I would like to apologize for a short diversion from fundamental thermodynamics, but this is a blog, after all. I'll get back to the second law shortly.
There are many opportunities for renewable energy but one stands out because it may inherently sequester carbon by enhancing a natural process that is already a major long term carbon sink.
Ocean Thermal Energy Conversion (OTEC) extracts solar energy through the temperature difference between warm surface water in the tropics and cold deep water: Warm water is used to heat a fluid with a relatively low boiling temperature and the vapor runs through a turbine to generate power. Below about 1,000 meters, water temperatures are just above freezing everywhere in the ocean, so the vapor can be condensed by cold water brought up from the deeps. There are many different approaches using different fluids, variations of heat exchanger and turbine technology and different platform and cold water pipe designs (most OTEC designs are floating platforms, "grazing" in the open ocean). OTEC has been demonstrated as a technically feasible method of generating energy – I worked on the design of “OTEC Early Ocean Test Platform 1” in the late 70’s, which demonstrated various parts of an OTEC system on a platform converted from a small oil tanker, nominally producing one megawatt – but it was not economical at the time.
The unique feature of OTEC is the cold water. The temperature difference between the warm and cold water is small, so OTEC extracts only a small amount of energy from each cubic meter of water and thus uses prodigious amounts of water. In one design, a thousand cubic meters of water per second are required to produce seventy megawatts of net output power.
The cold water is also laden with nutrients. In the tropics, the warm surface waters are lighter than the cold water and act as a cap to keep the nutrients in the deeps. This is why there is much less life in the tropical ocean than in coastal waters or in the Arctic, Antarctic and North Atlantic and Pacific. The tropical ocean is only fertile where some feature such as an island or a submerged canyon causes an upwelling of cold water. One such place is off the coast of Peru, where the Peru (or Humboldt) Current creates a nutrient laden waters near the Equator. In this area, with lots of solar energy and nutrients, ocean fertility is on the order of 1800 grams of carbon uptake per square meter per year, (mostly by microscopic photosynthetic phytoplankton). In the adjacent waters, and most of the rest of the ocean, fertility is typically well below 100 grams per square meter per year. This creates a rich fishery, but most of the carbon eventually sinks to the deeps in the form of waste products and dead microorganisms.
Throughout the entire world, these microorganisms currently sequester about forty billion metric tonnes of carbon per year. They are the major long term sink for carbon dioxide. A certain amount of carbon is converted to calcium carbonate, which eventually becomes limestone or other sedimentary rock and under very special circumstances, some of the organic material even becomes oil. Algae and other microorganisms that fed on them in ancient seas are the source of today’s oil.
We can make various estimates of fertility enhancement and sequestration, but a reasonable guess is that an OTEC plant designed to optimize nutrification might result in as much as 10,000 metric tonnes of carbon dioxide sequestration per year per megawatt. The recent challenge by billionaire Sir Richard Branson is to sequester one billion tonnes of carbon dioxide per year in order to halt global warming, so an aggressive OTEC program, hundreds of several hundred megawatt plants, might meet this.
In economic terms, optimistic guesses at OTEC plant costs are in the range of a million dollars per megawatt. Since a kilowatt-hour of electricity generated by coal produces about a kilogram of carbon dioxide, a carbon tax of one to two cents per kilowatt-hour might cover the capital costs of an OTEC plant in carbon credits alone. The equivalent in gasoline would be ten to twenty cents per gallon. With gasoline above two dollars per gallon and electricity above ten cents per kilowatt, these are not entirely unreasonable charges.
The actual effectiveness of OTEC in raising ocean fertility and thereby sequestering carbon has to be verified, and there has to be a careful examination of possible harmful environmental impacts – an old saying among engineers is "it seemed like a good idea at the time". An OTEC plant optimized for ocean fertility will also be different than one optimized to generate power, so any OTEC based carbon scheme has to include transfer payments of some sort – it won’t come for free. Finally, who owns the ocean thermal resource? Most plants will be in international waters, though these waters tend to be off the coasts of the developing world.
