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IDEAL-GAS MIXTURE

IDEAL-GAS MIXTURE. I am teaching Engineering Thermodynamics to a class of 75 undergraduate students. I went through these slides in one 90-minute lecture. Zhigang Suo, Harvard University. Plan.

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IDEAL-GAS MIXTURE

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  1. IDEAL-GAS MIXTURE • I am teaching Engineering Thermodynamics to a class of 75 undergraduate students. • I went through these slides in one 90-minute lecture. • Zhigang Suo, Harvard University

  2. Plan • Ideal gas, a review • PVT relation of ideal-gas mixture • Mixing (TV-representation) • Mixing (TP-representation) • Adiabatic mixing • Steady-flow, adiabatic mixing • Isentropic mixing. Stop generating entropy! Do work.

  3. Law of ideal gasesOscar Wilde: We are all in the gutter, but some of us are looking at stars.We all generate entropy, but some of us are doing work. mechanics chemistry P = pressure V = volume N = number of molecules t = temperature in the unit of energy geometry thermometry • Boyle (1662)-Mariotte (1679) law. PV = constant for a fixed amount of gas and fixed temperature. • Charles’s law (1780). V/t = constant for a fixed amount of gas and fixed pressure. • Avogadro’s law (1811). V/N = constant for all gases at a fixed temperature and fixed pressure. • Clapeyron (1834) combined the above laws into the law of ideal gases.

  4. Human folly To every beautiful discovery, we add many ugly ideas. Generating entropy is natural. The discovery Number of molecules mole Mass Ugly idea 1 Kelvin temperature kBT = t Boltzmann constant Ugly idea 2 Avogadro constant NAvogadro = 6.022 x 1023 Mole n = N/NAvogadro Universal gas constant Ugly idea 3 Specific gas constant

  5. Model a closed system as a family of isolated systems weight 2O vapor Isolated system closed system vapor liquid liquid fire • Each member in the family is a system isolated for a long time, and is in a state of thermodynamic equilibrium. The system can have many species of molecules. A state can have coexistent phases. • Change state by fire (heat) and weights (work). • 2independent variablesname all members of the family (i.e., all states of thermodynamic equilibrium). • 6 functions of state: TVPUSH • 4 equations of state. • The basic task: Obtain S(U,V) from experiment or theory. • Definition of temperature (Gibbs equation 1) • Definition of pressure (Gibbs equation 2) • Definition of enthalpy

  6. Law of ideal gases derived from molecular picture and fundamental postulate When molecules are far apart, the probability of finding a molecule is independent of the location in the container, and of the presence of other molecules. Number of quantum states of the gas scales with VN Definition of entropy S = kBlogW Gibbs equation 1: Gibbs equation 2:

  7. 4 equations of state Change state T0,V0 T,V 2 independent variables (T,V) name all states of thermodynamic equilibrium. 4 equations of state: PUSH

  8. Plan • Ideal gas, a review • PVT relation of ideal-gas mixture • Mixing (TV-representation) • Mixing (TP-representation) • Adiabatic mixing • Steady-flow, adiabatic mixing • Isentropic mixing. Stop generating entropy! Do work.

  9. Two species of molecules B A Number of moles of species A Number of moles of species B Number (mole) fraction of species A: Number (mole) fraction of species B: Algebra:

  10. Dalton’s law (1801) Dalton’s law: Partial pressures: Total pressure: T,V,nA T,V,nB T,V,,nA,nB PA PB PA + PB Boxes of the same volume and temperature

  11. Dry air (no water)

  12. Molecular picture of an ideal-gas mixture U,V,S,P,T,NA,NB,W When molecules are far apart, the probability of finding a molecule is independent of the location in the container, and of the presence of other molecules. Number of quantum states of the gas scales with volume as: Definition of entropy S = kBlog W Gibbs equation 2: Delton’s law: B A

  13. Plan • Ideal gas, a review • PVT relation of ideal-gas mixture • Mixing (TV-representation) • Mixing (TP-representation) • Adiabatic mixing • Steady-flow, adiabatic mixing • Isentropic mixing. Stop generating entropy! Do work.

