Consider a helium liquefier in which 1 mol s–1 of gas enters the Linde stage at Tin = 15K and at a pressure pin, 30 atm.
(a) Calculate the rate of liquefaction, in liter hr-1. Suppose (hat all the tiquefic.1 helium is withdrawn to cool an external experimental apparatus, releasing the boiled-o helium vapor into the atmosphere. Calculate the coaling load in watts sufficient to evaporate the helium at the rate it is liquefied Compare this with the cooling load obtainable if the liquefier is operated as a closed-cycle refrigerator by placing the apparatus into the liquid collection vessel of the liquefier, so that the still cold boiled-off-helium gas is retuned through the heat exchangers.
(b) Assume that the heat exchanger between compressor and expansion engine (Figure) is sufficiently ideal that the expanded return gas that leaves it with pressure pout is at essentially the same temperature Tc as the compressed gas entering it with pressure pc. Show that under ordinary liquefier operation the expansion engine must extract the work

Wg = H(Tc, pc) – H(Tin, Pin) - (1 – λ)[H(Tc, pout) – H(Tin, pout)] ≈ 5/2 R(Tc – Tin),

per mole of compressed gas. Here Tin, pin, pout, and λ have the same meaning as in the Linde cycle section of this chapter. Assume the expansion engine operates isentropicatly between the pressure-temperature pairs (Pc, Tc) and (Pin, Tin) from (12) and the given values of (pin, Tin) calculate (Pc, Tc).
(c) Estimate (lie minimum compressor power required to operate the liquefier, by assuming that the compression is isothermal from to pout it pc at temperature Tc = 50◦C. Combine the result with the Cooling loads calculated under
(a) Into a coefficient of refrigerator performance, for both modes of operation. Compare with the Carnot limit.