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.

(1)

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:

(2)

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

(3)

Upon integrating, equation 3 becomes

(4)

Further simplification yields

(5)

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

(6)

(7)

(8)

(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

(10)

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.

(11)

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:

(12)