Modeling of solid layer growth from melt for Taylor bubbles rising in a vertical crystallization tube

Chemical Engineering Science 58 (2003) 5257 – 5268

www.elsevier.com/locate/ces

Modeling of solid layer growth from melt for Taylor bubbles rising in a vertical crystallization tube Junxian Yun, Ziqiu Shen∗ School of Chemical Engineering, Dalian University of Technology, Dalian 116012, People’s Republic of China Received 25 February 2002; received in revised form 6 August 2002; accepted 15 May 2003

Abstract This paper presents a mathematical model for the prediction of solid layer growth from melt contained in a crystallization tube of a bubble column crystallizer. The model was solved by using the integral formulation approach together with the assumption of a second-degree polynomial approximation of temperature pro4le within the solid layer. In the model, the convective heat transfer coe6cient at solid layer–melt interface was calculated by analogy between heat and mass transfer from the models developed for mass transfer in the case of Taylor bubbles rising in a vertical liquid column. The predicted growth rates of solid layer were compared with experiments of caprolactam melt solidi4cation which were carried out in a 19 mm inner diameter crystallization tube at super4cial gas velocities of 0.04 and 0:08 m s−1 . A good agreement was obtained between experimental and predicted data. Di:erent characteristics of layer growth for slug units passing through di:erent growing stages were also described and discussed. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Solidi4cation; Phase change; Mathematical model; Solid layer growth; Taylor bubble; Caprolactam

1. Introduction Layer-based melt crystallization attracts the attention of researchers due to the increasing demand for high-purity organics in chemical industry over the past few years (Meyer, 1990; Wynn, 1992; Wellingho: and Wintermantel, 1994; Ulrich et al., 1996; Fukui and Maeda, 2000; Guardani et al., 2001). Bubble column crystallizer developed by Ruetgerswerke is one of the e:ective devices for layer-based crystallization (Stolzenberg et al., 1983; Wellingho: and Wintermantel, 1994; Ulrich et al., 1996). Generally, a bubble column crystallizer is commonly made up of several to hundreds of crystallization tubes, depending on the production scale and capacity needed. Crystallization in bubble column crystallizer is carried out batchwise. Melts to be puri4ed are fed into these tubes and an inert gas is introduced into the melts to produce gas slugs or Taylor bubbles. In the processes of gas slugs rising upward, the solid layer of puri4ed compound is grown on the inner tube

∗ Corresponding author. Tel.: +86-411-462-3092; fax: +86-411-363-3080. E-mail address: [email protected] (J. Yun).

0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2003.05.006

walls by circulating coolant outside these tubes. After a certain time of growth, the residual mother melts are drained o: and the solid layer is re-melted to get a high-purity product. In industrial applications, the growth rate of solid layer is very important and fundamental to the design of crystallization process in bubble column crystallizer. Several experimental investigations have been made to study the e:ects of growth rate, cooling temperature, gas Fow rate, composition of melts and sweating operations on the crystallization process (Shi et al., 1994; Ulrich et al., 1996; Wang et al., 1998; Yang et al., 1999). However, little attention has been devoted to the solid layer growth in bubble column crystallizer by considering the heat transfer characteristics. Ming and Shen (2000a) proposed a mathematical model for evaluating the solid layer growth in a short crystallization tube of a bubble column crystallizer. A linear temperature pro4le across the solid layer was assumed and the convective heat transfer coe6cient from melt to solid layer was calculated by dividing the slug unit into two zones: the falling liquid 4lm zone and the liquid slug zone. Different heat transfer characteristics in these two zones were considered. The present paper is aimed at detailed investigation of the mathematical model of solid layer growth from melt for

5258

J. Yun, Z. Shen / Chemical Engineering Science 58 (2003) 5257 – 5268

motion of Taylor bubbles and the heat transfer characteristics of slug units are negligible. 4. The wall temperature is constant because the Fow rate of coolant outside the tube is high. 5. No heat exchange takes place between Taylor bubbles and melt. 6. The solid layer is pure and its physical properties are constant. For a given gas Fow rate, the growing rates of solid layer are controlled by heat Fux entering and leaving solid layer and the released or absorbed heat Fux due to solid–liquid phase change at the solid layer–melt interface. Therefore, modeling of solid layer growth requires mathematical descriptions of heat conduction in solid layer, heat balance at the interface and convective heat transfer from melt to solid layer. 2.1. Energy integral equation for the solid layer growth The unsteady heat conduction equation in the solid layer is written as (Ming and Shen, 2000a)
 @Ts (t; r) 1 @Ts (t; r) 1 @ r = ; r @r @r  @t Fig. 1. Schematic diagram of solid layer growth in a crystallization tube of a bubble column crystallizer.

Taylor bubbles rising in a vertical crystallization tube of a bubble column crystallizer by taking into account the heat transfer kinetics. The solid layer growing rate of caprolactam at gas super4cial velocities of 0.04 and 0:08 m s−1 were measured experimentally and compared with the model predicted data. The characteristics at di:erent growing stages were analyzed.

R − s(t) 6 r 6 R;

t ¿ 0:

The solid layer thickness is assumed to be uniform along the tube axis and it is only a function of time. The solid layer is so thin that the heat accumulation within the layer is relatively small. Therefore, at any instant, the released or absorbed latent heat Fux at the solid layer–melt interface is balanced by the heat Fux transferred into and removed out of the interface, which yields 
dTs −ks dr r=R−s(t) = s [ + CpL (Tb − Tm )]

2. Model development The model was developed by combining the energy equation in the solid layer with the heat balance equation and the convective heat transfer coe6cient correlation at melt-solid layer interface. Fig. 1 shows a schematic diagram of solid layer growth in the crystallization tube. Temperature pro4les and coordinate system are also shown in this 4gure. To simplify the problem the following assumptions have been made: 1. The melt bulk temperature and physical properties (density, viscosity, surface tension, heat conductivity) are constant. 2. At a given super4cial gas velocity, velocities of Taylor bubbles and slug units are constant. 3. The layer thickness is small compared with the tube diameter and then the e:ects of solid layer growing on the

