Equations Governing Fire Simulation - Assignment Example

FDS assignment 1 Name Date Course Question 1: Mathematical model Part a: Equations governing fire simulation LES Formalism Large-eddy simulation (LES) is an equation that utilizes the concepts of low pass filter which is parametized by a width ∆ in the transport equation (McGrattan, et al, 2013). The transport equation used in the simulation includes that of mass, energy and momentum. This equation also considers the kinetic energy conservation central difference for momentum. The equation is represented as follows: Mass and species transport This equation mainly deals with the chemistry of fire and the transportation of the species. The products of hydrocarbon fuel during the reaction with oxygen to produce carbon dioxide and water is also considered. The equation can also be used for solving the transport equation for each of the species mass and density (McGrattan, et al, 2013). This is considering that the different species are involved in the burning process and they have different masses and density. The equation can be represented as follows: The equation can be further simplified as follows: Low Mach number approximation This equation is useful in terms of describing the airflow between compartments due to the pressure differences. An assumption that the spatially and temporally resolved air can be decomposed into a background pressure is usually made (McGrattan, et al, 2013). It therefore reduces the need for solving the detailed flow equation in the ventilation ducts. The concept of airflow is important in determining the spread of fire within a compartment. The equation can be represented as follows: Momentum transport Momentum transport is concerned with stagnation energy per unit mass (McGrattan, et al, 2013). It provides one of the most important equations that are used for the fire simulation. This is useful in terms of determining the rate at which the fire spreads within a compartment. Momentum transport can be represented with the following equation: Combustion and radiation The combustion of the materials usually takes place during the fire while the radiation is associated with spread of the fire through the hot air or surfaces (McGrattan, et al, 2013). The combustion process is useful as it leads to the analysis of the fire during the simulation. Combustion and radiation can be represented using the following equations: and Solution procedure The solution procedure is involved with the analysis of the background pressure, temperature, velocity, mass and density (McGrattan, et al, 2013). This procedure has two main aspects which includes the predictor and corrector. The solution procedure utilizes equations that are used during the computational process. Both the predictors and corrector have different equations which can be represented as follows: Predictor Corrector Boundary Conditions In order to solve the equations, the boundary conditions must be considered. The boundary conditions influence the temperature, species mass fraction and temperature. The typical boundary conditions may involve the solid boundaries, open boundaries and mesh interface boundaries. In the solid boundary, the convective heat flux at the surface is determined through the use of an empirical heat transfer coefficient (McGrattan, et al, 2013). In the open boundaries, the gases can flow freely in or out and has to be considered in solving the equations. Information has to be passed between meshes in mesh interface boundaries. During the calculations, the surface temperature may either be calculated or it may be provided. The heat transfer coefficients are also required during the calculations. Part b The regime of flow speed is one of the important aspects that distinguished the CFD model. The flow speed for which it is designed has to be relative to the speed of sound. As a result of this the codes are written in high speed solvers and low speed solvers are used. The high speed solvers mainly involves the compressibility effects and shock effects. The low speed solvers are mainly involved in the elimination of compressibility effects which in most cases gives rise to acoustic sound waves. According to the Navier-Stokes equation, the propagation of speed can be compared to the fluid flow and sound waves (McGrattan, et al, 2013). Small time steps are therefore required in order to ensure that the equations are solved. The practical simulations are however difficult due to the speed of sound waves and fluid flow. A student simulating a flying object at a velocity of 400m/s in air using FDS6 cannot obtain acceptable results. The propagation of speed in air is 300m/s. Even in such a condition, the simulation is still difficult since extremely small type steps are required. A speed of 400m/s is much higher and hence making it difficult to carry out the simulation and obtain acceptable results. The simulations may only be effective when the speed is less than that of air which is 300m/s. Part c The background pressure is a product of decomposition of the spatially and temporally resolved pressure. Different rooms can have different background pressures due to the influence of various factors. The differences between the airflows in the different rooms can result to different background pressures (McGrattan, et al, 2013). The airflow in the rooms can be influenced by various factors including the amount of open spaces and hence causing the differences. The formula below indicates how two different rooms can have different background pressures: The difference in background pressure can also be attributed to an isolation of volume within a computational domain. However, in such incidences, the leak paths as well as the ventilation ducts have to be considered. The background pressure ( ) is considered a