Compression of pure nitrogen: Calculating power and temperature

[Marcus10]

I am trying to find the theoretical power required to compress nitrogen from a pressure P1 to P2, as well as the final temperature T2 after compression.

I found no formula that were anywhere near complete, but I found this tool, which tells me that from 300K@1Bar, I need 271.4kw (271kj/s) to compress 1 kilogram (0.2194 cubic meters at 300K, 1 Bar) of pure nitrogen per second to 200 Bar. So I started with:

271.4kj/s * 1kg/s = 271.4kj/kg.​

The same tool says that the final temperature would be 1261.79K, but when I refer to http://www.nist.gov/data/PDFfiles/jpcrd39.pdf , I find that at 1200K @ 200 Bar, (the datasheet doesn't go beyond that temperature for that pressure) the enthalpy is 37.2324kj/mol, while at 300K @ 1 bar, enthalpy is 8.7166kj/mol, a difference of 28.5158kj/mol, or

28.5158kj/mol / (28.013g/mol * 1kg/1000g) =
28.5158kj/mol / (0.028013kg/mol) = 1,017.95kj/kg.​

So this doesn't add up. Even if I apply a 60% efficiency (i.e. break power of 271.4kj/kg / 0.60 = 452.33kj/kg), the enthalpy of the gas would have increased more than twice as much as the amount of energy consumed by the compressor.

Air being mostly nitrogen, I also tried this tool, which gives 374HP for 465cfm ft 2901 psi, i.e. 279kw for 1kg from 14.7psi to 200 bar, approximately the same as what I get with the first tool, still much less that 1,017.93kj/kg. It does not give any final temperature though.

So what went wrong? I don't want an "online tools" to do the work for me; I want to understand how it works and how to do it, and I want to know that the result I get is acceptable, i.e. not too much approximated. All formulae I fold rely on things like the Cp/Cv ratio being constant, or that PV=nRT, which are far from true in reality.

I tend to believe that the best way to get reliable results is to stick as much as possible to datasheets, rather than to general formula that give too inaccurate approximations. What is the proper procedure to determine the power required for compression, final pressure or final temperature given one of the others and the pressure and temperature of the gas before compression? Can I do that using only the data found in the datasheet?

Note that I am working with pure nitrogen only, not air nor a gas mixture. I also want to work with extreme temperature, at least as low as 100K and at least as high 800K, and also with high pressure differentials, so no PV=nRT or alike.

Can you help?

 

[SteamKing]

I am trying to find the theoretical power required to compress nitrogen from a pressure P1 to P2, as well as the final temperature T2 after compression.

I found no formula that were anywhere near complete, but I found this tool, which tells me that from 300K@1Bar, I need 271.4kw (271kj/s) to compress 1 kilogram (0.2194 cubic meters at 300K, 1 Bar) of pure nitrogen per second to 200 Bar.

The amount of power required is going to depend on several other factors.

How are you compressing the nitrogen? Most compressors don't generally compress at a ratio of 200:1 in a single stage, and the buildup of heat due to compression usually means that some sort of cooling is required between stages

https://en.wikipedia.org/?title=Gas_compressor

So I started with:

271.4kj/s * 1kg/s = 271.4kj/kg.​

The units in this calculation are kJ-kg/s², not kJ / kg

Also, the specific volume of nitrogen at 25 °C (298 K) and 1.013 bar is 0.8734 m³ / kg.

http://encyclopedia.airliquide.com/Encyclopedia.asp?GasID=5#GeneralData

 

[Marcus10]

Also, the specific volume of nitrogen at 25 °C (298 K) and 1.013 bar is 0.8734 m³ / kg.

Wow, it looks like I used the density at boiling point from earlier calculation. It works so much better with the right value. Now I get 1101.19, a difference of about 8%, which makes sense with the 61.79K difference between 1200K from the sheet and the actual temperature of 1261.79. Thank you!

The units in this calculation are kJ-kg/s², not kJ / kg

Give that equation, you are right. It should be a division; the equation is wrong; the result is correct. I am calculating the energy required to compress 1kg of nitrogen from power and mass flow. I will correct that. I would edit my post to correct it, but I can't find how. Is it possible to edit a post?

The amount of power required is going to depend on several other factors.

Can you expand on which factors you are talking about?

How are you compressing the nitrogen? Most compressors don't generally compress at a ratio of 200:1 in a single stage, and the buildup of heat due to compression usually means that some sort of cooling is required between stages

I want to understand how to work it out theoretically. Even though some cooling will be accounted for eventually and that multiple stages may be necessary, the matter of the question is how it works within a single stage without cooling. From then, I can split the process in multiple stages, and do some cooling in between as necessary.

I agree that a 200:1 ratio is huge, but it is just for sake of calculation. I figured that grater pressure differential would amplify errors and make them more obvious.

It will be a dynamic compressor for sure, most likely axial. Probably in different stages, depending on the pressure required and some other factors. I need to find out the pressures and temperatures required, then find out the best solution. Now that I corrected the density at 300K, 1bar, I am left with the core of my question:

How do I solve for Temperature, Pressure, Power after compression. I understand that I need only one of the three, just like the calculator takes only the pressure and gives the power and the temperature. I can use the table of properties to find one if I have the two others, but I am missing a way to find the second value.

