A Mass Transfer Coefficient for Radial Flow Adsorption and Ion

Research Note pubs.acs.org/IECR

A Mass Transfer Coefficient for Radial Flow Adsorption and Ion Exchange Dennis F. Hussey Electric Power Research Institute, Palo Alto, California, United States

Ashwini K. Pandey and Gary L. Foutch* School of Chemical Engineering, Oklahoma State University, Stillwater, Oklahoma, United States ABSTRACT: Radial-flow adsorption and ion exchange beds are considered for commercial water treatment processes on occasion, particularly for specialty applications such as nuclear reactor water cleanup. They have significant theoretical potential but some practical flow issues like short circuiting. For more common axial beds, at ultralow concentrations, a key design and operational performance factor is the mass transfer coefficient (MTC) that can accurately reflect the capabilities of the bed. MTC may be used as part of initial design but is most commonly used to monitor the status of fouling within the column. The question of whether MTC has the same meaning for both radial- and axial-flow is addressed. An analysis for ion exchange indicates that the method is still valid for MTC resin monitoring, and easily adapted to adsorption.

■

INTRODUCTION Ion exchange is a process that redistributes counterions between fluid and solid phases by diffusion. The ionic mass transfer coefficient (MTC), a property that characterizes the kinetic ability of the ion-exchange resin, is required for modeling and simulation of industrial bed performance. The MTC is used in a kinetic rate expression that is inserted into the column material balance and integrated. Its measurement as an axial bed property is common in the ultrapure water industry. This paper addresses whether an MTC measurement has similar meanings for radial bed operation. At low concentrations kinetics is controlled by external, film mass-transfer resistance.1,2 An estimate of a clean resin MTC is available from numerous dimensional analysis expressions.3 With resin age or damage, the MTC decreases, indicating greater chemical or physical resistance. This decrease is typically referred to as fouling. With an unfouled resin in an ultrapure water service cycle the trace ions in the effluent originate from the bed (equilibrium leakage). In an ideal bed the breakthrough curve is S-shaped with a long baseline followed by relatively rapid rise to the feed concentration near exhaustion. However, as resins foul some of the ions in the feedwater are not exchanged efficiently and move farther through the bed, and the initial concentration break occurs earlier. At high fluid velocity, or with severely fouled resins, some feedwater ions pass through the bed because the rate of bulk mass transfer through the bed is greater than the rate of ionic diffusion into the bead (kinetic leakage). Fouling results in poor water quality and increased operating cost for more frequent bed regenerations. Most plants that use ultrapure water monitor MTC to quantify the degree of fouling in order to know when resins should be replaced. Effectively, the MTC is a combination of the bed hydrodynamics and the resin diffusional resistances, and has the same units as velocity. Physically, MTC represents the © 2012 American Chemical Society

average velocity of an ion traveling through the stagnant water film surrounding the ion exchange beads. MTC is obtained experimentally with a simple column test resulting in an effective lumped-parameter of all possible resistances that slow the diffusion of ions from the solution to the exchange site. With the use of Harries’4 method, a short laboratory test column of resin is first regenerated and rinsed. A salt solution, usually containing sulfate for anionic resins and sodium for cationic resins, is fed at sufficient velocity to ensure that the sample demonstrates kinetic leakage. This can be checked by adjusting the flow rate: effluent concentration will remain constant with equilibrium leakage and will change with kinetic leakage. The effluent concentration is measured quickly after startup so that little capacity is used, and the entire bed depth can be assumed as the exchange zone. Harries’ method is based on a differential bed depth for the MTC test which is more theoretically valid; however, there are full-depth test beds that mimic plant operation and require higher flow rates to ensure kinetic leakage.5 The numerical value of MTC, when compared with that of new resin, identifies a resin problem; although the particular fouling mechanism is not defined. When resins are at their best, the MTC equals that predicted by hydrodynamics alone; typically greater than 2.0 × 10−4 m/s (anionic resin with sulfate) for moderate-flow systems and 3.0 × 10−4 for high-flow systems. Resins fouled severely may have MTC values an order of magnitude lower. Specific plant performance criteria, typically, the desired time between regenerations and the maximum allowable effluent concentrations, should be used to establish minimum MTC values. Received: Revised: Accepted: Published: 1429

