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Experimental and theoretical investigations of the fractal characteristics of frost crystals during frost formation process
Experimental Thermal and Fluid Science 36 (2012) 217–223
Contents lists available at SciVerse ScienceDirect
Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs
Experimental and theoretical investigations of the fractal characteristics of frost crystals during frost formation process Liu Yaomin, Liu Zhongliang ⇑, Huang Lingyan The Education Ministry Key Laboratory of Enhanced Heat Transfer and Energy Conservation, College of Environmental and Energy Engineering, Beijing University of Technology, Beijing 100124, China
a r t i c l e
i n f o
Article history: Received 1 April 2011 Received in revised form 14 September 2011 Accepted 26 September 2011 Available online 29 September 2011 Keywords: Frost Fractal Box-dimension
a b s t r a c t The frost crystals present needle-like shapes and dendrite growth patterns at the initial period of the frost formation on cold surfaces. They gradually turn into plane-like shapes with time disappear. After frost crystals cover the whole cold surface, the frost formation steps into the frost layer growth period. During this period, as the frost layer becomes thicker, its surface temperature can be high enough to result the frost crystals melt and fall down periodically. It is the growth pattern of the frost crystals that results in a multiple-step ascending tendency of the fractal dimension with time. The physical meaning of the fractal dimension and the volume fraction in describing the frost layer characteristics are also discussed in this paper and it is pointed out that it is necessary to introduce a fractal parameter in modeling frost formation process. The fractal dimension may well define the frost crystal shape and structure features; the volume fraction is directly proportional to the frost layer density, and has a very weak relation with the frost crystal shape and structure features. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Frost formation is a common phenomenon in nature and the engineering fields such as refrigeration, cryogenics and aeronautics. Frost deposition will occur once humid air is exposed to a cold surface whose temperature is below the freezing point. This frost accumulation is undesirable in most cases since it affects heat transfer due to the insulation of the frost layer and causes pressure loss by blocking the air flow. Therefore, the frost layer must be removed regularly from the cold surface, which increases the energy cost. Therefore it is of great practical significance to study the frost deposition phenomenon in theory and experiment. In the past several decades, a huge number of studies have been conducted to investigate the process of frost formation. During frost formation process, the shape and structure of frost crystals varies continuously with time. Based on the experimental observation of the frost crystal structure and variation in frost thickness and density, Hayashi [1] divided the frost formation process into three periods, that is, the crystal growth period, the frost layer growth period, and the frost layer full growth period. The shape and structure of frost crystals and frost layer presents different features in different growth periods. To simplify the mechanism, simple geometric bodies or their combinations were used to describe
⇑ Corresponding author. Tel./fax: +86 10 67391917. E-mail address: [email protected] (Z.L. Liu). 0894-1777/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2011.09.015
the frost structure [2,3]. However, as one may well understand, these simplified models could be anything but the accurate description of the real frost crystals and frost layers. Therefore, it is almost impossible for us to describe and analyze the shape and structure features of frost crystals quantitatively and their variation with time with the classical Euclidian geometry parameters. The new fractal theory provides a possibility to overcome the difficulty. Fractal theory is very effective in characterization of complex and irregular phenomena. It has been successfully used in many different fields of natural and social science. The most important concept of the fractal theory is so-called fractal dimension parameter. The fractal dimension is a parameter to describe the fractal features quantitatively, and it usually has particular physical meaning for a given fractal object. By investigating the fractal dimension, the characteristic properties and their variation rule of the fractal object might be discovered and deeper understanding of the physical phenomenon may also be acquired. For example, the change in the fractal dimension of the fracture surface demonstrates a wavy character and dispersion depending on the microstructural state of the tested steel [4]. Abnormal cancer cells can be distinguished from the normal ones according to their difference in fractal dimensions [5]. By analyzing the fractal dimension of the urban near-infrared images, the residential neighborhood can be obtained [6]. The work by Hao and Jose [7] and Hou et al. [8] indicated that the frost crystal presents fractal features. In this paper, the fractal characteristics of the frost crystals that are formed on cold surfaces will be further investigated based on the
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observation of the shape and structure features of frost crystals in different frost formation periods. And the physical meaning of the fractal dimension of the frost crystals also will be illuminated in detail and concretely. 2. Experimental apparatus The experimental system consists of two parts: a refrigeration unit and an image acquisition system, as shown in Fig. 1. In the refrigeration unit, a semiconductor thermoelectric refrigeration chiller is used to cool down the frosting surface which is made of copper and mounted on the refrigeration unit. The lowest temperature produced by this chiller can reach 26 °C, with a control accuracy of ±0.1 °C. A cold light illuminator unit is used to provide luminescence light for observation, having no appreciable thermal radiation to the frost layer. The temperature of the cold surface is measured by four T-type thermocouples installed at four different locations uniformly distributed on the surface. The image acquisition system mainly includes a microscope, a digital camera with high resolution and a computer for image processing. The microscope is used to observe the variations of frost crystal shapes and structures during the frost deposition process. The images obtained through the microscope are recorded by a digital camera. This experiment was conducted under natural convection condition and the frosting surface was laid horizontally. The experimental conditions were controlled as follows: the environmental temperature (T1) was 25.6 °C; the copper surface temperature (Tw) was 9.0 °C and the air relative humidity (RH) was 52.8%. 3. Background of fractal theory 3.1. Calculation of fractal dimension The fractal theory was proposed by Mandelbrot in the 1970s, which is used to describe the phenomena with properties of selfsimilarity and self-organization in nature [10]. An appropriate mathematical framework is provided by the fractal theory to study the irregular, complex shapes. Different from the classical Euclidean geometry, the objects with fractal features usually have nonintegral dimension. As an essential measurement parameter, fractal dimension has many different definitions, such as Hausdorff dimension, box-counting dimension, packing dimension and divider dimension. Among these definitions, the box-counting dimension is most popularly used and is defined by
DB ¼ lim d!0
lnðNd ðFÞÞ lnðdÞ
ð1Þ
where DB is the box-counting dimension; F is a non-empty and limitary set; Nd(F) is the minimum number of the measurement set used to cover F; d is the radius of the measurement set. For calculating the box-counting dimension, boxes of a fixed size can be assumed to cover the surface and the number (Nd(F)) of the boxes that cover the concerned area is counted with different boxes size (d). Then the box-counting dimension can be estimated from the logarithmic increasing rate of Nd(F) with d, when d ? 0. For a binary digital image, the pixel covering method can be used to calculate the box-counting dimension. The digital image is divided into many boxes by a matrix of k k, here k is equivalent to d in Eq. (1). Owing to the switch characteristic of binary images, it is easy to estimate whether a box covers the concerned area or not according to its pixel value. If all pixel values in a box are 0 assuming that 1 denotes the concerned area, then the box does not cover the concerned area and will not be taken into account. Otherwise the box covers the concerned area and should be taken into account. After checking all the divided boxes, the number of boxes that cover the concerned area is obtained. Change the box size (k) by the arithmetic-step method in which the step size increases in the manner of k = i (i = 1, 2, 3, . . .) [6], then the numbers of boxes (Nk) covering the concerned area can be obtained with different divided box sizes. Finally, the box-counting dimension of the binary image can be estimated from the linear least-square fit of ln Nk versus –ln k. We will use the box-counting dimension as the fractal dimension of frost crystals. For calculating the fractal dimension of frost crystals by above method, the original images acquired through digital camera must be transformed into binary images by digital image processing method. 3.2. Frost image processing The fractal dimension is usually calculated from some figures or images with different physical meaning. These figures and images are saved in magnetic disks in the form of matrix, calling digital image. In this study, the digital image used to calculate the fractal dimension is binary image. However the original frost crystal images are true color image taken by the digital camera whose brightness has a certain range. The high brightness part denotes the frost crystals and the low brightness part denotes the background. For distinguishing the frost crystals from the background,
1. Cooling water system 2. Power supply of refrigeration unit 3. Copper surface 4. Semiconductor thermoelectric refrigeration unit 5. T-type thermocouples 6. Data acquisition equipment 7. Microscope 8. Camera lens 9. Digital camera 10. Computer for image processing 11. Cables 12. Inlet of cooling water 13. Outlet of cooling water Fig. 1. Schematic diagram of experimental system (quoted from literature [9]).
