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Study of multilayer thermal insulation by inverse problems method O.M. Alifanov∗ , A.V. Nenarokomov, V.M. Gonzalez Moscow Aviation Institute, Department of Space System Engineering, 4 Volokolamskoe Sh., Moscow 125993, Russia Received 5 December 2007; accepted 10 March 2009 Available online 27 May 2009

Abstract The purpose of this paper is to introduce a new method in the research of radiative and thermal properties of materials with further applications in the design of thermal control systems (TCS) of spacecrafts. In this paper the radiative and thermal properties (emissivity and thermal conductance) of a multilayered thermal-insulating blanket (MLI), which is a screen-vacuum thermal insulation as a part of the TCS for perspective spacecrafts, are estimated. Properties of the materials under study are determined in the result of temperature and heat flux measurement data processing based on the solution of the inverse heat transfer problem (IHTP) technique. Given are physical and mathematical models of heat transfer processes in a specimen of the multilayered thermal-insulating blanket located in the experimental facility. A mathematical formulation of the inverse heat conduction problem is presented as well. The practical approves were made for specimen of the real MLI. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction The non-stationary state and non-linearity (considerable, at times) of heat transfer phenomena can be referred to the special features of thermal conditions of modern space structures and especially thermal control systems (TCS). These factors considerably reduce the possibility of using many traditional theoretical and experimental methods. So it became urgent to develop new approaches to thermal engineering studies. Among such approaches are methods based on the solution of inverse problems, in which it is required, through measurements of the system or process state, to specify one or several characteristics which cause this state (in other words, to find not causal–sequential, as in direct problems, but rather sequential–causal quantitative relations). The advantage of these methods is that they help to carry out

∗ Corresponding author.

E-mail address: [email protected] (O.M. Alifanov). 0094-5765/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2009.03.053

experimental studies under conditions very similar to full-scale tests or in the operation of the considered systems, in particular in flight tests. In addition, new information gained from these studies makes it possible to speed up the experimental methods and reduce the cost. This is very important for structures used in the space industry, and we observe this situation in the field of practical applications where the first formulations and methods of solving the inverse heat transfer problems have appeared. Experimental-and-computational methods based on solving coefficient inverse heat transfer problems form an intensively developing direction in the field of studies of heat transfer processes [1–4]. The need for a TCS is dictated by the technological/functional limitations and reliability requirements of all equipment used onboard of a spacecraft and, in the case of manned missions, by the need to provide to the crew with a suitable living/working environment. Almost all sophisticated equipment has specific temperature ranges in which it will function correctly. The role of the TCS is therefore to maintain the temperature and

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temperature stability of every item onboard the spacecraft within those pre-defined limits during all mission phases and thereby using a minimum of spacecraft resources. The overall thermal-control function can be split into several different sub-functions, one of them (and may be the most important) is connected with interaction with the environment. The external surfaces of a spacecraft may either give protection from the local environment or give improved interaction with it, involving • the reduction or increase of absorbed environmental fluxes, • the reduction or increase of heat losses to the environment. The problem of thermal control is among one of the most important problems arising in the development of space vehicles and supporting technology. One of the major problems of TCS design is that the spacecraft configuration is usually defined based on the physical accommodation of the various payload and basic subsystem (propulsion, solar arrays, etc.) elements. Only when the configuration is defined, the TCS designer brought in to assess whether all of the temperature requirements can be satisfied. Should this not be the case, a great deal of time (and money) then has to be spent in trying to re-accommodate equipment and find ad-hoc solutions, which are never resource-efficient. Concurrent engineering should be applied more often at all levels, from the equipment to the spacecraft design, to try to overcome these not infrequent problems [5]. And to choose a rational thermal control system, it is necessary to analyze the radiative properties of coating materials. The easiest way to modify a spacecraft external surface’s behaviour is to coat it with paint or layer of other suitable material. Coatings are characterized by their radiative and thermal properties: absorptivity, emissivity, reflectivity, transparency and thermal conductance. The main disadvantages of coatings are the degradation caused by the operating environment and the contamination induced by ground handling or space operations, the absorptivity being the parameter most affected. When a simple coating is insufficient to avoid great losses or gains for the surface, multi-layer insulation can be used. It consists of a certain number of layers of plastic material (normally Mylar or Kapton) coated with a layer of metallic material to reduce the radiation, and separated by sheets of spacer material (e.g. Dacron net) to avoid direct contact between adjacent foils. The external foil coating depends on the particular application: it can be painted or metallised, or can even consist of a

