Abstract
This paper introduces a mathematical model of a thermal protection system incorporating the Riemann–Liouville fractional derivative. The system is considered as a three-layer structure, where the temperature distribution in the first two layers follows the classical heat conduction equation. In contrast, the third layer, characterized by its porous nature, is modeled using a fractional-order heat conduction equation. The thermal contact resistances between the layers are taken into account. The external surface is subjected to a boundary condition of the second kind, incorporating an aerothermal heat flux derived from NASA Langley Research Center data, while the internal surface is governed by a Dirichlet boundary condition. Additionally, the temperature-dependent material properties are considered. A significant contribution of this study is the development of a numerical scheme for a three-layer thermal protection system model, in which one layer is porous and described using the Riemann–Liouville fractional derivative. The proposed approach allows for accurate simulation of heat conduction in systems with complex material structures. The influence of the fractional derivative order 
on the temperature profile was investigated, showing that variations in 
significantly affect the thermal response of the system. Furthermore, a mesh refinement study was conducted to assess the impact of spatial discretization on the numerical results. These findings establish the model as a valuable tool for computer simulations and provide a basis for further development and optimization of mathematical and computational approaches in the analysis of thermal protection systems.
Keywords: Thermal protection system, Riemann–Liouville fractional derivative, Aerothermal heat flux
Subject terms: Engineering, Aerospace engineering, Mechanical engineering
Introduction
Thermal protection systems are a crucial component of spacecraft, shielding them from damages caused by intense aero-thermal heat flux during atmospheric entry. Various classes of these systems have been developed and tested, including active, semi-passive, and passive solutions, each with its own advantages and limitations1. To enhance the protection of such vehicles, an integrated thermal protection system has been introduced. This system ensures both structural stability and thermal insulation of the spacecraft simultaneously. NASA’s technology development plans, formulated in 2015 and implemented in 2020, indicate that this topic remains relevant2,3. Effective thermal protection systems are essential and indispensable for future space missions. Therefore, special attention should be paid to the development of tools, modeling techniques, and simulations for such systems.
In recent times, numerous scientific papers on thermal protection systems have been published. Most often, these studies focus on describing modern materials used in the production of thermal protection systems. Another key area of research involves various computer and numerical simulations of the behavior of such systems. An example is the work by Le et al.4, which discusses layered systems applied in thermal protection. Similarly, Shi et al.5 presents research on a composite sandwich panel with a corrugated core, manufactured using the hot forming method to create an integrated thermal protection system. Meanwhile, Wang et al.6 describe an integrated thermal protection system that utilizes graded insulation materials and multilayer composite sandwich panels with a ceramic matrix.
Guo et al.7 focuses on analyzing the erosion resistance of selected composites used in thermal protection systems. Meanwhile, Xu et al.8 examine the impact of the porosity configuration of a carbonizing composite on the bondline temperature in a thermal protection system. The research presented by Hou et al.9 investigates a polysiloxane composite material, which is reinforced with carbon fiber. The material developed in a laboratory environment serves as an ablative outer layer for thermal protection systems. On the other hand, Li et al.10 focus on the properties of lightweight multifunctional composites reinforced with an integrated preform, designed for use in thermal protection systems. Chen et al.11 propose a bilayer lattice structure with a ceramic matrix. This matrix is filled with phase change and thermal insulation materials, intended for application in integrated thermal protection systems. The effectiveness of the research has been confirmed by numerical calculations.
Quite a lot of attention has been devoted in the scientific literature to the simulation, computational, and numerical methods of thermal processes and heat transfer in thermal protection systems. Fang et al.12 solve the direct problem of heat conduction in a multilayer thermal protection system containing phase-change doped materials. Research on similar protection systems, also incorporating phase change materials, is presented by Ren et al.13. The computational results showed that using the phase-change material is advantageous, as it helps reduce the temperature of each layer below the maximum allowable value. Zhang et al.14 applied the incremental differential quadrature element method to simulate the one-dimensional temperature distribution in multilayer thermal protection systems. Meanwhile, Chen et al.15 present a numerical analysis of heat transfer through conduction and radiation. This method uses a two-dimensional axisymmetric model. The authors applied and combined the finite-volume method with the discrete ordinate method. Another article where a numerical algorithm to compute the ablation of the metallic thermal protection system is presented by Peluchon et al.16. However, Song et al.17 propose an innovative texture-enhanced attention defect detection model that enables accurate, efficient and real-time defect detection of thermal protection materials.
