5.2 Pipe Friction, Moody Diagram & Darcy-Weisbach / Hazen-Williams Calculations

5.2 Pipe Friction, Moody Diagram & Darcy-Weisbach / Hazen-Williams Calculations

Accurate calculation of frictional pressure drop in piping networks is essential for sizing hydronic pumps, selecting pipe diameters, and ensuring balanced distribution across HVAC coils, chillers, cooling towers, and boilers. Frictional head loss represents the continuous conversion of mechanical flow energy into internal thermal energy due to fluid shear stress along the conduit walls and turbulent momentum exchange. The NCEES PE Mechanical exam tests both the universal Darcy-Weisbach method (and the associated Moody chart) and the empirical Hazen-Williams formulation.

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1. Flow Regimes & Reynolds Number ($Re$)

The flow regime inside a circular conduit is dictated by the dimensionless Reynolds Number ($Re$), which quantifies the ratio of inertial forces to viscous forces:

$Re = \frac{\rho V D}{\mu} = \frac{V D}{\nu}$

Where:

• $\rho$: Fluid density ($\text{lbm/ft}^3$ or $\text{kg/m}^3$)
• $V$: Average flow velocity ($\text{ft/s}$ or $\text{m/s}$)
• $D$: Internal pipe diameter ($\text{ft}$ or $\text{m}$)
• $\mu$: Dynamic (absolute) viscosity ($\text{lbm/(ft}\cdot\text{s)}$ or $\text{Pa}\cdot\text{s}$; $1\text{ cP} = 6.7197 \times 10^{-4}\text{ lbm/(ft}\cdot\text{s)}$)
• $\nu = \mu / \rho$: Kinematic viscosity ($\text{ft}^2/\text{s}$ or $\text{m}^2/\text{s}$; $1\text{ cSt} = 1.0764 \times 10^{-5}\text{ ft}^2/\text{s}$)

Practical GPM Shortcut Formulation for Water at $60^\circ\text{F}$

For liquid water at standard conditions ($\rho = 62.37\text{ lbm/ft}^3$, $\nu = 1.217 \times 10^{-5}\text{ ft}^2/\text{s}$), with flow rate $Q$ in $\text{GPM}$ and internal diameter $d$ in $\text{inches}$:

$Re \approx \frac{3160 \times Q\text{ (GPM)}}{d\text{ (inches)} \times \nu\text{ (cSt)}} \approx \frac{3790 \times Q\text{ (GPM)}}{d\text{ (inches)}}$

Flow Regime Boundaries

• Laminar Flow ($Re < 2300$): Fluid particles travel along smooth, parallel streamlines. Viscous forces dominate; momentum exchange is purely molecular. Friction factor depends solely on $Re$ and is completely independent of pipe wall roughness $\epsilon$.
• Transition Zone ($2300 \le Re \le 4000$): Flow alternates unpredictably between laminar bursts and turbulent eddies. Engineering calculations in this zone are inherently uncertain; designers generally avoid operating in this regime.
• Turbulent Flow ($Re > 4000$): Chaotic 3D velocity fluctuations and turbulent eddies dominate momentum transfer. Friction factor is governed by both Reynolds number $Re$ and relative pipe roughness $\epsilon / D$.

2. The Darcy-Weisbach Equation & Moody Diagram

The Darcy-Weisbach equation is the universally applicable, theoretically sound relationship for frictional head loss in any closed conduit carrying any single-phase fluid:

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

$\Delta P_f = \rho \frac{g}{g_c} h_f = f \frac{L}{D} \frac{\rho V^2}{2 g_c}$

Where:

• $h_f$: Frictional head loss ($\text{ft of fluid column}$)
• $\Delta P_f$: Frictional pressure drop ($\text{lbf/ft}^2$ or $\text{psi}$, where $\text{psi} = \Delta P_f / 144$)
• $f$: Darcy (Moody) friction factor (dimensionless; note that Fanning friction factor $f_{\text{Fanning}} = f / 4$)
• $L$: Length of pipe ($\text{ft}$)
• $D$: Internal pipe diameter ($\text{ft}$)
• $V$: Average flow velocity ($\text{ft/s}$)
• $g$: Acceleration due to gravity ($32.174\text{ ft/s}^2$)

