Ind. Eng. C h e m . R e s . 1988,27,481-485 Misek, T.; Berger, R.; Schroter, J. Standard Test Systems for L i p uid-Liquid Extractions; Institution of Chemical Engineers: Rugby,-England, 1985. Nemunaitis. R. R.: Eckert. J. S.: Foote. E. H.: Rollison. L. H. Chem. Eng. pro;. 1971,67(iij,60.’ Pratt, H. R. C. Znd. Eng. Chem. Process Des. Dev. 1975, 14, 74. Rathkamp, P. J.; Bravo, J. L.; Fair, J. R. Soluent Extr. Zon Exch. 1987,5,367. Row, S. B.; Koffolt, J. H.; Withrow, J. R. Trans. Am. Znst. Chem. Eng. 1941,37,559. Sakiadias, D. C.; Johnson, A. I. Znd. Eng. Chem. 1954,46, 1229. Schlichting, H. Boundary-Layer Theory, 6th ed.; McGraw-Hill: New York, 1960. Seibert, A. F. Ph.D. Dissertation, The University of Texas at Austin, Austin, 1986. Shenvood, T. K.; Evans, J. E.; Longcor, J. V. A. Znd. Eng. Chem. 1939,31, 1144.
481
Skelland, A. H. P.; Culp, G. L. Extraction Principles of Design; American Institute of Chemical Engineers: New York, 1982. Steiner, L.; Hartland, S. Handbook of Fluids in Motion; Ann Arbor Science: Ann Arbor, MI, 1983;Chapter 40. Sternling, C. V.; Scriven, L. E. AZChE J . 1959,5 , 514. Streiff, F. A.; Jancic, S. J. Ger. Chem. Eng. 1984,7, 178. Sulzer Brothers “Sulzer SMV Mixer-Packing for Extraction and Reaction Columns”. Technical Paper VT 10076/e,1982;Winterthur, Switzerland. Treybal, R. E. Mass Transfer Operations, 3rd ed.; McGraw-Hill: New York, 1980. Vermeulen, T.; Moon, J. S.; Hennico, A.; Miyauchi, T. Chem. Eng. Prog. 1966,62, 9.
Received f o r review April 6, 1987 Revised manuscript received September 29, 1987 Accepted October 16, 1987
Stationary Profiles for Periodic Cycled Separation Columns: Linear Case Bjarne Toftegird and Sten Bay Jerrgensen* Znstituttet for Kemiteknik, Technical University of Denmark, D t H , DK-2800Lyngby, Denmark
A linear mathematical model for periodic cycled binary separation columns with arbitrary feed location is formulated. By use of this model, the stationary concentration profiles in periodic cycled separation columns may be directly determined. A stage-to-stage calculation procedure for designing columns with arbitrary feed location is presented and a design example is given to illustrate the applicability of the two types of calculations. 1. Introduction Periodic cycled separation includes periodic cycled distillation as well as periodic cycled absorption or stripping. Periodic cycled distillation was introduced by Cannon (1961). The separation is performed in a tray column without downcomers. A cycle period consists of two parts: the vapor flow part (VFP) and the liquid flow part (LFP). In the VFP, the vapor flow up through the column and is collected in the condenser, while the liquid stays on the trays. In the LFP, bottom product is withdrawn from the reboiler, the liquid is shifted one tray down, ideally without being mixed with any other liquid, and the condenser holdup is partly filled on the top tray as reflux and partly taken out as distillate. Periodic cycled distillation can yield twice the column efficiency obtained in conventional distillation, as shown by, e.g., McWhirter and Lloyd (1963). As shown by, e.g., Furzer (1979), the ideal liquid draining is very important and may be difficult to obtain. Therefore, it is necessary to take special precautions against the liquid mixing. A periodic cycled separation column will be defined here to be stationary when the concentrations within a period follow identical trajectories from period to period. This condition is fulfilled when the concentrations at the beginning and at the end of a period are identical. Methods for determining stationary concentration profiles for a given column have been, mostly for special cases, published by, e.g., Chien et al. (1966), Duffy and Furzer (1978), and Baron et al. (1979). In the present paper, a general model is formulated in section 2. In section 3, a matrix formulation of the model is given and some solution procedures, taking advantage of matrix manipulation tools, are discussed. If matrix manipulation is used, the calculation becomes fairly easy and simple. The procedures discussed in section 3 are well suited for calculating the stationary
