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Validation of a new method for numerically modeling LTE plasmas and elemental demixing inPLASIMO

van Boxtel, J.

Award date:2013

Link to publication

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https://research.tue.nl/en/studentTheses/545f3051-3d7c-4429-adf7-2ac5896e88f1

Validation of a new method fornumerically modeling LTE plasmas and

elemental demixing in PLASIMO

J. van Boxtel

September 2013 EPG 13-08

Eindhoven University of TechnologyDepartment of Applied PhysicsElementary Processes in Gas discharges

Under supervision of:dr. ir. J. van Dijkir. K.S.C. Peerenboom

Contents

1 Introduction 31.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Definition and derivation of used quantities 42.1 Stoichiometric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Density relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Flux relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Analysis of the general LTE composition problem 73.1 Species continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1.1 Stefan-Maxwell equations of diffusion . . . . . . . . . . . . . . . . . . . . . 73.1.2 Mass conservation for fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Introducing local thermodynamic equilibrium . . . . . . . . . . . . . . . . . . . . . 83.3 Elemental continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 Solving locally determined LTE problems . . . . . . . . . . . . . . . . . . . . . . . 83.5 Solving LTE problems with elemental diffusion . . . . . . . . . . . . . . . . . . . . 9

4 Tests of LTE code on local LTE problems 124.1 Excitation problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.1.1 Deriving the analytical solution . . . . . . . . . . . . . . . . . . . . . . . . . 124.1.2 Setting up the numerical model . . . . . . . . . . . . . . . . . . . . . . . . . 124.1.3 Comparison of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Recombination-ionization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2.1 Deriving the analytical solution . . . . . . . . . . . . . . . . . . . . . . . . . 144.2.2 Setting up the numerical model . . . . . . . . . . . . . . . . . . . . . . . . . 144.2.3 Comparison of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.3 Eleven species air mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3.1 Reference results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3.2 Setting up the numerical model . . . . . . . . . . . . . . . . . . . . . . . . . 164.3.3 Comparison of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4 XeInI composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.1 Reference results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4.2 Setting up the numerical model . . . . . . . . . . . . . . . . . . . . . . . . . 184.4.3 Comparison of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Testing new LTE code on 1D diffusion problems 215.1 A neutral H-Ar mixture in a 1D cavity . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.1.1 Derivation of the analytical solution . . . . . . . . . . . . . . . . . . . . . . 215.1.2 Setting up the numerical model . . . . . . . . . . . . . . . . . . . . . . . . . 245.1.3 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2 Analysis of the total elemental multicomponent diffusion coeffcient . . . . . . . . . 26

6 General conclusions 30

A Detailed derivations 31A.1 Analysis of LTE problems with elemental diffusion . . . . . . . . . . . . . . . . . . 31

A.1.1 Solving for the elemental diffusion fluxes . . . . . . . . . . . . . . . . . . . . 32A.2 Excitation problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33A.3 Recombination-ionization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1

B Local composition calculation input files 35B.1 Excitation problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35B.2 Ionization-recombination problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35B.3 Eleven species air mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36B.4 XeInI composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

B.4.1 Input file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39B.4.2 In levels kurucz.in . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

C Diffusion calculation input files 46C.1 Neutral H-Ar mixture 1D input file . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2

1 Introduction

Our society consumes more and more energy every day at an exponential rate. More and morepeople come to rely on automated tasks in their daily lives and higher standards of luxury, nearlyall of which cost copious amounts of energy to maintain. However, we do not (yet) in a practicalsense have access to an infinte amount of energy to spend and must therefore be efficient in ouruse of energy. At the same time we must also take other factors like the environment into account.This is a challenge.

A very efficient energy user for its job is the High-Intensity Discharge lamp or HID lamp.Whenever a lamp with high intensity is required, for example stadium lights, the HID lamp isvery often used in large part due to its energy efficiency. However, a major contributor to thisenergy efficiency is the heavy metal mercury which is used in these lamps, but unwanted becauseof its environmental status. The most obvious solution to this is to find substitute elements formercury which are as efficient, but less environmentally destructive. This proves difficult, however.Numerical modeling can save time and effort supplementing experimental research by providingpredictions for properties of mixture compositions.

When modeling HID lamps, a major physical phenomenon that must be accounted for iselemental demixing. In general, non-vanishing diffusion of chemical elements causes elementsto be distributed across the lamp volume non-uniformly. This is called elemental demixing. Aphysical manifestation of elemental demixing is shown in figure 1: different chemical elements inthe lamp’s inner gas mixture radiate light at different wavelengths, causing a color gradient to beobserved.

1.1 Outline

This text is about validating a new method developed for modeling platform PLASIMO [2] basedon elemental diffusion under local thermodynamic equilibrium (LTE) and an article by Rini [1]for performing numerical simulations on gas mixtures in, for example, HID lamps. The methodconsists of a set of partial differential equations describing the elemental composition of a mixturein conjunction with a set of algebraic equations to describe species composition.

In the chapters 2 and 3, we will introduce the theoretical framework of the new modelingmethod. Chapter 4 will attempt to validate numerically generated results from PLASIMO bycomparing to analytical solutions and other reference material for problems where elemental dif-fusion is absent. Chapter 5 will do the same, but for problems that do involve elemental diffusion.Conclusions are draw in chapter 6.

Figure 1: Evidence of elemental demixing in a High-Intensity Discharge lamp

3

2 Definition and derivation of used quantities

Let us consider a gas mixture which consists of a set of Ns species. Of these species, Ne arechemical elements, and Nr = Ns−Ne are combined species: they are made up out of the elementsand are not elements themselves. The combined species have indices R = {1, . . . , Nr} and theelements have indices E = {Nr + 1, . . . , Ns}. The set S = R∪E contains the indices of all species.

2.1 Stoichiometric matrix

We define the matrix R− with elements R−es, s ∈ S and e ∈ E to be the number of elements e thespecies s contains. Matrix R+ with elements R+

rs, r ∈ R and s ∈ S will contain the stoichiometriccoefficients for the reaction ∑

s∈SνrsAs = 0, (r ∈ R) (1)

with as constraintsνpq = δpq, (p, q ∈ R) (2)

where As is the symbolic notation for species s and δpq is the Kronecker delta function witharguments p and q. If we now define matrix B with elements Ber to be the number of elementse ∈ E in combined species r ∈ R, then we can define the stoiciometric matrix R and infer itsstructure as

R =

[R+

R−

]=

[INr

−BT

B INe

], (3)

where In, n ∈ N \{0} is the identity matrix of dimension n.We consider the following example. Suppose a given gas mixture consists of elements N, O and

combined species N2, NO and O2. The partial matrix R− shows the amount of each element eachspecies contains. The other partial matrix R+ shows the stoichiometric coefficients for a reactionto create exactly one molecule of each combined species out of the elements. The two matricesare unified to create R below. The symbols for the chemical components are, of course, just forclarification.

R =

N2 NO O2 N O

N2 1 0 0 −2 0

NO 0 1 0 −1 −1

O2 0 0 1 0 −2

N 2 1 0 1 0

O 0 1 2 0 1

. (4)

Note that the above definition of the stoichiometric matrix does not allow for any chargedspecies in the mixture. The definition can be amended to account for ionized species by adding anelement for the electron. For every charged species an appropriate (positive or negative) amountof electrons can then be accounted for to reflect ionization. For instance, let us add the ionizedspecies N+ to the mixture presented above. To reflect this, we would amend the stoichiometricmatrix to be:

R =

N2 NO O2 N+ N O e

N2 1 0 0 0 −2 0 0

NO 0 1 0 0 −1 −1 0

O2 0 0 1 0 0 −2 0

N+ 0 0 0 1 −1 0 1

N 2 1 0 1 1 0 0

O 0 1 2 0 0 1 0

e 0 0 0 −1 0 0 1

. (5)

4

2.2 Density relations

Let ns be the species number density for the species corresponding to s, and n{e} be the elementalnumber density for the element corresponding to e. Mixture density is defined by n =

∑s∈S ns.

It is possible to relate species and elemental number densities as follows:

n{e} =∑s∈S

R−esns. (e ∈ E) (6)

If we define n = [n1, . . . , nNs ]T and {n} = [n{1}, . . . , n{Ne}]T , we can express this more elegantly:

{n} = R−n. (7)

Moreover, we can normalize the species number densities through dividing them by the mixturedensity n. This operation gives the species mole fractions xs = ns/n, for which trivially holdsthat

∑s∈S xs = 1.

Next, we can also derive with the masses ms for species s and m{e} for element e a similarexpression to (7) for the species and elemental mass densities, defined as ρs = msns and ρ{e} =m{e}n{e}, respectively.

ρ{e} =∑s∈S

m{e}

msR−esρs. (e ∈ E) (8)

Or, if we define ρ and {ρ} together with m and {m} completely analogously to their numberdensity counterparts above, we get:

diag−1({m}){ρ} = R−diag−1(m)ρ, (9)

or{ρ} = diag({m})R−diag−1(m)ρ. (10)

Note that species masses m can also be expressed in terms of R− and elemental masses {m},since the mass of the species is fully determined by the number of elements they contain and themasses of those elements. The relation can easily be verified to be:

m = (R−)T{m}. (11)

Both the species and elemental mass fractions are defined as ys = ρs/ρ and y{e} = ρ{e}/ρwith mixture mass density ρ =

∑s∈S ρs, that is, as their non-fractional counterparts divided by a

constant independent of any index. Therefore, when again vectors y and {y} have the analogousdefinitions, the same relation holds for the mass density fractions as does for the mass densities.So:

{y} = diag({m})R−diag−1(m)y. (12)

Note that, analogously to xs,∑s∈S ys = 1. Less trivially yet intuitively sound, though, the

elemental mass fractions also hold to∑e∈E y{e} = 1. This can be quite easily shown as follows.

