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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 40, NO. 8, AUGUST 2012

2065

Contrast the Effect of the Mass of Mercury
Between the Vertical and Horizontal
Mercury Discharge Lamps
M. B. Ben Hamida and K. Charrada

Abstract—This paper discusses the thermal behavior of a highintensity mercury lamp in a horizontal position, compared with
that of a vertical lamp when the pressure of the lamp varies.
The model adopted is 3-D, steady, and direct-current powered.
After the model validation, we keep the supply current equal to
3.2 A, and we vary the pressure of the lamp to 105 –106 Pa.
Then, we analyzed the temperature fields, the heat conduction
flow, the convective flow, and the accumulation of mercury behind
the electrodes for the case of the lamp in a horizontal position by
comparing it with those of a lamp in a vertical position.
Index Terms—Energy transfer, high-intensity mercury lamp,
local thermodynamic equilibrium, mass of mercury, vertical and
horizontal positions.

I. I NTRODUCTION

T

HIS PAPER is devoted to a 3-D modeling of highintensity discharge plasma. The proposed model is applied
to the study of energy transfer in a mercury discharge lamp with
high intensity.
The choice of this discharge as a medium for the application
of the model is due to, on one hand, the validation possibilities
that offer, according to numerous experimental and theoretical
studies, both, for which it serves as a support [1], [2]. On the
other hand, even if the lamp begins to be replaced by other
systems, it continues to be extensively studied because it is a
reference system.
In fact, virtually all discharge lamps contain significant
amounts of mercury (it is often the majority element in the
discharge) that sometimes plays the role of the buffer gas and, in
other times, the role of the active gas as well as other additives.
In addition, these lamps are characterized by high luminous
efficiency and a very good color rendering index and are used
today in a wide variety.Most studies have been devoted to the
study of lamps placed in vertical positions [3]–[9]. For example,
Charrada et al. [10]–[12] and Beks et al. [13] have studied the
phenomenon of convection in a vertical high-pressure mercury
lamp, supplied with a dc. Wendelstorf [14] investigated the convection and the interaction between the plasma of the discharge
with electrodes in a vertical lamp powered in a dc mode.
Manuscript received March 18, 2012; revised May 10, 2012 and
May 28, 2012; accepted June 2, 2012. Date of publication July 25, 2012; date
of current version August 7, 2012.
The authors are with the Unité d’Étude des Milieux Ionisés et Réactifs, Institut Préparatoire aux Etudes d’Ingénieurs de Monastir (IPEIM), 5019 Monastir,
Tunisia (e-mail: benhamida_mbechir@yahoo.fr).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPS.2012.2203614

For these studies, we can take advantage of the axial symmetry that can be satisfied with a 2-D model. However, this
symmetry is no longer valid in the case of the horizontal
position. That is why the 2-D configuration will no longer be
sufficient; therefore, we must necessarily use a 3-D model,
which naturally requires a longer calculation time.
In fact, the high-pressure mercury lamps are used in horizontal operating positions. Currently, very few 3-D models have
been developed for these systems [3], [15], [16]. Therefore,
we thought that it is appropriate to develop a 3-D numerical
code based on solving equations “fluid” governing the plasma
in the local thermodynamic equilibrium (LTE) state. The commercial simulation software COMSOL, which is based on the
finite-element technique, is used as the computational platform,
where customized application modules are used for solving the
physical models. We chose to study the influence of the mass
of mercury on transport phenomena of mass and energy in both
vertical and horizontal positions.
II. M ODEL
A. Simplifying Assumptions
The discharge lamp is supposed to be in the LTE state. The
fluid is considered an ideal gas, and all external forces other
than the gravity force are neglected. The classical equations of
the fluid mechanics described in the following are written for:
• a Newtonian fluid, which is single phase and homogeneous
for a 3-D flow;
• the flow, which is is assumed to be dominated by the
diffusion phenomena and, therefore, is laminar;
• The terms of the viscous dissipation in the energy equation, which are not taken into account;
• The compressibility in the viscous stress terms of the
momentum equations, which is neglected.
The plasma column is assumed to be independent from the
electrode properties or the arc attachment to the electrodes. This
assumption is valid as long as the electrode gap is large enough.
In that case, the properties of the plasma column can be treated
without taking the influence of the electrodes into account [17].
Therefore, in this paper, all phenomena at the electrode surface
and electrode regions are omitted. Thus, our model results can
be considered to be valid for a few mean free paths distant from
the electrodes. Note that, for short electrodes, this assumption
is already problematic.

