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Measuring the refractive index of crude oil using a capillary tube interferometer
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Optics & Laser Technology 35 (2003) 361 – 367 www.elsevier.com/locate/optlastec
Measuring the refractive index of crude oil using a capillary tube interferometer H. El Ghandoora , E. Hegazib;∗ , Ibraheem Nasserc , G.M. Beheryd b Laser
a Physics Department, Faculty of Science, Ain Shams University, Cairo, Egypt Research Section, Center for Applied Physical Science, Research Institute, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia c Physics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia d Mathematics Department, Faculty of Science, Mansoura University, Damietta, Egypt
Received 25 September 2002; received in revised form 20 January 2003; accepted 27 January 2003
Abstract A method for measuring the refractive index of low-transparent crude oils using a capillary tube interferometer is described. The method is based on analyzing the resulting transverse interference fringe patterns in terms of their positions with respect to the lens/capillary tube interferometer. The refractive indices of seven blended crude oils of low transparency were measured with accuracy of up to six decimal digits and were related to the API gravity of the oils. The ray tracing inside the capillary tube is explained and the transverse bell-shaped interference fringes are interpreted. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Interferometry; Refractive index; Non-fractionated crude oil
1. Introduction The refractive index is an important optical parameter for crude oils that is used in various calculations related to their speci7c compositions [1], in measuring oil slick thickness [2], and in a number of industrial applications such as predicting the onset of asphaltene precipitation [3] and measuring the solubility parameter [4]. However, there has been no formal endorsement for the refractive index to be used as a true identi7cation parameter for non-distilled (non-fractionated) crude oil. The reason behind this has to do with the low transparency and the high volatility characteristics of crude oils in addition to the lack of a simple measuring method that can report their refractive indices with adequate accuracy. Refractometric methods, for example, can attain accuracy in the vicinity of 10−4 only, and they are suitable only for the transparent fractions of crude oils (light fractions). They cannot be used, however, to measure the refractive indices of the non-fractionated crude oils directly because of the low-transparent nature of these oils. ∗
Corresponding author. CAPS, Research Institute, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia. Tel.: +966-3-860-4343; fax: +966-3-860-4281. E-mail address: [email protected] (E. Hegazi).
In such cases only estimates of these refractive indices can be given by extrapolating data from several oil/hydrocarbon mixtures of these crude oils [4]. To measure the refractive index of liquids with accuracy better than 10−4 , on the other hand, interferometric techniques must be used instead. The literature is rich of diGerent interference-based methods for measuring the refractive index of liquids [5–15]. However, the majority of these methods require a reference sample, and they are either not suitable for low transparent samples or they employ complicated setups that require a number of optical components and delicate alignments. In the present work we describe a simple interferometric method for measuring the refractive index of low-transparent non-fractionated crude oils, which require neither a reference sample nor an elaborate setup. The method was recently developed to examine optical 7bers [16,17], and has been modi7ed in this work to include capillary tubes 7lled with crude oil samples in place of the optical 7bers. It diGers from the other capillary-tube methods in that the laser beam impinges on the capillary tubes in a form of a sheet that covers the whole width of the capillary tubes as opposed to a single narrow beam hitting particular areas. Consequently, the transverse interference fringes will be formed in a vertical arrangement with locations directly
0030-3992/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0030-3992(03)00029-X
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H. El Ghandoor et al. / Optics & Laser Technology 35 (2003) 361 – 367
related to the angles of refractions and, hence, to the refractive index of the liquid inside the capillary tube. The method is applied on non-fractionated petroleum crude oils to measure their refractive indices with accuracy of up to 10−6 . The ray tracing inside the capillary tube is explained and the resulting interference patterns are interpreted. The diGerence between the observed interference patterns and the ones observed in the case of the optical 7bers is also highlighted.
