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17 August 2006
[Warning: Boring Theory] Thermodynamic Economies
A note of introduction. Given that I have put ‘boring theory’ stamps all over this, I feel no obligation to be zany or witty or anything of the sort. So I’m shooting for dry and boring. I did, however, want to point out why I like the whole fusing multiple field type theories. If one Author made everything up, it makes sense that His fingerprints are all over everything. And if you’re listening to one band, you hear the same guitar rifs multiple times. In fact, the better you know one song, the better you can recognize their artistry in another song. We draw a lot of false dichotomies between fields of study. Consider Aerobic Respiration. You can tie in Chemical Engineering, Thermodynamics, Structures, Statics, Dynamics, Electrical Engineering, Mechanical Engineering, and, of course, Biology in the way it works. We would never make a system with such efficient Systems Engineering, because we believe that EE has very little to do with Chemical Engineering. But, really, an electron in a covalent bond isn’t really that different from an electron excited up a few quanta. It’s all the same stuff, and truly elegant solutions break down a lot of these pre-existing boundaries that tell us what we can and can‘t do with the same stuff. Kind of like Unified Field Theory. But more a Unified Engineering Theory. So what the following theory is good for is to make a Figure of Merit for energy economy in a system that incorporates several sub-systems (or at least sub-functions.)
System Boundaries and Interactions. Thermodynamics, in its most direct form, is well adapted to dealing with one-dimensional energy systems, such as a Carnot cycle or a heat pump. Generally, though, large advanced systems incorporate many such systems, which may be pumping energy different directions through different media within the larger thermodynamic boundaries of the entire system. While not particularly useful for analyzing, say, a HVAC system, it would be tremendously useful to have a figure of merit on these internal energy flows for say, a urban area, or for an aircraft (which is, of course, why I care. Airplanes are cool. People who fly them are cool. And humble.)
Let’s boil this down by using an example. Take, for example, hmm… I know… the mighty Herk (C-130 to the uninitiated.) Which is awesome, but not exactly an example of thermodynamic elegance. So the Herk has this cool little thing called an Oil-to-Fuel heater. Hot oil, coming off the engine, goes through a heat exchanger, and heats the fuel before it goes into the engine. This does a couple of things, most importantly, keeping the fuel warm enough so there won’t be any problems with ice in the gas. From a thermodynamic viewpoint, though, it incorporates a degree of elegance. Waste heat that you’re trying to dump overboard is reincorporated to pre-heat (to a very small degree) the fuel, which allows a bit of thermodynamic recapture. You take something you’re going to throw out anyways and put it somewhere useful. There is a degree of stinginess in this, which to some degree reflects (to a very small degree) the high degree of thermodynamic recapture in natural designs. Despite multiple subsystems, most natural systems don’t throw overboard energy that would be useful somewhere else. That said, there is a cost on energy recapture. For the oil-to-fuel heater, there had to be a heat exchanger built into the engine, and more piping run. It was pretty low cost, because everything was there anyways, but that cost has to be subtracted from the benefit of the recaptured energy in order to find the energy ’profit’ from the recapture. Given the simplicity of the oil-to-fuel heater, and the high degree of usefulness, it is pretty safe to say that was a good idea. How, though, would we make a figure of merit to tell us if it would be worth doing thermodynamic recapture somewhere else, and where?
Let’s look at a bit more complex problem. The engine oil system takes hot oil from the engine and runs it through a ram air cooler to bring it back to a more manageable temperature. The thing is that the cooler increases drag, which means you have to kick up the power to maintain the same airspeed, which means you’re burning more gas. So you’re incurring an energy loss to throw heat overboard. Consider, now, the anti-icing system. This system uses hot air off of the engine, (bleed air) routes it through the leading edge of the wing to heat up the wing so ice can’t form. But, in drawing bleed air off the engine, you’re reducing engine power (ceteris parabus.) The actual thermodynamic flow is you’re drawing power off the turbine from the compressor, which comes from combustion, going back to costing you more gas. So you’re spending energy to make heat. Both of these systems could be on at the same time. Which then begs the question, why are we spending energy to both make and dump heat at the same time? Here’s the thing, though. There are two different fluids involved here (engine oil and air,) and there are going to be some losses associated with the energy transfer between two media. Also, you will have to add complexity and systems to the aircraft to do the heat exchange, which will add weight (losses) and lifecycle costs. This is not that different from a market, though. You make flour into bread. People will pay a certain amount for that bread. But, in order to sell them the bread, you have to pay for a store. And you have to work at that store. So you need to at least cover your costs (including overhead… fixed and variable) if its worth making your bread store. This is the same thing. But you have to have a means by which you can measure whether its worth building your energy store if you’re going to make your system energy stingy. Economics tells us when its worth running a market interaction. Thermodynamics tells us how energy works. Let’s put them together.
