2579
I n d . Eng. Chem. Res. 1992,31,257S2587 F o r k , G. S.; Coolidge, A. S. Relations between Distribution Ratio, Temperature, and Concentration in the System: Water, Ether, Succinic Acid. J. Am. Chem. Soc. 1919,41,150. Gmehling, J.; Onken, U.;h e y - N i e s , J. R. Vapor-Liquid Equilibrium Data Collection, Aqueous Systems, Supplement 2; DECHEMA: Frankfurt, 1988. Hafez, M.; Hartlaud, 5.Physical Properties of Three Ternary Systerne. Chem. Eng. Sci. 1976,31,247. Hen, W.; Lorentz, E. Physico-chemical Investigations in Dioxane. 2. Phys. Chem. 1929,140,406(in German). Kertes, A. S.; King, C. J. Extraction Chemistry of Fermentation Product Carboxylic Acids. Biotechnol. Bioeng. 1986,27, 269. King, C. J. Separation Processes, 2nd ed.;McGraw-Hilk New York, 1980,pp 115117. King, C. J.; Starr, J. N. Recovery of Carboxylic Acids by Precipitation from Organic Solutions. U.S.Patent 5,104,492.April 14, 1992. Korenman, Ya. I. Phenol Solvates in Organic Solvents. R w s . J. Phys. Chem. 1972,46,42-43.
Schriver, L.; Corset, J. Infrared Spectrometric Study of Molecular Aeeocitions among Thiocyanic Acid, Water and Methyl Ieobutyl Ketone. Application to the Extraction of Thiocyanic Acid. J. Chim. Phys. 1973,70,1483 (in French). Sorensen, J. W.; Arlt, W . Liquid-Liquid Equilibrium Data Collection. Binary System; DECHEMA: FmMurt, 1979;pp 380,467. Stan,J. N.; King, C. J. "Water Enhanced Solubility of Carboxylic Acids in Organic Solventa and Ita Applications to Extraction Processes"; Report No. LBL-31996,Lawrence Berkeley Laboratory, February 1992. Van Duyne, R.; Taylor, S. A.; Chrietian, S. D.; Affsprung, H. E. Self-Association and Hydration of Benzoic Acid in Benzene. J. Phys. Chem. 1967,71,3427. Wood, G. 0.; Mueller, D. D.; Christian, S. D.; Affsprung, H. E. Hydration of Benzoic Acid in Diphenylmethane. J. Phys. Chem. 1966,70,2691.
Received for review March 10, 1992 Accepted August 24,1992
Constrained Separations and the Analysis of Binary Homogeneous Separators Angelo Lucia* and Hailang Li Department of Chemical Engineering, Clarkson University, Potsdam, New York 13699-5705
Various properties of multistage equilibrium separation process models involving binary homogeneous mixtures are studied. The phase rule and mass and energy balance equations are used to derive analytical relations describing changes in compositions, total flow rates, and heat duties with respect to changes in stage temperatures in the column. These relationships are used to establish the number of steadystate solutions for a variety of column specifications. Using algebraic techniques, it is shown that those seta of specifications that constrain the separation made by the column admit unique steady-state solutions, while those that do not can have more than one solution. Numerical examples are used to provide clear illustration of the analysis while Ponchon-Savarit diagrams are used to give geometric interpretation to the algebraic analysis in order to make the results accessible at the undergraduate level. I. Introduction The past decade has seen somewhat of a renewed interest in analysis and numerical simulation issues surrounding the number of steady-state solution to multistage separation proceeses. The papers by Doherty and Perkins (1982) and Sridhar and Lucia (1989, 1990) all present proofs of solution uniqueness for equilibrium-staged distillation processes involving homogeneous mixtures (i.e., mixture which do not exhibit vapor- or liquid-phase instability for any conditions found in the column). Doherty and Perkins analyzed both binary multistage and multicomponent single-stage columns under constant molar overflow (CMO) conditions using linear stability theory and showed that the dynamic CMO model equations poeeeea a unique solution in either case. Sridhar and Lucia, on the other h d , analyzed the staady-state model M y using & i d fixed-point theory and relaxed the constant molar overflow assumption by including the energy balance equation for each stage in the column. In the binary case, solution uniqueness was established for multistage separators with (i) fixed temperature and pressure (TP)profiles,(ii) fmed heat duty and pressure (QP)profiles, and (i) specified reflux ratio and total bottom flow (RB). For multicomponent homogeneous mixtures, steady-state solution uniqueneee was only proved for fixed temperature and preasure profiles. In all casea,the only restrictions that
