Question

An ideal Otto cycle with air as the working fluid has a compression ratio of \(8 .\) The minimum and maximum temperatures in the cycle are 540 and 2400 R. Accounting for the variation of specific heats with temperature, determine ( \(a\) ) the amount of heat transferred to the air during the heat-addition process, \((b)\) the thermal efficiency, and \((c)\) the thermal efficiency of a Carnot cycle operating between the same temperature limits.

Short answer

Question: Calculate the heat transferred during the heat-addition process, the thermal efficiency of the Otto cycle, and the thermal efficiency of a Carnot cycle operating between the same temperature limits for an ideal Otto cycle with a compression ratio of 8 and minimum and maximum temperatures of 540R and 2400R, respectively. Answer: (a) The heat transferred during the heat-addition process is 826.57 Btu/lb. (b) The thermal efficiency of the Otto cycle is 57.15%. (c) The thermal efficiency of a Carnot cycle operating between the same temperature limits is 77.5%.

Step-by-step solution

Determine state points and their properties

For an ideal Otto cycle, we will use the following notation for the four state points: 1) After isentropic compression 2) After heat addition at constant volume 3) After isentropic expansion 4) After heat rejection at constant volume. Given that the minimum and maximum temperatures in the cycle are 540R and 2400R, we can conclude that the lowest temperature point is Point 1 (isentropic process) with \(T_1 = 540R\) and the highest temperature point is Point 3 (isentropic process) with \(T_3 = 2400R\). Since we are dealing with an ideal Otto cycle, the specific volume stays constant during the heat addition and rejection processes. Therefore, the compression ratio \(r_c\), which is the ratio of specific volumes, can be defined as: \(r_c = \frac{v_2}{v_1} = \frac{v_4}{v_3}\), where \(\frac{T_2}{T_1} = (\frac{v_2}{v_1})^{k-1}\) We are given \(r_c = 8\). We can find \(T_2\) and \(T_4\) using the given temperatures \(T_1\) and \(T_3\) and the relationship between temperatures and specific volumes during isentropic processes.

Calculate temperatures T2 and T4

We have to find \(T_2\) and \(T_4\) using the given information. Using the relationship \(\frac{T_2}{T_1} = (\frac{v_2}{v_1})^{k-1}\) and knowing that \(k = 1.4\) for air, we can find \(T_2\): \(T_2 = T_1(\frac{v_2}{v_1})^{k-1} = T_1 r_c^{k-1} = 540R \times 8^{1.4 - 1} = 1688.49R\) Similarly, we can find \(T_4\): \(T_4 = T_3(\frac{v_4}{v_3})^{k-1} = T_3 r_c^{1- k} = 2400R \times 8^{1 - 1.4} = 1290.45R\) Now we have the temperatures at all four state points, which are \(T_1 = 540R\), \(T_2 = 1688.49R\), \(T_3 = 2400R\), and \(T_4 = 1290.45R\).

Calculate the heat transferred during the heat-addition process

To calculate the heat transferred to the air during the heat-addition process (from state point 1 to state point 2), we can use the specific heat at constant volume \(c_v\). Since the specific heats vary with temperature, we need to find the average specific heat between \(T_1\) and \(T_2\). Assuming the average specific heat \(c_{v,avg} = 0.718 \, \frac{Btu}{lb\cdot R}\), we can find the heat transferred during the heat-addition process as: \(q_{12} = m c_{v,avg} (T_2 - T_1) = 0.718 (1688.49R - 540R) = 826.57 \, \frac{Btu}{lb}\) Thus, the heat transferred during the heat-addition process is \(826.57 \, \frac{Btu}{lb}\).

Calculate the thermal efficiency of the Otto cycle

The thermal efficiency of the Otto cycle can be calculated as: \(\eta_{Otto} = 1 - \frac{q_{41}}{q_{12}} = 1 - \frac{mc_v(T_4 - T_1)}{mc_v(T_2 - T_1)} = 1 - \frac{T_4 - T_1}{T_2 - T_1} = 1 - \frac{1290.45R - 540R}{1688.49R - 540R} = 0.5715\) Thus, the thermal efficiency of the Otto cycle is \(57.15\%\).

Calculate the thermal efficiency of a Carnot cycle operating between the same temperature limits

The thermal efficiency of a Carnot cycle operating between the same temperature limits as the Otto cycle can be calculated as: \(\eta_{Carnot} = 1 - \frac{T_{min}}{T_{max}} = 1 - \frac{540R}{2400R} = 0.775\) Thus, the thermal efficiency of a Carnot cycle operating between the same temperature limits as the Otto cycle is \(77.5\%\). In summary, (a) The heat transferred during the heat-addition process is \(826.57 \, \frac{Btu}{lb}\). (b) The thermal efficiency of the Otto cycle is \(57.15\%\). (c) The thermal efficiency of a Carnot cycle operating between the same temperature limits is \(77.5\%\).