Classical Mechanics of Fluids - Assignment Example

FV2001 ASSIGNMENT By Student’s name Course code and name Professor’s name University name City, State Date of submission 1. Classical Mechanics of Fluids 1.1 Navier-Stokes energy conservation equations Energy conservation equation; Where; Coordinates: (x,y,z) Time: t Pressure: p Heat Flux: q Velocity Components: (u,v,w) Density: ρ Stress: τ Reynolds Number: Re Total Energy: Et Prandtl Number: Pr (National Aeronautics and Space Administration, 2015) x, y and z are spatial coordinates and time which are dependent on each other. The other dependent variables pressure (p), density (r) and temperature (T) are contained in the total energy equation Et. This equation is derived based on the viscosity effects and Euler in that their processes are similar in derivation. Applying calculus, this differential equation can be solved in an analytical manner to come up with further simplifications to the equation a field known as computational fluid dynamics (National Aeronautics and Space Administration, 2015). ………………………………………………… (1.1.1) The Navier stokes equation for non-compressible flow is shown in 1.1.1 above and going by the norms, it is meant for measuring diffusion and convention mechanisms. In modelling for turbulent flow, the first variable considered is the Reynolds number effect. It should be noted that, while low Reynolds numbers signify smooth laminar flow, increasing it definitely leads to turbulent separation which may also be chaotic. Nonlinear eddy viscosity models consider molecular viscosity which is also considered for Reynolds stresses and velocity gradients (Jurij, 2007). Laminar flow is usually dominated by large scale object shapes and dimension while turbulent flow is challenging when it comes to computations since it occurs in small scales that evolve in eddies. Based on other challenges such as unsteadiness, dependency on initial conditions, three dimensional variations and a wide range of scales occurring as eddies, there arises a need to have a higher capacity computer for purposes of turbulent modelling. In order to achieve accuracy, these time accurate models require extremely fine grids (Jurij, 2007). Source Term The source term in energy conservation equation is defined as the sum of heat source from the equation. These include the per mass unit quantity φ thermodynamic internal energy e and the kinetic energy V2/2 to give a balanced energy equation 1.1.2 below; ……………….. (1.1.2) 1.2 Fire Fighting Risers Solution 2 Bernoulli’s equation applicable to this problem is in the form; Where = Height at two points = pressure Converting to N/m2 and solving equation above; Pressure = 11.8 Bars Given the following; Swamee-Jain equation for Darcy friction factor is stated as follows; Where=Darcy Friction factor =Pipe roughness =Internal pipe diameter =Reynolds number Considering laminar flow; 2. Dimensional analysis 2.1 Finding dimensions for; ………………………………………… (2.1.1) Where; ρ is the density in kg/m3 v is the velocity in m/s a and b are lengths in m μ is dynamic viscosity of the fluid in kg/(m·s) Substituting above units into equation 2.1.1 above; …………………………. (2.1.2) Multiplying units in equation 2.1.2 Units for equation 2.1.1 shall be 2.2 Dimensional analysis Obtaining the formula for dependence using the dimensional analysis of Kolmogorov scale of velocity, VT; In order to derive the dependency formula, we apply the five-thirds law where λd Read More

For wood, flames spread on the surface hence the definition of such fire as a sequence of ignitions. The heat release rate per unit area is described by the equation 3.1.4 below; On complete pyrolysing of the wood surface, the reaction heads towards the inner depth of the material. The rate at which heat propagates the pyrolysis front is faced with some resistance in form of charring. The charring rate however increases when density of the material is higher; a model that is only described by conservation of mass model (VTT Technical Research Centre of Finland Ltd., 2007). 3.

2 Reaction Rate of a fire The reaction rate of a fire is defined as the response of material to the components of fire which mainly are oxygen and heat to complete the fire triangle. This is the main determinant of ignition ability and production of various by-products which may include products of combustion. As such, it can be concluded that the factors that determine the rate at which a fire occurs are contained in the fire triangle in which oxygen, heat, fuel and nature of reaction whether exothermic or endothermic are closely studied (Kim & Lilley, 2000).

Fire reaction is also used to denote the flammability and combustion properties that define a material and which affect the early stages of fires till the flashover stage is experienced. Heat in this case is responsible for the initial ignition upon which the reaction rate is set as origin and during which the preheating, removal of moisture and warming of surrounding air occurs. Fuel is another determinant for fire reaction rate and is characterised by form e.g. gas, solid and liquid, size, shape etc.

Lastly oxygen presence is a paramount determinant of reaction rate as may be seen in any gaseous reactions where oxidation occurs. Fuel burning is obviously exposed to oxidation which in turn releases chemical products and other combustion products alongside heat (US. Forest Service, 2015). As way of example, the equation 3.2.1 shows a reaction in which ethane reacts with oxygen in a combustion process resulting to water carbon dioxide and heat. x4. Characteristics of Flames & Fire Plumes 4.1. Fire plumes Fire plumes are usually motion generated due to buoyancy existence when combustion occurs and is coupled with external momentum forces.

Due to various combinations of liquids and gaseous substances during combustion, buoyancy is sourced from glowing or flaming even when no external momentum is introduced (Heskestad, 1998). The characteristics of a fire plume and generalized smoke production rate and temperature calculations for axisymmetric plume model of a liquid fuel pool have been in the centre of investigations over time because of the dynamism involved. Figure 4.1.1: Development of an axisymmetric fire plume (Gant, 2010).

Plumes developing in cases where liquids are involved can be attributed to density gradients that are brought about by temperature changes and variations in density as fire develops. Whether the source is at the middle of the compartment or at the sides, effects of source conditions are lost as flow exhibits plume behaviour causing a transition region. This may however be affected by the nature of flow of the liquid that is in the centre of the compartment thus a buoyant fluid with momentum at its initial stage may lead to forced plumes also known as buoyant jets.

Since the transition exhibits jet-like behaviour in the far field densimentric Froude number Fr is deployed to distinguish the two flow regions (Gant, 2010). Computational fluid dynamics further divides axisymmetric plumes into intermittent, continuous and convectional regions. It is understood from experiments set to study these behaviours that air entrainment rates vary from one region to the other. Where flame regions are considered as complex, buoyancy induced turbulence and chemical reactions are considered to be at the centre of behavioural study of flume behaviours.

The correlations that have been studied over time do not necessarily agree to the effect of heat release rate, height and fuel source diameter on mass flow rates of the flame (Li & Chow, 2007).

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