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A closed form solution for falling cylinder viscometers
International Journal of Engineering Science 40 (2002) 605–620 www.elsevier.com/locate/ijengsci
A closed form solution for falling cylinder viscometers Nicolaie D. Cristescu *, Bryan P. Conrad, Roger Tran-Son-Tay Department of Aerospace Engineering, Institute of Mechanics and Engineering Science, University of Florida, 231 Aerospace Building, Gainesville, FL 32611-6250, USA Received 31 August 2001; accepted 31 August 2001 S) (Communicated by E. SOO
Abstract The paper presents a theory of the flow of a viscous fluid in a falling cylinder viscometer. The velocity profile for the flow in infinite tube and finite tube are obtained in finite form. That allows us to determine quite easily the influence of various parameters involved on the fluid flow and on the motion of the cylinder. Also a formula written in finite form is obtained for the determination of the viscosity coefficient. All these formulae contain a term that can describe the influence of a magnetic field on the motion of the falling cylinder. A comparison of the viscosities determined according to the present theory and with a cone-plate viscometers show a good agreement. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction and historical note There are many apparatuses used today to measure the viscosity of fluids and/or yield stress. Their principles and theories are presented in many books, as for instance in [1]. Generally all these devices need a significant volume of liquid in order to determine its viscosity. However, for some applications like biomedical, fluid supply can be extremely limited. For that reason, we have recently developed a falling cylinder viscometer that requires only a tiny amount of fluid (about 20 ll) in order measure viscosity. This viscometer is based on a falling ball viscometer of Tran-SonTay et al. [2]. The advantage of the falling cylinder versus the falling ball is that the generated shear rate is better defined. Falling cylinder viscometers have been used for over 75 years, but remarkably the problem of a cylinder of finite length falling inside a cylindrical tube has not been solved yet in simple closed form solution, involving a magnetic field as well.
*
Corresponding author. Tel.: +1-352-392-6747; fax: +1-352-392-7303. E-mail address: [email protected]fl.edu (N.D. Cristescu).
0020-7225/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 1 ) 0 0 0 9 4 - 5
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It appears that Pochettino [3] was the first to study experimentally the passage from the ‘‘solid state’’ to the ‘‘fluid state’’ using a method of a falling cylinder in order to study the mechanical properties of tar. Bridgman [4] used a falling cylinder to determine the ‘‘relative viscosities’’ of fluids subjected to high pressures. He devised the apparatus and analyzed the various possible experimental errors, mainly related to the ‘‘inertia effect’’, i.e., the time needed for the cylinder to reach a steady-state falling velocity. His apparatus did not give the absolute viscosity, but only the relative viscosity. The relative viscosities for a variety of fluids, for several temperatures and pressures were determined. It was found that viscosity increases with pressure. Using a falling cylinder type viscometer, viscosities of methane and propane at low temperatures and high pressures were determined by Huang et al. [5], and viscosities of methane, ethane, propane and n-butane by Swift et al. [6]. A theoretical analysis of the laminar fluid flow in the annulus of a falling cylinder viscometer was made by Lohrenz et al. [7]. The falling cylinder viscometer was also analyzed by Ashore et al. [8] for both Newtonian and non-Newtonian fluids; approximate expressions for axial non-Newtonian flow in the annuli was developed. They assumed that, since the annular slit is small, it can be regarded as a plane slit. A theoretical analysis for the falling cylinder viscometer for Bingham fluid and power law model, is due to Eichstadt and Swift [9], while the influence of the eccentricity on the terminal velocity of the cylinder was considered by Chen et al. [10]. A falling magnetic stainless steel slug was developed by Mc Duffie and Barr [11] to measure viscosities between 1 and 104 P at pressure up to 3500 kg=cm2 and temperature between )60 and 100 °C. The motion of the slug was determined with a differential transformer that moved along the tube. An automatic falling cylinder viscometer for high pressures was also developed by Irving and Barlow [12]; the sinker was either a solid cylinder or one with a central hole, and the fall time is detected inductively by a series of coils along the viscometer tube. Viscosity in the range 0.01– 3000 P was determined. Another kind of falling coaxial cylinder viscometer for lower viscosity fluids as paints, was constructed by Chee et al. [13], where a weighted rod is falling into a closedend concentric cylinder. A laser Doppler technique was used to measure the velocity of a falling-slug in a high pressure viscometer by Dandridge and Jackson [14]; the viscosities of two polyisobutenes have been determined as function of pressure. Viscosities in excess of 107 Pa/s have been determined by this method. An improved version of the viscometer was presented by Chan and Jackson [15]. Measurements of the relative viscosity of aqueous solutions for various temperatures and pressures up to 120 MPa are due to Tanaka et al. [16]; they found that viscosity increases almost linearly with pressure. Tanaka et al. [17] later used a laser beam that passed through a pair of sapphire windows to a phototransistor to determine the falling time of the cylinder. A special designed falling cylinder viscometer was used by Kiran and Sen [18] to measure high-pressure viscosities in the 10–70 MPa range and temperatures from 310 to 450 K, of n-butane, n-pentane, n-hexane and n-octane (see also [19]). Kiran and Gokmenoglu [20] also published viscosity values for Polyethylene solutions subjected to high pressures. Chen and Swift [21] analyzed the ‘‘entrance’’ and ‘‘exit’’ effects in a falling cylinder viscometer for creeping and non-creeping flow, giving a numerical correction for incompressible Newtonian fluid. End effects occurring in the falling cylinder viscometer were also analyzed by Wehbeh and Hussey [22]; experimental data for closed and open tubes were also presented. A theoretical and experimental study allowing for the prediction of end effects is due to Gui and Irvine [23]; the flow
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field is obtained numerically (see also [24]). A superposition technique was utilized by Park and Irvine [25] to account for the end effects of a flat ends falling cylinder viscometer; the technique is applicable for Newtonian fluids. Park and Irvine [26] also gave a method to simultaneously determine the density and viscosity of the liquid by using needles of three distinct densities. The derivation of the ‘‘exact’’ solution for the motion of a liquid flowing past a falling cylinder with a frontal spherical end was analyzed by Borisov [27] with some assumptions concerning the front shape of the cylinder and liquid velocity. The finite element study of a uniform flow past a needle in a cylindrical tube for materials having different constitutive equations is due to Phan-Thien et al. [28]. In the present paper we present a simple theory of the flow of a viscous fluid in a falling cylinder viscometer. The velocity profile is obtained in finite form for both open tube and closed tube. Also in finite form is obtained a formula for the determination of the viscosity coefficient. These formulae contain also a term describing the influence of a magnetic field on the motion of the falling cylinder and of the fluid. How this term is used in viscosity measurements will be described in future papers. Since all the obtained formulae are in finite form, a parametric study (determination of the influence of various factors involved) of the fluid flow in the falling cylinder viscometer can be done quite easily. The comparison of the determination of the viscosity parameter according to the present theory with viscosity determined with a cone-plate viscometer is quite good.
