Mass and Heat Transfer in the Problem of a Finite Thickness
Ablating Piston
L.M. de Socio, L. Marino
Dipartimento di Meccanica e Aeronautica
Via Eudossiana 18,1-00184 Roma
Universita "La Sapienza "
Abstract. The heat and mass transfer at the surface of a finite thickness ablating piston is considered. The piston is either
impulsively put in motion or its speed follows an assigned time law. In both cases the wall reaches an hypersonic speed and
its surface at the gas/solid interface is diffusive. The coupled problem of the temperature distribution in the two phases is
dealt with by solving the parabolic heat conduction equation in the slab by means of an integral boundary layer approach and
by adopting a Direct Simulation MonteCarlo Method for the gas phase. The solid and the gas regions are coupled by proper
boundary conditions. A time scale analysis was necessary to take into account the more rapidly changing gas characteristics
with respect to those of the solid.
INTRODUCTION
In this paper we consider the unsteady one-dimensional problem of a piston of finite thickness which is in motion at
high speed in a rarefied gas.
Since the temperature and the heat and mass transfer at the gas/solid interface are unknown and depend on the
solution for the thermal field in the entire gas/solid domain, the problem is of the conjugate type. Problems of this kind
present increasing interest in recent application as, for instance, in the emerging microtechnology devices (MEMS)
where the tiny dimensions of the solid parts make the solution of coupled situations unavoidable.
Most of the conjugate studies are treated in quasi-continuum flows, where the gas phase is studied by means of
the boundary layer approximation and analytic solutions are obtained [1]. However in the MEMS, cited before, the
characteristic scale of length and the gas pressure conditions are very often such that the gas rarefaction effects cannot
be neglected.
As far as the authors know, the rarefaction influence is taken into account in the literature only by imposing slip
flow and temperature jump conditions to the Navier-Stokes equations [2, 3].
A further example of conjugate problems is represented by the majority of the analytical and numerical solutions
concerning the aerospace re-entry problems where the available results are mainly obtained in the continuum case
although the physical circumstances, as the one treated in this work, would suggest consideration of the proper rarefied
affects.
Turning now to the ablation problem, a large number of references can be found on atmospheric re-entry and we
just cite here the very old analytic similar solution to the ablating problem in [4], the re-entry data in [5], and a most
recent paper [6] where an iterative procedure is proposed for a coupled gas/recessing front situation. In particular, in
[6] a complete re-entry profile is considered, the Navier-Stokes equations are adopted for the gas phase and attention
is paid to the different chemical fractions produced at the body surface. Here the analysis is focused to the first very
few moments (t = 0 -r- 10~4s) of the ablation process, when the rarefaction effects are most important and when
the molecular dynamics processes best help understanding the physical phenomena. In this respect, the simplest
possible model will be assumed for the molecular kinetics compatible with a realistic enough presentation of the
results. Molecular dissociation and radiation heat transfer will be neglected and the gas is Argon whose properties are
those reported in [7]
As we said, a kinetic approach is considered to the gas domain problem and the analysis is carried out by means of
a Direct Simulation based on a MonteCarlo algorithm. The code was adapted from the one originally reported in [7],
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
505
where the case of a specularly reflecting piston is discussed in the framework of a calculation of a strong shock wave
formation. The results in [7] were assumed as reference data for testing the generalized numerical program.
The temperature distribution in the slab was calculated from the energy equation and the analysis of the coupled
problem of the solid and gaseous phases is obtained introducing in the DSMC method an iterative procedure which
takes into account a variable surface temperature of the diffusive piston. In such a way the evaluation of the state
characteristics of the fluid and of the temperature distribution in the slab corresponds to solving a conjugated moving
boundary problem where the coupling conditions at the interface represent the coupling of the temperature distributions
in the gas and the solid and the conservation of the total energy flux. With reference again to paper [6] a case was run
which provided a favourable comparison of our results with those corresponding to the initial atmospheric impact
times of a re-entering body trajectory.
