 Good morning. I welcome you to this session of fluid machines. We have completed the discussions on fluid machines and now we will switch over to a new topic introduction to compressible flow. So, at the outset I must start with the definition of a compressible flow. What is meant by compressible flow? So, as you know the compressibility is a property of a fluid and it is characterized by a parameter known as bulk modulus of elasticity and physically the compressibility the property of the fluid is a measure of its change in volume or density with respect to the pressure. Now, if we look to the definition of elasticity the characteristic parameter for the compressibility of a fluid we will see the way it is defined the bulk modulus of elasticity is defined like this it is d p into v by d v where the v with a cut I used to represent the volume to distinguish it from the velocity or is equal to rho d p by d rho. So, therefore, you see the bulk modulus of elasticity is defined this way now a large value of bulk modulus of elasticity represents a large change in pressure required to cause a definite change in volume or density and for fluids whose bulk modulus of elasticity is very large are usually termed as incompressible fluid because a change in volume or density is very low as compared to the change in pressure. Similarly, for the fluids whose bulk modulus of elasticity is relatively very low that means which suffer a considerable change in volume or density for a given change in pressure are termed as compressible fluid. For an example I can tell you that for water at atmospheric pressure for water at atmospheric pressure for water at atmospheric pressure the value of e is equal to 2 into 10 to the power 6 kilo Newton per meter square as compared to that for air at atmospheric pressure for air at atmospheric for air at atmospheric pressure e is equal to 101 kilo Newton per meter square. So, you can very well see that water is almost incompressible practically because it is such value of e indicates a very large change in pressure is required to cause a little change in volume or density as compared to that that of air. Now, question comes this is the characteristic property of a fluid, but what is a compressible flow is it true that compressible flow means the flow of compressible fluid whose elasticity or coefficient of bulk modulus of elasticity is very low and flow of all incompressible fluids are incompressible flow it is not exactly so compressible fluids are defined in this way that the if the change in density brought about by the change in pressure due to the flow is very less those flows we treat as compressible flow. Now, the concept comes like that even if the fluid itself is compressible for example air whose bulk modulus of elasticity is very low if it flows in such conditions that the pressure differences the maximum value of the pressure difference due to the flow is such that it cannot change the density or volume in the flow very much then the flow can be treated as incompressible. So, therefore, a flow is whether incompressible or compressible depends upon whether the change in volume or density encountered in the flow is small or large. So, therefore, it is very much tagged with the flow condition because the change in pressure is not an arbitrary one. So, if the change in pressure is very low in the flow so that the change in volume and density is low those flows can be considered as incompressible. So, to have a criteria for an incompressible or compressible flow for the fluids we should confine ourselves with these deductions we see that a rough order of magnitude we can find out in this way that in any flow of fluid the pressure difference delta p the order of pressure difference can be written as like this it is in the order of the dynamic head where v is the velocity of flow we consider any flow through a duct any flow we can consider like that the delta p the maximum pressure difference or the order of the pressure difference in the flow which will be encountered will be in the order of the dynamic head half rho v square is true. Now, it is very simple manipulation now if we express this delta p in terms of the coefficient of modulus of elasticity from this expression you see that which we can rho we can write like this this is e delta rho by rho instead of delta p is in the order of half rho v square or we can write the order of del rho by rho is in the order of v square by e by rho well now after this therefore, I write it again that order of so we see that the order of del rho by rho is equal to the order of half rho v square now in any flow this in any flow situation through a duct it may be through a duct or it may be a flow over a body now this value e by rho represents the square of the velocity of sound in that flow at that condition where a is the velocity of sound velocity of sound velocity of sound or acoustic velocity velocity of sound or another name is acoustic velocity acoustic velocity in the fluid at that particular condition. So, this is the definition which probably you know is already derived in classical physics preliminary physics e by rho a square so therefore, if I use this definition we see that the order of the change in density to the density the ratio of change in density to the instantaneous density or the initial density whatever you call is half v square by a square now this ratio of v square by a square this is the ratio of the square of the velocity of flow to the square of the velocity of sound in that fluid at that condition. So, there is a non dimensional number known as mac number it is after the scientist