26  |  
 
Izvirni znanstveni članek 
TEHNIKA – 
procesna tehnika   
Datum prejema: 
27.10.2015 
ANALI PAZU 
5/ 2016/ 1-2: 26-32  
www.anali-pazu.si 
 
Adsorption equilibrium modelling for different 
temperature conditions and its influence on 
adsorption mass transfer 
Modeliranje adsorpcijskega ravnotežja v različnih 
temperaturnih pogojih in njegov vpliv na prenos snovi 
pri adsorpciji  
Teodor Štimec
1
, Matjaž Hriberšek
2
, Jure Ravnik
2
, Sani Bašič
1
, Matej Zadravec
2,3*
 
1
  Esotech d.d., Preloška cesta 1, SI-3320 Velenje, Slovenia 
2
  Faculty of Mechanical Engineer, University of Maribor, Smetanova 17, SI-2000 Maribor, Slovenia 
3
 Research Center Pharmaceutical Engineering GmbH, Inffeldgasse 13, Graz, Austria  
* 
Corresponding author: matej.zadravec@um.si 
Abstract: The paper deals with numerical modelling of adsorption kinetics in a narrow channel of 
a honeycomb type adsorber. The mass transfer model consists of solving the diffusion-convection 
transport equation for adsorbate species in the fluid phase. The adsorbate flux to the wall is 
computed implicitly by using the Boundary Element Method for the solution of the transport 
equation. Scalable boundary condition for the adsorbate concentration on the adsorbent walls is 
used, defined on basis of the ratio of the accumulated adsorbate mass and the equilibrium 
adsorbate mass for a given system temperature. As the state of equilibrium is dependent also on 
the system temperature, a sensitivity study of the adsorption process under different system 
temperatures is performed. Among the studied adsorption equilibria models, including the 
Freundlich model, Simplified local density (SLD) model and the Dubinin-Radushkevich model, 
the latter is proved as the best compromise between accuracy and computational cost and applied 
in mass transfer kinetics computations. The results of computations show, that the increase in 
system temperature results in a decrease of breakthrough times, as the equilibrium amounts of 
adsorbate at the adsorbent walls decrease with increasing temperature, therefore making the 
derived numerical model suitable for further use in detailed mass transfer kinetics computations.     
Key words: subdomain boundary element method, laminar viscous fluid flow, mass transfer, 
adsorption, honeycomb adsorber. 
Povzetek: V prispevku je predstavljen numer ični model za simulacijo adsorpcijskega 
procesa v kanalu, ki ima na svojih stenah nameščeno plast adsorbenta. V kanalu teče tok 
nosilnega plina z zelo nizko koncentracijo adsorbata, ki adsorbira v plast adsorbenta. Prenos snovi 
je simuliran z reševanjem difuzivno-konvektivne prenosne enačbe ohranitve mase adsorbata, 
diskretizirane z metodo robnih elementov. Adsorpcijski proces je modeliran s prilaganjem 
vrednosti robne koncentracije, ki je odvisna od razmerja med akumulirano ter ravnotežno 
množino adsorbata v plasti adsorbenta. Ker se ravnotežno stanje spreminja s temperaturo sistema 
je v prispevku prikazan vpliv le te na ravnotežno stanje. Ravnotežna množina adsorbata v plasti 
adsorbenta je določena s Freundlich. SLD in Dubinin-Radushkevich ravnotežnim modelom, kjer 
se slednji izkaže za najboljšo izbiro glede na natančnost rezultatov in računski čas. Rezultati 
izvedenih preračunov kažejo, da se s povečevanjem temperature v sistemu zniža čas pri katerem 
pride do preboja adsorbata, ker se ravnotežna količina adsorbata na stenah adsorbenta zmanjšuje s 
povečevanjem temperature sistema. Izdelana numerična rutina omogoča natančno simuliranje 
adsorpcijskih procesov v kanalu ter napovedovanje prebojne koncentracijske krivulje adsorbata. 
Ključne besede: podobmočna metoda robnih elementov, laminarni viskozni tok tekočine, 
prenos snovi, adsorpcija, adsorber s satovjem. 

