Event Date/Time: Jan 05, 2009 | End Date/Time: Jan 08, 2009 |
Registration Date: Jan 12, 2009 | |
Early Registration Date: Jan 10, 2009 |
Description
KEY WORDS:
composite materials, ablation, estimation, thermal properties.
ABSTRACT
The aim of this work is to give more
information on theinfluence of the experimental conditions to the estimated
thermal properties of ablative composite materials. Two kinds
of experiments with the carbon phenolic composite samples have been carried out. In the first, the sample has been exposed
to intensive thermal load from acetylene flame, and in the second, the sample has been incorporated in the rocket nozzle.
The obtained transient temperature responses are used to estimate unknown thermal properties
Transient one-dimensional partial differential equations with two moving boundaries and decomposition equations have
been used to describe complex process of heat and mass transport within material and at its surface. The set of equations
has been solved numerically. Newton-Raphson's and steepest descent methods have been combined to minimize the
difference between the model prediction and experimental response.
INTRODUCTION
Technology today creates many new kinds of composites used in many different applications. In this study ablative
phenolic composite designed for thermal protection is analyzed.
Property evaluation of ablative composites by method of nonlinear estimation is dependent of design and realization of
experiments and of mathematical description of the phenomena in the material. Especially, it is in the case of composites under
high heating loads, where complex heat and mass transfer processes are involved. The main objective of this study is to
examine the influence of the experimental conditions to the estimation of thermal properties of ablative composite
materials. In order to achieve this, two different experiments are analyzed and compared. The mathematical model of transient
processes of heat and mass transfer in composite and characteristics of nonlinear optimization model are presented.
Predicted and experimental transient temperature responses are given. The estimated parameters from the two experiments are
presented and compared.
NOMENCLATURE
A = sensitivity coefficient matrix
Bp = Arhenius constant, 1/s
c = specific heat, J/kgK
E = activation energy, J/kg
E = sum of squared differences, K2
Hc = specific ablation heat, J/kg
2 Copyright Â© 1999 by ASME
Hp = enthalpy of the gases from pyrolysis, J/kg
I = unitary matrix
k = reaction rate, 1/s
M = experimental temperature vector, K
mc = specific ablation mass flow rate, kg/m2s
mp = specific pyrolysis mass flow rate, kg/m2s
P = parameter values vector
s = zone boundary
T = calculated temperature vector, K
va = ablation velocity, m/s
vp = pyrolysis velocity, m/s
x = space coordinate, m
Î± = heat transfer coefficient, W/m2K
Î´ = thickness, m
Î”Hp = specific pyrolysis heat J/kg
Îµ = fraction of resin transformed to the gas
Î» = thermal conductivity, W/mK
Ï = density, kg/m3
Ï„ = time, s
Ïˆ,=Î¾ = transformed coordinates
MATHEMATICAL MODEL OF ABLATING COMPOSITE
Under the experimental conditions, in composite complex
processes of heat and mass transfer take place.
Î´1 Î´2 Î´3
s1
s2
s3
s4
Char layer Pyrolysis
zone
Initial
material Steel
q
.
.mp
mc
Figure 1. The zones within the ablating composite
When the composite is exposed to high-temperature and
high-velocity fluid stream, decomposition of resin and
formation of char layer begins at a critical temperature and, char
layer ablation at an another, higher temperature. The free
surface of the composite, under the influence of high
temperature stream is continuously spalled, and, depending on
the oxygen content in the gases, oxidized. After that, the two
processes take place simultaneously, so, three zones are formed
(Fig. 1.): the virgin material, the pyrolysis zone and the porous
char layer.
The mathematical model of in-depth ablation has been
developed on the basis of described processes (Kanevce, 1992).
The model is one-dimensional, with three layers: the char layer,
virgin material, and steel. The pyrolysis takes place at the
surface between them. A layer of steel exists behind the zone of
the virgin material. The material thermal properties in the zones
are assumed constant. Hence, the problem can be described by
the following transient partial differential heat conduction
equations:
âˆ‚ Ï„
âˆ‚
Ï
âˆ‚
âˆ‚
âˆ‚
âˆ‚
Î»
T
c
x
T
m c
x
T
2 p g 1 1
2
1 + = , 1 2 s < x < s , (1)
âˆ‚ Ï„
âˆ‚
Ï
âˆ‚
âˆ‚
Î»
T
c
x
T
2 2 2
2
2 = , 2 3 s < x < s , (2)
âˆ‚ Ï„
âˆ‚
Ï
âˆ‚
âˆ‚
Î»
T
c
x
T
2 3 3
2
3 = , 3 4 s < x < s , (3)
where, T is temperature, Î» denotes thermal conductivity, Ï is
density, and c is specific heat. Indexes 1, 2 and 3 refer to the
zones of char, virgin material and steel substructure
respectively, and index g to gases.
