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I just bought a small stirling engine which converts heat flow down a temperature gradient into circular motion of a little wheel. If I turn the wheel with my finger, will the engine generate a temperature gradient?

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Yes - it's a very common way of cooling detectors without using consumable liquid cryogens. Stirling cycle used ot be common, but the similar Giffordd-Mcmahon cycle is now used for practical reasons.

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Yes, you can use it for heating and cooling, if you supply mechanical energy, see

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Any heat engine can be reversed to make a heat pump. Going further, you can use any engine to create any gradient (thermal gradient, chemical gradient). But essentially what you are doing by turning the wheel by your finger is providing energy as work to the system and asking it to convert work into a heat flow. Now if you ask how efficiently can I do it that is a more involved question.

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Yes, driving a Stirling engine results in one chamber getting hot and the other becoming cold. With each revolution the temperature difference is progressively "captured" in the regenerator and very low temperatures can be achieved. Have a look at this and this.

This second part of the Philips film shows liquid nitrogen being created.

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