Content uploaded by Andrew Wright
Author content
All content in this area was uploaded by Andrew Wright on Feb 11, 2019
Content may be subject to copyright.
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
1 of 41
Copyright © 2019 Andrew Wright
Technical Report
Is Counter-Current Flow
Always Better than
Co-Current Flow for
Moving Bed Adsorption?
(Answer – No)
11th February 2019
Andrew Wright
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
2 of 41
Copyright © 2019 Andrew Wright
Chapter 1 – Executive Summary
In chemical engineering heat and mass transfer processes, designs typically employ counter-current
contacting as opposed to co-current as this results in better overall performance. Examples include
heat exchangers, distillation columns and absorption processes. However, is counter-current
operation always theoretically the best or can there be exceptions?
This query stems from preliminary investigatory work into a non-isothermal moving bed adsorption
process. Under certain circumstances operation in co-current mode surprisingly appeared to give
better results than when run in counter-current operation. It was therefore deemed of interest to
further explore the possible causes.
This report examines different scenarios involving a moving bed adsorption process and shows why in
some cases a co-current flow arrangement is an improvement over equivalent counter-current
operation. Even though heat transfer and isothermal mass transfer are both more efficient in counter-
current flow, if both heat and mass transfer are present at the same time then having good heat
transfer can actually negatively impact on the overall mass transfer process.
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
3 of 41
Copyright © 2019 Andrew Wright
Chapter 2 – Contents
Chapter 1 – Executive Summary ............................................................................................................. 2
Chapter 2 – Contents .............................................................................................................................. 3
Chapter 3 – Copyright, Warranties and Licenses .................................................................................... 4
Chapter 4 – Introduction ......................................................................................................................... 5
Chapter 5 – Isothermal Mass Transfer .................................................................................................... 6
Chapter 6 – Non-Isothermal Mass Transfer ............................................................................................ 9
Chapter 7 – Conclusions ........................................................................................................................ 12
Chapter 8 – Nomenclature .................................................................................................................... 13
8.1 Roman ................................................................................................................................... 13
8.2 Greek ..................................................................................................................................... 14
Chapter 9 – References ......................................................................................................................... 15
Chapter 10 – Isothermal, Trace Component Moving Bed Adsorption Model ...................................... 16
10.1 Rectangular Isotherm ............................................................................................................ 18
10.2 Linear Isotherm ..................................................................................................................... 19
10.2.1 Counter-Current Flow ................................................................................................... 21
10.2.2 Q*=-1 ............................................................................................................................. 22
10.2.3 Linear Isotherm Mass Transfer Profiles ........................................................................ 23
Chapter 11 – Moving Bed Adsorption Heat Transfer Model ................................................................ 27
11.1 Counter-Current Flow -1 < C* < 0 ......................................................................................... 29
11.2 Counter-Current Flow C* < -1 ............................................................................................... 30
11.3 Counter-Current Flow C* = -1 ............................................................................................... 31
11.4 Heat Transfer Profiles ........................................................................................................... 32
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
4 of 41
Copyright © 2019 Andrew Wright
Chapter 3 – Copyright, Warranties and Licenses
This work is made available under the Creative Commons Attribution-ShareAlike 4.0 International
Public License. The terms of this license as well as the warranty disclaimer and limitation of liability
can be found here:
https://creativecommons.org/licenses/by-sa/4.0/legalcode
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
5 of 41
Copyright © 2019 Andrew Wright
Chapter 4 – Introduction
At a previous time, a study was undertaken to evaluate the use of a moving bed adsorption process
for CO2 removal from flue gas. The envisioned design was similar to that of an absorption column with
gas flow upwards and the liquid replaced with solid adsorbent particles flowing downwards. The
feeding in of gas at one end of the unit and solid at the other end in counter-current mode is typically
found to give the best performance for chemical engineering processes. However, for some of the
adsorbents tested and operating modes chosen, it was found that feeding the solid and gas at the
same end of the column (co-current flow) resulted in greater removal of CO2.
At first, this result appeared to be counter-intuitive and was attributed to an error in the modelling
work. After further investigation though, the results held up to scrutiny and the assumption that
operation in counter-current mode always gives the best performance was found to be incorrect.
A separate enquiry was therefore undertaken to look for general rules for when co-current mass
transfer can be better than counter-current operation. The results of the work are presented in this
report.
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
6 of 41
Copyright © 2019 Andrew Wright
Chapter 5 – Isothermal Mass Transfer
As a starting point it is of interest to know how well does a moving bed adsorption process work under
isothermal conditions (i.e. ignoring heat effects). Figure 1 shows simple representations of a moving
bed adsorption process in co- and counter-flow arrangements in which solid adsorbent is contacted
with gas and an adsorbable component transferred between the two.
Figure 1 – Co-current versus counter-current operation in a moving bed adsorber
Figure 2 – Isotherm types
The amount of adsorbable component that will be removed from the gas phase by the adsorbent
depends upon the rate of mass transfer and the equilibrium capacity of the adsorbent. The
Gas
Out
Adsorbent
Out
Gas
In
Adsorbent
In
Gas
Out
Adsorbent
In
Gas
In
Adsorbent
Out
Co-Current
Operation
Counter-Current
Operation
Equilibrium
Loading
Mass Fraction of Adsorbate in Gas
Rectangular
Isotherm
Linear
Isotherm
Favourable
Isotherm
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
7 of 41
Copyright © 2019 Andrew Wright
equilibrium capacity depends on the isotherm for the adsorbent material and Figure 2 shows various
examples of such relationships with the amount of adsorbable component in the gas phase.
