Chapters_1_to_3_I_CM2103_II_2014

5/21/2018 Chapters_1_to_3_I_CM2103_II_2014

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ThemodynamicsI

CM2103

FallSemester

ITCR

2014

1fromthedeskofDr.BenitoA.StradiGranados


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Introduction Thermodynamics

power developed from heat

Modern Science: Thermondynamics deals withtransformations of energy of all kinds from one

form to another

Rules for those transformations

First and Second Law These laws are not derived mathematically

Their validity rests upon experience



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Introduction Thermodynamics does not allow for the

computation of the rates of chemical or physicalprocesses.

Classical thermodynamics cannot reveal themicroscopic (molecular) mechanisms of physicalor chemical processes.

Numerical results of thermodynamic analysis areaccurate only to the extent that the required dataare accurate.



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Introduction

Absence of experimental information

Correlations from limited data

System:

A particular body of matter (the matter itself or

volume that enclosed an assamble of matter).



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Dimensionsand

Units

The fundamental dimensions are primitives, these arequantities not definable in terms of anything simpler.

The second, symbol s, this is the duration of 9, 192, 631,770 cycles of radiation associated with a specifiedtransition of the cesium atom.

The meter, symbol m, this is the distance light travels in avacuum during 1/299, 792, 792, 458 of a second.

The kilogram, symbol kg, this is the mass of aplatinum/irridium cylinder kept at the International Bureauof Weights and Measures at Sres, France



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Dimensionsand

Units

The kelvin, symbol K, this is 1/273.16 of the

thermodynamic temperature of the triplepoint of water.

The mole, symbol mol, this is the amount ofsubstance represented by as many elementary

entities (e. g., molecules) as there are atoms in

0.012 kg of carbon12. This is equivalent to

the "gram mole" commonly used by chemists.



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DerivedUnits

The Newton, symbol N, the units are derived

from Newton's Second Law: F=ma

The Newton is defined as the force which when

applied to a mass of 1 kg produces an accelaration

of 1 m/s2, thus the Newton is a derived unit

representing 1 kg/m s2.

Weight refers to the magnitude of the force of

gravity on a body, and it is expressed in

newtons.



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DerivedUnits

&

English

Units

In the case of english units, force is measured

in pounds force. Consequently, anequivalence between primary units and

pound force is needed:

1lbf=32.1740lbmft/s2

Thisis

captured

in

the

term

gc:

gc=(32.1740lbmft/s2)/1lbf



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Temperature Temperature

Liquidinaglass thermometer is the most common method of temperaturemeasurement.

Numerical values are assigned to the various degrees of hotness by an arbitrary

definition.

Celsius scale:

Ice point (freezing point of water saturated with air at standardatmospheric pressure) is zero.

Steam point (boiling point of pure water at standard atmospheric pressure)is 100.

The distance of a thermometer between the two marks is divided into 100equal spaces called degrees.

The marks for zero and 100 degrees would correspond for all thermometerscalibrated this way, however intermediate points would not necessarily

coincide due to differences in the coefficient of expansion of liquids.

The temperature scale of the SI system, with its Kelvin unit, symbol K, is based on theideal gas as thermometric fluid. This is an absolute scale, and depends on theconcept of a lower limit of pressure where all ideal gases coincide.



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Temperature

Kelvin temperatures are given the symbol T. In Smith & Van

Ness, celsius temperatures are given the symbol t, the two

scales are related by :t (oC)=T (K)273.15

The absolute zero occurs at273.15oC.

International Practical Temperature Scale of 1968 (IPTS68) :

Temperatures measured on this scale closely approximate

idealgas temperatures; the differences are within the

limits of present accuracy of measurement.



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Temperature

The IPTS68 is based on assigned values of

temperature for a number of reproducibleequilibrium states (defining fixed points) and on

standard instruments calibrated at these

temperatures.Interpolation between the fixedpoint

temperatures is provided by formulas that

establish the relation between readings of thestandard instruments and values of the

international practical temperature.



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Thedefiningpointsarespecifiedphase

equilibriumstates

of

pure

substances

Substance Equilibrium State T68(K) t68(oC)

Hydrogen Solid,Liquid,Vapor (triplepoint) 13.81 259.34

Hydrogen Liquid, Vapor(P=33.3306

kPa) 17.042

256.108

Hydrogen Liquid, Vapor(Boilingpoint) 20.28 252.87

Neon Liquid,Vapor 27.102 246.048

Oxygen Solid, Liquid,Vapor(triplepoint) 54.361 218.789

Oxygen Liquid,Vapor 90.188

182.962

Water Solid,Liquid,Vapor(triple point)(noair) 273.16 0.01

Water Liquid,Vapor (Boilingpoint) 373.15 100.00

Zinc Solid,Liquid(Freezing point) 692.73 419.58

Silver Solid,Liquid(Freezing point) 1,235.08 961.93

Gold Solid, Liquid(Freezingpoint) 1,337.58 1,064.43



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Standardinstruments

Platinumresistancethermometer:259.34to630.74oC.

Thermocouple:platinum/(10%rhodiumplatinum

thermocouple)630.74

1064.43

oC.Pyrometer(Planck'sradiationLaw)1064.43oCand

above.

