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Experimental determination of the terminal settling velocity of a falling solid sphere Practical 16 for 2nd year BSc-ME students Course code WB2543 Photographs: 2 bowling balls with a diameter of 0.216 m entering water with an impact velocity of 7.6 m/s. The left ball is smooth, the right ball has a patch of sand roughness on its nose. Photographs taken from [P1]. Version 5 Last modified: November 16th, 2018 Author: Dr.ir. Wim-Paul Breugem / dr. René Delfos TU Delft, 3ME Process&Energy w.p.breugem@tudelft.nl Preface This manual has been developed for a practical on the determination of the terminal settling velocity of a falling solid sphere. It is intended for 2nd year BScME students at the Faculty of 3mE of TU Delft. The experimental setup was bought from G.U.N.T. Gerätebau GmbH in 2014. The present manual is in part based on the manual provided by G.U.N.T. for this setup (Report HM 135: Drag Coefficient for Spheres). I’m open for suggestions to improve this manual; please send your remarks to w.p.breugem@tudelft.nl . Wim-Paul Breugem Delft, October 29th, 2014 The manual has been revised based on the feedback from the students of last year. Among others, an appendix was added in which the working principle and use of a hydrometer is explained. Wim-Paul Breugem Delft, November 7th, 2015 I added a few remarks to clarify a few issues. The remarks are put in red. Wim-Paul Breugem Delft, November 30th, 2015 I added a paragraph on path instabilities of a freely falling sphere. Furthermore, a few typos were removed. Wim-Paul Breugem Delft, November 6th, 2017 Small structural changes were applied. René Delfos Delft, November 16th, 2018 [ 2 / 28 ] Contents 1. Motivation and learning goals ........................................................................................ 4 2. Theory ............................................................................................................................. 6 2.1 Drag coefficient ............................................................................................................ 6 2.2 Estimate of settling distance ......................................................................................... 8 2.3 Estimate of the terminal settling velocity ................................................................... 10 2.4 Path instabilities of freely falling spheres ................................................................... 11 3. Experimental setup........................................................................................................ 13 4. Carrying out the experiments ........................................................................................ 15 4.1 Material properties of the spheres ............................................................................... 15 4.2 Properties of the fluids ................................................................................................ 16 4.3 Experimental determination of the terminal settling velocity..................................... 17 4.4 Theoretical prediction of the terminal settling velocity .............................................. 19 5. Discussion ..................................................................................................................... 21 Literature ........................................................................................................................... 22 Photographs....................................................................................................................... 22 Appendix A. Determining the settling velocity solving a non-linear implicit equation. .. 23 Appendix B. Newton-Raphson algorithm......................................................................... 24 Appendix C. Dynamic viscosity of a water/glycerine solution ........................................ 25 Appendix D. Density of a water/glycerine solution.......................................................... 26 Appendix E. Principle of a hydrometer ............................................................................ 27 Appendix F. Principle of an Ostwald viscometer ............................................................. 28 [ 3 / 28 ] 1. Motivation and learning goals This practical is concerned with the determination of the terminal settling velocity of a falling solid particle. A precise prediction of the fall (or rise) velocity of a particle is important for many applications. A few of these applications are discussed below. Figure 1a shows the launch of a weather balloon with a radiosonde at KNMI in De Bilt. The radiosonde contains instruments to measure quantities such as the temperature, pressure, wind speed, etc., during its rise of 1-2 hours through the atmosphere. The balloon reaches typically an altitude of about 20 km before it pops due to the low atmospheric pressure at this height. During its journey the measured data is transmitted to a station on earth. It is important to choose the right initial diameter of the weather balloon, since the rise velocity depends on it and the sampling frequency should match with the rise velocity to properly sample the atmosphere. Furthermore, the journey till the top of the atmosphere should not take too long if otherwise the weather conditions will change too much in the meantime. Figure 1b shows the discharge of a dense sediment/water mixture or slurry from a pipe, which is used for instance to transport sediment from the hopper of a dredging vessel to land. The pipe flow is highly turbulent and need to provide enough energy to keep the sediment in suspension in order to prevent clogging of the pipe line. Roughly speaking, the higher the settling velocity of the sediments and the sediment volume concentration, the more energy is needed and thus a higher flow velocity and more pump power. Figure 1c shows another application from the dredging industry. It concerns the jetting of a water/sediment mixture (known as rainbowing) to create the “Maasvlakte 2”, an extension of the port of Rotterdam with about 2000 hectares of reclaimed land from the North Sea. The settling of the sediments causes elevation of the sea bed, which continues until the bed level is above sea surface level and land is formed. Finally, to mention a last example, figure 1d shows a Fluidised Bed Reactor (FBR). FBRs are used for chemical reactions. Solid catalyst particles are injected into the reactor to enhance the chemical reaction rate. To keep the catalyst particles in suspension, the upward gas flow through the reactor should be strong enough to prevent settling of the particles under gravity. The learning goals of this practical are that after this practical you are able to: a) explain the relevance of studying the settling of a particle under gravity for applications in practice; b) conduct an experiment to determine the terminal settling velocity of a solid sphere in a tube filled with a stagnant fluid; c) apply the particle force balance and appropriate correlations for the drag coefficient for an a priori prediction of the terminal settling velocity; d) explain possible causes for differences between the experimentally determined and the theoretically predicted values for the terminal settling velocity of solid spheres. [ 4 / 28 ] (a) (c) (b) (d) Figure 1. (a) Launch of a weather balloon at KNMI. Picture taken from [P2]. (b) Discharge of slurry from a horizontal pipeline. Picture taken from [P3]. (c) Jetting of a dense sediment/water mixture during creation of the “Maasvlakte 2”, an extension of the port of Rotterdam (2008-2012). Picture taken from [P4]. (d) Schematic of a fluidised bed reactor. Picture taken from [P5]. [ 5 / 28 ] 2. Theory The theory presented below is primarily based on the textbook of White [1], section 7.6. 2.1 Drag coefficient The drag force, , on an obstacle falling under gravity in a gas or liquid is defined as the total force1 that the flow exerts on it. The fall velocity of the obstacle is usually referred to as the settling velocity, . When the settling velocity has reached a constant value, it is called the terminal settling velocity. It is customary to parameterise the drag force according to: , (1) where is is the so-called frontal surface area, is the fluid ‘mass density’, usually simply called density, and is the so-called drag coefficient. The frontal surface area is the projected area of the body in the direction of the flow; for a . The drag coefficient is a sphere with diameter it is equal to function of the shape of the obstacle and the Reynolds number Re. For a perfect sphere the Reynolds number is defined as: Re , (2) is the dynamic viscosity of the fluid. The Reynolds number where characterises the importance of fluid inertial forces