 What we have studied so far regarding fluid systems and fluid systems are of utmost important to us are a model for a fluid called an ideal gas. This is an approximation for a real gas. What is the actual case? In an actual case, the actual gas, if you take it to appropriately high pressure, so low temperature, does not behave like an ideal gas. In fact, any material which we take, even the simplest of it, you will find it in three phases, solid, liquid and the vapor for the gas phase. These are known as phases and we know the general characteristics of each one of these. And we as engineers, particularly mechanical engineers, we work with all three phases. We work with solids, we work with liquids, we work with gases. So, when we come to gases which cannot be modded as ideal gases, what are the other options? For an ideal gas, we have a simple equation of state and we also know from Joule's law that u is a function only of temperature. The moment we define these two functions, the basic equation of state, the ppt relation and in some energy function, say either u as a function of t or h as a function of t. Thermodynamically, because the internal energy is a properly defined and primarily defined thermal quantity, we would prefer that u be defined as a function of t. But convenience of measurement and sometimes convenience of analysis forces us to specify h as a function of t. From this, almost everything else can be derived. When it comes to a real gas, a non-ideal gas, instead of real I should say non-ideal gas, because all gases are real. We will not have such a simple equation of state. We will have to have more complex equations of state. However, it is known that the deviation from ideality does not come up suddenly, slowly comes up as you go to higher pressures or lower temperatures. So, there are other proposed equations of state. The few of them we should know about. From the kinetic theory, it is known that a collection of molecules will behave like an ideal gas if the density is low and two things have to happen. Number one, in the system volume, the volume occupied by the molecules themselves has to be negligible. And number two, the interactions between molecules have to be only during their collisions with each other. And of course, the collisions of molecules with the wall, that is the interaction with the wall. As you go to higher pressures or higher densities, the volume of the system becomes lower and lower. And at some stage, the volume of the molecules starts becoming comparable or not an insignificant fraction of the system volume that has to be taken care of. Similarly, as the density increases, the typical intermolecular distance decreases. And hence, non-collision type of intermolecular interactions also come into operation. And there are a number of proposed equations of state which try to take into account these effects. Perhaps the simplest modification is the Clausius equation of state, which is P into V minus B equals RT. This is supposed to take care to some extent of the molecular volume as represented by B. The second modification is the famous Van der Waals equation of state. If you notice, the ideal gas equation of state has just one parameter R. Different gases when represented as an ideal gas will have different values of R. So, you can say that the ideal gas is a very simple single parameter of state. Whereas, when you come to the Clausius equation of state, now we have two parameters. So, for every gas which is modeled as a Clausius gas, you will have to specify R as well as B. Now, you will notice that for the Van der Waals gas, you will have three parameters. So, Van der Waals gas is not the only three parameter equation of state. For example, not the Berthelow equation of state is very similar to the Van der Waals equation of state. It is also a equation of state. So, as the behavior of gases needs to be modeled in a more complex way, you will notice that the number of parameters go up and a really complex fluid like water in its three phases will require a large number of parameters. In fact, in a general case, we write a Virial equation of state or Virial form of equation of state. Here, we write it as an infinite series. So, we can have any large number of parameters as large as we wish. Out of all these equations of state that we are going to look at, three equations of states are going to be of very great importance to us. The first one is the ideal gas equation of state, which will be using routinely. The second is the Van der Waals equation of state for the simple reason that some phenomena which we see in real fluids. For example, the critical point, the liquid vapour equilibrium zone. These things are seen and the Van der Waals model is the simplest model which shows that. So, from that point of view, this is important. Also, when you consider a kinetic theory, we have some significance attached to the values of A and B of the Van der Waals equation of state. However, for us, team, the same substance in other phases is of great importance. Hence, we should study the behavior of these, particularly that of the two fluid phases, steam and water, in great detail. That is what we are now going to study. This will bring us to our study of properties of steam or the so called study of steam tables. Let us consider first the phase diagram of water. Now, some explanation is needed about the world water. If you look at your chemistry book, they will say water is hydrogen oxide and chemical formula is given to us H2O. But if you look at the actual water molecules in nature, you will find that the natural abundance of water, natural water has two components H2O and D2O. D2O is a small fraction. It is of the order of I think 70 ppm. I am not so sure, but I think that is the order. So, when we consider this naturally occurring purest form of so called water with the appropriate mixture of H2O and D2O, we have given