Combustion, Explosion, and Shock Waves, Vol. 38, No. 1, pp. 65 70, 2002
Solving an Inverse Problem of Erosive Burning Rate Reconstruction
V. A. Arkhipov,1 E. A. Zverev,1 and D. A. Zimin1 UDC 536.46
Translated from Fizika Goreniya i Vzryva, Vol. 38, No. 1, pp. 73 79, January February, 2002.
Original article submitted January 16,2001.
A new method for an experimental study of burning processes in condensed sub-
stances is suggested, based on the statement and solution of inverse problems. An
inverse problem of reconstructing the erosive burning rate of solid propellants from
experimental data is formulated. The choice of an approach to solving the problem by
the joint application of well-known methods for inverse problem solution and specific
features of experimental studies of burning processes, in particular, erosive burning,
has been justified. The problem solution is illustrated by a numerical example. The
testing involves a comparative analysis of two optimization methods: although both
methods are characterized by an identical accuracy, the steepest descent method has
a higher rate of convergence for this class of problems than the conjugate gradient
method.
It still remains crucial to determine experimentally The effect of erosive burning has been the subject
internal ballistics parameters (e.g., propellant burning of intense experimental and theoretical studies since its
rate, dispersion of condensed particles, erosion factor, discovery by O. I. Leipunskii in 1942. At the same time,
etc.) both for the development and improvement of ar- only references [1 4] present results for sonic and super-
tillery and rocket systems and for the further develop- sonic flows, whereas, for practical reasons, it is crucial
ment of the combustion theory of condensed substances. to obtain new data exactly over this range. Rigorous
In particular, it is necessary to know the burning rate experimental conditions (high pressures and tempera-
dependence on the parameters of tangential flow (the tures, nonstationary processes, etc.) significantly limit
so-called erosive burning) to design solid rocket motors the application of direct measurements [4]. Methods of
(SRM), because flow velocities in the charge channels experimental data processing that involve time averag-
are rather high, especially in narrow slots, and they can ing fail to attain the required accuracy.
be supersonic in nozzleless SRM. The authors [5] described a series of experiments
The objective of the experimental study of ero- on erosive burning in a supersonic flow: a segment of a
sive burning is to obtain the dependence of the solid- plane diverging channel was considered (Fig. 1), where
propellant burning rate on some dimensionless parame- the solid-propellant sample studied formed one of the
ters characterizing flow conditions and use this informa- sides. At an initial time t0, a supersonic flow of com-
tion in designing solid-propellant power plants. There bustion products generated by a gas generator enters
is a number of well-known methods for obtaining this the channel. The compositions of combustion products
dependence, ranging from the simplest ones (burning from the gas generator and sample burning are assumed
quenching followed by measuring the thickness of the to be identical. The incoming flow ignites the solid-
burnt arc "e over the time of the sample burning tk) to propellant sample. Its burning changes the channel
such complex and expensive methods as rapid filming, configuration, flow parameters, and combustion char-
x-ray, capacitance, ultrasonic, and microwave meth- acteristics. The pressure and temperature in the gas-
ods providing continuous measurement of the burnt arc generator chamber can be measured in the course of
thickness "e(t). These methods were analyzed in [1] in burning. Then, at the time tk, quenching is performed,
considerable detail. and the sample burning is determined by measuring the
burnt arc thickness "e(tk) at equidistant cross-sections
1
of the sample with a 5-mm lengthwise step. The erosive
Institute of Applied Mathematics and Mechanics
at the Tomsk State University, Tomsk 634050;
burning rate is to be determined from these data.
[e-mail].
0010-5082/02/3801-0065 $27.00 © 2002 Plenum Publishing Corporation 65
66 Arkhipov, Zverev, and Zimin
(with a certain error) results of their indirect manifes-
tation are ill-posed. The reasons are:
 inaccurate specification of the measured value
resulting from measurement error;
 inaccurate specification of the operator defining
the relation of the measured value with the sought one,
which results from using idealized models.
