Classical Mechanics of Fluids - Report Example

FLUID MECHANICS Student’s Name Institutional Affiliation Date Fluid Mechanics 1. Classical Mechanics of Fluids 1.1 There is the equation of continuity, the energy equation, and the momentum equation under the Navier-Stokes equations (Anderson, 2011). Continuity: (Fielding, 2011) Momentum: X-momentum: Y-momentum: Z-momentum: Energy: +] (NASA, 2013) Where (x, y, z) =coordinates, ρ=density, t=time, p=pressure,, (u, v, w)=velocity components, q=heat flux, Re=Reynolds number, Pr=Prandtl number The momentum conservation equations The general equation for conservation of momentum for all coordinates is given by (Caretto, 2009) Where j is the coordinates. The conservation of energy equations The Bernoulli equation brings together the various terms of conservation of energy (Nave, 2014). The Bernoulli equation for conservation of energy (Nave, 2014) Where, Only those terms with Reynolds Number and Prandtl Number in the momentum equations need turbulence modelling. Turbulence affects the flow of fluids, and, as a result, turbulence modelling cater for this effect in fluid dynamics. In addition, it enabled fluid dynamic designers to predict the characteristics of a particular flow. The source term in the energy conservation equation is the constant total energy at any point within a system. For example, in a water system, the source term would be the sum of potential and kinetic energy of the system. 1.2. The Bernoulli equation governs pressure drop across a venture meter in its simplified form (Princeton University, 2013). Pressure before the constriction equals the pressure exerted by the water height in the tank. Properties of water before the constriction are denoted as part 1 while part 2 represents the smaller portion of the pipe. A simplified Bernoulli equation is given as; (Princeton Univeristy, 2013) According to the equation of continuity, mass flow rate is constant across the pipe irrespective of the size. Therefore, Where For a constant density, the mass flow rate equation becomes,. Hence, Substituting in the Bernoulli equation and simplifying further gives; 2. Dimensional analysis (25 marks) 2.1. Dimensions of terms a) b) c) d) e) There are no dimensionless terms among the terms above. 2.2. Derivation of a formula using dimensional analysis. Density has also been used in the derivation of the formula. Sorting terms with similar bases give Hence, Kolmogorov scale of velocity 3. Heat Transfer, Thermochemistry and Fluid Dynamics of Combustion 3.1. Burning process of PMMA PMMA burns in two stages; in the absence of oxygen (pyrolysis) and the presences of oxygen. In the absence of oxygen, PMMA decomposes thermochemically at high temperatures. The process involves a simultaneous and permanent change in chemical and physical phases of PMMA. However, when the polymer is ignited in the presence of oxygen, it decomposes thermally under temperature due to the heat produced during burning. Like in pyrolysis, the changes are permanent. The stoichiometric fuel-air ratio The atomic weights of elements involve in the reaction are: Molecular weight of oxygen Molecular weight of PMMA Therefore, the fuel-oxygen ratio Therefore, 0.521 kg of fuel would require 1kg of oxygen to burn. Since oxygen constitutes around 23.2 percent of mass in air, 1kg of oxygen would require As a result, 0.521kg of fuel require 4.31kg of air to burn. It gives a fuel-air ratio of Heat produced From the heat release rate given, 1kg of fuel produces 24.9MJ. Subsequently, 2.5kg of PMMA produce 3.2. Factors that affect a chemical reaction rate Factors that affect the reaction rate of a secondary-order chemical reaction include concentration of reactants, presence/absence of a catalyst, the nature of the reactants, and conditions of the reactions such as temperature and pressure. Concentration of reactants Concentration is vital in the reaction rate. The higher the concentration of the reactants, the faster the reaction will be. Lower concentration of reactants would result in a slow reaction. Presence/absence of catalysts A catalyst can either hasten or reduce the reaction speed. In most cases, catalysts are used to speed the reaction rate. As a result, a chemical reaction in the presence of a catalyst would proceed faster than that without a catalyst. Temperature The majority of chemical reactions proceeds faster at high temperatures than at low temperatures. An increase in temperature enhances molecules agitations, which increase the reaction rate. Pressure of the reactants The higher the pressure of the reactants, the faster the reaction rate. The reaction rate reduces as the pressure falls. High pressure increases the number of molecules within a space, which in turn increase the reaction rate. Nature of the reactants The properties of individual reactants affect the reaction rate. Such properties include the bond size and strength, and the state of matter for each reactant. For instance, gaseous reactants react faster than liquids and solids under the same conditions. A burning gaseous fuel For example, combustion of ethylene gas. (University College Cork) 4. Characteristics