I need a help to find the power coefficient (CP) using Double Multiple Streamtube model (DMST) for straight-balded, vertical axis wind turbine (VAWT) of Darrieus type. What is the mathematical expression for that.
Hi,
you can use the QBlade software. It includes modules for DMST analysis and blade design and should cover everything you need. However you need to get a bit involved with rotor design and simulation and read through the guidelines. Unfortunately there is no simple mathematical expression as all results depend on the specific rotor design and the operational regime.
Hi
If you want to know the equations for the double multiple streamtube model, I can recommend the book Wind Turbine Design: With Emphasis on Darrieus Concept by Ion Paraschivoiu (you can also check out his publications). The main principle behind a streamtube model is to use an empirical force model to calculate the forces for a blade from the local flow velocity. The flow velocity is obtained by momentum conservation in the streamtube, using the calculated force (hence you will have to iterate to find a solution). For a multiple streamtube model, you divide the cross-sectional area into many small streamtubes, and solve individually for each of them. The double multiple streamtube model means that you assume that you have atmospheric pressure in the center of the turbine, which allows you to use the same equations two times, one for the upwind side, and one time for the downwind side, where you use the velocities calculated upwind as inlets to the downwind streamtubes.
Note that the model by Ion Paraschivoiu uses straight streamtubes. If you want to include flow expansion, you can look at the model by Read and Sharpe. If you are interested in the local forces, expansion will be important, but if you only need the power coefficient, the use of straight streamtubes should probably be sufficient.
The tricky part with the streamtube model is all the extra models that you apply. First, if you are going to simulate tip speed ratios below about 4, you will need to use a dynamic stall model for the force calculations. These models are usually designed for pitching wings, and do not properly account for the curved motion of the blades, which causes differences between the upwind and downwind sides. You will also have effects due to the tip vortices for straight bladed VAWTs. There are correction models available (see e.g. the book by Parashivoiu), but I would say that more research is needed in this area to find good correction models. To calculate the power coefficient, you will also need to apply some model for the losses from the support arms. Note that if the support arms are mounted with an angle, you will also generate lift force on the support arms, which further complicates the situation. You may also need some additional loss model for attachments etc., as drag losses can have a significant impact on the power coefficient. Combined, all these additional models reduces the reliability of the double multiple streamtube model, and I would not expect the same accuracy for a VAWT, as you can get for a HAWT using the model.
Anders
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