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Applied 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 at very high velocities. Compressible flow combines 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 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 (푢) and flow energy (푃푣). For convenience these two terms are combined and termed as , ℎ = 푢 + 푃푣 (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

(1)

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 , , stagnation density, etc.). The stagnation state and the stagnation properties are indicated by the subscript 0. Fig. (1) Kinetic energy is converted to stagnation enthalpy represents the enthalpy of a enthalpy during a stagnation process. 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

or, (2)

The stagnation state is called the isentropic stagnation state when the fluid is brought to rest reversibly and adiabatically (i.e., isentropically). The 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 푐푝푇 and above equation of stagnation enthalpy can be Fig.(2) The actual state, actual stagnation written as state, and isentropic stagnation state of a fluid on an h-s diagram.

Or, (3)

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

(4)

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 .

1 푘 푘 By noting that 휌 = ⁄푣 and using the isentropic relation 푃푣 = 푃0푣0 , the ratio of the stagnation density to static density can be expressed as

(5)

Speed of Sound and

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 Fig. (3) Propagation of a small speed of c-dV. pressure wave along a duct.

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

Fig. (4) Control volume moving with the small pressure In absence of any heat and work interaction and change in potential wave along a duct. 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,

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

Or,

For an ideal gas, in an 푃푣 푘 = 푐표푛푠푡푎푛푡 푃 = 푐표푛푠푡푎푛푡 휌푘 where k is the specific heat ratio of gas. Taking logarithm of above equation yields, 푙푛푃 − 푘푙푛휌 = 푙푛푐표푛푠푡푎푛푡 Now differentiating the above equation yields

푑푃 푑휌 − 푘 =0 푃 휌 푑푃 푃 Or, = 푘 푑휌 휌 푑푃 푑푃 For isentropic process = ( ) 푑휌 푑휌 푠

푑푃 푃 ∴ ( ) = 푘 푑휌 푠 휌

푑푃 Since 푐2 = ( ) and 푃 = 푘푅푇 푑휌 푠

푐2 = 푘푅푇

(6)

푈푛𝑖푣푒푟푠푎푙 𝑔푎푠 푐표푛푠푡푎푛푡 Where, 푅 = 푐ℎ푎푟푎푐푡푒푟𝑖푠푡𝑖푐 푔푎푠 푐표푛푠푡푎푛푡 = 푀표푙푒푐푢푙푎푟 푤푒𝑖𝑔ℎ푡 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 (7) changes with temperature and varies with the fluid.

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 Fig. (6) The Mach number can be 5. when Ma≅1 different at different even if the velocity is the same. 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

(8)

Now, energy balance equation for steady flow process, without heat and work interaction and without change in potential energy, can be written as

푉2 Or, ℎ + = 푐푛푠푡푎푛푡 2 Differentiating the above expression, 푑ℎ + 푉푑푉 = 0 (9)

Second Tds equation can be written as 0 (for isentropic flow)

푇푑푠 = 푑ℎ − 푣푑푃 1 푑ℎ = 푣푑푃 = 푑푃 휌 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) (10)

Combining equations (8) and (10) gives

푑휌 1 Putting the value of = in above equation, yields 푑푃 푐2 (11)

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, 휌, and V are positive quantities. For subsonic flow (Ma<1), the term 1 − 푀푎2 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 − 푀푎2 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 휌푉 = − 푑푃⁄푑푉 from equation (10) into equation (11)

(12)

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.

Fig. (7) Variation of flow properties in subsonic and supersonic nozzles and diffusers. 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 푚̇ = 휌퐴푉 for steady flow, we see that the large decrease in density makes acceleration in the diverging section possible.

Fig. (8) We cannot obtain supersonic Fig. (9) The cross section of a nozzle at the velocities by attaching a converging smallest flow area is called the throat. section to a converging nozzle. Doing so converging–diverging nozzles are often will only move the sonic cross section called Laval nozzles. farther downstream and decrease the mass flow rate.

Property Relations for Isentropic Flow of Ideal Gases

The temperature T of an ideal gas anywhere in the flow is related to the stagnation temperature푇0 through eq. (3)

Or,

2 Noting that 푐푝 = 푘푅/(푘 − 1), 푐 = 푘푅푇, and 푀푎 = 푉/푐, we see that

Substituting yields,

(13)

Which is the desired relation between 푇0 and 푇. The ratio of the stagnation to static pressure is obtained by substituting eq. (13) into eq. (4)

(14)

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

(15)

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

(16)

(17)

(18)

The above ratios in eqs. (16) through (18) are called critical ratios (Fig. (10)) and 푇∗, 푃∗ and 휌∗are called critical temperature, critical pressure and critical density Fig. (10) When Mat = 1 the properties at the respectively. nozzle throat become the critical properties. 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, 푷풃 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 푃푟 and temperature 푇푟 . 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, 푃푏 at exit region of nozzle; 1. When 푃푏 = 푃1 = 푃푟 then there is no flow and the pressure distribution is uniform along the nozzle. 2. When the back pressure is reduced to 푃2, the exit plane pressure 푃푒 also drops to 푃2 . This causes the pressure along the nozzle in flow direction to decrease and velocity to increase. 3. When the back pressure is reduced to 푃3 ∗ (=푃 ) then exit plane pressure 푃푒 also drops ∗ to 푃3 (= 푃 ). This causes the pressure to decrease and velocity to increase along the flow direction and at exit of nozzle velocity Fig. (11) The effect of back pressure on the reaches to sonic velocity. As far as mass-flow pressure distribution along a converging is concern it reaches a maximum value and nozzle. the flow is said to be choked. 4. Further reduction of the back pressure to level 푃4 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,