ADSORPTION EQUILIBRIUM MODELLING FOR DIFFERENT TEMPERATURE CONDITIONS AND ITS INFLUENCE... 
|  27 
1. Introduction 
Adsorption based technologies are increasingly used 
in various fields of engineering. Their use range from 
the classical applications in gas cleaning (Dreisbach et. 
al. [1])., where the selective physical or chemical 
bonding of gas species on solid matrix of adsorbent is 
the most important effect, to applications in heat 
exchange applications, for example in domestic 
appliances. In both cases, adsorption is characterised by 
heat release and consequent increase in the system 
temperature, if no cooling is applied. The increase in 
temperature has a negative effect on the state of 
equilibrium between adsorbent and adsorbate, 
effectively reducing the maximum amount of adsorbate, 
that can be bound on the outer and inner surface of 
adsorbent.   A lot of research the work was already done 
in the field of adsorption equilibrium modelling 
(Rangarajan et. al. [2], Soule et. al. [3], Fitzgerald et. al. 
[4] ). In adsorption equilibrium modelling simple 
analytical models like Henry, Langmuir or Freundlich 
isotherms are predominantly used, although more 
accurate models are being developed, like Monte Carlo 
simulations (MC) and Molecular Dynamics simulations 
(MD), as noted by Fedorov [5]. For engineering 
calculations empirical models like Dubinin-
Radushkevich model are still the preferred choices over 
more complex MC and MD simulations (Fedorov [6], 
Valdés-Solís et al. [7]). Lately, new types of special 
adsorbers have been developed, among them a 
honeycomb adsorber where adsorbent material is 
attached to the honeycomb structure as a thin film on 
the surfaces of the channels (Valdés-Solís et. al. [8]). 
The main advantage of such configuration is fast mass 
exchange rate between the bulk flow and the adsorbent 
walls, enabling  very  short time intervals for cycling of 
adsorption/desorption processes, and  a low  pressure 
drop in the device (Wajima et. al. [9]), decreasing cost 
of operation.  
In the present paper, the influence of the increased 
temperature in a channel of a honeycomb type adsorber 
on mass transfer characteristics of the device is 
investigated. A special attention is given as well as to 
local mass transfer characteristics as to transient 
behaviour in form of a breakthrough curve. The paper 
builds on previous research, reported in work Štimec et. 
al [10] and [11]. 
The paper is organized as follows. After a short 
description of the adsorbent geometry and governing 
equations, Section 2, selected thermodynamic 
equilibrium models are described and evaluated in 
Section 3.  Details of the applied numerical model for 
the solution of the mass transfer kinetics in the channel 
are given in Section 4, where also main computational 
results and their analysis is presented. The paper closes 
with conclusions.  
 
2. Problem description and governing equations 
The configuration of adsorption device under 
consideration is a honeycomb structure adsorber with 
narrow channels. Since the channels are decoupled from 
one another by the solid walls, it is possible to study the 
adsorption kinetics only inside a single channel, shown 
in Figure 1. The channel surfaces are covered with 
adsorbent material, which adsorbs the adsorbate species, 
entering the channel. Since the mixture mass flow in a 
single channel is low enough, it is possible to consider 
flow as incompressible and laminar. Also, since the 
adsorption mass flux is negligibly small in comparison 
to the mixture mass flow, it is assumed that adsorption 
process does not affect the flow field. This allows 
decoupling the fluid flow from adsorption kinetics, 
leading to lower computational cost of simulations. 
Figure 1. Configuration of a narrow channel of a 
honeycomb adsorber.  
In our case, the flow field is considered as fully 
developed and the analytical solution of Chen [12] is 
used, as already reported in [10]:                            
(1)          
where      is dynamic viscosity,     channel width,      
channel height,         average inflow velocity,        and    
coordinates in the channel.  
The adsorption is accompanied by the heat release, 
which causes the change in the temperature in the 
channel.  Since the maximum amount of adsorbate, that 
can be adsorbed to the solid surface, changes with 
temperature, it is important to consider this in the 

