Combustion Theory and Modelling

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Modelling of candle burning with a self-trimmed wick

M. P. Raju a; J. S. T'ien a a Department of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, OH First Published on: 25 February 2008 To cite this Article: Raju, M. P. and T'ien, J. S. (2008) 'Modelling of candle burning with a self-trimmed wick', Combustion Theory and Modelling, 12:2, 367 — 388 To link to this article: DOI: 10.1080/13647830701824171 URL: http://dx.doi.org/10.1080/13647830701824171

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Combustion Theory and Modelling Vol. 12, No. 2, April 2008, 367–388

Modelling of candle burning with a self-trimmed wick M.P. Raju∗ and J.S. T’ien Department of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, OH 44106 (Received 22 June 2007; final version received 19 September 2007) As an example of a coupled gas-phase diffusion flame with porous media flow, a candle burning model with a porous wick is offered in this paper. The porous media analysis includes capillarity-induced liquid flow, liquid vaporisation, vapour motion and re-condensation and multi-phase heat transfer. Coupling with the gas phase flame is through the conservation of mass, momentum and energy at the wick surface. The steady state solutions obtained not only yield the flame structure but also the detailed flow pattern and saturation distributions inside the wick. One of the novel features of the present model is the capability to address the self-trimming phenomena of candle burning. The self-trimming wick length and the associated flame characteristics have been computed as a function of gravity level, wick permeability and wick diameter. Keywords: candle flame; porous wick; self-trimming; flame-porous-media-flow coupling

1. Introduction Candle flame is a representative example of wick-stabilised diffusion flames. In candle flames, the liquid fuel is supplied through the wick by capillary action towards the surface of the wick where evaporation occurs. The fuel vapour is a necessary reactant for the flame in the gas phase. The flame in return provides the heat feedback to the wick. The evaporation near the wick surface creates saturation gradients that sustain the liquid capillary motion. A self-sustained candle flame is thus the result of a balanced coupling between the processes in the gas phase and inside the porous wick. One of the interesting phenomena of candle burning is the self-trimming action of the wick. In ordinary candles, the wicks are made of materials that can be burnt away at the wick tip when the local temperature becomes sufficiently high. If we start a candle flame with too long a wick, the tip burns out and the wick is trimmed. If we start too short a length, the candle wick will reach a steady length as the candle wax is slowly consumed. Since the size of the candle flame depends strongly on the wick length, the normally observed steady candle flame is the result of the wick self-trimming. Detailed models of candle flames in the past have concentrated on the gas-phase processes [1–5]. In the most complicated model [4, 5], for example, complete Navier–Stokes equations were employed to investigate the gravity effect. Finite-rate one-step chemical reaction is assumed to enable the study of extinction phenomena. Flame radiation heat transfer is included to account for the heat loss and feedback from the flame. Parametric investigation on the effect of gravity,

∗

Corresponding author. E-mail: [email protected]

ISSN: 1364-7830 print / 1741-3559 online C 2008 Taylor & Francis DOI: 10.1080/13647830701824171 http://www.tandf.co.uk/journals