(19

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. 푑푚̇ = 0 푑푀푎

푘+1 3−푘 푘 − 1 2(푘−1) 푘 + 1 푘 − 1 2(푘−1) 푘 − 1 퐴푃 √푘/푅푇 [{1 + 푀푎2 } − 푀푎 ( ) {{1 + 푀푎2 }} ( ) 2푀푎] 0 2 2(푘 − 1) 2 2

= 0 푘+1 3−푘 푘−1 푘+1 푘−1 2(푘−1) Or, {1 + 푀푎2 }2(푘−1) = 푀푎2 ( ) {{1 + 푀푎2}} 2 2 2

푘−1 푘+1 1 + 푀푎2 = 푀푎2 ( ) 2 2

Or, 푀푎 = 1 So, the mass flow rate is maximum when Ma=1. Rewriting eqs (11) and (12) 푑퐴 푑푃 = (1 − 푀푎2) 퐴 휌푉2

푑퐴 푑푉 = − (1 − 푀푎2 ) 퐴 푉 Substituting Ma=1in any one of the above equation gives, 푑퐴 = 0 Or, 퐴 = 푐표푛푠푡푎푛푡

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):

(20)

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 푚̇ versus 푃푏 ⁄푃0 for a converging nozzle is shown in Fig. (12). Notice that the mass flow rate increases with decreasing 푃푏 ⁄푃0 , reaches a maximum at 푃푏 =P*, and remains constant for 푃푏 ⁄푃0 values less than this critical ratio. Also illustrated on this figure is the effect of back pressure on the nozzle exit pressure 푃푒 .

The effects of the stagnation temperature 푇0 and stagnation pressure 푃0 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- stagnation pressure ratio at the throat 푃푡 ⁄푃0. An increase in 푃0 (or a decrease in 푇0) will increase the mass flow rate through the converging nozzle; a decrease in 푃0 (or an increase in 푇0) will decrease it.

Fig. (13) The variation of the mass flow rate through a nozzle with inlet stagnation properties.

Fig. (12) The effect of back pressure 푃푏 on the mass flow rate and the exit pressure 푃 of a 푒 converging nozzle.

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

(21)

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

Ma

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

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

(22)

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 푃0. Now let us examine what happens as the back pressure 푃푏 is varied at the exit region of nozzle.

Fig. (15) The effects of back pressure on the flow through a converging–diverging nozzle.

1. When 푃푏 = 푃0 (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 푃0 > 푃푏 > 푃푐, 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 푃푏 = 푃푐, 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 푃푏 also reaches its maximum value. Furthering lowering 푃푏 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 푃푐 > 푃푏 > 푃푒 , 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 푃푏 is decreased, and it approaches the nozzle exit plane as 푃푏 approaches 푃퐸 .

When 푃푏 = 푃퐸 , 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 푃퐸 > 푃푏 > 0, the flow in the diverging section is supersonic, and the fluid expands to 푃퐹 at the nozzle exit with no normal shock forming within the nozzle. Thus, the flow through the nozzle can be approximated as isentropic. When 푃푏 = 푃퐹 , no shocks occur within or outside the nozzle. When 푃푏 < 푃퐹 , irreversible mixing and expansion waves occur downstream of the exit plane of the nozzle. When 푃푏 > 푃퐹 , however, the pressure of the fluid increases from 푃퐹 푡표 푃푏 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 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. 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. Fig. (16) Control volume for flow across a 2. Flow across the control volume is steady with no heat normal shock wave. and work interactions and no potential energy changes. 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: (23)

Or,

Conservation of energy: (24)

(25)

Rewriting eq. (10)

푚̇ Or, 푑푃 = − 푑푉 (∵ 푚̇ = 휌퐴푉) 퐴 Rearranging and integrating the above equation yields,

Conservation of momentum: (26)

2 2 푃1 + 휌1푉1 = 푃2 + 휌2푉2 퐼1 = 퐼2 Where 퐼 = 푃 + 휌푉2 is called impulse pressure.

Increase of entropy: (27)

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.

Fig. (17) The h-s diagram for flow across a normal shock. 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 푠2 > 푠1. 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(푇), 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. Fig. (18) Variation of flow properties (18)). across a normal shock.

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 푇01 = 푇02, we have

(28)

From the ideal-gas ,

Substituting these into the conservation of mass relation 휌1푉1 = 휌2 푉2 and noting that 푀푎 = 푉/푐 and 푐 = √푘푅푇, we have

(29)

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

(29)

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,

(30)

Or,

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

(31)

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. 푀푎2 (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:

(32)

Gas Table for Isentropic Flow:

The values of 푀푎 ∗, 퐴⁄퐴∗, 푃⁄푃0 , 휌⁄휌0 푎푛푑 푇⁄푇0 Ma Ma* 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.