profiles once the number of trays and the feed location are known. For design purposes however, an approach is desirable where the number of trays and the feed location are determined explicitly. Such a procedure is presented in section 4. The procedure is a stage-to-stage calculation, but as the concentrations vary within the period, the design cannot be performed graphically as with the MaCabeThiele method. In section 5, an example is given that illustrates the combined use of the procedures for design and calculation of stationary profiles. The linear models formulated in this paper are often reasonable for distillation of dilute solutions and for absorption and stripping processes. In cases where nonlinear models are necessary, e.g., if the vapor-liquid relationship is nonlinear, other algorithms should be used, as described, e.g., by Toftegkd and J~lrgensen(1987). 2. Model Formulation A relatively general model for a periodic cycled separation is formulated. First the model is formulated using dimensional variables. Then by use of appropriate scaling, the model is done dimensionless. The trays are numbered from the bottom tray upward. The assumptions used in the model derivation are (1)linear vapor-liquid equilibrium relationship, y = ax b; (2) constant reboiler concentration (or infinite reboiler holdup); (3) constant point efficiency; (4) liquid amount H, remains on the tray at the end of the LFP; the drained liquid mixes with the remaning H,, on the tray below; (5) same molar vapor flow throughout the column during the VFP; (6) constant boilup ratio during the VFP; (7) negligible entrainment and weeping; (8) feed is assumed to be boiling liquid. During the VFP, the compositions x i on the trays and the amount of volatile components in the condenser (Hex,) are changing. This variation may be described by mass
+
0 1988 American Chemical Society 0888-5885/88/262~-0~81~01.~0~0
482 Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988
balances, giving the following differential equations: reboiler, assuming infinite holdup: dx,/dt' = 0
(1)
stages:
i = 1, 2, .... N
Hi dx,/dt' = V(yi-1 - yi)
(2)
condenser: d(Hi,xc)/dt'= VyN
(3)
I
+
The period duration will be
T ' = (D +
(HN-
Ho))/V
(4)
In the LFP, the liquid is drained from one tray to the tray below. From mass balances over the reboiler, each tray, and the condenser, the following equations are obtained: reboiler: xr(0) = xr(T')
(5)
stages: Hixi(0) = Hoxi(T ')
+ (Hi+I- Ho)x;+,(T ') + Fxfs(i - if)
i = 1, 2, .... N (6)
... ... ...
.. .. .. ..
condenser: = x,
xd
H, = T ' V (Hcx,)(0) = 0
(9)
This liquid drainage model implies that the tray holdups are equal to H N (=H,- D)above the feed tray and equal to H N + F on and below the feed tray. The vapor compositions are found from the equilibrium relationship and the point efficiency:
+b
(10)
+ (1 - €)yi-I
(11)
yeq,j=
yi = ty,q,j Yr
=
or yi-1 =
taxi-1
+ (1.€)taxi+ + ... + (1 .€)"2€UX1 + (1 - c)~-'ux, + b
(13)
Dimensionless Model. A more general model is obtained by making the model dimensionless by scaling time with 7 = HN/ V and holdups and other amounts with HN. Thereby the vapor flow and the absolute tray holdups are eliminated. The variables used for scaling ( 7 and H N ) do not influence the concentration profiles but only the duration of the dimensioned cycling period. The following dimensionless variables are obtained: c = H,/HN; K = Ho/HN; hi = Hi/HN; T = T'/r (14) d = D/HN; f = F/HN; When the dimensionless variables in (1)-(3) are introduced, the equations for the VFP become dx,/dt = 0
hi dxi/dt = ~
i
-- yj~
8I 0 0
0 0 0 0
.. .. ...
.. .. .. ..
(12)
Yeq,r
i = 1, 2, ..., N
d(cx,)/dt = YN
(15) (16) (17)
Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988 483 Table 11. Matrix Equation for the LFP 0
0
...
0
0 0
0
0
0 0
...
...
E : :::
xq
-
Xr
0 0
0 0
...