0093-3813/$31.00 © 2012 IEEE

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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 40, NO. 8, AUGUST 2012

As has been said, in this paper, we use a time-dependent 3-D
code, thereby solving the coupled momentum, mass continuity,
energy, and electric field equations in the coordinate system
(x, y, z). It can be described by the following equations.

power of the arc. For each temperature range, the total
radiated power of the arc Urad has an expression. These
terms were actually used in (7) to obtain the results in
Figs. 3–18, i.e.,
T ≤ 2000

B. Equations of the Model
Taking into account these assumptions, the simplified system
of equations is written.
Mass conservation equation:



(ρu) +
(ρv) +
(ρw) = 0
∂x
∂y
∂z




(ρuu) +
(ρuv) +
(ρuw)
∂x
∂y
∂z








∂p
∂u

∂u

∂u
=−
+
η
+
η
+
η
.
∂x
∂x
∂x
∂y
∂y
∂z
∂z

(2)

(3)

U T l2 − U T l3
;
T l2 − T l3
UTl3 =

U T l2
2

where Tl2 = 4000, Tl3 = 2000, and UTl2 is the integral of
Uabs at a point in the middle of the wall of the lamp, i.e.,

AU1 =

U T l2 − U T l1
and BU1 = UTl1 − AU1 ∗ Tl1
T l2 − T l1

with Tl1 = 4500 and UTl1 = Uemit (Tl1) = 0.70 ×
10−10 Ng exp(−82 ∗ 103 /T l1), i.e.,
T > 4500
(4)

where V is the electric potential distribution
∂V
∂V
∂V
, jy = −σ
, and jz = −σ
.
∂x
∂y
∂z

AU2 =

where

In these equations, p is the mercury vapor pressure, g is the
gravity, and η is the dynamic viscosity.
Laplace’s equation:







∂V

∂V

∂V
σ
+
σ
+
σ
= 0 (5)
∂x
∂x
∂y
∂y
∂z
∂z

jx = −σ

with

4000 < T ≤ 4500 Urad = AU1∗ T + BU1

Momentum conservation equation according to z:



(ρuw)+ (ρvw)+ (ρww)
∂x
∂y
∂z








∂p
∂w

∂w

∂w
= − +
η
+
η
+
η
.
∂z
∂x
∂x
∂y
∂y
∂z
∂z

2000 < T ≤ 4000 Urad = AU2∗ T + BU2

BU2 = UTl2 − AU2 ∗ Tl2 and

Momentum conservation equation according to y:



(ρuv)+ (ρvv)+ (ρvw)
∂x
∂y
∂z







∂p
∂v

∂v

∂v
= − +
η
+
η
+
η
+ρg.
∂y
∂x
∂x
∂y
∂y
∂z
∂z

876π ∗ difV∗ I ∗ p
101325

where I is the electric current, i.e., the courant; p is the
mercury vapor pressure; and difV is the integral of the
electric potential at the center of the surface of the lower
electrode, i.e.,

(1)

where ρ is the density of mercury, and u, v, and w are the
velocities according to x, y, and z, respectively.
Momentum conservation equation according to x:

Urad = Uabs =

(6)

Energy conservation equation:



(ρuCpT )+ (ρvCpT )+ (ρwCpT )
∂x
∂y
∂z








∂T

∂T

∂T
=
λ
+
λ
+
λ
∂x
∂x
∂y
∂y
∂z
∂z

2
(7)
+ σE − Urad
where, Cp is the specific heat capacity at constant pressure,
σ is the electrical conductivity, E is the electric field, λ is
the gas thermal conductivity, and Urad is the total radiated

