See here for a similar type of problem, as well as hints for a solution.
Hello,
I am brazilian student looking for help at my calculus project. I am taking calculus right now and I can't figure out what this part of my project means:
The greatest risk of unintended discharge is when the tank is being filled or emptied through the valve (the diameter of which is approximately 9"), as through unintended contact or mechanical failure the valve could fail and allow unimpeded discharge through the pipe to which the valve is connected. In this case, it is known that the rate at which the height of the fuel in the tank will change is proportional to the ratio of the squares of the diameters of the valve and the tank and the square root of the height of the liquid in the tank, with constant of proportionality k=(2g)1/2 (where g is the acceleration due to gravity).
I have a really hard time interpreting those words. My project proposes to Analise how fast will the height drop inside the fuel tank and how tall should be a wall around it to prevent any loss of fuel. Thing is, I know what calculations I should do and I know how to do then. Yet, I can't understand the wording. Can someone show me in mathematics notation what the rate of change in height inside of the tank should look like?
So, can someone tell me if my reasoning is correct?
Using Bernoulli's equation:
v is the velocity that water flows out (I'll use meters per second)
g is gravity
z is how high the water is in relation to the valve
p is my pressure
P is my density (which in this case is "arbitrary", because there is no information about it. I'll just use a crude oil density for this one)
and then, on my RHS, there is a constant. Is this constant what the text calls "constant proportionality K=(2g)^1/2 ?
If so, it is clear by now that my Pressure and Velocity would change over time. But how would I describe this change over time? I'm still very confused.
So, I still had not any luck with my understanding of the project. I decided to do some more research and I got into this:
Representation
and this
Analysis
I could understand very well that analysis part but for one point. On the final derivation part, there is this part that dh over root h times dt becomes 2 root h. Can anyone explain that to me? I can't figure it out.