From a numerical point of view, the procedure adopted here represents a further example of the capabilities of the
DSMC to deal with non standard situations.
ANALYSIS AND RESULTS
Let x be the axis along which the piston moves at speed VP, x = 0 at the gas-solid interface, and the gas domain is for
x > 0, whereas the solid phase of the piston is for b < x < 0. At the time t = 0 the solid and the gas phase are both at
temperature T = TQ. The solid temperature is Ts and the surface temperature Ts i can reach with t the ablation value Ta
and a mass flux ma(t) of gaseous ablated material begins to leave the solid phase without passing through the liquid
one. In this case the heat flux q • which crosses the boundary from the gas to the solid corresponds in part to the latent
ablation heat, qa, and in part diffuses through the solid, qs t. Moreover, during the ablation process, the ablation front
moves at a speed x proportional to ma, while the initial thickness of the slab, b, decreases.
The solution of the thermal field in the solid phase, is obtained by considering the classical Fourier law and the
corresponding parabolic heat conduction equation
dTs
d2Ts
=a
~d7
^
(1)
for x < 0 and t > 0, where a is the thermal diffusivity.
Remark. We note that, due to the very short time and small length scales involved, also the hyperbolic law following
the Cattaneo-Vernotte [8] model could be adopted to analyse the thermal wave propagation in the solid phase.
Here the integral approach to solve equation (1) as proposed by Goodman [9] has been used.
In particular the concept of thermal layer 8(t) is introduced and one has that for b < x < 8(t) the solid is at TQ and
no heat is transferred beyond 8. Averaging equation (1) leads to the following form of the energy balance
(2)
where 9 = J05W Ts(x)dx. Taking into account that ^f (<5,/) = 0, ^(O,/) = F(t] and adopting a parabolic profile for
the spatial distribution of the thermal field, the surface temperature reduces to
while the thermal layer depth is given by
1/2
r
p
i
S(t) = V6a< [l/F(t)] \ F(t')dt' \
(4)
from which the speed dS/dt can also be evaluated.
Before describing the coupled problem, reference quantities have to be defined in both phases. Let b* be the
reference length scale in the solid and a/b* the corresponding reference speed. As a consequence, Z?* 2 /a is the
time scale for the slab. As a first step the initial value TQ is assumed as the reference temperature for both the solid and
the gas.
506
The choice of a proper reference length in the solid is not trivial. The temperature distribution in the slab (x G [—6, 0])
is constant for x < 8, and the solution, for 8 < b, is the same as in the case of a semi-infinite slab. Taking 8 as a reference
length might appear a good choice, but this quantity changes with the time. In this paper it was found convenient to
take &* = <5(A£*) as the length scale, where A£* is the time step over which the DSMC procedure samples the gas
flow. Therefore &* corresponds to the initial boundary layer displacement in the solid, from t — 0 to t — A£*, and,
in the cases considered later, where A/* is constant and dd/dt decreases with /, 6* is maximum with respect to the
subsequent variations A<5 = 8(t + A^*) — 8(t). We remark that the reference length in the solids depends upon a time
parameter which is adopted in the solution of the gas phase problem. If we now consider the gas side, the mean free
path lg of the molecules, the sound speed c = ^/jRT^ (with y the ratio of the specific heat coefficients and R the gas
constant) and the ratio lg/c provide meaningful measures of length, velocity and time.
For FpA£* < lg, the number of collisions between the solid surface and the gas molecules, is negligible in the time
step A£* and the gas behaves as if were in an almost free molecular regime. Therefore L = Vpkt* can be assumed as
the gas reference length, the Knudsen number Kn = lg/L, whereas Vp and A£* are, consequently, the velocity and time
references. The Mach number Ma = Vp/c gives a further parameter for the physical description of the problem.
