Hardy Cross Principles for Pipe Networks In the analysis of a closed network of pipes (e.g., using Hardy Cross or equivalent methods), which of the following statements is fundamentally correct about the energy balance around a loop and continuity at junctions?

Introduction / Context:
Closed water-distribution or pipeline networks are commonly analyzed using iterative methods such as the Hardy Cross technique. Two physical laws govern these systems: mass conservation at junctions and energy conservation around closed loops. Recognizing which condition applies where is essential to set up correct equations and converge to the true distribution of flows and heads.

Given Data / Assumptions:

• Incompressible steady flow of water in pipes.
• Head loss in each pipe determined by a formula such as Darcy–Weisbach or Hazen–Williams.
• Closed loops (circuits) and multiple junctions exist in the network.

Concept / Approach:

Two balances are enforced: (1) Continuity at each junction: the algebraic sum of discharges meeting at a node equals zero (inflows = outflows). (2) Energy conservation around any independent loop: the algebraic sum of head losses (pressure head + datum head drops; velocity head changes included in the loss terms) must equal zero when traced around a closed circuit back to the starting point. This loop condition is the cornerstone of the correction step in Hardy Cross iterations.

Step-by-Step Solution:

Verification / Alternative check:

When a converged solution is obtained, checking any additional loop not explicitly used in the iteration will also yield Σh_loss ≈ 0, confirming energy balance globally. Junction mass balance will show negligible residuals.

Why Other Options Are Wrong:

• The algebraic sum of discharges around a loop need not be zero; that condition applies at a junction, not around a circuit.
• Assuming hydraulic grade line elevations at junctions is only an initial guess, not a governing law.
• Replacing elementary circuits by equivalent single pipes is not part of standard network analysis methods and would distort hydraulic behavior.

Common Pitfalls:

Confusing node continuity with loop energy equations; inconsistent sign conventions for head losses; mixing units or loss formulas within the same loop.

Final Answer:

The algebraic sum of (pressure head + datum head) drops, i.e., total head losses, around each closed circuit is zero.