GAS POWER CYCLES...1 DERIVE AN EXPRESSION FOR CHANGE IN EFFICIENCY FOR A CHANGE IN COMPRESSION RATIO. IF THE COMPRESSION RATIO IS INCREASED FROM 6 TO 8 , WHAT WILL BE THE %AGE INCEASE IN EFFICENCY

To derive an expression for the change in efficiency of a gas power cycle with respect to a change in compression ratio, we can start by looking at the efficiency of the ideal Otto cycle, which is commonly used to analyze gas engines. The efficiency of the Otto cycle is given by the formula:

Efficiency of the Otto Cycle

The efficiency (η) can be expressed as:

η = 1 - (1 / r^(γ - 1))

Where:

• r = compression ratio (V1/V2)
• γ = specific heat ratio (Cp/Cv)

Understanding Compression Ratio

The compression ratio (r) is defined as the ratio of the volume of the combustion chamber before compression (V1) to the volume after compression (V2). Increasing the compression ratio generally leads to higher efficiency because it allows for more work to be extracted from the combustion process.

Deriving the Change in Efficiency

To find the change in efficiency when the compression ratio changes, we can differentiate the efficiency equation with respect to the compression ratio:

dη/dr = (γ - 1) / (r^(γ - 1) * ln(r))

This derivative tells us how the efficiency changes with respect to changes in the compression ratio. However, for practical purposes, we can directly calculate the efficiency at two different compression ratios and find the percentage increase.

Calculating Efficiency for r = 6 and r = 8

Let’s assume a typical value for γ, which is about 1.4 for air. We can calculate the efficiencies for the two compression ratios:

For r = 6:

η₁ = 1 - (1 / 6^(1.4 - 1)) = 1 - (1 / 6^0.4) ≈ 1 - 0.659 = 0.341 or 34.1%

For r = 8:

η₂ = 1 - (1 / 8^(1.4 - 1)) = 1 - (1 / 8^0.4) ≈ 1 - 0.574 = 0.426 or 42.6%

Calculating the Percentage Increase in Efficiency

Now, we can find the percentage increase in efficiency when the compression ratio is increased from 6 to 8:

Percentage Increase = ((η₂ - η₁) / η₁) * 100

Substituting the values:

Percentage Increase = ((0.426 - 0.341) / 0.341) * 100 ≈ (0.085 / 0.341) * 100 ≈ 24.9%

Summary

In summary, when the compression ratio is increased from 6 to 8, the efficiency of the gas power cycle increases by approximately 24.9%. This demonstrates how optimizing the compression ratio can significantly enhance the performance of gas engines, leading to better fuel efficiency and overall effectiveness.