A common approach in computer simulations is solving the so-called inverse problems. In engineering problems, these involve selecting appropriate model parameters (e.g., material data or initial-boundary conditions) in order to obtain the desired output values (e.g., temperature distribution). An example of such work is presented by Uyanna et al.18, where the authors focus on determining the aerothermal heat flux. The model considered in this study consists of three layers. Kumar et al.19 address a similar problem, but this time they consider a two-layer model. The selection of appropriate materials is also crucial, as demonstrated by Kumar et al.20, where the geometry and material properties for the thermal protection system are discussed. Another method used to solve the inverse problem in order to determine the heat flux is the Kalman filtering method coupled with the Rauch-Tung-Striebel smoother, as presented by Hong et al.21. Wen et al.22 use a modified Kalman method. Another example of solving the inverse problem in the context of a lightweight thermal protection system is presented by Wang et al.23, where the sequential function specification method is used.
The authors of this article have also addressed the issue of identification of heat fluxes in thermal protection systems24–26. In the first of these papers24, the simplest mathematical model was considered, assuming ideal contact between layers. In the next paper 25, a more complex mathematical model was examined, which included the temperature-dependent parameters of the material. The last of the mentioned articles26 also considered the non-zero thermal resistances at the contact between layers.
The next step in the research is an attempt to develop a mathematical model that would take into account the porosity of the materials from which the system is constructed (e.g. ceramic materials)5–8. It is also important to develop the computational methods appropriate for this issue. In the case of porous materials, the phenomenon of anomalous diffusion occurs27,28. In such cases, fractional derivatives are used in mathematical modeling. Examples of applying fractional derivatives in modeling can be found in the many works29–32. In the second of these papers30, a model of the heat conduction process in a three-layer composite carrier is presented, where the fractional derivative with respect to time is applied. Brociek et al.33 considered and compared several mathematical models that describe heat transfer in porous aluminum. Among the models considered were those based on fractional derivatives of Caputo and Riemann–Liouville. The best fit to the experimental data was obtained using the Riemann–Liouville fractional derivative with respect to space.
A current topic in the literature on fractional differential equations concerns the methods for their approximate solution. In particular, Shams and Alalyani34 presented an adaptive step size numerical scheme for solving the fractional ordinary differential equations. In turn, Pandey et al.35 obtained the approximate solution of space-time fractional-order reaction-diffusion equation using the homotopy perturbation technique and Laplace transform method. In the next paper, Pandey et al.36 use the variable-order Chebyshev collocation method to solve a nonlinear variable-order fractional reaction–diffusion equation with Mittag–Leffler kernel. Dwivedi et al.37 used the finite difference method and Fibonacci collocation method to solve the variable-order fractional reaction–advection–diffusion model in the heterogeneous medium. The collocation method was also used by Zhuang et al.38. Next, Zakaria and Moujahid 39 also used the finite difference method to solve the fractional time-space diffusion equation. Błasik40 present a numerical method for solving the two-dimensional subdiffusion equation with Caputo fractional derivative. The proposed method is an extension of the fractional Crank-Nicolson method, based on the discretization of the equivalent integral-differential equation. However, Sivalingam et al.41 applied the physics informed neural network for solving the fractional differential equations.
In this paper, a three-layer mathematical model of a thermal protection system is presented, where one of the layers is described using the Riemann–Liouville fractional derivative with respect to the spatial variable. A numerical procedure for solving the direct problem is also developed. Furthermore, the presented algorithm is tested for various meshes. The study includes tests for different values of the Riemann–Liouville fractional derivative. Non-zero thermal resistances are assumed between the layers. The outer surface is subjected to a boundary condition of the second kind with an aerothermal heat flux, which was generated at the NASA Langley Research Center. On the internal surface, the boundary condition of the first kind is applied. It is also assumed that the material parameters depend on temperature. The novelty of this paper lies in the application of Riemann–Liouville fractional derivatives in modeling heat conduction in the thermal protection system. The reason for applying this type of derivative is the heterogeneous (porous) structure of the third layer. Therefore, the use of fractional derivatives is justified27,28,33.
