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2719-9509
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01 Jan 1992
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# Numerical Simulation of the Burning Process in a King-Size Cigarette Based on Experimentally Derived Reaction Kinetics

###### Przyjęty: 22 Dec 2020
Informacje o czasopiśmie
Format
Czasopismo
eISSN
2719-9509
Pierwsze wydanie
01 Jan 1992
Częstotliwość wydawania
4 razy w roku
Języki
Angielski
INTRODUCTION

The modeling of the cigarette smoking process is a challenge because it involves complex chemical reactions, interactions of many thermo-physical processes, such as heat, mass and momentum transfer, and the filtration mechanism of the filter for aerosols. However, establishing a mathematical model of cigarette combustion and the indepth analysis of the smoking process are important for the design of cigarettes.

Since Egerton et al. (1) first tried to model the burning process of a cigarette, several mathematical models have been established in the last fifty years, as shown in Table 1 (2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18). Obviously, the modeling work has made significant progress. It is worth noting that during the first few decades almost all models published were restricted to a burning cigarette under free convection or steady draw conditions. In 2004 Saidi et al. (14) developed a three-dimensional model to simulate a burning cigarette during puffing. Afterwards more comprehensive models were reported, which could simulate the puff-smoldering cycles of a lit cigarette and provide more information about the cigarette burning behavior (17, 18).

The mathematical models of cigarette reported in the literatures.

Reference and year Author(s) Smoking conditions Geometry Model construction Simulation contents

Pyrolysis and char oxidation reaction kinetics Transport system Burning properties Products
(1) 1963 Egerton et al. Steady draw 1-D Temperature
(2) 1966 Gugan Smoldering 2-D Combustion cone Temperature
(3) 1977 Baker Smoldering 1-D Heat release rate O2 concentration CO, CO2
(4) 1978 Summerfield et al. Steady draw 1-D Using the kinetics parameters obtained by themselves (4) Burning rate Temperature Pressure
(5,6,7) 1979–1981 Muramatsu et al. Smoldering 1-D Using the kinetics parameters obtained by themselves (5, 6) Burn rate Temperature Density
(8) 2001 Miura et al. Smoldering 1-D Burning rate Temperature
(9) 2001 Yi et al. Smoldering 2-D Using the kinetics parameters obtained by Diblasi (10) Temperature Solid density Char density O2 concentration Water
(11) 2002 Chen Smoldering 1-D Using the kinetics parameters obtained by themselves (11) Temperature Density
(12) 2003 Rostami et al. Smoldering 2-D Using the kinetics parameters reported by Muramatsu et al. (5, 6) Burning rate Temperature O2 concentration
(13) 2004 Rostami et al. Smoldering and steady draw 2-D Using the kinetics parameters reported by Muramatsu et al. (5, 6) Temperature O2 concentration Pressure Flow velocity
(14) 2004 Saidi et al. Puffing 3-D Using the kinetics parameters for volatile species reported by Wojtowicz et al. (15) Burning rate Temperature Flow velocity O2 concentration CO, CO2 H2O Nicotine
(16) 2005 Eitzinger et al. Smoldering, puffing and steady draw 2-D Using the kinetics parameters obtained by themselves (16) Burning rate Temperature Flow velocity O2 concentration Combustion gas Water
(17) 2007 Saidi et al. Puff-smoldering cycles 3-D Using the kinetics parameters for volatile species reported by Wojtowicz et al. (15) Burning rate Temperature Flow velocity O2 concentration Char density CO, CO2 Volatile
(18) 2008 Saidi et al. Puff-smoldering cycles 3-D Using the kinetics parameters for volatile species reported by Wojtowicz et al. (15) Burning rate Temperature Flow velocity Char density CO, CO2 H2O

During the numerical simulation of a burning cigarette, the establishment of a tobacco pyrolysis and combustion kinetics model is a key step. The kinetics will determine the amount of tobacco involved in the reactions and the yield of char. At the same time, the heat released by char oxidation is transferred back to the tobacco, which forms a self-sustaining burning cycle of the cigarette. Finally, the reaction kinetics will affect the temperature and oxygen distribution, burning rate and smoke composition.

Table 1 shows that most literature published after 2003 applied the kinetics of pyrolysis and char oxidation reactions proposed by Muramatsu et al. (5, 6) and Wojtowicz et al. (15), who carried out their thermogravimetry (TG) analysis experiments at heating rates between 10 K·min−1 and 100 K·min−1. This heating rate was much lower than what is actually present in a cigarette during smoldering and puffing. “Tar” and CO are two important factors for cigarette design. In Table 1, we also found that only Saidi et al. (14, 17, 18) predicted the yields of CO in their model by using TG-FTIR data reported by Wojtowicz et al. (15). To the best of our knowledge, no one has yet predicted the yields of “tar” in the mathematical model of the cigarette burning process.

Therefore, in this work, we set up the kinetics of pyrolysis and char oxidation reactions by extending the heating rate to 800 K·min−1 and the oxygen-mass fraction ranging from 1% to 20%, respectively. The mathematical relationships of “tar” and CO at different temperatures and oxygen concentrations were obtained by a specifically designed tobacco pyrolysis and combustion experimental platform, in which the tobacco loading was 1 g compared to the TG experiment with a loading of mg. Thus, the experimental platform ensured that the release of “tar” and CO met the requirements of detection, and achieved data repeatability and accuracy. Furthermore, in order to predict the yields of “tar”, we applied the filtration mechanism model of aerosols in filters proposed by Du et al. (19) into the cigarette burning model. In the subsequent sections, the model is described, followed by a description of the governing equations and the boundary conditions.

Based on other contributions related to understanding and treatment of the cigarette burning process, this study aims to establish a relatively comprehensive model for prediction of the cigarette burning behavior and the yields of “tar” and CO, which includes the complex reactions, transport system, and the filtration effect of cigarette filter for “tar”. Finally, the computed results will be compared with experimental data.

MATERIALS AND METHODS
Material

The cigarette samples were obtained from China Tobacco Fujian Industrial Corporation (Xiamen, Fujian, P.R. China). For 48 h prior to analysis, the samples were conditioned in a chamber at 295 K and relative humidity of 60%.

Thermogravimetric experiment

The cut tobacco from the cigarette sample was pulverized into powder and then sifted through a 100-mesh sieve. 10 mg of tobacco powder were loaded evenly into an open ceramic pan. The pan was then placed on the sample holder of a SAT 449 F3 (Netzsch Feinmahltechnik GmbH, Selb, Germany). In the first stage, the tobacco powder was pyrolyzed under nitrogen atmosphere. The temperature was increased from room temperature to 873 K in pure nitrogen with a steady flow rate of 100 mL·min−1. The heating rates were 300 K·min−1, 400 K·min−1, 500 K·min−1, 600 K·min−1, 700 K·min−1 and 800 K·min−1, respectively. In the second stage, the obtained residual char was cooled to room temperature first, and then the mass loss of char at different oxygen concentrations was obtained by changing the concentration of oxygen in the carrier gas from 1% to 20%. The flow rate was 100 mL·min−1, and the temperature increased from room temperature to 873 K with a heating rate of 10 K·min−1.

Tobacco pyrolysis and combustion experiment

The schematic diagram of the pyrolysis and combustion experiment is shown in Figure 1. Approximately 1.0 g of tobacco sample was used. The quartz sample tube was flushed out by the carrier gas for 5 min. The applied carrier gas was 2% O2 + 98% N2, 10% O2 + 90% N2 and air, respectively. The tobacco sample was heated up from room temperature to the target temperature at the heating rate of 20 K·s−1. When the target temperature was reached, the tobacco sample was held for 5 min. The gas flow rate was maintained at 2.0 L·min−1, which proved high enough to eliminate the smoke from the tube quickly. The “tar” was trapped in a condensing system using a Cambridge filter pad, which was placed on the outlet side of the furnace, while CO was detected online by smoke analyzer (J2KN, rbr, Iserlohn, Germany). Each experiment was performed twice with good repeatability in order to obtain an average of “tar” and CO data.