Y.M. Liu et al. / Experimental Thermal and Fluid Science 36 (2012) 217–223
the original images are transformed into the grey images first and then into the binary images by threshold division method. It is worthwhile to note that the original image is ineluctably polluted by the noises which are generally considered as un-scaled white. Thus de-noise processing should be conducted to intensify the image before image transformation. The wavelet de-noising method [11] is used to reduce the noises, which is widely used for signal de-noising applications in signal and image processing. In this paper, Haar wavelet at level 2 was chosen for the wavelet transform which consists of three steps: (a) Discrete wavelet transform, (b) soft threshold, (c) inverse discrete wavelet transform. The wavelet transform is used to transform the noisy image into wavelet domain by decomposing the noisy image into difference levels of details coefficient. The soft threshold technique performs on the wavelet coefficient. The inverse wavelet transform inverses the threshold wavelet coefficients in wavelet domain back to time domain which is the de-noised image. During the threshold division process, an optimized threshold is selected as a critical value. The pixels whose grey level is not smaller than the critical value are set to be 1, and the others are set to be 0. The purpose of the threshold division is to extract the concerned area that contains special physical meaning from the original image. The optimized threshold can be calculated by the maximum variance difference method [12]. A brief introduction of this method is illuminated as follows. For an image with a grey level of 0–L, a threshold t is assumed to separate the image into two classes C1 and C2. In this method, one exhaustively searches for the threshold that maximizes the intraclass variance, defined as a weighted sum of variances of the two classes:
r2b ðtÞ ¼ x1 ðtÞr21 ðtÞ þ x2 ðtÞr22 ðtÞ
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Original image
Binary image (a) Magnified by 30 times
ð2Þ
where weights x1, x2 are the probabilities of the two classes separated by the threshold t, respectively and r21 , r22 are the variances of the two classes, respectively. One can see that if the intra-class variance r2b ðtÞ reaches the maximum, the two classes have the largest difference, which indicates that the division is an optimization one.
Original image
4. Fractal characteristics of frost crystals 4.1. Influence of microscope magnification on frost crystals fractal dimension Frost crystals grow continuously with time. To observe the frost crystals growth clearly through the microscope, the microscope magnification should be adjusted as the frost layer grows. The frost crystal images obtained through the microscope are of different magnification in different frost formation periods. Therefore, it is very important to determine if the magnification has any influence on the fractal dimension, since one may argue that the pictures obtained with the different magnification should be different. A series of tests are carried out to detect the possible influences of the microscope magnification on the fractal dimension deduction. The basic idea of the tests is that the pictures of the same frost layer are obtained with different microscope magnifications and the fractal dimensions of the obtained picture are calculated and compared. Fig. 2 shows the frost crystal images of different magnifications obtained at the same frosting time (48 min) during frost layer growth period. In Fig. 2, picture (a) was taken with magnification of 30 times and picture (b) was taken with magnification of 120 times corresponding to the region denoted by I in picture (a). The fractal dimensions of the pictures taken with the different magnification are given in Fig. 3. To compare the calculated results clearly, the fitted lines are plotted in one figure and the data are shifted along y (ln(Nk) axis. The fractal dimension of the frost
Binary image (b) Magnified by120 times Fig. 2. Frost crystal images of different magnifications.
crystals calculated from the picture taken with a magnification of 30 is 1.9217 and from the picture taken with a magnification of 120 is 1.9284. Therefore, the fractal dimensions calculated from the pictures taken with different magnifications are basically same, i.e., the influence of the magnification on the fractal dimension result of the frost crystals is trivial and can be neglected, if only the pictures are of good quality. This is in full agreement with the fractal theory. According to the theory, the fractal dimension measured
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from any small part truncated from the same object of strictly selfsimilar property is unique and is independent of the size of the computation region. However the objects in nature are characterized with self-similarity property only within a certain range of
scale and this characteristic is defined as statistical self-similarity property. The study by Liu et al. has shown that frost crystals are of such properties [9].
4.2. Variation of fractal dimension with time
Fig. 3. Estimation of fractal dimensions of the images shown in Fig. 2.
During frost formation process, the shape and structure of frost crystals varies with time continuously. So it should be well expected that the fractal dimension of the frost crystals also changes with time. To study the variation of the fractal dimension of the frost crystals with time, pictures are taken at different frost deposition times and these pictures are then used to calculate their fractal dimensions. Fig. 4 displays the original images and these images have to undertake the digital image processing mentioned before. As the transformation result of binary images is sensitive to the magnitude of the threshold, the critical threshold of each image is calculated by the maximum variance difference method to show if it is identical throughout the experimental process. The calculated result is listed in Table 1. As can be seen from Table 1, the critical threshold is varying with time. The mean value is selected as the critical threshold of each image in order to exclude the influence of the different thresholds on the measured results. The image transformation results are shown in Fig. 5, including the de-noised images, the grey images and the final binary images.
29min
35min
42min
48min
53min
71min
75min
88min
98min
113min Fig. 4. Images of frost crystals at different time.
