Polytropic Compression

-That the compressor performance deteriorates with decrease in polytropic efficiency can be seen by the following set of equations:

Gas Power = (Gas Mass flow*Polytropic Head) / Polytropic efficiency --------------(1)

For a given compressor, the polytropic head is nearly constant. Also, for an installed compressor the maximum power that can be supplied for compression is also fixed.

Re-arranging equation (1) gives:

Gas Mass flow = (Gas Power*Polytropic efficiency) / Polytropic Head --------(2)

From equation (2) it is quite clear that as polytropic efficiency decreases for a given polytropic head and a fixed power input the gas mass flow will decrease.

Relating polytropic efficiency to polytropic exponent by the following equation

ηpoly = (n / n-1) / (k / k-1) -------------(3)

Where:

ηpoly = Polytropic efficiency

n = polytropic exponent

k = isentropic exponent

From equation (3) for a nearly constant isentropic exponent (which is more gas property dependent) and an increased polytropic exponent (which is more machine performance dependent) the polytropic efficiency decreases -As a mathematical equation polytropic efficiency can be defined as follows:

ηpoly = n(K−1)K (n−1)

Where:

n = polytropic exponent, dimensionless

k = Cp/Cv =average (suction/discharge) ratio of specific heats, dimensionless

However, the tricky part is to calculate the polytropic exponent. For practical purpose as a first approximation an average polytropic efficiency of 75% (0.75) is considered per stage of compression & the polytropic exponent calculated based on the above equation knowing the average specific heat ratios.

Alternatively, if you know the discharge temperature (T2) of the compression stage based on field-measured values then you can calculate the exponent value as follows:

Exponent (n or gamma) = ln (P2/P1) / (ln (P2/P1) - ln (T2/T1))

Where:

P1 = stage suction pressure, absolute unitsP2 = stage discharge pressure, absolute unitsT1 = stage suction temperature, absolute units

T2 = stage discharge temperature, absolute units

Here again the pitfall is that if you don't know the discharge temperature from actual measurement then to calculate the discharge temperature you require the exponent value using the equation:

T 2 = T 1 ( P2

P1)n−1n

So if you are doing some sizing calculations it is always a practical approach to consider 75% polytropic efficiency for starting the calculations. -Before we move on to the why's of the compression process or compression efficiency type (adiabatic or polytropic) it is essential that the difference between the two be understood.

1. Adiabatic Compression: The terminology used by some compressor specialists is "Near-Adiabatic (Isentropic) Compression". Practically it is impossible to have a perfect adiabatic compression, which by the theoretical definition of "adiabatic" means that during the compression process no heat enters or leave the system when the system is represented as a perfectly insulated box from its surroundings. Positive displacement compressors come closest to adiabatic compression or near-adiabatic compression. It is important to note that even for dynamic compressors (centrifugal / axial) the compression process and the compression efficiency was defined as adiabatic more than 30 years ago

2. Polytropic Compression: The practical understanding of the polytropic process and its usage in defining the compression process and efficiency is a development of the last three decades. In a polytropic compression process infinitesimal changes occur in terms of heat absorbed or heat

removed during the entire compression process. If you plot a P-V curve for the compression process and represent the curves for all three compression processes viz. adiabatic, isothermal and polytropic, you will find the polytropic compression curve lays in-between the adiabatic and isothermal compression process. The polytropic process represents the real world compression process. Compressor manufacturers realized that the centrifugal compression process mimics the polytropic process more closely than the adiabatic process. There is a reason for it. In centrifugal compressors gas slippage occurring across the impeller(s) in the compressor casing is much more pronounced than the gas slippage encountered in a reciprocating (PD) compressor volume chamber. This gas slippage leads to mixing of compressed and slip gas, which are having different temperatures. This pronounced slippage in centrifugal machines mimics the polytropic compression process which as earlier mentioned is the infinitesimal change in heat gain or heat loss during the entire compression process. That is the reason that the compression process for centrifugal compressors is defined in terms of polytropic compression and polytropic efficiency.

