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Hybrid Rocket Fuel Regression Rate Data and Modeling

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Hybrid rocket fuel average regression rate is one of the most important values to accurately determine in the hybrid rocket design process and for rocket performance prediction. Yet there is no comprehensive theory that can be used to reliably predict this quantity. Additionally, regression rate data is difficult to measure. Measured data often contains a high degree of scatter, suffers from scale effects and is generally a closely held secret by those performing the experiments and therefore is unavailable for many propellant combinations. This paper presents a regression rate model that has been developed based on the results of several previous studies. The model is applicable to vaporizing fuels in a cylindrical grain configuration that do not form significant char or melt layers. It accounts for the presence of a pre-combustion chamber upstream of the fuel grain and also for variable gas properties (to a limited degree). The model is compared with existing published regression rate data. The results of the comparison are reasonable given the level of approximation in the model but additional work is required before models of this type will supplant regression rate measurements for rocket design purposes.
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AIAA 2006-4504
1
American Institute of Aeronautics and Astronautics
Hybrid Rocket Fuel Regression Rate Data and Modeling
Greg Zilliac
NASA Ames Research Center, Moffet Field, Mountain View, CA 94035
And
M. Arif Karabeyoglu
Stanford University, Stanford, CA 94305
Abstract
Hybrid rocket fuel average regression rate is one of the most important values to accurately
determine in the hybrid rocket design process and for rocket performance prediction. Yet there is no
comprehensive theory that can be used to reliably predict this quantity. Additionally, regression rate
data is difficult to measure. Measured data often contains a high degree of scatter, suffers from scale
effects and is generally a closely held secret by those performing the experiments and therefore is
unavailable for many propellant combinations. This paper presents a regression rate model that has
been developed based on the results of several previous studies. The model is applicable to vaporizing
fuels in a cylindrical grain configuration that do not form significant char or melt layers. It accounts for
the presence of a pre-combustion chamber upstream of the fuel grain and also for variable gas
properties (to a limited degree). The model is compared with existing published regression rate data.
The results of the comparison are reasonable given the level of approximation in the model but
additional work is required before models of this type will supplant regression rate measurements for
rocket design purposes.
Nomenclature
Research Scientist, NASA Ames Research Center/M.S. 260-1, Member AIAA.
Research Associate, Dept. of Aeronautics and Astronautics, Stanford University, Member AIAA.
a
= Regression rate coefficient
A
= Grouping of variables
p
c = Specific heat at constant pressure
D
= Port diameter
a
E = Activation energy
G = Local instantaneous mass flux
ox
G = Oxidizer mass flux
H
= Channel height
h = Enthalpy or heat transfer coefficient
v
h = Effective heat of gasification
k = Conductivity or D
Re exponent
L
= Grain length or characteristic length
l = Temperature ratio exponent
M
= Molecular weight
m
= Length exponent for a slab grain
m
& = Mass flow rate
ox
m
& = Oxidizer mass flow rate
= Flux exponent or
φ
exponent
O/F = Oxidizer to fuel ratio
Q
& = Rate of heat transfer
#
= Spatially-averaged quantity
#
ˆ = Temporally-averaged quantity
42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit
9-12 July 2006, Sacramento, California AIAA 2006-4504
This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
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#
ˆ = Averaged spatially and temporally
u
R = Universal gas constant
r
= Radial coordinate (see fig. 1)
r
& = Instantaneous local fuel regression rate
T
= Temperature
b
t = Time (burn duration)
v
u
,
= Axial and normal velocity, respectively
y
x
,
= Coordinate system (see fig. 1)
α
= Thermal diffusivity
β
= Blowing coefficient
δ
= Boundary layer thickness
ε
= Emissivity
η
= Boundary layer coordinate
θ
= Boundary layer momentum thickness
µ
= Absolute viscosity
σ
= Stefan-Boltzmann constant
ρ
f = Fuel density
Subscripts
b = Bulk property
bl = Boundary layer
c
= Pertaining to the flame
e
= Edge of the velocity layer
f = Final or fuel
i
= Initial
m
= Mean
o
= Reference or without blowing
ox
= Oxidizer
s
= Surface
I. Introduction
In a hybrid rocket engine, the fuel regression rate is the rate that the fuel surface recedes over the course
of a burn and this quantity has a first order impact on the configuration (i.e. combustion chamber length and
diameter) and therefore the performance of a motor. For example, since the specific impulse of many
hydrocarbon fuels burned with a given oxidizer are of similar level, a high regression fuel will result in a
combustion chamber design that is shorter and of greater diameter in comparison with a motor that employs a
low regression rate fuel (for a single port motor). In order to compare propellants, size a fuel grain, predict the
performance of a hybrid motor, and avoid burn-throughs, accurate regression rate data is of paramount
importance.
A majority of the existing theories used to analyze hybrid rocket fuel combustion employ a fuel
regression rate law that relates the local instantaneous fuel burning rate to the local instantaneous mass flux
through the fuel port in terms of an empirically-based power law. This has been the case since Marxman
derived such an expression in the first comprehensive theoretical treatment of hybrid rocket fuel combustion1
in 1963.
Marxman et al.1, 2 showed that the local instantaneous regression rate
r
&of a fuel slab submersed in an
oxidizer stream is related to the local instantaneous mass flux G by:
23.0
2.0
036.0
=
vc
e
fhh
u
u
GxG
rµρ
& (1)
Where
r
&is the instantaneous local fuel regression rate, G is the instantaneous local mass flux, x is the
distance along the port, ce uu is the velocity ratio of the gas in the main stream to that at the flame, v
hhis
the ratio of the total enthalpy difference between the flame and fuel surface to the effective heat of
vaporization of the fuel and
µ
is the viscosity main stream gas flow (note that the 0.036 coefficient is for
English units as originally derived). For a given propellant system, it is typically assumed that
r
&,G and
x
are variable and the remaining quantities in equation 1 are constants.
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The significance of Marxman’s theory is that it identifies many of the factors that influence fuel
regression rate and how they are related. Marxman modeled the steady state flow in the port of a hybrid rocket
motor as a diffusion flame within a turbulent boundary layer that forms over the regressing fuel surface.
Included in his analyses are the effects of conductive and convective heat transfer to the fuel surface and the
heat transfer limiting “blocking effect” that results from mass departing the fuel surface into the boundary
layer. A radiative heat transfer term (not shown in eq. 1) was also included in an ad hoc fashion but this term
is thought to be small for fuels that do not contain metals.
Over the years, the results of many regression rate tests have proven that the functional form of
Marxman’s regression rate law is valid. Unfortunately, the law as originally derived is not accurate enough for
rocket design purposes in that it predicts an averaged mass flux exponent that is too high and an axial
dependence that is too great. This is not surprising because the law was developed for a hybrid configuration
wherein the fuel is a slab and most practical hybrid motor designs employ a cylindrical fuel configuration.
Hence, the standard practice today is to invoke the form of Marxman’s regression rate law, augmented by
coefficients and exponents derived from subscale tests. Hence, the spatially and temporally averaged
regression rate law is written as:
m
n
ox xGarˆ
ˆ
ˆ=
&
(2)
In equation 2, the coefficient
a
and exponents n and m are propellant dependent constants that that are
determined experimentally. Note that the averaging process makes it possible to express the average regression
in terms of the oxidizer mass flux ox
G (an easily measured quantity) instead of the total mass flux G. This
expression assumes that the regression rate is pressure independent. (Note that the coefficient
a
is not
unitless therefore care must be taken to use consistent units. For the data presented herein, the units of the mass
flux ox
G are g/cm2-sec and the regression rate
r
& is in mm/sec.)
Regression rate data, used to determine the regression rate power law relationship, are typically obtained
from sub-scale testing of a fuel-oxidizer combination in a ground-based facility. Spatially and temporally
averaged regression rate, determined by measuring the fuel mass and port diameter change, are plotted against
the oxidizer mass flux and a nonlinear regression algorithm is then employed to compute the regression rate
law coefficient(s) and exponents. Multiple combustion tests of single-port grains are often required to
construct an average regression rate curve over a range of average oxidizer mass fluxes. In some modern
testing facilities, the instantaneous regression rate is measured using ultrasonic or x-ray techniques to measure
the instantaneous regression rate thus reducing the number of tests required. Using the traditional approach, a
minimum of two (or three if 0
m) tests are required to construct a regression rate curve but ten or more are
desirable for reasonable accuracy.
