Bernoulli equation with pressure tank

[iloveannaw]

Homework Statement

Water flows at a rate of 30ml/s from an opening in the bottom of a 4m high pressure tank (a tank with a plunger type lid). Calculate the flow rate when an extra 50 kPa of pressure is applied.

Homework Equations

Bernoulli's equation and $$\frac{V}{t} = Av$$

The Attempt at a Solution

My idea is to apply Bernoulli's equation to work out the area of the opening by finding the velocity of the water in the initial case and then to use Bernoulli again to find the speed of the same in the case where the extra pressure is applied. Then to use this together with the area to calculate the flow rate in the final case.

But I'm a bit uncertain of some of my assumptions. NOTE: subscript 2 refers to top of tank and subscript 1 refers to bottom.

Initial case (no extra pressure):
$$v_{2} << v_{1}$$, therefore $$v_{2} \approx 0$$
$$P_{2} = P_{1}$$
$$h_{1} = 0$$

therefore

$$v_{1} = \sqrt{2gh_{2}}$$

this means that area of opening is:

$$A = \left( \frac{V}{t} \right)_{initial} \frac{1}{\sqrt{2gh_{2}}}$$

Pressure case:

with application of additional pressure I did pretty much the same – using Bernoulli to work out the velocity of the out flowing water by considering the increased pressure of P₂ and then using the flow rate equation to work out the new flow rate.

However I'm not sure. How does the changing water level affect this? I'm guessing you can ignore this but am still unconvinced. Also in the initial case I've assumed the pressure difference between P₁ and P₂ is zero. I'm not sure if this is correct either, seeing as the top of the container is closed. thanks

 

[anathema]

for your post! It seems like you have the right idea in using Bernoulli's equation to solve this problem. However, I would recommend using the continuity equation (A1v1 = A2v2) instead of the flow rate equation (V/t = Av). This will allow you to find the velocity of the water at the opening in both cases, and then you can use Bernoulli's equation to solve for the flow rate.

In terms of the changing water level, you can ignore this as long as the opening is small compared to the size of the tank. The pressure difference between P1 and P2 should also be considered, as this will affect the velocity of the water at the opening.

Overall, your approach and assumptions seem reasonable, but I would suggest double checking your calculations and equations to make sure they are accurate. Best of luck!