Front. Chem. Sci. Eng. 2011, 5(4): 435–441 DOI 10.1007/s11705-011-1139-3

RESEARCH ARTICLE

Permeability analysis and seepage process study on crystal layer in melt crystallization with fractal and porous media theory Xiaobin JIANG, Baohong HOU, Yongli WANG, Jingkang WANG (✉) State Research Center of Industrialization for Crystallization Technology, School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2011

Abstract In this paper a porous media seepage model was applied to analyze the permeability and study the seepage process of crystal pillar formed in the preparation of electronic grade phosphoric acid (EGPA). By inspecting the seeping process, the structure parameter of crystal pillar could be obtained. Two basic ideal models (perfectly separated model and perfectly connected model) were presented and a characterized factor φ was introduced to modify the model. A good simulation result was obtained which met the experiment result well. The relationship between φ and permeability were also discussed. The characterized factor φ showed potential application on optimizing process. Keywords melt crystallization, porous media, fractal, permeability, hyperpure material, simulation

1

Introduction

Electronic grade phosphoric acid (EGPA) has been widely used as a hyperpure chemical in electronics industry. EGPA has exceptionally strict request for impurity ion concentration. Many researches indicate that melt crystallization will be an ideal method for EGPA preparation [1,2]. The product of melt crystallization is usually crystal layer (Fig. 1). The separation efficiency is strongly related to the permeability of crystal pillar. However, it is impossible to apply real time online particle analysis technology to detect the structure of crystal pillar in crystal

Received July 10, 2011; accepted October 9, 2011 E-mail: [email protected]

Fig. 1

Crystal pillar in the crystal tower (left) and product (right)

tower. So an effective model is imperative to represent the structure of crystal pillar, which has some features of porous media as shown in Fig. 1. The microstructure and permeability for porous media have drawn much attention due to practical applications on chemical engineering, such as soil science, oil production, polymer composite molding and heat pipes [3–6]. The microstructures of real porous media are complex disordered. Yu’s group introduced fractal theory to describe the structure of conventional porous media. A fractal model has also been developed for permeability and the effective thermal conductivity of bi-dispersed porous media by Yu and Cheng [7,8]. The fractal permeability model is a function of the tortuosity fractal dimension, pore area fractal dimension, sizes of particles and clusters, micro-porosity inside clusters, and the effective porosity of a medium. In this article, we introduce a characterized factor to modify this model.

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Front. Chem. Sci. Eng. 2011, 5(4): 435–441

2

Experiment

2.1

Materials

High quality analytical grade phosphoric acid solution (85 to 90 wt-%) was used for melt crystallization to prepare EGPA. Water used in this experiment was purified by UPW-20N ultra pure water. 2.2

Melt crystallization experiment

The experimental apparatus was shown in Fig. 2. The crystallize tower with three temperature recorders was maintained at desired temperatures by circulating water through the outer jacket from a water bath. After the material acid was added into the tower and cooled for 30 min, pasty seeds (0.1 wt-%) were added, which will spread in the crystallizer. The crystallizer was cooled according to a certain cooling curve set in advance, and hold the temperature for 1 h when the cooling procedure came to the end. Then the bottom valve was opened completely, and the time and quality of mother liquor were recorded simultaneously. When the velocity of mother liquid discharge was less than 0.5 g/min, the heating

procedure was started and the crystal product was collected. The quality and mass fraction of both material and product were measured. The porosity of crystal pillar was calculated based on the phase equilibrium data reported in Refs. [9,10].

3 Fractal porous media system model and modification 3.1

Fractal porous media system model

Yu and Cheng had developed a mature fractal model to simulate the structure of porous media with fractal properties [11–13]. In the fractal porous media setted in cylindrical-coordinate system, the fractal dimension of capillary areas in porous media Df is determined by [11] Df ¼ 2 –

lnf , l ln min lmax

(1)

where f is the porosity, lmin and lmax are the minimum and maximum diameter. The tortuosity fractal dimension DT can be obtained as following equation by [13] DT ¼ 1 þ

ln τ av , L ln lav

(2)

where τ av is the average tortuosity, L is the length of porous media, lav is the average diameter of pores. A simple geometrical model for tortuosity for flow path in porous media was derived by Yu and Li [12]. The model was based on the assumption that some particles in a porous media are unrestrictedly overlapped. The average tortuosity is expressed as s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 2 1 1 pffiffiffiffiffiffiffiffiffiffi – 1 þ 7 6 47 1–f 16 1 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 6 7: pffiffiffiffiffiffiffiffiffiffi τ av ¼ 61 þ 1–f þ 1–f 7 24 2 1– 1–f 5 (3)

