Summary of Thermo
from: first_law_rev_2005.mcd,
second_law_rev_2005.mcd,
availability.mcd
ref: van Wylen & Sonntag (eqn
#s) Woud (W nn.nn)
First Law
⌠
⌠
 1 dQ =  1 dW
⌡
⌡
first law for cycle
first law for system
change of state
Q1_2
Q1_2 = E2 − E1 + W1_2
E1
(5.2)
is the heat transferred TO system
E2
are intial and final values of
energy of system and ...
W1_2
(5.5)
is work done BY the system
δQ = dE + δW = dU + dKE + dPE + δW
Closed System
d
U = Q_dot − W_dot
dt
first law as a rate equation
dU = δQ − δW
(5.4)
d
d
d
d
d
d
d
Q = U + KE + PE + W = E + W
dt
dt
dt
dt
dt
dt
dt
H = U + p⋅ V
first law as a rate equation - for a control volume
∑
n
2



Vi
d
m_dot
⋅
h
+
+
g
⋅
z

i i
i  = Ec_v +
2


 dt
E = U + Ekin + Epot
Woud assuming energy
∑
n
2



Ve
d
m_dot
⋅
h
+
+
g
⋅
z

e e
e  + Wc_v
2


 dt
and ...
Ekin = Epot = 0




Vi
Ve
d
+ g ⋅ zi − m_dote⋅  h e +
+ g ⋅ ze
U = Q_dot − W_dot + m_doti ⋅  h i +
2
2
dt




2
2
steady state, steady flow process ...
open stationary
∑
m_dotin = ∑ m_doten
n
d
Qc_v +
dt
∑
n
2



Vi
m_dotin⋅  h i + 2 + g ⋅ zi  =



(5.31 and 5.32)
enthalpy defined - is
a property (5.12)
h = u + p⋅ v
d
Qc_v +
dt
(5.45)
E=U
N.B. dot => rate not d( )/dt
W (2.1)
m_dot = flow_rate (5.46)
n
2



Ve
d
m_doten⋅  he + 2 + g⋅ ze  + dt Wc_v



∑
n
(W 2.3)
m_dote = m_doti = 0
(5.47), (W2.8)
steady state steady flow ... - single flow stream
Vi
2
q + hi +
+ g ⋅ zi = h e +
2
10/6/2005
Ve
2
2
+ g ⋅ ze + w
1
this on per unit mass
basis q = Q/m_dot
(5.50)
uniform state, uniform flow process
Qc_v +
∑
n
2
 

Vi
min⋅  h i + 2 + g ⋅ zi  =
 

2



Ve
men⋅  h e + 2 + g ⋅ze  ...



∑
n
(5.54)
2
2




V2
V1
+ m2 ⋅  u 2 +
+ g ⋅ z2 − m1 ⋅  u 1 +
+ g ⋅ z1 + Wc_v
2
2




Second Law
Carnot cycle most efficient, and only function of temperature
Entropy
⌠
 1 dQ = 0
⌡
=> for all reversible
heat engines ...
 δQrev 
dS = 

T 
⌠


⌡
2
1
T
⌠


⌡
inequality of Clausius ...
reversible ...
⌠


⌡
1
T
T
TL
TH
integrals are cyclic
dQ ≤ 0
⌠


⌡
⌠
 1 dQ ≥ 0
⌡
=> all irreversibles
engines
dQ = 0
⌠
so as we did for energy E (e) in first law 

⌡
(7.2)
dQrev. = S2 − S1
1
η thermal = 1 −
1
T
1
T
dQ < 0
dQ is
independent of path in reversible process => is a porperty of
the substance. entropy is an extensive property and entropy
per unit mass is = s
(7.3)
two relationships for simple compressible
1
⌠
S2 − S1 ≥ 

