[American Institute of Aeronautics and Astronautics 9th AIAA/ASME Joint Thermophysics and Heat...

American Institute of Aeronautics and Astronautics

1

Numerical Simulation of Ablation for Reentry Vehicles

A. Ayasoufi1, R. K. Rahmani2, G. Cheng3, R. Koomullil4, and K. Neroorkar5

Dept. of Mechanical Eng., University of Alabama at Birmingham, 1530 3rd Ave. South, Birmingham, Alabama, 35294-4461

A computer program is developed for numerical simulation of ablation problems, taking into account in-depth pyrolysis; surface recession; non-equilibrium chemistry in the flow of pyrolysis gases through a variable porosity char; and thermal non-equilibrium between the char and the pyrolysis gases. The program provides a general and easy to use interface for coupling with a reacting flow solver and a radiation solver. The program is verified using available analytical solutions for heat conduction in non-receding and receding solids; the program will then be validated using available experimental data.

I. Introduction ANY modern aerospace systems experience temperatures higher than those for which usual materials function properly. Effective cooling of their parts, then, becomes necessary. The thermal protection

systems (TPS) can be designed in a broad range, containing active and passive cooling systems. Optimum selection of the TPS strongly depends on the application. For examples of the cooling systems, used in aerospace industries, accompanied by their inherent features, see [1]. It is common to categorize TPS in two broad categories of non-ablating and ablating systems. Non-ablating TPS are used on reusable hypersonic vehicles, on the leeward portions of hypersonic vehicles, and wherever the heating rate or duration of heating is insufficient for providing enough phase or chemical change for thermal protection of the body [2]. Another system, effective especially for reentry applications, works based on the usage of ablative heat shields. In such systems, cooling is provided by a material that undergoes phase changes or chemical reactions accompanied by removal of the product of the change. A major design criterion for reentry vehicles is the weight factor. Usually, the structural weight of the heat shield is significantly larger than the payload weight, e.g. by up to a factor of four for planetary probes [2]. By making the vehicle lighter, more space will be available to carry fuel (for longer distance travels), as well as scientific equipments. This goal is achieved by replacing heavy metallic structures by light weight modern materials. The thermal protection system, used for reentry applications, therefore, must not significantly reduce the advantages gained by usage of those light weight materials employed in the structures. The ablative thermal protection materials are usually categorized, (see [2] for example), as non-charring (no chemical reactions, such as carbon-carbon and silica) and charring ablators (such as phenolics, plastic resins, and structural ceramic ablators). The most common ablative TPS used in reentry applications involve charring ablators. Figure 1 shows schematic of a charring ablative heat shield.

1 Postdoctoral Researcher; 2 Postdoctoral Researcher; 3 Assistant Professor, AIAA Senior Member; 4 Assistant Professor, AIAA Member; 5 Research Assistant.

M

9th AIAA/ASME Joint Thermophysics and Heat Transfer Conference5 - 8 June 2006, San Francisco, California

AIAA 2006-2908

Copyright © 2006 by Anahita Ayasoufi. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


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A number of physical phenomena are involved in the ablation process. Since a planetary entry object is required to sufficiently reduce the entry velocities, it must be a high-drag vehicle, usually a blunt body. A detached bow shock, therefore, appears upstream of the body and causes a significant temperature increase near the vehicle surface. A boundary layer forms close to the body and interacts with the shock. Viscous dissipation in the boundary layer causes the wall temperature to rise. The heat is transferred to the shield by convection and radiation from excited particles in the flow. For explanation of these mechanisms of heat transfer to the surface, see [3]. Heat is then conducted from the surface through the solid. Once the virgin layer is heated enough, the ablative material starts changing state. Depending on the material, this can happen by a combination of melting, sublimation, and chemical reactions. Specifically, for charring ablative materials, decomposition (pyrolysis) reactions start and generate gaseous products plus a porous residue (char). A description of the pyrolysis mechanism can be found in [4].