Gas temperature measurement

The gas temperature measurement system is shown in Figure 2. Eight thermocouples were inserted into the center of the cigarette, their locations started at 22 mm from the lighting end of the cigarette. The temperature data at specific locations during the smoking process were collected online by software.

MODEL DESCRIPTION
Computational domains

The implementation of the model was carried out in a custom version of FLUENT's structured mesh solver, FLUENT 14.0 (28). The cigarette model was simplified as a two-dimensional model due to the cylindrical axis symmetric structure. The simulation domain was a central plane through the cigarette containing the longitudinal axis of the cigarette, as shown in Figure 3. There are four computational domains. The tobacco rod, cigarette paper and filter rod domains consisted of porous media. In the external environment, the pressure, temperature and gas composition were fixed at ambient conditions. A non-uniform structured mesh of 15854 control volumes was used in the domains, and the meshes near the surfaces were refined. Domain 1) was divided into 2790 grids (99 for the x-axis and 30 for the y-axis) while domains 2) and 3) were divided into 738 and 720 grids separately. A number of important parameters and values related to each domain are shown in Table 2.

Parameters and values related to each domain.

Domain Parameter Definition Unit Value
1) ρs0 Initial solid density kg·m−3 740 (12)
ϕ Porosity 1 0.7
Cp,s Specific heat of solid kJ·kg−1·K−1 1.043 (12)
Cp,g Specific heat of gas kJ·kg−1·K−1 1.004 (12)
ks Solid conductivity W·m−1·K−1 0.316 (12)
kg Gas conductivity W·m−1·K−1 0.0242 (12)
ɛ Emissivity of tobacco 1 0.98 (12)
dpore Pore diameter m 5.75 × 10−4 (12)
Hevaporation Water evaporation heat kJ·kg−1 −2.2572 × 103 (12)
Hcombustion Char combustion heat kJ·kg−1 1.757 × 104 (12)
v Flow velocity m/s 0
Kut Permeability of unburned tobacco m2 5.6 × 10−10 (18)
Kbt Permeability of burned tobacco m2 105 (18)
2) Kup Permeability of unburned cigarette paper m2 5 × 10−15
Kbp Permeability of burned cigarette paper m2 105 (18)
3) dp Aerosol particle diameter m 4.4 × 10−7 (19)
df Single fiber diameter m 2.51 × 10−5 (19)
Dt Total denier of filter g·(9000 m)−1 35000
Ds Denier of per single fiber g·(9000 m)−1 3
Cfiber Crimping ratio of fibers 1 0.17
Sfilter Cross-sectional area of filter rod m2 4.899 × 10−5
Tfilter Filter temperature K 288
Kfilter Permeability of filter m2 2.5 × 10−10 (18)
4) T Ambient temperature K 288
P Ambient gas pressure kPa 101.3
ρg0 Initial gas density kg·m−3 1.225
WO2 Mass fraction of O2 % 23
WN Mass fraction of N2 % 77

Domain 1) was the point source of energy and mass, which included the tobacco pyrolysis and combustion reaction kinetics. The tobacco was consumed and produced char, ash, and smoke, meanwhile releasing energy. Domain 2) affected the resistance to gas flow entering from the cigarette paper. Domain 3) covered the filtration model of the aerosols across the filter. Domain 4) is the external environment. All the exchange of heat, mass and momentum among the four domains were calculated based on the laminar flow model with energy equation and the species transport models with the effect of diffusion.

Initial, boundary, and ignition conditions

The initial pressure, temperature and gas composition were also set at ambient conditions within the entire computational domains. Tobacco and cigarette paper were initialized in the unburned state. The outside boundary of the Domain 4) was set as pressure outlet. The mouth end of the cigarette was set as the velocity inlet. The flow velocity was prescribed depending on the International Organization for Standardization smoking regime (ISO 3308) with 35-mL puffs of 2 s duration, and a rate of one puff every 60 s.

To ignite the cigarette, the temperature at the tip of the cigarette was raised to 1000 K, and 35 mL air was drawn for two seconds. After this ignition period, the pre-programmed smoking regime was applied.

Tobacco pyrolysis reaction kinetics

Accurate pyrolysis reaction kinetics are necessary to reasonably predict the yield of char, the fuel for the combustion process. The differential thermal gravity (DTG) curve of tobacco pyrolysis at 300 K·min−1 is presented in Figure 4. The pyrolysis DTG curve can be approximated by five Gaussian peaks (R1–R5), so the tobacco is considered as consisting of five precursors. They are moisture, volatiles, hemicellulose, cellulose and lignin, which is consistent with the results reported in (20). According to the area percentages, the mass fractions fp,j for each pre cursor are listed in Table 3.

Kinetic parameters of tobacco pyrolysis.

Parameters Unit R1 R2 R3 R4 R5
fp,j % 9.52 17.71 18.04 13.58 41.16
Ap,j min−1 1.47 × 105 1.48 × 108 1.82 × 1010 1.21 × 1013 0.4538
Ep,j kJ·mol−1 31.09 60.81 91.48 133.48 25.78
np,j 1.06 1.28 1.21 1.25 0.76
mp,j 1.24 1.54 1.48 1.49 −0.04
R2 = 0.9821

The pyrolysis of tobacco can be regarded as the parallel reactions of five precursors (21, 22). The pyrolysis kinetics of each precursor are expressed by Arrhenius equation: $∂αp,j∂t=Ap,jexp(−Ep,jRTs)(1−αp,j)np,j$ {{\partial {\alpha _{p,j}}} \over {\partial t}} = {A_{p,j}}\exp \left( { - {{{E_{p,j}}} \over {R{T_s}}}} \right){\left( {1 - {\alpha _{p,j}}} \right)^{{n_{p,j}}}} where αp,j (%) is the conversion ratio of the jth precursor, t (min) is the time, Ap,j (min−1) is the jth reaction pre-exponential factor, Ep,j (kJ·mol−1) is the jth reaction activation energy, Ts (K) is the solid temperature, R (8.314 J·mol−1·K−1) is the universal gas constant, np,j is jth reaction order.

For reactions that are heterogeneous and non-isothermal with β = dTs / dt, equation [1] can be rewritten as: $∂αp,j∂Ts=Ap,jβexp(−Ep,jRTs)(1−αp,j)np,j$ {{\partial {\alpha _{p,j}}} \over {\partial {T_s}}} = {{{A_{p,j}}} \over \beta }\exp \left( { - {{{E_{p,j}}} \over {R{T_s}}}} \right){\left( {1 - {\alpha _{p,j}}} \right)^{{n_{p,j}}}} where β (K·min−1) is the heating rate. A wide range of heating rates was applied, but mostly at much lower heating rates (10 K·min−1 – 100 K·min−1) than are actually present in a cigarette during smoldering and puffing. Therefore, the heating rate range was extended to 800 K·min−1 in this study. Figure 5 shows the approximated Gaussian peaks of five precursors at different heating rates. As seen from it, the effects of heating rate on pyrolysis reactions of five precursors are not the same. Hence, the pyrolysis kinetics of each precursor can be expressed by a modified Arrhenius which contains a calibration factor of the heating rate mp,j (20): $∂αp,j∂Ts=Ap,jβmp,jexp(−Ep,jRTs)(1−αp,j)np,j$ {{\partial {\alpha _{p,j}}} \over {\partial {T_s}}} = {{{A_{p,j}}} \over {{\beta ^{{m_{p,j}}}}}}\exp \left( { - {{{E_{p,j}}} \over {R{T_s}}}} \right){\left( {1 - {\alpha _{p,j}}} \right)^{{n_{p,j}}}} The conversion ratio of the jth precursor αp,j (%) is defined as: $αp,j=Wp,j0−Wp,jfp,j(Wp0−Wpf)$ {\alpha _{p,j}} = {{{W_{p,j0}} - {W_{p,j}}} \over {{f_{p,j}}\left( {{W_{p0}} - {W_{pf}}} \right)}} where the initial mass percentage of the jth precursor is Wp,j0 = 100%, the initial mass percentage of tobacco is Wp0 = 100%, the final mass percentage of tobacco is Wpf = 30.83%, measured by the TG experiment of tobacco pyrolysis, Wp,j (%) is the mass percentage of the jth precursor.

