 Good morning. I welcome you to this session. Today, we will discuss about the principles of physical similarity as applied to fluid machines. At the outset, I will discuss in a brief, I will discuss in brief the principle of similarity in general applied to any fluid flow problem. As you know, the solutions to engineering problems are determined mostly from experiments. Now, due to certain reasons, for example, economic reasons, due to economic conditions, saving of time and ease of investigations, it is not possible in a number of instances to perform the laboratory experiments under the identical conditions of the operating variables that exists in practice. So, what happens is that we have to perform the experiments in our laboratory under altered set of conditions from the actual problems existing in practice. These say conditions or the operating variables refer in case of fluid flow problems are geometrical dimensions, pressures, flow velocity like that. You know that geometrical dimensions, for example, we may not perform the experiments in laboratory of the full scale system as existing in practice because of the availability in floor space. Sometimes we may not cover the range of pressure or flow velocities as happen in practice because of the restrictions in the laboratory experiments. Similar may be the cases for using the particular fluid using the particular fluid, which is actually used in practice. So, therefore, we see that the laboratory tests are always performed under altered conditions of the operating variables that existing in practice. Now, the two pertinent questions now come or arise out of these situations. What are those questions? Number one is that how can you apply the results or the test results from laboratory experiments to the actual problems at a different set of conditions. Number two is that if the performance of a system is governed by a number of independent operating parameters, then a huge experiments, a huge experiments, a number of experiments are required to find out the influence of each and every independent operating parameters on the performance of the system. Now, is it possible by any way to reduce this number of experiments to a lesser one? For example, we can vary one or two independent operating parameters to predict the influence of all the operating parameters on the performance of the system to save huge time, energy and money. So, a positive clue in answering these two questions lies in the principles of physical similarity. So, it is the principle of physical similarity, which makes it possible and justifiable to apply the test results from laboratory under altered set of conditions to the actual problem in practice at a different set of conditions. And also to perform a lesser number of experiments with the variation of a lesser number of independent operating parameters to predict the influence of a large number of independent operating parameters on the performance of a system. So, how it is done? Let us see that if a process, for example, is governed by a number of variables, let we express a process as a functional relationship of all the variables, let a process is denoted or a process is expressed by m variables, m physical variables describe a process. Therefore, the process can be expressed as a functional and an implicit functional relationship like that. If these variables are expressed by n fundamental units, n is the number of fundamental units, n is the number of fundamental units, number of fundamental units. Then we know that by the use of dimensional analysis it can be proved that the functional relationship of all the dimensional variables of the problem can be expressed by m minus n number of dimensionless terms known as pi terms. This you know m minus n, that means we get m minus n is equal to the number of dimensionless number of dimensionless number of dimensionless terms known as pi terms. So, you see the number of independent variables were reduced from m to m minus n, where the variables are dimensionless variables and known as pi terms. So, these variables are a combination, some combinations, some specific combinations of the dimensional variables in a sense that each and every dimensionless variables which describe the problem now can be made like this which describe the problem is now dimensionless. So, therefore we see that this dimensionless numbers as independent variables are much less as compared to the number of dimensional variables and this pi terms each pi terms all the pi terms rather represents the condition of similarity represents the condition or criteria of similarity. What is meant by similarity? You see what is meant by similarity? As I told earlier that similarity is that clue which gives a positive answer to the question that how can you apply the test results under a altered set of conditions to the actual problem in practice. This can be done if for a particular problem that means when the physics of the problem is fixed that means for a particular problem if we maintain the similarity conditions how can it be maintained that if a particular class of problem started under a conditions where the entire physical similarity is maintained then we tell that the physical similarity between the problems are maintained. What are those similarities? Let us see these are geometrical similarity, geometrical similarity, kinematic similarity, similarity and dynamic similarity. Now, if we identify a particular class of problem then the geometrical similarity between two problems of the same class is attained when the ratio of one length of a system to the corresponding length of the other system bears a fixed ratio. Similarly, the kinematic similarity is attained when the motion or the velocity of a particle or a point of one system corresponding to the velocity of the same point of the other system or the corresponding point to the other system bears a fixed ratio. Similarly, dynamic similarity is the similarity of force when the force in one system the ratio of force in one system to the corresponding force of force to the corresponding point of the other system bears a fixed ratio. That means this means the geometrical similarity is the similarity of shape that means they are proportionate in shape. Kinematic similarity is the similarity of motion that means when the motions of the corresponding