Growth kinetics of ice films spreading on a subcooled solid surface

Separation and Purification Technology 39 (2004) 109–121

Growth kinetics of ice films spreading on a subcooled solid surface Frank G.F. Qin, Xiao Dong Chen∗ , Mohammed M. Farid Department of Chemical and Materials Engineering, The University of Auckland, Private Bag 92019, Auckland city, New Zealand

Abstract When an aqueous solution is brought into contact with a subcooled solid surface, an intermediate stage of ice crystallization is found between the surface nucleation and the growth of ice layer in the normal direction to the solid surface. During this period, ice is formed in thin films spreading along the solid surface until it is completely covered. This process can be found in the process of freeze concentration of aqueous solutions, where ice is formed and removed to concentrate the solutions. The growth rate of ice films in this period is controlled by the heat conduction from the ice into the solid surface. A mathematical model that describes the ice film growth kinetics is presented in this paper, which is expressed using the Modified Bessel Functions. The temperature distribution in the solid slab underneath the growing front of the ice films is derived from the model, and the growth rates of ice films on different solid surfaces are predicted using a numerical method. © 2004 Elsevier B.V. All rights reserved. Keywords: Ice crystallization; Growth kinetics; Ice film; Heat transfer; Subcooled solid surface; Freeze concentration

1. Introduction Ice growth on the normal direction of a subcooled solid surface can be found in some unit operations in industry where heat transfer is involved at subfreezing temperature, such as the manufacture of ice cream [1,2], freeze concentration of liquid foods [3], etc. The earliest scientific study of the growth of ice layer on a subcooled surface began in the field of geography by Lame, Clapyron, and Neumann et al. in the middle of the 19th century [4]. In their work, a mathematical model and its analytical solution for the ice growth, which is counter to the heat flux, was presented. This ∗ Corresponding author. Tel.: +64-9-373-7599x7004; fax: +64-9-373-7463. E-mail address: [email protected] (X.D. Chen).

model is known as the classical Neumann Problem, which is a one-dimensional solidification model assuming the presence of an initial layer. The schematic diagram of the problem is illustrated in Fig. 1. The Neumann problem is applicable for the freezing of pure water or diluted solution. For actual aqueous solution(s), Ratkje and co-workers [5,6] presented a layer growth model using the coupled heat and mass flux equations derived from irreversible thermodynamics. Nevertheless, [7] argued that the model presented by Ratkje and Flesland appeared to have ignored the temperature gradient on the ice side of the interface, which may lead to erroneous predictions of freezing roles that conflict with the predictions of the conventional heat–mass transfer and thermodynamics theory. In our previous experimental studies (Qin, Chen, Zhao, Russell, Chen, and Robertson, 29 September–3

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Nomenclature Symbols α (=λ/ρc) κ (=1/2α) δ η (= T − Tw /Tf − Tw ) ϕ λ µ θ ρ √ r = x 2 + z2 b c v H In Kn Q T Tw Tf Ts Tw (= Tf − Tw )

thermal diffusivity (m2 s−1 ) a variable defined for easing the treatment of Eq. (7) (m−2 s) thickness of the newly formed ice film at the growing front (m) dimensionless temperature an assumed variable in order to obtain the heat conduction Eq. (A.5) thermal conductivity (W m−1 K−1 ) intrinsic value of the ordinary differential equation with regard to the angle in the polar coordinate system (Rad) density (kg m−3 ) displacement from the origin to the point (x, z) at the X-Z plane (m) thickness of the solid wall (m) specific heat capacity (J kg−1 K−1 ) growth rate of ice films spreading along the subcooled solid wall (␮m s−1 ) fusion heat of freezing of water (= 334.11 × 103 J kg−1 ) (J kg−1 ) Modified Bessel Functions of the first kind of the nth order Modified Bessel Functions of the second kind of the nth order heat (J) temperature (K) solid wall temperature outside the heat affect zone (K) freezing point of the liquid (K) surface temperature of the solid wall used in the Neumann Problem (K) wall supercooling (K)

Subscripts i w n

ice solid wall the order of the Modified Bessel Equation and Function

Fig. 1. The temperature distribution in a partially frozen medium, which is described with the Neumann Problem presented at the right-hand side. (a) Temperature profile; and (b) the expression of the Neumann problem.