ANALI PAZU, 5/ 2016/ 1-2, str. 26-32                                                                                            Teodor ŠTIMEC et al. 
28  |  
numerical model.  It is therefore important to include in 
the numerical model an equilibrium model for 
adsorption, that involves also the temperature as a 
system unknown. The data on thermodynamic 
equilibrium is used in two ways, first in setting the 
upper limit of the amount of adsorbed species, 
computed as a sum of time varying species mass fluxes 
to the local wall area, 
 
 
where species mass flux is defined as 
 
 
where     is the diffusion coefficient of adsorbate in the 
carrying fluid and    is the outer surface area of the 
adsorbent layer.  
Second, the thermodynamic equilibrium data is used 
in setting up a scalable wall boundary condition, based 
on the ratio between the accumulated adsorbent mass 
and the equilibrium amount of adsorbent mass, 
  
which is then used in setting the wall boundary 
condition  
 
with C
i
 the concentration in the vicinity of the wall. 
This boundary condition is applied in the numerical 
solution of the diffusion-convection transport equation 
for adsorbate species, 
 
 
Due to very low species concentrations in the 
mixture, less than 1% of total mass in the flow, the 
mixture properties are assumed constant. In the above 
model for species transport buoyancy forces, thermal 
diffusion and inter diffusion effects are neglected.     
3. Adsorption equilibrium models 
Adsorption process in the channel stops when 
thermodynamic equilibrium is reached, and the 
numerical model must reflect this fact in order to be 
able to realistically describe mass transfer conditions 
inside the channel. Additionally, since the temperature 
of the system also increases as the adsorption 
progresses, a thermodynamic model must include also 
the system temperature as a state parameter. There are 
several thermodynamic equilibrium models, developed 
for the use in modelling of adsorption equilibrium. As 
stated in introduction, for engineering calculations and 
also computational mass transfer models empirical 
thermodynamic equilibrium models are still the 
preferred choices over more complex Monte Carlo and 
Molecular Dynamics simulations. As numerical 
simulations based on solving the mass transport 
equations in the fluid phase typically require dense 
computational grids, especially in the vicinity of solid 
walls, and hence long computational times, equilibrium 
model used should also offer low computational cost in 
order to reach feasible computational times in 
computing time dependent problems, like numerical 
determination of the breakthrough curve.    
In the present work, three adsorption equilibrium 
models for the case of adsorption of buthane from the 
air mixture on activated carbon are considered. First, the 
accuracy study of the models for a predefined and 
experimentally measured thermodynamic conditions is 
performed, followed by the application of the most 
favourable model in computation of mass transfer 
characteristics for the cases with different system 
temperature. 
The first considered model was the adsorption 
equilibrium model of Dubinin-Radushkevich [13], 
giving equilibrium amount of adsorbate per adsorbent 
volume as 
 
 
In the case of air-buthane activated carbon system, 
the                            is the maximum amount of 
adsorbate adsorbed at the saturation pressure,     is the 
gas constant,      is system temperature, 
is the vapour pressure     of      adsorbate    at      ambient    
conditions,         is concentration of adsorbate and 
affinity coefficient multiplied with energy parameter 
is                                   . The second equilibrium model 
was the three parameter Freundlich model: 
 
 
 
with the   and   and n are empirically determined 
coefficients with values       ,        ,                          
and               . The third model used was the Simplified 
local density (SLD) model, which enables determination 
of surface excess of adsorbate species on the solid 
active surface of adsorbent 
 
 
where Γ
ex
 is surface excess, ρ is adsorbate partial 
density at the   distance z    from the    active surface   of  
 (2) 
(3) 
(4) 
(5) 
(6) 
 
(9) 
(7) 
(8) 

ADSORPTION EQUILIBRIUM MODELLING FOR DIFFERENT TEMPERATURE CONDITIONS AND ITS INFLUENCE... 
|  29 
adsorbent and in the bulk of the fluid mixture The 
theoretical structure of the model with computational 
details is presented in [11]. With computed surface 
excess the equilibrium amount of adsorbate is computed 
as 
 
 
with             the adsorbent mass density,        the value 
of active surface relative to the adsorbent mass and M 
adsorbate molar mass. In the Table 1, parameters, used 
in the SLD model with Peng-Robinson equation of 
state, are listed. 
Table 1. Parameters used in SLD calculations.   
In order to assess the physical correctness of the 
described equilibrium models the adsorption isotherm, 
reported in Valdés-Solís et. al. [7], valid for system 
temperature                           was evaluated with all 
three equilibrium models. From the results of 
equilibrium curves,  shown in Fig. 2., it can be 
concluded that the SLD model and Dubinin-
Radushkevitch model give nearly identical results in the 
whole range of adsorbate concentrations, whereas the 
Freundlich model overshoots in the low concentration 
range. Due to the latter fact the Freundlich model was 
not selected for further investigation. The same 
conclusion was made  for the SLD model, which gives 
excellent results in terms of accuracy, but at a very high 
computational cost in comparison to Dubinin-
Radushkevitch model. The latter fact originates in 
numerical integration procedure and numerical solution 
of a nonlinear equation by the Newton's method in 
computing the adsorbate density profile in the vicinity 
of the active surface. 
 
