 we will consider the hydrogen compression and expansion. Now the hydrogen compression or expansion that will depend upon which thermodynamic process has been used and thus the final state of the gas will also be determined by the process used. In actual practice it is very difficult to identify the actual process that occurs in the compression and it requires a lot of efforts. So what is done is actual process is in fact mapped to a reference thermodynamic process which closely approximate the actual process. So we can consider the process to be either an isoentropic compression or we can consider it to be a isothermal compression. Now if the process of compression is considered to be an isothermal compression in that case the work of compression required is minimum. If it is isothermal compression then the compression work required is minimum and that we know from the knowledge of thermodynamics. If the process of compression is modelled to isoentropic process in that case the compression work is maximum. Now single stage or whether it is multi stage compression compressors when they are compressing hydrogen gas and cooling is involved or cooling is being considered during the compression process then they can be modelled as isothermal or the process can be modelled as isothermal process. However there is no cooling involved during the compression process or for moderate pressure ratios the process can be modelled as an isoentropic process and if there is no efforts being put in so as to remove the heat which is generated during the compression process then in that case this is a adiabatic process. So the process can be modelled according to the whether the process involves cooling whether it is it does not involve cooling. So accordingly the process can be modelled to the appropriate thermodynamic process. Now let us first consider the hydrogen compression process to be an isoentropic process. So as mentioned it is very difficult to find out what the actual process would approximate to however it can be modelled to be isoentropic process where the when the compression is done to a elevated pressure the work of compression we can calculate. If we consider the gas to be an ideal gas then the isoentropic work involved is given by the specific heat ratio gamma divided by gamma minus 1 R is the gas constant, T is the initial temperature, P2 is the discharge or the final pressure, P1 is the initial pressure or the inlet pressure and again to the power gamma minus 1 upon gamma minus 1. So this is the isoentropic work considering the gas to be an ideal gas. Here since the process is of compression so the initial pressure is lower than the delivery or the final discharge pressure. Now considering the gas to be diatomic because hydrogen is diatomic gas the specific heat ratio is 1.4. Now this is the process considering an isoentropic process which is reversible but in actual process the process is not a reversible process there are several irreversibilities involved. To account for those irreversibilities we can consider an isoentropic compressor efficiency. So this is considering the ideal process which is reversible but if in order to account for the irreversibilities involved in the process we can include the isoentropic compressor efficiency in the denominator. So the actual work done is the ratio of the two the isoentropic work and the isoentropic compression efficiency. This isoentropic compressor efficiency this lies in the range of 75 to 80 percent. Besides that we have to also include the efficiency of the motor which lies in the range of 90 percent. Ideal gas we know this is a theoretical concept where we know that we consider that the gases these are made up of hard molecules which have negligible volume solid spheres and they have no interaction between the gas molecules. However in actual case the real gases they have an interaction attractive or a repulsive interaction that exists. So the behavior of the gas as we have seen in the earlier class for a real gas it is different from the ideal gas. Now in order to calculate the work which is required for compression using an isoentropic process for a real gas we can find the work that is given by the gas constant R T1 the initial temperature, the compressibility factor Z1 and the initial pressure P1. So R T1 Z1 P1 raise to the power 1 minus K1 upon K. K1 is the isoentropic coefficient which relates the pressure volume equation of state and then there is an integration from initial pressure to the final pressure P2 P raise to the power K1 minus 1 over K1 integrated over the pressure Dp. Now we know that the exponent K1 also changes when pressure changes. So we know that it is a function of both temperature and pressure at a particular temperature let us say T when the pressure changes from P1 to P2 the exponent also changes. Now since the variation is linear we can take average of the exponent in these two states. So the K1 average we can use. So if at pressure P1 temperature T1 the value is K1 at temperature T2 pressure P2 the value and the two values and we can take its average. Now this average exponent K1 average divided by 1 minus K1 average R T1 Z1 P2 by P1 to the power K1 average minus 1 upon K1 average minus 1 this gives the isoentropic work for compression for a real gas. The another possibility is if the actual process is modeled to be isothermal process that means we are removing the heat during the compression. So whatever is the heat generated during the compression that is removed such that the temperature of the process remains constant. Now here again we can consider either the gas to be ideal gas if we consider the gas to be ideal gas