• Create Account

## ChExpress Blog - 5/14/13

Read the latest news from the chemical industry in Christa's blog.

## Featured Articles

Check out the latest featured articles.

## New Article

Product Viscosity vs. Shear

## Featured File

Heat Loss from Pipes

## New Blog Entry

Perforated Pipe Distributor Sizing Calculation- posted in Ankur's blog

# Psv Sizing

k factor

42 replies to this topic

### #1 vnpetroleum

vnpetroleum

Veteran Member

• Members
• 33 posts

Posted 14 October 2011 - 03:20 AM

Dear All,

I am very new in sizing PSVs. I have a question relating to k factor. k is calcualted
- for idea gas: k = Cp/(Cp-R)
- for real gas: k = Cp/Cv
As stated in API- k for ideal gas is used to PSV sizing. However, some engineering companies are using k for real gas (which is calculated from HYSYS).

Thanks - vnpetroleum

Edited by vnpetroleum, 14 October 2011 - 03:23 AM.

### #2 ankur2061

ankur2061

Gold Member

• Forum Moderator
• 1,998 posts

Posted 14 October 2011 - 04:07 AM

vnpetroleum,

Specific heat at constant volume (Cv) values for most gases is not directly available from thermodyanamic property tables found in books and literature. What is available, is the specific heat at Constant pressure (Cp). k values based on the ideal gas equation are:

k = Cp / (Cp - (R/MW))

where:

Cp = heat capacity at constant pressure, kJ/kg-K

R = universal gas constant, 8.314 (kJ / kmol-K)

MW = molecular weight, kg / kmol

'k' values based on ideal gas generally give lower values then 'k' values for real gases. When using it in sizing equations as per API 520 for relief valves for gas service a lower 'k' value provides a higher relief rate and relief area compared to a higher 'k' value.

What you get is a more conservative relief valve size with a 'k' value based on ideal gas. The trend today of doing optimum sizing and not conservative over design dictates the usage of the 'k' value based on real gas behaviour at elevated pressures and temperatures.

I still prefer to use the 'k' value based on the ideal gas behaviour and the relief valve sizing excel sheets I have used and also developed consider the 'k' value based on ideal gas behaviour.

However, I have posted a spreadsheet for calculating 'k' values based on 'real gas' behaviour on the forum. This spreadsheet is based on an empirical correlation and has its limitations in comparison to what a simulation software like HYSYS will calculate based on rigorous calculations using EOS like PR or SRK.

http://www.cheresour...0185#entry30185

Hope this helps.

Regards,
Ankur.

### #3 paulhorth

paulhorth

Gold Member

• ChE Plus Subscriber
• 306 posts

Posted 14 October 2011 - 04:14 AM

vnpetroleum,
Use the real gas k value as estimted by HYSYS. The ideal gas value is an approximation which was included in the API RP in the days before the real value was available in simulations, or, for those engineers who do not have access to a simulation package. The approximation is acknowledged in Appendix B of API RP 520.
The isentropic exponent k appears in the API capacity equation because k is part of the expression for pressure at sonic velocity ( in the throat of the PSV) as a function of upstream pressure, which is embedded in the API equation.
A good value for k can be obtained from k = ln (p1/p2)/ln (rho 1/rho 2), where subscript 2 refers to isentropic conditions at the throat (sonic velocity). I have checked this expression against HYSYS real k and they match.

Paul

### #4 ankur2061

ankur2061

Gold Member

• Forum Moderator
• 1,998 posts

Posted 14 October 2011 - 04:20 AM

Paul,

I wanted to give you a star for your posting and instead I clicked the wrong button and it shows a derating. I am truly sorry. My apologies. Any other member reading can probably remove the derating.

Regards,
Ankur.

