power sat energy payback.

[hkhenson]

At 12:00 PM 5/16/2008, Alberto Monteiro wrote:

>Keith wrote:
> >
> >>> There is plenty of energy around if we can figure out how to get
> >>> it.  For example, a solar power satellite repays the energy needed
> >>> to lift it to GEO in about a day (at 100% efficiency).  Five percent
> >>> efficient rockets would replay the lift energy in 3 weeks.
> >>
> >> This is almost surely wrong. Did you do the math?
> >
> > Yes.
> >
> > Specific orbital energy is u/2r, (398,600/42,000)/2 or -4.75Mj/kg
> >
>Ok
>
> > Potential is -9.5Mk/kg and kinetic is 4.75Mj/kg
> >
> > Potential at the earth's surface is -62.6 MJ/kg; the
> > difference is 53.1Mj/kg.
> >
>Ok.
>
> > Using a space elevator,
> >
>I didn't know you were invoking Magic. "If I had magical fairy
>godparents I would make 2 + 2 = fish".

Restoring what you cut (I need the numbers).

Using a space elevator, the rotation of the earth provides the 
kinetic energy.  Since a joule is a watt-second; 53,100 
kW-s/kg/3600kW-s/kWh is 14.75 kWh/kg

A kW/kg power sat repays its lift energy 14 hour and 45 minutes after 
being turned on.  A 2kg/kW power sat would take 29.5 hours.

That is close enough to a day for back of the envelope 
calculations.  Five per cent efficient rockets would take 20 times 
this long to reach payback--but try to get them!

There is a heck of a lot of sunlight out there, and you don't need 
much structure to capture it in zero g.

>Try to do the same exercise using chemical rockets :-)

Sure. First we need a rocket.  Here is a rocket: 
http://www.ilr.tu-berlin.de/koelle/Neptun/NEP2015.pdf

Neptune is about 3 times the capacity of a Saturn 5, and this design 
was done by some of the same people so it's solid engineering.

This vehicle delivers 350 mt to LEO, and 100 mt to lunar orbit or 
GEO. To lift 100 mt to GEO Neptune uses 3762-mt of propellant for the 
first stage, 1072 mt second stage and 249 mt for the third totaling 5077 mt.

SSME O2 to H2 ratio is 6 to 
1. 
http://www.pw.utc.com/vgn-ext-templating/v/index.jsp?vgnextrefresh=1&vgnextoid=75a0184c712de010VgnVCM100000c45a529fRCRD

I.e., 1 part in 7 of the propellant is LH. or about 725 mt of 
LH.  The launch site would make electrolytic hydrogen out of water 
(the only long term source). That costs about 50 kWh/kg plus another 
15 kWh to liquefy the H2.  Add 5 kWh for liquefying oxygen at 6 x .8 kWh/kg.

That would be 70 MWh per mt, or 70 GW hours for 1000 tons, or 50.8 
GWh for 725 mt. At a kg/kW, 100 tons of satellite produces 100,000 
kW, or .1 GW.  Thus it would take 508 hours to pay back the lift 
energy or 21.2 days, 42.4 days for 2kg/kW.

Rocket efficiency here would be 14.75/508 or 2.9%.

A nuclear tug stage shuttling between LEO and GEO might double this 
efficiency raising the payload from 100 mt to 200 mt.  That is the 
consequences of a 15 km/sec exhaust velocity.

It's the high cost of rocket hardware, not lift energy payback, that 
makes power sats expensive.

Keith

PS.  Don't count out a space elevator.  The theoretical strength of 
carbon nanotubes is up in the 170 GPa range and anything over about 
45 is good enough for a moving cable, step taper elevator.  There are 
persistent accounts of 20 GPa yarn having been produced by the 
University of Cambridge.