Based on equation [4], the mass loss rate of each precursor can be given by: $∂Wp,j∂Ts=−fp,j(Wp0−Wpf)∂αp,j∂Ts=−fp,j(Wp0−Wpf)Ap,jβmp,jexp(−Ep,jRTs)(1−αp,j)np,j$ {{\partial {W_{p,j}}} \over {\partial {T_s}}} = - {f_{p,j}}\left( {{W_{p0}} - {W_{pf}}} \right){{\partial {\alpha _{p,j}}} \over {\partial {T_s}}} = - {f_{p,j}}\left( {{W_{p0}} - {W_{pf}}} \right){{{A_{p,j}}} \over {{\beta ^{{m_{p,j}}}}}}\exp \left( { - {{{E_{p,j}}} \over {R{T_s}}}} \right){\left( {1 - {\alpha _{p,j}}} \right)^{{n_{p,j}}}} The overall mass loss rate would be the sum of each precursor mass loss rate: $∂Wp∂Ts=∑j=15∂Wp,j∂Ts$ {{\partial {W_p}} \over {\partial {T_s}}} = \sum\nolimits_{j = 1}^5 {{{\partial {W_{p,j}}} \over {\partial {T_s}}}} where Wp (%) is the mass percentage of tobacco.

Inserting equation [5] into equation [6]: $∂Wp∂Ts=−∑j=15fp,j(Wp0−Wpf)Ap,jβmp,jexp(−Ep,jRTs)(1−αp,j)np,j$ {{\partial {W_p}} \over {\partial {T_s}}} = - \sum\nolimits_{j = 1}^5 {{f_{p,j}}\left( {{W_{p0}} - {W_{pf}}} \right){{{A_{p,j}}} \over {{\beta ^{{m_{p,j}}}}}}\exp \left( { - {{{E_{p,j}}} \over {R{T_s}}}} \right){{\left( {1 - {\alpha _{p,j}}} \right)}^{{n_{p,j}}}}} In order to determine the kinetic parameters of each precursor, denoting the experimental data by $(∂Wp∂Ts)exp$ {\left( {{{\partial {W_p}} \over {\partial {T_s}}}} \right)^{exp }} and the calculated data by $(∂Wp∂Ts)calc$ {\left( {{{\partial {W_p}} \over {\partial {T_s}}}} \right)^{calc}} which can be calculated by equations [7] and equation [8]: $αp,ji+1=αp,ji−(∂Wp,ji∂Ts)calc∂Tsfp,j(Wp0−Wpf)(i=1,2,….N)$ {\alpha _{p,ji + 1}} = {\alpha _{p,ji}} - {{{{\left( {{{\partial {W_{p,ji}}} \over {\partial {T_s}}}} \right)}^{calc}}\partial {T_s}} \over {{f_{p,j}}\left( {{W_{p0}} - {W_{pf}}} \right)}}(i = {1},{2}, \ldots .N) where N is the number of experimental points, the initial conversion ratio of the jth precursor αp,j1 (%) is assumed to be 0%.

The unknown kinetic parameters (Ap,j, Ep,j, np,j, and mp,j) are searched at which the determination coefficient (R2) is maximal. $R2=1−∑1N((∂Wp∂Ts)exp−(∂Wp∂Ts)calc)2∑1N((∂Wp∂Ts)exp−(∂Wp∂Ts)exp¯)2$ {R^2} = 1 - {{\sum\nolimits_1^N {{{\left( {{{\left( {{{\partial {W_p}} \over {\partial {T_s}}}} \right)}^{{exp}}} - {{\left( {{{\partial {W_p}} \over {\partial {T_s}}}} \right)}^{calc}}} \right)}^2}} } \over {\sum\nolimits_1^N {{{\left( {{{\left( {{{\partial {W_p}} \over {\partial {T_s}}}} \right)}^{{exp}}} - \overline {{{\left( {{{\partial {W_p}} \over {\partial {T_s}}}} \right)}^{{exp}}}} } \right)}^2}} }} The corresponding kinetic parameters are listed in Table 3. The experimental DTG curves and the fitted DTG curves of tobacco pyrolysis at different heating rates are shown in Figure 6, and they are in a good agreement.

Char combustion reaction kinetics

The char is formed through the pyrolysis of tobacco and consumed in an oxidation reaction. Reasonable char combustion reaction kinetics are critical for the prediction of the heat generation as they are the driving force for the flameless combustion process.

The char combustion kinetics can be expressed by the Arrhenius equation with β = dTs/dt: $∂αc∂Ts=Acβexp(−EcRTs)(1−αc)nc(ρO2)no=Acβexp(−EcRTs)(1−αc)nc(ρg0WO2)no$ {{\partial {\alpha _c}} \over {\partial {T_s}}} = {{{A_c}} \over \beta }\exp \left( { - {{{E_c}} \over {R{T_s}}}} \right){\left( {1 - {\alpha _c}} \right)^{{n_c}}}{\left( {{\rho_{{O_2}}}} \right)^{{n_o}}} = {{{A_c}} \over \beta }\exp \left( { - {{{E_c}} \over {R{T_s}}}} \right){\left( {1 - {\alpha _c}} \right)^{{n_c}}}{\left( {{\rho _{g0}}{W_{{O_2}}}} \right)^{{n_o}}} where αc (%) is the conversion ratio of char, ρO2 (kg·m−3) is the oxygen density in gas mixture, ρg0 (kg·m−3) is the initial gas density, WO2 (%) is the mass fraction of oxygen, Ac (min−1) is the char oxidation reaction pre-exponential factor, Ec (kJ·mol−1) is the char oxidation reaction activation energy, nc is the reaction order of char, which is assumed to be 1, no is the reaction order of oxygen.

The conversion ratio of char αc (%) is defined as: $αc=Wc0−WcWc0−Wcf$ {\alpha _c} = {{{W_{c0}} - {W_c}} \over {{W_{c0}} - {W_{cf}}}} where the initial mass percentage of char is Wc0 = 100%, the final mass percentage of char is Wcf = 52.93%, measured by the TG experiment of char combustion, Wc (%) is the mass percentage of char.

Based on equation [11], the mass loss rate of char can be given by: $∂Wc∂Tc=−(Wc0−Wcf)∂αc∂Ts=−(Wc0−Wcf)Acβexp(−EcRTs)(1−αc)(ρg0WO2)no$ \matrix{{{\partial {W_c}} \over {\partial {T_c}}} &= - \left( {{W_{c0}} - {W_{cf}}} \right){{\partial {\alpha _c}} \over {\partial {T_s}}} \hfill\cr &= - \left( {{W_{c0}} - {W_{cf}}} \right){{{A_c}} \over \beta }\exp \left( { - {{{E_c}} \over {R{T_s}}}} \right)\left( {1 - {\alpha _c}} \right){\left( {{\rho _{g0}}{W_{{O_2}}}} \right)^{{n_o}}}} Inside a burning cigarette, the supply rate of oxygen is the dominant factor in determining the rate of char combustion and heat generation. Therefore, the char combustion kinetics need to be set up at different oxygen mass fractions WO2 (%). In this study, WO2 = 1%, 2%, 3%, 5%, 10%, 15% and 20% were considered.

In order to determine the kinetic parameters of char, denoting the experimental data by $(∂Wc∂Ts)exp$ {\left( {{{\partial {W_c}} \over {\partial {T_s}}}} \right)^{exp }} and the calculated data by $(∂Wc∂Ts)calc$ {\left( {{{\partial {W_c}} \over {\partial {T_s}}}} \right)^{calc}} which can be calculated by equation [12] and equation [13]: $αc,i+1=αc,i−(∂Wc,i∂Ts)calc∂TsWc0−Wcf(i=1,2,….N)$ {\alpha _{c,i + 1}} = {\alpha _{c,i}} - {{{{\left( {{{\partial {W_{c,i}}} \over {\partial {T_s}}}} \right)}^{{\rm{calc}}}}\partial {T_s}} \over {{W_{c0}} - {W_{cf}}}}(i = {1},{2}, \ldots .N) where the initial conversion ratio of char αc,1 (%) is assumed to be 0%.

The unknown kinetic parameters (Ac, Ec, and nO2 are searched at which the determination coefficient (R2) is maximal. However, the fitting results showing the reaction order of oxygen cannot be the same at different WO2. Therefore, the oxygen mass fraction is divided into three ranges (0% ≤ WO2 ≤ 2%, 2% < WO2 ≤ 10% and 10% < WO2 ≤ 23%) and the corresponding kinetic parameters are listed in Table 4.

Kinetic parameters of char combustion.

WO2 range 0% ≤ WO2 ≤ 2% 2 %< WO2 ≤ 10% 10% < WO2 ≤ 23%

Parameters Unit WO2 = 1%, 2% WO2 = 3%, 5%, 10% WO2 = 15%, 20%
Ac min−1 1.48 × 107 4.26 × 107 8.30 × 107
Ec kJ·mol−1 91.04 111.20 116.31
no 1.09 0.43 0.36
R2 0.9441 0.9574 0.9537

The experimental DTG curves and the fitted DTG curves of char combustion at different oxygen concentrations are shown in Figure 7, and they are in a reasonable agreement.

Transport system
Mass transport

The source terms of solid and gas species required in the mass transport equations could be obtained by calculating their density change rates.

The source term of all precursors, Sprecursors (kg·m−3·min−1), was calculated as the sum of five precursors consumed during the tobacco pyrolysis reaction. $Sprecursors=∂ρp∂t∑j=15∂ρp,j∂t$ {S_{precursors}} = {{\partial {\rho _p}} \over {\partial t}}\sum\nolimits_{j = 1}^5 {{{\partial {\rho _{p,j}}} \over {\partial t}}} where ρp (kg·m−3) is the total density of five precursors, ρp,j (kg·m−3) is the density of jth precursor, which can be calculated by: $ρp,j=fp,jρs0(1−αp,j)$ {\rho _{p,j}} = {f_{p,j}}{\rho _{s0}}\left( {1 - {\alpha _{p,j}}} \right) Based on equation (15), the density change rate of each precursor can be written as: $∂ρp,j∂t=−fp,jρs0∂αp,j∂t=−βfp,jρs0∂αp,j∂t$ {{\partial {\rho _{p,j}}} \over {\partial t}} = - {f_{p,j}}{\rho _{s0}}{{\partial {\alpha _{p,j}}} \over {\partial t}} = - \beta {f_{p,j}}{\rho _{s0}}{{\partial {\alpha _{p,j}}} \over {\partial t}} where $∂αp,j∂Ts=Ap,jβmp,jexp(−Ep,jRTs)(1−αp,j)np,jsee [3]$ \matrix{ {{{\partial {\alpha _{p,j}}} \over {\partial {T_s}}} = {{{A_{p,j}}} \over {{\beta ^{{m_{p,j}}}}}}\exp \left( { - {{{E_{p,j}}} \over {R{T_s}}}} \right){{\left( {1 - {\alpha _{p,j}}} \right)}^{{n_{p,j}}}}} \hfill & {{\rm{see\;[3]}} } \hfill \cr } Therefore, the source term of all precursors is solved in the following form: $Sprecursors=−βρs0∑j=15fp,jAp,jβmp,jexp(−Ep,jRTs)(ρp,jfp,jρs0)np,j$ S_{precursors} = - \beta {\rho _{s0}}\sum\nolimits_{j = 1}^5 {{f_{p,j}}{{{A_{p,j}}} \over {{\beta ^{{m_{p,j}}}}}}\exp \left( { - {{{E_{p,j}}} \over {R{T_s}}}} \right){{\left( {{{{\rho _{p,j}}} \over {{f_{p,j}}{\rho _{s0}}}}} \right)}^{{n_{p,j}}}}} Char is primarily formed through the pyrolysis reaction and then consumed in combustion reaction, therefore the source term of char, Schar (kg·m−3·min−1), consists of two terms, as shown in equation [18]. The first term represents the rate of formation, while the second term indicates the rate of consumption. $Schar=∂ρc′∂t=−fc∂ρp∂t+∂ρc∂t$ {S_{char}} = {{\partial {\rho _c^\prime}} \over {\partial t}} = - {f_c}{{\partial {\rho _p}} \over {\partial t}} + {{\partial {\rho _c}} \over {\partial t}} where $ρc′$ \rho _c^\prime (kg·m−3) is the overall char density, fc = 30.83% is the residual mass fraction of tobacco pyrolysis reaction measured by the TG experiment, and ρc (kg·m−3) is the combustible char density, which can be calculated by: $ρc=ρc,t(1−αc)$ {\rho _c} = {\rho _{c,t}}(1 - {\alpha _c}) where the total combustible char density after complete pyrolysis is ρc,t = fc ρs0 (kg·m−3).

Based on equation [19], the combustible char density change rate can be written as: $∂ρc∂t=−ρc,t∂αc∂t=−βρc,t∂αc∂Ts$ {{\partial {\rho _c}} \over {\partial t}} = - {\rho _{c,t}}{{\partial {\alpha _c}} \over {\partial t}} = - \beta {\rho _{c,t}}{{\partial {\alpha _c}} \over {\partial {T_s}}} where $∂αc∂Ts=Acβexp(−EcRTs)(1−αc)nc(ρO2)nosee [10]$ \matrix{ {{{\partial {\alpha _c}} \over {\partial {T_s}}} = {{{A_c}} \over \beta }\exp \left( { - {{{E_c}} \over {R{T_s}}}} \right){{\left( {1 - {\alpha _c}} \right)}^{{n_c}}}{{\left( {{\rho _{{O_2}}}} \right)}^{{n_o}}}} \hfill & {{\rm{see\;[10]}}} \hfill \cr } Finally, the source term of char is solved in the following form: $Schar=−fc∂ρp∂t−βρc,tAcβexp(−EcRTs)ρcρc,t(ρO2)no$ {S_{char}} = - {f_c}{{\partial {\rho _p}} \over {\partial t}} - \beta {\rho _{c,t}}{{{A_c}} \over \beta }\exp \left( { - {{{E_c}} \over {R{T_s}}}} \right){{{\rho _c}} \over {{\rho _{c,t}}}}{\left( {{\rho _{{O_2}}}} \right)^{{n_o}}} where $ρc=ρc,last'−fc∂ρp∂t∂t$ {\rho _c} = \rho _{c,last}^\prime - {f_c}{{\partial {\rho _p}} \over {\partial t}}\partial t $ρc,last′$ \rho _{c,last}^\prime (kg·m−3) is the overall char density at the last moment calculated by FLUENT, ρO2 (kg·m−3) is the oxygen density in the gas mixture.

From equation [21], we conclude that in order to calculate the rate of char combustion, the oxygen distribution needs to be known. Here, a single-step oxidation reaction leading to product species is assumed as proposed by Rostami et al. (12). Oxygen is the gas phase species being transported. The source term of oxygen, Soxygen (kg·m−3·min−1) can be directly obtained from the stoichiometric ratio of the combustible char and oxygen. $Soxygen=∂ρO2∂t=nO2'∂ρc∂t=−nO2'Acexp(EcRTs)ρc(ρO2)no$ {{S_{oxygen}} = {{\partial {\rho _{{O_2}}}} \over {\partial t}} = n_{{O_2}}^\prime{{\partial {\rho _c}} \over {\partial t}} = }{ - n_{{O_2}}^\prime{A_c}\exp \left( {{{{E_c}} \over {R{T_s}}}} \right){\rho _c}{{\left( {{\rho _{{O_2}}}} \right)}^{{n_o}}}} where the stoichiometric coefficient for oxygen consumption is $nO2′=1.65$ n_{{O_2}}^\prime = 1.65 (12).