368

M.P. Raju and J.S. T’ien

ambient oxygen percentage and wick size were performed. However, the model assumes a solid cylindrical wick with specified length and with its surface coated with liquid fuel at a uniform temperature. Thus, no heat and mass transport in the wick is included or needed. Clearly, this is a simplifying assumption. In the present work, two-phase (liquid and vapour) heat and mass transfer within the porous wick is treated in detail. Capillary motion is handled using Leverett’s function [6] on saturation and Darcy’s law. The formulation follows [7] but uses Gibbs function [8] to facilitate the resolution of the two-phase interface and phase change. The porous media equations are then coupled to the gas-phase equations from [4, 5]. This complete system is solved numerically including the detailed temperature and liquid/vapour fractions within the wick. Using this information, a criterion of wick trimming is then imposed based on the degree of dryness of the wick (to be discussed later). An iterative scheme is performed until both the gas and the porous phases reach convergence. The trimmed wick length and the associated gas flame and wick-phase profiles are then determined. 2. Literature review In addition to the gas-phase flame models mentioned previously [1–5], there are several porous media papers related to the present work. Experiments have been conducted [9] to study the characteristics of capillary-driven flow and phase change heat transfer in a porous structure heated with a permeable heating source at the top. Their results indicate the presence of a two-phase region and a sub-cooled liquid region below the two-phase region. Kaviany and Tao [10] performed transient experiments on the burning of liquids supplied through a wick. They have analysed the burning of a porous slab which is initially saturated with liquid fuel. The liquid is driven by capillarity and the effect of gravity is neglected. The effect of surface saturation, relative permeability and vapour flow rate on critical time has been studied. Modelling of two-phase flow inside the wick is complicated due to the presence of interface between the two-phase region and the single phase region. In a recent porous modelling effort, Benard et al. [8] have incorporated the three regions inside the porous media without the need to track the interface between the single phases and the two phase region. Using the thermodynamic Gibbs functions, the state of the porous media at any location can be determined. This greatly simplifies the modelling of two-phase flow inside the porous media. Recently Raju and T’ien [11] modelled a two phase flow inside an externally heated axisymmetric porous wick. Although there are quite extensive researches on premixed combustion inside inert porous media [e.g. 12, 13], there are very few modelling works that couples a bulk gas-phase nonpremixed flame with an adjacent porous medium. Some of the exceptions include flame spread over liquid-soaked sand [e.g. 14] and flame spread over solids involving in-depth pyrolysis [e.g. 15]. It should be noted that the combustion of this latter solid-fuel case is a very complicated one from the point of porous media transport since the porous structure created inside the solid is a part of the solution. A work similar to the present effort is the modelling of stagnation-point flame adjacent to a one-dimensional wick [16]. Because of the phase interface in the one-dimensional case only involves a single location, the problem was solved by an iterative process with the two-phase region governed by Clausius–Clayperon relation. Since the partial pressure of the vapour varies in the two-phase region, the equilibrium temperature is not uniform. 3.

Model formulation

Figure 1 shows the physical domain of the candle being modelled. The candle body is placed on a solid plate. A cylindrical wick is located at the top of the candle body. The diameter of the candle

Dw

369

x g Wick

H r


Combustion Theory and Modelling

Wax pool

x=r=0 20mm

Candle 5 mm Solid plate

Figure 1. Schematic of the candle and wick.

cylinder is 5 mm. The wick diameter, Dw , is 1 mm unless specified otherwise. The length of the candle wick is determined with the flame owing to the trimming action. On the top of the candle body a pool of molten liquid wax fuel is assumed to be present. The candle flame is located at the top and around the wick. Owing to capillary action, liquid fuel is induced from the base of the wick. The evaporated liquid wax supplies the fuel vapour to the flame and in turn the heat feedback from the flame provides the driving force for more evaporation. The heat and mass transfer taking place inside the porous wick is quite complex. Three possible regions could be expected inside the wick (single phase liquid region, two-phase region and single phase vapour region) depending on the heating conditions and the wick properties. The liquid is driven towards the surface of the wick due to capillary action induced by evaporation. The vapour in the two-phase region is driven by vapour pressure gradient induced by temperature gradient inside the wick. If the wick is sufficiently short, the wick can supply enough liquid to all parts of the wick through capillary action. As the candle wax is slowly consumed, longer wick is exposed. When the wick becomes too long, capillary action is not sufficient to supply the liquid to the top of the wick. The temperature of the dried wick increases quickly to reach a high value that causes the ‘burn-off ’ of this top portion of the wick. This phenomenon is normally referred to as ‘wick self-trimming’. Since it is related the degree of liquid saturation, detailed modelling of the heat and mass transfer processes inside the wick is needed to be able to predict self-trimming action and the length of the trimmed wick. 3.1.

