# How much energy does a bee/micro robot need per second of flight?

In my scifi setting small utility robot swarms are used for many purposes. I'm currently trying to estimate how much energy such a swarm would need to operate. Since biology, robotics and nanotechnology have pretty much merged into one thing in the setting I decided that taking bees as an example for a standard mircobot is as good as starting point as I'm going to get. So how much power ($$W$$) does a bee nee to fly?

Or to put a more hard-science spin on it, how much power ($$W$$) does a flying object of a mass ($$m$$) need to fly with a speed of ($$v$$) in an environment with an airpressure of ($$p$$), which might be anything from 0 to 25 bar, and a gravity of ($$g$$), which might be anything from 0 to 5 times that of Earth?

I know that a precise answer to this is probably extremely hard to give, but that isn't really my goal. I need to know the order of magnitude of energy these droneswarms will consume, so I can figure out if the powessource of the swarm should be batteries, chemical fuels or small nuclear reactors. This information is just here to give some context, it is irrelevant for the actual question.

A great answer would provide me with a formula and a right estimate for the bee-sized mircobot. The bee value is supposed to be a sanity check.

• Basically you are asking for a crash course in aerodynamics... It depends very very much on the size of the object and the speed. Air behaves very differently with respect to small and large objects; if the object is very small, it may even float without expending any energy whatsoever: that is, air may be viscous enough with respect to the object so that its terminal velocity is close to zero; this is why tiny droplets of water stay aloft to form clouds. Fluid dynamics is hard. The first question (power consumption of a bee in flight) is answerable; the second question is waaaaay too broad. – AlexP Nov 10 '19 at 14:11
• Based on all known laws of aviation... – weakdna Nov 10 '19 at 17:10
• In order to give a good value, you need to explicitly state the size of the microbots, in both linear dimensions and mass, as well as their static drag coefficient. There are many different species of bee... and some are more furry than others. – Monty Wild Nov 11 '19 at 1:33
• @AlexP, precisely; and for these reasons it is usually a lot easier (and more reliable) to approach such calculations from the opposide end: how much energy a bee consumes and how much it flies on that energy. The answers to these questions can be found in biology. – Zeus Nov 11 '19 at 5:03

This article might be of interest. It turns out energy expenditure is as much a factor of air temperature as anything else. Figure three seems to be pertinent to your question. As it becomes a matter of simple mathematics to convert from their units to yours, I leave that to you! • Please put the relevant part of the cited article in the answer. We want the answers to be self sufficient. – L.Dutch - Reinstate Monica Nov 10 '19 at 15:51
• As far as I'm concerned, the above constitutes the relevant portion of the cited article wrt to the given query. The graph demonstrates how much energy a bee uses per wingbeat. What remains is simply an off-topic exercise in mathematics that the OP can solve at their pleasure. – elemtilas Nov 10 '19 at 16:50
• I'd say the graph demonstrates some rather optimistic linear regression, but we're not here to critique the science or the stats so it seems OK to me. – Starfish Prime Nov 10 '19 at 20:45

Let's start the estimation with the drag: $$F_D = \frac{1}{2}\rho v^2 C_D A$$ Where

• $$C_D A$$ is going to be your form factor, depends on your drone design. Here are some example $$C_D$$ values
• $$\rho$$ is the air density. Here are some values, depending on temperature. If you increase the pressure to say 2x, the density also increases 2x.
• v is the velocity

If you increase gravity, the density also increases, I think the relationship is linear.

The power you need is: $$P= F_D s$$ Where

• s is the distance travelled

If you substitute: $$P(v, s) = \frac{1}{2}\rho v^2 C_D A s$$ or $$P(v, t) = \frac{1}{2}\rho v^3 C_D A t$$ Because $$s = vt$$ (in average of course).

This is the power you need only for the horizontal movement. For the vertical movement you need to calculate the potential energy difference.