Rankine

scale

(Absolute

Scale

in

English

Units): t(oF)=T(R)459.67

RankineandFarenheitScales

t(oF)=1.8t(oC)+32

RankineandKelvinScales

T(R)=1.8T(K)



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Ice

point

of

water

32

(o

F) Normalboilingpointofwater(212oF)

Substancesusedinthermometers

1.Mercury

2.Alcohols



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DefinedQuantities

Volume

Pressure

Work

Energy Heat



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Volume Productofthreelengths

Specificvolume

&

molar

volume

representthevolumeperunitmassorpermole

Density

Pressure

Forceperunitareaofsurface

v

1

A

FP



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Deadweightgauge

Ref.

http://www.gesensing.com/products/resources/datasheets/us

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Work Workisdonewheneveraforceactsthrougha

distance

dW=Fdl

Workthat

accompanies

achange

of

volume

of

afluid: Sayyouhaveacylinderwithafluid. Theexpansionor

compression

of

the

fluid

requires

work,

thusdW=Fd(V/A)where(dV)/Aaccountsforthelinearchangein

thefluidsize.



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Work

ifAisconstantthen

Thisequationcanbeintegrated

A

VPAddW

PdVdW

2

1

V

VPdVW



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Energy KineticEnergy:termintroducedbyLordKelvin(1856)thatis

writtenas:

Thisform

for

energy

is

derived

Newton's

second

law

dW=Fdl

dW=madl(becauseF=ma)

dW=mdu/dt

dl

(first

order

diferential

behave

as

adivision

operation)

dW=mdl/dtdu

dW=mudunowintegrating:

W=1/2mu2

2

2

1muEK



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Potential Energy:

Exchange of potential energy and kinetic

energy in absence of any losses:

Work & Energy:Work is energy in transit and is never regarded as

residing in a body.

When work is done and does not appear

simultaneously as work elsewhere, it is converted

into another form of energy.

mzgEp

0 PK EE



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Systemand

Surroundings

System: the body or ensamble on which

attention is focused is called the system.Everthing else is the surroundings.

When work is done, it is positive when the

work is done by the surroundings on the

system (compression of the boundary).

Energy is transferred to the system. Onlyduring this transfer is that the form of energy

known as work exists.



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HeatOne view, middle of the nineteenth century, heat

was seen as caloric: a weightless andundestructible substance.

Other view (from the seventeenth century):

particles or unknown medium penetrating allbodies (Francis Bacon, Newton, Robert Boyle,

Benjamin Thomson, Sir Humphrey Davy)



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HeatJosephBlack(17281799)

correctlyreconized

temperature

as

aproperty

which

mustbecarefullydistinguishedfromquantityofheat

demostratedexperimentallythatdifferentsubstances

ofthesamemassvaryintheircapacitytoabsorbheat

whenthey

are

warmed

through

the

same

temperature

range.

Hewasthediscovereroflatentheat.

Nowthe

matter

rested

until

near

the

middle

of

thenineteenthcentury.



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Energy

concept

of

heat:

moving

away

from

caloric Champions:Mohr,Mayer,Helmholtz,Colding

andJamesP.Joule.

Joulepresented

experimental

evidence

which

conclusivelydemonstratedtheenergytheory.

The

concept

of

heat

as

a

form

of

energy

is

nowuniversallyaccepted.

Observationsofthistheory:

Heatflows

from

ahigher

to

alower

temperature

Temperaturedifferenceisthedrivingforceforthe

transferofenergyasheat26

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Heat In the thermodynamic sense, heat is never regarded as being

stored within a body.

Like work, heat only exists as energy in transit from one bodyto another (or between a system and its surroundings)

When energy in the form of heat is added to a body , it isstored not as heat but as kinetic and potential energy of the

atoms and molecules making up the body.

Acalorieis4.1840Joules(J).

Awattisequalto1Joule/second=1J/s



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Chapter

2.

The

First

Law

and

Other

Basic

Concepts

Joule's Experiments

Joule was able to show conclusively that a quantitativerelationship exists between work and heat and, therefore,

heat is a form of energy.

He placed measured amounts of water in an insulated

container and agitated the water with a rotating stirrer. Theamount sof work done on the water was carefully noted. He

found that a fixed amount of work was required per unit of

mass of water for every degree of temperature rise caused

by the stirring. The original temperature of the water could then be

restored by the transfer of heat through simple contact with

a cooler object.



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Joule'sApparatus



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InternalEnergy

ExperimentbyJoule:

Energy is added to the water as work but is extracted

as heat.

What happens to this energy between the time it is

added to the water as work and the time it is

extracted as heat ?.

Mechanicalwork

(stirring)appliedto

water

Water

temperature

changes

Energyleavesthe

systemasheat



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InternalEnergy

This energy is stored in the water in another form called internal

energy.

Internal energy refers to the energy of the molecules making up the

substance:

Kinetic energy of translation (ceaseless motion) all molecules

mono and poliatomic. Kinetic energy of rotation and vibration (except for monoatomic

molecules).

The addition of heat to a substance increases this molecular

activity, and thus causes an increase in its internal energy. Work done on a substance also increases the molecular activity

through shear and causes an increase in its internal energy.



