Pipe Networks

Pipe networks represent interconnected piping systems where flow divides and recombines through multiple paths. HVAC hydronic systems typically consist of complex networks requiring systematic analysis to determine flow distribution, pressure losses, and pump requirements. Network analysis becomes essential for system balancing, troubleshooting, and design verification.

Physical Principles

Conservation Laws

Network analysis relies on fundamental conservation principles that govern all fluid systems:

Mass Conservation (Continuity) At any junction node, the sum of mass flow rates entering equals the sum leaving:

∑ṁ_in = ∑ṁ_out

For incompressible fluids (water in HVAC systems):

∑Q_in = ∑Q_out

This principle applies at every junction, mixing tee, and branch point in the network.

Energy Conservation

Around any closed loop, the algebraic sum of pressure changes must equal zero:

∑ΔP_loop = 0

This fundamental principle forms the basis of the Hardy-Cross method and all loop-based network analysis techniques.

Momentum Conservation

Pressure losses in pipe segments result from friction and local resistances:

ΔP = f(Q, L, D, ε, ρ, μ)

The relationship between flow rate and pressure drop is nonlinear, typically following:

ΔP = K·Q^n

where n ≈ 1.85 for turbulent flow in water systems.

Network Configuration Types

Series Pipe Systems

In series configurations, the total flow passes through each component sequentially.

Characteristics:

• Constant flow rate through all elements: Q₁ = Q₂ = Q₃ = … = Q_total
• Pressure drops are additive: ΔP_total = ΔP₁ + ΔP₂ + ΔP₃ + …
• Equivalent resistance: K_eq = K₁ + K₂ + K₃ + …

Analysis Approach:

1. Calculate individual pressure drops using Darcy-Weisbach equation
2. Sum all losses including pipe friction and fittings
3. Determine total system resistance curve

Series connections occur in:

• Main distribution piping between pump and first branch
• Terminal units with control valves and coils in series
• Primary-secondary decoupler piping

Parallel Pipe Systems

Parallel configurations provide multiple flow paths between common nodes.

Characteristics:

• Equal pressure drop across all parallel paths: ΔP₁ = ΔP₂ = ΔP₃ = … = ΔP_common
• Flow rates are additive: Q_total = Q₁ + Q₂ + Q₃ + …
• Equivalent resistance: 1/√K_eq = 1/√K₁ + 1/√K₂ + 1/√K₃ + …

Flow Distribution: Flow divides inversely with resistance. For two parallel pipes:

Q₁/Q₂ = (K₂/K₁)^(1/n)

where n ≈ 1.85 for turbulent flow.

Analysis Procedure:

1. Assume equal pressure drop across all branches
2. Calculate individual flow rates based on resistances
3. Verify total flow equals sum of branch flows
4. Adjust if necessary using iterative methods

Parallel configurations appear in:

• Multiple risers in high-rise buildings
• Redundant mains for reliability
• Distribution to multiple zones

Complex Networks

Most HVAC systems contain combinations of series and parallel elements forming complex networks with multiple loops and branches.

Network Complexity Factors:

• Number of loops (closed circuits in the piping)
• Number of nodes (junctions where pipes meet)
• Presence of fixed-pressure nodes (expansion tanks, pressure relief points)
• Variable flow devices (control valves, balancing valves)

Hardy-Cross Method

The Hardy-Cross method provides an iterative solution for flow distribution in looped networks. Developed in 1936, it remains a fundamental technique for manual calculations and forms the basis for many computer algorithms.

Method Fundamentals

Basic Principle: Start with assumed flow rates that satisfy continuity at all nodes, then iteratively adjust flows to satisfy energy conservation around all loops.