relative to fluid viscous forces; fluid inertial forces dominate for Re ≫ 1. If an obstacle is fully immersed in a fluid, then there are two contributions to the drag force , namely the friction (or viscous) drag and the pressure (or form) drag. Friction drag is the integral viscous shear stress that the fluid exerts on the obstacle surface area when it flows parallel to the surface. Pressure drag is the force that the fluid exerts due to the overall pressure difference between the front and the rear surface of an obstacle (relative to the flow direction). Friction drag is important at low Reynolds numbers and/or slender bodies (large ratio of total surface area over frontal surface area, such as is the case for big container ships). Conversely, pressure drag is important at high Reynolds numbers and/or for bluff bodies (small ratio of frontal surface area over total surface area, such as is the case for flow perpendicular to a thin circular disc). The drag force on the spheres shown on the cover page of this manual is almost entirely determined by pressure drag since in both cases the flow separation causes a very strong pressure difference over the sphere; flow separation is characteristic for high Reynolds numbers. 1 To be precise, the drag force is the total flow-induced force when the obstacle would fall at a constant velocity (i.e., it is the steady-state contribution of the flow to the force on the obstacle). When the obstacle accelerates or decelerates, other flow-induced forces exist, such as the added-mass force and history forces (the added-mass force will be discussed in section 2.2). [ 6 / 28 ] Figure 2 shows the drag coefficient as function of the Reynolds number for various three-dimensional obstacles. Focusing on the perfect sphere, we observe the following behavior for the drag coefficient [1,2]: Re ≲ 0.1, (3a) 0.44 5 ∙ 10 ≲ Re ≲ 10 . (3b) Figure 2. Drag coefficient as function of obstacle Reynolds number for various three-dimensional obstacles. Figure taken from [1]. Equation (3a) is known as Stokes’ drag law. In this so-called Stokes regime the flow field is mirror-symmetric with respect to the midplane of the sphere. Furthermore, the contribution from friction drag is a factor 2 larger than the contribution from pressure drag on the sphere. Substitution of Eq. (3a) into Eq. (1) yields 3 and thus the drag force increases linearly with the settling velocity for very low Reynolds numbers. Equation (3b) corresponds to the case of a turbulent wake past the obstacle as depicted for a bowling ball on the left photograph on the cover page. 0.17 : the drag force From Eq. (1) it follows that for this flow regime scales quadratic with the settling velocity. Notice furthermore that the drag force is now independent of the fluid dynamic viscosity, but scales linearly with the fluid density. The latter explains why speed skating world records are never ridden in the Netherlands since the altitude of the ice skating rink in Heerenveen is low 1.2 / is relatively high compared to, say, (sea level) and hence Salt Lake City at an altitude of about 1300 and 1.1 / . Notice also that the drag force scales now with the square of the diameter: speed skaters and racing cyclists know this very well and try to minimise their frontal surface area as much as they can in order to reduce the drag force acting on them. For Reynolds number up to about 6000 the following parameterisation appears to accurately capture the measured values of the drag coefficient as shown in Fig. 3 [2,3]: [ 7 / 28 ] 0.5407 0.1 ≾ Re ≾ 6000. (3c) Eq. (3c) is preferred over Eq. (3b) for Reynolds numbers up to 6000. Figure 3. Drag coefficient of a sphere as function of Reynolds number up to a Reynolds number of 6000. Symbols represent experimental data. The solid line is given by Eq. (3). Figure taken from [3]. 