UPLB Reviewer Part 2 PDF

UPLB REVIEWER Part 2  TABLE OF CONTENTS CHAPTER CONTENTS PAGES I Farm Irrigation and Drainage 1 – 17 II Soil and Water Conservation Engineering 1 – 14 III Mathematics...

UPLB REVIEWER Part 2  TABLE OF CONTENTS CHAPTER CONTENTS PAGES I Farm Irrigation and Drainage 1 – 17 II Soil and Water Conservation Engineering 1 – 14 III Mathematics 1 – 12 IV Basic Statistics 1 – 23 V Aquaculture 1 – 23 VI Agronomy 1 − 23 VII Animal Science 1 – 12 PSAE Region IV - Agricultural Engineering Board Review Materials I-1 Farm Irrigation and Drainage Engr. Leila T. Dominguez Instructor III Land and Water Resources Division Institute of Agricultural Engineering College of Engineering and Agro-Industrial Technology University of the Philippines Los Baños College, Laguna I. Introduction Irrigation is the application of water to create a condition in the soil that is favorable for plant growth. It provides the moisture needed by the crop that is not satisfied by rainfall and refers to the controlled application of water fro agriculture through man- made systems. Drainage, on the other hand, is the removal of excess water from the soil with the same aim of creating conditions favorable for the growth of crops. II. OBJECTIVE OF THE TOPIC Knowledge on farm irrigation and drainage will equip the individual with the understanding of the relation between crops and the amount and timing of both irrigation and drainage. III. DISCUSSION OF TOPICS A. Basic Soil and Water Relations In a soil system of total volume VT and total weight or mass WT: VT = Va +Vw +Vs VV = Va + Vw WT = Wa +Ww + Ws where: VT = total or bulk volume of the soil (volume units: cm3, m3) Va = volume occupied by air Vw= volume occupied by water Vs= volume occupied by soil particles VV =volume occupied by voids or void volume WT = total wet weight of the soil (weight units: g, kg) Wa = weight of air (negligible, assumed 0) Ww= weight of water Ws= dry weight of the soil particles Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials I-2 For the given soil system, the following properties can be defined: 1. Soil Texture - the relative proportion of primary particles (sand, silt and clay) in the soil 2. Soil Structure - the arrangements of primary particles in the soil into units or peds 3. Porosity (n) – the ratio of the void volume to the total soil volume (unitless) VV V + Vw V VS n= = a = T VT VT VT 4. Moisture contents (unitless) a. on a dry weight basis (mcw) – the ratio of the weight of water to the dry weight of the soil b. on a volume basis or volumetric moisture content (mcv) – the ratio of the volume of water to the total soil volume V mc v = w VT 5. Densities (units: g/cm3, kg/m3) a. Bulk density (ρB) – the ratio of the dry weight of the soil to the total soil volume W ρB = s VT b. Particle density (ρP) - the ratio of the dry weight of the soil to the volume of the soil particles W ρP = s Vs 6. Specific Gravities (unitless) a. Apparent Specific Gravity (As) – ratio of the bulk density of the soil with the density of water; it is the ratio of the weight of soil to the weight of water with volume equal to the total soil volume ρ Ws As = B = ρw ρ w VT b. Real Specific Gravity (Rs) – ratio of the particle density of the soil with the density of water; it is the ratio of the weight of soil to the weight of water with volume equal to the volume of the soil particles alone ρ Ws Rs = P = ρw ρ w VS 7. Depth of water (units: mm, cm, m, inch) a. present in the soil (dw)- the equivalent depth of water in the soil at a given condition dw = mcv x D dw = mcw x As x D where: D – depth of crop root zone or depth or height of soil column under consideration b. needed (dwn) - to increase the moisture content from an initial value (mci) to a final value (mcf) Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials I-3 dwn = (mcvf – mcvi) x D dwn = (mcwf-mcwi) x As x D where : mcvf - final soil volumetric moisture content mcvi - initial soil volumetric moisture content mcwf - final soil moisture content on dry weight basis mcwi - initial soil volumetric on dry weight basis 8. Volume of irrigation water (V iw) – volume of water to be applied to increase a the soil moisture content from an initial to final value (units: liters, cm3, m3) Viw = dwn x A where: A – area of the land in consideration NOTE: The density of water, ρw = 1 g/cm3 or 1,000 kg/m3 The following relationships hold true: mcv = mcw x As As = Rs ( 1-n) 9. Infiltration Rate - the time rate at which water will percolate into the soil and can be expressed in terms of the following empirical equations a. Lewis-Kostiakov Equation F = ct ft = dF/dt = ct-1 where: F - cumulative infiltration (mm) t - cumulative time c,  - constants ft - instantaneous infiltration (mm/hr or mm/min) b. Horton’s Equation ft = fc + (f0-fc) e –kt where: ft- instantaneous infiltration at time t fc- final or ultimate constant infiltration capacity f0- initial infiltration rate at the beginning of rain or a chosen moment constant c. Philip’s Method F = st1/2 +At ft = dF/dt = (st-1/2)/2 + A where: F - cumulative infiltration (mm) ft - instantaneous infiltration (mm/hr or mm/min) s - soil sorptivity, measure of the capillary or removal of water in a homogeneous soil A - soil parameter that depends on the ability of the soil to transmit water Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials I-4 5. Intake Rate - the rate of infiltration from a furrow into the soil. 6. Permeability - the velocity of flow into the soil caused by a unit hydraulic gradient in which the driving force is one kilogram per kilogram of water. Sample problem #1: A sharp edged cylinder 7.5 cm in diameter is carefully driven into the soil so that negligible compaction occurs. A 12-cm column of soil is obtained. The fresh and dry masses of the soil sample are 910.6 and 736.3 grams respectively. Assume that the particle density of the soil is 2.60 g/cm3. Determine: a. Apparent specific gravity b. Percent porosity c. Soil moisture content by dry weight d. Soil moisture content by volume e. Depth of water present in the soil column Solution: d 2 h  (7.5cm) 2 (12cm) a. VT = r 2 h = = = 530.144 cm 3 4 4 Ws 736.3g As = = = 1.39  wVT (1g / cm )(530.144cm 3 ) 3 Ws 736.3g b. Vs = = = 283.192 cm 3 p 3 (2.60 g / cm ) VV − VT VT − VS 530.144 cm 3 − 283.192 cm 3 n= = = = 46.58% VT VT 530.144 cm 3 Ww WT − Ws 910.6 g − 736.3g c. mcw = = = = 23.67% Ws Ws 736.3g d. mcv = mcw x As = 0.2367 x 1.39 = 32.90% e. d w = mcv x D = 0.3290 x 12 cm = 3.948 cm B. Soil Moisture Constants 1. Saturation Point – the amount of water the soil profile will hold when all its pore spaces are filled up with water 2. Field Capacity - (1) the amount of water a soil profile will hold against drainage by gravity at a specified time (usually from 24 to 48 hours) after a thorough wetting. (2) Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials I-5 the moisture content of the soil when gravitational water has been removed (after irrigation by flooding). It is usually determined few days after irrigation. The soil moisture tension at this point is normally between 1/10 to 1/3 atmosphere. 3. Permanent Wilting Point (or wilting coefficient) - the soil moisture content when plants permanently wilt. The soil moisture tension at this point is about 15 atmospheres. Permanent wilting percentage can be estimated by dividing the field capacity by a factor ranging form 2.0 top 2.4, with the value higher for soils with higher silt content. 4. Available Moisture (AM) - the difference in moisture content of the soil between field capacity and the permanent wilting point. 5. Readily Available Moisture (RAM) - that portion of the available moisture that is most easily extracted by plants; this is approximately 75% of the available moisture. 6. Computation of Moisture Content – the amount of moisture present in the soil (mc) given the percent AM content retained or used and the FC and PWP moisture contents of the soil can be computed using the equations: a. mc = FC – % AM used x (FC-PWP) b. mc = PWP + % AM retained x (FC-PWP) Sample problem #2: What is the present moisture content of a soil with 25% of its AM has been used if its FC is 24% and PWP is 16% by weight? Solution: mcw = FC – % AM used x (FC-PWP) = 0.24 - 0.25(0.24-0.16) = 0.24 - 0.25(0.08) = 0.24 - 0.02 = 0.22 = 22% by weight C. Irrigation Efficiencies 1. Water Conveyance Efficiency - the ratio between the water delivered to the farm and the water diverted from a river or reservoir expressed in percent. 2. Water Application Efficiency - the ratio between water stored in the soil root zone during irrigation and the water delivered to the farm expressed in percent. 3. Water-use Efficiency - the ratio of water beneficially used on the project, farm or field to the amount of water delivered to the farm expressed in percent. 4. Water Storage Efficiency - the ratio of water stored in the root zone during the irrigation to the water needed in the root zone prior to irrigation, expressed in percent. 