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Ind. Eng. Chem. Fundam. 1983, 22, 336-346
Equilibrium and Dissipative Effects in Cycling Zone Adsorption Kent S. Knaebelt and Robert L. Plgford’ Department of Chemical Engineering, Universiv of Delaware, Newark, Delaware 1971 1
Cyclic variation of the temperature of a dilute aqueous NaCl solution, fed to a fixed bed of thermally regenerable ion-xchange resin, leads to alternate purification and enrichment of the product. The effects of operating condins and staging of zones in series are examined experimentally. The resins used are Amberiite XD-2 and XD-5; the latter exhibits superior exchange capacity. A mathematical model, which incorporates the Langmuir isotherm and assumes local equilibrium, agrees with experimental results at low cycling frequency. A second model that accounts for transport resistances is found to agree with observations at high frequency.
Introduction Cyclic sorption of solute in a fixed bed of adsorbent has received attention in both experimental and theoretical aspects during the last decade. Such a separation may be induced by periodically changing a thermodynamic variable, e.g., temperature or pressure, that affects the equilibrium distribution of the adsorbed material between the fluid and stationary phases. As a result, there is a periodic variation in the composition in the interstitial fluid. Although the direction of flow may be varied, as in “parametric pumping”, several advantages are realized in employing steady, unidirectional flow, as in “cycling zone adsorption”. Two distinct methods of cycling temperature have been devised. In the “travelling-wave”mode of operation, heat is supplied and removed by alternately heating and cooling the feed. Thus, the temperature shift (or wave) advances gradually through the fixed bed at a velocity that depends on the thermal capacities of the fluid and components of the bed and the fluid velocity. Conversely, the “standing-wave”mode of operation involves heat transfer through the column wall. Ideally, this results in an instantaneous temperature change over the length of the adsorbent bed. These two methods are illustrated in Figures 1and 2, and they correspond to the “recuperative” and “direct” modes of parametric pumping, respectively. Two other schemes have been used which represent hybrids of the basic methods. The mixed mode employs heat transfer to the feed and through the column wall. Also, a step-wise standing wave apparatus has been used by Foo et al. (1980) to accurately control the propagation of the thermal wave for multicomponent separations. A wide range of cycling zone separations has been investigated and studied experimentally including aqueous acetic acid with activated carbon adsorbent and cycling temperature (Baker and Pigford, 19711, aqueous glucose and fructose using an ion-exchange cellulose and cycling pH (Busbice and Wankat, 1975; Dore and Wankat, 1976), an aqueous mixture of two dipeptides with high-pressure liquid chromatography with varying dichloroacetic acid concentration (Nelson and Wankat, 1976),and a mixture of enzymes using activated carbon adsorbent and cycling voltage (Lee and Kirwan, 1975). In addition, a series of investigations have dealt with purifying brackish water with thermally cycled ion-exchange resins (Ginde and Chu, 1972; Latty, 1974; Shih and Pigford, 1977). The results of Ginde and Chu, and Latty are, in essence, equivalent. The conclusion drawn from their experiments Department of Chemical Engineering, The Ohio State University, Columbus, OH 43210.
is that while mixed beds of ion-exchange resins are somewhat effective in purifying dilute salt water, they are inadequate when significant demineralization is necessary. Shih and Pigford, however, found a significant improvement through the use of a composite resin. A summary of the ranges and results of the previous experimental work is given in Table I, along with a summary of this work for comparison. Several mathematical models have also been developed for cycling zone separations. For example, Baker and Pigford (1971) developed a local equilibrium solution for a solute-adsorbent system following a Freundlich isotherm by using the method of characteristics. Gupta and Sweed (1971) presented a similar model, which applied to a linear isotherm. Similarly, Meir and Lavie (1974) developed a model that incorporated a linear isotherm with Arrhenius temperature dependence and sinusoidal, rather than square-wave, thermal cycling. Wankat (1973, 1974), Busbice and Wankat (1975), Nelson et al. (1978), and Foo et al. (1980) have developed equilibrium-state models based on discrete steps or continuous flow which apply to systems exhibiting linear, Freundlich, or Langmuir isotherms. Only Baker (1969) and Shih (1975)appear to have considered dissipative effects, in terms of film diffusion and intraparticle diffusion, respectively. The present work concerns purification of brackish water by cycling zone adsorption (CZA) with thermally regenerable ion-exchange resins. Two rather simple mathematical models are developed that account for local equilibrium following the Langmuir form and for intraparticle diffusion and axial dispersion with a linear isotherm, respectively. Notwithstanding, Rice and co-workers have developed the analysis of the parametric pumping process in which dissipative effects such as film-and solid-diffusion and axial dispersion were considered (Rice, 1973, 1975; Foo and Rice, 1975; Rice et al., 1979). Some of their methods are similar to those presented here but the process is different. Theory The basis of predicting the performance of a CZA process is the solute continuity equation, which accounts for the distribution between the fluid and adsorbent phases and conveyance through the interstices of the packed bed. The Langmuir equation adequately represents the solute distribution at equilibrium. Furthermore, since the process is periodic, the rate phenomena that affect solute transport act as constraints on the separation under certain conditions. The approach taken her is first to analyze the process neglecting transport resistances and kinetic effects in order to demonstrate realistically the influence of the nonlinear equilibrium behavior. Second, the effects of solute intraparticle diffusion and axial dispersion are taken
0196-4313/83/1022-0336$01.50/00 1983 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983 337 Table 1. Summary of Conditions and Results of
resin type mode of operation column length, m column diameter, m low temperature, high temperature, C low velocity, m/s high velocity, m/s feed NaCl concn, kmol/m3 maximum mean separation factor
“S
CZA Studies
Ginde and Chu (1972)
Latty (1974)
Shih and Pigford (1977)
this work
mixed mixed 0.40 0.044 20 75 0.0003 0.0007 0.025-0.050 1.4
mixed mixed 0.53 0.050 20 80 0.0 0.003 0.017-0.019 2.3
composite mixed 0.46 0.025 22 69-90 0.001 0.004 0.009-0.030 3.7
amphoteric travelling wave 0.50 0.051 25 70-95 0.0008 0.0045 0.030 9.3
r -
- 1
a-..:0 adsorbent
Th
1 ‘h
Heating h a l f - c y c l e
T&
4
Coolinq h a l f - c y c l e
Figure 1. Schematic diagram of travelling wave mode of operation.
