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

I would think that constantn volumetric flow would be hard to impossible to achieve in the sonic flow regime. As the fluid hits sonic velocities, it encounters a normal shock, compressing it, changing the volumetric flow rate downstream of the normal shock.

However, once steady state, steady flow conditions prevail, the mass flow balance should hold, even if the volumetric flow rate is different from input to output.

Think about what happens in the throat of a steam jet. There you have your sonic velocity, shock wave, pressure rise, compression, etc. Imagine your pipe as a jet with a long throat length, in other words, with no diffusing section to convert velocity energy to pressure.

rmw

Don't know how mathematical you want to get, but the answer may well be found by looking at how converging/diverging nozzles work, i.e. in gas turbines or Laval nozzles. Firstly you need to make a distinction between mass flow rate and volumetric flow rate and to establish that at subsonic velocities, the working fluid is considered to be incompressible whereas at supersonic velocity, the effect of compressibility cannot be ignored.

In a pipe of constant diameter and assuming no heat transfer, a subsonic flow velocity will be proportional to the pressure differential between entry and exit of the pipe. Increasing the pressure differential will increase the fluid mass flow rate and as we are assuming incompressibility, the volumetric flow rate will be proportional, i.e. no change in density of the fluid. Now, due to the implicit relationship between cross sectional area and the mach number there is a critical mass flow rate at which the fluid becomes sonic, Ma=1. At this point the pipe or nozzle is said to be choked and further increases of pressure differential result in no further mass flow rate increase at the local sonic velocity. A choked pipe cannot exceed its local sonic velocity. The fluid starts to compress and become denser but not faster. The downstream pressure will become higher causing a backpressure to the system with a noticable associated temperature increase. The only method of increasing the velocity of the fluid further is to allow a controlled expansion of the fluid by increasing the area ratio between throat or pipe and its exit, however the mass flow rate will now be constant regardless of the new higher exit velocity. If the fluid is allowed to expand too quickly, termed 'over-expansion' there will be a sudden deceleration and pressure increases seen as a result of the formation of a series of 'normal shocks' until the fluid has been expanded across these shocks to the exit conditions. Conversely, under-expansion will just decelerate the flow back to sonic or subsonic but with the possible formation of 'normal shock', prantyl meyer waves and viscous damping effects seen as rapid pressure fluctuations or pressure pulsing which will effect the pipe flow far upstream and will in fact cause fluctuations in the volumetric flow rate as the flow would tend towards transonic. To change the required constant mass flow rate of a system yet maintain sonic conditions would require a larger diameter pipe as determined by the area/mach number relationship.

So to answer your questions, subsonic mass flow rate will keep increasing as the pressure difference is increased until the pipe is choked at which point mass flow rate remains constant this will correspond to the a local fluid velocity of Ma=1 which cannot be exceeded. Further increase of the inlet pressure will simply start compressing the fluid and increasing the effective backpressure i.e. a change in fluid density. To accelerate the fluid to velocities greater than Ma=1 the pipe would need to change area as a function of the mach number and the effective back pressure controlled to maintain the flow without the formation of Normal Shocks causing rapid deceleration.

Tip: Design for constant mass flow not volumetric flow !!!!

You've asked several questions at once, which you probably realized. Your respondents have not directly answered your question about sonic/supersonic flow in pipes; they have refered to flow through nozzles (converging, or converg/diverg).

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.

"Axiomatrix" well-described some of the phenomena of nozzle flows. The main point being that the maximum mass flow per unit area is determined for a given set of upstream conditions. If the pressure ratio across a nozzle exceeds the "critical pressure ratio" (i.e. "choked" flow, or sonic velocity in the throat) , then the unit mass flow will NOT increase, but the exit velocity may increase beyond the sonic velocity if the nozzle is properly designed as a converging/diverging nozzle.