2. Experimental setup A schematic diagram of the experimental setup is shown in Fig. 1. Capillary tubes (inner and outer diameters 3 and 4 mm, respectively) 7lled with crude oils were placed on a 7xed holder, one at a time, and were illuminated by a laser sheet from a 50-mW HeNe laser in the z-direction. The sheet was formed by passing the laser beam through a microscope-objective lens, a collimating lens, and a cylindrical lens whose axis was parallel to the axis of the capillary tube (along the x-direction). The formed sheet had a length of approximately 5 cm, and a thickness wide enough to illuminate the whole width of the capillary tube. The resulting interference pattern was formed at the capillary tube itself and also in the region behind it where it was projected onto a 7xed screened whose plane was kept perpendicular to the z-axis. A digital image of the projected interference pattern was then taken using a CCD camera, stored in a computer and later retrieved for carrying out the analysis. Seven blended petroleum crude oils, named as A, B, C, D, E, F, and G were used in the experiment and they were so chosen because of the wide range in their grades. The grades were identi7ed in terms of the American Petroleum Institute (API) gravity, which is based on speci7c gravity values of the 250 –275◦ C (1 atm) and the 275 –300◦ C (40 mm) distillation fractions. The values of the API gravity of these crude oils were A: 49.32, B: 40.2, C: 40.0, D: 32.2, E: 29.4, F: 27.8, and G: 27.1, which corresponded to grades ranging
from Super Light for A (the highest API gravity) down to Heavy for F and G (the lowest API gravity). The thin capillary tubes made it possible for the interference patterns from the low transparent crude oils to be visible in transmission, while the design of the capillary-tube holder made it possible to record the interference patterns immediately after securing the crude oil sample in place, which negated any possibility of altering the temperature of the sample or volatizing its lighter aromatic compounds. The measurements on the oil samples were carried out independently from each other. Therefore the analysis of the resulting interference image of each crude oil sample was made on all the interference fringes to search for the correct interference order. This was done by consulting the refractive index values obtained using an Abbe refractometer for two of the crude oils, D and E. All samples were treated under the same exact experimental conditions of temperature (20◦ C) and duration of exposure. 3. Results and discussion Images of the resulting transverse interference patterns for all of the seven crude oils are shown in Fig. 2. The patterns comprises of only a few number of interference fringes, which were found to be typical for homogeneous liquids and could be explained in terms of two-beam type interference between pairs of laser rays passing through diGerent optical media. Fig. 3 shows the ray-tracing diagrams of a few of the possible combinations that can produce interference fringes. A fringe can be formed as a result of interference between R3, which passes entirely through the wall of the capillary tube, and R6, which passes perpendicularly through the middle of the capillary tube. Another fringe can be formed by the interference between a laser ray that enters the oil medium at an angle, i.e., R5, and one of the non-refracted rays that passes through the air on top of the capillary tube, i.e., R2, and a diGerent fringe can also be formed due to interference between R4, a ray that suGers total internal
Fig. 1. Schematic diagram of the experimental setup.
H. El Ghandoor et al. / Optics & Laser Technology 35 (2003) 361 – 367
363
Fig. 2. Raw images of the transverse interference patterns for all the seven crude oils as recorded using the CCD camera.
Fig. 3. Ray tracing of selected rays impinging on the upper hemisphere of the capillary tube. The locations of three possible fringes are shown on the screen at the right.
reLection at the glass/oil interface, and another ray from the group that passes on top of the capillary tube, i.e., R1. The interference patterns shown in Fig. 2 corresponds to the upper half of the capillary tube only; those corresponding to the bottom half are symmetrically inverted in shape and are not shown in the 7gure.
The reason that the interference fringes are bell-shaped has to do with two combined factors. The 7rst is the curved surface area of the capillary tube and the second is the fact that the combination of the lenses in the setup creates a thin laser sheet whose rays are parallel along the y-axis only but not along the x-axis. Therefore, along the x-axis, the
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H. El Ghandoor et al. / Optics & Laser Technology 35 (2003) 361 – 367
axis of the capillary tube itself, the laser sheet acts as a one-dimensional extended source whose rays travel through the media in a tilted, but symmetric, manner. This leads to the formation of tip, corresponding to the ray that is exactly perpendicular to the x-axis, and two symmetric wings, corresponding to pairs of rays having the same angle at each side of the perpendicular ray (in the x–z plane). The optical paths of these six rays between the tangential lines l1 and l2 de7ned in Fig. 3 can be expressed in terms of the refractive indices of the air (nai ), glass (ngl ), and liquid (nli ), without taking into consideration the phase change occurring at the media interfaces, as follows: OPL1 = 2nai Rou ;