Since the transport of oxygen depends critically on its ability to diffuse through the gas phase, it is important to include the correct dependence of diffusivity on temperature. The oxygen diffusivity is taken from (12): $D=D0(Tg273)1.75$ D = {D_0}{\left( {{{{T_g}} \over {273}}} \right)^{1.75}} $D0=0.677DgΦ1.18$ {D_0} = 0.677{D_g}{\Phi ^{1.18}} where D0 (m2·s−1) is the mass diffusivity in the porous media at 273 K and 101.3 kPa, Tg (K) is the gas temperature, the oxygen diffusion in nitrogen is Dg = 2 ×10−5 m2·s−1 and a tobacco rod filling with a total void fraction is Φ = 0.85 (12). Ash is produced by the char combustion reaction. The source term of ash, Sash (kg·m−3·min−1), can be written as: $Sash=∂ρash∂t=−fash∂ρc∂t=fashAcexp(−EcRTs)ρc(ρO2)no$ {{S_{ash}} = {{\partial {\rho _{ash}}} \over {\partial t}} = - {f_{ash}}{{\partial {\rho _c}} \over {\partial t}} }= {{f_{ash}}{A_c}\exp \left( { - {{{E_c}} \over {R{T_s}}}} \right){\rho _c}{{\left( {{\rho _{{O_2}}}} \right)}^{{n_o}}}} where ρash (kg·m−3) is the ash density, and fash = 52.93% is the residual mass fraction of char combustion measured by the TG experiment.

Using the tobacco pyrolysis and combustion experimental platform, the released amounts of “tar” and CO at different reaction conditions were obtained, as shown in Figure 8. Based on the data, the mathematical relationships of “tar” and CO at different temperatures and oxygen mass fractions were set up, as shown in Table 5.

Mathematical relationships of “tar” and CO at different temperatures and oxygen mass fractions.

 Temperature range WO2 range “Tar” (mg·g−1) 423 K ≤ T ≤ 623 K 0% ≤ WO2 < 5% Y1 = −0.0049T2 + 5.8838T−1617 (R2 = 0.8990) 5% ≤ WO2 < 15% Y1 = −0.0048T2 + 5.7621T−1580 (R2 = 0.9034) 15% ≤ WO2 ≤ 23% Y1 = −0.0061T2 + 7.1210T−1911 (R2 = 0.9734) 623 K T ≤ 1273 K 0% ≤ WO2 ≤ 23% 160 Temperature range WO2 range CO (mg·g−1) 423 K ≤ T ≤ 1273 K 0% ≤ WO2 < 5% Y2 = 0.2132T−89.19 (R2 = 0.9575) 5% ≤ WO2 < 15% Y2 = 0.2630T−107.32 (R2 = 0.9608) 15% ≤ WO2 ≤ 23% Y2 = 0.2516T−90.41 (R2 = 0.9333)

According to Y1 (mg·g−1 = 10−3) and Y2 (mg·g−1=10−3), the source terms of “tar”, Star (kg·m−3·min−1), and CO, SCO (kg·m−3·min−1), are given, respectively: $S“tar”=−10−3Y1∂ρs∂t$ {S_{"tar"}} = - {10^{ - 3}}{Y_1}{{\partial {\rho _s}} \over {\partial t}} $SCO=−10−3Y2∂ρs∂t$ {S_{CO}} = - {10^{ - 3}}{Y_2}{{\partial {\rho _s}} \over {\partial t}} where: $∂ρs∂t=∂ρp∂t+∂ρc'∂t+∂ρash∂t$ {{\partial {\rho _s}} \over {\partial t}} = {{\partial {\rho _p}} \over {\partial t}} + {{\partial \rho _c^\prime} \over {\partial t}} + {{\partial {\rho _{ash}}} \over {\partial t}} ρs (kg·m−3) is the density of solid species including five precursors, char and ash.

Energy transport

The solid and gas phase energy equation are: $∂((1−φ)ρsCp,sTs)∂t=∇→[ks,eff∇→Ts]+hAv(Tg−Ts)+Ssolid$ {{\partial \left( {\left( {1 - \varphi } \right){\rho _s}{C_{p,s}}{T_s}} \right)} \over {\partial t}} = \vec \nabla \left[ {{k_{s,eff}}\vec \nabla {T_s}} \right] + h{A_v}\left( {{T_g} - {T_s}} \right) + {S_{solid}} $∂(φρgCp,gTg)∂t+∇→[ρgCp,gTgv→]=∇→[kg,eff∇→Tg]+hAv(Ts−Tg)$ {{\partial \left( {\varphi {\rho _g}{C_{p,g}}{T_g}} \right)} \over {\partial t}} + \vec \nabla \left[ {{\rho _g}{C_{p,g}}{T_g}\vec v} \right] = \vec \nabla \left[ {{k_{g,eff}}\vec \nabla {T_g}} \right] + h{A_v}\left( {{T_s} - {T_g}} \right) respectively, where the porosity of tobacco rod is assumed to be a constant, ϕ = 0.7, ρg (kg·m−3) is the gas density, Cp,s (kJ·kg−1·K−1) is the specific heat of the solid phase, Cp,g (kJ·kg−1·K−1) is the specific heat of the gas phase. kg,eff (W·m−1·K−1) and ks,eff (W·m−1·K−1) are the effective heat conductivity in the gas phase / solid phase, which can be given by: $kg,eff=φkg$ {k_{g,eff}} = \varphi {k_g} $ks,eff=(ks+4εσTs3dpore)(1−φ)$ {k_{s,eff}} = \left( {{k_s} + 4\varepsilon \sigma T_s^3{d_{pore}}} \right)(1 - \varphi ) where kg (W·m−1·K−1) is the gas conductivity, ks (W·m−1·K−1) is the solid conductivity, ɛ is the emissivity of tobacco, dpore (m) is the pore diameter, Stefan-Boltzmann constant is σ =5.67 × 10−8 W·m−2·K−4.

The solid-gas interfacial area to volume is given as: $Av=AV=6φdpore$ {A_v} = {A \over V} = {{6\varphi } \over {{d_{pore}}}} h is the solid-gas heat transfer coefficient, which can be calculated by the following equation: $Nu=hdtkg=2+1.1 Re0.6Pr0.333$ Nu = {{h{d_t}} \over {{k_g}}} = 2 + 1.1\,{Re}^{0.6}{{Pr}^{0.333}} where Nu is the Nußelt number, Re is the Reynolds number, and Pr is the Prandtl number.

The heat source term of solid Ssolid represents the sum of heat of evaporation and char combustion. The heat of pyrolysis reaction was assumed to be very small and did not play a significant role: $Ssolid=Hevaporationdρwdt+Hcombustiondρcdt$ {S_{solid}} = {H_{evaporation}}{{d{\rho _w}} \over {dt}} + {H_{combustion}}{{d{\rho _c}} \over {dt}} where Hevaporation (kJ·kg−1) is the water evaporation heat and Hcombustion (kJ·kg−1) is the char combustion heat.

Momentum transport

The tobacco rod and cigarette paper are considered as porous media with known permeability. The source term of momentum represents the added pressure drop due to the presence of solid phase, which is given by: $S→momentum=−(μgv→Kmedia+12Cρs|v→|v→)$ {\vec S_{momentum}} = - \left( {{{{\mu _g}\vec v} \over {{K_{media}}}} + {1 \over 2}C{\rho _s}\left| {\vec v} \right|\vec v} \right) where $v→(m⋅s−1)$ \overrightarrow {\rm{v}} \left( {{\rm{m}} \cdot {{\rm{s}}^{ - 1}}} \right) is the specific velocity C is an empirical constant governing the magnitude of the inertial term, which is assumed zero.

μg (kg·s−1·m−1) is the gas dynamic viscosity, the effect of temperature on μg is accounted by using Sutherland's two-coefficient law: $μg=C1Tg32Tg+C2$ {\mu _g} = {{{C_1}T_g^{{3 \over 2}}} \over {{T_g} + {C_2}}} where C1 = 1.458 × 10−6 kg·s−1·m−1·K1/2 and C2 = 110.4 K (14).

Kmedia (m2) is the permeability of the porous media. For the permeability of tobacco Kt (m2), it was assumed to change linearly with the density of the unburned tobacco: $Kt=Kut(1−g)+Kbtg$ {K_t} = {K_{ut}}\left( {1 - g} \right) + {K_{bt}}g $g=−ρs−ρs0ρs0$ g = - {{{\rho _s} - {\rho _{s0}}} \over {{\rho _{s0}}}} where g is an interpolation factor, Kut (m2) is the permeability of unburned tobacco, Kbt (m2) is the permeability of burned tobacco.

