Pipe Flow

Pipe Flow

Friction Factor, Losses, Pump Matching

Scope

The tool treats steady, incompressible, fully developed flow of a liquid in a single pipe of constant diameter: the classic Darcy–Weisbach problem with the friction factor from the Colebrook equation, minor losses from fittings, the elevation change, the pump head and power, and the match between a pump curve and the system curve. Gases at low pressure drop (below about 10 % of the absolute pressure) can be treated the same way with their density at the mean pressure; compressible flow, two-phase flow, non-Newtonian liquids, open channels and pipe networks are outside its scope. For the sizing of a multi-segment hydronic water circuit against the usual velocity and friction-rate limits, use the HVAC area's Hydronic Pipe Sizing tool, which carries the same Colebrook friction factor through a table of segments.

Conventions and units

Lengths, diameters, flow rates, pressures and powers go through the site's unit dictionary (ft, in, gpm, psi, hp in USCS; m, mm, L/s, kPa, kW in SI, with the dropdown alternatives). Two inputs are plain numbers because the dictionary has no unit for them: the temperature (°F in USCS mode, °C in SI mode, not converted by the unit switch) and the dynamic viscosity of an "other liquid", entered in centipoise in both systems (1 cP = 1 mPa⋅s = 0.001 Pa⋅s = 2.089 × 10⁻⁵ lbf⋅s/ft² = 6.72 × 10⁻⁴ lbm/(ft⋅s)). The elevation rise is the outlet elevation minus the inlet elevation. All results are computed in SI and shown in the selected system.

Water properties

For water the density, dynamic viscosity and vapor pressure are interpolated linearly in a table of saturated-liquid values at every degree from the triple point (0.01 °C) to 100 °C, taken from the NIST Chemistry WebBook (IAPWS-95 equation of state for the density, IAPWS 2008 formulation for the viscosity). Below 100 °C the saturated-liquid values differ from those at atmospheric pressure by less than 0.01 %. A temperature outside the table is clamped to its nearest end with a note. The kinematic viscosity ν = μ/ρ is printed in ft²/s or m²/s. Moody's 1944 illustration took water at 60 °F with ν = 1.216 × 10⁻⁵ ft²/s from the tables of his day; the NIST value is 1.208 × 10⁻⁵ ft²/s, 0.7 % lower, which moves the Reynolds number by the same amount and the friction factor by far less.

Pipe dimensions and roughness

The nominal sizes give the inside diameter of ASME B36.10M carbon-steel pipe, outside diameter less twice the Schedule 40 or Schedule 80 wall (for example 2 in Sch 40: 2.375 − 2 × 0.154 = 2.067 in; 6 in Sch 40: 6.065 in). Copper tube, plastic pipe and ductile iron have other dimensions: enter their inside diameter as a custom size. The absolute roughness ε of new pipe follows Moody (1944) Fig. 2, as reproduced in White's Table 6.1: drawn tubing 0.000005 ft, commercial steel 0.00015 ft, asphalted cast iron 0.0004 ft, galvanized iron 0.0005 ft, cast iron 0.00085 ft; concrete (0.001–0.01 ft) and riveted steel (0.003–0.03 ft) are entered at the low end of their ranges and wood stave at 0.0006 ft. Moody himself warned that the roughness of a pipe in service is uncertain by far more than the chart's precision: tuberculation and scale in old water mains can multiply ε by ten, so a custom value from a field test or a design allowance is often the better input.

Reynolds number and regime

Re = ρVD/μ = VD/ν with the mean velocity V = Q/A. Below Re ≈ 2,300 the flow is laminar and the friction factor is exactly f = 64/Re (Hagen–Poiseuille), independent of roughness; above about 4,000 it is turbulent and follows the Colebrook equation; between the two it is transitional and no formula is reliable, the flow switching intermittently between the two states. The tool uses 64/Re in the laminar range, Colebrook above 4,000, and in the transition range the Colebrook value as the conservative (larger) choice, with a note.

