EPQ: Mathematical model of tea bag diffusion

[Planckian]

I've done some research, and I've looked at the derivation of the diffusion equation, but I don't want to reproduce even a more simple version of it. I'd like something that can I do almost exclusively myself. The point of this is not to plot some graphs, and extract some line of best fit or exponential equation, but to work toward a model (tea concentration as a function of time, probably) that can fits data, with improvements being made accordingly. I've tried the Rayleigh method of dimensional analysis as a start, but with few variables, I couldn't see it though. So, are there any mathematical techniques/methods that you guys suggest I could use, or anything else? I'm not looking at something exact, but maybe an approximation for a cylinder, or sphere, maybe. Thanks so much.

 

[Shinaolord]

I would solve for concentration left in the bag C(t) by solving the ODE
$\frac{dC}{dt} = -kC$ where k is a constant.
That is the simplest solution I can think of for the situation without the diffusion eq.

 

[Planckian]

I would solve for concentration left in the bag C(t) by solving the ODE
$\frac{dC}{dt} = -kC$ where k is a constant.
That is the simplest solution I can think of for the situation without the diffusion eq.

I'm in high/secondary school, so I'm know expert in solving, but I know what it involves. Anyway, I had considered that, but would it not just give something of the form c(t)=c(0)*e^(-kt)? Or are you saying that k is a function of all of my other variables (temp, volume..)?

 

[Shinaolord]

K would be based off of the data, say you had 60% of the tea left After 40 seconds. That would allow k to be found. With all of those variables, I don't think an OdE would be sufficient.

 

[Planckian]

K would be based off of the data, say you had 60% of the tea left After 40 seconds. That would allow k to be found. With all of those variables, I don't think an OdE would be sufficient.

Ah, I see, but having second thoughts, I don't thing it will work because it models a drop in concentration, whereas diffusing tea would result in conc increasing, and thereafter plateauing, right?

 

[Shinaolord]

Well, the concentration of the tea left in the bag is decreasing as it dilutes with the water. It would have a minimum, when the water has the same concentration as the bag has left. If we were talking about the waters concentration, you would be correct. ( it would increase and plateau)
Either way is a valid model.

 

[Planckian]

Well, the concentration of the tea left in the bag is decreasing as it dilutes with the water. It would have a minimum, when the water has the same concentration as the bag has left. If we were talking about the waters concentration, you would be correct. ( it would increase and plateau

Yeah, I;m talking about the water's conc

 

[Planckian]

Well, the concentration of the tea left in the bag is decreasing as it dilutes with the water. It would have a minimum, when the water has the same concentration as the bag has left. If we were talking about the waters concentration, you would be correct. ( it would increase and plateau)
Either way is a valid model.

I've been sitting here all day, not really sure where to go (well, I have a few ideas), but are you sure there isn't anything else you know of? It doesn't have to be simple

 

[Shinaolord]

If I knew pde'a possibly. I'm trying to find a good diff eq to model it.
Edited for spelling.

 

[Planckian]

If I knew pde'a possibly. I'm trying to find a gold diff eq to model it.

Not sure what that means, but, ok

 

[Shinaolord]

Something such as $\frac{dC}{dt} = -k(C-100)$ where I chose 100 because I wanted it to max at 100%.
Graph the solution of that. Does that look like you would expect? I can graph it for you if you can't.

pDE are partial differential equations, like the diffusion equation with $\partial$ derivatives. It's higher level math.

Edited for correct signs.
My previous post should've said good model not gold lol.

 

[Planckian]

Something such as $\frac{dC}{dt} = -k(y-100)$ where I chose 100 because I wanted it to max at 100%.
Graph the solution of that. Does that look like you would expect? I can graph it for you if you can't.

pDE are partial differential equations, like the diffusion equation with $\partial$ derivatives. It's higher level math.

Edited for correct signs.
My previous post should've said good model not gold lol.

Thanks, will look at it. What is y? Just a constant?

 

[Shinaolord]

Well y is the concentration of tea in the water. You can use c' and c if that makes it clearer for you.

Edit my bad. I see what you meant. I fixed my previous post.

 

[Planckian]

Well y is the concentration of tea in the water. You can use c' and c if that makes it clearer for you.

Edit my bad. I see what you meant. I fixed my previous post.

If I integrate, I get c(t)=kt(100-y). I already have a c term. So is y the same thing? Thanks

 

[Shinaolord]

So you know, the solution is $C =p*e^{-kt}+100$ where p is a constant, and e the natural base =2.7... . Since at t=0 there is no tea in the water, we have p*e^0t+100=0 which is the same as p+100=0. So p=-100.

 

[Shinaolord]

If I integrate, I get c(t)=kt(100-y). I already have a c term. So is y the same thing? Thanks

These integrations are more complex than that. Have you heard of wolfram alpha? It should give a step by step solution so you can see. P.s. At high school leve, this math is totally unnecessary. It's 2nd/3rd year college lvl.

 

[Planckian]

These integrations are more complex than that. Have you heard of wolfram alpha? It should give a step by step solution so you can see. P.s. At high school leve, this math is totally unnecessary. It's 2nd/3rd year college lvl.

For this project, I have to go beyond what I know

 

[Shinaolord]

Ok. Well the graph should look like this; is it what you expect?

 

[Planckian]

Ok. Well the graph should look like this; is it what you expect?
View attachment 72031

Wow. Thanks

 

[Shinaolord]

Your welcome. I hope that was what you're looking for! :)

 

[Planckian]

Wow. Thanks

Do you think there's an good way of incorporating variables like temp and volume of water?

 

[Shinaolord]

Not that I know of. You could wait until someone more knowledgeable replies though. I'm not sure if it's possible.

 

[Planckian]

Not that I know of. You could wait until someone more knowledgeable replies though. I'm not sure if it's possible.

Thanks so much. This was very nearly going to be a wasted day(!)

 

[Shinaolord]

Your welcome. Differential equations are fascinating, they make math wonderous in my opinion.

 

[Planckian]

Your welcome. Differential equations are fascinating, they make math wonderous in my opinion.

Agreed