Mathematical modelling of gas bubbles and oil droplets in liquid media using optical linear path projection

Flow Measurement and Instrumentation 21 (2010) 388–393

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Mathematical modelling of gas bubbles and oil droplets in liquid media using optical linear path projection Ruzairi Abdul Rahim a,∗ , Yusri Md. Yunos a , Mohd Hafiz Fazalul Rahiman b , Herlina Abdul Rahim a a

Faculty of Electrical Engineering, Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor, Malaysia

b

School of Mechatronic Engineering, Universiti Malaysia Perlis, 02600 Arau, Perlis, Malaysia

article

info

Article history: Received 13 April 2009 Received in revised form 7 March 2010 Accepted 27 April 2010 Keywords: Optical fibre Oil droplets Gas bubbles

abstract This paper describes mathematical modelling of oil droplets and gas bubbles in water. The light sources that are discussed are visible light radiation and infrared radiation. Gas bubbles or oil droplets less than 1 mm diameter are of interest in this investigation. Results show a good agreement between gas bubbles and an oil droplet of radius r < 0.5 mm. The importance of this modelling is that the bubbles approximately describe the characteristics or behavior of oil droplets. This paper shows why a gas bubble is used in this experiment instead of oil droplets. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction The major aim of the research is to overcome a process measurement problem in oil tankers, which after delivering their cargo of oil, flush out the empty tanks using water (or sea water). There are legal limitations which limit the permissible amount of oil to be contained in the discharged flushing water to 100 ppm. The aim of the project is to investigate the feasibility of using an optical tomography instrument for on-line measurement of gas bubble size and distribution in the process industry. The instrument will detect gas bubbles in water and measure their size, size distribution and velocity based on the interaction of gas bubbles in water with a light beam. A major problem exists in the off shore oil industry. When oil rises in a pipeline, it initially is a mixture of oil, water and dissolved gas. As it rises, pressure reduces and the gas forms bubbles which may coalesce. This shows why a gas bubble is used in this experiment instead of oil droplets. Mathematical modelling is important to predict the spatial and temporal behaviour of a process and it becomes more significant as the inherent complexity of a process increases [1]. This modeling has proven and justifies why gas bubbles were used in this project instead of oil droplets. Modelling was carried out based on two parameters affecting the measured output of the sensor for both an oil droplet and a gas bubble. These two parameters are given below:

∗

Corresponding author. E-mail addresses: [email protected] (R. Abdul Rahim), [email protected] (M.H. Fazalul Rahiman). 0955-5986/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.flowmeasinst.2010.04.010

(i) Optical attenuation due to changes in optical density within the pipeline. (ii) Path length of the sensing beam within the pipeline projections [2]. Generally, the following three important stages are involved in sensor modeling [3]. (i) Identify the mathematical model of the sensor and determine the governing equations and associated boundary conditions. (ii) Establish the geometric model of the sensor taking into consideration the significant aspects and special features of the problem domain so as to minimize the amount of data. (iii) Choose an efficient numerical method so as to realize a computer solution of the problem. In real practice, several projections are needed to reduce aliasing which occurs when two particles intercept the same view. The forward problem for the individual sensors is modeled to solve the inverse problem and derive the layergram back projection algorithm [4,5]. Optical tomography has become more popular for research due to its relatively low cost, potential resolution and a better dynamic response than all other measurement methods [6,7]. Optical sensors can be arranged to cover all the cross-sections of a flow rig, which should provide more accurate and higher resolution images than other non-radiation process tomography techniques [8]. Three methods of detecting optical particles/bubbles/droplets can be used. These are light absorption, light attenuation and light scattering. Scattering of light is very complex to describe mathematically due to the random shape and positioning of the particles

R. Abdul Rahim et al. / Flow Measurement and Instrumentation 21 (2010) 388–393

responsible [9], but absorption and attenuation may be quantified by relatively simple mathematical models [8]. Absorption occurs when incident light energy passes through a medium and is converted to heat causing attenuation. Some materials exhibit selective absorption of specific frequencies of light [8,9]. In a gas/liquid system for large bubbles, when the incident beam passes through the centre of the bubble, the major effect is optical attenuation, because the bubble, being gas, attenuates the optical energy less than water. In this model, the measurement section consists of a square, thin walled clear plastic section containing water. The path length between the incident beams entering the measurement section to the receiver is 100 mm. The path length for the beam when no gas bubble is present is 80 mm in water (water in a pipe of diameter 80 mm) and 20 mm in air. The output beam is attenuated according to Lambert–Beer’s law [10], Vm = Vin e(−αw lw −αa la )

Fig. 1. Equivalent thin lenses.

