Question

An Otto cycle with air as the working fluid has a compression ratio of \(10.4 .\) Under cold-air-standard conditions, the thermal efficiency of this cycle is \((a) 10\) percent (b) 39 percent \((c) 61\) percent \((d) 79\) percent \((e) 82\) percent

Short answer

Answer: Approximately 61 percent (option c).

Step-by-step solution

Identify the formula for thermal efficiency in an Otto cycle

Under cold-air-standard conditions, the thermal efficiency (η) of an Otto cycle is given by the formula: $$\eta = 1 - \frac{1}{(r^{(\gamma-1)})}$$ where \(r\) is the compression ratio and \(\gamma\) is the specific heat ratio (around 1.4 for air).

Substitute the given values into the formula

We are given a compression ratio of \(10.4\). Since the working fluid is air, we can assume the specific heat ratio \(\gamma = 1.4\). Substitute these values into the formula: $$\eta = 1 - \frac{1}{(10.4^{(1.4-1)})}$$

Calculate the thermal efficiency

Perform the calculation to find the thermal efficiency: $$\eta = 1 - \frac{1}{(10.4^{(0.4)})} = 1 - \frac{1}{(2.797)} = 1 - 0.3577 = 0.6423$$ The thermal efficiency of this Otto cycle is \(0.6423\), or \(64.23\) percent.

Compare the calculated value to the given options

Now, compare the calculated thermal efficiency to the given options: (a) \(10\) percent (b) \(39\) percent (c) \(61\) percent (d) \(79\) percent (e) \(82\) percent Comparing our calculated value of \(64.23\) percent, we find that it is closest to option (c).

Conclusion

The thermal efficiency of this Otto cycle under cold-air-standard conditions is approximately 61 percent (option c).