Ind. Eng. Chem. Process Des. Dev. 1985, 2 4 , 140-149
140
T a b l e X. Summary of P u l s e Test R e s u l t s and C o n t r o l l e r Settings
process gain process time constant, min process dead time, min ultimate gain ultimate frequency, rad/min ultimate period, min PI-Controller gain Ziegler-Nichols 1.408 design 1.189
column 1 column 2 12.857 4.864 2.22 14.3 0.1048 0.1495 3.096 30.76 15.27 10.55 0.412 0.596 Settings reset gain reset 0.343 13.843 0.537 0.103 13.482 0.163
column. These tray temperatures respond quickly, however, to heat input changes. Nomenclature B = bottoms flow rate, g-mol/h D = distillate flow rate, g-mol/h F = feed flow rate, g-mol/h NF = feed tray N T = total number of trays
P = column pressure, mmHg or psig Q, = condenser heat duty, lo6 cal/h Q R = reboiler heat duty, lo6 cal/h QT = total energy consumption, IO6 cal/h TF = temperature feed before preheater, OC T F p = temperature feed after preheater, “C THF = tetrahydrofuran Xu = azeotropic composition, mole fraction of THF XB = bottoms composition, mole fraction of THF XD = distillate composition, mole fraction of THF Z F = feed composition, mole fraction of THF AX = difference between azeotropic composition and distillate composition, mol % THF Registry No. THF,109-99-9.
Literature Cited Abu-Eishan, S. Ph.D. Thesis, Lehigh University, Bethlehem, PA, 1982. Hoffman, E. J. “Azeotropic and Extractive Dlstillation”; Wiley: New York, 1964. Luyben, W. L. Ind. Eng. Chem. Fundam. 1975, 14, 321. Shinskey, F. G. “Distillation Control”; McGraw-Hill: New York. 1977. Van Winkle, M. “Distillation”; McGraw-Hili: New York, 1967.
Received for review July 8, 1983 Accepted January 31, 1984
Rapid Procedures for the Prediction of Fixed-Bed Adsorber Behavior. 3. Isothermal Sorption of Two Solutes from Gases and Liquids Dlran BasmadNan and Concltantlne Karayannopoulos Depatfment of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, Ontario, Canada, M5S 1A4
We extend a simple method previously used to describe single-solute adiabatic sorption operations to binary isothermal systems. The method separates equilibrium and kinetic effects by first constructing the effective equilibrium pathways, followed by the application of solution charts for single-solute nonequilbrium sorption. Construction of the effective equilibrium curves involves the solution of a single nonlinear algebraic or differential equation. The solution charts are valid for both developing and broadening solute fronts. The procedure is used successfully in predlcting solute breakthrough from clean or preloaded isothermal adsorber and ion-exchange beds taking binary gaseous and liquid feeds. The relation to existing methods of describing column behavior is established and procedures are outlined for arriving at quick design estimates and analyzing the effect of various parameters.
Introduction In part 2 of this series (Basmadjian, 1980a),we had used “effective equilibrium curves” in conjunction with known solutions of single-solute isothermal nonequilibrium sorption to predict actual behavior of adiabatic columns taking a feed containing one sorbable component. The curves in question are in essence solutions of the equilibrium and conservation equations of equilibrium theory; they can be obtained in simple fashion by the numerical or graphical solution of algebraic or differential equations. Although the general validity of this approach still lacks a formal proof, we were encouraged by ita ability to predict adiabatic fixed-bed adsorption and regeneration for widely diverse systems with reasonable accuracy (Elasmadjian, 1980a,b; 1981). In addition to its simplicity, the method has the advantage of separating equilibrium and kinetic effects-a feature usually lacking in numerical solutions of the full PDE model-thus allowing the analyst to 0196-430518511124-Q740$07.50/0
identify sources of unsatisfactory performance and suggest parameter changes for process improvement. Once the effective equilibrium curves have been established for a particular feed/bed condition, design charts for the single-solute isothermal case can be applied in a few minutes’ work to derive profiles and breakthrough curves or required bed weight and regenerant. Changes in feed rate, bed length, cycle time, or particle size and diffusivity are accommodated by a simple adjustment of the dimensionless chart parameters. However, the method presently is not able to accommodate variable feed conditions and nonuniform initial bed distributions. Theory 1. Derivation of Effective Equilibrium Curves or Pathways. Equations for the effective equilibrium curves are derived from the solute conservation and equilibrium equations which apply to a column with prevailing local equilibrium. The relevant relations have been given in the 0 1984 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 1, 1985 141
classical studies of DeVault (1943) and Glueckauf (1949). For a system in which the solute fronts become discontinuities under equilibrium conditions (“self-sharpening” fronts) the relevant relations for a binary (1,2) system are Q l O - 41 ---q20 - 42
YlO - Yl
(1)
