High Temperatures-High Pressures, Vol. 37, pp. 61–70
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Measurement of mass diffusion
coefficients of saccharose solution
and dimethyl ether in air
using digital image holographic
interferometry system†
Mao-Gang He∗ , Qiu Zhong, Ying Zhang,
Zhi-Gang Liu and Xinxin Zhang
State Key Laboratory of Multiphase Flow in Power Engineering,
Xi’an Jiaotong University, Xi’an 710049, P. R. China.
Received: September 18, 2007. Revised: November 8, 2007. In Final form: November 30, 2007.
An experimental system based on digital image holographic interferometry and a novel diffusion cell are designed and constructed for measuring
mass diffusion coefficients. To verify the accuracy and reliability of the
system, the mass diffusion coefficients of KCl in aqueous solution at temperatures of 291.8 K, 294.7 K, 298.8 K, 299.8 K, 305.4 K, 308.5 K and
315.2 K and at concentration 0.33 mol·L−1 were measured. The results
show that the absolute average of relative deviations is 1.30% compared with the literature values. Then the mass diffusion coefficients of
saccharose in aqueous solution at temperatures of 288.15 K, 298.15 K,
303.15 K, 313.15 K and 333.15 K and at concentration 0.1 mol·L−1 were
measured. Furtherly, the mass diffusion coefficients of dimethyl ether
(DME) in air at temperatures of 296.45 K, 300.45 K and 303.25 K were
also measured.
Keywords: Digital image holographic interferometry, digital image processing,
dimethyl ether, DME, KCl, mass diffusion coefficients, saccharose.
∗ Corresponding author: E-mail: mghe@mail.xjtu.edu.cn
† Paper presented at the 8th Asian Thermophysical Properties Conference, August 21–24, 2007,
Fukuoka, Japan.
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1 INTRODUCTION
The mass diffusion coefficients express the diffusion capacity of matter.
It is introduced by the second Fick’s law [1], which is the theoretical
basis of experiment and theoretical estimation of mass diffusion coefficients.
Currently, the experimental methods of assessment of diffusion coefficients
mainly include [2–5]: diaphragm cells, Taylor’s dispersion, pulsed-field gradient nuclear magnetic resonance, dynamic light scattering and holographic
interferometry. Comparing with other methods, holographic interferometry
has many advantages such as short cycle, high accuracy and adopting it, the
absolute measurement can be carried out directly. Holographic interferometry
mainly includes twice exposal holographic interferometry [6], real-time
holographic interferometry [7] and digital image holographic interferometry [8–10]. Sensitivity plates are used in the first two methods, especially in
the real-time holographic interferometry which requires precise replacement
of the plate after exposure. On the one hand, it increases the difficulty of
the experimental operation; On the other hand, the accuracy of the experiments will be greatly reduced if the replacement of the plate is not precise.
In addition, in real-time holographic interferometry, once the first exposure
is completed, the reference time will be fixed. Therefore, it is difficult to
choose the best thermodynamic state of experiments and under this condition,
the experimental accuracy is impacted. The greatest improvement of digital image holographic interferometry is the abolition of sensitivity plate. The
object beam and reference beam join together at the prism and then expose
to the CCD camera directly. Digital image holographic interferometry gets
the coherent beam in the way of time division. With the passage of time, the
actual state of object changes and the interference fringes will change accordingly, which reflects the changes of the actual object’s state. Because there
is a relation of one-to-one correspondence between the refractive index and
concentration of the object, we can get the concentration changes by analyzing
interference fringes gotten at different time and then get the values of mass
diffusion coefficients. Over the past few years, with the developments of CCD
camera and the enhancement of computer transmission speed, digital image
holographic interferometry has gradually superseded the classic holographic
interferometry for measuring mass diffusion coefficients [11–13].
