Multiphase Interaction in Low Density Volumetric Charring Ablators

University of Kentucky UKnowledge

Theses and Dissertations--Mechanical Engineering Mechanical Engineering

2018

Ali D. Omidy University of Kentucky, [email protected] Digital Object Identifier: https://doi.org/10.13023/etd.2018.476

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Recommended Citation Omidy, Ali D., "Multiphase Interaction in Low Density Volumetric Charring Ablators" (2018). Theses and Dissertations--Mechanical Engineering. 128. https://uknowledge.uky.edu/me_etds/128

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REVIEW, APPROVAL AND ACCEPTANCE

The document mentioned above has been reviewed and accepted by the student’s advisor, on behalf of the advisory committee, and by the Director of Graduate Studies (DGS), on behalf of the program; we verify that this is the final, approved version of the student’s thesis including all changes required by the advisory committee. The undersigned agree to abide by the statements above.

Ali D. Omidy, Student

Dr. Alexandre Martin, Major Professor

Dr. Alexandre Martin, Director of Graduate Studies Multiphase Interaction in Low Density Volumetric Charring Ablators

DISSERTATION

A dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the College of Engineering at the University of Kentucky

By Ali D. Omidy Lexington, Kentucky

Director: Dr. Alexandre Martin, Professor of Mechanical Engineering Lexington, Kentucky 2018

Multiphase Interaction in Low Density Volumetric Charring Ablators

The present thesis provides a description of historical and current modeling methods with recent discoveries within the ablation community. Several historical assumptions are challenged, namely the presence of water in Thermal Protection System (TPS) materials, presence of coking in TPS materials, non-uniform elemental production during pyrolysis reactions, and boundary layer gases, more specifically oxygen, interactions with the charred carbon interface. The first topic assess the potential effect that water has when present within the ablator by examining the temperature profile histories of the recent flight case Mars Science Laboratory. The next topic uses experimental data to consider the instantaneous gas species produced as the ablator pyrolyzes. In this study, key gas species are identified and assumed to be unstable within the gas phase; thus, equilibrating to the solid phase. This topic investigates the potential effects due to the these process. The finial topic uses a simplified configuration to study the role of carbon oxidation, from diatomic oxygen, on the ablation modes of a TPS, surface versus volumetric ablation. Although each of these topics differ in their own right, a common theme is found by understanding the role that common pyrolysis and boundary layer gases species have as they interacts with the porous TPS structure. The main objective of the present thesis is to investigate these questions.

KEYWORDS: Atmospheric Entry, Material Response, Heat Transfer

Author’s signature: Ali D. Omidy

Date: December 6, 2018 Multiphase Interaction in Low Density Volumetric Charring Ablators

By Ali D. Omidy

Director of Dissertation: Alexandre Martin

Director of Graduate Studies: Alexandre Martin

Date: December 6, 2018 This document is dedicated to my family and ﬁanc´e.Without each of your love and support, I would not have survived graduate school. I thank each of you from the bottom of my heart. ACKNOWLEDGMENTS

I would like to express my gratitude to Dr. Alexandre Martin for serving as my Ph.D. advisor and Dr. JoséGraña-Oteroas my co-advisor. Without their guidance throughout my education I would not be in the position I am today. I would like to also thank them for their friendship over the years making my time at the university much more enjoyable. I would like to extend a special thanks to Dr. Martin for taking mr in as a “young” undergraduate student. His patience and motivation during this period gave me the confidence to pursue this degree. I would also like to express a great deal of gratitude to my mentors and friends at NASA Johnson Space Center and Ames Research Center, namely Brian Remark, Cooper Snapp, Stan Bouslog, Carlos Westhelle, Alvaro Rodriguez, Scott Coughlin, Ron Lewis, Adam Amar, Brandon Oliver, Nagi Mansour, and David Hash. Each of them have made my experiences at NASA very memorable and have all taught me so much. I would like to thanks my remaining committee members Dr. Kaveh Tagavi and Dr. Brad Berron for the time and expertise throughout this process. Lastly I would like to thank my friend/lab-mates, namely, Martin, Umran, Ra- gava, Rick, Olivia, Justin, Christen, Nick, and Chris. You all have made this experience memorable and enjoyable.

iii TABLE OF CONTENTS

Acknowledgments ...... iii

Table of Contents ...... iv

List of Figures ...... vi

List of Tables ...... viii

Chapter 1 Introduction to Atmospheric Entry ...... 1 1.1 Thermal protection materials ...... 2 1.2 Literature Review ...... 7

Chapter 2 Governing equations and material models ...... 11 2.1 Governing equations ...... 11 2.2 Boundary conditions ...... 13 2.3 Material models ...... 16 2.4 Eﬀective thermal conductivity model remarks ...... 19

Chapter 3 Effects of water phase change on the material response of low- density carbon-phenolic ablators ...... 21 3.1 Mars Science Lab entry data ...... 21 3.2 Effect of water on the material properties of PICA ...... 23 3.3 Concluding remarks on effects of water phase change on the material response of low-density carbon-phenolic ablators ...... 28

Chapter 4 Modeling multi-phase interactions in a charring ablator using KATS- US ...... 30 4.1 Motivation ...... 30 4.2 Formulation ...... 32 4.3 Model Veriﬁcation ...... 43 4.4 Concluding remarks on modeling multi-phase interactions in a charring ablator using KATS-US ...... 48

Chapter 5 Modeling carbon oxidation of a ﬁberForm plug ...... 51 5.1 Formulation ...... 53 5.2 Results ...... 63 5.3 Conclusion ...... 65

Chapter 6 Concluding remarks ...... 70 6.1 Summary ...... 70 6.2 Original contributions ...... 71 6.3 Future work ...... 72

iv Appendix A Modeling of the remain MISP’s for MSL ...... 74

Appendix B Oxidation effects from the Damköhlerand Pécletnumber . . . . 76 B.1 Damköhlernumber effects ...... 76 B.2 Pécletnumber effects ...... 77

Bibliography ...... 78

Vita ...... 104

v LIST OF FIGURES

1.1 Time progression footage of the atmospheric entry of the Hayabusa spacecraft and capsule (identiﬁed by the red arrow)...... 3 1.2 Recent NASA program heat shields...... 4 1.3 Material zone degradation for volumetric ablators heat shield ...... 5

2.1 Energy balance at the ablative material gas-solid interface...... 15 2.2 Graphical representation of a control volume of a typical porous decomposing ablator...... 16 2.3 Excerpt from the CMA manual (1) ...... 20

3.1 Instrumentation of the heat shield of the Mars Science Laboratory entry spacecraft ...... 22 3.2 Phase change of water as a function of temperature, pressure, and time at the location for the four thermocouples of MISP 2. The time at which pyrolysis reaction at the surface are significant is indicated on each graph by the symbol ...... 25 3.3 Thermocouple readings for MISP 2 using a modified thermal conductivity model accounting for the presence of water. The new results are compared to the previous results obtained using the estimated PICA conductivity from Mahzari et al. (4), as well as the MISP flight data...... 28

4.1 Gas pressure (black) and average molecular weight (blue) inside a closed wall domain as the temperature is increase linearly. As the gas equilibrium changes, the averaged molecular weight follows causing the pressure to increase non-linearly. Once the molecular weight nears a constant value, the pressure change becomes linear...... 43 4.2 Comparison of KATS-US (lines) with an equilibrium source term with Mutation++ (symbols). The relaxation time used for this problem ensures chemical equilibrium within the first time-step, τr = τf ...... 45 4.3 Effect of chemical relaxation time η = δt/τr. Four relaxation rates were selected with decreasing orders of magnitude: η = 5 ×10−1 (black), 5×10−2 (blue), 5×10−3 (green), and 5×10−4 (red). As the relaxation rate is decreased by orders of magnitude, the time necessary for the system to reach equilibrium increases by orders of magnitude...... 46 4.4 Effect of solid transfer sticking coefficient ζ. Four sticking coefficient were selected: ζ = 1×10−1, 1×10−2, 3×10−3, and 1×10−3. As the sticking coefficient is decreased, the time necessary for the system to equilibrate increases...... 47 4.5 Change in porosity as condensation proceeds within the material. . . . 47 4.6 Effect on the thermal transport properties as graphite is condensed to the solid matrix...... 49

vi 5.1 Two views of the topology of the inner structure of the porous porous plug. The ﬁrst, a preprocessed tomography slice of a carbon ﬁber plug (5). The second, a schematics of the geometry of the cross section normal to the tube’s axis. A(x) is the net area of carbon, or equivalently the volume per unit length of solid carbon, whereas Λ(x), represented by the orange lines, is net carbon perimeter, or equivalently the gas-carbon interfacial area per unit length. φ(x) is the porosity, given in terms of A(x) as φ(x) = 1 − A(x)/(π(D/2)2) ...... 55 5.2 Coverage of adsorbed atomic oxygen θ = [C(O)]/Γ and CO2 to CO ratio

as a function of temperature and partial pressure of oxygen pO2 as given by Eq. (5.8)...... 60 5.2 This figure presents the influence of temperature on the development of concentration of oxygen, normalized pressure and mass flux through the length of the plug...... 67 5.3 This figure presents the influence of temperature on the non-dimensional mass flux at the back face of the pellet (black) and the non-dimensional pressure at the front face (red)...... 68 5.4 This figure presents the influence of temperature on the species mass fraction at the front and back face of the pellet and the ...... 68 5.5 Here, the temperature influence on the in-depth oxygen depletion location is examined. The x-axis represents the non-dimensional plug length, while the y-axis relates the in-depth location in which O2 is exhausted within the plug...... 69

vii LIST OF TABLES

5.1 Kinetic mechanism proposed by Geier et al. (8)...... 57 5.2 Arrhenius rate constants for the heterogeneous reaction model proposed by Geier et al.(8) ...... 58

viii Chapter 1 Introduction to Atmospheric Entry

On the 12th of April, 1961 all odds were defied when the first man, Soviet cosmonaut Yuri Gagarin, was sent through low earth orbit aboard the Vostok spacecraft. This event helped to spark 55 years, and counting, of new missions, each with their own complexity. During these years, names such as A. Leonov, N. Armstrong and E. Aldrin, Jr, will forever be etched in history as pioneers and heroes for walking in space and on the moon. Since then, the Space Transportation System (STS), 1981 to 2011, helped to give birth to the International Space Station (ISS), 1998 to current, providing a test bed for countless microgravity experiments. Following the STS, capsule spacecrafts, such as Stardust (9, 10), 1999 to 2006, have reemerged for astronaut transport and have assisted memorable feats, such as the landing the Mars Curiosity Rover (11) in 2011. The final frontier, however, has not been exclusive to national space programs. Private sector companies, such as SpaceX, Sierra Nevada, and Blue Origin, have provided cheaper transport options to the ISS. Looking to the future, space transportation and discovery missions will continue to push the envelope of design. The National Aeronautics and Space Administra- tion, NASA, has recently pledged to land astronauts on Mars in 2030’s (12). SpaceX has also shown interest in landing humans on Mars and plans to land an unmanned capsule, Dragon, on Mars by 2018 (13). In order to achieve these boisterous goals, the supportive scientific community must make great strides. Respective to the At- mospheric Entry community, great efforts have been made in developing new high temperature capable materials and characterization of these materials to minimize payload weight while still achieving the overall safety of the spacecraft.

1 1.1 Thermal protection materials

An aspect which has made these accomplishments of the past and present more memorable is the safe return of the passengers and payloads. This would not be possible in the absence of Thermal Protection Systems (TPS), which are responsible for guarding the vehicle from extreme aeroheating. One example of the importance of TPS is the Hayabusa spacecraft which was launched in June 2005 by the Japanese Aerospace Exploration Agency (JAXA) (14). In 2010, the Hayabusa spacecraft’s 1.25 billion mile journey from the Itokawa comet came to an end as the capsule separated to make its final maneuvers through the Earth’s atmosphere and into Australian airspace. The atmospheric entry of the spacecraft and capsule was observed by 19 cameras aboard a NASA DC-8 airborne laboratory as a joint experiment between NASA, JAXA, and other space. During the entry process, high velocities, near 12 km/s, were reached. These high velocities generated a bow shock which exposed the face of the capsule to heating environments nearing 1,112 W/cm2 (15, 16). As the Hayabusa spacecraft encountered these extreme heat fluxes it quickly and violently dismembered the vehicle with the disintegrating parts aglow. The capsule, identified by the red arrow in Figure 1.1, however was equipped with a TPS material allowing for its safe return back home. The use of TPS materials are of common practice which dates back to some of our nation’s easiest missions. This includes but is not limited to Mercury (17), Gemini (18), and Apollo (19). There exist two types of TPS materials, ablative and non-ablative, with their usage depending on the expected heating environment at the specified location. It should be acknowledged that, more often than not, both materials are utilized in a single vehicle design. When relatively low heat fluxes are expected, non-ablative materials are used. Non-ablative materials gain their name from the obvious, the material class has little

2 Figure 1.1: Time progression footage of the atmospheric entry of the Hayabusa spacecraft and capsule (identiﬁed by the red arrow). to no geometric change in their optimal usage. These materials are used for their thermo-physical properties and are often applied as a heat soak. By far the most commonly known material in this class would be the LI-900 tiles (20) — also commonly known as shuttle tiles— that composed the underbelly of the STS shuttles. However, other materials exist (21). When high heating environments are expected, ablative materials are utilized. This material class has been used for high velocity entry missions, such as Apollo (3, 6, 7), Stardust (9, 10), MSL (22, 23), and EFT-1 (24, 25). Figure 1.2 shows the heat shields for the most recent NASA missions, MSL and EFT-1. These materials are usually composed of a highly porous, rigid structure which has been impregnated with a binding matrix and can often contain ﬁller additives to better control the material properties and overall behavior. A synonymous name for this material class

3 (a) Orion heat shield.

(b) Mars Science Laboratory Capsule.

Figure 1.2: Recent NASA program heat shields. is volumetric ablators with the name deriving from the behavior of the material as it is exposed to elevated temperatures. When temperatures are high enough, the

4 molecular structure of the binder, which is often a polymer, begins to breakdown thereby converting the solid mass into gas. As a result, the speciﬁc volume of this converted mass is increased causing a pressure gradient within the material which ultimately results in a ﬂow of mixed pyrolysis species through the pore structure of the material and out into the ambient. As this process proceeds, very distinct regions within the material becomes apparent. A generalized graphical representation of these regions can be found in Fig. 1.3 with a brief description of each region following in the text. It should be noted that these descriptions are not meant to fully inform the reader of all the complexities associated within these regions but guilde the reader to a broad understanding of the underlying physics. A later section, Sec. 1.2, will be dedicated to highlight seminal and recent works which address some of these complexities.

