Environ. Sci. Technol. 2003, 37, 5001-5007
Decontamination of Polyaromatic Hydrocarbons from Soil by Steam Stripping: Mathematical Modeling of the Mass Transfer and Energy Requirement OLIVER BRAASS,† CHRISTIAN TIFFERT,† JOACHIM HO ¨ HNE,‡ XING LUO,† AND B E R N D N I E M E Y E R * ,†,§ Institute of Thermodynamics, University of the Federal Armed Forces Hamburg, Holstenhofweg 85, 22049 Hamburg, Germany, GKSS Research Center Geesthacht, Institute of Coastal Research/Physical and Chemical Analysis, Max-Planck-Strasse, 21502 Geesthacht, Germany, and EADS Airbus GmbH, Water and Waste Systems, Kreetslag 10, 21129 Hamburg, Germany
For cleaning of contaminated soil from polyaromatic hydrocarbons (PAH), a thermal separation process is applied. The process uses superheated steam that is supplied through a nozzle together with a suspension (approximately 40% soil content) of the contaminated soil into a tube reactor. In the reactor, the soil suspension is vaporized, and the PAH are stripped from the soil at temperatures of 140-300 °C. In a cyclone, a solid-vapor separation is carried out, and after going through a condenser, a separation of the condensed water and the PAH is obtained. For improvement of the economical performance, a heat recovery is integrated. This is realized by preheating the water/stream supplied to the evaporator by cooling the vapor steam leaving the reactor. For the mathematical description of the process, the removal of the PAH from the soil is considered to take place by a desorption process. Sorption isotherms are measured by batch experiments and can be described by isotherms of Langmuir type. A dispersion model is used to describe the mass transfer of the process. The process is mathematically modeled for instationary and stationary operation. The simulation predicts the lowest energy consumption at a good cleaning performance at a steamto-suspension ratio of 5.
Introduction Cleaning of contaminated soils becomes more and more important in maintaining groundwater cleanness. Different technologies are available for soil decontamination, addressing diverse advantages and presumptions for their application. For a comprehensive overview, refer to the summary of an interdisciplinary cooperation for more than 12 yr comprising 25 groups (1). To clean the soils off organic pollutants, thermal processes can be applied. These technologies are fast and lower the * Corresponding author phone: +49 40 65 41 35 00; fax: +49 40 65 41 20 08; e-mail:
[email protected]. † University of the Federal Armed Forces Hamburg. ‡ EADS Airbus GmbH. § GKSS Research Center Geesthacht. 10.1021/es020564y CCC: $25.00 Published on Web 09/23/2003
2003 American Chemical Society
FIGURE 1. Types of PAH-soil accumulations and their separation mechanism. contaminant load efficiently. Some of these processes allow decontamination of the soil in situ without excavation (2). For other processes, the soil has to be excavated and is then treated in column or tube reactors (3-5). Generally steam stripping techniques have been developed successfully (6). A general obstacle of decontamination processes is the cleanup of fine-grained soils, which are often disposed as special refuse at present. Starting with 2005 in Germany these disposals have to be treated (7). For this reason, the processing of fine particles was the focus. As no technologically and economically efficient large-scale technique for decontamination is presently available, a scale-up of this process has to be reflected to fill in this gap.
Experimental Methods For the separation of PAH from solids, a thermal separation process employing a steam stripping technique has been developed (8, 9). Basic investigations for evaluating physicochemical conditions of different soil-contaminant systems were carried out discriminating relevant models for description and obtaining relevant parameters. For this reason, small-grained soil particles (dp < 100 µm) of various origin were spiked with different model PAHs (naphthalene, pyrene, anthracene, and benzo[a]pyrene). Investigations were carried out on a laboratory-scale reactor. The PAH analyses have been carried out by external, accredited laboratories according to U.S. EPA 610 and DIN ISO 13877. For the process development, a pilot-scale plant was developed. The contaminated matter is supplied into a tube reactor within a liquid suspension and fed into the tube reactor by a specially designed nozzle. The water added is superheated, and thus the organic contaminants are vaporized. The tube reactor is operated at temperatures of T ) 410-590 K and mass flow rates of m ˘ ) 20 kg/h (steam) and m ˘ ) 4 kg/h (suspension), respectively (10). For further data, refer to publications dealing with the experimental process development (5, 10, 11). Important experimental results influencing the modeling significantly are summarized: The adsorption mechanism of PAH onto a soil particle can be described in three ways (Figure 1): PAH and water are covering the soil as two liquid phases, the PAH is adsorbed on the wet soil, or the PAH is adsorbed on the surface of the dry soil. In case of the latter situation, the separation process can be treated as a sorption process, and the equilibrium conditions can be described by adsorption isotherms (12). Extensive investigations on plants of laboratory and technical scale (11) have shown that also the whole steam stripping process is mainly influenced by the desorption. The cleaning performance for various kinds of real contaminated matter reaches values of up to 99% on pilot scale (10). With these assumptions, an adsorption/desorption model was developed to describe the physicochemical process. This model allows us to simulate cost-determining parameters VOL. 37, NO. 21, 2003 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 2. Flow sheet of the desorption process with heat integration. such as the separation temperature and the energy input into the process (13). The reduction of energy consumption could be realized by the introduction of a heat exchange unit that preheats the water for the steam generator, which vaporizes and superheates (Figure 2). These technological units were integrated into the model, which allows the comparison of reduced operating costs (energy consumption) versus investment costs (integration of the heat exchange unit).
