Conjugated mass transfer in an inclined thermal-diffusion column for heavy water enrichment with plate aspect ratio variations

Progress in Nuclear Energy 66 (2013) 90e98

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Conjugated mass transfer in an inclined thermal-diffusion column for heavy water enrichment with plate aspect ratio variations Chii-Dong Ho*, Ho-Ming Yeh, Tung-Wen Cheng, Chiung-Jeng Wang Energy and Opto-Electronic Materials Research Center, Department of Chemical and Materials Engineering, Tamkang University, Tamsui, New Taipei 251, Taiwan

a r t i c l e i n f o

a b s t r a c t

Article history: Received 24 February 2012 Received in revised form 21 February 2013 Accepted 5 March 2013

A two-dimensional mathematical model was developed theoretically to predict the degree of separation of heavy water in an inclined flat-plate thermal-diffusion column. An orthogonal expansion technique for solving separation problems of heavy water enrichment under plate aspect ratio and inclination angle variations was carried out analytically. The effects of plate aspect ratio and inclination angle on the degree of separation have been investigated with a consideration of a fixed operating expense. Considerable improvements in device performance are obtained if the thermal-diffusion columns are inclined at the optimal angles as compared to the vertical orientation. Further improvement can be achieved if the flow-rate fractions of top and bottom products are suitably adjusted. Ó 2013 Elsevier Ltd. All rights reserved.

Keywords: Heavy water enrichment Thermal-diffusion column Graetz problems Inclination angle Plate aspect ratio

1. Introduction The classical thermal-diffusion column (Clusius and Dickel, 1938, 1939) investigated the cascading effect in convective currents for separating liquid or gas mixtures that are difficult to separate using traditional processes such as distillation or extraction. A complete theoretical presentation of the cascading effect analogized to the multistage effect in countercurrent extraction resulting in relatively large separation efficiency was derived by Jones and Furry (1946). Furry et al. (1939) applied the convective current concept in designing a practical separation system, and Yeh and Yang (1985a) and Yeh et al. (2002) studied theoretically and experimentally the separation efficiency improvement in heavy water enrichment. However, the convective currents in the ClusiuseDickel column are accompanied with the disadvantageous remixing effect across the column. Numerous researchers proposed investigations that emphasized either remixing effect suppression or cascading effect enhancement in aiming to modify the convective flow patterns such as inclined column (Washall and Molpolder, 1962; Chueh and Yeh, 1967), packed columns (Lorenz and Emery, 1959), rotary wired columns (Treacy and Rich, 1955; Yeh and Ho, 1975), permeable barrier columns (Ho and Guo, 2005) and impermeable barrier columns (Tsai and Yeh, 1986). The undesirable remixing effect could be * Corresponding author. Fax: þ886 2 26209887. E-mail address: [email protected] (C.-D. Ho). 0149-1970/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.pnucene.2013.03.007

reduced by tilting the thermal-diffusion column at various angles. Applications of titled devices with a proper adjustment on the intensity of convective currents have led to improved performance in separation processes. The mathematical model for thermal-diffusion was first proposed by Enskog (1919) using the kinetic theory of a mixture of gases. The theoretical formulations developed theoretically into a two-dimensional SturmeLiouville problem, which has been solved with the use of orthogonal expansion techniques (Singh, 1958; Brown, 1960; Nunge and Gill, 1966; Tsai and Yeh, 1985; Ho et al., 1998; Ebadian and Zhang, 1989). Since the infinite number of eigenvalues appearing in the solution of the problem converges rapidly, it is possible to solve the thermal-diffusion problem in a more straightforward bearing by utilizing the method of eigenfunction expansion technique and eigenfunctions expanding in terms of an extended power series. The convergence of the eigenfunction expansions used in the present study to solve them is briefly considered. The purpose of this work is to obtain the analytical solutions of simultaneous effects of inclined-angle and plate aspect ratio on the improvement of separation efficiency of heavy water enrichment in flat-plate thermal-diffusion column under consideration of the total expense kept unchanged. The device performance with flow-rate fraction of top product as an operating parameter is also discussed. A more direct method is described herein for determining that the flow-rate fraction of top product and plate aspect ratio for inclined device, which allows the specification set by the designer, of values directly dependent

C.-D. Ho et al. / Progress in Nuclear Energy 66 (2013) 90e98

Nomenclature a b B C b CC B D3 dmn emn Fm Gm g h H Jx-OD Jx-TD Jz-OD Keq Iq Iq* L m n r S Sm

constant defined by Eq. (21) constant defined by Eq. (22) column width, cm fraction mass concentration of heavy water in H2OeHDOeD2O system pseudo-product form of concentration for D2O ordinary diffusion coefficient, cm2/s ordinary diffusion coefficient of D2O, cm2/s constant defined by Eqs. (19a) and (19b) constant defined by Eqs. (20a) and (20b) eigenfunction function defined during the use of orthogonal expansion method gravitational acceleration, cm/s2 thermal-diffusion effect, 1/cm dimensionless of thermal-diffusion effect mass flux of heavy water in the x-direction due to ordinary diffusion, g/cm2s mass flux of heavy water in the x-direction due to thermal diffusion, g/cm2s mass flux of heavy water in the z-direction due to ordinary diffusion, g/cm2s mass-fraction equilibrium constant of H2OeHDOeD2O system performance improvement with inclination angle variations defined in Eq. (26) performance improvement with optimal inclination angle variations defined in Eq. (27) column length, cm number of eigenvalues terms of polynomial function flow-rate fraction of top product surface area, cm2 expansion coefficient

on separation efficiency, to be met with the fixed operating expense. 2. Mathematical formulations Fig. 1 illustrates the convective flows and fluxes prevailing in a continuous-flow thermal-diffusion column of the thickness W between hot and cold plates filled with water isotopes under variations of the top product to feed ratio (sT ¼ rsF). The thermogravitational column composed of length L is the feed introduced at the center with the mass flow rate sF and concentration CF, and top and bottom products are withdrawn at flow rates of rsF and (1  r)sF, respectively. The combination of the horizontal thermal-diffusion effect and vertical natural convective currents results in a countercurrent flow to create a resultant concentration gradient between the two ends of the column. Since the space between the hot and cold plates of the column is so small, we may assume that the convective flow produced by the density gradient is laminar and that the temperature distribution is determined by conduction in the x-direction only. After the following assumptions are made: the heat transfer in the space between the hot and cold plates is by heat conduction only; the fluid flow is purely laminar in both sections; the influences of ordinary and thermal diffusions, end effects, and inertia terms on the velocity are neglected; the ordinary diffusion in the vertical

T T U W x z

91

mean absolute temperature, K arithmetic mean value of T of hot wall and cold wall, K dimensionless velocity profile defined by Eq. (2) plate spacing of columns, cm coordinate in the horizontal direction, cm coordinate in the vertical direction, cm

Greek letters reduced thermal-diffusion constant for D2O in H2O eHDOeD2O system, <0 atern reduced thermal-diffusion constant for D2O and HDO 32 in ternary system bT thermal expansion coefficient, defined as vr=vT, g/cm3 K g defined by Eq. (3) D degree of separation in the whole column D* maximum degree of separation with optimal inclination angle in the whole column DT difference in temperature of hot and cold plates, K 2 dimensionless feed position defined by Eq. (3) xh plate aspect ratio coordinate in the horizontal direction l eigenvalue m absolute viscosity of fluid, g/cm s r mass density of fluid, g/cm3 s mass flow rate, g/s q inclination angle, degree q* optimal inclination angle, degree

a

Subscripts B in the bottom product stream c at cold plate e in the enriching section F in the feed stream h at hot plate s in the stripping section T in the top product stream

direction and the bulk flow in the horizontal direction are neglected; the external temperature field is applied in the horizontal direction to cause mass fluxes. The mass fluxes due to the thermal and ordinary diffusion are too small to affect the velocity profiles, and the convection velocity is assumed in the z-direction only. The hydro-dynamical equation for steady laminar flow is determined by applying the NaviereStokes relations in obtaining a cubic polynomial. Therefore, the physical reasoning is the same in both sections for assuming the same form in both sections accordingly. The mass balance equations and the velocity distributions for the enriching and stripping sections are obtained as follows:

Ui ðhÞ

vCi v2 Ci vHðhÞ ¼  ; 2 v vh vh2

i ¼ e; s

Ui ðhÞ ¼ u½hðh  1Þ ðh  gi Þ;

i ¼ e; s

in which

b g cos qW 4 DT 2z x ;h ¼ ;u ¼ T ; HðhÞ ¼ WhðxÞ; L 3mD3 L W atern Ci Cb i dT hðxÞ ¼ 32 dx T

2¼

(1) (2)

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gi ¼ 0:5 þ

36msi BW 3 rbT ðg cos qÞDT

(3)

The velocity sign will change at somewhere in the enriching section due to the natural convection associated with the thermaldiffusion effect, say h ¼ gi, and thus, the fluid velocity is zero at h ¼ gi , i.e. Ui j ¼ 0. Since the velocity distributions in two subchannels change sign over the interval in question, Eq. (1) is the special case of SturmeLiouville problem for which there may exist both positive and negative sets of real eigenvalues of having limit points þN and N, respectively. The pseudo concentration prodb in the Eq. (3) is defined as follows (Jones and Furry, 1946) ucts Ci C i

  b ¼ 0:10526C  0:10526  0:03981Keq C 2 Ci C i i i  1=2   Keq 3=2 1=2 Ci  0:07962Ci Keq 1  1  4

(4)

expansion technique with the eigenfunctions expanding in terms of an extended power series. An orthogonality relation for the sets of positive and negative sets of eigenfunctions associated with eigenvalues is required. By following the same mathematical treatment performed in the previous work (Ho et al., 2011), except the thermal-diffusion column with plate aspect ratio and inclination angle under consideration of a fixed plate surface area. A fixed charge and an operating cost are the main expenditure in making a separation by thermal diffusion. The fixed charge is roughly proportional to the equipment cost, say the working dimensions (heat transfer area), while the operating cost is taken into account the heating sources. Since the parallel flat plates are rectangular, the surface area S is the product of plate length L and plate width B. The plate aspect ratio was defined as

x ¼ L=B

The boundary conditions for solving Eq. (1) are

Then 1=2

vCe vCs ¼ ¼ HðhÞ vh vh

L ¼ ðSxÞ

at h ¼ 0

at h ¼ 1

(6)

Cs ¼ C T

at 2 ¼ 1

(7)

Ce ¼ CB

at 2 ¼ 1

(8)

Eqs. (5) and (6) denote the continuity of the mass fluxes on both hot and cold plates while Eqs. (7) and (8) refer to the imposed mixing zone condition at the column end. The analytical

1=2

(

le;m

h N Ss;m F 0 ðgs Þ els;m X s;m m¼1

(10) (11)

In obtaining the device performance for fixed expense with specified temperature gradient (DT/W) under the constant plate surface area, the degrees of separation, De and Ds, in the enriching and stripping sections, respectively, can be obtained solving Eqs. (1) and (2) to satisfy the nonhomogeneous boundary conditions, Eqs. (5)e(8), by superposition associated with the SturmeLiouville orthogonality properties along the h direction. A calculating procedure with the general expression for the expansion coefficients was carried out for evaluating the expansion coefficients in Eqs. (15) and (16) and the complete set of eigenfunctions associated with eigenvalues in Eqs. (17) and (18) to satisfy boundary conditions. The difference of the average concentrations of both column ends, De and Ds, in the enriching and stripping sections, respectively, are defined as

31 9 31 2 82 Z 1 Z ge > > > 6 7 > >4 > Ue ðhÞ dh5  4 Ue ðhÞ dh5 > i) > h > > > > ( > > l 0 g 0 e;m N Se;m F < = e g ð Þ 1  e X e e;m m¼1

Ds ¼ CF  CT ¼

¼ 2Le ¼ 2Ls

(5) B ¼ ðS=xÞ

vCe vCs ¼ ¼ HðhÞ vh vh

De ¼ CB  CF ¼

(9)

ls;m

> > > > > > > :

> > > > > > > ;

2

31 9 31 2 82 Z 1 Z gs > > > 6 7 > 4 > > Us ðhÞ dh5  4 Us ðhÞ dh5 > ii) > > > > > > > gs 0 < = 1 > > > > > > > :

solutions to Ce ¼ Ce(h, 2) and Cs ¼ Cs(h, 2) for this two-dimensional mathematical model were obtained using an orthogonal

2

> > > > > > > ;

(12)

(13)

The degree of the separation for the whole column can be obtained by summing Eqs. (12) and (13) as follows:

C.-D. Ho et al. / Progress in Nuclear Energy 66 (2013) 90e98

D ¼ CB  CT ¼ De þ Ds ¼

31 9 31 2 82 Z 1 Z ge > > > 6 7 > 4 > > Ue ðhÞ dh5  4 Ue ðhÞ dh5 > i) > h > > > > ( > > l 0 g 0 e;m N Se;m F < = e g ð Þ 1  e X e;m e m¼1

le;m

h N Ss;m F 0 ðgs Þ els;m X s;m

( þ

m¼1

ls;m

> > > > > > > :



b atern 32 Ce C e DT Fe;m ð1Þ  1 T le;m Fe;m ð1Þ

> > > > > > > :

b atern 32 Cs C s DT Fs;m ð1Þ  1 T ls;m Fs;m ð1Þ

bT g cos qW 4 DT b g cos qW 4 DT ge ; a2 ¼ T ðge þ 1Þ; 1=2 1=2 3mD3 ðSxÞ 3mD3 ðSxÞ b g cos qW 4 DT a3 ¼  T ; an ¼ 0; n  4 1=2 3mD3 ðSxÞ

(15)

(21)

0 ð1Þ vFs;m

(16)

vls;m

z

and

Fe;m ðhÞ ¼

N X

> > > > > > > ;

2

a0 ¼ 0; a1 ¼ 

vle;m



Ss;m ¼

(14)

31 9 31 2 82 Z 1 Z gs > > >4 6 7 > > > Us ðhÞ dh5  4 Us ðhÞ dh5 > i) > > > > > > > g 0 = < s 1



0 ð1Þ vFe;m

> > > > > > > ;

2

where

Se;m ¼

93

de;mn hn ; de;m0 ¼ 1ðselectedÞ; de;m1 ¼ 0

(17)

es;mn hn ; es;m0 ¼ 1ðselectedÞ; es;m1 ¼ 0

(18)

CT,

θ

σT

n¼0

Ls ws Flo



le;m 4n2

þ 2n

a0 de;mð2n1Þ þ a1 de;mð2n2Þ þ / 

þ a2n1 de;m0

de;mð2nÞ ¼

le;m

a0 de;mð2n2Þ þ a1 de;mð2n3Þ þ / 

þ a2n2 de;m0

es;mð2nþ1Þ ¼

ls;m 4n2 þ 2n



b0 es;mð2n1Þ þ b1 es;mð2n2Þ þ / 

þ b2n1 es;m0

es;mð2nÞ ¼

b0 es;mð2n2Þ þ b1 es;mð2n3Þ þ / 4n2  2n  þ b2n2 es;m0

(19a)

x W

C F,

σF

0

J x – OD H

4n2  2n



J z –O

C

de;mð2nþ1Þ ¼

ws Flo

in which

tive vec

n¼0

tive vec Con D

N X

Con

Fs;m ðhÞ ¼

(19b)

J x –TD Le

(20a)

C B,

ls;m

(20b)

σB

Fig. 1. Schematic diagram of a continuous flat-plate thermal-diffusion column with plate aspect ratio and inclination angle variations.