Turning now to the coupling between the phases, the solution must satisfy proper constraints at the interface, where
the energy balance can be expressed [10] as
?g,/ = 4S,/ + Qa
(5)
-J rr,
q • is the total heat flux on the gas side, qsi — — A?^ is the conduction heat flux in the slab, with Ay the thermal
conductivity and qa — hama is the heat ablation flux. hais the ablation enthalpy and ma = psx is the rate of the ablated
mass.
As we said, for the sake of simplicity, the radiation heat was not taken into account and the presence of evaporated
molecules was neglected in the gas side. This last assumption appeared plausible after the ablation rate was calculated.
When the reference quantities are introduced, the expression (5) reduces to the non dimensional form
~\ rj-i
(6)
where all quantities are from now on dimensionless and where two coupling parameters appear
_-Wa*.
_ hap,a/b*
The heat fluxes above were made non-dimensional with respect to pgVP^ , the speed of the ablation front is non
dimensional with respect to a/b*. In all the results reported here, the ablation temperature made dimensionless with
respect to ro, 6a, is 8.2.
A further constraint is satisfied by imposing that the molecules impinging on the solid at an unknown T • are reemitted from the wall at Ts^. Tg^ is the a priori unknown temperature parameter of a Maxwellian distribution at the
diffusive wall.
The coupled problem was solved at each simulation time t by Nt time steps A£*. At the end of each A£*, the
macroscopic state parameters of the gas (by the DSMC) and the thermal field in the solid (via equation (3)) are
obtained simultaneously. An iterative procedure is thus adopted to calculate the surface temperature of the piston Ts i
which satisfies, at each A£*, the interface constraints.
The results here reported were obtained for Np — 104 representative particles, Nc — 103 cells of non uniform width,
and Ns — 104 simulations steps. These numerical parameters were chosen on the basis of a compromise between
computational time and acceptable spatial resolution and statistical scatter. The time required for each run changed
between 1 and 100 hours for Kn between 1 and 10~3, respectively, on a PC with a 1.5 GHz processor.
The time step A£* was chosen once and for all cases after running a meaningful situation where a piston is
impulsively put in motion at Vp = 7000 w/s in Argon with lg — 0.001 m, TQ = 3QQK. As a consequence Kn — 0.1,
We consider two possible situations. The first one (a) corresponds to the impulsively moving piston and the second
one (b) to a piston changing its Mach number from an initial Main to a final Mafin.
507
30
25
20
Q- 15
10
5
0
/u
60
50
40^
30
20 :
10.
n
0.05
0.1
0.15
0.2
0.25
0.3
r\ '
0.4
1 —I — -
c)
^*^r~r~^
0.05
0.35
2 —x— _
3...^...
* * •*-- * - * -« - -r--4H^.-.*---ifcr^
0.1
0.15
0.2
0.25
0.3
0.35
x
FIGURE 1. Density distributions vs x at: (a) / = 2; (b) t = 6; (c) t = 10. Kn = 1 (1); Kn = 0.1 (2); Kn = 0.01 (3). Ma = 20.
FIGURE 2.
Velocity distributions vs x at: (a) t = 2; (b) f = 6; (c) / = 10. Kn = 1 (1); Kn = 0.l (2); ATw = 0.01 (3). Ma = 20.
508
FIGURE 3.
Temperature distributions vs x at: (a) t = 2; (b) t = 6; (c)t=\Q.Kn = l (1); toi = 0.1 (2); ATI = 0.01 (3). Ma = 20.
0.2
0.1
0-0.1
3 — *- _
-0.2
-0.3
-0.4
-0.5
0
0.2
0.4
13
0.6
14
15
16
17
FIGURE 4, Heat flux distributions vs x at: (a) t = 2;(b)t = 6;(c)t=lQ.Kn=l(\);Kn = Q.\ (2); Kn = 0.01 (3). Ma = 20.
509
TABLE 1. Case (a). Temperature jump at various t, for different Kn. IL - 0.05, Ma = 20.
