Non-ideal Carnot cycle based engine

Initial Stirling engine concept

For some time I was contemplating an idea for a Stirling engine that would work quite differently than the other designs that I came across. Whole internal volume of an engine may be viewed as a closed-loop pipe, trombone-like slide mechanism allows this pipe to change its length. Electrical fan replaces displacer and pumps working fluid throughout the engine constantly and in one direction. Some portion of the pipe is replaced with branching structure that consist of two heat exchangers (hot and cold) and regenerator. On both ends of this structure special valves are located that connect one of the structure’s parts to the main pipe, so that whole engine forms one closed-loop. While this happens, both of the unused parts are completely bypassed.

This design should have many advantages. Working gas is in constant motion and little energy is wasted on accelerating it. Gas may pass many times through heat exchangers, allowing for much better approximation of isothermal processes. Disconnecting unused heat exchangers from the rest of the engine should minimize volume of dead spaces.

There are some drawbacks to. Slide mechanism allows only very small changes of total volume (compression ratio is low), and that limits engine’s ability to operate with higher temperature differences. Piston, which is part of the slide mechanism, and especially valves, are heavy and this makes them unsuited for high speed operation, making the whole engine bulky. I also except that an area of all surfaces that are sliding against each other and produce friction would be higher than in traditional Stirlings.

Problems with Stirling cycle

Ideal Stirling cycle (Fig. 2) consist of two isothermal processes (constant temperature) during which work is performed, and two isochoric processes (constant volume, no work performed). Isochoric processes function is only to change temperature of the working gas, and this takes a lot of energy to do so (especially if gas with high heat capacity is used). Regular Stirling engines try to minimize this inefficiency with regenerator. But when I tried to analyze this process mathematically, I noticed that even regenerator with very high heat capacity cannot store more than half the energy needed to perform next isochoric process (see lower left part of the spreadsheets for more details).

Fig. 3 represents heat pump (or refrigerator) working in a modified Stirling cycle. Isochoric processes represent transfers of heat to and from the regenerator. Rest of the temperature change is performed by polytropic processes, during which heat flows to or from the heat exchangers and also work is performed. I chose to represent heat pump, instead of a engine because engine working with this modified cycle wold require isochoric processes to be performed midstroke (or transfers of heat to regenerator would have to be performed during polytropic processes, when working gas changes its temperature also due to work being performed).

Note that I am using word polytropic to indicate processes where gas changes its energy both due to heat transfers and work performed. When special cases of polytropic process are mentioned (isothermal, adiabatic, isochoric) I use their specific names.

Solving those problems with Carnot cycle

Legendary Carnot cycle consist of two isothermal processes (just like Stirling cycle), but changes of temperature are accomplished by the adiabatic processes (no heat transfers with the surroundings), instead of isochoric ones. This has huge advantage, because when energy is added to the working gas during adiabatic compression, all of it can be later recovered during adiabatic expansion.

Design I previously mentioned can easily be adapted to work as a Carnot engine. Only change that is necessary, is replacement of regenerator with the empty pipe (ADIABATIC PIPE in Fig. 4). When valves connect this empty pipe to the rest of the engine, there is no heat transfer between working fluid and thermal reservoirs. Any changes of engine volume modify temperature and pressure in an approximately adiabatic fashion.

To model what happens with the engine I used slightly modified Carnot cycle (Fig. 5). Just before adiabatic process end, short polytropic process begins. It is done so that loses associated with valves connecting both adiabatic pipe and heat exchanger to the main part of the engine for a short period of time can be better simulated. Heat pump working in this cycle is shown in Fig. 6.

Design details

Engine uses special valves, with an external shape resembling truncated cone. Smaller base of this cone faces main part of the engine, larger one heat exchangers. Inside there is a pipe, with one opening right in the center of smaller base. The other opening is located off-center on the larger base, so it can connect to the one of the heat exchangers or adiabatic pipe (which are situated just like chambers of a revolver’s cylinder in respect to each other).

