# Numerical Simulation Of Pressure Drop In Flow Over Fixed Porous Beds

Bachelor Thesis 2015 80 Pages

## Excerpt

## TABLE OF CONTENTS

ACKNOWLEDGEMENT

DEDICATION

ABSTRACT

LIST OF FIGURES

LIST OF TABLES

NOMENCLATURE

1.0 INTRODUCTION

1.1 BACKGROUND

1.2 Porous Bed Types and Formations

1.3 Porous Bed Properties

1.4 Fluid Properties

1.5 Flow Properties

1.6 Pressure and Pressure Drop

1.6.1 Pressure

1.6.2 Pressure Drop

1.7 Objectives

1.8 Scope of Study

1.9 Motivation and Justification

2.0 LITERATURE REVIEW

2.1 Empirical models

2.2 Porous Material

2.3 Phase Flow

2.4 Numerical Simulations

2.5 Experiments conducted with porous media

2.6 Other Studies

2.7 Brinkman Equations

3.0 Methodology

3.1 Governing Equations

3.1.1 Conservation of Mass

3.1.2 Conservation of Momentum

3.1.3 Darcy's Law

3.1.4 Brinkman Equations

3.2 Numerical Methods

3.2.1 Finite Element Method

3.3 COMSOL Multiphysics

3.3.1 Assumptions

3.3.2 Simulation

3.3.3 Model Navigator

3.3.4 Geometry Modelling

3.3.5 Parameters

3.3.6 Materials

3.3.7 Physics Setting

3.3.8 Domain/Sub-domain Settings

3.3.9 Boundary Conditions

3.3.10 Mesh Generation

3.3.11 Computing

3.3.12 Post-Processing and Visualization

4.0 Result and Discussion

4.1 Results

4.2 Discussion

5.0 CONCLUSIONS AND RECOMMENDATIONS

5.1 CONCLUSIONS

5.2 RECOMMENDATIONS

REFERENCES

APPENDIX

## LIST OF FIGURES

FIGURE 1.1: ILLUSTRATION OF A SCRUBBER. (SOURCE: GOOGLE IMAGES)

FIGURE 1.2: ILLUSTRATION OF A FILTER BED (SOURCE: GOOGLE IMAGES)

FIGURE 1.3: ILLUSTRATION OF A PACKED BED CHEMICAL REACTOR. (SOURCE: GOOGLE IMAGES)

FIGURE 3.1: ILLUSTRATION OF THE STRESSES ACTING ON A SMALL RECTANGULAR 3D ELEMENT. (SOURCE: ANDERSON, J., 2009, COMPUTATIONAL FLUID DYNAMICS AN INTRODUCTION, SPRINGER)

FIGURE 3.2: COMSOL START-UP NAVIGATION PAGE.

FIGURE 3.3: COMSOL SPACE DIMENSION SELECTION PAGE

FIGURE 3.4: COMSOL PHYSICS SELECTION PAGE

FIGURE 3.5: COMSOL STUDY SELECTION PAGE

FIGURE 3.6: GEOMETRY OF POROUS BED

FIGURE 3.7: COMSOL DOMAIN SETTINGS FOR MATERIAL PROPERTIES OF POROUS MATRIX

FIGURE 3.8: COMSOL DOMAIN SETTINGS FOR FLUID AND MATRIX PROPERTIES.

FIGURE 3.9: GEOMETRY OF POROUS BED WITH HIGHLIGHTED INLET BOUNDARY.

FIGURE 3.10: GEOMETRY OF POROUS BED WITH HIGHLIGHTED OUTLET BOUNDARY.

FIGURE 3.11: GEOMETRY OF POROUS BED WITH HIGHLIGHTED WALL BOUNDARY

FIGURE 3.12: COMSOL MESH SETTINGS WITH MESH GEOMETRY OF POROUS BED.

