3 replies to this topic

[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 does not occur at Mach = 1 under isothermal conditions and that for an ideal gas, sonic velocity can be approximated as c ~ sqrt(kRT/M) whatever the flow regime.
With:
k = Cp/Cv
T = Temperature
R = Gas constant
M = Molecular weight

On the contrary, in the reference nº2 (see above), Dr Ron Darby states that, because chocking always occur at Mach 1, the speed of sound equation of an ideal gas is different for an isentropic process than for an isothermal process:
- under isentropic conditions: c ~ sqrt(kRT/M).
- under isothermal conditions: c ~ sqrt(RT/M).

In addition, 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.bl...atio-k-in.html").

API concurs with Dr Darby's observation.

What is your opinion?

Kind regards

[djack77494]

sheiko,
I can only offer you my opinion as I have not performed the calculations and done a full comparison on which to base conclusions.

I have seen stated and I believe that reasonably accurate results are obtained using incompressible flow equations based on the upstream conditions IF the pressure loss is less than 10% of the upstream absolute pressure. Likewise, I believe you can extend the use of incompressible flow equations to less than 30% of the upstream absolute pressure if you can use the average properties of the gas. I do not endorse the use of ANY range of Mach Numbers as a determinant for this decision. In any case, you can verify these "rules of thumb" by subdividing a hydraulic circuit into ever smaller segments until you see that further reductions in size do not lead to a significant increase in accuracy. Use incompressible flow equations (e.g. Darcy) to perform the hydraulic calcs for the segments.

[Art Montemayor]

I don’t understand the introduction of the Mach Number criteria – unless we are discussing sonic or supersonic gas flow (which I understand we are NOT).

I concur with what Doug has commented and reinforce his statements with a direct quote from Crane’s Flow of Fluids – Tech Paper #410, found in page 1-7:

Principles of Compressible Flow in Pipe

"An accurate determination of the pressure drop of a compressible fluid flowing through a pipe requires knowledge of the relationship between pressure and specific volume; this is not easily determined in each particular problem. The usual extremes considered are adiabatic flow (p'Vka = constant) and isothermal flow (P'Va = constant). Adiabatic flow is usually assumed in short, perfectly insulated pipe. This would be consistent since no heat is transferred to or from the pipe, except for the fact that the minute amount of heat generated by friction is added to the flow.

Isothermal flow or flow at constant temperature is often assumed, partly for convenience but more often because it is closer to fact in piping practice. The most outstanding case of isothermal flow occurs in natural gas pipe lines. Dodge and Thompson show that gas flow in insulated pipe is closely approximated by isothermal flow for reasonably high pressures.

Since the relationship between pressure and volume may follow some other relationship (p'Van = constant) called polytropic flow, specific information in each individual case is almost an impossibility.

The density of gases and vapors changes considerably with changes in pressure; therefore, if the pressure drop between P1 and P2 in Figure 1-6 is great, the density and velocity will change appreciably.

When dealing with compressible fluids, such as air, steam, etc., the following restrictions should be observed in applying the Darcy formula:

1. If the calculated pressure drop (P1 – P2) is less than about 10% of the inlet pressure P1, reasonable accuracy will be obtained if the specific volume used in the formula is based upon either the upstream or downstream conditions, whichever are known.

2. If the calculated pressure drop (P1 - P2) is greater than about 10%, but less than about 40% of inlet pressure P1, the Darcy equation may be used with reasonable accuracy by using a specific volume based upon the average of upstream and downstream conditions; otherwise, the method given on page l - 9 may be used.

3. For greater pressure drops, such as are often encountered in long pipe lines, the methods given on the next two pages should be used."


I have always applied these principles and never run into any problems – from the design stage all the way through the operations stage.

HOWEVER: Note that Crane has an ERRATA here. P1 should really be P'1 (absolute pressure). One should always employ absolute pressures (P') when dealing with compressible fluids. Note the problem with the expression (P1 - P2) should P1 = 0,.0 psig.

[sheiko]

2/ Sonic velocity definition

In various books such as reference 1 (see above), we can read that chocking does not occur at Mach = 1 under isothermal conditions and that for an ideal gas, sonic velocity can be approximated as c ~ sqrt(kRT/M) whatever the flow regime.
With:
k = Cp/Cv
T = Temperature
R = Gas constant
M = Molecular weight

On the contrary, in the reference nº2 (see above), Dr Ron Darby states that, because chocking always occur at Mach 1, the speed of sound equation of an ideal gas is different for an isentropic process than for an isothermal process:
- under isentropic conditions: c ~ sqrt(kRT/M).
- under isothermal conditions: c ~ sqrt(RT/M).

In addition, 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").

API concurs with Dr Darby's observation.

What is your opinion?

Kind regards

Is anybody interested in item 2 of my query?

Regards