6. Flow-through packed beds.

6.1 Introduction
The flow of fluids through beds composed of stationary granular particles is a frequent occurrence in
the chemical industry. This flow will cause a pressure drop and therefore, expressions are needed to
predict the pressure drop across the bed due to the resistance caused by the presence of the particles.
Examples of fixed bed operations include catalytic reactors, adsorption columns, ion exchange columns,
and sand filters.
By definition, a packed bed is passing fluid between the pores of a particulate phase which is contained
in a bed. The fluid flow rate through the bed is Q (m 3/s) and the bed cross-sectional area is A (m2).
The superficial (empty bed) velocity Uo is then defined as the total flow rate divided by the bed cross-
sectional area.

6.2 Darcy’s Law

The first experimental work on this subject was done by Darcy (in 1830) when he examined the flow
rate of water through sand beds with various thicknesses. He concluded that the pressure drop per unit
length of the bed was proportional to the superficial water velocity, or
−∆𝑃 1
= 𝐵𝑢𝑜
𝑙
and B is the permeability coefficient. One would expect B to be a function of the particle size, as well
as the packing of the bed ().
The linear relation between the flow rate and the pressure difference leads to the assumption that the
flow was laminar. , is expected because the Reynolds number for the flow through the pore spaces in
a granular material is low since both the velocity of the fluid and the width of the channels usually are
small. Subsequent test work suggested that the viscosity of the fluid has an effect, and hence we can
write Darcy’s equation as
(−∆𝑃)
𝑢0 = 𝐵
𝜇𝐿

6.3 General expression for flow through a packed bed

6.3.1 Laminar flow
The analogy between streamline flow through a tube, and streamline flow through the pores in a bed
of particles, is a useful starting point for deriving a general expression.
Let us assume that the porous bed consists of several channels, each of the same length (lc), same
diameter (dc) and same linear velocity (uc). The Darcy Weisbach equation gives the pressure drop
through a single channel
−∆𝑃 𝑓𝑢𝑐2 𝜌 2
=
𝑙𝑐 2𝑑𝑐
If assumed that the flow through each channel is small enough such that there is only streamlined flow
(laminar flow), the friction factor f = 64/Re, therefore
−∆𝑃 32𝜇𝑢𝑐 3
=
𝑙𝑐 2𝑑𝑐2
The unknown variables in equation 3 have to be written in terms of easily measurable quantities.
For dc:
𝐹𝑟𝑒𝑒 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑏𝑒𝑑 4
𝜖=
𝑇𝑜𝑡𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑏𝑒𝑑
 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 5
𝑆𝐵 =
𝑇𝑜𝑡𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑏𝑒𝑑

Therefore
𝜖 𝐹𝑟𝑒𝑒 𝑉𝑜𝑙𝑢𝑚𝑒 6
=
𝑆𝐵 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠

The hydraulic mean diameter of a conduit is defined as

4 × 𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝑓𝑜𝑟 𝑓𝑙𝑜𝑤 4𝑆 7
𝑑𝐻𝑀 = =
𝑊𝑒𝑡𝑡𝑒𝑑 𝑤𝑎𝑙𝑙 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝐵

The cross-sectional area x length of the channel = Free (open) area

Wetted wall circumference x length of channel = Total surface area of particles
Thus
4𝑆𝐿 𝐹𝑟𝑒𝑒 𝑣𝑜𝑙𝑢𝑚𝑒 𝜖 8
𝑑𝑐 = =4 =4
𝐵𝐿 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 𝑆𝐵
For uc:
The solids will be packed in the bed with a specific porosity (which is usually an isotropic property, i.e.
same in all directions); hence the velocity through the porous (the interstitial velocity) is related to the
superficial velocity.
Assume there are n channels, then uc = Q/nAc
but nAcL = AL
therefore
𝑄 𝑢𝑜 9
𝑢𝑐 = =
𝜖𝐴 𝜖
For lc:
Assume that the ratio of the length of the tortuous flow channels to the total bed length is constant,
i.e. lc/L = constant (or lc = L x K)
Using all these relations in Equation 3, we get
−∆𝑃 (1 − 𝜖)2 𝜇𝑢𝑜 10
= 𝐾′
𝐿 𝜖3 2
𝑑𝑠𝑣
Carman & Kozeny experimentally found K’ = 180

6.3.2 Laminar and turbulent flow

Forcheimer (1930) suggested that the resistance to flow be considered in two parts,
- The viscous drag at the surface of the particles
- The loss in turbulent eddies and sudden changes in the cross-section of the channels.
i.e.,∆𝑃 = 𝛼𝑢𝑐 + 𝛼 ′𝑢𝑐𝑛
Based on this, Ergun developed a semi-empirical equation
−∆𝑃 (1 − 𝜖)2 𝜇𝑢𝑜 (1 − 𝜖) 𝜌𝑢𝑜2 11
= 150 3 2
+ 1.75
𝐿 𝜖 𝑑𝑠𝑣 𝜖3 𝑑𝑠𝑣
The above equation suggests that the only bed properties on which the pressure gradient depends, are
its specific surface, S (or particle size, d) and its porosity, ϵ. However, the structure of the bed
additionally depends on the particle size distribution, the particle shape, and how the bed has been
packed. The walls of the container and the nature of the bed support can considerably affect the way
the particles are packed. It will be expected, therefore, that experimentally pressure gradient values
differ considerably from values predicted by the equations.

6.3.3 Dependence on bed structure

6.3.3.1. Tortuosity.
Although it was implied during the derivation of equation 10 that a single value of the Kozeny constant
K’ applies to all packed beds, this assumption does not hold in practice. The tortuosity is a measure of
the fluid path length through the bed, compared with the actual depth of the bed, and K’ depends on
the shape of the cross-section of a channel through which fluid is passing.
In general, the values of the Carman – Kozeny constant (180), or the Ergun constant (150) hold. If
more accurate values are required, it should be determined experimentally. A graph of a modified
constant (R’ = 5 for Carman-Kozeny and 4.17 for Ergun) is shown in Figure 1.

Figure 1 Voidage vs modified flow resistance

6.3.3.2 Wall effect.

In a packed bed, the particles will be less dense in the region near the wall compared to the centre of
the bed. Therefore the actual resistance to flow in a bed of small diameter is less compared to an
infinite container for the same flow rate per unit area of bed cross-section. An experimental correction
factor fw is given by
1 𝑆𝑐 2
𝑓𝑊 = (1 + )
2𝑆
Sc is the surface of the container per unit volume of bed.

6.3.3.3 Sphericity and voidage

In the ideal case, there is a broad relationship between the sphericity of the particles and the voidage.
This relationship will depend on the packing of the bed, a bed which is tightly packed will have a much
lower voidage than a bed of densely packed particles. Figure 3 shows a general correlation which was
developed by Kunii and Levenspiel.
 Figure 2 Correlation of porosity vs sphericity

6.4 Determination of particle surface

The Carman–Kozeny equation relates the drop in pressure through a bed, to the specific surface of the
material. It is used to calculate S via experimental measurements in the pressure drop. This method is
only suitable for beds of uniformly packed particles and not for measuring the size distribution of
particles in the sub-sieve range. The experimental setup, developed by Lea and Nurse, is shown in
Figure 3. Air or any other suitable gas flows through the bed contained in a cell (25 mm diameter, 87
mm deep). The pressure drop is obtained from h1 and the gas flow rate from h2. This technique is
called the Blaine test using the Blaine air-permeability apparatus and is described in detail in ASTM
C204-18

Figure 3 Air permeability meter