778
I n d . E n g . C h e m . Res. 1989,28, 778-785
SEPARATIONS Experiments on Optimization of Thermal Swing Adsorption Mark M. Davist and M. Douglas Levan* Department of Chemical Engineering, University of Virginia, Charlottesuille, Virginia 22901
Thermal swing adsorption cycles in fixed beds are investigated through experiment and modeling to obtain the influence of process parameters on energy use and purge gas consumption. Cycle step times and regeneration conditions must be set to minimize costs. Objective functions involve heating and cooling demands, determining the extent of regeneration, and quantities of product produced. Experiments have been performed with a computer-controlled, pilot-scale, fixed-bed apparatus with rz-hexane adsorbed from air onto BPL-activated carbon. The results agree well with model predictions. Short regeneration times are found t o be efficient for energy and purge gas use. Proper timing of the cooling step can lead to significant energy savings. Adsorption processes involving gases are generally operated in a cyclic manner. Regeneration methods are based on raising the temperature of the adsorbent (thermal swing), reducing the total pressure in the bed (pressure swing), or both. Pressure swing adsorption is favored for adsorbates of fairly high volatility, while thermal swing adsorption is best for adsorbates of moderate volatility. For adsorbates of very low volatility, the adsorbent is replaced rather than regenerated in place, with spent adsorbent possibly reactivated in a furnace. Vapor-phase adsorption processes for recovery of solvents and removal of impurities frequently utilize thermal swing adsorption with hot purge gas regeneration or steam regeneration in cycles with three steps (adsorption, heating, and cooling) or two steps (adsorption and heating). The possibility of operating a cycle without a cooling step was first recognized by Basmadjian (1975a). Flow for the heating step is in the direction opposite to that for the adsorption step. Early design procedures for adsorption in fixed beds concentrated on solving for the adsorption breakthrough curve. Starting with a clean adsorption bed, the breakthrough time would be determined for the adsorption step. The bed would then be scaled and large safety factors added to meet process requirements. A lack of detailed knowledge of the process led to overly conservative design and operating procedures, especially for regeneration where crude enthalpy summations and overly conservative rules of thumb were relied upon. Yet, the major operating expenses for an existing adsorption process are almost completely associated with the regeneration of the adsorbent. Research on thermal swing adsorption can be classified into three areas: (1)studies of single cycle steps beginning with uniform conditions in the bed; (2) studies of single cycle steps beginning with nonuniform conditions in the bed; (3) studies of complete cycles. Research in the first two areas is concerned really with nonisothermal adsorption, rather than thermal swing adsorption per se. Only when the cycle exists is the operation truly thermal swing adsorption. Present address: Union Carbide Corporation, Tarrytown Technical Center, Tarrytown, NY 10591.
Considerable effort has been directed toward developing an understanding of single cycle steps beginning with uniform conditions. Elegant theory has been developed based on the assumption of local equilibrium (Rhee and Amundson, 1970; Pan and Basmadjian, 1971), and numerous rate studies have been carried out. Basmadjian et al. (1975a) and Kumar and Dissinger (1986) have considered the nonisothermal removal of adsorbed carbon dioxide from a fixed bed of 5A zeolite using a hot nitrogen purge. In both studies, a characteristic feed temperature is found for which energy consumption is a minimum; this temperature is the equilibrium reversal temperature, a t which complete desorption occurs in the fixed bed. These studies pointed out that, in general, increasing the temperature of the hot gas reduces purge gas requirements and that increasing the pressure for heating results in larger plateau concentrations but larger purge requirements on a molar basis. These studies consider complete regeneration of the adsorption bed. While this is possible for adsorbed light gases, it is economically prohibitive for heavier adsorbates. Papers reporting experimental results on thermal swing adsorption are few in number and have been restricted to the study of individual cycle steps (Basmadjian et al., 1975b; Chi, 1978; Friday and LeVan, 1985; Schork and Fair, 1988). With the exception of one experiment by Schork and Fair, in all of these studies, the bed was loaded uniformly at the start of heating. None of the studies considered the impact of the cooling step. There have been very few studies of cycle steps beginning with nonuniform initial loadings. Carter (1975) considered a heating step for a bed having a nonuniform initial loading. He compared the efficiencies for heating in the same direction as that for adsorption and in the direction opposite to that for adsorption, showing that the latter was preferable for initially nonuniformly loaded beds. There has also been little previous work on the analysis of cyclic behavior. Davis and LeVan (1987) have analyzed complete adiabatic adsorption cycles for benzene adsorbed from nitrogen onto activated carbon. Cycles with and without cooling steps were considered. The results show that, for cycles with a cooling step and flow for cooling in the same direction as that for heating, heating require-
0888-5885/89/2628-0778$01.50/0 0 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989 779 ments can be reduced significantly if the start of the cooling step is timed properly. Davis et al. (1988) have considered the periodic states that develop after repeated cycling for two-component adsorption with hot purge gas regeneration. Both components were recovered in the bed, so switching to the heating step occurred when the first component began to break through. Flows for adsorption and heating were in opposite directions. The results indicate that at modest or high regeneration pressures the heavy component does not accumulate in the bed over repeated cycles. Instead, the cycle operates with the bed enriched in the light component. This paper reports the results of an experimental and modeling study of complete thermal swing cycles; it is the first such publication on complete cycles to include an experimental component. We consider the n-hexaneair-BPL-activated carbon system. Experiments are carried out by using a computer-controlled pilot-scale thermal swing adsorption apparatus. We are concerned with regeneration cost. We focus on the definition and minimization of objective functions for optimal performance. Process parameters are varied to determine the effects of concentrations, pressures, and temperatures on optimal performance for cycles with and without cooling steps.
