AIAA JOURNAL
VOL. 5. NO. 6, JUNE 1967
1063
A Multicomponent Boundary Layer Chemically Coupled
to an Ablating Surface
ROBERT M. KENDALL,* ROALD A. RINDAL,! AND EUGENE P.
Downloaded by TOBB EKONOMI VE TEKNOLOJI on December 23, 2014 | http://arc.aiaa.org | DOI: 10.2514/3.4138
Aerotherm Corporation, Polo Alto, Calif.
Based upon an accurate approximation for binary diffusion coefficients, simplified equations are presented for a multicomponent boundary layer with unequal diffusion coefficients
for all species. Correlation equations are then presented for representing the multicomponent boundary layer in terms of convective transfer coefficients. These equations are
based upon analogy between the form of the present species conservation equation and that
for the case of equal diffusion coefficients. The form chosen for the correlation equations is
such that they should be valid for a wide range of boundary conditions. These equations are
then chemically coupled to an ablating surface, allowing arbitrary chemical composition of the
ablation material and boundary-layer-edge gas. Results of some numerical calculations
are presented which demonstrate transient response of charring materials in rocket nozzle
environments with and without the assumption of equal diffusion coefficients.
Nomenclature
B'
Be'
B0'
CH
normalized ablation rate, (pv)w/peUeCM
heat-transfer coefficient (Stanton number), qwf
peUe(Hr
-
hw)
number of k atoms in molecular species i
mass-transfer coefficient
D
reference diffusion coefficient [Eq. (10)]
DiT
thermal diffusion coefficient of species i
effective diffusion coefficient [Eq. (27)]
£>eff
3D»y = binary diffusion coefficient of species i into .7
Fi = diffusion factor for species i [Eq. (10)]
h
= static enthalpy, hc + hs
hc = chemical enthalpy, Z-, Kihi°
CM
CM
hi0
hs
Hr
/
ji
i
= heat of formation of species i at reference temperature
and 1-atm pressure
V*
/* T
= sensible enthalpy, 2^ Ki t CP dT
i
Jo
— recovery enthalpy
= total number of species i
— diffusional mass flux of species i by virtue of a concentration gradient
Ji
= total diffusional mass flux of species i [Eq. (12)]
;
k
= thermal conductivity
K
= total number of elemental species k
',
Ki = mass fraction of species i
Kk = total mass fraction of element k independent of molecular configuration
Kpi = equilibrium constant for formation of species i from gaseous elements
Le = Lewis number
m
= mass-flow rate per unit area
2flZ
= molecular weight
SflZjt = atomic weight of element k
Ni = chemical symbol for gaseous species i
Nk — chemical symbol for element k
Ni = chemical symbol for condensed species I
P
= pressure
Pi = partial pressure of species i
pr = Prandtl number
qr
= radiant heat flux incident to a point on the body
qw
= net surface heat flux
rc
= local radius of body curvature
r0
= local radius of body from centerline
R
= gas constant
s
= streamwise coordinate
T
= temperature
u
= velocity component parallel to surface
Ue
= velocity at edge of boundary layer
v
= velocity component normal to the surface
Xi
= mole fraction of species i
y
= coordinate normal to surface of body
Zi
= diffusion driving potential factor [Eq. (15a)]
Zk
= defined by Eq. (20)
Zk*
= defined by Eq. (29)
aki
= mass fraction of element k in species i
y
= mass transfer weighting factor defined by Eq. (29)
e
= emissivity
^
= viscosity
Mi> M2> MS = factors defined by Eq. (15)
p
= density
\//i
= mass generation of species i per unit volume as a
result of chemical reaction
Super scrip ts
= exponent equal to unity for axisymmetric bodies and
zero for two-dimensional bodies
Subscripts
a
= undecomposed plastic at initial temperature
c
= char or solid material
e
= evaluated at edge of boundary layer
g
= resin off-gas
i
= gaseous species
j
= gaseous species
k
= element
I
= condensed species
r
= condensed phase removal
w
= evaluated at wall
1.
T
Presented as Paper 65-642 at the AIAA Thermophysics Specialist Conference, Monterey, Calif., September 13-15, 1965; submitted September 20, 1965; revision received January 23, 1967.
The work was sponsored jointly by NASA under Contracts
NAS7-218 and NAS9-4559 and by the Air Force Systems Command under Contract AF 04 (611)-9073.
[3.06,6.08,11.14]
* Manager, Analytical Services Division. Member AIAA.
f Staff Engineer, Analytical Services Division. Member
AIAA.
Introduction
HE analysis of ablative-material thermal protection
systems for hyperthermal, chemically active environments requires theoretical techniques to characterize the
material thermal response and to represent the heatedsurface boundary condition. Analytical techniques are presently available for representing the transient thermal response of a variety of ablative material types and for representing boundary-layer transport phenomena necessary
for evaluating the heated-surface boundary condition.
Because an intimate coupling exists between boundary-layer
transport phenomena and ablative-material thermal response
Downloaded by TOBB EKONOMI VE TEKNOLOJI on December 23, 2014 | http://arc.aiaa.org | DOI: 10.2514/3.4138
1064
KENDALL, RINDAL, AND BARTLETT
for a number of material-environment combinations of current
interest, the need for a coupled solution is apparent. In some
cases coupling has been achieved by correlating the results
of a matrix of boundary-layer solutions in terms of boundarylayer-edge and material-response parameters. As illustrations of this approach, Fay and Riddell1 present correlations
to numerical solutions of the laminar boundary-layer equations at a stagnation point for a boundary-layer-edge gas of
air adjacent to a nonablating surface, and Scala and Gilbert2
present correlations for solutions of the laminar boundarylayer equations for an edge gas of air and an ablating carbon
surface. Of course, the correlations obtained from such results are restricted to the surface material and environmental
gas employed in the solution.