were placed on the homogeneous fluid phases involved were that they satisfy the Gibbs-Duhem equation and that they not exhibit retrograde behavior. In contract, there are a number of numerical studiea that show that multistage separators can exhibit steady-state solution multiplicity in certain circumstances. These include the numerical work of Magnussen et al. (1979), Prokopakis and Seider (1983), Kovach and Seider (1987), Venkataraman and Lucia (1988), Kingsley and Lucia (1988), Rovaglio and Doherty (1990),and, most recently, the paper by Jacobsen and Skogestad (1991). However, in all of these studies, with the exception of the last, the mixture is potentially heterogeneous (i.e., exhibits liquidphase splitting) for normal conditions encountered within the column, and thus it is at least widely believed, although unproven, that solution multiplicity in this situation is due to the heterogeneous nature of the liquid mixture. On the other hand, Jacobsen and Skogestad (1991) illustrate steady-state (output) multiplicity for specified reflux and boil-up flow rates with zero heat duties at all intermediate stages in two-product distillation columns using a binary mixture with ideal vaporliquid equilibrium. However, it should be noted that the analytical treatment of Jacobsen and Skogeetad is not rigorous because it is based on the assumption that specifications of distillate rate and vapor boil-up admit a unique steady-state solution. Furthermore, the numerical results presented in their paper in no way
0888-5885/92/2631-2579$03.00/00 1992 American Chemical Society
2580 Ind. Eng. Chem. Res., Vol. 31, No. 11, 1992
contradict the uniquemess results given in Sridhar and Luica (1989). In this paper, we complete the study of the number of steady-state solutions to multistage separation processes involving binary homogeneous mixtures initiated by Sridhar and Lucia (1989)by investigating a variety of product composition, heat duty and flow rate specifications. Accordingly, this paper is organized in the following way. In section 2, the phase rule and mass and energy balance equations are used to derive analytical expressions that describe the effect of stage temperature changes on other operating variables. These inequalities and various nonlinear maps are then used, in section 3, to establish steady-state solution uniqueness for direct or indirect product composition specifications and solution multiplicity for sgecifications of internal flows and/or heat duties. In section 4, the algebraic results are interpreted geomet r i d y using Ponchon-Sravarit dwams. Finally, in section 5, the practical utility of these specifications from the standpoint of process design, operation, and control are discweed. 2. Effects of Stage Temperature Changes on Column Ogersltion Consider the multistage separator shown in Figure 1. If the stage temperatures and pressures are chosen as the independent variables, then by a simple degrees of freedom analysis all other operating variables can be expressed as a function of T,, T2,..., T, and P,, P2,...,P,,. Furthermore, application of phase rule to the jth stage shows that the mole fractions in the liquid and vapor phases x . and y ’ respectively, and the corresponding molar entupies, and Hy,are functions of Ti and P . only. 2.1. Effwts of Temperature dhanges OB Compositions and Enthalpies. Because the phaae mole fractions of the most volatile component, x j and yj, are functions of T j only at constmt P, it follows that
a.Lj
Y j + l - Xn,
aa,
S;+,
-I-
-
av,
-o aTn, Yn, - zk-1 aTn,
- Yn, -av,
Vn, ayn. -> 0 (6) aTn, Yn, - xns-l aTn, Yn, - xh-1 aTn, 2.3. Effects of Temperature Changes on Heat Duties. The partial temperature derivatives of the heat duties can be obtained by differentiating the energy balance equations for the stages in the column. “he energy balance equation for the jth stage is Q j = L,H) + V ”, - F,Hf,j - Lj-lHfil- Vj+,Hy,, (7) YI
Differentiating eq 7 with respect to Th,for k = 1,2, ...,n,, gives
Ind. Eng. Chem. Res., Vol. 31, No. 11,1992 2681
"'
t
I
I
T . 8
Figure 2. Qualitative contours of L1and V,.