In adiabatic (no heat transfer) and steady state, if the critical pressure ration is exceeded, the downstream conditions will be choked (M=1). However, if the upsteam pressure is further increased the mass flow will increase. Further reducing the downstream pressure will not increase the mass flow.

instead of working w/ flow rates, you may want to explore mass rate which will constant at any cross section of the nozzle. With that in mind why care about the varying flow rates within the nozzle.

I think you would need to take a course in compressible flow in order to gain a good understanding of the physical mechanisms and avoiding having to reinvent the wheel.

For a compressible fluid such as steam flowing adiabatically ( no heat transfer) thru a fixed geometry orifice or pipe , the flow may choke acoustically thru a flow area minimum or it can also be choked frictionally ( a long pipe ,so-called Fanno flow).

The flow relationship will be fixed for a fixed area, geometry , inlet pressure,specific compressible fluid and inlet temperature. Once the flow parameters have been determined by a flow test or CFD analysis ( to account for oblique shock waves). For any flow for parameters other than the test parameters, the flow can be prorated up directly proportional to inlet pressure and up directly proportional to the square root of the absolute inlet temperature.

As others have stated, it is neccesary for the pressure drop to be greater than the critical drop in order to have choked flow, which can be predicted once the fluid's ratio of heat capacities is known ( R= Cp/Cv)

One important point that is often mis-stated and/or mis-understood:

When a gas flow is at "choked" conditions (i.e., at sonic velocity), the LINEAR VELOCITY IS AT A MAXIMUM. By linear velocity, I mean ft/sec or m/sec for example.

However, at "choked" conditions, the MASS FLOW RATE IS NOT AT A MAXIMUM. Increasing the upstream pressure will continue to increase the mass flow rate. By mass flow rate, I mean pounds/sec or kilograms/sec for example.

Milton Beychok
mbeychok@xxx.net (replace xxx with cox)
(Visit me at www.air-dispersion.com)

Some very interesting and informative information above.

One point that is missing is that sonic flow is not sustainable in straight pipe. Period. If you enter a pipe at sonic velocity (downstream of a choke for example), friction effects will begin to reduce the velocity immediately and the duration of M=1.0 is meters, not kilometers or even tens of meters.

The Continuity Equation explains that a system with one inlet and one outlet must have exactly the same mass flow rate at the inlet, at the outlet, and at every point between the inlet and the outlet. Any other answer would result in fluids "stacking up" at various points within the system -- a non-tenable situation. Therefore, as the pressure and temperature change across a system, the velocity will change, the volume-flow rate will change, and the mass flow rate will remain constant at every cross section in the system.

A non-zero "constant volume flow rate" with regard to time is not possible. Volume flow rate (in actual terms) is a function of pressure, temperature and compressibility. If pressure does not change with distance from a starting point, then there is zero flow. If you are talking about a "constant volume flow rate at standard conditions" then the "volume flow rate" is really a mass flow rate re-stated in volumetric units.

Supersonic flow is very difficult to acheive, is very energy-intensive to sustain, and will not occur without specifically engineered equipment. Take a look at the specifications for supersonic aircraft. Specifically look at the hp requirement and the fuel-consumption at subsonic and supersonic speeds. The difference is remarkable.

David Simpson, PE
MuleShoe Engineering
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips Fora.

The Plural of "anecdote" is not "data"

mbeychok (Chemical)
WRITES
However, at "choked" conditions, the MASS FLOW RATE IS NOT AT A MAXIMUM. Increasing the upstream pressure will continue to increase the mass flow rate. By mass flow rate, I mean pounds/sec or kilograms/sec for example.

I disagree. Choked flow in steady state refers to the maximum mass flux. For a perfect gas, this occurs at Mach=1.
For an adiabatic pipe, for which exit conditions are sonic, further increasing the upstream pressure will increase the mass flux, however, the exit condition will remain at Mach=1.
Similarly, if the flow is via a variable area, choking occurs at the max mass flux.
To calculate choked two phase single component flow from an adiabatic pipe, one calculates mass flux vs exit pressure. As exit pressurea drops mass flux increases until a max is reached.
That is the condition for choked flow.

zdas04 (Mechanical)WRITES
One point that is missing is that sonic flow is not sustainable in straight pipe. Period. If you enter a pipe at sonic velocity (downstream of a choke for example), friction effects will begin to reduce the velocity immediately and the duration of M=1.0 is meters, not kilometers or even tens of meters.

