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Thermodynamic Thermodynamic PropertiesProperties
Property TableProperty Table -- from direct measurement Equation of StateEquation of State --
any equations that relates P,v, and T of a substance
Ideal -Gas Equation of StateIdeal -Gas Equation of State Any relation among the pressure,
temperature, and specific volume of a substance is called an equation of state. The simplest and best-known equation of state is the ideal-gas equation of state, given as
where R is the gas constant. Caution should be exercised in using this relation since an ideal gas is a fictitious substance. Real gases exhibit ideal-gas behavior at relatively low pressures and high temperatures.
Universal Gas ConstantUniversal Gas ConstantUniversal gas constant is given on
Ru = 8.31434 kJ/kmol-K
= 8.31434 kPa-m3/kmol-k
= 0.0831434 bar-m3/kmol-K = 82.05 L-atm/kmol-K = 1.9858 Btu/lbmol-R = 1545.35 ft-lbf/lbmol-R = 10.73 psia-ft3/lbmol-R
ExampleExampleDetermine the particular gas constant for air and hydrogen.
Kkg
kJ0.287
kmolkg
28.97
KkmolkJ
8.1417
MRR u
air
Kkg
kJ4.124
kmolkg
2.016
KkmolkJ
8.1417
Rhydrogen
2
22
1
11
u
u
T
VP
T
VPV/N v T,Rv P
TNRPVRTPv
mRTPV
Ideal Gas “Law” is a simple Ideal Gas “Law” is a simple Equation of StateEquation of State
Percent error for applying ideal gas Percent error for applying ideal gas equation of state to steamequation of state to steam
Question …...Question …...
Under what conditions is it appropriate to apply the ideal gas equation of state?
Ideal Gas LawIdeal Gas Law Good approximation for P-v-T
behaviors of real gases at low densities (low pressure and high temperature).
Air, nitrogen, oxygen, hydrogen, helium, argon, neon, carbon dioxide, …. ( < 1% error).
Compressibility FactorCompressibility FactorCompressibility FactorCompressibility Factor The deviation from ideal-gas behavior
can be properly accounted for by using the compressibility factor Z, defined as
Z represents the volume ratio or compressibility.
Ideal Gas
Z=1
Real Gases
Z > 1 or Z < 1
Real GasesReal Gases Pv = ZRT or Pv = ZRuT, where v is volume
per unit mole.
Z is known as the compressibility factor. Real gases, Z < 1 or Z > 1.
Compressibility factorCompressibility factor What is it really doing? It accounts mainly for two things
• Molecular structureMolecular structure
• Intermolecular attractive forcesIntermolecular attractive forces
Principle of Principle of corresponding statescorresponding states
Principle of Principle of corresponding statescorresponding states
The compressibility factor Z is approximately the same for all gases at the same reduced temperature and reduced pressure.
Z = Z(PR,TR) for all gases
Reduced Pressure Reduced Pressure and Temperatureand Temperature
crR
crR T
TT ;
P
PP
where:
PR and TR are reduced values. Pcr and Tcr are critical properties.
Compressibility factor for ten substances Compressibility factor for ten substances (applicable for all gases Table A-3)(applicable for all gases Table A-3)
Where do you find critical-point properties? Where do you find critical-point properties? Table A-7Table A-7
Mol
(kg-Mol)
R
(J/kg.K)
Tcrit
(K)Pcrit
(MPa)
Ar 28,97 287,0 (---) (---)
O2 32,00 259,8 154,8 5,08
H2 2,016 4124,2 33,3 1,30
H2O 18,016 461,5 647,1 22,09
CO2 44,01 188,9 304,2 7,39
Reduced PropertiesReduced Properties This works great if you are given a
gas, a P and a T and asked to find the v.
However, if you are given P and v and asked to find T (or T and v and asked to find P), trouble lies ahead.
Use pseudo-reduced specific volume.
Pseudo-Reduced Pseudo-Reduced Specific VolumeSpecific VolumePseudo-Reduced Pseudo-Reduced Specific VolumeSpecific Volume
When either P or T is unknown, Z can be determined from the compressibility chart with the help of the pseudo-reduced specific volume, defined as
not vcr !