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InternalEnergy

Internal Energy is different from Kinetic and

Potential Energy.

Internal Energy does not include any energycoming from macroscopic position or

movement.



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Formulation

of

the

first

law

of

thermodynamics

First Law:

Although energy assumes many forms, the totalquantity of energy is constant, and when energy

disappears in one form it appears simultaneously

in other forms. This law is not proven by mathematical means. In this

sense, it is a primitive, this means that it cannot be

derived from other principles.

Without exception, all observations of ordinary

processes support it.



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Formulation

of

the

first

law

of

thermodynamics

The definition of the first law assumes the presence

of a system and its surroundingsThe system is the volume, mass, or ensemble of

interest

The surroundings comprise everything else The first law also is born from the interaction of the

system and its surrounding.

Consequently, the first law couples what happensinside the system with what is happening outside.



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Formulation

of

the

first

law

of

thermodynamics

In this light the first law implies the following

balance:

Changes may occur in the potential and kinetic

energy of the system and surroundings and they areincluded in the equation.

Heat and work refer to forms of energy in transit and

cannot be stored. Energy is stored in the form ofkinetic, potential, and internal energies.

In a closedsystem, no mass enters or leaves

the system.

0)()( gssurroundinofenergysystemtheofenergy



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Formulation

of

the

first

law

of

thermodynamics Suppose we have a system that has the following

characteristics

the system is closed

the surrounding can only exchange energy in the

form of heat or work

The heat is given the symbol Q and is positive

when it enters the system

The work is given the symbol W and is

computed with the equation

systemf

systemi

V

V

PdVW



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The work term is positive if work enters the

system and negative if it leaves (this means work

is done on the environment).

There is no flow even though the system may

already have its own potential and kineticenergies.

Formulation

of

the

first

law

of

thermodynamics

0)()( gssurroundinofenergysystemtheofenergy

PK EEU WQ



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Equating both sides of the equation

No flow

Steady state

Single substance (system)

Valid for reversible and irreversible processes

If the contribution from the kinetic and

potential energies is small then we write:

Formulation

of

the

first

law

of

thermodynamics

WQEEU PK

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The process can be reversible or irreversible.

The difference of Q and W generates a state property, even

though each one of them may be irreversible: a state property only depends of the initial and final conditions

an irreversible process depends on the path and not only on final and initial

conditions

Now if the process is reversible, there is a unique reversible

path for the work component and the transfer of heat to

occur reversibly then it is possible to write

There is no differential form for irreversible processes.

Formulation

of

the

first

law

of

thermodynamics

dWdQdU



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The

thermodynamic

state

and

state

functions

State functions

Those that only depend on initial and finalconditions.

At the initial and final conditions the property has

a fixed value that does not depend on the pathfollowed to achieve a specific state.

A thermodynamic state is characterized by

state functions with specific values for that

thermodynamic state.



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Internal energy is a state function because

once T, P or T, V are specified for a puresubstance the value of the internal energy can

be determined.

It is not important for the final value of the

internal energy how many heating/cooling cycles

were experienced before arriving to the final

conditions.

The U does not change with trajectory anddepends only on initial and final conditions

The

thermodynamic

state

and

state

functions



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UUUdUUU

122

1

The

thermodynamic

state

and

state

functionsA property of state can be integrated and the final result is the difference

between initial and final conditions:

12

2

21

2

xx

F

xx

F

This a not very useful definition because does not say anything about the case

where the property depends on T, P, or xi . Maxwell indicated that a property of

state has to satisfy that the mixed double derivatives be the same:

This says a change in the xdirection followed by a change in the ydirection has to

be equivalent to a change in the ydirection followed by a change in the x

direction.



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The

thermodynamic

state

and

state

functions

PV

U

VP

U

22

Thisistrueforinternalenergybecauseitisastateproperty.

P

v



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Heatandworkarenotpropertiesofstate

These quantities are trajectories dependent upon theparticularities of the problem.

Heat and work are path dependent and thus the mixed

double derivatives are not the same unless the Heat

Transfer or Applied Work occur reversibly.

The integral gives the reversible work. Irreversible

work does not have this or any specific functionality as the

dependence of P and V is not known or is described by

more than one path.

The

thermodynamic

state

and

state

functions

PdV



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Thethermodynamicstateandstatefunctions

P

V

PdVThe integral gives the

area under the curve

between the starts. Each

area is different, the

work value is different as

well. Work is not a

unique quantity unlessthe process is reversible.

Irreversible processes do

not have a PV

trajectory.1

2

A1

A2



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Experiments show that processes which accomplish

the same change in state by different paths in aclosed system require, in general, different amounts

of heat and work.

HOWEVER, THE DIFFERENCE QW IS THE SAME FOR

ALL SUCH PROCESSES. This means that each

quantity alone is not a state property but thedifference is a state property.




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Extensive properties: Those that depend on the

amount of mass present: volume.

Intensive properties: Independent of the amount of

mass: Pressure, temperature, density.

