1) This document discusses isentropic flow, including governing equations, stagnation relations, effects of area variation, nozzles, diffusers, and the effect of back pressure. 2) Key concepts covered are stagnation temperature, pressure and properties, how Mach number relates stagnation and static quantities, and how pressure and area change with Mach number in converging and diverging ducts. 3) Examples provided include calculating stagnation properties from flow conditions and sketching the steady flow adiabatic ellipse.

1. Chapter 4
Isentropic Flows
Course Title: Compressible Flow and Propulsion System
Course Code: (ME-417)

2. Governing Equation
Continuity Equation
𝒎 = 𝝆𝟏𝑨𝟏𝑽𝟏 = 𝝆𝟐𝑨𝟐𝑽𝟐
Momentum Equation
𝑭𝒑 = 𝒑𝟏𝑨𝟏 − 𝒑𝟐𝑨𝟐 = 𝝆𝟐𝑨𝟐𝑽𝟐
𝟐
− 𝝆𝟏𝑨𝟏𝑽𝟏
𝟐
Energy Equation
𝒉𝟏 +
𝑽𝟏
𝟐
𝟐
= 𝒉𝟐 +
𝑽𝟐
𝟐
𝟐
Second Law of thermodynamics
𝒔𝟏 = 𝒔𝟐

3. • The internal energy and the flow energy of a fluid are frequently combined into a
single term, enthalpy.
• Whenever the kinetic and potential energies of the fluid are negligible, as is often
the case, the enthalpy represents the total energy of a fluid.
• For high-speed flows, such as those encountered in jet engines, the potential
energy of the fluid is still negligible, but the kinetic energy is not.
• In such cases, it is convenient to combine the enthalpy and the kinetic energy of
the fluid into a single term called stagnation (or total) enthalpy.
• It is defined per unit mass as
Stagnation Relations
𝒉𝒕 = 𝒉 +
𝑽𝟐
𝟐
(1)

4. • When the potential energy of the fluid is negligible, the stagnation enthalpy
represents the total energy of a flowing fluid stream per unit mass.
• Thus the stagnation enthalpy indicates the enthalpy of a fluid when it is brought to
rest adiabatically.
• The properties of a fluid at the stagnation state are called stagnation properties.
• During stagnation process, since the kinetic energy of a fluid is converted to
enthalpy (internal energy or flow energy), the fluid temperature and pressure is
increased.
Stagnation Relations

5. Stagnation Relations
• Stagnation (or total) temperature is the temperature the gas attains when it is brought
to rest adiabatically.
• The term
𝑽𝟐
𝟐𝑪𝒑
corresponds to the temperature rise during such a process and is called the
dynamic temperature.
• For high-speed flows, the stagnation temperature is higher than the static (or ordinary)
temperature.
• For example, the dynamic temperature of air flowing at 100 m/s is about 5 K.
• Therefore, when air at 300 K and 100 m/s is brought to rest adiabatically (at the tip of a
temperature probe, for example), its temperature rises to the stagnation value of 305 K.
• The pressure a fluid attains when brought to rest isentropically is called the stagnation
pressure.
𝑻𝒕 = 𝑻 +
𝑽𝟐
𝟐𝑪𝒑
(2)

6. Stagnation Relations

7. • The stagnation and static properties can be related in terms of Mach numbers.
Stagnation Relations
𝑽𝟐 = 𝑴𝟐𝒂𝟐
𝒂𝟐
= 𝜸𝑹𝑻
The velocity of sound:
𝒉𝒕 = 𝒉 +
𝑴𝟐𝜸𝑹𝑻
𝟐
Now equation 1 can be written as:
But
𝑪𝒑 =
𝜸𝑹
𝜸 − 𝟏
𝒉𝒕 = 𝒉 +
𝑴𝟐(𝜸 − 𝟏)𝑪𝒑𝑻
𝟐
𝒉𝒕 = 𝒉 𝟏 +
𝑴𝟐 𝜸 − 𝟏
𝟐
𝑻𝒕 = 𝑻 𝟏 +
𝑴𝟐 𝜸 − 𝟏
𝟐

