About this Journal Submit a Manuscript Table of Contents
Journal of Combustion
Volume 2012 (2012), Article ID 138619, 8 pages
http://dx.doi.org/10.1155/2012/138619
Research Article

Modified Quasi-Steady Fuel Droplet Combustion Model

Department of Mechanical Engineering, University of Uyo, PMB 1017, Uyo, Akwa Ibom, Nigeria

Received 19 December 2011; Accepted 10 March 2012

Academic Editor: Evangelos G. Giakoumis

Copyright © 2012 Etim S. Udoetok. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The quasi-steady model of the combustion of a fuel droplet has been modified. The approach involved the modification of the quasi-steady model to reflect the difference in constant properties across the flame front. New methods for accurately estimating gas constants and for estimating Lewis number are presented. The proposed theoretical model provides results that correlate favorably with published experimental results. The proposed theoretical model also eliminates the need for unguided adjustment of thermal constants or the complex analysis of the variation of thermal properties with temperature and can serve as a basis for analysis of other combustion conditions like droplets cloud and convective and high-pressure conditions.

1. Introduction

Fuel droplet models are used to describe the influence of droplet size and ambient conditions on fuel combustion in devices such as diesel engines, rocket engines, gas turbines, oil fired boilers, and furnaces [1, 2]. The simple quasi-steady model, which is the focus of this paper, has its origin in the 1950s [3, 4] and is widely accepted as the theoretical model of fuel droplet combustion [5]. The important results of this theory are as follows [1, 47]:̇𝑚=4𝜋𝑘𝑔𝑟𝑠𝑐𝑝𝑔ln(1+𝐵),(1) where 𝐵=Δ𝑐/𝜐+𝑐𝑝𝑔𝑇𝑇𝑠𝑞𝑖𝑙+𝑓𝑔,𝑇𝑓=𝑞𝑖𝑙+𝑓𝑔𝑐𝑝𝑔([]1+𝜐)𝜐𝐵1+𝑇𝑠,𝑟𝑓𝑟𝑠=ln(1+𝐵),𝐷ln(1+1/𝜐)2=𝐷02𝑡𝐾𝑡,𝑑=𝐷02𝐾,(2) where 𝐾=8𝑘𝑔𝜌𝑙𝑐𝑝𝑔ln(1+𝐵)=2̇𝑚𝜋𝜌𝑙𝑟𝑠.(3) Note that though the simple quasi-steady model theory has its roots in the 1950s, it has been developed upon through the 1950s to the 1970s in order to arrive at the results presented above [17]. However, simple quasi-steady model provides unsatisfactory results in comparison with several experimental observations [1, 5, 812]. The simple quasi-steady model is best at predicting fuel mass flow rate ̇𝑚 and the 𝑑2 law but does not accurately predict the flame to droplet radius ratio 𝑟𝑓/𝑟𝑠 and flame temperature 𝑇𝑓 [5]. The simple quasi-steady model typically predicts 𝑟𝑓/𝑟𝑠 higher than observed values, and 𝑇𝑓 is usually lower than that experimentally observed. Experiments have also shown that the values of 𝑟𝑓/𝑟𝑠 and 𝑇𝑓 may not be constant. Law et al. [10] showed that fuel vapor accumulation causes transient effects in the values of 𝑟𝑓/𝑟𝑠 and 𝑇𝑓, but Raghunandan and Mukunda [5] later showed that the condensed-phase unsteadiness lasts for 20–25% of burning time and concluded that the discrepancies between experiments and simple quasi-steady model cannot be attributed to condensed-phase unsteadiness. A study by He et al. [8] revealed that the flame front motion has the effect of causing unsteadiness and variation of 𝑟𝑓/𝑟𝑠 and 𝑇𝑓 during combustion. The analytical model of Raghunandan and Mukunda [5] for quasi-steady droplet combustion with variable thermodynamic and transport properties, and nonunity Lewis number gave accurate prediction of 𝐾 and significant improvement in 𝑟𝑓/𝑟𝑠 and 𝑇𝑓 compared to the simple quasi-steady model. Puri and Libby [13] used a detailed expression for the heat transfer and transport properties and came up with a complex model of the quasi-steady fuel droplet combustion. Their model is best solved numerically. Filho [14] solved the quasi-steady fuel droplet combustion problem in a way similar to Puri and Libby’s solution. Filho’s solution was less complicated and involved the removal of nonlinearity in the heat transfer and transport properties coefficients. Imaoka and Sirignano [1517] solved the fuel droplet combustion problem for the case of a droplet cloud using unity Lewis number assumption. They acknowledged that the use of unity Lewis number significantly overestimates 𝑟𝑓/𝑟𝑠. Imaoka and Sirignano [1517] focused on the variation of constants from one droplet to another in the droplet cloud. The vaporization rate was found for each droplet in the droplet cloud because they assumed that the solutions for each droplet are not equal.

