AEE 3064 University of the District of Columbia Density Fluid Mechanics Lab Report

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Fluid Mechanics Lab, Experiment #1 1 EXPERIMENT #1 Properties of Fluids Background: The term "fluid" relates to both gases and liquids (e.g. air and water) and, although there are differences between them, they both have the same essential property that when acted upon by any unbalanced external force an infinite change of shape will occur if the force acts for a long enough time. Alternatively, one may say that if acted on by a force, a fluid will move continuously while a solid will distort only a fixed amount. If a shear force is applied to one surface of a volume of fluid, the layers of fluid will move over one another thus producing a velocity gradient in the fluid. For a given shear stress, a property called the viscosity determines the velocity gradient and hence the velocity of the fluid in the plane of the applied stress. The viscosity is a measure of the fluid's resistance to motion. Viscosity if a very important property in fluid mechanics since it determines the behavior of fluids whenever they move relative to solid surfaces. Liquids and gases both share the property of "fluidity" described above, but they differ in other respects. A quantity of liquid has a definite volume and if in contact with a gas it has a definite boundary or "free surface." Gases, on the other hand, expand to fill the space available and cannot be considered as having a definite volume unless constrained on all sides by fixed boundaries (e.g. a totally enclosed vessel). The volume of a liquid changes slightly with pressure and temperature, but for a gas these changes can be very large. For most engineering purposes liquids can be regarded as incompressible, meaning volume and density do not change significantly with pressure, whereas gases usually have to be treated as compressible. Similarly, the effects of varying temperature can often be ignored for liquids (except in certain special cases), but must be taken into account with gases. The engineer is often concerned with determining the forces produced by static or moving fluids and when doing this the above differences between liquids and gases can be very important. Generally it is much easier to deal with liquids because, for most purposes, it can be assumed that their volume and density do not change with pressure and temperature. In the study of hydrostatics we are primarily concerned with the forces due to static liquids. The forces result from the pressure acting in the liquid and at a given point this depends on the depth below the free surface. Density, or mass per unit volume, is a basic property which must be known before any calculations of forces can be made. When considering the interfaces between liquids, solids, and gases, there is a further property which can produce forces and this is called the surface tension. When a liquid/gas interface is in contact with a solid boundary, the edge of the liquid will be distorted upward or downward depending on whether the solid attracts or repels the liquid. If the liquid is attracted to, or "wets" the solid, it will move upward at the edge and the surface tension will cause a small upward force on the body of the liquid. If the liquid is in a tube the force will act all around the periphery and the liquid may be drawn up the tube by a small amount. This is sometimes called the capillarity effect or capilliary action. The forces involved are small and the effect need only be considered in a limited number of special cases. PART I - DENSITY Statement of Work: To determine the density of a liquid it is necessary to measure the mass of a known volume of liquid. = mass(g) 106 x (kg/ m3) volume(ml) 103 (1) The density of pure water at 20oC is 998.2 kg/m3 and this is sometimes rounded up to 1000 kg/m3 for engineering purposes. The experimental result should be within 1% of this value. The measurement of volume is not very precise and depends on the accuracy of the graduations on the beaker and this cannot be checked. Density Bottle: The problem of accurately measuring a volume of liquid can be overcome by using a ©Department of Aerospace, Physics and Space Sciences Florida Institute of Technology 2 Fluid Mechanics Lab, Experiment #1 special vessel with a known volume such as a density bottle. This is accurately made and has a glass stopper with a hole in it through which excess liquid is expelled. When the liquid is level with the top of the stopper, the volume of liquid is 25 cm3 (ml). Procedure for determination of density • Dry and weigh the bottle