As to this last question, there is an additional benefit: Another saying among engineers is "we aren’t trying to solve world hunger". In this case, though, we may have. Increased ocean fertility may enhance fisheries substantially. In addition, a problem of OTEC is that the energy is "stranded" far at sea, possibly on a drifting platform – a thousand-mile long extension cord is not an option. However, many OTEC advocates have suggested making nitrogen compounds for fertilizers at sea – this is an energy intensive process, now mostly using natural gas, so fertilizers are expensive. OTEC fertilizer could be sold to developing countries at a subsidy, where they would greatly enhance farm yields.
It would seem that the Branson Challenge is met, and more, but as much as I would like to claim the Branson prize, I cannot. I worked on OTEC for a contractor under the ERDA, in a program initiated by President Carter, who I am told, was an OTEC enthusiast. I’m sure Jimmy Carter will find a good use for Sir Richard’s check.
Monday, March 26, 2007
Friday, February 9, 2007
The 3.9 Laws Of Thermodynamics
“A theory is the more impressive the greater the simplicity of its premises are, the more different kinds of things it relates, and the more extended is its area of applicability, Therefore the deep impression that classical thermodynamics made upon me. It is the only physical theory of universal content concerning which I am convinced that, within the framework of applicability of its basic concepts, it will never be overthrown.”
(Albert Einstein, 1949)
Classical and statistical thermodynamics was one of the great intellectual achievements of physics of the last two centuries, and one of the least well known, especially statistical thermodynamics. Einstein’s interest in classical thermodynamics is especially remarkable in that it is an area of physics that he is very rarely associated with, especially now that classical thermodynamics is generally regarded as subject only worthy of juniors in mechanical engineering, not physicists. What is much less well known is that Einstein himself was not only interested in classical thermodynamics, but actually invented and patented the absorption cycle refrigerator. This device was once the dominant means of home refrigeration and will probably become an important tool in renewable energy, since it can be used to make relatively low temperature heat, such as solar energy, into cold. This in turn would greatly reduce the need for fossil fuels to generate electricity on hot days. Hot days, with lots of solar energy, are the very times when air cooling is most needed so this is potentially a match made in heaven.
However, we get ahead of ourselves, except to remark that absorption cooling is a good example of one key to making best use of solar energy. We have to understand the concept of availability, or quality, of energy, versus the required use of the energy to optimize our use of it. In the case of cooling, our energy need is to create relatively low temperature changes, at relatively normal temperatures. This is “low quality” energy, as opposed to electricity, for example, because it is at a low temperature difference. We should be able to use other low quality energy, such as the relative diffuse heat available from simple heat absorbing solar panels to satisfy this need. This type of efficiency is often referred to as “Second Law” efficiency, which is our cue to starting to discuss thermodynamics.
There are (sort of) four laws of thermodynamics.
The “Zeroth” law is needed, though it is obvious, because it is not derivable from the other laws, and because it is needed to form a basis for temperature measurement. It states that when two bodies are each at the same temperature as a third body, they are at the same temperature as each other. I guess its obviousness is why it is called the “Zeroth” law.
The next three are often described as “You can’t win, you can’t break even, and you can’t get out of the game”. The first two are pretty accurate statements of the first and second laws, but I’m not sure about the third.
The First Law of Thermodynamics
The first law is conservation of energy and mass, and says that energy (and mass) can’t be created or destroyed. It is generally stated in mathematical terms that mean the amount of energy and mass stored in a designated volume is the amount originally in the system, that plus going out, minus that going in. (The sign convention is that work by the system on the outside world is positive, and that done by the world on the system is negative.) There are a lot of things that are conserved in physics; mass, energy, momentum, spin, charge, current, voltage drop, (for subatomic particles called quarks) truth, beauty and charm, and for physics departments in general grants and other funding. It is often possible to analyze a system by writing a conservation law. (And in the old days of blue book exams, writing the appropriate conservation statement was generally worth at least partial credit.) It is a law of nature that we believe in, because we have never seen it violated, but we can’t prove it from other laws.