Governing equations and boundary conditions

The gas phase and the porous wick of the candle are modelled separately. The gas-flame model is taken from [4, 5]. The porous media model for the wick is formulated in this work. The two models are coupled at the wick surface through mass and energy balance relations. 3.1.1.

Gas phase

As shown in Figure 1, a cylindrical wick is inserted into an inert cylinder (to mimic the candle body) with a pool of liquid fuel at the top (the melting of the wax and the thermocapillary


370


motion in the liquid pool are not modelled). The wick is assumed to maintain its cylindrical shape, i.e. no bending during candle burning process. Gravity is in the negative x direction. The flow is assumed to be steady, laminar and axisymmetric. Gas phase model assumes a one-step, second order overall kinetics, variable specific heats and thermal conductivity, constant Lewis number for each species and ideal gas behaviour for component gases. The rate expression is given as W˙ = −Aρ 2 YF YO2 exp(−Ea /Ru T ), where W˙ is the fuel vapour reaction rate, A is the preexponential factor, Ea is the activation energy, YF and YO2 represent the mass fractions of the fuel and oxygen respectively, ρ is the density of the gas phase and Ru is the universal gas constant. The activation energy and pre-exponential factor [4] are taken as 30 kcal/gmol and 3 × 1012 cm3 /g.s respectively. The fuel is assumed to be a blend of 80% (by weight) of n-paraffin wax and 20% stearic acid [1–3]. The overall stochiometric combustion reaction of a candle and air can be described as follows C25 H52 + 0.31C18 H36 O2 + 46.06[O2 + 3.76N2 ] → 30.58CO2 + 31.58H2 O + 173.19N2 Gas phase radiation flux is solved using discrete ordinates method for the radiative transfer equation. The radiation treatment is described in detail in [4, 5]. CO2 and H2 O are treated as the participating media. The medium is treated as absorbing-emitting non-scattering medium with mean absorption coefficient equal to 0.4KP , where KP is the mixture Plank-mean absorption coefficient. All the condensed surfaces (wick and solid) are assumed to be radiatively diffuse with a total emmisivity ε = 0.9 and a total absorptivity α = 0.9. In the free ambient, ε = α = 1.0. The detailed radiation formulation and the solution procedure is described in detail in [4, 5].

3.1.2. Wick phase The axisymmetric steady state, two-phase flow inside the porous wick is described by the volume averaged continuity, momentum and energy equations as given by Whitaker [7]. The resultant simplified volume averaged equations are written down for an axisymmetric, steady state, twophase flow inside a porous wick. Continuity equation ∂ 1 ∂ 1 ∂ ∂ (ρl ul ) + (ρg ug ) + (rρl vl ) + (rρg vg ) = 0 ∂x ∂x r ∂r r ∂r

(1)

The individual terms represent the net mass flux of liquid and vapor at a point in x and r directions respectively. ρ represents the density at a point, u and v represent the velocity in the x and r directions respectively and subscripts l and g refer to the liquid and vapour respectively. Momentum equations The momentum equations are given by Darcy’s law. ul =

krl K µl

−

∂Pl − ρl g ∂x

∂Pg − − ρg g ∂x ∂Pl krl K − vl = µl ∂r

krg K ug = µg

(2) (3) (4)


Combustion Theory and Modelling vg =

krg K µg

371

∂Pg − ∂r

(5)

The liquid phase is treated as incompressible and the gaseous phase is treated as an ideal gas. P represents the pressure, k and K represent the relative and the absolute permeability of the porous wick respectively, µ is the dynamic viscosity and g represents the gravity. Energy equation ∂ 1 ∂ (ρl hl ul + ρg hg ug ) + (r(ρl hl vl + ρg hg vg )) ∂x r ∂r ∂ 1 ∂ ∂T ∂T = keff + rkeff ∂x ∂x r ∂r ∂r

(6)

Using the relations hl = cl T and hg = cg T + ifg , equation (6) becomes ∂ρl ul 1 ∂rρl vl ∂ 1 ∂ (ρl cl ul + ρg cg ug )T + (r(ρl cl vl + ρg cg vg ))T + ifg − − ∂x r ∂r ∂x r ∂r 1 ∂ ∂ ∂T ∂T keff + rkeff = ∂x ∂x r ∂r ∂r

(7)

The first two terms on the left- hand side of equation (7) represent the convective heat transport of liquid and vapour in the x and r directions. The third term represents the heat source term due to phase change taking place between the liquid and the vapour. The right-hand side of this equation represents the conductive heat transfer. h represents the enthalpy, c represents the specific heat, T represents the temperature, ifg represents the latent heat of wax and keff represents the effective thermal conductivity. The wick temperature is non-dimensionalised with the boiling point of wax, Tb . Capillary and permeability relations. The gas pressure is related to the liquid pressure using the capillary relation Pc (s) = Pg − Pl

(8)

The capillary pressure is related to the saturation by Leverett’s function [6], Pc =

σ [1.42(1 − s) − 2.12(1 − s)2 + 1.26(1 − s)3 ] (K/ε)1/2

(9)

In equation (9) s represents the saturation, σ represents the surface tension and ε represents the porosity of the wick. The relative permeability of the porous media is given by the following approximation [17] equation krl = s, krg = (1 − s)

(10a–b)

The non-dimensionalisation of the porous wick variables is carried out according to the variables indicated in Table 1. The non-dimensionalised variables are indicated by a ‘hat’ symbol on the top of the variable.


372

M.P. Raju and J.S. T’ien Table 1. Porous wick dimensionless variables.

Variable

Value

cˆg

cg cl g ρg (K/ε) 12 Dw σ if g cl Tb

gˆ iˆfg Krg Krl Pe Pˆg gˆ Tˆ uˆ g uˆ l ˆ rˆ x, µˆ g ρˆg kêff , kˆs

(1 − s) s

ε σ (K/ε) 12 α1 µ1 pg (K/ε) 12 σ ρ1 Rg To (K/ε) 12 σ T Tb u g Dw αl u l Dw αl x , Drw D µgw µ ρgl ρl keff ks , kl kl

Non-dimensionalised equations ∂ ∂ 1 ∂ 1 ∂ (uˆ l ) + (ρˆg uˆ g ) + (ˆr vˆl ) + (ˆr ρˆg vˆg ) = 0 rˆ ∂ rˆ rˆ ∂ rˆ ∂ xˆ ∂x ∂ Pˆl uˆ l = P ekrl − − gˆ ∂ xˆ P ekrg ∂ Pˆg uˆ g = − ρˆg gˆ − µˆ g ∂ xˆ ∂ Pˆl vˆl = P ekrl − ∂ rˆ P ekrg ∂ Pˆg vˆg = − µˆ g ∂ rˆ ∂ uˆ l 1 ∂ rˆ vˆl 1 ∂ ∂ (uˆ l + ρˆg cˆg uˆ g )Tˆ + (ˆr (vˆl + ρˆg cˆg vˆg ))Tˆ + iˆfg − − rˆ ∂ rˆ rˆ ∂ rˆ ∂ xˆ ∂ xˆ ∂ ˆ ∂ Tˆ ∂ Tˆ 1 ∂ = keff rˆ kêff + rˆ ∂ rˆ ∂ xˆ ∂ xˆ ∂ rˆ

(11) (12)

(13)

(14)

(15)

(16)

where Pe represents the Peclet number which is defined in Table 1. The variable αl defined in Peclet number is the thermal diffusivity of liquid wax.



