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  Sprinkler Irrigation Uniformity
by
Kenneth H. Solomon

CATI Publication #900803
© Copyright August 1990, all rights reserved

IRRIGATION UNIFORMITY
Irrigation is the artificial application of water to crops to ensure adequate moisture for growth. The ability of a sprinkler system to apply water uniformly throughout the irrigated area is a major factor influencing whether or not proper crop growth can be maintained. Specific quantitative study of sprinkler irrigation uniformity began with the pioneering work of J. E. Christiansen in 19421. The importance of this topic is indicated by the large body of related work done since then.

Ideally, an irrigation system would apply water in a completely uniform manner so that each part of the irrigated area receives the same amount of water. Unfortunately, there seems to be no way to achieve this. Even natural rainfall is not completely uniform. So the phrase "irrigation uniformity" actually refers to the variation, or non-uniformity, in the amounts of water applied to locations within the irrigated area. Significant effort in sprinkler irrigation system design and management is directed towards dealing with problems related to irrigation uniformity, or the lack of it.

AGRONOMIC IMPORTANCE
Whenever water is applied with less than perfect uniformity, some parts of the crop will receive more water than others. If the irrigation system is operated so that the part of the crop receiving the most water has its requirements met, then the remainder of the crop will be under-irrigated. If the system is operated so that the part of the crop receiving the least water has its requirements met, then the remainder of the crop will be over-irrigated. Thus, a non-uniform irrigation unavoidably results in some degree of under- or over-watering.

Irrigation uniformity is related to crop yields through the agronomic effects of under- or over-watering. Insufficient water leads to high soil moisture tension, plant stress and reduced crop yields. Excess water may also reduce crop yields below potential levels through mechanisms such as leaching of plant nutrients, increased disease incidence or failure to stimulate growth of the commercially valuable parts of the plant.

Irrigation uniformity is also inherently linked to the efficiency with which agricultural resources are used. To the extent that non-uniformity results in the application of excess water, several water related resources are lost. These include: energy for pumping the excess water; fertilizers, either applied with the irrigation water or leached by the excess water; other chemicals which may be applied with or washed away by the water; and capital losses due to the extra capacity designed into the irrigation and drainage systems to carry the excess water. To the extent that non-uniformity causes crop yield to fall below potential levels, agricultural inputs applied in anticipation of full yield are wasted.

ENGINEERING IMPORTANCE
Because irrigation uniformity relates to crop yield and the efficient use of resources, engineers regard it as an important factor to be considered in the selection, design and management of sprinkler irrigation systems. Various measures of uniformity are used as indices of performance by which sprinklers and sprinkler spacings are judged, and they may also be used to set hydraulic limitations on the sprinkler pipe network.

Irrigation uniformity is a key component in overall irrigation efficiency, and hence plays an important role in the scheduling of irrigations to meet crop moisture requirements. The two major losses that occur between the source and the crop components of irrigation efficiency are: the water root zone; and irrigation uniformity. Overall efficiency is, of course, a concern to the irrigation manager because it expresses the relationship between gross amount of water delivered by the system and the net amount of water that is effectively made available to the crop.

SPRINKLER IRRIGATION SYSTEMS
There are many types of sprinkler irrigation systems in use throughout the world today. For purposes of this paper, it will be convenient to consider just two types of systems: continuous moving and fixed grid systems. The most common continuous moving systems are the center pivot, the linear move, and the traveling gun sprinkler system.

In the center pivot system, sprinklers are placed along a lateral line approximately 400 meters long, which is pivoted at one end, and moved around the field in a circular fashion. The linear move system is similar, except that it is not pivoted at one end, and moves in a straight line fashion perpendicular to the direction of the lateral pipeline. Both systems are most often used to irrigate large areas, typically 50 hectares or more. Traveling gun systems use large volume (30 liters per second or more) sprinklers operating at relatively high pressures (6 Bars and up), mounted on a unit that is self-powered or towed through the field. In all three cases, the motion of the sprinkler is continuous.

Fixed grid sprinkler systems employ sprinklers that are somehow placed on a grid throughout the field, and that remain stationary during irrigation. Usually a line of sprinklers, or a block of such lines are operated at once. The next irrigation set would be with an adjacent line or block of lines. With solid set or permanent set systems, enough pipelines and sprinklers are placed in the field that going from one irrigation set to another involves little more than turning off one valve, and turning on another. With other systems, the sprinkler lines are moved manually or mechanically between irrigation sets. Over the full irrigation cycle, the field is irrigated by sprinklers located on a grid of positions, hence the name "fixed grid systems." This paper will concentrate on the irrigation uniformity of such fixed grid systems.