Commonly:

intensiveproperty=extensiveproperty/mass




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Enthalpy Itisdefinedby

H=U+PVwhere H isthetotalenthalpy

Uisthetotalinternalenergy

Pisthepressure

Visthetotalvolume

Bothontherightandleftoftheequalsignarestatefunctions,

consequentlyyoucanderivetheequationtogetadifferential

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Enthalpy VdPPdVdUPVddUdH )( There are two contributions of the product PV to the enthalpy

term: The contribution PdV will appear mainly for those cases

without flow and is associated with an expansion at

constant pressure.

The contribution VdP will appear mainly for those cases

with flow or for compression process where an

incompressible fluid is pumped.

The integrated forms are used for finite changes where theinitial and final states are clearly identified.

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The integrated form uses only finitedifferences.

PVVPUPVUH )(Enthalpy

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Steadystate:conditionsatallpointsinthe

apparatusare

constant

with

time:

Allratesmustbeconstant

Noaccumulationofmassorenergy

Massflow

rate

must

be

constant

An element of mass flowing along a tube receives aforce from the previous element that pushes and

delivers a force to push the element in front of it.

The

steady

state

flow

process

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The work is zero because the only way to lose

energy is with irreversibilities and that is notincluded in the model so far.

The

steady

state

flow

process

F F

F F

dl

0FdlFdl

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Only at the entrance and at the exit the Reaction and

the Action are balanced by an external body, those

two events coming IN and OUT of the volume appearin the energy balance because they are not balanced

in the control volume.

F F

F F

dl

The

steady

state

flow

process

IN OUT

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If both action and reaction are the same,

there still movement because they act upondifferent bodies.

The

steady

state

flow

process

V1

V2

U1

U2

Z1

Z2

Q

Ws

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Thefirstlawwithflowiswrittenas(Eq.2.3)

where W is the total work that comes in or is provided bythe system.

W is divided into shaft work and work providedpushing in and out the material from the controlvolume, thus W=Ws+P2V2P1V1.

The PV term is needed to account for pressure forcesas the element enters and leaves the volume.

The

steady

state

flow

process

WQEEU PK

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Wsiscalledshaftwork.

The term shaft work means work done by or onthe fluid flowing through a piece of

equipment and transmitted by a shaft which

protrudes from the equipment and whichrotates or reciprocates. The term represents

the work which is interchanged between the

system and its surroundings through this

shaft.

The

steady

state

flow

process

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WQEEU PK The

steady

state

flow

process

1122 VPVPWQEEU sPK

sPK WQEEVPUVPU 111222

sPK WQEEHH 12

sPK

WQEEH

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This is the first law applied to a flow process. Notice

that the effect of compression of the fluid appears in

the enthalpy and exchange through a shaft [rise andfall of weight] appear in the Wsterm.

In turbulent flow this is written as:

sPK WQEEH

The

steady

state

flow

process

sWQzguH 2

2

1

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Numerical values for enthalpy H are given in Tables. In the case ofwater, the Steam Tables contain that information.

The absolute value of H is not calculated from classicalthermodynamics.

Classical thermodynamics assigns a Zero Value at a set of referenceconditions. From those values Enthalpy changes are computed.

It is not until many years later that the midnineteenth century that

the absolute value for enthalpy and the other thermodynamicsquantities is established.

The third law thermodynamics establishes that at molecular level allmotion stops at273.15 oC ( 0 K) at which value all thermodynamicproperties have zero value [absolute zero]

For convenience other zero values are used because of the difficultyof having measures referred to a273.15 oC (0 K) temperature valueand at equilibrium.

The

steady

state

flow

process

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Equilibrium Driving forces favor a process to proceed.

Resistance favor a process to stop (or reducespeed).

Steadystate: there is no change with time

Equilibrium: there is no change with time and

there is no tendency to change with time

(require both Property)=0 andd(Property=0)).

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The

Phase

Rule Determines the number of intensive properties

(degrees of freedom) that can be specifiedseparately without affecting the properties(generally molar concentrations of thecomponents in the gas and liquid phases).

For nonreacting systems we have:

F=CP+2

P= # of phases

C= # of components

the +2 if for the temperature and pressure

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The

Phase

Rule ForthePhaseRuletoapplythesystemneedsto

beinequilibrium.

Aphaseisahomogeneousregionofmatter:Aboundarybetweentwophasesischaracterizedfor

anabrupt

change

in

properties

Aphasecanbecontinuous waterandoilinthecontainerfortwowelldefinedlayers

Aphasecanbediscontinuous small

drops

of

water

in

oil

define

two

phases

thephaseconstitutedbythedrops

thephaseconstitutedbytherestoftheliquid

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F=0,thesystemisinvariant. Thismeansthat

nointensive

property

can

be

modified

without

simultaneouslymodifyingtheotherintensive

propertiesofthesystem.

The

Phase

Rule

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The

Reversible

Process A process is reversible when its direction can

be reversed at any point by an infinitesimalchange in external conditions.

The piston confines the gas at a pressures just

sufficient to balance (no accelaration) the

weight of the piston and all that it supports.

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A piston that raises rapidly and oscillates prior tosettling will undergo a reversible process if there

are no losses, however in the absence of losses itwould oscillate indefinitely.

When losses activate, it then undergoes anirreversible process. Friction forces transformkinetic energy into internal energy.