Thermodynamics in Economics Language. So there are two laws of Thermo (really three, but the third is not useful to us here, so we’ll skip it.) In simple language, the first law, enthalpy, says that you can’t make or destroy energy (counting matter.) You end up with the energy you started with, you can just re-arrange it. The second law, entropy, tells you that energy tends to progress from more useful, or accessible, forms to less useful forms. Most people focus on the progression from order to disorder, but the useful part to us is the idea of inaccessible energy. The third law is boring and I already talked about it in the theory about quantum-level voltage and superconductors.
First law. So rocking the economics party on the first law, think back to the gold standard. When currency was tied to precious metals, there was always a finite amount of money in the economy. (Banks still multiplied it and all, but imagine that there are no banks. Like Old West style. Or Hazzard county.) Unless you were to dig up some more gold, you couldn’t really make more money. But you could use that money to do more or less useful things. You could gamble it all away, or you could build a store with it and take other people‘s money. In the same way, you can release about the same amount of energy and propel a car for a good ways or blow up your sister’s toys with M80s. The amount of energy didn’t change, but the usefulness of it did. (This is getting into the second law somewhat.) The important thing is that the amount of money out there is set, but that fixed amount of money doesn’t imply a fixed amount of utility.
Second law. Even if there’s set amount of energy out there, not all of it is as accessible. Say you’re a bank robber. You have a six-shooter. Say there’s some money just laying out on the ground. That money is easily accessible. The costs to pick up that money are very low, far below the revenue involved. So revenue minus costs equals profit. So now say there’s some money in a vault guarded by really mean dogs, the Terminator (from the first movie) and Vasquez from Aliens with that awesome Smart-Gun. Probably, the costs involved in getting that money are going to outweigh the money itself. The costs are more than the revenue, and you would make a loss. So probably you won’t try to get the money. That money is inaccessible, not because there’s no way to get to it, or because its not there, but because it would take more money to get that money than you would make from it. If it costs you a zillion dollars to make a widget, and you make a dollar from it, its not like there’s not still revenue. Its just not worth getting. That’s entropy. If you’re really cold, it doesn’t do you a whole lot of good that the planet Mercury is hot. It would take more energy to go to Mercury to get that heat than the energy you would get from that heat. That’s inaccessible energy: energy that costs you more to get than you get from it. So that’s thermo. Lets do econ now.
Economics in Thermodynamics Language. So the important thing from Econ are the supply and demand curves. (micro, at least. Hescher-Ohlin is cool and all, but not really helpful here. But, look at me, I’m smart because I mentioned it. Really. I can also say ’problematic’ and ’paradigm.’ That‘s what I learned with my vastly overpriced degree.) I’m going to use the single-firm startup/shutdown ATC/ATR/MR curves here.
Marginal Revenue, Average Total Revenue Curves. On our graph, the marginal revenue. This curve is the derivative of the total revenue curve, and basically tells you how much more money you can get for making one more thing. Put in energy terms, potential revenue is available energy. This curve is how much energy could be harvested per unit capacity (regardless of cost.) Imagine increasing a heat exchanger’s capacity between two systems, and seeing how much more energy you could reclaim. Hence, this curve is closest to enthalpy. This curve is also related to the Average Total Revenue curve, which is revenue divided by quantity. If you multiply the ATR curve by quantity, you get the total amount of revenue. Revenue is energy. So the one unit increase in capacity increases energy by MR there, but total energy reclaimed is the square carved out by ATR and Q (which is the same as the area under the MR curve.)
Average Total Cost, Marginal Cost Curves. All attempts to claim either revenue or energy inherently incur costs. The added energy gained from a compressor requires a turbine, which gives you a good return on investment, but it is an investment nonetheless. The additional costs incurred per unit capacity can be described as marginal cost, and the costs spread out through the capacity can be described as average total cost. Multiply that by capacity, and you get total cost. Cost has two big pieces: Variable and Fixed. Fixed is the easier of the two for us to describe. It is the overhead, the equipment costs. Whether or not your heat exchanger is exchanging heat, it is weighing something. It cost something to build. And it costs something to maintain. Variable Costs increase with the capacity on the heat exchanger. For our purposes, we will describe two different types of variable costs. The first, static variable costs, are increasing costs as the size of the heat exchanger increases. These are obviously fixed with the complete design, but they are variable during the design process, which will allow us to optimize the size of the heat exchanger. The second, dynamic variable costs, describe the losses inherent in a heat exchanger. In the transfer process, a good deal of the energy is lost. Dynamic variable costs, (like induced drag) are a function of energy reclamation. Dynamic variable costs must both be considered during the design process and be considered in subsequent analyses of the system. In this curve, we start to get at entropy, for an exchange for which cost exceeds revenue is an exchange with inaccessible revenue.
Profit. The area between the average total cost square and the average total revenue square is the profit. In energy terms, this is the reclaimed energy. An efficient heat exchanger maximizes this area. If there is no area to maximize (maximum profit is less than zero,) then there is no use in putting a heat exchanger between those two systems. And in this, we see entropy. This energy is inaccessible. Energy profit is the same as reclaimed losses.
