Mixing
How can i express concentration in this ODE ?
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So as you can see in this photo, i want to express ci (output concentration of mixture) as a function of time.
Qi, Cu, K, xv, V - known facts.
I know this is an easy work, but my math has become rusty.
Problem with expressing ci
differential-equations
asked Aug 6 at 19:17
user582165
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up vote
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down vote
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So as you can see in this photo, i want to express ci (output concentration of mixture) as a function of time.
Qi, Cu, K, xv, V - known facts.
I know this is an easy work, but my math has become rusty.
Problem with expressing ci
differential-equations
asked Aug 6 at 19:17
user582165
1
Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
– José Carlos Santos
Aug 6 at 19:21
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
So as you can see in this photo, i want to express ci (output concentration of mixture) as a function of time.
Qi, Cu, K, xv, V - known facts.
I know this is an easy work, but my math has become rusty.
Problem with expressing ci
differential-equations
asked Aug 6 at 19:17
user582165
1
So as you can see in this photo, i want to express ci (output concentration of mixture) as a function of time.
Qi, Cu, K, xv, V - known facts.
I know this is an easy work, but my math has become rusty.
Problem with expressing ci
differential-equations
asked Aug 6 at 19:17
user582165
1
asked Aug 6 at 19:17
user582165
1
asked Aug 6 at 19:17
user582165
1
asked Aug 6 at 19:17
user582165
1
1
Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
– José Carlos Santos
Aug 6 at 19:21
add a comment |Â
Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
– José Carlos Santos
Aug 6 at 19:21
Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
– José Carlos Santos
Aug 6 at 19:21
Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
– José Carlos Santos
Aug 6 at 19:21
add a comment |Â
1 Answer
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$$frac VQ_ifrac dc_idt+c_i=frac C_uKQ_iX_nu$$
Multiply both side by $frac Q_iV$
$$frac dc_idt+c_ifrac Q_iV=frac C_uKVX_nu$$
It's a first order differential equation
I use method of integration factor with $mu(t)= e^frac Q_itV$
Multiply both sides by $mu $
$$(c_i e^frac Q_itV)'=frac C_uKVX_nu e^frac Q_itV$$
Integrate both side
$$c_i e^frac Q_itV=int frac C_uKVX_nu e^frac Q_itV dt$$
Put the cionstants outside of the integral
$$c_i e^frac Q_itV=frac C_uKVX_nu int e^frac Q_itVdt$$
Evaluate the integral on the right side
$$c_i e^frac Q_itV=frac C_uKVX_nu e^frac Q_itVfrac VQ_i+C$$
Finally
$$c_i =left ( frac C_uKX_nuQ_i +Ce^frac -Q_itVright )$$
Where C is a constant of inetgration to be determinated at $t=0$ for example
Edit1
Note that you can write the differential equation simply this way
$$frac dc_idt+c_ifrac Q_iV=m $$
where
$$m=frac C_uKVX_nu$$
Since you only have a constant on the right side of the equation
answered Aug 6 at 20:02
Isham
10.8k3829
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
$$frac VQ_ifrac dc_idt+c_i=frac C_uKQ_iX_nu$$
Multiply both side by $frac Q_iV$
$$frac dc_idt+c_ifrac Q_iV=frac C_uKVX_nu$$
It's a first order differential equation
I use method of integration factor with $mu(t)= e^frac Q_itV$
Multiply both sides by $mu $
$$(c_i e^frac Q_itV)'=frac C_uKVX_nu e^frac Q_itV$$
Integrate both side
$$c_i e^frac Q_itV=int frac C_uKVX_nu e^frac Q_itV dt$$
Put the cionstants outside of the integral
$$c_i e^frac Q_itV=frac C_uKVX_nu int e^frac Q_itVdt$$
Evaluate the integral on the right side
$$c_i e^frac Q_itV=frac C_uKVX_nu e^frac Q_itVfrac VQ_i+C$$
