The aim of this paper is to evaluate the magnitude of the external magnetic field to be applied to a horizontal mercury discharge lamp such that the Lorentz forces counterbalance buoyancy forces and the hot region of the arc remains centered inside the lamp with the variation of six parameters of the lamp such as the external temperature of the lamp, envelope thickness, convective loss, Interelectrodeslength, pressure and current supply pointing to the influence of the parameters to the compensating magnetic field value. To achieve this objective, a commercial numerical software “Comsol Multiphysics” is used to implement the model that solves the equations of mass, energy and momentum for laminar compressible flow combined with the Laplace equation for the plasma in a three dimensional.

## I. INTRODUCTION

The study of the convection in mercury discharge lamp has been the center of interest of many researchers. For the case of the lamp operated in vertical position, we can be mentioned Zollweg,^{1} Wendelstorf,^{2} Flesch,^{3} Fischer,^{4} Beks,^{5} Charrada and Zissis.^{6–8} In their studies, they concluded that the arc has a symmetry that is why they have been content for a two-dimensional code.

This limitation applies only to the vertical configuration of the lamp. For all other positions, this code is no longer appropriate for a better study of the convection phenomenon inside the lamp.

Indeed, the development of a three-dimensional code is mandatory and has been recommended and used by some researchers^{9–13} who studied the phenomenon of convection in the discharge lamp in a horizontal position.^{10,14–16} In addition, these codes consider the wall temperature of the lamp uniform and constant. In their studies, they concluded that the arc was not symmetrical and the convection flow was driven by gravity force.

For the purpose of counteracting the buoyancy forces and centering the arc, a few researchers^{17–19} have managed numerically to do this by applying an externally magnetic field. Each study presents a single case of lamp with the hypotheses for simplifying their numerical code and found exactly the value of magnetic field applied without searching the parameters that affect this number which is the main objective of the present work. For this, we have developed a time-dependent three-dimensional code using a commercial software COMSOL Multiphysics which has solved all the equations required, the energy balance at the wall to calculate its temperature and added the JxB source term in the momentum equations.

## II. BASIC ASSUMPTIONS AND MODEL EQUATIONS

### A. Basic assumptions

The previously-presented equations are solved by considering the following simplifying hypotheses :

The fluid is supposed an ideal gas.

The discharge lamp is considered to be in the Local Thermodynamic Equilibrium state L.T.E.

All external forces other than the gravity force are not taken into account.

A Newtonian fluid, single phase and homogeneous for a three-dimensional flow.

The flow is assumed to be dominated by the diffusion phenomena i.e it is laminar.

In the energy equation, the terms of the viscous dissipation are neglected.

The fraction of the radiative power of the plasma absorbed by the wall is negligible

All phenomena at the electrodes surface and electrode regions are omitted.

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.^{20} Therefore, in this work all phenomena at the electrodes 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.

### B. Model equations

The used system of equations taking into account of the simplifying assumptions previously-mentioned are :

**Mass conservation equation:**

**Momentum conservation equation**

**Energy conservation equation for the plasma**

**Energy conservation equation for the wall:**

**The equation of state for ideal gas**

**The uniform electrical field in the positive column plasma**

The basic variables by the previous equations are the density of mercury *ρ*, the mercury vapor pressure p, the velocity vector $ u \u20d7 $, the dynamic viscosity *η*, the electric potential distribution V, the electric current density J, the specific heat capacity at constant pressure *Cp*, the electrical conductivity *σ*, the electric field E, the gas thermal conductivity *λ*, the total radiated power of the arc *U*_{rad}, the ideal gas constant *R*, the atomic mass M, the thermal conductivity of the quartz λ_{q}.

The LTE electrical conductivity is assumed to be given by the following equation with T_{ion} is 55820 K and σ_{0} is 12mho.cm^{−1}.K^{−0.75}.^{22}

The approximate net-emission coefficient U_{rad}, is given by :

With T^{*} is 86000 K, u_{0} is 2.14x10^{12} W.cm^{−1}.K^{−1} for the high pressure Hg discharge.^{22} The temperature dependent viscosity and thermal conductivities for the Hg are calculated based on the Lennard-Jones parametrization with only heavy neutral atom interactions included. No additional contributions from plasma and radiation transport are included here.