5. Consumptive Use Efficiency - the ratio of the normal consumptive use of water to the net amount of water depleted from the root zone soil. D. Pump Irrigation Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials I-6 1. Water horsepower - the power theoretically required to lift a given quantity of water each second to specified height. 2. Brake Horsepower - water horsepower divided by pump efficiency, in decimal. 3. Static Head - the difference in elevation of the water surface in a pond, lake, or river from which pumped water is taken, and the water surface of the discharge canal into which the water flows from a submerged discharged pipe. In pumping from groundwater source, static head is the difference in elevation between the water surface in the well and the water surface of the discharged canal. 4. Total Dynamic Head - the sum of total static head, pressure head, velocity head and friction head. 5. Drawdown - (in a well) is the difference in elevation between the groundwater table and the water surface at the well when pumping. 6. Characteristic Curve - graphs that show interrelations between speed, head discharge, and horsepower of a pump. 7. Specific Speed - expresses the relationship between speed in rpm, discharge in gpm, and head in feet. E. Irrigation Principles 1. Evapotranspiration - is the sum of transpiration and water evaporated from the soil, or exterior portions of the plants where water may have accumulated from irrigation, rainfall, dew, or exudation from the interior of the plants. Consumptive use is identical with evapotranspiration, for practical purposes. Consumptive use only includes water retained in the plant tissue. 2. Transpiration - the process by which water vapor escapes from living plants, principally by leaves, and enters the atmosphere. 3. Canal Capacity The ability of the stream to provide water determines the extent of the total service area of a national irrigation system. To compute for the service area, the stream’s dependable flow (to a certain percent of dependability) is divided by the diversion water requirement. In equation form it can be stated as: dependabl e s tream fl ow (m3 / s) s ervi c earea (ha) = t (m3 / s ha) di vers i onwater requi rem en The design of irrigation canals, whether main, secondary or tertiary is primarily based on its expected capacity. The canal capacity (Q) refers to the amount of water it can deliver at a unit time and is dictated by total area it serves and the area water requirement. In equation form: Q (m3/s) = area (ha) x water requirement (m 3/s-ha) Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials I-7 The capacity of a main and secondary canal is computed using the diversion water requirement while that of a tertiary canal is based on the farm water requirement. Both capacities are reflective of the area they serve. 4. Leaching Requirement (LR) - the fraction of the irrigation water that must be leached through the root zone to control soil salinity at specified level. Ddw EC(iw + r w) LR = = Diw + Dr w ECdw where: Ddw – depth of drainage water Diw - depth of irrigation water Drw - depth of rain water EC(iw+rw) - weighted average of the electrical conductivity of the irrigation and drainage water Diw x ECiw+ Dr w x ECr w EC(iw + r w) = Diw + Dr w ECiw - electrical conductivity of irrigation water ECiw- electrical conductivity of rain water ECdw- electrical conductivity of drainage water 5. Crop Water Requirement (CWR) The crop water requirement of an area grown to a certain crop refers to the amount of water used for the non-consumptive demands such as land soaking and land preparation, and for the consumptive demands such as evapotranspiration requirements of the crop during its entire growth period. Non-consumptive uses The amount of water needed for the non-consumptive uses depends on the type of crop being grown, type of soil, climatic conditions and farming practices and techniques. The non-consumptive uses include seepage and percolation losses. Seepage refers to the lateral movement of water along the soil profile while percolation which is highly dependent on soil texture, refers to the downward movement of water through a depth of soil. Consumptive uses The consumptive use of crop includes evaporation and transpiration, lumped together as evapotranspiration. Evaporation accounts for he quantity of water evaporating from soil and water surfaces as well as that from plant part surfaces. Transpiration the water absorbed by the plant through its roots and used in building plant tissues or released into the atmosphere. Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials I-8 6. Irrigation Water Requirement (IWR) The irrigation water requirement is the amount of water to be applied to the field as irrigation. It can be computed by deducting the effective rainfall (ER) from the total crop water requirement. In equation form it is: IWR = CWR-ER 7. Farm Water Requirement (FWR) From the tertiary canal to the field, application losses are incurred. Application losses include the seepage and percolation losses along the canals as well as losses due to evaporation. To account for these losses, the design farm requirement is computed with the application efficiency in consideration. Application efficiency is the ratio of the amount of water entering the tertiary canal and the amount of water that reaches the field. With the losses, the application efficiency (e a) of the tertiary canals becomes less than one. In equation form: Qin water enter ingthe ter tiar ycanal ea = = Qout water enter ingthe ter tiar ycanal (S + P + E) where: ea = application efficiency Qin = water entering the tertiary canal Qout = water exiting the tertiary canal or water entering the field S = seepage losses P = percolation losses E = evaporation losses The design farm water requirement can be computed with the equation: 8. Diversion Water Requirement (DWR) IWR FWR = ea Similar to that of tertiary canals, the main systems (main and secondary canals) also incur losses termed as conveyance losses. Seepage and percolation losses and evaporation along the conveyance canals comprise the conveyance losses. To account for these losses, the design diversion water requirement is computed with the conveyance efficiency in consideration. Conveyance efficiency is the ratio of the amount of water entering the main canal and the amount of water that reaches the tertiary canal. With the losses, the conveyance efficiency (ec) of the main systems becomes less than one. In equation form: Q in water enter ingthe main canal ec = = Q out water enter ingthe main canal (S + P + E) where: ec = conveyance efficiency Qin = water entering the main canal Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials I-9 Qout = water exiting the secondary canal or water entering the tertiary canal S = seepage losses P = percolation losses E = evaporation losses The design diversion water requirement, which is the quantity of water to be obtained from the source, can be computed by: FWR DWR = ec where: DWR = design diversion water requirement FWR = design farm water requirement ec = conveyance efficiency F. Modes of Irrigation 1. National Irrigation Systems (NIS) – irrigation systems that have relatively large service areas and are managed by government agencies 2. Communal Irrigation Systems (CIS) – managed and operated by farmers’ or irrigators’ associations 3. Shallow Tubewell Irrigation Systems (STW) – pipes vertically set into the ground that abstract groundwater to be used for irrigation, usually owned and operated by individual farmers G. Methods of Irrigation Irrigation water can be applied to the uplands in any of the following general ways: 1. by overhead irrigation, wherein the soil is moistened in much the same way as rain a. Watering can. It is the simplest piece of overhead irrigation equipment and is commonly used in small-scale upland farming. Since the water is carried by hand, this method is limited to small plots with easily accessible source of water. The size of the plot depends largely upon its distance from the source, and the time that it takes to fill the can at the source. b. Hose pipe. This method can be used if there is a piped water-distribution system where a hose pipe can be connected to a tap or outlet and there is enough pressure in the water as it emerges from the hose pipe. A major disadvantage of this system is likely to be the cost of the water supply. c. Sprinkler Irrigation. This method is the application of water to the surface of the soil in the form of spray, simulating that of rain. The spray is produced by the flow of water under pressure through small orifices or nozzles. Pumping usually provides the pressure. With careful selection of nozzle sizes, operating pressure, and sprinkler spacing, the amount of irrigation water required to refill the crop-root zone can be applied nearly uniformly at a rate to suit the infiltration rate of the soil to obtain efficient irrigation. 2. furrows, which wets only a part of the ground surface a. Furrow irrigation. It is accomplished by running water through small channels or furrows while it moves down or across the slope of the field. The water sips into the bottom and sides of t he furrows to provide the desired wetting. Careful land grading for uniform slopes is essential with this method. Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials I-10 b. Corrugation irrigation. It is a variation of the furrow method and it uses small rills or corrugations for irrigating closely spaced crops, such as small grains and pastures. The water seeps laterally through the soil, wetting the area between the corrugations. 3. by flooding, which wets the entire land surface a. Ordinary flooding. Water is applied from field ditches to guide its flow and it is difficult to attain high irrigation efficiency using this method. The chief advantage of this method is its low initial cost of preparing the land. b. Border-strip flooding. A field is divided into a series of strips by borders or ridges running down the predominant slope or on the contour. To irrigate, water is released into the head of the border. The water, confined and guided by two adjacent borders, advances in a thin sheet toward the lower end of the strip. The objective is to allow a sheet of water to advance down the narrow strip of land, allowing it to enter the soil as the water advances. c. Level-border or basin irrigation. Water is supplied to level plots surrounded by dikes or levees. This method is particularly useful on fine- textured soils with low permeability, it is necessary to hold the water on the surface to secure adequate penetration. d. Contour-ditch irrigation. It involves controlled flooding from field ditches along the contour of the land, which allows the water to flood down the slope between field ditches without employing dikes or other means that guide or restrict its movement. 4. by drip or trickle irrigation wherein the water is directed to the base of the plant. Water is applied to the soil through small orifices. The small orifices, often called emitters, are designed to discharge water at rates of 1 to 8 liters per hour. Water is delivered to the orifices through plastic pipelines, which are generally laid on the soil surface or buried. The rate of discharge is determined by the size of the orifice and the pressure in the pipelines. This method is particularly beneficial for young orchards, vineyards, closed-spaced perennials, and other crops of high value and in areas where water is scarce or has a high salt content. A highly efficient water utilization can be achieved with this method, bit it is very expensive. 5. by sub-irrigation, wherein the surface is rarely wet since the water is supplied from the soil underneath. This requires complete control of the water table so that the root zone is kept relatively free of excess water but is continually supplied with capillary moisture during the cropping season. H. Head computations for a Sprinkler Irrigation System The design capacity for sprinklers on a lateral with uniform spacing should be based on the operating pressure. From the chosen sprinkler operating pressure (from the catalog), the pressure or head Hn required at the junction of the lateral and the main can be computed using the equation: Hn = Ho + Hf ± He + Hrp Where: Hn = pressure or head required at the junction of the lateral and the main Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials I-11 H o= nozzle pressure at the farthest end of the line (sprinkler operating pressure) Hfl = friction head loss in the lateral He = maximum difference in elevation between the junction with the main and the farthest sprinkler on the lateral Hrp = riser height If the lateral is located downhill from the main, the elevation term is negative. If the lateral is located uphill from the main, the elevation term is positive. 1. Friction or head loss in pipes a. for main lines K sLQ1.9 Hfm = (4.1 0x1 06 ) D4.9 where: Hfm = total friction loss in the main line, in meters Ks = Scobey’s coefficient of retardation (0.32 for new Transite pipe, 0.40 for steel pipe or portable aluminum pipe and couplers, and 0.42 for portable galvanized steel pipe and couplers) L = length of the pipe in m Q = total discharge in L/s D = inside diameter of pipe in mm b. for laterals K s L Q1.9 H fl = (4.1 0x1 06 ) (F) D 4.9 where: Hfl = total friction loss in the main line, in meters F = correction factor from Table 1 Table 1. Correction factor F for Friction Losses in Aluminum Pipes with Multiple Outlets. Correction Factor, F Number of First Sprinkler One Sprinkler First Sprinkler One Sprinkler Sprinklers Interval from Main Interval from Main 1 1.00 1.00 2 0.63 0.51 4 0.48 0.41 6 0.43 0.38 8 0.41 0.37 12 0.39 0.36 16 0.38 0.36 20 0.37 0.35 30 0.36 0.35 Source: Schwab, G. O., et al.(1992) Soil and Water Conservation Engineering. 4th ed. Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials I-12 2. Pump head In selecting a suitable pump, it is necessary to determine the maximum total head against which the pump is working. Ht = Hn + Hfm + Hj + Hs Where: Ht = total design head against which the pump is working Hn = pressure or head required at the junction of the lateral and the main Hfm = total friction loss in the main line Hj = elevation difference between the pump and the junction of the lateral and the main Hs = elevation difference between the pump and the water supply after drawdown, all in m or kPa I. Drainage - removal of excess water in the soil to create conditions suitable for plant growth. Israelsen (1962) noted that adequate drainage: 1. improves soil structure 2. the productivity of soils 3. facilitates early plowing and planting 4. lengthens the crop growing season 5. increases the depth of root zone soil thereby provides more available soil moisture and plant food 6. improves soil ventilation 7. increases water infiltration into soils 8. favors growth of soil bacteria 9. leaches excess salts from the soil J. Sources/Causes of excess water: 1. rainfall 2. high water table 3. over-irrigation 4. runoff/seepage from adjacent farms K. Components of a Drainage System: 1. field drainage system (quarternary and tertiary or collector drains) 2. secondary drains or laterals 3. outfall 4. recipient water L. Types of Surface Field Drainage System or Open Drains 1. Bedding System. The soil type largely influences the width of bed to be used. The furrows drain to collection ditches. 2. Random Ditch System. This system is adapted to areas that have depressions which are too deep or too large to fill by land leveling. The ditches meander from one low spot to another, collecting the water and carrying it to an outlet ditch. 3. Interception or Cross-slope System. This resembles terracing in that the drainage ditches are constructed around the slope on a uniform grade according Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials I-13 to the land topography. The ditches should be constructed across the slope as straight and parallel s the topography permits. 4. Diversion or Parallel Ditch System. This is suitable on flat, poorly drained soils that have numerous shallow depresssions. In general, the ditches are 185 m to a maximum of 370 m apart (not necessarily equidistant) and the land in between the parallel ditches is sloped and smoothed to eliminate any minor depressions or obstructions to the overland flow of the water. M. Layout of a Tile-Drain System or Closed Drains 1. Natural System. This system is used in rolling topography where drainage is necessary only in small valleys. 2. Gridiron Layout. Used if the entire area is to be drained and is usually more economic. Laterals enter the submain from one side only to minimize the double drainage that occurs near the submain. 3. Herringbone Pattern. The submain is laid in a depression and the laterals join the submain from each side alternately. The land along the submain is double drained, but since it is in a depression, it probably requires more drainage. 4. Double-main System. This system is often used if the bottom of the depression is wide since it reduces the lengths of the laterals and eliminates the break in slope of the laterals at the edge of the depression. 