4.3 Flow Calculations
Key Takeaways
- DP flow through an orifice, venturi, or nozzle is proportional to sqrt(ΔP/ρ); applying square-root extraction twice or not at all is a classic loop error.
- Square-edge orifice Cd is nearly constant only at high Reynolds number (typically Re_D on the order of 10^4 and above); below that, Cd moves with Re.
- Preferred beta ratio is about 0.3–0.6: lower β raises ΔP and permanent loss; higher β cuts loss but needs more upstream pipe and loses accuracy.
- Gas and steam mass or standard-volume flow from a DP element need P-T (and often Z) compensation; incompressible liquids usually need temperature only if density swings.
- scfh already sits at a defined standard T and P; convert to lb/h with standard density ρ = P_std M / (R T_std) and do not re-apply flowing T and P.
4.3 Flow Calculations
NCEES Measurement 1.H lists element sizing, pressure and temperature compensation, mass/volume conversion, pressure drop, velocity, Reynolds number, and beta ratio. Almost every PE Control Systems flow item is a rearrangement of one statement: for a differential-pressure element, mass flow is proportional to Cd × ε × d² × sqrt(ρ ΔP). Volume at flowing conditions is that mass divided by flowing density. Standard volume is mass divided by standard density. If you keep those three bases separate, the algebra is short.
Beta ratio is β = d/D (orifice bore over pipe inside diameter). The velocity-of-approach factor is 1/sqrt(1 − β⁴). High β means a large bore, a small ΔP for a given flow, less permanent head loss as a fraction of ΔP, more sensitivity to the upstream profile, and a Cd correlation that is less comfortable. Low β means a small bore, a large ΔP, more permanent loss, and a more stubborn requirement that the process can spare the pressure. A practical sizing band is about 0.3 to 0.6; ISO 5167 allows a wider window, but the edges are where exam distractors live.
Reynolds number and what Cd actually does
Re_D = ρ V D / μ. In common US liquid units a working form is
Re_D ≈ 3160 × Q_gpm × SG / (μ_cP × D_inches).
Velocity in a circular pipe is V(ft/s) ≈ 0.4085 × Q_gpm / D_inches².
For a square-edge concentric orifice, the discharge coefficient Cd sits near 0.60–0.61 once the pipe Reynolds number is high (think Re_D ≳ 10⁴, and more comfortably 10⁵). In that region you may treat Cd as constant for PE arithmetic. As Re drops toward transitional and laminar flow, Cd changes with Re (the Reader-Harris/Gallagher ISO 5167 function is the industrial statement of that fact). The qualitative exam point is not the fifth decimal of Cd. It is: a low-Re orifice is not the same meter as the high-Re catalog plate, and you cannot “fix” laminar flow by applying a turbulent Cd.
Permanent pressure loss for an orifice is on the order of ΔP × (1 − β²)/(1 + β²). That is why a β = 0.2 plate that looks accurate on paper can still be a pump-horsepower problem, and why a β = 0.75 plate that “saves” ΔP may fail the straight-run and uncertainty test.
Square-root extraction
A DP transmitter’s 4–20 mA is linear with ΔP unless you configure square-root in the head. Flow is linear with sqrt(ΔP). Extract once, in the transmitter or in the DCS, not both. If ΔP is 25% of span, flow is 50% of span. Near zero, square-root magnifies noise, which is why loops need a low-flow cutoff. A single orifice’s flow rangeability is often about 3:1 to 4:1 (9:1 to 16:1 in ΔP); stacked DP transmitters extend that. Mag, vortex (with limits), ultrasonic, and Coriolis do not get a square-root block just because someone copied an orifice template.
Worked example A — Re, velocity, and beta
Water at 60°F (SG = 1.00, μ = 1.1 cP) flows at 200 gpm in 4-inch Schedule 40 pipe (D = 4.026 in). The orifice bore is d = 2.000 in.
β = 2.000 / 4.026 = 0.497 (in the preferred band).