Friction factor

Colebrook (1939) joined Prandtl's smooth-pipe law and von Kármán's fully rough law into one expression that fits the commercial-pipe data of Nikuradse and others through the transition between them:

1/√f = −2 log₁₀[ ε/(3.7 D) + 2.51/(Re √f) ]

Moody (1944) plotted it as the chart that carries his name. The equation is implicit in f; the tool iterates on x = 1/√f (x_n+1 = −2 log₁₀[ε/(3.7D) + 2.51 x_n/Re]), which converges in a dozen passes to a tolerance of 10⁻¹² starting from the Swamee–Jain value. Two explicit approximations are printed for comparison: Swamee and Jain (1976), f = 0.25 / [log₁₀(ε/(3.7D) + 5.74/Re^0.9)]², within about 1 % of Colebrook over most of the chart (up to 2.8 % at the low-Re, rough corner), and Haaland (1983), f = [−1.8 log₁₀((ε/(3.7D))^1.11 + 6.9/Re)]⁻², within 1.5 %. Either is accurate enough for design; the difference between them is a measure of how little the third digit of f means against the uncertainty of ε. The Moody chart on the results page is drawn from the same equation (Prandtl's law 1/√f = 2 log₁₀(Re √f) − 0.8 for the smooth line, 64/Re for the laminar line) with the pipe's point marked.

Head loss and pressure drop

The Darcy–Weisbach equation gives the friction loss of the straight pipe as h_f = f (L/D) V²/(2g), a head of the flowing liquid. Minor losses of fittings, valves, entrances and exits are h_m = ΣK V²/(2g) with each K a loss coefficient. The total h_L = h_f + h_m (plus the loss of any equivalent length) is the head the flow dissipates; the pressure drop in a horizontal pipe of constant diameter is Δp = ρ g h_L. With an elevation rise Δz between inlet and outlet, and equal pressures and velocities at the two free surfaces, the energy equation gives the head a pump must add as H = h_L + Δz (the total dynamic head); a drop larger than the losses means the flow runs by gravity. The hydraulic power is P = ρ g Q H and the shaft power P/η at the pump efficiency η.

Minor losses: the Crane method

Crane Technical Paper 410 expresses the loss coefficient of a fitting as a multiple of f_T, the friction factor of clean commercial steel pipe of the same size in fully turbulent flow: K = n f_T, with n = 30 for a standard 90° elbow, 16 for a 45° elbow, 14 for a 90° bend of r/d = 1.5, 20 and 60 for a tee with flow through the run and through the branch, 8 for a fully open gate valve, 340 for a globe valve, 150 for an angle valve, 3 for a full-bore ball valve, 45 for a butterfly valve of 2 to 8 in, 18 for a straightway plug valve, 100 for a swing check and 600 for a lift check valve. Crane's table of f_T (0.027 at ½ in down to 0.013 at 12 in) is the fully rough value [2 log₁₀(3.7 D/ε)]⁻² at ε = 0.00015 ft, which the tool evaluates at the pipe's actual inside diameter and uses for every material, as Crane does: the multiple n carries the size effect, the pipe's own roughness does not enter. Entrances and exits have fixed coefficients: inward-projecting 0.78, sharp-edged 0.5, well-rounded (r/d ≥ 0.15) 0.04, exit to a tank or the atmosphere 1.0 (the whole velocity head is lost). A coefficient from a manufacturer's data is entered directly, and a fitting given as an equivalent length of pipe is charged at the pipe's own friction factor. Valve coefficients vary widely between designs and with the degree of opening (a half-closed gate valve has K ≈ 2.7 rather than 8 f_T ≈ 0.15); the Crane values are for fully open valves of the conventional pattern.