1 balance; 2 beaker; 3 valve; 4,5,6 temperature recorders; 7 crystallizer;

In Ref. [8], Yu and Cheng described the pores in porous media by analogy with the islands or lakes on earth or spots on engineering surface, which means the cumulative size distribution of pores follow the following fractal scaling laws,   lmax Df N ðL³lÞ ¼ , (4) l

8 temperature control meter; 9 circulating water bath

Fig. 2 Schematic diagram of experimental apparatus

and

Xiaobin JIANG et al. Study on Crystal Layer in Melt Crystallization

D

f – dN ¼ Df lmax l – ðDf þ1Þ dl:

(5)

QA ði,lÞ ¼ –

Dividing Eq. (4) by Eq. (5) dN Df ¼ Df lmin l – ðDf þ1Þ dl ¼ f ðlÞdl: – Nt

¼J

(6)

lmax

!l

min

lmax

!l

437

qA ði,lÞdN ðlÞ

l2þDT – Df expð – Hl1þDT iÞdl,

(11)

min

D

So lav can be found from Eq. (6),     lmax Df lmin Df – 1 lav ¼ l : (7) lf ðlÞdl ¼ 1– lmin Df – 1 min lmax

!

f T where J¼ πgDf lmax L10 – DT =ð128Þ; H ¼ L g=ð32LD 0 Þ; Df N ðlÞ ¼ ðlmax =lÞ .

3.2.2

Perfectly connected model

Df and DT can be obtained based on above equations. 3.2

Model modification

When applying the fractal porous media model to melt crystallization process, some modification is necessary. During the mother liquid discharge process, the driving force is gravity instead of pressure, and thus continuously varies. So the flow rate through a single tortuous capillary is a function of pore canal diameter l and time i. The function could be expressed as followed based on the research in Ref. [9]: qði,lÞ ¼

π ΔPi l4 π L gLi l4 ¼ , 128 Lt ðlÞ  128 Lt ðlÞ 

(8)

where l is the hydraulic diameter of a single capillary tube, μ is the viscosity of the fluid, ΔP is the pressure gradient, and Lt is the length of the tortuous capillary tube ρL and g are the density of liquid phase and gravity acceleration, respectively. Li is the apparent liquid level in crystal pillar at i moment. Two extreme models were used to modify Li. 3.2.1

Assume that all the pore channels were completely connected each other and there were no diffusion resistance between two channels, so all the pore canals have the same Li which can be expressed by following formula LiB ¼ L0 –

Assume that each pore canal was completely separated with other ones,so we can completely neglect the diffusion rate of liquid phase between each pore canal. Li was expressed by following formula   π 2 i LiA ¼ L0 – 1= l $ qði,lÞdi : (9) 0 4

!

According to Eqs. (8) and (9), the flow rate through a single tortuous capillary in completely separated model is given by following equation   i  g V0 – ql ðiÞdi 0 l1þDT L qA ði,lÞ ¼ : (10) DT 32 L0

!

The total flow rate Q (diameter range from lmin to lmax) of all capillaries at i moment in completely separated model will be

i

!0QBði,lÞdi,

(12)

where A is the sectional area of crystallizer, f is the porosity of crystal pillar, and QB ði,lÞ is the total flow rate Q of all capillaries. According to Eqs. (1) and (12), the flow rate through a single tortuous capillary in completely connected model is given by following equation   i  g Vp – QB ði,lÞdi 0 πl3þDT L qB ði,lÞ ¼ , (13) DT 128 L0 Af

!

where Vp is the total porous volume and Vp ¼ L0 Af. So the total flow rate Q of all capillaries at i moment in completely connected model will be

!

Perfectly separated model

1 Af

D

f πL gDf lmin

lmax

lmax

!

