Falling film melt crystallization (II): Model to simulate the dynamic sweating using fractal porous media theory

Chemical Engineering Science 91 (2013) 111–121

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Falling film melt crystallization (II): Model to simulate the dynamic sweating using fractal porous media theory Xiaobin Jiang a,b, Baohong Hou a, Gaohong He b, Jingkang Wang a,n a

School of Chemical Engineering and Technology, State Research Center of Industrialization for Crystallization Technology, Tianjin University, Tianjin 300072, China b State Key Laboratory of Fine Chemicals, R&D Center of Membrane Science and Technology, School of Chemical Engineering, Dalian University of Technology, Dalian 116024, China

H I G H L I G H T S c c c c

Dynamic sweating was simulated with the fractal porous media theory. A characteristic factor j was introduced to modify the model with a good agreement. The structure and permeability variation of crystal layer during sweating was explained with the model. A model system of FFMC is expected to establish and wildly used in industrial crystallization.

a r t i c l e i n f o

abstract

Article history: Received 20 August 2012 Received in revised form 21 December 2012 Accepted 21 December 2012 Available online 17 January 2013

This paper describes the model development of dynamic sweating in falling film melt crystallization (FFMC) using fractal porous media theory. The crystal layer has the characteristics of a fractal porous medium. An overall mass balance is applied to determine the structural and characteristic parameters of the crystal layer. Two ideal hypothetical models are adopted to describe the dynamic change of flow rate in sweating under sweating conditions. The characteristic factor j is introduced to modify the model to describe the real process. The model is validated by experiment, and the simulated result agrees well with the experimental result. The model is then exploited to understand sweating behavior and the structure and permeability variations of the crystal layer in FFMC. More significantly, the sweating model is also an important part of the model system of the overall FFMC process. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Melt crystallization Fractals Porous media Separations Simulation Sweating

1. Introduction Sweating is defined as a temperature-induced purification step based on a partial melting of crystals or crystal layers by heating the cooled surface closely up to the melting point of the pure component (Ulrich and Bierwirth, 1995). As a consequence, the impurities adhering to the crystal surface and those contained in the pores of the crystal layer melt and are then discharged under the influence of gravity (Myerson, 2002). Sweating has received increasing amounts of attention as an effective technology for hyperpure material manufacturing (Jung et al., 2008), wastewater recovery (Veesler et al., 2010) and seawater desalination (Rich et al., 2012).

n

Corresponding author. Tel.: þ86 22 27405754; fax: þ86 22 27374971. E-mail address: [email protected] (J. Wang).

0009-2509/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2012.12.048

Wangnick and Ulrich (1994) focused on the sweating behavior and the impact factors of a solid-layer-type crystallization process. They described the purification efficiencies of each step using a dimensional analysis of the influencing parameters, which gave rise to a discussion on the separation effect of melt crystallization. The purity of the product was reported to be determined by the growth rate and the sweating conditions (Poschmann and Ulrich, 1996). To obtain a given product, research on the stage number was carried out, which facilitates research into the optimized melt crystallization strategy for industrial applica¨ tions (Ulrich and Neumann, 1997). Konig and Schreiner (2001) explored the purification potential of melt crystallization with sweating processes, which led to a high-purity product. This development inspired researchers to investigate sweating as an effective purification process when fine control was unapproachable in the crystal layer growth process. The purifying effect of

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sweating was influenced by the crystal layer growth conditions and the inclusion in the crystal layer (Kim and Ulrich, 2002a, 2002b). At the beginning of sweating, the impure liquid phase comprises the majority of the sweating liquid. The impure liquid phase will be transported out of the crystal layer under the temperature gradient. When the temperature is close to the melt point of the pure component, an increasing number of pure crystals melt, which leads to a reduction of the product. Therefore, it is necessary to develop a dynamic model for the sweating of melt crystallization that describes the sweating behavior and predicts the separation effect. There are various driving forces (temperature gradient, tension differences) of sweating liquid discharge, complex crystal–liquid phase surfaces and intricate pore distributions of the crystal layer. There is also a phase transition (crystal–melt–liquid) during the dynamic sweating. Thus, the application of multi-field theory is required to build the dynamic model. Yu (2008), Shou et al. (2010) and Cai and Yu (2011) reported that the seepage of fluid in porous media was strongly influenced by the structural and characteristic parameters of the porous medium. A model to simulate the seepage process in porous media with constant driving force and no phase transition was also developed by Xu et al. (2008) and Yun et al. (2009), which offered a fundamental basis for further theoretical and experimental research. In the work of Jiang et al. (2011), an existing fractal porous media model was applied to simulate the structure of the crystal pillar formed in static melt crystallization for hyperpure phosphoric acid separation. The model simulated the liquid discharge process with satisfactory results. Fractal porous media theory was deemed appropriate to describe the structure of the crystal layer in melt crystallization. Some early experiments were carried out on the dynamic sweating of falling film melt crystallization (FFMC) by Jiang et al. (2012a). The results showed that the average melt discharge rate increased with the overall average superheating degree. The structural parameters of the crystal layer also affected the sweating. As reported in the authors’ former work (Jiang et al., 2012b), the thickness of the crystal layer along the crystallizer can be obtained by the existing model with known operational conditions. The impure liquid entrapment and the formation of a branched-porous (B–P) structure occur inevitably in FFMC. Therefore, it is crucial to investigate the structural and characteristic parameters of the crystal layer. Fractal and porous media theory will be applied to describe the dynamic sweating in FFMC in this paper, which is a more challenging and complicated endeavor than that for static melt crystallization. This paper will focus on how the crystal layer structure changes during the dynamic sweating. Some critical parameters based on the porous media theory will be proposed to investigate the impact of sweating on the structure of the crystal layer. In addition to comparing the simulative and experimental results of sweating, this paper also evaluates the separation efficiency of sweating. All of these results will contribute to proving that the dynamic sweating model has potential applications in dynamic simulations and separation effect evaluations.

