What is the Temperature of Air at End of Compression Stroke?

Dear Mr.Kulas,

You are right. The value of 'n' will be 1.4 since the compression will be adiabatic and follow the equation P x V^1.4 = Constant.

As you said, the cylinder head will absorb some heat and release to the air entering. This just similar to the steam dryness factor will improve towards the end of the stroke in the steam engine.

DHAYANANDHAN.S

Yes, n>1 and is ~1.4 as pointed out by Original Poster. Nothing to see here.

As has been mentioned, the exhaust valve opens when the internal pressure is greater than the exhaust line pressure, so (unlike an internal combustion engine) the concept of a compression ratio is not applicable. V₂ (in the equation above) is not fixed, so my suggestion is not useful.

Maybe not for predicting the resultant temperature. However since little to no exhaust pressure drop exists wouldn't a pressurized air temperature measurement now be a way to make an indirect pressure measurement. Also wouldn't the use of a pressure differential exhaust valve render the original premise of this thread moot, the exhaust air temperature is the maximum temperature in the compressor.

"Also wouldn't the use of a pressure differential exhaust valve render the original premise of this thread moot, the exhaust air temperature is the maximum temperature in the compressor." - good point.

Emperature?? you must be Hinglish!!

All joking aside, I'd like to know how your probe test results compared with the calculated values, and if you got different results with different brands of compressors. My reason for asking is due to the typical reed or other valve arrangement on piston compressors, that is supposed to allow the compressed air to begin flowing out of the cylinder as it exceeds the pressure on the other side of the valve, the temperature should be essentially the same as the temperature immediately outside the cylinder head, minus any losses to the metal (especially when pumping up the tank, as opposed to a momentary high draw). This would give an idea of the efficiency of different brands.

Note too, that the calculation would have to be based on actual measured maximum cylinder pressure not outlet pressure, as the valve will be a restriction and thus a pressure loss. Have you managed to measure this?

The Wikipedia article on the polytropic process is very informative.

[Thanks Kulas for giving me the term to search Google with. A GA from me.]

What I'm curious about is why this useful to know. I'm sure that it is useful I'm just having a temporary block why. Also, in the case of a gasoline engine some of the energy of this temperature rise will go to vaporizing fuel. A diesel engine will presumably experience this temperature rise just prior to ignition. In the case of both engines the temperature quickly climbs after this compression point with ignition. With your stated case of an air compressor, there would be a rapid loss of temperature immediately after the outlet valve opened. I realize this is exactly why you wish to extrapolate back to the internal pressure temperature. I just wonder if there is another reason other than curiosity to know this.

Curiosity is always a good enough reason to seek knowledge.

A piston compressor doesn't experience the pressure drop at valve opening of an IC engine. The valves usually open based on pressure differential, so as soon as the cylinder pressure exceeds the opening force it then begins to act as a positive displacement pump for the rest of the stroke, with a minimal pressure increase or decrease due to the acceleration and deceleration of the piston throughout the cycle.

In a compressor the outlet valve opens when the cylinder pressure is over the line pressure. The differences required are usually small so that the compressed air flows with low pressure drop and thus the temperature drop is also low.

For the inlet valve the logic is the same. At compression begin the air temperature is almost the environment one and is less the cylinder wall temperature so that a heat transfer from wall to ait occurs. As pressure increases so does the temperature and it becomes greater than the wall temperature and the heat transfer does change its direction.

In fact convection is quite intensive due to high turbulence so that the process cannot be considered as adiabatic although the time is short.

In general the compression degree in one stroke is limited by the lubricant. Temperature MUST be lower as the one at which the lubricant molecules crack.

Wow, is this some sort of negotiation for cease fire Redfred? Peace!!! even you and europium MKII and Dearth Vader beats me to crying sometimes--nahh, just joking.

Makes me think and interested to know, how you and lyn get acquainted.

to Redfred #5. My special interest is in compressed air for breathing. The underlying reason for a formula to calculate the instant peak temperature is to assess the likelihood of producing toxic gases when the lubricating oil breaks down at high temperature. Carbon monoxide CO is the killer gas to watch.

Generally speaking, the 'trade' assumes the temperature of the compressor and the air, apparently kept low by cooling devices, is the one thing that is monitored and assumed to be 'safe' when controlled below the oxidation or ignition temperature of the oil.

