Progress in the Application of Fractal Porous Media Theory to

Review pubs.acs.org/IECR

Progress in the Application of Fractal Porous Media Theory to Property Analysis and Process Simulation in Melt Crystallization Xiaobin Jiang,*,† Jingkang Wang,‡ Baohong Hou,‡ and Gaohong He*,† †

State Key Laboratory of Fine Chemicals, R&D Center of Membrane Science and Technology, School of Chemical Engineering, Dalian University of Technology, Dalian 116024, People’s Republic of China ‡ National Engineering Research Center of Industrial Crystallization Technology, School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, People’s Republic of China ABSTRACT: The application of fractal porous media (FPM) theory in melt crystallization is a novel concept to be developed extensively, which has received wide attention in recent years. The primary advantage of FPM theory is the ability to describe the complex, random structure of porous crystal layers with fewer parameters. This review will introduce aspects of the property analysis of porous crystal layers grown in the heterogeneous thermal field. For the first time, the existing flow model of FPM theory will be outlined and an overview of the general issues, model modification, and validation of the separation process in melt crystallization will be discussed in detail. Kinetic and model research on separation evaluation and optimized operation will be mentioned to unfold the promising application field. Finally, the paper will conclude with perspectives on the future research directions, key challenges, and issues in this area.

1. INTRODUCTION Despite the maturity of the chemical industries, there is still a tremendous need for improving a variety of separation technologies, especially on the separation and preparation of hyperpure or heat-reactive chemicals.1 Traditional separation methods (such as distillation) usually are highly energyintensive. However, the major advantage of melt crystallization as a separation technology becomes obvious when comparing the required energy for the phase change of melt crystallization (solid/liquid) to that of distillation (liquid/vapor).2 In addition, investment and running costs of melt crystallization is relatively lower and the industrial scaleup of melt crystallization is easier than other traditional separation methods.3,4 Because of the advantages of industrial application, melt crystallization has been widely applied to the manufacture of pharmaceuticals, hyperpure chemical preparation,5 ionic liquid purification,6 and seawater desalination.7,8 During the crystal layer growth process of melt crystallization, the uncrystallized fluid will discharge at the end of the process (static melt crystallization) or migrate simultaneously during the crystal layer growth (dynamic melt crystallization, or falling film melt crystallization). The impure phase, entrapped during the crystal layer growth, leads to the porous structure of the crystal layer. This impure phase can be eliminated through the open pores during the sweating process.9 Thus, the terminal product of melt crystallization is the purified crystal layer, which is different from the conventional solution crystallization process. This difference results in the difficulty of online research on the process of melt crystallization. Moreover, it is difficult to simulate the separation process of melt crystallization due to the lack of an effective method to analyze the structure and relevant properties of the porous crystal layer. Thus, correlative research has been attracting a great deal of attention.10,11 Furthermore, when there are nonisothermal processes and phase transition processes (sweating processes, temperature oscillation oper© 2013 American Chemical Society

ations, etc), the simulation becomes more difficult. This is because the structure and property variations of the porous crystal layer are too complicated to describe with conventional crystallization models and Euclidean geometry theory. As a novel approach to describe the complex structure of the porous media, fractal porous media (FPM) theory and technology have been reported in the research of chemical engineering and other relative fields: Yu12−14 had proposed a series of fractal capillary models to analyze the permeability of non-Newtonian fluids through the porous media; Cai15 had developed the depth of extraneous fluid invasion by considering the fractal capillary pressure effect. The model could predict invasion by extraneous fluids into a permeable bed with good agreement. Heath16 had proposed a population balance model used to describe the aggregation/breakage kinetics in turbulent pipe flow with fractal theory. A fractal model for gas diffusivity in the porous media was recently derived and expressed as a function of microstructural parameters of porous media.17,18 Besides, applications of FPM theory on the heat conduction19 and other property20 were also reported. All the achievement of the application of fractal theory to engineering fields remind us that the crucial advantage of FPM theory is the ability to describe the complex, random structure of the porous media in diverse engineering field. Consequently, it is easy to anticipate the possible application of FPM theory and technology in melt crystallization: the structural and other property analysis of the porous crystal layer. In fact, the exploration on the crystal growth and fractal geometry had already started two decades ago. The fractal geometry concept was applied to analyze mechanisms occurring Received: Revised: Accepted: Published: 15685

July 9, 2013 October 16, 2013 October 16, 2013 October 16, 2013 dx.doi.org/10.1021/ie402182f | Ind. Eng. Chem. Res. 2013, 52, 15685−15701

Industrial & Engineering Chemistry Research

Review

Figure 1. Atomic force microscopy (AFM) height images showing the growth process of a seaweed crystal pattern at 278.15 K. (Reproduced from Ma et al.26 Copyright 2008, Elsevier, Amsterdam.)

during crystal growth or dissolution. It was found that specified reaction fractal dimensions (DR) were quite sensitive to the reaction conditions. The successful interpretation of reaction fractal dimensions enabled the evaluation of the crystallization behavior from relatively accessible experimental data.21 Krukowski22 had reported that the compact crystal grew under different surface cohesions presented various fractal dimensions. The crystal exhibited the channels or pores structure under different scale. For a sufficiently large area, the crystal growth rate obeyed the self-similarity relation, which indicated the fractal property. A particularly interesting finding had been reported by Marangoni23 was that the fractal dimension was directly related to the rate of nucleation, as well as being inversely related to the Avrami exponent. In addition, the exhibition of fractal property on crystal growth had been reported in some literatures, two dimensional (2D) or three dimensional (3D).24,25 The crystals grown under the ideal operations were observed to have a regular, self-similar fractal structure in polymer and inorganic systems. The fractal crystal pattern growth process was investigated by Ma (see Figure 1).26 He had synthesized single-crystal α-Fe2O3 with fractal nanostructures with micropine- and microsnowflake-like morphologies via a hydrothermal route. (See Figure 2.)27 All the research above indicates that the fractal property of the crystal grew under diverse crystallization configurations.

Figure 2. SEM images of α-Fe2O3 synthesized (a) without the surfactant and (b) with the surfactant NP-9. Also shown are highmagnification SEM images of (c) individual micropine and (d) microsnowflake structures. (Reproduced from He et al.27 Copyright 2008, Wiley, New York.)

The fractal-like structure was not only observed in the microscale crystallization: it was also observed in the macroscale crystallization process. The process of crystal growth in 15686

dx.doi.org/10.1021/ie402182f | Ind. Eng. Chem. Res. 2013, 52, 15685−15701


Review

Figure 3. The growth process of the H3PO4·0.5H2O crystal, relative to the cooling temperature. (Reproduced from Jiang et al.28 Copyright 2011, American Chemical Society, Washington, DC.)

