Modeling of a smoldering cigarette

Modeling of a smoldering cigarette

Ali Rostami a, Jayathi Murthy b, Mohammad Hajaligol a,�a Philip Morris USA, Research Center, P.O. Box 26583, Richmond, VA 23261, USA

b Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA

Abstract

A transient 2D model based on the first principles has been developed for the natural

smoldering of a cigarette. The domain is assumed to be 2D and axisymmetric. Heat transfer is

represented by the use of a two-temperature formulation whereby the solid and gas phases are

considered to have separate and distinct temperatures interacting through an inter-phase heat

exchange. The starting material first undergoes pyrolysis prior to oxidation of the remaining

carbonaceous residue. The pyrolysis is assumed to consist of a set of four reactions. Oxidation

of the carbonaceous residue is accompanied by the formation of gas-phase combustion

products, whose concentration is also computed. The change in the tobacco rod permeability

due to combustion is modeled using a single step dependence on the solid density. Calculations

under unsteady conditions are done for a variety of smoldering cases with varying operating

and boundary conditions. The model was validated by comparing the predictions with the

experimental data on the smoldering burn rate and the maximum gas and solid temperatures.

The computation captures the development of a steady combustion regime in which the burn

front moves at a constant rate. More information can be obtained from the model including

coal shape, gas and solid temperature profiles, product yields, solid density variations, and the

effects of ignition conditions. The results are shown to be very sensitive to the availability of

oxygen, and consequently to the oxygen diffusivity through the cigarette paper.

# 2002 Published by Elsevier Science B.V.

Keywords: Modeling; Smoldering; Combustion; Cigarette smoldering; Burn velocity

� Corresponding author. Tel.: �/1-804-274-2419; fax: �/1-804-274-2891

E-mail address: [email protected] (M. Hajaligol).

J. Anal. Appl. Pyrolysis 66 (2003) 281�/301

www.elsevier.com/locate/jaap

0165-2370/02/$ - see front matter # 2002 Published by Elsevier Science B.V.

PII: S 0 1 6 5 - 2 3 7 0 ( 0 2 ) 0 0 1 1 7 - 1

mailto:[email protected]

1. Introduction

Smoldering of a porous carbonaceous rod is normally controlled by two main

parameters: availability of oxygen to the combustion front and heat losses form it.

The velocity of the combustion front into the carbonaceous fuel after the ignition by

an external heat source and the peak temperature are two indicators of the

sustenance of combustion. It is a transient process which is controlled by acombination of endothermic and exothermic chemical reactions in the pyrolysis

and combustion zones, diffusion of oxygen to the combustion zone, diffusion of

reaction products away from the sources, and heat transfer as well.

In an attempt to model the steady state smoldering of a burning rod, Guan [1]

assumed that combustion of tobacco is controlled by oxygen diffusion and

developed a 2D model. Guan was able to predict the general features of the

temperature distribution. Kinbara et al. [2] used the same diffusion controlled

approach to model the smoldering of a cylinder. The model was one-dimensionaland the predicted behavior was in agreement with the measured burn rate. During

1973 and 1981, Muramatsu et al. [3�/5] introduced a 1D model and later

incorporated the radial effects on the transfer processes in the smoldering of a

cigarette. The 2D model required simultaneous solution of the differential equations

governing the heat and mass transfer characteristics in the combustion as well as

pyrolysis zones.

Muramatsu et al. [3�/5] assumed thermal equilibrium between the solid and

gaseous phases. The evaporation�/pyrolysis zone was represented in the model, witha four-step pyrolysis of tobacco. The virgin tobacco is considered as consisting of

four pyrolysis precursors and moisture. Four volatile pyrolysis products and water

vapor are produced as a result of heating of the solid tobacco, leaving two char

residues and ash behind. The model required the kinetic constants for the pyrolysis,

which were obtained in an earlier study by Muramatsu. The model was used to

predict the burn rate, peak temperature, and density variation in a smoldering

cigarette. Good agreement between the predictions and the experimental data on

these variables in the pyrolysis zone were reported.Simpson and Waymack [6] developed a 1D smolder model in which the pyrolysis

of four precursors and the oxidation of char were included. The model required the

specification of several parameters including the linear burn rate. Reasonable

agreement between the prediction of the density variation across the combustion

zone and the experimental data was observed.

Since the early 1980s, most of the work in the area was focused on the

generalization of combustion modeling in porous media, except for a few cases,

without a particular reference to a burning cigarette [7�/13]. The works that wererelevant to the burning of tobacco were concentrated on the steady draw or puffing.

