Fluid Mechanics and Hydraulic Machines by R K Bansal

16 Compressible Flow 16.1. Introduction. 16.2. Basic equations of compressible fluid flow. 16.3. Propagation of disturbances in fluid and velocity of sound. 16.4. Mach number. 16.5. Propagation of disturbances in compressible fluid. 16.6. Stagnation properties. 16.7. Area-velocity relationship and effect of variation of area for subsonic, sonic and supersonic flows. 16.8. Flow of compressible fluid through a convergent nozzle. 16.9. Variables of flow in terms of Mach number. 16.10. Flow through Laval nozzle (convergentdivergent nozzle). 16.11. Shock waves. Highlights—Objective Type Questions—Theoretical Questions—Unsolved Examples.

16.1. INTRODUCTION A compressible flow is that flow in which the density of the fluid changes during flow. All real fluids are compressible to some extent and therefore their density will change with change in pressure or temperature. If the relative change in density ∆ρ/ρ is small, the fluid can be treated as incompressible. A compressible fluid, such as air, can be considered as incompressible with constant ρ if changes in elevation are small, acceleration is small, and/or temperature changes are negligible. In other words, if Mach’s number U/C, where C is the sonic velocity, is small, compressible fluid can be treated as incompressible. • The gases are treated as compressible fluids and study of this type of flow is often referred to as ‘Gas dynamics’. • Some important problems where compressibility effect has to be considered are : (i) Flow of gases through nozzles, orifices ; (ii) Compressors ; (iii) Flight of aeroplanes and projectiles moving at higher altitudes ; (iv) Water hammer and accoustics. • ‘Compressibility’ affects the drag coefficients of bodies by formation of shock waves, discharge coefficients of measuring devices such as orificemeters, venturimeters and pitot tubes, stagnation pressure and flows in converging-diverging sections. 16.2. BASIC EQUATIONS OF COMPRESSIBLE FLUID FLOW The basic equations of compressible fluid flow are : (i) Continuity equation, (ii) Momentum equation, (iii) Energy equation, and (iv) Equation of state. 16.2.1. Continuity Equation In case of one-dimensional flow, mass per second = ρAV (where ρ = mass density, A = area of cross-section, V = velocity) Since the mass or mass per second is constant according to law of conservation of mass, therefore, ρAV = constant ...(16.1)

857

858

ENGINEERING THERMODYNAMICS

Differentiating the above equation, we get d(ρAV) = 0 or ρd(AV) + AVdρ = 0 or ρ(AdV + VdA) + AVdρ = 0 or ρAdV + ρVdA + AVdρ = 0 Dividing both sides by ρAV and rearranging, we get

dρ dA dV + + =0 ρ A V Eqn. (16.2) is also known as equation of continuity in differential form.

...(16.2)

16.2.2. Momentum Equation The momentum equation for compressible fluids is similar to the one for incompressible fluids. This is because in momentum equation the change in momentum flux is equated to force required to cause this change. Momentum flux = mass flux × velocity = ρAV × V But the mass flux i.e., ρAV = constant ...By continuity equation Thus the momentum equation is completely independent of the compressibility effects and for compressible fluids the momentum equation, say in X-direction, may be expressed as : ...(16.3) ΣFx = (ρAVVx)2 – (ρAVVx)1 16.2.3. Bernoulli’s or Energy Equation As the flow of compressible fluid is steady, the Euler equation is given as :

dp + VdV + gdz = 0 ρ Integrating both sides, we get

z z

dp + ρ

z

VdV +

z

...(16.4)

gdz = constant

dp V2 + + gz = constant ...(16.5) ρ 2 In compressible flow since ρ is not constant it cannot be taken outside the integration sign. In compressible fluids the pressure (p) changes with change of density (ρ), depending on the type of process. Let us find out the Bernoulli’s equation for isothermal and adiabatic processes. (a) Bernoulli’s or energy equation for isothermal process : In case of an isothermal process, or

pv = constant or

p = constant = c1 (say) ρ

(where v = specific volume = 1/ρ) ∴ Hence

ρ=

p c

z z z z dp = ρ

Substituting the value of

dp = p/c1

c1dp = c1 p

z

dp in eqn. (16.5), we get p

p V2 loge p + + gz = constant ρ 2

dharm \M-therm\Th16-1.pm5

FG H

dp p p = c1 loge p = loge p 3 c1 = p ρ ρ

IJ K

859

COMPRESSIBLE FLOW

Dividing both sides by g, we get

p V2 loge p + + z = constant ...(16.6) ρg 2g Eqn. (16.6) is the Bernoulli’s equation for compressible flow undergoing isothermal process. (b) Bernoulli’s equation for adiabatic process : In case of an adiabatic process, p = constant = c2 (say) ργ

pvγ = constant or

∴ Hence

z z dp = ρ

ργ = dp

p or ρ = c2

= = (c2 )1/ γ

( p/c2 )1/ γ

z

FG p IJ Hc K

1/ γ

2

1 p1/ γ

dp = (c2 )1/ γ

FG LM OP H p ( c ) ( p ) P = = (c ) M MM F − 1 + 1I PP F γ − 1I MN GH γ JK PQ GH γ JK FG IJ F γ I F pI H K = G H γ − 1JK GH ρ JK ( p) F I FG IJ F JJ ( p)H K γ IG p = G J G H γ − 1K G J Hρ K −

2

1/ γ

1 +1 γ

1/ γ

2

1/ γ

γ −1 γ

γ −1 γ

γ

=

1 γ

FG 1 + γ − 1IJ I ( p)H γ γ K

FG γ J H γ − 1K


z

p−1/ γ dp

F γ − 1IJ γ K

GH γ = (c2 )1/ γ ( p) γ −1

F3 GH

c2 =

I J ρ K p γ

γ −1 γ

1/ γ

γ ×

IJ K

z

ρ

=

FG γ IJ p H γ − 1K ρ

dp in eqn. (16.6), we get p

FG γ IJ p + V H γ − 1K ρ 2

2

+ gz = constant

Dividing both sides by g, we get

FG γ IJ p + V + z = constant H γ − 1 K ρg 2 g 2

...(16.7)

Eqn. (16.7) is the Bernoulli’s equation for compressible flow undergoing adiabatic process. Example 16.1. A gas with a velocity of 300 m/s is flowing through a horizontal pipe at a section where pressure is 78 kN/m2 absolute and temperature 40°C. The pipe changes in diameter and at this section, the pressure is 117 kN/m2 absolute. Find the velocity of the gas at this section if the flow of the gas is adiabatic. Take R = 287 J/kg K and γ = 1.4. (Punjab University)


860


Sol. Section 1 : Velocity of the gas, V = 300 m/s Pressure, p1 = 78 kN/m2 Temperature, T1 = 40 + 273 = 313 K Section 2 : Pressure, p2 = 117 kN/m2 R = 287 J/kg K, γ = 1.4 Velocity of gas at section 2, V2 : Applying Bernoulli’s equations at sections 1 and 2 for adiabatic process, we have

FG γ IJ p + V H γ − 1K ρ g 2 g

2 1

1

1

= z1 =


2 2

2

2

+ z2

[Eqn. (16.7)]

But z1 = z2, since the pipe is horizontal.

FG γ IJ p + V = FG γ IJ p + V H γ − 1K ρ g 2 g H γ − 1K ρ g 2 g 2 1

1

∴

2 2

2

1

2

Cancelling ‘g’ on both sides, we get

FG γ IJ FG p H γ − 1K H ρ FG γ IJ p FG 1 − p H γ − 1K ρ H ρ FG γ IJ p FG 1 − p H γ − 1K ρ H p

1

1

or, ∴

1

2

1

2

1

2

1

1

For an adiabatic flow :

IJ K ρ I V × J= 2 p K ρ I V × J= 2 ρ K

−

1

2

2

1

1

2

2

2

p1

ρ1γ


p2 V 2 V2 = 2 − 1 ρ2 2 2

=

p2

ρ2 γ

−

V12 2

−

V12 2

FG IJ H K

p ρ or 1 = 1 p2 ρ2


F I GH JK

ρ p1 or 1 = ρ2 p2

R| p F p I U| V V S|1 − p × GH p JK V| = 2 − 2 T W FG γ IJ p R|S1 − FG p IJ U|V = V − V H γ − 1K ρ | H p K | 2 2 T W FG γ IJ p R|S1 − FG p IJ U|V = V − V H γ − 1K ρ | H p K | 2 T W 1

2

1

1

1

2

1−

At section 1 :

γ

1 γ

ρ1 in eqn (i), we get ρ2


or,

...(i)

1

2

1

1

1

2

1

1

1 γ

2

1 γ

γ −1 γ

2

2

2

2

2 1

2

2 1

2 1

p1 = RT1 = 287 × 313 = 89831 ρ1 p2 117 = = 1.5, and V1 = 300 m/s p1 78

...(ii)

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COMPRESSIBLE FLOW

Substituting the values in eqn. (ii), we get

FG 1.4 IJ × 89831 R|S1 − (1.5) H 1.4 − 1K T|

1.4 − 1 1.4

U| V V| = 2 W 2

2

–

3002 2

V22 V2 – 45000 or – 38609.4 = 2 – 45000 2 2 = 12781.2 or V2 = 113.05 m/s. (Ans.)

314408.5 (1 – 1.1228) = V22

or,

Example 16.2. In the case of air flow in a conduit transition, the pressure, velocity and temperature at the upstream section are 35 kN/m2, 30 m/s and 150°C respectively. If at the downstream section the velocity is 150 m/s, determine the pressure and the temperature if the process followed is isentropic. Take γ = 1.4, R = 290 J/kg K. Sol. Section 1 (upstream) : Pressure, p1 = 35 kN/m2, Velocity, V1 = 30 m/s, Temperature, T = 150 + 273 = 423 K Velocity, V2 = 150 m/s R = 290 J/kg K, γ = 1.4 Section 2 (downstream) : Pressure, p2 : Applying Bernoulli’s equation at sections 1 and 2 for isentropic (reversible adiabatic) process, we have


2 1

1

F γ IJ p + V = G H γ − 1K ρ g 2 g

+ z1

1

2 2

2

+ z2

2

Assuming z1 = z2, we have

FG γ IJ p + V = FG γ IJ p + V H γ − 1K ρ g 2 g H γ − 1K ρ g 2 g 2 1

1

2 2

2

1

2

Cancelling ‘g’ on both the sides, and rearranging, we get

FG γ IJ p FG 1 − p H γ − 1K ρ H p 1

2

1

1

For an isentropic flow :

p1

=

ρ1γ

×

ρ1 ρ2

IJ = V K 2

p2

2

−

V12 2

...(i)

FG IJ H K

ρ1 p1 = ρ2 p2

or

ρ2 γ

2

γ

or

ρ1 = ρ2

ρ Substituting the value of 1 in eqn. (i), we have ρ2

R| p F p I U| V S|1 − p × GH ρ JK V| = 2 T W FG γ IJ p R|S1 − FG p IJ U|V = V H γ − 1K ρ | H p K | 2 T W


1

2

1

1

1

2

1−


1

2

1

1

1 γ

2

1 γ

2

2

2

−

V12 2

−

V12 2

Fp I GH p JK 1

2

1 γ

862



1

1

R| F p I S|1 − GH p JK T 2

γ −1 γ

1

Substituting the values, we get

R| F I S| GH JK T

1.4 p2 × 122670 1 − 1.4 − 1 p1

R| F p I S|1 − GH p JK T FG p IJ Hp K

0.2857

429345

2

1

or

1.4 − 1 1.4

2

=1–

1

F I GH JK

−

U| 150 V| = 2 W

0.2857

or Temperature, T2 : For an isentropic process, we have

2

2

V12 2

−

U| V| = 10800 W

2

or

∴ or

U| V| = V W 2

30 2 = 10800 2

FG3 H

IJ K

p1 = RT1 = 290 × 423 = 122670 ρ1

10800 = 0.9748 429345

p2 = (0.9748)1/0.2857 = (0.9748)3.5 = 0.9145 p1 p2 = 35 × 0.9145 = 32 kN/m2 (Ans.)

γ −1

1.4 − 1

T2 p2 γ = (0.9145) 1.4 = (0.9145)0.2857 = 0.9748 = T1 p1 T2 = 423 × 0.9748 = 412.3 K t2 = 412.3 – 273 = 139.3°C (Ans.)

