 So, you may recall that when we discussed moist air or rather define the specific enthalpy of moist air, we actually defined something like this. So, this is the enthalpy of the moist air mixture. And if you actually want specific enthalpy of moist air, then we divide both sides by the mass of the moist air and we are then led to something like this M A over M times H A plus M V over M times H V. So, this would be the specific enthalpy of moist air, it would be in units of say kilo joule per kg air. And we can then approximate H A using the calorically perfect assumption and H V may be approximated as H G of T. However, if you look at psychrometric applications, then you can the examples that we have done so far, you probably will notice that we get this combination of terms C P times T 2 plus omega times T 2 and C P times T 1 plus omega times T 1. So, you notice combinations like this. So, let me just show this with a different color. So, this is the second combination. So, if you look at the next example that we saw here too, you see the same thing. So, you see C P times T 2 plus omega times H 2 and you notice this combination C P times T 1 plus omega times H of T 1. So, you notice this combination. So, what you are saying is actually the specific enthalpy defined on a per dry air basis or per kg of dry air basis. Here also you see the same thing. This term, these two terms go together and then you again see that these terms go together. So, which means that we would like to define the specific enthalpy of moist air not on a kg of moist air basis. So, basically what I just wrote was something like this. So, H over M is M A dry air over mass times H A plus M V over M times H V. This of course, becomes C P times T, this is H G of T and this is nothing but the specific enthalpy of moist air in units of say kilo joule per kg air in the usual manner. But in the applications or examples that we have seen so far, we notice that it does not when you look at the energy equation that we have applied to different examples, we notice that the terms do not appear in this combination, but appear in a combination where the specific enthalpy is defined on a per unit mass of dry air. So, basically instead of dividing by M, we divide by mass of dry air. So, if you do that, then we end up with this combination, H A plus omega times H V, which is in units of kilo joule per kg dry air. There is nothing wrong with this. So, this is perfectly alright, but this is specific enthalpy on a per unit mass of moist air. But in applications when we apply energy balance to practical problems and examples that we have seen, this combination does not appear. We always see that this combination is what is seen. And this is specific enthalpy, which we have denoted H star specific enthalpy on a per kg dry air basis. This is far more useful in psychrometric applications than this, but there is nothing wrong with this also. And H star is what we will use in our examples from now onwards. But we will clearly indicate. So, this is actually H of moist air on specific enthalpy of moist air on a per kg moist air basis or per kg air. This is H star, which is specific enthalpy of moist air on a per kg dry air basis. Now, the new terms that we have introduced in psychrometry or like this. So, this is a list of new terms that we have introduced. First one is the partial pressure of water vapor, which may be written like this, omega over 0.62 plus omega times P, where P is the mixture pressure as always. And the mixture pressure during psychrometric operations, unit operations remains constant. Generally, we assume the mixture pressure to be constant. Furthermore, in all the examples that we are working out, we have assumed the mixture pressure to be 1 atmosphere. But mixture pressure is usually a constant, which means that P V depends only on omega. Now, relative humidity phi is nothing but P V over P sat of T. So, P V here depends on omega, P sat depends on T obviously. So, phi depends on both omega and T. So, we may write omega depends on temperature T and omega. By temperature, what we mean here is the dry bulb temperature. Now, H star may be written as H A plus omega H V and H A itself may be written as C P times T plus omega H G of T. Now, H G of T obviously is dependent on T. The first term here is linearly dependent on T. H G itself is dependent on T in some manner. We have not done a curve fit or anything like that, but we know that it is dependent on T and H R is dependent on omega as you can see in a linear fashion. So, in principle, this expression depends on T and omega. Now, the specific volume of dry air is nothing but the volume of the mixture divided by mass of dry air. Now, volume of mixture itself may be written like this. So, we may write for the water vapor, partial pressure water vapor times volume of mixture equal to M V or V times T. So, V itself may be written in terms of the mass of vapor and partial pressure of water vapor like this. And if you do that, it then simplifies to omega times R V times T over P V. And if I replace P V in favor of omega using this relationship, I get finally something like this. And as you can see V A depends on T and omega clearly. Now, this relationship, the next relationship comes from this one here. So, if you look at this one again, I can gather C P times T 2 plus this as H star of 2 and C P times T 1 plus this as H star of T 1 and then rewrite this. So, if I do that, I end up with this. And remember, we have already said that T 2 is T wet bulb, T 1 is temperature or T dry bulb. So, if you do that, we end up with this relationship. And again, prime of AC, this seems to depend on T and omega. And the dew point is nothing but T sat of P V and P V depends on omega. So, this depends on omega only. But the important point that emerges from looking at the list of expressions here is that all the quantities depend only on two independent properties T and omega, which is the basis for developing the psychrometric chart. So, basically the psychrometric chart has two axes, vertical axis is omega as you can see here. And the horizontal axis is T, which is the dry bulb temperature of course. So, the basis for that comes from this observation. Some of them interestingly depend only on omega. Let me denote this using a slightly different color. So, this depends only on omega. And this also depends only on omega. All the others seem to depend on T and omega. So, basically the psychrometric chart, as you can see has omega as the y axis as you can see from here and dry bulb temperature on the x axis. And lines of constant values of each one of these variable is depicted in the chart. So, basically the chart plots lines of constant PV, lines of constant phi, lines of constant H star, lines of constant VA, wet bulb temperature, lines of constant wet bulb temperature and dew point temperature. Now, if you look at this and this because these depend only on omega, that means if the axis is vertical, this can be shown very nicely in the axis itself. So, let me just erase this. So, what is shown here? Let me erase this also just. So, what is shown here? This is PV, but instead of showing it in kilopascal, this is shown in millimeters of mercury and the reference value 760 millimeters of mercury for