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Applied Thermodynamics

Module 4: Compressible Flow Introduction

Flows that involve significant changes in density is called compressible flows. They are frequently

encountered in devices that involve the flow of gases at very high velocities. Compressible flow combines

fluid dynamics and thermodynamics in that both are necessary to the development of the required

theoretical background. In this module, we develop the general relations associated with one-dimensional

compressible flows for an ideal gas with constant specific heats.

Stagnation Properties

Before going into details of stagnation properties, let’s first understand the way it comes.

In analysis of control volume, two terms are frequently encountered internal energy (𝑢) and flow energy

(𝑃𝑣). For convenience these two terms are combined and termed as enthalpy, ℎ = 𝑢 + 𝑃𝑣 (defined per unit

mass. 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, the potential energy of the fluid is still

negligible, but the kinetic energy is not. In the analysis of high-speed flows enthalpy and kinetic energy

appear frequently. For convenience these two terms are combined together and termed as stagnation (or

total) enthalpy ℎ0 , defined per unit mass as

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 it simplifies the thermodynamic analysis of high-speed flows.

Throughout this chapter the ordinary enthalpy h is referred to as the static enthalpy, whenever necessary,

to distinguish it from the stagnation enthalpy.

Stagnation process: A process in which flowing-

fluid is brought to rest adiabatically. The

properties of a fluid at the stagnation state are

called stagnation properties (stagnation

temperature, stagnation pressure, stagnation

density, etc.). The stagnation state and the

stagnation properties are indicated by the

subscript 0. stagnation enthalpy represents the enthalpy of a

fluid when it is brought to rest adiabatically. Consider the steady flow of a fluid through a duct where the flow takes place adiabatically and with no shaft or electrical work and no change in potential energy. Energy balance to the system yields If the fluid were brought to a complete stop, then the velocity at state 2 would be zero and above equation

becomes

Fig. (1) Kinetic energy is converted to

enthalpy during a stagnation process.

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or,

The stagnation state is called the isentropic stagnation state when the fluid is brought to rest reversibly and adiabatically (i.e., isentropically). The entropy of a fluid remains constant during an isentropic stagnation process. The actual (irreversible) and isentropic stagnation processes are shown on the h-s diagram in Fig. (2). Notice

that the stagnation enthalpy (and the stagnation temperature if the fluid is an ideal gas) of the fluid is the same for both cases. The stagnation processes are often approximated to be isentropic, and the isentropic stagnation properties are simply referred to as stagnation properties. stagnation (or total) temperature represents the

temperature an ideal gas attains when it is brought to

rest adiabatically.

When the fluid is approximated as an ideal gas with

constant specific heats, its enthalpy can be replaced by 𝑐𝑝𝑇 and above equation of stagnation enthalpy can be

written as Or, The term V2/2cp corresponds to the temperature rise during stagnation process and is called the dynamic

temperature. stagnation (or total) pressure represents the temperature an ideal gas attains when it is brought to rest

adiabatically. For ideal gases with constant specific heats, P0 is related to the static pressure of the fluid by From fig. (2) it may be noted that the actual stagnation pressure is lower than the isentropic stagnation pressure

because entropy increases during the actual stagnation process as a result of fluid friction.

By noting that 𝜌 = 1𝑣⁄ and using the isentropic relation 𝑃𝑣𝑘 = 𝑃0𝑣0

𝑘, the ratio of the stagnation density to

static density can be expressed as

Fig.(2) The actual state, actual stagnation

state, and isentropic stagnation state of a

fluid on an h-s diagram.

Speed of Sound and Mach Number

Consider a pipe that is filled with a fluid at rest, as shown in Fig.

(3). A piston fitted in the pipe is now moved to the right with a

constant incremental velocity dV, creating an infinitesimally small

pressure wave. The speed at which an infinitesimally small pressure

wave travels through the medium is known as speed of sound or

sonic speed. The wave front moves to the right through the fluid at

the speed of sound c and separates the moving fluid adjacent to the

piston from the fluid still at rest. The fluid to the left of the wave

front experiences an incremental change in its thermodynamic

properties, while the fluid on the right of the wave front maintains

its original thermodynamic properties, as shown in Fig. (3).

To simplify the analysis, consider a control volume that encloses

the wave front and moves with it, as shown in Fig. (4). To an

observer traveling with the wave front, the fluid to the right will

appear to be moving toward the wave front with a speed of c and

the fluid to the left to be moving away from the wave front with a

speed of c-dV.

Applying mass balance equation to the control volume or, Neglecting higher order terms, above equation reduces to

Now, applying energy balance equation to the control volume In absence of any heat and work interaction and change in potential energy above equation can be expanded to

Which yields where we have neglected the second-order term dV².

Since the variation in pressure and temperature are negligibly small and the change of state is so fast that the propagation of sonic wave is not only adiabatic but also very nearly isentropic. Then the second T ds relation reduces to

Or,

Fig. (3) Propagation of a small

pressure wave along a duct.

Fig. (4) Control volume

moving with the small pressure

wave along a duct.

