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Elsevier Editorial System(tm) for International Journal of Heat and Mass Transfer Manuscript Draft Manuscript Number: HMT-D-11-00350R1 Title: A TWO-ENERGY EQUATION MODEL FOR DYNAMIC HEAT AND MASS TRANSFER IN AN ADSORBENT BED USING SILICA GEL/WATER PAIR Article Type: Full Length Article Keywords: Adsorption; Cooling; Silica/gel; LDF; LTNE Corresponding Author: Dr. ismail solmuş, Ph.D. Corresponding Author's Institution: Middle East Technical University First Author: ismail solmuş, Ph.D. Order of Authors: ismail solmuş, Ph.D.; D. Andrew S. Rees; Cemil Yamalı; Derek Baker Abstract: In this study, the influence of the adsorbent bed dimensions, convective heat transfer coefficient between the cooling fluid and adsorbent bed and the thermal conductivity of the solid adsorbent material on the transient distributions of the solid and gas phase temperature difference, differences in the amount adsorbed predicted by the equilibrium and linear driving force (LDF) models, solid phase temperature, gas pressure and amount adsorbed inside the adsorbent bed of a solid sorption cooling system have been investigated numerically for a nearly isobaric adsorption process. Silica gel/water is selected as the working pair. A transient two-dimensional local thermal non-equilibrium model has been developed that takes into account both internal and external mass transfer resistances. The local volume averaging method has been used to derive the macro-scale governing conservation equations from the micro-scale equations. It has been found that generally, the effects of the parameters investigated on the transient distributions of the temperature difference between the phases, difference in amount adsorbed between the equilibrium and LDF models, and gas phase pressure gradients are negligible small. The thickness of the adsorbent bed for the given adsorbent bed length and thermal conductivity of the solid adsorbent material have a large influence on the transient distributions of the solid phase temperature and amount adsorbed. On the other hand, the transient temperature and amount adsorbed distributions are only slightly affected by the variation of the adsorbent bed length and convective heat transfer for the conditions studied.
Transcript of Elsevier Editorial System(tm) for International Journal of...
Elsevier Editorial System(tm) for International Journal of Heat and Mass Transfer Manuscript Draft Manuscript Number: HMT-D-11-00350R1 Title: A TWO-ENERGY EQUATION MODEL FOR DYNAMIC HEAT AND MASS TRANSFER IN AN ADSORBENT BED USING SILICA GEL/WATER PAIR Article Type: Full Length Article Keywords: Adsorption; Cooling; Silica/gel; LDF; LTNE Corresponding Author: Dr. ismail solmuş, Ph.D. Corresponding Author's Institution: Middle East Technical University First Author: ismail solmuş, Ph.D. Order of Authors: ismail solmuş, Ph.D.; D. Andrew S. Rees; Cemil Yamalı; Derek Baker Abstract: In this study, the influence of the adsorbent bed dimensions, convective heat transfer coefficient between the cooling fluid and adsorbent bed and the thermal conductivity of the solid adsorbent material on the transient distributions of the solid and gas phase temperature difference, differences in the amount adsorbed predicted by the equilibrium and linear driving force (LDF) models, solid phase temperature, gas pressure and amount adsorbed inside the adsorbent bed of a solid sorption cooling system have been investigated numerically for a nearly isobaric adsorption process. Silica gel/water is selected as the working pair. A transient two-dimensional local thermal non-equilibrium model has been developed that takes into account both internal and external mass transfer resistances. The local volume averaging method has been used to derive the macro-scale governing conservation equations from the micro-scale equations. It has been found that generally, the effects of the parameters investigated on the transient distributions of the temperature difference between the phases, difference in amount adsorbed between the equilibrium and LDF models, and gas phase pressure gradients are negligible small. The thickness of the adsorbent bed for the given adsorbent bed length and thermal conductivity of the solid adsorbent material have a large influence on the transient distributions of the solid phase temperature and amount adsorbed. On the other hand, the transient temperature and amount adsorbed distributions are only slightly affected by the variation of the adsorbent bed length and convective heat transfer for the conditions studied.
DEPARTMENT OF MECHANICAL ENGINEERING MIDDLE EAST TECHNICAL UNIVERSITY ANKARA, TURKEY
December 20, 2011
Dear International Journal of Heat and Mass Transfer Editors,
Please find the submission “A TWO-ENERGY EQUATION MODEL FOR DYNAMIC HEAT AND MASS
TRANSFER IN AN ADSORBENT BED USING SILICA GEL/WATER PAIR” uploaded to website for your
journal. I confirm that this submission is original, has not been published elsewhere, and is not
currently under review elsewhere and all authors have agreed to the manuscript submission.
Please let me know if you need any more information to complete the submission process.
Sincerely,
İsmail Solmuş
Cover Letter
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
A TWO-ENERGY EQUATION MODEL FOR DYNAMIC HEAT AND MASS TRANSFER IN
AN ADSORBENT BED USING SILICA GEL/WATER PAIR
İsmail Solmuşa, c, *
, D. Andrew S. Reesb, Cemil Yamalı
a, Derek Baker
a
aDepartment of Mechanical Engineering, Middle East Technical University, 06531 Ankara Turkey
bDepartment of Mechanical Engineering, University of Bath, Claverton Down, Bath BA2 7AY, U.K.
*Corresponding author: Tel.: +90 3122105276; Fax: +90 3122102536
E-mail address: [email protected]; [email protected]
cOn leave of absence from Department of Mechanical Engineering, Atatürk University, 25240
Erzurum Turkey
Title page with author details
Dear Prof. W. J. Minkowycz
Editor-in-Chief
International Journal of Heat and Mass Transfer
We are pleased to submit the revised paper: A TWO-ENERGY EQUATION MODEL FOR
DYNAMIC HEAT AND MASS TRANSFER IN AN ADSORBENT BED USING SILICA
GEL/WATER PAIR
.
Also, we would like to express our sincere thanks to the reviewers for their useful comments. Please
find our answers below to the reviewer queries.
We remain at your disposal for any other request.
Dr.İsmail Solmuş
[email protected] or [email protected]
*Detailed Response to Reviewers
Response to the reviewer # 1
1-x is defined as amount adsorbed. However, x is the amount of the adsorbed water per unit mass of
dry silica gel. Thus, it is concentration. It might be better to called x as adsorbate concentration, or
water concentration in silica gel.
Response to Comment 1: Amount adsorbed is replaced by the adsorbate concentration in the
manuscript.
2- The concept of external mass transfer resistance and non-uniform pressure approach has been
confused. We may have negligible external mass transfer resistance but a pressure gradient might exist
in the bed. The pressure gradient in the bed does not occur due to the external mass transfer resistance
around the adsorbent particle, it occurs due to different adsorption rate in different location of
adsorbent bed.
Response to Comment 2: First of all, we would like to make the definition of the external mass
transfer resistances more clear. The definition of the external mass transfer resistance is the resistance
to water vapor flow through the voids between the adsorbent particles (not around the adsorbent
particle)
The pressure drop inside the adsorbent bed during the adsorption process (pressure gradients) can be
eliminated for the bed by having a high permeability. When a pressure drop occurs in a region due to
the adsorption, gas flow from high pressure region to low pressure region will be fast for highly
permeable beds (having a negligible external mass transfer resistance) and as a result of this, the bed
suddenly reaches the nearly uniform pressure distribution, even if the rates of adsorption vary
spatially.
Yes you are right that pressure gradients inside the adsorbent bed might be in existence even if
external mass transfer resistances are negligibly small. However, these gradients are comparatively
small for a negligible external mass transfer resistance and thus uniform pressure assumption
throughout the adsorbent bed can be assumed without leading to any significant computational errors.
On the other hand, external mass transfer resistances are related to bed permeability and the bed
permeability is a function of the total porosity of the adsorbent bed and the particle diameter.
Therefore, the external mass transfer resistances can be negligible for a bed having a high permeability
and in this case, uniform pressure assumption can be meaningful inside the bed.
The bed permeability can be considered like a thermal conductivity. For example, a wall having a
uniform heat generation is cooled from the both sides by flowing air. If the wall material has a high
thermal conductivity and the convective heat transfer coefficient is low at the outer surfaces it most
likely to have negligible thermal gradients inside the wall. Therefore, external mass transfer resistance
can be neglected for high permeability beds and uniform pressure can be assumed.
3-The process should be shown on an isosteric chart of the considered silica gel-water pair. The
starting and ending conditions of the process should be defined on the chart.