Fig. 1. MLI.

different material (e.g. glass-reinforced cloth). The example of similar multilayered thermal-insulating (MLI) is presented in Fig. 1. MLI efficiency for steady state can be defined either in terms of the linear conductance through the blanket, or via the so-called “effective emittance”. In the first case, the thermal flux can be calculated as the product of the given value times the temperature difference between the external layer and the hardware covered by the blanket. In the second case, it is calculated as a radiative heat exchange using the effective emittance. This parameter has a very simple mathematical formulation, but it can have quite different physical meanings and the choice of definition depends on the modelling technique used. The factors affecting the efficiency are the physical composition of the blanket (number of layers, type of coatings, etc.), the average blanket temperature (usually the arithmetic mean between the two outermost layers), the eventual presence of air or humidity within the layers, and the pressure between them. A very important factor is the way in which the blanket is applied to the spacecraft surface: a single piece of blanket covering a large surface is more efficient than a number of small blankets covering the same surface. Generally speaking, the MLI’s efficiency is measured on relatively small samples, while the real efficiency of an MLI system is only known at the time of system-level thermal testing. Consequently, suitable safety factors have to be applied to the design phase. Very often steady state cases are insufficient for accurate numerical simulation of heat transfer processes in MLI. And requirements of thermal design of TCS drive us to a necessity of using a transient heat transfer mathematical model (particularly with lumped parameters). One of the main difficulties here is how to determine the coefficients of the mathematical model, providing

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its adequacy to real actions. Direct measurement of the most heat transfer characteristics is usually impossible, and their theoretical estimates are often far from being true and often contradictory. That is why a problem arises to determine the radiative and thermal properties of space structures by combining the calculations and the results of experiments. The algorithms suggested for specifying the unknown radiative and thermal parameters of MLI for spacecraft TCS are based on the methodology of inverse heat transfer problem (IHTP), which, at present, are widely used in the study of heat transfer processes. The computational methods for solving the boundary inverse heat conduction problems are now effectively used in experimental investigations of thermal processes which occurred between the solids and the environment. The inverse problem methods allow the use of mathematically proved search algorithms for unknown heat transfer characteristics, estimated from the results of indirect measurements. 2. Mathematical model Formulations of the transient heat transfer problem usually assume variations in the temperature both in time and in position. However, there are many engineering applications in which the variation of temperature within the medium can be neglected and temperature is then considered as a function of time only. Such a formulation of the problem, called a lumped system formulation, provides greater simplification in the analysis of transient heat transfer although the range of applicability is rather restricted. In this situation, a complex heat transfer in a spacecraft MLI is considered for a system of layers, which exchange thermal energy with the environment and between them. The basic heat transfer equation is obtained from the analysis of heat balance under the assumption that the MLI can be considered as a finite number L of isothermal layers: c1 (T )1 d1

ef f

+ 1,2 (T )(T24 −T14 )+k1,2 (T )(T2 − T1 )

∈ (min , max ]

Tl (min ) = Tl0 , l = 1, L

(3) (4)

where ef f

l−1,l (T ) =

l−1 (T )l (T ) l−1 (T ) + l (T ) − l−1 (T )l (T )

and where Tl is the temperature of the l-th layer, cl is the heat capacity of the l-th layer material, dl is the thickness of the l-th layer, l is the density of the l-th layer material, kl−1,l is the thermal conductance between elef f ements l − 1 and l, l−1,l is the effective emissivity, is the Stephan–Boltzmann constant, qs is the direct solar radiation, q R is the solar radiation reflected from the Earth, q is the Earth’s irradiation [6], A S is the absorptivity of the 1st layer, kin is the thermal conductance between the MLI and the internal wall. In the case when all layers are the same systems (1)–(4) are simplified to c1 (T )1 d1

dT 1 = As (T )(qs () + q R ()) d + 1 (T )q () − 1 (T )Tl4 + e f f (T )(T24 −T14 )+k(T )(T2 −T1 ) ∈ (min , max ]

cl (T )l dl

(1a)

dT 1 4 − 2Tl4 + Tl4 ) = e f f (T )(Tl−1 d + k(T )(Tl−1 − 2Tl + Tl+1 ) ∈ (min , max ], l = 2, L − 1

c L (T ) L d L

(2a)

dT 1 4 = e f f (T )(TL−1 − TL4 ) d +k(T )(TL−1 −TL )+kin (T )(Tin −TL ) (3a) (4a)

where (T ) 2 − (T )

