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
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 dif
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
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 init
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 o
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 pr
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
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)
Carr
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
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 thi
Some results from statistical mechanics
This is a good model of monatomic gases such as Helium, Neon,Argon, etc. Over
a very wide range of temperatures,
5
C p = - R
2
3
C v = - R
2
;
(2.104)
from near
Atmospheric models
a
2
#P
= & -'
$ #"%
(2.111)
s = cons tan t
For an ideal gas,
a
2
8P
= - = 8 RT
"
(2.112)
For a ow at velocity U the Mach number is
U
M = -8 RT
(2.113)
2.13 ATMOSPHERIC MODELS
In a s
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 equa
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 =
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
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
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
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 conveni
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 he
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 Ca
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 ca
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
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
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 i
Atmospheric models
1
-'8 1
&
gz
"
- = 9 1 ( 8 1 ) -:
2
"0
$
a %
(2.117)
0
In this model the atmospheric density decreases algebraically with height and
2
goes to zero (vacuum) when gz a 0 = 1 ( 8 1 )
Enthalpy - diatomic gases
The characteristic vibrational temperatures for common diatomic gases are
.v
O2
= 2238K ;
.v
N2
= 3354K
;
.v
H2
= 6297K .
(2.108)
The increasing values of . v with decreasing
Chapter 11
Area change, wall friction and heat
transfer
11.1
Control volume
Assume a calorically perfect gas, ( P = RT , Cp and Cv constant) is owing in a channel
in the absence of body forces. Viscou
Chapter 14
Thin airfoil theory
14.1
Compressible potential ow
14.1.1
The full potential equation
In compressible ow, both the lift and drag of a thin airfoil can be determined to a reasonable level of
Chapter 9
Quasi-one-dimensional ow
9.1
Control volume and integral conservation equations
In this chapter we will treat the very general stationary ow of a compressible uid in a
channel without body f