Cycle We consider a cycle with an adsorption step, a heating step, and possibly a cooling step. Times and flow directions for the individual steps are set as follows. The adsorption step lasts until breakthrough occurs to a specified concentration. Heating with solute-free gas is carried out for a fixed time. Flow for heating occurs in the opposite direction of that for the adsorption step. If a cooling step is used, the step ends when the highest temperature in the bed has just decreased to a specified value; this maximum allowed temperature is found at the bed outlet at the end of the step. Cooling is with solute-free gas, with flow in the same direction as for heating. This cycle is a popular one industrially. A measuring device at the outlet of a bed is often used to detect breakthrough, a t which time switching to the heating step occurs. Heating is carried out for a fixed time, with the flow direction reversed. Heating can be with the purified product gas or with the feed for the adsorption step. Most cycles use a cooling step. Flow is generally in the direction for heating if the cooling gas is clean and in the direction for adsorption if the cooling gas is the feed for the adsorption step. In the experimental system, there are short time intervals at the start of the heating and cooling steps during which flow is bypassed around the bed while the piping leading to the bed is heated or cooled. With the bed shut-in preceding the cooling step, some loss of energy from the bed occurs because the bed is not absolutely adiabatic. Treatment of this energy loss is included in the modeling. Objective Functions We have given considerable thought to the proper formulation of objective functions. From the outset, we sought objective functions that would be both simple and of some general significance. Of course, for a particular process, if all costs were known, then the proper objective function would be total cost and its minimization would be sought. However, not only is this usually not the case, but the use a detailed cost function lacks the generic character which we sought. Thus, we chose to focus our attention on the components of a cost function which would in most cases be dominant. For an existing ad-
sorption process, operating costs should be minimized. We focused on the energy input to the hot purge gas and the volumes of heating and cooling gases required.. Three objective functions are described briefly below, although only the first one is used in this paper to analyze experimental results. First, the energy added to the heating gas can be minimized. Since a bed can be regenerated with no temperature rise using adsorbate-free gas, constraints peculiar to a particular process must be available, such as bounds on flow rate and heating time or condensability of the effluent. Since, in many applications, the heating gas is ultimately passed through a condenser, if the amount of heating gas is minimized, the amount of gas that must be cooled to recover the adsorbate is also minimized. Second, circumstances can exist where heating is with the purified gas and where the allowable maximum temperature for heating is limited not by any economic considerations but by other factors (e.g., reactivity or safety). Since heating and cooling are often carried out using the product gas, it is often desirable to minimize the total volume of gas used for heating and cooling per unit volume of gas processed. This objective function is discussed in greater detail and used in analysis of experiments by Davis (1987). Third, for some solvent recovery applications, it may be desirable to minimize the energy added to the heating gas per unit quantity of solvent recovered in a condenser. Under such circumstances, the temperature a t which the condenser operates becomes an important varible. We have used such an objective function previously in theoretical analysis (LeVan et al., 1987). Other measures of performance can be calculated rapidly from these ratios, such as the enrichment in concentration of the adsorbate in the regeneration purge when compared to the adsorption feed. For example, an enrichment ratio can be defined as the ratio of the mean outlet concentration during regeneration (heating and cooling) to the adsorption feed concentration. This ratio can be shown to be the reciprocal of the second objective function described above.