A reversible process is that in which the transferof energy is slow and continuous (piston risesslowly).

The

Reversible

Process

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The piston rises slowly and the potentialenergy is accumulated by grains of sand thatare shelved along the way.

If the removal of powder (weight) from thepiston is stopped and the direction of transferof power is reversed, the process reverses

direction and proceeds backward along itsoriginal path.

The

Reversible

Process

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A reversible proces is frictionless. It is always only differentiallyremoved from equilibrium.

Traverses a succession of equilibrium states.

Driving forces are differential in magnitude.

The process direction can be reversed at any point by a differentialchange in external conditions.

If reversed, the process retraces its path leading to the restoration

of the initial state of the system and its surroundings.

The

Reversible

Process

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The Reversible Process


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Work inside a system and Work received by theenvironment.

For a process to undergo a reversible process, thereverse process has to reestablish the original state forboth the internal system and the environment (that is

the catch !).

Internally reversible process does not warantee an

externally reversible process. If the process isexternally irreversible then part of the work will betransformed to heat and lost.

TheReversibleProcess

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The work of compression or expansion of a gas caused by thedifferential displacement of a piston in a cylinder is given by theequation:

The work computed for the system and its surrounding is the same

if and only if the P has the same function P=f(V) for both.

Consequently, if the internal system has a reversible pressure, P, thesurrounding pressure can only be differentially different from the

internal pressure, and there can be no energy losses in thesurroundings.


PdVdW

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PdVdW PddW

System Surroundings

These two pressure have to be the same

When the pressures inside and outside are the same the process is in mechanical

equilibrium.

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Notation and Heat Capacities


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NotationandHeatCapacities

StartinginSection2.10,thecapitallettersU,

H,V

represent

the

specific

or

molar

property.

QandWremainrepresentingtotalheatand

totalwork.

Massand

moles

are

represent

with

the

letters

mandn,respectively.

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Heat Capacities


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HeatCapacities

Heatcapacitiesatconstantvolume

Heatcapacityatconstantpressure

V

VTUC

P

pT

HC

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Chapter 3 Volumetric Properties of Pure Fluids


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Chapter3.VolumetricPropertiesofPureFluids

ThePVTbehaviorofpuresubstances Thermodynamicproperties

InternalEnergy,

U

Enthalpy,H

GibbsFreeEnergy,G

Helmohltz,A

Entropy,S

Thereisnotdirectmeasurementofthesequantities,forexample

Uismeasuredthrough: CvT or

via

Q

W

TherearenoInternalEnergymeters.

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P

T

1

3

2

Critical Point

Solid Region Liquid region

Gas region

Fluid regionCritical Points

Triple point

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Chapter 3. Volumetric Properties of Pure Fluids


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Homogeneousfluids Liquidandgases

Thetwophaseshavedifferentphysicalpropertiesthatvarywithtemperatureandpressure.

Apurefluidintheliquidandvaporphasesimultaneouslyinthe

samevolume.

Thefusioncurvedeterminesthetemperaturesandpressuresatwhichtheliquidandsolidphasescoexist.

Thevaporizationlinedeterminedthelocioftemperaturesandpressuresatwhichtheliquidandvaporphasescoexist.


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Increasing temperature and pressure [of a closedsystem of constant total mass] cause the gas to

compress and the liquid may expand at very highpressures.

The liquid and vapor may coexist until theyreached a maximum of pressure and temperaturebeyond which phase separation disappears. The

temperature and pressure at which this occurs iscalled a Critical Point.

p p

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Criticalpoint:

the meniscus disappears WHILE BOTH PHASES ARESTILL PRESENT.

the liquid opalesces, this means it becomes turbid as

if a cloud were moving inside.

the densities and heat capacity of both phases

becomes equal.

Caution: just because a phase disappears, there is no

guarantee of achieving a critical point. The critical

point requires the presence of opalescense.

p p

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A critical point represents a limit of stability

Two phases are inminently formed by a differential

change in pressure and temperature

Mathematical critical points are stability limits of the

second order because they require, in general, that

p p

0,

inPT

G0

,

inTP

G0

,,,

ijjnPTin

G

0

,,,

2

2

ijji

nPTn

G0

,

2

2

inTP

G0

,

2

2

inPT

G

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Vapor

Liquid and VaporLiquid

P

V

Isotherm T=Tc

T1 Tc T2

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A sealed upright tube with a pure substance can be heated

In most cases the liquid simply will move to the gas phase

without going through criticallity.

At a specific filling of the tube at a given temperature and

pressure, the liquid in the tube can reach criticallity, for a

pure substance the set of conditions is unique.

Turbulence at criticality

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VolumetricPropertiesofPureFluids


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For the regions of the diagram where a single phaseexists, there may exist an equation connecting P, V, and

T which may be expressed:

An equation of state may be solved for any of the threequantities P, V, or T as a function of the other two.

For example, if V is considered a function of T and P,then V=V(T,P), and

0),,( TVPf

dPPVdT

TVdV

TP

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Thecoefficientsoftheequationhave

meaning:Thevolumeexpansivity

Theisothermalcompressibility

PT

V

V

1

TP

V

V

1

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we could rewrite our equation of state as:

The coefficients are measurable quantities. For liquids

they are small as the molar volume of a liquid does notchange significatively over small ranges of T and P.

