Aero Thermo Hydro Engineers Nexus Application

409.319A

Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University

Thermofluid Properties

https://fluids.eng.unimelb.edu.au/

409.319A



Independent Properties of the Pure Substance Model

Equilibrium Liquid-Vapor States

Equilibrium Liquid-Solid and Vapor-Solid States

Equilibrium Solid-Liquid-Vapor States

Gibbs Phase Rule

Energy Interactions During Changes of Phase

Phase Equilibrium

Thermodynamic Surfaces

Tabulation of the Thermodynamic Properties

Metastable States

Applications of the Pure Substance Model

A Quick Look

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Simple System1) no surface forces (from capillarity & surface tension)

2) no forces due to gravity, electric charge, magnetic fields

3) no shear forces

4) no bulk motion

p is uniform and hydrostatic at equilibrium is the only mode of reversible

work transfer

No chemical reactions

Pure Substance: a simple (sub)system invariant in chemical composition

pdV

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State Principle

Any two independent properties are sufficient to establish

a stable equilibrium thermodynamic state of a system

For each indentifiable departure from the requirements for

a simple system, one additional independent property is

required

Properties of Pure Substance Model

simple system

pure substance

state principle

one independent property

per reversible mode of

energy interactions

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Properties of Pure Substance Model 1st law

If irreversible, TdS > δQ by the same amount that pdV > δW

Intensive & Extensive Properties

intensive properties: mass independent -T, p, u, s, v

extensive properties: dependent on the extent of system U, S, V

Phase: All parts of a system which have identical and uniform values for each of

the specific properties as well as identical and uniform T and p are said to

constitute one phase

pdvTdsdumVpdmSTdmUd ,///

VSUUWQpdVTdSdU revrev ,,

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Properties of Pure Substance Model Saturation Temperature

temperature @ which two phases coexist in equilibrium with one another at a

given pressure

Saturation pressure

pressure @ which two phases coexist at a given temperature

Boiling, Evaporation, Vaporization

change in phase from sat. liquid to sat. vapor

Condensation, Liquefaction

change in phase from sat. vapor to sat. liquid

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Properties of Pure Substance Model Saturated Vapor

all saturation properties for which v > vcrit

Saturated Liquid

all saturation properties for which v > vcrit

sat.

liqsat.

vap

1 2

pcrit

vcrit

p

v

1 reaches sat. liquid

with positive δQ

at constant v

2 reaches sat. vapor

with positive δQ

at constant v

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Properties of Pure Substance Model Piston-cylinder Apparatus for Isothermal Process

pure

substance

HR

T=T1

F

dia-

thermal

crit.

2φ T1

T2

T3

Tcrit

T4

v

p

f1

g1

f2

g2

denser0

Tv

p

0

Tv

p

0

Tv

p

02

2

Tv

p

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Properties of Pure Substance Model Piston-cylinder Apparatus for Isothermal Process

supercritical

states

superheated

vapor2 phase state

(liq.+vap.)

subcooled

liquid

loci

of

sat

liq

.

loci of sat liq.

pcrit

vcrit

p

v

T

T

T

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Properties of Pure Substance Model Substance that Contracts upon Freezing

sat.

sol

sat.

liq

sat.

liq

sat.

sol

S

2 Φ S+V

L

2 Φ

L+V

2 Φ

S+L

3 Φ

S+L+V

p

v

ptp

p

T

TP(S+L+V)S+V

S+L

L+V

S

V

L

CP

const T

vL>v

S

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Properties of Pure Substance Model Substance Expands upon Freezing

p

T

TP(S+L+V)S+V

S+

L

L+VS

V

L

CP

const T

vL<v

Ssat l

iq.

sat

liq

.

sat

sol.

sat

sol.

S+L

LL+V

S+V

S

v

p

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Properties of Pure Substance Model Substance that Contracts & Expands upon Freezing

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Properties of Pure Substance Model Gibbs phase Rule

f: number of thermodynamic degrees of freedom

(number of independent intensive properties)

c: number of chemically independent species

ζ: number of phases

For a pure substance c=1

– 1 phase (f=2): p and T can be fixed arbitrarily. (divariant)

– 2 phase (f=1): p or T can be fixed arbitrarily (monovariant)

– 3 phase (f=0): (invariant)

» No intensive property can be arbitrarily fixed.

» Intensive state is automatically specified by virtue of equilibrium

cf 2

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Properties of Pure Substance Model Energy change (constant p phase from sat. L to sat. V)

f g f g

p

v

T

s

pdv

mWrev /

Tds

mQrev /

fgrevfg vvmpW ,

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Properties of Pure Substance Model Energy change (constant p phase from sat. L to sat. V)

1st law 2nd law

fgrevfg

fgfg

fgfgfg

mhQ

hhmQ

uumWQ

,

hfg:latent heat of vaporization

fgfg

fgfg

fgfg

fg

g

ffg

ggorTshTsh

hTs

mhmTs

ssmTTdsmQ

g: Gibbs free energy / mass

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Properties of Pure Substance Model Temperature-Entropy Diagram of Pure Substance

sat

S

sat

L

sat

L

sat V

TP line

V+L

V+S

S+L

S

L

V

Tcrit

s

T

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Properties of Pure Substance Model Tabulation of Data

