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working fluids
()*+,-./0123
• STUVW (equation of state)
p = f ( T, v )
• kl() (compressibility factor)
OPQR.STUVW
XYZR
XYZR.ST[
XYZR.ST\]
– XYZR
z = pv/(RT) = 1
– OPQR
z = 1 + B! + C!2 + !!
!"#$%&'
virial equation of state
!"#$%&'
virial equation of state
1.0
• z = 1 + B! + C!2 + !!
Tr=2
Z
T r=1.2
=1
c
T
/
T
T r=
0.2
0
1
pr=p/pc
7
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real fluids and corresponding state
real fluids and corresponding state
dbce
pc
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*Za3bc`
a/
:;>?
j
/
pr=p/pc
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:;>?
pr= p/pc
B: "#$%&'()*#+,-./0123
second virial coefficient
C: "4$%&'()
third virial coefficient
$%&'56789:;<=>?@ABC
+,-DEF=.()5GHIJKLMNC
pc
Z
/
i
hZ
g
f
Tr= T/Tc
Tc
*Za3bc`
a
/
j
/
fg
aR
i
#/e
mi
Zk=
fgh
Z/
:;B=
dbce
:;<=
!r=!/!c !c
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COYZ$[R
specific enthalpy
()*+
ideal gas
pv = RT
R = R0 / M
M [kg!kmol-1]:n'o[
R [kJ!kg-1!K-1] :ZRp)
R0 [J!mol-1!K-1]qn'ZRp)
rrrrrrrrrr*stZRp)3
R0 = 8.314472 J!mol-1!K-1
CDEF
specific heat capacity
"u‘’ dq = du + pdv
OYZ$[R\]^._`aLbcd
h [kJ!kg-1]= u + pv
efcd,
dh = du + pdv + vdp
S>23gh dp = 0
dh = du + pdv
dhhDFaijklmdJnDGHogc
6+GH
Thermophysical Properties of Fluids
"u‘’ dq = du + pdv
+IJKL.MN
dq = du
LICD cV=("q/"T)V=("u/"T)V
>?JKL.MN
dq = du + pdv = dh
L>CD cp=("q/"T)p=("h/"T)p
OP$QRST.UV
Law of equipartition of energy
• ZR+,.“”}~'•€6•.vw.–—˜
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1—˜…•žŸ6*WX #35¡BC
– ¢£,+,6yz.(x,y,z)<—˜…6 3q u=¤¥¦§¨RT
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—˜…6 ¥´*¥µu3=¶q u=¤¶¦§¨RT
– 4£,·ŠD¡B5—˜…665¡Bqu=¤¸¦§¨RT
ZR+,vwx
ZR+,.yzvwD{|F
vw}~'•€•‚ƒ„…D†‡ˆIKBC
u=(3/2)pV=(3/2)RT
„…u‰Š‹Œ•B.D(3/2)R.Ž•••
cV=(3/2)R
()*+%&.CDEF
specific heats at ideal gas state
dh
= du + d(pv)
= cV dT +RdT
= cp dT
cp = cV + R
CDCq " = cp / cV > u
()*+.%&pqs/012
3t
()*+.%&pqs/012
3t
change of states for ideal gas (quasi-static process)
change of states for ideal gas (quasi-static process)
™„\]*ÁV3
™k\]*ÁV3
™Â\]*ÁV3
ÃŽ\]*ÁV3$Ä%ÅÆ€Ç\]
™„\]*ÁV3
™k\]*ÁV3
™Â\]*ÁV3
ÃŽ\]*ÁV3$Ä%ÅÆ€Ç\]
SBpq
isothermal change
S>pq
isobaric change
• pv = RT = constant
dq = du + dw = cVdT + dw = dw = pdv
= (RT/v)dv
• q12 = w12 = RT (ln v2 # ln v1) = RT ln (v2/ v1)
= p1v1 ln (v2/ v1) = p2v2 ln (p1/ p2)
• ¼HžŽ[•½I¾¿D\ÀŒ›BC
SEpq
isochoric change
• p/T = constant = R/v
dw = pdv = w12 = 0
• dq = du + pdv = du = cv dT
q12 = u2 - u1
• XYZR.¹º6†Ž•sp»¡.<
q12 = u2 - u1 = cv (T2 - T1)
• v/T = constant = R/p
w12 = p (v2 # v1)
• dq = du + pdv = dh = cp dT r(vdp = 0)
q12 = h2 # h1
• XYZR.¹º6†Ž•sp»¡.<
q12 = h2 # h1 = cp (T2 # T1)
uDsSOYvw[Rtpq
adiabatic (isentropic) change
dq
= du + dw = 0
dw = -du
()*+.MNq w12 = -u12 = cv(T1-T2)
ÃŽ\].¹º6œÈÉ}~'•€\]•
¾¿D¡BC
uDsSOYvw[Rtpq
adiabatic (isentropic) change
uDsSOYvw[Rtpq
adiabatic (isentropic) change
vdp+ (cp / cv) pdv = 0
vdp+ " pdv = 0
*1/p3dp+ " (1/v)dv = 0
ln p + " ln v = constant
()*+.MN
pdv + vdp = RdT
dq = cv dT + pdv = cv(pdv + vdp )/R+ pdv = 0
cv(pdv + vdp ) + R pdv = 0
cv vdp+(cv + R ) pdv = 0
cv vdp+ cppdv = 0
vdp+ (cp / cv) pdv = 0
p v " = constant
p = RT/v JŸ
T v " -1 = constant
uDpq,x"vwRypq
adiabatic and polytropic changes
XYZR.ÃŽ\]
p v " = constantœ # =" :†Ž†
OPZR.ÃŽ\]
#
p v = constantœ# qÃŽÊ)
x"vwRypq
polytropic changes
ÃŽ\] n=#
p
isentropic exponent
OPZR.Ä%ÅÆ€Ç\]
p v n = constantœn qÄ%ÅÆ€ÇÊ)
w
polytropic exponent
x"vwRypq
polytropic changes
v
()*+.%&pq
change of states for ideal gas
n=0
p
n=%
n= 1
n= #
v
¾¿ w12 Ž[ q12
™„\]
RTln(v2/v1)
RTln(v2/v1)
™k\]
p(v2-v1)
h2 - h1
™Â\]
0
u2 - u1
ÃŽ\]
u1 - u2
0
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