(1)

e f f (T ) =

(2)

In the general case, the heat transfer process covered by Eqs. (1)–(4) is determined by the parameters of the boundary heat balance equations, by conductive and radiative heat transfer, by thermal properties, densities and thickness of layers, as well as by the system’s initial thermal state. Usually the corresponding parameters

dT 1 ef f 4 −T 4 ) = l−1,l (T )(Tl−1 l d + kl−1,l (T )(Tl−1 −Tl ) ef f

4 −T 4 )+k +l,l+1 (T )(Tl+1 l,l+1 (T )(Tl+1 −Tl ) l

∈ (min , max ], l = 2, L − 1

+k L−1,L (T )(TL−1 −TL )+kin (T )(Tin −TL )

Tl (min ) = Tl0 , l = 1, L

+1 (T )q ()−1 (T )Tl4

cl (T )l dl

dT 1 ef f 4 −T 4 ) = L−1,L (T )(TL−1 L d

∈ (min , max ]

dT 1 = As (T )(qs ()+q R ()) d

∈ (min , max ]

c L (T ) L d L

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are referred to as one so-called causal characteristics of heat transfer process under consideration. If we are going to calculate the thermal state (temperature–time dependence) of a system by given causal characteristics, such a calculation will be an objective of the direct heat transfer problem in the system. In the case that some of the causal characteristics are not known and it is required to define them using available information about the system thermal state, then we solve an inverse heat transfer problem. One should note that a simultaneous determination of all parameters of Eqs. (1)–(4) is possible only with an accuracy up to a constant multiplier. The uniqueness of solutions is extremely important in our studies involving the inverse problem solutions and the experimental merits ensure the determinacy of the solution. Moreover, in view of Tikhonov’s classical theorem the uniqueness theorems ensure the computation stability if the solution of inverse problems is sought on the compact set. The other approach to uniqueness of the corresponding problem solutions was analyzed by Nenarokomov [7]. 3. Numerical algorithm Let us suppose that in a real situation there are some unknown characteristics u i , i = 1, 2, . . . , N among elements of vectors {cl }1L , {l }1L , {kl,l+1 }1L (where k L ,L+1 = kin ). In addition, the results of temperature measurements in the system’s separate elements are available exp

Tl

(m ) = flm , m = 1, Ml , l = 1, L

(5)

One of the most promising directions in solving the inverse heat transfer problems is to reduce them to extremal formulations and apply numerical methods of the optimization in conjunction with the regularization theory [1,7–13]. In the exact extremal statement, the definition of functions u i , i = 1, 2, . . . , N corresponds to a minimization of the residual functional characterizing the deviation of temperature Tl (m ) calculated for certain estimates of u i , i = 1, 2, . . . , N from known (measured) temperature flm in the metric of space of the input data (the mean-square deviation of experimental flm and theoretical Tl (m ) temperatures can be used as a functional): u = arg min J (u) u∈L 2

where J (u) =

Ml L

(Tl (m ) − flm )2

l=1 m=1

(6)

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A theoretical temperature T is calculated using the mathematical model (Eqs. (1)–(4)). Such an inverse problem is known to belong to a class of ill-posed problems in their classical sense. More often the illposedness is stipulated by the instability of the problem solution with respect to small perturbations of the input data. Despite this feature, it is possible to solve an inverse problem using one or other regularization method. Among them an iterative regularization method is one of the most universal and efficient [1]. The method is based on gradient iterative algorithms, in which, and this is very important, the last iteration number is chosen according to the residual principle [1]. To solve the inverse problem (6), the following iterative method of unconstrained minimization can be used: u s = u s−1 + s g s , s = 1, . . . , s ∗ g s = −Ju + s g s−1 , g 0 = 0 s = Ju (u s ) − Ju (u s−1 ) L 2 /Ju (u s )2L 2 0 = 0

