International Journal of Heat and Mass Transfer 101 (2016) 267–273

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier.com/locate/ijhmt

Experimental measurements of the permeability of fibrous carbon

at high-temperature
Francesco Panerai a, Jason D. White b, Thomas J. Cochell c, Olivia M. Schroeder a, Nagi N. Mansour d,
Michael J. Wright e, Alexandre Martin a,⇑
a
Department of Mechanical Engineering, University of Kentucky, Lexington, KY 40506, USA
b
Advanced Technology & Systems Division, SRI International, Menlo Park, CA 94025, USA
c
Department of Chemical and Materials Engineering, University of Kentucky, Lexington, KY 40506, USA
d
NASA Advanced Supercomputing Division, NASA Ames Research Center, Moffett Field, CA 94035, USA
e
Entry Systems and Technology Division, NASA Ames Research Center, Moffett Field, CA 94035, USA

a r t i c l e i n f o a b s t r a c t

Article history: A series of experiments was performed to obtain permeability data on FiberFormÒ, a commercial carbon
Received 15 March 2016 preform used for manufacturing thermal protection systems. A porous sample was placed in a quartz
Received in revised form 3 May 2016 flow-tube heated by an isothermal furnace. The setup was instrumented to measure mass flow through
Accepted 5 May 2016
and pressure drop across the sample. The intrinsic permeability and the Klinkenberg correction, which
Available online 30 May 2016
accounts for rarefied effects, were computed from the experimental data. The role of the gas temperature
and pressure on the effective permeability is shown, and it is demonstrated that with proper data reduc-
Keywords:
tion, the intrinsic permeability is strictly a function of the micro-structure of the material. A function for
Porous media
Permeability
the effective permeability of FiberForm, dependent on temperature, pressure, pore geometry, and type of
Thermal protection systems gas is proposed. The intrinsic permeability was evaluated at K 0 ¼ 5:57  1011 m2, with a Klinkenberg
parameter of 8c=dp ¼ 2:51  105 m1 and a reference porosity of /y ¼ 0:87.
Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction carbon fibers (10 lm in diameter) and pores of 50 lm in diam-

eter [6]. The pores occupy nearly a 90% fraction of the volume of
The entry process into a planetary atmosphere requires space- the material, providing it with excellent insulation properties.
craft to be equipped with a thermal protection system (TPS). The Because of their high porosity, gases can easily flow within the
TPS protects the spacecraft from the high enthalpy and thermo- ablative materials. For example, pyrolysis gases produced by
chemical conditions of entry, during which the hypersonic flow decomposition of the phenolic resin travels through the charred
surrounding the vehicle generates strong aerothermal heating. structure – potentially reacting with the fibers – before exiting
An ablator is usually used as a TPS material for the harshest entry the material. Likewise, reactants from the boundary layer can enter
conditions due to the chemical and physical phenomena that take the material microstructure and flow within the pores. This gas
place when high heat fluxes are experienced. The ablator materials transport has a significant effect on the overall material response
significantly reduce the heat to the inner parts of the vehicle, pro- [7–9].
tecting the payload [1]. In recent years the focus has veered toward The flow behavior through a porous structure is characterized
a new class of low-density carbon/resin ablators, the most success- by the permeability, as it dominates the momentum transport
ful of which is NASA’s own phenolic-impregnated carbon ablator within the medium. Permeability is therefore a key material prop-
(PICA) [2], used in Earth return and Mars exploration missions erty when modeling porous media flow.
[3,4]. When the mean free path k of the gas molecules approaches the
PICA uses FiberFormÒ (Fiber Materials, Inc.) [5], a rigid carbon dimensions of the material pores, the gas flow within the material
fiber composite, as a substrate. As shown by the scanning electron is considered transitional between the continuum and Knudsen
micrograph in Fig. 1, its micro-structure is characterized by thin regimes. In this regime, slip effects become important.
A method for measuring the permeability of porous refractory
insulators was proposed by Marschall and Milos [10,11]
⇑ Corresponding author. and applied to various materials, such as silica-based tiles, PICA
E-mail address: Alexandre.Martin@uky.edu (A. Martin).