Ideal-Gas ApproximationIdeal-Gas Approximation The compressibility chart shows
the conditions for which Z = 1 and the gas behaves as an ideal gas:
(a) PR < 0.1 and TR > 1
Exercise 3-21 Exercise 3-21 Steam at 600 oC & 1 MPa. Evaluate the specific volume using the steam table and ideal gas law
Tsat = 180oC therefore is superheated steam (Table A-3)
Volume = 0,4011 m3/kg
TTooCC1MPa1MPa
vv
374374600600
180180
TcrTcr
Solution - page 1 Solution - page 1 Part (b) ideal gas law
Rvapor = 461 J/kgK
v = RT/P , v = 461x873/106 = 0.403 m3/kg
Steam is clearly an ideal gas at this state. Error is less than 0.5%
Check: PR = 1/22.09 = 0.05 & TR = 873/647 = 1.35
TEAMPLAYTEAMPLAY
Find the compressibility factor to determine the error in treating oxygen gas at 160 K and 3 MPa as an ideal gas.
Other Thermodynamic Properties:Isobaric (c. pressure) Coefficient
PTv
vv
TT
PP
0Tv
v1
P
Other Thermodynamic Properties:Isothermal (c. temp) Coefficient
TPv
vv
PP
TT
0Pv
v1
T
Other Thermodynamic Properties:
vdPvdTdPPv
dTTv
dvTP
We can think of the volume as being a function We can think of the volume as being a function of pressure and temperature, v = v(P,T).of pressure and temperature, v = v(P,T).Hence infinitesimal differences in volume are Hence infinitesimal differences in volume are expressed as infinitesimal differences in P and expressed as infinitesimal differences in P and T, using T, using and and coefficients coefficients
If If and and are constant, we can integrate for v: are constant, we can integrate for v:
000
PPTTvv
Ln
Other Thermodynamic Properties:Internal Energy, Enthalpy and Entropy
PvuP,Thh
v,Tuu
v,uss
Other Thermodynamic Properties:Specific Heat at Const. Volume
vTu
uu
TT
vv0
Tu
Cv
v
Other Thermodynamic Properties:Specific Heat at Const. Pressure
PTh
hh
TT
PP0
Th
CP
P
Other Thermodynamic Properties:Ratio of Specific Heat
v
p
C
C
Other Thermodynamic Properties:Temperature
ss
uu
vv0
us
T1
v
TT
11
Ideal Gases: u = u(T)Ideal Gases: u = u(T)
Therefore,
dvv
u dT
T
udu
Tv
0
dT)T(C dTT
udu v
v
We can start with du and We can start with du and integrate to get the change in u:integrate to get the change in u:
dT (T)C uuu
and dT,Cdu2
1
T
T v12
v
Note that Cv does change with temperature and cannot be automatically pulled from the integral.
Let’s look at enthalpy Let’s look at enthalpy for an ideal gas:for an ideal gas:
h = u + Pv where Pv can be replaced by RT because Pv = RT.
Therefore, h = u + RT => since u is only a function of T, R is a constant, then h is also only a function of T
so h = h(T)
Similarly, for a change in Similarly, for a change in enthalpy for ideal gases:enthalpy for ideal gases:
(T)dTChhh
and dT,Cdh
p
T
T 12
p
2
1
0P
h & )T(CC pp
Summary: Ideal GasesSummary: Ideal GasesSummary: Ideal GasesSummary: Ideal Gases
For ideal gases u, h, Cv, and Cp are functions of temperature alone. For ideal gases, Cv and Cp are written in terms of ordinary differentials as
gas idealp
gas idealv dT
dhC ;
dT
duC
For an ideal gas,For an ideal gas,
h = u + Pv = u + RT
R dT
du
dT
dh
Kkg
kJ Cp = Cv + R
RatioRatio of specific heats of specific heats is given the symbol, is given the symbol,
1CR
1C
C
vv
p
(T)(T)C
(T)C
C
C
v
p
v
p
Other relations with the Other relations with the ratio of specific heats which ratio of specific heats which
can be easily developed:can be easily developed:
1-R
C and 1-
RC pv
For monatomic gases,For monatomic gases,
constants. are both and
R2
3R , C
2
5C vp
Argon, Helium, and Neon
For all other gases,For all other gases, Cp is a function of temperature and it may be
calculated from equations such as those in Table A-5(c) in the Appendice
Cv may be calculated from Cp=Cv+R.
Next figure shows the temperature behavior …. specific heats go up with temperature.
Specific Heats for Some GasesSpecific Heats for Some Gases Specific Heats for Some GasesSpecific Heats for Some Gases
Cp = Cp(T) a
function of
temperature
Three Ways to Calculate Three Ways to Calculate Δu and ΔhΔu and Δh
Three Ways to Calculate Three Ways to Calculate Δu and ΔhΔu and Δh
Δu = u2 - u1 (table)
Δu =
Δu = Cv,av ΔT
(T) dT C2
1 v
Δh = h2 - h1 (table)
Δh =
Δh = Cp,av ΔT
(T) dT C2
1 p
Isothermal ProcessIsothermal ProcessIsothermal ProcessIsothermal Process Ideal gas: PV = mRT
For ideal gas, PV = mRT For ideal gas, PV = mRT We substitute into the We substitute into the
integralintegral
1
22
1 V
VnmRT
V
dV mRTW
b
dVV
mRT PdV W
b 2
1
2
1
Collecting terms and integrating yields:
Polytropic ProcessPolytropic Process Polytropic ProcessPolytropic Process
PVn = C
Ideal Gas Adiabatic Process and Reversible Work
What is the path for process with expand or contract without heat flux? How P,v and T behavior when Q = 0?