A complication that arises from experimentally determined regression rate laws based on a series of tests
is that often they contain effects of temporal and spatial averaging that can mask important local regression
rate behavior. Furthermore, the oxidizer-to-fuel ratio is usually not held constant from test to test hence the
data should be O/F ratio corrected. Reference 3 describes these effects and suitable regression rate data
reduction techniques to obtain an accurate and unambiguous average regression rate law.
Even though techniques exist to measure regression rate, the tests are often costly and typically the data
has a greater degree of scatter than is desirable. A minor change to propellant formulation often necessitates a
new test series.
An emerging route to obtaining fuel regression rate is through the use of computational fluid dynamic
codes coupled with modules that model the fuel pyrolysis and chemical reactions. Good agreement was
obtained by Serin4 between the measured and computed regression rate of HTPB burned with GOX using the
commercially available CFD-ACE code. Even so, currently, several limitations preclude the use of CFD in
lieu of combustion tests including turbulence modeling, and the complexities of finite rate chemical reactions
and fuel pyrolysis.
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American Institute of Aeronautics and Astronautics
In the current paper, Marxman’s regression rate law is revisited and used as a starting point in an attempt
to derive a regression rate model of higher fidelity. Grain configuration and port entrance effects are included
in the model. Factors that influence the magnitude of the average mass flux exponent
n
are considered.
Previously published regression rate data are then compared with the modified model.
II. Analysis
A. Dimensionless Parameters of Hybrid Combustion
Several non-dimensional numbers are of interest in the modeling of the diffusion flames and they are
summarized in Table 1.
Table 1. Dimensionless parameters.
Non-
dimensional
Number
Definition Physical Meaning Typical
Value Interpretation
Prandtl
g
ep
k
µc
=Pr momentum diffusivity
thermal diffusivity
1
Approximately the ratio of
the velocity boundary layer
thickness to the thermal layer
thickness.
Schmidt
12
D
µ
Sc e
ρ
= momentum diffusivity
mass diffusivity
1
Lewis
12
Pr DckSc
Le
pe
ρ
== thermal diffusivity
mass diffusivity
1
Note that in some references
Sc
Le Pr
=
Stanton
pee cu
hNu
St ρ
== PrRe
heat transferred
fluid thermal capacity
1
<<
Modified Nusselt number
Nusselt
k
hL
Nu = total heat transfer
conductive heat transfer
1
>>
Dimensionless temperature
gradient at the surface.
Damkohler
c
t
Da τ
τ
= fluid dynamic time
scale
chemical reaction time
scale
1
>>
1
>>
Da implies diffusion
controlled combustion
Reynolds
e
ee Lu
µ
ρ
=Re inertial force
viscous force
1
>>
B. Fuel Regression Model
As mentioned previously, Marxmans regression rate law generally over predicts the fuel regression rate.
An exception to this statement is for low regression rate fuels in a slab configuration. In this section, we will
make use of computation and experimental evidence from several sources to support a few modifications to
Marxman’s theory in order to improve the regression rate prediction for cylindrical fuel grain configurations.
The fuel-surface steady-state energy balance central to the model is shown schematically in figure 1.
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American Institute of Aeronautics and Astronautics
Fig. 1. Conceptualization of the flow and energy balance within hybrid rocket motor with a cylindrical port.
The essence of the model is that a thin flame sheet forms within a boundary layer on the fuel surface. The
flame sheet is fed from below by vaporized fuel and from above by the port oxidizer flow. The fuel is
vaporized primarily by convective heat transfer from the flame sheet to the fuel surface, although for some
fuels (e.g. fuels containing metals) radiation can contribute to the vaporization process. The flame sheet forms
at a location where the oxidizer to fuel mixture ratio is near but less than stoichiometric. It is assumed that the
reaction rate is infinitely fast (i.e. Da>>1) and therefore the chemical kinetics of the reaction are not explicitly
considered and the reaction rate is limited by diffusion of oxidizer and fuel into the flame sheet.
The flame-sheet approximation implies that the all chemical reactions are confined to an infinitely thin
sheet. In reality, the flame sheet has been observed to be approximately 10% of the boundary layer thickness.
Although the fuel port velocity profile is impacted by the fuel blowing, (i.e. transpiration from the fuel surface)
changes to the velocity profile caused by the presence of the flame sheet are negligible5. It is a fairly safe
assumption that the boundary layer flow in the port is turbulent from inception because of the transpiration of
fuel from the surface. To simplify the analysis, boundary and thermal layer similarity is assumed resulting in
fuel and oxidizer concentration profiles that are linearly dependent on the velocity profile.
At the fuel surface, the steady-state energy balance as shown in figure 1 is:
Q
&convection+Q
&radiation in=Q
&conduction out+Q
&phase change+Q
&radiation out
that can be written per unit surface area as:
4
0
4
0ssgf
y
ffg
y
gThr
y
T
kT
y
T
kσερσαε ++
=+
+ == &
(3)
Where g
h is the effective heat (enthalpy) of gasification. For many non-metalized fuels, radiation heat transfer
can be neglected. At the fuel surface, the rate-of-heat transfer per unit area convected from the flame sheet to
the surface is equal to that conducted , +
=
=0y
gsy
T
kQ
&. Therefore the simplified fuel surface energy
balance can be written as:
gfshrQ&
&ρ=
(4)
Determining the heat transfer rate s
Q
& is the crux of the regression rate problem. In general, s
Q
& is
dependent on many geometrical, combustion and flow related factors. An additional complication in a hybrid
rocket motor is that the vaporized fuel leaving the fuel surface substantially decreases the rate of heat
transferred to the surface and therefore the regression rate (aka the “blocking effect” caused by fuel blowing).
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American Institute of Aeronautics and Astronautics
Since a direct measurement of s
Q
& is nearly impossible, one has to resort to similitude and analogies to
estimate the heat transfer rate. Marxman employed the Reynolds analogy to estimate the heat transfer rate for a
turbulent boundary layer in the absence of blowing. He then developed an expression for the reduction in heat
transfer rate in a boundary layer with blowing. We will follow a similar course.
A Stanton number can be defined as huQSt mbsD=ρ
& where h
is the enthalpy difference between
the flame sheet and the fuel surface. The energy equation can then be written in terms of the Stanton number
as:
gfmbshrhuSt &
ρρ =
(5)
The goal now becomes to find a simple way to determine the Stanton number for the reacting flow
environment within the fuel port.
In Marxman’s original model, he assumed that
1
Pr
. This approximation allowed him to choose the
Reynolds analogy, 2/
f
cSt = over more complex correlations (e.g. Chilton-Colburn analogy,
2/Pr 3
2
f
cSt = that is valid for 60Pr6.0
<
<
). Implicit in the
1
Pr
assumption is that the thermal and
velocity boundary layers are of the same thickness. Data2,5 show that the flame sheet in a hybrid rocket motor
resides within the boundary layer (as shown schematically in figure 1) at a
δ
/y ranging from 0.1 to 1.0
(depending on the distance along the grain and the oxidizer choice) and therefore the thermal layer is thinner
than the velocity layer and
1
Pr
>
. Furthermore, the Reynolds and Chilton-Colburn analogies break down for
gas flows where there is a large temperature difference between the surface and the bulk flow resulting in
properties that vary significantly. A relation recommended by Gambill6 (eq. 6) that is applicable for heat flux
levels of 1600 kW/m2 or higher for a turbulent tube flow through a tube and is valid over a range of Prandtl
numbers is the most appropriate analogy identified and so this is the correlation that we will use. Defining a
Nusselt number as
b
x
DkDh
Nu =where x
h is the local heat transfer coefficient and a Reynolds number
b
mb
DDu
µ
ρ
=Re , then PrReD
D
DNu
St =and:
l
bs
k
D
.
DTT
.
