Thermo-Mechanical Model of an Axisymmetric Rocket Combustion Chamber Protection Using Ablative Materials ^†

Abstract

The integrity analysis of a combustion chamber that uses Ablative Thermal Protection Systems (ATPSs) is a process that requires the analysis of the thermal and mechanical behavior of the materials involved and their interaction. A 1D thermal model for multilayered combustion chambers of hybrid rocket engines and solid rocket motors is developed, taking into consideration the thermal behavior of charring ATPSs during phase change and the capability of implementing an ablation process. A stress model is also implemented to assess the structural integrity of the combustion chamber that undergoes pressure and thermal loads. A numerical finite-difference model is used to implement analytical models and simulate the behavior of the materials. Bibliographic data and finite element analysis tools are used to evaluate and verify the models developed. Lastly, six different materials are used as a case study, and a parametric optimization is applied to obtain the minimum-mass designs using the materials selected.

1. Introduction

The combustion chamber of a hybrid rocket engine is very similar to that of a solid rocket motor [1]. This cylindrical component serves as the housing where the chemical reaction of combustion occurs. For the case of the hybrid rocket engine, in this chamber, the solid fuel grains are stored, and the oxidizer is mixed with the fuel. After the oxidizer flow valve is opened, the combustion reaction starts with ignition, performed by the igniter. However, in solid rocket motors (Figure 1), both the fuel and oxidizer are mixed together in a solid grain, making the propellant of solid rocket motors highly reactive on its own. Therefore, the operation of a solid rocket motor only requires an initial ignition to be started. In both types of propulsion, the main chemical activity during operation occurs in this component.

Because the combustion chamber houses the chemical activity of the engine, this component is exposed to high values and gradients of pressure and temperature. Thus, it is necessary to protect this component from extreme combustion temperatures to prevent critical thermal failure of the engine and to reduce the thermal impact on the pressure limit of the chamber, since most engine structural materials lose strength as their temperature increases [3,4]. There are two main thermal protection methods for rocket engines [2]. The steady-state method, where the components reach thermal equilibrium during operation, and the unstable heat transfer method, where no equilibrium is reached and the temperature of the components increases during operation [2]. In the steady state method, the engine components can be cooled with a cooling fluid (regenerative cooling), or the components reach very high temperatures and are passively cooled by heat transfer through radiation (radiation cooling) [2]. In the case of the unsteady heat transfer method, the engine must be operated before the components reach their thermal operating limit [2]. To extend this limit, it is common to use ablative thermal liners inside the combustion chamber [2]. These ablative thermal liners absorb thermal energy from combustion while being ablated, reducing heat transfer from combustion to the structural parts of the combustion chamber.

Natali et al. [5] developed an article on the subject of Ablative Thermal Protection Systems (ATPSs). The authors sought to achieve a model that quantitatively represents the ablation rate of Elastomeric Heat Shielding Materials (EHSMs). For this study, a one-dimensional analytical model was developed to describe the insulation capabilities of the materials. The description of the ablation model can be seen in Figure 2.

To achieve the model, they use energy equations and material properties to determine the heat transfer and surface regression in each of the thermal liner’s layer states (char, pyrolysis zone and virgin material). The numerical results are compared with the experimental arc-jet tests, obtaining valid cohesive results and concluding that, for the properties of these materials, consideration of reversible and irreversible thermal expansion of the layers and char densification are essential to achieve a very precise estimate of the thermal protection behavior during engine operation.

The article Li et al. [6] also focused on developing a one-dimensional thermal conduction model of composite materials. Although the developed model did not consider the ablation of the material, it developed a pyrolysis model of the change in state of the composite material based on the Arrhenius equation. The authors also sought to validate their model using experimental results from tests they performed on the composite glass fiber/phenolic. A cylindrical sample of the material was heated with a constant heat flux, and the temperature was measured in time at certain distances from the heating surface. The authors then compared the measured data with the results simulated by their model.

Despite some inaccuracies being detected in certain moments of the heating process, the authors conclude that the results show a good agreement between the calculated and experimental temperatures. The simulated results do not show considerable deviation from the experimental results, which for a highly transient and nonlinear process, as is heat transfer between composite materials undergoing phase change, shows that the model achieved is of value.

Since ablative thermal liners are the most common type of thermal protection of the combustion chamber used in hybrid rocket engines [4], a method is needed to design the combustion chamber casing of a hybrid rocket motor (and similar solid [1]), taking into consideration two opposing objectives: achieving a combustion chamber design capable of sustaining the extreme conditions of motor operation while minimizing the mass of the motor, and of the rocket as a whole, to maintain a high mass efficiency.

2. Theoretical Background

The study of heat transfer through the combustion chamber during engine operation is a transient problem. A transient conduction process in a system is one in which the temperature varies in time and space.

When a transient conduction process is induced by changes in thermal boundary conditions, usually between a solid and a fluid through convection, there are two important dimensionless parameters [7] that characterize the transient process, and these are the Biot number ($Bi$ Equation (1)) and the Fourier number ($Fo$ Equation (2)).

$$Bi={\displaystyle \frac{h{L}_c}{k}},$$

(1)

where h is the heat transfer coefficient, k thermal conductivity and $L_c$ the characteristic length of the system ($L_c=V/A_S$, volume (V) by the area of surface ($A_S$)). The Biot number gives a correlation between the rate of heat transfer from the solid to the fluid through convection and the rate of heat transfer inside the solid through conduction [7].

$$Fo={\displaystyle \frac{{\alpha }_{d}t}{L_c^2}},$$

(2)

where ${\alpha }_d$ is the thermal diffusivity and t is the time. The Fourier number, on the other hand, is a dimensionless parameter of time [7].

It is a common practice when studying transient conduction problems to start by determining the two previous parameters. The values obtained can influence the choice of the solution technique to be used.

There are three solution techniques that are used to solve transient conduction problems [7]. The first is the Lumped Capacitance Method (LCM). This method is an approximation of the system under study and assumes that the temperature differentials within the solid are negligible [7]. This is equivalent to saying that the thermal conductivity of the solid is infinite. The LCM can be applied to systems that have $Bi<<1$.

For many systems, the values of the Biot number are too large to assume a uniform temperature distribution inside the solid. For these cases [7], the second solution technique might be the only possible approach that involves solving the exact solutions. The following is the usual and general form of the heat equation [6,8].

$$\left(\rho c_p\right){\displaystyle \frac{\partial T}{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left(k_x{\displaystyle \frac{\partial T}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left(k_y{\displaystyle \frac{\partial T}{\partial y}}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(k_z{\displaystyle \frac{\partial T}{\partial z}}\right)+q_g^{″},$$

(3)

where T is the temperature, $k_x$, $k_y$ and $k_z$ are the thermal conductivities in the different directions, $\rho$ the mass density, $c_p$ the specific heat capacity at constant pressure and $q_g^{″}$ the heat generated within the material.

The heat equation can be used to fully describe the temperature response inside a solid material, where conductivity is the only mode of heat transfer in action. Being a differential equation with second order in space and first order in time, obtaining the exact solution of the heat equation can be extremely difficult.

The third solution technique used is the Finite Difference Method (FDM) [5,7]. The FDM technique is useful for studying systems whose geometry is more complex than the ones assumed in simplifications of the exact solutions. This method solves different variations in the heat equation by performing discretizations in space and time. The following are the FDM discretization approximations for the one-dimensional space and time derivatives:

$${\displaystyle \frac{\partial T_m}{\partial t}}\approx {\displaystyle \frac{T_m^{p+1}-T_m^p}{\Delta t}},$$

(4)

$${\displaystyle \frac{\partial T_m}{\partial x}}\approx {\displaystyle \frac{T_{m+1}^p-T_{m-1}^p}{2\Delta x}}\mathrm{and}$$

(5)

$${\displaystyle \frac{{\partial }^2{T}_m}{\partial x^2}}\approx {\displaystyle \frac{T_{m+1}^p+T_{m-1}^p-2T_m^p}{{(\Delta x)}^2}},$$

(6)

the superscript (p) and subscript (m) indices of T are the indices representing the time step and radial step (position), respectively, and $\Delta t$ and $\Delta x$ are the time and position steps, respectively.

Considering now the mechanical behavior of the system, since the combustion chamber is assumed to be an axisymmetric pressure vessel, the equilibrium equation given by the infinitesimal strain theory for the case of axisymmetric geometries [9] is

$${\displaystyle \frac{d{\sigma }_r}{dr}}+{\displaystyle \frac{{\sigma }_r-{\sigma }_{\theta }}{r}}=0,$$

(7)

where the stress $\sigma$ is shown in the cylindrical coordinate system (subscripts $r,\theta ,z$), with the main components radial, hoop, and axial.

