Hydraulic Characteristics of Circular pipe section 

Note:- This theory is not applicable to Drinking Water pipelines as they flow under pressures.

In case of Bridge construction and Sewer line works, the hydraulic characteristics of flow through Circular sections plays important role.

Case 1:- Bridges 

    a) For Pipe culverts to calculate required number of pipes 

    b) When diversion is provided while bridge foundation works going on, then also we need to calculate number of pipes required to which can pass that discharge.

Case 2:- for sewer lines these are must calculations

General Partial filled pipe section Looks like as shown in image below

The depth at partial flow can be evaluated from equation -

$$ d = ({D\over 2}-{D\over 2}.cos{\alpha\over 2}) $$

Proportionate depth at partial flow can be evaluated from equation -

$$ {d\over D} = {1\over 2}(1-cos{\alpha\over 2}) $$

Area of cross section while running partially full 

$$ a = {\pi D^2\over 4}.{\alpha\over 360^\circ} -{D\over 2}.cos{\alpha\over 2}.{D\over 2}.sin{\alpha\over 2}$$

$$ a ={\pi D^2\over 4}({\alpha\over 360^\circ}-{sin\alpha\over 2\pi})$$

Proportionate area  = $$ {a\over A} ={\pi D^2\over 4}({\alpha\over 360^\circ}-{sin\alpha\over 2\pi})$$

Wetted perimeter, while running partially full 

$$P = \pi D{\alpha\over 360^\circ}$$

Proportionate perimeter

$$ {P\over P'} = {{ \pi D{\alpha\over 360^\circ}}\over {\pi D}} $$

$$ = {\alpha\over 360^\circ}$$

Hydraulic mean depth, while running partially full

$$ r = {a\over P} $$

So, proportionate hydraulic mean depth 

$$ {r\over R} = {D\over 4}.{(1-{{{360^\circ}sin\alpha}\over {2 \pi \alpha}})\over {D\over 4}}$$

$$ = (1-{{{360^\circ}sin\alpha}\over {2 \pi \alpha}}) $$

Velocity of flow is given by Manning's formula

Velocity at partial flow, 

$$ v = {1\over n}.r^{2\over 3}.s^{1\over 2}$$

Velocity when running full, 

$$ v = {1\over N}.R^{2\over 3}.s^{1\over 2}$$

Therefore proportionate velocity, as slope will remain constant

$$ {v\over v'} = {N\over n}.{{r^{2\over 3}}\over {R^{2\over 3}}}$$

also if v assume that the rugosity or roughness coefficient do not vary with depth as pipe material is homogeneous and uniform material then

proportionate velocity = $$ {v\over v'} = {{r^{2\over 3}}\over {R^{2\over 3}}}$$

$$ {v\over v'}= {(1-{{{360^\circ}sin\alpha}\over {2 \pi \alpha}})}^{2\over 3} $$

Discharge when pipe is running partially full,

q =av

Discharge when pipe is running full,

Q = A.v'

Proportionate discharge  = $$ {q\over Q} = {{a.v}\over {A.v'}}$$

=$$ {q\over Q} =({\alpha\over 360^\circ}-{sin\alpha\over 2\pi}).{(1-{{{360^\circ}sin\alpha}\over {2 \pi \alpha}})}^{2\over 3}$$