Mathematical model
The considered model of the protecting shell consists of three layers, and its scheme is presented in Fig. 1. The first layer, which is also the outer layer, is exposed to the heat flow. A constant temperature is assumed on the right boundary of the system. The model assumes that the thermal contact between the layers is not ideal, that is the thermal resistances occur there, and the material coefficients depend on temperature. A constant value of the thermal resistance is assumed in the calculations.
Fig. 1.
Scheme of the model of the protecting shell.
In the case of the first two layers, the temperature distribution is described by the heat conduction equation(42):
 |
1 |
, 
, where 
, 
, 
and 
, 
, mean, respectively, the specific heat, density, thermal conductivity coefficient and temperature of the s-th layer. Whereas, in the third layer, due to its porous nature, the temperature distribution is described by the equation with a derivative of fractional order43–45:
 |
2 |
, 
, 
, where 
is the scaled thermal conductivity [
], that is the thermal conductivity multiplied by the scaling constant 
. This constant is of numerical value equal to one, has unit [
] and is chosen so that the right and left units of the equation are the same33,46,47. Element 
denotes the thermal conductivity [
] of the third layer.
Fractional derivative used in Eq. (2) is the Riemann–Liouville fractional derivative defined as follows45,48:
 |
3 |
where 
means the gamma function.
The heat flux acting on the outer layer (
) goes partly to the interior of the system (
), and partly causes a heating of the surface that emits a heat flux by radiation (
). Hence, the energy balance on the outer surface (layer 1) can be written as follows
 |
4 |
The re-radiation term can be determined based on knowledge of temperature
 |
5 |
where 
denotes the surface emissivity, 
is the Stefan-Boltzmann constant, and 
describes the ambient temperature.
Therefore, in the mathematical model, on the outer surface of the first layer, a boundary condition of the second kind is adopted, of the following form
 |
6 |
Next, on the inner surface, i.e. at the end of the third layer, the first kind boundary condition is set
 |
7 |
To fully describe the mathematical model, some assumptions must also be made concerning the point of contact between the layers. The presented model assumes the interface boundary conditions with nonzero values of thermal resistances 
and 
. At the point where the first and second layers meet (point 
), this condition is as follows
 |
8 |
A similar condition is taken at the contact point of the second and third layers (point 
):
 |
9 |
At the initial moment, the temperature distribution is known and given by the relation given below
 |
10 |
At the common points, the compliance of relevant conditions is assumed.
Numerical procedure
To solve the problem, as in the previous paper26, the implicit scheme of the finite difference method is used42,49,50 with an appropriate approximation of the Riemann–Liouville derivative. The considered region is discretized by introducing an appropriate mesh. The layers are divided into 
, 
, subintervals, in result of which the following meshes in the subsequent layers are obtained:
 |
11 |
 |
12 |
 |
13 |
where 
, 
, 
. Two nodes are placed at the contact point of the layer. One applies to the end of the previous layer, and the other applies to the beginning of the next layer. This is a consequence of the assumption of nonzero thermal resistances at the contact points. The sum of the meshes for all three layers 
constitutes the spatial mesh for the entire considered region. An equal step is assumed in the time interval, creating a mesh composed of m equal subintervals:
 |
14 |
where 
. Consequently, the mesh for the entire domain of the considered unsteady heat flow problem is the Cartesian product of the meshes in the space and time domains: 
.
In the internal nodes of the first and second layers, an implicit scheme is used to discretize Eq. (1), obtaining a difference equation of the form
 |
15 |
where 
, 
, 
, 
and 
is the harmonic mean of the thermal conductivity coefficient in the temperature interval 
, and 
means the harmonic mean in the interval 
49,50.
The Riemann–Liouville derivative at point 
in moment 
is approximated in the following way51–53, 
:
 |
16 |
where
 |
17 |
Whereas, at point 
in moment , the approximation of the form:
 |
18 |
is received. Next, the backward quotient is used for the first component of the right side of Eq. (2):
 |
19 |
where 
. For the derivative with respect to time, the backward quotient is also used
 |
20 |
For the boundary condition of the second kind (4) defined on the outer surface, the following approximation is adopted
 |
21 |
where 
and 
.
For the condition at the contact of the first and second layers (8), the following two difference equations are obtained
 |
22 |
 |
23 |
where 
and 
.
Similarly, at the contact between the second and third layers (9), the following differential equations are get
 |
24 |
 |
25 |
where 
, and 
.