Finally
$$c_i =left ( frac C_uKX_nuQ_i +Ce^frac -Q_itVright )$$
Where C is a constant of inetgration to be determinated at $t=0$ for example
Edit1
Note that you can write the differential equation simply this way
$$frac dc_idt+c_ifrac Q_iV=m $$
where
$$m=frac C_uKVX_nu$$
Since you only have a constant on the right side of the equation
answered Aug 6 at 20:02
Isham
10.8k3829
add a comment |Â
up vote
0
down vote
$$frac VQ_ifrac dc_idt+c_i=frac C_uKQ_iX_nu$$
Multiply both side by $frac Q_iV$
$$frac dc_idt+c_ifrac Q_iV=frac C_uKVX_nu$$
It's a first order differential equation
I use method of integration factor with $mu(t)= e^frac Q_itV$
Multiply both sides by $mu $
$$(c_i e^frac Q_itV)'=frac C_uKVX_nu e^frac Q_itV$$
Integrate both side
$$c_i e^frac Q_itV=int frac C_uKVX_nu e^frac Q_itV dt$$
Put the cionstants outside of the integral
$$c_i e^frac Q_itV=frac C_uKVX_nu int e^frac Q_itVdt$$
Evaluate the integral on the right side
$$c_i e^frac Q_itV=frac C_uKVX_nu e^frac Q_itVfrac VQ_i+C$$
Finally
$$c_i =left ( frac C_uKX_nuQ_i +Ce^frac -Q_itVright )$$
Where C is a constant of inetgration to be determinated at $t=0$ for example
Edit1
Note that you can write the differential equation simply this way
$$frac dc_idt+c_ifrac Q_iV=m $$
where
$$m=frac C_uKVX_nu$$
Since you only have a constant on the right side of the equation
answered Aug 6 at 20:02
Isham
10.8k3829
add a comment |Â
up vote
0
down vote
up vote
0
down vote
$$frac VQ_ifrac dc_idt+c_i=frac C_uKQ_iX_nu$$
Multiply both side by $frac Q_iV$
$$frac dc_idt+c_ifrac Q_iV=frac C_uKVX_nu$$
It's a first order differential equation
I use method of integration factor with $mu(t)= e^frac Q_itV$
Multiply both sides by $mu $
$$(c_i e^frac Q_itV)'=frac C_uKVX_nu e^frac Q_itV$$
Integrate both side
$$c_i e^frac Q_itV=int frac C_uKVX_nu e^frac Q_itV dt$$
Put the cionstants outside of the integral
$$c_i e^frac Q_itV=frac C_uKVX_nu int e^frac Q_itVdt$$
Evaluate the integral on the right side
$$c_i e^frac Q_itV=frac C_uKVX_nu e^frac Q_itVfrac VQ_i+C$$
Finally
$$c_i =left ( frac C_uKX_nuQ_i +Ce^frac -Q_itVright )$$
Where C is a constant of inetgration to be determinated at $t=0$ for example
Edit1
Note that you can write the differential equation simply this way
$$frac dc_idt+c_ifrac Q_iV=m $$
where
$$m=frac C_uKVX_nu$$
Since you only have a constant on the right side of the equation
answered Aug 6 at 20:02
Isham
10.8k3829
$$frac VQ_ifrac dc_idt+c_i=frac C_uKQ_iX_nu$$
Multiply both side by $frac Q_iV$
$$frac dc_idt+c_ifrac Q_iV=frac C_uKVX_nu$$
It's a first order differential equation
I use method of integration factor with $mu(t)= e^frac Q_itV$
Multiply both sides by $mu $
$$(c_i e^frac Q_itV)'=frac C_uKVX_nu e^frac Q_itV$$
Integrate both side
$$c_i e^frac Q_itV=int frac C_uKVX_nu e^frac Q_itV dt$$
Put the cionstants outside of the integral
$$c_i e^frac Q_itV=frac C_uKVX_nu int e^frac Q_itVdt$$
Evaluate the integral on the right side
$$c_i e^frac Q_itV=frac C_uKVX_nu e^frac Q_itVfrac VQ_i+C$$
Finally
$$c_i =left ( frac C_uKX_nuQ_i +Ce^frac -Q_itVright )$$
Where C is a constant of inetgration to be determinated at $t=0$ for example
Edit1
Note that you can write the differential equation simply this way
$$frac dc_idt+c_ifrac Q_iV=m $$
where
$$m=frac C_uKVX_nu$$
Since you only have a constant on the right side of the equation
answered Aug 6 at 20:02
Isham
10.8k3829
edited Aug 6 at 20:21
answered Aug 6 at 20:02
Isham
10.8k3829
answered Aug 6 at 20:02
Isham
10.8k3829
answered Aug 6 at 20:02
Isham
10.8k3829
10.8k3829
add a comment |Â
add a comment |Â
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Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
– José Carlos Santos
Aug 6 at 19:21