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!