Discrete Liquid Flow Behavior in a 2D Random Packed Bed

Abstract

In many systems, the liquid flows in discrete rivulet/droplet form rather than continuous in non-wetting or low liquid flow rate conditions. A discrete liquid flow (DLF) theory has been used by a few researchers to describe these systems such as Ironmaking blast furnace. A few investigators have applied the discrete flow of liquid in structured packing where the particles are arranged in a particular pattern where void size and shape are fixed. However, in the real world, the packing system is random, for which the DLF theory has not been extended/verified. Also, DLF theory has not been verified for 2D structural packing in the absence of gas flow rigorously. In this article, this theory is not only validated for structural packing in the absence of gas flow but also extended and validated for 2D random packing. Random packing has been created using the Discrete Element Method. The void size and shape are determined using a novel graph-based algorithm in the random 2D bed to study the liquid flow. The liquid flow behaviour has been studied in various conditions, like changing the packing size and bed height. This study confirms that the bed topology plays an important role in dictating the liquid flow behavior in a randomly packed bed.

1. Introduction

The liquid flow through a packed bed is quite common in many chemical engineering applications like distillation,^1,2) stripping,^3,4) catalysis,^5,6,7) and in metallurgical processes like blast furnace,^{8,9,10,11,12)} heap leaching.^13,14,15) Many researchers have modelled the liquid flow in packed beds using continuum hypothesis^{16,17,18,19,20,21)} for both wetting and non-wetting conditions. However, from the physical observation made by various researchers,^{8,10,13,14,22,23,24)} in a non-wetting or low liquid flow system, it is found that the liquid flows not as a continuous stream but as a mixture of discrete rivulet(s) or droplet(s). Ilankoon and Neethling¹⁵⁾ examined the relationship between the liquid flow rate and its holdup in the packed beds by maintaining the flow rates consistent with that of heap leaching. At these flow rates, the system is unsaturated with the liquid flowing in the form of droplets and rivulets. George et al.¹¹⁾ established the criteria for the passage of slag through the coke particles (non-wetting system), taking into account the discrete nature of the liquid flow. Husslege et al.⁸⁾ studied the flow of hot metal and slag in the dripping zone of the blast furnace experimentally. They found that liquid drains from the bed in the form of droplets and small plugs of various sizes. Gupta et al.¹⁰⁾ simulated the liquid flow in the lower zone of the blast furnace experimentally. An X-ray technique is used to visualize the liquid flow.^25,26) Their study revealed that liquid percolates as a series of droplets and rivulets in a non-wetting system. So, the modelling of non-wetting liquid flow via the continuum hypothesis would be in serious contradiction to the observation when the system is non-wetting or when the liquid flow rate is low. Natsui et al.²⁷⁾ modelled the liquid flow for a structured packing of complete non-wetting particles by taking the contact angle as 180°, and by treating liquid droplets as particles. Ohno and Schneider²⁸⁾ considered liquid as discrete drops and used probabilistic model to determine the flow behaviour. The liquid flow in this study has been modelled using the force balance technique developed by Gupta et al.^10,22) and subsequently used by various other authors.^12,26,29,30) The earlier theory of discrete liquid flow proposed by Gupta et al.^10,22) was not only rigorously validated against experiments by Singh and Gupta,³¹⁾ but also modified it by including the liquid rupture phenomena. They also verified the influence of gas flow on the non-wetting liquid flow in structured packing. In structured packing, the particles are fixed in space in a particular pattern such as simple cubic, hexagonal cubic packing etc., so that the shape and size of the voids is same in the entire bed. As such, all the liquid flow results which have been shown by them were based on structured packing in order to avoid the bed related topologies such as the effect of bed structure on liquid flow. The serious drawback is that constant voidage (due to structural packing) is considered, which does not reflect actual bed structure in real-world systems which are random. Moreover, void shape and size dictate the liquid size and shape and hence the liquid flow behaviour. The main difference between the structured and random packed bed is that the former has a constant void fraction, and the particles’ positions are fixed in the space. Therefore, in this paper, the possibility of applying the same liquid flow theory is explored for the real-world system, i.e., in random packing, by determining the void shape and size and hence the liquid behaviour.

The whole study in this article is presented in the absence of gas flow, for which the theory has not been fully verified yet. Therefore, the theory of discrete liquid flow in the absence of gas flow is first verified in a structured bed by performing experiments. Then, this theory is extended to a randomly packed bed along with experimental verification. The Discrete Element Method (DEM) has been used to create particle packing.

2. Theory

The force balance technique developed by Gupta et al.^10,22) involves a balance among three major forces acting on a liquid volume: the body force (or liquid’s own weight), gas-liquid drag force and the solid-liquid drag force. In the present case, gas-liquid drag is not considered due to the absence of gas flow. Two types of packing are considered: (i) structural (to avoid the effect of bed related topology on liquid flow) and (ii) random packing of uniform sized particles. In this article, it is assumed that the bed consists of a single layer of spheres and particle to particle contacts are point contacts. Though the liquid is allowed to move down along the particle, assuming it to be a sphere, any force in the direction perpendicular to the packed bed is not considered, i.e. two-dimensional (2D) motion for liquid is assumed.

