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# non-recoverable losses at discharge of relief valves2

## non-recoverable losses at discharge of relief valves

(OP)
in determining the correct diameter of discharge piping from a relief valve used in steam service, how are the correct specific volume and temperature of the steam at the inlet of the discharge pipe calculated? I assume 10% of the valve set pressure is available at the inlet of the discharge pipe. So is the temperature and specific volume of the steam at that point determined by the available 10% pressure or are the temperature and specific volume based on the steam conditions at the inlet of the relief valve?

Thanks!
Replies continue below

### RE: non-recoverable losses at discharge of relief valves

You definitely want to use conditions downstream of your relief valve to determine fluid properties.  Perform an adiabatic expansion from the temperature, pressure conditions upstream of the relief valve to the pressure(s) downstream of the relief valve.  If you have a pressure-enthalpy diagram for steam (water), locate the point of temperature, pressure upstream of the relief valve and follow the constant enthalpy line to the pressure(s) downstream of your relief valve to find the downstream temperature and specific volume.  Or use other means you have available.

Assuming 10% of the valve set pressure is available at the inlet of the discharge pipe is reasonable but you should consider more than the conditions at the inlet.  If you are performing flow calculations following methods presented in Crane's Technical Paper 410 for compressible flow, seems I remember using the upstream or downstream fluid properties is recommended only when the pressure loss is 10% or less of the upstream absolute pressure.  Otherwise, use fluid properties at the average between the upstream and downstream conditions.  This means you should perform another adiabatic expansion to pressure conditions at the outlet of the discharge pipe.

There are limitations to using the average conditions (pressure loss up to about 40% of the upstream absolute pressure).  If you encounter this situation you may want to breakup your discharge line calculation into 2 or more segments.

I don't have my copy of Crane's handy so you should confirm the criteria I've listed above.  Of course if you are not following Crane's, then .... never mind.

### RE: non-recoverable losses at discharge of relief valves

Determining the pressure drop in a relief valve discharge line is really a trial-and-error procedure (although most people don't bother and do the calculation the way EGT01 explained). A relief valve is not an isenthalpic expansion but an isentropic expansion, unlike other valves. This is because the relief valve really contains a nozzle, not an orifice.

The rigorous way to do it is to assume a pressure at the discharge flange of the PSV and do the expansion isentropically. Get the new conditions. Now, do an isenthalpic expansion from this point to the tail pipe exit. You must check for choked flow at the pipe exit because this may make all the difference in the world. Once you have the exit conditions, work backwards to the discharge flange of the PSV. If you opt to use the isothermal equation, you can then just use the temperature and properties calculated at the exit. Note that there is a discontinuity between this temperature and the one you got leaving the relief valve nozzle. In reality, the two are resolved between the relief valve nozzle exit and the body flange. You can also use the adiabatic equations. In this case, the fluid properties change over the length of pipe. But as EGT01 points out, if the pressure drop is small, you can just use the properties at the tail pipe exit, you can take the average or break the pipe into segments.

If the pressure calculated at the relief valve discharge flange does not equal what you guessed, start with this new pressure and do it again.

The calculation done done this way is a pain, but it is the correct way to do it.

### RE: non-recoverable losses at discharge of relief valves

pleckner is correct that flow through a nozzle is more accurately considered an isentropic process.  Not only is the flow considered adiabatic (little time for heat transfer), the flow is considered frictionless so the process is reversible.  More appropriately then, for an isentropic expansion I believe the equation is
T1/T2 = (P1/P2)^((k-1)/k)

Where k is the isentropic exponent, I believe it is generally acceptable to assume ideal gas and use the specific heat ratio Cp/Cv though there are more rigorous means of determining the isentropic exponent.

pleckner also gives good advice for checking for sonic flow conditions at the end of the discharge pipe and working backwards to the relief valve.  To accurately determine the pressure loss in the outlet pipe it would require an iterative procedure but in some cases you may only want to check that a value is not exceeded.  In that case, if you assume max allowable backpressure at your relief valve and calculate the outlet loss to produce less, then you could say your line was adequately sized.  Further iteration is not necessary since it would only produce a result between your assumed max value and initial calculated value.  Certainly if your initial calculation produces a backpressure greater than assumed, further iteration is required.

