Classical Mechanics of Fluids - Assignment Example

University of Central Lancashire School of Forensic and Investigative Sciences 1. Classical Mechanics of Fluids (25 marks) 1.1. The Navier-Stokes equations govern fluid flow in fires and fire protection systems. It is the foundation for water flows, gas flows, and fire simulations and modelling. Please list the Navier-Stokes equations together with the equations of energy conservation.Explain the physical meanings of each term in the momentum conservation equations. Indicate what terms in the equation need turbulence modelling and give the reasons why turbulence models are necessary? Give an example of the source term in the equation of energy conservation. In Navier-Stokes equations there is description of motion of fluid through utilization of a set of partial differential equations. The Navier are used in the description of conservation of mass, momentum and energy flow in a fluid. Conservation of mass equation is Conservation of momentum Conservation of energy In the equation  represents pressure change in the fluid,  represents a small change in distance, represents a small volume change and u is velocity of fluid. In conservation of momentum equation t represents time is a constant. Also in conservation of energy  is a constant, while  is emissivity. It is important for there to be turbulence flow because in the equation there is assumption that particles are moving from outer layers to inner layers which is not a phenomenon experienced in laminar flow. [15 marks] 1.2. A pressure meter is calibrated using a Venturi meter attached to a small orifice at the bottom of a water tank. The cross-section areas of the wide and narrow parts of the Venturi meter are of 6 cm2 and 2 cm2 correspondingly. What should be the pressure drop between the wide and narrow parts of the Venturi meter, when the water level in the tank is 8.2 m? Please assume the Venturi meter is installed with pipes of cross-sectional area of 6 cm2 and the pipe finally discharges to open air at the outlet. [10 marks] In the wider section = pressure due to height of water = ρgh Where ρ (rho) represents the density of water g is gravity and h representing the height of water column By making the appropriate substitutions ρgh = 1000 x 9.81x 8.2 = 80442N/m2 Applying the principle of Continuity results to Q = A1V1 = A2V2 Velocity at wider section is given by V1 =  =  = 12.68m/s V2 = A1V1/A2 = 5.7x12.68/0.42 = 172.1m/s Using Bernoulli equation P1 – P2 =  =  = 500x29457.6 = 14728.8kN/m2 2. Dimensional analysis (25 marks) 2.1. Find the dimensions of the following terms. What term(s) are/is non-dimensional? [12 marks] 1.2 Demonstrate which of the following is the correct dimensionless Archiedes number a)  , b) , c)  , d), e)  Units for respective quantities v =kinematic viscosity m2/s, = density of body kg/m3 L= characteristic length of body m µ=dynamic viscosity kg/ms g = gravitational acceleration (9.81 m/s²),  = density of the fluid, ρ = density of the body, = dynamic viscosity, L = characteristic length of body, m The basic dimensions these elements are v = L2T-1 g=LT-2  = ML-3 ρ= ML-3 µ=ML-1T-1 L=L Substituting dimensions in place of variables in equation a) = LT-2 ML-3 ML-3L3/ ML-1T-1 ML-1T-1 = L-2T-2M2/M2L-2T-2 = L-2T-2M2M-2L2T2 = 1 this means the quantity is dimensionless Substituting dimensions in place of variables in equation c) = LT-2L/ ML-3 ML-1T-1 ML-1T-1 = LT-2L/ M3L-5T-2 = LT-2LM-3L5T2 = L-7M-3 This is not dimensionless Subs. for dimensions in place of variables in equation d) In order for the quantity to be dimensionless  need to be dimensionless Substituting dimensions in place of variables in equation in = ML-3 L/ LT-2ML-1T-1 ML-1T-1 = ML-3 L/ T-4M2 L-1 = ML-3 T4M-2 L1 = L-2 T4M-1 Which is not dimensionless Substituting dimensions in place of variables in equation e) So as for the quantity to be dimensionless  need to be dimensionless Substituting dimensions in place of variables in equation in v = L2T-1 g=LT-2  = ML-3 ρ= ML-3 µ=ML-1T-1 L=L =ML-3 LT-2 ML-1T-1 ML-1T-1 = ML-3 LT-2 M2L-2T-2 = M3L-2T-4 This is not dimensionless 2.2. Kolmogorov scale of velocity in homogeneous turbulence depends on the kinematic viscosity coefficient v [m2/s], specific dissipation rate [J/(kg s)] and, maybe, of fluid density [kg/m3]. Obtain the formula for this dependence using the dimensional analysis. [13 marks] indicial form of equation is  =  (1) The basic dimensions of the elements are  = LT-1 = MLT-2M-1T-1 = LT-3 v = L2T-1 Substituting dimensions in place of variables LT-1 = (L2T-1)a (LT-3) b By simplifying we have L1 T-1 = L2a Lb T-1a T-3b And through equating of powers T-1= T-1a T-3b -1= -1a-3b Or a = 1-3b L1 = L2aLb 1 = 2a +b 1 = 2(1-3b)+b 2b = 1 b= ½ 1 = 2a +1/2 a= 1/4 Through substitution