Analysis of a Continuous Immobilization Reactor - Industrial

Ind. Eng. Chem. Res. 1996, 35, 2889-2899

2889

Analysis of a Continuous Immobilization Reactor Karl F. Roenigk*,† and Rutherford Aris‡ 3M Home and Commercial Care Division, 3M Center, Building 250-3W-03, St. Paul, Minnesota 55144, and Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Avenue SE, 151 Amundson Hall, Minneapolis, Minnesota 55455

The second-order reaction by which a species is transferred from solution to a fixed site in a porous solid is considered. A web of the porous solid moves under a reservoir of the solution, which saturates as liquid percolates downward. The saturated web then passes through a zone in which there is no downward flow and on to a pair of rollers where an arbitrary fraction of the solution is physically expressed and the reaction continues in time. The salient behavior of this class of reactors is described in the hodograph plane. Accommodating optimized designs are detailed under constraints which arise naturally. Comparison is made with experiment for the application to chelation of sodium pyrithione to zinc sites in cellulose sponge. 1. Introduction This paper is dedicated to Eli Ruckenstein, a master in the art of the mathematical analysis of physical problems. It is a slight offering in comparison with his major contributions to chemical engineering science, but we hope he will enjoy its simplicity and the geometric insight nascent in the hodograph plane. We will refer to the reaction that binds a solute species to fixed sites in a porous solid as an immobilizing reaction. A common example of such a process would be dyeing, but there are others such as the immobilization of an enzyme, immobilized enzymatic reactions, and ion exchange. A recent particular application is the chelation of an antimicrobial agent in a spongelike material (Roenigk, 1994). Cross flow through a moving bed reactor, when the particles can be regarded as a moving porous solid, is another example. The literature contains several papers on cross-flow reactors (Fitzpatrick et al, 1986; Jowett, 1990; Marb and Vortmeyer, 1988; Richter et al., 1980; Wiegandt et al., 1978; Wigton, 1977; Wolff et al., 1994; Young and Yeh, 1993), but none seems to treat the peculiarly simple problem we consider here. The immobilizing reaction might be represented by:

RA + βB f C

(1)

where A is the immobilate, B the vacant binding site, and C the immobilized complex of A and B. The stoichiometric coefficients, in many cases, are both unity. The reactor works with a band, or web, of the porous material which moves from left to right with a velocity of vs. As shown in Figure 1, it is in the wetting section, between the points O and C, that the band is wet from above by solution that flows downward with a velocity vf and leaves, depleted of some of its solute, across BD. Note that the finiteness of the two velocities implies that the material is not wet below the line OB of slope vf/vs. Similarly, we note that the fluid that passes into the translation section DCEF all comes from the downward flow over PC. Emerging from this section, the material passes between two rollers, retaining only a fraction of its liquid. We assume that physical expression of liquid is a rapid process for now and neglect the reaction that takes place during this short time interval. The mate† ‡

3M Center. University of Minnesota.

S0888-5885(95)00693-2 CCC: $12.00

rial is then taken “off-line” and is assumed to be allowed to either dry or to rest until packaged possibly with little physical change from its freshly wrung state. Effects from any drying off-line are considered later. Ultimately, off-line, for reaction with addition of immobilate to sites, the bottom of the web will not be so heavily saturated by the immobilate as the top. For the opposite liberation reaction not considered here (Fitzpatrick, 1986), the reverse is true. Diffusion may play a more significant role off-line than in the wetting and translation sections but is neglected in this work. Since we shall neglect diffusion in the wetted material, each element of the wetted web in the translation and off-line sections behaves as an infinitesimal batch reactor. This behavior will be considered first, as it is with such dynamics that kinetic constants are likely to be determined. 2. Kinetics We consider a second-order reaction in which the rate of immobilization is proportional to the product of the concentration of the immobilate with that of the sites on which it is irreversibly laid down. Though it is possible to choose concentration units to avoid nonunit multipliers, it is convenient, and more symmetrical, to use the multipliers R and β. U(t) is the number of moles of the immobilate per unit total volume of the web and V(t) the number of vacant sites per unit total volume. Thus, in a batch situation, with U(t), the concentration of immobilate remaining, and V(t), the concentration of sites still vacant at time t,

U˙ ) -RkUV,

V˙ ) -βkUV,

U(0) ) A, V(0) ) B (2)

where k is a second-order rate constant. Clearly, W ) βU - RV is constant and so equal to its initial value βA - RB. If this constant is zero, the point UV in the phase plane moves along the line from the initial point to the origin, because the sites and immobilate are there in stoichiometric proportions, and so the sites are completely filled and the solution completely exhausted at the same time. If W < 0, there are more sites than available immobilate and U reaches 0 when V is still -W/R; similarly, if W > 0, there is an excess of the immobilate and its concentration is W/β when the sites are all occupied, i.e., V ) 0. If W is negative, -W/RB might be defined as the “incompleteness”, since it is the © 1996 American Chemical Society

2890 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

Figure 1. Reactor configuration.

