CALCULATION OF COLLISION INTEGRALS FOR ABLATION

CALCULATION OF COLLISION INTEGRALS FOR ABLATION SPECIES

A. Bellemans1 and T. Magin2 1

Aeronautics and Aerospace Department , von Karman Institute for Fluid Dynamics, 1640 Rhode-Saint-Gen`ese, Belgium, [email protected] 2 Aeronautics and Aerospace Department , von Karman Institute for Fluid Dynamics, 1640 Rhode-Saint-Gen`ese, Belgium, [email protected]

ABSTRACT Collision integrals are being computed for a 39species typical carbon-phenolic mixture for improving the calculation of transport properties such as the viscosity, diffusion and conductivity of a mixture obtained from ablative material recession. To calculate these integrals, classical potential models such as the Lennard-Jones potential for long range interactions at low temperatures, and the Born-Mayer potential for short range interactions at high temperatures are used. To better approximate experimental results, a sewing method is proposed to couple both the Lennard-Jones and Born-Mayer potential. The transport properties obtained with the new collision integrals are compared to the Chemical Equilibrium and Applications (CEA) database of NASA Ames Research Center.

This ablation is due to chemical reactions, sublimation, or erosion during re-entry. When the ablative material of the heat shield starts to recess, particles are introduced in the flow between the bow shock and the vehicle. The previously described ablation phenomena are visualized in Figure 1. The interaction of the ablative particles with the flow particles and each other have an important effect on the thermodynamics and transport properties of the flow. Measuring transport properties for ablative flows turns out to be a difficult task. A microscopic investigation studying binary collisions has to be carried out to determine a collision dataset for retrieving the transport properties. Unfortunately, accurate collision integrals are currently not available for ablation mixtures. The objective of the present paper is to give a first approximation of these collision integrals for a reduced ablation mixture. The ablation species under considshock  heated  gas  (T  ~  10,000K)  

Key words: plasma, collision integrals, ablation.

1.

INTRODUCTION

convec7ve     radia7on   heat  flux   flux    

hot   radia7ng   gas  

chemical   reac7on   species     products   diffusion  

boundary   layer  

mechanical   erosion  

porous  char  

Ablative heat shields are designed to protect a space vehicle during atmospheric re-entry. A new class of materials based on low density carbon/resin composites is considered for future space missions. The material is made up of carbon preform, injected with phenolic resin. Examples of such ablative materials are Phenolic Impregnated Carbon Ablator (PICA) developed by NASA Ames Research Center, and the European ASTERM developed by ASTRIUM [1]. During atmospheric re-entry, a part of the produced heat flux is injected into the heat shield. This causes the protective material to be destroyed by two ablation phenomena. The first phenomenon is the production of pyrolysis gases because of the carbonization of the phenolic resin in the material. The gases are transported in the flow by diffusion and convection, where they mix with air and change their chemical composition. The second phenomenon is the ablation of the carbon preform and carbonized resin.

surface   recession  

heat   transfer  

pyrolysis   gases  

OH  

pyrolysis    zone  

OH   virgin  material  

Figure 1: Ablation phenomena during re-entry: pyrolysis of phenolic resin and ablation of the carbon preform.

eration in this work are those present in a typical carbon phenolic decomposition: carbon, hydrogen and oxygen. When considering the species available in the Chemical Equilibrium with Applications (CEA), one ends up with 144 species. When also considering nitrogenous species, one obtains a large mixture of 201 species. Scoggins [2] proposes to reduced this

Table 1: Reduced ablation mixtures based on 3 tolerance levels Xi according to Scoggins [2]. Xi >  eC5 H CH4 C2 H C3 H H2 O O