it a technical name which is supposed to represent it in all its three phases and that is ordinary water substance. The word water substance is used to indicate that we are not looking at just water. Water traditionally is the liquid. What you have in this water is water. But if I boil it, the vapor of water is known as steam. That is the colloquial name. Similarly, if I put it in the freezer and freeze it, I will get ice which is a solid form of water. So, the common name for this water in all its three phases is water substance. And the word ordinary indicates that that is the typical mixture of H2O and D2O which we find in nature surrounding us on earth. The phase diagram is an important diagram. It is a projection of the state space on the Pt plane and this is plotted by taking a system containing ordinary water substance and by appropriate interactions bring it to the required pressure and required temperature. And the only observation we make is to find out whether it is solid, whether it is liquid or whether it is vapor. And we know how to distinguish between them. Solid means it will have a shape. You impose a force on it that shape may bend. If you remove the force, it may come back to it. But you leave it here, it will remain like a solid block like this. This is solid. I can bend it perhaps, but it does not flow. It retains its shape. A liquid if something when you fill it in a vessel may not fill the whole volume if the volume is not enough. And it will try to take the shape of the vessel and it is able to flow. I can make it flow out of it and I can think it. A solid cannot be drunk. A solid you can bite off and eat. I am not going to do it with my mobile phone. A vapor means you cannot see an interface. If you put a vapor inside this, it will occupy the whole volume. So, the air surrounding us, the vapor of water when we boil it, these are all gaseous phases. So, if we note down, we will find that the Tp state space is split into, I would say, approximately three parts, not exactly three parts. When you have low temperature, you will find that it is generally in the solid form. At high temperatures, but low pressures, you will find that it is in the vapor form and at other things in between, it is in the liquid form. And there are three lines which separate the solid from the liquid, liquid from the vapor and the solid from the vapor. So, this is the solid vapor. This is the solid vapor separation line. This is the liquid vapor separation line and this is the solid liquid separation line. When you have more than one phases, the phase rule from physical chemistry comes into operation. The phase rule says that if you have a system in which you have different chemically distinguishable components and also different phases, then this relation applies, where C is the number of components, different chemically identifiable substance. Here, we can say perhaps H2O plus D2O are two substances, so two components, but the D2O is small and during all our processing, we are not going to change the composition or the fraction of D2O. So, we can get away by considering it to be a single component. So, for us, the right hand side turns out to be three. On the left hand side, we have P is the number of phases and F is what is known as the degree of freedom. We can simply say that degree of freedom is the set of properties like pressure temperature, intensive properties which you are free to manipulate a bit and still retain that number of phases together. For example, you consider the liquid phase. Say, let us consider a state here. There is only one phase. So, the number of degrees of freedom is two. That means I am free to change the pressure a bit without changing the temperature and I will still be in the liquid phase. Similarly, I am independently free to change the temperature a bit and I am still in the liquid phase. So, this tells us that I have two degrees of freedom. The same thing happens if I am purely in the vapor phase or purely in the solid phase. But now, let us consider a situation where we have a system containing say a liquid and a vapor. So, let us say our system now contains a liquid and it is free to change its vapor. This is our system and we are maintaining it at a particular pressure and we are measuring its temperature. So, on the P space, this is solid, this is liquid, this is vapor. We are somewhere here at this point. The number of phases is two. F is the degrees of freedom. So, this will be components plus two. This gives us F equals 1. That means of the pair pressure and temperature only one we can change. The other will have to follow if you want to maintain the liquid vapor combination together. That means if we have a state here, I want to change the pressure a bit and still retain my liquid vapor in the system together in equilibrium. Then I will have to change the temperature appropriately. I cannot independently change the temperature or if I want to change the temperature and still maintain a liquid and vapor those two phases together in the system, I must change my pressure. So, that means when you have liquid plus vapor together, what is known as a two-phase situation or a state with two phases in it, we can select either pressure or temperature, but not both. That means on this line, which is known as the liquid vapor in the equilibrium line, we must consider that pressure must be defined as a function of temperature or temperature must be defined as a function of pressure. This line is known as the liquid vapor. A similar thing happens when you consider a situation a system containing solid and liquid together. We have this line, which is known as the solid liquid saturation line and this line is the solid vapor saturation line. Again let me sketch the diagram because this is one of the most important diagrams we should study. We have a very special point on the P T diagram and that is this point where all the