Another set of equations of interest that is obtained for the axisymmetric geometry is the strain–displacement relations [9], given as

$${\epsilon }_r={\displaystyle \frac{du}{dr}},{\epsilon }_{\theta }={\displaystyle \frac{u}{r}}\mathrm{and}{\epsilon }_z={\displaystyle \frac{dw}{dz}},$$

(8)

where $\epsilon$ is the strain, also expressed in cylindrical coordinates, u is the cross-sectional displacement, and w is the axial displacement.

3. Methodology

3.1. Thermal Model

In order to simulate the transient thermal behaviour of the heat transfer, through the thickness of the Ablative Thermal Protection Systems (ATPSs) and combustion chamber, the FDM solution technique was chosen. The choice of this technique was based on its generality, versatility with different materials and geometries, and due to the background support that exists, in the form of previously developed work, such as those performed by Barato et al. [4], Shi et al. [10], Li et al. [6] and Natali et al. [5].

Some assumptions are made to simplify the model in the study. Only heat transfer in the thickness direction is considered; therefore, a one-dimensional FDM model is used. It is assumed that the ATPS is a material that has constant physical properties in the thickness direction; therefore, any anisotropic material, such as composite materials, can be analyzed, as long as their properties are uniform in the thickness direction (transverse isotropy) [11], which is common for tubes made of fiber-reinforced matrices as seen on many ATPS materials. The thermal expansion of the materials is not modeled and is assumed to be negligible. Additionally, the thermal analysis of the ATPS assumes the worst-case scenarios, where the ATPS is exposed to the engine combustion during the entire burn time and is observed in the post-combustion chamber of a hybrid rocket engine.

Equation (3), which is the transient heat equation, can be simplified according to the model at use, into

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho h\right)={\displaystyle \frac{\partial }{\partial r}}\left(k_r{\displaystyle \frac{\partial T}{\partial r}}\right),$$

(9)

where r denotes the radial or thickness direction and h the specific enthalpy of the material. This is the energy equation that defines the heat transfer inside the material. When considering an ATPS charring material that undergoes phase change during pyrolysis, in which the virgin material changes into char material, the energy involved in this process is taken into consideration through two additional terms in the heat equation [5,6,8,10,12]

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho h\right)={\displaystyle \frac{\partial }{\partial r}}\left(k_r{\displaystyle \frac{\partial T}{\partial r}}\right)-Q{\displaystyle \frac{\partial \rho }{\partial t}}-{\dot{m}}_g{c}_g{\displaystyle \frac{\partial T}{\partial r}}.$$

(10)

The second term on the right side of the equation defines the energy involved in the decomposition of the material, being Q the heat of the decomposition. The third term on the right side of Equation (10) quantifies the energy absorbed by the gases released during pyrolysis (transpiration cooling), where ${\dot{m}}_g$ is the generated gas mass flow and can be calculated as

$${\dot{m}}_g=-{\int }_{r_i}^{r_f}{\displaystyle \frac{\partial \rho }{\partial t}}dr.$$

(11)

Equation (10) can be extended to its final form by expanding the left side of the equation and the first term on the right side of the equation, giving the following energy equation for the ATPS material:

$$\rho c{\displaystyle \frac{\partial T}{\partial t}}={\displaystyle \frac{k_r}{\partial r}}·{\displaystyle \frac{\partial T}{\partial r}}+k_r·{\displaystyle \frac{{\partial }^{2}T}{\partial r^2}}-(h_s-h_g){\displaystyle \frac{\partial \rho }{\partial t}}-{\dot{m}}_g{c}_g{\displaystyle \frac{\partial T}{\partial r}},$$

(12)

where $h_g$ is the enthalpy of the gases released during pyrolysis and $h_s$ is the enthalpy of the solid material. Both quantities can be defined, respectively, as [5,6,8]

$$h_g={\int }_{T_{ref}}^T{c}_{g}dT\mathrm{and}$$

(13)

$$h_s=Q+{\int }_{T_{ref}}^{T}cdT,$$

(14)

where c and $c_g$ are the specific heat capacities of the solid material and the released gases, respectively.

During the pyrolysis process, where the phase change can be observed, the virgin material is transformed into char material at a rate defined by the time derivative of $\rho$. To determine this rate, the Arrhenius degradation equation is used [4,5,6,8,10], which states that the degradation rate of the material due to thermal phase change can be expressed as an equation of the nth order of the form

$${\displaystyle \frac{\partial \rho }{\partial t}}=-A({\rho }_v-{\rho }_c){\left({\displaystyle \frac{\rho -{\rho }_c}{{\rho }_v-{\rho }_c}}\right)}^n{e}^{{\displaystyle \frac{-E_a}{RT}}},$$

(15)

where A is the Arrhenius preexponential factor, $E_a$ is the activation energy, ${\rho }_v$ and ${\rho }_c$ are the density of virgin material and charred material, respectively, and R is the universal gas constant. The variables in the Arrhenius equation depend on the type of material that is being degraded, and it is the most common way to study the phase change in charring materials that undergo pyrolysis [5,6,8,13]. As can be seen, the fraction to the power of n seen in the equation indicates the state of change that a portion of the material is in, taking the value of one when the material is still in the virgin state and taking a value of zero when the material is completely charred. This factor is therefore used in the mixture rule [6] to determine the thermal characteristics of an ATPS during decomposition:

$$k=\left({\displaystyle \frac{\rho -{\rho }_c}{{\rho }_v-{\rho }_c}}\right)k_v+\left[1-\left({\displaystyle \frac{\rho -{\rho }_c}{{\rho }_v-{\rho }_c}}\right)\right]k_c\mathrm{and}$$

(16)

$$c=\left({\displaystyle \frac{\rho -{\rho }_c}{{\rho }_v-{\rho }_c}}\right)c_v+\left[1-\left({\displaystyle \frac{\rho -{\rho }_c}{{\rho }_v-{\rho }_c}}\right)\right]c_c,$$

(17)

where $k_v$, $k_c$, $c_v$ and $c_c$ are the thermal conductivities and specific heat capacities of virgin and charred materials, respectively. To be precise, these are the thermal properties of the material both in its original state and after a complete thermal degradation, when the pyrolysis has ended and only charred residues remain.

Apply now the discretization methods mentioned in the FDM technique, in the form of Equations (4)–(6), and the approximated, discretized, second-order differential equation of energy becomes

$${\rho }_m{c}_m{\displaystyle \frac{T_m^{p+1}-T_m^p}{\Delta t}}={\displaystyle \frac{k_{m+1}-k_{m-1}}{2\Delta r}}·{\displaystyle \frac{T_{m+1}^{p+1}-T_{m-1}^{p+1}}{2\Delta r}}+k_m·{\displaystyle \frac{T_{m+1}^{p+1}+T_{m-1}^{p+1}-2T_m^{p+1}}{{(\Delta r)}^2}}-$$

$$-{\displaystyle \frac{\partial \rho }{\partial t}}{|}_m·[h_s-h_g]+\sum _{i=m}^k{\displaystyle \frac{\partial \rho }{\partial t}}{|}_i·\Delta r·c_g·{\displaystyle \frac{T_{m+1}^{p+1}-T_{m-1}^{p+1}}{2\Delta r}}.$$

(18)

Note that there are two FDM techniques, the explicit and implicit methods [7]. The difference between the methods consists of whether the temperature in node m at time $p+1$ is obtained through the previous temperature value $T_m^p$ (explicit method), or through the values of neighboring nodes $T_{m+1}^{p+1}$ and $T_{m-1}^{p+1}$ (implicit method). Although the explicit method can be easily solved step-by-step by solving the equation in each node, the implicit method can only be applied by solving the equation in all nodes simultaneously [7] through the use of matrix inversion solvers. The implicit method is chosen here for two main reasons. Firstly, the explicit method is conditionally stable and therefore depends on a stability criterion to remain stable and provide valid solutions. Using the implicit method, the stability of the solutions is safeguarded. Secondly, the lack of criterion limitation, and the use of matrix systems of equations, improve the computational performance of the solver.

