Compressible Fluid Flow

[sheiko]

Dear all,

I was wondering if you could help me clarify some ideas about compressible flow:

1/ Incompressible flow assumption:

I used to think that when the Mach number is below 0.3, one could use the incompressible flow equations when calculating pressure drop for compressible fluid flow (as shown in various books such as White F.M., "Fluid Mechanics", McGraw-Hill, 4th Ed., 1998).

But according the following sources we can read:

Reference 1: Green D.W., Perry R.H., "Perry's Chemical Engineers´Handbook", 8th Ed., 2008:
Compressibility effects are important when the Mach number exceeds 0.1 to 0.2. A common error is to assume that compressibility effects are always negligible when the Mach number is small. The proper assessment of whether compressibility is important should be based on relative density changes, not on Mach number.

Reference 2: Darby R., "Chemical Engineering Fluid Mechanics", 2nd Ed., 2001:
For gases, if the pressure change is such that the density does not change more than about 30%, the incompressible equation can be applied with reasonable accuracy by assuming the density to be constant at a value equal to the average density in the system.

Does this mean that the criteria based on Mach number should be considered as misleading?


2/ Sonic velocity definition

In various books such as reference 1 (see above), we can read that chocking occur in different conditions for isentropic and isothermal (i.e. chocking does not occur at Mach = 1 under isothermal conditions) and that sonic velocity is independent of the process, thus is unique. For ideal gas it can be approximated as c ~ sqrt(kRT/M).
With:
k = Cp/Cv
T = Temperature
R = Gas constant
M = Molecular weight

However, in the reference nº2 (see above), the author states that the speed of sound in an ideal gas is different for an isentropic process than for an isothermal process. Which would mean that, whatever the process conditions, chocking always occur at Mach 1. Thus,
- under isentropic conditions: c ~ sqrt(kRT/M).
- under isothermal conditions: c ~ sqrt(RT/M).

I think Dr Darby is correct. What about you?

For info., API Std 521 (Jan. 2007) has shown that by considering isothermal flow, the k factor should be ommited from the Mach number and critical pressure calculation (see "

http://webwormcpt.blogspot.com/2008/09/removal-of-specific-heat-ratio-k-in.html").

Kind regards

"We don't believe things because they are true, things are true because we believe them."
"Small people talk about others, average people talk about things, smart people talk about ideas and legends never talk."

 

[sailoday28]

For Item 1
Use of low mach no. as related to compressibility depends on how you are using it.
For example, sound speed in a fluid flowing at extremely low speeds, must take into account compressibility. Sqrt of Dp/dro at isentropic conditions.

For item 2. Sound speed for any process, such as isothermal, is defined by the sqrt(dp/droh) under isentropic conditions.

Regards

 

[sheiko]

On item 2/

Dr Darby states that the speed of sound is not the same for isothermal and isentropic conditions (the former does not include the factor k - the isentropic exponent - but the latter does). This seems in line with the recent changes appeared in API Std 521 (please read

http://webwormcpt.blogspot.com/2008/09/removal-of-specific-heat-ratio-k-in.html

and give your comments).

"We don't believe things because they are true, things are true because we believe them."

 

[sailoday28]

Consider adiabatic flow with friction, sound speed is still defined by sqrt dp/droh at isentropic conditions.
If Dr. Darby states otherwise, he is wrong.

 

[CJKruger]

sheiko, I used to think the same as you. But based on the discussion in another threat, I have changed my mind. My view now is:

Sonic velocity: By definition this occurs at constant entropy.

Adiabatic flow: Maximum velocity = Sonic velocity

Isothermal flow: Maximum velocity = Sonic velocity/sqrt(k)

 

[sheiko]

If you refer to thread378-215287, Yes - these are some interesting comments. I think some of them are hung up on "semantics" and "vague definitions". For example, if the speed of sound is defined ONLY under isentropic conditions, then of course it will be different if the flow is isothermal (neither of which will probably be truly achieved in practice). There are the "ideal" conditions and the "practical" conditions, which make some of the arguments moot with regard to actual conditions. For example, the formulas quoted are for an ideal gas, which works quite well for many gases under certain conditions, but is not exact for any of them. Also, there have been various attepts to dervie expressions for the speed of sound for two-phase mixtures, but there is not a definitive answer. Also, I'm sure that oblique shocks occur at sharp edges and corners, but whether the velocity at those points is the "classical" speed of sound seems to be debateable. With regard to choked flow, a maximum flow rate is exhibited for all kinds of compressible flows (single and two-phase) as the downstream pressure is lowered, but whether the velocity at this point is the same as the "classical" speed of sound can certainly be debated. Remember: the "speed of sound" is defined as the propagation rate of an INFINITESIMAL PRESSURE DISTURBANCE under EQUILIBRIUM (ISENTROPIC) CONDITIONS. This is an idealization (a limiting case) and perhaps should not be extrapolated too broadly.

"We don't believe things because they are true, things are true because we believe them."

 

[sailoday28]

sheiko (Chemical) I think you have given an excellent example of the sound speed "propagation rate of an INFINITESIMAL PRESSURE DISTURBANCE under EQUILIBRIUM (ISENTROPIC) CONDITIONS"
A snap of the fingers in a homogeneous fluid is a practical example of the above. The propagation of the resulting wave would be at the speed of sound.

 

[sheiko]

Thanks sailorday28 and CJKruger for previous contributions.

In addition,

In "The Dynamics and Thermodynamics of Compressible Fluid Flow" by Shapiro we can read about the definition of the sonic velocity (partial derivative of dP/drho at constant entropy):

"The variations in pressure AND TEMPERATURE are vanishingly small, and consequently, the process is nearly reversible.

Moreover, the comparative rapidity of the process, together with the SMALLNESS OF THE TEMPERATURE VARIATIONS makes the process nearly adiabatic.

In the limit, for an infinitesimal wave, the process may be considered both reversible and adiabatic, and therefore, isentropic."

Re-reading this quote, i have the feeling that sonic velocity definition also apply to isothermal conditions (...the SMALLNESS OF THE TEMPERATURE VARIATIONS...).
Why not?



"We don't believe things because they are true, things are true because we believe them."

 

[sailoday28]

sheiko (Chemical)
Re-reading this quote, i have the feeling that sonic velocity definition also apply to isothermal conditions (...the SMALLNESS OF THE TEMPERATURE VARIATIONS...).
Why not?
Try a perfect gas, with constant specific heats.
Take the partial derivatives of P with respect to rho for both isentropic and isothermal. In the limit there is a difference in sound speed.
Snap your fingers in an isothermal atmosphere. How fast does the sound wave travel? I believe Newton thought it was at the isothermal condition. However, the local speed is at isentropic conditions.