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The Hampson–Linde cycle is based on the Joule-Thomson effect and is used in the liquefaction of gases, especially for air separation. William Hampson and Carl von Linde independently filed for patent of the cycle in 1895.[1]

Similar to the Siemens cycle, this technique utilizes the temperature changes induced by compression and decompression of a gas. Where the Siemens cycle has the gas do external work to reduce its temperature, the Hampson-Linde cycle relies solely on work done against its own intermolecular forces. This is less efficient (any external work done would be an improvement) and particularly so at higher temperatures but has the advantage that the cold side needs no moving parts.[1]

The Hampson-Linde systems introduced regenerative cooling, in effect a self cascading heat exchanger arrangement that permits an absolute temperature difference (e.g. 0.27C/atm J-T cooling for air) to go beyond a single stage of cooling, and reach the low temperatures required to liquefy "fixed" gasses.
Siemens cycle sketch

In the Siemens cycle the gas is:

1. Heated - by compressing the gas - adding external energy into the gas, to give it what is needed for running through the cycle
2. Cooled - by immersing the gas in a cooler environment, losing some of its heat (and energy)
3. Cooled through heat exchanger with returning gas from next (and last stage)
4. Cooled further by expanding the gas, removing heat (and energy)

The gas which is now at its coolest in the current cycle, is recycled and sent back to be -

5. Heated - when participating as the coolant for stage 3, and then
6. resent to stage one, to start the next cycle, and be slightly reheated by compression.

In each cycle the net cooling is more than the heat added at the beginning of the cycle. As the gas passes more cycles and becomes cooler, reaching lower temperatures at the expanding cylinder (stage 4 of the Siemens cycle) becomes more difficult.
Hampson-Linde cycle sketch, different from the Siemens cycle only in stage 4

In the Hampson-Linde cycle the expansion of the gas in stage 4 is replaced with a Joule-Thomson orifice.

4. Cooled further by passing the gas through a Joule-Thomson orifice, removing heat, but conserving energy which is now potential energy rather than kinetic energy.

The rest of the cycle is the same: The gas which is now at its coolest in the current cycle, is recycled and sent back to be -

5 Heated - when participating as the coolant for stage 3, but this time retaining some of the potential energy, gained from the kinetic energy when exiting the capillary pipes of the Joule-Thomson orifice. the slightly hotter gas is finally
6. sent again to the beginning of the next cycle, to be slightly reheated by compression.