7.4 The T ds relations and entropy generation

The T ds relations and entropy generation are crucial concepts in thermodynamics. They help us understand how energy changes in different processes and why real-world systems aren't perfect. These ideas are key to grasping the second law of thermodynamics and its practical implications.

By exploring these concepts, we can see how entropy always increases in real processes. This explains why we can't create perfect machines and why some energy is always "lost" as waste heat. Understanding these principles is essential for designing efficient systems and solving real-world engineering problems.

T ds Relations for Thermodynamic Processes

Derivation and Application of T ds Relations

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• Derive the T ds relations, a set of equations that relate changes in temperature (T) and specific entropy (s) for different thermodynamic processes (constant volume, constant pressure, constant temperature, and adiabatic processes)
• Express the general form of the T ds relation as $T ds = du + P dv$, where $du$ is the change in specific internal energy and $P dv$ is the boundary work
• Apply the T ds relations to calculate changes in specific entropy for various thermodynamic processes, given the appropriate thermodynamic properties and process conditions

Simplified T ds Relations for Specific Processes

• Simplify the T ds relation for a constant volume process ($dv = 0$) to $T ds = du$, indicating that the change in specific entropy is directly proportional to the change in specific internal energy
• Modify the T ds relation for a constant pressure process to $T ds = dh - v dP$, where $dh$ is the change in specific enthalpy and $v$ is the specific volume
• Express the T ds relation for an isothermal process ($dT = 0$) as $T ds = dq$, where $dq$ is the heat transfer per unit mass
• Reduce the T ds relation for an adiabatic process ($dq = 0$) to $ds = 0$, implying that the specific entropy remains constant during the process

Entropy Generation and Real Processes

Concept and Implications of Entropy Generation

• Define entropy generation as the increase in the total entropy of a system and its surroundings due to irreversibilities in real processes
• Identify irreversibilities as deviations from ideal, reversible processes, including phenomena such as friction, heat transfer across finite temperature differences, mixing, and chemical reactions
• Recognize entropy generation as a measure of the lost potential for work in a system, representing the energy that cannot be converted into useful work due to irreversibilities
• Understand that the second law of thermodynamics states that the total entropy of an isolated system always increases or remains constant, and entropy generation is the mechanism by which this law is satisfied in real processes

Characteristics and Consequences of Entropy Generation

• Emphasize that entropy generation is always non-negative and is zero only for reversible processes; the greater the irreversibilities in a process, the higher the entropy generation
• Explain how the presence of entropy generation in real processes limits the efficiency of energy conversion devices (heat engines and heat pumps) by representing a loss of available energy
• Highlight the importance of minimizing entropy generation as a key objective in the design and optimization of thermodynamic systems to improve their efficiency and performance

Quantifying Entropy Generation

Entropy Balance Equation

• Introduce the entropy balance equation as a means to quantify entropy generation in a system, accounting for entropy changes due to heat transfer, mass flow, and internal irreversibilities
• Present the entropy balance equation for a closed system as $dS/dt = \sum(dQ/T) + \dot{S}_{gen}$, where $dS/dt$ is the rate of change of entropy in the system, $\sum(dQ/T)$ is the sum of entropy transfers due to heat transfer at the system boundaries, and $\dot{S}_{gen}$ is the rate of entropy generation within the system
• Extend the entropy balance equation for an open system to include additional terms for entropy transfer due to mass flow: $dS/dt = \sum(dQ/T) + \sum(\dot{m}_{in} \cdot s_{in}) - \sum(\dot{m}_{out} \cdot s_{out}) + \dot{S}_{gen}$, where $\dot{m}_{in}$ and $\dot{m}_{out}$ are the mass flow rates into and out of the system, and $s_{in}$ and $s_{out}$ are the specific entropies of the incoming and outgoing streams, respectively

Calculating Entropy Generation

• Demonstrate how to calculate the rate of entropy generation ($\dot{S}_{gen}$) by applying the entropy balance equation to a system and solving for $\dot{S}_{gen}$, given the appropriate heat transfer rates, mass flow rates, and specific entropies
• Express the entropy generation rate as a function of the system's irreversibilities (friction losses, heat transfer across finite temperature differences, or pressure drops)
• Obtain the total entropy generation in a process by integrating the rate of entropy generation over the duration of the process: $\Delta S_{gen} = \int(\dot{S}_{gen} \cdot dt)$

Entropy Generation and the Second Law

Relationship between Entropy Generation and the Second Law

• State the second law of thermodynamics: the total entropy of an isolated system always increases or remains constant during a process, and this increase is due to entropy generation caused by irreversibilities
• Identify entropy generation as the mechanism by which the second law of thermodynamics is satisfied in real processes, accounting for the increase in total entropy due to irreversibilities
• Explain that for a reversible process, entropy generation is zero, and the total entropy of the system and its surroundings remains constant, consistent with the second law of thermodynamics
• Emphasize that in an irreversible process, entropy generation is always positive, leading to an increase in the total entropy of the system and its surroundings, in accordance with the second law

Implications of the Second Law and Entropy Generation

• Discuss how the second law of thermodynamics places a fundamental limit on the efficiency of thermodynamic processes and devices by requiring that some entropy be generated due to irreversibilities, reducing the available energy for useful work
• Introduce the Clausius inequality, $\int(dQ/T) \leq 0$, as a mathematical statement of the second law of thermodynamics, which can be derived from the entropy balance equation and the non-negative nature of entropy generation
• Highlight the importance of minimizing irreversibilities in thermodynamic systems to improve their efficiency and comply with the fundamental limits imposed by the second law, emphasizing the relationship between entropy generation and the second law of thermodynamics