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Owing to the restriction of paper space, only two representative original images acquired at 29 min and 48 min are selected to display their transformation results, which are shown in Fig. 5a and b respectively. The calculated fractal dimensions of the frost crystals at different times are shown in Fig. 6. As one can see from this figure, the variation of the fractal dimension of the frost crystals with time has been detected as expected. Fig. 6 discloses that the fractal dimension presents a multiplestep ascending trend with time. The increasing tendency of the fractal dimension is resulted from the fact that the frost layer grows continuously and becomes denser as time went on. The multiple-step variation with time of the fractal dimension is due to the melting and falling-down phenomenon of the frost crystals which occurred regularly during the frost formation process. As shown in Fig. 4, the frost crystal presents a strong dendrite growth and needle-shaped crystals formed in the initial period of the frost formation, so the fractal dimension is small. As the frost deposition progresses, the frost crystals formed gradually turned into planetype and the frost layer becomes denser, this results in the fractal dimension increases with time. The surface temperature will increase with its growing. And if the surface temperature of the frost layer becomes equal or higher than the melting point of ice and the frost crystals on the frost layer surface that protrude out will begin to melt or fall down due to the heating effect of air. The melting water exists in small droplets and stays at the frost surface with the support of surface tension for a very short period of time. After that some of the water droplets suddenly penetrate into the frost layer and freeze there to increase the density of the frost layer, and some other melting water droplets may be frozen again to
Table 1 Calculated threshold of each image. Time (min) Threshold Time (min) Threshold Mean value
25 0.5226 71 0.5518
29 0.5255 75 0.5518
35 0.5216 88 0.5843
42 0.5216 98 0.4980 0.5210
48 0.5216 113 0.4863
53 0.5137 125 0.4871
61 0.5217 154 0.4863
221
form ice droplets. Both of the mechanisms will increase the density of the frost layer and thus result in a sharp increase in the fractal dimension. The above observations basically agree with the previous reports [1,13]. The densification effect of the frost layer will increase its thermal conductivity and thus its surface temperature will drop below the freezing point. The low surface temperature will promote the dendrite growth of the frost layer and new needle-shaped frost crystals begin to grow on the frozen droplets again. This will step into a new period during which the fractal dimension decreases with time. However, as one may understand, this tendency will not remain long. After the frost crystals turns into the plane-shaped, the period stops and the fractal dimension start to increase again. The cycle of frost crystal growing, thawing, falling down, and growing that can be repeated several times during the whole test period. The four local peak values of the fractal dimension denoted by A, B, C and D in Fig. 6 are calculated from the frost images acquired at 35 min, 53 min, 88 min and 113 min, respectively. The peak values of the fractal dimension are resulted from the thawing effects of the frost crystals at these four moments and this is supported by direct observation. Fig. 6 shows that the fractal dimension increases with time from 25 to 35 min. During this period, the frost crystals present strong dendrite growth and are needle-like as denoted by a, b and c in the picture obtained at 29 min shown in Fig. 4. However, only several minutes later, at 35 min, these frost crystals begin to thaw or even fall down as denoted by b0 , c0 in the picture obtained at 35 min shown in Fig. 4 and the shapes of other crystals gradually turned into plane-like as denoted by a0 in the picture obtained at 35 min shown in Fig. 4. The temporary thawing effect and the change of the frost crystal shape result in the increasing of fractal dimension during this short period of time. The densification increase the thermal conductivity of the frost layer, and this enhanced heat transfer through the frost layer lowers the frost layer surface temperature as mentioned before. Thus the frost crystals repeat its strong dendrite growth and turn into the needle-like shapes in the new initial growth period as shown in the picture obtained at 42 min in Fig. 4. As have been explained earlier, the strong dendrite growth corresponds to the smaller fractal dimension, so after that, from 35 to 42 min, the
De-noised image
De-noised image
Grey image
Grey image
Binary image (a) 29min
Binary image (b) 48min
Fig. 5. Transformation results of original images.
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Fig. 7. Frost crystals at different characteristic lengths.
Fig. 6. Fractal dimension varied with time.
fractal dimension decreases with time. After this, the thawing effect returns and the fractal dimension increases with time again and at 53 min it reaches its second peak value as denoted by B in Fig. 6 and shown by the picture obtained at 53 min in Fig. 4. Other peak values in Fig. 6 can be explained in the same way. Above discussions show that the frost layer fractal dimension may relate to the frost crystal shape and frost layer structure. The large fractal dimension means weak dendrite growth and dense frost layer structure, and the small fractal dimension means strong dendrite growth and loose frost layer structure. So the fractal dimension is an effective parameter to describe the shape and structure characteristics of the frost crystals and frost layer. 4.3. The fractal dimension and the volume fraction of the frost layer It could be found that the structure and density of the frost layer not only change with time as explained above, but also with space location and the same is true for the fractal dimension and volume fraction of the frost layer. That is to say, the two parameters both change with the distance to the cold surface. The volume fraction of the frost layer is defined here as the ratio of the volume of the ice frost occupied to the volume of the frost layer. As one may understand, this parameter is very difficult to calculate from the two-dimensional picture of the frost layer. However, if isotropic is supposed of the frost layer, then it can be proved that the volume fraction is equal to the area fraction of the frost layer that is defined as the ratio of the area that the frost crystals take up to the total area that the frost layer occupied in the same direction [14]. Therefore, we calculate the volume fraction of the frost layer from the images of the frost layer by measuring the area that is occupied by the frost crystals. The calculation are performed over all the images that we taken during the experiments and the similar results are obtained. The frost image acquired at 51 min is used to illustrate the idea. This image is divided into three regions (a), (b), (c) by the distance from the cold surface, as shown in Fig. 7. The fractal dimension and volume fraction of frost crystals of this three parts are calculated respectively and the results are listed in Table 2. The characteristic length or the characteristic thickness refers to the distance of the mid-line of the region from the cold surface. As can be seen from Table 2, the fractal dimension and volume fraction both decrease with the characteristic length of frost layer. At the bottom part (a) shown in Fig. 7, the fractal dimension is 1.9981, which is very close to the plane dimension that is two and the volume fraction is 0.9870. The two parameters both indi-
Table 2 The fractal dimension and volume fraction of frost crystals at different characteristic length. Region
a
b
c
Characteristic length (mm) Fractal dimension Volume fraction
0.24 1.9981 0.9870
0.68 1.9397 0.6746
1.02 1.9080 0.2002
cate that the frost crystals are dense and compacted at the bottom of frost layer. At the top part (c) shown in Fig. 7, the fractal dimension decreases to 1.9080 with a small variation range; however the volume fraction decreases to 0.2002 with a relative large variation range. Therefore it is deduced that the physical meanings reflected by the fractal dimension are different from that by the volume fraction. The fractal dimension mainly reflects the frost crystal shape and structure characteristics. As can be seen from part (c), the frost crystal presents plane-like shape, thus the fractal dimension of part (c) also approaches the plane dimension although frost crystals are much less than part (a). Distinguishing from the fractal dimension, the volume fraction mainly relates to the density of the frost layer, and has a very weak relation with the shape and structure features of the frost crystals. Therefore, the frost volume fraction of part (c) decreases by a larger scale than that of part (a), owing to the visibly decreasing of the frost crystal number.