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Combining Eqs. (a), (b), and (c) yields the desired expression for the speed of sound as

Or,

For an ideal gas, in an isentropic process

𝑃𝑣 𝑘 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑃

𝜌𝑘= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

where k is the specific heat ratio of gas. Taking logarithm of above equation yields,

𝑙𝑛𝑃 − 𝑘𝑙𝑛𝜌 = 𝑙𝑛𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 Now differentiating the above equation yields

𝑑𝑃

𝑃− 𝑘

𝑑𝜌

𝜌=0

Or, 𝑑𝑃

𝑑𝜌= 𝑘

𝑃

𝜌

For isentropic process 𝑑𝑃

𝑑𝜌= (

𝑑𝑃

𝑑𝜌)

𝑠

∴ ( 𝑑𝑃

𝑑𝜌)

𝑠

= 𝑘𝑃

𝜌

Since 𝑐2 = ( 𝑑𝑃

𝑑𝜌)

𝑠 and 𝑃 = 𝑘𝑅𝑇

𝑐2 = 𝑘𝑅𝑇

Where, 𝑅 = 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐 𝑔𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 =𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝑔𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑀𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑤𝑒𝑖𝑔ℎ𝑡

Note:

1. Lower molecular weight and higher value of k gives higher

sonic velocity at the same temperature as shown in Fig. (5)

2. For a specified ideal gas R is a constant. Ratio of specific heat

k, at most, could be a function of temperature. Hence the speed

of sound in a specified ideal gas is a function of temperature

alone (Fig. (5))

Mach Number: It is the ratio of the actual velocity of the fluid (or an

object in still air) to the speed of sound in the same fluid at the same

state.

Fig. (5) The speed of sound

changes with temperature

and varies with the fluid.

0 (for isentropic flow)

Note that the Mach number depends on the speed of sound, which

depends on the state of the fluid. Therefore, the Mach number of an

aircraft cruising at constant velocity in still air may be different at

different locations as shown in Fig. (6).

Fluid flow regime on the basis of Mach number, the flow is called

1. Sonic when Ma=1

2. Subsonic when Ma<1

3. Supersonic when Ma>1

4. Hypersonic when Ma>>1 (i.e. Ma≥5) and

5. Transonic when Ma≅1

Variation of Fluid Velocity with Flow Area

Consider the mass balance for a steady-flow process:

Differentiating and dividing the resultant equation by the mass flow rate, we obtain Now, energy balance equation for steady flow process, without heat and work interaction and without change

in potential energy, can be written as

Or, ℎ +𝑉2

2= 𝑐𝑛𝑠𝑡𝑎𝑛𝑡

Differentiating the above expression,

𝑑ℎ + 𝑉𝑑𝑉 = 0

Second Tds equation can be written as

𝑇𝑑𝑠 = 𝑑ℎ − 𝑣𝑑𝑃

𝑑ℎ = 𝑣𝑑𝑃 =1

𝜌𝑑𝑃

Here, it is worthwhile to mention that v denotes specific volume and V denotes velocity of flow. Putting

the value of dh in equation (9)

Combining equations (8) and (10) gives

Putting the value of 𝑑𝜌

𝑑𝑃=

1

𝑐2 in above equation, yields

Fig. (6) The Mach number can be

different at different temperatures

even if the velocity is the same.

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Fig. (7) Variation of flow properties in subsonic and supersonic nozzles and diffusers.

Above equation describes the variation of pressure with flow area. Two important points can be noted

from above equation.

1. It may be noted that A, 𝜌, and V are positive quantities. For subsonic flow (Ma<1), the term 1 −𝑀𝑎2 is positive; and thus dA and dP must have the same sign, i.e. P increases as A increases and P

decrease as A decreases. Thus, at subsonic velocities, the pressure decreases in converging ducts

(subsonic nozzles) and increases in diverging ducts (subsonic diffusers).

2. In supersonic flow (Ma>1), the term 1 − 𝑀𝑎2 is negative, and thus dA and dP must have opposite

signs, i.e. P decreases as A increases and P increases as A decreases. Thus, at supersonic velocities,

the pressure decreases in diverging ducts (supersonic nozzles) and increases in converging ducts

(supersonic diffusers).

Another important relation for the isentropic flow of a fluid is obtained by substituting 𝜌𝑉 = − 𝑑𝑃 𝑑𝑉⁄

from equation (10) into equation (11)

This equation governs the shape of a nozzle or a diffuser in subsonic or supersonic isentropic flow.

To accelerate a fluid, we must use a converging nozzle at subsonic velocities and a diverging nozzle at

supersonic velocities.

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Fig. (8) We cannot obtain supersonic

velocities by attaching a converging

section to a converging nozzle. Doing so

will only move the sonic cross section

farther downstream and decrease the mass

flow rate.

Fig. (9) The cross section of a nozzle at the

smallest flow area is called the throat.

converging–diverging nozzles are often

called Laval nozzles.

Note:

1. Maximum velocity that can be achieved by a converging nozzle is sonic velocity, which occurs at

the exit of the nozzle. If we extend the converging nozzle by further decreasing the flow area, in

hopes of accelerating the fluid to supersonic velocities, as shown in Fig. (8) then the sonic velocity

will occur at the exit of the converging extension, instead of the exit of the original nozzle, and the

mass flow rate through the nozzle will decrease because of the reduced exit area.

2. In order to accelerate a fluid from subsonic velocity to supersonic velocity we must add a diverging

section to a converging nozzle. The result is a converging–diverging nozzle. The fluid first passes

through a subsonic (converging) section, where the Mach number increases as the flow area of the

nozzle decreases, and then reaches the value of unity at the nozzle throat. The fluid continues to

accelerate as it passes through a supersonic (diverging) section. Noting that �̇� = 𝜌𝐴𝑉 for steady

flow, we see that the large decrease in density makes acceleration in the diverging section possible.

Property Relations for Isentropic Flow of Ideal Gases The temperature T of an ideal gas anywhere in the flow is related to the stagnation temperature𝑇0 through eq. (3)

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Or, Noting that 𝑐𝑝 = 𝑘𝑅/(𝑘 − 1), 𝑐2 = 𝑘𝑅𝑇, and 𝑀𝑎 = 𝑉/𝑐, we see that

Substituting yields,

Which is the desired relation between 𝑇0 and 𝑇. The ratio of the stagnation to static pressure is obtained by substituting eq. (13) into eq. (4)

The ratio of the stagnation to static density is obtained by substituting eq. (13) into eq. (5)

The properties of a fluid at a location where the Mach

number is unity (the throat) are called critical

properties. It is common practice in the analysis of

compressible flow to let the superscript asterisk (*)

represent the critical values. Setting Ma=1 in eqs. (13)

through (15) yields

The above ratios in eqs. (16) through (18) are called

critical ratios (Fig. (10)) and 𝑇∗, 𝑃∗ and 𝜌∗are called

critical temperature, critical pressure and critical density

respectively.