2.2 Estimate of settling distance Let us consider a solid sphere falling under gravity in a fluid at rest. The force balance (Newton’s 2nd law) for the sphere is approximately given by Morison’s heuristic equation [10]: , (4) is the sphere density, is the volume of the sphere and is where the gravitational acceleration. The meaning of the terms at the right-hand side of Eq. (4) is as follows: - Term 1 represents the gravitational force, which is the driving force for the settling of the sphere; - Term 2 represents the Archimedes force, the upward force from the displaced fluid; - Term 3 represents the so-called added-mass force and takes into account that when the sphere accelerates also the surrounding fluid of the sphere has to accelerate along with it; [ 8 / 28 ] - Term 4 represents the steady drag force when the sphere has reached its terminal (final) settling velocity. Equation (4) can be rewritten into the following form: , (5a) is a constant. The where and are constants provided that we assume that coefficients are given by, respectively: , and where , (5b) is the sphere-to-fluid density ratio. From Eq. (5a) the following analytical solution can be derived (see exercise P7.66 in [1]): tanh / , (6a) where tanh is the hyperbolic tangent function. The coefficient is the terminal and is a characteristic settling velocity that the sphere reaches for ≫ settling time that the sphere needs to reach the terminal settling velocity (for 2.65 , 0.99 ). The coefficients are given by, respectively: 1 , . (6b) (6c) at For this practical it is of special interest to estimate the settling distance which the sphere has reached 99% of its terminal velocity. The settling distance can be estimated by integrating Eq. (6b) from 0 to 2.65 : . . ln cosh 2.65 , (7a) where cosh is the hyperbolic cosine function. Assuming that (3b), this can be worked out and written as: 5.94 . 0.44, see Eq. (7b) Note that Eq. (7b) will likely overestimate the settling distance, since the drag coefficient is assumed here to be constant and equal to 0.44, while during the early falling stage it is actually larger than 0.44 (see Figs. 2 and 3). Nevertheless, Eq. (7b) is expected to yield a reasonable estimate for this practical as long as the Reynolds number based on the terminal velocity is much larger than 500. As 2860 / and 0.01 an example consider an aluminium sphere with falling in water with 1000 / . Equation (7b) then predicts that the 0.20 . As a second example, settling distance is approximately equal to consider a polyoxymethylene (POM) sphere with 1430 / and 0.005 falling in water. We get 0.057 , which is more than 3 times as small as in the previous example. [ 9 / 28 ] 2.3 Estimate of the terminal settling velocity When the terminal settling velocity is reached, the velocity of the sphere remains constant. Eq. (4) can then be rewritten in the following form: Re ∙ Re Ar 0 (8a) where Ar is the so-called Archimedes number. It is defined as: Ar 1 , (8b) ⁄ the kinematic fluid viscosity. The Archimedes number is a with dimensionless number that characterises the importance of the net gravitational force to the fluid viscous forces. Note that it only depends on the material properties of the fluid and the sphere. For the aluminium and the POM sphere in the previous section the Archimedes number is approximately equal to 1.82 ∙ 10 and 5.27 ∙ 10 (using 10 / and 9.81 / ), respectively, which indicates that friction drag is small compared to the net gravitational force on the spheres. To calculate the terminal settling velocity from Eq. (8a), we need to specify first the drag coefficient. In section 2.1 several correlations were given, see Eqs. (3a)-(3c). The problem is that we do not know the terminal settling velocity and hence the Reynolds number a priori. In Appendix A, a method using numerical mathematical techniques is presented to solve such an ‘implicit equation’. You will learn about those during the course Numerical Analysis in Q4. Here we follow a graphical method to determine the Reynolds number: First rewrite Eq. (8a) into an expression for the drag coefficient. This expression for the drag coefficient should be equal to Eq. (3c) for Reynolds numbers up to about 6000: 0.5407 . (11a) as Plot the left and the right-hand side of Eq. (11) both in one figure of function of Re. The Reynolds number for which force equilibrium exists, corresponds to the intersection point (zoom in to determine this accurately). Check whether the Reynolds number is in the valid range. Next, the terminal settling velocity can be determined from the definition of the Reynolds number: . (11b) The graphical procedure is illustrated in Fig. 4 for one of the POM spheres, i.e. for one specific value of Ar. We find that Re 1.23 ∙ 10 and 0.246 / . This new estimate for the Reynolds number is in the right range for which Eq. (3c) is valid. Furthermore, the estimate is about 3% more accurate than the 0.44 (the lower the Reynolds number, the previous one in step 2 based on larger the difference with the estimate from step 2). You may check yourself that the results found from the graphical procedure is the same as found with the Newton-Raphson procedure as presented in Appendix B. [ 10 / 28 ] 4Ar 3Re 24 Re 0.5407 1232 0.463 Figure 4. The graphical approach in which the Reynolds number is determined from Eq. (11a). The drag coefficient is plotted as function of Reynolds number. 