8. • The stagnation process is considered isentropic and following relations can be
used for pressure.
• The stagnation quantities Tt, pt etc. can be calculated from the actual conditions of
M, V, T, p, and ρ at a given point in a general flow field.
• The actual flow field itself may not have to be adiabatic or isentropic from one
point to the next.
• The isentropic process is only for definition of total conditions at a point.
Stagnation Relations
𝒑𝒕
𝒑
=
𝑻𝒕
𝑻
𝜸 (𝜸−𝟏)
= 𝟏 +
𝑴𝟐 𝜸 − 𝟏
𝟐
𝜸 (𝜸−𝟏)
𝝆𝒕
𝝆
= 𝟏 +
𝑴𝟐 𝜸 − 𝟏
𝟐
𝟏 (𝜸−𝟏)

9. Maximum Speed
• A gas attains its maximum speed when it is hypothetically expanded to zero
pressure.
• The static pressure at this state is also zero.
𝑻𝒕 = 𝑻 +
𝑽𝟐
𝟐𝑪𝒑
0
𝑽𝒎𝒂𝒙 = 𝟐𝑪𝒑𝑻𝒐
𝑽𝒎𝒂𝒙 = 𝟐
𝜸
𝜸 − 𝟏
𝑹𝑻𝒐 (3)

10. Critical Speed of Sound
• This is the speed of sound at the sonic state of a perfect gas, where M=1 and can
be given as
• From equation 2
• Eliminating T*
• The relation between these three reference speeds can be obtained via equations
𝑉∗
= 𝑎∗
= 𝛾𝑅𝑇
𝑉∗
=
2𝛾
𝛾 − 1
𝑅(𝑇𝑜 − 𝑇∗)
𝑉∗ =
2𝛾
𝛾 + 1
𝑅𝑇𝑜

11. Critical Speed of Sound
𝑎∗
𝑎𝑜
=
2
𝛾 + 1
𝑉
𝑚𝑎𝑥
𝑎𝑜
=
2
𝛾 − 1
𝑉
𝑚𝑎𝑥
𝑎∗
=
𝛾 + 1
𝛾 − 1
(4)
(5)
(6)

12. Critical Speed of Sound
• From eq. 2
• Using eq. 4, 5 and 6
𝑻𝒐 = 𝑻 +
𝑽𝟐
𝟐𝑪𝒑
(2)
2𝐶𝑝𝑇𝑜 = 2𝐶𝑝𝑇 + 𝑉2
2
𝛾
𝛾 − 1
𝑅𝑇𝑜 = 2
𝛾
𝛾 − 1
𝑅𝑇 + 𝑉2
𝑉
𝑚𝑎𝑥
2 = 2
𝑎2
𝛾 − 1
+ 𝑉2
𝑉
𝑚𝑎𝑥
2 =
2
𝛾−1
𝑎𝑜
2
(7)
• Dividing eq 7 by 𝑉
𝑚𝑎𝑥
2
1 =
2
𝛾 − 1
𝑎2
𝑉
𝑚𝑎𝑥
2 +
𝑉2
𝑉
𝑚𝑎𝑥
2
• Substituting eq. 8
(8)
𝑉2
𝑉
𝑚𝑎𝑥
2 +
𝑎2
𝑎𝑜
2 = 1
From eq. 5
𝑉2 =
2
𝛾 − 1
𝑎2 = 𝑉
𝑚𝑎𝑥
2 =
2
𝛾 − 1
𝑎𝑜
2 =
𝛾 + 1
𝛾 − 1
𝑎∗2
Equation A
Equation A is the kinematic form of the
steady, adiabatic energy equation
Equation B