4.4. More discussions on the fractal analysis of frost formation process In this paper, the fractal dimensions are calculated from the frost images acquired by the digital camera. In theory, we try to study one exactly vertical plane of frost crystals. However due to the focus depth of image, many frost crystals taken into account are not in the exact focus plane. During image processing, we have to treat those frost crystals of a certain brightness in front or back of the focus plane as the study objects, which results in a relative larger measured fractal dimension than that measured from a true focus plane. The study indicates that the high image acquisition technology and effective image processing method play a crucial role of ensuring the comparability of the measured results. The configuration of the frost crystals is influenced by many experimental conditions such as environmental temperature, relative humidity, cold surface temperature and convection condition. These changeable experimental conditions supply a multi-perspective to analyze the fractal characteristics of the frost crystals. For instance, the change of experimental temperature would result in obvious variation of frost formation process, especially during the initial period. The ice droplets formed during this period present fractal distribution feature [7]. The melting of the
Y.M. Liu et al. / Experimental Thermal and Fluid Science 36 (2012) 217–223
surface crystals occurs randomly in the view of time schedule, owing to the unpredictable tiny air current disturb, so the melting process also presents fractal feature. Studies have indicated that the images of frost crystals from the lateral view have fractal characteristics. Nevertheless from the theoretical point of view, the images of frost crystals from the vertical view should also have such characteristics, which show the fractal distribution of frost crystals in a horizontal plane, similarly the fractal distribution characteristic of the ice droplet at the initial period. The above analysis shows that the fractal dimension has an unambiguous physical meaning and it is a measurement for the self-similar and plane-type stretching properties of the frost crystals. When the crystals present needle-like shapes and dendrite growth patterns, the fractal dimension is small; As time elapses, when the crystals turn into plane-like shapes or the frost layer becomes thicker, the fractal dimension increases. Therefore it is considered that only one parameter of volume fraction is not enough to completely describe the configuration of the frost layer. So it is necessary introducing a fractal dimension parameter to reflect the shape features of the frost crystals and its variation with time when modeling frost formation process. Nevertheless the influence of the variation of fractal dimension to frost layer physical properties such as thermal resistance and density, and how to establish the mathematical correlation between each other is a key point to combine the fractal theory and thermodynamic parameters availably, especially at the initial period of frost formation. As in this period, the frost layer cannot be regarded as continuous porous medium since the frost crystals are sparse and discontinuous. Owing to this reason, the classical frost model is difficult to solve the initial frost formation problem while the fractal theory is exactly suitable to deal with such problem.
5. Conclusions The fractal characteristics of the frost crystals and frost layer are studied based on the observation of the frost crystal shape and structure during different frost growth periods. In the study, the fractal dimensions of the frost crystals and the frost layer are calculated from the images acquired at different time. The investigation indicates that the influence of the microscope magnification on frost crystals fractal dimension is trivial and can be neglected. The fractal dimension is closely related to the shape and structure characteristics of the frost crystals. The frost crystals present needle-like shapes and dendrite growth patterns at the initial period of the frost formation on cold surfaces when the fractal dimension is small. As time elapses, they gradually turn into plane-like shapes and the frost layer becomes thicker, thus the fractal dimension increases. After the frost formation steps into the frost layer growth period, its surface temperature is high enough to result the frost crystals melt and fall down cyclically. It is the growth pattern of the frost crystals that results in a multi-
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ple-step ascending of the fractal dimension with time. At the same time, the differences are pointed out between the fractal dimension and the volume fraction in describing the frost layer characteristics. The fractal dimension mainly reflects the frost crystal shape and structure features; however the volume fraction mainly relates to the density of the frost layer, and has a very weak relation with the frost crystal shape and structure features. Based on the above investigation, one conclusion is made that it is necessary to introduce a fractal dimension parameter to describe the shape features of the frost crystals and its variation with time when modeling frost formation process. In this paper, it is only an underway study of analyzing the frost phenomenon in fractal theory. This work remains need to be ulteriorly developed and consummated in the future, especially in aspects of how to combine the fractal dimension parameter with the traditional parameters as well as how to display the fractal characteristics visually in the computer when simulating frost formation process. Acknowledgements Project is supported by the National Natural Science Foundation of China (Grant No. 50376001), the Graduate Science and Technology Foundation Program of BJUT (Grant No. ykj-2011-4883). References [1] Y. Hayashi, Study of frost properties correlating with frost formation types, Journal of Heat Transfer 99 (2) (1977) 239–245. [2] H.W. Schneider, Equation of the growth rate of frost forming on cooled surface, International Journal of Heat and Mass Transfer 21 (1978) 1019–1024. [3] Z.L. Liu, Y.L. Pan, A physical and mathematical model of frost formation on a vertically cooled plate under free convective conditions, Journal of Dalian Marine College 11 (2) (1985) 35–44. [4] I. Dlouhy, B. Strnadel, The effect of crack propagation mechanism on the fractal dimension of fracture surfaces in steels, Engineering Fracture Mechanics 75 (2008) 726–738. [5] C. Timbo, L.A.R. Darosa, M. Goncalves, S.B. Duarte, Computational cancer cells identification by fractal dimension analysis, Computer Physics Communications 180 (2009) 850–853. [6] W.X. Ju, S.N. Nina, An improved algorithm for computing local fractal dimension using the triangular prism method, Computers and Geosciences 35 (2009) 1224–1233. [7] Y.l. Hao, I. Jose, Experimental study of initial state of frost formation on flat surface, Journal of Southeast University 35 (1) (2005) 149–153. [8] P.X. Hou, L. Cai, W.P. Yu, Experimental study and fractal analysis of ice crystal structure at initial period of frost formation, Journal of Applied Sciences 25 (2) (2007) 193–197. [9] Y.M. Liu, Z.L. Liu, L.Y. Huang, Fractal model for simulation of frost formation and growth, Science China—Technological Sciences 53 (3) (2010) 807–812. [10] B.B. Mandelbrot, Fractal: Form, Chance and Dimension, W.H. Freeman and Co., New York, 1977. [11] W. Zhou, Wavelet Analysis and its Applications Based on MATLAB, second ed., Xi dian University Press, Xi an, 2010. pp. 100–105. [12] O. Nobuyuki, A threshold selection method from grey-level histogram, Systems, Man and Cybernetics 9 (1) (1979) 62–66. [13] C.H. Cheng, K.H. Wu, Observations of early-stage frost formation on a cold plate in atmospheric air flow, Journal of Heat Transfer 125 (2003) 95–102. [14] F.J. Meng, The conceptual model of porous media and porosity and tortuosity of frost layer, Journal of Engineering Thermophysics 8 (1) (1987) 64–68.
