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Experiment #1 Density of Liquids Stephen Mirdo Performed on September 20, 2010 Report due October 4, 2010

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Page 1: Density of Liquids

Experiment #1

Density of Liquids

Stephen Mirdo

Performed on September 20, 2010

Report due October 4, 2010

Page 2: Density of Liquids

Table of Contents Object ………………………………………..………………………….………….…. p. 1 Theory …………………………………………………………………………..…pp. 1 - 2 Procedure ……………………………………………………………………..……..... p. 3 Results …................................................................................................................ pp. 4 - 6 Discussion and Conclusion …………………………………………………….....…... p. 7 Appendix ………………………………………………………..……...…..…... pp. 8 - 10

Page 3: Density of Liquids

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Object

The object of this experiment was to determine the density of fluids using the buoyant force principle and the maximum flexural stress and strain in a cantilever beam.

Theory

The density of a fluid is defined as its mass per unit volume and is a function of pressure and temperature. If a fluid undergoes thermal expansion, its molecules will be distributed more widely and will therefore be less dense. There will be fewer molecules, thus less mass, per a given unit of volume. Likewise, if a fluid experiences thermal contraction due to cooling, the kinetic energy of the molecules in the fluid will be a lesser value and they will be packed closer together. In this case, the fluid will have a higher mass per unit volume, therefore a higher density than the former case.

ρ = m/V (Equation 1)

Pressure likewise affects the density of a fluid. Consider a fluid with a free surface exposed to the atmosphere. If the pressure on the surface of the fluid were to increase, the density of the fluid would increase due to being compressed. The converse of this scenario is also true: a decrease in the pressure on the free surface of the fluid would decrease its density1. However, the pressure acting on the free surface of a fluid is beyond the scope of this experiment and will not be addressed.

A method for determining a fluid’s density is to assess the buoyant force the fluid exerts on an object and divide this value by the volume of fluid displaced by the object multiplied by gravitational acceleration. Rearranged algebraically, the buoyant force may be obtained by calculating the product of the density, volume and gravitational acceleration.

ρ = FB/gV FB = ρgV (Equation 2)

For this experiment, the flexural stress and strain induced by a buoyant force on a cantilever beam will be used to determine the density of a fluid. By Hooke’s Law, the stress acting in the beam is equal to the beam material’s modulus of elasticity, E, multiplied by the induced strain, ε, as seen in Equation 3 below. Because the stress is a flexural stress, meaning the material is bending due to the stress, the flexural stress equation must be used (Equation 4). The components of the flexural stress equation are as follows: M is the moment induced by a force, c is the distance from the neutral axis of the beam to the outer edge of its cross-section, and I is the moment of inertia of the cross-section of the beam.

σ = Eε (Equation 3)

σflexural = Mc / I (Equation 4)

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Substitute the flexural stress equation into Equation 3 and manipulate to solve for the strain, ε. The strain is the quantity to be measured in this experiment as seen below in Figure 1. The equation for strain is given as:

ε = σflexural / E ε = (Mc / I) / E (Equation 5) where:

M = FBL c = t/2 I = (1/12)bt3

Figure 1: Schematic of density experiment apparatus with forces and moment.

The moment M in Equation 5 is the product of the force F exerted on the beam, which is equivalent to the buoyant force, and the length L of the beam. By substituting the buoyant force in Equation 2 into Equation 5, the density variable ρ is inserted into the equation. The volume to be accounted for in this new equation is that of a sphere.

M = ρg(4/3)πR3*L (Equation 6)

Substituting Equation 6 into Equation 5 yields a more specific solution for the strain.

ε = (6ρg(4/3)πR3L) / (Ebt2) (Equation 7)

Because the strain is the quantity being measured in this experiment, the only

unknown value in Equation 7 is that of the density of the fluid. Therefore, Equation 7 can be algebraically manipulated to yield the equation to solve for the density of the fluid as seen in Equation 8.