(1)

+ hL (Tb − Tm );

ds(t) dt

r = R − s(t);

t ¿ 0:

(2)

The boundary conditions are Ts (R; t) = T0 ;

r = R;

Ts (R − s(t); t) = Tm ;

(3) r = R − s(t):

(4)

Chianese and Santilli (1998) has applied the integral formulation method by Goodman (1958) to solve the layer growth model in the case that the solid layer is grown on the outside of a cylinder. In this work, the solid layer is grown on the inner surface of a crystallization tube. The coordinate of the solid layer–melt interface is di:erent from that in the work of Chianese and Santilli (1998) or Parisi and Chianese (2001), as expressed in Eqs. (1), (2) and (4). But these equations can also be solved by the integral formulation method. Using the method, the energy integral equation for the layer growing for the present case

J. Yun, Z. Shen / Chemical Engineering Science 58 (2003) 5257 – 5268

is obtained
 hL @Ts Rks ds(t) =− (T − T ) − b m dt s  s  [R − s(t)] @r r=R  d R−s(t) ks − s  [R − s(t)] dt R

B = {4cs3 (t) + (7b − 12Rc)s2 (t) − [16Rb + 2(T0 − Tm )]s(t) + 8R(T0 − Tm )}=12: (13) 2.2. Convective heat transfer from melt to solid layer

× [Ts (r; t) − Tm ]r dr

(5)

where  = + CpL (Tb − Tm ). From Eq. (5), once the temperature within the layer is known, the growing rates can be determined. Since the heat accumulation within the solid layer is small, the temperature pro4le should be smooth and not abrupt (Chianese and Santilli, 1998). Assume that the time to establish the temperature pro4le across the layer is small compared with the solid layer growth time. At a 4xed growth time, temperature distribution across the layer is assumed to be the following second-degree polynomial form (Goodman, 1958) Ts (r; t) − Tm = b{r − [R − s(t)]} + c{r − [R − s(t)]}2 : (6) The following derivative condition from Eq. (4) together with Eq. (3) is needed to determine coe6cients b and c:  


@Ts ds(t) @Ts + − = 0: (7) @r r=R−s(t) dt @t r=R−s(t) Then



b=− f+ c=

 f2

8(T0 − Tm )ks − s2 (t) s 

 


b T0 − Tm − ; s2 (t) s(t)

2

ks s 

 ;

(8) (9)

where f=

2  hL (Tb − Tm ) − + : R − s(t) s  s(t)

(10)

Substituting Eqs. (6), (8), (9) and (10) into Eq. (5) leads to the following formulation for solid layer growth: ds(t) −1 = dt s  + ks (A + B)={[R − s(t)]}  Rks × hL (Tb − Tm ) + R − s(t) × [2(T0 − Tm )=s(t) − b] ; where

5259

(11)


  [2R − s(t)]s2 (t) s  2 A= + 2 24ks s (t) [R − s(t)]2


   8(T0 − Tm )ks 2 + f + + s2 (t) s3 (t) s  [R − s(t)]2  8(T0 − Tm )ks 2 (12) f − s2 (t) s 

The accurate correlation of the convective heat transfer coe6cient is crucial to the calculation of heat Fux from melt to solid layer. In the crystallization tube, heat transfer from melt bulk to the solid layer by convection is equivalent to that of wall–liquid heat transfer for Taylor bubbles rising upward through liquid column in vertical tubes. Considerable investigations on heat transfer for developed gas–liquid slug Fow in vertical tubes have been carried out and numerous correlations have been reported in the past several decades (Kim et al., 1999). In most of these previous experiments, liquid was heated and the super4cial liquid velocity used was relatively high. However, the slug Fow in the present crystallization tube is a special kind of two-phase Fow, in which the net liquid (or melt) Fow rate is zero, also known as zero net liquid Fow (ZNLF) (An et al., 2000; Liu and Scott, 2001). The wall–liquid heat transfer for this case has received little attention and there is a lack of available correlations to be directly used in crystallization tubes. In this work, the correlation for wall–liquid convective heat transfer coe6cient is obtained by Chilton–Colburn j-factors analogy (Coulson et al., 1990) from the mass transfer data. From the experimental results (Ming and Shen, 2000b; Yun and Shen, 2002), there are di:erent wall–liquid momentum and mass transfer characteristics in the falling 4lm zone, the wake region and the liquid slug for gas slugs rising upward through liquid contained in vertical tubes. Ghosh and Cui (1999) studied the wall–liquid mass transfer of slug Fow in a tubular membrane module. Three di:erent zones of the slug unit were distinguished in calculation of the wall–liquid coe6cient in their model: the falling liquid 4lm zone, the wake zone and the remaining liquid slug zone. Di:erent correlations for each of these zones were proposed based on the analysis of slug Fow hydrodynamics. Our previous work (Yun et al., 2002) suggested a modi4ed correlation for wall–liquid mass transfer for the present case, additionally taking account of the laminar liquid 4lm zone from the measured data by limiting di:usion current technique (Yun and Shen, 2002). The slug unit was divided into the following four di:erent zones (Yun et al., 2002): 1. the laminar falling liquid 4lm zone which covers the region from the nose of Taylor bubble to the transition location for falling 4lm from laminar to turbulent, 2. the turbulent falling liquid 4lm zone which covers the region from the transition location to the end of the Taylor bubble, 3. the highly turbulent wake zone, and 4. the remaining liquid slug zone.

5260

J. Yun, Z. Shen / Chemical Engineering Science 58 (2003) 5257 – 5268

r h1

     

x

h2

L TB TB

U TB

h3

  WZ 

L WZ L LS

h4

Based on the obtained data of heat transfer coe6cient by Chilton–Colburn j-factors method, the modi4ed coe6cient FrG CfL was 4tted as a function of gas Froude number √ (a ratio of inertial to gravitational forces; =UG = gd) and length ratio of this zone to the slug unit. The 4tting result is the same as that for mass transfer reported by Yun et al. (2002), which gives CfL = 0:48((LfL =LU )=(1 + FrG ))−0:91 .

