A Carnot engine between the Earth's poles and equator

[Prathyush]

TL;DR Summary

I am interested in a feasibility analysis of running a Carnot engine between poles and equator.

Earth absorbs and releases a massive amount of energy as a part of the day night cycles.(~100 Peta watts of solar radiation received) And the temperature difference between the poles and equator drive weather phenomenon, which happens at an energy scale beyond technology.

I am wondering if one can run a Carnot engine between from the pole to the equator. I am thinking of it as a massive infrastructure project, capable of meeting immediate energy needs of the planet.( Capable of generating atleast 5-10 terawatts of sustained power)

Here is a plausibility analysis, with some design choices (but feel free to design it differently).

1. I am thinking of using water for the required thermal mass. Water is abundant and has a high specific heat.
2. Water is contained in 2 separate tubes, one for hot side and one for the cold side. The tubes run from poles to equator. The tubes are constructed using a non corrosive thermally conductive metal and surrounded by thick insulation.
3. On the hot side, I am imagining simple solar concentrators to keep the water above 90 degrees. On the cold side, I am imagining a simple heat exchanger keeping the temperature below 10 degrees. Solar concentrators and heat exchangers have a very long life span and recyclable. I chose the number 10 degrees and 90 degrees because it seems broad enough to be useful, and not so extreme that it would be difficult to maintain the temperature.
4. I will do my calculatation per Terawatt. 1 Terawattday = 8.64e+13 Joules. Using this we can get an estimate of the thermal mass of water required. 1 Terawatt day/(Specific heat of water*50 degrees) = 400 million Kgs of water. The temperature difference of 50 degrees is a crude choice. This is affected by efficiency and ability to maintain the temperature differences.
5. An ideal engine operating between 0C(273K) and 90C(363K) has an efficiency of about 25 pc. I am assuming in practice we can achieve an efficiency of 10 pc. So I will multiply the thermal mass by a factor of 10 and 4 billion Kgs of water. To be on the safe side we can estimate the amount of water required 10 billion Kgs/terawatt. But well within the realm of possibility.
6. It can be easily built in a modular fashion.

Some other similar design ideas,

1. Use different material for thermal mass, instead of water. One can consider some kind of molten salt, to keep the hot side at a very high temperature, but it becomes close to some designs of concentrated solar power plants. And it does not seem useful(except perhaps for efficient gains), and water is abundant.
2. Heating and cooling form a large part of residential energy use. It is possible to directly transport heat, But last mile delivery seems tricky.
3. Another way is to completely do away with water and just use a insulated conducting material to transport heat. But that looks very expensive.

I am wondering if such a design is feasible

 

[Vanadium 50]

Isn't it simpler and easier to take advantage of the natural thermalizion - wind and hydro?

 

[Drakkith]

I am wondering if such a design is feasible

Nothing is feasible when the scale is so large. Not yet at least.

 

[DaveE]

Building pipelines all the way to the equator to cool to 10C is a lot of effort when you could cool down to 40C or less at the equator. How would your system compare to in situ thermal-solar power plants?

 

[Prathyush]

Isn't it simpler and easier to take advantage of the natural thermalizion - wind and hydro?

Perhaps, but I wanted to evaluate it.

Building pipelines all the way to the equator to cool to 10C is a lot of effort when you could cool down to 40C or less at the equator. How would your system compare to in situ thermal-solar power plants?

Concentrated solar power is a very promising direction.

While writing this post I realized there is a fatal and unfixable flaw. I had overestimated heat flow capacity of metals by many orders of magnitude. If you want to generate terawatt scale power, then you would need to transport terawatt scale heat from equator to poles.

Copper conductivity is 386 W/m/K. If we have to transport 1 terawatt heat over 10000Km, we would require material of unimaginable conductivity.

 

[russ_watters]

While writing this post I realized there is a fatal and unfixable flaw. I had overestimated heat flow capacity of metals by many orders of magnitude. If you want to generate terawatt scale power, then you would need to transport terawatt scale heat from equator to poles.

Copper conductivity is 386 W/m/K. If we have to transport 1 terawatt heat over 10000Km, we would require material of unimaginable conductivity.

