In calculating the shape ofa gravity-flow discharge chute that will minimize transittime

Chapter 10, Problem 10

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In calculating the shape ofa gravity-flow discharge chute that will minimize transittime ofdischarged granular particles, C. Chiarella, W. Charlton, and A. W. Roberts [CCR] solve the following equations by Newton's method: sin 0+i sin $ (i) (l-nwn+\) (\-iiw) = 0, for each n = 1, 2,..., Af I. fn+i vn (ii) /,7(6*1,,9N) = Ay tan 9/ - X = 0. where a. v* = vl + 2gnAy-2nAyjyj=\TT' for each " = 2 ' < ^ and COS 1/j b. wn Ayvn r , for each n = \,2,..., N. vfcosOj The constant uq is the initial velocity of the granular material, X is the x-coordinate of the end of the chute, /i is the friction force, N is the number of chute segments, and g = 32.17ft/s2 is the gravitational constant. The variable 6, is the angle ofthe /th chute segment from the vertical, as shown in the following figure, and v, is the particle velocity in the /th chute segment. Solve (i) and (ii) for 0 (0i,..., eNy with /x 0, X 2, Ay 0.2, N 20, and no 0, where the values for vn and iv can be obtained directly from (a) and (b). Iterate until \\9{k) 0 (A:_l)||oo < 10-2 .