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Ind. Eng. Chem. Res. 2001, 40, 2322-2334
Rapid Thermal Swing Adsorption Marc P. Bonnissel,* Lingai Luo, and Daniel Tondeur Laboratoire des Sciences du Ge´ nie Chimique du CNRS, ENSIC INPL 1, rue Grandville BP 451, 54001 Nancy Cedex, France
This article presents a new rapid thermal swing adsorption process. It is based on a composite adsorbent bed, composed of an arrangement of layers of activated carbon particles separated by sheets of a highly conductive graphite material. It also uses thermoelectric devices (Peltier elements) to alternatively heat and cool the adsorbent bed. The high effective conductivity of the bed and the fast dynamics of the thermoelectric elements allows cycles (cooling and adsorption/heating and desorption) to be run in 10-20 min. The operation is illustrated experimentally with the uptake and concentration of carbon dioxide from a helium flow. A model is developed that accounts for the usual mechanisms of equilibrium and heat- and mass-transfer kinetics in adsorption, coupled with the specific dynamic behavior of the thermoelectric elements. 1. Introduction Adsorption of a number of molecules such as VOCs from gaseous streams requires thermal desorption, and the corresponding processes belong to the family of temperature swing adsorption techniques. The heating and desorption are usually effected by flowing steam or hot gas through the bed and the cooling by flowing cold gas. This implies the flowing of large quantities of gas through the bed, and requires much time. Therefore, TSA cycles usually last for hours. The productivity of TSA cycles would be increased if one could heat and cool the adsorbent more rapidly. This work presents an attempt to approach such a goal in the context of smallscale applications, by simultaneously introducing two special characteristics in a TSA process: first, enhancement of the overall heat transfer in the particle bed by using “conductivity promoters”, here sheets of compressed natural graphite; and second, acceleration of the dynamics of temperature changes by using thermoelectric Peltier elements. This article presents the design of a prototype of such an apparatus, its full modeling, including the thermoelectric elements, and some examples of experimental results, compared to the numerical solution of the model. 2. Adsorption Apparatus and Operating Mode The adsorption apparatus (Figure 1) is composed of a parallelepiped of polypropylene with two aluminum plates on two opposite faces. Two thermoelectric devices are pressed onto these plates by two brass heat exchangers. The adsorption column consists of parallel and regularly spaced thin graphite sheets with high thermal conductivity (about 300 W m-1 K-1) separated by spacing slabs (permeable to gas flow) and enclosing activated carbon pellets. The apparent overall thermal conductivity of the adsorption column material is about 60 W m-1 K-1, and it allows the column temperature to be controlled through periodic variations of the wall temperature. These variations are produced by the thermoelectric devices, which can heat or cool alterna* Author to whom correspondence should be addressed. IMP, UP CNRS 8521, Universite´ de Perpignan, 52, avenue de Villeneuve, 66860 Perpignan, France. Tel.: 33 (0)4 68 66 21 10. Fax: 33(0)4 68 66 21 41. E-mail:
[email protected].
tively by reversing a DC current. Under operating conditions, the thermoelectric device pumps heat on one of its faces, while at the same time, the other face rejects heat. To have a temperature variation on the inner face (that in contact with the adsorbing bed), we keep the outer face at constant temperature with water circulation. Allowable operating temperatures are in the range from -20 to +80 °C. The polypropylene column is 11.5 cm long and 5.8 cm wide. The “useful” dimensions are given in Figure 1. Graphite sheets are 0.6 mm thick and separated by a 2-mm space that is filled with adsorbent particles. The particles are not pressed to avoid deformation of the graphite sheet. The adsorbent column (graphite and activated carbon particles) is 8.5 cm long in the direction of flow. The two thermoelectric devices used are Melcor CP 2-127-06L. Seven type-K thermocouples are placed through the graphite sheets in order to measure the temperature in the central activated carbon layer. Because of the 1-cm-long sensitivity zone of the thermocouple, the measured temperature is an average over about five graphite sheets and four activated carbon layers. These temperatures are called “average temperatures”. The operating mode is as follows: The adsorption column is continuously fed with a constant flow rate and composition of a He-CO2 mixture, while the column temperature is alternatively cooled and heated. Thus, no purge step with pure inert gas is used. Cooling produces adsorption while heating produces desorption. The adsorbed-species (CO2) concentration is measured at the column outlet by mass spectrometry (Sensorlab). 3. Modeling Assumptions Because of the stratified structure of the adsorption column, several zones can be distinguished, as shown in Figure 2. Because of the great thermal conductivity difference between graphite sheets, activated carbon particles, and gas, seven different temperatures can appear in the system in transient regime and seven heat fluxes can be distinguished: (1) transfer between the thermoelectric device and the graphite sheets (conduction), (2) conduction along the graphite sheets (conduction), (3) transfer between graphite and extragranular gas (convection), (4) transfer between graphite and
10.1021/ie000809k CCC: $20.00 © 2001 American Chemical Society Published on Web 04/12/2001
Ind. Eng. Chem. Res., Vol. 40, No. 10, 2001 2323
Figure 1. Overall view of adsorption column with thermoelectric devices. Outer polypropylene wall not shown.
Figure 2. Transfer mechanisms. White arrows are heat fluxes, and gray arrows are mass fluxes. Heat transfer: (1) thermoelectric device, (2) conduction through graphite, (3) gas/graphite convection, (4) graphite/grain conduction, (5) extragranular/intragranular gas convection, (6) extragranular gas/grain convection, (7) conduction through grain. Mass transfer: (I) gas flow, (II) extragranular/ intragranular diffusion, (III) adsorption.
activated carbon particles (conduction), (5) transfer between extragranular gas and intragranular gas (convection), (6) transfer between extragranular gas and activated carbon (convection), and (7) transfer between the external surface of the grain and its center (conduction). One can also consider heat transfer between the adsorbed phases and the activated carbon and between the adsorbed phases and the intragranular gas. For the mass transfers, we have: (I) mass transport by gas flow, (II) mass transfer between extragranular and intragranular gas, and (III) mass transfer between the intragranular gas and the adsorbed phases. Because of the great complexity of the system, we have made several primary simplifications based on geometric and physical approximations, and some arbitrary but usual assumptions are considered. (1) Activated carbon grains are considered to be spherical and regularly distributed along each elemental adsorption column. The particle diameter is taken as the mean hydraulic diameter. (2) The gas mixture is considered ideal in the inter-granular voids. (3) Inlet and outlet pressures are considered to be constant. (4) Gas flow is assumed to be uniform over a cross section. (5) The extragranular porosity is assumed to be the same in each elemental adsorption column. (6) Temperatures and concentrations are assumed to be uniform in the direction perpendicular to the flow.