(1)

where Vm — Voltage of the receiving sensor (V). Vin — Voltage of the receiver when no gas bubble present (V). αa — Absorption coefficient of air (mm−1 ). αw — Absorption coefficient of water (mm−1 ). la — Path length of air (mm). lw — Path length of water (mm). From experimental work [11], the absorption coefficients of air and water are αa = 0.00142/mm and αw = 0.004/mm respectively, whereas the coefficient of oil is αo = 0.032/mm for part of the IR spectrum and αo = ∞ for visible light.

ρw/a =

2. Use of visible radiation

ρw/o =

Fig. 2. The focal length and the power of a spherical surface.

n2 − n1

=

(2)

n2 + n1

where n2 is the refractive index of the first medium and n1 is the refractive index of the second medium. Thus for air and glass using Eq. (2)

ρa/g =



2

0.5

2

2.5

(3)

For water and glass

ρw/g =



2

1.5 + 1.33

 =

1.5 − 1.33 0.2 2.8

2

= 0.0051.

For water and air

(4)

2

= 0.02.

(5)



1.33 − 1.42

2

1.33 + 1.42 0.09

2

2.75

= 0.001.

(6)

At the other extreme when light is incident at the polarising angle, 15% of the component perpendicular to the plane of incidence is reflected and none of the parallel component. For a spherical droplet considered as a thick lens, the principal planes of the lens coincide at the middle of the droplet. Thus the oil droplet in water may be represented by an equivalent thin converging lens of focal length 5.93r, (Eq. (11)) placed at the centre of the bubble. In general, crude oil is black in colour, and opaque to visible light, but transmits in the infrared spectrum. The following analysis shows how the oil droplet will behave in visible light. All incident rays are assumed to be parallel to the principal axis; these rays of light after refraction by the bubble/oil droplet will pass through the second principal focus (f2 ) of the equivalent lens (Fig. 1). The focal length, f (m) and the power of a spherical surface F (Fig. 2) [13] is

f

= 0.04.

2

For water and oil

1

1.5 + 1

 =

1.5 − 1

0.33 2.33

=

2

1 − 1.33 1 + 1.33



This section investigates how a single gas bubble can act as an oil droplet and shows the relation between them. This investigation shows why a gas bubble may be used in this experiment instead of oil droplets. The incident light consists of a 1 mm collimated beam, which travels 100 mm (normally 20 mm air, 80 mm water) between source and sensors. Interfaces between phases can present many boundaries in the path of the radiation. On crossing such boundaries Fresnel [12] showed that the fraction of radiation reflected at normal incidence, irrespective of polarization, is ρ where;







2.1. Model for single spherical oil droplet

ρ=

389

=F =

n2 − n1 r

.

(7)

where n2 is the refractive index of the second medium, n1 is the refractive index of the first medium and r is the radius of the oil bubble. Using the sign convention shown in [13] for the first surface, the radius r is measured from a point to the right of where the bubble surface meets the principal axis (A) and is positive using the sign convention noil − nw F = r 1.42 − 1.33 0.09 F1 = = . (8) r r

390

R. Abdul Rahim et al. / Flow Measurement and Instrumentation 21 (2010) 388–393

Fig. 4. Absorption path lengths. Fig. 3. Beam spread by oil droplet.

Oil Droplet

1.2

The radius r is measured from a point to the left of where the bubble surface meets the principal axis (B) and is negative using the sign convention

Relative Intensity

1.33 − 1.42

1

0.09

. (9) r F1 and F2 are positive, so the lens is a converging lens. The equivalent thin lens power of the thick lens [12], Fe , is t Fe = F1 + F2 − F1 F2 n     2r 0.09 0.09 0.09 0.09 Fe = + − r r 1.42 r r F2 =

−r

0.18

Fe =

−

1 Fe

=

fe h0

5.93r

= 5.93r .

