Question

What is the compression ratio of the Otto cycle for a petrol engine with a cylinder bore of 50 mm, a stroke of 75 mm, and clearance volume of 21.3 cm\(^3\)?

• A. 7.9
• B. 6.9
• C. 5.9
• D. (4)9

Hint:

Ensure that all units are consistent before performing calculations. In this case, all dimensions were converted to centimeters to match the unit of the clearance volume. Remember the formula for the area of a circle and the volume of a cylinder.

Answer 1

The Correct Option is A

Step 1: Understand the definition of compression ratio in an Otto cycle.
The compression ratio (\(r\)) of an Otto cycle is defined as the ratio of the volume of the cylinder at the beginning of the compression stroke (maximum volume) to the volume of the cylinder at the end of the compression stroke (minimum volume). $$r = \frac{V_{max}}{V_{min}}$$ Where:
\(V_{max}\) is the maximum cylinder volume (swept volume + clearance volume).
\(V_{min}\) is the minimum cylinder volume (clearance volume).
Step 2: Calculate the swept volume of the cylinder.
The swept volume (\(V_s\)) is the volume displaced by the piston as it moves from one end of the cylinder to the other. It is calculated using the cylinder bore (diameter, \(d\)) and the stroke length (\(L\)). The area of the piston (\(A\)) is given by \(A = \frac{\pi d^2}{4}\). Given:
Cylinder bore \(d = 50\) mm = 5 cm
Stroke \(L = 75\) mm = 7.5 cm
Area of the piston:
$$A = \frac{\pi (5 \text{ cm})^2}{4} = \frac{\pi \times 25}{4} \text{ cm}^2 \approx 19.635 \text{ cm}^2$$ Swept volume:
$$V_s = A \times L = 19.635 \text{ cm}^2 \times 7.5 \text{ cm} \approx 147.26 \text{ cm}^3$$ Step 3: Determine the maximum and minimum cylinder volumes.
Minimum cylinder volume (\(V_{min}\)) is the clearance volume (\(V_c\)), which is given as 21.3 cm\(^3\).
$$V_{min} = V_c = 21.3 \text{ cm}^3$$
Maximum cylinder volume (\(V_{max}\)) is the sum of the swept volume and the clearance volume.
$$V_{max} = V_s + V_c = 147.26 \text{ cm}^3 + 21.3 \text{ cm}^3 = 168.56 \text{ cm}^3$$ Step 4: Calculate the compression ratio.
Now, we can calculate the compression ratio (\(r\)) using the formula: $$r = \frac{V_{max}}{V_{min}} = \frac{168.56 \text{ cm}^3}{21.3 \text{ cm}^3} \approx 7.9136$$ Step 5: Compare the calculated compression ratio with the given options.
The calculated compression ratio is approximately 7.9, which matches option (1).