In this work, an experimental system and a novel diffusion cell are designed
and constructed for measuring mass diffusion coefficients. The mass diffusion coefficients of KCl in aqueous solution at concentration 0.33 mol·L−1 ,
the standard solution for the mass diffusion coefficients measurement, were
measured to verify the accuracy and reliability of the system. Then the mass
diffusion coefficients of saccharose in aqueous solution at temperatures ranging from 288.15 K to 338.15 K and at concentration 0.1 mol·L−1 were also
measured, and the mass diffusion coefficients of dimethyl ether in air were
also measured.
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Mass Diffusion Coefficient Measurement
63
2 THEORY
Assuming that the diffusion is one-dimensional and the solution has clear
interface, the equation (1) can be gotten from the second Fick’s law:
∞
c2l − c2u
−z2
c(z, t) = c2u + √
exp
(1)
dz
4D12 t
2 π D12 t z =z
Here D12 is the mass diffusion coefficient, z is the direction of diffusion, c2l
and c2u are the concentration of the solution 2 at the lower half and at the
upper half of the diffusion cell, respectively.
The refractive index is a function of wavelength and solution concentration.
When the wavelength is fixed, the refractive index changes with the solution concentration. When the solution can be considered as infinite infinitely
diluted, the refractive index and concentration have linear relationship.
The relationship between the phase change of object beam and the change
of concentration is the following:
j = k
c2 l2π
λ
(2)
Here k is the ratio between the change of refractive index and that of concentration, λ is the wavelength of laser, l is the width of diffusion cell, and ϕ
is the distance between two extreme points of phase changes of object beam.
zm , the distance between two extreme points of concentration changes,
can be gotten and then the diffusion coefficient is expressed as:
2
D12 = zm
t0 /t1 − 1
8t0 ln(t0 /t1 )
(3)
Here, t0 and t1 are the times of getting two holograms. zm can be considered
as ϕ. Therefore the key factor is to determine ϕ. In this paper, the value
of ϕ is gotten by hologram’s digital image processing.
The digital image processing mainly includes: collection and storage of
images, pretreatment of hologram, frequency domain filtering of the digital
image, interference fringes rebuilding and post-processing of hologram, etc.
The common used image preprocessing methods include: image histograms
improvement, images smooth filtering and images sharpening, etc. Frequency
domain filtering is the key step of the digital image processing. After Fourier
transform of holographic images, the frequency spectrum information can
be gotten. Then the frequency spectrum information which reflects true
phase of the object beam can be reserved. After inverse Fourier transform
to the frequency spectrum information which is gotten at two different times,
the holograms which only contain the true phase of the object beam can be
obtained. Then the interference fringes are rebuilt. And the phase difference
information of object beam is also obtained. The post-processing of hologram
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mainly includes the unwinding of object beam’s phase difference and the true
phase difference will be acquired. The distance between two extreme points
of object beam’s phase difference can be acquired from above steps.
3 ESTABLISHMENT OF EXPERIMENTAL SYSTEM
Based on the digital image holographic interferometry, the experimental system for measuring mass diffusion coefficients was designed and constructed.
It consists of three parts: optical part for the interference, diffusion cell, and
water bath thermostatic system.
Figure 1 shows the optical system of the digital image holographic interferometry. The light source is supplied by a 633 nm laser. The beam is expanded
to a diameter about 40 mm by the optical beam expander. Then the expanded
beam goes through an iris diaphragm, and the central part of the beam which
has symmetrical intensity is saved. A beam-splitter prism separates the light
into a reference beam and an object beam. The object beam crosses the diffusion cell and falls onto another beam-splitter prism where the object and the
reference beam join together. Finally both the object beam and the reference
beam are projected onto the CCD camera and holograms collected and stored
by the software are automatically acquired. In the experiment, all the optical parts should be installed on an optical shockproof apparatus to reduce the
influence of outside shake.
A novel diffusion cell is designed and constructed. Figure 2 shows its
structure. In order to ensure a steady temperature of the experiment environment, a circulating water bath thermostatic system is designed (Figure 3).