Lj-l
I
I I
Lns - 1
Figure 1. Multistage separation process.
Application of eqs 2-6 to eq 8 gives
3. Algebraic Analysis of Binary Multistage Separators The partial derivativea presented in the previous section can be used to study the number of steady-state eolutions to multistage separators involving binary homogeneous mixtures. The strategy employed is to map the set of solutions for the specifications of fixed temperature and pressure profiles to the set of solutions for the sgecifications of interest and to analyze the determinant of the Jacobian matrix of that map. If the Jacobian matrix is nonsingular, the map is one-to-one and thus the specifications of intarest admit a unique solution. This is based on the fact that the uniqueness of solutions for the TP speciflation set has been rigorously eatablkhed by Sridhar and Lucia (1989). 3.1. Product Composition Specifications. In this subsection, we study the following specifications: DQxB, DYDQ,QBzB,YDQ& YDQZB, TiQTbY and RYDQ. BYDQzB, we mean the specification of a pressure profile, distillate flow rate, bottoms cornpaition, and fixed heat duties to all intermediate stages. The other specifications are defined similarly. Actually, the first six specification sets are equivalent. The equivalence of the first five sets can be shown easily by using the total and component mass balancea around the column. Application of the phase rule to the top and bottom stages gives the equivalence of TIQT and Y D Q z ~ Thus, it is sufficient to study, for examae, the specifications of TIQT,. The linearized map from the TP solutions to the T,QT,,, solutions can be represented by (AT~,AQ2,AQ3,...,AQ,i,AT~)T = J T , Q T , ( A T ~ , A T ~ , A T ~ , . . . , A T ~(10) -~,AT~)~ where the Jacobian matrix of the map is 0
ae2
aT2 aQj -I
aQ3
0
aTk
for j # k - 1, k, k + 1; where k # 1 and k # n,. These partial temperature derivatives show the effect of stage temperature changes on the operation of the column. We note here that eqs 4-6 and 9 provide tighter bounds than the inequalities given in Sridhar and Lucia (1989).
aT2 0
(11)
2582 Ind. Eng. Chem. Res., Vol. 31, No. 11, 1992
Elementary row operations and eq 9 show that the determinant is given by det [ J T ~ Q T , ,=~det
(12)
and has the following sign pattern
where aj,k = [ C ~ ~ o [ [ d ( C ~ = 2 8 i ) / a T k I ( ~ j , kd,AmT k) ,lmT+ l l / ATk, for k = 1, ..., n, and j = 2, ..., n, - 1. $ate that eq 15 involves no approximation, that, according to eq 9, it has the sign pattern
+ + - ... 0 0
1 0 0 0 '..O 0 0
det
Simple induction shows that the determinantof any matrix with the above sign pattern is positive, which implies that the matrix, JTlp,,,,is nonsingular. Thus,the specifications of T1QT, adrmt a unique solution in a local sense. However, do the specifications of TIQT, also admit unique solution in a global sense? Yea! To see this,note that the nonlinear map from the TP specification set to the specifications of TIQT, is given by
where n and ATkis are chosen such that ATk = & ATk,i, the ATk,l(shave the same sign as that of ATk, and the d(Eil2Qi)/dTk's are continuous on the corresponding integration intervals, for k = 1,..., n,,j = 2, ..., n, - 1,and 1 = 1, ..., n. It follows from mean-value theorem that
. . . * + 0 * + 0
0 0
0 0 0 0
. . . .
"'+ - *
...o + * ...o 0 1 n,-2xn,
and that simple induction shows that its determinant is always positive. Solution uniqueness for the specifications of RyDQcan be established in a similar way. 3.2. Internal Composition Specifications. Here the specification sets LITV,,,, LIT, and TV,, are studied. By LITV,, we mean the specification of a pressure profile, fixed reflux and boil-up flow rates, and specified temperatures at all intermediate stages. LIT is the specification of a pressure profile, fixed reflux flow rate, and specified temperatures at all stages except the first one. The definition of TV3 is similar to that of LIT. Of these t h seta ~ of ~ specificat~ons, LITV is the most complicated. The linearized map from the T 3 solutions to the LITVnI solutions is given by (bLi,ATz,AT3,...,AT~,-i,AV,I)T=
JL,TV,(AT~,AT~,AT~,...,AT~,-~,AT~,) (16) where the Jacobian matrix is
aT2
JLlrrY., = 10 ;
aT3
1
0
0
1
i
... 0 ... 0 *. . .
0
Ind. Eng. Chem. Res.,Vol. 31, No. 11,1992 2683
Cofactor expansion and eqs 4-6 show that the determinant is given by of JLITV, det[JL,TVnJ =
aL1 av,,
where the Jacobian matrix of this map is
aLl aV,,
-- - aT, aT, aTn, aT,
Use of eq 9 and some algebra gives
where aL1 aL1 aL1 ?Yl
which implies that the matrix, J L T V , is nonsingular. Thus, specifications of LITVn, adma a unique local steady-statesolution. Actually, the specifications of LITV, correspond to a unique global solution. To see this, suppose that there are two different temperature profiles that give the m e values of L1, T,, ..., T,+ V,, or that there exist two seta of T1and T, that give the same L1and V,, at constant T2,..., T,+ whose functionality can be written in the form L1
= gl(T,,Tn,),
Vn,
= g,(Tl,T,)
det 1= U3l-G)det
?Y2
?Y3
av, av, av, ?Yl
?Y2
(23)
?Y3
av3 av3 av3 ?Yl
aY2
?Y3
I-
1
(19)
For any specified values of L1 and V,, we can,on the basis of aL,/aT,, aL1/aT,, av,/aT,, and av,/aT , of the draw two qualitative curves for eq 19. See Figure 2. h there exist two seta of Tl and T,, that give the same L1 and V,,, then these curves must intersect at least twice, which, in turn,means that the slope of g, is simultanmusly greater than that of g2 at one point and less than that of g2 at another. However, eq 18 and application of implicit function theorem to eq 19 give that the slope of g1is always lea thanthat of g2 Thus, g1and g2can intersect only once, which means that the specifications of LITV, admit unique global solution. Similar analysis leads to the uniqueness of solutions for the LIT and TV, cases and again the analysis is nonlinear, involving no approximation. 3.3. Unconstrained Separation Specifications. Specifications of L,QV,, LIQ, and QV are studied in this section. The definitions of LIQV and QV, are similar to those for LITV,, L,T, an8 TV,. Jacobsen and Skogestad (1991) present an example that shows solution multiplicity for an eight-stage column involving an ideal binary mixture of methanol and 1-propanol with specifications of a pressure profile, reflux, and boil-up flows and adiabatic intermediate stages. Here we present a rigorous analysis of the reason for solution multiplicity for this specification set. Although the analysis that follows involves only a three-stage column with a single feed stream to stage 2, the strategy employed and conclusions obtained are applicable to any column with more than two stages, any number of feeds, and equipped with either a total or partial condenser. The map from the specifications of TP to the specifications of LiQZV3 is given by
"1 ?Y3
(24)
Here we are freeto fm F,let xf = (1/2)Cy2+ xJ, and choose, if possible, x 2 and x3 such that
and
,X1Q,
Note that x1 is free to vary between x 2 and y2 and yl, y2, and y3 depend on xl,x,, and x3, respectively, through the is a function conditions of equilibrium. Thus det [JLIGM] Of ~1 only. equation^ 25 and 26 can be manipulatsd w i l y using the equilibrium and enthalpy models presented in Jacobsen and Skogestad (1991). Substitution of eqs 4-6, 25, and 26 into eqs 23 and 24 and some algebra give
>0 (aiAQ2, ,
Av3)T
J~,~,v,(AT~,AT~,A (20) T~)~
(27)
2584 Ind. Eng. Chem. Res., Vol. 31, No. 11,1992
- 3.24191083720 - 02) x I.OD t 13
(ZI
Figure 3. Functionality of the determinant of the Jacobian matrix.
