ChemE 260 The Clausius Inequality & Entropy

ChemE 260 The Clausius Inequality& Entropy Dr. William Baratuci Senior Lecturer Chemical Engineering Department University of Washington TCD 7: A & BCB 6: 1 May 3, 2005

Hot Reservoir QH HER Wrev QC,rev Cold Reservoir The Clausius Inequality • Cyclic Integrals • Integrate through all the steps ina cycle and return to the initial state. • Inexact Differentials: Q & W • Used for path variables, Q and W • Evaluating Cyclic Integrals • Example 1: Carnot HE • Example 2: Carnot HE Baratuci ChemE 260 May 3, 2005

Hot Reservoir QH HEIrr Wirr QC,irr Cold Reservoir Clausius: Int. Rev. and Irrev. Cycles • Reversible Cycle, such as Carnot: • Kelvin: or: • Therefore: • Irreversible Cycles: • Definition of efficiency : • 1st Law : • All Cycles : Baratuci ChemE 260 May 3, 2005

Entropy • Cycle 1-A-B-2: • Cycle 1-A-C-2: • Subtract Eqns: • Does not depend on path ! • It is a state variable or property ! • Definitionof Entropy: Baratuci ChemE 260 May 3, 2005

S: Int. Rev. & Irrev. Processes • Change in Entropy: • Problem: Since Int. Rev. processes do not exist, how do we evaluate S ? • Special Case: Reversible, Isothermal Processes Especially useful for evaluating Sreservoir because Treservoir = constant Baratuci ChemE 260 May 3, 2005

Entropy of a Pure Substance • S = fxn( T, P, phase ) • NIST Webbook • Thermophysical Properties of Fluid Systems • Specific entropy is listed in the thermodynamic data tables • Observations: All substances at all T : IG and many real gases : • Sat’d Mixtures: • Subcooled Liquids: Where P* = vapor pressure = Psat Baratuci ChemE 260 May 3, 2005

T-S Diagram Baratuci ChemE 260 May 3, 2005

Carnot Cycle • Steps • 1-2: Isothermal expansion • 2-3: Adiabatic expansion • 3-4: Isothermal compression • 4-1: Adiabatic compression • Step 1-2: Isentropic ! • Step 2-3: • Step 3-4: Isentropic ! • Step 4-1: Baratuci ChemE 260 May 3, 2005

Heat, Work & TS Diagrams • 1st Law Cycle • QH = area under path for step 1-2 • QC = positive area under path for step 3-4 • W = area enclosed by the cycle ! Baratuci ChemE 260 May 3, 2005

Next Class … • Principle of Increasing Entropy & Entropy Generation • This will let us express the 2nd Law in terms of Entropy • This very powerful result will let us perform 2nd Law Analysis on processes as well as cycles. • Fundamental Property Relationships • These are sometimes called the Gibbs Equations • They show us how entropy is related to other properties, such as H, U, P, V and T • This will show us how the specific entropy values in the thermodynamic tables were determined. Baratuci ChemE 260 May 3, 2005

Example #1 • A piston-and-cylinder device contains saturated R-134a vapor at -5oC. This vapor is compressed in an internally reversible, adiabatic process until the pressure is 1.0 Mpa. Determine the work per kg of R-134a for this process.

Example #2 • Steam enters an adiabatic turbine at 5 MPa and 450oC and leaves at a pressure of 1.4 MPa. Determine the work output of the turbine per kg of steam flowing through the turbine if the process is reversible and changes in kinetic and potential energies are negligible.

Example #3 • Consider a process in which 1.00 kg of saturated water vapor at 100oC is condensed to a saturated liquid in an isobaric process by heat transfer to the surrounding air, which is at 25oC. What is the change in entropy of the water ? What is the change in entropy of the surroundings ? What is the change in the entropy of the universe ?

ChemE 260 The Clausius Inequality & Entropy