3.4 Pipe Flow

3.4 Pipe Flow

Introduction to Pipe Flow

• Pipe flow refers to the pressure-driven flow of fluid within closed conduits, forming the backbone of water supply, industrial processing, HVAC, and hydraulic systems.

• It is characterized by a balance between pressure forces, viscous forces, and gravity, resulting in a continuous energy loss along the pipe length.

• This section systematically covers the classification of pipe flow regimes, the governing equations for head loss, the graphical representation of energy lines, design principles, analysis of complex networks, and transient phenomena.

---

1. Types and Governing Equations

1.1 Classification Based on Flow Regime

• Laminar Pipe Flow:

  • Definition: Smooth, orderly flow in concentric layers. Highest velocity at the centerline, zero at the wall (no-slip condition).

  • Governing Parameter: Reynolds Number $\boldsymbol{Re = \frac{\rho V D}{\mu} = \frac{V D}{\nu}}$.

  • Regime: $\boldsymbol{Re \leq 2000}$.

  • Velocity Profile: Parabolic (Hagen-Poiseuille flow).

    $\boldsymbol{u(r) = \frac{1}{4\mu} \left( -\frac{dp}{dx} \right) (R^2 - r^2)}$

    where, $\boldsymbol{R}$ is the pipe radius, $\boldsymbol{r}$ is the radial coordinate.

• Turbulent Pipe Flow:

  • Definition: Chaotic, irregular motion with intense mixing. Velocity profile is much fuller (more uniform) across the core.

  • Regime: $\boldsymbol{Re \geq 4000}$.

  • Velocity Profile: Described by the logarithmic law of the wall or power-law approximations (e.g., $\boldsymbol{u / u_{max} \approx (y/R)^{1/n}}$, where $\boldsymbol{n}$ varies with $\boldsymbol{Re}$).

• Transitional Flow:

  • Definition: Unstable, intermittent flow switching between laminar and turbulent patterns.

  • Regime: $\boldsymbol{2000 < Re < 4000}$. Avoided in design due to unpredictability.

1.2 Governing Equations

• The primary equation for pipe flow analysis is the Extended Bernoulli Equation (Steady Flow Energy Equation) between two sections (1) and (2):

  $\boldsymbol{\frac{p_1}{\gamma} + \frac{V_1^2}{2g} + z_1 + h_p = \frac{p_2}{\gamma} + \frac{V_2^2}{2g} + z_2 + h_t + h_L}$

• For pipe flow without pumps or turbines ($\boldsymbol{h_p = h_t = 0}$), this simplifies to:

  $\boldsymbol{\left( \frac{p_1}{\gamma} + z_1 \right) - \left( \frac{p_2}{\gamma} + z_2 \right) = \frac{V_2^2 - V_1^2}{2g} + h_L}$

  The left side represents the available static head difference. This difference is used to overcome velocity head changes and all head losses $\boldsymbol{h_L}$.

• The continuity equation for incompressible flow is always used in conjunction:

  $\boldsymbol{Q = A_1 V_1 = A_2 V_2}$

---

2. Major and Minor Head Losses

• Head loss $\boldsymbol{h_L}$ is the irreversible conversion of mechanical energy into internal energy (heat) due to viscosity. It is expressed in units of length [$\boldsymbol{\mathrm{L}}$].

2.1 Major Losses (Friction Losses)

• Definition: Head loss due to friction along the straight, constant-area sections of pipe.

• Darcy-Weisbach Equation (Universal Form):

  $\boldsymbol{h_f = f \frac{L}{D} \frac{V^2}{2g}}$

  where,

  • $\boldsymbol{h_f}$ is the major head loss $\boldsymbol{[\mathrm{m \ or \ ft}]}$.

  • $\boldsymbol{f}$ is the Darcy friction factor (dimensionless).

  • $\boldsymbol{L}$ is the length of pipe $\boldsymbol{[\mathrm{m \ or \ ft}]}$.

  • $\boldsymbol{D}$ is the pipe diameter $\boldsymbol{[\mathrm{m \ or \ ft}]}$.

  • $\boldsymbol{V}$ is the average flow velocity $\boldsymbol{[\mathrm{m/s \ or \ ft/s}]}$.

2.1.1 Determining the Friction Factor $\boldsymbol{f}$

• For Laminar Flow ($\boldsymbol{Re \leq 2000}$): Theory gives an exact result.

  $\boldsymbol{f = \frac{64}{Re}}$

• For Turbulent Flow ($\boldsymbol{Re \geq 4000}$): $\boldsymbol{f}$ depends on $\boldsymbol{Re}$ and relative roughness $\boldsymbol{(\epsilon / D)}$.