Figure 4. Five stages in the construction of Sierpinski square. (Reproduced from Henderson et al.29 Copyright 2010, Elsevier, Amsterdam.)

the cooling crystallizer is shown in Figure 3. The phosphoric acid hemihydrate (H3PO4·0.5H2O) grew from the crystal seed to crystals with various sizes and shapes. The growth process exhibited a self-similar property that corresponded to classic fractal graphics (Figure 4, Sierpinski square). The pores (just like the black area in the square) became complicated and had the fractal property with the increasing of crystal numbers and crystal sizes (exactly like the increasing blank squares in the middle of the square). As a result, space in the crystallizer was continually segmented by the growing crystals until the crystals composed the fully formed crystal layer. For the crystals occupied the space in crystallizer with a fractal property, the residual space (or the pores and channels) presented selfsimilarity compulsorily. It was reasonable to conclude that the pores and channels in the maturely grown crystal layer had the fractal property and the entire porous crystal layer was a FPM. Note that the porous crystal layer was generally formed under a temperature gradient as the driving force. Because of the formation process under the inhomogeneous thermal field, the structure and relevant properties (permeability, thermal, etc.) of porous crystal layers vary from part to part. Thus, the application of FPM theory and technology to a porous crystal layer is an extremely challenging endeavor to the researchers in industrial crystallizations. Some critical modification on the existing FPM model should be implemented with adequate experimental results. In addition, the dynamic simulation of flow in this complicated porous crystal layer is also meaningful to evaluate the separation effect and efficiency of the relative process. There have been no reviews on the application of FPM theory and technology to melt crystallization yet. The research on the correlation between structure and relevant properties of the crystal layer with operation curves of crystal/crystal layer growth, which calls for developing new ideas and innovative

methods, will be discussed and highlighted in this review. Those who are interested in the establishment, development, and validation of fractal theory in porous media and the other possible application fields of FPM theory and technology are encouraged to read several excellent extensive review articles.30−33 In this review, we will first give a brief introduction of FPM theory. We then will summarize the development of the structure and the relevant properties (permeability, thermal conductivity, etc.) in analytical models of FPM. The possible application field of FPM theory and technology in melt crystallization will be outlined and, in the following discussion, recent progress on the thermal property analysis of crystal layer, the process model and separation evaluation of melt crystallization with FPM theory and technology will be introduced, the modification and newly defined parameters will be interpreted in detail. Some latest research results will also be proposed for in-depth interpretation of the application scope of the developing model. We will also present our perspectives on the future development and the key issue of FPM theory and technology application in melt crystallization.

2. OVERVIEW OF FRACTAL POROUS MEDIA (FPM) THEORY AND TECHNOLOGY 2.1. Principle of FPM. In fact, there are numerous objects in nature34,35 (such as coastlines, mountains, rivers, lakes, and islands) that are disordered and irregular. Because of the scaledependent measures of length, area, and volume, they do not follow the Euclidean description and Euclidean geometry theory. These objects are called fractals, because the dimensions of such objects are nonintegral and are defined as fractal dimensions. The measure of a fractal object M(L) is related to the length scale L through a scaling law, in the form of34 15687



M(L) ≈ LDf

Review

According to eq 4, the average pore diameter κav can be expressed as40

(1)

where M is the length of a line or the area of a surface or the volume of a cube or the mass of an object, and Df is the fractal dimension of an object. In the early work of Yu and Li,36 an unified model had been deduced to describe the fractal characters of the porous media. The results indicate that the proposed model is applicable to both the exact and statistically self-similar fractal porous media (FPM). More significantly, a statistical property of a porous medium and a criterion to determine whether a porous medium can be characterized by fractal theory were described. The fractal dimension (Df) is determined by Yu and Li:36 Df = 2 −

ln ϕ ln(κ min /κ max )

κav =

Following the work of Yu et al.,39,40 Cai,41,42 Yun,43 and Feng,44 the tortuosity and tortuous stream theory were developed for a wider application of this fractal theory and technology. It is conspicuous that the entire area of pores on the cross section of FPM is Ap = −

(2)

⎛ κ max ⎞ Df ⎜ ⎟ ⎝ κ ⎠

A=

(4)

f where f(κ) = DfκDmin κ−(Df + 1) is the probability density function of pore size. The porous media commonly found in the engineering field present fractal properties from two aspects. Besides the pore size distribution in the cross section (which defined by Df), the tortuosity of the porous channel along the porous media also shows a statistically fractal property. The tortuosity fractal dimension (DT) can be obtained as reported:39

ln τav ln(L /κav)

(5)

where τav and κav are the average tortuosity and the average pore diameter, respectively. L is the thickness of the porous media (or the apparent length of the porous channel). A simple geometrical model for the tortuosity of the flow path in the porous media was proposed by Yu and Li.40 The average tortuosity (τav) can be expressed as ⎡ ⎢ 1⎢ 1 τav = ⎢1 + 1−ϕ + 2 2 ⎢ ⎣⎢

(

1 1−ϕ

1−

2

)

−1

1−ϕ

+

1 4

1−ϕ

K= ⎤ ⎥ ⎥ ⎥ ⎥ ⎥⎦

λmax min

πDf κ max 2 1 2 πκ ( −dN ) = 4 4(2 − Df )

(8)

Ap ϕ

=

πDf κ max 2 4(2 − Df )ϕ

(9)

In summary, all of the work above had offered a theoretical foundation of describing FPM with few parameters, which improved the feasibility of FPM theory in the wide field of engineering. Research on the property analysis and mass transfer in FPM was explored. 2.2. Permeability, Seepage Process, and Thermal Conductivity Research on FPM. 2.2.1. Permeability and Seepage Process Research on FPM. As the basic characteristic parameter of the porous media, permeability research had drawn continuous attention during the recent decade.45 In fact, the principal target of former research was offering an easy and direct path to establish the permeability model and the flow process of FPM, which was significant to describe the multiphase mass transfer in FPM. To achieve this objective, a few meaningful attempts and work had been implemented. Sahimi46 first proposed the study on nonlinear transport processes in multiphase disordered porous media. The extension of the results to other types of nonlinearities is also discussed. Civan47 had found a permeability−porosity relationship, confirming Civan ’s power law flow units equation. Empirical formulations provide helpful insights into the mechanism of porosity and permeability variation by simpler lumped-parameter models. Yu48 derived new analytical expressions for the permeability and the semiempirical Kozeny−Carman constant based on the fractal geometry theory. A model that does not contain an empirical constant will be more meaningful for practical applications to the permeability analysis of multiphase (or unsaturated) porous media. Series work on the nonempirical constant model development then will be introduced. According to Darcy’s law, the permeability K of a porous medium can be expressed as

and

DT = 1 +

∫λ

the cross-sectional area of FPM (A) is given as

(3)

Df −(Df + 1) −dN (κ ) = Df κ max κ d κ = f (κ ) d κ

min

(7)

where ϕ is the porosity of the porous medium, κmin is the minimum diameter of the pores in the porous medium, and κmax is the maximum diameter of the pores in the porous medium. (κmin/κmax)Df = 0 is the criterion of a porous medium which can be characterized by fractal theory. Thus, the dimension is a function of porosity and pore size distributions. In the following work of Yu, Cheng, and co-workers,37,38 the cumulative pores number (N) whose sizes (L) are greater than or equal to κ have been proven to follow the fractal scaling law:

N (L ≥ κ ) =

∫λ

⎡ ⎛ κ ⎞ Df − 1⎤ ⎛ Df ⎞ ⎥ κf (κ ) d κ = ⎜ ⎟κ min⎢1 − ⎜ min ⎟ ⎢⎣ ⎥⎦ ⎝ Df − 1 ⎠ ⎝ κ max ⎠

λmax

μL0Q ΔPA

(10)

where μ is the viscosity of the fluid, L0 the length of the porous media, ΔP the pressure gradient, and A the cross-sectional area of the porous media. To express the flow rate Q, the flow rate through a single tortuous capillary is given by modifying the well-known Hagen−Poiseulle equation and introducing eq 1:

(6) 15688



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Table 1. Equations of Some Effective Permeability for Non-Newtonian Fluid through FPM effective permeability Ke

model power law

1−D

Ke =

Ellis

Ke =

T n2ϕL 2DT − 2Dfn − 1 (1 + 30n)D

(

⎡ ⎢1 − ⎣

⎥ ⎦

T

q(κ ) =

Ke =

1 − DT

ϕL 2DT − 4 D0 T

π ⎛ ΔP ⎞ κ 4 π ⎛ ΔP ⎞ κ 4 ⎜ ⎟ = ⎜ ⎟ 128 ⎝ L t(κ ) ⎠ μ 128 ⎝ L0DT ⎠ μ

∫κ

=−

=

κ max

( )

(11)

q(κ ) dN (κ )

min

1−D ⎞ Df π ⎛ ΔP ⎞ A ⎛ L0 T ⎞⎛ ⎟⎜ ⎟ ⎜ ⎟ ⎜ 128 ⎝ μ ⎠ L0 ⎝ A ⎠⎝ 3 + DT − Df ⎠ ⎡ ⎛ κ ⎞3 + DT − Df ⎤ ⎥ × κ max 3 + DT⎢1 − ⎜ min ⎟ ⎢⎣ ⎝ κ max ⎠ ⎦⎥

(12)

When the criterion of FPM is satisfied, eq 11 can be reduced to Q=

⎞ L0Df π ⎛ ΔP ⎞⎛ ⎜ D ⎟⎜ ⎟κ max 3 + DT 128 ⎝ μL0 T ⎠⎝ 3 + DT − Df ⎠

(13)

The driving force (ΔP) has an obvious influence on the flow discharge process. Moreover, ΔP is usually variable, decided by the operation condition in practical industrial processes, which makes the process simulation more complex. Thus, the permeability of FPM can be expressed as37,50 K =

μL0Q ΔPA

1−D ⎞ Df π ⎛ L 0 T ⎞⎛ ⎜ ⎟⎜ = ⎟κ max 3 + DT 128 ⎝ A ⎠⎝ 3 + DT − Df ⎠

14

oped mathematical models of effective permeability through fractal porous media are listed in Table1. The equations in Table 1 indicate that the permeabilities for non-Newtonian fluids are significantly different from those for Newtonian fluid flow in porous media. Here, n is the power exponent; α, ϕ0, and ϕ1 are material constants that describe the properties of fluids, and dp/dL is the pressure gradient. The parameters (DT, Df, ϕ, kmax) required to express the permeability for non-Newtonian fluids have been discussed in the former section. Thus, the application of fractal theory on the permeability for either Newtonian fluids or non-Newtonian fluids in porous media is achievable. 2.2.2. Thermal Conductivity Research on FPM. Fractal theory and technology also present their application to thermal property analysis.51,52 In fact, the porous media in the chemical engineering field usually have random-shaped porous channels filled with fluid. This complex multiphase construction makes the thermal conduction analysis difficult to chemical engineering designers. Ma and Yu reported an approximate fractal geometry model to simulate the thermal conductivity with thermal−electrical analogy technology.53 The proposed model is expressed as a function of porosity, the ratio of areas, and the ratio of component thermal conductivities. The recursive algorithm utilized to solve the proposed model is presented and found to be quite simple. Zhu et al. recently developed a fractal series-parallel model, which assumed that numerous capillary channels were parallel or perpendicular to the heat flow direction.54 Regarding the fractal parallel model, the thermal resistance of a single fluid channel (rg(κ)) can be expressed according to the Fourier’s law:

q(κ )f (κ ) dκ

κ max

13

⎭

3 + DT (2 − Df )κmax (3 + DT − Df )(1 − ϕ)

min

∫κ

12

( )

where κ and Lt are the hydraulic diameter and length of a single capillary, respectively. It is obvious that the flow rate Q is the radial integral of q(κ) and Q can be expressed as follows, using eq 4 and the work previously reported:45,48,49 Q =

)

ref

1 + DT (2 − Df )κmax (DT − nDf + 3n)(1 − ϕ)

D − 4 1 − DT ⎤ ⎡ α−1 ϕ (2D T − 2L1 − D T)α dp ϕ(2 − Df ) ϕ02 T L DT + 1 αD T + 1⎥ ⎢ κmax + α1 − κmax 1−ϕ ⎢ D T (α + 3)(αD T − Df + 3) dL ⎥⎦ ⎣ DT(DT − Df + 3) α−1 ⎫ ⎧ D D − ⎤ ⎡ D D D 2 1 T f − − T ⎪2 T ⎪ TDf ⎛ ⎞ dp L κmax ⎥⎬ ⎢1 − ⎜ κmin ⎟ ϕ0 + ϕ1⎨ − ⎪ ⎥⎦⎪ ⎢ DT dL D T − Df ⎣ ⎝ κmax ⎠

⎩

Bingham

n−1

DT / n − Df ⎤n − 1

( ) κmin κmax

n DT − nDf

rg(κ ) = (14)

L (κ ) 4L(κ ) = Aλ g πκ 2λg

(15)

where λg is the thermal conductivity of the fluid channels. The thermal resistance of fluid channels RN(κ) can be described by inserting eq 4:

Equation 14 indicates that the permeability is a function of the pore-area fractal dimension (Df), the tortuosity fractal dimension (DT), and structural parameters A, L0, and λmax. However, the seepage process of fluid transport through FPM is related to the fractal and structural parameters mentioned above. Considering that the flow in the melt crystallization involves non-Newtonian fluids (such as the organic polymer melt fluid phase with ultrahigh viscosity) at times, the effective permeability of the non-Newtonian fluids (such as power-law fluids, Ellis fluids, Bingham fluids) through FPM has been investigated theoretically and experimentally.12−14 The devel-

R N (κ ) = =

=

15689

rg(κ ) − dN 4L(κ ) Df −(Df + 1) πκ λg Df κ max κ dκ 2

4L0DT Df DT − Df πλg Df κ max κ dκ

(16)



Review

The effective thermal resistance of all fluid channels (Rg) can be expressed as follows, based on the thermal−electrical analogy principle, 1 = Rg

=

∫

κ max

κ min

2.3. Internal Relation between FPM Theory and Melt Crystallization. The objective of this section is to discuss the internal relationship between FPM theory and technology and melt crystallization process. In recent decades, researchers have been focusing on how the operating conditions impacted the separation effect and efficiency of melt crystallization (static and dynamic modes).56,57 Regarding static melt crystallization (SMC), the crystal tower is filled with a fully grown crystal layer and impure liquid entrapped in the porous channels. The impure phase can be separated from the crystal layer using two methods: the impure phase entrapped in large open pores is discharged directly through the porous crystal layer by opening the valve under the crystal tower, which is similar to the seepage process in FPM; the impure phase entrapped in closed pores or tortuous channels is discharged via a sweating process, which is a temperature-induced purification step and is analogous to the seepage process in FPM. The complex and tortuous channels in the crystal layers (before and after the sweating process) are clearly shown in Figure 5, which indicate that seepage fluid and sweating liquid phase both transport through a FPM-like crystal layer under a temperature-induced driving force.