All of these simplifications are independent of the heat- and mass-transfer mechanisms described in Figure 2. To have a simple model for the mass-transfer phenomena, we will use the LDF model and assume that the mass transfer between the intragranular gas and the adsorption surface is limited by intraparticular mass diffusion.11 The mass-transfer resistance between the extra- and intragranular gas is neglected. We also make some simplifications concerning heat transfer, assuming temperature equality for the activated carbon and the adsorbed phase and for the extragranular and intragranular gas. Figure 3 shows the three resulting temperature domains. The thermal conductivity of graphite is much higher than the other conductivities, and therefore, we consider only the graphite conduction term in the global energy balance. Heat transfer between the column wall and the graphite sheets is assumed to be without resistance because of the high deformability (low hardness) of the graphite sheets, which ensures good thermal contact. Direct conductive transfer between particles and graphite sheets is neglected relative to the transfer through the gas phase because of the small contact area of the particles with the sheets. 4. Model Equations The adsorption column is divided into five parts: the two thermoelectric devices, the two distribution zones and the adsorption column. The equations associated with each zone are different. For thermoelectric devices and distribution zones, there are heat-transfer equations only. The constitutive equations of the model are presented below. The mass, heat, and momentum equations for the adsorbent bed are relatively classical, and only their final forms are given here. Their derivations are presented in the Appendices. On the other hand, the modeling of the thermoelectric elements and their interface with the bed are discussed in more detail. 4.1. Adsorption Column. The constitutive equations presented below are derived in detail in Appendix A, where the detailled definition of the variables can also be found.
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Adsorption kinetics ∂q† †
) Stm(q*† - q†)
(1)
∂t
where q*† is the reduced equilibrium adsorbed quantity computed with the Langmuir-Freundlich isotherm equation.
Adsorbed gas mass balance
(
2 ∂x1 T† ∂q† 1 - ∂x1 1 ∂ x1 † β † +u + Ad †x2 † ) 1+ (2) ∂z Pem ∂z2 ∂t P ∂t
)
Global mass balance †
† †
†
Figure 3. Schematic representation of thermal model. Fluxes 3-5 similar to those in Figure 2.
∂2c† (3) 2 m ∂z
(1 + 1 - β) ∂c∂t + ∂c∂zu + Ad ∂q∂t ) Pe1 †
†
Graphite energy balance ∂Tgr† ∂t†
)
2 † 1 ∂ Tgr + RgNtgr(Tg† - Tgr†) Peth ∂z2
(4)
Activated carbon and adsorbed phases energy balance [(1 - )Rac + AdRadsq†]
∂Tac† ∂t†
)
L (1 - )RgNtac (Tg† - Tac†) (5) dp
pressure P, and the gas temperature Tg are linked by the ideal gas law
Global energy balance ∂Tgr† ∂t†
†
+
∂Tac 1-γ + (1 - )(Rac + RadsMadsq†) γ ∂t†
[(
]
† ∂Tg† 1-γ 1 - ∂Tg † Rgy 1 + β + u γ ∂z ∂t†
)
Figure 4. CO2 adsorption isotherms on ROX 0.8. O, measured points, s, fitting by eqs 8-10; isotherms at -25 and 80 °C are extrapolated.
)
2 † 1 ∂ Tgr 1-γ ∂q†∆hads(Tads) R Ad + ads Peth ∂z2 γ ∂t† Tocpads
[(
]
† ∂Pg† Po 1-γ 1 - ∂Pg † 1+ +u (6) β γ ToFgrcpg ∂z ∂t†
)
The local pressure P and the local velocity u are related by the pressure drop equation. We assume here the validity of Ergun’s equation. †
∂P ) ∂z Fouo (1 - )2 L 1 † † 1 - L †2 -150 µ u - 1.75 3 yu 3 Po d Re dp p p (7)
]
The system of differential equations consists of seven independent equations relating eight variables: q, x1, c, P, Tg, Tgr, Tac, and u. But the total concentration c, the
P† Tg†
4.2 Adsorption Isotherms. The activated carbon used is ROX 0.8 from Norit with a global porous volume of 1 mL g-1 and an apparent compact bed density of 390 kg m-3. CO2 adsorption isotherms were measured by a volumetric method and can be modeled by a Langmuir-Freundlich correlation
q* )
Pressure drop equation
[
c† )
qmbP11/n 1 + bP11/n
(8)
where qm and b are temperature-dependent parameters as follows
qm ) A + BT
(9)
D (RT )
(10)
b ) C exp
From experimental measurements, we have A ) 13.597 mol kg-1, B ) -1.235 × 10-2 mol kg-1 K-1, C ) 2.025 × 10-4, D ) -1.83 × 104 J mol-1, and n ) 1.371. Experimental data and this algebraic representation are shown in Figure 4. 4.3. Distribution Zones. The two distribution zones are located at the inlet and at the outlet of the adsorp-
Ind. Eng. Chem. Res., Vol. 40, No. 10, 2001 2325
effect at one copper/semiconductor junction, and a surface heat sink from the Peltier effect at the opposite copper/semiconductor junction. We have used a thermal equivalent of the thermoelectric device comprising three “effective media”, as shown in Figure 5. The thermal properties of each medium are computed by considering the corresponding materials placed in series:
Figure 5. (a) Structure of thermoelectric device and heat exchanger. (b) Equivalent thermal system with three media. Medium 1: brass heat exchanger, ceramic plate, copper block, interfacial Peltier heat source (or sink). Medium 2: semiconductor slabs with volumic Joule heat source. Medium 3: interfacial Peltier heat sink (or source), copper block, ceramic plate, aluminum plate.
