Packing Compression

Depending on the method of packing preparation, the particles arrange in a random loose packing configuration. However, in many practical situations it is necessary to increase the density of the packed bed, i.e. in chromatographic applications to maintain the column packing performance. The most common modes of packing consolidation and compaction are shaking or tapping of the column (Figure 3.13 A), compression by high flow velocities (Figure 3.13 B) or uniaxial mechanical compression by lowering a plunger (Figure 3.13 C).

38

Chapter 3. Physical Background

A

Shaking, Tapping

B

F_f

C

Fm

Figure 3.13.: Modes of packing compaction. (A)Shaking or tapping of the column,(B) com-pression by high flow velocities and drag forces Ff and (C)uniaxial compression by a plunger with Fm.

During compression, the initial volume of the packingV0 is reduced by a portionV0−V so that a compression factor can be defined [Stickel and Fotopoulos 2001]

λ= V0 −V V0

. (3.72)

In the case of uniaxial compression or flow compression, which is the usually applied in chromatographic packing preparation, the cross-sectional area of the packed bed remains constant so thatλ= (H0−H)/H0 with H being the height of the packing.

Assuming that the compression of the packing is mainly caused by a reduction of the external porosity and that the compression of the individual particle is negligible, the packing density φ and the compression factor λ are correlated by

φ= 1−0

1−λ, (3.73)

where 0 is the average initial external porosity of the packing in the uncompressed state. By definition, the average packing porosity in the compressed state is given by = (0−λ)/(1−λ).

For large packing compressions, i.e. during column packing, Keener et al. [2004a]

defined the packing compression as λ^∗ = (H₀ −H)/H referencing to the compressed state. This leads to

φ= (1−0)(1 +λ). (3.74)

Chapter 3. Physical Background

The compression of the packed bed in liquid chromatography due to hydrodynamic load is a general problem for chromatographers [Verhoff and Furjanic Jr. 1983]. The reduction of the interstitial void fraction due to packed bed compression, causes a non-linear increase in flow rate with pressure drop (see e.g. Equation (3.63)). Hence, the net force acting on any point of the packed bed is the sum of forces originating from the flow, the mechanical compression force caused by the end pieces of the column, the friction between the packing material and the column wall as well as the gravitational force due to density differences of the packing material and the liquid (Figure 3.14).

z

Figure 3.14.: Forces acting on a compressible packed bed. (A)Force balance of an axial segment of thicknessdzof the compressible packed bed of diameterD. (B)Forces acting on single particles of the packed bed. Ff denotes the drag force of the fluid, Fg is due to the weight of the particle, Fn,w

andFt,w are the normal and friction force of the wall,Fn,p andFt,p are the normal and friction force of neighboring particles.

A one-dimensional stress balance across a column segment results in

∂σ

∂z =−∂p

∂z − 4

Dµ_w ν

1−νσ+ ∆ρ(1−)g, (3.75) where σ is the axial solid stress, ∂p/∂z is the pressure gradient given by the Kozeny-Carman equation (Equation (3.57)), D is the column diameter, µ_w is the wall friction coefficient,ν the packing Poisson’s ratio,∆ρ is the density difference between the pack-ing material and the fluid,is the packed bed porosity andg the gravitational constant.

Generally, the stress transmitted from the fluid flow is assumed to be equal to the hy-drodynamic pressure drop [Östergren and Trägårdh 1999]. As is can be seen in Equation (3.75), the solid stress σ, due to the wall friction, is less than expected from the pres-sure drop and the gravitational force. The wall friction force always opposes the motion

40

Chapter 3. Physical Background

of the packing during compression and relaxation, which is known as the wall support.

The influence of the wall friction on the motion of the packing decreases with increasing column diameter. Thus, the shear stress originating from the wall friction is highest at the column wall and decreases towards the column axis, causing variations in the solid stressσ also in radial direction [Tiller and Lu 1972; Östergren and Trägårdh 1999].

In chromatographic applications, the stress originating from the gravitational force dσg

is usually small compared to the other stress contributions and can thus be neglected [Östergren and Trägårdh 1999].

4. Computational Background

The three-dimensional mechanistic modeling of the hydrodynamic behavior of chromato-graphic packed beds, being resolved in time and space, requires the explicit consideration of solid particulate phase whose behavior is governed by the interparticle micromechanics coupled to fluid mechanics. Therefore, a hybrid modeling approach was applied coupling Computational Fluid Dynamics (CFD) and the Discrete Element Method (DEM). In DEM, the packing is modeled by discrete particles of defined sizes and densities and the behavior of the particles is described by force and momentum equations. Coupled with CFD, the complex interaction between the packed bed and the percolating fluid can be modeled. The following chapter provides the basic principles of this coupled simulation approach.

4.1. Computational Fluid Dynamics (CFD)

The principles of computational fluid dynamics (CFD) are based on the conservation laws of mass, momentum, and energy, which are known as the Navier-Stokes equations [Versteeg and Malalasekera 2007]. Therein, the state variables of the fluid are calculated for a fluid element or a control volume in a fixed coordinate system (Eulerian method of approach). State variables can be the velocity, density, temperature of the fluid and concentrations of chemical species. The Navier Stokes equations form a coupled system of non-linear partial differential equations, which cannot be solved analytically. Numer-ical solutions require a discretization of the governing equations in space and time which transform the coupled system of differential equations in a system of linear equations.

Thus, every equation represents a balance of the state variables of the fluid at a specific point of the computational lattice that only depends on the state variables of the fluid of neighboring lattice points. The most common method of discretization used in computa-tional fluid dynamics calculations is theFinite Volume Method, in which the simulation domain and geometry is divided into non-overlapping finite control volumes. Here, the

Chapter 4. Computational Background

system of equations is solved on each point on a lattice, which represents the center of the control volume. In the following, a rough overview of the governing equations concerning the CFD part of the CFD-DEM coupling is given. Detailed information of numerical flow simulation and discretization schemes is given in literature [e.g. Patankar and Spalding 1980; Versteeg and Malalasekera 2007; Lecheler 2009; Schwarze 2013].