Download Modeling Problems in Hydrology-Modeling and Simulation-Lecture Slides and more Slides Mathematical Modeling and Simulation in PDF only on Docsity! 5/11/2011 1 & MODELING http://www.pieas.edu.pk/umarfaiz/cis308 SIMULATION http://www.pieas.edu.pk/umarfaiz/cis308 http://www.pieas.edu.pk/umarfaiz/cis308 Modeling Problems in Hydrology Applications of First‐order Differential Equations Mixing Problems Problem – Rawal Dam initially contains 500,000 gallons of unpolluted water. It has a spillway (an outlet) that releases 10,000 gallons of water per day. Many streams contribute to the dam level every day. One certain stream flows into the dam at 12,000 gallons per day containing water with a Mixing Problems Solution – Let x(t) be amount of pollutant in grams in the pond after t days. – We use a fundament property of rates: • Total Rate = Rate In ‐ Rate Out – To find the rate in we use 24000 1 12000 1 2 === x day gallonx gallon grams day grams Mixing Problems Solution (contd.) – There was initially 500,000 gallons of water in the lake and the water level is increasing at a rate of 2,000 gallons per day, the total number of gallons of water in the lake after t days is gallons = 500,000 + 2,000 t 24000 1 10000 000,2000,500 = + == x t x day gallonx gallon grams day grams t x 2500 10 + = grams per day Mixing Problems Solution (contd.) – Putting this all together, we get – This is a first order linear differential equation with t x dt dx 2500 1024000 + −= t xtp 2500 10)( + = 24000)( =tg )2500ln(5exp 2500 10exp tdt t += + = ∫μ 5)2500( t+= docsity.com 5/11/2011 2 Mixing Problems Solution (contd.) – Multiplying by the integrating factor and using the reverse product rule gives – Now integrate both sides to get 55 )2500(24000)')2500(( txt +=+= C65 )2500(2000))2500(( txt ++=+= 5)2500( )2500(2000 t Ctx + =+= Mixing Problems Solution (contd.) – Now we use initial conditions (t=0) 19 5 10x -3.125 )500( )500(2000 = == C Cx – For the conditions t=10 grams 218,07 )10(2500( 10 x 3.125-)10(2500(2000 5 19 = + =+= x x Mixing Problems Solution (contd.) – A graph is given below A Mixing Problem Example – A tank contains 50 gallons of a solution composed of 90% water and 10% alcohol. A second solution containing 50% water and 50% alcohol is added to the tank at the rate of 4 gallons per minute. As the second solution is being added, the tank is being drained at the rate of 5 gallons per http://www.pieas.edu.pk/umarfaiz/cis308 minute, as shown in figure. Assuming the solution in the tank is stirred constantly, how much alcohol is in the tank after 10 minutes? A Mixing Problem General Solution – Let y be the number of gallons of alcohol in the tank at any time t. We know that y=5 when t=0. Thee number of gallons of solution in the tank at any time is 50‐t and the tank loses 5 gallons of solution per minute – The amount of alcohol (in gallons) the tank must lose is http://www.pieas.edu.pk/umarfaiz/cis308 – The tank is gaining 2 gallons of alcohol per minute, the rate of change of alcohol in the tanks is given by y y ) 50 5( − y ydx dy ) 50 5(2 − −= 2) 50 5( = − + y ydx dy A Mixing Problem General Solution – Let – We obtain y y tP ) 50 5()( − = |50|ln5 50 5)( tdt t dttP −== − +∫ ∫ 5 455 )50( 2 50 )50(2 1 )50( 2 )50( tCty C t dt ty y −+ − = + − = −− ∫ docsity.com