As this engine engine will most likely operate under low RPMs, flywheel will have to be connected through a transmission with high gear ratio. The flywheel will have to be quite bulky as well. Of course configurations with multiple cylinders (what I’m describing here is basically just one of those) connected to one or more crankshafts should also be possible and they would definitively provide smoother engine operation.

Flywheel and crankshaft pose another challenge, as the pressure inside engine’s tubing will be higher than atmospheric, and at least space around the other side of the “piston” will have to be pressurized as well. If only the immediate surroundings will be pressurized, and gears, flywheel, electric motor/generator will be outside of the container, then very inefficient seal will have to be employed. If everything will be located inside the pressurized container, then fast moving parts will be working against dense gas. Pressure in this area ideally should have such a value, so that on every piston stroke energy is both added to the flywheel and extracted from it as to make energy storage requirements more manageable. This requirements may also be further reduced by employing some other energy storage form. And of course exchange of heat between main part of the engine and this pressurized section must be taken into consideration.

Then, there is problem of heat exchangers. They can be either large diameter pipes with fins, or smaller diameter ones that are densely packed (this seems to be most popular configuration in Stirling engines). As heat transfer with the outside of the engine takes around half of the piston stroke, it might be a good idea to employ two pumps of heat transfer fluid per each of the two heat exhangers. One would work constantly to transfer heat from some large heat reservoir, the other would transfer heat to the working fluid only when it is necessary. Another heat exchanger would have to be placed between those two circuits.

And there is also a question of the piston. Basic trombone-like version could be replaced by the something similar to the design drawn in Fig. 7, where two parallel tubes, connected by U-shaped piston, are replaced by the coaxial pipes with volute on the outer section that allows connection another parallel pipe. Piston itself has small tubular part attached, which allows working fluid to pass from outer section to the inner section. This design has only one high pressure seal, is possibly lighter, but fan that pumps working fluid will probably need to be more powerful. Problem of sideways motion of the piston must also be analyzed, and it can be either resolved with piston skirts or the crosshead. Interesting aspect of this problem is that while most frictional forces in the engine are mostly independent of the engine speed, forces associated with this sideways motion of the piston increase with engine rotational speed.

Another possible improvement would be making adiabatic pipe shorter than the heat exchangers, which would reduce volume of dead spaces.

Engine parameters

In case of a engine that uses helium as a working fluid, has minimum volume of 46.7 liters, maximum volume of 70 liters, operates at temperatures of -5°C and 37°C with the speed of 15 RPM you can expect that it will produce 1138W of power at 8.13% efficiency.

In the case of heat pump that operates at the same parameters you can expect that it will require 2300W of power and achieve COP 4.95

You can find calculations, diagrams and the detailed descriptions inside non-ideal_carnot_engine_1.03.zip  (mirror, SHA256: 040a5f21dbbf3898b26071355db6c46517ca406b954256afda236aee950a21e4)

Older version: non-ideal_carnot_engine_1.02.zip  (mirror, SHA256: 17f5354f623ed893e475feb43356db84098acea014557146a31a8984d3717261)

Possible applications

I started thinking about this concept when considering applications of thermal energy storage. In more northern (or southern) latitudes largest factor in domestic energy consumption is heating. So, at least in my opinion, any movement toward replacing fossil fuels with renewables should focus on this largest contributing factor. And this actually is quite fine because storing 1 kWh in a tank filled with water, rocks/concrete or simply in the ground is much cheaper than storing it inside lithium-ion batteries. And general idea behind this low temperature engine, was that some part of this large amount of energy stored to be later used to heat buildings, could be converted into electricity.

It should also be able to work in reverse as a heat pump or refrigerator.

NOTE: I originally posted description of this concept on Instructables’ forum/community section back in December of 2019. Unfortunately this section was deleted in 2021. To prepare for that I archived it, but later I found some mistakes in my original work (most notably wrong value of angular speed was calculated by FLYWHEEL CALCULATOR).