QB ði,lÞ ¼ – qB ði,lÞdN ðlÞ ¼ l2þDT – Df DT lmin l 128L0 Af min    l i max Vp – QB ði,lÞdi dl ¼ J l2þDT – Df dl

!0

–

lmax

!l

min

!l

min

l2þDT – Df

i

!0QBði,lÞdidlÞ,

(14)

where J is the same J in Eq. (11). Equations (11) and (14) can be solved by numerical integration method to get approximate function of QA ði,lÞ and QB ði,lÞ. Considering while i = 0, QA ði,lÞji¼0 ¼QB ði,lÞji¼0 ¼ J 3þD – D

J lmax T f ¼ 3 þ DT – Df

lmax

!l

1–

l2þDT – Df dl ! 3þD – D lmin T f min

(15)

3þD – Df

lmax T

For lmin =lmax £10 – 2 and ðlmin =lmax Þ3þDT – Df ffi 0, Eq. (15) can be reduced to

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Front. Chem. Sci. Eng. 2011, 5(4): 435–441

QA ði,lÞji¼0 ¼ QB ði,lÞji¼0 ¼

T πL gDf L10 – DT l3þD max : 128 3 þ DT – Df

(16)

Under actual situation, the porous channels should be partly connected to each other instead of totally separated or totally connected. So the real total flow rate QR ði,lÞ should be between QA ði,lÞ and QB ði,lÞ. We introduce a characteristic factor f to represent the similarity of actual situation to Model B and 1 – f is considered to represent the similarity of actual situation to Model A. So QR ði,lÞ can be QR ði,lÞ ¼ ð1 – φÞQA ði,lÞ þ φQB ði,lÞ:

(17)

In this article, lmin was set as 1.0  10–7 m. The average value of total flow rate at the initial time (the first 30 s, in this period the flow rate was kept at a high value and did not change sharply) measured in experiment and used as QR ði,lÞji¼0 . Considering DT and Df are also related to lmax, the iterative calculation was terminated until the error is smaller than 1.0  10–7 m. The characteristic factor f represents the extent of actual porous channels connected. Considering the porous channels are better connected with the higher porosity and the initial mass fraction also has impact on the connection of porous channels by generating the differences of crystal habits (as shown in Fig. 3, the single crystal of H3PO4$0.5H2O have Significantly different crystal habits on crystal size distribution, crystal shape and crystal surface, all of which will influence the structure and permeability). The initial mass fraction of H3PO4 can be considered as distribution density H3PO4$0.5H2O crystal seed and these seeds will grow up to block the porous channel when the temperature went down. So the saturation acid with mass fraction wsat = 91.6% was considered to be the upper limit of distribution density. To unify with the porosity, the mass fraction was transferred to volume fraction. When liquid phase transformed to solid phase, the density increased and the volume fraction of H3PO4$0.5H2O to whole system declined. So the density of H3PO4$0.5H2O crystal to will be introduced to adjust the variation. The impact factor of initial mass fraction on φ was expressed by following formula: IFðwini Þ ¼

wini =ini sat $ , wsat =sat C

(18)

where wsat = 91.6%, ρini, ρsat and ρC were the density of initial material acid, the acid density with the 91.6% mass fraction and the density of H3PO4$0.5H2O crystal, respectively. We use the following formula to express characteristic factor φ,

Fig. 3 Comparison of the H3PO4$0.5H2O single crystals with different wini. (a) wini = 85.48%; (b) wini = 88.86%; (c) wini = 89.56%

φ ¼ f=IFðwini Þ:

(19)

It should be noticed that Eq. (19) is just an empirical calculation formula, and its application extent should be verified by experimental data. The valuable significance of characteristic factor φ will be discussed later based on the simulation result and further study. In addition, when φ↕ ↓1, there is no application meaning to crystal engineering, and it is also impossible for a crystal to grow and build a stable solid framework. We consider Eq. (19) as an application meaning formula to describe the connection of porous channels in crystal engineering condition.

Xiaobin JIANG et al. Study on Crystal Layer in Melt Crystallization

Based on the definition formula of permeability and the modification in this model, the permeability K of crystal pillar can be expressed as followed: L0 Q L0 lmax qði,lÞ dN ðlÞ ¼ – ΔPi A lmin ΔPi ΔPi A   L0 lmax π ΔPi l4 ½ΔPi dN ðlÞ ¼– ΔPi A lmin 128 Lt ðlÞ 

!

