Chapter 1
Introduction to uid ow
1.1
Introduction
Compressible ows play a crucial role in a vast variety of natural phenomena and manmade systems. The life-cycles of stars, the creation of atmospheres, the sounds we hear,
the vehicles we ride and the syst

The entropy of mixing
where M w a, b refers to the molecular weights of the two distinctly different
gases. The entropy of the whole system is
S = ma sa + mb sb
(2.70)
Dene the mass fractions
ma
> a = -ma + mb
mb
> b = -ma + mb
(2.71)
The overall entropy

The entropy of mixing
2.8.2
ENTROPY CHANGE DUE TO MIXING OF DISTINCT GASES
The second law states that the entropy change of an isolated system undergoing
a change in state must be greater than or equal to zero. Generally, non-equilibrium
processes involve

Isentropic expansion
Pi > Pa
Pi
Pa
Ta
Ti
Pa
Tf
final state
initial state
As long as the gas is not near the wall where viscosity might play a role, the expansion of the gas parcel is nearly isentropic. The nal temperature is
81
-P a' 8
&
Tf
- = 9 - :
Pi %

Isentropic expansion
2.9 ISENTROPIC EXPANSION
2.9.1
BLOWDOWN OF A PRESSURE VESSEL
Consider the blowdown through a small hole of a calorically perfect gas from a
large adiabatic pressure vessel at initial pressure P i and temperature T i to the surrounding

Enthalpy
ds 1 # s # t
dh 1 # h # t
(2.29)
dP 1 # P # t
Obviously, we could do this with any spatial derivative as well. For example
1 # h v ( h, P ) # P
#s
- = - - - -T ( h, P ) # x T ( h, P ) # x
#x
1 # h v ( h, P ) # P
#s
- = - - - -T ( h, P ) # y T ( h

The entropy of mixing
S = kLogW
(2.78)
that is his most famous discovery.
Boltzmann showed that the entropy is equal to a fundamental constant k times the
logarithm of W which is equal to the number of possible states of the thermodynamic system with ener

Isentropic expansion
2.9.2
WORK DONE BY AN EXPANDING GAS
The gun tunnel is a system for studying the ow over a projectile at high speed in
rareed conditions typical of very high altitude ight. High pressure gas is used
to accelerate the projectile down a

Isentropic expansion
2.9.3
EXAMPLE - HELIUM GAS GUN
Suppose the tunnel is designed to use Helium as the working gas. The gas is introduced into the gun barrel and an electric arc discharge is used to heat the Helium
to very high pressure and temperature.

Isentropic expansion
where m gas is the mass of the gas expanding behind the projectile. The work integral (2.84) becomes
L
2
L1 8
2
2
1
& @ d -'
& -' dL .
- mU
-=
P
$ 4 % L 1$ L %
2
2
1
)
(2.86)
Carry out the integration
8
2 P L
2
1
1 1
& @ d -' - ( L 1

Some results from statistical mechanics
One of the most difcult challenges in the development of new power and propulsion systems is the accurate prediction of entropy changes in the system. Literally
billions of dollars are spent by manufacturers in the

The entropy of mixing
The initially separated volumes were each in a state of local thermodynamic equilibrium. When the partition is removed the gases mix and until the mixing is
complete the system is out of equilibrium. As expected the entropy increases

The entropy of mixing
2.8.1
SAMPLE PROBLEM - THERMAL MIXING
Equal volumes of an ideal gas are separated by an insulating partition inside an
adiabatic container. The gases are at the same pressure but two different temperatures. Assume there are no body f

The entropy of mixing
& m b'
T a + 9 -: T b
m a%
ma T a + mb T b
2T a T b
$
T final = - = - = - = 400K
( ma + mb )
Ta + Tb
m b'
&
9 1 + -:
m a%
$
(2.63)
2) What is the change in entropy per unit mass of the system? Express your result
in dimensionless for

Enthalpy
#s
-#h
P = cons tan t
1
= -T ( h, P )
#s
-#P
h = cons tan t
v ( h, P )
= -T ( h, P )
(2.23)
and for any homogeneous material we can write
2
#
# v ( h, P )
1
# s
- = - & -' = - & -'
# P $ T ( h, P )%
# h $ T ( h, P )%
#h#P
(2.24)
It is relatively

