Question

You are watching TV late at night and see an ad for the ACME Model #163 refrigeration and heating unit for only $99.99. The unit takes compressed air at 1.25 atm pressure and 75°F and produces two streams at atmospheric pressure, a refrigeration stream at -15°F and one at 90°F with no work or energy put into the system! Is this possible? [yes] Why or why not? [mass and energy balance work ] Prove your answer. Assume air is an ideal gas

Air Date: Tc = 238.5 R Pc = 37.2 atm Cp = 7 Btu/lbmole°F Sp. Gr. = 1.0

Answers are in brackets but I want to know why because I am a diligent student. lol.

Answer 1

Basis: 100 kg/s inlet stream at P = 1.25 atm (1.267 x 10⁵ Pa) and T = 75 F ( 297K)

Density = PM/RT = 1.267 x 10⁵ x (28.8 x 10^-3 )/(8.314 x 297) = 1.477 kg/m³

Let, flow rates of the refrigerated stream be r and the other stream be y.

From mass balance, 100 = r + y

From energy balance,

m1 x Cp x (T1 - Tr) = r x Cp x (T2 - Tr) + y x Cp x (T3 - Tr)

where, Tr is some reference temp. Also, for an ideal gas, Cp is independent of temp. Also, since mass is balanced, the effect of Tr is cancelled algebraically.

We would take all temp in Celcius. Thus,

75 F = 23.9 C; -15F = -26.1 C; 90F = 32.2 C

Thus, 100 x C x 23.9 = r x C x (-26.1) + y x C x 32.22

Solving the two equations, we get

r = 14.266 kg/s and y = 85.74 kg/s

We see that there exists a solution and hence it is theoritically possible. Proved.

(The values given in the question possibly require you to use in molar rates, but as you just saw, doing it in mass units is much simpler, also because the heat capacity 'C' is anyways cancelled out).

Hope it helped.