5. Intercepting Drain. This is used if the main source of excess water is drainage from hill lands. The drains are placed along the toe of the slope to protect the bottom land. 6. Arrangement to avoid trees. This system is adopted to minimize the exposure of the laterals to the hazard posed by root of trees which easily enter the open joints of underdrains. Mains and laterals were kept well away from trees. N. Drain Depths and Spacing In highly permeable soils underlain by an impervious layer or compact clay of low permeability, the groundwater flow is essentially horizontal towards the drains. To simplify the illustration, the source of water flowing toward the drain is considered a reservoir as shown in the following figure. Ground surface Drain open or closed Dwt Sandy soil Water Surface Saturated hf Sandy soil Water reservoir H (H+h) 2 h R Clay of low permeability Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials I-14 Linear flow of groundwater (for sandy soils over clay) toward drains spaced 2R in which the water table midway between the drains is H-h above the water surface drain. Adapted from Irrigation Principles and Practices by Israelsen and Hansen The water surface is maintained in the reservoir and adjoining soil a distance of H above the clay. Flow from the reservoir to the drain is steady, and it is assumed that the reservoir is the only source of water. Groundwater flows to the drain from both sides. Let 2q be the flow into a drain in length L. Then the groundwater flow from one side to the drain is q = av and from Darcy’s Law, hf H h v =k =k R R Consider the depth of saturated sand about midway between the reservoir and the drain as average; then the average area of saturated soil, in drain length L, through which the groundwater flows is:  H + h a=  L  2  and the quantity of flow from the reservoir to the drain H+h H h kL(H h)2 q= (Lxk) = 2 R 2R The quantity of flow to the drain from the reservoir on both sides would be: kL( H 2 − h 2 ) Q = 2q = R from which: kL( H 2 − h 2 ) R= Q and since drain spacing S equals 2R 2 kL( H 2 − h 2 ) S= Q Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials I-15 Some Irrigation Terms as defined by the Republic Act 8435 or The Agriculture and Fisheries Modernization Act of 1997 1. Communal Irrigation System (CIS) - an irrigation system that is managed by a bonafide Irrigators’ Association 2. Headworks - the composite parts of the irrigation system that divert water from natural bodies of water such as rivers, streams, and lakes 3. Irrigable Lands - lands which display marked characteristics justifying the operation of an irrigation system 4. Irrigated Lands - lands serviced by natural irrigation or irrigation facilities. These include land where water is not readily available as existing irrigation facilities need rehabilitation or upgrading or where irrigation water is not available year-round 5. Irrigation System - a system of irrigation facilities covering contiguous areas 6. Irrigators’ Association (IA) - an association of farmers within a contiguous area served by a National Irrigation System or Communal Irrigation System 7. Main Canal - the channel where diverted water from a source flows to the intended area to be irrigated 8. National Irrigation System (NIS) - a major irrigation system managed by the National Irrigation Administration 9. On-Farm Irrigation Facilities - composite facilities that permit entry of water to paddy areas and consist of farm ditches and turnouts 10. Secondary Canal - the channel connected to the main canal which distributes irrigation to specific areas 11. Shallow Tubewell (STW) - a tube or shaft vertically set into the ground for the purpose of bringing groundwater to the soil surface from a depth of less than 20 meters by suction lifting DRAIN SPACING EQUATIONS A. STEADY STATE: I. Drain Bottom Reaching Impermeable Layer (DBRIL) 1. with recharge, unconfined aquifer condition, drain levels of same elevation 4 k(H2 - D2 ) L= R where: L – drain spacing k- soil hydraulic concudtivity H- depth of water midway between the drains D- depth of water inside the drains R- net recharge into the area to be drained Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials I-16 2. no recharge, unconfined aquifer condition, drain levels of unequal elevation (unidirectional flow to lower level) k(h12 - h 22 ) L= 2q where: h1, h2 – water depths at the drains q- flow into the drain per unit drain width 3. no recharge, confined aquifer condition, drain levels of unequal elevation (unidirectional flow to lower level) bk(h1 - h 2 ) L= q where: b- thickness of confined auifer (land strip to be drained) 4. with recharge, unconfined aquifer condition, drain levels of equal elevation a. flow to left drain, qL RL (h 2 − h12 ) qL = − −k 2 2 2L b. flow to right drain, qR RL (h 2 − h12 ) qR = −k 2 2 2L c. elevation of the water table (h) at any point x Rx x h= (L − x) + (h2 2 −h12 ) + h12 k L d. stagnation point , xs( x where dh/dx = 0, also x where h= hmax) L (h 2 − h12 ) xs = +k 2 2 2RL e. maximum elevation of the water table (hmax) Rx s x hmax = (L − x s ) + s (h2 2 −h12 ) + h12 k L Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials I-17 II. Drain Bottom Not Reaching Impermeable Layer (DBNRIL) Hooghoudt’s Equation (PIPE DRAINS) 4 k1h2 + 8k 2 dh L= q where: k1- hydraulic conductivity of unsaturated soil k2 - hydraulic conductivity of unsaturated soil d – effective depth of pipe drains UNSTEADY STATE Glover-Dumm Formula  2 kd L=  reaction factor α has the following equation   1. 16 ho  1  = ln    h t  t where : μ – drainable pore space (m3/m3) ho – height of water midway between the drains at time=0 ht- height of water midway between the drains at time t t – time to drain References: Israelsen, O.W and V.E Hansen (1962). Irrigation Principles and Practices.3rd ed. JohnWiley and Sons, Inc. New York, U.S.A. Linsley, R. K. et al. Hydrology for Engineers. Mc Graw-Hill, Inc. U.S.A. Linsley, R.K. et al (1992). Water Resources Engineering.4th ed. Mc Graw-Hill, Inc. U.S.A. Schwab, G. O., et al.(1992) Soil and Water Conservation Engineering. 4th ed. JohnWiley and Sons, Inc. New York, U.S.A. Irrigation and Drainage PSAE Region IV - Agricultural Engineering Board Review Materials II-1 Soil and Water Conservation Engineering Engr. Jonathan P. Aguilar University Researcher Land and Water Resources Division Institute of Agricultural Engineering College of Engineering and Agro-Industrial Technology University of the Philippines Los Baños College, Laguna 4031 I. INTRODUCTION Agricultural Engineering is a profession that looks not only in the productivity of a farm but more importantly its long term sustainability. A good agricultural engineer takes into consideration factors that need to be addressed to achieve the optimum production without neglecting and putting in peril the needs of the future. And only by understanding these factors could someone harness its full potential and at the same time put the necessary intervention for its protection. Two of the most important resources in agriculture are the soil for which the plants grow and draw nutrient and the water for which the plants and practically all life forms are mostly made of. Soil and water conservation is a very broad subject matter. It touches many topics in hydraulics, irrigation and drainage because of the natural interaction of soil and water. The discussion will cover the following topics in the manner presented. A. Facts and Figures B. Hydrology 1. Hydrologic Cycle 2. Precipitation and Data Analysis a. Rainfall Occurrence b. Types of Rainfall c. Spatial Distribution d. Frequency Analysis C. Runoff 1. Runoff Components 2. Important Properties of Runoff 3. Factors Affecting Runoff 4. Hydrograph Analysis a. Baseflow Separation b. Direct Runoff Hydrograph Development 5. Streamflow Measurement a. Water Stage Measurement b. Discharge Measurement 6. Estimating Volume of Runoff a. Rational Equation b. Cook’s Method c. Curve Number d. Other Methods D. Erosion and Sediment Transport 1. Types of Erosion Soil and Water Conservation Engineering PSAE Region IV - Agricultural Engineering Board Review Materials II-2 2. Factors Affecting Erosion 3. (Modified) Universal Soil Loss Equation E. Erosion Control Measures 1. Biological/Vegetative 2. Engineering a. Terracing b. Conservation Structures 3. Vengineering F. Open Channel Flow 1. Types of Flow 2. Flow Measuring Devices/Structures G. Definition of Terms II. OBJECTIVES The general objective of this subject matter is to instill into the mind of an agricultural engineer that other than farm productivity, part of his/her responsibility is to conserve the very resource that is being tapped, soil and water. This can only be achieved by properly understanding its properties, the processes involved, the ways to maximize its potential and the proper measures to control or prevent the adverse effects it may cause. Specifically this aims to let the reviewer: 1. appreciate the topic by knowing some facts and figures; 2. understand the basic principle of hydrologic cycle and its processes; 3. understand precipitation and runoff considering its capacity to be beneficial and destructive at the same time; 4. learn to compute and predict precipitation, runoff and soil erosion as a prerequisite to harnessing its potential and/or averting its damaging effects; 5. learn the ways to control soil erosion; 6. understand some basic principles and computations in open channel hydraulics; 7. broaden his/her knowledge by knowing the some terms usually used in the soil and water conservation; and 8. enhance his/her problem solving capability through sample computations and problems. III. SOIL AND WATER CONSERVATION TOPICS A. Facts and Figures Inventory of water at the Earth’s surface and its Residence Time Approximate Volume Percent of Total Reservoir Residence Time (km3 x 10,000,000) (%) (years) Oceans 1370 97.25 Ice Caps and Glaciers 29 2.05 40 200 (shallow) Groundwater 9.5 0.68 10,000 (deep) Lakes 0.125 0.01 100 Soil Moisture 0.065 0.005 0.2 Atmosphere 0.013 0.001 Streams and Rivers 0.0017 0.0001 0.04 Biosphere 0.0006 0.00004 Soil and Water Conservation Engineering PSAE Region IV - Agricultural Engineering Board Review Materials II-3 B. Hydrology Important Processes in the Hydrologic Cycle 1. Evaporation 6. Percolation 2. Transpiration (Evapotranspiration) 7. Surface Runoff 3. Condensation 8. Interflow 4. Precipitation 9. Groundwater flow 5. Infiltration Importance of Some Hydrologic Data Precipitation –needed in estimating runoff, planning erosion control measures, planning for irrigation and drainage, and water conservation in low rainfall regions Runoff – needed in designing structures and channels that will handle natural flows of water Infiltration, Evaporation and transpiration –required in planning irrigation and drainage systems, moisture conservation practices, etc. Types of Precipitation 1. Convective Precipitation – caused by the rising of warmer, lighter air in colder, denser surroundings. 