4th International Conference on Integrating GIS and Environmental Modeling (GIS/EM4):
Problems, Prospects and Research Needs. Banff, Alberta, Canada, September 2 - 8, 2000.
Analysis of water flowing in a standpipe:
look at it go
Example Author One
Example Author Two
This report analyzes water flowing through an upright cylinder, also known as a
standpipe, at various velocities. Differential equations based on the balance law, and the
mechanical energy of the system are used to model the height of water in the pipe and the
rate at which water flows in and out of the pipe. MatLab version 5.3 was used to solve
the differential equations and to analyze experimental data obtained from a physical
standpipe system. LabVIEW was used to collect the experimental data from the physical
The body of this report begins by describing the standpipe system and the equations used
to model it in detail. The units of all the variables and constants in the differential
equations are discussed or derived. The equilibrium solutions of the two differential
equations used to model the standpipe system are found, and some of their constituent
functions are solved for when the system is at equilibrium. The system of differential
equations is linearized using Taylor’s theorem. This produces a linear system of
differential equations. The eigenvalues of this linear system are found analytically.
Numerical solutions to the system of differential equations used to model the standpipe
system are found and compared with the experimental data that was collected.
Ecological modeling, human-environment interactions, urban development impacts,
survey article, scientific review paper, spatial decision support systems (SDSS), problem
solving environments, parallel processing, object orientation, stochastic modeling
techniques, research opportunities.
The purpose of this report is to analyze the change, or lack of, in the amount of water in a
standpipe. The standpipe is a member of a larger system which pumps water into it, and
“drains” water out of it. The standpipe is essentially an upright cylinder with water
entering through a tube at “the top”, and draining through a tube at the bottom. The
water drains from the standpipe into a “catch basin” from which it can be reintroduced
into the system by a pump. The velocity of the water through the inflow pipe can be
controlled by a valve. Tightening the valve means water has to flow faster through,
hence the velocity of the water in the inflow pipe is increased. Opening or loosening the
valve has the opposite effect. These relationships are shown in figure 1 below.
Figure 1. The relationships between the parts of the standpipe sytem.
Since the standpipe in the system above is a cylinder, it can be easily described
mathematically. For instance, the volume of any cylinder is the product of the area of its
base and its height. Therefore, if we introduce a function h, which describes the height of
the water in the cylinder (measured from the bottom of the cylinder up) at anytime t, then
the volume of water in the cylinder can be described as a function of time. Symbolically
this can be stated as:
V(t) = Aph(t)
where Ap is the cross-sectional area of the standpipe. If it is assumed that water is an
incompressible fluid, the equation can be multiplied by the constant fluid density of
water, p, while keeping the equation a function of t alone. Since density is the quotient of
mass by volume, if all units are consistent, multiplying the equation by p will leave it in
units of mass instead of volume. Therefore the mass of water in the cylinder at anytime t,
is given by:
M(t) = pAph(t)
However, the amount of water in the standpipe is not going to be static, so it is of interest
to us to have an equation which describes the rate of change in the mass of water in the
standpipe. The balance law states that rate of change in any system where principles of
conservation apply to the material being modeling can be described by the difference of
the rate at which the material enters the system from the rate at which the material leaves
the system. Since we have a closed system, and the principles of conservation apply to
the mass of water, we can describe the rate of change of the mass of water in the
standpipe by the balance law. Let qin be the rate at which water enters the standpipe and q
be the rate at which water exits the standpipe in units of volume by time. This yields the
rate of change in volume of water in the standpipe = qin – q
However, for our purposes we would like to the equation to describe rate of change of
mass in the standpipe. Since qin and q are in units of volume by time, multiplying them
by p will produce an equation describing the rate of change of the mass of the standpipe.
rate of change in mass of water in the standpipe = pqin – pq
Since the equation we derived for the mass of water in the standpipe is a function of time,
we can say that its derivative with respect to time is qualitatively equal to pqin - pq.
pA p h(t ) pqin pq
Since p and Ap are constants we can rearrange the equation algebraically to yield:
dh qin q
Notice that q here is actually a function of time. At any given time t, q should give the
rate at which water is exiting the standpipe. Therefore, since q is unknown this equation
cannot be solved on its own.
Making a Two Dimensional system of equations
We can make a 2D system of equations with equation (1) and equation (2), which
describes the mechanical energy of the water exiting the standpipe.
dq At g fq 2
dt L 2 LAt
Description of the Two Dimensional system of equations
This equation describes the balance of momentum or mechanical energy of the
fluid in the tube underneath the standpipe. At is the cross-sectional area of the
tube, g is the acceleration of gravity, L is the length of the tube, and d is the
change in elevation from the top to the bottom of the tube. A derivation of this
equation is outside of the scope of this report.
Microsoft word generates terrible HTML code.
Bishop JM, Varmus HE. 1985. Functions and origins of statistical modeling techniques.
In: Strauss R, Tolland N, Varmint HE, Collins J, editors. Modeling theory and
practice. Seal Harbor, MA: Seal Harbor Laboratory Press. p 999-1019.
Fermi J. 1990. A treatise on distributed parameter modeling in a neutral decision space.
Belcher A, Condor M, translators and editors. Boulder: University of Colorado
Press. 709 p. Translation of the 1623 edition.
Example A. One, Chair, Department of Examples Sciences
University of Examples, 42 Example Street, Example, Example, Example 2T3 8M5.
Email:email@example.com, Tel: +1-403-456-7895, Fax: +1-403-456-7894.
Example A. Two, Chief Scientist, Division of Human Examples and Societal Examples
Example Scientific Academy, M512 Rue de Example, Example, Example N45237.
Email: firstname.lastname@example.org, Tel: +33-01-16-28-67-89, Fax: +33-01-16-28-67-83.