(1)
OPL2 = 2nai Rou ;
(2)
OPL3 = 2nai Rou (1 − cos A ) + ngl AB;
(3)
OPL4 = 2nai Rou (1 − cos C ) + 2ngl CD;
(4)
OPL5 = 2nai Rou (1 − cos F ) + 2ngl FG + nli GH;
(5)
OPL6 = 2ngl (Rou − Rin ) + 2nli Rin ;
(6)
where Rin and Rou are the inner and outer radii of the capillary tube, respectively, A , C , and F are the incident angles at points A, C, and F, respectively. (Explicit equations for these optical path diGerences in terms of the angles of incidence can be found in Ref. [18]). In the following we explain, mathematically, how the fringes appear as bell-shaped by deriving two equations of the optical path diGerence between two pairs of rays. The 7rst is the optical path diGerence between a ray that passes through the oil, i.e., Ray R5 and a reference ray, say R1, as a function of incident angle, and the second between a ray that passes through the wall of the tube only, i.e., R3, and the same reference ray R1. We assume that the diGraction and scattering eGects due to reLection are negligible. Consider a sheet of laser light is incident at point F with incident angle F , where −arcsin(ngl (Rin =Rou )) ¡ F ¡ arcsin(ngl (Rin =Rou )), where the origin of the Cartesian coordinates is at point O. At point F, the Cartesian coordinates (XF ; YF ) are then: YF = Rou sin F , and XF = −Rou cos F , and Snell’s law, applied at the air/glass interface, is: nai sin F = ngl sin ’F ;
(7)
where ’F is the angle of refraction at point F. Accordingly, the equation of a straight line that joins points F and G would be: Y = m1 X + b 1 ;
(8)
where m1 = −tan F , b1 = (YF − m1 XF ) and F = F − ’F . Therefore, the Cartesian coordinates of point G then become: −m1 b1 ± R2in + m21 R2in − b21 XG = (9) 1 + m21 and YG =
R2in − XG2 :
(10)
In Eq. (9), only the negative sign is of interest since XG ends up to the left of the origin O. For simplicity, we will consider FG ≈ HI ≈ (Rou − Rin ), which is the thickness of the capillary tube, Hence GH = (XG − XH )2 + (YG − YH )2 ≈ (XG − XH ) ≈ 2|XG |;
(11)
where Z is the intercept point with the y-axis. Therefore, Eq. (5) reduces to: OPL5 = 2Rou (1 − cos F ) + 2ngl FG + 2nli |XG |:
(12)
Following the same reasoning, the optical path length of ray R3 (OPL3 ) can be evaluated by expressing the Cartesian coordinates at point B (XB ; YB ) as: −m2 b2 + R2ou + m22 R2ou − b22 XB = ; (13) 1 + m22 YB = R2ou − XB2 ; (14) where m2 = −tan A , b2 = (YA − m2 XA ), and A = A − ’A , where this case in the range {−=2; −=2}; A will take the values −=2 ¡ A ¡ − arcsin(ngl Rin =Rou ) and arcsin(ngl Rin =Rou ) ¡ A ¡ =2. Fig. 4 shows plots of the optical path diGerences (OPL5 − OPL1 ) and (OPL3 − OPL1 ) a s functions of the incident angle in the range {−=2; =2} for the values R=Rou =Rin =1:8, ngl = 1:5 and nli = 1:477. The central bell-shaped curve, appearing between the solid vertical lines, is (OPL5 − OPL1 ), while the tails appearing to the left and the right of the solid vertical lines represent (OPL3 − OPL1 ). It is interesting to note that if the ratio R = Rou =Rin is less than ¡ 1:5, then the tails due to (OPL3 − OPL1 ) would disappear in Fig. 4 indicating that it would not be possible to observe interference fringes due to Rays R3 and R1. This is the reason why in Fig. 2 no such tails appear (R = Rou =Rin = 1:33 in our experiment). Before an accurate analysis can be made the images in Fig. 2 had to be digitally processed. This was done by using MATLAB-based software that was developed to conduct a search for the maximum and minimum brightness of the fringes and to report them as functions of position (x and y). This allowed highly contrasted constructive and destructive interference fringes to be exactly de7ned. Fig. 5 shows the outcome of this digital processing on the transverse interference patterns of crude oils D and F.
H. El Ghandoor et al. / Optics & Laser Technology 35 (2003) 361 – 367
Fig. 4. Optical path diGerences OPL5 − OPL1 and OPL3 − OPL1 plotted as functions of incident angle for the values R = Rou =Rin = 1:8, ngl = 1:5 and nli = 1:477. The curved pattern between the two middle solid lines simulates the bell-shaped interference fringe, which is due to interference between a ray passing through the glass and oil, i.e., R5, and a reference ray, e.g., R1. The tails outside the solid lines simulates an interference fringe of diGerent shape resulting from the interference between a ray passing through the glass wall of the capillary tube, i.e., R3, and R1.
The absolute value of the refractive index of the medium enclosed within the capillary tube can be determined by measuring the deLection angles (), de7ned as the angles between the R6 ray and the heights of the observed constructive (or destructive) interference fringes (y–z plane). To measure these deLection angles, in practice, it is convenient to employ Abel’s transform as follows. For a light rays de7ned as orthogonal projection from geometrical wave fronts, the equation relating the path of these rays to the local index of refraction is [19] d dr(s) n = ∇n; (16) ds ds where r(s) is the position vector of a point on the ray and s is the arc length of the ray. Choosing Cartesian coordinates system such that the axis of the capillary tube is in the x-direction and the incident plane monochromatic wave is along the positive z-axis, and using the paraxial
365
Fig. 5. Examples of highly contrasted interference patterns (crude oils D and F) produced by digitally processing the raw images.