For an incompressible fluid both coefficients are zero.

dPdTVdV

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TheVirialEquation


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The vapor or gas region has complexities

beyond that of the ideal gas model. How to deal with this

Add twobody interactions

Three body interactions, and so on

In practical terms express the compressibility

in terms of pressure or volume.

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TheVirialEquation


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...1 3'2'' PDPCPBRT

PVZ

...132

V

D

V

C

V

B

RT

PVZ

Both equations are called Virial Expansions and theparameter B', B, C, C', D, etc. are called virial coefficients.

B and B' are the second virial coefficients C and C' are the third virial coefficients

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TheVirialEquation


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The term B/V accounts for interactions betweenpairs of molecules.

The term C/V2 accounts for threebodyinteractions.

The contribution to Z of the successively higherorder terms fall off rapidly because secondorderinteractions are more common than threebodyinteractions and in turn these are more commonthan fourthorder interactions, and so forth.

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TheVirialEquation


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Thetwosetsofcoefficientsarerelatedas

follows:

RT

BB'

22

'RT

BCC

33

23'RT

BBCDD

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TheVirialEquation


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The graph of PV vs P has the same limiting value for all gases as

P>0.

PV=RT and the righthand side is a function of temperature only.

This means that there is a pressure and volume relationship totemperature that is independent of the material being used.

This relationship could be used for thermometry because it ispossible to have a temperature scale independent of the materialused.

Steps for an ideal gas thermometer are the following:

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TheVirialEquation


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Stepsforanidealgasthermometricscale:

Fix

the

functional

relationship

so

that

(PV)*

is

directlyproportionaltoT:

Assignavalueof273.16tothetemperatureofthe

triplepointofwater

(PV)*t(attriple

point)=R*273.16

Now,dividetheresultsoftheprevioustwosteps.

RTaPVPVP

*

0)(lim

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TheVirialEquation


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*

*

)(

)(16.273)(

tPV

PVKT

This is the ideal gas temperature scale.

The (PV)* values are experimentally accessible.

Why ideal gas: because as P>0, the volume of the gas molecules becomes

negligible.

because as P>0, molecular interactions die off and

interactions become zero An ideal gas is a substance made of molecules of zero

volume that do not interact between them.

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TheVirialEquation


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Can we use all this for something useful ?

Sure. For starters, the determination of the numerical

value of R is possible based on experimental values:

Since PVT data cannot in fact be taken at a pressureapproaching zero, data taken at finite pressures areextrapolated to the zeropressure state. Thecurrently accepted value of (PV)*tis 22,711.6 cm

3 barmol1.

K

PVR t

16.273

*

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TheVirialEquation


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ThevalueofRthenbecomes

11313

144.8316.273

6.711,22

KmolbarcmKmolbarcm

R

H2

N2

Air

O2PV/cm3barmol1

0 Pressure (atm)

lim(PV)t=(PV)*t=22,711 cm3 bar mol1

P>0

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TheIdealGas


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Internal energy of a real gas is a function oftemperature and pressure

Pressure dependence arises as a result of intermolecular forces.

Turn those forces off in an ideal gas.

Now the internal energy depends on temperature

only.

Intermolecular forces become negligible at lowpressure as the distance between molecules is

sufficiently large to die off (1/r6

) and not producean effect on properties.

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TheIdealGas


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At low pressure, we have ideal gases.

The compressibility factor is Z=1, and the equation of state

is PV=RT. The internal energy of an ideal gas then is a function of

temperature only.

This extends to enthalpy through:

H=U+PV=U+RT

Consequently, the enthalpy of an ideal gas is also a function

of temperature only.

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TheIdealGas


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Constantvolume process

notice that the integral form for W requires the process to bemechanically reversible.

Using now the definition of Cv:

But regardless of V and P, the change always has the same

value because U is a function of temperature only, and then Cv isalso a function of T only.

QPdVQWQU

V

V

CT

U

T

U


TheIdealGas


96/128

Consequently, regardless of the process thechange in internal energy for an ideal gas can be

computed with the use of Cv:

ConstantPressure (Isobaric) Process

Since enthalpy for an ideal gas is only a functionof temperature then we have

regardless of the process.

dTCQU v

dTCH P


TheIdealGas


97/128

Since enthalpy and internal energy are

functions of temperature only for an ideal gas

then

The heat capacities can vary but only in a waythat their difference adds up to the value of R.

RTdUdH

RdTdTCdTC vp RCC vp


TheIdealGas


98/128

ConstantTemperatureProcess

notice the use of differentials, that indicates

that the process is reversible and that the

internal pressure can be used in the

computation of work.

0 dWdQdU

dQdW

QW


TheIdealGas


99/128

V

dVRTPdVWQ

12

ln V

V

RTPdVWQ

since for the isothermal process, wecan also write:1

2

2

1

V

V

P

P

2

1lnP

PRTPdVWQ


TheIdealGas


100/128

AdiabaticProcess

now if the process is mechanically reversible(connected to the outside world) via a frictionlesspiston, and the transference of heat is also reversible,then we can use differentials.

