As you may expect, The solar thermal harvesting
component is the central part of all domestic heating and the electric
power production parts of our systems. This consists of a number of
solar collectors, a thermal storage tank, circulation pumps,
temperature sensors, the controller, and the plumbing to tie it into
the home systems.
Here's small tutorial by example on how to calculate
the collector sq footage, the flow rates and available power
relationships.
One thing about thermodynamics is that it has a HEAVY emphasis
on calculus. I’ve focused on discussions of the Carnot and
Rankin and have tried to simplify and clarify where I can. Everything
in [ square brackets ] is my inserted comments. In some instances
descriptions are out of sequence. I’ve transcribed them in the
order that I think would be a natural understandable order.
ηmax= maximum energy conversion efficiency
=1 –
(low temp. / high temp.)
condenser cool /
boiler
hot
Making the hot side even hotter increases efficiency.
Making the cold side even colder increases efficiency
Why
does Carnot efficiency
(ηmax) have that limitation? It's simply the portion
of thermal power available that can be harvested. It's useful. Don't
get hung-up on it.
Carnot
and Rankin cycles are noted as 2T systems. They MEAN two temperature
systems – a high boiler
temperature & a lower condenser temperature.
Here,
I’ll include two textbook problems with solutions to illustrate
the technique to determine the size of the solar thermal panels needed
to gather a particular power.
Example
1 – A Solar Engine
– from the book ‘Thermodynamics’
It is proposed that a solar energy be used to warm a large
“collector plate”; This energy would, in turn, be
transferred as heat to a fluid within a heat engine, and the engine
would reject energy as heat into the atmosphere. Experiments indicate
that about 200 Btu/hr-Ft2 of energy can be
collected when the plate is operating at 190°F. Estimate the
minimum collector area which would be required for a plant
producing 1 kw of useful shaft power. [ Notice that this doesn’t
specify the working fluid but the implication is that it is vapor to
drive the ‘engine’ at 190°F.]
We first estimate the maximum
energy conversion efficiency of this system using the Carnot efficiency
as the upper limit. The atmospheric temperature is assumed to be
70°F.
ηmax = 1 – (70°F + 460) / (190°F + 460) = 0.184
530°R / 650°R = .815
1
- .815 = .184 Carnot
efficiency value
18.4%
efficient
[The ‘+
460’ adjusts Fahrenheit to absolute
zero – called temperature Rankin – degrees Rankin - °R
The efficiency of any real heat engine operating between the collector
plate and atmospheric temperatures would be less than this, owing to
inefficiencies in real devices. The minimum rate
at which energy must be collected is related to the required
power output and the maximum energy conversion efficiency.
[
Ộmin symbolizes the minimum rate at which energy must be
collected ]
[
ẁ symbolizes work done ]
Ộmin =
ẁ / ηmax = 1 kw /
0.184 (maximum conversion efficiency)
= 5.44 kw * [ 1 kw = 3,413
btu hr ] = 18,600
btu hr
…so the minimum area required is :
Amin = 18,600 btu / 200 btu ft2= 93 ft2 of solar collector area,
minimum.
– say a 10’ x 10’
solar collector to get 1kw.
A real world system might be expected to need two or three times this
area since the actual efficiency would probably be considerably less
than 18.4%.
Example 2 – In the chapter on energy
analysis of thermodynamic systems.
A small solar engine for desert water pumping uses steam as the working
fluid. The hardware is shown in the figure below.
2
Óe
3 220 degrees F 30 psi
|
Boiler
solar collector panels
|
30 psi boost ẁf
pump
↓
turbine
work needed →
۞→ work
produced ẁt Work-Turbine
120 degrees F
↓
|
Condenser - tank -
liquid feed pump
|
1
Óc
4 1 atmosphere
Water enters the pump as a saturated liquid at 120°F
and is pumped up to 30 psia by a small centrifugal pump. The boiler
evaporates the water at 30 psia. The saturated vapor enters the turbine
at this pressure. The steam leaves the turbine with 6% moisture at
220°F at 1 atmosphere and is subsequently condensed. The flow rate
is 300 lbm / hr. [ lbm = pound mass ] The pump is driven by a ½
hp electric motor operating at full load.
Between points 1 and 2 the working fluid is liquid at constant
temperature. Pressure is increased.
Between points 3 and 4 gas at constant temperature. Pressure is
decreased.
Between points 2 and 3 liquid boiled to gas at constant pressure.
Temperature is increased.
Between points 1 and 4 gas condensed to liquid at constant pressure.
Temp. is decreased.
Determine
the net hp output of this plant,
Notice the English
units given – remember 1 btu is 1°F per pound. The heat added is sensible
temperature 220°F - 120°F = 100°F plus latent heat of vaporization
16 btu per pound to become steam. 300 * 16 = 4800 btu needed to swing
the water temperature 100 degrees - per pound and vaporize it.
The
flow rate is 300 lbs/hr ( /8.4 pounds per gallon = 36 gallons per
hour): 34,800 btu / hr
34,800 btu / hr = ( / 3,413 btu hr = 10.2 kw ) ( / 745 watts / hp
) = 13.7 hp [ẁt]
That minus the half hp [ẁf ] for the feed pump
nets out at 13.2 hp will be available.
Determine
the energy conversion efficiency [ net shaft-work output hp /
energy transfer to the fluid in the boiler = 1 – (low temp. /
high temp.)],
ηmax
= 1 – (120°F + 460) / (220°F + 460) =
580°R / 680°R
=
.853
1 - .853 =
.147
14.7% energy conversion efficiency
Estimate
the number of square feet of solar collectors which would be required,
assuming that the collectors can pick-up 250 btu / hr per square foot
of exposed surface.
34,800 but hr /
.147 = 236,734 btu hr needed.
At 250 btu hr per ft2
this requires 946 ft2 of solar collector area.
– say an 8’ x 135’
solar collector to get 10 kw all the time the sun shines.
And when it doesn't, run off storage -
either thermal or battery. Spend less than you make.
Working fluids that experience phase change at temperatures lower than
water are frequently called ‘organic’, being carbon based.
Things such as Freons, propane, butane, methyl chloride and also
nitrogen based ammonia.
It would be useful to have a chart which would show pressure on one
axis, heat content on another axis – drawing a line unique to
each type of working fluid showing where it’s solid, liquid, and
gas, That’s a Mollier diagram, a tool to get an idea of the range
of boiler-condenser pressures and temperatures for a working fluid.
These ‘Low Temperature Phase Change’ working fluids can be
expensive, poisonous, or flammable so it’s important that the
volume be minimized and containment is important but can be done - as
is done in refrigerators and heat pumps. One must isolate the heat and
cool by heat exchangers.
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