How does sonic flows maintain a steady flow rate if it does? 6

[SalvadorT]

I was wondering if anyone knows if a fluid that travels at sonic speeds in a pipe, or even supersonic speeds, maintain a constant flow rate, volume/time. If it does, can you please explain the mechanisms that it can. Also, I heard that steam, cannot be accelerated to speeds past Mach 1. If this is true or false, can you please explain this to me. I am researching the effects of a supersonic flow of steam for a steam generator, but I want to know the mechanism that are involved that ensure a steady flow rate, as I am told.
SalvadorT

 

[sailoday28]

davefitz (Mechanical) writes:
Flow in pipes cannot be supersonic. Flow in pipes is described in the section of most compressible flow texts under the heading of "flow in long ducts", or something to that effect. The common point being long channels of constant cross-section. Fanno Lines describe the non-dimensionalized relationships for such flow.

Flow in pipes CAN be supersonic. With M>1 the Mach no tends to decrease in the flow direction towards M=1. Refer to any compressible flow text AND THE FANNO LINE. Dependent upon friction and back pressure conditions, a shock can occur. The dowstream flow conditions are then subsonic with the Mach No. increasing towards M=1.

 

[sailoday28]

SalvadorT (Mechanical) writes: Also, I heard that steam, cannot be accelerated to speeds past Mach 1.
Mach number is a ratio of the local fluid velocity to that of the sound speed (of that medium of course). Of course steam can be accelerated to supersonic conditions.

Most of the thread discussion seems to be centered on steady state adiabatic flow.
It is interesting to note thata perfect gas, with constant specific heats under isothermal conditions,---- flow in a long duct will choke at a Mach No. of 1/sqrt(gamma. Where gamma is the ratio of the specific heats, Cp/Cv.

 

[poetix99]

This little tempest in a teapot is amusing - for a few posts, anyway.

"mbeychok":
If one increases the upstream pressure, one has changed the conditions of the flow "problem" under consideration. Of course the mass flow can be increased if the upstream pressure is increased, you are absolutely right, but missing part of the point that others have been making.

The phrase "choked flow" implicitly refers to fixed upstream conditions. This is not something to agree with, or disagree with; it is simply the convention for that expression.

Part of this depends upon one's point of view at a flow nozzle, orifice, etc. Terminology within a particular specialty of engineering might reflect a point of view that is either upstream or downstream, and seems to differently describe identical phenomena.

Some turbine manufacturers, for example, refer to flow through a row of (converging) airfoil nozzles as "restricted flow" if the throat velocity is at the local sonic velocity; the mass flow cannot be increased by increasing any of the downstream areas or pressures in the flow path because the sonic flow point represents a "restriction" in the flow FOR A GIVEN SET OF UPSTREAM CONDITIONS. Similarly, it is called "unrestricted flow" if the nozzle exit flow is at a velocity less than the local sonic velocity. One could think of it as "not yet limited" flow, rather than "unrestricted" flow. IT IS UNDERSTOOD THAT THE MASS FLOW COULD BE INCREASED - if the upstream pressure was increased, but that is not necessarily the objective.

 

[mbeychok]

Poetix99:

Quote from your posting of Nov.13:

"The phrase "choked flow" implicitly refers to fixed upstream conditions. This is not something to agree with, or disagree with; it is simply the convention for that expression."

Thank you for your very reasoned posting. Actually, I am not missing the point others make and I fully understand that choked flow implicitly refers to fixed upstream condition ... but only for some people wotking in some disciplines. Restriction orifices are very often used at choked velocity conditions as mass flow controllers. For example, when one wants to inject a constant mass flow of steam into a process vessel in a petroleum refinery or petrochemical plant and wants that flow to be constant even though the downstream pressure (i.e., inside the vessel) may not be constant. In that case, the upstream steam pressure is controlled at a pressure high enough to insure that choked conditions prevail in the flow through the orifice. If, for some reason, one wants to increase that constant mass flow amount (say from 100 kg/s to 150 kg/sec), then one simply controls the upstream steam at a higher pressure. In such such cases, the process designer does not implicitly assume any convention about choked flow referring to fixed upstream conditions. He knows that, at choked conditions, the mass flow is not at a maximum even though the gas velocity is at a maximum. That is common knowledge in everyday use for a process design engineer in the hydrocarbon processing industries (oil refineries, natural gas plants and petrochemical plants). As said by Flareman (Petroleum) in thread378-25153: "the actual pressure influences the true mass flow and there is no "maximum" flow, just a maximum velocity".

Please take just a few minutes to read the tutorial at

http://www.okcc.com/PDF/Choked.pdf

and you will see that there is an entirely different mindset in other disciplines.