In Ref. 3, Lees presents arguments for generalized boundarylayer correlations independent of boundary-layer-edge gas
and ablation material composition. This is accomplished
by assuming the diffusion coefficients of all species are equal.
Boundary-layer correlation equations that are general with
respect to edge gas and ablation material composition and
that include the effects of unequal diffusion coefficients in
the chemically reacting multicomponent boundary layer
have not been presented previously. Such correlation equations are developed and presented here. A technique for
coupling the correlations to a generalized ablative-material
thermal response solution is described, and some representative results for this coupled solution are presented. The
numerical results illustrate the importance of unequal diffusion coefficients and a coupled transient ablation solution.
The procedure described herein employs correlation equations that relate surface conditions directly to boundarylayer-edge conditions through the use of over-all convective
transfer coefficients. The form of these relations is suggested
by an explicit relation for diffusive mass flux made possible
by an approximation to binary diffusion coefficients introduced by Bird.4 It is shown that the errors in the diffusion
coefficients introduced by this approximation are generally
not greater than the uncertainties in the coefficients themselves.
This approximation also simplifies the multi component
boundary-layer equations, yielding significant computational
advantage in solving these equations and making practical
their solution with an indefinitely large number of molecular
species. To illustrate the complexity of the multi component
boundary-layer equations in the absence of this approximation, the largest number of molecular species considered
simultaneously to date appears to be 9.2 In the present
formulation, the boundary-layer equations are only slightly
more complex than their counterpart with assumed equal
diffusion coefficients, and the number of molecular species
has a minimal effect on computational time. Therefore,
although the primary purpose of this paper is to present the
correlation equations, the multi component boundary-layer
equations, modified to incorporate this approximation, are
also discussed briefly.
The approximation for binary diffusion coefficients and the
modified boundary-layer equations pertinent to hyperthermal
flow over an ablating surface are presented in Sec. 2. The
generalized correlation equations are proposed in Sec. 3.
In Sec. 4, the results of numerical calculations are presented
which demonstrate the transient ablation and internal response of some charring materials with and without the assumption of equal diffusion coefficients.
2. Conservation Equations for the Laminar
Multicomponent Boundary Layer, Including
an Accurate Approximation for Unequal
Diffusion Coefficients
The laminar boundary-layer equations for two-dimensional
or axisymmetric flow including thermal diffusion and unequal
AIAA JOURNAL
diffusion coefficients can be expressed as:
Continuity:
bpv =
_!
ro3
by
Momentum:
bu ,
bu
b \~ bu~]
dp
pu —— + pv — = — M — —
bs
by
by[_ ctyj
~ds~
(2)
Species:
bK<
pv -^—
by
as
—
by
(3)
Energy:
pu*jf
<y/2)
by
c)
+ pv(
—
oy
T,(ji + ^ *Ki\
by /
yy *
k_bh
cp by
£-£
w
where ji is the diffusional mass flux of species i.
The primary reason for the present apparent lack of practical computational schemes for solving these equations for an
indefinitely large number of species is the complex interdependence of the ji upon the species concentrations and
concentration gradients. In order to illustrate this complexity, let us consider the species conservation equation for
chemical elements. This results directly from Eq. (3) upon
application of the Shvab-Zeldovich transformation (multiplication by the mass of element k in molecule iy an, and summation over all species) .3
pu
bKk
bs
bKk
(5)
where
(6)
The diffusional velocity (in the absence of pressure diffusion
and body forces) is expressible via the Stefan-Maxwell relations5 as
XjXj
jj +
X
nbluT
by
i+
(7)
where 3)^ are the binary diffusion coefficients for species i
and j. Alternatively, the diffusional velocities may be expressed in terms of the multicomponent diffusion coefficients
Dti
b liiT
(8)
In the absence of thermal diffusion and with all S)^ equal,
Eq. (7) reduces to Fick's law, which relates the diffusion
mass flux explicitly to its mass fraction gradient
When the diffusion coefficients are not equal in a multicomponent mixture, utilization of the Stefan-Maxwell equations
[Eq. (7)] in conjunction with the species conservation equations is awkward even in the absence of thermal diffusion
effects, since the diffusional flux j» is expressed implicitly
in terms of mole fractions and their gradients. Hence, use
is often made of Eq. (8) together with the multicomponent
diffusion coefficients, for example, in Refs. 2, 6, and 7. How-
JUNE 1967
BOUNDARY LAYER COUPLED TO AN ABLATING SURFACE
ever, each of the (72 — 7) multicomponent diffusion coefficients Dij depends upon local concentrations and upon
(I2 — 7)/2 symmetric binary diffusion coefficients 3)^, where
7 is the total number of species being considered.
Approximation for Binary Diffusion Coefficients
An approximation to binary diffusion coefficients introduced by Bird4 and utilized herein permits explicit solution
of the Stefan-Maxwell relations for ji in terms of gradients
and properties of species i and of the system as a whole. The
approximation can be expressed in the form
Downloaded by TOBB EKONOMI VE TEKNOLOJI on December 23, 2014 | http://arc.aiaa.org | DOI: 10.2514/3.4138
3)^ - D/FiFj
Approximate Relations for Diffusive Flux
[Eq.
Substituting Eq. (10)
into the Stefan-Maxwell relation
(7) ] and rewriting in terms of mass fractions yields
by
pD \ 9fKi
diffusional fluxes:
InT/by)
m)
9%
where 3TC and 3H; represent mixture and species molecular
weights, and,
for convenience, a total diffusional mass flux
has been defined as the sum of the molecular and thermal
J The effect of pressure can be absorbed entirely into the D
since 3D»y is inversely proportional to pressure. It will be shown
thatjhe dominant temperature effect can also be absorbed into
the D so that the Fi are nearly constants for a given molecular
set.
