Classical Mechanics of Fluids, Heat Transfer, Thermochemistry and Fluid Dynamics of Combustion - Assignment Example

FLUID DYNAMICS OF FIRES АSSIGNMЕNT Name Institution Instructor Date 1. Classical Mechanics of Fluids 1.1 The list of Navier-Stokes equations is stated as follows: Equations of energy conservation are stated as follows (Bansal and Bansal, 2010): a) Gravitational potential energy, PE(grav) = mgh b) Elastic potential energy, PE(elastic) = (1/2)KX2 c) Kinetic Energy = (1/2) mv2 The equation of states: p V = n R T p = pressure (absolute) V = volume n = substance amount (in moles) R = the constant for an ideal gas T = Temperature (absolute) Explaining the meanings of terms in the energy conservation equation: Ki + Ui = Kf + Uf (Ki + Ui) Refers to the total energy spent (Kf + Uf) Refers to the total energy gained This conservation energy equation implies that energy can neither be destroyed nor created (Oteh, 2008). Conservation of Mass (dp/dt) + 𝛥pu = mbm (dp/dt) Refers to contraction or expansion 𝛥pu Refers to the flow of mass in and out mbm Refers to the amount of fluid that is generated through the system These equations can be solved through analytical methods and solutions. The methods that can be used in solving the equations for the equations that are involved in a typical compartment fire include fire modelling, thermal analysis and structural analysis (Shames, 2003). 1.2 Given: Cross-section area of the wide parts for the Venturi meter = 6 cm2 Cross-section area of the narrow part for the Venturi meter = 3 cm2 Water level in the tank is = 8.1 m Assumption: Installation of the Venturi meter is done using pipes having a cross-area of 6 cm2 Task: Determining the pressure drop between the wide and narrow parts of the Venturi meter 2. Dimensional analysis 2.1 The formula that is used in calculating the Reynolds number of a pipe flow is given as: Re = ρ u dh / μ = u dh / ν The parameters used in the formula that is used in calculating the Reynolds number as described as: Re= R Reynolds number ρ= Fluid density (kg/m3) dh = hydraulic diameter (m) ν = kinematic viscosity (10-6 m2/s ) u = velocity (ft/s) In demonstrating that the Reynolds number is non-dimensional using the dimensional analysis method, the primary dimensions used are: Mass M - (kilogram, kg) Length L - (metre, m) Time T - (second, s) The dimension of the parameters that are involved takes place as follows: Length [l] = L Velocity [v] = Gravitational Acceleration [g] = Density [ρ] = Pressure [p] = Coefficient of viscosity [μ] = Thus, dimensional analysis gives: 2.2 Obtaining the formula for Kolmogorov scale of velocity in homogeneous turbulence using the dimensional analysis: Assumptions: Where ki < k < kd And Where k > kd Substituting The Kolmogorov scale which is the inverse of (kd) is denoted as (ld) and is given as: 3. Heat Transfer, Thermochemistry and Fluid Dynamics of Combustion 3.1 Process of the burning of the PMMA (poly methyl meth acrylate ) The process of the burning of PMMA (poly methyl meth acrylate ) Involves burning with a bright flame that produces little or no smoke. The combustion mechanism and burning behaviour of poly methyl meth acrylate (PMMA) can be studied using cone calorimeter, gas chromatograph and a mass spectrograph (Sirignano, 2010). Under circumstances that are considered normal, pyrolysis and reaction with oxygen only produces water and carbon dioxide. Due to the chemical composition of its substance, which includes carbon, oxygen and hydrogen, there are no formation acutely toxic substances like acid vapours, sulphur dioxide and phosgene when PMMA burns. The combustion process of PMMA is decribed using the chemical reaction formula described as (Echekki and Mastorakos, 2011): Calculating the stoichiometric fuel-air ratio of poly methyl meth acrylate (PMMA) The chemical reaction formula for poly methyl meth acrylate (PMMA) is given by: Stoichiometric Air Requirement Mass of fuel (m f) = (12 x 5) + (1 x 8) + (16 x 2) = 100 g/mole Mass of air (m air) = (6 x 2 x 16) = 192 g/mole Therefore, the stoichiometric air-to-fuel ratio is: The amount of heat produced: Assuming that 1 volume of PMMA content is equivalent to 1 Kg Thus, 1 Kg of PMMA content produces 25 MJ when burnt Therefore, the amount of heat energy produced when 2 Kg of PMMA is burnt is given by: 3.2 Definition of the reaction rate of fire The reaction rate of a fire refers to the amount (in mass units or moles) per unit time in unit volume that is removed or formed through fire (Yeoh and Yuen, 2009). Factors that affect the reaction rate: Concentration There is usually an increase in the frequency of