mac who first discovered it mac number or introduced it is defined by v by a so therefore, mac number is a dimensionless number which represents the ratio of the velocity of fluid at any condition to the velocity of sound in the fluid medium at that condition this ratio of v by a is known as mac number. So, therefore, we can write in terms of a dimensionless number half m a square so you see the change in density as a fraction of the density itself rho is in the order of half m a square. So, if our criteria for incompressible flow is delta rho by rho is very very less than one for incompressible flow for flows to be for flows to be incompressible for flows to be incompressible for flows to be incompressible delta rho by rho should be very very less than one. So, therefore, the criteria is that half m a square should be very very less than one that means the mac number of flow should be such that half m a square should be very very less than one. So, that to make the delta rho by rho that is a change in density with respect to the density itself is very less now to have a definite quantitative criteria we set this delta rho by rho as or like this less than is equal to 0.05 which means that we can neglect a density variation of 5 percent of the initial one. So, a change in density of 5 percent a change in density of 5 percent or less than 5 percent can be ignored and the flow can be considered to be incompressible it is. So, then a quantitative criteria can be defined that half m a square should be very less than 0.05 or should be simply here simply less than 0.0 from which we can derive that m a should be less than is equal to 0.33. So, this is a very important conclusion you have to remember throughout your life whenever you deal with flows of fluids that when the mac number of flow is less than 0.33 the variation in density is 5 percent that 5 percent of the initial density or below the 5 percent at 0.33 it becomes 5 percent. So, mac number is equals or less than 0.33 the change in density equals to a less than 5 percent of the initial density and the flow can be considered to be incompressible flows are incompressible. So, therefore, we see whether a flow will be compressible or incompressible will depend upon this dimensionless parameter mac number just an example I am telling that flow of air at normal pressure and temperature you know that the speed of sound at that condition at NTP through air is 3 30 meter per second. So, with this criteria we can say that the velocity of air at this normal condition temperature and pressure if it is less than equal to 100 meter per second this is a thumb rule we tell that the flow of air is incompressible. That means in a situation where there is a flow of air at 50 meter per second we can tell the flow is incompressible. So, in that situation the pressure difference associated with that flow that is a flow of air at 50 meter per seconds at the atmospheric condition cannot bring about a change in volume or density which is more than 5 percent and we can neglect that change in volume and change in density in the flow and we can treat the flow to be incompressible all right now you see another interesting feature is that we have found out that del rho by rho del rho by rho is the order wise is in the order of half v square divided by e by rho. Now the bar bulk modulus of elasticity for incompressible flows are very large very large that means for all liquids which are treated as incompressible fluids because their bulk modulus of elasticity is very large other way the velocity of sound through that medium is very large. So, usually even for a very small velocity we get the value of delta rho by rho is very high. So, therefore a sorry very low sorry very low. So, therefore flows of all incompressible fluids are usually incompressible because even with very high velocity encountered in practice they cannot bring about a delta rho by rho more than 5 percent this is because of their very large values in e it is not practicable theoretically you can consider infinitely high velocity which can make, but it is not practicable. So, under all practical conditions flow of all incompressible fluids or flow of liquids are incompressible while the reverse is not the true that means flow of compressible fluids that is the flow of gases may be incompressible provided its velocity is low and that is not any absolute velocity it is relative to the velocity of sound and the criteria is the dimensionless parameter Mach number that if the velocity is such that it corresponds to a Mach number of flow less than 0.33 then the density change or volume change is lower than the 5 percent. So, therefore the flow can be considered as incompressible flow all right now before going to the next chapter we should recapitulate little bit of thermodynamics because the knowledge of thermodynamics and the property relations derived from thermodynamics will be very much applicable in the directions of compressible flows. So, first one or two lectures we will be recapitulating the basic laws of thermodynamics first and second law of thermodynamics and important property relations. So, therefore we must first start with the first law of thermodynamics. So, what is first law of thermodynamics first law of thermodynamics is basically the law of conservation of energy as you know the first law of thermodynamics is basically the law of conservation of energy. Now, if we keep aside the physical phenomena of conversion of mass to energy and energy to mass we can tell that the conservation of energy is that energy is neither created nor destroyed that we know since our childhood