### #5 paulhorth

paulhorth

Gold Member

• ChE Plus Subscriber
• 306 posts

Posted 14 October 2011 - 02:53 PM

Ankur,
Absolutely no problem - Thank you for your generous reaction, I appreciate it.
Paul

### #6 PaoloPemi

PaoloPemi

Gold Member

• Members
• 177 posts

Posted 15 October 2011 - 02:04 AM

a) that's a very important point which can introduce signifative differences in calculated area (consider the case of a real value 1.12 vs. 1.45 ideal for some fluids), several years ago I have had a long disussion with a valve's manufacturer showing the derivation of the API formula from ideal (not real) k, he also noticed that european translations (BS-4126 etc.) do not mention real k values, also there is the case of the values provided for water (used as test case).

actually you can calculate "real" cp/cv with a thermodynamic library which links directly with Excel, there is a free "student's" version of Properties available here

http://www.prode.com/en/properties.htm

which does the work with rigorous derivatives (H and V),
if ankur2061 gives the permission I can easily post the code which utilize the library for calculating rigorously all thermodynamic properties.

I use the "rigorous" values for design purposes while I prefer the "ideal" values when selecting a valve and comparing my calc's with psv manufacturer's data sheets

Edited by PaoloPemi, 15 October 2011 - 02:12 AM.

### #7 ankur2061

ankur2061

Gold Member

• Forum Moderator
• 1,998 posts

Posted 15 October 2011 - 07:16 AM

PaoloPemi,

I am not sure what permission you are asking from me. Do you want to edit the spreadsheet I have posted for Heat Capacity of Real Gases.

Can you please clarify as to what exactly you want from me.

Regards,
Ankur.

### #8 PaoloPemi

PaoloPemi

Gold Member

• Members
• 177 posts

Posted 15 October 2011 - 03:12 PM

Ankur,
editing is not required, I did intend to suggest a simple alternative for calculating rigorous cp, cv values, in the specific Properties adds methods for calculating cp and cv (gas and liquid phases) in Excel as macros for example =StrGCp(1) or =StrGCv(1) to get directly the values in Excel cells, see the manual for additional information.

### #9 sheiko

sheiko

Gold Member

• ChE Plus Subscriber
• 677 posts

Posted 15 October 2011 - 04:56 PM

vnpetroleum,
Use the real gas k value as estimted by HYSYS. The ideal gas value is an approximation which was included in the API RP in the days before the real value was available in simulations, or, for those engineers who do not have access to a simulation package. The approximation is acknowledged in Appendix B of API RP 529

The following article published in Chemical Engineering Magazine, November 2003 by Aubry Shackelford, "Using the Ideal Gas Specific Heat Ratio for Relief-valve Sizing", recommends using the ideal ratio. This recommendation has been aknowledged by API.

Edited by sheiko, 27 November 2011 - 09:03 AM.

### #10 ankur2061

ankur2061

Gold Member

• Forum Moderator
• 1,998 posts

Posted 16 October 2011 - 03:17 AM

PaoloPemi,

If you can provide an excel based calculation sheet that can provide gas 'k' values for a wide range of temperatures and pressures (real conditions) that has results matching with what is calculated using simulation software such as HYSYS you would be doing all of us a great service and specially to those who do not have access to simulation software.

The spreadsheet I have posted is quite reliable over a certain range of temperatures and pressures for many hydrocarbon gases. It does not however predict correctly the 'k' values for steam.

Regards,
Ankur.

### #11 PaoloPemi

PaoloPemi

Gold Member

• Members
• 177 posts

Posted 16 October 2011 - 11:52 AM

Ankur,
there are several Excel add-on tools which can provide "rigorous" properties for pure fluids and mixtures with accuracy equivalent to process simulators, I urilize Prode Properties (written in C++) and Nist Refprop (written in Fortran).
I prefer to utilize Excel (with these tools) for solving a large number of problems, even columns, not including recycle streams, while I use the simulator for solving a complete plant.

### #12 ankur2061

ankur2061

Gold Member

• Forum Moderator
• 1,998 posts

Posted 16 October 2011 - 12:53 PM

PaoloPemi,

Could you provide the links for these excel add-ins. That would be great for the readers and members of "Cheresources".

Regards,
Ankur.

### #13 functionlake

functionlake

Brand New Member

• Members
• 1 posts

Posted 17 October 2011 - 03:16 AM

Dear all,

I also cope with the similar thing. Can you summarize the main ideal for this issue? I have seen a lot of arguments before, but also have a lot of ideals and unclear destination. That makes confusing.

