Question 22: Thermodynamics & KTG

Question

A Carnot's heat engine works between the temperature $$527^{\circ} \mathrm{C}$$ and $$127^{\circ} \mathrm{C}$$. What amount of heat should it consume per second to deliver mechanical work at the rate of $$1.0 \mathrm{~kW}$$ ?

Accepted Answer

A Carnot heat engine operates between two temperatures: the hot reservoir temperature ($ T_h $) and the cold reservoir temperature ($ T_c $). To find the efficiency ($ \eta $) of a Carnot engine, we use the formula:
$$ \eta = 1 - \frac{T_c}{T_h} $$
where both temperatures ($ T_h $ and $ T_c $) must be in absolute temperature, i.e., Kelvin (K).
The given temperatures are $ 527^{\circ} \mathrm{C} $ and $ 127^{\circ} \mathrm{C} $. First, we convert these temperatures to Kelvin:
$$ T_h = 527^{\circ} \mathrm{C} + 273 = 800 \, \mathrm{K} $$
$$ T_c = 127^{\circ} \mathrm{C} + 273 = 400 \, \mathrm{K} $$
Next, we calculate the efficiency of the Carnot engine:
$$ \eta = 1 - \frac{400}{800} = 1 - 0.5 = 0.5 $$
The efficiency ($ \eta $) is equal to the fraction of heat converted to work. Therefore, the fraction of heat used to do mechanical work per second is:
$$ \eta = \frac{W}{Q_h} $$
where $ W $ is the work done per second (which is $ 1.0 \, 	ext{kW} $ or $ 1000 \, 	ext{W} $), and $ Q_h $ is the heat consumed per second.
Rearranging the formula, we get:
$$ Q_h = \frac{W}{\eta} $$
Substituting the given values:
$$ Q_h = \frac{1000 \, \mathrm{W}}{0.5} = 2000 \, \mathrm{W} $$
To convert the heat consumed per second from Watts to kcal/s, we use the conversion factor $ 1 \, \mathrm{kcal/s} = 4184 \, \mathrm{W} $:
$$ Q_h = \frac{2000 \, \mathrm{W}}{4184 \, \mathrm{W/kcal}} \approx 0.477 \, \mathrm{kcal/s} $$
Therefore, the amount of heat the Carnot engine should consume per second to deliver mechanical work at the rate of $ 1.0 \, 	ext{kW} $ is:
Option A: 0.477 kcal/s

Answer

0.477 kcal/s

Answer

4.77 kcal/s

Answer

447.7 kcal/s

Answer

0.2385 kcal/s

Thermodynamics & KTG

A Carnot's heat engine works between the temperature $527^{\circ} \mathrm{C} and 127^{\circ} \mathrm{C}. What amount of heat should it consume per second to deliver mechanical work at the rate of 1.0 \mathrm{~kW}$ ?

Choose the correct answer: