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This webpage shows that non-renewable electicity generating stations should be located well outside present or projected future metropolitan areas in order to limit the ambient temperature increase due to local heat dissipation. This web page further shows that distributed renewable electricity generators must extend far beyond metropolitan areas in order to meet the metropolitan area load.
A good indicator of the average standard of living is the usage of energy. Presently in Ontario the usage of electricity has an annual peak of about 2 KW per person. As non-fossil electricity gradually displaces fossil fuels in the transportation and heating sectors maintenance of this same standard of living will involve appproximate doubling of electricity usage to an annual peak of about 4 kW per person.
If the average standard of living of the world's population of 7 X 10^9 persons is to be brought up to the average standard of living presently enjoyed in Ontario, the world annual peak demand for electricity will be about:
7 X 10^9 persons X 4 kW / person = 28 X 10^9 kW.
The potential sources of energy for supplying this power can be divided into renewable energy (energy currently provided by the sun) and non-renewable energy (nuclear energy, tidal power, residual geothermal energy).
Use of renewable energy does not affect the average Earth surface temperature, whereas use of non-renewable energy increases the Earth's average surface temperature even if there is no change in the atmospheric greeenhouse gas concentration.
ELECTRICITY GENERATION HEAT DISSIPATION:
Present practical processes for generating electricity from nuclear fission have an overall efficiency of generation, transmission and distribution of about 0.2. The projected processes for generating electricity from nuclear fusion have an overall efficiency of generation, transmission and distribution of about 0.1. To produce 1 kW of electricity delivered to the load present fission based processes dissipate an additional 4 kW of heat either at the generator or within the transmission, storage and distribution system. Hence if the entire human electricity requirement is to be met via fission nuclear electricity the Earth must radiate into space an additonal thermal power Pt given by:
Pt = 28 X 10^9 kW X (1 / 0.2)
= 140 X 10^9 kW
and if the entire human electricity requirement is to be met via fussion nuclear electricity the Earth must radiate into space an additional thermal power Pt given by:
Pt = 28 X 10^9 kW X (1 / 0.1)
= 280 X 10^9 kW
SOLAR POWER CAPTURE:
Solar energy is continuously being captured and reradiated by the Earth. The solar thermal power flow Ps is given by:
Ps = (solar irradiance) X (1 - albedo) X (Earth cross sectional area)
The solar irradiance Ho at the Earth's orbit measured via satelite borne instruments is:
Ho = 1.361 kW / m^2.
The albedo Fr of the Earth measured via moonshine is:
Fr = 0.297
The circumference of the Earth is about 40,000 km. Hence the radius R of the Earth satisfies:
2 Pi R = 40,000 km
R = 40,000 km / (2 Pi)
(Earth cross sectional area) = Pi R^2
= Pi (40,000 km / (2 Pi))^2
= 4 X 10^8 km^2 / Pi
= (4 / Pi) X 10^14 m^2
Hence the solar thermal power flow Ps is given by:
Ps = 1.361 kW / m^2 X (1 - .297) X (4 / Pi) X 10^14 m^2
= 1.218 X 10^14 kW
CHANGE IN TEMPERATURE:
Energy balance on Earth requires the the infrared power emission Pe equal the solar power absorption Ps plus the thermal power Pt. As proved by Planck, the infrared power emission Pe takes the form:
Pe = Ke Te^4
where Ke is a constant and Te is the effective absolute emission temperature.
Differentiation of this formula gives:
dPe = 4 Ke Te^3 dTe
dTe = dPe / (4 Ke Te^3)
= (dPe Te) / (4 Pe)
In this case dPe = Pt, Pe ~ Ps and Te is the effective emission temperature as found by spacecraft infrared spectrometry measurements to be about Te = 275 kelvin.
Thus the increase in average Earth surface temperature due to stored energy dissipation with fission power is projected to be:
dTe = (Pt Te) / (4 Ps)
= (140 X 10^9 kW X 275 degrees) / (4 X 1.218 X 10^14 kW)
= .079 degrees kelvin
It must be emphasized that this is an average surface temperature increase which would only be physically true if the human population was uniformly distributed over the entire spherical surface of the Earth. The surface area of the Earth is given by:
As = 4 Pi R^2
= 4 X (Cross sectional area)
= 4 X 4 X 10^8 km^2 / Pi
= 5.093 X 10^8 km^2
The land area is about (1 / 3) of the spherical surface area, or:
Land Area = (1 / 3) X 5.093 X 10^8 km^2
= 1.698 X 10^8 km^2
If the temperature increase due to nuclear heat from fission power stations is just averaged over land area the average temperature increase over land is:
3 X .079 degrees kelvin
= 0.237 degrees K
The average human population density over the land area of the Earth is:
(7 X 10^9 persons) / (1.698 X 10^8 km^2)
= 41.22 persons / km^2
By comparison the Greater Toronto Area (GTA) has an average population density of about:
800 persons / km^2. If all the GTA's electricity generation was located within the GTA then the corresponding ambient temperature increase due to local heat dissipation would be:
[(800 persons / km^2) / (41.22 persons / km^2)] X 0.237 degrees C
= 4.6 degrees C
This ambient temperature increase due to local heat dissipation is mitigated in the GTA because two of the major electricity generating stations (Bruce and Darlington) are located well outside the GTA and dump their reject heat into Lake Ontario and Lake Huron, both of which extend well outside the GTA. The heat from the Pickering nuclear generating station is dumped into Lake Ontario at a location close to but outside the GTA. Hence the actual average temperature increase in the GTA due to local heat dissipation is less than 3 degrees C. However, the experimentally measured 3 degrees C summer temperature increase was sufficient to shift Toronto from a being non-airconditioned community to being an air conditioned community. It must be emphasized that the temperature increase due to local heat dissipation is additonal to the greenhouse gas and albedo change related temperature increases.