My philosophy:
I firmly believe that whenever one publishes a technical paper or posts a message in a technical forum such as this, one must realize that the paper or message must be understood by a universe of people in disciplines other than that of the writer. For example, a NASA engineer publishing a treatise on fluid mechanics on the Internet should keep in mind that people other than aeronautic engineers or rocket designers may be reading his treatise ... and not all of the readers accept the convention that choked flow refers to fixed upstream conditions. Nor should one use the gas constant R without explaining that it is the specific gas constant for air rather than the universal gas constant (since the specific constant R equals the universal gas constant divided by the gas molecular weight) ... his aeronautic engineering colleagues may understand what he has done, but other engineers will wonder why the molecular weight has disppeared because they deal with many gases other than air.


Milton Beychok
(Visit me at www.air-dispersion.com)

 

[sailoday28]

mbeychok (Chemical)writes:
The equation (in SI metric units) for calculating the mass flow rate of a gas across a restriction orifice at choked conditions is:

Q = C A P [ k M / ( R T ) ]1/2 [ 2 / ( k + 1 ) ] (k + 1) / (2k - 2)

where:
Q = mass flow rate of the gas, kg/s
C = discharge coefficient
A = discharge hole area, m2
P = absolute upstream pressure, Pa
k = Cp / Cv of the gas
M = gas molecular weight
R = Universal Gas Law constant = 8314.5 ( Pa ) ( m3) / ( kgmol ) ( deg K )
T = gas temperature, deg K

One should be careful on the use of what can be static and stagnation conditions.
The above formula P and T must be taken at stagnation conditions, not that of local static pressure and temperature. ---UNLESS THE FLOW IS FROM AN INFINITE RESERVOIR FEEDING DIRECTLY INTO A RESTRICTION FROM THE RESERVOIR. In that case stagnation conditions and static conditions are the same.

Of course the above results are for perfect gas, constant specific heats.

 

[mbeychok]

Sailoday28:

If you are going to copy and quote the equation that I posted for the choked flow of a gas through an orifice, then you should really learn how to display the superscript parameters (which you have not done). So I will again provide the equation in its correct form:

Q = C A P [ k M / ( R T ) ][sup]1/2[/sup] [ 2 / ( k + 1 ) ] [sup](k + 1) / (2k - 2)[/sup]

where:
Q = mass flow rate of the gas, kg/s
C = discharge coefficient
A = discharge hole area, m[sup]2[/sup]
P = absolute upstream pressure, Pa
k = Cp / Cv of the gas
M = gas molecular weight
R = Universal Gas Law constant = 8314.5 ( Pa ) ( m3) / ( kgmol ) ( deg K )
T = gas temperature, deg K

The pressure (P) and temperature (T) are very plainly and simply the conditions upstream of the orifice (i.e., at the inlet to the orifice). There is absolutely no need to use the terms "stagnation conditions" and "local static pressure". Anyone who can read English will readily understand "upstream" or "inlet". So why introduce unnecessary and confusing technical jargon ... especially when you don't define the meaning of the terms?

Yes, the equation was derived for an ideal gas and it has been in widespread use as such for the last 50 to 60 years to calculate the mass flow rate of a gas through an orifice at choked conditions. If non-ideality is of concern (and it rarely is), then it is very easy to introduce the compressibility factor (Z) of the gas into the equation.


Milton Beychok
(Visit me at www.air-dispersion.com)

 

[sailoday28]

I appologize for copying via edit the original mbeychok equation. It was a typo. My comments however were directed towards the use of his upstream conditions. Upsteam conditons of p and T must not be static. Stagnation conditions for pressure and temp must be used.
Stagnation is sometimes refered to as Total

alpha= (k-1)/2 M= Mach No.

Tstagnation=T(1+alpha*M^2)

Pstagnation= P(1+alpha*M^2)^[k/(k-1)]

Where P and T are the local static upstream conditions.
Use of static pressure and temperature will obviously give error. IF UPSTREAM MACH=0, EQUATION OF MBEYCHOK IS SATISFACTORY.
If fluid is flowing such as in a pipe to an orifice then stagnation conditions must be accounted for in mass flow equations. THEN SIMPLY SUBSTITUTE Pstagnation and Tstagnation for p and T.
See and text such as:
The Dynamics and Thermodynamics of Compressible Fluid Flow by Ascher H. Shapiro
Gas Tables by Keenan and Kaye
Elements of Gas Dynamics by Lepmann and Roshko.

 

[25362]

To those negating the existence of supersonic flows in ducts of constant cross-sectional area: how do you explain the existence of supersonic test wind tunnels ?

The test sections in these tunnels are supposed to have upstream and downstream combinations of converging/diverging nozzles to create and maintain supersonic velocities.

