stalk හ් බෝඵරායිනල මෙ coraçãoරා noises taking stagnation quantities is h plus the velocity v square by 2 the dynamic and with the aid of a perfect gas or an ideal gas as the system which obeys the ideal gas equation of state that p is equal to rho r t and for an isentropic process as you know p by rho to the power gamma gamma is the ratio of specific is at constant we derived the ratios between the stagnation temperatures t zero by t is one plus gamma this is this recapitulation of the earlier thing mac number square of the mac number where this mac number m e is defined as the flow velocity divided by the acoustic velocity or speed of sound which is nothing but root over gamma r t this is also for an ideal gas as the system of the flowing medium similarly for a perfect gas p zero by p the ratio of the stagnation pressure to the local pressure is given by gamma minus one by two m s square to the power gamma by gamma minus one and rho zero by rho the ratio of density stagnation density to the local density is one plus gamma minus one by two m s square one by gamma minus one so this have been derived with the aid of these two questions so constants for these definitions are like this the flow should be isentropic that means the stagnation temperature stagnation pressure not stagnation in stagnation temperature isentropic is not important but for stagnation pressure and the stagnation density the flow should be isentropically brought to rest okay now let us realize this situation physically for example if we have a large reservoir for example the flow commences from a large reservoir if we have a variable area duct the flow takes place through a variable area duct let this is the direction of flow and if this is a large reservoir a large reservoir and if this duct and the reservoir is insulated both are insulated that means q is equal to zero there is no q heat interactions neither comes out nor goes in and if we consider the flow to be inviscid flow that means the fluid flowing is in inviscid fluid that means free from frictional effect then this flow is an isentropic flow then for this kind of isentropic flow the conditions at the reservoir where the fluid is at rest are typically the stagnation conditions that means temperature corresponds to t zero the enthalpy in the reservoir when the fluid was at rest corresponds to h zero the pressure is p zero and the density is rho zero and the most important thing is that at any section where the pressure is p temperature is t corresponding density is rho and velocity is v the ratios are given by this this is the energy equation which is valid if there is no heat flow out or into the system the enthalpy plus the velocity head and the reservoir v is zero has to be constant so there is no restriction whether the flow will be reversible or not the frictional effect will be there or not so therefore this equation for this equation only constant is the adiabaticness of the flow that means the flow should be adiabatic and with the head of the perfect gas equation we can derive this equation this is because for a perfect gas we know h can be expressed as c p t plus some constant arbitrary constant c p times the temperature so c p times the temperature enthalpy specific enthalpy ok so with the help of this we can derive that so stagnation temperature does not require the condition for isentropicness but stagnation pressure and stagnation density must require the condition for isentropicness so stagnation properties as a whole are defined those properties which would arise if the fluid were brought to this isentropic that means in the flow situations in an isentropic flow the fluid where the fluid is at rest the situations or the parameters referred to stagnation parameters these are the stagnation parameters we can find out the velocity for example v square is 2 h 0 minus h or v is equal to root over 2 h 0 minus in a simple condition that means by virtue of the enthalpy difference for a perfect gas we can write c p t 0 minus t so therefore at any section the velocity v is achieved by virtue of its change in this stagnation temperature stagnation temperatures are simply the index of the energy or the enthalpy in case of an ideal gas ok another very important condition arises in this isentropic flow is the sonic condition that is another important condition what is meant by sonic condition that means at any section in the duct the flow situation be such that the velocity of flow becomes equal to the acoustic speed or velocity of sound at that section corresponding to the particular properties prevailing at that section that means the local properties of p t rho the velocity of sound velocity of sound is given by root over gamma art so the local section or the locality where v attains a is known as the sonic condition they are the mach number of flow becomes equal to 1 so sonic condition is a very important condition and sections where the fluid flow achieves the speed of sound with mach number 1 and the properties at that location is are referred as sonic properties and they are usually are conventionally denoted with an asterisk as the superscript for example p star is the known as sonic pressures so pressure p star that is the sonic condition sonic pressure what is meant by sonic pressure that is the pressure at the section where the sonic velocity is reached similarly t star similarly rho star just like p 0 t 0 rho 0 with 0 as the suffix conventionally represents the stagnation properties asterisk with a superscript represent the sonic properties that means these are the properties at the section where the sonic velocity that the velocity of sound is reached by the velocity of the fluid so to derive an expression of those quantity this is very simple that if we put m a mach number is equal to 1 that means if we put mach number is equal to 1 then the t correspond to t star then what is the value of t 0 by t star we can find out the ratio then then we can find out t 0 by t star becomes equal to if you put mach number is equal to 1 2 plus gamma minus 1 that means gamma plus 1 by 2 so simply it becomes gamma plus 1 by 2 because