Pyrolysis gas pressure then builds up against the virgin material and forces the gas to flow through the char into the boundary layer. As the gas mixture, flows through the porous char, it can go through further chemical reactions, such as cracking to smaller molecules and carbon generating reactions. The generated carbon is then deposited on the char, reducing its porosity. The gases can further interact with carbon from the char and consume it, producing another set of gaseous products. Change of the porosity of the char, in turn affects its conductivity and therefore the heat flow. The pyrolysis gases, then flow to the surface and are injected into the boundary layer. This process has the favorable effect of blocking, to some degree, the convective transfer of the heat from the free stream to the body. Moreover, the composition of the ablation products also affects the magnitude of radiative heating and its spectral distribution [3].

Further reactions can occur involving the outer surface of the char. The carbon from the char can be oxidized by the oxygen in the surrounding fluid. Moreover, the mechanical erosion can cause parts of the shield to be removed. Recession of the surface, not only, affects the heat transfer to the substrate, but also, can cause changes in the shape of the vehicle, which in turn affects the surrounding flow filed. In addition to the above phenomena, a melt layer may also form and affect the heat transfer.

Understanding the above physical phenomena is necessary for development of appropriate models in order to simulate the ablation process. Complicated physical phenomena involved in the ablation, makes the design of thermal protections systems, a difficult process. The design process deals with many scientific fields, such as materials, solid mechanics, aerothermodynamics, physical chemistry, chemical dynamics, and heat and mass transfer. Moreover, many factors (such as determination of the material properties in the actual entry conditions) introduce uncertainties.

Inclusion of the various physical phenomena in the ablation models, was performed, step by step, over a period of time starting with one-dimensional planar modeling of ablation by Landau [6], in 1950, and is still continuing. A number of early theoretical studies on the stagnation point ablation were referenced in [7]; those studies were mainly concerned with melting surfaces in which a layer of liquid flows over the surface. A simplified analysis of the stagnation point heat transfer was reported in [7]; this study assumed that the material changes state, from the solid, directly to the gaseous state.

The ability of formulating accurate mathematical models of ablation depends specially on the understanding of the phenomena occurring during gasification in the solid. As mentioned in [8], there are many gasification approaches mentioned in the literature: The simplest approach assumes that the solid

Substrate

Virgin Layer

Pyrolysis Zone

Char

x

Heated surface

Substrate

Virgin Layer

Pyrolysis Zone

Char

x

Heated surface

Figure 1. Schematic of an ablative heat shield.


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decomposes to gases directly at a critical temperature. The critical temperature is often assumed to be a fundamental material property which is invariant to changes in heating rate.

The second approach involves incorporating a kinetic mechanism for the degradation process of the plastic, often inferred from thermogravimetric analyses, which allows the solid to decompose over a characteristic temperature range [8]. This involves the use of one or more rate equations to model the rate of change in mass of a small sample as a function of remaining mass (or some mass-fraction scalar) and temperature. A representative selection of papers using this approach is to be found in [9.10.11.12] and [4].

Analytical approximate methods for ablation simulation were introduced by [13.14.15]. The most prominent of these are the integral methods.

Despite the extensive work mentioned above, a general ablation problem is still too complicated to be treated analytically. Accuracy of the approximate methods is limited. Further, experimental methods are not easily applicable to ablation problems, especially for reentry applications, due to the following two reasons:

• Flight tests are prohibitively expensive, • It is very difficult to simulate the actual reentry conditions in ground tests.

The most general treatment of ablation problems, therefore, remains in the realm of numerical solution of the governing PDE.

In 1965, a one-dimensional transient model for the study of a charring ablative TPS was presented in [16]. The computer program was called STAB II. This research took into account temperature dependent materials properties as well as regression of the ablating surface. In 1968, an improved analysis of the transient response of a thermally decomposing material was presented in [17]. The computer program was called the CMA (Charring Material Ablation) program. The problem was modeled in quasi one-dimensional form, in the sense that, the heat flow was considered one-dimensional in-depth but the cross sectional area (perpendicular to the heat flux) was allowed to vary arbitrarily with depth. Further, the program was coupled with a boundary layer solver. It was observed that, smooth solutions for gas generation rate could not be obtained easily in the fast decomposing materials. It is worthwhile to mention that, later in 2004, the effect of the thermal expansion was also added to the CMA program [18]. Parallel to the above work, [19] provided an improved analysis of the one-dimensional transient response of charring ablators. This work incorporated surface removal by oxidation and usage of a moving coordinate system. Listed in [20] are a number of references containing further early attempts of modeling surface reactions as well as those partially accounting for pyrolysis gas reactions.