Urad = Uemit (T )
= 0.70 × 10−10 Ng exp



−82∗ 103
T



where Ng is the gas density.
In (7), the first term for the left-hand side is the time
rate of change in energy, whereas the second one is related
to the plasma convective flow. The terms of the right-hand
side represent the plasma heat conduction and the source
term, respectively.
The electrical conductivity, thermal conductivity, and
viscosity included in this model are assumed to be described by the local temperature only and are calculated by
using the first approximation of the gas kinetic theory as
developed by Hirchfelderetal [18], assuming a Maxwellian
shape for the electron energy distribution function and
the Lennard–Jones’ interatomic potential. The value corresponding to a monoatomic ideal gas is used for the specific
heat Cp mentioned in [19].
The transport coefficients calculated in the frame of
the first-order approximation gives sufficient accuracy for
a large number of plasma references [2], [11] where the
basic data needed are available, whereas more sophisticated calculations such as those of Devoto [20] will require
unaffordable efforts due to missing fundamental data.

BEN HAMIDA AND CHARRADA: EFFECT OF MASS OF MERCURY BETWEEN DISCHARGE LAMPS

2067

TABLE I
B OUNDARY C ONDITIONS

TABLE II
C HARACTERISTICS OF THE L AMP

Fig. 1.

Geometry and boundary conditions.

An alternative to calculate the radiation energy
transport term is to replace it by a simple function
Urad = Uemit − Uabs representing the difference between
the emitted and absorbed radiation in the lamp [21]–[23].
The state equation of the ideal gas
ρ=

pM
RT

III. R ESULTS AND D ISCUSSION
(8)

where M is the atomic mass, and R is the ideal gas
constant.
The mass of mercury is calculated by integrating over the
entire volume of the lamp (except the electrodes) the density of
mercury in the steady state, i.e.,

mHg =

ρ(p, T )dx.dy.dz.

Fig. 2. Radial profile of temperature halfway of the electrodes.

(9)

A. Validation
Since we did not find, in the bibliography, the experimental
results concerning high-pressure mercury lamps operating in
horizontal positions, we were obliged to validate our 3-D model
by taking a vertical configuration. Fig. 2 shows the temperature
profile calculated by our 3-D model, in a midway section between the electrodes, compared with that measured by Zollweg.
We can notice the quite satisfactory agreement between the
calculated and measured values, thereby justifying the various
assumptions adopted in a 3-D model. We notice, however, that
our results are also in good agreement with the experimental
results of other authors not reported here [23].

C. Boundary Conditions
Due to the geometrical complexity of the arc tube, a simplified configuration is adopted, which is believed to be sufficient
to render the global effects of the convective mode on the arc
plasma behavior. The direction of the gravitational effect for the
lamp in a vertical position is according to the z-axis, whereas it
is according to the y-axis for horizontal position. This is shown
in Fig. 1.
Boundary conditions used in this model are summarized with
reference to Fig. 1 and Table I.



→ −



n. j = 0 mean, reWhere the terms −
n.( jp − je ) = 0 and −
spectively, the continuity and the electric insulation.
The characteristics of the studied lamp are given in Table II.

B. Influence of the Mass of Mercury
Here, we will deal with the influence of the mass of mercury
on the thermal behavior of a mercury lamp in a horizontal
position by comparing it with that of a lamp in a vertical
position while keeping the supplied current constant at 3.2 A.
For this, we vary the mass of mercury from 23 to 230 mg.
1) Spatial Distribution of the Temperature: The description
of a high-pressure discharge (105 − 106 Pa) is, in principle,
greatly simplified through the concept of LTE. According to
Griem, in the case of a discharge plasma where the transfer of
electrical energy is mainly collisional, to the detriment of the
radiation, all local characteristics of the plasma are expressed

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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 40, NO. 8, AUGUST 2012

Fig. 3. Variation of the temperature distribution as a function of the mass of
mercury in the discharge. (a) mHg = 23 mg. (b) mHg = 46 mg. (c) mHg =
103 mg. (d) mHg = 138 mg. (e) mHg = 184 mg. (f) mHg = 230 mg.