where ka and kd are the adsorption and desorption velocity constants, respectively. The equilibrium of the desorption is described by the Langmuir model:
∂c ∂c ∂2c 3F + u - Dax 2 ) km(c′ - c) ∂t ∂z rP ∂z
(1)
The mass balance for the solid phase, which is convectively transported through the reactor, results in 2
∂q ∂q ∂q + u - Dax 2 ) kac′(qmax - q) - kdq ) ∂t ∂z ∂z 3 - km(c′ - c) (2) rP The first terms on the left side of eqs 1 and 2 represent the accumulated matter. The second term in these equations considers the local change of the concentration due to convective transport phenomena, and the third term belongs to axial dispersion. The right side of the equations represent the limiting step occurring during the mass transfer from the inner particle surface to the bulk of the steam flow (abbreviations at the end of the paper). As the mixing and separation processes are rather fast (in the range of seconds) and the particles are small, isothermal and equilibrium operation conditions were assumed. Consequently, the enthalpy balances do not have to be considered to describe the separation process. The mass transfer from the PAH adsorbed on the solid particle to the vapor phase is described as ka
solid-PAH y\ z solid + PAH k d
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(3)
K ) ka/kd
(4)
where
Model Equations The process is simulated for soils containing the characteristic PAH naphthalene to calculate the heat recovery. The dispersion model is chosen to describe the convective multiphase flow through the horizontal tube reactor. A mass balance for the PAH in the fluid phase, neglecting radial dispersion, delivers the continuity equation:
qmaxKc 1 + Kc
q)
Applying eq 3, the experimental determination of the maximum PAH concentration of the solid qmax and the desorption equilibrium constant K can be obtained by batch sorption experiments. The time-dependent concentration of the PAH in the solid phase can be described as follows:
() (
q ) q0e-kdτ w kd ) ln
)
q0 1 q0 1 ) ln q τ (q0 - ∆q) τ
(5)
The desorption rate constant kd is received from experiments in the tube reactor taking eq 5 into account. By means of eq 4, the velocity constant of the adsorption ka is then determined. Under the constraints of short periods for the dependence of the desorption process on the temperature, an Arrhenius expression can be chosen (10, 11). The parameters are available from Figure 4:
( )
-ET ∆q ) k0qmax exp ∆τ RT
(6)
Using this set of equations, the phase transfer of the steam stripping process can now be described for varying temperatures and verified by experimental results (11). The partial differential equations are of second order and thus require two initial conditions and four boundary conditions: Initial conditions (t ) 0): q(0,z) ) 0 c(0,z) ) c0
at the start of process, no particles are in the reactor (introduction of soil suspension with beginning of steam stripping) known concentration of PAH in vapor phase (from steam generator)
The boundary conditions are the well-known Danckwerts boundary conditions (14): z ) 0 c - (1/Bo)(∂c/∂z) ) c0 z ) 0 q - (1/Bo)(∂q/∂z) ) q0
z ) L (dc/dz)(z,t) ) 0 z ) L (dq/dz)(z,t) ) 0
PAH concentration in vapor phase is changed due to axial dispersion PAH concentration in solid phase is changed due to axial dispersion of particles (particle back mixing) after outlet of tube reactor, flow pattern is assumed to be plug flow no particle back mixing after outlet of tube reactor
A parameter estimation is also necessary for fitting the mathematical model equations to the experimental data. The Reynolds number is defined as
Re )
ud ν
(7)
Re ) 12 222
(8)
is calculated reflecting a complete tubular flow regime in the reactor. The Bodenstein number indicates the degree of dispersion at operation conditions:
uL Bo ) Dax
(9)
)
3 × 107 1.35 + 0.125 ud Re2.1 Re
(10)
∆Tm )
(T′hot - T′′cold) - (T′′hot - T′cold) T′hot - T′′cold ln T′′hot - T′cold
(
)
(14)
