I.C. Engine Compression Ratio Calculation on Otto Cycle

This solution explains how to calculate the compression ratio for an internal combustion (I.C.) engine operating on the Otto cycle, given specific engine dimensions and clearance volume.

Given Parameters

• Number of cylinders = 6
• Engine Cycle = Otto cycle
• Bore ($d$) = 20 cm
• Stroke ($l$) = 40 cm
• Total Clearance Volume ($V_{c, \text{total}}$) = 9000 cm³ (Interpreted as total for all cylinders)

Understanding Compression Ratio

The compression ratio ($r$) in an engine is the ratio of the volume in the cylinder when the piston is at the bottom dead center (BDC) to the volume when the piston is at the top dead center (TDC). For an Otto cycle engine, it is expressed as:

$$r = \frac{V_{total}}{V_c} = \frac{V_s + V_c}{V_c} = 1 + \frac{V_s}{V_c}$$

Where:

• $V_s$ = Swept volume of one cylinder
• $V_c$ = Clearance volume of one cylinder

Calculating Swept Volume per Cylinder ($V_s$)

The swept volume ($V_s$) is the volume displaced by the piston during one stroke. The formula is:

$$V_s = \frac{\pi}{4} \times d^2 \times l$$

Substituting the given values:

$$V_s = \frac{\pi}{4} \times (20 \text{ cm})^2 \times (40 \text{ cm})$$

$$V_s = \frac{\pi}{4} \times 400 \text{ cm}^2 \times 40 \text{ cm}$$

$$V_s = \pi \times 100 \text{ cm}^2 \times 40 \text{ cm}$$

$$V_s = 4000 \pi \text{ cm}^3$$

Approximately:

$$V_s \approx 4000 \times 3.14159 \text{ cm}^3 \approx 12566.37 \text{ cm}^3$$

Determining Clearance Volume per Cylinder ($V_c$)

The question provides a clearance volume of 9000 cm³. Given the options, this value likely represents the total clearance volume for all 6 cylinders.

Therefore, the clearance volume per cylinder ($V_c$) is:

$$V_c = \frac{V_{c, \text{total}}}{\text{Number of cylinders}}$$

$$V_c = \frac{9000 \text{ cm}^3}{6}$$

$$V_c = 1500 \text{ cm}^3$$

Calculating the Compression Ratio ($r$)

Now, we can calculate the compression ratio using the formula:

$$r = 1 + \frac{V_s}{V_c}$$

Substitute the calculated values of $V_s$ and $V_c$:

$$r = 1 + \frac{4000 \pi \text{ cm}^3}{1500 \text{ cm}^3}$$

$$r = 1 + \frac{40 \pi}{15}$$

$$r = 1 + \frac{8 \pi}{3}$$

Using $\pi \approx 3.14159$:

$$r \approx 1 + \frac{8 \times 3.14159}{3}$$

$$r \approx 1 + \frac{25.13272}{3}$$

$$r \approx 1 + 8.37757$$

$$r \approx 9.37757$$

Final Result

The calculated compression ratio is approximately 9.38. This matches one of the provided options.

Comparison with Options

The calculated value $r \approx 9.38$ corresponds to option 3.