16.3. PROPAGATION OF DISTURBANCES IN FLUID AND VELOCITY OF SOUND The solids as well as fluids consist of molecules. Whereas the molecules in solids are close together, these are relatively apart in fluids. Consequently whenever there is a minor disturbance, it travels instantaneously in case of solids ; but in case of fluid the molecules change its position before the transmission or propagation of the disturbance. Thus the velocity of disturbance in case of fluids will be less than the velocity of the disturbance in solids. In case of fluid, the propagation of disturbance depends upon its elastic properties. The velocity of disturbance depends upon the changes in pressure and density of the fluid. The propagation of disturbance is similar to the propagation of sound through a media. The speed of propagation of sound in a media is known as acoustic or sonic velocity and depends upon the difference of pressure. In compressible flow, velocity of sound (sonic velocity) is of paramount importance. 16.3.1. Derivation of Sonic Velocity (velocity of sound) Consider a one-dimensional flow through long straight rigid pipe of uniform cross-sectional area filled with a frictionless piston at one end as shown in Fig. 16.1. The tube is filled with a compressible fluid initially at rest. If the piston is moved suddenly to the right with a velocity, a pressure wave would be propagated through the fluid with the velocity of sound wave.


863

COMPRESSIBLE FLOW

Piston

Rigid pipe

Wave front

V

dx = Vdt

C

(dL – dx) (dL = Cdt)

Fig. 16.1. One dimensional pressure wave propagation.

Let A = cross-sectional area of the pipe, V = piston velocity, p = fluid pressure in the pipe before the piston movement, ρ = fluid density before the piston movement, dt = a small interval of time during which piston moves, and C = velocity of pressure wave or sound wave (travelling in the fluid). Before the movement of the piston the length dL has an initial density ρ, and its total mass = ρ × dL × A. When the piston moves through a distance dx, the fluid density within the compressed region of length (dL – dx) will be increased and becomes (ρ + dρ) and subsequently the total mass of fluid in the compressed region = (ρ + dρ) (dL – dx) × A ∴ ρ × dL × A = (ρ + dρ) (dL – dx) × A ...by principle of continuity. But dL = C dt and dx = Vdt ; therefore, the above equation becomes ρCdt = (ρ + dρ) (C – V) dt or, ρC = (ρ + dρ) (C – V) or ρC = ρC – ρV + dρ . C – dρ . V or, 0 = – ρV + dρ . C – dρ . V Neglecting the term dρ.V (V being much smaller than C), we get

ρV ...(16.8) dρ Further in the region of compressed fluid, the fluid particles have attained a velocity which is apparently equal to V (velocity of the piston), accompanied by an increase in pressure dp due to sudden motion of the piston. Applying inpulse-momentum equation for the fluid in the compressed region during dt, we get dp × A × dt = ρ × dL × A (V – 0) (force on the fluid) (rate of change of momentum) dρ . C = ρV or C =

or, or,

dL Cdt V=ρ× × V = ρCV dt dt dp C= ρV Multiplying eqns. (16.8) and (16.9), we get dp = ρ

C2 =


dp dp ρV × = dρ ρV dρ

(3 dL = Cdt) ...(16.9)

864


∴

dp dρ

C=

...(16.10)

16.3.2. Sonic Velocity in Terms of Bulk Modulus The bulk modulus of elasticity of fluid (K) is defined as K=

dp

FG dvIJ H vK

...(i)

where, dv = decrease in volume, and v = original volume (– ve sign indicates that volume decreases with increase in pressure)

1 , or vρ = constant ρ Differentiating both sides, we get Also, volume v ∝

vdρ + ρdv = 0 or –

FG H

IJ K

dv dρ = v ρ

dv dp = from eqn. (i), we get v K

Substituting the value of –

K dp dρ dp = or = ρ K ρ dρ Substituting this value of

dp in eqn. (16.10), we get dρ C=

K ρ

...(16.11)

Eqn. (16.11) is applicable for liquids and gases. 16.3.3. Sonic Velocity for Isothermal Process

p = constant ρ Differentiating both sides, we get

For isothermal process :

ρ . dp − p . dρ ρ2 or,


or

dp p . dρ − =0 ρ ρ2

dp p . dρ = ρ ρ2

or

dp p = = RT dρ ρ

FG p = RT Hρ

...(16.12)

...equation of state

IJ K

dp in eqn. (16.10), we get dρ C=


=0

p = ρ

RT

...(16.13)

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COMPRESSIBLE FLOW

16.3.4. Sonic Velocity for Adiabatic Process For isentropic (reversible adiabatic) process :

p ργ

= constant

or p . ρ–γ = constant Differentiating both sides, we have p(– γ) . ρ–γ–1 dρ + ρ–γ dp = 0 Dividing both sides by ρ–γ, we get – p γ ρ–1 dρ + dp = 0 or dp = p γ ρ–1 dρ

dp p = γ = γRT dρ ρ dp Substituting the value of in eqn. (16.10), we get dρ or,

C=

FG3 H

p = RT ρ

IJ K

...(16.14)

γRT

The following points are worth noting : (i) The process is assumed to be adiabatic when minor disturbances are to be propagated through air ; due to very high velocity of disturbances/pressure waves appreciable heat transfer does not take place. (ii) For calculation of velocity of the sound/pressure waves, isothermal process is considered only when it is mentioned in the numerical problem (that the process is isothermal). When no process is mentioned in the problem, calculation are made assuming the process to be adiabatic. 16.4. MACH NUMBER The mach number is an important parameter in dealing with the flow of compressible fluids, when elastic forces become important and predominant. Mach number is defined as the square root of the ratio of the inertia force of a fluid to the elastic force. ∴ Mach number,

V2 = K /ρ

= i.e.

M=

ρAV 2 KA

Inertia force = Elastic force

M=

V C

V K /ρ

=

V C

[3

K /ρ = C ...Eqn. (16.11)]

...(16.15)

Velocity at a point in a fluid Velocity of sound at that point at a given instant of time Depending on the value of Mach number, the flow can be classified as follows : 1. Subsonic flow : Mach number is less than 1.0 (or M < 1) ; in this case V < C. 2. Sonic flow : Mach number is equal to 1.0 (or M = 1) ; in this case V = C. 3. Supersonic flow : Mach number is greater than 1.0 (or M > 1) ; in this case V > C. When the Mach number in flow region is slightly less to slightly greater than 1.0, the flow is termed as transonic flow. The following points are worth noting : (i) Mach number is important in those problems in which the flow velocity is comparable with the sonic velocity (velocity of sound). It may happen in case of airplanes travelling at very high speed, projectiles, bullets etc. Thus,


M=

866


(ii) If for any flow system the Mach number is less than about 0.4, the effects of compressibility may be neglected (for that flow system). Example 16.3. Find the sonic velocity for the following fluids : (i) Crude oil of specific gravity 0.8 and bulk modulus 1.5 GN/m2 ; (ii) Mercury having a bulk modulus of 27 GN/m2. Sol. Crude oil : Specific gravity = 0.8 (Delhi University) ∴ Density of oil, ρ = 0.8 × 1000 = 800 kg/m3 Bulk modulus, K = 1.5 GN/m2 Mercury : Bulk modulus, K = 27 GN/m2 Density of mercury, ρ = 13.6 × 1000 = 13600 kg/m3 Sonic velocity, Coil, CHg : Sonic velocity is given by the relation : C= ∴

Coil =

K ρ

[Eqn. (16.11)]

1.5 × 109 = 1369.3 m/s (Ans.) 800

27 × 109 = 1409 m/s (Ans.) 13600 Example 16.4. An aeroplane is flying at a height of 14 km where temperature is – 45°C. The speed of the plane is corresponding to M = 2. Find the speed of the plane if R = 287 J/kg K and γ = 1.4. Sol. Temperature (at a height of 14 km), t = – 45°C. T = – 45 + 273 = 228 K Mach number, M=2 Gas constant, R = 287 J/kg K γ = 1.4 Speed of the plane, V : Sonic velocity, (C) is given by, CHg =

γRT (assuming the process to be adiabatic)

C=

1.4 × 287 × 228 = 302.67 m/s

= M=

V C

or,

2=

V 302.67

or,

V = 2 × 302.67 = 605.34 m/s =

Also

...[Eqn. (16.14)]

...[Eqn. (16.15)]

605.34 × 3600 = 2179.2 km/h (Ans.) 1000

16.5. PROPAGATION OF DISTURBANCE IN COMPRESSIBLE FLUID When some disturbance is created in a compressible fluid (elastic or pressure waves are also generated), it is propagated in all directions with sonic velocity (= C) and its nature of propagation depends upon the Mach number (M). Such disturbance may be created when an object moves in a relatively stationary compressible fluid or when a compressible fluid flows past a stationary object.


867

COMPRESSIBLE FLOW

Consider a tiny projectile moving in a straight line with velocity V through a compressible fluid which is stationary. Let the projectile is at A when time t = 0, then in time t it will move through a distance AB = Vt. During this time the disturbance which originated from the projectile when it was at A will grow into the surface of sphere of radius Ct as shown in Fig. 16.2, which also shows the growth of the other disturbances which will originate from the projectile at every t/4 interval of time as the projectile moves from A to B. Let us find nature of propagation of the disturbance for different Mach numbers. Case I. When M < 1 (i.e., V < C). In this case since V < C the projectile lags behind the disturbance/pressure wave and hence as shown in Fig. 16.2 (a) the projectile at point B lies inside the sphere of radius Ct and also inside other spheres formed by the disturbances/waves started at intermediate points. Case II. When M = 1 (i.e., V = C). In this case, the disturbance always travels with the projectile as shown in Fig. 16.2 (b). The circle drawn with centre A will pass through B. Case III. When M > 1 (i.e., V > C). In this case the projectile travels faster than the disturbance. Thus the distance AB (which the projectile has travelled) is more than Ct, and hence

Ct

3 4 C

ZONE

t

3 4 C

ZONE t

1 4 C t

t

t

OF

B

A

1 2 C

A

t 1 C 4

1 2 C

Ct

B

ACTION

(a) M < 1 (V < C)

Mach cone

Subsonic motion Z O N E

OF

SILENCE

(b) M = 1 (V = C)

Sonic motion Z O N E

1 4 Ct

Ct

3 4 C t

1 2 C t

B

A

A C T I O N

a a = Mach angle V

C

a

OF

S I L E N C E

(c) M > 1 (V > C)

Supersonic motion Fig. 16.2. Nature of propagation of disturbances in compressible flow.


OF

868


the projectile at point ‘B’ is outside the spheres formed due to formation and growth of disturbance at t = 0 and at the intermediate points (Fig. 16.2 (c)). If the tangents are drawn (from the point B) to the circles, the spherical pressure waves form a cone with its vertex at B. It is known as Mach cone. The semi-vertex angle α of the cone is known as Mach angle which is given by,

Ct C 1 = = Vt V M

sin α =

...(16.16)

In such a case (M > 1), the effect of the disturbance is felt only in region inside the Mach cone, this region is called zone of action. The region outside the Mach cone is called zone of silence. It has been observed that when an aeroplane is moving with supersonic speed, its noise is heard only after the plane has already passed over us. Example 16.5. Find the velocity of a bullet fired in standard air if its Mach angle is 40°. Sol. Mach angle,

α = 40° γ = 1.4

For standard air : R = 287 J/kg K, t = 15°C or T = 15 + 273 = 288 K Velocity of the bullet, V : Sonic velocity, C = sin α =

Now, or,

sin 40° =

γRT =

1.4 × 287 × 228 = 340.2 m/s

C V 340.2 340.2 or V = = 529.26 m/s (Ans.) sin 40° V

Example 16.6. A projectile is travelling in air having pressure and temperature as 88.3 kN/m2 and – 2°C. If the Mach angle is 40°, find the velocity of the projectile. Take γ = 1.4 and R = 287 J/kg K. [M.U.] Sol. Pressure, p = 88.3 kN/m2 Temperature, T = – 2 + 273 = 271 K Mach angle, M = 40° γ = 1.4, R = 287 J/kg K Velocity of the projectile, V : Sonic velocity, Now, or,

C= sin α = V=

γRT =

1.4 × 287 × 271 ~ − 330 m/s

C 340 or sin 40° = V V 330 = 513.4 m/s (Ans.) sin 40°

Example 16.7. A supersonic aircraft flies at an altitude of 1.8 km where temperature is 4°C. Determine the speed of the aircraft if its sound is heard 4 seconds after its passage over the head of an observer. Take R = 287 J/kg K and γ = 1.4. Sol. Altitude of the aircraft = 1.8 km = 1800 m Temperature,

T = 4 + 273 = 277 K

Time,

t=4s


869

COMPRESSIBLE FLOW

From Fig. 16.3, we have tan α = But, Mach number, M = or,

V=

1800 450 = 4V V

...(i)

A

1 C = sin α V C sin α

tan α =

...(ii)

O

Fig. 16.3

450 450 sin α = (C/sin α) C

sin α 450 sin α C = or cos α = cos α C 450 But

B

AB = Vt = 4V

Substituting the value of V in eqn. (i), we get

or,

4V a

1800 m

Speed of the aircraft, V : Refer Fig. 16.3. Let O represent the observer and A the position of the aircraft just vertically over the observer. After 4 seconds, the aircraft reaches the position represented by the point B. Line AB represents the wave front and α the Mach angle.

