## Fluid Flow – General

### Pipe Friction (Major Losses)

The resistance to the incompressible flow of any fluid (head loss) in any pipe may be computed from the equation:

(3.A.1) $$h_f = f * {L \over D} * {v^2 \over 2g}$$

where:

• hf = Frictional resistance (head loss) in
• L = Length of pipe in
• D = Average internal diameter of pipe in
• v = Average velocity in pipe in
• g = Acceleration due to gravity
• f = Friction factor

The Colebrook Equation (portrayed below) offers a reliable means for computing the Darcy-Weisbach friction factor friction factor (f) to be used in Equation (3.1).

(3.A.2) $${1 \over \sqrt f} = -2 \log_{10}({\epsilon \over (3.7*D)} + [{2.51 \over Re \sqrt f}])$$

where:

• ε = absolute roughness, feet
• Re = Reynolds number for fluid flow

Another common form, which can be solved without iteration, is shown below.

(3.A.3) $${1 \over \sqrt f} = -2 \log_{10}({\epsilon/D \over 3.7} + {5.74 \over Re^{0.9}})$$

The Reynolds number (Re) is a non-dimensional ratio of inertial forces to viscous forces and is used to help scale data over a range of pipe sizes, fluid properties, and flow conditions. It is used as the basis for the Moody Diagram to determine friction factors and pressure/head losses.

The Reynolds number is defined as:

(3.A.4) $$Re = {ρvD \over μ} = {vD \over 𝜈} = {QD \over {𝜈A}}$$

where, for cylindrical pipes:

• ρ is fluid density in
• v is fluid velocity in
• D is pipe inner diameter in
• μ is dynamic viscosity in
• 𝜈 is kinematic viscosity in
• Q is volumetric flow rate in
• A is pipe cross-sectional area in

At Reynolds numbers less than about 2000, the flow tends to be laminar where it is traveling in a smooth, orderly manner with little mixing. At Reynolds numbers higher than about 4000, the flow is considered turbulent, with eddies forming and irregular motion.

### Pipe Roughness

Pipe roughness varies with pipe material, age, usage, fluid transport and lining. This table gives example values for some clean materials.

Roughness Values
Material Roughness, ε (ft)
Commercial steel or wrought iron 0.00015
Drawn tubing - glass, brass, plastic 0.000005
Galvanized iron 0.0005
Cast iron - uncoated 0.00085
Cast iron - asphalt dipped 0.0004
Concrete 0.001 - 0.01
Riveted steel 0.003 - 0.03
Wood stave 0.0006 - 0.003
Brass or copper (tubing) 0.000005
Fiberglass 0.0000164
Stainless steel 0.00005
Rubber, smoothed 0.000033
Carbon steel (cement-lined) 0.005
Water mains with tuberculations 0.004
Roughness Values
Material Roughness, ε (mm)
Commercial steel or wrought iron 0.0457
Drawn tubing - glass, brass, plastic 0.00152
Galvanized iron 0.152
Cast iron - uncoated 0.26
Cast iron - asphalt dipped 0.122
Concrete 0.305 - 3.05
Riveted steel 0.914 - 9.14
Wood stave 0.18 - 0.91
Brass or copper (tubing) 0.0015
Fiberglass 0.005
Stainless steel 0.015
Rubber, smoothed 0.01
Carbon steel (cement-lined) 1.5
Water mains with tuberculations 1.2

Determining the frictional roughness for old pipe is beyond this tutorial. Deterioration of pipes with age depends on the particular chemical properties of the fluid and the metal with which it is in contact. It is recommended that prior experience or testing be used to determine an accurate value. For commercial installations, it is recommended that 15 percent be added to the values shown above.

### Valves and Fittings (Minor Losses)

The resistance to flow (head loss) caused by a valve or fitting may be computed from the equation:

(3.A.5) $$h_f = K*{v^2 \over 2g}$$

where:

• hf = Frictional resistance (head loss) in of fluid
• v = Average velocity in in a pipe of corresponding diameter
• g = Acceleration due to gravity,
• K = Resistance coefficient for valve or fitting

Values of (K) for valves and fittings may be referenced below, and in Losses in Valves, Fittings, and Bends. Wide differences are found in the published values of (K) for all types of valves and fittings. The available data are inconclusive. As indicated in the above references section, flanged valves and fittings usually exhibit lower resistance coefficients than screwed valves and fittings. The resistance coefficients decrease with the increasing size of most valves and fittings.

