11.5: Maximum Lateral Inflow
- Page ID
- 44634
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The maximum inflow to a sprinkler lateral is limited by two conditions: the maximum permissible pressure variation and the maximum acceptable water velocity in the lateral pipe. The maximum permissible pressure variation along the lateral limits the maximum inflow as described in the previous section (Section 11.4). The maximum pressure variation along the lateral is 20% of the average operating pressure, therefore:
Max Pl = 0.20 x Pa
Using the Hazen-Williams equation for pressure loss the maximum inflow for the lateral can be determined:
\(\text{Max }P_i=4.56F\left(\dfrac{Q_{max}}{C}\right)^{1.852} \dfrac{L}{D^{4.866}} \) (11.11)
where F is the multiple outlet factor from Table 8.3 and all other terms are as previously defined. The above equations can be combined to yield an expression for the maximum inflow that is permissible for a lateral of given length and size (i.e., fixed diameter and C value):
\(Q_{max}=\left(\dfrac{0.2P_aD^{4.866} C^{1.852}}{4.56FL}\right)^{1/1.852} \) (11.12)
where Qmax is the maximum inflow to the lateral to maintain pressure variation less than 20% of the average pressure. The second factor that can limit the inflow rate to the lateral is the maximum allowable velocity in the pipeline. The danger of damage to the pipeline and its components due to pressure surges increases when the velocity of water in the pipeline increases. Sprinkler laterals can withstand higher velocities than mainlines because the sprinklers on the lateral allow water and air under high pressure to escape before damaging the pipe. However, there is still an upper limit to the velocity of water flow in sprinkler laterals. Commonly the upper limit is 7 feet per second, while 10 feet per second can be used if the valve closes gradually and the pipe is filled slowly. The maximum velocity can also determine the maximum inflow for the pipeline:
Qmax = 2.445 vmax D2
where vmax = the maximum water velocity in the lateral and all other terms are as previously defined.
Equations 11.12 and 11.13 can be combined to provide limits for the maximum inflow to the sprinkler lateral so that the velocity is below the maximum permissible and the pressure variation along the lateral is less than 20% of the average pressure. The smallest value from the two equations defines the maximum inflow. These equations were used to develop charts for the maximum inflow for aluminum pipe with couplers forty feet apart as shown in Figure 11.11. Similar relationships can be developed for other types of pipe material to use as general guidelines for laterals.
Solution of equation 11.13 for the maximum inflow at a velocity of seven feet per second gives flows of 410, 260, and 145 gpm for 5-inch, 4-inch, and 3-inch pipe respectively as shown in Figure 11.11.
Results in Figure 11.11 show that the velocity limit (7 ft/sec) determines the maximum inflow for the initial lengths of the lateral. As the lateral length increases, the friction loss limitation determines the maximum inflow rate. Results in Figure 11.11 can be used for laterals with different sprinkler outlet spacings because the velocity limits the inflow for short laterals. By the time friction loss becomes determinate a significant number of sprinklers will be included and the friction factor (F) for laterals will be nearly the same for either sprinkler spacing. Thus, the friction loss will be comparable since all other factors are the same for friction loss calculations.
Determine the maximum sprinkler discharge for a 5-inch aluminum pipe lateral that is 1,000 feet long where the average pressure is 50 psi. Sprinklers are spaced 40 feet apart along the lateral.
Given: Pa = 50 ps D = 5 in
L = 1,000 ft SL= 40 ft
C = 120 (aluminum pipe with couplers, Table 8.1)
qs = discharge of individual sprinklers
Solution
From Figure 11.11 the maximum lateral inflow is about 410 gpm.
For a lateral 1,000 ft long, 25 sprinklers would be needed if spaced at 40 ft.
Thus, each sprinkler could average up to 16.4 gpm, which is a high flow for most applications.
Figure 11.11. Maximum inflow for three diameters of aluminum sprinkler lateral pipe with outlets 40 feet apart (C value = 120), and three average pressures along the lateral. A maximum velocity of 7 feet per second is used for this chart.)
