CHE 220_INTRO TO TRANSPORT AND

MOMENTUM TRANSFER

MODULE 6_FLOW OF COMPRESSIBLE

FLUIDS
Prepared by: Engr. Francisco L. Carbon Jr.
 INTRODUCTION
• Many important applications of fluid dynamics require that density variations
be taken into account.
• The complete field of compressible fluid flow has become very large, and it
covers wide ranges of pressure, temperature, and velocity.
• Chemical engineering practice involves a relatively small area from this field.
• For in­compressible flow the basic parameter is the Reynolds number, a
parameter also important in some applications of compressible flow.
• In compressible flow at ordinary densities and high velocities a more basic
parameter is the Mach number.
• At very low densities, where the mean free path of the molecules is
appreciable in comparison with the size of the equipment or solid bodies in
contact with the gas, other factors must be considered.
 INTRODUCTION
The Mach number, denoted by NMa is defined as the ratio of
u, the speed of the fluid, to a, the speed of sound in the
fluid under conditions of flow,

By speed of the fluid is meant the magnitude of the relative

velocity between the fluid and a solid bounding the fluid or
immersed in it, whether the solid is considered to be stationary
and the fluid flowing past it or the fluid is assumed to be
stationary and the solid moving through it.
 INTRODUCTION
• Common in chemical engineering, is when the solid is considered to be
stationary and the fluid flowing past it.
• The opposite case is of great importance in aeronau­tics, for the motion of
missiles, rockets, and other solid bodies through the atmosphere.
• By definition the Mach number is unity when the speed of fluid equals that
of sound in the same fluid at the pressure and temperature of the fluid.
• Flow is called subsonic, sonic, or supersonic, according to whether the
Mach number is less than unity, at or near unity, or greater than unity,
respectively.
• The most interesting problems in compressible flow lie in the high-velocity
range, where Mach numbers are comparable with unity or where flow is
supersonic.
Simplifying assumptions:
1. The flow is steady.
2. The flow is one dimensional.
3. Velocity gradients within a cross section are neglected,
so that "α = β = 1 and V = u”.
4. Friction is restricted to wall shear.
5. Shaft work is zero.
6. Gravitational effects are negligible, and mechanical-
potential energy is neglected.
7. The fluid is an ideal gas of constant specific heat.
Basic relations used:
1. The continuity equation
2. The steady-flow total-energy balance
3. The mechanical-energy balance with wall friction
4. The equation for the velocity of sound
5. The equation of state of the ideal gas
Each of these equations must be put into a suitable form.
CONTINUITY EQUATION

For differentiation, this may be written in logar­ithmic form:

In p + ln S + ln u = const
Differentiating this equation gives
TOTAL-ENERGY BALANCE
 Consider a fluid in steady flow through a system, entering at station a with
velocity u0 and enthalpy H0 and leaving at station b with velocity ub and
enthalpy Hb.
 For the flow of m kilograms or pounds of material, heat in the amount of Q
joules or Btu must be added through the boundaries of the system to the
material flowing through it.
 Provided there is no significant change in elevation between stations a and b
and no work is done by the system on the outside or on the system from the
outside, the heat added to the fluid is given by the equation

This equation written differentially is

 MECHANICAL-ENERGY BALANCE

This equation may be written over a short length of

conduit in the following differential form:
MECHANICAL-ENERGY BALANCE
 VELOCITY OF SOUND
 The velocity of sound through a continuous material medium, also called
the acoustical velocity, is the velocity of a very small compression-
rarefaction wave moving adiabatically and frictionlessly through the
medium.
 Thermodynamically, the motion of a sound wave is a constant-entropy,
or isentropic, process.
 The magnitude of the acoustical velocity in any medium is shown in
physics texts to be

where the subscript S calls attention to the isentropic restraint on the process.
 IDEAL-GAS EQUATIONS
 Subject to assumptions 1 to 6, Eqs. (6.2) to (6.9) apply to any fluid.
 In fact, they may be used for incompressible flow simply by assuming that
the density p is constant.
 To apply them to compressible flow, it is necessary that the density be
related to temperature and pressure.
 The simplest relation, and one of considerable engineering utility, is the
ideal-gas law which for the present purpose may be written in the. form