SPRINKLER IRRIGATION FACTORS AFFECTING UNIFORMITY
A unique problem for sprinkler systems is that the water application pattern is susceptible to distortion by the wind. While wind speed and direction are not controlled variables, their effect on irrigation uniformity is significant, so that sprinkler system design must be done with anticipated wind conditions in mind.

While wind speed and direction are not controlled variables, their effect on irrigation uniformity is significant, so that sprinkler system design must be done with anticipated wind conditions in mind.

The water jet issuing from a nozzle may have velocities ranging from almost zero near the outside to a maximum velocity near the center of the stream. It is this variation in velocities that causes the initial breakup of the stream. In a sprinkler, this is mechanically aided by the rotation of the sprinkler and the interruption of the jet by the sprinkler arm. Further breakup is the result of the interaction of two opposing forces. The surface tension of the water tends to hold drops intact and in a spherical shape. Air resistance tends to distort the spherical shape of a drop by flattening the lead side of the drop. When the distortion due to air resistance exceeds the cohesive force of surface tension, the drop separates into two or more smaller drops.

Air resistance has the further effect of decreasing the velocity of the drops as they fall. This is why a sprinkler jet does not follow the parabolic trajectory of ballistic projectiles. The inertia of a drop is a function of its diameter cubed (mass) while the resistance of the air is a function of its diameter squared (cross sectional area), so that the smaller drops are slowed much more rapidly than the larger ones. Hence, the smaller drops tend to fall near the sprinkler, while the larger ones fall farther away. For the same reason, wind has a much greater effect on the smaller drops than on the larger ones.

Wind causes the breakup and distribution processes to change in two ways: (1) It affects the breakup of the water due to air resistance; and (2) it blows all the resulting drops around. For instance, when the nozzle is directed into the wind, air resistance is increased, so that some of the larger drops become unstable and disintegrate. The smaller drops resulting from this disintegration are slowed more rapidly by air resistance, tending to decrease the upwind radius of throw and, as the drops are falling, the wind will tend to blow them back toward the sprinkler. This same kind of interaction causes a slight decrease in the cross wind radius of throw as well. When the nozzle is pointing downwind, air resistance is less than normal, so that fewer large drops break down, and again, all drops are blown by the wind. The downwind radius is therefore lengthened. Because of the difference is the drop sizes, the wind shortens the upwind radius more than it lengthens the downwind radius. This is especially true for very high winds. Thus the maximum high wind diameter is somewhat shorter than the low wind diameter.

From the discussion above, it is clear that wind affects a sprinkler distribution pattern according to both wind speed and direction. There is a third parameter of the wind condition that also affects the distribution pattern, and therefore uniformity. This parameter involves the amount of change in a wind pattern over a period of time. If the wind is not constant, it is possible for areas of low precipitation due to one wind condition to receive heavy precipitation due to the next wind condition. The result is that areas of high and low precipitation are kept from becoming exaggerated. Therefore uniformity under varying wind conditions is usually higher than under steady wind conditions of a similar magnitude.

This explains how and why uniformity responds to a change in wind. Let us now examine some parameters of a sprinkler system response to this wind effect. To insure a common understanding of some ambiguous terms, let us define a low wind to be 0-7 kph, a moderate wind to be 7-14 kph, and a high wind to be 14 kph or more.

EQUIPMENT AND DESIGN FACTORS<
For a given wind condition, the primary factors affecting uniformity are nozzle type and size, operating pressure, and spacing. For a fixed grid system, there are two spacing dimensions, the distance between sprinklers on a lateral, and the distance between laterals. Rough rules of thumb for maximum spacings are given in Table 1. The spacings are given as a percentage of the sprinkler's wetted diameter.

Table 1. Maximum recommended sprinkler spacings
Wind Conditions Spacing
Low 60 - 65% of wetted diameter
Moderate 50% of wetted diameter
High 30 - 50% of wetted diameter

There is an interesting relationship between these recommended spacings and soil characteristics in sprinkler design. One of the tenants of sprinkler system design is that the application rate should not exceed the basic infiltration rate of the soil. Application rate is proportional to the flow rate and inversely proportional to the product of the two spacing dimensions. Now for a given pressure, increasing the nozzle size will increase both sprinkler flow rate and wetted diameter, but flow rate will increase considerably more than diameter. The increase in wetted diameter will permit slightly larger spacings, but the increase in flow rate overshadows this, so that for a fixed uniformity, increasing the nozzle size generally means an increased application rate as well.

Within the range of small to medium sized sprinklers, it is generally more economical to design the system with the largest sprinkler and spacings permissible. So the two factors that often determine sprinkler nozzle size and spacing are the desired uniformity, and the infiltration rate of the soil. When growing high value crops, where high uniformity is normally desirable, on fine textured soils, successful designs invariably employ small nozzles (3mm diameter) and close spacings (9m x 12m). On coarser textured soils with a higher infiltration rate, or where lower uniformities are acceptable, larger nozzle and wider spacings may be used.

Common operating pressures for these size sprinklers used to be in the range of 3 1/2 to 4 1/2 Bars, but with the high cost of energy, there has been a tendency to reduce the operating pressure. A variety of new nozzles, generally with non-circular orifices, have been specially designed for low pressure use. These nozzles use mechanical means to provide extra breakup of the water jet at low pressures. With such nozzles, operating pressures are often 1 Bar lower than with traditional nozzles.

There are other factors affecting sprinkler irrigation uniformity, not generally regarded as being as significant as nozzle, pressure and spacing, but important none the less. Several of these relate to the specific sprinkler being used.

Rapid rotation of a sprinkler may considerably affect the break up of the stream, and to this extent, it determines how wind will affect the pattern. A jet of water in the air tends to carry with it an envelope of air moving at a velocity approaching that of the jet. When this condition is achieved, air drag on the jet is at a minimum. If the jet is made to change position, it encounters a new mass of air that is essentially at rest, thereby providing resistance to the water. A rapidly rotating jet has no chance to develop an envelope of moving air, so it always encounters maximum drag and undergoes the most break up. Thus, rapidly rotating sprinklers are affected by wind more than sprinklers with lower rotation speeds.

The trajectory angle of the sprinkler (the angle above horizontal at which the water jet leaves the sprinkler) can influence the water pattern, and hence uniformity. In the absence of air drag, a 45o trajectory would give the maximum wetted diameter for a given nozzle and pressure. Due to the air resistance encountered by the water jet, the trajectory angle for maximum throw is actually less, perhaps just over 30o. In the presence of wind, however, high trajectory angles suffer the disadvantage that the water is in the air longer, and hence more susceptible to the wind. In an empirically derived compromise, many sprinkler manufacturers have settled on a trajectory angle of about 27o as "standard." It achieves near maximum throw in the absence of wind, yet does not suffer pattern distortion in wind to the extent that a 30o trajectory would. For sprinklers to be used in moderate to high wind conditions, lower trajectory angles are advised; 23o, 21o and even 18o trajectory angles are available for use in successively higher wind conditions. Even lower trajectory angles are available for special purpose uses.

An important question in sprinkler selection is whether to use sprinklers with a single nozzle or with dual nozzles. In most agricultural applications, the single nozzle is preferred, for the following reasons. For a given spacing, the application rate is determined by the sprinkler flow rate. As mentioned above, it is usually most economical to design for an application rate near the limit dictated by the soil type, so that spacings can be maximized. When selecting nozzles, then, it is presumed that the desired sprinkler flow rate is known. The nozzle choice becomes a question of whether one should put all the available water through a single nozzle, or use a slightly smaller main nozzle accompanied by a secondary or "spreader" nozzle. Water from the spreader nozzle is usually much finer and more diffuse than the spray from the main nozzle, so it is much more affected by the wind. Using the largest possible main nozzle will maximize wetted diameter and minimize wind distortion. Thus, unless the wind conditions are unusually calm, the single nozzle sprinkler will generally have the better coverage, the higher uniformity, and the superior resistance to wind.

An important question in sprinkler selection is whether to use sprinklers with a single nozzle or with dual nozzles.

In some sprinklers a special stream straightening device, or "vane" is placed behind (upstream of) the main nozzle. The purpose of the vane is to reduce turbulence in the water stream introduced during its passage through the sprinkler body. Less turbulence means that the stream is not broken up as much or as soon; hence more water is thrown farther. Any wind effects are reduced because the entire stream is more cohesive. Fewer drops are broken up and/or slowed by the wind, so the drop size distribution of a vaned sprinkler is affected less by the wind than that of a vaneless sprinkler.

The shape of the distribution curve for a vaned sprinkler is generally not as close to ideal as with a vaneless sprinkler, so under low wind conditions a vaned sprinkler may have a lower uniformity than a vaneless one. But the distribution pattern of a vaned sprinkler is less susceptible to distortion by the wind, so its uniformity is not as affected by wind. Above 15 kilometers per hour, the addition of a vane may improve uniformity.