See how important the use of reversibility becomesotherwise, the trajectory for both Q and W have to bespecified (seldom possible).

WQU

dWdQdU


TheIdealGas


101/128

TheIdealGas

Now,substitute and toget:

PdVdWdU This is only possible because is reversible

dTCdU v

PdVdTCv V

RTP

dVV

R

T

dTCv


dVV

R

T

dTCv

22 lnln V

VCR

TT


102/128

Needtodefine

then

11 VCT v

vv

v

v

P

C

R

C

RC

C

C

1

vv

P

C

R

C

C 1

v

C

R 1

1

2

1

2 ln)1(lnV

V

T

T

1

2

1

1

2

V

V

T

T


1

2

1

1

2

V

V

T

T

1


103/128

1

11

22

1

2

/

/

TP

TP

T

T

1

1

2

1

11

1

22

P

P

TT

TT

1

1

2

1

2

P

P

T

T

1

1

2

1

2

P

P

T

T


1

12

VV

TT

1

22

PT


104/128

21

VT

11

PT

1

2

1

1

1

2

V

V

P

P

2

1

1

1

2

V

V

P

P

2

1

1

1

2

V

V

P

P

2

1

1

2

V

V

P

P

1122 VPVP


nowWcanbeexpressedwithseveral

expressions:


105/128

p

we

can

substitute

Cv

in

terms

of

R:

IfV2isnotknown,itcanbeeliminatedandbe

leftinfunctionofpressures:

TCUW v

111

221121

VPVPRTRTTRTCW v

1

1

1

2

1

211

11

22112211 11

111

P

PPPVP

VPVPVPVPVPW


1

1

2

1

211

11

22112211

11111P

P

P

PVP

VP

VPVPVPVP

W


106/128

1P

1

1

1

211

1

1

211 11

11

P

PRT

P

PVPW

This is for ideal gases with constant heatcapacities. The process is mechanically reversible

as well as adiabatic. Processes which are

adiabatic but not mechanically reversible are notdescribed by these equations.


ThePolytropicProcess Thisisageneralcaseforwhichnospecificconditionsotherthanmechanical

reversibilityare

imposed.


107/128

Onlygeneralequationsappliedtoanidealgasinanonflowprocess. Foronemole,theseare:

ValuesofQcannotbedetermineddirectly,butmustbeobtainedthroughthefirstlaw. SubstitutionfordUanddWgives

SincethefirstlawhasbeenusedforthecalculationofQ,theworkmustbecalculateddirectlyfromtheintegral

The work

of

an

irreversible

process

is

calculated

by

a

two

step

procedure.

First,

W

isdeterminedforamechanicallyreversibleprocess thataccomplishesthesamechangeofstate. Second,thisresultismultipliedordividedbyanefficiencytogivetheactualwork.

WQUdWdQdU

PdvWPdvdW dTCUdTCdU vv dTCHdTCdH PP

PdvdTCdQ v

PdvdTCQ v


ApplicationoftheVirialEquation


108/128

Lowtomoderatepressures:

1.0

0.3

73.33 oC37.78 oC

204.44 oCZ=PV/R

T Compressibitygraph factor for methane (P, psia)

Pressure (psia)

1000 2000


The first approximation to low P behavior of non ideal gasesi ith th Vi i l E ti d th S d Vi i l C ffi i t



109/128

is with the Virial Equation and the Second Virial Coefficient:

This virial equation represents PVT behavior of most vapors

at subcritical temperatures up to a pressure of 15 bar.

At higher temperatures it is appropriate for gases over anincreasing pressure range as the temperature increases:

=> high pressure and low temperature should be cause of concern.

21 V

C

V

B

RT

PVZ


b b b b l b h l



110/128

For pressure above 15 bar but below 50 bar, the virial equationtruncated to three terms works well.

This equation is explicit in pressure, but cubic in volume (threevolume solutions).

Values of C and B depend on the identity of the gas and on thetemperature.

Virial equations with more than three terms are rarely usedbecause the number of roots for V also increases and an additionalscheme for selection is needed.

21V

C

V

B

RT

PVZ


P l i l i h bi i l l ff

CubicEquationsofState


111/128

Polynomial equations that are cubic in molar volume offer acompromise between generality and simplicity that is suitable tomany purposes.

Cubic equations are in fact the simplest equations capable ofrepresenting both liquid and vapor behavior.

The first general equation of state was proposed by J. D. van derWaals in 1873:

where a and b are positive constants.

2

V

a

bV

RTP



P


112/128

T1>TC

VVsat(liq) Vsat(vap)

TC

T2


113/128

The cubic equation oscillates between the molarvolume of the liquid and that of the vapor.

Measurements do not oscillate but rather follow thehorizontal line (green).

These nonequilibrium or metastable states ofsuperheated liquid and subcooled vapor areapproximated by those portions of the PV isothermwhich lie in the twophase region adjacent to the

saturatedliquid and saturated

vapor states.