tion column and are in contact with the thermoelectric devices. They are composed of only graphite, and we assume that no pressure drop occurs. The heat balances are therefore
medium 1
brass, ceramic, solder, and copper
medium 2
semiconductor
medium 3
copper, solder, ceramic, and aluminum
4.4.2. Equations. We first write a “classical” onedimensional heat diffusion equation in medium 2, which contains the two types of thermal sources or sink. Thus, we have (see Appendix B)
R2
∂T2† ∂t†
)
λ2† ∂2T2† + JouFe†I†2 Peth ∂z2
where Jou is a nondimensional Joule source term defined as
Jou )
graphite energy balance ∂Tgr† †
∂t
)
† 2 1 ∂ Tgr + RgNtgr(Tg† - Tgr†) (11) Peth ∂z2
and
global energy balance ∂Tgr† ∂t
†
+
[(
)
]
† ∂Tg† 1-γ 1 - ∂Tg † + u ) Rgy 1 + β γ ∂z ∂t† 2 † 1 ∂ Tgr (12) Peth ∂z2
4.4. Thermoelectric Devices. 4.4.1. System Description. The Peltier effect occurs at the junction of two different conducting materials fed by an electrical current. According to the direction of the electrical current, the junction rejects or pumps heat. The Peltier heat power is proportional to the current intensity, the absolute temperature, and the relative Seebeck coefficient R linked to the conductors materials. Metals have low Seebeck coefficients (Cu, 2.7 × 10-6 V K-1; constantan, -38 × 10-6 V K-1) compared to those of semiconductors (bismuth telluride, (200 × 10-6 V K-1 for N- or P-doped). The thermoelectric devices have a complex structure that is described in Figure 5.10 From left to right, we have a brass heat exchanger, which allows the outside face to be maintained at a constant temperature; a ceramic plate used to ensure mechanical resistance and electrical insulation; a thin layer of solder material; copper blocks; semiconductor material; and then the symmetric elements. The inside face of the device is pressed against an aluminum plate, which is also the wall of the adsorption column. There are three heat sources or sinks in the system: a volumic heat source due to Joule effect in the semiconductor, a surface heat source from the Peltier
(13)
2N2FeoImax2L
(14)
ωStot2FgrcpgruoTo
To take account of the Peltier effect, we consider the interface between medium 1 and medium 2 (and symmetrically, between medium 2 and medium 3). We write the heat conduction equation on each side of the interface and add the Peltier source effect, as described in Appendix B
[
Ra
]
dz1 dzb ∂T† L † + PelR†I†T† + Rb 2 2 ∂t Jou
[
(
)
Ldz1 † λ λ F †I†2 2 a gr e
Rgr λb† λa† † λa† λb† † ) T + T + T † uoTo dz2 i+1 dz2 dz1 i dz1 i-1
]
(15)
where Pel is a nondimensional Peltier source term defined by
Pel )
2NRoImaxL StotFgrcpgruo
(16)
4.5. Initial and Boundary Conditions. The adsorption column is initially at the water bath temperature and saturated by a He-CO2 mixture with a constant flow rate and composition. The inlet gas flow rate and composition remain constant during the experiments. The boundary conditions are summarized in Table 1. 5. Calculation and Experimental Results The values of the physical parameters in the previous equations are summarized in Table 2 for a sample of experiments, with a flow rate of 40 mL s-1 of a 30% CO2 mixture at 25 °C. The axial dispersion and heat-transfer coefficients are estimated from correlations, as presented in Appendix C. The values are assumed to be
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Table 1. Boundary Conditions for the Model Equations eq number
balance
inlet
outlet
adsorption kinetics adsorbed gas mass balance
1 2
none constant mole fraction xo
none
global gas mass balance
3
∂c constant gas )0 continuity concentration co ∂z
graphite energy balance
4
brass heat exchanger
∂T ) hw(T - Tw) ∂z
activated carbon energy balance global energy balance
5
adiabatic
adiabatic
6
constant gas temperature To
∂Tg† )0 ∂z
pressure drop equation
7
constant gas velocity uo
∂u† )0 ∂z
continuity
∂x )0 ∂z
Table 2. Numerical Values of Parameters Used in Simulationsa parameter
symbol
value
Pem Sc Dax Stm
163.36 4075 1.19 × 10-5 m2 s-1 1.088
Rep Pr Peth
0.92 0.31 12.9
Nugr Nuac
7 4.4
mass transfer Peclet number Schmidt number axial dispersion coefficient mass Stanton number heat transfer particle Reynolds number Prandtl number heat Peclet number Nusselt number graphite particle heat Stanton number graphite particle heat-transfer coefficient graphite/gas particle/gas heat-transfer number (graphite) heat-transfer number (activated carbon)
Stgr Stac
35.5 651.5
hgr/g hac/g Ntgr Ntac
4.69 W m-2 K-1 240 W m-2 K-1 70.3 316 565
adsorption number nondimensional Joule number nondimensional Peltier number
Ad Jou Pel
8.6 17 027 911
a
Flow rate 40 mL s-1, 30% CO2 at 25 °C.