collision with an increase in the concentration of reactants. This implies the existence of sufficient energy by the reactants in causing a reaction. Temperature The presence of high temperatures in a reaction leads to the existence of more energy in the system, which causes an increase in the rate of reaction. The reaction rate, then, leads to the doubling of the speed of the system every time the temperature increases by 10 degrees Celsius. Pressure An increase in pressure results in a corresponding increase in the rate at which gaseous reaction occurs. The increase in pressure also causes an increase in the gas concentration (Oppenheim, 2008). Light The rate of reaction is affected by light in the sense that it influences the course of the reaction. Reactions that take place in the absence of light occur at a low rate whereas those that take place in the presence of light occur at a high rate. Order The rate of reaction is majorly affected by its order. In most basic reactions, the order of reaction brings about an integer value. Catalyst The rate of reaction increases in the presence of a catalyst for forwards and reverse reactions through the provision of an option that requires lower energy of activation. Nature of the reactants Reactions that are involved in breaking and reforming of complex bonds take longer than those involved in the formation of simple bonds do. The rate of reaction is also affected by the positioning of the reactants in the series of reactivity. The parameter that is usually in control of the rate of combustion in a typical diffusion flame is the rate of diffusion of reactants. 4. Characteristics of Flames & Fire Plumes 4.1 Characteristics of a fire plume The characteristics of a fire plume include the following: Flame height The flame height refers to the height that exists when at least a portion of the flame has half of the probability in lying above it (Grimwood, 2008). The velocity distribution and the plume temperature The rate of entertainment for the air in the surrounding The nature of interactions with the compartments of the boundaries Axisymmetric plume model The generalisation of the plume model used in the calculation of the rate of smoke production and temperature along the fire plume axis takes the following form: The following assumptions are made in generalising the axisymmetric plume model: For a liquid fuel fire, the point source is equivalent to the virtual origin Small variations of density exist within the plume The velocity of air entrainment has a direct proportionality to the axial velocity; There are similarities between the profiles of the temperature and the vertical velocity at all heights. The radiated heat exists at a constant fraction. Thus, for the axisymmetric plume model the outcome for the excess temperature is given by (Drysdale, 2011): 4.2 Factors affecting the spread of the flame on the solid fuel surface These factors include the following: The fuel surface temperature The mass fraction of fuel vapour at fuel surface How heat is transferred from the flame to the solid surface The transfer of heat from flame to the solid surface involves the phenomenon of the profiles for the component of velocity that is normal to the edge of the flame. The process of heat transfer takes place within the range of the spread of stable flame. IN this case, the spread is independent of the shape associated with the leading edge of the flame (Sirignano et al, 1997). References Bansal, R. K., & Bansal, R. K. (2010). Text book of fluid mechanics: (in S.I units). Bangalore, Laxmi Publication Pvt Ltd. Drysdale, D. (2011). An introduction to fire dynamics. Chichester, West Sussex, Wiley. Echekki, T., & Mastorakos, E. (2011). Turbulent combustion modeling: advances, new trends and perspectives. Dordrecht [etc.], Springer. Grimwood, P. (2008). Euro firefighter. Lindley, Huddersfield, West Yorkshire, Jeremy Mills. Oppenheim, A. K. (2008). Dynamics of Combustion Systems. Berlin, Heidelberg, Springer-Verlag. Oteh, U. (2008). Mechanics of fluids. Bloomington, IN, AuthorHouse. Shames, I. H. (2003). Mechanics of fluids. Boston, Mass. [u.a.], McGraw-Hill. Sirignano, W. A. (2010). Fluid dynamics and transport of droplets and sprays. New York, Cambridge University Press. Sirignano, W. A., Merzhanov, A. G., De luca, L., & Zelʹdovich, I. B. (1997). Advances in combustion science in honor of Ya. B. Zelʹdovich. Reston, Va, American Institute of Aeronautics and Astronautics. Yeoh, G. H., & Yuen, K. K. (2009). Computational fluid dynamics in fire engineering theory, modelling and practice. Burlington, MA, Butterworth-Heinemann. Read More