Performance analysis for an inclined thermal diffusion column with side-stream operation for heavy water enrichment

Progress in Nuclear Energy 55 (2012) 61e67

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Performance analysis for an inclined thermal diffusion column with side-stream operation for heavy water enrichment Chii-Dong Ho*, Ho-Ming Yeh, Hsuan Chang, Sheng-Hung Chen Department of Chemical and Materials Engineering, Tamkang University, 151 Yingzhuan Road, Tamsui, New Taipei City 251, Taiwan

a r t i c l e i n f o

a b s t r a c t

Article history: Received 7 June 2011 Accepted 3 November 2011

Heavy water enrichment using an inclined ClusiuseDickel thermal diffusion column with side-stream operation has been investigated theoretically. The mathematical model to determine the separation efficiency and the best inclination angle have been developed. Significant improvement in heavy water enrichment can be obtained relative to a classic, vertical without side-stream, design. The study of the effects of design and operating variables points out the design provides greater improvement for lower feed flow rate and higher top and side product ratios. Ó 2011 Elsevier Ltd. All rights reserved.

Keywords: Thermal diffusion Inclined ClusiuseDickel columns Side-stream operations Heavy water enrichment

1. Introduction For the separation of isotopes, Clusius and Dickel (Clusius and Dickel, 1938, 1939) developed the thermal diffusion column. Furry et al. (Furry et al., 1939; Jones and Furry, 1946) further derived a complete theoretical presentation for the cascading effect of the thermal diffusion column. Besides the application on uranium separation (Murphy, 1955; Bebbing and Thayer, 1959), another important application of the ClusiuseDickel column is on heavy water enrichment and several theoretical and experimental studies have been reported by the authors (Yeh,1998; Ho and Chen, 2004; Ho et al., 2010). In a ClusiuseDickel column, the concentration gradient set up by the thermal diffusion due to two opposing vertical plates creates the horizontal convective currents and the temperature gradient can also induce density difference to cause two vertical convective currents, one flow upward near the hot plate and the other one flow downward near the cold plate. These convective currents construct a cascading effect similar to the multistage effect of countercurrent extraction. In combination with the thermal diffusion, the convective currents convey one of the components preferentially toward the top or the bottom and a concentration gradient across the column is established. This is advantageous to the separation efficiency. However, at the top and bottom of the column, part of the convective currents must flow backward toward the other side to supply the necessary currents. This unavoidable flow causes a remixing effect which is disadvantageous for the separation. This remixing effect could be reduced by tilting the column which results

* Corresponding author. Fax: þ886 2 26209887. E-mail address: [email protected] (C.-D. Ho). 0149-1970/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.pnucene.2011.11.003

in the adjustment of the convective strength. The desirable cascading effect and the undesirable remixing effect are the two conflicting consequences generated by the convective currents. Reported studies indicate that the cascading effect enhancement can override the remixing effect deterioration on the separation efficiency by adopting various modifications on the convective flow patterns, including inclination (Power and Wilke, 1957), rotation (Sullivan et al., 1955), moving wall (Ramser, 1957), packing (Sullivan et al., 1957) and winding a wire helix (Yeh and Ward, 1971). Stimulated by the scrubbing method, which is commonly utilized for removing particulates (Ebert and Buttner, 1996) or gaseous components (Lehner, 1998; Chien and Chu, 2000) from a gas stream, a side-stream adding design has been proposed for enhancing the separation efficiency of the thermal diffusion column in our previous work (Ho and Chen, 2004). In this paper, a design of the thermal diffusion column using inclined arrangement with sidestream operation is investigated. A theoretical model is developed for predicting the separation efficiency for heavy water enrichment and examining the effects of feed flow rate, feed location and product ratio on the optimal inclination angle, the degree of separation and performance improvement relative to that of the classical ClusiuseDickel thermal diffusion column. 2. Mathematical model Fig. 1 illustrates the convective flows and fluxes prevailing in a continuous-flow thermal diffusion column of the thickness 2w between hot and cold plates filled with water isotopes under variations of the top product to feed ratio ðrT ¼ sT =ðsT þ sB þ sP Þ ¼ sT =sF Þ with side product to feed ratio