Introduce the vector 1N as [1, 1, . . . , 1]T , 1N ∈ RN and verify its properties

1TNa =∑i

ai, 1TN diag(a) = aT . (a ∈ RN ,a = [. . . , ai, . . .]T ∀i) (13)

Now we can deduce using equations (12) and (11) that∑e∈E

y{e} = 1TNe{y},

= 1TNediag({m})R−diag−1(m)y,

= {m}TR−diag−1(m)y,

= 1TNsdiag((R−)T{m})diag−1(m)y,

= 1TNsy =

∑s∈S

ys = 1, (14)

which concludes the density definitions and relations.

5

2.3 Flux relations

We move on to define the fluxes in the mixture and the relations between them. Let column vectorvs with s ∈ S be the average diffusion velocity of the species with index s with respect to the bulkvelocity u. We can easily see that the number flux ws of that same species s equals

ws = nsvs. (15)

If we now group all the vectors vs ∀s into a matrix V and all vectors ws ∀s into a matrix W inthe following way:

V =

v1

T

v2T

...vNs

T

, W =

w1

T

w2T

...wNs

T

. (16)

We can compound all equations (15) into

W = diag(n)V . (17)

Given that the mass flux js of species s is defined by

js = msws, (18)

following the anologous definition for matrix J to above results in

J = diag(m)W . (19)

Now, we introduce the elemental number and mass fluxes of the mixture as w{e} and j{e} withe ∈ E , respectively. Their matrix representations are analogous to the ones in (16) and havesymbols {W} and {J}, respectively. Since w{e} adheres to

w{e} =∑s∈S

R−esws, (e ∈ E) (20)

we can express this in matrix notation as

{W} = R−W , (21)

from which we can deduce {J}:

{J} = diag({m}){W}. (22)

Finally, we observe that when combining the last equation with (19) and (21) the elemental massfluxes can be written in terms of the species mass fluxes as follows

{J} = diag({m})R−diag−1(m)J . (23)

Note that the linear transformation here is identical to the one in (12).

6

3 Analysis of the general LTE composition problem

To make accurate predictions for the composition of a mixture of multiple reacting species, wewill need a sound theoretical framework to describe the behaviour of such a mixture. The nextfew paragraphs will define and explain the theoretical framework as it will be used in the rest ofthe text.

3.1 Species continuity

In general, whenever the composition of a species mixture must be determined, the following setof continuity equations must be solved ∀s ∈ S:

∂

∂t(ρys) +∇ · (ρuys) +∇ · js = ω̇s. (24)

The equations above describe the balance between species convection and diffusion of mass onthe lefthand side and the local production and destruction of species mass due to chemical andradiative processes on the righthand side for each species s. The source term ω̇s denotes the specieslocal net rate of mass production/destruction.

3.1.1 Stefan-Maxwell equations of diffusion

To rewrite the species continuity equations (24) in a closed form, we must account for the speciesmass diffusion fluxes js or J : that is express them in terms of other known paramaters. This isachieved through the Stefan-Maxwell equations of diffusion stated below.

SW = ∂x, (25)

where matrix S is defined by elements Sij with i, j ∈ S

Sij = xiM/ (ρDijfij) , (i 6= j)

Sij =∑

s∈S\{i}

−xsM/ (ρDisfis) . (i = j) (26)

The symbol ∂x denotes the matrix:

∂x =d(. . . , xs, . . .)

d(. . . , ci, . . . ), (∀s ∈ S, i) (27)

where ci ∀i are the Cartesian spatial coordinates of the system. The parameters Dij = Dji are thebinary diffusion coefficients, which will be discussed when practically called for. The contributionsof the so-called Laguerre-Sonine polynomials fij = fji will be assumed to be unity in practice, inaccordance to the value used by PLASIMO for the specific problems that are discussed in this text.For generality, however, fij will still be explicitly mentioned in equations. As can be observed, theStefan-Maxwell diffusion equations relate species number fluxes (and therefore, through equation(19), species mass fluxes) to concentration gradients.

3.1.2 Mass conservation for fluxes

The Stefan-Maxwell law of diffusion (25), as can quite easily be observed by its definition, has

[1, 1, . . . , 1]T ∈ N (S). The null space vector implies that for any given ∂x, any solution W for

the number fluxes of (25) can have equal and arbitrary values added to all individual fluxes in W ,and it will be a solution again. The matrix S is therefore singular, non-invertable and thereforenot able to, by itself, determine the species fluxes uniquely. The root of the problem is physicallythat we have not yet constrained the system to preserve mass. Therefore, S will only be usablein the context of this text when ∑

s∈Smsws = 0 = W Tm. (28)

7

3.2 Introducing local thermodynamic equilibrium

Under assumption of the system being in local thermodynamic equilibrium (LTE), we can putmore constraints on the system through statistical mechanics alone. In LTE, for every r ∈ R:∑

s∈Sνrs lnxs = lnKr

p(T )− ln p∑s∈S

νrs , (29)

where Krp(T ) is the equilibrium constant in terms of partial pressures that goes with the reaction

for combined species r ∈ R as defined in 2.1. Combined with equation (12), the above equationsform a independent and complete set of equations which show that in general under LTE conditionsfor an s ∈ S:

ys = ys(T, p, . . . , y{e}, . . .). (∀e ∈ E) (30)

That is, the composition is only dependent on temperature, pressure and elemental composition.

3.3 Elemental continuity

By section 3.2, it is beneficial to find the elemental composition of the mixture, in order to find thespecies composition of the mixture. To accomodate this, we revisit the set of species continuityequations from section 3.1.

∂

∂t(ρys) +∇ · (ρuys) +∇ · js = ω̇s. (31)

By applying the transformation in equation (8) and recognizing that equations (12) and (23) areits equivalent in matrix notation, we get:

∂

∂t(ρy{e}) +∇ · (ρuy{e}) +∇ · j{e} =

∑s∈S

R−esω̇s = 0. (32)

Now, we have both gotten rid of the troublesome parameter ω̇s, since∑s∈S R

−esω̇s = 0, and

transformed the set of species continuity equations for each species into a set of elemental continuityequations for each element. The parameter ω̇s obviously no longer appears because, since thereare no nuclear reactions involved, chemical elements are conserved and therefore net elementalmass production per unit time is necessarily zero. Equations (32) have as an additional advantagethat they are few in number, when number of species is relatively large and number of elementsis relatively small, as is often the case.

3.4 Solving locally determined LTE problems

For a locally determined LTE problem, the elemental composition fixed in space and time. Itfollows that the elemental continuity equations (32) are trivially met, since je = 0: they nowdenote merely global mass conservation of the mass averaged velocity field. Former problemvariables y{e} are now effectively known constants, such that through (30):

ys = ys(T, p). (∀e ∈ E) (33)

This of course implies that the composition for the locally determined LTE problem is completelydetermined by only local temperature and pressure.

To solve the locally determined LTE composition problem, we will now take a closer look at achemical reaction between the species of the mixture:

a0X0 + a1X1 + . . .+ akXk + . . .� . . .+ blYl + . . . , (34)

where Xk is a symbolic notation for species with index k ∈ X and X ⊆ S contains all the indicesfor species on one side of the balance. The symbol ak denotes the stoichiometric coefficient for the

8

same species. Similarly, Yl with l ∈ Y and Y ⊆ S and bl are, respectively, the symbolic notation forspecies and stoichiometric coefficient on the other side of the balance. If we now assume there islocal thermodynamic equilibrium in the mixture, then the law of mass action is valid [4]. Togetherwith our earlier assumption of no convection or diffusion of species mass, we can use the followingequation:∏k∈X

nakk∏l∈Y

nbll=

∏k∈X

Qakk∏l∈Y

Qbll

(2πkBT

h2

) 32 (

∑k∈X

ak−∑l∈Y

bl)( ∏k∈X

makk∏

l∈Ymbll

) 32

exp

(−

∑k∈X

akEk −∑l∈Y

blEl

kBT

). (35)

The quantity Qs for s ∈ S is the partition sum of the species with index s, and is defined asfollows:

Qs =∑i

giZi exp(− EikBT

), (s ∈ S) (36)

where i is an element of the set of internal states of the species s. The degeneracy of the internalstate i is given by gi, and its internal partition sum by Zi (which is unity for atomic species).Finally, Ei is the energy of the internal state i relative to the internal ground state.

While equation (35) is useful, it obviously does not in general determine the entire composition.Since we have Ns species and therefore Ns unknowns initially, using (35) we are left with Ns−Nrunknowns. This lack of information can be remedied by applying mathematical constraints to thesystem based on knowledge about the composition, which can vary from problem to problem. Forinstance, when a total or partial pressure pZ exerted on the system is known as a result of the setof species Z ⊆ S, we can state that: ∑

s∈Zns =

pZkBT

. (37)

Also, when there are charged species in the mixture, there is an additional constraint that takeseffect. Since we are looking at mixtures that must be entirely determined by local properties,we will assume that there is no meaningful charge separation on the scale of the global mixture.Effectively, it is therefore everywhere electrically neutral when measured on a scale that is mean-ingful in the context of our particular problem. This is also called a quasi-neutral mixture. Thisassumption thus leads to the following constraint:∑

s∈Sqsns = 0. (38)

Equivalently, constraints on the elemental mass fractions can also be used to obtain a solution. Inthis case, mass conservation dictates that the sum of all elemental mass fractions be

∑e∈E y{e} = 1.

Also, when the mixture is quasi-neutral, then because of (38) and (12) the electron elemental massfraction must be y{e−} = 0.

3.5 Solving LTE problems with elemental diffusion

Although a locally determined LTE problem is a useful approximation of a physical situation withlittle diffusion, in general a mixture of chemically reacting species has significant variations inits composition and is therefore significantly affected by diffusion [1]. Thus, any model aimingto be widely useful for calculating mixture compositions cannot assume diffusion effects to benon-existent.

