Mixing problem

Group 3

박재무 윤동일 장석민

Con-tents

- Briefing conditions

- Problem 1-1) Determine the func-tion

- Problem 1-2) Approximation of the concentration- Problem 1-3) Compare with example 1 and problem

1

Briefing conditions

Briefing conditions

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 concentra-tion of salt in the tank is uniform

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

t

tx

1000

)(

t

tx

1000

)(5

Problem 1-1)

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

Concentration = substance/volume

?

?tan

volume

cesubs

Briefing conditions

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

5 L/min

tv

v

ctvdt

dv

1000

0)0(

156

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

t

tx

1000

)(

t

tx

1000

)(5

ttx

dtdx

1000)(5

6

x(t) = substance In tank at time t

Concentration = substance/volume

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

Using integrating factor

tx

dtdx

10005

6

5)1000ln()1000ln(51000

5

)1000(5

teee ttt

First linear dif-ferential equa-tion form

61000

5

xtdt

dx

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

cttx

cttxt

ttdtd

txt

tdtdx

t

xtdt

dx

5

655

55

555

)1000()1000(

)1000()1000(6)1000(

)1000(6)1000(

)1000(61000

5)1000()1000(

61000

5

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

5

6

65

5

)1000(

1000)1000(

)1000()1000(10000

0)0()1000()1000(

ttx

cc

xcttx

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

Concentration = substance/volume

Concentration =

tvolume

ttcesubs

1000

)1000(1000

)1000(tan5

6

6

65

6

t)(1000

10001

1000)1000(

1000)1000(

tt

t

Q&A

Problem 1-2)

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 Method300min

0.792822940271059

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 4300min

0.792823779888325

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 45300min

0.792823785611866

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

Method 300min error

Improved EulerSize 1

0.792822940271059

8.486959379716552e-007

Runge Kutta order 4Size 10

0.792823779888325

9.078672036366697e-009

ODE 45 0.792823785611866

3.355131061866246e-009

Analytic value 0.792823788966997

-300

t)(1000

10001 valueAnalytic

6

6

twith

Q&A

Problem 1-3)

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

5L/min

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

exampl 1 problem 1Volume – 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

t

tx

1000

)(

t

tx

1000

)(5

5

6

)1000(

1000)1000(

ttx

)1(1000 500/3tex

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 →∞

6

6

t)(1000

10001

)1( 500/3te

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 1 Problem 1

200 0.698159640000000 0.664938702088147

400 0.908892397075070 0.867098698559750

800 0.991699404689273 0.970574566707560

1600 0.999931100117488 0.996760014805278

3200 0.999999995252806 0.999817655195757

64000.999993904635251

Q&A

Thank You!