§ Equation (10) is exact for a ternary system.
(12)
<
Multiplying each side of Eq. (11) by 3Ki/FiJ summing over
all i, and noting that the sum of the diffusive fluxes is zero
and the sum of the mass fractions is unity, yields
J_fj
pD
Substituting Eq. (13)
pD
9fTCt- bxj
Fi
by
into Eq. (11)
?f f <13)
results in
9frc2 FiJi ^ K^
pD Sflffc j sr
bxi _ KiFi ^ y\lj bxj
by
mi j FJ by
(10)
with D(T, P) a property of the given multicomponent mixture and Fi(T)
a property of the ith species in the mixture.!
It is apparent when considering more than 3 species§ that
Eq. (10) is indeed approximate, since (72 — 7)/2 diffusion
coefficients £># are replaced by 7 diffusion factors Fi. Equation (10) should thus be viewed as a correlation equation
for actual binary diffusion coefficient data.
The accuracy of the correlation was investigated by Bird4
for a five-component mixture containing hydrogen and shown
to be surprisingly good, the maximum error in any SD^ being
4%. In order to establish more generally the adequacy of
the approximation, correlations have been performed for
several chemical systems including a 16-component (120 £>,-,/)
C-H-O-N system.8 Typical correlations are presented here
for two smaller systems (C-N-0 and 0-H).
Binary diffusion coefficients for the C-N-0
system are based on the
Lennard-Jones potential with the molecular properties
suggested in Ref. 9. The diffusion coefficients for the 0-H
system are not restricted to a single molecular potential
model, but are based on the collision cross sections suggested
by Svehla.10 Fi and D for these systems were obtained by
employing a least-squares fit to these diffusion coefficient
data. Utilizing these values, diffusion coefficients were
computed from Eq. (10) and compared to the original data.
The results of the computations are presented in Table 1.
It can be seen that substantial improvement over the equal
diffusion coefficient model is obtained in both cases, reducing
the average absolute error in 2X-y by more than an order of
magnitude. Even more significant, however, is the comparison with the original SD^. In the first case, the average
absolute error is 1.3%
and the maximum error in any one
single $)ij is 5.2%.
In the second, more severe case, the
average absolute error is 4.8%
and the maximum error in
any one diffusion coefficient is 11.3%. In the 16-component
system mentioned earlier, 98 of the 120 SD^ are represented
within 5%, the largest single error is 16.7%, and the average
error is 3.7%.
Similar results have been obtained for systems
containing other elements such as Cl, Ar, and Al. The
correlation equation thus seems to have rather general
applicability.
1065
(14)
At this point it is convenient to define several new quantities:
Fifi2
(15a)
(15o)
(15d)
Multiplying Eq. (15a)
to y yields
by /u2 and differentiating with respect
Substituting Eqs.
that
15, and 16) into Eq. (14)
(12,
and noting
Z Zj = 1.0
3
and that the Fi are functions of temperature only yields
T
by
311
by
Equation (17),
which replaces the more conventional multicomponent diffusion relation [Eq.
(8)], provides an explicit
representation of the diffusional mass flux of species i in
terms of properties and gradients of species i and of the
system as a whole, but not of other species. This is the
functional relationship that has been sought.
Consideration of the temperature dependence of Fi enables a further simplification to be obtained with little loss
of accuracy. Least-squares fits to diffusion coefficients were
performed to obtain Fi(T) for the C-N-0
system over a wide
temperature range. The results of the calculations are
shown in Table 2. If the Fi are assumed constant over the
temperature range from 4000° to 16,000°R, the maximum
error incurred is less than 1% for this system. Hence, it
may often be consistent with the accuracy of the approximation to consider the Fi as constants for a given molecular
set.
When this is done, Eq. (17) for diffusional mass flux
becomes
j
M2
Wl (18)
Simplified Boundary-Layer Equations
Substituting Eq. (18) into the boundary-layer elemental
species conservation equation [Eq.
