5.3 Duct Geometry and Worked Airflow Calculations

Duct Geometry and Worked Airflow Calculations

A traverse produces velocity at a defined plane; converting it to airflow requires the net internal area at that same plane. Record whether dimensions came from a drawing, label, or field measurement and whether they are outside or clear inside dimensions.

5. Duct Geometric Calculations and Net Internal Area

To compute total volumetric flow ($Q = A \times V$), technicians must calculate the precise net cross-sectional area of ductwork. External dimensions must be adjusted for internal insulation (acoustic liner).

   Rectangular Duct             Round Spiral Duct          Flat Oval Duct
  +------------------+             .--------.             .--------------.
  |  |============|  |           /  /======\ \           /  /==========\  \
  |  |   Net H    |  |          |  |  Net D | |         (  ( Net a  Net A ) )
  |  |============|  |           \  \======/ /           \  \==========/  /
  +------------------+             '--------'             '--------------'
         Net W

1. Rectangular Ducts

$$A\text{ (sq ft)} = \frac{W\text{ (in.)} \times H\text{ (in.)}}{144}$$

• Internal Liner Deduction: Acoustic duct liner lines all four interior walls. Subtract twice the liner thickness ($t_{liner}$) from each nominal dimension:

$$W_{net} = W_{nominal} - (2 \times t_{liner}), \quad H_{net} = H_{nominal} - (2 \times t_{liner})$$ $$A_{net}\text{ (sq ft)} = \frac{[W_{nominal} - 2t_{liner}] \times [H_{nominal} - 2t_{liner}]}{144}$$

2. Round Ducts

$$A\text{ (sq ft)} = \frac{\pi \times D^2}{4 \times 144} = \frac{\pi \times D^2}{576} \approx \frac{D^2}{183.35}$$

• Or in terms of radius $r = D/2$:

$$A\text{ (sq ft)} = \frac{\pi \times r^2}{144}$$

• Internal Liner Deduction:

$$D_{net} = D_{nominal} - (2 \times t_{liner})$$ $$A_{net}\text{ (sq ft)} = \frac{\pi \times (D_{nominal} - 2t_{liner})^2}{576}$$

3. Flat Oval Ducts

Flat oval ductwork consists of a rectangular center section flanked by two semicircular ends:

• Let $a$ = Minor internal axis (height/depth in inches)
• Let $A_{major}$ = Major internal axis (width in inches)

$$A\text{ (sq ft)} = \frac{\left( \frac{\pi \times a^2}{4} \right) + \left[ a \times (A_{major} - a) \right]}{144}$$

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6. Step-by-Step Worked Field Numerical Examples

Example 1: Standard Air Calculation in a Rectangular Supply Main

Scenario: A technician performs an equal-area Pitot-tube traverse on a $30\text{ in.} \times 16\text{ in.}$ unlined rectangular supply duct at standard conditions. Each point's velocity pressure has been converted to velocity before averaging, producing a representative mean velocity of $1,878.5\text{ FPM}$. Calculate total airflow.

1. Calculate internal cross-sectional area ($A$): $A = \frac{30 \times 16}{144} = \frac{480}{144} = 3.333\text{ sq ft}$
2. Use the point-by-point mean velocity: $V_{avg} = 1,878.5\text{ FPM}$
3. Calculate volumetric flow ($Q$): $Q = A \times V_{avg} = 3.333\text{ sq ft} \times 1,878.5\text{ FPM} = \mathbf{6,261\text{ CFM}}$

Do not average velocity pressures and then take one square root; the square-root conversion must be performed at each equal-area point before velocities are averaged.

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Example 2: Non-Standard Air Density Correction at High Altitude and Heating Temperature

Scenario: A discharge heating air duct measuring $36\text{ in.} \times 20\text{ in.}$ is traversed on a project in Denver, CO (elevation $5,280\text{ ft}$, measured barometric pressure $24.80\text{ in. Hg}$). The air temperature in the duct is $125^\circ\text{F}$. The measured velocity pressure is $V_p = 0.18\text{ in. w.g.}$ Calculate the actual airflow in CFM.

1. Calculate Internal Area ($A$): $$A = \frac{36 \times 20}{144} = \frac{720}{144} = 5.00\text{ sq ft}$$
2. Calculate Density Correction Factor ($K_d$): $$K_d = \left( \frac{530}{460 + 125} \right) \times \left( \frac{24.80}{29.92} \right) = \left( \frac{530}{585} \right) \times 0.82888 = 0.90598 \times 0.82888 = \mathbf{0.7509}$$
3. Calculate Actual Density ($\rho_{actual}$): $$\rho_{actual} = 0.075 \times 0.7509 = 0.05632\text{ lb/ft}^3$$
4. Calculate Density-Corrected Velocity ($V$): $$V = 4005 \times \sqrt{\frac{0.18}{0.7509}} = 4005 \times \sqrt{0.23971} = 4005 \times 0.48960 = \mathbf{1961\text{ FPM}}$$ (Note: The uncorrected standard formula would yield $4005 \times \sqrt{0.18} = 1699\text{ FPM}$, creating an uncorrected error of $13.4\%$!)
5. Calculate Total Actual Airflow ($Q$): $$Q = 5.00\text{ sq ft} \times 1961\text{ FPM} = \mathbf{9805\text{ CFM}}$$

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Example 3: Round Spiral Duct with Internal Acoustic Liner

Scenario: A $22\text{ in.}$ nominal diameter round supply duct is lined internally with $1.0\text{ in.}$ fiberglass acoustic insulation. An average traverse velocity of $1350\text{ FPM}$ is measured under standard air conditions. Determine the actual airflow.

1. Determine Net Internal Diameter ($D_{net}$): $$D_{net} = 22\text{ in.} - (2 \times 1.0\text{ in.}) = 20.0\text{ in.}$$
2. Calculate Net Cross-Sectional Area ($A_{net}$): $$A_{net} = \frac{\pi \times (20.0)^2}{576} = \frac{3.14159 \times 400}{576} = \frac{1256.64}{576} = 2.182\text{ sq ft}$$
3. Calculate Volumetric Flow Rate ($Q$): $$Q = A_{net} \times V = 2.182\text{ sq ft} \times 1350\text{ FPM} = \mathbf{2946\text{ CFM}}$$