and stopper. • Fill the bottle with liquid and replace the stopper. • Carefully dry the outside of the bottle with a cloth or tissue paper and remove any excess liquid from the stopper such that the liquid in the hole is level with the top of the stopper. • Re-weigh the bottle plus liquid and determine the mass of liquid and hence the density. • Calculate the Specific Gravity, SGT , of the liquid to be used for later comparison. This method should give an accurate result and is limited more by the accuracy of the balance than by the volume of liquid. Specific Gravity: Specific gravity, or relative density as it is sometimes called, is the ratio of the density of a fluid to the density of water. Typical values are 0.8 for paraffin; 1.6 for carbon tetrachloride; and 13.6 for mercury. Specific gravity should not be confused with density even though in some units (e.g. the cm/gram/sec. system) it has the same numerical values. SGT = density of fluid at a given temperature (T ) density of reference fluid (water for liquids and air for gases)at 60 o F ( 15.6 o C ) (2) where T is in degrees Fahrenheit. Specific gravity can be determined directly from the density of a liquid as measured, for example, by using a density bottle. The value is simply divided by the density of water to obtain the specific gravity. The density of water under Standard Temperature and Pressure conditions is considered to be 998kg / m 3 . A convenient alternative method is to use a specially calibrated instrument called a hydrometer. This takes the form of a hollow glass float which is weighted to float upright in liquids of various densities. The depth to which the stem links in the liquid is a measure of the density of the liquid and a scale is provided which is calibrated to read specific gravity. The sensitivity of the hydrometer depends on the diameter of the stem. A very sensitive hydrometer would have a large bulb and a thin stem (see Figure 1). NOTE: Density changes with temperature; therefore, measure the temperature of the liquid. PRECAUTION: The temperature of the hydrometer is the same as the temperature of the liquid. ©Department of Aerospace, Physics and Space Sciences Florida Institute of Technology Fluid Mechanics Lab, Experiment #1 3 Figure 1. Types of hydrometer Baume scale hydrometers (nonlinear) are used with liquids heavier than water, and American Petroleum Indrustries (API) hydrometers feature a linear scale for improved accuracy. Degrees Baume = 145 - 145 − Degrees API = 145 SG60o F / 60 F 141.5 (SG60 / 60 F) - 131.5 SG 60 = SG t + T - 60 3600 (3) (4) (5) where T is in degrees Fahrenheit. Procedure for determination of SG • Place one of the tall glass cylinders on the measuring surface, fill with a liquid, and allow air to rise to ©Department of Aerospace, Physics and Space Sciences Florida Institute of Technology 4 Fluid Mechanics Lab, Experiment #1 the top. • Carefully insert the hydrometer and allow it to settle in the center of the cylinder. • Take care not to let it touch the sides, otherwise surface tension effects may cause errors. • When the hydrometer has settled, read the scale at the level of the free water surface (i.e. at the bottom of the meniscus; see Figure 1 insert). • T, is the temperature of the liquid being tested. The reading from the Universal Hydrometer is the SGt . First calculate the SG60 in equation (5), then calculate the Degrees API in equation (4). Compare the result from equation (4) calculated against cited sources found in literature. • Compare the SGt value from the calculations done using the density bottles to the current SGt values found with the universal hydrometer. PART II – VISCOSITY, (T,P) The most familiar property of a fluid is the viscosity. The viscosity is responsible for drag on aerospace vehicles, friction in pipes, and the destabilization/stabilization of some laminar flows. Viscosity is a fluid property subject to changes in temperature and pressure. The viscosity of a liquid decreases with increasing temperature while the viscosity of a gas increases with increasing temperature. It is sufficient to recognize that viscous effects originate at the molecular level. In the presence of a velocity gradient normal to the mean flow, momentum exchange between adjacent fluid lamina result in a net decrease of momentum - fluid friction. This phenomenon occurs at the microscopic level and should not be confused with momentum transfer in turbulence, which occurs at the macroscopic level. Popular science characterizes the viscosity of a fluid by the degree of "thickness" or "resistance" to flow. Water drains out of a sink much faster than an