LECTURE 1 THE CONTENTS OF THIS LECTURE ARE AS FOLLOWS:


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1 LECTURE 1 THE CONTENTS OF THIS LECTURE ARE AS FOLLOWS: 1.0 INTRODUCTION TO FLUID AND BASIC EQUATIONS 2.0 REYNOLDS NUMBER AND CRITICAL VELOCITY 3.0 APPROACH TOWARDS REYNOLDS NUMBER REFERENCES Page 1 of 11
2 APPLICATION OF FLUID MECHANICS IN MINES 1.0 INTRODUCTION TO FLUID AND BASIC EQUATIONS First of all, the question arises about, what are fluids and why do we need to study application of fluid mechanics in mine ventilation? For defining fluid, we must know, how a solid and a fluid differ in their response when they are subjected to shear stress. Let me tell you that stress when applied parallel or tangential to the face of a body is called shear stress. It is given by: τ = F A, where F= force, A= area Let us refer to Fig.1 below: Fig. 1 Shear stress acting on a material (Solid or fluid) In Fig. 1, we can say that force F is being applied to the upper face of the material (Solid or fluid) and the force is parallel to the face of the material. A is the area of the face. H is the distance between the opposite faces of the material. D is the displacement or the deformation, the upper face of the Page 2 of 11
3 material undergoes. τ is the shear stress and is given by the relation, F A. Now, the material can respond in the following two ways: i. Shear Stress strain ( D ), or H ii. Shear Stress rate of strain ( V H i. e. D TH, T = time) The materials which follow the first relation are called solids and the material following the second are termed as fluids. Thus, we can say that fluid can be thought of as material which can undergo continuous deformation as long as the shear stress is applied, however small the shear stress may be. On the other hand, solid shows resistance to deformation or even if it undergoes deformation it comes to its original shape if the applied force is under elastic limit. Thus, if a fluid moves it means shear stress is acting over it. We are discussing about the application of fluid mechanics in mine ventilation because air is the chief element on which we will focus our attention when it comes to underground mine ventilation. Air falls under fluid and obeys all the basic laws of nature or science like conservation of mass, conservation of energy, Newton s law of motion, basic laws of thermodynamics, etc. Now, we will discuss in brief some of the basic equations without going in details about their derivation. (a) Newton s law of viscosity τ = μ du dy, N where μ = dynamic coefficient of viscosity ( m 2 s ) ; τ = shear stress ( N m 2) ; du dy = velocity gradient or rate of strain (1 s ) Fluid which follows the above relation is termed as Newtonian fluid. Let me tell you that viscosity is the property by which a fluid shows resistance in flow. This resistance is applied in between the layers of fluid and hence is internal in nature. Also, μ of fluid is related to density of fluid, ρ by the relation: ν = μ ρ, where ν = kinematic cofficient of viscocity (m2 s ) Page 3 of 11
4 (b) Hydrostatic law of fluid p = ρgh, where p = pressure ( N m2) ; ρ = density (kg m 3) ; g = accelaration due to gravity ( m s2) ; h = height of liquid column (m) (c) Law of mass conservation According to this, mass flowing in an isolated system remains constant. Let us consider Fig. 2. Fig. 2 From this figure we can write, M 1 = M 2 M = ρav, where v = velocity ( m ) ; ρ = density (kg s m 3) ; A = area(m2 ); M = mass flow rate (kg/s) For incompressible fluid, applying M 1 = M 2; we get ρa 1 v 1 = ρa 2 V 2 or, A 1 V 1 = A 2 V2 This is the equation of continuity in its simplest form and it explains that the mass of a fluid remains conserved in the course of its flow. (d) Bernoulli equation This is based on the law of conservation of energy. Bernoulli equation is a special case of Euler s equation and is given by: ρv 2 + ρgh + p = constant, where 2 Page 4 of 11
5 ρ = density ( kg m 3) ; v = velocity (m s ) ; g = acceleration due to garavity (m s 2) ; h = height or difference in elevation (m) p = pressure( N m 2) Thus, we can say that for an incompressible and nonviscous fluid the sum of the kinetic, potential and pressure energies per unit volume is constant. When a fluid is viscous, we apply law of conservation of energy taking energy loss due to friction into account. 2.0 REYNOLDS NUMBER AND CRITICAL VELOCITY Today, what we are going to discuss is familiar to most of you. How many of you have noticed a change in flow of water from a tap after loosening the knob of a tap gradually. Did you notice any change in the flow pattern in initial stage and after some time when you have loosened the knob significantly (Fig. 3). What difference do you notice? If we ask this question to a layman, he will say that initially it is simple and ordered and later on when we loosened the knob it is complex and no more ordered. Why does this change occur? Is it due to increase in velocity? Yes of course, it is. Again, you will notice that even if the flow from tap is maintained ordered as in the initial stage, the water after falling in the basin assumes a flow similar to what we see in the latter stage, i.e. the flow in the basin is complex and disordered. It means that along with velocity, it also depends on shape of opening through which it flows. Page 5 of 11
6 Fig. 3 Depiction of laminar and turbulent flow You would like to see that how this phenomena occurs and how the transition from simple ordered flow to complex disordered flow takes place. The theory behind these flow pattern was not developed in a day or two. It took years to explore it. First of all it was G. H. HAGEN ( ), who observed this and came out with a conclusion that, the flow pattern is a function of velocity and viscosity of the fluid. How many of you are familiar with the term viscosity? Viscosity is the inherent property of a fluid to resist its flow. Fluids include gases also, and therefore, this type of flow is shown by gases too. In 1880 s, Professor Osborne Reynolds carried out numerous experiment on fluid flow. We will now discuss the laboratory set up of his experiment. The experimental set used by Prof. Osborne Reynold is shown in Fig. 4. Page 6 of 11
7 Fig. 4 Laboratory experimental set up for Reynolds experiment As you can see from the figure, Reynolds injected dye jet in a glass tube which is submerged in the large water tank. Please see that the other end of the glass tube is out of water tank and is fitted with a valve. He made use of the valve to regulate the flow of water. The observations made by Reynolds from his experiment are given shown through Figures 5 to 7. Fig. 5 Sketch showing the flow to be simple and ordered at low velocity Fig. 6 The flow of dye forming wavy pattern at medium velocity Page 7 of 11
8 Fig. 7 The flow of dye is complex at higher velocity Now, it is important to know what velocity should we consider as low, medium or high for this phenomenon? The answer to this question can be known once we understand Reynolds number. 3.0 APPROACH TOWARDS REYNOLDS NUMBER Throughout the experiment, Reynolds thought that the flow must be governed by a dimensionless quantity. What he observed was that inertial force viscous force is unit less (dimensionless). Let us see the mathematical expression of inertial force and viscous force. Inertial force is the force due to motion i.e. which may be also called as kinetic force. Kinetic energy = mv2 2 where, v = velocity (m/s); m = mass (Kg) Inertial force = ρv2 2 where, ρ= density (kg/m3 ); v = velocity (m/s) Now, inertial force = kg m2 m 3 s 2 =kg m s 2 m m 3 = N/m2 = unit of pressure Thus, inertial force has unit of Newton per meter square. Viscous force = shear stress = µ du dy By Newtons law of viscosity (Newton per meter square) Thus, inertial force = ρ v2 dy viscous force µ du (as we are interested only in dimensionless quantity, so we are omitting 2 from the denominator) Page 8 of 11
9 Now, for a finite length we can write dy = l, and du = v This results in ρv2 l µ v viscosity; v = mean velocity. = ρ v l, where l = mean diameter of duct; µ = µ The expression (ρ v l)/µ is called REYNOLDS NUMBER and is designated as R e. Now, coming to CRITICAL VELOCITY, I would like to show you a graph which is depicted in Fig. 8. Fig. 8 Lower and upper critical velocities in the transition region of flow from laminar to turbulent We can see the change of flow from laminar (simple and ordered) to turbulent (complex and disordered). In between them, transition flow exists. The velocity at which flow changes from laminar to transition is called LOWER CRITICAL VELOCITY and that at which transition changes to turbulent is called UPPER CRITICAL VELOCITY. We can see that at lower speed, the flow is laminar and at higher speed it is turbulent. Page 9 of 11
10 Now let us go back to the expression of Reynolds number. R e = ρ v l µ Thus low velocity for simple and ordered flow indicate that R e should be low. Hence laminar flow(similar and ordered) takes place when ρ, v and l are small and µ is large. Reynolds concluded from his experiments that for: Laminar flow  0 Re 2000 Transition Re 4000 Turbulent flow  Re 4000 Now, we will discuss how the two different types of flow are related to velocity. Do they depend on velocity linearly or in some other way? The flow turns out to be turbulent from laminar if we increase the velocity, keeping all other parameters constant. Since REYNOLDS NUMBER is directly proportional to velocity, the flow should be a linear function of velocity. However, the pressure drop which these types of flow create while flowing in a pipe does not follow the linearity in case of turbulent flow. Δp u for laminar flow Δp u 2 for turbulent flow At this point let me tell you that in many text books you may find that it is mentioned Δp u But, we will use Δp u 2 for turbulent flow until or unless it is stated. REFERENCES Hartman, H. L., Mutmansky, J. M. & Wang, Y. J. (1982); Mine Ventilation and Air Conditioning ; John Wiley & Sons, New York. McPherson, M. J. (1993); Subsurface Ventilation and Environmental Engineering ; Chapman & Hall, London. Page 10 of 11
11 Vutukuri, V. S. & Lama, R. D. (1986); Environmental Engineering in Mines ; Cambridge University Press, Cambridge. Page 11 of 11
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