2. Orographic Precipitation – results from mechanical lifting of the air mass over mountain barriers. 3. Cyclonic Precipitation – associated with the movement of air masses from high- pressure to low-pressure regions. Non-frontal – air is lifted through horizontal convergence of the inflow into a low pressure area Frontal – lifting of warm air over cold air at the contact zone between air masses having different characteristics Spatial Distribution 1. Arithmetic mean – this involves the averaging arithmetically all the rainfall depths measured by the rain gages within the area. n P i P= i =1 n 2. Thiessen polygon – location of the rain gauges are plotted on the map of the area and stations are connected by straight lines. Perpendicular bisectors are constructed on each of the connecting lines thereby forming polygons enclosing each rain gage. The average rainfall over the area is then estimated as the area weighted average for all polygons. Thus, the average precipitation over a watershed is computed using the equation:  Pi Ai P= A Total 3. Isohyetal – station locations and amounts of rainfall are plotted on a suitable map, and contours of equal precipitation (isohyets) are then drawn. The equation used is similar to that of Thiessen method except that the area, Ai, is the area under one isohyet, Pi. Soil and Water Conservation Engineering PSAE Region IV - Agricultural Engineering Board Review Materials II-4 4. Grid-Point Method – computer aided method wherein it averages estimated precipitation at all points of a superimposed grid. 5. Inverse Distance Ratio Method – for rolling areas and non-uniform distribution of gages, distance factor fixed by location of gages is used for analysis using this method. It operates under the principle that the value of points close together in space are more likely to be similar than with points farther apart. Frequency analysis Conservation structures constructed, i.e. to convey runoff, should be designed to handle the maximum expected events. The maximum expected runoff, for example, can be calculated if the maximum expected rainfall is known. The maximum expected rainfall depends upon the frequency that is taken into consideration. This frequency or recurrence interval is defined as the number of years during which one storm of a given duration and intensity is expected to occur. Hydrologic processes are said to fall under stochastic, probabilistic or deterministic processes. Nevertheless, literature says that most hydrologic processes are stochastic in nature (more or less). Deterministic Process – one in which a definite law of certainty exists Probabilistic Process – governed by chance; time series INDEPENDENT Stochastic Process – governed by chance; time dependent For simplicity in dealing with hydrologic data, probabilistic process is assumed. Frequency Distribution / Probability Density Curve 1. Normal Distribution – data is more or less normally distributed 2. Log-Normal Distribution – used when there is skewness in the data distribution 3. Pearson Type III Distribution -- considers further the skewness of the logarithmically transformed hydrologic data; preferred for flood flow frequency; log-normal distribution is a special case of this method 4. Gamma Density Function – so far gives the best fit for most stations for 1-week, 2-week, 3-week and monthly rainfall totals of the country C. Runoff Runoff Components 1. Surface Runoff -- that which travels over the ground surface and through the channels to reach the basin outlet 2. Interflow – that which infiltrates into the soil surface and moves laterally through the upper soil horizons towards streams as perched groundwater above major groundwater level 3. Groundwater Runoff – groundwater discharge into a stream due to deep percolation of the infiltrated water into groundwater aquifers Other terms: a. Overland flow – that part of surface runoff that flows over the surface towards the stream channel b. Direct runoff – surface runoff plus interflow c. Subsurface runoff – same as interflow d. Baseflow – same as groundwater runoff Soil and Water Conservation Engineering PSAE Region IV - Agricultural Engineering Board Review Materials II-5 Important Properties of Runoff 1. Peak Flow (qp) – used to determine the magnitude of floods and a valuable consideration in the design of structures 2. Time to Peak (tp) – used for flood forecasting and water quality studies; watershed response time 3. Runoff Volume (V) – total water yield from a storm for a given watershed 4. Recession Time (tr) – time for surface and interflow to recede; duration of flooding 5. Base time (tb) – total of time to peak and recession time; (tb) = (tp) + (tr) 6. Base flow – low flow, dependable flow, groundwater discharge Factors affecting runoff Climatic factors ▪ Precipitation − Form − Intensity − Duration − Time Distribution − Areal Distribution − Direction of storm movement in relation to the orientation of the watershed − Frequency of occurrence − Antecedent precipitation ▪ Vegetation Interception of Rainfall ▪ Evapotranspiration Physiographic factors ▪ Watershed characteristics − Size − Shape − Slope − Orientation − Elevation − Stream Density − Land use and cover − Infiltration characteristics − Geologic Conditions (e.g. permeability, aquifer transmissivity and storage coefficient) − Topographic conditions (e.g. depression storage, lakes, swamps) − Artificial Drainage − Man-made modification (e.g. dams, terraces, etc.) ▪ Channel characteristics − Slope − Length − Cross-section slope − Roughness of channel bed − Tributaries − Storage Capacity (e.g. backwater effects) Soil and Water Conservation Engineering PSAE Region IV - Agricultural Engineering Board Review Materials II-6 Hydrograph Analysis A hydrograph is a graphical representation of the instantaneous runoff rate against time. The area under the hydrograph gives the runoff volume. Direct Runoff Hydrograph Development Basic Hydrograph – assumes that all hydrographs from all small watersheds (in the US) have similar forms -- it is plotted over 100 arbitrary units of flow and 100 arbitrary units of time Triangular Hydrograph -- approximation of the basic hydrograph -- developed mainly to simplify flood routing procedures in US Unit Hydrograph -- a hydrograph with a unit volume (e.g. 1cm) of direct runoff for a given storm duration -- represent the response of the basin on a given storm duration and characteristics -- enables one to synthesize hydrographs for complex storms by superimposing the hydrographs resulting from the individual components of the storm Synthetic Hydrograph -- prepared using data from a number of watershed to develop dimensionless unit hydrographs which are applicable to ungaged watersheds -- a dimensionless hydrograph is made from natural or unit hydrographs in which the time to peak (tp) and the peak runoff rate (qp) are considered an t/tp is plotted against q/qp Streamflow Measurement Water Stage Measurement 1. Staff Gage 2. Crest Stage Gage – provide record of the highest stage observed at a stream 3. Bubbler Gage -- record the pressure required to maintain a small flow of gas from an orifice submerged in the stream 4. Float-type Water-Stage Recorder – motion of a float is recorded on a graph Discharge Measurement 1. Float method – measures the velocity of a floating object preferably in a straight section of a stream 2. Current Meters a) Price (Cup-type) Meter b) Propeller-type Meter utilizes the equation or relation: V = a + bN Where: V – water velocity a – starting velocity or velocity required to overcome mechanical friction b – constant of proportionality N – revolution per seconds of the meter cups / propeller Soil and Water Conservation Engineering PSAE Region IV - Agricultural Engineering Board Review Materials II-7 Note: To get the average velocity of a stream, measurement of velocity must be taken at 1/5th and 4/5th of vertical depth below the surface for deep streams, or at 3/5th of the vertical depth for relatively shallow streams. 