Filling and emptying a barrel -- Bernoulli

In summary, the customer wants to reduce the time it takes to wash and drain the container. The mentor attempted to solve for the height of the container and found that at 12.5 GPM it would take 0.5 m of liquid to reach the top. They also mentioned that they need 10.5 GPM to keep from accumulating at 1 in.
  • #1
jzcrouse
5
1
<< Mentor Note -- thread moved from the schoolwork forums to ME for better views >>

1. Homework Statement

Not really homework as this is for work. We are washing a cylindrical container (15.5 cm) with only one (1.8" dia) hole at the bottom the rest is sealed. Our machine sticks a .5" pipe with a sprayer up through the hole with a 12.5 GPM output. The customer wants to reduce the time it takes to wash and drain the container. My main question is when the container begins filling with water (faster than the exit of water through the same hole), I need to find the rate the water level height increases. Need to know the height at 10 second increments up to 90 seconds. Also if I put a hollow tube spanning the length of the pipe sprayer so it is inside and outside the tank. Can I assume ATM pressure at the top of the container for the most part? Height of the container is 23 inches. All dimensions were converted to be in inches and seconds. Area of tank is ##A_{t}##, Area of hole is ##A_{h}## The flow in will go for 90 seconds (the container should be filled by then) We would prefer not to have it fill but understand that may not feasible if flow rate in doesn't change. The Volume accumulation rate in the tank should more than coming out from actual results

Homework Equations


##A_{h}V_{1}=A_{t}V_{2}##
##v=\sqrt{2g(h_{1}-h_{0})}## assuming atm at top of container. If not true finding pressure above water would be difficult i assume since it would change when the tank begins filling and ATM is pushing up while you only have hydrostatic + whatever pressure is there via ideal gas too much math since were running a simulation soon.

The Attempt at a Solution


Rate in - Rate out = Rate Accumulation

##12.5 GPM- v*A_{h}= \frac{dh*A_{t}}{dt}##

##\frac{12.5 gpm - \sqrt{2g(h)}*A_{h}}{A_{t}}=\frac{dh}{dt}##

##\int \, dt=\int\frac{ A_{t}}{12.5 gpm - \sqrt{2g(h)}*A_{h}}\, dh##

integrate and I end up with ##t=-\frac{2*188.69*(\sqrt{h}* 65.26 + 48.125* \log(\sqrt{h} *65.26 - 48.125))}{65.26^2}##

which doesn't work because Ill end up with a negative inside my log function. Perhaps I am just screwing up units but i tripled checked those. I was hoping to just input various times and solve for h.
 
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  • #3
And you may want to re-name your variable for the area of the hole... :wideeyed:

:smile:
 
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Likes Wrichik Basu
  • #5
crash course: Typing

#
#\Phi_in = 12.5## GPM

$$\Phi_{in}- v*A_{hole}= A_{tank} {dh\over dt} $$

Gives you

##\Phi_{in} = 12.5## GPM

$$\Phi_{in}- v*A_{hole}= A_{tank} {dh\over dt} $$And Berkie shouldn't worry about naming anymore this way.

Anyway, Ahole has to be corrected for Apipe (and a possible Atube) !
However, why not let the pipe spout in a water/air mixture: that should keep the internal pressure up for quick draining.

I can't complete your model:
You don't give the height of the container
You don't tell how long the 12.5 GPM (1GPM = 0.00079 m3/s) is sustained.
Your initial condition is probably h = 0 so I am not surprised it doesn't work as posted...

A ready-made calculator here says 0.5 m of liquid level (with ATM above) drains in 5 s
 
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Likes berkeman
  • #6
working on clarifying
 
  • #7
updated the initial post
 
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Likes berkeman
  • #8
Numbers are meaningless: can't see how they come about. What do you do with the 12.5 GPM ?

Tip: check the steady-state equation ( ##{dh\over dt} = 0## ) : what inflow do you need to maintain e.g. 30 cm liquid level ?

And: check with the link I gave
 
  • #9
12.5 gpm is 48.125 in^3/s. I can't change flow rate. The flow should stop before the container is full because the container will not be getting clean when it is full of water. at steady state I'm getting only about half an inch to maintain at 12.5 gpm. Using your program i am finding i need 10.5 gpm to for it to go in and out and not accumulate at 1 in. not sure how that helps me though cause I can't change flow rate... it was the first think i asked them

My integral came out to
##- \frac{2*A_{t}(\sqrt{h}*A_{h}+12.5gpm*\log(\sqrt{h}*A_{h}-12.5gpm))}{A_{h}^2}##
 
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  • #10
jzcrouse said:
Using your program i am finding i need 10.5 gpm to for it to go in and out and not accumulate at 1 in. not sure how that helps me though cause I can't change flow rate... it was the first think i asked them
Stream-of-consciousness here:
Without the vent (hollow tube) to atmosphere, the tank will pressurize and then will drain faster.
Which leads to the idea of pressurizing the tank.
Another possibility is to include a drain tube with a vacuum on it.
Or substantially increase the spray pressure so less wash liquid is needed.
Beyond that, maybe this is one of those jobs you need to walk away from. :frown:
 

1. How does Bernoulli's principle apply to filling and emptying a barrel?

Bernoulli's principle states that as the speed of a fluid increases, the pressure within the fluid decreases. When filling a barrel, the fluid (such as water) flows in with a high speed, creating a low pressure area at the top of the barrel. This allows air to escape through the top, making it easier for the barrel to fill. When emptying the barrel, the opposite occurs - the fluid flows out with high speed, creating a low pressure area at the bottom of the barrel, allowing air to enter through the bottom.

2. Does the shape of the barrel affect Bernoulli's principle?

Yes, the shape of the barrel can affect the speed of the fluid and therefore the pressure within it. A narrower barrel will cause the fluid to flow faster and create a greater pressure difference between the top and bottom of the barrel. A wider barrel will result in slower fluid flow and a smaller pressure difference.

3. How does the height of the barrel affect Bernoulli's principle?

The height of the barrel does not directly affect Bernoulli's principle, but it can indirectly affect the speed of the fluid and therefore the pressure within it. A taller barrel will have a greater pressure difference between the top and bottom, as the fluid has a longer distance to travel and can gain more speed. However, the shape and size of the barrel will also play a role in determining the pressure difference.

4. Can Bernoulli's principle be used to measure the amount of fluid in a barrel?

No, Bernoulli's principle cannot be used to measure the amount of fluid in a barrel. It only describes the relationship between fluid speed and pressure. Other factors such as the size and shape of the barrel, the type of fluid, and external forces can also affect the pressure and cannot be accurately measured using Bernoulli's principle alone.

5. Are there any real-world applications of Bernoulli's principle in filling and emptying barrels?

Yes, Bernoulli's principle is commonly used in industries such as oil and gas, where barrels are used to transport fluids. By understanding Bernoulli's principle, engineers can design barrels and pipes that maximize fluid flow and minimize pressure differences, making the transportation process more efficient. Bernoulli's principle is also used in hydraulic systems, where the principle is applied to pumps and valves to control fluid flow.

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