FIGURE 1
Optical layout of the digital image holographic interferometry system.
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Mass Diffusion Coefficient Measurement
Gas Outlet 1
Fluid Inlet 3 Honeycomb 2
65
Optical Glasses 5
Fluid Outlet 4
FIGURE 2
Diffusion cell.
FIGURE 3
Circulating water bath thermostatic system.
Experimental results show that the thermostatic bath temperature is accurate
to ±0.5 K.
4 TEST OF THE EXPERIMENTAL SYSTEM
To verify the accuracy and reliability of the system, the mass diffusion coefficients of KCl in aqueous solution at concentration 0.33 mol·L−1 from 291 K to
315.2 K were measured. Table 1 lists the experimental results of KCl in aqueous solution at the temperature of 298.2 K and at concentration 0.33 mol·L−1
and literature values [14]. Comparing the experimental results with literature
values, it can be seen that the absolute average of relative deviations is 1.30%.
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zm [mm]
t0 [s]
t1 [s]
D × 105 [cm2 ·s−1 ]
4.36
4.91
5.33
5.89
6.09
900
900
900
900
900
D ∗ × 105 (cm2 · s−1 )
D × 105 (cm2 · s−1 )
(|D ∗ −D |)/D (%)
1800
3000
5100
9000
10200
1.876
1.882
1.833
1.886
1.848
1.865
1.841
1.30
Note: D = experimental results; D ∗ = average of experimental results; D = litterature value.
TABLE 1
Mass diffusion coefficients of KCl in aqueous solution at the temperature of 298.2 K.
Choosing a certain time t0 as a reference, five holograms are shot at different times. The mass diffusion coefficients at those different times can be
gotten by comparing the hologram shot at t0 with the ones shot at those different times. Then the average of those five mass diffusion coefficients measured
at those five different times can be regarded as the mass diffusion coefficients
of KCl in aqueous solution. Table 1 lists five mass diffusion coefficients measured at different times and the average mass diffusion coefficient is 1.865 ×
10−5 cm2 · s−1. . The value provided by the literature is 1.841×10−5 cm2 ·s−1 .
The absolute average of relative deviations is 1.30%.
The experimental results of KCl in aqueous solution at temperatures of
291.8 K, 294.7 K, 298.2 K, 299.8 K, 305.4 K, 308.5 K and 315.2 K and at
concentration 0.33 mol·L−1 are listed in Table 2 and shown in Figure 4.
Through the measurement and by comparing the mass diffusion coefficients
of KCl in aqueous solution, the accuracy and reliability of both theory and
measurement system are verified.
T [K] D × 10−5 [cm2 · s−1 ]
291.8
294.7
298.2
299.8
305.4
308.5
315.2
1.597
1.624
1.876
1.937
2.413
2.831
3.437
TABLE 2
Mass diffusion coefficients of KCl in aqueous solution.
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Mass Diffusion Coefficient Measurement
4.500
Experimental Results
Polynomial Fit of Experimental Results
D*10-5 ( cm2⋅s-1)
4.000
3.500
3.000
2.500
2.000
1.500
290.0 295.0 300.0 305.0 310.0 315.0 320.0 325.0
T (K)
FIGURE 4
Mass diffusion coefficients of KCI in aqueous solution.
T [K]
D × 10−6
[cm2 · s−1 ]
D × 10−6
[cm2 · s−1 ]
288.1
298.3
303.3
313.2
333.5
4.725
5.694
6.064
8.027
10.372
4.62
5.56
5.89
7.66
9.87
(|D − D |)
(|D − D |)/
−6
2
−1
×10 [cm · s ]
D [%]
0.105
0.134
0.174
0.367
0.502
2.27
2.41
2.95
4.79
5.09
Note: D = experimental result; D = literature value [15].
TABLE 3
Mass diffusion coefficients of Saccharose in aqueous solution.