June 30, 2011 January 5, 2012 January 5, 2012 January 5, 2012 dx.doi.org/10.1021/ie201399g | Ind. Eng.Chem. Res. 2012, 51, 1429−1434

Industrial & Engineering Chemistry Research

Research Note

axial-flow beds may accumulate suspended solids rapidly, causing an increased pressure drop that may result in bed replacement before the available exchange capacity has been exhausted (Fejes et al.16) or require higher pumping cost. These authors proposed radial beds as an alternative that may minimize the pressure drop problem. Figure 1 shows the basic

While several resin properties may change with age or damage (capacity, water retention, zeta potential, etc.), for ultrapure water applications MTC is particularly sensitive. Commonly, a large MTC drop may have only a 1−2% decrease in exchange capacity. The fact that MTC correlates with the extent of resin deterioration and/or age and is relatively easy to obtain in a bench scale experiment makes it a valuable property measurement.

■

BACKGROUND Harries and Ray6 developed a simplified film mass transfer coefficient model to rate the performance of a cylindrical, axialflow mixed bed. The model assumes no radial or axial dispersion, zero interfacial concentrations, and no interactions resulting from dissociation equilibrium; however, it characterizes the removal efficiency of an ionic species with respect to bed geometry and service parameters. The relationship, in modified form, is

⎛ C eff ⎞ 6k f, i(1 − ε)(FR) πR o2L ln⎜⎜ i f ⎟⎟ = − dp F ⎝ Ci ⎠

(1) Figure 1. Inward flow radial packed bed.

7

Lee et al. and others have shown successfully the reduced MTC values of fouled resin by using the influent and effluent concentrations from simple bench-scale column experiments.

schematic of an inward flow radial column. The water enters the outer perimeter and flows with increasing velocity to the center of the vessel. Beds could also operate in outward flow.

■

■

RADIAL FLOW MODELING Most industrial ion exchange beds are cylindrical columns with one-dimensional axial flow. These systems have been modeled to obtain an overall MTC, and the results have been presented by several researchers4,6,8−10 with applications discussed by many others. In the 1980s radial-flow ion exchange technology was developed in Europe, tested in the US in 198911 and used in commercial application in 1997.12 Tsaur and Shallcross13 showed that radial ion exchange is more efficient than conventional flow under identical operating conditions for simulated geometries. The model developed by Lou14 indicated that radial flow geometry offers sharper profiles, which is favorable for separation and makes best use of resin capacity. Though this technology is limited commercially, it offers advantages over axial flow because velocity varies with bed depth. Since axial flow assumes that the fluid velocity is constant, the standard MTC method may not apply. As a result, a new derivation is presented here specific to this geometry. A factor limiting axial flow bed depth is the mechanical strength of ion-exchange resin beads. The beads at the bottom must withstand the weight of the entire bed. Bead crushing can result in increased pressure drop and the formation of resin fragments. In an annular ion exchange bed, the path length is horizontal between the inner and outer diameters of the bed and not vertical as in an axial bed. Therefore, it is possible to increase the flow path length through the ion exchange material by increasing the bed diameter.15 The variable velocity complicates mathematical modeling of the radial flow mixed bed. Lou’s14 model was a coupled, high-order system of differential equations; it included radial dispersion and used Patankar’s control volume method for numerical solution. The model produced reasonable results that were useful to rate radial bed performance; however, it was never implemented within commercial software. For systems with high concentration of suspended solids like nuclear reactor water clean up (RWCU)one-dimensional

RADIAL MTC DERIVATION To extend the model of Duong and Shallcross15 to handle fouled resins, a comparison of the laboratory generated MTC values should be evaluated. An analogous film MTC model for radial flow based upon the Harries’ model assumptions is presented here.17 The starting point is the reduced continuity equation for a radial-flow packed bed assuming that longitudinal and angular derivatives are equal to zero. ∂Ci u ∂C + R i − R* = 0 ∂t ε ∂R

(2)

The reaction term, R*, is equated to the change in resin phase concentration using a linear driving force model.

−R* =

1 − ε dqi = k f, ias(Ci* − Cib) ε dt

(3)

Equation 3 is substituted into eq 2 to yield the following continuity equation. The volume fraction of resin (FR) is added to handle both homogeneous and mixed resin bed samples.