Terms.
Reclaimable Energy is the sum total of energy that could theoretically be reclaimed between sub-systems. This term takes no account of recovery costs. Reclaimable energy is found by taking the sum of the absolute values of the sub-system energy flows (into and out of the total system, not counting internal transitions) and subtracting the sum of the sub-system energy flows. A loss-less system, which is impossible due to entropy, would have absolutely no difference between the sum and the absolute sum, as there would be total synchronization between the subsystems and the total system. Such a system would have no reclaimable energy.
Reclamation Cost is the cost to reclaim a given amount of energy. It is the total of fixed and variable costs for a given capacity heat transfer mechanism. It includes lifecycle costs, such as construction and maintenance, operations costs, such as weight, and thermodynamic costs, such as heat transfer losses.
Enthalpic Losses are lost energy that we can effectively reclaim with a heat exchanger. A first-order efficient system eliminates enthalpic losses, optimizing an existing design. The total amount of reclaimable energy minus recovery cost in all simultaneously possible profitable exchanges is the total amount of enthalpy losses. Specific enthalpic losses (referenced against total energy flow) is an overall figure of merit for first-order efficiency.
Entropic Losses are lost energy which would cost more to reclaim than the energy that would be gained. This energy, within a static design, is lost. But in a second-order efficient system, one that rethought its systems from the outset for thermodynamic efficiency. The difference between enthalpy losses and reclaimable energy is entropic losses. Specific entropic losses is a measure of second-order efficiency.
Recovered Energy is the difference between the reclaimable energy and the reclamation costs for a given combination of systems. This can be optimized by choosing the best combination of exchangers to maximize this term.
Reclaimable Energy Curve (Revenue.) We can construct a reclaimable energy curve for a given system by first deciding which two systems we are looking at. It makes sense, of course, to choose two systems with complimentary energy flows (one dumping, one creating heat.) Then, build the curve by determining how much energy will be transferred in a given set of conditions for a certain capacity system. Capacity is our independent variable, and energy is our dependent variable. Given that this curve is only for one possible interaction, it would do some good as an analytical tool to find the optimal points on a number of these curves and place them of a traditional supply/demand graph. This will allow the linkages between compliments and substitutes (substitutes would be multiple systems with complimentary flows. If waste heat has is reclaimed somewhere, it cannot be reclaimed somewhere else.) These can then be evaluated, and all the different individual graphs tied together to find a single optimized solution between all systems.
Reclamation Cost Curve (Costs.) There are three parts of reclamation costs. We can plot these on a traditional graph as Fixed costs and Variable Costs. Lifecycle costs and part of operational costs are fixed costs. Lifecycle costs include the cost of production, the purchase price, and the maintenance costs over the life of the unit. Fixed operational costs are the added weight and complexity that exists regardless of capacity, such as extra tubing and set parts of the heat exchanger. Variable costs include the remainder of operational costs, such as size of the heat exchanger, and the thermodynamic transfer losses.
Possible heat exchange mechanisms include:
Fluid <-> Fluid (Traditional Heat Exchanger)
Fluid <-> Electrical (Turbine/Fan)
Possible Pressure Differential Systems
Design Cycle. Once cost and revenue curves have been constructed, the designer can experiment with different combinations of individual transitions. With a bit of ingenuity, you can get a pretty good idea of where your energy is hiding and what you can get back. There is a mathematical way to do this, though, with Lagrangians, and using Supply/Demand graphs to plot out the possible interactions. If you can construct a web of total available energy, you can determine minimum reclamation cost, and then maximize total profit. The systems engineer should analyze these under different given conditions, for the different environments of the system (for an aircraft, ground, takeoff, cruise, landing. Or whatever.) You should give mind to surge capacity and emergency conditions. Note that this is a Systems Engineering methodology best applied as a finishing touch on an already nearly completed design. This methodology would do much to reduce enthalpy losses, but can do little to reduce entropic losses. This is first-order optimization, and achieves static efficiency for a design.
Second-order optimization, dynamic optimization, requires consideration of thermodynamic economies from the outset. This can modify the factors of production of the cost curve. By reducing the numbers of media used, or redesigning the system to be more unitary from the outset, much more energy can be reclaimed. A more unitary system reduces the number of necessary transitions, reducing losses from the outset. Reducing the number or types of media reduces the transition costs. Therefore, the tendency as systems become more and more thermodynamically efficient is for the systems to become more and more elegant, with less systems accomplishing more things synergistically. Sorry about the MBA word. One further offshoot of this is moving away from a large number of active pumps to more passive systems which regulate themselves by bypasses of available energy. These meta-stable systems, and their dynamic equilibria, more and more closely approximate biological designs. Which, of course, makes sense. About the best you can do is plagiarize God.
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