Equation (18) can be rearranged in a more useful form:

$${\displaystyle \frac{{\rho }_m{c}_m}{\Delta t}}{T}_m^p-{\displaystyle \frac{\partial \rho }{\partial t}}{|}_m·\left[h_s-h_g\right]=\left[{\displaystyle \frac{{\rho }_m{c}_m}{\Delta t}}+{\displaystyle \frac{2k_m}{{(\Delta r)}^2}}\right]T_m^{p+1}+\left[{\displaystyle \frac{k_{m-1}-k_{m+1}-4k_m}{4{(\Delta r)}^2}}+\sum _{i=m}^k{\displaystyle \frac{\partial \rho }{\partial t}}{|}_i·{\displaystyle \frac{c_g}{2}}\right]T_{m+1}^{p+1}+$$

$$+\left[{\displaystyle \frac{k_{m+1}-k_{m-1}-4k_m}{4{(\Delta r)}^2}}-\sum _{i=m}^k{\displaystyle \frac{\partial \rho }{\partial t}}{|}_i·{\displaystyle \frac{c_g}{2}}\right]T_{m-1}^{p+1}.$$

(19)

A similar equation can be obtained for the nodes in the casing. By disregarding the terms related to material phase change and assuming a uniform temperature distribution throughout the casing, Equation (20) can be obtained. Suppose a casing with uniform temperature distribution and consider the very low Biot number that is obtained for the boundary conditions of the casing, which are related to the casing materials’ high thermal conductivity.

$${\displaystyle \frac{\rho c}{\Delta t}}{T}_m^p=\left[{\displaystyle \frac{\rho c}{\Delta t}}+{\displaystyle \frac{2k}{{(\Delta r)}^2}}\right]T_m^{p+1}-{\displaystyle \frac{k}{{(\Delta r)}^2}}{T}_{m+1}^{p+1}-{\displaystyle \frac{k}{{(\Delta r)}^2}}{T}_{m-1}^{p+1}.$$

(20)

Additionally, four boundary conditions equations have to be defined in order to fully describe the system. The boundary conditions are divided into two solid/fluid boundaries and one solid/solid boundary. The solid/fluid boundaries are present in the inner and outer layers of the combustion chamber, where the material is in contact with the gases from combustion and the ambient air. The solid/solid boundary condition exists in the contact between the ATPS and the casing, where two equations are obtained, one for each boundary node of each material.

Equation (21) and Equation (22) are associated with node 1 and node n, respectively, where node 1 is part of the ATPS, while node n is part of the casing. The equations are obtained using the same process as Equations (19) and (20), only taking into account two additional energy terms, which come from heat transfer in the form of convection and radiation with the surrounding gases.

$${\displaystyle \frac{{\rho }_1{c}_1}{\Delta t}}{T}_1^p-{\displaystyle \frac{\partial \rho }{\partial t}}{|}_1·[h_s-h_g]+{\displaystyle \frac{h_i}{\Delta r}}(T_{fi}^p-T_1^p)+{\displaystyle \frac{{\epsilon }_i\sigma }{\Delta r}}({T_{fi}^p}^4-{T_1^p}^4)=$$

$$=\left[{\displaystyle \frac{{\rho }_1{c}_1}{\Delta t}}+{\displaystyle \frac{k_2}{{(\Delta r)}^2}}-\sum _{i=1}^k{\displaystyle \frac{\partial \rho }{\partial t}}{|}_i·c_g\right]T_1^{p+1}+\left[-{\displaystyle \frac{k_2}{{(\Delta r)}^2}}+\sum _{i=1}^k{\displaystyle \frac{\partial \rho }{\partial t}}{|}_i·c_g\right]T_2^{p+1}\mathrm{and}$$

(21)

$$\left[{\displaystyle \frac{\rho c}{\Delta t}}+{\displaystyle \frac{k}{{(\Delta r)}^2}}\right]T_n^{p+1}-{\displaystyle \frac{k}{{(\Delta r)}^2}}{T}_{n-1}^{p+1}={\displaystyle \frac{\rho c}{\Delta t}}{T}_n^p+{\displaystyle \frac{h_e}{\Delta r}}(T_{fe}^p-T_n^p)+{\displaystyle \frac{{\epsilon }_e\sigma }{\Delta r}}({T_{fe}^p}^4-{T_n^p}^4).$$

(22)

The boundary condition for the contact between the two materials, which is represented by nodes k and $k+1$, for the ATPS and the casing, respectively, is defined by an additional term that corresponds to the heat transfer coefficient between the two materials. This term has an equivalence with thermal circuits analysis, where we have two thermal resistances in series [7]. Equations (23) and (24) correspond to nodes k and $k+1$, respectively.

$${\displaystyle \frac{{\rho }_k{c}_k}{\Delta t}}{T}_k^p-{\displaystyle \frac{\partial \rho }{\partial t}}{|}_k·\left[h_s-h_g\right]=\left[{\displaystyle \frac{{\rho }_k{c}_k}{\Delta t}}+{\displaystyle \frac{k_{k-1}}{{(\Delta r)}^2}}+{\displaystyle \frac{1}{\frac{{(\Delta r)}^2}{2k_k}+\frac{{(\Delta r)}^2}{2k_{k+1}}}}-{\displaystyle \frac{\partial \rho }{\partial t}}{|}_k·c_g\right]T_k^{p+1}-$$

$$-{\displaystyle \frac{1}{\frac{{(\Delta r)}^2}{2k_k}+\frac{{(\Delta r)}^2}{2k_{k+1}}}}{T}_{k+1}^{p+1}+\left[-{\displaystyle \frac{k_{k-1}}{{(\Delta r)}^2}}+{\displaystyle \frac{\partial \rho }{\partial t}}{|}_k·c_g\right]T_{k-1}^{p+1}\mathrm{and}$$

(23)

$${\displaystyle \frac{\rho c}{\Delta t}}{T}_{k+1}^p=\left[{\displaystyle \frac{\rho c}{\Delta t}}+{\displaystyle \frac{k}{{(\Delta r)}^2}}+{\displaystyle \frac{1}{\frac{{(\Delta r)}^2}{2k_k}+\frac{{(\Delta r)}^2}{2k_{k+1}}}}\right]T_{k+1}^{p+1}-{\displaystyle \frac{k}{{(\Delta r)}^2}}{T}_{k+2}^{p+1}-{\displaystyle \frac{1}{\frac{{(\Delta r)}^2}{2k_k}+\frac{{(\Delta r)}^2}{2k_{k+1}}}}{T}_k^{p+1}.$$

(24)

According to energy conservation, the heat transferred from node k to node $k+1$ must be equal to that received from $k+1$, so the term is symmetric in both equations. Note that for all equations, the terms $h_s$ and $h_g$ are functions of the temperature in the respective node and are specific to the material of the ATPS. These functions are obtained by integrating the specific heat capacity in temperature [5,6,8]. Furthermore, the specific heat capacity of the gases of pyrolysis ($c_g$) is also a function of temperature.

Now that the equations of thermal behavior are defined for both materials and boundaries, a matrix system of equations can be obtained. The implicit method of FDM can be solved using a system of matrix equations, which allows the computation of the state of all nodes simultaneously. Starting by defining the matrix A as

$$A\equiv \left[\begin{array}{ccccccc}{a}_{11}& a_{12}& 0& \dots & 0& 0& 0\\ a_{j(-1)}& a_{j0}& a_{j1}& \dots & 0& 0& 0\\ \dots & \dots & \dots & \ddots & \dots & \dots & \dots \\ 0& 0& 0& \dots & a_{w(-1)}& a_{w0}& a_{w1}\\ 0& 0& 0& \dots & 0& a_{n(n-1)}& a_{nn}\end{array}\right],$$

where A is a tri-diagonal matrix of the terms of $T^{p+1}$, with the exception of the terms ${\rho }_m{c}_m/\Delta t$, which are defined in the matrix B. $a_{11}$ and $a_{12}$ and $a_{n(n-1)}$ and $a_{nn}$ are terms related to solid/fluid boundary conditions, as seen in Equation (21) and Equation (22), respectively. Equivalently, the entries of the matrix A with indices of k and $k+1$ (which correspond to the nodes that define the boundary between ATPS and casing) are related to the solid/solid boundary and are seen in Equations (23) and (24), respectively. Finally, the matrix entries $a_{j(-1)}$, $a_{j0}$, and $a_{j1}$ and the entries $a_{w(-1)}$, $a_{w0}$, and $a_{w1}$ are the terms of Equation (19) and Equation (20), respectively, with the subscripts 0, 1 and $-1$ indicating the terms of $T_m^{p+1}$, $T_{m+1}^{p+1}$ and $T_{m-1}^{p+1}$, respectively.