(11)

hi hi

(l − 5.93r )

The ratio of the detected intensity to the emitted intensity for bubbles less than 1 mm in diameter consists of two parts, one due I to the part of the beam that misses the bubble ( IR ) and the second o due to the effects of the bubble (AI × Fc × IB ). Ir

IR = + AI × Fc × IB (14) Ie Io where IB is the path length effect, Fc is a surface reflection effect and AI is a diverging effect due to the gas bubble. With no bubble present the intensity due to path length absorption, I, is related to the initial intensity I0 by

(l − 5.93r )h0 . 5.93r

 π 0.52 − π r 2 −0.00142×20−0.004×80 e I0 π 0.52   IR r2 = 1− e−0.00142×20−0.004×80 I0 0.52

(15)

IR = Io e−[0.004×80+0.00142×20] (12)

To compare the effects of small oil droplets and gas bubbles on a 1 mm diameter, collimated, visible light beam, where the principal axis of the bubble and droplet are centred on the collimated beam, the following analysis is carried out. Light which passes around the outside of the obstruction (droplet or bubble) (Fig. 4) is unaffected by the obstruction A0 = [π 0.52 − π r 2 ]. This light is attenuated due to the path length I = e−αa 20−αw 80 . There is no dispersion. The relative intensity at the receiver is IR

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Radius of Oil Droplet (mm)

where αa and αw are absorption coefficients of air and water while lw and la are path lengths in water and air respectively.

.

So R = hi =

0

IR = Io e−[αw lw +αa la ]

l − fe

=

0

2.2. Model for single spherical bubble (10)

Assume h0 mm is the radius of the oil droplet (Fig. 3), the radius of the image at the optical fibre receiver is, by similar triangles, ho

0.4

Fig. 5. Relative intensity versus radius at the optical sensor for oil droplet.

r

r where t is the thickness/diameter of the bubble/droplet and n is the refractive index of the droplet. The equivalent thin lens focal length enables a spherical lens to be considered as a simple thin lens (zero thickness) fe =

0.6

0.2

0.0114

r 0.169

Fe =

=

0.8



IR = 0.7058Io

(16)

which is the light arriving at the sensor with no droplet/bubble within the light beam. The proportion of light passing round the outside of the bubble is attenuated by the path length in water and gas. IR

=

π (0.52 − r 2 ) −0.00142×20−0.004×80 e . π 0.52

Io There are three effects due to the bubble:

(17)

(i) Path length effect IB . (ii) A surface reflection effect. (iii) A diverging effect due to the gas bubble.

=

(13)

where I0 is light coming into the measurement section and r is the oil droplet radius. This is the relative light intensity at the sensor when an oil droplet (assumed opaque) is present for different radii of the droplet (Fig. 5). When the oil droplet diameter equals the beam diameter, the relative intensity at the receiver is totally absorbed.

2.2.1. Path length effect IB The relative light intensity passing through the bubble is IB =

I0 e−[0.004lw +0.00142×la ]

0.7058I0 where lw = 80 − 2r la = 20 + 2r IB =

I0 e−[0.004(80−2r )+0.00142(2r +20)]

IB = e

0.7058I0 −(0.604+0.00284r )

.

(18)

R. Abdul Rahim et al. / Flow Measurement and Instrumentation 21 (2010) 388–393

391

Two cases exist: Case 1. Radius r < 0.5 mm Ir = 4(0.52 − r 2 )e−0.00142×20−0.004×80 Ie 1.25r 2

(0.96)13.2e−(0.604+0.00284r ) . (23) (l + 1.125r )2 Case 2. Radius r ≥ 0.5 mm If r ≥ 0.5 mm and is centred on the principal axis then no light +

passes round the bubble, so Fig. 6. Reflected due to TIR.

Ir Ie

=

1.25r 2

(l + 1.125r )2

(0.96)e−(0.604+0.00284r ) .

(24)

3. Use of infrared radiation 3.1. Model for single spherical bubble For infrared radiation, the gas bubble has the same optical characteristics as for visible light. The gas bubble is assumed to be modelled by Eqs. (23) and (24). The infrared equations for the oil droplet are discussed in the next section. 3.2. Model for oil droplet Fig. 7. Beam spread by small bubble.

2.2.2. Surface reflection losses The loss of intensity at each air/water boundary, assuming normal incidence, is 2% (Eq. (5)). For a single bubble there are two boundaries, as shown in Fig. 6. So the correction factor due to crossing boundaries is Fc = 0.96.

(19)

2.2.3. Optical effects of bubble A ring of light is reflected from the bubble due to Total Internal Reflection (TIR) (Fig. 6). By using the Snell law equation and at a critical angle θc , the losses from the bubble due to total internal reflection is due to the area: AL = π (r 2 − (0.75r )2 ).

=

l + 1.125r

1.125r (l + 1.125r )0.75r R= 1.125r h

AI =

π (0.75r )2 π [(l + 1.125r )2 × 0.672 ] (l + 1.125r )2

IR = 0.7058Io .

