Question

An ideal dual cycle has a compression ratio of 14 and uses air as the working fluid. At the beginning of the compression process, air is at 14.7 psia and \(120^{\circ} \mathrm{F}\), and occupies a volume of 98 in \(^{3}\). During the heat-addition process, 0.6 Btu of heat is transferred to air at constant volume and 1.1 Btu at constant pressure. Using constant specific heats evaluated at room temperature, determine the thermal efficiency of the cycle.

Short answer

The thermal efficiency of the ideal dual cycle is 12.04%.

Step-by-step solution

Determine initial and final temperatures during compression

Given the initial temperature \(T_1 = 120^{\circ} \mathrm{F}\), we need to convert it to absolute temperature scale. \(T_1 = 120 + 460 = 580\, \mathrm{R}\). The compression ratio is given as \(r_c = 14\), so the final temperature \(T_2\) during the compression process can be found by using the relation: $$ \frac{T_2}{T_1} = \left(\frac{V_1}{V_2}\right)^{\gamma - 1} \Rightarrow T_2 = T_1 \cdot r_c^{\gamma - 1} $$ where \(\gamma = \frac{c_p}{c_v}\) is the specific heat ratio for air and is approximately 1.4. Therefore, $$ T_2 = 580 \cdot 14^{(1.4-1)} \approx 1336.56\, \mathrm{R} $$

Determine heat addition at constant volume

The heat transferred at constant volume, \(q_{in1}\), equals 0.6 Btu. Convert it to Btu/lbm using the given volume and air properties (use the Ideal Gas Law: \(PV = mRT\)). The given data is: $$ P_1 = 14.7\, \mathrm{psia} = 14.7 \cdot 144\, \mathrm{psf} = 2116.8\, \mathrm{psf} $$ $$ V_1 = 98\, \mathrm{in^3} = \frac{98}{12^3}\, \mathrm{ft^3} = 0.056481\, \mathrm{ft^3} $$ The gas constant for air, \(R = 53.35\, \mathrm{ft.lb/(lbm.R)}\). Then we get the mass of air: $$ m = \frac{P_1 V_1}{RT_1} = \frac{2116.8 \cdot 0.056481}{53.35 \cdot 580} \approx 0.00407\, \mathrm{lbm} $$ Now we can find \(q_{in1}\) in Btu/lbm, $$ q_{in1} = \frac{0.6}{0.00407} \approx 147.394\, \mathrm{Btu/lbm} $$ Then, we can find the temperature at the end of the constant volume heat addition process: $$ T_3 = T_2 + \frac{q_{in1}}{c_v} $$ where \(c_v = 0.171\, \mathrm{Btu/(lbm.R)}\) is the specific heat of air at constant volume: $$ T_3 = 1336.56 + \frac{147.394}{0.171} \approx 2219.88\, \mathrm{R} $$

Determine heat addition at constant pressure

The heat transferred at constant pressure, \(q_{in2}\), equals 1.1 Btu. Convert it to Btu/lbm as before: $$ q_{in2} = \frac{1.1}{0.00407} \approx 270.188\, \mathrm{Btu/lbm} $$ We can find the temperature at the end of the constant pressure heat addition process: $$ T_4 = T_3 + \frac{q_{in2}}{c_p} $$ where \(c_p = 0.240\, \mathrm{Btu/(lbm.R)}\) is the specific heat of air at constant pressure: $$ T_4 = 2219.88 + \frac{270.188}{0.240} \approx 3353.17\, \mathrm{R} $$

Determine the work done during the cycle

Work done during the cycle \(W_{cycle}\) can be found by using the difference in heat addition and heat rejection. Heat rejection occurs during the constant volume heat rejection process (from \(T_4\) to \(T_1\)). Let \(q_{out}\) denote the heat rejected during this process: $$ q_{out} = c_v(T_4 - T_1) $$ Then, \(W_{cycle} = q_{in1} + q_{in2} - q_{out}\): $$ W_{cycle} = 147.394 + 270.188 - 0.171(3353.17 - 580) \approx 46.97\, \mathrm{Btu/lbm} $$

Calculate thermal efficiency

Finally, we can determine the thermal efficiency \(\eta_{th}\) of the cycle: $$ \eta_{th} = \frac{W_{cycle}}{q_{in1} + q_{in2}} = \frac{46.97}{(147.394 + 270.188)} = 0.1204 $$ Thus, the thermal efficiency of the cycle is 12.04%.