function of z which is the vertical spatial coordinate and t which is the time (McGrattan, et al, 2013). The changes may occur with height or time. Some effects that may also affect the volume include increase in pressure by fire, significance of the height of the domain or significance of pressure increase by HVAC. The atmospheric pressure stratification in such incidences serves as the initial boundary condition that governs the equation. Question 2: Turbulence Part a Turbulent model is the gradient diffusion required for closing the SGS momentum and scalar flux terms. The model is important as it is used for obtaining the turbulent transport coefficient. The coefficient model has two other aspects which involve the turbulent viscosity and turbulent diffusivity. Turbulence is thus important in CFD as it ensures the transport coefficient is determined. Turbulence model is also important in terms of ensuring that the concepts of burning are explained. The turbulence model is also important in terms of ensuring that the modeling of the chemical reactions is carried out and calculations made (McGrattan, et al, 2013). During the calculations, a number of values are required and it can be obtained through the use of the turbulent model. The rate of fuel consumption can also influence the movements. Large Eddy Simulation (LES) is one of the aspects that are involved in solving the momentum equation. LES is a description of turbulent mixing of the gaseous fuel and the products with the local atmosphere that surrounds the fire. The LES equation is usually derived by applying a low pass filter of width ∆ to the DNS equation (McGrattan, et al, 2013). The filtered field in the LES equation is considered as a cell means. LES is important in FDS as the kinetic energy conservation is applied together with the aspects of the closure turbulence. LES technique is used for extracting greater temporal and spatial fidelity from the fire simulations. Reynolds-Averaged Navier-Stokes (RANS) allows for the description of the fires in complex geometries through the application of a wide range of physical phenomena. RANS is also used as a time-averaged approximation for fluid dynamics in conservation equations. This however requires the introduction of large Eddy transportation coefficient in order to describe any unresolved fluxes of energy, mass and momentum. RANS model in most cases is applied in numerical computation through the use of implicit numerical techniques (McGrattan, et al, 2013). However, the use of the implicit numerical techniques is mainly used to take large time steps. Direct Numerical Simulation (DNS) involves the simulation where the numerical grid cells are on the order of 1 mm or less. DNS requires the approximation of thermal conductivity, viscosity and material diffusability from the kinetic theory. This is due to the temperature dependence of each combustion scenario. The LES parameters that can be used in the FDS6 software include the filter width and the cell volume. The finely meshed grids are also useful during the computational process and the process can be made faster through the use of the software. Part b Based on Kolmogorov’s turbulence theory, the mesh resolution required for LES is much lower than the DNS solving. LES has a multiple of Kolmogorov scale and the coarse reduction of meshing may be required in order to cut off the cascade spectrum of the turbulent kinetic energy and wave numbers. In the process of determining the mesh resolution both for LES and DNS the point where the Kolmogorov scale is lower and higher has to be determined. The behavior of SGS model has to be considered as the cell size for LES usually varies in most cases (McGrattan, et al, 2013). The other factor that needs to be considered includes the mixing time for diffusion, SGS advection as well as the buoyant acceleration. The differences between the SGS models have to be understood as they have a direct influence on the mesh size that has been chosen. When using the LES method, the mesh size must be in the inertia zone. This is for ensuring that it is smaller than the large scale phenomena and bigger than the smallest scale that can be filtered by the model. In most cases, a good LES tends to the DNS as the grid resolution tends to the smallest scale. An appropriate mesh resolution for the model is the one that matches the simulation results. There are also pros and cons in relation to the resolution of LES and DNS. One of the main advantages is that it is more economical with a minimal requirement in terms of the most energetic Eddies (McGrattan, et al, 2013). The transfer of energy from the larger to the smaller scale is also achieved and hence leading to efficiency. Obtaining the same order of Kolmogorov scale inside each of the unit mesh is also an important aspect that can be attained. This is useful in ensuring that configuration of the adequate refinement is required. However, it is also important to note that there are cons in relation to the process. The length of time scales associated with the reaction in some instances may be of orders of magnitude that are below the spatially and temporality resolved simulation. As a result of this, overlapping may occur and hence the needs for generalized approach. The process can also be affected by the physical processes such as high turbulence (McGrattan, et al, 2013). In the low strain fires, the inertia sub-range does not exists which affects the process. The height of the flame also limits the reaction timescale. Question 3: Combustion Part a The combustion model is required in fire simulations as it determines the mean chemical mass production rates of species per unit volume in the transport equation (McGrattan, et al, 2013). The transport equation is useful in the fire simulation model and it forms the