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# Theoretical Compressor Discharge Temperature Equations By Gpsa 13Th Edition I had mentioned in one of my earlier blog entries that the new GPSA Engineering Databook 13th Edition is now available and there have been many changes in it from the previous edition. Refer the link:

http://www.cheresour...ition-si-units/

Compressors have been a focal point of interest for me in the last couple of years and I try to keep myself updated on the subject. While going through Section 13 (Compressors & Expanders) of the new GPSA Databook, I found that they have now introduced new equations for compressor discharge temperature for both centrifugal and reciprocating compressors.

The old texts and literature for compressors used to express the compressor discharge temperature by means of the following equations:

T2 = T1*(P2 / P1)(k-1) / k -------------(1)

T2 = T1*(P2 / P1)(n-1) / n -------------(2)

where:

T2 = discharge temperature, K or R
T1 = suction temperature, K or R
P2 = discharge pressure, kPaa or psia
P1 = suction pressure, kPaa or psia
k = isentropic exponent (also known as the ratio of specific heats, Cp / Cv)
n = polytropic exponent and expressed in equation form as:

n / (n-1) = [k / (k-1)]*ηpoly

The GPSA Engineering Databook proposes a new set of equations for the compressor discharge temperature saying that these provide better results compared to equations (1) and (2). These equations can be written as follows:

ΔTisentropic = T1* [(P2 / P1)(k-1)/k - 1] / ηisen ----------------(3)

ΔTpolytropic = T1* [(P2 / P1)(n-1)/n - 1] / ηpoly ----------------(4)

where:
ηisen = isentropic efficiency expressed as a decimal
ηpoly = polytropic efficiency expressed as a decimal

T2 = T1 + ΔTisentropic ----------(5)

T2 = T1 + ΔTpolytropic ----------(6)

The polytropic efficiency for a centrifugal compressor can be found as a function of the inlet volume flow as per one of my earlier blog entries which correlates the polytropic efficiency as a function of the inlet volume flow at the following link:

http://www.cheresour...et-volume-flow/

Hope the readers of my blog and specially those who have interest in compressors like this blog entry. I hope to have some comments and observations on this blog entry from the readers of my blog.

Regards,
Ankur. mofidi

Just one remark and question that I would like to add;

The Isentropic exponent shall be calculated in the average suction and discharge temperature e.g. (Cp/Cv)@((Ts+Td)/2).

The question that I have in this regard,

In process simulation software's' like HYSYS, the exponent is calculated in the mean temperature or just suction temperature is considered?

If the mean temperature is used in HYSYS, then there should be a sort of iteration embedded in the compressor icon of HYSYS.

Regards,

Amir PaoloPemi

I have a different software (Prode Properties)

Prode has several methods to simulate compression stages including a proprietary method able to solve a compression stage with phase equilibria such as the case of wet gas,

for gas phase (without phase equilibria) generally I prefer the Huntington method which is a iterative method, you may find the details in the paper Evaluation of Polytropic calculation methods for turbomachinery performace (I did a copy and paste from Prode Properties operating manual), the other "classic" method is the ASME PTC10 (Schultz) procedure, results are quite close also if I consider Huntington (and Prode method) more accurate.

When a client requires a   formulation as that in GPSA manual I create a Excel page (Prode works directly in Excel) and (for GPSA formulation) I use the properties calculated at average values (p,t).

as you noted, the limits of these methods are that for real fluids properties as cp, cv density etc. are not constant, ASME PTC 10 introduces specific corrections while in Prode method values are recalculated at each step. chere

This equation is also used for predefining expander outlet temperature, but it assumes no liquid formation.

T2 = T1+T1* [(P2 / P1)(k-1)/k - 1] * ηisen Flyou

Hello Ankur,

Thanks a lot for your interesting article.

I am a process design engineer working on refining projects and I often specify compressors for various units.

I do not have the 13th edition of the GPSA, but only the 12th.

In the 12th edition, the delta temperature of reciprocating compressors is calculated with the adiabatic method where DT=T1*[r^((k-1)/k)-1]

For centrifugal compressor, the delta temperature can be calculated with 2 methods:

• Isentropic method where DT ideal= T1*[r^((k-1)/k)-1] and DT actual= DT ideal / ηis
• Polytropic method where DT poly= T1*[r^((n-1)/n)-1]

In the 13th edition, the formalism of the delta temperature calculations for the adiabatic and polytropic method seems to be inspired of the isentropic method of the 12th edition with a DT ideal and a DT actual.

The power consumed by the compressor can be calculated by: Power = W*H/η where W is the mass flow rate, η the efficiency and H the head which is directly proportional to DT.

In the 13th edition, has this formula changed to Power = W*H, without mentioning the efficiency that is already included in the head term (H)?

If not, the thing I would like to point out is that as the power consumed in a compressor is proportional to the delta temperature, the power calculated in the 13th edition will then be higher than in the 12th edition: Power 13th = Power 12th / η. Is it correct?

