Every foot of pipe, every elbow, every valve in a piping system adds resistance to flow. That resistance shows up as pressure drop, and it directly affects pump sizing, energy consumption, and system capacity. Undersized pipe means excessive pressure drop, higher pumping costs, and potentially insufficient flow at the point of use. Oversized pipe wastes capital on materials and installation.
The Darcy-Weisbach equation is the foundation of pressure drop calculations for incompressible flow in closed conduits. It works for any Newtonian fluid — water, oil, glycol, brine — and accounts for pipe diameter, length, flow velocity, fluid density, and the friction between the fluid and the pipe wall.
This guide covers the Darcy-Weisbach equation, the Moody chart for friction factor determination, equivalent length methods for fittings, and practical pipe sizing strategies used in plant engineering.
The Darcy-Weisbach Equation
The Darcy-Weisbach equation calculates friction head loss in a pipe: h_f = f x (L/D) x (V² / 2g), where h_f is head loss in feet of fluid, f is the Darcy friction factor (dimensionless), L is pipe length in feet, D is inside diameter in feet, V is fluid velocity in feet per second, and g is gravitational acceleration (32.2 ft/s²).
To convert head loss to pressure drop in PSI: ΔP = h_f x ρ / 144, where ρ is fluid density in lb/ft³. For water at 60°F, ρ = 62.4 lb/ft³, so 1 foot of head equals 0.433 PSI. For fluids denser than water (such as brine or heavy oil), the same head loss translates to a higher pressure drop in PSI.
The key variable in Darcy-Weisbach is the friction factor f, which depends on the Reynolds number (flow regime) and the relative roughness of the pipe wall. Getting the friction factor right is the critical step in any pressure drop calculation, and it separates a useful engineering estimate from a meaningless guess.
Note that velocity appears as V squared, meaning that doubling the flow velocity quadruples the friction loss. This is why maintaining reasonable fluid velocities is so important — the penalty for high velocity is steep and non-linear.
Friction Factor and Reynolds Number
The Reynolds number determines whether flow is laminar or turbulent: Re = V x D x ρ / μ, where μ is dynamic viscosity. Below Re = 2000, flow is laminar and the friction factor is simply f = 64 / Re. Above Re = 4000, flow is turbulent and f depends on both Re and pipe roughness. Between 2000 and 4000 is the transition zone, where flow is unpredictable.
For turbulent flow, the Colebrook-White equation gives the friction factor: 1/sqrt(f) = -2 log₁₀(ε/D/3.7 + 2.51/(Re x sqrt(f))). This is an implicit equation — f appears on both sides — and must be solved iteratively or with the Moody chart. The Moody chart plots f on the vertical axis against Re on the horizontal axis, with curves for different relative roughness values (ε/D).
Common pipe roughness values (ε) are: commercial steel = 0.0018 inches, drawn tubing = 0.00006 inches, cast iron = 0.0102 inches, concrete = 0.012 to 0.12 inches, PVC and HDPE = 0.00006 inches. As pipes corrode and scale, roughness increases. A 20-year-old carbon steel pipe may have an effective roughness of 0.01 inches or more, several times the new-pipe value.
For quick estimates, the Swamee-Jain approximation gives f explicitly: f = 0.25 / [log₁₀(ε/(3.7D) + 5.74/Re^0.9)]². This is accurate to within 1-2 percent of the Colebrook-White equation for Re between 5000 and 10^8 and ε/D between 10^-6 and 0.05.
Fitting and Valve Losses
Fittings (elbows, tees, reducers) and valves add pressure drop beyond straight pipe friction. The two common methods for accounting for these losses are the equivalent length method and the K-factor (resistance coefficient) method.
The equivalent length method converts each fitting to an equivalent length of straight pipe that would produce the same pressure drop. For example, a standard 90-degree elbow in 4-inch schedule 40 steel pipe is equivalent to roughly 11 feet of straight pipe (approximately 33 pipe diameters). A fully open gate valve is about 8 pipe diameters. These equivalent lengths are added to the actual pipe length in the Darcy-Weisbach equation.