  • Colebrook-White Equation (Implicit, most accurate):

    $\boldsymbol{\frac{1}{\sqrt{f}} = -2.0 \log_{10} \left( \frac{\epsilon / D}{3.7} + \frac{2.51}{Re \sqrt{f}} \right)}$

    Requires iterative solution.

  • Moody Chart: A graphical representation of the Colebrook-White equation. Used for manual determination.

  • Swamee-Jain Formula (Explicit Approximation):

    $\boldsymbol{f = \frac{0.25}{\left[ \log_{10} \left( \frac{\epsilon / D}{3.7} + \frac{5.74}{Re^{0.9}} \right) \right]^2}}$

    Valid for $\boldsymbol{10^{-6} \leq \epsilon/D \leq 10^{-2}}$ and $\boldsymbol{5000 \leq Re \leq 10^8}$.

2.2 Minor Losses (Local Losses)

• Definition: Head loss caused by flow disturbances at pipe fittings, bends, valves, expansions, contractions, and other appurtenances.

• General Form:

  $\boldsymbol{h_m = K_L \frac{V^2}{2g}}$

  where, $\boldsymbol{K_L}$ is the loss coefficient (dimensionless), typically based on the velocity in the smaller pipe or the downstream pipe.

• Common Minor Losses:

  • Sudden Contraction: $\boldsymbol{K_L \approx 0.5 \left(1 - \frac{A_2}{A_1}\right)}$ (approx.) or from standard tables.

  • Sudden Expansion (Borda-Carnot Loss):

    $\boldsymbol{K_L = \left(1 - \frac{A_1}{A_2}\right)^2}$

  • Pipe Entrance: Sharp-edged: $\boldsymbol{K_L = 0.5}$; Well-rounded: $\boldsymbol{K_L \approx 0.03}$.

  • Pipe Exit: $\boldsymbol{K_L = 1.0}$ (all kinetic energy is lost).

  • Bends, Elbows, Tees, Valves: $\boldsymbol{K_L}$ values are obtained from manufacturer tables or standard references (e.g., $\boldsymbol{K_L}$ for a 90° standard elbow $\boldsymbol{\approx 0.3}$, globe valve (fully open) $\boldsymbol{\approx 10}$).

2.3 Total Head Loss in a System

• The total head loss is the sum of all major and minor losses along the flow path.

  $\boldsymbol{h_{L_{total}} = \sum h_f + \sum h_m = \sum \left( f \frac{L}{D} \frac{V^2}{2g} \right) + \sum \left( K_L \frac{V^2}{2g} \right)}$

---

3. Hydraulic Grade Line (HGL) and Energy Grade Line (TEL)

• These are graphical tools for visualizing the energy distribution and pressure availability along a pipeline.

3.1 Definitions

• Energy Grade Line (EGL) or Total Energy Line (TEL):

  • Represents the total mechanical energy per unit weight of fluid.

    $\boldsymbol{\mathrm{EGL} = \frac{p}{\gamma} + \frac{V^2}{2g} + z}$

  • Slope: The slope of the EGL is the energy gradient $\boldsymbol{S_f = h_f / L}$, representing the rate of energy loss.

• Hydraulic Grade Line (HGL):

  • Represents the sum of pressure head and elevation head. It is the height to which fluid would rise in a piezometer (standpipe) attached to the pipe.

    $\boldsymbol{\mathrm{HGL} = \frac{p}{\gamma} + z}$

  • Relationship: $\boldsymbol{\mathrm{EGL} - \mathrm{HGL} = \frac{V^2}{2g}}$ (the velocity head).

3.2 Rules for Plotting HGL and EGL

1. The EGL always drops in the direction of flow due to head losses ($\boldsymbol{h_f + h_m}$).

2. A sudden drop in EGL occurs at locations of minor loss ($\boldsymbol{h_m}$).

3. The HGL also drops in the flow direction but can rise or fall more steeply than the EGL depending on changes in velocity head.

4. For a pipe of constant diameter (constant $\boldsymbol{V}$), the HGL and EGL are parallel ($\boldsymbol{\Delta(EGL) = \Delta(HGL)}$).

5. At a sudden enlargement, the HGL rises sharply due to recovery of pressure head from velocity head reduction.

6. Critical Condition: If the HGL falls below the pipe centerline, the pressure $\boldsymbol{p}$ becomes negative (below atmospheric), which can lead to cavitation or contamination ingress.

3.3 Practical Applications

• System Diagnosis: Identify locations of high pressure, low pressure, or cavitation risk.