But in reality, I think the temperature to watch is the actual instant peak temperature reached in the cylinder that will hit the oil vapour. Oil will ignite due to the temperature reached at certain pressure peaks as we know - otherwise diesel engines would not work.

In 'ideal' conditions when compressors are running cool, there is little evidence of the production of CO, but in practice 'ideal' conditions are assumed to prevail until such time the alarm sounds. Which to prevent unnecessary and spurious tripping are usually set 'high' at a level to protect the machines from damage.

When everything heats up in use, the working conditions drift away from 'ideal' to a point that, unwittingly and unknowingly, could allow the production of toxic gases.

People need to be aware that a gradual increase of the inlet temperature (the whole compressor heats up) and the inlet air heats up before it is compressed. Which from the formula can be seen to have a disproportional elevated effect on the outlet temperature.

Very important and hence the need to check the simple formula to give confidence in order to protect against this,

To JNB #4. My temperature measurement was primitive. I poked a thermocouple up the discharge pipe into the cylinder head. There was a rise in temperature to an 'average' - no peaks to speak of - but higher than the external contact temperature of the cylinder head.

PS to all. Apologies for the headline spelling error. But having said that, the spelling error is not with me. The 'T' somehow got lost in transit at CR4.

Have you looked into machines that do not require lube oil? There are advanced machines with materials (perhaps Teflon, perhaps other) that are basically self-lubricating as long as they are wetted, or mostly wetted. Not sure how that works with compression, but it works in a radial piston engine using these materials. If the water is pure, there is no residue, etc.

Obviously, for breathing air there are plenty of regulations, standards, calibrations, and preventative maintenance requirements. It is just an idea I tossed out there at you, so take and run with it. If the air is too moist, then I suspect some sort of cooler or refrigeration system might serve as a suitable dryer, allowing very clean packaged air for SCBA or SCUBA.

James S #17. There is not much to worry about if a dedicated breathing air system is being designed. Oil-free compressors can be selected and installed and maintained for the purpose. Generally the final air quality is dependent on purifying devices many of which work successfully with oily compressors.

The risk lies in uncontrolled random use of compressed airline breathing apparatus that can be plugged into any old compressed air system. It is not just the risk of compressor fires, but more so, is the contaminated ambient air drawn into the compressor inlet.

Obviously there are rules and regulations for setting up and maintaining a respiratory protection programme, but generally the 'policing' for prevention is poor or superficial - and often non-existent. Nothing will be done unless there is an accident - then an inquest - maybe prosecution and damages. Fortunately very few accidents resulting in death occur. As for poor health from breathing a cocktail of contaminants, it is anybody's guess.

Excellent!

I now understand your concern and how this could become an easily hidden problem. Assuming that the temperature cannot exceed the outlet measured temperature seems plausible only if every compressor used for producing breathable air used a differential exhaust valve. This might still be a dangerous assumption depending on the time delay for this "flap" valve to move. It is far safer to predict the peak temperature rise possible if no exhaust valve ever opens.

Your assumption is NOT acceptable since ALL compressors have a limiting fixture which opens the inlet valve if pressure reaches a predetermined level. The scope of such a system is to DELIVER air to the line/reservoir or any other consumer. Due to this the delivery pressure is obtained BEFORE end of stroke. If you consider that the valve does not open then compression goes to very high pressure much more than the one it has to supply. The so called "dead volume" at end of stroke is maintained by design small in order to obtain an as high as possible volumetric efficiency since after delivery, when the piston changes its direction , this volume -which is at delivery pressure- must first decompress to a level under the inlet pressure to open the inlet valve. Only after this position the outside air is aspired in the cylinder. Which means that part of stroke volume is lost for new air inlet.

I have somewhere the feeling that many did not have any contact with compressors during their career.

If you did not find the compressor diagram and are still interested to learn how it works then let me know and I shall supply one with all explanations.

The wiki link in #5 describes polytropic compression but does not suggest values for practical compressors. The usual figure in the industry is n = 1.3, k = (n - 1)/n = 0.23. Using this and efficiency 60 - 65% (depending on size) gives compressor shaft power. But using k = 1.4 or 1.3 in T₂ = T₁*r^k . doesn't give the correct outlet temperature, i.e. the manufacturer's stated figure. Actual rise is higher because most of the additional energy due to the inefficiency also goes into the compressed air.