this is the conservation of energy. That means if energy is transformed from one form to other or if energy is transferred from one system to other system in the same form in both the cases energy total energy remains constant it is neither created nor destroyed that means if energy disappears in one form it appears in other form this is simply the conservation of energy as simple as that and first law of thermodynamics is nothing but synonymous to this principle of conservation of energy. But in the applications of fluid flow and classical thermodynamics as applied to mechanical engineers or other engineering disciplines the same principle of conservation of energy we look from a view point where the heat is being converted into work or work is being converted into heat because heat and work this two types of energies are first described by classical thermodynamics at the energy in transit energy in transit that means the energy quantities which transfer from one system to other system are either in the form of heat or in the form of work. So, therefore, we are interested to define or receive the conservation of energy while applied to a system or applied to process where the heat and work energies are appearing as the energies in transit. So, therefore, if you recapitulate this we know the first law of thermodynamics written like that in any cyclic process executed by a system. So, this d q the cyclic integral of heat transfer is equal to the cyclic integral of work transfer here this cut I mean maintain to distinguish this d from the exact differential because as you know this q is a path function the heat flow and work flow or work transfer is also a path function. So, d cut where this d cut d u I simply will pronunciate is a d q is the infinite small heat transfer d w represents the infinite small work transfer hence forth. So, cyclic integral of d q is d w that means in any cyclic process that means if a system executes in a thermodynamic cycles executes a processes in a thermodynamic cycle that means in any thermodynamic property diagram there will be a closed loop the total heat transfer during the cycle that means heat may be coming out heat may be given in some processes there is no restriction in the direction work may come out in some process work may go in. So, as a whole if we make the accountability of the energy we will see the sum of all the heat transfer process in a cycle must equal to the sum of all the work transfer process in a cycle this is a mere recapitulation of your basic thing. So, if we write it in a different that way d q minus d w is 0 that means we can write cyclic integral d of q minus w is 0 this gives a very interesting thing that though the q and w are the path functions, but their difference becomes a point function because the cyclic integral of their difference is 0. That means if we represent this d q minus d w as some d x d cut x is 0 then we can tell that cut is not required for that because perfect differential of any quantity integrated over a cycle must be 0 that is the basic definition from mathematics. So, you see the difference between q and w over the cycle is 0 which means that q minus w can be expressed by a point function x. So, therefore, we can write that d cut q minus d cut w can be expressed as a perfect differential of a point function x what is that x this comes straight from the mathematical concept that this minus this over the cyclic integral is 0 that means d q minus d w can be expressed as a change of a point function where x is a point function and this point function and the property of a system. You know that any point function is known as property of a system property of a system and this way the birth of internal energy comes. So, this is the definition of internal energy. So, therefore, we can write d q minus d w can be expressed as a change of a property which is a point function known as internal energy. So, the physical implication of this mathematical statement comes like this that if we consider a process from one to two then this equation implying a infinite small process can be written like this if we integrate this d q from one to two minus d w from one to two is equal to d e from one to two as you know this q and w are the path functions and they cannot be integrated like this. So, therefore, we have represented this d with a cut this is not a exact differential. So, therefore, this is written as q one two that means the heat transfer in this process depends upon the path of the process does not depend only the state points. Similarly, the d w one to two to be represented as the work transfer during the process one two is usually written as w one two which depends upon the path of the process. Whereas e being a point function which is the internal energy by definition of the system it can be written as e two minus e one. So, simply the first law can be written as q one two is equal to w one two plus e two minus e one. This is also the conservation of energy applied to a system that if we consider the direction in this way that the heat added is positive simultaneously we will have to take the work out as the positive then we can interpret this physically the amount of it added to the system during it is change from a state one to state two by a process one two is equal to the work delivered by the system plus the change in its internal energy. So, this is precisely the first law which is written for a system. Now, this can be again written in a differential form rather I will tell this is written in a d q is d e in an infinite small process. So, differential form many people tell a differential