13. Critical Speed of Sound
• Equation B is the equation of ellipse and
is known as the steady flow adiabatic
ellipse.
• All the points on the ellipse have the
same energy.
• Each point differs from others owing to
the relative proportions of the thermal
and kinetic energy and thus corresponds
to different Mach Numbers.
• Mach Number can be obtained by
differentiating eq. B
𝑴 = −
𝟐
𝜸 − 𝟏
𝒅𝒂
𝒅𝑽

14. • Thus the change in slope from point to point indicates how the changes in the
Mach Number are related to the changes in speed of sound and velocity.
• Therefore, it is direct comparison of the relative magnitudes of thermal and kinetic
energies.
• As observed from figure, for low Mach Number flows. The changes in the Mach
number are mainly due to the changes in velocity.
• However, at high Mach numbers flows, they are due to the changes in the speed of
sound and the compressibility effects ate dominant.
• One more important point to note in the figure is the case corresponding to 𝑀 ≤ 3,
where the changes in the speed of sound are negligibly small. These flows are
considered to be incompressible.
Critical Speed of Sound

15. At a point in a flow passage, the velocity and temperature of air are 802 m/s and
400 K respectively,
a. Calculate the stagnation speed of sound.
b. Calculate the critical speed of sound.
c. Calculate the Mach number and the Mach number referred to critical conditions
at the given state.
d. If the air is accelerated in a suitable flow passage, find the maximum possible
velocity that can be attained. Determined the Mach number referred to critical
conditions corresponding to the state of maximum velocity.
e. Obtain the kinematic form of the adiabatic energy equation and sketch the
steady flow adiabatic ellipse.

16. Effects of Area Variation on Flow Properties in
Isentropic Flow
• Consider a one dimensional Compressible flow in an infinitesimal duct with
variable area as shown in figure.
• The infinitesimal variation of the cross-sectional area of the duct causes
infinitesimal changes in the flow properties.
• The changes may be related by using basic equations

17. Effects of Area Variation on Flow Properties in
Isentropic Flow
𝑚 = 𝜌𝐴𝑉 = (𝜌 + 𝑑𝜌)(𝐴 + 𝑑𝐴)(𝑉 + 𝑑𝑉)
By simplifying
𝑑𝑉
𝑉
+
𝑑𝐴
𝐴
+
𝑑𝜌
𝑑
= 0
Continuity Equation
Momentum Equation
𝑝𝐴 − 𝑝 + 𝑑𝑃 𝐴 + 𝑑𝑃 + 𝑑𝐹𝑝 = 𝑚 𝑉 + 𝑑𝑉 − 𝑚𝑉
• Where 𝑑𝐹𝑝 is the infinitesimal pressure forces acting in the x-direction on the side walls of the
control volume.
• On the side walls, the average pressure is 𝑝 +
𝑑𝑝
2
and the cross-sectional area of the side walls
perpendicular the flow direction is dA
𝑝𝐴 − 𝑝 + 𝑑𝑃 𝐴 + 𝑑𝑃 + (𝑝 +
1
2
𝑑𝑝) = 𝜌𝐴𝑉 𝑉 + 𝑑𝑉 − 𝜌𝐴𝑉2
(1)

18. Effects of Area Variation on Flow Properties in
Isentropic Flow
By simplifying
𝑑𝑝 = −𝜌𝑉𝑑𝑉
𝑑𝑝
𝜌
+ 𝑑
𝑉2
2
= 0
From equation 2
𝑑𝐴
𝐴
−
𝑑𝑝
𝜌𝑉2
+
𝑑𝜌
𝜌
= 0
(2)
−
𝑑𝑝
𝜌𝑉
= 𝑑𝑉
Put in equation 1
𝑑𝐴
𝐴
=
𝑑𝑝
𝜌𝑉2
−
𝑑𝜌
𝜌
𝑑𝐴
𝐴
=
𝑑𝑝
𝜌𝑉2
1 −
𝑑𝜌
𝑑𝑝
𝑉2
𝑑𝐴
𝐴
=
𝑑𝑝
𝜌𝑉2
1 −
𝑉2
𝑎2
𝒅𝑨
𝑨
=
𝒅𝒑
𝝆𝑽𝟐
𝟏 − 𝑴𝟐
𝒅𝑨
𝑨
= −
𝒅𝑽
𝑽
𝟏 − 𝑴𝟐
(A)
(B)