Boundary Layer Gas

Ablation Carbon Layer Removal Pyrolysis Gas Char Residual Layer Carbon

Pyrolysis Phenolic Layer Gasifcation

Virgin Raw Layer Pyrolysis Gas Material

Substructure

Figure 1.3: Material zone degradation for volumetric ablators heat shield

5 Virgin Layer

Volumetric ablators are composed of a rigid structure and binder. In this pure state, the material is often referred to as virgin material. With regards to a post-ﬂight heat-shield, the virgin layer constitutes a signiﬁcant majority of the overall material. The primary mode of energy transfer in this layer is transient heat conduction and thus the solid material properties are of great interest. When designing a TPS for a vehicle, the interface between the virgin layer and substructure, as seen in Fig. 1.3, must remain below a critical bond-line temperature to ensure the overall integrity of the adhesive and structure.

Pyrolysis Layer

When the virgin material reaches high enough temperatures, typically ∼ 500 - 900 K, decomposition reactions occur converting solid matter into a pyrolysis gas. This gas helps to protect the vehicle by 1) the generation of the pyrolysis gas which requires endothermic reactions and 2) the creation of a relatively cool film at the surface of the material. The pyrolysis layer is thin in nature, only occupying a small fraction of the overall material. In this layer, advection and the heat of pyrolysis, which is defined as the amount of energy required for reactions to proceed, plays a significant role in the thermal transport through this layer.

Char Layer

When the decomposition reactions complete, the reacting material is reduced down to a residual carbonaceous material, with traces of metallic components. The matrix is often composed of a material which can survive at these elevated temperatures and thus also exists within this layer. As the pyrolysis gas percolates through this layer, reactions can occur often leading to the deposition and/or even removal of solid carbon material.

6 Ablation Layer

In this layer, gas-surface interaction between the boundary layer gases and the outer- material-layer is of great importance. Mass diﬀusion is responsible for allowing the hot gases to penetrate into shallow depth of the material and can lead to oxidation of the solid material. These reactions compromise the structure of the ﬁbers and ultimately lead to mechanical removal (spallation).

1.2 Literature Review

In order to characterize and predict the behavior of the TPS, Material Response (MR) codes must be used. The development and application of MR codes began in the 1960’s. However, the idea behind these codes were initiated by development of rocket nozzles and InterContinental Ballistic Missiles (ICBM) nosecones a few years before (26, 27). The most notable MR codes is the one developed by Moyer and Rindal at Aerotherm, Charring Material Thermal Response and Ablation computer program (CMA) (28, 29, 30, 31, 32, 33). The assumptions used in this code has laid a foundation which many current MR codes are still built upon. Generally speaking, a 1-D assumption is valid for a significant portion of a heat shield acreage. However, the same can not be said for samples used for ground testing which are often of the length scale of centimeters. In this configuration, due to the small sample size used, two major phenomena play a significant role, lateral conduction (34) and advection through the side walls (35). Concerning the later, several momentum models have been used throughout the ablation community. By far the most widely used momentum model is Darcy’s Law (36) which has been used since the time of CMA. This model accounts for low velocity flow and is numerically inexpensive. Other codes, however, have deviated from this through the implementation of more complex models. Lachaud and Mansour (37) uses the Darcy-Forchheimer model (38)

7 which is applicable for both a high and low Darcy velocity regime, whereas, Weng and Martin (39) use an unsteady momentum equation, Darcy-Brinkman (40, 41) which not only accounts for both velocities regimes but also considers viscous dissipation. For any MR code, the boundary condition set at the interface between the solid material and the hypersonic boundary layer is of great importance as it is responsible for the energy generated from atmospheric entry. At this interface, an energy balance is present with an additional mass balance in the case of ablation (29). The rate of ablation of the material is driven by the chemistry at the surface. Thermochemical solvers are used to compute the equilibrium balance between the solid material and considered atmospheric species (42). These solvers generate tables – often referred to as B’ tables – which relate temperature, enthalpy, pressure, and dimensionless surface mass fluxes at the outer material layer (42). This approach has been validated using ground experiments, and usually leads to no more than 10% error when compared to experimental results (43). However, the approach loses accuracy at low temperature and pressure (43), and yielded high error for a recent flight case, Stardust (44, 45). Recently, Johnston (46) demonstrated this discrepancy can be reduced to 20% when a non-equilibrium model is used. It should also be noted that Chen and Milos (47) and Martin et al. (48) have also proposed similar models for phenolic-based materials. Building upon these works, several authors have investigated different modes which contribute to ablation, namely carbon oxidation and spallation. Experimen- tally, studies have been performed to better understand the mechanisms leading to surface ablation and fiber mass loss. Panerai et al. (49) observed that temperatures exceeding ∼ 900 K result in surface recession while lower temperatures lead to volumetric mass loss. They also observed that pressure plays a significant role in the development of the ablation front. Additional imagery from this study shows that carbon oxidation occurs in the form of pitting which contradicts the numerical model proposed by Lachaud et al. (50, 51) in which the fiber morphology is changed through

8 a uniform reduction of fiber radius. As the char of the material is oxidized the structural integrity of the material is compromised thus leading to the phenomena known as spallation or mechanical removal. Experimentally this has been observed in arcjets where particles on the order of microns are ejected into the flow field (52, 53, 54). On the numerical side, Davuluri et al. (55) investigated spalled particle trajectory based on partial size, initial velocity, and exit angle as well as the overall chemical effect imposed by the particles on the flow field. Aside from the physics present at the surface of the ablator, the composition of the gas generated though the pyrolysis process has drawn the attention of several researchers. Part of the CMA suite, a thermochemical solver, Aerotherm chemical equilibrium (ACE), was developed to solve for the chemical equilibrium balance of the pyrolysis gas at a specified temperature and pressure. Since then, several alter- native codes have been published (56, 57, 58). As in the case of surface chemistry, a non-equilibrium pyrolysis chemistry model has been proposed by Scoggins et al. (59). Experimentally, the pyrolysis generation has been studied through the use of thermogravimetric analysis and residual gas analyzers (60, 61, 62, 63). Although these experiments differ in configuration, the findings generally agree on the production of light molecular weight species H2, CO, CO2,H2O, and CH4 with traces of heavier molecules, such as phenol, xylene and cresol. In regards to the solid thermo-physical properties for specific TPS materials, such as PICA (64), information is scarce due to the international traffic in arms regula- tions. One way scientific advances have proceeded around this obstacle is through the use of Theoretical Ablative Composite for Open Testing (TACOT) (65) – a theoretical Phenolic Impregnated Carbon Ablator(PICA) similar material database – and

FiberForm® which is a precursor for PICA. Experimentally, the permeability with a slip eﬀect – which occurs when the gas is rariﬁed – was studied for various thermal

9 insulator materials (66, 67) and FiberForm® (68). As for the thermal conductivity of these materials, only a few studies (69, 70) were found and describe the radiation contribution to the eﬀective property. In these studies, FiberForm® is examined at the micro-scale. As for the strength of material, Agrawal et al. (71, 72) studied the mechanical failure of PICA at the micro-scale. On a similar note, Fu et. al. numerically studied the stress due to thermal expansion (73) through the uses of TACOT.

10 Chapter 2 Governing equations and material models

In this chapter, the numerical frame frameworkwork behind Kentucky Aerodynamic and Thermal-response System (KATS) (39) is thoroughly discussed. KATS is a three-deminsional solver which uses a finite volume numerical method and is built upon open source CFD framework freeCFD (74). It has been parallelized to increase computational power and verified (75, 76, 77). The KATS framework is currently composed of three codes: material response (MR), hypersonic, and universal solver (US), which simultaneously solves the MR and hypersonic problem as one. In the work presented here, several flavors of KATS are used. This section will attempt to not only inform the reader with an understanding of the generic framework which KATS is built upon but also to allude to future concepts which will be investigated in this work. A significant portion of the future chapters is dedicated to the analysis performed by material response codes. As such, the equations and models necessary to solve these problems are thoroughly described.

2.1 Governing equations

KATS solves the governing equations for mass (solid and gas), momentum, and energy. It should be noted here that KATS solves for n-number of solid and gas mass equations. These equations are organized in such a manner to share the same general form, presented in Eq. 2.1 in vector form.

∂Q + (F − F ) = S (2.1) ∂t d

11 In this representation, Q represents the conservative variables, F the convective

fluxes, Fd the diffusive fluxes, and S the source terms. By integrating Eq. (2.1) over this control volume, Eq. 2.2 is obtained.

∂ Z Z Z Q dV + ∇ · (F − Fd) dV = S dV (2.2) ∂t V V V The divergence theorem can then be applied to both the convective and diﬀusive terms, which takes the leads to:

∂ Z Z Z Q dV + (F − Fd) · n dA = S dV . (2.3) ∂t V A V If the control volume is assumed to be suﬃciently small, the integration of the volume can be approximated by a summation.

∂ X (Q V ) + (F − F ) · n A = S V (2.4) ∂t d faces Finally, in order to solve for the primitive variables P, the chain rule is applied to the conservative terms in Eq. 2.5.

∂Q ∂P X V + (F − F ) · n A = S V. (2.5) ∂P ∂t d faces The vectors and matrices used in this equation are:

      φ ρg1 p1 ωg1        .   .   .   .   .   .               φ ρ   p  ω   gngs   ngs   gngs               ρs1   ρs1   ωs1         .   .   .   .   .   .        Q =   , P =   , S =   . (2.6)  ρ  ρ  ω   snss   snss   snss               φ ρg u   u   Dx               φ ρ v   v   D   g     y               φ ρg w   w   Dz        φ Eg + Es T SD

12 For the conservative term (Q), φ is the open porosity — defined as the volume available to participate in porous flow— of the material, ρ is the density of the gas g and solid s, u, v, and w is the velocity in the x, y, and z direction, respectively. Finally E is the internal energy for the gas g and solid s. The primitive variables (P) are pressure p, solid density, gas velocity, and temperature T . As for the source term (S), ω represents the production/consumption of gas/solid mass, D; is the Darcy component in the x, y, and z direction, and SD is a source term which accounts for the porous effect to the overall thermal diffusivity.

    φ ρg1 u φ ρg1 v φ ρg1 w      .     .             φ ρ u φ ρ v φ ρ w     gngs gngs gngs             0 0 0         .     .   0      F =   , Fd =   . (2.7)      0 0 0         2    φ ρgu + p φ ρgu v φ ρgu w             φ ρ v u φ ρ v2 + p φ ρ v w     g g g         2     φ ρgw u φ ρgw v φ ρg w + p       φ ρg u H φ ρg v H φ ρg w H Fcond,x Fcond,y Fcond,z

In Eq. 2.7, H represents the solid enthalpy in the advective flux (F) and Fcond accounts for conduction — typically Fourier’s Law — in the x, y, and z direction in the diffusive flux (Fd) term. It should be pointed out that in the current representation of the momentum equation, there are no diffusive terms. Instead the restriction of

ﬂow caused by the pores placed inside the source term Di.

2.2 Boundary conditions

In order to solve Eq. 2.5, boundary conditions are necessary. In the following section various types of common boundary conditions for the ablation problem will be discussed.

13 Wall boundary condition

At a wall interface, the boundary is considered impermeable and bars the flow from passing through. As a result, the mass flux and the tangental velocity at the wall must be zero. Two different conditions can exist with regards to the perpendicular velocity: a so-called slip and non-slip boundary condition. In the case of a slip boundary condition, the flow is inviscid and results in a zero gradient at the surface. For the non-slip condition, the flow is viscid and the velocity at the surface must be 0. For the energy equation, one of two boundary conditions can be placed, a fixed temperature or a fixed heat flux.

Symmetry boundary condition

In the case of a symmetry boundary condition, a zero gradient is implemented for all primitive variables. This boundary condition is typically used to simplify geometries.

Outlet boundary condition

Unlike the wall boundary condition, flow can penetrate the interface for an outlet boundary condition, thereby bringing the necessity to define the velocity, density, and static pressure of the gas at this location. However, for the energy equation, there exists no difference between an outlet and wall boundary condition. Either a fixed temperature or heat flux can be specified at the interface.

Aerodynamic-heating boundary condition

The aerodynamic-heating boundary condition is a special scenario of the outlet boundary condition. At the TPS surface, the material is subjected to convective heating which in return leads to several cooling mechanisms: surface radiation, pyrolysis gas generation, surface recession, and advection cooling. These cooling processes only account for a margin of the onset heating resulting in a signiﬁcant portion of the heat

14 being conducted deep into the material. A graphical and mathematical representation of this energy balance can be seen in Fig. 2.1 and Eq. 2.8 with a deﬁnition of the contributing terms presented in Eq. 2.9. (78)

Convective Radiative Advective Heating (ch) Cooling (rc) Cooling (ac) Ambient Material Interface Material Conduction (c) Pyrolysis Ablation (a) Gas (pg)

Figure 2.1: Energy balance at the ablative material gas-solid interface.

00 00 00 00 00 00 − qc = qch + qrc + qac + qpg + qa (2.8) where, 00 Ch qch = ρe ue Ch0 (hr − hw) (2.9a) Ch0 00 4 4 qrc = σ T − T∞ (2.9b) 00 ~ qac = ρw V · ~n hw (2.9c)

00 qc = −k∇T · ~n (2.9d) 00 ~ qpg = ρpg V · ~n hpg (2.9e) 00 ~ qa = ρs V · ~n hs (2.9f)

In Eq. 2.9a, ρeueCh0 is the heat transfer coeﬃcient of the boundary layer gas at the edge (e), Ch/Ch0 is the Stanton number correction; while, hw and hr are the wall and recovery enthalpy, respectively. The heat transfer coeﬃcient and recovery enthalpy are usually determined using an hypersonic aerothermodynamic CFD code. The Stanton number correction is found empirically and is related by the following:

Ch = Ωhw Ωbtw , (2.10) Ch0

15 where Ωhw and Ωbtw represent the hot wall and blowing correction factor, respectively, and are dependent on the flow regime that it represents. For the hot wall correction factor, Cohen and Reshotko (79) found a correlation for the case of laminar flow while, Eckert and Drake (80) found a relation for turbulent flow. As for the blowing correction factor, a relationship is found by Crawford and Kays (81). The terms associated with this boundary condition are typically tabulated and often referred to as B’ tables.

2.3 Material models

As previously mentioned in Sec. 1.2, charring abalators are commonly composed of a rigid structure bound together by a decomposing matrix to produce a highly porous material. As such, the density of such material can be somewhat arbitrary depending on the control volume considered. Historically, the reference volume is of macro-scale and contains the matter of both the solid and gas, thereby being a property dictated by solid/gas ratio, temperature, and pressure.