basis for most of the mathematical calculations. The flame thickness is an important aspect in the fire simulations that can be explained through the combustion model. The heat release per unit volume has to be determined during the fire simulation. The combustion model provides detailed information about the heat release rate per unit volume. The heat release rate is considered an important factor in fire physics and it is also the largest contributor to the velocity divergence which is required in the fire simulation model (McGrattan, et al, 2013). The number of transport equations can also be reduced through the use of the combustion model which is required in the fire simulations. The model therefore makes it easy for the calculations to be performed. The fast as well as the slow chemistry mixing conditions can also be addressed through the combustion model and hence making it relevant in the fire simulation. The combustion model is also important in addressing the aspects of chemistry which is involved in the burning process. A chemical reaction usually takes place and it can be used for the purposes of obtaining values that can be used to calculate mass and volume of different components. The mixed is burnt assumption is usually made in a majority of FDS application. The mean chemical source term for the fuel is usually modeled through the use of Eddy Dissipation Concept (EDC) of Magnusson Hjertager (McGrattan, et al, 2013). The lumped mass fractions of fuel and air are also used in the simulation. A mass stoichiometric coefficient of air is also used during the process. It is assumed that the fuel consumption is proportional to the local limiting reactant concentration and the local rate of mixing. An assumption is also made that all the reactants are initially unmixed and the rate of chemical kinetics is infinite. The assumptions are useful during the simulation in order to ensure that some of the real situations are represented. Simple empirical rules are also used and it ignores the strain in order to predict the local extinction within a certain grid cell (McGrattan, et al, 2013). In the FDS application, it is assumed that the fuel and oxygen always react regardless of the local conditions for temperature. The local conditions involve strain, temperature and dilution. This assumption may however be effective in the event of a well ventilated fire. FDS6 cannot be used to simulate pre-mixed combustion and explosion. The pre-mixed combustion may contain suppressing agent such as carbon dioxide and water which may make it impossible for burning to occur (McGrattan, et al, 2013). The cell temperature may also be below the auto-ignition temperature and hence making it impossible for the combustion to take place. In most cases, the excess fuel acts as diluents although the excess air does not. This therefore means that the presence o excess fuel in the pre-mixture cannot burn. The reaction that usually takes place is complex and there are a lot of difficulties in modeling the chemical reactions and this therefore makes it difficult for the simulation of the pre-mixed combustion and explosion to take place. Part b YF - mass fraction of fuel = 0.59 YO - mass fraction of oxygen = 0.21 I - fuel mass fraction in the fuel stream = 0.93  - mass fraction of oxygen at infinity = 0.23 s - ratio of oxygen and fuel molecular weight in stoichiometric mixture = 0.63 The mass fraction for oxygen (YO), can be considered as 0.21. This is because air is considered as a lumped species and the group of primitive species only occurs in specific proportions which are known. The percentage of oxygen in air is 21% which therefore indicates that the mass fraction of oxygen is 0.21. It is assumed that the fuel fraction of air is 1.0. In order to obtain the mass fuel reaction, the following calculation can be used: YF = (Fuel mass fraction in fuel stream x 1 x s) Therefore, Mass of fraction fuel =0.93 x 0.63 =0.59 Z = =0.48 If mass fraction of fuel is 0.68, Z = =0.55 Question 4: Numerical techniques Part a Explicit scheme The use of the explicit scheme has its advantages as well as the disadvantages. One of the main advantages regarding the use of the explicit scheme in solving 1 D unsteady heat conduction equation is the simplicity of the algorithm (Hirsch, 2007). However, the main disadvantage involves the many steps that are required in carrying out the calculation. The stability constrains may also end up being imposed by the restrictions on ∆t. The following steps can be used during the process of solving the 1D unsteady heat condition equations. The time dependent heat equation in 1 D can be represented using the following equation  =k  ………………………………Equation 1 In explicit scheme, the temperature at a particular time n+1 is explicitly dependant on the temperature at time n. Therefore the explicit finite difference discretization of equation 1 above can be represented as follows. ………………………….Equation 2 In order to make the solution much simpler, the whole equation can be rearranged again. The rearrangement is also for the purposes of ensuring that all the quantities with n+1 is put on one side and the quantities at time n is put on the right hand side (Hirsch, 2007). After the rearrangement, it can be presented using the following equation. ………………………….Equation 3 As a result of the rearrangement, it is possible to carry out the calculations. This is considering that some of the values are already known. The values that are already known include,  and  This therefore makes it possible to calculate. The procedures therefore indicate that the use of this method can be relatively simple. However, one of the min challenges that can be encountered when using the explicit scheme is that the stable solution can only be obtained when the following condition is met. 0 Read More