It is not clear to this writer how in steady adiabatic flow the entrance Mach=1 for a pipe. Unless the pipe length is exactly zero length. Greater or less than M=1.0 is clear to me. Am I misunderstanding how the FANNO line works?

Sailorday128:

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

The same equation in customary USA units is:

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

where:
Q = mass flow rate of the gas, lb/s
C = discharge coefficient
A = discharge hole area, ft²
P = absolute upstream pressure, lb/ft²
g = gravitational acceleration of 32.17 ft/s²
k = Cp / Cv of the gas
M = gas molecular weight
R = Universal Gas Law constant = 1545.3 ( ft-lb ) / ( lbmol ) ( deg R )
T = gas temperature, deg R

As can be seen in the above equation, increasing the upstream gas pressure increases the mass flow rate (even at choked flow conditions).

A careful reading of the following web sites will confirm what I have said:

(1) (2) (3)
The above reference web site (2) presents the choked flow equation in an equivalent but different form, which still shows that increasing the upstream gas pressure increases the mass flow even at choked conditions.

The above reference web site (3) also presents the choked gas flow equation in an equivalent but different form and it also shows that increasing the upstream gas pressure increases the mass flow even at choked conditions.

I am sure that you can also find confirmation of what I said in any good chemical engineering textbook on fluid flow.



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

mbeychok (Chemical)
Q = C A P [ g k M / ( R T ) ]1/2 [ 2 / ( k + 1 ) ] (k + 1) / (2k - 2)
As can be seen in the above equation, increasing the upstream gas pressure increases the mass flow rate (even at choked flow conditions).

PLEASE NOTE: FIX THE UPSTREAM PRESSURE SUCH AS P IN THE ABOVE EQUATION. THEN DECREASE THE BACK PRESSURE UNTIL THE FLOW CHOKES--THAT IS REACHES A MAX. CHECK THE MACH NO AT THE CHOKED FLOW-IT SHOULD BE M=1. OF COURSE AN INCRESASE IN SOURCE PRESSURE WILL INCREASE THE FLOW, IF IT HAD BEEN PREVIOUSLY CHOKED. HOWEVER, AT THE MINIMUM AREA WHERE THE FLOW IS CHOKED AT A NEW FLOW RATE, THE MACH NO. IS STILL UNITY.

Sailoday28:

Quote from your response of Nov.11th:

"OF COURSE AN INCREASE IN SOURCE PRESSURE WILL INCREASE THE FLOW, IF IT HAD BEEN PREVIOUSLY CHOKED. HOWEVER, AT THE MINIMUM AREA WHERE THE FLOW IS CHOKED AT A NEW FLOW RATE, THE MACH NO. IS STILL UNITY."

Quote from your response of Aug.6th:

"If the critical pressure ratio is exceeded, the downstream conditions will be choked (M=1). However, if the upsteam pressure is further increased the mass flow will increase. Further reducing the downstream pressure will not increase the mass flow."

Quote from my posting of November 8th:

"When a gas flow is at "choked" conditions (i.e., at sonic velocity), the LINEAR VELOCITY IS AT A MAXIMUM. By linear velocity, I mean ft/sec or m/sec for example.

However, at "choked" conditions, the MASS FLOW RATE IS NOT AT A MAXIMUM. Increasing the upstream pressure will continue to increase the mass flow rate. By mass flow rate, I mean pounds/sec or kilograms/sec for example."

