8.3.1: Computing Losses Due to Friction
- Page ID
- 44578
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Several equations have been developed to calculate the friction loss in pipelines. A widely used empirical method is the Hazen-Williams Equation. The Hazen-Williams Equation for circular pipes is given by:
\(h_f = 1054 F\left(\dfrac{Q}{C}\right)^{1.852}\left(\dfrac{1}{d^{4.866}}\right) \) (8.11a)
or
\(P_f = 456 F\left(\dfrac{Q}{C}\right)^{1.852}\left(\dfrac{1}{d^{4.866}}\right) \) (8.11b)
where: hf = friction loss, ft of head/100 ft of pipe,
Pf = friction loss, psi/100 ft of pipe,
Q = flow rate (gpm),
d = inside diameter of the pipe (in),
C = roughness coefficient, and
F = outlet factor.
Friction loss increases as flow velocity increases. This fact is incorporated, but somewhat hidden in Equation 8.11. Equation 8.11 is applicable to essentially all pipelines used in surface and sprinkler irrigation. However, for small diameter pipelines, such as laterals that are used in microirrigation, a more appropriate equation is the Darcy-Weisbach equation which will be applied in Chapter 14. The roughness coefficient, C, accounts for the roughness of the wall of the pipe. Representative C values for different types of pipe materials are summarized in Table 8.1. As the roughness of the pipe wall increases C decreases. Of the materials in Table 8.1, steel pipe is the roughest material while PVC is the smoothest. Table 8.2a and b contain pressure losses due to friction for selected pipe materials and diameters based on the Hazen-Williams equation.
| Material C | |
|---|---|
| Aluminum pipe with couplers | 120 |
| Aluminum pipe with gates | 110 |
| Cement asbestos pipe | 140 |
| Galvanized steel pipe | 140 |
| Standard steel pipe | 100 |
| PVC | 150 |
| PVC pipe with gates | 130 |
| Q (gpm) | Pressure Loss Due to Friction (psi/100ft) | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Aluminum Sprinkler Pipe, 150 psi Rating, C=120 | PVC IPS Class 160, C=150 | ||||||||
| Nominal Diameter=2 in. | Nominal Diameter=3 in. | Nominal Diameter=4 in. | Nominal Diameter=6 in. | Nominal Diameter=2 in. | Nominal Diameter=2.5 in. | Nominal Diameter=3 in. | Nominal Diameter=4 in. | Nominal Diameter=6 in. | |
| Inside Diameter= 5.898 in. | Inside Diameter= 5.898 in. | Inside Diameter= 5.898 in. | Inside Diameter= 5.898 in. | Inside Diameter= 5.898 in. | Inside Diameter= 5.898 in. | Inside Diameter= 5.898 in. | Inside Diameter= 5.898 in. | Inside Diameter= 5.898 in. | |
