphery coming back into the hot central channel.The radial thermal fluxes in the median plane forincreasing mercury amounts from 10 to 190 mg (by stepsof 20 mg) are plotted in figure 4. One can see that,when the amount of mercury increases, the radial thermalflux decreases in the core of the arc and rises in theperiphery. This phenomenon is due to the arc broadeningwith increasing total amount of mercury in the burner. Thecorresponding thermal conduction loss as a function of theamount of mercury is shown in figure 5. We note that theaxial thermal conduction can be neglected (Charrada 1995).Convective transport of energy to the tube wall wasfound to be insignificant compared to other kinds of losses,for the amounts of mercury studied here. Figure 6 showsthe variation of the power loss by convection versus thesquare of the amount of mercury. It is clearly shown thatthis loss is not important even for 200 mg of mercury.Nevertheless, convective flows in arcs are responsible forconsiderably important effects concerning operating devicestability. 756  Convective flow in high-pressure mercury lamps Figure 4.  The radial component of thermal flux versus theamount of mercury in the burner; the amount of mercurywas varied from 10 mg (curve B) to 190 mg (curve A) bysteps of 20 mg. Figure 5.  Energy losses due to radial conduction as afunction of the amount of mercury. As shown previously in figure 3( b ), a mono-cellularconvection pattern appears in the discharge vessel. Thetransition from laminar to turbulent flow is characterized bya critical value of 1400 for the Reynolds number (Elenbaas1951). The mean value of the latter, calculated as inequation (7), is given in figure 7 as a function of the squareof the mass:Re = 2 R    R 0 rρvη d r.  (7)The variation in Re versus the amount of mercury isapproximated by  c 1  +  c 2 m 2 Hg , with  c 1  and  c 2  constants Figure 6.  Energy losses due to convection as a function ofthe amount of mercury. Figure 7.  The Reynolds number as a function of theamount of mercury. equal to 0.9 and 7  ×  10 − 4 mg − 2 respectively. If thevalidity of this formula is extrapolated to give values foramounts of mercury that we have not yet examined, the flowbecomes turbulent for unit length mercury mass greater than150 mg cm − 1 It is very important to know how the amount of mercury, initially introduced as a liquid into the burner,will be distributed over the three main discharge zonescited in the introduction of this paper. This distributionis a result of the temperature profile in the plasma. As wehave stated, the hot gas coming out of the arc channel heatsthe region behind the upper electrode. Thus, there should 757
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