### C. Boundary conditions

The geometry and the diagram of the studied lamp are located on Figure 1 and Figure 2. It consists of two big coaxial cylinders in horizontal position. The inner cylinder is divided into three zones. The lower electrode zone containing the anode is located to the left of Figure 1. The zone of the positive column is found between the two electrodes. The upper electrode zone containing the cathode is located to the right of Figure 1. The outer cylinder represents the envelope of the lamp. This simplified configuration retains the same convection relative to the real shape of the lamp. The gravitational direction is according to the opposite y axis.

The boundary conditions on temperature, velocities and electric potential distribution are detailed with reference to fig 2 in the table I.

Boundary number . | Condition on Temperature . | Condition on velocity . | Condition on potentiel . |
---|---|---|---|

1-4 | T = T(z) | $ u \u20d7 =0$ | $ n \u20d7 . j \u20d7 =0$ |

5-8 | $ n \u20d7 . q \u20d7 =0$ | Not defined | $ n \u20d7 . j \u20d7 =0$ |

9 | T = T _{elec} | $ u \u20d7 =0$ | $\u2212 n \u20d7 . j \u20d7 =I/S$ |

10 | T = T _{elec} | $ u \u20d7 =0$ | V = 0 |

11-12 | Not defined | Not defined | Not defined |

13-18 | $ n \u20d7 . ( q p \u20d7 \u2212 q w \u20d7 ) =0$ | $ u \u20d7 =0$ | $ n \u20d7 . ( j p \u20d7 \u2212 j e \u20d7 ) =0$ |

19-24 | $\u2212 n \u20d7 . q \u20d7 = q 0 $ | Not defined | $ n \u20d7 . j \u20d7 =0$ |

Boundary number . | Condition on Temperature . | Condition on velocity . | Condition on potentiel . |
---|---|---|---|

1-4 | T = T(z) | $ u \u20d7 =0$ | $ n \u20d7 . j \u20d7 =0$ |

5-8 | $ n \u20d7 . q \u20d7 =0$ | Not defined | $ n \u20d7 . j \u20d7 =0$ |

9 | T = T _{elec} | $ u \u20d7 =0$ | $\u2212 n \u20d7 . j \u20d7 =I/S$ |

10 | T = T _{elec} | $ u \u20d7 =0$ | V = 0 |

11-12 | Not defined | Not defined | Not defined |

13-18 | $ n \u20d7 . ( q p \u20d7 \u2212 q w \u20d7 ) =0$ | $ u \u20d7 =0$ | $ n \u20d7 . ( j p \u20d7 \u2212 j e \u20d7 ) =0$ |

19-24 | $\u2212 n \u20d7 . q \u20d7 = q 0 $ | Not defined | $ n \u20d7 . j \u20d7 =0$ |

$ q p \u20d7 =\u2212 \lambda p \u2207 \u20d7 T p + \rho p C p T p u \u20d7 $ and $ q w \u20d7 =\u2212 \lambda w \u2207 \u20d7 T w $ are the heat flux terms for the plasma and the wall with λp, T_{p}, ρ_{p}, C_{p} and $ u \u20d7 p $ are the thermal conductivity, temperature, density, heat capacity at constant pressure, and velocity of plasma, respectively. *T _{w}* is the wall temperature and λ

_{w}is the thermal conductivity, of the wall. The boundary condition $ n \u20d7 ( j p \u20d7 \u2212 j e \u20d7 ) =0$ specifies that the normal components of the electric current are continuous. $ n \u20d7 $ is the unit vector normal to the considered surface.

*I*,

*S*, and

*T*

_{elec}the current, section, and temperature of the electrode,

^{21}respectively.

The boundary condition for T at the surface of the arc tube is given by the nonlinear radiative and convective heat flux to the surrounding (T_{ext}):

Where ε_{q}, T_{ext} and σ_{s} are the emissivity of quartz, the ambient temperature and the Stefan–Boltzmann constant, respectively.

The convection loss is calculated by :

where C is a constant equal to 4.8 and L is the length of the lamp in centimeters.

## III. RESULTS AND DISCUSSION

### A. Validation

Unfortunately, the horizontal lamp measurements are not available. For this, we compared our three-dimensional model with the experimental measurements by Zollweg^{1} for the case of the lamp in a vertical position. These experimental measures relate the radial profile of temperature in a midway section between the electrodes. The characteristics of the lamp of Zollweg are given in table II.