Figure 2. Curves of constant τ and λ for the batch reaction in eq 7.

fraction of sites still left vacant when the immobilate has been stripped from solution. If W is positive, W/βA might be called the “excess”, since it is the fraction of the original solute that is not needed when the immobilization is complete. Ideally, one would like to do the batch process with the immobilate and the vacancies always in stoichiometric proportions, for then there would neither be excess nor incompleteness. It is tempting to use A and B to make the concentrations dimensionless, but we shall need the flexibility of varying initial conditions and therefore take two arbitrary concentrations, U* and V*, to be characteristic of U and V. Let

u(τ)/a ) w/(λa - b exp(-wτ)), v(τ)/b ) w/(λa exp(wτ) - b) (5) If we are considering a simple batch reaction, there is no reason why U* should not be identified with the feed concentration A and V* with the maximum immobilate concentration B. Then a ) b ) 1 and w ) (λ - 1). As we should expect, the solution simplifies to that of a second-order reaction when λ ) 1 and the two participants are there in stoichiometric proportions, i.e., where τ ) RBkt ) βAkt:

lim [u(τ)] ) v(τ) ) λf1

τ ) RkV*t,

λ ) βU*/RV*

u(τ) ) U(t)/U*, a ) A/U*,

(3)

v(τ) ) V(t)/V*

b ) B/V*,

w ) λu - v

u(τ) )

Then the equations become

dv/dτ ) -λuv,

u(0) ) a, v(0) ) b (4)

To solve this equation, we simply substitute v ) λu w and solve the equation for u by quadrature, rearranging to give

1+τ

(6)

There are two forms of the general solution as λ is greater than or less than 1, respectively:

1-λ

λ-1 ) λ-e

-(λ-1)τ

du/dτ ) -uv,

1

λ-1

v(τ) ) λe

(λ-1)τ

e

(1-λ)τ

(7)

-λ

1-λ )

-1

1 - λe-(1-λ)τ

When λ ) 1, both reduce to the true second order kinetic law. A useful presentation is that of the contours of τ and λ in the uv plane. These are shown in Figure 2. In this

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 2891

diagram the reaction paths along λ ) constant are straight lines from the point (1, 1), intersecting either the x-axis at u/a ) 1 - 1/λ if λ g 1 or the y-axis at v/b ) 1 - λ if λ e 1. Thus, the intersection of a straight line depending only on λ and a curve of constant τ gives the two concentrations for the chosen parameter λ and that value of τ. The parameter λ might be called the saturation ratio, since, when it is 1, the immobilate dissolved in a given volume of solution will exactly saturate the same volume of porous solid. For example, if w < 0, λ is the ultimate conversion of sites, and for w > 0, λ - 1 is the additional fraction of sites needed to immobilize excess immobilate. 2.1. Batch Reaction with Varying Volume. Concentration change occurs by chemical binding as well as potentially by volume change through drying or otherwise. Fortunately, since u appears linearly, this only distorts the time scale. The reaction path in the uv plane is still the line λu - v ) w, a constant, but the point representing the state moves along this path with a different speed. To see this, let Q(t) be the volume of solvent at time t in a unit volume and N(t) the number of moles of immobilate therein. Then:

d(N/N*) Q* N ) -RkV* v dt Q N*

(8)

N* Q* N dv d(Nv/Q*V*)) ) ) -βk v dt dt Q* Q N* where N* and Q* are later identified as initial values, but which for the present are only reference constants with U* ) N*/Q* and V* ) N*v/Q*. If q(t) ) Q(t)/Q* and τ ) ∫RkV* dt/q(t) and we use u for N(t)/N*, we regain the equations (4) and have the same solution. What is more, we do not really care about the warping of time as long as there is plenty of it, for the paths are λu - v ) w and the states of each element will proceed along them to eventual completion. This will certainly happen as the solvent dries out, for τ goes to infinity as q tends to zero. 3. Continuous Process. In the continuous process described earlier, the thickness of the band of the material to be impregnated is h, and the axis of the coordinate X lies in its upper surface. Between X ) 0 and X ) L, this upper surface Y ) 0 (Y being a depth coordinate into the material; i.e., Y ) h is the lower surface of the band) is irrigated with a solution of the species to be immobilized. Deposition of the solute takes place as the fluid flows downward through the porous band which carries it forward. In this work air entrainment is neglected and its complete displacement is assumed effected by the wetting front. At X ) L the cross flow ceases and the material is transported a distance L′ before being squeezed to some lesser degree of dampness. The streams of solution that penetrate through from the wetting section and which are expressed by the rollers are recycled. The X coordinate is the distance from the beginning of the wetting section in the direction of movement of the web, e.g., 0 < X < L, where L is the length of the section over which the solution enters. The dimensionless coordinate is x and 0 < x < ξ. Y is depth into the web of thickness h, so 0 < Y < h or in dimensionless form 0 < y < η. The same depth coordinate is used in the translation section, but a new length coordinate Z, the distance from the junction, is introduced with 0 < Z < L′, or dimensionlessly 0 < z < ζ. The letters u and