Legend Non-decompoing solid mater Decompoing solid mater Gas mater

Figure 2.2: Graphical representation of a control volume of a typical porous decomposing ablator.

16 The total density of the porous media can be expressed as,

ρ = φ ρg + ρs , (2.11) |{z} |{z} |{z} |{z} [Tot. mass] [Pore vol.] [Gas mass] [Sol. mass] [Ref. vol.] [Ref. vol.] [Pore vol.] [Ref. vol.] where, φ represents the porosity and is deﬁned a volumetric ratio of the “free” pore volume compared to that of the total control reference volume. This term is used to scale the gas density ρg, which by deﬁnition is contained by the pore volume, to the total reference volume. It should be made clear that the solid density ρs is the mass of the solid divided by the control reference volume and not that of the volume purely consumed by the solid mass. Furthermore, although these materials do depend on pressure, since the solid densities are typically much larger than that of the gas, this dependency is marginal in comparison to the change in solid mass as a result of pyrolysis.

Solid decomposition model

The solid decomposition model is derived from experimental data from ThermoGravi- metric Analysis (TGA). In these experiments, small material samples — which weigh the order of milligrams — are placed in a non-reactive crucible and subjected to a controlled heating source and shielding gas. The testing apparatus measures the mass of the material sample while a uniform, transient heating proﬁle is applied. Typical proﬁles consists of constant ramp-rates which range from ∼ 1 – 30 K/min, although higher rates are possible. During these experiments, hundreds, if not thousands, of chemical reactions occur, making the detailed modeling of the decomposition processes a feat in its own regard. A remedy to avoid these complex reactions is through the grouping of these complex reactions into a few semi-physical empirical reactions. In other words, the mass removed from the solid material, through pyrolysis, is combined to only a hand-full of species, which in principle can be as many as desired but is typically set to three.

17 As such, these component species occupy a portion of the total solid mass in the virgin state. Thus, the solid density can be found through the following relationship:

nss X ρs = Γi ρsi . (2.12) |{z} |{z} |{z} [Sol. mass] i=1 [Sol. vol.] [Sol. mass] [Ref. vol.] [Ref. vol.] [Sol. vol.]

In Eq. 2.12, ρsi represents the solid component species densities while Γi is defined as the volume ratio that these component species initially consumes. The component densities can be retrieved from experimental TGA data by the means of integrating through time — or temperature since a fixed temporal heating profile is often used — for each “reaction” rate. These rates can be modeled through a modified Arrhenius law: ψi ∂ρsi ρsi − ρsi,c −Tai /T = ωsi = −Ai ρsi,v e . (2.13) ∂t ρsi,v

In Eq. 2.13, Ai is the reaction rate constant with the units of 1/s, ρvi and ρci is the component virgin v and char c density, respectively, ψi is a unit-less exponential

factor, and Tai is the activation temperature with units in Kelvin. In order to obtain these Arrhenius coefficients, the defined system of relationships are optimized to back out experimental data. Thus, the empirical coefficients found here do not hold the same meaning as those used in a fundamental chemical reaction. In order to determine the amount of gas released though the pyrolysis reactions, the source terms are examined and presented in Eq. 2.14.

nss X ωg = − Γi ωsi (2.14) i=1 Here, it is clear that the total amount of mass released during pyrolysis is simply just the sum of the component solid mass source terms and the corresponding volume fraction that it consumes.

18 Decomposed material properties

As the solid material is transformed into gaseous species, the thermo-physical properties associated with the solid matter is altered. Resorting to the models proposed by CMA, which are still in use today in the atmospheric entry community, the instantaneous solid material properties can be modeled through the use of the so-called extent of reaction β. ρ − ρ β = v s (2.15) ρv − ρc The extent of reaction, deﬁned in Equation 2.15, presents a normalized density ratio such that when β = 1 the material as completely charred and when β = 0 the material is in its pure virgin state. The extent of reaction relationship can be applied to the porosity φ and permeability K to model the instantaneous values:

φ = (1 − β) φv + β φc , (2.16)

K = (1 − β)Kv + β Kc . (2.17)

Additionally, the extent of reaction has historically (28, 37, 39, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91) been applied to heat capacity, enthalpy, and thermal conductivity.

(1 − β)ρv βρc cp,s(T ) = cp,v(T ) + cp,c(T ) , (2.18) ρs ρs (1 − β)ρv βρc hs(T ) = hv(T ) + hc(T ) , (2.19) ρs ρs (1 − β)ρv βρc ks(T ) = kv(T ) + kc(T ) . (2.20) ρs ρs

2.4 Eﬀective thermal conductivity model remarks

Recent uncertainty quantification studies (4, 92, 93) have highlighted the effective thermal conductivity of the material to significantly change material response numerical solutions with small variations in the property. The current approached used to determine this effective property dates back to the design of Apollo capsule in the

19 1960’s, and was devised by Moyer and Rindal. (28) This model relies on thermal characterization of the so-called virgin — un-pyrolyzed — and char — fully pyrolyzed — material layers at various pressures. Using these measurements, the effective thermal conductivity is found through a simple linear interpolation using the instantaneous density to track the level of charring. The method is simple, and is a great first approximation, but has been dubbed as “convenient fiction”, as shown in Fig. 2.3, taken from the CMA manual. (1)

Figure 2.3: Excerpt from the CMA manual (1)

This methodology has served the community for a number of years and has aided the successfully design of many notable heat shields for the following programs: Mer- cury, Apollo, Stardust, MSL, and EFT-1. Looking toward the future, missions are becoming evermore demanding with the allowable weight of the heat-shield becoming more and more restricted, and the predictability of the behavior more demanding. This motive has sparked a plethora of new and exciting numerical and experimental studies geared to better predicting the thermal response of TPS materials during ﬂight. However, little attention within the community has been focused on the effective thermal conductivity of the ablator.With this said, the present work will not further investigate this topic. This mentioned historical background should be noted by the reader when proceeding to the coming chapters.

20 Chapter 3 Eﬀects of water phase change on the material response of low-density carbon-phenolic ablators

3.1 Mars Science Lab entry data

The atmospheric entry of the MSL spacecraft was of great importance to the scientific community, as the heat shield was equipped with the MSL Entry, Descent, and Landing Instrumentation (MEDLI) suite (120). MEDLI was composed of seven surface pressure ports, known as Mars Entry Atmospheric Data System (MEADS), as well as a combination of seven sets of thermocouples (TC) and a Hollow aErothermal Ablation and Temperature (HEAT) sensor (121), known as the MEDLI Integrated Sensor Plug (MISP). Each set of MISP used four TCs to measure temperature at specific depths within the heat shield, as well as the HEAT sensor to measure the propagation of an isotherm. The location of the MEADS and MISP sensors on the surface of MSL is shown in Figs. 3.1a and 3.1b, respectively. As documented in the literature (4, 120, 122, 123, 124, 125), the MISP data showed significant differences when compared with the predictive solutions of FIAT, the one- dimensional material response (MR) code used for the design of heat shields (126). Post-flight simulations performed using the MEDLI near-surface TC as a boundary condition (the so-called “TC driver method”) produced better results (4). These results suggest that variabilities in the atmospheric conditions and the simplifications used in the aerothermal boundary conditions are mostly responsible for the discrepancies (122). Moreover, uncertainty and sensitivity analysis demonstrated that varying the values of the material properties based on known uncertainties, as well as accounting for experimental errors resulted in better general agreements with the flight data (4). However, all of these studies still failed to model a specific temperature behavior observed in the data taken by the two deepest TCs. For these measure-

21 Ta b le 2. T u r b u le n t t e m p e r a t u re a u g m e n t a t io n ove r a t wo-seco n d p e r io d d e r i ve d f r o m T C 1 t e m p e r a t u re r a t es.

Plug 2 Plug3 Plug 6 Plug 7

(dT =dt)p r e ( K /s) 16 16 14 21 (dT =dt)post ( K /s) 61 60 48 56 Temperature augmentation (ratio) 3.81 3.75 3.43 2.80 quali⇥cation arcjet dataset, arcjet testing for some other materials than PIC A, and also for Mars Path⇥nder bondline thermocouples ight data. T his phenomenon is not well understood at present, but is believed to (a) Location of the MEADS sensors that mea- (b) Location of the MISP sensors that mea- be associated with suresom thee ty pressurepe of m ata theteri surfaceal- or ofin thestru heatmensuret-re thelat temperatureed process within. Cur theren heatt an shieldalysis tools are not able to model this behavshieldior; t –h theere colorfore, contourswe sho illustrateuld not heatingexpec–t thea m coloratch contoursbetwe illustrateen the d heatingata a (imagend model predictions for this part of the dat(imagea. from Bose et al. (122), with permis- from Bose et al. (122), with permission) sion) 550

500

450

400

350 TC1 300 Temperature (K) TC2 TC3 250 TC4 200

0 50 100 150 200 250 Time (s) (c) Temperature histories obtained by MISP 2, with the “hump” F ig u re 4. T h e \ h u m p " o bse r ve d i n T C 3 a n d T C 4 d a t a fo r p l u g 2 (see n also fo r o t h e r p l u gs). visible on TC3 and TC4 (image from Mahzari et al. (4), with permission)

Figure 3.1: Instrumentation of the heat shield of the Mars Science Laboratory entry spacecraft I V . D irect A nalysis

T he purpose ofments,the d theirec temperaturet analysis deviatesis to pe fromrform thea expectedn updat smoothed ana rise,lysi formings of the a “hump”MSL heating environment and T PS material response using the computational models that were employed in the design process. T hese near 300 K. The phenomenon is shown in Fig. 1(c) for the MISP2 sensor, highlighted model predictions are then compared with the ight data.

I V . A . A erothermal E nvironment P rediction 22 T he best-estimate trajectory based on the ight Inertial Measurement Unit (IM U) and M E A DS data was

Downloaded by Alexandre Martin on November 11, 2014 | http://arc.aiaa.org DOI: 10.2514/6.2013-185 not available in time for this study. T herefore, a pre-entry-simulated trajectory based on the latest orbital determination estimates is used in this work. T his trajectory is known as O D229 and was generated using the Program to Optimize Simulated Trajectories II (P OST2).21 F igure 5 shows plots for the altitude as a function of velocity and velocity as a function of time for the O D229 trajectory. T his trajectory has been compared with early versions of the MSL's best-estimate trajectory22 and there is good agreement between the two during the hypersonic region. Any dierences between the two trajectories are expected to have minor impact on aerothermal modeling. In order to predict the vehicle's aeroheating environment, C F D simulations are performed based on this trajectory using D PLR. D PLR is a modern, parallel, structured non-equilibrium Navier-Stokes ow solver developed and maintained at N ASA Ames Research Center.14 T he code employs a modi⇥ed Steger-Warming ux-splitting scheme for higher-order dierencing of the inviscid uxes, and is used here with 2n d order spatial accuracy and to steady-state 1st order in time. T he ow around the heatshield is modeled as thermochemical non-equilibrium ow, using the Mitcheltree and Gnoo 8-species 12-reactions Mars model (C O 2 , C O , N 2 , O 2 , 23 N O , C , N , and O ). T he Mars atmosphere is modeled as 97% C O 2 and 3% N 2 by mass. T he T PS surface is modeled as an unblown non-slip radiative equilibrium wall with constant emissivity ( = 0.85) and the

6 of 19

A merican Institute of A eronautics and Astronautics by dashed circles. It is important to point out that it was also observed in most of the arc-jet tests performed on the MISP plugs, flight TC data from the Mars Pathfinder entry (SLA-561V heat-shield), as well as in other arc-jet tested materials such as AVCOAT.(106, 127, 128) Since attempts at modeling the hump by modifying the thermal properties were unsuccessful in reproducing the phenomenon (4, 120, 122, 123, 124, 125), it is reasonable to assume a shortcoming of the current model at temperatures below 400 K. The present analysis provides a plausible pathway of investigation to fulfill the scientific interest in understanding the nature of this phenomenon. Moreover, since the hump occurs in the region near the back face of the heat shield, at the interface with the substructure (the “bond line”), a proper modeling and understanding of the hump might also reveal whether the phenomenon needs to be considered in future TPS design.

3.2 Eﬀect of water on the material properties of PICA

The thermal response of porous carbon/phenolic ablators, more specifically the heat transfer, can be significantly altered by the water content of the material. The presence of water could be due to (i) residual atmospheric absorbed moisture, or (ii) formation of H2O as a result of the decomposition of phenolic resin. The effects of the water content on the material properties of the ablator is not accounted for in traditional MR models. Process (i) assumes that water vapor from the atmosphere is absorbed by the material while the TPS is going through the manufacturing and assembly stages, or waiting on the launch pad in moisture heavy Florida (129). Since the heat shield of MSL was coated with a thin layer of low-outgassing silicon to prevent contamination (130), moisture might be adsorbed in the porous material as the vehicle travels to its destination. Even without the coating, the moisture could remain trapped within the

23 porous material and then solidify due to ﬂash freezing when exposed to the vacuum of outer space. Evidence of the presence of water due to process (ii) has been observed during experimental analysis of gaseous products resulting from phenolic decomposition (131,

132, 133, 134). The first species produced by the decomposition is H2O, appearing in significant quantity around 325 K, reaching a maximum concentration around 675 K, and essentially vanishing after 950 K. Molecular dynamics simulations also showed that H2O is formed through the pyrolysis of phenolic resin (135, 136). Because the pyrolysis gases are transported through the porous structure (117, 118, 119, 137, 138), part of the water vapor created in the pyrolysis zone travels toward the back of the ablator (138). In this region, the temperature is much lower, and the moisture condenses. Once the vapor has condensed, the liquid droplets remain trapped within the pores and do not travel to colder regions of the material, where they would freeze. It is reasonable to assume that process (i) is at-least partly accounted for in the thermogravimetric analysis (TGA) used to produce the PICA decomposition model (139). However, process (ii) is not considered since it is due to the transient and multi-dimensional nature of the temperature and pressure distribution within the ablator. For this reason, the present investigative analysis focuses on process (ii). During the entry phase, as the spacecraft travels through the upper atmosphere, the local temperature and pressure vary throughout the TPS. Figure 4.6 illustrates this variation for the four TCs of the second MISP. This MISP is chosen because it was located in the region of maximum heat flux, as shown in Fig. 3.1b. The local phase of water is obtained by superimposing the phase diagram of water on the time- dependent local thermodynamic state. The pressure used to generate the graph is the surface pressure predicted by a flow field simulation using reconstructed data from MEADS (122, 140, 141). The value used is likely lower than the pressure distribution within the ablator. (137)

24 106 300 LIQUID SOLID 250 104 200

102 150 Time, s

Pressure, Pa VAPOR 100 100 50

10-2 0 200 300 400 500 Temperature, K (a) TC1 depth = 0.106 in (b) TC2 depth = 0.203 in

Figure 3.2: Phase change of water as a function of temperature, pressure, and time at the location for the four thermocouples of MISP 2. The time at which pyrolysis reaction at the surface are signiﬁcant is indicated on each graph by the symbol .

