Lifting silicon wafers—M . Your supervisor has suggested the device shown in Fig. P2.17 for picking up delicate circular silicon wafers without touching them. A mass flow rate m of air of constant...

Lifting silicon wafers—M . Your supervisor has suggested the device shown in Fig. P2.17 for picking up delicate circular silicon wafers without touching them. A mass flow rate m of air of constant density ρ at a pressure p0 is blown down a central tube of radius r1, which is connected to a flange of external radius r2. The device is positioned at a distance H above the silicon wafer, also of radius r2, that is supposedly to be picked up. The air flows radially outwards with radial velocity ur(r) in the gap and is eventually discharged to the atmosphere at a pressure p2, which is somewhat lower than p0.

If the flow is frictionless and gravitational effects may be neglected:

(a) Obtain an expression for ur in terms of the radial position r and any or all of the given parameters. Sketch a graph that shows how ur varies with r between radial locations r1 and r2.

(b) What equation relates the velocity and pressure in the air stream to each other? Sketch another graph that shows how the pressure p(r) in the gap between the flange and wafer varies with radial position between r1 and r2. Hint: it may be best to work backwards, starting with a pressure p2 at radius r2. Clearly indicate the value p0 on your graph.

(c) Comment on the likely merits of the device for picking up the wafer.