(5)] yields the modified
boundary-layer conservation equation for chemical elements
in a multicomponent boundary layer with unequal diffusion
1066
KENDALL, RINDAL, AND BARTLETT
AIAA JOURNAL
Table 1 Comparison of binary diffusion coefficients for various selected species as computed by the present
correlation technique and from kinetic theory
£>ij from
kinetic theory,
ft 2 X 100/sec
Species
i
Downloaded by TOBB EKONOMI VE TEKNOLOJI on December 23, 2014 | http://arc.aiaa.org | DOI: 10.2514/3.4138
i
Fi
5)»y from
present
correlation,
2
ft X 100/sec
Error using
present
correlation, %
a) Species typical of those encountered in the boundary layer over graphite ablating in aira
02
5.6458
0.7399
5.5575
-1.6
N
7.3372
7.5274
2.6
N2
5.3837
5.3995
-0.3
CO
5.4662
5.4379
-0.5
4.3762
C02
4.4638
-2.0
8.0754
8.3663
c
3.6
5.0859
C3
5.1820
-1.9
CN
5.3697
5.3620
0.1
N
1.000
5.5695
5.6566
-1.5
3.9834
N2
3.9611
0.6
4.0235
CO
0.5
4.0028
3.2380
C02
2.3
3.1637
6.1902
-1.9
c
6.3129
3.7630
1.4
C3
3.7100
CN
3.9731
0.3
3.9623
N2
0.7383
5.3953
-0.6
5.4277
5.4496
-0.5
CO
5.4763
4.3857
-2.8
4.5136
C02
8.3844
5.2
7.9727
c
5.0969
-2.1
5.2069
C3
0.1
5.3813
CN
5.3784
0.1
1.0323
3.8977
3.8943
CO
3.1367
3.1114
0.8
C02
5.9967
-0.9
6.0528
c
3.6454
3.6214
0.7
C3
3.8488
-0.3
CN
3.8603
1.0220
3.1683
0.9
3.1390
C02
6.0570
6.1184
-1.0
c
3.6820
3.6584
0.6
C3
-0.2
3.8875
CN '
3.8938
1.2700
4.8745
4.9902
-2.3
C
2.9632
3.1
2.8753
C3
3.1286
0.1
CN
3.1245
C02
1 Q
0.6643
5.6649
c
5.7767
C3
-0.4
5.9811
c
CN
6.0033
1.0927
3.6359
0.2
CN
3.6276
C3
1.0350
CN
Average absolute error
1.3
b) Hydrogen-oxygen system6
0.2208
74.4024
10.1
67.6000
H2
H
27.0030
-4.7
28.3200
H20
H
30.8482
11.3
27.7200
H
0
22.5734
-8.1
24.5500
02
H
27.5549
-6.9
29.5900
OH
H
0.4
0.3034
19.6568
19.5800
H2O
H2
22.4560
-4.8
23.6000
0
H2
-4.4
16.4323
17.1900
02
H2
20.0586
-0.5
20.1600
OH
H2
-1.7
0.8360
8.1500
8.2950
0
H2O
4.4
5.9638
5.7150
02
H2O
1.9
7.2799
7.1450
OH
H20
-0.5
0.7317
6.8131
6.8500
02
0
-3.4
8.3166
OH
8.6060
O
9.6
1.0000
6.0857
OH
5.5520
02
0.8192
OH
4.8
Average absolute error
0
O
O
0
O
O
O
0
02
02
02
02
02
02
02
N
N
N
N
N
N
N2
N2
N2
N2
N2
CO
CO
CO
CO
C02
C02
'—— -L . ij
Error if all
5D;y are
assumed
equal, %
-16.8
-36.0
-13.0
-14.1
5.2
-41.9
-9.4
-12.4
-16.9
18.6
17.3
48.4
-25.6
26.6
18.4
-13.5
-14.2
4.0
-41.0
-9.8
-12.7
20.6
51.0
-22.4
29.7
21.7
49.7
-23.2
28.4
20.6
-5.9
63.4
50.3
-18.7
-21.8
29.5
24.2
-77.1
-45.4
-44.3
-37.1
-47.8
-21.1
-34.5
-10.1
-23.3
86.3
170.4
116.3
125.6
79.6
178.3
73.1
a
Temperature = 12,000°R, pressure = 1 atm (kinetic data from Svehla, Ref. 9).
6 Temperature = 12,000°R, pressure = 1 atm (kinetic data from Svehla, Ref. 10).
coefficients and thermal diffusion^:
VJLJL./G
pu ^—
H
bs
Kk = ^
.
+ pv •
where
by
by
\*y
(20)
The boundary-layer conservation equations for momentum
[Eq. (2)] and energy [Eq. (4)], together with the modified
t With the aid of Eq. (10), an approximate phenomenological model has recently been developed for the DiT which allows unequal
thermal diffusion coefficients to be included easily and with fair accuracy (see Ref. 8).
BOUNDARY LAYER COUPLED TO AN ABLATING SURFACE
Downloaded by TOBB EKONOMI VE TEKNOLOJI on December 23, 2014 | http://arc.aiaa.org | DOI: 10.2514/3.4138
JUNE 1967
elemental conservation equations [Eq. (19)], constitute a
set of differential equations to characterize the chemically
reacting, multicomponent boundary layer with unequal
diffusion coefficients. This set of equations represents a
major simplification to the set of equations previously employed to characterize the multi component, chemically reacting boundary layer with unequal diffusion coefficients. For
example, considering the H-C-N-0 system, typical of graphite-phenolic in air, the number of significant species can be
in the neighborhood of 30. The present technique permits
characterizing the diffusional processes in such a boundary
layer with 31 coefficients (30 diffusion factors Fi, and one D)
rather than 870 multi component diffusion coefficients, each
of which is dependent upon concentrations and upon 435
binary diffusion coefficients.
Since it is not practical to solve the boundary-layer problem for each and every engineering application, it is desirable
to have correlation equations that characterize solutions of
these equations in terms of bulk boundary-layer transfer
coefficients. In the next section such a set of equations is
proposed which permits the rapid prediction of ablation
material transient response while chemically coupled to
boundary-layer material interactions at the ablating surface.
Table 2
Variation of diffusion factors with temperature
Temperature, °R
Pressure, atm —»•
D X 102, ft a /sec ->
Percent error in
Fi for:
Fo = 0.7399
F0, = 1,0000
^'N - 0.7378
FN, = 1.0315
Fco = 1.0217
FCO* = 1-2690
Fc = 0.6692
Fez = 1.0928
FCN = 1.0343
a
1067
4000
1
0.6657
8000
1
2.0980
12,000
1
4.1160
16,000
1
6.6462
-0.06
-0.07
0.04
0.04
-0.04
-0.19
0.42
-0.15
0.03
-0.05
-0.05
0.03
0.03
0
-0.18
0.35
-0.16
0.04
0.04
0.05
0.11
0.12
0.07
0.12
-0.70
0.04
0.12
0.05
0.06
-0.19
-0.19
-0.03
0.25
-0.06
0.27
-0.17
SDi/ based on Lennard-Jones model.
appropriate for a number of gas mixtures that have been
considered to date. It has been observed8 that the second
term on the right-hand side of Eq. (21) is much smaller than
the first, that is
(Zk - Kk)b ]
3.