where R = molar gas-law constant, in units of mechanical energy per mole

per degree absolute
M = molecular weight
IDEAL-GAS EQUATIONS
IDEAL-GAS EQUATIONS
IDEAL-GAS EQUATIONS
IDEAL-GAS EQUATIONS
THE ASTERISK CONDITION
The state of the fluid moving at its acoustic
velocity is important in some processes of
compressible-fluid flow.
The condition where u = a and NMa,= 1 is
called the asterisk condition, and the
pressure, temperature, density, and enthalpy
are denoted by p *,T*,p*, and H* at this state.
STAGNATION TEMPERATURE
 The stagnation temperature of a high-speed fluid is defined as the
temperature the fluid would attain were it brought to rest adiabatically
without the development of shaft work.
 The relation between the actual fluid temperature, the actual fluid
velocity, and the stagnation temperature is

The stagnation enthalpy H, is defined by the equation

PROCESSES OF COMPRESSIBLE FLOW
 The individual processes to be
considered in this chapter are shown
diagrammatic­ally in Fig. 6.1.
 The individual processes to be
considered in this chapter are shown
diagrammatic­ally in Fig. 6.1.
 It is assumed that a very large supply
of gas at specified temperature and
pressure and at zero velocity and
Mach number is available.
 The origin of the gas is called the
reservoir, and the temperature and
pressure of the gas in the reservoir
are called reservoir conditions.
PROCESSES OF COMPRESSIBLE FLOW
 The reservoir temperature is a
stagnation value, which does not
necessarily apply at other points in
the flow system.
 From the reservoir the gas is assumed
to flow, without friction loss at the
entrance, into and through a conduit.
 The gas leaves the conduit at
definite temperature, velocity, and
pressure and goes into an exhaust
receiver, in which the pressure may
be independently controlled at a
constant value less than the
reservoir pressure.
 PROCESSES OF COMPRESSIBLE FLOW
1. An isentropic expansion. In this process the cross-sectional area of the
conduit must change, and the process is described as one of variable area.
Because the process is adiabatic, the stagnation temperature does not
change in the conduit. Such a process is shown diagrammatically in Fig.
6.la.
PROCESSES OF COMPRESSIBLE FLOW
2. Adiabatic frictional flow through a conduit of constant cross section. This
process is irreversible, and the entropy of the gas increases, but as shown by
Eq. (6.22), since Q = 0, the stagnation temperature is constant throughout
the conduit. This process is shown in Fig. 6.lb.
PROCESSES OF COMPRESSIBLE FLOW
3. Isothermal frictional flow through a conduit of constant cross-sectional area,
accompanied by a flow of heat through the conduit wall sufficient to keep the
temperature constant. This process is nonadiabatic and nonisentropic; the
stagnation temperature changes during the process, since T is constant, and by Eq.
(6.20), T, changes with u. The process is shown in Fig. 6.lc.
Flow through Variable-Area Conduits
 A conduit suitable for isentropic flow is called a nozzle.
 As shown in Fig. 6.la, a complete nozzle consists of a convergent
section and a divergent section joined by a throat, which is a short
length where the wall of the conduit is parallel with the axis of the
nozzle.
 For some applications, a nozzle may consist of a divergent section
only, and the throat connects directly with the receiver.
 The configuration of an actual nozzle is controlled by the designer,
who fixes the relation between S, the cross-sectional area, and L, the
length of the nozzle measured from the entrance.
 Nozzles are designed to minimize wall friction and to suppress
boundary­layer separation.
 Flow through Variable-Area Conduits
 The convergent section is rounded and can be short, since
separation does not occur in a converging channel.
 To suppress separation in the divergent section, the diverging
angle is made small, and this section is therefore relatively
long.
 The nozzle entrance is sufficiently large relative to the throat
to permit the velocity at the entrance to be taken as zero and
the temperature and pressure at the entrance to be assumed
equal to those in the reservoir.
 The purpose of the convergent section is to increase the
velocity and decrease the pressure of the gas.
 Flow through Variable-Area Conduits
 At low Mach numbers the process conforms essentially to the usual
Bernoulli relation for incompressible flow.
 In the convergent section flow is always subsonic, but it may become
sonic at the throat.
 Mach numbers greater than unity cannot be generated in a convergent
nozzle.
 In the divergent section, the flow may be subsonic or supersonic.
 The purpose of the divergent section differs sharply in the two
situations.
 In subsonic flow the purpose of the section is to reduce the velocity
and regain pressure, in accordance with the Bernoulli equation.
EQUATIONS FOR ISENTROPIC FLOW.
 Change in gas properties during flow.
 The density and temperature paths of the gas through any
isentropic flow are given by Eqs. (6.14) and (6.15).
 The constants are evaluated from the reservoir condition.
 This gives