I don't understand what you are getting at there. I thought your working fluid was water? Why do you need thermally conductive pipes? Seems to me you need un-conductive/insulated pipes.

What I'd be concerned with for this idea are:

• Construction cost.
• Pumping energy requirements.
• Heat loss on the journey.

 

[Vanadium 50]

Well, also this takes all the energy out of the hydrological cycle. If you want weather, you will need to create it.

 

[Baluncore]

I would look at the cost of insulating a 10 Mm length of pipe.
Might you make a smaller version work between sea level and high altitude, on the steep windward face of a mountain range.

 

[Keith_McClary]

Might you make a smaller version work between sea level and high altitude, on the steep windward face of a mountain range.

In many places the air temperature is seasonally much greater or less then the ground temperature.

 

[Prathyush]

I don't understand what you are getting at there. I thought your working fluid was water? Why do you need thermally conductive pipes? Seems to me you need un-conductive/insulated pipes.

What I'd be concerned with for this idea are:

• Construction cost.
• Pumping energy requirements.
• Heat loss on the journey.

I wanted to use water as thermal mass, I did not intend to flow it, because it would be very expensive. I imagined the pipes had conductive layer followed by a insulating layer.

For a rough estimate for every KG of water pumped from equator to pole can store maximum of 4200KJ (100 degrees temperature difference). Which is a measly 1.2 KWh.

I had implicitly assumed conduction will be fast enough. But I realized I was many orders of magnitude away.

I don't think it is a fixable idea anymore.

 

[Prathyush]

Might you make a smaller version work between sea level and high altitude, on the steep windward face of a mountain range.

This paper for instance discusses the idea

https://www.researchgate.net/publication/337382923_Thermoelectric_Energy_Harvesting_From_Gradients_in_the_Earth_Surface

It uses seebeck effect

 

[artis]

@Prathyush Now I might be wrong but I think the main problem is not so much technical as it is physical. You see a thermodynamic system's efficiency and power output is directly dependent on the difference in temperature between the "hot side" and the "cold side" as well as the maximum temperature at the hot side. I think you might have heard that designers of steam generators and power plants always try to maximize the outlet maximum temperature.
A typical nuclear plant of the PWR style I think has about 300 to 350 celsius output @Astronuc surely knows more precisely.

In your proposed system the water doesn't even reach 100 degrees c which you get in your home kettle.
Also given the length of the pipe system , how much energy will be spent by pumping that water up and down? I think quite a bit.

Also the reason why the Earth seems to get that much energy from the sun in total numbers is because of the large area that the sunlight covers , the problem is that this energy is spread out and mild and not focused , for energy production it is always better to have a focused source of energy. This is the reason why solar panels take up large areas simply because even though the total energy received from sun is huge but such is also the area that it is received upon.

 

[Prathyush]

I would look at the cost of insulating a 10 Mm length of pipe.

I don't know about costs but you can make a rough estimate for heat loss over 10000 Km.

If you assume the pipes are 20 m in circumference. And you use a material like styrofoam and use 1 meter thick insulation.
You get an estimate of 6-7 Gigawatts of power lost per degree of temperature difference. Heat losses are significant.

 

[artis]

wow a 20m circumference for a pipe , that is more like a tunnel and less a pipe. I think you see it yourself @Prathyush this is unrealistic

 

[Prathyush]

@Prathyush this is unrealistic

Yes, I had said so myself. It cannot be made to work. I was just exploring the idea.

I was willing to sacrifice efficiency if most of the system can be made from passive elements, except the turbine but that is also not possible.

 

[Prathyush]

@Prathyush this is unrealistic

But I want to be clear it is unrealistic mainly because heat transfer from poles to equator is very expensive. You cannot do it passively, you need an active mechanism like motion.

 

[russ_watters]

I wanted to use water as thermal mass, I did not intend to flow it, because it would be very expensive. I imagined the pipes had conductive layer followed by a insulating layer.

That makes no sense. If relying on conductivity the water would have to be conductive. Otherwise, why even have the water? I don't know what you think "thermal mass" means, but that isn't it.

I don't think it is a fixable idea anymore.

Agreed.

 

[artis]

That makes no sense. If relying on conductivity the water would have to be conductive. Otherwise, why even have the water? I don't know what you think "thermal mass" means, but that isn't it.