(26)

There are three effects due to the oil droplet: (i) Path length effect IB . (ii) A surface reflection effect. (iii) A diverging effect due to the oil droplet. 3.2.1. Path length effect The relative light intensity that passed through the oil droplet is I0 e−[0.004lw +0.032loil +0.00142×la ] IB = 0.7058I0 where lw = 80 − 2r, loil = 2r, la = 20 IB =

π h2 AI = π R2

1.25r 2

IR

+ AI × Fc × IB . (25) Ie Io The intensity due to path length absorption, Ir is the same as Eq. (16) because this modelling still has 80 mm for water and 20 mm for air. =

(21)

where R is the radius of the beam at the receiving fibre and h is the height of the beam on the bubble. From Eq. (17), for the light passing through the bubble, the fractional loss of intensity due to refraction is

AI =

Ir

(20)

The remaining area AP = π (0.75r )2 passes light through the bubble, which is diverged according to lens theory. The light transmitted through the bubble and diverged by it (Fig. 7) is R

The ratio of the detected intensity to the emitted case for an oil droplet less than 1 mm in diameter consists of two parts, one due to I the part of the beam that misses the oil droplet ( IR ) and the second o due to the effects of the oil droplet (AI × Fc × IB ).

.

I0 e−[0.004(80−2r )+0.032(2r )+0.00142(20)] 0.7058I0

IB = e

−(0.604+0.06684r )

.

(27)

3.2.2. Surface reflection losses The loss of intensity at each water/oil boundary, assuming normal incidence, is 0.1% (Eq. (6)). For a single oil droplet there are two boundaries. So the correction factor due to crossing boundaries is Fc = 0.998.

(28)

(22)

Combining these three effects gives the relative intensity at the receiver due to refraction by the bubble as AI × Fc × IB . The relative intensity at the receiver is the sum of the light passing round the I bubble and through it, IIr = IR + AI × Fc × IB . e

o

3.2.3. Optical effects of oil droplet From Eq. (6), R is the height of the infrared beam at the receiving fibre, r is the radius of the oil droplet and l is 50 mm (Fig. 3). R=

(l − 5.93r )h0 . 5.93r

(29)

392

R. Abdul Rahim et al. / Flow Measurement and Instrumentation 21 (2010) 388–393

If r = 2×5l .93 and l = 40 mm r = 0.084 l = 3.36 mm

Comparison of Oil Droplet and Gas Bubble (Visible Light) 1.2

π h20 AI = π R2

1 Relative Intensity

gives an image height 0.5 mm radius. Larger radii will not increase the amount of light energy. If r ≥ 0.084l then attenuation is according to the path length. For the infrared light passing through the oil droplet, the fractional loss of intensity due to refraction is

Oil Droplet (position 50mm from sensor)

0.6

Gas Bubble (position 40mm from sensor)

π r 2 × 5.932 r 2 π(l − 5.93r )2 r 2 35.2r 2

Gas Bubble (position 50mm from sensor) Oil Droplet (position 40mm from sensor)

0.4 0.2

where h0 = r AI =

0.8

0 0

.

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Normalized Radius (mm)

0.8

0.9

1

(30)

Fig. 8. Relative intensity at receiver due to effect of gas bubble and oil droplet using visible light source.

Combining these three effects gives the relative intensity at the receiver due to refraction by the oil droplet as AI × Fc × IB . The relative intensity at the receiver is the sum of the light passing I round the oil droplet and through it, IIr = IR + AI × Fc × IB .

Table 1 Comparison of relative intensities of oil droplets and gas bubbles in water (visible light).

(l − 5.93r )2

e

o

Two cases exist: Case 1. Radius r < 0.5 mm Ir = 4(0.52 − r 2 )e−0.00142×20−0.004×80 Ie 35.2r 2

(0.998)1.83e−(0.604+0.00284r ) . (31) (l − 5.93r )2 Case 2. Radius r ≥ 0.5 mm If r ≥ 0.5 mm and is centred on the principal axis then no light +

passes round the oil droplet, so Ir

=

(35.2)(0.5)2 (0.998)e−(0.604+0.00284r ) (l − 5.93r )2

Ie because the incident beam radius is fixed at 0.5 mm.