Mathematical Model Numerical simulations were used to model the full adsorption cycles over repeated cycles. For the two cycles studied, regeneration with cooling and regeneration without cooling, each of the individual steps was modeled, with the final state serving as the initial condition for the next step in the cycle. As mentioned above, the model also included bypass intervals in the cycle during which the bed was sealed and the approach to the bed was heated or cooled. Although there was no flow through the bed, there were heat losses from the bed during these intervals, which influenced the initial condition for the next cycle step. The model, similar in many respects to that of Schork and Fair (1988), included heat- and mass-transfer resistances. An order of magnitude analysis indicated that the heat-transfer rate was determined by external heat transfer, so a film coefficient was used. Thus, the internal resistance to heat transfer was neglected, giving particles at uniform temperature. Heat losses to the surroundings were also accounted for. Mass transfer was modeled by using a two-resistance model: a film coefficient for external transfer and the linear driving force approximation for intraparticle transfer (including a concentration dependence for surface diffusivity). An allowance was also made for axial dispersion. The transfer coefficients were calculated by using correlations for fixed beds. The mathematical model consists of conservation equations, rate
780 Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989
equations, and the adsorption equilibrium relationship. We assume that the bed operates isobarically and that the gas behaves ideally. Surface diffusion is widely believed to be important for diffusion in adsorbents. Yang (1987) gives two provisions that must be met for surface diffusion to be important: high specific surface area of the solid and high adsorbedphase concentrations. He also cites specific cases for which surface diffusion is important, and dominating, for light gases adsorbed on activated carbon and silica gel. In the present study, the ratio of adsorbed-phase concentration to gas-phase concentration is larger than in the systems cited by Yang, suggesting further that surface diffusion is the controlling mechanism for mass transfer within the adsorbent particles. The choice of surface diffusion as the controlling intraparticle resistance can be justified further by a detailed comparison of fluxes and length scales for diffusion in macropores and in microporous regions (Davis, 1987; Doong and Yang, 1987). Fluid-phase balances on the adsorbable component and energy are
~
_ ~ _
~
~_
_~
_
I
_
~_
_
+ . -+A+-
C
I* To C o mDpAuSt e8r
i
rJ4 T
i
(
E
d
Figure 1. Thermal swing adsorption apparatus. Sections: A, gas preparation; B, fixed bed; C, instrumentation. BPR, back-pressure regulator; CC, condenser/condensate collector; CV, control valve; DAS, data acquisition system; EF, electric furnace; F, flowmeter; GC, gas chromatograph; MV, metering valve; PT, pressure transducer tap; R, pressure regulator; S, sparging vessels; T, thermocouples. Shading indicates pipe or tube is wrapped with heating tape.
The stationary-phase film coefficient, k,, was calculated from 15 kq = --Dt?ff R,2 with (Sladek et al., 1974)
where
The interstitial velocity, u , was calculated assuming a constant inert flux in a dilute system. Rate equations for material and energy transfer are given by
where (7) The concentration at the fluid-phase/stationary-phase interface is calculated from the rightmost equality in eq 5, given the bulk concentrations, c and q, and the adsorption isotherm, q*(c*,T,). In eq ?, ha is the enthalpy of the adsorbable component as ideal gas at the actual temperature of the stationary phase; this enthalpy at the reference temperature is zero by definition. The thermodynamic path for calculating the internal energy of the stationary phase is heating the adsorbent and adsorbable component (as ideal gas) from the reference temperature .to the actual stationary-phase temperature and then allowing adsorption to occur isothermally. The fluid-phase film coefficients, k , and h, appearing in eq 5 and 6 were calculated from (Ruthven, 1984)
+ 1.1Reo.6Sc113 Nu = 2.0 + 1.1Reo,6Pr113 Sh = 2.0
(8)
(9)
Deff = Ds/rs
(11)
D, = D,, exp(-0.45X/RT)
(12)
In eq 10, Re is the effective radius of a domain for surface diffusion. Axial dispersion can be introduced into the model in several ways (Sundaresan et al., 1980; Nauman and Buffham, 1983). Fickian dispersion terms can be added to eq 1 and 2. Doing so gives a high degree of backmixing. On the other hand, eq 1 and 2 can be transformed into a mixing cell model with no backmixing. Many variations on these two basic approaches are available, which lead to more complicated descriptions of residence time distributions. We opted for a mixing cell model, obtaining it by writing the axial derivatives in eq 1 and 2 using backward differences. The equation set was then solved by using the Gear's method solver LSODE (Hindmarsh, 1980). The model contains two adjustable parameters. These were set so as to obtain reasonable agreement between experimental results and model predictions. The number of mixing cells was set to 20. The group 15Dso/(R,2r,), obtained by combining eq 10-12, was set to 100 s-l. The solution is not very sensitive to modest changes in these parameters; both correspond to nearly ideal performance. Twenty mixing cells give behavior approaching plug flow quite closely. The intraparticle diffusion parameter corresponds to a high rate of mass transfer; this group could be measured by independent experiments (e.g., Rodrigues and Costa (1986)). Experiments Fixed-Bed Apparatus. The apparatus is shown in Figure 1. It was constructed to run repeated cycles and to operate at pressures up to 50 psia and temperatures up to 130 "C. Complete flexibility of flow directions for all cycle steps is possible. Figure 1 is divided into three sections by dashed lines. Section A is the gas preparation
Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989 781
n
3-
+F‘
I
/I
Figure 2. Adsorption column. 1: Gas sampling lines. 2: Thermocouples. 3: Delrin end pieces. 4: Retaining screens. 5: Carbon adsorbent.