Interelectrodeslength (mm) | 80 |

Internaldiameter (mm) | 18 |

Length of the electrode (mm) | 10 |

Diameter of the electrode (mm) | 2 |

Current (A) | 3 |

Mass of mercury (mg) | 100 |

Pressure (Pa) | 5.66 |

Interelectrodeslength (mm) | 80 |

Internaldiameter (mm) | 18 |

Length of the electrode (mm) | 10 |

Diameter of the electrode (mm) | 2 |

Current (A) | 3 |

Mass of mercury (mg) | 100 |

Pressure (Pa) | 5.66 |

The results of experimental measurements by “Zollweg” and calculated by our studied model is shown in the figure 3.

We see that the values calculated using our model are in very good agreement with the experimental values of “Zollweg”. This confirms the suitable choice of the different coefficients and the simplifying assumptions used. Then, this model has been applied to reproduce the effect of several parameters on magnetic field applied in the x-direction. These results are shown in the following paragraphs.

### B. The effect of different parameters on a magnetic field applied

#### 1. Effect of the external temperature of the lamp

In this section, the burner has an electrode spacing of 72 mm with a total length of 90 mm. The diameter of arc tube is 18 mm. The radius and length of electrodes are 1 mm and 10 mm, respectively. The quartz thickness is 1 mm. The lamp is fed with a Direct Current of 4 A. The pressure of this lamp is 3x10^{5} Pascals. The convective loss is negligible. The characteristics of this lamp given in table III is the same for the lamp studied by Yan-Ming Li.^{22}

Interelectrodeslength (mm) | 72 |

Internaldiameter (mm) | 18 |

Externaldiameter (mm) | 20 |

Length of the electrode (mm) | 8 |

Radius of the electrode (mm) | 1 |

Pressure inside the lamp (Pa) | 3 |

Current (A) | 4 |

Interelectrodeslength (mm) | 72 |

Internaldiameter (mm) | 18 |

Externaldiameter (mm) | 20 |

Length of the electrode (mm) | 8 |

Radius of the electrode (mm) | 1 |

Pressure inside the lamp (Pa) | 3 |

Current (A) | 4 |

We varied the external temperature of the lamp T_{ext}. For this conditions, we plotted the temperature profile at the midsection of the lamp between electrodes for a horizontal lamp without both convective loss and magnetic field applied at three different external temperature of the lamp shown in Figure 4. From this figure, we notice that a temperature difference across the “down zone” of the lamp, the maximum temperature area and the upper wall of the lamp. To view a precise way this difference, we do a zoom in these areas shown in figures 5, 6, and 7. According to these figures, the convection flow causes large difference in arc-tube temperature distribution. Indeed, the wall temperature of the “down zone” is always less than the wall temperature of the “upper zone”, since the arc is directed to the “upper region” in contrast to the force of gravity. Also, it is observed for the case where the external temperature is 1000 K that the wall temperature of the “down zone” equal to 1070.9 K (Figure 6 and 8) and the wall temperature of the “upper zone” is equal to 1378 K (Figure 7). These values are found by Yan-Ming Li. Whereas for the case where the external temperature is 700 K and 300 K, the wall temperature of the “down zone” is equal to 972.37 K and 840.92 K (Figure 6) and the wall temperature of the “upper zone” is equal to 1261.4 K and 1105.8 K (Figure 7). This difference of the wall temperature between the two cases is due to the temperature gradient between the temperature of the envelope and the external temperature of the lamp.

In other words, an increase of the external temperature increases the walls temperatures of the lamp and the temperature throughout the “down zone” of the lamp.

In addition, it is showed for the case where the external temperature is 1000 K, 700 K and 300 K that the maximum temperature equal to 6160.24 K, 6200 K and 6233.5 K (Figures 5 and 8), respectively. We deducted that an increase in the external temperature decreases the maximum temperature in the positive column of the lamp. This is due to the large temperature gap of the radiative term in the equation (8).

By applying an external uniform magnetic field in the x direction, the exact value to counteract exactly the buoyancy force and centering the arc is equal to −2.3x10^{−4} Tesla.