FIGURE 3.13: 2D SURFACE PLOT OF VELOCITY

FIGURE 3.14 :2D CONTOUR PLOT OF PRESSURE

FIGURE 4.1.1: 2D SURFACE PLOT OF VELOCITY DISTRIBUTION OF FIXED POROUS BED. (FOR A POROSITY OF 0.2 AND SUPERFICIAL VELOCITY OF 4E-5 M/S USING WATER AS A FLUID)

FIGURE 4.1.2: 2D CONTOUR PLOT OF PRESSURE DISTRIBUTION OF FIXED POROUS BED (FOR POROSITY OF 0.2 AND SUPERFICIAL VELOCITY OF 4E-5 M/S USING WATER AS A FLUID)

FIGURE 4.1.3: 2D SURFACE PLOT OF PRESSURE DISTRIBUTION OF FIXED POROUS BED (FOR POROSITY OF 0.2 AND SUPERFICIAL VELOCITY OF 4E-5 M/S USING WATER AS A FLUID)

FIGURE 4.1.4: 1D LINE PLOT OF INLET AND OUTLET PRESSURE VALUES AGAINST POROSITY FOR FIXED POROUS BED (FOR POROSITY OF 0.2 AND SUPERFICIAL VELOCITY OF 4E-5 M/S USING WATER AS A FLUID)

FIGURE 4.1.5: 1D LINE PLOT OF THE GLOBAL PERMEABILITY VALUES OBTAINED AND USED BY THE CARMAN-KOZENY RELATIONSHIP.

FIGURE 4.1.6: 2D SURFACE PLOT OF VELOCITY DISTRIBUTION OF FIXED POROUS BED (FOR A SUPERFICIAL VELOCITY OF 4E-5 M/S AND POROSITY OF 0.1 USING AIR AS A FLUID)

FIGURE 4.1.7: 2D CONTOUR PLOT OF PRESSURE DISTRIBUTION OF FIXED POROUS BED (FOR A SUPERFICIAL VELOCITY OF 4E-5 M/S AND POROSITY OF 0.2 USING AIR AS A FLUID)

FIGURE 4.1.8: 2D SURFACE PLOT OF PRESSURE DISTRIBUTION OF FIXED POROUS BED (FOR A SUPERFICIAL VELOCITY OF 4E-5 M/S AND POROSITY OF 0.1 USING AIR AS A FLUID)

FIGURE 4.1.9: 1D LINE PLOT OF INLET AND OUTLET PRESSURE VALUES AGAINST POROSITY FOR FIXED POROUS BED (FOR POROSITY OF 0.2 AND SUPERFICIAL VELOCITY OF 4E-5 M/S USING AIR AS A FLUID)

FIGURE 4.2.1: GRAPH OF PRESSURE DROP AGAINST POROSITY FOR TABLE 1.0.

FIGURE 4.2.2: GRAPH OF PRESSURE AGAINST LENGTH (OVER THE LENGTH OF BED) FOR TABLE 6.0.

## LIST OF TABLES

TABLE 1.0: OF CALCULATED PRESSURE DROP OF FIXED POROUS BED USING WATER AS FLUID (FOR SUPERFICIAL VELOCITY OF 4E-5 m/s USING WATER AS A FLUID)

TABLE 2.0: CALCULATED PRESSURE DROP OF FIXED POROUS BED USING WATER AS FLUID (FOR SUPERFICIAL VELOCITY OF 4E-6 m/s USING WATER AS A FLUID)

TABLE 3.0: CALCULATED PRESSURE DROP OF FIXED POROUS BED USING WATER AS FLUID (FOR SUPERFICIAL VELOCITY OF 4E-7 m/s USING WATER AS A FLUID)

TABLE 4.0: CALCULATED PRESSURE DROP OF FIXED POROUS BED USING WATER AS FLUID (FOR SUPERFICIAL VELOCITY OF 4E-8 m/s USING WATER AS A FLUID)

TABLE 5.0: PRESSURE TAKEN AT VARYING LENGTHS OF 2 m FROM THE INLET TO THE OUTLET (FOR A POROSITY OF 0.2 FOR ALL VALUES OF SUPERFICIAL VELOCITY USING AIR AS A FLUID)

TABLE 6.0:PRESSURE TAKEN AT VARYING LENGTHS OF 2 m FROM THE INLET TO THE OUTLET (FOR A POROSITY OF 0.2 FOR ALL VALUES OF SUPERFICIAL VELOCITY USING AIR AS A FLUID)

## NOMENCLATURE

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## ACKNOWLEDGEMENT

I would like to appreciate almighty God who has been with me for all this time and His numerous blessings upon my life.

