CELE Hydraulics & Fluid Mechanics — Flow in PipesConcept Map

Concept

Flow Regime Classification

Sub Concepts

• Reynolds Number (Re = vD/ν)
• Laminar Flow (Re < 2000)
• Transitional Flow (2000 < Re < 4000)
• Turbulent Flow (Re > 4000)

Relationship To Central

Determines which friction factor equations apply; foundation for all pipe flow analysis

Concept

Major Head Losses (Friction Losses)

Sub Concepts

• Darcy-Weisbach Equation: hf = f(L/D)(v²/2g)
• Friction Factor f (Moody Chart; depends on Re and ε/D)
• Laminar Friction Factor: f = 64/Re
• Manning Equation (alternative for full pipes)
• Hazen-Williams Equation (common for water pipes)

Relationship To Central

Primary energy dissipation along pipe length; directly affects pump power and pipe sizing

Concept

Minor Head Losses (Local Losses)

Sub Concepts

• Loss Coefficient K: hm = K(v²/2g)
• Sharp Entrance (K ≈ 0.5)
• Exit Loss (K = 1.0)
• Elbow Loss (K ≈ 0.9 for 90°)
• Valve Loss (K varies)
• Sudden Expansion: h = (v₁ - v₂)²/2g
• Sudden Contraction

Relationship To Central

Energy loss at fittings and transitions; critical for accurate total head calculations

Concept

Pipe System Configuration

Sub Concepts

• Pipes in Series: Same Q, head losses add (ΣhL = h₁ + h₂ + ...)
• Pipes in Parallel: Same head loss, flows add (Q = Q₁ + Q₂ + ...)
• Branched Networks: Multiple loops, Hardy Cross method
• Equivalent Pipe Concept

Relationship To Central

Determines how head losses and flows combine; critical for network design

Concept

Friction Factor Determination

Sub Concepts

• Moody Diagram: f vs Re and ε/D
• Colebrook-White Equation (implicit)
• Explicit approximations (Swamee-Jain, etc.)
• Pipe Absolute Roughness ε values
• Relative Roughness ε/D

Relationship To Central

Essential input for Darcy-Weisbach equation; varies with flow regime and pipe roughness

Concept

Velocity and Discharge Relationships

Sub Concepts

• Continuity Equation: Q = A·v
• Cross-sectional Area A = πD²/4
• Velocity Head: hv = v²/2g
• Flow Rate Q in m³/s

Relationship To Central

Links geometry to flow behavior; fundamental to all calculations

Concept

Engineering Applications & Design

Sub Concepts

• Pump Sizing and Head Calculations
• Pipe Diameter Selection
• Water Supply Systems (gravity and pressure)
• Sewage and Storm Drainage Design
• Industrial Piping Networks

Relationship To Central

Practical use of pipe flow theory in system design and analysis

To

Flow Regime Classification

From

Reynolds Number (Re = vD/ν)

Strength

strong

Relationship

Re value determines whether flow is laminar, transitional, or turbulent

To

Friction Factor f

From

Flow Regime Classification

Strength

strong

Relationship

Laminar flows use f = 64/Re; turbulent flows require Moody chart or correlation

To

Darcy-Weisbach Equation

From

Friction Factor f

Strength

strong

Relationship

f is the primary variable in hf = f(L/D)(v²/2g); essential for major loss calculation

To

Friction Factor f

From

Relative Roughness ε/D

Strength

strong

Relationship

In turbulent flow, roughness significantly affects f; both Re and ε/D determine f from Moody

To

Velocity Calculation

From

Continuity Equation (Q = A·v)

Strength

strong

Relationship

Determines velocity in pipe; v = Q/(πD²/4); velocity needed for Re and head loss calculations

To

Darcy-Weisbach Equation

From

Velocity Head (v²/2g)

Strength

strong

Relationship

Head loss proportional to velocity head; appears in both major and minor loss equations

To

Minor Head Loss Equation

From

Velocity Head (v²/2g)

Strength

strong

Relationship

Minor losses hm = K(v²/2g) directly proportional to velocity head

To

Series Pipe Configuration

From

Major Head Losses (hf)

Strength

strong

Relationship

In series, major losses from each pipe add together for total system loss

To

Series Pipe Configuration

From

Minor Head Losses (hm)

Strength

strong

Relationship

Minor losses at all fittings in series are summed to obtain total minor loss

To

Total System Head Loss

From

Series Pipe Configuration

Strength

strong

Relationship

hL_total = Σ(hf) + Σ(hm) for all pipes and fittings in series

To

Head Loss Constraint

From

Parallel Pipe Configuration

Strength

strong

Relationship

Head loss must be equal across all parallel branches; flows add

To

Hardy Cross Method

From

Network Analysis

Strength

strong

Relationship

Multi-loop networks require iterative Hardy Cross procedure to solve

To

Manning Equation

From

Darcy-Weisbach Equation

Strength

moderate

Relationship

Both calculate head loss for full pipes; Manning is empirical alternative

To

Hazen-Williams Equation

From

Darcy-Weisbach Equation

Strength

moderate

Relationship

Both used for water pipes; Hazen-Williams empirical, coefficients differ

To

Minor Head Loss Equation

From

Loss Coefficient K

Strength

strong

Relationship

K value specific to each fitting type; used to calculate hm = K(v²/2g)

To

Minor Head Losses

From

Sudden Expansion Loss

Strength

moderate

Relationship

Special case using Borda-Carnot formula h = (v₁ - v₂)²/2g instead of K coefficient

To

Reynolds Number

From

Pipe Diameter D

Strength

strong

Relationship

Larger diameter increases Re; affects flow regime determination

To

Darcy-Weisbach Equation

From

Pipe Diameter D

Strength

strong

Relationship

Head loss inversely proportional to D; hf ∝ (1/D) in Darcy equation

To

Darcy-Weisbach Equation

From

Pipe Length L

Strength

strong

Relationship

Head loss directly proportional to L; hf ∝ L in Darcy equation

To

Reynolds Number

From

Viscosity (ν)

Strength

strong

Relationship

Re = vD/ν; higher viscosity decreases Re, promotes laminar flow

To

Relative Roughness ε/D

From

Pipe Absolute Roughness ε

Strength

strong

Relationship

ε/D ratio determines effect of roughness on friction factor in turbulent flow

To

Total System Head Loss

From

Pump Sizing Applications

Strength

strong

Relationship

Pump head must overcome total system losses (hL) plus elevation and pressure requirements

To

Major Head Losses

From

Pipe Diameter Selection

Strength

strong

Relationship

Larger diameter reduces head loss; D selection balances friction loss vs. capital cost

To

Head Loss (All Types)

From

Flow Velocity

Strength

strong

Relationship

Head loss proportional to v²; doubling velocity quadruples losses

To

Friction Factor f

From

Moody Diagram

Strength

strong

Relationship

Moody chart is primary tool to determine f for turbulent flow given Re and ε/D