We must now solve the system and in particular the elemental continuity equations (32) with-out the assumption of fixed and uniform elemental composition. This is accomplished by that thein general non-zero elemental diffusion fluxes je can be expressed as a function of gradients intemperature and elemental mass fractions, when pressure is uniform. The method of deduction isto use Stefan-Maxwell diffusion theory in 3.1 together with equilibrium constraints (29) and ele-mental mass fraction constraints (12) to link the desired temperature and elemental mass fractiongradients to diffusion fluxes.

9

As an intermediate step, we express the gradients of species mole fractions ∂x as found in (25)as a function of the desired gradients ∂T and ∂{y}. To do this, taking the gradient on both sidesof (29) is combined with van ’t Hoff’s law [5][6], which states that

dlnKrp(T )

dT=

∆hrRuT 2

, (39)

where ∆hr is the reaction enthalpy for the reaction corresponding to combined species r ∈ R andRu is the universal gas constant, to show

A+∂x = θ∂T, (40)

with for every r ∈ Rθr =

∆hrRuT 2

, (41)

andA+ = R+diag−1(x). (42)

Next, we express the gradients elemental mass fraction in terms of the species mole fractions.This can be done by using the fact that, derived from equation (12):

∂{y} = diag({m})R−diag−1(m)∂y, (43)

with matrices ∂{y} and ∂y analogously defined to ∂x. Secondly, we recognize that:

ys =ms

Mxs. (44)

Taking the gradient on both sides of this last equation, and combining the result with equation(43), ultimately yields

A−∂x = ∂{y}, (45)

withA− = diag({m})R−F (46)

and matrix F with elements Fij = 1M

(δij − ximj

M

)with i, j ∈ S.

We would now manipulate (25) into a form which explicitly has the elemental diffusion fluxes{W} rather than the species fluxes W . This is accomplished by splitting the Stefan-Maxwellcoefficient matrix S into matrices S1 ∈ RNr×Nr , S2 ∈ RNr×Ne , S3 ∈ RNe×Nr and S4 ∈ RNe×Ne

such that

S =

[S1 S2

S3 S4

]. (47)

We can find that by applying the following manipulation

SW = S

I︷︸︸︷[INr 0−B INe

][INr 0B INe

]W =

[S1 − S2B S2

S3 − S4B S4

][Wr

{W}

], (48)

the elemental mass fluxes {W} appear. The symbolWr represents the diffusion fluxes of combinedspecies only.

We will now split matrices A+ and A− into matrices A+1 ∈ RNr×Nr and A+

2 ∈ RNr×Ne and

matrices A−1 ∈ RNe×Nr and A−2 ∈ RNe×Ne , respectively, such that

A+ =[A+

1 A+2

], (49)

andA− =

[A−1 A−2

]. (50)

10

Using these split forms, combined with both constraints (40) and (45) and the elemental Stefan-Maxwell diffusion equations (48), we obtain a new system of equations

AWr + B{W} = θ∂T, (51a)

CWr + D{W} = ∂{y}, (51b)

where A = A+1 (S1 − S2B) + A+

2 (S3 − S4B), B = A+1 S2 + A+

2 S4, C = A−1 (S1 − S2B) +

A−2 (S3 − S4B) and D = A−1 S2 + A−2 S4. We can solve Wr from (51a) using straightforwardGaussian elimination [1], and substitute back into (51b) for(

D − C A −1B){W} = ∂{y}− C A −1θ∂T, (52)

which in turn can be expressed as a simple linear system

S {W} = β. (53)

with S = D − C A −1B and β = ∂{y} − C A −1θ∂T . Although coefficient matrix S is square,it is a singular, non-invertible matrix. Its singular character is inherited from the Stefan-Maxwelldiffusion equations (25), since the mass constraint discussed in 3.1 was not yet applied. The massconstraint, given by (28) can in terms of elemental diffusion fluxes be expressed as

{m}T{W} = 0. (54)

We introduce the new system of equations(S + ζ{m}{m}T

){W} = β, (55)

for any ζ ∈ R \ {0}, the solutions of which are the intersection of (53) and (54). The coefficient

matrix S̃ = S + ζ{m}{m}T is no longer singular and we can finally express the elemental massdiffusion fluxes explicitly as:

{J} = diag({m})S̃ −1∂{y}− diag({m})S̃ −1C A −1θ∂T. (56)

The fully-detailed derivation of this result can be found in appendix A.1.

11

4 Tests of LTE code on local LTE problems

In this section we will do testing on the newly developed LTE code in the PLASIMO environmentby using it to calculate compositions under certain conditions for a simple, locally determinedproblem. Specifically, such problems can be solved using the script example newlte local inPLASIMO paired with an input file which provides necessary data about the composition. Theexact input files used can be found in appendix B. We will compare the numerical results toan analytical derivation or other reference data for the same problem. The theory discussed inthe previous sections is used to derive the analytical solutions. Note that the new LTE code inPLASIMO can only deal with local constraints in the form of constraints on the elemental massfractions.

4.1 Excitation problem

First, we will turn our attention to a hydrogen excitation problem. This problem involves twospecies: the ground state of hydrogen H and the first excited state of hydrogen H∗, or symbolically

H � H∗. (57)

There is local thermodynamic equilibrium and no species convection or diffusion. Moreover, themixture is under a total pressure p and temperature T .

4.1.1 Deriving the analytical solution

We can therefore use the equations from the previous section and form a fully determined set ofequations for the number densities of the species as follows:

αE(T ) ≡ 4 exp(− EexckBT

),

β(T, p) ≡ p

kBT,

nH∗

nH= αE(T ), (58a)

nH + nH∗ = β(T, p). (58b)

Here, (58a) is derived from (35) and (58b) is derived from (37). The quantity Eexc = 10.2 eV isthe energy required for excitation. A detailed derivation of the solution can be found in appendixA.2. When solved for the mass fractions of the two species, the solutions are:

yH =1

1 + αE(T ), (59a)

yH∗ =1

1 + (αE(T ))−1 . (59b)

Note that the mass fractions are independent of pressure.

4.1.2 Setting up the numerical model

The input file for the model defines two atomic species, one representing the ground state and onefor the first excited state of atomic hydrogen. Each species has a defined and equal mass (the massof atomic hydrogen), and has the appropriate degeneracies and energies. Moreover, the chemicalreaction (57) is represented. The script was run with parameters such that the composition wascalculated for 1500 K ≤ T < 80000 K at increments of ∆T = 50 K. This temperature domainwas determined by trial-and-error, based on the criteria of ample change in the composition andsuccess of calculation (the numerical solver fails at T ≈ 1500 K and below). Finally, the elementalmass fraction of hydrogen was constrained to y{H} = 1.

12

4.1.3 Comparison of results

Below in figure 2 the numerical solution for the ground state atomic mass fraction generatedby PLASIMO using the new LTE code and equation (59a) are plotted both as a function oftemperature. In figure 3 the absolute value of the difference in the same quantity between thenumerical and analytical solution is plotted as a function of temperature. It can be observed thatthe absolute error in the mass fraction is extremely small at the lower temperatures (actually inthe order of 10−12) and then increases exponentially to about a 10−6 order at T ≈ 7000 K. Theactual maximum absolute error is |∆yH | ≈ 5.4 · 10−7.

Figure 2: Analytical and numerical excitation comparison graph

4.2 Recombination-ionization problem

Secondly, we take a look at another simple composition for which an analytical solution is quiteeasily derived. It is only slightly more complicated than the excitation problem: instead of theexcitation of a species, we will consider an ionization of, once again, hydrogen. The only reactionis, of course, given by

H � H+ + e. (60)

We will also once again assume local thermodynamic equilibrium, no convection or diffusion effects,and a known temperature T and pressure p. On top of that, we also assume quasi-neutrality, sincethis composition contains charged species and imposes all of the atomic hydrogen is in the groundstate, always.

13

4.2.1 Deriving the analytical solution

Using equations (35), (37) and (38) we derive the following system of equations for the ionizationcomposition

αI(T ) ≡

(2π(mH −me)mekBT

mHh2

) 32

exp(−EionkBT

),

β(T, p) ≡ p

kBT,

nH+nenH

= αI(T ), (61a)

nH + nH+ + ne = β(T, p), (61b)

nH+ − ne = 0. (61c)

where Eion = 13.6 eV is the ionization energy for atomic hydrogen. Solving the system for themass fraction of atomic hydrogen yH yields

yH =2

1 +(

β

β+2αI−2√α2

I+αIβ

) . (62)

A full derivation can once again be found in the A.3. We observe that, in contrast to the excitationproblem, the composition is dependent on both temperature and pressure.


The input file for the numerical ionization composition calculation contains three species: onecorresponding to atomic hydrogen, another for ionized hydrogen and one for the electron. Foreach species the appropriate energies, degeneracies, masses and particle radii are provided. Thereaction (60) is included. The parameters are such that the composition is calculated on 1500 K ≤T < 8000 K with ∆T = 50 K increments and 103 Pa ≤ p ≤ 105 Pa with ∆p = 9900 Pa increments.This domain was again chosen by trial-and-error based on calculability and ample visible change.Finally, the elemental mass fractions were constrained to y{H} = 1 and y{e} = 0.


Figure 4 is a surface plot is showing both the species mass fraction analytical solution and thenumerical solution for atomic hydrogen as a function of temperature and pressure. Figure 5shows the absolute difference between said quantities as a function of temperature and pressure.The latter figure shows that the error in the numerical values is relatively very small (again,order of 10−12 K) at all calculated pressures for temperatures outside roughly 4000 K < T <23000 K. Inside this temperature range for all calculated pressures the absolute difference growsexponentially to an order 10−6. The actual maximum absolute error is |∆yH | ≈ 7.7 · 10−7.

