Summary of the previous lecture

1. Plasma chemistry Plasma propulsion Plasma light Summary of the previous lecture Particles: Which species do we have how “much” of each Momentum: How do they move? Energy: What about the thermal motion internal energy What do we want/need to know in detail?

2. Particles Plasma Particles Energy Energy Momentum Momentum Particles: Plasma Chemistry Energy: Plasma Light Momentum: Plasma Propulsion

3. Hybride Quasi Free Flight mean free paths large mfp > L Sampling and tracking Transport Modes Fluid mean free paths small mfp << L There are many conditions for which some plasma components behave “fluid-like” whereas others are more “particle-like” Hybride models have large application fields

4. Source Discretizing a Fluid: Control Volumes Plasma Particles Particles Energy Energy Momentum Momentum For any transportable quantity  Transport via boundaries

5. How many species? Examples of transportables Densities Momenta in three directions Mean energy (temperature) Depends on Equilibrium departure As we will see: In many fluid/hybrid cases Energy: 2T: {e} and {h} Momentum: for the bulk Navier Stokes for the species: Drift Diffusion Species: the transport sensitive

6. Mean properties  Nodal Points Transport at boundaries  = Source, t  + = S Steady State Transient General structure:   = u -D  Convection Diffusion Nodal Point communicating via Boundaries Transport Fluxes: Linking CV (or NP’s)  -

7. Other Example: Poisson: .E = /o  = S   = u -D E = - V Simularities Thus: The Fluid Eqns: Balance of Particles Momentum Energy The Momenta of the Boltzmann Transport Eqn. Thus no “convection” Can all be Treated as  -equations

8. The  Variety  D S Temperature Heat cond Heat gen Momentum Viscosity Force Density Diffusion Creation Molecules atoms ions/electrons etc.

9. T Continuum t  + = S Tin Rod Tout x 0 + T = 0 Take k = Cst MathNumerics: a FlavorSourceless-Diffusion T = Cst T = - kT -T /k =T

10. Continuum Tin Rod Tout Discretized Intuition; T = Cst T2 = (T1 + T3)/2 1 2 3 4 Tin -2T1 + T2= 0 T1 - 2T2 + T3 = 0 T2 - 2T3 + T4 = 0 T3 - 2T4 + Tout = 0 2T2 = T1 + T3 Discretized

11. In matrix: M T = b A Sparce Matrix Many zeros Matrix Representation 1 2 3 4 -Tin 0 0 -Tout T1 T2 T3 T4 1 2 3 4 - 2 1 1 -2 1 1 -2 1 1 -2 =

12. Sourceless-Diffusion in two dimensions 1 1 – 4 1 1 N W P E S T5 = (T2 + T4 +T6 + T8 ) /4 Provided k = Cst !! In general:

13. If k Cst Convection Diffusion More general S-less Diffusion/Convection

14.   = u -D  = S   = -D If no “convection” Poisson Laplace - D = S S 0 S=0 - D = 0 - D = S Laplace and Poisson

15. Ohms’ law Space charge Ener balance Ion balance .E = /o .j = 0 j = E . E = 0 E = -V -.Dn = Ion - Rec -.kT = S -. V = /o -.V = 0 Examples Simularities !!

16. N Each V the Average of the two adjacent V d A Capacitor space-charge zero -. V = /o = 0 0 +V Basically a 1-D problem

17. +V 0 N Each V the Average of the two adjacent V d A resistor: Ohms law -.V = 0 1-D problem Provided  is Cst

18. Source of ions Example ions:  n+u+ = P+ - n+D+ Recombination Ordering the Sources  = S S = P - L L ~ D Source combination Production and Loss Large local - value in general leads to large Loss

19. t = Nt Concept disturbed Bilateral Relations  A proper channel  N f N b Equilibrium Condition: t/b << 1 or t b << 1 The escape per balance time must be small

20. N D t = Nt N D Mixed Channel   P - nD = n u The larger D The less important transport for  The more local chemistry determined Note D is more general than bin dBR: collective chemistry

21. Cells are needed To organize the field contributions Ingredients for non-fluid codes The more equilibrium is abandoned the more info we need Tracking the particles: Integrating Eqn of Motion F = ma Interaction by chance: Monte Carlo Field contructions: a) positions giving charge density  E b) motion giving current density  B