 

[mbeychok]

Sailoday28:

Thanks for your apology for mis-copying the equation that I posted, which you refer to as "the mbeychok equation". It is not my equation ... it has been available in "Perry's Chemical Engineers' Handbook" and other textbooks for the past 50-60 years. In my Sixth Edition of Perry's, it appears on page 5-14 as equation 5-21.

Milt Beychok

 

[sailoday28]

poetix99 (Mechanical) Nov 13, 2004 states
This little tempest in a teapot is amusing - for a few posts, anyway

Let the tempest pass. If Perry's Chemical Engineers' Handbook" uses an equation implying p and T as upstream pressures without qualifying these parameters as stagnation conditions, then the quoted equation is quoted wrong. With movement of the fluid upstream of the orifice, nozzle, etc,the total pressure and temp has to include the kinetic energy of the fluid.
That is my last word on proper use of static and stagnation conditions.

 

[mbeychok]

Sailoday28:

I am not trying to be facetious, but I simply don't understand what you said:

If "Perry's Chemical Engineers' Handbook" uses an equation implying p and T as upstream pressures without qualifying these parameters as stagnation conditions, then the quoted equation is quoted wrong. With movement of the fluid upstream of the orifice, nozzle, etc,the total pressure and temp has to include the kinetic energy of the fluid.

Let us assume that we have a gas in a pipe and the gas is flowing through a restriction orifice installed in the pipe. If we also have a pressure gauge installed in the pipe just upstream of the orifice and that pressure gauge tells us that the actual, measured pressure is 100 psig (114.7 psia), are you saying that we cannot use that pressure as the orifice inlet pressure unless we qualify it as the "stagnant" pressure or the "total pressure" or that it does or does not include the "kinetic energy" of the flowing gas? In my mind, that pressure read by the pressure gauge is the actual upstream pressure of the gas entering the restriction orifice ... and we need not qualify as anything but the actual pressure.

 

[sailoday28]

Upstream of the orifice, the pressure may be measured with a PITOT tube (a bent open tube which is inserted into the flow stream in a pipe-the open end points in the upstream direction and is connected to a pressure gage) or a STATIC pressure gage (the pressure gage is taped into the pipe and is normal to the flow direction. If the flow were incompressible, the Pitot tube measure total pressure or stagnation pressure p + 0.5U^2*rho, while the static gage would measure P. The difference of the readings is the velocity head. For compressible flow of perfect gas etc, the relation of Pstag to p is a function of the mach no. which I have given earlier in the thread.
In your example of 114.7 psia, if the pressure is static, via say a regular bordon tube gage and the gas air, k=1.4
Then p/pstag =0.99303 if M=.1 and T/Tstag=0.998
p/pstag=0.93947 if M=0.3 T/Tstag=0.98232
p/pstag=0.843 M=0.5 T/Tstag=0.952
The equation quoted from Chem Engrs Hand Book should be using Pstag and Tstag as the upstream press and temp.
Having the upstream Mach no. with a known static temp and press allows calc of pstag and Tstag. Derving the mass flow equation from consv of energy and mass will show that stagnation conditions satisfy the quoted equation.

 

[mbeychok]

sailoday28:

I think perhaps I am beginning to understand the problem you and I have communicating on the subject of this thread. In chemical manufacturing plants, steam generating plants, oil refineries, natural gas processing, and so forth there would rarely be gas flowing in pipes at velocities exceeding 120 ft/sec (36.6 m/sec). At gas velocities above that, the piping would undergo severe erosion/corrosion problems.

If, for example, we consider natural gas (essentially methane) flowing in a pipe at say 20 deg C, the sonic velocity in methane at that temperature is about 447 m/sec. Therefore, the Mach number of methane flowing at 36.6 m/sec would only be 36.6/447 = 0.08 and at that Mach number (according to your last posting) the difference between the pressure measured by a pressure gauge and what you call stagnation pressure would be a good bit less than 1 percent. So I guess that aeronautical engineers and rocket designers just work in an entirely different realm than we chemical engineers. We just don't deal with gases flowing at Mach numbers above 0.01 in designing industrial process plants. This not to say that I agree totally with you, but I am beginning to see that we just don't speak the same language.

Milton Beychok
(Visit me at www.air-dispersion.com)

 

[sailoday28]

With the flow of Mach nos in my previous posting, the error in mass flow rate using static, instead of stagnation pressure are as follow:
M flow %error
0.1 0.6
0.3 5.2
0.5 13.6

In the nuclear industry, one considers high energy line breaks and the calculations are required to be as accurate as possible.
With extremely low Mach Nos, the flow with LOW pressure drop can be approximated as incompressible. However, when dealing with compressible flow, one must be aware of the impact of compressibility.
For example:
----For transient flow, some enginers have thought that flow chokes at the critical pressure ratio used in steady state (which is not true)
----I don't believe placing Z in the Chem Eng. Hand Bk formula is not enough to compensate for High pressure and low temp gases. The orig derivation of the mass flow is based on an energy balance which requires input of enthalpy, which in turn is a function of Z.
If I were to deal with gases which deviated significantly from the perfect or with specific heats that varied significantly with Temp (Hand Bk assumes constant spec heats), my pencil, paper and table of integrals (for enthalph derivation) would come out. Possibly I would have to use numerical integration.