the work of isothermal compression is RT ln of V2 upon V1 we can have different models for real gas and considering the van der Waals gas model we can also get the isothermal work for the van der Waals gas as RT ln of V2 minus V upon V1 minus V plus alpha upon V2 minus alpha upon V1. So this is the required work of compression considering the process to be isothermal. However if we include the compressibility factor as a deviation between the real and ideal gas behavior we can write the isothermal compression work as RT integration over the initial pressure to final pressure compressibility factor Z upon P integrated over Dp. So RT integration P1 to P2 upon Pdp on integration we can get a value of RT Z average ln of P2 upon P1 so the pressure ratio. Now again this Z average is the average compressibility factor we have already seen the variation of compressibility factor for different temperature and pressure. Now in this case since it is an isothermal process the temperature remains constant. So the average compressibility factor is found as an average of the compressibility factor in the two different states where the pressure was different. In one state where the pressure was P1 in the another state when the pressure was P2 and then we can find out the isothermal work for the compressible gas. Now this figure shows that depending upon which gas model we have used the work of compression may differ. So whether we have used an ideal gas model whether we have considered it to be a real gas whether it is an isothermal process whether it is an isoentropic process whether we have considered a Wenderwals gas or we have considered it to be a compressible gas considering the compressibility factor the different curves are obtained. So for an ideal gas considering it to be an isoentropic process we are getting the highest amount of work of compression. Similarly for considering it to be a compressible gas with the compressibility factor the model which uses the compressibility factor and considering it to be isothermal the third one is considering it to be a Wenderwals gas and under isothermal conditions and finally considering it to be an ideal gas under isothermal conditions. This shows that depending upon the gas model used the work of compression varies with the pressure. Now important thing to be noted here is that other than the gas model the effect of the gas model used the curve is not linear that is to be noted it shows a parabolic nature. So the curve shows a parabolic nature and that is very important consideration because the work of compression its variation with the pressure the delivery pressure or the discharge pressure is showing a parabolic nature. That means it depends upon the suction pressure. How? Let us say if we have compressed it to 350 bar. So if we compare the work of compression which is required to compress from ambient pressure to 350 bar is much higher. If we see the variation this is much higher as compared to the amount of work which is required to compress it to from 350 to 700 bar. So this work is much higher compared to the amount of energy which will be required to compress it from 350 to 700 bar. And thus this makes it very important that the method of hydrogen production which is used whether it is reformer based hydrogen production or whether it is the hydrogen which is being produced is from electrolyzer whatever production route is giving hydrogen at a higher pressure we can save in terms of the work of compression or the process can get more and more energy efficient if the suction pressure for the compressor is available at a higher pressure. And that is the importance when the delivery from the production plant is at a higher pressure. Now as mentioned the theoretically the work required to compress hydrogen in an isothermal process is minimum and to compress it through an isoentropic process it is maximum. But the actual work of compression it lies between the two theoretical limits of isothermal and isoentropic compression. So the actual process of compression it is neither isothermal. The reason being as we compress the temperature of the gas increases on compression so we cannot keep it isothermal neither it is adiabatic because there will be heat exchange with the surroundings. Neither it is isoentropic because in isentropic process the process is considered to be reversible but the actual process is a non-reversible process. There will be losses associated because of the friction there will be viscous losses. So all these make the actual process neither isothermal nor adiabatic nor an isoentropic process. So some of the reports some of the researchers they have even modeled the process to be a polytropic process. Here in this curve the work of compression has been plotted so the same curve has been plotted as a percentage lower heating value that the how much amount of energy of hydrogen would be required energy content of hydrogen would be required in compressing the hydrogen to a higher pressure. Now as mentioned some researchers consider the compression process to be a polytropic process. So if we consider the process to be a polytropic process then it is pV to the power n is equal to constant and that n is the polytropic index. So the polytropic work is n upon n minus 1 n is the polytropic index RT1 the initial temperature gas constant times the P2 by P1 pressure ratio to the power n minus 1 upon n minus 1. So this is the work of compression which will be required if the process is considered to be a polytropic process. This is considering an ideal process however the actual work of compression will be higher and that can be accounted for by considering