section of the apparatus; it contains refrigerated sparging vessels in series, flowmeters, and an electric furnace. Section B contains the fixed bed and the valving and piping which directed the flow around and through the bed. Section C shows the instrumentation used to collect the data and control the experiment. Pipes and tubes wrapped with heating tape to prevent heat loss and/or condensation are shaded in Figure 1. The nominal pipe size was in. The valves which directed the flow were 1/2-in. air operated, spring return ball valves. A schematic diagram of the adsorption column is shown in Figure 2. The column was constructed from a copper tube with a wall thickness of 0.05 cm, an inside diameter of 7.62 cm, and a length of 120 cm. The top and bottom of the column were fitted with threaded brass rings, which were silver soldered to the copper tube. Threaded end pieces for the top and bottom were made from Delrin plastic. The carbon was supported in the column with a 60-mesh stainless steel screen at the top and bottom of the bed. The bottom screen was soldered ‘to the tube wall, while the top screen was held down securely on top of the packing by a ring and wedges made from Teflon. The supporting screen a t the top of the bed was necessary to prevent fluidization of the packing during pressure transients which occurred during cycle step transitions. The column was packed with carbon to a depth of 80 cm, which left 20 cm above and below the packing. Thermocouples were inserted along the axial centerline of the adsorbent packing a t 20-cm intervals. Gas could be sampled from the top or bottom of the column using a three-way valve shown in Figure 1. The column was insulated to minimize heat losses. The insulation was a wrap of alternating layers of Du Pont Hollofill Dacron insulation and metalized Mylar, with a thickness of approximately 9 cm. The.top and bottom of the column were also insulated with this material.
The activated carbon was 6- X 16-mesh BPL carbon (Calgon Corporation). It was sieved to 6 X 10 mesh to reduce the pressure drop through the bed. The instrumentation for the apparatus, shown in section C of Figure 1, consisted of a microcomputer (IBM PC-AT), a data acquisition system (Keithly DAS), a gas chromatograph, a pressure transducer, thermocouples, and various switching devices. The microcomputer performed the control and data logging functions. The data acquisition device performed the digital output and the analog-todigital conversion for the signals from other instruments. Gas compositions of both inlet and outlet streams were determined with the gas chromatograph fed by a heated gas sampling valve. A pressure transducer determined the total system pressure and pressure drop across the bed. Digital switching devices controlled heating elements and opened and closed valves. Operating Procedure. The procedure for conducting an experiment was as follows. The feed conditions for all inlet streams were set. A heating time was chosen. The adsorption apparatus would cycle through a two- or three-step cycle. The cycle was continued until a periodic state was achieved. A periodic state was determined by the convergence of adsorption breakthrough times. When the periodic state was achieved, the adsorption, heating, and cooling times were recorded. The heating time was then changed, and the system would then cycle to a new periodic state. Heating times ranged from 5 to 50 min. For the experiments, the adsorption cycle was actually composed of five distinct parts if cooling was performed and four if cooling was omitted: (1) adsorption, (2) countercurrent heating bypass to warm up pipes, (3) countercurrent heating, (4) countercurrent cooling bypass to cool down pipes (even if the cooling step is omitted), and (5) countercurrent cooling. During the cycle, variables were compared with threshold values of concentration, temperature, and time. Gas samples were injected into the gas chromatograph every minute from the bed outlet for the adsorption, heating, and cooling steps. Temperatures were also checked every minute. For the adsorption step, feed was fed into the bottom of the bed. Adsorption would continue until there was a concentration breakthrough at the bed outlet. The breakthrough concentration was always taken as 5% of the adsorption feed concentration. The cooling gas flowed through the bed, from top to bottom, until temperatures measured by all thermocouples in the bed were below 305 K. System Parameters. Adsorption equilibria for the hexane-activated carbon system were measured and correlated by Hacskaylo (1987). Data were described accurately by a modified Antoine equation (Hacskaylo and LeVan, 1985) of the form B’ l n p = A’C’+ T ~
with
+ In 0
(14)
+ b ( l - e)
(15)
A’ = A
B’ = B
C’=C
(16)
where A , B, and C are tabulated Antoine constants for the vapor pressure of a particular species, and 0 is the fractional filling of the pore space of the adsorbent. 0 is defined by
e = w/w,
(17)
where W is the specific volume of the adsorbate and W ,
782 Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989 Table I. System Parameters Physical Properties Pb, kg/m3 480 U,kJ/(s m2 K) c, kJ/(kg K) 1.56 d,, cm 0.20 Trer, K e 0.40
5r----
..____
0.001 15 298 298
,410
4380
-
Adsorption Equilibrium (Equations 13-17, p in MPa, T in K) A 6.989 46 Wo,cm3/kg 521.8 B 2737.59 b 3356.90 c -46.87
1350
43 2 0 Table 11. Cycle Step Conditions adsorption (upflow) F = 98 L/min STP P = 0.11 MPa T = 296 K c = 0.40, 1.13 mol/m3 (5.4, 15% saturated) breakthrough to 5% of feed concn heating (downflow) F = 198 L/min STP P = 0.13, 0.31 MPa 7’ = 373, 405 K heat for fixed time (5-50 min) cooling (downflow, if present) F = 198 L/min STP P = 0.13, 0.31 MPa T = 296 K cool until max temp in bed is 305 K
is the specific pore volume of the adsorbent. W was calculated assuming that the density of adsorbed hexane is the density of saturated liquid hexane a t the adsorption temperature. The liquid density was calculated by using the correlation of Campbell and Thodos (1984). This model for adsorption equilibria gives a proper Henry’s law slope at low loading and is consistent with vapor-liquid equilibrium at complete filling of the pore structure. The isosteric heat of desorption is defined by
In applying eq 18, we assumed, as in previous work (Hacskaylo and LeVan, 1985), that the derivative at constant q could be replaced by a derivative at constant 8. From eq 18, the heat of desorption is a function of temperature and adsorbed-phase concentration. It decreases with loading; for example, at 298 K, h = 71 kJ/mol at 8 = 0 and h = 32 kJ/mol at 6 = 1,the latter value being the heat of vaporization of liquid hexane. The parameters of the experimental system are summarized in Table I. The parameters cg, the effective heat capacity of the stationary phase (includes adsorbent, column wall, fittings, and insulation), and U , the overall heat-transfer coefficient, were evaluated by passing pure hot nitrogen through the bed. The effective heat capacity was found to be 49% greater than the heat capacity of clean activated carbon, which is 1.05 kJ/(kg K). Gas-phase heat capacities for air and n-hexane, needed to determine cpm and h,, were calculated by using the correlations in Felder and Rousseau (1978). Operating conditions for the various cycle steps are given in Table 11. Temperatures and solute concentrations are values for the feeds. The cooling gas, when used, was always at the same pressure as the heating gas.