The symbol on each panel of Fig. 4.6 indicates the local thermodynamic state of

H2O at each TC location, when the surface reaches 600 K. This temperature, occur- ring approximately at 50 s, promotes signiﬁcant pyrolysis decomposition. Although minor decomposition is seen between 400 and 600 K, mass loss is negligible in this range (131, 132, 133, 134, 142). As shown in Fig. 3.2a and 3.2d, after 50 s, any H2O being transported to the region between TC1 and TC2 would remain in vapor phase.

However, H2O reaching the location of TC3 and TC4 would condense in liquid phase. The occurrence of the hump, observed for TC3 and TC4 but not for TC1 and TC2,

25 is speculated to be caused by the condensation of H2O generated by the pyrolysis gas being transported within the ablator. To evaluate the effect of water condensation, an investigative model that modifies the local thermal conductivity of PICA based on the presence of water is implemented in the material response code PATO (143). PATO is a fully portable library for OpenFOAM1, an open-source finite-volume computational fluid dynamics (CFD) software released by OpenCFD Limited. The PATO library is specifically implemented to test innovative physics-based models for reactive porous materials subjected to high-temperature environments. In the present analysis, the state-of- the-art ablation models used for design (144) are used in PATO. When using these models, PATO reproduces accurately the results of FIAT (145). The investigative model used here, which could be used in most material response codes, modifies the local thermal conductivity k of the whole material according to the presence of water, as well as its phase. This is achieved using the parallel conductivity model (146), which simply adds a term to the conductivity of PICA:

k = kPICA + ψkH2O (3.1)

The value used for kPICA is taken from Mahzari et al. (4). The value of the conductiv-

ity of water kH2O depends on its phase, and is determined using the local temperature and pressure inside the ablator, as illustrated in the phase diagram shown in Fig. 4.6:

  0.0 if in gas phase kH2O = (3.2)  kH2O(l) if in liquid phase

The value of kH2O(l) depends on the local temperature and pressure (147, p. 6-1). As previously mentioned, the model assumes that the water vapor condenses to liquid 1The PATO library is not endorsed by OpenCFD Limited, the producer of the OpenFOAM software and owner of the OPENFOAM® and OpenCFD® trademarks. www.openfoam.org/ [retrieved 11 November 2014].

26 form before reaching the regions where it could turn to ice, which is why the solid phase is not considered in Eq. 3.2. It is to be noted that the presence of ice would increase the overall conductivity of the material even more. The parameter ψ can be seen as representing the fraction of water in the system. Because water is assumed to appear only through the pyrolysis process, this added conductivity is applied when the estimated surface temperature of the MISP reaches the temperature that promotes signiﬁcant pyrolysis reactions, approximated at 50 s. Therefore, ψ is speciﬁed according to the following:   0.0 for 0 ≤ t < 50 s ψ = (3.3)  ψw for 50 ≤ t ≤ 268 s

Figure 3.3 presents the results for the four thermocouples of MISP 2 for a simulation using the investigative model, using a value of ψw = 0.3. For these results, the TC driver boundary condition is used at the surface. As for the backface, an adiabatic boundary condition is applied. Due to the unavailability of material properties, the substructure below the heat shield material is not modeled. Other studies (4, 120, 123) performed the substructure modeling, and showed a better agreement at the end of the trajectory, especially for the two deepest thermocouples. As seen in the ﬁgure, the hump appears in the two deepest thermocouple measurements. The model was tested on all other MISP, and produced similar results (see

Fig. A.1 in Appendix (A). Various values of ψw were also tested, ranging from 0 (no water) to 0.80 (pores ﬁlled with water). It was observed that the size of the hump was directly proportional to the amount of water (see Fig. A.2 in Appendix (A). Other

0 parameters, such as heat capacity cp and enthalpy of formation ∆hf , were also tested using a similar approach, and produced similar behavior.

The value of ψw should not be regarded as an evaluation of the amount of water present in the TPS. This could only be achieved once a physical model is used. The

27 1400 Measured 400 PICA 350 PICA + Water 300 1200 250 200 1000 50 60 70 80 90 100 110 120

800

Temperature, K 600

400

200 0 50 100 150 200 250 Time, s Figure 3.3: Thermocouple readings for MISP 2 using a modified thermal conductivity model accounting for the presence of water. The new results are compared to the previous results obtained using the estimated PICA conductivity from Mahzari et al. (4), as well as the MISP flight data. present results simply point out that the hump appears if the thermal properties are modified according to the water content.

3.3 Concluding remarks on eﬀects of water phase change on the material response of low-density carbon-phenolic ablators

The hypotheses stating that the presence of water within the heat shield material may affect the material properties, and hence, the heating rates, is worth exploring further. Using a simple investigative model that modifies locally the thermal conductivity of the material according to the phases of water, a temperature hump that closely resembles the one observed in the flight data was generated. These preliminary results are encouraging and motivate the addition of a condensed water phase in a physics- based model for ablative materials. More than just conductivity, the new model will need to track other volume-averaged material properties that could be affected by

28 the presence of liquid water, such as density, heat capacity, porosity, and enthalpy of formation.

29 Chapter 4 Modeling multi-phase interactions in a charring ablator using KATS-US

4.1 Motivation

As the resin within the TPS is heated and decomposes, pyrolysis gas is produced. The exact composition of this gas depends on many factors, such as thermodynamic properties and the heating rate. (60, 148) Although it is difficult to model the decomposition from first principle, experimental techniques can help shed some light and quantify gas species composition. Current material models use a methodology (and often experimental results) dat- ing from the 1960’s. The methodology relies on the mass balance between the virgin material (initial state) and char material (final state). The mass difference between the two states is assumed to have been transformed into pyrolysis gas. For lack of better knowledge, the composition of the gas is assumed to be in equilibrium, which can lead to odd enthalpy values. To normalize the results, complementary data (116) is used to linearize the pyrolysis gas enthalpy and ensure the gas undergoes endothermic reactions. Although this methodology is sufficiently valid to design successful entry vehicles, it has several notable shortcomings. For instance, the methodology uses a virgin and char state as bounds to determine the overall gas composition. Experimen- tal findings have shown that the resin does not degrade uniformly between the two states. (62, 149, 150) In other words, species leave the solid surface at different activation temperatures. Therefore, the instantaneous elemental composition is likely not in a global equilibrium and composition state, which is assumed in the current methodology. Another inconsistency is the use of Ladacki et al.(116) pyrolysis gas enthalpy data

30 to support the development all material models. The material used in this experiment consists of a silica/phenolic composite at a (70/30) weight ratio. This material composition differs significantly from most common TPS materials and therefore, should not be used for this purpose. Additionally, although globally the decomposition of resin is endothermic, Sykes et al. (62) has shown that exothermic reactions do exist through the decomposition of phenolic, namely between the temperature ranges of ∼ 150-375 ◦C and 725-900 ◦C. Aside from the pyrolysis gas composition, other phenomena which depend on the species present within the gas phase have also shown reasons to investigate. One phenomenon, which is of particular interest, is the presence of condensation or deposition within the porous material. As shown in Chapter 3, water has been hypothesized to greatly effect the thermal response of the material within deep low- temperature regions of the ablator. With a limited model, characteristics from this phenomenon were loosely captured. Therefore, it is of interest to expend this study to reduce the limited nature of the previous work. Another instance of condensation/deposition, which has been observed in flight data, is coking. In 1970, Curry et al. (3) presented in-depth density data of the Apollo heat shield in which the density increased while approaching the outer material layer. The study hypothesized this finding to be due to coking from the break down of methane gas. Curry et al. proposed an Arrhenius rate to capture the effect on density; however, this study did not consider the effect this phenomenon had on other thermal transport properties. Additionally, the proposed rate is circumstantial and does not consider how changes to the ambient would influence this phenomenon. These phenomena are of interest in this study. Thus, the following text aims to propose models to capture these events, with test-case showing these models ability.

31 4.2 Formulation

In this study, the Kentucky Aerothermodynamic and Thermal response Solver (KATS) – Universal Solver (US), developed by Weng et al. (151), will be utilized. KATS-US is an extension of KATS-Material Response (35, 152, 153) and KATS-CFD (154, 155). The code is developed on a ﬁnite volume numerical framework which models the transport of mass, momentum, and energy in both a free and solid porous domain simultaneously. The numerical solver is second order spatial accurate and utilizes a ﬁrst order implicit backward Euler method for temporal integration. Although, KATS-US has the capability of solving both free and porous domains together, this work will only focus on that of the porous domain. This section will highlight the governing equations and physical models used in this study.

Governing equations

Here, the conservation of gas species mass Eq. (4.1a), solid species mass Eq. (4.1b), gas momentum Eq. (4.1c), and mixture energy Eq. (4.1d) are considered.

∂(φ ρ ) g,i + ∇· (ρ u) − ∇ · J = Ω˙ , (4.1a) ∂t g,i i g,i

32 Equation (4.1a) presents the transport of gas species by means of advection ρg,i u and diffusion Ji. Here, u represents the velocity vector in all Cartesian coordinate directions. Species diffusion Ji is Fickean in nature; however, this relationship must be modified for porous media. Thus, species diffusion for species i can be modeled through the following relationship

φ ρg Di ∇ yi Ji = ; τL where, yi represents the species mass fraction, Di is the species diffusion coefficient, and τL is the material dependent parameter, tortuosity. The species diffusion coefficient can be solve through multicomponent methods (156, 157, 158, 159, 160), as is done in Mutation++ (57). However, for this study constant species diffusion coefficients, calculated from a constant Lewis number, gas thermal conductivity, and the mixture translational-rotational specific heat at constant pressure, will be used for ˙ simplicity (154). The last term in Eq. (4.1a), Ωg,i, accounts for the source or sink of species i. This term is defined as

∂ ρ s,i = Ω˙ . (4.1b) ∂t s,i To account for decomposition of the ablator Eq. (4.1b) is considered. Here, multiple component solids are assumed to exist within the total solid matrix. In order to ˙ model the source term Ωs,i due to the decomposition of the ablator, the formulation in Equation (2.13) is used. Additional modiﬁcations must be applied when deviating from this formulation. A more detailed description of this will be provided in latter sections.

∂(ρ u/φ) ρ u uT g + ∇· g + p I = S (4.1c) ∂t φ2 d

33 The momentum through the porous plug is modeled through the use of the equation above. In this formulation, I is the identity matrix and Sd is the source term which accounts for the pore drag eﬀects. Weng et al. (151) provides a more detailed explanation of this equation and the terms it contains. The velocity used in this formulation is the averaged pore velocity and not that of the “true” velocity within the pore. These two velocities are related by u = φ up, where up is the pore velocity and u is the average velocity. It should be noted that for this analysis, the traditional viscous term ν∇2u is not considered in this analysis.

∂(φ Eg + Es) X + ∇ · [(E + p) u] −∇ · (J h ) − ∇ · F = S · u . (4.1d) ∂t g i gi d d i The last equation, Eq. (4.1d), accounts for the mixture energy of the gas and solid.

Here, Eg and Es are the internal energy of the gas and solid, respectively, and, hgi is the enthalpy of gas species i. The term Fd accounts for the transport of energy through the domain by means of conduction and is modeled by Fourier’s Law, k ∇T . The remaining ﬂux terms account for advective heating and enthalpy changes from diﬀusion. The source term Sd · u accounts for the work done by porous drag. In order to determine the internal energy of the gas and solid, the following relationships are used:

H = E + p (4.2)

h = e + p/ρ (4.3)

The gas internal energy can be solved through combining the diﬀerent number of energy modes nem (translational, vibrational, rotational, and electronic), the heat of formation, and the kinetic energy of the bulk gas, subscript g.

ngs nem ! X X 1 E = ρ y e + h◦ + ρ u2 + v2 + w2 (4.4) g g i i,j fi g 2 i j The enthalpy of the gas can be found one of two ways. The ﬁrst method is to solved Eq. (4.4) combined with the pressure as Eq. (4.2) suggests. The other method is to

34 simply use polynomial ﬁts, such as the NASA-9 or -7 format, which accounts for the previously stated energy modes for each molecule. The latter option is used in this analysis. Considering the polynomial expansion format, the following expressions is used to solve for the speciﬁc heat, enthalpy, and entropy of a species, all are a function of temperature:

8 Cp,i X = a T (4.5) R i u i=1 h R C dT i = p,i (4.6) RuT RuT s Z C dT i = p,i (4.7) Ru RuT

Thus, using these relationships, the gas internal energy can be calculated:

ngs X 1 E = ρ y h + ρ u2 + v2 + w2 − p . (4.8) g g i i g 2 i The solid internal energy is more straight forward since there is no kinetic energy or pressure needed:

E = ρ C T + h◦ = ρ h (4.9) s s p,s fi s s where the subscript s denotes that the property belongs to that of the solid.

Source terms

In this section a detailed description of three source terms is provided. The ﬁrst source term, which accounts for the production of pyrolysis species, is based on the CMA formulation. However, due to the multi-species approach, the model is modiﬁed. The second term will account for chemical equilibrium within the gas phase while, the third term accounts for solid condensation or deposition of the gas phase to the solid phase.

35 Pyrolysis species model

Revisiting the CMA model, Equations (2.13) and (2.14) provides a numerical pathway to convert solid species into a single gas product. The gas bulk source term must be

conserved for allω ˙ gi , and can be found by multiplying the mass fraction of pyrolysis gas species to the total gas source term, see Eq. (4.10):

ω˙ pyro = y ω˙ pyro (4.10) gi gi g

The value of the species mass fraction ygi is model speciﬁc. For this study, it is assumed that the available elemental composition of the gas is from that of the PICA data provided by Wong et al. (161). In their experiment, the species concentration was found by gas chromatography which also includes condensible species. This is an important consideration when considering the following section. Since it is assumed that the data provided by Wong et al.’s experiment is only temperature dependent and therefore can be mapped to the parameter regardless the heating rate or pressure. This assumption is planned to be revisited in future works. The last assumption made in this work, is that chemical equilibrium time associated with the pyrolysis generated gas is orders of magnitude smaller than the residence time. Furthermore, the composition of this equilibrium is that of the global Gibb’s Free-Energy minimum and not due to any pre-deﬁned reaction set. This is a common assumption when modeling TPS materials which dates back to the CMA model. This assumption is also valid for the models described in the next two sections.