approximation, ds ≈ d z this equation reduces to: d dr n = ∇n dz dz
(17)
where r = xi + yj + zk, and ∇n = (9n=9x)i + (9n=9y)j + (9n=9z)k. Since the impinging laser light into the capillary tube suGers refraction due to variation of refractive index in the z-direction, then at a particular height of the capillary tube n can be taken as independent of x. Then Eq. (17) becomes: d dy 9n n = : (18) dz dz 9y Integration of this equation with respect to z gives dy 1 9n d x: = no 9y dz
(19)
In two-dimensional cylindrical coordinates, we have r 2 = z 2 + y2 . Consequently, dz =
r dr r2
−
y2
; and
y 9r = : 9y r
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H. El Ghandoor et al. / Optics & Laser Technology 35 (2003) 361 – 367
Finally, it is interesting to note that, under the same experimental conditions, the number of interference fringes appearing in the images of the crude oils is found to be considerably fewer than those appearing in the case of optical 7bers [12]. This is due to the fact that in optical 7bers the refractive index changes gradually with the radius of the 7ber, while in the crude oil this change occurs in more discrete steps. In other words, if the inside of the 7ber were to be made out of crude oil then one would be able to describe the change in the refractive index simply in terms of discrete layers each of which has a constant refractive index. We estimate the widths of these layers to be ∼ 0:5 m for the samples used in this work. Fig. 6. A Plot of the oil’s refractive index as a function of the oil’s API gravity.
If is the angle of refraction of the rays that pass through the capillary tube, then +∞ dy 9n 9r 1 = = dz dz no −∞ 9r 9y +∞ r 9n y 1 dr = no −∞ 9r r r 2 − y2 dr 2y +∞ 9n = : (20) 2 no 0 9r r − y2 This is the form of Abel’s transform whose inverse is [20] n(r) − nf 1 f dy =− ; (21) nf i y2 − r 2 where i = r and f = ∞ are the limits of the phase object and nf is the index of refraction at f, which is ngl = 32 in our case. By using Eq. (21) the refractive indices of the seven crude oils were determined as A(1.446870), B(1.453822), C(1.460520), D(1.474833), E(1.481889), F(1.488885), and G(1.498860). Fig. 6 shows a plot of the refractive index as a function of the crude oils’ API gravity, where the line represents, arbitrarily, a polynomial 7t of second order. The plot demonstrates a decrease in refractive index with the increase of API gravity as expected. The accuracy of these data points is based on the parameters that de7ne the locations of the fringes i.e., ; y and r, which in turn correspond to displacements that are proportional to 12 , and is estimated to be 5 × 10−6 . It was important that one of the interference patterns be recorded with an actual length scale displayed in the background so that the magni7cation factor of the recorded images could be determined. Another important point to mention here is the accuracy of the digitally reported values of the brightness in the images. Care should be taken not to saturate the CCD camera with intense light, as this would compromise the validity of the reported brightness values.
4. Conclusion The work in this paper represents the 7rst attempt to measure the refractive index of the low transparent crude oil using a capillary tube interferometer. The oil-7lled capillary tubes were illuminated by a thick HeNe laser sheet and the resulting transverse interference patterns were projected on a screen whose plane was perpendicular to that of the laser sheet. As expected, the patterns showed bell-shaped fringes as in the case of optical 7bers except that the number of the fringes in this case was much smaller. The characteristic bell shape was demonstrated mathematically by calculating the optical path diGerence between a reference ray and a ray that passes through the oil sample. The refractive indices of the crude oil samples were determined with accuracy of 5 × 10−6 by measuring the deLection angles of the fringes and then by using Eq. (12). The high accuracy obtained makes this technique a valuable one for measuring the refractive index of crude oil without having to extrapolate it from the oil’s thermal distillates. It may also prove useful as a tool for 7ngerprinting crude oils. Acknowledgements The authors wish to acknowledge the support of the Research Institute and the Physics Department of King Fahd University of Petroleum and Minerals. References [1] Speight JG. The chemistry and technology of petroleum. New York: M. Dekker, 1980. [2] Otremba Z. Opt Express 2000;7:129–34. [3] Buckley JS. Energy Fuels 1999;13:328–32. [4] Buckley JS, Hirasaki GJ, Liu Y, Von Drasek S, Wang JX, Gill BS. Petroleum Sci Technol 1998;16:251–85. [5] Deng Y, Li B. Appl Opt 1998;37(6):998–1005. [6] Tarigan H, Neill P, Kenmore CK, Bornhop DJ. Anal Chem 1996;68:1762–70. [7] Menn ML, Lotrian J. J Phys D 2001;34:1256–65. [8] Marhic ME, Stein EP. Appl Phys Lett 1977;35:1678–82.
H. El Ghandoor et al. / Optics & Laser Technology 35 (2003) 361 – 367 [9] Saunder MJ, Gardner WB. Appl Opt 1977;16:2368–71. [10] Barakat N, Hamza AA, Gonied A. Appl Opt 1983;24:4383–6. [11] Barakat N, Hamed AM, El-Ghandoor H. Optik 1987;76(3): 102–5. [12] Barakat N, El-Ghandoor H, Hamed AM, Labib S. Exp Fluids 1993;16:42–5. [13] Barakat N, El-Ghandoor H, Hamed AM. Opt Laser Technol 1989;21(5):328–30. [14] Alexandrov SA, Chernyh IV. Opt Eng 2000;39:2480–6.
