SANITARY AND STORM SEWER DESIGN A Direct Algebraic Solution

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1 SANITARY AND STORM SEWER DESIGN A Direct Algebraic Solution
by A. M. Saatçi

2 Partially Flowing Pipe
D θ h

3 Manning’s Equation v = (1/n) R2/3S1/2 Q = (K/n) D8/3 S1/2

4 Given: D, D, n, h/D D, S, n, h/D Given θ = 2 cos-1(1-2h/D)
R=D(θ-sinθ)/4θ K= θ(1-sinθ/θ)5/3 Q = KD8/3S1/2 /n v = Q/A A= D2/8(θ-sinθ)

5 Given: D, S, n, Q D, S, n, Q Given Graphical Solutions Figure 1.
h/D, Aflow, v θ-2/3 (θ - sin θ)5/ n Q D-8/3 S-1/2 = 0 Trial & Error Solution:

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7 Geometry θ = 2 cos-1 [(1-2h)/D] A = D2 (θ-sin θ)/8
From geometry,  θ = 2 cos-1 [(1-2h)/D]  A = D2 (θ-sin θ)/8  K = θ-2/3 (θ -sin θ)5/3  θ-2/3 (θ - sin θ)5/ n Q D-8/3 S-1/2 = 0 θ-2/3 (θ - sin θ)5/ K = 0 Last two eqns can be solved for θ using trial and error methods.

8 Saatci Equation θ = (3π/2) 1-  1- πK K = (1/π) { 1- [1-(2θ/3π)2]2}2
It is suggested that an algebraic equation of the form: θ = (3π/2) 1-  1- πK which can also be written as: K = (1/π) { 1- [1-(2θ/3π)2]2}2 can give a quick solution of the water surface angle θ.

9 Trial and Error Solutions to solve for θ
Q, D, S, n Given K = Qn/(D8/3S1/2) θ-2/3 (θ - sin θ)5/ K = 0 Trial and Error Solutions to solve for θ Saatci's Eqn θ = (3π/2) 1-  1- πK Direct Solution Water Surface Angle θ A =( D2/8)(θ-sin θ) v = Q/A h/D=0.5(1-cos(θ/2)) R = (D/4)[(θ-sinθ) /θ]

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