constant and averaged over the temperature range. The model equations are then solved using the DASSL package.9 The column temperature is measured at several points with thermocouples placed through the graphite sheets. The resulting measured temperature is an average over the graphite, gas, and particle temperature, and we designate it as the “average temperature”. Some experiments are also conducted with measurements made in the gas, in the particle layer. This temperature is called the “gas temperature”. The model uses averaged properties in eqs 1-7. Only the thermoelectric device properties are considered to be temperature-dependent. Table 3 summarizes the values of the Seebeck coefficient, electric resistivity and thermal conductivity of the semiconductor material. The assumption of the absence of thermal resistance between the column wall and the graphite sheets entails an overestimation of the temperature of the graphite sheets during the heating period. The computed tem-
perature runs up to 120 °C (Figure 6), while the solder fusion temperature of the thermoelectric elements is 110 °C. On the other hand, we neglect the direct heat transfer between particles and graphite sheets, as well as radiative heat transfer. All of those assumptions lead to an overestimated graphite temperature and an underestimated activated carbon temperature during the heating periods. We do not take account of the column side walls in the thermal balance. They are made of polypropylene and are in contact with graphite sheets and gas. Consequently, the computed column temperature variations are expected to be slightly faster than measured. 5.1. Temperature Variations. Figure 6 shows the time variations of the measured and calculated temperatures in the adsorption column. The experiments consisted of adsorption periods of 10 min at a temperature set point of -10 °C and desorption periods of 5 min at a temperature set point of 60 °C. The upper graph in Figure 6 shows the variations of the average temperature measured through the graphite sheets and the corresponding calculated temperature and indicate rather good agreement. The lower part of the figure shows the calculated graphite sheets temperature (not measured) and the calculated and measured gas temperatures. Because of the high heat-transfer rate between gas and particles, the calculated activated carbon temperature practically coincides with the curve of the gas temperature. During the adsorption period, the graphite temperature decreases quickly to the set temperature. Because of the poor heat transfer between the gas and the graphite sheets, the gas temperature decreases slowly but remains well above the graphite temperature. Heating (desorption) periods are more complex. Because of the highly asymmetric heat flux in the thermoelectric devices and the process control based on the average temperature measurements, the graphite sheets are overheated to 120 °C at the beginning time of the heating period. The gas temperature increases slowly up to a final temperature somewhat lower than the setpoint temperature. The calculated activated carbon temperature varies between 5 and 35 °C. Again, the calculated and measured curves show relatively good agreement. 5.2. Electrical Variations. The system temperature is controlled by electric current variations passing through each thermoelectric device. The electric voltage is free and governed by
S V ) 2N F + R∆T L
(
)
(17)
Figure 7 shows the applied and computed electric currents to thermoelectric devices, and Figure 8 shows the resulting voltage measured and computed on the thermoelectric devices. Thermoelectric device 1 is situated at the column inlet, and device 2 at the outlet. During the cooling (adsorption) period, the current is set to the maximal allowed set point (10 A). During the heating (desorption) period, the electric current takes an initial value of -10 A for a very short time and then slowly increases. The differences between the inlet and outlet thermoelectric devices are due to the different duties of the elements. The inlet device has to heat both the adsorption column and the entering gas flow. The outlet device has only to heat the adsorption column.
Ind. Eng. Chem. Res., Vol. 40, No. 10, 2001 2327 Table 3. Electrical and Thermal Parameters of the Thermoelectric Elements as a Function of Temperature parameter name
symbol
a0
a1
a2
units
Seebeck coefficient electric resistivity thermal conductivity
R Fe λ
22 224 × 10-9 5112 × 10-10 62 605 × 10-4
930.6 × 10-9 163.4 × 10-10 -277.7 × 10-4
-0.9905 × 10-9 0.679 × 10-10 0.4131 × 10-4
V K-1 Ω m-1 W m-1 K-1
value ) ao + a1T + a2T2, with T in Kelvin
Figure 7. Electric current variations with time. Positive values correspond to the adsorption period and negative values to desorption. Computed and experimental values. TE device 1 is situated at the column gas inlet and TE device 2 at the gas outlet.
Figure 6. Experimental and calculated temperature variations with time; water temperature, 25 °C; adsorption, 10 min at -10 °C; desorption, 5 min at 60 °C.
Obviously, some discrepancy is observed between the calculated and measured values, especially for thermoelectric element 1. This leads to the conclusion that the model of the thermoelectric element is not accurate. 5.3. Adsorption. The column is steadily fed with a He-CO2 mixture of constant composition. This nonclassical operating mode without a purge step is used here not only as a simple test of the model, but also to prefigure a better-performing mode, cycling-zone adsorption, as explained in the next section. The temperature of the column is alternatively cooled and heated as described above. Figure 9 shows the measured and computed outlet carbon dioxide concentration with time, for an experiment with a 17% molar inlet concentration. During the cooling (adsorption) period, the CO2 concentration decreases rapidly and later slowly increases because of breakthrough of the adsorption front. During the heating (desorption) period, the CO2 concentration increases very quickly and then decreases, when the bulk of the CO2 has been desorbed.
Figure 8. Electric voltage variations with time. Positive values correspond to the adsorption period, and negative values to desorption. Computed and experimental values.
6. Augmenting the Separation by Cycling-Zone Adsorption The separation performance illustrated by Figure 9 is clearly limited by the operating mode chosen. This limitation is also partly due to the small bed length and probably to a poor chromatography quality. Also, the flow rate used here is on the order of 0.1-0.14 m3 h-1. A larger flow rate increases the spread of the residence time distribution and thus diminishes the chromatographic quality. These performances can be augmented by using a parallel-series arrangement of similar modules, as illustrated in Figure 11. The parallel setup allows for a larger flow to be processed at a similar gas velocity, and the series setup allows for an increase in the quality of separation, as discussed below. Cycling-zone-adsorption was introduced by Pigford1,4 in the late 1960s. It involves a periodic changes of tem-
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Figure 12. Principle of temperature shifts in five-column cyclingzone adsorption. H designates hot temperature; C designates cold temperature.
Figure 9. Adsorption-desorption cycles. Total flow rate, 30 mL s-1; 17% CO2, adsorption, 15 min at -10 °C; desorption. 7.5 min at 60 °C. Experimental and calculated mole fractions.
Figure 13. Simulation of outlet concentrations in five-column cycling-zone adsorption.
Figure 10. Breakthrough and elution curves, computed (line) and measured (dots). Flow rate, 40 mL s-1; temperature, 23 °C; CO2 step molar fraction, 16%.
effluent having a very low content of CO2 is obtained already in column 3. The effluent of the fifth column contains less than 0.1% for 20 min, while the effluent of the other 20 min is concentrated twice (average 40%). Notice that this operating mode (the so-called direct mode of cycling-zone-adsorption, whereby the columns are heated and cooled through the walls) is hardly practical by other means than those used here, which allow for a relatively fast temperature change in each column and the separate control of each column. 7. General Discussion and Conclusions
Figure 11. Series-parallel arrangement of adsorption modules.
perature of a series of columns, with a phase shift between successive columns. Figure 12 illustrates schematically the temperature distribution of a series of five columns at three different phases in such a process. We simulated the operation of such a series of five columns, operating under conditions comparable to those of Figure 9 (20% CO2; flow rate, 30 mL s-1; heating, 30 min at 60 °C; cooling, 10 min at -10 °C). The result is shown in Figure 13. It is clearly seen that the separation is amplified in each column, and an
With respect to the objectives of this work, namely, the acceleration of TSA cycles, a number of partial conclusions can be drawn, although further experimental work and further improvements are needed. (1) The overall, apparent thermal conductivity of the composite adsorbing block is indeed considerably increased with respect to that of a conventional uniformly packed bed of adsorbent particle. Independent thermal measurements2 quantify this very clearly. The graphite sheets do contribute significantly to increased heat transfer between the bed and the lateral walls. However, the heat efficiency is still strongly limited by the heat transfer between the graphite sheets and the adsorbent particles and the flowing gas. Probably only a global consolidation of the composite solid will overcome this bottleneck. Note that we have not evaluated the contribution of radiation between the graphite sheets and particles.
