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Summary of the previous lecture

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## Summary of the previous lecture

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**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?**Particles**Plasma Particles Energy Energy Momentum Momentum Particles: Plasma Chemistry Energy: Plasma Light Momentum: Plasma Propulsion**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**Source**Discretizing a Fluid: Control Volumes Plasma Particles Particles Energy Energy Momentum Momentum For any transportable quantity Transport via boundaries**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**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) -**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**The Variety** D S Temperature Heat cond Heat gen Momentum Viscosity Force Density Diffusion Creation Molecules atoms ions/electrons etc.**T**Continuum t + = S Tin Rod Tout x 0 + T = 0 Take k = Cst MathNumerics: a FlavorSourceless-Diffusion T = Cst T = - kT -T /k =T**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**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 =**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:**If k Cst**Convection Diffusion More general S-less Diffusion/Convection** = u -D** = S = -D If no “convection” Poisson Laplace - D = S S 0 S=0 - D = 0 - D = S Laplace and Poisson**Ohms’ law**Space charge Ener balance Ion balance .E = /o .j = 0 j = E . E = 0 E = -V -.Dn = Ion - Rec -.kT = S -. V = /o -.V = 0 Examples Simularities !!**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**+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**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**t = Nt**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**N D**t = Nt N D Mixed Channel P - nD = 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**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**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**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()**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**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**Source S**tn accumulation . nv and/or Efflux Simularities of the Boltzmann Transport equation tn + . nv = S Leads to**Generalization to 6-D phase space**Normal space tn + . nv = S Accumulation Transport Source tfi + .fiv = (tfi)CR Phase space The Boltzmann Transport Eqn**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**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**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**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**ti ui+ . iiuiui = -pi+ .i+ i g +ni**qiE+ niqiuiB + 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: ti ui+. iiuiui , niqiuiB and i g negligible p, ni qiEand Fij, dominant**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)}**Thus**ni (ui– u )= - (Di/ kTi ) pi +(ni i) E or ni (ui– u )= - Dini +(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,**E = -1 (j + i qi ipi)**or E = Ej + Eamb ne eqe qe e >> qii Ambipolar Diffussion ni(ui– u )= - (Di/ kTi ) pi +(ni i) E with i = qiDi/kTi mobility and = i nii qi conductivity j = i niqi (ui– u )= - i ipi +E In most cases the current density jis closely related to the external control parameter I; and E the result Eamb = -1i qi ipi Eamb = {kTe /qe}pe/pe**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!!**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**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**all Smass,i = 0**tni mi + .ni miui = Smass,i all all all Scharge,i = 0 The Composition Bulk in Mass tm + . mu = 0 gives with m = all nimi and u nimiui /m barycentric or bulk velocity Bulk in Charge tq + . j = 0 t ni qi + .ni qiui = Scharge,i With j = ni q1ui Current density**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**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**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} = RXij j=niuj Number of X nuclei in j Efflux of “j”**Removing 1 H atom**Equivalent: removing 3 H atoms Removing 1 H3 molecule {H} = 3 H3 In general: {H} = RHij**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 tm + . mu = 0 Total charge transport tq + . j = 0 tni qj + .ni qiuj = Scharge,i Total X-nuclei transport t {X} + . {X} = 0**=0**= 0 Charge neutrality Action = -Reaction The momentum balance on higher structure levels The elements: simple addition of the DD equation. The bulk: i { ti ui+ . iuiui = -pi+ .i+ i g +ni qiE+ ni qiuiB+ Fij } Navier Stokes tu+ . uu = -p+ .+ j B + g**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**If LSE is not established**CRM needed**+**1 p Collisional radiative models Continuum Free electron states Bound electron state In principle bound states Should we treat them all?**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