2. Microrheometer 2.1. Introduction The most common viscometers and rheometers presently in use are those based on the measurement of stress on a fixed surface while a parallel or opposing surface is moved with a known applied strain rate. However, as already mentioned, these devices require fairly large sample volumes. Often in medical applications it is difficult, if not impossible, to collect large volumes of samples for testing. In response to this challenge, Tran-Son-Tay et al. [2] developed an acoustically tracked falling ball rheometer (Microrheometer). The Microrheometer (Fig. 1) is a unique device that was designed to measure the viscosity and viscoelasticity of small samples of biological fluids by using a spherical steel ball that is concentrically located in a small cylindrical tube with a volume of about 20 ll. The ball falls either under the force of gravity or is levitated and oscillated with a force produced by a magnetic field. The position of the ball is tracked by ultrasonic pulseecho method. Furthermore, the small volume permits accurate temperature control and rapid temperature changes to be effected in the sample under study. Modifications have been made to the original setup so that a constant shear rate will be exhibited across the fluid, that is, a falling cylinder is used in place of a ball. 2.2. Apparatus A 20 ll sample is loaded, by retrograde injection, into a disposable glass tube with an inner diameter of 1.6 mm and a height of 10 mm. Once loaded, the tube is centered inside a cylindrical,
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Fig. 1. Schematic of Microrheometer (a – sample tube, b – cylinder, c – piezoelectric crystal, d – ultrasound transducer, e – water jacket, f – electromagnet).
Plexiglas, water-jacket chamber. The ultrasonic transducer is housed at the bottom center of the Plexiglas chamber. An O-ring forms a watertight seal with the base of the sample tube and with the transducer. A plastic cap is screwed into place at the top of the chamber and provides a seal that completely protects the sample. A schematic of the Microrheometer is given in Fig. 1. A small electromagnet coupled with a micromanipulator is used to position and drop a 1.1 mm diameter cylinder in the center of the tube, where the strongest echo occurs. The pulse-echo mode is used to locate and track the falling cylinder. A single sound pulse is transmitted into the fluid medium by pulsing an ultrasound transducer that also acts as a receiver. Any returning echoes from the cylinder cause a voltage rise across the transducer that is amplified by the ultrasonic pulser/receiver unit (Panametrics 5052 PR). Fig. 2 shows the main bang followed by a pulse reflected from the top of the cylinder. The time for the sound waves to travel to and back from the cylinder, and the speed of sound in the fluid are measured. From that information, the distance between the transducer and the cylinder, i.e., the location of the falling cylinder, is determined.
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Fig. 2. Main bang is the large signal on the left, the echo of the cylinder is the smaller signal on the right (tf time of flight, time for ultrasound pulse to travel to the cylinder and back to the transducer.).
2.3. Experiments Two types of measurements can be performed with our Microrheometer: (1) a speed-of-sound evaluation, and (2) a steady-state viscosity. The parameters of interest are c, the local speed of sound in the medium and g, the bulk suspension viscosity. 2.3.1. Speed of sound In order to measure the speed of sound in the fluid, the instrument needs to be calibrated. To accomplish this, the sample chamber is first filled to the top with distilled water and then capped with a glass cover slip to assure a fixed sample height h. This distance is given by h ¼ TfH2 O c0 =2; where TfH2 O is the measured time of flight for the sound to travel to the top of the sample chamber and back in distilled water, and c0 is the known speed of sound. Once the chamber height is determined, the water is removed and replaced by the fluid sample. The local speed of sound in the medium c, is then determined by the relationship c ¼ 2h=Tf ; where Tf is the measured time of flight in the fluid sample.
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2.3.2. Viscosity The fluid viscosity is determined from the speed of sedimentation of the falling cylinder. Details of the theory, including the use of a magnetic force to pull on the cylinder, are provided in the following section.
3. Theory 3.1. Formulation of the problem In what follows we use cylindrical coordinates (see Fig. 3) and the notations shown in the figure. We use: qc – cylinder density, q – fluid density, v1 – velocity of cylinder.
Fig. 3. Schematic of the falling cylinder viscometer and notations used.
