Question

A four-cylinder, four-stroke spark-ignition engine operates on the ideal Otto cycle with a compression ratio of 11 and a total displacement volume of 1.8 liter. The air is at \(90 \mathrm{kPa}\) and \(50^{\circ} \mathrm{C}\) at the beginning of the compression process. The heat input is \(1.5 \mathrm{kJ}\) per cycle per cylinder. Accounting for the variation of specific heats of air with temperature, determine \((a)\) the maximum temperature and pressure that occur during the cycle, \((b)\) the net work per cycle per cyclinder and the thermal efficiency of the cycle, \((c)\) the mean effective pressure, and \((d)\) the power output for an engine speed of \(3000 \mathrm{rpm}\)

Short answer

To find the maximum temperature and pressure during the cycle, net work, thermal efficiency, mean effective pressure, and power output in a four-cylinder, four-stroke spark-ignition engine with an Otto cycle, follow these steps: 1. Determine the initial state properties of the air using the ideal gas law equation and specific heat information. 2. Calculate the properties at state 2 after the isentropic compression using compression ratio and isentropic process equation. 3. Determine the properties at state 3 after the constant-volume heat input by adding heat input to the internal energy at state 2 and solve for the temperature. 4. Calculate the properties at state 4 after the isentropic expansion using the isentropic process equation. 5. Find the maximum temperature and pressure during the cycle by comparing the values at different states. 6. Calculate the net work per cycle per cylinder and the thermal efficiency of the cycle using heat input and heat rejected during the constant-volume cooling process. 7. Calculate the mean effective pressure using the net work and total displacement volume. 8. Compute the power output for an engine speed of 3000 rpm by multiplying the net work and total number of cycles per minute.

Step-by-step solution

Determine the initial state properties

We are given that the initial state of air is at \(P_1 = 90 \mathrm{kPa}\) and \(T_1 = 50^{\circ} \mathrm{C}\). We'll need to use the ideal gas law equation and the specific heat information to find the specific volume (\(v_1\)) and internal energy (\(u_1\)) at state 1.

Determine the properties at state 2 after the isentropic compression

Using the compression ratio (\(r = 11\)) and the relation \(v_2 = v_1/r\), we can find the specific volume at state 2. Then, using the ideal gas constant for air and the isentropic process equation \(T_2/T_1 = (v_1/v_2)^{k-1}\), we can find the temperature at state 2. Finally, using the specific heat information, we can find the internal energy at state 2.

Determine the properties at state 3 after the constant-volume heat input

We know that \(v_3 = v_2\). Given that the heat input per cycle per cylinder is \(1.5 \mathrm{kJ}\), we can find the internal energy at state 3 by adding the heat input to the internal energy at state 2, and then solve for the temperature at state 3 using the specific heat information.

Determine the properties at state 4 after the isentropic expansion

Using the relation \(v_4 = v_1\), we can find the specific volume at state 4. Then, using the ideal gas constant for air and the isentropic process equation \(T_4/T_3 = (v_3/v_4)^{k-1}\), we can find the temperature at state 4. Finally, using the specific heat information, we can find the internal energy at state 4.

Calculate the maximum temperature and pressure during the cycle

The maximum temperature occurs at state 3, which we have already determined in step 3. The maximum pressure will occur at either state 2 or state 3. We can find the pressure at both states using the ideal gas law equation and compare the values to determine the maximum pressure during the cycle.

Calculate the net work per cycle per cylinder and the thermal efficiency of the cycle

The net work per cycle per cylinder can be found using the relation \(W_{net} = Q_{in} - Q_{out}\), where \(Q_{in}\) is the heat input and \(Q_{out}\) is the heat rejected during the constant-volume cooling process between state 4 and state 1. The thermal efficiency of the cycle can be found using the relation \(\eta_{th} = W_{net}/Q_{in}\).

Calculate the mean effective pressure

The mean effective pressure (MEP) can be calculated as \(MEP = W_{net}/v_{1}\). In this case, we will use \(v_{1}\) as the total displacement volume, which was given as 1.8 liters.

Calculate the power output for an engine speed of 3000 rpm

Given an engine speed of 3000 rpm, we can calculate the power output as the product of the net work and the total number of cycles per minute (which is half of the engine speed for a four-stroke engine). The power output can then be expressed in the desired units (e.g., watts or horsepower).