3. Use of Manning’s Equation and similar empirical formula 4. Weirs and Flumes 5. Stage-discharge Relation (Rating Curve) Estimating Volume of Runoff Rational Method Q=C*I*A (Metric) 360 Q=C*I*A (English) Where: Q = peak runoff rate (cms or cfs) I = rainfall intensity (mm/h or in/h) for a duration equal to the time of concentration and for the given return period A = catchment area (ha or acres) C = runoff coefficient = ratio of the peak runoff rate to the rainfall intensity Time of concentration, 0.77 tc= 0.02 L S-0.385 (Metric) 0.77 = 0.0078 L S-0.385 (English) L = maximum length of flow (m or ft) S = watershed gradient (ft/ft or m/m) or the difference in elevation between the outlet and the most remote point of the watershed Cook’s Method q=PRF Where: q = peak runoff rate for a specified geographic location and a given recurrence interval P = peak runoff rate on a 10-year recurrence R = geographic rainfall factor F = recurrence interval factor Curve Number Method Q= (I − 0.2S )2 (I + 0.8S ) Where: Q = direct surface runoff depth (mm) I = storm rainfall (mm) Soil and Water Conservation Engineering PSAE Region IV - Agricultural Engineering Board Review Materials II-8 S = maximum potential difference between rainfall and runoff starting at the time the storm began 25400 S= − 254 CN CN = curve number ranging from 0 to 100; function of land use, antecedent moisture, soil type, hydrologic condition D. Erosion and Sediment Transport Effects of soil erosion: ▪ Loss of soil fertility ▪ Raising the beds of streams and rivers, thus reducing their capacity ▪ Silting of reservoirs, thus reducing their capacity and useful life ▪ Damage to agricultural lands due to soil deposition. Types of soil erosion: ▪ Water erosion − Splash / Raindrop Erosion – primarily caused by raindrop − Sheet erosion- a thin film of soil layer detached and transported by water flowing on the land surface. − Interrill erosion – combination and splash and sheet erosion − Rill erosion- finger-like rills appear on the soil surface. − Gully erosion- advanced stage of rill erosion. Rills when neglected develop in size and become gullies. − Stream bank erosion – erosion of stream banks by flowing water. − Coastal erosion- erosion caused by wave action on the seashore. ▪ Wind erosion – caused by high velocity winds moving over barren land surfaces. ▪ Slip erosion – land slides and slips due to saturation of steep hills and slopes. Factors affecting soil erosion − Climatic factors – precipitation, wind, temperature, humidity and solar radiation − Soil physical properties influence the extent to which soil can be dispersed and transported. − Soil structure, texture, organic matter content, moisture content, and bulk density. Vegetation acts in the following ways: − Interception of falling rain by the foliage helps in absorbing the energy of raindrops. − Transpiration, thus decreasing soil moisture and increasing storage capacity of the soil − Binding effect of root system helps retard erosion − Decayed roots provides cavities and pores which promotes infiltration − Increase surface friction reduces runoff velocity − Increased biological activity in the soil enhancing its good tilth Topography − Degree of slope − Shape and length of slope − Size and shape of the watershed Soil and Water Conservation Engineering PSAE Region IV - Agricultural Engineering Board Review Materials II-9 Universal Soil Loss Equation A = R* K* LS * C * P Where: A = Average soil loss for the given period R = Rainfall erosivity factor K = Soil erodibility factor C = Cropping management factor L = length of slope factor S = slope factor P = conservation practice factor E. Erosion Control Measures Vegetative Measure 1. Reforestation – replanting of forest trees species in the watershed area 2. Agroforestry – a practical adaptation of reforestation whereby the species planted have economic value, such as mango, pili and so on. 3. Strip cropping – is the practice of growing different crops in alternate strips across the slope to serve as barriers for soil erosion. 4. Mulching – the process of covering the land surface with plant residues, plastic or other materials appropriate to arrest loss of moisture through evaporation 5. Contour cultivation – consists of carrying out agricultural operations very nearly on the contour. It reduces the velocity of overland flow and retards soil erosion 6. Cropping systems –modification of copping system such as crop rotation and mixed cropping Engineering Measure 1. Terracing – construction of earth embankment or ridge and channel across the slope at an acceptable grade to control the flow of runoff as well as soil particles. 2. Grassed Waterways – establishment of natural waterways or construction of canals and planting it with grasses to make it stable and arrest soil erosion 3. Check dams or weirs – constructed along the channel or waterway to control the velocity of flowing water and encourage deposition of sediments carried by water 4. Farm Ponds / Water Impounding Dams – temporary detainment of water in farm pond and dams to mitigate the erosive capacity of water 5. Diversion Canal – a channel constructed around the slope and given a slight gradient to cause water to flow to a suitable and stable outlet. 6. Gabions – similar to stone check dams however the stones in this case are placed on rectangular wire mesh, piled-up as blocks and tied together to form a reinforced wall or dam structure. 7. Riprap – concrete structure made of stones constructed along steep embankments to prevent landslide or gully erosion 8. Stone wall – made of stones carefully and properly piled-up and arranged on steep embankments to protect from gully erosion or landslide Gully erosion and control Gully erosion usually starts as small rills and gradually develops into deeper crevices. The rate of gully formation depends on the amount of runoff, slope and the soil characteristics. Soil and Water Conservation Engineering PSAE Region IV - Agricultural Engineering Board Review Materials II-10 Control measures: ▪ Grassed waterways – is constructed across the slope above the gully. The grassed waterway intercepts the runoff coming from above the gully. ▪ Vegetation – can stabilize the slopes of the gully and hence reduce runoff velocities. ▪ Chute spillways, drops, and drop spillways are permanent structures that can reduce the energy of the runoff water. Terracing Functions: − Reduce sheet and rill erosion − Prevent gully formation − Moisture conservation Types of Terraces 1. Bench - construction of series of platforms along the contours cut into the hill slope in a step like formation - used for 25-30% slopes - used for maximum moisture conservation - used where land is at a premium - conventional - difficult to farm 2. Zingg or Conservation bench terrace - used for 9-24% slope - easier to farm than conventional - used for maximum soil and water conservation 3. Broad-based a. Graded or channel type - primary purpose is to remove excess water in such a way as to minimize erosion - reduce slope length, conducts intercepted runoff water to outlet at non- erosive velocity - outlet either surface or subsurface type b. Level or ridge type - no grade in channel - primary purpose is moisture conservation, erosion control is secondary - channel is normally closed at both ends to assure maximum detention - adopted to deep, permeable soils - more formable than bench types - used where outlets are a problem Terrace Design Vertical Interval, (VI)= aS + b for graded terraces (VI)= aS + 0.85b for level terraces Soil and Water Conservation Engineering PSAE Region IV - Agricultural Engineering Board Review Materials II-11 Horizontal Interval, (HI) = (VI x 100)/S = a(100) + b (100)/S Where: a -- constant for geographic location (value range from 0.3 – 0.8 depending on the intensity of rainfall) S – average slope above the terrace in percent b – constant for soil erodibility and cover conditions b = 1 erodible soil and poor cover b = 2 resistant soil and good cover Grassed Waterways Grassed waterways are open channels protected with suitable grasses constructed along the slope and act as outlet for terraces and graded bunds. They are also used to safely convey runoff from contour furrows, diversion channels and serve as emergency spillways in farm ponds. The velocity of flow through the grassed waterways is dependent upon the ability of the vegetation to resist erosion. For design purposes, an average of 1.5 m/s to 2 m/s is used. Conservation Structures Drop structures Structures constructed along a channel to dissipate safely the energy of water by letting the water fall freely for a certain height. Chute spillways Vengineering A term adapted to measures utilizing both or in combination with the biological and engineering measures. D. Open Channel Flow Types of Flow 1. Steady Flow – discharge is constant with respect to time 2. Unsteady Flow – discharge is not constant with respect to time 3. Uniform Flow - depth of flow is the same at every section of the prismatic channel 4. Varied Flow - depth changes along the length of channel 5. Rapidly Varied Flow - if the depth changes abruptly over a comparatively short distance such as in a hydraulic jump 6. Gradually