Putting the above equations together and taking into account the first kind condition at the boundary of the third layer, a system of linear equations of the form
 |
26 |
is obtained. Matrix 
contains the temperature-dependent material parameters, therefore it changes at each calculation step. The system of Eq. (26) has dimensions 
.
Numerical calculations
In the calculations, it is assumed that: 
m, 
m, 
m and 
m. The values of emissivity 
and temperature of the surrounding 
K are taken. It is also assumed that the temperature of the internal surface is equal to 
K. At the initial moment, the entire area had the same temperature: 
K, 
. On the outer surface, an aerothermal heat flux 
is applied (Fig. 2) that was generated at the NASA Langley Research Center18,19,54.
Fig. 2.
Aerothermal heat flux on the outer surface (measured by the NASA Langley Research Center54).
The following material data, independent of temperature, are applied for the calculations18,26: in the first layer 
[J/(kg K)], 
[kg/m
] and 
[W/(m K)], in the second layer 
[kg/m], and in the third layer 
[J/(kg K)] and 
[kg/m]. However, the conduction coefficient in the second and third layers depends on temperature, as does the specific heat of the third layer. The approximations of the data contained in Table1 are used in the calculations. The plots of these approximations are shown in Fig. 3.
Table 1.
Material properties dependent on temperature18.
| Temperature [K] |
 [J/(kg K)] |
 [W/(m K)] |
 [W/(m K)] |
|---|---|---|---|
 |
 |
0.0363 |  |
 |
|
0.0468 |  |
 |
1022.2 | 0.1063 | 0.203 |
| 1000 | 1064.7 | 0.1623 | 0.268 |
| 1050 | 1075.4 | 0.1788 | 0.284 |
| 1100 |  |
0.1963 |  |
Fig. 3.
Material properties dependent on temperature: (a) specific heat; (b) thermal conductivity coefficient.
Thermal resistances 
and 
between the layers can be determined using the relation50,55:
 |
27 |
where 
denotes the width of gap between the layers and 
is the thermal conductivity of the gap. The gap is assumed to be filled with dry air. The calculations use the thermal conductivity value of the air at a temperature of 
K, that is, the value 
W/(m K). The value of 
mm is taken as the width of the gaps.
The algorithm designed was implemented in the Wolfram language of Mathematica 14.0 package and the calculations were performed using a computer with an Intel Core i7-8565U, 1.80 GHz, 2.00 GHz processor, equipped with 16 GB RAM.
Various meshes
First, calculations were made for various meshes 
. Starting with a constant mesh over space 
and a variable mesh over time 
, the calculations were performed for the order of Riemann–Liouville derivative 
.
Figure 4 shows the temperature distribution at the end of the first layer (point 
) along with enlarging a fragment of the plot. A very similar result is obtained for each of the meshes, resulting in all curves overlapping. Only enlargement of a part of the plot illustrates the differences between the results obtained for different meshes. The differences themselves are plotted in Fig. 5a. The differences were calculated relatively to the densest mesh, i.e. the mesh for 
. In case of the mesh for 
the maximal difference at the end of the first layer is equal to 
K, for the mesh with 
the maximal difference decreases to 
K, for the mesh with 
it decreases to 
K, while for the mesh with 
the maximal difference reduces to 
K. The average differences for these variants of the mesh are equal to 
K, 
K, 
K and 
K, respectively. This data is also included in Table 2.
Fig. 4.
(a) Temperature at the end of first layer (point ) for various densities of time mesh; (b) zoom of the part of Figure (a).
Fig. 5.
Differences of temperature at the end of first layer (a) and in the middle of third layer (b) obtained for various densities of the time mesh.
Table 2.
Differences of temperature determined for various time meshes (calculated relatively to the densest mesh with , for 
).