Fig. 5. Variation of the isotherms as a function of the mass of mercury in
a vertical lamp. (a) mHg = 23 mg. (b) mHg = 58 mg. (c) mHg = 103 mg.
(d) mHg = 138 mg. (e) mHg = 184 mg. (f) mHg = 230 mg.

Fig. 4. Temperature profile in the positive column for different masses of
mercury.

only in a function of local temperature. Under these conditions,
to interpret the macroscopic transport phenomena, it suffices to
know the temperature distribution inside the discharge, which
regulates the spatial evolution of all other physical quantities.
Fig. 3 illustrates the spatial distribution of the temperature
in the discharge for different mass of mercury for a vertical
lamp. In this case, the discharge is axially symmetric. It is
clear that the narrowing of the arc near the lower electrode
increases with the increase in the mass of mercury in the burner.
The temperature distribution for a mercury mass of 23 mg
is largely independent of convection, whereas the convection
significantly changes the temperature distribution in the case of
a mass of mercury greater than 103 mg, particularly outside the
positive column of plasma.
Fig. 4 shows the profile of the temperature for different
masses of mercury.
Fig. 4 shows that, first, when the mass of mercury increases,
the temperature profile in the positive column increasingly
expands, whereas under the same conditions, the central value
of the temperature decreases.
Fig. 5 shows the variation of the isotherms according to the
mass of mercury.

Fig. 6. Profile temperature in the positive column at the midsection of the
lamp between electrodes for different mass of mercury.

Fig. 6 represents the temperature of the profile in the positive
column for different masses of mercury. From these figures, we
see that the positive column is not axially homogeneous, and
the only plane of symmetry is the vertical plane located halfway
between the electrodes in contrast to a vertical lamp.
The temperature of the profile is divided into two parts: the
left zone corresponding to a free arc and the right zone, which
is less extensive, corresponding to a wall-stabilized arc.
2) Flow of Conduction: Fig. 7 shows the variation of the
radial component of the heat flow of conduction as a function
of radial position for different mass of mercury, in the case of a
vertical lamp.

BEN HAMIDA AND CHARRADA: EFFECT OF MASS OF MERCURY BETWEEN DISCHARGE LAMPS

Fig. 7. Radial variation of the flow of conduction for different mass of
mercury as a function of the radial position.

2069

Fig. 9. Variation of radial heat flow at the midsection of the lamp between
electrodes for different masses of mercury (11 and 23 mg) according to the
radial position.

Fig. 8. Position of the maximum heat flow according to the mass of mercury
in the discharge.

We observe that, by increasing the mass of mercury in the
discharge, the heat flow decreases in the center and increases
on the edge. This confirms that the enlargement of the temperature profile becomes increasingly important when the mass
of mercury increases. As a result, the radial gradient of the
temperature decreases at the center and increases at the edge
of the discharge. In addition, we also find that the variation of
the radial flux according to the radius has a maximum. This
maximum is related to the fact that this conduction heat flow
is the product of the two competing terms: First, the thermal
conductivity that is at maximum at the center and low near the
wall and, second, the radial gradient of the temperature that
decreases at the center and increases near the wall.
Fig. 8 shows the radial position of the maximum of the heat
flow according to the mass of the mercury. It is clear that the
position of this maximum moves to the wall when the mass
of mercury increases due to the enlargement of the conducting
channel of the arc with the mass of mercury in the burner.
Figs. 9 and 10 show the variation of the radial component
of the conductive heat flow according to the radial position for
different mass of mercury.
We see from these figures that for the low mass of mercury
(mass less than 50 mg), there is a symmetry plane almost
passing through the center of the lamp.
The conduction heat flow is low on the edge of the lamp,
starts to increase by moving to the center, reaches a maximum,

