Stability and Bifurcation of Cathode Arc Attachment in ...

Presented at the COMSOL Users Conference 2006 Boston

Stability and Bifurcation of Cathode Arc Attachment in AC Operated HighIntensity Discharge Lamps Alan Lenef OSRAM SYLVANIA Lighting Research Center Beverly, MA 01915

Outline Electrode model for high-intensity discharge (HID) lamp Stability of attachment modes Example calculations with FEMLAB and comparison to stability theory Conclusions

COMSOL-Boston 22-24 Oct 2006

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Anatomy of a metal halide discharge lamp Arc tube: Alumina/quartz Vapor: Outer jacket:

Hg, Ar, Na, I, Dy, NaI,…

UV blocking glass

i

Mantel: hot radiating vapor Arc:

Molten salts:

T ≈ 4500 – 5000K

T ≈ 1000 – 1500K

ne ≈ 0.01 natoms

NaI, ScI3, TlI, DyI3,…

p ≈ 1 – 10 bar


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Some applications of HID lamps SYLVANIA METALARC® BRITELINE® LIGHTS UP BIG GAMES AT ARIZONA'S SUN DEVIL STADIUM

“Wöhrl City” in Nuremberg city centre

70 W OSRAM Powerball® HCI TC (ceramic)

2000 W Metal Halide Lamps COMSOL-Boston 22-24 Oct 2006

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Examples of arc attachment modes in HID lamps

Lamp 32: 0.7mm, 867 hrs. Cathode diffuse (L/R = 22.3)

Lamp 51: 1.1mm, 100 hrs. Cathode spot (L/R = 14.6)

(tungsten φw = 4.5 eV) COMSOL-Boston 22-24 Oct 2006

Data courtesy H. Adler (OSI) 5

Goals Understand different arc attachment modes in cathode phase

Constricted (spot) modes can damage electrodes and blacken lamps from excessive evaporation Determine conditions for various types of arc attachment

Demonstrate how highly non-linear heating of electrode in cathode phase leads to spatial mode changes

Use a boundary layer (BL) approach (Benilov, Bötticher, Neumann) Simple model: focus on conditions for uniform (diffuse) heating instability Show how anode to cathode transition (AC lamp operation) can further destabilize cathode diffuse mode COMSOL-Boston 22-24 Oct 2006

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Thermal model of electrode In electrode body with temperature distribution T ∇ ⋅ κ∇T = ρ c p

(

)

∫ dA ( ji + je )

S1+ S2

z=L

S1

∂T 4 = εσ SB T 4 − Tamb − qa ,c (T ,V ) ∂n I (t ) =

R

∂T ∂t

Boundary conditions on (S1) and S2 −κ

S2

Current I(t) fixes sheath potential V(t)

Boundary conditions on S0 T = T0 COMSOL-Boston 22-24 Oct 2006

z=0 T = T0

S0

je = electron current (A/m2) ji = ion current (A/m2) κ(T) = heat conductivity (W/m/K) cp = heat capacity (J/kg/K) ρ= density electrode (kg/m3) σSB = Stefan-Boltzmann constant (5.67 × 10-8 W m-2 K-4) ε = emissivity of electrode (0.37 for tungsten) 7

Cathode sheath description ionization

Thermionic emission (Schottky-enhanced)

[

je (T , Ec ) = ART 2 exp − e (φw − e′Ec ) k BT

+

]

e′ = e 4πε 0

ji je

Ion current in collisionless sheath (MacKeown Equation) 12

⎛ e ⎞ ⎟⎟ ji = ε 0 ⎜⎜ ⎝ 8mi ⎠

E c2 V1 2

Ionization energy balance jeV = jiVi


+ + + V + +

Electrode surface (-)

T = local surface temperature AR = Richardson coefficient (A/m2/K) (1.2 × 106 A/m2/K2) e = electron charge (C) ε0 = free-space permittivity (8.85 × 10-12 F/m) kB = Boltzmann’s constant (1.38 × 10-23 J/K) mi = ion mass (Kg) Ec = electric field at cathode surface (V/m) Vi = ionization energy (eV) V = sheath potential (V)

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Solutions to ionization energy balance equation •Literature models generally •Include detailed energy balance at sheath edge ⇒ saturated ion currents at high temperature •Often interpret appearance of spot as consequence of saturated jion

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10

V = 20V 10

2

Current density (A/m )

10

100% ionization

9

10

8

10

•Simple approach (this work)

7

10

jion

6

10

(V/Vi) je (2900K) (V/Vi) je (3500K)

5

10

Low field solution

(V/Vi) je (3600K)

4

10

6

10

7

10

8

10

9

10

10

10

11

10

•Neglect ionization limit, other electron energy loss ⇒ no saturation effects •Determine if different stable modes exist without saturated jion •Determine if saturation effects are important for driving spot formation

Esheath (V/m)


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Electrode heating qc (T ,V ) = ji (V + Vi − φ w ) − jeφ w

5k T⎞ ⎛ qa (T ,V ) = je ⎜ φw + B ⎟ 2 e ⎠ ⎝

ions

e

ions e

φw

(-)

φw

Ef

(+) Cathode

Ef

Anode

φw = 4.5 eV for pure tungsten, 2.5 – 3.5 eV for thoriated tungsten COMSOL-Boston 22-24 Oct 2006