1 dκ R N (κ )

⎡ DT + 1 πλLDf κ max ⎢1 − ⎣

DT − Df + 1 ⎤

( ) κ min κ max

⎥ ⎦

4L0DT(DT − Df + 1)

(17)

and the effective thermal resistance of the crystal phase (Rs) can be described as (1 − ϕ)Aλs 1 = Rs L0

(18)

where λs is the thermal conductivity of the solid phase. Thus, the thermal conductivity of a porous medium λs,par in the parallel model can be expressed as follows: λs,par =

=

L0 AR s L0 ⎛ 1 1 ⎞⎟ ⎜⎜ + A ⎝ Rs R g ⎠⎟

DT − Df + 1 ⎤ ⎡ κ DT − 1 λg ϕκ max (2 − Df )⎢1 − κ min ⎥ max ⎦ ⎣ = − 2 D ⎡ T⎤ κ L0DT − 1(DT − Df + 1)⎢1 − κ min ⎥ max ⎦ ⎣ + (1 − ϕ)λs

( ) ( )

(19)

Regarding the fractal perpendicular model, the series theory can be applied to modify the parallel model; the thermal conductivity of a porous medium (λs,per) in the perpendicular model is λs,per =

Figure 5. Crystal layer before (left) and after (right) the sweating process in SMC. (Reproduced from Jiang et al.28 Copyright 2011, American Chemical Society, Washington, DC.)

1

(φ/λg) + [(1 − φ)/λs]

(20)

The overall thermal conductivity of the porous media λs then is given as λs =

Regarding dynamic melt crystallization (or falling film melt crystallization, FFMC), the crystal layer frontier is grown under the falling liquid film. The crystal layer surface is rough (Figure 6), and a detailed branched-porous structure in the crystal layer is also observed in a slow gradient cooling experiment (Figure 7). The formation mechanism of branched-porous structure in crystal layer was discussed:58 the temperature gradient in the falling liquid film will increase with the cooling temperature during the crystal layer growth. When the gradient is greater than the equilibrium gradient, the steady growth state will breakdown. The unsteady growth frontier of the crystal layer then tends to form a branched-porous structure rather than a smooth surface. It is easy to conclude that the impure liquid is absorbed in the pores and channels of the crystal layer at the same time when the well-behavior crystallization frontier breakdown under the excessive temperature gradient. Besides the research on the separation process of SMC and FFMC above, Kim and Ulrich reported that a relatively gradual cooling route or a lower cooling temperature led to a lessporous crystal layer with better effective thermal conductivity.59

1

(ψ /λs,par + (1 − ψ )/λs,per) + ⎡⎣(1 − ψ )/λs,per ⎤⎦ (21)

where ψ is the ratio of the number of parallel channels to the total number of channels, with values ranging from 0 to 1. The prediction results from the fractal model proposed are compared with those calculated from nonfractal parameters models,55 and experimental data. The fractal model exhibited better agreement with the experimental data than the nonfractal model. Although the model has a reliable and clear theoretical foundation, there is still an unknown parameter, ψ, which is difficult to obtain under the chemical engineering conditions, such as the fouling layer formed in the heat exchanger or the porous crystal layer growing on the cooling interface of the crystallizer. This introducing parameter may limit the application of the model. 15690



Review

Figure 6. Branched-porous structure of the crystal layer in FFMC: (A) the outer surface and cross profile, (B) inner surface, and (C) detailed inner surface (panel (B) enlarged by a factor of 2.5). (Reproduced from Jiang et al.58 Copyright 2012, Elsevier, Amsterdam.)

3. PROPERTY ANALYSIS OF THE CRYSTAL LAYER IN MELT CRYSTALLIZATION 3.1. Porosity and Permeability of the Crystal Layer. Considering SMC, the crystal layer grown in the crystallizer is kept at a phase equilibrium state before the seepage process; thus, the solid−liquid component in the crystal layer can be easily obtained from the phase equilibrium diagram. The ratio of solid phase volume and pores volume (which is filled by the liquid phase) then can be obtained, which is utilized to calculate the porosity. As a dynamic process, the crystal layer in FFMC is observed, along with the discharge of an uncrystallized feed liquid phase. Therefore, the solid−liquid component in the crystal layer is determined by the overall cooling route and feed conditions rather than the terminal equilibrium temperature. Hence, the mass balance of solid and liquid phases in the crystal layer should be utilized to obtain the solid−liquid component. As mentioned in eq 14, the permeability is a function of the pore-area fractal dimension (Df), the tortuosity fractal dimension (DT), and structural parameters A, L0, and κmax. With the porosity and parameter of apparatus known, the parameters above can be obtained, except κmax. As a critical structure parameter and model boundary condition, κmax should be determined. Several approximate methods have been reported.28,37 According to eq 13, the initial driving force of seepage process was only influenced by the fluid density and the height of the crystal tower. Thus, the initial flow rate measured in the experiment was utilized for the calculations.60 The experimental results in a series test are listed in Table2. The porosity, permeability, and fractal property of porous crystal layers under different operation conditions were obtained. It was clear that the value of κmax increased with the porosity. The fractal property of the crystal layer became indistinctive (DT → 1; Df → 2) when the porosity increased. This result certified that the fractal property of porous channels in the crystal layer was influenced by the crystal layer growth

Figure 7. Details of the crystal layer grown during the slow gradient cooling experiment: (A) low-magnification image and (B) highmagnification image. (Reproduced from Jiang et al.58 Copyright 2012, Elsevier, Amsterdam.)

An empirical model to estimate the thermal conductivity of the porous crystal layer was developed, which inspired the researchers to investigate the influence of various operation conditions on the crystal layer structure and the thermal property, using the concept of the porous media and fractal theory. However, the lack of a connection between the operational condition and the crystal layer properties (FPM property, especially), and the absence of technology capable of measuring the structure parameters defined as the porous media are two major drawbacks that limit the direct application of existing models. Thus, the model development and modification is an urgent issue for expanding the application of FPM theory and technology to industrial crystallization. Meanwhile, FPM theory and technology will provide a novel viewpoint on the separation evaluation of melt crystallization.

Table 2. Experimental Result of Porosity ϕ, κmax, Df, and DT of the Crystal Layer in SMCa ϕ κmax [mm] Df DT a

Test No. 1

Test No. 2

Test No. 3

Test No. 4

Test No. 5

Test No. 6

Test No. 7

Test No. 8

Test No. 9

Test No. 10

0.289 0.810 1.861 1.042

0.392 1.320 1.901 1.031

0.569 1.690 1.942 1.019

0.590 2.010 1.947 1.018

0.601 2.380 1.949 1.017

0.625 2.540 1.954 1.016

0.635 2.770 1.956 1.015

0.664 3.090 1.960 1.014

0.715 3.470 1.968 1.012

0.783 3.880 1.977 1.009

Data taken from ref 60. 15691



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simulate the data reported in ref 61. with good agreement. However, it is clear that the difference-fractal model does not agree with the permeability data of the crystal layer no matter the method (SMC or FFMC). Thus, further theoretical research on the permeability evaluation of the porous crystal layer by considering the layer growth condition and the impact of crystallizer is required, especially when the porosity is smaller than 0.7, which is common in melt crystallization and other chemical engineering process. 3.2. Thermal Conductivity Analysis of the Porous Crystal Layer. As essential data for heat transfer research in industrial crystallization, the thermal conductivity of the porous crystal layer is difficult to obtain during the crystallization process. Thus, thermal conductivity analysis of the porous crystal layer is attractive work to the researchers. Meanwhile, the random distribution of pores in the crystal layer is not as “random” as the common FPM: The porosities of the crystal layers are different from the cooling surface to the growing surface of the crystal layers, to simulate the porosity variation of the crystal layer, and then the thermal conductivity of the porous crystal layer is analyzed. The following assumptions had been proposed by Jiang and Wang:64 (1) Heat transfer from the crystal/liquid (C/L) interface to the cooling surface is an ideal one-dimensional process under the experimental conditions (the experimental setup can be found in ref 65). (2) The porous crystal layer is at the phase equilibrium state, which is a principal assumption to calculate the porosity. Thus, the liquid concentration in any cross section corresponds to the saturation concentration at the respective equilibrium temperature. (3) The migration of the entrapped liquid occurs simultaneously with the layer growth. The concentration of impurity on the interface is assumed to be constant, since the mass of feed material is much greater than the mass of the crystal layer growth. A fractal slice model then was established (see the schematic diagram in Figure 9). The porosity, porous channel distribution, and effective thermal conductivity clearly changed continuously along these slices of the crystal layer. The porosity of the crystal slice increased from the cooling surface to the C/ L interface. Unlike the conventional model, the maximum

