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Consider the falling film of width H that we studied in recent lectures. A thin film of pure liquid water is flowing down an impermeable solid surface with velocity vx(y). The flow is laminar, steady and fully developed, and vy = vz = 0. In this problem, the Peclet number is large (Pe >> 1). For x

For x 0, solute species A (acetic acid) is adsorbed in the water at the air-water interface (y=0) at approximately a constant concentration C0 . As x increases, the solute species diffuses (with diffusion coefficient DA) across the film as it is carried along by the flow. Meanwhile, there is no flux of species A across the solid wall. Also, the local concentration CA(x,y) is steady in time and does not depend on the z coordinate.

a) Write the conservation equation for species A and corresponding boundary and initial conditions, that would be applicable for solving the general problem (i.e. not only in the entrance region). (Simplify as much as you can but do not solve.)

b) In the entrance region (small x), the concentration changes will be confined to a thin layer of width , where /H

c) Now find the concentration of species A in the water exiting the absorber, for the following case. The liquid is water (kinematic viscosity = 0.01 cm2 /s ; DA = 0.91 x 10-5 cm2 /s for species A (acetic acid) in water). The film has thickness H = 4 mm and a length L = 40 cm (beyond x=0). Near the air-water interface (y=0), the velocity of the water can be approximated as a constant value, v0 = 0.2 cm/s. The concentration of species A adsorbed from the air, C0 = 0.01 mol/liter.

The following expression is given for the average Sherwood number (Sh) in laminar film flow: Sh = (6/ * Re * Sc * H/L) 1/2 , where the Reynolds number is based on H.

What is the concentration of species A in the water exiting the absorber?

(Hint: for this problem, the parameter Y is defined as : Y = Sh / (Re * Sc * H/L) , where the Reynolds number is based on H, as mentioned in equation 6.4.8 of Middleman.)

H CA=0 CA=0 vx(y) X=0 CA=CO impermeable solid wall CA(x,y) air fluid DA y=H y = 0