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Table 1 Eigenvalues and expansion coefficients as well as the degree of separation in inclined flat-plate thermal-diffusion columns for CF ¼ 0.381, sF ¼ 0.1 g/h, x ¼ 12.2, r ¼ 0.5 and d ¼ 0.5. 0

0

q

n

le

ls

Se

Ss

Fe (1)

Fs (1)

Fe (ge)

Fs (gs)

D (%)

0

30 35 40 30 35 40 30 35 40 30 35 40 30 35 40

0.501 0.514 0.514 0.571 0.582 0.582 0.870 0.876 0.876 1.959 2.057 2.057 16.92 17.06 17.06

0.501 0.514 0.514 0.571 0.582 0.582 0.870 0.876 0.876 1.959 2.057 2.057 16.92 17.06 17.06

0.087 0.089 0.089 0.080 0.083 0.083 0.064 0.068 0.068 0.040 0.044 0.044 0.013 0.015 0.015

0.088 0.090 0.090 0.082 0.085 0.085 0.065 0.069 0.069 0.041 0.045 0.045 0.014 0.016 0.016

0.971 1.012 1.012 1.001 1.013 1.013 0.966 1.016 1.016 1.005 1.025 1.025 1.052 1.073 1.073

0.948 0.988 0.988 0.943 0.987 0.987 0.934 0.984 0.984 0.936 0.976 0.976 0.909 0.932 0.932

0.021 0.023 0.023 0.023 0.025 0.025 0.029 0.030 0.030 0.043 0.047 0.047 0.124 0.137 0.137

0.021 0.023 0.023 0.023 0.024 0.024 0.029 0.030 0.030 0.043 0.045 0.045 0.120 0.128 0.128

6.957 7.198 7.198 7.058 7.426 7.426 7.776 8.002 8.002 7.567 7.805 7.805 3.002 3.114 3.114

20

40

60

80

bT g cos qW 4 DT b g cos qW 4 DT gs ; b 2 ¼ T ðgs þ 1Þ; 1=2 1=2 3mD3 ðSxÞ 3mD3 ðSxÞ b g cos qW 4 DT b3 ¼  T ; bn ¼ 0; n  4 1=2 3mD3 ðSxÞ b0 ¼ 0; b1 ¼ 

(22) Making material balances around the whole column gives the concentration at the feed position

ðsT þ sB ÞCF ¼ sT CT þ sB CB

(23)

expression of the degree of separation (D) in terms of the inclination angle (q) and flow-rate fraction of top product (r) was obtained mathematically. The separation efficiency improvement Iq is defined as the ratio of improvement in the degree of separation of the column using the optimal inclination angle to that of the device without inclination under the same working dimensions and operating conditions. For the operation with inclination angle the separation efficiency improvement is defined as Eq. (26) by calculating the percentage improvement in performance operating at plate aspect ratio x ¼ 12.2. In other cases, a general form as Eq. (27) can be defined the optimal inclination angle, q*, for maximum heavy water enrichment, where the plate aspect ratio (x) must be specified with the flowrate fraction of top product as a parameter.

The top and bottom product stream concentrations are thus calculated from Eqs. (12)e(14) and (23) to give

CB ¼ CF þ

DðsT =sB Þ ð1 þ ðsT =sB ÞÞ

(24)

Iq ¼

Dðr; qÞ  Dðr; q ¼ 0Þ Dðr; q ¼ 0Þ



D CT ¼ CF  ð1 þ ðsT =sB ÞÞ

(25)

The degree of separation for whole column was obtained by solving Eq. (1) analytically with the use of Eqs. (5)e(8). The theoretical predictions of fraction mass concentration of heavy water in both enriching and stripping sections are expressed explicitly in terms of eigenvalues (le,m and ls,m), expansion coefficients (Se,m and Ss,m) and eigenfunctions (Fe,m and Fs,m). 3. Device performance analysis The device performance is affected by both design parameters of the plate aspect ratio (x) and inclination angle (q) and operating parameters of the mass flow rate (sF) flow-rate fraction of top product (r). The advantageous cascading effect can be strengthened and the undesired remixing effect can be reduced by tilting the ClusiuseDickel thermal-diffusion column, as mentioned by previous investigators (Power and Wilke, 1957). Therefore, an optimal inclination angle should be determined for the maximum degree of separation involving either remixing effect suppression or cascading effect enhancement. The

Iq*

(26)



D r; x; q*  Dðr; x; q ¼ 0Þ ¼ Dðr; x; q ¼ 0Þ

(27)

For a vertical column, q ¼ 0 , the expression for the degree of separation, D(r, q ¼ 0) and D(r, x, q ¼ 0) can be obtained from Eq. (14) with cos q ¼ 1 accordingly.