ρ = (Ebt2ε) / (6g(4/3)πR3L) (Equation 8)

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Procedure

Equipment: Strain Indicator Device Cantilever Beam Apparatus 4 Large Beakers filled with:

Water Glycerine Used Motor Oil Olive Oil

Experiment:

1) Turn on the strain indicator device. Set the gage factor to 2.5. Calibrate the device so that the strain display reads zero.

2) Select one of the four beakers filled with fluid and place it near the strain

indicator. Place the apparatus in the fluid so that the sphere is completely submerged as seen in Figure 1.

Note: It is important that the sphere does not touch the sides or bottom of the beaker, as this will interfere with accurate strain readings.

3) When the strain reading on the indicator no longer fluctuates, record the strain

value from the indicator. This value will be in με (x 10-6) and is unitless. 4) Remove the apparatus from the fluid and dry off the sphere. Recalibrate the strain

indicator device if necessary. 5) Repeat Steps 2 through 4 four times for each fluid and record the strain values. 6) Clean and dry the equipment as required.

Table 1: Known values to be used in Equation 8 for calculating density D = Diameter of Sphere 3.81 cm L = Length of Beam 18.03 cm b = Width of Beam 2.19 cm t = Thickness of Beam 0.0813 cm

E = Modulus of Elasticity of Beam Material 75.5 GPa

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Results

Table 2: Measured Strain and Calculated Density for Water Fluid: Water

Trial # με (x 10-6) ρ (kg/m3) 1 307 1092 2 313 1113 3 306 1088 4 310 1102

ρavg = 1099 kg/m3

Table 3: Measured Strain and Calculated Density for Glycerine Fluid: Glycerine

Trial # με (x 10-6) ρ (kg/m3) 1 381 1355 2 384 1366 3 386 1373 4 387 1376

ρavg = 1367 kg/m3

Table 4: Measured Strain and Calculated Density for Used Motor Oil Fluid: Used Motor Oil

Trial # με (x 10-6) ρ (kg/m3) 1 267 949 2 267 949 3 271 964 4 270 960

ρavg = 956 kg/m3

Table 5: Measured Strain and Calculated Density for Olive Oil Fluid: Olive Oil

Trial # με (x 10-6) ρ (kg/m3) 1 281 999 2 288 1024 3 287 1021 4 287 1021

ρavg = 1016 kg/m3

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The densities presented in Tables 2 through 5 were calculated using Equation 8. It was noted that the calculated average density for each fluid tested, excluding used motor oil, was not in complete agreement with an accepted value. The used motor oil has no accepted density as it is impossible to account for the composition of varying samples. For the purposes of this report, the value included in the materials laboratory manual will be used as it cannot be verified. All other accepted densities are from a verified source2. The accepted densities are as follows:

- Water: 1000 kg/m3 - Glycerine: 1259 kg/m3 - Used Motor Oil: 800 kg/m3 - Olive Oil: 920 kg/m3 To determine how far from the accepted values the experimental densities of the four tested fluids are, a percent difference analysis was performed.

Table 6: Percent difference of accepted and experimental densities

Material Accepted Density (kg/m3)

Experimental Density (kg/m3)

% Difference

Water 1000 1099 9.90% Glycerine 1259 1367 8.58% Used Motor Oil 800 956 19.50% Olive Oil 920 1016 10.43%

To determine if the results of the experiment were statistically significant, a Student’s t-Test was performed. Table 7: t-Test for all possible combinations of test fluid

t-Test Water Glycerine t-Test Water Used Motor Oil

Mean 1099 1367 Mean 1099 956 σ 11.2456 9.4087 σ 11.2456 7.3312

df = 6 df = 6

t stat = 0.251 t stat = 0.583 t crit = 2.447 t crit = 2.447

t-Test Water Olive Oil t-Test Glycerine Used Motor Oil

Mean 1099 1016 Mean 1367 956 σ 11.2456 11.3853 σ 9.4087 7.3312

df = 6 df = 6

t stat = 0.017 t stat = 0.348 t crit = 2.447 t crit = 2.447

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Table 7: (Continued)

t-Test Glycerine Olive Oil t-Test Used Motor Oil Olive Oil

Mean 1367 1016 Mean 956 1016 σ 9.4087 11.3853 σ 7.3312 11.3853

df = 6 df = 6

t stat = 0.268 t stat = 0.599 t crit = 2.447 t crit = 2.447

For each pair of fluids, the calculated densities are accepted by the Student’s t-Test, as for each pair, tstat < tcrit.

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Discussion & Conclusion

The buoyant force exerted by a fluid on a submerged object is a function of the fluid’s density. This concept is clearly outlined by Equation 2, where the buoyant force is the product of the density, displaced volume and gravitational acceleration. For a constant displacement of volume, the buoyant force will increase proportionally with the density of a fluid. This also implies that an increase in volume while density is held constant will yield an increase in the buoyant force.

However, for this experiment, the calculation of the buoyant force acting on the

submerged sphere neglects the force of weight exerted by the testing apparatus acting parallel and opposite. This weight acting opposite and parallel will cause for a decrease in the apparent buoyant force exerted on the sphere. This, perhaps, explains why the experimental values of the density are greater than the accepted values for each respective fluid. As seen in Equation 8, the components of the buoyant force reside in the denominator of the density expression. If the weight of the apparatus were to negate a portion of the buoyant force, a lesser denominator would be produced. Therefore, a higher calculated density would be yielded due to the error in the buoyant force components. Other sources of error in this experiment also interfered with calculating an experimental density that was in agreement with each respective accepted density. One such error was a possible contamination of the fluid samples used in the lab. From a visual inspection, it was clear that the samples had been cross contaminated from prior use. However, this contamination would most likely not interfere with the calculated values to a great extent.

Another source of error, which noticeably hampered density calculations, was the inability to remove all the mass of the fluid from the sphere between trials. Upon removing the testing apparatus from the fluid and drying it as thoroughly as possible, it was noted that the strain indicator displayed a non-zero value. This indicates that any residual mass of fluid on the apparatus will interfere with the next strain reading. The strain indicator device was recalibrated between trials to counteract this discrepancy. However, as indicated by Tables 2 through 5, for three of the four fluids tested, the indicated strain increased with each successive trial. Improvements for this experiment are in order. One such improvement would be to assess the cantilever beam material’s modulus of elasticity prior conducting the experiment. Although this value was provided with the material, not all materials have the exact values prescribed by the manufacturing process. Another improvement that would help target accepted density values for the fluids would be to take more data points for each sample. To get an accurate representation, more data points subjected to Chauvenent’s rule would yield more accurate results. Although the results are slightly skewed from accepted values, statistically speaking, the calculated densities are acceptable.

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Appendix

Data Usage Sample Calculation for Density of Olive Oil using Equation 8

ρ = (75.5e9 Pa * 0.0219m * (0.000813m)2 * 281με) = 999 kg/m3 (6 * 9.81m/s * (4/3) * π *(0.0381m/2)3 * 0.1803m)

Sample Calculation for the Average Density of Glycerine

[(1355 + 1366 + 1373 + 1376) kg/m3] / 4 = 1367 kg/m3 Sample Calculation for % Difference: Accepted and Experimental Density of Water

| 1000 kg/m3 – 1099 kg/m3 | / 1000 kg/m3 * 100 = 9.90%

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Bibliography

1. Introduction to Fluid Mechanics, 3rd Edition William S. Janna (1993)

2. The Engineering Toolbox: Liquids - Densities

http://www.engineeringtoolbox.com/liquids-densities-d_743.html

3. Materials Laboratory Manual, Fall 2010 University of Memphis, Department of Mechanical Engineering