For example, Ohlemiller [7] referred to a burning cigarette as forward smoldering, in

which the air flow is in the same direction as the travelling combustion front. A

comprehensive description of the mathematical formulation was presented by

Ohlemiller [7], but no attempt was made to develop a solution technique for the

multi-phase and multi-component system of equations. Gann et al. [14] reported a

A. Rostami et al. / J. Anal. Appl. Pyrolysis 66 (2003) 281�/301282

2D model that was developed in NIST. The model led to the temperature and

oxygen distributions that are unrealistically elongated in the axial direction.

We have recently embarked on an effort to develop comprehensive modeling

software for the prediction of flow, heat transfer and chemical reaction during

cigarette burning. In this paper we present the natural smoldering combustion in a

cigarette where no forced flow exists within the rod. A transient, 2D and two-

temperature formulation for the tobacco and the gas phase is then developed.Variation of permeabilities of paper and tobacco column as a result of combustion

are included. Furthermore, diffusion of oxygen and combustion product in the

porous medium and the paper as well as heat and mass transfer from the lateral

surface are accounted for. To capture the very basic behavior of the phenomena,

reactions of organic species in the gas phases for both pyrolysis and combustion

zones are ignored.

The computation is initiated by applying a thermal boundary condition at one end

of the cigarette for a short period of time. Then the lighting condition is removed,followed by self-sustaining smolder, leading to a steady state situation after about

100 s. In the subsequent sections, the model is described, followed by a description of

the governing equations and the boundary conditions. Then the parameters used as

input are introduced. Predictions of temperatures and species concentrations are

then presented. Finally, the computed results are evaluated and recommendations

are made. However, in the subsequent publications, we will present the results of

burning of a cigarette under a forced flow condition and for cases where the

reactions in the gas phase are taken into consideration.

2. Model description

The modeling of pyrolysis and combustion in a smoldering cigarette requires the

solution of flow, heat and mass transfer through porous media. The gas and solid

phases are treated separately. The tobacco constitutes the main porous zone; other

porous zones may be present as well, for example, the cigarette paper. These zones

may have different porosities and permeabilities, as well as different physical andchemical properties. Pyrolysis occurs in the virgin tobacco through a set of ‘n ’

pyrolysis reactions involving up to ‘n ’ pyrolysis precursors. The pyrolysis reactions

result in the formation of a series of products. The pyrolysis products leave the solid

phase, while the remaining carbonaceous residue is oxidized when exposed to high

temperatures and oxygen. Some species in the gas phase may undergo further

degradation reactions into other gas-phase species. In addition, we must also

account for the evaporation of water, other volatile materials, and ash formation

reactions as well.The combustion process is initiated by applying a high temperature or a high heat

flux at the front end of a cigarette for a fixed time. This initiates the pyrolysis

reactions in a very narrow, nearly 1D zone, followed by the oxidation of the

carbonaceous residue. It is followed by combustion reactions in a larger 2D cone,

sustained by the diffusion of oxygen from the surrounding air through the tobacco

A. Rostami et al. / J. Anal. Appl. Pyrolysis 66 (2003) 281�/301 283

under smoldering conditions. The burn front moves along the length of the cigarette,

so does the combustion cone. For a given smoldering condition, the burn front is

expected to travel at a constant velocity away from the lighted end of the cigarette.

2.1. Pyrolysis and oxidation reactions

The model adopted is similar to that of Muramatsu et al. [4], where the change in

the density of starting tobacco is represented by a set of four pyrolysis reactions.

Each of the four pyrolysis precursor consists of several unknown species. The

corresponding pyrolysis reactions are given by:

@rvi

@t��Zvi exp(�Evi=TS)rrn

vi (1)

@rv

@t�

X4

i�1

@rvi

@t(2)

Here rvi is the mass concentration of the ith pyrolysis precursor, Zvi is the pre-

exponential factor, Evi is the activation energy, R is the universal gas constant and

TS is the temperature of the tobacco. A first-order reaction is assumed, i.e., n�/1.

The total mass concentration of pyrolysis precursors is given by rv . The apparentkinetic parameters are listed in Table 1. Each of the four pyrolysis precursor is made

of many unknown species.

Table 1

Apparent kinetic parameters for tobacco pyrolysis, char oxidation and water evaporation reactions

(Muramatsu et al. [4])

Pyrolysis Precursor 1 Precursor 2 Precursor 3 Precursor 4

N 1 1 1 1

Evi (kcal (g mole)�1) 20.2 24.5 45.7 25.2

Zvi (1 s�1) 6.27�/107 1.69�/108 5.99�/1014 4.96�/106

Initial fraction 0.25 0.28 0.17 0.3

Oxidation Species 1 Species 2

Eci (kcal (g mole)�1) 19.5 38.0

Zci (1 s�1) 2.8�/106 1.15�/1011

Fraction f 0.5 0.5

Water evaporation

Ew (kcal (g mole)�1) 19.5

Zw (1 s�1) 2.8�/106


2.2. Formation and oxidation of carbonaceous residue

Two carbonaceous residue species are considered. The species are formed through

the pyrolysis of tobacco and consumed in oxidation reactions. It is assumed that an

equal fraction of the two solid carbonaceous residues are produced, i.e., f�/0.5.

@rci

@t��ncf

@rv

@t�Zci exp(�Eci=RTS)r

1=2O2

rci (3)

@rc

@t�

X2

i�1

@rci

@t(4)

Here rci is the mass concentration of the ith carbonaceous residue species, rc is

the total mass concentration of the carbonaceous residue species, and rO2

is the mass

concentration of oxygen in the gas phase. Zci and Eci are the pre-exponential factor

and the activation energy of the i th reaction, respectively, and nc is the

stoichiometric coefficient for the reaction. The kinetic parameters and the value of

nc are listed in Table 2. The right hand side of Eq. (3) consists of two terms. The first

term represents the rate of formation while the second term indicates the rate ofconsumption of the carbonaceous residue. If there is no oxygen available, then

rO2�/0 and residues are only produced.

Table 2

Important problem parameters

Parameter Value Definition

r 1.225 kg m�3 Gas density @ room temperature

rc0 0.0 Initial char density

rs0 740 kg m�3 Initial solid density

rw0 74 kg m�3 Initial water density

rv0 596.8 kg m�3 Initial density of pyrolysis precursors

nc 0.34 Stoichiometric coefficient for char formation

n02 1.65 Stoichiometric coefficient for oxygen consumption

na 0.33 Stoichiometric coefficient for ash formation

Cps 1.043 kJ (kg K)�1 Specific heat of solid

Cpg 1.004 kJ (kg K)�1 Specific heat of gas

R 4 mm Cigarette radius

L 5.7 cm Cigarette length

D0 1.12�/10�5 m2 s�1 Reference diffusion coefficient of oxygen

f 0.65 Porosity

ks 0.316 W (m K)�1 Solid conductivity

kg 0.0242 W (m K)�1 Gas conductivity

dp 0.0575 cm Pore diameter

Troom 300 K Room temperature

o 0.98 Emissivity of smoldering coal

DHw �/2.2572�/103 kJ kg�1 Heat of vaporization

DHc 1.757�/104kJ kg�1 Heat of oxidation


2.3. Water evaporation and ash formation

The volatilization of water and the formation of ash are modeled using first-order

reactions given by:

@rw

@t��Zw exp(�Ew=RTS)rw (5)

@rash

@t��nash

@rc

@t(6)

Here rw and rash are the mass concentrations of water and ash, respectively. The

values of the various kinetic parameters are given, as before, in Table 2. Eq. (6)

shows that the ash concentration does not need to be independently calculated from

an Arrhenius type of equation. It is directly obtained from the stoichiometric ratio of

the combustible char and ash.

2.4. Gas-phase species transport

Eq. (3) shows that in order to calculate the rate of char consumption, we need to

have the oxygen distribution. It can be determined by solving the gas transport

equations. Defining the mass fraction of gas-phase species Yi as

Yir�ri (7)

We may write the transport equations for the ith gas-phase species in the form:

@rfYi

@t�9(rVYi)�9 �(rDf9Yi)�Ri(1�f) (8)

Here f is the porosity of the tobacco matrix, r is the gas-phase mixture density, D

is the diffusion coefficient of the species in the mixture and Ri is the volumetric rate

of species production due to the oxidation reaction. The appearance of f in Eq. (8) isto correct for the gas density and the rate of reaction for the unit volume of the

computational cell. All terms involving velocity V will be omitted for the case of

smoldering. The species being transported in the gas phase result from the oxidation

of carbonaceous residue. A single step oxidation reaction leading to a single product

species is assumed:

C�nO2O2 0 npP (9)

The gas-phase species being transported are O2 and the product species.

Therefore, Eq. (8) has to be solved for only two gas species, O2 and P . The source

terms RO2

and Rp are given by:

RO2��nO2

X2

i�1

Zci exp(�Eci=RTS)r1=2O2

rci (10)


Rp�np

X2

i�1

Zci exp(�Eci=RTS)r1=2O2

rci (11)

The oxidation reaction is assumed to occur on the solid surface. It consumes gas-

phase oxygen and releases the product into the gas phase. The boundary layer

resistance to the transport of the gas-phase species from the solid to the gas is

assumed to be negligible.

2.4.1. Temperature dependence of diffusivity

Since the transport of oxygen to the burn front depends critically on its ability to

diffuse through the gas phase, it is important to include the correct dependence of

diffusivity on temperature.

D�D0(T=273)1:75 (12)

where D0 is the reference value of the mass diffusivity in the porous media at 273 K

and 1 atm. Its value is related to the void fraction of the media through [15].

D0�0:677DgF1:18

where Dg is the unrestrained diffusion coefficient of the gas in a binary mixture. For

the oxygen diffusion in nitrogen, Dg�/2�/10�5 m2 s�1 and a tobacco filling with a

total void fraction of F�/0.85 [5], D0�/1.12�/10�5 m2 s�1.The diffusivities of oxygen and product in the gas mixture are assumed unchanged

due to the change in permeability of tobacco as a result of pyrolysis and burning.

2.5. Solid and gas phase energy equations

The gaseous reactants and products are assumed not to be in thermal equilibrium

with the solid phase, consisting of burned and unburned tobacco. Hence, it is

necessary to deal with two energy equations, one for the solid and one for the gas

phase.

2.5.1. Solid phase

(1�f)rSCpS

@TS

@t�9 �(kS;eff (1�f)9Ts)�hS�g

�A

V

�(Tg�TS)�Ssolid (13)

2.5.2. Gas phase

@

@t(rfhg)�9 �(rVhg)�9 �(kgf9Tg)�hS�g

�A

V

�(TS�Tg)�Sgas (14)

In the above equations, Tg is the temperature of the gas, hg is the gas phase

sensible enthalpy, ks,eff is the effective solid conductivity, hs�g is the gas�/solid


interface heat transfer coefficient, and Ssolid and Sgas represent other source terms.

The surface area to volume ratio, A /V , depends on the assumed geometry of the

tobacco matrix. For the solid phase equation, the source term Ssolid would include

the heat of reactions:

Ssolid�X

k

(�DHk)@rk

@t(15)

Here DHk is the heat of reaction for either the oxidation or water vaporization. Itis assumed that the heat of pyrolysis reactions is small and does not play a significant

role in the energy equations.

2.5.3. Radiation effects

During oxidation, the tobacco burns with temperatures in excess of 1000 K and

radiative heat transfer can be very important. The effect of radiation on the solidtemperature equation is modeled using the Rosseland approximation, whereby the

solid thermal conductivity is augmented by a radiative conductivity:

ks;eff �ks�4osT3s dp (16)

where dp is the pore diameter and o is the emissivity of the tobacco. The gas phase is

assumed to be radiatively non-participating.

2.5.4. Interface heat exchange

The interface heat exchange between the solid and gas phases is prescribed

empirically by assuming a pore geometry. A number of correlations are available in

the literature depending on the geometry assumed. Wakao and Kaguei [16], for

example, proposed a correlation based on packed beds of spheres:

Nu�hdp

kg

�2�1:1Re0:6Pr0:333 (17)

This correlation can be used for smoldering by setting Re�/0. The area to volume

ratio corresponding to this geometry is given by:

A=V �6f=dp (18)

2.6. Porous medium model

The tobacco rod is treated as a porous medium with a known permeability. This

entails including a momentum sink (only for puffing conditions) in the momentum

equations given by:

S��mV

K�CrjV jV

�(19)

where m is the fluid viscosity, K is the permeability of the porous medium and C is


an empirical constant governing the magnitude of the inertial term; it is assumed

zero for all calculations done here. The medium is assumed isotropic.

2.7. Variation of tobacco permeability

The permeability of tobacco changes as the tobacco burns. In this work, we have

assumed that the permeability varies linearly with the density of the unburned solid:

K�Ku(1�f )�Kbf (20)

f ��rs � rsu

rsu

(21)

Here f is an interpolation factor and rs is the density of solid density, i.e., the total

density of all solids including pyrolysis precursors, carbonaceous residue, as well as

moisture content, and ash. rsu is the initial density of the unburned solid, and the

density of the completely burned solid is, of course, zero. The permeabilities of theburned and unburned tobacco are Kb and Ku respectively. Values of Ku and Kb are

given in Table 3.

3. Boundary and initial conditions

3.1. Inlet and outlet conditions

For smoldering, the internal flow in the porous media is assumed to be negligible.

The temperature for both the solid and gas phases at the inlet is held at T�/1000 K

until self-sustained combustion starts; this typically takes 10�/30 s depending on the

conditions. After this, the inlet temperature for both phases is set to T�/300 K,

because the energy provided by the combustion is either sufficient for the reactions

to sustain or to discontinue. All gas-phase species are assumed to have zero mass

fraction at the inlet except oxygen, which assumes a mass fraction of 0.23 and

nitrogen, which assumes a value of 0.77 throughout the smoldering process. At the

Table 3

Parameters related to tobacco and paper permeability

Parameter Value Definition

Ku 5�/10�10 m2 Permeability of unburned tobacco

Kb 1015 m2 Permeability of burned tobacco

Kpaper,u 10�20 m2 Permeability of unburned paper

Kpaper,b 1010 m2 Permeability of burned paper

Du 10�10 m2 s�1 Diffusivity of oxygen in unburned paper

Db 1.12�/10�5 m2 s�1 Diffusivity of oxygen in burned paper

Tp 450 8C Paper combustion temperature


outlet, a zero gradient condition is assumed for temperature and species mass

fraction.

3.2. Lateral boundary conditions

For smoldering, no flow boundary conditions are required. The gas and solid

phase temperatures are subjected to a convective boundary condition at the lateralcylindrical boundary. All gas-phase species except oxygen are subjected to zero

gradient conditions at the lateral boundary. For oxygen, different boundary

conditions have been used for smoldering calculations, ranging from fully imperme-

able, to having a specified mass transfer coefficient. The mass transfer resistance of

the paper has been combined with that of the surface convection resistance.

3.3. Initial conditions

For smoldering, a no-flow condition is specified at the start. Both the gas and solid

phase temperatures are set at 300 K throughout the cigarette. The mass fractions of

oxygen and nitrogen are set at their environmental values of 0.23 and 0.77,

respectively. The initial conditions for various tobacco species are given in Table 1.

These typically assign non-zero values for the pyrolysis precursors, while all other

solid species are assumed to have zero mass concentrations.

4. Numerical implementation

The implementation of the models described above is done in a custom version of

Fluent’s structured mesh solver, Fluent 4.5. The domain is discretized into structured

control volumes over which the conservation equations for mass, momentum, energy

and chemical species are discretized. Standard first- and second-order spatial

discretization schemes are used for convective operators, with a second-order

discretization of the diffusion terms. The discretization of the unsteady terms is doneusing first-order fully implicit scheme. For the solid species in the tobacco, no

diffusion or convection terms are present in the governing equations. In these cases,

the governing equation is discretized using a first-order implicit formulation, with a

suitable linearization of the reaction source terms to facilitate convergence. All

equations are solved sequentially and iteratively in keeping with the Fluent

algorithms. The equations described above are incorporated through the user-

subroutines available in Fluent, though some manipulations are not possible

through these subroutines and had to be done by making changes to the source.

5. Model geometry

The geometry of the calculation domain is shown in Fig. 1. A cylindrical

axisymmetric cigarette is assumed, with dimensions as shown. The paper resistance


to mass transfer is combined with the surface resistance, with a suitable mass transfer

boundary conditions to control oxygen transport into the domain.

A non-uniform structured mesh of 82�/22 control volumes is used in the x and r

directions, respectively; one control volume is reserved for the paper when present.

The tobacco region is set to be a porous zone in Fluent parlance. A variable time step

is used to do the calculations, with relatively large time steps of between 0.1 and 0.01

s during the initial passive heat-up stage and time steps of 0.001 s once combustion

commences. Typically 10�/30 iterations per time step are required. Several hundred

steps are required to cross a control volume. The size of the time step is controlled by

the time scale of the oxidation reaction which is very fast.

6. Results and discussions

Numerous calculations were performed to identify the effects of the appropriate

model parameters, numerical parameters and boundary conditions. A number of

results including the effects of the surface heat transfer coefficient, overall mass

transfer coefficient and the lighting conditions are reported here. The overall mass

transfer coefficient, to which the results are very sensitive, depends on the convective

coefficient on the surface as well as the diffusion coefficient in the paper. In practice,

the air velocity on the surface, hence the heat transfer and mass transfer coefficients,

are highly unpredictable and vary significantly depending on the conditions of the air

surrounding the cigarette. So is the diffusion coefficient through the paper, the

thickness and the permeability of which vary from one application to another.

Therefore, an accurate comparison between the experimental results and the

predictions was impossible, because most of the data available in the literature did

not provide sufficient information on either the operating conditions or the physical

properties.

Due to the limitations mentioned above, two criteria are used to check the validity

of the model. They are the smoldering burn rate and the maximum coal temperature.

Furthermore, the development of the coal shape may also be checked visually to

verify the reasonability of the predictions. Table 4 shows the variables used in the

calculations. Case A0 denotes the base conditions for which the overall mass transfer

coefficient on the surface and the total heat transfer coefficient, due to convection

and thermal radiation, are 0.008 kg (m2 s)�1 and 60 W (m2 K)�1, respectively.

Fig. 1. Geometry of computational domain.