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October 2002 [8]; Qin, Russell, Chen, and Robertson, 2003 [9]), a micro-cell with polished stainless steel cooling surface was made for the observation of the growth of ice film laterally. The temperature of the cooling surface was controlled by the coolant, the temperature of which was pre-set via a HAAKE unit (Gebruder HAAKE GmbH, Germany) with an accuracy of 0.1 ◦ C. A video camera (Vicam® CCD B/W camera, model MTC5C04S, Korea) was mounted on top of the microscope (Motic® B1 series biological microscope, Korea) for taking still or motion pictures, which were saved to the hard disk of a dedicated computer. Meanwhile, both the bulk and the cooling surface temperatures were measured with T-type thermocouples, respectively, and recorded to the computer through a Picolog TP801 data acquisition unit (Picolog Technology, UK). The micro-cell, microscope and video camera were all placed in the cabinet of a large refrigerator (White Refrigeration Ltd., NZ) to maintain a stable room temperature and to avoid moisture from condensing on the lens of the microscope and the window of the micro-cell. The temperature in the cabinet was pre-set to the freezing point, which is −0.5 ◦ C for 10 wt.% sugar solution. The field of view of the microscope was illuminated via a fiber optic cable from an external light-box (Micro Imaging Ltd., NZ). Ice formation on a subcooled surface can be examined from an overhead view (Fig. 2) and a side view as well (Fig. 3), in which the microscope can be oriented in a horizontal position in the refrigerated cabinet.

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Pictures of the video frames, which were taken during the ice formation, show that ice appears primarily at some spots and forms separate ice ‘islands’. These ice ‘islands’ grow larger until the subcooled surface was frozen-over by a thin film of ice. During this period, the thickness of these ice patches was almost unchanged (Figs. 4 and 5), indicating that the normal growth rate of ice on the solid surface is small enough to be ignored, if it is compared to the growth rate of ice along the solid surface. A solid surface can reduce the energy barrier of phase change, especially at those sites where the microstructure of the surface is not uniform. This may be the season why the nucleation sites are relatively distant one from another. Once ice embryos appear, the subcooled solid surface becomes a more favorite place for ice crystallization, because the fusion heat can be given to the solid surface directly, especially when the bulk solution temperature is close to the freezing point. The fact that the growth rate of ice along the cooling surface is much faster than the growth rate in the normal direction (i.e. perpendicular to the solid surface) results in the formation of ice films, and this characterizes an intermediate, transitional stage between the ice nucleation and the normal growth of the ice layer. After this transitional stage, the growth of ice layer starts. For ice layers growing from water, the Neumann Problem can be used to describe the process. For ice layers growing from aqueous solutions, concentration gradient of solutes will be formed in the vicinity of

Fig. 2. Experimental setup for the observation of ice formation on a subcooled solid surface. (a) Refrigerator cabinet (glovebox); (b) fibre optic illuminator; (c) microscope; (d) digital video camera; (e) thermocouple; (f) A/D converter (Picolog TP801); (g) refrigerator control panel; (h) gloves; and (i) computer for data acquisition.

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Fig. 3. The Micro-cell used for lateral observation of ice formation on the stainless steel surface. (a) Front view; and (b) section view.

the growing front, which can be observed and studied with laser holography [10]. In this case the coupling of the heat and mass transfer should be taken into consideration, and the process may be described with the irreversible thermodynamics layer growth model [6], or others [11,12]. However, the analysis regarding to the problem of the transitional stage is still missing in literature. In this paper, we aim to establish a mathematical model regarding the growth kinetics of ice films driven by the heat transfer on a subcooled, solid surface. The mathematical solution of this model is expressed with Modified Bessel Functions. From these the growth kinetics of ice film can be expressed in mathematical terms.