Given the choice between the temperature increase due to local heat dissipation and the further temperature increase due to increased atmospneric greenhouse gas concentration, the smaller temperature increase due to local heat dissipation is almost always preferable.
FUSION AND GEOTHERMAL ENERGY CONSTRAINT:
In the above analysis it was assumed that the stored energy source was fission nuclear and the corresponding end-to-end efficiency in generation, transmission and distribution of electricity was about 0.2 (20%). However, after the repeated loss of cooling accidents at Chernobyl,Three Mile Isand and Fukashima the general public has legitimate safety concerns about fission nuclear plants. The problem with all fission nuclear plants is that substantial heat production due to fission daughter decay continues for months after the plant is shut down.
A solution to this safety problem is use of a fusion rather than a fission power cycle. A fusion power cycle has the advantage that when the plant is turned off almost all the heat production immediately stops. However, a fusion power cycle has the disadvantage that about half the electricity generated is used to meet the plant parasitic load, so the net efficiency falls to about 0.1 (10%). Hence, if all the energy requirements of the GTA were met by fusion based power plants located within the GTA the ambient temperature increase due to local heat dissipation would be:
(0.2 / 0.1) X 4.6 degrees C = 9.2 degrees C
Clearly a 9.2 degrees C ambient temperature increase is unacceptable in most metropolitan areas, especially in the summer. The conclusion to be drawn is that future nuclear power plant sites should be located several hundred km outside present and projected future metropolitan areas to reduce the ambient temperature increase due to local heat dissipation.
Due to a relatively low working fluid temperature the net efficiency of a geothermal power plant is also only about 10%. Hence the same local heat dissipation calculations apply. A practical geothermal power plant should be located several hundred km outside a metropolitan area to reduce the ambient temperature increase due to local heat dissipation.
LIMIT ON WORLD POPULATION:
At some time in the future the Earth will run out of uranium and thorium for fuelling fission reactors and will have to rely on fusion reactors. Under those circumstances, again assuming a population of 7 X 10^9 persons and 4 kWe / person of electricity generation, the average temperature rise over land due to reactor heat dissipation will be:
2 X 0.237 degrees C = .474 degrees C
Note that this is an average which means that in the vicinity of power stations the temperature rise will be very much greater. The conclusion that can reasonably be drawn is that even with widespread application of fusion power, provision of the present standard of living in Ontario to the rest of the world population means that the world population must be capped at about 7 billion persons. As the polar ice caps melt the land area will decrease, so the world population will correspondingly have to decrease. It is reasonable to assert that the long term human carrying capacity of the Earth is probably about 4 billion people with an average standard of living comparable to what we in Ontario presently enjoy.
However, it is important to point out that if mankind loads the atmosphere with CO2 the Earth's ability to radiate heat will further diminish and hence the Earth's human population carrying capacity could easily drop to only two billion persons. Thus it is crucial for everyone to realize that our ability to solve the energy supply problem with nuclear energy is entirely dependent upon also capping and then gradually reducing the Earth's human population.
Thus, with suitable application of nuclear power it appears that an Earth population in the range of two billion to four billion people with a reasonable average standard of living is sustainable over the long term. However, economic models based on exponential population growth are no longer valid. Countries that presently have dense populations will have to find an orderly means of reducing these populations if the average standard of living of their citizens is to improve. There are numerous countries where cultural and religious opposition to birth control has yet to come to terms with scientific reality.
LIMITS ON RENEWABLE ENERGY:
Most of the major hydroelectric opportunities on Earth have already been exploited. Further hydroelectric generation will reduce fresh water available for agriculture. Similarly, biofuel development and solar energy development will both reduce food production capability. Agricultural waste is needed for production of synthetic hydrocarbon fuels. Hence, wind is the major opportunity for renewable energy development. However, there are significant constraints on wind energy development that are identified herein.
Wind energy has two components. Thermal expansion of dry air on the side of the Earth facing the sun and evaporation of water on the side of the Earth facing the sun. On the dark side of the Earth dry air contracts and water vapor condenses.
Wind energy has further constraints. For practical structural engineering reasons wind turbines have masts about 90 m high which limits the blades to being about 80 m long. Hence harvestable wind energy is confined to the altitude below 170 m above the ground. Even if it is possible to build the wind turbines taller there is the issue of interference with air navigation.
Wind energy is further constrained by the reality that it is impractical to build wind turbines over the deep ocean and it is impractical to locate wind turbines in coastal urban areas.
Dry air wind power can be regarded as a Stirling engine and is subject to Carnot efficiency limitations as well as turbine efficiency constraints.
The water vapor expansion component of air is limited by the latent heat of vaporization which severely reduces the efficiency of that energy mechanism.
Using these constraints we can make a reasonable estimate of the maximum average power that might reasonably be harvested from wind energy. This energy is not as large as some wind proponents assert.
This web page last updated May 18, 2013.
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