Ind. Eng. Chem. Res., Vol. 40, No. 10, 2001 2329
One could ask whether graphite sheets are any better than metallic sheets for the present purpose. This issue is discussed by Bonnissel.2 Let us merely state here that, for a given thermal conductivity λ, the heat capacity (Fcp) of graphite sheets is much smaller than that of metals, and therefore, the thermal diffusivity is larger (by a factor of approximately three with respect to that of copper). This property is essential for the dynamic aspect of TSA cycles. (2) The Peltier elements have fast dynamics and are very easy to control. The temperature switch occurs in a matter of seconds after inversion of the electric current feeding them. The thermoelectric devices used here have a thermal power on the order of 100-200 W depending on the temperature difference, which is just about the order of magnitude of the power pumped by the adsorbent composite, as limited by its apparent thermal conductivity. There is therefore strong bilateral coupling between the thermoelectric elements and the adsorbent bed. There are a number of distinctive features of thermoelectric elements, including their cost, their temperature limitations, but also their energy “dissymmetry”, resulting from the dissipative effects, as the heat released on the hot side is much larger than the heat pumped on the cold side. This dissymmetry has to be accounted for in the control of the water flow rate. Recall that the thermoelectric elements are essentially heat pumps, not sources. The energy source is the water flow, and the Peltier element either takes heat out of that flow to release it at a higher temperature to the adsorbent or conversely transfers heat from the adsorbent to the water flow. The excess heat stems from internal Joule effects. (3) The detailed modeling of these elements was not the aim of this paper, and we have observed, from Figures 7 and 8, that our model reflects the trends but is not very accurate. Because, on the other hand, the modeling of the adsorption/desorption is relatively accurate (see Figure 10), we tend to conclude that moderate discrepancies such as those observed in Figure 6 for the gas temperature, are essentially due to the overall model of the Peltier elements, together with their exchanges with the adsorbent bed. (4) The adsorption/desorption cycle tested here is not conventional: the mixture of CO2 and He is fed continuously to the adsorbent bed while the temperature is changed. There is neither flow reversal nor use of pure helium to purge the bed. This operating mode is inspired by the work of Jacob,5 who showed that, under certain conditions, a pure helium flow could be obtained with this operating mode. Such a result was not obtained here. The present experimental results merely illustrate that a concentration/depletion process is possible with short cycles and in one shallow column. Simulations of a cycling-zone-adsorption process with two or three columns shows that this separation can be considerably amplified. Cycling-zone adsorption is certainly an operating mode adapted to this technology. (5) The possibility of performing TSA cycles of about 10 min, over a bed thickness of about 10 cm, has been shown, with a temperature amplitude on the order of 50 °C. A full model of the system including the thermoelectric elements is able to predict properly the column dynamics for temperatures, concentrations, and electric parameters.
Nomenclature Roman Letters ap ) particle specific area, m2 m-3 A, B, C, D ) Langmuir constant parameters b ) Langmuir parameter, Pa-1/n c ) gas concentration, mol m-3 cp ) mass heat capacity, J kg-1 K-1 dp ) particle diameter, m Dax ) axial dispersion coefficient, m2 s-1 h ) heat-transfer coefficient, W m-2 K-1 hxx(Txx) ) mass enthalpy at T, J kg-1 I ) electric current intensity, A k ) internal mass-transfer coefficient, s-1 L ) column or thermoelectric element length, m M ) molar mass, kg mol-1 N ) number of thermoelectric pairs P ) pressure, Pa q ) adsorbed quantity, mol kg-1 q* ) adsorbed quantity at equilibrium, mol kg-1 R ) universal gas constant, J mol-1 K-1 S ) surface, m2 T ) temperature, K t ) time, s u ) interstitial gas velocity, m s-1 V ) voltage, V volume, m3 x ) mole fraction abscissa, m y ) reduced density Y ) mass fraction z ) reduced length Greek Letters R ) thermal diffusivity, m2 s-1 Seebeck coefficient, V K-1 β ) intragranular porosity ) extragranular porosity γ ) graphite volume fraction λ ) thermal conductivity, W m-1 K-1 µ ) viscosity, Pa s F ) density, kg m-3 Fe ) electrical resistivity, Ω m σ ) heat source (volumic or surfacic), W m-3 or W m-2 ω ) thermoelectric partial area divided by total device area Subscripts o ) reference state 1 ) adsorbable 2 ) not adsorbable ac ) activated carbon ads ) adsorbed phase bed ) particle bed g ) gas gr ) graphite i ) interstitial or generic species in gas phase int ) internal J ) Joule m, max ) maximal o ) initial, reference p ) Peltier, particle tot ) total v ) volume Superscripts * ) equilibrium † ) reduced quantity with respect to reference state (subscript o)