fractal porous media theory. The mass balance for each step should be carried out first. Mass balance for crystal layer growth: mF ¼ mCL þmML mF C F ¼ mCL C CL þmML C ML

ð1Þ

where mF, mCL and mML are the mass of the feed, crystal layer and mother liquid discharged from the crystallizer, respectively. CF, CCL and CML are the mass fraction of H3PO4 in each flow. Mass balance for impure liquid phase entrapment: mCL ¼ mC þ mLE mCL C CL ¼ mC C C þ mLE C LE

ð2Þ

where mC and mLE are the mass of the crystal phase and the mass of impure liquid entrapped in the crystal layer, respectively. CC and CLE are the mass fraction of H3PO4 in each phase. CLE is assumed to be the equilibrium concentration at the corresponding system temperature. Mass balance for sweating liquid discharge: During the sweating, the entrapped impure liquid phase will be transported out of the crystal layer from the open pore channels. Considering that some crystals will melt and discharge simultaneously, the sweating liquid is a mixture of the entrapped impure liquid and the melted crystal phase, which is mSw ¼ m0C þ m0LE mSw C Sw ¼ m0C C C þ m0LE C LE

ð3Þ

where mSw, m0C and m0LE are the mass of the sweating liquid, the melted crystal phase and the impure liquid phase discharge out of the crystal layer, correspondingly. CSw, CC and CLE are the mass fraction of H3PO4 in each phase. These three mass balances can be solved using experimental data. According to the definition of porosity, the initial porosity of the crystal layer at the beginning of sweating (also the terminal porosity of the crystal layer at the end of the layer growth process) is

fini ¼

VP mLE =rLE ¼ V CL mC =rC þmLE =rLE

ð4Þ

where rLE and rC are the densities of each phase, which can be obtained using known state temperature and mass fraction. As mentioned in authors’ former work (Jiang et al., 2012b), the composition of the crystal layer was determined by the layer growth conditions. Therefore, the structural parameter fini is not only the boundary condition of sweating but also the parameter that reflects the layer growth conditions (such as feed rate, feed mass fraction, cooling rate, etc.). fini is introduced into the sweating model instead of the multi-operation conditions in crystal layer growth. This substitution will vastly simplify the sweating model, allowing the study to focus on how the sweating operation conditions influence the process. The porosity of the crystal layer at the end of the sweating is

fFinal ¼

mLE =rLE þ m0C =rC V P þ DV P ¼ : V CL mC =rC þ mLE =rLE

ð5Þ

To simplify the developing model, the variation rate of the porosity a is assumed to be a constant, 2. Model development 2.1. Structural and characteristic parameters of the crystal layer as a fractal porous medium 2.1.1. Porous media structural parameters of the crystal layer As reported by Jiang et al. (2011), the structure of the crystal layer formed in static melt crystallization was analyzed using

Df ¼ fFinal fini ¼ at

ð6Þ

where t is the sweating duration. The variation rate of the porosity a is thought to be a criterion for the stability of sweating under different operation conditions. The effective porosity was defined as the ratio between the effective pore volume and the total volume of porous media. The effective pore volume is the connected volume in porous media,

X. Jiang et al. / Chemical Engineering Science 91 (2013) 111–121

which is equal to the volume of the liquid phase discharge. So,

feff

m0 =r m0 ¼ LE LE ¼ LE fini ¼ kfini V CL mLE

ð7Þ

where k is a criterion for the effect of the sweating process under different operation conditions. 2.1.2. Fractal structural parameters of the crystal layer As the fundamental fractal properties of the crystal layer (cylindrical cast), the fractal dimension of the crystal layer could be divided into two types: the dimension in the radial direction (capillary area fractal dimension) Df and the dimension in the axial direction (tortuosity fractal dimension) DT. The capillary area fractal dimension Df was determined by Yu and Li (2001): Df ¼ 2

lnf lnðlmin =lmax Þ

DT ¼ 1þ

where Q is the flow rate of the fluid through the porous medium, m is the viscosity of the fluid and @P=@x is the pressure gradient in the flow direction. If the pressure distribution in the flow direction is uniform, @P=@x ¼ DP=L. Regarding the sweating liquid discharge, the driving force varies with the crystal layer temperature as K¼

mL0 Q

where DPt is the pressure difference on the crystal layer at time t and L0 is the length of the crystal layer. The flow rate through a single tortuous capillary is a function of the pore channel diameter l and time t. Based on the research of Yu and Cheng (2002), this function could be expressed as follows:

p DPt l4 128 Lt ðlÞ m

qðt, lÞ ¼

lntav lnðL=lav Þ

ð9Þ

A simple geometrical model for the tortuosity of the flow path in porous media was proposed by Yu and Li (2004). The average tortuosity is expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi3 2   2

