Ideal Gas
P v = RT
P v̄ = R̄T
R = cp − cv
Cycle
I
Boundary Work–reversible process
Z
W
=
P dV
1 2
k = cp /cv
Work–polytropic process (P V n = constant)
 P V −P V
2 2
1 1

,
for n 6= 1

1−n
1 W2 =


P1 V1 ln(V2 /V1 ),
for n = 1
I
δQ =
δW
Closed System–Control Mass
Increase in Entropy Principle
∆Scm = S2 − S1
m(V22 − V12 )
+ mg(Z2 − Z1 ) + 1 W2
Q
=
U
−
U
+
1 2
2
1
2
Z
2
m(s2 − s1 ) ≥
1
δQ
=
T
2
Z
1
∆Ssurr = −
δQ
+ 1 S2gen
T
1 Q2
T0
∆Snet = ∆Scm + ∆Ssurr ≥ 0
Open System–Control Volume
X
ṁi −
X
ṁe =
e
i
Q̇cv − Ẇcv +
X
i
dmcv
dt
ṁ = V̇ /v = VA/v
X
dEcv
V2
V2
ṁi hi + i + gzi −
ṁe he + e + gze =
2
2
dt
e
X
X
X Q̇cv
X
X
X Q̇cv
dScv
≥
ṁi si −
ṁe se +
=
ṁi si −
ṁe se +
+ Ṡ gen
dt
T
T
e
e
i
i
Gibbs Relations
T ds = du + P dv = dh − v dP
Internal Energy, Enthalpy, and Entropy Change–Ideal Gas
Z 2
Z 2
u2 − u1 =
cv dT,
h2 − h1 =
cp dT,
1
Z
s2 − s1 =
1
2
1
P2
cp
dT − R ln
=
T
P1
Z
2
1
cv
v2
P2
dT + R ln
= s0T2 − s0T1 − R ln
T
v1
P1
Isentropic Process–Ideal Gas with constant specific heats
T2
=
T1
P2
P1
k − 1
k
=
v1
v2
k − 1
Shaft Work–reversible, neglect KE & PE
Z e
w=−
v dP
i
ME 301
Shaft Work–reversible, polytropic
w=−
n
(Pe ve − Pi vi )
n−1
DWM
Closed System–Exergy and Irreversibility
Combined first and second law
1 W2
= U1 − U2 − T0 (S1 − S2 ) +
X
j
T0
Qj 1 −
− T0 1 S2gen
Tj
Useful work
u
1 W2 = 1 W2 − P0 (V2 − V1 ) = U1 − U2 − T0 (S1 − S2 ) + P0 (V1 − V2 ) +
X
j
Exergy or nonflow availability per unit mass
T0
− T0 1 S2gen
Qj 1 −
Tj
Useful work
φ = (u − u0 ) − T0 (s − s0 ) + P0 (v − v0 )
u
1 W2
= m(φ1 − φ2 ) +
X
j
T0
Qj 1 −
− T0 1 S2gen
Tj
Irreversibility