The most generalized approaches to the solution
of ill-posed problems are the method of quasi-solutions
Fig. 1. Layout of the experimental device: 1) noz-
proposed by V. K. Ivanov and the regularization method
zle wall; 2) solid-propellant sample to be examined;
3) gas generator.
suggested by A. N. Tikhonov. The method of quasi-
solutions consists in an a priori specification of a com-
pact set, where the solution is sought proceeding from
The surface of the sample is rippled as is character-
physical assumptions (quantitative and qualitative) of
istic of erosive burning (the height of the ripples is com-
the solution character. The regularization method
parable with the burnout height itself at tk H" 0.1 sec),
requires some smoothness conditions of the solution.
which leads to a considerable error in measuring the de-
Therewith, the information on the experimental data
pendence "e(tk). On the other hand, a further increase
error is also used, which must be consistent with the
in the sample burning time aggravates the error due to
residual value [8]. The choice of the approach is based
time averaging of flow parameters (the pressure in the
on the specific subject of investigation, i.e., measure-
channel is a third of the initial pressure when the burn-
ment details, type of mathematical models employed,
ing time is tk H" 1 sec). These factors make it impos-
and additional a priori information about the solution
sible to attain reasonable accuracy in determination of
of the inverse problem.
the required characteristics. Therefore, to obtain inter-
IPIB are characterized by the following specific fea-
nal ballistics characteristic from indirect measurement
tures:
results and plot them versus the governing parameters
" high measurement accuracy of initial parameters;
in the general case, it is necessary to solve the corre-
" using simplified mathematical models to single
sponding inverse problem.
out an  elementary investigated process from an ag-
The methods for solving inverse problems are
gregate of complex processes;
widely used for interpretation of experimental data in
" rather large volume of a priori information (quan-
various fields of science and technology. These methods,
titative and qualitative) about the solution sought due
however, have not been adequately applied to intra-
to a sufficiently high level of accumulated knowledge
chamber processes, although the complexity and high
about the dependence of reconstructed internal ballis-
costs of firing tests make it crucial to enhance the infor-
tics characteristics on the governing parameters.
mational value of research and reliability of the results
Note that the residual of experimentally measured
obtained. The wide use of inverse methods is feasible
and calculated parameters minimized in solving IPIB is
due to the latest achievements in the theory of solv-
mainly determined by the simplifications of the adopted
ing ill-posed problems and emergence of more powerful
mathematical model as compared with the actual pro-
computer technologies that allow the numerical solution
cess rather than by measurement errors, which are not
of such problems. Direct methods, methods of the op-
very large and can easily be estimated. On the other
timization theory, and a direct search technology were
hand, by using evident physical restrictions on the solu-
proposed to solve ill-posed problems [6].
tion sought, a well-posedness class can be specified, i.e.,
The present paper suggests a new methodology for
a set on which the problem of obtaining the solution
the experimental research of burning processes, based
of the respective equation is well-posed (according to
on the solution of inverse problems of internal ballistics
Tikhonov) [7, 8]. Taking all these specific features into
(IPIB) and determines the scope of its applicability to
account, the approach associated with obtaining quasi-
processing experimental data on erosive burning in a
solutions appears to be the most natural for IPIB.
supersonic flow. The inverse problem of erosive burn-
In order to numerically implement the method of
ing rate reconstruction from experimental data is for-
quasi-solutions, an extreme formulation of the inverse
mulated. The approaches proposed for solving it are
problem is employed, and the problem is solved using
tested using a numerical solution of a model problem as
methods of the optimization theory. The problem of
an example.
reconstructing the erosive burning rate from measure-
As was noted in [7], inverse problems of obtaining
ment results in the extreme statement is to find a solu-
quantitative characteristics of processes from the known
Solving an Inverse Problem of Erosive Burning Rate Reconstruction 67
tion belonging to some specified domain of admissible the width of the burning surface, R is the gas con-
solutions, which provides the minimum functional of the stant, A is the open cross-sectional area of the chan-
residual of experimentally measured and calculated val- nel, H = E + p/Á = cpT is the specific enthalpy,
ues within the adopted model. The above-described E = cV T is the internal energy, cp and cV are the
layout of the experimental device from [5] was used as isobaric and isochoric heat capacities of the gas, re-
a physical model of the process. spectively, Hb = ÇcpTb (Tb is the propellant burning
The choice of the measured quantity (indirect man- temperature), Ç is the volume- and time-averaged co-
ifestation) and the respective functional of the residual efficient of heat losses in the combustion chamber, and
is based on how readily and accurately the selected pa- Áp is the propellant density.
rameter could be measured and how sensitive it is to The initial conditions are obtained by calculating
the sought parameter. The burnt arc thickness "e after system (2) for an unburned channel. The conditions
forced quenching of the sample versus the longitudinal at the left boundary are calculated for the input cross
coordinate x was taken as a measured quantity. The section of the channel, using gas-dynamic functions of
functional of the residual is stagnation parameters in the gas generator.