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Assumptions made are: A1. The flow is laminar and telescopic; the velocity components are vr ¼ vh ¼ 0; vz ¼ v ¼ f ðrÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi with r ¼ x2 þ y 2 . A2. The boundary conditions at the cylinder wall t P 0;
r ¼ R1 : vðR1 Þ ¼ v1 ;
ð3:1Þ
ð3:2Þ
i.e. the fluid adhere to the cylinder wall. A3. The boundary condition at the tube wall t P 0;
r ¼ R2 : vðR2 Þ ¼ 0;
ð3:3Þ
i.e., the fluid adhere to the tube wall. A4. The fluid is a Newtonian viscous fluid Tij0 ¼ 2gDij ;
ð3:4Þ
where Tij0 is the Cauchy stress deviator, Dij is the strain rate tensor and g the viscosity coefficient, may be depending on z, if the fluid is non-homogeneous. Taking into account (3.1) the only non-zero strain rate component is 1 Drz ¼ f 0 ðrÞ 2
ð3:5Þ
and from (3.4) we get for the stresses Trr ¼ Thh ¼ Tzz ¼ r;
Thz ¼ Thr ¼ 0;
Trz ¼ gðzÞf 0 ðrÞ:
ð3:6Þ
Since v is varying with r and assuming that this variation is of the kind shown in Fig. 3, we expect for Trz 8 positive if R1 < r < r0 ov < Trz ¼ gð zÞ ¼ negative if r0 < r < R2 or : 0 for r ¼ r0
ð3:7Þ
if at r ¼ r0 the velocity reaches a maximum and the stress Trz is zero. From the equilibrium equations written in cylindrical coordinates, since the problem is with cylindrical symmetry, follows for our case or or ¼ ¼ 0; or oh
oTrz or Trz þ þ þ qbz ¼ 0; oz or r
ð3:8Þ
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where bz is the body force component. Thus rðzÞ depends on z alone, and from the last equation (3.8) and (3.6) we get d rf 0 dr þ þ qbz r ¼ 0: g dr dz
ð3:9Þ
Integrating once with respect to r, we get dr r C1 þ : Trz ¼ þ qbz dz 2 r
ð3:10Þ
Integrating a second time with respect to r we obtain gf þ
2 dr r ¼ C1 ln r þ C2 : þ qbz 4 dz
ð3:11Þ
The integration constants can be determined from the boundary conditions A2 and A3 R2 R2 þ qbz 2 4 1 ; C1 ¼ ln RR21 R2 R2 2 þ qbz 2 4 1 dr R2 gv1 þ dr dz ln R2 : þ qbz C2 ¼ dz 4 ln RR2 gv1 þ
dr dz
ð3:12Þ
1
Thus the velocity distribution is obtained as gv ¼
dr þ qbz dz
R22 r2 4
2 1 dr R2 R21 r þ qbz þ R2 gv1 þ ln dz R2 4 ln R1
ð3:13Þ
if the viscosity coefficient is known and dr=dz is the pressure gradient along the tube. Let us write now a global equilibrium condition for the cylinder: the projection on the z-axis of all forces acting on the cylinder are zero; i.e. gravitational force + buoyancy force + shearing force on the cylinder wall ¼ 0. Thus Trz jR1 ¼
R1 gðqc q þ mÞ 2
ð3:14Þ
with m the ‘‘local density’’ of the magnetic force. With (3.9) this relation becomes gv1 ¼
dr þ qbz dz
R21 R2 R22 R21 R2 R2 ln þ 1 ln ½gðqc q þ mÞ ; 2 R1 4 2 R1
which is a relation between gv1 ,
dr dz
þ qbz and m.
ð3:15Þ
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3.2. Velocity profile for open tube First let us consider an open vertical tube (or an ‘‘infinite’’ tube) in which can move a heavy cylinder and a pressure gradient exists. Let us assign various values to the pressure gradient dr=dz in order to find out the various possible velocity profiles. Example 1. From (3.14) follows that if the pressure gradient satisfies the relation dr þ qb ¼ dz
R21 2
ln RR21 ½gðqc q þ mÞ R21 2
ln RR21
R22 R21 4
;
ð3:16Þ
the cylinder is stationary (v1 ¼ 0). In other words under this pressure gradient the shearing forces of the fluid flowing upwards will keep stationary the cylinder. Fig. 4 shows an example computed for g ¼ 1:005 N s=m2 (1005 cp) and dr þ qbz ¼ 9:36 104 Pa=m: dz
ð3:17Þ
The velocity distribution shown was obtained with (3.12). Example 2. If the pressure gradient is greater than that obtained from (3.16), the shearing force of the fluid flowing up is able to move upwards the cylinder. In order to give an exam, Fig. 5 is showing the velocity profile of the fluid for ðdr=dzÞ þ qbz ¼ 1:2 105 Pa=m: If this term reaches the value 1 106 Pa=m the velocity of the fluid in contact with the cylinder is equal with that of the cylinder itself. This case can be obtained from (3.12) by putting the condition that the maximum velocity of the fluid be reached for r ¼ R1 .
Fig. 4. Velocity of the fluid moving upward under the pressure gradient (3.16) which keeps the cylinder stationary.
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(a)
(b)
Fig. 5. Velocity profile for the fluid flowing upwards under a pressure gradient greater than (3.16); the fluid pushing the cylinder upwards: (a) the pressure gradient slightly bigger than (3.17); (b) pressure gradient bigger than (3.16) when the velocity of the cylinder is equal with the maximum velocity of the fluid (and a small plateau exists).
Example 3. This example considers the most common case when term ðdr=dzÞ þ qbz has a smaller value than that obtained from (3.15). Fig. 6 shows an example obtained with the value ðdr=dzÞ þ qbz ¼ 8:0 104 Pa=m and same g as before. In this case most of the fluid is moving upwards, but a thin layer of fluids neighboring the cylinder, the fluid is moving downwards. In the present example v1 ¼ 0:000528 m/s (down) is obtained from (3.14). Example 4. The last example corresponds to very small values for ðdr=dzÞ þ qbz . Let us consider the case when the pressure gradient is zero dr=dz ¼ 0: Introducing this value in (3.14) together with qg ¼ 9:807 103 ð kg=m2 s2 Þ and same value for g as above, we get v1 ¼ 0:00363 m=s (down). The velocity profile follows from (3.12). This time the falling cylinder
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Fig. 6. Under a small pressure gradient, part of the fluid is moving up, but part is moving down with the falling cylinder.
Fig. 7. The velocity profile in the fluid generated by the falling cylinder (pressure gradient is zero).
induces into the fluid a shearing stress which will put it to flow downwards (see Fig. 7). The velocity of the fluid in contact with the tube is certainly zero. 3.3. Velocity profile for closed tube Let us consider now the case when the bottom of the tube is closed and the pressure gradient is generated by the falling cylinder (Figs. 8 and 9). We can write down the condition that the volume of the fluid displaced by the falling cylinder is equal to the volume of the fluid flowing (up and down) between the cylinder and the tube Z
R2
R1
v2pr dr ¼ v1 pR21 :
ð3:18Þ
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Fig. 8. Schematic of the fluid flow in the case of closed tube.
Introducing here v ð¼ f Þ from (3.11) we get 3 Z 2 R2 dr r þ qbz þ C1 r ln r þ C2 r dr ¼ v1 R21 g R1 dz 4
ð3:19Þ
with the values of C1 and C2 from (3.14) and the notations 2 2
R2 R21 R42 R41 R22 R21 2 ln R2 1 1 R22 R22 R21 2 ln R1 R¼ R2 ln R2 ; R1 þ þ 4 4 16 2 2 4 2 4 ln RR2 8 ln RR2 1
1 1 1 ln R2 R2 R21 R21 2 ln R2 2 ln R1 þ ; V ¼ R2 R2 R1 þ R2 2 4 4 2 2 2 2 ln R1 ln R1
1
ð3:20Þ
we finally obtain dr V þ qbz ¼ gjv1 j; dz R
ð3:21Þ
which established another relationship between gv1 and ðdr=dzÞ þ qbz , obtained from a kinematic assumption.
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Fig. 9. Velocity profile and shear stress profile for the case of closed cylinder.
However, the term gv1 as obtained from (3.21) and from (3.15) must be equal. From this condition follows dr þ qbz ¼ dz
R21 2
ln RR21 ½gðqc q þ mÞ R V
R21 2
ln RR21 þ
R22 R21 4
:
ð3:22Þ
This relation determines ðdr=dzÞ þ qbz . The magnetic force is involved as a local density, i.e. if pushing down it is equivalent with a heavier cylinder. After calibration, this observation will be used to speed up some tests with very viscous fluids, and using the very same magnetic cylinder. Introducing this value into (3.21) we obtain a formula for the determination of the viscosity coefficient R dr þ qbz : ð3:23Þ g¼ V jv1 j dz We can consider the expression R a¼ V
R21 2 R V
R21 2
g ln RR21
ln RR21 þ
R22 R21 4
b
ð3:24Þ
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to be a viscometer constant. We have introduced b as a correction parameter, mainly because the cylindrical magnet has sometimes no perfect cylindrical shape, and also to take into account the end effects. Thus the formula for the determination of the viscosity coefficient (3.23) writes g¼a
qc q þ m : jv1 j
ð3:25Þ
3.4. Shearing stress distribution The shearing stress distribution in the layer of fluid is obtained from (3.10) with (3.12), as R2 R2 þ qbz 2 4 1 1 dr r gv1 þ dr dz þ : Trz ¼ þ qbz dz 2 r ln RR2
ð3:26Þ
1
The stress becomes zero there where the fluid velocity is maximum; Trz ¼ 0 or v ¼ vmax for r ¼ r0 . That is obtained from
dr r0 1 þ qbz þ C1 ¼ 0 dz r0 2
i.e. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2C1 r0 ¼ dr þ qbz dz
ð3:27Þ
which determines r0 . From the numerical examples given in this paper one can see that r0 depends strongly on the pressure gradient. Example 5. In order to check the prediction of the above formulae, we have performed experiments with a variety of standardized fluids available in our laboratory. The tests were done with a custom made falling cylinder viscometer. The viscometer data are: R1 ¼ 0:000505 m (cylinder radius) R2 ¼ 0:000805 m (container inner radius) qc ¼ 7228 kg=m3 (cylinder density) qf ¼ 1000 kg=m3 (fluid density) A sample of standard 1000 cp calibration fluid was measured to verify its viscosity using a coneplate viscometer (Brookfield, MA). We have tested 17 samples of this calibration fluid and determined a coefficient of viscosity g ¼ 1067 87 cp compared with 1005 cp as determined by the cone-plate viscometer. The experimental viscosity coefficient is 6.17% higher than the cone-plate value. Though the matching is quite good, the existing difference may be due either to the disregarding in the theory of the end effects, or/and to the non-perfect cylindrical shape of the falling
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Table 1 Summary statistics of all samples computed both experimentally and theoretically Fluid
Viscosity as measured on the Brookfield (cp)
Number of falling cylinder samples
Average falling cylinder velocity (m/s)
Average viscosity as determined from the theory (cp)
Standard deviation of theoretical values
Accuracy % ((theoretical valueBrookfield value)/ Brookfield value)
Variability % (standard deviation/ measured value)
1 2 3
1005 3085 9549
16 17 37
5.25E)04 1.71E)04 6.05E)05
1067 3274 9442
87 241 1564
6.169154229 6.126418152 )1.120536182
8.153701968 7.361026268 16.56428723
cylinder. However, for all practical measurements of viscosities, the accuracy is quite good. The data for all sample fluids is summarized in Table 1. 4. Conclusion A simple theory is presented in which the solution of the flow of a viscous fluid in a falling cylinder viscometer is expressed in a closed form. The formula giving the viscosity of the fluid is also obtained in finite form. Since all the formulae are obtained in finite forms, one can easily study the influence of all parameters involved [as densities (may be variable), geometry of cylinder-tube, pressure gradient, magnetic force, etc.] on the cylinder–fluid motion. The comparison of the viscosity coefficients, as determined by our experiments with the falling cylinder Microrheometer and using the present theory, and as determined with a classical cone-plate viscometers is quite good. It is important to note that the formulae for velocity, stress, viscosity, etc. contain a term that describes the influence of a magnetic field on the motion of the cylinder and thus on the flow of the fluid. However, the influence and use of a magnetic field in viscometer studies will be treated in a future paper. An extension of the analysis to non-homogeneous fluids, viscoplastic materials, and to the case when the fluid slides along the tube and/or cylinder will also be considered.