2. Theory and analysis 2.1. Coordinate system and assumptions

Fig. 4. The side view of the growing ice film on the subcooled, stainless steel surface of the micro-cell. The thickness of the ice film is about 250 ␮m in the range of the supercooling of 2–5 ◦ C. (a) Beginning of the observation; (b) at the 10th second; and (c) at the 20th second.

To facilitate the mathematical analysis of such a problem outlined above, a moving Cartesian coordinate system with its origin to be set at the heat source, i.e. the upper surface of the growing front of the ice film, is established (Fig. 6). In this coordinate system, the subcooled solid slab equivalently moves at the same velocity, but in the opposite direction of the growth of the ice film (Fig. 7). Correspondingly, the growing front of the ice film is still. Moreover, the following assumptions are made in order to arrive at the energy equation of the growing front of the ice film:

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1. Based on the above-mentioned experimental observation, the thickness of the ice film does not change before it covers the entire subcooled, solid surface. 2. The bulk solution temperature is at the freezing point, so it is a constant. This is an approximation for the case of freeze concentration of an aqueous solution when there is ice in the bulk solution [13,14]. 3. The fusion heat of freezing from the growing front of the ice film must dissipate into the solid slab to maintain further growth, particularly when the bulk solution is at the freezing point. 4. The heat dispersion from the growing front of the ice film into the subcooled, solid surface predominates the growth rate of the ice film. This is true for the ice growth from water or dilute aqueous solutions, in which there are abundant crystallizable water molecules in the vicinity of the ice–liquid interface, mass transfer at the ice surface is not a controlling step for ice crystallization [12,15]. 5. The subcooled solid slab has a constant, uniform wall temperature (Tw ) except in the small local region right underneath the growing front of the ice film, because the coolant flow on the other side is sufficient enough to maintain a stable average temperature in the solid slab, which is much thicker then the ice film. The temperature gradient outside this region is ignored (Fig. 7). The said small local region is assumed to have a higher temperature because of the input of the fusion heat of ice. We term this region a heat affect zone (or HAZ, similarly hereinafter) in this paper. 6. The solid surface right underneath the newly formed ice bud has the same temperature of the ice bud, i.e. at the freezing point (Tf ). 7. The ice film growth is not affected by the heat exchange between the liquid and the ice-free area of the solid surface outside the HAZ. This assumption is equivalent to ignore the heat exchange occurring on the solid–liquid interface outside the HAZ. 2.2. Energy equation of the growing front of the ice film Fig. 5. The overhead(bird’s) view of the ice film growing on a subcooled glass surface (a hemacytometer). The thickness of the ice film is about 150 ␮m in the range of the supercooling degree of 2–5 ◦ C. (a) Beginning of the observation; (b) at the 10th second; and (c) at the 20th second.

During a time interval from t to t + dt, the ice film has a small growth gain (dl) along the solid surface (Fig. 7). In a small element of heat transfer area right underneath the growing front, as shown in the semi-

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Fig. 6. Schematic diagram of the ice film spreading on a subcooled solid surface.

circle of Fig. 7, the heat input includes three parts: (1) a small part of ‘residual’ fusion heat of the ice film behind the growing front; (2) the fusion heat (Q) given by the newly formed growth gain (dl); and (3) the sensible heat from the unfrozen liquid in front of the growing ice film. Obviously, the transportation of the second part of heat (Q) is directly coupled with the spreading rate of the ice film (dl/dt) on the solid surface. Note that in this coordinate system, the medium (solid slab) is moving underneath the heat source (which is the growing front of the ice film), and the heat source is still. The energy equation in the solid slab can be written as [16]: ∂T ∂T ∂T ∂T + vx + vy + vz ∂t ∂x ∂y ∂z
2  2 2 ∂ T ∂ T ∂ T =α + + ∂x2 ∂z2 ∂z2

