AnalysisofaDirtyMotorPoolRichardJardine,MichaelKelley,JosephMyersAtopic of enduring concern to militaryoperators and logisticians is theenvironmental signature left bybothfield and garrison operations.Thispaperdetailsoneofthescenarioswhichweuse inourengineeringmathcoursetoexciteandmotivatefutureenvironmentalengineersaboutusesofmathematicsintheirdiscipline.Thisscenarioillustratestheuseofvectorcalculus,partial differential equations(PDE's),andnumerical methods,alongwithacomputeralgebra system (MathCad)and spreadsheet (Excel),inmodeling the advection and diffusion of an oil spill.Workingthroughthis scenarioexercisesthefollowingmathematical skills:Parameterization of space curves,using vector differential operators,and using the vector integraltheorems- Modeling using PDE's-Solvingthediffusion equationviaseparationofvariables-Making engineering value judgments (such as how much seepage is"appreciable",vs.nonzero)When and how to use the dominant eigenmode as a long-termapproximation-Refining amodel to incorporate neweffects-Numerically (via finite differences and a spreadsheet) solving morecomplexvariantsofthediffusionequation- Graphing solution curves and drawing inferencesScenarioYouare a Battalion ExecutiveOfficerstationed in Korea.Amongyourmanyduties,you are in chargeof vehiclemaintenanceand motor pool operations forthebattalion.TheAssistantDivisionCommanderforSupport (ADC(S))isflyingoveryourbattalionmotorpool areaonemorning,andhappenstonoticesomegrounddiscolorationnear theBrigadePOL (Petroleum, Oils,and Lubricants)TankFarm,forwhichyouhaveprimaryresponsibility.Helater callsyourcommanderandyouandtellsyoutoinvestigate.Yourinitial checkshowsthatone of the fuel storage tanks has started to leak at a seam. It is a large-areaground oil spillwith continued leakage(maintaining the surface at a constantlevelof contaminantconcentration).Thecontaminationisbeingadvectedbysurfacerunoff andis alsodiffusingdownwardtowardbedrocklevel.TheFacilityEngineersestimatethat itwilltake72hourstorepairthepipeandstoptheleak.ThelocalcivilianEnvironmentalOfficerimmediatelycallsanddemandstoknow41
41 Analysis of a Dirty Motor Pool Richard Jardine, Michael Kelley, Joseph Myers A topic of enduring concern to military operators and logisticians is the environmental signature left by both field and garrison operations. This paper details one of the scenarios which we use in our engineering math course to excite and motivate future environmental engineers about uses of mathematics in their discipline. This scenario illustrates the use of vector calculus, partial differential equations (PDE’s), and numerical methods, along with a computer algebra system (MathCad) and spreadsheet (Excel), in modeling the advection and diffusion of an oil spill. Working through this scenario exercises the following mathematical skills: - Parameterization of space curves, using vector differential operators, and using the vector integral theorems - Modeling using PDE’s - Solving the diffusion equation via separation of variables - Making engineering value judgments (such as how much seepage is “appreciable”, vs. nonzero) - When and how to use the dominant eigenmode as a long-term approximation - Refining a model to incorporate new effects - Numerically (via finite differences and a spreadsheet) solving more complex variants of the diffusion equation - Graphing solution curves and drawing inferences Scenario You are a Battalion Executive Officer stationed in Korea. Among your many duties, you are in charge of vehicle maintenance and motor pool operations for the battalion. The Assistant Division Commander for Support (ADC(S)) is flying over your battalion motor pool area one morning, and happens to notice some ground discoloration near the Brigade POL (Petroleum, Oils, and Lubricants) Tank Farm, for which you have primary responsibility. He later calls your commander and you and tells you to investigate. Your initial check shows that one of the fuel storage tanks has started to leak at a seam. It is a large-area ground oil spill with continued leakage (maintaining the surface at a constant level of contaminant concentration). The contamination is being advected by surface runoff and is also diffusing downward toward bedrock level. The Facility Engineers estimate that it will take 72 hours to repair the pipe and stop the leak. The local civilian Environmental Officer immediately calls and demands to know