For example, structural problems involving buckling can be very complicated, but one way of simplifying them is to look at energy conservation. Buckling occurs when the geometry becomes unstable, and the initial deflection under load into a shape that causes the structure to become less stiff, so that it deflects more, becoming even less stiff and so on until failure. Calculating the initial deflection is fairly simple these days with Finite Element Analysis, but looking at buckling requires repeatedly recalculating the stiffness and deflection and as the structure changes, and this is not as easy (this is a “geometrically non-linear analysis”), and some FEA packages don’t have this capability (and most that do charge extra for the capability as an add-on). However we can look at the first step and calculate the energy absorbed by the structure (the force it resists the load with times the distance it deflects) and the energy from the load (the load time the distance it deflects). If the structure absorbs more energy than the load yields, the First law tells us that the structure is therefore stable, without having to go into a non-linear analysis. First law approaches are very frequently used to get a "big picture" sort of approximation, without getting into the details.
Another example of this is proposals to use algae for sequestering carbon at power plants, generally as biofuels. A First law look at this suggests some problems without going into any details. Algae get the energy to make carbon dioxide into themselves from solar energy. A coal power plant is maybe 30% efficient, so it uses more than a ton of coal per megawatt, or hundreds of tons per hour. Peak insolation is around 400 Btu/ft2/hour, so after a bit of unit conversion and "swags" at efficiency, several square miles of sunlight will be required to power algae sequestration, which is probably not locally feasible at a powerplant. This is not to say that algae fuels will not have an important role in either replacing fossil fuels (after all, algae made them in the first place) or sequestering carbon (again, algae were responsible for changing Earth's primordial atmosphere into the one we enjoy now by sequestering carbon dioxide), but it will take more than bioreactors at a powerplant.
We have to realize that there are a lot of different appearances for energy, basically different units, but its all energy, and it’s all conserved. The term work is synonymous with energy, though it generally implies a mechanical form of energy. In mechanical terms, energy is force acting through a distance. The archetypical example is a raised weight. A weight weighing one pound one foot off the ground has one foot pound of energy. If we release it, it will fall, heating the air slightly through friction, possibly making a noise when it hits. Most of the energy will then become mechanical energy in the object it lands (and in itself) and deflects under impact. This mechanical energy will then swiftly degrade to heat. Energy can also be stored as chemical bonds in fuels or food (a calorie is a unit of energy, enough heat to raise the temperature of a kilogram of water one degree centigrade (specifically from 14.5 C to 15.5 C) , or as charge in a capacitor, or in any number of other forms . The thermal equivalent of mechanical energy is quite small. I was once worried about the disk brakes on my car rapidly becoming hot to the touch and computed that stopping repeatedly from twenty miles per hour should only heat them a few degrees. Sure enough, the parking brake was not releasing fully, and I had to replace the pads shortly thereafter. The chemical equivalent is even smaller; consider how far you have to run to burn off a small amount of food.
The English unit equivalent of a calorie is a British thermal unit or Btu, the amount of heat required to raise a pound of water 1 degree Fahrenheit, specifically from 59.5 F to 60.5 F. There are roughly four Btu in a calorie. The metric equivalent of a foot pound is the Joule (in the meter-kilogram-second, MKS, system a Joule is the force of one Newton acting through one meter) or the erg (centimeter-gram-second, CGS, system). You may sometimes see nutrition labels on Canadian food packages marked in kilojoules (kJ) instead of calories, because the Joule is part of the consistent metric MKS system.
(A note here for purists: I hereby acknowledge and affirm that “pound” is not properly a measure of mass, but a measure of force. A slug is the amount of mass that produces 32.2 pounds of force in the Earth’s gravity, which produces an acceleration of 32.2 feet per second squared. Thus the units of mass are actually pounds force times seconds squared per foot. However, “pound mass”, lbm, is frequently used to mean 1/32.2th of a slug, and “poundal” is sometimes used as the unit of force produced by a pound mass in the Earth’s gravity. In metric units, a kilogram is actually a unit of mass, and produces 9.805 newtons of force in Earths gravity acceleration of 9.805 meters per second squared. All of this is of some importance because it is frequently possible to figure out what is happening in physics because the units have to match.)
(Another note for purists: There is a proper style for capitalization of units. In general the abbreviations for units deriving from proper names such as newton, joule, or British are capitalized, even when used in the middle of a compound unit, like kW, but not the units themselves. There are some other rules which I forget frequently. I hope, in general, to get them right, though I reserve the right to make mistakes. I will nonetheless welcome comments correcting me.)
