Water tanks "in series" and steady state

Road to a Chemical Engineer: #2

Recap

　In the last article, we have learned a different way of looking at mass conservation, using water tank system as an example. We thought of an equation of water outflowing tank as:

$Q_{Accumulated}=-Q_{Out}$

　Then, we managed to figure out the same equation can be written in terms of change in height over measure time period. The equation that we derived is:

$A\frac{dh}{dt}=-\frac{\rho gh}{R}$

　Today, this article will consider about the tanks that are in align. In the previous article, we only focused on the tank that water outflows. However, how about the mass conservation is the tank that receives the water, or what if there is more tanks connected and so on. 

　This will be the last section for using the water tank example. So, let's cover this mass conservation up with this article!!

Water tanks in series

　Let's imagine a situation like above. Here, you can formulate multiple equations for the tanks' mass conservation.

For Tank A:

$A_1\frac{d{h}_1}{dt}=Q_0-\frac{\rho g{h}_1}{R_1}$

For Tank B:

$A_2\frac{d{h}_2}{dt}=\frac{\rho g{h}_1}{R_1}-\frac{\rho g(h_2-h_3)}{R_2}$

For Tank C:

$A_3\frac{d{h}_3}{dt}=\frac{\rho g(h_2-h_3)}{R_2}-Q_3$

　If you do not understand how these equations are brought up, please look back to the previous articles. This is just an application of ideas expanded to multiple water tanks. It is not easy at first but will get there eventually.

　So, if you imagine such situation, what can you say about this system? Can anything be drawn from this equation??

　One thing that can be thought is the situation where this system reaches steady state. Before we dig into the steady state and their equations, let's see what is steady state.

Steady state and Equilibrium

　People usually have some misunderstanding between equilibrium and steady state. This is because both indicate the state at which the system has no change in variable, height. However, their difference is quite simple. It is whether there is a flow in the system or not. In other words, whether Driving Force (DF) = 0 or not.

　Let's imagine a water tank where the water level is just at the outlet of hole. Will there be any water flow?? No, right. So, DF = 0, and we call such situation, 'equilibrium'. And, this occurs ing batch process, i.e. do not have continuous flow.

　This means 'steady state' is when there is no change in height with DF ≠ 0. Is such situation possible?? And, it is. Let's say there is water flowing into a tank, and water is also flowing out of the tank. What happens when the flow rates of inflows and outflows are the same. 

$A_1\frac{dh}{dt}=Q_0-Q_1 \{where Q_0=Q_1\}=0$

　You see that the rate of change in height is 0, right? Yet, there is a flow ongoing. And, that is the situation we call, 'steady state'.

Steady state and water tanks in series

　Now, we are back to the idea of water tanks in series. So, as we discussed now, since there is a continuous flow in this model, only steady state can be achieved. That is when all the rate of change of height in all tanks go to 0. Thus, for each tank, following should be achieved.

For Tank A:

$$Q0=ρgh1R1

For Tank B:

$\frac{}{}$ρgh1R1=ρg(h2-h3)R2

For Tank C:

$\frac{{\displaystyle \rho g(h_2-h_3)}}{{\displaystyle R_2}}=Q_3$

　Do you see anything surprising? Or, is it too obvious??

At steady state:

$Q_0=\frac{\rho g{h}_1}{R_1}=\frac{{\displaystyle \rho g(h_2-h_3)}}{{\displaystyle R_2}}=Q_3$

　This is what you can draw from this water tanks in series, at steady state. So, at steady state, it might be obvious but all the flow rates must be constant, and that can be formulated using equations!!


　Moreover, If you want to know the height at which tank A achieves steady state, you can simply, reforming and using the equation above, such that:

$h_1=\frac{Q_0{R}_1}{\rho g}$

　However, for other tanks, it is possible to find the difference between (h₂ - h₃) but not individually with my current knowledge. If you know how please share it with me, or otherwise, let's come back sometimes in the future with more knowledge on chemical engineering and mathematics.

Tank A at non-steady state

　So, we just saw what would happen at the steady state. What about at non-steady state? Is is possible to formulate something about it??

　It is actually possible with integration of equations that we formulated above for each tank. However, here, we will only focus on Tank A as it only consists h₁ as a variable. On the other hands, here again, for Tank B and Tank C, both consists h₂ and h₃, so it will require higher level of integration skills, which I will not cover for this article.

　Let's see what happens if we integration Tank A equation:

$A_1\frac{d{h}_1}{dt}=Q_0-\frac{\rho g{h}_1}{R_1}$

　In the integration of this equation, we assume that, at t = 0, h₁ = h₀, and at t = t, h₁ = h. However, I will not cover the method of integration here, so please try it out by yourself.

　It will be similar to the integration in last article but this time, we have Q₀ to consider. You will eventually get equation similar to the below:

$h=\frac{Q_0{R}_1}{\rho g}(1-e^{-\frac{\rho g}{R_1{A}_1}t})+h_0 e^{-\frac{{\displaystyle \rho g}}{{\displaystyle R_1{A}_1}}t}$

　"But... what is so special about this equation... this is just even more complex..."

　Don't worry.This is much more clear and meaningful when we plot on the graph. Let's see this.




　What do you see?? One thing that is clear is that, height of Tank A will eventually reach to the height for steady state (which we determined in above). In addition, from the values of Q₀ , R₁ , and h₀, it is also possible to know whether the height will start to decrease or increase, based on the comparison.

Endnote

　This article has covered what we call water tanks in series, and we determined that steady state is achieved when all the rates of flow are equal. Furthermore, we graphed what will the graph of Tank A be over time and checked that, it will eventually reach the steady state.

　However, at this moment, there is a lot that is yet to be detailed such as the similar graph for Tank B and Tank C. With those knowledge, it is possible to model a better model and have a better understanding of the mass conservation.

　Are you surprised how much we can dig into with just mass conservation?? However, this is just a scoop of what mass conservation is, and not even a lip of chemical engineering. Are you know more excited about chemical engineering??

　Please give me some comments and thank you for reading. Next article will be on still on mass conservation, but not water (Volumetric) anymore!! Molecular mass conservation for the next, so wait for more fun and more chemical engineering.


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