6 16 1 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 1þ 1f þ 1f tav ¼ 6 26 2 4

1 ffi pffiffiffiffiffiffiffi 1 1f

pffiffiffiffiffiffiffiffiffiffiffi 1 1f

þ 147 7 7 7 5

ð10Þ

ð16Þ

DPt

ð8Þ

where lmin and lmax are the minimum and maximum pore diameters in the porous medium. The tortuosity fractal dimension DT can be obtained as reported by Yu (2005):

lmin

lmin

According to Eqs. (16)–(18), the permeability K of the crystal layer can be expressed as follows: Z lmax mL0 Q qðt, lÞ ¼ mL0 dNðlÞ K¼ DP i DP i lmin ! Z lmax

l

and D

f dNðlÞ ¼ Df lmax l

ðDf þ 1Þ

dl ¼ f ðlÞdl

ð12Þ

Df lðDf þ 1Þ f ðlÞ ¼ Df lmin

is the probability density function of where the pore sizes. Thus, lav can be expressed as "  Df 1 # Z lmax Df l lav ¼ lf ðlÞdl ¼ lmin 1 min ð13Þ Df 1 lmax lmin and DT can be calculated by Eqs. (10) and (13). In the research conducted by Yu and Li (2001), these statements hold if and only if   lmin Df ¼0 ð14Þ

lmax

is satisfied. Eq. (14) implies that lmin =lmax r102 must be satisfied for a fractal analysis of a porous medium to be valid. 2.1.3. Characteristic parameter of the crystal layer The permeability K is a crucial parameter of porous media that characterizes the transit capacity of the fluid through the porous medium under a constant driving force. As defined in Darcy’s law, K can be expressed as Q ¼

K @P m @x

ð15Þ

ð17Þ

where l is the hydraulic diameter of a single capillary tube and Lt is the length of the tortuous capillary channel. As a fractal porous T medium, Lt ¼ LD 0 . The total flow rate Q (diameter range from lmin to lmax ) of all capillaries at time t in the completely separated model will be Z lmax   Z lmax Df Q ¼ qðt, lÞdN l ¼ qðt, lÞDf lmax lðDf þ 1Þ dl ð18Þ

¼ mL0

Yu and Cheng (2002) described the pores in porous media by analogy to the islands or lakes on Earth or the spots on engineering surface, indicating that the cumulative size distribution of the pores follow the following fractal scaling laws:   lmax Df ð11Þ N ðL Z l Þ ¼

113

lmin

¼

p

l4

128 mLDT

D

ðDf þ 1Þ

f l Df lmax

dl

0

T 3 þ DT pDf L1D lmax 0

128ð3 þ DT Df Þ

ð19Þ

Eq. (19) indicates that the permeability of the crystal layer is related to lmax, L0 and the structural property of the crystal layer. K is a constant when the liquid seepage occurs in porous media without a phase transition. However, the phase-transition during the dynamic sweating of FFMC has significant impact on the structure of the crystal layer and the sweating liquid discharge. Therefore, a dynamic model should be introduced to modify the phase transition during the process. Using fractal porous media theory, the structural and characteristic parameters of the crystal layer and the distribution of pore channels could be easily expressed by the dimensionless parameters. This expression offers a convenient approach to describing the sweating and the flow rate variation. 2.2. Development of the sweating model 2.2.1. Two ideal sweating models According to the work of Scholz et al. (1993), the crystal layer grown in FFMC can be divided into three regions (as shown in Fig. 1). Region 1 is close to the cooling surface. Because of the rapid growth rate of the crystal in this region, the crystal layer in this region has a relatively high porosity. As the layer grows, its structure becomes less porous with the decreasing temperature gradient, forming Region 2. Region 3 is in contact with the falling film and has a high impurity content. As to an ideal crystal layer, Regions 1 and 3 are very thin and their thicknesses can be neglected. Most of the impurities are enriched in Region 2 and will be discharged during the sweating. Also shown in Fig. 1, the pores are partially connected with each other and the sweating liquid discharge driving force (gravity, etc.) is much smaller than

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the pore channel resistance. Thus, the sweating liquid cannot discharge from the crystal layer. Nevertheless, the sweating liquid discharge driving force increases with increasing sweating temperature because the surface tension gradient increases sharply with temperature. On the other hand, the crystal network will partially melt and the discharge resistance will decrease significantly. To introduce the flow seepage model in porous media to the sweating in the crystal layer, some simplifications and assumptions are necessary:

(1) The structure of the crystal layer is assumed to be ideal. The pore channels in the crystal layer have a uniform distribution at a cross section. Close to the heating crystallizer inner wall, there is a series of parallel vertical channels in Region 1 for the sweating liquid discharge. (2) The driving force is mainly generated by the temperature difference in the liquid phase filling the pore channels from the heating surface to the outer surface of the crystal layer.