4. Results and discussion Table 1 shows some calculation results of the eigenvalues and their associated expansion coefficients, as well as the degree of separation for CF ¼ 0.381, sF ¼ 0.1 g/h, d ¼ 0.5 and x ¼ 12.2. The symbol n in Table 1 is the terms of the polynomial functions as shown in Eqs. (17) and (18). The number of the terms, n ¼ 35, of both enriching and stripping sections are considered necessary during the degree of separation calculation due to the rapid convergence, as shown in Table 1.Moreover, two eigenvalues for either enriching or stripping section including the zero eigenvalue, say m ¼ 0 and 1.The mean temperature of the mixture solution ðT ¼ 304 KÞ was used for the physical properties during the calculation procedure. A numerical example with some equipment

C.-D. Ho et al. / Progress in Nuclear Energy 66 (2013) 90e98

95

Table 2 Performance improvement with the inclination angle and flow-rate fraction of top product as parameters for x ¼ 12.2, d ¼ 0.5 and CF ¼ 0.381.

sF (g/h)

Iq (%)

q ¼ 10 0.01 0.04 0.08 0.1 0.2 0.4 0.8

q ¼ 30

q ¼ 50

r ¼ 0.1

r ¼ 0.3

r ¼ 0.5

r ¼ 0.1

r ¼ 0.3

r ¼ 0.5

r ¼ 0.1

r ¼ 0.3

r ¼ 0.5

32.20 10.29 3.78 0.94 0.28 0.19 0.00

30.41 9.52 3.54 0.84 0.28 0.06 0.00

28.11 9.34 3.27 0.80 0.27 0.08 0.00

100.13 35.86 12.13 8.66 1.36 0.36 0.04

95.13 32.61 10.54 7.32 1.30 0.34 0.03

91.45 30.15 8.60 6.87 1.26 0.30 0.01

196.21 56.55 19.06 24.39 2.36 1.27 0.56

178.41 49.14 15.06 16.40 2.26 1.25 0.51

152.68 42.10 11.58 13.77 2.01 1.20 0.40

Table 3 Performance improvement with the optimum inclination angle operation with plate aspect ratio and mass flow rate as parameters for CF ¼ 0.381.

sF (g/h)

Iq* (%)

x ¼ 12.2 0.01 0.04 0.08 0.1 0.2 0.4 0.8

x ¼ 15

x ¼ 20

r ¼ 0.1

r ¼ 0.5

r ¼ 0.9

r ¼ 0.1

r ¼ 0.5

r ¼ 0.9

r ¼ 0.1

r ¼ 0.5

r ¼ 0.9

481.43 161.61 65.10 43.65 20.31 9.36 3.18

357.13 121.20 44.11 13.77 5.38 2.36 0.78

481.11 161.21 64.60 43.20 20.05 9.16 3.02

401.43 136.78 45.13 30.15 14.38 5.36 1.43

281.29 81.61 24.61 9.76 3.28 1.31 0.35

401.05 136.15 45.00 30.01 14.01 5.11 1.38

313.82 100.55 33.26 23.84 10.37 3.76 0.40

117.41 40.54 14.06 3.22 1.87 0.66 0.00

313.28 100.15 33.08 23.57 10.04 3.61 0.38

Fig. 2. Effects of the aspect ratio x and inclination angle q on the eigenvalues in the enriching and stripping sections for CF ¼ 0.381 and r ¼ 0.5.

Fig. 3. Effects of plate aspect ratio x and flow-rate fraction of top product r on the eigenvalues in the enriching and stripping sections for CF ¼ 0.381 and q ¼ 0 .

