Question

An air-standard cycle, called the dual cycle, with constant specific heats is executed in a closed piston-cylinder system and is composed of the following five processes: \(1-2 \quad\) Isentropic compression with a compression ratio \(r=V_{1} / V_{2}\) \(2-3 \quad\) Constant volume heat addition with a pressure ratio, \\[ r_{p}=P_{3} / P_{2} \\] \(3-4 \quad\) Constant pressure heat addition with a volume ratio \\[ r_{c}=V_{4} / V_{3} \\] \(4-5 \quad\) Isentropic expansion while work is done until \(V_{5}=V_{1}\) \(5-1 \quad\) Constant volume heat rejection to the initial state (a) Sketch the \(P\) -V and \(T\) -s diagrams for this cycle. (b) Obtain an expression for the cycle thermal efficiency as a function of \(k, r, r_{c},\) and \(r_{p}\) (c) Evaluate the limit of the efficiency as \(r_{p}\) approaches unity and compare your answer with the expression for the Diesel cycle efficiency. (d) Evaluate the limit of the efficiency as \(r_{c}\) approaches unity and compare your answer with the expression for the Otto cycle efficiency.

Short answer

Based on the provided step-by-step solution, the short answer is: The dual cycle is an air-standard cycle that involves a combination of constant volume and constant pressure heat addition. To analyze this cycle, first sketch the P-V and T-s diagrams by illustrating the five processes involved (isentropic compression, constant volume heat addition, constant pressure heat addition, isentropic expansion, and constant volume heat rejection). Next, derive an expression for the thermal efficiency using the given ratios and the formula η = W_out / Q_in. Finally, evaluate the limits of efficiency as the pressure ratio (r_p) and volume ratio (r_c) approach unity, and compare these results with the efficiencies of the Otto and Diesel cycles.

Step-by-step solution

Sketch the P-V and T-s diagrams for the dual cycle

To sketch the P-V and T-s diagrams, we need to illustrate the five processes that compose the dual cycle: 1. Isentropic compression (1-2) 2. Constant volume heat addition (2-3) 3. Constant pressure heat addition (3-4) 4. Isentropic expansion (4-5) 5. Constant volume heat rejection (5-1) These can be represented graphically on P-V and T-s diagrams, using the given ratios for the processes. Note that isentropic processes appear as vertical lines in the T-s diagram, constant volume processes as horizontal lines in the P-V diagram, and constant pressure processes as horizontal lines in the T-s diagram.

Obtain an expression for the cycle thermal efficiency

To derive an expression for the thermal efficiency, we need to find the heat input (Q_in) and the work output (W_out) to determine the thermal efficiency using the formula: Thermal efficiency (η) = W_out / Q_in The heat input is the sum of the heat added during constant volume heat addition (Q_23) and the constant pressure heat addition (Q_34). The work output is the net work done by the system during the cycle, which can be determined as: W_out = W_12 + W_34 - W_51 Now, using the ratios r, r_c, and r_p, along with the temperature and pressure ratios, we need to derive expressions for Q_23, Q_34, and the work terms W_12, W_34, and W_51, and plug them into the formula for thermal efficiency to obtain an expression for η as a function of k, r, r_c, and r_p.

Evaluate the limit of efficiency as \(r_p\) approaches unity

To find the limit of the efficiency as the pressure ratio r_p approaches unity, we need to take the limit of the derived expression for the thermal efficiency as \(r_p \rightarrow 1\) and compare this result with the expression for the Diesel cycle efficiency.

Evaluate the limit of efficiency as \(r_c\) approaches unity

Similarly, to find the limit of the efficiency as the volume ratio r_c approaches unity, we need to take the limit of the derived expression for the thermal efficiency as \(r_c \rightarrow 1\) and compare this result with the expression for the Otto cycle efficiency. By following these steps, you will have sketched the P-V and T-s diagrams for the dual cycle, derived an expression for the thermal efficiency as a function of the given ratios, and compared the limits of the efficiency with those of Otto and Diesel cycles.

Key concepts

Isentropic Processes

When we talk about isentropic processes within thermodynamic cycles like the dual cycle, we refer to transformations where entropy remains constant. During such a process, the system is perfectly insulated, meaning no heat is exchanged with the surroundings. Isentropic processes in an ideal cycle occur during both compression and expansion phases; however, realizing them in actual systems is quite challenging due to inevitable heat losses.
In a dual cycle, isentropic compression and expansion appear as vertical lines on the temperature-entropy (T-s) diagram since entropy stays unchanged. Understanding isentropic processes is crucial as they represent idealized scenarios where the machine operates with no internal losses, giving us an upper bound for the efficiency we could potentially achieve.

Constant Volume Heat Addition

Constant volume heat addition is a phase in the thermodynamic cycle where heat is transferred to the working fluid without any change in volume. This is a crucial step in cycles like the Otto cycle, which is part of the dual cycle, and relates to the point where the fuel-air mixture ignites in an internal combustion engine.
On a pressure-volume (P-V) diagram, this process is depicted as a vertical line, as the volume remains constant while the pressure rises. The heat added during this process results in an increase in the internal energy of the system, which in turn increases the temperature and pressure. For students, visualizing this process helps in understanding why, despite no volume change, other properties of the system can still change dramatically.

Constant Pressure Heat Addition

Conversely, constant pressure heat addition is where the system receives heat at a steady pressure. This step is characteristic of the Diesel cycle, also incorporated into the dual cycle, and it simulates fuel injection in diesel engines.
On the T-s diagram, it's represented as a horizontal line, indicating temperature increase at constant entropy. The ability of the system to expand while receiving heat at constant pressure is what characterizes this process, leading to a significant volume increase. This is a pivotal phase within the dual cycle, seeing as it differentiates the behavior and efficiency determinants from merely constant volume processes.

Thermal Efficiency Formula

The thermal efficiency formula is a mathematical representation that compares the work output of a thermal system to the heat input required to generate this work. For any heat engine, thermal efficiency (\theta) is defined as \[\eta = \frac{W_{\text{out}}}{Q_{\text{in}}}\] where \(W_{\text{out}}\) is the work output and \(Q_{\text{in}}\) is the heat added to the system.
For students delving into thermal system performance, mastering this formula is essential as it allows for the evaluation of the effectiveness of the cycle in converting heat into usable work. It is instrumental in comparing different cycles, like the dual cycle to the Otto or Diesel cycles. The higher the efficiency, the more work is extracted per unit of heat added, leading to better performance and energy conservation.

Diesel Cycle Efficiency

The efficiency of the Diesel cycle is significant as it governs the performance of diesel engines. It is characterized by constant pressure heat addition, differing from the Otto cycle. The efficiency of the Diesel cycle is given by the formula:\[\eta_{Diesel} = 1 - \frac{1}{r^{\gamma - 1}} \left( \frac{r_p^{\gamma} - 1}{\gamma (r_p - 1)} \right)\]where \(r\) is the compression ratio and \(r_p\) is the pressure ratio, with \(\gamma\) representing the specific heat ratio.
Understanding the Diesel cycle's efficiency is critical for mechanical engineering students, as it highlights the impacts of higher compression ratios and pressures, which typically lead to better efficiency but also to the need for stronger and more robust engine components.

Otto Cycle Efficiency

The Otto cycle is idealized for gasoline engines and is solely characterized by constant volume processes. Its efficiency is defined by the compression ratio of the engine and is given by the relation:\[\eta_{Otto} = 1 - \frac{1}{r^{\gamma - 1}}\]where \(r\) is the compression ratio and \(\gamma\) is the specific heat ratio of the working fluid.
The significance of the Otto cycle efficiency comes from its simplicity and direct relationship with the compression ratio. Higher compression ratios signify a greater efficiency as they allow for a larger portion of the heat input to be converted into work. The Otto cycle sets a benchmark against which the efficiency of real engines and other cycles, such as the Diesel or dual cycles, can be measured.