Investigating the water flow of my water dispenser

The cylindrical mason jar on the left has been the water dispenser in my room for many years. I usually refill the water in it after 3 days, and when I do, I have noticed that towards the end when the water level decreases, the water flows out at a slower rate and it continues to do so until water does not flow out of the tap at all. As the tap is situated above the base of the jar, the water does not flow out once it passes the tap. After regularly noticing this relationship between water level and rate of flow, I wanted to apply math and predict how long it would take for the water to completely flow out.

Essentially, I will be investigating the flow rate. Flow rate is a concept in fluid dynamics that quantifies the volume of fluid, in this case, water, that passes through a specific point in time, in this case, the tap, per unit of time (Maram Ghadban, n.d.). It is commonly applied in the fields of engineering and physics. The importance of flow rate spans from aerospace engineering to the medical field. In the context of the aerospace engineering field, the flow rate is used while building spacecraft, for example, it is used to assess the movement of fluids in aircraft systems, hydraulic systems, fuel lines and cooling mechanisms (Ladd Howell, 2010). In the medical field, flow rate is used in the accurate delivery of fluids in medical devices and therapies. They ensure precise control of fluid flow rates down to the microliter per hour range, ensuring the effectiveness and safety of treatments. (News Medical, 2023)

This investigation aims to predict the time it would take for the water to fully drain out of the mason jar by using different models, then redesign the water dispenser to be more efficient and retain less water using calculus. Firstly, I will use three different models that show the relationship between the variables of time and height of water, which are linear, quadratic, and exponential. I used these three models because they fit the general trend of the data points; there is a clear negative correlation between both variables hence the linear trend is employed, and a negative decay relationship hence the quadratic and exponential. After fitting these models, I shall determine which model is best to use to predict when the water will fully drain out the mason jar through percentage error analysis, using the model which has the smallest value. This ensures that my prediction is accurate. Next, I will utilize calculus to redesign the water dispenser. I will use the same dimensions of the original cylinder and apply them to a different shape and find the volume of each, calculating the total volume. I will then calculate the volume of water left at the height when water stops flowing out to see how much water is left in each. By comparing the total volume and how much water remains, I will determine a shape which can be used to redesign the dispenser.

I used a ruler, a jar, and a timer to conduct this experiment. First, I added 10 inches of water, measuring it against my ruler. I measured the jar's radius (3.25in), the height of the tap from the ground (1.75 inches) then filled up the water using my ruler to measure 10 inches then opened the tap, let the water to flow for 10 seconds, and measured the height. I did this until the water did not flow out anymore. I tabulated my results into the following table. This experiment was done in warm conditions with no factors affecting the water flow. I assumed that the water would flow out of the tap consistently and that the cylinder was a perfect circle. I calculated the volume of water present with 10 inches of water through

\[ \begin{gathered} V=\pi r^2 h \\ V=\pi(3.25)^2(10)=331 \mathrm{in}^3 \end{gathered} \]

Therefore the total volume of 10 in of water in the cylinder is \(331 \mathrm{in}^3\)

From observing the data, I was able to notice the difference between changes in height and time. The change in height decreased as time increased, therefore telling me they are inversely proportional. However, to further determine the relationship between the height of the water and time, I will use different functions and determine which one is the most suitable to use to make predictions. For all my functions I will be having the domain of \(x=[0,200]\). This is because in the context of this investigation, it is reasonable to say that the water would hypothetically completely drain out of the cylinder between 0 to 200 seconds based on the general trend of the data. If there is 1.7 in of water left when the water is 130 s, it is logical that the remaining water would possibly drain out in some time within 80 seconds.

What is a linear function?
Linear models are functions where every singular input has exactly one output (Turito, 2023). The rate of change of \(x\) over in a linear function is also called the slope. An equation in slope-intercept form of a straight line includes the slope and the initial value of the function and the rate at which x is changing. The y-intercept or the initial value is the output value when zero is the input of a linear function. This can be mathematically represented as

\[ h(t)=at+b \]

Where a represents the slope, and \(b\) is the initial value of the function or \(y\)-intercept.

