Cascades and stages

Picture a base whose waste heat has to go into a loop at around 150 °C – for example because it is already 127 °C outside, as at night on Vulcan. Radiators only give off heat when their loop is hotter than the surroundings. A single air conditioner would therefore have to pump the heat from a room loop at 18 °C all the way into a loop at 150 °C. The larger this difference between inlet and waste, the further its TemperatureDifferentialEfficiency drops – it works poorly.

Two stages instead of brute force#

The air conditioner has three pipe connections. Gas at the inlet (Input) is brought to the target temperature Setting and leaves the device at the outlet (Output). On the waste network the opposite happens. In a cascade two devices work in a row, each lifting the heat one step higher:

Each stage only bridges about half of the temperature jump and so stays in its good efficiency range. Two small jumps achieve more than one big one.

Setting below the inlet temperature#

For the air conditioner Setting is target and direction: if it is above the temperature at the inlet, it heats the inlet and cools the waste. For a cascade stage that is fatal. At a cold start the intermediate loop may only be 20 °C warm. If stage 2 is fixed at 80 °C, it heats the intermediate loop – with heat it takes from the hot waste loop. It pumps at full power in the wrong direction.

The rule: the Setting of a stage is always below the temperature of its inlet.

sub set2 tMid MARGIN      # at least MARGIN below the inlet
min set2 set2 GOAL2       # but never above the target

If the intermediate loop is 95 °C warm, the 80 °C target applies. If it is only 20 °C warm, the Setting becomes 18 °C – the stage can then only cool, never heat.

Adding a stage#

Stage 1 runs as soon as the room loop is above its target. Stage 2 does not have to run all the time: the intermediate loop first takes up the waste heat of stage 1, like a buffer. Only when it gets too warm – so stage 1 alone is no longer enough – does the chip add stage 2. With hysteresis so it does not flutter:

sgt tmp tMid GOAL2        # still above the target?
and on2 on2 tmp           # otherwise off
sgt tmp tMid ON2          # above the switch-on threshold?
or on2 on2 tmp            # then on

On NaN both sgt give 0 – if the analyzer is missing, stage 2 stays off. Still, you never write the Setting as NaN but replace it with the target using snan and select.