three lines come together. This is known as the triple point. This triple point is a phase is a situation where you have in our system at a given pressure and temperature some solid, some liquid and the vapor all together at a given pressure, which is the pressure of the triple point, a given temperature which is temperature of the triple point. Why do I say that it has to be a point because if you apply the phase rule, the number of phases is still plus f, number of components is 1 plus 2 is part of the formula. So, this means f is 0 and that means we have no choice of selecting either the pressure or the temperature. These are fixed values. So, P and T no choice P and T fixed. The moment you take a particular type of material in this particular case ordinary water substance, if you bring your state containing such as ordinary water substance to a state where you have solid, liquid and vapor together in equilibrium. That means the same pressure and same temperature for all three phases then you have no choice about the pressure. It will occur at one unique value of pressure known as the triple point pressure, triple point T P. This is P T P and it will occur at one temperature that is the triple point temperature and this phenomena is used in our Kelvin scale, in many other temperature scales to define a fixed point, fixed state and fixed point. Now, before we go away from this, there are some terms which we should be familiar with. If I take a solid at a low enough pressure and start increasing its temperature by some mean, I will execute a process like this. As the state crosses the solid vapor saturation line, we will have a process of sublimation, solid going straight into vapor. Whereas, if I take that vapor at low pressure and try to reduce its temperature, then it will condense into a solid. If I take a solid at a reasonably high pressure, a pressure higher than the triple point pressure and raise its temperature, this will be a process of melting. Whereas, if I start with a liquid and cool it, I will end up with the solid state that will be the process of solidification or freezing. Suppose I take my liquid water or any other, this is nothing special about water because I have not talked about the actual data, water is an illustration. If I take a liquid and increase its temperature as it crosses the saturation line, it will convert itself into vapor that is the process of evaporation or boiling. The difference is only the rate at which the change takes place. If the change of state is slow, the rate of formation is pretty slow, we call it evaporation. If it is pretty fast, lots of bubbles in the water, we call it boiling. And similarly, if you take a vapor and reduce its temperature, we call it condensation. Notice that to convert a liquid into vapor, it is not always necessary to increase the temperature. I can take this liquid and reduce its pressure while keeping the temperature constant. Even then, when it crosses the saturation line, liquid vapor saturation line, it will evaporate. Technically, we call this flashing no change in temperature but reduction in pressure. And similarly, you can take a vapor and compress it, let me use some other color and you can compress it into a liquid. So, you can have condensation by compression. Now, there are a few things to be noted and this is typical of any material which we can have in three phases, water, ammonia, nitrogen, hydrogen, all of them. All of them have a triple point. All of them have a solid, liquid, liquid vapor and solid vapor equilibrium lines. Now, the liquid vapor equilibrium line is special in the sense that as you go to higher and higher pressures, the saturation temperature that is the temperature at which boiling will take place increases. But as you go to higher pressures and higher temperatures, the state property differences between liquid and vapor reduce and we come to a stage where we get what is known as a critical point. So, the red line here which is the liquid vapor equilibrium line is not a indefinite line, it ends at the critical point. The critical point is also an important point, although not as important as the triple point, but from an engineering point of view it is a very important point. The shape of these lines and their slope and their extent depends on the detailed thermodynamic properties of each material. It is a typical diagram for water, for ammonia for example, we will have a different diagram. The solid vapor line goes down to reasonably low pressures, there is no end found for that. Similarly, the solid vapor line goes indefinitely to higher and higher pressures, no end has been found experimentally on it. Sometimes the solid vapor line as shown here slopes towards the left, but there are materials for which the solid vapor line many other materials many other liquid for which the solid vapor line slopes towards the right. Later on when we study properties of fluids, we will realize why is it that water for water it slopes to the left and for some other liquids it slopes to the right. We will link it to the fact that the solid phase of water that is ice floats on water does not sink on water. We will demonstrate that if the solid floats on the liquid this like water this line will slope towards the left like this. Whereas, if the solid sinks in water like for many other liquids the line will slope to the right like this. Now the behavior of water is so complicated that we hardly ever have similar properties simple analytical expressions for its properties. So, the properties of water are tabulated or they can be even provided in a graphical form. Why do we do that? Because equations fitted to data have something like 400 constants in there or 400 parameters. The equation occupies something like 4 printed pages of this size. Maybe I should have brought and shown you those equations, but they are available. It is a standard international formulation