Problem 4 (35 points) The figure below shows a wetted-wall column, in which a thin film of an incompressible, Newtonian liquid with a constant viscosity of μ flows steadily down a plane wall, possibly for a gas-absorption study. The volumetric flow rate of the liquid is specified as Q per unit width of the wall (normal to the plane of the diagram). Assume that the width of the film (in the z direction) is W and the length of the film (in the x direction) is L. The length and width of the film are large compared to the thickness of the film (6). Assume that there is only one non-zero velocity component and that the air exerts negligible shear stress on the liquid film. a) Starting from a shell momentum balance, derive an expression for the velocity profile (as a function of p. u. g. y, and 8). (15 points) b) Starting from the equations of motions (Navier-Stokes equations), derive an expression for the velocity profile (as a function of p. H. g. y, and 8). Hint: your final expression for the velocity profile should be the same as what you derived in part a. (15 points) c) Assume the fluid is water at 25 °C and the film thickness (8) is 0.5 mm. Using graphical software (i.e., Excel or Mathematica), plot the velocity as a function of position. (5 points) Flow rate е g Wall Z 13 Liquid film Air