373

The effective thermal conductivity of the wick is function of saturation given by the expression [18] kêff = s + (1 − s) kˆs

(17)

where kˆs is the thermal conductivity of the solid wick material. Conditions for phase transition. The equilibrium thermodynamic state of candle wax (single phase liquid, single phase vapour, two-phase) can be determined for given liquid and gaseous pressures and temperature conditions, using the vapour pressure equilibrium data obtained from the thermodynamic Gibbs phase relationships. ˆ l = hˆ l − Tˆl ηˆ l G

(18)

ˆ g = hˆ g − Tˆg ηˆ g G

(19)

where G is the Gibbs potential per unit mass of the corresponding phase and symbol ‘hat’ denotes the non-dimensional value. h and η are respectively the enthalpy and the entropy of the corresponding phase. The dimensional expressions for the enthalpy and entropy are given by hl = cl (T − Tb )

(20)

hg = ifg + cg (T − Tb )

(21)

ηl = cl log(T /Tb )

(22)

ηg = ifg /Tb + cg log(T /Tb ) − R log (Pg /Pb )

(23)

where, P0 represents the atmospheric pressure. The thermodynamic equilibrium relations based on the minimisation of Gibbs function are given as follows ˆl < G ˆ g , no vapour phase (s = 1) State 1. G

(24)

ˆl = G ˆ g , liquid and vapour are in equilibrium (0 < s < 1) State 2. G

(25)

ˆl > G ˆ g no liquid phase is present (s = 0), State3. G

(26)

Equations (24)–(26) simply mean that whichever phase has the least Gibbs phase potential will dominate over the other phase. Equations (18)–(23) in conjunction with the phase equilibrium condition (equation (25)) yield the well known, Clausius–Clapeyron equation. 3.1.3. Boundary conditions Base of the wick. The wick is immersed in the candle liquid wax pool, which can be assumed to be at its melting temperature (subscript m represents the melting point). s = 1, Tˆ = Tˆm Cylindrical surface of the wick. Depending on the thermodynamic state of the wick on the surface, the boundary conditions will vary. In the case of either pure liquid or pure vapour, all the heat flux imposed on the surface is conducted into the wick. In the case of two-phase region, part of the heat supplied is used for evaporating the liquid on the surface of the wick and part of the


374


heat is conducted into the wick. The total pressure is assumed to be at one atmosphere. (subscript brepresents the boiling point) Liquid region: s = 1,

vˆl = 0,

qˆf = −kêff

Two - phase region: 0 < s < 1,

qˆf = −kêff

∂ Tˆ ∂ rˆ

Vapour region: s = 0,

vˆl = 0,

qˆf = −kêff

∂ Tˆ + vˆl iˆfg , Tˆ = Tˆb ∂ rˆ ∂ Tˆ ∂ rˆ

where qˆf represents the non-dimensional heat flux at the surface of the wick. Top of the wick. The boundary conditions are similar to those on the cylindrical surface except that the derivatives are with respect to x. Symmetry line. Symmetry boundary conditions are imposed along the symmetry line Coupling between the gas phase and the wick. The gas phase and the wick are coupled along the surface. The heat flux qf obtained from the gas phase is input to the wick and the fuel vapour flux normal to the wick surface is input to the gas phase. The species velocity of the non-condensable gases (gases other than fuel (F) like CO2 , H2 O, O2 , N2 ) are taken as zero at the surface. The effect of non-condensibles inside the wick is neglected. Although the presence of non-condensibles affect the temperature and the gas pressure distribution inside the wick, the thermodynamic relationships indicates that the drop in the boiling temperature of the fuel is low as long as the gaseous fuel concentration next to the surface of the wick is reasonably high (YF > 0.5). Mass balance on the wick surface. ˙ = (m ˙l +m ˙ g )wick m ∂YF ˙ − YF ) = m(1 ∂ nˆ ∂Yi ˙ i (i = O2 , CO2 , H2 O, and N2 ) ρDi = mY ∂ nˆ Yi = 1 (i = F, O2 , CO2 , H2 O and N2 )

−ρDF

˙ g are ˙ is the total mass flux supplied by the wick normal to the wick surface, m ˙ l and m where m the individual liquid and vapour mass flux normal to the wick surface. D represents the diffusion coefficient and Y represents the mass fraction and nˆ represents the direction normal to the wick surface. Energy balance at the wick surface. λ

∂T + qr,in = qwick + qr,out ∂ nˆ

where the first term on the left-hand side represents the heat conducted from the flame to the wick surface, qr,in and qr,out represents the incident radiation into the wick surface and the outward



375

radiation from the wick surface, qwick is the net heat supplied to the wick. λ represents the thermal conductivity in the gas phase.

