## 2 Critique of Generalized Momentum-Based Theories  
The generalized momentum-based theories rely on assumptions that only make mathematical sense and defy the law of conservation of momentum. Elementary mechanics reveals that if a given mass of air \(m\), moving with a speed \(U_{0}\) in the axial direction, impinges on a rotor of mass \(m_{R}\), it leaves the rotor with an axial speed of \(U_{R}\) with \(U_{R}<U_{0}\). Because of wake expansion, the speed of the air in the axial direction continues to reduce to \(U_{3}\) at the Trefftz plane with \(U_{3}<U_{R}\). Since the air loses linear momentum at the rotor due to the loss of translational KE, there is no linear momentum balance because the lost translational KE of the airstream is converted to rotational KE at the rotor, which results in an angular momentum on the rotor. The rotor does not move in the axial direction, viz, \(mU_{0}>m(U_{R}+U_{3})\). Instead, the rotor rotates about its axis with an angular momentum \(P^{\prime}{}_{R}=I_{R}\omega\). The angular momentum acts tangentially to the rotor. All actuator disk-based theories predicate on the scientific fallacy that the loss in momentum at the rotor results solely in an axial thrust, which is entirely borne by the air flowing past the WT rotor. This therefore means that no energy is imparted on the rotor; thus, the rotor does not rotate. It is worth noting that the rotor does not move in the axial direction--something that renders its linear momentum null. There are now two components of momentum at the rotor: the tangential component and the axial component. Just behind the rotor, the air exhibits a solenoidal flow that results from the swirl imparted on it by the rotor. If the air mass rotates with an angular speed \(\omega_{ar}\) just behind the rotor, its angular speed at the wake is \(\omega_{ar}\). Then, just behind the rotor, the momentum of the mass of air is \(P_{air}=mU_{R}\hat{\imath}+(I_{R}\omega_{ar}/R)\hat{k}\), where \(\hat{\imath}\) and \(\hat{k}\) are unit vectors along the x-axis and z-axis, respectively and \(I_{aR}\) is the moment of inertia of the air at the rotor. At the wake, the momentum of the mass of air is \(P_{w}=mU_{\hat{\imath}}\hat{\jmath}^{\ast}+(I_{aw}\omega_{aw}/r_{w})\hat{k}\), where \(I_{aw}\) is the moment of inertia of the air at the wake. The mere fact that the air is assumed to exert only an axial thrust or force on the rotor, which is idealized as a permeable disk, means that there is no torque on the rotor since there is no moment arm and hence no rotation on the rotor. Rotation can only occur if the rotor is directly coupled to a prime mover as is typical of propellers. WTs do not have such prime movers, as they are meant to generate power. The law of conservation of energy as stated in elementary mechanics warrants that the KE lost by the mass of air be equal to the rotational KE on the rotor; thus \(\frac{1}{2}mU_{0}^{2}=\frac{1}{2}(I_{R}\omega_{R}^{2}+mU_{3}^{2})\) or loss in KE is \(\frac{1}{2}m(U_{0}^{2}-U_{3}^{2})\). Clearly the KE lost by the air is linked to the rotation of the rotor. If the mass flow rate of the air is \(\dot{m}\), where an overdot denotes a time derivative, then the rate of loss of KE by the air stream is \(\frac{1}{2}\dot{m}(U_{0}^{2}-U_{3}^{2})\). This is the power extracted from the air stream by the rotor. [PERSON]'s momentum theory assumes that the axial thrust on the rotor is given by the rate of change of the axial momentum of the air; \(T=\dot{m}(U_{0}-U_{3})\) and the KE lost per unit time by the airstream at the rotor is \(TU_{R}\). Thus, \(\dot{m}(U_{0}-U_{3})U_{R}=\frac{1}{2}\dot{m}(U_{0}^{2}-U_{3}^{2})\), which upon simplifying, yields \(U_{R}=(U_{0}+U_{3})/2\). In his generalized momentum theory, [PERSON] included a tangential velocity on the actuator disk but still relied on the axial momentum theory in the axial direction [23]. The actuator disk and rotating annular stream-tube theories rely on the above theory, while the VRT does not.