To conclude, the performance and efficiency of a centrifugal compressor can be defined in terms of an adiabatic compression process but the polytropic compression process provides a more realistic performance of a centrifugal compressor compared to an adiabatic compression process -Simplified equation for power consumed by a centrifugal compressor considering polytropic compression path is:

P = M∗H poly

3.6∗106∗ηpoly

Where:

P = gas power required, kW

M = mass flow rate of gas, kg/h

H poly = Polytropic Head, N.m / kg

ηpoly = Polytropic efficiency expressed as a fraction

H poly - Which is a function of suction or inlet temperature. Lower the suction or inlet temperature lower will be the polytropic head. -Reducing the suction temperature will reduce the power consumption. For the gas with same molecular weight and based on a defined mass flow rate @ inlet pressure / temperature a decrease in the suction temperature will reduce the power consumption because the polytropic head will reduce. The equation for polytropic head is as follows:

H poly = ( 8314MW )*T 1*Zavg*( n

n−1 )*¿ -1)

Where:

H poly = Polytropic head, N.m/kg

MW = Molecular weight of the gas, kg / kg-mol

T 1 = Absolute temperature at inlet conditions, K

Zavg = Average compressibility factor

P1 = Inlet pressure, kPa (abs)

P2 = Discharge pressure, kPa (abs)

n = polytropic exponent

Note that in this equation T1 has a role to play. If T1 decreases H poly will decrease with other conditions remaining unchanged. If H poly decreases, the gas power required will also decrease. -Here is the equation for Polytropic head

H poly = ( 8314MW )*T 1*Zavg*( n

n−1)*¿ -1)

Where:

H poly = Polytropic head N-m/kg

MW = Molecular weight of the gas, kg/kg-mol

T 1 = Absolute temperature at inlet conditions, K

Zavg= Average compressibility factor, dimensionless= (Z1+Z2/2), Where 1 & 2 are inlet and discharge conditions respectively.

P1 = Inlet pressure in absolute units, kPa (a) or bar (a)

P2 = Discharge pressure in absolute units, kPa (a) or bar (a)

n = polytropic exponent

Now I think it would be obvious to you that what you have mentioned about polytropic head going down with increase in suction or inlet temperature is not correct. Obviously if the polytropic head is going to increase, so will the shaft power requirement

-Whichever standard or textbook I look at, the flow for a centrifugal compressor as defined in the problem statement is either at standard volumetric flow conditions (denoted as SCFM/SCFH/SCMH) or in terms of mass flow which means that both these values remain unchanged at the inlet and outlet of the compressor. What changes is the inlet volume flow rate (denoted as ACFM/ACFH/ACMH) defined as the volume flow rate at the inlet flange of the compressor which is a function of the inlet pressure and temperature (pressure and temperature of the gas being compressed at the inlet flange of the compressor) and the molecular weight of the gas being compressed.

Now coming to the density of the gas being compressed. Gas density is a function of pressure, temperature and molecular weight of the gas. A change in either of these changes the gas density, which in turn changes the inlet volume, flow (the ACFM/ACFH/ACMH) but neither the mass flow nor the standard volume flow.

For a fixed-speed centrifugal compressor there can be essentially no change in either the power drawn (gas horse power or gas kW), polytropic head (ft-lbf/lbm or N-m/kg) and the mass flow rate irrespective of the change in the inlet volume flow conditions. The effect of changing the inlet volume flow rate would manifest itself in the form of change in the differential pressure (Discharge Pressure - Suction Pressure) for the compressor. In other words either the suction pressure conditions or the discharge pressure conditions will change due to change in the pressure / temperature / molecular weight (gas density change) at the inlet of the centrifugal compressor. This can be proven easily mathematically:

Gas Horse Power or Gas kW = Polytropic Head*Mass flow rateNeither of these changes for a fixed-speed machine

Polytropic head is related to differential pressure as follows:

Differential Pressure = Polytropic head*Gas Density

Polytropic head remains unchanged for a fixed-speed machine, so a change in density will cause a change in the differential pressure.

For a variable-speed centrifugal compressor you can change the power drawn by changing the frequency of the motor and the affinity laws (speed vs. inlet volume flow rate, speed vs. polytropic head, speed vs gas horse power) as applicable to centrifugal pumps can be applied but with limitations compared to centrifugal pumps. Affinity laws can be applied between 80% and 105% of the speed range only, beyond which actual performance would be much different from the predicted performance. Also heavier gases (higher molecular weights) will give greater deviation from predicted performance.