barometric pressure is given there. So, this can be easily converted into kilo units of kilopascal. So, PV depends only on omega, since omega is the vertical axis, PV axis is also vertical. Dew point temperature as you can see here. So, this is TDP, since that also depends only on omega, that can also be shown in a vertical axis parallel to omega axis. So, this is the dew point. So, this is dew point 0, dew point 10 degree Celsius, 20 degree Celsius and so on. So, that is shown parallel to the omega axis. So, these two are straight forward. And so, we had denoted them using this green color. So, let me just show them again. So, this is in green. This is in green, they are dependent only on omega. So, we can show them in a vertical axis. Now, notice that the omega values here are given in grams of moisture per kg of dry air, not kg of vapor per kg of dry air, but rather grams of vapor per kg of dry air. So, the numbers are nicer that way in the chart. Now, let us look at these relations and then sort of try to get an idea of what these lines will look like on a T omega chart. Remember, omega is vertical and the dry boot temperature T is horizontal. Let us start with H star. Now, H star as you can see, the quantity whose dependence on temperature we do not really know is this H g. So, let us see. H g. Now, if you actually go to the property tables, let us say steam tables and for the range of temperatures that we are going to likely to encounter in psychrometric applications, we already mentioned that. So, for the range of temperature that we are likely to encounter in psychrometric application, H g varies very little, only by about 100 kilojoules per kilogram or so, out of a total of 1500 kilojoule per kilogram. I am sorry, out of 2500 kilojoule per kg, it varies only by about 100 kilojoule per kg, which is less than a 5 percent variation. So, what happens is, across the entire range of temperatures that we are seeing, H g varies by less than 5 percent, which means that essentially H g is a constant, more or less. Which means that all H star equal to constant lines will depend linearly on T and omega, which means the H star equal to constant line is a line with a negative slope, meaning in the second quadrant, straight line in the second quadrant. So, if we say that this is more or less constant, of course, when we draw this line, you know we are drawing accurately. But this argument that we are making is just to get an idea of what these lines will look like. There is constant, H star equal to constant, what does it look like on a psychrometric chart. So, you can see that this line, these are the constant lines. So, this is the H star axis. So, these are H star equal to constant lines, they go all the way. So, you can see that it is a straight line, more or less, with a negative slope, which is why we have shown the chart in the second quadrant. So, H star equal to constant line looks like this. So, these lines are, it has a negative slope. So, when the lines are drawn here, of course, you know, we would not have neglected H g, H g would still have been taken into account. But for qualitative idea of what these lines look like, we can see that or we can argue that H g is more or less a constant. So, H star equal to constant is going to appear more or less like a straight line with a negative slope in the psychrometric chart. Let us now move on to the next quantity, VA. So, let us indicate this in red. Now, VA on the face of it appears to depend both on t and omega. But if you look at the second term here, 0.622 plus omega, if you look at omega, remember omega in this expression has units of kg vapor per kg dry air. So, in those units, omega is typically a number of the order of about in 0.001 to maybe 0.01, 0.02 and very big in compared to 0.622, which means that essentially this term will not change much across the entire range of values that we are likely to encounter in in psychrotipical psychrometric applications. So, which means VA probably depends very weakly on omega. If VA does not depend on omega at all, then it will be a straight line just like PV and TDP vertical line. If the dependence is weak, then it is going to be nearly a vertical and not quite vertical, but nearly a vertical line and that is what we are likely to see. And the chart also shows that to be the case. Notice that so these are V equal to constant lines and you can see that they are very close to being vertical because the dependence on omega is very weak. So, the next expression that we are going to look at is a expression for wet bulb. Now, if you look at this expression again this quantity omega minus omega wet bulb is likely to be a very very small number. It is going to be a very small number. So, the second term probably is negligibly small, which means that lines of constant wet bulb temperature will coincide with the lines of constant h star. So, as you can see here lines of constant T wet bulb coincide with h star of T. So, this curved axis that you are seeing here is the axis corresponding to wet bulb temperature and you can see lines of constant wet bulb temperature here. So, for example, 20 degree Celsius, 15 degree Celsius, 10 degree Celsius. So, you can see that they are almost parallel to lines of constant h star, h star of T wet bulb and h star of T are almost parallel to each other. So, they almost coincide with each other. So, which is why we are seeing the wet bulb temperature constant axis being very very close to or the line of wet bulb temperature constant coincides almost with constant lines of h star. So, in summary, what we can say from here is that if this is indeed a small number, then h star of T wet bulb coincides with h star of T, which means lines of constant T wet bulb which would correspond to h star of T wet bulb being constant. So, lines of constant T wet bulb coincide with or almost parallel to lines of constant h star and that is what we are seeing in the chart also. So, basically these are the quantities, contours of which are depicted in the psychrometric chart. So, let us summarize what we have said so far. So, we have argued that lines of constant h star or straight lines almost straight lines with a negative slope in the second quadrant in T omega coordinate space. And we also argued that 0.622 plus omega depends very weakly on omega because omega itself is very small in psychrometric application. So, PV depends almost linearly on omega alone. The specific volume of dry air we argued depends weakly on omega and depends linearly on T. In other words, VA equal to constant lines are nearly vertical because it depends very weakly on omega and linearly on T. So, that is what we argued here. The quantity omega minus omega wet bulb is usually very small, which means that lines of constant wet bulb temperature almost coincide with lines of constant mixture specific enthalpy in the psychrometric chart. So, how does the psychrometric chart make things easy, make the calculations easy for us? Remember, we said that there was nothing wrong with the analysis that we have done so far, everything is fine. But the psychrometric chart makes the calculation process much simpler and much more quick in the actual, in the case of real life calculations. And let us see how that is made possible.