V, mg

L, mf

fgffgf

gf

g

g

gf

g

f

ggff

gf

xsssxvvv

mm

mx

vmm

mvv

vmvmmv

vvv

,

Steam table datafgffgffgf xhhhxuuu ,,

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Properties of Pure Substance Model Tabulation of Data

Ex.1 A system with a total volume of 3 m3 contains 2 kg of H2O at p=7104

N/m2. What is S?

sf s

gs

xsfg

sfg

s

T

f g

msS

xsss

vv

vvx

fgf

fg

f

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Properties of Pure Substance ModelMetastable States

In stable equilibrium states, a system can change from one stable equilibrium

to another only if there is a corresponding, finite, permanent change of state of

the environment.

In metastable states, a system can change states from metastable to some other

stable equilibrium state by means of a finite, but temporary, change of state of

the environment.

Example 1: very clean deaerated water may be heated at constant p in

a new unscratched glass container significantly above TBP(p) W/O the

appearance of bubbles ------ superheated liquid

– metastable: introduction of vapor will cause the rapid, explosive change to

a stable 2 phase state consisting of a liquid phase and a vapor phase in

mutual equilibrium

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409.319A



Thermodynamic Functions Define a canonical relation of thermodynamics for a closed system

mathematically

definitions of temperature and pressure in an abstract mathematical sense (at

hand prosuppose that the actual function u(s,v) is

Think of the independent variables(properties) as a set of coordinates, so that

various characteristic thermodynamic functions should be obtainable from one

another by means of a suitable transformation of coordinates

),( vsuupdvTdsdu

svsv v

up

s

uTdv

v

uds

s

udu

,,

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Thermodynamic FunctionsThe Legendre transformation for the canonical relation

To replace one of the independent variables in an exact differential with its

conjugate variable(property), subtract from the differential of the

dependent variable the differential of the product of the two conjugate

variables

sp

sp

s

hv

s

hT

dps

hds

s

hdh

pvuh

vdpTdsdh

,

Tv

Tv

v

fp

T

fs

dvv

fdT

T

fdf

Tsuf

pdvsdTdf

,

Tp

Tp

p

gv

T

gs

dpp

gdT

T

gdg

Tshg

vdpsdTdg

,

Enthalpy

h=h(s,p)

Helmholtz

free energy

f=f(T,v)

Gibbs

free energy

g=g(T,p)

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Maxwell Relations user the fact that the order of differentiation is immaterial. Given z=z(x,y) with an

exact differential dz

Apply to the canonical relation

xy

yxx

z

yy

z

x

pTpTTp

TvTvvT

vspssp

vsvssv

T

v

p

s

p

g

TT

g

p

v

s

T

p

T

f

vv

f

T

s

v

p

T

p

h

ss

h

p

s

p

v

T

v

u

ss

u

v

u=u(s,v)

h=h(s,p)

f=f(T,v)

g=g(T,p)

409.319A



The Clapeyron Relation (Along the sat. line)

fg

fg

fg

fg

fgfg

sat

fgfg

sat

ff

sat

sat

Tv

h

v

s

dT

dp

ggorg

dTdT

dpvsx

dT

dpvsdg

dTdT

dpvsdg

vdpsdTdgdTdT

dpdp

0

The Clapeyron relation

This result is general and holds for

any two phases in equilbiruim

P

S L

VTP

CP

Contract upon

freezing

T

P

SL

VTP

CP

Expand upon

freezing

T

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The Clapeyron Relation (Along the sat. line) obtain the equation for the vapor pressure curve by integrating:

Exercise: Alternate method of deriving the Clapeyron relation

2

112

2

1

2

1

T

Tfg

fg

fg

fg

T

dT

v

hpp

T

dT

v

hdp

fg

fg

vfg

fg

TvT vv

ss

T

p

vv

ss

v

s

T

p

v

s

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Incompressible Fluid: v=constant

0

.)(

ppT

TpT

p

p

pTT

p

p

p

pT

T

v

p

h

T

T

h

pc

p

constvdTcvdpdh

vT

vTv

p

sT

p

h

T

sTc

T

h

dTT

hdp

p

hdh

cp(T only)

2

1

2

1

2

1

2

1

2

1

1221

2112

12

1212

0,

T

T

v

v

T

T

pv

v

T

Tp

T

Tp

cdTmUUQ

pdvmWT

cdTss

cccT

u

dTTcuu

dTTcppvhh

v is constant rev. work transfer

associated with normal displacement

of the boundary is zero

409.319A



Ideal Gas: pv=RT, p=p(T,v), u=u(T,v)

0

pppT

pT

pv

sT

v

u

T

sTc

T

u

dvpv

sTdT

T

sTdu

dvv

udT

T

udu

v

TT

v

v

v

Tv

Tv

dTcdTcRdh

dTTcTTRhh

RTupvuh

dTTcuu

dTcdu

pv

T

Tv

T

Tv

v

)(

)()(

)(

2

1

2

1

1212

12

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More General Cases 2 Methods

p=p(v,T), u=u(v,T), h=h(v,T), s=s(v,T)

v=v(p,T), u=u(p,T), h=h(v,T), s=s(v,T)

