Thread: Energy in a temperature gradient

I'm trying to convert a given temperature gradient over a given area to watts (maximum). I tried using the Stefan-Boltzmann law combined with the Carnot Efficiency, but I think that's giving too big of numbers. I can't find any other equations though.

It's not clear exactly what you are doing. Mentioning Stefan-Boltzmann means you are doing a radiant heat transfer calculation. You shouldn't need the Carnot efficiency for that particular calculation. Mentioning a temperature gradient suggests conduction or convection rather than radiation. Can you provide a little more information?

I'm using the Stefan-Boltzmann law because I can't find anything else. I've seen some mention that conduction is linear in temperature, but I can't find any of the constants associated with that.

What I'm trying to do is get the maximum theoretical wattage extractable from a (very large) hot resevoir at a given temperature and a cold reservoir at a given temperature. Also, I know the surface area involved.

You are trying to calculate a rate of energy transfer (since wattage is a rate unit). If you know the temperature difference and the area the only other variable you need is the overall heat transfer coefficient. This can involve conduction, convection and/or radiation depending on the actual geometry of the setup, so it depends on what medium or media are between the source and the sink. You cannot calculate the rate based on just the properties of the source and sink because the rate will vary according whether the source and sink are connected by (say) a thin plate of silver, or a vacuum, or a flowing fluid and so on.

It will also depend on the nature of the source and sink, since as soon as you start removing heat from a source it will cool down a bit locally and you will start a temperature gradient inside the source. This part is sometimes ignored or assumptions are made to account for it.

I think the most reasonable assumption would be conduction with water or steam as a transfer mechanism, and that the source and sink are rock with essentially infinite heat capacity (so ignoring local effects).

Is there a table with the conduction coefficients for various materials anywhere?

Edit: Is this the right direction? http://en.wikipedia.org/wiki/Thermal_conductivity

Edit again: So averaging some rocks and soil gives a conductivity of around 2, and that'd be multiplied by the difference in temperatures, times the area, divided by the distance between them, and then multiplied by the Carnot efficiency? Err, I'm not sure how the properties of the device would enter the equation though (though since I just want to find the maximum possible, I can assume it'd be as efficient as possible).

Originally Posted by MagiMaster

So averaging some rocks and soil gives a conductivity of around 2, and that'd be multiplied by the difference in temperatures, times the area, divided by the distance between them, and then multiplied by the Carnot efficiency?

I think you need to provide a more complete description of the system you are trying to analyze. Conductivity really only stands alone if you are dealing exclusively with solids. If there is water or steam as you mentioned then convection will play a role. I still don't see where Carnot comes into it with the simple (but vague) model you have so far described.

Err, I'm not sure how the properties of the device would enter the equation though (though since I just want to find the maximum possible, I can assume it'd be as efficient as possible).

Device? What is the device? You've mentioned rocks, soil, water and steam. Is this some kind of green engine?

Originally Posted by MagiMaster

I'm trying to convert a given temperature gradient over a given area to watts (maximum). I tried using the Stefan-Boltzmann law combined with the Carnot Efficiency, but I think that's giving too big of numbers. I can't find any other equations though.

Bunbury is right. You need to provide a lot more definition of the problem.

For instance, the radiated energy is dependent on not only the temperature differences between two bodies, but on the absolute temperature itself. Knowing a gradient is insufficient.

The Stefan-Boltzman law, for instance, shows that radiated power is proportional to the fourth power of the absolute temperature. That has nothing to do with a gradient.

If you are looking for equations, and some accompanying explanation, get a good book on heat transfer.

Well, I don't exactly have more details, since I'm basically making it up as I go. I'm trying to calculate what the maximum wattage is that could be extracted from a very vaguely defined scenario.

I know that I have two natural surfaces, one hot and one cold. I know the area and temperature of the surfaces and, if needed, the distance between them. I want to put some imaginary heat-engine-type device between the two surfaces and then see what the maximum wattage it could produce would be.

If you're particularly interested, the imaginary device is something like a plant, only running on thermosynthesis(?) instead of photosynthesis. Since nothing like that actually exists, I can't really give many details. I'm trying to equate the two through wattage (first, at least), since it's something I can look up for sunlight.

Also, I'm not too interested in the detailed explaination, just this one equation. If there's no other way, I can try to find a good book, but I figured it'd be possible to get an answer without resorting to that.

You are confusing work and power. If you are starting with an infinite mass of hot rock and an infinite mass of cold rock you can theoretically convert that thermal energy into an infinite amount of work. But calculating the rate of conversion of thermal energy to mechanical (or chemical) energy, which is what you are after since you are talking in terms of watts, depends absolutely on what lies between the source and the sink. You question is unanswerable without details. If it’s something like photosynthesis then it depends on the reaction kinetics and a lot of other physical factors such as the size of the reactor, the operating temperature, the rate of waste removal, catalyst effectiveness and so on.

There is not one simple equation that can answer your question.

Well, I don't expect to get an exact answer, but isn't there a theoretical maximum? If there's no maximum, what would be some reasonable assumptions that would allow such a maximum to be calculated.

(Also, I don't think I mentioned work anywhere. As you say, I'm looking for rate.)

Originally Posted by MagiMaster

Well, I don't expect to get an exact answer, but isn't there a theoretical maximum? If there's no maximum, what would be some reasonable assumptions that would allow such a maximum to be calculated.

(Also, I don't think I mentioned work anywhere. As you say, I'm looking for rate.)

The rate equation is Q = U x A x DT (that D is a delta - I've forgotten how to do Greek symbols on here). Q is the heat transferred. A is the area which you say you know and DT (delta T) is the temperature difference which you also know. There is only one unknown: U, which is the overall heat transfer coefficient. U is the inverse of the sum of all the resistances in series between the source and the sink. So to get U you have to estimate and then combine the resistances due to all the intevening materials, solid, liquid, gas. These resistances individually are the inverses of the heat transfer coefficients which you can calculate from various semi-empirical methods if you have enough information to begin with. The calculations will involve conduction and convection, possibly boiling, and if the temperatures are high enough, thermal radiation.

You cannot do these calculations in a generic sort of way. You have to define the geometry of your system and model that into all the calculations, so while the above equation looks simple, it is U that will stymie you unless you define some geometric parameters of your system to at least make a first stab at it.

Hmm. I'm sticking with water-carbon life for the moment, and I can assume that evolution would work to maximize U. I suspect that the lifeforms in question would end up looking like cylinders stretching from the hot side to the cold side. I can't imagine any reason for increasing their surface area, just the surface in contact with the sides. A decreased overall surface area might help prevent losses, which would make them look like a soap bubble stretched between two circles of wire.

Did you mean that U is or ? I can't quite tell from the way you phrased that.

I think that starting with the U for a column of water would give a likely upper bound. I don't think that water-carbon life will be able to exceed that without creating internal metal structures, which doesn't seem to happen anywhere on Earth. Does that sound reasonable?

Originally Posted by MagiMaster

Hmm. I'm sticking with water-carbon life for the moment, and I can assume that evolution would work to maximize U. I suspect that the lifeforms in question would end up looking like cylinders stretching from the hot side to the cold side. I can't imagine any reason for increasing their surface area, just the surface in contact with the sides. A decreased overall surface area might help prevent losses, which would make them look like a soap bubble stretched between two circles of wire.

Did you mean that U is or ? I can't quite tell from the way you phrased that.

I think that starting with the U for a column of water would give a likely upper bound. I don't think that water-carbon life will be able to exceed that without creating internal metal structures, which doesn't seem to happen anywhere on Earth. Does that sound reasonable?

Bunbury is right.

Since you have now phrased the question in terms of power the issue is the rate of heat transfer, from which you can infer power if you postulate some sort of conversion efficiency. You might even simply postulate that all of the heat is converted into useful eneregy, although that is not achievable in practice (damn that thermodynamics).

But the rate of heat transfer is dependent on the method of heat transfer and the materials involved. If you assume that the bodies are connected by something with infinite heat conductivity then the heat transfer will be instantaneous and the associated power will be infinite.

This has nothing whatever to do with any supposed biological processes, which open another can of worms.

Your proposed problem is so open-ended and fantastic that you might as well ditch the science and just concentrate on the fiction.