Thermal equilibrium between gases produced in the
pyrolysis zone with a mass flow rate mp and the char material is
assumed in the Eq. (1).
The boundary conditions are:
g g ,eff a mpHp mcHc
x
(T âˆ’ T ) = âˆ’ T + +
âˆ‚
Î± Î» âˆ‚ 1 , x s (Ï„ ) 1 = , (4)
mp H p
x
T
x
T
âˆ’ = âˆ’ + Î”
âˆ‚
âˆ‚
Î»
âˆ‚
âˆ‚
Î» 1 2 , x s (Ï„ ) 2 = , (5)
x
T
x
T
âˆ‚
âˆ‚
Î»
âˆ‚
âˆ‚
Î» 2 3 âˆ’ = âˆ’ 3 x = s , (6)
= 0
x
T
âˆ‚
âˆ‚
4 x = s , (7)
where, Î±g is the heat transfer coefficient, Ta ablation
temperature, Tg,eff effective gas temperature obtained from the
recovery enthalpy.
The term mcHc represents the total heat of ablation, where
the specific heat of decomposition of the ablation surface char
layer is Hc, and the mass flow rate is mc.
The mass flow of the pyrolysis gases away from the surface
blocks the flux from the boundary layer toward the surface by
the effect mpHp included in the Eq. (4), where Hp is the enthalpy
of gaseous products.
3 Copyright Â© 1999 by ASME
The specific heat of decomposition of the phenolic resin is
Î”Hp. The term mpÎ”Hp in the second boundary condition (Eq.
(5)) represents the heat of decomposition in the pyrolysis zone.
As the ablating surface is at high temperature, significant
error may arise (up to 25%) if the radiative heat exchange with
surrounding solid surfaces at temperature T0 is ignored.
Therefore, in the boundary condition on the thermally loaded
surface, the heat transfer coefficient is replaced with its
effective value Î±g,eff :
g ,eff a
a
c gr
g ,eff g T T
)
T
) (
T
c (
âˆ’
âˆ’
= âˆ’
4 0 4
100 100
Îµ
Î± Î± (8)
where: cc is radiation constant of black body and Îµgr is
emissivity of the ablating surface.
The initial conditions are:
T( x,0 ) = T0 , (9)
mc =0, T( 0,Ï„ ) < Ta0 , (10)
where, Ta0 is the temperature of ablation beginning.
The ablation model involves three zones and two moving
boundaries, s1 and, s2. Third and fourth boundary, s3, and, s4 are
fixed. Corresponding equations for defining boundaries are:
dt
m
s ( ) = c
Ï„
Ï
Ï„
0 1
1 (11)
dt
m
s ( ) p
âˆ’
=
Ï„
Ï Ï
Ï„
0 2 1
2 (12)
s (Ï„ ) = const. 3 (13)
s (Ï„ ) = const. 4 (14)
To complete the system of equations of the in-depth
ablation model, it is necessary to include equations for
calculating the pyrolysis mass flow rate. The motion of the
pyrolysis zone is defined by kinetics of the phenolic resin
decomposition. It is assumed that the rate of decomposition may
be expressed by the first degree reaction:
k ( )
[ ( )]
g
g Ï Î²Ï‡ Îµ
âˆ‚Ï„
Ï Î²Ï‡ Îµ
= âˆ’
âˆ‚ âˆ’
âˆ’ 2
2 (15)
where: Î² denotes fraction of phenolic resin in the composite, Ï‡g
is the mass-fraction of the resin which may be decomposed at
temperature T in respect to total mass of the resin, Îµ is the mass
fraction of the resin transformed to the gas in respect to mass of
composite, and Ï2 is density of the composite. Therefore,
(Î²Ï‡g=âˆ’=Îµ)=represents the mass of decomposed resin in respect to
mass of virgin material. This factor multiplied by Ï2, defines
density of decomposing resin. The maximum fraction of resin
transformed to the gaseous state at given temperature is Îµmax
==Î²Ï‡g.=The reaction rate constant is given by Arrhenius form:
RT
E
k Bpeâˆ’ = (16)
From these, equations for calculation Îµ and mp follow:
RT
E
k( g ) Bp ( g )eâˆ’
= Î²Ï‡ âˆ’Îµ = Î² Ï‡ âˆ’Îµ
âˆ‚Ï„
âˆ‚Îµ (17)
Îµ ( x,Ï„ ) â‰¤Îµ max = Î²Ï‡ g (18)
m ( ) dx B ( ( x, ))e RT( x , )dx
E
p p g
Ï„
Î´ Î´
Ï Î² Ï‡ Îµ Ï„
âˆ‚Ï„
âˆ‚ Ï Îµ âˆ’
= = âˆ’
0
2
0
2 . (19)
For solving problems in this category, it is convenient to
use double transformation of coordinates in the zones separately
(Furzeland, 1980):
Î´
Ïˆ = x âˆ’ s (20)
1
2 1 2
+
= + âˆ’
R
Ïˆ Î¾ ( R )Î¾ (21)
The obtained system of double transformed partial
differential equations has been solved numerically using explicit
finite difference scheme.