L RLSZ

    RLSZ  

2.2.2. For the turbulent falling 5lm region Similarly, the wall–liquid convective heat transfer coe6cient h2 for the turbulent falling 4lm zone is expressed as h2 = 0:023Ref0:8 PrL0:33 kL Cft =de :

(17)

Fitting the analogous data, the modi4ed coe6cient for the turbulent falling liquid 4lm zone is given by Cft = Lft =LU −0:68 ) . 0:57( 1+Fr G Fig. 2. Four zones for heat transfer in a slug unit.

According to the analogy between heat and mass transfer, similar characteristics for heat transfer are expected in these zones. The slug unit for the case of heat transfer can also be divided into the same four zones, as shown schematically in Fig. 2. The corresponding heat transfer coe6cients for these zones h1 , h2 , h3 and h4 are then employed to replace hL in Eq. (11) during the solid layer growing. The analogous data for heat transfer under various conditions were obtained by Chilton–Colburn j-factors method (Coulson et al., 1990) from the experimental values of wall– liquid mass transfer (Yun and Shen, 2002; Yun et al., 2002) Nu Re PrL1=3

=

Sh Re ScL1=3

:

Eq. (14) is re-written as
1=3 KkL PrL h= : D ScL

(14)

(15)

From the measured values of mass transfer coe6cient K, values of heat transfer coe6cient h were obtained and employed to determine correlations for the corresponding zones. 2.2.1. For the laminar falling 5lm zone The falling 4lm liquid Fow in the 4rst zone is generally laminar. The wall–liquid convective heat transfer coe6cient h1 can be calculated by considering the falling liquid 4lm Fow as annular laminar Fow, which can be treated as laminar pipe Fow with an equivalent hydraulic diameter de (Coulson et al., 1990). However, this calculated heat transfer coe6cient is needed to be modi4ed, because the 4lm thickness in this region is not uniform and the e:ect of Taylor bubble rising is considerable. Here, a coe6cient CfL has been applied as h1 = 1:62(Ref PrL de =L)1=3 kL CfL =de :

(16)

2.2.3. For the wake region The low viscosity liquid Fow in the wake region is generally believed to be highly turbulent due to the falling liquid 4lm jetting, shearing and entrainment of small bubbles, which induce strong mixing, circulation of Fuid and irregular turbulent eddies (Campos and Guedes de Carvalho, 1988a,b; Pinto and Campos, 1996; Pinto et al., 1998). Campos and Guedes de Carvalho (1988a) studied the Fow pattern in the wake of a rising slug bubble with a moving camera. They found that there were three di:erent wake patterns: the closed axisymmetric wake, the closed unaxisymmetric wake and the opened wake with recirculatory Fow. For a long Taylor bubble, the wake pattern was found to depend mainly on the dimensionless parameter N = L gd3 =&L (Campos and Guedes de Carvalho, 1988a; Pinto and Campos, 1996). The opened wake with recirculatory Fow exists when this parameter is higher than. In our mass transfer experiments, N = 9485 ¿ 1500 (Yun et al., 2002), the Fow in wake zone is extensively turbulent. Therefore, the mass transfer coe6cient for the wake zone is higher than that in the remaining liquid slug region. The same trend is available for the corresponding heat transfer coe6cient h3 . The excess ratio of heat transfer coe6cient for the wake zone to the remaining liquid slug zone h3 =h4 is a function of Froude number, as same as that for mass transfer by Yun et al. (2002): h3 =h4 = 1:0 + 0:032 FrG−0:26 :

(18)

2.2.4. For the remaining liquid slug region Liquid Fow in the remaining liquid slug region is likely either laminar or turbulent depending on the liquid Reynolds number (Ghosh and Cui, 1999). For the present situation, small bubbles (spherical cap bubbles occasionally) are contained in liquid slug, which destroy the structure of boundary layer. Therefore, the wall–liquid convective heat transfer coe6cient cannot be calculated by the correlation of single phase directly and further correlations are needed. Actually, contributions to wall–liquid heat transfer in this zone can

J. Yun, Z. Shen / Chemical Engineering Science 58 (2003) 5257 – 5268 -1

correlation experimental

0.67

(h4/hLLS-1)/(Red PrL )

10

-2

10

-3

10

-4

10

2

10

3

4

10

10 2

5

10

2

Red FrG PrL

Fig. 3. Correlation of the wall–liquid heat transfer coe6cient using analogous data in the remaining liquid slug zone.

be divided into two parts: the 4rst is the forced convection mainly induced by the up-Fow of liquid slug, and the second is the e:ect due to the rising of small bubbles contained in the liquid slug. The former can be calculated using the widely used laminar or turbulent correlations for heat transfer in vertical tubes. The mechanism for the latter is similar to that in common bubble columns. According to the analysis by Deckwer (1980) for the wall–liquid heat transfer mechanism in bubble column, a Fuid element in front of a small bubble receives a radial momentum due to the bubble uprising, which induces an irregular radial mass Fow and causes heat exchange between liquid and wall. Deckwer found that Stanton number for wall–liquid heat transfer in this case is a function of Reynolds number, Froude number and Prandtl number of liquid: St = f(Reb ; Frb2 ; PrL2 ). Similarly, the correlation for wall–liquid convective heat transfer coe6cient for the remaining liquid slug zone was determined by 4tting the obtained heat transfer data by C analogy, as shown in Fig. 3: h4 Red PrL0:67 = 1 + 0:70 ; (19) hLLS (Red PrL2 FrG2 )0:74 where hLLS can be determined using the following correlation: hLLS = 1:62(Red PrL d=L)1=3 kL =d hLLS = 0:023Red0:8 PrL0:33 kL =d

for Red ¡ 2000

for Red ¿ 2000:

(20) (21)

