A1260974707Gscti

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Radiation flux density in short lamp arcs
David Wharmby Technology Consultant david_at_wharmby.
demon.co.uk
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

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Compact arcs need models for development
Short arcs have strong flows, no symmetry,
dominant electrode effects Time-dependence models
are needed Materials are very highly stressed
mm
Source Miguel Galvez, LS10 Toulouse, 2004
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Compact 3D arc models

• 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

Source M. Galvez, paper P-160 LS10 Toulouse, 2004
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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
resources Most approaches are unsatisfactory for
short HID lamps
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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 k(l,T)
• Total intensity in just one direction needs
  triple
• integration over s, ,l

plasma
Planck
optical depth
absorption coefficient
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Example calculation
Across diameter 100 torr Na plasma, parabolic
profile 4200K-1500K, Stark resonance broadening

• 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)

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Is plasma optically thin?

• Information needed for experiment and model
• Measurement of transmittance is unreliable in
  lamps
• tgt0.95 (say)
• Better guide
• compare measured spectral radiance with
  line-of-sight radiation transport calculation
  using assumed temperature distribution
• At given l plasma is thin when
• measured spectral radiance ltlt 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 measurement
2. Calculate radiance/BB radiance at Tmax,
   assuming T(r)

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How do we know that a plasma is optically thin?

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

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How do we know that a plasma is optically thin?

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

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Absorption coefficient k(l,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

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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?
Molecular absorption

• 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
Source A Gallagher in Excimer Lasers
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Molecular effects in the wing of the Tl line
Measurements 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 q and f
• This vector is radiation powerFRl through unit
  area at r (W m-2 nm-1)
• Total RFD FR (W m-2) needs 5th integration over l
  
• 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) 4peN
• difference between emission and absorption in
  element of plasma

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Treating radiation complexity in order of
increasing computer time
Remember RFD must be evaluated many times during
energy balance

• Ignore it by unphysical approximations
• t gtgt1 (diffusion)
• t ltlt1 (optically thin)
• Reduce complexity of RFD integral using symmetry
• e.g cylindrical
• Find a realistic way to express eN as a local
  quantity
• Find ways to pre-tabulate some of the integral
• Sevastyanenko (pre-tabulate integration over l)
• Galvez (pre-tabulate geometry)
• Use Monte Carlo methods
• Full-blooded numerical integration

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Treating radiation complexity in order of
increasing computer time
Remember RFD must be evaluated many times during
energy balance
often hopeless

• Ignore it by unphysical approximations
• t gtgt1 (diffusion)
• t ltlt1 (optically thin)
• Reduce complexity of RFD integral using symmetry
• e.g cylindrical
• Find a realistic way to express eN as a local
  quantity
• Find ways to pre-tabulate some of the integral
• Sevastyanenko (pre-tabulate integration over l)
• Galvez (pre-tabulate geometry)
• Use Monte Carlo methods
• Full-blooded numerical integration

20
Treating radiation complexity in order of
increasing computer time
Remember RFD must be evaluated many times during
energy balance
often hopeless

• Ignore it by unphysical approximations
• t gtgt1 (diffusion)
• t ltlt1 (optically thin)
• Reduce complexity of RFD integral using symmetry
• e.g cylindrical
• Find a realistic way to express eN as a local
  quantity
• Find ways to pre-tabulate some of the integral
• Sevastyanenko (pre-tabulate integration over l)
• Galvez (pre-tabulate geometry)
• Use Monte Carlo methods
• Full-blooded numerical integration

many examples Lowke, TUe
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Treating radiation complexity in order of
increasing computer time
Remember RFD must be evaluated many times during
energy balance
often hopeless

• Ignore it by unphysical approximations
• t gtgt1 (diffusion)
• t ltlt1 (optically thin)
• Reduce complexity of RFD integral using symmetry
• e.g cylindrical
• Find a realistic way to express eN as a local
  quantity
• Find ways to pre-tabulate some of the integral
• Sevastyanenko (pre-tabulate integration over l)
• Galvez (pre-tabulate geometry)
• Use Monte Carlo methods
• Full-blooded numerical integration

many examples Lowke, TUe
Jones Mottram?
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Treating radiation complexity in order of
increasing computer time
Remember RFD must be evaluated many times during
energy balance
often hopeless