whererunoffwill carrythecontamination,towhatdepthcontaminatedsoilwillhavetobeexcavated after theleak hasbeen stopped,and whatthe effect wouldbeonthespreadof contaminant iftheweatherisrainyoverthenextweek.TrackingAvectiveProcessesand RunoffYou decide to analyze the quickest acting process first: runoff.A quick survey ofthe terrain shows that the major runoff for the contaminant appears to be thestreamflowing from nearthespill (pointAon themapbelow)toward pointBneartheneighboringtown.Thevertical grid lines are1000mapart,thehorizontalgrid lines are75omapart,andcontour linesareatelevationsasmarked onthemapvelFigure1:Pathof streamfromatPointAtoB.Example 1:Mathematicallymodel thepath in orderto determine theflow of thestream.Solution:Webeginby using Mathcad toplot amodel the stream.We writeaparametricvectorfunctionforthepositionofamarkerparticle inthe streamas itmoves from A toB.A possibleparameterization of the streambed istheparametric equation r(t)=(e2t sin 7t+ sin15t)i+800t j+130t k, 0≤t≤1 (where t42
42 where runoff will carry the contamination, to what depth contaminated soil will have to be excavated after the leak has been stopped, and what the effect would be on the spread of contaminant if the weather is rainy over the next week. Tracking Avective Processes and Runoff You decide to analyze the quickest acting process first: runoff. A quick survey of the terrain shows that the major runoff for the contaminant appears to be the stream flowing from near the spill (point A on the map below) toward point B, near the neighboring town. The vertical grid lines are 1000m apart, the horizontal grid lines are 750 m apart, and contour lines are at elevations as marked on the map. Figure 1: Path of stream from at Point A to B. Example 1: Mathematically model the path in order to determine the flow of the stream. Solution: We begin by using Mathcad to plot a model the stream. We write a parametric vector function for the position of a marker particle in the stream as it moves from A to B. A possible parameterization of the streambed is the parametric equation ( ) ( sin 7 sin15 ) 800 130 , 0 1 2 t = e t + t + t + t £t £ t r i j k (where t
is measured in seconds and x,y,= are measured in meters).As graphicallyillustrated in Figure 2, this yields a good approximation to the path of the stream.Stream model0180-20060020X,Y,7Streammodel10050C200400600800X,Y,ZFigure2:ParameterizedrepresentationoftheAdvectingStreamExample2:Giventheabovepositionfunctionof the stream,withAatt=1andBat t=o,youmeasureand approximatethat thewater's velocityfield is infactv=0.li-0.2 j-0.1k.Find theflowof water(as defined in Finney/Thomas,Revised Printing,p.952)moving in theone-dimensional model of thestreambetweenAand B.Describewhat this meansphysically,includingthe signof theresult.43
43 is measured in seconds and x, y,z are measured in meters). As graphically illustrated in Figure 2, this yields a good approximation to the path of the stream. Figure 2: Parameterized representation of the Advecting Stream Example 2: Given the above position function of the stream, with A at t = 1 and B at t = 0 , you measure and approximate that the water's velocity field is in fact v = 0.1i - 0.2 j - 0.1k . Find the flow of water (as defined in Finney/Thomas, Revised Printing, p.952) moving in the one-dimensional model of the stream between A and B. Describe what this means physically, including the sign of the result. X,Y,Z Stream model 4 2 0 2 4 0 200 400 600 800 1005 X,Y,Z Stream model 4 2 0 2 4 0 200 400 600 800 50 100
Solution:Theflowof theadvectingstreamis calculated via the line integralJvdr for the given velocity field, yielding:CpJvdr = pfbv(t)-r(t)dt =-215.6m/ s.This isacombinedmeasurethatindicatesbothhowfarandhowfastthefluid ismoving.It isnegative,whichmeansthatthenetflowvelocityis inthedirectionfromAtoward B.The magnitude, obtained by dividing the total flowbythelength of the stream, is of about the correct order, as the Colorado River has aflowoffrom300to1000m/s.Wenowseek todeterminethe effect ofvorticity on contaminantmonitoringequipmentplaced inboreholesthat wedigatseveralspecified locationsalongthe length of the stream. Several boreholes have been dug near the spill siteand instruments placed in the wells to obtain data on the effect ofthe spill. (Seefiqurebelow).Eachholeiscylindricallyshaped,2metersdeepand20cmindiameter.In one oftheholes,the velocityfield ismeasured tobev=xi+yj-3zkm/s (withorigincenteredatthebottomof thehole)LandfillTestwellsKVAquiferFigure 3: Bore holes for contaminant measuring equipment.Example3:Atwhatrateiswaterenteringtheholefromabove?Solution:The flux of water into the hole is given byJJ v·ndA = J< x, J,-3z >-kdA = JJ- 3zdA = -3.2[JdA = -6A= -0.1885m3 /s4The sign isnegative sincenetflux is opposite in directiontothe outward normalvector; i.e., netflux is downward throughthe upper surface.Example4:Are there any sources or sinks in the interior of the hole? Describe.44