of a=1/2 b = 1/4 substituting for a and b in equation  =  3. Heat Transfer, Thermochemistry and Fluid Dynamics of Combustion (25 marks) 3.1. Burning of polymethylmethacrylate (PMMA) can be described by the following chemical reaction formula. Explain the process of the burning of the PMMA (pyrolysis and reaction with oxygen) and calculate the stoichiometric fuel-air ratio. If 2.5kg of PMM is burnt, how much heat is produced? [15 marks] Assuming that there is complete burning of polymethylmethacrylate the stoichiometric fuel ratio-to-air ratio (with the composition of air being 21% O2 and 79 % N2.) C2H8O2 + 6O2 5CO2 + 4H2O Putting into consideration the composition of air the equation changes to C2H8O2 + 6O2 + N2 5CO2 + 4H2O +22.57N2 From the calculations it is clear that for complete burning of one volume of Dimethylnitrosamine 6 +22.57= 28.57 volumes of air is required. This puts the fuel-to-air ratio requirement at 1:28.57 to achieve complete combustion and for one volume of C2H8O2 there is a maximum of two volumes of 5CO2. Given that PMM generates 24.9Mj/kg Then for 2.5kg the level of heat generated would be 24.9x2.5=62.25Mj. 3.2. Define the Reaction Rate of a fire, then, discuss the factors that affect the reaction rate in a general secondary-order A + B → C + D chemical reaction. Give an example of such a chemical reaction of burning a gaseous fuel. [10 marks] Reaction rate is the change of concentration of a substance over the time interval during which this change takes place One of the factors that affect the rate of reaction is the temperature. If the temperature of reactacts is increased the reaction will take place faster than at lower temperature. Presence of a catalyst can also affect the rate of reaction where when a catalyst is introduced the rate of reaction would be faster. The concentration of the reactants is also important in the determination of the rate of reaction where by having low concentration of reactants would result to low reaction rates while increasing concentration would increase the rate of reaction. A typical case of a general secondary order reaction is that of combustion of (PMMA) in air. 4. Characteristics of Flames & Fire Plumes (25 marks) 4.1. Fire plumes are important in fire dynamics. Using a solid fuel fire at the centre of a compartment as an example, explain the characteristics of a fire plume and generalise the axisymmetric plume model for calculating the smoke production rate and temperature along the axis of the fire plume. [15 marks] In a combusting plume, athere is complete burning of oxygen but there is insufficient mixing. 10x stoichiometric oxygen is required. The fire plumes normally will exhibit a centreline temperature as shown in Figure 1. Figure 2 SOURCE: Quintiere. J.G (2006). Fire plumes and jets. Hydrogen safety. Belfast. Large fire flame temperatures will be described as in figure 1. The temperature will increase as the radiation fraction Xf decreases, will increase with D and is not significant function of fuel. Figure 3 SOURCE: Quintiere. J.G (2006). Fire plumes and jets. Hydrogen safety. Belfast. 4.2 Diffusion flames are common in compartment fires. Assuming a solid fuel is ignited, discuss and analyse factors that affect the spread of the flame on the solid fuel surface. If the fuel is a gas or a liquid, how the flame will spread? [10 marks] Studies of fire spread by Lefevrre et al (2004) has shown that with high density the spread will be lower with a longer melting step of foam. Experiment have shown that a decrease in ignition delay time with an increase in oxygen concentration pressure. Experiment on how temperature of opposed flow affect the behaviour of flames has shown that there is a decrease in delay time and the rate of flame spread increases as the temperature increases in the opposed flow. For gaseous and liquid fuel once the fire has been ignited the spread of fire will depend on whether there is sufficient air to support the gaseous fuel or not. References Bhattacharjee, S.et al (2003). “Predicting of critical fuel thickness of flame extinction in a quiescent microgravity environment,” Combustion and flame, Vol. 132, pp. 523-532, 2003. Lefebvre, J., et al (2004). “Flame spread of flexible polyurethane foam: comprehensive study,” Polymer Testing, Vol. 3, pp. 281-290. Fujita, O., Takahashi, J., Ito, K.(2000). Proceeding of the Combustion Institute, Vol. 28, Pittsburgh, p. 2761-2767. Niioka, T. et al (1981). “Gas –phase ignition of a splid fuel in a hot stagnation point flow,” 18thSymposium (International) on Combustion, pp. 741-747. Magee, R. S., Mcalevy, R. F., “The mechanism of flame spread,” J. of Fire and Flammability, Vol. 2, pp. 271-282, 1971 Quintiere. J.G (2006). Fire plumes and jets. Hydrogen safety. Belfast. Read More