2.4 Path instabilities of freely falling spheres Freely falling spheres may exhibit path instabilities. This originates from instabilities of the particle’s wake. Whether a path instability will occur and which kind of instability will occur, depends on 2 parameters: (1) the particle/fluid / , and (2) the Archimedes number (Ar . This is indicated in density ratio the figure below. IV I VI II III V Regimes: I. Steady vertical trajectory II. Steady oblique trajectory III. Oscillating oblique trajectory (low frequency) IV. Oscillating oblique trajectory (high frequency) V. Periodic zigzagging trajectory VI. 3D chaotic trajectory VII. Alternating 3D chaotic and periodic zigzagging trajectory VII √ Figure 5. Path instability diagram for a freely falling sphere, from numerical simulations by Jenny et al. [11]. [ 11 / 28 ] For sufficiently small Ar (√Ar ≲ 150) the particle path is steady and vertical for all density ratios (regime I). Interestingly, in a limited range of Ar a regime exists where the particle path is still steady, but slightly inclined with respect to the vertical at an angle of approximately 5º (regime II). Note that in this case the sphere also rotates over a horizontal axis. At higher Ar the particle path undergoes several different instability regimes till its path becomes a 3D chaotic trajectory at large Ar. In the last regime the particle moves in the vertical direction on average, while for heavy particles the horizontal excursions tend to be very small. [ 12 / 28 ] 3. Experimental setup A schematic of the experimental setup is depicted in Fig. 6. It consists of 2 columns filled with different liquids: one is filled with tap water and the other one with an aqueous glycerine solution with approximately 50% glycerine by weight 20° this has a dynamic viscosity of around 6∙ 10 / ∙ and a (at density of 1125 / , see Appendix C and D, respectively). The tube cover on top (1) contains a hole to insert a sphere in the tube. The sphere will sink under the action of gravity. The settling velocity will be determined from the time the sphere takes to travel through the measurement section marked by 2 O-rings (4). At the bottom of the tube there is a sluice to collect a sphere after an experiment and to remove it without much loss of liquid (5 and 6). measurement section with height In total 10 spheres are provided for the experiments: - 2 x 2 aluminium spheres with 5 and 10 5 and 10 - 2 x 2 POM (Polyoxymethylene) spheres with - 2 Nylon (Polyamide 6.6) spheres with 10 3 Figure 6. A schematic overview of the experimental setup. 1: tube cover with inlet hole. 2: Pipe brackets. 3: O-rings. 4: measurement section marked by the off Orings. 5: upper chamber valve. 6: lower chamber valve. Picture from [4]. [ 13 / 28 ] Important remark: release only 1 sphere per experiment and first collect this sphere from the sluice before moving to the next experiment. This avoids the risk that multiple spheres get stuck in the sluice! For the experiments the following measurement instruments are provided: - a micrometer screw gauge (Dutch: schroefmicrometer) - a tape measure (Dutch: rolmaat) - a scale (Dutch: weegschaal) to measure the mass of the spheres - a thermometer - a hydrometer to measure the density of the aqueous glycerine solution (see Appendix E for explanation of the working principle and use) - an Ostwald viscometer to measure the viscosity of the aqueous glycerine solution (see Appendix F for explanation of the working principle and use) - a stopwatch (alternatively, you could use your cell phone when it contains a stopwatch) Important remark 2: Most of these instruments are sensitive and/or fragile. Please handle them with care, as if they cost 100 euro each (some do!). [ 14 / 28 ] 5. Discussion Task 11: The theory in section 2.3 was based on a single sphere falling in a fluid in an infinite medium. In the experiment the sphere falls in a column. Mass conservation requires that the fluid near the sphere has to move upwards