# Energy Balance Models of Downward Combustion of Parallel Thin Solid Fuels and Comparison to Experiments

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### Cites background from "Energy Balance Models of Downward C..."

...Since Kurosaki and Itoh’s model requires several parameters to predict the spread rate, Comas et al. attempted to generalize the model to be applicable for various situations by evaluating the flame geometry (Comas and Pujol, 2013)....

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...attempted to generalize the model to be applicable for various situations by evaluating the flame geometry (Comas and Pujol, 2013)....

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4 citations

##### References

356 citations

### "Energy Balance Models of Downward C..." refers background or methods in this paper

...In these cases, a model beyond the classical de Ris (1969) one is needed since radiation may play an important role....

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...The analytical model (A model in Figure 4), which generalizes the de Ris expression by including radiation, predicts higher values of the spread rate due to the positive contribution of the radiative terms in Eq. (15), as already expected. This analytical model matches the data at XO21⁄4 50% but fails to reproduce the observed behavior at greater atmospheric concentrations, underestimating by 19% the flame front speed at XO21⁄4 100%. The generalized model gives slightly greater values of Vf than the analytical model, but not higher enough to correctly report the measured data at high XO2. In comparison with de Ris Eq. (1), and for a single sheet, the differences with the generalized (I-G) model arise from the radiative flux from the flame (qrf1) and the radiative losses from the paper (qrs). This radiative correction may lead to a negative value for losses higher than the gain from the flame. However, this is not observed in our model due to the assumed flame shape. In our model, the flame is a rectangle parallel to the paper, separated from it by a length Lh (see Figure 3). This provides a view factor of the flame to a differential element of the preheated zone of the paper greater than it would be if a more realistic shape was assumed, which overestimates qrf1. Note also that the qrf1 value predicted by the I-G model is greater than the value obtained from the A model since it applies the Vf value instead of the Vf,deRis for determining the geometrical flame parameters. In contrast, Kurosaki et al. (1979) and Itoh and Kurosaki (1985) models clearly underestimate the experimental observations at moderate–high values of XO2, only correctly predicting the flame spread rate at environmental values XO21⁄4 21%, which correspond to the conditions that they were developed. Both K and I models shown in Figure 4 adopt a variable value of the flame temperature Tf as a function of XO2, being the same as that employed in A and I-G models. Figure 4 Downward burning rate Vf as a function of the oxygen molar fraction XO2 for a single sheet. Our experimental data is compared with the Kurosaki et al. (1979) model (K model), Itoh and Kurosaki (1985)...

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...For a single sample, the downward flame front speed in the thermal regime of a thin solid fuel follows the well-known de Ris’s expression (de Ris 1969): Vf ;deRis ¼ p 4 kg Tf;ad Tvap s=2ð Þcsqs Tvap T1 ð1Þ Received 21 June 2013; revised 25 August 2013; accepted 26 August 2013....

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...In contrast with the previous subsection, values of the parameters dg, Lh, Lf, and Le needed in qcv, qrf1, qrf2, and qre have been evaluated at Vf, deRis instead of at Vf. This makes Eq. (15) fully explicit. Note that in Eq. (15) we have neglected the contribution of the qrp term due to its small relevance in the total heat transfer rate as noted in Kurosaki et al. (1979). We point out that Eq....

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...The analytical model (A model in Figure 4), which generalizes the de Ris expression by including radiation, predicts higher values of the spread rate due to the positive contribution of the radiative terms in Eq. (15), as already expected. This analytical model matches the data at XO21⁄4 50% but fails to reproduce the observed behavior at greater atmospheric concentrations, underestimating by 19% the flame front speed at XO21⁄4 100%. The generalized model gives slightly greater values of Vf than the analytical model, but not higher enough to correctly report the measured data at high XO2. In comparison with de Ris Eq. (1), and for a single sheet, the differences with the generalized (I-G) model arise from the radiative flux from the flame (qrf1) and the radiative losses from the paper (qrs). This radiative correction may lead to a negative value for losses higher than the gain from the flame. However, this is not observed in our model due to the assumed flame shape. In our model, the flame is a rectangle parallel to the paper, separated from it by a length Lh (see Figure 3). This provides a view factor of the flame to a differential element of the preheated zone of the paper greater than it would be if a more realistic shape was assumed, which overestimates qrf1. Note also that the qrf1 value predicted by the I-G model is greater than the value obtained from the A model since it applies the Vf value instead of the Vf,deRis for determining the geometrical flame parameters. In contrast, Kurosaki et al. (1979) and Itoh and Kurosaki (1985) models clearly underestimate the experimental observations at moderate–high values of XO2, only correctly predicting the flame spread rate at environmental values XO21⁄4 21%, which correspond to the conditions that they were developed....

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183 citations

### "Energy Balance Models of Downward C..." refers background in this paper

...The opposed flow gas velocity Vg is assumed to vary accordingly to the expression (Frey and T’ien, 1979) Vg=Vg,ref1⁄4 [ag(Tf,ad T1)]=[ag,ref (Tf,ad,ref T1)], where the reference values correspond to those at XO21⁄4 21% with Vg,ref1⁄4 0....

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...The opposed flow gas velocity Vg is assumed to vary accordingly to the expression (Frey and T’ien, 1979) Vg=Vg,ref¼ [ag(Tf,ad T1)]1=3=[ag,ref (Tf,ad,ref T1)]1=3, where the reference values correspond to those at XO2¼ 21% with Vg,ref¼ 0.3ms 1....

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173 citations

### "Energy Balance Models of Downward C..." refers background in this paper

...(1) with a wide variety of experimental data (see, e.g., Fernández-Pello et al., 1981) corroborates that conduction through the solid-phase as well as radiative heat fluxes are of secondary importance when burning thin solid fuels far from the extinction limits....

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79 citations

### "Energy Balance Models of Downward C..." refers methods in this paper

...…# Taylor & Francis Group, LLC ISSN: 0010-2202 print=1563-521X online DOI: 10.1080/00102202.2013.839556 1820 where the p=4 term has been included after the exact solution derived by Delichatsios (1986) (see the Nomenclature for the description of the variables and parameters)....

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53 citations

### "Energy Balance Models of Downward C..." refers background or methods in this paper

...On the other hand, flame Lf, ember Le, and separation from flame to paper Lh lengths are calculated using Bhattacharjee et al.’s (2011) work:...

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...We assume that the relevant length scale where the convective heat flux qc is important corresponds to dg ag=(VgþVf), as it was determined in Bhattacharjee et al. (2005)....

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...In this case, the flame spread rate strongly depends on radiative effects (Bhattacharjee et al., 2005)....

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...We assume that the relevant length scale where the convective heat flux qc is important corresponds to dg ag=(VgþVf), as it was determined in Bhattacharjee et al. (2005). Therefore, we state that d1⁄4 dg in the exponential shape for the convective flux qc1⁄4 be ....

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...Our generalization of the I model employs the parameterizations of the flame geometrical dimensions of Bhattacharjee et al. (2011) and a variable convective flux (I-G model)....

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