Pipe and Pumping Formulas
Introduction
This calculation is valid for water flowing at typical temperatures found in municipal water supply systems. Our calculation is based on the steady
state incompressible energy equation utilizing Hazen-Williams friction losses as well as minor losses. The calculation can compute flowrate,
velocity, pipe diameter, elevation difference, pressure difference, pipe length, minor loss coefficient, and pump head (total dynamic head).
Piping Scenarios
Since boundary conditions affect the flow characteristics, our calculation allows you to select whether your locations 1 and 2 are within pipes, at the
surface of open reservoirs, or in pressurized mains (same as pressurized tank). If there is no pump between locations 1 and 2, then enter the pump
head (Hp) as 0.
Steady State Energy Equation used for this page
Back to Calculation
The first equation shown is the steady state energy equation for incompressible flow. The left side of the equation contains what we call the driving
heads. These heads include heads due to a pump (if present), elevation, pressure, and velocity. The terms on the right side are friction loss and minor
losses. Friction losses are computed using the Hazen-Williams friction loss equation. The energy equation is well-accepted in the field of fluid
mechanics and can be found in many references such as Cimbala and Cengel (2008), Munson et al. (1998), and Streeter et al. (1998), while the
Hazen-Williams equation for friction losses is well-established in the water supply literature and can be found in references such as Viessman and
Hammer (1998) and Mays (1999).
The Hazen-Williams equation (the hf =... equation) is empirical and requires that you use particular units as noted below. Though the other equations
are dimensionally correct, only units that can be used in all of the equations are shown below. Our calculation allows you to enter a variety of units
and automatically performs the unit conversions.
ft=foot, lb=pound, m=meter, N=Newton, s=second
A = Pipe cross-sectional area, ft2 or m2.
C = Hazen-Williams pipe roughness coefficient. See table below for values.
D = Pipe diameter, ft or m.
Driving Head (DH) = left side of the first equation (or right side of the equation), ft. This is not total dynamic head.
g = acceleration due to gravity = 32.174 ft/s2 = 9.8066 m/s2.
hf = Major (friction) losses, ft or m.
hm = Minor losses, ft or m.
Hp = Pump head (also known as Total Dynamic Head), ft or m.
k = unit conversion factor = 1.318 for English units = 0.85 for Metric units
Km = Sum of minor losses coefficients. See table below.
Pump Power (computed by program) = SQHp, lb-ft/s or N-m/s. Theoretical pump power. Does not include an inefficiency term. Note that 1
horsepower = 550 ft-lb/s.
P1 = Upstream pressure, lb/ft2 or N/m2.
P2 = Downstream pressure, lb/ft2 or N/m2.
Q = Flow rate in pipe, ft3/s or m3/s.
S = Weight density of water = 62.4 lb/ft³ for English units = 9800 N/m³ for Metric units
V = Velocity in pipe, ft/s or m/s.
V1 = Upstream velocity, ft/s or m/s.
V2 = Downstream velocity, ft/s or m/s.
Z1 = Upstream elevation, ft or m.
Z2 = Downstream elevation, ft or m.
All of the calculations on this page have analytic (closed form) solutions except for "Solve for V, Q" and "Q known. Solve for Pipe Diameter." These
two calculations required a numerical solution. Our solution utilizes a modified implementation of Newton's method that finds roots of the equations
with the result accurate to 8 significant digits. All of the calculations utilize double precision.
Table of Hazen-Williams Coefficients (C is unit-less)
Material
Asbestos Cement
C
140
Material
Copper
C
130-140
Brass
130-140
Galvanized iron
120
Brick sewer
100
Glass
140
Lead
130-140
Plastic
140-150
Cast-Iron:
New, unlined
130
10 yr. old
107-113
20 yr. old
89-100
Coal-tar enamel lined
145-150
30 yr. old
75-90
New unlined
140-150
40 yr. old
64-83
Riveted
110
Steel forms
140
Tin
130
Wooden forms
120
Vitrif. clay (good condition)
110-140
Centrifugally spun
135
Wood stave (avg. condition)
120
Steel:
Concrete/Concrete-lined:
Table of Minor Loss Coefficients (Km is unit-less)
Back to Calculation
Compiled from References
Fitting
Km
Valves:
Fitting
Km
Elbows:
Globe, fully open
10
Regular 90°, flanged
0.3
Angle, fully open
2
Regular 90°, threaded
1.5
Gate, fully open
0.15
Long radius 90°, flanged
0.2
Gate 1/4 closed
0.26
Long radius 90°, threaded
0.7
Gate, 1/2 closed
2.1
Long radius 45°, threaded
0.2
Gate, 3/4 closed
17
Regular 45°, threaded
0.4
Swing check, forward flow
2
Swing check, backward flow
infinity
180° return bends:
Tees:
Line flow, flanged
0.2
Line flow, threaded
0.9
Flanged
0.2
Branch flow, flanged
1.0
Threaded
1.5
Branch flow, threaded
2.0
Pipe Entrance (Reservoir to Pipe):
Pipe Exit (Pipe to Reservoir)
Square Connection
0.5
Square Connection
1.0
Rounded Connection
0.2
Rounded Connection
1.0
Re-entrant (pipe juts into tank)
1.0
Re-entrant (pipe juts into tank)
1.0
Common Questions
Back to Calculation
I took fluid mechanics a long long time ago. What is head? Why does it have units of length? Head is energy per unit weight of fluid (i.e. Force x
Length/Weight = Length).The program on this page solves the energy equation (shown below); we call energy "head."
Why is Pressure=0 for a reservoir? A reservoir is open to the atmosphere, so its gage pressure is zero.
Why is Velocity=0 for a reservoir? This is a common assumption in fluid mechanics and is based on the fact that a reservoir has a large surface area.
Therefore, the water level drops very little even if a lot of water flows out of the reservoir. A reservoir may physically be a lake or a large diameter
tank.
What is a "main" and a "lateral"? A "main" is a large diameter water supply pipe that has many smaller diameter "laterals" branching off of it to
supply water to individual residences, businesses, or sub-divisions. In fluid mechanics, we set V=0 for the main since it has a large diameter (relative
to the lateral) and thus a very small velocity. To further justify the V=0 assumption, the main's pressure is typically high, so the velocity head in the
main is negligible. The main is drawn such that it is coming out of your computer monitor.
Can I model flow between two reservoirs using either Scenario B or E? Yes, you can. If using Scenario E, just set P1-P2=0. Scenario B automatically
sets P1-P2=0.
Can I model flow between two mains using either Scenario B or E? Only if the pressure is the same in both mains.
How do I model a pipe discharging freely to the atmosphere? Use Scenario A, C, or F. Since P2=0 (relative to atmospheric pressure), P1-P2 that is
input or output will be P1.
What are minor losses? Minor losses are head (energy) losses due to valves, pipe bends, pipe entrances (for water flowing from a tank to a pipe), and
pipe exits (water flowing from a pipe to a tank), as opposed to a major loss which is due to the friction of water flowing through a length of
pipe. Minor loss coefficients (Km) are tabulated below. For our program, all of the pipes have the same diameter, so you can add up all your minor
loss coefficients and enter the sum in the Minor Loss Coefficient input box.
I'm confused about pumps. Only input Pump Head if the pump is between points 1 and 2. Otherwise, enter 0 for Pump Head.
Your program is great! What are its limitations? Pipes must all have the same diameter. Pump curves cannot be implemented. The fluid must be
water.
Where can I find additional information? References
What is Driving Head? See above definitions of variables. It is not total dynamic head. Hp is pump head (also known as total dynamic head).