Two types of liquid motion are postulated to take place within a void, depending on the liquid volume present at that location. If the liquid volume within a void is below a certain critical volume, then the liquid exists as a droplet (or a small rivulet) (Fig. 1), or else as a rivulet (large) (Fig. 2). The maximum droplet size is governed by the void size and liquid properties. When the liquid flows as a droplet or a small rivulet, it moves from one particle to another by a free fall motion, i.e. the liquid is not in contact with the solid surface when it falls from one particle to another (Fig. 1) (This is different from wetting flow in which the flow is in the form of film and liquid moves, always, in contact with the solid surface). However, when the liquid volume in the void is greater than the critical liquid volume (the volume of incoming liquid above which the liquid flows in a void as a large rivulet), the liquid is always in contact with the solid surface (Fig. 2(A)).

Fig. 1.

(A) Freely falling motion of liquid droplet in a random packing. In the first figure the liquid shifts right due to particle arrangement and in the second figure the liquid shifts left due to particle arrangement. (B) Freely falling motion of small rivulet under ‘No shift’ condition in a random packing.

Fig. 2.

(A) Liquid motion in a random packing. (B) The two incoming liquids have combined when they fall on A.

Figure 1(A) shows the free fall of a liquid in a random packing, and Fig. 1(B) shows the motion of a large rivulet in a structured packing. Figure 2 shows the motion of large rivulet in a random packing. The size and the shape of the liquid are determined using liquid physical properties by making simplifying assumptions of low contact angle hysteresis.³¹⁾ Contact angle hysteresis is the difference between the advancing contact angle and the receding contact angle while the droplet is moving or when the droplet is under the influence of driving force, like gravitational force.³²⁾

The solid resistance or the solid-liquid drag acting on liquid is, generally, quite large. When the liquid is moving in contact with the solid surface, the solid drag on the liquid tends to balance the liquid weight and as such the liquid droplet or rivulet moves down with a constant percolation velocity. In case of free fall of droplets or small rivulets, the liquid loses contact with the solid surface. This sudden loss of a resisting force on the liquid motion causes the liquid to accelerate downwards till it reaches the next particle top. It impacts the particle top and loses energy, and slides down with constant velocity. Thus, liquid moves as an accelerating and decelerating mass when the flow is in the form of small rivulet(s) or droplet(s). When the rivulets are large, the liquid may be in contact with some part of the solid and may move with a constant velocity magnitude. This is generally the case at the inlet where liquid flow rate is very high. Liquid velocity at top voids is chosen depending on the incoming liquid flow rate from the rotameter. The droplet flow is a complete free fall; small rivulet flow is also by free fall; however, its free fall distance is reduced depending on its own length, i.e. small rivulet stays in contact with solid for some time after entering the void and loses solid contact when rivulet length passes fully over the solid and then the rivulet goes into a free fall motion, as shown in Fig. 1(B). Thus, droplet and large rivulet flow represent two extremes of flow condition, and small rivulet flow is a sort of intermediate flow regime.

Thus, three flow regimes may be identified (Table 1):

Table 1. Flow regimes depending on the motion of the liquid.

Flow regime	Description
Flow as droplet	Liquid moves via free fall from one particle to another.
Flow as small rivulet	Liquid moves via free fall, but free fall length varies depending upon rivulet length
Flow as large rivulet	Liquid moves with constant velocity magnitude, and no free fall of liquid rivulet occurs

The motion of each discrete mass of liquid is described by the motion of its centre of mass.

2.1. Structured Bed

A 2D structural packing is considered where single vertical layer of particles are arranged in a structured pattern and the particles arrangement does not change with location. The liquid mass travelling from one particle to another may experience a gas drag (if gas flow is there) in one direction. In the presence of gas, the centre of mass of the liquid may not fall at the (geometrical) top of the bottom particle. This displacement of the liquid centre of mass is termed as ‘shift’. However, gas flow is not considered in this study. Thus, in the case of structured packing in the absence of gas flow, the liquid centre of mass may or may not coincides with the particle top exactly depending on the pattern of the structured packing. In the former case, the liquid divides into two equal streams and falls on either side of the particle, as shown in Fig. 1(B).

2.2. Random Bed

A single vertical layer (2D) of particles arranged in random order is considered. In contrast to structured packing, the size and the shape of the voids in random packing changes locally.³³⁾ The algorithm to determine the void position based on particle arrangement is described in section 4. The salient features of the random packing in terms of particle and void distribution is discussed here. Figure 3 shows the distribution of void locations and sizes for two different packings by keeping the bed size constant (17 × 15.5 cm). The size of the void is represented by a circle of equivalent radius, and the location of the void is represented by ‘*’. As the local void fractions and their configuration change, the pattern of liquid flow changes from one location to the other depending on the size, configuration of the voids and the amount of liquid that is coming into the void. Replicating a random packing theoretically and experimentally is impossible. Unlike, structured packing, there is no fixed arrangement of particles in random packing. Each time a bed is packed anew (keeping all other parameters same), the arrangement of particles change. This feature of random packing makes it difficult to replicate the random packing theoretically. However, there are stochastic models which can predict the probability distribution of the location of the particles. Using this, one can estimate the isotropic bulk properties of the packing like void fraction, permeability etc. Assuming average bulk properties can be useful in case of saturated flows where the average velocity of the flow and the pressure drop can be determined using the assumption of constant average bulk properties (void fraction, permeability etc.,). But this assumption cannot be made when the flow is unsaturated and the liquid flow rates are low. In these situations, local liquid flow behaviour depends on the local arrangement of the particles. Moreover, the study (in the manuscript) assumes a single layer of particles which are arranged randomly. This makes it impossible to replicate the random packing theoretically.

