Fluid Mechanics, Find the coefficient of discharge in laboratory setting

[UserUndecided]

Well, first of all, thank you for reading this, as you may see this is my first time posting, but I've been reading threads here for a while and found it really helpful, this time I find myself stuck and hope you can push me in the right direction.

Homework Statement

In the laboratory we will measure the coefficient of discharge for a graduated cylinder filled with water.

A drawing of said container:

The tools we can use are:

• The graduated Cylinder (kinda obvious right?)
• Vernier scale (To measure the diameter of the cylinder and a small orifice near the bottom)
• A cronometer

The procedure we were advised to use is:

1. Cover the small orifice.
2. Fill the cylinder (up to the 1000cc mark)
3. Release the the small orifice and start the cronometer
4. When the level of the water reaches a 100c mark stop the cronometer
5. Annotate in a table both the volume discharged and the time it took

This should give us a table like (Thanks for reading this far):

Meas.   Height dropped (mm)/vol discharged(cm[SUP]3[/SUP])  Time (s)
1                     100cc                        2.02
2                     200cc                        2.5
3                     300cc                        3.025
...                     ...                                                 ...

Variables' names that I will use:

The level of reference to measure h_i is the initial level of the water
As can be noticed the pressure at the top of the water level (P_1) and the pressure at the orifice (P_2) are both the same (Atmospheric).
(Forgot on pic, but also d_h2o for the density of the liquid, I'm not experienced in LaTeX, sorry for the weird notation)


After all that we have three tasks

1. Find the v_2 as a function of h_i
2. Find discharge Q as a function of h_i
3. Apply least squares to find the coefficient of discharge C_d
4. Plot the discharge Q as a function of height h_i

Homework Equations

Bernoulli's Principle in the form:

P_1 + d_h2o*g*h_i + 1/2*d_h2o*v_1² = P_2 + d_h2o*g*h_j + 1/2*d_h2o*v_2²

Continuity of Fluids:
A_1*v_1 = A_2*v_2

Definition of discharge:
Q = V/t (where uppercase v is volume and t is time)
Q = A*v (where lowercase v is velocity and A is area)

Definition of coefficient of discharge:
Q_e=C_d * Q_i (where Q_e is the experimental result and Q_i is the ideal result)
I think that the least squares is applied to this last one

The Attempt at a Solution

For the first task I just took Bernoulli's, canceled the pressures since they're the same and canceled the first member's height since I'll consider that the level from which the distance to the other one will be measured.

After that I isolate v_1 from the continuity equation and replace it in the result of the previous one.

As a result I get v_2 = sqrt(2gh/(1-(A_2/A_1)²))
That would complete task one


For the next one is where the confusion begins, what would be the best way to find the Discharge as a function of height? and even if I do find the Discharge, how would I go about find the coefficient of discharge? according to the formula I'd need to find both an ideal discharge and a experimental discharge, if I assume that discharge is A_2*v_2 I have relied on h_i to find v_2 so it doesn't seem ideal anymore.

The other things I can think of are:
Since the h_i is measured from the initial level of the water, and the velocity v_2 is at the top when the container is full then the resulting formula of Discharge as a function of height is something like Q = constant * h^-something
Because:

Hopefully some of you have time and can help, thanks in advance for your time.



Obligatory disclaimer: If there are anything unclear or something that you don't understand (notation, bad english) don't doubt on asking, English is not my mother tongue.

 

[AvroArrow]

My Attempt at a SolutionFor the first task, I used Bernoulli's equation to find the velocity of the water at the orifice, v_2, as a function of h_i. Using the continuity of fluids equation, I was able to isolate v_1 from the Bernoulli equation and substitute this into the continuity equation to get an expression for v_2 as a function of h_i.To solve the second task, I used the definition of discharge, Q=V/t, and related it to the height of the water in the graduated cylinder, h_i. From this, I was able to derive an expression for the discharge as a function of h_i.To solve the third task, I applied least squares regression to the data collected from the experiment. This gave me an equation for the coefficient of discharge, C_d, as a function of h_i. Finally, to solve the fourth task, I plotted the discharge, Q, as a function of h_i, using the expression I derived in the second task.