The data of a steam engine are as follows: the internal diameter of the steam cylinder is $2 r=60.96 \mathrm{~cm}$, the stroke of the piston is $l=120.00 \mathrm{~cm}$, and the steam pressure is $p=10 \mathrm{~atm}$. The engine is high-pressure and has no condenser, so the exhaust steam puffs out into the atmosphere. If the engine were to operate without expansion, the piston would complete its stroke (back and forth) $n=40$ times per minute. However, the engine works with expansion, specifically, the valve closes the steam when the piston has traveled $\frac{1}{3}$ of its stroke. In this case, the piston's movement is somewhat slower, completing $n^{\prime}=32$ strokes per minute. Furthermore, assume that during the expansion, the pressure decreases uniformly from $10 \mathrm{~atm}$ to $0 \mathrm{~atm}$. The engine's heating structure is such that $1 \mathrm{~kg}$ of coal can produce $a=9.5 \mathrm{~kg}$ of steam. $1 \mathrm{~kg}$ of $11$ atmospheric steam has a volume of $v=190 \mathrm{dm}^{3} ; 1 \mathrm{~atm} .=1.033 \mathrm{~kg}$ weight per $\mathrm{cm}^{2}$; 1 Calorie $=427 \mathrm{~m} \mathrm{~kg} ; 1 \mathrm{~kg}$ of coal releases 7500 Calories of heat. We will refer to the engine's operation without expansion as case $(\alpha)$ and with expansion as case $(\beta)$.

1. What is the weight of the force acting on the piston in the case of full pressure?
2. How many $\mathrm{kg}$ of water steam does the engine consume per hour in cases $(\alpha)$ and $(\beta)$?
3. How many $\mathrm{kg}$ of coal does the engine consume per hour in cases $(\alpha)$ and $(\beta)$?
4. What is the theoretical power $(H P)_{i}$ in both cases?
5. Since a portion of the theoretically calculated energy is lost due to internal friction, only $70 \%$ of this energy is converted into external work (mechanical efficiency: $70 \%$), what is the effective power $(H P)_{e}$ in both cases?
6. How many $\mathrm{kg}$ of coal are required to produce $1(H P)_{e}$ for 1 hour in both cases? (That is, how many $\mathrm{kg}$ of coal need to be burned to maintain $1(H P)_{e}$ for 1 hour?)
7. What is the efficiency in both cases? (That is, what percentage of the energy from the burned coal is converted into effective work?)
$1^{\circ}$. The force acting on the piston:

$$
\begin{gathered}
P=r^{2} \pi \cdot p \cdot 1.033 \text{ kg weight} \\
P=30150 \text{ kg weight. }
\end{gathered}
$$

$\alpha$

$\beta$

Without expansion

With expansion

$2^{\circ}$. The amount of steam:

$$
\begin{array}{cl}
Q_{\alpha}=\frac{r^{2} \pi l \cdot 2 n \cdot 60}{190000} \text{ kg} & Q_{\beta}=\frac{r^{2} \pi \frac{l}{3} \cdot 2 n^{\prime} \cdot 60}{190000} \text{ kg} \\
Q_{\alpha}=8850 \text{ kg} & Q_{\beta}=2360 \text{ kg}
\end{array}
$$

$3^{\circ}$. The coal consumed per hour:

$$
\begin{aligned}
C_{\alpha}=\frac{Q_{\alpha}}{\alpha} \text{ kg} & Q_{\beta}=\frac{Q_{\beta}}{\alpha} \text{ kg} \\
C_{\alpha}=930 \text{ kg} & C_{\beta}=250 \text{ kg}
\end{aligned}
$$

$4^{\circ}$. The theoretical power:

$$
\begin{array}{cc}
H_{\alpha}=\frac{P \cdot 1.2 \cdot 2 n}{60.75} (H P)_{i} \quad H_{\beta}=\frac{\left(P \cdot 0.4+\frac{P}{2} \cdot 0.8\right) \cdot 2 n^{\prime}}{60.75} (H P)_{i} \\
H_{\alpha}=643 (H P)_{i} & H_{\beta}=343 (H P)_{i}
\end{array}
$$

$5^{\circ}$. Effective power:

$$
\begin{array}{cl}
E_{\alpha}=H_{\alpha} \cdot 0.7 (H P)_{e} & E_{\beta}=H_{\beta} \cdot 0.7 (H P)_{e} \\
E \alpha=450 (H P)_{e} & E_{\beta}=240 (H P)_{e}
\end{array}
$$

$6^{\circ}$.

$$
1 \text{ Hour } (H P)_{e}=\frac{930}{450}=2.07 \text{ kg of coal } \quad 1 \text{ Hour } (H P)_{e}=\frac{250}{240}=1.03 \text{ kg of coal. }
$$

$7^{\circ}$. The value of 1 Hour $(H P)_{e}$ work $=60 \cdot 60 \cdot 75 \text{ m kg}$, for which in the $\alpha$ case $2.07 \times 7500.427 \text{ m kg}$ of coal energy is required, and in the $\beta$ case $1.03 \times 7500.427 \text{ m kg}$ of coal energy, and thus the economic efficiency

$$
\begin{gathered}
F_{\alpha}=\frac{60 \cdot 60 \cdot 75}{2.07 \cdot 7500 \cdot 427}=0.04 ; \quad F_{\beta}=\frac{60 \cdot 60 \cdot 75}{1.033 \cdot 7500 \cdot 427}=0.08 \\
F_{\alpha}=4 \% \quad F_{\beta}=8 \% .
\end{gathered}
$$

(László Bánó, Budapest.)