yzo - y2 where the subscript 0 refers to the known feed or initial bed conditions. For broadening (“proportionate pattern”) fronts the relations take the differential form
Local equilibrium is formally expressed through the binary
a
Peak or Plateau
2i Yio
isotherms Q1 =
41*(42, Yl, YZ)
(34
and qz = qz*(q1, Yl, YZ)
(3b)
Solution of eq 1 and 3a, b leads to the so-called shock curves (Basmadjian, 1980a) (71
= ql(Y1)
(44
42
= q2(Yz)
(4b)
which become the effective equilibrium curves for the system, provided they are of a form which leads to selfsharpening solute fronts. For this to be the case, the generated curves (4a, 4b) must be of type I (concave to Y axis) for an adsorbing solute and type 111(convex to Y axis) for a desorbing solute. If these conditions are violated, the differential equation (2) is used over the appropriate range instead of eq 1. It has been found, however, that in a large number of practical binary systems, both gaseous and liquid, both solutes propagate along self-sharpeningfronts. Even when this is not the case, the algebraic eq 1,3a, and 3b yield a good approximation to the exact solution described by eq 2. In the q-Y diagrams,Figure la, we show typical effective equilibrium curves for systems with self-sharpening fronts and no selectivity reversal. For the less adsorbed component 1, the equilibrium curves always consist of two branches: curve A which is the pure component (1)isotherm, represents the leading front of solute 1; curve B represents the rear. Intersection is at the point P, which for beds of sufficient length, leads to a plateau of constant concentration. The more strongly adsorbed component has only a single branch, curve C. The combination of these two sets of curves leads to the well-known “Langmuir” type solute fronts (Glueckauf, 1949) in which the light component, adsorbing along curve A, rises to a peak or plateau P normally exceeding the feed concentration and then desorbs along curve B to drop back to the inlet value. Simultaneously,the heavy component 2 breaks through, rising steadily to the feed concentration (Figure lb). For self sharpening fronts, derivation of the effective equilibrium curves involves simple substitution of eq 3a and 3b into the algebraic shock equation (1).The resulting nonlinear algebraic equation
G(Yi, Y2) = 0 (5) can be solved by standard root-finding methods for constant values of Yl or Yz,and the solutions back-substituted into the isotherm equations to yield the effective equilibrium curves 4a and b (see sample calculation given in the
b
t
Figure 1. (a) Effective equilibrium curves for binary sorption: (1) weakly adsorbed component; (2) strongly adsorbed component. (b) Typical binary concentration breakthrough curves generated from effective equilibrium curves. (c) Graphical construction of effective equilibrium curves.
Appendix). The procedure can also be carried out graphically on a ql-Yl diagram, as illustrated in Figure IC. The diagram contains the pure component (1)isotherm, as well as parametric curves Yz = constant which aid in locating the feed point F. Construction of the curve FP starts at the feed point and proceeds in a stepkse manner, each increment satisfying eq 1 or 2, the former usually being sufficient for all practical purposes. Parametric curves q2 = constant may be drawn in to aid in the procedure. Alternatively, the required qz values can be interpolated directly from tabulated equilibrium data. The method recommends itself in cases where only raw numerical values are available and one wishes to avoid fitting equations to the measured equilibrium data. A similar procedure has been outlined by Takeuchi et al. (1978). 2. Derivation of Binary Nonequilibrium Behavior. Use of Single-SoluteIsothermal Solution Charts. The second step in our method consists of the application of known solutions for single-component sorptionconveniently cast into “solution charts”-to the previously
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derived equilibrium pathways. In our previous treatment of adiabatic sorption (Basmadjian, 1980a) this was done without formal proof of its general validity. For binary isothermal systems, partial validity of the method has been established by Cooney and Strusi (1972) for solutes with fully developed constant-pattern fronts. Uncertainty remains with regard to both developing and broadening fronts, which we will attempt to remove by direct comparison with experimental data. The single-solute sorption solutions we use here are based on the well-known Hiester-Vermeulen treatment (1952a) of the Thomas solution. The operative expressions are their eq 23 and 24 for fluid and solid phase concentrations, respectively. We have recast their original solution charts (Hiester and Vermeulen, 1952b) into a more convenient form (Basmadjian, 1981, 1984), with the following dimensionless parameters
r=
( A Y / AYo)(1 - Aq / Aqo) (Aq/Aqo)(1 - A Y / A Y o )
(6)
applicable to adsorption along type I and desorption along type I11 equilibrium curves. The reference point for A Y and Aq differs depending on which front is being considered (see numerical calculations in the Appendix). For the latter case, the right side of eq 6 is inverted. number of reaction units NR
