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)
wrew
To find the lowest temperature and efficiency of a Carnot engine, we can use the Carnot's efficiency formula.
Carnot's efficiency formula:
Efficiency = 1 - (Tl/Th)
Where:
Efficiency = Efficiency of the Carnot engine
Tl = Lowest temperature
Th = Highest temperature
To calculate the lowest temperature, we can use the following equation:
Tl = Th / (ratio)^(γ-1)
Where:
Tl = Lowest temperature
Th = Highest temperature
γ = Specific heat ratio
Given data:
Initial temperature (Th) = 3270°C = 3270 + 273 = 3543 K
Initial pressure (Ph) = 12 atmospheres
Initial volume (Vh) = 1 liter
Expansion or compression ratio = 1:6
Specific heat ratio (γ) = 1.4
First, let's calculate the highest temperature (Th):
Th = Initial temperature + 273
Th = 3543 K
Next, let's calculate the final volume (Vl) using the expansion ratio:
Vl = Vh / (ratio)
Vl = 1 liter / 6
Vl = 0.1667 liter
Now, let's calculate the final pressure (Pl) using the ideal gas equation:
Ph * Vh / Th = Pl * Vl / Tl
Pl = (Ph * Vh * Tl) / (Vl * Th)
Plug in the known values:
Pl = (12 * 1 * 3543) / (0.167 * 3543)
Pl = 2556.89 atmospheres
Now, let's calculate the lowest temperature (Tl):
Tl = Th / (ratio)^(γ-1)
Tl = 3543 / (1/6)^(1.4-1)
Tl = 3543 / (6)^(0.4)
Tl = 3543 / 1.817
Tl = 1947.39 K
Finally, let's calculate the efficiency:
Efficiency = 1 - (Tl / Th)
Efficiency = 1 - (1947.39 / 3543)
Efficiency = 1 - 0.550
Efficiency = 0.450 (or 45%)
Therefore, the lowest temperature is approximately 1947.39 K and the efficiency of the Carnot engine is approximately 45%.