Additionally, the diagonal matrix B is defined as

$$B\equiv {\displaystyle \frac{1}{\Delta t}}\left[\begin{array}{cccccccc}{\rho }_1{c}_1& 0& \dots & 0& 0& \dots & 0& 0\\ 0& {\rho }_2{c}_2& \dots & 0& 0& \dots & 0& 0\\ \dots & \dots & \ddots & \dots & \dots & \dots & \dots & \dots \\ 0& 0& \dots & {\rho }_k{c}_k& 0& \dots & 0& 0\\ 0& 0& \dots & 0& \rho c& \dots & 0& 0\\ \dots & \dots & \dots & \dots & \dots & \ddots & \dots & \dots \\ 0& 0& \dots & 0& 0& \dots & \rho c& 0\\ 0& 0& \dots & 0& 0& \dots & 0& \rho c\end{array}\right]$$

where the diagonal entries change for each line related with nodes 1 to k, while for lines $k+1$ to n the material properties are constant, and therefore, the diagonal is fixed.

Finally, the vectors C, $T^{p+1}$, and $T^p$ are defined:

$$C\equiv \left[\begin{array}{c}{c}_1\\ c_2\\ \dots \\ c_k\\ 0\\ \dots \\ 0\\ c_n\end{array}\right],T^{p+1}\equiv \left[\begin{array}{c}{T}_1^{p+1}\\ T_2^{p+1}\\ \dots \\ T_{n-1}^{p+1}\\ T_n^{p+1}\end{array}\right]\mathrm{and}{T}^p\equiv \left[\begin{array}{c}{T}_1^p\\ T_2^p\\ \dots \\ T_{n-1}^p\\ T_n^p\end{array}\right],$$

where C is the vector of the constants of the equations of energy, taking into account the terms related to both material phase change and heat transfer with the exterior. Vectors $T^{p+1}$ and $T^p$ are the vectors of the temperature values in each node, for the current and following time steps, respectively.

Using Equation (19) to Equation (24), the following matrix system corresponds to the complete system of all equations of energy in all nodes:

$$T^{p+1}={(A+B)}^{-1}B{T}^p+{(A+B)}^{-1}C.$$

(25)

Equation (25) is the matrix function that gives the temperature value in all nodes in the next time step, knowing the temperature of all nodes in the current time step. This is the matrix solver that is computed to simulate the thermal behaviour of the system at each time step.

In order to estimate the heat convection coefficient inside the combustion chamber $h_i$, which controls the rate at which thermal energy is transferred from the combustion to the ATPS, Bartz’s equation (Equation (26)) [14,15] was used, which gives a semi-empirical correlation between the heat transfer coefficient and the flow properties and the geometry of the chamber.

$$h_i={\displaystyle \frac{0.026}{D_{*}^{0.2}}}\left({\displaystyle \frac{{\mu }^{0.2}{c}_p}{{\left(Pr\right)}^{0.6}}}\right){\left({\displaystyle \frac{p_c}{c^{*}}}\right)}^{0.8}{\left({\displaystyle \frac{A_{*}}{A}}\right)}^{0.9}{\left({\displaystyle \frac{D_{*}}{r_c}}\right)}^{0.1}\sigma ,$$

(26)

where $\sigma$ is given as:

$$\sigma ={\displaystyle \frac{1}{{\left[\frac{1}{2}\left(\frac{T_w}{T_0}\right)\left(1+\frac{\gamma -1}{2}{M}^2\right)+\frac{1}{2}\right]}^{0.8-m/5}{\left[1+\frac{\gamma -1}{2}{M}^2\right]}^{m/5}}},$$

(27)

where $D_{*}$ is the diameter of the nozzle throat, $r_c$ is the radius of curvature of the nozzle, $A_{*}$ and A are the areas of the nozzle throat and chamber, respectively, $c_{*}$ is the characteristic velocity of the engine, $Pr$ is the Prandtl number, $c_p$ is the specific heat capacity at constant pressure of the gases of the combustion reaction, $\mu$ is the dynamic viscosity of the gases, $p_c$ is the pressure inside the combustion chamber, $T_w$ and $T_0$ are the temperature of the wall in contact with the combustion and the temperature of the combustion gases, respectively, M is the Mach number, $\gamma$ is the specific heat ratio of the gases and finally m is the exponent of the viscosity dependence on temperature (assumed to be 0.6).

In order to reduce the complexity of the information required to run the simulation, some assumptions were made to acquire an approximation of the value of the flow Mach number. The velocity will be different throughout the engine length, and also due to the fact that the post-combustion chamber of a hybrid engine is the location where the flow conditions are more severe [16,17] (due to several factors such as, being the section where the ATPS is in direct contact with combustion during the entire burn, the flow has the highest velocity in the region before the nozzle, and in this zone, the oxidizer and liquid/gaseous fuel are well mixed, allowing the complete combustion reaction to occur), the location of the engine chosen as the simulation point is the entrance of the nozzle. This location allows us not only to approximately estimate the flow velocity using well-known propulsion equations but also to assume a simulation for a worst-case scenario, where the conditions of the combustion are more severe.

To determine the Mach number at the entrance of the nozzle, an analysis of the nozzle is performed, assuming that the flow through the nozzle is isentropic [2]. Additionally, it is assumed that the nozzle is ideally expanded; therefore, the pressure at the nozzle exit is equal to the ambient pressure, which is given as an input [2]. With these assumptions, the following analysis process can be executed to determine the velocity of the flow at the nozzle entrance.

Firstly, the flow velocity at the nozzle exit is determined by using the following correlation obtained from assuming an isentropic flow through the nozzle.

$${\displaystyle \frac{p_0}{p_e}}={\left[1+{\displaystyle \frac{\gamma -1}{2}}{M}_e^2\right]}^{{\displaystyle \frac{\gamma }{\gamma -1}}},$$

(28)

where $p_0$ and $p_e$ are the chamber pressure and the exit pressure of the nozzle (equal to the ambient pressure), respectively, and $M_e$ is the Mach number at the exit of the nozzle.

Then, the mass flow rate through the nozzle is calculated using the equation obtained from the analysis of the converging/divergent nozzle [2], with an isentropic flow with area variation, where the flow at the nozzle throat is assumed to be under sonic conditions.

$$\dot{m}=A_e\sqrt{{\displaystyle \frac{\gamma }{R{T}_0}}}{p}_0{M}_e{\left(1+{\displaystyle \frac{\gamma -1}{2}}{M}_e^2\right)}^{-{\displaystyle \frac{\gamma +1}{2(\gamma -1)}}},$$

(29)

where $A_e$ is the exit area of the nozzle and $T_0$ is the temperature inside the chamber.

Assuming a constant mass flow rate through the nozzle, Equation (29) can now be inverted to determine the Mach number at the nozzle entrance, using the area of the nozzle entrance, which is assumed to be equal to the area of the combustion chamber section. Note that inverting Equation (29) requires solving a nonlinear equation, for which no analytical solution can be obtained, and so the equation is solved numerically. Additionally, two solutions for the Mach number are obtained from solving the non-linear equation, one corresponding to the subsonic (converging) section of the nozzle and another to the hypersonic (diverging) section; therefore, only the subsonic solution is chosen as the result and applied to Bartz’s equation.

3.2. Mechanical Model

With the thermal model developed, the capability to estimate the temperature in each node of the casing at each time step of the burn was achieved. However, knowledge of the temperature alone is not sufficient to predict the occurrence of a critical engine failure. Although failure detection uses upper limit values of material temperature (related to the maximum service temperature of the casing material), additionally, a mechanical model of the combustion chamber is needed to predict a mechanical failure of the casing. To address this problem, a simple pressure vessel analysis would not be enough, since this type of analysis would not take into account the decrease in yield strength with temperature of the casing material. Knowledge of the temperature of the casing at each moment in time is also necessary in order to determine the mechanical properties of the casing at each moment and evaluate the capability of the casing to withstand the applied instantaneous pressure load.

Similarly to the thermal model, some assumptions were made to simplify the analytical model used. Firstly, only the elastic deformation of the material is considered [18,19], since, for the study of reusable rocket engines, it is essential that the structural integrity of the materials does not exceed the elastic range to prevent permanent deformations.

Secondly, only isotropic materials are considered for the casing. This assumption significantly simplifies the analytical models employed and is justified by a third assumption. The casing material has high thermal conductivity. More specifically, it is assumed that the Biot number at the interface between the casing and the external air is very low [7], which implies that the temperature distribution across the thickness of the casing is approximately uniform. Under this condition, thermal stress can be neglected, which is a desirable characteristic for combustion chamber casings [2]. The use of a metallic material with high thermal conductivity minimizes temperature gradients, thus substantially reducing thermal stresses. This helps prevent one of the primary causes of structural degradation in the casing: fissure formation [2].

When studying the use of metals as a material choice for the casing of the combustion chamber, the isotropic analytical equations for the mechanical model can be used, and the thermal stress loads can be neglected [9] (note that, as explained in the thermal model, the thermal expansion is also neglected), simplifying the calculations. The final assumption is made, which has already been used in the thermal model, which considers that the geometry of the combustion chamber is axisymmetric.

Following the work developed by Habib et al. [19] and Wang et al. [18], the development of the mechanical model starts with the stress equations for an uncoupled elastic steady-state system.

$$\left\{\begin{array}{c}{\sigma }_r={\displaystyle \frac{E}{1+v}}\left[{\epsilon }_r+{\displaystyle \frac{v}{1-2v}}\left({\epsilon }_r+{\epsilon }_{\theta }+{\epsilon }_z\right)\right]-{\displaystyle \frac{\alpha ET\left(r\right)}{1-2v}}\hfill \\ \\ {\sigma }_{\theta }={\displaystyle \frac{E}{1+v}}\left[{\epsilon }_{\theta }+{\displaystyle \frac{v}{1-2v}}\left({\epsilon }_r+{\epsilon }_{\theta }+{\epsilon }_z\right)\right]-{\displaystyle \frac{\alpha ET\left(r\right)}{1-2v}}\hfill \\ \\ {\sigma }_z={\displaystyle \frac{E}{1+v}}\left[{\epsilon }_z+{\displaystyle \frac{v}{1-2v}}\left({\epsilon }_r+{\epsilon }_{\theta }+{\epsilon }_z\right)\right]-{\displaystyle \frac{\alpha ET\left(r\right)}{1-2v}}\hfill \end{array}\right.$$

(30)

where the subscripts r, $\theta$, and z indicate the cylindrical coordinates of the stress, radial, hoop, and axial. The stress and the displacement are represented by $\sigma$ and $\epsilon$, respectively. The Young’s modulus is represented by E, while $\nu$ is the Poisson’s ratio. Finally, $T\left(r\right)$ and $\alpha$ are the temperature in function of the radius and the thermal expansion coefficient, respectively. Note that the assumption of a uniform temperature distribution across the casing gives a temperature function that is constant; therefore, T does not vary in the radial direction.

Starting by solving the equilibrium Equation (7), by replacing the stresses with the corresponding correlations given in Equation (30), the overall equilibrium equation is given as

$${\displaystyle \frac{d}{dr}}\left[{\displaystyle \frac{E{\epsilon }_r}{1+\nu }}+{\displaystyle \frac{E\nu }{(1+\nu )(1-2\nu )}}\left({\epsilon }_r+{\epsilon }_{\theta }+{\epsilon }_z\right)\right]+{\displaystyle \frac{E}{(1+\nu )r}}({\epsilon }_r-{\epsilon }_{\theta })=0.$$

Replace the strains in the previous equation with the correlations for the axisymmetric geometry given in Equation (8), we obtain the following equilibrium equation for axisymmetric vessels

$${\displaystyle \frac{E(1-\nu )}{(1+\nu )(1-2\nu )}}\left[r^2{\displaystyle \frac{d^{2}u}{d{r}^2}}+r{\displaystyle \frac{du}{dr}}-u\right]=0.$$

(31)

Equation (31) is a second-order differential equation. Notice that the format of this differential equation is of the type of Euler–Cauchy differential equations for which there are some known solutions. Since Equation (31) has two real roots, the solution of the differential equation is given as $y=c_1{x}^{m_1}+c_2{x}^{m_2}$, where $m_1$ and $m_2$ are the real roots. For Equation (31), the roots are given as $m_1=1$ and $m_2=-1$. Therefore, the solution to the Euler–Cauchy differential equation is the displacement function, given as

$$u\left(r\right)=Ar+B{r}^{-1},$$

(32)

where A and B are the integration constants to be determined.

Now, by replacing the strains in the system of equations in Equation (30) with the corresponding strain–displacement correlations given by Equation (8), the equations of stress can be presented as

$$\left\{\begin{array}{c}{\sigma }_r={\displaystyle \frac{E}{(1+\nu )(1-2\nu )}}\left[(1-\nu ){\displaystyle \frac{du}{dr}}+{\displaystyle \frac{\nu u}{r}}+\nu {\epsilon }_z\right]\hfill \\ \\ {\sigma }_{\theta }={\displaystyle \frac{E}{(1+\nu )(1-2\nu )}}\left[(1-\nu ){\displaystyle \frac{u}{r}}+\nu {\displaystyle \frac{du}{dr}}+\nu {\epsilon }_z\right]\hfill \\ \\ {\sigma }_z={\displaystyle \frac{E}{(1+\nu )(1-2\nu )}}\left[(1-\nu ){\epsilon }_z+\nu {\displaystyle \frac{du}{dr}}+{\displaystyle \frac{\nu u}{r}}\right]\hfill \end{array}\right.$$

(33)

Note that these are the stress values that correspond uniquely to the mechanical loads. Because the thermal stress load can be neglected due to the uniform temperature distribution, by applying the linear-elastic and uncoupled thermo-mechanical conduction conditions, the combined stress will be just the mechanical stress [18]. Using the obtained solution for the displacement function $u\left(r\right)$, and noting that $du/dr=A-(B/r^2)$, the previous system of equations can be rearranged, giving

$$\left\{\begin{array}{c}{\sigma }_r={\displaystyle \frac{E}{(1+\nu )(1-2\nu )}}\left[A+(2\nu -1){\displaystyle \frac{B}{r^2}}+\nu {\epsilon }_z\right]\hfill \\ \\ {\sigma }_{\theta }={\displaystyle \frac{E}{(1+\nu )(1-2\nu )}}\left[A+(1-2\nu ){\displaystyle \frac{B}{r^2}}+\nu {\epsilon }_z\right]\hfill \\ \\ {\sigma }_z={\displaystyle \frac{E}{(1+\nu )(1-2\nu )}}\left[2\nu A+(1-\nu ){\epsilon }_z\right]\hfill \end{array}\right.$$

(34)

with six unknown variables ${\sigma }_r$, ${\sigma }_{\theta }$, ${\sigma }_z$, A, B and ${\epsilon }_z$, and a system of three equations, the system is under-defined. Three additional equations are needed, which are obtained from the boundary conditions of the axisymmetric vessel under pressure load. These boundary conditions can be defined as

$$\left\{\begin{array}{c}{\sigma }_r=0,\mathrm{when}r=R_o\hfill \\ {\sigma }_r=-p,\mathrm{when}r=R_i\hfill \\ {\int }_{R_i}^{R_o}{\sigma }_{z}2\pi rdr=p\pi R_i^2\hfill \end{array}\right.$$

(35)

where p is the internal pressure of the combustion chamber, while $R_i$ and $R_o$ are the inner and outer radius of the casing, respectively. Note that the ATPS is assumed to have no structural utility, being simply applied as a thermal device, therefore, its contribution to the structural integrity of the engine is neglected, hence the use of the inner radius of the casing is chosen, and not the inner radius of the combustion chamber as a whole. By applying these boundary conditions, and after some solving and rearrangement of the system of equations, the values of A, B, and ${\epsilon }_z$ can be determined

$$\left\{\begin{array}{c}A=-{\displaystyle \frac{p{R}_i^2(2\nu -1)}{E(R_o^2-R_i^2)}}\hfill \\ \\ B={\displaystyle \frac{p{R}_i^2{R}_o^2(1+\nu )}{E(R_o^2-R_i^2)}}\hfill \\ \\ {\epsilon }_z={\displaystyle \frac{p{R}_i^2(1-2\nu )}{E(R_o^2-R_i^2)}}\hfill \end{array}\right.$$

(36)

Finally, by replacing the values obtained in the stress system of equations in Equation (34) and simplifying the results, the final equations of stress for the cylindrical coordinate system can be obtained

$$\left\{\begin{array}{c}{\sigma }_r={\displaystyle \frac{p{R}_i^2}{R_o^2-R_i^2}}\left(1-{\displaystyle \frac{R_o^2}{r^2}}\right)\hfill \\ \\ {\sigma }_{\theta }={\displaystyle \frac{p{R}_i^2}{R_o^2-R_i^2}}\left(1+{\displaystyle \frac{R_o^2}{r^2}}\right)\hfill \\ \\ {\sigma }_z={\displaystyle \frac{p{R}_i^2}{R_o^2-R_i^2}}\hfill \end{array}\right.$$

(37)

The mechanical model predicts the margin of safety that is expected for the engine design being studied from a structural point of view. For this, the formula for the mechanical margin of safety in stress loads is introduced as

$$MoS={\displaystyle \frac{{\sigma }_y}{FoS·{\sigma }_M}}-1,$$

(38)

where $FoS$ is the factor of safety, which is a design factor that defines the relation between the maximum applied stress load and the maximum stress the material can withstand. For ${\sigma }_y$, it is the strength of the material’s yield, the upper limit of the stress the material can withstand in the range of elastic deformation [3,9]. Lastly, ${\sigma }_M$ is the equivalent Von-Mises stress for the principal stresses of the system, given as

$${\sigma }_M=\sqrt{{\displaystyle \frac{1}{2}}\left[{\left({\sigma }_r-{\sigma }_{\theta }\right)}^2+{\left({\sigma }_{\theta }-{\sigma }_z\right)}^2+{\left({\sigma }_z-{\sigma }_r\right)}^2\right]}.$$

(39)

3.3. Materials

The first step in the process of testing and applying the developed model is the choice of materials to be studied and their corresponding properties. Six different materials were chosen as study subjects—three materials for the ATPS and three materials for the casing—providing a total of nine possible combinations of material designs.

