Any further decrease in pB has no effect on the velocity at the throat since the pressure information can no
longer propagate upstream into the reservoir. The fluid velocity out of the tank is the same as the speed of
the pressure wave into the tank so th
Conservation of Energy (COE)
W
on CV
Q
into CV
dx
d
e dV
h 1 V 2 gz
u
W
Q
2
dt
W
dA
rel
into CV
shaft,
other, on CV
on CV
CV
CS
where
(steady flow)
d
e dV 0
dt
CV
h
1
V gz
2
u
dA m h
1
1
V2 h
1
V2
d h
V2
2
2
rel
2
2
CS
m d h
1
V2
Note t
2nd Law of Thermodynamics (aka Entropy Equation)
Q
into CV
m s
m sd m s
dx
d
s dV
s
u
rel
dA
into CV
dt
T
CV
CS
CV
where
d
s dV 0
dt
CV
s u rel dA m s s ds m ds
CS
q
q
into CV
into CV
T
T
CV
Substitute and simplify.
(where dq
q
into CV
is the heat adde
4.
Speed of Sound
The speed of sound, c, in a substance is the speed at which infinitesimal pressure disturbances propagate through
the surrounding substance. To understand how the speed of sound depends on the substance properties, lets
examine the follo
Governing Equations for One-Dimensional, Steady Flow
Lets write our governing equations (COM, LME, COE, 2nd Law) for a one-dimensional, steady flow.
Conservation of Mass (COM)
VA
VA+d(VA)
dx
d
u rel dA 0
dV
dt
where
CV
CS
(steady flow)
d
dV 0
dt
CV
u
The Imperfect Gas
For most real gases, the compressibility factor is not equal to one, i.e. Z = pv/(RT) 1. However, in most
engineering applications the pressure ratio, p/pcr, is small enough and the temperature ratio, T/Tcr, large enough so
that the idea
Notes:
1. Gravitational effects have been neglected in the previous analysis since when dealing with gases,
gravitational effects are typically very small compared to other terms in the momentum equation.
2.
The DArcy friction factor, fD, is the friction
Linear Momentum Equation (LME)
(p+1/2dp)dA
V2A+d(V2A)
Let P be the pipe perimeter
V2A
pA+d(pA)
pA
Pdx
dx x
Since the flow is 1D, consider only the x-component of the LME:
d
u dV
u
u rel dA Fx, body Fx, surface
on CV
dt
CV
CS
where
d
(steady flow)
u dV
3.
One-Dimensional Flow
flow dimensionality
The dimension of a flow is equal to the number of spatial coordinates required to describe the flow.
For example:
0D flow:
u=u(t) or u=constant
1D flow:
u=u(t; x) or u=u(x)
2D flow:
u=u(t; r,) or u=u(r,)
3D flow
Relations for an Incompressible Substance
An incompressible system is one in which the density (or specific volume) of the system remains
constant:
d dv 0
(12.34)
Since the state of a simple system can be determined using two properties, we can write: u=u
Relations for an Ideal Gas
One particularly important class of substances that is very important in gas dynamics is the ideal
gas. An ideal gas is a model describing the behavior of real gases in the limit of zero pressure and
infinite temperature (i.e.,
st
nd
1 and 2 Law Considerations for an Ideal Gas (Gibbs Equation)
Now lets consider the 1st and 2nd laws for an ideal gas. Recall that the 1st Law (COE) for a system is:
de q
(12.19)
w
into system
on system
From the 2nd law we have:
qinto system
ds
(12
Notes:
1.
Recall that that for an ideal gas:
where h(T = 0) = 0
(12.29)
(12.30)
T
dh c p dT h T
cp
dT
0
u h pv h RT
Values for h and u can typically be found in tables given at the back of most thermodynamics texts.
2.
