I am involved in design of a HAWT. I would like to know the influence of number of blades in a wind turbine and its power output. Do share if you guys have an answer.
As a general rule, for a given tip speed, increasing the number of blades increases the maximum theoretical power coefficient (Cp) of a turbine. However, as the number of blades increases on a rotor, the gains in Cp get smaller and smaller. There is an equation for calculating this (It is given in Manwell's Wind Energy Explained book).
Also, when the number of blades increases, the blades become slenderer and structural considerations become important.
This should be a textbook topic in most books on wind energy engineering.
Typically the number of blades is expected to increase the starting couple of the turbine but makes the machine much heavier for very similar power output. For electrical power one prefers a fast tip speed and slow start than a complex heavy machine. Further the number of blades is chosen to be 3 for mechanical reasons (balance).
Pr. Avraham Hirschberg has a good comment about the mass, balance and starting couple of the turbine. The classical example for illustrating the use of the high starting couple of multiblade rotors is the agricultural water pumps actuated by wind turbines. They have multiple short blades and can extract a good amount of energy out of a relatively low wind.
If I'm not mistaken, adding blades also has a significant effect on thrust loads. Therefore, in combination with the heavier rotor, it results in a requirements fior a larger nacelle and tower. Since the additional material requirements increase the cost of the turbine (materials cost, manufacturing costs, transport costs, erection costs,...), the small gains in Cp eventually is largely exceeded by the additional burden of costs.
Another example of the fact that the main driving factor in wind turbine design is minimizing the cost of energy so that wind energy can remain competitive against conventional energy sources.
It is not actually a single equation, but an account of rotor solidity (ratio of blade area over swept area) in the optimisation problem posed for blade design.
Otherwise, Manwell suggests the following initial number of blades (N) based on tip speed ratio (L).