I think I know what he was thinking, he thought that instead of moving the hot fluid from the hot side towards the cold side as is done in heat engines , you simply heat up the fluid/water and the heat travels down the fluid and reached the other end , sort of like when you heat a piece of metal at one end and after a while the other end heats up as well due to thermal conduction of the material.

Anyway @Prathyush this works for small distances only , but any efficient means of heat transfer especially over larger distances always involves moving the very medium/fluid itself instead of just relying on passive conduction.

 

[Prathyush]

That makes no sense. If relying on conductivity the water would have to be conductive. Otherwise, why even have the water? I don't know what you think "thermal mass" means, but that isn't it.

By thermal mass I meant something with very high specific heat capacity. Something that can store 1 terawatt day of energy.

I did not worry about conductivity issues, which was a mistake.

 

[hutchphd]

I think turbines in the Gulf Stream might make more sense.

 

[artis]

By thermal mass I meant something with very high specific heat capacity. Something that can story 1 terawatt day of energy.

I did not worry about conductivity issues, which was a mistake.

Well you should have worried about conductivity issues because just as I said in my previous post , passive heat conduction is only efficient in good thermal conductors , so then water is worse than metals so you would need not a pipe but rather a solid metal rod , but the other problem that I also said is that passive heat conduction is very inefficient over larger distance and is normally ever only used over small distance.
Like for example heat exchange ducts within a ventilation system or a heat sink radiator for electronics.

The moment you need to bring some thermal energy from place A to B which is even as far as a bedroom away you need a thermal fluid/agent which is a substance that absorbs the heat and is then physically carried from one place to the other. This is how air conditioning systems work for example. You have a fluid/gas and a pump that moves the fluid/gas.

In your case for your idea you would need lots of water pumps and lots of energy to pump that water

 

[Prathyush]

The moment you need to bring some thermal energy from place A to B which is even as far as a bedroom away you need a thermal fluid/agent which is a substance that absorbs the heat and is then physically carried from one place to the other. This is how air conditioning systems work for example. You have a fluid/gas and a pump that moves the fluid/gas.

Yes, that's basically it.

 

[russ_watters]

By thermal mass I meant something with very high specific heat capacity. Something that can store 1 terawatt day of energy.

You are not trying to store heat, you are are trying to transfer it. Why do you think you need to store heat? What, exactly, does the water do?

I realize you've given up on your idea, but it would still be beneficial if you could fix whatever misunderstanding this is.

 

[russ_watters]

I think I know what he was thinking, he thought that instead of moving the hot fluid from the hot side towards the cold side as is done in heat engines , you simply heat up the fluid/water and the heat travels down the fluid and reached the other end , sort of like when you heat a piece of metal at one end and after a while the other end heats up as well due to thermal conduction of the material.

If so, there's a jumble of unrelated concepts in there. Storage isn't transfer, Transfer through the metal isn't transfer through the water, etc.

 

[Prathyush]

I realize you've given up on your idea, but it would still be beneficial if you could fix whatever misunderstanding this is.

I understood the mistake I made, I said so in the 5th post of this discussion, I over estimated thermal conductivity by many orders of magnitude. Only when I sat down to calculate, I realized my mistake.

While writing this post I realized there is a fatal and unfixable flaw. I had overestimated heat flow capacity of metals by many orders of magnitude

Copper conductivity is 386 W/m/K. If we have to transport 1 terawatt heat over 10000Km, we would require material of unimaginable conductivity.

 

[russ_watters]

I understood the mistake I made, I said so in the 5th post of this discussion, I over estimated thermal conductivity by many orders of magnitude. Only when I sat down to calculate, I realized my mistake.

That's not the mistake I'm referring to. You seem to think using water to store energy here is needed for...something. What, you haven't said. Let's assume you find a piping material with a thousand times the conductivity. Now then, what is the water for?

 

[Prathyush]

That's not the mistake I'm referring to. You seem to think using water to store energy here is needed for...something. What, you haven't said. Let's assume you find a piping material with a thousand times the conductivity. Now then, what is the water for?

I was imagining a significant delay between when(and where) the heat was stored/released, which is why I had originally considered a large heat capacity to be important. I calculated my estimates based on the ability to store a day's power output.

