Aero Thermo Hydro Engineers Nexus Application
Transcript of 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
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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 ,,
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
Properties of Pure Substance Model Substance that Contracts & Expands upon Freezing
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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 ,
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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 ,,
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
,,
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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)
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
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
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
Compressibility Charts
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
Generalized Compressibility Chart No.1
409.319A
Aero Thermo Hydro Engineers Nexus ApplicationDepartment of Nuclear Engineering, Seoul National University
Thermofluid Properties
Generalized Compressibility Chart No.2