K¼

¼ ¼

from experiment results. With the increasing of porosity and characterized factor φ, the permeability K of crystal pillar increased, too. So the following formulas were utilized to fit the relationship between K and f or φ, K ¼ afb ,

=

!

32D

Df : 3 þ DT – Df

K ¼ a# φb# :

(21)

K ¼ afb ¼ 3454:3f6:6889 :

(22)

Fit K with f,

Df π L10 – DT l3þDT 128 A 3 þ DT – Df max T L10 – DT l3þD max 2

439

(20)

(correlation coefficient equal to 0.950). Fit K with φ, #

K ¼ a# φb ¼ 760:64φ6:7454 :

4

Results and discussion

In Table 1, the experimental data showed that QR,m ði,lÞji¼0 (the volume flow rate were converted to quality flow rate for a better express) increased sharply when the porosity increased. lmax also increased with the increasing porosity. Pore area fractal dimension Df increased with increasing lmax because pore area fractal was losing its fractal feature progressively when lmax is bigger in a fixed cross area. Tortuosity fractal dimension DT increased with the decreasing lmax because the porous channel tortuosity fractal become more tortuous and acquired its fractal feature progressively with smaller lmax. φ also increased with the increasing porosity because more porous volume will offer more opportunity for porous channel connected. The real total flow rate QR,m measured in experiment and calculated from models (mass velocity, converted from volume velocity with known liquid phase density) were shown in Fig. 4. Figure 4 also showed how characteristic factor φ modified the deviation of models A and B’s results

Table 1

(23)

(correlation coefficient equal to 0.948). As shown in Fig. 5, f and φ hold almost the same fitting result to permeability. Because φ is also related to the initial mass fraction, Eq. (23) offers us a path to optimize the permeability by adjust the porosity and initial operation condition. So φ could be used as an optimized standard in the future research.

5

Conclusions

A porous media seepage model has been applied to represent the permeability of crystal pillar effectively. By introducing the characterized factor φ, this model obtained satisfactory results on the simulation of the seepage processing in melt crystallization which fitted the experimental ones well. The characterized factor φ possesses both theory and actuality meaning on the description of porous media. The characterized factor φ also has potential application on optimizing process.

QR,m ði,lÞji¼0 , lmax , DT, Df and φ with different wini and f QR,m ði,lÞji¼0 /(g$min–1) lmax /mm

Test number

wini /%

f

DT

Df

φ

1

89.56

0.2886

0.20

0.81

1.0419

1.8607

0.3669

K, md 0.9076

2

88.86

0.3924

1.80

1.32

1.0311

1.9014

0.4909

10.0037

3

85.48

0.5689

5.08

1.69

1.0189

1.9419

0.7163

26.6198

4

85.48

0.5900

12.05

2.01

1.0177

1.9468

0.7416

62.3883

5

84.97

0.6011

18.65

2.38

1.0170

1.9494

0.7593

110.9398

6

85.48

0.6249

28.75

2.54

1.0158

1.9536

0.7838

161.9138

7

85.48

0.6354

43.22

2.77

1.0154

1.9557

0.7965

199.8304

8

85.46

0.6638

68.20

3.09

1.0139

1.9604

0.8340

311.9058

9

85.68

0.7154

120.80

3.47

1.0115

1.9680

0.9003

507.2573

10

85.31

0.7825

179.42

3.88

1.0088

1.9768

0.9786

821.6206

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Front. Chem. Sci. Eng. 2011, 5(4): 435–441

: experiment data; ……(small dot): simulation result of model A;—(dash): simulation result of model B; ——: simulation result of model with

●

characteristic factor φ

Fig. 4 The comparison between real flow rate and model flow rate. (a) f = 0.7825, φ = 0.9786 (test No. 10); (b) f = 0.6638, φ = 0.8340 (test No. 8); (c) f = 0.6249, φ = 0.7838 (test No. 6); (d) f = 0.5689, φ = 0.7163(test No. 3); (e) f = 0.3924, φ = 0.4909 (test No. 2); (f) f = 0.2886, φ = 0.3669 (test No. 1)

Xiaobin JIANG et al. Study on Crystal Layer in Melt Crystallization

Fig. 5 The comparison between experiment data and fitting results

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