Table 4

Conditions for computation

Case Heat transfer coefficient,

W (m2 K)�1

Overall mass transfer coefficient

kg m�2 s�1

Case Heat transfer coefficient,

W (m2 K)�1

Overall mass transfer coefficient,

kg m�2 s�1

A0 60 0.008 B1 20 0.008

A1 60 0.005 B2 40 0.008

A2 60 0.007 B3 80 0.008

A3 60 0.010 B4 100 0.008

A4 60 0.0125 Ca 60 0.008

a Lighting condition for cases A and B is 1000 K/8.3 s and for case C is 1273 K/2.3 s at front end of cigarette.

A.

Ro

stam

iet

al.

/J

.A

na

l.A

pp

l.P

yro

lysis

66

(2

00

3)

28

1�

/30

12

92

Other cases refer to the conditions in which these quantities are varied from the base

conditions.

6.1. Validation

Fig. 1 shows the computational domain for a typical cigarette under smoldering

conditions. Under the base conditions, the temperature of the front end surface is

raised to 1000 K for 8.3 s to simulate the lighting. Subsequently, the carbonaceous

residue begins to burn vigorously and the front surface is set back to 300 K. Initially,

the temperatures and the concentrations of species are uniform everywhere.Fig. 2 shows the development of the gas phase temperature profile with time. We

see that the temperature distribution at small time contains very high maximums,

primarily as a result of the oxygen-rich initial condition and the lack of time for heat

dissipation. A cone of oxidation is clearly visible, with the maximum temperature

occurring on the axis. Oxidation persists even after the hot tip is reset to 300 K, and

it resembles the coal shape. About 100 s after the initiation of oxidation, a steady

state situation prevails, where the oxidation zone travels with a constant speed down

the length of the cigarette. The similarity between the coal shapes at times 120 and

300 s in Fig. 2 is an indication of the steady state condition.

Fig. 2. Gas temperature profiles development for case A0.


Fig. 3 shows the maximum gas and solid phase temperatures as a function of time.

The temperatures right at the end of lighting (8.3 s) are very high because of oxygen

enrichment of the combustion zone and the limited rate of heat loss. As the

temperature of the front end is set back to 300 K more heat is transferred from the

combustion zone to this surface, causing the peak temperatures drop sharply. After

about 100 s, a steady state condition is reached where the maximum gas and solid

phase temperatures remain at 1015 and 1035 K, respectively. These values are within

the range of experimental data, which varies from 1050 to 1080 K [4,17]. More

accurate comparison is not possible because the values of some of the parameters

needed for calculation are not specified in the experimental reports. The temperature

of the solid is slightly higher than the gas temperature. The difference is due to the

fact that heat is generated in the solid phase as a result of chemical reactions and has

to be transferred from the solid to the gas phase through the gas�/solid interface.

Therefore a solid-to-gas phase temperature gradient is required to keep the flow of

heat, which is caused by the thermal resistance associated with the interface. The

actual temperature difference between the two phases depends on the gas�/solid

interface resistance as well as the interface area, both of which are difficult to be

accurately evaluated. The predicted temperature difference is generally between 1

and 2% of the gas temperature. Under the base conditions, case A0, the maximum

solid temperature is generally 10�/20 K higher than the maximum gas temperature

depending on time and the distance from the front and/or surface. This result

confirms the assumption made by many investigators that during smoldering, the

solid and gas phases are in thermal equilibrium.

The steady state velocity of the burning coal is another criterion to check the

accuracy of the model. Smoldering velocity may be defined as the velocity of the

location of the maximum solid temperature. Fig. 4 shows the location of Tmax as a

function of time for two cases A0 and C, with the same conditions except for the

lighting conditions. For case A0, a temperature of 1000 K is applied to the front end

for a period of 8.3 s before it is set back to 300 K. The corresponding temperature

Fig. 3. Peak solid and gas temperatures.


and time for case C are 1273 K and 2.3 s, respectively. It is clear from Fig. 4 that the

steady state solution is independent of the lighting conditions. As long as thecombustion process is initiated, the eventual steady state smoldering conditions are

not influenced by the initial lighting conditions. The steady state smoldering velocity

from Fig. 4 is about 7 mm min�1 for the specified conditions, while the experimental

data varies from 3.6 to 7.2 mm min�1 depending, among other parameters, on the

packing density, tobacco moisture content, paper permeability and the cigarette

radius. More accurate evaluation of the predicted results requires knowledge of more

specific experimental conditions.