mach number is 1 it is free from mach number mach number is put 1 so similarly for a perfect gas we can write p 0 by p star is gamma plus 1 by 2 raise gamma by gamma minus 1 similarly rho 0 by rho star is equal to gamma plus 1 by 2 to the power 1 by gamma minus 1 so this defines the sonic properties the ratios of stagnation to sonic properties where the flow velocities has reached the acoustic velocities now what is the flow velocity v v star is equal to a star and is equal to gamma r t star all star or asterisk whatever you call we call it a star so with a star at the superscript means the condition where the mach number 1 has reached so at that condition the velocity of the fluid flow and the sound speed at that condition is also given as a asterisk mark star a star is root over gamma r t star alright so we can find out this t star is t 0 2 by gamma plus 1 so we can also write in terms of this stagnation property t 0 t star is t 0 into 2 by gamma plus 1 so one can express also these are little algebraic manipulation the v star or a star becomes equal to gamma r t 0 2 by gamma plus 1 so when the mach number 1 is reached the sonic properties are defined like this as a ratio of the stagnation properties like this so these are known as sonic properties now after this any question so after this we will see the very important thing will come to an important deductions or important problem that effects of area variation effects of you write this is the topic of the day effects of area variation which is very important and interesting area variation on flow properties on flow properties on flow properties in an isentropic flow in an isentropic flow in an isentropic flow effects of area variation on flow properties in an isentropic flow let us consider a general situation of an isentropic flow like this that means the duct is adiabatic and we consider an inviscid flow that means free from irreversibility that is a reversible adiabatic flow isentropic flow where the area is varying this is a varying area duct so we consider an analysis for one dimensional flow where velocities pressures all are functions of the direction of the flow but is uniform across uniform across a section so now you see that in this case if we write the equation of motion equation of motion equation of motion for an one dimensional inviscid flow simple equation of motion that is your Euler's equation Euler's equation which is the equation of motion for inviscid flow equation of motion for inviscid flow in one dimensional can be written in a differential form d p is equal to minus rho v d v in a differential form we can write rho v d v so this is the inertial term and this is the pressure this is the pressure this can be found out by making a force balance p p plus d p and taking this is the inertia force and this is the pressure force that is balanced by inertia force is equal to pressure force because there is no viscous force so d p is minus rho v d v if you recollect it is the differential form of your Euler's equation for one dimension flow that means the equation of motion for an inviscid fluid minus rho v d v now if we divide both the sides by rho v square we get d p by rho v square is very simple deduction is minus d v by now continuity if we write the continuity equation continuity means the integral form of the continuity not differential form the bulk continuity equation that is the bulk bulk form bulk continuity equation we can write the rho density into area into velocity that means we are considering the one dimensional approach that means at any section the velocity is uniform across the section area is the cross sectional area at that section and rho is density this is equal to constant this is the bulk continuity for a one dimensional flow that any section rho a v v is the uniform velocity at that section rho is the uniform density and area of cross section the product of this three is constant that means the mass flow rate this implies the mass flow rate now in a differential form this can be written d rho by rho plus d a by a is very simple deduction d v by v is 0 from continuity equation in this differential form we can substitute d v by v from here and we get d p by rho v square is equal to minus d v by v is d rho by rho plus d a by a that means d rho by rho plus d a sorry by a ok all right and next we can write this d p by rho this we can write d p if I take this d rho by rho d a by a d a by a in one side we can write d p by rho v square minus d rho or we can write d a by a or we can write is equal to d p by rho v square we take as common d p by rho v square into one minus v square divided by ok so again we can write d a by a or is equal to d p by rho v square now what is this value d p by d rho in an isentropic flow a square very good that is this is equal to a square and v square by a square is mac number square all right again we can write another equation or in terms of v d a by a this is d p by rho v square d p by rho v square is minus d v by v you see from the equation of motion d p by rho v square is minus d v by v so we can write so minus d v by v so this two equations are very very important equation this two this two this two equations these two equations are very important equations in compressible fluid flow so what do these equations indicate you see now these two equations indicate that when mac number is less than one that means subsonic flow subsonic flow when mac number is less than one that means in case of subsonic flow you see that d a and d p have the same sign d a and d p has the same sign and d a and d v has the opposite sign that means for d a greater than zero d a less than zero for example d p less than zero and d v greater than zero or d v less than zero d a greater than zero d p greater than zero and d v less than d a greater than zero means d v less than zero what does it mean that means when m a less than one is subsonic flow when area decreases that means d a less than zero d a d p is positive that means d a and d p are the same sign when area decreases that means area less than zero d p also less than zero that means the pressure also