Gas Table for Normal Shock Flow:

For different values of 푀푎1 and for 훾 = 1.4, the values of 푀푎2, 푃2 ⁄푃1 , 휌2⁄휌1 , 푇2⁄푇1 , 푃02 ⁄푃01 푎푛푑 푃02⁄푃1 can be computed from above eqs. (28) through (31) and tabulated as below.

휌 푃 푃2 2 푇2 푃02 02 푀푎 푀푎 1 2 휌 푇 푃 푃1 1 1 푃01 1

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.

Fig. (19) Percentage of error ([(푣푡푎푏푙푒 − 푣𝑖푑푒푎푙 )/푣푡푎푏푙푒)] × 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.

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,

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.

퐶2 = √2000(ℎ1 − ℎ2)

퐶2 = 44.72√(ℎ1 − ℎ2) , m/s Now ℎ1 and ℎ2 are enthalpy in kJ/kg at sections 1 and 2 respectively.

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

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’.

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

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 푝푣푛 = 푐표푛푠푡푎푛푡,

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 푣2 in above equation

Or,

Substituting pressure ratio 푝2⁄푝1 = 푟 in above equation

For given value of 푝1, 푣1 and 푛 there will be some value of throat pressure (푝2) or pressure ratio (푝2⁄푝1 = 푟) which offers maximum discharge per unit area. It can be obtained by differentiating expression of mass flow per unit area with respect to 푝2 or 푟 and equating it to zero. This pressure at throat for maximum discharge per unit area is also called critical pressure (푝푐 or 푝푡) and pressure ratio is called critical pressure ratio.

=

0

Or,

Or,

Critical pressure ratio at throat be given by 푝푐 or 푝푡

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.

Critical velocity

Rewriting the expression of velocity at exit with negligible inlet velocity

At throat

푝푡 Substituting the value of critical pressure ratio ⁄푝1

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 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,

퐴푐푡푢푎푙 ℎ푒푎푡 푑푟표푝 휂 = 푁표푧푧푙푒 퐼푑푒푎푙 ℎ푒푎푡 푑푟표푝

(∵ ℎ = 푐 푇 푓표푟 𝑖푑푒푎푙 푔푎푠 표푛푙푦) 푝

Fig. (22) T-s representation for expansion of gas through nozzle

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

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,

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.

퐿퐶푆: 푙푒푛푔푡ℎ 표푓 푐표푛푣푒푟푔𝑖푛푔 푠푒푐푡𝑖표푛

퐿퐷푆: 푙푒푛푔푡ℎ 표푓 푑𝑖푣푒푟푔𝑖푛푔 푠푒푐푡𝑖표푛 훼

훼: 푠푒푚𝑖 푐표푛푒 푎푛푔푙푒

Convergent Divergent

section section Length of diverging portion of nozzle can be empirically obtained as below

퐿퐷푆 = √15퐴푡 Where 퐴푡 is cross-sectional area of throat.

2. Nature of flow (laminar or 퐿퐶푆 퐿퐷푆 turbulent) and properties of fluid also determine the losses in diverging section. Fig. (24) convergent-divergent nozzle 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. 퐶 퐶표푒푓푓𝑖푐𝑖푒푛푡 표푓 푣푒푙표푐𝑖푡푦 = 푎푐푡푢푎푙 푎푡 푒푥𝑖푡 퐶𝑖푠푒푛푡푟표푝𝑖푐 푎푡 푒푥𝑖푡 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,

푚̇ 퐶표푒푓푓𝑖푐𝑖푒푛푡 표푓 푑𝑖푠푐ℎ푎푟푔푒 = 푎푐푡푢푎푙 푚̇ 𝑖푠푒푛푡푟표푝𝑖푐

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.

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

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 푝3 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 푝2 and 푝3 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”. 푝 퐷푒푔푟푒푒 표푓 푠푢푝푒푟푠푎푡푢푟푎푡𝑖표푛 = 3 푝3푠 Where 푝3 is limiting saturation pressure and 푝3푠 is saturation pressure at temperature 푇3 shown in T-s diagram

퐷푒푔푟푒푒 표푓 푢푛푑푒푟푐표표푙𝑖푛푔 = 푇3푠 − 푇3

Where 푇3푠 is saturation temperature at 푝3 and 푇3 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. 퐶2 ∝ √∆ℎ), exit velocity decreases slightly in supersaturated flow. 3. Specific volume is reduced, i.e. 푣4′ < 푣4 or 휌4′ > 휌4 (refer Fig. (26)) 4. Mass flow rate increases. 5. Supersaturated flow is an irreversible flow hence entropy increases. 6. Quality of steam increases. Fig. (26) p-v representation of steam expanding through nozzle

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 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 푛 2 of fluid pressure at throat to the inlet pressure is ( )푛−1 where n is the index of adiabatic 푛+1 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.