dP is simply the difference between upstream pressure and downstream pressure. If you are exhausting to atmosphere then dP is about equal to gauge pressure upstream (if you disregard the friction drop in the tail pipe, often a reasonable approximation). If you are trying to find the minimum upstream pressure that will give you choked flow, then you need to ask that question.
Katmar has a very good point--sonic velocity is incompatible with the need for a silencer. It is usually more economic to dump your pressure in stages and deal with smaller units of noise than to take it all at once and try to silence sonic flow.
The problem is a common one - choked flow thru a relief valve orifice plate(piping system area minimum- defines max flowrate)the expands thru exhaust pipe and finally exits thru the exhaust silencer.
The flow prediction and pressure distribution defined in the asme sect VIII appendix is based on the bechtel Liao method, but it does not directly recognize the possible restriction caused by the silencer. In my opinion, the problem would be solved as follows:
a) confirm the net open area of the silencer is much larger than the exhaust pipe between relief valve and silencer.
b) initially assume the system will provide a pressure at the relief valve elbow lower than required for full relief valve flow rating.
c) for that design flow , work backwards thru the silencer for the min required pressure entering the silencer. Then confirm that the the velocity of the fluid entering the silencer / exiting exhaust pipe is below sonic , ie mach number < 1.0.
d) for a mach no < 1.0 at the exit of the exhaust pipe, calculate the pressure drop in the exhaust pipe by breaking the pipe into 100 smaller segments ( ie divide total fL/d by 100) and incrementally work backwards to the min required pressure at the inlet of the exhaust pipe. exit of releif valve elbow.
e) confirm this elbow exhaust press is lower than required for full relief valve rating.
f) if the velocity calculated from step (c) is above mach no of 1.0, then there is a shock wave at the inlet to the silencer. Increase the pressure at the outlet of the exhaust pipe ( upstream of the shock wave) until that mach number = 1.0, then use the bechtel Liao method ( fanno curves) as outlined in sect VIII to compute pressures thru the piping system.
g) step d can alternately be computed using the fanno curves, but I'll leave that for a student homework assignment
davefitz (Mechanical)Your method is logical, however, Fanno is usually for constant flow area. My concern, is the proper model for the silencer, even with a larger area.
I doubt a mfgr would give that information for high Mach numbers.
The fanno relationships are used in the formulation of the sect VIII curves- the assumption is a constant area along the length of each component, in particular, the exhaust stack . However, the sect VIII curves limit were limited to low fL/d values- to extend this correlation to longer exhaust pipes one needs to directly use the fanno equations for frictionally choked flow .
davefitz - can you help me find these "curves"? Where in Section VIII are they? I looked at Appendix M and I looked in and around UG.135, but I don't see any specific calculations or curves. Maybe its right iin front of me?
Good stuff. On this topic, is there a useful practical reference for compressible flow in pipe for the practicing engineer? My text from college is great if one is designing rocket engine nozzles but pretty useless for the process engineer in a plant. Thanks!
I've bought a half dozen books that make a shot at real flows, but none of them are much use. I hope to someday write an undergraduate fluids book that might be useful to a practicing engineer. Both examples of choked flow in my undergraduate fluids book are using elaborate converging/diverging nozzles and you end up with M>1.0. Standard Conditions are mentioned in a problem in the introduction, and once in the fluid-statics section--never in relation to flows.
It really is a shame how much esoteric arithmetic is included and how much useful knowledge is omitted.
I guess textbooks are generally written by PhD's that have little experience in the real world.
Other choked flow situations where compressible fluid mechanics are used include:
a) relief/safety valve - flow thru the relief orifice
b) flow thru relief valve exhaust stack
c) reheater /LP steam dump to condenser - flow thru sparger holes into condenser steamspace
d) choked flow across valves,including turbine bypass valves, vent valves, turbine control valves during startup ops
e) drain flow to condenser drip inlets
f) inst air to pnuematic actuators during fast valve stroking
No rockets or jets or missiles involved in powerplants ( unless they are located in the mideast) , but most college profs that write books on compressible fluid mechanics have only been funded thru defense projects, so they only write about defense devices.
The pressure ratio (Downstream Pressure / Upstream Pressure)of a choked flow condition is almost equal to 0.5. You may try to test this by plotting a Fanno line graph and asses what the pressure ratio is at the maximum value of the curve. In plotting the fanno line you need to use very accurate enthalpy values to avoid large errors in the calculation of the flow velocity
Philip Oosthuizen
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