Results and Discussion The results were analyzed in terms of dimensionless cycle step times that are equal to the number of superficial bed volumes of gas passed into the bed, measured at a reference state. These times are 7,, 7h, and 7, for ad-
~
0
20
40
60
290 80
t (min)
Figure 3. Regeneration breakthrough curve for 41 min of heating followed by cooling. Adsorption: c = 0.40 mol/m3. Heating: P = 0.31 MPa, T = 405 K. Curves indicate model predictions.
sorption, heating, and cooling steps. Thus, an actual volumetric feed rate was converted to a volumetric feed rate at the reference state, which was then used to calculate dimensionless time by
The reference state was 0.1 MPa and 298 K. The operating conditions for the cycle were evaluated using the ratio 7h/r, (the ratio of column volumes of gas for heating to column volumes for adsorption to breakthrough, both measured at a reference state). This ratio corresponds directly to the first objective function described above. The periodic state was achieved after 2 or 3 cycles, both in the experiments and in the modeling. In the model, only the heating time was entered explicitly. The adsorption step was stopped in the model when breakthrough was occurring to 5% of the feed concentration. Similarly, the cooling step was stopped in the model when the maximum temperature in the bed was 305 K. Thus, we ran the model exactly as we ran the experiments. With the model, we attempted to predict the full cyclic performance. The regeneration breakthrough curve for a cycle with 41 min of heating followed by about 30 min of cooling is shown in Figure 3. The adsorption feed for the cycle was at a temperature of 296 K and contained n-hexane at a concentration of 0.40 mol/m3 (5.4% saturated). Heating and cooling were carried out a t a pressure of 0.31 MPa. The temperature of the feed for heating was 405 K. After about 3 min of heating, there is a rapid increase in concentration to plateau values of about c = 2.7 mol/m3 and T = 310 K. Once the plateau has exited the bed, there is a rapid decrease in concentration and further increase in temperature at the bed outlet as the major thermal breakthrough emerges. A t 41 min there are discontinuities in the breakthrough curves which occur during the bypass interval as pipes are cooled before the cooling gas is introduced into the bed. This interval lasts 25 min (for all experiments with a cooling step), during which time the sealed bed cools about 10 K. Cooling gas is then introduced into the bed. There is first a slight increase in temperature at the bed outlet as the hot profile created by the heating gas is transferred down the bed. Then the bed cools with the outlet finally reaching 305 K, a few minutes later than predicted by the model. Figure 4 shows the regeneration breakthrough curve for a cycle with 11 min of heating followed by cooling. Con-
Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989 783
t
1380
"
I 1
W'
1""
0low\ 5 0 0
20
I
I
40
60
290
80
t (min) Figure 4. Regeneration breakthrough curve for 11 min of heating followed by cooling. Adsorption: c = 0.40 mol/m3. Heating: P = 0.31 MPa, 2' = 405 K. Curves indicate model predictions.
