15. Compressible Flow

In thermodynamics, the “simple compressible” substance is referred to as a substance, in which the change of the specific volume is important. So far, the change of the specific volume is caused by a process associated with heat and work mixing accelerating or decelerating flow at

low speed However, the change of the specific volume due to the flow itself has not yet mentioned. In fluid mechanics, we always use the term “incompressible flow” to represent a low-speed flow. If the flow is accelerated to a certain limit, the change of the specific volume becomes significant.

15.1 Stagnation Properties

The isentropic stagnation state is the state of a flowing fluid if it undergoes a reversible adiabatic decelerating to zero velocity.

2

hh2

o

V

The subscript “o” denotes for the stagnation state. The stagnation properties are properties at the stagnation state: Stagnation pressure (Po) [Pa] Stagnation temperature (To) [K] Stagnation density (o) [kg/m3]

Note: The difference between the actual

and isentropic stagnation states is due to irreversibilities.

For most nozzles and diffusers, which are adiabatic, ho2=ho1 (and To2=To1 for ideal gases) due to the first law.

15.2 The Momentum Equation for a Control Volume

The general form of the continuity and momentum equation can be expressed:

Continuity equation

0 Ad dV t

AV

V

Momentum equation

BS

AV

F F Ad dV t

V V

Note: The continuity equation is the same

equation as the conservation of mass.

The conservation of momentum is indeed the Newton second law.

15.3 Forces Acting on a Control Surface

The summations of all forces in both x and y directions are

xxeoexioix RAPPAPPF

yyeoeyioiy RAPPAPPF

All of these forces will present the surface force in the RHS of the momentum equation.

15.4 Adiabatic, one-dimensional, steady-state flow of an incompressible fluid through a nozzle

Continuity equation:

eeiiie AAmm VV

The first law:

0g)ZZ(2

hh ie

2

i

2

eie

VV

From TdS = dh v dP = 0

)PP(vhh ieie

Thus, we have

0g)ZZ(2

)PP(v ie

2

i

2

eie

VV

This is the well-known Bernoulli equation derived from a combine first and second law.

15.5 Velocity of Sound in an Ideal Gas

When a pressure disturbance occurs in a compressible fluid, it travels at the sonic velocity or the velocity of sound.

The first law:

2

)d(c)dhh(

2

ch

22 V

0dcdh V

Continuity:

)dc(A)d(Ac V 0dcd V

Also, the relation among properties:

0dP

dhTdS

(isentropic flow)

Eliminating dh using the first law gives

0cddP

V

The above equation can be derived using the momentum equation:

)cdc(A

)cdc(mA)dPP(PA

VV

V

0cddP V

Eliminating dV using the continuity equation yields

2c d

dP

Since the isentropic process is assumed, the partial derivative should be used:

2

s

c P

Thus, c is a property as well as other properties. If the equation of state is known, the above expression can be solved. For an ideal gas,

const P/ Pv kk

Differentiate

Pk c

P

Pk

d

dP

dk

P

dP

2

s

Using the ideal-gas law: P = RT

or kRT c 2 kRT c The Mach number (M) [-]: the ratio of the actual velocity to the sonic velocity.

c

M V

When M < 1 the flow is subsonic M 1 the flow is transonic M > 1 the flow is supersonic

15.6 Reversible, Adiabatic, One-Dimensional Flow of an Ideal Gas through a Nozzle

A nozzle or a diffuser usually has a converging and diverging section.

The throat is the location, which has the minimum cross-sectional area.

The first law:

0 d dh VV (1)

The property relation

0dP

dhTdS

(2)

Continuity:

onstantc m A V

0 d

A

dAd

V

V (3)

Combining (1) and (2) gives

VVd dP

dh

0 d dP VV (4)

The above equation can be obtained by using the momentum equation.

Substituting (4) into (3) gives

2

1

)d/dP(

1dP

dd

A

dA

V

V

V

Since the flow is isentropic:

2

22

M c

d

dP

V

Thus

2

2M1

dP

A

dA

V (5)

There are some significant results from these equations For a nozzle, dV > 0 (accelerating flow) From (1), dh < 0 or dT < 0 From (2), dP < 0 From (5), If M < 1, dA < 0 If M > 1, dA > 0 For a nozzle, dV < 0 (decelerating flow) From (1), dh > 0 or dT > 0 From (2), dP > 0 From (5), If M < 1, dA > 0 If M > 1, dA < 0 Note: M = 1, dA = 0: the sonic velocity can be achieve only at the throat

Recall the first law: 2

hh2

o

V

1

T

T

1k

kRT2)TT(C2 o

oPo

2V

Since c2 = kRT

1

T

T

1k

c2 o

22V

2o M2

1k1

T

T

For an isentropic process:

P

P

T

T o

)1k/(k

o

and

o

)1k/(1

o T

T

Therefore,

)1k/(k

2o M2

1k1

P

P

)1k/(1

2o M2

1k1

Note: For a flow through a nozzle or a

diffuser, because ho is constant (from the first law) and s is constant for an isentropic process, the stagnation state is fixed. As a result, Po, To, and o remain constants for an isentropic nozzle or an isentropic diffuser.

