The transient temperature distribution of the ultrahightemperature ceramic (UHTC) thermal protection system (TPS) of hypersonic vehicles is calculated using finite volume method. Convective cooling enables a balance of heat increment and loss to be achieved. The temperature in the UHTC plate at the balance is approximately proportional to the surface heat flux and is approximately inversely proportional to the convective heat transfer coefficient. The failure modes of the UHTCs are presented by investigating the thermal stress field of the UHTC TPS under different thermal environments. The UHTCs which act as the thermal protection materials of hypersonic vehicles can fail because of the tensile stress at the lower surface, an area above the middle plane, and the upper surface as well as because of the compressive stress at the upper surface. However, the area between the lower surface and the middle plane and a small area near the upper surface are relatively safe. Neither the compressive stress nor the tensile stress will cause failure of these areas.
To obtain good aerodynamic characteristics, reentry and hypersonic vehicles require sharp nose cones and sharp leading edges, as well as nonablation thermal protection technique that can maintain the vehicle shapes. These vehicles usually fly in the atmosphere for long periods at speed of high Mach numbers and are often subjected to severe aerodynamic heating. This poses huge challenges for the thermal protection materials and structures of vehicles, such as the capability of withstanding ultrahigh temperature, chemical stability, and resistance to thermal shock, oxidation, and erosioncorrosion.
Ultrahightemperature ceramics (UHTCs) are candidate materials that can meet the above performance requirements and are receiving more and more attention [
Thermal management is one of the priority issues of hypersonic vehicles which are often subjected to severe aerodynamic heating because of the violent friction with air. Active cooling TPS has been proposed as alternative TPS to eliminate the aerodynamic heating and maintain the structural temperature within maximum operational temperature limit [
A schematic of the UHTC TPS with convective cooling is shown in Figure
Schematic of UHTC TPS with convective cooling.
The thickness is small in relation to the transverse dimensions (length and width) of the plate. Thus, conduction is assumed to occur exclusively in the
Since the material properties are temperaturedependent, a closed form solution for (
The computing grid of the FVM for the onedimensional transient heat conduction problem described in the preceding subsection is shown in Figure
Computing grid of FVM for onedimensional transient heat conduction problem.
Equation (
Following the tradition, the physical quantities on the control volume
The diffusion term at the faces of the control volume can be expressed by the central differencing scheme using the values of the nodes. Then (
How temperature changes over time should be given to calculate the timeintegral terms in (
Note that
The differential quotients in the thermal boundary conditions at the lower and upper surfaces can be replaced by the ahead difference quotient and the back difference quotient, respectively. Thus, (
Furthermore,
The fully implicit time integration scheme is unconditionally stable. That is, whatever the time increment is, oscillations of solutions will not appear. Taking
The coefficient matrix of (
The difference between the temperatures of node
Schematic of grid generation for UHTC TPS.
The transient temperature distribution obtained in the preceding subsection is a piecewise linear function of the node coordinate. The temperature in element
Hereto, the FVM for calculations of the transient temperature distribution and thermal stress filed of the UHTC TPS with convective cooling has been presented. One can see that it is very convenient to apply. The main work is to solve the system of linear algebraic equations whose coefficient matrix is a symmetric sparse matrix with bandwidth 2.
Taking the zirconium diboride (ZrB_{2}) ceramics as an example, the transient temperature distribution and the thermal stress field of the UHTC TPS of hypersonic vehicles are studied. The corresponding material properties are shown in Table
Temperaturedependent material properties of the ZrB_{2} ceramics.
Material properties  Values and expressions 






6.119 



Equation ( 

0.15 
In the following discussion, the temperature is limited to 2000°C because of the lack of experimental data and the significant plastic deformation.
Transient temperature distribution of UHTC plate for illustrative example 1. (Temperature increases with increased time.)
Transient temperature distribution of UHTC plate for illustrative example 2. (Temperature decreases with increased time.)
Transient temperature distribution of UHTC plate for illustrative example 3. (Upper surface temperature increases with increased time.)
Transient temperature distribution of UHTC plate for illustrative example 4. (Lower surface temperature decreases with increased time.)
The upper surface temperature at the balance has been calculated as functions of surface heat flux and convective heat transfer coefficient and shown in Figure
Upper surface temperature at the balance versus (a) surface heat flux and (b) convective heat transfer coefficient. (
Based on the discussion above, it can be seen that convective cooling increases the temperature gradient in the plate but decreases the change of the temperature gradient along the thickness. Convective cooling also enables a balance of heat increment and loss to be achieved. The temperature in the UHTC plate at the balance is approximately proportional to the surface heat flux and is approximately inversely proportional to the convective heat transfer coefficient. (See (
The thermal stress field of the UHTC plate that corresponds to illustrative examples 1–4 in the preceding subsection is calculated and shown in Figures
Thermal stress field of UHTC plate for illustrative example 1. (Upper surface is located by compressive stress which initially increases and then decreases with increased time.)
Thermal stress field of UHTC plate for illustrative example 2. (Lower surface is located by tensile stress which initially increases and then decreases with increased time.)
Thermal stress field of UHTC plate for illustrative example 3. (Upper surface is located by compressive stress which initially increases and then decreases with increased time.)
Thermal stress field of UHTC plate for illustrative example 4. (Lower surface is located firstly by tensile stress and then by compressive stress with increased time.)
Based on the discussion above, it can be seen that the UHTCs which act as the thermal protection materials of hypersonic vehicles can fail because of the tensile stress at the lower surface, in an area above the middle plane, and at the upper surface and because of the compressive stress at the upper surface, as shown in Figure
Failure modes of UHTCs which act as thermal protection materials. (Red is for tensile failure and blue is for compressive failure. Dash line indicates middle plane of the UHTC plate.)
In addition, it can be seen that the plate in Figure
The thermal stress near the upper surface, middle plane, and lower surface in Figure
The transient temperature distribution of the UHTC TPS with convective cooling was calculated using FVM and then the thermal stress field was obtained by combining the model of thermal stress field from literature [
Convective cooling increases the temperature gradient in the plate but decreases the change of the temperature gradient along the thickness.
Convective cooling enables a balance of heat increment and loss to be achieved. The temperature in the UHTC plate at the balance is approximately proportional to the surface heat flux and is approximately inversely proportional to the convective heat transfer coefficient.
The UHTCs which act as the thermal protection materials of hypersonic vehicles can fail because of the tensile stress at the lower surface, in an area above the middle plane, and at the upper surface as well as because of the compressive stress at the upper surface. The area between the lower surface and the middle plane and a small area near the upper surface are relatively safe. Neither the compressive stress nor the tensile stress will cause failure of these areas.
The thermal stress near the upper surface, middle plane, and lower surface in the UHTC plate under aerodynamic thermal environments can be reduced remarkably when the convective cooling is operated. That is, convective cooling can improve the TSR of the UHTC plate.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (no. 11172336), the Program for New Century Excellent Talents in University (no. ncet130634), the Fundamental Research Funds for the Central Universities (nos. CDJZR13240021 and CDJZR14328801), the Chongqing Natural Science Foundation (no. cstc2013jcyjA50018), and the Joint Fund of the National Natural Science Foundation of China and the China Academy of Engineering Physics (no. 11176035).