Thus, the trace of sweating liquid discharge is separated into two steps, the deep diffusion (radial direction) and the vertical seepage (vertical direction). In the deep diffusion step, the sweating liquid discharges through the pore channels to the heating crystallizer inner wall in the radial direction. In the vertical seepage step, the sweating liquid flows through the parallel vertical channels in Region 1. Based on these simplifications and assumptions, the flow rate is qðt, lÞ ¼

pDPðtÞl4 pDPðtÞl4 ffi 128mLxy 128mLDT

ð20Þ

where Lx  y is the total channel length of sweating liquid discharge, LDT is the length of channel length on the vertical direction (along the y axis), DP(t) is the pressure difference and t is the sweating duration. The main driving force of the sweating liquid discharge is the surface tension s, which is fitted by the temperature with a linear equation as follows:

s ¼ cbT

ð21Þ

where b and c are the parameters and T is the temperature of the liquid phase. Thus, the pressure difference DP(t) is expressed as follows:

DPðtÞ ¼

2Ds 4bDTðtÞ ¼ R l

ð22Þ

where R is the radius of the pore channel and DT(t) is the temperature difference of the liquid column. It is obvious that DT(t) is the main variable of the flow rate. Assuming that the temperature gradient in the liquid column is uniform when the crystal layer is thin,   lðtÞ DT ðtÞ ¼ vSw t ð23Þ l0

Fig. 1. Illustration of different layer characteristics in solid layer crystallization. ((Scholz et al., 1993), the figure had been redrawn by authors for better illustration).

where vSw is the heating rate during sweating; l(t) is the length of the liquid column at time t; and l0 is the original length of the liquid column, as the principal direction of driving force of sweating liquid is the radial direction. Thus, l0 is a function of the layer thickness s and the capillary area fractal dimension Df, which is l0 ¼ sDf =2 .

Fig. 2. Schematic diagram of the two ideal models: (A) completely separated model; (B) completely connected model (deep blue represents liquid phase). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

X. Jiang et al. / Chemical Engineering Science 91 (2013) 111–121

To describe the l(t), two ideal models are introduced to simulate the connection type of the pore channels in the crystal layer, as proposed by Jiang et al. (2011): A. Completely separated model. It is assumed that all pore channels are completely separated from the others, such that the sweating liquid cannot diffuse between adjacent channels. As shown in Fig. 2A, the top ring of each figure is the schematic of sweating liquid discharge and the bottom rectangle of each figure is the schematic of a fractal unit, which reveals how the series of pore channels are separated from each other. The four figures in Fig. 2A demonstrate how the liquid phase is distributed in the crystal layer and how the sweating liquid discharges with the sweating (from left to right). The sweating liquid discharges at different rates when the diameters of the pore channels are different. The flow rate of pore channels with smaller diameters is lower than that for the channels with larger diameters. Based on the model description, l(t) is not influenced by the adjacent channels, so DT(t) is expressed as ! Rt p 2 0 qA ðt, lÞdt= 4 l DT ðt Þ ¼ vSw t 1 ð24Þ sðDf =2Þ where qA(t,l) is the flow rate of a certain pore channel in the completely separated model. According to Eqs. (20), (22) and (24), qA(t,l) can be expressed as   Z t bvSw t p 2 ðDf =2Þ U U l s  q ðt, l Þdt ð25Þ qA ðt, lÞ ¼ A 2pmsDf l 4 0 For a certain pore channel, the diameter is a constant. Thus, the flow rate is only related to the time and defines the integral gross of flow for a certain pore channel as Z t qxA ðt, lÞdt ð26Þ GA ðtÞ ¼

series of pore channels are connected. The four figures in Fig. 2B demonstrate how the liquid phase distributes to the crystal layer and how the sweating liquid discharges with the sweating (from left to right). The level of the liquid column is uniform for the completely connected scenario. A circular ring-like distribution is obtained. Based on the model description, the l(t) is equal to one another and related to the total flow rate Q of all pore channels (diameter range from lmin to lmax ) and the sectional area of all pore channels. We should determine the surface area of all pore channels first, which is Z lmax  p 2 p Df  2Df 2D f ð30Þ l dNðlÞ ¼ Df lDmax l lmin f 4 2Df max lmin 4 when Eq. (14) is satisfied. There is Z lmax p 2 p Df l dNðlÞ ffi l2max 4 2Df lmin 4

dGA ðtÞ p 2 ¼ l sðDf =2Þ GA ðt Þ ðbvSw =2pmlsDf Þtdt 4

ð27Þ

which is an ordinary differential equation with initial conditions; thus, Gx  A(t) can be solved. The flow rate of a certain pore channel can be expressed as follows:   M qA ðt, lÞ ¼ Mltexp  lt 2 ð28Þ 2 where M ¼ bvSw =8msDf : The total flow rate Q of all pore channels (diameter ranging from lmin to lmax ) at time t in completely separated model will be Q A ðt, lÞ ¼ 

Z

lmax

lmin

D

f qA ðt, lÞdN ðlÞ ¼ MDf lmax t

Z

lmax

lmin

  M lDf exp  lt2 dl 2

ð29Þ Eq. (29) can be solved using a numerical method, and the thickness of the crystal layer s at different positions can be obtained from the model developed in the authors’ previous paper (Jiang et al., 2012b). B. Completely connected model. In this model, it is assumed that all the pore channels are completely connected to one another and that there is no diffusion resistance between adjacent channels. As shown in Fig. 2B, the top ring of each figure is the schematic of sweating liquid discharge and the bottom rectangle of each figure is a schematic of a fractal unit that reveals how the