Theoretical filtration model of aerosols across the filter

For single fibers, the filtration efficiency was calculated by the filtration mechanism model of aerosols in filters proposed by Du et al. (19), which includes four filtration efficiencies, such as the interception efficiency (EIN), the inertial impaction efficiency (EIM), the diffusion impaction efficiency (ED), and diffusion-inertial impaction efficiency (EID).

Interception efficiency (EIN) can be calculated by: $EIN=f(G)2Ku$ {E_{IN}} = {{f(G)} \over {2{K_u}}} $f(G)=11+G−(1+G)+2(1+G)ln(1+G)$ f(G) = {1 \over {1 + G}} - \left( {1 + G} \right) + 2\left( {1 + G} \right)\ln \left( {1 + G} \right) $G=dpdf$ G = {{{d_p}} \over {{d_f}}} $Ku=−lnαf2−34+αf−αf24$ {K_u} = - {{\ln {\alpha _f}} \over 2} - {3 \over 4} + {\alpha _f} - {{\alpha _f^2} \over 4} $αf=Dtπdf2(1+Cfiber)4DsSfilter$ {\alpha _f} = {{{D_t}\pi d_f^2\left( {1 + {C_{fiber}}} \right)} \over {4{D_s}{S_{filter}}}} where G is the dimensionless interception parameter, Ku is the Kuwabara hydrodynamic factor, αf is the solid fraction of filter, dp (m) is the particle diameter, df (m) is the single fiber diameter, Dt is the total denier of filter, Ds is the denier of per single fiber, Cfiber is the crimping ratio of fibers, Sfilter (m2) is the cross-sectional area of filter rod. Inertial impaction efficiency (EIM) can be calculated by: $EIM=(Stk)J2Ku2$ {E_{IM}} = {{\left( {Stk} \right)J} \over {2K_u^2}} $Stk=ρgdp2v18μgdf$ Stk = {{{\rho _g}d_p^2v} \over {18{\mu _g}{d_f}}} $J=(29.6−28αf0.62)G2−27.5G2.8$ J = \left( {29.6 - 28\alpha _f^{0.62}} \right){G^2} - 27.5{G^{2.8}} where Stk is the Stokes number, v (m·s−1) is the gas velocity, μg (kg·s−1·m−1) is the gas dynamic viscosity, which can be calculated by equation [38].

Diffusion impaction efficiency (ED) can be calculated by: $ED=2.7Pe23$ {E_D} = 2.7P{e^{{2 \over 3}}} $Pe=dfvDp$ Pe = {{{d_f}v} \over {{D_p}}} $Dp=KBTfilter3πμgdp$ {D_p} = {{{K_B}{T_{filter}}} \over {3\pi {\mu _g}{d_p}}} where Pe is the Peclet number, Dp (m2·s−1) is the particle diffusion coefficient, the Stefan-Boltzmann constant is KB = 1.38 × 10−23 J·K−1, Tfilter (K) is the temperature of filter. Diffusion-inertial impaction efficiency (EID) can be calculated by: $EID=1.23G23(KuPe)12$ {E_{ID}} = {{1.23{G^{{2 \over 3}}}} \over {{{\left( {{K_u}Pe} \right)}^{{1 \over 2}}}}} The combined single fiber efficiency can be calculated as equation: $εs=EIN+EIM+ED+EID$ {\varepsilon _s} = {E_{IN}} + {E_{IM}} + {E_D} + {E_{ID}} The overall filtration efficiency of a filter can be calculated from the single fiber efficiency by equation [23]: $ε=1−exp(−4αfεsLfπ(1−αf)df)$ \varepsilon = 1 - \exp \left( { - {{4{\alpha _f}{\varepsilon _s}{L_f}} \over {\pi \left( {1 - {\alpha _f}} \right){d_f}}}} \right) where Lf is the filter length, 0.027 m. The overall filtration efficiency at different positions of the filter can be calculated by: $dεdLx=4αfεsπ(1−αf)dfexp(−4αfεsLxπ(1−αf)df)$ {{d\varepsilon } \over {d{L_x}}} = {{4{\alpha _f}{\varepsilon _s}} \over {\pi \left( {1 - {\alpha _f}} \right){d_f}}}\exp \left( { - {{4{\alpha _f}{\varepsilon _s}{L_x}} \over {\pi \left( {1 - {\alpha _f}} \right){d_f}}}} \right) Therefore, the source term of “tar” in the filter, $S“tar”′$ S_{"tar"}^\prime (kg·m−3·min−1) can be written as: $S“tar”'=f“tar”ρgvdεdLx=f“tar”ρgv4αfεsπ(1−αf)dfexp(−4αfεsLxπ(1−αf)df)$ \matrix{S_{"tar"}^\prime &= {f_{"tar"}}{\rho _g}v{{d\varepsilon } \over {d{L_x}}} \hfill\cr &= {f_{"tar"}}{\rho _g}v{{4{\alpha _f}{\varepsilon _s}} \over {\pi \left( {1 - {\alpha _f}} \right){d_f}}}\exp \left( { - {{4{\alpha _f}{\varepsilon _s}{L_x}} \over {\pi \left( {1 - {\alpha _f}} \right){d_f}}}} \right) \hfill\cr} The source term of “tar” in the combustion of cigarette has been established in Section 3.5.1 as: $S“tar”=−10−3Y1∂ρs∂tsee [27]$ {S_{"tar"}} = - {10^{ - 3}}{Y_1}\matrix{{{{\partial {\rho _s}} \over {\partial t}}} \hfill & {{\rm{see\;[27]}}} \hfill \cr } According to the source term equations, the flow of “tar” and air over the entire cross section of the cigarette at the inlet of the filter can be obtained in FLUENT. The ratio of the two is taken as the mass fraction of the “tar” at the inlet of the filter, ftar (%).

The filtration efficiency of filter can be calculated by using the following equation: $η=min−moutmin$ \eta = {{{m_{in}} - {m_{out}}} \over {{m_{in}}}} where min (mg) is the “tar” released at the inlet of the filter before puffing, mout (mg) is the “tar” released at the outlet of the filter after puffing.

Numerical method

The four domains were discretized into structured control volumes over which the conservation equations for mass, momentum, energy and chemical species were discretized. A non-uniform structured mesh of 15854 control volumes was used in the domains, and the meshes near the surfaces were refined. All the exchange of heat, mass and momentum among the four domains were based on the laminar flow model with the energy equation and species transport models with the effect of diffusion. Standard second-order spatial discretization schemes were used for convective operators, with a second-order upwind discretization of the diffusion terms. Based on that, the gaseous emission towards the environment was rapid compared with the emission within the cigarette. The cigarette burning process was assumed to be free of gravity effects which allowed us to use symmetrical conditions, hence considerably shortening the computation time.

The length of the time interval was controlled by the time scale of the reaction. The number of time steps was set to 60,000, with a relative time step size of 0.01 s. Typically 10 iterations per time step were required. The discretization of the unsteady terms was carried out using a second-order implicit scheme.

In FLUENT, the Navier-Stokes equation was solved for conservation of mass, momentum and energy and other scalars such as turbulence and the chemical species using a pressure-based solver. A pressure-based solver requires pressure-velocity coupling, thus the Semi-Implicit Method for Pressure Linked equations (SIMPLE) (24) algorithm was recommended in order to improve the rate of convergence and to reduce simulation time. Basically, the governing equations were coupled to each other in a manner that the solution process required iterations wherein the entire set of governing equations were solved repeatedly until the solution converged.

All equations were solved sequentially and iteratively in keeping with the FLUENT algorithms. The equations described above were incorporated through the user-subroutines available in FLUENT, though some manipulations were not possible through these subroutines and had to be carried out by making changes to the source.

RESULTS AND DISCUSSIONS
Permeability of cigarette paper

The permeability of cigarette paper plays an important and critical role during the cigarette burning process, especially the permeability of the cigarette paper behind the char line, which determines the resistance to the gas flow entering from the cigarette paper during puffing and consequently the char combustion rate.

