Tc does seem like a bad assumption. I tried instead assuming a constant difference between the intake and the cold output, and the result surprised me. (The rest of this comment assumes this model holds exactly, which it definitely doesn’t).
Let Tr be the temperature of the room (also intake temperature for a one-hose model). Then at equilibrium,
Tr=(Tc+Th)/2
Tr=((Tr−Δ)+Th)/2
2Tr=Tr+Th−Δ
Tr=Th−Δ
i.e. no loss in cooling power at all! (Energy efficiency and time to reach equilibrium would probably be much worse, though)
In the case of an underpowered (Δ=15) one-hose unit handling a heat wave (Th=100), you’d get Tr=85 and Tc=70—nice and cool in front of the unit but uncomfortably hot in the rest of the room, just as you observed. Adding a second hose would resolve this disparity in the wrong direction, making Tr=Tc=85. So if you disproportionately care about the area directly in front of the AC, adding the second hose could be actively harmful.
Tc does seem like a bad assumption. I tried instead assuming a constant difference between the intake and the cold output, and the result surprised me. (The rest of this comment assumes this model holds exactly, which it definitely doesn’t).
Let Tr be the temperature of the room (also intake temperature for a one-hose model). Then at equilibrium,
Tr=(Tc+Th)/2
Tr=((Tr−Δ)+Th)/2
2Tr=Tr+Th−Δ
Tr=Th−Δ
i.e. no loss in cooling power at all! (Energy efficiency and time to reach equilibrium would probably be much worse, though)
In the case of an underpowered (Δ=15) one-hose unit handling a heat wave (Th=100), you’d get Tr=85 and Tc=70—nice and cool in front of the unit but uncomfortably hot in the rest of the room, just as you observed. Adding a second hose would resolve this disparity in the wrong direction, making Tr=Tc=85. So if you disproportionately care about the area directly in front of the AC, adding the second hose could be actively harmful.