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Even Flow Distribution and Maximising Pressure-Drop
21st February 2019
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Copyright © 2019 Andrew Wright
Technical Report
Is Maximising Pressure-
Drop over a Packed Bed
Synonymous with Achieving
the Best Flow Distribution?
(Answer – No)
21st February 2019
Andrew Wright
Even Flow Distribution and Maximising Pressure-Drop
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Copyright © 2019 Andrew Wright
Chapter 1 – Executive Summary
This technical report is a simple look at the theoretical distribution of gas through a packed bed where
the voidage varies across the diameter. The results show that if the bed flow resistance varies purely
because of the voidage, then in a bed with even flow distribution the overall pressure-drop is highest.
If there is maldistribution of flow then the overall pressure-drop reduces so that better flow
distribution is therefore synonymous with a higher pressure-drop.
If the bed flow resistance varies because of a factor other than voidage (e.g. particle diameter) then
an even flow distribution does not coincide with the maximum possible pressure-drop. Higher
pressure-drops can be achieved with maldistributed flow in this case. Therefore, there are situations
where designing a packed bed to achieve the maximum possible pressure-drop may not result in the
best flow distribution.
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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 – Packed Bed Flow Distribution and Pressure-Drop Model ................................................... 6
Chapter 6 - Packed Bed Flow Distribution and Pressure-Drop Model (Quadratic Velocity Resistance)
.............................................................................................................................................................. 11
Chapter 7 – Conclusions ........................................................................................................................ 13
Chapter 8 – Nomenclature .................................................................................................................... 14
8.1 Roman ................................................................................................................................... 14
8.2 Greek ..................................................................................................................................... 14
Chapter 9 – References ......................................................................................................................... 15
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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
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Chapter 4 – Introduction
A packed bed may not always be constructed homogeneously and this can lead to different flow
resistances in different sections of the bed. These can be due for example to variations in bed voidage
or particle size. A question therefore arises about what must be done to even up the flow distribution
through the packing construction where there are variations in bed flow resistance? For example, is
there anything that can be done to the local packing voidage to offset a difference in particle size. A
second question is then what is the resulting pressure-drop under the conditions of equalised flow
and does this result in a maximal possible pressure-drop through the system?
The purpose of the work in this report is to theoretically investigate these questions.
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Chapter 5 – Packed Bed Flow Distribution and Pressure-Drop Model
The part of the vessel of interest is assumed to be formed of two separate packed sections as shown
in Figure 1. The inlet and outlet pressures are assumed to be the same for the two sections and whilst
the total volumetric flow rate is known, the distribution of flow between the two parts needs to be
determined.
Figure 1 – Schematic Representation of the Packed Bed Model
The part of the vessel of interest is assumed to have a total volume , and the two sections each take
up half this total volume. There is also a total volume of packing, , that needs to be placed into the
vessel. It is assumed that the density of packing placed into the vessel varies between the two
sections, but that the density within each individual section does not vary radially or axially. The
voidage in the two sections can then be written as follows:
(1)
(2)
Voidage of the bed in packed section 1 (-)
Voidage of the bed in packed section 2 (-)
Volume fraction of total packing in packed section 1 (-)
Total volume of the vessel (m3)
Total volume of packing (m3)
For all the packing is in the first section, for all the packing is the second section and for
the packing is evenly distributed between the two sections. There are upper and lower limits
Fluid
Out
1
Fluid
In
2
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on acceptable values of limited by the constraint that the voidages in the two sections must lie
between 0 and 1.
The pressure-drop over the two sections can be written as follows using the Darcy equation [1]:
(3)
(4)
Pressure-drop over section 1 of packing (Pa)
Pressure-drop over section 2 of packing (Pa)
Length of the packed bed (m)
Flow resistance constant for the packing in section 1 (Pa m-2 s)
Flow resistance constant for the packing in section 2 (Pa m-2 s)
Function of the voidage in section 1 (-)
Function of the voidage in section 2 (-)
Superficial velocity of the fluid in section 1 (m s-1)
Superficial velocity of the fluid in section 2 (m s-1)
The Darcy equation is assumed to depend upon the voidage through a general function . It is
assumed that the overall volumetric flow that must be processed through the packed bed is known
and given by . The velocities through the two sections of the packed bed can then be related
through:
(5)
Total cross-sectional area over the radius of the packed bed (m2)
Overall volumetric flow through the vessel (m3 s-1)
This equation can then be rearranged and separated into the following expressions:
(6)
(7)
Volume fraction of flow through packed section 1 (-)
At steady-state, the pressure-drop over both sections will be the same and hence equations (3) and
(4) can be combined to give:
(8)
Making use of expressions (6) and (7) then gives:
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(9)
If the desire is for an even flow distribution then and this simplifies equation (9) to:
(10)
If it is further assumed that then:
(11)
i.e.
(12)
This means that where the only potential difference between the two sections is the voidage, to
achieve an even flow distribution the voidage between the two sections must be the same. If other
parameters can be varied, such as the particle diameter in the two sections (i.e. a different from
) then an even flow distribution requires the two sections to have different voidages to compensate.
Equation (9) can be rearranged to determine the actual split of volumetric flow between the two
sections of packing depending on the relative bed resistances:
(13)
Differentiating this equation gives the rate of change in with respect to :
(14)
Rate of change of with respect to (-)
Rate of change of with respect to (-)
Rate of change of with respect to (-)
The rate of change of the pressure-drop with respect to can be obtained by differentiating equation
(3) with respect to and substituting in equation (6) for :
(15)
as found from equation (14) can be used to rewrite this expression as follows:
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(16)
If the flow is evenly distributed between the two sections then and this reduces the previous
equation to:
(17)
As the flow is evenly distributed then equation (10) can be used to simply this expression to the
following:
(18)
Collecting terms then gives:
(19)
Taking out the common factor of 2 and using equation (10) again to eliminate the voidage function
ratio leads to:
(20)
The voidage function differential terms can be expanded as follows:
(21)
From equation (1) and (2) the differentials of and with respect to can be obtained:
(22)
Substituting this result back into equation (21) gives:
(23)
If then it was shown previously that for an evenly distributed flow, . If this result is
applied to equation (23) then
becomes zero, which means that the pressure-drop obtained
is at a maximum. Therefore, under the scenario, when the packing is distributed to achieve
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an equal flow distribution, the pressure-drop is at its highest. Deviations from an even packing
distribution result in a lower overall pressure-drop but maldistributed flow.
However, what happens for ? In this case equation (23) can be rearranged to:
(24)
The ratio of bed resistances can instead be written as follows using equation (10):
(25)
Taking out as a common factor leads to:
(26)
The voidage differential terms can then be written as differentials of logarithmic terms:
(27)
Only in the case that will the term in curled brackets equal zero (so that the pressure-drop
gradient with respect to equals zero) and this is only true in practice for . If and
hence then at even flow distribution the pressure-drop will not be maximal. There would be
a possible distribution of packing that resulted in a higher pressure-drop and also maldistributed flow.
So, depending upon the specific circumstances it is possible to have situations where the flow is evenly
distributed but the pressure-drop is not at its highest.
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Chapter 6 - Packed Bed Flow Distribution and Pressure-Drop Model
(Quadratic Velocity Resistance)
The previous chapter used the Darcy equation for the flow resistance model, but the same findings
are obtained when the flow resistance is quadratic with respect to velocity. In order to prove this, the
pressure-drop over the two sections of packed bed are now written as follows assuming a quadratic
relationship of velocity with pressure:
(28)
(29)
At steady-state, the pressure-drop over both sections will be the same and hence equations (28) and
(29) can be combined to give:
(30)
Making use of expressions (6) and (7) gives:
(31)
For an even flow distribution and this simplifies equation (31) to:
(32)
This finding is the same result as equation (10) when the Darcy assumption was used.
Equation (31) can be differentiated with respect to using the chain rule to give:
(33)
This equation can be rearranged and simplified using equation (32) to obtain the rate of change of
with respect to , :
(34)
Substituting equation (31) back into this expression allows a further simplification:
(35)
Substituting equation (6) into (28) allows the pressure-drop over section 1 to be written as follows:
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(36)
Differentiating this equation with respect to and applying the chain rule gives:
(37)
Substituting equation (35) into this expression and taking out as a common factor gives:
(38)
When the flow is evenly distributed then and this gives:
(39)
Which can be rearranged to:
(40)
Expanding the voidage function differential terms using the chain rule gives:
(41)
The voidage differentials can be found from equation (22) and this leads to the following expression
where the voidage function differentials have been reduced to logarithmic form:
(42)
This equation is analogous to (27) which was obtained for the model Darcy. Therefore, whether the
bed resistance is linear or quadratic with respect to velocity does not matter in terms of the findings
in this work. There is also no reason to believe that if the flow resistance was a combination of both
linear and quadratic velocity terms (e.g. Ergun equation [2]) that the general results would be
different.
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Chapter 7 – Conclusions
Work has been performed looking at how the voidage needs to vary inside a packed bed in order to
compensate for other flow resistance deviations such as changes in particle diameter. If even flow is
required and the packing material can be treated as homogeneous then the voidage in all sections
should be made the same. This also results in the maximum possible pressure-drop and whilst lower
pressure-drops can be achieved, they will result in flow maldistribution.
However, if the packing material varies between sections (e.g. differences in particle diameter) then
the voidage between sections will need to be changed to compensate so that even flow is obtained.
In this case, the pressure-drop obtained under even flow conditions will not be the maximum possible.
It would be possible to pack the bed in such a way as to achieve a higher pressure-drop but along with
flow maldistribution. Therefore, maximising pressure-drop in a design is not always synonymous with
achieving the best possible flow distribution.
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Chapter 8 – Nomenclature
8.1 Roman
Symbol
Description
Units
Total cross-sectional area over the radius of the packed bed
m2
Flow resistance constant for the packing in section 1
Pa m-2 s
Flow resistance constant for the packing in section 2
Pa m-2 s
Length of the packed bed
m
Pressure-drop over section 1 of packing
Pa
Pressure-drop over section 2 of packing
Pa
Overall volumetric flow through the vessel
m3 s-1
Superficial velocity of the fluid in section 1
m s-1
Superficial velocity of the fluid in section 2
m s-1
Total volume of the vessel
m3
Total volume of packing
m3
Volume fraction of total packing in packed section 1
-
Volume fraction of flow through packed section 1
-
Rate of change of with respect to
-
8.2 Greek
Symbol
Description
Units
Function of the voidage
-
Voidage of the bed in packed section 1
-
Function of the voidage in section 1
-
Rate of change of with respect to
-
Voidage of the bed in packed section 2
-
Function of the voidage in section 2
-
Rate of change of with respect to
-
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Chapter 9 – References
[1] Darcy, H. P. G. (1856). Les fontaines publiques de la ville de Dijon. Exposition et application à suivre
et des formules à employer dans les questions de distribution d’eau. Paris: Victor Dalmont.
[2] Ergun, S. (1952). Fluid flow through packed columns. Chem. Eng. Prog., 48(2), 89-94