C=

...(iii)

γRT , where C is the sonic velocity.

R = 287 J/kg K and γ = 1.4

...(Given)

∴ C = 1.4 × 287 × 277 = 333.6 m/s Substituting the value of C in eqn. (ii), we get cos α =

333.6 = 0.7413 450

1 − cos2 α = 1 − 0.74132 = 0.6712 Substituting the value of sin α in eqn. (ii), we get

∴

sin α =

V=

C 333.6 497 × 3600 = = 497 m/s = = 1789.2 km/h (Ans.) sin α 0.6712 1000

16.6. STAGNATION PROPERTIES The point on the immersed body where the velocity is zero is called stagnation point. At this point velocity head is converted into pressure head. The values of pressure (ps), temperature (Ts) and density (ρs) at stagnation point are called stagnation properties. 16.6.1. Expression for Stagnation Pressure (ps) in Compressible Flow Consider the flow of compressible fluid past an immersed body where the velocity becomes zero. Consider frictionless adiabatic (isentropic) condition. Let us consider two points, O in the free stream and the stagnation point S as shown in Fig. 16.4. Let, p0 = pressure of compressible fluid at point O, V0 = velocity of fluid at O, ρ0 = density of fluid at O, T0 = temperature of fluid at O,


870


and ps, Vs, ρs and Ts are corresponding values of pressure, velocity density, and temperature at point S. Applying Bernoulli’s equation for adiabatic (frictionless) flow at points O and S, (given by eqn. 16.7), we get

FG γ IJ p + V H γ − 1K ρ g 2 g 0

0

0

2

+ z0 =

FG γ IJ p + V H γ − 1K ρ g 2 g s

s

Streamlines

Body S (stagnation point)

O

2

+ zs

s

But z0 = zs ; the above equation reduces to

FG γ IJ p + V = FG γ IJ p + V H γ − 1K ρ g 2 g H γ − 1K ρ g 2 g 0

0

2

s

s

2

Fig. 16.4. Stagnation properties.

s

0

Cancelling ‘g’ on both the sides, we have


FG H

IJ K

V0 2 γ ps Vs2 + = γ − 1 ρs 2 2 0 At point S the velocity is zero, i.e., Vs = 0 ; the above equation becomes 0

+

FG γ IJ F p − p I = − V H γ − 1K GH ρ ρ JK 2 FG γ IJ p F 1 − p × ρ I = – V H γ − 1K ρ GH ρ p JK 2 FG γ IJ p F 1 − p × ρ I = – V H γ − 1K ρ GH p ρ JK 2

or, or,

0

s

0

s

0

s

0

0

s

0

0

s

0

0

0

s

2

0

0

0

2

2

...(i)

ρ0 p ρ γ p p For adiabatic process : 0γ = sγ or 0 = 0γ or = ρs p ρ ρ ρ0 s s s Substituting the value of

or,


1 γ

...(ii)

s

LM p F p I OP V MM1 − p × GH p JK PP = − 2 N Q FG γ IJ p R|S1 − FG p IJ U|V = – V H γ − 1K ρ | H p K | 2 T W LM F p I OP MM1 − GH p JK PP = – V2 FGH γ −γ 1IJK ρp N Q 0

s

0

0

0

s

1−

or,

0

ρ0 in eqn. (i), we get ρs

FG γ IJ p H γ − 1K ρ or,

Fp I GH p JK

0

s

0

0

s

1 γ

0

1 γ

2

0

γ −1 γ

0

2

2

0

V2 1+ 0 2

0 0

FG γ − 1IJ ρ H γ Kp

0 0

Fp I = G J Hp K s

0

γ −1 γ

...(iii)

871

COMPRESSIBLE FLOW

For adiabatic process, the sonic velocity is given by,

For point O,

γ

C0 =


FG3 H

p ρ

γRT = γ

C=

γp0 = C02 in eqn. (iii), we get ρ0

F I GH JK V Fp I 1+ (γ – 1) = G J 2C Hp K Fp I M 1+ (γ – 1) = G J 2 Hp K F p I = LM1 + FG γ − 1IJ M OP GH p JK N H 2 K Q p L F γ − 1IJ M OP = M1 + G p N H 2 K Q L F γ − 1IJ M OP p = p M1 + G N H 2 K Q 0

2

0

0

s

2

γ −1 γ

γ −1 γ

0

2

s

F3 GH

γ −1 γ

0

γ −1 γ

s

or,

IJ K

p0 p0 or C02 = γ ρ0 ρ0

V2 1 ps 1 + 0 (γ – 1) × = 2 p0 C02 or,

p = RT ρ

0

V02 C0

2

= M0 2

I JK

2

0

s

or,

0

0

or,

s

γ −1 γ 2

0

0

...(iv)

γ 2 γ −1

...(16.17)

Eqn. (16.17) gives the value of stagnation pressure.

Compressibility correction factor : If the right hand side of eqn. (16.17) is expanded by the binomial theorem, we get

OP LM Q N LM1 + γM F 1 + M + 2 − γ M + ...I OP JK PQ MN 2 GH 4 24 I p γM F M 2−γ M + ...J + 1+ + G 2 H 4 24 K

ps = p0 1 + γ M02 + γ M04 + γ (2 − γ ) M06 2 8 48 = p0

or,

ps = p0 But,

M02 =

0

0

V02 C0

2

=

0

V02 0

2

0

2

FG γp IJ Hρ K 0


2

0

=

V02 ρ0 γp0

4

2

0

4

...(16.18)

F3 GH

C0 2 =

γp0 ρ0

I JK

872


Substituting the value of M02 in eqn. (16.18), we get

F GH

I JK

p0 γ V02ρ0 M2 2 − γ M04 + ... × 1+ 0 + γp0 2 4 24

ps = p0 +

F GH

I JK

ρ0V02 M02 2 − γ + + M04 + ... 1 ps = p0 + 2 4 24

or,

...(16.19)

ρ0V0 2 (when compressibility effects are neglected) ...(16.20) 2 The comparison of eqns. (16.19) and (16.20) shows that the effects of compressibility are isolated in the bracketed quantity and that these effects depend only upon the Mach number. The Also, ps = p0 +

LM MN

bracketed quantity i. e.,

F1 + M GH 4

2 0

I OP may thus be considered as a compressibility JK PQ

2−γ M0 4 + ... 24

+

correction factor. It is worth noting that : l For M < 0.2, the compressibility affects the pressure difference (ps – p0) by less than 1 per cent and the simple formula for flow at constant density is then sufficiently accurate. l For larger value of M, as the terms of binomial expansion become significant, the compressibility effect must be taken into account. l When the Mach number exceeds a value of about 0.3 the Pitot-static tube used for measuring aircraft speed needs calibration to take into account the compressibility effects. 16.6.2. Expression for Stagnation Density (ρs) From eqn. (ii), we have

F I or ρ = F p I or ρ = ρ F p I GH JK GH p JK GH p JK ρ Fp I Substituting the value of G J from eqn. (iv), we get Hp K OP L ρ = ρ MRS1 + FG γ − 1IJ M UV MMT H 2 K W PP Q N L F γ − 1IJ M OP ρ = ρ M1 + G N H 2 K Q p ρ0 = 0 ps ρs

1 γ

s

s

0

0

1 γ

s

0

s

1 γ

0

s

0

s

or,

s

0

0

0

0

1 γ γ 2 γ −1

1 2 γ −1

16.6.3. Expression for Stagnation Temperature (Ts) The equation of state is given by :

p = RT ρ

For stagnation point, the equation of state may be written as :

ps 1 ps = RTs or Ts = ρs R ρs


...(16.21)

873

COMPRESSIBLE FLOW

Substituting the values of ps and ρs from eqns. (16.17) and (16.18), we get

Ts =

1 R

p0 ρ0

LM1 + FG γ − 1IJ M N H 2 K LM1 + FG γ − 1IJ M N H 2 K

0

0

LM FG IJ N H K p L F γ − 1IJ M 1+ G M H 2 K ρ N LM1 + FG γ − 1IJ M OP N H 2 K Q

OP Q OP Q

γ 2 γ −1

1 2 γ −1

1 p0 γ −1 M0 2 1+ = R ρ0 2 1 = R or,

Ts = T0

0

0

0

0

2

F

OPGH Q F OPGH Q

γ 1 − γ −1 γ −1

γ −1 γ −1

IJ K

IJ K

2

...(16.22)

F3 GH

p0 = RT0 ρ0

I JK

Example 16.8. An aeroplane is flying at 1000 km/h through still air having a pressure of 78.5 kN/m2 (abs.) and temperature – 8°C. Calculate on the stagnation point on the nose of the plane : (i) Stagnation pressure, (ii) Stagnation temperature, and (iii) Stagnation density. Take for air : R = 287 J/kg K and γ = 1.4. Sol. Speed of aeroplane, V = 1000 km/h =

1000 × 1000 = 277.77 m/s 60 × 60

Pressure of air,

p0 = 78.5 kN/m2

Temperature of air,

T0 = – 8 + 273 = 265 K

For air : R = 287 J/kg K, γ = 1.4

The sonic velocity for adiabatic flow is given by, C0 = ∴ Mach number,

M0 =

γRT0 =

1.4 × 287 × 265 = 326.31 m/s

V0 277.77 = = 0.851 C0 326.31

(i) Stagnation pressure, ps : The stagnation pressure (ps) is given by the relation,

L F γ − 1IJ M OP p = p M1 + G N H 2 K Q L F 1.4 − 1IJ × 0.851 OP p = 78.5 M1 + G N H 2 K Q s

or,

0

0

γ 2 γ −1

1.4 2 1.4 − 1

s

= 78.5 (1.145)3.5 = 126.1 kN/m2 (Ans.)


...[Eqn. (16.17)]

874


(ii) Stagnation temperature, Ts : The stagnation temperature is given by,

LM FG γ − 1IJ M OP ...[Eqn. (16.22)] N H 2 K Q L 1.4 − 1 × 0.851 OP = 303.4 K or 30.4°C (Ans.) = 265 M1 + Q N 2

Ts = T0 1 +

0

2

2

(iii) Stagnation density, ρs : The stagnation density (ρs) is given by,

ps = RTs or ρs ρs =

or,

ρs =

ps RTs

126.1 × 10 3 = 1.448 kg/m3 (Ans.) 287 × 303.4

Example 16.9. Air has a velocity of 1000 km/h at a pressure of 9.81 kN/m2 in vacuum and a temperature of 47°C. Compute its stagnation properties and the local Mach number. Take atmospheric pressure = 98.1 kN/m2, R = 287 J/kg K and γ = 1.4. What would be the compressibility correction factor for a pitot-static tube to measure the velocity at a Mach number of 0.8. Sol. Velocity of air, Temperature of air, Atmospheric pressure, Pressure of air (static),

V0 = 1000 km/h =

T0 = 47 + 273 = 320 K patm = 98.1 kN/m2 p0 = 98.1 – 9.81 = 88.29 kN/m2 R = 287 J/kg K, γ = 1.4

Sonic velocity,

C0 =

∴ Mach number,

M0 =

Stagnation pressure, ps : The stagnation pressure is given by,

γRT0 =

1.4 × 287 × 320 = 358.6 m/s

V0 277.78 = = 0.7746 C0 358.6

L γ − 1IJ M OP p = p M1 + FG N H 2 K Q L 1.4 − 1 × 0.7746 OP p = 88.29 M1 + N 2 Q s

or,

1000 × 1000 = 277.78 m/s 60 × 60

0

0

γ 2 γ −1

1.4 2 1.4 − 1

s

= 88.29 (1.12)3.5 = 131.27 kN/m2 (Ans.) Stagnation temperature, Ts :

LM FG γ − 1IJ M OP ...[Eqn. (16.22)] N H 2 K Q L 1.4 − 1 × 0.7746 OP = 358.4 K or 85.4°C (Ans.) = 320 M1 + Q N 2

Ts = T0 1 + or,


Ts

...[Eqn. (16.17)]

0

2

2

875

COMPRESSIBLE FLOW

Stagnation density, ρs : ps 131.27 × 103 = = 1.276 kg/m3 (Ans.) RTs 287 × 358.4 Compressibility factor at M = 0.8 :

ρs =

Compressibility factor

=1+

M0 2 2 − γ + M04 + ... 4 24

=1+

0.82 2 − 1.4 + × 0.84 = 1.1702 4 24

(Ans.)

Example 16.10. Air at a pressure of 220 kN/m2 and temperature 27°C is moving at a velocity of 200 m/s. Calculate the stagnation pressure if (i) Compressibility is neglected ;

(ii) Compressibility is accounted for.