of Flames & Fire Plumes 4.1. Characteristics of a fire plume When a solid fuel is ignited, the solid fuel is pyrolysed and starts burning, and releases smoke before the diffusion flame forms (Hartin, nd). Part of the gases released are burnt in the flames, and the remaining unburnt substances form part of the fire plume in the smoke layer. As the diffusion fire develops and spread, the fire plume also develops. A flow of hot gases is formed above and in the flame. As the fire propagates, the heat release from the exothermic reaction determines the subsequent characteristics of fire and fire plume as well. The higher the heat release rate, the faster the smoke and heat spread within the compartment. A fire plume is characterised by a temperature gradient that exist along flow of a hot gaseous stream. The temperature gradient also explains the density gradient along the hot gas stream. The gases in the upper part of the fire plume are at high temperatures and low density while the low temperature and high density gases form the lower part of the plume (Bandi, 2010). A fire plume is also divided into three zones depending on the velocity, temperature or the nature of flame. A continuous flame zone (I), intermittent flame zone (II), and far field zone (III) are the three zones of a fire plume (Bandi, 2010). Zone (I) consists of a flame of a continuous velocity. Zone (II) has a fluctuating flame while zone (III) gas velocity and temperature decreases with height of the fire. The max flux in the gas stream within the fire plume increases as the diffusion flame grows. In addition, the temperature and velocity of the gas decrease with the plume height. A generalised axisymmetric plume model Although there are quite a number of fire plumes, the buoyant axisymmetric plume is the most used model. The diffusion flame formed by a burning fuel form the axisymmetric plume. It is assumed an axis symmetry exist along the centre line of the fire plume (Karlsson and Quintiere, 2000). The air is entrained horizontally from all directions in the plume. The axisymmetric plume has a radius measured from the centreline to the outer boundary of the plume and is given in meters (m). Although the temperature of the plume differs with height, the highest temperature exists along the centreline of the plume and decreases towards the outer boundary of the plume. The velocity of the gases within the plume boundary in highest at the centreline and measured in meters per second (m/s). 4.2 Factors that affect the spread of the flame on the solid fuel surface The rate of spread of fire on solid fuel surfaces depends on the ambient conditions, ventilation, material’s thermal inertia, surface direction of fuel, and surface geometry of the solid fuel (National Institute of Standards and Technology (NIST), 2014; Bengtsson, 2011). Material’s Thermal inertia The material’s thermal inertia influences the spread of flame on its surface. Larger thermal inertia slows down the spread of flame on the fuel surface. As a result, the smaller the thermal inertia, the faster the rate of flame spread. Surface direction The direction of the solid fuel determines how fast the flame spread on its surface. The flame will spread faster in the vertical direction. Hence, the flame will spread faster upwardly. Surface geometry The interaction of the materials would enhance the spread of flame. The closer the interaction the solid fuels, the faster the spread of flame. In addition, materials in contact spread flame faster through conduction (NIST, 2014). The ambient conditions The ambient conditions around the solid fuel influence the spread of flame. The higher the temperature of the compartment, the faster the flame spread, and the flame spread reduces as the temperature falls. Flame spread for a liquid or gas fuel For the liquid and gas fuel, the flame spread faster than the solid fuel in a compartment. In liquid fuels, the flame spread mainly through convection, which is faster than conduction. Similarly, flame spread faster in gases through radiation. Bibliography Anderson, J.D., 2011. Governing equations of fluid dynamics. Available at: < http://www.springer.com/cda/content/document/cda_downloaddocument/9783540850557-c1.pdf?SGWID=0-0-45-621403-p173839306> [Accessed 27 March 2015] Bengtsson, L., 2011. Enclosure fires. Available at: [Accessed26 March 2015] Caretto, L.S., 2009. Equations of computational fluid dynamics. Available at: [Accessed 27 March 2015] Fielding, S., 2012. The basic equations of fluid dynamics. Available at: [Accessed 26 March 2015] Karlsson, B. and Quintiere, J.G., 2000. Enclosure fire. Florida:CRC Press LLC. National Institute of Standards and Technology (NIST), 2014. Fire dynamics. Available at: [Accessed 27 March 2015] Princeton Univeristy, 2013. Bernoulli’s equation. Available at: [Accessed 26 March 2015] University College Cork, 2013. Alkenes chemical properties. Availabble at: [Accessed 27 March 2015] Nave, R., 2014. Bernoulli equation. Available at: [Accessed 25 March 2015] National Aeronautics and Space Administration (NASA), 2013. Navier-Stocks equations. Available at: [Accessed 25 March 2015] Read More

A fire plume is characterised by a temperature gradient that exist along flow of a hot gaseous stream. The temperature gradient also explains the density gradient along the hot gas stream. The gases in the upper part of the fire plume are at high temperatures and low density while the low temperature and high density gases form the lower part of the plume (Bandi, 2010). A fire plume is also divided into three zones depending on the velocity, temperature or the nature of flame. A continuous flame zone (I), intermittent flame zone (II), and far field zone (III) are the three zones of a fire plume (Bandi, 2010). Zone (I) consists of a flame of a continuous velocity. Zone (II) has a fluctuating flame while zone (III) gas velocity and temperature decreases with height of the fire.

The max flux in the gas stream within the fire plume increases as the diffusion flame grows. In addition, the temperature and velocity of the gas decrease with the plume height. A generalised axisymmetric plume model Although there are quite a number of fire plumes, the buoyant axisymmetric plume is the most used model. The diffusion flame formed by a burning fuel form the axisymmetric plume. It is assumed an axis symmetry exist along the centre line of the fire plume (Karlsson and Quintiere, 2000).

The air is entrained horizontally from all directions in the plume. The axisymmetric plume has a radius measured from the centreline to the outer boundary of the plume and is given in meters (m). Although the temperature of the plume differs with height, the highest temperature exists along the centreline of the plume and decreases towards the outer boundary of the plume. The velocity of the gases within the plume boundary in highest at the centreline and measured in meters per second (m/s). 4.2 Factors that affect the spread of the flame on the solid fuel surface The rate of spread of fire on solid fuel surfaces depends on the ambient conditions, ventilation, material’s thermal inertia, surface direction of fuel, and surface geometry of the solid fuel (National Institute of Standards and Technology (NIST), 2014; Bengtsson, 2011).

Material’s Thermal inertia The material’s thermal inertia influences the spread of flame on its surface. Larger thermal inertia slows down the spread of flame on the fuel surface. As a result, the smaller the thermal inertia, the faster the rate of flame spread. Surface direction The direction of the solid fuel determines how fast the flame spread on its surface. The flame will spread faster in the vertical direction. Hence, the flame will spread faster upwardly. Surface geometry The interaction of the materials would enhance the spread of flame.

The closer the interaction the solid fuels, the faster the spread of flame. In addition, materials in contact spread flame faster through conduction (NIST, 2014). The ambient conditions The ambient conditions around the solid fuel influence the spread of flame. The higher the temperature of the compartment, the faster the flame spread, and the flame spread reduces as the temperature falls. Flame spread for a liquid or gas fuel For the liquid and gas fuel, the flame spread faster than the solid fuel in a compartment.

In liquid fuels, the flame spread mainly through convection, which is faster than conduction. Similarly, flame spread faster in gases through radiation. Bibliography Anderson, J.D., 2011. Governing equations of fluid dynamics. Available at: < http://www.springer.com/cda/content/document/cda_downloaddocument/9783540850557-c1.pdf?SGWID=0-0-45-621403-p173839306> [Accessed 27 March 2015] Bengtsson, L., 2011. Enclosure fires. Available at: [Accessed26 March 2015] Caretto, L.S., 2009. Equations of computational fluid dynamics.

Available at: [Accessed 27 March 2015] Fielding, S., 2012. The basic equations of fluid dynamics. Available at:

Read More