P/Po, T/To, and /o are given in Table A-12 for k = 1.40

The conditions are M = 1 is called the critical properties denoting by the superscript *. Thus, we have

1k/1

o

*

)1k/(k

o

*

o

*

1k

2

1k

2

P

P

1k

2

T

T

15.7 Mass Rate of Flow of an Ideal Gas through an Isentropic Nozzle

The mass rate of flow per unit area can be determined:

2

o

M2

1k1

R

k

T

PM

A

m V

Rewrite this equation in terms of stagnation properties:

)1k(2/)1k(

2o

o

M2

1k1

M

R

k

T

P

A

m

By setting M = 1, we have

)1k(2/)1k(

o

o

*

2

1k

1

R

k

T

P

A

m

Dividing both two equations gives

)1k(2/)1k(

2

*M

2

1k1

1k

2

M

1

A

A

The value of A/A* are also given in Table A.12.

We notice here that a subsonic nozzle is converging and a supersonic nozzle is diverging. Consider a convergent nozzle

The back pressure is the pressure outside the nozzle (Pb) whereas the pressure at the exit plan is denoted by PE.

Consider the variation of and Pm E /Po with PB/Po.

At a, PE = PB = Po and no flow. At b, PE = PB and there is subsonic

flow everywhere in the nozzle. At c, ME = 1, PE = PB = critical

pressure At d, ME = 1, PE = critical pressure,

but m and P E remain constants beyond point c, and PE ≠ PB. The nozzle is choked.

Consider a convergent-divergent nozzle

At a, PE = PB = Po and no flow. Between a and c, flow is subsonic

throughout the nozzle. At c, M* = 1 at the throat, and the

subsonic flow occurs in the divergent section.

Between c and d, the flow is discontinuous and is not entirely isentropic.

At d (the design condition), the flow is entirely isentropic and is supersonic in the divergent section.

Beyond d, PE ≠ PB and PE remains constant.

15.9 Nozzle and Diffuser Coefficients The nozzle coefficient (N) [-]:

EP same andconst sexit with nozzleat KE

exit nozzle at the KE ActualN

si0

ei0N hh

hh

Note: Generally, (N) varies from 90 to 99 percent.

The velocity coefficient (CV) [-]:

EP same andconst sexit with nozzleat

exit nozzle at the ActualVC

V

V

NVC

The coefficient of discharge (CD) [-]:

constsexit with nozzleat m

m ActualDC

Note: if the nozzle is not choked, the actual PB is used to calculate at the isentropic condition. If the nozzle is choked, m is based on the isentropic flow at the design condition.

m

For a diffuser, because the exit state of the diffuser is not necessary to be a stagnation state, the isentropic state (state 3) must be determined

The diffuser coefficient (D) [-]:

102

13

101

13

2

1

sD hh

hh

hh

hh

2/

h

V

Note: h01 = h02 due to the first law.

By assuming an ideal gas with constant specific heat, we have

2

1

k/)1k(

1o

2o2

1

D

M2

1k

1P

PM

2

1k1

15.10 Nozzles and Orifices as Flow-Measuring Devices

The rate of in a pipe is usually determined by measuring P across a nozzle or an orifice.

m

For incompressible fluids, by using the Bernoulli and continuity equations

02

)Pv(P2

1

2

212

VV

0

2

A/A)Pv(P

2

212

2

212

VV

2

12

122

A/A1

)Pv(P2

V

For an ideal gas, P across an orifice or nozzle is usually small. By using the first law, the isentropic relationship, and the binomial theorem, we obtain

02

)P(Pv2

i

2

eiei

VV

Thus, for an ideal gas with small P. The equation is identical to that of the incompressible fluids

The Pitot tube is an instrument for measuring the velocity of fluid.

For incompressible fluids, from the first law

o

2

h2

h V

If the flow is isentropic, the first law becomes

PPv2hh22 oo

2

V

PPv2 o V

For compressible fluids with an ideal-gas assumption, by using the ideal gas law, the isentropic relation for an compressible flow, and the binomial theorem, we obtain

2

o

2

o

o

c4

11

2/

PP

V

V

By comparing the result with the incompressible flow:

12/

PP2

o

o

V

Thus, the term 2

oc25.0 V/ represents the error involving incompressible assumption.