| 2 | 0.01 | ||||||||
| 4 | 0.04 | 0.01 | |||||||
| 6 | 0.08 | 0.03 | |||||||
| 8 | 0.13 | 0.04 | |||||||
| 10 | 0.20 | 0.03 | 0.07 | 0.03 | |||||
| 15 | 0.43 | 0.05 | 0.14 | 0.06 | |||||
| 20 | 0.73 | 0.09 | 0.24 | 0.09 | 0.04 | ||||
| 25 | 1.10 | 0.14 | 0.36 | 0.14 | 0.06 | ||||
| 30 | 1.54 | 0.20 | 0.51 | 0.20 | 0.08 | ||||
| 35 | 2.05 | 0.26 | 0.68 | 0.27 | 0.10 | ||||
| 40 | 2.63 | 0.34 | 0.87 | 0.34 | 0.13 | ||||
| 45 | 3.27 | 0.42 | 1.08 | 0.42 | 0.16 | ||||
| 50 | 3.97 | 0.51 | 0.12 | 1.31 | 0.52 | 0.20 | 0.06 | ||
| 55 | 4.74 | 0.61 | 0.14 | 1.56 | 0.62 | 0.24 | 0.07 | ||
| 60 | 5.57 | 0.71 | 0.17 | 1.83 | 0.72 | 0.28 | 0.08 | ||
| 65 | 6.46 | 0.83 | 0.20 | 2.13 | 0.84 | 0.32 | 0.10 | ||
| 70 | 7.41 | 0.95 | 0.22 | 2.44 | 0.96 | 0.37 | 0.11 | ||
| 75 | 8.42 | 1.08 | 0.25 | 2.77 | 1.09 | 0.42 | 0.12 | ||
| 80 | 9.49 | 1.21 | 0.29 | 3.12 | 1.23 | 0.47 | 0.14 | ||
| 85 | 10.61 | 1.36 | 0.32 | 3.49 | 1.38 | 0.53 | 0.16 | 0.02 | |
| 90 | 11.80 | 1.51 | 0.36 | 0.05 | 3.88 | 1.53 | 0.59 | 0.17 | 0.03 |
| 100 | 14.34 | 1.83 | 0.43 | 0.06 | 4.72 | 1.86 | 0.72 | 0.21 | 0.03 |
|
110 |
17.11 | 2.19 | 0.52 | 0.07 | 5.63 | 2.22 | 0.86 | 0.25 | 0.04 |
| 120 | 20.10 | 2.57 | 0.61 | 0.08 | 6.62 | 2.61 | 1.01 | 0.30 | 0.04 |
| 140 | 26.74 | 3.42 | 0.81 | 0.11 | 8.80 | 3.47 | 1.34 | 0.40 | 0.06 |
| 150 | 3.88 | 0.92 | 0.12 | 3.95 | 1.52 | 0.45 | 0.07 | ||
| 160 | 4.38 | 1.03 | 0.14 | 4.45 | 1.71 | 0.51 | 0.08 | ||
| 170 | 4.90 | 1.16 | 0.16 | 4.98 | 1.92 | 0.57 | 0.09 | ||
| 180 | 5.44 | 1.29 | 0.17 | 5.53 | 2.13 | 0.63 | 0.10 | ||
| 190 | 6.02 | 1.42 | 0.19 | 6.11 | 2.36 | 0.70 | 0.11 | ||
| 200 | 6.61 | 1.56 | 0.21 | 6.72 | 2.59 | 0.76 | 0.12 | ||
| 220 | 1.87 | 0.25 | 3.09 | 0.91 | 0.14 | ||||
| 240 | 2.19 | 0.30 | 3.63 | 1.07 | 0.16 | ||||
| 260 | 2.54 | 0.34 | 4.21 | 1.24 | 0.19 | ||||
| 280 | 2.92 | 0.39 | 4.83 | 1.43 | 0.22 | ||||
| 300 | 3.32 | 0.45 | 5.49 | 1.62 | 0.24 | ||||
| 320 | 3.74 | 0.51 | 6.18 | 1.83 | 0.28 | ||||
| 340 | 4.18 | 0.57 | 2.04 | 0.31 | |||||
| 360 | 4.65 | 0.63 | 2.27 | 0.34 | |||||
| 380 | 5.14 | 0.69 | 2.51 | 0.38 | |||||
| 400 | 5.65 | 0.76 | 2.76 | 0.42 | |||||
| 420 | 6.18 | 0.84 | 3.02 | 0.46 | |||||
| 440 | 6.74 | 0.91 | 3.29 | 0.50 | |||||
| 460 | 0.99 | 3.58 | 0.54 | ||||||
| 480 | 1.07 | 0.58 | |||||||
| 500 | 1.15 | 0.63 | |||||||
| 550 | 1.38 | 0.75 | |||||||
| 600 | 1.62 | 0.88 | |||||||
| 650 | 1.88 | 1.03 | |||||||
| 700 | 2.15 | 1.18 | |||||||
