An optical capillary flow viscometer

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An optical capillary flow viscometer

V. Teboul, J. M. StArnaud, T. K. Bose, and I. Gelinas

Citation: Rev. Sci. Instrum. 66, 3985 (1995); doi: 10.1063/1.1145405 View online: http://dx.doi.org/10.1063/1.1145405

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An optical capillary flow viscometer

V. Teboul,@ J. M. St-Arnaud, T. K. Bose, and I. Gelinas

Institut de Recherche sur l’hydrog&e, @partement de Phy&ue, Universite’ du Qukbec, Trois-Rivieres, Quebec GPA 5H7, Canada 1:

.,, i

(Received 13 January 1995; accepted for publication 15 April 1995)

A new capillary flow viscometer is $esented, based on optical: measurements of the refractive index.

The apparatus consisting mainly of :?vo pressure vessels, a caphlary, and a Michelson interferometer for refractive index measurements is described together with the experimental procedure. The viscosity is computed with the Poiseuille formula tacking into account perfect gas variations. The accuracy of the viscometer is then evaluated and tested using three different gases: carbon dioxide, methane, and sulfur hexafluoride. As this technique can easily be automated we think that it will be interesting for practical purposes. Moreover, this technique can be improved to produce results of higher accuracy and it can also be applied with a few modifications to other media like supercritical fluids. 0 1995 American Institute of Physics.

I. INTRODUCTION

In this paper we describe a new technique to measure optically the viscosity of gases. At the fundamental level, the measurement of the viscosity of gases has proved to be of interest as a direct source of thermophysical properties and as a probe for intermolecular potentials.‘-4 At the applied level, viscosity measurements are also of some importance.

For example, in the application of the supercritical extraction technique,5 the viscosity coefficient of the extraction fluid is an important parameter to design the process.

In the present study, a simple experimental method was devised to take advantage of the capabilities for accurate measurements of the refractive index of gases as a function of density. The experiment is based on the well-known capillary flow viscometer technique. Two optical cells, one containing the gas at a given reference pressure and the other emptied by a vacuum pumping device, are connected through a capillary. The originality of our viscometer is that the flow of the gas through the capillary was measured using the decrease of the refractive index in the optical cell which contained the gas at the initial time instead of measuring the pressure decrease in the cell.

Our paper’is organized as follows. In Sec. II the experimental setup and procedures used are described. In Sec. III, from the Poiseuille equation, a simple formula linking the viscosity coefficient with the experimental data is obtained.

In Sec. IV the precision of our viscometer is discussed, and as a test the viscosities of a few gases measured with our viscometer are compared with the literature values.

II. EXPERIMENTAL SETUP

The viscometer (see Fig. 1) is composed of a stainless steel optical cell of 30 cm3 internal volume, which is connected through a capillaryto another almost identical cell. At

‘)Permanent address: Universit6 d’Angers, Laboratoire des Propri6tBs Op- tiques des Matdriaux et Applications, 2 Boulevard Lavoisier 49000 Angers,

FWKe.

the beginning of the experiment the optical cell was filled with the gas studied and the other cell was emptied continu- ously by a vacuum pumping device. A Swagelok valve closed the capillary and another valve closed the optical cell.

The initial pressure was measured by a Texas Bourdon PPG 145-01 manometer. The capillary valve was then opened and the flow of gas through the capillary was measured using the decrease of density in the optical cell. We measured the change in density via the change in the optical path inside the cell, using an Hewlett-Packard Michelson ,,interferometer and an helium-neon laser. ‘v- 2

The capillary used was 3 m long, of stainless steel, with a 0.15 mm i.d. To ensure thermal stability the cells and capillary were immersed in a polyethylene glycol bath. The temperature of the bath was regulated, and controlled with two Cole-Parmer thermal resistors with a resolution of 0.01 K, previously calibrated with a platinum resistance thermom- eter. We never found differences higher than 0.02 K between the two measured temperatures and 0.04 K during the total expansion. The gases used were high purity grade and sup- plied by Union Carbide with less than 5 ppmv total impuri- ties.

The overall accuracy of our measurements is limited by the stability of the optical system, the number of fringes counted by the interferometer, preceding measurements -of the first optical refractive index virial coefficient,6 and the accuracy of the’ initial pressure measurement, although the initial pressure can be checked with the interferometer. The experimental device is tested at low pressures to ensure laminar flow. We estimate the accuracy of our test to be roughly 1.4% although’our mean relative measurements were only 0.8% different from previous results.7-*3 The accuracy is dependent on the refractive index of the gas used, and gases with higher refractive index will give better results.