(1)

where α (=λ/ρc) is the thermal diffusivity of the solid slab. λ, ρ, and c, the thermal conductivity, density, and specific heat capacity of the solid slab, respectively. vx , vy , and vz are the flow velocities of the medium (solid slab) at an arbitrary point (x, y, z) in the direction of x-, y- and z-axis. Here, the solid slab is treated as a fluid moving in the opposite direction of x-axis (i.e. vx = −v). The velocity of the solid slab at the direction of y-, z-axis is zero (i.e. vy = 0, vz = 0), and the temperature gradient along the y-axis dose not exit (i.e. ∂T/∂y = ∂2 T/∂y2 = 0), thus Eq. (14) can be re-written as:
2  ∂T ∂T ∂ T ∂2 T −v =α (2) + 2 ∂t ∂x ∂x2 ∂z The term, −v ∂T/∂x, on the left-hand side of Eq. (2) represents that the medium is moving in the opposite direction of x-axis at the velocity of v (v ≥ 0). The fusion heat released by the newly formed ice buds will be directly given into the solid slab, which originally had a uniform subfreezing temperature (Tw ) as narrated in assumption (5), where we defined a small heat affect zone right underneath the growing front wherein the released fusion heat can make a sensible contribution to impact the temperature distribution. Temperature out of the HAZ, in other words, will remain at Tw . This leads to the semi-finite approximation shown by Eq. (3). T |√ = Tw (x ≥ 0, z ≥ 0) (3) x2 +z2 →∞

Fig. 7. The cross sectional view of heat transfer from the newly formed ice bud (the white part) into the subcooled solid slab. The semi-circle of the dot line is the so-called heat affect zone (HAZ).

In a fixed wall supercooling with regard to the bulk solution (Tw = Tf − Tw ), the ice growth rate would stay constant, as such the heat production from the ice buds. The heat generation and the removal will

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quickly achieve a balance, resulting in a quasi-steady state in the HAZ, thus ∂T/∂t ≈ 0. Therefore, Eq. (2) can be re-written as: 
2 ∂T ∂ T ∂2 T −v (4) =α + ∂x ∂x2 ∂z2 At the solid surface right underneath the newly formed ice bud, assumption (5) and (6) lead to a boundary condition of the problem: T |√ (5) = Tf x2 +z2 →δ and θ=π/2

Eq. (4) will be more easily handled if we introduce a dimensionless temperature η = T − Tw /Tf − Tw , and let α = 1/2κ. Hence Eq. (4) will take the form of Eq. (6), and the boundary conditions of Eqs. (4) and (5) can both be made homogeneous, as expressed by Eqs. (7) and (8). −2κv

∂η ∂2 η ∂2 η = 2 + 2 ∂x ∂x ∂z (1/2κ = αw thermal diffusivity)

(6)

η|r→∞ = 0

(7)

η|r=δ,θ=π/2 = 1

(8)

The mathematical treatment of this problem is detailed in Appendix A. Its analytical solution is expressed as the following: K0 (κvr) η= exp(−κvx) (9) K0 (κvδ) or T = Tw +

Tw K0 (κvr)exp(−κvx), K0 (κvδ)

(z ≥ δ) (10)

where K0 is the Modified Bessel Function of the second kind of zero order [17]; v, the crystallization rate of the ice film spreading along the solid surface; δ, the thickness of the ice film, which can be, and has been measured experimentally in this study.