Chemical equilibrium model

In order to account for equilibrium chemistry eﬀects in the gas phase, a dedicated source term is applied. In this model, the advection and diﬀusion of gas species are considered from the last time step to accelerate the solution procedure. Thus, the

36 necessary source term can be written:

∂ρ yn+1 − yn ω˙ eq = g = ρ i i (4.11) gi ∂t g δt

The total bulk gas density is a conserved quantity in this formulation, and since no gas is being consumed or produced, the total gas source term,ω ˙ eq = P ω˙ eq, must be g gi n n+1 zero. In Eq. (4.11), yi is found from the current solution, while, yi is determined from equilibrating the current mixture. In the situation where chemical equilibrium has yet to be reached — i.e. the residence time is not signiﬁcantly larger than the chemical time— a chemical relaxation time τr can be introduced to alleviate this assumption. It should be noted that this model does not account for the transitional species that would arise in this situation.

n+1 δt eq n n yi = (yi − yi ) + yi (4.12) τr

Looking at the limits of the relaxation time, and its effect on Eq. (4.12), as τr −→ δt chemical equilibrium is immediately reached within the first time step (the characteristic time of the flow δt) thus causing the system to be advection/diffusion limited. If

τr is smaller than δt, than the solution will have reached chemical equilibrium within the ﬁrst time step. Conversely, when τr δt the gas reactions become negligible, causing the system to be reaction limited.

Condensation/deposition model

In order to model condensation/deposition inside the porous domain, species of interest must be identiﬁed. Following the work preformed in Chapter 3, the considered transfer species will be water H2O(L) and graphite C(gr). As mentioned in these chapters, the early rise and plateau phenomena, present in in-depth thermocouple readings, and the increased density, found in the char layer during the Apollo Pro- gram, has been hypothesized to be caused by these respective species.

37 In this model, it is assumed that the species of interest can be present in both the gas and solid phases. However, with enough time all species of interest in the gas phase are transferred to the solid pore structure. It is expected that once the transfer species has “stuck” to the solid pore structure, the solid constituents still contributes to the competing gas phase reactions. Therefore, the model described in the prior section, Chemical equilibrium model, is needed. It should be noted, that it is assumed that no other solid constituent will compete in the solid-gas interactions. This assumption is obviously not true, especially when considering the oxidation of the char layer which leads to surface recession. It is however, only used as a preliminary step and should be reﬁned in later works. Therefore, the source terms for the transfer species in solid and gas phase are

∂ρ yn+1 − yn ω˙ cond = ζ T = ζ ρ i i , (4.13) si ∂t T δt ∂ρ yn+1 − yn ω˙ cond = (1 − ζ) T = (1 − ζ) ρ i i , (4.14) gi ∂t T δt where, ζ is the condensation sticking coefficient, ρT is the conserved density and yi is the species mass fraction which both differs from the Chemical equilibrium model formulation. The condensation sticking coefficient is a non-dimensionalized term and is defined as the amount solid transfer species present within the gas phase which sticks to the pore structure. Thus, looking at the limits, as ζ −→ 0 all solid transfer species remain suspended in the gas phase; while, when ζ −→ 1, all solid transfer species immediately stick to the pore structure. In this work, it is assumed that ζ is a known value and is constant. However, this parameter is likely a function of other parameters, such as the open interfacial area, velocity, and species concentration. The conserved density, in the consideration of chemical reactions, is that of the solid and the gas. However, since the pre-existing matrix is assumed to not contribute the competing gas reactions, only the condensed species compose the solid. In this analysis, it is assumed that interaction between the solid and gas acts homogeneously

38 due to the large amount of available surface area. Therefore, the conserved density can be written as, nsts ngs X X ρT = ρsi + φ ρgi (4.15) i i thus, implying that the species mass fraction used in this section is found from the constituent density over the conserved density. Lastly, it should be noted that the sum of the source terms from the gas and solid transfer species should be equal to zero due to the deﬁned conserved mass.

Solid Material Model

Density

According to the CMA material model, the solid density can be expressed as

ρs = Γ (ρA + ρB) + (1 − Γ)ρC , (4.16) where subscripts A, B, and C are component densities. Additionally, Γ represents a unit-less volume ratio parameter which distributes the component densities during decomposition. The total density of the ablator can therefore be expressed as

ngs X ρ = ρs + φ ρgi . (4.17) i However, if condensation within the pores is considered, this model, does not hold and must be re-formulated. Before this can be done, an important assumption must be made concerning the formation of the char material. When performing TGA, a small sample of the ablative material, typically on the order of 10 mg, is heated at a fixed heating rate whilst being purged with an inert gas. Commonly, the applied flow rate of the purge gas is selected to be fast enough to shield against any gas-surface interactions, similar to the case of arc welding. During this type of experiment, if the flow is reduced, for the resin base materials considered

39 in this study, coking would be discovered within the chamber and is the motivation for applying the sticking coeﬃcient introduced in the prior section. In this formulation, it is assumed that the sticking coeﬃcient during the formation of the char material is near 0, as is the case for most TGA experiments. This implies that any carbonaceous material found within the material is strictly due to residual char from the material and not from any coking event. Therefore, the total density accounting for condensation or deposition of species can be expressed by Equation 4.18.

nsts ngs−nsts nsts X X X ρ = ρs + φ ζ ρgi + φ ρgi + φ (1 − ζ) ρgi (4.18) i i i | {z } | {z } Static Solids Transport Species

Here, ρs is the total solid density that is present in the CMA model, and abides by the corresponding thermogravimetric analysis data. In this formulation, the equation has been broken up into two parts: static and transport. The solid density attributed from condensation or deposition, can be simply found from multiplying the sticking coeﬃcient with the solid transport species density in the gas phase. It should become quickly apparent that when ζ = 0 the formulation reverts to the original CMA model. As mentioned in the prior section and seen in Eq. 4.18, the solid transport species are allowed to exist in the gas phase as long as ζ < 1.

Porosity

In the situation when ζ > 0, condensation occurs within pores thereby altering the porosity of the bulk solid material. In order to determine the porosity of the material, the existing model is examined. The porosity of a material can be found in the following forms

ρˆ φ = 1 + , (4.19a) ρ0 1 = φ + ε . (4.19b)

40 In Eq. (4.19a), the intensive density, or the density associated with a pure material, is represented by ρ0, while the extensive density, or the density with respect to the control volume, is represented byρ ˆ. In Eq. (4.19b), the volume ratio between the solid and the control volume is represented by ε. By joining these two equations, the following relationship for ε can be easily found

ρˆ ε = . (4.20) ρ0

Relating back to the CMA model for porosity, the property is found through the interpolation of the instantaneous solid density ρs between the virgin and char measurements. For the material TACOT, these values range between ∼ 0.85 – 0.9; thus, leading to ε values between ∼ 0.15 – 0.1. For simplicity, the solid volume ratio attributed to the decomposing matrix will be represented as E (ρs). Equation (4.19b) can thus be reformulated to include this model as well as the solid transfer of partic- ulates to the matrix. nsts X 1 = φ + E (ρs) + εi (4.21) i

In Eq. (4.21), εi represents the total volume ratio of solid particulate i which has transported onto the solid matrix. Using the relationship from Eq. (4.20), the control volume considered here will be that of the open pores; therefore, the density associated within this volume must be that of the gas,ρ ˆ ≡ ζ ρg. Using this frame of reference the species solid volume fraction εi can be found as the following

ζ ρgi εi ≡ 0 (4.22) ρi

0 3 The intensive densities ρi used in this analysis are 1,000 kg/m for H2O(L) and 2,000 kg/m3 for amorphous C(gr). These values are currently assumed to not be a function of temperature.

41 Speciﬁc heat and enthalpy of the solid material

Following the previous sections, the speciﬁc heat and enthalpy of the solid material must also adjust as gas species transfer to the solid state. Both of these properties are strictly dependent the amount of species present in the solid phase and has no dependency on the pore structure or connectivity through the material. Therefore, as was performed in the previous sections, the speciﬁc heat and enthalpy of the solid material can be found from the following relationships

nsts X c = C(ρ ) + ε c0 , (4.23a) ps s i pi i nsts X 0 hs = H(ρs) + εi hi . (4.23b) i In Eq. 4.23, C and H are the CMA-based solid density interpolated values for the speciﬁc heat and enthalpy, respectively.

Thermal Conductivity of the solid material

Unlike the previous sections, the effective thermal conductivity of a porous material is largely dependent on the configuration of the solid matrix. Even in the case of the CMA-model, for the effective thermal conductivity, no geometric configuration is considered. In this model, only the virgin and char layers are characterized, and when the material is decomposed between these states the property is interpolated through the current solid density. This model is currently used for all TPS materials, regardless of its matrix construct. The current CMA model resembles the parallel effective thermal conductivity model which is the most conservative model available when ks kg (162). Therefore, for the lack of better knowledge, the conservative approach is to be used in this model.

nsts X 0 ks = K(ρs) + εi ki (4.24) i

42 4.3 Model Veriﬁcation

To test the derived source terms and material model above, dedicated test-cases are presented below.

Test-Case 1: Chemical equilibrium

The ﬁrst test-case ensures that the equilibrium source term resorts back to the Mu- tation++ solution. For this case, a box with impermeable wall boundary conditions is considered. The box is pre-ﬁlled with an equilibrium gas at 300 K and 1 atm, with an elemental composition of C = 6, H = 6, O = 1. Due to the fact that the box is enclosed, the molar ratio of this gas as well as its density does not change, only the species composition, temperature, and pressure. The box is linearly heated to a temperature of 2,500 K and the contents within the box is examined and presented in Figs. 4.1 and 4.2. In this analysis, 7 gaseous species are allowed to form: CH4,

CO, CO2, H, H2,H2O, and C(gr).

14 1

0.8 13

0.6 12 0.4 11 Pressure, Mpa 0.2

0 10 Avg. Mol. Weight, g/mol 500 1000 1500 2000 2500 Temperature, K

Figure 4.1: Gas pressure (black) and average molecular weight (blue) inside a closed wall domain as the temperature is increase linearly. As the gas equilibrium changes, the averaged molecular weight follows causing the pressure to increase non-linearly. Once the molecular weight nears a constant value, the pressure change becomes linear.

43 Figure 4.1 presents the average gas molecular weight (blue line) and pressure (black line) inside the domain. As the temperature inside the box is increased from 300 K to ∼ 1,500 K, large changes within the species concentration are observed (Fig. 4.2). As a result of the change in species mole fractions, the averaged molecular weight of the gas also changes. Due to the fact that P ∝ 1/M w T , the pressure inside the box increases non-linearly. Beyond 1,500 K, changes in the species molar faction is still observed. However, for most species this change is minimal, the average molecular weight becomes almost constant. Monatomic hydrogen does not follow this trend due to the species low molecular weight. But since it has a low molecular weight, the average molecular weight remains constant, and the pressure becomes proportional the box temperature and increases linearly. It should be noted that if the box was increased to higher temperatures, the species concentration would change signiﬁcantly again as molecules begin to break down and create new species. However, for this study, the smallest possible species set was considered in-oder to keep numerical runtimes low. If the temperature was increased higher than this limit, additional species would be needed thereby increasing the number of equations. Figure 4.2 presents the mole fraction of the gas species, on a log-scale, as the domain is heated. An exact match between Mutation++ and KATS-US is expected since Mutation++ is fully coupled to KATS-US. This comparison, verify the implementation of the chemical equilibrium source term previously presented in Sec. 4.2. In Fig. 4.2, KATS-US is denoted by lines, while Mutation++ is designated by symbols.

tot tot As shown in Fig. 4.2 minimal error, εavg = 2.15% and εmax = 1.83%, is observed between KATS-US and Mutation++ thereby verifying the model.

44 100 C(gr) 10−1

CH4 KAT−US CO 10−2 2 Mutation++

10−3 CO H2O H2 H 10−4 Mole Fraction, mol/mol 10−5 500 1000 1500 2000 2500 Temperature, K

Figure 4.2: Comparison of KATS-US (lines) with an equilibrium source term with Mutation++ (symbols). The relaxation time used for this problem ensures chemical equilibrium within the ﬁrst time-step, τr = τf .

Test-Case 2: Chemical relaxation time

Test-case 2 presents a similar problem to test-case 1 in that an enclosed wall domain is considered. For this test-case, the box is assumed to be adiabatic at an initial temperature and pressure of 300 K and 1 atm, respectively. The box is ﬁlled with 5 species air, N, O, NO, N2, and O2, which is not in equilibrium at t = 0. The contents of the box in terms of species molar fraction is χN = 0.79 and χO = 0.21. During the test, the box is allowed to equilibrate according to the pre-assigned relaxation time. Figure 4.3 presents the results for a range of relaxation times assuming a constant numerical time step of δt = 1e-5 s. In Figure 4.3, four diﬀerent relaxation times were selected of increasing orders

−1 −2 −3 of magnitude, η = δt/τr = 5×10 (black), 5×10 (blue), 5×10 (green), and 5×10−4 (red). For each of these relaxation times, the system reaches the expected equilibrium mole fraction value of N2 for the designated temperature and pressure,

χN2 = 0.79. Although not shown, the similar ﬁnding was observed for N, O, and

O2. In this study, NO was not allocated an initial concentration and does not exist

45 0.8 0.7 0.6 0.5 0.4 0.3 η = 5x10−1 0.2 η = 5x10−2

Mole Fraction, mol/mol −3 2 0.1 η = 5x10 N η = 5x10−4 0 10−5 10−4 10−3 10−2 10−1 Time, s

Figure 4.3: Eﬀect of chemical relaxation time η = δt/τr. Four relaxation rates were selected with decreasing orders of magnitude: η = 5 ×10−1 (black), 5×10−2 (blue), 5×10−3 (green), and 5×10−4 (red). As the relaxation rate is decreased by orders of magnitude, the time necessary for the system to reach equilibrium increases by orders of magnitude. under the pre-deﬁned set conditions; therefore, this species mole fraction was 0 for the entirety of the test. As is apparent in Figure 4.3, when η increases, and therefore the relaxation time decreases with a constant δt, the time necessary to equilibrate the system is reduced by orders of magnitude. This trend follows the relaxation time limit study presented in Sec. 4.2.

Test-Case 3: Coking eﬀects

As was the case for the previous test, a closed box is considered. In this case, however, the contents placed inside the box is already in chemical equilibrium, and, the condensation source term is allowed to contribute to the solution. As time progresses the solid transport species, which are presently in the gas phase at t = 0, contributes to the rigid solid. Figure 4.4 presents this testcase for various sticking coeﬃcients ζ. As is shown in Fig. 4.4, all solutions approach the theoretical coking density limit

46 0.3 Theoretical Limit = 0.2786 3

0.2

−1 0.1 ζ = 1x10 ζ = 1x10−2 ζ −3 Coke Density, kg/m = 3x10 ζ = 1x10−3 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time, s Figure 4.4: Effect of solid transfer sticking coefficient ζ. Four sticking coefficient were selected: ζ = 1×10−1, 1×10−2, 3×10−3, and 1×10−3. As the sticking coefficient is decreased, the time necessary for the system to equilibrate increases.

