Maximum Flow Rate in Open-Channel Flow for a Circular Pipe - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Applications and Example Worksheets : Science and Engineering : applications/MaxFlowRatePartiallyFilledPipe

Maximum Flow Rate in Open-Channel Flow for a Circular Pipe

Introduction

This application determines the greatest attainable water flow rate in a partially filled circular pipe.

 

 

It uses the Manning formula to determine the flow rate in the open-channel flow of water:

Q = 1.49nAR23S__012,

where:

 

• 

Q is the flow rate

• 

n is an empirical coefficient

• 

A is the cross-sectional area of flow

• 

R is the hydraulic radius

• 

S0 is the incline of the channel

 

An equation that represents the hydraulic radius of a partially filled circular pipe is derived and substituted into the Manning formula. The resulting equation is then optimized to find the maximum flow rate.

 

Manning Formula for a Circular Pipe

restart:

 

Manning formula:

Q1.49nAR23S__012:

 

For a partially filled circular pipe, the flow area (the blue shaded area in the preceding diagram) is:

Aπr2r2θsinθ2:


The wetted perimeter is given by the following formula.

P2πrrθ:

 

Hence, the hydraulic radius, R, is:

RAP

Rπr2r2θ2sinθ22πrrθ

(2.1)

The Manning formula then becomes:

Q

1.49πr2r2θ2sinθ2πr2r2θ2sinθ22πrrθ23S__0n

(2.2)

Maximum Flow Rate for a Circular Pipe

Pipe radius:

r3:

Incline of the channel:

S__00.0001:

Roughness coefficient:

n0.013:

plotQ,θ=0.. 0.5 π,labels=θ,Flow rate,labeldirections=horizontal,vertical,labelfont=Calibri,title=Flow Rate in Circular Pipe,titlefont=Calibri,14,size=600,400,axesfont=Calibri,gridlines,color=ColorTools:-ColorRGB,0/255,79/255,121/255

 

resOptimization:-MaximizeQ,θ=0.. 0.5 π

res45.6796864427174,θ=1.00507814259573

(1)

The maximum flow rate is...

Q__maxflowres1

Q__maxflow45.6796864427174

(2)

...when θ is ...  

θ__maxflowrhsres2,1;

θ__maxflow1.00507814259573

(3)

... and the flow depth is:

hrcos0.5 θ__maxflow:h+r

5.62908731621774

(4)