2.3. Hydrodynamic parameters of Taylor bubbles and liquid slugs For a given super4cial gas velocity in a crystallization tube, gas slugs are assumed to rise steadily at a constant velocity and the slug unit length is assumed to be constant. Calculating the heat transfer coe6cient by Eqs. (16) to (21) requires parameters such as length of the laminar falling liquid 4lm zone LfL , length of the turbulent falling liquid 4lm

5261

zone Lft , length of Taylor bubble LTB , length of liquid slug LLS , length of slug unit LU (=LTB + LLS ), rise velocity of gas slug UTB and void fraction of the liquid slug LS . Numerous workers have investigated on modeling of fully developed vertical slug Fow (Taitel et al., 1980; Fernandes et al., 1983; Orell and Rembrand, 1986; Sylvester, 1987; Abdul-Majeed and Al-Mashat, 2000). Although the slug Fow for the present situation (ZNLF) is one of the basic two-phase Fow phenomena in vertical pipes (An et al., 2000; Liu and Scott, 2001), little attention has been given to the models for ZNLF in the past. In this work, the following model is used to determine parameters of slug Fow in Eqs. (16)–(21). Under the condition of ZNLF, the averaged velocity of gas–liquid mixture is equal to the super4cial gas velocity. Therefore, for turbulent liquid Fow the Taylor bubble velocity is expressed as (Nicklin et al., 1962) 1=2


L − G UTB = 1:2UG + 0:35 gd : (22) L For laminar liquid Fow, the Taylor bubble velocity is expressed as (Collins et al., 1978) 
2:39UG UTB = 2:0UG + 0:347(gd)1=2 ( : (23) (gd)1=2 The 4lm velocity Uf and its thickness ) at a distance x from the top of Taylor bubble can be calculated by numerically solving the following modi4ed equations of Barnea (1990):

 dUf 2fw Uf |Uf | g( L − G ) =− + ; dx L (UTB −Uf ) d(UTB −ULLS )(1 −LS ) (24) ) − d


2 (UTB − ULLS )(1 − LS ) ) ; = d 4(UTB − Uf )

Uf = ULLS

and

)=d

for x = 0

(25) (26)

where the liquid velocity in liquid slug for the present case ULLS is given by 1=4

LS g( L − G ), ULLS = UG − 1:53 : (27) 2 1 − LS L Sylvester (1987) proposed a correlation for the void fraction in the liquid slug. The correlation is employed to calculate the void fraction for the remaining liquid slug zone in the case of ZNLF: LS =

UG : 0:425 + 2:65UG

(28)

From these equations, the 4lm Reynolds numbers Ref (x) = 4 L |Uf |)=&L along the Taylor bubble and the mean 4lm Reynolds number Ref are obtained. The critical Reynolds number for falling 4lm Fow from laminar to turbulent is assumed to be 1600. The length LfL can then be estimated according to the liquid 4lm Reynolds numbers.

Lw/LLS

5262

J. Yun, Z. Shen / Chemical Engineering Science 58 (2003) 5257 – 5268

1.5

of wake zone to liquid slug zone is a function of gas Froude number, which is given by (Fig. 4)

1.2

LWZ = 0:187ln(FrG ) + 0:864: LLS

0.9

Therefore, the length of slug unit is obtained by LU = LTB + LLS and the length of the remaining liquid slug is given by LRLSZ = LLS − LWZ .

0.6 experimental data fitting data

0.3 0.0 0.0

0.6

1.2

1.8

2.4

(32)

3. Experiments

3.0

FrG Fig. 4. The length ratio of wake zone to liquid slug zone as a function of gas Froude number. The 4tting curve is represented by Eq. (32).

The length of Taylor bubble LTB can be determined by solving Eq. (24) until the following mass balance equation over a slug unit (deduced from Barnea (1990) under the condition of ZNLF) is satis4ed 1=4

LS g( L − G ), UG = 1:53 1 − LS 2L
  LTB UTB − ULLS UTB LTB − + dx : (29) Lu UTB − Uf 0

To validate the model, experiments of caprolactam layer growth were carried out in laboratory scale. The experimental apparatus is shown schematically in Fig. 5. A vertical stainless steel tube with diameter 19 mm and length 2500 mm (e:ective length of 2200 mm for solidi4cation) was used to simulate one of crystallization tubes in a bubble column crystallizer. Its upside is connected with a 100 × 200 × 400 mm3 rectangle gas-melt separator and its bottom is connected with a container section to assure the constancy of melt temperature. Temperature of the tube wall was controlled by circulating water through the jacket outside. The inner diameter of the jacket is 49 mm. Temperature of circulating water is maintained by several thermostat baths. Taylor bubbles were produced by injecting an inert gas into the tube through a 5 mm nose located at its bottom.

Then, the length Lft is determined from the relationship Lft = LTB − LfL . The liquid slug length was found to depend on several variables such as gas and liquid velocities, tube diameter and physical properties of Fuids (Sylvester, 1987). There is a broad scatter in the measured values of liquid slug length (Fernandes et al., 1983; Dukler et al., 1985; Orell and Rembrand, 1986; Sylvester, 1987). Zhang et al. (1991) suggested an empirical equation for the liquid slug length: LLS = [4:0 + 0:0526(Refnj )1=2 ]; d

(30)

where Reynolds number Refnj is based on the 4lm velocity at the end of Taylor bubble: Refnj = L (UTB − Ufn )d=&L :

(31)

For this case, Reynolds number de4ned in Eq. (31) can be obtained by solving Eq. (24) and the liquid slug length can then be estimated by Eq. (30). The length of the wake zone was found to be a function of the dimensionless parameter N (Campos and Guedes de Carvalho, 1988a,b; Pinto and Campos, 1996) for gas slugs rising through the vertical liquid column. In turbulent liquid Fowing, the length ratio of wake zone to tube diameter was about 5 (Pinto et al., 1998). Our previous work (Yun and Shen, 2002; Yun et al., 2002) suggested that the length ratio

Fig. 5. Schematic diagram of experimental apparatus: (1) crystallization tube, (2) jacket, (3) separator, (4) DC ampli4er, (5) A/D transformer, (6) computer, (7) thermostatic bath containing warming water, (8) thermostatic bath containing cooling water, (9) thermocouples, (10) rotameters, (11) valve, (12) pressure gauge and (13) gas supply.