Kn=\
t
AT
t
AT
t
AT
t
AT
2
4
6
8
10
64.86
60.96
48.65
42.65
37.82
2
4
6
8
10
43.86
29.13
22.85
19.35
17.05
2
4
6
8
10
28.08
18.11
14.87
12.26
11.34
2
4
6
8
10
25.08
17.07
13.59
11.35
9.94
Kn = Q.\
Kn = 0.01
Kn = 0.001
TABLE 2. Case (b). Temperature jump and speed of the ablation front at
various t, for different Kn. U{ = 0.005, U2 = 0.03, Main = 6, Ma//71 = 18.
Kn = 1
6
8
10
19.64
20.54
28.79
0
7.45
6.10
Kn = Q.l
6
8
10
Kn = 0.01
6
8
10
7.44
11.72
10.13
0
2.38
2.24
Kn = 0.001
6
8
10
10.66
16.34
14.40
7.19
10.37
9.00
0
4.50
4.00
0
3.00
2.75
Case (a). At Ma = 20, for II j = 0.05, the code was run for Kn = 1,0.1,0.01. In none of these conditions the ablation
temperature was reached. Figures 1-4 and Table 1 show the main results. In particular, Fig.l shows the dimensionless
gas density distributions with x for three different times.
Analogously Fig.s 2 and 3 show the velocity u and temperature T distributions vs x. Higher density cases show that
the changes of the thermofluidodynamic state are more rapid, reach higher values of p, T and u, and more rapidly
decay. Close to the interface the gas temperature is higher than the immediate surroundings but the formation of a
strong shock wave feeds a high temperature region which moves ahead of the piston. Table 1 present the temperature
jump at the interface AT" = T • — 7^ • for Kn = 1,0.1,0.01,0.001 vs t. As one would expect, at the same t, AT decreases
with Kn and, for Kn = const., decreases with t, since p increases, with t, at the interfaces.
Case (b). The piston moves at Main = 6 fort in the interval 0 -r 3, then at Ma = 12 for 3 < / < 6 and at Ma^n = 18.
Around t = 1 the solid wall temperature reaches the ablation value.
In all the cases Tll = 0.005, 1T2 = 0.03. Figures 5-8 and Table 2 show some significant results for Kn —
1,0.1,0.01,0.001. We note that as Kn increases tending to the continuum regime, all curves tend to be coincident
as t increases.
The results show that the temperature jump at the wall firstly decreases with t, then increases after the ablation
temperature is reached at the wall and then decreases again. Table 2 reports some values concerning the temperature
jump and the speed of ablation front at time around the instant when the ablation begins. After Case (a) was run for
air instead Argon, an error of little more than 20% was obtained in comparison with the initial results in [6], in spite
of the fact that dissociation and radiation were neglected.
510
0.4
0.6
0.8
1.2
FIGURE 5. Density distributions vs x at: (a) t = 2; (b) t = 6; (c) t = 10. Kn = 1 (1); Kn = 0.1 (2); Kn = 0.01 (3); Kn = 0.001
(4). 6<Ma< 18.
1.5
0.5
2.5
FIGURE 6. Velocity distributions vs x at: (a) f = 2; (b) f = 6; (c) ^ = 10. Kn = 1 (1); AT» = 0.1 (2); Kn = 0.01 (3); Kn = 0.001
(4). 6<Ma< 18.
511
1.5
0.5
2.5
FIGURE 7. Temperature distributions vs x at: (a) / = 2; (b) t = 6; (c) f = 10. ATn = 1 (1); Kn = 0. 1 (2); toi = 0.01 (3);
(4). 6<Ma< 18.
0.2
0.4
0.6
= 0.001
0.8
1.5
2.5
FIGURE 8. Heat flux distributions vs x at: (a) t = 2; (b) t = 6; (c) t = 10. Kn = 1 (1); Kn = 0.1 (2); Kn = 0.01 (3); Kn = 0.001
(4). 6<Ma< 18.
512
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