for the properties of speed. If you look at them you will find all the functions which you have studied in your advanced mathematics course are included there. When there is no way you can calculate it out none of us will have a patience to use a calculator and calculate it out. You will need a computer and a big program to crank out the properties. That is not very convenient and that is why the properties of water are tabulated and that brings us to the steam table. The steam tables provide us a discrete set of information about the properties of water. Let us go back to our phase diagram and see what information is provided in our steam tables. Take out your steam tables and now do not show me at all now till I say you can come to me. In this it is about 20-22 pages long not very big. This is a typical steam table. You are free to use any other. This happens to be my favorite for our favorite in IIT Bombay for so many years. It has improved over the years so we use it. First we locate the critical sorry the triple point. The triple point is and the information pertaining to the liquid vapor saturation line is provided in tables 1 and 2 and if I take a bigger version of this line this is one particular temperature and this is one particular pressure. You will find that on this liquid vapor saturation line if I fix the pressure I fix the temperature. If I fix the temperature I fix the pressure and that information is available in table 1 where for fixed temperatures you have the corresponding pressures. The most important point here is the point pertaining to 0.01 Celsius or 0.01 degree C and that is the point which is the triple point. At this point the pressure is 0.006112 bar. One thing you should note is this is 0.01 degree C by definition as we saw yesterday and this pressure this is T triple point and this is 0.006112 bar this is P triple point. Now one thing you should do is what I am going to do now. Take a scale and may be a reasonably thick pen and erase out this first line strike out the first line which pertains to 0 degree C. That line should not be there. Why? Because if you go to 0 degree C a temperature below the triple point temperature you do not have a liquid vapor equilibrium at all. You will either have a solid liquid vapor equilibrium or a solid liquid equilibrium but not a liquid vapor together. So remove the first line table 1 should start with 0.01 degree C. Now let us see more information about table 1. I am now showing only a part of the state space. Table 1 starts from the triple point and ends somewhere. Table 1 has first column in terms of temperature. The second column gives you a value of pressure if I select a temperature T it gives me a pressure at which water will boil and convert itself into vapor at that temperature. This is known as the saturation pressure corresponding to that temperature. For example if I take 30 degree C it means that at 30 degree C if I have to have water and its vapor together in equilibrium the pressure should be 0.04246 bar. You go to higher temperatures for example let us check out 100 degree C. This is on the next page, page 4. At 100 degree C if I want water and its vapor to be in equilibrium the pressure must be 1.01325 bar which we know is one standard atmospheric pressure. And as you go to higher and higher temperatures you will find that the pressure increases. Till you reach the critical point 374.15 degree C where the saturation pressure is 221.2 bar above this there is no distinction between liquid and vapor. So this is critical point and T critical point is 374.15 degree C and P critical point is 221.2 bar. So this is the information in table 1. First let us look at only the first two columns. Given temperature we can determine the saturation pressure at that temperature. Now let us go to table 2. In fact table 2 has the same information for table 1 except that table 2 is for rounded values of pressure. You can read the saturation temperature for a given pressure. So given a pressure if I want to know at what temperature will liquid vapor equilibrium be available that temperature is known as the saturation temperature at that pressure or we can say in colloquial language the boiling point at that pressure. Fortunately for us this starts at the proper pressure the triple point pressure 0.006112 bar temperature 0.01 degree C. This is an exact temperature by definition and you will notice that as you go to higher and higher pressures the temperature increases. Some points to note here come to a pressure of 1.01325 bar you will see this on page 6. This is one of the odd values of pressures which is tabulated because it is the standard atmospheric pressure at which the saturation temperature is 100 degree C means that at one atmospheric pressure water will boil at 100 degree C. You go to lower pressures for example if you go to a hill station and the water will boil at the pressure there high up in the air in a balloon is 0.9 bar the water will boil at a slightly lower temperature of 96.7 degree C. You go to higher pressures for example if you go to 2 bar the water will boil at a temperature of 120.2 degree C. Now this is the typical situation in a pressure cooker. In a pressure cooker I think the standard design of a pressure cooker keeps the internal pressure at roughly one atmosphere or one bar above the ambient and hence inside the pressure cooker when you cook your meat or dal or chawal whatever the temperature will 120 about 120 degree C. And this has relation to the rate at which food is cooked. Cooking food is typically a chemical reaction biochemical reaction in fact and many of these reactions have a characteristic that the rate of reaction typically doubles every 10 degree Celsius in approximate order of magnitude. So if something is cooked in 1 hour at 100 Celsius it will cook in half an hour at 100 and tells Celsius and it will cook in 15 minutes at 120 degree Celsius. So