In this paper, the simple quasi-steady fuel droplet combustion model is modified for higher accuracy by assuming discontinuity in the heat transfer and transport coefficients across the flame sheet and nonunity Lewis number for the inner and outer region. A method for estimating property constants for the two regions is recommended. Note that while the discontinuity in Imaoka and Sirignano [1517] solution is from droplet to droplet, the discontinuity assumed in this paper is from the inner region to the outer region of a burning droplet.

2. Method

In the derivation of the classical droplet combustion model, the following assumptions are made [1, 47].(i) Burning droplet is spherical and surrounded by a spherically symmetric flame in a quiescent infinite medium.(ii)Burning process is quasi-steady.(iii)Fuel is a single component and pressure is uniform and constant.(iv)Gaseous species are of 3 types: fuel vapor, oxidizer, and combustion products.(v)Stoichiometric proportions of fuel-oxidizer are at flame.(vi) Unity Lewis number is assumed.(vii)Radiation heat transfer is negligible.(viii)No soot or liquid water is present.(ix)Uniform species thermal constants: 𝑐𝑝 and 𝑘.

These assumptions are good, but the following assumptions changes will be made in order to improve the accuracy of the model.(i)Unity Lewis is assumed only at the source of diffusing species, and nonunity Lewis number is assumed in the outer and inner regions. This assumption is made because, at the sources of diffusing specie, the generation of the diffusing specie causes the thermal diffusivity to balance the mass diffusivity, while, away from the source of diffusing species, the thermal diffusivity and mass diffusivity have different values depending on the species concentration, species properties, and temperature profile. (ii)The property of the inner region is different from the property of the outer region. This assumption is made because the average temperature in the outer region is different from the average temperature in the inner region and the species composition in the outer region is different from the species composition in the inner region.

The new assumption that the property of the inner region is different from the property of the outer region is shown in Figure 1. This new assumption makes it necessary to have two average temperatures since there are two different sets of temperature extremes in the two regions.

138619.fig.001
Figure 1: Separation of inner region and outer region species properties.

In order to relate the outer constants to the inner constants, let 𝑍+=𝑛𝑍,(4) where 𝑛 is a constant and (4) implies that 𝑛=𝐿𝑒+𝑐+𝑝𝑔𝑘𝑔𝐿𝑒𝑐𝑝𝑔𝑘+𝑔.(5) The mass flow rate, ̇𝑚, is treated as a constant and independent of radius, 𝑟, since quasi-steady burning is assumed. In the inner region, Fick’s law can be presented in the form [1] ̇𝑚fuel=4𝜋𝑟2𝜌D1𝑌𝐹𝑑𝑌𝐹𝑑𝑟(6) with boundary conditions (BCs) 𝑌𝐹𝑟𝑠=𝑌𝐹,𝑠𝑇𝑠,𝑌𝐹𝑟𝑓=0.(7) Integration of (6) and application of BCs (7) gives 𝑌𝐹,𝑠=1exp𝑍̇𝑚/𝑟𝑠exp𝑍̇𝑚/𝑟𝑓,(8) where𝑍=1/4𝜋𝜌D=𝐿𝑒𝑐𝑝𝑔/4𝜋𝑘𝑔.