ð31Þ

Thus, DT(t) is expressed as ! Rt 0 Q B ðt, lÞdt DT ðtÞ ¼ vSw t 1 2 ðp=4Þlmax ðDf =2Df ÞsðDf =2Þ

ð32Þ

where QB(t,l) is the total flow rate Q of all pore channels (diameter ranging from lmin to lmax ) at time t in the completely connected model. According to Eqs. (20), (22) and (32), qB(t,l) is expressed as   Z t bvSw ð2Df Þ 3 p 2 Df ðDf =2Þ qB ðt, lÞ ¼ l t l s  Q ðt, l Þdt ð33Þ B 2 4 max 2Df 0 8msDf Df l max

Therefore, QB(t,l) will be Z lmax qðt, lÞdN ðlÞ Q B ðt, lÞ ¼  lmin

0

where GA(t) is the integral gross of flow at time t for a certain pore channel. Thus, Eq. (25) is transformed into

115

¼ Ht

Z

lmax lmin

l2Df dl

H V

Z

lmax

lmin

l2Df

Z 0

t

Q B ðt, lÞdtdl

ð34Þ

2 f =32msDf and V ¼ ðp=4Þlmax ðDf =ð2Df ÞÞsðDf =2Þ where H ¼ pbvSw Df lDmax Eqs. (29) and (34) can be solved using the numerical integration method. In the actual situation, the pore channels are partly connected to each other rather than completely separated or connected. The real total flow rate QR(t,l) should between QA(t,l) and QB(t,l); the more the pore channels are separated, the more similar QR(t,l) will be to QA(t,l), and the more the pore channels are connected, the more similar QR(t,l) will be to QB(t,l). A characteristic factor j is introduced to represent the extent of the actual porous channel interconnectivity, which is

Q R ðt, lÞ ¼ ð1jÞQ A ðt, lÞ þ jQ B ðt, lÞ

ð35Þ

2.2.2. Determination of lmax, j and Keff In this paper, lmin is set as 1.0  10  7m to satisfy Eq. (14). The total mass of the sweating liquid is based on known data, Z t Z t Z lmax  ð36Þ mSw ¼ rSw Q ðt, lÞdt ¼  rSw qðt, lÞdNðl dt 0

0

lmin

where rSw is the density of the sweating liquid. If the sweating liquid migrates slowly enough, the crystal melting process can be treated as a dynamic equilibrium state. rSw is equal to the density of saturated PA at the corresponding temperature, which can be calculated using the equation reported by Jiang et al. (2011). lmax can be calculated using the iteration method when the mass of sweating liquid is known (here, the iterative calculation was terminated after the error became less than 1.0  10  7 m).

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X. Jiang et al. / Chemical Engineering Science 91 (2013) 111–121

The characteristic factor j represents the connectedness of the pore channels (Jiang et al., 2011). j is related to the porosity and the mass fraction of the liquid phase and is expressed as follows:

j¼



ðDT þ Df =3Þ

f

ð37Þ

IFðwL Þ

where IF(wL) is the impact factor of the liquid phase mass fraction on j, IF ðwL Þ ¼

C LE =rLE rsat U C sat =rsat rC

ð38Þ

where CLE is the mass fraction of the liquid phase entrapped in the crystal layer; Csat ¼91.6% is the mass fraction of saturated PA; and rsat and rC are the density of the liquid phase of saturated PA and H3PO4  0.5H2O crystal (2.108 g/cm3), respectively. The impact factor of the liquid phase reflects the influence of the phase transition in the crystal layer on the connection of the pore channels, especially the variation of the pore volume when crystal melts into a liquid. The crystal layer is much thinner in FFMC than in static melt crystallization. The impact of the crystal melting on the variation of the crystal layer structure is non-negligible. The variation of the connectedness of pore channels is also non-negligible. Thus, j changes during sweating. The mass fraction of the liquid phase at time t can be calculated by the phase equilibrium formula, as can the density of the liquid phase. Thus, the characteristic factor j is the function of the system properties and operation conditions. According to Eq. (6), the porosity at time t is approximately

fðtÞ ¼ fini þ at

ð39Þ

If the variation of the crystal density is negligible, the mass of the melted crystal phase at time t can be expressed as follows: m0C ðtÞ ¼ arC t

ð40Þ

Assume that the ion impurity cannot imbed in the lattice of the crystal, so the mass fraction of ion impurity i in crystal cC,i is equal to 0. The mass fraction of the ion impurity i at the initial time is then expressed as follows: cCL,i ð0Þ ¼

mLE cLE,i mCL

Eq. (43) indicates that the effective distribution coefficient is related to the initial phase composition of the crystal layer, the rate of sweating liquid discharge and the melting rate of crystal phase. The initial phase composition of the crystal layer (mCL/mLE) is determined by the crystal layer growth conditions. A higher growth rate always leads to more of the impure liquid phase entrapped in the crystal layer (Kim and Ulrich, 2001a, 2001b). According to Eq. (43), the separation effect of sweating is better when more of the liquid phase is entrapped in the layer, which agrees with the findings of Kim and Ulrich (2002a, 2002b).