points between the two systems are same similarly the forces at the correspondence of the forces at the corresponding points between the two systems is same the similarity is known as dynamic similarity. Now, the question is that how do you know that this similarity prevails between the two systems of the same type of problem when this dimensionless terms remain the same. Now, you see the operating variables may vary, but the dimensionless terms for the two systems remain same then we ensure that this similarity is obtained and then the laboratory tests can be used to predict the actual performance and we can reduce the number of experiments to in predict the influence of a large number of parameters. Now, this can be well explained if we consider a pipe flow problem let us consider a pipe flow problem the flow takes place a pipe flow problem the flow takes place. Now, we know that delta p the pressure difference over a length l let the pressure difference over a length l delta p by l in this type of problem is a function of the flow velocity v let the v is the flow velocity well is the function of the diameter of the pipe function of the density of the liquid viscosity of the liquid. So, therefore, we see that the pressure drop per unit length in case of a pipe flow problem where the flow is governed by the pressure force and the viscous force depends on the flow velocity diameter of the pipe density of the pipe and the viscosity of the pipe. This is a very simple problem as we have already started already started earlier. So, that we can write that the problem can be described by like this that means the problem of pipe flow can be described by five dimensional variables that is pressure drop per unit length the flow velocity the diameter of the pipe the density of the liquid and the viscosity of the liquid. So, now if we apply the dimensional analysis to find out the pi terms as the criteria of similarity. So, first we see that one two three four five that means the number of variables m is equal to five. Now, these variables can be expressed by three fundamental dimensions that means the number of fundamental dimensions is equal to three. So, therefore, number of pi terms number of pi terms is equal to five minus three is equal to two. What are those pi terms if we find out by any method of dimensional analysis as you know there are two methods one is the bucking on pi theorem another is the Rayleigh's indicial method. So, if we take v rho and d as repeating variables as repeating variables pi over one comes like this delta p by rho v square into d by l and pi two comes as rho v d by me as you know this pi one is defined as the friction factor and pi two is defined as Rayleigh's number, but what we observe is that this number is a dimensionless number this number is also a dimensionless number. So, therefore, the pi flow problem you see is now expressed in terms of two dimensionless number like this function of delta p by rho v square into d by l and rho v d by mu. So, instead of five variables now these two variables pi one and pi two describe the problem. So, the one thing is very clear now if we make the experiments of pi flow problem in laboratory or if you consider the two systems of pi flow problem the pressure drop the velocity the diameter the length the density and viscosity of the liquid may vary to maintain the similarity what we have to do the pi terms that means pi one and pi two terms there is typical combinations of these two or the typical combinations of the variables in this fashion have to be maintained same. That means even if the variations are there in the two sets of experiments in respect of dependent independent dimensional variables, but the ranges of the non dimensional term should be made fixed to maintain the similarity. That means if you perform the experiments in pi flow problem in laboratory we will have to choose our density of the fluid viscosity of the fluid velocity of flow diameter of the pi in such a way that the typical combinations denoting the pi terms must be same or within the same range as it happens in actual practice. And another very important thing is that we can immediately show we can immediately show that the relationship can be expressed for example, this this functional relationship can be expressed as d by l that is it is a function of rho v d by mu. That means friction factor is a function of Reynolds number that means we can express by a single curve for a particular problem that is a laminar flow for example, or both laminar and turbulent flow combined with a flow in a smooth pipe. So, relationship between friction factor and Reynolds number now these relationship shows that the variation of friction factor with Reynolds number now the pressure drop in a pi flow problem may vary with different input parameters like velocity of flow diameter of pi density of the liquid viscosity of the liquid, but we may vary any one of them to show the influence of others. That means for example, in laboratory if we vary the velocity v we can change the Reynolds number and we can find out the corresponding friction factor for example, if this be the Reynolds number this is the friction factor. And we can choose the velocity of flow to be to vary to represent the variation of rho d mu, because in a laboratory it is very difficult to vary the diameter we have to go for different pipes. If we want to vary the density to show its influence then we have to take different liquids similar is the case for viscosity, but if we simply vary the velocity of flow which is done very easily by controlling a valve in a particular pipe using a fixed liquid then we can vary the Reynolds number and we can show its influence on f. And through that we can tell also the influence of rho and d for example, if velocity is doubled means that rho may be doubled d may be doubled or mu may be half. That means a change in Reynolds number may be brought by a change in any of the parameters. So, with a change in one parameter we can show the influence of other parameters. Now it is clear how the similarity between the two system of the same class of problem can be made, what are the criteria of similarity these are the pi terms they are found by a dimensional analysis. And then we can predict the influence of all the operating dimensional variables on the performance of the system by