Well, I wanted what amounts to hard sci-fi, so I'd prefer not to ditch it altogether.

I think a cylindrical column of either water or stone (probably water) may be a good starting place. (Although looking at the conductivity tables, carbon structures can get a whole lot higher than water.)

Water has a conductivity of 0.6 (according to Wiki), so the rate for a column of water would be 0.6*(area)/(length)*(hot - cold) watts, right? And the maximum work a heat engine could extract from that transfer would be that times the Carnot efficiency?

I guess water responds pretty slowly, so maybe that can be a decent lower bound. (Impure) diamond would be about 1500 times faster (and graphene/graphite even higher), so I might have to just guestimate it to be somewhere between the two. (The geometric mean of the two would be about 23, so I'd probably round to 25.)

There would be convection currents in water, which would probably be a lot more than the conduction, unless you are in zero gravity.

Err... Would that just raise the effective coefficient in that equation, or change it completely?

You'll notice that thermal conductivity has units of watts/M.degC so the equation as you wrote it is dimensionally inconsistent. You need another length term, which is the thickness of the conducting medium. As Harold says, though, if you're looking at water there will be convection. And the heat transfer coefficient depends on the geometry. At risk of being too repetitive, you have no hope of getting anywhere until you define the physical shape, size and materials you are using.

Sorry, but this is pointless. If you'd tell us exactly what it is you're trying to do we might be able to help.

(conductivity W/m/K) * (area m^2) / (distance or thickness m) * (hot-cold K) is inconsistent? Efficiency is dimensionless, so that doesn't change much.

And I did tell you exactly what I'm trying to do. I'm imagining a scenario where there is a very stable (millions of years) temperature gradient and there are carbon-water lifeforms that have evolved to use that as an energy source. Since such things don't actually exist, I really don't have a lot of concrete details. I can give an area they cover, and I can make up numbers for how far apart the walls are, and I can guess that the conductivity of these things will fall somewhere betwen that of water and carbon, and I can guess that they'll probably be roughly cylindrical (or maybe soap bubble shaped), and I reason that evolution will work like normal. But I've stated all that already. I don't know what else to say.

This scenario is sci-fi, but I want to keep it hard sci-fi, so I'm not going to just give up and make everything up. That said, if I can get an equation that gives results within a factor of 2 or 3, that's probably good enough. Is the above equation good enough then?

Sorry, I missed the /(length) so your units are not inconsistent. My error.

But still, let's look at your description.

I'm imagining a scenario where there is a very stable (millions of years) temperature gradient and there are carbon-water lifeforms that have evolved to use that as an energy source. Since such things don't actually exist, I really don't have a lot of concrete details.

The stable temperature gradient could be, say, the difference between the temperature of a star, and the temperature of space. The organisms could use this differential in the form of radiated energy to produce photosynthesis on the surface of a nearby planet, and reradiating excess heat back to the cold of space. The photosynthesis could produce food for many other life forms and as long as you didn't allow the balance of energy in versus energy out to be upset by changing the absorptive properties of the atmosphere too much it could be quite stable.

You could call this place Earth.

Ok, I think I've given enough details to narrow it down more than that, but just in case, here are a few more assumptions:
- Heat transfer from the hot side to the organism and from the organism to the cold side is by conduction. (This being the part where no such thing exists AFAIK. This is also the only assumption I want to make that's really sci-fi.)
- The gradient exists in between two surfaces. For the moment I'm assuming both surfaces are rock/soil/something similar.
- There is some fluid filling the space between the two surfaces. Very likely, this is either water or air.
- The organism is made of materials similar to Earth-life, so it probably won't have significant metal components.
- Evolution would tend to produce geometries that maximized heat transfer rate. I don't think the fluid around it will be significant compared to conduction between the sides, so I'm imagining it would end up as either a cylinder or a soap-bubble-surface-hyperbola-shape (whatever that's properly called). Feel free to prove me wrong by showing how more energy could be extracted by using the fluid around the creature too.
- For the moment, assume the organisms have plenty of water and other chemicals they need to grow. The only question I'm trying to answer is how energetic the gradient is compared to sunlight. (Not very, I know, but I need numbers.)

