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The design of thermal diffusion columns
Chemical Engineering Science, 1967, Vol. 22, pp. 18854887.
Pergamon Press Ltd., Oxford.
Printed in Great Britain.
The de&p of themml diffusion colmnm
(First received 3 April 1967; in revisedfirm
17 June 1967)
where
INTRODUCTION Tmr EXF%RMENTAL verification of the theory of liquid thermal diffusion has often been attempted. After one of the most exhaustive attempts, HOFFMANand EMIZ.RY[l] concluded that, “there are large experimental flaws in thermal diffusion work, possibly in the critical nature of the annular spacing”. It is the purpose of this note to point out that, if the convective flow rate is controlled, a much larger spacing between hot and cold plates than usual may be tolerated with no loss in separative duty. This results from the fact that, for a given column geometry and product rate, there exists an optimum convective flow rate at which the separation over the column and the separative work performed by the column, is at a maximum. The introduction of the concept of separative work is shown to make possible a more sensible design procedure than has previously been available.
u=P/H Y=HL/K. DLWUSION The relationship between P, Q and S is totally masked by the final expression for S, Eq. (2). This relationship is indicated in Fig. 1, from which it may be seen that for a given geometry and product rate, there exists a unique value of Q, Qopt. at which the separation factor is at a maximum value, S,... Attempts to demonstrate the existence of this maximum by analytic means have so far failed. The position of the maximum must therefore be found by graphical or computer methods.
THEORY For simplicity and ease of illustration, consider the case of steady-state thermal diffusion between parallel plane plates, where the variation of fluid properties with temperature is neglected. Following the conventional treatment of JONESand FURRY [2], but considering the flow rates due to convection and product removal separately and independently, the net rate of transport of the light component up the column is then given by
ml
L
(1) a01 I-
7 v)
where
H=a*$(&Q-fp) Kc=- 2w(j+Pz-fiPQ++#-Qz) BPD K,=2coBpD. 0.001 I_ W
Integrating leads to [3]
Eq. (1) and introducing
the separation factor
SJw+u+l)ewr+w-u-l (w+0-1)e”‘+w--tr+1
I”
Cmvection flow,
P,
p/see
Fro. 1. Variation of separation factor (shown as S-l) against convection flow rate, for 0=2-O X10-s ems se&, p=2*3 g cm-s, a=9*1 x 10-s, T=16O”C, AT=22O”C, L=750 cm, B=IOO cm.
(2) 1885
Shorter Communications TABLE 2. VARLM~ONOF SmaX, Qopt AND SD WITH
The significance of S,,, is best realized if the concept of separative duty [4] is introduced. This concept finds particular use in the field of isotope separation, but may be generalized to include any separation process. It is an excellent measure of the work necessary to perform a given separation, e.g. in distillation, the column volume and the rate of loss of availability are both directly proportional to the separative duty. The separative duty of a plant with i input or output streams is given by SD= CMi4i
COLUMNLENOTH L
(cm)
(3)
where Mt is the molar flow rate of an input/output stream, taken to be negative when input, positive when output and the separation potential of a stream, &, is given by
cji=(2c,-
1)lnC’ . (1 - Ci)
(4)
In Table 1, the variation of SmaX, f&t and SD is shown as a function of the gap width, where SD has been calculated for a single stage with a feed mole fraction of 00072. Table 2 shows similarly the variation in the same parameters with varying column length. Although these calculations have been performed for a liquid, essentially similar results would be obtained for gaseous diffusion, provided that turbulent effects could be neglected.
SD
Qopt S nW.X
(g/W
108 (g/=c cm)
SD/LX
x 10’
fg/&
100
1.0504
0.2
0.62
6.20
200
1.0713
0.3
1.23
6.15
500
1.1162
0.4
3.10
6.20
1000
1.1672
0.6
6.20
6.20
2000
1.2413
0.8
12.33
6.17
Conditions: D = 1.6 x 10-s cms/sec, a=1*86x 10-2, ?=16O’C, AT=l75”C, B=lOO cm, P=O*Ol g/set, c~=O*O15.
p =2*0 g/cm,, o=O*lOO cm,
From Table 1, it is apparent that the separative duty is inversely proportional to the gap width, under optimum conditions. Table 2 shows that the separative duty is proportional to the column length, under the optimum conditions and it can be shown similarly that SD is proportional to the column breadth, or, for that matter to column area.
TABLE1. VAIUATIONOF Sma,, Qopt AND SD WITH OAP WIDTH. CONDITIONS AS IN FIG. 1 P
Gap width, 2w (cm)
Qwt
SDx lo5
(glsec)
S msx
(g/s=)
Wsec)
0.15
0001
1.42
0.25
6.8
0.01
1.125
0.7
6.8
0.1
1.035
2.3
6.6
I
0.5
1.0152
1.0
1.0104
6.5 3.5
6.8 6.5
J
0.01
1.110
0.10
l-032
2.3
5.1
0.50
1.0141
3.6
5.0
1.oo
1.0097
7.0 0.7
4.9 5.0
0.10
1.029
2.3
4.2
0.50
1.0126
1.0
10088
7.0
3.9
2.0
10061
8.5 4.0
4.0 4.0
0.20
0.25
tFD x 105 205 (g/=1
x 10’ (g cm/&
1
6.7
1.0
5.0
1.0
4.0
1.0
i
i
t Because the calculation was performed by hand with few significant figures and &OX estimated graphically, there was some scatter in the computed values of SD and the computation of a mean value was felt to be justified.
1886
Shorter Communications However, neglecting other heat losses, these are precisely the ways in which the heat requirements of a column vary with the column geometry. Thus, under the optimum conditions, the heat load of the column is proportional to the separative duty. Because the separation factor is at a maximum at Qopt, the separative duty is also at a maximum and thus the heat requirement per unit separative duty is at a minimum. The column therefore operates at minimum heat cost. The separative duty also varies directly as the column area, under optimum conditions, so that the capital cost of the plant, which is also usually taken as proportional to the column area, is also proportional to the separative duty, and the capital cost per unit separative duty is minimised. Thus, provided the convective flow rate of a column can be controlled, any geometric design of column can be run under minimum cost conditions. Column design for a given process may be found by computing (&xz)/BL at Qopt for any chosen geometry and product rate and scaling-up to the required separative duty, SD’, keeping (SD’W’)/B’L’= (&u$/BL. This is a vastly more sensible design procedure than that suggested by KRASNY-ERGEN [5],as it leads to far more realistic results. The Krasny-Ergen procedure is equivalent to choosing the convective flow rate and finding the geometry for which the separative duty is maxim&d. For experimental columns, by controlling the convective flow rate, relatively wide gaps can be employed while maintaining significant separation factors. In studying the data of Table 1, it should be remembered that this was computed for a=9.1 x10-3. For many liquid pairs a>1 [6]. Several methods of controlling convective flow suggest themselves. For a parallel plate column, for instance, the column may be tilted off-vertical with the hot wall uppermost, or it may be laid flat and provided with external reflux pumps. P.J. LLOYD Atomic Energy Board Pelindaba South Africa
Acknowledgment-The author would like to thank Professor M. Benedict of the Department of Nuclear Engineering, Massachusetts Institute of Technology, for many helpful discussions during the course of this study. NOTATION
width of plates, cm mole fraction of light component mole fraction of light component in bottom CB product mole fraction of light component in feed CF CT mole fraction of light component in top product mass diffusion coefficient, cm2 set-r parameter, defined in Eq. (1) :: K Kc + K,, defined in Eq. (1) L length of plates, cm product rate, mole set-l e’ convective flow rate, mole set- l B c
S SD T V W
Y Z ct P z 4 0
separation factor of column =
REFERENCES HOFFMAND. T. and EMORYA. H., A.Z.Ch.E. JI 1963 9 653-659. JONESR. C. and FURRY W. H., Rev. Mod. Phys. 1946 18 151-91. ABELSONP. H.,.Liquid thermal diffusion, USAEC Report TID-5229 1958. BENEDICTM. and PIGFORDT. H., Nuclear Chemical Engineering, pp. 396-399, McGraw-Hill KRASNY-ERGENW., Phys. Rev. 1940 58 1078. VON HALLE E., USAEC Report K-1420 1959.
1887
cT(l
-
cB)
c.(l -CA separative duty, defined in Eq. (3) *’ temperature parameter, defined in Eq. (2) parameter, defined in Eq. (2) parameter, defined in Eq. (2) distance up column, cm thermal diffusion constant density, mole cm-3 net rate of transport of light component, mole -1 sei:ation potential, Eq. (4) half gap width of column, cm
1957.
Pergamon Press Ltd., Oxford.
Printed in Great Britain.
The de&p of themml diffusion colmnm
(First received 3 April 1967; in revisedfirm
17 June 1967)
where
INTRODUCTION Tmr EXF%RMENTAL verification of the theory of liquid thermal diffusion has often been attempted. After one of the most exhaustive attempts, HOFFMANand EMIZ.RY[l] concluded that, “there are large experimental flaws in thermal diffusion work, possibly in the critical nature of the annular spacing”. It is the purpose of this note to point out that, if the convective flow rate is controlled, a much larger spacing between hot and cold plates than usual may be tolerated with no loss in separative duty. This results from the fact that, for a given column geometry and product rate, there exists an optimum convective flow rate at which the separation over the column and the separative work performed by the column, is at a maximum. The introduction of the concept of separative work is shown to make possible a more sensible design procedure than has previously been available.
u=P/H Y=HL/K. DLWUSION The relationship between P, Q and S is totally masked by the final expression for S, Eq. (2). This relationship is indicated in Fig. 1, from which it may be seen that for a given geometry and product rate, there exists a unique value of Q, Qopt. at which the separation factor is at a maximum value, S,... Attempts to demonstrate the existence of this maximum by analytic means have so far failed. The position of the maximum must therefore be found by graphical or computer methods.
THEORY For simplicity and ease of illustration, consider the case of steady-state thermal diffusion between parallel plane plates, where the variation of fluid properties with temperature is neglected. Following the conventional treatment of JONESand FURRY [2], but considering the flow rates due to convection and product removal separately and independently, the net rate of transport of the light component up the column is then given by
ml
L
(1) a01 I-
7 v)
where
H=a*$(&Q-fp) Kc=- 2w(j+Pz-fiPQ++#-Qz) BPD K,=2coBpD. 0.001 I_ W
Integrating leads to [3]
Eq. (1) and introducing
the separation factor
SJw+u+l)ewr+w-u-l (w+0-1)e”‘+w--tr+1
I”
Cmvection flow,
P,
p/see
Fro. 1. Variation of separation factor (shown as S-l) against convection flow rate, for 0=2-O X10-s ems se&, p=2*3 g cm-s, a=9*1 x 10-s, T=16O”C, AT=22O”C, L=750 cm, B=IOO cm.
(2) 1885
Shorter Communications TABLE 2. VARLM~ONOF SmaX, Qopt AND SD WITH
The significance of S,,, is best realized if the concept of separative duty [4] is introduced. This concept finds particular use in the field of isotope separation, but may be generalized to include any separation process. It is an excellent measure of the work necessary to perform a given separation, e.g. in distillation, the column volume and the rate of loss of availability are both directly proportional to the separative duty. The separative duty of a plant with i input or output streams is given by SD= CMi4i
COLUMNLENOTH L
(cm)
(3)
where Mt is the molar flow rate of an input/output stream, taken to be negative when input, positive when output and the separation potential of a stream, &, is given by
cji=(2c,-
1)lnC’ . (1 - Ci)
(4)
In Table 1, the variation of SmaX, f&t and SD is shown as a function of the gap width, where SD has been calculated for a single stage with a feed mole fraction of 00072. Table 2 shows similarly the variation in the same parameters with varying column length. Although these calculations have been performed for a liquid, essentially similar results would be obtained for gaseous diffusion, provided that turbulent effects could be neglected.
SD
Qopt S nW.X
(g/W
108 (g/=c cm)
SD/LX
x 10’
fg/&
100
1.0504
0.2
0.62
6.20
200
1.0713
0.3
1.23
6.15
500
1.1162
0.4
3.10
6.20
1000
1.1672
0.6
6.20
6.20
2000
1.2413
0.8
12.33
6.17
Conditions: D = 1.6 x 10-s cms/sec, a=1*86x 10-2, ?=16O’C, AT=l75”C, B=lOO cm, P=O*Ol g/set, c~=O*O15.
p =2*0 g/cm,, o=O*lOO cm,
From Table 1, it is apparent that the separative duty is inversely proportional to the gap width, under optimum conditions. Table 2 shows that the separative duty is proportional to the column length, under the optimum conditions and it can be shown similarly that SD is proportional to the column breadth, or, for that matter to column area.