∂Ci u ∂C (1 − ε) ∂qi + R i + (FR) =0 ∂t ε ∂R ε ∂t

(4)

The equation, as written, is nonlinear because the fluid velocity, uR, is a function of the radius. However, the fluid velocity can be related to the radius with the following expression.18

RuR = ϕ

(5)

where ϕ is a constant relating flow rate and bed geometry. For the radial-flow system, the constant can be defined in terms of inward and outward flow. For outward flow, the constant is derived as 1430

dx.doi.org/10.1021/ie201399g | Ind. Eng.Chem. Res. 2012, 51, 1429−1434

Industrial & Engineering Chemistry Research

Research Note

F (6a) 2πh For inward flow, as the reference is measured in the radial direction from the center outward, the sign must be changed. F ϕ=− (6b) 2πh Substituting eq 5 into eq 4 yields,

k f, i (1 − ε)R ∂ξ = ∂R ϕ dp

ϕ=

Applying the chain rule to change the integration axes gives

∂Ci ϕ ∂Ci 1 − ε ∂qi + + (FR) =0 (7) ∂t εR ∂R ε ∂t Defining the solution and resin fraction dimensionless variables with respect to the individual ionic species feed concentrations and total capacity, xi =

yi =

(17)

∂xi ∂x ∂ξ ∂x ∂τ = i + i ∂R ∂ξ ∂R ∂τ ∂R

(18)

(8)

−

k f, i (1 − ε)R2 2ϕ dp

(10)

∂y ∂xi = − (FR) i ∂ξ ∂τ

− k f, iCif εR ∂τ = ∂R d pQ ϕ

(21)

Assuming the interfacial concentration is zero, which is a reasonable assumption for virgin resin used in ultrapure water applications, the right-hand side of eq 21 can be related to the particle exchange rate: a simple derivative. The work of Harries and Ray4 indicates for low concentrations, ion exchange is film diffusion controlled. In addition, the model is only applied at the beginning of service so that the bed depth can be assumed to be the exchange zone (L). Once ions develop exchange fronts, the depth of the exchange zone is unknown, which prevents an accurate prediction.

(11)

dyi (12)

dt

=

k f, ias Q

Ci

(22)

Changing the integration axis to dimensionless time yields,

∂yi ∂τ

=

∂yi ∂t k f, iasCi ⎛ d pQ ⎞ ⎟⎟ ⎜⎜ = ∂t ∂τ Q ⎝ k f, iCi ⎠

(23)

which reduces to the following function of xi:

k f, iCif

∂ξ =0 ∂t

(20)

Equation 20 collapses to the following partial differential equation.

The constant “2” is not in the velocity definition; however, it was needed to facilitate the reduction of the continuity equation to an ordinary differential equation. The choice of dimensionless variables is necessary to the solution of the equations. Their derivatives are required to scale the time and distance dimensions into dimensionless terms.

∂τ = ∂t d pQ

∂xi k f, iC f R ε ⎤ ⎥ + (FR) 1 − ε ∂τ d pQ ϕ ⎥⎦ ε

⎡ ∂yi k f, iCif ⎤ Q ⎢ ∂yi ⎥=0 + (0) ∂τ d pQ ⎥⎦ Cif ⎢⎣ ∂ξ

Equation 10 is a partial differential equation that has the fluid fraction varying with time and radius, and the resin fraction varying with time. The fluid fraction shares a common boundary condition in time and space: the feed concentration, Cif. This implies a combination of variables may be applied; however, the “R” term is still in the denominator of the radial variation of fluid fraction. The dimensionless variables chosen were found by substituting the velocity definition, eq 5, into the dimensionless time and distance used for cylindrical flow.