6.2. Effects of paper permeability

Since the smoldering process is mainly controlled by the diffusion of oxygen to the

combustion zone, paper permeability is expected to have a strong effect on thedevelopment and sustenance of the combustion process. This effect can be assessed

by changing the overall mass transfer coefficient on the cigarette surface, which

simulates the variation of paper permeability. The maximum steady state gas

temperature as a function of mass transfer coefficient, b , is shown in Fig. 5. For a

Fig. 5. Effects of paper permeability on the peak temperature.

Fig. 4. Location of maximum temperature.


small value of b�/0.002 kg (m2 s)�1, the temperature drops to 300 K in less than 80

s after initiation of lighting. For this case, the paper permeability is very low so that

the lateral supply of oxygen and the axial diffusion from the cigarette tip are too slow

to sustain smoldering, and the cigarette eventually dies out. The maximum

temperature generally increases with increasing paper permeability, or mass transfer

coefficient. The relation becomes linear when b exceeds 0.008 kg (m2 s)�1. For b�/

0.0125 kg (m2 s)�1, the temperature reaches 1340 K; much higher than the range of

values observed in practice. In practice, increasing the paper permeability increases

significantly the smoldering velocity, but does not affect the maximum temperature

considerably. The large predicted value may be attributed to the inaccuracy of

modeling of the energy balance and or the thermophysical and the kinetic

parameters used in the model.Fig. 6 shows the steady state smoldering velocity and the mass burn rate increasing

with the paper permeability. Again a change of slope in the velocity is observed at

b�/0.008 kg (m2 s)�1 with the largest velocity (13 mm min�1) occurring at b�/

0.0125 kg (m2 s)�1. This large value may also be due to the same reasons mentioned

above for the temperature. The change in the slope of Figs. 5 and 6 about b�/0.008

kg (m2 s)�1 may be explained by looking at the steady state coal shape develop-

ments, which are shown in Fig. 7 as oxygen concentration fields for b�/0.005, 0.008,

and 0.0125 kg (m2 s)�1. The region bounded by zero oxygen concentration

approximately defines the coal shape or the combusted region. The shapes are

similar for b�/0.008 and 0.0125 which indicate that the combustion zone extends to

the periphery of the rod. In other words, no virgin tobacco is left behind the coal.

However, for b�/0.005, a round combustion zone progresses along the axis of the

rod leaving an annular region of unburned tobacco behind. As a result, although the

smoldering velocity does not change significantly when b increases from 0.005 to

0.008 kg (m2 s)�1, the rates of burning of amounts of tobacco are considerably

different. This is clearly shown in Fig. 6, where the mass burn rate is almost linearly

increasing with the paper permeability for the entire range permeability. In fact, the

burn rate-smoldering velocity relations are not the same for these two cases.

Fig. 6. Effects of paper permeability on the smoldering velocity.


Fig. 7. Oxygen concentration for different paper permeabilities.


Similarly, the slow rate of change of coal temperatures for bB/0.008 kg (m2 s)�1 can

also be attributed to the difference in the coal shapes.

Consequently, the only way smoldering can proceed is through lateral diffusion of

oxygen through the paper. The oxygen diffusivity through the paper is therefore

critical in maintaining smoldering.

6.3. Surface heat losses

For a given set of kinetic parameters, the fate of combustion is tied to the oxygen

availability to as well as heat losses from the coal. Having discussed the oxygen

supply, we now turn to the heat transfer effects. The heat transfer mechanisms that

are relevant to the combustion zone are axial and radial conduction and radiation in

the porous tobacco and char as well as convection and thermal radiation form the

surfaces. The first two are incorporated in the governing equations. An overall heat

transfer coefficient is assigned to the lateral surface that includes the surfaceconvection and radiation. The value of overall heat transfer coefficient, h , is varied

from 20 to 100 W (m2 K)�1 as shown in Table 4.

Fig. 8 shows the effects of the overall heat transfer coefficient on the steady state

smoldering velocity and the gas maximum temperature, respectively. The smoldering

velocity reaches a high value of 16 mm min�1 for h�/20 W (m2 K)�1, while the

maximum temperature reaches 1660 K for the same conditions. As h increases to 100

W (m2 K)�1, they drop to 5.3 mm min�1 and 850 K, respectively. The ranges of

variations show that heat losses have considerable effects on the combustion process.Oxygen concentration fields for these two cases are shown in Fig. 9 for about 150 s

after initiation of lighting. For the larger heat transfer coefficient, the coal

temperature is low, which results in a short combustion cone barely touching the

lateral surface. In contrast, the shape is extraordinary elongated for h�/20

W (m2 K)�1 which is caused by the extended region of high temperature (Tmax�/

Fig. 8. Effects of heat dissipation on smoldering velocity and peak temperature.