Fig. (10) When Mat = 1 the properties at the

nozzle throat become the critical properties.

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Isentropic Flow Through Nozzles

In this section, effects of back pressure on exit velocity, mass flow rate and pressure distribution along the

nozzle are considered. Back pressure, 𝑷𝒃 is the pressure applied at the exit region of the nozzle.

Converging Nozzles

Consider the subsonic flow through a converging

nozzle as shown in Fig. (11). The nozzle inlet is

attached to a reservoir at pressure 𝑃𝑟 and temperature

𝑇𝑟 . Assuming that reservoir is sufficiently large so

that the nozzle inlet velocity is negligible and flow

through nozzle is isentropic. With these assumptions,

stagnation pressure and stagnation temperature of the

fluid are equal to the reservoir pressure and

temperature, respectively.

Following points can be noted by varying the back

pressure, 𝑃𝑏 at exit region of nozzle;

1. When 𝑃𝑏 = 𝑃1 = 𝑃𝑟 then there is no flow and

the pressure distribution is uniform along the

nozzle.

2. When the back pressure is reduced to 𝑃2 , the

exit plane pressure 𝑃𝑒 also drops to 𝑃2 . This

causes the pressure along the nozzle in flow

direction to decrease and velocity to increase.

3. When the back pressure is reduced to 𝑃3

(=𝑃∗) then exit plane pressure 𝑃𝑒 also drops

to 𝑃3 (= 𝑃∗). This causes the pressure to

decrease and velocity to increase along the

flow direction and at exit of nozzle velocity

reaches to sonic velocity. As far as mass-flow

is concern it reaches a maximum value and

the flow is said to be choked.

4. 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.

Mass flow rate

Under steady-flow conditions, the mass flow rate through the nozzle is constant and can be expressed as Substitution of T from eq. (13) and of P from eq. (14) in above equation gives,

Fig. (11) The effect of back pressure on the

pressure distribution along a converging

nozzle.

For a specified flow area, A and stagnation properties T0 and P0, the maximum mass flow rate can be

determined by differentiating eq. (19) with respect to Ma and setting the result equal to zero. 𝑑�̇�

𝑑𝑀𝑎= 0

𝐴𝑃0√𝑘/𝑅𝑇 [{1 +𝑘 − 1

2𝑀𝑎2 }

𝑘+12(𝑘−1)

− 𝑀𝑎 (𝑘 + 1

2(𝑘 − 1)) {{1 +

𝑘 − 1

2𝑀𝑎2 }}

3−𝑘2(𝑘−1)

(𝑘 − 1

2) 2𝑀𝑎]

= 0

Or, {1 +𝑘−1

2𝑀𝑎2 }

𝑘+1

2(𝑘−1) = 𝑀𝑎2 (𝑘+1

2) {{1 +

𝑘−1

2𝑀𝑎2}}

3−𝑘

2(𝑘−1)

1 +𝑘−1

2𝑀𝑎2 = 𝑀𝑎2 (

𝑘+1

2)

Or, 𝑀𝑎 = 1 So, the mass flow rate is maximum when Ma=1.

Rewriting eqs (11) and (12) 𝑑𝐴

𝐴=

𝑑𝑃

𝜌𝑉2(1 − 𝑀𝑎2)

𝑑𝐴

𝐴= −

𝑑𝑉

𝑉(1 − 𝑀𝑎2 )

Substituting Ma=1in any one of the above equation gives, 𝑑𝐴 = 0

Or, 𝐴 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Therefore, the only location in a nozzle where the Mach number can be unity is the location of minimum

flow area (the throat). Thus, the mass flow rate through a nozzle is a maximum when Ma=1 at the throat.

Denoting this area by A*, we obtain an expression for the maximum mass flow rate by substituting Ma=1

in eq. (19):

Thus, for a particular ideal gas, the maximum mass flow rate through a nozzle with a given throat area is

fixed by the stagnation pressure and temperature of the inlet flow.

.

A plot of �̇� versus 𝑃𝑏 𝑃0⁄ for a converging nozzle is shown in Fig. (12). Notice that the mass flow rate

increases with decreasing 𝑃𝑏 𝑃0⁄ , reaches a maximum at 𝑃𝑏 =P*, and remains constant for 𝑃𝑏 𝑃0⁄ values less

than this critical ratio. Also illustrated on this figure is the effect of back pressure on the nozzle exit pressure

𝑃𝑒 .

The effects of the stagnation temperature 𝑇0 and stagnation pressure 𝑃0 on the mass flow rate through a

converging nozzle are illustrated in Fig. (13) where the mass flow rate is plotted against the static-to-

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Fig. (12) The effect of back pressure 𝑃𝑏 on the

mass flow rate and the exit pressure 𝑃𝑒 of a

converging nozzle.

Fig. (13) The variation of the mass flow rate

through a nozzle with inlet stagnation

properties.

stagnation pressure ratio at the throat 𝑃𝑡 𝑃0⁄ . An increase in 𝑃0 (or a decrease in 𝑇0) will increase the mass

flow rate through the converging nozzle; a decrease in 𝑃0 (or an increase in 𝑇0) will decrease it.