v will denote the dimensionless concentrations of immobilate and vacancies, respectively, so that u(x,y) immediately announces itself as the concentration of immobilate at a general point of the wetting section and u(ξ,y,z) as the same concentration at a general point in the translation section. This point is important as u(ξ,y) gives the inlet conditions for the translation section. We would expect, on the physical argument of inaccessibility, that v(ξ,η,ζ) < v(ξ,y,ζ) for all y < η. 3.1. Wetted Section. Let U(X,Y) ) concentration of the immobilate per unit volume of the web in the neighborhood of the point (X, Y); Af is the concentration in the feed, expressed as a concentration per unit volume of the web (i.e., if ωl is the liquid volume fraction of the saturated web, Af/ωl, is the actual concentration in the liquid); V(X,Y) is the concentration of sites and Bf the concentration in untreated web, each again based on the total band volume. Then the balance on immobilate and sites gives the following equations and boundary conditions:

vs

∂U ∂U ∂V + vf ) -RkUV, vs ) -βkUV, ∂X ∂Y ∂X U(X,0) ) Af, V(Yvs/vf,Y) ) Bf (9)

The factors R and β have been retained for symmetry. Either of them may be 1 and, consequently, invisible. The rate constant is k and a time constant characteristic of the system is the reciprocal of RkV*, so we let

y ) YRkV*/vf, u ) U/U*, x ) XRkV*/vs, v ) V/V*, a ) Af/U*, b ) Bf/V* (10) giving

ux + uy ) -uv,

vx ) -λuv,

u(x,0) ) a, v(y,y) ) b (11)

We will transform these equations so that the characteristics are at right angles by letting

σ ) y,

F ) λ(x - y)

(12)

Then

uσ ) vF ) -uv u(0,F) ) 1,

v(σ,0) ) 1

(13)

There is a solution to this pair of partial differential equations in closed form. It can be found by first expressing u and v as the partial derivatives with respect to F and σ, respectively, of a potential function φ(σ,F). If u ) φF and v ) φσ, then the first equation uσ ) vF is satisfied automatically. Now φ satisfies the equation φσF ) -φFφσ, and its exponential ψ ) eφ can be shown after a little manipulation to satisfy the linear equation ψσF ) 0. Therefore, ψ must be the sum of separate functions of F and σ and, if it is to satisfy the boundary conditions, gives

u(σ,F) ) aeaF/{ebσ + eaF - 1}, v(σ,F) ) bebσ/{ebσ + eaF - 1} (14) In terms of the original orthogonal coordinates, x and y and with a ) b ) 1 (as is always possible for constant boundary conditions):


Figure 3. Contours of u(x,y) and v(x,y) per eq 15 at fixed λ. The contour u ) 1 is the x axis, and that for v ) 1 is the diagonal y ) x.

u(x,y) ) eλ(x-y)/{eλ(x-y) + ey - 1}, v(x,y) ) ey/{eλ(x-y) + ey - 1} (15) Contours of u and v are shown in Figures 3 and 4. The contours illustrate that constant u fans out from the wetting front diagonal, with slopes falling increasingly with λ, whereas constant v drops from along the y ) 0 axis again, with slopes increasing with λ. For both u and v, these lines emanate with decreasing value, the more distant from the origin. This solution for the wetting section allows us to calculate the amount of immobilate that escapes unreacted from the bottom of this section, BD, and the concentrations on the entrance plane of the final section, DC. On BD, y ) η ) RkBfh/vf, and x runs from x ) η ) RkBfh/vf to x ) ξ ) RkBfL/vs. Thus, the average concentration of the liquid exiting below the wetting section is:

u j (ξh,η) )