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[15] Zaiat SY, El-Henawi HA. Meas Sci Technol 1996;7(8):1119–23. [16] El-Ghandoor H, El-Ghafar EA, Hassan R. Opt Laser Technol 1999;31:481–8. [17] El-Ghandoor H, Nasser I, Rahman MA, Hassan R. Opt Laser Technol 2000;32:281–6. [18] Synovec RE. Ana Chem 1987;59:2877–84. [19] Born M, Wolf E. Principle of optics. New York: Pergamon, 1970. [20] Bracewell R. The Fourier and its applications. New York: McGraw-Hill, 1978. p. 262.
Optics & Laser Technology 35 (2003) 361 – 367 www.elsevier.com/locate/optlastec
Measuring the refractive index of crude oil using a capillary tube interferometer H. El Ghandoora , E. Hegazib;∗ , Ibraheem Nasserc , G.M. Beheryd b Laser
a Physics Department, Faculty of Science, Ain Shams University, Cairo, Egypt Research Section, Center for Applied Physical Science, Research Institute, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia c Physics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia d Mathematics Department, Faculty of Science, Mansoura University, Damietta, Egypt
Received 25 September 2002; received in revised form 20 January 2003; accepted 27 January 2003
Abstract A method for measuring the refractive index of low-transparent crude oils using a capillary tube interferometer is described. The method is based on analyzing the resulting transverse interference fringe patterns in terms of their positions with respect to the lens/capillary tube interferometer. The refractive indices of seven blended crude oils of low transparency were measured with accuracy of up to six decimal digits and were related to the API gravity of the oils. The ray tracing inside the capillary tube is explained and the transverse bell-shaped interference fringes are interpreted. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Interferometry; Refractive index; Non-fractionated crude oil
1. Introduction The refractive index is an important optical parameter for crude oils that is used in various calculations related to their speci7c compositions [1], in measuring oil slick thickness [2], and in a number of industrial applications such as predicting the onset of asphaltene precipitation [3] and measuring the solubility parameter [4]. However, there has been no formal endorsement for the refractive index to be used as a true identi7cation parameter for non-distilled (non-fractionated) crude oil. The reason behind this has to do with the low transparency and the high volatility characteristics of crude oils in addition to the lack of a simple measuring method that can report their refractive indices with adequate accuracy. Refractometric methods, for example, can attain accuracy in the vicinity of 10−4 only, and they are suitable only for the transparent fractions of crude oils (light fractions). They cannot be used, however, to measure the refractive indices of the non-fractionated crude oils directly because of the low-transparent nature of these oils. ∗
Corresponding author. CAPS, Research Institute, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia. Tel.: +966-3-860-4343; fax: +966-3-860-4281. E-mail address: [email protected] (E. Hegazi).
In such cases only estimates of these refractive indices can be given by extrapolating data from several oil/hydrocarbon mixtures of these crude oils [4]. To measure the refractive index of liquids with accuracy better than 10−4 , on the other hand, interferometric techniques must be used instead. The literature is rich of diGerent interference-based methods for measuring the refractive index of liquids [5–15]. However, the majority of these methods require a reference sample, and they are either not suitable for low transparent samples or they employ complicated setups that require a number of optical components and delicate alignments. In the present work we describe a simple interferometric method for measuring the refractive index of low-transparent non-fractionated crude oils, which require neither a reference sample nor an elaborate setup. The method was recently developed to examine optical 7bers [16,17], and has been modi7ed in this work to include capillary tubes 7lled with crude oil samples in place of the optical 7bers. It diGers from the other capillary-tube methods in that the laser beam impinges on the capillary tubes in a form of a sheet that covers the whole width of the capillary tubes as opposed to a single narrow beam hitting particular areas. Consequently, the transverse interference fringes will be formed in a vertical arrangement with locations directly
0030-3992/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0030-3992(03)00029-X
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H. El Ghandoor et al. / Optics & Laser Technology 35 (2003) 361 – 367
related to the angles of refractions and, hence, to the refractive index of the liquid inside the capillary tube. The method is applied on non-fractionated petroleum crude oils to measure their refractive indices with accuracy of up to 10−6 . The ray tracing inside the capillary tube is explained and the resulting interference patterns are interpreted. The diGerence between the observed interference patterns and the ones observed in the case of the optical 7bers is also highlighted.
