**Abstract:
**

The purpose of this experiment was to develop a differential equation
to model the flow rate as a function of orifice diameter. The
delivery system modeled was at a residential home. Our results
fit the logistics equation. The carrying capacity was equal to
the maximum flow rate of the system, which was estimated at 5.63
gallons per minute (GPM).

**Introduction:
**

The reason we chose this experiment is because we thought the
solution of the differential equation acquired would not be linear.
We expected the solution to be a function of orifice area, which
related to the square of the diameter. We hypothesized further
that we may approach the maximum flow rate of the delivery system.
This would cause the outcome to follow the logistics equation.

where R is the flow rate in GPM and D is the orifice diameter
in inches.

The logistics equation is a slight variation of a standard exponential.
The *(b-R)* term causes the solution to level off to the
carrying capacity instead of being unbounded as the independent
variable approaches infinity. The carrying capacity is therefore
an equilibrium since the slope approaches zero. We also expected
this equation to change with input pressure; therefore, we tried
to pick a system in which the pressure would be constant.

The logistics differential equation is solved in the following
manner:

The right side of the equation is easily integrated, but the left side needs to be simplified using partial fractions:

Upon integrating, equation 3 becomes

Further simplification yields

R is solved for using equations 6, 7, 8, and 9

**Description of the Experiment:
**

Equipment used:

1 five gallon bucket

1 water supply

1 drill with drill bits

8 orifice plates 1/16 "-1/2" diameter in increments of 1/16"

1 hose

1 stop watch

1 hose connector

The experiment began by placing the orifice with a drilled hole
of 1/16" diameter inside the hose connector and tightening.
The water supply was turned on and allowed to run for a few seconds
to ensure the system was stable before the trials began. The
hose was inserted into the bucket as low as possible. [See Figure
1]. The stop watch was started at the moment the hose went into
the bucket. It was stopped when the water reached the 5 gallon
mark. The orifice size and filling time were recorded in a two
column format. This process was performed for each orifice size.
The flow rate was determined by dividing 5 gallons by the recorded
time in minutes.

**Results and Discussion:
**

The experiment was originally set up to use a one gallon container,
but it was soon discovered that this would not be an adequate
size. The first set of data was not useful. The high velocity
of the water leaving the orifice caused an excessive amount of
turbulence. This turbulence caused air to be trapped in the water
already in the container and gave a false indication of the water
level. The process was refined using a 5 gallon bucket to get
more accurate times. The hose was lengthened so that it could
be inserted as low in the bucket as possible. This prevented
excessive air from entering the system and affecting the indicated
water level. Figure 2 shows a plot of the raw data, "GPM
versus Orifice Diameter". The raw data is supplied in Attachment
A.

The carrying capacity is estimated to be 5.7 GPM. If we return
to equation (5) and create a semi-log plot, [See Figure 3] we
can estimate the value of *ab* as the slope because the right
side of the equation fits the form of a straight line with *ab*
equal to the slope. The last two points in Figure 3 are not used
because the slope will continue to be zero. No further change
will be seen as the orifice diameter is increased because the
carrying capacity has been reached. During the semi-log graphing
process the carrying capacity estimate was adjusted to make the
slope fit a straight line. The new value of *b* equals 5.63
GPM. The value of the slope, *ab*, is 22.6 /inch.

Using these two constants equation (9) becomes

By using one of the values from the experiment, R(1/16)=.68, we
can solve for (k) = 0.03. Substituting into equation 10 produces
the final equation.

**Conclusions:
**

As expected the solution to the differential equation was not
linear. In fact, it produced an "S" curve that is characteristic
of a logistic equation. The results showed an unstable equilibrium
solution at R = 0 GPM and a stable equilibrium solution at R =
5.7 GPM. It also verified the hypothesis that increasing orifice
size beyond the carrying capacity of the system has no affect
on the flow rate. Attachment B is the plot of the raw data compared
with the plot of equation (11). It shows that the developed model
fits the experimental data. The value of k in equation (11) varies
when different initial values are used [See Attachment A]. The
initial value R(1/8)=2.44 yields a *k* value of .04 and provides
the best curve fit. The remainder of the error can be explained
as measurement error during the experiment. The best fitting
model is:

orifice size (in) | time (s) | GPM (R) | slope | ln | b | slope | k |

1/16 | 443 | 0.68 | 28.19 | -1.98974 | 5.63 | 22.59466 | 0.03331 |

1/8 | 123 | 2.44 | 30.54 | -0.26873 | 0.045363 | ||

3/16 | 69 | 4.35 | 13.19 | 1.221119 | 0.049027 | ||

1/4 | 58 | 5.17 | 9.55 | 2.425129 | 0.039814 | ||

5/16 | 52 | 5.77 | -1.74 | 3.724161 | -0.03556 | ||

3/8 | 53 | 5.66 | 1.74 | 5.227548 | -0.03895 | ||

7/16 | 52 | 5.77 | -1.74 | 3.724161 | |||

1/2 | 53 | 5.66 | 11.32 | 5.227548 |