In the outer region, Fick’s law in terms of the constant fuel mass flow rate can be presented in the form [1] ̇𝑚fuel=4𝜋𝑟2𝜌D𝜐𝑌𝑂𝑥𝑑𝑌𝑂𝑥𝑑𝑟(9) with BCs 𝑌𝑂𝑥𝑟𝑓𝑌=0,𝑂𝑥(𝑟)=1.(10) Integration of (9) and application of BCs (10) gives the relation between ̇𝑚 and 𝑟𝑓 as 𝑍exp+̇𝑚𝑟𝑓=𝜐+1𝜐.(11) Equation (4) in (11) gives exp𝑛𝑍̇𝑚𝑟𝑓=𝜐+1𝜐.(12) In order to find the temperature profiles in the inner and outer region, the Shvab-Zeldovich form of the energy equation [1] is used, that is, 𝑑𝑟2(𝑑𝑇/𝑑𝑟)𝑑𝑟=𝑍̇𝑚𝑑𝑇𝑑𝑟(13) with two sets of BCs for the inner and outer regions BC𝑇𝑟𝑠=𝑇𝑠𝑇𝑟𝑓=𝑇𝑓,BC+𝑇𝑟𝑓=𝑇𝑓,𝑇(𝑟)=𝑇.(14) Integration of (13) and application of inner region BCs (14) gives 𝑇=𝑇(𝑟)𝑠𝑇𝑓exp(𝑍̇𝑚/𝑟)+𝑇𝑓exp𝑍̇𝑚/𝑟𝑠𝑇𝑠exp𝑍̇𝑚/𝑟𝑓exp𝑍̇𝑚/𝑟𝑠exp𝑍̇𝑚/𝑟𝑓(15) for the inner region and 𝑇+𝑇(𝑟)=𝑓𝑇exp𝑍+̇𝑚/𝑟+𝑇exp𝑍+̇𝑚/𝑟𝑓𝑇𝑓exp𝑍+̇𝑚/𝑟𝑓1(16) for the outer region, and (4) into (16) gives 𝑇+=𝑇(𝑟)𝑓𝑇exp(𝑛𝑍̇𝑚/𝑟)+𝑇exp𝑛𝑍̇𝑚/𝑟𝑓𝑇𝑓exp𝑛𝑍̇𝑚/𝑟𝑓.1(17) At the surface of the droplet, the heat conducted to it balances the heat used to vaporize and heat up the droplet. Hence, the energy balance at the droplet surface [1] can be written in the form 𝑘𝑔4𝜋𝑟𝑠2𝑑𝑇|||𝑑𝑟𝑟𝑠=̇𝑚𝑓𝑔+𝑞𝑖𝑙.(18) Differentiating (15) and substituting into (18) give the energy balance at the droplet as 4𝜋𝑘𝑔𝑍𝑇𝑓𝑇𝑠𝑞𝑖𝑙+𝑓𝑔exp𝑍̇𝑚/𝑟𝑠exp𝑍̇𝑚/𝑟𝑠exp𝑍̇𝑚/𝑟𝑓+1=0.(19) At the flame sheet, the heat of combustion is conducted away by both the inner region and outer region gases. Therefore, the energy balance at the flame sheet can be written in the form [1]̇𝑚Δ𝑐=𝑘𝑔4𝜋𝑟𝑓2𝑑𝑇|||𝑑𝑟𝑟𝑓𝑘+𝑔4𝜋𝑟𝑓2𝑑𝑇+|||𝑑𝑟𝑟𝑓.(20) Differentiating (15) and (17) and substituting into (20) give the simplified energy balance at the flame sheet as1=4𝜋Δ𝑐𝑘𝑔𝑍𝑇𝑓𝑇𝑠exp𝑍̇𝑚/𝑟𝑓exp𝑍̇𝑚/𝑟𝑠exp𝑍̇𝑚/𝑟𝑓4𝜋Δ𝑐𝑘+𝑔𝑛𝑍𝑇𝑇𝑓exp𝑛𝑍̇𝑚/𝑟𝑓1exp𝑛𝑍̇𝑚/𝑟𝑓.(21) Solving (12), (19), and (21) for ̇𝑚, 𝑟𝑓, and 𝑇𝑓 gives ̇𝑚=4𝜋𝑘𝑔𝑟𝑠𝐿𝑒𝑐𝑝𝑔×1ln𝑘1+𝑔/𝑛𝜐𝑘+𝑔+𝑇𝑇𝑠𝐿𝑒𝑐𝑝𝑔/𝑘1+𝑔/𝑛𝜐𝑘+𝑔+Δ𝑐/𝑛𝜐𝑘+𝑔/𝑘𝑔+1𝑞𝑖𝑙+𝑓𝑔+1𝑛ln𝜐+1𝜐.(22) Let 1𝐴=𝑘1+𝑔/𝑛𝜐𝑘+𝑔𝜐+1𝜐1/𝑛,𝑇𝐵=𝑇𝑠𝐿𝑒𝑐𝑝𝑔/𝑘1+𝑔/𝑛𝜐𝑘+𝑔+Δ𝑐/𝑛𝜐𝑘+𝑔/𝑘𝑔+1𝑞𝑖𝑙+𝑓𝑔𝜐+1𝜐1/𝑛,(23) then ̇𝑚=4𝜋𝑘𝑔𝑟𝑠𝐿𝑒𝑐𝑝𝑔𝑇ln(𝐴+𝐵),(24)𝑓=𝑞𝑖𝑙+𝑓𝑔𝐿𝑒𝑐𝑝𝑔𝜐(𝐴+𝐵)𝜐+11/𝑛1+𝑇𝑠,𝑟𝑓𝑟𝑠=𝑛ln(𝐴+𝐵)[].ln(𝜐+1)/𝜐(25) The new mass flow rate from (24) was used to obtain burning constant and droplet life time as 𝐾=8𝑘𝑔𝜌𝑙𝑐𝑝𝑔𝐿𝑒𝑡ln(𝐴+𝐵),𝑑=𝐷02𝐾.(26)

