Question

Air in an ideal Diesel cycle is compressed from 2 to \(0.13 \mathrm{L},\) and then it expands during the constant pressure heat addition process to 0.30 L. Under cold air standard conditions, the thermal efficiency of this cycle is (a) 41 percent (b) 59 percent \((c) 66\) percent \((d) 70\) percent \((e) 78\) percent

Short answer

(a) 41 percent, (b) 59 percent, (c) 66 percent, (d) 70 percent, and (e) 78 percent.

Step-by-step solution

Understand the given information

We are given the following information: - Initial volume (V1): 2 L - Compressed volume (V2): 0.13 L - Expanded volume (V3): 0.30 L - Cold air standard conditions

Calculate the compression ratio and cutoff ratio

First, we need to calculate the compression ratio (r) and the cutoff ratio (rho). Compression ratio (r) is the ratio of initial volume to compressed volume: r = V1 / V2 = 2 / 0.13 Cutoff ratio (rho) is the ratio of expanded volume to compressed volume: rho = V3 / V2 = 0.30 / 0.13

Apply the ideal Diesel cycle formula for thermal efficiency

The thermal efficiency (η) for an ideal Diesel cycle operating under the cold air standard conditions can be calculated using the following formula: η = 1 - (1 / (r^(γ - 1))) * ((rho^γ - 1) / (γ * (rho - 1))) where γ (gamma) is the ratio of specific heats of the working fluid, which is approximately 1.4 for air. Now, we can substitute the compression ratio (r) and cutoff ratio (rho) calculated in Step 2, and the specific heat ratio (γ) to find the thermal efficiency.

Calculate the thermal efficiency and match it to the options

Now, we can calculate the thermal efficiency with the given values: η = 1 - (1 / (r^(γ - 1))) * ((rho^γ - 1) / (γ * (rho - 1))) After calculating the above expression, we get the thermal efficiency of the ideal Diesel cycle. Then, we need to match the result with the provided options (a) 41 percent, (b) 59 percent, (c) 66 percent, (d) 70 percent, and (e) 78 percent to find the correct answer.