Variation of flow area A relative to throat area A*

Combining eqs. (19) and (20) for the same mass flow rate and stagnation properties of a particular fluid. This Yields

From above equation it is obvious that 𝐴 𝐴∗⁄ is a function of Mach number Ma. Fig. (14) shows a

plot of 𝐴 𝐴∗⁄ vs. Ma (for air k=γ = 1.4). It can be seen that there is one value of A/A* for each value of the

Mach number, but there are two possible values of the Mach number for each value of A/A*—one for subsonic flow and another for supersonic flow.

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Local Velocity to the Sonic Velocity at the Throat, Ma*:

It can also be expressed as

Substitution of T from eq. (13) and T* from eq. (16) in above equation gives

Note that the parameter Ma* differs from the Mach number Ma. Ma* is the local velocity

nondimensionalized with respect to the sonic velocity at the throat, whereas Ma is the local velocity

nondimensionalized with respect to the local sonic velocity.

Converging–Diverging Nozzles

The highest velocity to which a fluid can be accelerated in a converging nozzle is limited to the sonic

velocity (Ma=1), which occurs at the exit plane (throat) of the nozzle. Accelerating a fluid to supersonic

velocities (Ma>1) can be accomplished only by Converging–Diverging Nozzles.

Consider the converging–diverging nozzle shown in Fig. (15). A fluid enters the nozzle with a low velocity at stagnation pressure 𝑃0. Now let us examine what happens as the back pressure 𝑃𝑏 is varied at the exit region of

nozzle.

Fig. (14) Area ratio 𝐴 𝐴∗⁄ as a function of Mach number Ma

Ma

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Fig. (15) The effects of back pressure on the flow through a

converging–diverging nozzle.

1. When 𝑃𝑏 = 𝑃0 (case A), there will be no flow through the nozzle. This is expected since the flow

in a nozzle is driven by the pressure difference between the nozzle inlet and the exit.

2. When 𝑃0 > 𝑃𝑏 > 𝑃𝑐 , the flow remains subsonic throughout the nozzle, and the mass flow is less

than that for choked flow. The fluid velocity increases in the first (converging) section and reaches

a maximum at the throat (but Ma<1). However, most of the gain in velocity is lost in the second

(diverging) section of the nozzle, which acts as a diffuser. The pressure decreases in the converging

section, reaches a minimum at the throat, and increases at the expense of velocity in the diverging

section.

3. When 𝑃𝑏 = 𝑃𝑐 , the throat pressure becomes P* and the fluid achieves sonic velocity at the throat.

But the diverging section of the nozzle still acts as a diffuser, slowing the fluid to subsonic

velocities. The mass flow rate that was increasing with decreasing 𝑃𝑏 also reaches its maximum

value. Furthering lowering 𝑃𝑏 does not influence the flow in the converging part of the nozzle or

the mass flow rate through the nozzle. But, it does influence the character of the flow in the

diverging section.

4. When 𝑃𝑐 > 𝑃𝑏 > 𝑃𝑒 , the fluid that achieved a sonic velocity at the throat continues accelerating to

supersonic velocities in the diverging section as the pressure decreases. This acceleration comes to

a sudden stop, however, as a normal shock develops at a section between the throat and the exit

plane, which causes a sudden drop in velocity to subsonic levels and a sudden increase in pressure.

The fluid then continues to decelerate further in the remaining part of the converging–diverging

nozzle. Flow through the shock is highly irreversible, and thus it cannot be approximated as

isentropic. Shock occur only when flow is supersonic and after the shock the flow becomes

subsonic. The normal shock moves downstream away from the throat as 𝑃𝑏 is decreased, and it

approaches the nozzle exit plane as 𝑃𝑏 approaches 𝑃𝐸 .

When 𝑃𝑏 = 𝑃𝐸 , the normal shock forms just at the exit plane of the nozzle. The flow is supersonic

through the entire diverging section in this case, and it can be approximated as isentropic. However,

the fluid velocity drops to subsonic levels just before leaving the nozzle as it crosses the normal

shock.

5. When 𝑃𝐸 > 𝑃𝑏 > 0, the flow in the diverging section is supersonic, and the fluid expands to 𝑃𝐹 at

the nozzle exit with no normal shock forming within the nozzle. Thus, the flow through the nozzle

can be approximated as isentropic. When 𝑃𝑏 = 𝑃𝐹 , no shocks occur within or outside the nozzle.

When 𝑃𝑏 < 𝑃𝐹 , irreversible mixing and expansion waves occur downstream of the exit plane of the

nozzle. When 𝑃𝑏 > 𝑃𝐹 , however, the pressure of the fluid increases from 𝑃𝐹 𝑡𝑜 𝑃𝑏 irreversibly in

the wake of the nozzle exit, creating what are called oblique shocks.

Normal Shocks

Shock waves that occur in a plane normal to the direction of flow are called normal shock waves. The flow process through the shock wave is highly irreversible and cannot be approximated as being isentropic.

Relationships for the Flow Properties Before and After the Shock

Consider a control volume that contains the shock wave as

shown in Fig. (16) and apply following equations to control

volume:

1. Conservation of mass

2. Conservation of momentum

3. Conservation of energy

4. Some property relations

Assumptions

1. The normal shock waves are extremely thin, so the

entrance and exit flow areas for the control volume

are approximately equal.

2. Flow across the control volume is steady with no heat

and work interactions and no potential energy

changes.

Fig. (16) Control volume for flow across a normal shock wave.

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Denoting the properties upstream of the shock by the subscript 1 and those downstream of the shock by 2, we have the following: Conservation of mass:

Or, Conservation of energy:

Rewriting eq. (10)

Or, 𝑑𝑃 = −�̇�

𝐴𝑑𝑉 (∵ �̇� = 𝜌𝐴𝑉)

Rearranging and integrating the above equation yields,

Conservation of momentum:

𝑃1 + 𝜌1𝑉12 = 𝑃2 + 𝜌2𝑉2

2

𝐼1 = 𝐼2 Where 𝐼 = 𝑃 + 𝜌𝑉2 is called impulse pressure.