By this time, the interest had started to be directed towards modeling of the chemical processes occurring within the char layer. One approach is based on using an effective specific heat, for the pyrolysis gases, which is adjusted to account for endothermic and exothermic processes. An example of a work that used this method can be found in [21]. This approach, however, relies strongly on the availability of experimental data. Another approach considers that the chemical reactions occurring in the char layer, i.e. cracking of the gases and gas-char interactions, can be taken into account using frozen, chemical equilibrium, or finite rate models. Definitions are given for each model in [22]. Frozen chemistry models assume that no chemical reaction is occurring within the char layer. This assumption disregards the heat absorbed by the endothermic reaction and could lead to underestimation of the performance of the ablator. However, if the residence time of the gases inside the char is small and temperature is such that the system can not provide enough activation energy for the reactions to start and progress, the frozen chemistry model can lead to good approximations. Chemical equilibrium models, on the other hand, assume that the chemical reactions are occurring infinitely fast and, therefore, this approximation gives the maximum amount of heat that can be absorbed by the predominantly endothermic reactions. This assumption could lead to overestimation of the performance of the ablator. However, if the residence time of the gases inside the char is large and temperature is such that the system provides abundant activation energy for the reactions to reach the equilibrium state, the chemical equilibrium would also lead to good approximations.

An experimental study ([23]) confirmed that there are temperature and flow rate ranges for which either chemical equilibrium or frozen chemistry models work fairly well. However, the study also showed that, there is a transition zone between the frozen and equilibrium conditions, where the non-equilibrium chemistry is needed for obtaining accurate estimations.

Experimental work by Clark, ([23.24]), also demonstrated that pyrolysis gases are not always in thermal equilibrium with the char. In fact, the temperature of the gas does not equilibrate with that of the char until the exit surface.


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Parallel to the above mentioned work, appeared [20] that presented a study of the one-dimensional reacting flow of pyrolysis products through the char layer of a low-density nylon phenolic resin charring ablator, using non-equilibrium chemistry model. The study demonstrated the shortcomings of frozen and equilibrium models in prediction of the behavior of the ablator.

The work that followed, ([25]), presented an analysis of a one-dimensional ablative thermal protection system undergoing stagnation ablation. This work was distinguished from the previous literature in that the pyrolysis gases were not assumed to be in thermal equilibrium with the char.

Numerical simulation of non-equilibrium chemistry is complicated due to the stiffness inherent in the source terms generated by the finite-rate chemistry; another complicating factor is the large difference between the time scales of the flow and the reactions. Methods for treating the above mentioned problems presented in [26], for CFD simulation of gas/liquid injectors used in rocket engine design.

Multidimensional ablation simulations were undertaken starting in 1970. For charring ablators, [27], referenced a number of test data suggesting presence of two-dimensional effects involving the lateral flow of pyrolysis gases within the char. They suggested that the two-dimensional phenomenon becomes important in regions of the heat shield that have relatively high curvature, high heating rate, and large surface pressure gradients. Another two-dimensional analysis was reported in [28]. A more recent reference, [29], used a method equivalent to effective heat of ablation method for studying two-dimensional ablation in Cartesian and cylindrical geometries. The research concluded that, unless the ablation is localized or non-continuous, the quasi one-dimensional results are adequately accurate.

Another important part of ablation simulation is the phenomena occurring at the ablating front, due to interaction of the ablator surface and the surrounding fluid. As mentioned in [3], there are two categories of methods for estimation of the convective heating:

1) Usage of the correlations and approximate methods available in the literature. 2) Coupling, fully, with continuum based Navier-Stokes computation simulations, e.g., program

LAURA (Langley Aerothermodynamic Upwind Relaxation Algorithm), which is a three-dimensional Navier-Stokes code for simulation of hypersonic non-equilibrium flow over blunt bodies [30.31.32].