Power is often confused with energy, but power is the rate at which energy is transmitted. A horsepower is 550 foot pounds per second (I guess the horse didn’t rate a proper name), a watt is one joule per second, and a horsepower is about 745 watts in the US (there is a European horsepower which is slightly different, but it’s rarely used).
One important transformation of energy in classical thermodynamics is pressure and volume to and from heat, since generally heating a gas causes it to expand, or increase in pressure, or both. If we hold the pressure constant, the volume increases as we heat a gas. For a finite area, pressure times area is force, and if the volume increases, we have the area moving, which is thus force times a distance, or work. This work is the transformed energy we added as heat. The reverse also works. If we apply enough pressure to a gas to cause it to decrease in volume, it gets hotter, which anyone who has touched a bicycle pump after it has been used to blow up a tire has noted. Because this exchange is so common, it is worth noting the equivalent units for work and heat energy here - one Btu is 778 foot pounds. Finally, note the order of “foot pound”. A pound foot is not the same as a foot pound. By convention, a pound foot is a unit of torque or twisting – a force of one pound applied at the end of a one foot wrench.
The issue of work and heat energy brings up another set of terminology. The internal energy of a substance (generally a gas or liquid) is the heat content. The enthalpy is the heat content plus the pressure of the gas times the volume per unit mass, or “specific volume”. Enthalpy is the total thermodynamic energy in a substance (relative to a standard condition) and is the quantity we are generally interested in for heat engines and similar systems. For this reason, enthalpy is generally the characteristic listed in tables of thermodynamic properties, and if you need internal energy, you generally have to compute it yourself from enthalpy, specific volume and pressure. (This is not, of course, the total energy of all types – hydrogen has a certain energy in terms of its temperature, pressure and specific volume but much more chemical energy if it was burned, and very, very much more if it undergoes a thermonuclear reaction.)
The Second Law of Thermodynamics
The Second Law is not really a law any more, in the sense that a physical law is something which we believe to be true based on never having seen it violated in nature, but cannot prove from other laws. Statistical thermodynamics can prove it, but classical thermodynamics was not able to describe the Second Law in terms of physical phenomena. There are numerous statements of the second law, the most important being the Clausius statement: “No device that operates in a cycle can produce no effect other than the transfer of heat from a cooler body to a hotter one.” We can reverse the sense of the useful energy flow and also say that this means we can’t have a refrigerator that doesn’t require an input of work. The Kelvin-Planck statement is equivalent, but from the viewpoint of work: No device can operate in a cycle and raise a weight (or do other work) by exchange of heat with only one reservoir. This last machine is a perpetual motion machine of the second kind, and has been occasionally proposed, usually as part of a scam. This machine doesn’t create energy from nowhere (that would be a First Law perpetual motion machine), but rather purports to take energy, usually heat, from some vast reservoir, and turn the heat into work without rejecting any heat to another reservoir at a lower temperature.
We can prove the equivalence of the two statements and the Second law by postulating two systems:
First, consider a system that included a Clausius (violation) heat pump, a cold reservoir and a conventional heat engine, (which we know works) connected to an external hot reservoir. (Note that the cold reservoir is within the system, and the hot one is outside). The Clausius heat pump would transfer heat to the hot reservoir from the cool one, and the heat engine device would derive work from transferring work to the cold reservoir from the hot reservoir. The cold reservoir would not get any colder, though, because the Clausius device would push heat back to the external cold reservoir. The system thus comprises a Kelvin-Plank (violation) heat engine, producing work from only a hot reservoir. This is not a violation of the first law, though – the net heat leaving the hot reservoir is equal to the work the system makes.
Second, consider a Kelvin-Planck heat engine and a conventional heat pump, with two external reservoirs, one hot, one cold. The Kelvin-Planck engine is only connected to the hot reservoir and produces just enough work (by taking just enough heat from the hot reservoir) to drives the conventional heat pump, the two together are a Clausius heat pump, moving heat without input of work.
Using this logic, the Second law can be expressed as noting that if we put a hot reservoir in contact with a cold one, heat flows from the hot one to the cold one, but not the other way around.