2. Experimental setup A schematic diagram of the experimental setup is shown in Fig. 1. Capillary tubes (inner and outer diameters 3 and 4 mm, respectively) 7lled with crude oils were placed on a 7xed holder, one at a time, and were illuminated by a laser sheet from a 50-mW HeNe laser in the z-direction. The sheet was formed by passing the laser beam through a microscope-objective lens, a collimating lens, and a cylindrical lens whose axis was parallel to the axis of the capillary tube (along the x-direction). The formed sheet had a length of approximately 5 cm, and a thickness wide enough to illuminate the whole width of the capillary tube. The resulting interference pattern was formed at the capillary tube itself and also in the region behind it where it was projected onto a 7xed screened whose plane was kept perpendicular to the z-axis. A digital image of the projected interference pattern was then taken using a CCD camera, stored in a computer and later retrieved for carrying out the analysis. Seven blended petroleum crude oils, named as A, B, C, D, E, F, and G were used in the experiment and they were so chosen because of the wide range in their grades. The grades were identi7ed in terms of the American Petroleum Institute (API) gravity, which is based on speci7c gravity values of the 250 –275◦ C (1 atm) and the 275 –300◦ C (40 mm) distillation fractions. The values of the API gravity of these crude oils were A: 49.32, B: 40.2, C: 40.0, D: 32.2, E: 29.4, F: 27.8, and G: 27.1, which corresponded to grades ranging
from Super Light for A (the highest API gravity) down to Heavy for F and G (the lowest API gravity). The thin capillary tubes made it possible for the interference patterns from the low transparent crude oils to be visible in transmission, while the design of the capillary-tube holder made it possible to record the interference patterns immediately after securing the crude oil sample in place, which negated any possibility of altering the temperature of the sample or volatizing its lighter aromatic compounds. The measurements on the oil samples were carried out independently from each other. Therefore the analysis of the resulting interference image of each crude oil sample was made on all the interference fringes to search for the correct interference order. This was done by consulting the refractive index values obtained using an Abbe refractometer for two of the crude oils, D and E. All samples were treated under the same exact experimental conditions of temperature (20◦ C) and duration of exposure. 3. Results and discussion Images of the resulting transverse interference patterns for all of the seven crude oils are shown in Fig. 2. The patterns comprises of only a few number of interference fringes, which were found to be typical for homogeneous liquids and could be explained in terms of two-beam type interference between pairs of laser rays passing through diGerent optical media. Fig. 3 shows the ray-tracing diagrams of a few of the possible combinations that can produce interference fringes. A fringe can be formed as a result of interference between R3, which passes entirely through the wall of the capillary tube, and R6, which passes perpendicularly through the middle of the capillary tube. Another fringe can be formed by the interference between a laser ray that enters the oil medium at an angle, i.e., R5, and one of the non-refracted rays that passes through the air on top of the capillary tube, i.e., R2, and a diGerent fringe can also be formed due to interference between R4, a ray that suGers total internal
Fig. 1. Schematic diagram of the experimental setup.
H. El Ghandoor et al. / Optics & Laser Technology 35 (2003) 361 – 367
363
Fig. 2. Raw images of the transverse interference patterns for all the seven crude oils as recorded using the CCD camera.
Fig. 3. Ray tracing of selected rays impinging on the upper hemisphere of the capillary tube. The locations of three possible fringes are shown on the screen at the right.
reLection at the glass/oil interface, and another ray from the group that passes on top of the capillary tube, i.e., R1. The interference patterns shown in Fig. 2 corresponds to the upper half of the capillary tube only; those corresponding to the bottom half are symmetrically inverted in shape and are not shown in the 7gure.
The reason that the interference fringes are bell-shaped has to do with two combined factors. The 7rst is the curved surface area of the capillary tube and the second is the fact that the combination of the lenses in the setup creates a thin laser sheet whose rays are parallel along the y-axis only but not along the x-axis. Therefore, along the x-axis, the
364
H. El Ghandoor et al. / Optics & Laser Technology 35 (2003) 361 – 367