(7)

where g S is a conjugate increment and, s is a parameter. The last iteration number s ∗ is chosen according to the iterative residual principle. As the input data measuring and registering may be performed using inaccurate technical equipment, so the data obtained may be approximate, and an inverse problem solution corresponding to the input data, already measured, may differ greatly from a true one and carry a pronounced oscillating nature. Also a discretization process inevitably leads to errors conditioned by the approximation of continuous functions through piecewise-polynomial dependences, as well as by approximation of differential and integral operators through difference and summation operators. Moreover, errors of rounding-off occur when performing arithmetic operations. All these errors give instability of results in approaching the minimum of the residual functional. It is possible to suppose that methods enabling effective initiation of the iterative process from distant approximation u i , i = 1, 2, . . . , N and the sharp slowdown in approaching the functional minimum would appear useful when solving inverse heat transfer problems. Such a method of instability damping when specifying the approximate solution for an ill-posed problem is based on the “viscous” properties of numerical optimization algorithms. It is necessary to keep in mind that as the number of iterations increases an inverse problem solution can worsen, gradually losing its smooth character. Any waviness appearing in u i , i = 1, 2, . . . , N , will gain in strength as fast as the increasing fluctuating errors burden the measured

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temperature data and the greater would be the sensitivity of u i , i = 1, 2, . . . , N to the temperature measurement. Here we suggest to stop the iterative process at certain iteration S ∗ , admitting no oscillation in the solution. The main question in such an approach is how to select a stopping criterion. With this in mind, let us introduce a complementary condition specifying an admissible degree of proximity of the approximation sought for the “exact” functions u i , i = 1, 2, . . . , N , which corresponds to the disturbed measured data. A condition as such can be a restriction to a residual level given as an aggregate error, including an error in temperature measurements and an approximation error (discretization) in the direct heat transfer problem. A theoretical analysis of different iterative methods has permitted the establishment of the following results. First of all, the gradient methods, such as the steepest descent, the minimal errors, the simple iteration and the conjugate gradients, generate the regularizing families of operators. That is, it becomes possible to choose a stable approximation to the unknown solution from the corresponding iterative sequence. Secondary, if iterations are stopped on the basis of residual criterion, these methods are the regularization algorithms, that is, they give stable approximate solutions whose accuracy increases steadily as the errors of the input data are reduced. These rigorous mathematical results were obtained for a linear case [1]. Thus, let us bound the iterative sequence (7) according to the condition ∗

J (u s ) 2f

(8)

where 2f is the mean-square temperature-measurement L Ml 2 error, namely 2f = l=1 m=1 lm . The descent parameter s is determined from a condition s = arg min (J (u s + g s )) ∈R +

The gradient of the minimized functional is computed using the solution of a boundary-value problem for an adjoint variable: Jcl k

=−

Ml

cl (T ) =

l ckl C k (T )

l (T ) =

Ml

+

k

k=1

⎞

ef f

⎝ ml (m )

Ml

*l *l,l+1 *l

4 ( )−T 4 ( )) l (T ( ))⎠ (Tl+1 m m l m k

( ml (m )(Tl+1 (m ) − Tl (m )) kkl (T (m ))

m=1

(11) where ml is the solution of the following adjoint problem: c1 1 d 1

d 1m * As (qs () + q R ()) = d *T ef f

+

*1,2 *1 (T )q ()−41 Tl3 + (T24 −T14 ) *T *T ef f

+ 41,2 (T23 − T13 ) +

∈ (min , max ]

*k1,2 (T2 − T1 ) *T (12)

ef f

*l−1,l d

4 − cl l dl ml = − Tl4 ) (Tl−1 d *T ef f

3 − Tl3 ) + + 4l−1,l (Tl−1

*kl−1,l (T )(Tl−1 − Tl ) *T

ef f

+ +

*l,l+1 *T

ef f

4 3 − Tl4 ) + 4l,l+1 (Tl+1 − Tl3 ) (Tl+1

*kl,l+1 (Tl+1 − Tl ) *T

∈ (min , max ], l = 2, L − 1

(13)

ef f

− cL L dL

* L−1,L d m L 4 = − TL4 ) (TL−1 d *T

*k L−1,L *kin (TL−1 − TL ) + (Tin − TL ) *T *T ∈ (min , max ]

Nkl

kkl kkl (T )

⎛

4 ( )−T 4 ( )) l (T ( ))⎠ (Tl−1 m m l m k

+

ekl kEl (T )

