 So, now let us come back to we had just looked at the critical point and we had looked at the saturation line relationship and the saturated liquid state its properties in dry saturated vapor state and its properties. Whenever I teach thermodynamics, I find it particularly useful to make the students read out certain properties at random. So, I ask the class to read out what is the saturation temperature at 3 bar, then saturation pressure at say 150 degrees C, then ask them what is the saturated liquid entropy at say 4 bar dry saturated vapor enthalpy at 200 degrees C and such things. So, that they get used to the steam tables and reading through the steam tables. This is necessary because unless such a practice is imposed, they will not get comfortable with the steam tables and they will waste time in the exams which are soon to follow. Now, after this we have now explored the liquid vapor saturation line or the interface between liquid and vapor. Now, it is time for us to go to other zones. The first zone which we will look at is the vapor zone. Let me start. Let us go to this zone. Here the vapor is at a temperature above its saturation temperature. Given a pressure, this is the saturation temperature. If this is P, this is T sat corresponding to that P and in this zone the temperature is higher than the saturation temperature. This zone is known as the zone of superheated vapor. All this is the zone of superheated vapor. In this zone, there is only one phase, the vapor phase and hence both pressure and temperature are independent variables. The property any property will be a function of two variables. The most convenient variables happen to be pressure and temperature. And these properties are tabulated in our steam table from table 3 onwards. And unlike table 1 and 2 which are I may call one dimensional tables, notice that you read something along a line. A different line is different item in the table. Here you have look at table 3. You have a two dimensional table. The title says superheated steam at different pressures and temperature. So, you have to know both the pressure as well as temperature to read out something. Let us take for example, a pressure of 2 bar and a temperature of 200 degrees C. So, notice that 200 degrees C is a column, 2 bar is a row. And where 200 degrees C and 2 bar come together, you have a tabulation of four properties. The four properties are specific volume V, specific internal energy U, specific enthalpy H and specific entropy S. And the values are 1.080 meter cube per kg, 265, 4.4 kilo joule per kg and so on. You go to a different temperature at 2 bar, you have a different set of values. Or at 200 degrees C you go to different pressures, you have a different set of values. So, this way you can read off for a given pressure and given temperature the values of the four properties V, U, H and S. This table goes over a number of pages, page 11, page 12, page 13 and so on. We will explore this in detail later, but let us come back to page 11. You will notice that part of this table is blank. When we were looking at 2 bar, notice that tabulations at 100 degrees C and 50 degrees C are blank. The reason for this is that at 2 bar, notice this 120.2 here and notice the heading. It is P at T S, P and T S. So, when the pressure is 2 bar, the saturation temperature is 120.2 degrees C. And for superheated vapour, the temperature has to be higher than 120.2. So, the tabulation begins at 150 C. But what is the state at 100 degrees C or 50 degrees C and the pressure of 2 bar? If you go to here, you will notice that if the pressure is 2 bar, the saturation temperature is 120.2 degrees C. So, you have tabulation at 150, 200, 250 and so on. But at 100 degrees C and 50 degrees C, you have liquid known as sub cooled liquid. Just to distinguish it from the saturated liquid. Sub cooled region is single phase, just like superheated vapour. And because it is single phase, the properties will be a function of pressure and temperature. The authors of this tabulation could well have tabulated those values here. And there are other more detailed tables. I will mention a few of them later. Have these tabulation in them. However, Messers Mathur and Mehta for some reason have decided not to tabulate sub cooled liquid properties in this zone. You will also notice that there is a column here which tabulates the properties of the dry saturated vapour at that pressure. The reason these are given is for convenience of interpolation when needed. Notice that we have tabulation at 150 degrees C, 200 degrees C, 250 degrees C. So, suppose you will have to calculate properties at say 170 degrees C, you will have to interpolate between the properties at 150 degrees C and those between 200 degrees C. But remember that the saturation temperature is 120.2 degrees C. So, it is possible that for some application you may need properties at say 135 or 140 degrees C. For that you will have to interpolate between properties at 150 degrees C which are those of superheated vapour and properties at 120.2 degrees C which is dry saturated vapour. It is for this interpolation, this tabulation is provided. Otherwise you can check from table 2 that at 2 bar these are the vapour properties already listed in table 2. So, this column here does not provide any additional information, this is information already provided in table 2 which is reproduced here for convenience. Let us explore this thing further. Notice that at low pressures on this page you have a pressure going up to 4 bar and the range of temperature is between 50 and 500 degrees C. You will notice that on the next page the range is 200 to 600 degrees C and the pressures go from 5 bar to 15 bar. On the next page range is from 250 to 700 degrees C and the pressures go up to 26 bar. On page 14 range again 250 to 700 degrees C, pressure goes up to 45 bar. On page 15 300 to 700 degrees C pressures up to 90 bar. Now come to page 16, it looks slightly different. There are two things, it is the same table extension of the same table, but the authors seem to have saved some space by changing the format slightly. Notice that here each block of properties had four properties, specific volume, specific