Mathematical Models, Turbulence, Combustion, and Numerical Techniques - Assignment Example

1 Mathematical models a Please list the governing equations of fire simulation and explain the physical meaning of each equation. To solve the equations, do we need to consider boundary conditions? Give three typical boundary condition examples. (12 marks). The governing equations of fire simulation consist of the Navier-Stroke equations, the transfer equation for the radiative intensity and the energy equation[Rex04]. The CFD model is based on the hydrodynamic model, which comprises different conservation laws. The hydrodynamics model is based on a system of five equations referred to as the governing equations[Rex04]. The five equations are as shown below: a) Conservation of mass +∇· ρu = ………………………………………..Equation 1 Using the fixed volume technique, in which the fluid flows through the volume Ω, the conservation of mass states that the time rate change of mass inside the volume is equivalent to the flow of mass over the surface of the volume. This can be illustrated by the following equation: …………………………………….Equation 2 The time rate change of mass inside the volume Ω is described by the term on the LHS. The mass flow through the surface S with the fluid velocity u is described by the term on the RHS. ρ is the density of the fluid. Using the Gaussian theorem, the above equation becomes[Rex04]: …………………Equation 3 For an infinitesimal volume Ω0, it gives ρ + ……………………………….Equation 4 This is the conservation of mass equation b) Conservation of momentum …………………….. Equation 5 The conservation of momentum is the Newton’s 2nd law, which states that the force on the fluid element is equal to its mass multiplied by the acceleration of the element[Rex04]. There are body forces such as magnetic and electric forces and gravitational force, acting directly on the mass of the volume element and forces acting on the surfaces of the volume caused by stress and shear distributions and the pressure distribution. The input of the body forces may be written as follows: ………………………………………Equation 6 Where f represents additional external forces and ρg is the gravitational force. The input of the surface forces may be written as follows: = - With the viscous stress tensor and pressure p. The conservation of momentum may now be written as follows: Ρ() = …………………….Equation 7 The spatial change of the velocity field and time rate are described by the term in brackets on the LHS. Forces acting on the volume are described by the term of RHS. c) Conservation of energy (ρh) +∇· ρhu = + − −∇· +ε ………………..........…Equation 8 The conservation of energy is simply the first law of thermodynamics which states that the rate of change of heat inside a fluid element is equal to the flux of heat into the element and the rate of working done by forces acting on the mass of the element or on the surface of the element[Rex04]. d) Equation of state p = ρR T W ………………………………………………………….Equation 9 e) Divergence constrain …………..Equation 10 The physical meaning of the divergence constrain may be illustrated with the moving volume element approach. The mass of the fluid volume is fixed but the size and the surface of the volume element changes as a result of the different densities at different locations. The divergence constrain then becomes the time rate of change of the size of volume element per unit volume[Rex04]. Boundary Conditions Every solid surface is assigned thermal boundary conditions, together with the information concerning the burning behaviour of the material. The boundary condition on the front surface is given by: −ks (0,t) = + …………………………..Equation 11 If the internal radiation is calculated for a solid, the radiation boundary condition cannot be used. The boundary condition on a solid wall is given by: = ε + || ……………………….Equation 12 Where, . The constraint indicates that only the incoming directions are considered when calculating the reflection. The boundary condition for a radiation intensity leaving a gray diffuse wall can be written as follows: = ε + …………………Equation 13 Where, is the intensity at the wall, is the wall surface temperature, and ε is the emissivity[Rex04]. b Explain the reasons why CFD codes are written in low speed solver and high speed solvers. A student is simulating an object flying at a velocity of 290m/s in the air using FDS6. Can this student obtain acceptable results and why? (7 marks) CFD codes are written in low speed solver and high speed solvers because some solvers are better at low speed flows while other are better at high speed flows such as supersonic/transonic[McG131]. In other words, some solvers perform better at low Mach, while others perform better at high Mach. Stimulating an object flying at a velocity of 290m/s in the air using FDS6 is not likely to yield acceptable results because the air velocity is too high. Large air velocity does not allow objects or flame to spread in FDS6. This can only happen maybe with some modifications. c What is background pressure used in FDS? Can different rooms have different background pressures? Please explain using a formula and define two pressure zones using FDS input instructions. (6 marks) The background pressure is can be described as simply a hydrostatic pressure in a region[Flo11]. Different rooms can have different background pressures because FDS decouples the pressure into a series of zone background pressures. The background pressure may change as a function of the energy and mass flows into or out of a pressure zone and is formulated by determining the sum of the divergence inside of a pressure zone with the volume flows in and out of the pressure zone[Flo11]. This can be achieved using the following formula: ………………………….Equation 14 Pressure zones definition using FDS input instructions: &ZONE XB= 0.0, 5.0, 0.0, 5.0, -1.0, 3.0 / Read More