ABSTRACT

We have modelled an open channel flow through a porous media (River). In the model, we considered water as an incompressible fluid; the flow as steady and uniform; the system is assumed to be isothermal and the flow, also a laminar flow. We have solved the resulting equation using analytical method. By some mathematical operations, the momentum partial differential equation (PDE) was reduced to ordinary differential equation (ODE) and the resulting equations are solved analytically using the technique for solving linear equations with constant coefficients-method of variation of parameters. The analysis of the result was done and plotted on graph using MATLAB to show the effect of permeability on flow parameters such as velocity, pressure gradient and the height of the channel. Recommendations were made to control and manage the flow of rivers.

 

 

 

 

TABLE OF CONTENTS

Title page ————————————————————————–  i

Certification page —————————————————————   ii

Dedication ————————————————————————   iii

Acknowledgement            ————————————————————–   iv

List of figures, tables and graphs ——————————————–    v

Abstract —————————————————————————- vi

Table of Contents —————————————————————-  vii-viii

Open Channel flow ————————————————————-   ix

CHAPTER ONE

1.1      Introduction ————————————————————–  1-3

1.2      Aims and Objectives of the study ————————————   3-4

1.3      Scope of the study ——————————————————- 4

1.4      Relevance of the study ————————————————-   4

1.5      Limitations of the study ————————————————   5

CHAPTER TWO

2.0      Literature Review ———————————————————            6-14

CHAPTER THREE

3.0      Theory of Channel flow

3.1      Properties of fluid ——————————————————-  15-26

3.2      Types of flow in Channels (structure) ———————————-          26-27

3.2.1  Steady flow and unsteady flow ————————————–   28-29

3.2.2  Uniform and Non-uniform (or varied flow) ———————-     29-30

3.2.3  Laminar flow and Turbulent flow ———————————-      30-31

3.2.4  Subcritical flow, critical flow and supercritical flow ————-    31-32

3.3      Volumetric flow rate —————————————————–            32-33

3.4      Permeability of fluid during flow ————————————– 33-35

CHAPTER FOUR

4.0      Model Equation in open channel flow over a permeable flow

4.1      Mass Conservation —————————————————–    36-39

4.2      Conservation of Momentum ——————————————  39-41

4.3      Equation of an open channel flow through porous media with porosity —————————————————————————————-            41-42

4.4      Method of solution of modelled equation ————————     43-51

CHAPTER FIVE                                                                                                   

5.0      Analysis and discussion of Results ———————————-     52-58

5.1      Conclusion —————————————————————-  59

5.2      Recommendations ——————————————————   60

REFERENCES ———————————————————————–            61-62

 

 

CHAPTER ONE

1.1 INTRODUCTION

As a result of the importance of open channel flow, and the occurrence of porous media in a wide variety of important practical applications, the overall objective of this work is to provide analytical technique for treating problems which involve flow of an open channel through porous media. Specifically flow through a permeable river bed.

The flow of water in an open channel is a familiar sight, whether in a natural channel like that of a river or an artificial channel like that of an irrigation ditch. Its movement poses a difficult problem when everything is considered especially with the variability of natural channels. However, in many cases the major features are expressed in terms of only few variables, whose behaviour can be described adequately by a simple theory.

The principal forces at work during fluid flow are those of inertia, gravity, viscosity and pressure gradient Calvert (2003).

In this study, water is considered as the fluid with a known density of 1g/cc and does not vary significantly for the temperature and pressures that would be considered in this work.

It has been observed that the total volume of water transported downstream by a river, is a combination of the free water flow together with a substantial contribution to flow through subsurface rocks and gravels that underlie the river and its flood plain. It has also been observed that some rivers flow intermittently. They only flow occasionally and can be dry for several years at a time. This has been attributed to geological conditions such as highly permeable river bed (Wikipedia, the free encyclopaedia).

Although almost all natural channels have permeable beds such as gravel Bed Rivers, not much research has been undertaken in order to study the effect of channel bed permeability on the mean and instantaneous flow. In common practice a permeable bed has usually been treated analogously to an impermeable bed, and flow resistance coefficient and velocity distributions derived irrespective of bed porosity. Thorsten et al (2007), observed that depending on the permeability of the subsurface, significant interaction processes occur between the flow above the porous bed and the subsurface area. The effects of this interaction are a non-zero velocity at the permeable boundary. They also observed that the driving force which is responsible for the exchange processes between the pore layer and the upper flow is the presence of local pressure gradients.

According to Radiom et al (2007), fluid flow in a porous media shows some of the characteristics of flow in the absence of rigid matrix and in such a flow regime, the inertia and fluid shear stress effects not included in the Darcy model becomes significant affecting the flow characteristics. Flow in solid media or flow in the absence of a rigid matrix is governed by some fundamental laws based on the conservation of mass, momentum and energy.

It has become pertinent to study open channel flow over a permeable river bed so that in managing or controlling a river to make it more useful or less disruptive to human activity, the effect of permeability would be considered.

When some rivers dry up as a result of permeability this will result to drop in energy supply, affect habitat conservation and some other uses of the river.

The permeability of a river may contribute to the overflow of its bank and consequently lead to over flooding.

1.2 AIM AND OBJECTIVE OF THE STUDY

River as an example of open channel flow is very important; they provide us with food; sometimes source of drinking water; source of energy; transportation etc.

The main purpose of this study is to provide further insight into open channel flow over permeable beds and to enhance the understanding of the effect of bed permeability on the mean and instantaneous flow, as well as other flow parameters such as viscosity, pressure gradient and height of the channel.

1.3 SCOPE OF THE STUDY

The study which is aimed at presenting the effect of permeability on the mean flow, pressure gradient, height and viscosity of a flowing river considers the flow of a river between the point where water from the mountain joins the channel and the point where the channel distributes water to the sea, ocean etc.

The study models the laminar open-channel flow over a permeable river bed, solves the resulting equation using analytical method.

Solution obtained is used to determine: the effect of permeability on the velocity of the flow; the effect of permeability on the ability of the river to resist flow; the effect permeability on the height of the river; etc. The chemistry of rivers which depends on inputs from the atmosphere is not considered in the study.

1.4 RELEVANCE OF STUDY

This study will help in the management and control of rivers to make them more useful, or less disruptive, to human activity.

1.5 LIMITATIONS OF THE STUDY

Time constraint will not allow for an in dept coverage of all the concepts connected with the topic under study; hence the study is limited to laminar not turbulent flow; steady and uniform flow; homogeneous and incompressible fluid. Finance is another constraint as experimental methods involving measurement of the flow parameters are generally costly.

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