2.1 TAB Mathematics, Units, Ratios, and Formula Discipline
Key Takeaways
- Write the known values with units before selecting a formula; units expose many setup mistakes before arithmetic begins.
- Percent of design equals measured divided by design times 100, while percent error and percent change answer different questions.
- Fan and pump flow varies with speed, pressure or head varies with speed squared, and power varies with speed cubed under the affinity-law assumptions.
- Square-root relationships require converting each velocity pressure before averaging velocities; averaging raw velocity pressures changes the result.
- A complete field calculation records inputs, units, formula, intermediate values, rounding, and a reasonableness check.
TAB Mathematics, Units, Ratios, and Formula Discipline
Why the setup matters more than the calculator
The current NEBB TAB Technician Seminar explicitly opens with a math and formula-chart review. A calculator can evaluate a square root, but it cannot decide whether 24 by 12 means inches, feet, duct dimensions, or a velocity grid. Most field calculation errors begin before the keys are pressed: a wrong unit, an outside rather than inside duct dimension, a gauge pressure used where differential pressure is required, or an inverted ratio.
Use a repeatable six-step method:
- State the quantity being solved and its required unit.
- List known values with units and the measurement condition.
- Select a formula whose variables match those quantities.
- Convert units before substituting.
- Carry enough intermediate digits and round only the reported result.
- Test whether the magnitude and direction make physical sense.
Core arithmetic patterns
| Pattern | Formula | TAB use |
|---|---|---|
| Ratio | $R = X_2/X_1$ | Speed, diameter, measured/design comparisons |
| Percent of design | $100(Q_{measured}/Q_{design})$ | Outlet, branch, fan, coil, and pump results |
| Percent change | $100[(X_2-X_1)/X_1]$ | Effect of an adjustment from an initial state |
| Average | $sum X/n$ | Equal-weight point values after required conversions |
| Area conversion | $in^2/144 = ft^2$ | Rectangular and round duct airflow |
| Pressure conversion | $1 psi approx 2.31 ft water$ | Hydronic gauge and head calculations for water |
Do not confuse the three percentage questions. If a 400 CFM outlet measures 360 CFM, it is at $360/400 × 100 = 90%$ of design. Its shortfall is 10% of design. If it began at 300 CFM and rose to 360 CFM, the increase from the initial condition is $60/300 × 100 = 20%$. All are mathematically correct, but they describe different relationships.
Direct, square, cube, and square-root relationships
For the same fan or pump operating in a comparable system regime, the affinity laws use a speed ratio $r = N_2/N_1$:
- Flow ratio: $Q_2/Q_1 = r$
- Fan pressure or pump head ratio: $P_2/P_1 = r^2$
- Power ratio: $BHP_2/BHP_1 = r^3$
If speed rises 10%, $r=1.10$. Flow is predicted to rise 10%, pressure or head about 21%, and power about 33.1%. The cubic result is why a modest airflow increase can exceed a motor or drive limit. These predictions assume the same machine geometry and a compatible system curve; they are not permission to exceed nameplate, VFD, sheave, impeller, vibration, or electrical limits.
Velocity from velocity pressure is a square-root relationship. At standard air conditions, the NEBB formula chart gives $V = 4005 × √VP$. If two points measure 0.04 and 0.16 in. w.g., their velocities are approximately 801 and 1,602 FPM. The correct average velocity is about 1,202 FPM. Averaging the two pressures first gives 0.10 in. w.g. and about 1,267 FPM—a different and incorrect result for an equal-area traverse.
Significant figures and resolution
Report precision supported by the instrument and method. A meter displaying 0.01 psi does not justify reporting 6.537 psi. Premature rounding can matter when a square or cube follows. Keep guard digits in intermediate work, then round to the project form's sensible resolution.
Also distinguish accuracy, resolution, and repeatability. More displayed digits mean finer resolution, not necessarily better accuracy. Three readings that agree closely may be repeatable but all biased by a zero error. A valid calculation cannot repair invalid input data.
A worked mixed example
A rectangular duct measures 30 by 18 inches internally. Area is $(30 × 18)/144 = 3.75 ft^2$. A valid traverse produces an average velocity of 1,280 FPM, so airflow is $3.75 × 1,280 = 4,800 CFM$. Design airflow is 5,000 CFM, so percent of design is $4,800/5,000 × 100 = 96.0%$.
If a speed adjustment is being considered, the flow ratio needed is $5,000/4,800=1.0417$. The simple affinity prediction is a 4.17% speed increase. Predicted power ratio is $1.0417³ ≈ 1.130$, or about a 13.0% increase. Before changing speed, verify motor load, drive limits, fan curve, system condition, controls, and authorization.
Field plausibility checks
Ask these questions before accepting a result:
- Does increasing speed increase—not decrease—the predicted flow?
- Is velocity compatible with the measured velocity pressure and air density?
- Does the sum of terminal flows reasonably agree with a main-duct traverse after leakage and measurement uncertainty?
- Does a hydronic pressure drop correspond to the correct valve setting and manufacturer chart?
- Did a result change because the system changed, or only because the method or instrument changed?
Good TAB math is traceable reasoning. Write enough that another technician can reproduce the result without guessing which number, unit, or assumption you used.
An outlet designed for 500 CFM measures 440 CFM. What is its percent of design?
A fan speed increases by 10% under affinity-law assumptions. What is the predicted power multiplier?
Why should velocity pressure readings be converted to velocity before averaging an equal-area traverse?