I found I could get pretty good agreement with manufacturer's data by a) calculating polytropic T₂ from T₂ = T₁*r^k. Then b) calculate theoretical polytropic compressor power, and subtract this from manufacturer's power figure, to get additional power. Calculate mass flow, and using air specific heat 1 kJ/kg/K, calculate "extra" temperature rise due to the additional power. c) Add this to the polytropic T₂ to get total outlet temperature.

That gives temperature of the air in the outlet pipe, but the instantaneous figure in the cylinder could be higher. But if you knew the actual figure, would it help much? Would you take the oil supplier's word for temperature at which the oil starts to break down? It might be worth doing a test, gradually raising output pressure, and measure temperature, but more important, analyse for nasties that might be produced (after cooling the air if necessary).

To Codemaster #23. Thanks for the detailed explanation. What formula would you use?

Please bear in mind that I am not trying to confirm manufacturers figures as such. Working temperatures of the delivered air (if given) usually relate to adequate cooling and operation at the average design temperatures and pressures.

It is very seldom that figures are given for compressors that are run at high temperatures near thermostatic protection levels. Control of risks of contamination are usually transferred to the air treatment people.

I am trying to draw attention to the fact that account needs to be taken of the peak discharge temperatures that could occur in normal use. For which I think a simple formula will suffice.

I appreciate more accuracy could be introduced by taking account of other factors, but these factors are not usually known. Unless you mean using n = 1.3 instead of n = 1.4. That would be fine - as long as it does not reduce the potential peak temperature by reliance on favourable working conditions.

Or put another way, what would the maximum peak air temperature that could be reached knowing only working pressure and ambient temperature - or whatever info is given on the compressor nameplate. Which all we have to go on.

The exponent 1.3 instead of 1.4 takes into consideration the heat transfer from the compressed gas to the cylinder wall. 1.4 means no heat transfer occurs which is as I explained in a previous comment false.

With 1.3 result will be nearer to reality.

Basically if you are afraid of oil vapors or oil burning it is better to use a lower compression ratio per step with intermediate cooling.As for the losses going into compressed air it all depends on the compressor design.If the compressor cylinder is air cooled its temperature will be higher and friction generated heat will go more into the air at start of stroke if the compressor is water cooled (as many big compressors are) then it less probable that the air will get a lot hotter than by compression only.

Nickname #25

Thanks for that. n = 1.3 makes sense, to allow for hot air to transfer heat to the cylinder walls, but if account of heat transfer is necessary, would that not suggest heat would also transfer from the cylinders walls to the cold inlet air so that we start compression with a higher temperature. Every degree added to the inlet is multiplied by r^k and added to the outlet.

You can see how important it is to ensure that compressors run cool. But in practice they run hot, often unattended until an alarm sounds. It is not only the average discharge air temperature that matters but more so, the peak temperature that burns the oil. Thermostatic trips are there to protect the compressor from mechanical damage.

With this in mind, is n = 1.3 realistic and likely to predict the maximum peak temperature. Or to be on the safe side, use n = 1.4.

Since this is a proposed theoretical calculation to achieve a reasonable safety margin with an(y) unspecified air compressor that has no expensive chemical analysis instrumentation constantly measuring the compressed air purity, I like using the safer number of 1.4.

If somebody wishes to pay for a gas chromatograph, tunable diode laser or other in-situ chemical analyzer to be mounted on the exhaust manifold to see if their glass seal scroll compressor can safely pressurize breathable air at any temperature they want then I say they should go for it and report back to us with their data.

To Redfred #27 and also to Codemaster for specific formulas.

Thanks. I think I will stick with n = 1.4 for the reason it will predict the maximum temperature, although in practice it would be useful to know the likely temperature. In which case I can use n = 1.3.

At the time of writing my OP I was not 100% sure whether r^k was the right relationship to use. It seems that it is.

Approached from a different angle, from the known properties of atmospheric air, assuming ideal conditions, what could be the maximum temperature achievable by compressing it from specified ambient conditions to a specified working pressure.

You may consider the alternative : when the compressed air quality is not a problem the efficiency is the main aspect, in your case since health is most important the lower efficiency will not be an important aspect.

The solution would be to use less compression per stage and so be sure that the maximal temperature will NEVER reach a dangerous level.

There is also as complement the possibility to use as lubricant special silicone based lubricants which will never generate CO or CO2.

The use of 1.3 or 1.4 is secondary the compressing ratio/stage is the main aspect.