(32)

4. Results 4.1. Comparison of the effect of visible light on an oil droplet and a gas bubble A comparison of the effect of visible light on an oil droplet and a gas bubble is shown in Fig. 8. These results show that very little light actually passes through small gas bubbles. The transmitted light appears to be attenuated by the lens effect of the bubble. The results (Table 1) for two different positions of the droplet/bubble relative to the sensor show that gas bubbles are a suitable model for oil droplets up to 1 mm diameter in water for well collimated light beams of 1 mm in diameter. With 1 mm radius (Table 1) the air bubble light level is only 2.4%/2.9% of the level when no bubble is present. This is the maximum error for bubbles/droplets up to 1 mm diameter. 4.2. Comparison of the effect of infrared radiation on an oil droplet and a gas bubble Table 2 shows a comparison of relative intensity at the receiving fiber due to oil droplets and gas bubbles in water using infrared radiation. The table shows very good agreement (better than 1%) between oil and gas for droplet radius up to r = 0.5 mm. The agreement is better than 3% up to r = 1.00 mm. Fig. 9 shows the relative intensity at the receiver due to the effect of an air bubble and an oil droplet for two different positions relative to the sensor. For radius r < 0.5 mm there is a good agreement between an oil droplet and a gas bubble. For r ≥ 0.5 mm the graph shows a rapid divergence between them.

Radius (mm)

50 mm

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 10.0

40 mm

Oil droplet

Gas bubble

Oil droplet

Gas bubble

1.000008 0.990008 0.960008 0.910007 0.840007 0.750006 0.640005 0.510004 0.360003 0.190002 0 0 0 0 0 0 0 0 0 0 0 0

1.000008 0.990067 0.960244 0.910539 0.840952 0.751485 0.642137 0.512907 0.363797 0.194806 0.005934 0.007182 0.008548 0.010033 0.011635 0.013355 0.015191 0.017142 0.019209 0.021390 0.023684 0.967372

1.000008 0.990008 0.960008 0.910007 0.840007 0.750006 0.640005 0.510004 0.360003 0.190002 0 0 0 0 0 0 0 0 0 0 0 0

1.000008 0.990082 0.960303 0.910671 0.841189 0.751854 0.642668 0.513631 0.364741 0.196000 0.007407 0.008962 0.010664 0.012512 0.014505 0.016642 0.018921 0.021343 0.023904 0.026605 0.029444 1.030806

Comparison of Oil Droplet and Gas Bubble (Infra Red Light)

1.2 1 Relative Intensity

AI =

0.8 Oil Droplet (position 50mm from sensor)

0.6

Gas Bubble (position 50mm from sensor) Oil Droplet (position 40mm from sensor)

0.4

Gas Bubble (position 40mm from sensor)

0.2 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized Radius (mm) Fig. 9. Relative intensity at receiver due to effect of gas bubble and oil droplet using infrared light source.

5. Conclusions The paper described mathematical modelling for oil droplets and gas bubbles in water. The light sources used were visible light radiation and infrared radiation. Gas bubbles and oil droplets less than 1 mm diameter are of interest in this investigation. From the

R. Abdul Rahim et al. / Flow Measurement and Instrumentation 21 (2010) 388–393 Table 2 Comparison of relative intensities of oil droplets and gas bubbles in water (infrared radiation). Radius (mm)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 2.00 3.00 5.74 10.0

50 mm

393

gas bubbles approximately describe the oil droplets’ characteristics and behavior. References

40 mm

Oil droplet

Gas bubble

Oil droplet

Gas bubble

1.000008 0.990042 0.960146 0.910322 0.840573 0.750902 0.641311 0.511803 0.362381 0.193048 0.003807 0.003854 0.003902 0.003951 0.004001 0.004052 0.004104 0.004157 0.004211 0.004266 0.004322 0.005733 0.007985 0.031932 0.091457

1.000008 0.990067 0.960244 0.910539 0.840952 0.751485 0.642137 0.512907 0.363797 0.194806 0.005934 0.007182 0.008548 0.010033 0.011635 0.013355 0.015191 0.017142 0.019209 0.02139 0.023684 0.091486 0.192076 0.525738 0.967372

1.000008 0.990062 0.960225 0.910504 0.840903 0.751428 0.642083 0.512876 0.363812 0.194898 0.00614 0.006238 0.006338 0.00644 0.006545 0.006652 0.006762 0.006875 0.006991 0.00711 0.007232 0.010531 0.016794 0.228897 0.021236

1.000008 0.990082 0.960303 0.910671 0.841189 0.751854 0.642668 0.513631 0.364741 0.196 0.007407 0.008962 0.010664 0.012512 0.014505 0.016642 0.018921 0.021343 0.023904 0.026605 0.029444 0.112045 0.230395 0.593957 1.030806

modelling and experiments, both visible light and infrared show good agreement between gas bubbles and oil droplets for radius r < 0.5 mm and this showed that for diameter less than 1 mm, the

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