# Balloon Design for Mars, Venus, and Titan Atmospheres
## 2 Theoretical Model
### Envelope Temperature  
The balloon flight is affected by radiant thermal energy as indicated in Figure 2, which shows involved radiant heat fluxes. The heat transfer model used in this work is from [37]. Figure 2: Radiant heat fluxes affecting the balloon flight. This figure is inspired from [23; 32; 37]. While the balloon ascends, the gas is expanding and the energy flux impinging the balloon envelope is constantly changing. The differential equation of the balloon envelope temperature \(T_{\rm e}\) is:  
\[\frac{dT_{\rm e}}{dt}=\frac{Q_{\rm e}}{c_{\rm e}m_{\rm e}}, \tag{17}\]  
where \(c_{\rm e}\) is the specific heat capacity of the balloon envelope and \(Q_{\rm e}\) is the total heat flux exchanged with the balloon envelope, which can be expressed as the sum of several heat fluxes:  
\[Q_{\rm e}=Q_{\rm sun+albedo}+Q_{\rm IR_{planet}}+Q_{\rm IR_{gas}}+Q_{\rm conv_{ int}}+Q_{\rm conv_{est}} \tag{18}\]  
The first term, \(Q_{\rm sun+albedo}\), accounts for direct solar heat and solar albedo. The term \(Q_{\rm IR_{planet}}\) accounts for the net infrared heat input from the planet. Next, \(Q_{\rm IR_{gas}}\) accounts for the net infrared heat flux between the gas inside the balloon and the envelope. The two terms \(Q_{\rm conv_{int}}\) and \(Q_{\rm conv_{est}}\) account for convective heat exchange with the internal gas and the external air, respectively. Using the laws of thermodynamics and the Stefan-Boltzmann law, the energies exchanged with the balloon envelope can be listed as follows:  
\[Q_{\rm e}=\bar{\alpha}_{\rm e}I_{0}(A_{\rm e}+F_{\rm bs}a_{\rm s }A)+\bar{\epsilon}_{\rm e}\sigma A(F_{\rm bs}T_{\rm s}^{4}-T_{\rm e}^{4})\\ +\bar{\epsilon}\sigma A(T_{\rm g}^{4}-T_{\rm e}^{4})+h_{\rm ge}A( T_{\rm g}-T_{\rm e})+h_{\rm ae}A(T_{\rm a}-T_{\rm e}) \tag{19}\]  
The effective solar absorptivity of the envelope is \(\bar{\alpha}_{\rm e}\) and can be calculated using the following formulation:  
\[\bar{\alpha}_{\rm e}=\alpha_{\rm e}\left[1+\frac{\tau_{\rm e}(1-\alpha_{\rm g })}{1-r_{\rm e}(1-\alpha_{\rm g})}\right], \tag{20}\]  
where \(\alpha_{\rm e}\) and \(\tau_{\rm e}\) are the solar absorptivity and solar transmissivity of the envelope respectively. \(\alpha_{\rm g}\) is the solar absorptivity of the lifting gas, and \(r_{\rm e}\) is the solar reflectivity of the envelope. Next, in Equation (19), \(I_{0}\) is the solar constant defined as a flux density measuring the mean solar radiation (solar irradiance) per unit area [37]. \(A_{\rm e}\) is the effective surface area. The albedo is dependent on the location of the balloon, time, and the presence of clouds and is given by \(a_{\rm s}\). In the current version of the integrated design and simulation tool the albedo for every planet is a constant number and can be changed by the user. The term \(\bar{\epsilon}_{\rm e}\) in Equation (19) is the effective infrared emissivity of the envelope and can be calculated using the following expression:  
\[\bar{\epsilon}_{e}=\bar{\epsilon}_{\rm e}\left[1+\frac{\tau_{\rm ei}(1-\epsilon _{\rm g})}{1-r_{\rm ei}(1-\epsilon_{\rm g})}\right], \tag{21}\]  
where \(\bar{\epsilon}_{\rm e}\) is the infrared emissivity of the envelope, \(\tau_{\rm ei}\) the infrared transmissivity of the envelope, \(\bar{\epsilon}_{\rm g}\) the infrared emissivity of the lifting gas, \(r_{\rm ei}\) the infrared reflectivity of the envelope, \(\sigma\) the Stefan-Boltzmann constant, and \(F_{\rm bs}\) the shape factor from the balloon to the planet, for which in this work the value 0.5 is used for all shapes [26]. Further, \(T_{\rm s}\) denotes the effective temperature of the ground surface as seen by the balloon, \(T_{\rm a}\) is the atmospheric temperature, and \(T_{\rm g}\) is the gas temperature.