Varied Flow - if the depth changes smoothly over a distance 7. Critical Flow -- Fr = 1 8. Supercritical Flow -- Fr > 1 9. Subcritical Flow -- Fr < 1 10. Laminar Flow - fluid moves in parallel layers with no cross-currents; Re 4000 12. Spatially Variable Flow – special case of flow in which the discharge varies with distance along the channel Flow Measuring Devices/Structures 1. Weirs (Rectangular, V-notch, Sharp/Broad -Crested, Proportional, etc.) 2. Parshall Flume 3. Orifice 4. Trajectory Definition of Terms: Hydrology – is a science that treats of the waters of the Earth, their occurrence, circulation, and distribuation, their chemical and physical properties, and their reaction with their environment, including their relation to living things. River Stage – is the elevation above some arbitrary zero datum of the water surface at a station along a river or stream. Staff Gage – a scale set at the river/stream so that a portion of it is immersed in the water at all times to measure river stage. Soil Erosion – is the detachment and transport of soil particles from the land by water or wind action. Erosivity – potential ability of the rain to cause erosion. It is a function of the physical characteristics of the rainfall. Erodibility – vulnerability or susceptibility of the soil to erosion. It is a function of both the physical properties of the soil and land management practices. Splash / Raindrop Erosion – is soil detachment and transport resulting from the impact of water drops directly on soil particles or on thin water surfaces. Interrill Erosion – combination of sheet and splash erosion Rill Erosion – detachment and transport of soil by concentrated flow of water creating shallow rills or furrows. Gully Erosion – detachment and transport of soil particles by concentrated flow of water creating channels larger and deeper than rills. As differentiated with rill, channels or gullies created cannot be obliterated by normal tillage operation. Thus gully erosion is an advanced stage of rill erosion. Stream Channel Erosion – consists of soil removal from stream banks or soil movement in the channel. Drainage coefficient - the depth of water in inches to be removed in a 24 hours period from the drainage area. Dc = ipd = in/day Soil and Water Conservation Engineering PSAE Region IV - Agricultural Engineering Board Review Materials II-13 Infiltration – the passage if water into the soil surface, and the rate is generally expressed in in/hr or cm/hr. Stilling Basin - structured device designed to hold a pool of water to cushion the impact and retard the flow of falling water as from an overflow weir, chute or drop. Strip Cropping - consist of growing alternate strip of clean cultivated and close-growing crops in the same field oriented across the slope. Multiple Cropping or Mixed Cropping - planting different crops simultaneously in the same field at the same cropping season. Relay Cropping - (crop rotation) planting different crops, one after the other each season. Flume - are specially shaped and stabilized channel sections which may also be used to measure flow and are generally less inclined than weirs to prevent floating debris and sediments from detention Culvert - closed conduit usually circular, square or rectangular in cross section, used for conveying water across and under an elevated roadway, embankments, or dike. Run-off Coefficient - the ratio of depth of run-off to depth of precipitation producing the run- off over a drainage area. Spillway - a structure for passing out water not needed for storage or diversion. Water Way - any open or close ground or surface for the passage of water. Watershed – a topographically delineated area which drains into a reference point in the stream. Percolation - downward movement of water within the soil. Seepage - the lateral flow of liquid through porous media; the lost of water from irrigation canals. Soil Texture - the relative proportion of the various size groups of individual soil grain. Soil - a natural body composed of mineral and organic material on the surface of the earth on which plants grow. Drop Structures - a structure which conveys water from a higher to a lower level, maybe inclined or vertical. Contour Line - an imaginary line of constant elevation on the surface of the ground. Soil Conservation - the application of engineering principles to the solution of soil management problems, any method used to fully utilize and conserve soil. Graded Terrace - constructed by cutting a shallow channel on the uphill-side and using only this soil to build the embankment. Vertical Interval – vertical interval between corresponding points on successive terraces or from top of slope to the bottom of first terrace. Soil and Water Conservation Engineering PSAE Region IV - Agricultural Engineering Board Review Materials II-14 Horizontal Interval - horizontal interval between corresponding points on successive terraces IV. REFERENCES Cabangon, R. J. (2000). Soil and Water Conservation Engineering: AE Board Review Material. College, Laguna, Philippines. Israelsen, O.W and V.E Hansen (1962). Irrigation Principles and Practices.3rd ed. JohnWiley and Sons, Inc. New York, U.S.A. Hauser, B. A. (1996). Practical Hydraulics Handbook. 2nd Ed. CRC Press. U.S.A. Linsley, R. K. et al. Hydrology for Engineers. Mc Graw-Hill, Inc. U.S.A. Linsley, R.K. et al (1992). Water Resources Engineering.4th ed. Mc Graw-Hill, Inc. U.S.A. Schwab, G. O., et al.(1992) Soil and Water Conservation Engineering. 4th ed. JohnWiley and Sons, Inc. New York, U.S.A. David, W. P. Undated. Terraces. Unpublished Technical Notes. David, W. P. (1990) Infiltration. Unpublished Technical Notes. David, W. P. (1990) Precipitation. Unpublished Technical Notes. David, W. P. (1990) Runoff. Unpublished Technical Notes. Aguilar, J.P. (1999) Engineering Measures for Controlling Soil Erosion. Paper Presented. Saplaco, S. R. (1999) Watershed Rehabilitation. Paper Presented. Soil and Water Conservation Engineering  Agricultural Engineering Board Review Materials Mathematics Prepared by Alleli C. Domingo Associate Professor, Mathematics Division Institute of Mathematical Sciences and Physics College of Arts and Sciences University of the Philippines at Los Baños College, Laguna September 2005 (Reproduction with Permission Only) PSAE Region IV – Agricultural Engineering Board Review Materials III- 1 Mathematics Alleli C. Domingo Associate Professor, Mathematics Division Institute of Mathematical Sciences and Physics University of the Philippines at Los Baños College, Laguna 4031 I. Introduction People in engineering must be familiar with an increasingly wide variety of mathematical tools The real world is full of problems where it is important to find the maximum or minimum value of some quantity. (For example, engineers want to cut the strongest beam from a log of wood.) Derivatives provide an efficient way of analyzing the quantitative behavior of functions and solving many optimization problems. Topics to be reviewed are: Formulas from algebra Algebraic problems Formulas from geometry Trigonometric problems Exponential growth or decay Basic differentiation formulas Related rates problems Maxima/minima problems Basic integration formulas Integral calculus II. Topics on Mathematics FORMULAS from ALGEBRA SPECIAL PRODUCTS 1. (x + a)(x + b) = x2 + (a + b)x + ab 2. (x + y)2 = x2 + 2xy + y2 3. (x - y)2 = x2 - 2xy + y2 4. (x - y)(x + y) = x2 –y2 5. (ax + by)(cx + dy) = acx2 + (ad +bc)xy + bdy2 6. (x + y)3 = x3 + 3x2y + 3xy2 + y3 7. (x - y)3 = x3 - 3x2y + 3xy2 - y3 FACTORING POLYNOMIALS 1. x2 + (a + b)x + ab = (x + a)(x +b) 2. x2 + 2xy +y2 = (x + y)2 3. x2 - 2xy +y2 = (x - y)2 4. x2 – y2 = (x - y)(x + y) 5. acx2 + (ad +bc)xy + bdy2 = (ax + by)(cx + dy) 6. ax + ay +az = a(x + y + z) 7. x3 + y3 = (x + y)(x2 – xy + y2) 8. x3 - y3 = (x - y)(x2 + xy + y2) Mathematics PSAE Region IV – Agricultural Engineering Board Review Materials III- 2 RULES ON EXPONENTS 1. Multiplication: an am = an+m an 2. Division: m = a n−m a n a an 3. Power of a Quotient:   = n b b 4. Power of a Product: (ab)n = anbn 5. Power of a Power: (an)m = anm 1 6. Negative Exponent: a −n = an 7. Zero as an Exponent: a0 = 1 , a  0. 1 8. Rational Exponent: a n =n a ; a m n = (n a ) m = (a ) n m RADICALS 1. n a  n b = n ab n a a 2. n =n ;b0 b b QUADRATIC FORMULA If a  0, the solutions of the equation ax2 + bx +c = 0 are given by − b  b 2 − 4ac x= 2a INEQUALITIES If a < b, and b < c, then a < c. If a < b, then a + c < b + c. If a < b, then a –c < b – c. If a < b, and c > 0, then ac < bc. If a < b, and c < 0, then ac > bc. If b > 0, x < b is equivalent to –b < x < b. If b > 0, x > b is equivalent to x > b or x < -b LOGARITHMS 1. y = log bx if and only if x = by. 2. log b1 = 0 3. log b b = 1 4. log b uv = log bu + lov bv u 5. log b = log bu – log bv v Mathematics PSAE Region IV – Agricultural Engineering Board Review Materials III- 3 6. log b un = nlog b u 7. ln x = log e x BINOMIAL THEOREM (a + b) n = n C 0 a n b 0 + n C1 a n −1b1 + n C 2 a n − 2 b 2 +....+ n C r a n −r b r +.....