Further changes in the heat energy of the steam stripping process occur at the condenser (Q˙ cool) and at the steam generator (Q˙ supply), those are
Bo ) 245
(11)
is found. According to ref 15, for Bodenstein numbers of Bo > 100, the axial dispersion coefficient becomes sufficiently low so that the influence of dispersion can be seen as neglectable. Due to marginal inhomogeneities between the flow characteristics of the steam and fine-grained, convectively transported particles, an axial dispersion coefficient of Dax ) 0.01 m2/s was assumed for both gas and particle flow. For stationary flow conditions the time-dependent term in the differential eqs 1 and 2 disappears. The balance for the fluid phase becomes
∂c ∂2c 3F u - Dax 2 ) km(c′- c) ∂z rP ∂z
(12)
and for the solid-phase we obtain
∂q ∂2q 3F - Dax 2 ) kac′(qmax - q)-kdq ) km(c′- c) ∂z rP ∂z
Q˙ suppIy ) m ˘ steam[h(Tsteam,psteam) - h(Tfeed,pfeed )]
(16)
The heat recovery of the whole process is
Q˙ rec ) βm ˘ steam[h(T′′cold,pcold) - h(T′cold,pcold)]
Combining eqs 9 and 10 and inserting the process parameters of flow velocity u, reactor diameter d, reactor length L ) 6.6 m, and the Reynolds number, a Bodenstein number of
u
with
Q˙ cool(m ˘ water + m ˘ steam)[h(T′′hot,phot) - h(Tcdns,pcdns)] (15)
For Reynolds numbers of Re > 2000, eq 10 can be considered to calculate the axial dispersion coefficient Dax (15):
(
stream to the liquid stream is
Q˙ ) kwA∆Tm
According to the process parameters of the flow velocity u ) 12 m/s, the reactor diameter d ) 5.5 × 10-2 m, and a kinematic viscosity of 54 × 10-6 m2/s, a Reynolds number of
Dax )
FIGURE 3. Sorption isotherms of naphthalene on soil fitted with the Langmuir model (solid lines) and the Freundlich equation (dashed lines).
(13)
The heat exchanger in Figure 2 is a double tube heat exchanger. So the amount of heat transferred from the vapor
(17)
where β is the ratio of the mass flow rate of condensed water flowing through the heat exchanger to that of the total condensed water. The numerical solution of the partial differential equations (eqs 1, 2, 12, and 13) are obtained by using the software Transient Analysis In Process Engineering (TAIPE) as previously presented (16, 17), which offers the numerical procedure for nonlinear transient transport processes. In TAIPE, the governing equation system is solved numerically with finite-difference method. The CrankNicholson scheme is used to discretize the partial differential equation system. The convection term is treated with the power law scheme that is a good approximation of the exponential scheme. Since the Crank-Nicholson scheme is implicit, an iteration procedure is required. For a steadystate process, the iteration procedure is similar. With assumed values of q, the approximate values of c can be obained by solving the discretized equation of eq 12 which is a tridiagnonal linear equation system. With the newly calculated values of c, the new values of q are calculated with the discretized form of eq 13, which is also a tri-diagnonal linear equation system. Repeat the above steps until the iteration deviation is less than the required accuracy.
Results and Discussion The measured values of the equilibrium concentrations of naphthalene in the solid and vapor phase at temperatures of 200 °C (473 K) and 300 °C (573 K) are fitted to isotherm VOL. 37, NO. 21, 2003 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 4. Temperature dependence of the solid-phase concentration of pyrene, anthracene, benzo[a]pyrene, and naphthalene.
FIGURE 5. Breakthrough curve of the naphthalene concentration in the solid phase at the reactor outlet.
FIGURE 6. Outlet concentrations after cleaning soil from naphthalene (mass flux: 35.3 kg/h): experimental data (b); simulation results (-). equations of Langmuir and Freundlich type. At 300 °C both curves fit well to the experimental data. At 200 °C the 5004
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measured data are fitted much better by the Langmuir isotherm than by the Freundlich isotherm (Figure 3).