C3 H C(s) CN C2 H2 (acetylene) C3 H3 (2-propynl) H+ N O+

10−5

C4 H CO+ C2 H 4 C3 H4 (propyne) H2 O 2 NO+

C3 H2 (singlet) C+ 2 C 2 H6 HCO N+ NO2

10−7

C5 H 5 CO+ 2 C3 H3 (1-propynl) C4 H6 (butadiene) HNCO O−

C− COOH C3 H4 (cyclo-) C6 H6 HNO2 OH+

10−3

Species 39 species C3 H2 (triplet) C CO C2 H2 (vinylidene) C4 H2 (butadiyine) HCN NH3 OH 69 species CH2 C− 2 CCN HCCN NCO NCN 104 species CH+ CH2 CO (ketene) C3 H5 (allyl) H− HNO3 OH−

ablative mixture by imposing a tolerance on the mole fractions of the different species. The mixture is still considered as long as the transport and thermodynamic properties are reproduced accurately as given by the complete mixture. Three different reduced mixtures of 39, 69, and 104 species are defined by considering tolerance levels of 10−3 , 10−5 , and 10−7 , respectively. A representation of the present species in the reduce mixture can be seen in Table 1. Collision integrals will be determined for the 39-species reduced ablation mixture. Over the last 30 years, considerable research has been carried out to provide collision data for various mixtures and atmospheres. Capitelli, Gorse, Longo and Giordano have studied and calculated collision integrals for high temperature air species between 50 and 100,000 K [3]. This publication has been revised in 2005 by Wright, Bose, Palmer and Levin with the publication of their paper gathering the best available transport data of high temperature 13-species air [4], and ion-neutral data for air-argon mixtures [5]. This revision was followed by another publication in 2007 [6] reviewing the transport properties of 17-species weakly ionized CO2 -N2 mixtures for Mars and Venus entries. Other data for Mars, and a review for Titan entries are published in the work of Andr´e et al. [7]. Bruno et al. have calculated the collision integrals and verified the transport properties for the Jovian atmosphere made up out of helium and hydrogen interactions [8]. Accurate collision data for ablation species is currently not available in literature. The present paper is structured as follows. Section

C4 C+ CO2 CNC C4 N2 HNC NO O2

C5 CH C2 C 2 N2 C6 H 2 H2 N2

CH3 C2 H3 (vinyl) C2 O HNO NH N2 O

CN− CH3 CN C3 H4 (allene) HO2 NH2 O3

CN+ C2 H5 C3 H6 (propylene) HCO+ H+ 2 O+ 2

CNN OCCN C 3 O2 HCCO HCHO (formaldehy) O− 2

2 studies binary collisions and gives an introduction to collision integrals and their role in the calculation of transport properties in high enthalpy flows. Section 3 presents the methodology for each interaction type. The collision integrals for neutral-neutral interactions are calculated by using a combination of the Lennard-Jones and Born-Mayer potential, a Langevin potential is considered for the ion-neutral interactions and a Coulomb potential is used for the charged interactions. The phenomenological potential is also presented as alternative method. In Section 4, the new collision integrals are presented and used for the calculation of transport properties of the 39-species reduced ablation mixture. Results are compared with CEA.

2.

COLLISION INTEGRALS

2.1.

General formulation

To calculate the transport properties, one needs to compute the transport coefficients starting from the collision integrals for every interaction (neutralneutral, electron-neutral, charged, ion-neutral). This quantity is obtained by solving a triple integral over the deflection angle, the impact parameter and the relative energy: s Ωl,s ij

=

2πkT µij

Z∞ Z∞ 0

0

2

2s+3 (1 − cosl χ)dbdγ (1) e−γij γij

The superscripts l and s are related to the LaguerreSonine polynomials of the spectral method. With γij the reduced relative velocity: r µij γij = gij (2) 2kT In Equation 2, µij is the reduced mass of the two colliding molecules, k the Boltzmann constant, and gij the initial relative velocity of the two molecules in a binary encounter. The angle of deflection, χ, is the only feature of a collision which enters into the formula for the transport coefficients and it is given by: Z∞ χ(b, g) = π − 2b

dr/r2 q

0

φ(r) 1 − ( 0.5µg 2) −

(3) b2 r2

From this equation we see that the deflection angle, and thus the collision itself, depends on two parameters: b, the impact parameter, which is the distance of closest approach in absence of the potential, and g which is the initial relative velocity as explained before. Both the deflection angle and the impact factor can be visualized on Figure 2.

