radiation flux density in short lamp arcs

1

Radiation flux density inshort lamp arcs

David WharmbyTechnology Consultant

[email protected]

COST 529 12-16 April 2005

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Outline

• Why is radiation flux density (RFD) important?• Line-of-sight radiation transport• Optical depth• Absorption coefficient• Radiation flux density• RFD in cylindrical geometry• Jones and Mottram net emission coefficient• Calculation of RFD in arbitrary geometry• Galvez’ method• Summary

3

Compact arcs need models for development

Tbottom= 1255K

= 1305KTtop

Tmax= 9095K

Tmin = 1325K

gas temperature velocity

outsidewall temperature

Short arcs have strong flows, no symmetry, dominant electrode effects

Time-dependence models are needed

Materials are very highly stressed

Source Miguel Galvez, LS10 Toulouse, 2004

~mm

4

• 50% of power may escape as radiation • High pressures mean that most of spectrum is at medium optical depth• But . . .LTE is usually OK, thankfully• Chemistry, conduction, convection and radiation must be included• Short arcs have radial and longitudinal temperature gradients, non-uniform

E field• Steady state energy balance

– all terms depend on temperature– solution gives the temperature field

• Satisfactory treatment of radiation flux density vector FR (W m-2) is critical because radiation is so dominant

Compact 3D arc models

Source M. Galvez, paper P-160 LS10 Toulouse, 2004

W m-3termsconvectionFdivFdivE CR _)()(2

conductionradiationpower in

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Role of radiation

• E field accelerates electrons• electrons collisions produce excited states• emitted photons may escape or may be absorbed • absorption determines excited state densities• photons can travel throughout the plasma & affect excited state

densities elsewhere• non-linear & non-local system

– electron and excited state densities depend exponentially on T– absorption and emission processes depend very strongly on

frequency– absorption depends on emission from rest of plasma

Radiation transport requires massive computer resourcesMost approaches are unsatisfactory for short HID lamps

6

Line of sight radiation transport

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Line of sight radiation transport• Spectral intensity (spectral radiance) of ray direction u at position r in the

plasma (W m-2 sr-1 nm-1)

• LTE (& Kirchhoff), no scattering, no incoming radiation

• Only data needed is local value of (,T)• Total intensity in just one direction needs triple

integration over s, ,

sdsdsTTTBsIs

s

s

so

))(,(exp),()()(

Planck

absorptioncoefficient

optical depth

s

plasma

I(s)

so

ss

u

r

8

Example calculation

• SR intensities are a guide to maximum temperature

– independent of oscillator strength, number density

– slightly dependent of T(r)

• SR dips can give some information about T(r)

Across diameter 100 torr Na plasma, parabolic profile 4200K-1500K, Stark & resonance broadening

0

2000

4000

6000

500 600 700 800

wavelength (nm)

spec

tral

rad

ianc

e

wall radiance

Planck

0

2000

4000

6000

585 590 595

wavelength (nm)

spec

tral

rad

ianc

e

wall radiance

Planck

centerradiance

0

2000

4000

6000

816 818 820 822

wavelength (nm)

spec

tral

rad

ianc

e

wall radiance

Planck

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Is plasma optically thin?

• Information needed for experiment and model

• Measurement of transmittance is unreliable in lamps

– t>0.95 (say)

• Better guide

– compare measured spectral radiance with line-of-sight radiation transport calculation using assumed temperature distribution

• At given plasma is thin when

– measured spectral radiance << Planck radiance at highest T

• For energy balance calculation

– radiation that is neither thick nor thin affects temperature profile

– needs full RFD calculation.

Source Griem “Plasma Spectroscopy”, 1964

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How do we know that a plasma is optically thin?

1. Make spectral radiance measurement2. Calculate [radiance/BB radiance] at Tmax, assuming T(r)

0.0

0.2

0.4

0.6

0.8

1.0

550 600 650wavelength (nm)

tra

nsm

itta

nce

0.00

0.01

0.02

rad

ian

ce fr

act

ion

spectralradiance (asfraction of BB)

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1. Make spectral radiance measurement2. Calculate ratio radiance/BB radiance at Tmax, assuming T(r)3. Calculate transmittance t = exp(-)

0.0

0.2

0.4

0.6

0.8

1.0


tra

nsm

itta

nce

0.00

0.01

0.02

rad

ian

ce fr

act

ion

plasmatransmittance

spectralradiance (asfraction of BB)

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1. Make spectral radiance measurement2. Calculate ratio radiance/BB radiance at Tmax, assuming T(r)3. Calculate transmittance t = exp(-)4. Where is t > 0.95

0.0

0.2

0.4

0.6

0.8

1.0


tra

nsm

itta

nce

0.00

0.01

0.02

rad

ian

ce fr

act

ion

transmittance

95%transmittance

spectral radiance(as fraction ofBB)

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Absorption coefficient (,T) data example

• Example– High pressure Hg at

8000K• Absorption from

– lines: resonance, van der Waals & Stark

– e-a and e-i Brems.– e-i recombination – molecular

• Omission of molecular wing of lines gives imperfect line profile

10

100

1000

10000

100000

1000000

540 580 620 660 700

wavelength (nm)

ab

sorp

tion

co

effi

cie

nt (

1/m

)

total

lines

electron-atom Brems.(Lawler)

electron-ion Brems.

