# Effect of Structural Materials on Monopropellant Thruster Propulsion Performance in Micro Scale
## 3 Numerical Comparative Performance Analysis  
### Governing Equations  
A conjugate heat transfer (CHT) analysis was conducted in a steady state to qualitatively and quantitatively understand the heat loss which led to performance loss, assuming the entry of a fully decomposed gas mixture from different wt% liquid phase hydrogen peroxide into the chamber. The conjugate heat transfer between the fluid and the solid is generally modeled by a coupling of Navier-Stokes equations in the fluid and a heat equation in the solid. The Navier-Stokes equations consist of continuity equations for the conservation of mass, momentum, and energy, complemented by the equation of state and constitutive relations. In this work, a two-equation-based turbulence model was added to the governing equations and solved along with Navier-Stokes equations. The set of governing equations in a time-dependent and conservative form is summarized below without the turbulence equations for simplicity.  
Conservation of mass:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{k}}(\rho u_{k})=0 \tag{1}\]  
Figure 1: Microthruster design profile [20] considered for numerical comparative performance study.  
Conservation of momentum:  
\[\frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial x_{k}}(\rho u_{ i}u_{k})=-\frac{\partial p}{\partial x_{i}}+\frac{\partial\tau_{ij}}{\partial x_{i}}+ \rho F_{i} \tag{2}\]  
Conservation of energy:  
\[\frac{\partial}{\partial t}\bigg{(}e+\frac{1}{2}u_{i}u_{i}\bigg{)}+\frac{ \partial}{\partial x_{k}}\bigg{[}\rho u_{k}\bigg{(}e+\frac{1}{2}u_{i}u_{i} \bigg{)}\bigg{]}=-\frac{\partial}{\partial x_{i}}(pu_{i})+\frac{\partial}{ \partial x_{i}}\big{(}\tau_{ij}u_{j}\big{)}+\frac{\partial}{\partial x_{i}} \bigg{(}k\frac{\partial T}{\partial x_{i}}\bigg{)}+\rho F_{i}u_{i} \tag{3}\]  
where  
\[\tau_{ij}=\mu\bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{ j}}{\partial x_{i}}\bigg{)}-\frac{2}{3}\mu\frac{\partial u_{k}}{\partial x_{k}} \delta_{ij} \tag{4}\]  
where \(e\) is the internal energy, \(k\) is the thermal conductivity, and \(\tau_{ij}\) is the viscous stress tensor. The heat equation governing heat conduction in the solid is as follows:  
\[\rho C_{P}\frac{\partial T}{\partial t}=\frac{\partial}{\partial x_{i}}\bigg{(} k\frac{\partial T}{\partial x_{i}}\bigg{)} \tag{5}\]  
A fluid flow in microsystems might fall in the free molecular flow regime when the characteristic length approaches the mean free path of molecules. To ensure the continuum assumption is valid in current simulations, the Knudsen number (\(Kn\)), was obtained using the following equation and checked to ensure it is within the continuum flow limit:  
\[Kn=\frac{M}{Re}\sqrt{\frac{\gamma\pi}{2}} \tag{6}\]  
where \(M\) is the Mach number, \(Re\) is the Reynolds number, and \(\gamma\) is the specific heat ratio. Two Knudsen numbers at the nozzle throat and exit were calculated for 90 wt% hydrogen peroxide adiabatic simulation without the heat transfer at the wall. The Knudsen numbers were \(2.964\times 10^{-4}\) and \(2.161\times 10^{-4}\) at the nozzle throat and exit, respectively, which were far lower than the continuum flow threshold of 0.01, as reported in [35]. The Knudsen numbers in other simulations were not significantly different from the adiabatic case, and accordingly, all simulations were within the continuum flow limit.  
A piece of commercial CFD software, ANSYS FLUENT, was used to solve the governing equations. The turbulent flow motions were described by the SST k-omega turbulence model since it can effectively model different turbulence scales at the near wall and free stream simultaneously.