Proposed Correlation Equations for a Multicomponent Boundary Layer over an Ablating
Material
In this section, simplified equations are proposed to correlate solutions of the multicomponent boundary-layer equations developed in Sec. 2. The form chosen for these correlation equations is based upon analogies between the form
of the present species conservation equation [Eq. (19)] and
that for the case of equal diffusion coefficients. The correlation equations selected are such that they should be valid
for a wide range of boundary conditions and include parameters appropriate to transient ablation for both charring and
homogeneous materials. The equations should be valid for
ablation material and boundary-layer-edge gas of arbitrary
chemical composition.
Species Conservation Equation
In the absence of thermal diffusion effects, the species
conservation equations [Eq. (19) ] take the form
pu-
pv
= by
^x
(21)
When all diffusion coefficients are equal, Fi = Fj = 1.0, D =
£>ijj (M2/Mi) = SfTC, ^ = Kk} and Eq. (21) reduces to the conventional laminar boundary-layer species conservation equation for equal diffusion coefficients:
bK
bK
pu -^—k + pv ——k
bs
by
by
(22)
Solution of Eq. (22) is often correlated by an expression relating the diffusion mass flux of element k independent of
molecular configuration, j^, to the product of a mass-transfer
coefficient and mass fraction difference
— Kke)
(23)
where the mass-transfer coefficient is related approximately
to the heat-transfer coefficient (Stanton number) by
CM = CH
Utilizing this approximation, the boundary-layer species
conservation equation becomes**
bKk
1
bs
bKk
b (
bZk\
; ^— ^ - (pSDef Tr-J
by
by \
by /
(26)
where an effective diffusion coefficient has bsen defined
(27)
The left-hand sides of Eqs. (22) and (26) represent the
flow of species due to convection, and the right-hand sides
that flow due to diffusion. Based on this observation, the
similarity of Eqs. (22) and (26), and the form of the diffusional
mass flux correlation equation for equal diffusion coefficients
[Eq. (23)], it seems reasonable to postulate the following
correlation equation to characterize solutions of the species
conservation equations when the diffusion coefficients are
not equal:
(28)
Jkw
where the previous relationship [Eq. (24)] between heatand mass-transfer coefficients remains valid; the Lewis
number is given by
Le = SDeff/a
by
jkw =
(25)
(24)
Equation (21) may be put in the form of Eq. (22) by
generalizing an approximation that has been found to be
where a is the thermal diffusivity.
The new variable 2k* is a slightly modified form of Zk
which accounts for the fact that the net mass transfer is influenced by the convective terms on the left-hand side of
Eq. (26) in most boundary-layer applications. In Couette
flow, Z]? would be equivalent to Z^ whereas in boundarylayer flow, the mass fraction retains significance. In order
to account for the relative mass transfer by diffusional
processes, the driving potential is defined on the basis of a
weighted average between Zi and K^:
=E
(29)
Based on analogy with the relative magnitudes of conductive
and convective terms in the energy equation, a value of 7 of
** It is seen that the Z gradient becomes the driving potential
for diffusional mass transfer. Observing that the Fi vary approximately as the square root of molecular weight, Eq. (15)
indicates that the Z fraction falls between a mass and mole
fraction. Thus lighter species will diffuse more rapidly than
heavier species.
KENDALL, RINDAL, AND BARTLETT
1068
Gaseous boundary-layer
edge
m K
r K
y=o-
r
Ablating material
surface
AIAA JOURNAL
appropriate to a charring material, and requiring that
energy be conserved at the ablating surface, yields
PeUeCB(Hsr
-
A»)
PeUeCM
Z
I
*c V
•ke
(pV)whw + 77i
a) Chemical elements
(33)
Gaseous boundarylayer edge
Employing the definition of a blowing parameter,
B' = (pv)w/PeUeCM
and expressing the sensible recovery enthalpy at the boundary-layer edge as the total recovery enthalpy less the chemical
enthalpy
i " t / ****
Ablatingmaterial
surface
T
m h
Isr = Hr-
Downloaded by TOBB EKONOMI VE TEKNOLOJI on December 23, 2014 | http://arc.aiaa.org | DOI: 10.2514/3.4138
b) Energy
Fig. 1
Flow terms at the surface of a charring ablation
material.
-f seems appropriate for boundary-layer flows and has been
adopted herein.
The elemental mass balance at the surface of a charring
ablation material is obtained by requiring conservation of
elements at the surface [see Fig. la]. Employing Eq. (28)
to express the diffusional flux yields
mcKkc + mgKkg = peUeCM(Zkw* - Z * +
(Pv)wKkw + mrKkr
An approximate correlation equation for the chemically
reacting, multicomponent boundary-layer energy equation
with unequal diffusion coefficients can also be rationalized
by comparing to correlation equations that have been employed in the case of equal diffusion coefficients.
The energy equation with equal diffusion coefficients for
a nonablating surface may be obtained from the arguments
proposed by Rosner11
— E jijit" = peUeCB(Hsr - h,m) +
(31)
PeUeCM E
the surface energy balance for a charring ablation material in
a chemically reacting boundary layer with unequal diffusion
coefficients is obtained:
PeUeCH<Hr
-
-
The terms on the left represent the total heat-transfer rate
to the material (the wall heat flux) by molecular conduction
and diffusion. The right-hand side represents the sensible
and chemical energy transfer in coefficient form, the former
being characterized by a heat-transfer coefficient and a
sensible enthalpy driving potential, and the latter being characterized by a mass-transfer coefficient, concentration driving
potentials, and the chemical enthalpies. For equal diffusion
coefficients, such an equation can be rationalized on a number
of semi theoretical bases or can be based on comparisons with
detailed numerical boundary-layer solutions, such as those
performed by Fay and Riddell.1 For unequal diffusion
coefficients, arguments analogous to those leading to Eq.
awqr —
mrhr
-—
dy
(Z™*
~ Ziv*)hf
(35)
Ablation Rate
In order to employ the species conservation equation [Eq.