conditions: when the crystal layer acquired sufficient growth driving force (enough supercooling degree, low enough terminal temperature, etc.), the layer structure was complex enough and acquired the self-similar property. Moreover, to evaluate the permeability of the crystal layer in SMC and FFMC, the calculated data were compared with the reported glass fiber (which was used as a filter).61 An obvious surplus (2−3 orders of magnitudes, which is 100−1000 times bigger) was shown in Figure 8. The excellent permeability of

Figure 8. Permeability comparison of crystal layer in SMC,60 FFMC,62 and other porous media,61 and the model described in ref 63.

the porous crystal layer explained the well separation performance of melt crystallization: the porous crystal layer should be beneficial to discharge the uncrystallized liquid phase even the porosity is limited. It was also shown that the permeability of the crystal layer in SMC was better than that in FFMC, even when the porosities of the crystal layers in SMC and FFMC are similar (also shown in Figure 8). This is because the pore volume distribution in the crystal layer of SMC and FFMC are different (shown in Figures 5 and 6, a further explanation is proposed in section 4.1). In addition, Shou63 established a difference-fractal model to predict the permeability of fibrous porous media as a function of area fractal dimension and porosity. The model could

Figure 9. Schematic diagram of fractal slice model in porous crystal layer. (Reproduced from Jiang et al.64 Copyright 2013, Wiley, New York.) 15692



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radius66 of the porous channel kmax can be easily calculated using a known porosity and cross-sectional area A when the experimental setup was determined. The heat flow rate is equal because of the ideal onedimensional heat conduction; that is,

the thermal resistance effectively, which had been applied in some industrial crystallization plants.

4. PROCESS SIMULATION OF MELT CRYSTALLIZATION The seepage process (discharge of uncrystallized liquid phase) and the sweating process (discharge of entrapped liquid phase and melt) of the porous crystal layer in melt crystallization are different from conventional seepage or fluid discharge through FPM: (1) The driving force of seepage and sweating process in melt crystallization is gravity and the interface tension (generated by the temperature gradient); (2) The driving force of each process varies with the discharge of fluid (or melt) and operational conditions; (3) The structure and relevant parameters (especially the permeability, connectivity) of the porous crystal layer will change with the melting of the crystal during the dynamic sweating process. 4.1. Primary Kinetic Research on the Sweating Process of SMC and FFMC with FPM Parameters. Different from the seepage process, the apparent driving force of the sweating process in melt crystallization is the temperature gradient on the crystal layer. The resistance factor of the sweating process is difficult to generalize, because of the complex structure in the crystal layer. Thus, a primary kinetic study on the sweating process had be carried out by Jiang et al.68 Two main possible resistance factors were considered: the average tortuosity of porous channels in the crystal layer (τav) and the porous probability on the crystal layer surface (p = ϕDf/3) The sweating discharge volume rate then could be expressed as

Q = λref ΔTref /Lref = λCL,1ΔT1/L0 ⋮ = λCL, iΔTi /L0 = λCL, i + 1ΔTi + 1/L0 ⋮ = λCL, nΔTn/L0 n

= λ′CL

∑ ΔTi /nL0 i=1

(22)

where ΔTi is the temperature gradient of each slice; n is the ′ is the effective slice number of the entire crystal layer; λCL thermal conductivity of the entire crystal layer. In addition, λref, Lref, and ΔTref are the thermal conductivity, thickness, and temperature difference of the reference sample, respectively. Then, the effective thermal conductivity of the entire porous crystal layer should be obtained as follows: ′ = λCL

⎞ nL0 ⎛ 1 1 ⎜⎜ n ⎟⎟ + n ∑i = 1 R C, i ⎠ A ⎝ ∑i = 1 RL, i

(23)

where RL,i and RC,i can be obtained with calculated λCL,i and ϕ. The thermal conductivity then can be calculated with proper boundary conditions and a step iterative method. The simulative results of fractal slice model agree with the measured data better than the nonfractal model, which indicates that the porous crystal layer has an extraordinarily complex structure and possesses the fractal properties.64 Thus, connection between crystal and liquid phase enhances the heat conduction that cannot be simulated by conventional Euclidean geometry model. This conclusion agrees with the result reported by Zhu et al.54 Moreover, the deviation of the fractal model decreases when the setting slice number becomes greater, which indicates that a fractal slice model with thinner slices (closer to the actual situation) can simulate the thermal property of porous crystal layer with better accuracy (the average deviation of the fractal slice model with 20 slices is 3.6%, and the average deviation of the fractal slice model with 2 slices is 4.6%, while that of a nonfractal model is 11.8%).64 This result is very important to the thermal balance during the layer growth model development; meanwhile, the thermal effect of fouling layer on the cooling surface can be evaluated by this fractal slice model, rather than the conventional empirical one.67 In addition, fractal slice analysis also offers the temperature distribution along the crystal layer under a certain cooling curve, which is very helpful for the enhancement of the thermal conduction. It is found that a purer and less porous crystal layer is more acceptable for smaller thermal resistance, which met the result reported in ref 59. The crystal layer closing the C/L interface comprises the major proportion of the overall thermal resistance. An internal scraper in the crystallizer can mitigate

Gsw = K swτ a avP b(S h − 1)gsw

(24)

where Sh is the degree of superheating during the sweating process, and a, b, and gsw were the orders of each parameter, respectively. Equation 24 was fitted with the experimental data, such that for SMC, Gsw,SMC = 0.703τav−3.917P−1.230(S h − 1)0.677

(25)

with a correlation coefficient of 0.999, and for FFMC, Gsw,FFMC = 0.717τav−6.158P−1.319(S h − 1)0.686

(26)

with a correlation coefficient of 1.000. Both the average tortuosity (τav) and the porous probability (P) clearly have an impact on the sweating discharge process. The average tortuosity τav (aSMC = −3.917 and aFFMC = −6.158) holds a greater resistant influence than the porous probability P (bSMC = −1.230 and bFFMC = −1.319). Regarding FFMC, the resistance effect of tortuous channels is considered as the major impact factor of the sweating liquid discharge. This conclusion is very meaningful for the development and simplification of the process model. Different from the reported mathematical model for the layer crystallization process,10,11 the introduction of structural parameters with fractal theory offers a convenient approach to illuminate how structure parameters (tortuosity, porosity, etc.) influence the sweating process. Thus, a separation property analysis of the crystal layers grown in diverse configurations and operation modes can be implemented by comparing the sweating kinetic equations: for instance, eqs 25 and 26 demonstrate that the sweating liquid 15693



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where R is the radius of the porous channel and ΔT(t) is the temperature difference of the liquid column. When the crystal layer is thin, the temperature gradient in the porous channel is uniform; thus,

discharges faster in the crystal layer of SMC than in that of FFMC with similar porosity and tortuosity. A possible interpretation had been proposed: it is apparently shown in Figure 10 that the left model (Figure 10a, less break angles on