⎝ ml (m )

m=1

Jkl = −

⎞

ef f

*l−1,l

ef f

k=1

kl,l+1 (T ) =

⎛

3 − TL3 ) + 4 L−1,L (TL−1

k=1 N El

d

m=1

N

Cl

dT l l

ml (m ) C (m ) k (T (m ))

m=1

J l = + ek

(9)

Accordingly in the approach suggested in [1] the unknown coefficients in Eqs. (1)–(4) can be approximated by some systems of basic functions (in particular B-splines), for example

Ml

(10)

(14)

cl l dl ( m+1,l (m ) − ml (m )) = 2(Tl (m ))

(15)

Ml +1,l (max ) = 0, l = 1, L

(16)

O.M. Alifanov et al. / Acta Astronautica 65 (2009) 1284 – 1291

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and for the case of systems (1a)–(4a) the adjoint problem is transformed as c1 1 d1 +

*As d 1m = (qs () + q R ()) d *T

*e f f *1 (T )q () − 41 Tl3 + (T24 − T14 ) *T *T

+ 4e f f (T23 − T13 ) +

*k (T2 − T1 ) *T

∈ (min , max ] −cl l dl

(12a)

*e f f d ml 4 4 = − 2Tl3 + Tl+1 ) (Tl−1 d *T + 4 +

ef f

3 (Tl−1

− 2Tl3

+

Fig. 2. A testing scheme for specimens A and B: 1—electrical heating element; 2—specimen A; 3—specimen B; 4—elements of thermal-insulating holder using as 1-D calorimeters for specimens; T1 , T2 , T3 , T4 , T5 —thermocouples at the specimens.

3 Tl+1 )

*k (T )(Tl−1 − 2Tl + Tl+1 ) *T

∈ (min , max ], l = 2, L − 1

(13a) Table 1 A-priori known information.

d

*e f f 4 − cL L dL m L = − TL4 ) (TL−1 d *T

Material

Aluminized polymer

Spacer material

Number of layers of material in blanket Emissivity, Material’s layer thickness, m Density of materials, kg/m3 Heat capacity of materials, J/(kg K)

14

13

0.06 7.5 × 10−6 1460 1600

n/a 25.4 × 10−6 288 1200

3 + 4e f f (TL−1 − TL3 )

*kin *k L−1,L (TL−1 − TL ) + (Tin − TL ) *T *T ∈ (min , max ] +

(14a)

cl l dl ( m+1,l (m ) − ml (m )) = 2(Tl (m ) − f ml ) m = 1, Ml , l = 1, L Ml +1,l (max ) = 0, l = 1, L

(15a) (16a)

4. Practical approves The purpose of this study is to estimate radiative and thermal properties (emissivity and thermal conductance) of a multi-layered thermal-insulating blanket. Consider a physical model of heat transfer process in the specimen located in the experimental facility (Fig. 2). For the given tests a heating element was used in the experimental facility, which was in the form of a refractory stainless steel foil of 180 mm in length, 150 mm in width and 0.1 mm in thickness. The experimental specimen of the blanket is a multi-layer slab in the form of a rectangular parallelepiped with the dimensions 150×150×5 mm. Such ratio of specimen’s dimensions as well as the application of a symmetrical heating scheme of two identical specimens and a corresponding test procedure provide in the course of testing the formation the almost uniform heating over the surface. The tested MLI was a set of similar aluminized

polymers with spacers. At zero time, the uniform temperature distribution is realized in the specimen. The initial data for estimating the radiative and thermal properties of MLI are formed based on the results of heat flux and temperature measurements and include external and internal heat fluxes time–temperature dependence at both surfaces of the specimen. Data on the a-priori known properties of materials composing a considered MLI are given in Table 1. Based on the given physical model, a corresponded mathematical model of heat transfer process in the material’s specimen can be considered. For this case the direct problem of heat transfer can be presented as c(T )d

dT 1 = q1 () + e f f (T )(T24 − T14 ) d + k(T )(T2 − T1 ) ∈ (min , max ]

(1b)

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c(T )d

dT 1 4 = e f f (T )(Tl−1 − 2Tl4 + Tl+1 ) d + k(T )(Tl−1 − 2Tl + Tl+1 ) ∈ (min , max ], l = 2, L − 1