To nickname #28. Thanks. All you have said about selection is very relevant at the design stage. But in practice the problem arises when we drop in on an existing works compressed air system.

Generally speaking, all compressors are made to a compromise between efficiency reliability and cheapness. A works engineer of any merit will go for efficiency and reliability because compressed air is just about the most expensive routine power source there is.

Air quality is related to what is suitable for machinery (no liquids or solids), and since the quality of atmospheric is an unknown factor anyway, compressor manufacturers (and suppliers) discharge any liability by transferring the quality risk to air treatment people.

As long as the compressors deliver air continuously at the rated flow, pressure and temperature, what comes out of the discharge pipe in terms of quality, is what you get. You then select suitable filters, dryers, purifiers, etc to get the final quality you want.

If (often as an afterthought) it is used to feed respiratory protective devices then consideration of the effect of peak pressure is important to the air treatment suppliers. Respiratory protection selection procedures require a risk assessment of the worst- case scenarios. Hence the need for a simple formula to predict peak temperature because this figure is hardly ever known.

Best-case performance figures are much easier to come by; they can be found in the sales literature.

No, you can modify an existing compressor in order to decrease its compression rate.The problem is NOT to know how high is the temperature but how to obtain compressed air at a temperature low enough so that what you are afraid of does not occur.If you have to fill air bottles at high pressure you cannot obtain the final value in only one stage, it is necessary to have several in any case.If the compression is considered as adiabatic the ratio per step is about 4.4, if the 1.3 exponent is used it is 6.2 or most of single stage compressors go up to 6 bar which validates the 1.3 value. This is, as I mentioned, limited by the oil maximal acceptable temperature around 180°C. If you will ever want to compute the full cycle consider that for expansion an exponent of 1.2 is used in order to take into consideration the heat transfer from the hot air in the dead volume to the wall.If you are willing to reduce risk the ratio MUST be reduced so that the maximal temperature comes around 100 to 125°C so that you should limit the ratio/stage to 2.8...3.6. To modify the ratio is very simple even on existing compressors.

To Nickname #31. Modifying or adapting a compressor at the design and selection stage is fine, but it is not an option for an existing installation.

The final air quality is dependent on proper selection and maintenance of suitable air treatment devices. The supplier of these products must assess the worst quality likely to be discharged by the compressor. Then to supply products able to remove this contamination in the foreseeable working conditions.

The temperature formula I am looking for is to assist in a risk assessment so that air treatment products are not undersized.

To answer various posts, using k = 1.4 does not give a safe i.e. high discharge temperature. To take a particular example, Roots blower, inlet pressure atmospheric, inlet temperature 20°C, discharge pressure 2 bara (maximum for a Roots blower) using n = 1.3 gives discharge temperature 70.8°C, using n = 1.4 gives 84.2°C. Aerzen say discharge temperature = 116°C.

For my calculation I assume outlet temperature rise due to compression T₂ = T₁*rⁿ, where n = 0.23. I worked out compression power by W = (P₁*V₁)/n*((P₂/P₁)ⁿ - 1). Shaft power = W/η where η = efficiency. So extra power dissipated in the gas = W/η - W = W*(1/η - 1). Mass flow of air q_m = inlet flow (m³/s) * density at inlet conditions ρ ~ 1.2 kg/m³). Using specific heat 1kcal/kg/K, additional ΔT = W/q_m/ρ *(1/η - 1). Total discharge temperature = T₂ + ΔT.

My approach for the Roots blower, using η = 60% (to agree with Aerzen power figure for a particular flow) comes to 113°C. It varies a bit for different sizes, it's not an exact science, but I think that's fair agreement. I looked into this as needed an estimate on a project a couple of years ago, but I haven't tried it on higher pressure machines. As you pointed out, if the supplier states an outlet temperature this could be after some cooling so doesn't help much. That's why I said temperature of the air in the outlet pipe, the instantaneous figure in the cylinder could be higher. The Aerzen figure is in the pipe, but without cooling.

Then again, it depends on the type of compressor. I believe some oil-lubricated screw types have an oil cooler, which might reduce the peak air temperature.

Thanks to Codename #33. Roots blowers are usually oil-free and being oi-free there is 'leakage' past the lobes (that gets worse as pressure increases) where there is a degree of re-compressing already hot air, and that explains the higher working temperatures.