Both the predictors and corrector have different equations which can be represented as follows: Predictor Corrector Boundary Conditions In order to solve the equations, the boundary conditions must be considered. The boundary conditions influence the temperature, species mass fraction and temperature. The typical boundary conditions may involve the solid boundaries, open boundaries and mesh interface boundaries. In the solid boundary, the convective heat flux at the surface is determined through the use of an empirical heat transfer coefficient (McGrattan, et al, 2013).

In the open boundaries, the gases can flow freely in or out and has to be considered in solving the equations. Information has to be passed between meshes in mesh interface boundaries. During the calculations, the surface temperature may either be calculated or it may be provided. The heat transfer coefficients are also required during the calculations. Part b The regime of flow speed is one of the important aspects that distinguished the CFD model. The flow speed for which it is designed has to be relative to the speed of sound.

As a result of this the codes are written in high speed solvers and low speed solvers are used. The high speed solvers mainly involves the compressibility effects and shock effects. The low speed solvers are mainly involved in the elimination of compressibility effects which in most cases gives rise to acoustic sound waves. According to the Navier-Stokes equation, the propagation of speed can be compared to the fluid flow and sound waves (McGrattan, et al, 2013). Small time steps are therefore required in order to ensure that the equations are solved.

The practical simulations are however difficult due to the speed of sound waves and fluid flow. A student simulating a flying object at a velocity of 400m/s in air using FDS6 cannot obtain acceptable results. The propagation of speed in air is 300m/s. Even in such a condition, the simulation is still difficult since extremely small type steps are required. A speed of 400m/s is much higher and hence making it difficult to carry out the simulation and obtain acceptable results. The simulations may only be effective when the speed is less than that of air which is 300m/s.

Part c The background pressure is a product of decomposition of the spatially and temporally resolved pressure. Different rooms can have different background pressures due to the influence of various factors. The differences between the airflows in the different rooms can result to different background pressures (McGrattan, et al, 2013). The airflow in the rooms can be influenced by various factors including the amount of open spaces and hence causing the differences. The formula below indicates how two different rooms can have different background pressures: The difference in background pressure can also be attributed to an isolation of volume within a computational domain.

However, in such incidences, the leak paths as well as the ventilation ducts have to be considered. The background pressure ( ) is considered a function of z which is the vertical spatial coordinate and t which is the time (McGrattan, et al, 2013). The changes may occur with height or time. Some effects that may also affect the volume include increase in pressure by fire, significance of the height of the domain or significance of pressure increase by HVAC. The atmospheric pressure stratification in such incidences serves as the initial boundary condition that governs the equation.

Question 2: Turbulence Part a Turbulent model is the gradient diffusion required for closing the SGS momentum and scalar flux terms. The model is important as it is used for obtaining the turbulent transport coefficient. The coefficient model has two other aspects which involve the turbulent viscosity and turbulent diffusivity. Turbulence is thus important in CFD as it ensures the transport coefficient is determined. Turbulence model is also important in terms of ensuring that the concepts of burning are explained.

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