3. Results and discussions 3.1. Growth rates of ice films In addition to the energy balance and the temperature profile, the spreading velocity of the growing

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front of the ice film is also interest, which is termed as the growth rate of ice films in this model and is coupled with the rate of fusion heat release of the growing front. The dissipation of the fusion heat determines the growth rate of the ice film, which is the moving velocity of the solid slab in this problem. In a very short time period (dt), suppose that the ice film has a growth gain (dl), as shown by Fig. 7, the fusion heat given off by the ice bud can be written as: (dl)δρi H. The heat dissipated away in this period (dt) from the growing front into the solid slab, according to the Fourier’s law, is written as: 
  ∂T λ − ds (dt) (dl)δρi H = HAZ ∂r  π
  ∂T = −λ r dθ (dt) (11) ∂r r=δ 0 where ds (= rdθ) is the area element surrounding the heat source. Note in the assumption made in earlier part, we have assumed that the heat exchange in the ice-free area does not influence the growth rate of ice films. Re-arranging Eq. (11) yields:  π
 λ ∂T v=− dθ (12) ρ H 0 ∂r r=δ where v(= dl/dt) is the growth rate of ice films spreading on a subcooled solid surface. The expression of (∂T/∂r)r=δ can be obtained from Eq. (10) by using the substitution  π of x = r cos θ. Details of the integral process of 0 (∂T/∂r)r=δ dθ are presented in Appendix B. Because the integral on the right hand-side of Eq. (12) is still the function of v, the expression of v cannot be further simplified in the form v = f(λ, ρ, δ, Tw ), but can be expressed with implicit functions, in which v is incorporated in the argument of the Modified Bessel Functions [18]. This is shown in the following: K1 (κvδ)I0 (−κvδ) ρH + I1 (−κvδ) − =0 K0 (κvδ) πλκTw (13) In other words, for a given solid material (thus λ and κ = ρc/2λ is fixed) and a given supercooling condi-

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Fig. 8. The growth rate of ice films on metal surfaces.

tion of the solid surface (Tw , δ), the ice film growth rate (v) is determined by Eq. (13). Note for many commonly used materials, ρ, c, and λ are available in relevant handbooks [19].

Fig. 9. The growth rate of ice films on non-metal surfaces.

3.2. Effects of different materials Another issue of the model is the possibility to predict the growth rate of ice films on different solid surfaces. However, the thickness of ice films (δ) in Eq. (13) must be given for an analytical calculation. As described earlier, the thicknesses of ice films growing on glass and stainless steel surfaces were determined, respectively, which varied in a range from 150 to 250 ␮m (Figs. 4 and 5). Compared to the thickness of the solid wall, which was 1.5 mm (i.e. 1500 ␮m) in the micro-cell, ice films were thin. In addiction, ice films growing on the glass surface were found thinner than that on the stainless steel surface at the same supercooling. This may attribute to the growth rate of ice on stainless steel surfaces is higher than that on glass surfaces. For illustration purpose, it is assumed that the thickness of ice films is 200 ␮m either on metal surfaces or non-metal surfaces in all the analytical calculations of this paper, which is applicable from Figs. 7–12. On the other hand, since the Modified Bessel Functions are included in Eq. (13), computations for obtaining the v from other given variables are generally complicated and tedious. To use the computer-aided calculation, we can let Y = Eq. (13). Hence for a given supercooling (Tw ) on a specified material (thus λ, ρ, c, and δ are fixed), the approximation of v, which may satisfy Y = 0 with a preset acceptable error, can

Fig. 10. The temperature distribution in the HAZ of stainless steel at the supercooling of T = 2 ◦ C. (a) 3D view in temperature contours in the HAZ; and (b) projection of the temperature contours on the ZX plane.

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Fig. 11. The temperature distribution in the HAZ of stainless steel at the supercooling of Tw = 5 ◦ C. (a) 3D view in temperature contours in the HAZ; and (b) projection of the temperature contours on the ZX plane.