But in my (ex) line of business I was more interested in a formula for medium pressure lubricated compressors that could have problems with burning oil.

And in general, as with most compressor manufacturers, performance figures are based on ideal average conditions. My belief is that in those conditions the peak temperature of each piston stroke will be higher than the average discharge temperature.

It is analogous to an AC generator where the voltage is usually the average or RMS based on a meter reading, and OK for most situations, but peak voltage is much higher which is necessary to know in certain circumstances. Any competent person involved in AC circuits will know this.

But in compressed air circles it is not well known and nearly always ignored. A simple formual to predict peak temperature will help.

Codename? Not to be confused with Nickmaster .

OK, there is some slip with a Roots blower, but that's not the whole story. If I remember right typical volumetric efficiency is about 80% but overall efficiency is lower (it cannot be higher of course) so there are additional losses.

For a piston compressor (which you referred to in original post) volumetric efficiency is probably better than Roots but still < 100% due to dead volume and leakage. So I'd still be surprised if k = 1.4 gives the maximum temperature, but I could be wrong. I don't remember seeing a figure quoted for the discharge before cooling, let alone the maximum in the cylinder; I think you're right, that will be higher. But will there be much oil in the air? I'm sure the oil has to changed at maintenance intervals, but I doubt if much of it goes out with the compressed air. Am I right thinking it's not the oil as such you're worried about, as that can be thoroughly filtered out, but CO etc which is harder to remove?

Codemster #35. Sorry about the mix-up with your name. You've guessed where it came from. Senior moment for me.

Generally speaking, compressors have become well designed and built to high standards where oil carry-over is minimal in design operating conditions. Even so, the amount discharged is often too high for most production processes, and anything can be sucked in from the atmosphere anyway, so reliance is placed on filtration devices.

In this respect an increase in temperature is critical, simply because hot air holds more contamination than cold air. Filters get over-loaded.

But assuming filters are maintained properly, the air will be oil-free in the technical sense. ie, below the specified quality limits. Which for most production purposes is often well below the aerosol (water, oil and dirt) quality limits needed for respiratory protection.

Thus for breathing air, it is the gases and vapours that are of concern, because these pass through most oil filters. Usually activated charcoal is used for the oil vapours in breathing air applications - these remove most combustion products that cause smell and taste.

Yes, carbon monoxide is of concern because it can easily produced by over-heating compressors. CO passes through activated charcoal. Thus in a situation with a properly working activated charcoal filter, the smell, which is one warning of burning oil, has been removed, thus unwittingly exposing the user to CO poisoning.

CO filter technology is well developed, but it has bad economic press because catalyst suffers from water vapour saturation (which is always there) (and worse when the air is hot) and annoying economically because the running cost of CO removal is related to drying the air to low dewpoints (a cost that is still incurred when no CO is present).

The economics of CO removal will change when new ISO standards emerge where air drying limits will be specified. This is not because dry is needed for breathing itself as suc, but to protect the breathing apparatus against freezing and corrosion.

It is my belief that production of 'burning' oil products is related to the peak temperature and it would be useful to have an acceptable formula that would predict this.

Why not use pressure instead temperature sensors? Since you have the relationship already. Peak readings will do. The pic below shows a port available for instrumentation, typically ideal for pressure sensor each cylinder.

Sample is below McNaught Indicator. This could easily be substituted by a pressure transducer and a pc interphase.

The object of my request is to find a rule-of-thumb formula that anyone can use to predict the peak temperature that air can reach in the end of the pressure compression stroke for any compressor - piston - screw -axial -whatever, when only the pressure and temperature of the ambient and working conditions are known.

An elaborate mechanism to carry out an actual test is out of the question. But even if a peak pressure could be measured easily, we still have to rely on a formula to calculate the temperature.

With progress in science there might be a temperature probe that can instantly respond to rapid changes in temperatures. But again, I doubt if anybody is going to bother to fit them.

So back to the theory. The maximum pressure is likely to be a little above working pressure depending on the pressure drop across the outlet valves, as is the inlet pressure a little lower than ambient for the same reason.

The inlet temperature is assumed to be ambient for favourable cases, but likely to be higher because (as time passes) it will warm up as it fills the hot cylinder especially when it mixes with some hot air left over from the previous stroke (or much hotter if mixing with hot air leaking back through faulty outlet valves or blocked inlet filters that reduce the inlet pressure giving rise to a higher pressure ratio).

