Determine the minimum size smooth rectangular duct with an...

Determine the minimum size smooth rectangular duct with an aspect ratio of 3 that will pass $1 \mathrm{m}^{3} / \mathrm{s}$ of $10^{\circ} \mathrm{C}$ air with a head loss of $25 \mathrm{mm}$ of water per $100 \mathrm{m}$ of duct.

Key Concepts

Darcy–Weisbach Equation

This fundamental relation in fluid mechanics links the head loss (i.e., energy loss per unit weight of fluid) due to friction along a duct to parameters such as flow velocity, hydraulic diameter, and friction factor. It is essential for determining the pressure drop in ducts and thereby guiding the sizing of duct systems under given performance constraints.

Head Loss

Head loss represents the amount of energy per unit weight of fluid lost due to frictional effects in the duct system. Expressed in units such as millimeters of water, it is a key design parameter that determines the energy needed to drive the flow and is directly linked to the frictional characteristics of the duct.

Volumetric Flow Rate and Velocity Relationship

The volumetric flow rate is related to the average flow velocity and the duct’s cross-sectional area through the continuity equation. This relationship is used to compute the necessary dimensions of the duct to accommodate a given flow rate while simultaneously satisfying the required friction loss criteria.

Hydraulic Diameter

For non?circular ducts, the hydraulic diameter (defined as four times the cross-sectional area divided by the wetted perimeter) serves as a representative length scale for friction calculations. It allows design equations originally developed for circular pipes to be applied to rectangular ducts by characterizing the effective dimension that influences flow and friction losses.

Friction Factor

The friction factor is a dimensionless parameter that quantifies the resistance to flow due to friction along the duct walls. In smooth ducts, it is typically determined from empirical correlations or charts (such as the Moody chart) as a function of the Reynolds number. This factor is critical to linking flow conditions with head loss in the Darcy–Weisbach equation.

Aspect Ratio

The aspect ratio for a rectangular duct, defined as the ratio of its longer side to its shorter side, influences the duct geometry and its hydraulic diameter. For an aspect ratio of 3, the relationship between the dimensions is fixed, and this has direct consequences on both the flow distribution and the pressure drop characteristics.

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