Heat capacities
Gibbs equation is the key to understanding the thermodynamic behavior of compressible uid ow. Its usefulness arises from the fact that the equation is
expressed in terms of perfect differentials and therefore correctly describes the
evolut

Ideal gases
An innitesimal amount of heat is added to the system causing an innitesimal
rise in temperature. There is an innitesimal change in volume while the piston is
withdrawn keeping the pressure constant. In this case the system does work on
the out

The Carnot Cycle
Q2
1
2
T
K
3
0
Q1
s
J
kg-K
Figure 2.4 T-S diagram of the Carnot cycle
A concrete example in the P-V plane and T-S plane is shown in Figure 2.3 and
Figure 2.4 above. The working uid is nitrogen cycling between the temperatures
of 300 and 5

Ideal gases
de = C v ( T )dT ;
dh = C p ( T )dT .
(2.44)
In this course we will deal entirely with ideal gases and so there is no need to distinguish between the standard enthalpy and enthalpy and so there is no need to
use the distinguishing character .

Ideal gases
nR u T
P = -V
(2.38)
where n is the number of moles of gas in the system with volume, V . The universal gas constant is
R u = 8314.472 Joules ( kmole K )
(2.39)
It is actually more convenient for our purposes to use the gas law expressed in
te

Constant specific heat
The van der Waals equation provides a useful approximation for gases near the
critical point where the dilute gas approximation loses validity.
2.7 CONSTANT SPECIFIC HEAT
The heat capacities of monatomic gases are constant over a wi

Constant specific heat
( 8 1 )
& s 2 s 1'
& T 2' & " 2'
.
exp 9 - : = 9 - : 9 -:
Cv %
T 1% $ " 1%
$
$
(2.52)
We can express Gibbs equation in terms of the enthalpy instead of internal energy.
dp
dT
1
dh
ds = - & - ' dp = C p - R - .
p
T
T $ "T%
(2.53)
Int

The Carnot Cycle
this point. Consider the piston cylinder combination containing a xed mass of a
working uid shown below and the sequence of piston strokes representing the
four basic states in the Carnot cycle.
T1
T2
T1
T1
T2
0
1
adiabatic compression
3

The entropy of mixing
The relations in (2.56) are sometimes called the isentropic chain.
Note that when we expressed the internal energy and enthalpy for a calorically
perfect gas in (2.50) we were careful to express only changes over a certain temperatur

Isentropic expansion
& L 1'
T 2 = T 1 9 -:
L 2%
$
81
0.1
= 2000 & - '
$ 2%
23
= 271 K .
(2.96)
The corresponding speed of sound is
a1 =
8RT 2 = 968 M/Sec .
(2.97)
The main assumption used to solve this problem is embodied in the use of (2.85)
to determine

The third law of thermodynamics
8gz
-2
a0
altitude
km
" " 0 or
T T0
Figure 2.14 Density and temperature of the standard atmosphere.
How accurate are these models? A comparison is shown in Figure 2.14. Note that
the two models tend to bracket the actual be

Chapter 6
Several forms of the equations of
motion
6.1
The Navier-Stokes equations
Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a
linear, thermally conductive medium, the equations of motion for compressible ow becom

Chapter 7
Entropy generation and
transport
7.1
Convective form of the Gibbs equation
In this chapter we will address two questions.
1) How is Gibbs equation related to the energy conservation equation?
2) How is the entropy of a uid aected by its motion?

Chapter 5
The conservation equations
5.1
5.1.1
Leibniz rule for dierentiation of integrals
Dierentiation under the integral sign
According to the fundamental theorem of calculus if f is a smooth function and the integral
of f is
I (x) =
Z
x
f x0 dx0
(5.1)

Chapter 10
Gasdynamics of nozzle ow
A nozzle is an extremely e cient device for converting thermal energy to kinetic energy.
Nozzles come up in a vast range of applications. Obvious ones are the thrust nozzles
of rocket and jet engines. Converging-divergi