• Ignore it by unphysical approximations
• t gtgt1 (diffusion)
• t ltlt1 (optically thin)
• Reduce complexity of RFD integral using symmetry
• e.g cylindrical
• Find a realistic way to express eN as a local
  quantity
• Find ways to pre-tabulate some of the RFD
  integral
• Sevastyanenko (pre-tabulate integration over l)
• Galvez (pre-tabulate geometry)
• Use Monte Carlo methods
• Full-blooded numerical integration

many examples Lowke, TUe
Jones Mottram?
control of approx?
23
Treating radiation complexity in order of
increasing computer time
Remember RFD must be evaluated many times during
energy balance
often hopeless

• Ignore it by unphysical approximations
• t gtgt1 (diffusion)
• t ltlt1 (optically thin)
• Reduce complexity of RFD integral using symmetry
• e.g cylindrical
• Find a realistic way to express eN as a local
  quantity
• Find ways to pre-tabulate some of the integral
• Sevastyanenko (pre-tabulate integration over l)
• Galvez (pre-tabulate geometry)
• Use Monte Carlo methods
• Full-blooded numerical integration

many examples Lowke, TUe
Jones Mottram?
control of approx?
will examine in detail
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Treating radiation complexity in order of
increasing computer time
Remember RFD must be evaluated many times during
energy balance
often hopeless

• Ignore it by unphysical approximations
• t gtgt1 (diffusion)
• t ltlt1 (optically thin)
• Reduce complexity of RFD integral using symmetry
• e.g cylindrical
• Find a realistic way to express eN as a local
  quantity
• Find ways to pre-tabulate some of the integral
• Sevastyanenko (pre-tabulate integration over l)
• Galvez (pre-tabulate geometry)
• Use Monte Carlo methods
• Full-blooded numerical integration

many examples - Lowke
Jones Mottram?
control of approx?
will examine in detail
useful for checking
out of sight
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Infinite cylindrical geometry
wall

• treated by Lowke for Na arcs
• shows contribution from various rays to RFD in
  blue element

q
s(f)
f
r
J J Lowke JQRST 9, 839-854, 1969
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Infinite cylindrical geometry

• shows contribution at r to RFD from ray in
  direction u
• reduce evaluation of to 4 integrations by
  projecting q variation onto horizontal plane
  using pre-tabulated function G1(t(s(f))
• FR only has a radial component

Sodium arc
Jones Mottram FR after Lowke (1969)
eN (1/4p)div(FR)
J J Lowke JQRST 9, 839-854, 1969
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Jones and Mottram - eN as an approximate local
function

• Guess temperature to start to energy balance and
  calculate RFD FR exactly
• Calculate eN(r) from div FR
• Represent eN(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 eNfit to represent radiationuntil energy
  balance converges
• Recalculate eNfit
• Converge energy balance again

For HPS in cylindrical geometry requires only 3
RFD evaluations
Jones BF Mottram DAJ J. Phys. D Appl. Phys.
14, 1183-94, 1981
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Jones and Mottram - eNfit as an approximate local
function

• Makes eN 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 eNfit, 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 eNfit(T(u), P, FR) to FR one or more
  directions u

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
temperature contours
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5
A
4
1
3
2

• Green cell A receives radiation from all other
  cells(e.g. n 1 . .6)
• Amount of radiation from cell n is k(l,Tn)
  Bl(Tn)
• Absorption at A depends k(l,TA)
• These depend on local values of temperature
• The heavy computation occurs because the spectrum
  emitted cell n is selectively absorbed in the
  path to 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

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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

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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

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Galvez method geometrical precalculation

• 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

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 4p/N
  4p/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
For each ray in from each start cell
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)

Geometrically complicated integral for t 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

Inner Wall Temperature Distribution
Molten salts
Molten salts
Source S Juengst, D Lang, M Galvez, LS10
Toulouse, Paper I-14 2004
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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

Source M Galvez, LS10 Toulouse, Paper P-137 2004
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The future?

• Galvez method will prove to be method of choice
  for asymmetric arcs
• Iteration will be further speeded up by Jones
  Mottram semi-empirical eN 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

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