FIGURE 7. Optimization of desorption performance. Influence of the volume ratio of solid-vapor on the outlet concentrations: τ(1 + 1/F) (dashed lines), F (solid lines). Although the maximum concentration qmax of naphthalene changes slightly from 61 mg/kg at 200 °C to 21 mg/kg at 300 °C, the Langmuir isotherm seems to be much better suited to describe the desorption process. The sorption equilibrium constant of the Langmuir isotherm was calculated as 1.4 × 10-3 kg/mg for 200 °C and 1.8 × 10-4 kg/mg for 300 °C, respectively, and applied for the simulations. The temperature-dependent desorption kinetic is presented in Figure 4. The measured values of pyrene, anthracene, benzo[a]pyrene, and naphthalene as well as the modeled data according to eq 6 (dotted lines) are shown. The concentration of the solid phase is given in dependence of the reciprocal temperature. The fit shows a good correlation to the concentration-temperature behavior of these four PAH. With these parameters, an instationary simulation of the separation of naphthalene from a soil is carried out (Figure 5). Process parameters are the above-mentioned, and a mass flow of the solid suspension m ˘ susp ) 4 kg/h, mass flow of the water vapor m ˘ vapor ) 35 kg/h, initial concentration of naphthalene in the solid q0 ) 20 mg/kg dry matter are considered for the calculations. After a residence time of τ ) 0.5 s at 200 °C and T ) 0.4 s at 300 °C, the first soil particle reach the reactor outlet. Then after 0.3 s at 200 °C and 0.2 s at 300 °C, steadystate conditions with concentrations of qmax ) 1.57 mg/kg (200 °C) and qh ) 1.35 mg/kg (300 °C) are obtained.
A higher temperature in the reactor leads to a better desorption of PAH (simulated for naphthalene) from the soil. This is caused by the higher vapor volume at this temperature, which results in a higher driving force for the separation of the PAH from the soil. In contrast, higher vapor volumes lead to a higher flow velocity and a shorter residence time at this temperature, which counteracts good separation results. A higher slope of the breakthrough curve indicates a smaller influence of the dispersion at 300 °C. A mathematical fit of the model eqs 12 and 13 to a fixed mass flux of m ˘ vapor ) 35.3 kg/h is compared to experimental data of naphthalene outlet concentrations on soil particles processed at various temperatures to validate the mathematical model. A decreasing outlet concentration of naphthalene on the soil with rising temperature is predicted by the simulation results of the stationary operated process, as shown in Figure 6 , and is confirmed by the experimental data. The spreading of the measurement points results both from highly complex systems of contaminated soil characterized by, for example, a biological matrix, a wide range of contaminant distribution, inhomogeneities of the solid phase, and the treatment process described by strongly varying transport phenomena at the reactor inlet and through the reactor; by nonuniform flow pattern within the reactor for both the soil particles and the vapor flow, and by separation mechanisms varied according operation conditions. A simulation of the separation process of naphthalene with an initial load of q0 ) 0.5qmax shows the solid-phase outlet concentrations q, which become smaller with a falling value of volume ratio of solid-vapor, F. The same result occurs if the residence time τ in the reactor rises, as indicated in Figure 7. If the effects of both solid-vapor volume ratio and residence time are considered, an optimization has to be realized. A higher vapor stream (smaller solid-vapor volume ratio) causes a higher driving potential for the phase transition of the PAH leading to an improved separation, while a shorter residence time caused by the increased velocity in the reactor yields to a lower time of decontamination decreasing the final desorption level. This optimization requirement is indicated by the dotted lines in Figure 7 presenting a minimum function. The simulation of the heat recovery (Figure 8) shows that more energy is saved at higher temperatures (lower slope of the long dotted lines in comparison to the short dotted lines). It is also seen that the heat energy introduced into the process is also higher with rising temperature, for the process both