For air take R = 287 J/kg K, γ = 1.4. Sol. Pressure of air, p0 = 200 kN/m2 Temperature of air, T0 = 27 + 233 = 300 K Velocity of air, V0 = 200 m/s Stagnation pressure, ps : (i) Compressibility is neglected : ps = p0 + where ρ0 =

ρ0V0 2 2

p0 220 × 103 = = 2.555 kg/m3 RT0 287 × 300

2.555 × 2002 × 10–3 (kN/m2) = 271.1 kN/m2. Ans. 2 (ii) Compressibility is accounted for : The stagnation pressure, when compressibility is accounted for, is given by, ∴

ps = 220 +

ps = p0 + Mach number,

Whence,

M0 =

V0 = C0

F GH

I JK

ρ0V02 M2 2 − γ M04 + ... 1+ 0 + 2 4 24 200 = γRT0

200 1.4 × 287 × 300

F GH

...[Eqn. (16.19)]

= 0.576

2.555 × 2002 0.5762 2 − 1.4 −3 × 10 1 + + × 0.5764 ps = 220 + 2 4 24

I JK

ps = 220 + 51.1 (1 + 0.0829 + 0.00275) = 275.47 kN/m2 (Ans.)

or,

Example 16.11. In aircraft flying at an altitude where the pressure was 35 kPa and temperature – 38°C, stagnation pressure measured was 65.4 kPa. Calculate the speed of the aircraft. Take molecular weight of air as 28. (UPSC, 1998) Sol. Pressure of air, p0 = 35 kPa = 35 × 103 N/m2 Temperature of air, T0 = – 38 + 273 = 235 K Stagnation pressure, ps = 65.4 kPa = 65.4 × 103 N/m2


876


Speed of the aircraft, Va : p0V = mRT0 = m × where

FG R IJ T H MK 0

0

or ρ0 =

m p0 M = V R0T0

R = characteristic gas constant, R0 = universal gas constant = 8314 Nm/mole K. M = molecular weight for air = 28, and ρ0 = density of air. Substituting the values, we get ρ0 =

(35 × 103 ) × 28 = 0.5 kg/m3 8314 × 235

Now, using the relation : ps = p0 + Va =

or,

ρ0Va2 2

...[Eqn. (16.20)]

2( ps − p) = ρ0

2(65.4 × 103 − 35 × 103 ) = 348.7 m/s (Ans.) 0.5

16.7. AREA-VELOCITY RELATIONSHIP AND EFFECT OF VARIATION OF AREA FOR SUBSONIC, SONIC AND SUPERSONIC FLOWS For an incompressible flow the continuity equation may be expressed as : AV = constant, which when differentiated gives

dA dV =− A V But in case of compressible flow, the continuity equation is given by, ρAV = constant, which can be differentiated to give ρd(AV) + AVdρ = 0 or ρ(AdV + VdA) + AVdρ = 0 ρAdV + ρVdA + AVdρ = 0 Dividing both sides by ρAV, we get AdV + VdA = 0 or

or,

dV dA dρ + + =0 V A ρ or,

...(16.23)

...(16.24)

dρ dA dV =− – ρ A V The Euler’s equation for compressible fluid is given by,

...[16.24 (a)]

dp + VdV + gdz = 0 ρ Neglecting the z terms the above equation reduces to,

dp + VdV = 0 ρ

This equation can also be expressed as :

dρ dp × + VdV = 0 or dρ ρ dp = C2 dρ

But ∴


C2 ×

dρ + VdV = 0 ρ

or C2

dp dρ × + VdV = 0 dρ ρ

dρ = – VdV or ρ

...[Eqn. (16.10)]

dρ VdV =– ρ C2

877

COMPRESSIBLE FLOW


dρ in eqn. (16.24), we get ρ

VdV dV dA + – =0 V A C2 VdV dA dV dV = – = A V V C2

or,

FV GH C

dA dV = (M2 – 1) A V This important equation is due to Hugoniot. ∴

2 2

I JK

−1

FG3 H

M=

V C

IJ K

...(16.25)

FG dA IJ for the flow of incompressible and comH AK F dA IJ and FG dV IJ are respectively fractional variations in pressible fluids respectively. The ratios GH HVK AK Eqns. (16.23) and (16.25) give variation of

the values of area and flow velocity in the flow passage. Further, in order to study the variation of pressure with the change in flow area, an expression similar to eqn. (16.25), as given below, can be obtained. dp = ρV2

F 1 I dA GH 1 − M JK A

...(16.26)

2

From eqns. (16.25) and (16.26), it is possible to formulate the following conclusions of practical significance. (i) For subsonic flow (M < 1) :

dV dA >0; < 0 ; dp < 0 (convergent nozzle) V A dV dA <0; > 0 ; dp > 0 (divergent diffuser) V A

V1

V2 > V1 p2 < p1 r2 < r1 T2 < T1

V2

(a) Convergent nozzle.

V1

V2

V2 < V1 p2 > p1 r2 > r1 T2 > T1

(b) Divergent diffuser.

Fig. 16.5. Subsonic flow (M < 1).

(ii) For supersonic flow (M > 1) :

dV >0; V dV <0; V


dA > 0 ; dp < 0 (divergent nozzle) A dA < 0 ; dp > 0 (convergent diffuser) A

878


V1

V2 > V1 p2 < p1 r2 < r1 T2 < T1

V2

V1

(a) Divergent nozzle.

V2 < V1 p2 > p1 r2 > r1 T2 > T1

V2

(b) Convergent nozzle.

Fig. 16.6. Supersonic flow (M > 1).

(iii) For sonic flow (M = 1) :

Throat A2 = A1

dA = 0 (straight flow passage A

since dA must be zero) and dp = (zero/zero) i.e., indeterminate, but when evaluated, the change of pressure p = 0, since dA = 0 and the flow is frictionless.

Fig. 16.7. Sonic flow (M = 1).

16.8. FLOW OF COMPRESSIBLE FLUID THROUGH A CONVERGENT NOZZLE Fig. 16.8 shows a large tank/vessel fitted with a short convergent nozzle and containing a compressible fluid. Consider two points 1 and 2 inside the tank and exit of the nozzle respectively.

Large tank

V1 = 0

p1

Convergent nozzle

Let p1 = pressure of fluid at the point 1, V1 = velocity of fluid in the tank (= 0),

r1

T1 = temperature of fluid at point 1,

1

2 V2

ρ1 = density of fluid at point 1, and p2, V2, T2 and ρ2 are corresponding values of pressure, velocity, temperature and density at point 2.

T1

Assuming the flow to take place adiabatically, then by using Bernoulli’s equation (for adiabatic flow), we have

Fig. 16.8. Flow of fluid through a convergent nozzle.

FG γ IJ p + V H γ − 1K ρ g 2 g 1

2 1

1

+ z1

p2, r2, T2

F γ IJ p + V = G H γ − 1K ρ g 2 g

2 2

2

2

+ z2

But z1 = z2 and V1 = 0 ∴

or,

FG γ IJ p + V H γ − 1K ρ g 2 g FG γ IJ L p − p O = V or γ LM p H γ − 1K MN ρ g ρ g PQ 2 g γ − 1 Nρ γ p1 = γ − 1 ρ1 g 1

1


2

2

2

2 2

2

2

2

1

1

−

p2 ρ2

OP Q

V22 2

[Eqn. (16.7)]

879

COMPRESSIBLE FLOW

FG H

IJ K

or

V2 =

2γ p1 p2 − (γ − 1) ρ1 ρ2

or

V2 =

p1 p ρ 2γ 1− 2 − 1 ρ2 p1 (γ − 1) ρ1

p2 p1 p1 γ = ρ γ or p2 ρ1 2

For adiabatic flow :

1

γ

I JK

...(1)

F I GH JK

ρ p or 1 = 1 ρ2 p2

2

ρ1 in eqn. (1), we get ρ2


or

F GH Fρ I =G J Hρ K

LM F GH MM N LM F p I MM1 − GH p JK N

V2 =

2γ p1 p p1 1− 2 × p1 p2 (γ − 1) ρ1

V2 =

2γ p1 (γ − 1) ρ1

γ −1 γ

2

1

I JK

1 γ

OP PP Q

OP PP = Q

1 γ

...(i)

LM F I OP MM GH JK PP N Q

2γ p1 p2 1− p1 (γ − 1) ρ1

1−

1 γ

...(16.27)

The mass rate of flow of the compressible fluid, m = ρ2A2V2, A2 being the area of the nozzle at the exit = ρ2A2

or

m = A2

From eqn. (i), we have

2γ p1 (γ − 1) ρ1

2

γ −1 γ

2

Fp I GH p JK 2

γ −1 γ

2

1

F I GH JK

ρ1 p = ρ1 2 ρ2 = p1 ( p1/ p2 )1/ γ ρ22 = ρ12

∴

LM F I OP MM GH JK PP , [substituting V N Q LM F p I OP × ρ M1 − G J MN H ρ K PPQ

2γ p1 p2 1− p1 (γ − 1) ρ1

2 γ

1 γ

1

Substituting this value in the above equation, we get m = A2

= A2


2γ p1 × ρ12 γ − 1 ρ1

2γ p1ρ1 γ−1

Fp I GH p JK

LMF p I MMGH p JK N 2

1

2 2 γ 1

2/ γ

LM F p I MM1 − GH p JK N

γ −1 γ 2 1

Fp I −G J Hp K 2

1

γ −1 2 + γ γ

OP PP Q

OP PP Q

2

from eqn. (16.27)]

880


m = A2

LMF p I MMGH p JK N

2γ p1ρ1 γ−1

Fp I −G J Hp K

2/ γ

2

2

1

γ +1 γ

1

The mass rate of flow (m) depends on the value of point 1).

OP PP Q

...(16.28)

p2 (for the given values of p1 and ρ1 at p1

p2 for maximum value of mass rate of flow : p1 d (m) = 0 For maximum value of m, we have p d 2 p1 Value of

FG IJ H K

p2 are constant p1

As other quantities except the ratio

LMF p I F p I OP d PP = 0 (m) = MGH p JK − GH p JK p I F MN Q dG J Hp K F I 2Fp I =0 G J – FGH γ +γ 1IJK GH pp JK γ Hp K F p I = FG γ + 1IJ F p I or F p I = γ + 1 F p I GH p JK GH p JK G J H 2 K GH p JK 2 Hp K FG p IJ = FG γ + 1IJ F p I H 2 K GH p JK Hp K FG p IJ = FG γ + 1IJ or FG p IJ = FG γ + 1IJ H 2 K Hp K H 2 K Hp K F p I = FG 2 IJ GH p JK H γ + 1K Fp I = F 2 I GH p JK GH γ + 1JK 2/γ

∴

2

2

2

1

1

γ +1 γ

1

or,

2 −1 γ

2

1

or,

2

1

2 −1 γ

2

2−γ

γ

2

1

γ

2

1

2

1 γ

1

1

2

1− γ

γ

2

1

γ −1

or,

2

2−γ γ

1

2 − γ −1

or,

1 γ

1

1

or,

γ +1 −1 γ

2

γ

2

1

or,

γ γ −1

2

1

...(16.29)

Eqn. (16.29) is the condition for maximum mass flow rate through the nozzle. It may be pointed out that a convergent nozzle is employed when the exit pressure is equal to or more than the critical pressure, and a convergent-divergent nozzle is used when the discharge pressure is less than the critical pressure. For air with γ = 1.4, the critical pressure ratio,

p2 = p1


FG 2 IJ H 1.4 + 1K

1.4 1.4 − 1

= 0.528

...(16.30)

881

COMPRESSIBLE FLOW

Relevant relations for critical density and temperature are :

ρ2 = ρ1

FG 2 IJ H γ + 1K

1 γ −1

...[16.30 (a)]

T2 2 = T1 γ +1 Value of V2 for maximum rate of flow of fluid : Substituting the value of

V2 =

=

or

V2 =

...[16.30 (b)]

p2 from eqn. (16.29) in eqn. (16.27), we get p1

LM F I MM GH JK N FG γ + 1 − 2 IJ = H γ +1 K

2γ p1 2 1− γ − 1 ρ1 γ +1 2γ p1 γ − 1 ρ1

OP 2γ p F 2 I PP = γ − 1 ρ GH 1 − γ + 1JK Q 2γ p F γ − 1I G J γ − 1 ρ H γ + 1K

γ γ −1 × γ −1 γ

1

1

1

1

2γ p1 (= C2) γ + 1 ρ1

...(16.31)

Maximum rate of flow of fluid through nozzle, mmax : Substituting the value of

mmax = A2

= A2 For air, γ = 1.4,

∴

mmax = A2 = A2

or

p2 from eqn. (16.30) in eqn. (16.28), we get p1

FG 2γ IJ p ρ LMFG 2 IJ H γ − 1K MMH γ + 1K N FG 2γ IJ p ρ LMFG 2 IJ H γ − 1K MMH γ + 1K N

2 γ × γ +1 γ

1 1

2 γ −1

1 1

2 × 1.4 p1ρ1 (1.4 − 1)

LMF 2 I MMGH 1.4 + 1JK N

F 2 IJ −G H γ + 1K

F 2 IJ −G H γ + 1K

2 1.4 − 1

γ γ +1 × γ −1 γ

γ +1 γ −1

OP PP Q

F 2 IJ −G H 1.4 + 1K

1.4 + 1 1.4 − 1

OP PP Q

OP PP Q

7 p1ρ1 (0.4018 − 0.3348)

mmax = 0.685 A2

p1ρ1

Variation of mass flow rate of compressible fluid with pressure ratio

...(16.32)

FG p IJ : Hp K 2

1

A passage in which the sonic velocity has been reached and thus in which the mass flow rate is maximum, is often said to be choked or in choking conditions. It is evident from eqn. (16.28) that for a fixed value of inlet pressure the mass flow depends on nozzle exit pressure.