Is Real K used for sizing PSV? Are you sure using Real K will not establish a undersized PSV?

In my point, I need a truly clear explanation. Could you state why it happened based on thermodynamic principle aspect?

Thanks,

Lake

Gold Member

• ChE Plus Subscriber
• 596 posts

Posted 17 October 2011 - 04:50 AM

1. Theoretically, k-real should be used.
2. Practically, can use either one.My recommendation, use the values published by API RP 520 (Table 7).
3. This is because, after calculating the orifice area required, we always select the next higher "standard size".
4. Sometime, we need also to use our judgement. For example, the need to have two PSV instead of only one PSV.
5. This is to avoid chattering when discharging at much lower rate.
6. For example, the PRV capacity is 100,000#/h, and one of the emergency scenario (e.g. loss of cooling water) discharging flowrate is only 20,000#/h, then chattering will occur under that scenario.

### #15 PaoloPemi

PaoloPemi

Gold Member

• Members
• 177 posts

Posted 17 October 2011 - 05:03 AM

Ankur,
there is a free (limited database) version of Excel add on for thermodynamic calc's (including cp,cv) at www.prode.com.
NIST has another good tool (Refprop), also for this tool there is a free version (but database has only a few components).
I agree with you that it is very useful to have access from Excel to rigorous thermodynamic properties, you can export data from a simulator but when that is not available a library is a much more competitive solution.

functionlake
there are frequent discussions about use of ideal or real k when sizing psv, there are many reasons to use ideal k (original formulation, conversions from air or steam for rated capacity etc.), as said I prefer real k for preliminary design and ideal k when selecting a psv but that is my personal point of view

### #16 ankur2061

ankur2061

Gold Member

• Forum Moderator
• 1,998 posts

Posted 17 October 2011 - 05:29 AM

PaoloPemi,

First of all thanks for the resources for the thermodynamic properties. I will definitely have a look at it.

As I had earlier mentioned and which I reiterate, I prefer to use the ideal 'k' value for reasons of having a conservative design compared to real 'k' value.

Regards,
Ankur.

### #17 paulhorth

paulhorth

Gold Member

• ChE Plus Subscriber
• 306 posts

Posted 17 October 2011 - 01:49 PM

Dear all,

Ankur is right to say that in most cases, the real k is higher than the ideal k value, and using the ideal k value in the API equation results in a larger calculated area, which is thus conservative.
However,about 12 years ago I was looking into this subject and I found that this is not always the case. I checked the data for propane vapour, and it turns out that the real k value for propane, over a range of pressures and temperatures, is less than the ideal value, and in fact is less than 1.0 . Thus, a relief valve for propane vapour would be undersized if the ideal k value was used. The undersizing was between 2% and 8% over the range 10 to 40 bar, my old results show.

I also tried various blends of propane and methane, and the effect was still there, though smaller, with a 70% propane - 30% methane mix.

The Hysys version available to me at the time did not calculate the real Cp/Cv, so I derived the isentropic exponent k using a property table and the definition k = ln(p1/p2)/ln (rho 1/rho 2)isentropic.

The same effect may also be true for other vapours near the dewpoint, but I did not check any other cases. Likewise, I don't know the comparison for non-hydrocarbon gases.

I did some relief valve sizing calculations for the propane cases using the derived real k values and the API equation, and as a check I also used the HEM method for sizing, which uses rigorous thermodynamics explicitly, without an input for k, and the API method came out very close, about 2% to 3% higher in calculated area than HEM. This difference can be attributed to the API discharge coefficient of 0.975, whereas in the HEM, I used 1.0.

HEM was developed for two-phase flow, but the principle is to treat the two phases as a single phase (Homogeneous Equilibrium), so applying the same method to a single phase is quite consistent.

Imperial College has verified HEM experimentally in a research programme in 1998 and 1999, I have the tables of results. The discharge coefficient of 1.0 for propane is taken from these. Basically this means that the measured flows exactly matched the HEM calculations.