FIGURE 8. Optimization of the heat recovery: energy consumption without heat recovery (small dashes), energy with heat recovery (long dashes), outlet concentration (solid lines). VOL. 37, NO. 21, 2003 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 9. Simulation of the heat-integrated process: energy consumption without heat recovery (dashed lines) and with heat recovery (thin lines); outlet concentration (thick lines). with and without heat recovery (positive slopes of both types of lines). Additionally the requirement of heat energy rises with vapor mass flows. The outlet naphthalene concentration of the solid phase also rises with higher vapor fluxes; an optimum is reached at a vapor flow of m ˘ vapor ) 20 kg/h. These curves also indicate that a stripping temperature higher than T ) 300 °C is not necessary for the separation of this model compound because of the no longer decreasing naphathalene concentration of the solid phase at higher temperatures. The mathematical simulation of the whole steam stripping process shows a similar optimization problem like the simulation of the heat recovery (Figure 9). With rising vapor flow and temperature the required heat energy of the process rises, regardless of whether energy integration is realized or not (the corresponding straight lines indicating higher temperatures are located above the lower temperature lines). The heat energy that can be saved rises with increasing temperature of the process (difference in the slopes of the corresponding straight lines increase with higher temperatures). Also more energy can be saved if the vapor flow is higher (all straight lines have a positive slope). On the other hand, the cleaning performance of the process runs through a minimum, which is moving toward smaller vapor flows with rising temperature. For naphthalene, optimal process operation conditions seem to be between vapor flows of m ˘ vapor ) 15 and 20 kg/h and at a process temperature of T ) 300 °C. Here the required heat energy with integrated heat recovery is only slightly higher than at lower temperatures, but the cleaning performance is significantly improved. The major advantage in running the process at these conditions is a smaller requirement of freshwater for the process. At operating conditions of m ˘ vapor ) 20 kg/h and T ) 300 °C, the total energy consumed by the process is Q˙ ) 17.3 kW, respectively. Simulation results (Figure 9) show that the energy consumption with heat recovery is Q˙ ) 15.2 kW. This is a reduction of 12.7% and improves the economical performance of the steam stripping process regarding the operating costs. The numerical modeling allows the evaluation of operational conditions at reduced experimental efforts. The simulation of the steam stripping shows that low soil outlet concentrations can be obtained at energy saving conditions. The required cleaning performance is directed by the subsequent use of the solid after the cleaning process, by 5006
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governmental laws, by alternative disposal possibilities and their costs, and by process requirements for solid reuse, respectively. From the process simulation can be shown that higher operating temperatures raise the cleaning performance of the steam stripping process. They lead to a higher energy requirement. This means a higher operating cost, but also the amount of energy to be saved by heat integration increases. The absolute value of the heat energy that can be saved is small and can only be obtained if larger heat transfer areas are introduced into the process. This increases the investment costs. Whether a heat recovery is useful or not has to be considered by balancing investment cost like additional piping, pumps, and heat exchangers against operating costs, which are mainly energy costs. This is a site-dependent economical optimization.
Abbreviations A
Area of heat transfer (m2)
Bo
Bodenstein number (-)
c
bulk PAH concentration in the fluid (mol/m3 of vapor) (mg/kg of vapor)
c′
PAH concentration of surface layer in the fluid (mol/m3 of vapor) (mg/kg of vapor)
c0
initial PAH concentration (mol/m3 of vapor)
d
reactor diameter (m)
dp
particle diameter (m)
Dax
axial dispersion coefficient (m2/s)
ET
energy of phase transfer (J/mol)
F
volume ratio solid-vapor (m3/m3)
k0
preexponential factor of Arrhenius (s-1)
ka
sorption rate constant of the adsorption [m3/ (mol s-1)] [kg/(mg s-1)]
kd
sorption rate constant of the desorption (s-1)
km
mass transfer coefficient of boundary layer (m/ s)
kw
