IDEAL-GAS MIXTURE I am teaching Engineering Thermodynamics to a class of 75 undergraduate students....

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

• 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.

2

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.

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mechanics

geometry

chemistry

thermometry

• Boyle (1662)-Mariotte (1679) law. PV = constant for a fixed amount of gas and fixed temperature.• Charles’s law (1780). V/ = 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.

P = pressureV = volumeN = number of molecules = temperature in the unit of energy

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mole Mass

Ugly idea 1Kelvin temperature kBT = Boltzmann constant

Ugly idea 2Avogadro constantNAvogadro = 6.022 x 1023

Mole n = N/NAvogadro

Universal gas constant

Ugly idea 3Specific gas constant

Gas Formula Molar mass, M kg/kmol

R kJ/kgK

Air 0.2870

Steam H2O 18 0.4615

Hydrogen H2 2 4.124

Human follyTo every beautiful discovery, we add many ugly ideas.

Generating entropy is natural.

Number of moleculesThe discovery

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Isolated system

weight

fire

Model a closed system as a family of isolated systems

• 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).• 2 independent variables name 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

closed system2O

liquid

vapor

liquid

vapor

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 = kBlog

Gibbs equation 1:

Gibbs equation 2:

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Law of ideal gases derived from molecular picture and fundamental postulate

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4 equations of state

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

T,VT0,V0

Change state

Plan


Do work.

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Two species of molecules

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A

B

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:

Dalton’s law (1801)

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Dalton’s law:

Partial pressures:

Total pressure:

Boxes of the same volume and temperature

T,V,nA T,V,nB T,V,,nAnB

PA PB PA + PB

Dry air (no water)

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name formula Molar mass M kg/kmol

number fraction y

nitrogen N2 28 0.78

oxygen O2 32 0.21

Molecular picture of an ideal-gas mixture

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

Gibbs equation 2:

Delton’s law:

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U,V,S,P,T,NA,NB,

A

B

Plan


Do work.

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Energy and entropy of mixing

Ta,Va,nA

T,V,nAnB

Tb,Vb,nB

mix

Internal energy of an ideal-gas mixtureAt a fixed temperature, mixing two ideal gases do not change internal energy.

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T,Va,nA

T,V,nAnB

T,Vb,nB

Mix at a constant temperature

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Ta,Va,nA

T,Vb,nB

T,V,nA,nB

T,Va,nA

Tb,Vb,nB

Internal energy of mixing

Change state of pure A

Change state of pure B

Mix atconstant temperature

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Entropy of an ideal-gas mixtureAt a fixed volume and a fixed temperature, mixing two ideal gases do not change entropy

T,V,nB

T,V,nA,nB

T,V,nAMix atconstant temperatureconstant volume

Isentropic mixing

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Ta,Va,nA

T,V,nB

T,V,nA,nB

T,V,nA

Tb,Vb,nB

Entropy of mixing



Mix atconstant temperatureconstant volume

Enthalpy of an ideal-gas mixture

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Ideal-gas mixture (using mole)

4 independent variables (T,V, nA, nB) name all states of thermodynamic equilibrium.4 equations of state: PUSH

T,V,nAnB

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Ideal-gas mixture (using mass)

Plan


Do work.

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Ta,Va,nA

Pa

T,V,nB

Pressure = yBP

T,V,nA,nB

P

T,V,nA

Pressure = yAP

Tb,Vb,nB

Pb

Entropy of an ideal-gas mixture



Mix at constant entropy

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Ideal-gas mixture (TP-representation)

Entropy of mixingat constant temperature and pressure

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P,T,VA,nA

Thermostat, T

P,T,VB,nB P,T,V,nA,nB

Plan


Do work.

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Adiabatic mixingTa,Pa,nA

T,P,nAnB

Tb,Pb,nB

Adiabatic mixing

• 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.

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Conservation of energyTa,Pa,nA

T,P,nAnB

Tb,Pb,nB

Adiabatic mixing

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Entropy of mixingTa,Pa,nA

T,P,nAnB

Tb,Pb,nB

Adiabatic mixing

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.

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Steady-flow, adiabatic mixing

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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.

Conservation of energy

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Adiabatic chamber

Generation of entropy

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Adiabatic chamber

Plan


Do work.

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Isolated systemWhen confused, isolate.

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Isolated system

IS

Isolated system conserves mass over time:

Isolated system conserves energy over time:

Isolated system generates entropy over time:

Define words:

Carnot: “The steam is here only a means of transporting the caloric (entropy).”

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Engine

High-temperature source, TH

Low-temperature sink, TL

Isolated system = source + sink Isolated system = source + sink + engine + generatorThermal contact transports and generates entropy Reversible engine transports but does not generate entropy

Low-temperature sink, TL

High-temperature source, TH

Q W

W = QH - QL

QH

QL

Generator

The world according to entropy

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• Irreversible process transports and generates entropy. Natural process. Non-equilibrium process. e.g., Friction, mixing, conduction.

• Reversible process transports 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.

Isentropic mixing and separationBalance osmosis with external force.

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AirPressure = PTemperature = TNumber fraction = yN2,yO2

Pure nitrogenPressure = yN2PTemperature = T

P = yN2P + yO2P Semipermeable membranePermeable to nitrogenImpermeable to oxygen

Weight

Direct mixing generates entropy Isentropic mixing transports entropy

Pure nitrogen

P

EquilibriumWeight = A (P –yN2P)

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Summary

IDEAL-GAS MIXTURE I am teaching Engineering Thermodynamics to a class of 75 undergraduate students....

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