Modified frazier-scheme thermal-diffusion columns with column length varied at a constant ratio

Modified Frazier-scheme thermaldiffusion columns with column length varied at a constant ratio Ho-Ming Yeh and Yue-Tung Wang Department of Chemical Engineering, Republic of China

Tamkang University, Tamsui, Taiwan,

The separation theory of the modified Frazier-scheme thermal-diffusion columns with column length varied at a constant ratio has been investigated. The equations for estimating the best ratio of column length and the best column number as well as the maximum degree of separation have been derived. It is found that about 10% of improvement can be achieved by employing the present type of modified Frazier scheme rather than by using the classic Frazier scheme.

Keywords:

thermal diffusion; modified Frazier scheme

Introduction Thermal diffusion occurs when a temperature gradient in a mixture of two gases or liquids gives rise to a concentration gradient with one component concentrated near the hot wall and the other component concentrated near the cold wall. This process can be used to separate mixtures that are difficult to separate by conventional methods such as distillation and extraction. It was the great achievement of Clusius and Dickel’ to introduce the termogravitational thermal diffusion column, in which convective currents were created to produce a cascading effect, analogous to the multistage effect of countercurrent extraction, giving a relatively large separation. A more detailed study of the mechanism of thermodiffusion separation in the Clusius-Dickel column indicates that the convective currents actually have two conflicting effects: the desirable cascading effect and the undesirable remixing effect. It appears, therefore, that proper control of the convective strength might effectively suppress this undesirable remixing effect while still preserving the desirable cascading effect, and thereby lead to improved separation.3-‘8 Frazier19q20 proposed a scheme to connect several vertical thermodiffusion columns of same column length, in which the sampling streams do not pass through but move outside the columns. Theory of

Address reprint requests to Dr. Yeh at the Department of Chemical Engineering, Tamkang University, Tamsui, Taiwan, Republic of China. Received 10 December 1991; accepted 12 May 1992

192

Sep. Technol.,

1992, vol. 2, October

thermal-diffusion separation according to the Frazier scheme was given on the basis of the transport equation for a binary mixture.2’-Z3 This scheme has been employed for producing lubricating oil. A modified Frazier-scheme thermal-diffusion column in which column length varies at a constant ratio is introduced in the present study. The effect of column length ratio and column number on the separation efficiency of the modified Frazier-scheme thermal-diffusion columns with fixed-column material (i.e., the sum of column length is kept constant) is also investigated.

Separation theory Figure I shows the modified Frazier-scheme thermaldiffusion columns with forward transverse sampling streams and with column length varied at a constant ratio while Figure 2 shows the single-column type of the Frazier scheme. The delivery of supply o with concentration Co is accomplished at the upper and lower ends in a plane thermal-diffusion column with gap (20), length L, and width B, where both streams have the same direction. Sampling of the product is carried out at the ends opposite to the supply entrance. An analytic solution to Frazier-scheme thermal-diffusion columns of same column length was given by Rabinovich.21-23 Because the column lengths in the present modified Frazier-scheme thermal-diffusion columns are not the same but varied at a constant ratio, the theory developed by Rabinovich must be modified and extended. If the column width, the plate spacing, and the temperature difference in each column are all the same, and we 0 1992 Butterworth-Heinemann

Modified

Frazier-scheme

thermal-diffusion

assume that the physical properties are constant in all columns, the transport equation for nth column is r,, = HC,(l

=H

[

Wang (8)

c Y.ll- CCC I = Cs,n-l - c&n

dC, I

(1)

Substitution sults in (aXEO

H = @rpg(2w)3WA T)’ 6!/.LT, T)’

Yeh and Yue-Tung

Making material balance around the nth column yields

where

K = P$s2(2w)‘WA 9!Dp

Ho-Ming

Z, = Hz,IK

dC, - C,) - Kz

C,(l-C,)-dz

columns:

(2) + 2DwBp

Z = HzIK

(3)

9 re-

= (%),_,,

C p,n - C,,,, = aZ,, - (uZJH)(C,,,, C e,n - C,,, = aZ,, - ((+Z,,/H)(C,v,,-j

Combination C,( 1 - C,) = a (constant)

(9)

6 and 7 into Equation

Thus, (dC,,ldz) may be assumed to be constant through the entire column. Integrating Equations 6 and 7, we have

(4)

If

of Equations

(5)

of Equations

C e.n - C,s., = au z, Z,-tu

is assumed, the material balances for the top and bottom of the nth column are

- C,,,-l>

(10)

- C,,,,)

(11)

10 and 11, one obtains + &CC,.,-

n

I - c&n- I)

(12) where

= 2 (C,,, - Cc,,- ,I

(6)

u = 2Hla

(13)

Setting (7)

(14)

A, = C,,, - C,,, Equation

in which

12 becomes

Notation :

c,u - CJ

column width C fractional mass concentration of component 1 in a binary mixture C e,n C in the top product stream of nth column C0 C in the feed streams C s,n C in the bottom product stream of nth column binary Fickian diffusion coefficient D gravitational acceleration $ defined by Equation 2 I (A~.max - A&,,JA~,,,, JX.0o mass flux of component 1 in x direction owing to ordinary diffusion JX_TDmass flux of component 1 in x direction owing to thermal diffusion J,.Oo mass flux of component 1 in z direction owing to ordinary diffusion K defined by Equation 3 L total column length N column number N* the best column number R column length ratio R* the best column length ratio absolute temperature T

TtPl arithmetic mean value of T of hot plate and u 2

cold plate 2Hl(r axis parallel to the surfaces in parallel plate column

Z

Hz/K

Greek letters

;r A All A, A: AT P P

5 a! 7, 6J

thermal diffusion constant - (apld T)P evaluated at T,n degree of separation obtained in the ClusiusDickel column degree of separation, C,,, - C,,, An obtained in the Nth column AN obtained in a system of constant column length difference in temperature of the hot and cold plate absolute viscosity mass density (TL/~K mass flow rate transport of component I along z-direction in nth column one half of the plate spacing of the column

Sep. Technol.,

1992, vol. 2, October

193

Modified

Frazier-scheme

thermal-diffusion

columns:

Ho-Ming

Yeh and Yue-Tung

Wang

-Cc.

,N

, ue

I

co,oe t

Ze,N

Ze,l

-c

SIN

-L-c

CS,n-I Figure 1

AtI+

The modified

Frazier-scheme

An_, = ..A

thermal-diffusion

n

, 0s

,

us

,Ln

columns with column length varied at a constant ratio.

(15)

AN

=

ce,N

-

cS,N

zNzN

ZN

Equation 15 is the first-order difference equation of A,, whose solution is

- I

ZN+U'(ZN+U)(ZN_~+U)"" &,,z,_, +(ZN+

. . .

.z,

u)(ZN_, + u). . . .(Z, + U)1

(16)

in which AN denotes the degree of separation obtained in modified Frazier-scheme thermal-diffusion columns. If the column length ratio for every two neighboring columns (R) is kept the same, and the sum of column length of N columns in the Frazier scheme (L) is fixed, i.e., 22 -c-z

23

ZI

z2

L

= z, =

ZN

*.*=-=*__z-=Gl G-I + z2 +

R

(17)

ZN-I

. . . + z,

+

. . , +

ZN

zr(l + R + R2 + . . . + RN-')

(18)

then

1

I

3-

Figure 2

194

The single-column

Sep. Technol.,

type of the Frazier-scheme.

1992, vol. 2, October

Figure 3

----

log

(N*)

log (N,‘)

The best column number.

I

I

Modified

-l-

I

*a

Frazier-scheme

thermal-diffusion

columns:

Ho-Ming

Yeh and Yue- Tung Wang

-

$ -2 -

-3t,‘,,““““‘.“‘.,“.“‘,“.““‘I,“.“’,”.,,. 0 1

Figure 4

2

3

log (5)

4

5

The best column length ratio.