∫

ξ 1 u(x,η) dx ) η ξ-η 1 [ln{eλ(ξ-η) + eη - 1} - η] ) λ(ξ - η) 1 ln{eλξ-(λ+1)η + 1 - e-η} (16) λ(ξ - η)

j (ξh,η) ) e lim u

tions in the wetting section and with three variables ξ, y, and z as the concentrations in the translation section, then the web of material at the rollers (EF) has liquid concentration u(ξ,y,ζ) for y between 0 (E) and η (F). Similarly, the vacant sites are present in concentration v(ξ,y,ζ), 0 e y e η. The average concentration in the liquid at the section EF is:

u j (ξ,η j ,ζ) )

(17)

ξfη

which is just the concentration along the wetting front line where v ) 1. 3.2. Translation Section. On the plane DE, x ) ξ and y runs from 0 to η. Although one might consider the case where fluid drains to dewet the web, we will treat here the simplification where capillary action retains all liquid fed between PC and thus the matrix is fully saturated in DCEF as it is in OBDC. Each infinitesimal volume now moves with constant y and behaves like a batch reactor for the time ζ that it takes to go from CD to EF. Thus, if we define u, v, and w with two variables x and y as the respective concentra-

∫0ηu(ξ,y,ζ) dy/η

(18)

3.3. Off-Line Problem. After expression of liquid by wringing or otherwise, reaction continues off-line in the wetted volume of the matrix. Reduction of liquid content by the fraction χ has opposite effects on the immobilate and sites concentrations. For example, reduction in the liquid content increases the solid content relative to the liquid, thus reducing the immobilate concentration in the wetted volume, which is carried by the liquid. Since, on the other hand, sites are carried by the solid matrix, their concentration in the wetted volume is now increased as the relative solid content is increased. We will introduce two parameters φi and φs representing these effects for immobiliate and sites, respectively. Each is the ratio of respective wetted volume concentrations after to that before liquid removal by wringing or otherwise. It is straightforward to show that in general:

φi )

It is straightforward to show that in the limit: -η

Figure 4. Contours of u and v as a function of λ (u emanate off diagonal, v off x axis).

Fl Fs - Fd χ ) FT - Fd Fl - FT + Fs 1 -

Fl 1 φs ) ) Fl - FT + Fs 1 -

(19)

where FT is the density of saturated matrix, Fs is the density of the wetted matrix after squeezing, Fd is the density of the dry mass of the matrix on a per total matrix volume basis in the final wrung state (e.g., mass upon drying per total original damp volume matrix), Fl is the density of the liquid wetting the matrix, χ is the fraction of original liquid remaining after squeezing, and is the porosity (void volume fraction) characteristic of the final squeezed and wetted product. It is straightforward to show that φi < 1 for Fl > FT - Fs and similarly φs > 1. The actual attainable φi and φs are arbitrary if other than physical squeezing is employed (e.g., forced


Figure 5. Reactor in the hodograph plane.

air drying, etc.). Note that the natural upper limit of χ becomes 1 - . The importance of φi and φs becomes apparent now when considering the dynamics and scaling of the kinetics equations. Let

u° ) U°/U*,

v° ) V°/(φsV*),

τ ) RkφsV*t, λ° ) λ/φs (20)

then balance equations for the off-line become

du°/dτ ) -u°v°,

dv°/dτ ) -λ°u°v°

(21)

with initial conditions

j (ξ,ηj ,ζ) u°(y,τ)0) ) φiu

(22)

v°(y,τ)0) ) v(ξ,y,ζ)

(23)

These combined definitions allow a convenient depiction of the relation of u and v from the wetting section to their ultimate off-line destiny in the hodograph plane. 3.4. Hodograph Plane. Before going on to the method of design, it is useful to present the complete solution from the wetting and translation regions to the off-line problem in the hodograph plane, as it is called on account of its utility with the first-order equations of fluid mechanics. It is a mapping of the reactor in the plane of u and v and illustrates the relation of u with v rather than u or v with position. As show in Figure 1 at point O, both u and v are 1 and u ) 1 on OC since the liquid feed comes in over this section, while v ) 1 on OB since the web is not wetted below this line. B is thus the point (u, v) ) (e-η, 1) and C the point (1, e-λξ). Point A corresponds to that directly above the bottom of the web where liquid first penetrates. Points C and A coincident implies that only sufficient liquid is used to wet the web (i.e., ξ ) η). The states on the bottom of the web, y ) η, along BD all lie on the line u + (1 - e-η)v ) 1, which seeks to join B to the (1, 0) corner of the square, as illustrated in Figure 5. The states in a vertical section of the web at CD lie on the contour of constant x ) ξ. The family of constant x curves is shown in Figure 6 with the constant y which are straight lines emanating from (u, v) ) (0, 1). Point

Figure 6. uv phase plane for constant x and y (x are solid).

D is the intersection of the ξ member of the constant x family and the η member of the constant y. No combination of ξ and η can push the segments BD and DC beyond the diagonal of the square, for it is clear from (15) that u + v - 1 ) 1/(eλ(x-y) + ey - 1) > 0. Thus, a translation section is necessary if we are to approach on-line the desirable region near the origin. In the translation section DCEF the web is merely translated, and each element of it acts as a batch reactor starting with the concentrations u(ξ,y) and v(ξ,y) that it has on DC, z being the dimensionless reaction time and ζ the time at which it passes virtually instantaneously through the rollers. From the batch equations we have the formulas given below in the recap equations (24)-(26). We have seen that all batch paths in the uv plane are lines of constant w ) λu - v, so that E and F must lie on parallel lines of slope λ through C and D, respectively. Were the translation section to be very long, the state of the web would approach the projected intersection of the u and v axes of lines DF and CE, respectively. In such a case, a most deplorable product is obtained since it would have a very wasteful excess of immobilate on the top surface. The action of the rollers is to eliminate this excess and so translate the state of the web from EF to GH. Since their action is rapid, it is assumed that v changes negligibly. But since


their action is violent, the distribution of u across the web is made uniform at a fraction φi of the mean concentration immediately before the squeeze, i.e., the mean concentration of the section EF. The broken intermediate line connects the mean values of u and v at several cross sections. The line representing the state of the web is therefore translated between the horizontal parallels v ) v(ξ,0,ζ) at E and v ) v(ξ,η,ζ) at F and becomes the vertical GH. States off-line then move down the parallels GJ and HK and can be made to finish as a segment of the u and v axes with tolerable excess and incompleteness. From eq 22 u at GH is just the average at EF scaled down by φi, and by definition v is left unchanged. Now off-line the parallels HK and GJ are of a lesser slope λ/φs )λ° compared to the translation section. The action of the rollers is therefore of great importance in minimizing the imperfection of the final product by removal of excess reactant, thus repositioning u and v off-line trajectories for ultimate targets near the origin. 4. General Design 4.1. Design Equations. The basic formulas may be usefully recapitulated here. For the wetting section we have

u(x,y) )

v(x,y) )

eλ(x-y) λ(x-y)

e

+ ey - 1 ey

eλ(x-y) + ey - 1

(24)

w(x,y) ) λu(x,y) - v(x,y) Similarly, for the translation section we have

w(ξ,y) E(ξ,y,ζ) u(ξ,y,ζ) ) u(ξ,y) λu(ξ,y) E(ξ,y,ζ) - v(ξ,y) (25) w(ξ,y) v(ξ,y,ζ) ) v(ξ,y) λu(ξ,y) E(ξ,y,ζ) - v(ξ,y) E(ξ,y,ζ) ) exp(w(ξ,y,) ζ) w(ξ,y,ζ) ) λu(ξ,y,ζ) - v(ξ,y,ζ) ) w(ξ,y) Finally for the off-line problem in the time domain we have

u°(ξ,y,ζ,τ) ) φiu j (ξ,ηj ,ζ)

w°(ξ,y,ζ) E°(ξ,y,ζ,τ) λ°φiu j (ξ,η j ,ζ) E°(ξ,y,ζ,τ) - v(ξ,y,ζ)

(26) w°(ξ,y,ζ) v°(ξ,y,ζ,τ) ) v(ξ,y,ζ) λ°φiu j (ξ,ηj ,ζ) E°(ξ,y,ζ,τ) - v(ξ,y,ζ) E°(ξ,y,ζ,τ) ) exp(w°(ξ,y,ζ) τ) w°(ξ,y,ζ) ) λ°φiu j (ξ,ηj ,ζ) - v(ξ,y,ζ) ) λ°u°(ξ,y,ζ,τ) - v°(ξ,y,ζ,τ) where τ was defined for the off-line in eq 20. With formulas of the simplicity of these, the design problem begs to be done on a spreadsheet. We shall describe

one that works very well in Microsoft Excel version 5 or 4 but whose principles will be the same for all major spreadsheet software. 4.2. Optimization Constraints. At this point it is useful to introduce objectives of interest in the design of the reactor configuration. Of immediate interest is the ultimate product state, and it will, in general, be desirable to achieve a product whose greatest incompleteness is less than υ relative to the initial number of sites and whose excess immobilate is less than µ relative to the initial liquid feed concentration. Thus ultimately, by virtue of scaling choices in the off-line variables, we seek:

u°∞ ) max[u°(y,τf∞)] ) u°(y)0,τf∞) < µφi v°∞ ) max[v°(y,τf∞)] ) v(y)η,τf∞) < ν

(27)