St )/(Re0210
Pr 60 =
(6)
In equation 6, D
St ,
Pr
and D
Re are the local values (i.e. at
x
) with properties evaluated at the local
bulk gas temperature b
T. The bulk temperature is an energy-averaged fluid temperature across a tube that will
be approximated as the film temperature
(
)
2/
scbTTT +. In reference 6, 2.0
=
k and
)(0019.029.0Dxl+= but in the current context, kand l will remain a free parameters during the
derivation. Fundamentally, equation 6 relates the convective heat transfer between a gas and a surface to the
fluid friction for a flow through a tube. Equation 6, like all convective heat transfer correlations for turbulent
flow, is empirical. It should be noted that if bsTT =, equation 6 is nearly equivalent to the Chilton-Colburn
analogy for flow in a tube which in turn is nearly equivalent to the Reynolds analogy when
1
Pr
=
. The
denominator of equation 6 compensates for variable gas properties caused by large temperature gradients.
In most hybrid rocket motors of practical interest, the grain configuration is cylindrical with length-to-
port diameter ratios in the range of 5 to 50 and a fuel grain fore-end geometry that resembles a forward-facing
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step. Therefore in the Reynolds number range of 104 to 106 (based on port diameter) typically encountered in
hybrid rocket motors, it is expected that entrance effects should play a role. The HTPB-GOX cylindrical-port
hybrid computational result shown in figures 2-4 illustrates this point. These results were obtained using an
equilibrium-chemistry Navier-Stokes code on a channel configuration with a pre-combustion chamber.
Clearly, the flow is not fully developed until 2.41/
=
Hx which is close to the port exit. Therefore, equation
6 must be modified to account for entrance effects.
Fig. 2. Computed port velocity distribution, 4.3
0=
=x
u m/sec (From reference 4).
Fig. 3. Computed port temperature distribution, 331
0=
=x
T K (From reference 4).
Fig. 4. Computed port density distribution 3.43
0=
=x
ρkg/m3 (From reference 4).
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In reference 7, experiments were performed to investigate the impact of a separated zone on the heat transfer in
a pipe. These experiments are of interest because of their similarity to the flow at the inlet of a cylindrical fuel
grain. In the experiments, a restriction was used within a heated pipe to create a separated zone. The effects of
the restriction on the heat transfer were measured over a range of Nusselt, Reynolds and Prandtl numbers for a
few restriction ratios. Presented in figure 5 is the ratio of local Stanton numberx
St to the fully developed
Stanton number D
St versus distance downstream of the restriction (note that the Stanton number ratio shown
in figure 5 was calculated from Nusselt number data presented in reference 7). The data plotted is for a
restriction diameter ratio of 3/2/
=
Ddwhich is considered to be representative of the entrance effects (i.e.
right trends but magnitude is too high at 2/
Dx) expected near the fore-end of a cylindrical fuel grain. As
one can see in figure 5, a minor separation zone substantially increases the heat transfer for Dx/ less than
about 10 over a range of Reynolds numbers. It should be noted that the boundary layer is turbulent and fully
developed upstream of the restriction, and therefore the increased heat transfer is related to the restriction and
not laminar or transitional flow.
Fig. 5. Tube entrance effects on Stanton number (Pr=3, Nusselt number data source is reference 7).
It was difficult to find a simple function that would fit the data of figure 5 across the complete Dx/
range. Equation 7 was chosen because it is well behaved at large Dx/and does a reasonable job for Dx/
greater than 1. For the comparison shown in figure 5, 6.34
1=C but based on data shown in other references
and also the fact that the fuel port entrance becomes rounded during the burn, it is believed that 0.2
1=C will
better represent the actual geometrically-induced entrance effects encountered in a fuel port and so this value
will be used in the regression rate model.
Dx
D
D
xeC
St
St /4.0
22.0
1Re1
+=
(7)
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In the paragraphs that follow, we will make use of several results that were derived for a zero pressure
gradient turbulent boundary layer (e.g. flat plate boundary layer) with a thickness
δ
and a characteristic
velocity e
u at the boundary layer edge (subscript bl is used to distinguish boundary layer quantities from
those expressed in terms of other characteristic velocities and length scales, e.g. a tube flow). In flows through
cylinders without mass addition, the characteristic velocity typically employed is the mean velocity m
u which
along with D
Re are assumed to be constant. In a hybrid fuel grain port, the velocity increases, the density
decreases and the temperature increases with increasing
x
resulting in D
Re that is not constant with
x
. For
instance, D
Re , based on port centerline quantities, decreases with
x
for the computational results presented
in figures 2-4.
Marxman1 defined a non-dimensional blowing coefficient 2/
)(
fee
s
bl cuv
ρρ
β= to quantify the mass
addition to a boundary layer on a flat plate. He then derived a very fundamental result, namely, an expression
for the velocity profile of a fully-developed turbulent boundary layer with mass injection. Boundary layer
similarity is assumed and the result is valid for any boundary layer with a no-blowing velocity profile that
follows a power law of the form n
e
bl u
uηφ== where 1
<<
n. Typically, for a flat plate, 71=n with
δ
η
/y
=
and for a tube, 91=nwith Dy/2
=
η
when 0
=
β
. The velocity profile with blowing that
Marxman derived is:
+
+= 2
1
2
1bl
n
bl
n
bl β
η
β
ηφ. The velocity ratio desired for flow through a
cylinder is
m
e
bl
mu
u
u
uφφ == . Using the definition of bl
φ and 91=n, it can be shown that
22.1
meuu for a tube-flow velocity profile with mass addition in the range of 1005
<
<
β
where e
u
is the tube centerline velocity. For a tube, assuming the density does not vary significantly with
y
, the
blowing coefficient can be defined as 2/
)(
fmb
s
cuv
ρρ
β= , therefore bl
m
e
bl u
uβββ 22.1== . Hence, for a
tube:
2
22.1
2
22.122.1 9
1
9
1
β
η
β
η
φ+
+
= (8)
As mentioned previously, fuel blowing reduces the heat transfer to the fuel. This effect can also be
quantified in terms of a Stanton number ratio
o
St
St where o
St is the Stanton number without blowing.
Marxman2 invoked the Reynolds analogy coupled with the von Karman momentum integral equation to show
that
( )
5
1
5
4
1
1ln
+
+
=bl
o
bl
bl
o
II
St
St ββ βwhere η
ρρ
δ
θd
u
u
u
u
I
eee
== 1
1
0
for a turbulent boundary
layer with mass injection.1,2,8 In performing the integration, a linear density variation was assumed between the
surface and the edge of the boundary layer. He also showed that this result can be approximated by the simpler
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expression 77.0
2.1
bl
o
St
St βfor 1005
<
<
β
. He then used the Reynolds analogy to show
that
gc
e
bl hh
u
u
=β. This is a very useful result because it quantifies the reduction of heat transfer to the fuel
surface caused by blowing in a very succinct expression that is only a function of a velocity ratio and two
thermodynamic properties. Chiaverini et al.9 deduced a similar o
St
St correlation based on the results of
HTPB-GOX motor tests (when slightly different blowing coefficient definitions are reconciled with each
other). For a tube, 77.0
4.1
β
o
St
St and
gc
mhh
u
u
=49.1β. The Marxman result will be used in the
present analysis even though it was derived assuming that
1
Pr
=
. Since this term is a ratio of Stanton
numbers, inclusion of the Prandtl number in the derivation would result in a quantity o
Pr
Pr that should be
close to one for most situations and therefore have a minimal impact on o
St
St .
So the Stanton number s
St in equation 5 should be replaced by the product
oD
x
DSt
St
St
St
St where
o
St is the Stanton number without blowing, o
St
St is the heat transfer decrement caused by blowing and
D
xSt
St is the entrance effect. Combining equations 5 and 6 results in:
g
m
f
b
oD
x
k
D
.
l
mshh
u
St
St
St
St
TT .
r
=ρ
ρ
RePr
)/(0210 60
&
(9)
The local total mass flux is mbuGρ=. Applying the variable definition m
c
cu
u
=φ, substituting
77.0
1
03.1
gc
ohh
St
St φ and equation 7 into equation 9 leads to the following local instantaneous
regression rate expression:
1
77.0
23.0
/4.0
22.0
1
60 1Pr
)/(022.0 +
+
=k
c
g
Dx
m
k
m
.
l
msf
G
hh
e
GD
C
D
TT
rφ
µµ
ρ
&
(10)
The above equation shows, among other things, that the position of the flame sheet relative to the fuel
surface (as indicated c
φ) has a first order impact of the heat transfer rate to the fuel and therefore the fuel
regression rate. Marxman2 used boundary layer integral methods along with species continuity equations to
derive an expression for the velocity ratio for which the tube flow equivalent is
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( )
( )
g
oxox
g
c
hh
KFOK
hh
FO
ee
++
=/
/22.1
φ where FO/ is the oxidizer to fuel ratio at the flame and e
ox
Kis the
oxidizer mass fraction at the edge of the boundary layer (usually unity). This relation is not immediately
helpful because the local FO/ is unknown but it does show the factors that influence flame stand-off
distance including oxidizer choice.
