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Further study on heavy water enrichment by thermal diffusion in the countercurrent-flow Frazier scheme
Journal of the Taiwan Institute of Chemical Engineers 42 (2011) 908–913
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Journal of the Taiwan Institute of Chemical Engineers journal homepage: www.elsevier.com/locate/jtice
Further study on heavy water enrichment by thermal diffusion in the countercurrent-flow Frazier scheme Ho-Ming Yeh *, Ching Chun Hsu Energy and Opto-Electronic Materials Research Center, Department of Chemical and Materials Engineering, Tamkang University, New Taipei city, Taiwan
A R T I C L E I N F O
A B S T R A C T
Article history: Received 9 November 2010 Received in revised form 13 May 2011 Accepted 5 June 2011 Available online 6 September 2011
The separation theory for heavy water enrichment from water–isotope mixture in the countercurrentflow Frazier scheme of thermal diffusion columns has been modified by the method of least squares. The results show that if the degree of separation is calculated from the modified separation equation, a serious error will be prevented, especially for employing the Frazier scheme with larger number of thermal diffusion columns. ß 2011 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: Heavy water enrichment Thermal diffusion Frazier scheme Modified theory
1. Introduction In addition to being the indirect fuel of fusion energy, heavy water is also the most feasible moderator and coolant for fission reactors to furnish excess neutrons which may be absorbed in materials other than uranium. Between 1940 and 1945, four heavy water production plants we placed in operation by the US Government under Manhattan District Program [1,2]. Thermal diffusion process is one of the feasible methods for separation of isotope mixtures. For separation of hydrogen isotopes, this method is particularly attractive because of the large ratio in molecular weights. It has been shown that heavy water can be enriched in a thermogravitational thermal diffusion column, called the Clusius–Dickel column [3–5]. The complete theory of separation in the C–D column was first presented by Furry et al. [6] and Jones and Furry [7]. Later, the enrichment of heavy water from water–isotope mixture was intensively studied both for operation in the C–D column [8,9] and for operation in the Frazier scheme [10–12], in which many C–D columns are connected in series [13,14]. However, the separation equations derived in previous works were obtained with the consideration that the concentration of heavy water in the pseudo product form was roughly taken as the feed concentration. This may lead to somewhat error when the degree of separation is predicted, especially for operating in the Frazier scheme with many C–D columns connected. It is the purpose of present study to modify the
* Corresponding author. Tel.: +886 2 26215656x2601; fax: +886 2 26209887. E-mail address: [email protected] (H.-M. Yeh).
separation theory for separation of heavy water from water– isotope mixture by thermal diffusion in the countercurrent-flow Frazier scheme. 2. Separation theory Fig. 1 shows the scheme proposed by Frazier, in which N flatplate thermogravitational thermal diffusion columns with column width B, column length L and plate spacing 2v, are connected in series with transverse sampling streams. The delivery of supply s with concentration CF is accomplished at the upper end of the first column and the bottom end of the last column where both streams have the opposite direction. Sampling of the product is carried out at the ends opposite to the supply entrances. When a horizontal temperature gradient (DT/2v) is applied between the two surfaces of a flat-plate thermogravitational thermal diffusion column for separation of water–isotope mixture, the following two effects will arise: (i) a flux of heavy water relative to the other is brought about by thermal diffusion toward the cold surface, and (ii) natural convective currents are produced parallel to the surfaces and down the cold surface. The combined result of these effects is to produce a concentration diffusion of heavy water between the two ends of the column. Fig. 2 illustrates the flows and fluxes prevailing in the ith thermal diffusion column with transverse sampling streams. 2.1. Separation equation The steady transport equation for enrichment of heavy water (D2O) from water–isotope mixture (H2O–HDO–D2O) in the (i + 1)th thermal diffusion column of the countercurrent-flow
1876-1070/$ – see front matter ß 2011 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jtice.2011.06.001
H.-M. Yeh, C.C. Hsu / Journal of the Taiwan Institute of Chemical Engineers 42 (2011) 908–913
Frazier scheme is [8]
Nomenclature A a 1, a 2 B C Ci CB,i CT,i CF D E g H K Keq L N Tm
909
a constant defined by Eq. (6) integration constant defined by Eq. (14), by Eq. (15) column width (cm) fractional mass concentration of heavy water C in ith column Ci in bottom-product stream of ith column Ci in top-product stream of ith column C in the feed streams ordinary diffusion coefficient of D2O in H2O–HDO– D2O system (cm2/s) deviation of DF from D gravitational acceleration (cm2/s) transport coefficient defined by Eq. (4) (g/s) transport coefficient defined by Eq. (5) (g cm/s) mass-fractional equilibrium constant of H2O– HDO–D2O system column length (cm) column number of a Frazier scheme average absolute temperature (K)
dC iþ1 dz
(1)
t iþ1 ¼ s ðC B;iþ1 C B;iþ2 Þ
(2)
t iþ1 ¼ AðHÞ þ K
while a material balance for D2O around the ith column is
s ðC T;i C T;iþ1 Þ ¼ s ðC B;iþ1 C B;iþ2 Þ
(3)
where the transport coefficients are defined by H¼
arbgð2vÞ3 BðDTÞ2 <0 6!mT m
½aðthermal diffusion constantÞ < 0 for H2 OHDOD2 O system (4)
rb2 g 2 ð2vÞ7 BðDTÞ2 K¼ 9!Dm2
(5)
In obtaining Eq. (1), assumptions were made that the concentrations were locally in equilibrium (H2O + D2O , 2HDO) with mass equilibrium constant Keq (=3.793 at 25 8C and 3.80 at 30.5 8C), and that the product form of concentration was assumed to be a constant [8] 1=2
A ¼ C½0:05263 ð0:05263 0:0135K eq ÞC 0:027Keq C 1=2
Greek symbols a thermal diffusion constant b (1/r) (@r/@T)p evaluated at Tm (1/K) a1, b1, g1 constant defined by Eq. (19), by Eq. (20), by Eq. (21) D CB,1 CT,N DF D obtained with A taken as AF DT difference in temperature between hot and cold plates (K) m absolute viscosity (g/cm s) r mass density evaluated at Tm (g/cm3) s mass flow rate (g/s) plate spacing, distance between hot and cold plates (2v) (cm) ti transport of heavy water along z-direction in ith column (g/h)
(6)
By integrating Eq. (2) through the (i + 1)th column from z = 0 (Ci+1 = CB,i+1) to z = L (Ci+1 = CT,i+1), the following equation is obtained: AH s ðC B;iþ1 C B;iþ2 Þ L (7) þ C T;iþ1 ¼ C B;iþ1 þ K K By replacing i with (i 1), Eq. (7) becomes AH s ðC B;i C B;iþ1 Þ L þ C T;i ¼ C B;i þ K K
(8)
Substitution of Eqs. (7) and (8) into Eq. (3) to eliminate CT,i and CT,i+1 yields sL ¼0 ðC B;i 2C B;iþ1 þ C B;iþ2 Þ 1 þ K or C B;iþ2 2C B;iþ1 þ C B;i ¼ 0
Fig. 1. Schematic diagram of countercurrent-flow Frazier scheme.
(9)
H.-M. Yeh, C.C. Hsu / Journal of the Taiwan Institute of Chemical Engineers 42 (2011) 908–913
910
2.2. Improvement in separation prediction In the all previous works [10–12] the product form of concentration A was assumed as a constant, in which the concentration was roughly taken as the feed concentration CF: 3=2
AF ¼ a1 C F b1 CF2 g 1 CF
(18)
where
a1 ¼ 0:05263
(19)
b1 ¼ 0:05263 0:0135K eq
(20)
1=2 g 1 ¼ 0:027Keq
(21)
and 2AF ðHÞLN ðN þ 1ÞK þ s L
DF ¼
(22)
For thermal-diffusion operation in the Frazier scheme with many columns connected in series, this will lead to somewhat error. For more precise prediction, therefore, the method of least squares may be carried out from Eq. (6) by finding the appropriate choice of constant A for the function E¼
Z
C B;1 C T;N
2
½ða1 C b1 C 2 g 1 C 3=2 Þ A dC
(23)
to have a minimum. In performing the integration of Eq. (23), the upper and lower limits are substituted by Eqs. (24) and (25): C B;1 ¼ C F þ
D
(24)
2
Fig. 2. Flows and fluxes prevailing in the ith thermal-diffusion column.
C T;N ¼ C F Eq. (9) is a second-order difference equation, whose general solution is C B;i ¼ a1 þ a2 i;
OiNþ1
(10)
in which a1 and a2 are arbitrary constants to be determined. Substitution of Eq. (10) into Eq. (8) results in sL AHL C T;i ¼ a1 þ a2 i þ (11) K K Applying two inlet conditions of feed at the top and bottom to Eqs. (11) and (10), respectively, i ¼ 0;
C T;i ¼ C F
i ¼ N þ 1;
(12)
C B;i ¼ C F
Z
C F þD=2 C F D=2
(14)
a1 2
"
" # # D 2 b D 3 D 3 CF þ CF CF 1 2 2 3 2 2 " 5=2 5=2 # 2g 1 D D CF þ CF AD 5 2 2 CF þ
D
2
or A ¼ a1 C F
AHL=K ðN þ 1Þ þ ðs L=KÞ
(15)
D ¼ C B;1 C T;N
(16)
NAHL=K ðN þ 1Þ þ ðs L=KÞ
2NAðHÞL ðN þ 1ÞK þ s L
(26)
¼0
Finally, the equation of separation is readily obtained from Eqs. (10) and (11) with the use of Eqs. (14) and (15):
D¼
ða1 C b1 C 2 g 1 C 3=2 AÞ dC ¼ 0
After integration, Eq. (26) becomes
(13)
ðN þ 1ÞAHL=K ; a1 ¼ ðN þ 1Þ þ ðs L=KÞ
D ¼ CF
(25)
2
Eqs. (24) and (25) are obtained from the overall mass balance: 2CF = CT,1 + CB,N with the use of the definition: D = CB,1 CT,N, shown in Eq. (16). By taking (@E/@A) = 0, we have
one obtains
a2 ¼
D
CF þ
2g 1 5D
b1
2 ð12CF2 þ D Þ
12 " # D 5=2 D 5=2 CF þ CF 2 2
Since kðk 1ÞX 2 kðk 1Þðk 2ÞX 3 2! 3! kðk 1Þðk 2Þðk 3ÞX 4 4! kðk 1Þðk 2Þðk 3Þðk 4ÞX 5 5!
ð1 XÞk ¼ 1 kX þ
NAHL=K ðN þ 1Þ þ ðs L=KÞ
(17)
(27)
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911
Table 1 Comparison of heavy water enrichments, DF and D, in countercurrent-flow Frazier scheme with N = 10.
s (g/h)
CF = 0.1
0.1 0.2 0.4 0.8 1.6 3.2 6.4 12.8
CF = 0.3
CF = 0.5
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
8.523 6.964 5.098 3.319 1.955 1.073 0.564 0.289
5.162 4.693 3.932 2.905 1.856 1.055 0.561 0.289
65.11 48.37 29.66 14.27 5.37 1.68 0.47 0.12
16.825 13.747 10.064 6.553 3.860 2.118 1.114 0.571
9.769 8.930 7.546 5.638 3.636 2.078 1.108 0.571
72.23 53.94 33.37 16.23 6.17 1.93 0.54 0.14
18.065 14.760 10.805 7.036 4.144 2.274 1.196 0.614
11.028 10.016 8.377 6.176 3.938 2.238 1.190 0.613
63.81 47.35 28.99 13.92 5.23 1.63 0.46 0.12
14.030 11.463 8.392 5.464 3.218 1.766 0.929 0.477
9.541 8.531 6.962 4.986 3.109 1.747 0.926 0.476
47.05 34.37 20.53 9.58 3.52 1.08 0.30 0.08
5.637 4.605 3.371 2.195 1.293 0.710 0.373 0.191
4.678 4.025 3.117 2.119 1.277 0.707 0.373 0.191
20.49 14.40 8.17 3.61 1.28 0.39 0.11 0.03
since D < 1 and D4 D2, Eq. (29) may reduce to " !# b1 g1 D2 þ A ¼ AF 1=2 12 32C
2kðk 1Þðk 2ÞX 3 3! 2kðk 1Þðk 2Þðk 3Þðk 4ÞX 5 þ þ 5!
k
ð1 þ XÞk ð1 kÞ ¼ 2kX þ
Substitution of Eq. (30) into Eq. (17) yields:
5D 2C F
D ¼ DF
3
5D 64CF3
þ
!
5
3D 4096CF5
þ
! (28)
þ
Eq. (25) becomes
5=2
10; 240CF "
b1
12
b1
12
þ
!#
g1
þ
D2
1=2
32CF
!
3g 1
A ¼ AF
"
D4
g1 1=2
32CF
!#
D2
DF AF
"
b1
12
þ
g1 1=2
32CF
!#
D2
1 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1=2 1 þ 4ðDF =AF Þ½ðb1 =12Þ þ ðg 1 =32CF Þ
(31)
1=2
2ðDF =AF Þ½ðb1 =12Þ þ ðg 1 =32CF Þ
3. Calculation
3
2ð5=2Þð3=2Þð1=2ÞðD=2C F Þ þ 3! 5 2ð5=2Þð3=2Þð1=2Þð1=2Þð3=2ÞðD=2C F Þ þ þ 5!
3=2
or
D¼ D 5=2 D 5=2 5 D 1 ¼2 1þ 2 2C F 2C F 2C F
A ¼ ða1 C F b1 CF2 g 1 CF Þ
(30)
F
one has
CF = 0.9
DF (%)
and
¼
CF = 0.7
!
3g 1 5=2
10; 240CF
D4
(29)
It should be noted that since Keq = 3.793 (at Tm = 30.5 8C) and 0 CF 1, one has [(b1/12) + (g1/32CF1/2)] = 0.1644 and (3g1/10, 240CF5/ 2 ) = 4.876 103 for CF = 0.1, while [(b1/12) + (g1/32CF1/ 2 )] = 1.732 103 and (3g1/10240CF5/2) = 2.005 105. Further,
3.1. Numerical example For the purpose of illustrating the improvement in precise prediction for the degree of separation by the present modified theory, let us employ some experimental data of previous work [8] for separation of heavy water from water–isotope mixture in the thermal diffusion columns: L = 177 cm; B = 10 cm; (2v) = 0.0406 cm; DT = 47 14 = 33 8C; Keq = 3.793 at Tm = 303.65 K (30.5 8C); H = 1.473 104 g/s = 0.53 g/h; K = 1.549 103 g cm/s = 5.58 g cm/h. From these values, the degrees of separation, DF and D, are calculated from Eqs. (22) and (31), respectively. The results are presented in Tables 1–3 and Figs. 3–8. 3.2. Results and discussion As expected, the degrees of separation, DF and D, increase with the number N of thermal diffusion columns, but decrease as the flow rate increases. Further, higher degree of separation is obtained as the feed concentration CF approaches 0.5, as shown in Tables 1–3 and Figs. 3–8.
Table 2 Comparison of heavy water enrichments, DF and D, in countercurrent-flow Frazier scheme with N = 20.
s (g/h)
0.1 0.2 0.4 0.8 1.6 3.2 6.4 12.8
CF = 0.1
CF = 0.3
CF = 0.5
CF = 0.7
CF = 0.9
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
9.995 8.835 7.170 5.208 3.366 1.971 1.078 0.565
5.508 5.242 4.763 3.984 2.936 1.869 1.060 0.563
81.47 68.54 50.54 30.71 14.62 5.45 1.69 0.47
19.730 17.440 14.155 10.281 6.644 3.891 2.128 1.116
10.383 9.912 9.055 7.642 5.697 3.662 2.087 1.110
90.02 75.96 56.32 34.52 16.63 6.26 1.95 0.54
21.184 18.726 15.198 11.038 7.134 4.178 2.285 1.198
11.776 11.201 10.166 8.490 6.243 3.967 2.248 1.193
79.90 67.18 49.49 30.02 14.26 5.31 1.64 0.46
16.452 14.543 11.803 8.573 5.540 3.245 1.774 0.931
10.308 9.717 8.678 7.068 5.045 3.133 1.755 0.928
59.61 49.67 36.00 21.29 9.82 3.57 1.09 0.30
6.610 5.843 4.742 3.444 2.226 1.304 0.713 0.374
5.214 4.798 4.117 3.174 2.146 1.287 0.710 0.374
26.78 21.78 15.17 8.50 3.71 1.30 0.39 0.11
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Table 3 Comparison of heavy water enrichments, DF and D, in countercurrent-flow Frazier scheme with N = 30.
s (g/h)
0.1 0.2 0.4 0.8 1.6 3.2 6.4 12.8
CF = 0.1
CF = 0.3
CF = 0.5
CF = 0.7
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
10.605 9.704 8.294 6.427 4.431 2.734 1.548 0.829
5.630 5.445 5.101 4.500 3.587 2.484 1.496 0.820
88.36 78.20 62.61 42.80 23.52 10.05 3.43 1.01
20.936 19.156 16.373 12.687 8.748 5.396 3.055 1.636
10.600 10.273 9.660 8.581 6.911 4.841 2.939 1.617
97.51 86.47 69.50 47.84 26.57 11.48 3.95 1.16
22.478 20.568 17.579 13.621 9.392 5.794 3.281 1.757
12.040 11.641 10.895 9.600 7.637 5.277 3.175 1.740
86.69 76.69 61.35 41.89 22.98 9.79 3.34 0.98
17.457 15.973 13.652 10.579 7.294 4.500 2.548 1.364
10.584 10.168 9.407 8.124 6.282 4.218 2.492 1.355
64.94 57.09 45.14 30.21 16.11 6.67 2.23 0.65
7.014 6.418 5.485 4.250 2.931 1.808 1.024 0.548
5.415 5.114 4.588 3.779 2.757 1.764 1.015 0.547
29.51 25.50 19.55 12.48 6.28 2.48 0.81 0.23
25
20 N = 10 σ = 0.1 g/h
N = 20
ΔF Δ
ΔF Δ
σ = 0.1 g/h
Degrees od separation, (%)
Degrees od separation, (%)
18
CF = 0.9
16 14 12 10 8 6 4
20
15
10
5
2 0 0.0
0.2
0.4
0.6
0.8
0 0.0
1.0
0.2
0.4
0.6
0.8
1.0
CF
CF Fig. 3. Comparison of the degrees of separation for N = 10 and s = 0.1 g/h.
Fig. 5. Comparison of the degrees of separation for N = 20 and s = 0.1 g/h.
The value of the product form of concentration calculated with feed concentration AF is higher than that from the modified one A, and their difference increases with the degree of separation as, from Eq. (30) " !#
increases when the number N of thermal diffusion columns in a Frazier scheme increases, or when the flow rate s decreases, as shown in Tables 1–3 and Figs. 3–8. The deviation of DF from D may be defined as
AF A ¼
b1
12
þ
g1
1=2
32CF
D2
(32)
Accordingly, the degree of separation DF estimated with AF, is larger than that from the present modification D. This fact is also evidentable from Eqs. (17) and (22). Therefore, the deviation of DF from D increases with the degrees of separation, as well as
E¼
N = 20
ΔF Δ
ΔF Δ
σ = 0.8 g/h
Degrees od separation, (%)
Degrees od separation, (%)
12 N = 10 σ = 0.8 g/h
6 5 4 3 2
10
8
6
4
2
1 0 0.0
(33)
D
One many also find in Tables 1–3 and Figs. 3–8 that E increases as the feed concentration CF approach 0.3. The deviation even seriously reaches 97.51% (DF = 0.20936 and D = 0.106) when N = 30, CF = 0.3 and s = 0.1 g/h. It should be mentioned that in this
8 7
jD DF j
0.2
0.4
0.6
0.8
1.0
CF Fig. 4. Comparison of the degrees of separation for N = 10 and s = 0.8 g/h.
0 0.0
0.2
0.4
0.6
0.8
1.0
CF Fig. 6. Comparison of the degrees of separation for N = 20 and s = 0.8 g/h.
H.-M. Yeh, C.C. Hsu / Journal of the Taiwan Institute of Chemical Engineers 42 (2011) 908–913
25
Degrees od separation, (%)
N = 30 σ = 0.1 g/h
However, this is a non-linear difference-and-differential equation, whose solution is difficult and cumbersome to be obtained even by numerical integration.
ΔF Δ
20
4. Conclusion 15
10
5
0 0.0
0.2
0.4
0.6
0.8
1.0
CF Fig. 7. Comparison of the degrees of separation for N = 30 and s = 0.1 g/h.
14
N=30
ΔF Δ
σ = 0.8 g/h
Degrees od separation, (%)
913
The separation theory for separation of heavy water from water–isotope mixture in the countercurrent-flow Frazier scheme of thermal diffusion columns, has been modified with the value of the concentration product form determined by the method of least squares, instead of that the concentration in the product form is just taken as the feed concentration, as performed in the previous works [10–12]. A numerical example for calculating the degrees of separation with various number of thermal diffusion columns and operating conditions, is illustrated. The results indicate that one may prevent a serious error if the performance of heavy water enrichment are calculated by the present modified equations, especially when employing the Frazier scheme with larger number of thermal diffusion columns. It is also found that the deviation of previous results from the present modified results increase with the number of thermal diffusion columns, especially for lower flow-rate operation.
12
References
10
[1] Murphy GM. Production of heavy water. National Nuclear Energy Series. III-4F. New York: McGraw-Hill; 1995. [2] Bebbington WP, Thayer VR. Production of heavy water. Chem Eng Prog 1959;55(9):70–8. [3] Clusius K, Dickel G. New process for separation of gas mixture and isotopes. Naturwiss 1938;26:546–52. [4] Clusius K, Dickel G. The separation-tube process for liquid. Naturwiss 1939;27:379–450. [5] Korsching H, Wirtz K. Separation of liquid mixture in the Clusius separation tube. Naturwiss 1939;27:367–73. [6] Furry WH, Jones RC, Onsager L. On the theory of isotopes separation by thermal diffusion. Phys Rev 1939;55:1083–95. [7] Jones RC, Furry WH. The separation of isotope by thermal diffusion. Rev Mod Phys 1946;18(2):151–224. [8] Yeh HM, Yang SC. The enrichment of heavy water in a batch-type thermal diffusion column. Chem Eng Sci 1984;36:1277–82. [9] Yeh HM. Effects of inclination angle and plate spacing on the enrichment of heavy water in continuous-type flat-plate thermal diffusion columns. Ann Nucl Energy 2009;36:787–92. [10] Yeh HM. Enrichment of heavy water by thermal diffusion in countercurrentflow Frazier scheme inclined for improved performance. Sep Sci Technol 2001;36(13):3015–26. [11] Yeh HM. Enrichment of heavy water in spiral wired thermal diffusion columns of the Frazier scheme. J Chin Inst Chem Eng 2002;33(2):1–7. [12] Yeh HM. Recovery of deuterium from water–isotope mixture by thermal diffusion in countercurrent-flow flat-plate Frazier scheme with optimum plate spacing. Prog Nucl Energy 2010;52:443–8. [13] Frazier D. Analysis of transverse-flow thermal diffusion. Ind Eng Chem Process Des Dev 1962;1:237–40. [14] Grasselli R, Frazier D. A comparative study of continuous liquid thermal diffusion systems. Ind Eng Chem Process Des Dev 1962;1:241–8.
8 6 4 2 0 0.0
0.2
0.4
0.6
0.8
1.0
CF Fig. 8. Comparison of the degrees of separation for N = 30 and s = 0.8 g/h. 1=2
2
case: AF = 0.00709, ½ðb1 =12Þ þ ðg 1 =32CF ÞD ¼ 3:504 105 and 4 1=2 ð3g 1 =10240CF ÞD ¼ 3:945 108 . Therefore, the simplification of A from Eq. (29) to Eq. (30) is reasonable. Mathematically, the theoretical predictions of present study may be improved if the governing equation, Eq. (2), is precisely rewritten as 3=2
2 þ g 1 Ciþ1 þ K ðHÞ½a1 C iþ1 þ b1 Ciþ1
dC iþ1 ¼ s ðC B;iþ1 C B;iþ2 Þ dz
(34)
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Journal of the Taiwan Institute of Chemical Engineers journal homepage: www.elsevier.com/locate/jtice
Further study on heavy water enrichment by thermal diffusion in the countercurrent-flow Frazier scheme Ho-Ming Yeh *, Ching Chun Hsu Energy and Opto-Electronic Materials Research Center, Department of Chemical and Materials Engineering, Tamkang University, New Taipei city, Taiwan
A R T I C L E I N F O
A B S T R A C T
Article history: Received 9 November 2010 Received in revised form 13 May 2011 Accepted 5 June 2011 Available online 6 September 2011
The separation theory for heavy water enrichment from water–isotope mixture in the countercurrentflow Frazier scheme of thermal diffusion columns has been modified by the method of least squares. The results show that if the degree of separation is calculated from the modified separation equation, a serious error will be prevented, especially for employing the Frazier scheme with larger number of thermal diffusion columns. ß 2011 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: Heavy water enrichment Thermal diffusion Frazier scheme Modified theory
1. Introduction In addition to being the indirect fuel of fusion energy, heavy water is also the most feasible moderator and coolant for fission reactors to furnish excess neutrons which may be absorbed in materials other than uranium. Between 1940 and 1945, four heavy water production plants we placed in operation by the US Government under Manhattan District Program [1,2]. Thermal diffusion process is one of the feasible methods for separation of isotope mixtures. For separation of hydrogen isotopes, this method is particularly attractive because of the large ratio in molecular weights. It has been shown that heavy water can be enriched in a thermogravitational thermal diffusion column, called the Clusius–Dickel column [3–5]. The complete theory of separation in the C–D column was first presented by Furry et al. [6] and Jones and Furry [7]. Later, the enrichment of heavy water from water–isotope mixture was intensively studied both for operation in the C–D column [8,9] and for operation in the Frazier scheme [10–12], in which many C–D columns are connected in series [13,14]. However, the separation equations derived in previous works were obtained with the consideration that the concentration of heavy water in the pseudo product form was roughly taken as the feed concentration. This may lead to somewhat error when the degree of separation is predicted, especially for operating in the Frazier scheme with many C–D columns connected. It is the purpose of present study to modify the
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separation theory for separation of heavy water from water– isotope mixture by thermal diffusion in the countercurrent-flow Frazier scheme. 2. Separation theory Fig. 1 shows the scheme proposed by Frazier, in which N flatplate thermogravitational thermal diffusion columns with column width B, column length L and plate spacing 2v, are connected in series with transverse sampling streams. The delivery of supply s with concentration CF is accomplished at the upper end of the first column and the bottom end of the last column where both streams have the opposite direction. Sampling of the product is carried out at the ends opposite to the supply entrances. When a horizontal temperature gradient (DT/2v) is applied between the two surfaces of a flat-plate thermogravitational thermal diffusion column for separation of water–isotope mixture, the following two effects will arise: (i) a flux of heavy water relative to the other is brought about by thermal diffusion toward the cold surface, and (ii) natural convective currents are produced parallel to the surfaces and down the cold surface. The combined result of these effects is to produce a concentration diffusion of heavy water between the two ends of the column. Fig. 2 illustrates the flows and fluxes prevailing in the ith thermal diffusion column with transverse sampling streams. 2.1. Separation equation The steady transport equation for enrichment of heavy water (D2O) from water–isotope mixture (H2O–HDO–D2O) in the (i + 1)th thermal diffusion column of the countercurrent-flow
1876-1070/$ – see front matter ß 2011 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jtice.2011.06.001
H.-M. Yeh, C.C. Hsu / Journal of the Taiwan Institute of Chemical Engineers 42 (2011) 908–913
Frazier scheme is [8]
Nomenclature A a 1, a 2 B C Ci CB,i CT,i CF D E g H K Keq L N Tm
909
a constant defined by Eq. (6) integration constant defined by Eq. (14), by Eq. (15) column width (cm) fractional mass concentration of heavy water C in ith column Ci in bottom-product stream of ith column Ci in top-product stream of ith column C in the feed streams ordinary diffusion coefficient of D2O in H2O–HDO– D2O system (cm2/s) deviation of DF from D gravitational acceleration (cm2/s) transport coefficient defined by Eq. (4) (g/s) transport coefficient defined by Eq. (5) (g cm/s) mass-fractional equilibrium constant of H2O– HDO–D2O system column length (cm) column number of a Frazier scheme average absolute temperature (K)
dC iþ1 dz
(1)
t iþ1 ¼ s ðC B;iþ1 C B;iþ2 Þ
(2)
t iþ1 ¼ AðHÞ þ K
while a material balance for D2O around the ith column is
s ðC T;i C T;iþ1 Þ ¼ s ðC B;iþ1 C B;iþ2 Þ
(3)
where the transport coefficients are defined by H¼
arbgð2vÞ3 BðDTÞ2 <0 6!mT m
½aðthermal diffusion constantÞ < 0 for H2 OHDOD2 O system (4)
rb2 g 2 ð2vÞ7 BðDTÞ2 K¼ 9!Dm2
(5)
In obtaining Eq. (1), assumptions were made that the concentrations were locally in equilibrium (H2O + D2O , 2HDO) with mass equilibrium constant Keq (=3.793 at 25 8C and 3.80 at 30.5 8C), and that the product form of concentration was assumed to be a constant [8] 1=2
A ¼ C½0:05263 ð0:05263 0:0135K eq ÞC 0:027Keq C 1=2
Greek symbols a thermal diffusion constant b (1/r) (@r/@T)p evaluated at Tm (1/K) a1, b1, g1 constant defined by Eq. (19), by Eq. (20), by Eq. (21) D CB,1 CT,N DF D obtained with A taken as AF DT difference in temperature between hot and cold plates (K) m absolute viscosity (g/cm s) r mass density evaluated at Tm (g/cm3) s mass flow rate (g/s) plate spacing, distance between hot and cold plates (2v) (cm) ti transport of heavy water along z-direction in ith column (g/h)
(6)
By integrating Eq. (2) through the (i + 1)th column from z = 0 (Ci+1 = CB,i+1) to z = L (Ci+1 = CT,i+1), the following equation is obtained: AH s ðC B;iþ1 C B;iþ2 Þ L (7) þ C T;iþ1 ¼ C B;iþ1 þ K K By replacing i with (i 1), Eq. (7) becomes AH s ðC B;i C B;iþ1 Þ L þ C T;i ¼ C B;i þ K K
(8)
Substitution of Eqs. (7) and (8) into Eq. (3) to eliminate CT,i and CT,i+1 yields sL ¼0 ðC B;i 2C B;iþ1 þ C B;iþ2 Þ 1 þ K or C B;iþ2 2C B;iþ1 þ C B;i ¼ 0
Fig. 1. Schematic diagram of countercurrent-flow Frazier scheme.