when the sphere is moving downwards. The averaged upward fluid velocity at the sphere midplane is equal to: , , (15a) where is the inner tube diameter and is the absolute , terminal settling velocity in a fixed frame of reference. The relative terminal settling velocity of the sphere, , , relative to the surrounding fluid at the sphere mid-plane, is thus equal to: , (15b) , , As a crude model to predict , (since this is the velocity you have actually measured in the experiments), we could replace , by the terminal settling velocity that a sphere would have when falling in a : quiescent fluid in an infinite medium, , , 1 , , (15c) can be estimated from the procedure detailed in section where , 2.3. Experimental data for the fall velocity of drops showed that Eq. (15c) is fairly accurate [6]. Measure the (inner) diameter of the tube. Based on Eq. (15c), do you expect that the tube confinement effect did have a strong effect on the terminal settling velocity in the experiments? Task 12: Optional. In applications where sedimentation occurs such as shown in Fig. 1c where large amounts of sand deposit on the sea floor, we need to deal with a similar effect as above. Due to the settling of the particles, a return flow exists, which implies that the effective settling velocity of the sediment is higher than the absolute settling velocity. This effect is known as hindered settling. Define the sediment volume concentration (or fraction) as (it has a value between 0 and 1). In a similar way as above, try to incorporate the effect of the return flow on the terminal settling velocity of the spheres. Show that this model predicts that: 1 , (16) , , with 1 and where can be estimated from the , procedure detailed in section 2.3. Experimental data shows that the exponent is actually a function of the particle Reynolds number and generally significantly larger than 1 (see the data of [7], which suggests that 2.39 for ≳ 500). A more detailed discussion is beyond the scope of the present practical. [ 21 / 28 ] Literature [1] F.M. White. Fluid Mechanics. McGraw-Hill, Boston, 4th edition, 1999. [2] R.B. Bird, W.E. Stewart and E.N. Lightfoot. Transport Phenomena. John Wiley and Sons, New York, 2nd edition, 2002. [3] F.F. Abraham. Functional dependence of drag coefficient on a sphere on Reynolds number. Physics of Fluids, vol. 13, pp. 2194-2195, 1970. [4] Experiment instructions: HM 135 Drag Coefficients for Spheres. Publicationno.: 917.000 00 A 135 02 (A), G.U.N.T. Gerätebau GmbH, Barsbüttel, Germany, 2013. [5] N.-S. Cheng. Formula for the viscosity of a glycerol-water mixture. Ind. Eng. Chem. Res., vol. 47, no. 9, pp. 3285-3288, 2008. [6] J.R. Strom and R.C. Kintner. Wall effect for the fall of single drops. A.I.Ch.E. Journal, vol. 4, no. 2, pp. 153-156, 1958. [7] J.F. Richardson and W.N. Zaki. Sedimentation and fluidisation: part I. Trans. Inst. Chem. Eng., vol. 32, pp. 35-53, 1954. [8] J.B. Segur and H.E. Oberstar. Viscosity of glycerol and its aqueous solutions. Industrial and Engineering Chemistry, vol. 43, no. 9, pp. 2117-2120, 1951. [9] Taken from the website of Dow Chemical at www.dow.com. [10] J.R. Morison, M.P. O’Brien, J.W. Johnson and S.A. Schaaf. The forces exerted by surface waves on piles. AIME, Petroleum Transactions, vol. 189, pp. 149-154, 1950. [11] M. Jenny, J. Dušek and G. Bouchet. Instabilities and transition of a sphere falling or ascending freely in a Newtonian fluid. J. Fluid Mech., vol. 508, 201-239, 2004. Photographs [P1] F.M. White. Fluid Mechanics. McGraw-Hill, Boston, 4th edition, 1999. [P2] http://www.knmi.nl/cms/content/32978/weerballon_of_radiosonde [P3] http://www.engineeringtoolbox.com/slurry-transport-velocity-d_236.html [P4] http://www.nationalgeographic.nl/fotografie/foto/rainbowing-1 [P5] http://en.wikipedia.org/wiki/Fluidized_bed_reactor [P6] http://fr.wikipedia.org/wiki/Hydrometre [ 22 / 28 ] Appendix A. Determining the settling velocity solving a non-linear implicit equation. Below a fairly simple solution procedure is presented in 3 steps. 