### RE: non-recoverable losses at discharge of relief valves

As an added note to my previous post regarding the isentropic expansion, I believe that P2 in the equation will be the greater of the relief valve backpressure or the critical flow pressure (Mach 1).

### RE: non-recoverable losses at discharge of relief valves

The flow is isentropic "up to" the end of the nozzle. At that point you get a lot of non-reversible flow effects (shock , etc.) and the adiabatic model applies (but not the isentropic one).

The piping codes (b31) cover calculation methods with worked examples for the builtup back pressure.

The discharge pressure calc. is only and estimate. As the previous advice suggests, you have to assume a discvharge condition and work backwards.

### RE: non-recoverable losses at discharge of relief valves

Pleckner is right on the money.  Having been involved in Process Safety for numerous years re-verifing existing installations as well as designing many new exhaust systems and common discharge headers with multiple reliefs feeding them; the process Pleckner describes is the correct approach.

Note: this is the one most common mis-understood procedures in relieving systems.  I have seen many grave mistakes made, the most common is not correcting specific volume and temperature.

Don Coffman

### RE: non-recoverable losses at discharge of relief valves

Hacksaw,

Was that B31.l Power Piping where the examples are found?

### RE: non-recoverable losses at discharge of relief valves

In case anyone was wondering, that was a typo in my last post.  I meant to show B31.1 Power Piping but I was being lazy using copy and paste which didn't work like I expected.

Also, I did happen to come across the following from Perry's 7th edition, Section 26 - Process Safety, Vent System Flow Capacity,
"The presence of both liquid and vapor phases in the vent stream is normally treated as a vapor-liquid mixture at equilibrium. .... In practice little error is introduced by carrying out the flash computation at constant enthalpy.  With this simplification, the flash temperature-pressure-composition history can be established before starting the actual flow calculations, thus eliminating the need for repetitive flash calculations at each step in the integration."

### RE: non-recoverable losses at discharge of relief valves

Calculation method is covered in B31.1 Appendix II. I do not have it before me, but that is the reference I have.

The method is not a precise estimate of the pressure distribution but one that produces a conservative estimate for use in piping design.

### RE: non-recoverable losses at discharge of relief valves

The B31.1 and section VIII div 2 approach is based on the method by Liao ( Bechtel). It is based on sizing the vent pipe to avoid blowback in the case where one has a open drip pan at the transition ofrom exhaust elbow to the stack. As such , it is applicable to the releif valves which use a drip pan and are not hardpipe to the exhaust.

The flow is isenthalpic ,, except for the inital instant after valve pop . It is not truly isentropic as large shocks form at the relief orifice , primirily oblique shocks , as sometimes accoutned for by the constant Xt in the ISA compressible flow sizing equations.

In most cases the outlet from the exhaust pipe to atmosphere is choked . The B31.1 approximate equations for steam can be used to estimate this outlet pressure , and if the pressure is above about 30 psia, the flow is choked. If it is choked, the standard Fanno curve analysis can predict the pressure at the inlet to the exhaust stack transition ( ie, from releif valve outlet flange to stack). The B31.1 curves can be extended as per:
fL/d = (1-M2)/(rM2) +((r+1)/(2r)*Ln{(r+1)*M2/2/(1+(r-1)/2@M2)}
r= Cp/Cv
M= mach no,  M2 = M^2
fL/d = sum of equivalent fL/d include bends, accel, etc .

