Friction Head Loss in Center-Pivot Laterals with the Lateral Divided into Several Reaches

Qdd(i)=flow rate in the pipe at the segment i; Qcd(s)=flow rate in the pipe at the coordinate s; Qe=discharge of end gun sprinkler; Qei=discharge at the beginning or end of the reach i; Q0=total inflow rate in lateral; qj=discharge of sprinkler j; Rir= radius of the irrigated basic circle; Rbig= big radius of the irrigated annulus area; Rles= less radius of the irrigated annulus area; r =variable of integration with respect to distance; rj=distance from pivot to sprinkler at point j; r0=radius of dry area; Sf=friction slope; ssp=spacing between sprinklers; Tmr=minimal time period required for one revolution of the system; α= exponent of velocity


Nomenclature
A=annulus area; All the above references assumed equal spacing between outlets and between the first open outlet and the pivot. Usually, the distance from the pivot to the first open outlet is longer than the distance between outlets due to the presence of several closed outlets. Moreover, when the centerpivot lateral has an end gun, the friction head loss in the distance between the last sprinkler and the end gun is neglected by several investigators.
Tabuada [8] developed closed form expressions and analytical expressions (the last based on the Hypergeometric function) characterizing the friction head loss of center-pivot laterals with and without closed outlets at the beginning of the center-pivot lateral and with and without end gun sprinkler where he considered friction head loss also in the distance between the last outlet and the end gun sprinkler.
For a center-pivot lateral divided into several reaches the existing literature does not present expressions to solve this problem.
In this paper, we propose general expressions, based on the discrete outflow distribution, for the computation of the friction head loss and to obtain the friction correction factors in center-pivot laterals with and without closed outlets at the beginning of the center-pivot lateral and with and without end gun sprinkler.
These expressions are based on the discharge required by each sprinkler and each reach to irrigate an annular area.
The friction head loss based on the discrete outflow distribution is compared with the values obtained based on the continuous outflow distribution.

Sprinkler discharge
The center-pivots can use different kinds of sprinklers. The choice of sprinklers is based on the soil infiltration rate and the shape of the water application rate profile.
Sprinklers can be installed using equidistant or variable spacing. In the case of equidistant spacing, the water flow per sprinkler increases from the pivot to the end of the lateral. When variable spacing is used, the distance between sprinklers decreases from the pivot to the end of the lateral while the water flow per sprinkler is kept constant.
To avoid problems associated with large nozzles, sometimes the lateral is divided into three or four reaches, and a different uniform spacing is used in each reach [9].
Tabuada [8] showed that the discharge (q j ) for each sprinkler located at the point j where the radius of the center-pivot for this point is r j (Figure  1), can be obtained by: Where 2π radians is the angle described by the center-pivot, T mr is the minimal time period required for one revolution of the system, N dw is the number of irrigation days per week, a E is the irrigation application efficiency, s sp is the spacing between sprinklers, and h Max represents the maximum irrigation depth.
If the irrigation interval is greater than seven days then 7/N dw = 1.
Equation (1) is an improved expression and is similar to an equation proposed by Heermann and Hein [10] and later by Mohamoud et al. [11].

End-gun discharge
Many center-pivots are designed with an end gun at the end the lateral to extend the irrigated area, therefore the discharge of an end gun sprinkler (Q e ) is the discharge required to irrigate an annular area at the end of the lateral.
The discharge of the end gun sprinkler (Q e ) is a fraction (ψ) of the total system flow rate (Q 0 ). They can be related as follows: In the Figure 1 is shown a schematic diagram of a lateral for the discrete outflow.

Flow rate at any point along a lateral
In order to calculate the friction head loss we need to know the flow rate at any point along the lateral. This can be done by the discrete outflow distribution (DOD) or continuous outflow distribution (COD).
Discrete approach: 1) with single diameter and constant spacing along the lateral: For a discrete distribution of the outflow, the flow inside every segment i (Figure 1), in a lateral with n segments (or n open outlets), is given by [8]: Also for the various segments, Tabuada [8] presented the following equations: Figure 1: Schematic diagram of a lateral divided into n segments where is shown the spacing between sprinklers (S sp ), outflow along the lateral (q 1 ,…,q n ), inflow (Q 0 ) and outflow (Q e ) that represents the discharge of the end gun, the flow rate [Q dd (i)] into segment i (in this case i= 4) and the distance between the last sprinkler and the end gun sprinkler (d eg ): without an end gun; with an end gun.
At the beginning of the lateral there is, usually, one or two (sometimes three) closed outlets, therefore the first segment (r 1 , Figure 2) is longer than the remaining segments. This creates a dry area. This segment (r 1 ) is correlated with the distance between sprinklers, s sp , that is: where ξ is the coefficient that must be multiplied by s sp to obtain r 1 .   (7) we have for this case:

2) With single diameter but with the lateral divided into several reaches where the spacing between sprinklers is different but in each
The flow inside every segment in a center-pivot lateral with three reaches, can be obtained, for each reach by: where Q 0 , Q e1 and Q e2 represent the inflow at the inlet of each reach and Q e1 , Q e2 and Q e3 represent also the outflow at the outlet of each reach. They are related between themselves, that is: In this study we assumed that: 1)-the irrigated area by each sprinkler consists of all the points whose distance to the pivot is between r i -s sp /2 and r i +s sp /2 ( Figure 4); 2)-the irrigated area by each reach is also an annulus; 3)-the length of the basic irrigated circle for a lateral without an end gun sprinkler is )-there is no overlap between the areas irrigated by each sprinkler; 5)-the distance between the last sprinkler of reach i and the first sprinkler of reach i+1 is 2 2 So it is easier, for this case, to compute the discharge of each sprinkler and of each reach as: For the first reach, for example, 0 r R les = The irrigated area for each reach is given by: for the first, second, and third, respectively.
Considering the assumptions referred above, the equations (9 b), (9 c) and (9 d) can be rewritten as:  Hence, we can obtain the friction head loss and the friction correction factor for each reach. The internal flow in each segment can be obtained by: a) For the first reach: Substituting (13 a) and (8) in (14) this becomes: Substituting (13 b) and (8) c) For the third reach: Substituting (13 c) and (8) On the other hand, the lateral length with a single diameter and constant spacing along the lateral is related with (7) and also with n segments through the expression: In the case of a lateral divided into three reaches, the lateral length ( where dρ is the infinitesimal spacing and c 1 is given by equation (5). For a center-pivot lateral with closed outlets at the beginning of the lateral, the discharge [ Q cd (s)] in the main line at distance r can be given by: Substituting (2) in (24) and after (24) in (23) this becomes: ii) With single diameter but with the lateral divided into several reaches where the spacing between sprinklers is different but in each reach the spacing is kept constant: For this situation, since we are assuming no overlap between the area irrigated by sprinkler i and the area irrigated by sprinkler i+1, equation (25) can still be used.

Friction head loss:
In this study the friction head loss at any segment along a lateral is calculated by the expression: In (26), β is the roughness factor, which we assume to be independent of s, D represents the internal diameter of the main line, and γ and γ are exponents obtained from the formula used to compute f S .

Discrete approach: i) with single diameter and constant spacing along the lateral:
For a single diameter, taking into account (6) and (26), the head loss can be computed segment by segment [8] as: So the total friction head loss between the beginning of segment 1 and the end of segment n, is given by: If 0 ≠ ψ , in equation (28), then the center-pivot lateral has an end gun sprinkler and then we have to consider the friction head loss in the distance between the last sprinkler and the end gun (d eg , Figure 4) given by

ii) With single diameter but with the lateral divided into several reaches where the spacing between sprinklers is different but in each reach the spacing is kept constant:
Following a similar procedure as before we have: a) First reach: So the total friction head loss between the beginning of segment 1 and the end of segment m 1 , is obtained by: For the friction correction factor, after substituting (21) in (30) we have: The total friction head loss, in the second reach, between the beginning of segment 1 ( Figure 3) and the end of segment m 2 is given by: [ ] And for the friction correction factor, after substituting (21) for inflow inlet equal to Q 0 .
Taking into account the inflow at the inlet in each reach (Q 0 , Q e1 and Q e2 ) or the inflow at the inlet in the first reach (Q 0 ), the friction head loss in each reach can be obtained by: The solution of (39) depends on the α exponent value, α=2, as in the Darcy-Weisbach or Chézy equation, or α=1.852, as in the Hazen-Williams equation.
In this study we only consider the case of α=non integer real value since when α is equal to 2 the integration of the equation (39) yields an analytical expression.
Tabuada [8] studied the friction head loss for α=2 and for α ≠ non integer real value and for several cases: center-pivot laterals with and without an end gun sprinkler with and without close outlets at the beginning of the center-pivot lateral and he showed that for this situation, α ≠ non integer real value and for a center-pivot lateral with an end gun sprinkler, the friction head loss is given by: (1 ) 0.5, ;1.5; (1 ) (1 ) 0.5, ;1.5; (1 ) represent the Hypergeometric function.
ii) With single diameter but with the lateral divided into several reaches where the spacing between sprinklers is different but in each reach the spacing is kept constant: It follows from assumption 4) that equation (40) can also be used in this case.

Comparative Analyses
To illustrate the proposed methods we consider a center-pivot lateral with an end gun sprinkler and with the following characteristics: The friction head loss computed using equation (40) (COD) and using equations (30), (32) and (35) (DOD) is presented in the Figure 6. To obtain the values of the friction head loss using the Hypergeometric function we can use any mathematical software such as Mathematica. Figure 6 shows that the difference of the friction head loss calculated with COD and with DOD is neglected. Tabuada (2011) verified that to irrigate a similar area with a center-pivot without an end gun sprinkler the difference between the values of the friction head loss obtained for a center-pivot with end gun and without end gun sprinkler is small. This is verified when the discharge of the end gun sprinkler is similar to the discharge distributed by m sprinklers (q n+1, +....,q n+m) to irrigate a similar area. On the other hand to irrigate the similar area without end gun sprinkler it is necessary to increase the number of sprinklers (or outlets) on the lateral and therefore the friction correction factor (F cf ) decreases as well as the friction head loss.

Summary and Conclusions
This paper presents efficient closed form expressions to calculate the friction head loss and the friction correction factor using the discrete outflow distribution.