J. Yun, Z. Shen / Chemical Engineering Science 58 (2003) 5257 – 5268

Copper-constantan thermocouples were used to measure the wall temperature of the crystallization tube (mounted in the tube wall), the bulk temperature of circulating water in the jacket and the temperature of melt bulk. The voltage signals from these thermocouples were simultaneously ampli4ed by a linear DC ampli4er, fed to a 32-channel A/D transformer and recorded by a PC. For a given super4cial gas velocity, measurements of the solid layer growth thickness increasing with time were conducted as follows: 1. The caprolactam melt was put into the crystallizer and nitrogen gas was injected continuously through the gas-inlet nozzle into the melt to produce gas slugs. The mixture was pre-heated to 345 –347 K by circulating warm water in the jacket. 2. Cooling water was injected into the jacket of the crystallization tube and circulated by a centrifugal pump with constant rate of 197 l=h to ensure the constancy of the desired wall temperature. The solid layer was then grown on the inner surface of the tube. 3. After the desired time for layer growth was attained, the gas injection was stopped. The layer growth was terminated suddenly draining the melt from the bottom of the crystallizer. The solid layer was re-melted, poured out, weighted and analyzed. Repeating these steps, the solid layer growth data at different growing times were then obtained. The layer of caprolactam was found to be so compact that its porosity can be neglected (Chianese and Santilli, 1998). Therefore, the mean thickness of layer was determined by measuring the produced mass (Chianese and Santilli, 1998; Ming and Shen, 2000a), which gives 
1=2  4V s = 0:5 d − d2 − ; (33) 1Le

5263

where V is the layer volume calculated by the ratio of its mass to its density at the mean temperature of solid layer and Le is the e:ective tube length for layer growth. 4. Results and discussion Variations of the thickness of caprolactam solid layer versus time at super4cial gas velocities of 0.04 and 0:08 m s−1 were obtained experimentally and predicted by the model. From the results calculated by the slug Fow model, laminar Fow covers the entire falling 4lm region at super4cial gas velocities of 0.04 and 0:08 m s−1 . Therefore, in the model, the length of laminar falling 4lm was set to be equal to Taylor bubble length and only three zones (laminar falling 4lm zone, wake zone and the remaining liquid slug zone) existed under the present conditions. Therefore, h1 , h3 and h4 were used to calculate the solid layer–melt convective heat transfer coe6cient at the corresponding zones. Physical properties of caprolactam used in the calculation are as follows: Tm = 342:5 K, CpL = 2117 J kg−1 K −1 , kL = 0:129 W m−1 K −1 , L = 1027 kg m−3 , &L = 0:0123 Pa s, ,L = 0:038 N m−1 , = 1:089 × 105 J kg−1 for melt (at 343:2 K) and s = 1096 kg m−3 , Cps = 1420 J kg−1 K −1 , ks = 0:163 W m−1 K −1 for solid layer. The average temperatures of melt bulk and tube wall are Tb = 344:7 K, T0 = 334:5 K at UG =0:04 m s−1 and Tb =344:9 K, T0 =336:9 K at UG = 0:08 m s−1 . 4.1. Solid layer growth rate Fig. 6(a) and (b) show typical trends of predicted and experimental solid layer thickness versus time. The solid layer grows quite rapidly at the beginning, then its growth rate decreases until its thickness tends to a constant value. From the results, the agreement between predicted and experimental data is satis4ed except the layer growing rate is

0.006

0.004

0.003

s (m)

s (m)

0.004

prediction, by second-degree polynomial form of Ts prediction, by linear form of Ts experimental data

0.002

0.000 (a)

0

1000

2000 t (s)

3000

0.002

0.001

prediction experimental data

0.000

4000

0 (b)

500

1000

1500

2000

t (s)

Fig. 6. Comparison between predicted and experimental results for layer thickness against time (a) UG = 0:04 m s−1 (Tm = 342:5 K, Tb = 344:7 K, T0 = 334:5 K) and (b) UG = 0:08 m s−1 (Tm = 342:5 K, Tb = 344:9 K, T0 = 336:9 K).

5264

J. Yun, Z. Shen / Chemical Engineering Science 58 (2003) 5257 – 5268

overestimated slightly by the model at the beginning of growth. This is mainly due to the di:erence between the actual wall temperature in the experiment and the wall temperature used in the model at this stage. Fig. 7 shows the experimental data of the wall temperature and the melt bulk temperature with run time. From this 4gure, the actual melt

bulk temperature is nearly constant in the growing procedure, but the wall temperature decreases quite quickly from the initial values of 347–348 K to 334 –335 K at the beginning of the growth. Since the actual wall temperature in this period is higher than the constant value T0 set in the model, the experimental layer growth rate is lower than that predicted by the model. Furthermore, during the period for the wall temperature decreasing from the initial value to Tm , no solid layer grows on the tube wall. Therefore, this time interval was ignored in the calculation of the solidi4cation time. In the model, the wall temperature T0 is set as the averaged value obtained in experiments. Fig. 6(b) shows e:ects of uncertainty of T0 on the prediction for UG = 0:08 m s−1 . The solid line in the 4gure represents the predicted results using averaged value of T0 , while the dash lines represent the prediction by maximum and minimum of T0 obtained in di:erent experimental runs respectively. The maximal deviation between these predictions is about 12–15%. In Fig. 6(a), the prediction of layer growth using linear pro4le of temperature across the solid layer, as that used by Ming and Shen (2000a), is compared with the prediction based on the present second-degree polynomial form and with the experimental data simultaneously. In the case that the time is small and the solid layer is thin, a linear form of

360

T (K)

350

Tb

340 T0

330

320

0

400

800

1200

1600

2000

t (s) Fig. 7. Measured temperatures of the tube wall and the melt bulk at UG = 0:04 m s−1 .

