Problem — Fridge with the door open

Problem

A friend suggests cooling the kitchen in summer by leaving the refrigerator door open. What actually happens to the average temperature of the kitchen? Explain by reasoning on the balance of heat extracted from the compartment and heat discharged by the rear coil, taking into account that the compressor dissipates electrical energy. Which device would actually cool the room, and why?

Solution

Refrigerator balance. The refrigerator extracts heat $Q_C$ from the inner compartment and, via the compressor powered by electrical work $L_\text{el}$, discharges it from the rear coil. In a cycle $\Delta E_\text{int} = 0$, so the first law gives

$Q_H = Q_C + L_\text{el} > Q_C$

Kitchen balance. With the door open, both the compartment (which absorbs $Q_C$) and the coil (which emits $Q_H$) are inside the same room. The net heat put into the kitchen air is

$Q_\text{netto} = Q_H - Q_C = L_\text{el} > 0$

Conclusion. The kitchen heats up: the heat extracted from the compartment is put back into the room by the coil, and on top of that the electrical energy $L_\text{el}$ dissipated by the compressor is added. The open-door refrigerator is, in the overall balance, a small heater.

$\ev{Q_H > Q_C \Rightarrow \text{the kitchen heats up by } L_\text{el} \text{ each cycle}}$

What would actually cool it. An air conditioner, whose condenser (the hot coil) is outside the room: it expels $Q_H$ outdoors and absorbs $Q_C$ from indoors, with a net balance for the room of $-Q_C < 0$. Only by separating the hot source from the environment to be cooled can real cooling be achieved.

Links

Topics: Macchine termiche Concepts: Frigorifero e pompa di calore · Primo principio della termodinamica · Secondo principio della termodinamica Skills: Conservazione dell’energia Objects: Frigorifero