1

2

ideal gas

@vT

v

T1

T2

v1

v2

v

1

2

T

T1

T2

ρρ2

ρ1

1

2

T

T1

T2

pp2

p1

Compute the difference in internal energy between states 1&2

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More General Cases p=p(T,v)

2

22

1

1

112

TT

v

vT

T

Tv

TT

v

vT

T

v

dvpT

pTdTc

dvpT

pTuu

dvpT

pTdTcdu

p (T,v) and cv are two complex to be

integrated in closed form

evaluated numerically

TvsTTvhg

TvsTTvuf

TvpvTvuh

dvv

R

T

p

dvv

R

T

p

dTT

c

v

vRss

dTT

cdv

v

Rdv

v

R

T

pds

v

vTTv

v

vTTv

T

T

v

v

v

,,

,,

,,

ln

2

2

1

1

2

11

212

409.319A



More General Cases v=v(T,p)

2

2

1

1

2

1

0

0

01

212 ln

TT

p

p

TT

pp

p

T

Tp

p

p

dpT

v

p

R

dpT

v

p

R

dTcp

pRss

dTT

cdp

p

Rdp

T

v

p

Rds

),(),(),(

),(),(),(

),(),(),(

2

22

1

1

1

0

0

12

TpsTTphTpg

TpsTTpuTpf

TpvpTphTpu

dpT

vTvdTc

dpT

vTvhh

dTcdpT

vTvdh

TT

p

p

T

Tp

TT

pp

p

p

409.319A



Special Formulation of the p-v-T Data van der Waals Equation of State

suggested by J.D. van der Waals (1873)

represent the p-v-T characteristics of both L+V

has a critical state

approaches the ideal gas behavior as p0

the simplest model of a substance explaining the departure from the ideal gas

behavior

a molecular model in which the molecules have a finite volume and exert long

range attractive forces on one another

2v

a

bv

RTp

b: volume excluded by the dimensions of molecules

a: accounts for attractive forces between molecules

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Special Formulation of the p-v-T Data van der Waals Equation of State

result 1 not all of the physical volume of the container is available to the

molecules of the gas 2 the force that the molecules exert on the container wall

is reduced by the attractive force exerted on a molecule by its neighbors

Example

0)(

8

1,

64

27

0

23

22

2

2

abavvRTbppv

p

RTb

p

TRa

v

p

v

p

c

c

c

c

TTcc

fm

o

ng

v

p

T=T1

Area(f-m-o)=Area(o-m-g)

n-g: metastable vapor

f-m: metastable liquid

m-n: gas is mechanically

unstable

0

Tv

p

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Special Formulation of the p-v-T Data Virial Equation of State

f1, f2, f3 = functions (T only) 2nd, 3rd, 4th virial coefficient

first proposed by Clausius as an improvement over the ideal gas model

virial coefficients account for interaction forces among molecules

magnitudes of interaction forces depend on the nature of the microscopic

model used to describe the forces of interaction

3

3

2

21 )()()(1

v

Tf

v

Tf

v

Tf

RT

pv

409.319A



Special Formulation of the p-v-T Data Generalized Equation of State

First define reduced properties:

All substances follow the same equation of state expressed in terms of

reduced properties

compressibility factor

c

r

c

r

c

rv

vv

T

TT

p

pp

rrrr pTfvTfRT

pvZ

RT

pvZ

,,

1

21

409.319A



Summary Incompressible Fluid

dT

T

Tcss

dTTcppvhh

dTTcuu

cccconstv

T

T

T

T

T

T

pv

2

1

2

1

2

1

12

1212

12

.

-Not 100% accurate

-Pretty good for subcooled

(compressed) liquid

-mechanical / thermal aspects

only weakly coupled

409.319A



Summary Ideal Gas

Rcc

dTT

Tc

p

pRss

dTTchh

dTTcuu

RTpv

vp

T

T

p

T

Tp

T

Tv

2

1

2

1

2

1

1

212

12

12

ln

-In fact u2-u1 holds

regardless of whether

v remains constant or not

409.319A



Summary of Thermodynamic Relation

vdpsdTdgpTggTshg

pdvsdTdfvTffTsuf

pdvTdsdhpshhpvhh

pdvTdsduvsuuuu

),(

),(

),(

),(

The characteristic thermodynamic functions

Property Function Total Derivative

The First partial derivatives of the characteristic

functions

Ts

pv

Ts

pv

pgphv

TgTfs

vfvup

shsuT

)/()/(

)/()/(

)/()/(

)/()/(

The mixed second partial derivatives of the

characteristic functions, the Maxwell relation

pT

vT

ps

vs

T

v

p

s

pT

g

T

p

v

s

vT

f

s

v

p

T

ps

h

s

p

v

T

vs

u

2

2

2

2

409.319A



Compressibility Charts

409.319A



Generalized Compressibility Chart No.1

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Generalized Compressibility Chart No.2

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Aero Thermo Hydro Engineers Nexus Application