Understanding Duct Pressure Loss Calculations

When air flows through a duct system, it encounters resistance, causing a drop in its total pressure. This pressure loss needs to be overcome by the fan or air handling unit to ensure adequate airflow. There are two main types of pressure losses:

• Friction Loss: Occurs due to the friction between the moving air and the inner surfaces of the ductwork. The longer the duct and the rougher its material, the higher the friction loss.
• Dynamic Loss (or Minor Loss): Caused by changes in the direction or velocity of the airflow. Fittings like elbows, reducers, enlargements, and branches create turbulence, which dissipates energy and results in pressure loss.

Key Concepts and Definitions:

• Air Density: \(\rho\) [kg/m³]. Standard: \(1.204\ \text{kg/m}^3\). Mass of air per unit volume (e.g., kg/m³). Standard air density: 1.204 kg/m³.
• Air Viscosity \(\mu\) [Pa·s]: Air's resistance to flow (e.g., Pa·s). Standard air viscosity: \(1.825 \times 10^{-5}\ \text{Pa·s}\).
• Airflow Rate \(Q\) [m³/s or CFM]: Volume of air moving through the duct per unit time (CFM or m³/s).
• Duct Area \(A\) [m²]: Cross-sectional area of the duct.
  • For circular ducts with diameter \[A = \pi \left(\frac{D}{2}\right)^2\]
  • For rectangular ducts with width \(W\) and height \(H\): \[A = W \times H\]
• Air Velocity \(V\): Speed of air movement. Calculated as: \[V = \frac{Q}{A}\].
• Hydraulic Diameter ($D_h$): Used for non-circular ducts to convert them into an equivalent circular duct for calculation.
  • For circular ducts: \(\;D_h = D\)
  • For rectangular ducts: \(\;D_h = \frac{2WH}{W+H}\)

Friction Loss Calculation: The Darcy-Weisbach Equation

The primary formula for calculating friction loss in ducts is the Darcy-Weisbach equation:

\[\Delta P_\text{friction} = f \cdot \frac{L}{D_h} \cdot \left(\frac{\rho V^2}{2}\right)\]

Where:

• \(\Delta P_\text{friction}\) = Pressure loss due to friction [Pa]
• \(f\) = Darcy friction factor [dimensionless]
• \(L\) = Length of the duct [m]
• \(D_h\) = Hydraulic diameter [m]
• \(\rho\) = Air density [kg/m³]
• \(V\) = Air velocity [m/s]

Determining the Friction Factor (\(f\))

The friction factor depends on two key parameters:

1. Reynolds Number (Re): This dimensionless number determines the flow type.

   \[\mathrm{Re} = \frac{\rho V D_h}{\mu}\]

   • If (\(\mathrm{Re} \leq 2300\)), the flow is laminar \[f = \frac{64}{\mathrm{Re}}\].
   • If (\(\mathrm{Re} > 4000\)), the flow is turbulent.
2. Relative Roughness \(\varepsilon / D_h\): The ratio of duct material's absolute roughness \(\varepsilon\) to the hydraulic diameter \(D_h\).

For turbulent flow, the tool uses the Swamee-Jain equation for the friction factor:

\[f = \frac{0.25}{\left[\log_{10}\!\left(\frac{\varepsilon}{3.7 D_h} + \frac{5.74}{\mathrm{Re}^{0.9}}\right)\right]^2}\]

Dynamic Loss Calculation: The K-Factor Method

Dynamic losses (minor losses) are calculated using loss coefficients (K-factors), which are empirical values for different fitting types.

\[\Delta P_\text{dynamic} = K \cdot \left(\frac{\rho V^2}{2}\right)\]

Where:

• \(\Delta P_\text{dynamic}\) = Pressure loss due to fittings [Pa]
• \(K\) = Loss coefficient (K-factor) for the fitting [dimensionless]
• \(\rho\) = Air density [kg/m³]
• \(V\) = Reference air velocity [m/s] (from the first duct section)

Total Pressure Loss

The total pressure loss is the sum of all friction losses and dynamic losses:

\[\Delta P_\text{total} = \sum \Delta P_\text{friction} + \sum \Delta P_\text{dynamic}\]

Results are converted from Pascals [Pa] to Inches of Water Column (in. w.c.) using: \[1~\text{Pa} \approx 0.00401463~\text{in. w.c.}\]