October 14, 2005

Refining The Model: Equations Of State 

In the industrial, chemical and engineering practice, it is necessary to know many properties of fluids (gases and liquids) and occasionally solids, often to a good degree of accuracy.

Density of fluids is of fundamental importance for adsorption studies (the work I'm doing at the moment) but also for almost any operation involving handling of fluids; heat capacity and thermal conductivity must be known for anything regarding heat exchange; viscosity determinates how a fluid flows (and in the case of fluids like molten polymers, the situation is pretty complex); distillation requires knowledge of the liquid/vapor equilibria of mixtures; membrane separation is based on diffusivity differences; and more unusual applications require the knowledge of more exoteric properties (enthalpy, compressibility etc). All these properties vary greatly with temperature and pressure.

If you trade in compressed and liquified gases; you want to know at what temperature the pressure in your oxygen cylinders will get close to the design limit - for obvious safety reasons. Motor oil is a high-tech fluid, and among other properties its viscosity must be as constant as possible in a wide range of temperature in order to facilitate cold starts and retain good lubricating properties when the engine runs hot. If you want to design a steam turbine, you need to know the whole shebang: liquid/vapor equilibrium, density, viscosity, heat capacity, enthalpy content and probably even more of steam - which is quite a bitch of a fluid. Finally, if you want to cause the nuclear detonation of a mass of plutonium, you must know waht pressure is required to compress it above its critical density.

Most of these properties can be measured experimentally, but it easily becomes an exceedingly difficult, expensive and time-consuming job. And sometimes it's not possible.

What would be very useful is a mathematical model, an equation into which you can simply feed your temperature and pressure, and it will churn out the value of the property you want to know. Fortunately, these equations exist, and they are called equations of state (EOS).

The first of the kind, now known as ideal or perfect gas equation of state (IG EOS) was proposed in the 18th, 19th at latest century and it correlates Pressure, Temperature and molar volume in this fashion:

Pv = RT

R is a constant called the gas constant; it is necessary for the equation to work but it also constitutes a sort of indication of the energy content of a gas (science is full of constants of this kind)*. This equation is very simple, ad calculations can be done with just paper and pen - or even mentally for those versed in mental arithmetic. The molar volume of gases at 0 C and 1 bar is 22.4 liters - this is one of the results of this EOS.
Molar density is the inverse of molar volume (1/v), so it's pretty straightforward to calculate it, and all other properties can be calculated with manipulations of this basic equation and using other concepts of thermodynamics.

However, pretty soon people realized that real gases do not quite behave like ideal ones; hydrogen and helium are pretty close to being ideal, but oxygen and carbon dioxide (for example) are sensibly different. Not to mention hydrocarbons. Using the IG EOS to predict their properties will result in noticeable errors. So, scientists began seeking to improve the model and have more accurate predictions.

It's not the case to enter in the mathematical details, but the method used was to add a corrective factor to the equation, a new parameter to be calculated fitting the equation to experimental data - for example, a weighed amount of gas is placed into a strong sealed container, and the pressure is recorded while varying the temperature. Or you can work at constant temperature and vary the volume of your chamber. Then you have a series of data, and with proper techniques you can calculate the parameter(s) of your EOS - parameter(s) which sometimes are temperature-dependant themselves. Basically, each new equation was a bit, sometimes even a lot, more accurate than the previous ones.

The first alternative equations of state had one or two parameters characteristic for each substance; some have three or four. One of the most popular EOS, the Peng - Robinson (the two guys who developed it) has two parameters, one temperature-dependant. This equation is used extensively for hydrocarbons and their mixtures. Although not very accurate, it is pretty simple to implement with any calculus package (Mathematica, Matlab etc).

For the case of carbon dioxide - the fluid I work with - not even the PR EOS is enough for very accurate work. Two German researchers, Span and Wagner, did a monumental work of compiling thousands of experimental data for the properties of CO2 and using all of them to fit a monstrous equation, with literally 30 or more parameters. But their efforts paid off, because now theirs is the state-of-the-art, international standard equation of state for carbon dioxide.

Problem is, it's a nightmare to encode; but fortunately institutions such as NIST have developed software packages that do the job. So I just had to buy the software, and now I can happily copy & paste my experimental data into it, and obtain high accuracy density values in just a few seconds. Marvels of this age.

Update - last night I was in haste, so I forgot a couple of things...

* The units used for all the quantities in the equation must be consistent; in the SI system, pressure is in Pa, volume in m3, temperature in K and quantity in mol. R thus is 8.314 J / K*mol. In the fractional system... I don't even want think about that perversion!

The IG EOS is valid only for gases, while other equations of state can be used for liquids too - and most importantly, for liquid-vapor equilibria. Also solids have their own equations of state, but I've never seen one, actually. Aside from nuclear technology, the modification of solids at high pressure is of interest in the field of geology, given that in the depths of the Earth pressure reaches unbelievable extremes.


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