0.00014

0.000788

0.00013

WZ

TB

RLSZ

0.000786

TB

WZ

48.30

48.45

RLSZ

s (m)

s (m)

0.00012 0.00011 0.00010

0.5

(a)

0.6

0.7

0.8

0.9

0.000778 48.15

1.0 (b)

t (s)

TB

WZ

48.75

t (s)

0.0025314

RLSZ

s (m)

s (m) (c)

48.60

0.0025320

0.001758

0.001757

0.000782 0.000780

0.00009 0.00008 0.4

0.000784

0.001756

TB

WZ

RLSZ

0.0025308

0.001755

0.0025302

0.001754 340.3 340.4 340.5 340.6 340.7 340.8 340.9

0.0025296 1216.7 1216.8 1216.9 1217.0 1217.1 1217.2 1217.3

t (s)

(d)

t (s)

Fig. 8. The predicted solid layer growth in (a) the second slug unit, (b) the 100th slug unit, (c) the 700th slug unit and (d) the 2500th slug unit at UG = 0:08 m s−1 . TB: the falling 4lm zone; WZ: the wake zone; RLSZ: the remaining liquid slug zone (Tm = 342:5 K, Tb = 344:9 K, T0 = 336:9 K).

J. Yun, Z. Shen / Chemical Engineering Science 58 (2003) 5257 – 5268

2000

15000 WZ

TB

RLSZ

qs , qc or qb (W m )

-2

-2

qs , qc or qb (W m )

qs

9000 qc

6000 qb

3000

0.5

0.6

0.7

0.8

0.9

WZ

qc qb

400

48.30

48.45

48.60

48.75

t (s)

800

TB

1000

RLSZ

WZ

RLSZ

-2

qs , qc or qb (W m )

-2

800

1200 TB

qs , qc or qb (W m )

qs

(b)

1000

qs

600 qb

400 qc

800 qb

600

t (s)

qs

400 200 0

qc

-200 1216.7 1216.8 1216.9 1217.0 1217.1 1217.2 1217.3

0 340.3 340.4 340.5 340.6 340.7 340.8 340.9 (c)

RLSZ

1200

0 48.15

1.0

t (s)

(a)

200

WZ

TB

1600

12000

0 0.4

5265

(d)

t (s)

Fig. 9. Variations of heat Fuxes qs , qc and qb at the solid layer–melt interface: (a) the second slug unit, (b) the 100th slug unit, (c) the 700th slug unit and (d) the 2500th slug unit at UG = 0:08 m s−1 . TB: the falling 4lm zone; WZ: the wake zone; RLSZ: the remaining liquid slug zone (Tm = 342:5 K, Tb = 344:9 K, T0 = 336:9 K).

Ts in the model gives a reasonable prediction compared with experimental data. However, with the increase of time, the deviation between the prediction based on the linear form of Ts and the experimental data increases. At t = 3400 s, the relative deviation exceeds 30%. The results show that the second-degree polynomial form of Ts in the model gives a more reasonable prediction than that by a linear one. 4.2. The solid layer growing characteristics in di:erent zones of the slug unit The model was used to predict the solid layer growing characteristics at the beginning, the middle and the residual growing stages. From an overall view of the solidifying duration, the layer grows continuously with time until its thickness attains to a constant value. However, from a local view in a slug unit, di:erent characteristics exist for layer growing in the falling 4lm zone, the wake zone and the remaining liquid slug zone when a slug unit passes through these stages, as shown in Fig. 8. By calculating the layer growth rates for these zones at different growing stages, the following characteristics were revealed.

At the beginning stage of growing, the solid layer increases rapidly with increase in time due to the thin solid layer. Fig. 8(a) shows the calculated solid layer growth in the process of the second slug unit passing through at UG = 0:08 m s−1 . The result shows that the layer growth rate in the falling 4lm zone is almost the same as that in the other two zones. Fig. 9(a) shows variations of the heat Fux removed out of the solid layer–melt interface across the solid layer by conduction qs (= − ks (dTs =dr)r=R−s(t) ), the convective heat Fux from melt bulk to the interface, qb (=hL (Tb − Tm )) and the released or absorbed latent heat Fux at the interface, qc (= s [ + CpL (Tb − Tm )] ds(t)=dt), by taking into account the cooling of the melt in these zones. At this stage, the layer is relatively thin and its heat resistance is low, which causes a relatively high temperature gradient within solid layer and corresponding high heat Fux from the layer to the cooling water outside the crystallization tube, as shown in Fig. 9(a). Therefore, high growth rates are expected everywhere, falling 4lm zone, wake zone and the remaining liquid slug zone. At the middle growth stage, however, the solid layer growth rate is lower than that at the beginning stage.