The pressure at the inlet to the stack is a function of the stack exhaust pressure P* and inlet mach number M:
P/P* = 1/M*{(r+1)/(2+(r-1)M2)}

### RE: non-recoverable losses at discharge of relief valves

The B31.1 and section VIII div 2 approach is based on the method by Liao ( Bechtel). It is based on sizing the vent pipe to avoid blowback in the case where one has a open drip pan at the transition ofrom exhaust elbow to the stack. As such , it is applicable to the releif valves which use a drip pan and are not hardpipe to the exhaust.

The flow is isenthalpic ,, except for the inital instant after valve pop . It is not truly isentropic as large shocks form at the relief orifice , primirily oblique shocks , as sometimes accoutned for by the constant Xt in the ISA compressible flow sizing equations.

In most cases the outlet from the exhaust pipe to atmosphere is choked . The B31.1 approximate equations for steam can be used to estimate this outlet pressure , and if the pressure is above about 30 psia, the flow is choked. If it is choked, the standard Fanno curve analysis can predict the pressure at the inlet to the exhaust stack transition ( ie, from releif valve outlet flange to stack). The B31.1 curves can be extended as per:
fL/d = (1-M2)/(rM2) +((r+1)/(2r)*Ln{(r+1)*M2/2/(1+(r-1)/2*M2)}
r= Cp/Cv
M= mach no,  M2 = M^2
fL/d = sum of equivalent fL/d include bends, accel, etc .

The pressure at the inlet to the stack is a function of the stack exhaust pressure P* and inlet mach number M:
P/P* = 1/M*{(r+1)/(2+(r-1)M2)}

### RE: non-recoverable losses at discharge of relief valves

One my last reading of App. II, it primarily is used to determine the reaction forces for extended safety valve discharge piping where discharge in the immediate vicinity of the valve is not possible.

"A Method for Determining the Design Pressures and Velocities in the Discharge Elbow and Vent Pipe to Calculate Reaction Forces."

I am not familar with versions prior to 1996. The short comings as an exact pressure calculation are obvious.

### RE: non-recoverable losses at discharge of relief valves

To davefitz: I hope you are not trying to say the calculation of a relief valve is the same as a control valve as you seem to imply with your reference to the ISA calculatons and the use of the constant Xt. The relief valve is most definitely a nozzle, not an orifice and it is isentropic (with some inefficiencies) and not isenthalpic. These inefficiencies are accounted for in the k value which is obtained by direct test in an ASME approved laboratory on an ASME approved test stand.

The flow in the piping after the relief valve (starting at the outlet flange) is isenthalpic, however.

To EGT01: Careful about that statement in Perry's. If it is implying the flash across the relief valve can be treated as an isenthalpic expansion with little error, I will beg to differ as it all depends. Taking steam; if the conditions of the steam just at relief are too close to the saturation line, then an isentropic expansion (assuming we agree it is isentropic; if not then no need to continue the discussion) may very well put you into two-phase flow. An isenthalpic expanson won't. I don't see that as a minor error.

### RE: non-recoverable losses at discharge of relief valves

pleckner:
No , I am not suggesting that one size the relief valve using the ISA equations, only trying to emphasize that there are large increases in entropy at the valve vena contracta, and these are usually oblique shocks. A measure of the magnitude of the importance of oblique shocks is via the Xt factor. I agree that  only the code approved method of sizing relief valves must be used.

### RE: non-recoverable losses at discharge of relief valves

Pleckner,

Your comments are appreciated and your words of caution are duly noted.  I do agree that flow through a nozzle is isentropic (that is well established thermodynamic theory) and I continue this discussion only to further my understanding of what I've seen in the literature.

Since my last post, I've discovered 2 more articles that indicate or imply that isenthalpic flash from relieving pressure to downstream backpressure would be sufficient.  However, there may be a pattern to these claims and it seems to be associated with two-phase flow conditions.

