Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

346 Fluid Mechanics, Thermodynamics of Turbomachinery
TABLE 10.1. BEM method for evaluating a and a¢
Step Action required
1 Initialise a and a¢ with zero values
2 Evaluate the flow angle using eqn. (10.38)
3 Evaluate the local angle of incidence, a = f - b
4 Determine C L and C D from tables (if available) or from formula
5 Calculate a and a¢
6 Check on convergence of a and a¢, if not sufficient go to step 2, else go to step 7
7 Calculate local forces on the element
tediously.) An outline of the algorithm, called the BEM method, is given in Table 10.1,
which is intended to become an important and useful time-saving tool. Further extension
of this method will be possible as the theory is developed.
The blade element momentum method
All the theory and important definitions to determine the force components on a blade
element have been introduced and a first trial approach has been given to finding a solution
in Example 10.4. The various steps of the classical BEM model from Glauert are
formalised in Table 10.1 as an algorithm for evaluating a and a¢ for each elementary
control volume.
Spanwise variation of parameters
Along the blade span there is a significant variation in the blade pitch angle b, which
is strongly linked to the value of J and to a lesser extent to the values of the lift coefficient
CL and the blade chord l. The ways both C L and l vary with radius are at the discretion
of the turbine designer. In the previous example the value of the pitch angle
was specified and the lift coefficient was derived (together with other factors) from it.
We can likewise specify the lift coefficient, keeping the incidence below the angle of
stall and from it determine the angle of pitch. This procedure will be used in the next
example to obtain the spanwise variation of b for the turbine blade. It is certainly true
that for optimum performance the blade must be twisted along its length with the result
that, near the root, there is a large pitch angle. The blade pitch angle will decrease with
increasing radius so that, near the tip, it is close to zero and may even become slightly
negative. The blade chord in the following examples has been kept constant to limit
the number of choices. Of course, most turbines in operation have tapered blades
whose design features depend upon blade strength as well as economic and aesthetic
considerations.
Example 10.6. A three-bladed HAWT with a 30m tip diameter is to be designed to
operate with a constant lift coefficient C L = 0.8 along the span, with a tip–speed ratio
J = 5.0. Assuming a constant chord of 1.0m, determine, using an iterative method of
calculation, the variation along the span (0.2 £ r/R £ 1.0) of the flow induction factors
a and a¢ and the pitch angle b.