This consideration gives us Carnot efficiency, which is the theoretical best efficiency of a heat engine. A heat engine operating in accordance with the Second law takes heat from a hot reservoir, rejects some of it to a cold reservoir, and what is left is useful work, which is required by the First law, since we can’t make energy vanish, any more than we can make it appear. Second law efficiency is basically about making best use of energy and suiting the sources to the use. Kenneth Deffeyes, in Hubbert's Peak: The Impending World Oil Shortage (Princeton University Press, 2001, ISBN 978-0-11625-9) compares poor Second law efficiency to making every bit of a cow into hamburger, instead of first cutting off the steaks, and he also gives an example in his own home – his furnace captures more than 90% of the heat of the fuel it burns and uses it to warm his house. But, the fuel burns at a very high temperature, so this high temperature could have been used in a heat engine for something more valuable, like making electricity, then the heat rejected by the heat engine would still be warm enough to heat his house. As a matter of fact, it is likely he could have heat his house with even less fuel by using a heat engine to run a heat pump, plus using the waste heat. Barry Commoner, in The Poverty of Power (Knopf, 1976, ISBN 0-394-40371-1) bases much of the book on this point.
The Third Law is basically that there is a zero temperature.
We call the Second and Third laws of Thermodynamics laws of nature, because prior to the advent of statistical mechanics, we had to take this on faith, because we have never, ever, seen it violated just as we have never seen energy or matter mysteriously appear from nowhere, or because we have always seen two things that are at the same temperature as one thing the same temperature as each other. However classical thermodynamics didn’t have any way to really prove this from other principles.
By looking at the Second Law in various complicated ways, the concept of entropy, another property of a substance based on its temperature, specific volume and pressure was developed. Entropy is the change in enthalpy (energy content) divided by the absolute temperature. (Absolute temperature is degrees Rankine or Kelvin. Rankine is degrees F plus 460, since absolute zero is -460 F. Kelvin is degrees C plus 273.) This is derived by calculations involving the Second law and Carnot cycle engines to develop a temperature scale, but it never had a very good physical basis under classical thermodynamics. Whereas you can point to the vibration of the molecules in a substance and say that is heat, internal energy, or enthalpy, it wasn’t clear what entropy was. Popular literature often refers to entropy as disorder or something like that, but that still doesn’t give us a real handle on it.
However, statistical thermodynamics, especially when combined with a core concept of quantum mechanics, that energy only comes in discrete packages, which though small, cannot be subdivided, gives a wonderful answer and puts us on solid ground understanding entropy, which turns out to be a major key to the universe. (Steven Hawking is known for having derived a very simple equation that calculated the entropy of a black hole by combining quantum mechanics and general relativity, thus leading him to believe he was near a Grand Universal Theory of Everything, the Holy Grail of theoretical physics. Alas …) Modern thermodynamics is really based on the Second law, which turns out to be not really a law after all.
More in the next episode. “Same Bat-time, same Bat-station.”
(Albert Einstein, 1949)
Classical and statistical thermodynamics was one of the great intellectual achievements of physics of the last two centuries, and one of the least well known, especially statistical thermodynamics. Einstein’s interest in classical thermodynamics is especially remarkable in that it is an area of physics that he is very rarely associated with, especially now that classical thermodynamics is generally regarded as subject only worthy of juniors in mechanical engineering, not physicists. What is much less well known is that Einstein himself was not only interested in classical thermodynamics, but actually invented and patented the absorption cycle refrigerator. This device was once the dominant means of home refrigeration and will probably become an important tool in renewable energy, since it can be used to make relatively low temperature heat, such as solar energy, into cold. This in turn would greatly reduce the need for fossil fuels to generate electricity on hot days. Hot days, with lots of solar energy, are the very times when air cooling is most needed so this is potentially a match made in heaven.
However, we get ahead of ourselves, except to remark that absorption cooling is a good example of one key to making best use of solar energy. We have to understand the concept of availability, or quality, of energy, versus the required use of the energy to optimize our use of it. In the case of cooling, our energy need is to create relatively low temperature changes, at relatively normal temperatures. This is “low quality” energy, as opposed to electricity, for example, because it is at a low temperature difference. We should be able to use other low quality energy, such as the relative diffuse heat available from simple heat absorbing solar panels to satisfy this need. This type of efficiency is often referred to as “Second Law” efficiency, which is our cue to starting to discuss thermodynamics.
There are (sort of) four laws of thermodynamics.