0 0
0 0
K/h1 I-K/hl
... ... ...
0 0 0
0 0 0
0
0
...
K/hl
0
0 0 0
0
...
0
0
0
0 0
...
0
...
- 0 0
0 -
l/hl
0
0 0 0 0
- -
-0 0
-
0 0 0
0 0 0
-Zub
‘ z ’ = ( l + f - K ) / [ T U + ( f - d ) ]*.Z ” = ( T b ) / [ T a + ( f - d ) ] .
For the LFP, (5)-(9) and the condition for product outlet
drainage models may be used just by modifying D and e. From (29) it is possible to find a CP and a cp; hence, the variation of the concentrations during the vapor flow period is given by
x(t) = Wt)x(O) + 44t) For t = T, (30) and (31) may be combined: (I - D @ ( T ) ) x ( O = ) Dq(T) + exf + f
The dimensionless cycling period duration is T = d + l - ~ (25) Instead of formulating (16) as a function of y,it may be formulated as a function of x by using (10) and (11): hi dxi/dt = -taxi - eb tyi-l (26)
+
where yt-l is given as in (13). Finally, a mass balance over the reboiler at stationary operation is needed: x, =
Tu
hl - K Tb + (f - d )xl(T) - T u (f- d )
+
(27)
3. Stationary Concentration Profiles A matrix method for finding the concentration profile at a specific time within the period, e.g., at the start of the period, is described, as well as aspects of using the method. Matrix description gives a much simpler description and, having matrix manipulation tools, a easy way to make the calculations. Defining the liquid-phase composition vector
x(t)= ( x r ~ l , x ~ , * * . , x T~ , x c ) (28) the differential equations (15), (26), and (17) for the VFP can be formulated as the matrix differential equation dx(t)/dt = Ax(t) + b (29) where A and b contain information about the vapor-liquid equilibrium, holdups, and point efficiency as shown in Table I. The equations (18)-(23) and (27) for the LFP can be formulated as the algebraic matrix equation x(0) = D x ( T ) + exf + f (30) where D and e contain information about the liquid drainage. D and f contain information for determining the reboiler concentration at the end of the LFP. The elements in (30) are shown in Table 11. Other liquid
(31) (32)
From (321, x ( 0 ) can be calculated and then x ( T ) can be determined by using (31). Chien et al. (1966) formulated A, b , D, and f for a general column, where either b is zero and the tray holdups are equal from tray to tray or t is one. They found CP and cp by using Laplace transformation. Duffy and Furzer (1978) formulated a simpler model for investigating different liquid drop distances. The equilibrium relationship was limited to the case where b = 0 and the feed could only be introduced on the top tray, that is, having equal holdup on all trays. CP may then be found by Laplace transformation. Baron et al. (1979) also described an analytical calculation of stationary profiles in order to compare Cannon’s (1961) realization of periodic cycled distillation form with their realization. The model was limited to total CP was found by reflux and b equal to zero. “straightforwardintegration” (ie., Laplace transformation). Using Laplace transformation becomes very cumbersome when the holdups are different from tray to tray and the point efficiency t is different from one. However, alternative (and easier) solution methods are possible for determining the CP matrix in (31) than using Laplace transformation. Two methods will be mentioned here. CP can generally be written as
Q(T) = exp(AT) (33) One possible solution method is to use a suitable eigenvalue algorithm, finding the eigenvalues and eigenvectors of A. This decoupled system of equations is now easy to integrate. After integration the eigenvectors are used to map the decoupled states to the original states. One must be aware of the existence of multiple eigenvalues. For the model shown in Table I, there are two eigenvalues at zero: if at - t a / h and N - if at -ea. Another possible solution method is to use the definition of the exponential matrix function: exp(AT) = I
1 1 + AT + -AT + -61A 2 T 2 + ... + ?AnTn + ... (34) 2 n.
