(3.38) and (3.39). The next chapters will present also relations for other elements, such as pumps, motors, and linear actuators.2 Flow lawThe flow law applies at any junction of pipes, and it was already presented as a direct consequence of the conservation of mass principle:(3.41) Equation (3.41) can be seen as the equivalent of the Kirchhoff's current law in the electric domain. Essentially, this equation states that the sum of the flow rates entering a junction has to be equal to the sum of the flow rates exiting it (Figure 3.16a).
3 Pressure lawThe pressure law can be seen as the equivalent of the Kirchhoff's voltage law in the electric domain. It states that the overall pressure drop around any closed circuit has to be null:(3.42)
Figure 3.16 Graphical representation of flow law (a) and pressure law (b).
Example 3.1 Series and parallel hydraulic connections
The pressure drop–flow rate relation for three different pipes is known to be linear, as shown in the figure below. Find the pressure drop–flow rate relation for different configurations of the three pipes: (a) series; (b) parallel; and (c) series–parallel.
Given:
The linear characteristic of three pipe sections:
ΔpA = RA · QA; ΔpB = RB · QB; ΔpC = RC · QC
Find:
The equivalent hydraulic resistance for the three cases:
Δpseries = Rseries · Q; Δpparallel = Rparallel · Q; Δpseries/parallel = RC · Q
Solution:
Case (a) series
This case can be solved by considering the quantities shown in the figure below:
Therefore,
which means
Case (b) parallel
The approach is similar to case (a). With reference to the figure below,
Considering the flow law,
which gives
or also
which means
Case (c) series–parallel
In this case, the reference schematic is shown below:
Pipe A is in series with B and C, which are in parallel. Using the relations derived above:
which means