By cons, where the external temperature is 700 K and 300 K, the magnetic field value for centering the arc is B_{x} = − 2.65x10^{−4} Tesla and −3.1x10^{−4} Tesla, respectively. Therefore, we deduce that the external temperature has a great influence on the magnetic field value applied. In other words, a decrease of the external temperature requires increasing the external magnetic field (in absolute value) for centering the arc of a discharge lamp in a horizontal position.

#### 2. Effect of the convective loss

In this section, we keep the same characteristics of the lamp quoted in Table III. To appear the effect of convective loss, we plotted the figure 9 which shows the temperature profile at the midsection of the lamp between electrodes for the horizontal lamp without and with convective loss in the absence of magnetic field.

According to this figure, we note that there is a temperature difference in the “down zone” of the lamp, the maximum temperature area, the upper and down wall of the lamp. To see with an accurate way this difference, we do a zoom in these areas shown in figures 10 and 11.

From these figures, it is observed for the horizontal lamp with convective loss that the wall temperature of the “down zone” equal to 970.9 K (Figure 10) and the wall temperature of the “upper zone” is equal to 1278 K (Figure 11). Against by a slight increase of the maximum temperature compared to the maximum temperature in the positive column for the case of the horizontal lamp without convective loss. We deduce that the presence of the convective loss cooled the walls of the lamp and increases the temperature in the positive column of the lamp. The magnetic field value for centering the arc is B_{x} = − 2.5x10^{−4} Tesla. Therefore, we note that the presence of convective loss has an influence on the magnetic field value applied ie increase the external magnetic field (in absolute value) for centering the arc of a discharge lamp in a horizontal position.

#### 3. Effect of the envelope thickness

In this section, we keep the same characteristics of the lamp cited in table III. To show the effect of external diameter of the lamp, we varied the external diameter of the lamp (20 mm and 22 mm) and we plotted the figure 12 which sees the temperature profile at the midsection of the lamp between electrodes for two different envelope thickness (1 mm and 2 mm) in the absence of magnetic field.

From this figure, it is observed almost the same temperature profile. We conclude that the envelope does not have a great influence on the phenomenon of convection within the lamp and thereafter on the magnetic field strength applied to center the arc.

#### 4. Effect of the Interelectrodeslength

In this section, we keep the same characteristics of the lamp cited in table III. For two different interelectrodeslength (72 mm and 57.6 mm), we plotted the figure 13 which sees the temperature profile at the midsection of the lamp between electrodes for two different interelectrodeslength (ratio interelectrodeslength equal to 1 and 0.8) in the absence of magnetic field. According to this figure, we note that there is a temperature difference only in the “down zone” of the lamp. In fact, the decrease of the interelectrodeslength enables the decrease of the temperature in the “down zone”. The magnetic field value for centering the arc is B_{x} = − 2.4x10^{−4} Tesla in the case where the interelectrodeslength is 57.6 mm. We deduce that the interelectrodeslength has a small effect on the magnetic field value applied. Indeed, a decrease of the interelectrodeslength requires increasing the external magnetic field (in absolute value) for centering the arc of a discharge lamp in a horizontal position.

#### 5. Effect of the pressure

In this section, we keep the same characteristics of the lamp given in Table III but the pressure is varied (2x10^{5}, 3x10^{5} and 4x10^{5} Pa) for a constant current equal to 4 A.

Figure 14 shows the temperature profile at the midsection of the lamp between electrodes for the 3 A Hg arc and a three pressures of 2x10^{5} Pa, 3x10^{5} Pa and 4x10^{5} Pa in a horizontal lamp without magnetic field.

According to this figure, we note that the positive column is not axially homogeneous. Indeed, the arc is not centered and it is located near the “upper” wall for the case of the horizontal lamp without applied magnetic field. Also, we see that 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. In addition, the maximum temperature moves away from the center of the lamp by increasing the pressure inside the lamp. This is the physical basis of arc bowing. We notice that the distance between the tube center and the maximum plasma temperature increase with the increasing the pressure. Indeed, this distance is 2.6 mm, 3.3 mm and 4.3 mm for horizontal lamp at three pressures of 2x10^{5} Pa, 3x10^{5} Pa and 4x10^{5} Pa, respectively, when the arc current equal to 4 A.

Figure 15 and 16 show the profile of wall temperature at the midsection of the lamp between electrodes for the 4 A Hg arc and a three pressures of 2x10^{5} Pa, 3x10^{5} Pa and 4x10^{5} Pa in a horizontal lamp without magnetic field. According to these figures, we see that the down wall temperature increases when the pressure decreases (figure 15) and the upper wall temperature decreases with the decreasing of the pressure inside the lamp as shown in figure 16.