Assuming Conductors can flow heat, keeping hot and cold reservoirs at a fixed temperature. The reservoirs should themselves need high thermal capacity right ? (The reservoirs are heated at the equator and cooled at the poles.)

I will rethink this also however.

 

[gmax137]

A typical nuclear plant of the PWR style I think has about 300 to 350 celsius output @Astronuc surely knows more precisely.

Yes a typical PWR reactor outlet temperature is about 620F (327 C). The pressure is 2250 psia (15.5 MPa). The steam generator pressure is typically 900 to 1000 psia (6.2 to 6.9 MPa).

 

[russ_watters]

I think I know what he was thinking, he thought that instead of moving the hot fluid from the hot side towards the cold side as is done in heat engines , you simply heat up the fluid/water and the heat travels down the fluid and reached the other end , sort of like when you heat a piece of metal at one end and after a while the other end heats up as well due to thermal conduction of the material.
[separate post]
Well you should have worried about conductivity issues because just as I said in my previous post , passive heat conduction is only efficient in good thermal conductors , so then water is worse than metals so you would need not a pipe but rather a solid metal rod...

Ok, yeah, he's making a mess of things mixing-up storage and transfer, but yeah if you wanted to use passive transfer, then you don't have the working fluid, you just have the rod. And depending on the type of heat engine (Stirling engine?) you may not need water at all.

I was imagining a significant delay between when(and where) the heat was stored/released, which is why I had originally considered a large heat capacity to be important. I calculated my estimates based on the ability to store a day's power output.

What delay? Are you talking about the propagation time of heat from one end to the other? Once up and running in steady-state, you ignore that. You certainly don't want to add to it by adding storage. You just use the energy as its collected.

Assuming Conductors can flow heat, keeping hot and cold reservoirs at a fixed temperature. The reservoirs should themselves need high thermal capacity right ? (The reservoirs are heated at the equator and cooled at the poles.)

No, the equatorial environment and polar ice cap are the reservoirs. You don't need to add your own.

Consider if you have a bucket of ice and a pot of boiling water and you wanted to put a Stirling engine between them. Two copper rods is all you need/want. A copper pipe full of stagnant water is a much worse way to conduct heat between them. Again: you want conduction, not storage. They are two very different things.

You may benefit from buying a cheap Stirling engine from Amazon/Ali Baba to experiment with.

 

[Prathyush]

What delay?

I was thinking about day night cycles for example, or changes power demand. You are right, I am mixing storage and conduction. Conduction is the problem not storage.

Again: you want conduction, not storage. They are two very different things.

I will reevaluate it once more.

 

[artis]

I was imagining a significant delay between when(and where) the heat was stored/released, which is why I had originally considered a large heat capacity to be important. I calculated my estimates based on the ability to store a day's power output.

Assuming Conductors can flow heat, keeping hot and cold reservoirs at a fixed temperature. The reservoirs should themselves need high thermal capacity right ? (The reservoirs are heated at the equator and cooled at the poles.)

I will rethink this also however.

Well @Prathyush here is the main difference I want you to understand. Think of a central heating system within a city. You have a power/heating plant , some pipes , water in them and then the radiators in the apartments and houses. You heat the water at the plant then use pumps to circulate it and then it gives off it's heat along the pipes and in the radiators. The heat lost in the pipes is considered loss.
This is considered active heat transfer using a physical medium (water in this case) to store and transport heat/energy.

A passive heat transfer system in its purest form is basically a metal rod. The problem is this. Even if the rod is of the best heat conducting metal , if made long enough it will not be efficient in fact it will not be able to make a heat engine work at all. Imagine having a 100 metre long rod, even if you heat one end up to red hot glowing temperature the other end will be room temperature, why? Because passive heat transfer means the heat is directed within the material evenly due to random motion of atoms/molecules , this process is not lossless or ideal. As you go along the length of the rod the temperature starts to decrease gradually and if the rod is long enough at some point you won't even be able to detect the heat supplied to the other side of the rod, why? Because the heat will have managed to dissipate away along the rod.
No matter how great your insulation is , it will still dissipate.