(9)
H.-M. Yeh, C.C. Hsu / Journal of the Taiwan Institute of Chemical Engineers 42 (2011) 908–913
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2.2. Improvement in separation prediction In the all previous works [10–12] the product form of concentration A was assumed as a constant, in which the concentration was roughly taken as the feed concentration CF: 3=2
AF ¼ a1 C F b1 CF2 g 1 CF
(18)
where
a1 ¼ 0:05263
(19)
b1 ¼ 0:05263 0:0135K eq
(20)
1=2 g 1 ¼ 0:027Keq
(21)
and 2AF ðHÞLN ðN þ 1ÞK þ s L
DF ¼
(22)
For thermal-diffusion operation in the Frazier scheme with many columns connected in series, this will lead to somewhat error. For more precise prediction, therefore, the method of least squares may be carried out from Eq. (6) by finding the appropriate choice of constant A for the function E¼
Z
C B;1 C T;N
2
½ða1 C b1 C 2 g 1 C 3=2 Þ A dC
(23)
to have a minimum. In performing the integration of Eq. (23), the upper and lower limits are substituted by Eqs. (24) and (25): C B;1 ¼ C F þ
D
(24)
2
Fig. 2. Flows and fluxes prevailing in the ith thermal-diffusion column.
C T;N ¼ C F Eq. (9) is a second-order difference equation, whose general solution is C B;i ¼ a1 þ a2 i;
OiNþ1
(10)
in which a1 and a2 are arbitrary constants to be determined. Substitution of Eq. (10) into Eq. (8) results in sL AHL C T;i ¼ a1 þ a2 i þ (11) K K Applying two inlet conditions of feed at the top and bottom to Eqs. (11) and (10), respectively, i ¼ 0;
C T;i ¼ C F
i ¼ N þ 1;
(12)
C B;i ¼ C F
Z
C F þD=2 C F D=2
(14)
a1 2
"
" # # D 2 b D 3 D 3 CF þ CF CF 1 2 2 3 2 2 " 5=2 5=2 # 2g 1 D D CF þ CF AD 5 2 2 CF þ
D
2
or A ¼ a1 C F
AHL=K ðN þ 1Þ þ ðs L=KÞ
(15)
D ¼ C B;1 C T;N
(16)
NAHL=K ðN þ 1Þ þ ðs L=KÞ
2NAðHÞL ðN þ 1ÞK þ s L
(26)
¼0
Finally, the equation of separation is readily obtained from Eqs. (10) and (11) with the use of Eqs. (14) and (15):
D¼
ða1 C b1 C 2 g 1 C 3=2 AÞ dC ¼ 0
After integration, Eq. (26) becomes
(13)
ðN þ 1ÞAHL=K ; a1 ¼ ðN þ 1Þ þ ðs L=KÞ
D ¼ CF
(25)
2
Eqs. (24) and (25) are obtained from the overall mass balance: 2CF = CT,1 + CB,N with the use of the definition: D = CB,1 CT,N, shown in Eq. (16). By taking (@E/@A) = 0, we have
one obtains
a2 ¼
D
CF þ
2g 1 5D
b1
2 ð12CF2 þ D Þ
12 " # D 5=2 D 5=2 CF þ CF 2 2
Since kðk 1ÞX 2 kðk 1Þðk 2ÞX 3 2! 3! kðk 1Þðk 2Þðk 3ÞX 4 4! kðk 1Þðk 2Þðk 3Þðk 4ÞX 5 5!
ð1 XÞk ¼ 1 kX þ
NAHL=K ðN þ 1Þ þ ðs L=KÞ
(17)
(27)
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Table 1 Comparison of heavy water enrichments, DF and D, in countercurrent-flow Frazier scheme with N = 10.
s (g/h)
CF = 0.1
0.1 0.2 0.4 0.8 1.6 3.2 6.4 12.8
CF = 0.3
CF = 0.5
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
8.523 6.964 5.098 3.319 1.955 1.073 0.564 0.289
5.162 4.693 3.932 2.905 1.856 1.055 0.561 0.289
65.11 48.37 29.66 14.27 5.37 1.68 0.47 0.12
16.825 13.747 10.064 6.553 3.860 2.118 1.114 0.571
9.769 8.930 7.546 5.638 3.636 2.078 1.108 0.571
72.23 53.94 33.37 16.23 6.17 1.93 0.54 0.14
18.065 14.760 10.805 7.036 4.144 2.274 1.196 0.614
11.028 10.016 8.377 6.176 3.938 2.238 1.190 0.613
63.81 47.35 28.99 13.92 5.23 1.63 0.46 0.12
14.030 11.463 8.392 5.464 3.218 1.766 0.929 0.477
9.541 8.531 6.962 4.986 3.109 1.747 0.926 0.476
47.05 34.37 20.53 9.58 3.52 1.08 0.30 0.08
5.637 4.605 3.371 2.195 1.293 0.710 0.373 0.191
4.678 4.025 3.117 2.119 1.277 0.707 0.373 0.191
20.49 14.40 8.17 3.61 1.28 0.39 0.11 0.03
since D < 1 and D4 D2, Eq. (29) may reduce to " !# b1 g1 D2 þ A ¼ AF 1=2 12 32C
2kðk 1Þðk 2ÞX 3 3! 2kðk 1Þðk 2Þðk 3Þðk 4ÞX 5 þ þ 5!
k
ð1 þ XÞk ð1 kÞ ¼ 2kX þ
Substitution of Eq. (30) into Eq. (17) yields:
5D 2C F
D ¼ DF
3
5D 64CF3
þ
!
