A carnot’s engine with air as working substance is initially at 3270C and pressure of 12 atmospheres. The volume is one litre to start with. The expansion or compression ratio is 1:6. Find the lowest temperature and efficiency. (γ= 1.4)
58.28%
what is lowest temperature answer and what is procedure for that
To find the lowest temperature and efficiency of the Carnot engine, we can use the Carnot efficiency formula and the gas laws.
1. Find the lowest temperature (T_low):
The lowest temperature is the temperature at the end of the adiabatic expansion process in the Carnot cycle. To find this, we need to calculate the final temperature after the expansion.
First, let's find the initial pressure (P1) and initial volume (V1).
Given:
Initial temperature (T1) = 3270°C = 3543.15 K
Initial pressure (P1) = 12 atmospheres
Initial volume (V1) = 1 liter
We can convert the initial temperature from Celsius to Kelvin by adding 273.15.
T1 = 3270 + 273.15 = 3543.15 K
According to the gas law, PV^γ = constant, where γ is the adiabatic index.
For the adiabatic expansion process, we have:
P1 * V1^γ = P2 * V2^γ
The expansion ratio is given as 1:6, which means V2 = 6 * V1.
Substituting the values into the equation, we have:
12 * (1^γ) = P2 * (6^γ)
Solving for P2:
P2 = (12 * 1^γ) / (6^γ) = 12 / 6^γ
Now, we can calculate the final temperature (T2) using the relation:
P1 * V1 / T1 = P2 * V2 / T2
Substituting the values, we get:
(12 * 1) / 3543.15 = (12 / 6^γ) * (6 * 1) / T2
Simplifying the equation:
1 / 295.26 = 2 / T2
Solving for T2:
T2 = 2 * 295.26 = 590.52 K
Therefore, the lowest temperature (T_low) is 590.52 K.
2. Calculate the Carnot efficiency (η):
The Carnot efficiency is given by the formula:
η = 1 - (T_low / T_high)
Given:
T_high = T1 = 3543.15 K
T_low = 590.52 K
Substituting the values, we get:
η = 1 - (590.52 / 3543.15) = 1 - 0.1665 = 0.8335
Therefore, the Carnot efficiency is 0.8335, which is equivalent to 83.35%.
In summary:
- The lowest temperature (T_low) is 590.52 K.
- The Carnot efficiency (η) is 0.8335, or 83.35%.