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The improvement of the heavy water enrichment under countercurrent-flow Frazier scheme and flow-rate fraction variations
Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 555–562
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The improvement of the heavy water enrichment under countercurrent-flow Frazier scheme and flow-rate fraction variations Chii-Dong Ho *, Ho-Ming Yeh, Yu-Cheng Su, Jia-Jan Guo, Jr-Wei Tu Department of Chemical and Materials Engineering, Tamkang University, 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 17 December 2008 Received in revised form 23 February 2009 Accepted 24 February 2009
The desirable cascading effect and undesirable remixing effect are the two conflicting effects produced in a thermogravitational thermal-diffusion column. A new design is proposed in the present study to obtain equilibrium separation efficiency improvement associated with a remixing-effect suppression and cascading effect enhancement. The separation equation for the heavy water enrichment by inclined thermal-diffusion columns in countercurrent-flow Frazier schemes under flow-rate fraction variations has been developed theoretically. The concentration-product term in the transport equation is treated in the enriching and stripping sections separately for more accurate analysis. It was found that the undesirable disadvantage of the remixing effect could be reduced effectively by suitably tilting the column and controlling the flow-rate fraction at both top and bottom product streams, leading to a considerable separation efficiency improvement. Numerical examples were given to illustrate the best inclination angle and flow-rate fraction on separation efficiencies for H2O–HDO–D2O system. ß 2009 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: Thermal-diffusion column Heavy water Frazier scheme Inclination angle
1. Introduction The importance of the thermal-diffusion columns has been enhanced by concentrating isotopes and rare gases successfully which are not possibly separated by conventional separation techniques such as extraction and distillation. This powerful separation technique has attracted a great attention for more detailed studies on improved thermal diffusion devices that consider the two conflicting effects of the convective currents: the advantage of the cascading effect and disadvantage of the remixing effect. In a thermal-diffusion column, a horizontal temperature gradient is provided to produce the thermal diffusion in the direction of the temperature gradient and the natural convection of the fluid in the upward direction near the hot plate or downward near the cold plate. These two convective currents cause a cascading effect, which is analogous to the multistage effect of a countercurrent extraction, resulting in a enhancement of separation (Chueh and Yeh, 1967; Jones and Furry, 1946). However, a remixing effect also occurs to diminish the separation since the convection current takes down the fluid of rich in one component at the top of the column to the bottom of the column where the fluid is rich in other component. An optimum design with proper adjustment on the convective strength by selecting the design and operating parameters was obtained effectively to
* Corresponding author. Tel.: +886 26215656x2724; fax: +886 2 26209887. E-mail address: [email protected] (C.-D. Ho).
reduce the undesirable remixing effect and preserve the desirable cascading effect, resulting in a substantial separation efficiency improvement. The enrichments obtained from some improved thermal-diffusion columns are somewhat better than the classical Clusius–Dickel column (Clusius and Dickel, 1938, 1939) such as the inclined column (Power and Wilke, 1957), wired column (Yeh and Ward, 1971), inclined moving-wall columns (Ramser, 1957), rotary columns (Sullivan et al., 1955), packed columns (Sullivan et al., 1957) and permeable and impermeable barrier columns (Ho et al., 2002, 2004). Recently, the practical applications of thermaldiffusion columns are commonly used to purify tritium in production and recover hydrogen isotopes in fusion nuclear fuel cycle (Kobayashi et al., 2002, 2003; Yamakawa et al., 1999a,b), and several theoretical model and experimental runs have been presented (Ho et al., 2002, 2004a,b; Ho and Chen, 2003; Yeh, 1998) for the enrichment of heavy water in the thermal-diffusion columns. The separation rate of the transverse-flow system, the so-called Frazier-scheme thermal-diffusion columns proposed by Frazier (1962) and Grasseli and Frazier (1962), is superior to that in the classical Clusius–Dickel column and particularly for obtaining high separation levels. A complete theory of separation in the Frazierscheme thermal-diffusion columns was developed by Rabinovich (1976) and Suvorov and Rabinovich (1981) for binary mixtures. The mathematical theory on the separation efficiency of thermaldiffusion columns in modified Frazier scheme for the enrichment of heavy water under the optimal operation and design parameters have been developed (Ho and Chen, 2003; Ho et al., 2004). The
1876-1070/$ – see front matter ß 2009 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jtice.2009.02.008
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Nomenclature A B C CB,i CF CT,i C Cˆ D g H Ir Ir,u Iu J x-OD J x-TD J z-OD K Keq n r Sn S0n T T¯ x z
ˆ defined in Eq. (11) C C, column width (cm) fraction mass concentration of heavy water in H2O–HDO–D2O system C in the bottom product stream of ith column C in the feed stream C in the top product stream of ith column pseudo product form of concentration for D2O defined by Eq. (6) ordinary diffusion coefficient (cm2/s) gravitational acceleration (cm/s2) transport coefficient defined by Eq. (3) (g/s) improvement of the degree of separation defined by Eq. (33) improvement of the degree of separation defined by Eq. (31) improvement of the degree of separation defined by Eq. (32) mass flux of heavy water in the x-direction due to ordinary diffusion (g/(cm2 s)) mass flux of heavy water in the x-direction due to thermal diffusion (g/(cm2 s)) mass flux of heavy water in the z-direction due to ordinary diffusion (g/(cm2 s)) transport coefficient defined by Eq. (4) (g/(cm s)) mass-fraction equilibrium constant of H2O–HDO– D2O system total column number flow-rate fraction total degree of separation of n columns for countercurrent-flow operation total degree of separation of n columns for concurrent-flow operation mean absolute temperature (K) arithmetic mean value of T of hot wall and cold wall (K) coordinate in the horizontal direction (cm) coordinate in the vertical direction (cm)
Greek symbols reduced thermal diffusion constant for D2O in H2O–HDO–D2O system, <0 bT (@r/@T) evaluated at reference temperature (g/(cm3 K)) Dn degree of separation of nth column, C B;n C T;n DT difference in temperature of hot and cold plates (K) m absolute viscosity of fluid (g/(cm s)) r mass density of fluid (g/cm3) s mass flow rate of top or bottom product (g/s) ti transport of heavy water along z-direction in ith column (g/s) v one-half of the plate spacing of columns (cm) u inclination angle (8)
purpose of the this study is to investigate theoretically the separation efficiency of heavy water enrichment on inclined Frazier scheme with mass flow-rate fraction variation, feed flow rate and feed concentration fraction as parameters under countercurrent-flow operations. 2. Inclined thermal-diffusion column in Frazier scheme The scheme proposed by Frazier (Frazier, 1962; Grasseli and Frazier, 1962) to connect several flat-plate thermogravitational thermal-diffusion columns with countercurrent-flow with inclination angle and flow-rate fraction variations is shown in Fig. 1. All flat-plate columns have the same plate-spacing 2v, length L, width B, and transverse-flow streams of opposite directions. The mass flow rates rs and (1 r)s with feed concentration CF is accomplished at the upper and lower ends, respectively, with the product end and the supply entrance on the opposite sides. Fig. 2 illustrates the flows and fluxes prevailing in the (i + 1)th column. The transport equation of D2O for thermal-diffusion columns in countercurrent-flow Frazier schemes was derived by Ho et al. (2004a,b). For the inclined column, the transport equation is modified by replacing the gravitational acceleration g to g cos u. Accordingly, the transport equation of the top and bottom parts in the ith column of countercurrent-flow inclined Frazier scheme are t i ¼ HC Cˆ cos u þ K cos2 u
dC i ¼ r s ðC T;i1 C T;i Þ dz z¼L
(1)
t i ¼ HC Cˆ cos u þ K cos2 u
dC i ¼ ð1 rÞs ðC B;i C B;iþ1 Þ dz z¼0
(2)
H¼
abT rgð2vÞ3 BðDTÞ2 < 0 for a < 0 6!mT¯
(3)
rg 2 b2T ð2vÞ7 BðDTÞ2 9!m2 D
(4)
and K¼
In a thermal-diffusion column, the mass transport of one component is a result of the combination of the thermal diffusion and ordinary diffusion, as shown in the first and second terms of the Eq. (1) or Eq. (2), respectively. Hence, the transport coefficients H and K represent the transport ability of the component by the thermal diffusion and ordinary diffusion, respectively. The material balances around the ith column yield:
a
r s ðC T;i1 C T;i Þ ¼ ð1 rÞs ðC B;i C B;iþ1 Þ
(5)
The pseudo concentration product C Cˆ is the product of the concentration and can be defined as C Cˆ ¼ Cð1 CÞ. The value of C Cˆ is a function of the mass equilibrium constant Keq in a heavy-water thermal diffusion system, as shown in Eq. (6) (Jones and Furry, 1946): ( ˆ C C ¼ C 0:05263 ð0:05263 0:0135K eq ÞC 0:027 1=2 ) K eq C CK eq 1 1 4
(6)
in which the mass equilibrium constant Keq for the following equilibrium relation: H2 O þ D2 O @ 2HDO
(7)
C.-D. Ho et al. / Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 555–562
557
Fig. 1. Schematic diagram of the countercurrent-flow inclined Frazier-scheme thermal-diffusion columns with flow-rate fraction variations.
is K eq ¼
½HDO2 19 19 ½H2 O ½D2 O 18 20
(8)
Keq does not vary sensitively within the operating temperature range. For instance, the values of the equilibrium constant are Keq = 3.80 and 3.793, respectively, at T = 25 8C and 30.5 8C (Standen, 1978).
The appropriate value of C Cˆ ¼ A (=constant) in Eqs. (1) and (2) may be obtained from the simultaneous solution of this set of nonlinear equations. A rather reasonable approximation for C Cˆ ¼ A in inclined Frazier-scheme thermal-diffusion columns (IFSTDC) is determined using the least square method in the previous works (Yeh et al., 2002). Accordingly, minimizing the following integral equation: R¼
Z
CB
ðC Cˆ AÞ2 dC
(9)
CT
i.e., dR ¼ dA
Z
CB
2ðC Cˆ AÞ dC ¼ 0
(10)
CT
one obtains A¼
1 CB CT
Z
CB
C Cˆ dC
(11)
CT
The appropriate value of A, thus obtained in Eq. (11), is exactly the average value of C Cˆ in the concentration range. Substitution of Eqs. (1) and (2) into Eq. (5) gives dC dC ¼ HC Cˆ cos u þ K cos2 u i (12) HC Cˆ cos u þ K cos2 u i dz z¼L dz z¼0 or dC i dC i dC i ¼ constant ¼ ¼ dz z¼L dz z¼0 dz
(13)
Eq. (13) indicates that the concentration gradient in the z direction is constant through the entire column. 2.1. Countercurrent-flow operations
Fig. 2. The one-column inclined Frazier-scheme thermal-diffusion column.
2.1.1. One-column operations The separation efficiency of a thermal-diffusion column can be defined as the composition difference between the top and bottom product and it is so-called the degree of separation Dn ¼ C T;n C B;n . The degree of separation D can be derived by the transport equations, Eqs. (1) and (2). For one-column operations, i = 1, C T;0 ¼ C F and C B;2 ¼ C F , and substituting the parameters into Eqs. (1) and
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(2) gives C T;1 C B;1 ¼
i HL cos u h rs Aþ ðC F C T;1 Þ H cos u K cos2 u
(14)
C T;1 C B;1 ¼
HL cos u ð1 rÞs A þ ðC C Þ F B;1 H cos u K cos2 u
(15)
HL cos u K cos2 u
s rð1 rÞ Aþ C F C T;1 þ C B;1 C F H cos u
s Lrð1 rÞ þ K cos2 u HLA cos u S1 ¼ K cos2 u K cos2 u
S2 ¼
(16)
(17)
where S1 is the total degree of separation of one-column operations which is equals to the degree of separation D1 as S1 ¼ C B;1 C T;1 ¼ D1 . The best angle of inclination u 1 for the maximum separation in a Frazier scheme is obtained by differentiating Eq. (17) with respect to u and setting @S1 =@u ¼ 0. The result is "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# s Lrð1 rÞ (18) u1 ¼ cos1 K 1Þ < 1
Since cos1 ðu and s < K=½Lrð1 rÞ, substitution Eq. (18) into Eq. (17) gives the equation for calculating the maximum separation as
S1;maxðuÞ ¼
HLA cos u 1 s Lrð1 rÞ þ K cos2 u1
(19)
2.1.2. Multi-column operations For two-column operations, i ¼ 2, C T;0 ¼ C F and C B;3 ¼ C F . Thus, the degree of separation is obtained from Eqs. (1)–(3) for the first and second columns separately C T;1 C B;1 ¼
i HL cos u h rs Aþ ðC F C T;1 Þ H cos u K cos2 u
(20)
C T;1 C B;1 ¼
HL cos u ð1 rÞs A þ ðC C Þ B;1 B;2 H cos u K cos2 u
(21)
C T;2 C B;2
i HL cos u h rs ¼ Aþ ðC T;1 C T;2 Þ 2 H cos u K cos u
C T;2 C B;2
HL cos u ð1 rÞs ¼ A þ ðC C Þ F B;2 H cos u K cos2 u
HL cos u K cos2 u s rð1 rÞ Aþ ðC F C T;1 þ C B;1 C B;2 Þ H cos u
(22)
(23)
HL cos u K cos2 u s rð1 rÞ Aþ ðC T;1 C T;2 þ C B;2 C F Þ H cos u
(27)
in which D1 = CB,1 CT,1, D2 = CB,2 CT,2 and S2 = CB,1 CT,2. For further multi-column operations, the degree of separation is readily obtained if we perform the calculations of Eqs. (20)–(27) recursively. The detail derivations of the degree of separation for the multi-column operations were presented in Appendix. The resultant expressions of multi-column operations for heavy water enhancement in countercurrent-flow thermal-diffusion column are as follows: n X HL cos u s rð1 rÞ Di ¼ nA þ S (28) n H cos u K cos2 u i¼1 or Sn ¼
" # n H cos u HL cos u X D þ nA i s rð1 rÞ K cos2 u i¼1
(29)
in which Dn = CB,n CT,n and Sn = CB,1 CT,n. 2.2. Concurrent-flow operations The calculation procedure for a concurrent-flow Frazierscheme thermal-diffusion column (Ho and Chen, 2003) of the same working dimension is rather simpler than that for a countercurrent-flow device. The degree of separation for ISFTDC under concurrent-flow operation is n HLA cos u Ls rð1 rÞ S0n ¼ 1 (30) 2 Ls rð1 rÞ þ K K cos u 3. The separation efficiency improvement The improvement of the countercurrent-flow Frazier-scheme thermal-diffusion columns by operating under the optimal inclination angle is best illustrated by calculating the percentage increase in separation efficiency based on the vertical countercurrent-flow Frazier-scheme thermal-diffusion columns with the flow-rate fraction equal to 0.5 as Ir;u ð%Þ ¼
Sn ðr; un ¼ un Þ Sn ðr ¼ 0:5; un ¼ 0Þ 100 Sn ðr ¼ 0:5; un ¼ 0Þ
(31)
and the separation efficiency improvement based on the concurrent-flow Frazier-scheme thermal-diffusion columns with the optimal inclination angle as
Iu ð%Þ ¼
Sn ðun ¼ u n Þ S0n ðun ¼ u n Þ 100 S0n ðun ¼ u n Þ
(32)
Similarly, the flow-rate fraction improvement for the concurrentflow device with the optimal inclination angle may be defined as (24)
and C T;2 C B;2 ¼
H cos u HL cos u ðD1 þ D2 Þ þ 2A 2 s rð1 rÞ K cos u
Multiplying Eqs. (20) and (22) by (1 r) and Eqs. (21) and (23) by r, we add these equations to get C T;1 C B;1 ¼
(26)
or
Multiplying Eq. (14) by (1 r) and Eq. (15) by r, and adding the results one gets ðC T;1 C B;1 Þ ¼
Combination of Eqs. (24) and (25) gives HL cos u s rð1 rÞ D1 D2 ¼ 2A þ S2 2 H cos u K cos u
Ir ð%Þ ¼
Sn ðr; un ¼ un Þ S0n ðr ¼ 0:5; un ¼ u n Þ 100 S0n ðr ¼ 0:5; un ¼ u n Þ
(33)
Some equipment parameters and physical properties of the H2O– HDO–D2O system (Standen, 1978; Yeh and Yang, 1984) used to calculate the separation efficiency improvement as a numerical example for the separation of heavy water are given as follows: (25)
H ¼ 1:473 104 g=s ¼ 0:53 g=h
C.-D. Ho et al. / Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 555–562
Fig. 3. Effect of flow-rate fraction variation on the optimal inclination angle with column number as parameters; s = 0.15 g/h and CF = 0.381.
K ¼ 1:549 103 g cm=s ¼ 5:5763 g cm=h K eq ¼ 3:793; L ¼ 122 cm; B ¼ 10 cm; s ¼ 2:1 g=h; 2v ¼ 0:04 cm
DT ¼ 35 K; T¯ ¼ 30:5 K; C F ¼ 0:381; DT=2v ¼ 875 K=cm 4. Results and discussions 4.1. Optimal inclination angle for best performance The study for improving the device performance in IFSTDC under flow-rate fraction variations countercurrent-flow operations is to suppress the remixing effect and to enhance the cascading effect. Fig. 3 shows the optimal inclination angle versus flow-rate fraction with feed fraction concentration and feed flowrate as parameters for one through fifth column operations. It is found in Fig. 3 that the optimal inclination angle for the countercurrent-flow inclined Frazier-scheme thermal-diffusion columns increases with increasing number of columns and with flow-rate fraction values away from 0.5. 4.2. Degree of separation in countercurrent-flow inclined Frazier-scheme thermal-diffusion columns For the countercurrent-flow Frazier-scheme thermal-diffusion columns with the optimal inclination angle and for the vertical column, the top and bottom product fraction concentrations verus
559
Fig. 4. Effect of flow-rate fraction variation on the product concentration of bottom with optimal inclination angle and vertical column with column number as parameters; s = 0.15 g/h and CF = 0.381.
r, with column number as a parameter, are given in Figs. 4 and 5. It is seen in Figs. 4 and 5 that the product fraction concentration in the bottom increases with increasing the number of columns and product fraction concentration for the countercurrent-flow Frazier-scheme thermal-diffusion column. However, the product fraction concentration at the top increases with increasing flowrate fraction but it decreases as the number of columns increases. Figs. 6 and 7 show the comparisons of the degree of separation of the optimal inclination angle ðun Þ and u n ¼ 0 in the countercurrent-flow Frazier-scheme thermal-diffusion columns with flow-rate fraction, feed flow rate and column numbers as the parameters. It is seen in Figs. 6 and 7 that the maximum degree of separation in countercurrent-flow Frazier-scheme thermal-diffusion column increases with increasing the number of columns and the flow-rate fraction values getting away from 0.5 but decreases as feed mass flow rate increases. These results would suggest that the devices with flow-rate fraction variations and inclination angle could enhance the separation efficiency in Frazier-scheme thermal-diffusion columns. 4.3. Separation efficiency improvement in countercurrent-flow operations The separation efficiency improvements, Ir;u , Iu and Ir , after n columns operations were calculated from Eqs. (31)–(33). Table 1 shows that the separation efficiency improvement, Ir;u , of the optimal inclination angle in the countercurrent-flow Frazierscheme thermal-diffusion columns is much larger than that of
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560
Fig. 5. Effect of flow-rate fraction variation on the product concentration of top with optimal inclination angle and vertical column with column number as parameters; s = 0.15 g/h and CF = 0.381.
Fig. 6. Effect of flow-rate fraction variation on the degree of separation with optimal inclination angle with column number as parameters; s = 0.15 g/h and CF = 0.381.
concurrent-flow vertical Frazier-scheme thermal-diffusion columns with the operation at r = 0.5. It is also found that the separation efficiency improvement increases with the flow-rate fraction values getting away from 0.5 but decreases as feed mass flow rate and the number of columns increase. Tables 2 and 3 show the separation degree improvements, Iu and Ir, in the countercurrent-flow Frazier-scheme thermal-diffusion columns with optimal inclination angle and flow-rate fraction as a parameter based on a concurrent-flow Frazier-scheme thermal-diffusion column with the same working dimensions and the same mass
flow rate at both ends. The minimum degree of separation is at r = 0.5 and under concurrent-flow devices, as concluded from Eq. (30). Therefore, the separation degree improvements, Iu and Ir, defined by Eqs. (32) and (33), respectively, and calculated with respect to the operation at u ¼ un or r = 0.5 to show the percentage increase of separation efficiency, and thus the zero value is obtained at n = 1 and r = 0.5. It is seen in Table 2 that the separation efficiency improvement, Iu, increases with increasing the number of columns and the flow-rate fraction at 0.5. However, the separation efficiency improvement, Ir, increases with increasing
Table 1 The separation efficiency improvement with the flow-rate fraction variation, feed flow rate, optimal inclination angle and column number as parameters; CF = 0.381. Ir,u (%) r
n=1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 a
s (g/h).
n=2
n=5
n = 10
n = 20
0.10a
0.15a
0.10a
0.15a
0.10a
0.15a
0.10a
0.15a
0.10a
0.15a
80.00 30.00 10.93 2.47 0.00 2.47 10.93 30.00 80.00
80.00 30.00 10.93 2.47 0.00 2.47 10.93 30.00 80.00
64.05 23.97 8.40 1.65 0.00 2.42 10.55 28.41 75.56
66.41 24.78 8.75 1.79 0.00 2.43 10.73 28.69 76.01
42.58 15.56 4.68 0.37 0.00 1.65 10.14 26.74 70.70
48.78 17.87 5.72 0.77 0.00 1.94 10.43 27.91 73.13
22.14 6.80 0.61 0.21 0.00 1.36 9.03 24.22 57.79
31.11 10.69 2.48 0.37 0.00 1.57 9.34 24.54 59.96
12.42 1.35 0.52 0.17 0.00 0.75 7.45 22.87 46.86
12.51 2.19 1.67 0.20 0.00 0.93 8.20 22.25 50.57
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561
Table 3 The efficiency degree improvements by operating countercurrent-flow devices with optimal inclination angle and number of columns as parameters; CF = 0.381, r = 0.5 and s = 0.15 g/h. Ir (%) r
n=1
n=2
n=3
n=4
n=5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
66.67 25.00 9.11 2.06 0.00 2.06 9.11 25.00 66.67
68.72 26.66 12.09 5.73 4.00 6.14 13.06 28.65 69.92
70.60 28.65 13.18 7.40 5.99 8.23 15.10 30.55 71.70
72.36 30.15 13.62 8.35 7.22 9.58 16.43 31.82 73.06
73.22 31.42 13.73 8.91 8.04 10.53 17.41 32.82 74.11
5. Conclusions
Fig. 7. Effect of feed flow-rate on the degree of separation with optimal inclination angle and vertical column with column number as parameters; s = 0.15 g/h and CF = 0.381.
the number of columns and the flow-rate fraction values getting away from 0.5. Tables 2 and 3 show the achievable maximum separation degree of countercurrent-flow Frazier-scheme thermaldiffusion columns is larger than that of concurrent-flow operations under the same feed rate when comparing the optimal inclination angle and flow-rate fraction between these two flow-type operations. The degree of separation in countercurrent-flow operations is larger than that in concurrent-flow operations and hence the less operating cost is needed.
The separation efficiency for heavy water enrichment in countercurrent-flow inclined Frazier-scheme thermal-diffusion columns has been investigated theoretically with the number of columns and flow-rate fraction as parameters. It is shown in Fig. 3 that the optimal inclination angle for the countercurrent-flow inclined Frazier-scheme thermal-diffusion columns increases with increasing the number of columns and the flow-rate fraction values getting away from 0.5. The degree of separation decreases almost with the mass flow rate but increases with increasing the number of columns and the flow-rate fraction values getting away from 0.5, as indicated in Figs. 6 and 7. The separation efficiency improvements by operating at the optimal inclination angle is illustrated for both concurrent- and countercurrent-flow Frazierscheme thermal-diffusion columns, as represented in Tables 1–3. The countercurrent-flow Frazier-scheme thermal-diffusion columns can enhance the separation degree of heavy water compared with that in concurrent-flow Frazier-scheme thermal-diffusion columns under the same working dimensions and operating parameters. Considerable improvement in separation efficiency is obtainable by employing such a countercurrent-flow Frazierscheme thermal-diffusion device, instead of using the concurrentflow one. Also, there exists the separation efficiency enhancement by suitable adjusting the inclination angle and flow-rate fractions of two product streams. This is the valuable contribution, in the economic sense, of the present study in designing countercurrentflow Frazier-scheme thermal-diffusion columns with the less number of columns. Acknowledgement The authors wish to thank the National Science Council of the Republic of China for its financial support.
Table 2 The separation efficiency improvements by operating countercurrent-flow devices with the flow-rate fraction variation, optimal inclination angle and number of columns as parameters; CF = 0.381 and s = 0.15 g/h.
Appendix A
Iu (%)
For the multi-column operations, the degree of separation for each column can be derived as
r
n=1
n=2
n=3
n=4
n=5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
1.41 2.48 3.30 3.82 4.00 3.82 3.29 2.45 1.35
1.91 3.53 4.83 5.68 5.99 5.71 4.87 3.59 1.94
2.17 4.11 5.72 6.81 7.22 6.88 5.83 4.24 2.31
2.32 4.44 6.30 7.56 8.04 7.69 6.57 4.72 2.58
C T;1 C B;1 ¼
i HL cos u h rs Aþ ðC F C T;1 Þ 2 H cos u K cos u
(A-1)
C T;1 C B;1 ¼
HL cos u ð1 rÞs A þ ðC C Þ B;1 B;2 H cos u K cos2 u
(A-2)
C T;2 C B;2 ¼
i HL cos u h rs Aþ ðC T;1 C T;2 Þ 2 H cos u K cos u
(A-3)
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C T;2 C B;2 ¼
HL cos u ð1 r Þs Aþ ðC B;2 C B;3 Þ H cos u K cos2 u
C T;N C B;N ¼
i HL cos u h rs Aþ ðC T;N1 C T;N Þ H cos u K cos2 u
(A-5)
C T;N C B;N ¼
HL cos u ð1 rÞs Aþ ðC B;N C F Þ 2 H cos u K cos u
(A-6)
(A-4)
Multiplying Eqs. (A-1), (A-3) and (A-5) by (1 S r) and Eqs. (A-2), (A-4) and (A-6) by r, and adding the results one gets
HL cos u C T;1 C B;1 ¼ D1 ¼ K cos2 u s rð1 rÞ Aþ ðC F C T;1 þ C B;1 C B;2 Þ H cos u
HL cos u C T;2 C B;2 ¼ D2 ¼ K cos2 u s rð1 rÞ Aþ ðC T;1 C T;2 þ C B;2 C B;3 Þ H cos u
HL cos u C T;N C B;N ¼ DN ¼ K cos2 u s rð1 rÞ Aþ ðC T;N1 C T;N þ C B;N C F Þ H cos u
(A-7)
(A-8)
(A-9)
Combining Eqs. (A-7)–(A-9), one can get the formulation of degree of separation for the multi-column operations, as shown in Eqs. (28) and (29) References Chueh, P. L. and H. M. Yeh, ‘‘Thermal Diffusion in a Flat-Plate Column Inclined for Improved Performance,’’ AIChE J., 13, 37 (1967). Clusius, K. and G. Dickel, ‘‘Neues Verfahren zur Gasentmischung und Isotopentrennuug,’’ Naturwissenschaften, 26, 546 (1938). Clusius, K. and G. Dickel, ‘‘Grundlagen Eines Neuen Verfahrenz zur Gasentmischung und Isotopentrennung Durch Thermodiffusion,’’ Z. Phys. Chem. B, 44, 397 (1939).
Frazier, D., ‘‘Analysis of Transverse-Flow Thermal Diffusion,’’ Ind. Chem. Eng. Proc. Des. Dev., 1, 237 (1962). Grasseli, R. and D. Frazier, ‘‘A Comparative Study of Continuous Liquid Thermal Diffusion System,’’ Ind. Chem. Eng. Proc. Des. Dev., 1, 241 (1962). Ho, C. D., H. M. Yeh, and J. J. Guo, ‘‘An Analytical Study on the Enrichment of Heavy Water in the Continuous Thermal Diffusion Column with External Refluxes,’’ Sep. Sci. Technol., 37, 3129 (2002). Ho, C. D. and Y. T. Chen, ‘‘The Study of Continuous Thermal-Diffusion Columns on Modified Frazier-Scheme for the Enrichment of Heavy Water with Column Length Varied at a Constant Ratio,’’ Sep. Sci. Technol., 38, 2425 (2003). Ho, C. D., H. Chang, and J. J. Guo, ‘‘Effects of Inclined Angle and Aspect Ratio on Heavy Water Separation Efficiencies in Double-Flow Thermal-Diffusion Columns with External Refluxes,’’ Sep. Sci. Technol., 39 (5), 975 (2004a). Ho, C. D., Y. C. Su, J. J. Guo, and S. H. Chen, ‘‘Theoretical and Experimental Studies on the Heavy Water Enrichment by Thermal Diffusion for Improved Performance under Countercurrent-Flow Frazier Scheme and Flow-Rate Fraction Variations,’’ Sep. Sci. Technol., 39, 2299 (2004b). Jones, R. C. and W. H. Furry, ‘‘The Separation of Isotopes by Thermal Diffusion,’’ Rev. Mod. Phys., 18, 151 (1946). Kobayashi, N., R. Hirano, Y. Enokida, and I. Yamamoto, ‘‘Isotope Separation of H2–HD Gas Mixture with ‘‘Cryogenic-Wall’’ Thermal Diffusion Column with a Heated Tube,’’ J. Nucl. Sci. Technol., 39, 431 (2002). Kobayashi, N., R. Hirano, Y. Enokida, and I. Yamamoto, ‘‘H2–HT Separation Using ‘‘Cryogenic-Wall’’ Thermal Diffusion Ccolumn with Heated-Tube,’’ Fusion Sci. Technol., 44, 415 (2003). Power, J. E. and C. R. Wilke, ‘‘Separation of Liquid by Thermal Diffusion Column,’’ AIChE J., 3, 213 (1957). Rabinovich, G. D., ‘‘Theory of Thermodiffusion Separation According to the Frazier Scheme,’’ Inzh. Fiz. Zh., 31, 514 (1976). Ramser, J. H., ‘‘Theory of Thermal Diffusion under Linear Fluid Shear,’’ Ind. Eng. Chem., 49, 155 (1957). Standen, A., Encyclopedia of Chemical Technology, Wiley, New York, USA (1978). Sullivan, L. J., T. C. Ruppel, and C. B. Willingham, ‘‘Rotary and Packed Thermal Diffusion Fractionating Columns for Liquids,’’ Ind. Eng. Chem., 47, 208 (1955). Sullivan, L. J., T. C. Ruppel, and C. B. Willingham, ‘‘Packed Thermal Diffusion Column,’’ Ind. Eng. Chem., 49, 110 (1957). Suvorov, A. V. and G. D. Rabinovich, ‘‘Theory of a Thermal Diffusion Apparatus with Transverse Flows,’’ Inzh. Fiz. Zh., 41, 231 (1981). Yeh, H. M. and H. C. Ward, ‘‘The Improvement in Separation of Concentric Tube Thermal Diffusion Columns,’’ Chem. Eng. Sci., 26, 937 (1971). Yamakawa, H., K. Matsumoto, Y. Enokida, and I. Yamamoto, ‘‘Numerical Analysis of Dynamic Behavior at Start-Up of the Thermal Diffusion Column for Hydrogen Isotope Separation,’’ J. Nucl. Sci. Technol., 36, 198 (1999a). Yamakawa, H., K. Matsumoto, Y. Enokida, and I. Yamamoto, ‘‘Effect of Operational Pressure on Periodic Oscillation of Secondary Circulation near the Bottom of the Thermal Diffusion Column,’’ J. Nucl. Sci. Technol., 36, 297 (1999b). Yeh, H. M. and S. C. Yang, ‘‘The Enrichment of Heavy Water in a Batch-Type Thermal Diffusion Columns,’’ Chem. Eng. Sci., 39, 1277 (1984). Yeh, H. M., ‘‘Enrichment of Heavy Water by Thermal Diffusion,’’ Chem. Eng. Commun., 167, 167 (1998). Yeh, H. M., C. D. Ho, and Y. L. Yen, ‘‘Further Study on the Enrichment of Heavy Water in Continuous-Type Thermal-Diffusion Columns,’’ Sep. Sci. Technol., 37, 1179 (2002).
Contents lists available at ScienceDirect
Journal of the Taiwan Institute of Chemical Engineers journal homepage: www.elsevier.com/locate/jtice
The improvement of the heavy water enrichment under countercurrent-flow Frazier scheme and flow-rate fraction variations Chii-Dong Ho *, Ho-Ming Yeh, Yu-Cheng Su, Jia-Jan Guo, Jr-Wei Tu Department of Chemical and Materials Engineering, Tamkang University, 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 17 December 2008 Received in revised form 23 February 2009 Accepted 24 February 2009
The desirable cascading effect and undesirable remixing effect are the two conflicting effects produced in a thermogravitational thermal-diffusion column. A new design is proposed in the present study to obtain equilibrium separation efficiency improvement associated with a remixing-effect suppression and cascading effect enhancement. The separation equation for the heavy water enrichment by inclined thermal-diffusion columns in countercurrent-flow Frazier schemes under flow-rate fraction variations has been developed theoretically. The concentration-product term in the transport equation is treated in the enriching and stripping sections separately for more accurate analysis. It was found that the undesirable disadvantage of the remixing effect could be reduced effectively by suitably tilting the column and controlling the flow-rate fraction at both top and bottom product streams, leading to a considerable separation efficiency improvement. Numerical examples were given to illustrate the best inclination angle and flow-rate fraction on separation efficiencies for H2O–HDO–D2O system. ß 2009 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: Thermal-diffusion column Heavy water Frazier scheme Inclination angle
1. Introduction The importance of the thermal-diffusion columns has been enhanced by concentrating isotopes and rare gases successfully which are not possibly separated by conventional separation techniques such as extraction and distillation. This powerful separation technique has attracted a great attention for more detailed studies on improved thermal diffusion devices that consider the two conflicting effects of the convective currents: the advantage of the cascading effect and disadvantage of the remixing effect. In a thermal-diffusion column, a horizontal temperature gradient is provided to produce the thermal diffusion in the direction of the temperature gradient and the natural convection of the fluid in the upward direction near the hot plate or downward near the cold plate. These two convective currents cause a cascading effect, which is analogous to the multistage effect of a countercurrent extraction, resulting in a enhancement of separation (Chueh and Yeh, 1967; Jones and Furry, 1946). However, a remixing effect also occurs to diminish the separation since the convection current takes down the fluid of rich in one component at the top of the column to the bottom of the column where the fluid is rich in other component. An optimum design with proper adjustment on the convective strength by selecting the design and operating parameters was obtained effectively to
* Corresponding author. Tel.: +886 26215656x2724; fax: +886 2 26209887. E-mail address: [email protected] (C.-D. Ho).
reduce the undesirable remixing effect and preserve the desirable cascading effect, resulting in a substantial separation efficiency improvement. The enrichments obtained from some improved thermal-diffusion columns are somewhat better than the classical Clusius–Dickel column (Clusius and Dickel, 1938, 1939) such as the inclined column (Power and Wilke, 1957), wired column (Yeh and Ward, 1971), inclined moving-wall columns (Ramser, 1957), rotary columns (Sullivan et al., 1955), packed columns (Sullivan et al., 1957) and permeable and impermeable barrier columns (Ho et al., 2002, 2004). Recently, the practical applications of thermaldiffusion columns are commonly used to purify tritium in production and recover hydrogen isotopes in fusion nuclear fuel cycle (Kobayashi et al., 2002, 2003; Yamakawa et al., 1999a,b), and several theoretical model and experimental runs have been presented (Ho et al., 2002, 2004a,b; Ho and Chen, 2003; Yeh, 1998) for the enrichment of heavy water in the thermal-diffusion columns. The separation rate of the transverse-flow system, the so-called Frazier-scheme thermal-diffusion columns proposed by Frazier (1962) and Grasseli and Frazier (1962), is superior to that in the classical Clusius–Dickel column and particularly for obtaining high separation levels. A complete theory of separation in the Frazierscheme thermal-diffusion columns was developed by Rabinovich (1976) and Suvorov and Rabinovich (1981) for binary mixtures. The mathematical theory on the separation efficiency of thermaldiffusion columns in modified Frazier scheme for the enrichment of heavy water under the optimal operation and design parameters have been developed (Ho and Chen, 2003; Ho et al., 2004). The
1876-1070/$ – see front matter ß 2009 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jtice.2009.02.008
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Nomenclature A B C CB,i CF CT,i C Cˆ D g H Ir Ir,u Iu J x-OD J x-TD J z-OD K Keq n r Sn S0n T T¯ x z
ˆ defined in Eq. (11) C C, column width (cm) fraction mass concentration of heavy water in H2O–HDO–D2O system C in the bottom product stream of ith column C in the feed stream C in the top product stream of ith column pseudo product form of concentration for D2O defined by Eq. (6) ordinary diffusion coefficient (cm2/s) gravitational acceleration (cm/s2) transport coefficient defined by Eq. (3) (g/s) improvement of the degree of separation defined by Eq. (33) improvement of the degree of separation defined by Eq. (31) improvement of the degree of separation defined by Eq. (32) mass flux of heavy water in the x-direction due to ordinary diffusion (g/(cm2 s)) mass flux of heavy water in the x-direction due to thermal diffusion (g/(cm2 s)) mass flux of heavy water in the z-direction due to ordinary diffusion (g/(cm2 s)) transport coefficient defined by Eq. (4) (g/(cm s)) mass-fraction equilibrium constant of H2O–HDO– D2O system total column number flow-rate fraction total degree of separation of n columns for countercurrent-flow operation total degree of separation of n columns for concurrent-flow operation mean absolute temperature (K) arithmetic mean value of T of hot wall and cold wall (K) coordinate in the horizontal direction (cm) coordinate in the vertical direction (cm)
Greek symbols reduced thermal diffusion constant for D2O in H2O–HDO–D2O system, <0 bT (@r/@T) evaluated at reference temperature (g/(cm3 K)) Dn degree of separation of nth column, C B;n C T;n DT difference in temperature of hot and cold plates (K) m absolute viscosity of fluid (g/(cm s)) r mass density of fluid (g/cm3) s mass flow rate of top or bottom product (g/s) ti transport of heavy water along z-direction in ith column (g/s) v one-half of the plate spacing of columns (cm) u inclination angle (8)
purpose of the this study is to investigate theoretically the separation efficiency of heavy water enrichment on inclined Frazier scheme with mass flow-rate fraction variation, feed flow rate and feed concentration fraction as parameters under countercurrent-flow operations. 2. Inclined thermal-diffusion column in Frazier scheme The scheme proposed by Frazier (Frazier, 1962; Grasseli and Frazier, 1962) to connect several flat-plate thermogravitational thermal-diffusion columns with countercurrent-flow with inclination angle and flow-rate fraction variations is shown in Fig. 1. All flat-plate columns have the same plate-spacing 2v, length L, width B, and transverse-flow streams of opposite directions. The mass flow rates rs and (1 r)s with feed concentration CF is accomplished at the upper and lower ends, respectively, with the product end and the supply entrance on the opposite sides. Fig. 2 illustrates the flows and fluxes prevailing in the (i + 1)th column. The transport equation of D2O for thermal-diffusion columns in countercurrent-flow Frazier schemes was derived by Ho et al. (2004a,b). For the inclined column, the transport equation is modified by replacing the gravitational acceleration g to g cos u. Accordingly, the transport equation of the top and bottom parts in the ith column of countercurrent-flow inclined Frazier scheme are t i ¼ HC Cˆ cos u þ K cos2 u
dC i ¼ r s ðC T;i1 C T;i Þ dz z¼L
(1)
t i ¼ HC Cˆ cos u þ K cos2 u
dC i ¼ ð1 rÞs ðC B;i C B;iþ1 Þ dz z¼0
(2)
H¼
abT rgð2vÞ3 BðDTÞ2 < 0 for a < 0 6!mT¯
(3)
rg 2 b2T ð2vÞ7 BðDTÞ2 9!m2 D
(4)
and K¼
In a thermal-diffusion column, the mass transport of one component is a result of the combination of the thermal diffusion and ordinary diffusion, as shown in the first and second terms of the Eq. (1) or Eq. (2), respectively. Hence, the transport coefficients H and K represent the transport ability of the component by the thermal diffusion and ordinary diffusion, respectively. The material balances around the ith column yield:
a
r s ðC T;i1 C T;i Þ ¼ ð1 rÞs ðC B;i C B;iþ1 Þ
(5)
The pseudo concentration product C Cˆ is the product of the concentration and can be defined as C Cˆ ¼ Cð1 CÞ. The value of C Cˆ is a function of the mass equilibrium constant Keq in a heavy-water thermal diffusion system, as shown in Eq. (6) (Jones and Furry, 1946): ( ˆ C C ¼ C 0:05263 ð0:05263 0:0135K eq ÞC 0:027 1=2 ) K eq C CK eq 1 1 4
(6)
in which the mass equilibrium constant Keq for the following equilibrium relation: H2 O þ D2 O @ 2HDO
(7)
C.-D. Ho et al. / Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 555–562
557
Fig. 1. Schematic diagram of the countercurrent-flow inclined Frazier-scheme thermal-diffusion columns with flow-rate fraction variations.
is K eq ¼
½HDO2 19 19 ½H2 O ½D2 O 18 20
(8)
Keq does not vary sensitively within the operating temperature range. For instance, the values of the equilibrium constant are Keq = 3.80 and 3.793, respectively, at T = 25 8C and 30.5 8C (Standen, 1978).
The appropriate value of C Cˆ ¼ A (=constant) in Eqs. (1) and (2) may be obtained from the simultaneous solution of this set of nonlinear equations. A rather reasonable approximation for C Cˆ ¼ A in inclined Frazier-scheme thermal-diffusion columns (IFSTDC) is determined using the least square method in the previous works (Yeh et al., 2002). Accordingly, minimizing the following integral equation: R¼
Z
CB
ðC Cˆ AÞ2 dC
(9)
CT
i.e., dR ¼ dA
Z
CB
2ðC Cˆ AÞ dC ¼ 0
(10)
CT
one obtains A¼
1 CB CT
Z
CB
C Cˆ dC
(11)
CT
The appropriate value of A, thus obtained in Eq. (11), is exactly the average value of C Cˆ in the concentration range. Substitution of Eqs. (1) and (2) into Eq. (5) gives dC dC ¼ HC Cˆ cos u þ K cos2 u i (12) HC Cˆ cos u þ K cos2 u i dz z¼L dz z¼0 or dC i dC i dC i ¼ constant ¼ ¼ dz z¼L dz z¼0 dz
(13)
Eq. (13) indicates that the concentration gradient in the z direction is constant through the entire column. 2.1. Countercurrent-flow operations
Fig. 2. The one-column inclined Frazier-scheme thermal-diffusion column.