Sign Convention:

• Clockwise flows in a loop: positive
• Counterclockwise flows: negative
• Pressure drops in flow direction: positive
• Pressure gains opposite to flow: negative

Solution Algorithm

Step 1: Initial Flow Assumption

Assume initial flow rates in each pipe that satisfy continuity (∑Q = 0) at every node. Reasonable initial guesses accelerate convergence:

• Estimate based on load distribution
• Use preliminary hydraulic calculations
• Assume equal division at branches

Step 2: Loop Correction Calculation

For each loop, calculate the flow correction:

ΔQ = -∑(K·Q^n) / [n·∑(K·|Q|^(n-1))]

where:

• ΔQ = flow correction to be applied to all pipes in the loop
• K = resistance coefficient for each pipe
• Q = current flow rate estimate
• n = flow exponent (typically 1.85-2.0)

Step 3: Flow Adjustment

Apply corrections to pipe flows:

• Add ΔQ to all pipes in the loop (accounting for flow direction)
• For pipes common to multiple loops, algebraically sum corrections from all loops
• Update flow rates for next iteration

Step 4: Convergence Check

Continue iterations until:

|∑(K·Q^n)|_loop < tolerance

Typical tolerance: 0.1-1.0 Pa (0.00001-0.0001 psi) for HVAC systems.

Practical Implementation

Resistance Coefficient Determination:

Express pressure drop as K·Q^n:

For Darcy-Weisbach friction: K = (f·L·8)/(π²·g·D⁵)

For Hazen-Williams: K = (10.67·L)/(C^1.85·D^4.87)

Units must be consistent throughout analysis.

Common Pipe Treatment:

When pipes appear in multiple loops:

1. Apply correction from each loop containing the pipe
2. Respect sign conventions based on flow direction relative to each loop
3. Net correction = algebraic sum of all loop corrections

Pseudo-Loops for Branch Networks:

Create artificial loops by:

• Connecting terminal branches to a common reference node
• Assigning zero resistance to fictitious pipe
• Treating as regular loop in analysis

| Iteration | Loop A ΔQ (gpm) | Loop B ΔQ (gpm) | Max |∑ΔP| (Pa) | Status | |———–|—————–|—————–|—————|——–| | 1 | -12.4 | +8.7 | 3,250 | Continue | | 2 | -3.8 | +2.1 | 420 | Continue | | 3 | -0.9 | +0.5 | 75 | Continue | | 4 | -0.2 | +0.1 | 12 | Converged |

Nodal Analysis Method

Nodal analysis solves for pressures at network nodes rather than flows in pipes. This approach proves more efficient for large networks and integrates readily with computer solution techniques.

Methodology

Node Pressure Variables:

Assign pressure as the primary unknown at each node. Fixed-pressure nodes (pump discharge, expansion tank) serve as boundary conditions.

Flow-Pressure Relationships:

Express pipe flow as a function of nodal pressures:

Q_ij = √[(P_i - P_j)/K_ij]

where:

• Q_ij = flow from node i to node j
• P_i, P_j = pressures at nodes i and j
• K_ij = resistance of pipe connecting nodes

Nodal Equations:

At each free node, apply continuity:

∑Q_in - ∑Q_out + Q_external = 0

This generates a system of nonlinear equations equal to the number of unknown node pressures.

Solution Techniques

Newton-Raphson Iteration:

Linearize the nonlinear system and solve iteratively:

[J]·{ΔP} = -{F}

where:

• [J] = Jacobian matrix of partial derivatives
• {ΔP} = pressure corrections
• {F} = residual vector (continuity errors)

Convergence Criteria:

Continue until maximum residual < tolerance:

max|F_i| < ε

Typical tolerance: 0.01 gpm or 0.001 L/s at any node.

Advantages Over Hardy-Cross:

• Fewer equations for typical HVAC networks
• Easier incorporation of pumps and control valves
• Direct pressure calculations at critical points
• Better numerical stability for large systems
• Natural handling of pressure-dependent demands

System Curves and Operating Points

Network System Curve

The system curve represents total pressure drop vs. flow rate for the entire network at a specific valve/damper configuration.

Development Procedure:

1. Select a range of flow rates from 0% to 150% of design
2. For each flow rate, solve network for flow distribution
3. Calculate total pressure drop from pump discharge to suction
4. Plot ΔP_system vs. Q_total

Mathematical Form:

System curves typically follow:

ΔP_system = ΔP_static + K_system·Q²

where:

• ΔP_static = static head (elevation change, system pressure)
• K_system = effective system resistance coefficient

Factors Affecting System Curve:

• Valve positions (balancing and control valves)
• Coil fouling and filter loading
• Pipe roughness degradation over time
• Number of parallel devices in operation

Pump-System Interaction

Operating Point Determination:

The operating point occurs where pump curve intersects system curve:

ΔP_pump(Q) = ΔP_system(Q)

This determines actual flow rate and pressure delivered.