882


Fig. 16.9. depicts the variation of actual and theoretical mass flow rate versus

p2 . Following p1

mmax A2

Actual

m/A2

points are worthnoting : Choked Subsonic (i) The flow rate increases with a decrease in flow flow p2 the pressure ratio and attains the maximum p1 p value of the critical pressure ratio 2 = 0.528 for Theoretical p1 air. (ii) With further decrease in exit pressure below the critical value, the theoretical mass flow rate decreases. This is contrary to the actual re0.528 sults where the mass flow rate remains constant p2 Pressure ratio p after attaining the maximum value. This may be 1 explained as follows : Fig. 16.9. Mass flow rate through At critical pressure ratio, the velocity V2 at a convergent nozzle. the throat is equal to the sonic speed (derived below). For an accelerating flow of a compressible fluid in a convergent nozzle the velocity of flow within the nozzle is subsonic with a maximum velocity equal to the sonic velocity at the throat : Thus once the velocity V2 at the throat has attained the sonic speed at the critical pressure ratio, it remains at the same value for all the values of

FG p IJ less than critical pressure ratio, since the flow Hp K 2

1

in the nozzle is being continuously accelerated with the reduction in the throat pressure below the critical values and hence the velocity cannot reduce. Thus, the mass flow rate for all values of

FG p IJ less than critical pressure ratio remains constant at the maximum value (indicated by the Hp K 2

1

solid horizontal line in Fig. 16.9). This fact has been verified experimentally too. Velocity at outlet of nozzle for maximum flow rate : The velocity at outlet of nozzle for maximum flow rate is given by,

FG 2γ IJ p H γ + 1K ρ F 2 IJ = G H γ + 1K

1

V2 =

Now pressure ratio,

∴

For adiabatic flow :


p2 p1

p1 =

p1

ρ1γ

=

...[Eqn. (16.31)]

1

γ γ −1

p2

FG 2 IJ H γ + 1K p2

ρ2 γ

γ γ −1

or

= p2

FG 2 IJ H γ + 1K

FG IJ H K

ρ1 p1 = ρ2 p2

γ

γ γ −1

ρ1 or = ρ2

Fp I GH p JK 1

2

1 γ

Fp I = G J Hp K 2

1

−

1 γ

883

COMPRESSIBLE FLOW

∴

ρ1 = ρ2

Fp I GH p JK

−

2

1 γ

1

or ρ2

FG 2 IJ H γ + 1K

1 γ × γ −1 γ

or ρ2

FG 2 IJ H γ + 1K

1 1− γ

Substituting the values of p1 and ρ1 in the above eqn. (16.31), we get

V2 =

=

FG 2γ IJ × p FG 2 IJ H γ + 1K H γ + 1K 2

FG 2γ IJ × p H γ + 1K ρ FG 2γ IJ × p H γ + 1K ρ

2

2

or i.e.,

V2 =

γ 1− γ

2

2

R| 1 F 2 I ×S ×G |T ρ H γ + 1JK

1 γ −1

2

F 2 IJ ×G H γ + 1K FG γ + 1IJ = H 2 K

1 γ + 1− γ γ −1

=

U| V| W

FG 2γ IJ × p FG 2 IJ H γ + 1K ρ H γ + 1K

−1

2

2

γp2 = C2 ρ2

V2 = C2 ...(16.33) Hence the velocity at the outlet of nozzle for maximum flow rate equals sonic velocity.

16.9. VARIABLES OF FLOW IN TERMS OF MACH NUMBER In order to obtain relationship involving change in velocity, pressure, temperature and density in terms of the Mach number use is made of the continuity, perfect gas, isentropic flow and energy equations. For continuity equation, we have ρAV = constant Differentiating the above equation, we get ρ[AdV + VdA] + AVdρ = 0 Dividing throughout by ρAV, we have

dV dA dρ =0 + + V A ρ From isentropic flow, we have

p dp dp = constant or = γ γ p p ρ

For perfect gas, we have p = ρRT or From energy equation, we have cpT + Differentiating throughout, we get cpdT + VdV = 0 or

dρ dT dp = + T ρ p V2 + constant 2

FG γR IJ H γ − 1K

dT + VdV = 0

γR dT dV + =0 2 V γ −1 V

or,

Also, sonic velocity,


C=

FG3 H

cp =

IJ K

γR γ −1

...(i)

γRT

∴ γR =

C2 T

884


C2 in eqn. (i), we get T

Substituting the value of γR =

C2 dT dV × + =0 V (γ − 1)T V 2

or,

FG3 H

1 dT dV × + =0 2 T V (γ − 1) M From the Mach number relationship M=

V γRT

(where

M=

V C

IJ K

γRT = C)

dM dV 1 dT = – M V 2 T Substituting the value of

...(16.35)

dT from eqns. (16.34) in eqn. (16.35), we get T

LM N

dV dM dV 1 − × (γ − 1) M 2 = – V M V 2

=

dV 1 dV + × (γ – 1) M2 V 2 V

dM dV = M V

or,

...(16.34)

OP Q

1 LM1 + γ − 1 M OP or dV = N 2 Q V LM1 + FG γ − 1IJ M OP N H 2 K Q 2

2

dM M

...(16.36)

Since the quantity within the bracket is always positive, the trend of variation of velocity and Mach number is similar. For temperature variation, one can write

LM MM F MN GH

2

OP PP PQ

− (γ − 1) M dT dM = ...(16.37) γ −1 T M 2 1+ M 2 Since the right hand side is negative the temperature changes follow an opposite trend to that of Mach number. Similarly for pressure and density, we have

IJ K

OP dM LM − γM MM γ − 1 PP M N1+ 2 M Q LM OP −M dM dρ P = M γ − 1 MM 1 + FG IJ M PP M ρ N H 2 K Q For changes in area, we have O LM − (1 − M ) P dM dA = M A MN 1 + γ 2− 1 M PPQ M dp = p

2

2

...(16.38)

2

and,

2

...(16.39)

2

2


...(16.40)

885

COMPRESSIBLE FLOW

The quantity within the brackets may be positive or negative depending upon the magnitude of Mach number. By integrating eqn. (16.40), we can obtain a relationship between the critical throat area Ac, where Mach number is unity and the area A at any section where M > < 1

LM MN

A 1 2 + (γ − 1) M 2 = Ac γ +1 M

OP PQ

γ +1 2( γ − 1)

...(16.41)

Example 16.12. The pressure leads from Pitot-static tube mounted on an aircraft were connected to a pressure gauge in the cockpit. The dial of the pressure gauge is calibrated to read the aircraft speed in m/s. The calibration is done on the ground by applying a known pressure across the gauge and calculating the equivalent velocity using incompressible Bernoulli’s equation and assuming that the density is 1.224 kg/m3. The gauge having been calibrated in this way the aircraft is flown at 9200 m, where the density is 0.454 kg/m3 and ambient pressure is 30 kN/m2. The gauge indicates a velocity of 152 m/s. What is the true speed of the aircraft ? (UPSC) Sol. Bernoulli’s equation for an incompressible flow is given by,

ρV 2 = constant 2 The stagnation pressure (ps) created at Pitot-static tube, p+

ρ0V0 2 (neglecting compressibility effects) 2 p0 = 30 kN/m2, V0 = 152 m/s, ρ0 = 1.224 kg/m3

ps = p0 + Here

...(i) ...(Given)

2

1.224 × 152 ×10–3 = 44.139 kN/m2 2 Neglecting compressibility effect, the speed of the aircraft when ρ0 = 0.454 kg/m3 is given by [using eqn. (i)], ∴

ps = 30 +

44.139 × 103 = 30 × 103 +

0.454 × V02 2

∴

(44.139 − 30) × 103 × 2 = 62286.34 0.454 V0 = 249.57 m/s

Sonic velocity,

C0 =

Mach number,

M=

V02 =

or

γRT0 =

∴ True speed of aircraft Hence true speed of aircraft


p0 = ρ0

1.4 ×

V0 249.57 = = 0.82 C0 304.16

30 × 103 = 304.16 m/s 0.454

F M I , neglecting the terms containing higher powers GH 4 JK F 0.82 IJ = 1.168 = G1 + H 4K

Compressibility correction factor = 1 + of M0 (from eqn 16.19).

γ

=

0

249.57

2

= 230.9 m/s 1.168 = 230.9 m/s (Ans.)

886


Example 16.13. (a) In case of isentropic flow of a compressible fluid through a variable duct, show that

A Ac

L + 1 γ − 1) M 1 M1 2 ( = M M MN 12 ( γ + 1)

2

OP PP Q

γ +1 2( γ − 1)

where γ is the ratio of specific heats, M is the Mach number at a section whose area is A and Ac is the critical area of flow. (b) A supersonic nozzle is to be designed for air flow with Mach number 3 at the exit section which is 200 mm in diameter. The pressure and temperature of air at the nozzle exit are to be 7.85 kN/m2 and 200 K respectively. Determine the reservoir pressure, temperature and the throat area. Take : γ = 1.4. (U.P.S.C. Exam.) Sol. (a) Please Ref. to Art. 16.9. (b) Mach number, M=3 Area at the exit section, A = π/4 × 0.22 = 0.0314 m2 Pressure of air at the nozzle, (p)nozzle = 7.85 kN/m2 Temperature of air at the nozzle, (T)nozzle = 200 K Reservoir pressure, (p)res. :

F

From eqn. (16.17),

or,

(p)res. = (p)nozzle

(p)res. = 7.85 Reservoir temperature, (T)res. : From eqn. (16.22),

or,

Throat area (critical), Ac :

or, or,

IJ K

2

γ γ −1

2

1.4 1.4 − 1

IJ K

= 288.35 kN/m2 (Ans.)

LM FG γ − 1IJ M OP N H 2 K Q L F 1.4 − 1IJ × 3 OP = 560 K = 200 M1 + G N H 2 K Q 2

(T)res. = (T)nozzle 1 + (T)res.

From eqn. (16.41),

LM1 + FG γ − 1IJ M OPGH N H 2 K Q F LM1 + FG 1.4 − 1IJ × 3 OPGH N H 2 K Q

2

LM OP MN PQ LM 2 + (14. − 1) 3 OP MN 1.4 + 1 PQ

A 1 2 + (γ − 1) M 2 = Ac M γ +1 0.0314 1 = Ac 3 Ac =

(Ans.)

γ +1 2( γ − 1)

1.4 + 1 2 2 (1.4 − 1)

or

0.0314 1 = (2.333)3 = 4.23 Ac 3

0.0314 = 0.00742 m2 (Ans.) 4.23

16.10. FLOW THROUGH LAVAL NOZZLE (CONVERGENT-DIVERGENT NOZZLE) Laval nozzle is a convergent-divergent nozzle (named after de Laval, the swedish scientist who invented it) in which subsonic flow prevails in the converging section, critical or transonic conditions in the throat and supersonic flow in the diverging section.


887

COMPRESSIBLE FLOW

Let p2 (= pc) = pressure in the throat when the flow is sonic for given pressure p1. — When the pressure in the receiver, p3 = p1, there will be no flow through the nozzle, this is shown by line a in Fig. 16.10 (b).

Throat Inlet (p1) 1

Flow

Exit Receiver

2 3 (a) a b c d

p1

pc

e

p3

f

Critical pc = 0.528 p1

Shock wave fronts

Inlet

Throat

j Exit

(b) Fig. 16.10. (a) Laval nozzle (convergent-divergent nozzle) ; (b) Pressure distribution through a convergent-divergent nozzle with flow of compressible fluid.