the Born-Mayer potential at high temperatures, and the Lennard-Jones potential at low temperatures. The Lennard-Jones and Born-Mayer potential can be combined through other methods as presented in the work of Sokolova (sewing method) [11] and the upcoming work of Schwenke [12]. An example of the sewing method can be seen in Figure 4. Another potential that can be considered is the phenomenological potential based on the polarizability of the neutral collider. The phenomenological potential can be written in function of three parameters (See Equation 6). The first parameter, φ0 is the reference potential, re is the reference length, and β is a special parameter ranging from 6 to 10, depending on the hardness of colliding electronic distribution densities. The m parameter equals 4 for ion-neutral interactions and 6 for neutral-neutral interactions. m 1 n(x) 1 φ = φ0 · [ · ( )n(x) − · ( )m ] (6) n(x) − m x n(x) − m x with x = r/re

(7)

2

(8)

n(x) = β + 4x

The ESA report, ESA-STR-256, proposes fits for the collision integrals based on this phenomenological potential in function of the β parameter, the reference potential and the temperature. This fitting equation is given in Equation 9, and uses the reduced temperature as given in Equation 10. lnΩ(l,s)∗ = [a1 (β) + a2 (β)x]·

b

(9)

exp[(x−a3 (β))/a4 (β)] exp[(x−a3 (β))/a4 (β)]+exp[(a3 (β)−x)/a4 (β)]

χ

exp[(x−a6 (β))/a7 (β)] +a5 (β) exp[(x−a6 (β))/a 7 (β)]+exp[(a6 (β)−x)/a7 (β))]

Figure 2: Elastic binary collision between 2 particles with b, the impact factor and χ the deflection angle.

Another important parameter that shows up in this last relation is the spherical intermolecular potential φ(r). Depending on the temperature and the range of the interaction between molecules, different potential models are used. When the distance between the particles is small, the interaction forces are due to the crossing and repulsion of the electron clouds. For those short range interaction forces, the exponential repulsive potential of Born-Mayer [9] can used to describe neutral-neutral interactions: 0

φ(r) = φ0ij e−r/σij

(4)

This Born-Mayer potential performs well at high temperature where short range interactions dominate. On the other hand, low temperatures are dominated by long range interactions. Such long interaction forces between neutral molecules can be modeled by a Lennard-Jones potential: φ(r) = φ0ij ((

0 0 σij σij )12 − ( )6 ) r r

(5)

The Tang-Toennies potential [10] is a compromise between the previous two potentials. It follows

kT (10) φ0 The fit coefficients for Equation 9 are tabled for every collision integral, from Ω(1,1) to Ω(4,4) , as a function of c and β (Equation 11). x = lnT ∗ = ln

ai (β) =

2 X

cj β j

(11)

j=0

To obtain the dimensional collision integral σ 2 Ω(l,s)∗ these ESA-STR-256 fits must be multiplied with σ 2 as expressed in Equation 12 and 13. The ξ coefficients for both ion-neutral and neutral-neutral interactions are given in Table 2. σ 2 = (re · x0 (β))2

(12)

x0 (β) = ξ1 β ξ2

(13)

with

Table 2: ξ coefficients for the calculation of x

ion-neutral neutral-neutral

m 4 6

ξ1 0.7564 0.8002

ξ2 0.064605 0.049256

5

(l,s)

10

With ΩijRS the collision integral for a rigid sphere:

4

10

s 3

(l,s) ΩijRS

10

2

10

=

kT (s + 1)! 1 + (−1)l (1 − 0.5 )πσ 2 (15) 2πµ 2 1+l

1

ϕ / ϕ0

10

Considering these reduced expressions one can express the various transport properties in function of the collision integrals. The viscosity for instance, is (2,2)∗ a function of the viscosity collision integral Ωii and given by: √ 5 πkT mi (16) ηi = 16 Ω(2,2)∗ ii

1.0 0.5 0.0 -0.5 -1.0

I

II

III

-1.5 0

1

2

r/σ

60

The binary diffusion coefficient is written in function (1,1)∗ of the diffusion collision integral Ωij and is given by the following relation: s 1 3πkT µij (17) Dij = n 8Ω(1,1)∗

50

3.

ij

3.1.