Stated differently, these conditions insist on reactor operation which makes maximum use of reactive sites v while at the same time minimizing potentially detrimental unreacted immobilate. Thus ultimate states K and J are sought, represented by a segment of the v-axis with 0 e v°∞ e υ and a segment of the u-axis with 0 e u°∞ e µφi, respectively. Optimization of the design may be sought for varieties of scenarios. We construct here an approach which is natural as a result of the dimensionless parameters and practical operation. For example, it is natural to let η be fixed, since typically the product performance fixes V*, the product format or the production capacity and profitability requirements fix h, the physical properties of the matrix fix vf, and the intrinsic kinetics and stoichiometry fix Rk. At a minimum, ξ will be at least η, and it is impractical to realize ξ ) η without risking incomplete wetting of the bottom portion of the matrix due to typical process variability. Thus, it is reasonable to allow ξ to take some multiple of η, where it is understood that ξ ) η is a limiting and perhaps academic case. The function of the wetting section is 2-fold: first, it is to provide adequate immobilate for the reaction to proceed to its ultimate desirable destiny in the combined translation and off-line sections; second, and very important, its function is to enhance the uniformity of immobilate and sites concentrations. The second function is appreciated by study of the hodograph plane since nonuniformity of ultimate vacancies is maximum when ξ ) η, and all paths from B collapse (u, v) toward the (1, 0) point. However, reactor capital requirements and operating costs which are proportional to ξ must be considered, but without ζ, the design and operation to achieve objectives µ and υ is subject to φi and the off-line dynamics discussed below. The translation section value must be weighed depending on the product uniformity requirements, relative zone costs, and practical off-line dynamics constraints. Such are considered in the following. To begin, it is sensible to assume ζ should be at least η as it is not likely the web is wetted right up to the rollers. But neither should ζ be so long as to meet objective υ at the top of the web on-line, since it is inevitably more efficient to traverse this objective off-line and thus consume residual immobilate in doing so. Neither should ζ be so long as to achieve µ at the bottom of the web on-line and so incur excessive starvation of the reaction there off-line. In general, both objectives µ and υ should be traversed off-line to both respect and take advantage of the exhaustion there that naturally wants


to occur. These constraints place indirect upper bounds on ζ and take the form:

u(ξ,η,ζ) > µ,

v(ξ,0,ζ) > ν

(28)

One might be tempted to allow violation of these constraints on-line but must at the same time appreciate that practical variability in reactor operation may land one at an unacceptable frequency on the u ) 0 position with considerable incompleteness in the former case and in the latter at the v ) 0 position with grossly excess immobilate. However, the theoretical possibilities may still be of interest and do considerably relax design latitude. We consider now a minimum practical value for ζ apart from the fundamental minimum of ζ > η. As mentioned, a particular function of the wetting and translation sections has to do with dynamics. For example, it is conceivable that the off-line domain alone could be used to achieve µ and υ even in the ξ ) η case. However, because the reaction naturally slows after the rollers (i.e., λ° < λ), the real time required for traversal of µ and υ might then be prohibitive, and thus ξ > η and ζ > η may be of interest if only to speed up reaction progress. Therefore, to avoid prohibitive off-line time, one might constrain traversal there relative to the online time ξ + ζ. For example, we will define τ95 to be the time at which 95% of u° or v° reduction is traversed. This traversal time is found by solution of the dynamic equations in (26) for traversal beyond either u°95 or v°95, e.g., letting:

u°95 ) 0.95u°∞ + 0.05φiu j (ξ,ηj ,ζ) (29)

v°95 ) 0.95v°∞ + 0.05v(ξ,η,ζ)

We find then two values for τ95; a lower value from u and an upper value from v:

τu95 )

{

ln τv95 )

}

u°95v(ξ,0,ζ)

/w°(ξ,0,ζ,τ)

φiu j (ξ,η j ,ζ)[λ°u°95 - w°(ξ,0,ζ,τ)]

(30)

{

ln

}

v(ξ,η,ζ)[v°95 + w°(ξ,η,ζ,τ)] λ°φiu j (ξ,ηj ,ζ) v°95

/w°(ξ,η,ζ,τ)

On the opposite end of the off-line time scale and again in appreciation of inherent reactor operation variability, it is also important to avoid the situation where the reaction instantaneously completes itself once off-line. To avoid this, one might stipulate that the lower τ95 estimate is at least as great as ξ + ζ. It is natural then to let the upper value of τ95 be constrained by some multiple ϑ of ξ + ζ. Since time is inflated by φs offline, the complete dynamics constraint on ξ + ζ takes the form:

1
η, as well as dynamics constraints on ξ + ζ through τ95 and ϑ offline through equations in (31). 4.3. Objective Surface. Optimization can be achieved in more than one manner. However, with constraints as prescribed, minimization of only quadratic departure from ultimate µ and υ leads to strongly correlated parameter sets. This correlation can be eliminated with additional application specific constraints. These would go beyond the scope of this work and unnecessarily obscure the essence of reactor operation. But this need not be done as the answer lies in the mundane matter of manufacturing cost. For example, from a reactor capital and operating cost perspective, it is not unlikely that the minimization of the wetting section is of foremost interest. Thus, it is natural to choose an objective which assumes the reactor is optimal whose ξ is minimal, i.e., taking the form:

Φ)ξ

(32)

With this surface determined from the design equations (24)-(26), the problem is handled by the Solver analysis tool in Excel, subject to the constraints described above for ξ and ζ. Several examples follow for thin to thick web cases. 4.4. Optimization of Thin to Thick Webs. Parts a-e of Figure 7 show results for η ranging from 0.1 to 0.48. It is interesting to note the progression of translation section significance with the constraints employed as η is decreased and, conversely, that the wetting section becomes increasingly important for thicker webs. These findings are sensible as it is expected that, in the limit of η f 0, the need for ξ > η to reduce top to bottom differences in u is diminished. Thus, because of dynamics constraints on ξ + ζ, ζ takes greater value in relation to ξ. Without the dynamics constraints, one would find otherwise that the reaction could be carried out almost entirely off-line. As η increases, greater ξ is required to eliminate top to bottom variations in setting up for the off-line trajectories. These optimizations displayed in parts a-e of Figure 7 can be reduced to a single plot for corresponding λ, ξ, and ζ as shown in Figure 8. It is important to note that, for constraints described, optimization below η ) 0.1 leads to correlation between λ and ζ since top to bottom variations vanish. On the



Figure 7. (a) Optimized immobilization reactor hodograph: (a) η ) 0.1, (b) 0.12, (c) 0.2, (d) 0.4, and (e) 0.48.

5. Comparison with Experiment

Figure 8. Summary optimization parameter space from parts a-e of Figure 7.

other end, above η ) 0.48, correlation again sets in between λ and ξ as no feasible solutions exist which do not violate constraints. Unique solutions can then only be achieved by selective relaxation of constraints.

Limited experiments were undertaken for a system represented by the proposed salient model involving the immobilization of sodium pyrithione on distributed zinc sites within cellulose sponge. Such a system is new and was recently disclosed in detail (Roenigk, 1994). Dimensionless parameters employed in these tests included φi ) 0.25 and φs ) 2.5. Kinetics were determined in separate laboratory experiments with aqueous slurries of the zinc complex for R ) 2 and β ) 1. Of particular interest in these studies were the off-line dynamics. Prediction of experimental data is shown in parts a and b of Figure 9 for two separate cases with replication of the second. Transient data were obtained by sampling the sponge immediately as it exited the rollers, followed by timed water rinsing of samples which stopped the reaction by removal of excess reactant. Chelated levels of pyrithione were then determined by a compleximetric extraction technique. The zinc levels were determined by acid digestion of samples followed by inductively coupled plasma analysis. Vari-


to comprise up to three regions including (1) a wetting region where immobilate permeates downward through the moving web, (2) a translation region in which the web only carries fluid, and (3) an off-line section after excess liquid is removed where the reaction continues to its ultimate destiny. Behavior of this class of reactors is captured succinctly in a hodograph mapping, relating states of immobilate and sites through all regions of reaction space. Optimization is elucidated with constraints which arise naturally, and optimal and unique parameter sets are identified. Predictions of the chelation of sodium pyrithione to zinc sites in cellulose sponge show close agreement with observed on-line extent and off-line dynamics. Simulations suggest improvements in predictive value of the model may be realized with incorporation of diffusion limitations online and off. In general, additional value may be added by more correctly accounting both for imperfect wetting of the reactive fluid in the solid matrix with air entrainment, as well as for potential drainage in the translation section. Other significant kinetic schemes might be explored as well having intermediates and attendent selectivity concerns. It is hoped the development offered here provides the reaction engineer a groundwork from which such studies may be fruitfully pursued for this interesting and industrially significant class of reactors. Figure 9. Prediction of experimental observations for (a) λ ) 10.2, η ) 0.021, ξ ) 0.065, and ζ ) 0.019 and (b) λ ) 15.6, η ) 0.014, ξ ) 0.042, and ζ ) 0.012 using lab kinetics, with 80% accessible sites.

