- No tags were found...

Simulation of heat and momentum flow in a quartz mercury ... - Comsol

Excerpt from the Proceed**in**gs **of** the COMSOL Multiphysics User's Conference 2005 BostonThomas Dreeben**Simulation** **of** **heat** **and** **momentum** **flow** **in** a **quartz** **mercury**-filledHID lamp **in** vertical operationAbstract FEMLAB is used to provide a benchmarksimulation for a classical problem **in** high-**in**tensitydischarge light**in**g: the pure **mercury**-filled **quartz**HID (high-**in**tensity discharge) lamp. A briefdescription **of** how an HID lamp works is provided.A steady 2-dimensional axi-symmetric formulationestimates the operat**in**g temperatures, electricalpotential, **and** gas velocities for an HID lamp runn**in**g**in** vertical orientation. Govern**in**g equations **and** theirapproximations are described: These **in**clude thecompressible cont**in**uity equation, the Elenbaas-Heller (**heat** balance) equation, the current cont**in**uityequation for electrical potential, **and** **momentum**equations **in** the radial **and** vertical directions. Lamppower **of** 175 W is imposed by adjust**in**g theelectrode voltage so that the specified power isobta**in**ed. This uses the **in**tegration coupl**in**g variablefeature **in** FEMLAB, as the lamp power is a volume**in**tegratedquantity. Heat conduction through the**quartz** arc-tube wall is also **in**cluded, with theexternal boundary condition governed by radiationfrom the exposed surface. The model providesestimates **of** the temperature pr**of**iles **in** the gas **and** **in**the arc tube, electrical potential **in** the gas, **and**buoyancy-driven velocity **in** the gas. Results give usa wealth **of** basel**in**e **in**formation on a work**in**g HIDlamp.1 How a **mercury** HID lamp worksA typical **mercury** HID lamp 1 is comprised **of** 10 –50 mg **of** **mercury** enclosed **in** a transparent arc tube,usually made **of** **quartz**. The arc tube has a tungstenelectrode sealed **in** each end. A ballast supplieselectrical power to the electrodes **in** the form **of**alternat**in**g current. This electrical power raises thetemperature **of** the **mercury** to the po**in**t that it turnscompletely **in**to vapor. The current then ionizes thevapor **in** a path between the electrodes called the arc.In steady operation, the temperature **in** the arc istypically around 6000 K, **and** many **mercury** ionsthere are **in** excited states. A photon is emitted everytime an excited ion reduces its energy level to a lowerstate – this is how the lamp gives **of**f light. Theelectrical power that is not converted to light is lost as**heat**, most **of** which conducts to the arc-tube wall **and**radiates from the outer surface accord**in**g to theStefan-Boltzmann law.2 Problem formulationThe computational doma**in** is a 2-d axisymmetricslice **of** an arc tube **in** vertical orientation as shown **in**Figure 1.Keywords FEMLAB 3.1 – HID lamp, buoyancy,natural convection, Elenbaas-Heller, compressibleT. DreebenOSRAM SYLVANIATel: 978-750-1688Fax: 978-750-1792E-mail: thomas.dreeben@sylvania.comFigure 1: HID lamp computational doma**in**.Axisymmetric coord**in**ates are normalized to thearc-tube **in**ner radius.

