ES-F341 F04

Lab 4

Stability of a Floating Body

Prepared by Abdul Sayed, awsayed@alaska.edu

Group members:

Ryan Johnson, rjohnson15@alaska.edu

Mi Chin Yi, myi5@alaska.edu

Grant Cummings gkcummings@alaska.edu

Cecilia Hull cahullz@alaska.edu

Date performed: 10/9/2014

Date due 10/16/2014

IntroductionThe gravimetric hydraulic bench is a device which can be used to measure the flow rate

of water entering a system. It functions by using a pivot arm to balance a water bucket against a

weight. The device serves as a control volume. A control volume is a concept which assumes

that for a given volume, the flow rate entering the control volume will equal the flow rate leaving

the control volume.

The problem of measuring flow rate effectively and the relationship between flow rate

and the steady state height of water accumulation inside the control volume is an important

consideration when conducting this experiment.

To assess the validity of this assumption, an experiment will be set up in which the flow

rate of water leaving a control volume will be measured using a gravimetric hydraulic bench

under the principals of conservation of volumetric flow. The amount of water in the control

volume will be represented by the steady state height of the water in the control volume. Using

lengths of moment arms on the hydraulic bench, moment balance will be applied to obtain the

mass of the fluid. This information, along with the density of the fluid (water) will be sufficient

for obtaining the flow rate of the fluid.

The extent of this experiment will be limited to 8 different steady state heights and a mass

of water entering the gravimetric hydraulic bench equal to 6 kg. Other limitations include the

assumption that the density of water is uniform, random quantum effects, and unforeseen acts of

God.

Equipment List

Materials:

Gravimetric Hydraulic Bench Model Model # H1D, Serial # S5014/21

Metal load

Stopwatch

Ruler

Figure 11: Gravimetric Hydraulic Bench

Pivot

Load arm

Load

Control Volume

Water tank

Water stopper

Water LevelPump

Valve

Load arm (Water side)Load arm

(Load side)

Direction of flow

Figure 2: Control Volume part of Gravimetric Hydraulic Bench

Procedure1. Record the ratio of the arm lengths to the pivot in the Observations section of the lab.

2. Turn on the gravimetric hydraulic bench.

3. Adjust the flow rate with the control knob until the control volume reaches a steady

state.

4. Record the height of water in the control volume in the Observations section.

5. After the load arm lifts up, place the 2 kg weight on the load hanger and begin the

stopwatch timer.

6. After the load arm again returns to the upward position, stop the stopwatch.

7. Record the time on the stopwatch in the Observations section.

8. Remove the 2 kg weight.

9. Adjust the flow rate to a higher amount with the control knob.

10. Repeat steps 4 – 8 for a total of 8 steady state heights.

ResultsObservations:

Table A: Raw data collected during procedure

Trial # Steady State Height (mm) t (s)1 230 33.092 180 35.823 215 33.774 315 27.975 339 27.066 250 31.507 135 42.978 302 28.76

Table A presents the steady state heights as well as time to equilibrium for eight trials.

rarm = 3 : 1 = 3

Definitions:

mwater = mass of the water in the water tank (kg)

mload = mass of the load on the load arm (kg)

rarm = ratio of arm lengths from the pivot (m)

Vwater = volume of water in the water tank (m3)

t = elapsed time for the water to equilibrate (s)

Q = flow rate of the water in the system (m3/s)

Steady state height = the height in the control

volume when Qin = Qout. When flow rate is

changed, the steady state adjusts to reach these

conditions automatically

Formulas:

mwater =mload*rarm Vwater = mwater/ρwater Q = Vwater/t

Table B: Calculated data

Trial # Steady State Height (mm)

mwater (kg) Vwater (m3) Q (m3/s)

1 230 6 0.006 1.81 E-4

2 180 6 0.006 1.68 E-4

3 215 6 0.006 1.78 E-4

4 315 6 0.006 2.15 E-4

5 339 6 0.006 2.22 E-4

6 250 6 0.006 1.90 E-4

7 135 6 0.006 1.40 E-4

8 302 6 0.006 2.09 E-4

Table B presents volumetric flow rate calculated from recorded dependent variables such as mass and volume of water for the eight trials.

Discussion:

Figure 3: Plot of flow rate vs the square root of steady state height

Figure 3 shows the relationship between the flow rate and steady state height. The

expected linear trend was not seen. The resulting data is largely inconclusive. This is perhaps due

to the narrow scope of the experiment, which includes only 8 data points. Also, imprecise

observations and readings probably heavily impacted data acquisition.

Questions:

1. Question: Why do we multiply the mass of the load by the ratio of the arm lengths to

get the mass of the water? Derive and show your work.

Answer: This is to calculate the mass of the load. In order for the sum of the moments about

the device’s pivot to be zero, the product of the mass of the water bucket and its

perpendicular distance to the pivot must be equal to the product of the mass of the loads

and three times the perpendicular distance between the bucket and pivot. Stated

differently;

∑MA = 0; (Mass of Bucket)(“L”) - (Mass of Load)(“3L”) = 0

2. Question: In step 6b, why do we wait for the load arm to begin to lift up before we put

the load on?

Answer: This is to ensure that the load reaches equilibrium with the water weight.

3. Question: Why do we call the equipment used the “gravimetric water bench”?

Answer: The driving force for this experiment is gravity. The product of acceleration due to

gravity and the masses provide the weights that hover in balance in this bench.

4. Question: Discuss the sources of experimental error in this experiment.

Answer: Error was introduced into the experiment by shortcomings of those conducting the

experiment. Notable examples are lack in perfect ability to ascertain steady state height, add

on weights at the instant required, and stopping the timer the precise instant the load arm

began to lift up. Besides this “human error”, there may have been further error introduced

from any possible leaks or other equipment issues.

Conclusion

In this experiment flow rate of water was calculated using a Gravimetric Hydraulic

bench. The times to equilibrate as well as steady state heights were observed for a total of eight

trials. The measured data was taken and used to compute volumetric flow rate, which was plotted

against the square root of steady state height. Surprisingly, this plot did not demonstrate a linear

correlation. Future experiments should be designed to take a wider array of data and address the

lack of accuracy that comes with human data recording.