The K-factor method calculates fitting loss directly: h_f = K x V²/(2g). Each fitting type has a K value that depends on size, geometry, and sometimes Reynolds number. Common K values include: 90° standard elbow = 0.75, 90° long-radius elbow = 0.45, standard tee (flow through branch) = 1.8, gate valve (fully open) = 0.17, globe valve (fully open) = 6.0 to 10.0, check valve (swing type) = 2.0 to 3.5.
Globe valves deserve special attention because of their extremely high resistance. A fully open globe valve in a 4-inch line can produce as much pressure drop as 100+ feet of straight pipe. Wherever possible, use gate valves or butterfly valves for isolation service and reserve globe valves for throttling applications where the high resistance is intentional.
Pipe Sizing and Velocity Limits
Industry practice establishes recommended velocity ranges for different services. For water in general plant piping, target 4 to 8 feet per second. Suction piping to pumps should be limited to 3 to 5 fps to prevent cavitation. Steam lines run 4,000 to 6,000 fpm for saturated steam and up to 10,000 fpm for superheated steam. Compressed air systems typically size for 20 to 30 fps in headers and 30 to 50 fps in branch lines.
To size a pipe, start with the required flow rate and an initial velocity assumption. Calculate the required inside diameter: D = sqrt(4Q / (π x V)), where Q is volumetric flow rate. Select the nearest standard pipe size and check the actual velocity. If it exceeds the recommended range, go up one size.
For long runs and critical systems, calculate the total pressure drop including fittings and compare it to the available pressure. If the pump or compressor cannot deliver the required pressure at the design flow, either increase the pipe size or reduce the number of fittings (for example, by using long-radius elbows instead of standard elbows, or by rerouting the pipe to eliminate unnecessary direction changes).
Economic pipe sizing balances capital cost (bigger pipe costs more) against operating cost (bigger pipe saves pumping energy). The optimum diameter is the one that minimizes the total lifecycle cost. For most plant piping, this analysis leads to velocities in the 5 to 7 fps range for water and 15 to 25 fps for compressed air.
Worked Example and Rules of Thumb
Consider a 200 GPM water flow through 300 feet of 4-inch schedule 40 steel pipe (ID = 4.026 inches) with six standard 90-degree elbows, two gate valves, and one check valve. Water temperature is 70°F (viscosity = 0.658 x 10^-5 ft²/s, density = 62.3 lb/ft³).
First, calculate velocity: V = Q / A = (200/449) / (π/4 x (4.026/12)²) = 0.445 / 0.0884 = 5.04 fps. Reynolds number: Re = V x D / ν = 5.04 x (4.026/12) / (0.658 x 10^-5) = 257,000 — well into turbulent flow. Using ε = 0.0018 inches for clean steel, ε/D = 0.000447. From the Swamee-Jain equation, f = 0.0188.
Straight pipe head loss: h_f = 0.0188 x (300/0.3355) x (5.04²/64.4) = 0.0188 x 894 x 0.394 = 6.62 feet. Fitting losses using equivalent lengths: 6 elbows x 11 ft + 2 gate valves x 2.7 ft + 1 check valve x 20 ft = 91.4 equivalent feet. Fitting head loss: 0.0188 x (91.4/0.3355) x 0.394 = 2.02 feet. Total: 8.64 feet = 3.74 PSI.
Rules of thumb for quick estimates: pressure drop in clean steel water piping runs roughly 1 to 4 PSI per 100 feet at typical velocities. Add 50 percent to straight-pipe calculations to account for fittings in a normally routed system. For compressed air, a well-designed distribution system should have no more than 1 PSI total drop from compressor discharge to the most remote use point. More than that indicates undersized piping or excessive fittings.