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C.-D. Ho et al. / Progress in Nuclear Energy 55 (2012) 61e67

H ¼

abT rgð2wÞ3 BðDTÞ2 < 0 for a < 0 6!mT

(6)

and 2

K ¼

rg2 bT ð2wÞ7 BðDTÞ2 þ 2wrDB 9!m2 D

(7)

For steady state operation, the mass transport, se and ss, and the mass flow rates, sT and sB, are constant, Eqs. (1) and (2) with the use of Eqs. (4) and (5) become, respectively,

 dCs 1 0 s ðCs  CT Þ þ cos q Cs C^s ¼ dz’ cos2 q T 1 ½s0 ðCs  CT Þ þ cosqðas þ bs Cs Þ ¼ cos2 q T

(8)

 dCe 1 0 sB ðCB  Ce Þ þ cos q Ce C^e ¼ 2 dz’ cos q 1 ½s0 ðCB  Ce Þ þ cos qðae þ be Ce Þ ¼ cos2 q B

(9)

with the boundary conditions as follow:

Ce ¼ Cs ¼ CP Fig. 1. Schematic diagram of a continuous thermal diffusion column with inclination and side-stream operation.

ðrP ¼ sP =ðsT þ sB þ sÞP ¼ sP =sF Þ. The feed is introduced at the position Ls from the top of the column and the side-stream is withdrawn at the opposite side of the feed position, which is a modification from the classical ClusiuseDickel column (Yeh et al., 2002). The combination of the horizontal thermal diffusion flux and vertical natural convective currents results in a concentration difference between the two ends of the column. The transport equations may be written as follows (Ho and Chen, 2004)

se ¼ sB CB ¼ sB Ce þ Hcos qCe C^ e  Kcos2 q

dCe dz

dCs dz

at

z0 ¼ L0s

(11)

Ce ¼ CB

at

z0 ¼ L0e

(12)

in which the dimensionless variables are

s s Hz 0 Ls L0 ; sT ¼ T ; s0B ¼ B ; z ¼ ¼ s0 ; L0s ¼ L0 z; K H H L L L0e ¼ L0 ð1  zÞ z0 ¼

(13)

Ds ¼ cp  cT ¼ as þ bs cp $ bs exp

  s0T L0 z  0 q s cos þ b þ s T cosq cos2 q   0    L ðzÞ  $ exp bs cos q þ s0T  1 cos2 q ! (     6as;0 cT þ cp bs  ¼ þ 12b cT þ cp þ as;0 þ  s;0 2 D2 



s

$ðcF þ rT $Ds  ð1  rT  rP Þ$De Þ !   " 6as;0 cT þ cp  þ 12bs;0 2 (3)

Ds

L0 z  exp cos2 q h

i rT $sF q Hcos

(4)

þ

(5)

$ exp

"

and

Ce C^e ¼ ae þ be Ce

Cs ¼ CT

(2)

and is assumed to be a linear approximation, i.e.,

^ s ¼ as þ bs Cs Cs C

(10)

(1)

for the stripping section. In obtaining the above equations, the pseudo concentration ^ e and Cs C ^ s defined in Eq. (3) for enriching and stripproducts, Ce C ping sections, respectively, are defined as

8 <   ^ ¼ C 0:05263  0:05263  0:0135Keq C CC : 9   1 =   2 Keq C CKeq  0:027 1  1  ; 4

z0 ¼ 0

Following similar mathematical treatment in the previous work (Ho and Chen, 2004), the degrees of separation, De and Ds, in the enriching and stripping sections, respectively, can be obtained by integrating Eqs. (8) and (9) with the use of Eqs. (10)e(12). The results are

for the enriching section and

ss ¼ sT CT ¼ sT Cs þ Hcos qCs C^ s  Kcos2 q

at

The values of the equilibrium constant are Keq ¼ 3.80 and 3.793, respectively, at T ¼ 25  C and 30.5  C (Standen, 1978). The constants in Eqs. (1) and (2) are defined as

þ



  6as;0 cT þ cp

!!