Key concepts

Ideal Dual Cycle

When discussing thermal cycles used in high-efficiency engines, one of the concepts encountered is the ideal dual cycle. This theoretical thermodynamic cycle is a combination of both the Otto and Diesel cycles, designed to optimize the performance characteristics of internal combustion engines. It consists of two isentropic processes (adiabatic compression and expansion), one constant volume heat addition process, and one constant pressure heat addition process. The intention behind combining these two heat addition processes is to take advantage of both the efficiency increase provided by the constant volume heat addition (like in the Otto cycle) and the increased work output facilitated by the constant pressure heat addition (like in the Diesel cycle). The thermal efficiency of this cycle, therefore, is influenced by both the amount of heat added and the manner in which it is added, regulated by the specific heat ratio and compression ratio of the working fluid— typically air for internal combustion engines.

Constant Volume Heat Addition

In the context of the ideal dual cycle, constant volume heat addition occurs immediately after the air is compressed to its smallest volume. It is at this stage that the air-fuel mixture is ignited, causing an increase in the temperature and pressure while maintaining the volume constant. Since this is similar to the heat addition phase in the Otto cycle, the thermodynamic properties of working fluid change dramatically due to the energy input, but the volume remains unchanged. This characteristic significantly affects the cycle's efficiency since heat addition at constant volume allows for the maximum increase in temperature leading to a higher thermal efficiency. It also plays a critical role in the calculation of the state properties, as seen in the provided exercise, where it determines the final temperature after heat addition.

Constant Pressure Heat Addition

Following the constant volume heat addition in the ideal dual cycle comes the constant pressure heat addition. During this phase, the heat is added while allowing the volume to increase, maintaining constant pressure. As seen in the Diesel cycle, this process results in additional work output as the piston moves in response to volume changes, a scenario where the engine can perform work while still receiving heat. The inclusion of this process in the dual cycle is an attempt to balance efficiency with work output, as it reduces the peak pressure compared to the Otto cycle, which only has constant volume heat addition, while potentially improving efficiency over a Diesel cycle, which only has constant pressure heat addition. In the exercise, this process allows further calculation of state properties and eventually the cycle's thermal efficiency.

Specific Heat Ratio

The specific heat ratio, symbolized by the Greek letter gamma \(\gamma\), is the ratio of the specific heat capacity at constant pressure \(c_p\) to that at constant volume \(c_v\). It is a crucial parameter in thermodynamics as it affects the thermal efficiency of thermodynamic cycles. For any ideal gas, the value of \(\gamma\) influences several processes within the cycle, such as the temperature change during adiabatic processes. The standard value for air is approximately 1.4. When dealing with constant volume or constant pressure processes, this ratio is vital for determining the temperature change associated with the respective heat addition, as detailed in the given solution. In engine performance, a higher \(\gamma\) generally relates to a higher thermal efficiency since it means that a larger portion of the total heat added is converted into useful work.

Compression Ratio

The compression ratio \(r_c\) of an engine is defined as the ratio of the volume of its combustion chamber from its largest capacity to its smallest capacity. It is a key factor in determining an engine’s performance and efficiency, with higher compression ratios typically leading to higher efficiencies. This is because higher compression ratios allow the engine to extract more mechanical energy from a given mass of air-fuel mixture. The compression ratio affects the thermal efficiency of the cycle by influencing both the amount of work done during the compression stroke and the temperature at which the heat addition process begins. In the provided exercise, a compression ratio of 14 is relatively high, indicating a potential for high efficiency in the theoretical cycle being considered. Calculating the final temperature after the compression process, as demonstrated in the solution steps, is crucial in assessing the thermal efficiency of the ideal dual cycle.