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A closed form solution for falling cylinder viscometers Nicolaie D. Cristescu *, Bryan P. Conrad, Roger Tran-Son-Tay Department of Aerospace Engineering, Institute of Mechanics and Engineering Science, University of Florida, 231 Aerospace Building, Gainesville, FL 32611-6250, USA Received 31 August 2001; accepted 31 August 2001 S) (Communicated by E. SOO
Abstract The paper presents a theory of the flow of a viscous fluid in a falling cylinder viscometer. The velocity profile for the flow in infinite tube and finite tube are obtained in finite form. That allows us to determine quite easily the influence of various parameters involved on the fluid flow and on the motion of the cylinder. Also a formula written in finite form is obtained for the determination of the viscosity coefficient. All these formulae contain a term that can describe the influence of a magnetic field on the motion of the falling cylinder. A comparison of the viscosities determined according to the present theory and with a cone-plate viscometers show a good agreement. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction and historical note There are many apparatuses used today to measure the viscosity of fluids and/or yield stress. Their principles and theories are presented in many books, as for instance in [1]. Generally all these devices need a significant volume of liquid in order to determine its viscosity. However, for some applications like biomedical, fluid supply can be extremely limited. For that reason, we have recently developed a falling cylinder viscometer that requires only a tiny amount of fluid (about 20 ll) in order measure viscosity. This viscometer is based on a falling ball viscometer of Tran-SonTay et al. [2]. The advantage of the falling cylinder versus the falling ball is that the generated shear rate is better defined. Falling cylinder viscometers have been used for over 75 years, but remarkably the problem of a cylinder of finite length falling inside a cylindrical tube has not been solved yet in simple closed form solution, involving a magnetic field as well.
*
Corresponding author. Tel.: +1-352-392-6747; fax: +1-352-392-7303. E-mail address: [email protected]fl.edu (N.D. Cristescu).
0020-7225/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 1 ) 0 0 0 9 4 - 5
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It appears that Pochettino [3] was the first to study experimentally the passage from the ‘‘solid state’’ to the ‘‘fluid state’’ using a method of a falling cylinder in order to study the mechanical properties of tar. Bridgman [4] used a falling cylinder to determine the ‘‘relative viscosities’’ of fluids subjected to high pressures. He devised the apparatus and analyzed the various possible experimental errors, mainly related to the ‘‘inertia effect’’, i.e., the time needed for the cylinder to reach a steady-state falling velocity. His apparatus did not give the absolute viscosity, but only the relative viscosity. The relative viscosities for a variety of fluids, for several temperatures and pressures were determined. It was found that viscosity increases with pressure. Using a falling cylinder type viscometer, viscosities of methane and propane at low temperatures and high pressures were determined by Huang et al. [5], and viscosities of methane, ethane, propane and n-butane by Swift et al. [6]. A theoretical analysis of the laminar fluid flow in the annulus of a falling cylinder viscometer was made by Lohrenz et al. [7]. The falling cylinder viscometer was also analyzed by Ashore et al. [8] for both Newtonian and non-Newtonian fluids; approximate expressions for axial non-Newtonian flow in the annuli was developed. They assumed that, since the annular slit is small, it can be regarded as a plane slit. A theoretical analysis for the falling cylinder viscometer for Bingham fluid and power law model, is due to Eichstadt and Swift [9], while the influence of the eccentricity on the terminal velocity of the cylinder was considered by Chen et al. [10]. A falling magnetic stainless steel slug was developed by Mc Duffie and Barr [11] to measure viscosities between 1 and 104 P at pressure up to 3500 kg=cm2 and temperature between )60 and 100 °C. The motion of the slug was determined with a differential transformer that moved along the tube. An automatic falling cylinder viscometer for high pressures was also developed by Irving and Barlow [12]; the sinker was either a solid cylinder or one with a central hole, and the fall time is detected inductively by a series of coils along the viscometer tube. Viscosity in the range 0.01– 3000 P was determined. Another kind of falling coaxial cylinder viscometer for lower viscosity fluids as paints, was constructed by Chee et al. [13], where a weighted rod is falling into a closedend concentric cylinder. A laser Doppler technique was used to measure the velocity of a falling-slug in a high pressure viscometer by Dandridge and Jackson [14]; the viscosities of two polyisobutenes have been determined as function of pressure. Viscosities in excess of 107 Pa/s have been determined by this method. An improved version of the viscometer was presented by Chan and Jackson [15]. Measurements of the relative viscosity of aqueous solutions for various temperatures and pressures up to 120 MPa are due to Tanaka et al. [16]; they found that viscosity increases almost linearly with pressure. Tanaka et al. [17] later used a laser beam that passed through a pair of sapphire windows to a phototransistor to determine the falling time of the cylinder. A special designed falling cylinder viscometer was used by Kiran and Sen [18] to measure high-pressure viscosities in the 10–70 MPa range and temperatures from 310 to 450 K, of n-butane, n-pentane, n-hexane and n-octane (see also [19]). Kiran and Gokmenoglu [20] also published viscosity values for Polyethylene solutions subjected to high pressures. Chen and Swift [21] analyzed the ‘‘entrance’’ and ‘‘exit’’ effects in a falling cylinder viscometer for creeping and non-creeping flow, giving a numerical correction for incompressible Newtonian fluid. End effects occurring in the falling cylinder viscometer were also analyzed by Wehbeh and Hussey [22]; experimental data for closed and open tubes were also presented. A theoretical and experimental study allowing for the prediction of end effects is due to Gui and Irvine [23]; the flow
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field is obtained numerically (see also [24]). A superposition technique was utilized by Park and Irvine [25] to account for the end effects of a flat ends falling cylinder viscometer; the technique is applicable for Newtonian fluids. Park and Irvine [26] also gave a method to simultaneously determine the density and viscosity of the liquid by using needles of three distinct densities. The derivation of the ‘‘exact’’ solution for the motion of a liquid flowing past a falling cylinder with a frontal spherical end was analyzed by Borisov [27] with some assumptions concerning the front shape of the cylinder and liquid velocity. The finite element study of a uniform flow past a needle in a cylindrical tube for materials having different constitutive equations is due to Phan-Thien et al. [28]. In the present paper we present a simple theory of the flow of a viscous fluid in a falling cylinder viscometer. The velocity profile is obtained in finite form for both open tube and closed tube. Also in finite form is obtained a formula for the determination of the viscosity coefficient. These formulae contain also a term describing the influence of a magnetic field on the motion of the falling cylinder and of the fluid. How this term is used in viscosity measurements will be described in future papers. Since all the obtained formulae are in finite form, a parametric study (determination of the influence of various factors involved) of the fluid flow in the falling cylinder viscometer can be done quite easily. The comparison of the determination of the viscosity parameter according to the present theory with viscosity determined with a cone-plate viscometer is quite good.