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.05.016
0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
268 F. Panerai et al. / International Journal of Heat and Mass Transfer 101 (2016) 267–273

Nomenclature

Symbols Greek symbols

R universal gas constant [J/(K mol)] k mean free path [m]
df frequency resolution [s1] l viscosity [kg/(m s)]
DP pressure difference [Pa] / porosity [m3/m3]
m_ mass flow rate [kg/s] q density [kg/m3]
A area of the flow-tube [m2]
b permeability slip parameter [Pa] Superscripts
c proportionality constant y values scaled at / ¼ 0:87
d diameter [m] ⁄ values scaled at T ¼ 298 K
F resistive force [N]
K permeability [m2] Subscripts
L length of the sample [m] 0 intrinsic
M molar mass [kg/mol] 1 port P1
P pressure [Pa] 2 port P2
T temperature [K] avg average across sample
t time [s] eff effective
u gas velocity [m/s] f furnace
x spatial coordinates [m] fib fiber
p pore
s surface

the tube by interference fitting, and positioned in the center of

the furnace, using a plastic dowel rod. As discussed in the related
literature [10,11], the axial geometry of the porous material plays
a major role in the permeability.
Because of its manufacturing method, FiberForm is an orthotro-
pic material. More specifically, it is transverse isotropic, since
most of the fibers are oriented within ±15° of the compression
plane. The direction perpendicular to this plane is defined as
‘‘Through-Thickness” (TT) and that parallel as ‘‘In-Plane” (IP). The
bulk of the experiments described here were performed on sam-
ples machined with a TT orientation, in which the carbon fibers
are preferentially aligned perpendicular to the gas flow direction.
One experiment was also performed with a sample oriented in
the IP direction. A dedicated mass flow controller (Aalborg Model:
UFC 8160) calibrated to nitrogen trifluoride was controlled by a
Tylan RO-28 Readout/Control Box to feed the argon gas at fixed
flow rates ranging between 10 and 100 sccm. The system was evac-
uated by means of an Alcatel R301B Roots pump using FomblinÒ oil
Fig. 1. Scanning electron micrograph of FiberForm. and backed by an Alcatel BF ADP 81 dry pump. The pumping man-
ifold was outfitted with a copper mesh to collect particulates that
might be emitted during the experiment. The outlet of the flow-
(in virgin and charred from), ceramics, and to a lesser extent, Fiber-
tube was connected to the vacuum system through a manual
Form. Data on FiberForm in Ref. [10] were obtained up to 300 K on
bellows-angle valve, fully opened during the experiments.
an older, less dense version of the material with large observed
The main tube was equipped with both an upstream (P1) and
variabilities in the samples.
downstream (P2) port from the furnace, which were connected
The experiments documented in the current work provide an
to a manifold of calibrated differential pressure transducers mea-
updated set of FiberForm permeability values at temperatures
suring the pressure loss (P 1  P 2 ) across the sample. A separate
ranging from 298 to 1500 K, in inert atmosphere. The data gener-
set of pressure gauges were also used to monitor absolute pressure
ated also constitute a baseline for the numerical rebuilding of
conditions.
effective reactivity data from experiments on the high-
A valving manifold was used to control gas flows and normalize
temperature decomposition of FiberForm [12].
pressure in the system when starting experiment operations. One
set of these valves was used as a by-pass to prevent the formation
2. Experiment of strong pressure gradients across the sample during evacuation
or venting operations that could potentially move the sample from
A high-temperature flow-tube setup (Fig. 2) was assembled to the desired initial position. Thermocouple (TC) sensors were
perform gas/material interaction experiments on porous samples. installed at different strategic positions along the tube as depicted
The system consisted of a 129.5 cm long, 22 mm inner diameter in Fig. 2. Two Type-K thermocouples were used to measure the
quartz tube positioned inside of an open-ended furnace providing temperature T 1 and T 2 at the pressure ports P1 and P2, respec-
temperatures up to 1675 K by means of a radiative ceramic ele- tively, and two other Type-K TCs monitored the temperature T in
ment. The cylindrical plug samples (FiberForm) were inserted in and T out at the inlet and outlet of the furnace. A Type-S and
 F. Panerai et al. / International Journal of Heat and Mass Transfer 101 (2016) 267–273 269

Fig. 2. Schematic of flow-tube setup at SRI International.


