So you’ve built a solar balloon. But will it even fly? And how high will it go? This post outlines a method to model the buoyancy and 1D trajectory (elevation vs. time) of a solar balloon. Quite a few formulas go into the complete the model, so a full description is left to a longer PDF report, which is available here. Additionally, the code written to compute the trajectory is available here.
Modeling solar balloon buoyancy is an exercise in heat transfer. Generally, the balloon is heated by the radiation it absorbs and cooled by the fluid flow around it and infrared emission. The sources of radiation absorbed by the balloon include direct solar radiation, reflected and diffuse solar radiation, and infrared radiation emitted by the earth and sky:
A contour plot of the radiation intensity falling upon a balloon vs. time and elevation on Nov 1 is shown below. The amount of direct solar radiation reaching the balloon increases with elevation due to the thinning atmosphere, however, the overall radiation intensity is highest near to the ground due to the IR radiation emitted by the Earth.
The other important source is heat transfer is due to fluid flow. As described in the report, there are fairly accurate engineering level correlations which can be used to determine the rate of heat transfer from the surface of the balloon to interior and to the exterior atmosphere. In conjunction with our model of the radiation environment, we use these correlations to determine the balloon buoyancy and ascent rate.
An example of the trajectory of a solar balloon 5 m in diameter with a 3 kg payload is shown below. The balloon takes off slightly after dawn (0 hrs), reaches a peak elevation of ~15 km, and then descends after sunset (indicated by the red line).
One interesting prediction of the 1D trajectory model is that certain configurations of solar balloons should be able to fly at night, powered by the infrared flux from the Earth alone. There is precedent for this, as French scientific balloons powered by solar/terrestrial radiation alone have achieved flights of up to 70 days, circumnavigating the globe several times.
There is still a good deal of work to be done on this model. Most importantly a validation against real solar ballon flight data is needed. Additionally, there are a number of issues to be addressed, such as modeling the effects of non-spherical balloon shape and finding a convenient way to determine the optical properties of the balloon film (such as solar absorptivity). We eventually plan on combining this model with rNomads to compute 3D balloon trajectories.