V = 0.4085 × 200 / (4.026)² = 81.70 / 16.21 = 5.04 ft/s (a reasonable liquid velocity, not an erosion special).
Re_D = 3160 × 200 × 1.00 / (1.1 × 4.026) = 632,000 / 4.429 = 1.43 × 10⁵.
That Reynolds number is firmly turbulent. Cd is in the flat high-Re region, so you do not apply a laminar correction. If a distractor sized β = 0.75 “to save pressure drop,” ΔP would fall but ISO straight-run requirements and Cd uncertainty would rise. If a distractor sized β = 0.20, Re at the bore would still be turbulent here, but the permanent loss fraction of ΔP would climb toward ~90% and the pump would pay for a plate that is easy to calculate and hard to live with.
Worked example B — scfh to lb/h
Convert 2,500 scfh of a hydrocarbon gas, MW = 18.0, ideal at the US standard 14.696 psia and 60°F (520°R), to lb/h. Use R = 10.73 psia·ft³/(lbmol·°R).
Standard density: ρ_std = P M / (R T) = (14.696 × 18.0) / (10.73 × 520) = 264.53 / 5,580 = 0.0474 lb/scf.
Mass flow: ṁ = 2,500 scf/h × 0.0474 lb/scf = 119 lb/h.
The flowing line happens to be 80 psig and 90°F. Do not use those numbers. Standard cubic feet are already referred to 14.696 psia and 60°F. Re-applying P/T would be the actual-cubic-foot conversion.
Check the trap with actual cubic feet: 2,500 acfh at 94.7 psia and 550°R would be ρ = (94.7 × 18.0) / (10.73 × 550) = 0.289 lb/ft³ and ṁ = 2,500 × 0.289 = 722 lb/h — a different problem with a different answer.
P-T compensation: when the sqrt(ρ ΔP) term moves
For an ideal gas, ρ ∝ P M / T (insert Z if the problem gives it). Mass flow from a DP element therefore scales as
ṁ / ṁ_des = sqrt( (ΔP/ΔP_des) × (P/P_des) × (T_des/T) × (Z_des/Z) × (M/M_des) ).
If flowing temperature rises at fixed P and ΔP, mass flow falls. If flowing pressure rises, mass flow rises. Saturated steam uses steam-table density from pressure (and quality if wet). Superheated steam needs P and T. Liquids are incompressible for PE purposes: skip pressure compensation unless the problem is a near-critical fluid; apply temperature compensation when density actually moves (hot hydrocarbons, wide T span).
| Measured / reported basis | Compensate DP for P and T? | Notes |
|---|---|---|
| Liquid volume, nearly constant T | No | Density is stable |
| Liquid mass, or hot hydrocarbon volume | T (density) | P usually negligible |
| Gas mass, or scfh, from orifice/venturi/nozzle | P and T (and Z if given) | ṁ ∝ sqrt(ρ ΔP), ρ ∝ P/(Z T) |
| Saturated steam mass | P (steam table) | Quality if the steam is wet |
| Superheated steam mass | P and T | Superheat from both |
| Mag meter volume reported as mass | Density (T or composition) | Mag is volumetric |
| Coriolis mass | Not for mass | Already mass; density is a measured extra |
Element sizing on the exam is usually “is β reasonable, is Re high enough for a constant Cd, and did you compensate the right basis?” rather than a full ISO 5167 iteration. Do the Re and β checks first, then the unit conversion, then compensation.
A DP orifice transmitter outputs 4–20 mA linear with differential pressure. The DCS must report volumetric flow. Which statement is correct?
Water at 60°F (SG = 1.00, μ = 1.1 cP) flows at 200 gpm in 4-inch Sch 40 pipe (D = 4.026 in) through a 2.000 in orifice. What do Re_D and β imply?
2,500 scfh of ideal gas (MW = 18.0) is referred to 14.696 psia and 60°F. R = 10.73 psia·ft³/(lbmol·°R). The flowing line is 80 psig and 90°F. Mass flow is closest to which value?