 

[25362]

The formulas brought by sailoday28 for the estimation of the properties at stagnation originated from the assumption that the process is isentropic, ie, adiabatic and reversible, which, of course, is only a useful approximation.

The stagnation properties can also be estimated from the free-stream velocity V[sub]o[/sub], as follows:

T[sub]stag[/sub]=T[sub]o[/sub] + V[sub]o[/sub][sup]2[/sup]/(2g[sub]c[/sub]c[sub]p[/sub]J)

p[sub]stag[/sub]=p[sub]o[/sub]+([ρ][sub]o[/sub]V[sub]o[/sub][sup]2[/sup]/2g[sub]c[/sub])[1 + Ma[sub]o[/sub][sup]2[/sup]/4 + (2-[γ]) Ma[sub]o[/sub][sup]2[/sup]/24 +...]

From the second formula one sees that for incompressible fluids, the stagnation pressure at low Mach numbers is reducible to the sum of the static pressure and the velocity pressure which chem. engineers are well acquainted with.

In order to get an idea of the values of Ma[sub]o[/sub], the sonic velocity of dry and superheated steam is in the order of 2000 fps.

Notation:

Ma[sub]o[/sub]: the free-stream Mach number. The Mach number is also a measure of the ratio of the inertial force to the elastic force and a measure of the kinetic energy to the internal energy.
g[sub]c[/sub]: gravitational conversion factor
c[sub]p[/sub]: specific heat
J: the mechanical equivalent of heat
[ρ][sub]o[/sub]: the free-stream gas density; the stagnation density can be estimated from the stagnation pressure and temperature [ρ][sub]stag[/sub]=p[sub]stag[/sub]/RT[sub]stag[/sub], where R is the specific gas constant.
[γ]: Cp/Cv

BTW, there are indeed industrial uses for sonic velocities, as when designing safety relief valves, control valves, flares, steam generating plants, steam ejectors, reducing orifices, etc.

As an aside, in hypersonic wind tunnels, air flows at speeds roughly in the range of 5 to 15 times the speed of sound !

 

[sailoday28]

25362 (Chemical) writes:
pstag=po+(?oVo2/2gc)[1 + Mao2/4 + (2-?) Mao2/24 +...]
I don't disagree with this approximation. However, I believe that definition relating static to stag pressure is for isentropic flow. Perhaps I am wrong.
I'm interested in seeing how this approximation is derived for non-isentropic flows.

 

[25362]

To sailoday28, you're right, the formula results from assuming a reversible adiabatic (or isentropic) process and is equivalent to the second formula in your post of Nov. 15th repeated herebelow, which, BTW, also results from assuming isentropic conditions,:

p[sub]t[/sub] = p[sub]0[/sub] [1 + Ma[sub]0[/sub][sup]2[/sup] ([γ] -1)/2][sup][γ]/([γ] -1)[/sup]​

The above "series-type" approximation formula was taken from Chapter 2, "Fluid Flow", under the heading: Stagnation Properties in Fan Engineering by the Buffalo Forge Company, eighth (1983) edition. Edited by Robert Jorgensen.

 

[sailoday28]

What's the best way of inputing formuals? The edit-coppy doesn't work very will for me.

 

[Latexman]

Sailoday,

Left click the Process TGML link and read.

Good luck,
Latexman

 

[sailoday28]

My email to McGraw-Hill Book Co.
Sent: Thursday, February 24, 2005 4:06 PM
To: Reed, Dorise
Cc: cab4@nrc.gov; fred moody; pvanstol@bechtel.com
Subject: handboook error


Ms. Reed,
Please refer to the Standard Handbook for Mechanical Engineers-Baumister& Marks, Seventh Edition pg 4-62

The reference page includes formulas for orifice computations. The formulas for the mass flow, m, in particular for choked flow (being independent of downstream pressure) indicate dependence on pressures of p(subscript1) and T(subscript1) where the subsripts refer to the static pressure and temperature and a particular section.

The pressure and temperature refered to in the handbook should be refered to as stagnation pressure and temperatures at the corresponding section of upstream flow.

The formulas are for a perfect gas with constant specific heat. Using the static conditions in place of stagnation will lead to error in computation of mass flow rates for choked flow.

I believe the same type of formula with error is given in your Chemical Engineers Handbook and more recent editions of the Mechanical Engineers Handbook.

Sincerely