5266

J. Yun, Z. Shen / Chemical Engineering Science 58 (2003) 5257 – 5268

Furthermore, the growth rate in the falling 4lm zone is lower than that in the other two zones of the slug unit. Fig. 8(b) shows the layer growth for the 100th slug unit passing through at UG = 0:08 m s−1 . Fig. 9(b) shows the corresponding variations of the heat Fuxes qs , qc and qb at the solid layer–melt interface. In this case, the conducted heat Fux out of the crystal–melt interface across the solid layer qs is decreased due to the increasing of the solid layer thickness, while the heat Fux from the melt bulk to the crystal–melt interface qb in the falling 4lm zone is higher either than that in the wake region or in the remaining liquid slug zone, which causes a corresponding lower layer growth rate. At the residual growth stage, the total layer growth rate is almost zero and the thickness of the solid layer attains a constant value. Figs. 8(c) and (d) show the predicted layer growth for the 700th and 2500th slug units passing through at UG = 0:08 m s−1 , respectively. Figs. 9(c) and (d) show the corresponding variations of qs , qc and qb at the solid layer–melt interface. In this case, the layer thickness is kept constant or decreases in the falling 4lm zone, while it increases slightly in the other two zones with the increase in time. Actually, when the balance between the heat Fux removed through the solid layer and transferred from melt to the interface arrives in the falling liquid 4lm zone, the layer growth is stopped and its thickness becomes constant. However, this heat balance does not arrive in the other two zones at the same time. When the liquid slug passes by, the solid layer still grows which then destroys the heat balance established in the former falling 4lm zone. Consequently, as the next falling 4lm comes, the heat Fux from the melt to the interface is higher than that removed out of the interface through the solid layer; thus, the exceeded heat is absorbed by the solid layer which causes the re-melting of the layer. In this case, qb ¿ qs and then a negative qc exists as shown in Fig. 9(d). Therefore, a negative value of the layer growth rate was observed in this zone. The processes of re-melting in the falling liquid 4lm zone and solidi4cation in the other two zones continues until a new balance is established. When the rate of re-melting is equal to that of solidi4cation, the layer growth in each slug units is almost stopped. 5. Conclusions The proposed model can be used to describe the solid layer growth in a crystallization tube of bubble column crystallizer. There was good agreement between predicted and experimental data of caprolactam layer growth. The layer grows with the increase of time until its thickness reaches an almost constant value. From the view in a slug unit, different characteristics exist for layer growing in the falling 4lm zone, the wake zone and the remaining liquid slug zone when a slug unit passes through the beginning, the middle and the residual growing stages.

Notation A; B b; c; f CpL Cps CfL ; Cft d db de D Frb FrG g h hL hLLS kL ; ks K L Le LfL Lft LWZ LRLSZ LU N Nu PrL qb qs qc r R Re Ref Reb Red s ScL Sh

terms de4ned in Eqs. (12) and (13) terms de4ned in Eqs. (8), (9) and (10) speci4c heat of melt and solid layer, W kg−1 K −1 modi4ed factors in Eqs. (16) and (17) inside tube diameter, m bubble diameter in liquid slug, m equivalent hydraulic diameter of falling liquid 4lm (=4)(1 − )=d)), m di:usion coe6cient, m2 s−1 gas Froude √ number based on bubble diameter (=UG = gdb ), dimensionless gas Froude number based on tube diameter √ (=UG = gd), dimensionless acceleration due to gravity, m s−2 convective heat transfer coe6cient, W m−2 K −1 convective heat transfer coe6cients at the solid layer–melt interface, W m−2 K −1 convective heat transfer coe6cient de4ned in Eq. (20) or (21), W m−2 K −1 heat conductivity coe6cients of melt (or liquid) and solid layer, W m−1 K −1 mass transfer coe6cient, m s−1 tube length, m e:ective tube length for solid layer growth, m length of the laminar falling 4lm zone, m length of the turbulent falling 4lm zone, m length of the wake zone, m length of the remaining liquid slug zone, m length of a slug unit, m dimensionless parameter Nusselt number (=hd=kL ), dimensionless Prandtl number of liquid (=CpL &L =kL ), dimensionless convective heat Fux from melt bulk to the solid layer–melt interface, W m−2 conducted heat Fux out of the solid layer–melt interface, W m−2 released or absorbed latent heat Fux at the solid layer–melt interface, W m−2 radial coordinate, m inside tube radius, m Reynolds number (=U d=&), dimensionless Reynolds number of falling 4lm(=4Uf L )=&L ), dimensionless Reynolds number based on bubble size (=UG L db =&L ), dimensionless Reynolds number of liquid slug (=UG L d=&), dimensionless layer thickness, m Schmidt number of liquid (=&L = L D), dimensionless Sherwood number (=Kd=D), dimensionless

J. Yun, Z. Shen / Chemical Engineering Science 58 (2003) 5257 – 5268

St t Tm Tb Ts To U UG ; UL V

Stanton number (=h=CpL L UG ), dimensionless time, s temperature of melting, K temperature of melt bulk, K temperature in solid layer, K wall temperature, K velocity, m s−1 super4cial gas and liquid velocities, m s−1 volume of solid layer, m3

Greek letters  LS

 ) & ,

heat di:usivity, m2 s−1 void fraction, dimensionless heat of solidi4cation, J kg−1 latent heat taking into account of the cooling of melt, J kg−1 thickness of falling liquid 4lm, m viscosity, Pa s density, kg m−3 surface tension of liquid or melt, N m−1

Subscripts b f fL ft G L LLS LS s TB 1 2 3 4

bubble falling liquid 4lm laminar liquid 4lm turbulent liquid 4lm gas liquid or melt liquid in liquid slug liquid slug crystal or solid layer Taylor bubble the laminar falling 4lm zone the turbulent falling 4lm zone the wake zone the remaining liquid slug zone

Acknowledgements The support of the National Natural Science Foundation of China (grant No. 50176007) is gratefully acknowledged.

References Abdul-Majeed, G., Al-Mashat, A.M., 2000. A mechanistic model for vertical and inclined two-phase slug Fow. Journal of Petroleum Science and Engineering 27, 59–67. An, H.J., Langlinais, J.P., Scott, S.L., 2000. E:ect of density and viscosity in vertical zero net liquid Fow. ASME Journal of Energy Resources Technology 122, 49–55. Barnea, D., 1990. E:ect of bubble shape on pressure drop calculations in vertical slug Fow. International Journal of Multiphase Flow 16, 79–89.

