Three-reservoir systems
Department of Civil and Environmental Engineering...New Mexico State University
Contents
1 Learning objectives
- Understand flow behavior in multi-reservoir pipe systems
- Apply energy and continuity principles at a junction
- Solve for unknown junction head and pipe discharges
- Interpret flow directions and physical meaning of results
2 System description
- Three reservoirs with known water surface elevations
- Pipes connect each reservoir to a common junction
- Flow is driven by head differences
- Unknowns:
- Junction head $H_J$
- Discharges $Q_1$, $Q_2$, $Q_3$
3 Physical picture
- Water flows from higher head to lower head
- Junction acts as a mixing point
- Key idea:
- The system adjusts so that energy losses match head differences
4 Governing equations
- Continuity at junction: \[ Q_1 + Q_2 + Q_3 = 0 \]
- Energy equation for each pipe: \[ H_i - H_J = h_{L,i} \]
- Head loss model (Darcy–Weisbach): \[ h_{L,i} = f_i \frac{L_i}{D_i} \frac{V_i^2}{2g} \]
5 Expressing discharge
- Using $Q = A V$: \[ h_{L,i} = K_i Q_i^2 \]
- Where: \[ K_i = \frac{8 f_i L_i}{g \pi^2 D_i^5} \]
- Then: \[ Q_i = \pm \sqrt{\frac{H_i - H_J}{K_i}} \]
6 Sign convention
- Assume flow direction initially
- If computed $Q_i < 0$:
- Actual flow is opposite
- This is not an error
- It is a solution insight
7 Solution strategy
- Unknown: $H_J$
- Steps:
- Assume $H_J$
- Compute $Q_i$ for each pipe
- Check continuity: $Q_1 + Q_2 + Q_3$
- Adjust $H_J$
- Repeat until balanced
8 Graphical interpretation
- Plot $Q_i(H_J)$ for each pipe
- Sum of flows should be zero
- Intersection point gives:
- Correct junction head
- Consistent flow distribution
9 Physical insight
- If $H_J$ is too high:
- Outflows dominate
- If $H_J$ is too low:
- Inflows dominate
- Correct $H_J$ balances inflow and outflow
10 Special cases
- Equal pipe resistances:
- Symmetric flow behavior
- One reservoir much higher:
- Dominant supply source
- One pipe very large:
- Low resistance, carries most flow
11 Common mistakes
- Ignoring sign of $Q$
- Using wrong head difference direction
- Forgetting continuity at junction
- Mixing units in $K_i$
12 Example problem (conceptual)
- Given: \[ H_1 > H_2 > H_3 \]
- Questions:
- Which reservoirs supply flow?
- Which receives flow?
- Can flow reverse in one pipe?
13 Key points
- Three-reservoir systems are nonlinear
- Junction head is the central unknown
- Continuity + energy → complete solution
- Negative discharge = reversed flow
- Iteration is essential