Fluid Dynamics of Fire - Case Study Example

1.1. a) A hydraulic press has the working cylinder radii ratio / 100 1 2 R R . What mass should be put on the smaller cylinder in order to balance the Challenger 2 tank of 62.5 tonnes? (3 marks) Solution: To balance the 62.5 tonnes in a hydraulic press with a working cylinder radii ratio of R1/R2 = 100 is b) Flow velocity V measured by the Pitot-static tube depends on the flow density ρ and pressure drop Δp in the tube relevant to the undisturbed flow. Use dimensional analysis to determine the Pitot-static tube law for velocity V . (10 marks) Solution: In order to derive the Pitot-Static tube law for velocity, it is necessary to determine the flow density and pressure drop in P using the schematic diagram found below. From Bernoulli’s equation, V2/2 + gh + p/ ρ = constant In dimensional equivalents are V[L/T], g[L/T2], h[L], p[LM/T2L2], ρ[M/L3] From this, we form the dimensionless variable H = pV2/ ρ to get the Pitot-Static tube Law which is , The equation we are looking for is 1.2. a) Define laminar and turbulent flow. What is the transitional regime and critical Reynolds number? Provide at least two examples. (3 marks) Answer: Laminar flow occurs when the flow of the fluid layers are in parallel to each other with no form of disturbance on its flow. An example of a laminar flow is the flow of oil through a thin tube (AHOLM, 2007). Turbulent flow on the other hand is a three dimensional flow of fluid characterized by vortices or irregular and short-lived rotation of the flow currents in space and time. An example of a turbulent flow is sea currents (Klabunde, 2007). Transitional regimes occur when the conditions for annular-mist, slug, and bubble regimes are not met. It is characterized by chaotic flow characteristic and is very difficult to describe. An example of a transitional regime is steam-water mixture flows from a geothermal well. The Reynolds number is a dimensionless constant that expresses the relationship between resistant forces and viscous forces. A high Reynolds number indicates less viscous fluid (TET, 2005). b) An aircraft of the cross-section area A 1.47 m2 is equipped with an engine developing steady thrust of 48.29 kN. Estimate the cruising speed of the aircraft in air at sea level assuming it is subject to the Newton's aerodynamic drag with the aerodynamic coefficient 0.73 d C . (9 marks) Solution: From the equation 2. Heat Transfer 2.1. a) Describe what a dimension of a physical parameter is and give at least two examples of physical parameters and of their dimensions (4 marks). Answer: The dimension of a physical parameter is the unit of measurement of any physical quantity which is a combination of mass [M], time [T], length [L], speed [L/T], temperature [Θ], charge [Q], and the likes. The concept of dimension of physical parameter is similar to but not necessarily the same as the ‘unit’ of measurement. b) Derive the formula for the time of temperature equalization in a solid body of length l and thermal diffusivity κ using dimensional analysis (10 marks). Solution: The dimensional equivalent for the physical properties of thermal conduction are t[T], κ[L2/T], l[L]. From this, we form the dimensionless variable and derive the temperature of equalization. or the equation we are looking for is t = L2/K 2.2. a) Explain the radiation heat flux formula. (4 marks) Answer: The formula for heat flux is which means that the heat flux is the net flow of energy hitting the surface and the energy radiated by the very same surface. b) A flat steel plate of total area 1.47 m2 at temperature 631 CO radiates 27 KJ of energy per second. Calculate its emissivity coefficient? (7 marks) Solution: From the Stefan-Boltzmann Law, one can derive 3. Fluid Dynamics of Combustion 3.1. a) Explain what is the normal laminar flame velocity. (4 marks) Answer: The normal laminar flame velocity is the velocity of the laminar flame in combustion process relative to the velocity of the flame during combustion. b) Estimate the normal flame velocity in a pure stoichiometric mixture of propane and oxygen at temperature 48oC and 1.87 of normal atmospheric pressure using formulas and data from the handouts. (7 marks) Solution: The normal flame velocity in a pure stoichiometric mixture of propane and oxygen can be estimated in two manners, one using temperature estimate and the other pressure estimate. The estimates for the normal flame velocity are as follows 3.2. a) Explain Damköhler's definition of the turbulent burning velocity. (5 marks) Answer: Damkohler denies the general assumptions for turbulent flows as having well defined properties that depends only on local mean properties but instead proposed that turbulent flows are limited by (a) the magnitude of scale of turbulence and (b) the thickness of laminar premixed flame, coming up with a more realistic mathematical interpretation of turbulence in flames. b) For the average amplitude of flow velocity fluctuations u’=33 cm/s, compare the turbulent burning velocities ST of wrinkled flames calculated using the original Damköhler formula and correlations by Schelkin, Klimov and Clavin & Williams. Use SL calculated in the problem 3.1.b. (9 marks) Solution (Using SL=33.54): For Damkohler: For Schelkin, For Klimov For Clavin and Williams 4. Diffusion flames 4.1. a) Explain the Richardson and Froude numbers. (4 marks) Answer: Richardson Number is a dimensionless number that expresses the ratio between potential and kinetic energies. It is primarily used to measure expected air turbulence (Weissten, 2007). Froude Number of the other hand is another dimensionless number that compares inertial force and gravitational force. Its main use is in quantifying resistance of objects moving in fluids (TET, 2005). b) Evaluate the Richardson number through the estimation of orders of magnitude of the forces involved. (8 marks) Answer: To evaluate the Richardson number, consider the two cases where u1= 20m/s and u2 =10m/s with h1 =15 and h2 = 8 respectively. From the Richardson Number, we get . We can clearly see that the closer the object is to the ground, regardless of its initial velocity, the greater the Richardson number. This is consistent with the concept of gravitational force where the body closest to the earth experiences a stronger pull compared to a body farther from the earth (TET, 2005). 4.2. a) The laminar diffusion flame height f L depends on the volumetric fuel flow rate Q & and diffusion coefficient D . Obtain a formula for f L using dimensional analysis. (9 marks) Solution: Q[L3/T], D[L2/T] Thus, the equation we are looking for is Lf=Q/D b) Using Roper & Roper model of the laminar diffusion flame at temperature 2186 f T K, calculate its height for the diffusivity coefficient 3 1.21 10−D m2 /s, volumetric flow rate 012 . 0 Q & m3 /s and stoichiometric molar oxidizer-fuel ratio S 2 . Fuel and oxidizer temperatures are equal to 44 CO . (4 marks) Solution: From the Roper & Roper equation, we get the height of the laminar flame as REFERENCES Aviation History On-Line Museum. Laminar Flow Airfoil. Aircraft Theory. March 18, 2007. http://www.aviation-history.com/theory/lam-flow.htm [Retrived February 26, 2009] Klabunde, Richard. Turbulent Flow. Cardiovascular Physiology Concepts. April 10, 2007. http://www.cvphysiology.com/Hemodynamics/H007.htm [Retrived February 26, 2009] The Engineering Toolbox. Reynolds Number. 2005. http://www.engineeringtoolbox.com/reynolds-number-d_237.html [Retrived February 26, 2009] The Engineering Toolbox. Froude Number. 2005. http://www.engineeringtoolbox.com/froude-number-d_578.html. [Retrived February 26, 2009] Weisstein, Eric. Richardson Number. Wolfram. 2007 http://scienceworld.wolfram.com/physics/RichardsonNumber.html [Retrived February 26, 2009] Read More