Leaky-Bucket Experiment

Haefner/Powell
BIOL/MATH 4230 - BioMath and Modeling

February 4, 2002

Lab Dates: February 4 and 11 Rough Draft (Group) due Febuary 15

A draining ``bucket'' during the February 5 lab.

Leaky Bucket - Set Up

This exercise has two parts: a model formulation and calibration phase and a model validation or test phase. In the first, students create a model in addition to that based on Torricelli's Law and choose the placement and shape and sizes of holes in buckets to parameterize their models. In the second, an ``Evil Genius'' creates buckets with holes for students to test their models against.

The following materials are needed.

1. 1-2 quart plastic jugs such as those containing milk or juice for use as the leaky buckets
2. Drill and bits to create circular holes
3. Scapels for cutting non-circular holes and removing burrs
4. Stop watches
5. Graduated cylinders for volume measurements
6. Plastic dish washing tubs for water if a large sink is not available
7. Ring stands and plates for elevating the buckets

It is up to the students to ensure that they have plans to measure all the parameters needed in their models. This may involve different levels of ingenuity, flexibility, and special equipment from the instructors, depending on the models used.

A creating a ``bucket'' during the February 5 lab.

The Evil Genius will play by a few rules to ensure that unreasonable test buckets are not created. For example, if E.G. wishes to cut more than one hole, he should cut the holes at the same vertical level. In particular, similar plastic jugs will be used. Holes on more than one level will not be used. Otherwise, shape, size, and number of holes will be freely created.

This exercise can be tuned for more or less difficulty depending on instructors' aims and students' abilities. For example:

• Use a bucket with a regular, but vertically-varying cross-section.

• Put holes at different levels to illustrate problems with coupled models.

• Vary the fluid or change the surface tension by adding a drop or two of dish-washing liquid.

• Alter the local gravity field, or induce a time-varying gravity field (just joking).

Leaky Bucket - Student Expectations

Students collecting data during the February 5 lab.

The general purpose of this lab exercise is to

1. Introduce the concepts of a mathematical model of a physical system,

2. Instill in you the value of multiple working hypotheses and alternative models,

3. Force you to confront the messy attributes of real world data as they relate to quantitative predictions, and

4. Give you more practice in technical, scientific report writing.

The following specific objectives of the leaky bucket model will determine what kinds of models you build and how you go about testing them.

1. The question of interest is: ``Will the poor students correctly predict the time required for the bucket to empty, and thereby save their lives?''

2. To solve the problem, the students will create 2 models (or more, but 2 is plenty) which will allow the students to determine the emptying time. The models must be ``signficantly different'' from each other.

3. The students will have the opportunity to calibrate their models (i.e., estimate parameters) on data they collect.

4. The models, however, must be applicable to different containers which were not available when the models were created and calibrated. These new containers will differ from the first by having holes of (slightly) different sizes, number, and shapes. The container shapes will be similar. In other words, the models must have a minimal degree of ``generality'' in the sense that they will work on these new containers.

The tasks to complete are:

1. Define the models
2. Define the experimental protocol needed to estimate the parameters
3. Do the measurements and estimate the parameters
4. Verify that the model does ``acceptably well'' (to be defined by the modelers) on the original containers
5. Apply the models to the new containers supplied by the Evil Genius
6. Answer the questions: ``Did we survive the guillotine?'' and ``Which model did best? Why?''

The Torricelli Model

Derivation

The Torricelli model for the draining bucket is based on two physical concepts:

1. The Bernoulli relationship between pressure, $P$, density, $\rho$ and speed, $u$ for fluid along a streamline:
   
   $$\Delta P = \frac12 \rho u^2,$$
   
   

2. The hydrostatic relationship between changes in pressure over the height of a fluid column and gravity (acceleration $g$), fluid density $\rho$ and the height $h$ of the column:
   
   $$\Delta P = \rho g h.$$
   
   

Equating these two gives a relationship between height of the fluid above the hole and the speed at which it spews forth,

$$u = \sqrt{2 g h}.$$

The because the volume of fluid lost from the bucket must equal the flux of fluid through the bucket's hole (with area $a$),

$$\frac{dV}{dt} = - u a,$$

and using the fact that (for a bucket with regular sides and constant cross section in height) $V=A h + V_0$,

$$- u a = \frac{dV}{dt} = \frac{d}{dt} \left[A h + V_0 \right] = A\frac{dh}{dt} .$$

Substituting in the relationship $u=\sqrt{2gh}$ from above gives a differential equation for height of the fluid, $h$, as a function of time:

$$\frac{dh}{dt} = -\frac{a\sqrt{2 g}}{A} h^\frac12. \hspace{1.5in} (*)$$

Solution

To solve equation ($*$) we use separation of variables. First, divide both sides by $\sqrt{h}$ and multiply by $dt$ to separate $h$ from $t$:

$$\frac{dh}{\sqrt{h}} = -\frac{a\sqrt{2 g}}{A} dt,$$

and now integrate both sides with respect to their variables. We will need to use initial conditions as the bottom limits of integration, so let the initial height at time zero be $h_0 = h(t=0)$. Then:

$$\begin{eqnarray*} \int_{h_0}^h h^{-\frac12} dh & = & - \int_0^t \frac{a\sqrt{2 g... ... h & = & \left(\sqrt{h_0} - \frac{a\sqrt{2 g}}{2A} t \right)^2. \end{eqnarray*}$$

As a natural consequence of this solution one can calculate the predicted time for the bucket to empty by setting $h=0$ above, which gives

$$\sqrt{h_0} - \frac{a\sqrt{2 g}}{2A} t = 0,$$

or

$$t_{\mbox{\tiny empty}} = \frac{2A\sqrt{h_0}}{a\sqrt{2 g}} = \frac{A\sqrt{2 h_0}}{a\sqrt{g}}.$$

Testing the Torricelli model during the February 5 lab.