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Assuming that there is no mass-transfer resistance between the intragranular and the extragranular gas, Yig ) Yigint, and introducing the molar concentrations with eq 20, we have
Nondimensional Numbers Ad )
Fbedqo* Facqo* ) (1 - ) ) adsorption number co co
Bi )
hdp ) thermal Biot number λp
[
1+
Jt ) Chilton-Colburn factor 2
2
Jou ) Nu )
2N FeoImax L ωStot2FgrcpgruoTo
) nondimensional Joule coefficient
hL ) thermal Nusselt number λg
]
∂ci ∂ciu FbedMads ∂q (1 - ) β Mads + Mads + ) ∂t ∂x ∂t 2 ∂ ci MadsDax 2 (21) ∂x
Equation 21 can be written using volume fractions as
SgrL ) heat-transfer number Ntgr ) Stthg/gr V uoL Pem ) ) mass column Peclet number Dax
[
1+
](
)
∂xi ∂xiu (1 - ) ∂c ∂cu β c + xi + xi + +c ∂t ∂t ∂x ∂x
Peth )
uo L ) thermal column Peclet number Rgr
cpµ ) Prandtl number λ Fouodp ) particle Reynolds number Rep ) µ pouox Rex ) ) local Reynolds number µ Focp R) ) reduced heat capacity Fgrcpgr (see eqs 51, 56, 57, and 62)
[
1+
]
(1 - ) ∂c ∂cu Fbed ∂q ∂2 c β + + ) Dax 2 (23) ∂t ∂x ∂t ∂x
Substituting eq 23 into eq 22, we obtain for the adsorbable gas with index 1
[
]
∂x1 Fbed ∂q ∂2x1 (1 - ) ∂x1 β +u + x2 ) Dax 2 (24) 1+ ∂t ∂x ∂t ∂x
µ ) Schmidt number FDm
Stm )
)
For a binary mixture of an adsorbable gas and a nonadsorbable gas, the total material balance is
Pr )
Sc )
(
Fbed ∂q ∂2xi ∂2c ) Dax c 2 + xi 2 (22) ∂t ∂x ∂x
uodp Pemp ) ) mass particle Peclet number Dax
kL ) mass Stanton number uo
and for the nonadsorbable gas with index 2
†
Nu λgo ) heat Stanton number Stth ) Peth Rg
[
1+
A. Adsorption Column Equations A.1. Mass Balance. The mass balance is written for the two gases. For an adsorbed ga, we designate by Yi the mass fraction of species i in the gas and by q the adsorbed quantity per unit mass of activated carbon. The mass balance is then
DYig DYigint ∂q Fg + (1 - )βFint + FbedMads ) Dt Dt ∂t Dax∇2FgYig (18)
]
∂x2 Fbed ∂q ∂2x2 (1 - ) ∂x2 β +u - x2 ) Dax 2 (25) ∂t ∂x ∂t ∂x
The sum of eqs 24 and 25 is identically equal to zero. A.2. Ergun Equation. The Ergun equation relates the pressure drop to the gas velocity as follows:
(1 - )2µ (1 - )2F ∂P ) -150 3 2 u - 1.75 3 2 u2 ∂x d d p
(26)
p
For a one-dimensional problem, we have
∂FgYig ∂FgYigu (1 - ) ∂FintYigint FbedMads ∂q + + β + ) ∂t ∂x ∂t ∂t ∂2FgYig Dax (19) ∂x2 The partial density follows the relation
FgYig ) Fig ) cigMads
(20)
A.3. Global Internal Energy Balance. The global internal energy balance is the sum of the internal energy balance of each component (gas, graphite, adsorbed phases, and activated carbon) multiplied by the volume fraction of each component. Therefore, we have a γ factor for terms concerning the graphite, a (1 - γ)(1 - ) factor for the activated carbon terms , a (1 - γ) factor for the extragranular gas terms, and a (1 - γ)(1 - )β factor for the intragranular gas terms. The global balance is then
Ind. Eng. Chem. Res., Vol. 40, No. 10, 2001 2331
γFgrcpgr
∂Tgr ∂Tac + (1 - γ)(1 - )Faccpac + ∂t ∂t ∂FgTg + ∇(FguTg) + (1 - γ)cpg ∂t ∂FgintTgint (1 - γ)(1 - )βcpgint + ∂t ∂Tads Fbed qcpads (1 - γ) Mads ∂t
[
]
2
[
) (1 - γ)(1 - ) λac
) γλgr∇ Tgr + (1 - γ)(1 - )λac∇ Tac + 2
[
Dax∇2(FgYig) hi(Tgint) ∑ i)1
2
∇Jigcpg(Tgint - Tg) + ∑ i)1
cpg(Tgint - Tads)] + (1 - γ)
DPg DPgint (1 - γ) + (1 - γ)(1 - )β (27) Dt Dt
)
2
]
2
λgr ∂ Tgr 1 - γ λac ∂ Tac ) + + (1 - ) Fgrcpgr ∂x2 γ Fgrcpgr ∂x2
]
)
[
∂2Tac ∂x2
]
+ hac/gap(Tg - Tac) (32)
(33)
u† )
u uo
(34)
q† )
q qo
(35)
z)
x L
(36)
cpg cpo
(37)
cpg† )
2 1 - γ λg ∂ Tg 1 - γ FbedMads ∂q ∆h (T ) + γ Fgrcpgr ∂x2 γ Fgrcpgr ∂t ads ads ∂Pg 1-γ 1 1 - ∂Pg β +u 1+ (28) γ Fgrcpgr ∂t ∂x
[(
]
t tuo t† ) ) τ L
Faccpac FacMadscpac ∂Tac ∂Tgr 1 - γ + q + (1 - ) + ∂t γ Fgrcpgr Fgrcpgr ∂t ∂Tg (1 - ) ∂Tg 1 - γ Fgcpg 1+ β +u γ Fgrcpgr ∂t ∂x
[(
]
Sgr/ac (Tgr - Tac) (31) V
A.4. Nondimensional Form. Let us consider a reference state defined by a temperature To, a pressure Po, a gas velocity uo, and a gas concentration co. These conditions are typically the initial and/or inlet conditions of the experiment. With this reference state, we can calculate the adsorbed quantity qo* using the adsorption isotherm. We define reduced variables as follows:
Introducing total molar concentration instead of the gas density and assuming the equality of intragranular and extragranular gas concentrations and temperatures, we obtain
