May 8, 2009
Thermal & Fluids 2 Summary Sheet
Carl Hansen
Types of Energy :
Static Energies:
Formula
Dynamic Energies
Formula
Kinetic Energy
KE = ½m*
v
2
Work
w = ∫ F∙dx
w = ∫ P*d
V
Gravitational Potential Energy
GPE = m*g*h
Heat Transfer
Q = m*C*ΔT
Strain Energy
SPE = ½k*x
2
Electrical power
dw/dt = i*V
Internal Thermal Energy
ΔU = m*C*ΔT
Chemical & Nuclear Energy
1
st
law of Thermodynamics: ΔE = Q – w
“increase in system energy = heat added – work done by system.”
ΔE = Δ(U + PE + KE)
Model: heat engine system
Gases:
Ideal Gas law: P*V = m*
K
M
R
T
*
°
;
where M is molar mass,
R
= 8.31434 kJ/(kmol * °K) = 0.08205
K
*
mol
atm
*
L
°
Expansion Work: w = ∫ P*d
V
M
air
= 28.95 kJ/(kmol*°K)
Special Cases:
Constant Volume (d
V
/dt = 0): w = 0
Constant Pressure (dP/dt = 0): w = P(
V
1
–
V
2
)
Isothermal (dT/dt = 0, or P*
V
=const):
w = P
1
*
V
1
*
(
29
1
2
V
V
ln
Polytrophic (P*
V
n
=const):
Adiabatic(no heat transfer):
n
1
V
*
P
V
*
P
w
1
1
2
2


=
n = k = C
p
/C
v
k
air
= 1.4
More for Adiabatic, QuasiEqlbrm, Ideal gas, constant specific heat
(
29
k
V
V
P
P
2
1
1
2
=
(
29
1
k
V
V
T
T
2
1
1
2

=
(
29
k
1
k
P
P
T
T
1
2
1
2

=
Specific Heats of ideal gases:
c
p
: if it has constant pressure
c
v
: if it has constant volume
Relation: c
p
= c
v
+
M
R
Fluid Dynamics:
(a)
Conservation of mass:
volumetric flow rate definition:
A
*
v
V
avg
a
=
2
2
2
1
1
1
A
*
v
*
A
*
v
*
a
a
ρ
=
ρ
(b) Conservation of energy:
Bernoulli Equation:
2
2
2
2
1
2
1
1
z
*
g
2
v
P
z
*
g
2
v
P
+
+
ρ
=
+
+
ρ
(isothermal, incompressible, frictionless, zero work, adiabatic)
For general open energy system:
∑
∑
+
+

+
+
+

=
∆
out
o
2
o
o
o
in
i
2
i
i
i
out
,
cv
in
,
cv
cv
gz
2
v
h
m
gz
2
v
h
m
W
Q
E
(c)
Conservation of momentum:
∑
∑
∑

+
=
in
in
in
out
exit
exit
cv
v
*
m
v
*
m
dt
B
d
F
a
Two Phase Properties:
Quality of a gas/liquid mixture:
L
g
L
g
L
g
V
V
V
V
m
m
m
x


+
=
=
Specific volume: v = v
L
+ x *(v
g
– v
L
) =
v
L
+ x*v
gL
Specific Internal Energy: u = u
L
+ x*(u
g
– u
L
) = uL + x*u
gL
Specific Enthlapy: h = h
L
+ (h
g
– h
L
)*x = h
L
+ x*h
gL
Subscripts labeled “
gL
” = latent heat of vaporization.
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 Spring '08
 Kaminski
 Thermodynamics, Energy, Heat, Heat Transfer, Summary sheet, Carl Hansen

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