3 of 0.2786 kg/m . This limit can be found by a simple relationship: ρcoke = φ ρg yCgr, at t = 0. As ζ is increased, the time necessary to condense all of the solids within the gas phase is reduced.

3 0.3 0.8 0.25 0.79998 3 /m 0.2 0.79996 3 0.79994 0.15 0.79992 0.1 0.7999 0.05 0.79988 Porosity, m 0 0.79986 Coke Density, kg/m 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Time, s Figure 4.5: Change in porosity as condensation proceeds within the material.

As the gas species are condensed to the solid state, the solid material properties are changed and shown in Figs. 4.5 and 4.6. Figure 4.5 presents the change in porosity as graphite is condensed to the material surface. The intensive density used in this

0 3 study is that of amorphous carbon, ρC(gr) = 2, 000 kg/m . As condensation takes

47 place, the constituent density (black line) reduces the amount of available volume for gas to ﬂow. This eﬀect reduces the porosity. However, in this case, the amount of

3 available graphite for condensation is ρc = 0.2786 kg/m , and thus, when compared to the intensive density, only a minimal change to the porosity is observed. As the amount available C(gr) increases a more pronounced effect should be expected. Similar to the porosity, the effective thermal conductivity and specific heat of the solid material is affected by condensation. Figures 4.6a and 4.6b presents the change in these properties, as the density of the material is altered. For this analysis, the thermal conductivity is assumed to be constant. However, it could easily be made a function of temperature. The specific heat however, is temperature dependent and computed using the NASA-9 polynomials. The thermal conductivity used in this analysis is 5 W/m-K. By contrast, the thermal conductivity of pure graphite is reported to be ∼ 142 W/m-K (163). However, it is assumed when condensed, the structure of the new solid is fairly disorganized thereby reducing the intensive value. Thus, a fraction of this reported value is used. Lastly, Fig. 4.6c shows that as the density and specific heat is increased, the thermal diffusivity is also increased, allowing for energy to pass through the material more easily.

4.4 Concluding remarks on modeling multi-phase interactions in a charring ablator using KATS-US

This chapter is motivated by phenomena present in ﬂight data from notable missions such as Mars Science Laboratory and Apollo. The present study aimed at investigat- ing the composition of the pyrolysis gas. This was accomplished by accounting for pyrolysis production as a function of elemental composition, chemical equilibrium of the mixture, and condensation of solids within the gas phase, such as H2O(L) and C (gr). For each of these phenomena, source terms were proposed with controlling parameters such as the chemical relaxation time and the sticking constant. Results from

48 879.32 879.30 879.28 879.26 879.24 879.22 879.20

Solid Heat Capacity, J/kg−K 0 0.02 0.04 0.06 0.08 0.1 Time, s (a) Heat Capacity

0.04690

0.04689

0.04688

0.04687

0.04686 0 0.02 0.04 0.06 0.08 0.1 Thermal Conductivity, W/m−K Time, s (b) Thermal Conductivity

/s 1.6158 2 1.6156 1.6154 1.6152 1.6150 1.6148

Thermal Diffusivity, mm 1.6146 0 0.02 0.04 0.06 0.08 0.1 Time, s (c) Thermal Diﬀusivity

Figure 4.6: Eﬀect on the thermal transport properties as graphite is condensed to the solid matrix.

49 this analysis veriﬁed these source terms and displayed the limits of the controlling parameters.

50 Chapter 5 Modeling carbon oxidation of a ﬁberForm plug

As a result of the thermochemical decomposition reactions, a char layer is formed near the gas–material interface of the TPS and is comprised of carbon fibers as well as residual char generated by the pyrolysis process (3). As the boundary layer interacts with the charred material, oxidation occurs weakening the structural integrity of the fibers leading to surface recession. The recession of the TPS remains a key design parameter as it is directly linked to the safety of the spacecraft. Historically, ablation has been modeled (37, 39, 83, 89) through equilibrium surface energy balance (164), accounting for incoming heat fluxes and blowing rates of the material. This method has proven itself through extensive ground testing, and usually leads to no more than 10% error when compared to experimental results (43, 83). However, it has been observed that the method is less accurate at low pressures (43). Moreover, the approach has also yielded high error margins for a recent flight case, Stardust. In 2008, Kontinos and Stackpoole (44) observed that the current material response models over-predicted the surface recession by 70%. Recently, Johnston (46) showed this error is significantly reduced, overshoot of 20% of measured results, when a finite-rate model is used. Researchers have also used a microscopic approach (50, 51, 165) in an attempt to increase the fidelity of these models. Experimentally, studies have been performed to better understand the mechanisms leading to surface ablation and fiber mass loss. Panerai et al. (49, 166) observed that temperatures exceeding ∼900 K result in surface recession while lower temperatures lead to volumetric mass loss. It was also found that pressure plays a significant role in the development of the ablation front. Aside from the ablation community, carbon oxidation is a prevalent phenomena which can be found in many engineering disciplines. For instance, in energy produc-

51 tion technologies carbon oxidation has been studied for decades now, first beginning in the early 50’s with the pioneering work by P. L. Walker (167). Since then, a simplistic view of surface oxidation of carbon has been established and agreed on amongst most present researchers. This view is that the phenomena is a result of four unique processes acting on the microstructure: (a) oxygen surface adsorption, either dissociative or non-dissociative with later dissociation, (b) surface diffusion of adsorbed oxygen, (c) surface reactions producing adsorbed CO and CO2, and finally (d) desorption of these two species (8). Despite this newfound understanding, uncertainties in the evaluation of oxidation rates still remains. In an effort to bridge this gap, several researchers have proposed semi-detailed heterogeneous reaction mechanisms that characterize the behavior of carbon oxidation of crushed coal combustion (168, 169, 170, 171). In these studies, the rate constants for the proposed mechanisms were selected in order to match results of targeted experimental measurement sets (172, 173) which include the production of CO2/CO. It has be shown that the proposed models are only valid under certain conditions and completely break down when considering the dependency of O2 concentration on the production of CO2/CO (8); therefore, these models lack the ability to be used for predictive modeling purposes. Recently, a model proposed by Geier et al. (8) considers a semi-detailed model, in which the microscopic structure is introduced through the use of the number of active sites which suggest pitting mechanisms of the fibers. The relevance of these magnitudes is well known since the seminal papers by Radovic et al. (174, 175), Sun et al. (176), and Larciprete et al. (177). This is a distinct difference between that of Campbell and Mitchell (169), who models the microstructure evolution of the fiber as a uniform radial recession according to the Bhatia–Perlmutter model (178). Oxidation SEM studies of the material (49, 166), however, clearly show that oxidation of the fibers occurs in the form of pitting. This model, proposed by Geier et al. (8),

52 represents a compromise between the previously mentioned seminal papers and the very detailed model of Campbell and Mitchell (169). As such, the model has shown to have an excellent agreement with experimental gasification CO2/CO results from Tognetti et al. (173) not only for a wide range of temperatures but also for varying concentration of oxygen. In the present work, the semi-detailed oxidation model proposed by Geier et al. will be used to investigate the experimental behavior of a porous carbon plug which is exposed to an oxygen rich flow (49). To model this experimental work, governing equations will be presented and non–dimensionalized to uncover the non-dimensional Damköhlernumber which relates the gasification reaction rate to the residence flow time. Through this investigation, the scale of these key oxidation parameters will be assessed for future implementation in full-scale ablation problems.

5.1 Formulation

Here, we consider a long tube with a cylindrical cross section of diameter D, in which a porous plug with a length L ∼ D is inserted. The plug completely fills the tube’s cross section, so that the flow ahead of the plug must pass through the plug in its entire length. This incoming flow is pure oxygen, with of a uniform velocity profile of velocity U∞ far ahead from the plug, and is taken from the ambient at a temperature

T∞, and a pressure p∞. Typical values considered are in the range of T∞ ∼ 700 –

1,000 K, p∞ ∼ 1kPa, and U∞ ∼ 0.01 – 1 m/s, such that the typical Reynolds number,

Re∞ = U∞D/ν∞, based on the tube diameter D ∼ 1 cm is of order unity. The carbon fiber materials typically used for atmospheric entry applications are fabricated by stacking plane layers, with a ±15◦ uncertainty, of randomly oriented fibers. The resulting materials have overall orthotropic properties, which are near uniform in planes parallel to the layers but different from those along the axis normal to these planes. It is assumed that the porous plug is introduced into the tube such

53 that the plane layers are oriented normal to tube axis, and therefore to the flow. Consequently, the physicochemical properties of the plug are assumed uniform in planes normal to the tube axis. However, they can vary along the axis of the tube as a result of the gasification. The microscopic structure of the porous plug is characterized by two parameters: the total volume of solid carbon and the interfacial area shared between the solid and gas. The gasification rate is proportional to this exposed surface area. The characteristic time for the gasification of the carbon, however, is proportional to the total amount of carbon. As the carbon plug gasifies, these two properties begin to vary along the axis of the tube. Thus, it is convenient to define them in per unit length. This can be accomplished by differentiating with respect to the coordinate x (along the axis of the tube). The total volume of the solid carbon per unit length is defined by A(x) and has the dimension of surface; where as, the interfacial surface of carbon per unit length is defined by P (x) and has the units of length. These two magnitudes are accessible experimentally through tomography quantification (5). Figure 5.1 illustrates an example of such imagery while also showing a schematic representing the two baseline parameters. The total volume V of carbon can be calculated by integrating A(x) over the length L of the plug:

Z L V = A(x)dx (5.1) 0

Similarly, the total interfacial area Ae is obtained from Ps(x):

Z L Ae = P (x)dx (5.2) 0 The porosity φ through the plug varies along the tube axis due to its dependency on A(x). Since the porosity is deﬁned as the ratio of the void over the total volume, the porosity of the plug can be written as φ(x) = 1−A(x)/(π(D/2)2). Typical values

54 of carbon ﬁbers-based materials used in TPS applications are of the order of 0.8, that is, only 20% of the cross section is occupied by solid carbon ﬁbers which have a diameter of 10 – 20 µm.

Figure 5.1: Two views of the topology of the inner structure of the porous porous plug. The ﬁrst, a preprocessed tomography slice of a carbon ﬁber plug (5). The second, a schematics of the geometry of the cross section normal to the tube’s axis. A(x) is the net area of carbon, or equivalently the volume per unit length of solid carbon, whereas Λ(x), represented by the orange lines, is net carbon perimeter, or equivalently the gas-carbon interfacial area per unit length. φ(x) is the porosity, given in terms of A(x) as φ(x) = 1 − A(x)/(π(D/2)2)

Quasi-steady ﬂow

TPS systems are designed to protect the spacecraft during the entry stage and can last several minutes (3, 122, 179). Therefore, the total gasification time of the TPS, which is proportional to its total mass of carbon, must be at least of the order of magnitude of the duration of the entry phase. The total gasification times obtained through lab-scale experiments (see Panerai et al. (166)) are consistent with flight time scales, and are of order of minutes. On the other hand, the transit time across the porous plug is of order L/U∞, typically ranging between a hundredth to a tenth of a second. Clearly, this is much smaller than the total gasification time, and the gas

55 ﬂow through the porous plug can be assumed quasi–steady. With this in mind, it is possible to derive the continuity, the momentum, and the mass conservation equations, averaged over each cross section x:

∂p˜ µ = −φ u˜ (5.3a) ∂x K ∂(φρ˜ u˜) π(D/2)2 g = P˜(x) m˜˙ 00 (5.3b) ∂x s ∂ ∂Y π(D/2)2 φρ˜ uY˜ − D i = P˜(x) m˜˙ 00 (5.3c) ∂x g i i ∂x s,i

Here, the momentum equation was reduced down to Dacy’s Law form; where, the permeability, K, and kinetic viscosity, µ, help to dissipate the ﬂow macroscopically through the plug (36). In these system of equations,p ˜,u ˜, and Yi are solved for and represent the instantaneous pressure, velocity, and the mass fraction of each of the species (O2, CO and CO2), respectively. Once solved, the gas density,ρ ˜g, can be solved for through the use of the state equation. The source term present in the mass and mass conservation equations can be found through the use of the interfacial area

˜ ˙ 00 of the ﬁber, P (x), and the mass production rate of species m˜ s,i which are per unit time and unit surface, due to the surface oxidation of the carbon ﬁbers. The total mass production rate of the system, is simply the net gaseous mass production rate,

˙ 00 P ˙ 00 m˜ s = i m˜ s,i. Finally, the effective diffusion coefficient of species i in the porous material is represented as Di, and typically holds a value which is different from the same species in a homogeneous gaseous mixture.

Solid phase reactions

To investigate the oxidation of carbon ﬁbers, the semi-detailed heterogeneous reaction model proposed by Geier et al. (8) is considered.

In this mechanism, C() and Cs represent the surface carbon sites, active (ready to accept adsorbed oxygen) and inactive, respectively. Additionally, C(O) represents a carbon site which has already adsorbed an oxygen atom. The ﬁrst reaction, Re-

56 Table 5.1: Kinetic mechanism proposed by Geier et al. (8).

kAd 2 C() + O2 −−→ 2 C(O) (Ad)

kCO,O2 Cs + C() + O2 −−−−→ CO + C(O) (PCO,O2 )

kCO2,O2 Cs + C(O) + O2 −−−−→ CO2 + C(O) (PCO2,O2 )

kCO Cs + C(O) −−→ CO + C() (PCO)

kCO2 Cs + 2 C(O) −−−→ CO2 + 2 C() (PCO2 ) action (Ad), represents the pure dissociative adsorption of oxygen. The following

reactions, Reactions (PCO,O2 ) – (PCO2 ), provide routes for the desorption of both CO and CO2, and therefore the gasiﬁcation of solid carbon. The net production rates of carbon monoxide and dioxide, per unit area of solid carbon material and time,

˙ 00 ˙ 00 are given by m˜ s,CO = MCO($ ˜ 2 +$ ˜ 4) and m˜ s,CO2 = MCO2 ($ ˜ 3 +$ ˜ 5), whereas the ˙ 00 rate of carbon gasification is m˜ s,C = MC ($ ˜ 2 +$ ˜ 3 +$ ˜ 4 +$ ˜ 5) where$ ˜ i is the ith reaction rate, in units of moles/(area · time). It should be noted that the last four reactions also provide a path for the activation of inactive sites. There is, however, no deactivation mechanism in the scheme other than the complete removal of carbon material. Therefore, in time, all of the initial solid carbon mass will eventually be gasified if oxygen is available. Following Geier et al. (8), the total number of active sites per unit of solid carbon surface can be represented by the molar concentration, denoted as [ · ], of adsorbed and available sites, Γ = [C(O)] + [C()]. The coverage of the material is represented as θ and can be defined as the ratio between the molar concentration of adsorbed sites, C(O), compared to the total number of sites Γ. By combining these two definitions, the following relationships can be obtained: [C(O)] = θΓ and [C()] = (1 − θ)Γ.