Response to Comment 3: The manuscript has been revised according to this comment as follow;
A schematic view of the isobaric adsorption process on a Clapeyron diagram is shown in Fig. 2. At
the starting point of the adsorption process for all simulations (A), the adsorbate concentration, the
solid and gas phase temperatures, and the pressure throughout the adsorbent bed are uniform and are
equal to 0.2 kgwkgad-1
, 348.5 K and 1.2282 kPa, respectively. At this point, the temperatures for the gas
and solid phases were calculated by means of the regeneration temperature of the adsorbent bed (Tre),
condenser pressure (Pco), and evaporator pressure (Pe) given in Table 2. The adsorption capacity of
the solid adsorbent was assumed to be constant as the pressure inside the adsorbent bed was decreased
from the condenser (Pco) to the evaporator (Pe) pressure. The adsorption process was terminated at
point B where the adsorbent bed nearly reached thermal, mechanical and chemical equilibrium
conditions and the values of the solid phase temperature, pressure and the adsorbate concentration
throughout the adsorbent bed at this point (B) are presented in the Figures 5, 6 and 7 (3th column) for
various adsorbent bed lengths, respectively.
Fig.2 A schematic view of the adsorption process on a Clapeyron diagram.
Tc Th
Pco
B
-1/Ts
lnP
Desorption
XminXmax
PeAAdsorption
Tre
4-Parameters in Table 2 were not written.
Response to Comment 4: I confirmed that these parameters are in Table.2 in the original word file,
but they must not have been printed when the table was converted to PDF.
5- Reynolds number has been defined in the text, but no information has been given on the water
vapor velocity for determination of Reynolds number. What is the value of vapor velocity?
Furthermore, is the constant heat transfer coefficient used or local heat transfer coefficient is used?
Response to Comment 5: Water vapor velocity is ranging from 0.017 m/s to 0 (impermeable walls)
when the process time is nearly equal to 300 sec and it decreases as the time goes on. In the model,
local heat transfer coefficient is used.
6-The concept of interparticle mass transfer resistance and external mass transfer resistance is
confused. We have interparticle and intraparticle mass transfer resistance concepts. The interparticle
mass transfer resistance should not be confused with external mass transfer resistance.
Response to Comment 6: Yes we are agree with you there are two types of mass transfer resistances,
interparticle and intraparticle. However, interparticle and intraparticle mass transfer resistances are
also called as external and internal mass transfer resistances respectively, in the literature.
The following definitions are already in existence in the manuscript
The overall performance of thermal powered adsorption cooling (TPAC) systems is typically
limited by heat and mass transfer limitations inside the adsorbent bed due to the poor thermal
conductivity of the solid adsorbent, and internal (intraparticle) and external (interparticle)
mass transfer resistances.
The adsorbate gas flow from a solid adsorbent particle surface to inner points of the particle
and through the voids between the solid adsorbent particles are generally referred to as internal
and external mass transfer, respectively.
7- It will be more proper to use the term of "Instantaneous Equilibrium Model", instead of
"Equilibrium Model".
Response to Comment 7: Instantaneous Equilibrium Model is used instead of Equilibrium Model in
the manuscript.
8- Particle size and porosity are the most important parameter in design of adsorbent bed; however
they are not analyzed in this study.
Response to Comment 8: We are agree with you that the particle size and porosity are the most
important parameters in design of the adsorbent bed. However, the adsorbent particle diameter, which
is related to adsorbent density, and total porosity are mutually dependent variables and hence, one
cannot be held constant while the other is varied. This is the reason why we didn’t analyze the effect
of these parameters on the transient heat and mass transfer distribution inside the adsorbent bed.
9-Number of figures is too much. They should be reduced by somehow.
Response to Comment 9: we agree that there are a large number of figures but we also believe that
the conclusions drawn in this study can be strongly supported by keeping all the figures presented in
the manuscript. Therefore, we don’t want to lose any information and lead to misunderstandings.
However, you still insist that number of the figures needs to be reduced we can.
10-Some results were generalized. There are many parameters for the considered problem, hence the
generalization of results may not be correct.
Response to Comment 10: Conclusions part of the manuscript has been revised as follow;
Based on the numerical model developed and for the conditions explored, the conclusions about the
current study can be summarized as follows:
11- Thermal conductivity of silica gel was changed and its effect was analyzed. Is it possible to change
the thermal conductivity of silica gel? The change of silica gel thermal conductivity means that we
have new type silica gel since specific heat, isotherm, diffusivity and other properties will also be
changed.
Response to Comment 11: The thermal conductivity of the solid adsorbent material (but not effective
thermal conductivity) was varied by considering the adsorbent material to have various thermal
conductivity values. Although, this approach is not realistic, it shows that response of the adsorbent
bed to the varying thermal conductivity of the adsorbent material clarify the effect of the thermal
conductivity of the adsorbent material on the heat and mass transfer distributions.
Thermal conductivity of the bed can also be changed by using fins, conductive filler material, etc.
A TWO-ENERGY EQUATION MODEL FOR DYNAMIC HEAT AND MASS TRANSFER IN
AN ADSORBENT BED USING SILICA GEL/WATER PAIR
İsmail Solmuşa, c, *
, D. Andrew S. Reesb, Cemil Yamalı
a, Derek Baker
a
aDepartment of Mechanical Engineering, Middle East Technical University, 06531 Ankara Turkey
bDepartment of Mechanical Engineering, University of Bath, Claverton Down, Bath BA2 7AY, U.K.
*Corresponding author: Tel.: +90 3122105276; Fax: +90 3122102536
E-mail address: [email protected]
cOn leave of absence from Department of Mechanical Engineering, Atatürk University, 25240 Erzurum
Turkey
ABSTRACT
In this study, the influence of the adsorbent bed dimensions, convective heat transfer coefficient between
the cooling fluid and adsorbent bed and the thermal conductivity of the solid adsorbent material on the
transient distributions of the solid and gas phase temperature difference, differences in the adsorbate
concentration predicted by the instantaneous equilibrium and linear driving force (LDF) models, solid
phase temperature, gas pressure and adsorbate concentration inside the adsorbent bed of a solid sorption
cooling system have been investigated numerically for a nearly isobaric adsorption process. Silica
gel/water is selected as the working pair. A transient two-dimensional local thermal non-equilibrium
model has been developed that takes into account both internal and external mass transfer resistances. The
local volume averaging method has been used to derive the macro-scale governing conservation equations
from the micro-scale equations. It has been found that generally, the effects of the parameters investigated
on the transient distributions of the temperature difference between the phases, difference in adsorbate
concentration between the instantaneous equilibrium and LDF models, and gas phase pressure gradients
are negligible small. The thickness of the adsorbent bed for the given adsorbent bed length and thermal
conductivity of the solid adsorbent material have a large influence on the transient distributions of the
solid phase temperature and adsorbate concentration. On the other hand, the transient temperature and
*Manuscript
2
adsorbate concentration distributions are only slightly affected by the variation of the adsorbent bed
length and convective heat transfer for the conditions studied.
Keywords: Adsorption; Cooling; Silica/gel; LDF, LTNE
Nomenclature
area of gas-solid interface per unit volume, m-1
specific heat, J kg-1
K-1
equivalent diffusivity in the adsorbent particles, m2 s
-1
reference diffusivity, m2 s
-1
average diameter of the adsorbent particle, m
activation energy of surface diffusion, J mol-1
convective heat transfer coefficient between the adsorbent bed and cooling fluid, W m-2
K-1
interfacial convective heat transfer coefficient, W m-2
K-1
permeability, m2
mass transfer coefficient within the adsorbent particles, s-1
thermal conductivity, W m-1
K-1
length of the adsorbent bed, m
Nusselt number
pressure, kPa
Prandtl number
heat of adsorption, J kgw-1
universal gas constant, J mol-1
K-1
Reynolds number
specific gas constant for water vapor, J kg-1
K-1
inner diameter of the adsorbent bed, m
outer diameter of the adsorbent bed, m
radial coordinate, m
temperature, K
time, s
gas phase velocity in radial direction, m s-1
gas phase velocity in axial direction, m s-1
adsorbate concentration, kgwkgad-1
equilibrium adsorption capacity, kgwkgad-1
axial coordinate, m
3
Greek Symbols
viscosity, Ns m-2
density, kg m-3
total porosity
bed porosity
particle porosity
effective thermal conductivity for the gas phase, W m-1
K-1
effective thermal conductivity for the solid phase, W m-1
K-1
Subscripts
cooling
condenser
evaporator
gas phase
regeneration
initial
solid phase
saturation
4
1. INTRODUCTION
Traditional air-conditioning technologies can have numerous adverse impacts due to both the energy they
consume and the refrigerants used including the following: contributing to environmental problems such
as climate change, ozone depletion, and pollution [1-6]; contributing to infrastructure problems by
stressing the electric generating, transmission and distribution infrastructure by increasing the peak
demand for electricity [2]; contributing to economic problems through the cost of supplying this energy
[2, 3]; contributing to reducing energy security problems if energy must be imported to meet this demand
[5]. Thermally driven cooling systems powered using solar energy or waste heat such as absorption,
adsorption, and desiccant systems have the potential to reduce or eliminate many of these problems [4, 6].