3.1.4. Wick-trimming criterion As mentioned previously, wick self trimming occurs when the wick tip becomes dried up. The lack of liquid prevents evaporative cooling action to occur which keeps the wick temperature down. Without evaporative cooling the wick tip temperature increases quickly. With little oxygen under the flame, the wick tip is pyrolysed when the pyrolysis temperature is reached (typically 600–800 K for many wick materials). Experimentally, this is manifested by the red coloured tip. The details of the pyrolysis reaction can be complex and depend on the particular material used for the wick. Since the portion of the wick that undergoes pyrolysis is small compared with the entire length of the wick, the following approximation will be made in the present model. We will assume that the wick is trimmed as soon as the local saturation in the wick reaches zero, i.e. becomes dried. By ignoring the pyrolysis portion, physical details are simplified without losing much accuracy on the computed length of the trimmed wick. A second approximation has to do with the numerical model. If we start with a cylindrical wick with a flat top as sketched in Figure 1, one would expect that the first part that will be dried is the corner rim of the top surface. This will lead to a rounded wick at the top. The exact shape of the rounded portion will be a part of the solution. This complication is, however, neglected in the present work. The wick top is assumed to be flat all the time. Trimming is assumed to occur when the rim (corner) saturation reached zero.

4. Solution procedure The differential equations in the gas phase and the wick are descretised using conventional finite volume differencing technique over a non-uniform mesh. In the gas phase a grid size of 110 × 85 (see Figure 2(a)) is chosen for a physical domain of 50 × 40 cm. The detailed solution procedure is described in Alsairafi [5]. In the wick region, only half of the domain is computed taking advantage of the symmetry. Note that gravity is in the negative x direction. The candle shoulder is situated at the left bottom corner of the grid with the wick situated right of the candle shoulder. A grid size of 80 × 40 (see Figure 2(b)) is chosen for the wick. The input parameters for the wick region are presented in Table 2. The set of non-linear equations obtained from the descretisation is solved using Newton’s formulation. A direct solver based on multifrontal technique (UMFPACK, Unsymmetric MultiFrontal PACKage [19–21]) is used solve the linear system obtained during each Newton iteration.

5. Computed results 5.1. Flow profiles in the wick and associated flame structure for a self-trimmed candle flame At normal gravity, 21% oxygen and one atmosphere pressure for a wick with diameter 1 mm, the self-trimming wick length is found to be 4.34 mm. Figure 3(a) shows the saturation profiles for a self-trimmed wick. The saturation profiles indicate the presence of two-phase region and liquid phase region separated by an interface (s = 1). Figure 3(b) shows the enlarged view of the two-phase region. The saturation at the corner of the tip of the wick surface reaches zero. Since the saturation contours are not flat, the trimming of the wick will change the shape of the wick tip. But as mentioned before, this is being neglected in this work since we expect that overall

M.P. Raju and J.S. T’ien 45 40 35

r (cm)

30 25 20 15 10 5 0

0

10

20

30

40

50

X (cm) 80x40

g

0.5 0.4

r (mm)


376

0.3 0.2 0.1 0

0

1

2

X (mm)

3

4

5

Figure 2. Computational grids (half domain) for the (a) gas phase and (b) candle wick.

distribution of the saturation curves will not be affected much from the change of the shape of the wick tip. Figure 3(c) shows the non-dimensional temperature contours inside the wick. The slight temperature gradient at the base of the wick indicates some heat is being lost to the wax pool. Figure 3(d) shows the enlarged view of temperature distribution inside the two-phase region. The variation of temperature inside the two-phase region is extremely small. This temperature gradient, even though small, causes a vapour pressure gradient (as determined by the Gibb’s equilibrium relations) in the two-phase region. Figure 4 shows the non-dimensional pressure distribution inside the wick. This vapour pressure gradient causes the vapour to move inwards. The liquid pressure gradient drives the liquid from the wax pool to the surface of the wick. Figure 5 shows the liquid and the vapour mass fluxes inside the wick. The liquid and the vapour appear to move in countercurrent fashion inside the two-phase region of the wick. The liquid is evaporated from