⎛ L (t ) ⎞ ΔT (t ) = vSwt ⎜ ⎟ ⎝ L0 ⎠

(31)

where vSw is the heating rate during sweating, L(t) is the length of the liquid column in the porous channel at time t, and L0 is the original length of the liquid column. As the sweating liquid discharges in the radial direction, L0 is a function of the layer thickness s: L0 = s Df /2

Thus, eq 27 can be translated to62 Figure 10. Different porous channel microstructures with similar porosity and tortuosity. (Reproduced from Jiang et al.68 Copyright 2012, American Chemical Society, Washington, DC.)

q(t , κ ) =

π ⎛ ΔP(t ) ⎞ κ 4 ⎜ ⎟ 128 ⎝ Lt (κ ) ⎠ μ

Q (t , κ ) =

π ⎛ ρgL(t ) ⎞ κ 4 ⎜ ⎟ 128 ⎝ Lt (κ ) ⎠ μ

(27)

(28)

(29)

where b is the constant and T is the temperature of the liquid phase. Thus, ΔP(t) can be expressed as ΔP(t ) =

4bΔT (t ) 2Δσ = κ R

κ max min

q(t , κ )f (κ ) dκ = −

∫κ

κ max

q(t , κ ) dN (κ )

min

It is obvious that the flow rate varies with the duration of the process. Moreover, the flow rate will be distinctly influenced by the different crystal layer structure with various pores size distributions, which were formed under different thermal fields and configurations. 4.2.2. Two Basic Hypothetical Models. It is clearly that L(t) is the key variable for both the seepage and sweating processes. The variation of L(t) is determined by the connection of porous channels, which raise the simulation of the porous crystal layer structure. Hence, two basic hypothetical models were proposed: (A) Completely Separated Model: In this model, all porous channels are completely separated from the others and the fluid cannot diffuse to adjacent channels (Figure 11). As shown in Figure 11 (left), the L(t) values in the porous channels with different diameters are different. The flow rate of a single porous channel qA(t,κ) at moment t is only related to the diameters κ and t. (B) Completely Connected Model: In this model, all the porous channels are completely connected to each other and there is no diffusion resistance between adjacent channels (Figure 11). As shown in Figure 11 (right), the L(t) values in the porous channels with different diameters are same. The flow rate of a single porous channel qB(t,κ) at moment t is related to the diameter, κ, t, and the pore size distribution f (which is given as f (κ) = Df κDmin κ(Df+1)). According to eqs 27, 32, and 33, the deduction results on two basic hypothetical models and to the seepage process and sweating process are listed in Table 3. The equations in Table 3 can be solved using the numerical integration method. Two basic hypothetical models offer a perspicuous method to simulate the kinetic flow process of FPM (or porous crystal layer in this section) with fractal structures, which are rarely seen under realistic conditions. In fact, the main purpose of proposing the two basic hypothetical models is to confirm the possibility boundary (from total separated channels to total connected ones) of the porous channel connected in the

Regarding the sweating process, the driving force is the difference of surface tension (Δσ). The surface tension can be fitted approximatively by the temperature with the following linear equation: σ = c − bT

∫κ

(33)

Thus, the principal work is simulating the driving force variation ΔP(t) in eq 27 during the process. Regarding the seepage process, the driving force is gravity; eq 27 then can be translated to28 q(t , κ ) =

(32)

The average flow rate Q(t,κ) in the seepage and sweating processes then can be expressed as

the tortuous melt transport path) is easier for melt discharge than the right model (Figure 10b). It is believed that the inner structure of the porous crystal layer in SMC mode is similar to the model on the left (Figure 10a), which had been proved by photographing the crystal layer image of SMC and FFMC modes (shown in Figures 5 and 6). This result demonstrates that the crystal layers grown under different conditions possess various pore distributions and channel connections, even though the porosity and tortuosity parameters are similar. This result also certifies that the structural analysis and process models for porous crystal layer are more complex than that of conventional FPM, which is due to the reason mentioned above: the different thermal fields and mass-transfer processes created by the diverse operation modes and configurations. 4.2. Modification of the Process Model of Melt Crystallization and the Simulation Results. 4.2.1. Discussion on the Process Driving Force and the Flow Rate. The seepage and sweating process both involves fluid discharge via the porous crystal layer (or FPM). The flow rate (Newtonian fluid) of a single porous channel is a function of diameter of a porous channel κmax and time t, as follows: q(t , κ ) =

π ⎛ bL(t ) ⎞⎛ κ 3 ⎞ ⎜ ⎟⎜ ⎟vSwt 32 ⎝ s Df ⎠⎝ μ ⎠

(30) 15694



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Q R (t , k) = (1 − φ)Q A(t , k) + φQ B(t , k)

(34)

It is easy to understand that φ is related to the porosity ϕ and the structural property of FPM. Regarding the porous crystal layer, the porosity ϕ reflects the terminal operation conditions (terminal cooling temperature, terminal crystal−liquid ratio, etc.) of crystal layer growth; while the structure property is also related to the initial condition during crystallization process (initial mass fraction wini, initial supersaturation, etc). Jiang and co-workers28 had proposed a possible formula to express the characteristic factor φ which considered the effect of wini: ⎛ ϕ ⎞ DT + Df /3 φ=⎜ ⎟ ⎝ IF(wini) ⎠

(35)

and ⎛ wini /ρ ⎞ ρ ini ⎟⎟ sat IF(wini) = ⎜⎜ ⎝ wsat /ρsat ⎠ ρC

(36)

where wsat is the mass fraction of solute in pure crystal (wsat = 91.6%, as H3PO4 in H3PO4·0.5H2O crystal, for instance), ρini is the density of the initial solution, ρsat is the density with wsat, and ρC is the density of pure crystal. When wini → 0%, there is a tendency toward no crystal growth in the system and the porous channel tend to be totally connected; in contrast, when wini → wsat, all the porous channels tend to be totally separated. The characteristic factor φ had linked the two basic hypothetical models to express the realistic process. Moreover, φ offers a possible approach to describe the structural property of porous crystal layer by involving the initial and terminal operation conditions (such as wini, ϕ), which is very important to industrial application. For instance, a temperature oscillating operation had been reported as effective fine crystal destruction technology in batch crystallization, which is expected to improve the separation effect.69,70 Jones71 had investigated the effect of this method on crystal size distribution on continuous crystallizer and batch crystallizer. Zipp72 proved that fine destruction technology did reduce the quantity of fine crystal. This temperature oscillating

Figure 11. Schematic diagram of two basic hypothetical models (A and B) and the realistic model (R).

porous crystal layer. The following part work involves modifying the two basic hypothetical models to the process simulation in the realistic porous crystal layer. 4.2.3. Introduction and Interpretation of Characteristic Factor φ. Under the realistic situation, a single porous channel is partly separated and partly connected to other channels, rather than being completely separated or connected. Thus, the flow rate in the realistic model QR(t,κ) should between QA(t,κ) and QB(t,κ) (shown in Figure 11); the more the pore channels are separated, the more similar QR(t,κ) will be to QA(t,κ); in contrast, the more the pore channels are connected, the more similar QR(t,κ) will be to QB(t,κ). As shown in Table 3, at the initial moment (t = 0), QA(t,κ) = QB(t,κ). Therefore, a characteristic factor φ is introduced to represent the extent of the actual porous channel interconnectivity, which is

