Final Exam Formula Sheet

M.P.
Bradley
11
April
2012
€
EP271
Formula
Sheet
1 ST 
LAW
OF
THERMODYNAMICS:
Delta‐Form:
 ΔE = ΔKE + ΔPE + ΔU = Q −W
Differential
Form:
 dE = dKE + dPE + dU = δQ −δW
dE
Time
Rate
Form:
dt = dKE + dPE + dU = Q ˙ −W ˙
dt dt dt
€
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where
Q
=
positive
heat
INPUT
to
system
and

 W
=
positive
work
OUTPUT
from
system
€
WORK
Expansion
Work:
 δW = pdV
Generalized
work:
 δW = pdV −σd(Ax) −εdZ − E ⋅ d VP
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HEAT
TRANSFER:
€
Conduction: Q ˙
x
Radiation:
= κA dT
dx
˙ Q rad
= εσAT 4
( ) − µ 0
H ⋅ d VM
( ) + ...
Special
case:
 € Q ˙
rad
= εσA(T 4 4
− T environment
)
NET
radiative
heat
transfer
to
environment
€
Convection:
€
Q ˙
c
= hA(T − T fluid
)
ENTHALPY: H = U + PV 
SPECIFIC
ENTHALPY:
 h = u + pv
TWO‐PHASE
LIQUID‐VAPOUR
SYSTEMS
€
€
m
Vapour
“Quality”
Factor
 χ ≡
vapour
m liquid
+ m vapour
Specific
volume
of
two‐phase
mixture:
 v = ( 1− χ)v f
+ χv g
= v f
+ χ( v g
− v f )
Specific
internal
energy
of
two‐phase
mixture:
€
u = ( 1− χ)u f
+ χu g
= u f
+ χ( u g
− u f )
€
€

M.P.
Bradley
11
April
2012
€
€
€
€
Specific
enthalpy
of
two‐phase
mixture:
SPECIFIC
HEATS:
€
⎛
Specific
heat
at
constant
volume
 c v
≡ ∂u ⎞
⎜ ⎟
⎝ ∂T ⎠
v
⎛
Specific
heat
at
constant
pressure
 c p
≡ ∂h ⎞
⎜ ⎟
⎝ ∂T ⎠
h = ( 1− χ)h f
+ χh g
= h f
+ χ( h g
− h f )
Specific
heat
ratio:
 k ≡ c € p
(Note
that
k
is
also
often
called
“gamma”
 γ )
c v
€
IDEAL
GAS
MODEL:
€
€
Gas
Compressibility
Factor
 Z = pv
R T
For
an
ideal
gas,
 Z =1.0
and
therefore:
€
pv = R
€
T 
(per
unit
mole,
 R = 8.314
J/mol.K)
pv = RT 
(per
unit
mass)
€
Pressure‐Volume
Relationship
for
a
Polytropic
process:

 pV n = constant
Work
done
in
polytropic
expansion
(2
cases):
W = p 2V 2
− p 1
V 1
(valid
for
 n ≠1)
1− n
⎛
W = p 1
V 1
ln V ⎞
2
⎜ ⎟ 
(valid
for
 n =1.0)
⎝ V 1 ⎠
€
CONTROL
VOLUME
(CV)
ANALYSIS
€
Mass
Flow
Rate
Balance:

 ∑ m ˙ i
= ∑ ˙
Energy
Rate
Balance:
€
CARNOT
EFFICIENCY
€
⎛
η MAX
= W ⎞
OUT
⎜ ⎟
⎝ ⎠
Q H
MAX
dE CV
dt
=1− T C
T H
i
e
m e
p
€
where
 m ˙ = ρAV
= Q ˙
CV
− ˙ ⎛
W € CV
+ m ˙ i
h i
+ V 2
i
2 + gz ⎞ ⎛
∑ ⎜
i⎟
− m ˙ e
h e
+ V 2
e
⎝
⎠
2 + gz ⎞
∑ ⎜
e⎟
⎝
⎠
i
e
€

M.P.
Bradley
11
April
2012
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€
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ENTROPY
⎛
Clausius
Formula
for
Entropy
Changes:
 dS = δQ ⎞
⎜ ⎟
⎝ T ⎠
⎛ 2δQ⎞
S 2
= S 1
+ ⎜ ∫ ⎟
⎝
1
T ⎠
internally reversible €
Boltzmann
Formula
for
Absolute
Entropy:
 S = k B
lnΩ
KINETIC
THEORY
OF
GASES
Maxwell
Boltzmann
Distribution
of
Molecular
Speeds
€
3 /
⎛ m ⎞
g(v) = ⎜ ⎟
24πv 2 e − mv 2
2k B T
⎝ 2πk B
T ⎠
Characteristic
Atomic/Molecular
Speeds
in
a
Gas
Most
Probable
Atomic/Molecular
Speed
v MP
= 2 k T B
m ≈1.41 k BT /m
Mean
Atomic/Molecular
Speed
v MP
=
8 k B
T
π m ≈1.60 k BT /m
RMS
Speed
€
Maxwell
Boltzmann
Energy
Distribution
1
( k B T ) 3 / 2 Ee −E / k B T
g E
(E) = 2 π
Atomic/Molecular
Collision
Theory
Mean
Free
Path
in
a
Gas:
λ =
1
2σn =
k B T
2σP
FREE
ENERGY
FUNCTIONS
€
€
v MP
=
3 k B T
m ≈1.73 k BT /m
Gibbs:

 G = H − TS
Helmholtz:

 Ψ = U − TS
€
Equilibrium
condition
at
constant
Temperature
&
pressure:

 dG] T ,p
= 0
(i.e.
at
equilibrium
the
Gibbs
Free
Energy
Function
has
its
minimum
value
)
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M.P.
Bradley
11
April
2012
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THERMODYNAMIC
RELATIONS
TdS
Equations:
TdS = dU + pdV
TdS = dH −Vdp
Maxwell
Relations:
⎛ ∂T ⎞ ⎛
⎜ ⎟ = − ∂p ⎞
⎛ ∂p ⎞ ⎛
⎜ ⎟ 
























 ⎜ ⎟ = ∂s ⎞
⎜ ⎟
⎝ ∂v ⎠
s
⎝ ∂s ⎠
v
⎝ ∂T ⎠
v
⎝ ∂v ⎠
T
⎛ ∂T ⎞ ⎛
⎜ ⎟ = ∂v ⎞
⎛ ∂v ⎞ ⎛
⎜ ⎟ 

























 ⎜ ⎟ = − ∂s ⎞
⎜ ⎟
⎝ ∂p ⎠ ⎝ ∂s⎠
s
p € ⎝ ∂T ⎠
p ⎝ ∂p⎠
T
USEFUL
CONSTANTS:
€
Boltzmann’s
Constant:
kB
=
1.38
x
10 ­23 
J/K
Avogadro’s
Number:
NA
=
6.022
x
10 23 
particles/mole
Molar
Gas
Constant:
 R = 8.314
J/mol.K
Stefan‐Boltzmann
Constant:

σ
=
5.67
×
10 ‐8 
W/m 2 •K 4
€