METHODOLOGY Neutral-neutral Interactions

40

Q

(1,1)

2

[A ]

Figure 3: Potential models for the O2 − O2 interaction: – Tang-Toennies, - - Born-Mayer, · · · LennardJones.

As a first approximation, the collision integrals between neutral species have been calculated using a Lennard-Jones potential as presented in Equation 18. The expression for the potential uses two important parameters: the reference potential energy φ0ij [J] 0 and the reference potential length σij [m].

30

20 0

2500

5000

7500

T [K]

10000

12500

15000

φ(r) = φ0ij ((

Figure 4: Sokolova sewing method. Q11 collision integral for the CO2 -CO2 interaction. [13]

Neutral-ion interactions use a Langevin potential based on the polarizability. For collision between charged particles a Coulomb potential is considered.

0 0 σij σij )12 − ( )6 ) r r

(18)

These reference Lennard-Jones parameters can be found in the literature for many common species. Unfortunately, most of them are missing for the reduced ablation mixture we want to study. Combination rules have been set up to calculate the missing interactions starting from the pure species interactions (indices ii and jj). The combination rule for the potential energy is given by the Lorentz rule:

2.2.

Transport Properties φij =

As mentioned in the introduction, the mixture transport properties can be retrieved from the collision integrals, Ωl,s ij , for each colliding pair. Therefore a new dimensionless quantity has been derived, namely the reduced collision integral:

1 (φii + φjj ) 2

(19)

The Berthelot rule gives the combination rule for the potential length: √ σij = σii σjj (20)

(l,s)

(l,s)∗ Ωij

=

Ωij

(l,s)

ΩijRS

(14)

Other combination rules can be found back in the work by Kong for both the Lennard-Jones and More

Table 3: Polarizability values by R. Jaffe.

potential [14] and the DymondAlder potential [14]. For polyatomic species Andr´e and Aubreton [7] propose a variant of these combination rules. An example is given below for CO2 :

C3 H C 3 H2 C4 C5 CH C2 H C2

φCO2 −CO2 = ((φC−C )3 + (φ3O−O ) + (φO−O )3 )1/3 (21) √ σCO2 −CO2 = σC−C σO−O σO−O 1/3 (22) Lennard-Jones potentials are mainly used to model long range interactions. For the short range interactions we consider a Born Mayer potential. The missing potential parameters have also been retrieved by the use of combination rules. As a first approximation, both potentials will be combined by using the Sokolova sewing method [11]. The sewing will be compared with collision integrals obtained with the phenomenological potential in a second step.

3.3.

C2 H2 CNC C2 N2 C3 C 3 H3 C4 H2 C6 H2

α [a.u.] 3.33 15.28 14.74 17.02 18.64 20.48 27.85

Charged Interactions

The charged interactions have been modeled using a classic repulsive Coulomb potential.

3.4. 3.2.

α [a.u.] 23.82 19.5 40.37 24.14 9.54 8.41 37.64

Results

Neutral-ion Interactions

The reduced mixture we are considering includes three ions: C+ , H+ , and O+ . Ion-neutral interactions have been modeled with Langevin potentials based on the polarizability of the neutral collision partner (Equations 24 to 32). Simple relations have been derived in the paper by Bruno [8] to express all collision integrals in function of the ion charge, z, and the polarizability, α. The expressions of the collision integrals can be written in function of the Ω(1,1) collision integral. Finding polarizability values for the ablation species in our reduced mixture turned out to be a difficult task. Some can be found in literature (Mason [15], CRC Handbook of Chemistry and Physics [16]), but most of them had to be recalculated. Richard Jaffe from NASA Ames Research Center computed the missing polarizability values for this work (Table 3). The resonant charge transfer should also be added for interactions between parent species: Ω(l,s) =

q (l,s) (l,s) Ωelastic + Ωexchange

σ 2 Ω(1,1)∗ = 424.443z

p

(23)

