 In the previous class, we have studied underground hydrogen storage. Underground hydrogen storage is meant for large scale storage. It can provide grid stability, it can ensure like the seasonal storage and the capacities which usually we consider when we look at the underground hydrogen storage is above 5 gigawatt hour. It can also provide discharge or withdrawal of hydrogen for several days. However, if the requirement is a small scale hydrogen storage for stationary or portable applications for vehicular applications, in that case like whether the storage is required for daily storage or in onboard tanks, in that case hydrogen can be stored in the compressed state or it can be stored in liquid state. Now what typical technology will be used for hydrogen storage that is decided based on the end use application, the boundary conditions, the requirements of the end use applications. So today we will start up the hydrogen compression and expansion. We will first look at some of the fundamentals, the thermodynamics of hydrogen compression and expansion process. These concepts you would have earlier studied in your previous classes and we are just going to revise those concepts. To start with let us first look at the Joule-Thomson effect. Now when there is an adiabatic throttling at a constant enthalpy through an insulated porous plug then the if the expansion is known as Joule-Thomson expansion. Now depending upon whether the temperature of the fluid or gas it increases, decreases or stays constant the effect is known as Joule-Thomson effect and the Joule-Thomson coefficient it is given by the change in temperature with the change in pressure at a constant enthalpy. We can also find out the temperature change at a constant enthalpy as the temperature T2 at enthalpy H when the pressure is P2 the final pressure is P2 minus the temperature T1 the initial temperature enthalpy constant and the initial pressure. Now this can also be obtained as integrating over say pressure P1 to P2 the Joule-Thomson coefficient which is a function of temperature and pressure times Dp. Now if we consider ideal gas case then during an iso enthalpy process there is no change in temperature of the gas. So for an ideal gas the del T upon del P at constant enthalpy this Joule-Thomson coefficient remains 0. However when we consider a real gas in contrast to the ideal gas where there is an attractive interaction or a repulsive interaction that acts between the molecules of the gas the change in temperature occurs when there is a change in pressure. Now depending upon how the gas molecules interact if they interact attractively in that case the if there is an expansion that means the pressure if it reduces or the volume increases in that case the potential energy increases. Now to keep the internal energy constant the kinetic energy of the gas decreases and thus the temperature also reduces. So if it is attractive interaction then it could result into cooling of the gas. However if the molecules of the gas it interact repulsively in that case when there is an expansion pressure decreasing volume increasing then the potential energy of the gas it decreases. To keep the internal energy constant the kinetic energy of the gas or the temperature of the gas increases and in that case we can observe heating inside the gas heating in the gas. Now there is a temperature which is known as inversion temperature this inversion temperature it depends upon enthalpy and pressure. So this is a function of enthalpy and pressure. Now depending upon the thermodynamic state which thermodynamic state it is we can have Joule-Thompson coefficient either positive or negative like if we consider one of the thermodynamic state where the Joule-Thompson coefficient is positive that means if there is a pressure decrease on expansion there is a positive change in the Joule-Thompson coefficient that means if the pressure decreases in that case the temperature also decreases. So on expansion there is a cooling effect which is being observed or the gas cools down in case of an iso enthalpy expansion process. There could be another thermodynamic state where the Joule-Thompson coefficient is negative that means if there is a decrease in pressure because of the expansion the temperature increases that means the gas gets heated up. So in the expansion of the gas the temperature of the gas increases. Now this inversion temperature depends on enthalpy and pressure for hydrogen this inversion temperature is 202 Kelvin that means if we want to cool down hydrogen to liquefy it during the liquefaction process using a Joule-Thompson wall or a throttle wall in that case we have to reduce the temperature of hydrogen below 202 Kelvin else there will be a negative Joule-Thompson effect observed. At room temperature the coefficient Joule-Thompson coefficient this is negative for hydrogen it has a value of minus 0.3 Kelvin per megapascal for hydrogen this is for normal hydrogen. Now the another thing let us revise is the equation of state. There are different independent state variables which are related and can be expressed in terms of equation of state. This equation of state is essential or it is the fundamental in providing or describing the information about the nature of the gas how it is going to behave. Another example if we consider an ideal gas we know the ideal gas law where the pressure volume and temperature are related as it is M R i t where M is mass R i is the specific gas constant T is temperature P is pressure and V is volume or we can also write it as molar mass times the universal gas constant and temperature. We can also write the isobaric specific heat for an ideal gas this is a function of temperature this is the isobaric specific heat. The isochoric specific heat we can write it as isobaric specific heat minus the gas constant. We can define for an ideal gas an isoentropic exponent which is the ratio of isobaric specific heat to the isochoric specific heat. At the same time we can find out the enthalpy of the gas for an ideal gas the enthalpy we can find out by integrating from some reference temperature let us say it is T i 2 of final temperature T we can integrate the isobaric specific heat with temperature and then we can have some initial enthalpy and considering that enthalpy initial enthalpy to be 0 this term can becomes 0. For an ideal gas we can also find out the internal energies again the some reference temperature integrated between the limits of that reference temperature and temperature T the isochoric specific heat and some reference value of that specific internal