For a centrifugal compressor for example, the polytropic efficiency is often close to 76%. It means that the calculated power of the compressor will be (1/0.76=1.32) 32% higher when calculated with the 13th edition compared to the 12th. It seems to be a very huge difference for the same compressor! In my opinion, this will lead to an unnecessary very large oversizing of the driver for the service required.

Thanks ankur2061

Flyou,

I had only extracted bits of information in my blog entry.

Here is what exactly the 13th Edition of GPSA has to say about discharge temperature for centrifugal compressors:

Isentropic Process:

The approximate theoretical discharge temperature can be
calculated from:

ΔTideal = T1*[(P2/P1)(k-1)/k – 1]

T2ideal = T1 + ΔTideal

The actual discharge temperature can be approximated:

ΔTactual = T1*[(P2/P1)(k-1)/k – 1] / ηis

T2 = T1 + ΔTactual

GP = (w)*(His) / (ηis)*(3600)

Polytropic Process:

The approximate theoretical discharge temperature can be
calculated from:

ΔTideal = T1*[(P2/P1)(n-1)/n – 1]

T2ideal = T1 + ΔTideal

The actual discharge temperature can be approximated:

ΔTactual = T1*[(P2/P1)(n-1)/n – 1] / ηp

T2 = T1 + ΔTactual

GP = (w)*(Hp) / (ηp)*(3600)

The various terms are defined as follows:

P1 = Pressure at compressor inlet, kPa(abs)

P2 = Pressure at compressor outlet, kPa(abs)

T1 = Temperature at compressor inlet, K

T2 = Temperature at compressor outlet, K

k = isentropic exponent (Cp / Cv)

n = polytropic exponent

ηis = isentropic efficiency

ηp = polytropic efficiency

GP = Gas Power excluding mechanical losses, kW

w = mass flow of compressor, kg/h

His = Isentropic Head, kN.m / kg

Hp = Polytropic Head, kN.m / kg

Hope this clarifies what is mentioned in the GPSA 13th edition.

Regards,

Ankur. Flyou

Thank you Ankur for your clarifications.

My point is that if we take for example the polytropic method for the calculation of a centrifugal compressor, as you said, the Delta Temperature of the 13th edition is higher than the Delta Temperature of the 12th edition:

ΔT(13th)= ΔT(12th) / ηp

As the polytropic Head of the compressor is proportional to ΔT, it means that:

Hp(13th)= Hp(12th) / ηp

Finally, as the Gas Power is proportional to Hp, it means that:

GP(13th)= GP(12th) / ηp

For the example of a centrifugal compressor with a polytropic efficiency ηp of 0.76, the Gas Power calculated (but also the ΔT and the Head) with the 13th edition will be (1/0.76=1.32) 32% higher than when calculated with the 12th edition. This makes a very large difference in the consumed power for a same compressor calculated either with the 12th or 13th edition.

Is this correct? Do you get my point?
Thank you for your answer, sorry if the first post was not so clear :-D ankur2061

Flyou,

The polytropic head equation does not take into account ΔT, the only input to the head equation whether polytropic or isentropic for temperature is the inlet or suction temperature of the gas being compressed.

So what you are proposing is not correct. Refer the equation below:

Hpoly = (8314 / MW)*T1*Zavg*(n / n-1)*[(P2/P1)(n-1)/n – 1]

Regards,

Ankur. Flyou

Ankur,

Sorry to insist but I do not totally agree with you since the polytropic head equation you have just written above can also be written as follows:

Hpoly = (8314 / MW)*Zavg*(n / n-1)*T1*[(P2/P1)(n-1)/n – 1]

Hpoly = (8314 / MW)*Zavg*(n / n-1)*ΔTideal (13th) = (8314 / MW)*Zavg*(n / n-1)*ΔTactual (12th)

Correct me if i am wrong but if i well understand what is said in the 13th edition is that the Gas Power and the Polytropic Head is the same as in the 12th edition whereas the ΔTactual is different:

ΔTactual(13th) = ΔTactual(12th) / ηp = ΔTideal(13th) / ηp

If it is the case, then i do not understand how the discharge temperature could be higher with the same energy consumed? ankur2061

Flyou,

I do not have the 12th edition of GPSA engineering databook. Before I bought the 13th Edition I was refering the 11th Edition, so I am unable to comment on what is written in the 12th edition of GPSA databook. I have written what is there in the 13th edition and you have the 12th edition with you. Hence you can come to  a better understanding better than me for comparing the 12th and 13th edition.

Regards,

Ankur. Flyou

I think that the 12th edition GPSA engineering databook polytropic equations are the same than in the 11th edition and also the same than in the old texts and literature for compressors you are talking about at the beggining of your blog article:

ΔT = T1*[(P2/P1)(n-1)/n – 1] ------------------------------------------------------(1a)

Hp = (8314 / MW)*Zavg*(n / n-1)*T1*[(P2/P1)(n-1)/n – 1] ----------------------(2a)

GP = (w)*(Hp) / (ηp)*(3600) -------------------------------------------------------(3a)

For the 13th edition, the actual outlet temperature equation seems to be different whereas the Polytropic Head and the Gas Power equations remain the same:

ΔTideal = T1*[(P2/P1)(n-1)/n – 1] ------------------------------------------------- (1b1)

ΔTactual = T1*[(P2/P1)(n-1)/n – 1] / ηp ------------------------------------------ (1b2)

Hp = (8314 / MW)*Zavg*(n / n-1)*T1*[(P2/P1)(n-1)/n – 1] ----------------------(2b)

GP = (w)*(Hp) / (ηp)*(3600) -------------------------------------------------------(3b)

My only concern was that i did not not understand how the discharge temperature can be higher in the 13th edition while the gas power calculated is exactly the same than in the old texts.

