Mathematical and Physical Journal
for High Schools
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# Problem P. 5018. (March 2018)

P. 5018. If fuel is burnt in a heat engine and a heat pump is operated with the heat engine then more heat can be transferred into the flat than in the case when the fuel is burnt in a stove. Let the flat be the low temperature heat reservoir of the heat engine and the high temperature heat reservoir of the heat pump. The cold temperature heat reservoir of the heat pump can be the air of the street. Suppose that the efficiency of the heat engine is $\displaystyle \eta_1$, and that the efficiency of the heat pump if it was operated as a heat engine would be $\displaystyle \eta_2$. Calculate the factor by which the heat transferred to the flat by means of the heat engine and the pump is greater than the heat transferred to the flat by burning the fuel in the stove.

(5 pont)

Deadline expired on April 10, 2018.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. A tüzelő elégetésekor felszabaduló $\displaystyle Q$ hőből $\displaystyle W=\eta_1 Q$ hasznos munka nyerhető, $\displaystyle (1-\eta_1)Q$ pedig az alsó hőtartályt, vagyis a lakást melegíti.

A hőszivattyú esetében $\displaystyle W$ munkával $\displaystyle Q'=W/\eta_2$ hőt lehet a felső hőtartályba ,,pumpálni'', hiszen fordított irányban (hőerőgépként) működtetve a berendezést $\displaystyle Q'$ leadott hőből $\displaystyle W=\eta_2Q'$ munkát lenne képes végezni.

A lakásba jutó hő összes hő:

$\displaystyle Q_\text{hasznos}=\left(1-\eta_1\right) Q+\frac{\eta_1}{\eta_2}Q=Q+\eta_1\left(\frac{1}{\eta_2}-1\right)Q,$

és ez valóban több, mint $\displaystyle Q$.

### Statistics:

 20 students sent a solution. 5 points: Absur Khan Siam, Bartók Imre, Elek Péter, Marozsák Tóbiás , Olosz Adél. 4 points: Csire Roland, Fekete Balázs Attila, Garamvölgyi István Attila, Kolontári Péter, Molnár Mátyás, Sal Dávid. 2 points: 9 students.

Problems in Physics of KöMaL, March 2018