ξ=

(19)

⎡ ⎡ ⎛ k C ⎞⎤ k (1 − ε)R ⎢ ∂xi (0) + ∂xi ⎜ f, i f ⎟⎥ + ϕ ⎢ ∂xi f, i ⎜ ⎟ ⎢⎣ ∂ξ ∂τ ⎝ d pQ ⎠⎥⎦ εR ⎢⎣ ∂ξ ϕd p

and dividing eq 7 by (CifQ)/Q gives the continuity equation with dimensionless fluid and resin fractions, and dimensional time and radius.

k f, iCif ⎛ R2 ε ⎞ ⎟ ⎜t − τ= d pQ ⎝ 2ϕ ⎠

∂yi ∂ξ ∂y ∂τ + i ∂ξ ∂t ∂τ dt

Substituting the derivatives into eq 10 and expanding yields

(9)

∂xi ϕ ∂xi 1 − ε Q ∂yi + + (FR) =0 ∂t εR ∂R ε Cif ∂t

=

∂t

qi Q

∂xi ∂x ∂ξ ∂x ∂τ = i + i ∂t ∂ξ ∂t ∂τ ∂t

∂yi

Ci Cif

(16)

∂yi ∂τ

(13)

= d pasxi

(24)

Equation 24 is substituted into eq 21 to give an ordinary differential equation that can be integrated by separation of variables.

(14)

∂xi = − (FR)d pasxi ∂ξ

(15) 1431

(25)

dx.doi.org/10.1021/ie201399g | Ind. Eng.Chem. Res. 2012, 51, 1429−1434

Industrial & Engineering Chemistry Research

Research Note

and the exchange will be more effective since new resin has less mass-transfer resistance. The terms in eqs 1 and 32 have physical significance; V/F is the superficial residence time of a fluid element in the empty bed, while the remainder of the right-hand side represents the rate (units of inverse time) at which ions are transferred into a specific volume of resin. It applies to both outward and inward flow radial beds. Figure 2 shows the impact of MTC on the concentration profile through a 0.02 m bed of resin in radial flow. This

The boundary conditions are

Ci = Cif

@R = R o

Ci = Cieff @R = R i xi = 1 @ ξ = ξ o xi = xieff

@ξ = ξ i

Separating the variables yields,

∂xi = − (FR)d pas∂ξ xi

(26)

Integrating, x

ξ

ln(xi)|1 i,eff = − (FR)d pas|ξi

(27)

⎛ C eff ⎞ ln⎜⎜ i f ⎟⎟ − ln(1) = (FR)d pas(ξo − ξi) ⎝ Ci ⎠

(28)

o

Inserting the dimensionless definitions to transform eq 28 back to dimensional distance,

ξo =

ξi =

Figure 2. Radial concentration profiles for various MTC values in 0.02 m slices of ion-exchange resin with radial geometry.

k f, i(1 − ε)R o2 2ϕd p

geometry was selected to match the amount of resin typical in an axial differential test bed (0.1 m). Results show clearly that, at lower MTC, bed performance is reduced. For 2 × 10−4 m/s, typical of new resin at moderate flow rates, the exchange zone is less than half of the total resin depth. Reducing MTC by half moves the concentration profile significantly through the bed; however, the effluent concentrations appear nearly the same on a rectilinear plot. For the case of extreme fouling (for example, near 1 × 10−5 m/s) the bed is able to remove only a fraction of the ions at the same operating conditions. Because ultrapure water typically has extreme effluent concentration requirements, the presentation in Figure 2 is somewhat misleading. The distinction between 2 × 10−4 and 1 × 10−4 m/s MTC is easier to see in a semilog plot, Figure 3. In

(29)

k f, i(1 − ε)R i 2 2ϕd p

(30)

and reducing yields eq 31.

⎛ C eff ⎞ k f, i(1 − ε)as ln⎜⎜ i f ⎟⎟ = (FR) (R o2 − R i 2) 2ϕ ⎝ Ci ⎠

(31)

Finally, substituting the definitions of specific surface area, as, and ϕ yields the expression in final form. Note in this derivation, eq 6b is applied for ϕ.

⎛ C eff ⎞ − 6k f, i(1 − ε) π(R o2 − R i 2)h ln⎜⎜ i f ⎟⎟ = (FR) dp f ⎝ Ci ⎠

(32)

Comparing eq 32 for radial flow with eq 1 for axial-flow, it is easy to see the only difference is the volume of the container πRo2L for the axial flow bed and π(Ro2 − Ri2)h for the radial flow bed. The MTC values calculated from each expression are identical for equal volume and flow rate systems.