This page seem to indicate that finding the temperature raise from the change in pressure would be the first step, but it uses this formula:

T₁V₁^cp/cv-1=T₂V₂^cp/cv-1

It says that it is "is valid for a reversible adiabatic expansion or compression of an ideal gas." Cp and Cv both changes with both temperatures and pressures. If I dealt only with temperatures that are not too far from room temperature, perhaps it could do, but I need temperatures ranging from at least 100K to 600K.

Worst case would be trial and error, e.g. guessing a temperature, then lookup the table of properties energy to somehow find out if that guess makes sense, but I don't know how to do that either.

 

[SteamKing]

The amount of work it takes to compress a quantity of gas is going to depend on how the gas is being compressed.

Since you are looking to compress the nitrogen with an axial compressor, then an analysis of the work required using that method of compression will be what you are looking for:

https://en.wikipedia.org/?title=Axial_compressor

Be advised that the compression ratio at each stage is quite modest, anywhere from 5%-20%, so you'll have to work out the number of stages required for a 200:1 compression and calculate the work consumed at each stage.

 

[Marcus10]

The amount of work it takes to compress a quantity of gas is going to depend on how the gas is being compressed.

Since you are looking to compress the nitrogen with an axial compressor, then an analysis of the work required using that method of compression will be what you are looking for

Do you mean that each compressor has its own inefficiencies due to factors related to how they work, or that the theoretical power required for compressing the gas depends on the nature of the force compressing that gas? The first makes sense to me, but I don't see how the second one can make sense.

I think I should ignore the inefficiencies of the compressor itself since I am only (at first) interested in the work on the gas itself. I am looking for the theoretical power, not the actual power. I agree that inefficiencies makes things different and more complex, and they are sure real, but before I account for those, I must have a reliable way to calculate the core values, which I assume to be independent of the type of compressor. Am I misunderstanding something?

 

[SteamKing]

Do you mean that each compressor has its own inefficiencies due to factors related to how they work, or that the theoretical power required for compressing the gas depends on the nature of the force compressing that gas? The first makes sense to me, but I don't see how the second one can make sense.

In my experience, the thermodynamic analysis of a positive displacement compressor is somewhat different from a similar analysis for an axial compressor, for example.

The article linked on axial compressors in the post above shows how axial compressors are analyzed.

I think I should ignore the inefficiencies of the compressor itself since I am only (at first) interested in the work on the gas itself. I am looking for the theoretical power, not the actual power. I agree that inefficiencies makes things different and more complex, and they are sure real, but before I account for those, I must have a reliable way to calculate the core values, which I assume to be independent of the type of compressor. Am I misunderstanding something?

An ideal analysis can be done for whatever method of compression you choose. Just make sure that the analysis matches the cycle used for the compression.

 

[Marcus10]

The article linked on axial compressors in the post above shows how axial compressors are analyzed.

The best solution I have so far for solving for temperature from pressure is this:

T₁V₁^Cp/Cv-1=T₂V₂^Cp/Cv-1​

I would be find with solving anyone of the Temperature, Power and Pressure given one other. That formula, however, assumes that Cp and Cv are constant and thus only gives an approximation.

Is there anything on that wikipedia page that does any better and I am missing it?

 

[SteamKing]

The best solution I have so far for solving for temperature from pressure is this:

T₁V₁^Cp/Cv-1=T₂V₂^Cp/Cv-1​

I would be find with solving anyone of the Temperature, Power and Pressure given one other. That formula, however, assumes that Cp and Cv are constant and thus only gives an approximation.

Is there anything on that wikipedia page that does any better and I am missing it?

I think as long as the compression ratio at each stage of an axial compressor is kept reasonably low, the ratio of C_P to C_V can be treated as a constant value, which is sometimes called k or γ. For nitrogen, k or γ = 1.40:

https://en.wikipedia.org/wiki/Heat_capacity_ratio

 

[Marcus10]

I think as long as the compression ratio at each stage of an axial compressor is kept reasonably low, the ratio of C_P to C_V can be treated as a constant value, which is sometimes called k or γ. For nitrogen, k or γ = 1.40:

That looks very inaccurate, given an error margin would be amplified at each stage over so many stages required. The ratio is 1.87 at 800K, 100 bar, that is a huge difference.

Isn't there a way to find the corresponding P1, T1, P2, T2 from matching the properties from the table? Something like matching the densities at both pressures, then add or subtract the difference in enthalpy or internal energy to that of the gas at (Px, Tx)?

 

[Marcus10]

Would the Carnot theorem be useful?

 

[SteamKing]

I don't design turbomachinery, so it was just a guess, perhaps a poor one.

There are books on how to design such machinery. You'll have to do some more research.

 

[SteamKing]

Would the Carnot theorem be useful?

Only insofar as calculating what the ideal efficiency of the cycle would be.