⌡
2
1
T
equality holds when
substance - Gibbs equations
reversible and when
appicable to rev & irrev processes
irreversible, the change of
entrpy will be greater than
T⋅ dS = dU + p ⋅ δV (7.5)
T⋅ ds = du + p ⋅ δv
the reversible
T⋅ dS = dH − V⋅ dp (7.6)
T⋅ ds = dh − v ⋅ dp
dQ
1
second law for a control volume
d
Sc_v +
dt
Q_dot c_v
∑ (
m_dote⋅se)
− ∑ (
m_doti⋅si)
≥∑
n
n
T
c_v
steady state, steady flow process
d
Sc_v = 0
dt
∑
(
m_dote⋅se)
− ∑ (
m_doti⋅si)
≥∑
n
n
(7.49)
= when reversible
(7.50)
Q_dot c_v
(7.51)
T
c_v
= when reversible
uniform state, uniform flow process
t
m2 ⋅ s2 − m1 ⋅ s1 +
∑ (
me⋅se)
− ∑ (
)
n
10/6/2005
⌠ Q_dot
c_v

mi⋅ si ≥
dt

T
⌡
n
0
2
(7.56)
= when reversible
(7.7)
Availability
reversible work (maximum) of a control volume that exchanges heat with the
surroundings at To
2
2
 
  

 
Vi
Ve

 ...
Wrev =
min⋅  hi − To ⋅si + 2 + g⋅ zi  −  men⋅  h e − To⋅se + 2 + g ⋅ ze   (8.7)
 




n
n
 latter [..] is total for
2
2
 


 
V2
V1
c.v.
+ − m2 ⋅  u 2 − To ⋅s2 +
+ g ⋅ z2 − m1 ⋅  u 1 − To ⋅s1 +
+ g ⋅ z1 
2
2
 



∑
∑
system (fixed mass
Wrev_1_2
m
2


V1
= wrev_1_2 =  u 1 − To ⋅s1 +
+ g ⋅ z1 −
2


2


V2
u
−
T
⋅s
+
+
g
⋅
z
 2
o 2
2
2


(8.8)
steady-state, steady flow process - rate form
W_dot rev =
2
 
 
Vi

m
⋅
h
−
T
⋅s
+
+
g
⋅
z
 in  i
o i
i −
2
 


∑
n
∑
n
2


 
Ve

m
⋅
h
−
T
⋅s
+
+
g
⋅
z
 en  e
o e
e 
2




(8.9)
2


Ve
= wrev = h i − To ⋅ si +
+ g ⋅ zi −  h e − To ⋅ se +
+ g ⋅ ze
m_dot
2
2


Vi
W_dot rev
single flow of fluid
2
(8.10)
availablity
steady state, steady flow process ...(e.g. single flow ...availability (per unit mass flow)
2


Vo
ψ = h − To ⋅s +
+ g ⋅ z −  h o − To ⋅so +
+ g ⋅ zo
2
2


Vi
2
(8.16)
reversible work between any two states = decrease in availablity between them
(
)
(
wrev = ψ i − ψ e = h 1 − To ⋅s1 − h 2 + To ⋅s2 = h 1 − To ⋅s1 − h 2 + To ⋅s2 = h 1 − h 2 − To ⋅ s1 − s2
can be written for more than one flow ...
W − dotrev =
∑
 min⋅ψ in − ∑ 
men⋅ψ en
n
10/6/2005
(8.18)
n
3
)
(8.17) extended
availability w/o KE and PE per unit mass of system
(
) (
)
(
)
(
φ = u + p o ⋅v − To ⋅s − u o + p o ⋅v o − To ⋅so = u − u o + p o ⋅ v − v o − To ⋅ s − so
)
(8.21)
and reversible work maximum between states 1 and 2 is ...
2
(
)
wrev_1_2 = φ1 − φ2 − p o ⋅ v 1 − v 2 +
10/6/2005
V1 − V2
2
2
(
+ g ⋅ z1 − z2
4
)
(8.22)