As is known, in the range of positive erosion, ex-
L tk
2
perimental results can be approximated by a linear
J = "e(x) - ub(x, t)dt dx, (1)
two-parameter
"dependence on the Vilyunov parameter
I = ÁÅ/Ápu0 cf [10] with satisfactory accuracy, and
0 t0
the erosive burning rate is given by the expression
where L is the channel length and ub is the erosive burn-
ub = u0[1 - k(I - I0)],
ing rate, which is specified algorithmically, and its cal-
culation is based on the given flow model.
where u0 = u1(p/pa)½ is the burning rate in the absence
The model of the flow in a supersonic channel,
of the incoming flow, pa is the atmospheric pressure,
which relates the measured final burnout with the
and cf is the friction coefficient. This function can be
sought erosive burning rate, is based on the following
treated as the sought function in obtaining the empirical
assumptions:
parameters k and I0.
" The solid-propellant sample is ignited instantly,
At the same time, it would be of interest to find
the ignition and nonstationary burning processes are
data on erosive burning in a more generic form. It is,
not taken into account;
therefore, natural to solve the inverse problem under
" The flow is assumed to be one-dimensional and
less rigorous restrictions on the set of functions µ. It is
quasi-stationary;
suggested that the erosion coefficient be given as
" Heat losses, volume heat release, and friction are
µ = E(p/pa)·()µ, (3)
ignored (model of an ideal gas).
The system of equations describing a gas flow in
where  = Å/a" is the reduced flow velocity and a" is
the ignited channel with allowance for gas inflow from
the critical velocity of sound. The coefficient µ being
the sample has the following form in the context of the
represented as a power dependence on dimensionless pa-
above-mentioned assumptions [9]:
rameters is typical of processing experimental data on
erosive burning and makes it possible to show the results
"
(ÁÅA) = Ápbub, deviating from the linear dependence without involving
"x
methods of functional optimization. Then the sought
erosive burning rate is written as ub = µu0.
" "p
(ÁAÅ2) = -A ,
The burning rate is positive and finite and increases
"x "x
with the increase in pressure and flow velocity. Conse-
quently, the parameters E, ·, and µ are positive and
" Å2
ÁÅA H + = ÁpbubHb, (2)
limited from above. The set comprising a class of pos-
"x 2
sible solutions is compact and unites dependences on a
finite number of limited real parameters [11]. This en-
"A
= bub,
ables one to overcome ill-posedness by using method of
"t
quasi-solutions suggested by V. K. Ivanov.
p = ÁRT. Therefore, the inverse problem consists in find-
ing the parameters E, ·, and µ, corresponding to the
Here p, Á, T , and Å are the pressure, density, tempera- minimum of the residual functional (1) under restric-
ture, and gas velocity in the channel, respectively, b is tions (2).
68 Arkhipov, Zverev, and Zimin
Fig. 2. Projection of the descent trajectory onto the response surface of the residual functional (1) versus erosive
burning rate parameters: µ = 0.8 (a), 
= 0.8 (b), and · = 0.1 (c).
Å» Å»
The steepest descent method and the conjugate of the most efficient optimization method are illustrated
gradient method [12] were employed as optimization by a three-stage solution of a model problem.
methods for obtaining the minimum of functional (1). Stage 1. The direct problem (2) was solved with
Their employment is based on regularizing properties of given values of erosive burning rate parameters, which
gradient methods, which make it possible to initiate ef- were further treated as an  exact solution: 
= 0.8,
ficiently the iterative process from a distant initial esti- · = 0.1, and µ = 0.8. The direct problem was solved for
Å» Å»
mate and slow significantly as the functional approaches the following parameters: p0 = 6.35 MPa, Tb = 2360 K,
the minimum. The Fibonacci method was applied as a tk = 0.84 sec, b = 10 mm, L = 110 mm, angle of
method of one-dimensional sweep, as it also has a high channel expansion ą = 10ć%, nozzle-throat area of the
rate of convergence. gas generator A" = 40 mm2, cp = 1.5 kJ/(kg · K), ratio
The verification of the method proposed, selection of specific heats Å‚ = 1.245, R = 413 J/(kg·K), Áp = 1.6·
of the optimum calculation parameters, and the choice
103 kg/m3, u1 = 0.37·10-3 m/sec, ½ = 0.82, and Ç = 1.