22. Fluid versus particles (swarms) Particle codes Directly binary interacting individual particles Bookkeeping Position/velocity Each indivual part Particle in cell interaction via self-made field Sampling Distribution Over r and v Hybride particles in a fluid environment Distribution function Known in shape Continuum

23. A quasi free flight example Radiative Transfer • Ray-Trace Discretization spectrum. • Network of lines (rays) • Compute I (W/(m2 .sr.Hz) along the lines • Start outside the plasma with I() = 0. • Entering plasma I() grows afterwards absorption. • dI()/ds = j- k()I()

24. Ray Tracing

25. General Procedure ?? Fluid Swarm Collection {h} {i} {e} {} {h} {i} {es} {ef} {} E {h} {i} {es}{ef} {} E {h} {i} {es}{ef} {} E {h} {i} {es}{ef} {} Pressure

26. The BTE: basic form The BTE deals with fi(r,v,t) defined such that fi(r,v,t)d3rd3v  the number of particles of kind “i” in a volume d3r of the configuration space centered around r with a velocity in the velocity space element d3v around v. Examples e A, A+, A*, A**, etc, N, N2, etc. NH, H2 Note that “i” may refer to an atom in a ground state the same atom in an excited state An ion or molecule, etc. etc. The BTE states that: tfi + .fiv + v.fia = (tfi)CR

27. Source S tn accumulation . nv and/or Efflux Simularities of the Boltzmann Transport equation tn + . nv = S Leads to

28. Generalization to 6-D phase space Normal space tn + . nv = S Accumulation Transport Source tfi + .fiv = (tfi)CR Phase space  The Boltzmann Transport Eqn

29. The BTE: general form Use the divergence in the 6 dim  space (r x v). tfi + .fiv = (tfi)CR BTE: Shorthand notation This is a  equation in  space Representing tfi + .fiv + v.fia = (tfi)CR

30. I Multiply BTE for “i”with a function g(v) and integrate over v. For each species: specific balances of mass; charge; current; momentum; energy Higher structures units: Elements; Bulk: mass; charge; current; momentum; energy II From Micro to Macro ordering using BTE Fluid approach: assume shape of f is known Procedure

31. g(v) tni + .niui = Spart, i Particle 1 0-mom Mass mi tni mi + .ni miui = Smass,i Momentum BTE miv tfi + .fiv = (tfi/dt)CR qi Charge tni qi+ .ni qiui = Scharge,i, 1-mom tni  i+ .ni iui = Senergy,i  =1/2miv2 2-mom Energy tni miui+ .ni miuiui = Smom, i, The momenta of the BTE: Specific balances Note:  is in configaration space solely u is systematic velocity in configuration space Smom contains p and .: This approach is questionable

32. tni + .niui = Spart, i Particle 1 Mass mi tni mi + .ni miui = Smass,i qi Charge tni qi+ .ni qiui = Scharge,i The zero order momenta Note that Smass,i = mi Spart,i Scharge,i = qi Spart,i

33. ti ui+ . iiuiui = -pi+ .i+ i g +ni qiE+ niqiuiB + Fij, or Drift/Diffusion equation: 0 = -p +ni qiE+ Fij, Simplifications for specific mom balance ui is omnipresent: simplifications of the origen: mom bal tni miui + .ni miuiui = Smom, i, In many cases: ti ui+. iiuiui , niqiuiB and i g negligible p, ni qiEand Fij, dominant

34. 0 = -pi + ni qiE+ Fij, F = Ffric + Fthermo Fijfric = (pipj/pDij) (uj– ui) Case one dominant species with udom= u p = pdom pi = ni kTi ni (ui –u)= - (Di/ kTi ) pi + (ni qiDi/ kTi ) E, Drift Diffusion continued Mostly: Ffric >> Fthermo Fijfric = -(pipdom /pDij) (ui - u) Fijfric = -(pi/Di) (ui- u) Fijfric = -(kTi /Di) {ni (ui- u)}

35. Thus ni (ui– u )= - (Di/ kTi ) pi +(ni i) E or ni (ui– u )= - Dini +(ni ) E With i = qiDi/kTi If T  0 Einstein relation Drift - Diffusion II Normally: ni (ui– u )= - (Di/ kTi ) pi + (ni qiDi/ kTi ) E,