performing a lesser number of experiments by varying a lesser number of independent operating parameters. Now the same physical principle now if we apply to a fluid machines let us find out if we apply to a fluid machines. Now therefore, we will have to recognize first the variables. So physical variables describing physical variables in a fluid machines that means describing the problem of fluid machine. So we have to first find out the physical variables in a fluid machines what are those this is d let us first d which is equal to a characteristic a characteristic a characteristic dimension a characteristic dimension of the machine a characteristic dimension of fluid machine of the machine a characteristic dimension of the machine which is usually the rotor diameter which is usually the usually the rotor diameter well the characteristic dimension then comes q volume flow rate through the machine volume flow rate n rotational speed rotational speed then h the head across the machine head across the machine head across the machine then density the fluid property let density of the fluid let density of the fluid these are the rheological properties of the fluid then mu viscosity of the fluid viscosity of the fluid then e the coefficients of elasticity then g comes always acceleration due to gravity acceleration due to gravity and then p power transferred power transferred between fluid and rotor. Now we see that these are the physical variables in general in a fluid machine handling a compressible fluid let us first talk about compressible fluid d a characteristic dimension of the machine which is usually the rotor diameter q is the volume flow rate through the machine n is the rotational speed h is the head across the machine that means this is the head in case of a turbine this is the head given up by the fluid or in case of a pump or compressor this is the head developed by the fluid that means the head across the machine then these are the liquid property or fluid property is the density the viscosity and coefficient of elasticity which comes when the compressibility of the fluid comes into consideration which is not beyond the scope of this class of course we are discussing the fluid machines handling incompressible fluid but general coefficient of elasticity comes when the compressibility is taken care of or the fluid handles the compressible fluid machine handles the compressible fluid g the acceleration due to gravity and p is the power transferred between fluid and rotor that means the difference between p and h is taken care of by the hydraulic efficiency it is the power transferred between fluid and rotor and this is the head across the machine you must understand the difference between these two that means in case of turbine it is the head that is being given that is being given by the fluid to the rotor and this is the power which is being obtained by the rotor. So, difference is there in terms of the hydraulic efficiency similarly in case of pump this is the power which is being received by the rotor and head h is the head developed by the fluid machines now in almost all cases in fluid machines the free surface does not exist. So, the variation or the influence of g is neglected because the influence of g in any problem comes when there is a free surface. So, when there is no free surface the problems without a free surface the variation of g we do not take care and this g is rather coupled here with g h instead of g we take g h as the variable rather than g. So, therefore, we arrive at this number of variables this variables d q n g h that means energy per unit mass across the machine it is not the head g h then rho then mu then e then p then how many variables we do have one two three four five six seven eight. So, therefore, we see that eight variables describe the problems in fluid machines that means now we can write this the functions of this eight variables that means d q n g h rho mu e p is zero now to seek for the pi terms as the criteria of similarity what we have to do we have to apply the one we have to apply the dimensional analysis here the number of variables is eight and number of fundamental dimensions to express these variables are three mass length and time. So, therefore, the number of pi terms m minus n is equal to five that means number of pi terms number of pi terms is equal to five well if we take d n and rho as the repeating variables then we get the pi terms like that pi one as q by n d q pi two g h by n square d square pi three rho n d square by mu by the typical analysis of backing on pi theory we get those terms pi four is equal to p by rho n cube d five and pi five as e by rho n square d square. So, we get this five distinct pi terms q by n d q pi two is g h by n square d square pi three is rho n d square by mu pi four is p by rho n cube d five and pi five is e by rho n square d square now let us see let us try to recognize the physical interpretations or significance of these pi terms what are the physical significances of these pi terms let us first consider the first pi term because all these pi terms represent now the principle of similarity or the similarity criteria that means if we make a test on fluid machines under all that set of conditions we will have to make these pi terms same with the actual cases. So, let us see the physical significances of the pi terms let us consider the first pi term pi one it is q by n d q which can be written in this fashion q by d square divided by n d now this q by d square can be written as this that q is the volumetric flow rate and d square is the area. So, that the q by d square represents a characteristic fluid velocity. So, we can write it as a characteristic a characteristic a characteristic fluid velocity a characteristic fluid velocity let v and n d is the characteristic rotor velocity very good a characteristic rotor velocity rotor velocity very good. So, therefore, the pi term represents let you the ratio is proportional to the ratio of characteristic fluid velocity to rotor velocity. So, what is then pi two terms pi two is equal to g h by n square d square what is this pi two term. So, g h by n square d square now if we see the numerator is the energy total energy of the fluid either gained or given and denominator represent the square of the rotor velocities to get a more clear idea if we multiply