673 623 573

(2b)

dT L 4 c(T ) L d L − TL4 ) = e f f (T )(TL−1 d + k(T )(TL−1 − TL ) − q2 () ∈ (min , max ] Tl (min ) = T0 , l = 1, L

523

1 3 2 4

473 423 373

(3b)

323

(4b)

273 0

200

400

600

800

1000 1200 1400

where e f f (T ) =

(T ) 2 − (T )

There are three unknown parameters in Eqs. (1b)–(4b): c, , k. For complimentary information needed for solving the inverse problem prescribed are the results of temperature measurement at the two surfaces of the specimen: exp

Tl

(m ) = flm , m = l, Ml , l = 1, L

(5)

The uniqueness conditions of IHTP solution usually determine a minimum wanted volume of measurements necessary in one experiment. For simultaneous determination of c, , k it is necessary to measure a heat flux passing through the specimen surface differing from zero at least at one boundary and perform unsteady temperature measurements not less than in three timepoints. In the process of unsteady heating of specimens by means of an automatic system, recording of temperatures inside the specimen in places of thermocouple positioning, heater’s temperature and also electric power released on it were performed: Q electr = U ∗ I

(17)

where U is the electrical voltage on the heater, I is the magnitude of the current passing through the heating element. The heat flux supplied to a specimen due to symmetry is determined as [14,15]

*T1 Q electr q1 () = 2 (18) − ch k dh A * where A is the heater’s surface area; h =7900 kg/m3 is the stainless steel density; dh = 0.0001 m is the heating element thickness; ch =(450+0.57 T)J/(kg K)—specific heat capacity of stainless steel. The back heat flux q2 () was calculated using the temperature measurements on the surfaces of the thermal-insulated holder from a

Fig. 3. Comparing the calculated and measured temperatures (specimen A): 1—measured temperature at the heated surface, 2—calculated temperature at the heated surface, 3—measured temperature at the internal surface, 4—calculated temperature at the internal surface.

Table 2 The estimated parameters. Parameter

Aluminized polymer

Spacer material

c, J/(m K )

1721 0.061 0.012

1752 0.062 0.011

k, W/(m2 K)

Table 3 The deviation of the calculated temperatures. Specimens

Least-square temperature deviation (K)

Maximum temperature deviation (K)

A B

3.54 7.45

10.5 12.8

solution of the direct problem for the insulated slab of 30 mm thickness with known thermal properties and the first kind (temperature) boundary condition on the surfaces. Comparison of the calculated and measured temperatures on the specimens’ A surfaces is presented in Fig. 3. The result of estimating the parameters presented in Table 2. Table 3 includes the obtained values of the least squares and the maximum deviation of the calculated temperatures from that measured in the experiments. 5. Conclusions This paper aims to describe the algorithm developed to process the data of thermal experiments and hence to

O.M. Alifanov et al. / Acta Astronautica 65 (2009) 1284 – 1291

find the radiative and thermal properties of MLI. The algorithm is suggested for estimation of these unknown properties of the surface as a solution of the nonlinear IHTP in extreme formulation. The illustrative examples are presented to make a judgment on the convergence for real technical problems. Acknowledgements A portion of this work was performed while the authors held Grant no. NSh 3056.2006.08 from the Russian Government and Grant no. 05-02-17309 of Russian Foundation of the Basic Research. References [1] O.M. Alifanov, Inverse Heat Transfer Problems, Springer, Berlin, 1994. [2] I. Anderson, Inverse problem of the calculus of variations for ordinary differential equations, American Mathematical Society, Providence, USA, 1992. [3] D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, Heidelberg, 1992. [4] M.P. Menguc, P. Dutta, Scattering tomography and its application to sooting diffusion flames, Journal of Heat Transfer—Transactions of the ASME 116 (1) (1994) 144–151. [5] M.N. De Parolis, W. Pinter-Krainer, Current and future techniques for spacecraft thermal control 1, Design drivers and current technologies, ESA Bulletin No. 87, ESTEC Publ., Noordwijk, The Netherlands, 1996. [6] L.V. Kozlov, M.D. Nusinov, A.I. Akishin, et al., Simulation of Spacecraft Thermal States, Mashinostroenie Publ., Moscow, 1971 (in Russian). [7] A.V. Nenarokomov, Extreme methods of the flight vehicles thermal state analysis, Thesis for Doctor of Science Degree, Moscow Aviation Institute, 1995 (in Russian).

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