19. Effects of Area Variation on Flow Properties in
Isentropic Flow

20. • Equation describes the variation of pressure with flow area. For subsonic flow (M
< 1), dA and dp must have the same sign.
• That is, the pressure of the fluid must increase as the flow area of the duct
increases and must decrease as the flow area of the duct decreases.
• Thus, at subsonic velocities, the pressure decreases in converging ducts (subsonic
nozzles) and increases in diverging duct (subsonic diffusers).
• In supersonic flow (Ma >1), dA and dp have opposite signs.
• Two common devices involving area change are nozzle and diffuser.
• The same piece of equipment can operate as either a nozzle or a diffuser,
depending on the flow regime.
• Thus a device is called a nozzle or a diffuser because of what it does, not what it
looks like.
Effects of Area Variation on Flow Properties in
Isentropic Flow

21. • A nozzle is a device that converts
enthalpy (or pressure energy for the
case of an incompressible fluid) into
kinetic energy.
• From Figure we see that an increase
in velocity is accompanied by either
an increase or decrease in area,
depending on the Mach number.
One dimensional isentropic flow

22. • A diffuser is a device that converts kinetic energy into enthalpy (or pressure
energy for the case of incompressible fluids).
• The highest velocity we can achieve by a converging nozzle is the sonic velocity,
which occurs at the exit of the nozzle.
• To accelerate a fluid, we must use a converging nozzle at subsonic velocities and a
diverging nozzle at supersonic velocities.
• Based on Equation B, which is an expression of the conservation of mass and
energy principles, a diverging section must be added to a converging nozzle to
accelerate a fluid to supersonic velocities.
• The result is a converging– diverging nozzle.
• The fluid continues to accelerate as it passes through a supersonic (diverging)
section. A large decrease in density makes acceleration in the diverging section
possible.
One dimensional isentropic flow

23. • If the mass flow rate is fixed, there is a minimum cross-sectional area required to
pass this flow and this phenomenon is known as chocking.
• For a given reduction in area, there is a maximum initial Mach number which can
be maintained in a subsonic flow, while there is a minimum initial Mach number
which can be maintained steadily in a supersonic flow.
One dimensional isentropic flow
Throat

24. Effect of Back Pressure
One dimensional isentropic flow
• Consider the subsonic flow through a converging
nozzle as shown in Figure.
• The nozzle inlet is attached to a reservoir at pressure
pr and temperature Tr.
• The reservoir is sufficiently large so that the nozzle
inlet velocity is negligible.
• The fluid velocity in the reservoir is zero and the
flow through the nozzle is approximated as
isentropic,
• The stagnation pressure and stagnation temperature
of the fluid at any cross section through the nozzle
are equal to the reservoir pressure and temperature,
respectively.

25. One dimensional isentropic flow
• Now we begin to reduce the back pressure and observe the resulting effects on
the pressure distribution along the length of the nozzle, as shown.
• If the back pressure pb is equal to pt, which is equal to pr, there is no flow and the
pressure distribution is uniform along the nozzle.
• When the back pressure is reduced to p2, the exit plane pressure pe also drops to
p2. This causes the pressure along the nozzle to decrease in the flow direction.
• Now suppose the back pressure is reduced to p3 (= p*), which is the pressure
required to increase the fluid velocity to the speed of sound at the exit plane or
throat).
• The mass flow reaches a maximum value and the flow is said to be choked.
• Further reduction of the back pressure to level p4 or below does not result in
additional changes in the pressure distribution, or anything else along the nozzle
length.