Component (minor) losses can be summed together with the pipe losses to determine an overall frictional loss for the system, producing the equation

(3.A.6) $$h_f = {({fL \over D} + ΣK) * {v^2 \over 2g}}$$

where:

• f = pipe friction factor
• L = pipe length in
• D = pipe inside diameter in
• ΣK = sum of the minor losses, which includes losses across valves

Resistance coefficients for pipe bends with less than 90 degree deflection angles as reported by Wasielewski(g) 13 are shown in Section IIIB, Fig. 4. The curves shown are for smooth surfaces but may be used as a guide to approximating the resistance coefficients for surfaces of moderate roughness such as clean steel and cast iron. Figs. 3 and 4 in Section IIIB are not reliable below R/D = 1, where R is the radius of the elbow in feet. The approximate radius of a flanged elbow may be obtained by subtracting the flange thickness from the center-to-face dimension. The center-to-face dimension for a reducing elbow is usually identical with that of an elbow of the same straight size as the larger end.

The resistance to flow (head loss) caused by a sudden enlargement may be computed from the equation:

(3.A.7) $$h_{f} = K{{(v_1 - v_2)^2} \over {2g}}$$

$$= K(1 - {A_1 \over A_2})^2 {v_1^2 \over 2g}$$

$$= K[1 - ({D_1 \over D_2})^2]^2{v_1^2 \over 2g}$$

$$= K[{({D_2 \over D_1})^2 - 1 }]^2{v_2^2 \over 2g}$$

where:

• h = Frictional resistance (head loss) in
• v1 = Average velocity in in the smaller (upstream) pipe
• A1 = Internal cross-sectional area of the smaller pipe in
• D1 = Internal diameter of the smaller pipe in
• v2, A2, D2 = Corresponding values for the larger (downstream pipe)
• g = Acceleration due to gravity,
• K = Resistance coefficient, usually taken as unity since the variation is almost always less than ±3 per cent.

Equation (3.A.7) is useful for computing the resistance to flow caused by conical increasers and diffusers. Values of (K) for conical increasers based on data reported by Gibson(g) 14 are given in Section IIIB, Fig. 5 or may be computed by the equation:

(3.A.8) $$K = 3.50{(\tan(\theta/2)})^{1.22}$$

where:

• θ = total conical angle of the increaser in degrees

Equation (3.8) applies only to values of θ between 7.5 and 35 degrees. Noteworthy is the fact that above 50 degrees a sudden enlargement will be as good or better than a conical increaser. Values of (k) for conical diffusers as reported by Reference 11 of Section VI are shown in Section IIIB, Fig. 5. The values shown include the entrance mouthpiece which accounts in part for the increase over Gibson's values for conical increasers.

### Friction Factor Diagrams

As previously stated, the resistance to the incompressible flow of any fluid (head loss) in any pipe may be computed from equation (3.1)

$$h_f = f * {L \over D} * {v^2 \over 2g}$$

Values of (f) may be obtained directly from Fig. 3.A.2 based on Reynolds Number (Re) and Relative Roughness (ε/D) of the pipe. Values of (ε) may be obtained from the Roughness Values table in Section 3.A where the pipe is new clean asphalt-dipped cast iron, new clean commercial steel or wrought iron, or other materials shown. There will be probable variations in (ε) for these materials.

Values of the kinematic viscosity (v) at various temperatures are given in Fig. IIIA-5 for a number of different fluids. The Reynolds Number assistanceis calculated using Equation 3.4. It is absolutely essential that viscosities obtained from sources be expressed in the correct units, typically sq ft/sec or stokes or centistokes. Kinematic viscosities measured in stokes or centistokes may be converted to v in sq ft/sec by the formula:

(3.A.9) $$v = 0.00107639 * stokes$$ $$= 0.0000107639 * centistokes$$

Further information on viscosity can be found here.

If the Reynolds Number is less than 2000, the flow is laminar and the friction factor for any fluid in any pipe is given by the equation:

(3.A.10) $$f = {64 \over Re}$$

If the Reynolds Number is above 4000, the flow will usually be turbulent and the Moody Diagram pictured below can be used to determine the friction factor. The range Re = 2000–4000 is called the critical zone in which the flow may be highly unstable and the friction factor indeterminate.

### Moody Diagram [1]

A Frictional Loss Calculator is available to determine the pipe friction according to this methodology.

### References

1. L.F. Moody, "Friction factors for Pipe Flow", Trans. A.S.M.E., Vol 66, 1944

Last updated on May 12th, 2020