energy. Now this reference specific internal energy is given by the gas constant and that initial temperature. Considering ideal gas we are able to explain many of the phenomena many of the cases and we can still explain those with very good accuracy. But when it comes to considering high pressure or low temperature case in that case there is a deviation from the ideal gas behavior and if we still use the ideal gas equation of state then we may result we may end up in getting Erranias results or deviations. Now it is observed that as we move towards the critical point these deviations increase or the change in the deviations is higher when we approach the critical point. Now in order to account for this deviation of a real gas from an ideal gas we can include a dimensionless quantity which is known as the compressibility factor. Now if we include that compressibility factor in the equation of state then we can rewrite the equation of state for real gas as P v is equal to m z R i t or it can be n z R t. So this compressibility factor it is also not constant it also depends upon the temperature and pressure. Now to know the behavior of real gases there are several models which are there existing in the literature which can predict the behavior of the real gas. Now this compressibility factor the dimensionless quantity which tells us the difference or the deviation between the real and ideal gas behavior that itself changes with temperature and pressure. So it is changing with temperature and pressure for a constant temperature if we plot the compressibility factor in that case we can see that it is showing a linear behavior at a particular temperature. Now as the temperature increases so this is a higher temperature the slope is found to decrease for the variation of compressibility factor with the pressure. Now this is what is representing how the how accurate will be if we consider the ideal gas behavior or under what conditions we need to consider the real gas behavior. For example if we consider a storage tank let us say having a geometric volume of 1 meter cube and the temperature condition is say 20 degree centigrade and we have compressed the gas at 250 bar. Now if we use the ideal gas equation to find out the mass stored and if we use the compressibility factor in that case we find that using the compressibility factor the mass stored comes out to be 18 kg while considering the ideal gas behavior it comes out to be 21 kg. So that means if we consider the ideal gas behavior then we have over estimated the mass of hydrogen stored in that storage vessel by about 14% and that tells us that under conditions of higher pressures we need to consider the real gas equation of state. Now there are different real gas equations of state available in the literature that can be used to find out the compression as well as expansion work which will be required that we will see if we consider the process, a process which is isoentropic that means the entropy of the system remains constant. So for an isoentropic process the entropy of the system remains constant. Now here we need to consider an isoentropic exponent which along with the different state variables the independent state variables we can make different we can have different equations of state with the different independent variables considering the isoentropic exponent and then using that we can describe the different processes involved in the different we can understand better the compression and expansion of the different gases. Now if we consider like the independent variable as pressure and volume in that case the exponent is k so pv to the power k this is constant this is one of the equation which relates pressure and volume. If we consider this for an ideal gas then the exponent remains almost constant we can relate volume and pressure such that v to the power k pressure to the power 1 minus k this is constant we can also relate the volume and temperature. So for ideal gas the relationship between the state variables the pressure volume and temperature and isoentropic exponent is given by these expressions. However if we consider the case of a real gas in that case the exponent which is used for different state variables for different equation each of these equation that exponent also differs. So in case of a real gas we can write the expression as 1 minus k2 is equal to constant. So for the first equation we have used k1 for the second equation we have used k2 and for the third equation k3. So these exponents differ depending upon which state variables these are connecting and these exponent this exponent also changes with temperature and pressure. Now if we want to find out the final temperature using these exponent in that case we can consider for example let us consider the first equation which was pressure volume to the power k1 is equal to constant. So if on compression we want to find the final temperature T2 then that is related to the initial temperature T1 by the ratio of the compressibility factor z2 upon z1 pressure ratio P2 upon P1 raise to power k1 minus 1 over k1. So this exponent k it depends upon the nature of the gas which type of gas we are considering and what is the temperature and pressure. So it behaves differently at different temperatures and pressure. Now this is a variation for the isoentropic coefficient k1 which we have used for the state equation relating pressure and volume. So the k1 exponent if we see how it varies with different pressure different temperature we can see that there is a linear variation. Now in when we will study the work of compression in that case we will see that the isoentropic exponent could have a different value at different states. Now when we consider the variation to be linear the process the calculation becomes simpler and we can take the arithmetic mean in the 2 different states to find out the exponent at a particular state. Also we can see that as the temperature increases so this is the highest temperature for this graph the slope also decreases for the isoentropic exponent. To summarize this part we have seen the basic thermodynamics of the compression and expansion process the various state equations that we will be using for finding out the work of compression and explaining the hydrogen compression in more detail. We have seen the Joule-Thompson coefficient the different state equations what is the isoentropic exponent and how is it related to the different state variables in this class. In the next class we will calculate the work of compression which is required for a compressed hydrogen storage. Thank you.