 So, welcome back after having seen the property changes that happens in the subcooled or compressed liquid region when you go on compressing the subcooled liquid to higher and higher pressures we had seen that what is this percentage change that happens in this properties. So, on this slide I can again show you the same table where we saw that with the compression of the liquid subcooled liquid from its PCAT value which is 0.15 MPa to going to a very high pressure in fact above the critical pressure of water we saw that the value of V changes by only 1.21 percent compared to its VF value or VF that is a value at the saturation pressure of water alright corresponding to its saturation value if you see these changes are with respect to the VF value which lies on the two phase region or LV line. So, if I compare this property changes at higher and higher pressure with VF, UF and SF these are just close to 1 percent just above 1 percent I should say. As far as H is concerned the variation is little more than other properties so what do I conclude from here? It is very important conclusion that can be drawn from these figures from these numbers. So, we can see that if I just concentrate on the first column or the second column the property column which is specific volume we can say that the specific volume at 25 MPa can be approximately be equal to specific volume at 5 MPa or for that matter 10, 20 MPa and also all of these are equal to VF at 0.15 that means specific volume at 0.15 MPa or on the LV line on the two phase curve alright. So, these changes whatever are happening because they are happening in the fifth decimal place or the changes are just above 1 percent we possibly can neglect those changes even after compressing from 0.15 to 25 MPa these changes are approximately negligible if I assume this we can say approximately V25 is equal to V5 is equal to VF 0.15. We can conclude one more thing that specific volume which is a function of pressure and temperature I can say that specific volume in this region subcooled or compressed liquid region is not a function of pressure that is what we have seen here even if I went on compressing I could not see a effect a significant effect because of this compression process. That means p parameter here possibly is missing and I will say in this region in the subcooled or compressed liquid region specific volume does not depend on pressure it depends only on temperature. When I say it depends only on temperature I have not shown this but you can verify it using the steam table alright. Similarly now if I look at other properties if I look at thermal energy you can see this percentage change or hardly again matters so much and again can conclude that Upt is approximately equal to Uft so at higher and higher pressure these properties is the same as what it was on the saturation pressure line on the LV line alright. So I will say that again it is independent of pressure it does not depend so much on the pressure similarly I can see for the S also or for the entropy also because this variation also is negligible I say spt in the subcooled or compressed liquid region does not depend on pressure it is approximately equal to Sft that means p is not there at all what do I say from this I will say that the weak dependence on pressure in this region the properties V, U and S have weak dependence on pressure and why is this why am I making such a statement because if you remember right in the first lecture we had tried to compress liquid alright if you remember the compressibility of liquid we had said that liquid is incompressible so even if I compress liquid we had said that it does not get compressed so much the density changes the volume changes were hardly anything then as compared to that for the gas. Gas is compressible but liquid we had concluded was can be considered as incompressible so this is the reason the weak dependence on pressure is basically we say that liquid is incompressible so in this region of subcooled or compressed liquid because we know that liquid is incompressible even if we go on compressing liquid it makes hardly any changes in the properties of this subcooled liquid or a compressed liquid at higher and higher pressures and that is why we say that Vpt is equal to Vft, Upt is equal to Uft and spt is equal to Sft P is missing from these values because in this region of subcooled liquid this fluid is incompressible alright pressure will not play a significant role in this case. Now I am talking only about three properties well the fourth property was enthalpy and enthalpy I can write HPT enthalpy as you know is H is equal to U plus PV enthalpy is equal to U plus PV so if I write dependence of enthalpy so H depends on P and T generally is equal to Upt plus P and Vpt as we know but in this region we have already concluded that HPT is equal to Upt is Uft so if I go on writing those values Upt is equal to Uft and Vpt is equal to Vft because I know that the dependence in this region on pressure is negligible I can write Vpt as Vft and Upt as Uft so if I see overall enthalpy dependence in the subcooled region I will be getting governed by this equation now Uft plus P into Vft what do you mean by this what do you see from this if I go back to the table again I see that the enthalpy dependence on the pressure if not very very significant it is still there the changes had been of the order of 3 to 4 percent as against 1 to 1.5 percent in other properties it does show that as you go on compressing this liquid further there is some dependence on the pressure and why does it show it shows because of this presence of P here if I come back to the equation you can see that this is the pressure which plays important role because even if Upt is equal to Uft and Vpt is equal to Vft there is a P term that appears there which directly relates Hpt or the enthalpy to the pressure term. So if I go on increasing the pressure the pressure will show up on enthalpy because of this formulation alright in the other case there is a weak dependence but there is the pressure dependence in this case that is why you see in a table that for enthalpy I cannot assume that Hpt is equal to Hft I cannot do this assumption well I can do the same for V, U and S but not for H because of the presence of this P term here in this particular equation for enthalpy. So approximation I can do with volume with respect to volume thermal energy entropy for subcooled liquid may be taken same as P sat itself if I want to find out the properties the values of this properties at higher and higher pressure I did not go to higher and higher pressures I can do the same thing at P sat value only knowing that higher and higher pressure because this subcooled liquid is incompressible the properties will not change so much and therefore properties like V, U and S can be taken directly at P sat itself yeah this is the reason V, U, S at Pt is equal to Vf, Uf, Sf at P sat t alright. However enthalpy has to be calculated at the particular point as it is pressure dependent which is what we have seen that Hpt cannot be approximated as Hf P sat t which is what we could do because of the basic formulation of the enthalpy itself as we have done this formulation I have shown this formulation earlier alright. So properties can be taken V, U, S at P sat but not the enthalpy enthalpy can be directly computed from this formulation why did I do all these things it was first of all to show you that such an approximation holds good for V, U and S but importantly there are lot many steam tables which do not give all the properties the one which we are giving you it gives the properties at higher and higher pressures only no problems you can go into the details and find out all the properties at higher and higher pressures there are some steam tables which do not give the complete data for subcooled or compressed liquid regions they may give you properties at 0.15 for example and the next property may appear at 20 MPa state away or 15 MPa state away it is expected that you find out the value at 0.15 MPa or P sat and you can take the same values for 1 MPa, 2 MPa, 3 MPa with the approximation that that subcooled liquid is incompressible alright. In our table as far as the table that we have given that is not a problem because you got the values at 0.15, 0.16, 0.17, 0.2 etc where all these values can directly be taken from the steam tables but there are some steam tables which do not give the complete data for subcooled or compressed liquid in such cases properties at higher pressures may be approximated using P sat values why I need to take only the values at P sat and assuming that the liquid is incompressible I use the same values for further calculations hope you are clear with this approximation which we may not use but I expect that all the students of thermodynamics should know about this approximation just because of the fact that subcooled liquid is incompressible thank you very much.