Fig. 12. The temperature distribution in the HAZ of glass at the supercooling of T = 2 ◦ C. (a) 3D view in temperature contours in the HAZ; and (b) projection of the temperature contours on the ZX plane.

be obtained using the computerized approaching calculation [20]. For example, stainless steel has a thermal conductivity λss = 17.45 Wm−1 C−1 , a density ρss = 7900 kg m−3 and a specific heat capacity css = 502.4 J kg−1 C−1 , giving a thermal diffusivity αss = 4.4 × 10−6 m2 s−1 or κ = 113756 m−2 s. Therefore, the growth rates (v) on stainless steel surface at a given supercooling (Tw ) can be obtained by solving Eq. (14) with above-mentioned method. As such, the growth rates of ice films on different solid surfaces can also be calculated individually, as shown in Fig. 8 for metal surfaces and Fig. 9 for non-metal surfaces. Note this model does not consider the influence of surface properties, such as the roughness and the interfacial tension, on the growth rate of ice.

It is obvious that the thermal conductivity has a prime influence upon the growth rate of the ice film as shown by Eq. (12). A higher value of the thermal conductivity gives a faster growth rate because the fusion heat transfers into the solid wall faster. This determines that the growth rate of ice films on a metal surface is normally faster than that on a non-metal surface. Besides the thermal conductivity, the density (ρ) and specific heat capacity (c) of the solid wall also impact the spread of ice films. The product of ρ and c represents the volumetric heat capacity of the solid wall. However, the value ρc seems to have a dual nature. First, a higher value of ρc denotes that the wall material can take up more heat for increasing every Celsius degree, which retards the temperature increasing rate

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Table 1 The effect of thermal properties on the growth rate of ice films

Copper (Cu) Aluminum (Al) Stainless steel Titanium (Ti) Zinc (Zn) Graphite Cast iron Low carbon steel Lead (Pb) PVC Polyethylene Perspex Glass Wood

◦ C)

λ (W m−1 K−1 )

ρ (kg m−3 )

c (J kg−1 K−1 )

κ (m−2 s)

v(2

401 203.53 17.45 21.9 116 129 62.80 45.36 34.89 0.163 0.256 0.188 0.74 0.087

8,960 2,670 7,900 4,500 7,140 2,620 7,220 7,850 11,400 1,400 920 1,200 2,500 550

380 921 502 520 390 710 502.4 460.5 129.8 1842.2 2219 1465.4 670 2721

4245 6042 113,756 53,425 12,003 7,210 28,880 39,854 21,204 7,920,000 3,989,455 4,666,667 1,131,571 8,579937

8054 3790 364 403 2221 2236 1281 924 585 3 2.7 3.3 12.7 1.5

(␮m s−1 )

Note: in calculating the growth rate of ice films at 2 ◦ C supercooling, the thickness of ice films is assumed to be 200 ␮m. The roughness of the solid surface and its surface properties, such as interfacial tension, are not considered.

in the HAZ (i.e. in the solid slab right underneath the growing front of the ice film), and encourages a higher growth rate. Second, a higher value of ρc straightly results in a greater value of κ (because κ = ρc/2λ), where ρc is the numerator, but 2λ the denominator. This may in turn be interpreted that ρc has the opposite effect of λ, implying it may slow down the film spread. Any way, the contribution of ρc to the film growth is mathematically determined by Eq. (13). The typical film growth rates on different selected materials at 2 ◦ C of supercooling are shown in Table 1 together with their thermal properties that affect the film growth. 3.3. Temperature distribution in HAZ The temperature distribution in a cross-section of the HAZ in a solid slab has been described by Eq. (10). Note Eq. (10) represents a serial temperature contours above the XZ plane of the coordinate system defined in Figs. 6 and 7. Taking stainless steel and PVC plastic for instances, as we assumed earlier for the purpose of illustrating that the thickness of the ice film is 200 ␮m, when the wall supercoolings are Tw = 2 ◦ C and Tw = 5 ◦ C, respectively, the growth rates (v) of ice films can be calculated, respectively, from Eq. (13). Therefore, with known v, δ, ρ, c, and Tw , the temperature distribution in the HAZ becomes computable according to Eq. (10), and the graphics are shown in Figs. 10 and 11 for stainless steel, and in Fig. 12 for PVC (they are generated by the TableCurve 3D® ,