The permeability of the cigarette paper increased almost exponentially with the temperature (25). We have reported that the pyrolysis temperature range of cigarette paper was 473 K – 623 K, and the temperature of the cigarette paper behind the char line could reach 623 K (26). Normally, the permeability of cigarette paper is measured at room temperature, but the actual permeability of the cigarette paper behind the char line was difficult to obtain. Hence, in this model, when the temperature of the cigarette paper was < 473 K, the permeability of the unburned cigarette paper was Kup = 5 × 10−15 m2, (≈ 30 CU, the thickness of the cigarette paper was about 0.06 mm), when the temperature of the cigarette paper was > 623 K, the permeability of burned cigarette paper was Kbp = 105 m2 (18). The cigarette paper behind the char line was set locating at the position where the temperature ranged from 473 K to 623 K, and its permeability was Kp.

In order to obtain the proper Kp in this model, four cases were considered, Kp = 5 × 10−15 m2, 10−9 m2, 1.5 × 10−9 m2 and 10−3 m2, namely cases 1, 2, 3 and 4. Figure 9 shows the char density fields in the four cases during puffing (240 s, 241 s, and 242 s). The development of the coal shape could be checked visually to verify the adequacy of the simulations. When Kp = 5 × 10−15 m2, most char was consumed and could not form the combustion coal, resulting from most of the air flowing into the combustion center, leading to the acceleration of the char combustion reaction. In contrast, when Kp = 10−3 m2, the combustion coal was more obtuse and the shape nearly remained the same, which was due to most of the air entering from the cigarette paper behind the char line, resulting in the deceleration of the char combustion reaction.

These results show the two cases are unrealistic. When Kp = 10−9 m2 and 1.5 × 10−9 m2, the shapes of combustion coals were conical, which was similar to the realistic situation.

The proper value of Kp was determined by matching the numerical and experimental results. Table 6 shows the numerical and experimental yields of “tar” and CO in the mainstream. It can be seen that case 3 has produced a good match with the experimental data. The predicted yields of “tar” and CO were 12.2 mg/cig and 14.6 mg/cig, and the experimental values were 11.2 mg/cig and 13.2 mg/cig, respectively, with relative deviations of 8.9% and 10.6%, respectively. Therefore, Kp = 1.5 × 10−9 m2 was used in the numerical simulations.

Comparison of the numerical and experimental results.

Case 1 Case 2 Case 3 Case 4 Experimental
“Tar” (mg/cig) 15.8 14.1 12.2 5.1 11.2
Relative deviation 41.1% 25.9% 8.9% 54.5%
CO (mg/cig) 21.1 17.3 14.6 5.6 13.2
Relative deviation 59.8% 31.1% 10.6% 57.6%

In practice, a full comparison of the experimental results with all predictions of the model is difficult. In order to check the validity of the proposed model, five criteria were chosen. These were: puff number, temperatures, flow velocity, filtration efficiency of the filter and the yields of “tar” and CO under different puff intensities.

Validity of puff number

Figure 10 shows the development of char density with time in case 3. We see that the combustion cone has formed at 60 s. As the combustion progresses, the shape of the combustion center becomes gradually more conical and moves in the smoldering direction, indicating that the cigarette can keep burning.

Figure 11 shows the permeability changes of the cigarette paper at different times in case 3. It can be noticed that the changes in permeability of the cigarette paper were closely synchronized with the changes in char density, which indicates setting Kp = 1.5 × 10−9 m2 when the surface temperature in the range of 473 K – 623 K was reasonable.

From the ignition point to the point located 3 mm from the tipping paper (the length of the tipping paper of the actual cigarette was 31 mm), the experimental puff number was 6.8. This model terminated smoking at 420.5 s when the char line was located at 50.0 mm, the predicted puff number was 7.3. The agreement between the predicted and experimental puff numbers suggests that the burning speed predicted by the cigarette model is basically consistent with the actual cigarette burning speed.

Validity of temperature

Figure 12 shows the gas temperature fields during puffing (240 s, 241 s, and 242 s) in case 3. It can be seen that the gas temperature increased rapidly, and the maximum gas temperature of the center of the cigarette combustion cone could rise up to 1292 K at 242 s. In order to verify the accuracy of the predicted cigarette combustion temperature, the temperatures at specific positions were measured. Eight thermocouples were inserted into the center of the cigarette, at 22 mm, 24 mm, 26 mm, 28 mm, 30 mm, 32 mm, 34 mm and 36 mm from the lighting end of the cigarette, respectively. After igniting, the cigarette kept smoldering. When the char line migrated at 26 mm, puffing started. The initial gas temperature at the 26-mm position before puffing was 933 K as shown in Figure 13. In the cigarette combustion model, the cigarette also kept smoldering after ignition. When the predicted temperature at 26 mm reached 933 K, a puff was simulated using the model. The real-time temperatures of the center of the cigarette at the corresponding positions were monitored.

Figure 13 compares the predicted gas temperature at each position from 0 s to 350 s with the experimental data. Table 7 shows the standard root mean square error (NRMSE) of the predicted gas temperatures and experimental gas temperatures for eight positions, and the overall deviation was less than 18%. The results show that this model has reproduced the basic features of the temperature distribution, and is also in good agreement with the experimental data quantitatively.

NRMSE of the predicted gas temperatures and experimental gas temperatures for eight locations.

Location 22 mm 24 mm 26 mm 28 mm 30 mm 32 mm 34 mm 36 mm
NRMSE 16.0% 17.5% 11.0% 8.5% 13.9% 16.0% 15.8% 11.9%
Validity of flow velocity

Figure 14 shows the flow velocity fields during puffing (240 s, 241 s, and 242 s) in case 3. It can be seen that the flow velocity increased at the first second, reached the maximum flow velocity at 241 s, and then dropped significantly. From 241 s to 242 s, the flow velocity near the cigarette paper area was higher than in the central region of the combustion cone. A similar tendency was experimentally observed by Li et al. (27). Because of the increase in the permeability of cigarette paper behind the char line, a large amount of flow avoids the center of the combustion cone and enters from the cigarette paper area.

The density fields of “tar” and CO during puffing (240 s, 241 s, and 242 s) in case 3 are shown in Figures 15 and 16. It is clear from the figures that both “tar”- and CO-release amounts increased significantly during puffing. This situation could be explained from the above simulations based on the char density field, temperature field and flow velocity field. The main difference between smoldering and puffing is the increase in the flow velocity (Figure 14), which resulted in an increase of oxygen supply. Therefore the char combustion reaction rate increased while more heat was released and the temperature increased (Figure 12). After a temperature increase, the tobacco pyrolysis would be accelerated too and more tobacco would be involved in the reactions and produce more char for combustion. Thus, the rapid increase in “tar” and CO release can be attributed to the acceleration of the reactions due to the increase in flow velocity and temperature.

Validity of filtration efficiency

Figure 17 shows the release amounts of “tar” at the inlet and outlet of the filter rod and the filtration efficiency during puffing (240 s – 242 s) in case 3. The overall filtration efficiency varied on a degree of about 44.9%. There were two kinds of effects during this period. On the one hand, the flow velocity was high, which lead to a decrease in the filtration efficiency. On the other hand, the concentration of aerosol particles grew higher, which resulted in an increase in the filtration efficiency. Accordingly, these two effects tended to compensate each other and the filtration efficiency remained stable.

The total “tar” amounts released at the inlet and outlet of the filter rod were calculated to be 22.63 mg/cig and 12.19 mg/cig, respectively, and the predicted filtration efficiency was 46.1%. In this study, nicotine retention efficiency was chosen to represent the experimental filtration efficiency of the filter. The experimentally determined filtration efficiency for nicotine was 44.5%.

Validity of “tar” and CO yields by changing the puff intensity

In order to reinforce the validity and robustness of the model, case 5 was numerically simulated in which just 25 mL air was drawn for 2 s, all other conditions were kept constant. In this case, the model terminated smoking at 421 s when the char line was at 50.0 mm. The predicted puff number was 7.5, and the actual puff number was 7.