The “Zeroth” law is needed, though it is obvious, because it is not derivable from the other laws, and because it is needed to form a basis for temperature measurement. It states that when two bodies are each at the same temperature as a third body, they are at the same temperature as each other. I guess its obviousness is why it is called the “Zeroth” law.
The next three are often described as “You can’t win, you can’t break even, and you can’t get out of the game”. The first two are pretty accurate statements of the first and second laws, but I’m not sure about the third.
The First Law of Thermodynamics
The first law is conservation of energy and mass, and says that energy (and mass) can’t be created or destroyed. It is generally stated in mathematical terms that mean the amount of energy and mass stored in a designated volume is the amount originally in the system, that plus going out, minus that going in. (The sign convention is that work by the system on the outside world is positive, and that done by the world on the system is negative.) There are a lot of things that are conserved in physics; mass, energy, momentum, spin, charge, current, voltage drop, (for subatomic particles called quarks) truth, beauty and charm, and for physics departments in general grants and other funding. It is often possible to analyze a system by writing a conservation law. (And in the old days of blue book exams, writing the appropriate conservation statement was generally worth at least partial credit.) It is a law of nature that we believe in, because we have never seen it violated, but we can’t prove it from other laws.
For example, structural problems involving buckling can be very complicated, but one way of simplifying them is to look at energy conservation. Buckling occurs when the geometry becomes unstable, and the initial deflection under load into a shape that causes the structure to become less stiff, so that it deflects more, becoming even less stiff and so on until failure. Calculating the initial deflection is fairly simple these days with Finite Element Analysis, but looking at buckling requires repeatedly recalculating the stiffness and deflection and as the structure changes, and this is not as easy (this is a “geometrically non-linear analysis”), and some FEA packages don’t have this capability (and most that do charge extra for the capability as an add-on). However we can look at the first step and calculate the energy absorbed by the structure (the force it resists the load with times the distance it deflects) and the energy from the load (the load time the distance it deflects). If the structure absorbs more energy than the load yields, the First law tells us that the structure is therefore stable, without having to go into a non-linear analysis. First law approaches are very frequently used to get a "big picture" sort of approximation, without getting into the details.
Another example of this is proposals to use algae for sequestering carbon at power plants, generally as biofuels. A First law look at this suggests some problems without going into any details. Algae get the energy to make carbon dioxide into themselves from solar energy. A coal power plant is maybe 30% efficient, so it uses more than a ton of coal per megawatt, or hundreds of tons per hour. Peak insolation is around 400 Btu/ft2/hour, so after a bit of unit conversion and "swags" at efficiency, several square miles of sunlight will be required to power algae sequestration, which is probably not locally feasible at a powerplant. This is not to say that algae fuels will not have an important role in either replacing fossil fuels (after all, algae made them in the first place) or sequestering carbon (again, algae were responsible for changing Earth's primordial atmosphere into the one we enjoy now by sequestering carbon dioxide), but it will take more than bioreactors at a powerplant.
We have to realize that there are a lot of different appearances for energy, basically different units, but its all energy, and it’s all conserved. The term work is synonymous with energy, though it generally implies a mechanical form of energy. In mechanical terms, energy is force acting through a distance. The archetypical example is a raised weight. A weight weighing one pound one foot off the ground has one foot pound of energy. If we release it, it will fall, heating the air slightly through friction, possibly making a noise when it hits. Most of the energy will then become mechanical energy in the object it lands (and in itself) and deflects under impact. This mechanical energy will then swiftly degrade to heat. Energy can also be stored as chemical bonds in fuels or food (a calorie is a unit of energy, enough heat to raise the temperature of a kilogram of water one degree centigrade (specifically from 14.5 C to 15.5 C) , or as charge in a capacitor, or in any number of other forms . The thermal equivalent of mechanical energy is quite small. I was once worried about the disk brakes on my car rapidly becoming hot to the touch and computed that stopping repeatedly from twenty miles per hour should only heat them a few degrees. Sure enough, the parking brake was not releasing fully, and I had to replace the pads shortly thereafter. The chemical equivalent is even smaller; consider how far you have to run to burn off a small amount of food.
