Question

A Carnot cycle operates between the temperature limits of 300 and \(2000 \mathrm{K},\) and produces \(600 \mathrm{kW}\) of net power. The rate of entropy change of the working fluid during the heat addition process is \((a) 0\) (b) \(0.300 \mathrm{kW} / \mathrm{K}\) \((c) 0.353 \mathrm{kW} / \mathrm{K}\) \((d) 0.261 \mathrm{kW} / \mathrm{K}\) \((e) 2.0 \mathrm{kW} / \mathrm{K}\)

Short answer

Question: Calculate the rate of entropy change of the working fluid during the heat addition process in a Carnot cycle operating between two temperature limits, 300 K and 2000 K, producing 600 kW of net power. Answer: The rate of entropy change of the working fluid during the heat addition process in this Carnot cycle is 0.353 kW/K.

Step-by-step solution

Find the heat addition

First, let's find the heat added to the system during the heat addition process. We know that the efficiency of the Carnot cycle can be expressed as: Efficiency = \(\frac{T_{H} - T_{L}}{T_{H}}\), where \(T_{H}\) is the high temperature limit (2000 K) and \(T_{L}\) is the low temperature limit (300 K). Now, we can calculate the heat addition, \(Q_{H}\), by using the formula: Efficiency = \(\frac{W_{net}}{Q_{H}}\), where \(W_{net}\) is the net work output (600 kW).

Calculate Efficiency

Calculate the efficiency of the Carnot cycle using the formula from step 1: Efficiency = \(\frac{2000 - 300}{2000} = \frac{1700}{2000} = 0.85\)

Calculate Heat Addition

Now, calculate the heat addition using the formula from step 1, using the efficiency found in step 2: \(Q_{H} = \frac{W_{net}}{\text{Efficiency}} = \frac{600\mathrm{kW}}{0.85} = 705.88\mathrm{kW}\)

Calculate the rate of entropy change

Finally, calculate the rate of entropy change during the heat addition process using the formula: Rate of entropy change = \(\frac{\Delta S}{\Delta t} = \frac{Q_{H}}{T_{H}\Delta t}\) Rearranging this formula, we get: Rate of entropy change = \(\frac{Q_{H}}{T_{H}}\) Now, substitute the values of \(Q_{H}\) and \(T_{H}\): Rate of entropy change = \(\frac{705.88\mathrm{kW}}{2000\mathrm{K}} = 0.353 \mathrm{kW} / \mathrm{K}\) So, the correct answer is (c) 0.353 kW/K.

Key concepts

Entropy Change

Entropy is a measure of disorder or randomness in a system, and understanding its change is crucial in thermodynamics, particularly when studying the Carnot cycle. In a Carnot cycle, which is a theoretical idealization, entropy change occurs during heat transfer processes.

During the heat addition phase of the Carnot cycle, entropy increases because heat energy is entering the system, which tends to increase disorder. Conversely, during the heat rejection phase, entropy decreases because the system is losing heat energy. In a perfect Carnot cycle, the total change in entropy over one complete cycle is zero, because the cycle is reversible, and any entropy increase from heat addition is balanced by the entropy decrease during heat rejection.

To calculate the rate of entropy change during heat addition, as done in the given exercise, we use the relationship \(\frac{Q_{H}}{T_{H}}\), where \(Q_{H}\) is the heat added at the constant high temperature reservoir, \(T_{H}\). This relation implies that the rate of entropy change is directly proportional to the heat added and inversely proportional to the temperature at which heat is added. Thus, for an effective and efficient Carnot engine, managing the change in entropy is instrumental in maximizing output.

Heat Addition

Heat addition is a significant stage in the Carnot cycle, impacting the system's efficiency and performance. It occurs at a constant high temperature when the working fluid (such as a gas) absorbs energy, causing an expansion. This stage is crucial because it represents the input of energy into the thermodynamic cycle, which later translates into work done by the system.

In the context of the Carnot cycle example, we calculated the total amount of heat added, \(Q_{H}\), by using the known values of net work output and the efficiency of the cycle. It's important to understand that the efficiency relates to how well the cycle converts heat into work. The heat that is not converted into work is rejected to a lower temperature reservoir. By maximizing \(Q_{H}\), while keeping the temperature limits constant, we can increase the net work output, which is the primary goal of any engine or heat pump operating on a Carnot cycle.

Carnot Efficiency

Carnot efficiency is the theoretical maximum efficiency that a heat engine can achieve during the conversion of heat into work while operating between two temperature reservoirs. It serves as a benchmark for measuring the performance of real engines, which can never reach Carnot efficiency due to practical limitations such as friction and non-reversible processes.

The formula for calculating Carnot efficiency is \(1 - \frac{T_{L}}{T_{H}}\), where \(T_{L}\) and \(T_{H}\) are the absolute temperatures of the cold and hot reservoirs, respectively. In our problem, the efficiency calculation provided a value of 0.85, indicating that 85% of the heat addition could theoretically be converted into work. Any deviation from this ideal scenario in real-world engines is a consequence of irreversibilities and indicates the potential areas where improvements can be made to improve the efficiency of heat engines. Understanding Carnot efficiency is not only academically important but also has practical implications for energy conservation and environmental impacts.