I would also wish to extend my sincere gratitude to Prof. O.M. Oyewola, my project supervisor, for giving me this topic and also for his immense direction and support over the course of this project.

My appreciation also extends to Engr M.O Petinrin for his assistance and advice in using COMSOL Multiphysics as a tool in performing this project. My appreciation also goes to the project coordinator of the undergraduate project programme, Engr O.O. Ajide for the inspiration he has been throughout the project period.

To my parents Prof. & Mrs A. Olatunbosun, I thank them for their support and sacrifices, which they have made for me over the years most especially during my five years in the university. To my siblings, Mr J.S. Olatunbosun & Mr P.O. Olatunbosun, I appreciate them for their helping hand in time of need.

I would also like to appreciate my friends and classmates, Mr S.A. Ajiboye and Mr O.O. Ogunsolu for their encouragement in pursuing and achieving good results.

To Ms Tinuola Osho, I appreciate her lasting contributions over these long years and outstanding help.

## DEDICATION

This project is dedicated to God almighty.

## ABSTRACT

The study of porous beds has attracted a wide range of attention in recent years. Researchers have studied various effects and attempted to solve various problems in porous beds. But one issue that has remained standing is that of accurate modelling of the behaviour of pressure drop. This project attempts to add to the existing knowledge in the area by performing a simple numerical simulation of pressure in flow over fixed porous beds.

In performing this simulation, the model was governed by two fluid flow principles, the conservation of mass and conservation of momentum using the Brinkman Equations. COMSOL Multiphysics was employed in helping to solve this model because it uses a discretization of finite elements in solving a set of partial differential equations that help to promote accuracy of results. The model involved a rectangular porous bed measuring (25 m × 2 m) for porosity values of 0.20.9 at intervals of 0.1 and air and water were used as working fluids.

From the simulation performed, the velocity of the porous bed increases for all porosity values by 8.25%. For the range of porosity values between 0.2 and 0.9, the maximum pressure at the inlet of the porous bed was1134.42 Pa. for water and 20.38 Pa for air which occurred at a velocity of 4×10-^{5} m/s and the minimum pressure at the outlet of the porous bed was 0.045 Pa for water and 8.16×10-^{4} Pa for air which occurred at a velocity of 4×10-^{8} m/s.

It can be concluded that a safe range of porosity values in minimizing pressure drop can be found at higher porosity values. This is useful in the design of industrial equipment such as packed bed reactors as deemed fit.

CHAPTER ONE

## 1.0 INTRODUCTION

### 1.1 BACKGROUND

A porous bed is a medium that consists of a solid matrix with pores (spaces) that permits the flow of a fluid or fluids. The pores serve as a transportation route or passage for the fluid that occupies them to flow through the porous bed. Water flowing through the soil is a common example of this process. A fixed porous bed is therefore one that is not oscillating, translating, or rotating about any axis (i.e., its position remains constant along any axis).

Porous beds have a wide range of applications in the world and most importantly in engineering today. Some of those applications include the following:

i. Scrubber: Scrubbers are industrial waste cleaners that help in reducing the amount of toxic waste produced from industrial processes, such as combustion in plants producing energy. The scrubbers serve as devices to control air pollution and are usually of two types; the wet scrubber and the dry scrubber. Packed beds are usually found only in wet scrubbers because these use a scrubbing liquid to wash the contaminated exhaust gases from the industrial combustion process. The purpose of these packed beds is to increase the surface area of the scrubbing liquid to ensure its effective delivery (usually in droplets) on the contaminated gas and also to increase the efficiency of the removal of toxic waste. A graphical illustration of a scrubber is shown in figure 1.1.

illustration not visible in this excerpt

FIG 1.1: Illustration of a Scrubber. *Taken from http://croll.com/wetscrubbers.html*

ii. Filter Bed: Filter beds can be defined as a medium of material that purify a substance as it flows through it by absorbing the unwanted materials that could be found in that substance. An example is the purification of water. They are also used in the cleaning of industrial wastewater. The material for filter beds could be sand, pebbles, peat, etc. The filter bed itself can be used to define the porous bed as it is porous and allows for fluid flow through. The difference between a filter bed and a porous bed is that a filter bed is concerned with purification or the removal of unwanted material from a fluid stream where as a porous bed just allows the passage of fluid without regard to the absorption of any unwanted material. A good example is a fluidized bed which is designed for chemical purification in which a high surface area is required; it involves liquid and gas and is not applicable to solids. Figure 1.2 shows the graphical illustration of a filter bed used in purifying water.

