Because industrial, residential, or agricultural activities use energy, they will produce heat. Every human produces an average of 100 Watts of heat as well. This heat eventually needs to be dissipated into the environment, or the temperature will rise and the settlement become too hot. Heat can be transported via convection, conduction, and radiation, or by mass transfer and phase change.
Agriculture in particular requires large amounts of light, typically between 400 and 600 W/m2 to be productive. This light turns into heat, that must be removed from the greenhouse or grow room. A large part of the heat is transferred to water vapor by the plants by evapo-transpiration. This water must be removed from the air by condensation. This is a form of mass transfer heat transportation.
Convection on Mars is minimal due to its very thin atmosphere, with even fan-assisted convection appearing to be less mass efficient than radiative cooling.
the convection equation is Q=h*A*dT, where Q is the power (Watts), A is the area(m2), h is the experimental coefficient for convective heat transfer (J/m2*°K) and dT is the temperature difference (°K) between the surface and the convecting fluid.
Conduction into Mars regolith or megaregolith (soil or bedrock) may be feasible, since the ground's average temperature is around -60C. On Earth ground-source heat pumps are feasible for cooling. On Mars, depending on the ground conditions, sufficient cooling may be available via the building's foundation alone, or this could be augmented with cooling channels, which could be combined with existing utility trenches used for power or materials.
Challenges include the low temperature of the ground requiring a careful choice of working fluid, and interior humidity may deposit frost, and will certainly condensate, on cooling panels.
Some regolith, such as dry dust or loose rock, may have poor thermal conductivity, requiring either additional conduction area such as drilled cooling pipes or channels, or a soil treatment such as water injection to increase thermal conductivity by filling the soil pore voids with ice.
Radiative cooling is a standard solution for spacecraft, since the large temperature difference between outer space (around 3K) and human habitable areas (around 300K) gives substantial radiative cooling from high emissivity surfaces. However, near the terrestrial planets and closer to the sun the actual 'space temperature' is usually set at 200K.
The Stefan-Boltzmann law describes the thermal emission of a black body radiator as q=e*σ*dT4, where the radiated unit power q (Watts/m2) is equal to the surface emissivity e (between 0 and 1), a constant σ (Stephen Boltzman constant = 5,67 e-8 W/m2°K), and the fourth power of thermodynamic temperature difference dT (°K). For a surface at 293K (about 20C) with emissivity 0.8, the black body radiative cooling can reach 334 W/m2 when facing the cold dark of space, but is closer to 260 W/m2 when the average temperature of the environment is taken into account.
On Mars, during the nighttime a structure's roof could be used for radiative cooling, which could be as simple as a high-emissivity coating applied to the existing roof. However, the thermal resistance of the roof construction will reduce conduction considerably, and the actual surface temperature may be much lower than the interior temperature. In particular, the thick radiation shielding required for the settlement usually has a low coefficient of conduction. So some form of mass transfer of energy will probably be required between the interior of the settlement and the radiators at the surface.
One challenge with radiative cooling is keeping sunlight from warming the radiator panels. A possible mitigation is a careful arrangement of mirrors to reflect sunlight away, or a paint with high visible reflectance and high thermal emittance.
The sun has most of its light in higher frequencies (visible light), while radiative heat is at a much lower frequency (infra red light). So a paint can reflect the light from the sun (reflectance) and absorb little heat, while having a high thermal emittance.
Mass transfer and phase change
The mass transfer equation is Q=m*Cp*dT where Q is the power in Watts, m is the mass flow in kg/s, Cp is the specific heat in Joules/kg/°K and dT is the temperature difference of the moving flow between the source and the heat sink.
The phase change equation is Q=m*Phe where m is the mass flow in kg/s and Phe is the phase change energy in J*/kg. A fluid can either gain energy through phase change (melting, evaporation and sublimation) or lose energy (solidification, condensation)
- von Arx and Delgado, "Convective heat transfer on Mars", AIP Conference 1991 https://aip.scitation.org/doi/abs/10.1063/1.40133?journalCode=apc
- Lunarpedia Lunar Radiator https://lunarpedia.org/w/Lunar_Radiator