■

ANALYSIS OF THE MODEL The equations were programmed into Excel and typical results were obtained in order to visualize the effects of eqs 1 and 32 graphically. A differential laboratory radial-flow column was simulated for the operation of a homogeneous anionic resin bed with the following operating conditions: flow rate = 1.667 × 10−4 m3/s, bed depth = 0.02 m, influent concentration = 100 mg/L, particle diameter = 5.9 × 10−4 m, and inner column radius = 0.1 m. Concentration was obtained as a function of radius for different MTC values. As MTC drops the amount of ionic removal decreases emphasizing less exchange because of increased kinetic resistance. For new resins, MTC will be higher

Figure 3. Semilogarithmic plot of concentration profile with radial position in 0.02 m ion-exchange resin.

practice, even for clean, new resin, there is likely to be some equilibrium leakage (very low residual impurity concentrations released from the resin) for ultrapure water, and the lower concentration curves will become horizontal with this limitation. In addition, there may also be analytical limits to measurements at very low concentrations that can only be interpreted as a “less than” level on the profile plot. However, 1432

dx.doi.org/10.1021/ie201399g | Ind. Eng.Chem. Res. 2012, 51, 1429−1434

Industrial & Engineering Chemistry Research

Research Note

obtain. The advantage is that the continuity equation, in cylindrical coordinates, results in an analytical solution requiring Bessel functions (which often have infinite series solutions when solving for the constants of integration) and numerical software models of the cylindrical case can be adapted to radial-flow with minimal changes in the algorithm.

on the basis of kinetic leakage alone, doubling the MTC value results in about a 2.5 log effluent concentration difference within this small test bed. In a plant the higher MTC resin can withstand some level of concentration spike in the feedwater. However, when the MTC is halved a spike in feed solution concentration would result in nearly immediate ionic passage through the resin bed. Depending on the application this would require immediate action for the protection of downstream materials. For this reason, microelectronics plants avoid the problem of resin fouling with repeated cycles by using nonregenerable polishing bottles as the last water treatment step. Another way to view the importance of MTC on ionexchange resin performance is to consider the efficiency of ionic removal. Figures 4 and 5 present comparative plots for similar

■

CONCLUSIONS On the basis of the results presented it is possible to follow resin fouling in a radial-flow bed with the same MTC methods developed by Harries for axial beds. The only difference in the derived expressions is redefining the path length through the bed, and this factor is independent of the resin properties, in particular, the MTC. Once an experimental MTC is available then a model, such as that by Duong and Shallcross, can be used to analyze radial bed performance.

■ ■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

Figure 4. MTC observed for specific flow rate and removal efficiency in radial flow.

Figure 5. MTC observed for specific flow rate and removal efficiency in axial flow.

lab scale columns using eqs 32 (radial flow column) and 1 (cylindrical flow column), respectively. As flow rate increases the film thickness through which ions must travel decreases and the MTC goes up. For these plots, with the same flow rate and separation efficiency the MTC of radial operation is higher than that of axial flow. For example, at 20 L/min with a 99% efficiency the radial MTC is 1.6 × 10−4 m/s compared with 1.4 × 10−4 m/s for axial flow. Equation 32 is useful for evaluating the performance of a radial flow mixed bed ion exchange column (or any radial flow fixed-bed adsorption columns with a constant fluid density), and is a limited variation of the derivation of eq 21 focused specifically on experimental MTC. While eq 21 is more general, the caveat is that the MTCs must be calculated as a function of radial distance to account for the varying fluid velocity. Equation 21 is exactly the same form as the well-known cylindrical fixed-bed adsorption equation that can be solved with many numerical methods (e.g., Euler’s method or Gear’s backward difference method); the only difference being the definition of the dimensionless time and distance variables. A relationship for the mass transfer rate would be more difficult to

■

NOMENCLATURE as = specific surface area of resin bead (m2/m3), asdp = 6 for spherical particles Ci = fluid phase concentration (equiv/L) Cif = fluid phase concentration in feed (equiv/L) Cieff = fluid phase concentration in effluent (equiv/L) Ci* = concentration at the liquid−solid interface (equiv/L) Cib = bulk fluid phase concentration (equiv/L) dp = sample resin harmonic mean size (m) F = volumetric flow rate (m3/sec) FR = volume fraction of sample resin (m3/m3 bed) h = height of radial flow bed (m) kf,i = average film mass transfer coefficient of ion “i” (m/s) L = cylindrical bed depth (m) qi = resin phase concentration of an individual specie (equiv/ L) Q = total ion-exchange resin capacity (equiv/L) R = radius of column (m) R* = reaction term Ri = inner radius of column (m) Ro = outer radius of column (m) t = time (s) ur = fluid velocity (m/s) xi = dimensionless bulk solution fraction of ion i y = dimensionless resin fraction of ion i ε = bed porosity, (m3/m3 bed) ξ = dimensionless distance τ = dimensionless time ϕ = constant REFERENCES

