Things cool down because of conduction, convection and
radiation. Space
ships are not in contact with anything,
so conduction does not work. There is
no atmosphere or
ocean outside a space ship, so convection is not
available.
The only option is radiation. Radiation
is proportional to surface area. From
the picture,
the area is:
- 250m radius disk (both sides):
2xπx250² = 400000m²
- 100m diameter cylinder 500m long:
500xπx100 = 160000m²
- two 50m diameter cylinders 500m long:
2x500xπx50 = 160000m²
- total: about
720000m²
Now lets take a look at how much heat is generated:
There are one 1.5GWe and two 0.5GWe nuclear power plants
for a total of 2.5GW
of electricity. To get that
2.5GW of electricity, the nuclear power plants
create
5GW of waste heat. Lets pretend all the electricity is
used for
propulsion and that all the waste heat from
propulsion goes into heating the
propellent.
(Apart from the 100MW laser, all the other proposed
uses for the
electricity end up heating the space
ship.)
The magic formula we need
is:
T = (P/σA)¼
- T: temperature in
Kelvin.
- P: power in Watts.
- σ: Stefan–Boltzmann
constant =
5.67x10-8Wm-2K-4
- A: area
in m²
Plug the numbers in, and the hull temperature is:
591K.
Subtract 273 to get 318°C, or 605°F.
Oops, we fried the crew. Try that
again without using
the saucer section for cooling: 451°C. Still not
good.
That rather generous 33% efficiency for nuclear
thermal to electricity
conversion assumed a massive
source of cold water (big power stations are
coastal
for a reason). Without the cold source, there will
be much more than
5GW of waste heat when generating
2.5GW of electricity.
Quick! Turn on
the air-con. Part of electricity
generation is involves letting heat flow from
a
hot place to a cold place. Air conditioning is the
reverse process - using
electricity to pump heat from
a cold place to a hot place. If air conditioning
and
power generation are as efficient as possible, they
cancel each other out.
In the real world, the
electricity used by the air conditioning is more than
the extra generating capacity provided by the lower
temperature.
I
would love a return trip to Mars, but not on this
Enterprise.
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