Increase of entropy:

Fanno line: It is the locus of states that have the same value of stagnation enthalpy and mass flux (mass

flow per unit flow area). When two equations, conservation of mass and energy equations, are combined

into a single equation and plotted on h-s diagram by using property relation then the resultant curve is called

Fanno line.

Rayleigh line: It is the locus of states that have the same value of impulse pressure and mass flux (mass

flow per unit flow area). When two equations, conservation of mass and momentum equations, are

combined into a single equation and plotted on h-s diagram by using property relation then the resultant

curve is called Rayleigh line.

Both these lines, Fanno and Rayleigh lines are shown on the h-s diagram in Fig. (17). Following points may be noted:

1. The Fanno and Rayleigh lines intersect at two points (points 1 and 2), which represent the two states.

One of these (state 1) corresponds to the state before the shock, and the other (state 2) corresponds to the state after the shock.

2. At both the points (state 1 and state 2), all three conservation equations are satisfied. 3. The points of maximum entropy on these lines (points a and b) correspond to Ma=1. The state on the

upper part of each curve is subsonic and on the lower part supersonic. 4. Note that the flow is supersonic before the shock and subsonic afterward. Therefore, the flow must

change from supersonic to subsonic if a shock is to occur.

5. The larger the Mach number before the shock, the stronger the shock will be. In the limiting case

of Ma=1, the shock wave simply becomes a sound wave.

6. Notice from Fig. (17) that 𝑠2 > 𝑠1. This is expected since the flow through the shock is adiabatic

but irreversible. The conservation of energy principle (Eq. (25)) requires that

the stagnation enthalpy remain constant across the shock; ℎ01 = ℎ02. For ideal gases ℎ = ℎ(𝑇), and thus

That is, the stagnation temperature of an ideal gas also

remains constant across the shock. Note, however, that the

stagnation pressure decreases across the shock because of

the irreversibilities, while the thermodynamic temperature

rises drastically because of the conversion of kinetic energy

into enthalpy due to a large drop in fluid velocity (see Fig.

(18)). We now develop relations between various properties before and after the shock for an ideal gas with constant specific heats. Relation between stagnation temperature and thermodynamic temperature at state 1 & 2.

Dividing the first equation by the second one and noting that 𝑇01 = 𝑇02 , we have

Fig. (17) The h-s diagram for flow across a normal shock.

Fig. (18) Variation of flow properties

across a normal shock.

(28)

(29)

(29)

(30)

(31)

From the ideal-gas equation of state,

Substituting these into the conservation of mass relation 𝜌1𝑉1 = 𝜌2 𝑉2 and noting that 𝑀𝑎 = 𝑉/𝑐 and 𝑐 =

√𝑘𝑅𝑇, we have

Combining Eqs. (28) and (30) gives the pressure ratio across the shock:

Above equation is a combination of the conservation of mass and energy equations; thus, it is also the

equation of the Fanno line for an ideal gas with constant specific heats. A similar relation for the Rayleigh

line can be obtained by combining the conservation of mass and momentum equations.

Rewriting eq. (26)

However,

Thus,

Or,

Combining eqs. (29) and (30) yields,

(32)

This represents the intersections of the Fanno and Rayleigh lines and relates the Mach number upstream of

the shock to that downstream of the shock. 𝑀𝑎2 (the Mach number after the shock) is always less than 1

and that the larger the supersonic Mach number before the shock, the smaller the subsonic Mach number

after the shock.

The entropy change across the shock is obtained by applying the entropy change equation for an ideal gas

across the shock:

Gas Table for Isentropic Flow:

The values of 𝑀𝑎 ∗, 𝐴 𝐴∗⁄ , 𝑃 𝑃0⁄ , 𝜌 𝜌0⁄ 𝑎𝑛𝑑 𝑇 𝑇0⁄

computed for an ideal gas

k=1.4 for various Mach

number Ma from the eqs.

(22), (21), (14), (15) and

(13) respectively are given

below in tabulated form.

These may be used with

advantage for

computations of problems

of isentropic flow.

Ma Ma*

𝑀𝑎1 𝑀𝑎2 𝑃2

𝑃1

𝜌2

𝜌1

𝑇2

𝑇1

𝑃02

𝑃01

𝑃02

𝑃1

Gas Table for Normal Shock Flow:

For different values of 𝑀𝑎1 and for 𝛾 = 1.4, the values of 𝑀𝑎2 , 𝑃2 𝑃1⁄ , 𝜌2 𝜌1⁄ , 𝑇2 𝑇1⁄ ,𝑃02 𝑃01⁄ 𝑎𝑛𝑑 𝑃02 𝑃1⁄ can be computed from above eqs. (28) through (31) and tabulated as below.

Steam Nozzle:

The error involved in treating water vapor as an ideal gas is calculated and plotted in Fig. (19). It is clear

from this figure that at pressures below 10 kPa, water vapor can be treated as an ideal gas, regardless of its

temperature, with negligible error (less than 0.1 percent). At higher pressures, however, the ideal gas

assumption yields unacceptable errors, particularly in the vicinity of the critical point and the saturated

vapor line (over 100 percent). Since pressure of steam, when flow through nozzles or blade passages in

steam turbines, is moderate or high therefore most of the relations developed so far in this module are not

applicable to steam nozzles or steam turbines.

Let us consider expansion of steam through a nozzle between sections 1 and 2. Neglecting change in potential energy and heat interaction and applying steady flow energy equation yields,

Fig. (19) Percentage of error ([(𝑣𝑡𝑎𝑏𝑙𝑒 − 𝑣𝑖𝑑𝑒𝑎𝑙 )/𝑣𝑡𝑎𝑏𝑙𝑒)] × 100) involved

in assuming steam to be an ideal gas, and the region where steam can be

treated as an ideal gas with less than 1 percent error.