57 Therefore, the rate for each reaction can be written in terms of θ as

2 2 $˜ 1 = k1(T )Γ (1 − θ) [O2] (5.4a)

$˜ 2 = k2(T )Γ(1 − θ)[O2] (5.4b)

$˜ 3 = k3(T )Γθ[O2] (5.4c)

$˜ 4 = k4(T )Γθ (5.4d)

2 2 $˜ 5 = k5(T )Γ θ (5.4e)

with the molar concentration of oxygen given by [O2] = pO2 /(RuT ) = ρYO2 /MO2 .

0 2 These reaction rates can be scaled by the factor$ ˜ 1 = k1Γ from the ﬁrst reaction.

0 The normalized reaction rates $i =$ ˜ i/$˜ 1, are thus dimensionless for i = 1,..., 3, and have the same units as a concentration. These factors can be expressed in

−TA /T an Arrhenius form, $0i = Aie i , where the rate constants, Ai, and activation

temperature, TAi , for all reactions are listed in Table 5.2. Thus, the normalized reaction rates are given as the following:

2 $1 = (1 − θ) [O2] (5.5a)

$2 = $02(1 − θ)[O2] (5.5b)

$3 = $03θ[O2] (5.5c)

$4 = $04θ (5.5d)

2 $5 = $05θ (5.5e)

Table 5.2: Arrhenius rate constants for the heterogeneous reaction model proposed by Geier et al.(8)

Reaction Ai (1/s) TAi (K) Reaction (Ad) 1.00 × 100 0 8 Reaction (PCO,O2 ) 7.48 × 10 20,241 4 Reaction (PCO2,O2 ) 1.64 × 10 13,250 8 Reaction (PCO) 4.26 × 10 20,207 7 Reaction (PCO2 ) 1.20 × 10 15,997

58 The rate of mass production/consumption of CO, CO2, and O2 per unit surface of ˙ 00 ˙ 00 ˙ 00 carbon and time, m˜ s,CO, m˜ s,CO2 and m˜ s,O2 , can be written in terms of these reaction rates as:

˙ 00 0 m˜ s,CO =$ ˜ 1MCO ($02(1 − θ)[O2] + $04θ) (5.6a)

˙ 00 0 2 m˜ s,CO2 =$ ˜ 1MCO2 $03θ[O2] + $05θ (5.6b) ˙ 00 0 2 m˜ s,O2 =$ ˜ 1MO2 (1 − θ) + $02(1 − θ) + $03θ [O2] (5.6c)

Finally, the ratio of mass production of CO2 to CO, which is a magnitude often measured experimentally, can be easily obtained.

m˜˙ 00 $ [O ]θ + $ θ2 M s,CO2 = 03 2 05 CO2 (5.7) ˙ 00 m˜ s,CO $02[O2](1 − θ) + $04θ MCO

Steady state approximation

Within the current application, the intermediate species, C() and C(O), are assumed to be in steady state. The rationale behind such assumption is that the time scales of the microscopic processes, such as oxygen absorption and desorption, are much shorter than the macroscopic time scales, such as gas residence time in the porous plug and solid recession rate. Therefore, the microscopic scale processes can be assumed to respond instantaneously to the external changes of the macroscopic state, such as a change in the local temperature or of the partial pressure of oxygen. As a result, the number of active sites with or without adsorbed oxygen, C(O) or C() respectively, are in equilibrium at the given macroscopic state.

Thus, setting equal to zero the net production rate of C() or of C(O), 2$1 + $2 −

$4 − 2$5 = 0, gives

2 −TA4 /T −TA5 /T 2 pO2 $04θ + 2$05θ A4e θ + 2A5e θ [O2] = = = (5.8) 2 2 −TA /T RuT 2(1 − θ) + $02(1 − θ) 2(1 − θ) + A2e 2 (1 − θ) which is a relationship between the equilibrium value of the coverage, θ, and the

oxygen concentration written in terms of the partial pressure of oxygen pO2 = XO2 P .

59 In this expression, XO2 represents the molar fraction of oxygen in the gas phase. Figure 5.2 graphically illustrates this relationship.

1 104 PO = 100 kPa 0.9 2 3 P = 100 kpa 10 PO = 10 kPa 0.8 O2 2 P = 1 kPa 2 O2 0.7 10 P = 0.1 kPa O2 0.6 101

0.5 / CO P = 10 kpa 2 0 0.4 O2 10 CO Coverage, [-] 0.3 10-1 0.2 P = 1 kpa -2 O2 10 0.1 P = 0.1 kpa O2 0 10-3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Temperature, [1000 K] Temperature, [1000 K]

Figure 5.2: Coverage of adsorbed atomic oxygen θ = [C(O)]/Γ and CO2 to CO ratio

as a function of temperature and partial pressure of oxygen pO2 as given by Eq. (5.8).

The relationship presented in Eq. (5.8) can now be used to obtain the coverage

in terms of [O2] and of the temperature. Inserting this coverage back into Eq. (5.6)

provides the mass production rates of CO and CO2, and the consumption rate of oxygen in terms of the partial density of oxygen.

Non-dimensional form

Though the use of previously deﬁned reference parameters, Equations (5.3a–5.3c)

can be non–dimensionalized using the following relationships: ξ = x/L, ρg =ρ ˜g/ρ∞,

u =u/U ˜ ∞, and p =p/p ˜ ∞. The dimensionless pressure p and density ρ are based,

respectively, on the ambient pressure p∞, and on ρ∞ = p∞MO2 /RuT , the density

60 that would correspond to pure oxygen at the ambient pressure.

∂p m = −φ λ g (5.9a) ∂ξ ρ ∂ 1 ∂Y φ m Y − ρ O2 = −Da P (x)m ˙ 0 (5.9b) ∂ξ g O2 P e ∂ξ s s,O2 ∂ d ∂Y φ m Y − ρ CO CO = Da P (x)m ˙ 0 (5.9c) ∂ξ g CO P e ∂ξ s s,CO ∂(φm ) g = Da P (x)m ˙ 0 (5.9d) ∂ξ s s,g

YO2 + YCO + YCO2 = 1 (5.9e) X p = ρ Yi MO2 /Mi (5.9f)

The dimensionless parameter present in the momentum equation, λ, accounts for pressure dissipation through the plug due to porous effects. This non–dimensional parameter is defined as λ = (U∞ L µ)/(p∞ K) and has been determined experimentally (166) to be of the order of ∼0.7. The Pécletnumber P e is defined as

P e = (LU∞)/DO2 , and the dimensionless diffusion coefficients di refer to the diffusion coefficient or the considered species to that of oxygen; therefore, dCO is the ratio

DCO/DO2 of the effective diffusion coefficient of carbon monoxide to that of oxygen.

0 The dimensionless mass production ratesm ˙ s,i of species i = O2, CO, CO2, or g for 0 ˙ 00 ˙ 00 the net gaseous production rate, is obtained asm ˙ s,i = m˜ s,i/m˜ 0, where the reference ˙ 00 ˙ 00 2 ˙ 00 ˜ mass flux m˜ 0 is defined as m˜ 0 = (π (D/2) m˜ ∞)/(P0 L), obtained straightforwardly from (5.3b) using L as the reference length, P0 as the reference interfacial area of ˙ 00 carbon per unit length, and m˜ ∞ = ρ∞U∞ as the reference mass flux. The rates of consumption of oxygen and of production of carbon dioxide masses are given by

0 2 m˙ s,O2 = (1 − θ) + $02(1 − θ) + $03θ [O2] (5.9g)

0 MCO m˙ s,CO = ($02(1 − θ)[O2] + $04θ) (5.9h) MO2

61 ˙ 00 with the Damk¨ohlernumber Das = (L/m˜ ∞)MO2 $01, and the coeﬃcients $0i are

−TA /T $0i = Aie i with values for Ai and Ti listed in the Table 5.2.

Boundary conditions

The incoming gaseous ﬂow through the tube is, far upstream from the plug, pure

oxygen so the total mass ﬂuxm ˙ g is that of oxygen,m ˙ O2 =m ˙ g, whereas the mass

ﬂuxes of CO and CO2 are zero. Thus, since no reactions are considered in the gas phase, these are also the ﬂuxes at the front face, ξ = 0, of the porous plug:m ˙ g(0) =

m˙ O2 (0) = 1,m ˙ CO(0) = 0 andm ˙ CO2 (0) = 0. Therefore, taking into account that the

0 mass ﬂux can be written asm ˙ i =m ˙ gYi −(di/P e)Yi , the mass fractions of each species at ξ = 0 must be such that:

0 m˙ gYO2 (0) =m ˙ g + (dO2 /P e)YO2 (0) (5.10a)

0 m˙ gYCO(0) = (dCO/P e)YCO(0) (5.10b)

0 m˙ gYCO2 (0) = (dCO/P e)YCO2 (0) (5.10c) where the prime denotes the derivative with respect to the axial coordinate ξ. If the gasiﬁcation reactions are fast enough, oxygen will be completely depleted inside the porous plug, say at ξ = ξd < 1. Since no gas phase reactions are being considered, and the surface reactions can not proceed without oxygen, and clearly the mass ﬂuxes of CO and CO2 must remain constant downstream of ξd. This is easily seen to be equivalent to the conditions that the gradients of all the species is null,

0 that is Yi = 0 at ξ = ξd < 1. The value of ξd is determined by the condition that m˙ CO2 = 0. The same conditions can also be applied at the exit of the plug, ξ = 1, when oxygen has yet to be depleted.

62 5.2 Results

It is clear from the formulation, Section 5.1, that multiple parameters inﬂuence the overall response of the presented system. These terms are as following: temperature

(T ), Damköhlernumber (Das), Pécletnumber (Pe), ambient pressure (p∞), Darcy dissipation (λ), porosity (φ), and exposed interfacial edge (Λ). The evolution of the of porosity and interfacial area as the plug gasifies is unclear and therefore will not be the focus of this study but left for further work. By solving the system of equations, Eq. 5.9a – 5.9f, the following results can be obtained as a function of the non-dimensional plug length (ξ); where, ξ = 0 represents the front face or inlet of the pellet and ξ =1 is the back face or exit. The influence of temperature on the numerical system is studied by ranging the temperature from 500K to 1,300 K while fixing the remaining parameters. The values of the fixed parameters are as following: p∞ = 10 kpa, Pe = 1, Das = 10, λ = 0.7, φ = 0.9, and Λ = 1. Figure 5.2 presents the solution for the oxygen concentration, non-dimensional pressure and mass flux, respectively, through the plug. As temperature is increased, the oxygen concentration is reduced not only through the length of the plug but also at the face. Although it may be trivial to see why the oxygen concentration is depleted through the length of the plug, the same can not be

said at the face. At low temperature, the rates for Reactions Ad – PCO2,O2 are small and result in minimal gasification through the plug. As the temperature is increased, the fiber reactivity follows leading to an intense gasification of reactants at the surface of the pellet. This process effectively reduces the oxygen partial pressure ahead of the pellet minimizing the amount of oxygen which can reach the solid mass. Thus, as the temperature is increased, the concentration of oxygen at the face is reduced. The same reasoning is also applied to the trends found in the pressure at ξ = 0 and the mass flux at ξ = 1. In order to characterize the influence of temperature on the

63 non-dimensional pressure and mass flux, the flow is analyzed at the inlet and outlet. Figure 5.3 presents this analysis for the pressure at the inlet and for the mass flux which is located at the outlet. Clearly seen in Fig. 5.3, as temperature is increased from 500 K minimal change is observed until ∼ 700 K where the reactions are initiated and the non-dimensional mass flux and pressure increase rapidly. As these reactions progress from ∼ 700 K to ∼ 1,100 K these trends continue. Beyond this point, the model shows that a region is formed where the mass flux and pressure has reached a maximum. To gain an understanding of the species gasification behavior at the face and through the length of the pellet, the species concentration at the face and exit are studied and presented in Fig. 5.4. Here, the color red, green, and blue are used to represent species O2, CO, and CO2, respectively; while, the solid lines represent the concentration of these species at the face of the pellet while the dashed line represent the values at the back face. The mass fraction of the species is measured using the y-axis while the x-axis represents the set iso-thermal temperature of the system. As observed in Fig. 5.4, under the predefined conditions, the global reactions are initiated or become “significant enough” around ∼ 700 K. As defined in Table 5.2,

Reactions PCO2,O2 and PCO2 contains the lowest activation temperatures and have

CO2 producing mechanisms. This is reﬂected in Fig. 5.4. A purely CO2 gasiﬁcation regime is present between ∼ 700 – 750 K. Beyond this regime, CO is produced with

CO2 and O2 being depleted at 1,300 K. Under these conditions, the maximum CO2 production occurs near ∼ 900 K; while the the maximum production of CO occurs at the highest considered temperature. Through the length of the pellet, the maximum oxygen consumption occurs near ∼ 900 K and is a result of two balancing effects. That is, as temperature is increased the rate at which oxygen is consumed is also increase. The second balancing effect is that as temperature is increased the amount of available oxygen for heterogamous reactions are reduced. This observation reflects

64 an experimental finding by Panerai et al. that showed two oxidation regimes exist for this problem. The first occurs at relatively low temperatures and oxidized the pellet through its length or volumetrically. The other regime exists at higher temperatures and focuses the oxidation at the inlet of the pellet. To further investigate the transition between these two oxidation regimes, the location within the pellet at which oxygen is exhausted is examined and presented in Fig. 5.5. In Fig. 5.5, the transition between the two oxidation regimes begins around ∼ 950 K and appears to develop exponentially through the length of the pellet. Like most transition regimes, the exact point of transition is a bit subjective. For this study, 10% of the pellet original length will be used to signify the development of the surface ablation regime. This means that, under the previously defined fixed parameters, surface ablation occurs at any temperature grater than ∼ 1175 K. As the pre-defined parameters change, the specifics of these findings are altered; however the trends of the underlining system behavior remains. It has been observed and shown in Appendix B.1 and B.2 that as the Damköhlernumber is increased the onset temperature of the heterogeneous reactions are decreased, and when the Péclet number is increased the opposite is found.