Fig. 10. Variation of radial heat flow at the midsection of the lamp between
electrodes for different masses of mercury (57, 103, 138, and 184 mg) according
to the radial position.

and then decreases to its minimum value near the center. This
phenomenon is similar to what was observed in the case of a
vertical lamp [12]. Indeed, for the masses of mercury greater
than 50 mg, we can see in Fig. 10 that the conduction heat flow
is no longer symmetrical, and the thermal stresses on the wall
become strongly inhomogeneous, which will necessarily lead
to a nonuniform wall temperature.
In addition, we note that this conduction heat flow increases
with the mass of mercury at the hot face of the wall. Whereas at
the cold face, it decreases at the beginning, and then it increases.
This explains that the cold zone heats increasingly when the
mass of mercury increased. This warming is mainly due to the
convection that tends, of course, to make uniform the temperature in the plasma. This phenomenon becomes important with
the mass of mercury.
C. Convective Flow
Fig. 11 shows the radial variation of the axial component of
velocity in the positive column for different mass of mercury
for a vertical lamp.
Fig. 12 confirms that the central value of the axial velocity
increases almost linearly with the mass of mercury. This finding

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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 40, NO. 8, AUGUST 2012

Fig. 14. Variation of the losses of mercury behind the electrodes according to
the total mass of mercury.
Fig. 11. Radial profiles of the axial velocity for different mass of mercury for
a vertical lamp.

Fig. 12. Variation of the axial velocity in the center of the positive column
according to the mass of mercury for a vertical lamp.

that for the masses of mercury below 50 mg, the radial profiles
of the axial velocity have a certain symmetry compared with the
axis of the discharge through the two electrodes: zero on the
wall and maximum at the center of the lamp. In this range of
pressures of mercury, the maximum of the axial velocity goes
up, increasing the pressure of mercury. We also note that this
component is always positive, which means that there are no
recirculation zones that lengthen with the axis of the discharge.
However, for the mass of mercury greater than 50 mg, the
radial profiles of the axial velocity are not symmetrical because
of the force of gravity that becomes important enough to
influence the plasma flow so that the maximum of the axial
velocity no longer coincides with the center of the discharge.
This maximum of the axial velocity does not increase with the
mass of mercury.
IV. ACCUMULATION OF M ERCURY
B EHIND THE E LECTRODES

Fig. 13. Radial profiles of the axial velocity at the midsection of the lamp
between electrodes for different values of the pressure of mercury.

is no longer valid in the case of a high-pressure sodium lamp.
Indeed, Ben Ahmed et al. [24] showed that the axial velocity
at the center of the positive column of a high-pressure sodium
lamp is almost independent of the mass of sodium because of
the significant effect of the mass of sodium on the variation of
the temperature in the discharge.
The velocity field for the mercury horizontal lamp differs
fundamentally from a vertical lamp [12].
In Fig. 13, we represent the radial profiles of the axial
velocity for different values of the mass of mercury. We note

The mass of mercury, which does not participate in the
discharge and which remains behind the electrodes, can be
calculated from the temperature distribution in the discharge.
Fig. 14 shows the amount of mercury accumulated behind the
two electrodes as a function of the initial charge of mercury for
a vertical lamp.
It is clear that the amount of mercury lost behind the lower
electrode is higher than that hidden behind the upper electrode.
The difference between the two regions is even more important
that the mercury load is high. This difference results from the
direction of the circulation of fluid inside the tube (see Fig. 15).
Fig. 16 shows the variation of the losses of mercury behind
the electrodes according to the total mass of mercury initially
introduced.
According to Fig. 16, we note that the amount of mercury
lost behind the left electrode is almost the same as that hidden
behind the right electrode. These mercury losses increase when
the total mass of mercury increases.
This is explained by the fact that the discharge and, consequently, the velocity field in the plasma have a plane of symmetry through the middle between the two electrodes. There is no
cold zone behind the left electrode and no hot zone behind the
right electrode.

BEN HAMIDA AND CHARRADA: EFFECT OF MASS OF MERCURY BETWEEN DISCHARGE LAMPS

2071

Fig. 17. Circulation of gas inside the discharge lamp in a horizontal position.