It is this work that is my basis for recommending the use of the real k value in the API equation. If the ideal k is known to be higher correction- LOWER, then that value can be used, but if the resulting larger area then triggers the next standard size of orifice to be selected, that outcome is not necessarily optimum engineering, as the piping and block valves will all end up a lot larger.

Paul

Edited by paulhorth, 17 October 2011 - 05:06 PM.

### #18 ankur2061

ankur2061

Gold Member

• Forum Moderator
• 1,998 posts

Posted 17 October 2011 - 03:51 PM

Paul,

I did a quick check for 'k' values for Propane vapor using HYSYS and my spreadsheet and with increasing temperature and pressure. The 'k' values start decreasing close to the point where there is a phase change from vapor to liquid as shown by HYSYS in the case of 473.15 K. There were certainly two funny values for 'k' with HYSYS which I have highlighted in bold red font in the attached excel sheet.

I am not sure what to make of it. Let me know your comments.

Regards,
Ankur.

Edited by ankur2061, 18 October 2011 - 05:17 AM.

### #19 paulhorth

paulhorth

Gold Member

• ChE Plus Subscriber
• 306 posts

Posted 18 October 2011 - 04:37 AM

Ankur,
Thank you for the data, this is most interesting. But I feel this discussion may be getting too lengthy for this forum.

Your values for k are different from mine - but I think the explanation may be that we are tabulating different parameters.
As practising engineers we sometimes need reminding that the isentropic exponent k is equal to Cp/Cv FOR IDEAL GASES ONLY. For real gases, and particularly for vapours near the dewpoint, k firstly is not constant, and secondly is not necessarily equal to the real Cp/Cv., because such gases do not exactly follow the law p(v)^k = constant even for an isentropic process.
Allow me to quote from my thermodynamics textbook:

For a vapour, there is no simple and unique relation between p and v which applies to every isentropic expansion and compression. Nevertheless it is sometimes convenient to assume an approximate relation of the polytropic form p(v)^k = constant. k then mecomes the index of isentropic expansion although it is not necessarily equal to the ratio of specific heats. The value of k can be found by substituting the values of p and v at the end states in the expression p1(v1)^k = p2(v2)^k. There is no unique value for k, it varies with the end states.

Rogers & Mayhew, "Engineering Thermodynamics, Work and Heat Transfer", 1967.

My vaules for k for propane were calculated as described in this quote. You have tabulated the real Cp/Cv from Hysys, and indeed the two properties are different. Your values are mainly in the range 1.1 to 1.2, whereas my k values are from 1.05 to as low as 0.85 for the same range of conditions.

Part of the difference may be due to the inaccuracy of the equation of state (I used Peng Robinson) in the region near to the dewpoint. Your "funny" values are also in this region.

However, when applied to the API equation, these low k values yield results which match HEM (also employing simulated data from Peng Robinson). This demonstrates that the "approximate relation" of (p(v)^k)isentropic = constant (basis of the API equation) matches the enthalpy change equals kinetic energy relaton which is the basis of HEM. Values calculated by HEM have been matched to experimental results, but I don't remember which equation of state Imperial College used in this work.

Conclusion? When using the API equation to size the relief orifice, the engineer can be generally confident in using the real or ideal k value, but in the case of a vapour close to the dewpoint, should check the sensitivity to different k and should use a simulator to generate a value for k as defined above, not as Cp/Cv.

Paul

### #20 ankur2061

ankur2061

Gold Member

• Forum Moderator
• 1,998 posts

Posted 18 October 2011 - 05:16 AM

Paul,

The excel table I generated with 'k' values presented from HYSYS also uses P-R EOS. I have updated this excel table by adding another column with ideal gas 'k' values. Also there was some error in the 'my spreadsheet' column values because I did not provide the correct input for the Cp values which is also a function of temperature. The revised excel workbook is attached and the old workbook is deleted.

Regards,
Ankur.

#### Attached Files

Edited by ankur2061, 18 October 2011 - 05:18 AM.