heat transfer coefficient [W/(m2 K)]
K
sorption equilibrium constant (m3/mol) (kg/mg)
L
reactor length (m)
m ˘
mass flow (kg/s)
q
PAH concentration or load at the solid (mg/kg)
q0
initial PAH concentration or load (mg/kg)
qmax
maximum PAH concentration or load (mg/kg)
Q˙
heat energy flow (W)
Q˙ cool
heat energy flow of the cooling process (W)
Q˙ rec
recovered heat energy flow (W)
Q˙ supply
supply heat energy flow (W)
rP
equivalent particle radius (m)
R
general gas constant [J/(mol K)]
Re
Reynolds number (-)
t
time (s)
Tm
average temperature for heat energy transfer (K)
T
temperature (K)
u
flow velocity (m/s)
z
length scale (m)
β
ratio of liquid flow through heat exchanger and through bypass valve (-)
ν
kinematic viscosity (m2/s)
τ
residence time (s)
Literature Cited (1) Stegmann, R.; Brunner, G.; Calmano, W.; Matz, G., Eds. Treatment of Contaminated SoilsFundamentals, Analysis, Applications; Springer-Verlag: Berlin, 2001. (2) Lord, A. E., Jr. Geotech. Test. J. 1995, 18, 32-40. (3) Brouwers, H. J. H. J. Hazard. Mater. 1996, 50, 47-64. (4) Rodriguez-Maroto, J. M.; Go´mez-Lahoz, C.; Wilson, D. J. Sep. Sci. Technol. 1995, 30, 317-336. (5) Ho¨hne, J.; Eschenbach, A.; Niemeyer, B. Minimization of Waste by Sludge Treatment Applying the Steam Stripping Process. In Waste Management: The Challenge for Asian CitiessSearch for a Sustainable Future; Poon, C. S., Lei, P. C. K., Eds.; Proceedings of the ISWA-International Symposium & Exhibition on Waste Management in Asian Cities, Hong Kong, China, October 2326, 2000; Vol. 2, 292-297. (6) Niemeyer, B. Application of Physical-Chemical Desorption Technology for Soil Decontamination. In Treatment of Con-
taminated SoilsFundamentals, Analysis, Applications; Stegmann, R., Brunner, G., Calmano, W., Matz, G., Eds.; Springer, Berlin, 2001; Chapter 32, pp 519-529. (7) Dritte Allgemeine Verwaltungsvorschrift zum AbfallgesetzTechnische Anleitung zur Verwertung, Behandlung und sonstigen Entsorgung von Siedlungsabfallen-TA Siedlungsabfall, May 14, 1993; BAnz S. 4967 u. Beilage. (8) Wietstock, P.; Wilichowski, M.; Niemeyer, B. Chem.-Ing.-Tech. 1998, 70, 1167-1168. (9) Niemeyer, B.; Ho¨hne, J.; Eschenbach, A. Steam Strippings Influence of Process Parameters on the Decontamination of FineGrained Particles; CHISA 2000; 14th International Congress on Chemical and Process Engineering, Praha, Czech Republic, August 27-31, 2000; CD-ROM Lecture Group I4, Contribution 529. (10) Ho¨hne, J.; Niemeyer, B. Elimination of Hazardous Components from Fine-Grained Particles and Sludges by the Application of the Steam Stripping Process. In Treatment of Contaminated SoilsFundamentals, Analysis, Applications; Stegmann, R., Brunner, G., Calmano, W., Matz, G., Eds.; Springer-Verlag: Berlin, 2001; 531-546. (11) Ho¨hne, J. Entwicklung eines Verfahrens zur Abtrennung organischer Schadstoffe aus feinkornigen Bodenmaterialien durch gekoppelten Einsatz von Wasserdampfdestillation und Desorption; VDI Verlag: Duesseldorf, 2002; ISBN 3-18-323815-2. (12) Pignatello, J. J. Sorption Dynamics of Organic Compounds in Soils and Sediments In Reactions and Movement of Organic Chemicals in Soils; Sawhney, B. L., Brown, K., Eds.; Proceedings of a Symposium of the Soil Science Society and the American Society of Agronomy, Atlanta, November 30-December 1, 1989; SSSA Special Publication 22. (13) Braass, O.; Ho¨hne, J.; Luo, X. Reduction of the Energy Requirement of an Environmental Process by Modelling the Heat Recovery of an Existing Plant; Proceedings of the International Colloquium on Modelling for Saving Resources, May 17-18, 2001, Riga, Lativa; 100-106. (14) Danckwerts, P. V. Chem. Eng. Sci. 1953, 2, 1-13. (15) Baerns, M.; Hofmann, H.; Renken, A. Lehrbuch der technischen Chemie; Bd.1-Chemische Reaktionstechnik; Georg Thieme Verlag: Stuttgart, 1987. (16) Luo, X.; Niemeyer, B. Modelling and Simulation of Transient Transport Processes Using Axial Dispersion Model. In Scientific Computing in Chemical Engineering II; Keil, F., Mackens, W., Voss, H., Werther, J., Eds.; Springer: Berlin, 1999, 167-174. (17) Niemeyer, B.; Luo, X. A Diffusion Model for Desorption of Hazardous Components from Solids; Proceedings of the International Scientific Colloquium Modelling of Material Processing, May 28-29, 1999, Riga, Lativa; (CD-ROM), 140-145.
Received for review January 29, 2002. Revised manuscript received April 2, 2003. Accepted June 9, 2003. ES020564Y
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