-The discharge pressure is normally an input and not a calculated result. However, you can calculate it as a function of the inlet and outlet pressure as follows:

T 2 = T 1 ( PdP s )n−1n ...........................(1)

Where:

T 2 = Absolute discharge temperature, KT 1 = Absolute suction temperature, KPd = Absolute discharge pressure, kPaPs = Absolute suction pressure, kPan = polytropic exponent

n, can be calculated as follows:

n−1n = K−1

(k∗ηpoly) .........................(2)

Where:

k = Cp / Cv at the inlet conditions

ηpoly= Polytropic efficiency, %

In your case the suction and discharge temperatures are known, the suction pressure Ps is known and you can calculate the polytropic exponent as per equation 2. Now using equation 1 you can calculate the discharge pressure Pd. As I mentioned earlier for compressor calculations the discharge pressure is an input.

The formula for a stage discharge temperature for a centrifugal compressor is:

T 2 = T 1 ( P2

P1)n−1n

Where:

T2 = stage discharge temperature, KT1 = stage suction temperature, KP2 = stage discharge pressure, kPaP1 = stage suction pressure, kPan = polytropic exponent

(n - 1) / n = (κ - 1) / κ*ηpoly

Where:

κ = specific heat ratio = Cp / Cv (1.4 for air and 1.407 for hydrogen)

ηpoly = Polytropic efficiency (normally for large volumetric capacity centrifugal compressors it ranges from 75 to 80% (0.75 to 0.80))

From above it is clear that the discharge temperature is dependent on the pressure ratio (P2 / P1), the specific heat ratio κ and the polytropic efficiency.

If you compare the specific heat ratio of air and hydrogen they are quite similar. Considering that that the polytropic efficiency is not affected much, than the theoretical discharge temperature for a fixed pressure ratio for both hydrogen and air will be quite similar.

When I check your case considering a pressure ratio of 3.33 and a suction temperature of 40°C, I get almost the same discharge temperature of 216°C for both hydrogen and air.

To conclude, the discharge temperature is not changing much between hydrogen and air but the power required for the same pressure ratio for hydrogen would be much higher compared to air. This is evident from the fact that the compressor inlet volume flows for hydrogen for a unit mass flow rate (1 kg/h) at 15°C and atmospheric pressure (101.325 kPa) is 5.75 m3/h whereas for air for a unit mass flow rate it would be 0.40 m3/h. This is a 14-fold increase in inlet volume flow rate for hydrogen compared to air. The reason is obvious - the large difference in the molar mass of hydrogen and air (2.015 versus 28.97). Notice that the volume flow rate increase for hydrogen is in the same ratio as the ratio of the molar masses of air to hydrogen.

-After a 2-month hiatus I am back on my blog and with something really special.

Since the past 1 year or so the subject of compressors has intrigued me and led me to a quest in understanding more of this fascinating

subject which all chemical engineers have to deal at one time or the other in their professional careers. A lot of reading on the subject went into this and subsequently led to the development of a spreadsheet on "Centrifugal Compressors Head & Power Calculations" which is on sale at the "Cheresources" online store. This spreadsheet has seen a good response from professional engineers who probably don't have access to simulation software such as HYSYS or from engineers who have ventured to understand the basic thermodynamic equations governing compression of gases rather than depending on simulation software where the backend calculations are not provided. Mind you, I have not provided a treatise on thermodynamics of gas compression but a simple calculation tool, which is fairly in agreement with the simulation software existing today.

There are some novel features in the spreadsheet, which are not there in some software (e.g. HYSYS), such as calculation of number of stages and two different methods of calculating the rotational speed of the compressor

In the spreadsheet the polytropic efficiency as a function of inlet volume flow has been picked up from a table as published in the book "Pipeline Rules of Thumb Handbook" by E.W. McAllister.