In classic diffusion flame theory, the flame sheet resides at the location where the FO/ is
stoichiometric. In hybrid rocket motors, the diffusion flame is immersed in a turbulent boundary layer with
mixing and transport aided by the eddy viscosity of the turbulent boundary layer. In addition, the flame sheet
is observed to reside closer to the fuel surface (at a location where FO/ is less than stoichiometric) in
comparison to a classic diffusion flame. In Marxman’s original modeling of PMMA burned with oxygen, the
flame sheet was positioned at a fuel rich local FO/ ratio of 1.5, (i.e. slightly less then the PMMA
stoichiometric FO/ of 1.92) without rigorous justification, (based on some measured results). In the current
model, the flame sheet FO/ ratio will be the stoichiometric value.
Equation 10 can be simplified by grouping variables that are approximately constant. It can be seen in figure 3
that the flame and surface temperatures are constant along the grain. Furthermore it can be assumed that the
flame sheet resides at the same local flame FOalong the grain, therefore c
φ is constant. These
considerations make it possible to assume that c
gm
shh
T
Tφµ,,,Pr, and f
ρ are independent of
x
and
t
and
are roughly constant. Let:
77.0
23.0
60
Pr
)/(
0220 c
g
.
l
ms
k
mhh
TT
.
Aφ
µ
=
(11)
Therefore the final form of the local instantaneous regression rate expression in terms of the total mass flux for
a cylindrical grain is:
kkDx
mf
DGe
GD
C
A
r1/4.0
22.0
1
1+
+µρ
&
(12)
Since the total mass flux G is a quantity that is difficult to measure, it is desirable to express the regression
rate law in terms of the oxidizer mass flux ox
G. It takes several steps to accomplish this goal.
In equation 12, ox
GG in the term Dx
m
e
GD
C/4.0
22.0
1
µ because at the fore end of the grain ox
GG =
and for increasing
x
, 0
/4.0
Dx
e. Therefore:
kkDx
m
ox
f
DGe
DG
C
A
r1/4.0
22.0
1
1+
+µρ
&
(13)
Conservation of mass written between the port entrance and
x
dictates that.
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+= x
fox dxr
D
D
GxG
0
2
4
)( &
ρ
π
π
(14)
Lets assume that fuel is added uniformly along the grain. We can then write the mass flux ratio o
GG in
terms of the
(
)
Lx
FO=at the end of the grain as:
( )
L
x
FOK
G
G
Lx
ox
ox =
+=1
(15)
We can substitute equation 13 into equation 14 and then use equation 15 to get:
( )
+
=
+
++xk
Lx
ox
Dx
m
ox
k
k
ox
ox
dx
L
x
FOK
e
DG
C
D
AG
G
G
0
1
/4.0
22.0
1
111
4
1µ
(16)
The integral in equation 16 can be integrated after using the binomial expansion
( ) ( )
( )
L
x
FOKk
L
x
FOK
Lx
ox
k
Lx
ox
=
+
=
+
+
+1
11
1
and neglecting small terms (note that 11
<
+
k and
1/>
=lx
FO ). The result is:
( ) ( ) ( )
( )
+
+
+
=15.2
21
41 /4.0
22.0
1
2Dx
m
ox
Lx
ox
k
ox
ox
e
DG
L
D
C
L
x
L
x
FOK
k
D
L
DGA
G
Gµ
(17)
Comparing equation 15 and 17 and neglecting small terms, it can be seen that:
( ) ( )
L
D
DGA
K
FOk
ox
ox
Lx4
=
(18)
Equation 17 can be substituted into 13 to obtain the desired result of the local instantaneous regression rate in
terms of the oxidizer mass flux ox
G:
( )( )
k
k
ox
k
ox
Dx
m
ox
f
DG
D
L
DGkAe
DG
C
A
r1
/4.0
22.0
1141 +
++
+µρ
&
(19)
Where
A
is defined by equation 11.
Most measured regression rate results are temporally and spatially averaged results. Several approaches to
averaging measured data are in common use and the techniques and the various pitfalls related to how the
average oxidizer mass flux is defined are described in reference 3. As will be seen, it is difficult to obtain an
exact closed-form solution for the modeled average regression rate and therefore similar techniques to that
used for measured regression rate are invoked. A spatially and temporally averaged regression rate can be
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defined as dtdxr
Lt
rL
t
b
b
=00
1
ˆ&& . It is not possible to obtain an exact solution of this integral. The integration
can be performed numerically, or for short burn times, it can be solved approximately. The approach taken
here is to first perform the spatial average assuming a linear axial
r
& variation. A spatially-averaged oxidizer
mass flux can be written as 2
4DmG oxox π
&
= where
(
)
2/
0Lxx DDD == += . The integral is:
( )
( )
dxDG
D
L
DGkAe
DG
C
A
L
rk
k
ox
k
ox
Dx
m
ox
f
L1
/4.0
22.0
1
0
141
1+
++
+µρ
&
(20)
Therefore, the spatially averaged regression rate is:
( )
( )
k
k
ox
k
ox
m
ox
f
DG
D
L
DGkA
L
D
DG
C
A
r1
22.0
1145.21 +
++
+µρ
&
(21)
A space-time averaged regression rate is given by
(
)
bif tDDr2
ˆ
=
&and a space-time averaged port
diameter can be defined as 2/)(
ˆ
fi DDD += , therefore the spatially and temporally averaged regression
rate is given by:
( )
k
k
ox
k
ox
m
ox
f
DG
D
L
DGkA
L
D
DG
C
A
rˆ
ˆ
ˆ
ˆ
ˆ
14
ˆ
ˆ
ˆ
5.21
ˆ1
22.0
1+
++
+µρ
&
(22)
Where
(
)
2
16
ˆfioxox DDmG += π
& and
A
is defined by equation 11.
As previously mentioned, the
(
)
l
msTT term of equation 6 was included to compensate for variable
properties that arise from the extreme temperature gradients normal to the surface. It has been recognized by
many that better high heat-flux correlations result with the explicit inclusion of a temperature dependency
when properties are evaluated at the bulk temperature. The property most greatly impacted is viscosity which
is strongly dependent on temperature. Physically, when the surface is cooler than the freestream, the boundary
layer velocity profile is fuller in comparison to the isothermal flow because the viscosity is higher than that
calculated using the equivalent bulk temperature. For most fuel grains, DL/ is of the order 20 and therefore
we can neglect the small dependency on Dx/ in the original exponent )(0019.029.0Dxl+= .
Furthermore, equation 6 was established without considerations of mass addition and variable species, both of
which have a major impact on local properties. Nevertheless, 29.0
=
l will be used in the current model, and
all properties related to the correlation will be evaluated at the bulk temperature.
The kexponent in equation 10 can be traced back to the D
Re exponent of the heat transfer correlation
implemented (i.e. equation 6). This exponent has been confirmed to be -0.2 by many heated-wall tube-flow
experiments. Even so, it is a bit of a stretch to expect that the correlation will work perfectly for a fuel port
containing a chemical reaction, a flame zone and mass addition from the walls. Typical values of k found
experimentally from regression rate tests range between -0.50 and -0.25 for fuels without metal additives and
for some fuels containing metals, k approaches 0. All else being equal, the difference between 2.0
=
k and
5.0
=
k can result in a factor of 10 in regression rate. Therefore the accuracy of this exponent is critical to
the success of a regression rate theory.
