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
I
δQ =
ηth =
δW,
reversible
Q̇H
TH
=
TL
Q̇L
Work–polytropic process (P V n = constant)
 P V −P V
2 2
1 1

,
for n 6= 1

1−n
W
=
1 2


P1 V1 ln(V2 /V1 ),
for n = 1
Ẇnet
Q̇L
Q̇H
, β=
, β0 =
Q̇H
Ẇin
Ẇin
Increase in Entropy Principle
Closed System–Control Mass
∆Scm = S2 − S1
m(V22 − V12 )
Q
=
m(u
−
u
)
+
+ mg(Z2 − Z1 ) + 1 W2
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
V2
dEcv
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
cp
P2
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 200
Shaft Work–reversible, polytropic
w=−
n
(Pe ve − Pi vi )
n−1
DWM