3. Experimental In this study, the experimental apparatus was same as in the authors’ previous research (Jiang et al., 2012b). The same feed phosphoric acid was utilized in every test. The crystal layer was grown under similar conditions to ensure that the structure of the crystal layer was the same as that of the others (cooling rate, 0.067 K/min; feed rate, 10–15 g/min). The only difference between each test was the terminal cooling temperature, which is helpful for obtaining crystal layers with different porosities. The sweating experiments were carried out at the end of the crystal layer growth process. When the temperature reached the terminal cooling temperature, the beaker was set to hold the sweating liquid and the set heating curve was begun. The time and the mass of the sweating liquid were recorded. Examples of the sweating liquid were analyzed by inductively coupled plasma mass spectrometry (ICP-MS) and volumetric titration every 30 min. At the end of the experiment, the crystal layer was melted. The mass of the melt was measured and then analyzed by ICP-MS and volumetric titration. The experiments were implemented with different heating rates and terminal sweating temperatures. The temperature of the experimental environment was kept synchronized with the sweating curve.

4. Results and discussion 4.1. Structural and characteristic parameters of the crystal layer

ðmLE mSw ðtÞ þ m0C ðtÞÞcLE,i ðmLE m0LE ðtÞÞcLE,i ¼ mCL mSw ðtÞ mCL mSw ðtÞ

ð42Þ

where cLE,i is the mass fraction of the ion impurity i in the liquid phase entrapped in the crystal layer. According to Eqs. (2), (36), (41) and (42), the effective distribution coefficient of the ion impurity i Keff,i is K eff,i ðt Þ ¼

ð43Þ

ð41Þ

During the sweating, the mass fraction of the ion impurity i at time t in the crystal layer is expressed as follows: cCL,i ðt Þ ¼

!   mC m0C ðtÞ mCL mCL mC arC t ¼ ¼ U 1 U 1 Rt mLE mLE mCL mSw ðtÞ mCL  0 rSw Q ðt, lÞdt

cCL,i ðtÞ mCL mLE mSw ðtÞ þ m0C ðtÞ ¼ U cCL,i ð0Þ mLE mCL mSw ðtÞ

The terminal sweating temperature was defined as the temperature at which the crystal layer fractured and fell down. Thus, the terminal sweating temperature reflected the stability of the crystal layer structure around the melt point. The experimental results and process parameters were listed in Table 1. The crystal layer was clearly more stable and could resist higher sweating temperatures when the sweating was implemented slowly. The initial porosity (fini) of crystal layer was

Table 1 The experimental results and parameters. Test no.

La, m

Heating rate, K/min

Tendb, K

fini

fFinal

DT,ini

Df,ini

lmax, mm

104 a, min  1

k

K , mdc

Q exp , g/min

A B C D

1.5 1.5 1.0 1.0

0.105 0.072 0.087 0.055

301.55 302.35 301.95 302.85

0.212 0.554 0.148 0.386

0.705 0.851 0.616 0.580

1.364 1.442 1.101 1.187

1.750 1.693 1.905 1.834

2.91 2.90 2.80 2.65

0.284 0.177 0.327 0.138

0.757 0.849 0.745 0.939

223.00 321.16 244.97 207.92

0.774 0.904 0.801 0.670

a b c

L¼ The length of crystallizer. Tend ¼Terminal sweating temperature. md¼1.02  10  9 m2.

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determined by the former crystal layer growth operation, and the initial structural parameters (DT,ini, Df,ini) were different as well. fFinal was strongly influenced by the sweating conditions. As shown in Table 1, the porosity variation rate (a) increased with the heating rate in different crystallizers as expected, although the initial porosities of the crystal layer were different. In addition, k represented the volume fraction of liquid with the impurities discharged during sweating and was related to the initial structural property of the crystal layer and the sweating conditions. In general, the liquid discharged effectively when the heating rate was low and the initial porosity was ideal, such as in

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Test D. The proper porosity offered sufficient pore connections, as will be discussed later. Concerning the porous medium property of the entire crystal layer, the average permeability K calculated by Eq. (19) and the measured average sweating flow rate Q exp during the sweating experiment were also listed in Table 1. The flow rate Q exp was clearly larger when the permeability K of the crystal layer was optimal, which was coincident with porous media theory. To investigate the variation of K with sweating time under various experimental conditions, the calculative results of K vs sweating time were shown in Fig. 3. It was apparent that the initial K was principally determined by fini, whereas the variation velocity of K was strongly influenced by the heating rate. For example, in Test C, the initial K was insignificant because the initial porosity fini was 0.148. However, K increased rapidly under the conditions of a quick heating process (0.087 K/min), which indicated that the crystal layer structure changed dramatically due to the melted crystal discharged with the sweating liquid. 4.2. Flow rate of dynamic sweating

Fig. 3. Permeability K vs sweating time.

The flow rates of two ideal models, QA(t,l) and QB(t,l), and the real flow rate model QR(t,l), which was modified by j, were shown in Fig. 4. The experimental results were also compared in this figure. The variation of j in each test was noted in the figure. The initial value of j was chiefly determined by the initial porosity fini, which was related to the layer growth conditions. The variation rate of j was influenced by the sweating condition, as expected: j increased more quickly in Test A and C than in Test B and D, which were coincident with the heating rate and porosity variation rate.

Fig. 4. Comparison between real flow rate and model flow rate. J, experimental data; (red dot), simulation result of model A, QA(t,l); (blue dot), simulation result of model B, QB(t,l); ———— (black dot), simulation result of model with characteristic factor j, QR(t,l). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 5. Diagram of crystal layer in sweating: (A) the beginning of sweating; (B) crystal layer getting loose; and (C) crystal layer fracture.

