Let's go through each of the calculations step by step.
4.1.1 Stroke Volume in m³
The stroke volume (V_s) for a single cylinder is calculated using the formula:
\[ V_s = \pi \left(\frac{d}{2}\right)^2 \cdot L \]
Where:
- \( d \) = diameter of the cylinder = 180 mm = 0.18 m (convert mm to m)
- \( L \) = stroke length = 2500 mm = 2.5 m (convert mm to m)
First, calculate the piston area:
\[ \text{Area} = \pi \left(\frac{0.18}{2}\right)^2 = \pi \times (0.09)^2 = \pi \times 0.0081 \approx 0.025446 \text{ m}^2 \]
Now calculate the stroke volume for one cylinder:
\[ V_s = \text{Area} \times L = 0.025446 , \text{m}^2 \times 2.5 , \text{m} \approx 0.063615 , \text{m}^3 \]
Since there are three cylinders, the total stroke volume (V_t) is:
\[ V_t = 3 \times V_s = 3 \times 0.063615 \approx 0.190845 , \text{m}^3 \]
4.1.2 Mass of Air in kg/min
First, we need to calculate the mass flow rate of air using the ideal gas equation. However, we first need to find the effective volume flow rate and then the mass flow rate.
The stroke volume per minute gives the volume of air delivered:
- Number of strokes per minute (for one cylinder) = 240
- Total strokes per minute (for three cylinders) = \( 240 \times 3 = 720 \text{ strokes/min} \)
Now, calculate the volume flow rate:
\[ \text{Volume flow rate} = V_t \times \text{strokes/min} = 0.063615 , \text{m}^3 \times 240 = 15.274 , \text{m}^3/\text{min} \]
Now we need to calculate the density of air (\(\rho\)) at the given conditions. Use the ideal gas law:
\[ \rho = \frac{P}{R \cdot T} \]
Where:
- \( P = 101325 , \text{Pa} \)
- \( R = 0.289 , \text{kJ/kg.K} = 289 , \text{J/kg.K} \)
- \( T = 20 , \text{°C} = 293.15 , \text{K} \) (convert °C to Kelvin)
Calculate the density:
\[ \rho = \frac{101325}{289 \cdot 293.15} \approx \frac{101325}{84709.35} \approx 1.195 , \text{kg/m}^3 \]
Now we can calculate the mass flow rate:
\[ \text{Mass flow rate} = \text{Volume flow rate} \times \rho = 15.274 \times 1.195 \approx 18.271 , \text{kg/min} \]
4.1.3 Volumetric Efficiency
Volumetric efficiency (\(\eta_v\)) is given by:
\[ \eta_v = \frac{\text{Actual volume flow rate}}{\text{Theoretical volume flow rate}} \times 100% \]
The theoretical volume flow rate (\(Q_t\)) can be calculated based on the volume of the air in the cylinder:
Using the clearance volume \(V_c\)
\[ V_c = 0.05 \times V_s = 0.05 \times 0.063615 \approx 0.00318075 , \text{m}^3 \]
The total volume for one cycle (considering clearance volume) for one cylinder would be:
\[ V_t + V_c = V_s + V_c = 0.063615 + 0.00318075 \approx 0.06679575 , \text{m}^3 \]
Considering 3 cylinders:
\[ Q_t = 3 \times (V_s + V_c) \times \text{strokes/min} = 3 \times 0.06679575 \times 240 = 48.084192 , \text{m}^3/\text{min} \]
Now calculate volumetric efficiency
\[ \eta_v = \frac{15.274}{48.084192} \times 100% \approx 31.8 % \]
Final Answers
- 4.1.1 Stroke Volume (m³): \(0.190845 , \text{m}^3\)
- 4.1.2 Mass of Air (kg/min): \(18.271 , \text{kg/min}\)
- 4.1.3 Volumetric Efficiency: \(31.8%\)