— When the receiver pressure is reduced, flow will occur through the nozzle. As long as the value of p3 is such that throat pressure p2 is greater than the critical pressure 0.528 p1, the flow in the converging and diverging sections will be subsonic. This condition is shown by line ‘b’. — With further reduction in p3, a stage is reached when p2 is equal to critical pressure pc = 0.528 p1, at this line M = 1 in the throat. This condition is shown by line ‘c’. Flow is subsonic on the upstream as well the downstream of the throat. The flow is also isentropic. — If p3 is further reduced, it does not effect the flow in convergent section. The flow in throat is sonic, downstream it is supersonic. Somewhere in the diverging section a shock wave occurs and flow changes to subsonic (curve d). The flow across the shock is non-isentropic. Downstream of the shock wave the flow is subsonic and decelerates. — If the value of p3 is further reduced, the shock wave forms somewhat downstream (curve e). — For p3 equal to pj, the shock wave will occur just at the exit of divergent section. — If the value of p3 lies before pf and pj oblique waves are formed at the exit. Example 16.14. A large tank contains air at 284 kN/m2 gauge pressure and 24°C temperature. The air flows from the tank to the atmosphere through a convergent nozzle. If the diameter at the outlet of the nozzle is 20 mm, find the maximum flow rate of air. Take : R = 287 J/kg K, γ = 1.4 and atmospheric pressure = 100 kN/m2. (Punjab University)


888


p1 = 284 kN/m2 (gauge) = 284 + 100 = 384 kN/m2 (absolute) Temperature in the tank, T1 = 24 + 273 = 297 K Diameter at the outlet of the nozzle, D = 20 mm = 0.02 m Sol. Pressure in the tank,

π × 0.022 = 0.0003141 m2 4 R = 287 J/kg K, γ = 1.4 (Two points are considered. Point 1 lies inside the tank and point 2 lies at the exit of the nozzle). Maximum flow rate, mmax : ∴

Area, A2 =

Equation of state is given by p = ρRT or ρ =

p RT

p1 384 × 103 = = 4.5 kg/m3 RT1 287 × 297 The fluid parameters in the tank correspond to the stagnation values, and maximum flow rate of air is given by, ∴

ρ1 =

mmax = 0.685 A2

...[Eqn. (16.32)]

p1ρ1

384 × 103 × 4.5 = 0.283 kg/s

= 0.685 × 0.0003141 Hence maximum flow rate of air = 0.283 kg/s (Ans.)

Example 16.15. A large vessel, fitted with a nozzle, contains air at a pressure of 2500 kN/ m2 (abs.) and at a temperature of 20°C. If the pressure at the outlet of the nozzle is 1750 kN/m2, find the velocity of air flowing at the outlet of the nozzle. Take : R = 287 J/kg K and γ = 1.4. Sol. Pressure inside the vessel, p1 = 2500 kN/m2 (abs.) Temperature inside the vessel, T1 = 20 + 273 = 293 K Pressure at the outlet of the nozzle, p2 = 1750 kN/m2 (abs.) R = 287 J/kg K, γ = 1.4 Velocity of air, V2 : V2 =

where,

ρ1 = =

FG 2γ IJ p H γ − 1K ρ

1

1

FG H

LM F p I MM1 − GH p JK N 2

γ −1 γ

1

OP PP Q

...[Eqn. (16.27)]

p1 p From equation of state : = RT RT1 ρ

IJ K

2500 × 103 = 29.73 kg/m3 287 × 293

Substituting the values in the above equation, we get V2 =


FG 2 × 1.4 IJ × 2500 × 10 H 1.4 − 1K 29.73

3

LM F 1750 I MM1 − GH 2500 JK N

1.4 − 1 1.4

OP PP Q

889

COMPRESSIBLE FLOW

=

7 × 84090 (1 − 0.903) = 238.9 m/s

V2 = 238.9 m/s (Ans.)

i.e.,

Example 16.16. A tank fitted with a convergent nozzle contains air at a temperature of 20°C. The diameter at the outlet of the nozzle is 25 mm. Assuming adiabatic flow, find the mass rate of flow of air through the nozzle to the atmosphere when the pressure in the tank is : (i) 140 kN/m2 (abs.), (ii) 300 kN/m2 Take for air : R = 287 J/kg K and γ = 1.4, Barometric pressure = 100 kN/m2. Sol. Temperature of air in the tank, T1 = 20 + 273 = 293 K Diameter at the outlet of the nozzle, D2 = 25 mm = 0.025 m Area, A2 = π/4 × 0.0252 = 0.0004908 m2 R = 287 J/kg K, γ = 1.4 (i) Mass rate of flow of air when pressure in the tank is 140 kN/m2 (abs.) :

p1 140 × 103 = = 1.665 kg/m3 RT1 287 × 293 p1 = 140 kN/m2 (abs.) p2 = atmospheric pressure = 100 kN/m2

ρ1 = Pressure at the nozzle,

p2 100 = = 0.7143 p1 140 Since the pressure ratio is more than the critical value, flow in the nozzle will be subsonic, hence mass rate of flow of air is given by eqn. 16.28, as

∴ Pressure ratio,

m = A2

= 0.0004908 = 0.0004908 or

LMF p I F p I OP MMGH p JK − GH p JK PP N Q LM FG 2 × 14. IJ × 140 × 10 × 1665 . H 1.4 − 1K MN(0.7143)

2γ p1ρ1 γ −1

2

2 γ

1

2

γ +1 γ

1

3

2 1.4

−

1.4 + 1 (0.7143) 1.4

OP PQ

1631700 (0.7143)1.4285 − (0.7143)1.7142

m = 0.0004908 1631700 (0.6184 − 0.5617) = 0.1493 kg/s (Ans.) (ii) Mass rate of flow of air when pressure in the tank is 300 kN/m2 (abs.) : p1 = 300 kN/m2 (abs.) p2 = pressure at the nozzle = atmospheric pressure = 100 kN/m2

p2 100 = = 0.33. p1 300 The pressure ratio being less than the critical ratio 0.528, the flow in the nozzle will be sonic, the flow rate is maximum and is given by eqn. (16.32), as ∴ Pressure ratio,

mmax = 0.685 A2 p1ρ1 where, ∴


ρ1 =

p1 300 × 103 = = 3.567 kg/m3 287 × 293 RT1

mmax = 0.685 × 0.0004908

300 × 103 × 3.567 = 0.3477 kg/s (Ans.)

890


Example 16.17. At some section in the convergent-divergent nozzle, in which air is flowing, pressure, velocity, temperature and cross-sectional area are 200 kN/m2, 170 m/s, 200°C and 1000 mm2 respectively. If the flow conditions are isentropic, determine : (i) Stagnation temperature and stagnation pressure, (ii) Sonic velocity and Mach number at this section, (iii) Velocity, Mach number and flow area at outlet section where pressure is 110 kN/m2, (iv) Pressure, temperature, velocity and flow area at throat of the nozzle. Take for air : R = 287 J/kg K, cp = 1.0 kJ/kg K and γ = 1.4. Sol. Let subscripts 1, 2 and t refers to the conditions at given section, outlet section and throat section of the nozzle respectively. Pressure in the nozzle, p1 = 200 kN/m2 Velocity of air, V1 = 170 m/s Temperature, T1 = 200 + 273 = 473 K Cross-sectional area, A1 = 1000 mm2 = 1000 × 10–6 = 0.001 m2 For air : R = 287 J/kg K, cp = 1.0 kJ/kg K, γ = 1.4 (i) Stagnation temperature (Ts) and stagnation pressure (ps) :

V12 2 × cp

Ts = T1 +

Stagnation temperature,

= 473 +

ps = p1

Also,

FT I GH T JK s

170 2 = 487.45 K (or 214.45°C) (Ans.) 2 × (1.0 × 1000)

γ γ −1

1

F 487.45 IJ =G H 473 K

1.4 1.4 − 1

= 1.111

∴ Stagnation pressure, ps = 200 × 1.111 = 222.2 kN/m2 (Ans.) (ii) Sonic velocity and Mach number at this section : C1 =

Sonic velocity,

γRT1 =

1.4 × 287 × 473 = 435.9 m/s (Ans.)

V1 170 = = 0.39 (Ans.) C1 435.9 (iii) Velocity, Mach number and flow area at outlet section where pressure is 110 kN/m2 : ...(Given) Pressure at outlet section, p2 = 110 kN/m2 Mach number,

M1 =

LM FG IJ OP From eqn (16.17), N H K Q 222.2 L F 1.4 − 1IJ M OP = M1 + G 110 N H 2 K Q F 222.2 IJ = 1.222 (1 + 0.2 M ) = G H 110 K . − 1I F 1222 M = G H 0.2 JK = 1.05 ps γ −1 M22 = 1+ p1 2

0

or,

2 2

γ γ −1

1.4 2 1.4 − 1

= (1 + 0.2 M02)3.5

1 3.5

1/ 2

or,


2

(Ans.)

891

COMPRESSIBLE FLOW

T2 = Ts

Also, or,

Fp I GH p JK 2

γ −1 γ

s

F 110 IJ = G H 222.2 K

1.4 − 1 1.4

= 0.818

T2 = 0.818 × 487.45 = 398.7 K Sonic velocity at outlet section, C2 =

γRT2 =

1.4 × 287 × 398.7 = 400.25 m/s

∴ Velocity at outlet section, V2 = M2 × C2 = 1.05 × 400.25 = 420.26 m/s. Ans. Now, mass flow at the given section = mass flow at outlet section (exit) ......continuity equation ρ1A1V1 = ρ2A2V2 or

i.e.,

p2 p1 A1V1 = A2V2 RT RT1 2

∴ Flow area at the outlet section, A2 =

p1 A1V1T2 200 × 0.001 × 170 × 398.7 = = 6.199 × 10–4 m2 T1 p2V2 473 × 110 × 420.26

Hence, A2 = 6.199 × 10–4 m2 or 619.9 mm2. Ans. (iv) Pressure (pt), temperature (Tt), velocity (Vt), and flow area (At) at throat of the nozzle : At throat, critical conditions prevail, i.e. the flow velocity becomes equal to the sonic velocity and Mach number attains a unit value. From eqn. (16.22),

2

or,

t

t

Hence Also, or,

LM FG IJ OP N H K Q 487.45 L F 1.4 − 1IJ × 1 OP = 1.2 or T = 406.2 K = M1 + GH T 2 K N Q Ts γ −1 Mt 2 = 1+ Tt 2

Tt = 406.2 K (or 133.2°C). Ans.

F I GH JK

pt Tt = Ts Ts

γ γ −1

pt or = 222.2

FG 406.2 IJ H 487.45 K

1.4 1.4 − 1

= 0.528

pt = 222.2 × 0.528 = 117.32 kN/m2. Ans. Sonic velocity (corresponding to throat conditions), Ct =

γRTt =

1.4 × 287 × 406.2 = 404 m/s

∴ Flow velocity, Vt = Mt × Ct = 1 × 404 = 404 m/s By continuity equation, we have : ρ1A1V1 = ρt AtVt

pt p1 A1V1 = AtVt RT RT1 t

or,

∴ Flow area at throat, At = Hence,


p1 A1V1Tt 200 × 0.001 × 170 × 406.2 = = 6.16 × 10–4 m2 T1 pt Vt 473 × 117.32 × 404

At = 6.16 × 10–4 m2 or 616 mm2 (Ans.)

892


16.11. SHOCK WAVES Whenever a supersonic flow (compressible) abruptly changes to subsonic flow, a shock wave (analogous to hydraulic jump in an open channel) is produced, resulting in a sudden rise in pressure, density, temperature and entropy. This occurs due to pressure differentials and when the Mach number of the approaching flow M1 > 1. A shock wave is a pressure wave of finite thickness, of the order of 10–2 to 10–4 mm in the atmospheric pressure. A shock wave takes place in the diverging section of a nozzle, in a diffuser, throat of a supersonic wind tunnel, in front of sharp nosed bodies. Shock waves are of two types : 1. Normal shocks which are almost perpendicular to the flow. 2. Oblique shocks which are inclined to the flow direction. 16.11.1. Normal Shock Wave Consider a duct having a compressible sonic flow (see Fig. 16.11). Let p1, ρ1, T1, and V1 be the pressure, density, temperature and velocity of the flow (M1 > 1) and p2, ρ2, T2 and V2 the corresponding values of pressure, density, temperature and velocity after a shock wave takes place (M2 < 1). Normal shock wave

p1, r1, T1

p2, r2, T2

V1 > C1

V2 < C2 M>1

M<1

Fig. 16.11. Normal shock wave.

For analysing a normal shock wave, use will be made of the continuity, momentum and energy equations. Assume unit area cross-section, A1 = A2 = 1. ...(i) Continuity equation : m = ρ1V1 = ρ2V2 Momentum equation : ΣFx = p1A1 – p2A2 = m (V2 – V1) = ρ2A2V22 – ρ1A1V12 for A1 = A2 = 1, the pressure drop across the shock wave, p1 – p2 = ρ2V22 – ρ1V12 ...(ii) p1 + ρ1V12 = p2 + ρ2V22 Consider the flow across the shock wave as adiabatic.