Redlich/Kwong Equation (1949) started modern



114/128

Redlich/KwongEquation(1949)startedmodernequationsofstate:

VaporVolumes:

The

ideal

gas

provides

a

suitable

initial

value,

V0=RT/P

)(2/1 bVVT

a

bV

RTP

)(

)(2/11

bVPVT

bVab

P

RTV

ii

ii


Li id l



115/128

Liquidvolumes:

aniterationschemeresultswhenthisiswritten

where

foran

initial

value,

take

V0=b.

02/12/1

223

PT

abV

PT

a

P

bRTbV

P

RTV

2/123

1

1

PT

abV

P

RTV

cVi

2/1

2

PT

a

P

bRTbc


F i l bi i f d b



116/128

For simple cubic equations of state, a and b

can come from estimates using TCand PC.

Two equations and two unkowns from:

ThevanderWaalsequation:

0

,

2

2

,

critTcritT V

P

V

P

C

C

PTRa

6427

22

C

C

PRTb8


the Redlich/Kwong equation



117/128

theRedlich/Kwongequation

alsowecanfindtheconstantfittingvaluesto

theequationsofstate.

C

C

P

TR

a

5.2242748.0

C

C

P

RTb

08664.0


Higherorderequationsofstate

2

32

000 1/

VecaabRTTCARTBRT

P


118/128

2236321 e

VTVVVVVP

where

A0,

B0,

C0,

a,

b,

c,

,andareallconstantsforagivenfluid.


GeneralizedCorrelationsforGases

RTVbydividedKwongdlichh

h

bRT

a

hZ /Re

11

151


119/128

hbRTh 11 5.1

need to eliminate a and b from theseequations using the a and b definitions asfunctions of Tcand Pc.

The resulting equation is in function ofreduced properties.

ZRTbP

PZRTb

Vbh

/



hZ

9340.41


120/128

hTh

Zr 11

5.1

r

r

ZT

Ph

08664.0

The resulting equation is a function of

reduced properties.

C

rT

TT

C

rP

PP


Generalized equation: it is that which expresses Z



121/128

Generalized equation: it is that which expresses Zas a function of Trand Pr.

Generalized charts: those which predict Z as afunction of Tr and Pr (this instead of using an

equation of state).

Twoparameter theorem of corresponding states:all gases, when compared at the same reducedtemperature and reduced pressure, haveapproximately the same compressibility factor,

and all deviate from ideal

gas behavior to aboutthe same degree.


Generalized equation of state work well with argon,



122/128

q g ,krypton, and xenon.

Serious problems for other more complex molecules. Solution: include the effect of molecular structure in

another parameter. This parameter should measurethe difference in behavior between a simple fluid (SF)

and a complex molecule. Appreciable improvement results from the

introduction of a third correspondingstates parameter,characteristic of molecular structure; the most popular

such parameter is the acentric factor w, introduced byK. S. Pitzer and coworkers.


What is thiswsupposed to measure ?. Interactions, yes, but how ? . Therei l l f !



123/128

are no intermolecular force meters yet!.

The tendency of remaining in the liquid phase is evidenced by a low vaporpressure. So for condensable gases, a good measure of intermolecularforces is the vapor pressure.

Also in order to measure a quantity, there is a need to have a reference or

some form of metric.

Pitzer noted that all vaporpressure data for the simple fluids (Ar, Kr, Xe) lieon the same line when plotted as log(Pr

sat) vs. 1/Tr and that the line passesthrough log(Pr

sat)=1 at Tr=0.7.

Data for other fluids can be referred to simple fluids (Argon, Krypton,Xenon) and use the conditions above to determinew.


Theacentricfactorisdefinedas



124/128

This definition of w makes its value zero for argon,krypton, and xenon, and experimental data yieldcompressibility factors for all three fluids that are

correlated by the same curves when Z is represented asa function of Tr and Pr.

Threeparameter theorem of corresponding states: all

fluids having the same value of w have the same valueof Z when compared at the same Tr and Pr.

7.0log0.1

rT

sat

rPw


The correlation for Z developed by Pitzer and



125/128

The correlation for Z developed by Pitzer and

coworkers takes the form:

where Z0 and Z1 are complex functions of both

Tr and Pr. For w=0, Z=Z0, this is the compressibility factor

for argon, krypton, and xenon.

10wZZZ


The Generalized Correlations for Gases with the Pitzer



126/128

acentric factor work well for nonpolar materials, and

work no so well for polar and associating substances.

Use of graphs is outdated and modern computingprograms can predict compressibility factors based onregressed parameters.

For low pressures we can attempt to refer back to the

virial equation to give functional value to the Z0 and Z1compressibility factors.


Pitzer proposed a second correlation based on



127/128

Pitzerproposedasecondcorrelationbasedon

thesimplestformofthevirialcoefficient:

nowdefine

andsubstitute

in

the

previous

equation

r

r

C

C

T

P

RT

BP

RT

BPZ

11

10wBB

RT

BP

C

C

r

r

r

r

T

PwB

T

PBZ

101 127

fromthedeskofDr.BenitoA.StradiGranados


comparingtermbyterm:


128/128

r

r

T

PBZ

00 1

r

r

T

PBZ

11

whereB1 andB0 havethefollowingdefinitions

6.1

0 422.0083.0rT

B 2.4

1 172.0139.0rT

B


Documents

Chapters_1_to_3_I_CM2103_II_2014