Table 3. Deduction Results of Seepage and Sweating Process on Two Basic Hypothetical Models deduction results

ref Seepage Processa

π

qA (t , κ ) = 4 κ 2L0Kκ 1 + DT exp(− Kκ 1 + DTt ) Q A(t , κ ) =

qB(t , κ ) = Q B(t , κ ) =

Df πDf Kκmax L0

4

28

κ

∫κ max κ 2 + DT − Df exp(−Kκ1 + DTt )dκ min

πKκ 3 + DT (L0Aϕ 4Aϕ

t

− ∫ Q B(t , κ ) dt ) 0

Df κ πDf Kκmax L0 ( max κ 2 + DT − Df 4 κmin

∫

dκ − ∫ κ

κmax

t

κ 2 + DT − Df ∫ Q B(t , κ ) dt 0

min

dκ ) Sweating Processb M

(

62

)

qA (t , κ ) = Mκt exp − 2 κt 2 Df Q A(t , κ ) = − MDf κmax t∫

κmax −D κ f

κmin

qB(t , κ ) = Q B(t , κ ) =

(

(

M(2 − Df ) 3 π 2 D κ t 4 κmax 2 − fD s Df /2 2 Df κmax f Df π MDf κmax

4

t∫

κmax

κmin

κ

2 − Df

M

)

exp − 2 κt 2 dκ

dκ −

t

− ∫ Q B(t , κ ) dt 0

M(2 − Df ) 2 − Df Df /2 κmax s

∫κ

κmax

min

κ

)

2 − Df

t

∫0 Q B(t , κ ) dt

dκ a

D

K = (ρg)/(32μL0 T). bM = (bvSw)/(8μsDf). 15695



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Figure 12. Comparison between the experimental results and simulative results (realistic model) of seepage process in SMC. (Data taken from ref 28.)

indicating that the connectivity increases in the porous channels. 4.2.4. Simulation Result of Seepage Process in SMC. The discharge of uncrystallized fluid is a critical process to SMC. Thus, the seepage process has a decisive impact on the sweating process and will determine the terminal separation effect. The simulative (realistic model) and experimental results of the seepage process are listed in Figure 12. The simulative results met the flow rate result along the single seepage process with good agreement. The forecast on the flow rate in different porous crystal layers with various porosities also held a stable agreement. This result verifies the effectiveness and stability of the developing fractal porous media model. In addition, the initial flow rate of the seepage process declines sharply with the decreasing of the porosity, indicating that the migration of uncrystallized fluid gets more difficult when the crystal branches grow more complicated and the fractal property of the crystal layer gets more significant. To reveal the characteristic factor φ further, the flow rate of model A (completely separated model), model B (completely connected model), and the model with φ in three sample tests are shown in Figure 13 (the respective characteristic factor φ in each model is also listed). The modification function to the fractal model with φ is clearly shown in Figure 13: the initial flow rate in each sample increases sharply with the increasing φ, and this indicated a greater permeability of the crystal layer, which is crucial to the seepage and sweating processes in melt crystallization. The enhancement of permeability is due to the connectivity improvement of porous channels in the crystal layer. It is obvious that the decisive reason for connectivity improvement is the various operation conditions (cooling rate, initial mass fraction, etc.) during the crystal layer growth and then it is easy to understand that the realistic model (with φ) involved in these operation conditions will obtain a satisfactory simulation result. In addition, it should be pointed out that φ in the seepage process of SMC is a constant, since the overall liquid seepage occurs under isothermal conditions without a phase transition. This assumption makes the model of the seepage process relatively simple.

operation had been applied to refine the connection of porous channels in the crystal layer, which will be helpful for the sweating process late on. In one stage of “temperature oscillates”, the crystal layer growth was suspended when a certain amount of fine crystals appeared. The temperature was raised slightly to melt the fine crystal and then the temperature was adjusted to maintain the crystal layer growth. So there are two crystal growth steps: the number of porous channels with smaller diameter and high tortuosity (which mostly structured by the fine crystals) will be partly diminished. The connection and structure of the crystal layer are believed to be refined by this technology. The temperature oscillating operation can be simulated easily by using φ: regarding the one-stage temperature oscillation, there are two impact factor of wini in optimized processing: ⎛ wini,1/ρini,1 ⎞ ρ ⎟⎟ sat IF(wini,1) = ⎜⎜ ⎝ wsat /ρsat ⎠ ρC

(37a)

⎛ wini,2 /ρini,2 ⎞ ρ ⎟⎟ sat IF(wini,2) = ⎜⎜ ⎝ wsat /ρsat ⎠ ρC

(37b)

where wini,1 and wini,2 are the initial mass fraction of each crystal growth step; ρini,1 and ρini,2 are density of initial acid at each crystal growth step, respectively. Thus, the expression of φ in the one-stage temperature oscillating process is ⎛ ⎞(DT + Df )/3 ϕ ⎟⎟ φosc = ⎜⎜ ⎝ IF(wini,1)IF(wini,2) ⎠

(38)

By this analogy, the characteristic factor φ in two-stage oscillating should be ⎛ ⎞(DT + Df )/3 ϕ ⎟⎟ φdouble‐osc = ⎜⎜ ⎝ IF(wini,1)IF(wini,2)IF(wini,3) ⎠

(39)

Considering the relation between w and ρ, IF(wini) is always smaller than 1, so φosc is bigger than φ with the same ϕ and wini, 15696



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Figure 13. Comparison between experimental and model flow rates. Legend: (○) experimental data; the red dotted line represents the simulation result of model A; the blue dotted line represents the simulation result of model B; and the black dotted line represents the simulation result of the realistic model with φ. (Data taken from ref 28.)

Figure 14. Comparison between the real flow rate and the model flow rate. Legend: (○) experimental data; red dotted line represents the simulation result of model A; blue dotted line represents the simulation result of model B; and the black dotted line represents the simulation result of the realistic model with φ. (Data taken from ref 62.)

4.2.5. Simulation Result of the Sweating Process in FFMC. As listed in Table 3, the model of the sweating process has more parameters than that of the seepage process. The most meaningful parameter is the heating rate during sweating (vSw). It means that the model simulates the dynamic sweating process as a nonisothermal process and the phase transition (crystal-sweating phase) should be regarded. Simulative and experimental results of the seepage process are listed in Figure 14. Other than the seepage process, the crystal layer is heated to create the driving force of sweating phase discharge and the crystal will partly melt with the increasing temperature. This difference leads to an inconvenient truth: the flow rate of sweating phase gets greater when the melted crystal afflux into the sweating flow and the structure of the porous crystal layer is getting more porous and connected due to the loss of the crystal framework. Nevertheless, practical results require more intensive model by considering all the key factors. The developing model (realistic model with φ) had simulated the sweating phase flow rate with a good agreement with the experimental ones. The complicated fluid discharge and phase transition process leads to an interesting simulation result: the realistic flow rate model QR(t,κ) (represented by the black dotted line in Figure 14) attains a high value and then decreases to a valley floor value before increasing again at a much faster rate. This phenomenon is noteworthy when the variation range of φ is great (see Figure 14B, for instance). In addition, the control model of sweating discharge switches from model A to model B through the sweating process: as shown in Figures 14A−C, the realistic model and experimental curves follow the tendency of model A (red dotted line) in the early period of the sweating process, then change to follow the tendency of model B (blue