Excerpt from the Proceed**in**gs **of** the COMSOL Multiphysics User's Conference 2005 BostonThe doma**in** **in**cludes two electrodes withproperties **of** tungsten, an arc tube with properties **of****quartz**, **and** an enclosed gas with properties **of****mercury** vapor.A mesh **of** 2484 first-order Lagrangian elementsis used **and** shown **in** Figure 2. Ref**in**ement aroundthe electrodes is needed to help assure convergence,as the spatial temperature gradients are steepest there.Figure 2: Mesh for the gas, electrodes, **and** arctubeSome approximations are used here to simplifythe problem:• Although HID lamps run on alternat**in**gcurrent, the FEMLAB model isconstructed for direct current so that asteady simulation can be performed.• Radiation is characterized as a localtemperature-dependent loss term **in** theElenbaas-Heller (energy) equation. Thenon-local effect **of** absorption isneglected **in** the model.• Heat transfer between the gas **and** theelectrodes is mischaracterized **in** themodel as pure **heat** conduction. In HIDlamps, there is a non-neutral plasmas**heat**h near each electrode **in** whichmany more processes govern the **heat**transfer – all **of** these additionalprocesses are neglected **in** the model.• Spatial pressure variation is **in**corporatedfor buoyancy, but neglected **in** the idealgas law.• Although compressibility is **in**cluded **in**the mass cont**in**uity equation, it isneglected **in** the viscous stress terms **of**the **momentum** equations. We assumethat stresses associated with dilitation aresmall compared with stresses associatedwith shear.The ma**in** aspects **of** the mathematical formulation**of** the problem are as follows: The govern**in**gequations are those **of** natural convection coupledwith **heat** transfer **and** electrical current cont**in**uity 2,3,4 .For velocity vector u r , gas density ρ , **and** pressurep , the steady **momentum** equations **in** the radial ( r )**and** vertical ( z ) directions are expressed **in** vectorform asr r r r r r r rρu ∇ u =− ∇{p −{ρg +∇ ( μ∇u).123 14243 (1)convectionpressuregradientbuoyancyviscous stressThe steady, compressible mass cont**in**uity equation isr r∇ ( ρu ) = 0.(2)Velocity boundary conditions are no-slip **and**impermiability at all solid surfaces, **and** naturalsymmetry conditions along the axis. Because thespatial pressure variation is smaller than the lamppressure by 5 orders **of** magnitude, pressure isassumed to be contant **in** the ideal gas law which thenpecomesρ T = constant.(3)For electrical potential φ **and** temperature T , theenergy balance **in** the gas is expressed by theElenbaas-Heller equation:r r r r rρcu∇ T=∇ κ∇ T + σ ∇φ− q2( ) {radp14243 14243 123convection conduction electrical**heat** sourceradiation**heat** s**in**k(4)Heat also conducts through the arc tube **and**electrodes with the steady **heat**-condcution equation,us**in**g thermal conductivities that are appropriate to**quartz** **and** tungsten respectively. Heat leaves thesystem with the Stefan-Boltzmann law as theboundary condition on the outer arc-tube surface. Asmall amount **of** **heat** also conducts away from theouter electrode ends, us**in**g a semi-empirical mixedboundary condition. The source **and** s**in**k terms **of**Eq. (4) are very strong functions **of** temperature,because **of** the temperature dependence **of** electricalconductivity σ **and** volumetric radiation qrad. For apure **mercury** lamp runn**in**g at 3.78 atm **of** pressure,these terms are approximated **in** the model with thefollow**in**g semi-empirical expressions: 50.75 −55820/Tσ = 1070 T e ,142.69×10(5)−86000/Tq = e .radTThe electrical potential φ (which is also voltage) isdeterm**in**ed from the current cont**in**uity equation

Excerpt from the Proceed**in**gs **of** the COMSOL Multiphysics User's Conference 2005 Bostonr r∇ ∇ =( σ φ) 0,(6)with a coupl**in**g to the temperature through theelectrical conductivity σ . The boundary conditionson φ are no-flux along the **in**ner arc-tube surface **and**lamp axis, **and** constant values at the electrodes.These electrode values impose a voltage differenceacross the lamp which drives the Elenbaas-Hellerequation through the **heat**-source term **of** Eq. (4).Mathematically, the electrode values **of** φ aredeterm**in**ed by allow**in**g them to float to whatevervalue enables the satisfaction **of** the global **in**tergralconstra**in**tr 2P= σ ∇φdV,∫Lamp(7)where P = 175 W is the specified power **of** thelamp.Equations (1)-(6) are recast **in** a form with alldimensionless variables, **and** then implemented **in**FEMLAB us**in**g its general form. This dimensionlessformulation is straightforward except for thetreatment **of** the potential φ . Here a dimensionlessvariable φ % is def**in**ed by scal**in**g the potential with itsunknown value at the electrodes. This way, theelectrode boundary conditions on the dimensionless% . Then% φ are the simple Dirichlet condition φ =± 1the **in**tegral constra**in**t **of** Eq. (7) is imposed us**in**g the**in**tegration coupl**in**g variable feature, **and** is then usedto determ**in**e the dimensionless parameter thatprecedes the **heat**-source term **in** the dimensionlessversion **of** Eqs. (4). The dimensionless system coded**in**to FEMLAB is a full mathematical equivalent toEqs. (1) - (7).3 Convergence **and** model outputBecause **of** the coupl**in**g **and** strong temperaturedependence **of** the electrical conductivity σ **and** theradiation qrad, some care is required **in** the **in**itialconditions to obta**in** convergence. Convergence isobta**in**ed by build**in**g up the computation **in** 3 stages,each comput**in**g a larger portion **of** the fully-coupledproblem. All **of** this is programmed **in** a MATLAB.m file. The **in**itial fields are a quiescent velocityfield, a gas temperature pr**of**ile that is parabolic **in** theradial direction, **and** a potential pr**of**ile that is l**in**ear**in** the axial direction. The first stage is to solve Eq.(6) (current cont**in**uity) for the potential φ with allother variables held fixed. In the second stage, theoutput **of** the first stage is fed **in**to a coupled,unsteady simulation **of** Eqs. (4) **and** (6) together, withthe **momentum** **and** mass cont**in**uity equations stillheld fixed with zero velocity field. This pair **of**equations is run until the solution for temperature T**and** potential φ is approximately steady. The thirdstage is to solve the full coupled set **of** equations **in** asteady simulation, start**in**g from the output **of** thesecond stage. For this stage, the model must beconverted from the general form to the weak form, toassure that FEMLAB **in**cludes Eq. (7) when itcomputes the Jacobian. This nonlocal Jacobianmakes the simulation rather computationally**in**tensive. Nevertheless, the simulation with all 3stages reaches a fully-coupled steady solution **in** justunder 2 m**in**tues.The primary output **of** the simulation is atemperature **and** velocity map, as shown **in** Figure 3.Figure 3: Temperature pr**of**ile with velocityvectors. Spatial dimensions are normalized by thearc-tube **in**ner radius.The radial temperature gradient gives rise to a densitygradient, which makes the gas **in** the center muchlighter than the gas near the walls. This creates abuoyancy force which makes the gas rise **in** the center**and** fall along the arc-tube wall.Halfway between the electrodes, the temperature**and** velocity pr**of**iles are shown **in** Figure 4 **and**Figure 5 respectively.

