> Anders Sandberg gives a calculation of the thermodynamic efficiency of the
> dyson shell in his (draft) Jupiter Brain paper, and concludes that a
> smaller dyson shell is more efficient than a larger one. But I think his
> reasoning is slightly flawed. I'll show that the reverse is true, and that
> dyson shells may have a much lower temperature than previously thought
I have spent this afternoon arguing with my office mate, filling the
whiteboard with equations and having him look through thermodynamics
textbooks. Our current consensus is that you are right and I was
wrong.
> A heat engine works between two heat reservoirs with a temperature
> difference. A dyson shell works between the Sun which has a surface
> temperature of 5800 K and the temperature of its external radiators which
> is initially at 3 K, the temperature of the cosmic background radiation.
This has a maximal Carnot efficiency of eta_ultima=(5800-3)/5800
~0.9995. So any dyson shell cannot become better than this.
My mistake was that I calculated the efficiency of the shell to be
(T_shell - 3)/T_shell, which is wrong since it is not the temperature
difference between the cooling system of the shell and outer space
that is used to do work.
If we look at the system star+shell as a single heat engine, its
maximal efficiency is eta_max=(5800-T_shell)/5800, where T_shell is
the temperature of the shell radiators.
It seems that our derivations of T_shell differ, since I assumed that
the available work was expended inside the sphere, while you assumed
it was expended outside in interstellar space. That resulted in my
estimate of T_shell to [S/4*pi*r^2*sigma]^(1/4), and yours to
(S/(4*pi*r^2*sigma*Th))^(1/3). Doing the work outside of course makes
the system more efficient since the waste heat of the work in my case
would heat up the radiators more and lower the efficiency of the
entire shell.
But you are right, the larger the shell becomes, the more efficient it
is.
Thanks for pointing out my error and making me learn more thermodynamics.
-- ----------------------------------------------------------------------- Anders Sandberg Towards Ascension! asa@nada.kth.se http://www.nada.kth.se/~asa/ GCS/M/S/O d++ -p+ c++++ !l u+ e++ m++ s+/+ n--- h+/* f+ g+ w++ t+ r+ !y