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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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