Excerpt from the Proceed**in**gs **of** the COMSOL Multiphysics User's Conference 2005 BostonFigure 4: Temperature pr**of**ile versus normalizedlamp radius, halfway between the electrodes.Figure 6: Electrical potential (Voltage) throughoutthe lamp.The predicted voltage across the electrodes **of** 106.5V is lower than the real lamp voltage by 20%, due tothe model’s neglect **of** electrode plasma effects **and**their associated cathode fall. The slight verticalasymmetry **in** potential is an **in**direct result **of** theconvection: Convection causes a slight verticalasymmetry **in** the temperature pr**of**ile **of** Figure 3.This **in**fluences the electrical potential through theappearance **of** electrical conductivity σ **in** Eq. (6).A pressure map, referenced to zero at the lowerelectrode tip, is shown **in** Figure 7.Figure 5: Vertical velocity pr**of**ile versusnormalized lamp radius, halfway between theelectrodes.Historically, measurement **of** temperature pr**of**iles **in****mercury** lamps show that the pr**of**iles such as Figure4 are generally accurate to with**in** 5 or 10%. Thereare no direct measurements **of** gas velocity pr**of**iles,so Figure 5 gives us an approximation that would beunavailable without model**in**g: The maximumvelocity for a lamp **of** these dimensions is **of** order 10– 20 cm /sec. Note that the velocity upward **in** thecenter greatly exceeds the velocity downward nearthe walls. This is partly because the gas **in** the centeris much less dense, **and** partly because the gas **in** thecenter occupies less volume than the gas near thewall due to the axisymmetry **of** the arc tube.The electrical potential is shown **in** Figure 6.Figure 7: Pressure referenced to zero at the lowerelectrode tipThis characterizes the scale **of** the buoyancy forces**and** shows that the pressure difference required tosupport the weight **of** the gas is about 2 Pa.ConclusionF**in**ite-element analysis us**in**g FEMLAB is anefficient method for obta**in****in**g basel**in**e **in**formationon the work**in**g **of** a **mercury**-filled HID lamp.

Excerpt from the Proceed**in**gs **of** the COMSOL Multiphysics User's Conference 2005 BostonReferences1. Elenbaas W., The High Pressure MercuryVapour Discharge, North-Holl**and**,Amsterdam, 19512. Zollweg, R. J., Convection **in** vertical highpressure**mercury** arcs, J. Appl. Phys., 49(3),1077-1091 (1978).3. Lowke, J. J., Calculated properties **of**vertical arcs stabilized by naturalconvection, J. Appl. Phys., 50(1), 147-157,(1979).4. Shyy, W., Effects **of** convection **and** electricfield on therm**of**luid transport **in** horizontalhigh-pressure **mercury** arcs, J. Appl. Phys.,67(4), 1712-1719, (1990).5. Li, Yan-M**in**g, personal communication