In the linear model it is assumed that the rate of water flow is constant in only one direction hence the time taken for the water to drain is directly proportional to the height of water remaining (Turito, 2023).

Using GeoGebra, I was able to use the FitLine tool to find an appropriate linear equation. GeoGebra was the best option because it allowed for accuracy and a clear presentation of the fit of the function.

I opted to form my function in the form of \(h(t)=h_0-kt\) so that my variables of time and height are consistent and clear, Where k is a constant representing rate of flow, and \(h_0\) being the initial height when \(t=0\). My function is \(h(t)=9.15-0.06x\)

The line of best fit above does not encompass all the points, however, this is expected as the line of best fit passes through the approximation of the set of data, not all points. The line of best fit shows a strong negative correlation between height of water and time.

To confirm my findings, I will use the coefficient of determination or \(R^{2}\) to test whether it is a good fit. This is a measure that provides information about the goodness of fit of a model. In the context of regression, it is a statistical measure of how well the regression line approximates the actual data. It is important when a statistical model is used either to predict future outcomes or in the testing of hypotheses, like in this investigation. As it is a percentage it will take values between 0-1 (Coefficient of Determination, R-squared, n.d.). To find the \(R^{2}\) value I input my values into my GDC and found the \(R^{2}\) and \(R\) values. This is because it is a simple and easy method of finding the value while ensuring that the answer is accurate. The \(R^{2}\) equals 0.944. This means that 94.4% of the variability of Y is explained by X. The correlation \((R)\) equals 0.891. This means that there is a very negative relationship between X and Y. However, the strength of the correlation is easily seen through the line of best fit. The more important value is \(R^{2}\), which clearly shows that the linear fit is relatively good in showing that there is a negative correlation.

To verify this models accuracy, I will calculate the percentage error and pick the one that has the least to decide which to make predictions.

Percentage error is a measure of the difference between an estimated or measured value and the actual or true value of a quantity, expressed as a percentage (Byjus, 2020). It is calculated by taking the absolute difference between the estimated/measured value and the actual/true value, dividing it by the actual/true value, and multiplying by 100.

I will use the formula

\[\text{percentage error} = \frac{\text{measured value} - \text{true}}{\text{true}} \times 100\]

Where I will call the measured value 'Predicted graphical value' and the true value 'Measured true value'. I will use the different models I have and input the \(t\) values from the graph and calculate the output (y value) which I will get. That will be my 'predicted graphical value'. Then, I will use the formula and calculate the percentage error.

Using the linear function \(h=9.15-0.06t\)

\[\begin{gathered}\text{When }t=10\\h=-0.06(10)+9.15=8.55\\\text{percentage error}=\frac{8.55-8.8}{8.8}\times 100=-2.84\end{gathered}\]

The percentage error of the linear function is high at 149.60%, thus I will not use this to make my prediction and model it using a different function. The large percentage error stems from the assumption of linear models, which is that the rate of water flow is consistent, which is not the case. This is why there is a large deviation between the predicted graphical values and the measured values from the experiment; the points on the graph indicate a curve while the linear function is straight.

I have opted to use a quadratic function because there is a clear curve of the data, which means that a quadratic can be fitted to it.

A quadratic function is a polynomial function where the highest exponent of the variable is 2 (Cuemath, n.d.). This function forms a parabola when plotted. The parent from of the function is the from \(h(t)=x^{2}\), and transformations are applied on this function to introduce other variables, \(b\) and \(c\), convert the parent function into the form

\[h(t)=at^{2}+bt+c\]

Where \(a, b\), and c are real numbers with \(a \neq 0\).

The function assumes that there is a changing rate of flow. I assume this will work well with my data values as there is a changing rate of flow because of the relationship between the area and pressure of the cylinder. The assumptions of the quadratic model are the same as the assumptions of the exponential model, mainly because the assumptions are based on the changing rate of flow, which is applicable to this model as well.