Source Lawler, J. Phys. D: Appl. Phys. 37, 1532-6, 2004 (e-a data for Hg)Hartel, Schoepp & Hess, J. App. Phys 85, 7076-7088, 1999 (line broadening)

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8.0

8.5

9.0

546

436

405

6 3,1D 1S0

365 group

313 group

297

X

G

D

F

E

B A

185

Molecular absorption

Source A Gallagher in “Excimer Lasers”

• Green transitions give absorption in UV 254nm resonance line

• Blue transitions affect extreme wing profile of non-resonance lines

• Upper levels are completely unknown

• Generally bb, ff, fb and bf emission from molecules will be important

254

?

15

100

80

60

40

20

0

relative intensity %

wavelength (nm)

530 534 542538

Molecular effects in the wing of the Tl lineMeasurements of Tl resonance line broadened by Tl and Hg

Note strongly curtailed red wing

Time dependent spectra on 50Hz operation

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Radiation flux density

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Radiation flux density (RFD)

• Integrate intensity (vectorially) passing through area at r in directions u

• Two more integrations over and

• This vector is radiation powerFRthrough unit area at r (W m-2 nm-1)

• Total RFD FR(W m-2) needs 5th integration over

• For a uniform element, div(FR) (W m-3) gives radiation power (W m-3) in element for calculation of energy balance

• net emission coefficient div(FR) = 4N

– difference between emission and absorption in element of plasma

durIrF R ),()(

r

I(r,u2)

FR(r)

I(r,u1)

I(r,u4)

I(r,u3)

plasma

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Treating radiation complexity – in order of increasing computer time

• Ignore it by unphysical approximations

– >>1 (diffusion)

– <<1 (optically thin)

• Reduce complexity of RFD integral using symmetry

– e.g cylindrical

• Find a realistic way to express N as a local quantity

• Find ways to pre-tabulate some of the integral

– Sevast’yanenko (pre-tabulate integration over )

– Galvez (pre-tabulate geometry)

• Use Monte Carlo methods

• Full-blooded numerical integration

Remember RFD must be evaluated many times during energy balance

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– >>1 (diffusion)



– e.g cylindrical








often hopeless

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– >>1 (diffusion)



– e.g cylindrical








many examples – Lowke, TUe

often hopeless

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– >>1 (diffusion)



– e.g cylindrical









Jones & Mottram?

often hopeless

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– >>1 (diffusion)



– e.g cylindrical


• Find ways to pre-tabulate some of the RFD integral







Jones & Mottram?

control of approx?

often hopeless

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– >>1 (diffusion)



– e.g cylindrical









Jones & Mottram?

control of approx?will examine in detail

often hopeless

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– >>1 (diffusion)



– e.g cylindrical








many examples - Lowke

Jones & Mottram?

control of approx?will examine in detail

useful for checking

out of sight

often hopeless

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Infinite cylindrical geometry • treated by Lowke for Na arcs• shows contribution from various rays to RFD in

blue element

J J Lowke JQRST 9, 839-854, 1969

r

s()

wall

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-5

0

5

10

15

20

25

0.0 0.2 0.4 0.6 0.8 1.0

radial position-5

0

5

10

15

20

F R (MW m-2) N (MW m-3 sr-1)

Infinite cylindrical geometry • shows contribution at r to RFD from ray in

direction u • reduce evaluation of to 4 integrations by

projecting variation onto horizontal plane using pre-tabulated function G1(s)

• FR only has a radial component

J J Lowke JQRST 9, 839-854, 1969

Sodium arc

Jones & Mottram FR

after Lowke (1969)

N = (1/4)div(FR)

r

u

s()

wall

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Jones and Mottram - N as an approximate local function

• Guess temperature to start to energy balance and calculate RFD FR exactly

• Calculate N(r) from div FR

• Represent N(r) as a function (T)- Emission part depends on depends on upper state number density- Absorption part depends on FR and lower state number density

• Use Nfit to represent radiation

until energy balance converges

• Recalculate Nfit

• Converge energy balance again

))(/exp()())(/exp()(

)]([rTdrcrTba

rT

rTpRNfit F

Jones BF & Mottram DAJ J. Phys. D: Appl. Phys. 14, 1183-94, 1981

For HPS in cylindrical geometry requires only 3 RFD evaluations

-5

0

5

10

15

1000 2000 3000 4000 5000

temperature (K)

net

emis

sion

coe

ffici

ent

Jones and Mottram data

estimated from empirical formula

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• Makes N seem local as long as conditions do not change too much

• The closer the arc temperature profile is to the guessed profile used to

calculate Nfit, the more rapid the solution of energy balance

• Particularly applicable to calculating effect of

– a sequence of changes of pressure or power

– time-dependent solutions of energy balance

because from previous input values can be used

• Can this be used in 3D???