centrations, temperatures, and pressures are the same as for Figure 3. The overall trends in Figure 4 are somewhat similar to those shown in Figure 3. Concentrations and temperatures at the bed outlet initially increase rapidly. Plateau concentrations for the two cases are nearly identical. In Figure 4,the discontinuity at 11 min, caused by the bypass interval, is pronounced for the concentration breakthrough curvp. There are significant differences between thermal waves for Figures 3 and 4. In Figure 4, a t the end of the heating step, the temperature at the bed outlet is still the plateau value; the major thermal breakthrough has not occurred. It is only after the onset of the cooling step that the energy stored in fairly hot regions within the bed appears at the bed outlet. The temperature at the bed outlet rises early during the cooling step and reaches about 340 K before the bed outlet cools. The cooling time is predicted accurately for this case. For cycles in which the cooling step was omitted, the heating step was very similar to those shown in Figures 3 and 4. Adsorption steps (countercurrent to heating) in all experiments showed no evidence of premature concentration breakthrough, resulting from the splitting of waves due to nonisothermal behavior. A slight degree of leakage for a short time was noted in both modeling and experiment for cycles with a short heating time and no cooling. This was due to unsubstantial regeneration and the influence of the transfer resistances. For a cycle with a moderate heating time and no cooling, the inlet end of the bed for heating is well regenerated. For our operating conditions, if the entire bed were regenerated completely, then on the ensuing adsorption step a pure thermal wave would propagate ahead of a single adsorption front; the temperature to which the bed is heated by this thermal wave is independent of the initial bed temperature (Pan and Basmadjian, 1970; Rhee and Amundson, 1970). Our bed is not regenerated completely, but the wave interactions are such that a nearly pure thermal wave forms in the well-regenerated region, thus avoiding premature breakthrough (Davis and LeVan, 1987). Figure 5 shows the objective function, Th/Ta, for cycles with and without cooling for the conditions of Figures 3 and 4. The curves in this figure (and in the following figures) indicate predictions of the model. For the curve with cooling, the 41-min heating step of Figure 3 corresponds to the point 7 h / 7 , = 0.68 a t 7 h = 2400, and the 11-min heating step of Figure 4 corresponds to the point 7 h / 7 , = 0.27 at 7h = 640. The curve increases monotonically, showing no minimum because the cooling gas is very effective in regenerating the heated bed. The minimum
0
0
1000
2000
3000
Ttl
Figure 5. T ~ / T for , cycles with and without cooling. Adsorption: c = 0.40 mol/m3. Heating: P = 0.31 MPa, T = 405 K.
Without Cooling
l o t
With Cooling
t
0 0
1000
2000
3000
rh
Figure 6. r h / s , for cycles with and without cooling. Adsorption: c = 1.13 mol/m3. Heating: P = 0.13 MPa, T = 405 K.
in the curve for a cycle without cooling corresponds to an optimal heating time (about 11min) for a cycle operated in this mode. For short heating times, heating requirements exceed significantly those for a cycle with cooling. For 11min of heating, the energy input (above the reference temperature of 298 K) for the cycle without cooling is approximately 60% greater than that for the cycle with cooling. Figure 6 resembles Figure 5 but corresponds to a higher adsorption feed concentration (c = 1.13 mol/m3, 15% saturated) and lower pressure for regeneration (0.13 MPa). Model predictions agree well with experimental results. The three-step cycle again shows no minimum in the objective function. The minimum in the curve for the twostep cycle again OCCUIS near 11min. Values of the objective function 7h/Ta are higher here than in Figure 5, due largely to a decrease in 7 , caused by the increased feed concentration for the adsorption step. Note that in both Figures 5 and 6 the curves for cycles with and without cooling approach one another at large Sh; for a cycle with a long heating time, the cooling step removes little additional adsorbate from the bed. The influence of regeneration pressure on the objective function is shown in Figure 7 for cycles with cooling. The adsorption feed concentration for these curves is 1.13 mol/m3 (15% saturated). Heating is with air at 405 K. The curves increase monotonically, showing no minima. As the pressure for heating is reduced, the maximum concentration leaving the bed is also reduced. At 0.13 MPa, this concentration is only 1.3 mol/m3 (compare to about 2.7 mol/m3 in Figures 3 and 4 for 0.31 MPa).
784 Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989
cantly. (See also Davis and LeVan (1987).) An optimal cycle for solvent recovery using hot purge gas regeneration should involve high-pressure heat-up to create high-plateau concentrations followed by low-pressure purge to reduce molar flow rates and sensible heat loads in the condenser. Acknowledgment This research was supported by the National Science Foundation under Grant CBT-8417673. 1-
3
i
”
01
I
I
c
1000
I
I
2000
I 3000
th Figure 7. Influence of regeneration pressure on Th/?. for cycles with cooling. Adsorption: c = 1.13 mol/m3. Heating: T = 405 K. 30-i17
ic 2
I
01
I
01 0
1
I
I000
1
I
2000
3000
th Influence of feed temperature for heating on
Figure 8. T h / T a for cycles with cooling. Adsorption: c = 1.13 mol/m3. Heating: P = 0.13 MPa.
Nevertheless, the low-pressure feed is more effective (on a molar basis) in removing hexane from the bed because of the greater real volumetric throughput at low pressures. (However, if a solute is to be recovered downstream by condensation, a concentrated vapor would be desirable, favoring a high-pressure heating step.) Figure 8 shows the effect of the feed temperature for heating on the objective function for cycles with cooling. The adsorption feed concentration is 1.13 mol/m3 and the pressure for heating and cooling is 0.13 MPa. The higher temperature results in the development of higher concentrations within the bed (Basmadjian et al., 1975a; Friday and LeVan, 1985) which are swept out by the purge gas. Thus, the higher temperature is more effective in regenerating the bed.
