| From: | Adam Getchell <acgetchell@*******.EDU> |
|---|---|
| Subject: | Re: About suncell on a larger scale ... warning: Physics! |
| Date: | Tue, 17 Nov 1998 12:28:39 -0800 |
>you describe
>the energy production of a single unit of Suncell as it gets closer and
>farther away from the Sun (or other star).
The correct answer would be to base power as a function of the radiant flux
arriving at the receiver.
The irradiance E is defined as the radiant flux (power in watts) per unit
area (square meter) incident on the solar cell. E for the sun is 1.35 x
10^3 W/m^2; this is denoted as the solar constant.
The radiant flux arriving at the solar cell is a function of the solid
angle and area da of the emitter:
P = (const) da dOmega(alpha) = (const) da da'/R^2 [W]
where d is understood to be the differential operator and multiplication is
implicit.
The radiance L is defined as the radiant flux per unit area and solid angle.
L = radiant flux/ (area)*solid angle [W/m^2-sr]
where sr is understood to be non-dimensional solid angle steradians.
The solid angle of an entire sphere is 4*pi. Assuming small solid angles, a
spherical section may be approximated by a flat section with the small
angle approximation. Then the expression for a solid angle is:
dOmega(alpha) = (pi * (R sin(alpha))^2)/ R^2 = pi * sin(alpha)^2
Assuming that the area of the emitter and receiver is small compared to the
distance between them, and their normal vectors are colinear the expression
for power becomes:
P = L da pi sin(alpha)^2
Then, using the expression for irradiance:
E = P / da'
with the expression for Power we get:
E = L dOmega(alpha ')
To get L for the sun, divide E (the solar constant) by the solid angle from
the earth. At earth distance the sun subtends alpha = 0.25 deg = 0.004 rad,
then
dOmega(alpha') = pi sin(0.25)^2 = pi (0.004)^2 = 6 x 10E-5 sr
using the small angle approximation sin (theta) = theta
and L = E/dOmega(alpha') = 1.35x10E3 / 6 x 10E-5 = 2.25 x 10E7 W/m^2-sr
Note solid angle occurs twice, once for the emitter and once for the
receiver -- the analysis above considers the emitter. Of course, the solid
angle subtended by solar receivers is very small, proportional to 1/(4 pi
R^2), and the efficiency of solar cells is something like 0.10 so the final
amount of energy is quite small.
At any rate, both of the preceeding analyses were fairly inaccurate. But
it's up to you as far as how accurate you want things to be. Myself, I
don't think spreadsheets are terribly hard to set up, so these equations
don't really bother me.
>Basically, in the end I am looking at having spacefaring sailing vessels.
Dr. Robert L Forward's "Rocheworld" series should be of interest, as the
appendices have some hard numbers on interstellar solar sail craft. The
short of it is that solar sails aren't good much past Jupiter without power
boosting, which he provides in the form of laser propulsion. There are also
macromatter catch-sail schemes which might be more efficient due to matter
streams not falling off with the square of the distance, but such streams
really require nanotech self-guided swarms to achieve peak efficiency.
>-Mike
--Adam
acgetchell@*******.edu
"Invincibility is in oneself, vulnerability in the opponent." --Sun Tzu
