The physical properties of gases
Physicists are interested in the properties of every substance but engineers are primarily interested in air, water and a few other substances that are either gaseous or in their vapour phase at commonly used ranges of pressure, volume and temperature. Many substances that might be very useful to engineers are not used because they are toxic, inflammable or corrosive or just banned by diktat from politicians. Air is the gas, or more correctly mixture of gases, that must be the principal object of our interest and, if we understand the behaviour of air, it is not hard to deal with other gases.
For a quantity of gas held in a rigid container there seems to be no doubt that we can write:- where is the absolute pressure of the gas, is the volume, is the absolute temperature, is the mass and is a constant for the gas.
There is more to this simple expression than is obvious. It is not just a mathematical expression; it represents a gas with volume, pressure and temperature. It interests me that, provided that the mass is known, when the pressure and temperature are known the volume can be calculated, (and presumably checked by measurement) regardless of the history of the gas. This tells us something about the nature of gases. I described the molecular structure of gases in chapter 6 and the key observation is that all the never-ending collisions going on in the gas are perfectly elastic. This means that the state of the gas is not dependent on its history and that, as pressure and temperature are both manifestations of the intensity of the molecular activity, there must be an unalterable relationship between pressure, temperature and the volume of a given mass of gas. We are lucky that it is so simple but it would not have been had we used a different scale of absolute temperature.[1]
I have met one or two people who can immediately put a physical significance to these expressions but most of us need some sort of mental image that has some spatial element to it. This means having a graph and that means having to put a size to it and, because we live on this planet, there are practical ranges to all these quantities. It makes no sense to ignore this and plot grossly distorted graphs that are totally deceiving. We need a graph relating pressure, volume and temperature for air over the ranges of pressure and temperature that we are forced to use in engineering. As a guide Diesel engines and gas turbine engines work up to temperatures of about 1,000°C and pressures of about 60 bar. These pressures and temperatures are achieved in different ways in that air is compressed through a compression ratio of about 11/1 in a diesel engine and then increased by combustion partly at nearly constant volume and partly at constant pressure whereas modern gas turbine engines have compression ratios of 18/1 and combustion is at constant pressure. The lowest temperature that occurs naturally on Earth is about -50°C[2] in the Polar Regions where we might choose to operate piston engines and in the upper atmosphere where we might fly supersonic aeroplanes with gas turbine engines.
We need some starting data for air. Physicists work in moles and the universal gas constant but engineers, who really have just one or two gases to deal with, work in kilograms and a value of R for each gas. So, for us, R for air is 287 J/kg.°K. The density of air at 1.293 kg/m3 at 0°C and 1 atmosphere. 1 atmosphere is equal to 101,325 N/m2 and for most practical purposes in engineering it is good enough to put 1 atmosphere equal to 1 bar.
In graph 13-1 I have plotted pressure against
volume for 1 kilogram of air at absolute temperatures of 200°K
(-73°C),
400°K,
600°K,
800°K,
1,000°K,
1200°K,
and 1,400°K
(1127°C). I
have let the graph range up to 80 bar and between 0.025 and 1 cubic metre when
the volume of the air at 0°C
and 1 bar is 1/1.293 cubic metres = 0.773 cubic metres. The lower dotted line
in red is for 1 bar, ie atmospheric pressure
It is all too obvious that the possible states of the air for this very wide range of pressures and temperatures are all crammed into the corner. This is not at all what engineers trying to design reciprocating engines want to see because any air breathing engine will have to function of some closed cycle within this family of graphs and the area within the cycle on this p-V plane will represent work done. Useful values of work done will go with large volumes and high pressures, in other words big, strongly-built engines. It is a stroke of good luck that gas turbine engines can work in this miserably small area and produce large powers with practical pressures and temperatures even if it has taken 50 years to get there.
A quantity of gas will have energy stored in its molecular structure and this is called internal energy and it can obviously be altered. All you have to do is heat or cool the gas or change its volume in some arbitrary way. But this is not much use to engineers and we need some way of relating the internal energy to the state of the gas. If we take a quantity of gas held in a rigid container and supply heat to the gas, its pressure and its temperature will rise but the gas will do no work during the heating. It follows that all the heat has gone into the internal energy of the gas. This gives rise to the use of a specific heat for the gas. It is defined by the following expression:-
where is the heat supplied, is the mass of gas, is the rise in temperature and is the specific heat at constant volume.
We also use another specific heat for a gas and this one is for heating at constant pressure. Then, during the heating process, work will be done and more heat must be supplied to produce the same rise in temperature. This leads to the defining of a specific heat at constant pressure from :-
where is the heat supplied, is the mass of gas, is the rise in temperature and is the specific heat at constant pressure.
Now, if and are the same in both processes, the increase of internal energy is the same in each case and equal to . It follows that the work done equals:-
.
The question then arises about whether this work done depends on the properties of the gas. The answer is yes because the work done is equal to where is the constant pressure and is the increase in volume and the increase in volume depends on the molecular structure of the gas.
This has a further consequence in that the ratio of is also a property of the gas. It is given a special symbol . This statement will require more careful examination later in this text.
For air =718 J/kg.°K =1,005 J/kg.°K and =1.4