α/T

(24)

2

(1,1)∗

(25)

2

(1,1)∗

(26)

σ 2 Ω(1,4)∗ = 0.656σ 2 Ω(1,1)∗

(27)

2

σ Ω

(1,2)∗

2

(1,3)∗

σ Ω 2

σ Ω

(1,5)∗

2

(2,2)∗

= 0.833σ Ω = 0.729σ Ω 2

(1,1)∗

(28)

2

(1,1)∗

(29)

σ 2 Ω(2,3)∗ = 0.761σ 2 Ω(1,1)∗

(30)

σ Ω 2

= 0.870σ Ω

(1,1)∗

(31)

σ 2 Ω(3,3)∗ = 0.842σ 2 Ω(1,1)∗

(32)

σ Ω

(2,4)∗

= 0.602σ Ω

2

= 0.685σ Ω

A numerical code for calculating collision integrals has been developed at the von Karman Institute for Fluid Dynamics by Professor T. Magin during his Ph.D. thesis [13]. The coded uses the potential parameters (Lennard-Jones or Born-Mayer) as input, and performs the corresponding triple integration to obtain the collision integrals over a predefined temperature range. In a second step, the collision data is fitted to a third order polynomial in function of their reduced temperature:

Ω(l,s) = exp(ax3 + bx2 + cx + d) x = ln(Treduced ) T · kB Treduced = φ0

(33) (34) (35)

As we use the reduced expression for the collision integral, we can use the same fit coefficients for all interactions and consequently correct for the potential parameters φ0 and σ0 . More explicitly, the corresponding φ0 should be used in the expression of the reduced temperature, and the collision integral should be multiplied with σ0 /σref . The σref is the reference potential length for the collision integral of the first fitted and calculated interaction. In the following example, we show how to obtain the fits for CN-CO starting from the calculation of the CN-CN interaction: • We first perform a calculation to obtain the collision integrals for CN-CN using the LennardJones potential in the code of Magin. • The collision integrals for this CN-CN interaction are fitted in function of their reduced temperature to a third order exponential expression: 3 2 Ωl,s CN −CN = exp(ax + bx + cx + d).

• By using the previous fitting coefficients (a, b, c, and d) we can calculate the collision integrals for the CN-CO interaction: σCN −CO 2 3 2 Ωl,s CN −CO = ( σCN −CN ) exp(ax + bx + cx + d) T ·kB with x = ln φCN . −CO

Table 4 and Table 5 contain the Lennard-Jones and Born-Mayer potential parameters of the pure species interactions used for the calculation of the collision integrals for the reduced phenolic ablation mixture. They can be found back in the Appendix of this work. Most of the time, combination rules have been used to calculate the potential parameters as they were hard to find back in literature. For the missing potentials of polyatomic species, the combination rules of Andr´e et al. [7] have been used. A comparison of the different potentials has been provided in Figure 5 for the CO2 pure species interaction. In this plot, the Lennard-Jones potential has been replaced by the (m,6) potential used in the work of Magin [13]. It has been combined with the Born-Mayer potential to obtain a sewing according to Sokolova [11].

Figure 6: Viscosity for the 39 species reduced ablation mixture at equilibrium. Pressure: 0.1 atm. Temperature range: 200:6,000 K. Composition C:H:N:O = 0.2838:0.5749:0.0085:0.1328.

CO2−CO2 100 Born−Mayer (m,6) Magin Sewing

90

80

Omega11, 10−20 m

70

60

50

Figure 7: Conductivity for the reduced ablation mixture at equilibrium. Pressure: 0.1 atm. Temperature range: 200:6000 K. Composition C:H:N:O = 0.2838:0.5749:0.0085:0.1328.

