661\, \mathrm{K})}{0.113\, \mathrm{m^3/kg}} = 4,191\, \mathrm{kPa}\)
Step 5: Calculate the Temperature after Heat Addition
#tag_title# Calculate the Temperature after Heat Addition #tag_content# We know that heat is added at constant volume, so we can calculate the temperature after the heat addition process using the relation:
\(q_{in} = C_v (T_3 - T_2)\)
Where \(C_v\) is the specific heat at constant volume for air, which is \(0.718 \, \mathrm{kJ/kg \cdot K}\). Now, we can calculate the temperature after heat addition:
\(T_3 = T_2 + \frac{q_{in}}{C_v} = 1661\, \mathrm{K} + \frac{750\, \mathrm{kJ/kg}}{0.718\, \mathrm{kJ/kg \cdot K}} = 2,711\, \mathrm{K}\)
Step 6: Calculate the Pressure after Heat Addition
#tag_title# Calculate the Pressure after Heat Addition #tag_content# We can calculate the pressure after heat addition using the ideal gas law:
\(p_3 = \frac{RT_3}{v_2}\)
\(p_3 = \frac{(0.287\, \mathrm{kJ/kg \cdot K})(2,711\, \mathrm{K})}{0.113\, \mathrm{m^3/kg}} = 6,866\, \mathrm{kPa}\)
Step 7: Calculate the Net Work Output
#tag_title# Calculate the Net Work Output #tag_content# The net work output for an ideal Otto cycle can be calculated using the relationship:
\(w_{net} = q_{in} - C_v (T_4 - T_1)\)
We need to find temperature after the expansion process \(T_4\):
\(\frac{T_4}{T_3} = \left(\frac{v_4}{v_3}\right)^{k-1} = \left(\frac{v_1}{v_2}\right)^{k-1} = \frac{T_1}{T_2}\)
\(T_4 = T_1 \times \frac{T_3}{T_2} = 300\, \mathrm{K} \times \frac{2,711\, \mathrm{K}}{1,661\, \mathrm{K}} = 489\, \mathrm{K}\)
Now we can find the net work output:
\(w_{net} = q_{in} - C_v (T_4 - T_1) = 750\, \mathrm{kJ/kg} - (0.718\, \mathrm{kJ/kg \cdot K})(489\, \mathrm{K} - 300\, \mathrm{K}) = 493\, \mathrm{kJ/kg}\)
Step 8: Calculate the Thermal Efficiency
#tag_title# Calculate the Thermal Efficiency #tag_content# The thermal efficiency of the ideal Otto cycle can be calculated using the relationship:
\(\eta_{th} = \frac{w_{net}}{q_{in}} = \frac{493\, \mathrm{kJ/kg}}{750\, \mathrm{kJ/kg}} = 0.657 = 65.7\%\)
Step 9: Calculate the Mean Effective Pressure
#tag_title# Calculate the Mean Effective Pressure #tag_content# The mean effective pressure (MEP) is a measure of the average pressure exerted during the power stroke of an engine. It can be calculated using the relationship:
\(MEP = \frac{w_{net}}{v_1 - v_2} = \frac{493\, \mathrm{kJ/kg}}{0.905\, \mathrm{m^3/kg} - 0.113\, \mathrm{m^3/kg}} = 638\, \mathrm{kPa}\)
Now we have calculated all the required parameters for this ideal Otto cycle:
- Pressure after heat addition: \(p_3 = 6,866\, \mathrm{kPa}\)
- Temperature after heat addition: \(T_3 = 2,711\, \mathrm{K}\)
- Net work output: \(w_{net} = 493\, \mathrm{kJ/kg}\)
- Thermal efficiency: \(\eta_{th} = 65.7\%\)
- Mean effective pressure: \(MEP = 638\, \mathrm{kPa}\)