Qout
Heating h a l f - c y c l e
Cooling h a l f - c y c l e
Figure 2. Schematic diagram of standing wave mode of operation.
Amundson, 1970; Aris and Amundson, 1973; Sherwood et al., 1975). The technique has also been applied in the analysis of thermally cycled processes. For example, Pigford et al. (1969) and Aris (1969) have developed the theory for parametric pumping, while Baker and Pigford (1971) and Meir and Lavie (1974) have analyzed the CZA process, as mentioned previously. The method of characteristics is applied here to the CZA process subject to square-wave thermal cycling and a Langmuir isotherm. All dissipative effects, including film and intraparticle diffusion and radial and axial dispersion, are neglected. In some respects, though, this approach is an extension of the earlier work by Baker and Pigford and by Aris and Amundson. Since the separation is induced by a thermal shift within the fixed bed of ion-exchange resin, which is defined by T(z,t) = To + A@q[w(t - z / v ~ ) l it is natural to redefine the independent variables that appear in eq 1with respect to the thermal wave velocity i.e. 7 = t - z/vT; f =z The thermal-wave velocity is related to bed properties by
into account, but the isotherm is linearized. The continuity equation for a single solute in a continuous rod-like flow and a fixed bed system is
Note that the three solute concentrations, C, E, and Q*, represent the state of the interstitial fluid, the fluid in the intraparticle pores (averaged over the particle radius), and the resin-phase, which is assumed to be at equilibrium with the pore fluid at each point along the radius. The principal distinction between the two techniques employed here to solve this solute balance equation lies in the treatment of the particle-averaged solute content, cpC + (1- cp)prQ*. The first approach assumes that local equilibrium exists inside the particles, i.e., E = c = C. Conversely, when intraparticle diffusion is taken into account, the local concentrations are incorporated as 3 + (1 - cp)prQ*= -JRr2(rpc + (1- tp)pr4*] dr R3 0
(2)
The local pore-fluid concentration is then related to the interstitial fluid concentration as in Fick’s law. Method of Characteristics Solution The method of characteristics has been applied elsewhere to isothermal or adiabatic adsorption and chromatography (Acrivos, 1956; Rhee et al., 1970a,b; Rhee and
where Doand DIare the outer and inner diameters of the column. The terms in the denominator represent the ratios of the thermal capacity of each component in the fued bed to that of the interstitial fluid. This equation represents the naturally occurring thermal velocity in the travelling wave mode of operation. In other modes of operation this velocity is overridden by an externally imposed profile. The separation is governed by the form of the isotherm which may be written as (4)
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This first-order, quasilinear, partial differential equation may be solved by the method of characteristics. The results are two simultaneous ordinary differential equations: one describes the characteristic curves in the {-7 plane; another prescribes the concentration along those characteristics. The equation for the characteristic lines is
\
2.3E
4
vT
vC
where the concentration-wave velocity is V v, = (1 - d e p (1 - ep)PrA 1+ (l + Y)‘b Eb(1 + ’)’)(I+ BC)2
+
(7)
The second differential equation, which must be solved along the characteristics in order to determine the concentration, is
[K+
-1
dC = 0 (8)
A
(9)
The terms in square brackets in eq 8 represent partial derivatives of an exact function with respect to the variables A, B, and C. Thus, concentration is constant along a characteristic in an isothermal region because A and B depend only on temperature. At a thermal shift, however, the concentration change may be determined from KC q* = KC, go* = F (10)
where vi?,
1 < -7-10 -
