Question

During the isothermal heat addition process of a Carnot cycle, \(900 \mathrm{kJ}\) of heat is added to the working fluid from a source at \(400^{\circ} \mathrm{C}\). Determine \((a)\) the entropy change of the working fluid, \((b)\) the entropy change of the source, and \((c)\) the total entropy change for the process.

Short answer

Question: Determine the entropy change of the working fluid, the entropy change of the source, and the total entropy change during an isothermal heat addition of 900 kJ in a Carnot cycle with a source temperature of 400°C. Answer: The entropy change of the working fluid is approximately 1337.15 J/K, the entropy change of the source is approximately -1337.15 J/K, and the total entropy change is 0 J/K.

Step-by-step solution

Convert the source temperature to Kelvin

To convert the temperature from Celsius to Kelvin, we add 273.15 to the given temperature in Celsius. $$ T_{source} = 400 + 273.15 = 673.15\,K $$

Calculate the entropy change of the working fluid

To calculate the entropy change of the working fluid, we can use the following formula for an isothermal heat addition process: $$ \Delta S_{fluid} = \frac{Q_{added}}{T_{source}} $$ Where \(\Delta S_{fluid}\) is the entropy change of the working fluid, \(Q_{added}\) is the heat added to the working fluid, and \(T_{source}\) is the temperature of the source. Plugging in the values: $$ \Delta S_{fluid} = \frac{900\,\mathrm{kJ}}{673.15\,K} = \frac{900\,000\,\mathrm{J}}{673.15\,K} \approx 1337.15\, \frac{\mathrm{J}}{\mathrm{K}} $$

Calculate the entropy change of the source

The entropy change of the source can be found using the same formula, with a negative sign because the heat is transferred from the source to the working fluid: $$ \Delta S_{source} = -\frac{Q_{added}}{T_{source}} $$ Using the values: $$ \Delta S_{source} = -\frac{900\,000\,\mathrm{J}}{673.15\,K} \approx -1337.15\, \frac{\mathrm{J}}{\mathrm{K}} $$

Calculate the total entropy change

The total entropy change is the sum of the entropy changes of the working fluid and the source: $$ \Delta S_{total} = \Delta S_{fluid} + \Delta S_{source} $$ Plugging in the values: $$ \Delta S_{total} = 1337.15\, \frac{\mathrm{J}}{\mathrm{K}} + (-1337.15\, \frac{\mathrm{J}}{\mathrm{K}}) = 0\, \frac{\mathrm{J}}{\mathrm{K}} $$ So the total entropy change is zero, which is expected for a Carnot cycle. To summarize: (a) Entropy change of the working fluid: \(\Delta S_{fluid} \approx 1337.15\, \frac{\mathrm{J}}{\mathrm{K}}\) (b) Entropy change of the source: \(\Delta S_{source} \approx -1337.15\, \frac{\mathrm{J}}{\mathrm{K}}\) (c) Total entropy change: \(\Delta S_{total} = 0\, \frac{\mathrm{J}}{\mathrm{K}}\)

Key concepts

Isothermal Heat Addition

The isothermal heat addition is a critical process in a Carnot cycle where heat is transferred to the working fluid without a change in temperature. During this process, the working fluid absorbs heat energy from a high-temperature reservoir. This process occurs at constant temperature, which is a defining characteristic of the isothermal expansion in the Carnot cycle.

In simple terms, imagine boiling water in a pot; as long as the water is boiling, it remains at a constant temperature despite the continuous input of heat from the stove. This steady temperature in a Carnot cycle allows us to apply specific thermodynamic formulas that relate heat transfer to entropy change.

Entropy Change Calculation

Entropy can be thought of as a measure of disorder or randomness in a system. The concept of entropy is vital as it relates to the second law of thermodynamics, which states that in an isolated system, entropy tends to increase over time.

For example, if you dye a glass of water by dropping in ink, the ink will spread out until it evenly colors the water, resulting in an increase in entropy. When calculating entropy change during the isothermal heat addition, the formula \(\Delta S = \frac{Q}{T}\) is used. This equation takes the heat added to the system, denoted by \(Q\), and divides it by the thermodynamic temperature, \(T\), in Kelvin. It's important to use Kelvin because entropy is a measure of absolute disorder, and Kelvin is an absolute thermodynamic scale.

Thermodynamic Temperature Conversion

Temperature conversion is a fundamental step in entropy calculations because thermodynamic equations require temperature to be in Kelvin. Celsius can be converted to Kelvin by simply adding 273.15.

For instance, if you're given a temperature of 25 degrees Celsius and need to convert it to Kelvin to use in an equation, simply add 273.15, so the temperature in Kelvin would be 298.15 K. It's crucial to make this conversion because thermodynamics is based on absolute measurements, and Kelvin provides a scale starting from absolute zero, where theoretically all molecular motion stops.

Carnot Cycle Efficiency

The Carnot cycle demonstrates the most efficient engine theoretically possible, one that operates between two heat reservoirs. The efficiency of a Carnot engine is a measure of the work output divided by the heat input, and it is determined by the temperatures of the hot and cold reservoirs.

The formula for efficiency \(\eta\) in a Carnot engine is given by \(\eta = 1 - \frac{T_{cold}}{T_{hot}}\), where \(T_{cold}\) and \(T_{hot}\) are the absolute temperatures of the cold and hot reservoirs, respectively. Notably, no actual engine can reach this efficiency due to practical limitations such as friction, and the fact that real processes are not truly reversible.