Rapid flow countercurrent heat exchanger

Efficiency considerations are crucial to the TrombePump or any heat engine operating on low ΔT heat. For example, operating on 30K ΔT with a 300K hot heat source (and therefore a 270K cold heat sink) means at most 10% of the heat accepted from the hot heat source can be converted to work by a heat engine operating on a closed cycle, while at least 90% must be rejected to the cold sink.

Consider a cycle using isothermal expansion, isobaric cooling and compression, isothermal compression, and isobaric heating and expansion of the working gas. The system includes a store of internal energy. For the cycle to close, the working gas must return to the initial pressure, volume, and temperature, and the internal energy store must contain exactly as much energy as it started with. The isobaric steps cancel out, absorbing or rejecting the same amount of heat and increasing or decreasing the internal energy by the same amount. The isothermal expansion converts all of the input heat to internal energy of the system, while the isothermal compression converts 90% of the internal energy to heat and rejects it. The unused 10% is output as useful work in order to return the system to the initial state.

In this cycle, the working gas repeatedly moves between two pressures and volumes. The expansion step absorbs enough heat to allow the gas to reach the final low pressure and high volume, storing the work done as internal energy, while the compression step uses enough internal energy to return the gas to the initial high pressure and low volume. The isobaric steps change the temperature and volume while leaving the pressure constant. The isobaric cooling and compression step uses internal energy to do work on the gas, while the isobaric heating and expansion step stores the work done by expanding gas as internal energy.

The above describes a piston engine. In a continuous cycle engine such as the TrombePump, some working gas is expanding while some is being compressed, some is warming and some is cooling. The isobaric steps exchange heat with each other but not with the external heat source and sink. In that case, ideally the system is in a steady state where the amount of gas in each condition is constant and the amount of heat in the heat exchanger is constant. Actually, the situation is more complicated, but results for the ideal case should be good enough.

When the processes are inefficient, the expansion step converts less than 100% of the input heat to internal energy, and the compression step uses more internal energy than the expected 90% of the input heat to return the working gas to the high pressure and low volume. Now when the system returns to its initial state, the working gas must be at the initial pressure, volume, and temperature, and the internal energy must return to its initial value. This last adjustment is made by letting the system do work if it has excess internal energy, or by doing work on the system if it has a deficit.

Well, it can easily happen that after recompressing the working gas, the internal energy of the system is lower than the starting point. This is known as operating at a loss. The system does not produce any work, but instead consumes work.

In the TrombePump system, losses can occur at many places. Archimerged is particularly worried about what happens to the kinetic energy of rapidly moving water when it flows out of the narrow trompe into the wide heat exchanger and decelerates. He now thinks the energy must be going to heat without doing any useful work, and therefore has concluded that a design change is necessary to eliminate this loss.

He also thinks he ought to calculate how much energy this would be, but is too lazy just now.

The new principle for TrombePump V0.2 will be that the water runs at constant speed all the way around the cycle, through tubes with a constant cross sectional area devoted to water. When air and water flow together, the cross section will be larger to allow for the air volume. Because it takes a long time for heat to flow, the countercurrent heat exchanger must be very long so that rapidly moving water will remain in the exchanger long enough to be warmed or cooled.

There will be no baffles inside the water channels of the heat exchanger. Recall that the heat exchanger is also a series of temperature reservoirs, and uses gravity feed heat pipes to prevent heat flow in undesired directions. The diameter of the tubes will need to be rather low so that water in the center will also exchange heat. It may be necessary to add a polymer to the water to reduce friction losses and achieve laminar flow. Also, the heat exchanger might be wide and thin instead of cylindrical, and the heat pipe might be integrated so that a single metal wall separates the water from the refrigerant.

… expect revisions to this post…

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