Problem Data for Illustrative Example stale
W
I
F2 (kmollmin)
X I
1
2
9.363640
3
6.286440
- 03 - 03
3.246570
- 02
2.196480
- 02
2.091470
- 02
1.0
/ 2,Y
Figure 5. Enthalpy-compition diagram.
(11
- 3.24150 - 02) x 1.00 + 06
Figure 4. Functionality of the terms of the determinant of the Jacobian matrix.
and lim det 2 = lim XI-%
Xl-xl
-
Ind. Eng. Chem. Res., Vol. 31, No. 11, 1992 2688 w.m
LI = 2539.49 kmol/min W.W
:
0.m
j
,,,,,,,1,,,/,,,,,,,, /,,,,,,,, ,,,,,,,,, D.00
0.20
0.b
0.b
0.b
t .m
2,Y
Figure 8. Ponchon-Savarit diagram for example with multiple solutions. ordinate of
MD
Figure 6. Qualitative relation between n, and the ordinate of MD.
L, (kmol/min)
Figure 7. Bifurcation diagram for a two-stage distillation.
JLlQnVS singular, at some x1 (see Figures 3 and 4). Therefore, it may be possible to find more than one steady-state solution for the specifications of LIQzV,. Solution multiplicity for the specificationsof LIQ and QV, can be shown in the same way. 4. Graphical Analysis of Binary Multistage Separators In this section, we use Ponchon-Savarit diagrams to analyze binary multistage separators in an effort to make the results of the previous section more geometric and somewhat more accessible to undergraduate chemical engineering students. We choose two specification sets, YDQxBand LIQ,for illustrative purposes. 4.1. Deeign Specifications of yDQxB.The standard deeign specifications for Ponchon-Savarit diagrams include fired product compositions and adiabatic trays for all stages except the condenser and reboiler. Consider then the Ponchon-Savarit diagram shown in Figure 5. For known feed conditions,the point F is fixed and specifying YD and ZB fmm the vertical lines y = YD and x = X W Note that the points MD and MB,which are given by M D = HD + Q,/D and MB HB - QJB, lie on the lines y = Y D and x = xB, reepectively, and the straight line connecting these two points must pass through the point F in order to satisfy the energy balance around the column. From Figure 5, it follows from plane geometry that the higher
the point MD,(1)the lower the point MB and (2) the fewer are the number of stages needed to accomplish a given separation. The combination of these two statements shows that the number of stages needed to accomplish a given separation decreases monotonically as the ordinate of MD increases. This is illustrated qualitatively in Figure 6. Reversing the above arguments, it follow that specifying the number of stages and the product compositions uniquely fixes the point MD. Furthermore, for each MD there is one and only one way to draw the (horizontal) equilibrium tie lines and the (more or less vertical) energy balance tie linea that repreeent the stagea in the separator. This follows from simple plane geometry (see Figure 51, and is sufficient to establish that the (graphical) solution is unique for specificationsof ~DQxBwith adiabatic trays. However, we note here that specifyingthe number of stagea and the product compositionsmakes the Ponchon-Savarit method a trial and error procedure. In particular,one must guess the location of the point MD, draw the overall energy balance line connecting the points MD, F,and MB,and step off the specified number of stages. If the cornposition of the liquid leaving the bottom tray does not coincide with the vertical line 1: = xB, then a new value of MDmust be estimated and the graphical procedure repeated until the liquid leaving the k t stage is equal to the specified bottom composition. 4.2. Specifications of L I Q . As noted in section 3.3, specifications of reflux flow rate and reboiler duty can exhibit multiple steady-state solutions. Here we use a numerical example to illustrate multiplicity. Consider a two-stage separator involving the homogeneous binary mixture of methanol and 1-propanolwith a single feed of Fz = 1 kmol/min and nf = 0.18 to the second stage. Furthermore, assume that the liquid and vapor phases are ideal, that the relative volatility is constant at 3.55, and that the saturated phase enthalpies are described by the models given in Jacobsen and Skogestad (1991). For a specified reboiler duty of Q2 = 1.039914 X lo4 kJ/min and parametric values of reflux flow rate, we constructed the bifurcation diagram shown in Figure 7. Note that there is a regular turning point at an approximate reflux flow rate of L1 = 2539.45 kmol/min, and for all reflux flows greater than this value, but less than 2539.50 kmol/min, there are two steady-state solutions for the same specifications of L1 and Q2. Note also that the reflux flow for this illustration is very high. Figure 8, on the other hand, show the relevant sections of the Ponchon-Savarit diagrams for both solutions corresponding
