Practical applications of this topic include sizing relief valve outlet laterals and low-pressure compressor suction lines.  These pose a special challenge as the velocities and pressure changes are high.

Adiabatic Flow of a Compressible Fluid Through a Conduit

Flow through pipes in a typical plant where line lengths are short, or the pipe is well insulated can be considered adiabatic.  A typical situation is a 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 downstream pressure.  As the line gets longer friction losses increase and the following occurs:

a)      Pressure decreases
b)     Density decreases
c)      Velocity increases
d)      Enthalpy decreases
e)      Entropy increases

The question is “will the velocity continue to increasing until it crosses the sonic barrier?”  The answer is NO.  The maximum velocity always occurs at the end of the pipe and continues to increase as the pressure drops until reaching Mach 1.  The velocity cannot cross the sonic barrier in adiabatic flow through a conduit of constant cross section.   If an effort is made to decrease downstream pressure further, the velocity, pressure, temperature and density remain constant at the end of the pipe corresponding to Mach 1 conditions. The excess pressure drop is dissipated by shock waves at the pipe exit due to sudden expansion. If the line length is increased to drop the pressure further the mass flux decreases, so that Mach 1 is maintained at the end of the pipe.

Analyzing the adiabatic flow using energy and mass balance yields the following analyses along with this nomenclature:

Analysis One

This analysis derives the relationship between the stagnation temperature, flowing temperature, and the Mach number for a flowing ideal gas.  Stagnation temperature is the temperature a flowing gas rises to when it is brought isentropically to rest, thereby converting its kinetic energy into enthalpy.

Conservation of energy requires that the energy balances:

Energy Balance

For adiabatic flow, no shaft work and for gases:  Q=0, Ws=0 and dz=negligible....or:

Enthalpy per unit mass of an ideal gas is defined H = C_p T

The gas, at rest, has no kinetic energy and is at its stagnation temperature (Tst), while the moving gas has kinetic energy and is at another temperature (T).  The energies are therefore:
energy at rest, per unit mass = 0 + C_p T_st
energy in motion, per unit mass = v²/2 + C_p T

Equating the energy at rest and in motion:

hst= h+v²/2

or h= hst-v²/2

or

This implies:
a)      Stagnation enthalpy of the fluid during adiabatic flow is constant.  For an ideal gas, this implies the stagnation temperature is constant.
b)      Enthalpy of the gas drops and kinetic energy increases in the direction of flow.
c)      For as given mass flux the enthalpy and density are related to each other.

A useful way of looking at this relationship is by fanno lines.  The fanno lines are lines of constant mass flux plotted on enthalpy/entropy diagram:

C_p T_st = v²/2 + C_p T

To make this equation useful, we must replace C_p and v by terms containing only constants and the Mach number.

Also for an ideal gas

and

Substituting yields:

or

(1)

Thus we see that for an ideal gas the temperature decreases as velocity increases.

If the gas is flowing adiabatically, then no energy has been added or subtracted from it and Tst is constant along the length of the pipe.  Knowing Tst, then the above equation can be used to find the flowing temperature from the Mach number, (or vice versa) at any position along the pipe.

Analysis Two

This analysis uses the principles of conservation of energy and mass to derive a relationship between pressure and Mach number at up and downstream conditions,  for adiabatic flow in a pipe of constant cross-sectional area.

The conservation of mass requires the mass flux to be the same at any position along a pipe. Mass flux at any of these positions can be expressed in terms of density and velocity :

(2)

For an ideal gas:

and

Substituting for density and velocity, we obtain Equation 3 which relates Mach number, mass flow rate and flowing pressure and temperature:

(3)

or

Substituting for T from Equation 1:

G is same at inlet (1) and outlet(2), so:

which leads to:

(4)

This implies that pressure decreases as the Mach number increases.  A similar analysis for temperature gives:

(5)

This implies that temperature decreases as the Mach number increases.  However, this is true for ideal gases only.  For real gases temperature may increase!

Analysis Three

Now the momentum equation is introduced to incorporates the losses due to friction. The derivation is available in any standard textbook for compressible flow In summary the final result is:

(6)

where

f= Average Darcy friction factor
L= Equivalent length of line
d= I.D. of the line

Thus this equation relates losses due to friction to inlet and outlet velocities.  Solving for the unknown parameter requires a trial and error approach and is suitable for an Excel spreadsheet using the “Goal Seek” or “Solver” tools.  Depending on the number of unknowns one or all three of the following equations need to be solved simultaneously:

Mass balance Equation 2
Energy balance Equation 4 or 5
Momentum balance Equation 6.

In cases where the outlet velocity is defined as Mach 1, then the equation can be solved for the maximum length, which can be used to flow a certain amount of fluid through a line of known diameter.  Beyond this length choked flow condition occurs and, as explained above, any further increase in pipe length will cause the flow to decrease in such a manner that velocity at the end of the pipe is still sonic ( Mach=1). This particular application is of considerable practical use in sizing blowdown lines or relief valve outlet lines relieving to the atmosphere.

Recall that the above equations have assumed that the gas is ideal.   One can compensate for non-ideality to an extent by incorporating the Z factor. A rigorous approach implies solving simultaneously the momentum, energy, and mass balance equation numerically.  An analytical approach, as given above for ideal gases, is useful most of the time and the results are valid for engineering purpose.

Isothermal Flow

In isothermal flow, the temperature of the gas remains constant.  This simplifies matters considerably.

Energy Balance equation :

Mechanical Energy Equation

Multiplying both sides by r²:

Rearranging and integrating gives:

(7)

When the temperature change over the conduit is small Equation 7 can be used instead of the adiabatic Equation 6.  Adibatic flow below Mach 0.3 follows Equation 7 closely.

If Equation 7 is differentiated with respect to r_b   to obtain a maximum G then:

(8)

and the exit Mach number is:

(9)

This apparent choking condition for isothermal flow is not physically meaningful, as at these high speeds, and rates of expansion, isothermal conditions are not possible.

References:

1) Unit operations of Chemical Engineering- Mccabe, Smith and Hariott; McGraw-Hill

2) Perry’s Chemical Engineers’ Handbook’ McGraw-Hill.

Guest Author:

Rajiv Narang, Principal Process Engineer, Worley Parsons
Email: rxnarang "at" gmail.com