Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

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Flow Through Branched Pipes
 

In several practical situations, flow takes place under a given head through different pipes jointed together either in series or in parallel or in a combination of both of them.

Pipes in Series

  • If a pipeline is joined to one or more pipelines in continuation, these are said to constitute pipes in series. A typical example of pipes in series is shown in Fig. 36.1. Here three pipes A, B and are joined in series.

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

In this case, rate of flow remains same in each pipe. Hence,

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

  • If the total head available at Sec. 1 (at the inlet to pipe A) is H1 which is greater than H2, the total head at Sec. 2 (at the exit of pipe C), then the flow takes place from 1 to 2 through the system of pipelines in series. 
  • Application of Bernoulli's equation between Secs.1 and 2 gives

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

where, hf is the loss of head due to the flow from 1 to 2. Recognizing the minor and major losses associated with the flow, hf can be written as

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev        (36.1)

The subscripts A, B and refer to the quantities in pipe A, B and C respectively. Cc is the coefficient of contraction.

  • The flow rate Q satisfies the equation

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev      (36.2)

Velocities VA, VB and VC in Eq. (36.1) are substituted from Eq. (36.2), and we get

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev      (36.4)

Equation (36.4) states that the total flow resistance is equal to the sum of the different resistance components. Therefore, the above problem can be described by an equivalent electrical network system as shown in Fig. 36.2

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev


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

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev                    (36.5)

where, is the total flow rate and Qand QB 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 - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

Equating the above two expressions, we get -

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev             (36.6)

 

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev            (36.7)

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering 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 - Applications of Viscous Flows Through Pipes Mechanical Engineering 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 - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRevat a node (Junction)                (36.9)

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev in a loop                        (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 - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

Then according to Eq. (36.10)

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev      (36.13a)

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev      (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 - Applications of Viscous Flows Through Pipes Mechanical Engineering 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 - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev     (36.15)

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

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

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

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev     (36.16)

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

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering 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 - Applications of Viscous Flows Through Pipes Mechanical Engineering 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.

 

Solve a Problem

Problem

A pipe network with two loops is shown in Fig. 36.5. Determine the flow in each pipe for an inflow of 5 units at the junction A and outflows of 2.0 units and 3.0 units at junctions and respectively. The resistance for different pipes are shown in the figure.

 

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

 

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

Flow direction is assumed positive clockwise for both the loops ABD and BCD . The iterative solutions based on Hardy-Cross method has been made. The five trials have been made and the results of each trial is shown in Fig. 36.5; for each trial, d is calculated from Eq. (36.17). After fifth trial, the error d is so small that it changes the flow only in the third place of decimal. Hence the calculation has not been continued beyond the fifth trial.

First Trial

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev
 

Second Trial

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

Third Trial

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

Fourth Trial

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

Fifth Trial

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

Flow Through Branched Pipes - Applications of Viscous Flows Through Pipes Mechanical Engineering Notes | EduRev

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