|
|
|
|
|||||
|---|---|---|---|---|---|---|---|---|
| Max | Mean | Max | Mean | Max | Mean | Max | Mean | |
 |
8.359 | 1.665 | 3.860 | 0.775 | 1.641 | 0.331 | 0.545 | 0.110 |
|
8.218 | 1.663 | 3.789 | 0.776 | 1.613 | 0.332 | 0.536 | 0.111 |
 |
5.628 | 0.990 | 2.908 | 0.466 | 1.361 | 0.200 | 0.480 | 0.067 |
 |
0.501 | 0.211 | 0.236 | 0.099 | 0.102 | 0.042 | 0.034 | 0.014 |
 s |
8.228 | 3.897 | 3.796 | 1.802 | 1.616 | 0.768 | 0.537 | 0.255 |
 s |
0.649 | 0.207 | 0.306 | 0.094 | 0.132 | 0.040 | 0.044 | 0.013 |
 s |
1.276 | 0.691 | 0.589 | 0.319 | 0.252 | 0.137 | 0.084 | 0.045 |
 s |
5.349 | 2.642 | 2.468 | 1.216 | 1.061 | 0.522 | 0.354 | 0.174 |
The graph of the differences of the determined temperatures inside the third layer (point 
) is displayed in Fig. 5b. When moving away from the external surface, the differences for each mesh decrease. This is, of course, a consequence of applying the first kind boundary condition on the inner surface (see the first part of Table 2). By observing the results collected in Table 2, we can observe that the temperature differences determined for various time meshes decrease with the movement towards the inside of the thermal shell, which is consistent with the course of the process, as the situation stabilizes inside the shell away from the influence of external factors.
In turn, Figures 6, 7 present the temperature distributions in the region at moments s and s obtained as a result of calculations performed for different mesh densities over time (Figure a). The differences in temperatures obtained in result of calculations performed for different meshes are also presented there (Figure b). The differences were calculated relatively to the densest mesh ().
Fig. 6.
(a) Temperature in moment of time s for various densities of the time mesh; (b) differences of temperature in moment of time s obtained for various densities of the time mesh.
Fig. 7.
Differences of temperature in moments of time s (a) and s (b) obtained for various densities of the time mesh.
In the case of mesh for , the maximal differences change from the value of 
K to the value of 
K, whereas the average differences vary from the value of 
K to 
K (see Table 2). For the mesh with , the maximal differences vary from 
K to 
K and the average ones range from 
K to 
K. At each analyzed moment of time, with increasing time mesh density, temperature differences decrease, which was to be expected. The smallest differences are recorded for the mesh with . The maximal difference does not exceed the value of 
K, and the average difference is not bigger than 
K.
In the next step, a constant mesh is established over time, assuming 
. Whereas, the mesh with respect to space changes taking 
(20, 20, 40), (40, 40, 80), (80, 80, 160), (120, 120, 240), 
. This time the results for 
are presented. For the similar results were obtained.
Figure 8 plots the temperature at the end of the first layer (). Again, the results obtained are not much different from each other (Figure a). Only enlarging a part of the plot reveals differences between the temperatures determined for different grids (Figure b). These differences are plotted in Fig. 9 and were counted, each time, relatively to the densest grid which is 
. At the end of the first layer, the maximal difference decreases from the value of 
K for the mesh (10, 10, 20) to the value of 
K for the mesh (150, 150, 320). Whereas, the average differences decrease respectively from the value of 
K to the value of 
K (see Table 3). At the end of the second layer () the differences are the largest. For mesh (10, 10, 20) they amounted to 
K (maximal) and 
K (average). For the mesh (150, 150, 300) the differences decreased to the values of 
K and 
K, respectively. Temperature differences on the external surface () and in the middle of the third layer (), obtained by calculations performed for different mesh densities in space, are presented in Fig. 10. They are decreasing rapidly again: at point from the value 
K to the value 
K, and at point from the value 
K to the value 
K (see Table 3).
Fig. 8.
(a) Temperature at the end of first layer (point ) for various densities of spatial mesh; (b) zoom of the part of Figure (a).
Fig. 9.
Differences of temperature at the end of first layer obtained for various densities of the spatial mesh.
Table 3.
Differences of temperature in selected points of the region determined for various spatial meshes (calculated relatively to the densest mesh , for ).