internal energy, specific enthalpy, specific entropy. On this page last page of table 3, 16 you have just three values and the three values are specific volume, specific enthalpy and specific entropy. Specific internal energy is not tabulated just perhaps to save space, but we know h is u plus p v. So, you can always be computed as h minus p v and another thing you will notice if you go to the bottom of the previous page that here the specific volumes have become really low 0.0 something. So, rather than tabulate this 0.0 every time what is tabulated on this page is specific volume into 10 raise to 2. This has to be brought to the notice of students, because otherwise they will blindly use these values. So, if you need specific volume at say 130 bar and 500 degrees C, the specific volume is not 2.450 meter cube per kilogram, but 2.450 into 10 raise to minus 2, because what is tabulated is specific volume into 10 raise to 2. So, the actual value of specific volume is 2.450 into 10 raise to minus 2 kilo joule per kilogram. This table ends at a pressure just below the critical pressure and the highest temperature is I think 700 degrees C. Let us go to the next table. Here the authors claim to have tabulated the thermodynamic properties of supercritical steam. The format is very similar to the previous page pressure going from 230 bar to 1000 bar pretty high pressure. Temperatures going from 350 to 800 degrees C. Notice that the format is very similar. Now, what is this supercritical steam? We have this critical point and the temperature of the critical point is TCR is 374.15 degrees C. You can read it from the table 1 or table 2, the last entry in table 1 or table 2. Similarly, the critical pressure P C is 221.2 bar. Any state in which the temperature is 221.2 bar above critical or the pressure is above critical is known as supercritical state. So, this zone out here is tabulated in the so called supercritical vapor or supercritical steam. It is a single phase zone and hence the tabulation will be similar to that of superheated vapor and in our steam tables in table 4 page 17 you have the tabulation and you can read off the values as necessary. Now, you will notice why I have selected this table and sort of imposed it as a common document for all of you. So, that we can discuss the same thing together and I can tell you something and you can read it off from the table. Other steam tables would be similar may have slightly lesser detail or slightly more detail than this. We find this to be good enough for our purpose. Now, with this we have looked at the zones in the state space saturation line everything to the right of the saturation line. We have explored the saturation line both on the liquid side and the vapor side. We have explored the superheated vapor zone and the so called supercritical vapor zone. What remains now is this zone of the liquid phase which is to the west and north of the saturated liquid vapor saturation line. This zone is generally known as subcooled liquid or sometimes is also known as compressed liquid. You can reach this zone from the saturated liquid line either by reducing the temperature that is what gives it the name subcooled liquid or by increasing the pressure at the same temperature that gives it the name compressed liquid. The subcooled liquid and the compressed liquid mean the same thing and let us see whether we have a tabulation of this in our steam tables. One of the reasons for selecting this is one of the few short tables used by students which has some information about this and come to table 5 compressed liquid water. For some reason they have changed the format here and you will see that this is only a single page table. On the next page you have the saturated ice vapor line. So, that is not of use to us but it you can understand it. Here in the compressed liquid water you will find that you have pressure 50, 100, 150, 200, 300 and 500 bar a wide range but a very crude tabulation you know differences of 50, 100 and here 200 bar and at every pressure from the from 0 degrees C to some 0 degrees C. Maximum of 340 degrees C out here 380 degrees C the values of specific volume, specific internal energy H and S are tabulated. So, this is very similar to the superheated steam table but in a slightly different format. Given pressure, given temperature you can read off the 4 values. They could have used the same tabulation, same format as in the previous case. Notice something here and that is we start only from 50 bar we have no data below 50 bar and this is very crude 50 bar, 100 bar, 150 bar. So, you will have to do significant interpolation but our problem arises is quite often we have to look at pressure significantly below 50 bar. Many of our industrial boilers work at pressures of 5 bar, 10 bar, 20 bar, 30 bar and we do not have data for that. Quite often we pump liquids over a few floors may be 10 floors, 20 floors. The pump needs to increase the pressure from atmosphere which is approximately 1 bar to may be 3 bar, 4 bar. If you raise it by 1 bar, typically 1 to 2 bar you can typically pump it to may be 3 floors. If you raise it to 10 bar may be you can pump it to 20 or 30 floors. So, from that point of view we need to know the properties of steam or water substance. At pressures below 50 bar you can pump and in the subcooled liquid zone in this zone, subcooled liquid zone below 50 bar. How do you do that? For that we notice the following. This is crude but you will notice here that let us take at 20 degree C or 40 degree C. You will notice that as you go from 50 bar to 100 bar the specific volume changes but changes very little. The specific internal energy changes but also changes very little. Specific entropy also changes very little but there is some difference in the specific enthalpy and this happens crudely at all temperatures. Why does this happen? This happens because what we said at the beginning of this lecture that water, liquid water being like any other liquid is not really compressible. It is almost incompressible. I have almost said impossible. It is possible but it is almost incompressible. So, let us see what is the trick which we can use to obtain these of subcooled