FG H

IJ K

V12 γ p2 V22 + = 2 γ − 1 ρ2 2 1 (z1 = z2, duct being in horizontal position) Energy equation :

or,

1

FG H

+

IJ K

...[Eqn. (16.7)]

γ p2 p1 V 2 − V22 − = 1 γ − 1 ρ2 ρ1 2 Combining continuity and momentum equations [refer eqns. (i) and (ii)], we get p1 +


(ρ1V1)2 (ρ V )2 = p2 + 2 2 ρ1 ρ2

...(iii)

...(16.42)

893

COMPRESSIBLE FLOW

This equation is ‘known as Rankine Line Equation. Now combining continuity and energy equations [refer eqns. (i) and (iii)], we get

γ γ −1

F p I + (ρ V ) GH ρ JK 2ρ 1 1 2 1

1

1

2

γ γ −1

=

F p I + (ρ V ) GH ρ JK 2ρ 2

2

2 2 2 2

2

...(16.43)

This equation is called Fanno Line Equation. Further combining eqns. (i), (ii) and (iii) and solving for

p2 = p1

FG γ + 1IJ ρ − 1 H γ − 1K ρ FG γ + 1IJ − ρ H γ − 1K ρ

p2 , we get p1

2

1

2

1

Solving for density ratio

...(16.44)

ρ2 , the same equations yield ρ1 V ρ2 = 1 = V2 ρ1

FG γ + 1IJ p H γ − 1K p FG γ + 1IJ + p H γ − 1K p

1+

2 1

...(16.45)

2

1

The eqns. (16.44) and (16.45) are called Ranking-Hugoniot equations. One can also express

T2 p2 V2 ρ2 , , and in terms of Mach number as follows : T1 p1 V1 ρ1 2 γ M12 − (γ − 1) p2 = (γ + 1) p1

...(16.46)

V1 (γ + 1) M12 ρ = 2 = V2 (γ − 1) M12 + 2 ρ1

...(16.47)

T2 [(γ − 1) M12 + 2] [2γ M12 − (γ − 1)] = T1 (γ + 1)2 M12

...(16.48)

By algebraic manipulation the following equation between M1 and M2 can be obtained. M22 =

(γ − 1) M12 + 2 2γM12 − (γ − 1)

...(16.49)

Example 16.18. For a normal shock wave in air Mach number is 2. If the atmospheric pressure and air density are 26.5 kN/m2 and 0.413 kg/m3 respectively, determine the flow conditions before and after the shock wave. Take γ = 1.4. Sol. Let subscripts 1 and 2 represent the flow conditions before and after the shock wave. Mach number, Atmospheric pressure, Air density,


M1 = 2

p1 = 26.5 kN/m2

ρ1 = 0.413 kg/m3

894


Mach number, M2 : M22 = = ∴ Pressure, p2 :

(γ − 1) M12 + 2 2γM12 − (γ − 1)

...[Eqn. (16.49)]

(1.4 − 1) × 2 2 + 2 3.6 = = 0.333 2 × 1.4 × 22 − (1.4 − 1) 11.2 − 0.4

M2 = 0.577 (Ans.)

p2 2 γM12 − (γ − 1) = p1 (γ + 1) =

...[Eqn. (16.46)]

2 × 1.4 × 2 2 − (1.4 − 1) 11.2 − 0.4 = = 4.5 (1.4 + 1) 2.4

p2 = 26.5 × 4.5 = 119.25 kN/m2 (Ans.)

∴ Density, ρ2 :

ρ2 (γ + 1) M12 = ρ1 (γ − 1) M12 + 2 =

(1.4 + 1)22 9.6 = 2.667 = 2 +2 1 . 6 (1.4 − 1)2 + 2

ρ2 = 0.413 × 2.667 = 1.101 kg/m3 (Ans.)

∴ Temperature, T1 : Since p1 = ρ1 RT1,

...[Eqn. (16.47)]

∴ T1 =

p1 26.5 × 103 = = 223.6 K or – 49.4°C (Ans.) ρ1 R 0.413 × 287

Temperature, T2 :

T2 [(γ − 1) M12 + 2] [2γM12 − (γ − 1)] [(1.4 − 1)22 + 2] [2 × 1.4 × 22 − (1.4 − 1)] = = T1 (γ + 1)2 M12 (1.4 + 1)2 22 (16 . + 2)(11.2 − 0.4) = 1.6875 23.04 ∴ T2 = 223.6 × 1.6875 = 377.3 K or 104.3°C (Ans.) Velocity, V1 : =

C1 = Since

γRT1 =

1.4 × 287 × 223.6 = 299.7 m/s

V1 = M1 = 2 ∴ V1 = 299.7 × 2 = 599.4 m/s (Ans.) C1

Velocity, V2 : C2 = Since


γRT2 =

1.4 × 287 × 377.3 = 389.35 m/s

V2 = M2 = 0.577 ∴ V2 = 389.35 × 0.577 = 224.6 m/s (Ans.) C2

895

COMPRESSIBLE FLOW

16.11.2. Oblique Shock Wave As shown in Fig. 16.12, when a supersonic flow undergoes a sudden turn through a small angle α (positive), an oblique wave is established at the corner. In comparison with normal shock waves, the oblique shock waves, being weaker, are preferred.

Shock Supersonic flow

The shock waves should be avoided or made as weak as possible, since during a shock wave conversion of mechanical energy into heat energy takes place.

a

Sharp corner

Fig. 16.12. Oblique shock wave. 16.11.3. Shock Strength The strength of shock is defined as the ratio of pressure rise across the shock to the upstream pressure.

i.e.

Strength of shock

=

p2 − p1 p2 = –1 p1 p1

=

2 γM12 − (γ − 1) 2 γM12 − (γ − 1) − (γ + 1) − 1= γ+1 γ+1

=

2 γM12 − γ + 1 − γ − 1 2 γM12 − 2 γ 2γ = = (M12 – 1) γ+1 γ+1 γ +1

Hence, strength of shock =

2γ (M12 – 1) γ +1

...(16.50)

Example 16.19. In a duct in which air is flowing, a normal shock wave occurs at a Mach number of 1.5. The static pressure and temperature upstream of the shock wave are 170 kN/m2 and 23°C respectively. Determine : (i) Pressure, temperature and Mach number downstream of the shock, and (ii) Strength of shock. Take γ = 1.4. Sol. Let subscripts 1 and 2 represent flow conditions upstream and downstream of the shock wave respectively. Mach number, M1 = 1.5 Upstream pressure, p1 = 170 kN/m2 Upstream temperature, T1 = 23 + 273 = 296 K γ = 1.4 (i) Pressure, temperature and Mach number downstream of the shock :

p2 2 γM12 − (γ − 1) = p1 γ +1 = ∴

2 × 1.4 × 1.52 − (1.4 − 1) 6.3 − 0.4 = = 2.458 1.4 + 1 2.4

p2 = 170 × 2.458 = 417.86 kN/m2. Ans.

T2 [(γ − 1) M12 + 2] [2γM12 − (γ − 1)] = T1 (γ + 1)2 M12


...[Eqn. (16.46)]

...[Eqn. (16.48)]

896


= ∴

[(1.4 − 1) × 1.52 + 2][2 × 1.4 × 1.52 − (1.4 − 1)] 2

(1.4 + 1) × 1.5

2

=

2.9 × 5.9 = 1.32 12.96

T2 = 296 × 1.32 = 390.72 K or 117.72°C. Ans. M22 = =

(γ − 1) M12 + 2 2γM12 − (γ − 1)

...[Eqn. (16.49)]

(1.4 − 1) × 1.52 + 2 2 × 1.4 × 1.52 − (1.4 − 1)

=

2.9 = 0.49 5.9

∴ M2 = 0.7. Ans. (ii) Strength of shock : Strength of shock

=

p2 – 1 = 2.458 – 1 = 1.458. Ans. p1

HIGHLIGHTS 1. 2.

A compressible flow is that flow in which the density of the fluid changes during flow. The characteristic equation of state is given by :

p = RT ρ

3.

where p = absolute pressure, N/m2, ρ = density of gas, kg/m3, R = characteristic gas constant, J/kg K, and R = absolute temperature (= t°C + 273). The pressure and density of a gas are related as :

p = constant ρ

For isothermal process :

p

For adiabatic process : 4.

ργ

= constant.

The continuity equation for compressible flow is given as : ρAV = constant

dA dV dρ + + =0 A V ρ 5.

... in differential form.

For compressible fluids Bernoulli’s equation is given as :

p V2 loge p + + z = constant ρg 2g

FG γ IJ p + V 2 H γ − 1K ρg 2 g 6.

+ z = constant

... for isothermal process

... for adiabatic process.

Sonic velocity is given by : C=


dp = dρ

K ρ

... in terms of bulk modulus

897

COMPRESSIBLE FLOW

7.

Mach number, (i) Subsonic flow : (ii) Sonic flow : (iii) Supersonic flow :

C=

p = ρ

C=

γp = γRT ρ

RT

... for isothermal process ... for adiabatic process.

V C M < 1, V < C... disturbance always moves ahead of the projectile M = 1, V = C... disturbance moves along the projectile M > 1, V > C... The projectile always moves ahead of the disturbance.

M=

C 1 = . V M The pressure, temperature and density at a point where velocity is zero are called stagnation pressure (ps), temperature, (Ts) and stagnation density ρs. Their values are given as :

Mach angle is given by : sin α =

8.

γ

ps = po

LM1 + FG γ − 1IJ M02 OP γ − 1 N H 2 K Q 1

ρs = ρo

Ts = To 9.

LM1 + FG γ − 1IJ M02 OP γ − 1 N H 2 K Q LM1 + FG γ − 1IJ M02 OP N H 2 K Q

where p0, ρ0 and To are the pressure, density and temperature at any point O in the flow. Area-velocity relationship for compressible fluid is given as :

(i) Subsonic flow (M < 1) :

(ii) Supersonic flow (M > 1) :

(iii) Sonic flow (M = 1) :

dA dV = (M2 – 1) V A dV dA >0; < O ; dp < 0 V A dV dA <0; > 0 ; dp > 0 V A

(convergent nozzle)

dV dA >0; > 0 ; dp < 0 V A

(divergent nozzle)

dV dA <0; < 0 ; dp > 0 (convergent diffuser) V A dA =0 (straight flow passage since dA must be zero) A

zero i.e. indeterminate, but when evaluated, zero the change of pressure dp = 0, since dA = 0 and the flow is frictionless. Flow of compressible fluid through a convergent nozzle : (i) Velocity through a nozzle or orifice fitted to a large tank :

dp =

10.

(divergent diffuser)

V2 =


γ −1 F 2γ I p1 LMM1 − F p2 I γ OPP GH γ − 1JK ρ1 M GH p1 JK P N Q

898


(ii) The mass rate of flow is given by : m = A2

(iii) Value of

2 γ +1 FG 2γ IJ p1ρ1 LMMFG p2 IJ γ − FG p2 IJ γ OPP H γ − 1K MH p1 K H p1 K P N Q

F p2 I for maximum value of mass rate of flow is given by : GH p1 JK F p2 I = F 2 I γ γ− 1 = 0.528 GH p1 JK GH γ + 1JK

(when γ = 1.4)

(iv) Value of V2 for maximum rate of flow of liquid is given as, V2 =

FG 2γ IJ p1 H γ + 1K ρ 1

(= C2)

(v) Maximum rate of flow of fluid through nozzle,

mmax = A2

2 γ +1 FG 2γ IJ p1ρ1 LMMFG 2γ IJ γ − 1 − FG 2 IJ γ − 1 OPP H γ − 1K MH γ + 1K H γ + 1K P N Q

For air, substituting γ = 1.4, we get mmax = 0.685 A2

11.

p1ρ1

If the pressure ratio is less than 0.528, the mass rate of flow of the fluid is always corresponding to the pressure ratio of 0.528. But if the pressure ratio is more than 0.528, the mass rate of flow of fluid is corresponding to the given pressure ratio. Whenever a supersonic flow (compressible) changes to subsonic flow, a shock wave (analogous to hydraulic jump in an open channel) is produced, resulting in a sudden rise in pressure, density, temperature and entropy. p1 +

γ γ −1

One can also express

(ρ V )2 (ρ1V1)2 = p2 + 2 2 ρ2 ρ1

F p1 I + (ρ1V1)2 = γ F p2 I + (ρ2V2 )2 GH ρ1 JK 2ρ21 γ − 1 GH ρ2 JK 2ρ22 FG γ + 1IJ ρ2 − 1 U| p2 γ − 1K ρ 1 H || = p1 FG γ + 1IJ − ρ2 || H γ − 1K ρ1 |V F γ + 1I p2 | 1+ G V1 ρ2 H γ − 1JK p1 || = = V2 ρ1 FG γ + 1IJ + p2 || H γ − 1K p1 |W

... Fanno line Equation

...Rankinge-Hugoniot Equations

p2 V2 ρ2 T , , and 2 in terms of Mach number as follows : p1 V1 ρ1 T1 2γM12 − (γ − 1) p2 = γ +1 p1


... Ranking Line Equation

...(i)

899

COMPRESSIBLE FLOW

V1 ρ2 (γ + 1) M12 V2 = ρ = (γ − 1) M 2 + 2 1 1

...(ii)

[(γ − 1) M12 + 2] [2 γM12 − (γ − 1)] T2 = (γ + 1) 2 M12 T1

M22 =

Also

(γ − 1) M12 + 2

2γM12 − ( γ − 1)

...(iii)

.