TABLE1. VAIUATIONOF Sma,, Qopt AND SD WITH OAP WIDTH. CONDITIONS AS IN FIG. 1 P
Gap width, 2w (cm)
Qwt
SDx lo5
(glsec)
S msx
(g/s=)
Wsec)
0.15
0001
1.42
0.25
6.8
0.01
1.125
0.7
6.8
0.1
1.035
2.3
6.6
I
0.5
1.0152
1.0
1.0104
6.5 3.5
6.8 6.5
J
0.01
1.110
0.10
l-032
2.3
5.1
0.50
1.0141
3.6
5.0
1.oo
1.0097
7.0 0.7
4.9 5.0
0.10
1.029
2.3
4.2
0.50
1.0126
1.0
10088
7.0
3.9
2.0
10061
8.5 4.0
4.0 4.0
0.20
0.25
tFD x 105 205 (g/=1
x 10’ (g cm/&
1
6.7
1.0
5.0
1.0
4.0
1.0
i
i
t Because the calculation was performed by hand with few significant figures and &OX estimated graphically, there was some scatter in the computed values of SD and the computation of a mean value was felt to be justified.
1886
Shorter Communications However, neglecting other heat losses, these are precisely the ways in which the heat requirements of a column vary with the column geometry. Thus, under the optimum conditions, the heat load of the column is proportional to the separative duty. Because the separation factor is at a maximum at Qopt, the separative duty is also at a maximum and thus the heat requirement per unit separative duty is at a minimum. The column therefore operates at minimum heat cost. The separative duty also varies directly as the column area, under optimum conditions, so that the capital cost of the plant, which is also usually taken as proportional to the column area, is also proportional to the separative duty, and the capital cost per unit separative duty is minimised. Thus, provided the convective flow rate of a column can be controlled, any geometric design of column can be run under minimum cost conditions. Column design for a given process may be found by computing (&xz)/BL at Qopt for any chosen geometry and product rate and scaling-up to the required separative duty, SD’, keeping (SD’W’)/B’L’= (&u$/BL. This is a vastly more sensible design procedure than that suggested by KRASNY-ERGEN [5],as it leads to far more realistic results. The Krasny-Ergen procedure is equivalent to choosing the convective flow rate and finding the geometry for which the separative duty is maxim&d. For experimental columns, by controlling the convective flow rate, relatively wide gaps can be employed while maintaining significant separation factors. In studying the data of Table 1, it should be remembered that this was computed for a=9.1 x10-3. For many liquid pairs a>1 [6]. Several methods of controlling convective flow suggest themselves. For a parallel plate column, for instance, the column may be tilted off-vertical with the hot wall uppermost, or it may be laid flat and provided with external reflux pumps. P.J. LLOYD Atomic Energy Board Pelindaba South Africa
Acknowledgment-The author would like to thank Professor M. Benedict of the Department of Nuclear Engineering, Massachusetts Institute of Technology, for many helpful discussions during the course of this study. NOTATION
width of plates, cm mole fraction of light component mole fraction of light component in bottom CB product mole fraction of light component in feed CF CT mole fraction of light component in top product mass diffusion coefficient, cm2 set-r parameter, defined in Eq. (1) :: K Kc + K,, defined in Eq. (1) L length of plates, cm product rate, mole set-l e’ convective flow rate, mole set- l B c
S SD T V W
Y Z ct P z 4 0
separation factor of column =
REFERENCES HOFFMAND. T. and EMORYA. H., A.Z.Ch.E. JI 1963 9 653-659. JONESR. C. and FURRY W. H., Rev. Mod. Phys. 1946 18 151-91. ABELSONP. H.,.Liquid thermal diffusion, USAEC Report TID-5229 1958. BENEDICTM. and PIGFORDT. H., Nuclear Chemical Engineering, pp. 396-399, McGraw-Hill KRASNY-ERGENW., Phys. Rev. 1940 58 1078. VON HALLE E., USAEC Report K-1420 1959.
1887
cT(l
-
cB)
c.(l -CA separative duty, defined in Eq. (3) *’ temperature parameter, defined in Eq. (2) parameter, defined in Eq. (2) parameter, defined in Eq. (2) distance up column, cm thermal diffusion constant density, mole cm-3 net rate of transport of light component, mole -1 sei:ation potential, Eq. (4) half gap width of column, cm
1957.