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ρn
where n is some polytropic process. The adiabatic efficiency can then be represented by Tadc -「，-1-1「，「n-1 -1「 (3-33)
， n
P2tP2t
P1tP1t
The isentropic efficiency of the turbine can be written in terms of the total enthalpy change
Actual work h3t -h4 ' t
Tadt -- (3-34)
Isentropic work h3t -h4t
This equation can be rewritten for a thermally and calorifically perfect gas in terms of total pressure and temperature
匹
T4 ' t
1 -
T3t
Tadt -P4t ，-1，(3-35) 1 -
P3t
Polytropic Efficiency
Polytropic efficiency is another concept of efficiency often used in com-pressor evaluation. It is often referred to as small stage or infinitesimal stage efficiency. It is the true aerodynamic efficiency exclusive of the pressure-ratio effect. The efficiency is the same as if the fluid is incompressible and identical with the hydraulic efficiency
， -1
，

匹 dP2t
1 +-1
P1tTpc -匹 n-1n(3-36)dP2t
1 +-1
P1t
which can be expanded assuming that
dP2t
..1
P1t
.eglecting second-orderterms， the following relationship is obtained: ，-1Tpc -， (3-37)n -1 n
From this relationship， it is obvious that polytropic efficiency is the lim-iting value of the isentropic efficiency as the pressure increase approacheszero， and the value of the polytropic efficiency is higher than the correspond-ing adiabatic efficiency. Figure 3-6 shows the relationship between adiabatic and polytropic efficiency as the pressure ratio across the compressor increases. Figure 3-7 shows the relationship across the turbine.
Another characteristic of polytropic efficiency is that the polytropic effi-ciency of a multistage unit is equal to the stage efficiency if each stage has the same efficiency.
.imensional Analysis
Turbomachines can be compared with each other by dimensional analysis. This analysis produces various types of geometrically similar parameters. Dimensional analysis is a procedure where variables representing a physicalsituation are reduced into groups， which are dimensionless. These dimen-sionless groups can then be used to compare performance of various types of machines with each other. Dimensional analysis as used in turbomachinescan be employed to: (1) compare data from various types of machines一it isa useful technique in the development of blade passages and blade profiles，
(2) select various types of units based on maximum efficiency and pressurehead required， and (3) predict a prototype"s performance from tests con-ducted on a smaller scale model or at lower speeds.
95 
90 
85 
80 
75  90 % Add. Eff 
85 % Add. Eff 
70  80 % Add. Eff 
65  75 % Add. Eff 
70 % Add. Eff 
60  65 % Add. Eff 
55 
50 
45 
40 
Figure 3-6. Relationship between adiabatic and polytropic efficiency.

 

1611 16 21 26 31 36 4146 Pressure Ratio

Dimensional analysis leads to various dimensionless parameters， which are based on the dimension"s mass (M)， length ( L)， and time ( T). Based onthese elements， one can obtain various independent parameters such as density (ρ)， viscosity ( μ)， speed ( N)， diameter ( D)， and velocity ( V). The independent parameters lead to forming various dimensionlessgroups， which are used in fluid mechanics of turbomachines. Reynolds number is the ratio of the inertia forces to the viscous forces
ρVD
Re -(3-38)
μ
where ρis the density of thegas， Vthevelocity， D the diameter of theimpeller， and μ the viscosity of the gas.
The specific speed compares the head and flow rate in geometrically similar machines at various speeds
....
 
NQ
Ns -(3-39)
　
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