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.
n this case, rate of flow Q remains same in each pipe. Hence,
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
H1 - H2 = hf
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
Friction loss Loss due to
entry to pipe B
in pipe AFriction loss
in pipe B
The subscripts A, B and C refer to the quantities in pipe A, B and C respectively. Cc is the coefficient of contraction.
Velocities VA, VB and VC in Eq. (36.1) are substituted from Eq. (36.2), and we get
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.
Pipes In Parallel
Q = QA + QB (36.5)
where, Q is the total flow rate and QA And QB are the flow rates through pipes A and Brespectively.
Equating the above two expressions, we get -
Equations (36.5) and (36.6) give -
The flow system can be described by an equivalent electrical circuit as shown in Fig. 36.4
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:
The principles 3 and 4 can be written analytically as
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
Then according to Eq. (36.10)
in a loop (36.13a)
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
Where dh (= h - h' ) is the error in pressure equation for a path. Again from Eq. (36.12a), we can write
Substituting the value of dh from Eq. (36.15) in Eq. (36.14) we have
Considering the error dQ to be the same for all hydraulic paths in a loop, we can write
he Eq. (36.16) can be written with the help of Eqs (36.12a) and (36.12b) as
The error in flow rate dQ is 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
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.