2586 Ind. Eng. Chem. Res., Vol. 31, No. 11,1992
to a reflux flow rate specification of L1= 2539.49 kmol/ min. Each set of vertical and horizontal lines (i.e., solid and dashed) correspondsto a different steady-state solution. Moreover, because of the high reflux flow rate, the points M D and M Bare located well off the graph and, as a consequence, the energy balance tie lines appear to be vertical. However, the important point to note is that specifications of reflux flow rate and reboiler duty do not uniquely constrain the separation made by the column and thus give rise to the two solutions illustrated in Figure 8. 5. Concluding Remarks
In this paper, we have used the idea of constrained separation to classify the effects that various specification sets have on the number of steady-state solutions for binary homogeneous separators. Many sets of specifications uniquely fix the separation made by a binary multistage column, largely due to the fact that, at fixed pressure, the phase rule provides little freedom for variation at the single-stage level. Clearly, both direct and indirect product composition specifications do this and therefore uniquely fix the steady-state solution for the column. In a similar but less obvious way, specifications of either a fixed heat duty profile or reflux ratio and bottoms flow constrain the separation made by any column. Thus the corresponding steady-state solution for these specification seta is also unique. On the other hand, if the specifications are such that the column can make any arbitrary separation, then the appearance of steady-state multiplicity should come as no surprise. In this regard, it is particularly important to point out the intrinsic difference between fixing say the reflux and/or reboil rates and the reflux and/or reboil ratios. While specificationsof reflux or reboil flows only fix an internal flow within the column, specified reflux or reboil ratios fix the relative amounts of vapor and liquid leaving the end stages of the separator and have a normalizing effect on column behavior. Whether viewed in terms of a McCabeThiele or Ponchon-Savarit diagram, the latter seta of specifications fix slopes of operating lines (independent of the basis of flows, mass or molar), whereas fixing single internal flows does not, and this seemingly simple difference has a profound effect on the behavior of the column. In light of this, it seems evident that specifications that do not fix the separation made by the column should be avoided. Clearly, this position is supported by well-established control concepts such as material balance and composition control that date back to the 1970s or earlier (see, for example, Shinskey, 1977; Buckley et al., 1985; Deshpande, 1985). For example, all acceptable singlecomposition column control strategies based on (direct or indirect) material or energy balance control constrain the separation made by the column by adjusting either the reflux or reboil ratio. In dual composition control, both reflux and reboil ratio are used. In no case is the manipulation of reflux or reboil flow rate independentof product withdrawal, since there are inventory effects that are limited by capacities. Furthermore, even if flows such as reflux or reboil flow rates are used aa manipulated variables for the purpose of control, accompanying tray temperature measurements are also usually used to infer compositions and this removes any ambiguity, even near regular turning points since intermediate solutions in situations of multiplicity are always unstable dynamically (open loop). For example, for the eight-stage methanol-propanol separator studied by Jacobsen and Skogestad, a feed rate of 1.0 kmol/min and specified values of molar reflux flow (L, = 4.78 kmol/min) and reboil rate (Vn,= 4.60 kmol/min), the