3. Estimation of Thermal Property Constants

The species properties for the inner region are estimated as follows:𝑇𝑇=0.5𝑠+𝑇𝑓,𝑘(27)𝑔=0.4𝑘𝐹𝑇+0.6𝑘𝑂𝑥𝑇𝑐,(28)𝑝𝑔=𝑌𝐹𝑐𝑝𝐹𝑇+𝑌𝑂𝑥𝑐𝑝𝑂𝑥𝑇.(29) Equation (27) is average temperature in the inner region. Equation (28) has been directly adapted from law and William’s suggestion [9, 18] since it was experimentally derived, and it replaces the complex estimation of thermal conductivity for a mixture of gas. Equation (29) is the specific heat constant of the gaseous mixture in the inner region. The mass fractions in (29) are estimated by assuming a linear fuel mole fraction from the fuel surface to the region close to the flame where there is a stoichiometric mixture of fuel and oxidizer. Therefore, at the fuel droplet surface, 𝜒𝐹,𝑠𝜒1,𝑂𝑥,𝑠0.(30) At the region close to the flame and for the case of hydrocarbon fuel droplet combustion in air, the reaction equation is typically C𝑥H𝑦+𝑦𝑥+4O2+3.76N2𝑥CO2+(𝑦/2)H2𝑦O+3.76𝑥+4N2.(31) The mole fractions can be estimated from the reactants as 𝜒𝐹,𝑟𝑓=1,𝜒1+𝑥+𝑦/4𝑂𝑥,𝑟𝑓=𝑥+𝑦/4.1+𝑥+𝑦/4(32) Therefore, the average mole fractions for the inner region are𝜒𝐹𝜒=0.5𝐹,𝑠+𝜒𝐹,𝑟𝑓=0.5+0.5,𝜒1+𝑥+𝑦/4𝑂𝑥𝜒=0.5𝑂𝑥,𝑠+𝜒𝑂𝑥,𝑟𝑓=0.5𝑥+𝑦/8.1+𝑥+𝑦/4(33) The mole fractions can then be used to estimate the mass fractions𝑌𝐹=𝜒𝐹𝑀𝐹𝜒𝐹𝑀𝐹+𝜒𝑂𝑥𝑀𝑂𝑥,𝑌𝑂𝑥=𝜒𝑂𝑥𝑀𝑂𝑥𝜒𝐹𝑀𝐹+𝜒𝑂𝑥𝑀𝑂𝑥.(34) And the specie constants for the outer region are estimated as follows: 𝑇+𝑇=0.5+𝑇𝑓,𝑐+𝑝𝑔=𝑐𝑝𝑂𝑥𝑇+,𝑘+𝑔=𝑘𝑂𝑥𝑇+.(35) The oxidizer (which is usually air) mainly dominates the outer region, so the outer region constants are evaluated directly from the oxidizer properties. Additionally, it is reasonable to assume that 𝑞𝑖𝑙=0 and 𝑇𝑠=𝑇boil, since the droplet is burning vigorously after an initial transient heat up.

4. Estimation of Lewis Number

For the estimation of Lewis number in the two regions, unity Lewis number is assumed at the source of diffusing specie. Therefore, in the inner region where fuel diffuses from the droplet surface, unity Lewis number is assumed at the droplet surface 𝐿𝑒𝑇𝑠=1.(36) By definition, 𝐿𝑒=𝛼D.(37) Assuming ideal-gas behavior, the pressure and temperature dependence of diffusion coefficient [19] is given as 𝑇D3/2𝑃(38) which implies that 𝑇D=𝐶3/2𝑃=𝐶𝑇3/2,(39) where is 𝐶 is a constant to be found and 𝑃 has been absorbed into the constant because the combustion takes place at constant pressure. By applying the boundary condition 𝐿𝑒(𝑇𝑠)=1, 𝐶 is found as 𝛼𝐶=𝑇𝑠𝑇𝑠3/2.(40) Combining (40), (39), and (37) gives 𝐿𝑒=𝛼𝑇𝛼𝑇𝑠𝑇𝑠𝑇3/2,(41) where 𝛼𝑇𝑠=𝑘𝐹𝑇𝑠𝜌𝐹vapor𝑇𝑠𝑐𝑝𝐹𝑇𝑠.(42) Fuel dominates at the vapor-surface interface, so density is estimated as 𝜌𝐹vapor𝑇𝑠=𝑃𝑇𝑠𝑅𝐹=𝑃𝑇𝑠𝑅𝑢/𝑀𝐹,𝛼𝑇=𝑘𝑔𝜌𝑐𝑝𝑔,(43) where 𝑘𝑔 and 𝑐𝑝𝑔 are given by (28) and (29), and 𝜌 is estimated as 𝜌=𝑃𝑇𝑅=𝑃𝑇𝑅𝑢/𝑀,(44) with 𝑀=𝑌𝐹𝑀𝐹+𝑌𝑂𝑥𝑀𝑂𝑥.(45) Secondly, in the outer region where the combustion products diffuse from the flame sheet, so unity Lewis number is assumed at the flame sheet 𝐿𝑒+𝑇𝑓=1.(46) Using estimation method similar to that done for the inner region gives 𝐿𝑒+=𝛼+𝑇+𝛼+𝑇𝑓𝑇𝑓𝑇+3/2.(47) At the flame sheet, the mixture fractions can be estimated from the reaction equation ((31) for the combustion of CxHy droplet in air). αs in (47) are estimated as 𝛼+𝑇𝑓=𝑘Pr𝑜duct𝑇𝑓𝜌Pr𝑜duct𝑇𝑓𝑐𝑝Pr𝑜duct𝑇𝑓,𝛼+𝑇+=𝑘+𝑔𝜌+𝑐+𝑝𝑔.(48) For the case of combustion in air, N2 dominates the product and outer region, and the property of air can be used to estimate αs𝐿𝑒+𝛼Air𝑇+𝛼Air𝑇𝑓𝑇𝑓𝑇+3/2.(49) Equation (49) provides a valid approximation for Lewis number of the outer region and is recommended, since tabulated values of 𝛼Air are readily available [1, 2023].

5. Results and Discussions

As an example, calculation of combustion variables for the case of 𝑛-heptane (C7H16) droplet combustion in air was done. Both the simple quasi-steady model and the proposed new model (modified quasi-steady model) were used. Ambient conditions were used, that is, 𝑃=1atm and 𝑇=298K. It was assumed that 𝑇𝑠=𝑇boil, and droplet heating is negligible, that is, 𝑞𝑖𝑙=0. 𝑇𝑠=𝑇boil is assumed because it has been experimentally observed that the droplet boils vigorously during the combustion after an initial and brief heatup, and the heat used to heat the droplet from its initial temperature, 𝑞𝑖𝑙, is usually negligible and has negligible effect on the model result [1, 2]. Initial guess used for 𝑇𝑓 is 2100 K since tabulated values of adiabatic flame temperature of common hydrocarbon fuels are approximately 2000 K.