Ill. MATHEMATICAL DEVELOPMENT

The quantities measured are the initial pressure of the gas, the temperature of the whole system, and the evolution of the number of interference fringes seen by the Michelson

Rev. Sci. Instrum. 66 CI), July 1995 0034-6740/95/66(7)13985/4/$6.0ij 0 1995 American Institute of Physics 3985

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YALVE

MlCHELSON lNERFEROMarER

FIG. 1. Schematic representation of the apparatus.

interferometer as a function of time. We will fiow try to obtain a simple formula linking these data to the viscosity coefficient.

Using the capillary flow viscometer we first had to cali- brate our viscometer with a reference gas or equivalently to make relative measurements. So we have recorded the fringe numbers as a function, of time for several gases CH4, SF,, and CO2 at the same temperature 300 K and the same initial pressures 0.3 ,and 0.15 bar, respectively.

The number of fringes is connected to the pressure via the density. For.the same ratio of the total number of fringes on the first virial coefficient, the pressure is almost the same for different gases. To be more accurate one’has to take into account the second virial coefficient. Although the measurement of two different pressures on these curves are sufficient in principle to obtain the relative viscosity of the gases in- volved, by’ measurement of the entire curves one can obtain more accurate results. We have used this method, computing the megn viscosity from the different results obtained with the whole set of two points on the recorded curves.

Frok the Poiseuille equation for laminar flow expansions and compressible gases we obtain the pressure depen- dence with time3’14

J,= - (m4/8 g)pdP/dx. (1)

In this equation J, tepresents the mole fldw rate of gas through the capillary, r is the capillary radius, 71 the viscosity coefficient, P Andy p, respectively, the pressure and molar density at a distance x along the capillary.

Using the virial expansion of pressure and neglecting the third-order term

P=RTp(l+Bpp), (2)

where T is the temperature of the gas, R the coefficient of perfect gases, and B, the second pressure virial coefficient.

Then replacing the pressure in Eq. (1) by its expansion we find

J,= - ( n-r418 v)RTp( 1 +2Bpp)dpldx. (3) By integrating this equation along the capillary length d:

I

J&x=-(m-“/81;I)RT p(l+2Bpp)dp

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3966 Rev. Sci. Instrum., Vol. 66, No. 7, July 1995

and then remembering that J, is constant along the capillary length, and writing J, as a function of density p1 and volume V, ‘of the cell containing the gas

d Vldpl/dt=-(m4/16~jRTp~(1+4BPp1/3),

is)

we now can use the virial expansion of the refractive index II to obtain a relation between the number of fringes and the density. Up to the second’order we have

hK/4L=(n-l)=A,pI+B,p;+... . (6)

K is the number of fringes at density p1 taking the vacuum measurement as a reference, A, and B, are virial coefficients defined in a preceding papery6 X is the wavelength of the laser, and L is the internal length of the optical cell containing the gas. If the density is low enough one can invert this relation and obtain

p1=(XK/4A,L)-B,/A,(XK/4A,L)2+... . (7) Replacing this equation in Eq. {S) we then obtain the number of fringes evolving with time

dK/dt(X/A,4L)( 1 -2K/A,2(X/4L)2B,/A,)

= (m4RT/16 ~dVIj((XK/4A,L)2

-2(XK/4A,Lj3B,/A,)(1+4B,(XK/4A,Lj+~~~ . (8) Up to the second order in density this reduces linally to

‘dK/dt(X/4A,L)= ( rr4RT/16 ~dV1j(XK/4AnL j2[ 1 +(XK/4AnL)(4Bp/3-2B,/A,)].

(9) Then by integrating this equation over the time of expansion At:

At=(16~dVI/ar4RT)(4A,L/X) dK/K2[1 I

+(XK/4A,L)(4B,/3+2B,/An)]. (10)

From Eq. (9) or Eq. (10) one can obtain absolute measurements of the viscosity, but due to the lack of accuracy arising from the uncertainty of the capillary radius r, absolute measurements are scarcely obtained from capillary flow viscometers. InStead, the apparatus constants are measured with a gas whose viscosity is chosen as a standard, or equivalently one can measure viscosity ratios of different gases. As a test of laminar flow one can compare the fringe number evolution of the gas chosen as standard with the calculated value. We found for SF6:

AK/Atmeasured=1.2X10-2 fringes/s and AK/Atcalcu’ated

=1.06X 10d2 fringes/s, which differs by an amount consis- tent with the 10% tolerance on our capillary radius.

If density is low enough, Eq. (10) leads to two times measurements involving two gases (I and II) at the same temperatures, with the same initial and final fringe numbers (K, and Kz)

Capillary flow viscometer

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TABLE I. Viscosities measured with our apparatus for COz and CHH, using SFs as a standard compared with preceding measurements.