To develop an expression to the adiabatic process is necessary employ:
1. Reversible work mode: dW = PdV2. Adiabatic hypothesis: dQ =03. Ideal Gas Law: Pv=RT4. Specific Heat Relationships5. First Law Thermodynamics: dQ-dW=dU
Ideal Gas Adiabatic Process and Reversible Work (cont)
First Law:
Using P = MRT/V
Integrating from
(1) to (2)
dTvMCPdV0
dUdWdQ
TdT
RC
VdV
11
V
1
1
2
1
2VV
TT
Ideal Gas Adiabatic Process and Reversible Work (cont)
2
1
1
2VV
PP
1
2
1
1
2VV
TT
Using the gas law :
Pv=RT, other
relationship amid
T, V and P are
developed
accordingly:
1
1
2
1
2PP
TT
Ideal Gas Adiabatic Process and Reversible Work (cont)
An expression for
work is developed
using PV =
constant.
i and f represent
the initial and final
states
1
PVPV
VV1
PVW
V
dVPVPdVW
if
1i
1f
i
i
Ideal Gas Adiabatic Process and Reversible Work (cont)
The path representation are lines where Pv = constant.
For most of the gases, 1.4
The adiabatic lines are always at the righ of the isothermal lines.
The former is Pv = constant (the exponent is unity)
PP
vv
Q = 0
Q = 0T=const.
T=const.
ii
ffff
Polytropic ProcessPolytropic ProcessA frequently encountered process for gases is the polytropic process:
PVn = c = constant
Since this expression relates P & V, we can calculate the work for this path.
2
1
V
Vb PdVW
Polytropic ProcessPolytropic Process
Constant pressure 0 Constant volume Isothermal & ideal gas 1 Adiabatic & ideal gas k=Cp/Cv
Process Exponent n
Boundary work for a gas which Boundary work for a gas which obeys the polytropic equationobeys the polytropic equation
111
11
12
1 2
1
2
1
2
1
n n
VV
-n
Vv
v c
V
dV cPdV W
nn-n
nb
We can simplify it We can simplify it furtherfurther
The constant c = P1V1n = P2V2
n
11
1
1122
1111
1222
, nn
VPVP-n
VVP)(VVP Wnnnn
b
Polytropic ProcessPolytropic Process
1
11
1
2
1122
2
1
2
1
, nV
VnPV
, nn
VPVP
dVV
c PdVW
n
b
Exercise 3-11Exercise 3-11A bucket containing 2 liters of water at 100oC is
heated by an electric resistance.
a) Identify the energy interactions if the system boundary is i) the water, ii) the electric coil
b) If heat is supplied at 1 KW, then how much time is needed to boil off all the water to steam? (latent heat of vap at 1 atm is 2258 kJ/kg)
c) If the water is at 25oC, how long will take to boil off all the water (Cp = 4.18 J/kgºC)
Solution - page 1Solution - page 1Part a)
If the water is the system then Q > 0 and W = 0. There is a temperature difference between the electric coil and the water.
If the electric coil is the system then Q < 0 and W < 0. It converts 100% of the electric work to heat!
Solution - page 2Solution - page 2
The mass of water is of 2 kg. It comes from 2liters times the specific volume of 0.001 m3/kg
To boil off all the water is necessary to supply all the vaporization energy:
Evap = 2285*2=4570 KJ
The power is the energy rate. The time necessary to supply 4570 KJ at 1KJ/sec is therefore:
Time = 4570/1 = 4570 seconds or 1.27 hour
Part b)
Solution - page 3Solution - page 3The heat to boil off all the water initially at 25oC is the sum of : (1) the sensible heat to increase the temperature from 25oC to 100oC, (2) the vap. heat of part (b)
The sensible heat to increase from 25oC to 100oC is determined using the specific heat (CP = 4.18 kJ/kgoC)
Heat = 4.18*2*(100-25)=627 KJ
The time necessary to supply (4570+627)KJ at 1KJ/sec is :
Time = 5197/1 = 5197 seconds or 1.44 hour
Part c)
Exercise 3-12Exercise 3-12A bucket containing 2 liters of R-12 is left outside
in the atmosphere (0.1 MPa)
a) What is the R-12 temperature assuming it is in the saturated state.
b) the surrounding transfer heat at the rate of 1KW to the liquid. How long will take for all R-12 vaporize?