in a pressure cooker when you increase the internal pressure by closing the valve putting the weight on it from 1 bar to 2 bar by boiling it boiling water or a water or something containing water inside it. The temperature will rise by 20 degrees and hence the rate of cooking will go up by a factor of 4 and the time required for cooking will come down by a factor of 4. This is a typical order of magnitude analysis. Now you will notice that just the way table 1 ended at the critical point which was by temperature 274.15 Celsius and by pressure to 21.2 bar. The table 2 which is a more exhaustive table and goes over a number of pages on page 10 at the end of table 2 you will find the last pressure tabulated is 221.2 bar. The corresponding temperature is 374.15 Celsius. Now let us come to the next level of appreciation. Again we are going to study further the behavior of the liquid phase and the vapor phase when the state is on the saturation line. We can label it either using its temperature or using its pressure. If we want to use temperature as our base use table 1. If we want to use pressure as the base we use table 2. Now at this state liquid and vapor are together in equilibrium Pt linked through the saturation relationship first two columns of tables 1 and 2. But the liquid will have a different density than the vapor. The liquid will have a different set of specific properties compared to the vapor. So if you look at the state approaching it from the liquid side and if you look at the state approaching it from the vapor side you will have two distinct sets of properties. When a liquid and vapor are together we say in equilibrium we say that we have a state of saturation. The liquid part is known as saturated liquid and the vapor part traditionally is known as dry saturated vapor. The word dry is used to emphasize that the vapor has no liquid droplets or liquid parts associated with it. That is why dry saturated vapor. It is a technical name but if you call it saturated vapor that is perfectly okay. At least thermodynamically it is okay. Maybe from a steam engineering point of view it may not be okay. So the properties of the liquid phase properties of the liquid phase are tabulated in tables 1 and 2. Properties of the vapor phase are also tabulated in tables 1 and 2. For the saturated vapor phase the subscript used is G. For the saturated liquid phase the subscript use is F. That is the tradition which is used. So now come back to our ELMO. Let us look at any one of the two tables. Let us look at table 2 and let us look at our 100 degrees C or 1 atmosphere data. Now look at the other components here which are tabulated. We have 1, 2 columns pertaining to specific volume. We have 2 columns pertaining to specific internal energy. We have 3 columns pertaining to specific enthalpy and 3 columns pertaining to specific entropy. We have not yet studied what entropy is but at this stage let us say that entropy is some property which is tabulated. Similar to enthalpy internal energy but has a different set of characteristics. Columns 3 and 4 tabulate the specific volume of saturated liquid named VF and specific volume of dry saturated vapor VG. Similarly, columns 5 and 7 have the properties pertaining to the saturated liquid and saturated vapor internal energy. Next 3 columns have in fact column number 1, 2, 3 and 4 where VF VG 5 and 6 are UF, UG, C, C, C. 7 and 9 are HF, HG enthalpies and 10 and 12 are SF and HG entropies. So, again if I show only a part of this Pt properties of the saturated liquid phase that is VF, UF, HF and SF are tabulated. And from this side properties of the so called dry saturated vapor phase VG, UG am I back here? Yes VG, UG, HG and SG tabulated. And if I show only a part you can read these either by selecting the pressure in which case you will be using table 2 or by selecting the temperature use table 1. Select either table 1 or table 2 according to your convenience. So, tables 1 and 2 tabulate P and T in some order and the further columns are VF, VG, UF, UG, HF, HG and SF, SG. Now, there are other columns for example there is a column here known as HFG and there is another column which is SFG. Actually these two columns just provide no additional information. They provide a convenient tabulation of the column HFG and SFG. They provide just differences SFG is defined as SFG is defined as SG minus SF which is tabulated, HFG is defined as HG minus HF and it is also tabulated. HFG and SFG provide no additional information. It is only for convenience that those two columns are tabulated. If tomorrow paper becomes too costly we might as well drop those columns and save that much paper. Let us go to the end of this table. Just the way we can go to the end of table 1, the critical point we can go to the end of table 2 on page 10 I suppose. Yes, 221.2 bar 374.515 Celsius. You will notice that at the critical point notice something funny. VF equals VG, UF equals UG, HF equals HG. So, HFG is 0 and SF equals SG. Hence SFG is 0. This indicates that since the properties of liquid and vapor including the specific volume are the same there is no distinction between liquid and vapor. Traditionally this is known as the critical point and if you go beyond the critical point again coming back to our phase diagram. If you go to a pressure above the critical pressure you will not notice when you go from the liquid zone to the vapor zone. In fact, above the critical pressure what we have it is simply a fluid. We should not really call it a liquid or a vapor and you can go from solid to liquid and then you will remain in the fluid form. The density will keep on continuously reducing, but you will never see a bubble being formed. These states are known as states of super critical fluid. Now, it is time for us to move to the other zones in the steam tables. I will sketch this, but then I will take a break for a few minutes after one hour of my lecture.