Gas/Steam Medium

If the medium is already a gas/steam, the phenomenon that the flow medium will change from the liquid to the gas phase does to occur, thus no cavitation has to be considered. For lower temperature, the sonic velocity is low. In order to avoid supersonic and sonic flows the flow velocity has to lower. But the flow velocity may be desired to be higher. Above sonic velocity(supersonic flow) there is an occurrence of shock waves which represent losses.

The nearness of the flow velocity to the sonic velocity a may be expressed using the Mach Number. IN general, flow conditions where M=1 and normally also those with M>1 (supersonic flow) have to be avoided. The sonic velocity of a gas is: Since for a given gas k and R are constant, the locally prevailing temperature determines the sonic velocity a, the danger or reaching sonic velocity is greatest where the temperature is the lowest. In general, this is at the suction end of the turbomachine.

Compressors The sonic velocity is lowest at the suction end of the impeller, in the case of multi-stage compressors, at the suction end of the first impeller as here the temperature is lowest. The criteria to limit the locally highest velocity below the sonic velocity may be expressed as follows: Similarly to avoiding cavitation, an optimum angle βoa can be determined: The lower value refers to impeller with thick vanes The higher value to those with thin vanes Where Wmax = locally highest velocity in the vane channel near suction end Experimental value: λ ≈ 0.2 to 0.3

Assuming no pre-rotation (δr=1) and the value of λ the optimum angle βoa is: The optimum angle which avoids sonic velocity best has value of nearly twice of optimum angle avoiding cavitation best By applying the above criterion the velocity in the channel near the suction edge is kept lowest. As the velocity decreases the frictional loss also decreases which results a higher hydraulic efficiency.

The Suction Diameter-The Inlet Number ε and the Discharge Number ε2

The suction diameter has to be chosen so that βoa obtains its desired value with regard to avoiding cavitation or sonic velocity or with regard to obtaining lowest friction loss at the vane suction edge. The following optimum values of βoa were mentioned in the previous chapters:

The suction diameter D1a and the angle βoa are interrelated to each other by the velocity Coma as follows A relation between Com and D1a can be found from the velocity triangle at point o: Cross sectional area at point o Perpendicular to Com

It follows and

Knowing the value of ε, ε2 , the suction diameter Ds follows from The suction diameter Ds may also be determined using the following dimensionless numbers: Knowing the value of ε, ε2 , the suction diameter Ds follows from The values of ε and ε2 are functions of the shape Number Nshape as can be noted from the following derivation: Pumps and compressors Inlet Number Discharge Number turbines

and where

The Inlet number for pumps and compressors, assuming δr=1 and k = 1-(dn/Ds)2≈0.8 and the common range βoa =14 to 380 is As far as the slow-running rotor with Nshape < 0.1 and k =1 is concerned the value of ε may be taken independent of Nshape as

The values δr and k of water turbines of normal designs are δr≈1 Francis, Kaplan Turbine K ≈ 0.8, Kaplan Turbine, k ≈1 Francis Turbine The following table shows some values of ε2 taken from actual designs. Francis Turbines Kaplan T Dim. Nshape 0.063 0.065 0.123 0.210 0.34 0.52 0.70 1 ns 70 94 141 237 387 592 797 (metric) nq 21 28 41 114 174 234 ε2 0.032 0.048 0.096 0.152 0.331 0.486 βoa 30 23.5 21.3 18.6 19.4 19.2 degree

Faster running turbines have higher values of ε2 Faster running turbines have higher values of ε2. In case of the Kaplan Turbines ε2 may account up to 0.5. This means that the kinetic energy Co2 /2 with which the water is discharged from the rotor is equal to 50 % of the available energy Y. All this kinetic energy would be lost if no draft tube were provided. A draft tube, however, will ‘recover’ part of this kinetic energy. As ε2 of fast running turbines is large, these turbines have to be designed with very effective draft tubes. For this reason Kaplan turbines are mostly designed with elbow type draft tubes.

Number of Vanes

If the number of vanes is small, each vane is loaded much and, hence the pressure difference between both sides of the vane is high. This leads to non uniform velocity distribution in the vane channel and very high local velocities may occur. Thus also cavitation or sonic velocity is more likely to occur. If the number of vanes is high the vane channel becomes narrow and consequently the friction loss is high. Thick vanes do not allow to install many vanes. Generally the thickness of the vanes should be as small as possible observing, however the strength of the vane material, the vibration of the vane and proper profiling if desired.

Number of vanes Axial- Flow Pumps and water turbines have mostly cast vanes, radial-flow blowers have sheet metal vanes. Axial- Flow The above formulas are also applicable to determine the number of vanes of guide vanes. where e = length of the mean stream line in meridian section measured between vane in-and outlet rm = radius of the middle of the line e k = empirical factor = 5 to 6.5 cast vanes = 6.5 to 8(to 12) sheet metal vanes