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An ideal gas with heat capacity ratio of 2 is used in an ideal Otto-cycle which operates between minimum and maximum temperatures of 200 K and 1800 K. What is the compression ratio of the cycle for maximum work output?
- a)1.5
- b)2
- c)3
- d)4
Correct answer is option 'C'. Can you explain this answer?
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An ideal gas with heat capacity ratio of 2 is used in an ideal Otto-cy...
Heat capacity ratio, γ = Cp/Cv = 2
For maximum work output for Otto cycle:
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An ideal gas with heat capacity ratio of 2 is used in an ideal Otto-cy...
To find the compression ratio of the ideal Otto cycle for maximum work output, we need to consider the heat capacity ratio (γ) of the gas and the given minimum and maximum temperatures.
Given:
Heat capacity ratio (γ) = 2
Minimum temperature (T1) = 200 K
Maximum temperature (T3) = 1800 K
The compression ratio (r) is given by the equation:
r = (V1/V2) = (T3/T2)^(1/(γ-1))
where V1 and V2 are the initial and final volumes of the gas, and T2 is the temperature at the end of the adiabatic compression process.
To find the compression ratio, we need to determine the value of T2. To do this, we can use the equation for adiabatic process:
T1/T2 = (V2/V1)^(γ-1)
Rearranging the equation, we get:
T2 = T1 * (V2/V1)^(γ-1)
Now we can substitute the given values to find T2:
T2 = 200 K * (V2/V1)^(2-1)
Next, we substitute the values of T1, T2, and T3 into the equation for the compression ratio:
r = (1800 K / T2)^(1/(γ-1))
Substituting the value of T2, we have:
r = (1800 K / (200 K * (V2/V1)))^(1/(γ-1))
Simplifying further:
r = (9 / (V2/V1))^(1/(γ-1))
Since the work output of the Otto cycle is maximum when the compression ratio is maximum, we need to find the maximum value of r.
As the heat capacity ratio (γ) is given as 2, the maximum value of r is obtained when the ratio of V2/V1 is minimum. In other words, V2 should be as small as possible compared to V1.
Therefore, the compression ratio for maximum work output is when V2 = V1/3, which gives:
r = (9 / (1/3))^(1/(2-1)) = 3
Hence, the correct answer is option C) 3.
Given:
Heat capacity ratio (γ) = 2
Minimum temperature (T1) = 200 K
Maximum temperature (T3) = 1800 K
The compression ratio (r) is given by the equation:
r = (V1/V2) = (T3/T2)^(1/(γ-1))
where V1 and V2 are the initial and final volumes of the gas, and T2 is the temperature at the end of the adiabatic compression process.
To find the compression ratio, we need to determine the value of T2. To do this, we can use the equation for adiabatic process:
T1/T2 = (V2/V1)^(γ-1)
Rearranging the equation, we get:
T2 = T1 * (V2/V1)^(γ-1)
Now we can substitute the given values to find T2:
T2 = 200 K * (V2/V1)^(2-1)
Next, we substitute the values of T1, T2, and T3 into the equation for the compression ratio:
r = (1800 K / T2)^(1/(γ-1))
Substituting the value of T2, we have:
r = (1800 K / (200 K * (V2/V1)))^(1/(γ-1))
Simplifying further:
r = (9 / (V2/V1))^(1/(γ-1))
Since the work output of the Otto cycle is maximum when the compression ratio is maximum, we need to find the maximum value of r.
As the heat capacity ratio (γ) is given as 2, the maximum value of r is obtained when the ratio of V2/V1 is minimum. In other words, V2 should be as small as possible compared to V1.
Therefore, the compression ratio for maximum work output is when V2 = V1/3, which gives:
r = (9 / (1/3))^(1/(2-1)) = 3
Hence, the correct answer is option C) 3.
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An ideal gas with heat capacity ratio of 2 is used in an ideal Otto-cycle which operates between minimum and maximum temperatures of 200 K and 1800 K. What is the compression ratio of the cycle for maximum work output?a)1.5b)2c)3d)4Correct answer is option 'C'. Can you explain this answer?
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