form, but I will tell for an infinite small process because q and w cannot be expressed as in a differential these are the path function. That means either in a differential form will automatically mean in that case that d is the differential of internal energy, but q and w are the infinite small amount of work and heat that takes place for an infinite small process. So, this is the outcome of the first law of thermodynamics as applied to a process executed by a system involving heat and work transfer. Now, we come to a definition of a property enthalpy which is very important enthalpy how do you define the enthalpy what is the definition of enthalpy please what is the enthalpy how it is defined h good h enthalpy is a property which is defined as u plus p into v good. So, the very first line of definition of enthalpy is like this the first line the very first definition of enthalpy comes from its mathematical statement that h is equal to u plus p v where is u u is the intermolecular energy. Now, before that I tell you that this is the total internal energy total internal energy. So, if I write the internal energy of any system it comprises several types of energies that can be stored in the system internal energy of a system is the energy that is stored in the system and it is a point function it depends upon the state of the system. So, it is a point function. So, therefore, internal energy are those energy which can be stored in the system at a given state. So, it comprises first the intermolecular energy which is the kinetic energy and potential energy of the molecules which depends upon the state of the system precisely the temperature. Similarly, the system itself may have velocities that means the macroscopic particles of the system may move within the system. Even if the system is a closed system there may be a substantial motion of the system that is the different particles of the system may be motion. So, therefore, the kinetic energy is an energy which may be contained by the system and another type of energy may be stored or contained by the system that is known as potential energy. What is that energy this is the energy by virtue of the state of the system or the position of the system in a conservative force field. So, there may be number of conservative force fields magnetic force field electrical force field in which the system is exposed or system is placed. If all conservative force fields are relieved the gravitational force field is there. So, at least there will be gravitational potential energy or simply potential energy. So, this kinetic energy of the particles of the system the potential energy and the intermolecular energy are the total contributions are comprising the total internal energy. So, if I write the internal energy general symbol u is the intermolecular energy. So, the kinetic energy of the particles plus the let us consider only gravitational force field that the potential energy. Let us write the mass of the m g z the total potential energy total kinetic energy m and the total internal energy. So, this is the internal energy total internal energy. Now, in a closed system in equilibrium the kinetic energies are not appearing because the system is at rest the particles is at rest and if we neglect the potential energy not because of its absolute value because you know the absolute value of potential energy to ascribe the absolute value of potential energy is very difficult we also we always measure it in terms of its change. So, if you neglect the change in potential energy of the system between different states we can neglect this m g z the potential energy part. So, we can tell the internal energy for a closed system or a stationary system simply comprises the intermolecular energy. So, u is the intermolecular energy. So, therefore, this typical combinations of u p and v where p is the pressure and v is the volume defines the term enthalpy. Now, you see u is a point function p is a point function v is a point function. So, therefore, enthalpy is a property and it is a point function. Another interesting thing is that the dimension of enthalpy is the dimension of energy because u is the internal energy its dimension is energy the product of p and v this dimension is energy. So, enthalpy is a property and its dimension is energy. So, it is something similar to energy. So, very first line of definition of it does not give us any physical concept, but immediately the query comes why such a combination is defined as a property. So, you know you start with many properties first we start with measurable properties first we start with observable properties that we can see the mass we can feel the temperature then comes with the measurable properties that volume we cannot see, but we can measure pressure we cannot see, but we can measure. So, therefore, we see that afterwards in thermodynamics we make several combinations out of this preliminary properties or primary properties to define other properties, but why such definition is required and why such a particular combination is made. So, that query is satisfied if we go little further to see the physical significance of such a combination to yield the definition of another property. For example, this enthalpy if we see the physical significance of this enthalpy parameter of this enthalpy this property enthalpy we have to extend our first law to a to an open system or a steady flow system. So, let us do that we consider a steady flow system or a open system. So, before that I think I should tell you the system different types of system. So, how do you define a system and there are two types of