4.3 Eleven species air mixture

In order to test the new code on a more complicated mixture, we turn our attention to a composi-tion calculation performed in the article by Rini [1]. The article considers an ordinary air mixtureconsisting of eleven species and three elements: nitrogen, oxygen and the electron. The speciesconsidered are N, O, N+, O+, N2, O2, NO, N+

2 , O+2 , NO+ and e. The model disregards convection

and diffusion, assumes LTE and quasi-neutrality. Furthermore, the elemental mass fraction foroxygen is fixed at y{O} = 0.23, total pressure is p = 1 atm and the composition is calculated fortemperatures 0 K ≤ T ≤ 15000 K. Finally, we assume that the species are all perpetually in theirrespective ground states. The reactions which are active in the mixture are given by the reactionsas defined in (1).

14

Figure 3: Absolute difference numerical and analytical excitation results

Figure 4: Analytical and numerical ionization comparison graph

15

4.3.1 Reference results

Because of the high number of species and reactions involved in this model, it is not at all feasibleto pursue an analytical solution as was done with the excitation and ionization-recombinationproblems. The calculation of such a solution would be rather complicated, and the result wouldbe unwieldy at best. The only further result reference material available for this particular com-position calculation is the result graph in the article. Since the amount of composition calculationproblems with reference material for results and appropriate known parameters (constraints on theelemental mass fractions, exact calculation parameters, detailed information about species used,etc.) is scarce, it is worth the time to compare results from the new LTE code by visual inspectionto the available result graph.


The input file for this model contains all the species and their ground state degeneracies, energies,particle radii and masses. All reactions according to (1) for this particular model are included.The parameters for the script are such that 1000 K < T ≤ 15000 K with increments ∆T = 50 Kand p = 1 atm. It is not possible to calculate the composition for a lower bound temperature of0 K, since the solver will fail for temperatures below T ≈ 1000 K. Elemental mass fractions areconstrained to y{O} = 0.23, y{e} = 0 and y{N} = 1− y{O} − y{e} = 0.77. Finally, output is set tomole fractions.


In figure 6 the composition as calculated by Rini [1] is shown. Figure 6 shows the results of thecalculation performed by the new LTE code. Although it is obvious that the general trend of bothresult sets is the same, there are also some notable differences. The composition as calculated byPLASIMO seems rather stretched along the temperature axis relative to the reference graph. Also,the NO species is quite underrepresented in the PLASIMO calculation. These huge discrepancies,when viewed in the light of the maximum errors of the previous test cases which were in the orderof roughly 10−6 , are probably caused by the assumption that all species are perpetually in theirground states. The original description for the mixture makes in Rini [1] makes no mention ofthis assumption, and it could very well be that multiple excited states were accounted for in thecalculation for the reference graph. Since we cannot possibly hope to find out the exact statesused, we cannot reliably find more information about the validity of the new LTE code using thiscomposition. Note that the molecular ion results are left out both in the reference graph and theresults of the PLASIMO calculation, since they do not provide a significant contribution to thecomposition.

4.4 XeInI composition

As a final test of local LTE composition calculation, we take a look at a complicated mixture ofcompounds of the elements Xe, In and I. This test will use quantitative reference material fora complex mixture, as opposed to the eleven species air mixture calculation. The exact speciesconsidered are Xe, In, I, Xe+, In+, In+, I+, I−, I2, InI, InI2 and InI3. This composition is oneof many that has been under research as a potential efficient HID-lamp. We once again disregardconvection and diffusion, and assume both LTE and quasi-neutrality. As opposed to the previousmodels, however, this model does take both ground as well as some excited states and molecularstates into account. The details are made clear by the input file corresponding to this model inappendix B.4 in conjunction with PLASIMO documentation. The composition must be calculatedfor elemental constraints y{Xe} = 0.5, y{In} = 0.125 and y{I} = 0.375. Finally, pressure p = 1 barand the range of temperatures is 0 ≤ T ≤ 30000 K. The reactions which are active in the mixtureare given by the reactions as defined in (1).

16

Figure 5: Absolute difference numerical and analytical ionization results

Figure 6: Reference results for eleven species air mixture

17

4.4.1 Reference results

The reference results for this test are a composition calculation of the same composition performedusing existing MATLAB code by J.F.J. Janssen from the Elementary Processes in Gas dischargesgroup at the Eindhoven University of Technology. The input for the MATLAB script was basedon the input file also used for this test.


The input file for this numerical model contains energies, masses, radii and other parameters ofvarious atomic ground and excitated states. Also, more advanced modeling of some molecularstates with corresponding parameters is requested. The reactions are according to the standarddefinition in (1). The parameters for the script are such that 1500 K < T ≤ 30000 K withincrements ∆T = 100 K and p = 1 bar. The lower bound for T is once again determined bylimitations of the non-linear solver used. Elemental mass fractions are constrained to y{Xe} = 0.5,y{In} = 0.125, y{I} = 0.375 and y{e} = 0. Finally, output is set to mole fractions.


Figure 8 and figure 9 show reference MATLAB script and PLASIMO calculated mole fractions ofthe relevant species in the mixture as a function of temperature, respectively. Note that the molefractions for both InI2 and InI3 were left out of the graph since they were negligable and invisible.By inspection, reference and calculated results are very similar: the graphs do not seem to differin any features whatsoever. In figure 10 the absolute difference between reference and calculatedmole fractions are plotted per species. The maximum absolute difference in the mole fraction is|∆x| ≈ 2.0 · 10−4: this is definitely higher than the comparisons to analytical solutions with theexcitation and ionization-recombination problems.

18

Figure 7: Numerical composition calculation results for eleven species air mixture

Figure 8: Reference composition data for the mole fractions of the XeInI composition againsttemperature

19

Figure 9: Composition data calculated by PLASIMO for the mole fractions of the XeInI compo-sition against temperature

Figure 10: Absolute difference between reference and PLASIMO calculated data for the molefractions of the XeInI composition against temperature

20

5 Testing new LTE code on 1D diffusion problems

In this chapter, a simple physical model of a mixture in a cavity will be introduced designed tofacilitate the derivation of an analytical solution to the equations in the last chapter. This willenable us to analytically calculate the composition of the mixture in the model. Then, we willrun the model in PLASIMO using the new LTE code and analyze the differences between thenumerical and analytical sets of results.

5.1 A neutral H-Ar mixture in a 1D cavity

The test model consists of a neutral mixture of two atomic species, here arbitrarily chosen to beatomic hydrogen and atomic argon. The mixture is confined to a one dimensional line cavity,between x = 0 and x = L, where x is the Cartesian coordinate of the one dimensional coordinatesystem. At the edges of the cavity, the composition is known. Furthermore, the mixture is ina stationary state, the mass averaged velocity u = 0 and the pressure p and temperature T areuniform and fixed over time. Also, there will be no chemical reactions.

5.1.1 Derivation of the analytical solution

When applying the constraints of the proposed model to the system of equations presented in lastchapter, it is greatly simplified. From equations (32) and the used constraints, we gather

∇ · j{H} = 0, (63)

∇ · j{Ar} = 0. (64)

Also, the expressions for the remaining elemental diffusion fluxes, given by (126), are simplifiedmainly by the fact that Nr = 0. The expression that remains is

{J} = diag({m})(D + ζ{m}{m}T

)−1∂{y}, (65)

with D = A−2 S4 = A−S. Also, since R− = INs , the definition of A− in (46) is simplified toA− = diag({m})F . We first calculate this matrix using the definition of F :

A− = diag({m})F =

[mH 0

0 mAr

][1M

(1− mHxH

M

)1M

(−mArxH

M

)1M

(−mHxAr

M

)1M

(1− mArxAr

M

)] ,=

1

M2

[mH 0

0 mAr

][M −mHxH −mArxHmHxAr M −mArxAr

],

=mHmAr

M2

[xAr −xH−xAr xH

], (66)

using that m{H} = mH , m{Ar} = mAr and mHxH +mArxAr = M . Next, we calculate D :

D = A−S =mHmAr

M2

[xAr −xH−xAr xH

]M

ρDH,ArfH,Ar

[−xAr xHxAr −xH

],

=mHmAr

ρDH,ArfH,ArM

[−x2Ar − xHxAr x2H + xHxArx2Ar + xHxAr −x2H − xHxAr

],

=mHmAr

ρDH,ArfH,ArM

[−xAr xHxAr −xH

], (67)

where the properties Dij = Dji, fij = fji and xH + xAr = 1 were used. According to (65),we must now regularize and invert D using the method discussed in the previous chapter. Letχ = mHmAr

ρDH,ArfH,ArM6= 0, then:

S̃ = D + ζ̃{m}{m}T = χ

[−xAr xHxAr −xH

]+ ζ̃

[m2H mHmAr

mHmAr m2Ar

],

= χ

[−xAr + ζm2

H xH + ζmHmAr

xAr + ζmHmAr −xH + ζm2Ar

](68)

21

where ζ̃ ∈ R\{0} and we have chosen ζ = ζ̃/χ. To invert S̃ we will first establish its determinant:

det(S̃ ) = χ2

∣∣∣∣ −xAr + ζm2H xH + ζmHmAr

xAr + ζmHmAr −xH + ζm2Ar

∣∣∣∣ ,= χ2

[(−xAr + ζm2

H)(−xH + ζm2Ar)− (xH + ζmHmAr)(xAr + ζmHmAr)

],

= −ζχ2(m2HxH +mHmArxH +mHmArxAr +m2

ArxAr),

= −ζχ2 [mHxH(mH +mAr) +mArxAr(mH +mAr)] ,

= −ζχ2M(mH +mAr), (69)

such, that the inverse becomes

S̃ −1 = − χ

ζχ2M(mH +mAr)

[−xH + ζm2

Ar −xH − ζmHmAr

−xAr − ζmHmAr −xAr + ζm2H

],

=1

ζχM(mH +mAr)

([xH xHxAr xAr

]+ ζ

[−m2

Ar mHmAr

mHmAr −m2H

]). (70)