Taking ambient and working pressures and temperatures at face value, my formula predicts the lowest peak temperature in favourable working conditions. It is this temperature that hits the oil - and consideration must be given to the possibility to cause oxidation - burning or ignition - from whence cometh toxic gases - that could persist unnoticed for some time. From a machinery point of view, running hot does not normally matter until the thermostat stops the compressor, but if someone is using the air for respiratory protection it could result in death or poisoning.

Left unchecked, in some cases working practices enable a build up of oil deposits that could detonate with devastating effect - probably not unnoticed. Google compressor fires and explosions for coffee time reading.

So I use a simple formula to help with risk assessments. Feedback seems to suggest it is fit-for-purpose.

For that I thank all contributors.

It seems to me that the way to deal with issues like this is in the design phase on in the operation phase. Usually during the design phase, someone has modeled the process very accurately, and determined the window of operation for not producing dangerous levels of contaminants in the breathing air.

Various laser, or infrared techniques can be utilized to "see" inside the compression volume and rapidly measure the temperature of a zone during all points of the compressor "stroke", or rotation cycle. Since the instruments are considered to be at least one order of magnitude or more faster than the cycle, it is clear that instantaneous to near-instantaneous temperature is the true observable. The compressor engine is then ranged through all the range of its output, and data is gathered and assessed. Then the head of the engineering staff works out the final solution that will be offered to management for production, etc, etc.

It does appear to be that such "problems" as may exist with lubricant pyrolysis products, the place to prevent these entering into the product air is during compression (while in contact with the compression engine), not after the fact, when in general there is a high energy penalty for removal of contamination from a process stream. Thus a fair amount of research should (and as far as I know is placed) be placed in avoiding improper conditions during production, than in attempting to clean up bad production. That is why oil-less compressors may be more popular in this arena.

To James #36.

In an ideal world these things would be thrashed out at the design stage, but unless things have changed since I have been retired, normal practice is to select compressors to economically produce air to a quality that suits the bulk production process. Incidental use for specialised applications is dealt with separately by additional air treatment devices.

These devices have to cope with whatever contamination is present, for which a risk assessment is required, which in line with guidance for designing and implementing a respiratory programme, the worst case scenario has to be considered.

To be clear on this, it is the 'programme' that must accommodate to worst case, not necessarily the 'filter'. A high temperature alarm to 'stop work' might prove to be suitable.

And whilst there is merit in using oil-free compressors, apart from oil, the inlet air in an industrial environment is likely to be contaminated in excess of the respiratory protection limits, so air treatment is required anyway.

I am pretty confident no works engineer is going to carry in-depth investigations on air temperature, because whatever the peak temperature might be, the discharge average is still too high for practical use. The air must be cooled before it hits the filters (and dryers and purifiers).

This market is well established and well developed, so the there is no practical problem of using general works air as a source of breathing air, providing the air treatment device.contains the appropriate stages sized to continuously remover all foreseeable toxic contaminants to meet the health and safety standards.

I tend toward agreement with all you are saying. I know that the local fire deparment here where I live has their own breathing air system for use with SCBA when they are in full turn-out. They do have a very rigidly monitored QC/QA program in place for that, and have the most state-of-the-art equipment available.

One factor I do not remember seeing in this thread is the speed of compression as relates to cylinder cooling vs. adiabatic compression. As I recall, the object is not to use near adiabatic compression in this case, but machine speed is regulated to provide a more controlled cylinder temperature.

To James #39

Thanks. It's comforting to think I am not entirely on my own.

As an after-thought, I should have made it clear that I am talking of general works air compressors - nominal 10 bar max.

Usually fire brigades have dedicated breathing air systems for filling cylinders. They use high pressure compressors - circa 300 bar - that are designed. maintained and monitored accordingly for this purpose.

For high pressure compressors the speed is typically 1,500 rpm for 50Hz supplies. The maximum running temperature is usually kept down by using low compression ratios in multi-stage machines with effective inter-coolers. However the risk of high temperature increases rapidly if the cooling system fails.

As a thought exercise for total cooling failure, the compressor behaves as a single stage machine in terms of temperature rise, r = 300. Probably not for long. Things catch fire. But as I said it is a thought exercise and high pressure cooling failure mode was not the intention of this thread.

The discussion has been very useful for me and I would like to thank everybody for their input.