Power & Biology
In the biological realm, there are various empirical relations associating an organism’s dimensional parameters and other characteristics. The general form is shown below. For example, if bone strength is limited in biological structures, we would expect the cross-sectional area of bones to scale with the mass or volume to the 2/3rd power. Other relations include heart rate and cruising flight speed , each with their own underlying physical explanations.
Kleiber’s law [^1] [^2]relates the mass or volume of an organism with its power rating. Figure 14 shows how organism power rating is constrained in a fashion consistent with a maximum surface heat flux boundary condition. Organisms must dissipate heat and remain at safe temperatures, and they are constrained by their surface area and the temperature limits of Earth-based biological processes. The power law varies somewhat between insects, mammals, birds, and trees, but the overall trend in the figure below has between .6 and .85, meaning that specific power decreases with mass. This is slightly better than a constant heat flux boundary condition. This basically means organisms can only give off so much energy per unit area without overheating. Solving for the maximum heat flux for spherical plants and animals yields 22 and 83 W/m2, and HTC between 1-20 W/m2K. The greater cooling capabilities of animals and birds could be due to evaporative cooling, internal liquid cooling, higher flow rates, and higher thermal conductivity.
Another way to think about Kleiber's law is in terms of power density. Considering the mass is proportional to the volume, we can recompute Kleiber's law for power density by dividing the y-axis by the x-axis. Smaller organisms have higher power density.
Kleiber's law indicates that mice have much higher power density than elephants. This is significant, because power density is a key driver of performance and cost.
Why might evolved structures have developed this relation between power and volume? We can speculate that it is the most survivable and successful way to design a power system because it more or less retires the risk of overheating.
The existence of Kleiber’s Law in evolved systems inspires its application to man-made thermal power systems that aim for passive safety and reduced risk through reduced consequence. How might these heat flux boundary conditions apply to mechanical systems like nuclear reactors? The temperature limits of typical biological processes are about 40 °C – much lower than metal and ceramic based systems. The bodily flows homogenize temperature across the body so that the maximum temperature gradients are at most a few degrees. Low surface heat flux is likely due to low temperature difference between the outer wall and the ambient. A mechanical system using metals and ceramics will be able to achieve far higher centerline temperatures and wall temperatures and a larger heat flux boundary condition.
A nuclear reactor can operate at whatever desired power rating you want. You just have to be able to extract the heat fast enough and also make sure you can do so during accident conditions. How have we picked the power rating for our nuclear reactors? We were inspired by Kleiber’s law to constrain the power according to the size, ensuring we can maintain low temperatures during accident conditions using only passive cooling mechanisms. The result is the Micro Modular Reactor, which is, in some ways, a bio-inspired nuclear reactor.
[1] Banavar et al., “Form, Function, and Evolution of Living Organisms.”
[2] Ballesteros et al., “On the Thermodynamic Origin of Metabolic Scaling.”