Torricelli's Law of Fluids: Solving a Differential Equation for Drainage Time

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In summary, the conversation discusses working on a project involving differential equations and seeking help with a problem. The person also shares some extra information about the project and their progress so far. They mention using technology to solve the problem and provide detailed steps for solving different parts of the project.
  • #1
mikky05v
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I am working on this project for Diff eq's and I seem to be stuck. i think I am messing something up but I need a fresh pair of eyes to show me where it is.

imgur: the simple image sharer
imgur: the simple image sharer (this is the same page as ^ just scrolled down)
imgur: the simple image sharer

Extra information: He sent us this email which really lost me
Project C: you can get an approximate answer for part (d) by using the formula r = 3h/5 in part (c) [so A(h) = Pi*(3h/5)^2] you really should be able to solve that diff eq in (c) with that info. (note that the "actual" time is a bit longer if you solve it the "correct" way using r = .59h + .5). For part (e)--you first need the formula r = -3/5 h +30 which of course gives you the radius at any given height. This makes your A(h) = Pi* (-3/5 h + 30)^2. When you put this into the differential equation and divide by the square root of h and then integrate you should get an equation with h^5/2 and h^3/2 and h^1/2 and t (and of course coefficients on all of those!). Once you find your constant by inputting your initial condition [h(0)=50] you can solve the resulting implicit equation BUT you can't do it by hand, you have to use technology! (I used desmos by inputting "x" for my "t" and "y" for my "h") the graph then showed me the time to drain the other tank is around 10 minutes (600 seconds). I'll leave it to you to find the exact value.

my work so far:


l did part a by integrating twice.

b. I am not entirely sure what this section wanted. I put A (h) dh/dt= -a(sqrt(2gh))

C. r=3h/5 according to email
A (h) = pi (3h/5)^2
a= pi (1/2) ^2 = pi/4
g= 98.1 cm/s^2
Giving the seperable differential equation
Pi (3h/5)^2 dh/dt = -pi/4 sqrt (2×98.1h)
That simplifies to .0257012h^(3/2) dh=dt

D. Integrating both sides I got t= .0102805h^(5/2) + c
I thought to solve for c by taking t=0 and h=50 but I get c=-181.7352816 so I know I did something wrong.
That's as far as I've gotten.
 
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  • #2
For part b), we are being told to consider that:

\(\displaystyle \frac{dV}{dt}=A(h)\frac{dh}{dt}=a\cdot v(t)\)

Now, from part a), we found:

\(\displaystyle v(t)=-\sqrt{2gh}\)

And so the differential equation given follows.

For part c) if $r$ is the radius of the circular surface $A(h)$ and $h$ is the depth of the water, both of these at time $t$, we know by similarity, we must have:

\(\displaystyle \frac{r}{h}=\frac{3}{5}\implies r=\frac{3}{5}h\)

And so we have:

\(\displaystyle A(h)=\pi\left(\frac{3}{5}h \right)^2=\frac{9\pi}{25}h^2\)

\(\displaystyle a=\pi\left(\frac{1}{2} \right)^2=\frac{\pi}{4}\)

Hence, the ODE becomes:

\(\displaystyle \frac{9\pi}{25}h^2\,\frac{dh}{dt}=-\frac{\pi}{4}\sqrt{2gh}\)

Multiplying through by \(\displaystyle \frac{100}{\pi\sqrt{h}}\) we may now write the IVP:

\(\displaystyle 36h^{\frac{3}{2}}\,\frac{dh}{dt}=-25\sqrt{2g}\) where \(\displaystyle h(0)=50\)

I would suggest solving this without using decimal approximations, or substituting for $g$.
 

FAQ: Torricelli's Law of Fluids: Solving a Differential Equation for Drainage Time

1. What is Torricelli's law of fluids?

Torricelli's law of fluids, also known as Torricelli's theorem, states that the speed at which a fluid exits a container is directly proportional to the height of the fluid above the exit point.

2. Who discovered Torricelli's law of fluids?

Italian physicist Evangelista Torricelli is credited with discovering this law in the 17th century while studying the flow of water through pipes.

3. How is Torricelli's law of fluids used in real-world applications?

Torricelli's law is used in many practical applications, such as designing water fountains, calculating the flow rate of liquids in pipes, and predicting the behavior of fluids in hydraulic systems.

4. What are the limitations of Torricelli's law of fluids?

One limitation of this law is that it assumes the fluid is ideal and does not account for factors such as viscosity, turbulence, and friction. Additionally, it only applies to fluids under the influence of gravity.

5. How is Torricelli's law related to Bernoulli's principle?

Torricelli's law is a special case of Bernoulli's principle, which states that the total energy of a fluid remains constant as it flows through a pipe or tube. Torricelli's law specifically focuses on the relationship between fluid speed and height, while Bernoulli's principle considers the effects of pressure and velocity changes on the fluid's energy.

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