The calculation results using the simple quasi-steady model are given in Table 1. Law and Williams’s suggestion was used to evaluate species properties for use in the simple quasi-steady model [1, 9, 18], and the results for each iteration step are presented. Iteration was repeated till the solution converged to a difference of 2 K or less between the guess flame temperature and the calculated flame temperature. Similarly, results obtained by using the proposed new model are presented in Table 2. Additionally, calculation results for the case of combustion of hexane droplet in ambient conditions using the proposed new model are shown in Table 3. The sample calculations showed that proposed new model predicts realistic flame temperature, evaporation constant, and flame to droplet radius ratio compared to the simple quasi-steady model. In the next section, the results of the proposed new model will be compared to experimental results published.

tab1
Table 1: Simple quasi-steady model 𝑛-heptane results.
tab2
Table 2: Proposed model 𝑛-heptane results.
tab3
Table 3: Proposed model 𝑛-hexane results.

6. Comparison with Published Experimental Results

The proposed new model calculation results are compared favorably with published experimental results on the combustion of 𝑛-heptane. These experimental results and other models calculation results for the combustion of an 𝑛-heptane droplet in air are summarized in Table 4.

tab4
Table 4: Proposed model 𝑛-heptane results compared with published experimental observations.

The proximity between calculated and measured values has been greatly improved by the new model proposed in this paper. The proposed new model accurately predicts 𝐾 and 𝑟𝑓/𝑟𝑠. It predicts 𝑇𝑓 slightly higher and yet closer to the experimentally observed range compared to previous models. The proposed new model also predicted flame temperature closer to the values predicted by the 1999 and 1991 models compared to the original simple quasi-steady model of the 1950s. The 𝐾 and 𝑟𝑓/𝑟𝑠 values obtained for the combustion of 𝑛-hexane and 𝑛-heptane were approximately equal, while 𝑇𝑓 for 𝑛-heptane was slightly higher than that for hexane.

The estimated flame temperature seems to have the greatest error, and the most probable source of error is in the estimation of 𝑘𝑓.𝑘𝑓 at 𝑇 was estimated by extrapolation, which is not good for estimating thermal conductivity of vapor, because the available data used ranged up to 1000 K only. However, since the estimation of flame temperatures by the newest models is approximately 2600 K, the error may come from experimental error since it may be more difficult to capture the spiked temperature of the flame sheet.

Most of the available tabulated fuel vapor thermal conductivities ranges up to 500 K, and this points out the need for having thermal conductivities tables or curve fits that ranges up to 1500 K or higher in order to use and achieve results with less error.

7. Conclusions

The simple quasi-steady model of a fuel droplet was modified to reflect the difference in constant properties across the flame sheet. Two average temperatures were used: one for the inner region and the other for the outer region. The two average temperatures were used to evaluate the assumed constant specific heat and thermal conductivities for the two regions. Nonunity Lewis number was assumed for the two regions while unity Lewis number was assumed at the source of diffusing species, which implies that unity Lewis number was assumed at the flame sheet for the outer region and at the liquid-vapor interface for the inner region. The Lewis numbers obtain in the sample calculation falls within the range that has been observed experimentally [5]. Sample calculations and comparison with experimental results showed that the new model accurately modeled the droplet combustion than the simple quasi-steady model. The new model performance shows that the quasi-steady model of fuel droplet combustion when appropriately applied is a good approximation of the combustion results. The new model also eliminates the need for unguided adjustment of thermal constants and eliminates the need for complex analysis of specific heat and thermal conductivity variation with temperature. The proposed new model is slightly more complex than the original simple quasi-steady model; however, it does not require complex numerical computation for its solutions. The result of the theoretical models of the droplet combustion is best estimated by iteration as shown in the sample calculations. The new model was derive following the process used to derived the old model; hence, it can be noted that if 𝑛=1, 𝑐+𝑝𝑔=𝑐𝑝𝑔=𝑐𝑝𝑔, 𝑘+𝑔=𝑘𝑔=𝑘𝑔, and 𝐿𝑒+=𝐿𝑒=1 are substituted into the new model equations, the old quasi-steady model will be obtained. The new model can serve as a basis for analysis of other droplet combustion conditions like droplet cloud and convective and high pressure conditions.