However, how the Polytropic efficiency is related to the inlet volume flow was something that was still not very clear to me. Further study of the subject revealed that there are charts available in some chemical engineering texts, which relate polytropic efficiency to the inlet volume flow. Following are the known sources for such charts:

1. Figure 3.6, Page 83 in Coulson And Richardson Volume 6

2. Figure 7.27 (a), Page 158 in Chemical Process Equipment - Selection & Deign by Stanley M. Walas

3. Figure 2 in the article titled "What Process Engineers Need to Know About Compressors" by William Dimoplon, "Hydrocarbon Processing", May 1978

As I had mentioned earlier in one of my blogs I like to use math more than reading charts and I decided to find out if there was a way that the inlet volume flow (m3/h or cfm)(x-axis) versus polytropic efficiency (y-axis) chart could be regressed into an equation form for ease of use.

Further searching led me to a published source from a company standard wherein the relationship between inlet volume flow and polytropic efficiency was expressed as an empirical equation. However, the units used in this equation were metric units and there was no direct means available to find the corresponding equation in English units. In order to find the corresponding equation in English units I regressed the data from the metric unit equation in an excel spreadsheet with the corresponding conversion of flow rate from m3/h to cfm. Next I introduced a trend line on the excel spreadsheet and using the equation option generated the corresponding English unit equation.

The purpose of this entry was to share these equations with all of you. Below are the equations:

Metric Units:

ηpoly = 0.0992 +0.2463*log10Q1-0.02167*(log10Q1 ¿2

Where:

ηpoly= Polytropic efficiency

Q1 = Inlet volume flow, m3/h

English Units:

ηpoly= 0.027*logeQ1+0.4984

Where:

ηpoly = Polytropic efficiency

Q1 = Inlet volume flow, ft3/min

Note: Above equations provide fairly good results for inlet volume flow rates ranging from 1000 cfm to 100,000 cfm (1700 m3/h to 170,000 m3/h)

I am sure that there will be some questions and further debate on this matter and I am looking forward to it from our knowledgeable forum members.

-The polytropic constant or polytropic exponent is related to the adiabatic exponent through the polytropic efficiency. In equation form this can be represented as follows:

n/(n-1) = (k/(k-1))*ηpoly

Where,

n = Polytropic exponent k = Adiabatic exponentηpoly = Polytropic efficiency

The polytropic efficiency may be considered approximately as 77-78% for a wide range of inlet volume flows to the compressor. Alternatively polytropic efficiency may be approximated from the following empirical correlation:

ηpoly = 0.0992 +0.2463*log10Q1-0.02167*(log10Q1¿2

Where,

ηpoly = Polytropic efficiency

Q1 = inlet volume flow, m3/h (at inlet conditions of pressure and temperature)

The above should be able to provide you the polytropic exponent n.

-The compressor stage discharge temperature can be calculated as follows:

T2 = T1*(r p )( n−1n )

or

T2 = T1*(r p )(k−1k )

Where,

T2 = stage discharge temperature, K

T1 = stage suction temperature, K

r p= Pressure ratio (Discharge Pressure / Suction Pressure)

n = polytropic exponent

k = isentropic exponent = CpC v

Relation between n & k is given as below:

n-1 / n = ( k-1 / k)*(1 / ηpoly)

Where,

ηpoly = Polytropic efficiency

As you will notice that the key to calculating the accurate stage discharge temperature is the accurate evaluation of the k-value. If you use k-values based on the suction conditions this will provide a more conservative (read higher temperature). k-values normally decrease during compression.

- What you are calling as polytropic index is more commonly known as polytropic exponent "n". Following is the relationship between polytropic efficiency Ep and polytropic exponent n: Ep = (n/n-1) / (κ/κ-1) Or

Ep = n*(κ-1) / κ*(n-1) Where: Ep = polytropic efficiencyn = polytropic exponentκ = isentropic exponent or specific heat ratio When performing power or head calculations the polytropic efficiency is used as an input to calculate the polytropic exponent.

A fair approximation of polytropic efficiency can be made from the inlet volume flow also commonly abbreviated as ACFM or ACMH.

-Yes, ideally a reciprocating compressor is described as following a thermodynamic adiabatic and reversible process – which makes it an ISENTROPIC process. This, of course means that the entropy remains constant through the process and is a handy means to calculate the brake work required for the compression.

