Diesel Engine Volume Calculation: Understanding Key Terms

To effectively solve this problem, it's essential to understand the key terms related to an internal combustion engine's cylinder volumes:

• Stroke Volume (\(V_s\)): This is the volume displaced by the piston as it moves from one extreme position (either top dead center or bottom dead center) to the other. It's also known as displacement volume. In this question, the stroke volume is given as 450 cc.
• Clearance Volume (\(V_c\)): This is the volume of the combustion chamber when the piston is at its Top Dead Center (TDC). It is the smallest volume occupied by the gases in the cylinder. This is what we need to calculate.
• Total Volume (\(V_t\)): This is the maximum volume of the cylinder when the piston is at its Bottom Dead Center (BDC). It is the sum of the stroke volume and the clearance volume, i.e., \(V_t = V_s + V_c\).
• Compression Ratio (\(r\)): The compression ratio is a fundamental characteristic of an engine. It is defined as the ratio of the total volume to the clearance volume. For this single-cylinder diesel engine, the compression ratio is 16.

Compression Ratio Formula and Calculation

The compression ratio \(r\) for an internal combustion engine is mathematically expressed as:

\[r = \frac{\text{Total Volume}}{\text{Clearance Volume}} = \frac{V_t}{V_c}\]

Since the total volume \(V_t\) is the sum of the stroke volume \(V_s\) and the clearance volume \(V_c\), we can substitute this into the formula:

\[r = \frac{V_s + V_c}{V_c}\]

This formula can also be rearranged as:

\[r = \frac{V_s}{V_c} + \frac{V_c}{V_c}\]

\[r = 1 + \frac{V_s}{V_c}\]

Solving for Clearance Volume (\(V_c\))

Now, let's plug in the given values from the problem statement:

• Compression ratio (\(r\)) = 16
• Stroke volume (\(V_s\)) = 450 cc

Using the derived formula \(r = 1 + \frac{V_s}{V_c}\):

Step 1: Substitute the given values into the formula.

\[16 = 1 + \frac{450 \text{ cc}}{V_c}\]

Step 2: Subtract 1 from both sides of the equation.

\[16 - 1 = \frac{450 \text{ cc}}{V_c}\]

\[15 = \frac{450 \text{ cc}}{V_c}\]

Step 3: Rearrange the equation to solve for \(V_c\).

\[V_c = \frac{450 \text{ cc}}{15}\]

Step 4: Perform the division.

\[V_c = 30 \text{ cc}\]

Therefore, the clearance volume of the single-cylinder diesel engine is 30 cc.