 |
|
|
|
|
|
|---|---|---|---|---|---|
| (10, 10, 20) | Max | 31.127 | 32.605 | 444.022 | 11.986 |
| Average | 17.590 | 20.859 | 372.339 | 5.109 | |
| (20, 20, 40) | Max | 25.733 | 26.811 | 287.874 | 6.052 |
| Average | 10.594 | 12.410 | 247.837 | 2.560 | |
| (40, 40, 80) | Max | 19.030 | 19.852 | 175.025 | 2.833 |
| Average | 6.118 | 7.072 | 147.615 | 1.202 | |
| (80, 80, 160) | Max | 9.509 | 9.888 | 115.596 | 1.056 |
| Average | 2.929 | 3.348 | 69.844 | 0.449 | |
| (120, 120, 240) | Max | 3.853 | 3.993 | 56.868 | 0.388 |
| Average | 1.261 | 1.438 | 30.191 | 0.166 | |
| (140, 140, 280) | Max | 1.765 | 1.827 | 28.068 | 0.180 |
| Average | 0.619 | 0.708 | 15.219 | 0.077 | |
| (150, 150, 300) | Max | 0.847 | 0.877 | 14.108 | 0.092 |
| Average | 0.326 | 0.374 | 8.290 | 0.040 |
Fig. 10.
Differences of temperature on the outer surface (a) and in the middle of third layer (b) obtained for various densities of the spatial mesh.
In turn, Fig. 11 shows the temperature curves for moments of time 
s and s obtained on the way of calculations made for different mesh densities in space. Next, Table 4 contains the maximal and average differences of temperature at selected moments of time obtained by calculations performed for different mesh densities in space. They are counted relatively to the densest mesh, that is . In the case of mesh (10, 10, 20), the maximal differences do not exceed 
K, and the average differences do not exceed 
K. The smallest differences occurred for mesh (150, 150, 300). The maximal difference is not bigger than 
K, and the average one is at most equal to 
K. The results are presented only for few selected points in the region, but similar results were obtained for other points. By observing the results in both tables, that is in Tables 3 and 4, one can see that the increase in mesh density relative to the spatial variable results in a clear decrease in temperature differences, i.e. an improvement in the accuracy of the results.
Fig. 11.
Temperature in moment of time s (a) and s (b) for various densities of the spatial mesh.
Table 4.
Differences of temperature in selected moments of time determined for various spatial meshes (calculated relatively to the densest mesh , for ).
|
 |
 |
 |
 |
|
|---|---|---|---|---|---|
| (10, 10, 20) | Max | 320.264 | 380.052 | 414.360 | 322.042 |
| Average | 48.896 | 53.949 | 60.434 | 67.762 | |
| (20, 20, 40) | Max | 231.82 | 247.469 | 266.685 | 236.919 |
| Average | 32.328 | 29.749 | 32.540 | 44.570 | |
| (40, 40, 80) | Max | 159.440 | 142.505 | 149.191 | 175.025 |
| Average | 21.347 | 15.452 | 16.185 | 29.870 | |
| (80, 80, 160) | Max | 102.301 | 65.434 | 61.905 | 97.229 |
| Average | 12.876 | 6.675 | 6.296 | 16.296 | |
| (120, 120, 240) | Max | 43.082 | 31.154 | 23.041 | 36.418 |
| Average | 6.231 | 3.691 | 2.801 | 7.147 | |
| (140, 140, 280) | Max | 19.182 | 19.874 | 10.301 | 16.095 |
| Average | 2.509 | 2.052 | 1.139 | 2.858 | |
| (150, 150, 300) | Max | 9.096 | 13.843 | 8.470 | 7.610 |
| Average | 1.370 | 1.641 | 0.805 | 1.472 |
Separately, each of the differential schemes used in the method (for ordinary derivatives and the Riemann–Liouville derivative) are convergent52,56. The calculations carried out were intended to show that the connection of these approaches is also convergent. The optimal choice seems to be to apply the mesh 
. The use of a denser mesh only slightly changes the results, but the computation time is unnecessarily extended.
In the case of mesh the calculations took 
s, whereas for mesh 
the computation time was equal to 
s. Therefore, increasing the mesh density no longer makes sense, as it does not significantly improve the results and increases the computation time.
Various orders of fractional derivative
The next calculations are aimed at verifying the convergence of the investigated model to the model with ordinary derivatives, when the order 
of the Riemann–Liouville derivative tends to one. The model with ordinary derivatives was used earlier to solve the inverse problem26.
This time, all calculations were performed for mesh 
. The order of Riemann–Liouville derivative changed from 0.5 to 0.9 with step 0.1 and additionally the values 0.95, 0.99, 0.995, 0.999, 0.9995 and 0.9999 were also used.