  14. Energy and entropy of mixing Ta,Va,nA T,V,nA,nB mix Tb,Vb,nB

  15. Internal energy of an ideal-gas mixtureAt a fixed temperature, mixing two ideal gases do not change internal energy. T,Va,nA T,V,nA,nB Mix at a constant temperature T,Vb,nB

  16. Internal energy of mixing Change state of pure A T,Va,nA Ta,Va,nA Mix at constant temperature T,V,nA,nB Change state of pure B Tb,Vb,nB T,Vb,nB

  17. Entropy of an ideal-gas mixtureAt a fixed volume and a fixed temperature, mixing two ideal gases do not change entropy Mix at constant temperature constant volume T,V,nA Isentropic mixing T,V,nA,nB T,V,nB

  18. Entropy of mixing Change state of pure A Mix at constant temperature constant volume Ta,Va,nA T,V,nA T,V,nA,nB Change state of pure B Tb,Vb,nB T,V,nB

  19. Enthalpy of an ideal-gas mixture

  20. Ideal-gas mixture (using mole) T,V,nA,nB 4 independent variables (T,V, nA, nB) name all states of thermodynamic equilibrium. 4 equations of state: PUSH

  21. Ideal-gas mixture (using mass)

  22. Plan • Ideal gas, a review • PVT relation of ideal-gas mixture • Mixing (TV-representation) • Mixing (TP-representation) • Adiabatic mixing • Steady-flow, adiabatic mixing • Isentropic mixing. Stop generating entropy! Do work.

  23. Entropy of an ideal-gas mixture Change state of pure A Ta,Va,nA Pa T,V,nA Pressure = yAP Mix at constant entropy T,V,nA,nB P Change state of pure B Tb,Vb,nB Pb T,V,nB Pressure = yBP

  24. Ideal-gas mixture (TP-representation)

  25. Entropy of mixingat constant temperature and pressure P,T,VA,nA P,T,VB,nB P,T,V,nA,nB Thermostat, T

  26. Plan • Ideal gas, a review • PVT relation of ideal-gas mixture • Mixing (TV-representation) • Mixing (TP-representation) • Adiabatic mixing • Steady-flow, adiabatic mixing • Isentropic mixing. Stop generating entropy! Do work.

  27. Adiabatic mixing Ta,Pa,nA T,P,nA,nB Adiabatic mixing Tb,Pb,nB • Know the initial states in the two boxes (Ta,Pa,nA) and (Tb,Pb,nB) • Also know the pressure of the mixture, P. • Assume the mixing is adiabatic. • Determine the temperature of the mixture, T. • Determine the entropy of mixing, Smix.

  28. Conservation of energy Ta,Pa,nA T,P,nA,nB Adiabatic mixing Tb,Pb,nB

  29. Entropy of mixing Ta,Pa,nA T,P,nA,nB Adiabatic mixing Tb,Pb,nB

  30. Plan • Ideal gas, a review • PVT relation of ideal-gas mixture • Mixing (TV-representation) • Mixing (TP-representation) • Mixing at constant temperature and pressure • Adiabatic mixing • Steady-flow, adiabatic mixing • Isentropic mixing. Stop generating entropy! Do work.

  31. Steady-flow, adiabatic mixing Adiabatic chamber • Know the inlet conditions • The pressure at the outlet is the same as that at the inlets, P. • The mixing chamber is adiabatic. • Determine the temperature at the outlet, T. • Determine the entropy generation.

  32. Conservation of energy Adiabatic chamber

  33. Generation of entropy Adiabatic chamber

  34. Plan • Ideal gas, a review • PVT relation of ideal-gas mixture • Mixing (TV-representation) • Mixing (TP-representation) • Adiabatic mixing • Steady-flow, adiabatic mixing • Isentropic mixing. Stop generating entropy! Do work.

  35. Isolated system IS Isolated systemWhen confused, isolate. Isolated system conserves mass over time: Isolated system conserves energy over time: Isolated system generatesentropy over time: Define words:

  36. Carnot:“The steam is here only a means of transporting the caloric (entropy).” High-temperature source, TH QH High-temperature source, TH Engine Generator Q W W = QH - QL Low-temperature sink, TL QL Low-temperature sink, TL Isolated system = source + sink Isolated system = source + sink + engine + generator Thermal contact transports and generates entropyReversible engine transports but does not generate entropy

  37. The world according to entropy • Irreversible processtransports and generates entropy. Natural process. Non-equilibrium process. e.g., Friction, mixing, conduction. • Reversible processtransports but does not generate entropy. Idealized process. Quasi-equilibrium process. Isentropic process. e.g. Carnot cycle, Stirling cycle, a frictionless pendulum. • Impossible process. Entropy of an isolated system can never decrease over time. • Equilibrium. A system isolated for a long time reaches a state of thermodynamic equilibrium, and maximizes entropy. • Every reversible process (i.e., natural process) is an opportunity to do work.

  38. Isentropic mixing and separationBalance osmosis with external force. Air Pressure = P Temperature = T Number fraction = yN2,yO2 Equilibrium Weight = A (P –yN2P) Pure nitrogen Pressure = yN2P Temperature = T P Pure nitrogen P = yN2P + yO2P Weight Semipermeable membrane Permeable to nitrogen Impermeable to oxygen Direct mixing generates entropyIsentropic mixing transports entropy

  39. Summary

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