OBJECTIVE TYPE QUESTIONS 1.

2.

3.

4.

Choose the Correct Answer : All real fluids are (a) incompressible (b) compressible to some extent (c) compressible to any extent (d) none of the above. A change in the state of a system at constant volume is called (a) isobaric process (b) isochoric process (c) isothermal process (d) adiabatic process. A process during which no heat is transferred to or from the gas is called an (a) isochoric process (b) isobaric process (c) adiabatic process (d) isothermal process. An adiabatic process is one which follows the relation (a) (c)

5.

6.

7.

8.

9.

10.

p = constant ρ p ρn

= constant (n ≠ γ)

(b)

p ργ

= constant

(d) v = constant.

An isentropic flow is one which is (a) isothermal (b) adiabatic (c) adiabatic and irreversible (d) adiabatic and reversible. Indica upto what Mach number can a fluid flow be considered incompressible ? (a) 0.1 (b) 0.3 (c) 0.8 (d) 1.0. Which of the following is the basic equation of compressible fluid flow ? (a) Continuity equation (b) Momentum equation (c) Energy equation (d) Equation of state (e) All of the above. The velocity of disturbance in case of fluids is ...... the velocity of the disturbance in solids. (a) less than (b) equal to (c) more than (d) none of the above. Sonic velocity (C) for adiabatic process is given as (a) C =

γRT 3

(b) C =

(c) C =

γ 2 RT

(d) C = γRT.

γRT

where γ = ratio of specific heats, R = gas constant, T = temperature. The flow is said to be subsonic when Mach number is (a) equal to unity (b) less than unity (c) greater than unity (d) none of the above.


900 11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.


The region outside the Mach cone is called (a) zone of action (b) zone of silence (c) control volume (d) none of the above. A stagnation point is the point on the immersed body where the magnitude of velocity is (a) small (b) large (c) zero (d) none of the above. A convergent-divergent nozzle is used when the discharge pressure is (a) less than the critical pressure (b) equal to the critical pressure (c) more than the critical pressure (d) none of the above. At critical pressure ratio, the velocity at the throat of a nozzle is (a) equal to the sonic speed (b) less than the sonic speed (c) more than the sonic speed (d) none of the above. Laval nozzle is a (a) convergent nozzle (b) divergent nozzle (c) convergent-divergent nozzle (d) any of the above. A shock wave is produced when (a) a subsonic flow changes to sonic flow (b) a sonic flow changes to supersonic flow (c) a supersonic flow changes to subsonic flow (d) none of the above. The sonic velocity in a fluid medium is directly proportional to (a) Mach number (b) pressure (c) square root of temperature (d) none of the above. The stagnation pressure (ps) and temperature (Ts) are (a) less than their ambient counterparts (b) more than their ambient counterparts (c) the same as in ambient flow (d) none of the above. Across a normal shock (a) the entropy remains constant (b) the pressure and temperature rise (c) the velocity and pressure decrease (d) the density and temperature decrease. A normal shock wave (a) is reversible (b) is irreversible (c) is isentropic (d) occurs when approaching flow is supersonic. The sonic speed in an ideal gas varies (a) inversely as bulk modulus (b) directly as the absolute pressure (c) inversely as the absolute temperature (d) none of the above. In a supersonic flow, a diffuser is a conduit having (a) gradually decreasing area (b) converging-diverging passage (c) constant area throughout its length (d) none of the above. Choking of a nozzle fitted to a pressure tank containing gas implies (a) sonic velocity at the throat (b) increase of the mass flow rate (c) obstruction of flow (d) all of the above. A shock wave which occurs in a supersonic flow represents a region in which (a) a zone of silence exists (b) there is no change in pressure, temperature and density (c) there is sudden change in pressure, temperature and density (d) velocity is zero. Which of the following statements regarding a normal shock is correct ? (a) It occurs when an abrupt change takes place from supersonic into subsonic flow condition (b) It causes a disruption and reversal of flow pattern (c) It may occur in sonic or supersonic flow (d) None of the above.


901

COMPRESSIBLE FLOW

26.

For compressible fluid flow the area-velocity relationship is (a)

dV dA = (1 – M2) V A

(b)

dV dA = (C2 – 1) V A

dV dV dA dA = (M2 – 1) (d) = (1 – V2). V V A A The sonic velocity is largest in which of the following ? (a) Water (b) Steel (c) Kerosene (d) Air. Which of the following expressions does not represent the speed of sound in a medium ?

(c) 27.

28.

29.

(a)

K ρ

(c)

K

p ρ

(b)

γRT

(d)

dp . dρ

The differential equation for energy in isentropic flow is of the form (a)

dp dV dA + + =0 ρ V A

(c) 2VdV +

(b) VdV +

dp =0 ρ

dp =0 ρ

(d) dp + d (ρV2) = 0.

30.

Which of the following statements is incorrect ? (a) A shock wave occurs in divergent section of a nozzle when the compressible flow changes abruptly from supersonic to subsonic state (b) A plane moving at supersonic state is not heard by the stationary observer on the ground until it passes him because zone of disturbance in Mach cone trails behind the plane (c) A divergent section is added to a convergent nozzle to obtain supersonic velocity at the throat (d) none of the above.

1. 8. 15. 22. 29.

(b) (a) (c) (a) (b)

ANSWERS 2. 9. 16. 23. 30.

(b) (b) (c) (d) (c).

3. 10. 17. 24.

(c) (b) (c) (c)

4. 11. 18. 25.

(b) (b) (b) (a)

5. 12. 19. 26.

(d) (c) (b) (c)

6. 13. 20. 27.

(b) (a) (d) (b)

7. 14. 21. 28.

(e) (a) (d) (c)

THEORETICAL QUESTIONS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Differentiate between compressible and incompressible flows. Give the examples when liquid is treated as a compressible fluid. When is the compressibility of fluid important ? What is the difference between isentropic and adiabatic flows ? What is the relation between pressure and density of a compressible fluid for (a) isothermal process (b) adiabatic process ? Obtain an expression in differential form for continuity equation for one-dimensional compressible flow. Derive an expression for Bernoulli’s equation when the process is adiabatic. How are the disturbances in compressible fluid propagated ? What is sonic velocity ? On what factors does it depend ? What is Mach number ? Why is this parameter so important for the study of flow of compressible fluids ?


902 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.


Prove that velocity of sound wave in a compressible fluid is given by : C = k/ρ , where k and ρ are the bulk modulus and density of fluid respectively. Define the following terms : (i) Subsonic flow (ii) Sonic flow (iii) Supersonic flow, (iv) Mach cone. What is silence zone during the disturbance which propagates when an object moves in still air ? What is stagnation point of an object immersed in fluid ? What is stagnation pressure ? What are static and stagnation temperatures ? Derive an expression for mass flow rate of compressible fluid through an orifice or nozzle fitted to a large tank. What is the condition for maximum rate of flow ? What is the critical pressure ratio for a compressible flow through a nozzle ? On what factors does it depend ? Describe compressible flow through a convergent-divergent nozzle. How and where does the shock wave occur in the nozzle ? What do you mean by compressibility correction factor ? How is a shock wave produced in a compressible fluid ? What do you mean by the term “Shock strength” ?

UNSOLVED EXAMPLES 1.

2.

3.

4.

5.

6. 7.

8.

9.

A 100 mm diameter pipe reduces to 50 mm diameter through a sudden contraction. When it carries air at 20.16°C under isothermal conditions, the absolute pressures observed in the two pipes just before and after the contraction are 400 kN/m2 and 320 kN/m2 respectively. Determine the densities and velocities at the two sections. Take R = 290 J/kg K. [Ans. 4.7 kg/m3 ; 3.76 kg/m3 ; 39.7 m/s ; 198.5 m/s] A gas with a velocity of 300 m/s is flowing through a horizontal pipe at a section where pressure is 60 kN/m2 (abs.) and temperature 40°C. The pipe changes in diameter and at this section the pressure is 90 kN/m2. If the flow of gas is adiabatic find the velocity of gas at this section. Take : R = 287 J/kg K and γ = 1.4. [Ans. 113 m/s] An aeroplane is flying at 21.5 m/s at a low altitude where the velocity of sound is 325 m/s. At a certain point just outside the boundary layer of the wings, the velocity of air relative to the plane is 305 m/s. If the flow is frictionless adiabatic determine the pressure drop on the wing surface near this position. Assume γ = 1.4, pressure of ambient air = 102 kN/m2. [Ans. 28.46 kN/m2] A jet propelled aircraft is flying at 1100 km/h. at sea level. Calculate the Mach number at a point on the aircraft where air temperature is 20°C. Take : R = 287 J/kg K and γ = 1.4. [Ans. 0.89] An aeroplane is flying at an height of 20 km where the temperature is – 40°C. The speed of the plane is corresponding to M = 1.8. Find the speed of the plane. Take : R = 287 J/kg K, γ = 1.4. [Ans. 1982.6 km/h] Find the velocity of bullet fired in standard air if its Mach angle is 30°. [Ans. 680.4 m/s] Air, thermodynamic state of which is given by pressure p = 230 kN/m2 and temperature = 300 K is moving at a velocity V = 250 m/s. Calculate the stagnation pressure if (i) compressibility is neglected and (ii) compressibility is accounted for. Take γ = 1.4 and R = 287 J/kg K. [Ans. 313 kN/m2, 323 kN/m2] 2 A large vessel, fitted with a nozzle, contains air at a pressure of 2943 kN/m (abs.) and at a temperature of 20°C. If the pressure at the outlet of the nozzle is 2060 kN/m2 (abs.) find the velocity of air flowing at the outlet of the nozzle. Take : R = 287 J/kg K and γ = 1.4 [Ans. 239.2 m/s] Nitrogen gas (γ = 1.4) is released through a 10 mm orifice on the side of a large tank in which the gas is at a pressure of 10 bar and temperature 20°C. Determine the mass flow rate if (i) the gas escapes to atmosphere (1 bar) ; (ii) the gas is released to another tank at (a) 5 bar, (b) 6 bar. [Ans. (i) 0.183 kg/s ; (ii) 0.183 kg/s ; 0.167 kg/s]


COMPRESSIBLE FLOW

10.

11.

12.

13.

14.

903

Air is released from one tank to another through a convergent-divergent nozzle at the rate of 12 N/s. The supply tank is at a pressure of 400 kN/m2 and temperature 110°C, and the pressure in the receiving tank is 100 kN/m2. Determine : (i) The pressure, temperature, and Mach number in the constriction, (ii) The required diameter of constriction, (iii) The diameter of the nozzle at the exit for full expansion, and the Mach number. [Ans. (i) 210 kN/m2 ; 319 K, (ii) 43.5 mm ; (iii) 48 mm ; 1.56] Oxygen flows in a conduit at an absolute pressure of 170 kN/m2. If the absolute pressure and temperature at the nose of small object in the stream are 200 kN/m2 and 70.16°C respectively, determine the velocity in the conduit. Take γ = 1.4 and R = 281.43 J/kg K. [Ans. 175.3 m/s] Air at a velocity of 1400 km/h has a pressure of 10 kN/m2 vacuum and temperature of 50.16°C. Calculate local Mach number and stagnation pressure, density and temperature. Take γ = 1.4, R = 281.43 J/kg K and [Ans. 1.089 ; 192.358 kN/m2 ; 1.708 kg/m3 ; 399.8 K] barometric pressure = 101.325 kN/m2. A normal shock wave occurs in a diverging section when air is flowing at a velocity of 420 m/s, pressure 100 kN/m2, and temperature 10°C. Determine : (i) The Mach number before and after the shock, (ii) The pressure rise, and (iii) The velocity and temperature after the shock. [Ans. (i) 1.25 ; 0.91 ; (ii) 66 kN/m2, (iii) 292 m/s ; 54°C] A normal shock wave occurs in air flowing at a Mach number of 1.5. The static pressure and temperature of the air upstream of the shock wave are 100 kN/m2 and 300 K. Determine the Mach number, pressure and temperature downstream of the shockwave. Also estimate the shock strength. [Ans. 0.7 ; 246 kN/m2 ; 396.17 K ; 1.46]


Fluid Mechanics and Hydraulic Machines by R K Bansal

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