1 I2

= 1 W2rev − 1 W2 = 1 W2u,max − 1 W2u = T0 1 S2gen = T0 (S2 − S1 ) −
X Qj
j
Tj

Exergy balance
Φ2 − Φ1 = m(φ2 − φ1 ) = − [1 W2 − P0 (V2 − V1 )] +
X
j
T0
− T0 1 S2gen
Qj 1 −
Tj
Control Volume–Exergy and Irreversibility
Combined first and second law
X
X
X
Ve2
T0
V2i
+ gzi − T0 si −
+ gze − T0 se +
ṁe he +
Q̇j 1 −
− T0 Ṡgen
Ẇcv =
ṁi hi +
2
2
Tj
e
j
i
Flow availability or exergy per unit mass
ψ = (h + V2 /2 + gz − T0 s) − (h0 + gz0 − T0 s0 ).
Reversible work
X
X
X
T0
rev
Q̇j 1 −
Ẇcv
=
ṁi ψi −
ṁe ψe +
Tj
e
i
j
Irreversibility
"
rev
I˙cv = Ẇcv
− Ẇcv = T0 Ṡgen = T0
X
ṁe se −
X
e
i
ṁi si −
X Q̇cv
#
T
Exergy balance
Ψ̇2 − Ψ̇1 =
X
e
ME 301
ṁe ψe −
X
i
ṁi ψi = −Ẇcv +
X
j
T0
Q̇j 1 −
− T0 Ṡgen
Tj
DWM
ME 301
Notes
Gas Mixtures
Composition
• Gravamatric analysis (mass basis)
m = m1 + m2 + m3 + · · · =
X
mi
i
1=
or with ci ≡ mi /m, then
P
X mi
m2
m3
m1
+
+
+ ··· =
m
m
m
m
i
ci = 1.
• Molar analysis (mole basis)
n = n1 + n2 + n3 + · · · =
X
ni
i
1=
or with yi ≡ ni /n, then
P
X ni
n2
n3
n1
+
+
+ ··· =
n
n
n
n
i
yi = 1.
• Molecular weight
Mi =
mi
ni
m
=
M=
n
P
X
n i Mi
=
yi Mi
n
• Conversion
mole basis → mass basis
ci =
mi
n i Mi
ni Mi /n
yi M i
=P
=P
=P
m
nj M j
nj Mj /n
yj Mj
mass basis → mole basis
yi =
ni
mi /Mi
mi /(Mi m)
ci /Mi
=P
=P
=P
n
mj /Mj
mj (Mj m)
cj /Mj
Ideal Gas Relationships
P V = nR̄T = mRT
X
R̄ = 8.314
ni = n
and
X
mi = m
kJ
ft lbf
Btu
= 1545
= 1.986
kmol K
lb mol R
lb mol R
and
R = R̄/M
ME 301
Notes
• Dalton Model - components behave as an ideal gas at T and V of mixture
Pi = ni R̄T /V = mi Ri T /V
ni R̄T /V
ni
Pi
=
=
= yi
P
n
nR̄T /V
=⇒ Pi = yi P
X
∴
Pi =
X
yi P = P
X
yi = P
X
yi = V
• Amagat-Leduc Model - components behave as an ideal gas at T and P of mixture
Vi = ni R̄T /P = mi Ri T /P
ni R̄T /P
ni
Vi
=
= yi
=
V
n
nR̄T /P
• Note that R =
P
=⇒ Vi = yi V
X
∴
Vi =
X
yi V = V
ci Ri
Properties
X
U=
X
U = nū =
c̄v ≡
∂ ū
∂T
=
X
yi
H=
c̄p ≡
∂ h̄
∂T
X
=
∂ ūi
∂T
X
=
X
yi
p
Hi
∂ h̄i
∂T
X
Si
=
or
or
yi c̄vi
or
X
X
yi h̄i
cv ≡
Hi = ni h̄i
or
or
ni s̄i =⇒ s̄ =
X
X
yi c̄pi
where
X
Si = ni s̄i
∂h
∂T
or
dPi
dT X
−
yi R̄
T
Pi
or
ds =
Mi =
ci
v
∂ui
∂T
ci ui
=
X
X
ci hi
=
X
ci cvi
v
X
mi hi =⇒ h =
=
X
ci
p
∂hi
∂T
ci cpi
p
Si = mi si
S = ms =
X
X
X
Hi = mi hi
or
and
=
H = mh =
cp ≡
mi ui =⇒ u =
∂u
∂T
yi s̄i
ū
h̄
s̄
c̄v
c̄p
= = =
=
u
h
s
cv
cp
X
or
yi c̄pi
Ui = mi ui
U = mu =
p
X
M=
X
yi ūi
where
S = ns̄ =
ds̄i =
X
Ui = ni ūi
v
ni h̄i =⇒ h̄ =
S=
ds̄ =
where
ni ūi =⇒ ū =
v
H = nh̄ =
Ui
X
mi si =⇒ s =
dsi =
X
ci cpi
X
ci si
dPi
dT X
−
ci Ri
T
Pi
ūi
h̄i
s¯i
c̄v
c̄p
=
=
= i = i
ui
hi
si
cvi
c pi