1. Assume 0.44 In this case the terminal settling velocity and Reynolds number are equal to, respectively: 1.74 1 1.74√ , (9a) . (9b) It is remarked that Eq. (9a) is actually the same as Eq. (6b) in this case. 2. Check the Reynolds number The constant value of the drag coefficient is only valid when the Reynolds number is in the range of 5 ∙ 10 ≲ ≲ 10 , see Eq. (3b). Furthermore for ≲ 6000, Eq. (3c) will yield a more accurate estimate than the constant of 0.44. Let us check this for the aluminium and the POM sphere. From Eq. (9b) we 1.27 ∙ 10 for calculate that 7.44 ∙ 10 for the aluminum sphere and the POM sphere. We conclude that the Reynolds number is in the right regime for the aluminium sphere and we estimate from Eq. (9a) the terminal settling velocity by 0.74 / . On the contrary, the Reynolds number for the POM sphere is such that we expect a more accurate value of the terminal settling velocity when the drag coefficient is estimated by Eq. (3c). This is discussed in the next step. 3. Redo the calculation for another correlation of the drag coefficient, if necessary For the POM sphere we expect a more accurate estimate of the terminal settling velocity when the drag coefficient is estimated by Eq. (3c). Equation (8a) can then be rewritten as: 18.12 82.09 4.56 0, (10) where is a 4th-order polynomial function of √ . The polynomial roots of can be found numerically by using an iterative procedure such as for example the Newton-Raphson method. In the software program MATLAB you could use the function “roots” to find the polynomial roots of Eq. (10), though you may also want to program it yourself in whatever computer language you like. A schematic overview of the Newton-Raphson method is shown in Appendix B. [ 23 / 28 ] Appendix B. Newton-Raphson algorithm Below a schematic overview is given of the Newton-Raphson algorithm for solving the terminal settling velocity from equation (10). 0 (initialisation, is the iteration step) 1.74√ (estimate from Eq. (9b)) 18.12 82.09 4.56 4 54.36 164.18 (derivative of to ) (new estimate based on tangent line to ) | | | | 1 (next iteration) 18.12 82.09 4.56 4 54.36 164.18 (new estimate) It is recommended to put the threshold value smaller than10 (less than 0.1% error in the old estimate compared to the new one). Don’t forget to check whether the Reynolds number is the correct range for which Eq. (3c) is valid. [ 24 / 28 ] Appendix C. Dynamic viscosity of a water/glycerine solution The table below is taken from [8].1 centipoise [ 25 / 28 ] 10 / ∙ . Appendix D. Density of a water/glycerine solution The table below is taken from [9]. [ 26 / 28 ] Appendix E. Principle of a hydrometer A picture of a hydrometer is shown below [P6]. The principle of a hydrometer is to determine the density of a liquid from the balance between the downward gravitational force and the upward Archimedes force. To measure the density of a liquid, stick the hydrometer in a tube filled with the respective liquid. Due to the mass in the bottom of the hydrometer the hydrometer will sink until the downward gravitational force balances the upward Archimedes force from the displaced liquid. The density can be read from the scale in the upper part of the hydrometer and is equal to the level indicated at the meniscus. scale value of liquid mass density = value indicated on scale at meniscus mass in bottom part [ 27 / 28 ] Appendix F. Principle of an Ostwald viscometer A picture of a so-called Ostwald viscometer is shown below. The liquid of which the viscosity has to be determined, is poured in the left arm of the U-shaped glass tube till the 2 liquid menisci are approximately located at positions 1 and 2. Due to the hydrostatic weight of the liquid in the left arm, the meniscus at position 1 will gradually move downwards and the meniscus at 2 will gradually move upwards. The flow rate is set by the balance between the net hydrostatic weight in the Utube on the one hand and the friction drag in the tube on the other hand. Assuming laminar flow inside the tube, the flow rate depends linearly on the kinematic viscosity (that is the dynamic viscosity divided by the density of the liquid). The kinematic viscosity, , can thus be determined by measuring the time, ∆ → , it takes for the meniscus in the right arm to travel from position 3 till position 4: (D.1) ∙∆ → where is a constant (dependent only on the geometry of the viscometer), ∆ → / . For the Ostwald is measured in seconds and is determined in units of viscometer shown below the value of is engraved in the left arm and equal to 0.03023 ∙ 10 / . Remark: cleaning of the viscometer is a delicate thing with the fragile glasswork. Please do NOT try yourself but ask one of the assistants! flow direciton 1 / right arm left arm Constant C (in 10 4 3 2 [ 28 / 28 ]