decreases but velocity increases similarly when area increases that d a greater than zero then pressure also increases and accordingly these two are the opposite signs so accordingly the velocity decreases and it is the usual happenings that we already know which takes place in case of an incompressible flow so this qualitative trend with which we are which we are acquainted with in case of incompressible flow remains the same in case of subsonic flow when Mach number less than one but what happens in case when Mach number is greater than one from this we can write when Mach number is greater than one when just the reverse you see when Mach number is greater than one that is supersonic flow supersonic flow you just see from the equation when Mach number is greater than one we see that when d a is less than zero d p is because this is negative Mach number greater than one so when d a is less than zero d p is greater than zero that means when d a is less than zero d p is greater than zero d p is less than zero because this is negative similarly when this is negative when d a is less than zero d v is greater than zero that means d v is greater than zero and d v is less than zero I am sorry d v is less than zero similarly when d a is greater than zero just the opposite from here we can write but again we see when d a is greater than zero so it has to be positive this is negative d p is less than zero so d p is less than zero and d v is greater than zero that means it is just the reverse when area decreases the pressure increases and the velocity decreases well when the area increases then what happens the pressure decreases and the velocity increases just the reverse from that of the subsonic flow let us then see that in therefore we see that the change in area in case of subsonic flow and supersonic flow has the reverse effect has the reverse effect now we know the device nozzle what is the nozzle by definition in the flow of a fluid in the fluid flow nozzle is a device where d v velocity increases d v is greater than zero and d p is less than zero where the pressure is decreased and velocity is increased now in case of a subsonic flow we see the nozzle action in fluid flow takes place that means the velocity increases and pressure decreases when area decreases that means a convergent duct that means a convergent duct a convergent duct this is the in case of mach number less than one that means subsonic flow that means in a subsonic flow a nozzle is a convergent duct where d a less than zero this is a convergent duct convergent duct but if you make a convergent duct for a supersonic fluid will not act as a nozzle in case of a supersonic flow you see the nozzle action that means the pressure decrease of pressure and increase of velocity will take place when d a is greater than zero that means in case of a sorry in case of a supersonic flow that means when m a m a greater than one this is the direction of flow a divergent duct a divergent duct a divergent duct act as a that means in case of supersonic flow act as a nozzle therefore we see that while a convergent duct act as a nozzle in case of subsonic flow a divergent duct where the area increases act as a nozzle in case of a supersonic flow the reverse happen in case of diffusers diffusers are those ducts where the velocity of the fluid decreases and the pressure of the fluid increases that means d p greater than zero and d v less than zero this is the process of diffusion where the pressure of the fluid increases while the velocity decreases and the duct where it happens so is known as diffuser now in case of a subsonic flow you see that the increase in pressure and decrease in velocity is associated with an increase in area that means a diffuser is a divergent duct in case of a subsonic flow that means this is the direction of flow that means in case of a subsonic flow subsonic flow subsonic flow so a diffuser is a divergent duct divergent duct where d a greater than zero here it is also d a greater than zero so d a but in case of a supersonic flow you see that process of diffusion where d p is greater than zero and d v is less than zero is associated with d a less than zero that means in case of a supersonic flow this is the direction of flow that means when m a greater than one that means in case of a supersonic flow that means in case of a supersonic flow d a less than zero is that means a convergent duct acts as convergent duct acts as a diffuser so therefore we see as our convention for the incompressible flow this holds good equally for a subsonic flow that where a convergent duct is a nozzle and divergent duct is a diffuser but for a supersonic flow a divergent duct act as a nozzle and a convergent duct is acting as a or acts as a diffuser convergent duct act as a diffuser in case of a supersonic flow so therefore we see for a supersonic flow if we have to have a nozzle that nozzle action then we have to have a divergent duct now a situation where the fluid has to for example increase its velocity continuously from a reservoir a stagnation condition continuously up to the supersonic region that means initially the mac is equal to zero if i want a fluid at stagnant from a stagnant condition to reach a supersonic velocity that means continuous increase of fluid velocity takes place with a continuous decrease in pressure then what happens in the subsonic region to make the nozzle action we will have to make a convergent duct but when the fluid velocity will reach sonic after that if we want still expansion that means decrease in pressure and increase in velocity we will have to provide a divergent duct because we know that in case of supersonic flow in case of supersonic flow a divergent duct for here acts as a where it is here acts as a nozzle in case of supersonic flow nozzle is a divergent duct in case of subsonic flow it is a convergent duct so therefore from a very low velocity or from exactly zero mac zero v almost zero if we want to continuously increase the velocity up to a supersonic level that means a greater than one we will have to provide both convergent and divergent duct in the convergent portion