2. Microrheometer 2.1. Introduction The most common viscometers and rheometers presently in use are those based on the measurement of stress on a fixed surface while a parallel or opposing surface is moved with a known applied strain rate. However, as already mentioned, these devices require fairly large sample volumes. Often in medical applications it is difficult, if not impossible, to collect large volumes of samples for testing. In response to this challenge, Tran-Son-Tay et al. [2] developed an acoustically tracked falling ball rheometer (Microrheometer). The Microrheometer (Fig. 1) is a unique device that was designed to measure the viscosity and viscoelasticity of small samples of biological fluids by using a spherical steel ball that is concentrically located in a small cylindrical tube with a volume of about 20 ll. The ball falls either under the force of gravity or is levitated and oscillated with a force produced by a magnetic field. The position of the ball is tracked by ultrasonic pulseecho method. Furthermore, the small volume permits accurate temperature control and rapid temperature changes to be effected in the sample under study. Modifications have been made to the original setup so that a constant shear rate will be exhibited across the fluid, that is, a falling cylinder is used in place of a ball. 2.2. Apparatus A 20 ll sample is loaded, by retrograde injection, into a disposable glass tube with an inner diameter of 1.6 mm and a height of 10 mm. Once loaded, the tube is centered inside a cylindrical,
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Fig. 1. Schematic of Microrheometer (a – sample tube, b – cylinder, c – piezoelectric crystal, d – ultrasound transducer, e – water jacket, f – electromagnet).
Plexiglas, water-jacket chamber. The ultrasonic transducer is housed at the bottom center of the Plexiglas chamber. An O-ring forms a watertight seal with the base of the sample tube and with the transducer. A plastic cap is screwed into place at the top of the chamber and provides a seal that completely protects the sample. A schematic of the Microrheometer is given in Fig. 1. A small electromagnet coupled with a micromanipulator is used to position and drop a 1.1 mm diameter cylinder in the center of the tube, where the strongest echo occurs. The pulse-echo mode is used to locate and track the falling cylinder. A single sound pulse is transmitted into the fluid medium by pulsing an ultrasound transducer that also acts as a receiver. Any returning echoes from the cylinder cause a voltage rise across the transducer that is amplified by the ultrasonic pulser/receiver unit (Panametrics 5052 PR). Fig. 2 shows the main bang followed by a pulse reflected from the top of the cylinder. The time for the sound waves to travel to and back from the cylinder, and the speed of sound in the fluid are measured. From that information, the distance between the transducer and the cylinder, i.e., the location of the falling cylinder, is determined.
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Fig. 2. Main bang is the large signal on the left, the echo of the cylinder is the smaller signal on the right (tf time of flight, time for ultrasound pulse to travel to the cylinder and back to the transducer.).
2.3. Experiments Two types of measurements can be performed with our Microrheometer: (1) a speed-of-sound evaluation, and (2) a steady-state viscosity. The parameters of interest are c, the local speed of sound in the medium and g, the bulk suspension viscosity. 2.3.1. Speed of sound In order to measure the speed of sound in the fluid, the instrument needs to be calibrated. To accomplish this, the sample chamber is first filled to the top with distilled water and then capped with a glass cover slip to assure a fixed sample height h. This distance is given by h ¼ TfH2 O c0 =2; where TfH2 O is the measured time of flight for the sound to travel to the top of the sample chamber and back in distilled water, and c0 is the known speed of sound. Once the chamber height is determined, the water is removed and replaced by the fluid sample. The local speed of sound in the medium c, is then determined by the relationship c ¼ 2h=Tf ; where Tf is the measured time of flight in the fluid sample.
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2.3.2. Viscosity The fluid viscosity is determined from the speed of sedimentation of the falling cylinder. Details of the theory, including the use of a magnetic force to pull on the cylinder, are provided in the following section.
3. Theory 3.1. Formulation of the problem In what follows we use cylindrical coordinates (see Fig. 3) and the notations shown in the figure. We use: qc – cylinder density, q – fluid density, v1 – velocity of cylinder.
Fig. 3. Schematic of the falling cylinder viscometer and notations used.
