In Eq. (30), Ω is the impeller rotational speed. Note that in the case where there are an infinite number of blades, then View the MathML source is the unit vector normal to the blade camber surface (or camber line in 2D). For the case where the number of blades is finite, some flow deviation exists between the blade camber surface and the streamline that the mean flow follows, and this deviation must be modeled. As an example of such a model, the traditional cascade deviation model shown in Fig. 12 may be used, which states that the flow deviation from the blade geometry is a function of flow incidence at the blade leading-edge. With respect to the flow in a cross-flow fan, it is seen from Fig. 15 and Fig. 34 (taken from CFD simulations) that a large portion of the flow in region A is well behaved (low flow incidence and deviation), and the cascade turning and loss model is appropriate. However, as seen from these figures, the cascade model is not appropriate for regions B and C.
If the blades are replaced by a body-force field denoted by View the MathML source (force per unit mass) and viscous loss modeled by a body-force field denoted by View the MathML source, then the governing equations to be solved are the continuity and momentum equations. The equations can then be written in cylindrical coordinates as follows: