Suppose there is steady, laminar flow of a suspension of spherical particles through a parallel-plate channel, as in Fig. P5.7. The plate spacing is h, the channel length is L, the mean fluid velocity along the channel is u, and each particle has a constant downward velocity U. As can be shown using the methods of Chapter 7, the local fluid velocity is

(a) Confirm that u is indeed the average of v_x for 0 ≤ y ≤ h.

(b) Calculate the trajectory of a point-size particle that enters the channel at a distance y₀ from the top. That is, determine x(t) and y(t) when x(0) = 0 and y(0) = y₀.

(c) Suppose that L is just large enough to allow all particles to reach the bottom before the suspension exits the channel. Plot the trajectories for two entering heights, y₀/h = 0 and y₀/h = 1/2.

(d) A simpler but less accurate approach is to assume that all particles move horizontally at the mean velocity u (as in Example 3.3-2). Compare the resulting trajectories with the ones from part (c). How would the simpler approach affect the calculated minimum value of L needed for all particles to settle?

Transcribed Image Text:

= 6u [G) - ()²]. vx(y) = би y X v₂ = 0 = V₂. h (P5.7-1) L Figure P5.7 Particles settling in a parallel-plate channel. The particle size is greatly exaggerated. g