(30)] and surface energy equation [Eq. (35)] to assess the
material ablation rate, it is necessary to consider the degree
of chemical equilibrium at the surface, since the terms Ziw*,
hsw, and hw depend strongly on the molecular composition of
the gases at the surface. The results presented thus far
are valid independent of the degree of chemical equilibrium
achieved in the boundary layer and at the ablating surface.
In the paragraphs that follow, this degree of generality will
be sacrificed in order to indicate how solutions to the equations may be obtained for the limiting case of chemical
equilibrium. Introducing into Eq. (30) the definitions of
Eqs. (15), and utilizing Dalton's law, yields an expression for
the wall condition in terms of quantities at the boundarylayer edge and in the ablating material:
E
B'
+
Bc'Kkc -
(36)
where Ck% is the number of k atoms in molecular species i,
Pi is the partial pressure of species i, and
tf = P,UeCH(Hsr - hsw) +
PeUeCM Z
-o-€wJV = 0
Equation (35) is valid for homogeneous materials as well, for
which case the gas generation rate in depth, mg = 0, and the
char mass-recession rate, mC) is replaced with the material
mass ablation rate.
(31) yield
«. = ka —— - E
oy w
i
hs
Z
(30)
Energy Equation
= kg -—
(34)
(32)
where it is noted that chemical enthalpy transport is now
characterized by the Z-driving potential of mass transfer.
The energy balance at the surface of an ablating material
may be written utilizing Eq. (32) to express the boundarylayer heat transfer by conduction and diffusion. Referring
to Fig. Ib, which depicts the primary energy transfer terms
ft f ==
Q
ft r ==
mc
It is noted that only (K — 1) of the foregoing equations are
unique. The Kth relation is obtained by requiring the sum
of partial pressures to equal the system pressure
Z Pi = P
(37)
BOUNDARY LAYER COUPLED TO AN ABLATING SURFACE
JUNE 1967
Chemical equilibrium relations may be written considering
formation reactions of each gaseous species i from the elemental gaseous species k. Thus there are I — K reactions
of the form
CkiNk -* Nt
1069
5500
SUR =ACE
TEM PERATUR E
5000
/ ^
(38)
EQUAL DIFFUSIC)N COEFF CIENTS
_
^1
F^^
UNEQL AL DIFFU SION COEFFICIENT s
4500
TIAL GEOMETRY- 0.3-INCH- DIAMETE R NOZZL : THROAT
Similarly, for the formation of condensed phase species
from the gaseous elements
3500
(39)
Downloaded by TOBB EKONOMI VE TEKNOLOJI on December 23, 2014 | http://arc.aiaa.org | DOI: 10.2514/3.4138
The equations of chemical equilibrium corresponding to
Eqs. (38) and (39) may be written in terms of the equilibrium
constant KP(T) for each reaction. For each gas-phase
formation reaction [Eq. (38)], the following equilibrium
relation applies:
(40)
£
cr
H 3000
ce
UJ
Q_
1
BOUNDARY-LAYER-EDGE GAS COMPOSITION
K He = 0.2284, KN = 0.6I9I, K 0 = O.I525
RECOVERY ENTHALPY = 3000 BTU/LB
HEAT-TRANSFER COEFFICIENT, /> e U e C H = 0.39 L 3/FT 2 -SE C
@ TIME ZERO TO 0.226 @ 30 SECONDS
£ 2500
UJ
EQU AL DIFFU SION ^
COE FFICIENT
§-^"^
-"-I
|
^^
I 2000
^
If chemical equilibrium is achieved between the gas phase
.and the surface material, the following equilibrium relation
mav be written:
Ckl InP* ^
1500
JRFACE
ICESSION
i
^
/^ UNEQUAL DIFFUSIO ISI
COEFFICI ENTS
^
1000
^
5 § § 8
k
Cki lnPk =
SERIAL - 60-PERCE'NT GRAP HITE
40-PERC iNT PHEHJOLIC
^
(41)
SURFACE RECESSION, IN. X 10
Z CklNk -+ Ni
4000
1/.,
500 ^^
The equality in Eq. (41) implies the existence of condensed
species 1} if the condensed phase is not present, the inequality
applies. When considering simple systems such as the ablation of graphite in air, the surface is most certainly carbon;
however, when considering ablation of a composite such as
.silica-phenolic, equilibrium considerations may dictate the
presence of Si02, Si, SiC, or C, depending on the freestream
oxidation potential, the mechanical surface removal rate,
and the relative char and resin off-gas ablation rates.
The simultaneous solution of Eqs. (36, 37, 40, and 41)
yields the surface temperature and molecular composition
of the gases adjacent to the surface for specified ablation
rates (Bgf and Bc') and total pressure.ft In addition, the
equality in Eq. (41) indicates which of the possible condensed
species is at the surface.
By specifying a parametric array of pressure, Bc' and Bgf,
a map of boundary conditions satisfying the mass balances
and equilibrium constraints is obtained. This map may be
employed in conjunction with the surface energy equation
[Eq. (35)] and relations describing the "in depth" response
of the ablation material to obtain coupled solutions of the
ablation problem. Some results for a typical charring
ablation material are presented in the following section.
