Flow Rates vs. Equilibrium

We are always in constant interaction with water in daily life. The nature by which water flows is a peculiar thing and having an understanding of it is most important. Each time you go to the bathtub and look at it draining the realization of a drop of the level is noted. However as time progress and the accompanying decrease in the height of water results in the slow nature of the flow. This is interesting, and though you might think the time taken in draining is increasing due to the impatience developed, Physics has an explanation for this. The same concept also applies to swimming pools. This paper aims at investigating the relationship that exists between the flow rates of water at different heights. Though I would not be able to have a whole bathtub and for convenience purposes, I will be using a beaker to get volumes of water and also a beaker with holes at different levels. With an increase in height of the water, we expect the flow rate to increase.

Variables

As said before this experiment will aim at determining the relationship between flow rates of water at a different height; therefore, we have two variables. We will be using a beaker to determine our volumes and for the other beaker, we will make a hole in the side. Water will be filled on the beaker and allowed to drain. It is always difficult determining the variables with accuracy and for this practical our independent variable will be the flow rate of water into the column measured in ml/s. This is established by finding the total volume of water that has been poured into the beaker. The dependent variable for our experiment will be taken as the difference in equilibrium height of water and the height from the external point in the column wall. The dependent variable is established by measuring the quantity of water drained in a given time, say 5.0 seconds.

A control is necessary in experiments to ensure that we are doing the right thing. . One control for this experiment is the temperatures of the water used. Temperature has an effect on the water density and, therefore, has a great possibility of altering the experiment, and the variable has to be kept constant.

Another control for this practical is the size of the hole in the beaker. It is mandatory to ensure that a hole of constant diameter is used. The increase in size of the hole will result in an increase in flow rate, and this will affect our results. Likewise, the use of many holes for this experiment will pose a challenge and we ought to ensure that the water drains at a constant so with many holes faster draining will occur.

Not to forget in our control is the pressure. We have to maintain a constant pressure on the water in the beaker. A thought of applying a cap will result in pressure changes with the continuous draining of water. With an open top of beaker we can achieve a constant pressure of 1 atmosphere and the overall room pressure also has to be kept constant.ApparatusRuler ( 100 + or - 0.1 cm)

Stop watch as a timer

500 ml beaker

500 ml beaker with hole

250 ml beaker

100 ml graduated cylinder

Water and adjustable height stand

Method/ procedure

First part

Pour water into the 500 ml beaker and ensure the readings are at 500 ml.

1. Using a ruler measure a given height and set a stand for the beaker with a hole.

2. From the beaker pour the water into the beaker with a hole and ensure that the hole is sealed to with finger to prevent water loss.

3. Drain the water into a 250 ml beaker and this should be simultaneous with the start of the stopwatch (unseal at 0.00 seconds and 5.00 seconds reseal).

4. Using a 100 ml graduated beaker to measure the water drained.

5. Record your value.

6. Replace the water drained to the original volume of 500 ml.

7. Repeat the steps 3-7 for five trials and record your values.

Second Part

Pour water into the 500 ml beaker and ensure the readings are at 500 ml.

1. Using the ruler set the height of the beaker with the hole to 3 cm.

2. From the beaker pour the water into the beaker with a hole and ensure that the hole is sealed to with finger to prevent water loss.

3. Drain the water into a 250 ml beaker and this should be simultaneous with the start of the stopwatch (unseal at 0.00 seconds and 5.00 seconds reseal).

4. Using a 100 ml graduated beaker to measure the water drained.

5. Record your value.

6. Replace the water drained to the original volume of 500 ml.

7. Repeat the steps using the heights of the beaker as 4.0 cm. 5.5 cm, 8.5 cm, and 17.5 cm respectively.

8. For each height record your value.

Data Collection

Equilibrium height +(+/- 1,0cm) Tap setting Avg. Flow rate (ml s-1) Trial 1 Trial 2 Trial 3 1 6 3.0 3.5 3.0 2 16 4.0 5.0 4.0 3 23 5.5 5.0 5.5 4 41 8.5 8.5 8.0 5 68 17.5 18.0 17.0 Representation of the data on graph

Flow Rate of water vs. height of the water

From the data obtained in the excel file, we can infer with our thoughts of the daily that the results obtained are true. The data obtained clearly shows that it has the best fit of a square root curve. The x-axis has been measured in cm while our y-axis which is the flow rate of water has been measured in milliliters per second. A keen look and extension of our line project its x and y-intercepts to (0, 0). This shows that flow rate equals to zero milliliters per when there is no water past the hole. Proof that my result is accurate and with minimal error is that if the beaker has no water to drain, then I will have a flow rate of zero.