Multiple Pump Configurations:

For pumps in parallel:

• Add flow rates at each pressure
• System operates at higher flow, same pressure

For pumps in series:

• Add pressures at each flow rate
• System operates at higher pressure, same flow

Variable Flow Operation

System Curve Shift:

As control valves modulate:

• System resistance increases
• System curve becomes steeper
• Operating point moves along pump curve

Pump Curve Modification:

Variable speed pumping shifts pump curve:

Q_new = Q_old·(N_new/N_old) ΔP_new = ΔP_old·(N_new/N_old)²

Target Operating Line:

Ideal variable flow operation follows a design target:

ΔP_target = ΔP_static + K_design·Q²

Differential pressure setpoint reset maintains efficient operation across load range.

Balancing and Valve Coefficient Analysis

Hydronic Balancing Fundamentals

Objective:

Distribute flow to match design requirements within acceptable tolerance (±5-10%).

Balancing Valve Sizing:

Required valve authority α determines necessary valve Cv:

α = ΔP_valve,wide-open / (ΔP_circuit + ΔP_valve,wide-open)

Target authority: α = 0.3-0.5 for stable control.

Valve coefficient:

Cv = Q / √(ΔP_valve)

where Q in gpm, ΔP in psi.

Proportional Balancing Method

Procedure:

1. Measure flow at all terminals with balancing valves wide open
2. Calculate percent of design flow at each terminal
3. Identify index circuit (lowest % flow)
4. Leave index circuit valve wide open
5. Throttle other circuits proportionally to achieve design flows

Balancing Sequence:

Work from index circuit outward:

• Balance terminal units on each branch
• Balance branches on each riser
• Balance risers on main distribution

Temperature Balancing

For systems without flow meters, use temperature measurements:

Q = ṁ·c_p·ΔT = (ṁ_design·c_p·ΔT_design)

At design conditions:

ΔT_actual/ΔT_design = Q_design/Q_actual

Adjustment:

• Low ΔT indicates excess flow → throttle balancing valve
• High ΔT indicates insufficient flow → open valve or investigate restrictions

Network Analysis Software Tools

Commercial Solutions

Modern HVAC design relies on computational tools for complex network analysis:

Capabilities:

• Automatic network topology recognition
• Simultaneous solution of thousands of nodes
• Integration with CAD/BIM platforms
• Transient analysis capabilities
• Optimization algorithms for pump selection

Common Packages:

• PIPE-FLO (Engineered Software)
• AFT Fathom (Applied Flow Technology)
• EPANET (EPA, public domain)
• Trane TRACE Piping
• Elite Software Chvac

Solution Algorithms

Gradient Methods:

Minimize objective function representing continuity and energy errors:

F = ∑(continuity errors)² + ∑(loop energy errors)²

Use gradient-based optimization to find minimum.

Sparse Matrix Solvers:

Network equations form sparse matrices (most entries zero). Specialized algorithms:

• Gauss-Seidel iteration
• Successive over-relaxation (SOR)
• Conjugate gradient methods
• Direct sparse matrix factorization

Exploit matrix structure for computational efficiency.

Design Considerations

Pipe Sizing Strategy

Velocity Limits:

Standard HVAC practice limits velocities to control noise and erosion:

Pressure Drop Guidelines:

ASHRAE recommendations for design pressure drop rates:

• Distribution mains: 1-4 ft H₂O/100 ft (100-400 Pa/m)
• Branch piping: 2.5-4 ft H₂O/100 ft (250-400 Pa/m)
• Risers: 0.5-1.5 ft H₂O/100 ft (50-150 Pa/m)

Higher pressure drops reduce first cost but increase pumping energy.

Diversity and Safety Factors

Diversity Factors:

Not all terminals operate at full load simultaneously. Apply diversity factors to reduce pipe sizing in mains:

Q_main = F_div · ∑Q_terminal

Typical diversity factors:

• Office buildings: 0.7-0.8
• Hotels: 0.6-0.7
• Hospitals: 0.9-1.0 (conservative)

Safety Factors:

Account for uncertainties in load calculations and future modifications:

• Friction factor: use aged pipe roughness values
• Load calculations: 10-15% safety margin for critical applications
• Pump head: 5-10% margin for future fouling

Do not stack multiple safety factors; results in oversizing and control problems.