here the subsonic nozzle it is subsonic nozzle subsonic nozzle where subsonic nozzle where mac number is less than one and this is known as this is subsonic nozzle this is known as supersonic nozzle where mac number is greater than one and the area in between which is the minimum area where the area remains constant and becomes minimum this is known as the throat of this nozzle throat portions where mac number is equal to one is reached mac number is equal to one is reached upstream of this side the convergent duct where the nozzle action takes place that means d v greater than zero and d p less than zero which is the subsonic nozzle and the downstream of the throat sections the mac from mac number one the mac number increases this is the mac number less than one region that means supersonic flow here also d v greater than zero and d p less than zero so this type of duct where fluid at a velocity very low that means mac number much low corresponding to subsonic region it may be even zero from a stagnation condition or situation increases continuously up to supersonic velocity is a convergent divergent duct and it is known as convergent any question you can ask convergent divergent and it is known as divergent nozzle d p greater than less than zero d v greater than zero I am sorry d v greater than zero very good d v greater than zero here yes all right this is known as convergent divergent nozzle this is sometimes known as d level nozzle this d level is the name of a scientist who first introduced his name is carl g p d level so carl g p d level is the man or scientist d level who first introduced this type of nozzle in relation to a steam turbine it was first used according to the late nineteenth century he introduced this so according to his name this is known as d level nozzle or convergent divergent now this section throat section where the area remains constant the sonic condition is achieved how we can prove this this very simple we can prove please ask any questions if you want to ask please what what happened any question please you ask me now if you see that from this equation it is clear that when mach number equal to one d a by a is zero so now from this equation it is clear that d a becomes zero when mach number equal to once or when d v equals to zero that means there is no change in the flow velocity so mach number essentially becomes one when d a is equal to zero that means the from this we can tell that in case of a convergent divergent nozzle this is the section where mach number one is achieved associated with this d a is equal to zero this is the section but at the same time we see that this d a zero may be achieved without the mach number becoming one that means mach number may be less than one greater than one when d v is equal to zero achieved what is the physical significance of it that means a throat area may be there even without reaching the mach number one but satisfying the condition d v is equal to zero these are these these are very simple things let us consider completely a supersonic a subsonic flow with a convergent divergent duct a convergent let us consider a convergent divergent duct convergent divergent duct in a subsonic flow in a subsonic in a subsonic flow that means m a less than one so this is the flow direction so if we see the graph for m a you will see like this let let this is the m a is equal to one line m a is equal to one so initially the velocity increases in this duct that means the mach number initially increased the mach number this is one the velocity increase then what will happen the velocity will reach its maximum here mach number and then it will go on decreasing that means it is either the graph of mach number or velocity to a scale so this is the qualitative trend of the mach number so mach number will be always below one so this convergent divergent duct will act as a nozzle come diffuser that means the velocity will increase and decrease what will be the pressure just the reverse pressure will first decrease if we draw the pressure graph so pressure will decrease and reach the minimum value and we will increase like that so this is the pressure graph so flow velocity and mach number will follow like that that means it will the convert first convergent part the first upstream convergent part will act as a nozzle so this part is the nozzle where the velocity or the mach number increases and the pressure decreases mach number is not necessarily reaching one entirely it is a subsonic flow then the maximum velocity is attained at the throat that means dv is 0 is associated with d a is 0 well and then the rest the last rest downstream part which is the divergent duct the velocity decreases and the pressure increases this typical section is known as when churimeter this is also the name of the scientist who first devised this type of duct in measuring the flow of fluid in a fluid circuit you know the venturi meter is one of the very accurate flow measuring instruments so venturi meter convergent and divergent duct the similar thing happens in case almost in case of a convergent divergent duct for example a convergent divergent duct convergent divergent duct in a supersonic yes well in a supersonic flow where a me greater than one please yes here here this figure previous case where deliver nozzle yes one that is yes yes yes no it cannot flow in the subsonic region that I will come afterwards so whenever the mach one is reached then if you apply a divergent duct then from mach one a divergent duct will always expand that means the velocity will always increase that means from mach one it cannot reach other way in a subsonic region that means you want to tell why not from mach one it will go to the subsonic velocities that means it will act as the diffuser it cannot go like that it will be because mach one is reached here when the mach one is reached here mach one is reached d a by a is 0 so when d a by a increased from mach one it goes into the I will discuss that afterwards in isentropic flow calculations you will see it cannot come back again to the subsonic region it will automatically go to the