three steady-state solutions have distillate compositions of methanol of 0.9971,0.9958,and 0.9600,respectively, and are not too far removed from the regular turning point at (Ll, V,,) = (4.73,4.50). However, the composition of the bottoms stream (and thus the bottoms flow rate and temperatures in the bottom of the column) vary significantly. The respective bottoms compositionsare 0.4610,0.2053, and 0.0161, with corresponding bottom tray temperatures of 352.7,361.5, and 369.6 K. The corresponding bottom flows are 0.9272,0.6272,and 0.4872 kmol/min. Moreover, the rates of change in these variables in the bottom of the column with respect to reflux or reboil flow rate are significant. Thus small changes in reflux or reboil rate produce noticeable changes in the composition, product flow, and temperature in the bottom of the column, regardless of whether the column is operated in open or closed loop. On the other hand, the corresponding reflux and reboil ratios behave monotonically for all solutions. In particular, for these three solutions, the correspondingreflux ratios are 65.63,12.82, and 9.34 and the reboil ratios are 4.85,7.18, and 9.24, respectively. Acknowledgment This work was supported by the Office of Basic Energy Science, Division of Chemical Sciences, U.S.Department of Energy, under Grant DE-FG02-86ER13552. Nomenclature a = element of a matrix B, L, = total molar flow rate of bottom product D, V , = total molar flow rate of distillate F = total molar feed rate g = internal flow rate functionality H = molar enthalpy, parti@ molar enthalpy J = Jacobian matrix L = total liquid molar flow rate M = corrected molar enthalpy of product streams nc = component number in the mixture n, = number of stages P = pressure Q = heat duty R = reflux ratio T = temperaturem+, + Tfiird= Ti + c1.1ATl,l,.-Tk-l ~K~lATk-l,I,Tk+l cziATk+l,l,...Tn,+ CKiATn,,I (eqs 13 and 14) V = total vapor molar flow rate r = liquid molar composition y = vapor molar composition Subscripts and Superscripts B = bottoms c = condenser D = distillate f = feed j , k = stage index L = liquid phae I , m,n = integration path index r = reboiler T = transpose V = vapor phase Greek Letters A, AT = perturbation or difference, in temperature 4 = mean-value theorem variable
Literature Cited Buckley, P. S.;Luyben,W. L.; Shunta, J. P. Design of Distillation Column Control System; I S A Rseearch Triangle Park, NC, 1985; pp 3-24.
I n d . Eng. Chem. Res. 1992,31, 2587-2593 Deahpande, P. B. Distillation Dynamics and Control; IS& Fkaearch Triangle Park, NC, 1986; pp 359-389. Doherty, M. F.; Perkine, J. D. On the Dynamics of Distillation Pro~ ~ 8 8 8 % .VI: Uniqueneee and Stability of the Steady State in HomogeneousContinuous Distillations. Chem. Eng. Sci. 1982,37, 381-392. Jacobaen, E. W.;Skogmtad, 5. Multiple Steady-States in Ideal Two-Product Distillation. AZChE J. 1991,37,499-511. Kingeley, J. P.; Lucia, A. Simulation and Optimization of ThreePhase Distillation Promsees. Znd. Eng. Chem. Res. 1988,27, 1900-1910. Kovach, J. W.,ID, Seider, W.D. Heterogeneous Azeotropic Distillation: Homotopy-Continuation Methods. Comput. Chem. Eng. 1987,11 (6), 593-605. M a g n w n , T.;Michelsen, M. L.; Fredenslund, Aa. Azeotropic Distillation Using UNIFAC. Znst. Chem. Eng. Symp. Ser. 1979,56, 4-211-4.2119.
2687
Prokopakk, G. J.; Seider, W.D.Feasible Specificationsin Azeotropic Distillation. AZChE J. 1983,29 (11,49-60, Rovaglio, M.; Doherty, M. F. Dynamics of HeterogeneousAzeotzopic Distillation Columns. AIChE J. 1990,36,39-52. Shinekey, F. G. Distillation Control; McGraw-Hill: New York, 1977; pp 93-121,167-194. Sridhar, L. N.;Lucia, A. Analysis and Algorithms for Multistage Separation Processes. Znd. Eng. Chem. Res. 1989,28,793-803. Sridhar, L. N.; Lucia, A. Analysis of Multicomponent, Multistage Separation Procesees: Fixed Temperature and Pressure Profilea. Znd. Eng. Chem. Res. 1990,29,1668-1675. Venkataraman, 5.;Lucia, A. Solving Distillation Problems by Newton-Like Methods. Comput. Chem. Eng. 1988,12 (l),55-69.