Fig. (21) T-s and h-s diagram for the expansion of steam through a nozzle

Fig. (20) P-V diagram for the expansion of steam through a nozzle

Velocity at exit from nozzle:

For negligible velocity at inlet above equation can be written as

Where ℎ1 and ℎ2 are enthalpy in J/kg at sections 1 and 2 respectively.

𝐶2 = √2000(ℎ1 − ℎ2)

𝐶2 = 44.72√(ℎ1 − ℎ2) , m/s Now ℎ1 and ℎ2 are enthalpy in kJ/kg at sections 1 and 2 respectively.

Expansion of steam on T-s and h-s diagram for superheated steam and wet steam is shown by 1–2 and 3–4

respectively under ideal conditions whereas 1-2’ and 3-4’ represent actual expansion of steam through

nozzle (Fig. (21)). In this figure isentropic heat drop shown by 1–2 and 3–4 is also known as ‘Rankine heat

drop’.

Mass flow through a nozzle can be obtained from continuity equation between sections 1 and 2.

Mass flow per unit area;

Differential form of steady flow energy equation can be written as

Or,

also as

Or, For the expansion through a nozzle being governed by process 𝑝𝑣𝑛 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡,

or, velocity at exit from nozzle

For negligible inlet velocity,

Mass flow rate per unit area,

From expansion’s governing equation,

Or,

Substituting the value of 𝑣2 in above equation

Or,

Substituting pressure ratio 𝑝2 𝑝1⁄ = 𝑟 in above equation

For given value of 𝑝1 , 𝑣1 and 𝑛 there will be some value of throat pressure (𝑝2) or pressure ratio (𝑝2 𝑝1⁄ = 𝑟) which offers maximum discharge per unit area. It can be obtained by

differentiating expression of mass flow per unit area with respect to 𝑝2 or 𝑟 and equating it to zero. This

pressure at throat for maximum discharge per unit area is also called critical pressure (𝑝𝑐 or 𝑝𝑡) and

pressure ratio is called critical pressure ratio.

Or,

Or,

Critical pressure ratio at throat be given by 𝑝𝑐 or 𝑝𝑡

Here subscript ‘c’ and ‘t’ refer to critical and throat respectively. The maximum discharge per unit area can be obtained by substituting critical pressure ratio in expression for

mass flow per unit area at throat section.

=

0

Critical velocity Rewriting the expression of velocity at exit with negligible inlet velocity

At throat

Substituting the value of critical pressure ratio 𝑝𝑡

𝑝1⁄

Hence, critical velocity

Critical pressure ratio value depends only upon expansion index and so shall have constant value. Value of adiabatic expansion index and critical pressure ratio are tabulated below;

Flow Through Steam Nozzles

Effects of back pressure on exit velocity, mass flow rate and pressure distribution along the steam nozzle

are same as that of gas nozzle which are already elaborated in this module.

Effect of Friction on Nozzle

Due to friction prevailing during fluid flow through nozzle the expansion process through nozzle becomes

irreversible. Expansion process since occurs at quite fast rate and time available is very less for heat transfer

Fig. (22) T-s representation for

expansion of gas through nozzle

Fig. (23) T-s and h-s representation for steam expanding through nozzle

to take place so it can be approximated as adiabatic. Non ideal operation of nozzle causes reduction in

enthalpy drop. This inefficiency in nozzle can be accounted for by nozzle efficiency. Nozzle efficiency is

defined as ratio of actual heat drop to ideal heat drop.

Nozzle efficiency,

𝜂𝑁𝑜𝑧𝑧𝑙𝑒 =𝐴𝑐𝑡𝑢𝑎𝑙 ℎ𝑒𝑎𝑡 𝑑𝑟𝑜𝑝

𝐼𝑑𝑒𝑎𝑙 ℎ𝑒𝑎𝑡 𝑑𝑟𝑜𝑝

Due to friction the velocity at exit from nozzle gets modified by nozzle efficiency as given below. Ideal velocity at exit,

Or ideal enthalpy drop,

In case of nozzle with friction the enthalpy drop, gives velocity at exit as,

Or actual enthalpy drop,

(∵ ℎ = 𝑐𝑝𝑇 𝑓𝑜𝑟 𝑖𝑑𝑒𝑎𝑙 𝑔𝑎𝑠 𝑜𝑛𝑙𝑦)

Efficiency of the nozzle becomes

For negligible inlet velocity Nozzle efficiency,

Thus it could be seen that friction loss will be high with higher velocity of fluid. Generally frictional losses

are found to be more in the downstream after throat in convergent-divergent nozzle because of simple fact

that velocity in converging section upto throat is smaller as compared to after throat. Expansion upto throat

may be considered isentropic due to small frictional losses.

Note:

1. In order to avoid flow separation, angle of divergence (or semi cone angle) of divergent section

is kept small (typically in the range of 10º to 25º). This small divergence angle causes greater

length of diverging section in convergent-divergent nozzle. Greater length of diverging

section means fluid will be exposed to larger surface area, consequently friction will be more.

Thus a compromise is made in selecting angle of divergence. Very small angle is desirable

from flow separation point of view but undesirable due to long length and larger frictional

losses point of view.

Length of diverging portion of nozzle can

be empirically obtained as below

𝐿𝐷𝑆 = √15𝐴𝑡

Where 𝐴𝑡 is cross-sectional area of throat.

2. Nature of flow (laminar or

turbulent) and properties of fluid

also determine the losses in

diverging section.

3. Nozzle material, shape and size also

play role in frictional losses

coefficient of velocity:

The ‘coefficient of velocity’ or the ‘velocity coefficient’ can be given by the ratio of actual velocity at exit

and the isentropic velocity at exit.

𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =𝐶𝑎𝑐𝑡𝑢𝑎𝑙 𝑎𝑡 𝑒𝑥𝑖𝑡

𝐶𝑖𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 𝑎𝑡 𝑒𝑥𝑖𝑡

Convergent

section

Divergent

section

𝐿𝐶𝑆 𝐿𝐷𝑆

𝛼

Fig. (24) convergent-divergent nozzle

𝐿𝐶𝑆: 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑖𝑛𝑔 𝑠𝑒𝑐𝑡𝑖𝑜𝑛

𝐿𝐷𝑆: 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑑𝑖𝑣𝑒𝑟𝑔𝑖𝑛𝑔 𝑠𝑒𝑐𝑡𝑖𝑜𝑛

𝛼: 𝑠𝑒𝑚𝑖 𝑐𝑜𝑛𝑒 𝑎𝑛𝑔𝑙𝑒

Fig. (25) Super saturated expansion of steam in nozzle

Coefficient of discharge: The ‘coefficient of discharge’ or ‘discharge coefficient’ is given by the ratio of actual discharge and the discharge during isentropic flow through nozzle. Mathematically,

𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒 =�̇̇�𝑎𝑐𝑡𝑢𝑎𝑙

�̇�𝑖𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐

Supersaturated Flow or Metastable Flow in Steam Nozzle

As the steam expands in the nozzle, its pressure and temperature drop, and ordinarily one would expect the

steam to start condensing when it strikes the saturation line. However, this is not always the case. Owing

to the high speeds, the residence time of the steam in the nozzle is small (of the order of 0.01 sec), and there

may not be sufficient time for the necessary heat transfer and the formation of liquid droplets. Consequently,

the condensation of the steam may be delayed for a little while. This phenomenon of delayed condensation

is known as supersaturation, and the flow is termed as supersaturated flow or metastable flow. The steam

that exists in the wet region without containing any liquid is called supersaturated steam. Supersaturation

states are nonequilibrium states.

Let us understand this complete phenomenon (supersaturation) with the help of T-s and h-s plots.

Isentropic expansion of steam (without super-saturation) is represented by process 1-4. This

expansion is in thermal equilibrium from state 1 to all the way upto state 4. In this expansion,

Superheated steam undergoes continuous change in its state and becomes dry saturated steam at

state 2 and subsequently wet steam leaving the nozzle at state 4. At every point along expansion

line there exists a mixture of vapour and liquid in equilibrium.

In super saturated expansion, superheated steam expands isentropically upto state 2. Beyond state

2, Steam continue to expand in unnatural superheated state untit state 3.

State 3 is achieved by extension of the curvature of constant pressure line 𝑝3 from the superheated

region which strikes the vertical expansion line at 3 and through which Wilson line (a line of 94-

95% dryness fraction) also passes.

At state 3 (metastable state) density reaches about eight times that of the saturated vapour density

at the same pressure. When this limit (also called supersaturation) is reached then the steam will

condense suddenly & irreversibly along constant enthalpy line (3-3’) to a normal state 3’.

At any pressure between 𝑝2 and 𝑝3 i.e., within the superheated zone, the temperature of the vapour

is lower than the saturation temperature corresponding to that pressure.

3’-4’ is again isentropic, expansion in thermal equilibrium.

Metastable flow is characterized by parameters called “degree of supersaturation” and “degree of

undercooling”.

𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑠𝑢𝑝𝑒𝑟𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛 =𝑝3

𝑝3𝑠

Where 𝑝3 is limiting saturation pressure and 𝑝3𝑠 is saturation pressure at temperature 𝑇3 shown in T-s

diagram

𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑢𝑛𝑑𝑒𝑟𝑐𝑜𝑜𝑙𝑖𝑛𝑔 = 𝑇3𝑠 − 𝑇3

Where 𝑇3𝑠 is saturation temperature at 𝑝3 and 𝑇3 is supersaturated steam temperature at point 3 which is

the limit of supersaturation. All these temperatures and pressure are shown in T-s diagram.

Effect of super saturated flow:

1. Enthalpy drop in super saturated flow is less,

i.e. (ℎ1 − ℎ4′) < (ℎ1 − ℎ4).

2. Since exit velocity is proportional to square

root of enthalpy drop (i.e. 𝐶2 ∝ √∆ℎ), exit

velocity decreases slightly in supersaturated

flow.

3. Specific volume is reduced, i.e. 𝑣4′ < 𝑣4 or

𝜌4′ > 𝜌4 (refer Fig. (26))

4. Mass flow rate increases.

5. Supersaturated flow is an irreversible flow

hence entropy increases.

6. Quality of steam increases.

Video links for further help on compressible flow

https://www.youtube.com/watch?v=xk2PnH9vh-I&list=PLuxNLGEvT0tWfwY0yti8KsVCUxrZDfR4R&index=1 https://www.youtube.com/watch?v=cVmK3dLeCZ4&list=PLuxNLGEvT0tWfwY0yti8KsVCUxrZDfR4R&index=2 https://www.youtube.com/watch?v=sgvzcqHj9c4&list=PLuxNLGEvT0tWfwY0yti8KsVCUxrZDfR4R&index=3 https://www.youtube.com/watch?v=DLzdz97XkmQ&list=PLuxNLGEvT0tWfwY0yti8KsVCUxrZDfR4R&index=4 https://www.youtube.com/watch?v=5CPydaSn15M&list=PLuxNLGEvT0tWfwY0yti8KsVCUxrZDfR4R&index=5 https://www.youtube.com/watch?v=WKcur4vXb_s&list=PLuxNLGEvT0tWfwY0yti8KsVCUxrZDfR4R&index=6

https://www.youtube.com/watch?v=nl3RuJRToig https://www.youtube.com/watch?v=c3-93PW7S08 https://www.youtube.com/watch?v=3AWQRixSe5c

Fig. (26) p-v representation of steam

expanding through nozzle

Module 4: Assignment

Note:

1. Solution is to be shared with me and not in group.

2. Those who will share in the group will be awarded zero marks.

3. You must write your name, registration number and roll number before and then share the

solution.

