OK: A comparison with Dinorwig
Energy stored at Dinorwig: 3.2x10¹³J
Volume of Marchlyn Mawr: 6700000m³
Maximum power output: 1.8GW.
Imagine a liquid air storage facility built according to the description:
When it is windy, air gets pulled in through some gravel. The surface of the gravel is ambient temperature. As you get lower, the temperature falls. At some depth (the boundary) the temperature is -190°C. Below the boundary, the temperature remains at -190°C to the bottom of the gravel pit. This gives you a source of gaseous air almost at boiling point, but taking this air lowers the level of the boundary. When the boundary reaches the bottom of the gravel pit, you have reached the limit of its capacity. When it is not windy, you take air that was boiled in a turbine and push it into the bottom of the gravel pit. This raises the boundary layer and restores the system back to its initial state.
When it is windy, you use a heat pump to pump energy out of your cold air to liquefy it. The heat you pump out of the cold air goes into the atmosphere. The atmosphere is the place where the energy is actually stored, but because it is so big you will not be able to measure the increase in temperature. When it is not windy, you use heat from the atmosphere (or preferably waste heat from a power station of data centre) to boil the liquid air and drive a turbine which spins a generator.
Lets start by matching the maximum power output of Dinorwig: 1.6GW electrical. If the losses at Dinorwig are split equally between the pumps and the generators, that makes the generators 87% efficient, so to get 1.6GW electrical, we need 1.84GW mechanical. The maximum possible efficiency of converting heat to mechanical energy is: 1-(Tcold/Thot). Using -190°C and +25°C gives an efficiency of 72% so we need 2.55GW thermal from the atmosphere. There aren't any data centres that big. For that much heat, you need all the waste heat from a big nuclear reactor. If you do not have one handy, Thot will be less than 25°C, the efficiency falls and you need even more thermal energy from the atmosphere (and more liquid air) to get 1.84GW mechanical out of your turbine.
We just got 2.55GW of waste heat for free from our nuclear power station, but only put out 1.84GW mechanical from the turbine. The rest: 0.71GW boils liquid air. If you run out of liquid air, Tcold gets bigger, the efficiency falls and you need more nuclear power stations to get enough waste heat. Last time we calculated boiling liquid air requires 202J/kg, so we need 3500000 kg/second of liquid air. That is 4000m³ per second. Dinorwig can run full power for over 4.9 hours. The vacuum flask needed to store this liquid air is over ten times the volume of Marchlyn Mawr (the upper reservoir for Dinorwig).
The boiled air from our turbine is still cold, and we have to store that cold in a gravel pit ready for the recharge cycle. The cold air gets heated at constant pressure from -190°C to ambient (say 10°C) by the gravel. Wakapedia tells us the constant pressure specific heat capacity of air is 29.07J/mol/K (a mol of air is about 28.8g). I could not find the specific heat capacity of gravel, but it should be similar to that of glass: 840J/kg/K. The 3500000 kg/second of boiling air will cool 4060000 kg/second of glass. The density of glass is about 3700kg/m³, but we need space for the air to get through, so call it 2.7kg/m³. That is a mild 1500m³/second. To last as long as Dinorwig we need a thermally insulated gravel pit four times the volume of Marchlyn Mawr.
But we have not finished yet. We still have to do the recharge cycle. We need to take cold gaseous air and use electricity to pump heat out of it so it liquefies. Pumping heat from a cold place (-190°C) to a hot place (atmosphere say 10°C) requires at least (Thot-Tcold)/Tcold times as much mechanical energy as the thermal energy you want to pump. We only need 3x10¹³J of mechanical energy (yes less than the 3.2x10¹³J we are storing. We get the difference from the waste heat from our nuclear reactor). Assuming the electric motors are 87% efficient we need 3.45x10¹³J of electrical energy to refill the vacuum flask with liquid air.
To match the capacity of Dinorwig, a liquid air energy storage facility would need a volume 14 times the size of Dinorwig's upper reservoir and the waste heat from a large nuclear power station. The best possible efficiency is 92.5% (assuming the best theoretically possible heat pumps) compared to Dinorwig's 75% (with real world turbines). Although Dinorwig is big, it is only there to handle fluctuations in demand. Its peak power is nothing like the base load. Most winters, the wind is calm over the whole of Europe for 5 days - not 5 hours. If your energy plan is mostly windmills then you have to convince people that Scotland or Wales is mostly pumped storage - or you could cover England with vacuum flasks and gravel pits.