5 MASS DIFFUSION COEFFICIENTS OF SACCHAROSE
SOLUTION
To further verify the accuracy and reliability of the system, the mass diffusion
coefficients of saccharose in aqueous solution at temperatures of 288.1 K,
298.3 K, 303.3 K, 313.2 K and 333.5 K and at concentration 0.10 mol·L−1
were also measured. The results of the experiments and literature values [15]
were listed in Table 3 and shown in Figure 5.
Table 3 and Figure 5 show that as experimental temperature rises, the values
of mass diffusion coefficients increase rapidly.
Comparing the experimental results with literature values, it can be seen
that the experimental results of mass diffusion coefficients measured at the
lower experimental temperatures such as 288.1 K, 298.3 K and 303.3 K have
the smaller deviation. Whereas, the experimental results measured at the upper
temperatures of 313.2 K and 333.5 K have the bigger deviation. From what has
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Mao-Gang He et al.
13.00
Literature Values
Experimental Results
D*10-6 /( cm2·s-1)
12.00
11.00
10.00
9.00
8.00
7.00
6.00
5.00
4.00
280.0 290.0 300.0 310.0 320.0 330.0 340.0
T( K)
FIGURE 5
Mass diffusion coefficients of saccharose in aqueous solution.
been discussed above, it can be concluded that a small temperature disturbance
can cause a big deviation in the experimental result.
From what has been analyzed above, the deviation of the experiments
mainly comes from the following aspects: 1. the fluctuation of the interference
fringes caused by vibration and the influence of indoor and outdoor light can
reduce the definition of the holograms and infect the processing of images;
2. the instability of surrounding environment such as air flow will cause the
drop of imaging quality.
6 MASS DIFFUSION COEFFICIENTS OF DIMETHYL ETHER
IN AIR
Using the above experimental system, the mass diffusion coefficients of
Dimethyl Ether in air at the temperature of 296.45 K, 300.45 K and 303.25 K
were also measured. The experimental results are listed in Table 4 and Figure 6.
T [K]
D × 10−6 [cm2 · s−1 ]
296.45
300.45
303.25
2.3923
2.9568
3.2827
TABLE 4
Mass diffusion coefficients of dimethyl ether in air.
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Mass Diffusion Coefficient Measurement
3.8
Experimental Results
Polynomial Fit of Experimental Results
D*10-6 /( cm2·s-1)
3.6
3.4
3.2
3.0
2.8
2.6
2.4
2.2
296
298
300
302
304
306
T(K)
FIGURE 6
Mass diffusion coefficients of dimethyl ether in air.
7 CONCLUSION
An experimental system based on digital image holographic interferometry
and a novel diffusion cell were designed and constructed to measure mass
diffusion coefficients. The theory of digital image holographic interferometry
system and the processing of the digital image are introduced in detail. By the
experimental system, the mass diffusion coefficients of KCl in aqueous solution at temperatures of 291.8 K, 294.7 K, 298.8 K, 299.8 K, 305.4 K, 308.5 K
and 315.2 K and at concentration 0.33 mol·L−1 , the standard solution for mass
diffusion coefficients measurement, were measured to verify the accuracy and
reliability of the system. The measurement results show that the absolute average of relative deviations is 1.30% compared with the literature values. Then
the mass diffusion coefficients of saccharose in aqueous solution at temperatures of 288.1 K, 298.3 K, 303.3 K, 313.2 K and 333.5 K and at concentration
0.10 mol·L−1 were also acquired, and finally the mass diffusion coefficients
of dimethyl ether in air at the temperature of 296.45 K, 300.45 K and 303.25 K
were also measured.
ACKNOWLEDGEMENTS
This work was supported by National Nature Science Fund Committee
(NSFC No. 50676069), Program for New Century Excellent Talents in
University (NCET-04-0925) and NSFC Fund for Creative Research Groups
(No.50521604). The authors gratefully acknowledge them for financial support of this work.
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