What's the highest efficiency a Solarthermal power system could have?
This document was inspired by two comments in
the Introduction to Solar energy by John J. Hopfield and Jerry Gollub (1978),
in which they explain the fundamental limits to the efficiency of
photovoltaic systems. (PV can't be better than 45% efficient, they show.)
J.H. and J.G. say:
The atmosphere is essentially a large heat engine that converts
incident solar energy to mechanical energy with an
efficiency of about 2%.
And they point out that solar thermal electricity production (in which sunlight is
first converted to heat) is
far less efficient than an ideal photovoltaic system.
I thought it would be fun to make a model, in the style of Hopfield and Gollub,
pinning down the maximum possible efficiency of a solarthermal power system.
Then we can assess how impressive the 2% figure achieved by the atmosphere is.
The maximum efficiency also gives insights into the
huge `solar chimneys' being considered in Australia.
(A prototype in Spain
generated about 1W per m^2.)
We start with a system that generates no mechanical work or
electricity.
A black body sits in sunlight.
Let's assume the body's front face is perfectly directed at the sun,
that its back and sides are perfectly insulated (so that all heat is lost
through the same face where the heat comes in)
and that in all other directions where the sun ain't, there be
empty space at temperature zero.
What's the equilibrium temperature of the body?
The flux in (per unit area) is
σ T_{sun}^{4} (r_{sun}/R_{sun})^{2},
where
r_{sun} is the radius of the sun,
R_{sun} is the distance to the sun,
T_{sun} is the temperature of the sun,
and
σ is the StefanBoltzmann constant.
The flux out (per unit area) is
σ T^{4}
where T is the
equilibrium temperature of the body.
So
T = T_{sun} (r_{sun}/R_{sun})^{1/2}
[
T_{sun} = 5000K,
r_{sun} = 7x10^5 km,
R_{sun} = 1.5x10^8 km,
gives
T = 342K.
]
If the body only sees sunlight a fraction f = one quarter of the time
(a fair approximation of the effect of the earth's rotation)
then we get a reduced equilibrium temperature of
T = T_{sun} f^{1/4} (r_{sun}/R_{sun})^{1/2}
= 241K.
Now, let's add an ideal thermal power plant on the back side of the
body, which takes heat from the hot body and sends it
to a cold body and extracts mechanical work along the
way.
If the hot body's temperature is
T_{hot}
and the cold sink's temperature is
T_{cold},
then a heat flux Q per unit area from the hot body
produces a work flux W
given by
W = Q (1  T_{cold}/T_{hot})
Sucking heat out of the back of the hot body leaves
less energy to come out of its front.
So the more heat is sucked out of the hot body,
the lower its equilibrium temperature is.
Our task now is to optimize the temperature of the hot body so
as to maximize the work done.
For convenience, let's redefine Q to be the ratio of
the heat going into the heat engine to the total flux incident on
the black body.
And keep the definition
W = Q (1  T_{cold}/T_{hot}).
Then W is the efficiency, that is, the
fraction of the incoming solar energy that gets turned into work.


W(Thot,Tcold,Tmax) = (Tmax**4  Thot**4 ) * ( 1.0  Tcold/Thot ) / Tmax**4 (eq.1)


The first graph
shows the efficiency W as a function of the hot body's temperature
(horizontal axis),
scaled such that the maximum possible temperature T is 1.
The efficiency depends on the temperature of the cold sink (which varies
by equal steps from curve to curve).
Where's the optimum?
It's a pretty good approximation, as long as T_{cold} >
0.1 T, to say that
the optimal T_{hot} is half way between
the maximum possible temperature T and T_{cold}.
At this sweet spot, the efficiency is, to leading order,
(1 T_{cold}/T)^{2}
[This approximation is accurate to within 10%]
We are now ready to bound the efficiency of
a range of solarthermal devices.
Our model omitted the greenhouse effect produced by
having an atmosphere in front of the black body,
but for the moment let's fudge this in by simply
positing a few numbers.
The atmosphere as a heat engine
hot packets of air move slowly and adiabatically to colder locations,
where they are cooled down. As the packets drift around, they
do mechanical work.
Not a Carnot cycle, but if it were,
its efficiency would be
(1 T_{cold}/T)^{2}.
Here, T is the average equilibrium temperature of a black body
in windless conditions  say 300K?,
and T_{cold}.... hmm, should we pick
a polar temperature, or a stratospheric one?
Let's say T_{cold} = 230K, a good Canadian temperature.
Then the maximum efficiency, if the wind whooshes at the
right speed, is
(1 T_{cold}/T)^{2} = 0.055.
Solar thermal system with concentrators
The most promising design for a massive solar power station
seems to be 'concentrating solar power' (CSP).
Mirrors are used to make it feel, from the point of view of the
black body, that it is surrounded on all sides by hot sunshine.
With perfect spherical mirrors tracking the sun, the
factor
(r_{sun}/R_{sun})^{1/2}
is eliminated.
The factor f is also eliminated if we remember
to run the system only when it's sunny. (An option
not available to the weather system.)
The cold side of the heat engine is air or water at a temperature
something like T_{cold} = 290K.
Assuming perfect optics we have
a maximum possible equilibrium temperature T
equal to T_{sun} = 5000K,
and the efficiency could be
(1 T_{cold}/T)^{2} = 0.89.
However, most materials out of which power stations could be made
would fall apart at the corresponding hot working temperature of 2500K.
If we plug in plausible hot and cold temperatures of
900K and 300K instead, we get an efficiency of about 66% (eq.1).
Thus a solar thermal power station with perfect concentrating
mirrors could deliver a power per unit
area of at most 2/3 times the solar constant, at midday.
David MacKay
Last modified: Tue Mar 3 22:11:30 2009