Since 1992 work has been done on coupling of the material response with solvers for viscous shock layer, parabolized Navier-Stokes, and full Navier-Stokes sets. An early attempt for coupling the Navier-Stokes calculations to the material thermal response was published in [34]. In this effort, axisymmetric geometries were considered and the recession of the surface was allowed. However, the ablation product was considered a single species, which was not allowed to react with the air flow. Another coupling of an axisymmetric Navier-Stokes program with a one-dimensional CMA program was presented in [35]. In this program, the ablation products were allowed to react with a non-equilibrium air flow, but the surface was not allowed to recess. In [34], a similar approach was used, combined with coupling to a flowfield radiation program. A similar effort, but also allowing the receding of the surface can be found in [36]. More recently, [37] presented coupling of a CMA program with a two-dimensional flow solver. They took into account thermochemical non-equilibrium in the boundary layer (but not in the char layer), and turbulence produced due to injection of the pyrolysis gases into the boundary layer.

In 1999, Park ([38]) summarized past attempts for reproducing flight data in the entry flights. The data, e.g., the flight data obtained during the entry of the Galileo probe into the Jovian atmosphere, indicated substantial differences, in the char thickness, compared to pre-flight calculations. A conclusion is drawn, therefore, that despite huge body of research accomplished to date, the design methodology for ablative heat shields still demands improvements. Although the uncertainties inherent in the problem, such as determining material properties in the actual flight conditions, may render accurate and robust predictions distant, an improvement in the accuracy of the available analyses would be of great value. The following contains two lines of possible improvements that may be suggested based on the historical background.

1) The literature survey above sections has revealed that different models have employed various approximation/simplification to account for different sets of physical phenomena and ignoring the others. Although each model presents arguments on the order of importance of the included and ignored phenomena, different reports show disagreement on this matter. To the best of our knowledge, the phenomena, for which the importance is disagreed upon, e.g. thermal non-equilibrium between the pyrolysis gases and the char, are not incorporated in their entirety in either a single research publication or in an accompanying computer program. A new model incorporating an increased number of physical phenomena, can be useful in both improving


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accuracy of the existing predictions, and providing a numerical means of studying the order of importance of the incorporated phenomena.

2) Although at the present time there exists highly effective methods for individual numerical modeling of phenomena, such as non-equilibrium chemistry, advanced numerical schemes are not generally used in modeling such phenomena coupled within a general ablation problem. Employing these methods could also improve the accuracy and efficiency of the predictions.

This paper contains the report of the preliminary phase of a research attempting to improve the accuracy of ablation predictions, by following the above mentioned guidelines. A computer program is designed for prediction of the transient response of an ablative heat shield. The capabilities of the program include: 1) 1-D and quasi 1-D models, 2) variable porosity for the char, 3) temperature-dependent thermodynamic properties of char and pyrolysis gases, 4) finite-rate chemistry for modeling reactions of the pyrolysis products, 5) thermal non-equilibrium between the char and the pyrolysis gas mixture, 6) surface recession, 7) in-depth pyrolysis, and 8) coupling to a flow solver through the ablation front boundary condition.

II. Analysis For a compressible mixture of ideal pyrolysis gases through a variable porosity char, there are 2 ng + 5

unknowns where ng denotes the total number of gaseous species in the mixture. These unknown are: ng mass fractions of the species, one velocity of the gaseous mixture, two temperatures (one for the char and one for the gaseous mixture), ng reaction rates, one pressure, and one char porosity. There are also 2 ng + 5 equations: ng species continuity equations, one momentum equation, two energy equations, ng rate equations, one state equation, and one porosity equation.

As mentioned in [20] and [25], the general rate equation for each species can be written as

( )∑ ∏∏= =

′

=

′

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−−==

rip

jk

rj

k

m

j

m

k

ppkrj

m

k

rrkfjijij

ki AkAkrp

dtAd

1 1

)(

1

)(

)()(

]~[]~[]~[

ω (1)

where iω represents the molar rate of production of species i, measured in sm

mol3 ; ijp denotes the

stoichiometric coefficient of the species i as a product of reaction j; ijr denotes the stoichiometric

coefficient of the species i as a reactant of reaction j; rim is the total number of reactions involving species

i, (for example, for i = 3 in the above Table, (i.e. H2), we have 8=rim ); )(jm represents the total number

of species whose concentrations, i.e. ]~[ )(kA , appear in the corresponding rate law; superscripts (p) and (r)

refer to products and reactants of reaction j, respectively; kp′ and kr′ are orders of reaction with respect to

species )(~kA , in the reverse and forward reactions, respectively.

