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Mixing problem Group 3 박재무 윤동일 장석민

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Contents - Briefing conditions - Problem 1-1) Determine the function - Problem 1-2) Approximation of the concentration - Problem 1-3) Compare with example 1 and problem 1

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Briefing conditions

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Volume – t = 0, 1000 L – Inputting = (6 L/min) – Outputting = (5 L/min) 5 L/min ※ The brine solution in the tank is kept well stirred, let’s assume the concentration of salt in the tank is uniform

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Briefing conditions Substance – t = 0, x = 0 kg – Inputting = (6 L/min)(1 kg/L) = 6 kg/min – Outputting = (5 L/min)( kg/L) = kg/min 5 L/min

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Problem 1-1)

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Problem 1-1) Determine the concentration of salt in the tank as function of time Concentration = substance/volume

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Briefing conditions Volume – t = 0, 1000 L – Inputting = (6 L/min) – Outputting = (5 L/min) 5 L/min

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Briefing conditions Substance – t = 0, x = 0 kg – Inputting = (6 L/min)(1 kg/L) = 6 kg/min – Outputting = (5 L/min)( kg/L) = kg/min 5 L/min x(t) = substance In tank at time t Concentration = substance/volume

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Problem 1-1) Determine the concentration of salt in the tank as function of time Using integrating factor First linear differential equation form

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Problem 1-1) Determine the concentration of salt in the tank as function of time

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Problem 1-1) Determine the concentration of salt in the tank as function of time

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Problem 1-1) Determine the concentration of salt in the tank as function of time Concentration = substance/volume Concentration =

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Q&AQ&A

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Problem 1-2 )

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Problem 1-2) Use the improved Euler’s method and fourth order Runge-Kutta method to approximate the concentration of salt In the tank after 300 minutes Improved Euler’s Method 300min 0.792822940271059

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Problem 1-2) Use the improved Euler’s method and fourth order Runge-Kutta method to approximate the concentration of salt In the tank after 300 minutes Runge Kutta order 4 300min 0.792823779888325

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Problem 1-2) Use the improved Euler’s method and fourth order Runge-Kutta method to approximate the concentration of salt In the tank after 300 minutes ODE 45 300min 0.792823785611866

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Problem 1-2) Use the improved Euler’s method and fourth order Runge-Kutta method to approximate the concentration of salt In the tank after 300 minutes Method300minerror Improved Euler Size 1 0.79282294027105 9 8.486959379716552e-007 Runge Kutta order 4 Size 10 0.79282377988832 5 9.078672036366697e-009 ODE 45 0.792823785611866 3.355131061866246e-009 Analytic value 0.79282378896699 7 -

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Q&AQ&A

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Problem 1-3)

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Problem 1-3) Compare to example 1, what can you tell about the concentration of salt in the tank when t →∞ 5L/min

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Problem 1-3) Compare to example 1, what can you tell about the concentration of salt in the tank when t →∞ exampl 1 problem 1 Volume – t = 0, 1000 L t = 0, 1000 L – Inputting = (6 L/min) Inputting = (6 L/min) – Outputting = (6 L/min) Outputting = (5 L/min) v(t) = 1000 v(t) = 1000 + t Substance – t = 0, 1000 L t = 0, 1000 L – Inputting = (6 L/min)(1 kg/L) = 6 kg/min – Outputting = (6 L/min) Outputting = (5 L/min)( kg/L) = kg/min

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Problem 1-3) Compare to example 1, what can you tell about the concentration of salt in the tank when t →∞ exampl 1 problem 1 concentration – Analytic, Both converge to 1, when t →∞

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Problem 1-3) Compare to example 1, what can you tell about the concentration of salt in the tank when t →∞ Numerically, Both also converge to 1, when t →∞ concentration of salt(Kg/L) Time(min)Example 1Problem 1 200 0.6981596400000000.664938702088147 400 0.9088923970750700.867098698559750 800 0.9916994046892730.970574566707560 1600 0.9999311001174880.996760014805278 3200 0.9999999952528060.999817655195757 6400 0.999993904635251

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Q&AQ&A

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9.3 Separable Equations.

9.3 Separable Equations.

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