2.3 Entropy Changes in Various Processes

Entropy changes in various processes are crucial for understanding the Second Law of Thermodynamics. This topic explores how entropy behaves in reversible and irreversible processes, and how it affects the universe as a whole.

We'll dive into entropy changes for specific processes like isothermal and adiabatic, as well as entropy generation in systems. Understanding these concepts is key to grasping the broader implications of entropy in thermodynamics.

Entropy Changes in Processes

Calculating Entropy Changes in Reversible and Irreversible Processes

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• Entropy is a state function, and the change in entropy depends only on the initial and final states of the system, not the path taken between the states
• For a reversible process, the entropy change of a system can be calculated using the equation $\Delta S = \int \frac{dQ}{T}$, where $dQ$ is the heat transfer and $T$ is the absolute temperature at which the heat transfer occurs
• In an irreversible process, the entropy change of the system is greater than the integral of $dQ/T$. The inequality $\Delta S > \int \frac{dQ}{T}$ holds for irreversible processes
• The entropy change of the surroundings for a reversible process is equal to the negative of the entropy change of the system, maintaining a net entropy change of zero for the universe
• For an irreversible process, the entropy change of the surroundings is less than the negative of the entropy change of the system, resulting in a net increase in the entropy of the universe
• The second law of thermodynamics states that the entropy of an isolated system always increases for irreversible processes and remains constant for reversible processes

Entropy Changes in the Universe

• In a reversible process, the entropy change of the universe (system + surroundings) is zero, as the entropy change of the system is equal and opposite to the entropy change of the surroundings
• For an irreversible process, the entropy change of the universe is always positive, as the entropy change of the system is greater than the negative of the entropy change of the surroundings
• The increase in the entropy of the universe for an irreversible process is a measure of the irreversibility of the process and the lost potential for work
• Examples of irreversible processes that increase the entropy of the universe include spontaneous heat transfer from a hot object to a cold object, the mixing of two gases, and the expansion of a gas into a vacuum

Entropy Changes for Specific Processes

Isothermal and Adiabatic Processes

• In an isothermal process, the temperature remains constant, and the entropy change can be calculated using the equation $\Delta S = \frac{Q}{T}$, where $Q$ is the heat transfer and $T$ is the constant absolute temperature
• For an ideal gas undergoing an isothermal process, the entropy change is given by $\Delta S = nR \ln \frac{V_2}{V_1}$, where $n$ is the number of moles, $R$ is the universal gas constant, and $V_1$ and $V_2$ are the initial and final volumes, respectively
• In an adiabatic process, there is no heat transfer between the system and the surroundings ($Q = 0$). As a result, the entropy change of the system is zero ($\Delta S = 0$) for a reversible adiabatic process
• For an irreversible adiabatic process, the entropy of the system increases ($\Delta S > 0$) due to internal irreversibilities, such as friction or turbulence

Polytropic Processes

• A polytropic process is characterized by the equation $PV^n = \text{constant}$, where $P$ is pressure, $V$ is volume, and $n$ is the polytropic exponent
• The entropy change for a polytropic process can be calculated using the equation $\Delta S = nC_v \ln \frac{T_2}{T_1}$, where $C_v$ is the specific heat at constant volume and $T_1$ and $T_2$ are the initial and final temperatures, respectively
• Special cases of polytropic processes include isothermal ($n = 1$), isobaric ($n = 0$), isochoric ($n = \infty$), and adiabatic ($n = \gamma = C_p/C_v$) processes
• The polytropic exponent $n$ determines the slope of the process curve on a $PV$ diagram, with higher values of $n$ corresponding to steeper curves

Entropy Generation in Systems

Causes and Effects of Entropy Generation

• Entropy generation occurs due to irreversibilities in real-world systems, such as friction, heat transfer across finite temperature differences, mixing, and chemical reactions
• The Gouy-Stodola theorem states that the lost work due to irreversibilities is directly proportional to the entropy generated multiplied by the ambient temperature ($W_{\text{lost}} = T_0 S_{\text{gen}}$, where $T_0$ is the ambient temperature and $S_{\text{gen}}$ is the entropy generated)
• Entropy generation leads to a decrease in the efficiency and performance of real-world systems, such as heat engines, refrigerators, and power plants
• Minimizing entropy generation is crucial for improving the efficiency and sustainability of energy systems. This can be achieved through better design, use of advanced materials, and optimized operating conditions

Exergy and Entropy Generation

• The concept of exergy (the maximum useful work that can be obtained from a system in a given environment) is closely related to entropy generation
• Exergy destruction is proportional to the entropy generated in a process, as given by the equation $\dot{Ex}_{\text{dest}} = T_0 \dot{S}_{\text{gen}}$, where $\dot{Ex}_{\text{dest}}$ is the rate of exergy destruction, $T_0$ is the ambient temperature, and $\dot{S}_{\text{gen}}$ is the rate of entropy generation
• Analyzing exergy destruction in a system helps identify the locations and magnitudes of irreversibilities, guiding efforts to improve system efficiency
• Examples of systems where exergy analysis is useful include power plants (Rankine and Brayton cycles), refrigeration systems (vapor-compression and absorption cycles), and heat exchangers

Entropy Balance for Systems

Closed Systems

• The entropy balance for a closed system is given by $\Delta S_{\text{system}} = \int \frac{dQ}{T} + S_{\text{gen}}$, where $\Delta S_{\text{system}}$ is the change in entropy of the system, $\int \frac{dQ}{T}$ represents the entropy transfer due to heat, and $S_{\text{gen}}$ is the entropy generated within the system due to irreversibilities
• In a reversible process, the entropy generation term is zero ($S_{\text{gen}} = 0$), and the entropy change of the system is equal to the entropy transfer due to heat
• For an irreversible process in a closed system, the entropy generation term is positive ($S_{\text{gen}} > 0$), and the entropy change of the system is greater than the entropy transfer due to heat
• The entropy balance for a closed system can be applied to various processes, such as the heating or cooling of a substance, the mixing of two substances, and the compression or expansion of a gas

Open Systems

• For an open system, the entropy balance equation includes an additional term to account for the entropy transfer associated with mass flow: $\frac{dS_{\text{system}}}{dt} = \sum \frac{Q_j}{T_j} + \sum \dot{m}_i s_i - \sum \dot{m}_o s_o + \dot{S}_{\text{gen}}$, where $Q_j$ is the heat transfer rate at the boundary where the temperature is $T_j$, $\dot{m}_i$ and $\dot{m}_o$ are the mass flow rates into and out of the system, respectively, and $s_i$ and $s_o$ are the specific entropies of the inlet and outlet streams, respectively
• The entropy balance equation can be applied to various real-world systems, such as power plants, heat exchangers, and combustion processes, to analyze their performance and identify sources of irreversibility
• In steady-state processes, the rate of change of entropy within the system is zero ($\frac{dS_{\text{system}}}{dt} = 0$), simplifying the entropy balance equation
• The entropy balance can be combined with other conservation laws (mass and energy) to solve problems involving the analysis and design of thermodynamic systems