Entropy and Thermodynamics Concepts

ABE 210 - Lecture 7: Entropy

Objectives

• Understand and apply the second law of thermodynamics to various processes.
• Define and quantify entropy to understand its effects in thermodynamics.
• Establish and explain the principle of the increase of entropy.
• Calculate the changes in entropy during processes involving pure substances, incompressible substances, and ideal gases.

Clausius Inequality

• Definition: The cyclic integral of \frac{dQ}{T} is always less than or equal to zero.

  \oint \frac{dQ}{T} \leq 0

• Interpretation: Indicates the directionality of heat transfer in thermodynamic cycles.

Carnot Heat Engine Cycle

• Relationship between heat transfers and temperatures:
  \frac{QH}{QL} = \frac{TH}{TL}
• Thus,
  \frac{QH}{TH} = \frac{QL}{TL}

Entropy Changes

• In internally reversible systems, the change in entropy can be defined as:
  \delta S = \int1^2 \frac{dQ{rev}}{T}
• Overall change in entropy:
  ΔS = S2 - S1
• For processes,
  ΔS = \int \frac{dQ_{rev}}{T}

Internally Reversible Isothermal Heat Transfer

• For isothermal processes where the temperature remains constant:
  ΔS = \int1^2 \frac{dQ{rev}}{T0} = \frac{Q}{T0}
• Example: Analyze a piston-cylinder device at 300K (water mixture)

Increase of Entropy Principle

• For a cycle (1-2 arbitrary and 2-1 internally reversible):
  rac{dQ}{T} + S1 - S2 \leq 0
  S2 - S1 \geq \int1^2 \frac{dQ}{T} + S{gen}
• Entropy Generation: Due to irreversibilities present in the process.

For Isolated Systems

• ΔS{gen} = \int1^2 \frac{dQ}{T} + S_{gen} \geq 0

• Entropy will never decrease in isolated systems.

• Each component contributes to overall system entropy, thus never less than zero.

Remarks on Entropy

1. Processes have specific directions based on entropy principles, violating them is not possible.
2. Entropy is nonconserved; it conserves only in ideal reversible processes.
3. Irreversibilities degrade system performance and can be quantified by entropy generation.

Heat Transfer Example

• Evaluate entropy generation during heat transfer:
  • Source at 1000 K transferring 1000 J to sinks at varied temperatures (500 K, 250 K, 100 K).
  • More irreversible transfer results in higher entropy generation.

Entropy Change for Pure Substances

• Fixed entropy value once the state is established:
  ΔS = mΔs = m(s2 - s1)
• Where: s is specific entropy in kJ/kg imes K.

Property Diagrams Involving Entropy

• Key diagrams: (T-S) and (h-s) used for illustrating thermodynamic processes.

Isentropic Processes

• Definition: Processes with constant entropy, achieved through reversible and adiabatic conditions.
• Analyze example with water compressed and determine temperature and work output.

Tds Relationship

• Using energy equations:
  \delta Qi - \delta Wi = dU
• Resulting in variations of enthalpy and state changes.

Entropy Change of Liquids and Solids (Incompressible)

• For incompressible substances, changes are expressed as:
  s2 - s1 = cp \ln \left(\frac{T2}{T_1}\right)

Entropy Generation in Open Systems

• Entropy transfer equations account for heat, work, and mass flow:
  S{gen} = S{in} - S{out} + S{gen}

Example of Entropy Generation During Compression

• Analyze nitrogen being compressed in an adiabatic context, yield entropy generation results.

Summary of Key Concepts

• Clausius inequality outlines the limits for cyclic processes; understanding entropy shifts crucial for thermodynamics.
• Focus on properties and practical applications of entropy in energy systems, including implications for work and efficiency in systems.