Flow of a film on the outside of a circular tube (see Fig. 2B.6), in a gas absorption experiment a viscous fluid flows upward through a small circular tube and then downward in laminar flow on the outside. Set up a momentum balance over a shell of thickness Ar in the film, as shown in Fig. 2B.6. Note that the "momentum in" and "momentum out" arrows are always taken in the positive coordinate direction, even though in this problem the momentum is flowing through the cylindrical surfaces in the negative r direction.

(a) Show that the velocity distribution in the falling film (neglecting end effects) is

(b) Obtain an expression for the mass rate of flow in the film.

(c) Show that the result in (b) simplifies to Eq. 2.2-21 if the film thickness is verysmall

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Velocity distribution inside tube V₂ L z-Momentum into shell of thickness Ar K Velocity distribution outside in film H Har Ar z-Momentum out of shell of thickness Ar Gravity force acting on the volume 2πτΔrL aR = PER [ 1 - (A)² + 2² In (F)] Αμ R R