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Several factors appear to have an impact on kincluding fuel composition, oxidizer choice, radiation and grain
configuration (i.e. slab or cylindrical). Yet as the theory stands, kis a constant regardless of these factors. In
reference 23, it was found that the pyrolysis Arrhenius constant eaRTE, where a
E is the activation energy,
was strongly correlated with the oxidizer mass flux exponent as determined from numerical solutions of a
chemically reacting laminar boundary layer. A correlation of this type is not surprising because fuel production
is related to the chemical degradation of the solid polymer even though the rate of the combustion chemical
reaction is controlled by the interplay of heat transfer to the surface and diffusion of fuel and oxidizer into the
flame zone. In the current model, without much of a physical basis at this point, the following relationship will
be used (note that the flux exponent 1
+
=
kn):
08.0005.0
= su
a
TR
E
k
(23)
This expression was determined by looking at
n
, a
E and s
Ttrends in the measured data of different
propellant combinations.
Several points can be made concerning equations 19 and 22. Most importantly is that regression rate
decreases with scale. For instance, increasing the diameter by a factor of ten while maintaining constant
ox
Gand DL/results in a regression rate decrease of 50%. A similar but not as pronounced scale effect was
obtained computationally in reference 11. It should be noted that radiation (not included in the present model)
should offset the regression rate scale effect to some degree. A second point to observe about equations 19 and
22 is that when equation 23 is implemented, the oxidizer flux exponent is around 0.7. This level is in line with
that obtained experimentally for several common propellant combinations (and is less than the classical
exponent of 0.8 that Marxman obtained). A final point is that both the instantaneous and average regression
rate laws depend on a priori knowledge of the FO/ at the flame sheet, (or alternately, c
φ). This is the only
quantity in the model that is not readily obtainable. It is expected that at some point, a theory will be developed
that predicts the local FO/ (and therefore, c
φ) at the flame sheet location for a diffusion flame in a
boundary layer. Until this occurs empiricism must be relied on to specify this quantity.
It should be reiterated that the model developed within this paper is most applicable to fuels that do not
form significant melt or char layers. A char layer acts like an insulating blanket on the fuel surface that alters
the heat transfer rate to the fuel. It may be possible to include the effects of char via a Stanton number ratio
based on char number similar to the approach used to account for the blowing effect. The melt layer that forms
on the surface of some high regression rate fuels (i.e. n-alkanes such as paraffin) results in a fuel entrainment
mechanism into the flame zone that alters the basic fuel surface energy balance of equation 3.
Additional factors (aside from radiation effects) known to impact regression rate that are not incorporated
in the model include the effects of fuel surface roughness, combustion chamber pressure (for some fuels) and
combustion efficiency. Furthermore, finite rate chemical kinetics, additional aspects of fuel pyrolysis, and
flame sheet curvature may be important for some fuel systems.
The local bulk temperature b
T is required in order to specify e
µ. It can be seen in figure 3 that the
temperature is highly variable throughout the domain with the temperature along the symmetry axis varying
between ox
T at the inlet and c
T. The bulk temperature is computed approximately as
(
)
2/
scbTTT +
where c
T is the adiabatic flame temperature computed by an equilibrium chemistry code. An additional
complication is that the gas is a mixture of several species. During the fuel pyrolysis process, the polymer
breaks down into low molecular weight volatile fuel species near the fuel surface. For HTPB burned with
oxygen, the most prevalent species found between the surface and the flame zone are C4H6, C2H4, CH4 and
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American Institute of Aeronautics and Astronautics
CO. Above the flame, the main species are O2, CO2 and H2O. Therefore determining the viscosity of the port
gas is difficult. The approach taken here is to use the approximate formula22 (from the Kinetic Theory of
Gasses):
7
21069.26
×
=σ
µb
eMT secmkg
(24)
Where
M
is the molecular weight, and
σ
is the hard-sphere diameter in Angstroms and b
T is specified in K.
As a rough approximation, it is assumed that on average,
(
)
2
Lxox MMM =
+= and 0.5
=
σ
Angstroms
(an average value). Fortunately, the viscosity in
A
of the regression rate expressions is raised to
approximately the power of 0.2, so a factor of two change in b
T results in only an 8% difference in m
µ. For
averaged regression rate calculations, b
T is that found at 2/Lx
=
.
The effective heat of gasification g
h is the difference between the enthalpy of the solid in the initial state
at ambient temperature and pressure, o
T and o
P and the enthalpy of the volatile thermally decomposed
products at s
T and c
P:
+++= s
T
Tvdfpg hhhdTch
0
(25)
Where f
h is the enthalpy of fusion, d
his the enthalpy associated with fuel thermal degradation and v
h
is the enthalpy of vaporization of the decomposed products. The effective heat of gasification is g
h is a
quantity that can be measured in a constant heat flux gasification device or a flaming calorimeter. These
measurements should be done under conditions that match the fuel surface regression rate.
In the model, the quantity h
is the enthalpy change of the gas between the flame and the fuel surface.
This quantity is dependent on the local FOin addition to temperature. This quantity can be calculated using:
(
)
(
)
os
s
poc
c
pscTTcTTchhh ==
(26)
C. Fuel Regression Rate Data
The average regression rate for various commonly used hybrid rocket fuels burned with oxygen is shown
in Figure 7 and data related to the burning tests that the regression rate curves were derived from are shown in
Table 2. The curves shown were obtained by applying a nonlinear regression to measured data from many
sources in the open literature over the oxidizer mass flux range encompassing the tests. Only data from
cylindrical combustion chamber configurations are shown and no additional corrections beyond those of the
original source have been applied.
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Fig. 7. Measured average fuel regression rate with oxygen.
Table 2. Data from regression rate tests.
Results summary of the average regression rate with oxygen for various fuels
No. Fuel a n No.
of
Tests
Chamber
Pressure
Range
(MPa)
Average
O/F
Ratio
Range
Data
Reduction
Technique
Oxidizer
Mass Flux
Range
(g/cm2-sec)
Ref.
1 Paraffin, SP1A 0.488 0.62 65 1.1-6.9 1.0-4.0 DA 1.6-36.9 15
2 HTPB, (Thiokol) 0.146 0.681 16 - - - 3.8-30.2 16
3 HTPB+19.7%AL 0.117 0.956 2 1.2 - OA 5.1-23.0 17
4 HTPB 0.304 0.527 3 2.0 - OA 6.2-31.0 17
5 HTPB+20%GAT 0.473 0.439 5 - - - - 18
6 PMMA 0.087 0.615 8 0.3-2.6 - - 3.3-26.6 19
7 HDPE 0.132 0.498 4 0.7-1.3 3.8-5.9 DA 7.7-26.1 20
8 PE Wax, Marcus
200 0.188 0.781 4 0.5-1.2 2.2-3.2 DA 4.8-15.8 20
9 PE Wax, Polyflo
200 0.134 0.703 3 0.6-1.2 1.6-1.7 DA 4.4-16.3 20
10 HTPB 0.194 0.670 6 - - OA 17.5-32.0 21
11 HTPB+13% nano
Al 0.145 0.775 12 - - OA 16.5-34.2 21
12 Paraffin, FR5560
+ 13% nano Al 0.602 0.730 8 - - OA 14.5-29.0 21
13 Paraffin, FR5560 0.672 0.600 4 - - OA 6.3-12.3 21
14 Paraffin, FR4550 0.427 0.748 3 0.7-? 1.3-1.8 DA 4.3-11.9 20
Regression rate equation: m
n
oxGar=
& with 0
=
m
For use with Go with units of gm/cm2-sec, produces an average regression rate in mm/sec.
DA: Diameter Averaged, FA: Flux Averaged, AA: Area Average, OA: Other averaging technique applied
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A couple of disclaimers are in order concerning the data presented in Fig. 7. Many factors including
scale, O/F ratio, combustion chamber configuration, oxidizer injector design, fuel composition (i.e. trace
additives) and processing, ignition and thrust termination transients, data reduction and experimental technique
greatly impact the accuracy of reported regression rate data. This is one reason for the disparity between
reported regression rate data for seemingly similar propellant combinations. An attempt was made to choose
data from reliable sources, nevertheless, the data should be used for design purposes only and the actual
regression rate of a fuel and oxidizer combination for any given application should be independently verified.
Fuel composition and processing can greatly affect the regression rate. This is particularly true for
polymer-based fuels such as HTPB. Typical HTPB fuels are long-chain hydrocarbons that result from the
mixing of a resin with a hardener and often a plasticizer, anti-oxidant, dispersant and an opacifier. The actual
recipe and fuel processing techniques used varies widely and this information is usually closely held by
companies as trade secrets.