Both [20] and [25] considered the pyrolysis zone to be an infinitesimally thin plane. The decomposition (pyrolysis) reactions, therefore, are not considered in the char zone. For the case of in-depth pyrolysis, on the other hand, decomposition reactions must also be considered. However, as mentioned in [17], the decomposing constituents are usually selected to match a particular thermogravimetric analysis (TGA) data set, often without being related to the actual decomposing molecules. The constituents are assumed to decompose in a manner independent of one another. It is known, (see [17] and [4]), that the rate equation for these reactions can be modeled realistically using an Arrhenius type expression as follows

im

V

CiV

ii

i

TRE

Bt ⎟

⎟⎠

⎞⎜⎜⎝

⎛ −−−=

∂

∂

ρρρ

ρρ

)(exp (2)


6

where t denotes time; iρ is density of the pyrolyzing constituent i; subscripts V and C refer to the original

(virgin) and final (char) densities of a pyrolyzing component, respectively; iB is called the pre-exponential

factor for the decomposition reaction of constituent i, RE i relates to the activation energy of the

decomposition reaction of constituent i (selected to match the corresponding TGA data), and im is the

reaction order of the decomposition reaction of constituent i. Note that, as mentioned previously, iB ,

RE i , and im are selected to match the corresponding TGA data.

In many cases, it has been shown that it is sufficient to use three different decomposing constituents, [17]. It is also mentioned (see [17]) that, most of the times, even one pyrolyzing constituent provides good simulations of the corresponding TGA data.

The governing equations can be derived for quasi one-dimensional compressible flow of gaseous species through a porous char layer with variable porosity. Since the outer surface of the heat shield, recedes due to ablation, the governing equations derived for a fixed coordinate system go through a change of variable in order to take the surface regression into account. Figure 2 depicts this variable change.

The equation for mass conservation of each gaseous species, is

( ) ( ) ( ) ( )Az

sAAuz

At idiiii ηρηωωηρηρ

∂∂

+++∂∂

−=∂∂

&

(3)

where the porosity, η , is defined as the void fraction of the char layer; u is the velocity of the gas mixture, estimated from the mass flux which is determined by integrating the decomposition generation rate. The subscript i refers to each gaseous species, and A denotes the total cross sectional area. Further, iω and diω represent mass generation terms due to the gas mixture reactions and the decomposition reactions, respectively.

For the solid species, i.e. carbon, it is assumed that, carbon is deposited where generated. In other words, for the purpose of simplification, transport of carbon in the form of solid particles floating in the gas flow is neglected. The species continuity equation for solid species, therefore, can be obtained from the above equation by requiring u = 0. Further, since the decomposition reactions are assumed to produce only gaseous species, diω will also be equal to zero for the solid. Moreover, if the density of a solid species is considered constant, the final equation becomes

( ) ( )Az

sA

At i

i ηρηω

η∂∂

+=∂∂

& (4)

where i refers to solid species involved. Note that the above equation is in fact our porosity equation.

Neglecting energy transfer due to diffusion, p-v work, and radiation, the energy equation for the pyrolysis gas mixture becomes

x

zs (t)

Original position of the ablative surface

Position of the ablative surface at time t

x

zs (t)

Original position of the ablative surface

Position of the ablative surface at time t

Figure 2. Change of variable for surface recession.


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( ) ( )

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

++−

−−+∂∂

+

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂

∂∂

+⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

−=⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

∑= 2

22

2

1carbon

ousheterogenecarbon

carbonshomogeneou

carbon

22

ueAz

shAAh

AhTTAHAuz

zT

Akz

ueAuz

ueAt

gg

n

iidi

gsAxx

gggggg

g

ηρωηηω

ηωητ

ηηρηρ

&

(5)

where ge denotes the internal energy of the gaseous mixture; subscripts s and g refer to solid and gas, respectively; and, as explained in [39] and used by [25], HA, i.e. the volumetric coefficient for convective energy transfer between the gases and the char, due to thermal non-equilibrium between them, is calculated from

( )β

ρ∑==

gn

iipi

A

cuH 1 (6)

where the specific heat ipc is measured in Kkg

J.