For the case of the ATPS, the chosen materials are:

• Glass fiber phenolic composite;
• Carbon fiber epoxy composite;
• Twaron fiber EPDM composite.

The glass and carbon fiber composites are categorized as fiber Reinforced Polymeric Ablators (FRPAs), while the twaron fiber composite is categorized as an Elastomeric Heat Shielding Material (EHSM). The glass fiber phenolic was chosen due to the available data and studies used in this article, with an emphasis on the work developed by Li et al. [6], whose analytical thermal model was one of the bases for the work developed in Section 3.1, and the experimentation data performed by the author using glass fiber phenolic composites are valuable for model validation. In terms of the carbon fiber epoxy, it was chosen because it is a very common composite in the aerospace industry. Composite materials using carbon fibers have been extensively studied, developed and used in a wide range of applications during the past years [20]. It is therefore of interest to study its applicability for this specific function. Lastly, the twaron EPDM composite is chosen for a state-of-the-art comparison with the rest of the materials. EPDM matrix-based composites have been used for many years as the go-to advanced material for ATPS in solid rocket motors [5,21], and are therefore chosen to provide an industry standard comparison with the remaining materials.

As for the materials chosen for the casing of the combustion chamber, these are:

• Stainless steel, AISI 304;
• Aluminum, Al6061-T6;
• Titanium, Ti-6Al-4V.

Note that only metals were chosen for the casing material. This is due to the assumptions and simplifications that were applied in the development of the mechanical model, mentioned in Section 3.2. One of the design requirements for the casing material is a high maximum operating temperature, a characteristic in which most composite materials cannot compete with metals. Moving to the materials themselves, the stainless steel alloy AISI 304 was chosen because it is one of the most common stainless steel alloys available in the industry. The wide availability of this alloy will be used as the strength point for this material, providing a low-cost alternative to compare with the other metals. The 6061-T6 aluminum alloy has improved mechanical properties compared to other aluminium alloys [22]. Due to its high yield strength and maximum operating temperature, this alloy is chosen to represent a high-performance choice among aluminum alloys. Finally, titanium alloy, similar to the choice of EPDM twaron, is the state-of-the-art choice for the casing [23]. Of the three choices, the material that possesses the better balance of the design requirements is the material with the highest yield strength and the second highest maximum operating temperature. However, it is also the most expensive of the materials. While, on average, the aluminum alloy can be about twice as expensive as the stainless steel per unit of mass, the titanium alloy can be ten times more expensive than the aluminum alloy, even more for other advanced alloys [24]. This specific alloy of titanium was chosen because it is one of the most commonly used titanium alloys in engine components in the aerospace industry [25].

3.4. Materials’ Properties

Since the ATPS is considered to have minimal impact on the combustion chamber’s structural integrity, as outlined in the mechanical model, only its thermal characteristics are relevant. Due to ablation and phase changes during engine operation, additional thermal parameters are required, including the heat of decomposition and the degradation behavior described by the Arrhenius equation (see Section 3.1). These thermal properties are derived from studies by Li et al. [6], Tranchard et al. [26] and Natali et al. [5].

Table 1. Glass fiber/phenolic thermal properties [6].

Property	Value	Unit
${\rho }_v$	1810	$\mathrm{kg}/{\mathrm{m}}^3$
${\rho }_c$	1440	$\mathrm{kg}/{\mathrm{m}}^3$
$k_v$	$0.804+2.76\times {10}^{-4}\mathrm{T}$	$\mathrm{W}/(\mathrm{m}·\mathrm{K})$
$k_c$	$0.955+8.42\times {10}^{-4}\mathrm{T}$	$\mathrm{W}/(\mathrm{m}·\mathrm{K})$
$c_v$	$1.089+1.09\times {10}^{-3}\mathrm{T}$	$\mathrm{kJ}/(\mathrm{kg}·\mathrm{K})$
$c_c$	$0.870+1.02\times {10}^{-3}\mathrm{T}$	$\mathrm{kJ}/(\mathrm{kg}·\mathrm{K})$
$c_g$	9.63	$\mathrm{kJ}/(\mathrm{kg}·\mathrm{K})$
$E_a$	$2.60\times {10}^5$	$\mathrm{J}/\mathrm{mol}$
n	6.3	-
A	$8.16\times {10}^{18}$	$1/\mathrm{s}$
Q	$-234.0$	$\mathrm{kJ}/\mathrm{kg}$

and

Table 2. Carbon fiber/epoxy thermal properties [26].

Property	Equation	Unit
${\rho }_v$	1575	$\mathrm{kg}/{\mathrm{m}}^3$
${\rho }_c$	1165	$\mathrm{kg}/{\mathrm{m}}^3$
$c_v$	$2.8773\mathrm{T}+687.31$ with $20 °\mathrm{C}\le \mathrm{T}\le 300 °\mathrm{C}$	$\mathrm{J}/(\mathrm{kg}·\mathrm{K})$
$c_c$	$5.1327\times {10}^{-7}{\mathrm{T}}^3-2.0761\times {10}^{-3}{\mathrm{T}}^2+2.599\mathrm{T}+662.53$	$\mathrm{J}/(\mathrm{kg}·\mathrm{K})$
$c_g$	$3.5977\times {10}^{-7}{\mathrm{T}}^3-9.2485\times {10}^{-4}{\mathrm{T}}^2+1.0610\mathrm{T}+1256.6$	$\mathrm{J}/(\mathrm{kg}·\mathrm{K})$
$k_{v//}$	$7.4675\times {10}^{-3}\mathrm{T}+2.7811$ with $20 °\mathrm{C}\le \mathrm{T}\le 150 °\mathrm{C}$	$\mathrm{W}/(\mathrm{m}·\mathrm{K})$
$k_{v\perp }$	$1.1113\times {10}^{-3}\mathrm{T}+0.61391$ with $20 °\mathrm{C}\le \mathrm{T}\le 150 °\mathrm{C}$	$\mathrm{W}/(\mathrm{m}·\mathrm{K})$
$k_{d//}$	$-3.5481\times {10}^{-6}{\mathrm{T}}^2+6.4898\times {10}^{-3}\mathrm{T}+2.3005$	$\mathrm{W}/(\mathrm{m}·\mathrm{K})$
$k_{d\perp }$	$7.4228\times {10}^{-10}{\mathrm{T}}^3-4.1903\times {10}^{-7}{\mathrm{T}}^2$	$\mathrm{W}/(\mathrm{m}·\mathrm{K})$
	$+2.3397\times {10}^{-4}\mathrm{T}+7.7211\times {10}^{-2}$
Q	$-152.22$	$\mathrm{kJ}/\mathrm{kg}$
A,	$5.3309\times {10}^8$	1/s
$E_a,n$	$146.674;2.0825$	$\mathrm{kJ}/\mathrm{mol}$,-

show the values of density, thermal conductivity, specific heat capacity, Arrhenius parameters, and heat of decomposition for glass fiber/phenolic and carbon fiber/epoxy composites, as given by the bibliography. For the case of carbon fiber/epoxy, the values of thermal conductivity used are in the perpendicular direction, corresponding to the radial direction of the 1D thermal model. As for the case of the Twaron/EPDM,

Table 3. Twaron/EPDM thermal properties [5].