| 750 | 2.45 | 1.34 | |||||||
| 800 | 2.76 | 1.51 | |||||||
| Q (gpm) | Pressure Loss Due to Friction (psi/100ft) | ||||||
|---|---|---|---|---|---|---|---|
| Aluminum Gated Pipe, 0.051 Wall, C=110 | PVC IPS Class 125, C=150 | ||||||
| Nominal Diameter=6 in. | Nominal Diameter=8 in. | Nominal Diameter=10 in. | Nominal Diameter=6 in. | Nominal Diameter=8.5 in. | Nominal Diameter=10 in. | Nominal Diameter=12 in. | |
| Inside Diameter= 5.898 in. | Inside Diameter= 7.898 in. | Inside Diameter= 9.898 in. | Inside Diameter= 5.766 in. | Inside Diameter= 7.658 in. | Inside Diameter= 9.572 in. | Inside Diameter= 11.486 in. | |
| 240 | 0.34 | ||||||
| 260 | 0.40 | 0.25 | |||||
| 280 | 0.46 | 0.29 | |||||
| 300 | 0.52 | 0.33 | |||||
| 320 | 0.59 | 0.14 | 0.37 | ||||
| 340 | 0.66 | 0.16 | 0.41 | ||||
| 360 | 0.73 | 0.18 | 0.46 | ||||
| 380 | 0.81 | 0.19 | 0.51 | 0.13 | |||
| 400 | 0.89 | 0.21 | 0.56 | 0.14 | |||
| 420 | 0.97 | 0.23 | 0.61 | 0.15 | 0.05 | ||
| 440 | 1.06 | 0.26 | 0.09 | 0.66 | 0.17 | 0.06 | |
| 460 | 1.15 | 0.28 | 0.09 | 0.72 | 0.18 | 0.06 | |
| 480 | 1.24 | 0.30 | 0.10 | 0.78 | 0.20 | 0.07 | |
| 500 | 1.34 | 0.32 | 0.11 | 0.84 | 0.21 | 0.07 | |
| 550 | 1.60 | 0.39 | 0.13 | 1.01 | 0.25 | 0.09 | |
| 600 | 1.88 | 0.45 | 0.15 | 1.18 | 0.30 | 0.10 | 0.04 |
| 650 | 2.18 | 0.53 | 0.18 | 1.37 | 0.34 | 0.12 | 0.05 |
| 700 | 2.50 | 0.60 | 0.20 | 1.57 | 0.39 | 0.13 | 0.05 |
| 750 | 2.84 | 0.69 | 0.23 | 1.79 | 0.45 | 0.15 | 0.06 |
| 800 | 3.20 | 0.77 | 0.26 | 2.01 | 0.51 | 0.17 | 0.07 |
| 850 | 3.58 | 0.86 | 0.29 | 2.25 | 0.57 | 0.19 | 0.08 |
| 900 | 3.98 | 0.96 | 0.32 | 2.50 | 0.63 | 0.21 | 0.09 |
| 950 | 4.40 | 1.06 | 0.35 | 2.77 | 0.70 | 0.23 | 0.10 |
| 1000 | 4.84 | 1.17 | 0.39 | 3.04 | 0.76 | 0.26 | 0.11 |
| 1050 | 5.30 | 1.28 | 0.43 | 3.33 | 0.84 | 0.28 | 0.12 |
| 1100 | 5.77 | 1.39 | 0.46 | 3.63 | 0.91 | 0.31 | 0.13 |
| 1150 | 6.27 | 1.51 | 0.50 | 3.94 | 0.99 | 0.33 | 0.14 |
| 1200 | 6.78 | 1.64 | 0.55 | 4.26 | 1.07 | 0.36 | 0.15 |
| 1250 | 1.77 | 0.59 | 1.16 | 0.39 | 0.16 | ||
| 1300 | 1.90 | 0.63 | 1.24 | 0.42 | 0.17 | ||
| 1350 | 2.04 | 0.68 | 1.33 | 0.45 | 0.19 | ||
| 1400 | 2.18 | 0.73 | 1.43 | 0.48 | 0.20 | ||
| 1450 | 2.33 | 0.78 | 1.52 | 0.51 | 0.21 | ||
| 1500 | 2.48 | 0.83 | 1.62 | 0.55 | 0.23 | ||
| 1550 | 0.88 | 0.58 | 0.24 | ||||
| 1600 | 0.93 | 0.62 | 0.25 | ||||
| 1650 | 0.99 | 0.65 | 0.27 | ||||
| 1700 | 1.04 | 0.69 | 0.28 | ||||
| 1750 | 1.10 | 0.73 | 0.30 | ||||
| 1800 | 1.16 | 0.77 | 0.32 | ||||