The English unit equivalent of a calorie is a British thermal unit or Btu, the amount of heat required to raise a pound of water 1 degree Fahrenheit, specifically from 59.5 F to 60.5 F. There are roughly four Btu in a calorie. The metric equivalent of a foot pound is the Joule (in the meter-kilogram-second, MKS, system a Joule is the force of one Newton acting through one meter) or the erg (centimeter-gram-second, CGS, system). You may sometimes see nutrition labels on Canadian food packages marked in kilojoules (kJ) instead of calories, because the Joule is part of the consistent metric MKS system.
(A note here for purists: I hereby acknowledge and affirm that “pound” is not properly a measure of mass, but a measure of force. A slug is the amount of mass that produces 32.2 pounds of force in the Earth’s gravity, which produces an acceleration of 32.2 feet per second squared. Thus the units of mass are actually pounds force times seconds squared per foot. However, “pound mass”, lbm, is frequently used to mean 1/32.2th of a slug, and “poundal” is sometimes used as the unit of force produced by a pound mass in the Earth’s gravity. In metric units, a kilogram is actually a unit of mass, and produces 9.805 newtons of force in Earths gravity acceleration of 9.805 meters per second squared. All of this is of some importance because it is frequently possible to figure out what is happening in physics because the units have to match.)
(Another note for purists: There is a proper style for capitalization of units. In general the abbreviations for units deriving from proper names such as newton, joule, or British are capitalized, even when used in the middle of a compound unit, like kW, but not the units themselves. There are some other rules which I forget frequently. I hope, in general, to get them right, though I reserve the right to make mistakes. I will nonetheless welcome comments correcting me.)
Power is often confused with energy, but power is the rate at which energy is transmitted. A horsepower is 550 foot pounds per second (I guess the horse didn’t rate a proper name), a watt is one joule per second, and a horsepower is about 745 watts in the US (there is a European horsepower which is slightly different, but it’s rarely used).
One important transformation of energy in classical thermodynamics is pressure and volume to and from heat, since generally heating a gas causes it to expand, or increase in pressure, or both. If we hold the pressure constant, the volume increases as we heat a gas. For a finite area, pressure times area is force, and if the volume increases, we have the area moving, which is thus force times a distance, or work. This work is the transformed energy we added as heat. The reverse also works. If we apply enough pressure to a gas to cause it to decrease in volume, it gets hotter, which anyone who has touched a bicycle pump after it has been used to blow up a tire has noted. Because this exchange is so common, it is worth noting the equivalent units for work and heat energy here - one Btu is 778 foot pounds. Finally, note the order of “foot pound”. A pound foot is not the same as a foot pound. By convention, a pound foot is a unit of torque or twisting – a force of one pound applied at the end of a one foot wrench.
The issue of work and heat energy brings up another set of terminology. The internal energy of a substance (generally a gas or liquid) is the heat content. The enthalpy is the heat content plus the pressure of the gas times the volume per unit mass, or “specific volume”. Enthalpy is the total thermodynamic energy in a substance (relative to a standard condition) and is the quantity we are generally interested in for heat engines and similar systems. For this reason, enthalpy is generally the characteristic listed in tables of thermodynamic properties, and if you need internal energy, you generally have to compute it yourself from enthalpy, specific volume and pressure. (This is not, of course, the total energy of all types – hydrogen has a certain energy in terms of its temperature, pressure and specific volume but much more chemical energy if it was burned, and very, very much more if it undergoes a thermonuclear reaction.)
The Second Law of Thermodynamics
The Second Law is not really a law any more, in the sense that a physical law is something which we believe to be true based on never having seen it violated in nature, but cannot prove from other laws. Statistical thermodynamics can prove it, but classical thermodynamics was not able to describe the Second Law in terms of physical phenomena. There are numerous statements of the second law, the most important being the Clausius statement: “No device that operates in a cycle can produce no effect other than the transfer of heat from a cooler body to a hotter one.” We can reverse the sense of the useful energy flow and also say that this means we can’t have a refrigerator that doesn’t require an input of work. The Kelvin-Planck statement is equivalent, but from the viewpoint of work: No device can operate in a cycle and raise a weight (or do other work) by exchange of heat with only one reservoir. This last machine is a perpetual motion machine of the second kind, and has been occasionally proposed, usually as part of a scam. This machine doesn’t create energy from nowhere (that would be a First Law perpetual motion machine), but rather purports to take energy, usually heat, from some vast reservoir, and turn the heat into work without rejecting any heat to another reservoir at a lower temperature.