36. E = -1 (j + i qi ipi) or E = Ej + Eamb  ne eqe qe e >> qii Ambipolar Diffussion ni(ui– u )= - (Di/ kTi ) pi +(ni i) E with i = qiDi/kTi mobility and  = i nii qi conductivity j = i niqi (ui– u )= - i ipi +E In most cases the current density jis closely related to the external control parameter I; and E the result Eamb = -1i qi ipi Eamb = {kTe /qe}pe/pe

37. For the ions ni(ui– u )= - (Di/ kTi ) pi + niDi{Te/Ti } {qi /qe}pe/pe, For the electrons neqeue = j Beware of the signs!!

38. Since Smass = m Spart Reaction Conservatives I : mass AB  A+ B mAB = mA + mB Reactions Each creation of couple A and B associated with disappearence AB SAB = - SA = -SB Thus SAB mAB + mASA + mBSB =0 More general all mi Spart, i = 0 or all Smass, i = 0

39. Since Scharge = q S Reaction Conservative II: charge AB  A+ + B- qAB = qA + qB Reactions SAB = - SA = -SB Each creation of couple A and B associated with destruction AB Thus qAB SAB + qASA+ qBSB=0 More general all qi Si = 0 or all Scharge,i = 0

40. all Smass,i = 0 tni mi + .ni miui = Smass,i   all all all Scharge,i = 0 The Composition Bulk in Mass tm + . mu = 0 gives with m = all nimi and u  nimiui /m barycentric or bulk velocity Bulk in Charge tq + . j = 0 t ni qi + .ni qiui = Scharge,i With j = ni q1ui Current density

41. Reaction Conservatives III: Nuclei In general: species can be composed e.g. NH3 is composed out of one N nucleus and three H We say R of N in NH3 = 1 or RN(NH3) = 1 and R of H in NH3 = 3 or RH(NH3) = 3 or Ri = 3 with “i” = NH3 and “” = H Now consider NH3 N + 3H RH(NH3) Spart(NH3) = RH(N) Spart(N) + RH(H) 3 Spart(H) In general: all RH(i) Spart,i = 0

42. all RHj Spart, j= 0 tni RHi + .niRHiuj = Smass,i  all Elemental transport of H t {H} + . {H} = 0 gives With  ni RHi = {H} and {H} =  niRHiuj In steady state: . {H} = 0

43. In general t {X} + . {X} = 0 The change in time of the number density of nuclei of type X Equals minus the efflux of these nuclei; The efflux {X}of X is the weigthed sum: {X} = RXij j=niuj Number of X nuclei in j Efflux of “j”

44. Removing 1 H atom Equivalent: removing 3 H atoms Removing 1 H3 molecule {H} = 3 H3 In general: {H} = RHij

45. all mj Spart,j= 0 all RHj Spart,j = 0 tni mi + .ni miuj = Smass,i tni RXi + .niRXiuj = Smass,i    all all all all qj Spart,j = 0 Simularities Total mass transport tm + . mu = 0 Total charge transport tq + . j = 0 tni qj + .ni qiuj = Scharge,i Total X-nuclei transport t {X} + . {X} = 0

46. =0 = 0 Charge neutrality Action = -Reaction The momentum balance on higher structure levels The elements: simple addition of the DD equation. The bulk: i { ti ui+ . iuiui = -pi+ .i+ i g +ni qiE+ ni qiuiB+ Fij } Navier Stokes tu+ . uu = -p+ .+ j  B + g

47. Metal Halide Lamp LTE or LSE is present (??) Still not uniform LTE: at each location the composition prescribed by the Temperature and elemental concentration Convection and diffusion results in non-uniformity 10 mBar NaI and CeI in 10 bar Hg

48. If LSE is not established CRM needed

49. + 1 p Collisional radiative models Continuum Free electron states Bound electron state In principle  bound states Should we treat them all?

50. 1 + CRM Black Box: the ground state as entry CRM as a Black Box With two entries Typical Ionizing system Generation of efflux of photons and radicals As a result of input at entry 1 Response on influx largerly depends on ne and Te