Determining the Friction Factor ($f$)

+-----------------------------------------------------------------------------------------+
| MOODY DIAGRAM SCHEMATIC OVERVIEW                                                        |
+-----------------------------------------------------------------------------------------+
| Friction                                                              Relative Roughness|
| Factor f                                                                     epsilon / D|
|  0.10 +----+---------------------------------------------------------------+ 0.05       |
|       |    | Laminar Flow: f = 64 / Re                                     |            |
|       |\   +-------------------+                                           | 0.02       |
|       | \                      | Transition                                |            |
|  0.04 |  \                     | Zone                                      | 0.005      |
|       |   \                    |                                  ======+  |            |
|  0.02 |    \                   |         Smooth Pipe Line =======          | 0.001      |
|       |     \                  |                 ================          | 0.0001     |
|  0.01 +------+-----------------+-----------------+-------------------------+ Smooth     |
|       10^3            10^4              10^5              10^6             10^8         |
|                                    Reynolds Number (Re)                                 |
+-----------------------------------------------------------------------------------------+

1. Laminar Flow Regime ($Re < 2300$): $f = \frac{64}{Re}$

2. Turbulent Smooth Pipe Regime (Blasius Equation for $4000 < Re < 10^5$): $f = \frac{0.3164}{Re^{0.25}}$

3. General Turbulent Regime (Colebrook Implicit Equation): $\frac{1}{\sqrt{f}} = -2.0 \log_{10} \left( \frac{\epsilon / D}{3.7} + \frac{2.51}{Re \sqrt{f}} \right)$

4. Haaland Equation (Explicit Approximation of Colebrook): Because the Colebrook equation requires iterative root-finding, S.E. Haaland developed an explicit formulation accurate to within $1.5%$ of Colebrook across $4000 \le Re \le 10^8$ and $10^{-6} \le \epsilon/D \le 0.05$: $\frac{1}{\sqrt{f}} = -1.8 \log_{10} \left[ \left( \frac{\epsilon / D}{3.7} \right)^{1.11} + \frac{6.9}{Re} \right]$ $f = \left( -1.8 \log_{10} \left[ \left( \frac{\epsilon / D}{3.7} \right)^{1.11} + \frac{6.9}{Re} \right] \right)^{-2}$

5. Fully Rough / Wholly Turbulent Regime (Right Plateau of Moody Chart): At very high Reynolds numbers, the laminar viscous sublayer becomes thinner than the surface asperities $\epsilon$. In this regime, friction is entirely independent of $Re$ and depends solely on relative roughness: $\frac{1}{\sqrt{f}} = -2.0 \log_{10} \left( \frac{\epsilon / D}{3.7} \right) = 1.14 - 2.0 \log_{10}\left(\frac{\epsilon}{D}\right)$

Pipe Absolute Roughness Values ($\epsilon$)

3. The Hazen-Williams Empirical Method (Water Only)

The Hazen-Williams equation is an empirical formulation widely used in civil fire protection (NFPA 13) and domestic plumbing design. It relates water flow rate to friction loss without calculating Reynolds numbers or reading the Moody diagram.

Hazen-Williams Head Loss Formulas (IP Units)

$h_f = 0.002083 \cdot L \cdot \left( \frac{100}{C} \right)^{1.852} \cdot \frac{Q^{1.852}}{d^{4.8655}}$

$\Delta P_{\text{per 100 ft}} = \frac{4.52 \cdot Q^{1.852}}{C^{1.852} \cdot d^{4.8655}}$

Where:

• $h_f$: Head loss ($\text{ft of water}$)
• $\Delta P_{\text{per 100 ft}}$: Pressure loss per 100 ft of pipe ($\text{psi/100 ft}$)
• $L$: Total equivalent pipe length ($\text{ft}$)
• $Q$: Volumetric water flow rate ($\text{GPM}$)
• $d$: Inside pipe diameter ($\text{inches}$)
• $C$: Hazen-Williams roughness coefficient (higher $C$ = smoother pipe = lower friction)

Hazen-Williams Roughness Coefficients ($C$)

Critical Engineering Limitations of Hazen-Williams

[!WARNING] When Hazen-Williams CANNOT Be Used on the PE Exam:

1. Non-Water Fluids: Glycol mixtures (ethylene/propylene glycol), oils, and liquid refrigerants MUST use Darcy-Weisbach because Hazen-Williams embeds standard water viscosity.
2. Non-Standard Water Temperatures: Water temperatures significantly above $75^\circ\text{F}$ (e.g., heating hot water at $180^\circ\text{F}$ or boiler feedwater at $220^\circ\text{F}$) have much lower viscosity, causing Hazen-Williams to overestimate head loss by $20% - 35%$.
3. Laminar Flow: Hazen-Williams is completely invalid for $Re < 4000$.
4. Air & Steam Distribution: Gases must always be evaluated using Darcy-Weisbach or dedicated compressibility charts.

4. Hydraulic Diameter for Non-Circular Conduits

For fluid flow through non-circular passages—such as rectangular HVAC ducts, flat-oval ductwork, or annular spaces in double-pipe heat exchangers—the characteristic dimension in $Re$ and Darcy-Weisbach is replaced by the Hydraulic Diameter ($D_h$):

$D_h = \frac{4 A}{P_{\text{wetted}}}$

Where:

• $A$: Cross-sectional flow area ($\text{ft}^2$ or $\text{in.}^2$)
• $P_{\text{wetted}}$: Wetted perimeter in direct contact with the moving fluid ($\text{ft}$ or $\text{in.}$)

Standard Geometry Hydraulic Diameters

+-----------------------------------------------------------------------------+
| HYDRAULIC DIAMETERS FOR COMMON HVAC GEOMETRIES                              |
+-----------------------------------------------------------------------------+
| 1. Rectangular Duct (Width a, Height b):                                    |
|    Area A = a * b                                                           |
|    Perimeter P = 2 * (a + b)                                                |
|    D_h = (4 * a * b) / [2 * (a + b)] = (2 * a * b) / (a + b)                |
|                                                                             |
| 2. Annular Space (Outer Pipe ID = D_i, Inner Pipe OD = d_o):                |
|    Area A = (pi / 4) * (D_i^2 - d_o^2)                                      |
|    Perimeter P = pi * (D_i + d_o)                                           |
|    D_h = [4 * (pi/4) * (D_i^2 - d_o^2)] / [pi * (D_i + d_o)] = D_i - d_o   |
+-----------------------------------------------------------------------------+

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5. Worked Engineering Examples

Worked Example 1: Darcy-Weisbach Friction Loss with Haaland Equation

Chilled water at $45^\circ\text{F}$ (density $\rho = 62.42\text{ lbm/ft}^3$, kinematic viscosity $\nu = 1.54 \times 10^{-5}\text{ ft}^2/\text{s}$) flows through $L = 300\text{ ft}$ of 4-inch Schedule 40 commercial steel pipe ($D = 4.026\text{ in.} = 0.3355\text{ ft}$, absolute roughness $\epsilon = 0.00015\text{ ft}$) at a volumetric flow rate of $Q = 200\text{ GPM}$. Calculate the Darcy friction factor $f$, the friction head loss $h_f$, and the frictional pressure drop $\Delta P_f$ in $\text{psi}$.

Given:
- Fluid: Water at 45°F (rho = 62.42 lbm/ft3, nu = 1.54 x 10^-5 ft2/s)
- Pipe: 4" Sch 40 Steel (D = 0.3355 ft, epsilon = 0.00015 ft, L = 300 ft)
- Flow: Q = 200 GPM

Step 1: Calculate flow area and velocity: $A = \frac{\pi}{4} D^2 = \frac{\pi}{4} (0.3355\text{ ft})^2 = 0.08841\text{ ft}^2$

$Q = 200\text{ GPM} \times 0.002228\text{ ft}^3/(\text{s}\cdot\text{GPM}) = 0.4456\text{ ft}^3/\text{s}$

$V = \frac{Q}{A} = \frac{0.4456\text{ ft}^3/\text{s}}{0.08841\text{ ft}^2} = 5.040\text{ ft/s}$

Step 2: Calculate Reynolds number and relative roughness: $Re = \frac{V D}{\nu} = \frac{5.040\text{ ft/s} \times 0.3355\text{ ft}}{1.54 \times 10^{-5}\text{ ft}^2/\text{s}} = 109,802 \approx 1.10 \times 10^5 \quad (\text{Turbulent})$

$\frac{\epsilon}{D} = \frac{0.00015\text{ ft}}{0.3355\text{ ft}} = 0.0004471$

Step 3: Calculate friction factor $f$ using the Haaland equation: $\frac{1}{\sqrt{f}} = -1.8 \log_{10} \left[ \left( \frac{0.0004471}{3.7} \right)^{1.11} + \frac{6.9}{109,802} \right]$

$\left( \frac{0.0004471}{3.7} \right)^{1.11} = (0.00012084)^{1.11} = 4.400 \times 10^{-5}$

$\frac{6.9}{109,802} = 6.284 \times 10^{-5}$

$\text{Sum} = 4.400 \times 10^{-5} + 6.284 \times 10^{-5} = 1.0684 \times 10^{-4}$

$\log_{10}(1.0684 \times 10^{-4}) = -3.9713$

$\frac{1}{\sqrt{f}} = -1.8 \times (-3.9713) = 7.1483 \implies \sqrt{f} = 0.13989 \implies f = 0.01957 \approx 0.0196$

Step 4: Calculate head loss and pressure drop: $h_f = f \frac{L}{D} \frac{V^2}{2g} = 0.01957 \times \left( \frac{300\text{ ft}}{0.3355\text{ ft}} \right) \times \left( \frac{(5.040\text{ ft/s})^2}{2 \times 32.174\text{ ft/s}^2} \right)$

$h_f = 0.01957 \times 894.19 \times 0.39474\text{ ft} = 6.908\text{ ft} \approx 6.91\text{ ft}$

$\Delta P_f = \frac{\rho h_f}{144} = \frac{62.42\text{ lbm/ft}^3 \times 6.908\text{ ft}}{144\text{ in.}^2/\text{ft}^2} = 2.994\text{ psi} \approx 3.00\text{ psi}$

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6. NCEES Reference Handbook Navigation & Exam Tips

• Moody Diagram Location: In the Reference Handbook under Fluid Mechanics $\rightarrow$ Friction Factor / Moody Chart. To read the chart, first calculate relative roughness $\epsilon/D$ on the right vertical axis, follow the curve to the calculated Reynolds number on the horizontal axis, and read $f$ on the left vertical axis.
• Haaland Formula in Handbook: The explicit Haaland equation is listed under Fluid Mechanics $\rightarrow$ Colebrook & Haaland Equations. Using this formula on your TI-36X Pro or Casio fx-115ES is significantly faster and more accurate than visually interpolating the Moody chart.
• Nominal vs. Actual Pipe Diameters: Always use actual internal diameter (e.g., $4.026\text{ in.}$ for 4-inch Schedule 40 steel, $3.068\text{ in.}$ for 3-inch Schedule 40) rather than nominal diameter ($4.00\text{ in.}$ or $3.00\text{ in.}$). Because diameter appears to the 5th power ($D^5$) in pressure drop formulas, small diameter discrepancies create massive errors.