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Answer :
The clearance volume of one cylinder is 34.57 cubic cm (2.11 cubic inches). The bore is 8.69 cm (3.42 inches), and the stroke is 9.21 cm (3.63 inches). The average piston speed is 9.83 m/sec (32.24 ft/sec).
To determine the required parameters, we start with the given data:
- Engine displacement: [tex]\(2.4 \, \text{L}\)[/tex]
- Number of cylinders: 4
- Compression ratio: 9.4:1
- Bore and stroke relationship: [tex]\(S = 1.06 B\)[/tex]
- Connecting rod length: [tex]\(r = 18 \, \text{cm}\)[/tex]
- Engine speed: 3200 RPM
Step 1: Calculate the total displacement volume [tex]\(V_d\)[/tex] of the engine.
[tex]\[V_d = 2.4 \, \text{L} = 2400 \, \text{cm}^3\][/tex]
Since this is the total for four cylinders, the displacement per cylinder [tex]\(V_d\)[/tex] is:
[tex]\[V_d = \frac{2400 \, \text{cm}^3}{4} = 600 \, \text{cm}^3\][/tex]
Step 2: Determine the clearance volume [tex]\(V_c\).[/tex]
The compression ratio (CR) is given by:
[tex]\[CR = \frac{V_s + V_c}{V_c} = 9.4\][/tex]
Where [tex]\(V_s\)[/tex] is the swept volume per cylinder. Therefore,
[tex]\[V_s = \frac{600 \, \text{cm}^3}{\frac{9.4}{1} - 1} = 600 \, \text{cm}^3 \cdot \frac{1}{8.4} \approx 71.43 \, \text{cm}^3\][/tex]
Then,
[tex]\[V_c = \frac{600 \, \text{cm}^3}{9.4} \approx 63.83 \, \text{cm}^3\][/tex]
Step 3: Calculate the bore [tex]\(B\)[/tex] and stroke [tex]\(S\).[/tex]
Using the relationship [tex]\(S = 1.06 B\)[/tex] and the formula for the swept volume of a cylinder:
[tex]\[V_s = \pi \left(\frac{B}{2}\right)^2 S\][/tex]
[tex]\[600 = \pi \left(\frac{B}{2}\right)^2 (1.06 B)\][/tex]
Solving for [tex]\(B\):[/tex]
[tex]\[600 = \pi \left(\frac{B}{2}\right)^2 \cdot 1.06 B\][/tex]
[tex]\[600 = \pi \cdot 0.265 \cdot B^3\][/tex]
[tex]\[B^3 = \frac{600}{0.834} \approx 719.18\][/tex]
[tex]\[B \approx \sqrt[3]{719.18} \approx 8.97 \, \text{cm}\][/tex]
[tex]\[S = 1.06 \cdot 8.97 \approx 9.51 \, \text{cm}\][/tex]
Step 4: Convert bore and stroke to inches.
[tex]\[B \approx 8.97 \, \text{cm} \times \frac{1 \, \text{inch}}{2.54 \, \text{cm}} \approx 3.53 \, \text{inches}\][/tex]
[tex]\[S \approx 9.51 \, \text{cm} \times \frac{1 \, \text{inch}}{2.54 \, \text{cm}} \approx 3.74 \, \text{inches}\][/tex]
Step 5: Calculate average piston speed.
Average piston speed [tex]\(v\)[/tex] is given by:
[tex]\[v = 2 \times S \times \text{RPM} / 60\][/tex]
[tex]\[v = 2 \times 9.51 \, \text{cm} \times 3200 / 60 \approx 101.44 \, \text{cm/sec} \approx 10.14 \, \text{m/sec}\][/tex]
In feet per second:
[tex]\[v \approx 10.14 \, \text{m/sec} \times 3.281 \approx 33.26 \, \text{ft/sec}\][/tex]
Stepwise Calculation Summary
1. Displacement per cylinder [tex]\(V_d\)[/tex] is 600 cm³.
2. Using the compression ratio, calculate the clearance volume [tex]\(V_c \approx 71.43 \, \text{cm}^3\).[/tex]
3. Solve for bore [tex]\(B \approx 8.97 \, \text{cm}\)[/tex] and stroke [tex]\(S \approx 9.51 \, \text{cm}\).[/tex]
4. Convert bore and stroke to inches: [tex]\(B \approx 3.53 \, \text{in}\), \(S \approx 3.74 \, \text{in}\).[/tex]
5. Calculate average piston speed: [tex]\(v \approx 10.14 \, \text{m/sec} \approx 33.26 \, \text{ft/sec}\).[/tex]
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