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(a) Formulate a system of differential equations and initial conditions for the oil thickness in the first three cells. Take S = 50 gallons/min, which was roughly the spillage rate for the Mississippi River incident, and take w = 200 ft, d = 25 ft, and v= 1 mi/h (which are reasonable estimates for the Mississippi River). Take L = 1000 ft. (b) Solve for si(t). [Caution: Make sure your units are consistent.] (c) If the spillage lasts for T seconds, what is the maximum oil layer thickness in cell 1? (d) Solve for s₂ (1). What is the maximum oil layer thickness in cell 2? (e) Probably the least tenable simplification in this analysis lies in regarding the layer thick- ness as uniform over distances of length L. Reevaluate your answer to part (c) with L reduced to 500 ft. By what fraction does the answer change?

(a) Formulate a system of differential equations and initial conditions for the oil thickness in the first three cells. Take S = 50 gallons/min, which was roughly the spillage rate for the Mississippi River incident, and take w = 200 ft, d = 25 ft, and v= 1 mi/h (which are reasonable estimates for the Mississippi River). Take L = 1000 ft. (b) Solve for si(t). [Caution: Make sure your units are consistent.] (c) If the spillage lasts for T seconds, what is the maximum oil layer thickness in cell 1? (d) Solve for s₂ (1). What is the maximum oil layer thickness in cell 2? (e) Probably the least tenable simplification in this analysis lies in regarding the layer thick- ness as uniform over distances of length L. Reevaluate your answer to part (c) with L reduced to 500 ft. By what fraction does the answer change?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I need these questions answered as concisely as possible. I am very confused about where to us L and I don't understand how to change my units for part b. 

On part c people said it is 2.5, but I am not getting that. 

Please also solve for part d and e as well, making sure to show all the steps, please. 

 

(a) Formulate a system of differential equations and initial conditions for the oil thickness in the first three cells. Take S = 50 gallons/min, which was roughly the spillage rate for the Mississippi River incident, and take w = 200 ft, d = 25 ft, and v= 1 mi/h (which are reasonable estimates for the Mississippi River). Take L = 1000 ft. (b) Solve for s₁ (1). [Caution: Make sure your units are consistent.] (c) If the spillage lasts for T seconds, what is the maximum oil layer thickness in cell 1? (d) Solve for s₂ (1). What is the maximum oil layer thickness in cell 2? (e) Probably the least tenable simplification in this analysis lies in regarding the layer thick- ness as uniform over distances of length L. Reevaluate your answer to part (c) with L reduced to 500 ft. By what fraction does the answer change?
Transcribed Image Text:(a) Formulate a system of differential equations and initial conditions for the oil thickness in the first three cells. Take S = 50 gallons/min, which was roughly the spillage rate for the Mississippi River incident, and take w = 200 ft, d = 25 ft, and v= 1 mi/h (which are reasonable estimates for the Mississippi River). Take L = 1000 ft. (b) Solve for s₁ (1). [Caution: Make sure your units are consistent.] (c) If the spillage lasts for T seconds, what is the maximum oil layer thickness in cell 1? (d) Solve for s₂ (1). What is the maximum oil layer thickness in cell 2? (e) Probably the least tenable simplification in this analysis lies in regarding the layer thick- ness as uniform over distances of length L. Reevaluate your answer to part (c) with L reduced to 500 ft. By what fraction does the answer change?
A Oil Spill in a Canal In 1973 an oil barge collided with a bridge in the Mississippi River, leaking oil into the water at a rate estimated at 50 gallons per minute. In 1989 the Exxon Valdez spilled an estimated 11,000,000 gallons of oil into Prudhoe Bay in 6 hours, and in 2010 the Deepwater Horizon well leaked into the Gulf of Mexico at a rate estimated to be 15,000 barrels per day* (1 barrel = 42 gallons). In this project you are going to use differential equations to analyze a simplified model of the dissipation of heavy crude oil spilled at a rate of S ft³/sec into a flowing body of water. The flow region is a canal, namely a straight channel of rectangular cross section, w feet wide by d feet deep, having a constant flow rate of ft/sec; the oil is presumed to float in a thin layer of thickness s (feet) on top of the water, without mixing. In Figure 2.12, the oil that passes through the cross-section window in a short time At occu- pies a box of dimensions s by w by At. To make the analysis easier, presume that the canal is conceptually partitioned into cells of length L ft. each, and that within each particular cell the oil instantaneously disperses and forms a uniform layer of thickness s; (t) in cell i (cell 1 starts at the point of the spill). So, at time t, the ith cell contains s; (t)wL ft³ of oil. Oil flows out of cell i at a rate equal to s; (t) wv ft2/sec, and it flows into cell i at the rate s;-1(1) wv; it flows into the first cell at S ft³/sec. Figure 2.12 Oil leak in a canal.
Transcribed Image Text:A Oil Spill in a Canal In 1973 an oil barge collided with a bridge in the Mississippi River, leaking oil into the water at a rate estimated at 50 gallons per minute. In 1989 the Exxon Valdez spilled an estimated 11,000,000 gallons of oil into Prudhoe Bay in 6 hours, and in 2010 the Deepwater Horizon well leaked into the Gulf of Mexico at a rate estimated to be 15,000 barrels per day* (1 barrel = 42 gallons). In this project you are going to use differential equations to analyze a simplified model of the dissipation of heavy crude oil spilled at a rate of S ft³/sec into a flowing body of water. The flow region is a canal, namely a straight channel of rectangular cross section, w feet wide by d feet deep, having a constant flow rate of ft/sec; the oil is presumed to float in a thin layer of thickness s (feet) on top of the water, without mixing. In Figure 2.12, the oil that passes through the cross-section window in a short time At occu- pies a box of dimensions s by w by At. To make the analysis easier, presume that the canal is conceptually partitioned into cells of length L ft. each, and that within each particular cell the oil instantaneously disperses and forms a uniform layer of thickness s; (t) in cell i (cell 1 starts at the point of the spill). So, at time t, the ith cell contains s; (t)wL ft³ of oil. Oil flows out of cell i at a rate equal to s; (t) wv ft2/sec, and it flows into cell i at the rate s;-1(1) wv; it flows into the first cell at S ft³/sec. Figure 2.12 Oil leak in a canal.
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