Transcript SANITARY AND STORM SEWER DESIGN A Direct Algebraic …

SANITARY AND STORM
SEWER DESIGN
A Direct Algebraic
Solution
by
A. M. Saatçi
Partially Flowing Pipe
D
θ
h
Manning’s Equation
v = (1/n)
Q = (K/n)
2/3
1/2
R S
8/3
D
1/2
S
Given: D, D, n, h/D
Given
D, S, n, h/D
θ = 2 cos-1(1-2h/D)
R=D(θ-sinθ)/4θ
K=0.0496 θ(1-sinθ/θ)5/3
Q = KD8/3S1/2 /n
v = Q/A
A= D2/8(θ-sinθ)
Given: D, S, n, Q
Given
D, S, n, Q
Graphical
Solutions
Figure 1.
h/D, Aflow, v
Trial & Error Solution:
θ-2/3 (θ - sin θ)5/3 - 20.161 n Q D-8/3 S-1/2 = 0
h/D, Aflow, v
Geometry
From geometry,
θ = 2 cos-1 [(1-2h)/D]
A = D2 (θ-sin θ)/8
K = 0.04968θ-2/3 (θ -sin θ)5/3
θ-2/3 (θ - sin θ)5/3 - 20.161 n Q D-8/3 S-1/2 = 0
θ-2/3 (θ - sin θ)5/3 - 20.161 K = 0
Last two eqns can be solved for θ using trial and
error methods.
Saatci Equation
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.
Given
Q, D, S, n
K = Qn/(D8/3S1/2)
Saatci's Eqn
θ-2/3 (θ -
sin
θ)5/3
θ = (3π/2) 1-  1- πK
- 20.161 K = 0
Trial and Error Solutions to solve for θ
Direct Solution
θ
Water Surface Angle
A =( D2/8)(θ-sin θ)
h/D=0.5(1-cos(θ/2))
v = Q/A
R = (D/4)[(θ-sinθ) /θ]