axis of the capillary tube itself, the laser sheet acts as a one-dimensional extended source whose rays travel through the media in a tilted, but symmetric, manner. This leads to the formation of tip, corresponding to the ray that is exactly perpendicular to the x-axis, and two symmetric wings, corresponding to pairs of rays having the same angle at each side of the perpendicular ray (in the x–z plane). The optical paths of these six rays between the tangential lines l1 and l2 de7ned in Fig. 3 can be expressed in terms of the refractive indices of the air (nai ), glass (ngl ), and liquid (nli ), without taking into consideration the phase change occurring at the media interfaces, as follows: OPL1 = 2nai Rou ;
(1)
OPL2 = 2nai Rou ;
(2)
OPL3 = 2nai Rou (1 − cos A ) + ngl AB;
(3)
OPL4 = 2nai Rou (1 − cos C ) + 2ngl CD;
(4)
OPL5 = 2nai Rou (1 − cos F ) + 2ngl FG + nli GH;
(5)
OPL6 = 2ngl (Rou − Rin ) + 2nli Rin ;
(6)
where Rin and Rou are the inner and outer radii of the capillary tube, respectively, A , C , and F are the incident angles at points A, C, and F, respectively. (Explicit equations for these optical path diGerences in terms of the angles of incidence can be found in Ref. [18]). In the following we explain, mathematically, how the fringes appear as bell-shaped by deriving two equations of the optical path diGerence between two pairs of rays. The 7rst is the optical path diGerence between a ray that passes through the oil, i.e., Ray R5 and a reference ray, say R1, as a function of incident angle, and the second between a ray that passes through the wall of the tube only, i.e., R3, and the same reference ray R1. We assume that the diGraction and scattering eGects due to reLection are negligible. Consider a sheet of laser light is incident at point F with incident angle F , where −arcsin(ngl (Rin =Rou )) ¡ F ¡ arcsin(ngl (Rin =Rou )), where the origin of the Cartesian coordinates is at point O. At point F, the Cartesian coordinates (XF ; YF ) are then: YF = Rou sin F , and XF = −Rou cos F , and Snell’s law, applied at the air/glass interface, is: nai sin F = ngl sin ’F ;
(7)
where ’F is the angle of refraction at point F. Accordingly, the equation of a straight line that joins points F and G would be: Y = m1 X + b 1 ;
(8)
where m1 = −tan F , b1 = (YF − m1 XF ) and F = F − ’F . Therefore, the Cartesian coordinates of point G then become: −m1 b1 ± R2in + m21 R2in − b21 XG = (9) 1 + m21 and YG =
R2in − XG2 :
(10)
In Eq. (9), only the negative sign is of interest since XG ends up to the left of the origin O. For simplicity, we will consider FG ≈ HI ≈ (Rou − Rin ), which is the thickness of the capillary tube, Hence GH = (XG − XH )2 + (YG − YH )2 ≈ (XG − XH ) ≈ 2|XG |;
(11)
where Z is the intercept point with the y-axis. Therefore, Eq. (5) reduces to: OPL5 = 2Rou (1 − cos F ) + 2ngl FG + 2nli |XG |:
(12)
Following the same reasoning, the optical path length of ray R3 (OPL3 ) can be evaluated by expressing the Cartesian coordinates at point B (XB ; YB ) as: −m2 b2 + R2ou + m22 R2ou − b22 XB = ; (13) 1 + m22 YB = R2ou − XB2 ; (14) where m2 = −tan A , b2 = (YA − m2 XA ), and A = A − ’A , where this case in the range {−=2; −=2}; A will take the values −=2 ¡ A ¡ − arcsin(ngl Rin =Rou ) and arcsin(ngl Rin =Rou ) ¡ A ¡ =2. Fig. 4 shows plots of the optical path diGerences (OPL5 − OPL1 ) and (OPL3 − OPL1 ) a s functions of the incident angle in the range {−=2; =2} for the values R=Rou =Rin =1:8, ngl = 1:5 and nli = 1:477. The central bell-shaped curve, appearing between the solid vertical lines, is (OPL5 − OPL1 ), while the tails appearing to the left and the right of the solid vertical lines represent (OPL3 − OPL1 ). It is interesting to note that if the ratio R = Rou =Rin is less than ¡ 1:5, then the tails due to (OPL3 − OPL1 ) would disappear in Fig. 4 indicating that it would not be possible to observe interference fringes due to Rays R3 and R1. This is the reason why in Fig. 2 no such tails appear (R = Rou =Rin = 1:33 in our experiment). Before an accurate analysis can be made the images in Fig. 2 had to be digitally processed. This was done by using MATLAB-based software that was developed to conduct a search for the maximum and minimum brightness of the fringes and to report them as functions of position (x and y). This allowed highly contrasted constructive and destructive interference fringes to be exactly de7ned. Fig. 5 shows the outcome of this digital processing on the transverse interference patterns of crude oils D and F.
H. El Ghandoor et al. / Optics & Laser Technology 35 (2003) 361 – 367
Fig. 4. Optical path diGerences OPL5 − OPL1 and OPL3 − OPL1 plotted as functions of incident angle for the values R = Rou =Rin = 1:8, ngl = 1:5 and nli = 1:477. The curved pattern between the two middle solid lines simulates the bell-shaped interference fringe, which is due to interference between a ray passing through the glass and oil, i.e., R5, and a reference ray, e.g., R1. The tails outside the solid lines simulates an interference fringe of diGerent shape resulting from the interference between a ray passing through the glass wall of the capillary tube, i.e., R3, and R1.