Concluding Remarks Both theoretical and experimental results indicate that short regeneration times are efficient for energy and purge gas use. For cycles without a cooling step, there is a heating time that minimizes q , / r g . It occurs as the temperature is increasing rapidly at the bed outlet during the major thermal breakthrough. For a cycle with a cooling step, no minimum in this ratio was found over the range of the experimental variables. For short heating times, a large fraction of the energy input to the bed is used to supply energy for desorption rather than sensible heating of the bed and elution of hot gas. With short heating times, the cooling step has a pronounced effect on the performance of the cycle. If timed properly, heating requirements can be reduced signifi-
Nomenclature a = specific surface area of adsorbent, m-l c = gas-phase concentration of solute, mol/m3 c * = value of c at particle surface cpm = heat capacity of gas phase, kJ/(kg K) c, = heat capacity of adsorbent, kJ/(kg K) d, = particle diameter, m D, = surface diffusion coefficient, m2/s F = volumetric flow rate, m3/s h = film heat-transfer coefficient, kJ/(s m2 K) ha = enthalpy of solute in gas phase, kJ/mol hf = enthalpy of gas phase, kJ/kg k , = gas-phase mass-transfer coefficient, m/s k , = stationary-phase mass-transfer coefficient, s-l p = partial pressure of solute, MPa P = total pressure, MPa q = adsorbed-phase concentration, mol/kg q* = value of q in equilibrium with c* R = gas constant R, = radius of column, m R, = effective radius of microporous domain, m t = time, s T = temperature, K Tamb= ambient temperature, K Tref= reference temperature, K uf = internal energy of gas phase, kJ/kg u, = internal energy of stationary phase, kJ/kg CT= overall heat-transfer coefficient for heat losses, kJ/(s m2 K) u = interstitial velocity, m/s Vbed = volume of bed, m3 W = specific volume of adsorbate, cm3/kg z = axial coordinate, m Greek Letters t = void fraction of packing 8 = fractional filling of pore volume, eq 17 X = heat of desorption, kJ/mol Pb = bulk density of packing, kg/m3 pf = density of gas phase, kg/m3 T = dimensionless time T, = tortuosity Subscripts a = adsorption step c = cooling step f = fluid phase h = heating step s = stationary phase
Literature Cited Basmadjian, D. On the Possibility of Omitting the Cooling Step in Thermal Gas Adsorption Cycles. Con. J . Chem. Eng. 1975,53, 234.
Basmadjian, D.; Ha, D.; Pan, C. Y. Nonisothermal Desorption by Gas Purge of Single Solutes in Fixed-Bed Adsorbers. I. Equilibrium Theory. Ind. Eng. Chem. Process Des. Deu. 1975a, 14, 328. Basmadjian, D.; Ha, D.; Prouix, D. P. Nonisothermal Desorption by Gas Purge of Single Solutes in Fixed-Bed Adsorbers. 11. Experimental Verification of Equilibrium Theory. Ind. Eng. Chem. Process Des. Deu. 1975b, 14, 340.
I n d . Eng. Chem. Res. 1989,28, 785-793 Campbell, S. W.; Thodos, G. Saturated Liquid Densities of Polar and Nonpolar Pure Substances. Ind. Eng. Chem. Fundam. 1984,23, 500. Carter, J. W. On the Regeneration of Fixed Absorber Beds. AIChE J . 1975, 21, 380. Chi, C. W. A Design Method for Thermal Regeneration of Hydrated 4A Molecular Sieve Bed. AIChE Symp. Ser. 1978, 74 (No. 179), 42. Davis, M. M. Ph.D. Dissertation, University of Virginia, Charlottesville, 1987. Davis, M. M.; LeVan, M. D. Equilibrium Theory for Complete Adiabatic Adsorption Cycles. AIChE J. 1987, 33, 470. Davis, M. M.; McAvoy, R. L., Jr.; LeVan, M. D. Periodic States for Thermal Swing Adsorption of Gas Mixtures. Ind. Eng. Chem. Res. 1988,27, 1229. Doong, S. J.; Yang, R. T. Bidisperse Pore Diffusion Model for Zeolite Pressure Swing Adsorption. AIChE J. 1987,33, 1045. Felder, R. M.; Rousseau, R. W. Elementary Principles of Chemical Processes; Wiley: New York, 1978; pp 535-536. Friday, D. K.; LeVan, M. D. Hot Purge Gas Regeneration of Adsorption Beds: Experimental Studies. AIChE J. 1985,31, 1322. Hacskaylo, J. J. Ph.D. Dissertation, University of Virginia, Charlottesville, 1987. Hacskaylo, J. J.; LeVan, M. D. Correlation of Adsorption Equilibrium Data Using a Modified Antoine Equation: A New Approach for Pore Filling Models. Langmuir 1985, 1, 97. Hindmarsh, A. C. LSODE and LSODI, Two New Initial-Value Ordinary Differential Equation Solvers. ACM-SIGNUM Newslett. 1980, 15(4), 10. Kumar, R.; Dissinger, G. R. Non-Equilibrium, Nonisothermal Desorption of Single Adsorbate by Purge. Ind. Eng. Chem. Process Des. Deu. 1986, 25, 456. LeVan, M. D.; McAvoy, R. L., Jr.; Davis, M. M.; Dolan, W. B.