• Pump/Turbine Siting: Determines required pump head (rise in EGL) or available turbine head (drop in EGL).

• Siphon Analysis: Ensures pressure in the siphon crest remains above vapor pressure to avoid breaking.

• Water Hammer: Initial HGL level determines the potential pressure surge magnitude.

---

4. Pipe Flow Design

• Involves selecting pipe diameter, material, and layout to meet flow requirements while considering economics and constraints.

4.1 Types of Design Problems

• Type I (Analysis): Given $\boldsymbol{D, L, \epsilon, Q}$ (or $\boldsymbol{V}$), find head loss $\boldsymbol{h_f}$ and required pump head.

• Type II (Sizing): Given $\boldsymbol{L, \epsilon, Q, h_f}$ (available head), find pipe diameter $\boldsymbol{D}$. Requires iteration due to implicit $\boldsymbol{f(D)}$.

• Type III (Flow Rate): Given $\boldsymbol{D, L, \epsilon, h_f}$, find flow rate $\boldsymbol{Q}$. Requires iteration (e.g., using the Colebrook equation or Moody chart).

4.2 Economic Design and Optimization

• Basis: Minimize total annual cost = Capital Cost (pipe, installation) + Operating Cost (energy for pumping).

• Capital Cost: Increases with diameter $\boldsymbol{D}$.

• Operating Cost: Decreases with diameter $\boldsymbol{D}$ (due to lower velocity and lower $\boldsymbol{h_f}$ for the same $\boldsymbol{Q}$).

• The economic diameter is found at the minimum point of the total cost curve.

4.3 Pipe Material and Roughness Selection

• Material Choice Criteria: Cost, corrosion resistance, strength, life expectancy, jointing method.

• Typical Roughness Values ($\boldsymbol{\epsilon}$):

  • Drawn tubing, glass: $\boldsymbol{\epsilon \approx 0.0015 \ \mathrm{mm}}$

  • Commercial steel, welded: $\boldsymbol{\epsilon \approx 0.046 \ \mathrm{mm}}$

  • Cast iron: $\boldsymbol{\epsilon \approx 0.26 \ \mathrm{mm}}$

  • Concrete: $\boldsymbol{\epsilon \approx 0.3-3.0 \ \mathrm{mm}}$

  • Riveted steel: $\boldsymbol{\epsilon \approx 0.9-9.0 \ \mathrm{mm}}$

4.4 Design for Non-Circular Conduits

• Use the Hydraulic Radius $\boldsymbol{R_h = A / P}$, where $\boldsymbol{P}$ is the wetted perimeter.

• For turbulent flow, use the equivalent diameter $\boldsymbol{D_h = 4 R_h}$ in the Darcy-Weisbach and Reynolds number equations.

---

5. Pipe Network Problems

• Systems of interconnected pipes forming loops and branches, commonly found in municipal water distribution.

5.1 Governing Principles

• Two basic laws must be satisfied at every node and for every loop.

  1. Continuity at Nodes (Mass Conservation): The algebraic sum of flows into any junction (node) is zero.

     $\boldsymbol{\sum Q_{in} - \sum Q_{out} = 0}$

  2. Energy Conservation Around Loops: The algebraic sum of head losses around any closed loop must be zero (conservation of energy).

     $\boldsymbol{\sum_{loop} h_{L,i} = 0}$

     where head loss for each pipe is $\boldsymbol{h_{L,i} = K_i Q_i^n}$, with $\boldsymbol{n \approx 2}$ for turbulent flow and $\boldsymbol{K_i = \frac{8f_i L_i}{g \pi^2 D_i^5}}$.

5.2 Analysis Methods

• Hardy Cross Method (Loop Method):

  • An iterative, manual method for solving networks.

  • Correction Formula for a Loop: For turbulent flow ($\boldsymbol{h_L = K Q^2}$), the flow correction $\boldsymbol{\Delta Q}$ for a loop is:

    $\boldsymbol{\Delta Q = -\frac{\sum h_{L,i}}{2 \sum \frac{|h_{L,i}|}{Q_i}}}$

  • Procedure: Assume initial flows satisfying node continuity. Compute head losses and corrections. Apply corrections (clockwise positive convention). Iterate until $\boldsymbol{\Delta Q}$ is negligible.

• Nodal Method: Solves for unknown pressures at nodes instead of flows in pipes. Uses the continuity equation at each node in terms of pressure differences.

• Computer Software: Modern analysis uses specialized software (EPANET, WaterGEMS, etc.) that employs more efficient matrix methods (Linear Theory, Newton-Raphson).