 

[Keith_McClary]

wow a 20m circumference for a pipe , that is more like a tunnel and less a pipe.

Saudi Arabia’s Saline Water Conversion Corporation (SWCC) has completed the longest water pipeline tunnel project in the world, with a diameter of 8.4 meters and a distance of 12.5 km.
https://idadesal.org/swcc-completes-worlds-longest-water-pipeline-tunnel-project/

 

[berkeman]

https://idadesal.org/swcc-completes-worlds-longest-water-pipeline-tunnel-project/

Wow. I wonder if they coat/line the interior of the tunnel with anything to lower loss of the water into the walls of the tunnel, or if they rely on it having been bored out of mostly solid rock...

 

[Prathyush]

As you go along the length of the rod the temperature starts to decrease gradually and if the rod is long enough at some point you won't even be able to detect the heat supplied to the other side of the rod, why? Because the heat will have managed to dissipate away along the rod.
No matter how great your insulation is , it will still dissipate.

If you want to transport 1 GW of heat across a temperature difference of 10C using a copper conductor assuming perfect insulation, over a distance of 1KM. You will need a surface area of 500m*500m.

If you lower the surface area to something modest like 1m*1m over a 1 Km same 10C, you will transport about 4 watts of heat.

and area of 1m*1m over 1 Km and 1000C temperature difference, you will transport 400 watts of heat.

This is assuming perfect insulation. Conduction is really bad over long distances, even with perfect insulation.

 

[sophiecentaur]

I think turbines in the Gulf Stream might make more sense.

I used to be in favour of water turbines, driven by tides (or anything) but stuff grows on stuff under the sea and soon kills the efficiency. Annual (or more often) cleaning of underwater moving parts would be a major item, as opposed to wind turbines which are subject to much less surface coverings.

Someone already mentioned the existing thermal to mechanical Energy transfer that the weather achieves for us and efficiency is much less relevant because we don't have to build that system.

 

[sophiecentaur]

I used to be in favour of water turbines,

As it happens, I heard a BBC News item that they have just installed a tidal turbine in the North of Scotland where the tides are fierce so somebody must think it's worth while. It's not very big though. I must look for a hard copy of the info.

 

[Tom.G]

I must look for a hard copy of the info.

(with video)
https://www.bbc.com/news/uk-scotland-north-east-orkney-shetland-57991351

And from April 2021:
https://www.energylivenews.com/2021...idal-turbine-ready-to-make-waves-in-scotland/

 

[NTL2009]

I was thinking there must be a much easier way to find a cool heat sink for the solar collectors to drive a Stirling engine. How about the colder ocean water? Instead of getting your 'cool' from 10,000km away, you could go just a single km down, and find a massive heat sink at just ~ 4~5 C.

https://www.amnh.org/learn-teach/curriculum-collections/deep-sea-vents/global-ocean-circulation-and-deep-sea-temperatures

In fact, no matter how warm it is up top, by the time the sub has sunk to a depth of just 1,000 meters (3,280 feet), the water temperature is about 40ºF. From there it continues to drop until, 7,200 feet down, where the smokers are, the temperature is only about 2º to 3ºC (35ºF)—just above freezing!

And I figured I was not the first to think of this.

https://en.wikipedia.org/wiki/Stirling_engine

Heat sink
The larger the temperature difference between the hot and cold sections of a Stirling engine, the greater the engine's efficiency. The heat sink is typically the environment the engine operates in, at ambient temperature. In the case of medium- to high-power engines, a radiator is required to transfer the heat from the engine to the ambient air. Marine engines have the advantage of using cool ambient sea, lake, or river water, which is typically cooler than ambient air.

 

[FinBurger]

Saudi Arabia’s Saline Water Conversion Corporation (SWCC) has completed the longest water pipeline tunnel project in the world, with a diameter of 8.4 meters and a distance of 12.5 km.
https://idadesal.org/swcc-completes-worlds-longest-water-pipeline-tunnel-project/

What about this? https://www.water-technology.net/projects/new-york-tunnel-3/
60 miles of tunnel to funnel clean water into New York City.

 

[bob012345]

It's just a shame there is no way to capture a little of the Earth's rotational energy other than tidal power. Or is there?