)
Sac/g (Tg - Tac) + V
Fbed ∂Tac ) qcpads Mads ∂t
(1 - ) λac
Fbed ∂q (1 - γ) [hgint(Tads) - hads(Tads) Mads ∂t
+ hac/g
Introducing the specific area ap of the particles and using eq 30, we have
(1 - γ)λg∇2Tg - (1 - γ)
(
∂x
2
hgr/ac
(1 - )Faccpac +
2
∂2Tac
y)
F Fo
(38)
A.3.1. Graphite Energy Balance. The graphite energy balance is given by
Tx† )
Tx To
(39)
∂Tgr ∂2Tgr Sgr/g ) γλgr (Tg - Tgr) + + γhgr/g 2 ∂t V ∂x Sgr/ac γhgr/ac (Tac - Tgr) (29) V
λx† )
λx λgr
(40)
γFgrcpgr
Assuming that the heat flux between the graphite sheet and the activated carbon grains is negligible with respect to the gas-graphite heat flux, we obtain
For the gas thermal conductivity, we use a reference the gas conductivity defined at the inlet of the adsorption column. Thus, we have
λg λgo λg ) ) λgo†λ† λgr λgr λgo
2
∂Tgr ∂ Tgr Sgr/g + hgr/g Fgrcpgr ) λgr (Tg - Tgr) (30) 2 ∂t V ∂x A.3.2. Activated Carbon and Adsorbed-Phase Energy Balance. The activated carbon and adsorbed-phase energy balance is given by
[
]
Fbed ∂Tac qcpads (1 - γ) (1 - )Faccpac + Mads ∂t
(41)
A.4.1. Gas-Phase Mass Balance. The adsorbed gas mass balance is
(
1+
2 x2 ∂q† ∂x1 1 - ∂x1 1 ∂ x1 β † + u† + Ad † † ) (42) ∂z Pem ∂z2 ∂t c ∂t
)
with the Peclet and adsorption numbers defined by
2332
Ind. Eng. Chem. Res., Vol. 40, No. 10, 2001
uoL Dax
(43)
Facqo* Fbedqo* ) (1 - ) co co
(44)
Pem ) Ad )
The global mass balance is then given by †
† †
†
∂2c† 2 m ∂z
(1 + 1 - β) ∂c∂t + ∂c∂zu + Ad∂q∂t ) Pe1 †
†
∂Tgr† ∂t†
)
∂q† †
) Stm(q*† - q†)
[(
Peth )
with the Stanton number
kL uo
(48)
)
uoLFgrcpgr λgr
A.4.3. Graphite. The energy balance of the graphite
∂t
)
(56)
coMadscpads Fgrcpgr
(57)
†
1 ∂ Tgr + RgNtgr(Tg† - Tgr†) Peth ∂z2
(49)
with
(55)
Faccpac Fgrcpgr
Rac ) 2
]
The other nondimensional groups are the reduced heat capacities of the activated carbon Rac, of the gas Rg previously defined in eq 51, and of the adsorbed phase Rads.
is
∂Tgr† †
]
)
In eq 54, we can isolate several nondimensional numbers such as the adsorption number Ad and the axial mass Peclet number Pem. We define a thermal Peclet number Peth as
∂t
Stm )
[(
† Po ∂Pg† 1-γ 1 - ∂Pg † β 1+ + u (54) γ ToLFgrcpgr ∂z ∂t†
(46)
(47)
)
∂2Tg† λg 1-γ γ uoLFgrcpgr ∂z2 1 - γ Fbedqo*Mads ∂q† ∆hads(Tads) + γ ToLFgrcpgr ∂t†
(45)
The nondimensional form is
(
Faccpac FacMadsqcpads ∂Tac† 1-γ + + (1 - ) γ Fgrcpgr Fgrcpgr ∂t† † ∂Tg† 1 - γ Faccpac 1 - ∂Tg † 1+ +u β γ Fgrcpgr ∂z ∂t†
λgr ∂2Tgr† 1 - γ ∂2Tac† λac + + (1 ) uoLFgrcpgr ∂z2 γ uoLFgrcpgr ∂z2
A.4.2. Adsorbed Phase Mass Balance. The mass balance of the adsorbed phase is given by the kinetic equation
∂q ) k(q* - q) ∂t
+
Rads )
The reduced global energy balance is then
Ntgr ) Stthg/gr Rg )
SgrL Nug/grλg† ) V Peth Rg
(50)
Fgocpgo Fgrcpgr
(51)
∂t†
A.4.4. Activated Carbon and Adsorbed Gas Energy Balance. Assuming that the particles of activated carbon are spherical, the energy balance of the activated carbon and the adsorbed gas is given by †
[(1 - )Rac + AdRadsq ] 2
∂Tac† ∂t†
∂Tgr†
)
†
+
∂Tac 1-γ (1 - )(Rac + RadsMadsq†) + γ ∂t†
[(
)
∂ Tac L 1 (1 - ) λac† + (1 - )RgNtac (Tg† - Tac†) Peth dp ∂z2 (52)
2 † 1 ∂ Tgr 1-γ ∂q† ∆hads(Tads) + RadsAd † 2 Peth ∂z γ Tocpads ∂t
[(
]
† Po ∂Pg† 1-γ 1 - ∂Pg † β 1+ + u (58) γ ToFgrcpg ∂z ∂t†
)
B.1. Medium 2. The equation is a “classical” onedimensional heat equation with heat source
F2cp2
with
SacL Nug/acλg† ) V Peth Rg
)
b. Equations for the Thermoelectric Elements
†
Ntac ) Stthg/ac
]
† ∂Tg† 1 - ∂Tg 1-γ † +u Rgy 1 + β γ ∂z ∂t†
(53)
A.4.5. Global Energy Balance. The reduced global energy balance is
∂T2 ∂2T2 σj + ) λ2 ∂t V ∂x2
(59)
The σj term is the volume density of the Joule heat source, given by
2NFeI2 dx 2N2FeI2 σj ) ) V 2Splot(Stot dx) ωStot2
(60)
Ind. Eng. Chem. Res., Vol. 40, No. 10, 2001 2333
[
The nondimensional energy balance is
R2
Ra
∂T2†
λ2† ∂2T2† 2N2FeI2L ) + Peth ∂z2 ∂t† ωStot2FgrcpgruoTo
(61)
where R2 is the reduced heat capacity defined by
F2cp2 R2 ) Fgrcpgr
(62)
2
2
Jou )
2N FeoImax L
(63)
ωStot2FgrcpgruoTo
(
]
)
λa† λa† λb† λb† Rgr ) T + T + T -1 (71) dz2 i+1 dz2 dz1 i dz1 i uo
∂T2†
R2
∂t†
)
λ2† ∂2T2† + JouFe†I†2 Peth ∂z2
(64)
B.2. Medium 1/Medium 2 Interface. The order-2 Laplace development of the temperature at node i in medium 1 is
( )
Ti-1 ) Ti - ∆x1
∂T ∂x1
2
+
i1
∆xi 2
( ) ∂T ∂x1
(65) i1
Thus
( ) 2
∂T ∂x12
)
i1
[
( )]
2 ∂T Ti-1 - Ti + ∆x1 2 ∂x ∆x1 1
(66)
i1
The heat conduction equation is verified on both sides of the interface. Thus, assuming that there is a Joule heat source in medium 1, we have
( )
σj ∂T ∂2T + ) R1 2 ∂t ∂x i F1cp1S dx1
(67)
A similar equation can be written in medium 2. In addition, the heat flux conservation equation is
∂T ∂T S ) λ2 S + σp λ1 ∂x i1 ∂x i2
( )
( )
(68)
The σp term is the Peltier heat source. The energy balance at node i is then