2.1.1. One-column operations The separation efficiency of a thermal-diffusion column can be defined as the composition difference between the top and bottom product and it is so-called the degree of separation Dn ¼ C T;n C B;n . The degree of separation D can be derived by the transport equations, Eqs. (1) and (2). For one-column operations, i = 1, C T;0 ¼ C F and C B;2 ¼ C F , and substituting the parameters into Eqs. (1) and
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(2) gives C T;1 C B;1 ¼
i HL cos u h rs Aþ ðC F C T;1 Þ H cos u K cos2 u
(14)
C T;1 C B;1 ¼
HL cos u ð1 rÞs A þ ðC C Þ F B;1 H cos u K cos2 u
(15)
HL cos u K cos2 u
s rð1 rÞ Aþ C F C T;1 þ C B;1 C F H cos u
s Lrð1 rÞ þ K cos2 u HLA cos u S1 ¼ K cos2 u K cos2 u
S2 ¼
(16)
(17)
where S1 is the total degree of separation of one-column operations which is equals to the degree of separation D1 as S1 ¼ C B;1 C T;1 ¼ D1 . The best angle of inclination u 1 for the maximum separation in a Frazier scheme is obtained by differentiating Eq. (17) with respect to u and setting @S1 =@u ¼ 0. The result is "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# s Lrð1 rÞ (18) u1 ¼ cos1 K 1Þ < 1
Since cos1 ðu and s < K=½Lrð1 rÞ, substitution Eq. (18) into Eq. (17) gives the equation for calculating the maximum separation as
S1;maxðuÞ ¼
HLA cos u 1 s Lrð1 rÞ þ K cos2 u1
(19)
2.1.2. Multi-column operations For two-column operations, i ¼ 2, C T;0 ¼ C F and C B;3 ¼ C F . Thus, the degree of separation is obtained from Eqs. (1)–(3) for the first and second columns separately C T;1 C B;1 ¼
i HL cos u h rs Aþ ðC F C T;1 Þ H cos u K cos2 u
(20)
C T;1 C B;1 ¼
HL cos u ð1 rÞs A þ ðC C Þ B;1 B;2 H cos u K cos2 u
(21)
C T;2 C B;2
i HL cos u h rs ¼ Aþ ðC T;1 C T;2 Þ 2 H cos u K cos u
C T;2 C B;2
HL cos u ð1 rÞs ¼ A þ ðC C Þ F B;2 H cos u K cos2 u
HL cos u K cos2 u s rð1 rÞ Aþ ðC F C T;1 þ C B;1 C B;2 Þ H cos u
(22)
(23)
HL cos u K cos2 u s rð1 rÞ Aþ ðC T;1 C T;2 þ C B;2 C F Þ H cos u
(27)
in which D1 = CB,1 CT,1, D2 = CB,2 CT,2 and S2 = CB,1 CT,2. For further multi-column operations, the degree of separation is readily obtained if we perform the calculations of Eqs. (20)–(27) recursively. The detail derivations of the degree of separation for the multi-column operations were presented in Appendix. The resultant expressions of multi-column operations for heavy water enhancement in countercurrent-flow thermal-diffusion column are as follows: n X HL cos u s rð1 rÞ Di ¼ nA þ S (28) n H cos u K cos2 u i¼1 or Sn ¼
" # n H cos u HL cos u X D þ nA i s rð1 rÞ K cos2 u i¼1
(29)
in which Dn = CB,n CT,n and Sn = CB,1 CT,n. 2.2. Concurrent-flow operations The calculation procedure for a concurrent-flow Frazierscheme thermal-diffusion column (Ho and Chen, 2003) of the same working dimension is rather simpler than that for a countercurrent-flow device. The degree of separation for ISFTDC under concurrent-flow operation is n HLA cos u Ls rð1 rÞ S0n ¼ 1 (30) 2 Ls rð1 rÞ þ K K cos u 3. The separation efficiency improvement The improvement of the countercurrent-flow Frazier-scheme thermal-diffusion columns by operating under the optimal inclination angle is best illustrated by calculating the percentage increase in separation efficiency based on the vertical countercurrent-flow Frazier-scheme thermal-diffusion columns with the flow-rate fraction equal to 0.5 as Ir;u ð%Þ ¼
Sn ðr; un ¼ un Þ Sn ðr ¼ 0:5; un ¼ 0Þ 100 Sn ðr ¼ 0:5; un ¼ 0Þ
(31)
and the separation efficiency improvement based on the concurrent-flow Frazier-scheme thermal-diffusion columns with the optimal inclination angle as
Iu ð%Þ ¼
Sn ðun ¼ u n Þ S0n ðun ¼ u n Þ 100 S0n ðun ¼ u n Þ
(32)
Similarly, the flow-rate fraction improvement for the concurrentflow device with the optimal inclination angle may be defined as (24)
and C T;2 C B;2 ¼
H cos u HL cos u ðD1 þ D2 Þ þ 2A 2 s rð1 rÞ K cos u
Multiplying Eqs. (20) and (22) by (1 r) and Eqs. (21) and (23) by r, we add these equations to get C T;1 C B;1 ¼
(26)
or
Multiplying Eq. (14) by (1 r) and Eq. (15) by r, and adding the results one gets ðC T;1 C B;1 Þ ¼
Combination of Eqs. (24) and (25) gives HL cos u s rð1 rÞ D1 D2 ¼ 2A þ S2 2 H cos u K cos u
Ir ð%Þ ¼
Sn ðr; un ¼ un Þ S0n ðr ¼ 0:5; un ¼ u n Þ 100 S0n ðr ¼ 0:5; un ¼ u n Þ
(33)
Some equipment parameters and physical properties of the H2O– HDO–D2O system (Standen, 1978; Yeh and Yang, 1984) used to calculate the separation efficiency improvement as a numerical example for the separation of heavy water are given as follows: (25)
H ¼ 1:473 104 g=s ¼ 0:53 g=h
C.-D. Ho et al. / Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 555–562
Fig. 3. Effect of flow-rate fraction variation on the optimal inclination angle with column number as parameters; s = 0.15 g/h and CF = 0.381.
K ¼ 1:549 103 g cm=s ¼ 5:5763 g cm=h K eq ¼ 3:793; L ¼ 122 cm; B ¼ 10 cm; s ¼ 2:1 g=h; 2v ¼ 0:04 cm
DT ¼ 35 K; T¯ ¼ 30:5 K; C F ¼ 0:381; DT=2v ¼ 875 K=cm 4. Results and discussions 4.1. Optimal inclination angle for best performance The study for improving the device performance in IFSTDC under flow-rate fraction variations countercurrent-flow operations is to suppress the remixing effect and to enhance the cascading effect. Fig. 3 shows the optimal inclination angle versus flow-rate fraction with feed fraction concentration and feed flowrate as parameters for one through fifth column operations. It is found in Fig. 3 that the optimal inclination angle for the countercurrent-flow inclined Frazier-scheme thermal-diffusion columns increases with increasing number of columns and with flow-rate fraction values away from 0.5. 4.2. Degree of separation in countercurrent-flow inclined Frazier-scheme thermal-diffusion columns For the countercurrent-flow Frazier-scheme thermal-diffusion columns with the optimal inclination angle and for the vertical column, the top and bottom product fraction concentrations verus
559
Fig. 4. Effect of flow-rate fraction variation on the product concentration of bottom with optimal inclination angle and vertical column with column number as parameters; s = 0.15 g/h and CF = 0.381.
r, with column number as a parameter, are given in Figs. 4 and 5. It is seen in Figs. 4 and 5 that the product fraction concentration in the bottom increases with increasing the number of columns and product fraction concentration for the countercurrent-flow Frazier-scheme thermal-diffusion column. However, the product fraction concentration at the top increases with increasing flowrate fraction but it decreases as the number of columns increases. Figs. 6 and 7 show the comparisons of the degree of separation of the optimal inclination angle ðun Þ and u n ¼ 0 in the countercurrent-flow Frazier-scheme thermal-diffusion columns with flow-rate fraction, feed flow rate and column numbers as the parameters. It is seen in Figs. 6 and 7 that the maximum degree of separation in countercurrent-flow Frazier-scheme thermal-diffusion column increases with increasing the number of columns and the flow-rate fraction values getting away from 0.5 but decreases as feed mass flow rate increases. These results would suggest that the devices with flow-rate fraction variations and inclination angle could enhance the separation efficiency in Frazier-scheme thermal-diffusion columns. 4.3. Separation efficiency improvement in countercurrent-flow operations The separation efficiency improvements, Ir;u , Iu and Ir , after n columns operations were calculated from Eqs. (31)–(33). Table 1 shows that the separation efficiency improvement, Ir;u , of the optimal inclination angle in the countercurrent-flow Frazierscheme thermal-diffusion columns is much larger than that of
C.-D. Ho et al. / Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 555–562
560
Fig. 5. Effect of flow-rate fraction variation on the product concentration of top with optimal inclination angle and vertical column with column number as parameters; s = 0.15 g/h and CF = 0.381.
Fig. 6. Effect of flow-rate fraction variation on the degree of separation with optimal inclination angle with column number as parameters; s = 0.15 g/h and CF = 0.381.
concurrent-flow vertical Frazier-scheme thermal-diffusion columns with the operation at r = 0.5. It is also found that the separation efficiency improvement increases with the flow-rate fraction values getting away from 0.5 but decreases as feed mass flow rate and the number of columns increase. Tables 2 and 3 show the separation degree improvements, Iu and Ir, in the countercurrent-flow Frazier-scheme thermal-diffusion columns with optimal inclination angle and flow-rate fraction as a parameter based on a concurrent-flow Frazier-scheme thermal-diffusion column with the same working dimensions and the same mass
flow rate at both ends. The minimum degree of separation is at r = 0.5 and under concurrent-flow devices, as concluded from Eq. (30). Therefore, the separation degree improvements, Iu and Ir, defined by Eqs. (32) and (33), respectively, and calculated with respect to the operation at u ¼ un or r = 0.5 to show the percentage increase of separation efficiency, and thus the zero value is obtained at n = 1 and r = 0.5. It is seen in Table 2 that the separation efficiency improvement, Iu, increases with increasing the number of columns and the flow-rate fraction at 0.5. However, the separation efficiency improvement, Ir, increases with increasing
Table 1 The separation efficiency improvement with the flow-rate fraction variation, feed flow rate, optimal inclination angle and column number as parameters; CF = 0.381. Ir,u (%) r
n=1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 a
s (g/h).
n=2
n=5
n = 10
n = 20
0.10a
0.15a
0.10a
0.15a
0.10a
0.15a
0.10a
0.15a
0.10a
0.15a
80.00 30.00 10.93 2.47 0.00 2.47 10.93 30.00 80.00
80.00 30.00 10.93 2.47 0.00 2.47 10.93 30.00 80.00
64.05 23.97 8.40 1.65 0.00 2.42 10.55 28.41 75.56
66.41 24.78 8.75 1.79 0.00 2.43 10.73 28.69 76.01
42.58 15.56 4.68 0.37 0.00 1.65 10.14 26.74 70.70
48.78 17.87 5.72 0.77 0.00 1.94 10.43 27.91 73.13
22.14 6.80 0.61 0.21 0.00 1.36 9.03 24.22 57.79
31.11 10.69 2.48 0.37 0.00 1.57 9.34 24.54 59.96
12.42 1.35 0.52 0.17 0.00 0.75 7.45 22.87 46.86
12.51 2.19 1.67 0.20 0.00 0.93 8.20 22.25 50.57
C.-D. Ho et al. / Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 555–562
561
Table 3 The efficiency degree improvements by operating countercurrent-flow devices with optimal inclination angle and number of columns as parameters; CF = 0.381, r = 0.5 and s = 0.15 g/h. Ir (%) r
n=1
n=2
n=3
n=4
n=5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
66.67 25.00 9.11 2.06 0.00 2.06 9.11 25.00 66.67
68.72 26.66 12.09 5.73 4.00 6.14 13.06 28.65 69.92
70.60 28.65 13.18 7.40 5.99 8.23 15.10 30.55 71.70
72.36 30.15 13.62 8.35 7.22 9.58 16.43 31.82 73.06
73.22 31.42 13.73 8.91 8.04 10.53 17.41 32.82 74.11
5. Conclusions
Fig. 7. Effect of feed flow-rate on the degree of separation with optimal inclination angle and vertical column with column number as parameters; s = 0.15 g/h and CF = 0.381.
the number of columns and the flow-rate fraction values getting away from 0.5. Tables 2 and 3 show the achievable maximum separation degree of countercurrent-flow Frazier-scheme thermaldiffusion columns is larger than that of concurrent-flow operations under the same feed rate when comparing the optimal inclination angle and flow-rate fraction between these two flow-type operations. The degree of separation in countercurrent-flow operations is larger than that in concurrent-flow operations and hence the less operating cost is needed.
The separation efficiency for heavy water enrichment in countercurrent-flow inclined Frazier-scheme thermal-diffusion columns has been investigated theoretically with the number of columns and flow-rate fraction as parameters. It is shown in Fig. 3 that the optimal inclination angle for the countercurrent-flow inclined Frazier-scheme thermal-diffusion columns increases with increasing the number of columns and the flow-rate fraction values getting away from 0.5. The degree of separation decreases almost with the mass flow rate but increases with increasing the number of columns and the flow-rate fraction values getting away from 0.5, as indicated in Figs. 6 and 7. The separation efficiency improvements by operating at the optimal inclination angle is illustrated for both concurrent- and countercurrent-flow Frazierscheme thermal-diffusion columns, as represented in Tables 1–3. The countercurrent-flow Frazier-scheme thermal-diffusion columns can enhance the separation degree of heavy water compared with that in concurrent-flow Frazier-scheme thermal-diffusion columns under the same working dimensions and operating parameters. Considerable improvement in separation efficiency is obtainable by employing such a countercurrent-flow Frazierscheme thermal-diffusion device, instead of using the concurrentflow one. Also, there exists the separation efficiency enhancement by suitable adjusting the inclination angle and flow-rate fractions of two product streams. This is the valuable contribution, in the economic sense, of the present study in designing countercurrentflow Frazier-scheme thermal-diffusion columns with the less number of columns. Acknowledgement The authors wish to thank the National Science Council of the Republic of China for its financial support.
Table 2 The separation efficiency improvements by operating countercurrent-flow devices with the flow-rate fraction variation, optimal inclination angle and number of columns as parameters; CF = 0.381 and s = 0.15 g/h.
Appendix A
Iu (%)
For the multi-column operations, the degree of separation for each column can be derived as
r
n=1
n=2
n=3
n=4
n=5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
1.41 2.48 3.30 3.82 4.00 3.82 3.29 2.45 1.35
1.91 3.53 4.83 5.68 5.99 5.71 4.87 3.59 1.94
2.17 4.11 5.72 6.81 7.22 6.88 5.83 4.24 2.31
2.32 4.44 6.30 7.56 8.04 7.69 6.57 4.72 2.58
C T;1 C B;1 ¼
i HL cos u h rs Aþ ðC F C T;1 Þ 2 H cos u K cos u
(A-1)
C T;1 C B;1 ¼
HL cos u ð1 rÞs A þ ðC C Þ B;1 B;2 H cos u K cos2 u
(A-2)
C T;2 C B;2 ¼
i HL cos u h rs Aþ ðC T;1 C T;2 Þ 2 H cos u K cos u
(A-3)
562
C.-D. Ho et al. / Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 555–562
C T;2 C B;2 ¼
HL cos u ð1 r Þs Aþ ðC B;2 C B;3 Þ H cos u K cos2 u
C T;N C B;N ¼
i HL cos u h rs Aþ ðC T;N1 C T;N Þ H cos u K cos2 u
(A-5)
C T;N C B;N ¼
HL cos u ð1 rÞs Aþ ðC B;N C F Þ 2 H cos u K cos u
(A-6)
(A-4)
Multiplying Eqs. (A-1), (A-3) and (A-5) by (1 S r) and Eqs. (A-2), (A-4) and (A-6) by r, and adding the results one gets
HL cos u C T;1 C B;1 ¼ D1 ¼ K cos2 u s rð1 rÞ Aþ ðC F C T;1 þ C B;1 C B;2 Þ H cos u
HL cos u C T;2 C B;2 ¼ D2 ¼ K cos2 u s rð1 rÞ Aþ ðC T;1 C T;2 þ C B;2 C B;3 Þ H cos u
HL cos u C T;N C B;N ¼ DN ¼ K cos2 u s rð1 rÞ Aþ ðC T;N1 C T;N þ C B;N C F Þ H cos u
(A-7)
(A-8)
(A-9)
Combining Eqs. (A-7)–(A-9), one can get the formulation of degree of separation for the multi-column operations, as shown in Eqs. (28) and (29) References Chueh, P. L. and H. M. Yeh, ‘‘Thermal Diffusion in a Flat-Plate Column Inclined for Improved Performance,’’ AIChE J., 13, 37 (1967). Clusius, K. and G. Dickel, ‘‘Neues Verfahren zur Gasentmischung und Isotopentrennuug,’’ Naturwissenschaften, 26, 546 (1938). Clusius, K. and G. Dickel, ‘‘Grundlagen Eines Neuen Verfahrenz zur Gasentmischung und Isotopentrennung Durch Thermodiffusion,’’ Z. Phys. Chem. B, 44, 397 (1939).
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