The really ideal compression process would be one where the compression step could be carried out in an isothermal fashion. But this is a thermodynamic daydream because this is not a feasible or practical application in real life. There simply is no manner - nor mechanical method - that can be devised (to date) that allows for continuous, differential cooling of the gas as it is being compressed. If you can invent or come up with a feasible method to do this, you will become a super billionaire by revolutionizing the application of thermodynamics. The ONLY method that has been applied is a vain and “cosmetic” attempt to use a built-in cooling jacket on reciprocating cylinders. The compression cooling obtained in this manner is only a “best effort” type – you get what is available due to the mechanical size and constraints. I have operated reciprocating compressors since 50 years ago – some were cooled, others were not. For example I operated reciprocating compressors handling air, CO2, Oxygen, Nitrogen, Argon, Acetylene, Nitrous Oxide, Hydrogen, natural gas, etc., etc. and most of these were jacket-cooled with cooling water. The acetylene application is an exceptional one because it is a special case where the cylinder and intercoolers are all immersed in a cooling water box. Acetylene is VERY special and has its own rules to follow. I read, in Hydrocarbon Processing, an article in the 1960s that opposed the use of cooling jackets – for various practical reasons. I tried operating my compressors without cooling water in the

jackets and found – to my surprise – that the article was correct: there was NO NEGATIVE EFFECT on the compressor’s performance and operation when the jackets were run dry. It was then that I realized that I had been naïve in thinking that such a small cooling area could have a marked effect on a gaseous chamber that involved an inherently poor gas film heat transfer coefficient. You will recall that the worse film coefficients are those of gases. In fact, a static gas film is used as a natural insulator. Nature does this in giving us body hair – much as animals, such as Polar bears, can be insulated with only a fur coat. Our engineering insulation materials all depend on the effect of a static gas film.

In summary, it is my experience that a cooling jacket on a reciprocating compressor cylinder cannot be relied upon to lend any marked, positive effect on the efficiency or operability of the compressor. It exists only as a token attempt to lend some heat removal – but it is so inefficient as to be practically negligible. The same effect takes place in a reactor or process vessel when one considers a cooling jacket and tries to “optimize” the cooling/heating effect with external coils, dimples, etc., etc. The net effect is negligible because you are limited to the vessel size. You get what you get – and no more. A process reactor or vessel is designed for the reaction and the process capacity – not for the heat transfer. The heat transfer is an after-thought and is a best effort process.

Further, and direct to your query is the fact that since the actual, field compression process does not transfer any significant compression heat, it should then be considered adiabatic. And here, my design and process experience in the field has also confirmed the advice of many authors – specifically the NGPSA – in recommending the use of an isentropic compression with enthalpy data when calculating the thermodynamic results of a gas compression in a reciprocating machine. My experience has been

that when I assume an isentropic process and use accurate enthalpy data – especially on pure gases – I get excellent design results on the work requirement and the discharge temperatures of each stage in a reciprocating machine. All this confirms for me that the performance of a reciprocating compressor is very close to the presumed isentropic process. I have never employed polytropic calculations with a reciprocating compressor simply because I had good results with the simpler and more direct isentropic assumptions. The centrifugal compressor is another animal altogether different and more complex in its calculations – something a practical engineer would expect taken that this machine is so mechanically simple. The expected trade off in a more complex process calculation given a simpler mechanical design is experienced.

A centrifugal machine is vastly more inefficient and limited in control than the reciprocating model. This is yet another trade off. Reciprocating machines are not only more thermodynamic efficient, but their process calculations are much simpler and the machine has much more positive capacity control than the centrifugal. Where the reciprocating machine cannot compete is in size, capacity, and price.

Note that when you increase the number of stages (and associated intercoolers) in a given reciprocating compressor, you are practically applying one version of isothermal compression. You could, theoretically, achieve isothermal compression if you had infinite stages and intercoolers. This is the positive effect a user profits from when multiple stages are employed in a given compression operation. The multiple stages result in a lower total horsepower compression requirement. Of course, more stages mean more capital monies; but an optimization exists.