systems one is the control mass system another is the control volume system. How do you define a system? System is a definite quantity of mass within a fixed boundary that means the two very interesting characteristic of system a definite quantity of mass and a content or separated from the outside by a definite boundary. So, quantity of mass quantity of mass quantity of mass and definite boundaries definite boundaries are the two characteristic feature of a system. So, system is defined just like that it is some quantity of mass a definite quantity of mass at any instant and bounded by definite boundaries. Now, the system basically is divided into two ways two categories one is control mass system another is control volume system what is a control mass system in a control mass system the mass which is content within the system by its system boundaries. So, therefore, we can see a system is a definite quantity of mass and this is the system boundary. So, if the if the mass identity remains the same within the system boundary that means there is no flow of mass either in or out that means d m is 0 from the system boundary that means the same mass not only the amount, but with the identity remains the same then we call the system as the control mass system which is usually told as closed system. So, the characteristics of closed system the additional characteristics apart from the definite quantity of mass within definite boundary is that the identity of the mass remains same that means in other way there is no mass flow in or out from the system. So, if you take some mass flow out and make an inflow of the same amount to make the mass of the system remains same that we will not satisfy the characteristics of a control mass system or closed system because in that case though the mass remains same, but the identity of the mass changes that means the same identity has to be there. While on the other hand the control volume this the system where we allow the mass flow that means d m not is equal to 0 that may be mass in there may be mass out there may be mass in there may be mass out, but the restriction is that there is the volume of the system remains same that means the boundary is fixed fixed boundary. Now you can ask me that then what is the difference here the difference for the closed system that system boundary may move system boundary is not fixed at any instant there should be a boundary of the system, but there is no restriction that the boundary of the system has to be fixed that means the volume of the system may change while the mass and the identity will remain the same that is why it is control mass system. Whereas in the control volume system the fixed boundary the boundary will not move there is no displacement in the boundary the volume of the system is controlled in a control mass system the boundary remains fixed the boundary may move, but control volume system the boundary remains the volume remains fixed the boundary will be fixed the volume remains same. So, this is known as control volume system it is simply known as control volume or open system this is known as open system or simply control volume the system we do not use. So, we see there are two types of system one is the closed system another is the open system or control volume an example of closed system is your reciprocating pump you have seen that one of the boundary that is the piston which is moving. So, at any instant the boundary is defined, but instant to instant one of the boundary the piston which is moving that means in a closed system there is a there may be a displacement of the boundary. So, boundary may expand or boundary may contract. So, that the volume of the system may change whereas a control volume system is the system where the boundary is rigid that means the volume is same, but through the boundary the fluid can go out or the sorry the mass system mass can go out can come in. So, this is the control volume system. So, now one difference is that if you see in a control volume system if the same amount of mass comes in and the same amount of mass goes out that means the net rate of mass inflow or mass outflow is 0 and the total mass remains same. So, in which way it differs from that of a closed system is that the mass identity is changed. So, therefore, at steady state a control volume system differs from that of a closed system is that though the mass remains same in both the cases the identity of the mass is same in closed system whereas the identity of the mass is not same in the control volume system. So, control volume system we simply tell as control volume or an open system whereas a control mass system we call as closed system or simply system. When we call system this is henceforth you know that system means is a closed system or a control mass system sometimes these are not used. So, system implies the control mass system or the closed system and with all its characteristics. Similarly, a control volume means a control volume system or another name is the open system that means where the mass flow out or mass flow in is applicable or allowed that means mass flux across the system by a boundary is possible. But the restriction is that the boundary should be rigid that means the boundary should be strict that means there is volume boundary should not move or should not displace that means the volume of the system should remain same that is the control volume or open system. So, if you see the application of first law to an open system then you will come to the physical significance of enthalpy I think the time is up. So, next class I will discuss it any query please.