The equations for pyrolysis mass flow rate calculation, are
not in transformed coordinates, so their numerical integration
has been realized by inverse transformation of Î¾ to x. Euler's
method has been applied for solving these equations. This
method is satisfactory, because the time step is sufficiently
small to achieve the necessary accuracy. Common forward or
backward space finite differences have been applied to the
boundary conditions.
EXPERIMENTALY
Two kinds of experiments have been carried out in order to
estimate unknown thermal characteristics. The experiments
have been conducted with a phenolic-composite samples
exposed to intensive thermal load. The samples have been
equipped with in-depth located thermocouples.
At the first experimental setup, for the simulation of the
heat fluxes met under working conditions of ablative
composites, an acetylene flame jet is used as a heat source, with
axis of the hot stream normal to the sample ablating surface.
The burner is mounted on a motorised stand that can be moved
by remote control into and out of the position, as required.
When working position is achieved, a micro switch signals the
acquisition system that test has begun.
The design of the carbon fibre phenolic composite sample
is shown in Fig. 2. The cylindrical sample 30 mm in diameter
4 Copyright Â© 1999 by ASME
and 19.18 mm thick has been fitted with a central plug 5.35 mm
in diameter, equipped with thermocouples.
Figure 2. Scheme of the acetylene setup
Fast-response Chromel-Alumel thermocouples (0.1 mm in
diameter) have been located close to each other, near the
exposed surface. The whole sample assembly has been mounted
on a water cooled stand. The heat flux applied to the sample
surface has been measured by the same setup with copper
sample equipped with thermocouple at the back side.
Figure 3. Scheme of the rocket nozzle setup
In the second experiment composite sample has been
incorporated in rocket nozzle wall and, in working conditions,
exposed to high velocity and high temperature stream along to
the sample surface.
The design of the experimental section is presented in Fig
3. The annular 9.1 mm thick composite sample has been
equipped with Chromel-Alumel thermocouples (0.1 mm in
diameter) located close to each other, near the exposed surface.
Beside this measuring part, the annular, 7 mm thick graphite
part has been placed. The graphite sample with incorporated
thermocouples has been used for heat flux determination.
Thickness of the samples and the distances of the
thermocouple locations from the thermal loaded surface are
presented in Table 1.
Table 1. Thermocouple locations from the free surface
Exp.
No.
Sample
Thickness
[mm]
Thermocouples Location [mm]
No.1 No.2 No.3
1 19.18 1.22 4.44 9.79
2 9.1 1.1 1.66 3.22
On the basis of measured sample thickness before and after
experiments, and experiment duration, the ablating velocities
have been calculated. The average ablating velocities of va1 =
1.19Â·10-4 m/s, and va2 = 2.57Â·10-4 m/s have been obtained for the
first and second experiment respectively.
The heat flux and the heat transfer coefficient for the first
experiment have been determined as boundary inverse problem
with copper sample of known thermal characteristics by using
nonlinear estimation model presented here and analytical
solution of ordinary heat transfer equation for copper sample.
Obtained values of the heat transfer coefficient and gas
temperature have been 3300 W/m2K and 3370 K.
During the second experiment, two phases with different
heat fluxes have been realized. The duration of the first phase
has been 2 s and the duration of the second phase 3.4 s. The
heat flux and heat transfer coefficient have been determined as
boundary inverse problem with known thermal characteristics
of graphite sample in the nozzle wall, and also with applying an
integral model of the boundary layer (Kanevce, 1992). The
obtained heat transfer coefficient and gas temperature have been
19695 W/m2K and 3170 K for the first phase and, 6200 W/m2K
and 2910 K for the second phase.