5
3D 4096CF5
þ
! (28)
þ
Eq. (25) becomes
5=2
10; 240CF "
b1
12
b1
12
þ
!#
g1
þ
D2
1=2
32CF
!
3g 1
A ¼ AF
"
D4
g1 1=2
32CF
!#
D2
DF AF
"
b1
12
þ
g1 1=2
32CF
!#
D2
1 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1=2 1 þ 4ðDF =AF Þ½ðb1 =12Þ þ ðg 1 =32CF Þ
(31)
1=2
2ðDF =AF Þ½ðb1 =12Þ þ ðg 1 =32CF Þ
3. Calculation
3
2ð5=2Þð3=2Þð1=2ÞðD=2C F Þ þ 3! 5 2ð5=2Þð3=2Þð1=2Þð1=2Þð3=2ÞðD=2C F Þ þ þ 5!
3=2
or
D¼ D 5=2 D 5=2 5 D 1 ¼2 1þ 2 2C F 2C F 2C F
A ¼ ða1 C F b1 CF2 g 1 CF Þ
(30)
F
one has
CF = 0.9
DF (%)
and
¼
CF = 0.7
!
3g 1 5=2
10; 240CF
D4
(29)
It should be noted that since Keq = 3.793 (at Tm = 30.5 8C) and 0 CF 1, one has [(b1/12) + (g1/32CF1/2)] = 0.1644 and (3g1/10, 240CF5/ 2 ) = 4.876 103 for CF = 0.1, while [(b1/12) + (g1/32CF1/ 2 )] = 1.732 103 and (3g1/10240CF5/2) = 2.005 105. Further,
3.1. Numerical example For the purpose of illustrating the improvement in precise prediction for the degree of separation by the present modified theory, let us employ some experimental data of previous work [8] for separation of heavy water from water–isotope mixture in the thermal diffusion columns: L = 177 cm; B = 10 cm; (2v) = 0.0406 cm; DT = 47 14 = 33 8C; Keq = 3.793 at Tm = 303.65 K (30.5 8C); H = 1.473 104 g/s = 0.53 g/h; K = 1.549 103 g cm/s = 5.58 g cm/h. From these values, the degrees of separation, DF and D, are calculated from Eqs. (22) and (31), respectively. The results are presented in Tables 1–3 and Figs. 3–8. 3.2. Results and discussion As expected, the degrees of separation, DF and D, increase with the number N of thermal diffusion columns, but decrease as the flow rate increases. Further, higher degree of separation is obtained as the feed concentration CF approaches 0.5, as shown in Tables 1–3 and Figs. 3–8.
Table 2 Comparison of heavy water enrichments, DF and D, in countercurrent-flow Frazier scheme with N = 20.
s (g/h)
0.1 0.2 0.4 0.8 1.6 3.2 6.4 12.8
CF = 0.1
CF = 0.3
CF = 0.5
CF = 0.7
CF = 0.9
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
9.995 8.835 7.170 5.208 3.366 1.971 1.078 0.565
5.508 5.242 4.763 3.984 2.936 1.869 1.060 0.563
81.47 68.54 50.54 30.71 14.62 5.45 1.69 0.47
19.730 17.440 14.155 10.281 6.644 3.891 2.128 1.116
10.383 9.912 9.055 7.642 5.697 3.662 2.087 1.110
90.02 75.96 56.32 34.52 16.63 6.26 1.95 0.54
21.184 18.726 15.198 11.038 7.134 4.178 2.285 1.198
11.776 11.201 10.166 8.490 6.243 3.967 2.248 1.193
79.90 67.18 49.49 30.02 14.26 5.31 1.64 0.46
16.452 14.543 11.803 8.573 5.540 3.245 1.774 0.931
10.308 9.717 8.678 7.068 5.045 3.133 1.755 0.928
59.61 49.67 36.00 21.29 9.82 3.57 1.09 0.30
6.610 5.843 4.742 3.444 2.226 1.304 0.713 0.374
5.214 4.798 4.117 3.174 2.146 1.287 0.710 0.374
26.78 21.78 15.17 8.50 3.71 1.30 0.39 0.11
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Table 3 Comparison of heavy water enrichments, DF and D, in countercurrent-flow Frazier scheme with N = 30.
s (g/h)
0.1 0.2 0.4 0.8 1.6 3.2 6.4 12.8
CF = 0.1
CF = 0.3
CF = 0.5
CF = 0.7
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
DF (%)
D (%)
E (%)
10.605 9.704 8.294 6.427 4.431 2.734 1.548 0.829
5.630 5.445 5.101 4.500 3.587 2.484 1.496 0.820
88.36 78.20 62.61 42.80 23.52 10.05 3.43 1.01
20.936 19.156 16.373 12.687 8.748 5.396 3.055 1.636
10.600 10.273 9.660 8.581 6.911 4.841 2.939 1.617
97.51 86.47 69.50 47.84 26.57 11.48 3.95 1.16
22.478 20.568 17.579 13.621 9.392 5.794 3.281 1.757
12.040 11.641 10.895 9.600 7.637 5.277 3.175 1.740
86.69 76.69 61.35 41.89 22.98 9.79 3.34 0.98
17.457 15.973 13.652 10.579 7.294 4.500 2.548 1.364
10.584 10.168 9.407 8.124 6.282 4.218 2.492 1.355
64.94 57.09 45.14 30.21 16.11 6.67 2.23 0.65
7.014 6.418 5.485 4.250 2.931 1.808 1.024 0.548
5.415 5.114 4.588 3.779 2.757 1.764 1.015 0.547
29.51 25.50 19.55 12.48 6.28 2.48 0.81 0.23
25
20 N = 10 σ = 0.1 g/h
N = 20
ΔF Δ
ΔF Δ
σ = 0.1 g/h
Degrees od separation, (%)
Degrees od separation, (%)
18
CF = 0.9
16 14 12 10 8 6 4
20
15
10
5
2 0 0.0
0.2
0.4
0.6
0.8
0 0.0
1.0
0.2
0.4
0.6
0.8
1.0
CF
CF Fig. 3. Comparison of the degrees of separation for N = 10 and s = 0.1 g/h.