Contents lists available at SciVerse ScienceDirect
Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs
Experimental and theoretical investigations of the fractal characteristics of frost crystals during frost formation process Liu Yaomin, Liu Zhongliang ⇑, Huang Lingyan The Education Ministry Key Laboratory of Enhanced Heat Transfer and Energy Conservation, College of Environmental and Energy Engineering, Beijing University of Technology, Beijing 100124, China
a r t i c l e
i n f o
Article history: Received 1 April 2011 Received in revised form 14 September 2011 Accepted 26 September 2011 Available online 29 September 2011 Keywords: Frost Fractal Box-dimension
a b s t r a c t The frost crystals present needle-like shapes and dendrite growth patterns at the initial period of the frost formation on cold surfaces. They gradually turn into plane-like shapes with time disappear. After frost crystals cover the whole cold surface, the frost formation steps into the frost layer growth period. During this period, as the frost layer becomes thicker, its surface temperature can be high enough to result the frost crystals melt and fall down periodically. It is the growth pattern of the frost crystals that results in a multiple-step ascending tendency of the fractal dimension with time. The physical meaning of the fractal dimension and the volume fraction in describing the frost layer characteristics are also discussed in this paper and it is pointed out that it is necessary to introduce a fractal parameter in modeling frost formation process. The fractal dimension may well define the frost crystal shape and structure features; the volume fraction is directly proportional to the frost layer density, and has a very weak relation with the frost crystal shape and structure features. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Frost formation is a common phenomenon in nature and the engineering fields such as refrigeration, cryogenics and aeronautics. Frost deposition will occur once humid air is exposed to a cold surface whose temperature is below the freezing point. This frost accumulation is undesirable in most cases since it affects heat transfer due to the insulation of the frost layer and causes pressure loss by blocking the air flow. Therefore, the frost layer must be removed regularly from the cold surface, which increases the energy cost. Therefore it is of great practical significance to study the frost deposition phenomenon in theory and experiment. In the past several decades, a huge number of studies have been conducted to investigate the process of frost formation. During frost formation process, the shape and structure of frost crystals varies continuously with time. Based on the experimental observation of the frost crystal structure and variation in frost thickness and density, Hayashi [1] divided the frost formation process into three periods, that is, the crystal growth period, the frost layer growth period, and the frost layer full growth period. The shape and structure of frost crystals and frost layer presents different features in different growth periods. To simplify the mechanism, simple geometric bodies or their combinations were used to describe
⇑ Corresponding author. Tel./fax: +86 10 67391917. E-mail address: [email protected] (Z.L. Liu). 0894-1777/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2011.09.015
the frost structure [2,3]. However, as one may well understand, these simplified models could be anything but the accurate description of the real frost crystals and frost layers. Therefore, it is almost impossible for us to describe and analyze the shape and structure features of frost crystals quantitatively and their variation with time with the classical Euclidian geometry parameters. The new fractal theory provides a possibility to overcome the difficulty. Fractal theory is very effective in characterization of complex and irregular phenomena. It has been successfully used in many different fields of natural and social science. The most important concept of the fractal theory is so-called fractal dimension parameter. The fractal dimension is a parameter to describe the fractal features quantitatively, and it usually has particular physical meaning for a given fractal object. By investigating the fractal dimension, the characteristic properties and their variation rule of the fractal object might be discovered and deeper understanding of the physical phenomenon may also be acquired. For example, the change in the fractal dimension of the fracture surface demonstrates a wavy character and dispersion depending on the microstructural state of the tested steel [4]. Abnormal cancer cells can be distinguished from the normal ones according to their difference in fractal dimensions [5]. By analyzing the fractal dimension of the urban near-infrared images, the residential neighborhood can be obtained [6]. The work by Hao and Jose [7] and Hou et al. [8] indicated that the frost crystal presents fractal features. In this paper, the fractal characteristics of the frost crystals that are formed on cold surfaces will be further investigated based on the
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observation of the shape and structure features of frost crystals in different frost formation periods. And the physical meaning of the fractal dimension of the frost crystals also will be illuminated in detail and concretely. 2. Experimental apparatus The experimental system consists of two parts: a refrigeration unit and an image acquisition system, as shown in Fig. 1. In the refrigeration unit, a semiconductor thermoelectric refrigeration chiller is used to cool down the frosting surface which is made of copper and mounted on the refrigeration unit. The lowest temperature produced by this chiller can reach 26 °C, with a control accuracy of ±0.1 °C. A cold light illuminator unit is used to provide luminescence light for observation, having no appreciable thermal radiation to the frost layer. The temperature of the cold surface is measured by four T-type thermocouples installed at four different locations uniformly distributed on the surface. The image acquisition system mainly includes a microscope, a digital camera with high resolution and a computer for image processing. The microscope is used to observe the variations of frost crystal shapes and structures during the frost deposition process. The images obtained through the microscope are recorded by a digital camera. This experiment was conducted under natural convection condition and the frosting surface was laid horizontally. The experimental conditions were controlled as follows: the environmental temperature (T1) was 25.6 °C; the copper surface temperature (Tw) was 9.0 °C and the air relative humidity (RH) was 52.8%. 3. Background of fractal theory 3.1. Calculation of fractal dimension The fractal theory was proposed by Mandelbrot in the 1970s, which is used to describe the phenomena with properties of selfsimilarity and self-organization in nature [10]. An appropriate mathematical framework is provided by the fractal theory to study the irregular, complex shapes. Different from the classical Euclidean geometry, the objects with fractal features usually have nonintegral dimension. As an essential measurement parameter, fractal dimension has many different definitions, such as Hausdorff dimension, box-counting dimension, packing dimension and divider dimension. Among these definitions, the box-counting dimension is most popularly used and is defined by
DB ¼ lim d!0
lnðNd ðFÞÞ lnðdÞ
ð1Þ
where DB is the box-counting dimension; F is a non-empty and limitary set; Nd(F) is the minimum number of the measurement set used to cover F; d is the radius of the measurement set. For calculating the box-counting dimension, boxes of a fixed size can be assumed to cover the surface and the number (Nd(F)) of the boxes that cover the concerned area is counted with different boxes size (d). Then the box-counting dimension can be estimated from the logarithmic increasing rate of Nd(F) with d, when d ? 0. For a binary digital image, the pixel covering method can be used to calculate the box-counting dimension. The digital image is divided into many boxes by a matrix of k k, here k is equivalent to d in Eq. (1). Owing to the switch characteristic of binary images, it is easy to estimate whether a box covers the concerned area or not according to its pixel value. If all pixel values in a box are 0 assuming that 1 denotes the concerned area, then the box does not cover the concerned area and will not be taken into account. Otherwise the box covers the concerned area and should be taken into account. After checking all the divided boxes, the number of boxes that cover the concerned area is obtained. Change the box size (k) by the arithmetic-step method in which the step size increases in the manner of k = i (i = 1, 2, 3, . . .) [6], then the numbers of boxes (Nk) covering the concerned area can be obtained with different divided box sizes. Finally, the box-counting dimension of the binary image can be estimated from the linear least-square fit of ln Nk versus –ln k. We will use the box-counting dimension as the fractal dimension of frost crystals. For calculating the fractal dimension of frost crystals by above method, the original images acquired through digital camera must be transformed into binary images by digital image processing method. 3.2. Frost image processing The fractal dimension is usually calculated from some figures or images with different physical meaning. These figures and images are saved in magnetic disks in the form of matrix, calling digital image. In this study, the digital image used to calculate the fractal dimension is binary image. However the original frost crystal images are true color image taken by the digital camera whose brightness has a certain range. The high brightness part denotes the frost crystals and the low brightness part denotes the background. For distinguishing the frost crystals from the background,
1. Cooling water system 2. Power supply of refrigeration unit 3. Copper surface 4. Semiconductor thermoelectric refrigeration unit 5. T-type thermocouples 6. Data acquisition equipment 7. Microscope 8. Camera lens 9. Digital camera 10. Computer for image processing 11. Cables 12. Inlet of cooling water 13. Outlet of cooling water Fig. 1. Schematic diagram of experimental system (quoted from literature [9]).