We are both saying the same thing. On Nov.8th, I said that at choked conditions, the linear velocity is at a maximum (i.e. the linear velocity is at sonic velocity which is M=1) but increasing the upstream pressure will still increase the mass flow. In other words, the meters/sec or feet/sec is at a maximum but increasing the upstream pressure will still increase the kilograms/sec or pounds/sec.

Isn't that exactly what you said on Aug. 6th and Nov. 11th? So I am at a loss as to why you disagreed with what I said on Nov. 8th.

I think the problem is that mechanical engineers tend to use the terminology "mach number=1" which means that the linear velocity is at the speed of sound, and we chemical engineers use the terminology "sonic velocity" which also means the linear velocity is at the speed of sound.

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

mbeychok (Chemical)
My disagreement is with the statement that "However, at "choked" conditions, the MASS FLOW RATE IS NOT AT A MAXIMUM.
Choked flow means that mass flux is a maximum. Further increasing upstream pressure, etc. will increase flux, however, with the new source pressure, lowering back pressure will not further increase flux.

Sailoday28:

One more time and then I quit!! Choked flow does not mean that the mass flow (kg/sec) is at a maximum ... it means that the linear velocity (m/sec) is at a maximum (i.e., the linear velocity is at Mach=1). Yes, lowering the downstream pressure will not increase the mass flow but increasing the upstream pressure will do so. The very fact that you can increase the mass flow by increasing the upstream pressure proves that the mass flow is not at a maximum.

Mass flow equals the gas linear velocity times the cross-sectional area times the gas density ... (m/s)(m²)(kg/m³] = kg/s ... increasing the upstream pressure increases the gas density and that increases the mass flow even though the linear velocity is at a maximum.

Milton Beychok
mbeychok@xxx.net (replace xxx with cox)
(Visit me at www.air-dispersion.com)

My last word on this thread for choked flow:from the below nasa website

The compressibility effects on mass flow rate have some unexpected results. We can increase the mass flow through a tube by increasing the area, increasing the total pressure, or decreasing the total temperature. But the effect of increasing velocity (Mach number) is a little harder to figure out. If we were to fix the area, total pressure and temperature, and graph the variation of mass flow rate with Mach number, we would find that a limiting maximum value occurs at Mach number equal to one. Using calculus, we can mathematically determine the same result: there is a maximum airflow limit that occurs when the Mach number is equal to one. The limiting of the mass flow rate is called choking of the flow. An equation for the choked mass flow rate is given below the box.

mdot = (A* * pt/sqrt[Tt]) * sqrt(g/R) * [(g+1)/2]^-[(g+1)/(g-1)/2]

Sailoday28:

As I said in my very first posting to this thread on Nov. 8th, the point I made is often mis-stated and/or mis-understood ... and that includes NASA as well as many others. Go back through all of my posts and think it through for yourself!!

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

Nozzles don't have the limitations of cylindrical pipes, limitations caused by their constant cross-sectional areas. As said above by various contributors properly designed nozzles enable the interchange of internal and kinetic energy of a fluid as a result of the changing cross-sectional area.

Steam turbines or, for example, ammonia or ethylene expanders, have alternate sets of nozzles and rotating blades through which vapor or gas flow in a steady-state expansion process whose overall effect is the efficient conversion of the internal energy of a high-pressure stream into shaft work.

For subsonic flows in a converging nozzle pressure decreases and velocity increases as the cross-sectional area diminishes. The speed of sound, the maximum attainable speed, is reached at the throat. To allow the speeds to become supersonic and the pressure to drop further would require an increase in cross-sectional area, a diverging section, as in the diffusers of steam ejectors.

To SalvadorT. In short: a converging nozzle can be used to deliver a constant flow rate (at a given P1 upstream pressure) of a compressible fluid into a region of variable pressure by reducing the downstream pressure P2, so the ratio to upstream pressure is below a critical value, which for steam is ~0.55 at moderate temperatures and pressures.

The flow remains constant, and the velocity in the throat sonic, regardless of the P2/P1 ratio as long as it stays below the critical value.