Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev

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Civil Engineering (CE) : Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev

The document Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev is a part of the Civil Engineering (CE) Course Fluid Mechanics - Notes, Videos, MCQs & PPTs.
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Pipes In Parallel

  • When two or more pipes are connected, as shown in Fig. 36.3, so that the flow divides and subsequently comes together again, the pipes are said to be in parallel.
  • In this case (Fig. 36.3), equation of continuity gives

 

Q =  QA + QB                            (36.5)             

 

 

where, is the total flow rate and QA  And Q are the flow rates through pipes and Brespectively.

  • Loss of head between the locations 1 and 2 can be expressed by applying Bernoulli's equation either through the path 1-A-2 or 1-B-2. 
  • Therefore, we can write

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev

 

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev

 

 

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev  

 

Equating the above two expressions, we get -

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev                          (36.6)                      


 

where,

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev  

 

Equations (36.5) and (36.6) give -

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev                                   (36.7)            

 

Where 

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev                                                            (36.8)

 

The flow system can be described by an equivalent electrical circuit as shown in Fig. 36.4

 

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev

From the above discussion on flow through branched pipes (pipes in series or in parallel, or in combination of both), the following principles can be summarized:

  1. The friction equation must be satisfied for each pipe.
  2. There can be only one value of head at any point.
  3. Algebraic sum of the flow rates at any junction must be zero. i.e., the total mass flow rate towards the junction must be equal to the total mass flow rate away from it.
  4. Algebraic sum of the products of the flux (Q2) and the flow resistance (the sense being determined by the direction of flow) must be zero in any closed hydraulic circuit.

The principles 3 and 4 can be written analytically as

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev                          (36.9)     

 


 

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev                             (36.10)

 

While Eq. (36.9) implies the principle of continuity in a hydraulic circuit, Eq. (36.10) is referred to as pressure equation of the circuit.

 

Pipe Network: Solution by Hardy Cross Method

  • The distribution of water supply in practice is often made through a pipe network comprising a combination of pipes in series and parallel. The flow distribution in a pipe network is determined from Eqs(36.9) and (36.10). 
  • The solution of Eqs (36.9) and (36.10) for the purpose is based on an iterative technique with an initial guess in Q
  • The method was proposed by Hardy-Cross and is described below: 
    • The flow rates in each pipe are assumed so that the continuity (Eq. 36.9) at each node is satisfied. Usually the flow rate is assumed more for smaller values of resistance and vice versa. 
    • If the assumed values of flow rates are not correct, the pressure equation Eq. (36.10) will not be satisfied. The flow rate is then altered based on the error in satisfying the Eq. (36.10). 
  • Let Q0 be the correct flow in a path whereas the assumed flow be Q. The error din flow is then defined as

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev                                                  (36.11)

Let Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev                                                 (36.12a)

And Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev                                          (36.12b)

 

Then according to Eq. (36.10)

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev             in a loop                                         (36.13a)                                       


 

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev          in a loop                                              (36.13b)

 

Where 'e' is defined to be the error in pressure equation for a loop with the assumed values of flow rate in each path. 
From Eqs (36.13a) and (36.13b) we have

 

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev                                                     (36.14)

 

Where dh (= h - h' ) is the error in pressure equation for a path. Again from Eq. (36.12a), we can write

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev                                                 (36.15)

Substituting the value of dh from Eq. (36.15) in Eq. (36.14) we have

 

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev                                                         

Considering the error dQ to be the same for all hydraulic paths in a loop, we can write

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev                                                        (36.16)

he Eq. (36.16) can be written with the help of Eqs (36.12a) and (36.12b) as

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev                                                   (36.17)

The error in flow rate dis determined from Eq. (36.17) and the flow rate in each path of a loop is then altered according to Eq. (36.11).


The Hardy-Cross method can also be applied to a hydraulic circuit containing a pump or a turbine. The pressure equation (Eq. (36.10)) is only modified in consideration of a head source (pump) or a head sink (turbine) as 

Flow Through Branched Pipes (Part - 2) Civil Engineering (CE) Notes | EduRev                                       (36.18)

 

where ΔH  is the head delivered by a source in the circuit. Therefore, the value of  ΔH  to be substituted in Eq. (36.18) will be positive for a pump and negative for a turbine.

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