Fig. 15. Circulation of gas inside the discharge lamp in a vertical position.

Fig. 16. Mercury loss versus total mass of mercury.

To confirm this result, we see the circulation of gas inside
the discharge lamp in a horizontal position in Fig. 17 and the
velocity fields in a cross-sectional plane in Fig. 18.
From these figures, we see that in the case of the horizontal
lamp and in a cross-sectional plane, the velocity field is composed of two symmetrical cells. We observe the same effects in
the case of a vertical lamp but in a longitudinal section plane.
Therefore, there is no cold area behind the lower electrode and
a hot zone behind the upper electrode.
V. C ONCLUSION
This paper has been devoted to a 3-D modeling of highpressure discharge plasma. The proposed model is applied
to the study of energy transfer in a mercury discharge lamp
with high intensity. The designed model makes it possible to

Fig. 18. Radial sections of the spatial distribution of temperature and velocity
field of the horizontal lamp at: (a) the left electrode, (b) center, and (c) the right
electrode.

solve the coupled system of the equations of energy, mass, and
momentum, as well as the equation of Laplace for the plasma.
The results obtained from the 3-D model for a vertical configuration of the discharge perfectly reproduce the evolutions of
the characteristics of bibliographic results. This allowed us to
validate the different assumptions and the relations adopted in
our model by comparing its calculations with the corresponding
experimental results. Finally, we examined the effect of the
variation in mass of mercury on the behavior of the discharge.
The main results obtained for the case of a mercury lamp in
a vertical position are as follows.
When the mass of mercury increases in the burner, the
following occurs.
• The narrowing of the arc near the lower electrode
increases.
• The temperature profile in the positive column increasingly widens and the central value of the temperature
decreases. This confirms that the temperature profile near
the lower electrode is completely different from that of
the positive column essentially for the important masses
of mercury.
• The heat flow decreases in the center and increases on
the edge.
• The central value of the axial velocity increases almost
linearly with the mass of mercury.
• The amount of mercury lost behind the lower electrode
is greater than that hidden behind the upper electrode. The
difference between the two regions is even more important

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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 40, NO. 8, AUGUST 2012

that the mercury load is high. This difference results from
the direction of circulation of fluid inside the tube.
• The distribution of the temperature for the mass of mercury of 23 mg is largely independent of convection,
whereas convection significantly changes the temperature
distribution in the case of a mass of mercury in excess of
103 mg, particularly outside of the positive column of the
plasma.
The main results obtained for the case of a mercury lamp in
a horizontal position are:
When the mass of mercury increases in the burner, the
following occurs.
1) The heat flow of conduction increases at the hot zone of
the wall, whereas in the cold zone, it decreases at the
beginning, and then it starts growing.
2) The losses of mercury behind the electrodes increase,
whereas the mass of mercury lost behind the left electrode is almost the same as that hidden behind the right
electrode.
3) For the masses of mercury less than 50 mg, the maximum
axial velocity goes up increasing the load of mercury.
However, for mercury with masses greater than 50 mg,
the maximum of the axial velocity does not increase with
the mass of mercury in the burner.
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pp. 093502-1–093502-12, Sep. 2011.

M. B. Ben Hamida was born in Moknine, Tunisia,
on July 7, 1977. He received the Ph.D. degree
from the National Engineering School of Monastir,
Monastir, Tunisia, in 2012.
From 2002 to 2011, he was technologist at the
Higher Institute of Technological Studies of Sousse.
Since January 2012, he has been a Master Technologist with the Higher Institute of Technological
Studies, Zaghouan, Tunisia. He has published four
journal articles and three international communications. His research activities focus on plasma and
discharge lamp applications in model 3D.

K. Charrada was born in Jemmel, Tunisia, on
February 5, 1967.
He is a Professor with the Preparatory Institute for
Engineering Studies, Monastir, Tunisia, where he is
also the Director of the Unit for the Study of Ionized
Gases and Reagents.


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