• These equations apply both to frictionless subsonic and

supersonic flow, but they must not be used across a shock front.
EQUATIONS FOR ISENTROPIC FLOW.
EQUATIONS FOR ISENTROPIC FLOW.
A Mach-number form of Eq. (6.26) is convenient. It is derived by
substituting
u2 from Eq. (6.2) into the first equality of Eq. (6.18) and eliminating
p/ p 0 by
substitution from Eq. (6.23). This gives
EQUATIONS FOR ISENTROPIC FLOW.
EQUATIONS FOR ISENTROPIC FLOW.
EQUATIONS FOR ISENTROPIC FLOW.
EQUATIONS FOR ISENTROPIC FLOW.
EQUATIONS FOR ISENTROPIC FLOW.
EQUATIONS FOR ISENTROPIC FLOW.
 Adiabatic Frictional Flow
 Flow through straight conduits of constant cross section is adiabatic when heat
transfer through the pipe wall is negligible.
 The process is shown diagrammatically in Fig. 6.l b.
 The typical situation is a long pipe into which gas enters at a given pressure and
temperature and flows at a rate determined by the length and diameter of the pipe and
the pressure maintained at the outlet.
 In long lines and with a low exit pressure, the speed of the gas may reach the sonic
velocity.
 It is not possible, however, for a gas to pass through the sonic barrier from the direction
of either subsonic or supersonic flow; if the gas enters the pipe at a Mach number
greater than 1, the Mach number will decrease but will not become less than 1.
 If an attempt is made, by maintaining a constant discharge pressure and lengthening
the pipe, to force the gas to change from subsonic to supersonic flow or from supersonic
to subsonic, the mass flow rate will decrease to prevent such a change.
 This effect is called choking.
 Adiabatic Frictional Flow
THE FRICTION PARAMETER.
• The basic quantity that measures the effect of friction is the friction parameter
fL/rH.
• This arises from the integration of Eq. (6.8).
• In adiabatic frictional flow, the temperature of the gas changes.
• The viscosity also varies, and the Reynolds number and friction factor are not actually
constant.
• In gas flow, however, the effect of temperature on viscosity is small, and the effect of
Reynolds number on the friction factor f is still less.
• Also, unless the Mach number is nearly unity, the temperature change is small.
• It is satisfactory to use an average value for f as a constant in calculations.
• If necessary, f can be evaluated at the two ends of the conduit and an arithmetic
average used as a constant.
EQUATIONS FOR ADIABATIC FRICTIONAL FLOW.
EQUATIONS FOR ADIABATIC FRICTIONAL FLOW.
EQUATIONS FOR ADIABATIC FRICTIONAL FLOW.
 EQUATIONS FOR ADIABATIC FRICTIONAL FLOW.

For maximum conduit length,Lmax:

Corresponding equations for p/ p*, T/ T*, and p/ p* are found from Eqs.
(6.39), (6.43), and (6.44).
 EQUATIONS FOR ADIABATIC FRICTIONAL FLOW.

Since, for constant-area flow, G is independent of length, the mass

velocity can be evaluated at any point where the gas properties are known.
Normally the conditions at the entrance to the conduit are used.
ADIABATIC FRICTIONAL FLOW.
ADIABATIC FRICTIONAL FLOW.
ADIABATIC FRICTIONAL FLOW.
ADIABATIC FRICTIONAL FLOW.
Isothermal Frictional Flow
 The temperature of the fluid in compressible flow through a conduit of
constant cross section may be kept constant by a transfer of heat through the
conduit wall.
 Long, small, uninsulated pipes in contact with air transmit sufficient heat to
keep the flow nearly isothermal.
 Also, for small Mach numbers, the pressure pattern for isothermal flow is
nearly the same as that for adiabatic flow for the same entrance conditions,
and the simpler equations for isothermal flow may be used.
 The maximum velocity attainable in isothermal flow is
Equations for Isothermal Frictional Flow
Isothermal Frictional Flow
Isothermal Frictional Flow
Problems
Reference:
 UnitOperations of Chemical Engineering 7th Ed,,
McCabe et. al., McGraw-Hill International Editions