(1) Frisch, N. W.; Kunin, R. Kinetics of Mixed-Bed Deionization: I. AIChE J. 1960, 6, 640. (2) Helfferich, F. G. Ion Exchange; McGraw-Hill: New York, 1962. (3) Chowdiah, V. N.; Foutch, G. L.; Lee, G. C. Binary Liquid-Phase Mass Transport in Mixed-Bed Ion Exchange at Low Solute Concentrations. Ind. Eng. Chem. Res. 1993, 42, 1485. (4) Harries, R. R. Ion Exchange Kinetics in Condensate Purification. Chem. Ind. (London U.K.) 1987, 4, 104. (5) ASTM D6302-98, Standard Practice for Evaluating the Kinetic Behavior of Ion Exchange Resins. ASTM: West Conchohocken, PA, 2009, DOI: 10.1520/D8602-98, www.astm.org.

1433

dx.doi.org/10.1021/ie201399g | Ind. Eng.Chem. Res. 2012, 51, 1429−1434

Industrial & Engineering Chemistry Research

Research Note

(6) Harries, R. R.; Ray, N. J. Anion Exchange in High Flow Rate Mixed Beds. Effluent Water Treat. J. 1984, 24, 131. (7) Lee, G. C.; Arunachalam, A; Foutch, G. L. Evaluation of Mass Transfer Coefficient Data for New and Used Ion Exchange Resins. React. Funct. Polym. 1997, 35, 55. (8) Harries, R. R. Water Purification by Ion Exchange Mixed Bed. Ph.D. Dissertation, Loughborough University of Technology, Loughborough, U.K., 1986. (9) Haub, C. E.; Foutch, G. L. Mixed-Bed Ion Exchange at Concentrations Approaching the Dissociation of Water, 1. Model Development. Ind. Eng. Chem. Fundam. 1986, 25, 373. (10) Haub, C. E.; Foutch, G. L. Mixed-Bed Ion Exchange at Concentrations Approaching the Dissociation of Water. 2. Column Model Applications. Ind. Eng. Chem. Fundam. 1986, 25, 381. (11) Berg, A. ABB Atom AB, Vesterås, Sweden. Personal Communication, 1992. (12) Schneider, H. M.; Allen E.; Woodling R.; Barnes, R. Method for removing contaminants from water. U.S. Pat. 5,597,489, 1997. (13) Tsaur, Y.; Shallcross, D. C. Comparison of Simulated Performance of Fixed Ion Exchange Beds in Linear and Radial Flow. Solvent Extr. Ion Exch. 1997, 15, 689. (14) Lou, J. Modeling of Boron Sorption Equilibrium and Kinetic Studies of Ion Exchange with Boron Solution. M.S. Thesis, Oklahoma State University, Stillwater, OK, 1997. (15) Duong, H. M.; Shallcross, D. C. Modeling Radial-Flow IonExchange Bed Performance. Ind. Eng. Chem. Res. 2005, 44, 3681. (16) Fejes, P. P.; Heldin, G.; Samuelsson, A. Experience with Clean Up Systems at BWRs. Water Chem. Nucl. React. Syst. 1989, 5, 241. (17) Hussey, D. F. Development of a Multicomponent Film Diffusion Controlled Mixed Bed Ion Exchange Column Model Applicable to Variable Influent Systems. Ph.D. Dissertation, Oklahoma State University, Stillwater, OK, 2000. (18) Bird, R. B.; Stewart, W.; Lightfoot, E. Transport Phenomena, 2nd ed.; John Wiley & Sons: New York, 2002.

1434

dx.doi.org/10.1021/ie201399g | Ind. Eng.Chem. Res. 2012, 51, 1429−1434