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the original images are transformed into the grey images first and then into the binary images by threshold division method. It is worthwhile to note that the original image is ineluctably polluted by the noises which are generally considered as un-scaled white. Thus de-noise processing should be conducted to intensify the image before image transformation. The wavelet de-noising method [11] is used to reduce the noises, which is widely used for signal de-noising applications in signal and image processing. In this paper, Haar wavelet at level 2 was chosen for the wavelet transform which consists of three steps: (a) Discrete wavelet transform, (b) soft threshold, (c) inverse discrete wavelet transform. The wavelet transform is used to transform the noisy image into wavelet domain by decomposing the noisy image into difference levels of details coefficient. The soft threshold technique performs on the wavelet coefficient. The inverse wavelet transform inverses the threshold wavelet coefficients in wavelet domain back to time domain which is the de-noised image. During the threshold division process, an optimized threshold is selected as a critical value. The pixels whose grey level is not smaller than the critical value are set to be 1, and the others are set to be 0. The purpose of the threshold division is to extract the concerned area that contains special physical meaning from the original image. The optimized threshold can be calculated by the maximum variance difference method [12]. A brief introduction of this method is illuminated as follows. For an image with a grey level of 0–L, a threshold t is assumed to separate the image into two classes C1 and C2. In this method, one exhaustively searches for the threshold that maximizes the intraclass variance, defined as a weighted sum of variances of the two classes:
r2b ðtÞ ¼ x1 ðtÞr21 ðtÞ þ x2 ðtÞr22 ðtÞ
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Original image
Binary image (a) Magnified by 30 times
ð2Þ
where weights x1, x2 are the probabilities of the two classes separated by the threshold t, respectively and r21 , r22 are the variances of the two classes, respectively. One can see that if the intra-class variance r2b ðtÞ reaches the maximum, the two classes have the largest difference, which indicates that the division is an optimization one.
Original image
4. Fractal characteristics of frost crystals 4.1. Influence of microscope magnification on frost crystals fractal dimension Frost crystals grow continuously with time. To observe the frost crystals growth clearly through the microscope, the microscope magnification should be adjusted as the frost layer grows. The frost crystal images obtained through the microscope are of different magnification in different frost formation periods. Therefore, it is very important to determine if the magnification has any influence on the fractal dimension, since one may argue that the pictures obtained with the different magnification should be different. A series of tests are carried out to detect the possible influences of the microscope magnification on the fractal dimension deduction. The basic idea of the tests is that the pictures of the same frost layer are obtained with different microscope magnifications and the fractal dimensions of the obtained picture are calculated and compared. Fig. 2 shows the frost crystal images of different magnifications obtained at the same frosting time (48 min) during frost layer growth period. In Fig. 2, picture (a) was taken with magnification of 30 times and picture (b) was taken with magnification of 120 times corresponding to the region denoted by I in picture (a). The fractal dimensions of the pictures taken with the different magnification are given in Fig. 3. To compare the calculated results clearly, the fitted lines are plotted in one figure and the data are shifted along y (ln(Nk) axis. The fractal dimension of the frost
Binary image (b) Magnified by120 times Fig. 2. Frost crystal images of different magnifications.
crystals calculated from the picture taken with a magnification of 30 is 1.9217 and from the picture taken with a magnification of 120 is 1.9284. Therefore, the fractal dimensions calculated from the pictures taken with different magnifications are basically same, i.e., the influence of the magnification on the fractal dimension result of the frost crystals is trivial and can be neglected, if only the pictures are of good quality. This is in full agreement with the fractal theory. According to the theory, the fractal dimension measured
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from any small part truncated from the same object of strictly selfsimilar property is unique and is independent of the size of the computation region. However the objects in nature are characterized with self-similarity property only within a certain range of
scale and this characteristic is defined as statistical self-similarity property. The study by Liu et al. has shown that frost crystals are of such properties [9].
4.2. Variation of fractal dimension with time
Fig. 3. Estimation of fractal dimensions of the images shown in Fig. 2.
During frost formation process, the shape and structure of frost crystals varies with time continuously. So it should be well expected that the fractal dimension of the frost crystals also changes with time. To study the variation of the fractal dimension of the frost crystals with time, pictures are taken at different frost deposition times and these pictures are then used to calculate their fractal dimensions. Fig. 4 displays the original images and these images have to undertake the digital image processing mentioned before. As the transformation result of binary images is sensitive to the magnitude of the threshold, the critical threshold of each image is calculated by the maximum variance difference method to show if it is identical throughout the experimental process. The calculated result is listed in Table 1. As can be seen from Table 1, the critical threshold is varying with time. The mean value is selected as the critical threshold of each image in order to exclude the influence of the different thresholds on the measured results. The image transformation results are shown in Fig. 5, including the de-noised images, the grey images and the final binary images.
29min
35min
42min
48min
53min
71min
75min
88min
98min
113min Fig. 4. Images of frost crystals at different time.