Figure 5

-- R-lL ‘1 - RN _ 1 ZN+R

(19)

j,L,Z _ RN-‘(R - 1) L IRN- 1

(20)

and from Equation 8 z N

= RN-‘(R - I)& RN- 1

(21)

where HL 5=Ku=z

crL

(22)

Substituting Equation 2 1 into Equation 16, one obtains the separation equation in terms of column length ratio R and column number N as RN-~

AN = au

1 -’

RN-~ -’ RN-~ -’ 1[ 1 -’ +... ’+ RN_1 .., ’ + RN-‘(R

RN-‘(R - 115

- l)<

’ + RN-*(R - l)[

RN-‘(R - l){

=

au{fW, R, 5)) (23)

For the case that the length of each column in the Frazier scheme is the same, R = 1, and Equation 23 reduces to that derived by Rabinovich,20-22

The maximum

degree of separation.

Ak=au{&[l(‘)“‘j} =

4foWo, 5))

(24)

For a single-column type of the Frazier scheme, as shown in Figure 2, N = 1, Equation 24 reduces to A;=, = C, - C, = au-

5 5+1

(25)

For a single-column type of the Clusius-Dickel scheme, the feed with flow rate 2a is introduced at the center of the column, and the products with flow rate u are withdrawn from the top and the bottom of the column. The separation equation of the Clusius-Dickel column was given by Jones, Furry and Onsager’ as A=C,-C,=au[l

-exp(-{)I

(26)

When the value of 5 becomes large, i.e., 5 B 1, both Equations 25 and 26 reduce to A$=,=A=au

(27)

Best column number and best column length ratio The best column number N* and best column length ratio R * for maximum degree of separation A,,,,ax in a modified Frazier scheme are obtained by employing the method of two variable univariant search in Equation 23 with 5 as parameter. Once N* and R* are known A,,,,, is calculated from Equation 23. The results are presented in Figures 3-5. It is found from these figures that all the results are nearly straight lines, and thus the following expressions are reached N* = [1.35355’.‘]

(28)

R* = 1.44545-“,5 + 1

(29)

Similarly, the best column number and the maximum Sep. Technol.,

1992, vol. 2, October

195

Modified

Frazier-scheme

thermal-diffusion

columns:

degree of separation in the Frazier scheme in which the column length of all columns is the same, have been also obtained. The results are also presented in Figures 3 and 5 and can be expressed by the following: N,* = [1.126{“.5]

(31)

1AR max= 0 6042 Jo.5

(32)

au

’

Note that the values N* and ZV$calculated from Equations 28 and 31 must be taken as integers, and that AN,tTUlX =

1.0935

AkBrnaX

(33)

Improvement in separation The improvement in separation by employing the modified Frazier scheme is best illustrated by calculating the percentage increase in maximum separation based on the Frazier scheme I=

AN,max

-

&ma,

4Lll?tX

Substitution of Equations 30 and 32, or Equation into Equation 34 results in

33

I = 9.35% Therefore, about 10% of improvement can be achieved by employing the present type of modified Frazier scheme rather than by using the classical Frazier scheme. For the purpose of illustration, let us employ the equipment parameters for the separation of a benzone and n-heptane mixture given in the previous work as follows:4 B = 20 cm, 2w = 0.09 cm, AT = 2OS”C, T = 328 K, Co = 0.5, C,(l - C,,) = 0.25. The transport coefficients thus obtained from the experimental data are: H = 0,845 g/min and K = 419 (g)(cm)/min. Here we assign the total length of thermal-diffusion columns as L = 50m. Substitution of the preceding values into Equations 5, 13,28,29,30,31, and 32, the best column number and the best column length ratio as well as the maximum degree of separation are calculated from the corresponding equations. The results are presented in

Ho-Ming Table 1

Yeh and Yue-Tung

Wang

Comparison of separation

obtained in the columns of

constant and varied column length R=l

R#l

g:s

N;