The duration of the first and second experiment has been
30 s and 5.4 s respectively.
The both experimental systems provide accurate
temperature data acquisition with sampling intervals in the
range 50 ms.
5 5
1. Sample
2. Measuring part
3. Water cooled stand
4. Thermocouples
5. Water
Ã¸5,35 2 1 3
4
Acetylene
flame jet
Ï† 36
4 mm
3
1
1. Composite sample 2
2. Graphite sample
3. Thermocouples
Hot gases
5 Copyright Â© 1999 by ASME
NONLINEAR ESTIMATION
The nonlinear optimization problem is concerned with
finding the minimum of the sum of squared differences between
experimental and predicted temperatures:
E = (T âˆ’M(P))T (T âˆ’M(P)) (22)
where, the vector P = [p1, p2, â€¦ , pn]T represents the parameter
values, the vector M = [ M1, M2 , â€¦, Mm] contains the
measured temperature values, the vector T = [T1, T2, â€¦Tm]T
contains corresponding calculated values at the thermocouple
locations, n is the total number of parameters and m is the total
number of experimental temperature values i.e. the total
number of calculated temperatures corresponding to the time
and location of the measured temperature values.
To find the minimum of E (Eq. (22)), Marquardt's
combination (Fletcher, 1971) of Newton-Raphson's and the
steepest descent method has been used. The solution for P can
be achieved using the iterative scheme
P(i +1) = P(i) + Î”P(i) (23)
where i indicates iteration number and Î”P(i) is solution of the set
of linear equations:
[Î³ I + AT (i)A(i) ]Î”P(i) = AT (i) (T âˆ’M(i) ) (24)
The A matrix represents the sensitivity coefficient matrix
(Beck, Arnold, 1977) and it is defined as:
âˆ‚
âˆ‚
âˆ‚
âˆ‚
âˆ‚
âˆ‚
âˆ‚
âˆ‚
âˆ‚
âˆ‚
âˆ‚
âˆ‚
âˆ‚
âˆ‚
âˆ‚
âˆ‚
âˆ‚
âˆ‚
=
n
m
2
m
1
m
n
2
2
2
1
2
n
1
2
1
1
1
p
... T
p
T
p
T
p
... T
p
T
p
T
p
... T
p
T
p
T
A
M
(25)
The adjustable parameter Î³ is chosen at every iteration so
as to follow the Gauss-Newton method to as large an extent as
possible, whilst retaining a bias towards the steepest descent
direction to prevent divergence.
CALCUATIONS AND RESULTS
As it is mentioned before, responses of two carbon fibre
phenolic samples from two different experiments are studied.
The duration of the first experiment has been 30 s. The total
number of 180 temperature readings has been taken in this
experiment for the comparison with estimated temperature
responses. The numerical calculation of temperature responses
by using presented in-depth ablation model has been realized
with 24 grid points. The duration of the second experiment has
been 5.4 s. In this experiment 312 temperature readings have
been taken. The numerical calculation has been realized with 80
grid points.
Model assumes thermophysical properties of the virgin
material as known parameters. Different methods can be applied
for thermal property estimation of composites (Dowding et al.,
1995, Garnier et al., 1992). In this work, the room temperature
values of thermal conductivity, Î»2== 0.76 W/m/K, density, Ï2==
1340 kg/m3 and the specific heat, c2 = 1249 J/kg/K have been
obtained by conventional methods (Maglic, Perovic, 1988).
The fraction of phenolic resin in the composite used is
Î²=0.48. For Ï‡g=(T), the following relation is valid (Kanevce,
1992):
T T K
T K
T T T
g
g
( ) 0.55 , 1255 ,3
0.12527 , 1255 ,3
( ) 0.21322 10 6 2 0.80560 10 3
= â‰¥
âˆ’ <
= âˆ’ â‹… âˆ’ + â‹… âˆ’
Ï‡
Ï‡
(26)
The nonlinear estimation of three and seven parameters has
been analyzed.
Firstly, the estimation of the three most influential
properties, thermal conductivity, Î»1, density, Ï1 and specific
heat, c1, in the first zone has been conducted. The values of the
other parameters have been taken from the references for the
composites of the same kind. Good agreement between
experimental and calculated temperature has been obtained.