+ n C n a 0 b n n! where n Cr = (n r)! r! ARITHMETIC PROGRESSION nth term: t n = t1 + (n − 1)d n(t1 + t 2 ) Sum of the first n terms: Sn = 2 FORMULAS from ANALYTIC GEOMETRY DISTANCE FORMULA: The distance between two points P1(x1,y1) and P2(x2,y2) is P1 P2 = ( x2 − x1 ) + ( y 2 − y1 ) 2 MIDPOINT FORMULAS: If M(x, y)is the midpoint of the line segment from P1(x1,y1) and P2(x- 2,y2), then x1 + x 2 y1 + y 2 x= and y= 2 2 EQUATION OF A CIRCLE: The circle with center at (h,k) and radius r has an equation (x-h)2 + (y-k)2 = r2 ( x − h) 2 + ( y − k ) 2 = r 2 SLOPE OF A LINE: If P1(x1,y1) and P2(x2,y2) are any two distinct points on a non-vertical line, then the slope of the line is m, given by y y 2 − y1 m= = x x2 − x1 EQUATIONS OF A LINE: The point-slope form of a line having a slope m and through a point P(x1,y1) is y - y1 = m(x - x1) The slope-intercept form of a line having slope m and y-intercept b is y = mx + b The general 1st degree equation is Ax + By + C = 0 Mathematics PSAE Region IV – Agricultural Engineering Board Review Materials III- 4 FORMULAS from TRIGONOMETRY Let a , b, and c represent the sides of a triangle , and let , , and  be the measures of the angles opposite a, b and c respectively. LAW OF SINES: a b c = = sin  sin  sin  LAW OF COSINES c 2 = a 2 + b 2 − 2ab cos  FORMULAS from PLANE and SOLID GEOMETRY The following symbols are used for the measure: r – radius; l-length; w-width;; b-base; h-altitude A – area; B- Area of base; C - circumference ;V-volume CIRCLE: A =  r2 ; C= 2 r 1 TRIANGLE: A = bh 2 RECTANGLE : A = lw PARALLELOGRAM: A = bh 1 TRAPEZOID: A = (a + b)h 2 RIGHT CIRCULAR CYLINDER: V =  r2h ; S = 2rh 1 RIGHT CIRCULAR CONE: V = πr 2 h 3 4 3 SPHERE: V = πr ; S = 4  r2 3 PRISM (with parallel base): V= Bh 1 PYRAMID: V = Bh 3 BASIC DIFFERENTIATION FORMULAS 1. Let f be a differentiable function. If h(x) = C f(x) , where C is a constant, then h ‘(x) = C f ‘(x). 2. Let f and g be two differentiable functions. If h(x) = f(x) + g(x), then h’(x) = f’(x) + g’(x), 3. Let f and g be two differentiable functions and suppose h(x) = f(x)g(x), then h’(x) = f’(x)g(x)+f(x)g’(x), Mathematics PSAE Region IV – Agricultural Engineering Board Review Materials III- 5 4. Let f and g be two differentiable functions. such that g(x)  0. Then the function defined by f (x h( x ) = is differentiable and we have g ( x) f ' ( x) g ( x) − f ( x) g ' ( x) h' ( x ) = g ( x)2 5. The Chain Rule If f and g are differentiable, then so is the function h = f g, In addition, h’(x) = g’[f(x)] f’(x), BASIC INTEGRATION FORMULAS 1.  du = u + C 2  adu = a du 3.  (du + dv) =  du +  dv u n +1 4.  u n du = n +1 + C , n  −1  u = ln u + C du 5. au 6. a)  eu du = eu + C  b)au du = ln a +C VERBAL PROBLEMS: 1. What three consecutive even numbers have their sum equal to 384? 2. Find three consecutive odd numbers whose sum is 1,323. 3. The sides of a right triangle form an arithmetic progression with a common difference of 6. Find the sides of the triangle 4. A student takes a part-time job paying a starting salary of 5,000 a month and is promised a fixed raise each month. How much is his monthly raise if he receives P7,000 on the ninth month? 5. A well driller charges P100 for the first 50 feet and P10 less for every 50 feet there after. How much would a 350-ft deep well cost? 6. Daniel is constructing a ten - level tower. He puts ten blocks on the bottom level. If each level after that there is one block less than the level below it. The completed tower has how many blocks? 7. One computer can do a job twice as fast as the other. Working together, both computers can do the job in 2 hours. How long would it take each computer, working alone, to do the job? Mathematics PSAE Region IV – Agricultural Engineering Board Review Materials III- 6 8. A pipe can fill a swimming pool in 10 hours. If a second pipe is open, the two pipes together can fill the pool in 4 hours. How long will it take the second pipe alone to fill the pool? 9. How much money do we have to invest at 3 percent compounded annually if we want to have 750 thousand pesos in the bank after ten years? 10. Immaya is 11 years old today. If her favorite aunt is 31 years old, how many years from now will the aunt be twice as old as she? 11. How many liters of a 15% solution of alcohol should be added to 3 liters of the 30% solution to get a 20 percent solution? 12. Ms. Sison invests part of P4,000 at 3% and the and the balance at 4% per annum. How much did she invest at each of these rates if she earns P135 in one year? 13. Andre and Chris are traveling to a business conference. Andre travels 110 km in the same time that Chris travels 140 km. Chris travels 15 km per hour faster than Andre. Find the average rate of each person. 14. A man wants to make an open box from apiece of metal that is 12 inches wide and 14 inches long by cutting equal squares out of the corners and folding up the sides. How large a square must be cut out of each corner if the area of the bottom of the box is 80 square inches? 10  1  15. Find the 4 term of  2 x − th y.  4  PROBLEMS IN ANALYTIC GEOMETRY: 1. Find two values of x if the distance between (x,2) and (6,6) is 5. 2. Determine if the three points (0,-3), (1,4) and (2, 1) are collinear. 3. Prove that the three points (2,4) (1,-4) and (5,-2) are the vertices of a right triangle. 4. Find the slope and y intercept of the line having the equation 2x –5y –10 = 0 5. Find an equation of the line passing through the point (1, -3) and having a slope of ½. 6. Find an equation of the circle with center at (3, -5) and radius 2. 7. Find the equation of the circle having a diameter with endpoints at (3, -4) and (1,2). 8. Find the center and radius of the circle x2 + y2 –6x –8y +9 =0. 9. Draw a sketch of the graph of the equation y = x2 +3. 10. Locate the vertex and focus of the parabola 9y2 + 2x – 24y – 96 = 0. 11. Find the equation of the line tangent to the curve y2 –2x-4y –1=0 at (2 , -1). Mathematics PSAE Region IV – Agricultural Engineering Board Review Materials III- 7 12. Find the equation of the line normal to the curve xy+ 2x–5y–2=0 at (3 , 2). 13. Find the equations of the lines tangent to the curve y = x3 –6x + 2 and parallel to the line y =6x – 2. 14. Find the equation of the line normal to the curve xy + 2x –y = 0 and parallel to the line 2x + y = 0. 15. Does the line tangent to the curve y = x3 at the point (1,1) intersect the curve t any other point? If so, find the point. 16. Sketch the ellipse 9x2 + y2 = 9. 17. Sketch the hyperbola 9x2 - 4y2 =36. EXPONENTIAL GROWTH /DECAY A function defined by an equation of the form f(t) = Bekt where B and k are positive constants, is said to describe exponential GROWTH. Such a function results when the rate of growth of a quantity is proportional to its size. A function defined by an equation of the form f(t) = Be-kt where B and k are positive constants, is said to describe exponential DECAY. Exponential decay occurs when the rate of decrease of a quantity is proportional to its size. EXPONENTIAL GROWTH /DECAY PROBLEMS: 1. In a particular bacterial culture, if f(t) bacteria are present at t minutes, then f(t) = Be 0.04t where B is a constant, If there are 1,500 bacteria present initially, how many bacteria will be present after 1 hour? 2. If f(t) grams of radioactive substance are present after t seconds, then f(t) =ke -0.3twhere k is a constant. If 100 grams of the substance are present initially, how much is present after 5 seconds? 3. If V(t) pesos is the value of a certain equipment t years after its purchase, then V(t) = Be— 0.20t where B is a constant, If the equipment was purchased for 8,000pesos, what will be its value in 2 years? 4. If P(h) pounds per square foot s the atmospheric pressure at a height h feet above sea level, then P(h) = ke-0.00003h, where k is a constant. Given that the atmospheric pressure at sea level is 2116ft/lb2, find the atmospheric pressure outside of an airplane that is 10,000 ft high. 5. The population of a particular town is increasing at a rate proportional to its size. If the rate is 6 percent, and the population after t years is P(t), then P(t) = ke0.06t, where k is a constant. If the current population is 10,000, what is the expected population a) after 10 years? b) after 20 years? 6. Carbon 14, also known as radiocarbon, is a radioactive form of carbon that is found in all living plants and animals. After a plant or animal dies the radiocarbon disintegrates. Scientists can determine the age of the remains by comparing the amount of radiocarbon with the amount present in living plants and animals. This technique is known as carbon dating. The amount of radiocarbon present after t years is given by y = y0e-(ln2) (1/5700)t , Mathematics PSAE Region IV – Agricultural Engineering Board Review Materials III- 8 where y0 is the amount present in living plants and animals. Find the half-life of radiocarbon. TRIGONOMETRIC PROBLEMS: 1. If the angle of elevation of the sun is 42O, what is the length of the shadow on the level ground of a man who is 6.1 ft tall? 2. A tower is 150 ft high, and from its top, the angle of depression of an object on the ground is 3