Type-K TC were used to measure the temperature of the sample T s k
K eff ¼ K 0 1 þ 8c ð1Þ
and the temperature of the furnace T f , respectively, at the position dp
of the carbon plug. A two-color pyrometer (Mikron M90-R2)
where K 0 is the value of the permeability in the limit of continuum
pointed at the upstream surface of the sample was also used as a
flow regime, and c is a proportional constant. Accounting for pres-
redundant temperature measurement for the tests at temperature
sure and temperature, the mean free path can be expressed as:
above 1200 K. All of these temperature measurements agreed
rffiffiffiffiffiffiffiffiffiffiffiffi
within 10 K. lðTÞ p RT
Pressure and temperature measurements were acquired at 4 s k¼ ð2Þ
P 2 M
intervals by a dedicated acquisition card (NI Model USB-6210). A
customized LabView interface recorded each parameter directly To simplify the notation, it is convenient to define parameter
to a computer. bðTÞ according to
The protocol for the experiments followed that of Marschall and rffiffiffiffiffiffiffiffiffiffiffiffi
8c p RT
Milos [10]. First, length and mass of the FiberForm samples were bðTÞ ¼ lðTÞ ð3Þ
measured, and sample densities were calculated. The samples were dp 2 M
then inserted into the Quartz tube, the system was evacuated to a thus obtaining the Klinkenberg expression for the effective
base pressure P < 13:33 Pa, and the background temperature was permeability:
stabilized to a steady target while supplying a constant 10 sccm

Ar flow to the system. b
K eff ¼ K 0 1 þ ð4Þ
The permeability of virgin char in an Ar environment was mea- P
sured as follows. Absolute pressure, differential pressure, and tem-
Because the flow-tube configuration produces a well-defined
perature data were collected at a frequency resolution of
and well-characterized flow, it is possible to analyze the flow field
df ¼ 0:25 Hz until a steady state was reached. Gas flow was
using an analytical approach and extract the permeability param-
stopped following the measurement.
eters from the experimental results. By combining Eq. (4) with
The furnace was then cooled, and the FiberForm samples were
the conservation of mass, ideal gas law, the geometry of the sam-
removed by pushing from the backside of the plug toward the
ple, and Darcy’s Law, the following relationship can be derived
pyrometer. Post-testing mass and length measurements were per-
for one dimensional, laminar and isothermal flows:
formed on the samples. Negligible mass loss and length changes
 " 2 # 
were measured compared with pre-testing measurements. PM pD K eff dP
_
m ¼ quA ¼  ð5Þ
RT 4 l dx
3. Permeability in the slip regime
Z Z
_
4lmRT L P2
The permeability of FiberForm was determined by measuring dx ¼ K 0 ðP þ bÞdP ð6Þ
the pressure gradient DP across the sample for a given combination
pD2 M 0 P1

_ and gas mixture including vis-

of temperature T, mass flow rate m,
_
4lmRTL
cosity l and molar mass M. Klinkenberg derived an expression that F¼ ¼ K 0 ðP avg þ bÞ ð7Þ
accounts for non-continuum effects in porous media [13]. This pD2 MDP
equation describes the permeability as a function of the Knudsen In these equations, L and D are length and diameter of the sam-
number Kn ¼ k=dp . Here, k is the mean free path of gas molecules ple, and Pavg ¼ 0:5ðP1 þ P2 Þ is the average pressure in the sample.
and dp is the mean pore diameter of the material, assumed to be As for F, it is a force that results from the material permeability,
the characteristic length of the porous medium. This Klinkenberg and is only dependent on known constants or measured experi-
expression takes the form of an effective permeability mental values.
270 F. Panerai et al. / International Journal of Heat and Mass Transfer 101 (2016) 267–273

For the conditions of the current experiments, the argon flow

within the porous medium is in the rarefied regime. For tempera-
tures between 1000 and 2000 K, and pressures between 1 and
10 kPa, the Knudsen number remains between 0.06 and 2. This
wide variation supports the use of the Klinkenberg equation to cor-
rect the permeability.