26. • The effect of back pressure on the nozzle exit pressure pe is:
One dimensional isentropic flow

27. A converging nozzle with an exit cross sectional area of 0.001 m2 is
operated with air at a back pressure of 69.5 kPa, as shown. The nozzle is
fed from a large reservoir where the stagnation pressure and temperature
are 100 kPa and 60o C respectively. Determine the Mach number and
temperature at the nozzle exit. Also, find the mass flow rate through the
nozzle. Assume one dimensional steady isentropic flows.

29. • Now repeat the previous assumptions for
converging-diverging nozzle and study
the effects of back pressure.
• Consider the valve at the exit of the
nozzle is closed, then the pressure is
constant throughout the nozzle. Such that
pe=pb=pt=po.
• As represented by horizontal line
One dimensional isentropic flow
Flow in converging diverging nozzle
No Flow conditions

30. One dimensional isentropic flow
No Flow conditions
• When the back pressure in the discharge
reservoir is slightly decreased by opening
the valve then the flow is subsonic
throughout the converging-diverging
nozzle. As shown as (i) in fig.
• The static pressure of the fluid decreases
from the entrance to the throat, where it
reaches its minimum value and then
increases in the diverging section.
• Therefore, the converging part of the
nozzle acts as a sub sonic nozzle while the
diverging part acts as a subsonic diffuser
and the passage behaves like a
conventional venturi-tube.

31. One dimensional isentropic flow
Chocking conditions
• The flow is chocked at the throat where
the Mach number is unity. As shown by
(ii) in figure.
• The throat pressure is equal to the critical
pressure at the exit and is equal to the
back pressure.
• The flow is subsonic at every point except
at the throat so that the diverging part
again acts as a subsonic diffuser.

32. One dimensional isentropic flow
Non Isentropic Flow Regime
• Flow pattern as indicated by (iii) is a
typical representation of this flow regime,
which is not isentropic.
• A non-isentropic phenomenon, known as
shock transforms the supersonic flow in
the diverging section into a subsonic one.
• This will be covered in chapter 5.

33. One dimensional isentropic flow
Exit Plane shock conditions
• In this case, the shock phenomenon
moves to the exit plane of the converging
diverging nozzle as represented by (iv) in
figure.

34. One dimensional isentropic flow
Overexpansion Flow regime
• A typical representation of this flow is given
pattern (v).
• The flow is sonic at the throat where the
Mach number is unity, and supersonic in the
entire diverging section of the nozzle.
• The mass flow rate is invariant with respect
to the back pressure since the flow is choked
at the throat.
• The gas is overexpanded at the exit plane,
because the pressure at the exit plane is lower
than the back pressure.
• The compression, which occurs outside the
nozzle, involves non-isentropic oblique
compression waves which cannot be treated
with one-dimensional, while the flow
throughout the nozzle is isentropic.

35. One dimensional isentropic flow
Design conditions
• Flow pattern (vi) in figure represents the
condition for which the converging-
diverging nozzle is actually designed.
• The flow is entirely isentropic within and
outside the nozzle such that the exit plane
pressure is identical with back pressure.
• Owing to the chocking at the throat, the
flow is supersonic in the entire diverging
section of the nozzle.
• As long as pe=pb, the shape of the jet
leaving the nozzle is cylindrical.

36. A converging-diverging nozzle with a throat area of 0.0035 m2 is
attached to a very large tank of air in which the pressure is 125
kPa and the temperature is 47o C as shown in figure. The nozzle
exhausts to the atmosphere with a pressure of 100 kPa. If the
mass flow rate is 0.9 kg/s, determine the exit area and the Mach
number at the throat. The flow in the nozzle is isentropic.

38. At a point upstream of the throat in a converging-diverging
nozzle, the velocity, temperature and pressure are 172 m/s, 22o C
and 200 kPa, respectively as shown in figure. If the nozzle,
operating at its design condition has an exit area of 0.01 m2 and
discharges to the atmosphere with a pressure of 100 kPa,
determine the mass flow rate and the nozzle throat area.