For an ideal gas we also have:
dT
T
COM:
d
u rel dA 0
dV
dt
where
CV
CS
d
dV 0
dt
CV
u rel dA cA c V
A
CS
Substitute and simplify.
cA
c
V A
(12.52)
c c V c V
c
V
LME:
d
u dV
u
u rel dA Fx, body Fx, surface
dt
CV
CS
where
on CV
on CV
d
u dV 0
x
dt
CV
u x u rel dA m c m c V m
For a sound wave, the changes across the wave are infinitesimal so that:
c 2 lim
pdp
(12.54)
p 1 p
d
We also need to specify the process by which these changes occur. Since the changes across the wave are
infinitesimal, we can regard the wave as a r
Lets interpret equation (12.88). Consider the following the cases:
Ma < 1 (subsonic flow)
dA < 0 dV > 0
(decreases in area result in increases in velocity)
dA > 0 dV < 0
(increases in area result in decreases in velocity)
A subsonic nozzle should have a d
Since the mass flow rate must be a constant in 1D, steady flow, we can write:
m AV * A*V *
where the * quantities are the conditions where Ma=1. Lets re-arrange this equation and substitute the
isentropic flow relations we derived previously.
*
A
*
V
V
Choked Flow
Consider the flow of a compressible fluid from a large reservoir into the surroundings. Let the pressure of the
surroundings, called the back pressure, pB, be controllable:
throat conditions
stagnation
conditions
p0, T0, 0
A ,p ,V
th
th
th
bac
Mollier (aka h-s) Diagrams
Mollier diagrams are diagrams that plot the enthalpy (h) as a function of entropy (s) for a process. They are often
useful in visualizing trends.
Notes:
1.
Sketches of constant pressure and constant volume (or density) curves ar
6.
Effects of Area Change on Steady, 1D, Isentropic Flow
Mass conservation states that for a steady, 1D, incompressible flow, a decrease in the area will result in an increase
in velocity (and visa-versa). This is not necessarily true, however, for a comp
Stagnation and Sonic Conditions
It is convenient to choose some significant reference point in the flow where we can evaluate the constants in
equations (12.70)-(12.72). Two such reference points are commonly used in compressible fluid dynamics. These
are
Sonic Conditions
Another convenient reference point is where the flow has a Mach number of one (Ma=1). Conditions where
the Mach number is one are known as sonic conditions and are typically specified using the superscript *.
Equations (12.73)-(12.76) eva
The Mach Cone
Consider the propagation of infinitesimal pressure waves, i.e., sound waves, emanating from an object at rest.
The waves will travel at the speed of sound, c.
object
(V=0)
c(t)
location of sound pulse after
time t
c(3t)
c(2t)
Now consider an
Lastly, consider an object travelling at supersonic speeds, V>c:
The object out runs the pressure pulses it
generates.
V(3t)
The locus of wave fronts forms a cone
which is known as the Mach Cone. The
V(2t)
object cannot be heard outside the Mach
V(t)
Cone
5.
Adiabatic, 1D Compressible Flow of a Perfect Gas
Now lets consider the 1D, adiabatic flow of a compressible fluid. Recall that from the energy equation we have:
h 1 V constant
(12.68)
2
For a perfect gas we can re-write the specific enthalpy in terms o
6. The Mach number, Ma, is a dimensionless parameter that is commonly used when discussing
compressible flows. The Mach number is defined as:
V
Ma
(12.61)
c
where V is the flow velocity and c is the speed of sound in the flow.
Notes:
a.
Compressible flow
5.
Equation (12.55) can also be written in terms of the bulk modulus. The bulk modulus, E of a substance is a
measure of the compressibility of the substance. It is defined as the ratio of a differential applied pressure to the
resulting differential chan
du c dT
(12.10)
v
Integrating both sides and noting that the specific heat can be a function of temperature in general:
T
(12.11)
cv T dT
u uref
T
ref
The enthalpy is also only a function of temperature for an ideal gas as shown below:
p
hu
h h(T )
(12.1
caloric (aka energy) equation of state
Before discussing the caloric equation of state, we must first define the property known as specific heat.
specific heat
Recall that earlier it was mentioned that internal energy, sensible energy in particular, is
re
.(15.5)
By comparing eqns. (15.4) and (15.5), we find that I is analogus to, Q, dV is analogous to dt
and R is analogous to the quantity
dx
kA
. The quantity dx
kA
is called thermal conduction resistance
(Rth)cond. i.e.,
(Rth)cond. =
dx
kA
The reciproca