D. Comparison of Model with Measured Data
The space-time average fuel regression rate predicted by the model developed in the preceding section is
shown in figure 8 for three propellant combinations. The model was exercised using fuel grain dimensions
equivalent to that of the test article. The Prandtl number in the model has been assumed to be equal to one
uncertainty in the flame FO ratio and port gas viscosity makes it difficult to determine a value of greater
accuracy. Data supplied to the model is listed in Table 3 with the data source referenced where applicable.
Several of the parameters used in the model were calculated using the thermo-chemistry code CEA that was
developed at the NASA Glenn Research Center.
It was found that the model is very sensitive to specification of the fuel surface temperature s
T. Small
adjustments in this temperature can easily result in a regression rate change of a factor of two. The surface
temperature has an impact on the bulk temperature that is used in property determination and also on the
kexponent, so it is understandable why this quantity is important. Since s
T is an input to the model and
reliable measurements of s
T are difficult to obtain, (they must be performed under equivalent heat transfer
rates in a specially designed test rig) some liberty was taken in the s
Tvalue choice to produce a favorable
regression rate match. The slope of the modeled results in figure 8 does not match the measured data very well.
Since the slope is directly related to the oxidizer mass flux exponent
n
, the poor agreement is an indication
that equation 23 could be improved. Finally, the port gas viscosity has a significant impact on the modeled
results and this value has been determined using a very simple relationship (i.e. equation 24) that is probably
not as accurate as desired.
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Fig. 8. Modeled average fuel regression rate with oxygen.
Table 3. Data used to determine the modeled fuel regression rate (with oxygen).
PMMA HDPE HTPB
Average Formula (C5H8O2)n (C2H4)n (C7.337H10.982O0.058)n
MW of repeat unit, (g/mol) 100 28 100
f
h, Heat of Formation (kcal/mole) -102.9 -53.8 13 -2.97
O/FStoic 1.92 3.0 2.7
O/Fopt (at optimal Ispvac) 1.7 2.7 2.5
Tc ( at O/F, K) 3483 3626 3701
Ts (K) * 500 840 935
ex
M ( at O/F, g/mol) 27.03 24.1 24.63
s
p
c (gas at surface, J/g K ) 1.548 11 3.0 2.386 11
c
p
c (gas at flame, J/g K ) 7.16 8.02 7.88
h
(eq. 26, J/g) 22306 24734 25296
g
h (J/g) 966 11 2200 12 1812 11
f
ρ (solid, kg/m3) 1100 11 959 12 930 11
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American Institute of Aeronautics and Astronautics
m
µ (gas, eq. 24, kg/m sec) at m
T 1.29×10-4 1.34×10-4 1.37×10-4
c
φ(calculated using definition) 0.65 0.73 0.72
a
E* (kJ/mol) 160 264 203
D
^ (cm) 2.08 1.45 3.00
DL^ 12.21 21.0 13.3
^ Values determined from grain configuration of test motor used in comparison
Calculated using CEA thermo-chemistry code, NASA Glenn.
* from measured data under similar average regression rate conditions
III. Conclusions
This paper presents a regression rate model that has been developed based on the results of several
previous studies. The model is applicable to vaporizing fuels in a cylindrical grain configuration that do not
form significant char or melt layers. It accounts for the presence of a pre-combustion chamber upstream of the
fuel grain and also variable gas properties (to a limited degree). The model is compared with existing
published regression rate data and the comparison is reasonable given the level of approximation in the model.
The modeled oxidizer mass flux exponent is too high in comparison to that obtained by curve fits of measured
data but it is closer to measured values than the exponent predicted by Marxman’s classical regression rate
theory.
The model is very sensitive to several parameters including the average fuel surface temperature and port
gas absolute viscosity. The modeled regression rate can be off by a factor of two or more depending on the
accuracy of the data used in the model. A major short coming of the model is the necessity of specifying the
oxidizer-to-fuel ratio a priori (specified as stoichiometric even though the actual value may be less for some
propellant combinations). Nevertheless, the model serves as a good starting point in assessing the factors that
influence hybrid fuel regression rate.
Acknowledgment
This paper contains data and analysis originated under NASA cooperative agreements NAG3-2615 with
the NASA Glenn Research Center and agreements NCC2-1172 and NCC2-1300 with the NASA Ames
Research Center.
References
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Symposium (International) on Combustion, Academic Press, 1963, pp 371-372.
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3 Karabeyoglu, A., Cantwell, B.J. and Zilliac, G., “Development of Scalable Space-Time Averaged Regression
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and Regression Rate Evaluation” AIAA 2003-4462, Huntsville, Al, 2003.
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American Institute of Aeronautics and Astronautics
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American Institute of Aeronautics and Astronautics
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... U-net consistently provides low error image segmentations to allow accurate computation of the regression rate of the fuel. important measurement since it directly defines the motor thrust and geometry (e.g., a higher regression rate results in smaller combustion chambers) [54]. Low fuel regression rates correspond to low fuel mass flow rates, leading to excess oxidizer passing through the motor, which reduces the available thrust and therefore reduces the overall efficiency [21]. ...
... Another drawback to weighing the fuel grain is that it does not provide data of local and instantaneous regression rates which is useful for validating regression rate models [54]. Other methods of regression rate measurement include using combustion chamber pressure [33], x-ray radiography [10], initial and final port diameter [26,46,54], and ultrasonic transducers [7,9,32]. ...
... Another drawback to weighing the fuel grain is that it does not provide data of local and instantaneous regression rates which is useful for validating regression rate models [54]. Other methods of regression rate measurement include using combustion chamber pressure [33], x-ray radiography [10], initial and final port diameter [26,46,54], and ultrasonic transducers [7,9,32]. More recently, [4] developed an imaging-based regression rate measurement for a 2D slab burner experiment. ...
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This study presents an imaging-based deep learning tool to measure the fuel regression rate in a 2D slab burner experiment for hybrid rocket fuels. The slab burner experiment is designed to verify mechanistic models of reacting boundary layer combustion in hybrid rockets by the measurement of fuel regression rates. A DSLR camera with a high intensity flash is used to capture images throughout the burn and the images are then used to find the fuel boundary to calculate the regression rate. A U-net convolutional neural network architecture is explored to segment the fuel from the experimental images. A Monte-Carlo Dropout process is used to quantify the regression rate uncertainty produced from the network. The U-net computed regression rates are compared with values from other techniques from literature and show error less than 10%. An oxidizer flux dependency study is performed and shows the U-net predictions of regression rates are accurate and independent of the oxidizer flux, when the images in the training set are not over-saturated. Training with monochrome images is explored and is not successful at predicting the fuel regression rate from images with high noise. The network is superior at filtering out noise introduced by soot, pitting, and wax deposition on the chamber glass as well as the flame when compared to traditional image processing techniques, such as threshold binary conversion and spatial filtering. U-net consistently provides low error image segmentations to allow accurate computation of the regression rate of the fuel.
... It is calculated by semiempirical expressions that depend generally on Re and the rugosity along the wall, and vary from one geometry to another. To model the entrance effects of the flow in the channel on the skin friction, the law provided by Zilliac and Karabeyoglu [28] in their modeling of the physics of a HRE [Eq. (11)] has been employed. ...
... Moreover, the comparison with the 2-D numerical resolution is considered to be good enough too when taking into account the simplicity of our model for its application to the engine predesign phases.The next verification step for a non-reactive flow case has been focused on the effect of surface blowing on a flat plate. The analytical solution of the von Karman's integral equation for a zero pressure gradient over a porous wall with mass injection is presented in Eq.(28), where the relation of C f with the blowing parameter given in Eq. (13) for a low mass flow injection has been considered. In this case, the skin friction coefficient without blowing appearing in Eq.(13) corresponds to that of a flat plate, which is defined by C f 0 = 0.059Re −0.2x[42]. ...