, and β , which has units of length and is called the

heat transfer length of the solid, is calculated from

)1(6Pr

ηηβ−

=sA

d (7)

where Pr is the Prandtl number; sA is a dimensionless constant that is characteristic of the solid; d is a characteristic length in the porous solid.

Note that the fifth and sixth terms in Eq. (5) relate to the transport of energy due to mass transfer. The former of those two terms is the lost enthalpy due to lost carbon atoms because of carbon deposition reactions. The latter takes into account the gained enthalpy due to extraction of carbon atoms from the char through carbon consumption reactions. The last term, takes into account the energy transport due to mass transfer from the char because of the decomposition reactions. This term is in fact a source term produced due to the in-depth pyrolysis. The char energy equation can be written as

[ ] ( )

( )[ ]

[ ]ss

n

iidipd

n

i

nn

jiijiijji

gsAs

sss

hAz

shAAh

hrhpMA

AhAh

TTAHz

TAk

zhA

t

g

t

)1()1(

)1()1(

1

1 1

carbonousheterogene

carboncarbonshomogeneou

carbon

ousheterogeneshomogeneou

ηρωηηω

ωη

ηωηω

ηηρ

−∂∂

+−−Δ−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−

++

−−⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

∂∂

=−∂∂

∑

∑ ∑

=

=

+

=

&

(8)


8

where h is the specific enthalpy (measured in J/kg); tn is the total number of species (gaseous species plus

solid carbon); rtn is the total number of reactions, and phΔ is the heat of pyrolysis of the plastic. Note that, the last two terms takes into account the heat absorbed by the decomposition reactions (in-depth pyrolysis), and the enthalpy lost due to mass transfer from the char to the pyrolysis gases, respectively.

III. Numerical approach The grid, with the exception of the last (innermost) ablating node, consists of equally spaced grid nodes

along the z axis. The last ablating node has a variable thickness that changes due to surface recession. The last node is eliminated when its thickness falls below a predefined threshold, see [17] for more details on the grid system.

Mass conservation equations are treated, numerically, using an upwind differencing method, explicit with respect to temperature. As suggested in [17], the energy equations, however, are treated implicitly with respect to temperature. For treatment of the decomposition, more stable solutions are reported in [17], to be achievable if, instead of Eq. (2), its integrated form (with respect to time) is used, see [17] for the integrated form and derivation details. The integrated form can then be used for estimating the value of

ti

∂

∂ ρ, for the purpose of computing the decaying density of the solid. Further, the mass rate of generation

of gaseous pyrolysis products, due to decomposition of the solid, can be calculated as follows

∑ ∂

∂−=

i

id t

ρω (9)

where subscript d denotes decomposition. The total value of dω is then partitioned between the gaseous species known to exist as products of the pyrolysis process of interest, based on the known initial composition extracted from experiments. This partitioning procedure, then, provides mass rates of generation due to decomposition reactions, ( diω ), for each of the gaseous species. Moreover, integration of the above rate can also be used in estimation of the mass flux of the gaseous products, which in turn, provides a means for calculation of the velocity of the mixture.

As mentioned in Analysis section, for the finite-rate chemistry, the productions rates of the involved species are calculated from Eq. (2), which are the species continuity equations, neglecting convection and diffusion. For treatment of the stiffness inherent in these equation, an implicit method, (implicit with respect to species concentrations), was successfully used in [26] for solving fluid flow with chemical reactions, see [26] for derivation details.

IV. Verification Cases Case1. Heat conduction in a finite slab, constant temperature boundary conditions: This case was used

for verification of the heat conduction features in the program. A finite slab with unit thermal conductivity, initially at a uniform temperature of T = 2000 K, is subject to a step change in the back temperature at time t = 0. Front and back temperatures are held at T = 2000 K and T = 3000 K, respectively. Figure 4 shows evolution of the transient solutions for temperature and comparison of the steady state profile to the analytical solution.


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Case2. Heat conduction in a finite slab, insulated boundary condition: This test case is used for verifying the front boundary condition, which is programmed using an energy balance (see [17] for details). The case features a finite slab with unit thermal conductivity, initially at a uniform temperature of T = 2000 K. At time t = 0, the back face is subject to a step change in the temperature. The back temperatures is raised to T = 3000 K and held at that temperature while the front face is insulated. Figure 3 shows the transient temperature profile for t = 1 s. The expected vanishing slope, evident from this figure, confirms the zero heat flux applied at the front surface.