ability indicated represents the measured standard deviation as a result of analytical method variability, process fluctuations and their effect on values from multiple samples over extended duration of reactor operation, and inherent top to bottom differences. Model curves in parts a and b of Figure 9 reflect an empirically derived assumption of only 80% of total sites accessible due to poisoning. The model would otherwise overpredict reaction extent on-line and off-line based on total sites. Evidence from other studies suggests adverse poisoning effects on kinetics as well, but such pursuits are left off here. Overestimation of the reaction extent by the model on-line may also be expected if diffusion limitations are significant between the pore liquid and the swollen mass regimes, since the incoming connate liquid would then necessarily retard extent by dilution. Some overestimation of the reaction extent should also be expected off-line since the model neglects both diffusion limitations within the tortuous solid matrix as well as the finiteness of the sites clusters. The clusters are in particulate form in this specific application which necessarily results in significant heterogeneous local depletion. What is significant is that without much correction the model appears to approximate this data closely, albeit over a limited range. In spite of its simplicity, the model has proven predictive in broader studies of on-line extents, dynamics, and requisite reactant feed when coupled with a complete system mass balance. 6. Conclusions and Recommendations The cross-flow reactor with applications in chelation, decoating, dyeing, and ion exchange has been modeled assuming bimolecular kinetics between convected immobilate and fixed matrix sites. The reactor is assumed

Acknowledgment Generous support from the 3M Home and Commercial Care Division Laboratory and its O-Cel-O manufacturing unit is gratefully acknowledged. The authors also thank R. Brozo and K. Dunbar of the 3M Corp. for their extensive effort in execution and analysis of experiments. Nomenclature k ) bimolecular rate constant N, N* ) moles of immobilate Nv ) moles of sites Q, Q* ) volumes q ) relative volume U, V ) on-line concentrations U°, V° ) off-line concentrations U*, V* ) reference concentrations u, v ) dimensionless on-line concentrations u°, v° ) dimensionless off-line concentrations u°∞, v°∞ ) ultimate dimensionless off-line concentrations u°95, v°95 ) u°, v° at 95% traversal to ultimates u°∞, v°∞ vs, vf ) web (transverse) and wetting front (downward) velocities X, Y, Z ) positions x, y, z ) dimensionless positions Greek Symbols R ) stoichiometric coefficient for immobilate β ) stoichiometric coefficient for sites η ) dimensionless vertical thickness of web ξ ) dimensionless wetting section length ζ ) dimensionless translation section length λ, λ° ) saturation ratios on-line and off-line χ ) fraction of liquid remaining after wringing by rollers ) porosity (void volume fraction) φi, φs ) ratios of concentrations after to before wringing Fs, Fd, FT, Fl ) densities τ ) dimensionless time

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 2899 τ95, τu95, τv95 ) dimensionless 95% traversal time, function of u° or v° ωl ) volume fraction of liquid in a saturated matrix σ, F ) transformed coordinates warped from x, y φ ) potential function ψ ) eφ µ ) tolerable excess (maximum u°∞) υ ) tolerable incompleteness (maximum v°∞) ϑ ) maximum relative time off-line Φ ) objective function for optimization

Literature Cited Fitzpatrick, N. P.; McCubbin, J. G.; Evans, J. F. Fundamentals of Decoating of Organically Coated Aluminum Scrap. Proc. 115th TMS 1986, 2, 793-796. Jowett, G. E. Improvements to solid/liquid reactions using a continuous belt of sponge material.GB2192403, 1990. Marb, C. M; Vortmeyer, D. Multiple Steady States of a Crossflow Moving Bed Reactor: Theory and Experiment. Chem. Eng. Sci. 1988, 43, 81. Richter, E.; Knoblauch, K.; Ju¨ngten, H. Design of Cross-flow Moving Bed: Flue Gas Desulfurization by means of Activated Coke. Verfahrenstechnik 1980, 14, 338.

Roenigk, K. F. Durable Antimicrobial Agents. International Application PCT/US93/10316 1994. Wiegandt, H. F.; Von Berg, R. L.; Patel, P. R. The Crossflow Piston Bed. Desalination 1978, 25, 303. Wigton, H. F. Mathematical model of a cross-flow moving bed granular filter. In Proceedings of the 5th International Conference on Fluidized Bed Combustion. Developmental Activities; Bliss, C., Williams, B. M., Eds.; Dec 12-14, 1977, Vol. 3, pp 583-605. Wolff, E. H. P.; Veenstra, P.; Chewter, L. A. A novel Circulating Cross-flow Moving Bed Reactor System for Gas-Solids Contacting. Chem. Eng. Sci. 1994, 49, 5427. Young, R.; Yeh, J. T. Computer Simulation of a flue Gas Desulfurization Moving-Bed Reactor. Environ. Prog. 1993, 12, 200.

Received for review November 16, 1995 Accepted March 18, 1996X IE9506936

X Abstract published in Advance ACS Abstracts, August 15, 1996.

Analysis of a Continuous Immobilization Reactor - Industrial

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