 
 
 
 
4. Computational results 
The adsorption case, studied in Valdés-Solís et. al. 
[7], was selected as the test case for the study of the 
influence of temperature on adsorption characteristics. 
Adsorption equilibrium for different system 
temperatures was modeled with Dubinin – 
Radushkevich equation. A range of temperatures from 
273.15K to 333.15K was selected. The computed 
Dubinin – Radushkevich equilibrium isotherms are 
shown on Figure 3. One of the important facts is that the 
geometrical shape of the curves is more or less 
comparable throughout the computed concentration 
range, a fact that will also help explain the mass transfer 
conditions in the moving concentration front along the 
wall. 
In Table 2, values of equilibrium amounts of 
adsorbate per volume of adsorbent are given, computed 
with  Dubinin-Radushkevich model for the case of 
adsorbate concentration                              . Clearly, 
with increasing the temperature the equilibrium amount 
of adsorbate is decreased, leading to the need for shorter 
operation periods of an adsorber.   
(10) 
Parameter Value Units 
 
2.725·10
-10
 m 
 
2.8·10
-10
 m 
M 0.05812 kg/mol 
 
647.1 K 
 
2.206·10
7
 Pa 
w 0.34486   
Figure 3. Adsorption equilibrium for different system 
temperatures, Dubinin-Radushkevich model. 
Figure 2. Adsorption isotherms for the case of : FR - 
Freundlich model , R-D Dubinin-Radushkevitchevim 
model and SLD – SLD Peng-Robinson model. 

ANALI PAZU, 5/ 2016/ 1-2, str. 26-32                                                                                            Teodor ŠTIMEC et al. 
30  |  
Table 2. Specific equilibrium amounts of adsorbate           
for adsorbate concentration C=0,007358 kg/m
3  
at different 
system temperatures. 
 
Next, mass transfer kinetics inside the narrow 
channel of the honeycomb adsorber was studied. The air 
mixture considered had constant physical properties 
density                                      and kinematic viscosity      
                                   .   In the channel, the species 
of buthane are transported within the air mixture by 
means of convection and diffusion, and are adsorbed or 
desorbed on the wall. Due to lack of accurate data the 
species diffusivity                                          was 
selected for T=293.15K and was considered as constant, 
leading      to the          Schmidt              number     value                          
                                  and the Reynolds number value    
 
was set to 2.92. At the inflow the constant species 
concentration                                           was prescribed. 
On the outflow of the channel the convective boundary 
condition was set, allowing the unhindered outflow of 
the concentration fronts. Adsorbate concentration on the 
channel wall        was set to 0 only in the first time step 
of the simulations. 
Computational scheme based on the subdomain 
Boundary Element Method [10] was used for the 
numerical simulation of the adsorption process in a 
honeycomb adsorber. A diffusion-convection transport 
equation (6) for species transport was solved for the 
mass transfer in the fluid with implementation of 
dynamic boundary condition for adsorbent 
concentration at the wall (5). Adsorption equilibrium 
was calculated with the use of the Dubinin-
Radushkevich equilibrium model. Initial validation of 
the numerical scheme was done with the information 
provided in the work of Valdés-Solís et. al. [7]. Based 
on this the computational grid with 8x8x378 elements 
and 546215 nodes and the time step            were 
selected.  
Figure 4. Computational domain with highlighted 
centerline (1) along the channel, which was used for 
analysis of mass transfer characteristics, and outlet plane 
(2), where concentration kinetics                was monitored.   
In Fig. 5 the computed breakthrough curve for the 
case of T=293.15K and its comparison with results of 
Valdés-Solís et. al. [7] is shown. A very good 
agreement can be observed, although there are some 
differences in the velocity of the concentration front, 
which can be attributed to the use of two different 
diffusion coefficients of adsorbate species for radial and 
axial direction and also to the art of experimental 
evaluation of concentration values in the multichannel 
configuration, both reported in [7].  
In Figure 6, the computed breakthrough curves for 
different values of system temperature are given. The 
effect of increasing the temperature is shortening the 
time of the concentration front breakthrough, meaning 
that the adsorber has a decreased operational time 
period, which is a consequence of the decrease in the 
equilibrium amount of adsorbate (see Table 2). 
Figure 6. Average concentration           at the outlet of the 
channel for different  system temperatures.  
The adsorbate mass transfer characteristics were 
studied based on the evaluation of local values of the 
Sherwood number, defined as  
 
 
with    the local concentration,      the inlet 
concentration,        the concentration gradient in 
direction normal to the wall, computed implicitly by the 
BEM based computational algorithm [10], and   the 
characteristic length, equal to the channel width.   
T 
 
273.15 K 322.34 kg/m
3
 
293.15 K 300.71 kg/m
3
 
313.15 K 279.16 kg/m
3
 
333.15 K 257.89 kg/m
3
 
(11) 
Figure 5. Simulation results for               the 
breakthrough curve for T=293,15K:  o Valdes-Solis, - - - 
present model. 

ADSORPTION EQUILIBRIUM MODELLING FOR DIFFERENT TEMPERATURE CONDITIONS AND ITS INFLUENCE... 
|  31 
In terms of mass transfer nondimensional variable, 
the Sherwood number, the same effect as in 
breakthrough curves can be observed. In Figure 7 local 
values of the Sherwood number along the line (1) at the 
time instant of t=460s are shown.  
Figure 7. Local values of the Sherwood number along 
line (1)  at the                      . 
Each curve with non-zero point values denotes the 
area along the line (1), where mass transfer (adsorption) 
is taking place.  Along each Sh curve, the area to the left 
of the non-zero values corresponds to the saturated 
conditions, where adsorption had already reached 
equilibrium, and the area to the right corresponds to the 
unsaturated conditions along the wall, where no contact 
with incoming mixture flow has yet been established, 
see also Figure 8 for the distribution of concentration 
values and Figure 9 for distribution coefficient k(t) 
values.   
Figure 8. Local values of  concentration        along the 
line (1) at                       . 
The position of the Sh curve (Figure 7) for the 
highest temperature, the k(t)=1 values (Figure 9) as well 
as                   values have advanced furthest along the 
direction of the flow (to the right), proving, that the 
process of adsorption is decreasing in strength as the 
temperature of the system increases. The similar 
magnitude of the curve peaks and strong similarity in 
the shapes (Figure 7) is the result of the adsorption 
isotherm data at different temperature levels, which also 
show a very high degree of similarity, as depicted in 
Figure 3. 
 
 
 
 
5. Conclusions 
In the presented work the effect of different system 
temperatures on the adsorption characteristics in a 
narrow channel of a honeycomb type adsorber has been 
studied.  The applied numerical method, based on the 
Boundary Element Method computational algorithm. 
Three different adsorption equilibrium models for the 
use in the numerical model were tested, including the 
Freundlich model, Dubinin-Radushkevich model and 
Simplified Local Density model. As the Dubinin-
Radushkevich model proved to offer accurate results 
with low computational cost, it was selected as the 
model for study of mass transfer kinetics at different 
system temperatures. The computed results indicate that 
the effect of increasing the temperature is shortening the 
time of the concentration front breakthrough, meaning 
that the adsorber has a decreased operational time 
period, which is a consequence of the decrease in the 
equilibrium amount of adsorbate, that can be adsorbed 
at the walls of the channel. This is an expected result, 
which proves, that the developed physical and 
numerical model is capable of solving complex time-
dependent analysis of mass transfer kinetics at varying 
system temperatures. This also indicates, that the model 
can be extended to simulation of non-isothermal cases, 
where release of heat during adsorption locally 
increases temperature, which negatively affects the 
thermodynamic equilibrium at the wall. The latter 
consideration will be considered in the scope of the 
future work. 
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Figure 9. Local values of distribution coefficient     
along the line (1) at                     . 

ANALI PAZU, 5/ 2016/ 1-2, str. 26-32                                                                                            Teodor ŠTIMEC et al. 
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