Problem 5.50 - U A a 2 1 4 e x 2 a e x a or e x 2 a 4 x max...

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Problem 5.50 [Difficulty: 4] Given: Data on pollution concentration Find: Plot of concentration; Plot of concentration over time for moving vehicle; Location and value of maximum rate change Solution: t z w y v x u Dt D Basic equation: (Material Derivative) Assumption: Concentration of pollution is a function of x only Sensor travels in x-direction only For this case we have uU v0 w0 cx () A e x 2a e x a Hence Dc Dt u dc dx U x Ae x 2a e x a d d UA a e x a 1 2 e x 2a We need to convert this to a function of time. For this motion u = U so xU t Dc Dt UA a e Ut a 1 2 e Ut 2a The following plots can be done in Excel 0 10 20 30 21 0 6 41 0 6 61 0 6 81 0 6 11 0 5 x (ft)
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0 0.1 0.2 0.3 0.4 0.5 1 10 4 11 0 4 21 0 4 31 0 4 41 0 4 t (s) Dc/Dt (ppm/s) The magnitude of the rate of change is maximized when d dx Dc Dt d dx UA a e x a 1 2 e x 2a
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Unformatted text preview: U A a 2 1 4 e x 2 a e x a or e x 2 a 4 x max 2 a ln 4 ( ) 2 3 ft ln 4 ( ) x max 8.32 ft t max x max U 8.32 ft s 70 ft t max 0.119 s Dc max Dt U A a e x max a 1 2 e x max 2 a Dc max Dt 70 ft s 3 10 5 ppm 1 3 ft e 8.32 3 1 2 e 8.32 2 3 Dc max Dt 4.38 10 5 ppm s Note that there is another maximum rate, at t = 0 (x = 0) Dc max Dt 70 ft s 3 10 5 ppm 1 3 ft 1 1 2 Dc max Dt 3.50 10 4 ppm s...
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Problem 5.50 - U A a 2 1 4 e x 2 a e x a or e x 2 a 4 x max...

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