# Towards a Heat-Resistant Tethered Micro-Aerial Vehicle for Structure Fire Sensing
## 4 Thermal Analysis
### Heat Transfer Model  
Once a viable heat insulation has been picked, the next step is to evaluate whether this system can satisfy the design and operational requirements specified in Section 3.1 using a detailed transient heat transfer model. The change in temperature of a specific body can be obtained from the heat balance equation:  
\[C\frac{dT}{dt}=\dot{Q}_{in}+\dot{W}_{in}-\dot{Q}_{out}-\dot{W}_{out}, \tag{1}\]  
where \(C=mc\) is the heat capacity of the body, expressed as a function of its mass \(m\) and specific heat capacity \(c\); \(\dot{Q}_{in}\) and \(\dot{Q}_{out}\) are the heat fluxes in and out of the system, respectively; and \(\dot{W}_{in}\) and \(\dot{W}_{out}\) are other energy fluxes in and out of the system. The main heat fluxes that need to be considered are radiation, convection, and conduction. Figure 4: Sandwich structure of the lateral thermal insulation and duct insulation. Figure 3: Sandwich structure of the main frame thermal insulation. Deriving one equation like Equation (1) for each thermal component, the heat transfer model can be expressed as eight time-dependent differential equations:  
\[m_{F}c_{F}\frac{dT_{F}}{dt}= \dot{Q}_{rad_{A_{1},F}}+\dot{Q}_{conv_{A_{1},F}}-\dot{Q}_{cond_{F,C_ {1}}},\] \[m_{C_{1}}c_{C_{1}}\frac{dT_{C_{1}}}{dt}= -\dot{Q}_{rad_{C_{1},C_{2}}}-\dot{Q}_{cond_{C_{1},H}}+\dot{Q}_{ cond_{F,C_{1}}},\] \[m_{H}c_{H}\frac{dT_{H}}{dt}= \dot{Q}_{cond_{C_{1},H}}-\dot{Q}_{cond_{H,C_{2}}},\] \[m_{C_{2}}c_{C_{2}}\frac{dT_{C_{2}}}{dt}= \dot{Q}_{cond_{H,C_{2}}}+\dot{Q}_{rad_{C_{1},C_{2}}}-\dot{Q}_{ rad_{C_{2},A_{2}}}-\dot{Q}_{conv_{C_{2},A_{2}}},\] \[m_{D_{p}}c_{D_{p}}\frac{dT_{D_{p}}}{dt}= \dot{Q}_{rad_{A_{1},D_{p}}}+\dot{Q}_{conv_{A_{1},D_{p}}}-\dot{Q}_ {cond_{D_{p},D_{p}}},\] \[m_{D_{p}}c_{D_{p}}\frac{dT_{D_{p}}}{dt}= -\dot{Q}_{rad_{D_{p},A_{2}}}-\dot{Q}_{conv_{D_{p},A_{2}}}+\dot{Q}_ {cond_{D_{p},D_{p}}},\] \[m_{L_{F}}c_{L_{F}}\frac{dT_{L_{F}}}{dt}= \dot{Q}_{conv_{A_{1},L_{F}}}+\dot{Q}_{rad_{A_{1},L_{F}}}-\dot{Q}_ {rad_{L_{F},A_{2}}}-\dot{Q}_{conv_{L_{F},A_{2}}},\] Figure 4:0 Figure 4:1 Figure 4:2  
where subindices \({}_{F}\), \({}_{C_{1}}\), \({}_{H}\), \({}_{C_{2}}\), \({}_{D_{F}}\), \({}_{D_{F}}\), \({}_{L_{F}}\), \({}_{A_{1}}\), \({}_{A_{2}}\), \({}_{E}\) represent the outside carbon felt, CFRP 1, Kevlar honeycomb, CFRP 2, duct polycarbonate, duct felt, lateral felt, outside air, inside air, and electrical components, respectively. In heat-flow terms, subindices \({}_{rad_{e}}\), \({}_{core}\), \({}_{core}\) represent radiation, convection, and conduction heat fluxes, respectively. The last equation of the system models the heat balance due to the electronic components inside the system. The total electrical power consumption (added as maximum consumptions indicated in the technical datasheet of each manufacturer) is introduced into the system as heat, while the electronic components transfer heat to the air inside the isolated volume. The heat conduction flux from material \(i\) to \(i+1\) is obtained from  
Figure 4:3  
where \(A\) is the contact surface area, \(T\) the material temperature, \(\delta\) its thickness, and \(\lambda\) its heat conduction. For the thermal junction, the worst-case scenario is assumed, for which there is no thermal resistance in the junction between material \(i\) and \(i+1\). The radiation term between materials is calculated as  
Figure 4:4  
where \(\sigma\approx 5.6704\cdot 10^{-8}\) W/m\({}^{2}\)K is the Stefan-Boltzmann constant, \(A_{i}\) is the radiative material surface, and \(\varepsilon_{i}\) is its emissivity. Since most radiative fluxes involve air, the worst-case scenario is again assumed, where the target material absorbs all the emitted radiation.