40

30

20

10

0

0

1000

2000

3000

4000 Temperature

5000

6000

7000

8000

Figure 5: Comparison of three potentials for the Ω11 collision integral for CO2 . blue: Born-Mayer potential, green: (m,6) potential, x-x: sewing

Transport properties such as the dynamic viscosity and the mixture equilibrium thermal conductivity have been compared to the NASA CEA database and the previous database of MUTATION++ . The results can be visualized on both Figure 6 and Figure 7. The red curve represents the data of the new MUTATION++ database, where the collision integrals for ablation species have been calculated with a Lennard-Jones potential. The green curve represents the old database of MUTATION++ where the ablation species have been calculated by B. Moore during his internship at NASA Ames Research Center.

The dynamic viscosity (Figure 6) is now close to the CEA database. As can be seen on this viscosity plot, the data of the old MUTATION++ has been overestimated starting from 1,500 K. The equilibrium thermal conductivity of the reduced ablation mixture is underestimated compared to CEA. The old database was overestimating it. Of course these results are still an approximation as no elaborated quantum calculations have been carried out to determine these collision integrals. This work shows that Lennard-Jones potentials can already give good results.

4.

CONCLUSION

A new set of collision integrals for a reduced ablation mixture considering 39 species has been calculated using a combination of the Lennard-Jones and Born-Mayer potential. Both potentials have been combined with the Sokolova sewing method. In the

future, this potential will be compared with the phenomenological potential which is based on the polarizability of the neutral species in the interaction. Combination rules have been used to calculate the potential parameters of the interactions that could not be found in literature. For polyatomic species the combination rules of Andr´e have been used. Transport properties such as the dynamic viscosity and the mixture equilibrium thermal conductivity have been compared with the Chemical Equilibrium and Applications (CEA) database of NASA Ames Research Center. All data is available in the officially released plasma library, MUTATION++ , of the von Karman Institute for Fluid Dynamics [18].

ACKNOWLEDGMENTS

The authors wish to acknowledge R. Jaffe from NASA Ames Research Center for providing polarizabilities for the ablation species. The research of A. Bellemans, is sponsored by the Belgian FRIA-FNRS grant. The work of Prof. T.E. Magin is sponsorded by the European Research Council, Starting Grant No. 259354.

Table 4: Lennard-Jones parameters for pure species.

C O2 O N N2 Ar CH4 H C2 H C4 HCN CO C2 N2 CN NO H2 C3 C2 H H2 O C5 C6 H3

C O2 O N N2 CH4 H C2 H C4 HCN CO C2 N2 CN NO H2 C3 C2 H H2 O C5 C6

C O2 O N N2 CH4 H C2 H C4 HCN CO C2 N2 CN NO H2 C3 C2 H H2 O C5 C6

σ [A] 3.078 2.850 4.140 2.680 3.160 3.043 3.009 3.043 3.078 2.917 3.470 3.078 3.160 2.840 3.260 3.220 3.078 3.043 3.651 3.078 3.078

φ [eV] 359.20 820.00 1410.00 86.00 2290.00 359.55 32 .50 452.62 570.19 360.92 4740.00 452.56 2290.00 360.84 2160.00 246.00 518.06 452.62 1412.49 614.22 652.71

Reference [25] [25] [25] [25] [25] [7] [25] [7] [7] [7] [25] [7] [25] [7] [25] [25] [7] [7] [7] [7] [7]

REFERENCES 1. Helber, B., Asma, C. O., Babou, Y., Hubin, A., Chazot, O. & Magin, T. E. (2014). Material response characterization of a low-density carbon composite ablator in high-enthalpy plasma flows. J. Mater. Sci., 49(13), 4530 4543.

APPENDIX

C O2 O N N2 Ar CH4 H C2 H C4 HCN CO C2 N2 CN NO H2 C3 C2 H H2 O C5 C6 H3

Table 5: Born-Mayer parameters for pure species.

σ [A] 2.000 3.433 2.660 2.980 3.681 3.504 4.010 2.400 3.243 3.550 3.630 3.690 3.913 3.798 3.856 3.599 2.827 3.245 3.243 2.726 3.207 4.054 2.616

φ [K] 100.00 113.00 70.00 119.00 91.50 117.70 143.80 40 .00 538.00 353.45 569.10 91.70 78.80 71.40 75.00 91.00 59.70 535.30 538.00 356.00 25.16 10.06 3.93

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