Khalid Fluid Dynamics - Assignment Example

Khalid Fluid Dynamics Name: Course: Instructor: Institution: Date of Submission: 1. Classical Mechanics of Fluids 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 (not just the names but also the mathematical expressions) [4 marks]. Explain the physical meanings of each term in the momentum conservation equations [3 marks]. Indicate what terms in the equation need turbulence modelling [2 marks]. Discuss the reasons why turbulence models are necessary by analyzing the scale of vortices in fluid flow and the required capacity of computers? [5 marks] Give an example of the source term in the equation of energy conservation Navier-Stokes Equations govern the motion of fluids, which is perceived as the second law of motion of Newton for liquids. The compressible Newtonian fluid/ compressible Navier-Stokes Equation is expressed as: U = fluid velocity P = fluid pressure Ƿ = fluid density µ = dynamic viscosity of the fluid The terms developed as presented relate to the inertial forces such as the pressure forces, the viscous forces, and the eternal forces of the fluid. The Navier-Stokes Equations are attained from the Poisson, Navier, Stokes, and the Saint Venant that occurred between 1827 and 1845. The equation is solved for the continuity equation as presented below (Wikie & Morgan, 2009, 553). The equations display the momentum conservation representing the mass conservation (Richardson, 1989 75). The simplified Navier-Stokes Equations to the Linear Momentum Conservation is presented as (a) = Transient (b) = convection (c) = gravitational (d) = pressure (e) = viscous (a) and (b) = inertial terms (f) = Stress term If Re is high >>1 then the viscous term, which is negligible to the inertial terms. The inertial terms need the turbulence modelling in the equation below. With a high RE, the Reynolds Navier-Stokes equations should be as follows (Richardson, 1989, 75): Turbulence models are essential in arithmetical simulations due to the perceived inconvenience of calculating the turbulent motion of scales. The computer storage should be larger than the capacity of the most powerful computers today to resolve the scales. The computational speed of the current computers is too slow, unless it handles simple problems. Thus, the necessity of the turbulence models is to simplify the practical computations. The turbulence models depend on the turbulent scales included in the model (Marvin & Coakley, 1989, pg 1). The source term refers to an internal production or consumption of property such as energy among others. In an equation of energy conversation, a source equivalence is presented as below (Demirel, 2012, 147). Storage + Outflow - Inflow = Source (Demirel, 2012, 148) 1.2 a rise of 100mm in diameter is provided to a building to facilitate firefighting. The inlet of the riser is 1m above ground level. The outlet of the riser is 21m above the ground level. A hose of 45m in length and 80mm in diameter is connected to the outlet of the riser and horizontally laid on the floor. The hose has the same height as the riser outlet and its nozzle is open. Assuming the a fire engine is connected to the inlet of the riser and maintains a constant pressure of 12 bar at the inlet of the riser, please perform the following: (1) draw a diagram to show the system; (2) Assuming energy loss along the pipes can be omitted, calculate the pressure at the outlet of the riser where the diameter is 100mm using Bernoulli’s equation and the equation of continuity; Bernoulli’s Equation PA/eg + VA2/2g + ZA =PB/eg + VB2/2g + ZB Where PA and PB are pressure at section A & B respectively VA and VB are velocity at section A & B ZA and ZB are distances above the ground Continuity Equation e1A1V1 = e2A2V2 Where A1 and A2 are areas at section 1 and 2 respectively Since the density of water at point A and B are similar e1=e2 show that the riser has the same diameter at section A & B leading to equal area of A1 & A2 Thus V1 = V2 or VA = VB Applying continuity equation е1 A1 V1 = е2 A2 V2 Since е1 = е2 A1 V1 = A2 V2 A1 = ΠD2/4 = Π/4 (100)2 A2 = ΠD2/4 = Π/4 (80)2 Π/4 (100)2 V1 = Π/4 (80)2 V2 V1 = (80)2/ (100)2 V2 V1 = 0.64 V2 Applying Bernoullis equation to calculate flow velocity PA/eg + V2A/2g + ZA = PB/eg + V2B/2g + ZB The same height is attained = ZA =ZB PA/eg + V2A/2g = PB/eg + V2B/2g 12 * 105/9.81 * 103 + V2A/19.62 = 10.038 *10.5/ 9.81 * 103 + V2B/19.62 122.3241 + (0.64V2/19.62)2 = 102.3241 + V22/19.62 V22 =33.8753 V2 = 5.8203 m/s Therefore, flow velocity 5.8203m/s (3) assuming the pipe surface roughness is ε=0.1mm, density of water is 