The next step is linearization of the data. The linearized value will have a representation of the z variable. From our equation, we can observe a square root correlation between the flow rate and the height of water. To linearize the data we use the following:

Linearized X = z = height

By use of the value of 3.0 cm, we have to calculate the error obtained in height. With an allowance of + or -1 cm, we have minimal errors from this practical. We can have a figure of 0.075 cm as the outlier for the practical. Using that as the error result we can have the following table

Linearized Height vs. Flow Rate

Linearized Height (cm) Error in height (cm) Flow rate(ml/sec) Error in flow rate(ml/sec)

9.0 0.075 6.00 0.20

16.00 0.040 16.00 0.60

30.25 0.030 23.00 0.90

72.25 0.010 41.00 1.40

35.55 0.700 68.00 1.90

We also have to determine the maximum and the minimum slope. This is possible by application of the uncertainties. For maximum slope we use both the highest and the lowest uncerertainities. The overall formula for this calculation is:

( xmin + uncert, ymin - uncert) and (xmax uncert, ymax + uncert )

For our data the points will be;

(9.0 +0.075, 6-0.20) and (35.5 0.70, 68 +1.9). The values give us the points (9.075, 5.80) and (34.8, 69.9). To find the equation for the slope we use the formula m=y2-y1/x2-x1, therefore from our data;

M=69.9-5.80/34.8-9.075

=64.1/25.725

=2.492

M= 2.492 ml/seconds*meter

For minimum slope the following data will be used.

(9.0 -0.075, 6 + 0.20) and (35.5 + 0.70, 68 -1.9). The same formula for calculating the slope applies. (8.925, 6.20) and (36.2, 66.1)

m=y2-y1/x2-x1

=59.9/27.275

=2.196

M= 2.196 ml/seconds*meter

Both the information will give the graphs shown below.

Height vs. Flow Rate

Theory

From the linearized graph, we can affirm that height a flow rate. From both the values obtained from the regular graph and the linearized graph there is a correlation between them, and this shows that the experiment was very accurate.

Torricellis Law applies to our data. The equation for the law is achieved through a sequence of derivations. As this applies to fluids we will first apply the law of conservation of energy.

Evaluation

In our experiment, we have a number of things to note that may not wholly apply to Bernoullis equation.

P

Y2h

Y1v1

The height (h) of water is given by y2-y1. Density is constant for the water and therefore we can say that P=P0. With these conditions we can have a simpler version of Bernoullis equation.

(v1)2 + gy1=gy2

When the above is rearranged we come up with the following equation

(v1)2= 2gh

V1 = 2gh

That is a representation of Torricellis law and we can say our data verifies with the law of fluid dynamics due to the equation flow rate a h.

As a confirmation that flow rate is directly proportional to h I sampled out some data from the table. The formula above can be rearranged into flow rate/ height = constant. Selection of different points we will expect to have a constant.

Flow rate (ml/sec) height(meter ) Flow rate

height

68.00 17.50 3.89

16.00 4.50 3.95

23.00 5.50 4.14

From the trials that we selected and divided the flow rate by the heights square root a rough figure of 4 is achieved. The slight variation is due to errors of the experiment. In this experiment we used flow rates and we did not measure the velocity. Owing to the uncertainty of the hole size, we could not use the velocity of the liquid.

For the experiment, we have less to evaluate regarding the error. There is much-improved precision in the practical, and further accuracy can be achieved by the use of the averages. The procedure allowed us to find average flow rates, and this should also be applied to the heights.

Sources of Error

From the look at my graph, it shows minimal proportionality. This is a sharp indicator that there may have been an error in the course of the experiment. A major source of error for the experiment is the height drop. The 1700 ml trial was approximately 100 ml, and this could have had some errors on the values collected. This error causes the skewed nature of the graphs and becomes worse with further increase in height. A simple way to have the error minimized is ensuring a reduction in the draining time. However, this comes with more challenges and it is not advisable decreasing the drainage time as it will interfere with human reaction time which is termed as a random error. We term this a s a balancing problem, and it is upon us to see which is to be preferred.

The second though a minor cause of error is the sealing and unsealing of the hole by the finger. With electrostatic forces always present some drops will be attracted to the finger. Though they appear minimal, they are substantial to cause an error. This greatly affects the uncertainty for the volume as we are dealing with volumes over 100 ml. A correction of this error would be used of a non-charged material to prevent the water from draining. This could be a piece of wood or metal.

Another error that could have resulted in the faults of the experiment is the water that was left in the beaker each time before the next measurement was done. This in a way could contribute to a positive increase of water or a decrease. Topping up of an extra amount of water in most cases will result in error due to the poor readings of the water left. A simple way to avoid this error would be by ensuring that each time before the next experiment we should empty the beaker to 0 ml and fill it afresh.

In conclusion, the experiment has ensured that the dynamics of fluid that pertain Torricelli's law have been fully emphasized. The objectives of the experiment which included having to find a correlation between the rate of flow and the heights of water have been proven and a formula that explains more of it derived. Aspects of design and participation in the experiment were put in place, and it ensures the concepts stick in our minds, and we can engage those who lack the knowledge on the relation of Physics and the daily life. From the inference and hypothesis, we had for our lab we can now confirm to it and say that water obeys the laws for flow rates...