pressures. Say up to. So, we are looking at a zone and this zone. Our approximation is very good up to 50 bar. We will do up to 100 bar. After 100 bar it is going to be very poor. Here we use the approximation that subcooled compressible liquid liquid this is our approximation. Now, we go back to our state postulate and see what this leads us to. Notice that a compressible fluid two-way work mode is one that of compression expansion. Hence, number of properties needed to determine the state of a system is two. And we have seen in the superheated steam zone and even in the subcooled liquid zone the two properties which we select for convenience is pressure and temperature or temperature and pressure. But now when we say that our liquid is incompressible we cannot compress it nor will it expand if we allow it to expand because it is incompressible. So, the number of two-way work modes is zero. So, a system containing a liquid becomes essentially a rudimentary system because there is no two-way work mode. And that means the number of properties needed to fix the state of this system is one and which is the most convenient property for this temperature. Because remember one sort of a not a logical, but a common sense deduction from the zeroth law of thermodynamics would be that you take any system. So, long as it is capable of having a heat interaction with some other system will have to have a property called temperature. So, that some stage it will be able to come in thermal equilibrium with some other system in which case the two systems will have the same temperature by definition. So, the most convenient property we select this is temperature. And that means for an incompressible liquid the basic properties would be a function of temperature. For example, specific volume will be only a function of temperature specific internal energy would only be a function of temperature specific entropy would only be a function of temperature and notice that this is what we approximately noticed here we took I think 40, but we can take 60 degrees C. You go from 50 bar to 100 bar and the specific volume went up or changed from 0.000, 0.001015 to 001013 a difference of 2 in a 1000. Similarly, you changed 250.23 to 249.36. And let us look at the difference between that at 60 degrees C what was the saturated liquid value let us come down to an earlier page 60 degrees C 0.001017 the pressure 0.1994 bar. So, from 0.1994 bar to 50 bar and then to 100 bar the differences was a few parts in a 1000 to page 18. So, 1017 to 1015 to 1013 well our incompressible liquid is an approximation, but you will notice that it is a reasonably good approximation because the specific volume changes by a few parts in a 1000 a fraction of a percent. So, that is why we say it is an approximation. So, that means what does this mean this means that if so this means that determine properties of say P naught T naught some pressure and temperature we do the following. Let me draw a sketch and then we will write the procedure this goes up to critical point this is pressure we are in the sub cool zone so our P naught and T naught is somewhere here this is our point naught and that is sub cooled notice that from point 0 to this point 1 consider these two states. So, 0 is P naught T naught sub cooled liquid state 1 is saturated liquid at T naught let us look at this state in this zone because it is liquid if we use the incompressible liquid approximation in this zone these properties V U and S will be functions of temperature only that means U 0 will be U f at T 0 because this is U 1 similarly V 0 this is U 1 actually I should write U 1 first I write it here V 0 will also be V 1 which is V f at T 0. Similarly, S 0 will also be equal to V 0 equal to S 1 will be S f and notice that this is an approximation of very good approximation lower the P naught better the approximation and that is why I said earlier that this approximation will be good up to a pressure of 50 bar maybe you can stretch it to 100 bar, but do not try to stretch it beyond 100 bar you are likely to get into trouble because the assumption that we have an incompressible liquid that assumption itself will break down. Now, notice that whenever I talked of these properties I wrote here that basic properties will be functions of temperature and I listed V U and S why so and why am I calling them basic properties because V is a geometric property it is a physical property U is a thermodynamic property comes out of first law S is a thermodynamic property we will extract it out of second law there is one useful property which I have not listed and that is enthalpy. Now, what happens to enthalpy I should specifically bring it to your notice that enthalpy H is defined as U plus P V P is specifically included in enthalpy. So, you have to be careful with enthalpy let us see what happens to calculate H naught write H naught as U naught plus P naught V naught and then use this this is going to be U is U f at T naught plus P naught will remain P naught V naught will become V f at T naught. So, that means H is going to be H naught to depend on the pressure do not be under the impression that for a sub cool liquid the enthalpy is independent of pressure even if you assume it to be and real incompressible liquid enthalpy will not be independent of pressure in fact it will linearly increase with pressure because of the presence of P naught. And that is what you will notice here that you look at the enthalpy values the enthalpy values go on increasing as the pressure increases whereas, other values they do not go on changing significantly as the pressure goes on changing. So, remember that our incompressible liquid assumption which is very good up to 50 bar can be stretched up to 100 bar is to be applied for the basic properties specific volume specific internal energy and specific entropy it is not to be applied for enthalpy because of the direct presence of the pressure here. So, the enthalpy of an incompressible liquid will be a function of temperature as well as a function of pressure, but a study of steam table is not complete unless we start using it and that brings me to a stage where I would request you to take out your exercise sheet.