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Assumptions made are: A1. The flow is laminar and telescopic; the velocity components are vr ¼ vh ¼ 0; vz ¼ v ¼ f ðrÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi with r ¼ x2 þ y 2 . A2. The boundary conditions at the cylinder wall t P 0;
r ¼ R1 : vðR1 Þ ¼ v1 ;
ð3:1Þ
ð3:2Þ
i.e. the fluid adhere to the cylinder wall. A3. The boundary condition at the tube wall t P 0;
r ¼ R2 : vðR2 Þ ¼ 0;
ð3:3Þ
i.e., the fluid adhere to the tube wall. A4. The fluid is a Newtonian viscous fluid Tij0 ¼ 2gDij ;
ð3:4Þ
where Tij0 is the Cauchy stress deviator, Dij is the strain rate tensor and g the viscosity coefficient, may be depending on z, if the fluid is non-homogeneous. Taking into account (3.1) the only non-zero strain rate component is 1 Drz ¼ f 0 ðrÞ 2
ð3:5Þ
and from (3.4) we get for the stresses Trr ¼ Thh ¼ Tzz ¼ r;
Thz ¼ Thr ¼ 0;
Trz ¼ gðzÞf 0 ðrÞ:
ð3:6Þ
Since v is varying with r and assuming that this variation is of the kind shown in Fig. 3, we expect for Trz 8 positive if R1 < r < r0 ov < Trz ¼ gð zÞ ¼ negative if r0 < r < R2 or : 0 for r ¼ r0
ð3:7Þ
if at r ¼ r0 the velocity reaches a maximum and the stress Trz is zero. From the equilibrium equations written in cylindrical coordinates, since the problem is with cylindrical symmetry, follows for our case or or ¼ ¼ 0; or oh
oTrz or Trz þ þ þ qbz ¼ 0; oz or r
ð3:8Þ
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where bz is the body force component. Thus rðzÞ depends on z alone, and from the last equation (3.8) and (3.6) we get d rf 0 dr þ þ qbz r ¼ 0: g dr dz
ð3:9Þ
Integrating once with respect to r, we get dr r C1 þ : Trz ¼ þ qbz dz 2 r
ð3:10Þ
Integrating a second time with respect to r we obtain gf þ
2 dr r ¼ C1 ln r þ C2 : þ qbz 4 dz
ð3:11Þ
The integration constants can be determined from the boundary conditions A2 and A3 R2 R2 þ qbz 2 4 1 ; C1 ¼ ln RR21 R2 R2 2 þ qbz 2 4 1 dr R2 gv1 þ dr dz ln R2 : þ qbz C2 ¼ dz 4 ln RR2 gv1 þ
dr dz
ð3:12Þ
1
Thus the velocity distribution is obtained as gv ¼
dr þ qbz dz
R22 r2 4
2 1 dr R2 R21 r þ qbz þ R2 gv1 þ ln dz R2 4 ln R1
ð3:13Þ
if the viscosity coefficient is known and dr=dz is the pressure gradient along the tube. Let us write now a global equilibrium condition for the cylinder: the projection on the z-axis of all forces acting on the cylinder are zero; i.e. gravitational force + buoyancy force + shearing force on the cylinder wall ¼ 0. Thus Trz jR1 ¼
R1 gðqc q þ mÞ 2
ð3:14Þ
with m the ‘‘local density’’ of the magnetic force. With (3.9) this relation becomes gv1 ¼
dr þ qbz dz
R21 R2 R22 R21 R2 R2 ln þ 1 ln ½gðqc q þ mÞ ; 2 R1 4 2 R1
which is a relation between gv1 ,
dr dz
þ qbz and m.
ð3:15Þ
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3.2. Velocity profile for open tube First let us consider an open vertical tube (or an ‘‘infinite’’ tube) in which can move a heavy cylinder and a pressure gradient exists. Let us assign various values to the pressure gradient dr=dz in order to find out the various possible velocity profiles. Example 1. From (3.14) follows that if the pressure gradient satisfies the relation dr þ qb ¼ dz
R21 2
ln RR21 ½gðqc q þ mÞ R21 2
ln RR21
R22 R21 4
;
ð3:16Þ
the cylinder is stationary (v1 ¼ 0). In other words under this pressure gradient the shearing forces of the fluid flowing upwards will keep stationary the cylinder. Fig. 4 shows an example computed for g ¼ 1:005 N s=m2 (1005 cp) and dr þ qbz ¼ 9:36 104 Pa=m: dz
ð3:17Þ
The velocity distribution shown was obtained with (3.12). Example 2. If the pressure gradient is greater than that obtained from (3.16), the shearing force of the fluid flowing up is able to move upwards the cylinder. In order to give an exam, Fig. 5 is showing the velocity profile of the fluid for ðdr=dzÞ þ qbz ¼ 1:2 105 Pa=m: If this term reaches the value 1 106 Pa=m the velocity of the fluid in contact with the cylinder is equal with that of the cylinder itself. This case can be obtained from (3.12) by putting the condition that the maximum velocity of the fluid be reached for r ¼ R1 .
Fig. 4. Velocity of the fluid moving upward under the pressure gradient (3.16) which keeps the cylinder stationary.
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(a)
(b)
Fig. 5. Velocity profile for the fluid flowing upwards under a pressure gradient greater than (3.16); the fluid pushing the cylinder upwards: (a) the pressure gradient slightly bigger than (3.17); (b) pressure gradient bigger than (3.16) when the velocity of the cylinder is equal with the maximum velocity of the fluid (and a small plateau exists).
Example 3. This example considers the most common case when term ðdr=dzÞ þ qbz has a smaller value than that obtained from (3.15). Fig. 6 shows an example obtained with the value ðdr=dzÞ þ qbz ¼ 8:0 104 Pa=m and same g as before. In this case most of the fluid is moving upwards, but a thin layer of fluids neighboring the cylinder, the fluid is moving downwards. In the present example v1 ¼ 0:000528 m/s (down) is obtained from (3.14). Example 4. The last example corresponds to very small values for ðdr=dzÞ þ qbz . Let us consider the case when the pressure gradient is zero dr=dz ¼ 0: Introducing this value in (3.14) together with qg ¼ 9:807 103 ð kg=m2 s2 Þ and same value for g as above, we get v1 ¼ 0:00363 m=s (down). The velocity profile follows from (3.12). This time the falling cylinder
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Fig. 6. Under a small pressure gradient, part of the fluid is moving up, but part is moving down with the falling cylinder.