Figure 12 presents the distribution of temperature at the end of the first layer (point ) obtained for different values of the order of derivative. In this case the differences are small. They can be seen only by zooming a part of this plot (see Fig. 13). The results obtained for various values of order of the Riemann–Liouville derivative begin to differ more and more when approaching to the third layer and in this layer itself. This is, obviously, related to the fact that in the third layer the equation with Riemann–Liouville derivative is given. In Fig. 14 the courses of temperature at the end of the second layer are displayed. Also, as the time runs, the differences in the results obtained for different values of order of the Riemann–Liouville derivative increase. Figure 15 shows the results obtained for the moment s. The differences in the results obtained for different orders of the fractional derivative are visible in the first half of the considered region, close to the outer layer, and in the second half they stabilize. Whereas, in Fig. 16 the results obtained for the moment of time 
s are presented. The differences observed here are larger, but the closer the order of the fractional derivative approaches one, the closer the temperature curve comes to the temperature curve obtained from the classical model. Similar results were also obtained for other points in the considered region.
Fig. 12.
Distribution of temperature at the end of first layer (point ) for various orders of fractional derivative.
Fig. 13.
Zoom of the part of Figure 12.
Fig. 14.
Distribution of temperature at the end of second layer (point ) for various orders of fractional derivative.
Fig. 15.
Distribution of temperature in moment s for various orders of fractional derivative.
Fig. 16.
Distribution of temperature in moment s for various orders of fractional derivative.
The average absolute and relative difference for the entire mesh between the solution obtained for the classical model and the solution for the model with Riemann–Liouville derivative decrease as tends to one. In case of , these differences are equal to 
K and 
%, respectively, and next these differences are: for : 
K and 
%, for 
: 
K and 
%, for 
: 
K and 
%, while for 
they are equal to 
K and 
%.
The greater differences in the temperature distributions obtained for different values of the order , visible in Figs. 12, 14, are a consequence of the location of the point for which the temperature course is plotted. Figure 12 shows the temperature distribution at point , i.e. at the point of contact between the first and second layers. However, Fig. 14 shows the temperature distribution at point , i.e. at the point of contact between the second and third layers. The temperature at this point is therefore also influenced by the processes taking place in the third layer, and there is already a fractional model applied there, hence the greater differentiation depending on the order of derivative is noticed. The point of contact between the first and second layers is further away from the third layer, hence the dependence on the processes taking place in the third layer (i.e. the fractional model) is much smaller.
The obtained results show that when the order of Riemann–Liouville derivative tends to one, the solution of the model with Riemann–Liouville derivative converges to the solution of the model with classical derivatives. Therefore, we can say the same about the model with Riemann–Liouville derivative itself.
Conclusions
The paper presents the mathematical model of the thermal protection system in which the Riemann–Liouville fractional derivative is used. It was assumed that the system consists of three layers. In the first two layers, the temperature distribution was described by means of the classical heat conduction equation. Whereas, in the third layer, due to its porous nature, the temperature distribution was described with the aid of fractional order equation with the Riemann–Liouville type derivative. The non-zero thermal resistances were assumed between the layers. A constant value of the thermal resistance was assumed in the calculations. The outer surface was under the influence of the second kind boundary condition with aerothermal heat flux, which was generated at the NASA Langley Research Center. Whereas, on the internal surface, the first kind boundary condition was applied. It was also assumed that the material parameters depend on temperature.
Additionally, the implicit differential scheme for the considered model was presented, which allowed to determine the temperature distribution in the system. Calculations were made for various space-time meshes, which confirmed the convergence of the results with increasing mesh density. The optimal choice, taking into account the accuracy of the results and the calculation time, is mesh grid. Calculations were also performed for various values of order of the Riemann–Liouville derivative. The results showed that when the order tends to one, the solution of the problem with fractional derivative converges to the solution of the problem with only classical derivatives. A certain limitation of the method is the computation time, which for the most optimal mesh amounted to s.
In the future, it is planned to use the presented model to solve the inverse problem in which the aerothermal heating of the thermal protection systems of space vehicles will be reconstructed. The obtained results will be also compared with the results of the inverse problem for a model with classical derivatives24–26.
Author contributions
Conceptualization – D.S.; methodology – R.B., E.H., D.S.; software – D.S.; validation – D.S; analysis of results – R.B., E.H., D.S. All authors wrote the main manuscript text. All authors reviewed the manuscript.
Data availability
Data are available on request from authors. Requests for materials should be addressed to D.S.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
Data are available on request from authors. Requests for materials should be addressed to D.S.