Sizing a multi-branch pumping network solved by Newton-Raphson

The branched-network problem

When a single pump feeds several demand points at once, a problem arises that sizing an isolated segment never captures: the branch flow rates are interdependent. Opening or closing one branch changes the pressure at the take-off node, and that change redistributes flow among all the others. There is no such thing as “the flow of branch 3” computable in isolation — there is an equilibrium state of the entire network that must be found all at once.

The common intuition — split the pump flow evenly, or in proportion to the pipe area — is wrong. Flow distributes according to the hydraulic resistance of each path: short, wide, low-elevation branches carry more; long, narrow, high ones carry less. What is equal among them is not the flow rate, it is the pressure at the point where they branch off.

The two physical principles

The solution rests on two classic pillars, the same ones used in water-supply network analysis:

• Mass conservation at the nodes. At every internal node, the sum of the inflows equals the sum of the outflows (Σ Q_in = Σ Q_out). This is the hydraulic form of Kirchhoff’s first law. An incompressible fluid does not accumulate.
• Energy conservation along the segments. Along each pipe, the piezometric head drops by the Darcy-Weisbach head loss: h_f = f·(L/D)·(v²/2g) + ΣK·(v²/2g). The friction factor f follows the Colebrook-White method (Serghides in the turbulent regime, laminar below Re 2000, cubic transition between them).

The catch complicates the problem: because h_f ∝ Q², the equations are nonlinear. There is no closed-form formula for the flow split. That is why an iterative method is required.

How Newton-Raphson solves the network

The solver assembles an unknown vector holding the flow rate of each segment and a residual vector F(Q) that gathers all the equations that must hold at equilibrium:

1. Continuity residuals — one per internal node: Σ Q_in − Σ Q_out. When it is zero, mass is conserved there.
2. Pressure residuals — one per leaf branch: H_node(Q) − (z + P_min). When it is zero, the consumer receives exactly the required pressure.

At each iteration, the algorithm:

1. Propagates head from the pump: H_out = z_level + H_pump(Q_tot) − h_suction(Q_tot), and descends the network by subtracting the loss of each segment down to every consumer. The pump curve H_pump(Q) comes from a 3-point Lagrange fit of the catalog.
2. Evaluates the residuals F(Q) with the current flow rates.
3. Assembles the Jacobian J = ∂F/∂Q by central finite difference, perturbing each flow rate up and down.
4. Solves the Newton step J·ΔQ = −F by Gaussian elimination with partial pivoting and updates the flow rates.
5. Damps the step (line search): if the full step does not reduce the residual norm, the factor is shrunk until it does — ensuring robustness when the initial guess is far off.

The iteration stops when ‖F‖ < 1e-7, which usually happens in 5 to 15 steps. Because Q_total is one of the unknowns, the pump slides along its own curve during convergence: the actual operating point is not imposed, it emerges from the joint solution of pump and network.

Unreachable branches and balancing

Not every branch receives flow. If a consumer sits too high or too far for the available head, the solution would tend toward a negative flow rate — physically impossible. The solver detects these branches (flow dropping to near zero), marks them as unreachable, and recomputes the equilibrium with the remaining branches, instead of producing a meaningless result. In practice this signals that the pump is undersized or that the point needs its own booster.

At the other extreme is the favored branch, which “steals” flow from the rest. The fix is hydraulic balancing: insert a deliberate local loss — an orifice plate or balancing valve — in the favored branch, sized (via its K) to dissipate the excess pressure and push flow back to the disadvantaged branches. The result is a network where each point receives the design flow, not whatever the accident of geometry would deliver.

Practical design considerations

• The bottleneck sets the TDH. The pump’s total head is governed by the most unfavorable branch (high, long, narrow), not by the average. Size the pump to guarantee minimum pressure there.
• Reference velocities: keep the main line between 1.5 and 3.0 m/s and the branches between 1.0 and 2.5 m/s — this controls head loss, noise, and water hammer.
• The main-line loss is shared: it depends on the sum of the branch flows. That is why you cannot size branch by branch independently.
• Elevation matters as much as length: two branches with identical piping but different elevations receive different flow rates — the geometric energy enters the balance directly.
• Re-evaluate when the configuration changes: switching branches on/off changes Q_total, shifts the pump operating point, and redistributes everything. The calculation must be redone, not scaled.

In short: sizing a multi-branch network means finding the equilibrium state in which mass is conserved at every node and energy is conserved in every pipe, simultaneously. Newton-Raphson delivers that state — the flow of each branch, the pressure at each node, and the pump’s actual operating point — and reveals where balancing is needed so the design does what it promises.