Problem i [20 points. A cattle watering trough (width B-0.5 m, depth H = 0.3 m,
ID: 115564 • Letter: P
Question
Problem i [20 points. A cattle watering trough (width B-0.5 m, depth H = 0.3 m, length L-12 m) was filled with stagnant quiescent water. A cow pushed hay into the one end of the trough, and the hay sank to the bottom in the middle of trough width. The trace metal selenium was sorbed to the hay at a mass fraction of 4.5 mgkg and enough hay was present Selenium desorbed from the hay and entered the trough wa a distribution coefficient K 0.6 Lg. Diffusion within the trough was homogeneous but anisotropic, with horizontal diffusion constant D-0.002 m2/s and vertical diffusion constant D,-0.001 m2/s. Assume that the selenium otherwise behaved conservatively, and did not volatize, react, or sorb to the trough lining. t that this content remained approximately steady despite subsequent desorption. ter in a rapid and reversible process, characterized by a) What was the concentration of dissolved selenium in the trough immediately adjacent to the hay? b) How long did it take the selenium to mix across the trough width? depth? length? c) Sketch the selenium concentration over trough length after 5 (i) 12 minutes and (ii) 24 hours. d) Aftler 12 minutes, how much selenium had entered the trough from the hay? (Hint: think about theg o average concentration within a boundary layer.) 10 12 Distance from hay, x (m) hayExplanation / Answer
ANSWER:
A theoretical investigation is made into the spreading of a very viscous liquid, with positive spreading coefficient, on a quiescent water surface. When the boundary-layer drag on the spreading layer due to the water is assumed negligible, the equations determining the motion are found by extending the use of lubrication theory to a higher order than that necessary for the situation where boundary-layer drag is important. In this manner the various systems of equations and their regimes of validity are found for situations ranging from those where the effect of boundary-layer drag is dominant to those where it is negligible. Finally various analytic solutions are found for the spreading equations for the case where boundary-layer drag is negligible. Numerical values of layer velocities and sizes are found for typical examples
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