The present work is focused on adsorption cooling systems. Dieng and Wang provide an excellent
overview of this technology [7]. The heart of this technology is a thermal compression process using an
adsorbent bed that replaces the mechanical compression process in a vapor-compression cycle. This
adsorbent bed is alternately cooled and heated, which in turn alternately causes the bed to adsorb
refrigerant at a low pressure and desorb refrigerant at high pressure, thus producing a thermally powered
compression process.
The overall performance of thermal powered adsorption cooling (TPAC) systems is typically limited by
heat and mass transfer limitations inside the adsorbent bed due to the poor thermal conductivity of the
solid adsorbent, and internal (intraparticle) and external (interparticle) mass transfer resistances.
Therefore, many attempts have been made to improve the heat and mass transfer characteristics of the
adsorbent beds for a TPAC system. However, reducing heat transfer resistances inside a bed tends to
increase the mass transfer resistances and vice versa. Therefore, heat and mass transfer conditions inside
the adsorbent bed need to be understood well to design a high performance adsorbent bed. Over the past
few decades, various mathematical models have been developed to understand the heat or heat and mass
transfer mechanism inside the adsorbent bed of TPAC systems. These models are summarized in Table.1
in terms of their important characteristics and their most important characteristics in terms of the present
research are as follows. The equations were proposed for the heat transfer within a porous medium
typically assuming a mobile gas (vapour) phase, an immobile solid phase (adsorbed adsorbate +
5
adsorbent ), and Local Thermal Equilibrium (LTE) between the gas and solid phases. However, LTE
assumption is no longer valid in some circumstances [8] and thus, an energy equation for each phase
needs to be developed, i.e. Local Thermal Non- Equilibrium (LTNE). The adsorbate gas flow from a solid
adsorbent particle surface to inner points of the particle and through the voids between the solid adsorbent
particles are generally referred to as internal and external mass transfer, respectively. The resistance to
internal mass transfer is typically predicted using the Linear Driving Force (LDF) or Solid Diffusion (SD)
models. Occasionally, resistance to this type of flow is neglected and adsorption equilibrium is assumed.
This assumption is reasonably good for adsorbent particles with small diameters. Darcy’s or Ergun’s
equation is used widely to account for the resistance to external mass transfer. On the other hand, in these
models external mass transfer resistances are often ignored and a uniform pressure is assumed, especially
if the permeability of the bed or/and working pressure of the adsorbate is high.
Table 1
These previous studies show that even though the performance of TPAC units have been widely studied
for different design parameters and operating conditions [9-24], little attention has been focused on
investigating the transient heat and mass transfer behavior of the adsorbent bed of the TPAC units [25-
29].
Mhimid [25] studied the heat and mass transfer in a zeolite bed during water desorption using the LTE
and LTNE models and the results showed that the LTE assumption is not valid in regions with high rates
of heat transfer (at the wall where external heating occurs, in the region where the vaporization takes
place). Jemni and Nasrallah [26] investigated transient heat and mass transfer in a metal-hydrogen
reactor and they concluded that the LTE model is not valid in the whole reactor. Guilleminot and Meunier
[27] investigated numerically and experimentally heat and mass transfer in a non-isothermal fixed bed
solid adsorbent reactor and they concluded that the uniform pressure model is more realistic than the
uniform temperature models proposed previously. Yong and Sumathy [28] compared heat transfer only
and combined heat and mass transfer models for transport processes in an adsorbent bed and they
proposed two general criteria to perform an order of magnitude analysis to determine when the simpler
6
heat transfer only model is appropriate. Demir et.al. [29] performed a numerical study to investigate the
effects of porosity on heat and mass transfer in a granular adsorbent bed and they found that the
distributions of temperature and adsorbate concentration are strongly influenced by the bed porosity.
The objective of the present work is to perform a numerical study to investigate the dynamic behavior of
the adsorbent bed of an adsorption cooling unit during the adsorption process. A transient two-
dimensional LTNE model that accounts for both internal and external mass transfer resistances has been
developed using the local volume averaging method. Most of the models proposed previously assume
LTE between the vapour and solid phases without fully justifying the validity of this assumption.
Therefore, one of the innovative points of this study is to compare the predictions resulting from the
LTNE and LTE models to better understand when LTE can be assumed.
2. DESCRIPTION OF THE ADSORBENT BED
A schematic view of the adsorbent bed using the silica-gel/water working pair is shown in Fig. 1. All the
dimensions of the adsorbent bed and the thermo-physical properties of the silica-gel/water pair are
presented in Table.2. The primary components of the adsorbent bed are an inner vacuum tube, a mass
transfer tube, a larger tubular shell and a single top cover for the vacuum tube and the tubular shell.
Silica-gel granules are packed in the annulus between the vacuum and mass transfer tubes and a vapor
gap is left at the top of the vacuum tube to allow better vapor transfer in the axial direction. The vacuum
tube is inserted into the larger tubular shell and a heat transfer fluid circulated between the shell and
vacuum tube to cool down the adsorbent bed during the adsorption process. Refrigerant vapor enters the
vacuum tube through the top cover and flows both in the radial and axial directions. The right hand side
from the center line of the adsorbent bed represents the plots in the results.
Figure 1
3. MATHEMATICAL MODELING
7
The 2D domain modeled is labeled as Modeling Domain in Fig. 1. The right and bottom edges are in
thermal contact with the heat transfer fluid and are referred to as the heat transfer boundaries. The top and
left edges are assumed to be well-insulated and are modeled as being adiabatic.
The adsorbent bed consisting of vapour adsorbate, adsorbed adsorbate, and solid adsorbent is a
discontinuous medium and thus, it is not suitable for theoretical modeling. For this reason, the local
volume averaging method has been used to model this discontinuous medium as a nearly continuous one.
The details of this method can be found in references [8, 30-32]. The governing macroscopic conservation
equations are derived by taking the average of the microscopic ones over the averaging volume.
In this study, the combination of the adsorbed adsorbate and solid adsorbent are modeled as a single solid
and are referred to collectively as the solid phase. The adsorbed adsorbate is considered as being
immobile and in thermal equilibrium with the solid adsorbent, and its volume fraction is assumed
negligible. The resulting model is therefore two-phase (vapour and solid) with single phase flow (vapour).
The mathematical model proposed in this study is primarily based on the following assumptions and
simplifications:
• the size of the adsorbent particles and the bed porosity are spatially uniform;
• the adsorbate’s vapor phase is assumed to be an ideal gas;
• radiative heat transfer, viscous dissipation and the work done by pressure changes are neglected;
• the surface porosity is considered to be equal to the total porosity;
• physical properties such as thermal conductivities, specific heat capacities and viscosity are not a
function of temperature;
• the thickness of the outer walls for the vacuum tube are assumed to be very thin and hence, its thermal
resistance is neglected.
3.1 Mass conservation equation
8
The macro-scale mass conservation equation for the adsorbate gas is written as:
(1)
The dispersive transport term was neglected since it is generally very small compared with the convective
term and this is based on the order of magnitude analysis given by [30]. The local volume-averaged
velocity is assumed to be equal to the superficial or Darcy’s velocity. The density of the vapor adsorbate
was considered to be uniform within the averaging volume [8] and hence, the intrinsic phase average
density was replaced with the point density while the equation above was derived from its micro scale
form.
The volume fraction of the gas phase, εg , is assumed to be equal to the total porosity, εt, and is evaluated
using [9],
(2)
Finite internal mass transfer rates are modeled using the LDF model [9]. The adsorption rate is assumed
equal to this internal mass transfer and therefore adsorption equilibrium is not assumed.
(3)
Here km is the internal mass transfer coefficient given by
(4)
and De is the equivalent diffusivity in the adsorbent particles which is expressed as
9
(5)
The equilibrium adsorption capacity of the adsorbent to the adsorbent’s temperature and the adsorbate’s
pressure, i.e., X = f (P, Ts), may be evaluated using the following modified Dubinin-Astakhov (D-A)
equation [33].
(6)
3.2 Momentum equation
External mass transfer resistances are included in the present model, which can lead to significant bulk
pressure gradients. Darcy’s equation including only viscous effects is used to describe the velocity field
of the vapor adsorbate in the computational domain. The inertial effects accounted for by Ergun’s
equation are ignored because of the very low speed of the vapor adsorbate flowing through the voids
between the adsorbent particles.
(7)
Here K is the permeability which can be calculated by the following semi-emprical Blake-Kozeny
equation,
(8)
3.3 Energy conservation equations
10
During the energy transport inside the adsorbent bed, it was considered that local non thermal equilibrium
between the gas and solid phases is significant. Therefore, two different energy conservation equations
are developed to determine the separate temperature fields of the gas and solid phases.
3.3.1 Energy conservation equation for the gas phase
The macro scale energy conservation equation for the gas phase is written as:
(9)
The area-averaged temperatures are represented by volume-averaged temperatures. The intrinsic phase
averages of density and temperature are represented by the point density and temperature, since the
variation in the density and temperature over the averaging volume is considered to be very small [31].
3.3.2 Energy conservation equation for the solid phase
The local volume-averaged macroscopic energy conservation equation for the solid phase is derived in
the same way as for the gas phase. It is given by
(10)
The fluid–solid specific surface area for spherical particles is determined by [25]
(11)
The interfacial heat transfer coefficient for the spherical particle is evaluated by [25]
11
(12)
where, , ,
The equation of state for the adsorbate vapour phase is written as
(13)
3.4 Initial and Boundary Conditions
Temperature, pressure and adsorbate concentration gradients are investigated where T = T(t,r,z), P =
P(t,r,z), and X = X(t,r,z).
The temperatures (solid and gas), pressure and adsorbate concentration distributions in both directions
inside the adsorbent bed are initially considered to be uniform.
, , (14)
Referring to the analysis domain in Fig. 1, at the r=Ri (left) and z=L (top) boundaries it is assumed that
the vapour pressure is equal to the evaporator pressure and the temperature gradients for both the solid
and gas phases are zero (i.e., adiabatic boundaries).
(15)
(16)
12
(17)
(18)
At the r=Ro (right) and z=0 (bottom) boundaries the pressure gradient is zero since the walls are
impermeable and a convective heat transfer boundary conditions exists for the solid and gas phases.
(19)
(20)
(21)
(22)
(23)
(24)
4. SOLUTION PROCEDURE
The nonlinear coupled governing partial differential equations under consideration were solved
numerically using the finite difference technique. The central differencing, first order upwind scheme,
and forward differencing were used to discretize the second order spatial derivatives, convective, and
unsteady terms, respectively. The resulting set of nonlinear algebraic equations was solved iteratively by
13
the combination of the alternating direction implicit (ADI) method, the Newton-Raphson iteration scheme
and a block tridiagonal matrix solver algorithm (Thomas algorithm). The grid distribution in the
computational domain is uniform and nine algebraic equations were solved at each grid point. A
computer simulation program based on the numerical procedure above was written in Matlab to perform
the parametric investigation. In the simulation program, at each time step, iterations were terminated
when the calculated difference between the two successive iterations of any dependent variable was less
than 10-6
. The main simulation parameters used in the computer simulation program are given in Table 2.
Table 2
The influence of the number of grid points and time steps on the solid phase temperature at nearly thermal
equilibrium case and various locations in the computational domain is shown in Table 3. It can be seen
from Table 3 that the difference between the results obtained for two different grid sizes (5*25 and
10*50) and time steps (0.5 and 1 s) is quite small. Therefore, the number of grid points is varied in the
range between 20*80 and 25*100 and the time step is selected as 0.25 s to ensure the reliability of the
numerical computations.
Table 3
5. RESULTS AND DISCUSSION
The effect of the adsorbent bed dimensions in the axial and radial directions, the convective heat transfer
coefficient and the thermal conductivity of the adsorbent material on the transient distributions of the
temperature difference between the phases, the solid phase temperature, the pressure, the adsorbate
concentration and the difference in the adsorbate concentration calculated from the instantaneous
equilibrium and LDF models have been investigated numerically, and the results are presented below.
For each parametric study all the parameters given in Table 2 are kept constant except the parameter
investigated i.e. adsorbent bed length.
14
A schematic view of the isobaric adsorption process on a Clapeyron diagram is shown in Fig. 2. At the
starting point of the adsorption process for all simulations (A), the adsorbate concentration, the solid and
gas phase temperatures, and the pressure throughout the adsorbent bed are uniform and are equal to 0.2
kgwkgad-1
, 348.5 K and 1.2282 kPa, respectively. At this point, the temperatures for the gas and solid
phases were calculated by means of the regeneration temperature of the adsorbent bed (Tre), condenser
pressure (Pco), and evaporator pressure (Pe) given in Table 2. The adsorption capacity of the solid
adsorbent was assumed to be constant as the pressure inside the adsorbent bed was decreased from the
condenser (Pco) to the evaporator (Pe) pressure. The adsorption process was terminated at point B where
the adsorbent bed nearly reached thermal, mechanical and chemical equilibrium conditions and the values
of the solid phase temperature, pressure and the adsorbate concentration throughout the adsorbent bed at
this point (B) are presented in the Figures 5, 6 and 7 (3th
column) for various adsorbent bed lengths,
respectively.
Figure 2
5.1 Temperature difference between the phases for the LTNE model
The temperature difference between the solid and gas phase inside the adsorbent bed was not taken into
account in most of the studies published previously (see Table.1). Typically, in these studies, a single
energy equation was used by assuming LTE between the phases without showing the validity of this
assumption. However, it is stated in the literature [8] that this assumption may not be correct in some
circumstances and it may lead to unrealistic simulation results. Therefore, one of the main objectives of
this study is to investigate the validity of this assumption for the parameters investigated. For this
purpose, two different energy conservation equations were developed to determine the separate
temperature fields of the gas and solid adsorbent phases.
A parametric study of the transient distributions of the temperature difference between the solid and gas
phases was performed for the following parameters and ranges: adsorbent bed length (0.05 to 0.2 m) and
thickness (0.005 to 0.015 m); HTF convective heat transfer coefficient (25 to 400 Wm-2
K-1
); and, thermal
15
conductivity of the solid adsorbent material (0.2 to 1.5 Wm-1
K-1
). For brevity only the condition of Ro =
0.015 m at t = 50 s is presented in Fig.3 as this is representative of the other conditions. The heat transfer
fluid flows past the right hand and bottom sides of Fig.3, which results in large rates of heat transfer and
temperature gradients at these boundaries. Conversely, the left and top sides are modeled as adiabatic
boundary conditions and therefore the temperature gradients go to zero at these boundaries. Generally, the
temperature difference between the phases is only significant (> 4 oC) close to the outer boundaries
exposed to the HTF, especially, during the first few seconds. However, this temperature difference
typically becomes less than 4 oC for process times greater than 50 s and decays toward zero as the time
progress. The temperature difference between the phases at a given time increases with decreasing values
of the adsorbent bed thickness and convective heat transfer coefficient and increasing values of the
thermal conductivity of the solid adsorbent material, but is relatively insensitive to changes in adsorbent
bed length. Therefore, it can be concluded that decreasing conductive and increasing convective thermal
resistances resulting in low Biot number increases the temperature difference between the solid and gas
phases. Consequently, the results obtained in this part of the study indicate that LTE assumption is
reasonable for the range of conditions explored and thus, this assumption can be used for the simplicity.
Figure 3
5.2 Adsorbate concentration difference between the instantaneous equilibrium and LDF models
The difference in the adsorbate concentration distributions for the instantaneous equilibrium and LDF
models for the various values of the adsorbent bed length and thickness, convective heat transfer
coefficient and thermal conductivity of the solid adsorbent material are investigated and the range of
values explored is the same for section 5.1. The results presented in Fig. 4 are representative of the results
for the other conditions and for brevity these other results are discussed but not presented graphically. The
difference in adsorbate concentration between the instantaneous equilibrium and LDF models for all the
parameters studied is generally negligible small and thus, typically, the instantaneous equilibrium
adsorption model can be used instead of the LDF models for the modeling of the adsorbate concentration
without leading to any significant computational errors under the given conditions. The difference in
16
adsorbate concentration between the instantaneous equilibrium and LDF models decreases with
increasing process time. This can be explained by the fact that the rate of adsorbate concentration
calculated from the instantaneous equilibrium adsorption model is quite high at the early time steps of the
process due to the high heat transfer rate and negligible pressure gradients and it decreases as time goes.
On the other hand, the rate of adsorbate concentration computed from the LDF model is relatively slow at
all times because of the internal mass transfer resistances. Therefore, initially, the adsorbate concentration
difference between the instantaneous equilibrium and LDF models is relatively large and this difference
decreases as time increases.
Figure 4
5.3 Adsorbent bed length
Isotherms of the solid phase, isobars and contours of the adsorbate concentration for adsorbent bed
lengths of 0.05, 0.1 and 0.2 m and at times of 300, 900 and 1800 seconds are shown in Fig.5, Fig.6 and
Fig.7, respectively. It is clear that the temperature, pressure and adsorbate concentration distributions
inside the adsorbent bed is only slightly affected by variations in the adsorbent bed length for the given
adsorbent bed thickness, especially for adsorbent bed lengths greater than 0.1 m. The temperature and
adsorbate concentration gradients in the axial direction are only significant near the bottom boundary at
the early stages of the process. However, in the radial direction these gradients are not only considerable
near the right heat transfer boundary but also throughout the rest of the bed. Temperature and adsorbate
concentration gradients in both directions gradually become insignificant as the time progress. It can be
concluded that the heat and mass transfer inside the adsorbent bed can be approximated as one
dimensional in the r direction for large values of adsorbent bed length, but the two-dimensional effects
cannot be neglected for aspect ratios nearly equal to one. The pressure gradients throughout the adsorbent
bed are generally negligible. The resistance to adsorbate vapor flow in the axial direction is higher than
that in the radial direction as a result of the high aspect ratio. Initially, the pressure at the heat transfer
boundaries decreases suddenly due to increasing value of the adsorbate concentration since there is a
sharp temperature drop at these boundaries. Thus, the adsorbate vapor in the interior of the adsorbent bed
17
starts to move from the high pressure to low pressure regions. As a result of this, the adsorbate
concentration at the outer boundaries increases suddenly. After half an hour, the adsorbent bed nearly
reaches thermal, mechanical and chemical equilibrium conditions.
Figure 5
Figure 6
Figure 7
5.4 Adsorbent bed thickness
The influence of the adsorbent bed thickness on the distributions of the solid phase temperature, pressure
and adsorbate concentration are presented in Fig.8, Fig.9 and Fig.10, respectively. The adsorbent bed
thickness has a great influence on the temperature and adsorbate concentration distributions. The heat and
mass transfer inside the adsorbent bed depend strongly on the radius when the adsorbent bed thickness is
decreased for the given adsorbent bed length. However, the pressure distribution is only slightly affected
by the variations in the adsorbent bed thickness, and generally the uniform pressure assumption is valid.
The bed approaches thermal, mechanical and chemical equilibrium conditions after 600, 1800 and 3700
sec for adsorbent bed thickness 0.005, 0.01 and 0.015 m, respectively. The reason behind this result is that
the thermal resistance inside the adsorbent bed increases with increases in the adsorbent bed thickness and
this results in longer times to equilibrium. Therefore, the adsorbent bed thickness should be kept as small
as possible to improve the heat transfer conditions and as result, reduce the process time. However, this
conclusion is based on the assumption that the mass of the shell is neglected. Additionally, it is quite
obvious in Fig.8 that the adsorbent bed thickness (Ro – Ri) should not be greater than 0.01 m for good
heat transfer characteristics. Improving the heat transfer conditions makes it possible to reach the
chemical equilibrium condition in a shorter time as well since the adsorbate concentration is mainly a
function of temperature and pressure.
Figure 8
Figure 9
18
Figure 10
5.5 Convective heat transfer coefficient
The variation of the solid phase temperature, pressure, and adsorbate concentration inside the adsorbent
bed for values of the convective heat transfer coefficient between the adsorbent bed and heat transfer fluid
of 25, 200 and 400 Wm-2
K-1
at times of 600, 1200, and 2000 seconds are presented in Fig.11, Fig.12 and
Fig.13, respectively. The simulations were terminated after 2000 seconds as the system was close to
equilibrium. The temperature distribution, and thus pressure and adsorbate concentration distributions as
well, are nearly insensitive to the variation of the convective heat transfer coefficient due to the low
thermal conductivity of the adsorbent material. The pressure distribution throughout the adsorbent bed is
generally uniform except at early time steps of the process. This is due to the fact that the temperatures of
the solid and gas phases near the heat transfer boundaries decreases under the given boundary conditions
and as a result of this the adsorbate concentration increases and the pressure decreases. Consequently, the
pressure gradients inside the adsorbent bed becomes significant and adsorbate vapor flows from high
pressure to low pressure regions and thus the pressure also decreases at the adiabatic boundaries. In terms
of the thermal response of the system little benefit is seen in increasing the value of the convective heat
transfer coefficient above 200 Wm-2
K-1
since there is not any significant difference between the 200 and
400 Wm-2
K-1
in terms of temperature, pressure and adsorbate concentration distributions.
Figure 11
Figure 12
Figure 13
5.6 Thermal conductivity of the solid adsorbent material
The variation of the solid phase temperature, pressure and adsorbate concentration inside the adsorbent
bed at 150, 300 and 500 sec for thermal conductivities of the solid adsorbent material of 0.2, 0.75 and 1.5
Wm-1
K-1
are shown in Fig.14, Fig.15 and Fig.16, respectively. It is clear in Fig.14 that heat transfer
19
conditions inside the adsorbent bed are affected positively by increases in the solid phase thermal
conductivity. Temperature gradients in the both directions decrease when the thermal conductivity of the
solid adsorbent material is increased. The thermal equilibrium condition (maximum temperature
difference within the bed is less than 3 oC) is nearly satisfied for the adsorbent bed having a thermal
conductivity of 1.5 Wm-1
K-1
for process times equal to 500 sec. However, at this time instant, temperature
gradients are comparatively high for a thermal conductivity of 0.2 Wm-1
K-1
. Therefore, the specific
cooling power of the adsorption cooling system can be improved considerably by eliminating the limiting
effects of the heat transfer rate inside the adsorbent bed. Pressure distributions for the three cases are
generally uniform but the pressure distribution for the k=1.5 Wm-1
K-1
is slightly less than the others at
early time steps of the process due to the high adsorption rate. This is due to the fact that the heat transfer
rate is comparatively high and this results in a sudden temperature drop inside the adsorbent bed. As a
result of this, the adsorption rate increases and the pressure decreases, and this pressure drop is not
compensated for by an increase in the vapor diffusion flux. Adsorption equilibrium condition inside the
adsorbent bed is achieved in a comparatively short period of time when the thermal conductivity of the
solid adsorbent material is high.
Figure 14
Figure 15
Figure 16
6. CONCLUSIONS
Based on the numerical model developed and for the conditions explored, the conclusions about the
current study can be summarized as follows:
• The temperature difference between the solid and gas phases are generally insignificant. Hence the local
thermal equilibrium assumption is generally valid. The exception is during the early stages of the process.
20
• The adsorbate concentration difference between the instantaneous equilibrium and LDF models is small
and varies only slightly over the range of parameters investigated. Hence, the internal mass transfer
resistances are negligibly small for these conditions.
• The distribution of the gas phase pressure inside adsorbent bed does not vary significantly for all the
cases and thus, a spatially isobaric bed assumption can be accepted without leading to any significant
computational errors.
• The heat and mass transfer inside the adsorbent bed occurs almost exclusively in the radial direction
when the ratio of the adsorbent length over adsorbent bed thickness is greater than 10.
• The adsorbent bed thickness has a large impact on the heat and mass transfer inside the adsorbent bed.
The adsorbent bed thickness should not be more than 0.01 m for good heat transfer rates and hence,
shorter cycle times. The heat and mass transfer gradually becomes one dimensional (almost only in the r
direction) when the adsorbent bed thickness is decreased for a fixed adsorbent bed length.
• The heat and mass transfer inside adsorbent bed is nearly insensitive to variations in the heat transfer
fluid convective heat transfer coefficient since the thermal conductivity of the solid adsorbent material is
low. Therefore, based on heat and mass transfer considerations the value of the convective heat transfer
coefficient does not need to be more than 200 Wm-2
K-1
since increasing value of the convective heat
transfer coefficient (greater than 200 Wm-2
K-1
) does not lead to any significant decrease in the
temperature, pressure and adsorbate concentration gradients.
• The solid phase thermal conductivity has a strong influence on the heat and mass transfer conditions
inside the adsorbent bed and both conditions can be improved considerably by using highly conductive
adsorbent materials, fins, or other thermal enhancements.
21
ACKNOWLEDGEMENT
İsmail Solmuş would like to thank The Scientific & Technological Research Council of Turkey
(TÜBİTAK) and the Middle East Technical University for supporting him with a fellowship during his
study at the University of Bath in the United Kingdom.
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Reference Dimension Internal Mass Transfer
Resistance Model
External Mass Transfer
Resistance Model
Energy
Equation
[9-11] 3D LDF Darcy’s equation LTE
[12,13] 1D LDF Uniform pressure LTE
[12,13] 1D SD Uniform pressure LTE
[12-14] 1D Adsorption equilibrium Uniform pressure LTE
[15-19] 2D LDF Darcy’s equation LTE
[20,21] 2D Adsorption equilibrium Ergun’s equation LTE
[23] 2D LDF Uniform pressure LTE
[24] 0D (Lumped) LDF Uniform pressure LTE
[25,26] 2D LDF Darcy’s equation LTE and LTNE
[22,27] 2D Adsorption equilibrium Uniform pressure LTE
[28] 1D Adsorption equilibrium Darcy’s equation LTE
[29] 1D LDF Darcy’s equation LTE
Table 1
Table
Parameter Value Unit Reference
Cpg
1800 Jkg-1
K-1
Cps
924 Jkg-1
K-1
[34]
dp 5e-4
m [34]
Do 2.54e-4
m2 s
-1 [34]
Ea 4.2e4 Jmol
-1 [34]
h 100 Wm-2
K-1
kg
0.024 Wm-1
K-1
ks
0.198 Wm-1
K-1
[34]
L 0.1 m
Pco
4.246 kPa
Pe
1.228 kPa
Ri 0.01 m
Ro 0.02 m
Tc
40 oC
Tre
100 oC
εb 0.37 [34]
εp
0.42 [34]
Q 2693 kJkg-1
[34]
μg
1.5e-5
kgm-1
s-1
[20]
ρs
2027 kgm-3
[34]
Table 2
Table
Table 3
Δt = 0.5 s Δt = 1 s
5*25 10*50 5*25 10*50
r, z
(m
)
0.012, 0.088 314,329935 314,379638 314,334036 314,383828
0.018, 0.088 313,802534 313,845411 313,804846 313,847829
0.012, 0.012 313,998231 314,032621 314,001147 314,035600
0.018, 0.012 313,623038 313,652964 313,624663 313,654667
Table
Table 1. Classification of the existing mathematical models in terms of their important characteristics.
Table 2. Main simulation parameters.
Table 3. Effects of the number of grid points and time steps on the solid phase temperature (K) at
nearly thermal equilibrium case and various locations in the computational domain.
Table
Figure 1
Annulus
Vacuum tube
Vacuum tube is fixed to the shell
Shell
Inlet of the HTF
Top cover
Outlet of
the HTF
Silica-gel
Silica-gel
Mass transfer
tube (Ri) )
Shell
Z
R
Vacuum
tube (Ro)
Refrigerant inlet
Figure
Figure 2
Tc Th
Pco
B
-1/Ts
lnP
Desorption
XminXmax
PeAAdsorption
Tre
Figure
Figure 3
r (m)
z (m
)
0.01 0.0105 0.011 0.0115 0.012 0.0125 0.013 0.0135 0.014 0.0145 0.0150
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.5
1
1.5
2
Ro = 0.015 m
t = 50 s
Figure
Figure 4
z (m
)
0.0
0431
0.0
0609
0.0
0786
0.0
0964
0.01 0.012 0.014 0.016 0.018 0.020
0.01
0.02
0.03
0.04
0.05
0.0
0392
0.0
0335
0.0
0279
0.0
0222
0.0
0166
0.01 0.012 0.014 0.016 0.018 0.020
0.01
0.02
0.03
0.04
0.05
0.0
0202
0.0
0169
0.0
0135
0.0
0102
0.0
0068
0.0
0035
0.01 0.012 0.014 0.016 0.018 0.020
0.01
0.02
0.03
0.04
0.05z
(m)
0.0
0431
0.0
0609
0.0
0787
0.0
0965
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
0.0
0391
0.0
0335
0.0
0279
0.0
0222
0.0
0166
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
0.0
0202
0.0
0169
0.0
0135
0.0
0102
0.0
0068
0.0
0035
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
z (m
)
0.0
0431
0.0
0609
0.0
0787
0.0
0964
0.01 0.012 0.014 0.016 0.018 0.020
0.05
0.1
0.15
0.2
r (m)
0.0
0391
0.0
0335
0.0
0278
0.0
0222
0.0
0166
0.01 0.012 0.014 0.016 0.018 0.020
0.05
0.1
0.15
0.2
r (m)
0.0
0202
0.0
0169
0.0
0135
0.0
0102
0.0
0068
0.0
0035
0.01 0.012 0.014 0.016 0.018 0.020
0.05
0.1
0.15
0.2
Figure
Figure 5
z (m
)
33
9.4
33
5.1
33
0.7
32
6.4
32
2.1
31
7.8
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
32
7.0
32
4.7
32
2.4
32
0.1
31
7.8
31
5.5
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
31
8.1
31
7.3
31
6.4
31
5.6
31
4.8
31
3.9
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
z (m
)
33
9.4
33
5.1
33
0.7
32
6.4
32
2.1
31
7.8
0.01 0.012 0.014 0.016 0.018 0.020
0.05
0.1
0.15
0.2
r (m)
32
7.0
32
4.7
32
2.4
32
0.1
31
7.8
31
5.5
0.01 0.012 0.014 0.016 0.018 0.020
0.05
0.1
0.15
0.2
r (m)
31
8.1
31
7.3
31
6.4
31
5.6
31
4.8
31
3.9
0.01 0.012 0.014 0.016 0.018 0.020
0.05
0.1
0.15
0.2
z (m
)
33
9.4
33
5.1
33
0.7
32
6.4
32
2.1
31
7.8
0.01 0.012 0.014 0.016 0.018 0.020
0.01
0.02
0.03
0.04
0.05
32
7.0
32
4.7
32
2.4
32
0.1
31
7.8
31
5.5
0.01 0.012 0.014 0.016 0.018 0.020
0.01
0.02
0.03
0.04
0.05
31
8.1
31
7.3
31
6.4
31
5.6
31
4.8
31
3.9
0.01 0.012 0.014 0.016 0.018 0.020
0.01
0.02
0.03
0.04
0.05
Figure
Figure 6
r (m)
z (m
)
1.2
26
7
1.2
25
1
1.2
23
6
1.2
22
1
1.2
20
5
1.2
19
0.01 0.012 0.014 0.016 0.018 0.020
0.05
0.1
0.15
0.2
r (m)
1.2
27
8
1.2
27
3
1.2
26
9
1.2
26
4
1.2
26
1.2
25
5
0.01 0.012 0.014 0.016 0.018 0.020
0.05
0.1
0.15
0.2
r (m)
1.2
28
1.2
27
9
1.2
27
7
1.2
27
6
1.2
27
4
1.2
27
3
0.01 0.012 0.014 0.016 0.018 0.020
0.05
0.1
0.15
0.2
z (m
)
1.2
26
7
1.2
25
1
1.2
23
6
1.2
22
1
1.2
20
6
1.2
19
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
1.2
27
8
1.2
27
3
1.2
26
9
1.2
26
4
1.2
26
1.2
25
5
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
1.2
28
1.2
27
9
1.2
27
7
1.2
27
6
1.2
27
4
1.2
27
3
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
z (m
)
1.2
26
7
1.2
25
1
1.2
23
6
1.2
22
1
1.2
20
6
1.2
19
0.01 0.012 0.014 0.016 0.018 0.020
0.01
0.02
0.03
0.04
0.05
1.2
27
8
1.2
27
3
1.2
26
9
1.2
26
5
1.2
26
1.2
25
6
0.01 0.012 0.014 0.016 0.018 0.020
0.01
0.02
0.03
0.04
0.05
1.2
28
1
1.2
27
9
1.2
27
8
1.2
27
6
1.2
27
5
1.2
27
3
0.01 0.012 0.014 0.016 0.018 0.020
0.01
0.02
0.03
0.04
0.05
Figure
Figure 7
z (m
)
0.2
22
0.2
32
0.2
41
0.2
51
0.2
61
0.2
70
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
0.2
57
0.2
63
0.2
70
0.2
76
0.2
83
0.2
89
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
0.2
82
0.2
84
0.2
87
0.2
89
0.2
91
0.2
94
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
z (m
)
0.2
22
0.2
32
0.2
41
0.2
51
0.2
61
0.2
70
0.01 0.012 0.014 0.016 0.018 0.020
0.05
0.1
0.15
0.2
r (m)
0.2
57
0.2
63
0.2
70
0.2
76
0.2
83
0.2
89
0.01 0.012 0.014 0.016 0.018 0.020
0.05
0.1
0.15
0.2
r (m)
0.2
82
0.2
84
0.2
87
0.2
89
0.2
91
0.2
94
0.01 0.012 0.014 0.016 0.018 0.020
0.05
0.1
0.15
0.2
z (m
)
0.2
22
0.2
32
0.2
41
0.2
51
0.2
61
0.2
70
0.01 0.012 0.014 0.016 0.018 0.020
0.01
0.02
0.03
0.04
0.05
0.2
57
0.2
63
0.2
70
0.2
76
0.2
83
0.2
89
0.01 0.012 0.014 0.016 0.018 0.020
0.01
0.02
0.03
0.04
0.05
0.2
82
0.2
84
0.2
87
0.2
89
0.2
91
0.2
94
0.01 0.012 0.014 0.016 0.018 0.020
0.01
0.02
0.03
0.04
0.05
Figure
Figure 8
z (m
)
33
2.3
32
9.2
32
6.1
32
3.1
32
0.0
0.01 0.011 0.012 0.013 0.014 0.0150
0.02
0.04
0.06
0.08
0.1
32
4.6
32
2.7
32
0.9
31
9.0
31
7.1
0.01 0.011 0.012 0.013 0.014 0.0150
0.02
0.04
0.06
0.08
0.1
31
7.5
31
6.8
31
6.1
31
5.4
31
4.6
0.01 0.011 0.012 0.013 0.014 0.0150
0.02
0.04
0.06
0.08
0.1
z (m
)
33
2.5
32
9.3
32
6.1
32
2.9
31
9.7
31
6.4
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
32
4.8
32
2.9
32
0.9
31
9.0
31
7.0
31
5.1
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
31
8.1
31
7.3
31
6.4
31
5.6
31
4.8
31
3.9
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
z (m
)
33
2.4
32
9.2
32
6.0
32
2.8
31
9.6
31
6.4
0.01 0.015 0.02 0.0250
0.02
0.04
0.06
0.08
0.1
r (m)
32
4.7
32
2.8
32
0.8
31
8.9
31
7.0
31
5.0
0.01 0.015 0.02 0.0250
0.02
0.04
0.06
0.08
0.1
r (m)
31
7.9
31
7.1
31
6.3
31
5.5
31
4.7
31
3.9
0.01 0.015 0.02 0.0250
0.02
0.04
0.06
0.08
0.1
t= 300 sect= 150 sec
t= 1050 sec t= 1800 sect= 600 sec
t= 3700 sect= 2200 sect= 1300 sec
t= 600 sec
Figure
Figure 9
z (m
)
1.2
27
7
1.2
27
1.2
26
3
1.2
25
5
1.2
24
8
1.2
24
0.01 0.011 0.012 0.013 0.014 0.0150
0.02
0.04
0.06
0.08
0.1
1.2
28
1.2
27
7
1.2
27
3
1.2
26
9
1.2
26
5
1.2
26
2
0.01 0.011 0.012 0.013 0.014 0.0150
0.02
0.04
0.06
0.08
0.1
1.2
28
1
1.2
28
1.2
27
9
1.2
27
7
1.2
27
6
1.2
27
5
0.01 0.011 0.012 0.013 0.014 0.0150
0.02
0.04
0.06
0.08
0.1
z (m
)
1.2
27
5
1.2
26
8
1.2
26
1
1.2
25
4
1.2
24
7
1.2
24
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
1.2
27
8
1.2
27
5
1.2
27
1
1.2
26
7
1.2
26
4
1.2
26
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
z (m
)
1.2
27
5
1.2
26
8
1.2
26
1.2
25
3
1.2
24
6
1.2
23
9
0.01 0.015 0.02 0.0250
0.02
0.04
0.06
0.08
0.1
r (m)
1.2
27
8
1.2
27
4
1.2
27
1.2
26
6
1.2
26
2
1.2
25
80.01 0.015 0.02 0.0250
0.02
0.04
0.06
0.08
0.1
r (m)
1.2
28
1.2
27
9
1.2
27
7
1.2
27
6
1.2
27
4
1.2
27
2
0.01 0.015 0.02 0.0250
0.02
0.04
0.06
0.08
0.1
1.2
28
1.2
27
9
1.2
27
7
1.2
27
6
1.2
27
4
1.2
27
3
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
t = 150 sec
t = 600 sec t = 1050 sec
t = 3700 sec
t = 600 sec
t = 1800 sec
t = 2200 sect = 1300 sec
t = 300 sec
Figure
Figure 10
z (m
)
0.2
41
0.2
50
0.2
59
0.2
67
0.2
76
0.2
84
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
0.2
63
0.2
68
0.2
74
0.2
79
0.2
85
0.2
90
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
0.2
82
0.2
84
0.2
87
0.2
89
0.2
91
0.2
94
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
z (m
)
0.2
44
0.2
52
0.2
61
0.2
70
0.2
79
0.2
87
0.01 0.015 0.02 0.0250
0.02
0.04
0.06
0.08
0.1
r (m)
0.2
65
0.2
70
0.2
75
0.2
80
0.2
86
0.2
91
0.01 0.015 0.02 0.0250
0.02
0.04
0.06
0.08
0.1
r (m)
0.2
83
0.2
85
0.2
88
0.2
90
0.2
92
0.2
94
0.01 0.015 0.02 0.0250
0.02
0.04
0.06
0.08
0.1
z (m
)
0.2
31
0.2
36
0.2
40
0.2
45
0.2
50
0.2
54
0.01 0.011 0.012 0.013 0.014 0.0150
0.02
0.04
0.06
0.08
0.1
0.2
55
0.2
59
0.2
63
0.2
68
0.2
72
0.2
76
0.01 0.011 0.012 0.013 0.014 0.0150
0.02
0.04
0.06
0.08
0.1
0.2
79
0.2
81
0.2
84
0.2
86
0.2
89
0.2
91
0.01 0.011 0.012 0.013 0.014 0.0150
0.02
0.04
0.06
0.08
0.1
t = 3700 sect = 2200 sect = 1300 sec
t = 600 sec t = 1050 sec t = 1800 sec
t = 600 sect = 300 sect = 150 sec
Figure
Figure 11
z (m
)
33
5.1
33
1.7
32
8.2
32
4.7
32
1.3
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
32
5.9
32
3.8
32
1.8
31
9.7
31
7.7
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
z (m
)
33
1.7
32
8.6
32
5.5
32
2.4
31
9.3
31
6.2
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
32
2.1
32
0.6
31
9.1
31
7.6
31
6.1
31
4.6
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
31
6.6
31
6.0
31
5.4
31
4.8
31
4.3
31
3.7
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
z (m
)
33
2.0
32
8.8
32
5.7
32
2.6
31
9.4
31
6.3
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
32
2.4
32
0.9
31
9.3
31
7.8
31
6.2
31
4.7
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
31
6.7
31
6.1
31
5.5
31
4.9
31
4.3
31
3.7
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
31
9.2
31
8.2
31
7.2
31
6.2
31
5.2
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
Figure
Figure 12
z (m
)
1.2
27
4
1.2
26
7
1.2
25
9
1.2
25
1
1.2
24
4
1.2
23
6
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
1.2
27
8
1.2
27
4
1.2
27
1
1.2
26
7
1.2
26
3
1.2
25
9
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
1.2
28
1.2
27
8
1.2
27
7
1.2
27
5
1.2
27
3
1.2
27
1
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
z (m
)
1.2
27
5
1.2
26
8
1.2
26
1
1.2
25
4
1.2
24
8
1.2
24
1
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
1.2
27
9
1.2
27
6
1.2
27
3
1.2
27
1.2
26
7
1.2
26
5
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
1.2
28
1
1.2
28
1.2
27
9
1.2
27
8
1.2
27
6
1.2
27
5
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
z (m
)
1.2
27
5
1.2
26
8
1.2
26
2
1.2
25
5
1.2
24
8
1.2
24
1
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
1.2
27
9
1.2
27
6
1.2
27
4
1.2
27
1
1.2
26
8
1.2
26
5
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
1.2
28
1
1.2
28
1.2
27
9
1.2
27
8
1.2
27
7
1.2
27
6
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
Figure
Figure 13
z (m
)
0.2
35
0.2
44
0.2
53
0.2
63
0.2
72
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
0.2
61
0.2
66
0.2
72
0.2
78
0.2
83
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
0.2
79
0.2
82
0.2
85
0.2
87
0.2
90
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
z (m
)
0.2
43
0.2
51
0.2
60
0.2
68
0.2
76
0.2
85
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
0.2
69
0.2
74
0.2
78
0.2
83
0.2
87
0.2
92
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
0.2
86
0.2
87
0.2
89
0.2
91
0.2
93
0.2
94
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
z (m
)
0.2
44
0.2
52
0.2
60
0.2
68
0.2
77
0.2
85
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
0.2
70
0.2
74
0.2
79
0.2
83
0.2
87
0.2
92
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
0.2
86
0.2
88
0.2
89
0.2
91
0.2
93
0.2
95
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
Figure
Figure 14
r (m)
z (m
)
330.0
328.2
326.4
324.6
322.8
320.9
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
322.3
321.3
320.3
319.3
318.3
317.2
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
317.5
317.0
316.5
316.1
315.6
315.1
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
z (m
)
335.6
332.6
329.7
326.7
323.7
320.7
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
327.9
325.9
323.9
321.9
319.9
317.9
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
321.6
320.4
319.3
318.1
317.0
315.8
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
z (m
)
342.4
337.6
332.9
328.1
323.3
318.6
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
339.4
335.1
330.7
326.4
322.1
317.8
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
334.7
331.2
327.6
324.0
320.4
316.8
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
Figure
Figure 15
r (m)
z (m
)
1.2
24
7
1.2
21
3
1.2
17
8
1.2
14
3
1.2
10
8
1.2
07
4
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
1.2
26
5
1.2
24
7
1.2
23
1.2
21
2
1.2
19
5
1.2
17
7
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
1.2
27
4
1.2
26
5
1.2
25
7
1.2
24
9
1.2
24
1.2
23
2
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
z (m
)
1.2
24
9
1.2
21
5
1.2
18
2
1.2
14
9
1.2
11
6
1.2
08
2
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
1.2
26
5
1.2
24
8
1.2
23
1
1.2
21
4
1.2
19
7
1.2
18
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
1.2
27
3
1.2
26
4
1.2
25
5
1.2
24
5
1.2
23
6
1.2
22
7
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
z (m
)
1.2
25
3
1.2
22
5
1.2
19
6
1.2
16
7
1.2
13
9
1.2
11
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
1.2
26
7
1.2
25
1
1.2
23
6
1.2
22
1
1.2
20
6
1.2
19
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
1.2
27
3
1.2
26
5
1.2
25
6
1.2
24
7
1.2
23
9
1.2
23
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
Figure
Figure 16
r (m)
z (m
)
0.2
37
0.2
42
0.2
46
0.2
50
0.2
54
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
0.2
61
0.2
64
0.2
67
0.2
70
0.2
73
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
r (m)
0.2
78
0.2
79
0.2
81
0.2
83
0.2
85
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
z (m
)
0.2
26
0.2
31
0.2
37
0.2
42
0.2
47
0.2
53
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
0.2
48
0.2
53
0.2
59
0.2
64
0.2
69
0.2
74
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
0.2
67
0.2
71
0.2
75
0.2
79
0.2
83
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
z (m
)
0.2
12
0.2
20
0.2
28
0.2
35
0.2
43
0.2
51
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
0.2
22
0.2
32
0.2
41
0.2
51
0.2
61
0.2
70
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
0.2
35
0.2
45
0.2
54
0.2
63
0.2
72
0.2
82
0.01 0.012 0.014 0.016 0.018 0.020
0.02
0.04
0.06
0.08
0.1
Figure
Fig. 1 A schematic view of the cylindrical adsorbent bed.
Fig.2 A schematic view of the adsorption process on a Clapeyron diagram.
Fig.3 The distribution of the temperature difference, K, between the solid and gas phases for the Ro = 0.015 m at
t = 50 s.
Fig.4 The distribution of the adsorbate concentration difference between the equilibrium and LDF models,
kgwkgad-1
. [1st row L=0.05 m, 2
nd row L=0.1 m, 3
th row L=0.2 m and 1
st column t=300 s, 2
nd column t=900 s, 3
th
column t=1800 s]
Fig.5 Transient solid phase temperature, K, distribution for various adsorbent bed lengths. [1st row L=0.05 m, 2
nd
row L=0.1 m, 3
th row L=0.2 m and 1
st column t=300 s, 2
nd column t=900 s, 3
th column t=1800 s]
Fig.6 Transient pressure, kPa, distribution for various adsorbent bed lengths. [ 1st row L=0.05 m, 2
nd row L=0.1
m, 3
th row L=0.2 m and 1
st column t=300 s, 2
nd column t=900 s, 3
th column t=1800 s ]
Fig.7 Transient adsorbate concentration, kgwkgad-1
, distribution for various adsorbent bed lengths. [1st row
L=0.05 m, 2nd
row L=0.1 m, 3th
row L=0.2 m and 1st column t=300 s, 2
nd column t=900 s, 3
th column t=1800 s]
Fig.8 Transient solid phase temperature, K, distribution for various adsorbent bed thicknesses. [1st row Ro=0.015
m, 2nd
row Ro=0.02 m, 3th
row Ro=0.025 m]
Fig.9 Transient pressure, kPa, distribution for various adsorbent bed thicknesses. [1st row Ro=0.015 m, 2
nd row
Ro=0.02 m, 3th
row Ro=0.025 m]
Fig.10 Transient adsorbate concentration, kgwkgad-1
, distribution for various adsorbent bed thicknesses. [1st row
Ro=0.015 m, 2nd
row Ro=0.02 m, 3th
row Ro=0.025 m]
Fig.11 Transient solid phase temperature, K, distribution for three convective heat transfer coefficients and
times. [1st row h=25 Wm
-2K
-1, 2
nd row h=200 Wm
-2K
- 1, 3
th row h=400 Wm
-2K
-1and 1
st column t=600 s, 2
nd
column t=1200 s, 3th
column t=2000 s]
Fig.12 Transient pressure, kPa, distribution for three convective heat transfer coefficients and times. [1st row
h=25 Wm-2
K-1
, 2nd
row h=200 Wm-2
K- 1
, 3th
row h=400 Wm-2
K-1
and 1st column t=600 s, 2
nd column t=1200 s, 3
th
column t=2000 s]
Fig.13 Transient adsorbate concentration, kgwkgad-1
, distribution for three convective heat transfer coefficients
and times. [1st row h=25 Wm
-2K
-1, 2
nd row h=200 Wm
-2K
- 1, 3
th row h=400 Wm
-2K
-1and 1
st column t=600 s, 2
nd
column t=1200 s, 3th
column t=2000 s]
Figure
Fig.14 Transient solid phase temperature, K, distribution for various adsorbent material conductivity. [1st row
ks=0.2 Wm-1
K-1
, 2nd
row ks=0.75 Wm-1
K-1
, 3th
row ks=1.5 Wm-1
K-1
and 1st column t=150 s, 2
nd column t=300 s,
3th
column t=500 s]
Fig.15 Transient pressure, kPa, distribution for various adsorbent material conductivity. [1st row ks=0.2 Wm
-1K
-1,
2nd
row ks=0.75 Wm-1
K-1
, 3th
row ks=1.5 Wm-1
K-1
and 1st column t=150 s, 2
nd column t=300 s, 3
th column t=500
s]
Fig.16 Transient adsorbate concentration, kgwkgad-1
, distribution for various adsorbent material conductivity. [1st
row ks=0.2 Wm-1
K-1
, 2nd
row ks=0.75 Wm-1
K-1
, 3th
row ks=1.5 Wm-1
K-1
and 1st column t=150 s, 2
nd column
t=300 s, 3th
column t=500 s]
Conflict of interest
The source of the funding for this research is The Scientific & Technological Research
Council of Turkey (TÜBİTAK) and the Middle East Technical University as given in the
Acknowledgments. None of the authors presently have or have had in the past any personal
or financial relationships that may compromise the integrity of this work.
İsmail Solmuş
*Conflict of Interest Statement