1660 K) and availability of oxygen. The coal shape shrinks (not shown in the Figure)

at later times, when it approaches the rear end surface of the cigarette, due to the low

temperature boundary condition imposed on this surface. For this case, the

temperature on the lateral surface is high enough to sustain the combustion on a

large portion of this surface, whereas for the case of h�/100 W (m2 K)�1 thecombustion zone is mainly confined to the interior of the rod.

7. Conclusions

Smoldering combustion of a cigarette has been computed using Fluent 4.5. A

variety of models have been incorporated for pyrolysis and oxidation as well as for

Fig. 9. Effects of heat losses on the coal shape.


heat transfer in porous media, including a two-temperature model for thermal non-

equilibrium between gas and solid. Predictions of temperature and species

concentration have been made. The results establish that the overall physics of the

smoldering process is captured by the simulation, and the actual values of the gas

and solid temperatures and the rate of burn are in reasonable agreement with the

experimental data. The accuracy of the results depends strongly on a number of

empirical parameters. The most significant parameters are the lateral mass transferboundary condition for oxygen, or the paper permeability, and the surface heat

transfer coefficients. The kinetic parameters are also key players in the overall

pyrolysis and combustion process. The effects of these parameters are being studied

and will be reported later. The numerics have proved to be stable and convergent

and allow us to do computations with relative ease.

Though the effort has been successful in capturing the broad features of

smoldering, as discussed above a number of improvements to the current model

needs to be made.

Acknowledgements

Authors wish to thank Dr. Sung Yi and M. Subbiah for their helpful technical

discussions and computer modeling support. The authors are also grateful to Philip

Morris, USA management for their support of this fundamental research.

References

[1] K. Guan, Natural smolder in cigarette, Combust. Flame 10 (1966) 161�/164.

[2] T. Kinbara, H. Endo, S Sega, Downward propagation of smoldering combustion through solid

materials, 11th Symposium Combustion, Pittsburgh, Pennsylvania, (1967) 525�/531.

[3] M. Muramatsu, S. Umemura, T. Okada, Combustion of oxygen and heat evolved during natural

smolder of a cigarette, J. Chem. Soc. Jpn Chem. Ind. Chem. (1978) 1441�/1448.

[4] M. Muramatsu, S. Umemura, T. Okada, A mathematical model of evaporation�/pyrolysis processes

inside a naturally smoldering cigarette, Combust. Flame 36 (1979) 245�/262.

[5] M. Muramatsu, Studies on the transport phenomena in naturally smoldering cigarette, Sci. Pap.

Cent. Res. Inst. Jpn Tobacco Salt Monop. Corp. 123 (1981) 9�/77.

[6] D.L. Simpson, B.E. Waymack, A one dimensional model of the smoldering cigarette, 31st Tobacco

Conference, Greensboro, NC, 1977.

[7] T.J. Ohlemiller, Modeling of smoldering combustion propagation, Prog. Energy Combust. Sci. 11

(1985) 277�/310.

[8] T.J. Ohlemiller, D.A. Lucca, An experimental comparison of forward and reverse smolder

propagation of permeable fuel beds, Combust. Flame 54 (1983) 131�/147.

[9] S.V. Leach, G. Rein, J.L. Ellzey, O.A. Ezekoye, J.L. Torero, Kinetics and fuel property effects on

forward smoldering combustion, Combust. Flame 120 (2000) 346�/358.

[10] J.L. Torero, A. Fernandez-Pello, Forward smolder of polyurethane foam in a forced air flow,

Combust. Flame 106 (1996) 89�/109.

[11] J. Buckmaster, D. Lozinski, An elementary discussion of forward smoldering, Combust. Flame 104

(1996) 300�/310.


[12] T.J. Ohlemiller, J. Bellan, F. Rogers, A model of smoldering combustion applied to flexible

polyurethane foams, Combust. Flame 36 (1979) 197�/215.

[13] C. Di Blasi, Modeling and simulation of combustion processes, of charring and non-charring solid

fuels, Prog. Energy Combust. Sci. 19 (1993) 71�/104.

[14] R.G. Gann, R.H. Harris, J.F. Krasny, R.S. Levine, H.E. Mitier, T.J. Ohlemiller, The effect of

cigarette characteristics on the ignition of soft furnishings, National Bureau of Standards, Technical

Note, 1241, Section 5, (1988) 139�/206.

[15] J. Szekely, J. Evans, H. Sohn, Gas�/Solid Reactions, Academic Press, New York, 1976.

[16] N. Wakao, S. Kaguei, Heat and Mass Transfer in Packed Beds, Gordon and Breach, New York,

1982.

[17] R.R. Baker, Temperature variation within a cigarette combustion coal during the smoking cycle,

High Temp. Sci. 7 (1975) 236�/247.


Documents

Modeling of a smoldering cigarette