illustration not visible in this excerpt

FIG. 1.2: Illustration of a Filter Bed *. Taken from http://images.patrika.com/ mediafiles/2015/10/05/rain-water1-56122564cd663_l.jpg*

iii. Chemical Reactor: A chemical reactor can be defined as a specially designed vessel/container where chemical reactions take place. The term porous bed applies to chemical reactors in relation to tubular reactors in which these tubes are packed with solid particles that perform the role of catalysts in the reaction (i.e., increasing the rate of the reaction and favouring the process of the reaction to the end product). One of the reasons for using packed beds as catalysts is that they enable a higher conversion rate of the reactants. An increase in the amount of particles of the packed bed is more important than the volume (i.e., where more density would mean very low porosity). Figure 1.3 shows the graphical illustration of packed bed chemical reactor.

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FIG. 1.3: Illustration of a Packed Bed Chemical Reactor. *Taken from http://4.bp.blogspot.com/-YgFQWJ-POPc/T9Z936_4UCI/AAAAAAAAAE8/-Ql- YQsvGAw/s400/FIXEDBED.png*

iv. A porous bed can also be used to store heat temporarily in large industrial systems where this stored heat can be used to preheat fluids passing through.

### 1.2 Porous Bed Types and Formations

Porous beds can be made from a variety of materials depending on the intended purpose. In filtration processes, sand and rock particles can be used as materials while in chemical reactors, Raschig rings can be used.

The formation of the packing or arrangement of porous beds, is usually of two types:

i. Uniformly packed: In this type, the porous bed is uniform in its packing, i.e., the structure or pattern is well defined throughout.

ii. Randomly Packed: In this type, the arrangement does not follow any structure and the components are located haphazardly in no set direction.

### 1.3 Porous Bed Properties

In porous beds, various properties exist that affect the nature of the flow, the flow and regimes, and ultimately the pressure drop. Principally, in describing flow in porous beds, the Darcy equation is one of the first pioneering equations, as follows;

illustration not visible in this excerpt

Where: qx - is the component of volume flux in the x-direction

[illustration not visible in this excerpt]- is the specific weight of the fluid

[illustration not visible in this excerpt] - dynamic viscosity of the fluid

Kx - permeability in the x-direction

Lx - distance in the x-direction

[illustration not visible in this excerpt] - difference in pressure (pressure drop)

[illustration not visible in this excerpt] - difference in elevation.

The basic most important properties of porous beds are porosity, permeability, and superficial velocity.

Porosity can be defined as the ratio of the void volume (volume of void spaces) to the volume of the body.

Permeability can be defined as a property that measures the flow through a porous bed that is subject to a pressure drop ([illustration not visible in this excerpt]).

Superficial Velocity for porous beds can be defined for fluid flow as the volumetric flow rate for a cross-section of unit area.

### 1.4 Fluid Properties

The fluid properties that are usually in question for porous beds are density and dynamic viscosity.

Density is the ratio of the mass of a substance (fluid) to the total volume occupied by that substance (fluid).

Dynamic viscosity refers to the shear force per unit area that is required to drag a layer of fluid over another layer a unit distance away from it in the fluid.

### 1.5 Flow Properties

In defining the flow in porous beds, not only the Reynolds number is used in defining the flow regimes but rather a Modified Reynolds number Re1. When this modified Reynolds number Re1 must be used, laminar flow for a porous bed is defined as having a modified Reynolds number less than 1 (Re1 < 1) and turbulent flow is defined as having a modified Reynolds number greater than 2 (Re1 >2). This means that transition starts for a modified Reynolds number greater than 1 and less than 2 (1 > Re1 < 2).

### 1.6 Pressure and Pressure Drop

#### 1.6.1 Pressure

In relation to fluid flow, pressure is defined as the normal force exerted by the fluid to a unit surface area. Pressure is measured in Bars (bar), Pascals (pa), Newtons per metre square (N/m^{2} ), Atmospheres (atm), and Pounds per square inch (psi). All these have their relative numerical values. Pressure is also defined as the way in which it acts and the relativity with which it is measured. These include the following:

Absolute Pressure: this is defined as the pressure exerted on a boundary wall

Atmospheric Pressure: this is the pressure exerted by the atmosphere (in the environment).

Gauge Pressure: this is measured as the difference in absolute and atmospheric pressure of a system for a given set of conditions.

#### 1.6.2 Pressure Drop

In any hydraulic system in which a fluid flows, cross-sections of various forms exist that affect the flow. These cross-sections allow for a change in the mechanical energy of the fluid between potential and kinetic as it flows through the system. A pressure drop ([illustration not visible in this excerpt]) is the difference in pressure between any point in the system and a set reference point. The pressure drop of a system is measured using the continuity equation and the equation of fluid motion (i.e., the momentum equation).

Pressure drops in porous media (porous beds) occur due to the viscous friction forces of the flow as well as the resistance of the porous media to the flow of the fluid. The porous media, due to their porosity, tends to constrain the fluid area through which the fluid flows. As a result, the interstitial velocity (velocity within the fluid) increases and in turn, gives rise to a drop in pressure ([illustration not visible in this excerpt]).

In measuring the pressure drop ([illustration not visible in this excerpt]) for porous media (porous beds), the fluid equations of motion are used (momentum equations) which in their differential form is the Navier-Stokes equation.

### 1.7 Objectives

The primary objective of this project is to perform a numerical simulation of flow in fixed porous beds to determine the pressure drop.

The specific objectives are as follows:

i. To vary two key parameters of the porous bed and observe the effect on the pressure drop.

ii. To perform the simulation with more than one fluid.

I. To analyse the behaviour of flow through the porous media by carefully studying the velocity and pressure gradients

### 1.8 Scope of Study

This study will be numerical in nature for Newtonian fluids. It will include both compressible and incompressible flow for flow through a porous medium for a singular porous matrix (porous material) and the porous medium is isotropic and only occupied by one fluid at a time. In addition, the flow will be assumed stationary and not time dependent.

### 1.9 Motivation and Justification

The reason for this project is to increase the amount of knowledge and information as regards porous media, which have a wide use of applications and for which interest in the engineering world has recently increased. The project is also concerned with validating or questioning some of the models that have been used in analysing porous media empirically and hopes to provide professionals designing porous media with data they can use.

CHAPTER TWO

## 2.0 LITERATURE REVIEW

### 2.1 Empirical models

In recent years, extensive research has been carried out in the area of porous media. The Darcy equation has been the existing empirical relationship that has helped in determining the pressure drop in porous media (1856). Furthermore, scientists and engineers have performed experiments to test the validity of the Darcy (1857) model to come up with better descriptions of the flow that occurs in porous media. One of the researchers was Carman (1997) who focused on the work of Blake and Kozeny (1927). In analysing the work performed by Blake, he justified the postulation that the dimensionless groups used for fluid flow through granular beds can be correlated for flow through a bed of spherical grains. He stated that the Kozeny extension of the Darcy law does not hold true for two sizes of spherical particles as well as pointing out other shortcomings in their work. He was, however, able to propose a variation to the equation developed by Kozeny, hence the Carman-Kozeny equation, which is an expression for pressure drop under viscous flow (Kareem, 2009). Nield (2000) discussed different models commonly used in modelling fluid flow and heat transfer in a saturated porous medium. In his work, he discussed the Brinkman Forcheimer equation with its various key terms in the modelling of a porous medium in which emphasis on the Brinkman equation was made. In addition, he also discussed viscous dissipation, the effect of magnetic fields, the effect of rotation, and non- Newtonian fluid.

### 2.2 Porous Material

Various materials can be used to represent a packed bed. But in engineering, we are concerned with modelling of real-life situations. Therefore, porous bed materials are best obtained from practical applications. In addition, porous beds can have a wide range of media and fluid combinations. Researchers have had to experiment with various materials for porous beds from catalytic material to metal foam. Orodu et al (2012) performed an experiment using a capillary tube model for a range of particles which included Ballotini, non-spherical beads and standard mesh gravel of various sizes. In addition, Kareem (2009) carried out experiments with several materials, such as gravel, marble, and glass spheres, to mention a few, which had diameters within a range of 0.2 to 8.89 cm, porosity from 0.3 to 0.47 bed height from 26.03 to 55.88 cm and bed diameter from 7.62 to 15.24 cm. Cruz et al (2013) used indigenous seed (acai seeds) as the porous bed material with particle diameter of 1.16x cm, 0.98y cm, 0.97z cm. Vesicular rocks and fibre mats have also been used as porous media as seen in the work of Costa (2006). Even rectangular cylinders arranged in a pattern have been used to model porous beds Liu et al (2007).

### 2.3 Phase Flow

In considering the type of flow through the porous bed, Darcy's (1857) semi-empirical formula can be applied to fluid transport for single-phase flow. Sochi (2009) performed an elaborate evaluation for this single-phase flow through porous media using four approaches namely: continuum models, pore-scale network, numerical methods, and the bundle of tubes model, all with a non-Newtonian fluid as the working fluid. He concluded that network modelling was the most realistic method for modelling porous media but failed to disprove any of the continuum models used in determining the pressure drop.

### 2.4 Numerical Simulations

In addition, Hassanizadeh and Gray (1987) carried out a numerical study on high velocity flow in porous media. The aim of this study was to derive non linear relationships that had been observed experimentally between the pressure head gradient and fluid velocity. In their approach, they made use of continuum mechanics using an equation of motion. From their study, it was concluded that there is a non linear relationship of drag forces at flow with high velocities at a reynolds number about 10.

Bernsdorf et al. (2000) performed a numerical analysis of the pressure drop in porous media flow with a numerical tool, a lattice Boltzman (BGK) automata. In their work, they performed simulations for a catalytic porous matrix and a SiC (silicon carbide) porous matrix using a 3D geometrical model. From their results, it was concluded that the lattice Boltzman is very efficient in accurately determining the pressure drop in a packed bed.

Marek (2014) carried out one of the few computational studies that have been performed to help understand flow in porous beds. In his work, a numerical simulation was performed using Raschig rings as the porous geometry for an incompressible gas for which the inlet into the system was at the top and the outlet at the bottom. The velocity and pressure distributions were obtained and compared with existing empirical results and there was some correlation. Hellstrom and Lundstrom (2006) performed a numerical simulation of flow through porous media for moderate Reynolds numbers. In their simulation, they used ANSYS CFX 10.0, modelling their design for flow between parallel cylinders for a moderate set of Reynolds numbers and comparing their results with the empirically derived Ergun equation; there was a level of correlation between the two results.

In addition, Sobieski et al (2012) carried out a numerical study for predicting the tortuosity of airflow through packed beds consisting of randomly packed spherical particles. In their calculation, the discrete element method was used for the arrangement of the porous bed and a commercial discrete element package was used for the simulation. The results were well correlated with the result of a carbon testing done on the porous media. It should be noted that there has been a limited amount of simulations done for airflow in porous media.

### 2.5 Experiments conducted with porous media

Numerous experiments have been carried out on porous beds to validate or justify empirical and semi-empirical methods over the years. Of these, Orodu et al (2012) performed an experiment to validate the capillary tube model. They concentrated more on non-spherical particles and were able to compute the tortuosity of the particles. In their results, it was observed there were correlations between their curves with the curves obtained by Ergun. Furthermore, Ribeiro et al. (2010) investigated experimentally the mean porosity and pressure drop measurements in packed beds of monosized spheres. They conducted their experiment using glass beads for a distinct range of diameters using columns of Perspex glass in which the working fluid used was distilled water. The components of the experiment included a liquid tank, a feed pump, a valve, the packed bed, and a manometer to measure the pressure drop. The results of their experiment showed that there were correlations with previously carried out experiments and an increase in the pressure gradient with an increase in the Reynolds number.

Cruz et al (2013) carried out an experiment to evaluate the pressure drop in the flow over a fixed porous bed. Their experiment was based on an analytical solution of the Ergun equation (1952). It involved the use of a special type of palm tree seeds called acai seeds, which were used in the packing of the fixed porous bed and set in a wind tunnel apparatus in which air was blown through the tunnel with the aid of a fan. The length of the bed was varied for the experiment over a range of discrete values which were related to the diameter of the tube attached to the tunnel. Their experiment was defined in relation to gas reactors for the process of gasification because the acai seeds are used as biomass. From their results and findings the experiment, though successful, failed to conform with the Ergun formula, hence it can be inferred from their work that the Ergun formulation does not principally apply for fixed porous beds that involve downdraft gasifiers.

Krishna and Murthy (2013) carried out very extensive experimental research for a wide range of flow regimes with different sizes of aggregates as well as different sizes for the porous medium. The aim of the experiment was to obtain a new set of equations without the use of empirical constants. They used a series of formulations for laminar and turbulent flow as some formulations are suitable only for laminar flow based on experiments carried out (Darcy's Law, Forcheimer Equation) while the other formulations used are capable of handling both laminar and turbulent flow regimes (Ergun equation). The experiment was for a parallel flow through coarse granular media and glass spheres. The set up of the experiment was very elaborate; the components included a permeameter of 6 m in length, pressure measuring devices, a sump, porous media, several valves (inlet and outlet), and a constant supply of fluid (water). The results of the experiment produced new equations for both Darcy and Forcheimer regimes using a relation between the velocity and hydraulic gradient for all the regimes.

### 2.6 Other Studies

Brownell et al. (1956) expanded the already existing body of knowledge by studying pressure drop in the flow over porous media for consolidated beds (i.e., relating to the packing structure of the bed) over a range of varied porosity between 12 and 37%. From their study, they were able to develop three empirical equations that can be used to compute the pressure drop in their specific case for which these equations have been ground breaking in their approach.

Furthermore, Abbood (2009) undertook an analytical model study in which he investigated the viscous flow in porous media using hydrodynamic modelling for five filter media to estimate the pressure drop for water filters. The study was performed for a Reynolds number range of 100 to 800. He also made use of the capillary model to help predict pressure gradients. From the results of the study, empirical relationships were evaluated to measure this pressure drop as well as the friction factor for the range of Reynolds numbers used. One thing we can obtain from this study is that it was limited to measuring the pressure drop for water in porous media and the range of Reynolds numbers is not very large.

### 2.7 Brinkman Equations

The Brinkman model is an extension of Darcy's law developed by H.C. Brinkman in 1947. Durlofsky and Brady (1987) carried out an analysis of the Brinkman equations that describe the flow in porous media. Their study involved using the Green's function of the solution of the Brinkman equation, to compare it with the fundamental solution of the Brinkman equation. From the completed analysis, it was discovered that for volume fractions (porosity) below 0.05, the Brinkman equation is valid but after this value it no longer accurately predicts the behaviour of flow through the porous bed. A limitation of this work was that simulations were carried out only for porosity up to 0.2. Furthermore Nield (2000) observed that the Brinkman viscous term in the Brinkman Forcheimer equation is not completely valid when the porosity of the bed is close to 1 and the effective viscosity is uncertain. However, Fetecau et al. (2011) carried out an analytical study using Fourier sine transforms on Stokes' problem for fluid of the Brinkman type. The results showed that the functions satisfied the governing equations with all imposed boundary and initial conditions.

CHAPTER THREE

## 3.0 Methodology

### 3.1 Governing Equations

In Fluid Dynamics (FD), to analyse fluid flow, there are a set of governing equations that must be solved to determine the nature of the flow as well as the effect of certain macroscopic flow parameters which are common to all fluid flows. These parameters include Velocity, Pressure, Density, Temperature, and Energy.

These parameters are dealt with using a set of governing equations that provide solutions for each of the following conservation principles.

i. Conservation of mass

ii. Conservation of momentum

iii. Darcy's Law

iv. Brinkman Equations

These conservation principles are those that will be considered in solving the analysis of the packed beds.

#### 3.1.1 Conservation of Mass

For a fluid flowing through a fixed porous bed to obey the physical principle of mass conservation, an equation that justifies this phenomenon must be employed. This equation is popularly known in fluid dynamics as the continuity equation.

illustration not visible in this excerpt

#### 3.1.2 Conservation of Momentum

The conservation principle extends from elementary physics, specifically Newton's second law of motion (the rate change of momentum of an object is directly proportional to the force causing the change and that change is in the same direction as the force) applying it to fluid motion.

illustration not visible in this excerpt

FIGURE 3.1: Illustration of the Stresses acting on a Small Rectangular 3D Element.

**[...]**