4. High-temperature permeability measurements

Since all quantities in the left-hand side of Eq. (7) are known or
measured, K 0 and b can be obtained by a linear least-squares fit of
F ¼ FðPavg Þ. The slope provides the value for K 0 , and depends on the
material micro-structure only. The abscissa at zero ordinate
divided by K 0 provides b and depends on the flow temperature,
the type of gas, as well as on the material micro-structure through
the average pore diameter.
Fig. 3. Measured permeability values in Ar flow for sample TT07, as a function of
An example of permeability measurements in Ar flow, at tem- temperatures.
peratures from 310 to 1320 K, is shown in Fig. 3 for sample
TT07. F is a linear function of P with constant slope. To build a rep-
resentative database for the permeability of FiberForm, multiple
samples were tested at various conditions. The results from these Table 1
tests are presented in Table 1. The full set of measured data is pro- Temperature-dependent permeability data.
vided in Table 4 of the Supplementary Material A.
The dependency of parameter b on the temperature and the
type of gas can be removed by normalizing the data to a standard
condition for the gas, here chosen to be 298 K. Replacing
temperature-dependent terms with an asterisk, Eq. (7) becomes
 
F  ¼ K 0 ðP avg þ b Þ, where b is calculated by normalizing Eq. (3) as
rffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b l p RT 2 M
 ¼  ð8Þ
b l 2 M p RT 
Here, l is the viscosity for Ar at reference temperature T  = 298 K.
The molar mass ratio M  =M is equal to 1, since only Ar is used. Using
the process, the curves of Fig. 3 collapse onto a single curve shown
with circles in Fig. 4.
New curve fits can be calculated for the scaled data, and the
8c=dp term in Eq. (1) can be calculated for each sample. These val-
ues of 8c=dp are listed in Table 2.
Fig. 4 also illustrates the strong transverse isotropic properties
of FiberForm with a comparison to the IP configuration. This geom-
etry allows the gas to flow more easily along the axial direction of
the planar alignment of the fibers, and results into a higher perme-
ability than in the TT direction. Both curves have different slopes,
which results into distinctive intrinsic permeability values K 0 .
The numerical values, presented in Table 2, also show that the rar-
efied term, 8c=dp does not vary with direction, as expected, since it
is only a function of the average pore size.
Table 2 shows a non-negligible scatter in the measured density
of the samples due to the method of fabrication of FiberForm that
generates inhomogeneities, as can be seen in Fig. 1. To further nor-
malize the samples, F  becomes F y using a factor /y =/ that
accounts for the deviation of the density of each sample from the
nominal density.
In this factor, the porosity / is calculated according to: 1
In-Plane orientation: all other samples are Through-Thickness.
q
/¼1 ð9Þ
qC deviations to Darcy’s law at high velocities [14,15]. As was also the
where qC is the density of the fibers. A value of qC ¼ 1400 kg/m3 is case for other classes of porous ablators [10], the scatter is most
calculated by using the average density of the samples pronounced at high mass flow rates, reinforcing the Forchheimer
qy ¼ 183:6 kg/m3 and the average reported open porosity of Fiber- effect hypothesis. Nevertheless, a parametric curve can be fitted
to the data, and a single expression for the permeability, based
Form is /y ¼ 0:869 [5]. The new normalized results using F y are
on the normalization and Eqs. (1) and (2), can be obtained:
plotted in Fig. 5.
rffiffiffiffiffiffiffiffiffiffiffiffi!
Despite this density normalization, some scatter remains, likely 8c lðTÞ p RT /
due to either the non-uniformity of the specific micro-structure of K eff ¼ K 0 1þ ð10Þ
dp P 2 M /y
each sample, or the Forchheimer effects which account for inertial
 F. Panerai et al. / International Journal of Heat and Mass Transfer 101 (2016) 267–273 271

Fig. 4. Normalized permeability data for sample TT07 (Through-Thickness) and Fig. 5. Summary of FiberForm permeability measurements.
sample IP01 (In-Plane) highlighting the transverse orthotropic properties of
FiberForm. Fitting curves are displayed in black and red for Through-Thickness eters listed in Table 1, are also plotted in the same figure. The fit
and In-Plane orientation, respectively. (For interpretation of the references to colour based on Eq. (4) is closer to the experimental values since it was
in this figure legend, the reader is referred to the web version of this article.) generated using these. Outside of the pressure range where mea-
surements were collected, the fit is less accurate but still within
acceptable errors, and the use of Eq. (10) is recommended for mod-
eling the permeability in numerical simulations.
Table 2
Normalized permeability data.
5. Error analysis