equivalent volume of molasses. In fact, this observation has been exploited by the petroleum industry as a means to quantify viscosity. For example, SAE 30 motor oil means that it took 30 seconds for a given quantity of the oil at a specified temperature to drain from a container (the container in this case is more appropriately a "Saybolt viscometer"). However, this is an oversimplification as viscosity does not have the units of seconds. Viscosity is better defined as that property which relates an applied strain rate to the resulting shear stress and vice versa. It is usually beyond the scope of an undergraduate fluid mechanics course to develop a general relation between stress and strain since this requires the use of tensor analysis. However, students should recognize that a linear relation between stress and strain is a special case in fluid mechanics. In general, the relation between stress and strain in a fluid is written as  = f ( ) (6) where  is the shear stress and  is the strain rate. There is not an a priori reason to suppose a linear relationship. In fact, viscosity may be a function of strain rate itself. Effects of temperature Gases: Viscosity increases with increasing temperature as seen in Sutherland's formula: ©Department of Aerospace, Physics and Space Sciences Florida Institute of Technology Fluid Mechanics Lab, Experiment #1 = b T 1+ S T b = 1.458 x 10-6 5 (7) kg ms K (8) where µ is the coefficient of absolute viscosity, T is in K(elvin), and S = 110.4K. Liquids: Viscosity decreases with increasing temperature  = Ae B T For water, A = -1.94 B = -4.80 [F.M. White, 1994] (9) A Newtonian fluid is defined as a substance in which the shear stress in linearly proportional to the strain rate. Consider the flow between parallel plates as shown in Figure 2. A parcel of fluid is strained by the moving upper plate; hence, shear stresses are produced. For a Newtonian fluid, the relation between shear and stress reduces to du du where dy is twice the strain rate, 2  . (10) = dy du Note that this special geometry results in a linear velocity distribution; hence, dy is constant. The deformation of a fluid parcel is represented by element A which, as it moves to the right, deforms to element B in Figure 2. This deformation is associated with the strain rate. Figure 2. Strain in a fluid due to a moving boundary. Non-Newtonian fluids may also be characterized by their stress-strain behavior. Table 1 contains a description of various rheological classifications, and Figure 3 contains a comparative plot of stress versus strain rate for a variety of rheological classifications. ©Department of Aerospace, Physics and Space Sciences Florida Institute of Technology 6 Fluid Mechanics Lab, Experiment #1 RHEOLOGICAL CLASSIFICATION Classification Characteristics Newtonian: Stress is linearly proportional to strain. Bingham Plastic: Yield—Newtonian; stress is linearly proportional to strain after initial applied yield strain. Dilatant: Stress increases with increasing strain rate. Pseudo Plastic: Stress decreases with increasing strain rate. Rheopectic*: Stress increases with time – constant strain rate. Thixotropic*: Stress decreases with time – constant strain rate. *not shown in plot Table 1. Rheological Classification Figure 3. Stress-strain behavior for various fluids. Given an unknown fluid, the above classifications could be determined by imposing a strain rate and qualitatively sensing the resistance (stress). Slowly pouring the fluid out of a cup would simulate a relatively low strain rate while vigorous stirring would correspond to a high strain rate. For example, if the resistance (shear stress) to stirring "appeared" to decrease with more vigorous stirring (strain rate), the fluid would be classified a pseudoplastic. The phenomenon can be characterized by an apparent viscosity,  , which is defined as =  du dy ©Department of Aerospace, Physics and Space Sciences Florida Institute of Technology (11) Fluid Mechanics Lab, Experiment #1 7 Figure 4. Definition of Apparent Viscosity. The definition is applied as indicated in Figure 4. Measurement of viscosity of liquids Rotational viscometer: The absolute viscosity of liquids can be determined by using a rotational viscometer or a falling ball viscometer (see Figure 5). ©Department of Aerospace, Physics and Space Sciences Florida Institute of Technology 8 Fluid Mechanics Lab, Experiment #1 Figure 5. Rotational Viscometer The apparatus is designed such that the velocity profile between the cylinder and wall is linear; therefore, the strain rate is constant for a given rotation speed of the inner cylinder. If the cylinder is