Just conjecture on my part but maybe it is the presence of liquid that would allow such approach without much error.  Looking at a pressure-enthalpy diagram, for systems that remain liquid or are low in vapor fraction, the change in entropy with change in pressure doesn't appear to be as great compared to systems that remain vapor or are high in vapor fraction.  Also, maybe since a nozzle seems to be less "efficient" for liquid flow than for vapor flow (as evidenced by typical valve discharge coefficients) has some bearing on how closely the process follows the theory.

Anyway, additional comments by all are welcomed.  Below are the additional references and I've included a full excerpt from Perry's.

From Perry's 7th edition, Section 26 - Process Safety, Vent System Flow Capacity,
"The presence of both liquid and vapor phases in the vent stream is normally treated as a vapor-liquid mixture at equilibrium conditions.  The adiabatic flashing of the stream as the pressure falls along the flow path is usually computed by conventional flash distillation methods. In principle, the flash path should be isentropic for flow in devices exhibiting low friction losses (nozzles and short pipes). For friction flow, the sum of the stream enthalpy, kinetic energy, and potential energy is held constant along the path. In practice, little error is introduced by carrying out the flash computations at constant enthalpy.  With this simplification, the flash temperature-pressure-composition history can be established before starting the actual flow calculations, thus eliminating the need for repetitive flash calculations at each step in the integration."

Leung, J.C., "Easily Size Relief Devices and Piping for Two-Phase Flow", Chem.Eng.Progress, p.33, Dec-1996.  From the section discussing Nozzle Flow -
"For two-phase flashing flow, the P-v relation should be given by a constant entropy flash calculation.  However, in practice, the result is almost indistinguishable when a constant enthalpy (isenthalpic) flash calculation is used; the latter seems to be much more common with commercially available flash routines."

Later in the same article there is some discussion about the accuracy of the above assumption.
"This deviation is due to the difference between the assumed isenthalpic and the actual isentropic expansion processes."

Fauske, H.K., "A Practical Approach to Capacity Certification", Chemical Processing, Feb-2003.  This article can be accessed from the internet
http://www.chemicalprocessing.com/Web_First/CP.nsf/ArticleID/CBOH-5JQR8Y/
In the section of the article discussing sizing of the outlet piping the following example is given for a 1 percent steam-water mixture with a set pressure of 35 psig.
"The maximum allowable backpressure, P1 = 35 x 0.1 + 14.7 = 18.2 psia. Considering an isenthalpic flash to this pressure from 53.2 psia results in x1 = 0.075."

From API RP-520 Part I, 7th edition, Jan-2000, when following the "Omega" method for sizing for two-phase flow, the following is noted:
"v9 = specific volume evaluated at 90% of the PRV inlet pressure Po (ft3/lb). When determining v9, the flash calculation should be carried out isentropically, but an isenthalpic (adiabatic) flash is sufficient."

### RE: non-recoverable losses at discharge of relief valves

EGT01:

Well done with your references. I have all these as well. Leung's article should be the bible for two phase flow (in my opinion).

Yes, I too have used isenthapic flashes in the past based on these references...until I came up against a steam relief that was near the saturation line. Then I learned to do it the more correct and rigorous way. With today's simulators, there is no reason not to do an isentropic flash, as I pointed out earlier on. On a personal note, if it is practical, I get anal about doing things the technically correct way and not use simplifications; but this is me.

### RE: non-recoverable losses at discharge of relief valves

returning to the original question, in no way can the flow thru the entire exhaust system be deemed isentropic. At best, the flow local to the nozzle throat may be approximated as isentropic , but once shock waves are formed, the entropy increases radically.

For purposes of computing the pressure at the outlet of the  nozzle ( or inlet of the exhaust system) , isenthalpic processes are used( along the exhaust system) , as outlined in the B31.1 appendix ( Liao method). One works backwards from the known bounday condition of outlet choked flow back to the pressure at the inlet of the exhaust system.

### RE: non-recoverable losses at discharge of relief valves

To davefitz:

Allow me to quote a sentence from my July 15 response to you, "The flow in the piping after the relief valve (starting at the outlet flange) is isenthalpic, however."

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