Fig. 7. The velocity profile in the fluid generated by the falling cylinder (pressure gradient is zero).
induces into the fluid a shearing stress which will put it to flow downwards (see Fig. 7). The velocity of the fluid in contact with the tube is certainly zero. 3.3. Velocity profile for closed tube Let us consider now the case when the bottom of the tube is closed and the pressure gradient is generated by the falling cylinder (Figs. 8 and 9). We can write down the condition that the volume of the fluid displaced by the falling cylinder is equal to the volume of the fluid flowing (up and down) between the cylinder and the tube Z
R2
R1
v2pr dr ¼ v1 pR21 :
ð3:18Þ
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Fig. 8. Schematic of the fluid flow in the case of closed tube.
Introducing here v ð¼ f Þ from (3.11) we get 3 Z 2 R2 dr r þ qbz þ C1 r ln r þ C2 r dr ¼ v1 R21 g R1 dz 4
ð3:19Þ
with the values of C1 and C2 from (3.14) and the notations 2 2
R2 R21 R42 R41 R22 R21 2 ln R2 1 1 R22 R22 R21 2 ln R1 R¼ R2 ln R2 ; R1 þ þ 4 4 16 2 2 4 2 4 ln RR2 8 ln RR2 1
1 1 1 ln R2 R2 R21 R21 2 ln R2 2 ln R1 þ ; V ¼ R2 R2 R1 þ R2 2 4 4 2 2 2 2 ln R1 ln R1
1
ð3:20Þ
we finally obtain dr V þ qbz ¼ gjv1 j; dz R
ð3:21Þ
which established another relationship between gv1 and ðdr=dzÞ þ qbz , obtained from a kinematic assumption.
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Fig. 9. Velocity profile and shear stress profile for the case of closed cylinder.
However, the term gv1 as obtained from (3.21) and from (3.15) must be equal. From this condition follows dr þ qbz ¼ dz
R21 2
ln RR21 ½gðqc q þ mÞ R V
R21 2
ln RR21 þ
R22 R21 4
:
ð3:22Þ
This relation determines ðdr=dzÞ þ qbz . The magnetic force is involved as a local density, i.e. if pushing down it is equivalent with a heavier cylinder. After calibration, this observation will be used to speed up some tests with very viscous fluids, and using the very same magnetic cylinder. Introducing this value into (3.21) we obtain a formula for the determination of the viscosity coefficient R dr þ qbz : ð3:23Þ g¼ V jv1 j dz We can consider the expression R a¼ V
R21 2 R V
R21 2
g ln RR21
ln RR21 þ
R22 R21 4
b
ð3:24Þ
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to be a viscometer constant. We have introduced b as a correction parameter, mainly because the cylindrical magnet has sometimes no perfect cylindrical shape, and also to take into account the end effects. Thus the formula for the determination of the viscosity coefficient (3.23) writes g¼a
qc q þ m : jv1 j
ð3:25Þ
3.4. Shearing stress distribution The shearing stress distribution in the layer of fluid is obtained from (3.10) with (3.12), as R2 R2 þ qbz 2 4 1 1 dr r gv1 þ dr dz þ : Trz ¼ þ qbz dz 2 r ln RR2
ð3:26Þ
1
The stress becomes zero there where the fluid velocity is maximum; Trz ¼ 0 or v ¼ vmax for r ¼ r0 . That is obtained from
dr r0 1 þ qbz þ C1 ¼ 0 dz r0 2
i.e. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2C1 r0 ¼ dr þ qbz dz
ð3:27Þ
which determines r0 . From the numerical examples given in this paper one can see that r0 depends strongly on the pressure gradient. Example 5. In order to check the prediction of the above formulae, we have performed experiments with a variety of standardized fluids available in our laboratory. The tests were done with a custom made falling cylinder viscometer. The viscometer data are: R1 ¼ 0:000505 m (cylinder radius) R2 ¼ 0:000805 m (container inner radius) qc ¼ 7228 kg=m3 (cylinder density) qf ¼ 1000 kg=m3 (fluid density) A sample of standard 1000 cp calibration fluid was measured to verify its viscosity using a coneplate viscometer (Brookfield, MA). We have tested 17 samples of this calibration fluid and determined a coefficient of viscosity g ¼ 1067 87 cp compared with 1005 cp as determined by the cone-plate viscometer. The experimental viscosity coefficient is 6.17% higher than the cone-plate value. Though the matching is quite good, the existing difference may be due either to the disregarding in the theory of the end effects, or/and to the non-perfect cylindrical shape of the falling
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Table 1 Summary statistics of all samples computed both experimentally and theoretically Fluid
Viscosity as measured on the Brookfield (cp)
Number of falling cylinder samples
Average falling cylinder velocity (m/s)
Average viscosity as determined from the theory (cp)
Standard deviation of theoretical values
Accuracy % ((theoretical valueBrookfield value)/ Brookfield value)
Variability % (standard deviation/ measured value)
1 2 3
1005 3085 9549
16 17 37
5.25E)04 1.71E)04 6.05E)05
1067 3274 9442
87 241 1564
6.169154229 6.126418152 )1.120536182
8.153701968 7.361026268 16.56428723
cylinder. However, for all practical measurements of viscosities, the accuracy is quite good. The data for all sample fluids is summarized in Table 1. 4. Conclusion A simple theory is presented in which the solution of the flow of a viscous fluid in a falling cylinder viscometer is expressed in a closed form. The formula giving the viscosity of the fluid is also obtained in finite form. Since all the formulae are obtained in finite forms, one can easily study the influence of all parameters involved [as densities (may be variable), geometry of cylinder-tube, pressure gradient, magnetic force, etc.] on the cylinder–fluid motion. The comparison of the viscosity coefficients, as determined by our experiments with the falling cylinder Microrheometer and using the present theory, and as determined with a classical cone-plate viscometers is quite good. It is important to note that the formulae for velocity, stress, viscosity, etc. contain a term that describes the influence of a magnetic field on the motion of the cylinder and thus on the flow of the fluid. However, the influence and use of a magnetic field in viscometer studies will be treated in a future paper. An extension of the analysis to non-homogeneous fluids, viscoplastic materials, and to the case when the fluid slides along the tube and/or cylinder will also be considered.
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