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Assume the following setup:

Input -> First    ----------> Second    ----------> etc.
Air      heatpump   hot air   heatpump    hot air
T_0        |        T_h1>T_0     |        T_h2>T_h1
           v                     v
        cold air              cold air
        T_c0<T_0              T_c1<T_h1

or

enter image description here

Ts are temperatures; the darker the red, the warmer. The barriers between the areas are isolating materials. The throughput is smaller for nested pumps so that the most inner, hottest area is quite small and heated up slowly in comparison to the out ones. The heat pumps would operate with different fluids suited for the temperature differences. The output temperature (after etc.) is limited by the availability of the right fluid and the construction of a pump which can withstand the heat.

The only question to decide whether such a machine could operate efficiently, i.e. provide more energy than the pumps consume, is how much do the heat pumps consume? I'm searching for figures to estimate the consumption of thermal heat pumps (let's assume they're electric).

I'm pretty sure that it's not a perpetuum mobile of the second kind because the system is fed with the energy taken out of the cold air which leaves the first (left) heat pump.

The figures for the energy consumption would show whether such a machine can be constructed and provide (heat) energy rather than consume more than it provides. Example use cases would be geo-/climate-engineering and production of electricity or heat for heavy engineering.

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    $\begingroup$ Lots about heat pump thermodynamics is available on the interwebs, try searching for "coefficient of performance heat pump" and see what you get. $\endgroup$ – niels nielsen Jun 25 '18 at 20:57
  • $\begingroup$ This is a staged(multistage) system. They are very common. The idea is to get to the desired endpoint via the path of least expense. Sometimes you want outputs at different temps, and sometimes you have heatsources/heatsinks at different temps. $\endgroup$ – Phil Sweet Jun 25 '18 at 22:22
  • $\begingroup$ So your diagram is showing the mass flow (air in / air out), and the temperature flow (hot air here, cold air here), but you are not showing the energy flow. Where do you think energy is being "consumed" and where do you think energy is being "provided"? I think if you draw the energy flow explicitly, it will help. i.e. I'm about 90% sure that what you are thinking about violates the 1st law of thermodynamics, but I'm not 100% sure because you talk about "energy" in the text of the question, whereas it is not explicit in the diagram... $\endgroup$ – Daniel K Jun 27 '18 at 0:24
  • $\begingroup$ @DanielKiracofe Drawing the sketch made me realize that the heat pump need to be nested like shown in the new sketch rather than in a chain. The nested concept simply uses the energy flow of a heat pump and recursively applies it to itself for the nested pumps. The outer (left) pump is cooling it's environment which means an energy plus inside the machine which is then multiplied while the quantity of hotter air is reduced in each step. $\endgroup$ – Karl Richter Jun 27 '18 at 1:10
  • $\begingroup$ Yes but you are still missing something. Where is the energy flow? Heat does not equal energy. You can extract energy from hot air, it is true, but you will extract less energy from it than you used in heating the air up in the first place. i.e. there is no such thing as a free lunch. If your intent is just to create some hot air, you have met your goal. If you are trying to get energy out of that hot air, it won't work. $\endgroup$ – Daniel K Jun 27 '18 at 1:37
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Here is the diagram that I was trying to lead you to in the comments

       Energy In E1          Energy out E2
           |                     ^
           V                     |
Input -> First    ---------->  Heat    ----------> etc.
Air      heatpump   hot air    engine    warm air
T_0        |        T_h1>T_0     ^       T_c1<T_h2<T_h1
           v                     |
        cold air              cold air
        T_c0<T_0              T_c1<T_h1

First thing, is that heat pumps require external energy input to run. If you don't believe me, go look at your refrigerator or air conditioner. Both of these are heat pumps, and I guarantee that they are plugged into an external electricity source, and will cease to function if unplugged.

The second thing is that just generating hot air does not automatically get you energy. You need to extract the energy from that hot air using a Heat engine. This could take many forms. E.g. a stirling engine might work well here. Note also that the heat engine must have not only a source of hot air, but also a source of cold air too.

Now I know what you are thinking, that you can just use the energy E2 extracted from the heat engine in order to drive the heat pump. But the problem is that E2 will always be less than E1. You need to put more energy into this system than you will get out of it. Therefore, this system does not produce any net energy.

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Did you ever get a real answer to your question? I have been considering this question for years and I have no real answer. Many responses I see show me that people are a bit confused as to what you are trying to do. In a simplistic explanation you are trying to move thermal energy from a 70 degree environment into a say 75 degree environment which is very efficient but you cannot then get enough energy out of that 75-70 degree difference to recoup the energy you used to move the thermal energy. You need several hundred degree difference before any any heat engine can be efficient. So, the you do the same thing again. Move thermal energy from a 75 degree environment into an 80 degree environment. Then again and again and again until you have say a 400 degree environment and your original 70 degree environment from which you can then extract the thermal energy efficiently. You never need to use high pressures or any special equipment/techniques except to make sure what you are using can function at 400 degrees.

What is not efficient is going straight from the 70 degree environment to the 400 degree environment. This seems to be what most people think we are talking about. But, I have yet to see why moving energy from a 395 degree environment into a 400 degree environment is inefficient.

It helps me to look at the gas laws and think of it as joules rather than temperature. Put it into a spread sheet and see what happens. But, I am no physics and I don't know what I am doing wrong.

There are many mechanisms other than a traditional heat pump which could perform the desired function if it were efficient.

Another thing to consider is that you need a steep thermal gradient in order to get close to the 45% theoretical max extraction. So, why just go hot? Why not go hot and cold? I am switching to Celsius now but if it were 20 degrees outside why not go up 100 degrees and down 100 degrees. So, you have a 120 vs -80 which should be sufficient for say an ethanol filled heat engine. It should be easier and more efficient to not just move thermal energy from one environment to another but instead separate the thermal energy from the two environments.

Anyhow, did you ever get an answer?

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    $\begingroup$ I planned the different steps of tempature differences (which equal levels of nesting in setup) because I assume that cooling machines work on condensation and that the effect only works a certain difference around the temperature of condensation. The idea was based on the comparison of the energy one needs to be operating a steel pot with 1200°C in comparison to a few heat pumps. You probably can't win usable energy from it, but concentrated thermal energy - be that for a melting pot or a pizza oven - might work since it already does with a nesting level of 0 for heat pumps for buildings. $\endgroup$ – Karl Richter Dec 31 '20 at 18:15

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