In order to understand the difference between co-current operation and counter-current operation,
Chapter 10 provides analytical solutions for how the adsorbable component varies in concentration
along the length adsorber and how much is loaded on the adsorbent. The analysis is performed for a
single, trace adsorbable component under isothermal conditions and assuming the isotherm is either
rectangular or linear. More generalised models could be created with increased complexity, but this
is not really necessary for the goal at hand. It will also be assumed in this analysis that the adsorbent
always enters the adsorber in a completely regenerated state so that the particulars of the desorption
process step can be ignored.
The models show that under isothermal conditions and a rectangular isotherm, it does not matter
whether co-current operation or counter-current operation is employed. For a given size of adsorber
and inlet conditions, the amount of adsorbable component removed from the gas phase will be the
same no matter which direction the adsorbent and gas flow. For a linear isotherm however, counter-
current operation gives better removal performance than an equivalently sized co-current design.
The simple reason for this can be seen from Figure 3. In counter-current operation the adsorbent is
making its way towards the gas inlet where the concentration of adsorbable component (adsorbate)
is highest. This means that at the point of leaving the adsorber it has a high uptake (concentration) of
adsorbate on the adsorbent. This therefore minimises the amount of adsorbent that must be supplied
to remove the adsorbable component from the gas phase to a sufficiently low enough level.
Figure 3 – Equilibrium loading at inlet and outlet of adsorber for different isotherm types
In co-current operation the adsorbent is leaving with the gas flowing out of the adsorber where the
concentration of adsorbent in the gas phase is much lower than at the feed end. If the adsorbent
exhibits a rectangular isotherm, then this has no impact because the capacity of the material leaving
the adsorber is exactly the same as if it left at the gas inlet end. It is though, a serious problem if the
isotherm is linear because the equilibrium capacity of the adsorbent at the gas outlet end is much
lower than it would be if it was leaving at the gas inlet. The low adsorbent capacity at the outlet means
that a much higher flow rate of adsorbent must be supplied to remove the same amount of adsorbable
component in co-current operation than in counter-current operation.
Equilibrium
Loading
Mass Fraction of Adsorbate in Gas
Rectangular
Isotherm
Linear
Isotherm
Gas In End
Gas Out End
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
8 of 41
Copyright © 2019 Andrew Wright
The key learning from this particular exercise is that the performance of the moving bed adsorption
process depends primarily on the capacity of the adsorbent at the point it leaves the adsorber. In
order to minimise the amount of adsorbent that needs to be supplied (and subsequently regenerated)
the design needs to maximise the capacity at the adsorbent outlet.
These results also show that under isothermal conditions, operating the adsorber in counter-current
mode is equal to or better than using co-current operation. From simple observation of Figure 4 it can
be seen that the performance difference between the two modes depends on the shape of the
isotherm with isotherms more favourable for adsorption showing smaller differences. For example,
in Figure 4 the favourable isotherm has an equilibrium loading at the gas exit end that is quite close
to that at the gas feed end. Therefore, even though co-current operation would be worse for this
material, it would not be as bad as for a linear isotherm.
Figure 4 – Equilibrium loading at inlet and outlet of adsorber for a favourable isotherm compared
to rectangular and linear types
Equilibrium
Loading
Mass Fraction of Adsorbate in Gas
Rectangular
Isotherm
Linear
Isotherm
Gas In End
Gas Out End
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
9 of 41
Copyright © 2019 Andrew Wright
Chapter 6 – Non-Isothermal Mass Transfer
It was shown in the previous chapter that operating an isothermal moving bed adsorption process
should be done counter-currently rather than co-currently in order to minimise the adsorbent flow
rate required. However, for a favourable isotherm the difference in performance between the two
modes of operation may not be that great and predominantly depends on the equilibrium capacity of
the adsorbent as it leaves the adsorber. In isothermal operation, the equilibrium capacity depends
solely on the concentration of the adsorbable component in the gas phase, but in non-isothermal
operation temperature comes into play. If the adsorbent leaving the adsorber was hotter in counter-
current operation than co-current operation then this could dramatically change the optimum design.
Within the adsorber the heat transfer between the gas and solid phases obeys the same rules as for a
standard heat exchanger. This means in principle that heat transfer will be most effective through
counter-current operation rather than co-current operation. However, it is not obvious that good
heat transfer is necessarily desirable from the perspective of the effectiveness of the mass transfer
process and therefore needs to be better understood.
In a moving bed adsorption process there is also the question of what happens to the heat that is
liberated during the uptake of the adsorbable component? If this heat increases the temperature of
the adsorbent phase then this will also lower the equilibrium capacity that can be achieved as the solid
leaves the adsorber. It possible, it would be preferable to transfer the heat generated by adsorption
to the gas phase and keep the solid as cool.
The reason why heat transfer and the resulting temperature of the adsorbent is important for the
performance of the moving bed adsorption process can be seen from Figure 5. Each of the three
diagrams contains a solid line isotherm which represents the adsorbent at the gas outlet conditions of
the adsorber and a dotted line isotherm which represents the adsorbent at the gas inlet conditions.
At the gas inlet the concentration of adsorbate is higher than that at the outlet, but it has also been
assumed that the temperature at the gas inlet is higher than that at the outlet. Counter to the results
for the isothermal case, with a higher temperature at the gas inlet it is possible for the equilibrium
capacity to be lower than at the gas outlet. This could in turn lead to poorer performance in counter-
current operation than co-current operation depending on the temperature difference between the
two ends, the desired product mass fraction and the isotherm characteristics.
For an isotherm that is linear, achieving a lower capacity at the inlet versus the outlet is unlikely to be
achieved in practice because the effect of the adsorbate mass fraction is substantial. However, for
isotherms that are strongly favourable or even rectangular, it could be possible to achieve a lower
capacity at the gas feed end compared to the gas product end. This would therefore mean that
counter-current operation would perform less well than co-current operation due to the adsorbent
having a lower capacity as it left the adsorber. This would then lead to more adsorbent being required
to remove the same amount of adsorbate from the gas stream. It is therefore of interest to know if it
is possible to achieve a higher adsorbent outlet temperature in counter-current flow than co-current
flow and if so, under what conditions.
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
10 of 41
Copyright © 2019 Andrew Wright
Figure 5 – Equilibrium loading at inlet and outlet of adsorber for a favourable isotherm compared
to rectangular and linear types with non-isothermal counter-current flow
In Chapter 11, models are presented for co-current and counter-current heat transfer in a moving bed
adsorption process. These models include heat generated by adsorption in a uniform manner over
the length of the adsorber. This is a gross simplification and in a real process the rate of heat
generation at any point in the adsorber will depend on the rate of adsorption occurring at that point.
Equilibrium
Loading
Mass Fraction of Adsorbate in Gas
Equilibrium
Loading
Mass Fraction of Adsorbate in Gas
Equilibrium
Loading
Mass Fraction of Adsorbate in Gas
Rectangular
Isotherm
Favourable
Isotherm
Linear
Isotherm
Gas Outlet
Gas Inlet
Gas Outlet
Gas Outlet
Gas Inlet
Gas Inlet
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
11 of 41
Copyright © 2019 Andrew Wright
However, this simplified work still brings up some interesting observations without having to delve
into the more realistic, but more complex, coupled heat and mass transfer.
If there is no heat of adsorption then the heat transfer results for the adsorber are equivalent to those
of a simple two-stream heat-exchanger operating in either co-current or counter-current operation.
If the gas stream enters at a temperature below the adsorbent temperature then you would want to
operate the adsorber counter-currently. This maximises the heat transfer and brings the adsorbent
temperature down as low as possible. A low adsorbent temperature means a high capacity and
therefore less adsorbent is required to make the process work. If the gas enters at a higher
temperature than the adsorbent then it would be beneficial to minimise the heat transfer so that the
adsorbent is not heated up. Therefore, even for this simple scenario where the heat of adsorption is
excluded from the analysis, co-current operation could actually be better than counter-current.
Where the heat of adsorption is substantial then transferring as much energy as possible from the
adsorbent to the gas is generally more desirable than worrying about simple heating and cooling based
on inlet temperatures. The results in Chapter 11 show that the resulting temperature rise during co-
current flow operation is mitigated by the combined heat capacity of the gas and adsorbent. In
counter-current flow, the results obtained depend on whether the heat capacity of the gas or the
adsorbent is greater. If the flowing heat capacity of the gas is greater than the adsorbent, then the
heat of adsorption is transferred to the gas phase. This keeps the temperature of the adsorbent as
low as possible, maximising its capacity and therefore minimising the amount of material needed for
the adsorption process. If the adsorbent phase has the higher flowing heat capacity then the heat of
adsorption is retained within the adsorbent. The cooling effect of the gas stream is reduced and the
temperature rise of the adsorbent over the adsorber can then be higher than for the co-current flow
case. This means that the adsorbent leaves at a higher temperature, potentially resulting in a lower
capacity in counter-current mode than co-current mode.
It is also interesting to note from Chapter 11 that when the adsorbent flowing heat capacity is higher
than the gas flowing heat capacity, then both no heat transfer and infinite heat transfer gave the same
resulting temperature rise over the adsorbent (equations (96) and (99)). Therefore, improving heat
transfer between the adsorbent and gas phase may seem like a good idea in terms of getting the
adsorbent temperature down, but this is not always beneficial in counter-current mode as the overall
impact could be the opposite of that intended.
An argument could be made that even though the outlet temperature may be higher in counter-
current mode, that the overall performance may be better because of higher mass transfer driving
forces over the length of the adsorber. However, the analysis undertaken shows that in counter-
current mode, the temperature of the adsorbent over the adsorber length can be higher on average
than in co-current operation. If the equilibrium capacity is more strongly dependent on temperature
than mass fraction (i.e. a strongly favourable isotherm) then this is can lead to lower mass transfer
rates within the adsorber itself. Therefore, whether the analysis looks simply at outlet temperatures
alone or is extended to the average temperature of the adsorbent within the adsorber, it can still be
shown why counter-current operation may lead to worse performance than co-current operation.
The results from the heat transfer analysis in Chapter 11 are based on the assumption that heat
generation by adsorption is at a constant rate along the adsorber. However, even if this is a poor
assumption, the general findings are unlikely to change qualitatively. The way the heat of adsorption
distributes between the gas and adsorbent streams is still going to depend on the relative flowing heat
capacities. Where the adsorption actually takes place along the adsorber and the heat is released is
unlikely to have a huge impact because ultimately it is all summed together to give the outlet
conditions which predominantly determine the overall performance of the unit.
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
12 of 41
Copyright © 2019 Andrew Wright
Chapter 7 – Conclusions
Work has been performed looking at generalised cases of heat and mass transfer in a moving bed
adsorber. The results show that for isothermal processes, counter-current contacting of the gas and
adsorbent streams results in better performance than co-current contacting. However, when heat
effects are taken into account the overall results can be much more complex. It is quite possible for
scenarios to arise in which counter-current operation actually results in worse performance than co-
current operation. Whilst this should not be expected to occur in most cases, it is certainly something
to be aware of and to check on. The likelihood of co-current operation giving better results increases
where the adsorption isotherm is strongly favourable for adsorption (close to rectangular) rather than
linear and that the temperature rise due to the heat of adsorption is considerable.
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
13 of 41
Copyright © 2019 Andrew Wright
Chapter 8 – Nomenclature
8.1 Roman
Symbol
Description
Units
Total surface area between the gas and adsorbent particles
m2
Surface area per unit axial length between the gas and adsorbent particles
m2 m-1
Heat capacity ratio between the adsorbent stream and the gas stream
-
Specific heat capacity of the adsorbent
J kg-1 K-1
Specific heat capacity of the gas
J kg-1 K-1
Difference between and at any point along the length of the adsorber
-
Fractional loading of adsorbate on the adsorbent at a point along the adsorber
-
Fractional loading of adsorbate on the adsorbent at the bottom of the adsorber
-
Fractional loading of adsorbate on the adsorbent at the top of the adsorber
-
Fraction of adsorbate entering in the feed still remaining in the gas phase
-
Fraction of adsorbate still in the gas phase at the bottom of the adsorber
-
Fraction of adsorbate still in the gas phase at the top of the adsorber
-
Difference between and at the bottom of the adsorber
-
Difference between and at the inlets of the adsorber
-
Difference between and at the inlets of the adsorber
-
Heat generated per unit length of adsorber due to adsorption
W m-1
Heat of adsorption for the adsorbate
J kg-1
Mass transfer rate constant between the gas and adsorbent
s-1
Constant term in equation (65)
K
Length of the adsorber
m
Mass flowrate of adsorbent particles
kg s-1
Mass flowrate of gas along the adsorber
kg s-1
Number of heat generation units per unit axial distance along the adsorber
K m-1
Number of heat transfer units per unit axial distance along the adsorber
m-1
Number of mass transfer units per unit length of adsorber
m-1
Local loading of adsorbate on the adsorbent
kg kg-1
Loading of adsorbate on the adsorbent in equilibrium with the gas phase conditions
kg kg-1
Ratio of maximum adsorbent carrying capacity to feed adsorbate flow rate
-
Loading of adsorbate on the adsorbent in equilibrium with the feed gas phase conditions
kg kg-1
Temperature difference between the gas and adsorbent at a point along the adsorber
K
Temperature of the adsorbent
K
Temperature of the adsorbent at the bottom of the adsorber
K
Temperature of the adsorbent at the top of the adsorber
K
Temperature of the gas
K
Temperature of the gas at the bottom of the adsorber
K
Temperature of the gas at the top of the adsorber
K
Temperature difference between the gas and adsorbent at the bottom of the adsorber
K
Temperature difference between the gas at the bottom of the adsorber and the
adsorbent at the top
K
Temperature difference between the gas at the top of the adsorber and the adsorbent
at the bottom
K
Overall heat transfer coefficient between the gas and the adsorbent
W m-2 K-1
Distance along adsorber
m
Mass fraction of adsorbate in the feed to the adsorber
kg kg-1
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
14 of 41
Copyright © 2019 Andrew Wright
8.2 Greek
None
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
15 of 41
Copyright © 2019 Andrew Wright
Chapter 9 – References
None
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
16 of 41
Copyright © 2019 Andrew Wright
Chapter 10 – Isothermal, Trace Component Moving Bed Adsorption
Model
For this work the moving bed adsorption unit is modelled as a simple one-dimensional process as
shown in Figure 6. In this particular case gas shown flowing up the column along with adsorbent from
position 0 at the bottom to length at the top. The figure shows the gas and adsorbent particles
separated by an interface, but they could equally be considered as radially well mixed. Mass and heat
exchange occur between the gas and the outside surface of the adsorbent particles as they move
along the length of the adsorber. It is assumed that there is no axial mixing of the gas or adsorbent.
Figure 6 – Simplified representation of a moving bed adsorber
A material balance on the adsorbent phase at any point along the adsorber can be written as follows:
(1)
Mass flowrate of adsorbent particles (kg s-1)
Local loading of adsorbate on the adsorbent (kg kg-1)
Distance along adsorber (m)
Mass transfer rate constant between the gas and adsorbent (s-1)
Surface area per unit axial length between the gas and adsorbent particles (m2 m-1)
Loading of adsorbate on the adsorbent in equilibrium with the gas phase conditions (kg kg-1)
where:
(2)
Gas
Out
Adsorbent
Out
Gas
In
Adsorbent
In
0
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
17 of 41
Copyright © 2019 Andrew Wright
Total surface area between the gas and adsorbent particles (m2)
Length of the adsorber (m)
The left-hand-term in equation (1) represents the change in the amount of adsorbate loaded on the
adsorbent as it moves along the adsorber. The right-hand-term is the mass transfer rate from the gas
to the adsorbent. A linear driving force model is used to model this mass transfer process.
The local loading can be written as a fraction of the equilibrium loading based on the gas inlet feed
conditions as follows:
(3)
Loading of adsorbate on the adsorbent in equilibrium with the feed gas phase conditions (kg
kg-1)
Fractional loading of adsorbate on the adsorbent at a point along the adsorber (-)
The material balance for the gas phase can be written as follows noting that the mass of adsorbate
transferred to the adsorbent must equal the loss in mass from the gas phase.
(4)
Mass flowrate of gas along the adsorber (kg s-1)
Mass fraction of adsorbate in the feed to the adsorber (kg kg-1)
Fraction of adsorbate entering in the feed still remaining in the gas phase (-)
The term on the left-hand-side of equation (4) is the rate of change in adsorbate mass along the length
of the adsorber. This term assumes that the amount of adsorbate in the feed gas is small so that the
total mass flowrate of gas moving through the adsorber can be assumed to be constant.
For simplification of the equations the following two quantities are defined:
(5)
Number of mass transfer units per unit length of adsorber (m-1)
(6)
Ratio of maximum adsorbent carrying capacity to feed adsorbate flow rate (-)
The mass flow rate of gas and adsorbent are vector quantities and can therefore take positive or
negative values. For the adsorbent and gas flow in the same direction (i.e. co-current flow).
A negative value of means that the adsorbent and gas are flowing in opposite directions (i.e.
counter-current flow). If or then a minimum amount of adsorbent is being supplied
such that if the exit loading was in equal to the gas feed conditions, then all the adsorbate would be
removed from the gas phase. For an insufficient amount of adsorbent has been
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
18 of 41
Copyright © 2019 Andrew Wright
supplied to remove all the adsorbate from the gas phase, even if it reached equilibrium with the gas
feed conditions.
10.1 Rectangular Isotherm
With a rectangular isotherm, the equilibrium loading in equation (1) can be written as follows for any
position within the adsorber:
(7)
Combining equations (1), (3), (5) and (7) leads to:
(8)
This can be rearranged and integrated from the inlet of the adsorber to any position along it as follows:
(9)
(10)
Fractional loading of adsorbate on the adsorbent at the bottom of the adsorber (-)
This equation is a model for the loaded adsorbate as a function of the distance along the adsorber. It
does not matter whether the adsorber is run in co-current or counter-current operation. However, it
is assumed in this case that the adsorbent flows upwards as is likely to be a known quantity.
Combining equations (4) and (6) gives:
(11)
If co-current operation is employed so that the concentration of adsorbate at the bottom of the
adsorber is known then this equation can be integrated as follows:
(12)
(13)
Fraction of adsorbate still in the gas phase at the bottom of the adsorber (-)
Equation (13) uses the result from equation (10) and applies to both co-current and counter-current
gas flow, but it is unlikely that the value of will be known in counter-current flow. The value at
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
19 of 41
Copyright © 2019 Andrew Wright
the gas inlet, , will typically be available instead. This can be found from equation (13) by applying
it to the entire length of the adsorber as follows:
(14)
Eliminating from equation (13) using equation (14) then gives:
(15)
This equation is better suited for counter-current operation and determining the amount of adsorbate
in the gas phase over the length of the adsorber.
A constraint though in the use of these equations for the rectangular isotherm case is that they can
lead to negative gas phase concentrations. In reality the adsorbate would be completely removed
from the gas phase and at this point no more mass transfer will occur. The required length of adsorber
to completely remove the adsorbate from the gas phase is given by the following equation for both
co-current and counter-current operation:
(16)
An adsorber longer than this will have some sections where there is no adsorbate in the gas phase and
no mass transfer.
10.2 Linear Isotherm
With a linear isotherm the equilibrium loading at any point along the adsorber can be written as
follows:
(17)
This can be substituted into equation (1) and following further rearrangement and simplification the
adsorbent mass balance reduces to:
(18)
Following a similar procedure, the gas phase material balance can be written as follows:
(19)
The term, , is now defined as follows:
(20)
Difference between and at any point along the length of the adsorber (-)
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
20 of 41
Copyright © 2019 Andrew Wright
which allows equation (19) to be subtracted from (18) and written as follows:
(21)
Integrating this equation from the bottom of the adsorber to any position within the adsorber gives:
(22)
(23)
(24)
Difference between and at the bottom of the adsorber (-)
Equation (24) can now be substituted back into equation (18) to obtain the profile of fractional loading
on the adsorbent over the length of the adsorber:
(25)
(26)
(27)
The same can be done with equation (19) and equation (24) for the gas phase adsorbate profile.
(28)
(29)
(30)
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
21 of 41
Copyright © 2019 Andrew Wright
10.2.1 Counter-Current Flow
Equation (27) and (30) work for both co-current and counter-current flow. However, they require that
is known and this is only likely to be true for co-current flow where the inlet conditions at the
same end of the adsorber are known. For counter-current flow, an expression for is required
based on the known inlet conditions for the gas and adsorbent which will be at opposite ends of the
adsorber. One approach to obtaining such an expression is to take equation (27) and apply it to the
length of the adsorber:
(31)
Fraction of adsorbate still in the gas phase at the top of the adsorber (-)
Subtracting from both sides of this equation then gives:
(32)
where:
(33)
Difference between and at the inlets of the adsorber (-)
Rearranging equation (32) then gives:
(34)
Another approach to finding an expression for is to take equation (30) and write it over the whole
adsorber as follows:
(35)
Fractional loading of adsorbate on the adsorbent at the top of the adsorber (-)
Subtracting both sides of the equation from gives:
(36)
where
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
22 of 41
Copyright © 2019 Andrew Wright
(37)
Difference between and at the inlets of the adsorber (-)
Rearranging equation (36) then gives:
(38)
If then it is best to specify the adsorbent inlet to be at the bottom and the gas inlet to
be at the top of the adsorber. This will require the use of equation (34) to find . However, if
and
is positive then using this equation could prove difficult to solve numerically. This is
because the exponential terms in the equations could be very large and result in overflow errors in
the calculations. The approach to use in such a situation is to specify that the adsorbent inlet is at the
top of the adsorber and the gas inlet is at the bottom. This will make
a negative number and
the exponential terms will be easier to evaluate. In this case the value of should be calculated
from equation (38).
10.2.2 Q*=-1
For the special case that then equation (21) becomes:
(39)
Integrating this equation gives:
(40)
(41)
Substituting back into the adsorbent and gas phase material balances gives:
(42)
(43)
(44)
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
23 of 41
Copyright © 2019 Andrew Wright
(45)
(46)
As this is a counter-current flow case then is unlikely to be known and so an expression is required
based on the inlet gas and adsorbent conditions at the opposite ends of the adsorber. Assuming the
adsorbent is fed in at the bottom and the gas at the top, the material balance over the adsorber for
the gas phase can be written as:
(47)
Subtracting from both sides gives:
(48)
Rearranging gives:
(49)
which can be substituted back into equations (44) and (46) to give:
(50)
(51)
10.2.3 Linear Isotherm Mass Transfer Profiles
For the co-current case (equations (27) and (30)) the gas phase concentration and adsorbate loading
tend to equilibrium with each other at the outlet. If at the inlet and , then as
then the outlet values of and become:
(52)
If the goal is for a low concentration of adsorbate in the outlet gas, then the value of must be very
large. Figure 7 shows example profiles along an adsorber of and with co-current flow, ,
and :
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
24 of 41
Copyright © 2019 Andrew Wright
Figure 7 – Co-current flow results with Q* = 5
Figure 8 – Counter-current flow results with Q* = -5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5 6 7 8 9 10
fgand fa(-)
N'MTUx(-)
fg
fa
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5 6 7 8 9 10
fgand fa(-)
N'MTUx(-)
fg
fa
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
25 of 41
Copyright © 2019 Andrew Wright
Figure 8 shows the equivalent results for the same size adsorber as in Figure 7, but with counter-
current flow (for with ). The gas flows from left to right in this figure, whilst the adsorbent is
moving from right to left. The adsorbent leaves the adsorber with a higher loading than in the co-
current flow case and the gas leaves with a lower concentration of adsorbate. Counter-current flow
therefore gives better performance than co-current flow under the same conditions. The low loading
of adsorbate on the adsorbent is due to the large excess in adsorbent flow compared with the
theoretical minimum. The following figure shows the results for a more optimal .
Figure 9 – Counter-current flow results with Q* = -1
The amount of adsorbate removed from the gas phase is less than in the scenario, but still
greater than for the co-current flow case. The adsorbent though is closer to saturation as it leaves the
adsorber, making better use of the available adsorbent. A longer adsorber gives even better results
as shown in Figure 10:
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5 6 7 8 9 10
fgand fa(-)
N'MTUx(-)
fg
fa
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
26 of 41
Copyright © 2019 Andrew Wright
Figure 10 – Counter-current flow results with Q* = -1 and N’MTUL = 20
In counter current flow for as
then the outlet tends to the inlet . If the
adsorbent enters fully regenerated, then this means that the adsorbate mass fraction in the gas tends
to zero at the outlet and how close it gets depends on
for the adsorber.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 2 4 6 8 10 12 14 16 18 20
fgand fa(-)
N'MTUx(-)
fg
fa
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
27 of 41
Copyright © 2019 Andrew Wright
Chapter 11 – Moving Bed Adsorption Heat Transfer Model
The heat transfer model for the adsorber starts with the same simple one-dimensional model as
described in Chapter 10. Only this time, it is heat being transferred between the gas and adsorbent
rather than mass. Assuming only trace component adsorption and constant physical properties the
steady state heat balance for the gas phase can be written as follows:
(53)
Specific heat capacity of the gas (J kg-1 K-1)
Temperature of the gas (K)
Temperature of the adsorbent (K)
Overall heat transfer coefficient between the gas and the adsorbent (W m-2 K-1)
The heat lost from the gas phase is transferred to the adsorbent. The adsorbent is also heated up due
to mass transfer and the heat of adsorption.
(54)
Specific heat capacity of the adsorbent (J kg-1 K-1)
Heat generated per unit length of adsorber due to adsorption (W m-1)
Assuming that the heat generated due to adsorption is evenly distributed along the length of the
adsorber,
can be calculated from the following:
(55)
Heat of adsorption for the adsorbate (J kg-1)
In practice the heat of adsorption is not released uniformly along the length of the adsorber and this
means that the heat and material balances are coupled. However, it will be shown that the results
from this analysis can still be used to help understand the general rules for more realistic scenarios.
The following terms are used to help simplify some of the equations above:
(56)
Temperature difference between the gas and adsorbent at a point along the adsorber (K)
(57)
Number of heat transfer units per unit axial distance along the adsorber (m-1)
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
28 of 41
Copyright © 2019 Andrew Wright
(58)
Number of heat generation units per unit axial distance along the adsorber (K m-1)
(59)
Heat capacity ratio between the adsorbent stream and the gas stream (-)
Using the simplifying terms above, the following heat balance equations can be written for the gas
and adsorbent phases:
(60)
(61)
Subtracting equation (61) from (60) gives:
(62)
This can be integrated from the inlet of the adsorber to any point within it as follows:
(63)
(64)
(65)
where:
(66)
Temperature difference between the gas and adsorbent at the bottom of the adsorber (K)
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
29 of 41
Copyright © 2019 Andrew Wright
(67)
Constant term in equation (65) (K)
Substituting equation (65) back into (60) gives:
(68)
This can then be integrated to give the gas phase temperature profile.
(69)
(70)
Temperature of the gas at the bottom of the adsorber (K)
Similarly, substituting equation (65) into (61) allows the adsorbent phase temperature profile to be
obtained after integration.
(71)
(72)
(73)
Temperature of the adsorbent at the bottom of the adsorber (K)
Equation (70) and (73) apply generically to both co-current and counter-current flow arrangements,
but they require and to be known which is only likely to be true for the co-current scenario.
In order to use these equations in counter-current situations where only one of and may be
known, a value for must be found.
11.1 Counter-Current Flow -1 < C* < 0
One way of calculating is to apply equation (70) to the entire length of the adsorber:
(74)
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
30 of 41
Copyright © 2019 Andrew Wright
Temperature of the gas at the top of the adsorber (K)
Subtracting and adding to both sides of this equation gives the following expression:
(75)
where:
(76)
Temperature difference between the gas at the top of the adsorber and the adsorbent at the
bottom (K)
Equation (75) can then be rearranged to give:
(77)
11.2 Counter-Current Flow C* < -1
An alternative expression for can be obtained by applying equation (73) across the entire adsorber
to give:
(78)
Temperature of the adsorbent at the top of the adsorber (K)
Adding to both sides of the equation and subtracting them from gives:
(79)
where:
(80)
Temperature difference between the gas at the bottom of the adsorber and the adsorbent at
the top (K)
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
31 of 41
Copyright © 2019 Andrew Wright
Rearranging equation (79) then gives:
(81)
Using this expression for when is preferable because then
is also negative. This
means that the exponential term is of a negative number and this can be computed without having to
worry about potential overflow errors.
11.3 Counter-Current Flow C* = -1
For the special case that , equation (62) simplifies down to:
(82)
This can be integrated easily to obtain the temperature difference over the length of the adsorber.
(83)
(84)
Substituting this result back into the gas and adsorbent differential heat balance equations allows the
gas and adsorbent temperature profiles to be determined.
(85)
(86)
(87)
(88)
(89)
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
32 of 41
Copyright © 2019 Andrew Wright
(90)
As this is for a counter-current flow case then is unlikely to be known upfront. It can instead be
found by applying equation (87) over the entire adsorber:
(91)
Subtracting from both sides of the equation then gives:
(92)
which can be rearranged to end up with:
(93)
11.4 Heat Transfer Profiles
Figure 11 shows an example of heat transfer between the adsorbent and gas when the adsorber is
operated in co-current mode and there is no heat generation by adsorption. For this case,
, ,
,
, , and . The results are simply those
expected for two streams undergoing co-current heat transfer with the temperatures approaching
each other to a final temperature that is a weighted average of the inlet temperatures. With ,
the average temperature is 80% of the adsorbent temperature and 20% of the gas temperature.
Figure 12 shows what happens in the same case when heat generation by adsorption is introduced,
.
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
33 of 41
Copyright © 2019 Andrew Wright
Figure 11 – Co-current heat transfer with no heat generation by adsorption
Figure 12 – Co-current heat transfer with heat generation by adsorption
18
20
22
24
26
28
30
32
34
36
38
40
42
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Temperature (°C)
L(m)
Gas Temperature
Adsorbent Temperature
18
20
22
24
26
28
30
32
34
36
38
40
42
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Temperature (°C)
L(m)
Gas Temperature
Adsorbent Temperature
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
34 of 41
Copyright © 2019 Andrew Wright
The steady heat generated by adsorption over the length of the adsorber causes the temperatures to
continuously rise. The temperature difference between the gas and adsorbent gets smaller along the
adsorber length, but heat is continuously generated within the adsorbent and this is partially
transferred over to the gas phase. The temperature of the adsorbent is therefore continually being
mitigated by the heat transferred over to the gas phase.
Figure 13 shows what happens with the same adsorber design as for Figure 11 expect that the path
taken by the adsorbent has been reversed so that it enters on the right-hand-side of the chart and
exits on the left. With no heat of adsorption in counter-current mode and these conditions, the
adsorbent leaves the process at a slightly lower temperature than it does in co-current mode. This
would be favourable for the adsorption process because it would mean that the adsorbent would have
a higher capacity as it leaves purely resulting from the lower operating temperature, never mind the
higher concentration of adsorbate at the gas entrance.
Figure 13 – Counter-current heat transfer with no heat generation by adsorption
Figure 14 on the other hand shows what can happen when the heat of adsorption is taken into account
in counter-current mode,
. Compared with Figure 12, the adsorbent is now leaving the
adsorber at a higher temperature than in co-current mode and this higher temperature may actually
cause the equilibrium capacity of the adsorbent to be lower in counter-current flow. The reason for
this change in results is that the heat of adsorption in co-current mode is spread between the
adsorbent and the gas phase, whilst in counter-current mode the heat of adsorption is mostly retained
within the adsorbent. This means that the temperature rise of the adsorbent in counter-current mode
is higher due to the heat of adsorption. The cooling effect of the gas is unable to counteract the
temperature rise due to adsorption and the overall effect is to increase the outlet adsorbent
temperature.
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Temperature (°C)
L(m)
Gas Temperature
Adsorbent Temperature
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
35 of 41
Copyright © 2019 Andrew Wright
Figure 14 – Counter-current heat transfer with heat generated by adsorption
The adsorbent temperature inside the adsorber is also higher on average than in the co-current flow
case. This means that equilibrium capacities and hence mass transfer driving forces within the
adsorber could be lower in the counter-current flow case. It would of course depend on whether the
temperature or the adsorbate mass fraction has a bigger impact on the equilibrium capacity. For
highly favourable (i.e. near rectangular) isotherms the temperature effect could be greater.
It may be thought that an increase in heat transfer between the adsorbent and the gas in counter-
current mode would be beneficial as more of the heat of adsorption could then be transferred over
to the gas. This would then cool the adsorbent and increase its capacity. However, this may not
necessarily be advantageous.
Figure 15 for example shows what happens in the case in Figure 12 when the rate of heat transfer is
increased by an order of magnitude,
so that the gas and adsorbent temperatures close in
to each other. Figure 16 then shows what happens when this system is run in counter-current mode.
Even though the heat transfer rate is an order of magnitude higher than in Figure 14, the adsorbent
actually leaves with a higher temperature at the bottom of the adsorber. The higher heat transfer
rate is causing the gas to be cooled down as it passes along the adsorber and this heat is ending up
with the adsorbent. Therefore, a higher heat transfer rate between the adsorbent and gas could
actually be damaging in terms of the adsorber mass transfer performance due to lower adsorbent
capacities.
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Temperature (°C)
L(m)
Gas Temperature
Adsorbent Temperature
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
36 of 41
Copyright © 2019 Andrew Wright
Figure 15 – Co-current heat transfer with heat of adsorption and high heat transfer rate
Figure 16 – Counter-current heat transfer with heat of adsorption and high heat transfer rate
18
20
22
24
26
28
30
32
34
36
38
40
42
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Temperature (°C)
L(m)
Gas Temperature
Adsorbent Temperature
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Temperature (°C)
L(m)
Gas Temperature
Adsorbent Temperature
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
37 of 41
Copyright © 2019 Andrew Wright
As a general observation regarding this result, combining equation (78) with (81) gives:
(94)
With and assuming , as
, then the above equation reduces to:
(95)
(96)
This is the temperature rise of the adsorbent over the adsorber if there is infinite heat transfer with
the gas phase. Solving equation (94) with
gives:
(97)
and noting that L’Hôpital’s rule as
gives:
(98)
means that the overall result is the same as that of equation (96):
(99)
Therefore, whether the heat transfer rate is infinite or zero, the temperature rise over the adsorber
for the adsorbent is the same. Compared to the extremes, intermediate heat transfer rates give lower
temperature rises for the adsorbent.
In the counter-current analyses performed above the heat capacity of the adsorbent was taken to be
equal to four times that of the gas (). This means that the heat of adsorption was
preferentially picked up and transported along the adsorber by the adsorbent. This has a big impact
on the outlet temperature of the adsorbent from the adsorber. If instead the gas heat capacity is
higher than the adsorbent heat capacity then the heat of adsorption is preferentially carried out by
the gas. This changes the results for the counter-current flow heat transfer case.
Figure 17 and Figure 18 are equivalent to those of Figure 14 and Figure 16 except that the flowing heat
capacity of the two streams has been reversed (). This means that instead of the
adsorbent having four times the flowing heat capacity of the gas, the gas has four times the flowing
heat capacity of the adsorbent. In the new figures the adsorbent leaves the bottom of the adsorber
at a noticeably lower temperature as the heat of adsorption is instead carried away by the gas leaving
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
38 of 41
Copyright © 2019 Andrew Wright
from the top of the adsorber. Whilst this is clearly a more beneficial scenario, the relative heat
capacity of the adsorbent to the gas is not something that can easily be manipulated but it does show
that a substantial change in fundamental operating principles can occur if such a swap can be
obtained.
Figure 17 – Counter-current flow with heat generation by adsorption and high gas heat capacity
18
20
22
24
26
28
30
32
34
36
38
40
42
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Temperature (°C)
L(m)
Gas Temperature
Adsorbent Temperature
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
39 of 41
Copyright © 2019 Andrew Wright
Figure 18 – Counter-current flow with heat of adsorption, high gas heat capacity and high mass
transfer rate
If the adsorbent and gas have approximately the same flowing heat capacity, then this can lead to
some rather unusual results. Figure 19 shows the results for the following counter-current case;
, ,
,
, and . The heat of adsorption
is transferred out of the adsorber by both the adsorbent and the gas, although due to the low rate of
heat transfer, most of the heat is retained within the adsorbent. Figure 20 shows that as the heat
transfer rate between the adsorbent and gas is increased to
the outlet adsorbent and gas
temperatures close in and the heat of adsorption is equally distributed between the two streams.
However, the internal temperatures within the adsorber notably increase. This can be seen with
further clarity in Figure 21 where the heat transfer rate has been increased to
. A higher
heat transfer rate results in the heat of adsorption being trapped inside the adsorber when is close
to -1 and this builds up the temperatures of the streams. This of cause would not happen in a real
adsorber because the assumption of constant adsorption rate and heat generation would be very poor
under these conditions, but it is an interesting observation in such an idealised case.
18
20
22
24
26
28
30
32
34
36
38
40
42
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Temperature (°C)
L(m)
Gas Temperature
Adsorbent Temperature
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
40 of 41
Copyright © 2019 Andrew Wright
Figure 19 – Counter-current heat transfer with C* = -1 and N’HTU = 0.25
Figure 20 – Counter-current heat transfer with C* = -1 and N’HTU = 2.5
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Temperature (°C)
L(m)
Gas Temperature
Adsorbent Temperature
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Temperature (°C)
L(m)
Gas Temperature
Adsorbent Temperature
Counter-Current versus Co-Current Moving Bed Adsorption
11th February 2019
41 of 41
Copyright © 2019 Andrew Wright
Figure 21 – Counter-current heat transfer with C* = -1 and N’HTU = 25.0
18
68
118
168
218
268
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Temperature (°C)
L(m)
Gas Temperature
Adsorbent Temperature