Page 319 - Fluid Mechanics and Thermodynamics of Turbomachinery

300 Fluid Mechanics, Thermodynamics of Turbomachinery
geometrically similar machines. The primary reason for this is that it is not possible
to establish perfect dynamical similarity between turbomachines of different size.
In order to obtain this condition, each of the the dimensionless terms in eqn. (1.2)
would need to be the same for all sizes of a machine.
To illustrate this consider a family of turbomachines where the loading term,
2
2
2
D gH/N D is the same and the Reynolds number, Re D ND / is the same for
every size of machine, then
2
gH N D 4 gHD 2
2
Re D Ð D
2
N D 2 v 2 v 2
must be the same for the whole family. Thus, for a given fluid ( is a constant),
a reduction in size D must be followed by an increase in the head H. A turbine
model of 1 the size of a prototype would need to be tested with a head 64 times that
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required by the prototype! Fortunately, the effect on the model efficiency caused by
changing the Reynolds number is not large. In practice, models are normally tested
at conveniently low heads and an empirical correction is applied to the efficiency.
With model testing other factors effect the results. Exact geometric similarity
cannot be achieved for the following reasons:

(a) the blades in the model will probably be relatively thicker than in the prototype;
(b) the relative surface roughness for the model blades will be greater;
(c) leakage losses around the blade tips of the model will be relatively greater as a
result of increased relative tip clearances.

Various simple corrections have been devised (see Addison 1964) to allow for the
effects of size (or scale) on the efficiency. One of the simplest and best known is
that due to Moody and Zowski (1969), also reported by Addison (1964) and Massey
(1979), which as applied to the efficiency of reaction turbines is

n
1 p D m
D (9.23)
1 m D p
where the subscripts p, m refer to prototype and model, and the index n is in
the range 0.2 to 0.25. From comparison of field tests of large units with model
tests, Moody and Zowski concluded that the best value for n was approximately
0.2 rather than 0.25 and for general application this is the value used. However,
Addison (1964) reported tests done on a full-scale Francis turbine and a model made
to a scale of 1 to 4.54 which gave measured values of the maximum efficiencies of
0.85 and 0.90 for the model and full-scale turbines, respectively, which agreed very
well with the ratio computed with n D 0.25 in the Moody formula!
EXAMPLE 9.5. A model of a Francis turbine is built to a scale of 1/5 of full
size and when tested it developed a power output of 3 kW under a head of 1.8 m of
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water, at a rotational speed of 360 rev/min and a flow rate of 0.215 m /s. Estimate the
speed, flow rate and power of the full-scale turbine when working under dynamically
similar conditions with a head of 60 m of water.
By making a suitable correction for scale effects, determine the efficiency and
the power of the full-size turbine. Use Moody’s formula and assume n D 0.25.