5.3 Conclusion

Oxidation of carbon fibers plays a key role when considering the recession of a heat shield material under re-entry conditions. The present work presents a simplified model to better understand the O2 effects on porous carbon preform. This is done through assumptions concerning the material structure, flow parameters, and chemical behavior. This model is then non-dimensionalized to further reduce the complexity of the problem. A sample reference pressure and a Damköhlernumber are selected to isolate the temperature effects. It is observed that the overall fiber reactivity increases when the temperature increases. This reactivity increment causes gasification

65 products to diffuse far upstream, and reduces the amount of oxygen which can travel to the plug surface. Furthermore, two physical regimes can be identified based on the temperature: volumetric ablation (oxygen arrives at the exit) and surface ablation (oxygen is depleted within the plug). For the low temperature regime, volumetric ablation, the mass flux and the surface pressure are largely affected by an increase in temperature. However, for the converse situation – high temperature regime – volumetric ablation causes an increase in mass flux but the pressure at the surface begins to decrease as temperature increases.

66 T = 600 K T = 825 K T = 900 K T = 800 K T = 850 K T = 1000 K 1

0.8

0.6 Increasing Temperature 0.4

0.2 Oxygen Mass Fraction 0

1.8

1.6

Increasing Temperature 1.4

1.2

Non-Dimensional Pressure 1

1.8

1.6

Increasing Temperature 1.4

1.2

1 Non-Dimensional Mass Flux 0 0.2 0.4 0.6 0.8 1 Length Along Plug, (x/L)

Figure 5.2: This figure presents the influence of temperature on the development of concentration of oxygen, normalized pressure and mass flux through the length of the plug.

67 1.8 1.8 1.7 1.75 1.6 1.5 1.7 1.4 1.65 1.3

1.2 1.6 1.1 Non-Dimensional Pressure Non-Dimensional Mass Flux 1 1.55 500 600 700 800 900 1000 1100 1200 1300 Temperature, [K]

Figure 5.3: This figure presents the influence of temperature on the non-dimensional mass flux at the back face of the pellet (black) and the non-dimensional pressure at the front face (red).

0.8

O2 (inlet) 0.6 CO (inlet) CO2 (inlet) O2 (outlet) 0.4 CO (outlet) CO2 (outlet) 0.2 Specie Mass Fraction

0 500 600 700 800 900 1000 1100 1200 1300 Temperature, [K]

Figure 5.4: This ﬁgure presents the inﬂuence of temperature on the species mass fraction at the front and back face of the pellet and the .

68 1300 1250 1200 1150 1100 1050

Temperature, [K] 1000 950 900 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Length Along Plug, [x/L]

Figure 5.5: Here, the temperature inﬂuence on the in-depth oxygen depletion location is examined. The x-axis represents the non-dimensional plug length, while the y-axis relates the in-depth location in which O2 is exhausted within the plug.

69 Chapter 6 Concluding remarks

6.1 Summary

In this work, the interaction between gas and solid in the field of atmospheric entry is explored. This is accomplished through the use and modification of the pre- existing modeling framework, Kentucky Aerodynamic and Thermal- response System (KATS). In this work KATS Material Response and Universal Solver were utilized. In addition, an ODE code was developed in MATLAB. Each of these solvers are capable of modeling flow through porous material as well as the interaction key species have with the solids. At the beginning this work Chapter 1 provides a brief introduction to atmospheric entry. In this introduction, the physics surround thermal protection systems is provided with a thorough review of relevant literature within this field. The next chapter, Chapter 2, focuses on the numerical framework used to model such material. This chapter highlights the governing equations used in KATS-MR and provides a detailed description of the solid model which dates back to CMA, which is one of the first MR codes. Chapter 3 uses open literature to accumulate a central gathered location of historical data on Avcoat. This chapter uses this information to highlight the information needed to model such problems, material response during atmospheric entry. In addition, this chapter highlights key assumptions that are frequently used or neglected form material models. The next chapter, Chapter 4, explores one of these neglected assumption. Chapter 4 the effect that water would have if present within an ablator. This chapter uses the Mar Science Laboratory thermocouple data to investigate this topic. The next chapter, Chapter 5, further investigates this topic by laying out a mathematical framework to tack and condense key species, such as water, within the ablator. Once condensed, this chapter provides a model to pre-

70 dict eﬀective material properties. Lastly Chapter 6 investigates the role of diatomic oxygen has on the ablation process of TPS. To simplify this problem, the chapter considers a porous carbon plug which is subjected to pure O2 ﬂow.

6.2 Original contributions

1. Development of an open Avcoat-similar material model, VISTA (Chap- ter 3) Although open material databases exist, such as TACOT, these databases provide little information to the extent of the assumptions made when developing the database. These assumptions can often be just as important as the reported values. This work provides little original contribution to the research community. However, what this chapter oﬀers is re-opening several key assumptions which are often neglected, knowingly or unknowingly, within the ﬁeld. In doing so, this chapter helps to provide further motivation for the future chapters in this document as well as topics for future work.

2. Effects of water phase change on the material response of low-density carbon-phenolic ablators (Chapter 4) This chapter uses a ready-to-go material response code from NASA-ARC, PATO. No contribution was made to this code. This work adjusted material property inputs, namely the virgin thermal conductivity, mid-simulation as a first-approximation to model the effect water would have if present within the ablator during flight. This work justifies the necessity for higher fidelity models which would take into consideration the species present as well as how these species would instantaneously change the effective material properties.

3. Modeling multi-phase interactions in a charring ablator using KATS- US (Chapter 5) This chapter uses the pre-existing numerical framework KATS Universal Solver as well as the chemical kinetic solver Mutation++.

71 KATS Universal Solver already contained the ability to solve multi-species flow in the free and porous domain. This work expanded this code by first fully coupling the framework with Mutation++. This allows KATS-US to preform chemical equilibrium and gas mixture property calculations on-the-fly without the need to import any additional information. This work also expands the capability of KATS-US to incorporate solid decomposition and force the gas mixture into chemical equilibrium. Lastly, this work expands on the CMA material model by accounting for the solid transfer from the gas phase. These models are important as they could potentially account for observable phenomena such as the early rise and plateau in thermocouple data and the carbon densification at the outer material layer.

4. Modeling carbon oxidation of a fiberForm plug (Chapter 6) This chapter uses MATLAB to model a 1-D steady flow through a porous domain. This work makes use of pre-existing functions within MATLAB which can solve a system of ODE’s. The reaction set used in this work was taken from Geier et al. (8). This chapter highlights the assumptions used to derive a set of governing equations and homogeneous chemical interactions. A Damköhlernumber and Péclet number where also derived. It was shown that the Pécletnumber controls the

solution by the means of coverage of O2. The eﬀect of temperature on the solution was also explored. It was shown that as temperature is increased the ablation mechanism proceeded towards surface ablation. This work is of importance due to its ability show the transition regimes between surface and

volumetric ablation as well as the production of CO2.

6.3 Future work

1. Validation of the CMA solid interpolation model As brieﬂy discussed at the end of Chapter 2, the model used for interpolating properties between

72 the virgin and char state dates back to CMA. This model was recognized as a ﬁrst-approximation and is not validated using experimental data. It would be of great importance to quantify the conservatism of this model for various TPS materials. Furthermore, if large discrepancies exist it would be of interest for formulate material speciﬁc property models for the decomposition regime.

2. Exploration of the solid sticking coefficient As discussed in Chapter 5, the solid sticking coefficient was assumed to be constant for convenience. It is therefore of interest to relax this assumption. Preliminary experimental results have shown that the sticking coefficient is likely a function of flow-rate and available surface area. It is of interest to determine all parameters which influence this coefficient, and how this coefficient might be changed during flight conditions.

73 Appendix A Modeling of the remain MISP’s for MSL

Measured Measured 1200 PICA 1200 PICA PICA + Water PICA + Water

1000 TC1 1000 TC1

800 TC2 800 TC2

600 600 Temperature, K Temperature, K TC3 TC3

400 400 TC4 TC4

200 200

0 50 100 150 200 250 0 50 100 150 200 250 Time, s Time, s (a) MISP 1 (b) MISP 3

Measured Measured 1200 PICA 1200 PICA PICA + Water PICA + Water

1000 TC1 1000 TC1

800 800 TC2 TC2

600 600 Temperature, K Temperature, K TC3

400 400 TC4

200 200

0 50 100 150 200 250 0 50 100 150 200 250 Time, s Time, s (c) MISP 4 (d) MISP 5

Measured Measured 1200 PICA 1200 PICA PICA + Water PICA + Water

1000 TC1 1000 TC1

800 TC2 800 TC2

600 600 Temperature, K Temperature, K

TC3 400 400 TC4

200 200

0 50 100 150 200 250 0 50 100 150 200 250 Time, s Time, s (e) MISP 6 (f) MISP 7

Figure A.1: Comparison of the investigative model, the baseline estimated model (from Mahzari et al. (4)), and the measured ﬂight data for all MEDLI Integrated Sensor Plug (MISP) of Mars Science Laboratory (MSL), except MISP 2, which is discussed in the main text

74 550 Measured ψ=0.0 500 ψ=0.1 ψ=0.3 450 ψ=0.5 TC3 ψ=0.8 400 TC4 350

Temperature, K 300

250

200 50 60 70 80 90 100 110 120 Time, s Figure A.2: Comparison of the measured ﬂight data for MISP2 and the investigative model applied to k, using a volume fraction of water ψw ranging from 0 to 0.8

75 Appendix B Oxidation effects from the Damköhlerand Péclet number

B.1 Damk¨ohlernumber eﬀects

To investigate the inﬂuence of the Damk¨ohlernumber three values are chosen, Das

= 0.1, 1, and 10, while ﬁxing the following parameters: p∞ = 10 kpa, Pe = 10, λ = 0.7, φ = 0.9, and Λ = 1. The results for this analysis is presented in Fig. B.1. Like Fig. 5.3 the non-dimensional mass ﬂux and pressure are analyzed.

1.8 1.8 Da = 0.1 1.7 s Da = 1 1.75 1.6 s Das = 10 1.5 1.7 1.4 1.65 1.3

1.2 1.6 1.1 Non-Dimensional Pressure Non-Dimensional Mass Flux 1 1.55 500 600 700 800 900 1000 1100 1200 1300 Temperature, [K]

Figure B.1: This figure presents the influence of the Damköhlernumber on the non- dimensional mass flux at the back face of the pellet (black) and the non-dimensional pressure at the front face (red) as a function of temperature.

From Fig. B.1 it is clear to see that as the Damk¨ohlernumber is increased the onset temperature by which the reactions are activated is reduced. This should not be surprising considering the Damk¨ohlernumber is a multiplier in the species equation’s source term.

76 B.2 P´ecletnumber eﬀects

To investigate the inﬂuence of the P´eclet number three values are chosen, Pe = 0.1,

1, and 10, while ﬁxing the following parameters: p∞ = 10 kpa, Pe = 10, Das = 10, λ = 0.7, φ = 0.9, and Λ = 1. The results for this analysis is presented in Fig. B.2. Like Fig. 5.3 the non-dimensional mass ﬂux and pressure are analyzed.

1.8 1.8 Pe = 0.1 1.7 Pe = 1 1.75 1.6 Pe = 10 1.5 1.7 1.4 1.65 1.3

1.2 1.6 1.1 Non-Dimensional Pressure Non-Dimensional Mass Flux 1 1.55 500 600 700 800 900 1000 1100 1200 1300 Temperature, [K]

Figure B.2: This figure presents the influence of the Pécletnumber on the non- dimensional mass flux at the back face of the pellet (black) and the non-dimensional pressure at the front face (red) as a function of temperature.

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103 Vita

Ali D. Omidy

Education

Bachelor of Science in Mechanical Engineering 2008-20013 University of Kentucky | Lexington, KY

Awards

• Awarded best poster in the 2016 and 2017 Department of Mechanical Engineer- ing Poster Session at the University of Kentucky

• Won Outstanding Graduate Student for the 2016 Summer NASA JSC Intern- ship from a total internship class size of 130 students

Grants/Scholarships

• Recipient of the NASA Kentucky Space Grant Graduate Fellowship 2015, 2016, and 2017

• Recipient of the NASA Kentucky Undergraduate Scholarship 2013

• Recipient of the Undergraduate Summer Research Grant (USRG) at Texas A&M University 2012

Publications in Referred Journals

• Gra˜na-Otero,J., Omidy, A. D., Martin, A., “Ablation of Carbon-based Porous Materials I. Formulation”, Fuel, Submitted.

104 • Gra˜na-Otero,J., Omidy, A. D., Martin, A., “Ablation of Carbon-based Porous Materials II. Results”, Fuel, In Preparation.

• Rostkowski, P., Venturi, S., Panesi, M., Omidy, A. D., Weng, H., Martin, A., “Calibration and Uncertainty Quantiﬁcation of VISTA Ablator Material Database Using Bayesian Inference”, Journal of Thermophysics and Heat Trans- fer, Accepted.

• Omidy, A. D., Cooper, J. M., Fu, R., Weng, H., Martin, A., “Development of VISTA, an open-source Avcoat material database”, Journal of Thermophysics and Heat Transfer, Accepted.

• Omidy, A. D., Panerai, F., Lachaud, J., Mansour, N. N., Martin, A., “Eﬀects of water phase change on the material properties of lightweight charring ablators”, Journal of Spacecraft and Rockets, Vol. 30, No. 2, Apr. 2016, pp. 472–477.

Refereed Reports

• Omidy, A. D., Panerai, F., Lachaud, J., Cozmuta, I., Mansour, N. N., Mar- tin, “Code-to-Code Comparison, and Material Response Modeling of Stardust and MSL using PATO and FIAT”, Contractor Report NASA/CR-2015-218960, NASA Ames Research Center, Moﬀett Field, CA, 2015.

Conference Articles

• Omidy, A. D., Weng, H., Martin, A., Gra˜na-Otero, J., “Modeling Gasiﬁcation of Carbon Fiber Preform in Oxygen Rich Environments”, AIAA Aerospace Sci- ences Meeting, No. AIAA-2017-3686, Denver, CO, 05-09 June 2017, Submitted.

• Omidy, A. D., Cooper, J. M., Fu, R., Weng, H., Martin, A., “Development of VISTA, an open-source Avcoat material database”, AIAA Aerospace Sciences Meeting, No. AIAA-2017-3356, Denver, CO, 05-09 June 2017, Submitted.

105 • Smith, D. L., Omidy, A. D., Weng, H., White, T. R., Martin, A., “Eﬀects of Water Presence on Low Temperature Phenomenon in PICA”, AIAA Aerospace Sciences Meeting, No. AIAA-2015-2505, Dallas, TX, 22-26 June 2015, Submit- ted.

106