After many numerical tests using a three-dimensional code, we managed to find the values of the uniform magnetic fields along the x axis for three pressures of 2x10^{5} Pa, 3x10^{5} Pa and 4x10^{5} Pa when the arc current equal to 4 A. With theses applied values shown in table IV, the magnetic force can balance the buoyancy force, pushing the arc back to the center of the arctube.

Pressure (Pa) . | Distance between the tube center and the maximum plasma temperature (mm) . | The value calculated of the magnetic field (Tesla) . |
---|---|---|

2 x10^{5} | 2.6 | −1.8x10^{−4} |

3 x10^{5} | 3.3 | −2.3x10^{−4} |

4 x10^{5} | 4.3 | −3.9x10^{−4} |

Pressure (Pa) . | Distance between the tube center and the maximum plasma temperature (mm) . | The value calculated of the magnetic field (Tesla) . |
---|---|---|

2 x10^{5} | 2.6 | −1.8x10^{−4} |

3 x10^{5} | 3.3 | −2.3x10^{−4} |

4 x10^{5} | 4.3 | −3.9x10^{−4} |

We concluded that the absolute value of applied magnetic field is increased by increasing the pressure inside the lamp.

#### 6. Effect of the current supply

In this section, we keep the same characteristics of the lamp given in Table III but the current is varied for a constant pressure of the lamp equal to 3x10^{5} Pa.

Figure 17 shows the temperature profile at the midsection of the lamp between electrodes for the pressure of 3x10^{5} Pa and a three current of 2 A, 3 A and 4 A in a horizontal lamp without magnetic field. According to this figure, we note that the temperature in the left zone increases and the temperature in the right zone decreases when the current increases. Then, the down wall temperature increases when the current increases as shown in figure 18. Moreover, the upper wall temperature decreases with the increasing of current as shown in figure 19. In addition, the distance between the tube center and the maximum plasma temperature increase when the current inside the lamp decreases. Indeed, this distance is 3.3 mm, 4 mm and 4.45 mm for horizontal lamp at three current of 4 A ; 3 A and 2 A, respectively, when the pressure of the lamp equal to 3x10^{5} Pa.

After many numerical tests using a three-dimensional code, we managed to find the values of the uniform magnetic fields along the x axis for three current of 4 A ; 3 A and 2 A when the pressure equal to 3x10^{5} Pa shown in table V.

Current (A) . | Distance between the tube center and the maximum plasma temperature (mm) . | The value calculated of the magnetic field (Tesla) . |
---|---|---|

2 | 4.5 | −4.3x10^{−4} |

3 | 4 | −3x10^{−4} |

4 | 3.3 | −2.3x10^{−4} |

Current (A) . | Distance between the tube center and the maximum plasma temperature (mm) . | The value calculated of the magnetic field (Tesla) . |
---|---|---|

2 | 4.5 | −4.3x10^{−4} |

3 | 4 | −3x10^{−4} |

4 | 3.3 | −2.3x10^{−4} |

For the purpose to center the arc, we noted that the absolute value of applied magnetic field is decreased by increasing the current supply of the lamp.

## IV. CONCLUSION

This paper focuses on a three-dimensional code of mercury discharge lamp in a horizontal position. This model solve the coupled system of the equations of mass, energy and momentum, as well as the equation of Laplace for the plasma. It has been extended to include effects associated with magnetic fields. Using this model, we confirmed theoretically that an externally imposed magnetic field is an effective way to straighten and center the arc and we managed to find exactly the applied magnetic field values for the horizontal lamp operating with different case of the lamp. Also, we study the effect of the external temperature of the lamp, the convective loss, the envelope thickness, the interelectrodeslength, the pressure of the lamp and the current supply on magnetic field applied. For centering the arc of a discharge lamp in a horizontal position, it was pointed out that :

A decrease of the external temperature requires increasing the external magnetic field.

The presence of convective loss increase the external magnetic field.

The envelope thickness does not have a great influence on the magnetic field strength.

A decrease of the interelectrodeslength requires increasing the external magnetic field.

The applied magnetic field increase by increasing the pressure inside the lamp.

When the current increases, the applied magnetic field decreases.