supersonic region it will automatically go it cannot come back again to the subsonic region when the critical condition is reached it depends upon the pressure here if you put the this pressure will be such that it will go on expanding that means this pressure will be less than this pressure always in this case because this will be designed in such a way that this pressure will be less than this pressure and it will go on expanding that means the pressure will go on decreasing and the velocity will go on increasing this will be made clear afterwards when I will discuss the isentropic flow situation it is a good question understand but whenever the critical condition is reached then if you apply or if you provide a divergent duct the flow will be always reaching the supersonic supersonic situation there is no other way out it cannot go back to again to the subsonic region this will be clear when I will discuss the isentropic flow situation then what we are discussing well then convergent divergent duct in a supersonic flow when m a is greater than 1 this is also very simple that means if totally the flow is totally the flow is m a is greater than 1 in the then the flow direction is like this then it will be like this if m a 1 is this line m a is equal to 1 the flow is initially what will be there though this will be this will act as a diffuser that means the velocity will be decreasing and this part will act as a nozzle that means this will be accelerating. So, initially there will be a decelerating flow this is the Mach number graph let this is the Mach number and this is the Mach number graph. So, velocity will decrease initially and then a divergent duct velocity will increase entirely in the subsonic region because this is the Mach 1 supersonic region I am sorry supersonic region and similarly the pressure will follow like that when this velocity will increase initially the pressure decrease the pressure will increase because initially it is a diffuser and then the pressure will decrease. So, this is the for example, this is the pressure. So, this is the pressure graph so that it is not necessarily that in a convergent divergent duct the critical condition has to be reached at the throat here d a 0 is associated with d v 0 d v 0 or d p 0. That means the maximum or minimum of the velocities are achieved depending upon whether the flow is subsonic or supersonic similarly the minimum or maximum accordingly just with the river sign pressure is associated with, but when the flow is changes from supersonic to subsonic then d a is equal to 0 is associated with m a is equal to 0. That means the Mach number reaches its maximum there that means the Mach number is equal to 1 not reaches its maximum I am sorry the Mach number is equal to 1 sorry d a is 0 the Mach number reaches 1. That means its well the sonic condition yes the interesting question is that it always when you go give a divergent duct depending upon the design pressure maintained here it will go on that you will be clear when you will be dealing with the isentropic flow that entirely depends at this pressure that is known as the design pressure when the back pressure this critical condition corresponds to certain pressure here you understand. So, to make the flow through it this pressure is very important so this pressure cannot be more than this. So, this pressure is less than that and this pressure has to be set in such a way that there should be undisturbed expansion or undisturbed acceleration in the supersonic region through this duct there is no other way out it cannot go to a lower velocity or in the subsonic region this will be clear I will discuss this in discussing the isentropic flow well any question please. Yes no so definitely not minimum area again I am telling this from mathematics. So, minimum area where d a is 0 the throat corresponding to again I am telling this is a very important concept that at the minimum area throat section which is associated mathematically with d a 0 this is achieved when either mac is 1 or d v or d p is 0 well either mac is 1 or d v or d p is 0 that means at the throat section either mac number will be 1 either mac number will be 1 that means fluid is completely accelerating that means from a very low velocity it is totally accelerating that means in the initially it is acceleration in the subsonic region then supersonic region in between at the throat mac number 1 or it is total diffusion from a supersonic flow first in convergent duct the diffusion takes place that means a deceleration takes place then mac number 1 reaches then deceleration takes place in the divergent duct in the subsonic region with mac number 1 at the throat this is one situation that corresponds to the mathematical condition that d a is 0 associated with m a is equal to 1 otherwise d a will be 0 without achieving or attaining m a is equal to 1 either d v is 0 d v d p is 0 either not both d v and d p will be 0 simultaneously. So, this is the case when mac number is not 1 that means either it is less than 1 or it is greater than that means in case of less than 1 subsonic flow the throat area is associated with the maximum velocity or minimum pressure that is a typical venturi meter first acceleration then deceleration nozzle and diffuser or in case of a supersonic fluid is just the reverse it is first diffuser and then nozzle that means first deceleration and then acceleration with the minimum velocity or maximum pressure at the throat that is the situation mathematically that is the situation that refers mathematically to the situation that d a is equal to 0 achieved when d v or d v is 0. But the most interesting question that I will explain to you while we will be deriving the isentropic flow equations that with a convergent divergent duct and the concept of choking that is the very important thing then it will be clear that mac number 1 if you reach from a subsonic flow through a convergent duct mac number 1 at the throat that means at the minimum area then if you go on increasing the area it cannot go to the subsonic fluid it will always go to the supersonic flow. Any question? Thank you.