AISN Software Inc.). The z-axis in these charts point to the inner direction of the subcooled stainless steel slob (same with Figs. 6 and 7). It starts with 200 ␮m (not zero), which is the thickness (δ) of the ice film, due to the definition zone of z in Eq. (10). Therefore, we use another axis in parallel with the z-axis, but starts with zero, as the measurement of the distance from the solid surface. It can be found that the shape of those temperature contours looks similar at different supercoolings, but they become more elliptic when the growth rate of the ice film is faster (see Figs. 10(b) and 11(b)). As expected, when the wall supercooling (Tw ) is the same, a non-metal solid slab will have a greater temperature gradient (in HAZ) than that of a metal one, because a non-metal slab generally has a lower thermal conductivity (λ) than a metal slab has. This can be seen by comparing Figs. 10 and 12. Both are at 2 ◦ C of supercooling, but the latter shows a sharper peak in the temperature profile. The thickness of the solid slab is presumed to be infinite in the model of this paper. In the case of the finite thickness solid slab (i.e. T |r→b = Tw , where, b is the thickness of the solid wall), the temperature away from the HAZ will approach to Tw faster than that in the case of infinite thickness, as if the HAZ is ‘compressed’. Nevertheless, the authors do not intend to deal with such a problem in this paper. In summary, based on this study and previous work (Qin, Zhao, Russell, Chen, Chen and. Robertson, 2002

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[8]; Qin, Russell, Chen and Robertson, 2003 [9]), ice crystallization in the form of thin films spreading along a subcooled solid surface is an intermediate, transitional period between the surface nucleation and normal growth of ice layers. The sequences of ice formation in an aqueous solution brought into contact with a subcooled solid surface can be described as: • Nucleation. Random occurrence of newly formed crystal embryos of ice on a subcooled solid surface, which is the start of phase transition. • Film growth. Those newly formed, separate ice embryos quickly spread along the subcooled solid surface to form thin films of ice until it is completely covered. This is a transitional step. Experimental observation shown that the thickness of ice films vary in a range of 150–250 ␮m depending on the material. • Layer growth. The ice film starts to grow thicker in the normal direction on the solid surface to form a visible ice layer after the transitional period. The heat transfer resistance then increases as a result, and the ice formation rate starts to decline afterward.

η|r→∞ = 0

(A.2)

η|r=δ,θ=π/2 = 1

(A.3)

An empirical method for solving Eq. (A.1) is to assume the solution has the form of η = ϕ(x, z)exp(−κvx)

∂2 ϕ ∂2 ϕ + 2 − (κv)2 ϕ = 0 ∂x2 ∂z

(A.5)

Now that the problem of seeking the solution for Eq. (A.1) has become the problem of seeking the solution for Eq. (A.5). Re-writing (A.5) in the form of polar coordinate system gives 1 ∂2 φ ∂2 φ 1 ∂φ + + − (κv)2 φ = 0 r ∂r ∂r 2 r 2 ∂θ 2

(A.6)

In order to use the variable-separation method to solve this equation, we assume (A.7)

This generates two ordinary differential equations:

The growth kinetics of ice films in the above-mentioned transitional step can be described by the heat conduction model presented in this paper. Since the establishment of the moving coordinate system at the growing front of the ice film, the unsteady-state heat transfer problem has been simplified into a quasisteady-state one, which is expressed with the Modified Bessel Equation. Furthermore, the analytical solution of the problem has been worked out by solving the Modified Bessel Equation. Application of the analytical solution leads to the following predictions possible: (1) the growth rate of ice films on different materials, and (2) the temperature distribution in the solid slab right underneath the growing front of the ice film. Appendix A

Φ (θ) + µΦ(θ) = 0

(A.8)

r 2 R (r) + rR (r) − [(κv)2 r 2 + µ]R = 0

(A.9)