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Electrode heat fluxes from plasma

•Lower currents (for given R) ⇒Anode cools relative to cathode

2

q (W/m )

1E9

V = 10 V (cathode) V = 20 V (cathode) anode (I = 6A, R = 1 mm) Cathode temperature @ I = 6A, R = 1 mm

1E8

•Higher currents ⇒Anode heats relative to cathode

Cathode Anode 1E7

2400

2600

2800

3000

3200

3400

3600

3800

Electrode temperture T (K)


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Intuitive explanation of mode instability q

Δq

λchar

r

κ κ ⎛ ∂q0 ⎞ ΔT ≈ ΔT ⇒ ⎜ ⎟ ΔT = Δq > R β mp λchar ⎝ ∂T ⎠

Instability

•The onset of diffuse mode instability → occurs when change that increases heat flux from a local temperature fluctuation ΔT cannot be balanced by dissipation through corresponding induced temperature gradients •Amplification factor dq0/dT increases with: • Higher sheath potential (greater ion energy) • Increased electrode temperature (greater ion current density) COMSOL-Boston 22-24 Oct 2006

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Analytic theory of mode instability Let temperature distribution evolve up to time t0 – example is just after transition from anode to cathode Examine stability of eigenmodes of linearized BL model at t > t0 Example: only top surface S2 interacts with discharge

T1 (t ) =

Uniform T0(t) on S2 is exact diffuse mode solution Perturbation in heat flux excites stable or unstable spatial modes

∑ Am, p J m (α mp r )cos(mθ ) m, p

)

(

(

2 × sinh α mp − λmnp z exp − Dλmnpt

α mp = β mp R

(

)

′ β mp = 0 Jm

⎛ 2 2 − R 2λmnp coth⎜ β mp − R 2λmnp β mp ⎝

) =

Thermal diffusion coefficient

D ≈ κ ρc p COMSOL-Boston 22-24 Oct 2006

R ∂q0 (t − t0 ) ≡ Ks ∂T κ S2

m= p≠0

Eigenvalue equation for λmnp determines diffuse mode stability 13

L⎞ ⎟ R⎠

Rod with DC current + heat flux perturbation – bifurcation behavior 2800

1.8 mm (edge) 1.8 mm (center) 2.0 mm (edge) 2.0 mm (center)

Temperature (K)

2780 2760 2740 2720 2700

1.8/2.0 mm dia x 5 mm length Perturbation 2 A DC, T0 = 1000K

2680 2660 0.0

0.1

0.2

0.3

0.4

0.5

Time (s)

•(Blue) 1.8 mm diameter has instability factor Ks = 3.6316 •(Olive) 2.0 mm diameter has instability factor Ks = 3.9335. •Critical instability factor for ring mode is Ks ≈ β01 = 3.832 COMSOL-Boston 22-24 Oct 2006

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Movies of rod with perturbation at 0.1 s

Stable diffuse (1.8 mm dia) COMSOL-Boston 22-24 Oct 2006

Unstable diffuse (2.0 mm dia) 15

Anode → cathode transition causing instability 1.8 mm dia x 5 mm long rod with 6 A square-wave (DC stable case) Radial heat flux perturbation: 0.24 – 0.25 s f = 20 Hz; I.C.: DC 6 A cathode solution; T0 = 1000K 16

Center Edge

3100

Central spot

14

cathode

12

3000

Ring

10

2900

Ks

Temperature (K)

Center Edge

8 6

2800 4 2

2700

anode

0 -2

2600 0.0

0.1

0.2

0.3

0.4

0.5

0.0

0.1

0.3

0.4

Time (s)

Time (s)

Instability factor

Temperatures

Ks > 3.83 for ring mode

Anode phase much cooler ⇒ Large sheath potential on next ½ cycle COMSOL-Boston 22-24 Oct 2006

0.2

16

0.5

Movie of anode induced-instability


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Simulated “anode cooling” effect on cathode stability for hemispherical tip

60

3300

3100

Cathode Sheath Potential (V)

Temperature (K)

3200

Edge (1.50A) Center (1.50A) Edge (1.59472A) Center (1.59472A) Edge (1.59480A) Center (1.59480A)

3000

2900

2800

1.50A 1.59472A 1.59480A

50

40

30

2700 1E-4

1E-3

0.01

0.1

20 1E-4

1E-3

0.01

0.1

Time (s)

Time (s)

Instability border: At t = 0s, switch cathode current from I = 0.75 A to 1.50A, 1.59472A, and 1.59480A (1.0 x 7.0 mm rod with hemispherical tip; T0 = 1000K)


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Movie of anode cooling for hemispherical tip: 0.75 → 1.50 A


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Conclusions Used FEMLAB to develop 2D time-dependent model to study bifurcations and stability of BL electrode model for HID lamps Simulations agree with stability theory for very simple model Results show how diffuse mode instability occurs by amplification of local heat flux perturbations in cathode sheath → drives formation of localized spatial modes Ion current saturation and additional energy losses in ionization layer often determine the final spatial mode (attractor) but do not drive instability Results show additional modes can exist without saturation effects Acknowledgements: I would like to thank Y. M. Li (Osram Sylvania) and W. Graser (Osram) for many discussions. Also, I would like to thank Y. M. Li for his work on the original electrode FEMLAB code. COMSOL-Boston 22-24 Oct 2006

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