1000kg/m3, and viscosity μ=0.9×10-3 kg/(m∙s), calculate the flow velocity and volume flow rate at the outlet of the hose use the Swamee-Jain equation for the friction factor; Pipe surface roughness Ɛ =0.1mm Density e = 1000 kg/m3 Viscosity = μ=0.9×10-3 kg/(m∙s) Question : calculate the flow of velocity and volume flow rate Swamee Jain Equation F = 0.25 {10g10 (Ɛ/3.70 + 5.74/Re 0.9)}2 Reynold’s number Re = eVD/µ = 1000 * 5.8203 * (80 *10-3) / 0.9 * 10 -3 Re = 517360 f= 0.25 [(10g10 (0.1 * 10 -3 / 3.7 * 10 * 10-3 + 5.74/517360 0.9)] 2 f=0.25 {10g10 (3.3784 * 10 -4 + 4.13521 *10 -5)} 2 f= 0.25 {11.704} f = 2.92605 Applying Bernoulli’s equation PA/eg + V2A/2g + ZA = PB/eg + V2B/2g + ZB Since VA = VB PA/eg + ZA = PB/eg + ZB Pressure at section A is 12 bar = 12 * 105 NM3 12*105 /1000 * 9.81 + 1 = PB/ 9.81 * 103 + 21 122.3241 + 1 = PB/9.81 * 103 +21 PB/9.81 * 103 = 123.3241 -21 PB/9.81 * 103 = 102.3241 PB = 10.038 *105 NM3 Thus, the pressure at the outlet of the riser is 10.038 bar ΔP = 8efLQ2 / Π2 Δ5 Q2 = Π2Δ5ΔP / 8 e f L Q2 = Π2 * (80 * 10 -3)5 (12- 10.038) / 8 * 1000 * 2.92605 * 45 = 0.34525 / 1053.3783 Q2 = 6.02371 * 10-3 Q = 0.07761 m3l5 2. Dimensional Analysis 2.1 Find the dimensions of the following term. 𝜌𝑣𝑎𝑏 𝜇2 Where ρ is the density, v is the velocity, a and b are lengths, and µ is dynamic viscosity of the fluid. P = density V = velocity a & b = length µ = dynamic viscosity Let f = pvab/µ2 Substituting dimensions for each term MaLbTc = [ML-3] [LT-1][L][ML-1T-1]2 Ma= M1+2 a= 3 Lb= L-3+1+1+1-2 b=-2 Tc = T-1 -2 C= -3 M3L-2T-3 Therefore = M3/L2T3 2.2 Kolmogorov scale of velocity T  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. Formula for this dependence using dimensional analysis Kolmogorov scale of velocity T T = f ( v k e) There are four variables and 3 basic dimensions M, L, and T. Therefore, the solution should yield one Π term Π = va kb ec T Substituting dimensions for the variables, we have Mo Lo To = [L2/T] a [L2/T3] b [M/L3] c [L/T] Equating the exponents for M, L, & T M: M0 = Mc > c = 0 L: L0 = L 2a + 2b + 3c + 1 2a = - 2b – 1 T: T0 = T – a – 3b – 1 2a = -6b – 2 Solving the two equations simultaneously a = -¼ b= - ¼ Π1 = V -¼ K - ¼ e0 T Therefore Π = T/ 4 VK 3. Heat Transfer, Thermochemistry and Fluid Dynamics of Combustion 3.1 Explain the process of the burning of wood (pyrolysis process, charring, flaming, and heat transfer process should be discussed). A timber beam is ignited in a compartment fire. After the beam burns for a period of time, a layer of char is formed on the surface of the beam, using the principle of heat transfer, explain how this layer of char affect the burning of the beam. The process of burning wood occurs through the combustion occurring, which is a reaction between oxygen and gases released from the material. However, in charred wood oxygen reacts directly with carbon. Wood produces substances that react with oxygen that causes high propensity of wood to ignite and burn. The burning and ignition of wood occur through the pyrolysis process also identified as the thermal decomposition process of cellulose. An increase in temperature steers to cellulose to pyrolyse. The decomposition materials remain inside, or the gases are released that react with oxygen. Consequently, ignition occurs and a large amount of heat induces pyrolysis and combustion. Flaming is perceived when the temperature of the fuel is reached, and combustion begins. The heat released during the combustion process is the force that drives to the occurrence of fire. If the heat is large, the fire spreads fast and with the hotter gases, the limited surfaces of fire are enclosed. The essential quantities of describing the burning of materials are the rate of heat release identified as , which is expressed as MW or kW. The specific kW or MW values for different materials cannot be provided. The important part is identifying the effect the external factors have on to determine the net heat flux. The internal properties of a material that affect the heat release derive from heat combustion ∆Hc, the heat gasification Lv, and heat capacity C. Rate of heat release of burning materials occurs as in the equation: Charring occurs when wood burns at a constant rate of heat release. The frontier between the pyrolysed and intact wood is that the pyrolysis anterior ensues to the wood in depth bearing. The pyrolysing wood can be reflected to as char where the charring rate β relates to the spread rate of the pyrolysis anterior (VTT, n.d.). The charring rate of wood has important factors such as density ρ, which is also considered as the external heat flux. The moisture content w denoted as , and the moisture content. The rate of charring is affected by an increase in density as it decreased to the power law presented as . The rate of charring upsurges linearly with the exterior heat flux, . An estimated association amid the charring rate and moisture content is presented in the equation. 