4. Some Coupled Solutions for the Transient
Ablation and Internal Response of Charring
Materials
The equations presented in the previous section have been
programmed for solution with tabular output in a form convenient for characterizing the boundary conditions requisite
to obtaining a solution of the in-depth response of charring
materials. The differential and finite difference equations
for the subsurface solution are based upon the model proposed in Ref. 12 and are developed and presented in Ref. 8.
The means of obtaining the coupled solution are described
in Ref. 13.
The results of several coupled solutions are presented here
which illustrate 1) the difference in ablation rate with and
without the assumption of equal diffusion coefficients in the
ft Equations (36, 37, 40, and 41) yield K-l, 1, I-K, and 1
equations, respectively, to solve for (K) Pks, (I-K) Pi's, and
(1) Tw unknowns.
12
16
T I M E , SECONDS
24
Fig. 2 Comparison of predicted transient ablation response with and without assumption of equal diffusion
coefficients.
boundary layer, and 2) the importance of a transient solution
to characterize the chemical ablation rate of charring
materials.
Comparison of Transient Solutions with and without
the Equal Diffusion Coefficient Assumption
Predicted surface temperature and recession histories are
presented for a graphite phenolic material subjected to a
high-temperature gas mixture of helium, nitrogen, and
oxygen, tt Thermodynamic data and material properties
employed for the calculations are presented in Ref. 15.
Approximately 70 molecular and atomic species were considered in the calculation. The computer program is written
so that all possible species for a given combination of chemical
elements are selected from a large catalog of thermochemical
data. For these particular calculations, 7 was taken as -f-,
and the diffusion factors Fi were presumed to be proportional
to the square root of molecular weight, a supplemental
approximation discussed earlier in this paper.
The results of the two transient ablation calculations are
presented in Fig. 2, where the surface temperature and recession histories are shown as functions of time for boundarylayer transport processes characterized by equal and unequal
diffusion coefficients based upon the methods previously proposed. The ablated depth assuming equal diffusion coefficients is 23% greater than that when consideration is given to
diffusional transport characterized by unequal diffusion coefficients. This result is rationalized from the fact that the
primary oxidizing species have significantly larger molecular
weights and, therefore, smaller diffusion coefficients than the
majority of the boundary-layer species. As a result, relatively
less oxidizing species are transported to the surface than if
all diffusion coefficients were equal.
Jl This gas mixture was employed in an arc-plasma generator
ablative material test to simulate certain aspects of a solidpropellant rocket environment as described in Ref. 14.
KENDALL, RINDAL, AND BARTLETT
1070
*
12
20
28
36
44
A graphite-phenolic and a silica-phenolic nozzle material at
the 7.8-in.-diam throat of an N204-N2H4/unsymmetrical
dimethyl hydrazine (UDMH) propellant rocket engine
operating at a chamber pressure of 100 psia are considered.
The boundary-layer heat-transfer coefficient is evaluated
employing the simplified equation proposed by Bartz16
modified to a boundary-layer enthalpy potential rather than
a temperature potential. Details of the material properties
and thermodynamic data are reported in Ref. 13. The
Lewis number for relating the heat- and mass-transfer coefficients was based on the effective diffusion coefficient £>eff
[Eq. (27)]. A portion of the results from solution of the
equilibrium surface thermochemistry equations (36, 40 and
41) is shown for the graphite phenolic and silica phenolic
materials in Figs. 3a and 3b, respectively, where the normalized char recession rate Bcf is shown for each material as a
function of surface temperature, with the normalized resin
decomposition off-gas rate Bgf as a parameter. The results
shown in the figure include only thermochemical ablation;
that is, no mechanical erosion of the carbonaceous char or the
liquid silica surface is considered [rar = 0 in Eq. (36)]. All
possible chemical reactions including vaporization and
sublimation are considered. The significant effect of the
resin off-gas rate on the char consumption rate is apparent.
The predicted transient ablation rates (Bgf, Bcf, and B')
for each of the materials are shown in Figs. 4a and 4b. Also
shown are the steady-state ablation rates, which are approached asymptotically. Both of the predictions are for
100-sec firings, but the degree of approach to steady-state
conditions for the two is vastly different. The graphitephenolic material has effectively reached steady state after
60 sec, whereas the silica-phenolic requires over 800 sec to
reach 95% of its steady-state ablation rate. Hence, adoption of the steady-state approximation to characterize the
52
Downloaded by TOBB EKONOMI VE TEKNOLOJI on December 23, 2014 | http://arc.aiaa.org | DOI: 10.2514/3.4138
SURFACE TEMPERATURE °R x IO" 2
a) Graphite phenolic, 60-40% by mass
0.3 0.1 0
4
12
20
28
36
44
SURFACE TEMPERATURE °R x IO" 2
52
AIAA JOURNAL
60
b) Silica phenolic, 67-33% by mass
Fig. 3 Normalized throat char-recession rate as a function of surface temperature with decomposition off-gas
rate as a parameter. Chemical equilibrium, environment:
N2O4 - N2H4/UDMH, O/F = 2.0, Pc = 100 psia, throat
diameter = 7.8 in.
The Significance of a Transient-Coupled Solution
Steady-state ablation is a very popular assumption since
it leads to great simplification. In particular, evaluation of
the steady-state ablation rate does not require consideration
of the in-depth response of the material; rather, only the
boundary-layer surface interaction (dependent upon the
virgin material composition and the boundary-layer-edge gas)
need be considered. For steady-state ablation, the following
simplifications apply to the species conservation and energy
conservation equations [Eqs. (36) and (35), respectively]:
Be' Kk
m
c
B'
—— B'c
. STEADY>STATE ~
VALUES
0.2
O.I
40
60
80
100
TIME FROM IGNITION, SEC
120
140
160
a) Graphite phenolic—50 sec required to reach 95% of
steady-state ablation
Bg'Kka =
and
0.04
mchc — kc(c)T/dy)\w
= maha
Therefore, it is pertinent to investigate the validity of this
approximation.