– do full RFD calculation using Galvez or other method

– fit Nfit(T(u), P, FR) to FR one or more directions u

Jones and Mottram - Nfit as an approximate local function

This could be a powerful aid but needs to be tested

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Radiation flux in asymmetric 3D plasma

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Radiation flux in asymmetric 3D plasma

Green cell A receives radiation from all other cells(e.g. n = 1 . .6)

• Amount of radiation from cell n is TnB(Tn)

• Absorption at A depends TAThese depend on local values of temperature

The heavy computation occurs because the spectrum emitted cell n is selectively absorbed in the path to A

temperature contours

1

2

3

4

56

A

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Galvez method – geometrical precalculation

• 2D picture of 3D process, showing finite volumes in calculation mesh• From a starting cell (green) take rays to other parts of the plasma

wallwall

32



wallwall

33



wallwall

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• 2D picture of 3D process, showing finite volumes (FV) in calculation mesh• From a starting cell (red) take rays to other parts of the plasma

• For each ray tabulate – which FV is emitting ray– which FV the ray crosses– Distance traversed in each FV– Which FV is the exit volume

wallwall


geometry only!

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How many rays are needed?

• FV mesh

• Rays emitted from a single cell chosen at random

• Let cell emits N rays isotropically

• N increased until the rays visit at least 95% of the cells

• N used by Galvez is typically 100

• So solid angle element for each ray – 4/N =4/100

• Repeats checks using other FV for emission confirm 100 is about enough for a good representation of the radiation field

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Pre-tabulate following

wall

ray 3

9

2345678

10j =

3 4 5 6 7 8 9210k =

Start cell s(j,k) s(4,3) Ray number r 3 Cells visited n(j,k) n(3,4) n(3,5) n(2,6) n(2,7) n(1,7) n(1,8) n(1,9) n(0,9) Distance in cell d(j,k) d(3,4) d(3,5) d(2,6) d(2,7) d(1,7) d(1,8) d(1,9) d(0,9) Exit cell e(j,k) e(0,9)

For each ray in from each start cell

s

s kj

kjdkjTsdsTss,

),()],(,[))(,(),,(

Geometrically complicated integral for then becomes a simple sum based on pre-tabulated geometry & absorption coefficients

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Results

• Galvez at LS10 gave an example of a MH short arc energy balance calculation with

– 7400 finite volumes

– 100 rays per finite volume

– 19000 wavelengths

• Used in design study of ceramic metal halide lamps

– calculates inner wall temperature

– bulge shape avoids corrosion at corners of cylinder

– salt temperature is more independent of orientation – better color

– bulge shape reduces mechanical stress by factor 2

Source S Juengst, D Lang, M Galvez, LS10 Toulouse, Paper I-14 2004

Inner WallTemperature Distribution

Molten salts

Molten salts

Inner Wall Temperature Distribution


Molten salts

Molten salts



Molten salts

Molten salts

Inner Wall Temperature Distribution

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Advantages of Galvez method

• Straightforward evaluation of div(FR)

– In the course of energy balance FR is updated every 10 to 20

iterations

– Look up tables for geometry and absorption coefficients used to calculate non-local part of integral

• Spectral flux from finite volumes adjacent to wall

– summed to give the spectral flux (power) distribution

– this can be compared with measurements in an integrating sphere

• Accuracy approaches that of full integration as number of rays N increases

• As with many RFD calculations it can be parallelized

• Applicable to time-dependent solutions

Source M Galvez, LS10 Toulouse, Paper P-160 2004

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Conclusions

• Ray tracing precalculation of Galvez is a major advance

– provides a practical solution to RFD calculations in arbitrary geometry• Computational speed means that it can be applied to arbitrary geometry with

convection • Has been applied to time-dependent calculations

• Example is ultra high mercury pressure video projection lamp showing gas temperature, combined with electrode sheath model

Tmax = 9095K

Tmin = 1325K

Source M Galvez, LS10 Toulouse, Paper P-137 2004

3684K

3602K

3556K

3511K

3465K

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• Galvez method will prove to be method of choice for asymmetric arcs

• Iteration will be further speeded up by Jones & Mottram semi-

empirical N approach (or something like it)

• This will make radiation modelling of time-dependent short arcs practical

• But– Better data on high temperature absorption coefficients will

be needed, especially for bb, bf, ff molecular processes

The future?

radiation flux density in short lamp arcs

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