### #21 PaoloPemi

PaoloPemi

Gold Member

• Members
• 177 posts

Posted 18 October 2011 - 05:35 AM

paulhorth.
my point of view is that for real gases the expansion (chocked flow) takes place at constant entropy,
while API formulation assumes ideal, constant cp/cv, that's not a big problem when you compare these values with test values and introduce a safety factor as usual, but if you change the reference system in some cases large errors may be generated...

Ankur,
I am not sure to understand your VLE data for propane, according my data critical temperature is 369.842 K, does your simulator calculate a liquid phase above critical temperature for a pure fluid (not a mixture) ? Instead I would expect a dense phase.
Anyway, you can compare k values generated by your simulator with those calculated by Prode Properties (I assume you have Peng Robinson and base Alpha correlation) results are not much different, note that for Properties at 50 Bar.a you are in dense gas phase (not liquid) being a pure fluid above critical temperature.

T (K) P(Bar.a) k
373.15 40 2.2812
373.15 45 15.44587
373.15 50 3.56443

the large cp near critical point is quite normal, if you go closer (to the critical point) you'll get larger values...
For your reference I calculated also the values at 323.15 K , well below critical temperature, Prode Properties calculates a saturation pressure of 17.089 Bar.a, I calculated a value (17.088 Bar.a) very close to saturation point
T (K) P(Bar.a) k
323.15 17 1.3661
323.15 17.088 1.369689

as you see for this specific case there are not large differences even very close to phase boundary

Edited by PaoloPemi, 18 October 2011 - 05:45 AM.

### #22 paulhorth

paulhorth

Gold Member

• ChE Plus Subscriber
• 306 posts

Posted 18 October 2011 - 02:23 PM

Paolopemi,

my point of view is that for real gases the expansion (chocked flow) takes place at constant entropy,

I agree completely, this was the basis for all of my posts, I hope I made that clear. That is why k is the ISENTROPIC exponent, for gases which follow the expression p.(v)^k = constant for a constant entropy change.

but if you change the reference system in some cases large errors may be generated...

I don't understand what you mean here. What reference system?

Paul

### #23 PaoloPemi

PaoloPemi

Gold Member

• Members
• 177 posts

Posted 19 October 2011 - 01:44 AM

"my point of view is that for real gases the expansion (chocked flow) takes place at constant entropy,
while API formulation assumes ideal, constant cp/cv, that's not a big problem when you compare these values with test values and introduce a safety factor as usual, but if you change the reference system in some cases large errors may be generated..."

I don't understand what you mean here. What reference system?
Paul

Paul
for real gases k (cp/cv) is not constant, it depends from operating conditions, if you go close to the critical point large values for cp (and k) are found as noticed by Ankur.
Ideally one should integrate from inlet pressure to discharging pressure but that's require a series of complex calc's.
On the other hand API formulation which consider a constant (ideal) cp/cv has been proven successful and it's a " Acknowledged Standard" so why risk to undersize a valve and go into troubles ?
I think you are correct to verify both cases but (for the mentioned reasons) I would suggest to be prudent when selecting the orifice.

Gold Member

• ChE Plus Subscriber
• 596 posts

Posted 19 October 2011 - 03:47 AM

1. API RP 520 recommends k=1 for unknown case. This gives C = 315 in the equation.
2. This gives conservative orifice size
3. It is ok to oversize, except for chattering which is easy to resolve
4. However, may result in "explosion" if undersize
5. In process design, we sometime need to use some element of "art" under the supervision of "good judgement"
6. We may spend a lot of time "refining" the value of k. However, at the end the day, we may select the "next bigger standard size"

Edited by S.AHMAD, 19 October 2011 - 03:55 AM.

### #25 paulhorth

paulhorth

Gold Member

• ChE Plus Subscriber
• 306 posts

Posted 19 October 2011 - 03:57 AM

Paolo,

for real gases k (cp/cv) is not constant, it depends from operating conditions, if you go close to the critical point large values for cp (and k) are found as noticed by Ankur.

Yes, I know that, I said this in an earlier post. Furthermore, k (the isentropic exponent) is not equal to Cp/Cv for a real gas or vapour. I would propose to take the average k value calculated between the inlet pressure and the throat pressure.

I still do not understand what you meant by "reference system".