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Mixing problem

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  1. Mixing problem Group 3 박재무 윤동일 장석민

  2. 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

  3. Briefing conditions

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

  5. 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

  6. Problem 1-1)

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

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

  9. 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 x(t) = substance In tank at time t Concentration = substance/volume 5 L/min

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

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

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

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

  14. Q&A

  15. Problem 1-2)

  16. 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

  17. 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 RungeKutta order 4

  18. 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

  19. 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

  20. Q&A

  21. Problem 1-3)

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

  23. 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

  24. 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 →∞

  25. 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 →∞

  26. Q&A

  27. Thank You!