5267

Campos, J.B.L.M., Guedes de Carvalho, J.R.F., 1988a. An experimental study of the wake of gas slugs rising in liquids. Journal of Fluid Mechanics 196, 27–37. Campos, J.B.L.M., Guedes de Carvalho, J.R.F., 1988b. Mixing induced by air slugs rising in narrow columns of water. Chemical Engineering Science 43, 1569–1582. Chianese, A., Santilli, N., 1998. Modelling of the solid layer growth from melt crystallization the integral formulation approach. Chemical Engineering Science 53, 107–111. Collins, R., De Moraes, F.F., Davidson, J.F., Harrison, D., 1978. The motion of large gas bubble rising through liquid Fowing in a tube. Journal of Fluid Mechanics 89, 497–514. Coulson, J.M., Richardson, J.F., Backhurst, J.R., Harker, J.H., 1990. Chemical Engineering, Vol. 1, 4th Edition. Pergamon Press, Oxford, p. 491. Deckwer, W.D., 1980. On the mechanism of heat transfer in bubble column reactors. Chemical Engineering Science 35, 1341–1346. Dukler, A.E., Maron, D.M., Brauner, N., 1985. A physical model for predicting the minimum stable slug length. Chemical Engineering Science 40, 1379–1385. Fernandes, R.C., Semiat, R., Dukler, A.E., 1983. Hydrodynamic model for gas-liquid slug Fow in vertical tubes. A.I.Ch.E. Journal 29, 981–989. Fukui, K., Maeda, K., 2000. Melt layer solidi4cation of fatty acids in a rectangular cell. Journal Chemical Physics 112, 1554–1559. Ghosh, R., Cui, Z.F., 1999. Mass transfer in gas-sparged ultra4ltration: upward slug Fow in tubular membranes. Journal of Membrane Science 162, 91–102. Goodman, T.R., 1958. The heat-balance integral and its application to problems involving a change of phase. Transaction of the ASME 80, 335–342. Guardani, R., Neiro, S.M.S., BQulau, H., Ulrich, J., 2001. Experimental comparison and simulation of static and dynamic solid layer melt crystallization. Chemical Engineering Science 56, 2371–2379. Kim, D., Ghajar, A.J., Dougherty, R.L., Ryali, V.K., 1999. Comparison of 20 two-phase heat transfer correlations with seven sets of experimental data, including Fow pattern and tube inclination e:ects. Heat Transfer Engineering 20, 15–40. Liu, L., Scott, S.L., 2001. Investigation on liquid holdup in vertical zero net-liquid Fow. Chinese Journal of Chemical Engineering 9, 284–290. Meyer, D.W., 1990. Using crystallization for organic separations. Chemical Processing 53, 50–60. Ming, P.W., Shen, Z.Q., 2000a. Solid layer growth of bubble column crystallizer. Journal of Chemical Industry and Engineering (China) 51, 481–484 (in Chinese). Ming, P.W., Shen, Z.Q., 2000b. Wall transfer of a gas plug rising in a vertical short tube. Journal of Chemical Industry and Engineering (China) 51, 303–307 (in Chinese). Nicklin, D.J., Wilkes, M.A., Davidson, J.F., 1962. Two-phase Fow in vertical tubes. Transactions of the Institution of Chemical Engineers 40, 61–68. Orell, A., Rembrand, R., 1986. A model for gas-liquid slug Fow in a vertical tube. Industrial Engineering and Chemical Fundamentals 25, 196–206. Parisi, M., Chianese, A., 2001. The solid layer growth from a well-mixed melt. Chemical Engineering Science 56, 4245–4256. Pinto, A.M.F.R., Campos, J.B.L.M., 1996. Coalescence of two gas slugs rising in a vertical column of liquid. Chemical Engineering Science 51, 45–54. Pinto, A.M.F.R., Coelho Pinheiro, M.N., Campos, J.B.L.M., 1998. Coalescence of two gas slugs rising in a co-current Fowing liquid in vertical tubes. Chemical Engineering Science 53, 2973–2983. Shi, S.Z., Zhao, Z.Y., Guo, C.T., 1994. Study on crystallization in the slug Fow crystallizer. Chemical Engineering (China) 22, 52–58 (in Chinese). Stolzenberg, K., Koch, K.H., Marrett, R., Ruetgerswerke, A.G., 1983. Device and method for the mass separation of a liquid mixture by fractional crystallization. DE Patent, 3203818.

5268

J. Yun, Z. Shen / Chemical Engineering Science 58 (2003) 5257 – 5268

Sylvester, N.D., 1987. A mechanistic model for two-phase vertical slug Fow in pipes. ASME Journal of Energy Resources Technology 109, 206–213. Taitel, Y., Barnea, D., Dukler, A.E., 1980. Modeling Fow pattern transitions for steady upward gas-liquid Fow in vertical tubes. A.I.Ch.E. Journal 26, 345–354. Ulrich, J., Bierwirth, J., Henning, S., 1996. Solid layer melt crystallization. Separation and Puri4cation Methods 25, 1–45. Wang, H.C., Ma, L.Q., Dong, X.Q., Yang, X.X., Tan, Q., 1998. New method for re4ning naphthalene by bubble column crystallization. Journal of Chemical Industry and Engineering (China) 49, 495–499 (in Chinese). Wellingho:, G., Wintermantel, K., 1994. Melt crystallization: theoretical premises and technical limitations. International Chemical Engineering 34, 17–27.

Wynn, N.P., 1992. Separate organics by melt crystallization. Chemical Engineering Progress 88, 52–60. Yang, J.H., Wang, G.M., Zhang, C.Y., Li, Y.H., 1999. Separation of p-chloro-toluene from mixed chloro-toluene using slug Fow crystallizer. Petrochemical Technology 28, 380–384 (in Chinese). Yun, J.X., Shen, Z.Q., 2002. An experimental study of wall-liquid transfer characteristics for gas slugs rising in a vertical tube. Journal of Chemical Engineering of Chinese Universities 16, 275–280 (in Chinese). Yun, J.X., Shen, Z.Q., Ming, P.W., 2002. Wall-liquid mass transfer for Taylor bubbles rising through liquid column in a vertical tube. Chinese Journal of Chemical Engineering 10, 404–410. Zhang, H.Q., Lu, M.J., Lian, C.G., Zhu, S.L., 1991. The hydrodynamic model for gas liquid slug Fow in vertical tubes. Journal of Chemical Engineering of Chinese Universities 5, 102–111 (in Chinese).