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C.-D. Ho et al. / Progress in Nuclear Energy 66 (2013) 90e98

parameters and physical properties (Standen, 1978; Yeh and Yang, 1985b) of the mixture for the heavy water enrichment is analyzed:

H0 ¼ 1:473  104 g=s; K 0 ¼ 1:549  103 g$cm=s; Keq ¼ 3:793; m ¼ 0:0126 g=cm$s; B ¼ 10 cm; L ¼ 177 cm; W ¼ 0:04 cm; D ¼ 1:98  106 cm2 =s; r ¼ 1:1 g=cm3 ; Tc ¼ 287:15 K; Th ¼ 320:15 K; CF ¼ 0:381; sF ¼ 0:01e0:8 g=h; x ¼ 1e40; q ¼ 0e80

The results listed in Tables 2 and 3 allow the summary of the following for the improvement of separation efficiency with optimal inclination angle and over that of vertical arrangement, i.e. Iq or Iq* :  The performance improvement is higher for lower feed flow rate.  When the mass flow rate is large enough, the optimal inclination angle is 0, i.e. vertical orientation, and hence, the device performance improvement is zero.  The lower the plate aspect ratio, the greater the performance improvement.  The performance improvement increases as flow-rate fraction of top product r goes away from 0.5.  The performance improvements range from 0 to several hundreds. Fig. 4. Effects of plate aspect ratio and flow-rate fraction of top product on degree of separation for vertical orientation q ¼ 0 .

Fig. 5. Effects of inclination angle and flow-rate fraction on degree of separation for plate aspect ratio x ¼ 12.2.

Fig. 6. Effects of inclination angle and plate aspect ratio on degree of separation under the same top and bottom product rates.

C.-D. Ho et al. / Progress in Nuclear Energy 66 (2013) 90e98

The theoretical predictions of the optimal inclination angle and plate aspect ratio under fixed expense for the degree of separation are given in Figs. 2e7.Figs. 2 and 3 show the eigenvalues in the enriching and stripping sections under flow-rate fraction of top product and inclination angle variations for x ¼ 1e40 and CF ¼ 0.381. It is found that the eigenvalues in the stripping section increases with increasing plate aspect ratio, flow-rate fraction of top product and inclination angle but the eigenvalues in the enriching section are inversely. There many parameters influence the device performance in the thermal-diffusion columns. Among them, flow-rate fraction of top product (r), mass flow rate (sF), inclination angle (q) and plate aspect ratio (x) may be the most important factors. The degree of separation (D) decreases with increasing the mass flow rate (sF), because the resident time of fluid decreases with increasing the mass flow rate, as indicated by Figs. 4e7 as well as in Tables 3 and 4.Fig. 4 shows the degree of separation with the plate aspect ratio as parameter for q ¼ 0 while Fig. 5 with inclination angle as a parameter for x ¼ 12.2. It is concluded that a better degree of separation is obtained with increasing the plate aspect ratio and inclination angle as the flow-rate fraction of top product r goes away from 0.5. The effects on the maximum degree of separation determined are given in Figs. 6 and 7. As shown in Fig. 6, for fixed flow-rate fraction of top product, when the inclination angle increases, the maximum degree of separation occurs at a smaller value of plate aspect ratio. The curves also show that the inclination angle is an important factor affecting the maximum degree of separation. The results illustrated in Fig. 7 indicate that when the plate aspect ratio is larger, the optimal inclination angle is smaller, i.e. closer to the

97

Table 4 Summary of the effect analysis. Performance improvement (I: Iq or Iq* ) Feed flow rate Y I[ Plate aspect ratio [ IY Flow-rate fraction of top product I [ r goes away from 0.5 Optimal inclination angle (q*) q* Y (90 / 0 ) Feed flow rate [ Plate aspect ratio [ q* Y Flow-rate fraction of top product q* [ r goes away from 0.5 Enhancement in degree of separation (D*  D) Feed flow rate [ ( D*  D) Y Plate aspect ratio [ (D*  D) Y (if flow-rate fraction of top product r goes away from 0.5) Flow-rate fraction of top product D* [ r goes away from 0.5