This inverse is only valid on a domain which equals the range of S = D . The range of D isspanned by [1,−1]T . The null space of the matrix on the lefthand side of the addition operatorin (70) is also spanned by this vector.1 Therefore, for all allowed domain vectors, this matrix willnot have any influence on the image vector of S̃ −1 and can therefore be omitted entirely. Thisleaves a reduced (70):

S̃ −1 =ζ

ζχM(mH +mAr)

[−m2

Ar mHmAr

mHmAr −m2H

],

=ρDH,ArfH,Ar

mHmAr(mH +mAr)

[−m2

Ar mHmAr

mHmAr −m2H

](71)

And all that remains to be done is to see that

diag({m})S̃ −1 =ρDH,ArfH,Ar

mHmAr(mH +mAr)

[mH 0

0 mAr

][−m2

Ar mHmAr

mHmAr −m2H

],

=ρDH,ArfH,ArmH +mAr

[−mAr mH

mAr −mH

]. (72)

And, finally, therefore

{J} =ρDH,ArfH,ArmH +mAr

[−mAr mH

mAr −mH

]∂{y}. (73)

A quick reality check shows that mass conservation is indeed applied, since jH + jAr = 0. Now,to solve the system, we will explicitly move on to a one-dimensional problem. This gives fromequations (64) and (73)

d

dx

(ρDH,ArfH,ArmAr

mH +mAr

dy{H}

dx+−ρDH,ArfH,ArmH

mH +mAr

dy{Ar}

dx

)= 0. (74)

Because jH = −jAr, the second equation from (64) does not add any information to the systemas both equations can trivially be shown to be entirely equivalent by substituting (73) into andscalar multiplying either of them left and right by −1. The system can be reduced even furtherdue to the presence of only two elements, which implies that y{H} + y{Ar} = 1 and thereforedy{H}/dx = −dy{Ar}/dx. And finally, there are no combined species, so elemental mass fractionsequal corresponding species mass fractions: y{H} = yH and y{Ar} = yAr. This all leads to

d

dx

(ρDH,ArfH,Ar

dyHdx

)= 0. (75)

1This is of course no coincidence, but we will not go into the details of this type of inverse any further in thistext.

22

Since we want to calculate the hydrogen fraction in this case (this choice is arbitrary, of course)we eliminated the argon fraction here. The diffusion coefficient DH,Ar for neutral-neutral particleinteraction is calculated in PLASIMO as follows:

Dij =9

16

√kBT

8πµH,Ar

1

(rH + rAr)2

1

n, (76)

where the reduced mass µH,Ar = mHmAr/(mH + mAr) and rH and rAr are the atomic radii ofhydrogen and argon, respectively. A hard sphere approximation of the collision integral normallyfound in the expression for the diffusion coefficient was used to obtain this result. Note that onlyn is spatially dependent here, and for the purposes of solving equation (75) we may as well leavethe other symbols out when substituting the expression for the diffusion coefficient into (75). This,and the fact that fH,Ar = 1 (see section 3.1.1), gives:

d

dx

(ρ

n

dyHdx

)= 0, (77)

and since

ρ

n=mHnH +mArnAr

nH + nAr,

= mH1

1 + nAr

nH

+mAr1

1 + nH

nAr

,

= mH1

1 + mHyAr

mAryH

+mAr1

1 + mAryHmHyAr

,

= mHmAryH

mAryH +mH(1− yH)+mAr

mH(1− yH)

mH(1− yH) +mAryH,

=mHmAr

(mAr −mH)yH +mH, (78)

this givesd

dx

(mHmAr

(mAr −mH)yH +mH

dyHdx

)= 0⇒ d

dx

(1

yH + γ

dyHdx

)= 0, (79)

where γ = mH/(mAr −mH). All that remains to be done to obtain the analytical solution is tosolve the above ODE and impose boundary conditions. To solve the ODE, let u = yH(x) + γ,then:

d

dx

(1

u

du

dx

)=

d2lnu

dx2= 0, (80)

orln(yH(x) + γ) = C1x+ C2, (81)

where C1, C2 ∈ R are constants of integration. If we impose the boundary conditions at the edgesof the cavity yH(x = 0) = yH,0 and yH(x = L) = yH,L then we can derive

ln(yH,0 + γ) = C2, (82)

ln(yH,L + γ) = C1L+ C2. (83)

and, as can be easily derived from the above linear system in C1 and C2:

C1 =1

Lln

(yH,L + γ

yH,0 + γ

), (84)

C2 = ln(yH,0 + γ). (85)

Substituting above equations into (81) we obtain

ln

(yH(x) + γ

yH,0 + γ

)= ln

(yH,L + γ

yH,0 + γ

)x

L, (86)

23

and finally the analytical solution explicitly expressed in yH(x):

yH(x) = (yH,0 + γ)

(yH,L + γ

yH,0 + γ

)x/L− γ, (87)

where γ = mH/(mAr −mH).


Since this is a problem involving diffusion, we cannot use the example newlte local script fornumerical calculation as was done in 4: this script was designed to do calculations for 0D-problemsonly. Instead, we will make use of the PLASIMO environment together with a more extensiveinput file. The input file not only defines the properties of species involved as was done previously,but also links to external library dependencies and sets desired numerical accuracy. Moreover, itdefines a spatial grid on which to calculate and defines the values of thermodynamic properties,transfer coefficients and boundary conditions. The full input file can be found in appendix C.1and can be interpreted in detail through the PLASIMO documentation [2]. In the paragraphsbelow, however, the meaning of the conceptually and numerically relevant sections of the inputfile is also discussed.

Figure 11 shows how the grid is generated from the input file data. The input file specifies that agrid of cartesian coordinates be composed between 0.0 m ≤ x ≤ 1.0 m; this implies that L = 1.0 m.The grid points, or the points where composition data actually will be calculated, are generatedon top of a structure of cells. Cells are appointed consecutive regions of the entire space insidethe grid domain. The amount of cells is determined by the CellVector and RefinementPower

properties in the input file (although BaseRefinenementPower performs a nearly equivalent role, itwas not used and therefore is kept at value 0: it does not have an effect). The grid was configuredsuch that the amount of cells #cells equals

#cells = 7 · 2P , (88)

where P is the value of RefinementPower. The property P is kept as a variable here since itwill serve as such later on. The grid points are placed on top of the cell structure as follows, bydefinition in the input file: there will be a grid point in the center of each cell and the edges of thedomain will each have one grid point. The cartesian coordinates of the grid points therefore are:

x = 0.0 m, x = 1.0 m, x =1.0− 0.0

#cells(i+ 1/2) m, (i ∈ N, i < #cells) (89)

and the total number of grid points #gp is:

#gp = #cells + 2 = 2 + 7 · 2P . (90)

The species defined in the input file are the ground state of hydrogen and the ground state ofargon, where the degeneracy, energy and atomic radii are included as data. Two additional speciesnot found in the analytical model, however, were also included: ground state Ar+ and the electron.The pragmatic reason for this is that, as PLASIMO is designed to model plasmas, there has tobe at least one ionized species in the mixture, of which there is in principle none in this modelof a neutral mixture. To ensure that the additional species only negligably influences the resultsof the calculation, a high dummy value for the energy of Ar+ was defined (EAr+,g = 20.0 eV).This ensures that the balance of the amount of argon and ionized argon swings heavily in favorof the former, leaving an essentially neutral mixture still. An ionization/recombination reactionfor argon obviously also needs to be included where there would ordinarily be no reactions in thismodel: this reaction has been added as well.

Boundary conditions for the elemental mass fractions have been set such that y{e} = 0 atx = 0.0 m and x = 1.0 m for practical reasons (note that ionized species are of negligable quantityin this problem, making y{e} = 0 largely inconsequential): the constraint ensures y{H}+y{Ar} = 1

24

remains true. Furthermore, y{H} = 0.999 at x = 0.0 m and y{H} = 0.4 at x = 1.0 m semi-

arbitrarily: i.e.∣∣y{H}∣∣x=0.0 m

− y{H}∣∣x=1.0 m

∣∣ should be high enough as to be able to discern

exponential behaviour of the solution. Note that y{H}∣∣x=0.0 m

6= 1. This is because a value of

1 for the fraction of hydrogen would imply y{Ar}∣∣x=0.0 m

= yAr|x=0.0 m = 0, which would causeproblems as PLASIMO does not deal well with absence of one of the species in the mixture.

The numerical model will be run for RefinementPower values P = 0, . . . , 8 and compared tothe analytical solution to determine how the maximum absolute error relates to number of gridpoints.