The results of the estimation of the seven thermal
parameters: thermal conductivity, Î»1, density, Ï1 and specific
heat, c1, of the first zone, heat of ablation, Hc, heat of pyrolysis,
Î”Hp and Arhenius constants, Bp, and E; figuring in the ablation
model are presented in this paper. The values of estimated
parameters are divided in three groups depending of their
contribution to the agreement between experimental and
calculated values.
Since the agreement between experimental and predicted
temperatures mainly depend of the thermal conductivity, density
and specific heat of the first zone, the estimated values of these
parameters are shown firstly, in Table 2. In consideration of
significantly different experimental conditions in the two
represented experiments, reasonable match of the estimated
property values is obtained.
Table 2. Estimated thermal properties of char layer
Exp. No. Î»1=[W/m/K] Ï1=[kg/m3] c1 [J/kg/K]
1 2.957 1089 1105
2 2.407 1019 1978
Table 3. contains evaluated values for heat of ablation, and
heat of pyrolysis obtained from acetylene and rocket nozzle
experimental data.
6 Copyright Â© 1999 by ASME
Table 3. Estimated heat of ablation and pyrolyses
Exp. No. Hc [J/kg] Î”Hp [J/kg]
1 2.3â‹…107 4.8â‹…106
2 3.2â‹…106 7.5â‹…106
Table 4. Estimated Arhenius constants
Exp. No. E/R [K] Bp [1/s]
1 2.53â‹…104 5.6â‹…106
2 2.30â‹…104 9.9â‹…106
The values of Arhenius constants are presented in Table 4.
The influence of parameters in Table 3. and Table 4. to the
temperature responses is of lower degree so the agreement of
presented parameter values for two experiments could be
accepted as good.
Comparison of experimental and predicted temperature
responses presented in Fig. 4. and Fig. 5.
Experiment 1
0
200
400
600
800
1000
1200
1400
0 5 10 15 20 25 30 35
Ï„[s]
T[0C]
exp
model
1,22mm
4,44mm
9,79mm
Figure 4. Experimental and predicted temperature responses for
acetylene flame jet experiment
Experiment 2
0
200
400
600
800
1000
1200
1400
0 1 2 3 4 5 6
Ï„[s]
T[0C] exp
19,10mm model
19,66mm
21,22mm
Figure 5. Experimental and predicted temperature responses for
rocket nozzle experiment
In consideration of severe experimental conditions (very
intensive thermal load, high temperature gradients in samples
especially in the first zone, short experimental duration, fast
complex chemical and physical changes in composite) a good
agreement have been achieved in both experiments.
The obtained agreement confirms that the mathematical
model is suitable for matching the thermocouple responses for
the range of test conditions studied in this work.
CONCLUSIONS
The two kind of experiments with the same phenoliccomposite,
equipped with in-depth located thermocouples,
exposed to high temperature and high velocity fluid stream,
have been carried out and presented. In the first, the sample has
been exposed to intensive thermal load from acetylene flame,
and in the second, the sample has been incorporated in the
rocket nozzle.
The obtained match between experimental and predicted
temperatures in both experiment enables use of presented
mathematical model of ablating composite for transient
temperature prediction in ablating fibre phenolic composite.
The applied method of nonlinear estimation enabled rapid
and sure convergence. However, to reduce influence of
measurement noise to the results of calculations, the presented
method, in next improvements, ought to include the
regularization.
The coordinate transformations, and the applied explicit
method enable solving proposed mathematical model with
sufficient accuracy, using relatively small number of grid points,
and consequently short computing time.
Obtained experimental data, calculated parameters and
proposed mathematical model can be successfully used for
transient temperature prediction in ablating fibre phenolic
composite.
The main difference between conducted experiments is in
applied thermal load. The greater thermal load has been
accomplished in the second experiment. The applied high
velocity and high temperature stream has been normal to the
free sample surface at the first experiment and parallel with the
free surface at the second experiment. The thickness of the
examined samples has been different too.
Although conducted experiment have been different,
estimated values of thermal properties in two experiments,
especially that of great influence, show reasonable agreement.
Therefore it can be concluded that the first experiment can be
used for investigation of ablative composite behavior in real
conditions. The second experiment ought to be applied for more
precise analysis.
In further investigations the analysis of sensitivity
coefficients and optimal experiment design (Takt), 1993)
ought to be included. In this stage sensitivity coefficients of
three the most influenced parameters for thermal conductivity,
specific heat and density have been analysed. It has been shown
that these three sensitivity coefficients have been independent
(cenov)2007
Venue
Restrictions
Members Only