Article
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A pseudo 2-D model for the cylindrical combustion chamber of a hybrid rocket is developed in this article with the purpose of its integration in a system design tool used for engine predesign phases. A compromise of simplicity and accuracy is thus required. Integral transient boundary layer equations are included and coupled with the mass and energy balances at the fuel surface, where fuel pyrolysis is described by an Arrhenius law. Studies assessing the model in the non-reactive case provide satisfactory results regarding the description of the boundary layer. Concerning the hybrid case, the computed steady-state fuel regression rates are found to be in the low region of the reference values coming from semiempirical laws for several oxidizer/fuel couples and oxidizer mass fluxes, producing errors up to 50% for certain sources. The effect of mass flux and motor size is correctly simulated. Eventually, the whole engine operation is assessed through the comparison of the computed results with three experimental tests realized at our laboratory and using a 87.5% Hydrogen Peroxide/High-Density Polyethylene couple. Corresponding results lead to regression rate errors varying in a wide range (from 10 to 40%). Such differences, although high, remain acceptable for our engine predesign phase application.
... Specific impulse (usually abbreviated Isp) is a measure of how efficiently a reaction mass engine (a rocket using propellant or a jet engine using fuel) creates thrust. For engines whose reaction mass is only the fuel they carry, specific impulse is exactly proportional to the effective exhaust gas velocity [23,24]. ...
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The paper aims at understanding the various parameters of hybrid rocket motors such as thrust control, specific impulse, burn rate, efficiency, and specific fuel consumption through the study of various research papers. Thrust vector control (TVC) is a vital part of space/aerospace manoeuvrings. The specific impulse of a rocket propellant is a rough measure of how fast the propellant is ejected out of the back of the rocket. The burn rate is the pace at which a new company is running through its start-up capital ahead of it generating any positive cash flow. Efficiency results from ground tests of single-port hybrid rocket engines show measured characteristic velocity efficiency ranging from 55 to 97 percent. Specific fuel consumption (TSFC) is the fuel efficiency of an engine design with respect to thrust output.
... The research in this domain follows mainly two directions, one to find/develop new fuels, pure or compounds, with higher regression rates; and the second one to refine/develop the experimental measurement techniques for the regression rates with better accuracy in space and time. For instance, ref. [1] presents a regression rate model which is valid for vaporizing fuels in a cylindrical grain configuration without char or melt strata, the comparison with existing pub-lished data is reasonable; ref. [2] HTPB polymer has been characterized through its ballistic properties, mechanical tests, and thermochemical evaluations. The authors of ref. [3] used a pressurized combustion chamber to determine the regression rate of a hybrid fuel using two flow measuring methods, the throttling to measure the flue gas flow and the weight loss method to determine the O/F ratio; the authors of ref. [4] wrote a review regarding the experimental measuring methods to evaluate the regression rates, based on 104 references, with special attention given to measurements of very fast processes, i.e., direct high speed photography, ultrasound techniques and X-ray radiography, and microwave and laser methods with high spatial and temporal resolution of the transient regression rate. ...
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The paper presents a useful engineering model for the design of preliminary hybrid rocket engines. This model involves the experimentally obtained equation regarding the speed of the burning interface between solid fuel and gaseous phase, called the regression rate. This regression rate characterizes the mass rate of the burning fuel and additionally the mass rate of the oxi-dizer through the imposed ratio of these mass rates during combustion. The preliminary design is applied to combustion using pure HTPB and gaseous oxygen and was developed for three cases, constant regression rate, constant oxygen mass flow rate and constant O/F (ratio of mass rates of oxygen and of fuel). The design evaluates the initial fuel port geometry, the initial mass of fuel and oxygen, the combustion time, the thrust at sea level, and the time-dependent functions of regression rate, of fuel and oxygen mass rates, and of thrust. It was assumed that the regression rate formula applies at any time of combustion. These evaluations can also be applied for other fuel/oxygen couples by knowing the correct formula for the regression rate.
... The fuel and oxidizer are in two different states: typically the fuel is a solid and the oxidizer is a gas/liquid, providing a highly dense fuel source similar to a bi-propellant solid motor but with the throttle control of a liquid motor [3][4][5]. The fuel regression rate (̇) corresponds to the rate that the fuel recedes over the course of a burn and is an important measurement since it directly defines the motor thrust and geometry (e.g., a higher regression rate results in smaller combustion chambers) [6]. Low fuel regression rates correspond to low fuel mass flow rates, leading to excess oxidizer passing through the motor, which reduces the available thrust and therefore reduces the overall efficiency [7]. ...
Article
This study presents an imaging-based deep learning tool to measure the fuel regression rate in a 2D slab burner experiment for hybrid rocket fuels. The slab burner experiment is designed to verify mechanistic models of reacting boundary layer combustion in hybrid rockets by the measurement of fuel regression rates. A DSLR camera with a high intensity flash is used to capture images throughout the burn and the images are then used to find the fuel boundary to calculate the regression rate. A U-net convolutional neural network architecture is explored to segment the fuel from the experimental images. A Monte-Carlo Dropout process is used to quantify the regression rate uncertainty produced from the network. The U-net computed regression rates are compared with values from other techniques from literature and show error less than 10%. An oxidizer flux dependency study is performed and shows the U-net predictions of regression rates are accurate and independent of the oxidizer flux, when the images in the training set are not over-saturated. Training with monochrome images is explored and is not successful at predicting the fuel regression rate from images with high noise. The network is superior at filtering out noise introduced by soot, pitting, and wax deposition on the chamber glass as well as the flame when compared to traditional image processing techniques, such as threshold binary conversion and spatial filtering. U-net consistently provides low error image segmentations to allow accurate computation of the regression rate of the fuel.
Article
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The low regression rate of Hybrid Rocket Engines (HREs) is one prominent characteristic that is addressed in most abstracts concerning hybrid propulsion. Over the years, researchers developed and investigated numerous ways to tackle the low regression rate problem of HREs. This article is a collection and assessment of these diverse methods and designs. It allows for a quick overview of the different mechanisms that are being employed and can serve both as information and inspiration. The enhancement ideas are grouped together as (a) adjustments to the solid fuel chemical properties, (b) advanced injection methods and concepts and (c) improving the combustion chamber design. These different techniques are discussed and their individual impact on the regression rate is assessed both qualitatively and quantitatively. All methods that are presented come with a different set of advantages and disadvantages, making the regression rate enhancement a trade-off problem. In our view, the most promising designs and methods are those that only call for minor adjustments to the HRE design, as they can be also added to already existing engines. Above all, it is to be said that regression rate enhancing techniques that change the unique features of HREs (namely safety, simplicity and low cost) are to be employed with caution. Only if the achievable regression rate increase is justifying the implications for the HRE in the envisioned use-case, these concepts represent promising alternatives to the status quo.
Article
This paper presents a closed-loop thrust control method for laboratory-scale hydrogen peroxide and polyethylene hybrid rocket motors. In this paper, the thrust is used as the feedback signal, and the oxidant flow adjustment is achieved by variable area cavitating venturi. The hot test based on thrust closed-loop control was carried out. The closed-loop control significantly improves the thrust control accuracy as demonstrated by the hot test with constant thrust. To obtain thrust closed-loop control parameters and improve control accuracy, a nonlinear increment proportional-integral-derivative algorithm is proposed. With the help of valve test parameters and thermodynamic calculation results, the method can directly obtain the control parameters of the thrust closed-loop control, which avoids the need for a large number of ground hot tests to debug the control parameters, and greatly reduces the test cost. For the purpose of improving thrust control accuracy, a linear variable area cavitating venturi was selected as the flow regulator and hydrogen peroxide and polyethylene were used as the propellant, which made the thrust approximately proportional to the flow rate. The experimental results show that the target thrust values of three stages in the test process are 400 N, 600 N and 400 N, the actual average thrust values are 399.8 N, 599.8 N and 400.8 N respectively, and the average thrust control accuracy is better than 0.5%.
Article
This paper presents the status of developments worldwide regarding use of hybrid rocket motors for space transportation. Historical roots are presented and reasons for revisiting hybrid technology after a few decades of limited interest are examined. Modern developments in sounding rockets, reusable suborbital systems and launch vehicles are discussed with particular focus on propellant technology. Various propellant combinations include use of liquid oxygen, hydrogen peroxide, nitrous oxide and nitrous oxide-oxygen mixtures as oxidizers. Different fuels are considered, taking into account performance, as well as inter alia obtainable regression rates. Results of preliminary calculations for vehicles using different propellant combinations are presented and analysed. This is compared with proposed configurations of hybrid rockets worldwide. Unresolved problems and several unknowns are pointed out, including hybrid rocket motor scalability issues, large motor combustion instabilities, combustion efficiency of metalized fuels, propellant volumetric performance and mass of fuel residuals in case of wagon wheel grain geometry. It is discussed whether new-space hybrid launch vehicles, while typically with limited stage reusability, may be cost-competitive in regard to other chemical rocket propulsion system developments. The paper is summarized with a list of potential future advances and technical opportunities. The main purpose of the conducted research is to provide a comparison between different hybrid propulsion technologies available, or currently under development, worldwide.
