1. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ introduction ]
It is easy for us to imagine that few pipe systems are as simple as a single pipe connecting two points
From water distribution networks to the veins and arteries in our bodies, pipe networks can evolve into complex architectures
In the simplest analysis of pipe networks, we first typically assess if pipes are connected in parallel or in series
Fluid Mechanics

2. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ introduction ]
parallel multiple pipe connection
series multiple pipe connection
Fluid Mechanics

3. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ introduction ]
As we have discussed before, an interesting analogy exists between fluid and electrical circuits
We see in Ohm’s law, that voltage, e, is a product of the electric current, i, and the conducting resistance, R
- [1]
In a fluid circuit there exists a balance between the pressure drop, Dp, the flow or velocity, Q or V, and the resistance given in terms of the friction factor and minor losses, f and KL
- [2]
Fluid Mechanics

4. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ introduction ] or
- [3]
where R is a measure of resistance to the flow, (proportional to f)
Where the electrical analogy starts to break down is when we consider that Ohm’s law is linear, and the fluid equations are generally non-linear, (that is, doubling the pressure drop does not double the flow rate unless we have laminar flow)  so we appreciate that the analogy is a loose one
Fluid Mechanics

5. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ introduction ]
Referring back to the first of our two basic multipipe architectures:
parallel multiple pipe connection
In parallel multipipe connections, the fluid travelling from A to B may take any path to get there, flows are additive in these networks, and the headloss experienced by a fluid particle taking any path from A to B is the same i.e.:
- [4]
- [5]
Fluid Mechanics

6. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ introduction ]
This situation is different for series connections
series multiple pipe connection
In series multipipe connections, the fluid travelling from A to B may take travels through the same pipes, thus the flow (not necessarily the velocity) is the same in each, and the headloss experienced is summed through each pipe i.e.:
- [6]
- [7]
Fluid Mechanics

7. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ introduction ]
A third architecture exists, “the pipe loop”
multiple pipe loop
In the loop Q1=Q2+Q3 and hL2=hL3
Fluid Mechanics

8. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ the three reservoir problem: an example ]
Often for pipe networks that appear simple, they are not always that straight forward or intuitive to assess
For example, in the figure below, it is obvious that water must flow from A to B and C, but how will the flow behave between B and C (this will be dependent on the elevations of B and C as well as pipe roughness, length, and diameter)
three reservoir system
Fluid Mechanics

9. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ the three reservoir problem: an example ]
GIVEN:
f = 0.02, for all pipes
assume no minor losses
REQD:
Determine the flowrate into or out of each reservoir
Fluid Mechanics

10. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ the three reservoir problem: an example ]
SOLU:
f=0.02, for all pipes
assume no minor losses
1. We start by attempting to assign flow directions and then writing the energy equation between A, B, and C
when we look at pipe (2) we can not be sure of its flow direction, so we assume it flows from the reservoir, then we can say that (from continuity):
- [E1]
Fluid Mechanics

11. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ the three reservoir problem: an example ]
SOLU:
f=0.02, for all pipes
assume no minor losses
2. Write the energy equation from A to C (pipes (1) and (3))
- [E2]
Fluid Mechanics

12. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ the three reservoir problem: an example ]
SOLU:
f=0.02, for all pipes
assume no minor losses
simplifying, we can write
- [E3]
Fluid Mechanics

13. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ the three reservoir problem: an example ]
SOLU:
f=0.02, for all pipes
assume no minor losses
plugging in some numbers we then arrive at
- [E4]
- [E5]
Fluid Mechanics

14. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ the three reservoir problem: an example ]
SOLU:
f=0.02, for all pipes
assume no minor losses
3. Now let’s write it from B to C (pipes (2) and (3))
- [E6]
- [E7]
Fluid Mechanics

15. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ the three reservoir problem: an example ]
SOLU:
f=0.02, for all pipes
assume no minor losses
plugging in some numbers we then arrive at
- [E8]
Fluid Mechanics

16. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ the three reservoir problem: an example ]
SOLU:
f=0.02, for all pipes
assume no minor losses
4. Attempt to solve [E8], [E5], and [E1] simultaneously
we discover that these two equations yield no positive, real velocities
- [E8]
we therefore conclude that we assumed the wrong flow direction for pipe (2)
- [E5]
Fluid Mechanics

17. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ the three reservoir problem: an example ]
SOLU:
f=0.02, for all pipes
assume no minor losses
5. Now, let us change the assumed direction of flow in (2), we now say
- [E9]
Fluid Mechanics

18. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ the three reservoir problem: an example ]
SOLU:
f=0.02, for all pipes
assume no minor losses
6. We write the energy equation between A and B, and A and C
- [E10]
- [E11]
Fluid Mechanics

19. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ the three reservoir problem: an example ]
SOLU:
f=0.02, for all pipes
assume no minor losses
these reduce to
- [E12]
- [E13]
Fluid Mechanics

20. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ the three reservoir problem: an example ]
SOLU:
f=0.02, for all pipes
assume no minor losses
we subtract [E12] from [E13] we acquire
- [E14]
so we can rewrite [E12] as
- [E15]
Fluid Mechanics

21. VISCOUS FLOW IN CONDUITS: MULTIPLE PIPES
[ the three reservoir problem: an example ]
SOLU:
f=0.02, for all pipes
assume no minor losses
solving for V2 we acquire suitable roots:
which yields the following flow distribution and direction
- [ans]
Fluid Mechanics