Symbols Used

𝐴:Smaller transfer number (~1)
𝐵:Transfer or Spalding number
𝑐𝑝𝑔:Specific heat constant of gas [J/kgK]
𝐶:Constant
𝐷:Droplet diameter [m]
D:Mass diffusivity [m2/s]
𝑓𝑔:Latent heat of vaporization [J/kg]
𝐾:Evaporation rate constant [m2/s]
𝑘𝑔:Thermal conductivity of gas [W/mK]
𝑀:Molecular weight [kg/kmol]
̇𝑚̇:Fuel mass flow rate [kg/s]
𝐿𝑒:Lewis number
𝑛:Constant: Z ratio
𝑃:Pressure [atm]
𝑞𝑖𝑙:Interface to liquid heat transfer per unit mass (droplet heating) [J/kg]
𝑅:Gas constant [J/KgK]
𝑟:Radius [m]
𝑇:Temperature [atm]
𝑡:Time [s]
𝑡𝑑:Droplet life time [s]
𝑥:Number of carbon atoms in fuel molecule
𝑌:Mass fraction [kg/kg]
𝑦:Number of hydrogen atoms in fuel molecule
𝑍:𝐿𝑒𝑐𝑝g/4𝜋𝑘g[ms/kg]
Δ𝑐:Enthalpy of combustion [J/kg]
α:Thermal diffusivity [m2/s]
𝜐:Oxidizer-to-fuel stoichiometric mass ratio [kg/kg]
𝜌:Density [kg/m3]
𝜒:Mole fraction [kmol/kmol].
Superscripts
+:Outer region
−:Inner region.
Subscripts
0:Initial condition
:Free stream—far from surface
boil:Boiling point
𝑑:Droplet
𝑓:Flame
𝐹:Fuel
𝑔:Gas
𝐺:Guess
𝑖:Interface
𝑙:Liquid
𝑂𝑥:Oxidizer
𝑠:Droplet surface
𝑢:Universal.