Solution - page 1Solution - page 1Part a)
From table A-2, at the saturation pressure of 0.1 MPa one finds:
• Tsaturation = - 30oC
• Vliq = 0.000672 m3/kg
• vvap = 0.159375 m3/kg
• hlv = 165KJ/kg (vaporization heat)
Solution - page 2Solution - page 2Part b)
The mass of R-12 is m = Volume/vL, m=0.002/0.000672 = 2.98 kg
The vaporization energy:
Evap = vap energy * mass = 165*2.98 = 492 KJ
Time = Heat/Power = 492 sec or 8.2 min
Exercise 3-17Exercise 3-17
A rigid tank contains saturated steam
(x = 1) at 0.1 MPa. Heat is added to
the steam to increase the pressure to
0.3 MPa. What is the final
temperature?
Solution - page 1Solution - page 1PP
0.1MPa0.1MPa
vv
99.6399.63ooCC
Tcr = 374Tcr = 374ooCC
0.3MPa0.3MPa133.55133.55ooCC
superheatedsuperheated
Process at constant volume, search at the superheated steam table for a temperature corresponding to the 0.3 MPa and v = 1.690 m3/kg -> nearly 820oC.
Exercise 3-30Exercise 3-30 Air is compressed reversibly and adiabatically
from a pressure of 0.1 MPa and a temperature of 20oC to a pressure of 1.0 MPa.
a) Find the air temperature after the compressionb) What is the density ratio (after to before
compression)c) How much work is done in compressing 2 kg of
air?d) How much power is required to compress 2kg
per second of air?
In a reversible and adiabatic process P, T and v follows:
PP
vv
Q = 0
Q = 0T=const.
T=const.
ii
ffff
Solution - page 1Solution - page 1
2
1
1
2VV
PP
1
2
1
1
2VV
TT
1
1
2
1
2PP
TT
Solution - page 2Solution - page 2
Part a) The temperature after compression is
Part b) The density ratio is
)oC293( K566
1.01
293TPP
TT4.14.0
2
1
1
212
179.51.0
1PP
VV 4.11
1
21
1
2
2
1
1
2
Solution - page 3Solution - page 3Part c)
The reversible work:
Part d)
The power is:
KW391
1TTRM
dtdW
P 12
KJ391
4.02935662872
1TTRM
1
PVPVW 1212
REV
Exercícios – Capítulo 3Propriedades das Substâncias
Puras
Exercícios Propostos: 3.6 / 3.9 / 3.12 / 3.16 / 3.21 / 3.22 / 3.26 / 3.30 / 3.32 / 3.34
Team Play: 3.1 / 3.2 / 3.4
Ex. 3.1) Utilizando a Tabela A-1.1 ou A-1.2, determine se os estados da água são de líquido comprimido, líquido-vapor, vapor superaquecido ou se estão nas linhas de líquido saturado ou vapor saturado. a) P=1,0 MPa; T=207 ºC b) P=1,0 MPa; T=107,5 ºC c) P=1,0 MPa; T=179,91 ºC; x=0,0 d) P=1,0 MPa; T=179,91 ºC; x=0,45 e) T=340 ºC; P=21,0 MPa f) T=340 ºC; P=2,1 MPa g) T=340 ºC; P=14,586 MPa; x=1,0 h) T=500 ºC; P=25 MPa i) P=50 MPa; T=25 ºC
a) Vapor superaquecido
b) Líquido comprimido
c) Líquido saturado
d) Líquido-Vapor
e) Líquido comprimido f) Vapor superaquecido
g) Vapor saturado
h) Vapor superaquecido - Fluido
i) Líquido comprimido - Fluido
Ex. 3.2) Encontre o volume específico dos estados “b”, “d” e “h” do exercício anterior.
b) P=1,0 MPa; T=107,5 ºC vvl=0,001050(utilizar referência T=107,5ºC)
d) P=1,0 MPa; T=179,91 ºC; x=0,45 v=0,08812[v=(1-x)vl+x(vv)]
h) T=500 ºC; P=25 MPa v=0,011123 m3/kg(Tabela A-1.3 Vapor Superaquecido)
Ex. 3.4) Amônia a P=150 kPa, T=0 ºC se encontra na região de vapor superaquecido e tem um volume específico e entalpia de 0,8697 m3/kg e 1469,8 kJ/kg, respectivamente. Determine sua energia interna específica neste estado.
u =1339,345 kJ/kg(u = h - P v)