There are three ultimate sources for energy on the earth, none of them renewable. One of them, solar nuclear fusion, deposits so much energy on the earth that its consequences are left behind for millions or even billions of years. Whether a sources is renewable or not is really whether these consequences can provide meaningful amounts of energy without causing environmental consequences or using the resource faster than it's produced. I'll get into the two ultimate sources first.
Nuclear fission: All chemical elements have some degree of of nuclear instability and spontaneously decompose, emitting sub-atomic particles and energy. Leo Szillard figured out how to use the sub-atomic particles to create a self sustaining chain reaction, generating a lot of heat. At controlled densities, this heat can be used to run power generating equipment, such as steam turbines. At uncontrolled densities, it's an atomic bomb. The atoms involved are destroyed, but there are plenty of them around so renewability of the supply is not a serious problem. But the process generates a lot of dangerous radioactive waste. Presently, there are no really good solutions for dealing with this waste, although there are some that are not extremely terrible.
Geothermal compression. Gravity causes the materials in the interior of the earth to be compressed, which causes them to heat up. Most of this stays deep inside the earth, but a little comes out in volcanoes, geothermal vents and a few other things. This heat can be used to run power generating equipment, such as steam turbines. The total supply of geothermal energy is finite, but far larger than present consumption rates could use for millions of years. As far as I know, there's no real downside to using geothermal energy, but there just aren't very many places where it's near enough to the surface to be of much help to energy needs. In order to drill our own geothermal wells into the earth's mantle would require some big breakthroughs, none of which show any signs of happening. For the time being, we pretty much have to rely on earthquakes and volcanoes to do it for us.
The third, and far and away the most important, is Nuclear fusion in the sun, causing lighting and heating of the earth. The sun is finite, and will eventually burn itself out, but this won't be for billions of years. Radiation from the sun strikes the earth at about 1 kilowatt per square meter. average US energy consumption is about 3 billion kilowatts. The US is about 10 trillion square meters, so theoretical capacity is about 3000 times what we need. No means of collecting that energy is very efficient, but that's a big multiplier. Ultimately, the question of renewability is whether we can extract energy in a way that doesn't wreck the environment. Here are a few:
There are numerous ways of using solar energy fairly directly.
Photovoltaic cells (mostly silicon but there are others) can be exposed to the sun and turn about 15% of the energy received into electricity. This are almost completely renewable and will work just fine for as long as they aren't damaged or obstructed, although some of the chemistry in making the cells is nasty.
Solar heating: There are several ways of doing this. The sun can be used to heat blackened tubes to generate hot water, which is a useful commodity, and can be used for bathing, washing, or radiant heating. The sun can be focused and used to generate something hot enough to run generating equipment, such as steam turbines. These are completely renewable, but a little fussy to manage.
Hydroelectric power: the sun heats bodies of water, including the ocean, and evaporates it. Some of this later precipitates and fills other bodies of water, at higher elevations, where it can be run through a turbine to generate electricity as it falls to lower elevations. Modulo the environmental damage involved in redirecting the water, this is completely renewable.
Wind Energy: Differential heating of the earth causes air to move from warmer places to cooler ones. We call this air movement "Wind". Windmills can be put up to collect electrical (or mechanical) energy from this. This too is completely renewable, modulo the environmental consequences of building the windmills. There is also a tiny but real risk that we affect the movement of air. Windmills have also been known to injure flying animals, such as birds and bats, that can't recognize the fast moving blades. The harm from this is considerably less than from fossil fuel plants.
Biofuel. Ultimately, biofuel uses energy from the sun to cause plants to grow. The complex molecule chlorophyll uses the suns energy to convert carbon dioxide from the air, and minerals and water from the ground, into plant material. There are many biofuels. The most familiar is wood. Trees can be harvested and burned, generating heat directly, or used to run power generating equipment, such as steam turbines. Whether this is renewable or not is whether the trees are harvested at a rate that damages the forest or allows it to continue to be as healthy and large. There is a lot of wood that is cut down and thrown awaybecause it's the easiest thing to do with it. A lot of this could be turned into energy. There are many wood-products industries that generate all the energy they need from wood waste.
There are numerous other biofuels. The one in largest use is Ethanol, which in the US is mostly from corn. Because the government subsidizes corn farmers, some of the surplus is fermented and distilled into ethanol. Most of this gets added to gasoline. This technically renewable but it is a terrible waste of agricultural resources and taxpayer money. Professor Patzek calculates that it takes between 4 and 7 times as much fossil energy to create the ethanol as it produces, which makes it a big new loser on the renewable scale.
Other biofuels include converting dirty cooking oil into biodiesel. This is certainly renewable: the cooking oil is an agricultural product, which probably was sun and carbon dioxide less than a year prior to it being turned into biodiesel, but the total supply is very small.
Another which seems promising until you do the math is microbes which secrete, or can be turned into, some sort of petroleum-like product. Again, this is completely renewable, on short cycles, but chlorophyll is a very inefficient converter of the sun's energy to usable energy: around 1/10th of 1% at the very best. If we were to use the entire area of the US (including alaska) to generate biofuel this way, it would not be sufficient for present energy needs despite that amazing 3000:1 multiplier. Their advantage is once the process is started, it's very low maintenance, but Photovoltaic is way more efficient.
Fossil fuels are technically biofuels and renewable, but the time it takes to renew it is in the hundreds of millions of years. Plant matter in large quantities lived, grew, and died, over thousands or millions of years creating big piles. When they're in the process of being created, they tend to be called things like marshes or peat bogs. Eventually they get covered up by hundreds or thousands of feet of earth and over millions of years they turn into fossil fuels. Whether it's coal, petroleum or natural gas depends on the exact temperature, pressure and chemical composition. Coal stays where it was formed, but petroleum and gas are lighter than earth and stone, and if there's a path to the surface, they float up and eventually are lost into the atmosphere where the sun causes them to break down into CO2 and water. When there's a geologic formation on top that causes it to be trapped, we can drill into it and recover the petroleum or gas. Most petroleum was created around 300 million years ago, but over hundreds of millions of years. By burning fossil fuels, we're emitting hundreds of millions of years of CO2 storage in a few decades. Not renewable.
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