Regarding the completely separated model (red dotted line in each figure), the flow rate peaked during the process and then decreased to a very small value because the integral average value of the sweating liquid discharge driving force in completely separated pore channels reached a maximum and then declined to nearly zero. In the completely connected model (blue dotted line in each figure), the flow rate curve increased with the increasing sweating temperature due to the continuously increasing average driving force of the porous media (crystal layer). The above two phenomena led to an interesting simulation result: the real flow rate model QR(t,l) (black dotted line in each figure) peaked and then decreased to a valley floor value before increasing again at a much faster rate than before. This behavior was due to the modification of j between the two ideal models. Fig. 4 showed that the simulation results agreed very well with the experimental results. These experimental results validated the reliability of the developed models. The simulation and experimental results could be explained by the variation of the crystal layer structural properties in detail:

(1) At the beginning of sweating, there was no driving force generated, and the sweating liquid flow rate was zero. There was also no crystal melted from the layer because the temperature was far below the melting point; the flow rate curve reflected the completely separated model. (2) The length of the liquid column decreased with the sweating liquid discharge, whereas the discharge driving force increased with the heating temperature. (3) Then, the flow rate of the sweating liquid peaked and the driving force of the sweating liquid discharge decreased, with the fraction of the crystal melted in sweating increasing as the temperature approached the melting point. (4) The temperature increased persistently, and more and more of the crystal melted. The connection between pore channels in the crystal layer was intensified. The flow rate curve reflected the completely connected model. (5) Apparently, the flow rate did not increase indefinitely. When too much crystal melted occurred, the crystal layer became unstable due to the increasing connection of pore channels.

Fig. 6. Variation of Keff,i vs sweating time.

Finally, the crystal layer fractured with excessive sweating temperature and the separation system broke down (as shown in Fig. 5). The variation of Keff,i was simulated as a function of sweating time, the results were shown in Fig. 6. According to the report of Kim and Ulrich (2002a, 2002b), ‘the more the liquid phase inside the layer exists, the faster removal of impurity in sweating operations occurs’’. At the beginning of the sweating, the crystal layer with a higher porosity did exhibit a better separation effect (smaller Keff,i). With the increasing crystal layer temperature, an increasing amount of the crystal phase melted. This phenomenon was more noteworthy in the crystal layer, which entrapped more of the liquid phase, which was why the Keff,i curve plateaued (Test A) or decreased more slowly than in the initial period (Test B). Keff,i exhibited a minimum (such as in Tests B and D). After that point, as the effect of the liquid phase immigration was subordinate compared to pure crystal phase loss, Keff,i increased (Tests A, B and D).

X. Jiang et al. / Chemical Engineering Science 91 (2013) 111–121

The experimental Keff,i values for the sweating step were listed with the simulated values in Table 2. The Keff,i of different ion impurities were similar and agreed well with the simulated values. In addition, the separation effects of different ions during sweating were not as varied as those during the crystal layer growth due to the accurate phase separation and good distribution of ions during the well controlled sweating operation. This finding also indicates that sweating can separate impurities with various diffusivities, which is an advantage over the single crystal layer growth stage. In addition, Keff,i of the overall FFMC process was simulated by the models developed in this paper and a previous paper (Jiang et al., 2012). The simulated results were listed with the experimental data in Table 3. The deviations between the simulated and experimental data were clearly greater than the single sweating step. As mentioned above, a smaller initial porosity of the crystal layer indicates a smaller amount of entrapped impure liquid. The crystal layer is expected to achieve a desirable purification efficiency when the B–P structure exhibits good behavior. A possible strategy is maintaining a moderate cooling rate during the crystal layer growth process. This operation will shorten the overall operating time and increase the separation effect of the subsequent sweating stage.

4.3. Variation of sweating liquid components The ion impurity concentrations (Ci), phosphoric acid (PA) mass fraction of the sweating liquid during the entire process and the terminal product of the experiment in Test D were listed in Fig. 7. The results agreed with our proposed explanation of the structure variation of the crystal layer: the PA mass fraction of sweating liquid increased with the sweating time and temperature due to the melted crystals with a higher mass fraction of PA mixing in thing sweating liquid. Furthermore, the crystals were pure enough and lacked ion impurities. Thus, the Ci of sweating liquid decreased quickly when melted crystals were mixed in. At the end of the sweating, the Ci reached an approximate equilibrium value. Using CNa as an example, the CNa in the sweating

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liquid and the product were almost the same, which indicated that most of the sodium impurities had already been discharged and the following sweating was inefficient at removing additional sodium impurities. Regarding calcium and ferrum, the deviations of Ci between the sweating liquid and product were more notable than that of sodium, which indicated that they were much more difficult to separate than sodium was. This conclusion coincided with the result reported by Jiang et al. (2012c). In addition, the crystal layer became structurally fragile when the deviations of Ci between the sweating liquid and product became stable. This occurrence was believed to be the critical point for terminating the sweating. It is clear that the initial structural properties of the crystal layer and the sweating operation conditions both affect the separation effect and the sweating efficiency. Therefore, a fully controlled crystal layer growth process will lead to a crystal layer with well connected pore channels, which is conducive to the full exertion of sweating. As shown in these experiments, the ion impurity concentration was reduced from 4.51 ppm to 0.75 ppm and the H3PO4 mass fraction was 91.4% (91.6% as pure crystal), which is a good separation result. In summary, the crystal layer growth and sweating could be simulated with satisfactory results based on the model developed in this paper and the authors’ former work (Jiang et al., 2011, 2012b). An optimized FFMC with precisely controlled crystal layer growth and sweating operation is a very promising separation approach. A model system is expected to be established with further research. A reliable optimized operation route for the industrial application of FFMC is expected to be improved by the utilization of the aforementioned model system.