Predictions of the transient thermochemical response of
two charring ablation materials subjected to boundary conditions characteristic of a liquid-propellant rocket engine
throat are presented which demonstrate that the transient
nature of the subsurface response can play a very significant
role in establishing the ablation rate of charring materials.
It is concluded that the assumption of steady-state ablation,
even when the boundary conditions are invariant with time,
may lead to gross errors.
0.02
40
60
80
100
TIME FROM IGNITION, SEC
120
140
160
b) Silica phenolic—860 sec required to reach 95% of
steady-state ablation
Fig. 4 Transient ablation calculation of ablative material
response at 7.8-in.-diam throat of N2O4-N2H4/UDMH
rocket engine. P = 100 psia.
JUNE 1967
BOUNDARY LAYER COUPLED TO AN ABLATING SURFACE
ablation rate for an exposure time even as long as 100 sec
can lead to gross overprediction of the total ablated depth.
Downloaded by TOBB EKONOMI VE TEKNOLOJI on December 23, 2014 | http://arc.aiaa.org | DOI: 10.2514/3.4138
5.
Closure
A technique has been devised which substantially reduces
the complexity of the equations required to characterize the
multicomponent, chemically reacting boundary layer with
unequal binary diffusion cofficients for all species present.
Approximate correlation equations are then proposed to
characterize the transfer of heat and mass for this case.
Employing the proposed boundary-layer correlations, equations are presented to characterize heat and mass transfer
at the surface of an arbitrary ablation material in an environment of arbitrary chemical composition. A computer code
has been programmed for the numerical solution of the indepth transient response of charring materials with the surface
boundary condition coupled to the approximate treatment
of the chemically reacting boundary layer, within the constraints of chemical equilibrium at the ablating surface.
Results of this transient-coupled solution are presented
for several charring materials in two high-temperature
environments. The very significant transient nature of
the solution illustrates the magnitude of errors that may be
incurred if steady-state ablation is assumed. Also, the predicted ablated depth is shown to be significantly changed
when the assumption of equal diffusion coefficients in the
boundary layer is removed.
No attempt has been made to present experimental data
that will correlate properly with the theory presented herein.
Such data are in existence along with at least a like amount
of data that would be poorly correlated. The techniques
proposed herein constitute only a partial theory, a basic
building block for characterizing that portion of ablative
materials behavior attributable to thermochemical ablation,
in general. When consideration is given to particular material-environment combinations, the present technique may
have to be modified to include such factors as heterogeneous
and homogeneous chemical kinetics, micro-spallation, and
the removal of liquid from the surface.
References
1
Fay, J. A. and Riddell, R. F., "Theory of stagnation-point
heat transfer in dissociated air," J. Aeronaut Sci. 25, 73-85, 121
(1958).
1071
2
Scala, S. M. and Gilbert, L. M., "Sublimation of graphite at
hypersonic speeds," AIAA J. 3, 1635-1644 (1965).
3
Lees, L., "Convective heat transfer with mass addition and
chemical reactions," Third AGARD Colloquium on Combustion
and Propulsion (Pergamon Press, New York, 1959).
4
Bird, R. B., "Diffusion in multicomponent gas mixtures,"
25th Anniversary Congress of the Society of Chemical Engineers
(Japan), November 6-14, 1961; published in abbreviated form
in Kagaku Kogaku 26, 718-721 (1962).
5
Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B., Molecular
Theory of Gases and Liquids (John Wiley & Sons Inc., New York,
1954).
6
Libby, P. A. and Pierucci, M., "Laminar boundary layer
with hydrogen injection including multicomponent diffusion,"
AIAA J. 2, 2118-2126 (1964).
7
Pallone, A. J., Moore, J. A., and Erdos, J. L, "Nonequilibrium nonsimilar solutions of the laminar boundary-layer equations," AIAA J. 2,1706-1713 (1964).
§ Kendall, R. M., Bartlett, E. P., Rindal, R. A., and Moyer,
C. B., "An analysis of the coupled chemically reacting boundary
layer and charring ablator," Aero therm Corp., Rept. 66-7
(March 1967).
9
Svehla, R. A., "Estimated viscosities and thermal conductivities of gases at high temperatures," NASA TR R-132 (1962).
10
Svehla, R. A., "Thermodynamic and transport properties
for the hydrogen-oxygen system," NASA SP-1011 (1964).
11
Rosner, D. E., "Similitude treatment of hypersonic stagnation heat transfer," ARS J. 29, 215-216 (1959).
12
Kratsch, K. R., Hearne, L. F., and McChesney, H. R.,
'Theory for the thermophysical performance of charring organic
heatshield composites," Lockheed Missiles and Space Co., Rept.
LMSC-803099 (October 1963).
13
Rindal, R. A., Flood, D. T., and Kendall, R. M., "Analytic
and experimental study of ablation material for rocket-engine
application," Itek Corp., Vidya Div., Rept. NASA CR-54757
(May 1966).
14
Flood, D. T. and Schaefer, J. W., "Simulation of rocketnozzle environments with an arc-plasma generator," AIAA J.
3 1361-1363 (1965).
15
McCuen, P. A., Schaefer, J. W., Lundberg, R. E., and
Kendall, R. M., "A study of solid-propellant rocket motor exposed materials behavior," Itek Corp., Vidya Div., Rept. 149
(February 1965).
16
Bartz, D. R., "A simple equation for rapid estimation of
rocket nozzle convective heat transfer coefficients," Jet Propulsion 27, 49 (1957).