Property	Value	Unit
${\rho }_v$	960	$\mathrm{kg}/{\mathrm{m}}^3$
${\rho }_c$	123	$\mathrm{kg}/{\mathrm{m}}^3$
$c_g$	9.63	$\mathrm{kJ}/(\mathrm{kg}·\mathrm{K})$
$E_a$	$2.164\times {10}^5$	$\mathrm{J}/\mathrm{mol}$
n	1.55	-
A	$5.55\times {10}^{14}$	$1/\mathrm{s}$
Q	$-4.86\times {10}^5$	$\mathrm{J}/\mathrm{kg}$

shows the constant thermal properties obtained from [5].

The model considers only three properties for the casing materials: thermal conductivity, specific heat capacity, and yield strength. These materials are assumed to remain in a single phase as long as the maximum operating temperature remains below their phase-change thresholds. Since all three properties vary with temperature, they were defined using polynomial fits based on experimental data. This data was sourced from ANSYS Granta EduPack (version 2024 r2), a comprehensive academic database of material properties.

4. Results and Discussion

The thermal model validation is performed with the use of experimental bibliographic data. Li et al. [6] sought to develop a thermal model to analyze transient thermal conduction in composite materials under phase change. To validate its model, the author performed experimental tests with the composite material being studied, the glass fiber phenolic composite. Using a secondary graphical function to process the data obtained from the thermal model alone, it is possible to compare the results of the thermal model with the experimental data points obtained from Li et al. [6]. Figure 3 and Figure 4 show the temperature in the nodes of the ATPS that are 1 mm, 5 mm and 10 mm from the flame. Additionally, the crossed points are the experimental points obtained from the bibliography.

The simulation parameters defined as inputs were chosen to match the test environment. A burn of 600 s was defined, with only one layer of 30 mm of ATPS with the boundary condition on the side opposed to the flame defined as adiabatic. Phase change was observed, but the ablation was defined as non-existent, similar to what was observed in the test, since the flame of the test torch did not create enough shear force to remove material.

Similarly to the results obtained from the author’s own thermal model, the simulation seems to overestimate the temperature values during the first part of the burn, and then the values start to show an underestimation. The source of this deviation can be explained by different factors, such as the fact that the thermal model ignores the effect of thermal expansion in the transient conduction, or the somewhat inaccurate properties of the charred ATPS for temperatures that are well above the pyrolysis temperature. The presence of impurities in the composite or even small bubbles of air can also affect the conduction of the material, not only in its virgin state but also on the gases released during pyrolysis. Additionally, some other assumptions of the model can also be a source of error, like the one-dimensional model, since in reality the transfer of heat and gases in the directions perpendicular to the radial direction can be a factor. The effects of this assumption, in particular, can be very well minimized in systems with axisymmetric geometries, where the heat transfer is closely uniform in all of the surfaces. However, in an experimental test, the dimensions of the sample and setup might not allow us to fully disregard this effect.

However, it can be concluded that the thermal model is able to more accurately describe the temperature profile of the ATPS in locations where the transient behavior of the burn is more gradual. The simulated data obtained for the node at 1 mm from the flame provided the results with the largest deviation from the experimental data, with the highest relative error of 16.7% at a time of about 150 s. On the other hand, the simulated data from the node at 10 mm from the flame, which had a much slower heating process, displayed a maximum relative error of 10.5% at around 500 s. Despite these sources of inaccuracy that are connected with the transient thermal conduction model and the model of material degradation, the overall thermal behavior of the system is captured by the simulated model, with the temperature curves being able to fit reasonably well with the experimental data points.

The verification of the mechanical model is performed with the objective of verifying the analytical equations that were derived in Section 3.2 and that compose the integrity of the mechanical analysis of the system. This verification is achieved through the use of a commercial FEM software.

The results of the FEM tool are used as verification values, to which errors are calculated and the accuracy of the 1D model is evaluated.

A combination of three different geometries and three different loads was used. In the FEM tool, the three different geometries were modeled as cylinders with the respective internal radius and thickness, and a length of one meter. Two loads were applied in the FEM model, an internal pressure load applied on the inner surface of the cylinder, and a force applied on the top of the cylinder, equal to the axial load given by $p\pi R_0^2$.

A constraint was applied on the bottom of the cylinder, blocking the movement of the elements in the axial direction, in order to counteract the top load and prevent rigid body motion. The three geometries were meshed with solid brick elements, which were radially extruded from quadrilateral plate elements formed on the inner surface of the cylinder. The mesh was refined to a sufficient level to allow accuracy, with the number of elements in each geometry varying from 400 to 5000. Figure 5 shows one of the meshes used in the FEM simulations, while Figure 6 shows a result of the Von-Mises stress.

The finite element mesh was constructed using quadrilateral plate elements with a side length of 1 mm. To represent the casing’s thickness, each plate element was extruded radially through six iterations, forming solid brick elements. This meshing strategy provided sufficient refinement to capture the thermo-mechanical behavior of the system under operational loads, while maintaining computational efficiency.

Table 4. Maximum equivalent Von-Mises stress and relative error for different geometries and loads from mechanical model and FEM software.

Internal Radius (mm)	Thickness (mm)	Internal Pressure (bar)	Maximum Von-Mises Stress (MPa)		Relative Error
			Mech. Model	FEM
100	1	20	89.120	89.144	0.0269%
100	1	50	222.80	222.87	0.0314%
100	1	100	445.60	445.87	0.0606%
250	3	20	74.747	74.738	0.0120%
250	3	50	186.87	186.85	0.0107%
250	3	100	373.74	373.70	0.0107%
500	5	20	89.191	89.157	0.0381%
500	5	50	222.98	222.89	0.0404%
500	5	100	445.96	445.79	0.0381%

shows the results of both the FEM software and the mechanical model, and the corresponding relative error for different geometries and applied loads.

The results obtained from the FEM models are in accordance with the values gathered from the developed mechanical model. As can be seen in Table 4, the relative error between the mechanical model and the FEM tool is small in relation to the absolute value of the Von-Mises stress, which shows that the assumptions made and the system of equations deduced in Section 3.2 for the mechanical behavior of the system are verified by the tools in existence developed for this type of mechanical analysis.

Sensitive Analysis

In order to evaluate the design of the combustion chamber, a sensitivity analysis (parametric optimization) must first be defined. The case study on hand is the design of the combustion chamber of a hybrid rocket engine that uses nitrous oxide as an oxidant and paraffin as fuel. The engine is intended to be a small propulsive device with high thrust; therefore, a designed burn time of 5 s is applied. For simplification, transient loads are defined as step functions, with the internal pressure assuming the value of 40 bar and the internal temperature having the value of 2750 K, which is the reaction temperature of nitrous oxide with paraffin, at 40 bar, with an oxidizer/fuel ratio of 7, as given by the web application CEARUN (https://cearun.grc.nasa.gov/index.html, accessed on 8 July 2025), which is based on the Chemical Equilibrium with Applications (CEA) [27] computer programme developed by NASA. The simulation time is set at 6 s, which corresponds to 5 s of burn and an additional second of engine shutdown, which is implemented to take into account the soak-back heat that can be expected after engine shutdown. The remaining geometry, performance, and simulation parameters can be seen in

Table 5. Input parameters of the design optimization.

Parameter	Value	Units
Internal Radius ($r_0$)	50	mm
Nozzle Throat Diameter ($D_t$)	27	mm
Nozzle Exit Diameter ($D_e$)	66	mm
Nozzle Throat Radius of Curvature ($r_c$)	5.16	mm
Simulation Time ($t_f$)	6	s
Time Step ($dt$)	$5\times {10}^{-4}$	s
Radial Step ($dr$)	$5\times {10}^{-5}$	m
Ablation Rate ($\dot{r}$)	0.2	mm/s
Radiation Emissivity of Combustion (${\epsilon }_1$)	0.8
Radiation Emissivity of Casing (${\epsilon }_2$)	0.2
Convection Coefficient of Exterior Boundary ($h_2$)	10	W/(m2 K)
Initial Temperature ($T_0$)	294	K
Initial Internal Pressure ($p_0$)	101,325	Pa

.

The only input parameters missing are the material properties of the casing materials that were not specified before, namely the density and the maximum operating temperature. For stainless steel, aluminium, and titanium, respectively, the densities are 8060 kg/m³, 2733 kg/m³ and 4451 kg/m³, and the maximum operating temperatures are 1200 K, 430 K and 700 K [22,24,28,29].

The margin of safety is the main output of the integrated model, calculated using finite-difference methods applied to both thermal and mechanical analyses. Due to the complexity of the model, it is not possible to analytically determine the input ranges that produce a specific safety margin, which prevents the use of mathematical optimization.