dotted line) in the later period of the sweating process. All the results demonstrate that the inner structural variation of the porous crystal layer framework has a profound influence on the nonisothermal, dynamic sweating process and the migration of an impure fluid phase. 4.3. Application to the Evaluation of the Separation Process and Optimized Operation. It should be noted that the principal aim to simulate the seepage and sweating processies (which are all separation processes) is the evaluation of the separation process and optimized operation. As reported in refs 69, 70, and 73, the temperature oscillation has improved on the separation effect. By applying the temperature oscillation technology to the static melt crystallization, the fine crystals in the porous crystal layer were mingled, then the seepage process was enhanced due to the decrease of the porous channel resistance. The introduction of characteristic factor φ proved that the porous channel connection had been improved with the temperature oscillation (shown in Figure 15, when the temperature oscillation was implemented, φ became greater than the one without temperature oscillation, despite the similar porosities of each crystal layer). As a further clarification to evaluate the separation effect improvement of the porous crystal layer, three comparison experiments were implemented: the seepage process of three crystal layers with similar porosities but different growth operations (outlined by a red dotted rectangle in Figure 15) were investigated. The results are shown in Figure 16. To evaluate the terminal time of the seepage process, an empirical flow rate criterion was proposed: When the flow rate is smaller than that of 2 g/min (the red dotted line in Figure 16 (left)), the uncrystallized fluid is believed to discharge adequately and the separation efficiency 15697



Figure 15. Comparison on φ between temperature operation and the original one. Legend: black dot, original red dot, one-stage temperature oscillation; blue dot, temperature oscillation. (Reproduced from Jiang et al.28 2011, American Chemical Society, Washington, DC.)

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(1.58%) decreases almost 30%, compared to the original operation (2.23%); regarding two-stage temperature oscillation, the amount of impure liquid phase (1.29%) had decreased more than 18%, compared to the one-stage operation. Less entrapped impure liquid phase is beneficial to enhance the stability of the porous crystal layer, which is important for the following sweating process. That is because the meltingpoint of the impure phase in melt crystallization is usually lower than that of the pure crystal; impurities contained in the closed pores of crystal layer is inclined to lead the collapse of the porous crystal layer, especially when the sweating temperature is close to the melt point. As a further consideration, the effective distribution coefficient Keff, sw in the sweating process was simulated based on the developing process model. The results are shown in Figure 17 and agree with the proposed conclusion of Kim and

oscillation operation; two-stage Copyright

of consecutive seepage process is very low. In other words, the rest of the liquid phase is entrapped in the closed pores or highresistance channels and should be separated during the sweating process. Also shown in Figure 16 (left), the terminal time of the seepage process in one-stage temperature oscillation (23.3 min) decreases more than 44% compared to the original operation (42.2 min); regarding two-stage temperature oscillation, the terminal time (19.5 min) had decreased more than 16%, compared to the one-stage operation. In addition to enhance the separation efficiency by shortening the seepage time, the temperature oscillationg operation also decreases the ratio of liquid phase entrapped in the closed pores or high resistance channels at the end of the seepage process (as shown in Figure 16, right). As reported by Parisi74 and Myasnikov,75 the liquid phase entrapped in the closed or high-discharge-resistance pores was concentrated with an impure component. If these impure liquids melted with the crystal layer during the sweating process, the product purity became lower. Thus, the key advantage of temperature oscillation operation is enhancing the separation effect: the amount of liquid phase with high concentration impurity

Figure 17. Simulation results of Keff in the sweating process. (Data taken from ref 62.)

Ulrich: “the more the liquid phase inside the layer exists, the faster removal of impurity in sweating operations occurs”.76,77 However, with increasing temperature, an increasing amount of the crystal phase melted and the separation effect became worse. This phenomenon was more noteworthy in the crystal layer with high porosity. That was the reason for the separation effect recede (Keff, sw decreased more slowly), compared to the

Figure 16. (Left) Comparison of seepage flow rate versus time under different temperature oscillation operation stage and the original one. (Right) Comparison of the discharged fluid ratio versus time under different temperature oscillating operation stage and the original one. (solid black line, original operation; solid red line, one-stage temperature oscillating; solid blue line, two-stage temperature oscillation; red dotted line, the flow rate of 2 g/min; blue dotted line, the ratio of 100%.) (Data taken from ref 28.) 15698



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similar to the thermal field in ideal crystal tower utilized in melt crystallization.

initial period. Keff, sw exhibited a minimum (Test A, blue line in Figure 17). As a comparison, Test B (black line in Figure 17) with a relatively small initial porosity (0.15) and a fast heating rate (vsw = 0.087 K/min) also did not provide a satisfactory separation effect. This result indicates that the initial porosity (determined the initial permeability of the crystal layer) and the heating rate (determined the stability of the crystal layer framework) are two key parameters of the sweating process. Thus, Test C (red line in Figure 17) could be an optimized sweating operation (initial porosity = 0.28, vsw = 0.055 K/min). Thus, the conventional process model on the crystal layer growth, impurity distribution, inclusion migration, and sweating process simulation (for more information, the reader is referred to the excellent fundamental work by Chianese and Santilli10 Xu and McHugh,78 Kumashiro et al.,79 and Beierling and Ruether80) should be applied with FPM model referenced above to optimize the entire melt crystallization process. The reasonable approach is linking the structure and fractal property of the crystal layer with the operation condition (which involves the definition and interpretation of characteristic factor φ in porous crystal layer), then simulating and optimizing the relative separation process with the known structure of the crystal layer and the basis of the hydromechanical and crystallization theories.

■

AUTHOR INFORMATION

Corresponding Authors

*Tel.: +86 411-84707892. Fax: +86 411-84986291. E-mail: [email protected]. *Tel.: +86 411-84707892. Fax: +86 411-84986291. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Fund for Distinguished Young Scholars of China (No. 21125628), National Natural Science Foundation of China (Nos. 21306017, 21206014, 21006008), China Postdoctoral Science Foundation funded project (No. 2013M530126), and the Fundamental Research Funds for the Central Universities of China (No. DUT12RC(3)43).

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REFERENCES

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5. PERSPECTIVES AND CONCLUSIONS Introducing and applying FPM theory and technology to melt crystallization can promisingly lead to significant innovations in property analysis of porous crystal layer and process simulation of the relative separation processes (seepage, sweating process, etc). Thus, the description of the thermal property, permeability to the crystal layer, the research on the structure of the crystal layer with various operation conditions, and the evaluation on the optimized separation operation can be implemented with innovative FPM theory and technology. Authors of this review are strongly convinced that some of the most interesting developments are actually related to the possibility of integrating the novel FPM theory analysis and process model together with the structure and relevant properties of porous crystal layer and the separation evaluation of melt crystallization. Despite the great potential of FPM theory and technology applying in industrial crystallization, the existing model and theory are still not enough to fulfill all the expectations and requirement. The application of FPM theory in melt crystallization has started recently. More efforts dedicated to developing the model which is stable and reliable to different operation profiles are needed. In addition, the new application of FPM theory and technology should be explored by using the physical and material theory in-depth to conquer the existing barriers. In this review, the definition and interpretation of the porous channels connectivity in porous crystal layer are the crucial aspect to be addressed in further research. Any definition formula should involve the operational parameters and possess reliable and stable performance when applied to the actual simulation, especially the nonisothermal and phase transition process (the sweating process, for example) in melt crystallization. Moreover, thermal property analysis should be developed to describe the crystal layer grown not only in the one-dimensional thermal field but also in the two-dimensional and three-dimensional thermal field; the research can be implemented under the cylindrical coordinate, which is quite 15699



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