Received for review March 5, 1992 Accepted July 28,1992
GENERAL RESEARCH Radical Scavenging by Hydroaromatics in the Presence of Oxygen Junichi Kubo Central Technical Research Laboratory, Nippon Oil Company, Chidori-cho 8, Naka-ku, Yokohama, 231 Japan
The radical scavenging abilities of hydroaromatics toward DPPH (N&-diphenyl-N'-picrylhydrazyl) in the presence of oxygen were investigated. Tetralin, octahydrophenanthrene (OHP), synthetic H/D (a multicomponent additive containing various hydroaromatics produced by hydrogenation of decrystallized anthracene oil derived from coal tar), and HHAP (a multicomponent additive produced by hydrogenation of a highly aromatic heavy fraction from petroleum) were examined by heating at 50 O C for 3 h with DPPH in air. It was confirmed by the changes in color of the solution and the ESR spectra that those hydroaromatics have obvious radical scavenging abilities and that OHP, synthetic H/D, and HHAP have higher abilitiea than tetralin. From these experimental results, the additive effects of these hydroaromatics on the deterioration of petroleum products, rubbers, and plastics reported in our previous papers can be attributed to their radical scavenging abilities. Introduction Pure hydrocarbons without functional groups containing heteroatoms such as 0, N, P, and S have not been applied to date to inhibit the deterioration of hydrocarbon products (except in very exceptional cases). In conventional autoxidation studies, results showing that the addition of hydroaromatics accelerates the absorption of oxygen (Larsen et al., 1942; Robertson and Waters, 1948; Yasutomi and Sakurai, 1976) have been reported. It seems that hydroaromatics, typically represented by tetralin, have been considered to be easily oxidized, and they have never been used as inhibitors to deterioration, despite some exceptional results reported to be effective in limited conditions such as the additive effect of tetralin on the oxidation of cumene (Russell, 1955),tetralin with sulfur compounds on the oxidation of light hydrocarbons (Yamaji, 1960), and tetralin hydroperoxide (Thomas and Tolman, 1962). On the other hand, hydrogen-donating hydroaromatics (abbreviated as hydmaromatics) have been widely used in such pmcessea as coal liquefaction and heavy oil upgrading (Carbon et al., 1958; Fisher et al., 1982; Kubo et al., 1988) serving as radical scavengers to reduce coke formation in a reductive atmosphere. In view of these facts, hydroaromatic-type additives were examined with regard to the deterioration of hydrocarbon products, and it was found that they were obviously effective against the thermal and oxidative deterioration of petroleum products (Kubo,
1991a),rubbers (Kubo, 1991b) and plastics (Kubo, 1991b) and also against radiation degradation of polyolefins (Kubo and Otsuhata, 1992). The inhibiting effects of the hydroaromatic-typeadditives seem to be due to their radical scavenging abilities, as judged from the results of the thermal deterioration tests (Kubo, 1991a). To confirm this, the radical scavenging abilities of tetralin, octahydrophenanthrene, and two multicomponent additives, both of which exhibited noticeable effects on the deterioration of hydrocarbon products, were examined with DPPH (NJV-dipheny1-N'picryhydrazyl). In this paper, the relation of theae results to those from the thermal deterioration tests is discussed. Experimental Section (Kubo et al., 1991) The radical scavenging abilities of the following substances toward DPPH, which contains relatively stable radicals at room temperature, were examined (1)synthetic H/D (Kubo, 1991a);(2)HHAP (Kubo, 1991b);(3) tetralin; (4) OHP (1,2,3,4,5,6,7,&o&ahydrophenanthrene); (5)naphthalene; (6) phenanthrene; (7)decalin; (8) blank (without addition). Synthetic H/D, which exhibited noticeable effects on the deterioration of petroleum products, rubbers, and plastics, was produced by the hydrogenation of decrystallized anthracene oil obtained from a coal tar fraction.
0808-5885/92/2631-2687$03.00/0(8 1992 American Chemical Society