vertical orientation. However, when the plate aspect ratio is higher, the inclination angle shows little effect on the maximum degree of separation. The difference between the degree of separations of optimal inclination angle and inclination angle variations, i.e. the enhancement, is getting smaller when increasing mass flow rate and plate aspect ratio, as examined from in Tables 2 and 3.For high enough mass flow rate, the enhancement vanishes. The effect of flow-rate fraction of top product is significant, especially for the flow-rate fraction of top product r deviating from 0.5. The enhancement of the operation with smaller or larger flow-rate fraction of top product is much higher compared to that of equal flow-rate fraction of both top and bottom products. However, for operating at vertical column and smaller plate aspect ratio, as shown in Figs. 4 and 5, the flow-rate fraction of top product on the degree of separation is small. Moreover, when adopting the optimal inclination angle, decreasing the plate aspect ratio results in considerable increase of the device performance improvement, as confirmed in Table 3. The parametric study results are also summarized in Table 4. 5. Conclusions

Fig. 7. Effects of plate aspect ratio and inclination angle on degree of separation under the same top and bottom product rates.

The equation of the flat-plate thermal-diffusion columns for heavy water enrichment has been formulated theoretically and solved analytically by the use of the orthogonal expansion technique with the eigenfunction expanding in terms of an extended power series. The mathematical model of mass transfer was developed based on ignoring the influences of ordinary and thermal diffusions on the fluid velocity. However, the effects of ordinary and thermal diffusions on the fluid velocity increase with decreasing the fluid velocity (the mass flow rate), as indicated in Tables 2 and 3. The theoretical predictions of the separation degree and device performance improvement for heavy water enrichment in thermal-diffusion columns was also studied with the mass flow rate, plate aspect ratio, inclination angle and flow-rate fraction of both top product as parameters. Figs. 4e7 show the influences of inclination angle, plate aspect ratio and flow-rate fraction of top product on the separation degree while Tables 2 and 3 show the improvements on device performance. The results indicate that the separation degree increases while the plate aspect ratio and mass flow rate decreases and the flow-rate fraction of top product moves away from r ¼ 0.5. The design offers greater performance improvement for a thermal-diffusion column with smaller mass flow rate and plate aspect ratio and operating at larger inclination angle.

98

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Acknowledgments

the eigenfunctions associated with the corresponding eigenvalues are well defined by Eq. (A2).

The authors wish to thank the National Science Council of the Republic of China for its financial support. Appendix In order to apply the orthogonality theory to the SturmeLiouville system, Eq. (1), and nonhomogeneous boundary conditions for Eqs. (5), the use of the SturmeLiouville orthogonality properties along the h direction was made in aiming to transfer nonhomogeneous boundary conditions into homogeneous forms. The nonhomogeneous boundary conditions are thus removed using the method based on an expansion of the solution Ce in terms of the eigenfunctions of a purely homogeneous problem. Applying separation of variables in the form

Ci ðh; 2Þ ¼

N X

Si;m Fe;m ðhÞGi;m ð2Þ þ YðhÞ;

i ¼ e; s

(A1)

m¼0

to Eqs. (1), (5) and (6) leads to 00 Fi;m ðhÞ  li;m Ui ðhÞFi;m ðhÞ ¼ 0

(A2)

Ge;m ð2Þ ¼ ele;m ð1þ2Þ

(A3)

Gs;m ð2Þ ¼ els;m ð12Þ

(A4)

0 ð0Þ ¼ 0 Fi;m

(A5)

0 ð1Þ ¼ 0 Fi;m

(A6)

YðhÞ ¼

Z h 0

HðhÞ dh

(A7)

where the prime on Fi,m denotes the differentiation with respect to h. Without losing generality, the eigenfunction Fi,m(h) is assumed as a polynomial function as shown in Eqs. (17) and (18). Substituting Eqs. (14) and (18) into Eq. (A2), all of the coefficients were obtained in terms of the eigenvalues le,m and ls,m after using the boundary conditions, Eqs. (A5) and (A6), and Eqs. (19)e(22). Hence, the eigenvalues le,m and ls,m, thus calculated, include a set of real values with the limit points N to þN. The eigenvalues indicated in Table 1 are set dominant in the system and

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