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Unformatted text preview: Sprinkle & Trickle Irrigation Lectures Page 241 Merkley & Allen Lecture 22 Numerical Solution for Manifold Location I. Introduction • In the previous lecture it was seen how the optimal manifold location can be determined semi-graphically using a set of non-dimensional curves for the uphill and downhill laterals • This location can also be determined numerically • In the following, equations are developed to solve for the unknown length of the uphill lateral, x u , without resorting to a graphical solution . Definition of Minimum Lateral Head • In the uphill lateral, the minimum head is at the closed end of the lateral (furthest uphill location in the subunit) • This minimum head is equal to: II n l fu u h ' h h x S = − − (394) where h n ’ is the minimum head (m); h l is the lateral inlet head (m); h fu is the total friction loss in the uphill lateral (m); x u is the length of the uphill lateral (m); and S is the slope of the ground surface (m/m) • Note that S must be a positive value x m (h ) fd 2 (h ) fd 1 x u h fu manifold location h ’ n h ’ n • In the downhill lateral, the minimum head may be anywhere from the inlet to the outlet, depending on the lateral hydraulics and the ground slope • The minimum head in the downhill lateral is equal to: ( ) ( ) n l fd fd m 1 2 h ' h h h x S = − + + (395) Merkley & Allen Page 242 Sprinkle & Trickle Irrigation Lectures where (h fu ) 1 is the total friction loss in the downhill lateral (m); (h fu ) 2 is the friction loss from the closed end of the downhill lateral to the location of minimum head (m); and x m is the distance from the manifold (lateral inlet) to the location of minimum head in the downhill lateral (m) • Combining Eqs. 1 and 2: ( ) ( ) ( ) fu u m fd fd 1 2 h S x x h h + + − + (396) III. Location of Minimum Head in Downhill Lateral = • The location of minimum head is where the slope of the ground surface, S, equals the friction loss gradient, J’: S J' = (397) where both S and J’ are in m/m, and S is value of S) • he friction loss gradient in the downhill positive (you can take the absolute Using the Hazen-Williams equation, t lateral (at the location where S = J’) is: ( ) − ⎡ ⎡ ⎤ + 1.852 S f ⎤ ⎛ ⎞ − − ⎢ ⎥ = ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ 10 4.87 a u m e e e e q L x x J' 1.212(10) D S 3,600S C (398) the distance from the manifold to the location of minimum head in D is the lateral inside diameter (mm); • The value of 3,600 is to convert q a units from lph to lps • Note that x d = L - x u , where x d is the length of the downhill lateral • Note that q a (L - x u - x m )/(3,600 S e ) is the flow rate in the lateral, in lps, at the location of minimum head, x m meters downhill from the manifold • Combining the above two equations, and solving for x...
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- Fall '03
- Fluid Dynamics, Trigraph, Darcy–Weisbach equation, Merkley, Irrigation Lectures