Lesson Plan: Conservation of Mass
Summer 2006
David Vacco
441 Deike Bldg.
Penn State Ice and Climate Exploration
Dept. of Geosciences
Pennsylvania State University
University Park, PA 16802
dvacco@geosc.psu.edu
Title: How fast is water flowing into the tank?
Purpose: The purpose of this lesson plan is to lead students to the theory of Conservation
of Mass as a means of predicting the flow of mass, and give them some intuition
about the reason for needing such a law.
Objectives: In this exercise, the students will:
1. Students will verbally state the Law of Conservation of Mass, and express the
Law in equation form.
2. Students will quantify the concept of “flux”, and correctly demonstrate that
conservation mass is the same as conservation of volume if the density is
constant (incompressible fluid), by solving for the volume influx of water into
the tank.
3. Explicitly state (via discussion) that all forms of measurement are simply a
comparison to a reference quantity.
Set up: Students will be presented with a tank of water that has a small hole in the
bottom (figure 1). There will be water flowing into the (closed) tank from the top,
and water flowing out of the hole in the bottom. The rate of water influx is to be
slightly slower than the rate of out-flux, so that the volume of water in the tank
decreases at a measurable rate (but slow enough that the students are in no rush to
figure out what they are doing).
Qin, to be calculated by the
students
Water filled tank
tank
H
Qout, to be measured by
the students
W ~ 0.5 m
Figure 1. The tank of water of study. The students will not be able to measure the influx of water
(Qin), but can measure the volume of the tank, the flux out water out (Qout), and dH/dt. The height
of water (H) in the tank will decrease at a measurable rate, slow enough that the students feel no
pressure to rush. If they take too long, Qout can always be stopped until the tank fills up again.
Student Activity: Calculate how fast water is flowing into the tank. You may NOT open
the tank and directly measure the water flow. You may measure the volume of
water in the tank, and the rate it is flowing out of the hole in the bottom.
Steps:
1. The students will first have to articulate how to measure water flow rate.
** The goal is to lead them to explicitly verbalizing that measurable flux is
volume per time, or mass per time. Since density is constant (here), they will
be lead to measuring density * volume/time = mass flux out of the tank.
2. What else can they measure? The dimensions of the tank, and the change in water
height in the tank over the same time period they measure the flow out. Since the
area of the tank is constant, that will lead them to volume/time = Area * dH/dt.
3. During their discussing how to do this,
** I will pose the question: as the volume in the tank decreases, will Qout
remain constant over time? The answer I am looking for is no, and why. Why
is water flowing out of the tank at all? There is a force on the water at the hole in
the tank in the form of pressure gradient.
On the inside, the weight of the water is pressing, on the outside, nothing. This
pressure gradient decreases through time as the volume of water in the tank
decreases. Can we calculate the rate? YES!! This is a preview of something that
will come later in class – conservation of momentum (i.e. force).
** The point of me asking is to lead them to the discovery that in order to
accurately calculate the influx of water in the tank they must measure the
change in volume of water in the tank during the same time period they
measure the out-flux.
4. Next, the ultimate question of this lesson: how to calculate the influx in the tank?
Thus the crux of the whole lesson: Change of volume in the tank per time = the
flow rate in – flow rate out. Or much more formally stated:
(1) Time rate of change of mass in a fixed volume = mass rate in – mass rate out.
** This is a verbal statement of the Law of Conservation of Mass, which the
students just intuited quantitatively.
Once they have calculated what they think is the water influx, we will
remove the lid from the tank, and measure Qin to see if they got it right (which
they did with my guidance).
5. Finally I will summarize the whole concept with a formal derivation. The verbal
statement (1) is the Law, but how can we write in terms of measurable quantities?
(2)

Vol   Qin  Qout ,
t
Where Q = mass flow per unit time (like kg/hour),  = density, and ∂/∂t is the
time derivative, which represents the continuous version of the discrete (and
therefore measurable) Mass/t.