R1
[
]
σp σj † dx1 dx2 ∂T + R2 + λ R 2 2 ∂t SFgrcpgr 2S a gr †
(
†
†
)
†
]
The Peltier heat is given by
(70)
The heat created by Joule effect is given by eq 60. The nondimensional form of the energy balance is then
0.5 9.7 1+ Re Sc
(72)
with the Schmidt number defined by
Sc )
µ FDm
(73)
The column Peclet number is related to the particle Peclet number by
Pem )
L Pe dp mg
(74)
In our case, the column Peclet number used was about 13 for a 30 mL s-1 flow rate of a 30% CO2-He mixture at 25 °C. C.2. Mass-Transfer Coefficient. To determine the kinetic coefficient appearing in eq 46, we performed breakthrough and elution experiments using a step of the CO2 concentration in a constant feed of He at constant temperature. Figure 10 shows the breakthrough and elution curve for a flow rate of 40 mL s-1 and a temperature of 23 °C. By fitting the previously described model to these particular experimental results, we can evaluate the kinetic coefficient k. In our case, we find k ) 0.25 s-1. The mass Stanton number is given in that case by
Stm )
kL ) 0.816 uo
(75)
C.3. Gas-Graphite Heat Transfer. The gasgraphite coefficient evaluated by the Nakayama correlation8 for an isolated plate surrounded by a saturated porous medium is
Nux )
λa λa λb λb Ti+1 + Ti + T -1 Rgr (69) dx2 dx2 dx1 dx1 i
σp ) 2NRIT
C.1. Axial Dispersion Coefficient. The axial particle Peclet number is given by the Edwards and Richardson correlation3
0.73 1 ) + Pemp Re Sc
then eq 61 transforms into
)
[
N2FeoImax2L2dz1 † Rgr † †2 F I λa u oT o e Stot2ω
C. Model Parameter Estimation
If we let
[
]
2NRoImaxL † † † dz1 dz2 ∂T† L † + RIT + Rb 2 2 StotFgrcpgruo ∂t
xπ Pr
1/2
Rex1/2
(76)
which gives, in our case, a value of 6 for Nux for an 8.5cm-long adsorbent column. On the other hand, for laminar heat transfer in rectangular tubes, another correlation is available7
Nu )
5.385 q2 1 - 0.346 q1
(77)
2334
Ind. Eng. Chem. Res., Vol. 40, No. 10, 2001
where q1 and q2 are the heat fluxes for the two plates. In our case, these heat fluxes are equal, so we have Nu ) 8.23. In the simulations, we used an average value of 7 for Nu. C.4. Heat Transfer between Gas and Particles. The heat-transfer coefficient between gas and particles have been widely studied. One of the simplest correlations is from Yoshida.12
for 0.01 e Rep e 50
Jt ) 0.904Rep-0.51 (78)
for 50 e Rep e 1000
Jt ) 0.613Rep-0.41 (79)
where
Jt )
Nu Rep Pr1/3
(80)
is the Chilton-Colburn Jt factor. A more accurate correlation is given by Kramers.6
Nu ) 2 + 1.3Pr0.13 + 0.66Pr0.31 Rep0.5
(81)
The value found from eq 81 in the present case is 4.43 for Nut at a flow rate of 40 mL s-1. Literature Cited (1) Baker, B.; Pigford, R. L. Cycling Zone Adsorption: Quantitative theory and experimental results. Ind. Eng. Chem. Fundam. 1971, 10 (2), 283-292.
(2) Bonnissel, M.; Luo, L.; Tondeur, D. Compacted exfoliated natural graphite as heat conduction medium. Carbon, 2001, to be published. (3) Edwards, M.-F.; Richardson, J.-F. Gas dispersion in packed beds. Chem. Eng. Sci. 1968, 30, 261. (4) Grevillot, G. Principles of Parametric Pumping. In Handbook of Heat and Mass Transfer; Chereminisoff, N., Ed.; Gulf Publishing Company: Houston, TX, 1986; Chapter 36, pp 14291472. (5) Jacob, P.; Tondeur, D. Adsorption non isotherme de gaz en lit fixe. IIIsE Ä tude expe´rimentale des effets de guillotine et de focalisation. Se´paration n-pentane-isopentane sur tamis 5A. Chem. Eng. J. 1983, 26, 143-156. (6) Kramers, H. Heat transfer from spheres to flowing media. Physica 1946, 12, 61. (7) Lundberg, R. E.; Reynolds, W. C.; Kays, W. M. Heat transfer with laminar flow in concentric annuli with constant and variable wall temperature and heat flux; Technical Report NASA-TN-D1972; Stanford University: Stanford, CA, August 1963. (8) Nakayama, A.; Kokudai, T.; Koyama, H.; Nondarcian boundary layer flow and forced convective heat transfer over a flat plate in a fluid-saturated porous medium. J. Heat Transfer 1990, 112 (1), 157-162. (9) Petzolt, L.R. A description of DASSL: A differential/algebric system solver; Report SAND82-8637; Sandia National Laboratories: Albuquerque, NM, 1982. (10) Spokojny, M.; Galev, V. Mathematical modeling of unsteady operating conditions of thermoelectric modules and their automatic design. The 9th International Conference on Thermoelectrics; IEEE: Piscataway, NJ, 1990; pp 160-162. (11) Yang, R. T. Gas Separation by Adsorption Processes; Butterworth: Woburn, MA, 1987. (12) Yoshida, R.; Ramaswami, D.; Hougen, O. A. Temperature and partial pressure at the surface of catalyst particles. AIChE J. 1962, 8 (1), 5-11.
Received for review September 12, 2000 Revised manuscript received February 8, 2001 Accepted February 8, 2001 IE000809K