Measured viscosities in ,uP First virial refractive

(Micro Poiseuillej index coefficient

taking SF, as a Measurements of used in the standard other authors in P calculation

(300 K) (300 K) (cm3/mol)

co, 151.4 150.5 (Ref. 9) 9.975 (Ref. 6)

149.9 (Ref. 8) 150.1 (Ref. 10) 150.1 (Ref. 11)

cH.$ 112.0 111.8 (Ref. 7) 9.990 (Ref. 6)

112.4 (Ref. 12)

SF, 153.3 (standard) 153.3 (Ref. 13) 17.01 (Ref. 6) 152.4 (Ref. 11)

153.9 (Ref. 12)

-2B,n/A;,X/4LA;- (4B’,/3

ill) If we chose on the curves K(t), K, approaching K, , we can neglect the term which is proportional to ln(KaIK,) to avoid the second term calculation which will decrease the accuracy of the measurements and find

~“I~‘= (A~~At=>l(A;At’). w

In a first approximation one can then directly obtain the relative viscosities by a ratio of the times of expansions. To obtain mean measurements for the viscosity coefficient using E$. (12), one can compute for a small chosen variation of fringe number, the ratio of the flow durations, for the same

initial fringe number in each gas describing the curve K(t), the fringe numbers being corrected from the first refractive index virial coefficient. Then to increase the accuracy small corrections have to be taken into account.7’14-1” These corrections are dependent on the experimental apparatus, gases, and thermodynamic conditions used.

IV. RESULTS AND PRECISION OF THE VISCOMETER We have applied this technique to three gases, chosen for their high refractive index, first virial coefficient, or for the number of experimental data available to be compared with our results. We have measured the dynamic viscosity of carbon dioxide, methane, and sulfur hexatluoride at room temperature, and low pressures. Our results are listed in Table I, together with the reference data and values of the refractive index coefficients used for the calculations. These results are in close agreement with the reference values. The accuracy of measurements increases as the gas is (heavier) and has higher refractive index. Although the agreement is quite good between our measurements and the previously published experimental data, the real accuracy of our measurements is lower than it would appear from this comparison.

Let us now scan the different sources of inaccuracies.

A. Inaccuracy in the jirst repactive index virial coefi- cient: We estimate this inaccuracy to be around 0.2% and it leads to an accuracy of 0.4% in the ratio of the two gases compared. This inaccuracy cannot be removed in this technique, but if one uses higher pressures or higher refractive index, such as supercritical state elements, an expansion between two closed cells containing the element at different pressures will eliminate the need to know the refractive index coefficient. In this case one cell can be placed on each arm of the Michelson interferometer and a different development of expression (1) has to be used.3

3. The interferometer precision: Although our interferometer was able to measure 0.1 fringe which leads to an accuracy of A/40, we found it rather inaccurate below 0.4 fringes (X/10), this leads in our case to 1% error. Using re- peated expansions this error was decreased to 0.5%.

C. Optical stability during the expansion: Due to vibra- tions the optical alignment can move during the experiment, this leads to a variation in the fringe number; therefore the optical path inside and outside the cell is increased. In our expansions of several hours long, we found a maximum variation of 1.0 fringe. However, due to the kind of calculation performed to obtain the viscosity from the fringe number variation, using very short time differences, and eliminat- ing singular points, the error arising from these relatively brutal changes was decreased to roughly 0.4%.

D. Temperature variations of the air during the experiment: The temperature and pressure variations of the air during the experiment, partly compensated by the variation of the path length in the other branch of the interferometer which is in the air too, lead to an inaccuracy of approxi- mately 0.1% which can be mainly corrected if one records the temperature and pressure variation during the experiment.

E. Time measurements: Time was measured in our case with an electronic clock with a relative precision we estimate to be better than 0.02%

F: Pressure measurements: Pressure was measured with a Texas Bourdon manometer. The accuracy of this measurement was 0.1%.

Calculating the total relative precision of our measurements we then find 1.4%. As shown in Table I the results obtained with the three gases tested agree with preceding measurements well within this value. We think that this precision can be highly improved using more precise instru- ments. The main remaining error will be the inaccuracy on the first virial refractive index coefficient, leading to an overall accuracy of 0.5% for most gases. As the accuracy in the first refractive index is being improved by new measurements, this technique could become very accurate.

ACKNOWLEDGMENTS

We would like to thank Dr. Ahmed Houri and Dr. Rich- ard Okambawa for fruitful discussions.

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3988 Rev. Sci. Instrum., Vol. 66, No. 7, July 1995 Capillary flow viscometer

An optical capillary flow viscometer

An optical capillary flow viscometer

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