Assuming you have a loop, with the area $S = W \cdot L$, like in an dc motor, and $L$ keeps constant, then the term
$$B\frac{\partial S}{\partial t}\cos(\alpha)$$
simplifies to
$$B\frac{\partial W}{\partial t}\cdot \cos(\alpha) \cdot L = B \cdot v \cdot \cos(\alpha) \cdot L$$
where $v$ is the velocity.
