Question

The compression ratio of an ideal dual cycle is 14. Air is at \(100 \mathrm{kPa}\) and \(300 \mathrm{K}\) at the beginning of the compression process and at \(2200 \mathrm{K}\) at the end of the heat-addition process. Heat transfer to air takes place partly at constant volume and partly at constant pressure, and it amounts to \(1520.4 \mathrm{kJ} / \mathrm{kg} .\) Assuming variable specific heats for air, determine \((a)\) the fraction of heat transferred at constant volume and \((b)\) the thermal efficiency of the cycle.

Short answer

Answer: To determine the fraction of heat transferred at constant volume and the thermal efficiency, follow these steps: 1. Calculate initial specific volume and final specific volumes during the compression process using the compression ratio and ideal gas law. 2. Calculate the heat added during constant-volume and constant-pressure processes, and find the ratios q_v / q_p and T_3 / T_4. 3. Calculate the fraction of heat transferred at constant volume using the ratio q_v / q_p and the given total heat transfer. 4. Calculate the thermal efficiency of the cycle using the calculated values for heat transfer and heat rejected during the constant volume process.

Step-by-step solution

Calculate initial specific volume and final specific volumes during the compression process

Using the compression ratio and the ideal gas law, we can determine the initial specific volume (\(v_1\)) and final specific volume (\(v_2\)) during the compression process. The compression ratio is given as: Compression ratio = \(\frac{v_1}{v_2} = 14\) Also, using the ideal gas law, we know that: \(p_1v_1 = RT_1\) and \(p_2v_2 = RT_2\) Since the compression process is adiabatic: \(v_1^\gamma = v_2^\gamma \frac{T_1}{T_2}\) Thus, we can use the given values for \(p_1\), \(T_1\), \(T_2\), and the compression ratio to find \(v_1\) and \(v_2\).

Calculate the heat added during constant-volume and constant-pressure processes

Let the heat added during the constant volume process be \(q_{v}\), and during the constant pressure process be \(q_{p}\). We're given that the total heat added is: \(q_\text{total} = q_v + q_p = 1520.4 \, \text{kJ/kg}\) We can use the following equation for constant volume process (\(\text{Process } 1\to2\)) and constant pressure process (\(\text{Process } 3\to4\)), respectively: \(q_v = mC_v(T_3 - T_2)\) and \(q_p = mC_p(T_4 - T_3)\) From these equations, we can find the ratios \(q_v / q_p\) and \(T_3 / T_4\). The ratio \(T_3 / T_4\) can be obtained from the given \(T_2\) and equation for constant volume: \(v_2T_2^\text{ratio} = v_3T_3^\text{ratio}\)

Calculate the fraction of heat transferred at constant volume

Using the ratio \(q_v / q_p\) and the given total heat transfer, we can solve for the fraction of heat transferred at constant volume: \(\frac{q_v}{q_\text{total}} = \frac{q_v}{q_v + q_p}\)

Calculate the thermal efficiency of the cycle

The thermal efficiency (\(\eta_\text{cycle}\)) of the cycle can be found by: \(\eta_\text{cycle} = 1 - \frac{q_\text{out}}{q_\text{in}}\) In this case, \(q_\text{in}\) is the total heat transfer, and \(q_\text{out}\) is the heat rejected during the constant volume process from \(5 \to 1\) as: \(q_\text{out} = mC_v(T_5 - T_1)\) Using the calculated values for \(q_\text{in}\) and \(q_\text{out}\), find the thermal efficiency of the cycle. Finally, we have the results for the fraction of heat transferred at constant volume and the thermal efficiency of the cycle.

Key concepts

Understanding Compression Ratio

Compression ratio is a crucial factor in thermal engines that directly affects their performance and efficiency. It represents the ratio of the volume of its combustion chamber from its largest capacity to its smallest capacity. A higher compression ratio in an ideal dual cycle indicates that the engine compresses the air-fuel mixture more before igniting it, leading to more power out of each explosion within the cylinder.

In the given problem, we have a compression ratio of 14, which means the volume is being reduced 14 times. Understanding this concept is essential for students as it directly relates to the efficiency of an engine. The thermal efficiency increases with a higher compression ratio, up to a point where other factors limit further gains. It is important, however, to understand that practical engines have limits on the compression ratio, influenced by fuel properties and the tendency to knock or self-ignite prematurely.

Specific Volume and its Role in Thermal Cycles

Specific volume plays a pivotal role in analyzing thermal cycles such as the ideal dual cycle. It is defined as the volume per unit mass of a substance and is the inverse of density. In the context of the exercise, specific volume helps us understand the changes in volume during the compression process within the engine cycle.

By applying the ideal gas law, we can determine the initial and final specific volumes in the cycle using the given temperatures and compression ratio. The significance of specific volume becomes evident when considering volume changes throughout the cycle; it allows for the calculation of work done during the processes. In thermodynamics, understanding how to manipulate specific volume, temperature, and pressure is crucial to predict the behavior of a system and to calculate the work and heat transfer for each process within the cycle.

Heat Transfer in Dual Cycle Engines

Heat transfer is a fundamental process in any thermal cycle, and understanding it is critical for determining the efficiency of engines. In a dual cycle engine, heat transfer occurs in two primary ways: at constant volume and at constant pressure.

The amount of heat added to the system during these two processes determines the temperature rise and the amount of work the engine can perform. In the exercise, we are given that heat transfer to the air is 1520.4 kJ/kg, split between these two processes. For students, grasping how heat transfer differs between constant volume and constant pressure processes is key to analyzing an ideal dual cycle and, more broadly, how different types of work and heat interactions can affect the performance of the cycle.

Thermal Efficiency and its Calculation

Thermal efficiency is the measure of how well an engine converts heat into work. It is a critical concept because it reflects the performance of the cycle by indicating the ratio of useful work output to heat input. The higher the thermal efficiency, the more cost-effective the engine is, since it uses less fuel to produce the same amount of work.

In the context of the dual cycle, the thermal efficiency is calculated by taking into account the heat added to the system and the heat rejected. The exercise required us to determine the thermal efficiency by finding the difference between these two quantities. To improve student comprehension, it's critical to convey the idea that thermal efficiency serves as an indicator of an engine's overall energy conversion effectiveness, thereby directly affecting its economic and environmental impact.