@joe-shmo
I don't like to beat a dead horse, but I've done some back of the envelope calculations to better explain my take on the engineering challenges associated with a system that utilizes only a pump, refrigerant, and the special material.
Its a closed loop that consist of a cooling coil ( in contact with the air of your home) and a condensing coil ( the special material radiating heat to space), and a pump to ensure uniform temperature of the system while the system is removing heat.
The pump is necessary to move heated fluid to the special material heat exchanger. If it was not there the system would need to develop a large temperature gradient across the exchangers for heat to flow by free convection. Meaning, the coil that is supposed to be actively cooling your house, would heat up to a temperature close to that of room temperature before very much heat exchanging occurred with space. Once it does that the rate of heat it can draw from the room will be very small. So it would very slowly cool your room, so do you wish your room to be cool within 15 minutes or sometime next week?
Perhaps there is some argument that can be made for climates that are hot 24/7 365 days a year that this unit just cools the home endlessly once installed. But I don't really know of such a climate, and I believe most users would want to turn the thing on or off with changing temperatures.
All of that above does not really turn out to be the big issue as far as I can tell. The problem with system ( pump or no pump ) is the steady state temperature of the working fluid. The mode of heat transfer of the cooling coil ( inside the home ) is predominantly free/forced convection ( free or forced depending on if you wish to add an intake fan to the design )
Q_in = h*A_in*(T_in - T_f)
"h" is the convection coefficient and from personal experience free air passing over a coil "h" at best will be 100 W/(m^2*K)
"A_in" is the surface area of the cooling coil (interior to the home)
"T_in" is the temperature of the space we wish to maintain ( I chose 20 C for the analysis )
"T_f" is the working fluid temperature.
The special materials mode of heat transfer is radiation to space, hence
Q_out = ε*σ*A_out*(T_f)^4
"ε" is the emissivity of the radiating body. For a Black Body ( perfect emitter over all wavelengths ε = 1) A physicist might correct me on this, but I would think this material specifically designed to emit at certain wavelengths to take advantage of the "space window" will have an emissivity much lower than 1. for the analysis I chose ε = 0.5
"σ" is Boltzmann's constant
"A_out" is the radiating materials surface area
"T_f" is again the temperature of the working fluid.
From there it is setting up an energy rate balance for the mass of working fluid Assuming all the heat is "sensible", meaning it goes into changing the temperature of the fluid and not its phase. The steady state solution for the working fluid temperate is:
ε*σ*A_out*(T_f)^4 - h*A_in*(T_in - T_f) = 0
Next, I had to decide what a pretty unreasonable amount of special material would be to install up on roof in an apartment building ( seeing as how this roof must be shared by all tenants with there own material for there own systems). Also, the cost of this amount of material I expect to be very high. I concluded that it would be A_out = 15 m^2 of material. Then I chose a cooing coil surface area, and plotted a solution for "T_f". Once I had a solution I recalculated the power being drawn from the home from
Q_in = h*A_in*(T_in - T_f)
Then I manually ( by iteration of the solution) just maximized that power with respect to the area of the cooling coil. With my values the power output of the system maximized around 3.1 kW, when the cooing coil surface area was 10 m^2. You may think, look its producing 3 kW, we did it!
Then I went over to my "window unit" - the whole system fitting entirely within the frame of my window ( which is also 3 kW ) and estimated the surface area of its cooling coil to be 0.27 m^2.
And that is the kicker. The system would need around 40 of my standard window units coils strewn about the house , and a half of a single family home roof space to produce the same cooling effect as my single window AC unit that cost me $100, and $40 a month to operate.
Please feel free to argue any points, correct the analysis, etc...