Pressure Class Selection

Piping Pressure Ratings:

Select pipe and fitting pressure classes based on maximum operating pressure plus safety margin:

P_rating ≥ F_s · (P_static + P_pump)

where F_s = safety factor (typically 1.5-2.0).

ASHRAE and Code References

Applicable Standards

ASHRAE Handbooks:

• ASHRAE Handbook—Fundamentals, Chapter 22: Pipe Sizing
• ASHRAE Handbook—HVAC Systems and Equipment, Chapter 13: Hydronic Heating and Cooling
• ASHRAE Handbook—HVAC Applications, Chapter 47: Design and Application of Controls

ASHRAE Standards:

• ASHRAE 90.1: Energy Standard for Buildings (pump power limits)
• ASHRAE 189.1: Standard for High-Performance Green Buildings

Industry Standards

ASME Codes:

• ASME B31.9: Building Services Piping
• ASME B16.5: Pipe Flanges and Flanged Fittings

Other References:

• SMACNA HVAC Systems—Duct Design
• AHRI Standard 430: Performance Rating of Central Station Air-Handling Units
• Hydraulic Institute Standards (pump application)

Authority Having Jurisdiction Requirements

Check local amendments to:

• International Mechanical Code (IMC)
• International Plumbing Code (IPC)
• NFPA 54: National Fuel Gas Code (for hot water boiler piping)

Local codes may impose stricter requirements for:

• Seismic bracing and supports
• Pipe pressure ratings
• Backflow prevention
• Water treatment

Best Practices

Design Phase

1. Establish Clear Hydraulic Boundaries: Define system limits, interface points, and boundary conditions before detailed analysis.

2. Minimize Fittings and Valves: Each component adds resistance; optimize routing to reduce losses.

3. Provide Adequate Valve Authority: Size control valves for α = 0.3-0.5 to ensure stable modulation.

4. Design for Balancing: Include balancing valves at strategic locations; plan measurement points.

5. Consider Future Expansion: Provide spare pump capacity and oversized mains in areas likely to expand.

6. Avoid Common Piping: Use primary-secondary or hydraulic decoupling to separate variable and constant flow systems.

Analysis Phase

1. Verify Model Inputs: Check all pipe lengths, sizes, fittings, and elevation changes against drawings.

2. Use Conservative Assumptions: Apply aged pipe roughness; assume partially loaded filters and coils.

3. Perform Sensitivity Analysis: Vary key parameters (pipe roughness, load diversity) to assess impact.

4. Check Multiple Operating Scenarios: Analyze full load, part load, and extreme conditions.

5. Validate Against Rules of Thumb: Compare results to experience-based velocity and pressure drop guidelines.

Construction and Commissioning

1. Document As-Built Conditions: Record actual pipe routes, fittings, and valve locations.

2. Pre-Balance Checkout: Verify all valves operational, strainers clean, air purged before balancing.

3. Progressive Balancing: Balance in stages from terminals toward pump.

4. Performance Testing: Measure actual pump curves under operating conditions; compare to predicted system curves.

5. Create O&M Documentation: Provide marked-up drawings showing balancing valve settings, flow measurements, and adjustment procedures.

Advanced Topics

Transient Analysis

Water hammer and surge analysis for:

• Rapid valve closure
• Pump startup and shutdown
• Power failure events

Critical for systems with:

• Long pipe runs (>300 ft)
• High velocities (>6 ft/s)
• Quick-closing valves

Optimization

Minimize lifecycle cost:

LCC = C_first + PW(C_energy + C_maintenance)

Optimize:

• Pipe sizes (larger pipes = higher first cost, lower pumping cost)
• Pump selection (efficiency, speed control)
• Network configuration (series vs. parallel, primary-secondary)

Cavitation Prevention

Maintain adequate net positive suction head (NPSH):

NPSH_available = P_suction/γ + V²/(2g) - P_vapor/γ

Require: NPSH_available ≥ NPSH_required + margin (3-5 ft typical)

Design suction piping for minimal losses; avoid high points and air pockets.

Related Topics:

• Pump Selection and Application
• Control Valve Sizing and Authority
• Primary-Secondary Pumping
• Variable Flow System Design
• Water Treatment and Corrosion Control