References

  1. S. R. Turns, An Introduction to Combustion: Concepts and Applications, McGraw Hill, 2nd edition, 2000.
  2. M. Goldsmith and S. S. Penner, “On the burning of single drops of fuel in oxidizing atmosphere,” Jet Propulsion, vol. 24, pp. 245–251, 1954.
  3. G. A. E. Godsave, “Studies of the combustion of drops in a fuel spray-the burning of single drops of fuel,” Symposium (International) on Combustion, vol. 4, no. 1, pp. 818–830, 1953. View at Publisher · View at Google Scholar · View at Scopus
  4. D. B. Spalding, “The combustion of liquid fuels,” Symposium (International) on Combustion, vol. 4, no. 1, pp. 847–864, 1953. View at Publisher · View at Google Scholar · View at Scopus
  5. B. N. Raghunandan and H. S. Mukunda, “The problem of liquid droplet combustion-A reexamination,” Combustion and Flame, vol. 30, no. C, pp. 71–84, 1977. View at Scopus
  6. J. W. Aldred, J. C. Patel, and A. Williams, “The mechanism of combustion of droplets and spheres of liquid n-heptane,” Combustion and Flame, vol. 17, no. 2, pp. 139–148, 1971. View at Scopus
  7. H. Wise, J. Lorell, and B. J. Wood, “The effects of chemical and physical parameters on the burning rate of a liquid droplet,” Symposium (International) on Combustion, vol. 5, no. 1, pp. 132–141, 1955. View at Publisher · View at Google Scholar · View at Scopus
  8. L. He, S. D. Tse, and C. K. Law, “Role of flamefront motion and criterion for global quasi-steadiness in droplet burning,” Symposium (International) on Combustion, vol. 2, pp. 1943–1950, 1998. View at Scopus
  9. S. Kumagai, T. Sakai, and S. Okajimah, “Combustion of free fuel droplets in a freelly falling chamber,” Symposium (International) on Combustion, vol. 30, pp. 778–785, 1971.
  10. C. K. Law, S. H. Chung, and N. Srinivasan, “Gas-phase quasi-steadiness and fuel vapor accumulation effects in droplet burning,” Combustion and Flame, vol. 38, no. C, pp. 173–198, 1980. View at Scopus
  11. C. K. Law, “Recent advances in droplet vaporization and combustion,” Progress in Energy and Combustion Science, vol. 8, no. 3, pp. 169–199, 1982. View at Scopus
  12. F. A. Williams, Combustion Theory, Benjamin/Cummings, Menlo Park, Calif, USA, 2nd edition, 1985.
  13. I. K. Puri and P. A. Libby, “The influence of transport properties on droplet burning,” Combustion Science and Technology, vol. 76, pp. 67–80, 1991.
  14. F. F. Filho, “An analytical solution for the quasi-steady droplet combustion,” Combustion and Flame, vol. 116, no. 1-2, pp. 302–306, 1998. View at Publisher · View at Google Scholar · View at Scopus
  15. R. T. Imaoka and W. A. Sirignano, “Transient vaporization and burning in dense droplet arrays,” International Journal of Heat and Mass Transfer, vol. 48, no. 21-22, pp. 4354–4366, 2005. View at Publisher · View at Google Scholar · View at Scopus
  16. R. T. Imaoka and W. A. Sirignano, “A generalized analysis for liquid-fuel vaporization and burning,” International Journal of Heat and Mass Transfer, vol. 48, no. 21-22, pp. 4342–4353, 2005. View at Publisher · View at Google Scholar · View at Scopus
  17. R. T. Imaoka and W. A. Sirignano, “Vaporization and combustion in three-dimensional droplet arrays,” in Proceedings of the 30th International Symposium on Combustion, vol. 30, pp. 1981–1989, July 2004. View at Publisher · View at Google Scholar · View at Scopus
  18. C. K. Law and F. A. Williams, “Kinetics and convection in the combustion of alkane droplets,” Combustion and Flame, vol. 19, no. 3, pp. 393–405, 1972. View at Scopus
  19. R. H. Perry, D. W. Green, and J. O. Maloney, Perry’s Chemical Engineers Handbook, McGraw-Hill, New York, NY, USA, 6th edition, 1984.
  20. T. L. Bergman, A. S. Lavine, F. P. Incropera, and D. P. Dewitt, Fundamentals of Heat and Mass Transfer, John Wiley and Sons, 7th edition, 2011.
  21. American Society of Heating, Refrigeration and Air-Conditioning Engineers, ASHRAE Handbook: Fundamentals, ASHRAE, 1981.
  22. N. B. Vargaftik, Tables of Thermophysical Properties of Liquids and Gases, Hemisphere Publishing, New York, NY, USA, 2nd edition, 1975.
  23. A. Kamei, S. W. Beyerlein, and R. T. Jacobsen, “Application of nonlinear regression in the development of a wide range formulation for HCFC-22,” International Journal of Thermophysics, vol. 16, no. 5, pp. 1155–1164, 1995. View at Publisher · View at Google Scholar · View at Scopus
  24. S. Kumagai and H. Isoda, “Combustion of fuel droplets in a falling chamber,” Symposium (International) on Combustion, vol. 6, no. 1, pp. 726–731, 1957. View at Publisher · View at Google Scholar · View at Scopus
  25. M. T. Monaghan, R. G. Siddall, and M. W. Thring, “The influence of initial diameter on the combustion of single drops of liquid fuel,” Combustion and Flame, vol. 12, no. 1, pp. 45–53, 1968. View at Scopus
  26. J. W. Aldred and A. Williams, “The burning rates of drops of n-alkanes,” Combustion and Flame, vol. 10, no. 4, pp. 396–398, 1966. View at Scopus
  27. M. Goldsmith, “Experiments on the burning of single drops of fuel,” Jet Propulsion, vol. 26, pp. 172–178, 1956.