Can you tell the direction of flow of a box fan just by looking at the frame?

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This is another question I posted in a discussion board for my online junior level fluid mechanics class.

Below are two photographs of a box fan showing views from the right and from the left. The question is, knowing what you do about angular momentum, which direction does the air flow when the fan is turned on (right to left or left to right)?

View from the left view from the right

As with many of the posts during the COVID semester, there was only a limited response. The responses were generally correct on the direction but occasionally a little off on the explanation.

The only information we have is that there are little feet on the right hand side of the fan and not on the left hand side.

**Explanation:**

We start by drawing a control volume around the fan blade assuming that the flow is from right to left. We assume that the outflow is a horizontal jet of flow rate U_{o}A where U_{0} is the velocity out of the control volume and A is the outlet area circle formed by rotating the tip of the fan blades. The flow rate in is the same. The inflow will come in from all directions. However, it is only the horizontal component that contributes to the inflow volume flow rate so the horizontal component of the inflow velocity is the same as that of the outflow (U_{in.x}=U_{0}). We then write the angular momentum equation (sometimes called moment of momentum) about the bottom right hand corner of the control volume (also where the feet are on the box fan). Taking anti-clockwise as positive we get

On the intake side the only component that creates a moment about (a) is the horizontal component. Therefore, assuming that the inflow and outflow are approximately axisymmetric about the fan blade axis, the right hand side sums to zero. That leaves

Applying Bernoulli’s equation along a horizontal stream line from far upstream to the fan intake we get

For a horizontal streamline and a stationary ambient at zero gauge pressure the elevation terms cancel and the other two terms on the left hand side are both zero. Therefore, the inlet pressure is

Therefore there is a moment about (a) given by

Where R is the radius of the fan blades. This negative moment must be balanced by a moment applied by the fan blade to the flow. The net effect of this is that a positive anti-clockwise (counter-clockwise) moment must be applied to the fan blade by the casing (box). This moment is applied by the two small feet on the right hand side at the base of the fan box. Therefore, our initial assumption of the flow going from right to left is correct.

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