Classical Mechanics of Fluids - Assignment Example

FV2001 Assignment Brief – Resit Name: Course: Professor: Date: 1. Classical Mechanics of Fluids 1.1 The Navier-Stokes Equations The Navier-Stokes equations govern the behavior of viscous heat conducting fluids. These equations are momentum equation, continuity equation, energy equation, and also equation of fluid motion. The equations apply Newton’s Law of Motion to an element of a fluid. i. Momentum Equation: + . .u) + g ii. Continuity Equation: ) = 0 iii. Energy Equation (First Law Of Thermodynamics): + . u = 0 iv. Equation of fluid motion = + F + u Physical meaning of terms in Navier-Stokes Equations: = pressure – Temperature – Velocity of the fluid - Gravitational acceleration - Fluid density - External force per unit mass - Thermodynamic energy - Dynamic viscosity of the fluid - Coefficient of heat conduction The Navier-Stokes equations show the conservation of momentum, and the continuity equation is a representation of mass conservation. These equations form the center of fluid flow modelling. Solving these equations for given boundary conditions enables the prediction of fluid velocity and pressure in the geometry of flow. Due to the complexity of the equations, only a small number of analytical solutions. The pressure of the fluid, and the velocity are the two terms which require turbulence modelling because they vary with the conditions of fluid flow. Modelling provide solutions for more complex flow geometries and turbulence. Turbulence modelling is required in engineering applications because in practical cases, fluid flows are turbulent in nature. It enables the prediction of turbulent flows. Analytical methods with linear equations cannot be used to predict these kind of flows. This is because it is very challenging to capture all scales of flow and one would require very powerful and fast computers to predict turbulent flows (Quartapelle, 2013). 1.2 Calibrating a pressure meter using a Venturi meter Solution: Pressure of water in the tank = Where: - Height of water (m) – Density of water (kg/m3) – Acceleration due to gravity Pressure of water () = = 88.29 kPa (This is the upstream pressure in the Venturi meter) Velocity of water through the upstream of the meter is given by: = = = = 13.29 m/sec. From the continuity equation: Where: – Area of the upstream of the Venturi – Area of the downstream of the Venturi – Velocity of water at upstream (13.29m/sec) – Velocity of water at downstream Therefore, = = 170.85 m/sec Pressure drop ( ) = = = 14.507 mPa This is the pressure drop between the wider and the narrow section of the Venturi meter (Kreith, 1999). 2. Dimensional analysis 2.1 Kolmogorov scale of velocity There three variables in Kolmogorov scale of velocity () equation are: density of the fluid (), kinematic viscosity () and specific dissipation (). Their relationship between these terms can be written as: = [1] Writing this in dimensional terms, we have; [L/T)= [L2/T] a [L2/T3] b [M/L3] c [2] Where: , and are constants. To get what these constants represent, we compare the terms on both sides of the dimensional equation: L T M Given that Equations & can be solved simultaneously to obtain the value of and. Thus, Substituting these values in equation [1], we have;This is the Kolmogorov velocity scale. 2.2 Archimedes dimensionless number Archimedes number is a ratio (dimensionless number). To see which of the above the correct Archimedes number is, we have to find whether or not they have units. a. = = = No units b. = c. = d. = e. = Conclusion: Formula (a) is the correct Archimedes number. 3. Heat Transfer, Thermochemistry and Fluid Dynamics of Combustion 3.1 Thermal radiation The scenario described in the question can be visualized as shown in the figure below, assuming the heat source is rectangular with a height and a width. Figure 1: Thermal radiation from a rectangular flame. Given that: Width (a) = 0.2 m Height (b) = 1m Distance (x) = 3m Temperature (Tr) = 579ºC Emissivity = 0.9 (assumption) Emitted radiation () = Where: – Emmited radiation (kW/m2) – Emissivity (0.9) - Stefan-Boltzmann Constant (5.6703 10-8 (W/m2K4)) – Temperature (Kelvin) 3.2 Stoichiometric fuel-air ratio of gasoline Considering octane as a content of gasoline, below is the stoichiometric equation of combustion of octane. C8H18 + 25 O2 → 16 CO2 + 18 H2O Fuel-to-air ratio = 1/14.7 Air-fuel-ratio = 1/ (1/14.7) = 14.7 (Air-to-fuel ratio) stoichiometric = = = = 103.16 Equivalence ratio (ϕ) = = = 0.0097 < 1 (Lean mixture) Equivalence ratio of its lean mixture with the fuel-air ratio 1/25 is obtained as follows: Air-fuel ratio = 1/ (1/25) = 25 = = = 175.44 Equivalence ratio (ϕ) = = = 0.0057 < 1 (Lean mixture) The mass fraction of gasoline in this mixture is given by: (1/26)*100% = 3.8% 4. Characteristics of Jet and Buoyant Flames & Fire Plumes 4.1. The main characteristics of fire plumes and two other types of flows encountered in fire environments. A fire plume is divided into three main regions; persistent flame region at the base of the flame, followed by intermittent zone, and lastly, the plume zone. Of the three regions, the persistent zone is characterized by air entrainment and chemical reactions occurring in this region, and therefore, it is the most area of interest in terms of flame establishment, growth, stabilization as well as mass formation. Heat release also occurs in the persistent zone, with a light blue flame that appears almost laminar. The high heat release rate in this region induces an increment in the gas temperature and velocity. The characteristics of fire flame are generated by the basic mechanisms of heat transfer; convection and radiation. Within a plume zone, temperature and velocity decrease with increasing plume height. Plume temperature – A fire plume is normally characterized by centerline temperature, radial temperature – some distance from the centerline, and mean temperature. These plume temperatures show the thermal characteristics of a plume. In an ideal axi-symmetric plume, the temperature is maximum at the centerline and reduces radially as you move to the edge of the plume. The centerline temperature in the persistent region is roughly constant and represents the average temperature (Kaminski & Jaupart, 2003). As more ambient air continues to be entrained into the plume, there is a sharp decrease in temperature of the plume. Plume velocity – This is the rate at which a plume is moving away from the source of fire. Like temperature, the plume velocity is maximum at the centerline and reduces towards the edge of the plume. The velocity increases as height increases within a continuous flame region. As the height increases above the flame, the velocity reduces progressively because of entrainment of more air which cools the plume. Types of Flows Encountered In a Fire Environment There are two types of flows commonly encountered in a fire environment depending on the boundary conditions. These are laminar flow and turbulent flow. Laminar Axisymmetric Plume – Laminar plumes are characterized by a constant velocity as the plume rises above the source. In practical experience, most fire plumes are turbulent in nature, but there are times will remain laminar for some reasonable height, especially when the viscosity of the fluid is large. Laminar flow depends on Prandtl numbers. Turbulent flow – This type of flow is experienced as the plume rises, and grows from a laminar cross flow. The level of turbulence depends on the rate of entrainment, and the difference in temperature and velocity between the plume and the ambient air. Atmospheric conditions, especially wind and temperature, play a very significant factor on the flow of a plume as it rises from the source. In plume jets, turbulent flow depends on the geometry of the domain and the external forces acting on the plume. Plume flows can also flows due to buoyancy and momentum effects. These are described below. Buoyant flow – This flow occurs due to the differences in temperature between ambient air and the stack plume. The outer edge of a fire plume is more susceptible to effects of buoyancy since the velocity of the plume in this region is lower. This effect results in a buoyant flow. Momentum flow – This is flow due to continuous injection, and the velocity of exit of effluents in a plume or plume emissions. 4.2. The main characteristics of jet and buoyant flames Jet and buoyant flames are fire plumes driven with buoyancy and/or additional propulsion momentum. Jet and buoyant flames are largely characterized by temperature and velocity of flow, which are the main parameters that affect their diffusion in other fluids. The stabilization of jet and buoyant flames depends on the Reynolds’s number and the cross-flow speeds. Temperature – Jet flames are produced when a flame is discharged in the quiescent environment through a narrow conduit. When a jet flame enters another fluid or ambient air, a velocity shear forms between the ambient fluid and the entering fluid. The temperature of a jet flame decreases further away from the center of the jet flame (Yao, et al., 2007). On the other hand, the temperature of buoyant flames is unsteady and fluctuates from ambient to the maximum. Temperatures are higher at the center (near the reaction zone) and reduces towards the edge of the flame. Speed of flow – The speed of flow of jet and buoyant flames is largely affected by the boundary conditions e.g. the rate of entrainment, ambient temperature, humidity, and pressure. Near the source, the velocity is maximum at the centerline of the flame, and fluctuates in the outer edges. References Read More