1. An aircraft is cruising in still air at 5°C at a velocity of 400 m/s. The air temperature at the nose of

the aircraft where stagnation occurs is

(a) 5º C (b) 25º C (c) 55º C (d) 80º C (e) 85º C

2. Consider a converging nozzle with a low velocity at the inlet and sonic velocity at the exit plane.

Now the nozzle exit diameter is reduced by half while the nozzle inlet temperature and pressure are

maintained the same. The nozzle exit velocity will

(a) remain the same (b) double (c) quadruple (d) go down by half (e) go down to

one-fourth

3. Air is approaching a converging–diverging nozzle with a low velocity at 20°C and 300 kPa, and it

leaves the nozzle at a supersonic velocity. The velocity of air at the throat of the nozzle is

(a) 290 m/s (b) 98 m/s (c) 313 m/s (d) 343 m/s (e) 412 m/s

4. Argon gas is approaching a converging–diverging nozzle with a low velocity at 20°C and 120 kPa,

and it leaves the nozzle at a supersonic velocity. If the cross-sectional area of the throat is 0.015

m2, the mass flow rate of argon through the nozzle is

(a) 0.41 kg/s (b) 3.4 kg/s (c) 5.3 kg/s (d) 17 kg/s (e) 22 kg/s

5. Consider gas flow through a converging–diverging nozzle. Of the five following statements, select

the one that is incorrect:

(a) The fluid velocity at the throat can never exceed the speed of sound.

(b) If the fluid velocity at the throat is below the speed of sound, the diversion section will act like

a diffuser.

(c) If the fluid enters the diverging section with a Mach number greater than one, the flow at the

nozzle exit will be supersonic.

(d) There will be no flow through the nozzle if the back pressure equals the stagnation pressure.

(e) The fluid velocity decreases, the entropy increases, and stagnation enthalpy remains constant

during flow through a normal shock.

6. Shocks can occur

(a) anywhere in convergent divergent nozzle

(b) in convergent section

(c) in the divergent section when flow is supersonic

(d) in the divergent section when flow is subsonic

7. Which of the following statements are correct for compressible fluid flow:

1. Maximum velocity that can be achieved by a converging nozzle is sonic velocity and to

accelerate a fluid from subsonic velocity to supersonic velocity we must use converging–diverging

nozzle.

2. For a particular ideal gas, the maximum mass flow rate through a nozzle with a given throat area

is fixed by the stagnation pressure and temperature of the inlet flow.

3. There are two possible values of the Mach number for each value of A/A*—one for subsonic

flow and another for supersonic flow.

4. The flow must change from supersonic to subsonic if a shock is to occur and stagnation pressure

and stagnation temperature increases after shock.

(a) 1, 2 & 3 (b) 2, 3 & 4 (c) 1, 3 & 4 (d) 1, 2 & 4 (e) 1, 2, 3 & 4

8. Consider the following statements:

1. All the equations, developed for isentropic flow of ideal gases are equally applicable for

isentropic flow of steam also.

2. Friction in steam nozzle cause reduction in enthalpy drop.

Of these statements

(a) only 1 is correct (b) only 2 is correct (c) both are correct (d) none is correct

9. Consider the following statements related to super saturated flow:

1. High speed of steam through nozzle causes super saturated flow.

2. Steam starts condensing as it hits the saturation line.

3. It is a reversible phenomenon.

4. At metastable state density reaches about eight times that of the saturated vapour density at the

same pressure.

5. Mass flow rate increases and dryness fraction decreases.

Of these statements

(a) only 1 & 2 are correct (b) only 1 & 3 are correct (c) only 1 & 4 are correct (d) only 1 & 5 are

correct.

10. Air at 1 MPa and 600°C enters a converging nozzle with a velocity of 150 m/s. Determine the mass

flow rate through the nozzle for a nozzle throat area of 50 cm² when the back pressure is (a) 0.7

MPa and (b) 0.4 MPa.

11. Air enters a converging–diverging nozzle at 1.0 MPa and 800 K with a negligible velocity. The

flow is steady, one-dimensional, and isentropic with k = 1.4. For an exit Mach number of Ma = 2

and a throat area of 20 cm², determine (a) pressure, temperature, density & velocity at the throat

(b) pressure, temperature, density, velocity and exit area at the exit plane, and (c) the mass flow

rate through the nozzle.

12. Show that the point of maximum entropy on the Fanno line for the adiabatic steady flow of a fluid

in a duct corresponds to the sonic velocity, Ma =1.

13. If the air flowing through the converging–diverging nozzle of Q. No.-11 experiences a normal

shock wave at the nozzle exit plane, determine the following after the shock: (a) the stagnation

pressure, static pressure, static temperature, and static density; (b) the entropy change across the

shock; (c) the exit velocity; and (d ) the mass flow rate through the nozzle. Assume steady, one-

dimensional, and isentropic flow with k = 1.4 from the nozzle inlet to the shock location.

14. Prove that the maximum discharge of fluid per unit area through a nozzle shall occur when the ratio

of fluid pressure at throat to the inlet pressure is (2

𝑛+1)

𝑛

𝑛−1 where n is the index of adiabatic

expansion. Also obtain the expression for maximum mass flow through a convergent-divergent

nozzle having isentropic expansion starting from rest.

15. In a steam nozzle steam expands from 16 bar to 5 bar with initial temperature of 300ºC and mass

flow of 1 kg/s. Determine the throat and exit areas considering (i) expansion to be frictionless and,

(ii) friction loss of 10% throughout the nozzle.