5.3 Network Components and Operation

• Pumps: Modeled as a known head addition ($\boldsymbol{h_p}$) or by a pump curve ($\boldsymbol{h_p}$ vs $\boldsymbol{Q}$).

• Valves: Modeled as a variable minor loss ($\boldsymbol{K_L}$) or as pressure-reducing/ sustaining devices.

• Reservoirs/Tanks: Constant pressure head nodes.

• Demands: Known outflow rates at specific nodes.

---

6. Unsteady Flow in Pipes and Relief Devices

• Also known as transient flow or water hammer. Occurs when flow conditions change rapidly (e.g., valve closure, pump start/stop).

6.1 Causes and Consequences

• Causes: Rapid valve closure/opening, pump start-up or trip, rapid filling/emptying of lines.

• Consequences:

  • Pressure Surges: Can exceed steady-state pressure by many times, risking pipe rupture (water hammer) or collapse (cavitation).

  • Column Separation: Extreme low pressure can vaporize water, forming cavities that subsequently collapse violently.

6.2 Governing Wave Speed (Celerity)

• The speed at which a pressure wave travels through the pipe-fluid system.

  $\boldsymbol{a = \sqrt{\frac{K / \rho}{1 + \frac{K D}{E e} C}}}$

  where,

  • $\boldsymbol{a}$ is the wave speed $\boldsymbol{[\mathrm{m/s}]}$.

  • $\boldsymbol{K}$ is the bulk modulus of the fluid.

  • $\boldsymbol{E}$ is the Young's modulus of the pipe wall.

  • $\boldsymbol{e}$ is the pipe wall thickness.

  • $\boldsymbol{C}$ is a anchorage factor ($\boldsymbol{C=1}$ for pipes anchored at one end, $\boldsymbol{C=1-\mu^2}$ for pipes anchored throughout).

• For rigid pipes or very slow events, $\boldsymbol{a \approx \sqrt{K/\rho}}$ (speed of sound in the fluid).

6.3 Maximum Pressure Rise (Joukowsky Equation)

• For instantaneous valve closure (or rapid closure time $\boldsymbol{T_c < 2L/a}$):

  $\boldsymbol{\Delta p = \rho a \Delta V}$

  or in terms of head:

  $\boldsymbol{\Delta h = \frac{a \Delta V}{g}}$

  where, $\boldsymbol{\Delta V}$ is the change in flow velocity.

• Interpretation: The pressure surge is directly proportional to the wave speed and the magnitude of the velocity change.

6.4 Relief Devices and Protection Methods

• Purpose: To limit pressure surges to safe levels.

• Common Devices:

  • Surge Tanks: Open or closed standpipes connected to the pipeline. They absorb surge pressure by allowing fluid to enter/exit, providing a large surface area for wave reflection.

  • Pressure Relief Valves (PRVs): Open at a set pressure to divert flow and limit maximum pressure.

  • Air Chambers/Anti-water Hammer Arrestors: Contain a cushion of compressed air that compresses to absorb the surge.

  • Bypass Lines: Allow flow to recirculate during pump shutdown.

  • Flywheels: Increase pump rotational inertia to slow down rate of change during trip.

• Operational Controls:

  • Slow Valve Closure: Ensure closure time $\boldsymbol{T_c > 2L/a}$ (more than one pipe period) to allow wave reflections to mitigate the surge.

  • Soft Starters/Variable Frequency Drives (VFDs): For pumps, to ramp up/down speed gradually.

6.5 Analysis Methods

• Rigid Water Column Theory: Assumes fluid is incompressible and pipe is rigid. Simpler, valid for very slow transients.

• Elastic Water Hammer Theory (Method of Characteristics): Accounts for fluid compressibility and pipe elasticity. The standard method for detailed transient analysis, solved numerically on a space-time grid.

---

Key Relationships and Summary

• Core Equation: $\boldsymbol{h_L = h_f + h_m = f \frac{L}{D} \frac{V^2}{2g} + \sum K_L \frac{V^2}{2g}}$.

• Flow Regime: Determined by $\boldsymbol{Re}$. Defines friction factor relation.

• Design Philosophy: Balance hydraulic requirements (flow, pressure) with economic costs (pipe, energy).

• Network Solution: Satisfy continuity at nodes and zero net head loss around loops.

• Transient Control: Understand wave speed $\boldsymbol{a}$ and Joukowsky head $\boldsymbol{\Delta h}$. Use protective devices and operational care to manage rapid changes.

Mastery of pipe flow principles enables the safe, efficient, and reliable design of the vast network of conduits that underpin modern civilization's water and energy infrastructure.

---