Table 2 The experimental and simulative results of Keff,i of the sweating step. Test no.

Keff,i Simulation

A B C D

0.920 0.577 0.615 0.433

Experiment Ca2 þ

Fe3 þ

Na þ

0.842 0.586 0.558 0.406

0.912 0.533 0.561 0.378

0.860 0.555 0.573 0.351

Fig. 7. Ion impurity concentration (Ci) and PA mass fraction (wPA) of sweating liquid and product during the whole process. (’, CCa of sweat; , CFe of sweating liquid; , CNa of sweating liquid; , wPA of sweating liquid; ————, CCa of the product; , CFe of the product; , CNa of the product; , wPA of the product). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 3 The experimental and simulative results of Keff,i of the overall FFMC process. Test no.

Keff,i Ca2 þ

A B C D

Fe3 þ

Na þ

Simulation

Experiment

Simulation

Experiment

Simulation

Experiment

0.365 0.315 0.348 0.235

0.437 0.322 0.379 0.237

0.243 0.208 0.231 0.154

0.258 0.250 0.219 0.105

0.208 0.144 0.156 0.088

0.269 0.198 0.170 0.050

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5. Conclusions

s

DT(t) A model to simulate the dynamic sweating of FFMC was developed in this paper using fractal porous media theory. The model could simulate the sweating liquid discharge and separations with satisfactory results under various sweating conditions. Meanwhile, the sweating model reflected the structural changes of the crystal layer and offered a credible explanation of the changes of the crystal layer permeability and sweating liquid components. In addition, the effective distribution coefficient Keff,i of sweating could also be obtained from the resulting model. This dynamic model is a crucial part of the model simulation system for the overall FFMC purification process. The FFMC model system is expected to be applied to a wide range of industrial crystallizations, especially for trace amount multi-ion impurity separation and hyperpure chemical manufacturing.

Nomenclature a b CC CCL CF CLE CML CSw Csat c cCL,i cLE,i Df DT K K Keff,i k L0 Lt l(t) l0 mC m0C mCL mF mLE m0LE mML mSw DPt Q QA(t,l) QB(t,l) QR(t,l) Q exp q(t,l) qA(t,l) qB(t,l) R

variation rate of porosity, s  1 parameter of surface tension with temperature, N/(m K) H3PO4 mass fraction of crystal H3PO4 mass fraction of crystal layer H3PO4 mass fraction of feed H3PO4 mass fraction of liquid phase entrapped H3PO4 mass fraction of mother liquid discharged H3PO4 mass fraction of sweat mass fraction of saturated phosphoric acid parameter of surface tension, N/m mass fraction of ion impurity i in crystal layer mass fraction of ion impurity i in the liquid phase capillary areas fractal dimension tortuosity fractal dimension permeability of porous media, md average permeability of crystal layer during sweating, md effective distribution coefficient of ion impurity i ratio of effective porosity vs initial porosity length of crystal layer, m length of the tortuous capillary channel, m length of liquid column at t moment, m original length of liquid column, m mass of crystal phase in crystal layer, kg mass of melted crystal, kg mass of crystal layer, kg mass of feed added, kg mass of liquid entrapped in crystal layer, kg mass of liquid phase discharge out of crystal layer, kg mass of mother liquid discharged, kg mass of sweating liquid, kg the pressure difference on the crystal layer at t moment, Pa flow rate of fluid through the porous media, m3/s the total flow rate of all pore channels in completely separated model (model A), m3/s the total flow rate of all pore channels in completely connected model (model B), m3/s real total flow rate in model, m3/s average sweating liquid flow rate, m3/s flow rate through a single tortuous capillary, m3/s flow rate of certain pore channel in completely separated model (model A), m3/s flow rate of certain pore channel in completely connected model (model B), m3/s radius of pore channel, m

t VCL Vp vSw

thickness of crystal layer, m temperature difference on liquid column at t moment, K time of sweating, s total volumn of crystal layer, m3 porous volumn in crystal layer, m3 heating rate during sweating, K/s

Greek letters

j feff fFinal fini lav lmax lmin

rC rLE rsat rSw tav m

characteristic factor effective porosity final porosity of crystal layer initial porosity of crystal layer average diameter of pores, m maximum diameter of pores, m minimum diameter of pores, m density of crystal, kg/m3 density of liquid entrapped, kg/m3 density of saturated phosphoric acid, kg/m3 density of sweating liquid, kg/m3 average tortuosity viscosity of fluid, Pa s

Subscripts A av B C CL eff exp F ini LE max min ML R sat Sw

completely separated model (model A) average completely connected model (model B) crystal crystal layer effective experiment feed initial liquid entrapped in crystal layer maximum minimum mother liquid real saturated sweating

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