DPZ NR = 15--f(r) R2 u or (7)
dimensionless time TR
Dm TR = 15-tf(r) R2 or
D, Pf AYO 15- tf(r) -R2 Pb AQO reproduced in Figure 2 for the fractional fluid phase concentrations 0.01, 0.1, 0.5,0.9, and 0.99. Additional charts for solid phase concentrations appear in previous publications (Basmadjian, 1981, 1984). The correction term f ( r ) is that suggested by Vermeulen for the reaction kinetic rate expression (Perry 1963, p 16-14). The separation factor r is analogous in concept to the relative volatility and serves to express the shape of the equilibrium curve. It is always 1for broadening fronts. r = 1corresponds to the special case of a linear equilibrium curve. Transport resistance is expressed in terms of overall D,) which are incorporated pore or solid diffusivities (Dw, in the dimensionless parameters NR and TR. When NR equals TR,as it does above NR = 100, transport resistance becomes negligible and bed behavior is solely dictated by equilibrium considerations, i.e., the value of r. To reproduce a breakthrough curve, one first determines r for the previously constructed equilibrium curves. For reasonably symmetrical curves, a single r value, usually computed at the midpoint ( A Y / A Y o = 0.5), will suffice. NRis next computed according to eq 7 using known bed, particle, and fluid parameters, with AqO/AYorepresenting
the slope of the line connecting initial bed loading or feed concentration to the plateau. Finally, breakthrough time t is calculated from the parameter TR read from the solution charts (Figure 2). A comment needs to be made about the derivation of the leading solute breakthrough curve. The two branches of this curve are generated from the two equilibrium pathways A and B of Figure la, which intersect at the plateau point P. For systems with relatively high diffusivities (gaseous feeds, ion-exchange beds), the full plateau concentration is usually attained within a reasonably short distance from the bed inlet, and the two transfer zones merge neatly into the horizontal plateau portion of the breakthrough curve (see Figures 3 and 4). The situation differs in systems involving the removal of large organic solutes onto granular carbon. Here the low diffusivity of the solute molecules often combines with relatively large particle size to bring about a partial or complete erosion of the plateau at short distances from the bed inlet. In this case the two transfer zone breakthrough curves intersect each other at a peak discontinuity which does not fully reproduce the continuous form of the eroded plateau. We will show later that in most instances this leads to only minor deviations from actual breakthrough behavior. The use of the solution charts and the significance of the various parameters are illustrated further by means of a numerical example give in the Appendix. Comparison with Experimental Data. We compared the predictions of our two-step method-construction of the effective equilibrium curves and application of single solute solution charts-with experimental binary breakthrough curves reported in the literature. The main features of the systems tested are summarized in Table I, and additional properties are given in Table 11. The four cases involing liquid feeds include a ternary ion-exchange system which is equivalent to binary sorption and hence amenable to the same type of analysis. As well, several breakthrough curves from other preloaded adsorption columns were analyzed. These latter systems are of particular relevance to commercial processs such as ESSO’SEnsorb which use isothermal displacement in the desorption step (Asher et al., 1969; Chi and Lee, 1969). Our predictions, together with available numerical simulations, are compared with the experimental data in Figures 3-5. Also shown are the effective equilibrium curves for the two solutes (three in ion exchange) for each system. Comments on several features follow. (1) Breakthrough Curves for Binary Gaseous Systems. The three sets of measurements shown comprise, to our knowledge, the only gaseous systems for which independent equilibrium and particle diffusivity data from the same investigators were available. Diffusion coefficients were in all cases derived from pure component kinetics and assumed to be constant. Independent computer solutions of the full PDE model were available only for the system carbon dioxide-water4A molecular sieves (Carter and Husain, 1974). Our predictions matched these and the experimental data satisfactorily (Figure 3a). Of particular note here is the good prediction of the early breakthrough of the leading COz front which evidently has not fully attained a constant pattern. The reason for this successful prediction lies in the general validity of the solution charts, Figure 2, which are capable of accomodating both developing and broadening solute fronts. Good agreement of our predictions with experimental data is also evident in the case of the benzene(g)-hexane(g)-carbon system investigated by Shen and Smith
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 1, 1985 143 AYIAY, = 0.50
NR 1W 80
60 40
20
2-
TR