5.1.3 Discussion of results

In figure 12 the results for the elemental mass fraction of hydrogen calculated by PLASIMO usingthe input file discussed in the above section with grid variable P = 4 (#gp = 114) are shownin the same graph as analytical results. The analytical results were calculated using equation(87) with parameters and boundary conditions in accordance with the input file. For clarity,these parameters and boundary conditions are taken to be L = 1.0 m, yH,0 = 0.999, yH,L = 0.4,mH = 1.008 amu and mAr = 39.95 amu. By inspection alone, one can see that analytical andnumerical results at the very least show reasonable similarity. We would, however, quantify thissimilarity more thoroughly. To this purpose, we look at the maximum absolute error in hydrogenelemental mass fraction ∆ymax(P ) for each numerical result of refinement power P :

∆ymax(P ) = maxj |yH,ana(xj,P )− yH,num(xj,P ;P )| . (91)

where xj,P are the grid points for refinement power P , yH,ana(x) is the aforementioned analyticalequation (87) with the above mentioned parameters and yH,num(xj,P ;P ) are the numerical resultsat the grid points and for refinement power P . We will look at ∆ymax(P ) plotted against the gridpoint separation ∆xj(P ):

∆xj(P ) =1.0 m

#cells(P ), (92)

as a function of refinement power P , for all P = 0, . . . , 8. All the consecutive inner grid pointsare spatially seperated by ∆xj(P ), which makes it the relevant scale of length for error analysis.Since this error analysis is essentially an analysis of discretization error, it is highly likely that∆ymax(P ) will depend on a certain power of ∆xj(P ), or:

∆ymax(P ) = C(∆xj(P ))b. (C, b ∈ R) (93)

To verify this, we will plot ∆ymax(P ) against ∆xj(P ) double logarithmically, yielding a linearrelationship, as can be seen by taking the logarithm on both sides of the above equation:

log10(∆ymax(P )) = b log10(∆xj(P )) + log10 C. (94)

The power b in the double logarithmic plot is equivalent to the slope of the line, and can beretrieved by means of linear fitting. In figure 13, the double logarithimic plot and linear fit areshown. The slope has a value of b = (1.53± 0.05), implying that:

∆ymax(P ) ∼ (∆xj(P ))1.53. (95)

According to [3], we would expect to find b = 2. Contributing factor to the discrepancy is the errorin the solver for local problems, which is used in numerically calculating diffusion problems as well.In section 4, this error was shown to be in the order of 10−6, which limits the accuracy of thiscalculation. Figure 13 tends toward log10(∆ymax(P )) = −6 for low grid separation, thus flatteningthe slope of the outcome of the linear fit. Moreover, equation (93) only holds for low ∆xj(P ),whereas for the lower P calculated is close to the order of the domain, e.g. ∆xj(1) ∼ 0.07 m. Thevalues of ∆ymax(P ) range from ∆ymax(0) = 3.3 · 10−3 to ∆ymax(8) = 1.0 · 10−6, which is decentaccuracy when roughly referenced to yH ≈ 1.

25

5.2 Analysis of the total elemental multicomponent diffusion coeffcient

One of the intermediate results that can be easily obtained from the PLASIMO environment arethe total elemental multicomponent diffusion coefficients for each element DTot

e e ∈ E . We will nowderive an analytical solution for DTot

H or the total elemental multicomponent diffusion coefficientfor hydrogen in the H-Ar mixture and use this to determine the accuracy of the numerical resultfor the same property in PLASIMO. DTot

H is defined to be in 1D:

j{H} = −ρDTotH

dy{H}

dx, (96)

or the proportionality constant between the elemental mass fraction gradient and the elementalmass flux, divided by −ρ. Since PLASIMO provides easier access to ρDTot

H , we will verify theaccuracy of this value instead ofDTot

H directly. The first row in the matrix equation (73) interpretedfor one dimension tells us that

j{H} =ρDH,ArfH,ArmAr

mH +mAr

dy{H}

dx+−ρDH,ArfH,ArmH

mH +mAr

dy{Ar}

dx. (97)

Since there are only two elements, it has been shown that −dy{H}/dx = dy{Ar}/dx. Therefore:

j{H} = −ρDH,ArfH,Ardy{H}

dx. (98)

It follows from the definition of DTotH that

ρDTotH = ρDH,ArfH,Ar. (99)

Now we take the standard fH,Ar = 1 and substitute equation (76)

ρDTotH =

9

16

√kBT

8πµH,Ar

1

(rH + rAr)2

ρ

n. (100)

Finally, substitute equation (78) to find ρDTotH (x):

ρDTotH (x) =

9

16

√kBT

8πµH,Ar

1

(rH + rAr)2

mHmAr

(mAr −mH)yH(x) +mH, (101)

which is the explicit analytical expression we wanted to find, since yH(x) is known. In figure 14the numerical and analytical result for DTot

H are plotted against the position x. By inspectionalone, they seem to agree. In figure 15 the absolute error is plotted against the position. The

maximum absolute error is∣∣∣∆DTot

H

∣∣∣ ≈ 1.0 · 10−7.

26

#cells

x = 0.0 m x = 1.0 m

Figure 11: Graphical representation of the grid structure for the 1D diffusion problem. Blackboxes are cells, red dots are grid points.

Figure 12: Analytical and numerical results for the neutral H-Ar mixture mass fractions as afunction of position

27

Figure 13: Double logarithmic plot of maximum mass fraction error against grid point separationwith linear fit

Figure 14: The numerical and analytical result for DTotH plotted against the position x

28

Figure 15: The absolute error for DTotH plotted against the position x

29

6 General conclusions

We have attempted to validate the new method for numerically modeling LTE plasmas and ele-mental demixing in PLASIMO.

In chapters 2 and 3 we introduced a theoretical, mathematical framework to describe neutralmixtures in local thermodynamic equilibrium, using a set of algebraic species equations and aset of partial differential equations of elemental continuity. We also showed how to solve theseequations both when elemental diffusion was absent, but also in general, using an expression forelemental diffusion in terms of gradients in temperature and elemental mass fractions.

In chapter 4 the numerical implementation of the new method in PLASIMO was tested onproblems which do not involve elemental diffusion. First, numerical results were compared tothose of a pair of simple problems for which an analytical solution was known. The PLASIMOmodel was then tested against results from another implementation of the new LTE modelingmethod in MATLAB. A fourth test was proven to be unreliable. The most reliable tests againstanalytical solutions yielded maximum absolute errors in species mass fractions of approximately10−6. Comparing the two numerical models yielded maximum absolute errors in species molefractions of order 10−4.

Finally, chapter 5 tested the new PLASIMO implementation on another problem with analyt-ical solution, but this time involving elemental diffusion in 1D. Specifically, an analytical solutionwas compared to several numerical solutions with varying amount of grid separation. The maxi-mum absolute error of species mass fractions were shown to be a power of approximately 1.53 ofgrid seperation, which is less favorable then the expected power of 2. Some possible explanationswere given for the discrepancy. Finally, an analytical solution for the total elemental multicom-ponent diffusion coefficient was tested against numerical results for the same value at one of thespecific values for the grid separation, yielding a maximum absolute error of order 10−7.

30

A Detailed derivations

A.1 Analysis of LTE problems with elemental diffusion

We will first attempt to express the species concentration gradients ∂x as a function of gradientsof temperature ∂T and elemental mass fractions ∂{y}, respectively. Recognize that, in LTE, forevery r ∈ R: ∑

s∈Sνrs lnxs = lnKr

p(T )− ln p∑s∈S

νrs , (102)

where Krp(T ) is the equilibrium constant in terms of partial pressures that goes with the reaction

for combined species r ∈ R as defined in 2.1. Taking the gradient on both sides and imposinguniform pressure, this becomes ∑

s∈S

νrsxs∇xs =

dlnKrp(T )

dT∇T. (103)

However, van ’t Hoff’s law states that dlnKrp(T )/dT = ∆hr/(RuT

2), where ∆hr is the reactionenthalpy for the reaction corresponding to combined species r ∈ R and Ru is the universal gasconstant. Using this, we gather that∑

s∈S

νrsxs∇xs =

∆hrRuT 2

∇T. (104)

And finally, we rewrite these equations in matrix notation

R+diag−1(x)∂x = θ∂T, (105)

with for every r ∈ Rθr =

∆hrRuT 2

, (106)

and we defineA+ = R+diag−1(x). (107)

At the same time, the elemental mass fractions are constrained by equation (12), and theirgradients ∂{y} therefore by

∂{y} = diag({m})R−diag−1(m)∂y, (108)

The definition of the species mass fractions leads to an expression in terms of the species molefractions

ys =ρsρ

=msns∑

j∈Smjnj

=msxs∑

j∈Smjxj

=ms

Mxs, (109)

where M =∑j∈S mjxj is the mixture mass. From this we can express the gradients of species

mass fractions in terms of the gradients of species mole fractions.

∇ysms

=1

M∇xs −

xsM2∇M =

1

M∇xs −

xsM2

∑j∈S

mj∇xj . (110)

If we rewrite this to fit a translation into matrix notation we get

∇ysms

=1

M∇xs −

xsM2

∑j∈S

mj∇xj =∑j∈S

1

M

(δsj −

xsmj

M

)∇xj . (111)

Now define matrix F with elements Fij = 1M

(δij − ximj

M

)with i, j ∈ S. Then

diag−1(m)∂y = F ∂x. (112)

31

If we combine the last result with equation (108), we finally obtain

diag({m})R−F ∂x = ∂{y}, (113)

and we defineA− = diag({m})R−F . (114)

Now, recall the Stefan-Maxwell equations of diffusion (25). For mathematical convenience, wedefine matrices S1 ∈ RNr×Nr , S2 ∈ RNr×Ne , S3 ∈ RNe×Nr and S4 ∈ RNe×Ne such that

S =

[S1 S2

S3 S4

]. (115)

Further, we define matrices A+1 ∈ RNr×Nr and A+

2 ∈ RNr×Ne such that

A+ =[A+

1 A+2

]. (116)

Using these definitions and equations (21) and (3) we can derive

S

I︷︸︸︷[INr 0−B INe

][INr 0B INe

]W =

[S1 S2

S3 S4

][INr

0−B INe

][Wr

{W}

]=

[S1 − S2B S2

S3 − S4B S4

][Wr

{W}

],

(117)where Wr contains the transposed column vectors of the number fluxes of the combined speciesonly. The Stefan-Maxwell diffusion law (25) tells us that the matrix expression that is given aboveequals species concentration gradients. Using this fact, equations (105), (107) and (116) then showthat

A+SW = θ∂T =[A+

1 A+2

][S1 − S2B S2

S3 − S4B S4

][Wr

{W}

], (118)

orAWr + B{W} = θ∂T, (119)

where A = A+1 (S1 − S2B) + A+

2 (S3 − S4B) and B = A+1 S2 + A+

2 S4. We can, of course,

analogously use (117) and (25) in equations (113) and (114). To do this, we first define A−1 ∈RNe×Nr and A−2 ∈ RNe×Ne such that

A− =[A−1 A−2

]. (120)

Now, using equations (117), (25), (113) and (114), we reveal

A−SW = ∂{y} =[A−1 A−2

][S1 − S2B S2

S3 − S4B S4

][Wr

{W}

], (121)

orCWr + D{W} = ∂{y}, (122)

where C = A−1 (S1 − S2B) +A−2 (S3 − S4B) and D = A−1 S2 +A−2 S4.