driven by a falling weight as shown, the applied stress to the fluid will be constant (and known). Further, the strain rate, which is proportional to the rotation rate,  , will be known. Hence, by applying different weights, the apparent viscosity for each strain rate can be determined. For this circular Couette flow, we can determine the viscosity from the following relation: 2 mg( Ro - Ri ) R  2  L  U = r where  r m Ro Ri L U = = = = = = = 3 i (12) dynamic viscosity radius of the cylinder pulley mass which produces a steady velocity U outer radius of cylinder inner radius of cylinder length of the cylinder Terminal velocity of fall of the mass m Procedure • Fill the gap between the stationary and rotating cylinders with the fluid whose viscosity is to be determined. Be sure the rotating cylinder is completely immersed in the fluid. • Apply different weights to the weight hanger and, in each case, measure the velocity of the fall (terminal velocity) by timing the fall through a known height. ©Department of Aerospace, Physics and Space Sciences Florida Institute of Technology Fluid Mechanics Lab, Experiment #1 9 • In order to calculate the terminal velocity of the dropped mass compare the time it takes the mass to travel a pre-determined distance offset from the release point. Repeat this process for the same mass and 2-3 different distances so as to find the suitable constant / terminal velocity of the mass, which shall be used in the calculation of the dynamic viscosity. • Note the dimensions of the cylinders and pulley, etc. • Repeat the procedure for other fluids as necessary. Falling Ball Viscometer: Thermo Fisher Scientific introduced another technique to perform and understand the fluid viscosity. The equipment is called HAAKE Falling Ball Viscometer. It is used to measure the viscosity of transparent Newtonian liquids. This viscosity is correlated to the time a ball requires to fall a defined distance. The rolling and sliding movement of the ball through the sample filled into a slightly inclined cylindrical measuring tube is described by means of the fall time. The test results are given as the dynamic viscosity using the internationally standardized absolute unit of “milli Pascal seconds” (mPa∙s). The heart of the instrument is the measuring tube made of glass ① and a ball ②. The tube has two ring marks A and B, which are spaced 100 mm apart and which limit the measuring distance. The ring mark C is in the middle between A and B. The measuring tube is placed inside of an outer glass tube ③, which encloses a room to be filled with a temperature-controlled liquid. The measuring tube is fastened to the stand in such a way that its axis is inclined with respect to the vertical by 10° during the measurement. The measuring tube together with the outer-shell tube may be pivoted in order to turn upside down to let the ball return to the initial position. The measuring tube is closed on both ends by two stoppers, one of which ⑬ contains a capillary and a small reservoir. This stopper prevents undesirable changes of pressure in the liquid sample and has a passage for air bubbles when the temperature is being changed. The viscometer encloses all samples completely to prevent volatilization and film forming. The stand may be leveled by means of its water level [a] and the levelling screws [b]. As standard, the Falling Ball Viscometer is supplied with a thermometer in the range of -10℃ to +26℃. Figure 6. HAAKE Falling Ball Viscometer Apparatus – part 1 ©Department of Aerospace, Physics and Space Sciences Florida Institute of Technology 10 Fluid Mechanics Lab, Experiment #1 Figure 6. HAAKE Falling Ball Viscometer Apparatus – part 2 ©Department of Aerospace, Physics and Space Sciences Florida Institute of Technology 11 Fluid Mechanics Lab, Experiment #1 HAAKE Falling Ball Viscometer comes with a set of 6 different balls, which pass through the measuring tube of an inner diameter of approximately 15.94 ± 0.01 mm. The properties of each ball are given in the table below. This includes the pre-calibrated constant K value. The ball used for this experiment is to be provided by your GSA. (Ball Gauge is provided to differ between ball 1 and 2; calipers are required to ensure correct ball selection) Table 2. Given Balls Properties Data Sheet Figure 6. HAAKE Falling Ball Viscometer Ball Box & Extra Parts Calculation HAAKE Falling Ball Viscometer had pre-calibrated constant K values to simplify all the equipment properties, such as 10 degrees inclination, rolling, sliding, and Couette flow since the gap between the ball and the measuring tube’s wall is small. Hence the dynamic viscosity (mPa∙s) equation yields: 𝜇 = 𝐾 (𝜌1 − 𝜌2 ) ∙ Δ𝑡 where: K : 𝜌1 : 𝜌2 : Δt : (13) ball constant in 𝑚𝑃𝑎∙𝑠∙𝑐𝑚3/𝑔∙𝑠 density of the ball in 𝑔/𝑐𝑚3 density of the liquid to be measured at the measuring temperature in 𝑔/𝑐𝑚3 falling time of the ball in seconds ©Department of Aerospace, Physics and Space Sciences Florida Institute of Technology 12 Fluid Mechanics Lab, Experiment #1 Procedure Note: Safety Instruction and Detailed Equipment Setup & Maintenance are provided in the HAAKE Falling Ball Viscometer. It must be read and understand before the experiment. Contact your GSA/ Lab representative for more instruction. • Loading Fluid Sample: All parts of the viscometer being in direct contact with the sample must be kept clean and dry. 1. A sample volume of approximately 45 cm3 is poured into the measuring tube ① up to 20mm below the rim of the tube. 2. The ball ② is placed into the tube and the hollow stopper ⑬ is used to close the tube. * The liquid should reach a level just beyond the capillary ⑮ and the sample fluid in the tube must be free of air bubbles. Only then does the measurement begin. Figure 7. HAAKE Falling Ball Viscometer Apparatus – part 3 • Measuring of Falling Time: 1. The outer tube snaps into a defined 10° - position at the bottom of the instrument; by turning over the outer tube, the ball is set to the measuring position. 2. Use the stop-watch to measure the falling time of the ball. 3. Turn the outer tube 180 degrees again to set initial position for the ball and repeat as much trials as desired. * The time period starts when the lower periphery of the ball touches the ring mark A, which must appear as a straight line. The falling time ends when the lower periphery of the ball touches the ring mark B, which again must appear as a straight line. If one uses the distance A to C or C to B to reduce long falling times for high viscous liquids, the double of the measuring time period must be considered. ©Department of Aerospace, Physics and Space Sciences Florida Institute of Technology 13 Fluid Mechanics Lab, Experiment #1 FALLING SPHERE VISCOMETER Ambience Temperature: ________________ Run Ball # Constant K (𝑚𝑃𝑎 ∙ 𝑠 ∙ 𝑐𝑚3 /𝑔 ∙ 𝑠) Ball Density (g/𝑐𝑚3 ) Solution Density (g/𝑐𝑚3 ) Falling Time t (s) Viscosity  (𝑚𝑃𝑎 ∙ 𝑠) Table 3. Suggested Presentation of Experimental Data Suggested Presentations • Calculate the viscosity using Equations [13] • How does the different ball influence the measurement? • Compare the measured viscosity values with standard source values. • How do your values of viscosity compare between the two methods and with the accepted values for the given fluids found in your textbooks? (The instructor should be able to give the accepted viscosity values.) • Discuss the accuracy of the results and possible sources of errors. ©Department of Aerospace, Physics and Space Sciences Florida Institute of Technology 14 Fluid Mechanics Lab, Experiment #1 NOTES ©Department of Aerospace, Physics and Space Sciences Florida Institute of Technology Part 1 (Use equation1,2) Mass(g) Vegetable oil Motor oil (SAE 5W20) Mass1 23.599±0.05 21.899±0.05 Mass2 23.699±0.05 21.699±0.05 Mass3 23.719±0.05 21.699±0.05 Average Volume(ml) Mass(g) Vegetable oil 25 23.719 Motor oil 25 21.769 Part 2 (Compare SGt with Part 1) Vegetable oil SGt 1 SGt 2 SGt 3 SGt avg Motor oil 0.925±0.05 0.860±0.05 0.925±0.05 0.860±0.05 0.923±0.05 0.860±0.05 Density(kg/ m3 ) SGt Part3 degree Baume(use equation below) SGt = 140 130 + degree Baume Vegetable oil(°B) SGt 1 SGt 2 SGt 3 SGt avg SGt Motor oil(°B) 20±0.5 34±0.5 21±0.5 34±0.5 19±0.5 34±0.5 20±0.5 34±0.5 Instructor: Fall 2019 Lab #: AEE 3064: Fluid Mechanics Laboratory (email: Students: _/10 Formatting: /2 Title Page: Course Number and Title, Name of Student, Recipient of Report Number and Title of Experiment, Dates of Experiment and Submission /2 Table of contents Proper Section Titles, Correct Page Numbers Listed and pages numbered /3 No typos or grammatical errors, easily readable English /1 Standard or Specified Mathematical Symbols tabulated /2 All external references listed /3 Abstract: /1 Statement of Problem (What was investigated?) /1 Comprehensive summary of the experiment (How was the investigation Performed?) ./1 Major Results and Conclusions /5 Introduction: /3 Objectives Stated /1 Hypothesis Stated /1 Major Fluid Mechanics Principles Mentioned 15 Theory: _/5 Theory for Specific Experiment explained /2 Free Body Diagrams /2 Assumptions Stated /2 Experimental and Ambient Conditions listed /4 Equations and Sample Calculations presented (10 Facility and Apparatus: -/3 Specific Experimental Apparatus Depicted /2 Components Labeled /5 Description of Experimental Setup 10 Procedure: /8 All Vital Steps Listed /2 Proper Chronological Order /25 Results: /6 Presented calculated results and plots correct based on data /10 Data Analysis Discussion and Comparison to Theory /5 Relevant Plots presented and properly formatted _/4 Unusual or Unexpected Results Identified and Investigated (15 Statement of Uncertainty: _/5 Qualitative Statement of Uncertainty with Sources /10 Calculus-Based Error Propagation /7 Conclusion: /5 Major Results Restated and Physically Meaningful Conclusions Drawn /2 Recommendations for Further Research or Improvements Total: /100 Any Additional Comments:
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Title Page

Table of Contents
Abstract..................................................................................................................................... 3
Introduction ............................................................................................................................... 3
Theory....................................................................................................................................... 3
Facility and Apparatus............................................................................................................... 4
Procedure ................................................................................................................................. 5
Density Bottle ....................................................................................................................... 5
Hydrometer ........................................................................................................................... 6
Results ...................................................................................................................................... 6
Statement of Uncertainty .......................................................................................................... 6
Conclusion ................................................................................................................................ 6
References ............................................................................................................................... 6

Abstract
- Statement of the problem
- Comprehensive summary of the experiment
- Major results and conclusions

Introduction
Density is a basic yet crucial parameter of a material or substance in many
applications. In engineering, density is used in many calculations and estimations, particularly
in fluid mechanics. This quantity comes up in many key fluid mechanics equations and
calculations such as equations for pressure gradients and buoyancy, Bernoulli’s, Euler’s and
Navier-Stokes equations, just to name a few.
Density is simply a measure of mass per unit volume. Hence, it can easily be obtained
given an estimate of the mass and volume of a sample substance. Historically however, the
measure of mass and volume did not come easy and convenient. Several apparatus that
measure density, called hydrometers, have been developed over the centuries. It is believed
that these instruments date back to Archimedes in the 3 rd century BC (Hornsey, 2003), who
was the first recorded person to described the principle of buoyancy.
In modern laboratory and industry settings, the mass and volume of fluid samples can
easily be measured with high precision and accuracy using modern instrumentations.
Hydrometers, however, still do exist and are used. In this experiment, density of common
fluids were estimated using these methods. The objective was to compare and contrast the
accuracy and precision of the measurements produced.

Theory
- Theory for specific experiment explanation
- Free body diagrams
- Assumptions
- Experimental and ambient conditions list
- Equations and sample calculations
Density is simply a measure of mass per unit volume. Hence, it can easily be
calculated by dividing the mass of a sample by its volume (Eq. 1). Units for density are
therefore generalized as[𝑀][𝐿]−3, where M and L are mass and length units, respectively.
Density measurements are also commonly reported relative to other fluids. Specific gravity
one such measure which is defined as the ratio of the density of a substance to the density of
pure water under standard temperature and pressure. Because density is related to volume,
its quantity is a function of temperature and pressure, hence the importance to define them in
density measurements.
Density, 𝜌 =

𝑚 𝑀
[ ]
𝑉 𝐿3

(1)

Hydrometers are instruments that measure specific ...


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