We can prove the equivalence of the two statements and the Second law by postulating two systems:
First, consider a system that included a Clausius (violation) heat pump, a cold reservoir and a conventional heat engine, (which we know works) connected to an external hot reservoir. (Note that the cold reservoir is within the system, and the hot one is outside). The Clausius heat pump would transfer heat to the hot reservoir from the cool one, and the heat engine device would derive work from transferring work to the cold reservoir from the hot reservoir. The cold reservoir would not get any colder, though, because the Clausius device would push heat back to the external cold reservoir. The system thus comprises a Kelvin-Plank (violation) heat engine, producing work from only a hot reservoir. This is not a violation of the first law, though – the net heat leaving the hot reservoir is equal to the work the system makes.
Second, consider a Kelvin-Planck heat engine and a conventional heat pump, with two external reservoirs, one hot, one cold. The Kelvin-Planck engine is only connected to the hot reservoir and produces just enough work (by taking just enough heat from the hot reservoir) to drives the conventional heat pump, the two together are a Clausius heat pump, moving heat without input of work.
Using this logic, the Second law can be expressed as noting that if we put a hot reservoir in contact with a cold one, heat flows from the hot one to the cold one, but not the other way around.
This consideration gives us Carnot efficiency, which is the theoretical best efficiency of a heat engine. A heat engine operating in accordance with the Second law takes heat from a hot reservoir, rejects some of it to a cold reservoir, and what is left is useful work, which is required by the First law, since we can’t make energy vanish, any more than we can make it appear. Second law efficiency is basically about making best use of energy and suiting the sources to the use. Kenneth Deffeyes, in Hubbert's Peak: The Impending World Oil Shortage (Princeton University Press, 2001, ISBN 978-0-11625-9) compares poor Second law efficiency to making every bit of a cow into hamburger, instead of first cutting off the steaks, and he also gives an example in his own home – his furnace captures more than 90% of the heat of the fuel it burns and uses it to warm his house. But, the fuel burns at a very high temperature, so this high temperature could have been used in a heat engine for something more valuable, like making electricity, then the heat rejected by the heat engine would still be warm enough to heat his house. As a matter of fact, it is likely he could have heat his house with even less fuel by using a heat engine to run a heat pump, plus using the waste heat. Barry Commoner, in The Poverty of Power (Knopf, 1976, ISBN 0-394-40371-1) bases much of the book on this point.
The Third Law is basically that there is a zero temperature.
We call the Second and Third laws of Thermodynamics laws of nature, because prior to the advent of statistical mechanics, we had to take this on faith, because we have never, ever, seen it violated just as we have never seen energy or matter mysteriously appear from nowhere, or because we have always seen two things that are at the same temperature as one thing the same temperature as each other. However classical thermodynamics didn’t have any way to really prove this from other principles.
By looking at the Second Law in various complicated ways, the concept of entropy, another property of a substance based on its temperature, specific volume and pressure was developed. Entropy is the change in enthalpy (energy content) divided by the absolute temperature. (Absolute temperature is degrees Rankine or Kelvin. Rankine is degrees F plus 460, since absolute zero is -460 F. Kelvin is degrees C plus 273.) This is derived by calculations involving the Second law and Carnot cycle engines to develop a temperature scale, but it never had a very good physical basis under classical thermodynamics. Whereas you can point to the vibration of the molecules in a substance and say that is heat, internal energy, or enthalpy, it wasn’t clear what entropy was. Popular literature often refers to entropy as disorder or something like that, but that still doesn’t give us a real handle on it.
However, statistical thermodynamics, especially when combined with a core concept of quantum mechanics, that energy only comes in discrete packages, which though small, cannot be subdivided, gives a wonderful answer and puts us on solid ground understanding entropy, which turns out to be a major key to the universe. (Steven Hawking is known for having derived a very simple equation that calculated the entropy of a black hole by combining quantum mechanics and general relativity, thus leading him to believe he was near a Grand Universal Theory of Everything, the Holy Grail of theoretical physics. Alas …) Modern thermodynamics is really based on the Second law, which turns out to be not really a law after all.
More in the next episode. “Same Bat-time, same Bat-station.”
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