The absolute value of the refractive index of the medium enclosed within the capillary tube can be determined by measuring the deLection angles (), de7ned as the angles between the R6 ray and the heights of the observed constructive (or destructive) interference fringes (y–z plane). To measure these deLection angles, in practice, it is convenient to employ Abel’s transform as follows. For a light rays de7ned as orthogonal projection from geometrical wave fronts, the equation relating the path of these rays to the local index of refraction is [19] d dr(s) n = ∇n; (16) ds ds where r(s) is the position vector of a point on the ray and s is the arc length of the ray. Choosing Cartesian coordinates system such that the axis of the capillary tube is in the x-direction and the incident plane monochromatic wave is along the positive z-axis, and using the paraxial
365
Fig. 5. Examples of highly contrasted interference patterns (crude oils D and F) produced by digitally processing the raw images.
approximation, ds ≈ d z this equation reduces to: d dr n = ∇n dz dz
(17)
where r = xi + yj + zk, and ∇n = (9n=9x)i + (9n=9y)j + (9n=9z)k. Since the impinging laser light into the capillary tube suGers refraction due to variation of refractive index in the z-direction, then at a particular height of the capillary tube n can be taken as independent of x. Then Eq. (17) becomes: d dy 9n n = : (18) dz dz 9y Integration of this equation with respect to z gives dy 1 9n d x: = no 9y dz
(19)
In two-dimensional cylindrical coordinates, we have r 2 = z 2 + y2 . Consequently, dz =
r dr r2
−
y2
; and
y 9r = : 9y r
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H. El Ghandoor et al. / Optics & Laser Technology 35 (2003) 361 – 367
Finally, it is interesting to note that, under the same experimental conditions, the number of interference fringes appearing in the images of the crude oils is found to be considerably fewer than those appearing in the case of optical 7bers [12]. This is due to the fact that in optical 7bers the refractive index changes gradually with the radius of the 7ber, while in the crude oil this change occurs in more discrete steps. In other words, if the inside of the 7ber were to be made out of crude oil then one would be able to describe the change in the refractive index simply in terms of discrete layers each of which has a constant refractive index. We estimate the widths of these layers to be ∼ 0:5 m for the samples used in this work. Fig. 6. A Plot of the oil’s refractive index as a function of the oil’s API gravity.
If is the angle of refraction of the rays that pass through the capillary tube, then +∞ dy 9n 9r 1 = = dz dz no −∞ 9r 9y +∞ r 9n y 1 dr = no −∞ 9r r r 2 − y2 dr 2y +∞ 9n = : (20) 2 no 0 9r r − y2 This is the form of Abel’s transform whose inverse is [20] n(r) − nf 1 f dy =− ; (21) nf i y2 − r 2 where i = r and f = ∞ are the limits of the phase object and nf is the index of refraction at f, which is ngl = 32 in our case. By using Eq. (21) the refractive indices of the seven crude oils were determined as A(1.446870), B(1.453822), C(1.460520), D(1.474833), E(1.481889), F(1.488885), and G(1.498860). Fig. 6 shows a plot of the refractive index as a function of the crude oils’ API gravity, where the line represents, arbitrarily, a polynomial 7t of second order. The plot demonstrates a decrease in refractive index with the increase of API gravity as expected. The accuracy of these data points is based on the parameters that de7ne the locations of the fringes i.e., ; y and r, which in turn correspond to displacements that are proportional to 12 , and is estimated to be 5 × 10−6 . It was important that one of the interference patterns be recorded with an actual length scale displayed in the background so that the magni7cation factor of the recorded images could be determined. Another important point to mention here is the accuracy of the digitally reported values of the brightness in the images. Care should be taken not to saturate the CCD camera with intense light, as this would compromise the validity of the reported brightness values.
4. Conclusion The work in this paper represents the 7rst attempt to measure the refractive index of the low transparent crude oil using a capillary tube interferometer. The oil-7lled capillary tubes were illuminated by a thick HeNe laser sheet and the resulting transverse interference patterns were projected on a screen whose plane was perpendicular to that of the laser sheet. As expected, the patterns showed bell-shaped fringes as in the case of optical 7bers except that the number of the fringes in this case was much smaller. The characteristic bell shape was demonstrated mathematically by calculating the optical path diGerence between a reference ray and a ray that passes through the oil sample. The refractive indices of the crude oil samples were determined with accuracy of 5 × 10−6 by measuring the deLection angles of the fringes and then by using Eq. (12). The high accuracy obtained makes this technique a valuable one for measuring the refractive index of crude oil without having to extrapolate it from the oil’s thermal distillates. It may also prove useful as a tool for 7ngerprinting crude oils. Acknowledgements The authors wish to acknowledge the support of the Research Institute and the Physics Department of King Fahd University of Petroleum and Minerals. References [1] Speight JG. The chemistry and technology of petroleum. New York: M. Dekker, 1980. [2] Otremba Z. Opt Express 2000;7:129–34. [3] Buckley JS. Energy Fuels 1999;13:328–32. [4] Buckley JS, Hirasaki GJ, Liu Y, Von Drasek S, Wang JX, Gill BS. Petroleum Sci Technol 1998;16:251–85. [5] Deng Y, Li B. Appl Opt 1998;37(6):998–1005. [6] Tarigan H, Neill P, Kenmore CK, Bornhop DJ. Anal Chem 1996;68:1762–70. [7] Menn ML, Lotrian J. J Phys D 2001;34:1256–65. [8] Marhic ME, Stein EP. Appl Phys Lett 1977;35:1678–82.
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