Density Lab v3c

8/6/2019 Density Lab v3c

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Chapter 3

Uncertainty Analysis and Density

Measurement

3.1 Measures of Mass

When dealing with fluids and fluid flow the absolute measure of mass (kg, slugs, or lbm) can beinconvenient. There are a variety of indirect methods for describing property of mass in thermo-dynamics and fluid mechanics. These indirect measures of mass include specific gravity, specificvolume, specific weight, and density.

Specific Volume: The specific volume of a fluid is the volume per unit mass and is typicallydenoted as v:

v =volume

mass L3

M

Specific volume is commonly expressed as m3/kg in the SI system.

Density: The density of a fluid is the mass per unit volume and is typically denoted as :

=mass

volume M

L3

Density is equal to the inverse of the specific volume: = 1v and is commonly expressed inunits of slug/ft3 in the British Gravitational System and in lbm/ft3 in the English EngineeringSystem. In the SI system density is expressed as kg/m3 and as g/cm3 in the CGS system.Density of liquids is relatively insensitive to pressure; water increases by only 1% for a 220-fold increase in pressure. Thus, most liquids are treated as incompressible in engineeringanalysis.

Specific Weight: The specific weight of a fluid is the weight per unit volume and is typically

denoted as :=

weight

volume F

L3 M

L2t2

Using the definitions of Force and Density, the Specific Weight can be expressed as = g.

Specific Gravity: The specific gravity is the ratio of the fluid density to a standard reference ;water for liquids and air for gases. Specific Gravity is dimensionless:

SG =

ref

11


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12 CHAPTER 3. UNCERTAINTY ANALYSIS AND DENSITY MEASUREMENT

For liquids and solids, distilled water at 4 C and 101.325 kPa is used as the reference density;ref = 998 kg/m

3 = 1.936 slugs/ft3.

Additional information on these properties can be found in Cengel and Boles [2], Munson et al. [3],and Turns [4] as well as any other engineering thermodynamics or fluid mechanics textbook.

3.2 Uncertainty AnalysisUncertainty analysis is a method for estimating the total uncertainty in experimental result basedon the uncertainties associated with each of the measured parameters. For example, in this lab thedensity of an unknown fluid will be determined by measuring the mass and volume of the fluid.The uncertainty in the density will depend on the combined measured uncertainty of the mass andthe volume. Uncertainty analysis is thoroughly discussed in the assigned readings [5, sections 3.2 -3.5].

3.3 Objectives

The objectives of this lab are to:

To determine the density of an unknown fluid by measuring mass and volume.

To use uncertainty analysis to determine the amount of error in the derived values.

3.4 Experiment

The experiment uses three pieces of equipment.

CP-1500 Model Balance capability of measuring 1500 grams

1 Beaker filled with unknown fluid

1 Graduated Cylinder

3.4.1 Procedures

1. Familiarize yourself with the various parts of the set-up, controls and instrumentation. TheT Function located on the bottom right hand corner of the balance controls can be usedto tare, or zero, the mass reading.

2. Note the uncertainties associated with the balance and the graduated cylinder.

3. Turn on the balance and let the balance warm up for 5 minutes in order to stabilize theelectronic circuitry.

4. Zero the reading on the balance.

5. Measure the mass of the graduated cylinder and record the value.

6. With the graduated cylinder still in place, zero the electronic balance again.

7. Carefully pour fluid from the beaker into the graduated cylinder until it is nearly full, butnot filled above the highest measure.

8. Record the fluid volume and measured mass.

9. Carefully pour approximately 1 ml of fluid from graduated cylinder back into the beaker.

10. Repeat steps 89 until the graduated cylinder is empty.

11. repeat steps 710 until data table is full.


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3.4. EXPERIMENT 13

3.4.2 Data Analysis

1. Plot mass versus volume in EES and perform a linear regression on the data by hand and inEES. Determine the correlation coefficient for your data.

2. Compare the slope of the line from the linear regression to the mean value of the calculateddensity. Discuss the relationship between these two values.

3. Derive the expression for the uncertainty in the density measurement. What is the uncertaintyof the measured density? Use any one of your data points for the calculation.

4. Perform an uncertainty analysis in EES with your data. How does this calculation compareto the calculation in 3?

5. Include all hand calculations and data sheets with your lab report.


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14 CHAPTER 3. UNCERTAINTY ANALYSIS AND DENSITY MEASUREMENT

3.5 Data Table Name:

Date:

Lab Partners:Bias Error for Balance:

Bias Error for Grad. Cylinder:

Mass of Graduated Cylinder:

Mass Volume Density Specific Weight SpecificTrial (g) (ml) (kg/m3) (kg/m2s2) Gravity

1

2

3

4

56

7

8

9

10

11

12

13

14

15

Density

Mean

Median

Mode

Standard Deviation


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Bibliography

[1] F-Chart Software. EES Manual. www.fchart.com, v7.663 edition, 2006.

[2] Yunus A. Cengel and Michael A. Boles. Thermodynamics: An Engineering Approach. McGraw-Hill, 5th edition, 2006. ISBN 0-07-288495-9.

[3] Bruce R. Munson, Donald F. Young, and Theodore H. Okiishi. Fundamentals of Fluid Me-chanics. John Wiley & Sons, Inc., 5th edition, 2006. ISBN 0-471-67582-2.

[4] Stephen R. Turns. Thermodynamics: Concepts and Applications. Cambridge University Press,40 West 20th Street, New York, NY 10011-4211, USA, 2006. ISBN 0-521-85042-8.

[5] J.P. Holman. Experimental Methods for Engineers. McGraw-Hill, 7th edition, 2001. ISBN0-07-366055-9.

[6] Duane Abata. ME223 Thermodynamics Laboratory Manual. Mechanical Engineering - En-gineering Mechanics Department, Michigan Technological University, Houghton, Michigan49931, 2nd edition, 1983.

[7] Frank M. White. Fluid Mechanics. McGraw-Hill, 2003. ISBN 0-07-283180-4.

[8] Michael J. Moran, Howard N. Shapiro, Bruce R. Munson, and David P. DeWitt. Introduction toThermal Systems Engineering: Thermodynamics, Fluid Mechanics, and Heat Transfer. JohnWiley & Sons, Inc., 2003. ISBN 0-471-20490-0.

[9] N.V. Suryanarayana and Oner Arici. Design and Simulation of Thermal Systems. McGraw-Hill, Inc., 2003. ISBN 0-07-249798-X.

[10] Faye C. McQuiston and Jerald D. Parker. Heating, Ventilating, and Air Conditioning. JohnWiley & Sons, Inc., 2nd edition, 1982. ISBN 0-471-08259-7.

[11] Michael J. Moran and Howard N. Shapiro. Fundamentals of Engineering Thermodynamics.John Wiley & Sons, Inc., 5th edition, 2004. ISBN 0-471-27471-2.

[12] J.A. Goff. Standardization of thermodynamic properties of moist air. Transactions ASHVE,55, 1949.

[13] Richard E. Sonntag, Claus Borgnakke, and Gordon J. Van Wylen. Fundamentals of Thermo-dynamics. John Wiley & Sons, Inc., 5th edition, 1998. ISBN 0-471-18361-X.

[14] Jim Lally and Dan Cummiskey. Dynamic Pressure Calibration. TN-15-0205, PCB Piezotronics,Inc., Depew, NY 14043 USA, 2005.

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Density Lab v3c

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