Death Star Math: Return(ing) with a (possible) Solution

Solving the greatest (well maybe not but definitely one of its less sexy issues) engineering problem of science fiction, managing the surplus heat produced by all of the fanciful technologies imagined in far off worlds. 

When doing the math for the Death Stars exhaust port calculations it started to become more and more apparent* that if any civilization was developing warp drives, planet destroying super lasers, matter to energy converting teleportation platforms, and other crazy innovations that I would love to see during my life time, they would also need to be developing a way to keep these systems within some kind of operation temperature.  Previous entries on this blog on system cooling have hopefully created a rather deep explanation as to why heat management is such an issue in building your technological wonder, unfortunately for we hard science buffs, there isn’t much of a solution when it comes to applying traditional 3 spatial dimension based mathematics to radiative cooling.  Hopefully nerdier readers will most likely have latched on to the concept of using 3 spatial dimensions and started to ask themselves what I might be suggesting.  Right now I would ask those who saw this idea as critical, to take 5 seconds to imagine what solution I am inferring in the previous statement.

Anyone who honestly clued in and said to themselves “Hey, couldn’t you just add a new dimension for the heat to go out into” or something of that overall gist, please leave a comment bragging about this and I will happily add two points into your arbitrary cosmic score card that I have running in the back of my mind for most of humanity (not really).  Those of you who have come up with an alternative solution that you think makes even more sense or less sense really, I look forward to seeing your suggestions as well (if anyone ever reads this), also if you could make it a workable solution for cooling technologies like the transporters in Star Trek, I would be eternally grateful.    

Back to mathing.  The physics of black body emissions as a means of maintaining thermal equilibrium have been rather established with work starting with Ben Franklin (or earlier really I don’t know the history of physics that well).  Under our paradigms in physics a surface will transfer heat based upon the properties defined in the Stefan-Boltzmann Law, where the exposed area of the surface will emit some percentage of an ideal body’s energy flux, as a dependent on the relationship of the temperature of the objects surface to the ambient environment with both of those values raised to the fourth power. 

Within the physics of narratives like Star Trek, Star Wars, Babylon 5 etc… we are told that advanced species have discovered additional dimensions that are used for transport and communication.   For a power nerd like me, these additional dimensions seem like a perfect place to dump all of our spaceship’s surplus heat.

Long before Darth Sideous asked his various minions to begin developing designs for the Death Star, or before Captain.  James. T. Kirk seduced his way across the galaxy.  Some rather bright scientists and engineers got together and developed a technology that would allow them to make a radiator that cools 4 dimensionally.  No longer would only the exposed surface of your vessel be your only means of keeping your reactors operational.  These innovators created a solution that would allow each individual atomic layer of the cooling system to dump heat into a fourth spatial dimension and they would do it all with a really contrived acronym.  A smooth cube, one meter on a side, would go from having an exposed surface area of 6 sq m to 5.97E09 sq m of radiative surface area (this is actually a minimal estimate as I’m not super-duper certain as to how many liberties I can take with estimations.)  While not an exact comparison for various reason, one rather contrived analogy is saying that it would be like going from riding a horse from point a to b to traveling at the speed of light.  Additionally it is discovered that the scientists have found that subspace or hyperspace, whatever this fourth spatial dimension is called, is roughly as cold as the vacuum of space in the majority of our universe.  This critical property will allow you to reduce the heat transfer equation to the following.

\

As I mentioned earlier the area of the radiator is where the math gets the fuzziest, the estimated value of 5.97E09 square meters per cubic meter was calculated by finding the inter-planar spacing of graphene on Wikipedia, listed as 0.335 nm or 3.35E-10m.  Graphene was chosen because it is awesome and for how arbitrary calculations are here it worked for me.  The cube can then be thought of as 2.985E09 individual atom thick layers capable of radiating their waste heat into the fourth dimension.  Each layer is somehow able to simultaneously transfer energy between layers and emit energy from both sides, a trait of the fourth dimensional special phsysics.  It should be noted, that by my logic it might make “more sense” to treat each individual atom as a four dimensional radiator and/or eliminate the concept of treating the radiator as a solid system, why not a 4D cooling plasma, and to that I say, I’m already spending too much time on theoreticals and not enough finding work (I wish it was otherwise).  Anyways all of this logic adds up to us creating a new function for balancing out the requirements of the cooling system.  Depending on how the variables are solved you then plug them back into the unit conversion shown below.

 

In the case of the super radiator material developed by the Death Star Development teams, one capable of operating at 20,000K and as a perfect black body.  The Death Star would also need to be at least      99.9999975563% efficient as a system for the radiator to fit within the exterior envelope of the Battle Station as in both have an estimated volume of 2.1447E+15 cubic meters. (before the theoretical detractor says that if I’m putting the radiators magic into a 4^th spatial dimension why should I even worry about its volume and to that I simply say, “eh, it shows scale and system complexity”) At 99.9999999% efficiency the radiator system would be about 4% of the volume of the non-moon.  I hope that someone found this interesting.