For Eq. (A.8), the physical aspect of this problem shows that Φ(θ) must be a periodical function with 2π as its cycle, which results in µn = n2 (n = 0, 1, 2, 3, . . . ). µn is the so-called intrinsic value of Eq. (A.8) and the corresponding intrinsic functions are: a0 (constant) (A.10) Φ0 (θ) = , 2 Φn (θ) = an cos nθ + bn sin nθ

(A.1)

(A.11)

Putting µn = n2 into Eq. (A.9) with the substitution of ζ = κνr, we convert Eq. (A.9) into a typical Modified Bessel Equation ([18]): ζ 2 F  (ζ) + ζF  (ζ) − (ζ 2 + n2 )F(ζ) = 0

Re-write Eqs. (6)–(8): ∂η ∂2 η ∂2 η −2κv = 2 + 2 ∂x ∂x ∂z (1/2κ = αw thermal diffusivity)

(A.4)

where ϕ(x,z) is an unknown function to be found. Putting the expression (A.4) into (A.1) and performing the re-arrangement, we find

ϕ = R(r)Φ(θ) 4. Conclusion

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(A.12)

where F(ζ) = R(ζ/κv). The solutions of Modified Bessel Eq. (A.12) are known as Bessel functions with imaginary argument. Hence the general solution of Eq. (A.12) is expressed as:

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Fn (ζ) = AIn (ζ) + BKn (ζ), (A, B are constants)

(A.13)

or substituting ζ = κνr back to (A.13) to gives: (A.14) Rn (r) = AIn (κvr) + BKn (κvr) where In (x) is known as the Modified Bessel Functions of the first kind of order n, and Kn (x) the Modified Bessel Functions of the second kind of order n. So the solution of Eq. (A.1) can be expressed as: η = Φ(θ)[AIn (κvr) + BKn (κvr)] exp(−κvx) (A.15) The physical nature of this problem in our modeling have stipulated that the temperature is bounded as r → ∞. However, In (κvr)exp(−κvx)|r→∞ and x≤0 → ∞, unless x > 0, i.e. the boundary condition of Eq. (A.2) forces A = 0. The monotropic decrease of temperature in the HAZ makes the order of Modified Bessel Eq. (A.14) to be zero, i.e. n = 0. Thus, Eq. (A.14) becomes R0 (r) = BKn (κvr) (A.16) Substituting the Eqs. (A.10) and (A.16) into (A.7), and then putting (A.7) into (A.15) yields: η = CK0 (κvr) exp(−κvx)

or it can be written as: Tw K0 (κvr)exp(−κvx) T = Tw + K0 (κvδ)

(B.2)

0



π

cos θ exp(−κvδ cos θ)dθ = πI1 (−κvδ)

(B.3)

0

therefore,  π
 ∂T dθ ∂r r=δ 0

 π  Tw =− κvK1 (κvδ) exp(−κvδ cos θ)dθ K0 (κvδ) 0  π  cos θ exp(−κvδ cos θ)dθ + κvK0 (κvδ) 0

Tw =− [κvK1 (κvδ)πI0 (−κvδ) K0 (κvδ) + κvK0 (κvδ)πI1 (−κvδ)]   K1 (κvδ)I0 (−κvδ) = −πκvTw + I1 (−κvδ) K0 (κvδ) (B.4)

(A.17)

Where C(=a0 B/2) is a constant to be determined. Applying the boundary condition (A.3), note x = r cos θ, we have:   1 1  = (A.18) C=  −κvrcos(θ) K0 (κvδ) K0 (κvr)e r=δ, θ=π/2 Therefore, the solution of Eq. (A.1) is: K0 (κvr) exp(−κvx) η= K0 (κvδ)

According to [17]:  π exp(−κvδ cos θ)dθ = πI0 (−κvδ)

(A.19)

(A.20)

Appendix B Take the differential of T against r for Eq. (11), and substitute r = δ into it, we obtain:
 ∂T Tw  =− κvK1 (κvδ)exp(−κvδ cos θ) ∂r r=δ K0 (κvδ) + κvK0 (κvδ)cos θ exp(−κvδ cos θ) (B.1)

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