-The differential head (delta P) (discharge pressure - suction pressure) in feet does not change with the change in gas

characteristics is what Lieberman-has stated - and I stand in agreement with it. It is a function of both vapor density and polytropic head. Lieberman’s statement essentially means that if the vapor density decreases, the polytropic head increases & vice-versa; i.e., if the vapor density increases the polytropic head decreases - thus keeping the delta P as constant.

Differential head = vapor density * polytropic head

This is exactly what I have said in my post: that the molecular weight (vapor density) strongly affects the polytropic head.

I hope you are not confusing polytropic head with the differential head - as obviously both are not the same -Many filed/operations engineers (even experienced ones) confuse between differential head and polytropic head when dealing with centrifugal compressors by considering them to be the same. Polytropic head is nothing but the change in enthalpy of the gas from state 1 (suction) to state 2 (discharge) whereas the differential head is the change in pressure of the gas from state 1 (suction) to state 2 (discharge). Gas characteristics do not affect the differential head if the machine configuration remains the same but the change in MW of the gas does have an impact on the polytropic head.

As regards to your question:

Delta P (N/m2 or Pascal) = vapor density (kg/m3)*polytropic head (N-m/kg)

-Polytropic compression process is generally considered for centrifugal (dynamic) compressors. For Positive Displacement (PD) compressors (Reciprocating/Rotary) the compression process is generally considered as adiabatic for most engineering

calculations

-Polytropic ProcessA Polytropic Process is one in which changes in gas characteristics during compression are considered. This is considering that an Adiabatic Process is one during which there is no heat added to or removed from the system (Q = 0), and Isentropic Process is one wherein the entropy remains constant (Delta S = 0), and an Isothermal Process is one in which there is no change in temperature (Delta T = 0). A Polytropic Compression Process is typical of a dynamic-type compressor – such as a centrifugal compressor. A reciprocating compressor typically follows an Adiabatic Compression Process very closely.

A Polytropic Process is defined in Thermodynamics as an internally reversible process, which conforms to the relation

PV n = Constant.

If you plot the compression curves for Isothermal, Adiabatic, and Polytropic compressions on a Pressure versus Volume graph, you will find that the Polytropic curve falls between the other two, but closer to the Adiabatic curve.

Polytropic HeadThe Polytropic Head is an expression used for dynamic compressors to denote the foot-pounds of work required per pound of gas.

-I like your query. It challenges my explanation and allows me to digress further on the subject of gas compression – on a practical level, and not on a theoretical basis. I didn't explain the frailties of Isothermal Compression in my prior post because it didn't enter into the basic question. I hate it when professors in

thermodynamics start throwing thermo processes at you, expecting a student to be able to know inherently what is fiction and what is realistic – and not pointing out the fact that some thermodynamic processes presented in the classroom are simply not feasible from a practical point of view. Isothermal Compression is one of those, thank you.

I defy any human being reading this to devise a machine that can compress a real gas without raising its temperature – as measured in the discharge port of such a machine. If you can imagine, you would need an infinite number of finite, differential stages – each with an intercooler – compressing the gas. The greater the number of stages, the less the noticeable discharge temperature. However, this is totally impractical and impossible to build due to mechanical and economic limitations. It would prove nothing from a practical aspect and only serve to reinforce what we already know theoretically.

ALL gas compressors – whether centrifugal, reciprocating, screw, vane, lobe, or whatever other type can be built – will deliver a compressed gas with an increased discharge temperature. You are absolutely practical and engineering-like in asserting this fact. Keep up your practical thermo outlook and insist on a thorough and detailed explanation of what is occurring thermodynamically and why. This type of probing thought can only make you a better engineer.

-When you reduce the suction temperature of the gas to any compressor, the following takes place:

• The density of the gas increases i.e. mass/unit volume.• The compressor continues to displace the rated volumetric

capacity i.e. compressors recognize volume and not mass• Since the density has increased, the rated volumetric capacity is

now related to a higher mass throughput.• Compressors are like any fluid transport equipment that does

work on fluids by transporting mass/unit time.• If more mass is being transported per unit of time, MORE

horsepower (kW) are required to carry out the increased load - not less.

Therefore, by reducing the temperature of the suction gas, you will require more power fed to the compressor