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“Studies on the Optimal Thermal Regeneration of Adsorption Beds”. In Fundamentals of Adsorption; Liapis, A. I., Ed.; Engineering Foundation: New York, 1987; pp 349-358. Nauman, E. B.; Buffham, B. A. Mixing in Continuous Flow S y s t e m ; Wiley-Interscience: New York, 1983. Pan, C. Y.; Basmadjian, D. An Analysis of Adiabatic Sorption of Single Solutes in Fixed Beds: Pure Thermal Wave Formation and Its Practical Implications. Chem. Eng. Sci. 1970, 25, 1653. Pan, C. Y.; Basmadjian, D. An Analysis of Adiabatic Sorption of Single Solutes in Fixed Beds: Equilibrium Theory. Chem. Eng. Sci. 1971, 26, 45. Rhee, H. K.; Amundson, N. R. An Analysis of an Adiabatic Adsorption Column. Chem. Eng. J. 1970, 1, 241, 279. Rodrigues, A. E.; Costa, C. A. “Fixed Bed Processes: A Strategy for Modelling”. In Ion Exchange: Science and Technology; Rodrigues, A. E., Ed.; NATO AS1 Series E No. 107; Martinus Nijhoff Dordrecht, The Netherlands, 1986; pp 271-287. Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley-Interscience: New York, 1984. Schork, J. M.; Fair, J. R. Parametric Analysis of Thermal Regeneration of Adsorption Beds. Ind. Eng. Chem. Res. 1988, 27, 457. Sladek, K. J.; Gilliand, E. R.; Baddour, R. F. Diffusion on Surfaces. 11. Correlation of Diffusivities of Physically and Chemically Adsorbed Gases. Ind. Eng. Chem. Fundam. 1974,13, 100. Sundaresan, S.; Amundson, N. R.; Aris, R. Observations on FixedBed Dispersion Models: The Role of the Interstitial Fluid. AIChE J . 1980,26, 529. Yang, R. T. Gas Separation by Adsorption Processes; Butterworth, Stoneham, MA, 1987. Received for reuiew September 19, 1988 Reuised manuscript receiued February 2, 1989 Accepted February 23, 1989
Particle Classification for Dilute Suspensions Using an Inclined Settler Robert H. Davis,* Xiaoguang Zhang, and J. P. Agarwala Department of Chemical Engineering, University of Colorado, Boulder, Colorado 80309-0424
Rectangular settling channels with walls inclined from the vertical may be used to classify suspended particles by size due t o differential sedimentation. We consider two operating modes of a single settling channel: steady-state continuous operation where a feed suspension is divided into a coarse fraction (underflow) and a fine fraction (overflow), and transient operation where multiple steps are used to sequentially remove particles in different size ranges. The theory to predict the particle concentration and size distribution in each fraction is developed for each of the two operating modes, with the restriction that the suspension is dilute and the particles settle according t o Stokes’ law. Experiments were performed with dilute suspensions of polystyrene beads, and the results are in good agreement with the theory. In a variety of applications involving suspensions of small particles in fluids, it is desirable to classify or separate the particles based on the differences in physical characteristics such as size, shape, and density. Common commercial equipment capable of effecting particle classification includes hydrocyclones, elutriators, centrifuges, cone classifiers, spiral classifiers, and sieves or screens. Particle classification may also be accomplished by hydrodynamic chromatography, field-flow fractionation, inertial deposition/filtration, and electrophoretic or magnetic mobility. An especially desirable classification method is the exploitation of differences in particle settling velocities under the action of gravity; gravitational classifiers are both simple in design and exert very little shear on the particles. However, gravitational sedimentation may be quite slow, especially if the particles are small. An attractive possibility for greatly increasing the rate of gravity sedimentation involves the use of lamella settlers. *To whom inquiries should be addressed. 0888-588518912628-0785$01.50/0
Lamella settlers are composed of one or more narrow channels that are inclined from the vertical. As such, the particles settle onto the upward-facing side of each plate and then slide down to the bottom of the settler where they are collected. The operation of these inclined settlers, along with mathematical models and experimental verification, is described in the review by Davis and Acrivos (1985). Most analyses of inclined settlers have been restricted to monodisperse suspensions where all of the particles have the same settling velocity. Exceptions include the works of Davis et al. (1982) and Schaflinger (1985), which considered polydisperse suspensions and showed that a spatial segregation of the particles occurs, with only the slower settling particles remaining in the upper portion of an inclined settling channel. Also, Law et al. (1988) have studied the settling behavior of bidisperse suspensions of heavy and buoyant particles in an inclined channel. The work of Davis et al. (1982) suggests that lamella settlers may be used to classify particles according to 0 1989 American Chemical Society