A.1.1 Solving for the elemental diffusion fluxes

From equations (119) and (122), we can explicitly express the elemental diffusion fluxes in termsof θ∂T and ∂{y}. We can solve Wr from (119) using straightforward Gaussian elimination [1],and substitute back into (122) for(

D − C A −1B){W} = ∂{y}− C A −1θ∂T, (123)

which in turn can be expressed as a simple linear system

S {W} = β. (124)

32

with S = D − C A −1B and β = ∂{y}− C A −1θ∂T .Although S ∈ RNe×Ne is verifiably square, it can also be shown to be a singular and therefore

non-invertible matrix with nullity(S ) = 1. This problem stems from the nature of the S matrixin the Stefan-Maxwell equations of diffusion.

Ideally, we would have a new, invertible linear system whose solution set is the intersection ofthe mass constraint and the singular system solution sets for every β ∈ RNe in the range of S .This linear system is given below:(

S + ζ{m}{m}T){W} = β, (125)

for any ζ ∈ R \ {0}. Finally, we can express the elemental mass diffusion fluxes explicitly in termsof gradients of temperature and elemental mass fractions as follows

{J} = diag({m})S̃ −1∂{y}− diag({m})S̃ −1C A −1θ∂T, (126)

with S̃ = S + ζ{m}{m}T .

A.2 Excitation problem

We must derive the correct mass fractions for yH and yH∗ from the determined set of equations

αE(T ) ≡ 4 exp(− EexckBT

),

β(T, p) ≡ p

kBT,

nH∗

nH= αE(T ), (127a)

nH + nH∗ = β(T, p). (127b)

From combining equations (127a) and (127b) we gather

nH(1 + αE(T )) = β(T, p). (128)

Also,

yH = ρH/ρ =mHnH

mHnH +mH∗nH∗

=nH

nH + nH∗= nH/β(T, p). (129)

since mH∗ = mH . Substituting this last result into equation (128) in favor of yH , and noticingthat yH∗ = 1− yH , we obtain the end result:

yH =1

1 + αE(T ), (130a)

yH∗ =1

1 + (αE(T ))−1 . (130b)

A.3 Recombination-ionization problem

We must derive the correct mass fraction of hydrogen from the determined set of equations

αI(T ) ≡

(2π(mH −me)mekBT

mHh2

) 32

exp(−EionkBT

),

β(T, p) ≡ p

kBT,

nH+nenH

= αI(T ), (131a)

nH + nH+ + ne = β(T, p), (131b)

nH+ − ne = 0. (131c)

33

We work towards an expression for yH explicit in nH :

yH = ρH/ρ =mHnH

mHnH +mH+nH+ +mene, (132)

which together with equation (131c) and mH = mH+ +me becomes:

yH =mHnH

mHnH + nH+(mH+ +me)=

nHnH + nH+

. (133)

Combining equations (131c) and (131b) tells us that

nH+ =1

2(β − nH), (134)

such that

yH =nH

nH + 12 (β − nH)

=2

1 + ( βnH

). (135)

This is an expression for yH explicitly in terms of nH , T and p. All that remains to be done is toexpress nH explicitly in T and p.

Substituting equation (131c) into (131a) yields

n2H+

nH= αI . (136)

Now, we substitute equation (134) into this result and rearrange terms to be left with

1

4(β − nH)2 − αInH = 0. (137)

The remaining problem has now been reduced to solving a quadratic equation for nH :

(β − nH)2 − 4αInH = 0,

n2H − (2β + 4αI)nH + β2 = 0,

(nH − (β + 2αI))2 − 4αIβ − 4α2

I = 0,

nH = β + 2αI − 2√α2I + αIβ, (138)

since β + 2αI ≥ nH because β = n ≥ nH and αI ≥ 0. Substituting this result into (135) yieldsthe end result:

yH =2

1 +(

β

β+2αI−2√α2

I+αIβ

) . (139)

34

B Local composition calculation input files

In this appendix, all the input files which were used to calculate compositions without diffusion inthis text are presented in full. The files are meant to serve as input for the example newlte local

script in PLASIMO, which takes simplified input files compared to the regular PLASIMO modelinput file as can be loaded through the GUI. Together with these files, the script requires a numberof parameters. These parameters are loosely discussed in section 4, and for exact usage of theseparameters should be referred to calling the script itself with the help parameter. As for the inputfiles: their syntax is discussed in PLASIMO documentation [2].

B.1 Excitation problem

Species {

Particle {

State {

Type Atom

Name HGround

Weight 2

Energy -13.61*eV

}

Name H

Radius 0.53*Ang

}

Particle {

State {

Type Atom

Name Hexc

Weight 8

Energy -3.41*eV

}

Name Hexc

Constitution H

Radius 0.53*Ang

}

IsotopeFile mixture/isotopes.in

}

Relations {

Relation {

Name "H exc"

Format "H <=> Hexc"

}

Type List

}

NewComposition {

}

B.2 Ionization-recombination problem

Species {

Particle {

State {

Type Atom

Name H+Ground

Weight 1

35

Energy 0*eV

}

Name H+

Radius 0.53*Ang

}

Particle {

State {

Type Atom

Name TheOneAndOnlyElectron

Weight 2

Energy 0*eV

}

Name e

Radius 0.001*Ang

}

Particle {

State {

Type Atom

Name HGround

Weight 1

Energy -13.61*eV

}

Name H

Radius 0.53*Ang

}


}

Relations {

Relation {

Name "H ionization"

Format "H + e <=> H+ + e + e"

}

Type List

}

NewComposition {

}

B.3 Eleven species air mixture

Species {

Particle {

State {

Type Atom

Name NGround

Weight 4

Energy 0.0*eV

}

Name N

Radius 1.05*Ang

}

Particle {

State {

Type Atom

Name OGround

36

Weight 9

Energy 0.0*eV

}

Name O

Radius 1.02*Ang

}

Particle {

State {

Type Atom

Name N+Ground

Weight 9

Energy 14.60*eV

}

Name N+

Radius 1.05*Ang

}

Particle {

State {

Type Atom

Name O+Ground

Weight 4

Energy 13.68*eV

}

Name O+

Radius 1.05*Ang

}

Particle {

State {

Type Atom

Name N2Ground

Weight 1

Energy -9.79*eV

}

Name N2

Radius 1.80*Ang

}

Particle {

State {

Type Atom

Name O2Ground

Weight 3

Energy -5.16*eV

}

Name O2

Radius 1.75*Ang

}

Particle {

State {

Type Atom

Name NOGround

Weight 2

Energy -6.54*eV

}

Name NO

37

Radius 1.78*Ang

}

Particle {

State {

Type Atom

Name N2+Ground

Weight 2

Energy 5.85*eV

}

Name N2+

Radius 1.80*Ang

}

Particle {

State {

Type Atom

Name O2+Ground

Weight 4

Energy 6.98*eV

}

Name O2+

Radius 1.75*Ang

}

Particle {

State {

Type Atom

Name NO+Ground

Weight 1

Energy 2.79*eV

}

Name NO+

Radius 1.78*Ang

}

Particle {

State {

Type Atom


Weight 2

Energy 0.0*eV

}

Name e

Radius 0.001*Ang

}


}

Relations {

Relation {

Name "N dissoc"

Format "N2 <=> 2 N"

}

Relation {

Name "O dissoc"

Format "O2 <=> 2 O"

}

Relation {

38

Name "NO dissoc"

Format "NO <=> N + O"

}

Relation {

Name "N ion"

Format "N + e <=> N+ + e + e"

}

Relation {

Name "O ion"

Format "O + e <=> O+ + e + e"

}

Relation {

Name "N2 ion"

Format "N2 + e <=> N2+ + e + e"

}

Relation {

Name "O2 ion"

Format "O2 + e <=> O2+ + e + e"

}

Relation {

Name "NO ion"

Format "NO + e <=> NO+ + e + e"