The uncertainty associated with the calculation of F depends on

the uncertainty contained in the measured and calculated param-
_ R; T; L; D; M; DPg. If the
eters found in Eq. (7), which are xi ¼ fl; m;
uncertainty given by the variables dxi is small and there is no
covariance between them, the error contained in F can be written
as:
"
 #12
dF XN
@F
2
¼ d2xi ð13Þ
F i¼1
@xi
1
In-Plane orientation: all other samples are Through-Thickness. It should be noted that, since l and T are dependent variables,
the null covariance condition for the Taylor series expansion is
not strictly satisfied; however, it is estimated that considering l
and T as independent variables does not significantly affect the val-
where K 0 ¼ 5:57  1011 m2 , 8c=dp ¼ 2:51  105 m1 and /y ¼ 0:87. ues of the calculated uncertainty. In order to simplify Eq. (13), F can
Eq. (10) can therefore be directly used in Material Response codes be expressed as:
when modeling FiberForm, and is valid at any temperature and
pressure in the range covered in the experiment. Y
N
a
F¼ xi i ð14Þ
The proportionality constant c can be evaluated using the mean i¼1
pore diameter for the FiberForm fibrous structure obtained from
the porosity. Eichhorn [16] gives the following expression for the
average In-Plane pore diameter of a fibrous 2D material:
pffiffiffiffi

p p
dp;IP ¼  1 dfib ð11Þ
2 2lnð1=/Þ
Using an average fiber diameter dfib of 11 lm, a value of
dp;IP  96 lm is estimated. From Ref. [17], In-plane pore diameter
 for cylindrical fibers, can be related using
dp;IP and pore height h,
the following expression:
 pffiffiffiffi
h /
¼ ð12Þ
dp;IP 2

from which h   45lm. Therefore, with dp ¼ h,

 a value of c ¼ 1:6 is
obtained.
Fig. 6 compares the experimental results of the permeability
with the values obtained using Eq. (10) for sample TT07. While
good agreement is shown, there is no perfect match, since Eq.
(10) uses all the dispersed experimental data to generate the curve Fig. 6. Comparison permeability values obtained using Eqs. (10) and (4), and the
fit. The permeability values obtained using Eq. (4), with the param- experimental data.
272 F. Panerai et al. / International Journal of Heat and Mass Transfer 101 (2016) 267–273

Table 3 heterogeneous surface reactions with a hot char. Furthermore, a

Determined uncertainty of experimental parameters. standard practice has been defined for handling porous materials
Observed variable Uncertainty (%) in a flow-tube, which can be applied to oxidation experiments with
l ±3 reactive gas species. In our protocol, the acquisition of permeability
m_ ±7 data at both room temperature and at oxidation temperature, prior
R ±0.1 to the start of the reactive gas/material interaction and material
T ±2 recession phase, can be used as a tool for comparison to verify
L ±1
the absence of leakage in the interference fitting of the sample in
D ±0.5
M ±0.1 the tube.
DP ±2 Finally, by defining temperature- and pressure-independent
F ±8.2 permeability parameters, the work presented here provides an
improvement to the material property databases used in high-
X
N fidelity Computational Fluid Dynamics and Material Response
ln F ¼ ai ln xi ð15Þ codes, which are of the utmost importance as the need for more
i¼1
accurate spacecraft re-entry simulations increases.
Thus, the variance of F becomes:
Conflict of interest
X
N
d2ln F ¼ a2i d2ln xi ð16Þ
i¼1 None declared.
If sufficiently small (620 %), the standard deviation of the nat-
ural logarithm of a random variable is approximately equal to Acknowledgments
the relative standard error, i.e., dln xi ¼ dxi =xi . Therefore, Eq. (13)
can be approximated with sufficient accuracy as: Financial support for this work was provided by NASA Award
NNX14AI97G. The authors are grateful to J. Marschall for initiating
"
2 #12 this project and engaging in useful discussions, as well as to F.S.
X N
dF 2 dxi
¼ ai ð17Þ Milos and Y.-K. Chen for reviewing the manuscript and providing
F xi
i¼1 constructive comments.
Expanding the series for all parameters, the expression
becomes: Appendix A. Supplementary data
"
2
2
2
2
2
2
2
2 #12
dF
¼
dl
þ
dm_
þ
dR
þ
dT
þ
dL
þ
dD
þ
dM
þ
dDP
ð18Þ
Supplementary data associated with this article can be found, in
F l m_ R T L D M DP the online version, at http://dx.doi.org/10.1016/j.ijheatmasstransfer.
2016.05.016.
In Eq. (18), dR and dM are negligible, and dL and dD are very small.
Therefore, the main contributions to errors on F are associated with
mass flow rate, temperature, and consequently gas viscosity and References
pressure measurements. A list of the determined uncertainties
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