Figures 18 and 19 compare the released puff-by-puff amounts of “tar” and CO in the mainstream smoke under different puff intensities. As the puff intensity decreased, “tar” and released CO amounts also decreased. The predicted yields of “tar” and CO were 9.9 mg/cig and 11.6 mg/cig in case 5, and the experimental values were 9.3 mg/cig and 11.0 mg/cig, with the relative deviations of 6.5% and 5.5%, respectively. The numerical results were in good agreement with the experimental ones.

CONCLUSIONS

The cigarette burning process has been simulated using FLUENT. In contrast to previously published models, the kinetics of pyrolysis and char oxidation reactions were established at high heating rates and different oxygen concentrations and in addition, the filtration efficiency of the cigarette filter was taken into consideration. Furthermore, an experimental platform for tobacco pyrolysis and combustion reactions was designed to obtain the source terms of “tar” and CO at different reaction conditions, which could supply more repeatable and accurate data compared to a TG experiment. The cigarette burning properties such as the density fields, temperature field and flow velocity field were predicted by the model. The puff number, the temperatures at specific locations, the filtration efficiency and the yields of “tar” and CO under different puff intensities were calculated and compared with the experimental data, which showed a good agreement. Authors of future work in this field will be encouraged to use this model for a prediction of the cigarette burning process for different cigarette paper and filter parameters.

#### Kinetic parameters of char combustion.

WO2 range 0% ≤ WO2 ≤ 2% 2 %< WO2 ≤ 10% 10% < WO2 ≤ 23%

Parameters Unit WO2 = 1%, 2% WO2 = 3%, 5%, 10% WO2 = 15%, 20%
Ac min−1 1.48 × 107 4.26 × 107 8.30 × 107
Ec kJ·mol−1 91.04 111.20 116.31
no 1.09 0.43 0.36
R2 0.9441 0.9574 0.9537

#### NRMSE of the predicted gas temperatures and experimental gas temperatures for eight locations.

Location 22 mm 24 mm 26 mm 28 mm 30 mm 32 mm 34 mm 36 mm
NRMSE 16.0% 17.5% 11.0% 8.5% 13.9% 16.0% 15.8% 11.9%

#### Kinetic parameters of tobacco pyrolysis.

Parameters Unit R1 R2 R3 R4 R5
fp,j % 9.52 17.71 18.04 13.58 41.16
Ap,j min−1 1.47 × 105 1.48 × 108 1.82 × 1010 1.21 × 1013 0.4538
Ep,j kJ·mol−1 31.09 60.81 91.48 133.48 25.78
np,j 1.06 1.28 1.21 1.25 0.76
mp,j 1.24 1.54 1.48 1.49 −0.04
R2 = 0.9821

#### Mathematical relationships of “tar” and CO at different temperatures and oxygen mass fractions.

 Temperature range WO2 range “Tar” (mg·g−1) 423 K ≤ T ≤ 623 K 0% ≤ WO2 < 5% Y1 = −0.0049T2 + 5.8838T−1617 (R2 = 0.8990) 5% ≤ WO2 < 15% Y1 = −0.0048T2 + 5.7621T−1580 (R2 = 0.9034) 15% ≤ WO2 ≤ 23% Y1 = −0.0061T2 + 7.1210T−1911 (R2 = 0.9734) 623 K T ≤ 1273 K 0% ≤ WO2 ≤ 23% 160 Temperature range WO2 range CO (mg·g−1) 423 K ≤ T ≤ 1273 K 0% ≤ WO2 < 5% Y2 = 0.2132T−89.19 (R2 = 0.9575) 5% ≤ WO2 < 15% Y2 = 0.2630T−107.32 (R2 = 0.9608) 15% ≤ WO2 ≤ 23% Y2 = 0.2516T−90.41 (R2 = 0.9333)

#### The mathematical models of cigarette reported in the literatures.

Reference and year Author(s) Smoking conditions Geometry Model construction Simulation contents

Pyrolysis and char oxidation reaction kinetics Transport system Burning properties Products
(1) 1963 Egerton et al. Steady draw 1-D Temperature
(2) 1966 Gugan Smoldering 2-D Combustion cone Temperature
(3) 1977 Baker Smoldering 1-D Heat release rate O2 concentration CO, CO2
(4) 1978 Summerfield et al. Steady draw 1-D Using the kinetics parameters obtained by themselves (4) Burning rate Temperature Pressure
(5,6,7) 1979–1981 Muramatsu et al. Smoldering 1-D Using the kinetics parameters obtained by themselves (5, 6) Burn rate Temperature Density
(8) 2001 Miura et al. Smoldering 1-D Burning rate Temperature
(9) 2001 Yi et al. Smoldering 2-D Using the kinetics parameters obtained by Diblasi (10) Temperature Solid density Char density O2 concentration Water
(11) 2002 Chen Smoldering 1-D Using the kinetics parameters obtained by themselves (11) Temperature Density
(12) 2003 Rostami et al. Smoldering 2-D Using the kinetics parameters reported by Muramatsu et al. (5, 6) Burning rate Temperature O2 concentration
(13) 2004 Rostami et al. Smoldering and steady draw 2-D Using the kinetics parameters reported by Muramatsu et al. (5, 6) Temperature O2 concentration Pressure Flow velocity
(14) 2004 Saidi et al. Puffing 3-D Using the kinetics parameters for volatile species reported by Wojtowicz et al. (15) Burning rate Temperature Flow velocity O2 concentration CO, CO2 H2O Nicotine
(16) 2005 Eitzinger et al. Smoldering, puffing and steady draw 2-D Using the kinetics parameters obtained by themselves (16) Burning rate Temperature Flow velocity O2 concentration Combustion gas Water
(17) 2007 Saidi et al. Puff-smoldering cycles 3-D Using the kinetics parameters for volatile species reported by Wojtowicz et al. (15) Burning rate Temperature Flow velocity O2 concentration Char density CO, CO2 Volatile
(18) 2008 Saidi et al. Puff-smoldering cycles 3-D Using the kinetics parameters for volatile species reported by Wojtowicz et al. (15) Burning rate Temperature Flow velocity Char density CO, CO2 H2O

#### Parameters and values related to each domain.

Domain Parameter Definition Unit Value
1) ρs0 Initial solid density kg·m−3 740 (12)
ϕ Porosity 1 0.7
Cp,s Specific heat of solid kJ·kg−1·K−1 1.043 (12)
Cp,g Specific heat of gas kJ·kg−1·K−1 1.004 (12)
ks Solid conductivity W·m−1·K−1 0.316 (12)
kg Gas conductivity W·m−1·K−1 0.0242 (12)
ɛ Emissivity of tobacco 1 0.98 (12)
dpore Pore diameter m 5.75 × 10−4 (12)
Hevaporation Water evaporation heat kJ·kg−1 −2.2572 × 103 (12)
Hcombustion Char combustion heat kJ·kg−1 1.757 × 104 (12)
v Flow velocity m/s 0
Kut Permeability of unburned tobacco m2 5.6 × 10−10 (18)
Kbt Permeability of burned tobacco m2 105 (18)
2) Kup Permeability of unburned cigarette paper m2 5 × 10−15
Kbp Permeability of burned cigarette paper m2 105 (18)
3) dp Aerosol particle diameter m 4.4 × 10−7 (19)
df Single fiber diameter m 2.51 × 10−5 (19)
Dt Total denier of filter g·(9000 m)−1 35000
Ds Denier of per single fiber g·(9000 m)−1 3
Cfiber Crimping ratio of fibers 1 0.17
Sfilter Cross-sectional area of filter rod m2 4.899 × 10−5
Tfilter Filter temperature K 288
Kfilter Permeability of filter m2 2.5 × 10−10 (18)
4) T Ambient temperature K 288
P Ambient gas pressure kPa 101.3
ρg0 Initial gas density kg·m−3 1.225
WO2 Mass fraction of O2 % 23
WN Mass fraction of N2 % 77

#### Comparison of the numerical and experimental results.

Case 1 Case 2 Case 3 Case 4 Experimental
“Tar” (mg/cig) 15.8 14.1 12.2 5.1 11.2
Relative deviation 41.1% 25.9% 8.9% 54.5%
CO (mg/cig) 21.1 17.3 14.6 5.6 13.2
Relative deviation 59.8% 31.1% 10.6% 57.6%

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