}

Type List

}

NewComposition {

}

B.4 XeInI composition

B.4.1 Input file

Species {

Include @PLASIMO_INPUTDATA_DIR/ideas/gnybida/Data/In_levels_kurucz.in

Particle {

State {

Type Atom

Name Xe_ground

Weight 1

Energy 0*eV

}

Name Xe

Radius 2.28*Ang

}

Particle {

State {

Type Atom

Name Xe+_ground_state

Weight 4

Energy 12.12*eV

}

Name Xe+

Radius 2.28*Ang

}

39

Particle {

State {

Type Atom

Name I_ground

Weight 4

Energy 0*eV

}

Name I

Radius 1.86*Ang

}

Particle {

State {

Type Atom

Name I_ion

Weight 5

Energy 10.45*eV

}

Name I+

Radius 1.86*Ang

}

Particle {

State {

Type RotVib

Name I2_ground

Weight 1

Energy -1.56662*eV

B 0.368*cm1

Omega 286*cm1

RedMass 19.46*amu

}

Name I2

Radius 3.19*Ang

}

Particle {

State {

Type Atom

Name In_ion

Weight 1

Energy 5.7863*eV

}

Name In+

Radius 2.02*Ang

}

Particle {

State {

Type Atom

Name I_negative_ion

Weight 1

Energy -3.0*eV

}

Name I-

Radius 1.86*Ang

}

Particle {

40

State {

Type Atom

Name e

Weight 2

Energy 0*eV

}

Name e

Radius 0.001*Ang

}

Particle {

ASDF {

Type Boltzmann

}

State {

Type SpecParam

Name InI-X1

RedMass 60.27*amu

Energy -3.057*eV

Weight 1

D 3.057*eV

Omega 166*cm1

B 0.0359*cm1

Omega_x 0*cm1

Omega_y 0*cm1

Alpha 0*cm1

Re 2.791*Ang

}

State {

Type SpecParam

Name InI-X2

RedMass 60.27*amu

Energy -3.057*eV

Weight 1

D 3.487*eV

Omega 170*cm1

B 0.0359*cm1

Omega_x 0*cm1

Omega_y 0*cm1

Alpha 0*cm1

Re 2.807*Ang

}

State {

Type SpecParam

Name InI-A

RedMass 60.27*amu

Energy -0.186*eV

Weight 6

D 0.448*eV

Omega 141*cm1

B 0.0364*cm1

Omega_x 0*cm1

Omega_y 0*cm1

Alpha 0*cm1

Re 2.771*Ang

41

}

State {

Type SpecParam

Name InI-B

RedMass 60.27*amu

Energy -0.105*eV

Weight 6

D 0.105*eV

Omega 130*cm1

B 0.0359*cm1

Omega_x 0*cm1

Omega_y 0*cm1

Alpha 0*cm1

Re 2.789*Ang

}

State {

Type SpecParam

Name InI-3

RedMass 60.27*amu

Energy 3.091*eV

Weight 1

D 0.123*eV

Omega 168*cm1

B 0.0383*cm1

Omega_x 0.209*cm1

Omega_y 0*cm1

Alpha 0*cm1

Re 2.7*Ang

}

Name InI

Radius 3.32*Ang

Contains In 1

Contains I 1

ChargeNr 0

}

Particle {

State {

Type RotVib

Name InI2_ground

Weight 1

Energy -5.1199*eV

B 0.0383*cm1

Omega 168.00*cm1

RedMass 60.27*amu

}

Name InI2

Radius 4.10*Ang

}

Particle {

State {

Type RotVib

Name InI3_ground

Weight 1

Energy -6.9129*eV

42

B 0.0383*cm1

Omega 168.00*cm1

RedMass 60.27*amu

}

Name InI3

Radius 4.47*Ang

}

IsotopeFile @PLASIMO_INPUTDATA_DIR/input/mixture/isotopes.in

}

Relations {

Relation {

Name Xe_ionization

Format "Xe + e <=> Xe+ + 2 e"

}

Relation {

Name InI_dissociation

Format "In + I <=> InI"

}

Relation {

Name InI3_dissociation

Format "InI3 <=> InI2 + I"

}

Relation {

Name InI2_dissociation

Format "InI2 <=> In + I2"

}

Relation {

Name In_ionization

Format "In <=> In+ + e"

}

Relation {

Name I_ionization

Format "I <=> I+ + e"

}

Relation {

Name I2_dissosiation

Format "I2 <=> 2 I"

}

Relation {

Name I_negative_ion_formation

Format "I2 + e <=> I- + I"

}

Type List

}

NewComposition {

}

B.4.2 In levels kurucz.in

Particle {

Name In

Radius 1.66*Ang

ASDF {

Type Boltzmann

43

}

State { # 0

Type Atom

Energy 0.000000*eV

Weight 2

Name 0

}

State { # 1

Type Atom

Energy 0.274341*eV

Weight 4

Name 1

}

State { # 2

Type Atom

Energy 3.022055*eV

Weight 2

Name 2

}

State { # 3

Type Atom

Energy 4.078378*eV

Weight 4

Name 3

}

State { # 4

Type Atom

Energy 4.081267*eV

Weight 6

Name 4

}

State { # 5

Type Atom

Energy 4.336970*eV

Weight 2

Name 5

}

State { # 6

Type Atom

Energy 4.466311*eV

Weight 4

Name 6

}

State { # 7

Type Atom

Energy 4.501139*eV

Weight 2

Name 7

}

State { # 8

Type Atom

Energy 4.643757*eV

Weight 6

Name 8

44

}

State { # 9

Type Atom

Energy 4.841721*eV

Weight 4

Name 9

}

State { # 10

Type Atom

Energy 4.847907*eV

Weight 6

Name 10

}

State { # 11

Type Atom

Energy 5.038650*eV

Weight 2

Name 11

}

State { # 12

Type Atom

Energy 5.187381*eV

Weight 4

Name 12

}

State { # 13

Type Atom

Energy 5.190523*eV

Weight 6

Name 13

}

State { # 14

Type Atom

Energy 5.296782*eV

Weight 2

Name 14

}

State { # 15

Type Atom

Energy 5.440896*eV

Weight 2

Name 15

}

}

45

C Diffusion calculation input files

C.1 Neutral H-Ar mixture 1D input file

Plugins {

PluginLibs {

Type Common

LoadFile @PLASIMO_DIR/ideas/newmixture/ltemixture/libltelinsysmixture_1d.la

}

PluginLibs {

Type Common

LoadFile @PLASIMO_DIR/ideas/jan/lte/libnewlte_local.la

}

}

Include @PLASIMO_INPUTDATA_DIR/input/plplugins-1d.cnf Plugins

MainIterConfig {

iter_max 10000000

iter_log 100

accuracy 1e-9

data_path data/

data_ext .test

init no

init_path data/

init_ext .in

}

EM {

Sigma {

Function 1/(Ohm*m)

}

Type Null

}

Discretizer {

Type SteadyState

}

PlasmaRegion {

Grid {

Geometry {

CellVector {

100000002

}

Material {

Id 0

Name Plasma

}

Material {

Id 1

Name Left

}

Material {

Id 2

Name Right

}

BaseRefinementPower 0

RefinementPower 1

46

}

Stretch {

Type No_Stretch

}

Type cartesian

Name DefaultCartesianGrid

xmin 0*m

xmax 1*m

ymin 0*m

ymax 1*m

}

Flow {

MassFluxDensFunc {

Function {

Function 0*kg/(m^2*s)

}

}

MassFluxDensFunc {

Function {


}

}

MassFluxDensFunc {

Function {


}

}

PressureFunc {

Function {

Function 100000*Pa

}

}

Type Fixed

}

Temperature {

IterConfig {

MatrixSolver {

Type BiCGstabSolver

maxiter 1000

Preconditioner Diagonal

}

urf 1.0

urf_factor 0.0

}

BoundaryConditions {

BndConfig {

BndCond {

Type ConstDirichlet

Value 5000*K

}

ApplyAt Left All

ApplyAt Right All

}

}

47

INIT_VAL 5000*K

Fixed yes

IgnoreViscDiss yes

}

RadTrans {

Transfer {

Type Default

}

Urf 1.0

}

Transport {

ThermalCond {

Calculator {

Type StdThermalCondLTE

}

URF 1.0

}

SpecificHeat {

Calculator {

Type SpecificHeatLTE

}

URF 1.0

}

ElectricalCond {

Calculator {

Type StdElectricalCond

}

URF 0.9

}

ViscosityHP {

Calculator {

Type ViscosityWilke

}

URF 1.0

}

Diffusion {

Calculator {

Type AmbipolarDiffusion

}

URF 1.0

}

Type LTE

}

Mixture {

Composition {

Type GuldbergWaage

}

IterConfig {

MatrixSolver {

Type SuperLUSolver

}

urf 1.0

urf_factor 0.0

}

48

ElementBC {

LinsysElementBndConfig {

LinsysElementBndCond {

WallFraction {

Element Ar

Fraction 0.001

}

WallFraction {

Element H

Fraction 0.999

}

WallFraction {

Element e

Fraction 0.0

}

}

ApplyAt Left All

}

LinsysElementBndConfig {

LinsysElementBndCond {

WallFraction {

Element Ar

Fraction 0.6

}

WallFraction {

Element H

Fraction 0.4

}

WallFraction {

Element e

Fraction 0.0

}

}

ApplyAt Right All

}

}

Species {

Particle {

State {

Type Atom

Name Ar+Ground

Weight 1

Energy 20.0*eV

}

Name Ar+

Radius 0.53*Ang

}

Particle {

State {

Type Atom


Weight 2

Energy 0*eV

}

49

Name e

Radius 0.001*Ang

}

Particle {

State {

Type Atom

Name HGround

Weight 2

Energy 4.46*eV

}

Name H

Radius 0.53*Ang

}

Particle {

State {

Type Atom

Name ArGround

Weight 1

Energy 0*eV

}

Name Ar

Radius 8.8e-11*m

}


}

Relations {

Relation {

Name "Ar ionization"

Format "Ar + e <=> Ar+ + e + e"

ApplyDB Yes

}

Type List

}

InitFraction {

Element Ar Value 0.001

Element H Value 0.999

Element e Value 0.0

}

Radiation {

}

CrossSections {

DefaultNonAveragedCrossSec {

Model NonAvStandardCrossSec

}

DefaultAveragedCrossSec {

Model AvStandardCrossSec

}

DefaultCollInt {

Model Standard

}

}

ConservationRules {

}

Type LTE

50

}

Type LTE

}

Type ModelLTE-1D

Author "Kim Peerenboom"

Version ""

51

References

[1] P. Rini, D. Vanden Abeele and G. Degrez, Closed form for the equations of chemically reactingflows under local thermodynamic equilibrium, Physical Review E 72, 2005

[2] Eindhoven University of Technology, PLASIMO Manual and Source Documentation, Availableat http://plasimo.phys.tue.nl/

[3] J.H.M. ten Thije Boonkkamp, J. van Dijk, L. Liu, K.S.C. Peerenboom, Extension of the com-plete flux scheme to systems of conservation laws, Journal of Scientific Computing, 53, No. 3,p. 552-568, 2012

[4] H.W.P. van der Heijden, Modelling of radiative transfer in light sources, PhD thesis, EindhovenUniversity of Technology, Eindhoven, 2003

[5] W.G. Vincenti and C.H. Kruger, Introduction to physical gas dynamics, Wiley, New York, 1965

[6] J.N. Butler and R.S. Brokaw, Journal of Chemical Physics, 26, 1636, 1957

52

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