Case3. Decomposition verification: This case was designed for testing the modules calculating decomposition, as well as treatment of the continuity equation. Consider a passage of constant porosity and uniform cross-sectional area, consisting of a material that decomposes to only one gaseous species. Assumption of a fictitious constant temperature process (T = 3000 K) leads to a problem that can be treated analytically. As long as the decomposing material remains, the analytical solution for the decaying density of the solid, which is derived by direct integration of Eq. (2) for mi = 1, becomes

Ci tCC ρρ +−= )(exp 21

Figure 4. Transient and steady state temperature profiles for case 1.

Figure 3. Transient temperature profile for case 2.

Table 1. Reaction specifications for case 4.

General reaction: ∑∑== ←

→ pr n

iii

r

fn

iii Bb

k

k

Aa11

, where ( )TEAk /~exp −=)

Reaction index # Reaction Rate law

A)

(Frequency

factor)

E~ (Activation energy, K)

1 2624 21

21 HHCCH +→ ][ 1Ak f 14106.7 × 410775.4 ×

2 24262 HHCHC +→ ][ 1Ak f 151014.3 × 410019.3 ×

3 22242 HHCHC +→ ][ 1Ak f 81075.2 × 510157.1 ×

4 222 2 HCHC +→ 21][Ak f 101014.2 × 410009.2 ×


10

where )(exp2 TRE

BC ii

−= and C1 depends on the

initial density of the solid. Figure 6 shows the solid density decay with time compared to the analytical solution.

Case4. Effect of time step on the finite rate chemistry results: This case was designed for testing the effect of time step on the calculation of chemical generation terms. Consider a five-species mixture of ideal gases containing Methane (CH4), Ethane (C2H6) Hydrogen (H2), Ethylene (C2H4), and Acetylene (C2H2), going through four reactions specified in Table 1. The reaction specifications are extracted from a broader test case mentioned in [25]. Figure 5 shows the generation rate of one of the species calculated using different time steps. As can be seen, for this set of reactions, for the result to be independent of time step, step sizes of the order of 10-4 should be used. For reaction sets featuring more sever stiffness, however, the required step size may reduce to much smaller values.

Case5. Semi-infinite receding solid: This test case features a semi-infinite receding solid with unit diffusivity and constant properties, which is initially at a uniform temperature of T0 = 2000 K. At time t = 0, the front boundary is subject to a step change in temperature and a constant, non-vanishing surface recession rate. The front temperature is raised to Te = 2010 K and held constant. The surface recession rate is set to s. = 0.2 m/s. This case can be used to verify surface recession feature in combination with heat conduction. An analytical similarity solution is available for this case in quasi-steady form, see [17]. This problem is numerically solved using the developed program. Figure 7 depicts the comparison of the numerical and analytical temperature profiles for this case. The agreement is excellent.

Case6. Heat convection verification: This test case is one of the cases that are both numerically and experimentally investigated in [20]. The case features the flow of a pyrolysis mixture through a nylon phenolic resin char having a constant porosity of 0.8. The chemistry is assumed frozen and pyrolysis is considered planar. The mixture composition is given in Table 2. Front and back temperatures are constant and set equal to 1090 K and 533 K, respectively. Material properties are temperature dependent and are available in [20]. Figure 8 shows the comparison between the results of the present code with those from [20], for the temperature profile. Good agreement is achieved.

Table 2. Pyrolysis mixture composition for case 6.

Species Mole % Phenol 6.2 Water 48.9

Carbon Dioxide 1.1 Carbon Monoxide 3.7

Methane 6.7 Hydrogen 33.4

Totals 100.0

Figure 6. Decaying density of the solid for case 3.

Figure 5. Time step effect for case 4.


11

V. Conclusion Various features of the computer program developed for numerical simulation of ablation problems were verified using available analytical and numerical results available in the literature. Hitherto, verification and validation process of the developed program has confirmed correctness and accuracy of the implementation of the features of interest.

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Figure 7. Temperature profile for test case 5. Figure 8. Temperature profile for test case 6.


12

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