MANAGEMENT FACTORS
Uniformity can be influenced not only by the irrigation equipment in the system, but by how that system is managed. The key management factors affecting uniformity are discussed below.

The length of the irrigation time can affect uniformity. As mentioned earlier, variations in wind speed and direction can improve uniformity relative to the case of a constant wind. Longer irrigation times create more change for this wind variation to occur, and hence generally have higher uniformities than systems using short irrigation sets. The time of day of the irrigation can also have an effect, particularly in areas with prevailing winds. It is best to plan your irrigation so that the same parts of the field are not irrigated at the same time of day each time they are irrigated. This will give an opportunity for natural changes in wind speed and direction to balance out, improving the uniformity of application over consecutive irrigation events.

The practice of offsetting laterals (also called "alternate sets") by one half the lateral spacing for every other irrigation can improve uniformity at little cost. Suppose, for example, the lateral spacing is 18m, and lateral positions are at distances A, B, C, ... The practice of alternate sets improves uniformity because the light and heavy application areas of one set tend to fall on the heavy and light areas, respectively, of the alternate set. If uniformity is measured according to Christiansen's Uniformity Coefficient (UCC) (see later in this paper) then the following simple formula may be used to estimate the improvement in uniformity due to alternate sets: UCC with alternate sets = Square Root (NCC without alternate sets). For example, a uniformity of 0.75 without alternate sets might be improved to 0.86 with alternate sets.

A final management practice that can improve uniformity is the practice of irrigating blocks of several adjacent laterals at once. A beneficial micro-climate develops within the block, minimizing wind distortion and losses due to wind drift and evaporation. Numerous field experiments have documented an improvement in uniformity due to this block effect.

SPRINKLER TESTING AND UNIFORMITY ESTIMATION
Estimation of sprinkler uniformity in the field is a two-part process. First, the water application pattern of a single sprinkler is determined empirically. This pattern is then offset and overlapped upon itself to represent the pattern of sprinklers on the grid of locations in the field. This later process of overlapping is done quickly and efficiently by potential spacings may be considered. To determine sprinkler performance in the absence of wind, tests are often run indoors. A series of rain gauges are set out along a radial leg, and the application amount is determined as a function of distance from the sprinkler. In the absence of wind, the sprinkler pattern may be assumed to be symmetrical, so this single radial leg test determines the pattern characteristics. To determine the sprinkler pattern under windy conditions, tests must be run outdoors under the wind conditions of interest. The pattern for conditions of varying wind may be estimated by adding up (summing, position by position) the patterns for the various pure wind conditions involved in the total design wind condition. Readers interested further in this aspect of sprinkler performance evaluation are invited to contact the author.

CROP RESPONSE TO WATER AND UNIFORMITY
The response of a crop to applied water can be summarized in a water yield function. This is an equation by which the yield can be calculated from the seasonal water application. It is convenient to express both yield and applied water in relative or dimensionless terms. Relative yield (y) is defined as the ratio of actual yield to maximum yield, and relative applied water (w) is defined as the ratio of actual applied water to that amount of applied water corresponding to maximum yields. If w is taken to include effective rainfall and soil moisture stored at the beginning of the season, the yield function will be fairly general and can be representative of more than one location or year. If the yield function is adjusted so that w refers only to the water applied by the irrigation system, the significance of various irrigation options is more apparent, though some generality is lost. The shape of the yield function also depends on the irrigation scheduling procedure, but it is usually assumed that a yield function is valid for most "reasonable" irrigation schedules.

A particular yield function for sugarcane is given below. It is based on data from a number of sources, and assumes that rainfall and moisture stored in the root zone at the beginning of the season amounts to 20% of the water necessary for maximum yields, and that sensitivity to excess water is relatively low.

y(w) = 0.05 + 2.47w - 2.19w2 + 0.77w3 - 0.10w4

where

y(w) = relative yield as a function of w only

w = relative seasonal irrigation application

Table 2 shows how the relative sugarcane yield changes with the relative seasonal irrigation application.