Fig. 3.

Two different packings are created keeping the bed size (17 × 15.5 cm), particle size (5.8 mm) and number of particles (600) constant. The figure shows Void size and distribution for two different packings (A) and (B). The size of the circle represents the size of the void. (Online version in color.)

In the case of random packing, the centre of mass of the liquid may not fall at the (geometrical) top of the bottom particle, even in the absence of gas flow. In this case, liquid falling from the top particle may shift depending upon the arrangement of the particles with respect to neighbouring particles, as shown in Fig. 1(A). Figures 2(A) and 2(B) show the liquid motion in a random packing. The centre of masses of liquids coming from particle B and particle D (Figs. 1(A) and 2(A)) does not coincide with the geometrical top of particle A when they fall on it. The two incoming liquids combine to form a liquid whose volume is the sum of incoming liquid volumes. In random packing, liquid may enter from various points in a void depending on the shape and size of the void, making it a very challenging problem in contrast to a structured bed. The problem is challenging in terms of liquid input and output points in a void and then determining whether liquid can flow as a droplet or rivulet or in combination depending on the void size.

2.3. Modeling of Liquid Flow

At every point in the bed, the voids either contain a certain amount (volume) of liquid, or they might be dry. If the liquid falls on the (geometrical) top of the particle below, it mixes and distributes into two equal streams and falls from either side of that particle (see Fig. 1(B)). In this case, liquid from ‘A’ and ‘B’ falls on top of ‘C’ and distributes equally on both sides of ‘C’. But in case of shift, the whole liquid falling from top moves to the particular side of the particle below, on which it falls (Fig. 1(A)). Thus the flow of liquid in case of shift depends completely upon the position of the centre of mass of the falling liquid with respect to the particle on which it is falling (see Fig. 1(A)). The point at which the liquid detaches from the upper particle and falls on the lower particle is governed by the moment equation.²⁶⁾ As long as moment of the liquid (about the detachment point) moving on the particle is less than zero, the liquid stays in contact with the particle; otherwise, it detaches and falls on the lower particle.

In order to determine the critical liquid volume, interfacial solid-liquid contact area and other parameters like gas-liquid contact area, a simple shape of the liquid is assumed, which has been determined based on simplifying assumptions:

Droplet: spherical shape

Rivulet: cylinder with spherical ends

Typical rivulet and droplet shapes are shown in Fig. 4.

Fig. 4.

Shape of (A) liquid rivulet is assumed to be cylindrical with hemispherical ends, whereas the shape of (B) droplet is assumed to be spherical; θ is (π-contact angle) at the gas-liquid-solid interface, L is length of the rivulet, r is the radius and W is the diameter of the droplet in contact with solid surface and the width of the rivulet in contact with solid surface.

At each void in the bed, the liquid velocity is determined depending on the liquid volume contained in the void. The liquid in the void may exist and flow as a droplet or a rivulet, depending on the void size. If the void size is large, liquid properties determine whether liquid exists as a droplet or a rivulet. But first, the nature of the flow is described. Liquid flows via free fall from one particle to another if it exists as a droplet or a small rivulet. But if it is always in contact with some part of solid then it is flowing as a large rivulet. This distinction is based on the volume of the liquid in the void. Droplet flow can occur via free fall only. For a droplet under free fall, the maximum possible radius of the droplet (assuming the droplet to be spherical) is determined by two governing factors, void size (volume of the void. The description of it is given in section 4) and liquid properties as given below.

If voids are small, then the maximum droplet size (defined by its radius r_MAX) is restricted by void dimensions, and for equal sized particles in a structured packing where void is surrounded by three particles, this maximum value of radius is given by   

$$r_{MAX}=\frac{\left(2-\sqrt{3}\right)d_p}{2\sqrt{3}}$$(1)

Where, d_P is the particle diameter.

If void size is large, liquid properties limit the maximum droplet size. For free fall conditions (droplet flow occurs via free fall from one particle top to other) in the presence of still air:³⁴⁾   

$$r_{MAX}=\frac{3}{16g}\frac{\left(\frac{{\rho }_g}{{\rho }_L}\right)\sigma C_{D_{gl}}}{\left(1-\frac{{\rho }_g}{{\rho }_L}\right){\rho }_g\left(\frac{0.32}{2\pi }+\frac{1}{8}\right)}$$(2)

In case of random packing, the value of r_MAX is different for different void locations as the size of the void varies locally. Where σ, $C_{D_{gl}}$ are surface tension constant (N/m) and coefficient of drag between liquid and gas, respectively. The calculation method used in deriving the maximum droplet size is based on the assumption of 2D packing and spherical shape of the droplet.

2.4. Liquid Volume Calculation

If θ is (π-contact angle) at the gas-liquid-solid interface, then it may be easily shown that rivulet volume (see Fig. 4) is:   

$$v_o=L{r}^2\left(\pi -\theta +sin\theta cos\theta \right)+{\frac{2\pi r}{3}}^3\left(1+cos\theta +\frac{1}{2}si{n}^2\theta cos\theta \right)$$(3)

Where, L denotes the rivulet length, and r denotes its radius. Liquid exists as a droplet only when it fails to exist as a rivulet. Thus, the liquid volume below which liquid will exist as a droplet is found by putting L = 0 and r = r_MAX in the expression for rivulet volume. The effective rivulet length is equal to L_eff = L + 2r_MAX sinθ.

Similarly, for a large rivulet, as shown in Fig. 2(A), the rivulet volume is:   

$$\begin{array}{l}{v}_o=\pi r^2\left(L-x+\frac{2r}{3}\right)+{\frac{\pi r}{3}}^3\left(1+cos\theta +\frac{1}{2}si{n}^2\theta cos\theta \right)\\ \hspace{1em}+x{r}^2\left(\pi -\theta +sin\theta cos\theta \right)\end{array}$$(4)

The width of a rivulet or the base diameter of a droplet is given by W, where W = 2r sinθ.

Thus, for a large rivulet, the liquid-solid contact area could be written as   

$$a_{sl}=\left(x+r_{MAX}sin\theta \right)W$$(5)

Where, r is the droplet radius given by:   

$$r={\left[\frac{v_o}{\frac{2\pi }{3}\left(1+cos\theta +\frac{1}{2}si{n}^2\theta cos\theta \right)}\right]}^{1/3}$$(6)

The critical liquid volume is found by putting the effective length of the rivulet during free fall as the void length. If the effective rivulet length is more than this void length, then the liquid is always in contact with solid during falling, and hence it is the case of a large rivulet.

2.5. Velocity Calculation

When the liquid is flowing in contact with the solid surface, the two major forces acting on it are balanced and assumed to give it a constant velocity magnitude. However, as soon as liquid loses solid contact fully, it experiences a loss of resistance and accelerates to the next particle during free fall. Thus, acceleration terms appear in the equation of motion of free-falling liquid drops or rivulets.

CASE 1: Volume of liquid within a void is more than or equal to the critical liquid volume v_critial:

In this case, the forces acting on the liquid volume are bed resistance and gravitation. Vector force balance gives:   

$$\frac{1}{2}{C}_{D_{sl}}{a}_{sl}{\rho }_L\left|{\overrightarrow{V}}_s-{\overrightarrow{U}}_L\right|\left({\overrightarrow{V}}_s-{\overrightarrow{U}}_L\right)-{\rho }_L\overrightarrow{g}{v}_o=0$$(7)

Where (${\overrightarrow{V}}_s-{\overrightarrow{U}}_L$) is the relative velocity of the liquid with respect to particles in m/s. When the liquid is going to drop from one particle to another, then just before entering the next void, it has no radial velocity, and its velocity vector is directed vertically downwards.

CASE 2: Volume of liquid within a void is less than the critical liquid volume v_critial (free fall condition):

In this case, the only force that is acting on the liquid volume is gravitation. The balance of forces gives:   

$${\rho }_L{v}_{o}g={\rho }_L{v}_o\frac{d{\overline{U}}_L}{dt}$$(8)

Where ${\overline{U}}_L$ is the velocity of the liquid in m/s. Liquid moves in the horizontal direction solely due to the body force with solid resisting the motion.

The criterion for the detachment of droplet and small rivulet from the solid surface during the free fall is given in Singh and Gupta,³¹⁾ and the same has been adopted in this work. Similarly, the criterion of rupturing of rivulet has been developed by the previous investigators,³¹⁾ and the same has been used here. The first criterion (detachment from the solid surface) was developed by considering the balance between the surface tension, centrifugal and body forces. And, for rivulet rupture, it was developed considering the body, gas drag, solid drag and surface tension forces. However, in the present case, the gas drag term has been excluded due to the absence of gas flow.

3. Creating a Randomly Packed Bed

The randomly packed bed is created by modeling the motion of the particles using the soft sphere approach and a linear spring model for modeling inter-particle and wall particle interactions.³⁵⁾ The linear and angular momentum balance for a particle′ i’ is given in Eqs. (9) and (10)   

$$m_i\frac{d\overrightarrow{v_i}}{dt}=m_i\overrightarrow{g}+{\sum }_{j=1}^N\left({\overrightarrow{F}}_{C,ij}+{\overrightarrow{F}}_{D,ij}\right)$$(9)

  

$$I_i\frac{d\overrightarrow{{\omega }_i}}{dt}={\sum }_{j=1}^N{\tau }_{ij}$$(10)

Where, ${\overrightarrow{F}}_{C,ij}$ is the contact force between two colliding or interacting particles, ${\overrightarrow{F}}_{D,ij}$ represents the damping force and comes into picture to account for dissipation. The term $\sum _{j=1}^k{\overrightarrow{\tau }}_{ij}$denotes the sum of torque acting on particle ‘i’ due to particle ‘j’ at their contact points. All other symbols are defined in nomenclature. The detailed description of the model may also be found in the work of other researchers.³⁶⁾

A randomly packed bed is created by allowing particles, which are initially randomly distributed in space, to settle down under gravity in a bed of a given dimension. Equations (9) and (10) are solved for each particle, and a randomly packed bed is created. Within one-second real time, the particles are settled down in the empty bed, and random packing of particles is created. This initial solid particle packing is used to study liquid flow behaviour in the bed.

4. Void Detection

The size and shape of all voids were the same for the structured packing simulation; however, they are different for random packing. Determining the shape and size of voids for a random packing is challenging. Some researchers^37,38) have used the image analysis technique to determine the void network inside packed beds. But the literature with respect to determining the position and size of the voids geometrically is lacking. For this purpose, a novel graph-based recursive Depth First Search (DFS) algorithm is developed. The particles in a bed can be considered vertices of a graph, and the lines joining the particles and their neighbours can be treated as edges (see Figs. 5(A) and 5(B)). For each of the particles, a graph is traversed to determine the location of the voids. The algorithm searches for polygons formed by the particles, i.e. the particles cluster forms a closed loop (represented with red edges in Fig. 5(B)). These polygons should have no particles inside them. When such polygons are identified, the void position is determined by finding the centroid of the polygon. The portion occupied by particles inside the polygon is subtracted from the polygon (area of the polygon multiplied by thickness of the 2D layer of particles) to get the size of the void. The recursive DFS starts at the root vertex (the particle from which the DFS. subroutine is called) and continues traversing along each branch (by recursively calling the DFS subroutine for each of the visited vertices) as far as possible before tracing back (if the vertex is neighbour to the root vertex or if there are no neighbours to the final vertex). Figure 5(A) Shows a random cluster of particles inside the bed. These particles can be treated as vertices of a graph, as shown in Fig. 5(B). The numbers on the particles represent their identities. The position, size and configuration of particles surrounding the void are used in the discrete liquid flow solver to determine the type of liquid flow (droplet/rivulet) and the motion of the liquid inside the void.

Fig. 5.

(A) The particles and their connectivity can be treated as a graph data structure. The figure shows a sample cluster of particles inside a packed bed. The numbers on the particles are the particle identities. (B) Schematic of graph for the sample cluster of particles. The particles are treated as vertices of a graph and the connection betweeen the vertices as edges. (Online version in color.)

5. Computational Procedure

Discrete liquid flow equations were (Eqs. (1), (2), (3), (4), (5), (6), (7), (8)) solved at each void location, depending upon the amount of liquid being fed to the void. Based on liquid volume descending into a void, the forces acting on liquid were calculated, and the location of the centre of mass of liquid rivulet or droplet falling on the downstream particle of the void was evaluated. Finally, liquid velocity is determined by solving Eqs. (7) and (8). This procedure was carried out for all void locations maintaining the mass balance in each void. This ensures mass of the liquid coming into the void is the same as the mass of the liquid leaving the void.

All the codes required for the simulation are developed in-house. A DEM solver developed in C for particle packing inside a bed and GPU parallelized using CUDA is used. The particle packing code is run on NVIDIA GeForce RTX 3090. It took an average runtime of around 12 mins to create a packing of particles (660 particles) of size 5.8 mm in a bed of height 0.176 m and width 0.14 m. The liquid flow solver, written in C, is run on Intel i7 9th generation processor with 16GB RAM. It took around 3 s of CPU time for each run.

6. Experiment

Experiments were conducted in pseudo 2D³⁹⁾ structured and random beds with single and multiple liquid inlet points at the top of the bed. The structured packing was prepared using 4.8 mm spherical plastic beads. The bed height is 120 mm, width is 176 mm and thickness is 59 mm. 10 mm thick Perspex sheet was used for fabricating the experimental setup. Wall effects are minimized by keeping bed thickness to particle diameter ratio more than 4 as stated by Scheffel et al.⁴⁰⁾ and followed by others^38,41) in random bed. In a structured bed, the wall effect is expected even lower. Bed thickness to particle diameter ratio is more than 12 in the current study. Dry and clean beads were carefully glued at the point of contact and arranged into the bed to get a structured packing of height 125 mm. Barium chloride in water solution, with a concentration of 0.09 g BaCl₂/mL is introduced into the packed through a single rotameter at a flow rate of 0.04 L/min. The condition is maintained for 30 mins to reach the steady state.^10,42) X-ray radiography was used to visualize the flow in this packed bed once the flow reaches the steady state. The flow visualization technique exploits the principle of X-ray absorption to trace the liquid flow path. A medical X-ray unit (Stenoscop 9000) is used for this purpose. The image intensifier in the stenoscop has a maximum resolution of 58 lp/cm and contrast ratio of 21:1. The X-ray unit is capable of scanning the transient phenomena i.e., the motion of liquid movement.⁴³⁾ The bed was divided into small regions, which were imaged separately. The actual picture of the packed bed under liquid flow was reconstructed from the images recorded during the experiment using Image Pro Software. More details about the experiments are given elsewhere.^42,43)

7. Results and Discussion

The unknown parameter for the simulation includes the interfacial drag coefficient term (solid-liquid drag), and it was taken based on the results of Liu et al.²⁸⁾ for C_D,sl. Based on this input parameter and a contact angle of 92° for liquid over solid, the simulation runs were carried out. The parameters used for the simulation are shown in Table 2.

Table 2. Experimental conditions and parameters used in the simulation.

Packed bed	Parameters
Material	Plastic beads
Bed width (m)	0.176
Bed height (m)	0.155, 0.14, 0.11, 0.10
Particle diameter dP (m)	0.0058, 0.0021 ,0.002, 0.003, 0.0048, 0.004
Liquid
Contact Angle	92 deg
Flow rate at inlet (L/min)	0.04, 0.07
Densityρl (kg/m3)	1000
Viscosity (Pa s)	0.001

7.1. Liquid Flow in a Structured Bed (Experimental Validation)

Figure 6 shows a comparison of the liquid flow path, as obtained from the simulation (Fig. 6(B)) and from the reconstructed X-ray image (Fig. 6(A)). This figure shows the flow of liquid through one inlet in a structured packed bed. The height of the packing is 12 cm and width of the packing is 17.6 cm. One can see that the liquid is meandering down vertically in a straight line which is well captured by the developed model. As such, the liquid stream has broadened a little when it is meandering down on the packing, which is also well predicted by the model. This phenomenon has been captured in the model using liquid shift, detachment and rupture criteria. It can be seen from this figure that the liquid stream is a little zigzag in nature which is not captured by the model. There could be many reasons for this. One factor could be the structural effects during the experiment. The beads/particles were joined using adhesive, and as such, some improper joining might give rise to a zigzag path in the flow of liquid. There could be other effects contributing to the zigzag nature of the liquid flow, and that is molecular dispersion, which is not captured by the simulations. The effect of mechanical dispersion had been minimized during the experiments by use of a structured packing. Any of the above mentioned factors may contribute to the deviations referred to above. Nevertheless, these results lend good support to the developed theory and model which can be extended to study the liquid flow in random packing.

Fig. 6.

Descent of liquid through the structured packing of particle size 4.8 mm; The height of the packing is 12 cm and width of the packing is 17.6 cm. (A) Liquid flow lines at 0.04 L/min in structured bed (X-ray image); (B) Simulation results showing liquid velocity vectors for conditions of (A).

7.2. Liquid Flow in a Randomly Packed Bed

As mentioned before under the random packing section that theoretically two similar random beds cannot be created as each time the bed is packed the configuration of the bed changes. As the liquid flow pattern has a strong dependence on bed topology, quantitatively validating the random bed modelling results with experiments is hard. Due to this reason, the model is validated qualitatively.

Figure 7 shows liquid flow through a randomly packed bed of packing height 0.14 m and width 0.176 m with particle packing of 5.8 mm spherical plastic beads. Three different randomly packed beds (initial particle distribution) are created using 5.8 mm particles. A liquid flow study is carried out theoretically in these beds. The liquid inlet (shown by a downward pointing arrow on the top) in all three cases is at the same point, as shown in Figs. 7(A)–7(C). The liquid velocity vectors are shown in black arrows. The liquid flow model in a randomly packed bed is a simple extension of the model developed and validated in structured packing. The difference now lies in the fact that the voids are of different shape and size.

Fig. 7.

Liquid flow for different particle packings. In each case ((A) to (C)) the bed is packed anew to change the particle distribution (all other parameters are kept constant). The figure shows that the liquid flow behavior depends on the particle distribution. The Arrows depict liquid flow lines. Arrow at the top represents inlet position. (A) Liquid flow through a randomly packed bed of packing height 0.14 m and width 0.176 m with 5.8 mm spherical plastic beads; liquid flow rate is 0.04 L/min. (B) Liquid flow through a randomly packed bed of packing height 0.14 m and width 0.176 m with particle packing of 5.8 mm spherical plastic beads; liquid flow rate is 0.04 L/min. (C) liquid flow through a randomly packed bed of packing height 0.14 m and width 0.176 m with particle packing of 5.8 mm spherical plastic beads; liquid flow rate is 0.04 L/min. (Online version in color.)

It can be observed from the figures that a change in packing results in a change in the liquid flow paths. In Fig. 7(A), the liquid flows in a zig zag manner till it reaches a particular height of the bed where it splits into two streams which continue till the bottom of the bed.

In Fig. 7(B), liquid flow continues till the bottom without splitting. Near the bottom, the liquid stream slightly broadens due to spreading. Figure 7(C) clearly shows the deviation of liquid flow from the straight path revealing the effect of bed topology on the pattern of liquid flow. It is apparent from the above figures that a change in packing structure has led to change in the liquid flow paths. What is generally termed as the stochastic component^12,29) of the liquid flow in modelling may be thought to be arising due to this bed topology factor. A change in the bed topology and hence the void location and size dictate the liquid flow in the bed.

Figure 8 shows the liquid flow through a randomly packed bed of packing height 0.14 m and width 0.176 m with particle packing of 5.8 mm. The horizontal liquid inlet position for Fig. 8(A) is at 0.88 m, while it is at 0.11 m for Fig. 8(B) from the left wall of the bed(the vertical location of the lnlet is not changed). The figure shows that changing the position of the inlet point also affects the liquid flow path as the topology of the bed changes locally.

Fig. 8.

Simulated results of liquid flow profile for the same packing but with inlet at different horizontal positions (the vertical location of the inlet is kept constant). Arrow at the top represents inlet position. (A) liquid flow through a randomly packed bed of packing height 0.14 m and width 0.176 m with particle packing of 5.8 mm spherical plastic beads. The horizontal inlet location at 0.88 m from the left wall of the bed. Liquid flow rate at the inlet is 0.04 L/min. (B) liquid flow through a randomly packed bed of packing height 0.14 m and width 0.176 m with particle packing of 5.8 mm spherical plastic beads. The horizontal inlet location at 0.11 m from the left wall of the bed. Liquid flow rate at the inlet is 0.04 L/min. (Online version in color.)

Figure 9 shows the liquid distribution at the bottom of the packed bed for 3 different initial packings of height 0.14 m and width 0.176 m, constructed from 5.8 mm particles. The figure shows the amount of liquid collected in a minute at the bed bottom. The bottom of the bed is divided into 33 sampling points or collectors (each 5.3 mm in width). The liquid flow at the top is maintained at 0.04 L/min in all the cases (the arrow, pointing downwards, on the top shows the liquid inlet location at the bed surface). It is observed that liquid collected is maximum at different points for different initial packings. But away from this point, the amount of liquid collected in the collectors decreases.

Fig. 9.

Liquid distributions at the bed bottom for three different initial packings. The packing height is 0.14 m and width is 0.176 m with particle packing of 5.8 mm spherical plastic beads. Liquid flow rate at the inlet is 0.04 L/min.

This observation is in line with the experimental observations of Gupta et al.¹⁰⁾ and other researchers.^33,41) Due to the different bed structures, the liquid flow collected at the bed bottom may vary along the bed width, but the basic nature of the collection or flow is similar. An average of the above 3 different data or more may give an indication to the average flow profile or liquid distribution at the bed bottom. Similar conclusion may be drawn by analyzing liquid distribution at the bed bottom from multiple liquid inlet points, as shown in Fig. 10.

Fig. 10.

Liquid collection at the bed bottom for multiple liquid inlets for 2.1 mm plastic bead packing. The simulation results are compared with the experiments. The packing height is 0.14 m and the width is 0.176 m. The liquid flow rate is 0.04 L/min. (Online version in color.)

Figure 10 shows liquid distribution at the bed bottom when liquid is coming from multiple sources (two sources), as shown by the downward pointing arrows at the top in the figure. The packing height of 0.10 m is built of 2.1 mm plastic beads. All the liquid inlet points irrigate the bed with liquid coming at a flow rate of 0.04 L/min. Three different packings were employed (for simulation results), and liquid flow from multiple sources was studied. As commented earlier, a change in packing leads to a different flow profile in the bed. As such, there is a different liquid distribution at the bed bottom (collected in 17 collectors, each 10 mm in width) in all three simulation runs. To validate the results, experiments were carried out to study liquid distribution at the bed bottom, employing above mentioned parameters. One can visibly see different peaks of liquid collection points. At the left inlet, the deviation of the horizontal position of peaks for simulation from that of the experiment is due to the bed topological factor. This distribution is again in qualitative agreement with the experimental results of Gupta et al.¹⁰⁾ and other researchers^33,41) and shows a good match with our experimental distribution results. In Gupta et al.,¹⁰⁾ the authors show that the exit local superficial velocity is for different packings hovers around a mean value though the liquid distribution profiles change from packing to another. Similarly, it can be noticed from Fig. 10 that the distribution hovers around a mean value of 40 cc as shown by the orange dashed line.

Figure 11 shows the comparison of liquid out flow fraction at the bottom of the packing with the published results. The results of the current work (DLF model) is compared with the results of Fernando et al.³⁸⁾ (experiments and simulation) for a liquid flow rate of 0.07 L/min in a 8 mm glass particles randomly packed in a bed of 0.266 × 0.2 m. Liquid out-flow fraction is calculated as the fraction of liquid collected in particular bin/channel for 60 s. Fernando et al.³⁸⁾ used Random-Walker based simulation²⁾ to get the liquid distribution profile. It can be observed from the figure that the distribution for the simulations (both the DLF model and the RW simulation) is different from that of the experiments. This is due to the local variation in the particle arrangement for random packing. The DLF model is able to capture the overall distribution trend of the experiments and the RW simulations. Not only is the current model able to match the trend, also it is closer to the experimental observations. Also, the current work has the advantage of being a deterministic model as compared to the RW simulations, which is based on probability distribution. This is also in agreement with the observations made in the current work (manuscript) where the liquid distribution trend of simulations matches with the experiments (Fig. 10).

Fig. 11.

Comparison of out-flow fraction at the bottom of the packing with the published results. The packing height is 0.2 m and the width is 0.266 m. The liquid flow rate is 0.07 L/min.

Figure 12 shows the effect of packing particle size on liquid distribution or spreading. Simulation results for 3 different sizes of plastic beads packing are shown in a bed of width 0.176 m and a single liquid inlet point (as marked by an arrow on the horizontal axis, liquid flow rate maintained at 0.04 L/min). The plot shows the amount of liquid collected in a minute in different collectors (17 in total) at the bed bottom. The result for liquid distribution at the bed bottom, as shown for different particles in Fig. 12, clearly indicates that the spreading of liquid is more for larger particle size packing. The three packing sizes employed are 2, 3 and 4 mm spherical plastic beads. Liquid spreading is more pronounced in packings where particle size is smaller. This is because void volumes are lesser in packings with smaller particles, which makes liquid to spread out more. This feature is adequately captured in the current study.

Fig. 12.

Liquid collection, at the bed bottom for different particle size packing. The packing height is 0.14 m and the width is 0.176 m. The liquid flow rate is 0.04 L/min.

From the above simulations for the single-phase flow of liquid in a randomly packed bed, one may conclude that the model captures the experimental features qualitatively quite well. The so-called stochastic nature of liquid flow may in part be attributed to the topology of the packed bed.

This study deals with the discontinuous flow of liquid in random packing, which could be used to understand real-world problems such as rainwater flow over windowpanes, water flow on plant leaves (in droplet form), volcanic eruptions (rivulets/droplets), and discontinuous liquid flow in many engineering disciplines^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)} etc. as discussed in the introduction section.

8. Limitations and Future Work

The current model builds on the work of Gupta et al.^10,22) and Singh et al.³¹⁾ The model is extended to pseudo-2D random-packed beds by integrating it with the graph-based DFS algorithm for void detection. As the particle sizes are smaller, the capillary effects are not so significant and the gravitational forces dominate the flow (because of low bond number). So, the model doesn’t consider the effect of capillary pressure on the fluid flow. Also, the model assumes liquid flow in a 2D packed bed. The liquid flow phenomena can be better understood by a proper 3D study. As the liquid descends the bed, some part of the flowing liquid is held between the particles as static holdup. The current model assumes that the liquid doesn’t stick to the particles i.e., the amount of liquid that is entering the packing must be same as the amount of liquid leaving. Future work on this model should be extended to 3D packing by incorporating the effect of static holdup on the liquid flow.

9. Conclusions

Previously developed Discrete Liquid Flow (DLF) theory for structured bed is extended to random bed which occurs in real world. In random packing, the description of DLF is lacking. Therefore, in the current study, the discrete liquid flow model, when incorporated along with Discrete Element Model and Depth First Search (DFS) algorithm to create the random packing and detect the void size and shape respectively, is able to predict reasonably the liquid flow profile in the absence of gas flow. The model has been validated against the structured packed bed in a carefully designed experiment using the X-ray radiography technique. When void shape and size are considered in the DLF theory in a random 2D bed, significant effect of bed topology and particle size on the liquid distribution, which is in the line of the experimental study carried out in this work and results obtained by other researchers, is observed. It is concluded that stochastic nature of the liquid flow in a random packed bed may in part be attributed to the bed topology. The results obtained in randomly packed bed are non-reproducible, i.e., for each packing, unique liquid flow profile is obtained due to a change in local bed topology.

Funding

The financial aid, to support the part of this work, provided by Council of Scientific and Industrial Research (CSIR), India, under the project number 22(0437)/07/EMR-II and Department of Science and Technology (Project No. C.R.G./2019/000292) is gratefully acknowledged.

Competing Interests

The authors have no relevant financial or non-financial interests to disclose.

Author Contributions

All authors contributed to the development of the model. Discrete Liquid Model was initially developed by Prof. G. S. Gupta. The work is extended to random packed bed by G. V. A. Chaitanya. The first draft of the manuscript was written by G. V. A. Chaitanya and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Acknowledgements

We would like to thank Dr V. Singh, S. Kamble, N. Gupta (from Indian Institute of Science, Bangalore) and Prof. M. R. Lollchund (University of Mauritius) in providing their inputs during discussion of this work. The financial aid, to support the part of this work, provided by Council of Scientific and Industrial Research (CSIR), India, under the project number 22(0437)/07/EMR-II and Department of Science and Technology (Project No. C.R.G./2019/000292) is gratefully acknowledged.

Notations

a: effective contact area between two phases, m²

C_D: drag coefficient, dimensionless

dp: particle diameter, m

F: force, N

F_R: bed resistance force, N

F_G: gravity force, N

g: acceleration due to gravity, ms⁻²

L: length of a rivulet, m

m: mass, kg

N: Number of particles

r: radius, m

r_MAX: maximum possible droplet radii, m

t: time, s

U_y: Vertical velocity of liquid, ms⁻¹

vo: liquid volume, m³

V: velocity, ms⁻¹

V_l: liquid velocity, ms⁻¹

Vs: solid velocity, ms⁻¹

Subscripts

‘sl’: solid–liquid

‘gl’: gas–liquid

Greek letters

θ: π - liquid phase contact angle, rad

θ_L: droplet detachment angle, rad

ρ_g: gas density, kgm⁻³

ρ_l: liquid density, kgm⁻³

ω: angular velocity, rads⁻¹

I: moment of inertia, kgm²

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