| 1850 | 1.22 | 0.81 | 0.33 | ||||
| 1900 | 1.28 | 0.85 | 0.35 | ||||
| 1950 | 1.34 | 0.89 | 0.37 | ||||
| 2000 | 1.41 | 0.93 | 0.38 | ||||
| 2050 | 0.40 | ||||||
| 2100 | 0.42 | ||||||
| 2150 | 0.44 | ||||||
| 2200 | 0.46 | ||||||
| No. of Outlets | F |
|---|---|
| 1 | 1.0 |
| 2 | 0.634 |
| 3 | 0.528 |
| 4 | 0.480 |
| 5 | 0.451 |
| 6 | 0.433 |
| 7 | 0.419 |
| 8 | 0.410 |
| 9 | 0.402 |
| 10 | 0.396 |
| 11 | 0.392 |
| 12 | 0.388 |
| 13 | 0.384 |
| 14 | 0.381 |
| 15 | 0.379 |
| 16 | 0.377 |
| 17 | 0.376 |
| 18 | 0.373 |
| 19 | 0.372 |
| 20 | 0.370 |
| 22 | 0.368 |
| 24 | 0.366 |
| 26 | 0.364 |
| 28 | 0.363 |
| 30 | 0.362 |
| 35 | 0.359 |
| 40 | 0.357 |
| 50 | 0.355 |
| 100 | 0.350 |
| >100 | 0.345 |
|
* F = 0.54 for center pivots without end guns F = 0.56 for center pivots with end guns |
|
A four-inch aluminum sprinkler lateral is 1280 feet long. Sprinklers are spaced at 40-foot intervals. The lateral goes up (rises) 12 feet in elevation along its length. Each sprinkler on the lateral discharges 5 gpm.
Given: L = 1280 ft
sprinkler spacing = 40 ft
rise = 12 ft q = 5 gpm
q = 5 gpm
Find: Pressure loss due to friction in the lateral in psi. If the inlet pressure to the lateral is 60 psi, what is the pressure at the downstream end of the lateral? Ignore minor losses.
Solution:
There are 33 sprinklers on the lateral (1280/40).
The inlet flow rate is then 165 gpm (i.e., 5 gpm x 33).
From Table 8.3, the multiple outlet factor is 0.36.
Interpolating from Table 8.2a, the pressure loss due to friction is 1.1 psi/100 ft.
\(P_f=F \times (P_f/100 \text{ ft}) \times L \)
\(P_f= \dfrac{0.36 \times 1.1 \times 1280\text{ ft}}{100 \text{ ft}} = 5.1 \text{ psi}\)
The pressure at the downstream end of the lateral can be determined using the concepts shown in Figure 8.4.
\(P_2=P_1 - P_f - P_m - 0.433 \times Rise \)
\(P_2=60 - 5.1 - 0.433 \times 12 = 49.7 \text{ psi} \)
A pipeline with outlets, such as a lateral where water is removed by sprinklers, gates, or emitters, has a lower friction loss than a conveyance pipe because the velocity decreases with distance along the pipe. To correct for the effect of the outlets a multiple outlet factor F is used. F = 1.0 for a pipeline without outlets. For laterals with constant spaced outlets, and nearly the same discharge per outlet, use Table 8.3. With center pivots, sprinkler discharge increases with distance from the pivot point. Outlet factors for pivots are given at the bottom of Table 8.3.