Solving an Inverse Problem of Erosive Burning Rate Reconstruction 69
Stage 2. An inverse problem of erosive burning
rate reconstruction was solved, where the dependence
"e(x) obtained from the solution of the direct problem
and  perturbed by the error generator was used as an
input ( experimental ) parameter.
Stage 3. The burning rate values obtained at stage
No. 2 were compared with the  exact solution, and the
systematic error of calculation was estimated.
The results of solving the model problem are plot-
ted in Fig. 2, which shows the projections of the descent
trajectory onto the response surface of the residual func-
tional (1) versus the parameters of Eq. (3) for the speci-
fied  exact values of the following parameters: µ = 0.8
Å»
(Fig. 2a), 
= 0.8 (Fig. 2b), and · = 0.1 (Fig. 2c).
Å»
Fig. 3. Residual functional versus the number of iter-
The numbers in the figure indicate the values of the
ations in seeking the functional minimum: n = 5 (1),
residual functional. It can be seen from Fig. 2 that the
10 (2), 25 (3), and 50 (4).
functional is convex and has a single minimum and no
specific characteristics making it difficult to obtain it.
the functional approaches its minimum. A similar man-
The descent trajectory illustrates the applicability of
ifestation of the instability effect near the minimum is
the steepest descent method to this particular problem.
also observed for n = 25 and 50, but oscillations emerge
The main problem of application of gradient meth-
much later, and their amplitude is significantly smaller
ods is to develop a resource-sparing procedure for cal-
than for lower values of n. On this basis, the stabil-
culating the gradient of the objective functional. The
ity domain of an iterative optimization algorithm can
most general (although the least efficient) method for
be found, whose boundaries are determined by initial
gradient calculation is a finite-difference approximation
manifestation of oscillations, and the condition of the
of the gradient "J/"z in the vicinity of the considered
non-decreasing residual functional can be chosen as a
approximation of z. Here z = (z1, . . . , zk) is a vec-
termination criterion:
tor approximating the sought solution z under problem
Ji+1 - Ji > 0.
discretization. In the simplest case, we have
The stability domain of the method determines its
"J 1
= J(z1, . . . , zi + "zi, . . . , zk)
minimum systematic error and, accordingly, its accu-
"zi "zi
racy. In this respect, the value n = 25 is also optimum,
- J(z1, . . . , zi, . . . , zk) ,
and its further increase does not reduce the error of the
method.
where "zi is the increment of ith component of the
During the refining of the method, its tolerance to
vector z.
input data variation was also studied. It was found
In addition, repeated calculations of the objec-
that a 10 15% error in obtaining "e(x) (correspond-
tive functional necessary to calculate the gradient, the
ing to the experimental data of [5]) makes it possible
choice of parameters "zi also presents serious difficul-
to reconstruct the burning rate parameters with a sys-
ties. Since the error of numerical estimation of the
tematic error of H"(3 4)% as compared with the  exact
derivative is a sum of truncation errors and condi-
solution for all unknown parameters, which proves the
tion errors; the consequences of finite-difference inter-
stability of the method.
val variation are opposite for them [12]. The investiga-
A comparison of the steepest descent method with
tions showed that the optimum value is "zi = 10-12
the conjugate gradient method (Fig. 4) shows that the
(i = 1, 2, 3).
steepest descent method provides a higher rate of con-
To obtain the optimum value of the one-
vergence and, therefore, is more effective for solving this
dimensional search parameter (with n Fibonacci num-
problem: the respective estimated computer time for
bers), a series of calculations was performed with var-
the numerical solution of this problem decreases by a
ious values of n. The calculation results are shown in
factor of 3. This can be attributed to the conjugate gra-
Fig. 3. It is clear that the value n = 25 is sufficient,
dient method requiring greater accuracy of determining
and its further increase does not improve the rate of
the direction and the step than the steepest descent
convergence or accuracy of the obtained solution of the
methods. Due to the accumulation of the computation
inverse problem. At n = 5 and 10, characteristic insta-
error, the directions cease to be conjugate [12].
bility effects are observed, manifested as oscillations as
70 Arkhipov, Zverev, and Zimin
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