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Owing to the restriction of paper space, only two representative original images acquired at 29 min and 48 min are selected to display their transformation results, which are shown in Fig. 5a and b respectively. The calculated fractal dimensions of the frost crystals at different times are shown in Fig. 6. As one can see from this figure, the variation of the fractal dimension of the frost crystals with time has been detected as expected. Fig. 6 discloses that the fractal dimension presents a multiplestep ascending trend with time. The increasing tendency of the fractal dimension is resulted from the fact that the frost layer grows continuously and becomes denser as time went on. The multiple-step variation with time of the fractal dimension is due to the melting and falling-down phenomenon of the frost crystals which occurred regularly during the frost formation process. As shown in Fig. 4, the frost crystal presents a strong dendrite growth and needle-shaped crystals formed in the initial period of the frost formation, so the fractal dimension is small. As the frost deposition progresses, the frost crystals formed gradually turned into planetype and the frost layer becomes denser, this results in the fractal dimension increases with time. The surface temperature will increase with its growing. And if the surface temperature of the frost layer becomes equal or higher than the melting point of ice and the frost crystals on the frost layer surface that protrude out will begin to melt or fall down due to the heating effect of air. The melting water exists in small droplets and stays at the frost surface with the support of surface tension for a very short period of time. After that some of the water droplets suddenly penetrate into the frost layer and freeze there to increase the density of the frost layer, and some other melting water droplets may be frozen again to
Table 1 Calculated threshold of each image. Time (min) Threshold Time (min) Threshold Mean value
25 0.5226 71 0.5518
29 0.5255 75 0.5518
35 0.5216 88 0.5843
42 0.5216 98 0.4980 0.5210
48 0.5216 113 0.4863
53 0.5137 125 0.4871
61 0.5217 154 0.4863
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form ice droplets. Both of the mechanisms will increase the density of the frost layer and thus result in a sharp increase in the fractal dimension. The above observations basically agree with the previous reports [1,13]. The densification effect of the frost layer will increase its thermal conductivity and thus its surface temperature will drop below the freezing point. The low surface temperature will promote the dendrite growth of the frost layer and new needle-shaped frost crystals begin to grow on the frozen droplets again. This will step into a new period during which the fractal dimension decreases with time. However, as one may understand, this tendency will not remain long. After the frost crystals turns into the plane-shaped, the period stops and the fractal dimension start to increase again. The cycle of frost crystal growing, thawing, falling down, and growing that can be repeated several times during the whole test period. The four local peak values of the fractal dimension denoted by A, B, C and D in Fig. 6 are calculated from the frost images acquired at 35 min, 53 min, 88 min and 113 min, respectively. The peak values of the fractal dimension are resulted from the thawing effects of the frost crystals at these four moments and this is supported by direct observation. Fig. 6 shows that the fractal dimension increases with time from 25 to 35 min. During this period, the frost crystals present strong dendrite growth and are needle-like as denoted by a, b and c in the picture obtained at 29 min shown in Fig. 4. However, only several minutes later, at 35 min, these frost crystals begin to thaw or even fall down as denoted by b0 , c0 in the picture obtained at 35 min shown in Fig. 4 and the shapes of other crystals gradually turned into plane-like as denoted by a0 in the picture obtained at 35 min shown in Fig. 4. The temporary thawing effect and the change of the frost crystal shape result in the increasing of fractal dimension during this short period of time. The densification increase the thermal conductivity of the frost layer, and this enhanced heat transfer through the frost layer lowers the frost layer surface temperature as mentioned before. Thus the frost crystals repeat its strong dendrite growth and turn into the needle-like shapes in the new initial growth period as shown in the picture obtained at 42 min in Fig. 4. As have been explained earlier, the strong dendrite growth corresponds to the smaller fractal dimension, so after that, from 35 to 42 min, the
De-noised image
De-noised image
Grey image
Grey image
Binary image (a) 29min
Binary image (b) 48min
Fig. 5. Transformation results of original images.
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Fig. 7. Frost crystals at different characteristic lengths.
Fig. 6. Fractal dimension varied with time.
fractal dimension decreases with time. After this, the thawing effect returns and the fractal dimension increases with time again and at 53 min it reaches its second peak value as denoted by B in Fig. 6 and shown by the picture obtained at 53 min in Fig. 4. Other peak values in Fig. 6 can be explained in the same way. Above discussions show that the frost layer fractal dimension may relate to the frost crystal shape and frost layer structure. The large fractal dimension means weak dendrite growth and dense frost layer structure, and the small fractal dimension means strong dendrite growth and loose frost layer structure. So the fractal dimension is an effective parameter to describe the shape and structure characteristics of the frost crystals and frost layer. 4.3. The fractal dimension and the volume fraction of the frost layer It could be found that the structure and density of the frost layer not only change with time as explained above, but also with space location and the same is true for the fractal dimension and volume fraction of the frost layer. That is to say, the two parameters both change with the distance to the cold surface. The volume fraction of the frost layer is defined here as the ratio of the volume of the ice frost occupied to the volume of the frost layer. As one may understand, this parameter is very difficult to calculate from the two-dimensional picture of the frost layer. However, if isotropic is supposed of the frost layer, then it can be proved that the volume fraction is equal to the area fraction of the frost layer that is defined as the ratio of the area that the frost crystals take up to the total area that the frost layer occupied in the same direction [14]. Therefore, we calculate the volume fraction of the frost layer from the images of the frost layer by measuring the area that is occupied by the frost crystals. The calculation are performed over all the images that we taken during the experiments and the similar results are obtained. The frost image acquired at 51 min is used to illustrate the idea. This image is divided into three regions (a), (b), (c) by the distance from the cold surface, as shown in Fig. 7. The fractal dimension and volume fraction of frost crystals of this three parts are calculated respectively and the results are listed in Table 2. The characteristic length or the characteristic thickness refers to the distance of the mid-line of the region from the cold surface. As can be seen from Table 2, the fractal dimension and volume fraction both decrease with the characteristic length of frost layer. At the bottom part (a) shown in Fig. 7, the fractal dimension is 1.9981, which is very close to the plane dimension that is two and the volume fraction is 0.9870. The two parameters both indi-
Table 2 The fractal dimension and volume fraction of frost crystals at different characteristic length. Region
a
b
c
Characteristic length (mm) Fractal dimension Volume fraction
0.24 1.9981 0.9870
0.68 1.9397 0.6746
1.02 1.9080 0.2002
cate that the frost crystals are dense and compacted at the bottom of frost layer. At the top part (c) shown in Fig. 7, the fractal dimension decreases to 1.9080 with a small variation range; however the volume fraction decreases to 0.2002 with a relative large variation range. Therefore it is deduced that the physical meanings reflected by the fractal dimension are different from that by the volume fraction. The fractal dimension mainly reflects the frost crystal shape and structure characteristics. As can be seen from part (c), the frost crystal presents plane-like shape, thus the fractal dimension of part (c) also approaches the plane dimension although frost crystals are much less than part (a). Distinguishing from the fractal dimension, the volume fraction mainly relates to the density of the frost layer, and has a very weak relation with the shape and structure features of the frost crystals. Therefore, the frost volume fraction of part (c) decreases by a larger scale than that of part (a), owing to the visibly decreasing of the frost crystal number.