Fig. 5. Comparison of the degrees of separation for N = 20 and s = 0.1 g/h.
The value of the product form of concentration calculated with feed concentration AF is higher than that from the modified one A, and their difference increases with the degree of separation as, from Eq. (30) " !#
increases when the number N of thermal diffusion columns in a Frazier scheme increases, or when the flow rate s decreases, as shown in Tables 1–3 and Figs. 3–8. The deviation of DF from D may be defined as
AF A ¼
b1
12
þ
g1
1=2
32CF
D2
(32)
Accordingly, the degree of separation DF estimated with AF, is larger than that from the present modification D. This fact is also evidentable from Eqs. (17) and (22). Therefore, the deviation of DF from D increases with the degrees of separation, as well as
E¼
N = 20
ΔF Δ
ΔF Δ
σ = 0.8 g/h
Degrees od separation, (%)
Degrees od separation, (%)
12 N = 10 σ = 0.8 g/h
6 5 4 3 2
10
8
6
4
2
1 0 0.0
(33)
D
One many also find in Tables 1–3 and Figs. 3–8 that E increases as the feed concentration CF approach 0.3. The deviation even seriously reaches 97.51% (DF = 0.20936 and D = 0.106) when N = 30, CF = 0.3 and s = 0.1 g/h. It should be mentioned that in this
8 7
jD DF j
0.2
0.4
0.6
0.8
1.0
CF Fig. 4. Comparison of the degrees of separation for N = 10 and s = 0.8 g/h.
0 0.0
0.2
0.4
0.6
0.8
1.0
CF Fig. 6. Comparison of the degrees of separation for N = 20 and s = 0.8 g/h.
H.-M. Yeh, C.C. Hsu / Journal of the Taiwan Institute of Chemical Engineers 42 (2011) 908–913
25
Degrees od separation, (%)
N = 30 σ = 0.1 g/h
However, this is a non-linear difference-and-differential equation, whose solution is difficult and cumbersome to be obtained even by numerical integration.
ΔF Δ
20
4. Conclusion 15
10
5
0 0.0
0.2
0.4
0.6
0.8
1.0
CF Fig. 7. Comparison of the degrees of separation for N = 30 and s = 0.1 g/h.
14
N=30
ΔF Δ
σ = 0.8 g/h
Degrees od separation, (%)
913
The separation theory for separation of heavy water from water–isotope mixture in the countercurrent-flow Frazier scheme of thermal diffusion columns, has been modified with the value of the concentration product form determined by the method of least squares, instead of that the concentration in the product form is just taken as the feed concentration, as performed in the previous works [10–12]. A numerical example for calculating the degrees of separation with various number of thermal diffusion columns and operating conditions, is illustrated. The results indicate that one may prevent a serious error if the performance of heavy water enrichment are calculated by the present modified equations, especially when employing the Frazier scheme with larger number of thermal diffusion columns. It is also found that the deviation of previous results from the present modified results increase with the number of thermal diffusion columns, especially for lower flow-rate operation.
12
References
10
[1] Murphy GM. Production of heavy water. National Nuclear Energy Series. III-4F. New York: McGraw-Hill; 1995. [2] Bebbington WP, Thayer VR. Production of heavy water. Chem Eng Prog 1959;55(9):70–8. [3] Clusius K, Dickel G. New process for separation of gas mixture and isotopes. Naturwiss 1938;26:546–52. [4] Clusius K, Dickel G. The separation-tube process for liquid. Naturwiss 1939;27:379–450. [5] Korsching H, Wirtz K. Separation of liquid mixture in the Clusius separation tube. Naturwiss 1939;27:367–73. [6] Furry WH, Jones RC, Onsager L. On the theory of isotopes separation by thermal diffusion. Phys Rev 1939;55:1083–95. [7] Jones RC, Furry WH. The separation of isotope by thermal diffusion. Rev Mod Phys 1946;18(2):151–224. [8] Yeh HM, Yang SC. The enrichment of heavy water in a batch-type thermal diffusion column. Chem Eng Sci 1984;36:1277–82. [9] Yeh HM. Effects of inclination angle and plate spacing on the enrichment of heavy water in continuous-type flat-plate thermal diffusion columns. Ann Nucl Energy 2009;36:787–92. [10] Yeh HM. Enrichment of heavy water by thermal diffusion in countercurrentflow Frazier scheme inclined for improved performance. Sep Sci Technol 2001;36(13):3015–26. [11] Yeh HM. Enrichment of heavy water in spiral wired thermal diffusion columns of the Frazier scheme. J Chin Inst Chem Eng 2002;33(2):1–7. [12] Yeh HM. Recovery of deuterium from water–isotope mixture by thermal diffusion in countercurrent-flow flat-plate Frazier scheme with optimum plate spacing. Prog Nucl Energy 2010;52:443–8. [13] Frazier D. Analysis of transverse-flow thermal diffusion. Ind Eng Chem Process Des Dev 1962;1:237–40. [14] Grasselli R, Frazier D. A comparative study of continuous liquid thermal diffusion systems. Ind Eng Chem Process Des Dev 1962;1:241–8.
8 6 4 2 0 0.0
0.2
0.4
0.6
0.8
1.0
CF Fig. 8. Comparison of the degrees of separation for N = 30 and s = 0.8 g/h. 1=2
2
case: AF = 0.00709, ½ðb1 =12Þ þ ðg 1 =32CF ÞD ¼ 3:504 105 and 4 1=2 ð3g 1 =10240CF ÞD ¼ 3:945 108 . Therefore, the simplification of A from Eq. (29) to Eq. (30) is reasonable. Mathematically, the theoretical predictions of present study may be improved if the governing equation, Eq. (2), is precisely rewritten as 3=2
2 þ g 1 Ciþ1 þ K ðHÞ½a1 C iþ1 þ b1 Ciþ1
dC iþ1 ¼ s ðC B;iþ1 C B;iþ2 Þ dz
(34)