L0 z cos2 q

 rT $sF Hcos q

rT $s Hcos q F   6as;0 cT þ cp þ



D2s

D2s

!!

! þ 12bs;0 cos q

#)  ! þ 12bs;0 cos q

# 1

(14)

C.-D. Ho et al. / Progress in Nuclear Energy 55 (2012) 61e67

63

Table 1 Performance improvement with the optimum inclination angle operation for equal top and bottom product rates (z ¼ 1/2).

sF (g/hr)

Iq (%) CF ¼ 0.1 r¼0

r¼1

r ¼ 10

r¼0

r¼1

r ¼ 10

r¼0

r¼1

r ¼ 10

162.29 50.29 18.84 7.38 2.37 0.36 0.00

213.49 50.51 18.84 7.36 2.35 0.36 0.00

468.21 53.35 20.33 7.97 2.90 0.59 0.00

176.43 51.68 19.13 7.46 2.38 0.36 0.00

206.13 51.60 19.11 7.45 2.38 0.36 0.00

498.40 54.05 20.60 8.41 2.96 0.61 0.00

176.22 51.54 19.06 7.42 2.37 0.36 0.00

232.41 51.54 19.06 7.42 2.37 0.36 0.00

513.86 54.12 20.58 8.39 2.94 0.61 0.00



 0   L ð1  zÞ  0 q s cos  b e B cos2 q   0    s0B L ð1  zÞ  0 q s $ exp cos   1 b þ e B cos q cos2 q ! (     6ae;0 cB þ cp be  ¼ þ 12b cB þ cp þ ae;0 þ  e;0 2 D2

Table 3 Performance improvement with the optimum inclination angle operation with rT ¼ 0.2 and CF ¼ 0.381.

De ¼ cB  cp ¼ ae þ be cp $  be exp

sF (g/hr)

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

e

De

  "

L0 ð1  zÞ cos2 q



L0 ð1  zÞ cos2 q

 

!!

ð1  rT  rP Þ $sF Hcos q

in which, ae;0 ¼

1

De

Z

CB CP

D2e

! þ 12be;0 cos q

#) ð1  rT  rP Þ $sF Hcos q !   6ae;0 cB þ cp  þ 12b e;0 cos q 2

ð1  rT  rP Þ $sF Hcos q

$ exp

  6ae;0 cB þ cp 

þ

!!

De

r¼2

47.11 20.48 11.02 6.46 3.90 2.33 1.31 0.66

58.52 25.79 13.02 6.69 3.24 1.34 0.37 0.02

65.00 28.11 13.68 6.58 2.79 0.87 0.09 0.00

52.39 22.51 11.35 5.99 3.11 1.50 0.61 0.16

64.72 29.04 14.81 7.66 3.72 1.55 0.43 0.02

70.39 31.13 15.69 7.88 3.61 1.32 0.24 0.00

66.93 29.96 15.67 8.46 4.44 2.14 0.85 0.21

77.56 35.87 19.25 10.60 5.64 2.72 1.06 0.24

149.85 109.22 96.33 91.37 89.32 88.61 88.56 88.87

= 1/2

80 (15)

70

Ce C^e dCe and be;0 ¼

1

D3e

Z

CB

CP

r=0

Ce2 C^e dCe . Eqs. (14)

sF CF ¼ ðsT þ sB þ sP ÞCF ¼ sT CT þ sB CB þ sP CP

r=1 60

r=2 50

r = 10

40

(16)

30

or Table 2 Performance improvement with the optimum inclination angle operation for rT ¼ 0.1 and CF ¼ 0.381. (g/hr)

r¼1

CF = 0.381

and (15) give Ds and De in the implicit form and the values of Ds and De may be determined by the successive iteration method. The appropriate values of ae, be, as and bs may be determined separately from CP to CB in the enriching section and from CT to CP in the stripping section by the least square method, except that the concentration at the feed position is determined by the material balance around the entire column with different mass-flow rates at both top and bottom product streams, i.e.

sF

r¼0

90

#

1

IrT ;q (%)

z ¼ 1/3 z ¼ 1/2 z ¼ 2/3 z ¼ 1/3 z ¼ 1/2 z ¼ 2/3 z ¼ 1/3 z ¼ 1/2 z ¼ 2/3

$ðcF þ rT $Ds  ð1  rT  rP Þ$De Þ !   " 6ae;0 cB þ cp   þ 12be;0 2  exp

CF ¼ 0.5

*

0.01 0.04 0.08 0.12 0.16 0.20 0.24

CF ¼ 0.381

IrT ;q (%) r¼0

r¼1

20

10

r¼2

z ¼ 1/3 z ¼ 1/2 z ¼ 2/3 z ¼ 1/3 z ¼ 1/2 z ¼ 2/3 z ¼ 1/3 z ¼ 1/2 z ¼ 2/3 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

50.01 30.20 21.68 16.06 11.90 8.73 6.29 4.41

51.67 25.94 15.10 9.15 5.43 3.03 1.49 0.56

55.67 23.64 11.48 5.50 2.30 0.68 0.07 0.00

47.36 27.57 19.49 14.36 10.62 7.76 5.55 3.84

51.00 24.94 14.15 8.39 4.89 2.63 1.24 0.41

56.37 23.73 11.43 5.40 2.22 0.63 0.05 0.00

45.35 24.91 17.10 12.42 9.09 6.60 4.67 3.18

50.67 24.10 13.33 7.64 4.31 2.23 0.98 0.27

57.46 24.11 11.51 5.38 2.19 0.57 0.03 0.00

0 0.0

0.2

0.4

0.6

0.8 F

1.0

1.2

1.4

1.6

g/h

Fig. 2. Effect of feed flow rate and side product to top product ratio on optimal inclination angle when the top and bottom product rates are equal and feed position is at center.

64

C.-D. Ho et al. / Progress in Nuclear Energy 55 (2012) 61e67

CP ¼ CF þ rT Ds  ð1  rT  rP ÞDe

(17)

Combination of Eqs. (14) and (15) yields the degree of the separation for the whole column as

D ¼ CB  CT ¼ De þ Ds

(18)

For a vertical column, q ¼ 0 , the expression for the degree of separation, (D0), can be obtained from Eqs. (14) and (15) with cosq ¼ 1 accordingly.

inclination under the same working dimensions and operating conditions. For the operation with side-stream under the condition that top and bottom product streams are equal in mass flow rate, the separation efficiency improvement is defined as Eq. (19). In other cases, a general form as Eq. (20) can be defined, where the top product to feed ratio (rT) must be specified.



D r; q*  Dðr; q ¼ 0Þ Iq ¼ Dðr; q ¼ 0Þ

3. Device performance analysis The device performance is affected by both design parameters and operating parameters. The former include the feed position (z) and inclination angle (q). The latter include the top product to feed ratio (rT), the side product to top product ratio (r ¼ rp/rT), feed flow rate (sF) and feed concentration (CF). The intensity of the convective flow can be adjusted and the undesired remixing effect can be reduced by tilting the ClusiuseDickel thermal diffusion column, as mentioned by previous investigators (Power and Wilke, 1957). Hence, an optimum inclination angle should be determined for the maximum degree of separation. It is too complicated to express the inclination angle (q) and degree of separation (D) in terms of other design and operating parameters mathematically. Therefore, numerical approach is adopted. The separation efficiency improvement is defined as the ratio of improvement in the degree of separation of the column using the optimum inclination angle to that of the device without

90

IrT ;q

(19)



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

(20)

4. Results and discussion A device with the following dimensions and parameters (Standen, 1978; Yeh and Yang, 1984) are analyzed:

H ¼ 1:473  104 g=s; K ¼ 1:549  103 g$cm=s; Keq ¼ 3:793 L ¼ 122 cm; B ¼ 2pR ¼ 10 cm; 2w ¼ 0:04 cm; DT ¼ 35 K T ¼ 303:5 K; CF ¼ 0:1; 0:381; 0:5; DT=2w ¼ 875 K=cm

14

CF = 0.381

0

r =1

80

12

rT = 0.1

CF=0.381

r=0 r = 10 r= 0 r = 10

rT = 0.2

70

10

= 1/3

*

60

8

= 1/2

50

40

6

= 2/3

30

4

20

CF=0.5 2

10

0 0.0

0 -1.0 0.2

0.4

0.6

0.8 F

1.0

1.2

1.4

1.6

(g/h)

Fig. 3. Effect of feed flow rate, feed position and top product to feed ratio on optimal inclination angle when the side product and top product rates are equal.

-0.8

-0.6

log

-0.4

-0.2

0.0

(g/h)

F

Fig. 4. Effect of feed flow rate, feed concentration and side product to top product ratio on degree of separation when the top and bottom product rates are equal and the feed position is at center.

C.-D. Ho et al. / Progress in Nuclear Energy 55 (2012) 61e67

The results listed in Tables 1e3 allow the summary of the following for the improvement of separation efficiency with optimal inclination angle over that of vertical arrangement, i.e. Iq or IrT ;q :  The performance improvement is higher for lower feed flow rate.  When the feed flow rate is large enough, the optimal inclination angle is 0, i.e. vertical orientation. Therefore, the performance improvement is zero.  The effect of feed concentration on performance improvement is relatively small compared to other variables.  The higher the side product ratio, the greater the performance improvement.  The increase of top to feed ratio results in the increase of performance improvement.  The performance improvements range from 0 to several hundreds. The analysis of the optimal inclination angle and the degree of separation are given in Figs. 2e6. The effects on the optimal inclination angle determined are given in Figs. 2 and 3. As shown in Fig. 2, for fixed side to top product ratio, when the feed flow rate increases, the optimal inclination angle varies from near 90 e0 , i.e. from almost horizontal to vertical (without inclination). The curves also show that the side product ratio is an important factor affecting the optimal

100

r=2 = 0.1 g/h F

90

= 1/3 = 1/2 = 2/3

80

70

60

50

40

CF=0.381

30

CF=0.5 CF=0.1

20

10

0 0.00

25

65

0.05

0.10

0.15

0.20

rT

= 1/2 = 0.1 g/h F

Fig. 6. Effect of top product to feed ratio on the degree of separation when employing the optimal inclination angle.

20 inclination angle. The results illustrated in Fig. 3 indicate that when the feed position is lower, the optimal inclination angle is smaller, i.e. closer to the vertical orientation. However, when the top product ratio is higher, the feed position shows little effect on the optimal inclination angle. Examining the results in Fig. 4 shows that the enhancement, i.e. the difference between the degree of separations of vertical and optimal inclination angle, is smaller when increasing feed flow rate. For high enough feed flow rate, the enhancement vanishes. The effect of side product ratio is significant. The enhancement of the configuration with high side product ratio is much higher compared to that of no side stream. For vertical

CF =0.381

15 CF =0.5

10

Table 4 Summary of the effect analysis. Performance improvement (I:Iq or IrT ;q )

5

Feed flow rate Y

CF =0.1 0

0

2

4

r

6

8

10

Fig. 5. Effect of feed concentration and side product to top product ratio on degree of separation when the top and bottom product rates are equal and feed position is at center.

I[

Side product ratio [ I[ Top product ratio [ I[ Optimal inclination angle (q*) q* Y (90 /0 ) Feed flow rate [ Feed position Y (lower position) q* Y (unimportant if high top product ratio) Top product to feed ratio [ q* Y Enhancement in degree of separation (D* e D) Feed flow rate [ (D* e D) Y Side product ratio [ (D* e D) [ (unimportant to D for vertical arrangement) D* Y Top product to feed ratio [

66

C.-D. Ho et al. / Progress in Nuclear Energy 55 (2012) 61e67

arrangement, as shown in Fig. 5, the effect of side product ratio on the degree of separation is small. However, when adopting the optimal inclination angle, increasing the side product ratio results in considerable increase of the degree of separation. Besides, for the feed concentration analyzed, the middle-value concentration (CF ¼ 0.318) gives the highest degree of separation. In Fig. 6, when the top product to feed ratio increases, the degree of separation with optimal inclination angle decreases. However, when the ratio is greater than about 0.03, the degree of separation decreases to a stable level. The parametric study results are summarized in Table 4. 5. Conclusions The performance of the proposed inclined with side-stream arrangement thermal-diffusion column for heavy water enrichment has been investigated theoretically in this study. Considerable improvement of the degree of separation relative to the vertical without side-stream arrangement can be obtained. The analysis of the effects of the feed flow rate, feed position and product ratio on the relative and absolute performance improvement as well as the optimal inclination angle provides the criteria for applying the inclination with side-stream design. The design offers greater performance improvement for a column with lower feed flow rate and using higher side and top product ratios. The optimal inclination angle decreases with increases of feed flow rate or top product ratio and with lower feed position. Acknowledgment The authors thank the National Science Council of the Republic of China for the financial support. Nomenclature

ae,as be, bs B C CB Ce CF CP CS CT ^ CC ^e Ce C ^s Cs C D H Iq IrT ;q K Keq L Le Ls 0 L L0e

Constants defined by Eqs. (4) and (5), respectively (-) Constants defined by Eqs. (4) and (5), respectively (-) Circumference length 2pR (cm) Mass fraction of heavy water in the H2O-HDO-D2O system (-) Concentration of the bottom product (-) Concentration in the enriching section (-) Concentration of the feed (-) Concentration of the side product (-) Concentration in the stripping section (-) Concentration of the top product (-) Pseudo product form of concentration for D2O defined by Eq. (3) (-) ^ in the enriching section (-) CC ^ in the stripping section (-) CC Ordinary diffusion coefficient (cm2/s) Transport coefficient defined by Eq. (6) (g/s) Performance improvement with the optimum inclination angle operation as defined by Eq. (19) (-) Performance improvement with the optimum inclination angle operation as defined by Eq. (20) (-) Transport coefficient defined by Eq. (7) (g/s cm) Mass fraction equilibrium constant of the H2O-HDO-D2O system (-) Column length (cm) Length of enriching section (cm) Length of stripping section (cm) Dimensionless coordinate defined by Eq. (13) (-) L0 in the enriching section defined by Eq. (13) (-)

L0s r rp rT T T W z 0 z

0

L in the stripping section defined by Eq. (13) (-) Side product to top product ratio, r ¼ rp/rT (-) Side product to feed ratio (-) Top product to feed ratio (-) Mean temperature (K) Arithmetic mean temperature of hot wall and cold wall (K) Thickness of the region (cm) Coordinate in the vertical direction (cm) Dimensionless coordinate defined by Eq. (13) (-)

Greek Symbols Thermal diffusion constant for D2O in H2O-HDO-D2O system, <0 (-) vr bT ð Þ evaluated at T (g/cm3 K) vT d Thickness of the barrier (cm) D Degree of separation, CBCT (-) D* Maximum degree of separation (-) De Degree of separation, CBCP (-) Ds Degree of separation, CPCT (-) DT Temperature difference of hot and cold plates (K) D0 Degree of separation without inclination q ¼ 0 (-) ε Permeability of the barrier (-) z Dimensionless feed position defined by Eq. (13) (-) m Viscosity of fluid (g/cm s) r Density of fluid (g/cm3) r Density evaluated at T (g/cm3) sB Mass flow rate of bottom product (g/s) s0B Dimensionless mass flow rate of the bottom product (-) sF Mass flow rate of feed (g/s) sP Mass flow rate of side product (g/s) sT Mass flow rate of top product (g/s) s0T Dimensionless mass flow rate of the top product (-) q Inclination angle (deg.) Optimum inclination angle (deg.) q* se Transport of heavy water along z-direction in enriching section (g/s) ss Transport of heavy water along z-direction in stripping section (g/s) w One-half of the plate spacing of column (cm)

a

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