In calculating the shape of a gravity-flow discharge chute that will minimize transit time of discharged granular particles, C. Chiarella, W. Charlton, and A.W. Roberts [CCR] solve the following equations by Newton's method:
(i)

(ii) fN (θ1, . . . , θN) = Δy ˆ‘Ni=1 tan θi ˆ’ X = 0, where
a.

b.

The constant v0 is the initial velocity of the granular material, X is the x-coordinate of the end of the chute, μ is the friction force, N is the number of chute segments, and g = 32.17ft/s2 is the gravitational constant. The variable θi is the angle of the ith chute segment from the vertical, as shown in the following figure, and vi is the particle velocity in the ith chute segment. Solve (i) and (ii) for θ = (θ1, . . . , θN)t with μ = 0, X = 2, Δy = 0.2, N = 20, and v0 = 0, where the values for vn and wn can be obtained directly from (a) and (b). Iterate until || θ(k) ˆ’ θ(kˆ’1) ||ˆž

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