Table 2. Water yield relationship for sugarcane
Relative irrigation w Relative yield y(w)
0.25 0.54
0.50 0.83
0.75 0.96
1.00 1.00
1.25 0.98
1.50 0.92
1.75 0.85

In a recent study, Solomon reviewed published data on yield response and presented general yield functions for the following crops:
Alfalfa Grapefruit Safflower
Banana Grapes Sorghum-grain
Barley Mustard Sorghum-silage
Bean Onion Soybeans
Berseem Paddy Sugarbeet-root
Cabbage Pasture Sugarbeet-top
Chili pepper Pea Sugarcane
Citrus Peanuts Sunflower
Corn-grain Pepper Tobacco
Corn-silage Potato Tomato
Cotton Rice Watermelon
General Russian thistle Wheat
Gram forage
Because plants respond to water, they respond to how uniformly the water is applied. Suppose, for example, that sugarcane is irrigated so that 60% of the land receives the yield maximizing amount, but 20% of the land receives only 0.75 times this amount, and 20% of the land receives 1.25 times this amount. You would naturally expect the overall yield to be:

Relative Yield = (0.2) y(0.75) + (0.6) y(1.00) + (0.2) y(1.25)

= (0.2)(0.96) + (0.6)(1.00) + (0.2)(0.98)

= 0.99

The small degree of non-uniformity in the irrigation causes only a 1% decrease in yield. But suppose the irrigation is much less uniform: 35% of the land receives only half the yield maximizing amount, 30% receives the correct amount, and 35% receives 1.5 times the proper amount. In this case

Relative Yield = (0.35) y(0.50) + (0.30) y(1.00) + (0.35) y(1.50)

= (0.35)(0.83) + (0.30)(1.00) + (0.35)(0.92)

= 0.91

The larger degree of non-uniformity causes a 9% reduction in yield.

The general formula for estimating crop yields from non-uniform irrigation is (4.5):

Relative Yield = (f1) y(w1) + ... +(fi) y(wi) + ...

where the f1, f2, ... are the fractions of the land that receive relative irrigation amounts, w1, w2, ... , plus y(wi) is the relative yield associated with wi by the yield function. In this way the crop response to various degrees of non-uniformity can be estimated.

UNIFORMITY COEFFICIENTS

Christiansen defined a uniformity coefficient UCC by:

UCC = 1 - (D/M)

where D = Average absolute deviation of irrigation amounts

M = Average of irrigation amounts

Based on the yield function given previously, the relationship between Christiansen's uniformity coefficient (UCC) and sugarcane yield is as shown in Table 3 (assuming irrigation amounts are normally distributed).

Table 3. Uniformity yield relationship for sugarcane
Irrigation uniformity UCC Sugarcane relative yeild
1.00 1.00
0.95 1.00
0.90 0.99
0.85 0.98
0.80 0.97
0.75 0.95
0.70 0.93
0.65 0.90
0.60 0.86
0.55 0.82
0.50 0.77

Other uniformity coefficients have been suggested which incorporate the standard deviation of the irrigation amounts. The two most common ones are UCW and UCH. These are defined by:


UCW = 1 - (S/M)

UCH = 1 - (0.798)(S/M)

where S = Standard deviation of irrigation amounts

It can be shown that for yield functions like the one given for sugarcane (a polynomial of degree four), the yield depends on just four statistical parameters of the distribution of irrigation amounts:

1. Average
2. Standard deviation
3. Skewness
4. Kurtosis

For quadratic yield functions (polynomials of degree two), only the average and standard deviation are required to estimate the yield from non-uniform irrigation. Since UCW and UCH directly incorporate these two factors, they are the uniformity coefficients that relate most directly to the physical and economic significance of crop responses to non-uniform irrigation.

REFERENCES
Christiansen, J. E.. 1942. Irrigation by Sprinkling. California Agricultural Experiment Station Bulletin 670, University of California, Berkeley, CA.

Hart, W. E. and Reynolds, W. N. 1965. Analytical Design of Sprinkler Systems. Transactions of the American Society of Agricultural Engineers 8, 1:83-85, 89.

Redditt, W. M. 1965. Factors Affecting Sprinkler Uniformity. Sprinkler Irrigaton Engineering Manual, Hawaiian Sugar Planters Association, Honolulu, Hawaii.

Solomon, K. H. 1983. Irrigation Uniformity and Yield Theory. Ph.D. Dissertation, Department of Agricultural and Irrigation Engineering, Utah State Unversity, Logan, Utah.

Solomon, K. H. 1984. Yield Related Interpretations of Irrigation Uniformity and Efficiency Measures. Irrigation Science 5: 161-172.

Wilcox, J. C. and Swailes, G. E. 1947. Uniformity of Water Distribution by Some Under-Tree Orchard Sprinklers. Scientific Agriculture 27, 11:565-583.