 

[Vanadium 50]

It's just a shame there is no way to capture a little of the Earth's rotational energy other than tidal power.

A. That's not what this thread is about.
B. Be careful what you wish for. The Earth's rotational kinetic energy is only enough to power civilization for about 13 years.

 

[NTL2009]

...
B. Be careful what you wish for. The Earth's rotational kinetic energy is only enough to power civilization for about 13 years.

That's fascinating. I would have guessed much more. And the phrase that popped into my head was "And where she stops, nobody knows!" :) The price of real estate at the horizons would go nuts, everywhere else would be freezing or hot as Hades!

 

[FinBurger]

A. That's not what this thread is about.
B. Be careful what you wish for. The Earth's rotational kinetic energy is only enough to power civilization for about 13 years.

Rotational energy of Earth = 2e29 J
Energy consumption of humanity = 1.2E20 J/yr
Rotational energy would last for >1e9 year. (give or take - that was a really loose calculation)

 

[sophiecentaur]

Rotational energy of Earth = 2e29 J
Energy consumption of humanity = 1.2E20 J/yr
Rotational energy would last for >1e9 year. (give or take - that was a really loose calculation)

I haven’t done any sums but @Vanadium ‘s figure is hard to credit. 13 years’ energy would make loads of daft projects possible and I’ve always felt safe from such nonsensical futures. 1e9 makes me feel much safer.

 

[bob012345]

I haven’t done any sums but @Vanadium ‘s figure is hard to credit. 13 years’ energy would make loads of daft projects possible and I’ve always felt safe from such nonsensical futures. 1e9 makes me feel much safer.

I don't wish to make you feel less safe but at the current rate of energy which is about $6 x 10^{20}J$, I figure we only have 369 million years.

 

[Vanadium 50]

Energy consumption of humanity = 1.2E20 J/yr

I stand corrected. My Googling gave me J/s and not J/year, so I am off by π x 10⁷.

 

[Prathyush]

I was thinking there must be a much easier way to find a cool heat sink for the solar collectors to drive a Stirling engine. How about the colder ocean water? Instead of getting your 'cool' from 10,000km away, you could go just a single km down, and find a massive heat sink at just ~ 4~5 C.

You are right, is that is (possibly) a valid sink. It has different challenges though. 1 KM deep you are talking about 100 atmospheres of pressure.

 

[hutchphd]

And the planet keeps it cold
https://www.ontrack-media.net/gateway/science8/g_s8m3l10s5.html

 

[NTL2009]

You are right, is that is (possibly) a valid sink. It has different challenges though. 1 KM deep you are talking about 100 atmospheres of pressure.

Yes, but I'm not sure (emphasis on "not sure") that's a huge problem. 1500 PSI is pretty routine for small diameter tubing (obviously, this would need to be large for any meaningful power), and Nitrogen cylinders are about 2200-2400 PSI, and in common use. But I'm not sure that you would have that pressure difference - I'm thinking the working fluid would also be that deep, and have the same pressure in and out? In the same way that we don't crush under the ~ 14.7 PSI atmosphere.

The challenge with manned vessels is that something close to normal atmosphere pressure must be maintained inside, so there is a pressure delta. And this manned vessel went down to 10,911 meters, way back in 1960.

https://en.wikipedia.org/wiki/Trieste_(bathyscaphe)

That doesn't mean it would be practical for a Stirling Engine, but I'm thinking that pressure issue may not be a barrier.

 

[Prathyush]

obviously, this would need to be large for any meaningful power

This is a rough estimate for quantity of fluid transported want to generate 1 GW of electricity.

Assuming a heat capacity of 4KJ per Kg per degree. We have heat capacity of 200KJ over 50 degrees.
If we generate electricity at 10 pc efficiency, we need to transport 10 GW of heat.
Which amounts to transport 50000 Kg of working fluid per second.
At a density of 1000Kg/m^3 we have to transport 50m^3/s of working fluid.
With a flow velocity of 10 m/s, we get a cross section area of 5m^2.I am not sure what kind of heat exchangers would be required to absorb/dissipate heat. But if the pressure difference is too high, we need thick heat exchangers, decreasing the rate of heat exchange.
But it gives a rough idea with regard to the scale/nature of the problem.