Lecture 3
Principle of Similarity and Dimensional Analysis in Rotodynamic Machines
The principle of similarity is a consequence of nature for any physical phenomenon. The concept of similarity and dimensional analysis related to the problems of fluid flow, in general, has been discussed in Chapter 6. By making use of this principle, it becomes possible to predict the performance of one machine from the results of tests on a geometrically similar machine, and also to predict the performance of the same machine under conditions different from the test conditions. For fluid machine, geometrical similarity must apply to all significant partsof the system viz., the rotor, the entrance and discharge passages and so on. Machines which are geometrically similar form a homologous series. Therefore, the member of such a series, having a common shape are simply enlargements or reductions of each other. If two machines are kinematically similar, the velocity vector diagrams as inlet and outlet of the rotor of one machine must be similar to those of the other. Geometrical similarity of the inlet and outlet velocity diagrams is, therefore, a necessary condition for dynamic similarity.
Let us now apply dimensional analysis to determine the dimensionless parameters, i.e., the terms as the criteria of similarity. For a machine of a given shape, and handling compressible fluid, the relevant variables are given in Table 15.1
Table 15.1 Variable Physical Parameters of Fluid Machine
Variable physical parameters Dimensional formulaD = any physical dimension of the machine as
a measure of the machine’s size, usually the
rotor diameter L
Q = volume flow rate through the machine L3T-1
N = rotational speed (rev/min.) T -1
H = difference in head (energy per unit weight) across
the machine. This may be either gained or given by
the fluid depending upon whether the machine is a
pump or a turbine respectively L
= density of fluid ML -3
= viscosity of fluid ML -1T -1
E = coefficient of elasticity of fluid ML -1T -2
g = acceleration due to gravity LT -2
p = power transferred between fluid and rotor (the
difference between p and H is taken care of by
the hydraulic efficiency ML 2T -3
e = Modulus of elasticity
In almost all fluid machines flow with a free surface does not occur, and the effect of gravitational force is negligible. Therefore, it is more logical to consider the energy per unit mass gHas the variable rather than H alone so that acceleration due to gravity dose not appear as a separate variable. Therefore, the number of separate variables becomes eight: D, Q, N, gH,,E and P. Since the number of fundamental dimensions required to express these variable are three, the number of independent terms (dimensionless terms), becomes five. Using Buckingham’s theorem with D, N and as the repeating variables, the expression for the terms are obtained as,
We shall now discuss the physical significance and usual terminologies of the different terms.
All lengths of the machine are proportional to D, and all areas to . Therefore, the average flow velocity at any section in the machine is proportional to. Again, the peripheral velocity of the rotor is proportional to the product ND. The first term can be expressed as
Thus, represents the condition for kinematic similarity, and is known as capacitycoefficient ordischarge coefficient. The second term is known as the headcoefficient since it expresses the head H in dimensionless form. Considering the fact that ND rotor velocity, the term becomes , and can be interpreted as the ratio of fluid head to kinetic energy of the rotor, Dividing by the square of we get
The terms can be expressed as and thus represents the Reynolds number with rotor velocity as the characteristic velocity. Again, if we make the product of and, it becomes which represents the Reynolds’s number based on fluid velocity. Therefore, if is kept same to obtain kinematic similarity, becomes proportional to the Reynolds number based on fluid velocity.
The term expresses the power P in dimensionless form and is therefore known as power coefficient. Combination of and in the form of gives. The term represents the of total energy given up by the fluid, in case of turbine, and gained by the fluid in case of pump or compressor. Since P is the power transferred to or from the rotor. Therefore becomes the hydraulic efficiency for a turbine and for a pump or a compressor.From the fifth term, we get
Multiplying, on both sides, we get
Therefore, we find that represents the well known Mach number, Ma.
For a fluid machine, handling incompressible fluid, the term can be dropped. Moreover, if the effect of liquid viscosity on the performance of fluid machines is neglected or regarded as secondary, (which is often sufficiently true for certain cases or over a limited range) the term can also be dropped. Then the general relationship between the different dimensionless variables ( terms) can be expressed as
(15.14)
or, with another arrangement of the terms,
(15.15)
If data obtained from tests on model machine, are plotted so as to show the variation of dimensionless parameters with one another, then the graphs are applicable to any machine in the same homologous series. The curves for other homologous series would naturally be different.
Therefore one set of relationship or curves of the terms would be sufficient to describe the performance of all the members of one series.
The performance or operating conditions for a turbine handing a particular fluid are usually expressed by the values of N, P and H, and for a pump by N, Q and H. It is important to know the range of these operating parameters covered by a machine of a particular shape (homologous series). Such information enables us to select the type of machine best suited to a particular application, and thus serves as a starting point in its design. Therefore a parameter independent of the size of the machine D is required which will be the characteristic of all the machines of a homologous series. A parameter involving N, P and H but not D is obtained by dividingby . Let this parameter be designated by as
(15.16)
Similarly, a parameter involving N, Q and H but not D is obtained by divining by and is represented by as
(15.17)
Since the dimensionless parameters and are found as a combination of basic terms, they must remain same for complete similarity of flow in machines of a homologous series. Therefore, a particular value of or relates all the combinations of N, P and H or N, Q and H for which the flow conditions are similar in the machines of that homologous series. Interest naturallycenters on the conditions for which the efficiency is a maximum. For turbines, the values of N, P and H, and for pumps and compressors, the values of N, Q and H are usually quoted for which the machines run at maximum efficiency.
The machines of particular homologous series, that is, of a particular shape, correspond to a particular value of for their maximum efficient operation. Machines of different shapes have, in general, different values of. Thus the parameter is referred to as the shape factor of the machines. Considering the fluids used by the machines to be incompressible, (for hydraulic turbines and pumps), and since the acceleration due to gravity dose not vary under this situation, the terms g and are taken out from the expressions of and. The portions left as and are termed, for the practical purposes, as the specific speedfor turbines of pumps. Therefore, we can write,
(specific speed for turbines) = (15.18)
(specific speed for turbines) = (15.19)
The name specific speed for these expressions has a little justification. However a meaning can be attributed from the concept of a hypothetical machine. For a turbine, is the speed of a member of the same homologous series as the actual turbine, so reduced in size as to generate unit power under a unit head of the fluid. Similarly, for a pump, is speed of a hypothetical pump with reduced size but representing a homologous series so that it delivers unit flow rate at a unit head. The specific speed is, therefore, not a dimensionless quantity.
The dimension ofcan be found from their expressions given by Eqs. (15.18) and (15.19). The dimensional formula and the unit specific speed are given as follows:
Specific speed Dimensional formula Unit (SI)(turbine) M1/2T-5/2L-1/4 kg1/2s5/2m1/4
(pump) L3/4T-3/2 m3/4s3/2
The dimensionless parameter is often known as the dimensionless specific speed to distinguish it from. The values of specific speed (for maximum efficiencies) for different types of turbines and pump will be discussed later.