&&ax (%I

0.05 0.1 0.2 0.4 0.6 0.8 1.0

5 7 10 13 17 19

36.05 25.46 18.00 12.73 10.36 9.00

2.0 4.0 6.0 8.0 10

21 30 43 52 60 67

8.05 5.69 4.02 3.28 2.85 2.55

N*

6 8 11 16 20 23 26 36 51 63 72 81

R” 1.3415 1.2415 1.1708 1.1208 1.0986 1.0854 1.0764 1.0540

1.0382 1.0312 1.0270

1.0242

A N,max

(%I

39.42 28.84 19.68 13.92 11.33 9.84 8.80 6.22 4.40 3.58 3.11 2.78

obtained as shown in Equations 28-30. About 10% improvement can be achieved by employing a modified Frazier-scheme thermal-diffusion column rather than by using the classic one. In the given illustration, the best column number and best column length ratio for maximum degree of separation have been calculated. It is found that as the flow rate increases, the best column number increases, while the best column length ratio decreases.

References

13.

Clusius, K. and Dickel G. Nature 1938,26, 546 Furry, W.H., Jones, R.C. and Onsager, L. Phys. Rev., 1939, 55, 1083 Powers, J.E. and Wilke, C.R. AIChE J., 1957, 3, 213 Chueh, P.L. and Yeh. H.M. AIChE J.. 1967.13. 37 Washall, T.A. and Molpolder, F.W. Ind. Eng. Ch>m. Process Des. Dev. 1982, 1, 26 Yeh, H.M. and Ward, H.C. Chem. Eng. Sci. 1971, 26, 937 Ramser, J.H. Znd. Eng. Chem. 1957, 49, 155 Yeh, H.M. and Tsai, C.S. Chem. Eng. Sci. 1972, 27, 2065 Sullivan, L.J., Ruppel, T.C. and Willingham, C.B. Ind. Eng. Chem. 1957. 49, 110 Yeh, H.M. and Chen, S.M. Chem. Eng. Sci. 1973, 28,1803 Yeh. H.M. and Tsai. SW.. Sep. Sci. Technol. 1981, 16. 63 Yeh, H.M. and Tsai, S.W., Sep. Sci. Technol. 1982,17, ‘1075 Lorenz, M. and Emergy, A.H., Jr. Chem. Eng. Sci. 1959,11,

Table 1.

14.

16 Yeh, H.M. and Chu, T.Y. Chem.

Discussion and conclusion

15. 16. 17.

The modified Frazier-scheme thermal-diffusion columns in which the column length is varied at a constant ratio, has been introduced. The separation theory of the modified Frazier-scheme thermal-diffusion columns has been derived. The equations of the best column number, the best column length ratio and maximum degree of separation in the modified columns are

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Sep. Technol.,

1992, vol. 2, October

1.

2.

3. 4. 5. 6.

7. 8. 9.

10. 11.

12.

18. 19. 20.

21. 22. 23.

Eng. Sci. 1974, 29, 1421 Yeh, H.M. and Ho, F.K. Chem. Eng. Sci. 1975, 30,‘1381 Yeh. H.M. and Tsai. S.W. J. Chem. Enp. Jon. 1981. 14. 90 Yeh, H.M. and Tsai, S.W. and Lin, C.S:AiChE J. 1486: 32, 971 Tsai, S.W. and Yeh, H.M. Znd. Chem. Eng. 1986, 25, 367 Frazier, D. Ind. Chem. Eng. Proc. Des. Dev. 1962, 1, 237 Grasselli R. and Frazier D. Ind. Chem. Eng. Proc. Deu. 1962, 1, 241 Rabinovich, G.D. Inzh. Fiz. Zh. 1976, 31, 21 Rabinovich, G.D. Inzh. Fiz. Zh. 1976, 31, 514 Suvorov, A.V. and Rabinovich, G.D. Inzh. Fiz. Zh. 1981,41, 231