or divided by pi two by pi one square what we get it is g h by n square d square. And what is that pi one square is pi one is q by d square by n d that means q by d square by n d sorry it is divided by that means n square d square divided by q by d square whole square all right. That means we can tell that pi two by pi one square this is equal to g h divided by q by d square whole square which is proportional to total fluid energy total fluid energy divided by the kinetic energy. Now here you see that this pi two term alone represents the total fluid energy divided by some energy representative of the rotor velocity. So, therefore, we have to make a manipulations with the pi one terms. So, any combinations of the pi terms also represent another pi term that is a corollary of the pi theorem as a similarity parameter. So, pi two by pi one square is also a similarity parameter which represents the total fluid energy to the kinetic energy of the fluid. Now we come to the third pi term third pi term represents rho n d square by n what is the physical significance very simple it is rho n d we take separately d by n. So, you see that n d is the characteristic rotor velocity. So, this is a shot of Reynolds number Reynolds number based on rotor velocity based on u the rotor velocity we can make the Reynolds number based on fluid velocity if you multiply with pi one. That means rho n d d divided by mu if you just multiply with pi one what is pi one q by d square by n d that means q by n d q. So, in that case it becomes rho q by d square d by mu that means it becomes the Reynolds number Reynolds number based on fluid velocity. That means pi three is the representation of the Reynolds number based on rotor velocity or you make a combination with pi one like this we get the Reynolds number based on fluid velocity. Then we come to pi four what is the physical significance of pi four this combination p by rho n cube d five p is very straight forward thing that is the power transferred between the fluid and the rotor, but what is rho n cube d five as such it does not come out to be a physical does not give any physical concept rho n cube d five, but what we can make if we make this combination pi four divided by pi one pi two. Then what we get p by the rho n cube d five what is pi one q by n d q. So, n d cube by q and what is pi two g h by n square d square that means n square d square by g h. So, therefore, we see this n and d almost cancels. So, we get p by rho q divided by mu that means g h. So, therefore, now we see that if we make a manipulations with pi one and pi two then we get this is equal to what p by rho q g h that means power transferred between fluid and the rotor divided by the total head available by the fluid or gain by the fluid that means in case of turbine it is nothing, but eta h in case of turbines. Obviously because p is the power available by the rotor or transfer to the rotor and this is the head available by the fluid that means head given up by the fluid or one by eta h in case of pumps in case of pumps one by eta h because in case of pumps it reads that p is the input power to the rotor and rho q g h is the head developed by the fluid. So, therefore, pi four by pi one pi two represents the hydraulic efficiency or one by hydraulic efficiency depending upon whether it is a turbine or a pump. Now, we come to the last one pi five what is pi five as we have straight forward obtained is e by rho n square d square. Now, if we make this pi one by root over pi five that means we can write this way e by rho square root with little adjustment and then n d well and what is pi one q by d square sorry it is the other way I am sorry I am sorry I am sorry pi one by root pi that means it will be n d denominator will be root over e by rho and what is pi q by d square and n d that means n d as a whole cancels this become equal to this becomes this is equal to q by d square by root over e by rho. Now, q by d square is proportional to the characteristic fluid velocity and what is root over e by rho can you tell what is root over e by rho e by rho is the velocity of sound in that medium that means velocity of sound in that fluid medium which we can write as a acoustic velocity in the fluid medium and this ratio v by a is known as mach number. So, therefore, we see the implication of pi five is the ratio of fluid velocity to the sound velocity which can be directly obtained if we make a manipulation of pi one by root over pi five. Now, let us come back again to the basic pi terms. So, therefore, now we see the physical implications of each and every pi terms as the similarity parameters that means if this terms or the combinations of the basic variables in this fashions are maintained same in two kinds of investigation the entire physical similarity is obtained and we have found the physical significances of these parameters. In a fluid machine handling incompressible fluid we can get rid of these parameters because there the compressibility is not coming into picture and moreover it has been found out that the influence of viscosity liquid viscosity in a fluid machine is very much negligible as compared to the influences of other parameters. So, therefore, the variation of this third pi term pi three is also discarded. So, therefore, we only take or only we are left only with the pi terms pi one pi two pi four pi pi one pi two pi four only this three pi term. So, therefore, the problem of fluid machine handling incompressible fluids are defined in terms of the dimensionless pi terms as this three pi terms only pi one pi two and this is pi one this is pi two and this I give that nomenclature pi four as it came after this pi three. So, pi one pi two pi four so this three dimensionless pi terms or dimensionless terms of three pi terms define the problem of fluid machines handling incompressible fluid that means handling liquid. So, therefore, we see that if we express the functional relationship in terms of this three parameters the similarity in the fluid machines are maintained. That means a particular fluid machines of a particular geometrical shape will behave in a similar conditions provided this pi one pi two pi four are maintained same in all the machines in all the machines even if the variables q n d p h vary. This is the principle of similarity I will discuss it further in the next class. Thank you.