When a suitable algorithm is used, only one matrix multiplication is needed for each term in the series. The maximum error by truncating after n terms is
Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988
484
(- TA,i,)
*+I
(35)
=
( n + l)! where A, is the smallest (greatest negative) eigenvalue of A. When b is as in Table I, the rp in (31) will be emax
p
= Tb
(36)
The two latter methods for finding CP are well-known in other connections but are also well suited for determining stationary profiles cycled separation columns. 4. Design Algorithm For design purposes, an algorithm where the number of trays and the feed location are the unknown variables may be useful. Such a tray-to-tray calculation can be derived directly by using the Laplace transform solution when the reboiler concentration is specified. The resulting algorithm is presented below for the simple case where K = 0 and 6 = 1. The equations for the general algorithm are presented in the Appendix section. (The proof is given by ToftegArd (1988).) initialization: i = l dl = eaT/h x1
=
c S , ~
=
(Qr ~1
(37)
(*)
+ ( h - T)xr)/h + ( b - Yr)/a
(38) (39)
stripping section: repeat i=i+l 1-1
cs,~= C C 8 , k d r - k
(40)
k=l
(41)
x , = c,,, - ( b - % ) / a
+
d, = eaT/h(-uT/h)L+l/(i l)!
(*)
(42)
+ (b- &)/a)
(43)
until I,> xf feed tray: Cr,1
= hc,,, - ( h -
N X f
if=i-l; i = l rectifying section: repeat i = i + l fi,k
=
hl-L@i~,l/h,l
k = 1, ..., i,
+ g,,L,l,l/h) (44)
(*)
=
ePT ]=Ii ( - 1 ) l - j (
l+k-E-1 k-1-1
i-1
ei = d i ( h = 1)
a’-’(p - q)’-’-kT’-’/b - l)!
(*) (45)
ir
(47)
(*)
where x i is the same as until x i > 3td End of algorithm N=i,+j
reboiler concn concn on tray 1 concn on tray 2 concn on tray 3 concn on tray 4 concn on tray 5 concn on tray 6 feed concn exceeded concn on tray 6 concn on tray 7 top concn exceeded
0.000 100
0.000 173 0.000418 0.001 012 0.002 37 0.005 47 0.012 52
0.01380 0.0652
The equations indicated with an asterisk (*) do not have to be recalculated if the algorithm is used more than once, e.g., with different reboiler concentrations. Note that, as for a MaCabe-Thiele construction, the concentrations found are not the exact ones. The exact concentration may be found from the algorithm described in section 3, where the number of trays and the feed location are determined. In the general case when K # 0 or t # 1, xl,d,, c ~ , fk,l, ~ , and g are then determined differently. The algorithm is, however, the same, and the modified expressions are shown in the Appendix section. 5. Design Example To illustrate the usage of the design algorithms, a example is given for purification of a dilute liquid solution in a distillation column. The vapor-liquid equilibrium relationship is y = 2.00~.The liquid is desired to be purified from a liquid molar fraction of 0.01 to that of O.OOO1. The point efficiency E is 1.0, and ideal liquid drainage is assumed ( K = 0.0). Eighty percent of the feed should be taken out as purified bottom product. The top product concentration therefore will be above 0.0496. To reduce the column size, a reflux ratio of 10 is accepted. By use of the design algorithm described in section 4, the calculations proceed as shown in Table 111. The feed composition is exceeded on tray 6, which is chosen as the feed tray. The calculated top composition is exceeded on tray 7; thus seven trays should be sufficient. The stationary concentration profile, e.g., just after the LFP, for the designed column can then be determined by using the algorithms described in section 3. Equation 34 is used with 15 terms, givintg a error of less than 1.6 X lo4. The results using first (32) and then (31) are x ( 0 ) = (0.000037,0.000158,0.000382,0.000897,0.00207,0.00473, 0.0108, 0.0499, 0.0499)T.
where gk,l,p,q
Table 111. Progression of the Tray-to-TrayDesign Algorithm, Calculating the Molar Fractions
xi
in (40)-(42),
(48)
For comparison, a conventional distillation column with the same specifications would required 13 trays with feed stream located on tray 9; that is, the number of trays is reduced to half by using periodic cycled distillation. 6. Conclusion Two algorithms are described for calculation of periodic cycled separation of binary mixtures with linear thermodynamics equilibrium relationships. The first algorithm is a generalization of earlier presented algorithms for determining stationary concentration profiles. The solution method is, however, different from that of earlier investigators. The second algorithm is an algorithm proposed for design purposes based upon a tray-to-tray calculation. This algorithm is especially easy to use for ideal columns with no intertray liquid mixing. The distillation design example illustrates the design algorithm and the complimentary usage of the two algorithms.