44 Solution: The flow of the advecting stream is calculated via the line integral ò × C v dr for the given velocity field, yielding: ( ) ( ) 215.6 / . 1 0 d t t dt m s C r òv ×r = rò v ×r¢ = - This is a combined measure that indicates both how far and how fast the fluid is moving. It is negative, which means that the net flow velocity is in the direction from A toward B. The magnitude, obtained by dividing the total flow by the length of the stream, is of about the correct order, as the Colorado River has a flow of from 300 to 1000 m3 /s. We now seek to determine the effect of vorticity on contaminant monitoring equipment placed in bore holes that we dig at several specified locations along the length of the stream. Several boreholes have been dug near the spill site and instruments placed in the wells to obtain data on the effect of the spill. (See figure below). Each hole is cylindrically shaped, 2 meters deep and 20 cm in diameter. In one of the holes, the velocity field is measured to be v = xi + yj - 3zk m/s (with origin centered at the bottom of the hole). Landfill Test wells Aquifer Figure 3: Bore holes for contaminant measuring equipment. Example 3: At what rate is water entering the hole from above? Solution: The flux of water into the hole is given by òò × = òò< - > × = òò- = - × òò = - = - A A A A dA x y z dA zdA dA A m s 3 v n , , 3 k 3 3 2 6 0.1885 . The sign is negative since net flux is opposite in direction to the outward normal vector; i.e., net flux is downward through the upper surface. Example 4: Are there any sources or sinks in the interior of the hole? Describe
Solution:Fromthevelocityfield of thefluid intheborehole,we calculatethatthe divergenceatall points is V·v=1+1-3=-1 per sec.Since thedivergenceis negative at all points, this indicates that there is a sink at every point in thehole.Thismightoccurif thedensityofthefluiddecreasedafterit entered thehole,perhapsduetowarmingbythe equipment inthehole.Example5:Atwhat rate is water entering thehole from all sides (including theporoussidesand bottom,aswell asthetop)?Solution:Thetotalflux offluid intotheholecanbecalculated bytransformingtheclosedsurface integral v·ndo intothevolumeintegral JjVvdv viaTStoke's Theorem, yielding fff-ldV =-1.V ; this gives a net flux into the hole of-.06275m2 / s.In another of thetest wells,ground waterrotates aroundthe center of the hole(within the plane z = 1) with velocity v= o(yi - xi), where , the angularvelocity,is 0.25rad/s.If the circulation in the well exceeds 15 m2/s,the datacollected by instruments inthewell will becorrupted.Example6:Will theempirical results obtained fromthatholebevalid?Solution:Theflow circulation around the midpointplaneof the cylindrical holecanbefound bytransformingtheline integral v·drintothedoubleintegralJJV×vdV via Gauss'Theorem. This yields:AJ<0,0,-20>·kdA=-20A=-0.0157m2/sThe circulation ismuch lessthanthe critical valueoftheequipment;therefore,collected datawillbereliable.Calculating DownwardDiffusionYou now turn your attention to the contamination of the ground underneath thespill duetodiffusion. Youbeginwiththefollowing assumptions:The spill is uniform over an area large enough such thatthere isdiffusion only in the vertical direction.-Thesoil ishomogeneouswithadiffusionrateof0.3ft/hr.-Thereisnoadvection.- The leak is such that it maintains a constant concentration of 5000 g/ft3overthesurface45
45 Solution: From the velocity field of the fluid in the borehole, we calculate that the divergence at all points is Ñ ×v = 1+ 1- 3 = - 1 per sec. Since the divergence is negative at all points, this indicates that there is a sink at every point in the hole. This might occur if the density of the fluid decreased after it entered the hole, perhaps due to warming by the equipment in the hole. Example 5: At what rate is water entering the hole from all sides (including the porous sides and bottom, as well as the top)? Solution: The total flux of fluid into the hole can be calculated by transforming the closed surface integral òò × s v nds into the volume integral òòòÑ × V vdV via Stoke’s Theorem, yielding dV V V òòò- 1 = - 1× ; this gives a net flux into the hole of - .06275 / 3 m s. In another of the test wells, ground water rotates around the center of the hole (within the plane z = 1) with velocity v = w ( yi - xj) , where w , the angular velocity, is 0.25 rad/s. If the circulation in the well exceeds 15 m2 /s, the data collected by instruments in the well will be corrupted. Example 6: Will the empirical results obtained from that hole be valid? Solution: The flow circulation around the midpoint plane of the cylindrical hole can be found by transforming the line integral ò × C v dr into the double integral òòÑ ´ A vdV via Gauss’ Theorem. This yields: òò< - > × = - = - A 0,0, 2w kdA 2wA 0.0157m 2 /s. The circulation is much less than the critical value of the equipment; therefore, collected data will be reliable. Calculating Downward Diffusion You now turn your attention to the contamination of the ground underneath the spill due to diffusion. You begin with the following assumptions: - The spill is uniform over an area large enough such that there is diffusion only in the vertical direction. - The soil is homogeneous with a diffusion rate of 0.3 ft2 /hr. - There is no advection. - The leak is such that it maintains a constant concentration of 5000 g/ft3 over the surface