The thickness of the ATPS and the casing ($t_1$ and $t_2$) were defined in a range sufficiently large to allow visualization of the possible outcomes (from 0.1 mm to 5 mm), each assuming 100 different values in that range, giving a total of 10⁴ simulations to be run for each combination of materials of interest. With nine different possible combinations to be made from the three materials of choice for ATPS and casing, a total of 9.10⁴ simulations were run to obtain the parametric optimization results required.

Having obtained the results for each material combination, two sets of designs were extracted with some post-processing. One set of results has as a design requirement a margin of safety of 0.5, while another has 2. Both sets evaluated results that fall within a variation of 4% of the margin of safety. The values presented in these sets are the values for which the mass is minimum.

Table 6. Minimum mass values for a design margin of safety of 0.5 within a variation of 4%.

MOS = 0.5; VAR = 4%
Casing Material	ATPS Material	Casing Thickness (mm)	ATPS Thickness (mm)	Mass (kg/m)	MoS
Al 6061-T6	GF/Phenolic	1.05	3.90	3.2855	0.5090
	CF/Epoxy	1.05	4.10	3.0968	0.4977
	Tw/EPDM	1.00	2.25	1.5997	0.4863
AISI 304	GF/Phenolic	1.25	3.55	5.5197	0.4896
	CF/Epoxy	1.30	3.50	5.3574	0.4871
	Tw/EPDM	1.10	2.45	3.7095	0.4847
Ti-6Al-4V	GF/Phenolic	0.45	3.30	2.6120	0.4953
	CF/Epoxy	0.45	3.50	2.4685	0.4956
	Tw/EPDM	0.40	2.20	1.2643	0.4935

and

Table 7. Minimum mass values for a design margin of safety of 2 within a variation of 4%.

MOS = 2; VAR = 4%
Casing Material	ATPS Material	Casing Thickness (mm)	ATPS Thickness (mm)	Mass (kg/m)	MoS
Al 6061-T6	GF/Phenolic	2.10	4.05	4.3832	1.9349
	CF/Epoxy	2.10	4.30	4.2151	1.9450
	Tw/EPDM	2.05	2.30	2.5868	1.9827
AISI 304	GF/Phenolic	2.40	3.60	8.7813	1.9242
	CF/Epoxy	2.40	3.80	8.6365	1.9234
	Tw/EPDM	2.20	2.40	6.7018	1.9233
Ti-6Al-4V	GF/Phenolic	0.80	3.70	3.3922	2.0014
	CF/Epoxy	0.80	3.90	3.2198	1.9911
	Tw/EPDM	0.70	2.40	1.7739	1.9326

show both sets of results for all different combinations of materials under study.

The results provided give the dimensions needed for the required MoS for each of the nine possible combinations of materials. The thickness of the ATPS is more variable because the thermal model of the system depends on many other variables, unlike the mechanical model, which depends mainly on the inner radius of the casing and its thickness.

Glass fiber and carbon fiber composites have more similar properties, both categorized as FRPAs [30]; however, their properties can be useful in different situations. While the glass fiber composite is better in terms of thermal conductivity and pyrolysis rate [6], showing that it can achieve a thermal MoS using a lower thickness than the carbon fiber composite, the latter option has the advantage in terms of density [26]. Looking at all designs using glass fiber and carbon fiber composites in Table 6 and Table 7, the carbon fiber composite always requires a greater thickness, but the overall mass of the combustion chamber is smaller than when using the glass fiber composite.

As presented in Table 6 and Table 7, carbon fiber composites consistently require greater wall thickness than their glass fiber counterparts. However, the resulting combustion chamber mass is lower, making carbon fiber/epoxy a generally superior choice for mass optimization. However, in configurations where the inner radius is tightly constrained, particularly in compact propulsion systems—glass fiber/phenolic may be advantageous, allowing for reduced wall thickness, simplified manufacturing, and only a marginal increase in mass.

The third material used as ATPS is the state-of-the-art material, which is shown as a comparison benchmark. The twaron/EPDM composite is an EHSM that is better in every property than the other two composites [5]. It has a lower virgin density, a lower thermal conductivity, and a higher specific heat capacity [5], which means that it weighs less, transmits less heat across its thickness, and requires more heat to be transferred to it in order to raise its temperature. These are the key performance properties required for an ATPS material [21], and that is why this EHSM and other variants of this composite are state-of-the-art materials used as thermal liners and insulators in solid and hybrid rocket engines developed in the industry.

The optimized designs show that using twaron/EPDM can reduce the mass of the combustion chamber by almost half of the second best choice, the carbon fiber composite. This is a very significant performance improvement, especially for large-scale launchers, where this difference can represent a mass savings of hundreds of kilograms in the total rocket mass. The only downside of this material is its difficult commercial access. Of the three composite materials, the twaron/EPDM is the only one whose commercial cost is unknown, and no research results were found on this value.

A direct cost comparison between the three ATPS materials is challenging. However, given that glass fiber/phenolic costs approximately EUR 25/kg and carbon fiber/epoxy around EUR 50/kg [24], the mass savings offered by carbon fiber/epoxy may not justify its higher cost, depending on the application. In the case of Twaron/EPDM, the lack of pricing data precludes a definitive assessment. However, considering its substantial mass reduction potential, its use is recommended whenever the material is accessible and budget constraints allow.

Among the casing materials analyses, the comparison between aluminum and titanium is more relevant. Despite being more expensive than stainless steel, when compared with titanium, aluminum can still hold the position of a low-cost alternative with wide availability. Looking at the design results, when comparing the designs using aluminum with those using titanium, for the same ATPS material, it shows that the use of titanium can lead to a mass saving of approximately 20%.

This mass save comes from both the higher yield strength/density of the titanium and its higher maximum operating temperature, reducing the needed thickness of the ATPS. However, once again, looking at the mass in Table 6 and Table 7, the performance improvement of this state-of-the-art material is not as appealing as that seen from the use of twaron/EPDM. Unlike the composite, there are commercial costs available for this titanium alloy, with values being one order of magnitude higher than those of aluminum.

Considering that designs using titanium allow for a casing thickness that is approximately 40% of the thickness of the aluminum, but considering also that the titanium has a density that is 65% higher than the density of the aluminum [22,29], it can be roughly estimated that designs using titanium casings would have a casing mass that is approximately 66% of the casing mass of designs using aluminum (noting that for inner radius values much higher than the thickness, the cross-section area of the casing is approximately directly proportional to the thickness).

This means that the use of titanium as a casing material would provide a mass savings, for the casing alone, of approximately 33% while increasing the cost of the casing almost seven times. In many applications, this disparity between the cost and the mass gain can be seen as not cost-efficient enough to justify the use of titanium as a casing material.

The thermal and physical properties of this state-of-the-art material cannot be denied, having a range of elastic deformation that is considerably higher than the other two materials. However, if financial viability is a parameter that has considerable weight during the project design, then it is necessary to reflect on the sustainability of using titanium as a casing material in a rocket combustion chamber.

5. Conclusions

The development of a thermo-mechanical tool to design the combustion chamber of rocket engines is presented. A finite-difference method was applied, using equations that describe both the transient heat transfer for the thermal model and the quasi-static deformation for the mechanical model.

The proposed study assumes that the chamber is axisymmetric, the casing material undergoes only elastic deformation, thermal expansion is negligible, and heat transfer on the inner surface is uniform. Under these conditions, the developed model shows strong agreement with commercially available finite element method (FEM) software. The comparison reveals minimal error under quasi-static conditions and provides accurate stress predictions within the elastic regime. Furthermore, analysis and comparison with documented experimental databases confirm the model’s effectiveness in simulating ATPS and combustion chamber behavior.

The materials researched and chosen for the ATPS and casing achieve a minimum mass for the combustion chamber. Of these two materials, although the use of the Twaron/EPDM composite as ATPS is considered an optimal choice when possible, the use of Titanium as a casing material is seen with some more skepticism, as it allows a mass savings of approximately 33% compared to aluminum, while having a cost that is approximately seven times higher. The use of common alloys of stainless steel as a casing material is highly discouraged, since the safety range for elastic deformation that this material allows is very similar to that of aluminum, while being twice as heavy. Aluminum is considered a strong choice for casing material and offers the best weight-to-cost ratio. Carbon fiber/epoxy is a better alternative than glass fiber/phenolic as an ATPS material when only considering mass optimization. If minimizing the thickness of the ATPS is a design requirement, glass fiber/phenolic should be preferred.

Future work should expand the mechanical model to account for non-isotropic casing materials, develop an ablation model that predicts the ablation rate based on combustion and system properties, and include a fatigue analysis to more accurately estimate the combustion chamber’s lifespan under different operating cycles.