Quick question. If the hot surface has a conduction coefficient of U1, the organism as a whole has U2, and the cold surface U3, how can I calculate the conduction coefficient of the whole system. The two surfaces don't have any depth that I can make sense of.

So I'm envisaging a pool of water with a hot rock face on one side. like a vertical cliff, say, and a cold rock face on the other side. We don't know how big the pool is - width or depth - but let's say it's one metre wide between rock faces and ten metres deep and for all practical purposes it's infinitely long. The organisms are floating in this water.

The first thing that will happen is you'll get circulation due to convection. Water will rise up the hot face and fall down the cold face. The heat transfer from and to the rock will be much higher than by conduction alone. The hot rock surface will cool somewhat and the cold rock surface will warm and an equilibrium will be found. The organisms will circulate with the water, unless they're anchored to something. If they are anchored they can use the velocity of water flowing past them to increase the film coefficient to their own bodies, so perhaps they'd evolve suckers to anchor themselves. Perhaps they'd anchor themselves to the hot rock at one end and the cold rock at the other end and generate electricity like a thermocouple; or they could evolve rotating appendages, like turbines, to generate electric power. There are some microorgansims that have rotating appendages, which creationists use to "prove" intelligent design. Just rambling on here. Am I finally on the right track?

Yeah. That's very similar to what I was imagining, although I had the thing turned sideways in my head (an air pocket/cave in the rock between an ocean and a magma pocket for example).

I didn't consider free-floating things making use of the temperature, but now that you mention it, I can't see any reason to exclude them.

Originally Posted by MagiMaster

Yeah. That's very similar to what I was imagining, although I had the thing turned sideways in my head (an air pocket/cave in the rock between an ocean and a magma pocket for example).

I didn't consider free-floating things making use of the temperature, but now that you mention it, I can't see any reason to exclude them.

Might I suggest continuing this thread in the sci-fi forum?

Well, I'm trying to ask a very specific physics question. If that'd be better answered in the sci-fi section, then I wouldn't mind.

Speaking of specific physics questions, I can't figure out how you'd calculate the thermal conductivity of a stack of alternating layers of water and graphite as the number of layers increased to infinitiy, assuming the total height of the stack is a constant. (Actually, I guess I don't even know how to calculate the conductivity of a graphite-water-graphite sandwich of a given thickness, even ignoring convection.)

Resistances in series - same as in an electric circuit. Add up the reciprocals of the conductivity of each layer and take the reciprocal of the sum (ignoring convection, which is not necessarily a good idea).

So the resistance in this case would be the conductivity times the thickness?

So, assuming graphite has about the same conductivity as graphene (Wiki doesn't list graphite), a 1 meter thick slab with a 1/2 meter layer of water and 1/2 meter layer of graphite would be:
- 1/2 * 0.6 = 0.3, and 1/2 * 4000 = 2000
- 1/(1/0.3 + 1/2000) = 0.299955

For four layers at 1/4, it'd be:
- 1/4 * 0.6 = 0.15, 1/4 * 4000 = 1000
- 1/(1/0.15 + 1/1000 + 1/0.15 + 1/1000) = 1/(2/0.15 + 2/1000) = 0.0374944

If I'm doing that right, that very quickly goes to 0. It doesn't seem right that alternating layers of different materials really block heat that effectively, so I figure I must be doing something wrong.

Originally Posted by MagiMaster

So the resistance in this case would be the conductivity times the thickness?

Does that seem reasonable? Think about it - is resistance going to increase as conductivity increases? I'm about done with this. Q & A on here isn't the way to get educated. Start by googling heat conduction.

Well, no. As I said in my last sentence, it doesn't seem reasonable.

Rereading the part about conductance and resistance on Wikipedia, it looks like I had the right idea, but I need to use area/length instead of length. Doing that gives me a constant regardless of how many layers there are, which makes a lot more sense. For half water, half graphite, I get a final conductivity of nearly 1.2.

Another question. (One I've tried to look up, unsuccessfully.) What is the Carnot efficiency supposed to be applied to?

(And asking questions isn't the way to learn things? That doesn't make much sense, especially since part of the reason I asked in the first place was because I didn't know what terms to look for on Google or Wikipedia.)