Article
Hydroxyl-terminated polybutadiene (HTPB) is a common ingredient in rocket propellants, but its thermochemical properties (composition, density, and heat of formation) are not well defined. A survey of the literature and thermochemical databases indicated wide ranges for these properties, especially heat of formation. Six group additive schemes were used to estimate the heat of formation of HTPB and analyze the effects of hydroxyl functionalization and curing reactions. Good agreement is observed between the methods for HTPB isomer units, but cumulative differences result in significant disparities for larger, practical polymers. Increased hydroxyl functionality and the curing reaction are predicted to yield nonnegligible decreases in the heat of formation. The heat of formation of propellant-grade, isophorone-diisocayante-cured HTPB R-45M (C213.8H323.0O4.6N2.3) was computed as 342 kJ/mol or 114 kJ/kg. Chemical equilibrium analyses were completed for solid propellants composed of ammonium perchlorate and HTPB, and for hybrid rocket engines based on HTPB burning with liquid oxygen or nitrous oxide, where the heat of formation of HTPB was varied within a reasonable range. The chemical equilibrium analysis computations indicated that combustion gas properties and theoretical propellant performance can vary up to 5% within practical operating conditions for the range of HTPB heats of formation implemented.
Article
In this study, impact of the port diameter and length changes of a solid fuel on its performance parameters was investigated to propose design considerations for the solid fuel for hydrogen peroxide/high-density polyethylene hybrid rockets. A laboratory-scale hydrogen peroxide/high-density hybrid rocket was designed to perform a parametric study of the solid fuel through experiments. The ratio of the nozzle throat area to the fuel port area and ratio of the fuel length to the fuel port diameter were used to design the port diameter and length of the solid fuel. Altering the fuel port diameter affected the characteristic velocity efficiency, oxidizer mass flux, velocity-based erosive burning, oxidizer-to-fuel ratio, and regression rate. Altering the fuel length affected the oxidizer-to-fuel ratio and characteristic velocity efficiency and had little effect on the regression rate because it did not significantly contribute to the change in the oxidizer mass flux. No improvement was observed in the characteristic velocity efficiency after the ratio of the fuel length to the fuel port diameter reached 20. Finally, design considerations for the port diameter and length of the solid fuel for a hydrogen peroxide/high-density polyethylene hybrid rocket were suggested based on the experimental results considering the changes in the ratio of the nozzle throat area to the fuel port area and ratio of the fuel length to the fuel port diameter.
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Recent research at Stanford University has led to the identification of a class of paraffin-based fuels that burn at surface regression rates that are three to four times that of conventional hybrid fuels. The approach involves the use of materials that form a thin, hydrodynamically unstable liquid layer on the melting surface of the fuel. Entrainment of droplets from the liquid-gas interface substantially increases the rate of fuel mass transfer, leading to much higher surface regression rates than can be achieved with conventional polymeric fuels. Thus, high regression rate is a natural attribute of the fuel material, and the use of oxidizing additives or other regression rate enhancement schemes is not required. The high regression rate hybrid removes the need for a complex multiport grain, and most applications up to large boosters can be designed with a single port configuration. The fuel contains no toxic or hazardous components and can be shipped by commercial freight as a nonhazardous commodity. At the present time, grains up to 0.19 m [19.1 cm (7.5 in.)] in diameter and 1.14 m [114.8 cm (45.2 in.)] long are fabricated in a general-purpose laboratory at Stanford University. To further demonstrate the feasibility of this approach, a series of scale-up tests with gaseous oxygen have been carried out using a new Hybrid Combustion Facility (HCF) at NASA Ames Research Center. Data from these tests are in agreement with the small-scale, low-pressure, and low mass flux laboratory tests at Stanford University and confirm the high regression rate behavior of the fuels at chamber pressures and mass fluxes representative of commercial applications.
Article
Experiments were performed to determine the effect of flow separation on the heat-transfer characteristics of a turbulent pipe flow. The flow separation was induced by an orifice situated at the inlet of an electrically heated circular tube. The degree of flow separation was varied by employing orifices of various bore diameters. Water was the working fluid. The Reynolds number and the Prandtl number, respectively, ranged from 10,000 to 130,000 and from 3 to 6. The measurements show that the local heat-transfer coefficients in the separated, reattached, and redevelopment regions are several times as large as those for a fully developed flow. For instance, at the point of reattachment, the coefficients were 3 to 9 times greater than the corresponding fully developed values. In general, the increase of the heat-transfer coefficient owing to flow separation is accentuated as the Reynolds number decreases. The point of flow reattachment, which corresponds to a maximum in the distribution of the heat-transfer coefficient, was found to occur from 1.25 to 2.5 pipe dia from the onset of separation.
Conference Paper
A regression rate equation is developed for hybrid motors in which convective heat transfer to the grain is the rate-limiting mechanism. Transient as well as steady-state phases of the operation are considered. The theoretically predicted flame position and regression rate for a flat-plate flow are compared with measurements from a Plexiglas-oxygen slab burner. The regression rate equation is also applied to a cylindrical port and compared with results from a tube burner. Good agreement is demonstrated in both cases.
Article
A theoretical treatment of the turbulent boundary layer with heat transfer, mass transfer, and chemical reactions is presented. The analysis differs from earlier ones, primarily in the way the Spalding mass-transfer number B is related to the thermodynamic characteristics of the flow and in the treatment of the heat-transfer "blocking effect" caused by wall mass addition. The dual role of the mass-transfer number as a similarity parameter of the boundary-layer flow, and as a thermochemical parameter characterizing the reactants, is discussed. The Howarth-Dorodnitsyn coordinate transformation accounts for the effect of variable fluid properties, and, in the present treatment, a relatively easy measurement of the boundary-layer thickness serves to define the reference state. The analysis provides good agreement with experimentally observed heat and mass-transfer rates even at relatively large values of B, where earlier analyses using a "thin-film" approximation are not very accurate. Diffusion-limited turbulent boundary-layer combustion calls for certain modifications in the classical diffusion-flame model. The analysis shows that the flow field of the turbulent boundary layer may substantially affect the mixture ratio of the flame. Data relevant to flame position and stoichiometry are cited. It is found that the theory and measurements are qualitatively entirely consistent, and the quantitative agreement is remarkably good. For example, both the theory and the data indicate that the oxidizer-fuel ratio of a Plexiglas-oxygen flame approaches a value of about 1.4 at high Reynolds Numbers, whereas the stoichiometric ratio is 1.92. More complete experimental information will be required to establish clearly the quantitative limitations of the theory, but the available data support the general conclusion that the combustion mixture ratio depends to a significant degree on the Reynolds Number of the flow.
Article
An experimental investigation of the regression-rate characteristics of hydroxyl-terminated polybutadiene (HTPB) solid fuel burning with oxygen was conducted using a windowed, slab-geometry hybrid rocket motor. A real-time, x-ray radiography system was used to obtain instantaneous solid-fuel regression rate data at many axial locations. Fuel temperature measurements were made using an array of 25-mu m fine-wire embedded thermocouples. The regression rates displayed a strong dependence on axial location near the motor head-end. At lower mass Bur levels, thermal radiation was found to significantly influence the regression rates. The regression rates were also affected by the addition of activated aluminum ponder. A 20% by weight addition of activated aluminum to HTPB increased the fuel mass flux by 70% over that of pure HTPB. Correlations were developed to relate the regression rate to operating conditions and port geometry for both pure HTPB and for HTPB loaded with certain fractions of activated aluminum. Thermocouple measurements indicated that the fuel surface temperatures for pure HTPB were between 930 and 1190 Ii. The HTPB activation energy was estimated at 11.5 kcal/mole, suggesting that the overall regression process is governed by physical desorption of high-molecular weight fragments from the fuel surface.