Spreading model

Notice that all the above dependencies assume that i is proportional to C and g independent of C, that is the chain length must be sufficiently long that a nucleation and spreading model is still valid. At very low molecular weights process (iv) will be important and may predominate the effects considered here. 

Thus, for opposed flow spread, the steady state thermal flame spread model appears valid. In wind-aided flame spread, it seems appropriate to modify our governing equation for the thermally thin case as... 

Various calculation tools and other types of fire models are available that do not fit neatly into the above categories. Among these are multicalculation packages, flame spread models, and glass breaking simulations. 

The Spreading Model. Considering the discrete version of Equation 1, and bearing in mind that all intervening functions are of finite length, then one may write ... 

Microscopically, the interface reaction during crystal growth may be through various mechanisms. One mechanism is called the continuous model. Two other models are layer-spreading models. 

There are two layer-spreading models. In these models, the crystal surface is atomically flat except at screw dislocations or steps of a partially grown surface layer. If there are screw dislocations, growth would continue on the screw... 

Figure 3.10. Schematic to show the layer-by-layer growth model due to two-dimensional nucleation. This figure assumes the mode of nucleation to be the mononuclear model. Other models, such as the poly-nuclear or birth and spread models, as explained in the text, may also be considered.

A birth and spread model, which allows nucleation and advancement of one growth layer at a time on one surface. 

Hakkarainen T, Kokkala MA. Application of a one-dimensional thermal flame spread model on predicting the rate of heat release in the SBI test. Fire Mater. 2001 25 61-70. 

This type of model works well at high applied heat flux levels, where the pyrolysis front is thin. Simplicity is its advantage it is not necessary to specify any parameters related to the decomposition kinetics. A large body of flame spread modeling work has applied this type of model, but there is a tendency to focus with great detail on gas-phase phenomena (i.e., full Navier-Stokes, detailed radiation models, multistep combustion reactions) and treat the condensed-phase fuel generation process in an approximate manner. 

FIGURE 20.7 Comparison of measured and modeled HRR in room/corner test with wood lining materials. (Adapted from Carlsson, J., Computational strategies in flame-spread modelling involving wooden surfaces—An evaluation study, Lund University Department of Fire Safety Engineering, Lund, Sweden, Report 1028, 2003.)... 

Hietaniemi et al. [76] used a prerelease version of FDS4 to model lire spread on several materials in several different configurations and compared the calculated results with experimental data. This is one of the most comprehensive (in terms of the number of materials and the number of different configurations simulated) large-scale flame spread modeling studies conducted to date. The materials simulated include spruce timber (SBI, room/corner, and 6 m cavity), medium density fiber board (SBI and room/corner), PVC wall carpet on gypsum board (SBI, room/corner), upholstered furniture (furniture calorimeter and ISO room), and polyethylene-sheathed cables in 6 m cavity. 

Zone model based fire spread model... 

Field model based fire spread models... 

Improved fire spread models should be developed to determine the effects of street arrangement, street canyon aspect ratio, building heights, etc. on fire propagation. CFD appears to be a valuable tool in investigating these phenomena systematically. 

The birth and spread model and its modifications predicts that the growth rate G increases with increasing supersaturation and increasing temperature. The dependence of growth rate on these variables is not a simple function of supersaturation and the model does not have the obvious problems of the other two models. For this reason, some semiempirical relations obtained from Eq. (2.41) are sometimes used to correlate experimental growth data and obtain the needed constants. 

In the mononuclear model, the limiting step is the formation of a nucleus. Once one is formed, the subsequent growth spreading across the crystal surface is infinitely rapid. For the polynuclear model, the spreading velocity is taken as zero and the crystal surface can only be covered by the accumulation of a sufficient number of nuclei. These two growth models represent two extreme cases. A third model, known as the birth-and-spread model, allows for formation of nuclei and their subsequent growth at a finite rate. In this case, new nuclei can form on top of uncompleted layers. 

The low molecular weight of NO may allow rapid movement in aqueous environments. Molecules larger than NO have much lower diffusion coefficients and rates of spread. Models to describe the kinetic and concentration profiles for NO have been developed, based on diffusion models used to describe heat conduction in solids. These models show NO concentrations to be a function of the rate of formation, diffusion coefficient, distance from the source and time. Close to a production source, perhaps within a radius of 10/rm, concentrations reach a steady state within a few hundred ms of release. This is an order of magnitude faster than the measured biological half-life of NO, so that metabolism of NO in vivo should not significantly affect these concentration gradients... 

Although spreads may be viewed as a function of default risk and recovery risk, spread models do not attempt to break down the spread into its default risk and recovery risk components. 

First generation pricing models for credit spread options may use models as described in the section on spread models. The key market parameters in a spread option model include the forward credit spread and the volatility of the credit spread. 

A key issue with credit spread options is ensuring that the pricing models used will calibrate to the market prices of credit risky reference assets. The recovery of forward prices of the reference asset would be a constraint to the evolution of the credit spread. More complex spread models may allow for the correlation between the level of the credit... 

The birth and spread model (B+S) describes the formation of critical nuclei on a smooth crystal surface and their subsequent growth. The so-called nucleus above nucleus model leads to... 

Practitioners increasingly model credit risk as they do interest rates and use spread models to price associated derivatives. One such model is the Heath-Jarrow-Morton (HJM) model described in chapter 4. This analyzes interest rate risk, default risk, and recovery risk—that is, the rate of recovery on a defaulted loan, which is always assumed to retain some residual value. 

Equations (20.24) and (20.25) were developed for spills of constant volume, constant surface tension, and low viscosity on calm water. The effects of wind and currents on spreading rates are not well studied and are difficult to estimate. Therefore, the quantifiable uncertainty in the spreading rate lies in the estimation of the parameters used in Eqs. (20.24) and (20.25). The transition from a viscous spread, i.e., Eq. (20.25) to a surface tension spread, i.e., Eq. (20.23) occurs rapidly for most spills, and the spreading rate is described by Eq. (20.24). Since the density and viscosity of water can be estimated fairly confidently, most of the uncertainty in the spreading rate lies in the estimation of the net surface tension, specifically in the estimation of the air-oil surface tension and the oil-water surface tension. There is also an uncertainty in the applications of the slick-spreading model to a cross-sectional nonuniform velocity profile, where the nonuniformities would add to the spreading. In this case, the slick would experience a longitudinal dispersion in addition to the water. This phenomenon is not a component of the sensitivity analysis. 

Figure 6 A schematic of the birth and spread model of crystal growth (after Mullin ).

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