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. The reaction rate of fire is the time that the fire takes combustion to occur. The factors that affect the reaction rate in a general secondary-order A + B → C + D chemical reaction include the fact that the sum of the exponents is equal to two according to the rate law. It is also affected by the nature of the reaction, the concentration where the reaction rate increases with the concentration due to an increase in the collision frequency increases. It is also affected by pressure; a gaseous reaction increases with pressure, which is equivalent to the gas concentration increase attained. The order also affects the second order reaction as it controls the effect of the concentration on the reaction rate. Temperature also affects the reaction rate of second order, including solvents. Catalyst and isotopes among others. A second order reaction rate relies on the concentrations of the order reactant or two first order reactants. An example of a second order reaction rate can be presented as: 4. Characteristics of Flames and Fire Plumes 4.1 Fire plumes are important in fire dynamics. Using a liquid fuel pool fire at the center of a compartment as an example, explain the characteristics of a fire plume and generalize the axisymmetric plume model for calculating the smoke production rate and temperature along the axis of the fire plume. The characteristics of a fire plume derive from the process of entrainment, which is also liable for flame height. One key characteristic of fire plume is that the flow is turbulent or laminar. The laminar and turbulent flow is perceived through projecting the flame on a screen with a collimated beam of light that shines through it. High turbulent frequency is illustrated through dark and light images. Thomas et al. regarding large and small fire equation present the axisymmetric plume equation for smoke production and temperature. The small fire equation for smoke production and temperature presented through (Quintiere, 1998, 131): A1=2 f zf (H – d)/2, zf > (H- d) and z ≤ H – d ρa = Density Cp = Specific Heat Ta = Temperature of the Ambient H = Ceiling Height P = Fire Perimeter d= Depth of Layer of Hot Gas Beneath Ceiling Af = Area of Fire Smoke production equation is derived from the total heat output equation, which is: Q = Btu/sec; that is Q*Btu/Sec 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? The factors that affect the spread of the flame on solid fuel surface include the nature of the substance. The substances must be vaporized or melted or pyrolysis. Heat is delivered to the fuel, which later produces vapors. The surface area of a given mass also affects the spread of the flame where thin substances allow flames to spread faster unlike solid blocks of the same materials. High surface areas have combustible materials that allow the quick spread of flames on the material. The density of the materials also affects the spread of the flame. High-density materials conduct energy away while low-density materials ignite fast and allow a quick spread of flames on the materials. On liquids, the flames movement depends on the flash point of the liquid where at a low flash point temperature, there is sufficient vapor to spread the flame quickly on the surface. On the other hand, gases require a low mass for ignition and quick spread of flames. However, the combustible gases must be in a gaseous state (NFPA, 1998). References Demirel, Y., 2012. Energy: Production, Conversion, Storage, Conservation, and Coupling. New York: Springer Science & Business Media. Marvin, G. J. & Coakley, J. T., 1989. Turbulence Modeling for Hypersonic Flows. NASA Technical Memorandum, pp. 1- 46. NFPA, 1998. Inter Fire Online: Chemistry of Combustion. [Online] Available at: http://www.interfire.org/res_file/9213-1.asp [Accessed 15 3 2016]. Quintiere, G. J., 1998. Principles of Fire Behavior. New York: cengage learning. Richardson, M. S., 1989. Fluid Mechanics. New York: CRC Press. VTT, n.d. Burning of Wood. [Online] Available at: http://virtual.vtt.fi/virtual/innofirewood/stateoftheart/database/burning/burning.html [Accessed 10 3 2016]. Wikie, A. C. & Morgan, B. A., 2009. Fire Retardancy of Polymeric Materials. Second Edition ed. New York: CRC Press. Read More