a polytropic efficiency, eta P and this polytropic efficiency is related to both the polytropic index and the specific heat ratio as n is the polytropic index n upon n minus 1 is equal to gamma upon gamma minus 1 eta P where gamma is the specific heat ratio eta P is the polytropic efficiency. Now considering the process of compression the temperature the final temperature of the gas on compressing it from pressure P1 to P2 can be obtained as T2 will be the final temperature. So T2 upon the initial temperature T1 is the pressure ratio P2 upon P1 to the power n minus 1 upon n. Usually in the compression process the temperature increases on compression we know that now since the temperature increases if we cool down the gas on compression the increase in the temperature and we reduce the temperature of the gas the required work of compression will also reduce. So if we cool the gas on compression the work of compression reduces. Now ideally what is required is as we know that the isothermal process if we make the process to be completely isothermal an ideal isothermal process is very difficult to attain. The reason being by means of isothermal process an ideal isothermal process is we remove the heat from the gas or during the compression process at the same rate as it is being produced in the process or we consider that the temperature is uniform throughout the compression. So ideally it is very difficult to achieve it is a very complicated cooling system will be required. So actually we can approximate it by considering a multi-stage compression or including an intercooling stage so that the temperature at each stage of the gas attains its initial temperature. So if this intercooling which is introduced between the different stages if it is perfect in that case the temperature of the gas will become same it will be identical after each stage of compression. Considering a two-stage compression considering an ideal gas undergoing an isoentropic compression including two stage of compression if the initial pressure or the suction pressure is P1 the delivery pressure is let us say P3 or the discharge pressure is P3 and there is an intermediate or an optimal pressure. This is an intermediate pressure or an optimal pressure. Considering that the isoentropic work of compression is given by gamma RT1 upon gamma minus 1 times the pressure ratio P2 upon P1 to the power gamma minus 1 upon gamma minus 1 this is corresponding to the first stage plus gamma RT1 the temperature remains constant after the first stage compression upon gamma minus 1 times the pressure ratio P3 upon P2 to the power gamma minus 1 upon gamma minus 1 where the pressure P2 which is the intermediate optimal pressure is given by under root of P3 and P1. So, ideally it is considered that the pressure ratio should be same between the different stages. Now, if we consider a multi-stage compression with intercooling in between so what we are doing is after each stage we are cooling down the gas to its initial temperature by introducing an intercooler and using that it is not becoming ideal isothermal process but we are trying to keep the temperature same after every stage and using that we can reduce the work of compression. Now, if we consider the electrical work which is required for compression as against the variation in the pressure for 2 different suction pressure this is being plotted in this curve. So, for 2 different pressures 35 bar and 2 bar considering different stages of compression for 2 bar there have been 5 stages which has been considered considering 1 stage considering 2 stage 3 stage and 5 stage compression for 35 bar 3 stages has been considered 1 stage 2 stage and 3 stage. Now, from this curve we can see that introducing an additional stage with intercooling in between it reduces the amount of work of compression as we go from stage 1 to stage 5 the work of compression required reduces. At the same time this curve also emphasizes that the importance of the suction pressure. So, if the initial pressure for compression is higher the required amount of work for hydrogen compression will be lower. Usually like the there is a tradeoff between the number of stages that we can add to the amount of energy being saved. Like if we add more number of stages then there will be a requirement of infrastructure and additional cost associated with it and that needs to be considered against the energy which is being saved or energy work of compression which is being saved by adding an additional stage of compression. To summarize this part we have seen that it is difficult to approximate the actual process it requires a lot of efforts as such the actual process could be approximated by a reference thermodynamic process it could be an isothermal process or an isoentropic process depending upon whether the compression is done with cooling or whether the heat is not removed during the compression process. We know from the basic thermodynamics that the work of compression required is minimum if the process is an isothermal process however it is maximum when the process is an isoentropic process. So, we have in this class also seen that work of compression considering whether the gas is ideal gas or whether it is a real gas which type of thermodynamic process we have modeled and also we have looked at the compression work which is required when it is multi-stage compression with intercooling. Multi-stage compression with intercooling has an advantage that it can save the work of compression and it can approximate it to be a isothermal nearly isothermal process such that the temperature of the process is same after every stage of compression. Thank you.