4.4. More discussions on the fractal analysis of frost formation process In this paper, the fractal dimensions are calculated from the frost images acquired by the digital camera. In theory, we try to study one exactly vertical plane of frost crystals. However due to the focus depth of image, many frost crystals taken into account are not in the exact focus plane. During image processing, we have to treat those frost crystals of a certain brightness in front or back of the focus plane as the study objects, which results in a relative larger measured fractal dimension than that measured from a true focus plane. The study indicates that the high image acquisition technology and effective image processing method play a crucial role of ensuring the comparability of the measured results. The configuration of the frost crystals is influenced by many experimental conditions such as environmental temperature, relative humidity, cold surface temperature and convection condition. These changeable experimental conditions supply a multi-perspective to analyze the fractal characteristics of the frost crystals. For instance, the change of experimental temperature would result in obvious variation of frost formation process, especially during the initial period. The ice droplets formed during this period present fractal distribution feature [7]. The melting of the
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surface crystals occurs randomly in the view of time schedule, owing to the unpredictable tiny air current disturb, so the melting process also presents fractal feature. Studies have indicated that the images of frost crystals from the lateral view have fractal characteristics. Nevertheless from the theoretical point of view, the images of frost crystals from the vertical view should also have such characteristics, which show the fractal distribution of frost crystals in a horizontal plane, similarly the fractal distribution characteristic of the ice droplet at the initial period. The above analysis shows that the fractal dimension has an unambiguous physical meaning and it is a measurement for the self-similar and plane-type stretching properties of the frost crystals. When the crystals present needle-like shapes and dendrite growth patterns, the fractal dimension is small; As time elapses, when the crystals turn into plane-like shapes or the frost layer becomes thicker, the fractal dimension increases. Therefore it is considered that only one parameter of volume fraction is not enough to completely describe the configuration of the frost layer. So it is necessary introducing a fractal dimension parameter to reflect the shape features of the frost crystals and its variation with time when modeling frost formation process. Nevertheless the influence of the variation of fractal dimension to frost layer physical properties such as thermal resistance and density, and how to establish the mathematical correlation between each other is a key point to combine the fractal theory and thermodynamic parameters availably, especially at the initial period of frost formation. As in this period, the frost layer cannot be regarded as continuous porous medium since the frost crystals are sparse and discontinuous. Owing to this reason, the classical frost model is difficult to solve the initial frost formation problem while the fractal theory is exactly suitable to deal with such problem.
5. Conclusions The fractal characteristics of the frost crystals and frost layer are studied based on the observation of the frost crystal shape and structure during different frost growth periods. In the study, the fractal dimensions of the frost crystals and the frost layer are calculated from the images acquired at different time. The investigation indicates that the influence of the microscope magnification on frost crystals fractal dimension is trivial and can be neglected. The fractal dimension is closely related to the shape and structure characteristics of the frost crystals. The frost crystals present needle-like shapes and dendrite growth patterns at the initial period of the frost formation on cold surfaces when the fractal dimension is small. As time elapses, they gradually turn into plane-like shapes and the frost layer becomes thicker, thus the fractal dimension increases. After the frost formation steps into the frost layer growth period, its surface temperature is high enough to result the frost crystals melt and fall down cyclically. It is the growth pattern of the frost crystals that results in a multi-
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ple-step ascending of the fractal dimension with time. At the same time, the differences are pointed out between the fractal dimension and the volume fraction in describing the frost layer characteristics. The fractal dimension mainly reflects the frost crystal shape and structure features; however the volume fraction mainly relates to the density of the frost layer, and has a very weak relation with the frost crystal shape and structure features. Based on the above investigation, one conclusion is made that it is necessary to introduce a fractal dimension parameter to describe the shape features of the frost crystals and its variation with time when modeling frost formation process. In this paper, it is only an underway study of analyzing the frost phenomenon in fractal theory. This work remains need to be ulteriorly developed and consummated in the future, especially in aspects of how to combine the fractal dimension parameter with the traditional parameters as well as how to display the fractal characteristics visually in the computer when simulating frost formation process. Acknowledgements Project is supported by the National Natural Science Foundation of China (Grant No. 50376001), the Graduate Science and Technology Foundation Program of BJUT (Grant No. ykj-2011-4883). References [1] Y. Hayashi, Study of frost properties correlating with frost formation types, Journal of Heat Transfer 99 (2) (1977) 239–245. [2] H.W. Schneider, Equation of the growth rate of frost forming on cooled surface, International Journal of Heat and Mass Transfer 21 (1978) 1019–1024. [3] Z.L. Liu, Y.L. Pan, A physical and mathematical model of frost formation on a vertically cooled plate under free convective conditions, Journal of Dalian Marine College 11 (2) (1985) 35–44. [4] I. Dlouhy, B. Strnadel, The effect of crack propagation mechanism on the fractal dimension of fracture surfaces in steels, Engineering Fracture Mechanics 75 (2008) 726–738. [5] C. Timbo, L.A.R. Darosa, M. Goncalves, S.B. Duarte, Computational cancer cells identification by fractal dimension analysis, Computer Physics Communications 180 (2009) 850–853. [6] W.X. Ju, S.N. Nina, An improved algorithm for computing local fractal dimension using the triangular prism method, Computers and Geosciences 35 (2009) 1224–1233. [7] Y.l. Hao, I. Jose, Experimental study of initial state of frost formation on flat surface, Journal of Southeast University 35 (1) (2005) 149–153. [8] P.X. Hou, L. Cai, W.P. Yu, Experimental study and fractal analysis of ice crystal structure at initial period of frost formation, Journal of Applied Sciences 25 (2) (2007) 193–197. [9] Y.M. Liu, Z.L. Liu, L.Y. Huang, Fractal model for simulation of frost formation and growth, Science China—Technological Sciences 53 (3) (2010) 807–812. [10] B.B. Mandelbrot, Fractal: Form, Chance and Dimension, W.H. Freeman and Co., New York, 1977. [11] W. Zhou, Wavelet Analysis and its Applications Based on MATLAB, second ed., Xi dian University Press, Xi an, 2010. pp. 100–105. [12] O. Nobuyuki, A threshold selection method from grey-level histogram, Systems, Man and Cybernetics 9 (1) (1979) 62–66. [13] C.H. Cheng, K.H. Wu, Observations of early-stage frost formation on a cold plate in atmospheric air flow, Journal of Heat Transfer 125 (2003) 95–102. [14] F.J. Meng, The conceptual model of porous media and porosity and tortuosity of frost layer, Journal of Engineering Thermophysics 8 (1) (1987) 64–68.