Finally, i think i have found the answer i wanted: the actual outlet temperature equation of the 13th edition is only used to determine a conservative value of the outlet temperature. Nevertheless, the calculation of the polytropic head and gas power does not takes into account this actual outlet temperature but only the ideal outlet temperature otherwise the polytropic head equation for the 13th edition would have been:

Hp = (8314 / MW)*Zavg*(n / n-1)*T1*[(P2/P1)(n-1)/n – 1]ηp ------------------(2c)

And the power consumed would have been 1/ηhigher than in the old texts which is not the case.

Regards,

Flyou rando

Please Help Me, I want to Conclude the size of header of air compressor with air velocity aproxximately 57 m/s and pressure inlet to header is 6,5 bar,and flow rate is about 5773 m3/s SyedAhmed

Could any1 can explain that how to calculate the discharge temperature at block discharge case and fire case for PSV sizing. I go through the API 520 & 521 but block discharge case relieving temperature is not discussed. raj1

Hello ankur,

I have one doubt. Actually i was trying to use the equation

Delta T = T1 * [(P2/P1)^(K-1)/K -1]/poly effici to fit in DCS for my compressor surge control. For that , i manually try to estimate the discharge temperature in excel spreadsheet. But i ended up in high discharge temperature which is not matching with my design value. I have doubt on K. The specific heat ratio which i have calculated in hysys based on suction condtion (Temperature, pressure). S it correct? or do u recommend some other condition. ankur2061

Hello ankur,

I have one doubt. Actually i was trying to use the equation

Delta T = T1 * [(P2/P1)^(K-1)/K -1]/poly effici to fit in DCS for my compressor surge control. For that , i manually try to estimate the discharge temperature in excel spreadsheet. But i ended up in high discharge temperature which is not matching with my design value. I have doubt on K. The specific heat ratio which i have calculated in hysys based on suction condtion (Temperature, pressure). S it correct? or do u recommend some other condition.

For polytropic process (centrifugal compressors follow this) the following formulas are used as per GPSA 13th edition:

ΔTactual = T1*[(P2/P1)(n-1)/n – 1] / ηpoly

and actual discharge temperature T2 as:

T2 = T1 + ΔTactual

'n' is called as the polytropic exponent and can be calculated as follows:

n / (n-1) = [k / (k-1)]*ηpoly

Choosing the right value of ηpoly is very important. For determining ηpoly refer the following link:

http://www.cheresour...et-volume-flow/

Hope this helps.

Regards,

Ankur. raj1

Thanks ankur. One more thing i want to add up. The formula u have mentioned here is almost matching with my results for stage compression ratio 1.6. But, when i was trying to find the discharge temperature of gas for the compressor stage whose design stage compression ratio 13.6. The above formula found inconsistent (values are not matching). S there any solution? i know the compression ratio which i have mentioned is very high. But, compression ratio value is real. ankur2061

raj1,

You want to have a compression ratio of 13.6 in one stage. Sorry, you can't do this my friend because the temperatures would be abnormally high. You provide multi-stage compression for this very reason.You work with a temperature to fix the stage compression ratio. As a rule of thumb you fix the stage compression ratio in such a manner that the outlet temperature from the stage does not exceed 150 deg C. When moving from one stage to another,you provide inter-stage coolers to reduce the temperature of the gas which is entering the next stage.

For gases like hydrogen and acetylene, the stage outlet temperature is maintained even lower than 150 deg C.

Hope this explains that a compression ratio of 13.6 per stage
would not be practically possible to implement.

Regards,
Ankur raj1

Dear ankur, i agree with ur answer. But in my case inlet temperature is very less (as its is the refrigerant). So for the same power, i can be able to pressurise more for the required power and outlet temperature is coming around 35.5°F. I have attached  table for your reference.

Parameter UOM Stage 1 Stage 2 Stage 3 Stage 4 Inlet Pressure psig 0.2 55.4 92.1 150.8   psia 14.9 70.1 106.8 165.5 Outlet Pressure psig 55.4 92.1 150.8 244.4   psia 70.1 106.8 165.5 259.1 Inlet Temperature °F -135 -78 -24.2 32.2 Outlet Temperature °F 35.5 -22.3 32.2 93.5 Polytropic Efficiency % 78.1 77 77.6 76.2 Power hp 628 1,429 1,576 1,733 Flow LBM 487 3,685 3,838 3,705   ACFM 3,968 7,079 5,459 3,831 Pressure Ratio   4.7 1.52 1.55 1.57

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