The three problems

Given the pipe and the flow (or the velocity), the head loss follows directly. Given the head loss and asked for the flow, V is unknown and so are Re and f; the tool iterates: starting from the fully rough f, V = √[2 g h_L / (f (L + L_eq)/D + ΣK)] is evaluated, Re and f recomputed, and the loop repeated until f stops changing (the passes are tabulated). With friction only the iteration is unnecessary: White (7th ed., eq. 6.51) rewrites Colebrook in terms of the parameter ζ = g D³ h_f / (L ν²), which contains no unknown, so Re = −(8ζ)^½ log₁₀[ε/(3.7D) + 1.775/√ζ] directly; the page prints this value beside the iterated one. Given the flow and the head loss and asked for the diameter, the loss (f (L + L_eq)/D + ΣK) 8Q²/(π² g D⁴), with f and the Crane coefficients re-evaluated at each trial diameter, decreases monotonically with D, and the tool finds D by bisection; Swamee and Jain's explicit formula, D ≈ 0.66 [ε^1.25 (L Q²/(g h_f))^4.75 + ν Q^9.4 (L/(g h_f))^5.2]^0.04, good to about 2 %, is printed when there are no fittings, together with the nearest Schedule 40 size at or above the required diameter.

Pump matching and NPSH

The system curve H_sys(Q) = Δz + h_L(Q) rises roughly with Q² (exactly so only when f is constant; the tool recomputes Colebrook at every flow). A centrifugal pump's head falls with flow; where the two curves cross is the operating point, the flow the pump will actually deliver in this system. Paste the pump curve as flow–head pairs read from the manufacturer's chart; the tool fits a natural cubic spline through them, which passes exactly through the points and is smooth between, and finds the crossing by bisection. A design flow far from the operating point means the pump is the wrong size, or a throttling valve will be doing the work. The net positive suction head available, NPSH_a = (p_atm − p_v)/(ρ g) − z_lift − h_suction, is the absolute head at the pump inlet above the liquid's vapor pressure, with the suction lift positive when the pump sits above the liquid surface (negative for a flooded suction) and the suction-line losses entered separately. It must exceed the pump's NPSH required by a margin, typically 0.6 m (2 ft) or 10 %, or the liquid flashes in the impeller eye and the pump cavitates. For water the vapor pressure comes from the NIST table at the entered temperature; for another liquid enter it.

Reading the results and their precision

Moody's own illustration (200 ft of 6-in asphalted cast-iron pipe at 6 ft/s, water at 60 °F) read f ≈ 0.02 from the chart and gave a loss of 4.5 ft; the Colebrook equation gives f = 0.0199 and 4.46 ft, and White, who uses the same case as his Example 6.7, quotes a pressure drop of about 280 lbf/ft². The difference is the precision of a chart reading. Moody put the uncertainty of f at about ±5 % for smooth tubing and ±10 % for commercial steel pipe because of the roughness, and much more for pipe in service; the third significant figure the tool prints is for checking, not a claim of accuracy. The Reynolds number tells whether the laminar formula or Colebrook applies, the relative roughness whether the pipe is near the fully rough regime where f no longer depends on Re, and the comparison of friction and fitting losses shows which the design is sensitive to: in a short run with many valves the minor losses dominate and the roughness barely matters; in a long pipeline the reverse. The hydraulic power is what the liquid receives; the shaft power at the entered efficiency is what the motor must supply at the design point, and the operating-point power is what it will actually draw.

Limits

Single pipe of constant diameter; Newtonian liquid of constant properties; fully developed flow (entrance lengths of the order of 10 D in turbulent flow are neglected); fittings treated as independent (closely spaced fittings interact, usually to a smaller total than the sum); water properties from 0.01 to 100 °C; the Colebrook equation above Re = 4,000 and 64/Re below 2,300, the band between flagged; no pressure difference between the two free surfaces other than through the elevation term (add it to the elevation rise as an equivalent head if needed); pump curve interpolated, not extrapolated, so the operating point must lie within the pasted flow range.