Nomenclature a = constant in vapor-liquid equilibrium relationship
I n d . Eng. C h e m . Res. 1988,27, 485-491
A = matrix; see (29) b = constant in vapor-liquid equilibrium relationship b = vector; see (29) c = x + (b-yo)/a c = dimensionless condenser holdup d = variable in (37) d = dimensionless distillate amount D = distillate amount D = matrix; see (30) e = vector; see (30) e = truncation error f = vector; see (30) f = dimensionless feed amount F = feed amount h = dimensionless tray holdup H = tray holdup Ho= part of liquid staying on tray I = unit matrix N = number of trays s = Laplacian variable t' = real time t = dimensionless time T ' = real period duration time T = dimensionless period duration time V = vapor flow x = liquid molar fraction y = vapor molar fraction
485
Appendix The modified expressions are
(ea-;)
Greek S y m b o l s
0'- l)!
6 = pulse function: 6(0) = 1 and 6 ( n ) = 0 for n # 0 t
= point efficiency
K
= dimensionless part of liquid staying on tray
-1
-I-k+j Tj-1
Literature Cited
X = eigenvalue r = scaling time 'p = vector; see (31)
Baron, G.; Waje, S.; Lavie, R. Chem. Eng. Sci.
1979, 35, 859-865.
Cannon, R. M. Ind. Eng. Chem. 1961,53, 629. Chien, H. H.; Sommerfeld, J. T.; Schrodt, V. N.; Parisot, P. E. Sep. S C ~1966, . 1, 281-317. Duffy, G. J.; Furzer, I. A. Chem. Eng. Sci. 1978, 33, 897-904. Furzer, I. A. AIChE J. 1979,25, 600-609. McWhirter, J. R.; Lloyd, W. A. Chem. Eng. B o g . 1963,59(6), 58-63. ToftegArd, B. Ph.D. Thesis, Instituttet for Kemiteknik, Technical University of Denmark, 1988. Tofteglrd, B.; Jsrgensen, S. B. Ind. Eng. Chem. Res. 1987, 26, 1041-1043.
= transfer matrix; see (31)
Subscripts
b = bottom product c = condenser f = feed d = distillate, top product i = tray i N = top tray r = reboiler r = rectifying section in c, s = stripping section in c,
Received for review February 13, 1987 Revised manuscript received September 9, 1987 Accepted September 25, 1987
Sulfur Solubility in Pure and Mixed Organic Solvents Steven F. Sciamanna and Scott Lynn* Department of Chemical Engineering, University of California, Berkeley, California 94720
Data on the solubility of sulfur in organic solvents were needed for the development of the University of California, Berkeley, Sulfur Recovery Process. Sulfur solubility was measured in a variety of poly(glyco1 ethers), in Nfl-dimethylaniline and quinoline, and in mixtures of some of these solvents. Special techniques were developed for making these measurements. The effect of water on sulfur solubility in selected poly(glyco1 ethers) was also investigated. The results indicate that, of the poly(glyco1 ether) solvents tested, the best solvents for sulfur are the dimethyl ethers of di(and higher)ethylene glycol. The production of elemental sulfur by the liquid-phase reaction of hydrogen sulfide and sulfur dioxide is the basis of the University of California, Berkeley, Sulfur Recovery Process (UCBSRP). Information on the physical interactions of sulfur and the process solvent is necessary in deciding how to handle sulfur in the process. For instance, if sulfur is insoluble in the process solvent, unwanted precipitation of sulfur may occur in less-than-suitable 0888-5885/88/2627-0485$01.50/0
points in the process. Furthermore, if the sulfur-solvent separation is accomplished by melting and decanting, data on solvent solubility in sulfur are necessary for estimating potential solvent losses. On the other hand, if sulfur is sufficiently soluble in the process solvent, then sulfur crystallization can be controlled in a suitable processing scheme. Thus, information on the temperature dependence of sulfur solubility in the process solvent is required 0 1988 American Chemical Society