377

Parameter

Unit

Value

cg cl ifg ke (s = 0) ke (s = 1) K Rg Tm Tb ε αl ρl µl µg σ

J/kg-K J/kg-k J/kg W/m-K W/m-k m2 J/kg-K K K

1430 2452 8.8 × 105 6.40 6.31 1.09 × 10−11 25.2 330 620 0.55 3.4 × 10−6 770 5 × 10−4 1.1 × 10−5 0.035

m2 /s kg/m3 Pa s Pa s N/m Saturation contours

0.6 0.5 0.4 0.3 0.2 0.1 00

0. 68

1.000

1

2

X/Dw

3

34 0.1 .242 0 0.320 1

r/Dw

Table 2. Porous wick numerical values.

0.43

3

4

5

(a) Saturation contours

0.6

r/Dw

0.5

0.007 0.022

0.134

0.4

0.198

0.3 0.2 4.2

4.25

4.3

4.35

X/Dw

4.4

(b) 0.6 0.5 0.4 0.3 0.2 0.1 00

Non-dimensional temperature contours

0.554 210

r/Dw

0.7 54 63 5

0.9

0.9 20 22 9

1

74

2

63

0.999918 1

0.999083

X/Dw

3

4

5

(c) Non-dimensional temperature contours (expanded in two-phase region) 0.99

99

0.9

99

9980 0.999990

95

00

99 0

0.5 0.4 0.999774 0.3 0.999143 0.2 0.1 03 3.5

0.

r/Dw



4 X/Dw

4.5

5

(d) Figure 3. Plot of (a) saturation profiles (b) saturation profiles expanded in the two-phase region (c) nondimensional temperature (non-dimensionalised by 330 K) profiles and (d) non-dimensional temperature profiles (expanded in the two-phase region) inside the porous wick for a self-trimmed candle flame at normal gravity.

M.P. Raju and J.S. T’ien Non-dimensional liquid pressure contours

0.6

6.39099

r/Dw

X/Dw

4

Non-dimensional capillary pressure contours

0.6

0 0.47 038

27

9

6 72

0.2

0.4114

1

61

0.3

010

06

0.3

51

47 0.1

0.4

0.36204

0.0 1

0.5

r/Dw

3

962

2

6.0249

1

6.06356

0

6.09508

0

6.19095

0.1

6.22968

0.2

6.28962

0.3

6.13331

6.15851

0.4

5.97

0.5

6.3398

0.1 0

0

1

2

3

X/Dw

4

Non-dimensional vapor pressure contours

0.6 0.5

1 36 6.4 8 058 6.4 92 6.310

0.4

r/Dw

0.3 0.2 0.1 0

0

1

2

3

X/Dw

4

r/D w

Figure 4. Plot of non-dimensional pressure contours: liquid pressure (top) capillary pressure (middle) and gas pressure (bottom) inside the porous wick for a self-trimmed candle flame at normal gravity. 0.5 0.4 0.3 0.2 0.1 0

2

10 kg/m s

0

1

2

3

4

X/D w (a)

r/D w


378

0.5 0.4 0.3 0.2 0.1 0

2

0.1 kg/m s

1

2

3

4

X/D w (b)

Figure 5. Plot of (a) liquid mass flux vectors (b) vapour mass flux vectors inside the porous wick for a self-trimmed candle flame at normal gravity.

379

18 16 14

qwick (W/cm2)



12 10 8 6 4 2 0

0

0.25

0.5

0.75

1

X/Dw Figure 6. Plot of net heat flux supplied by the candle flame along the lateral cylindrical surface of the wick.

all along the cylindrical surface and the tip of the wick. Part of the vapour, which is evaporated at the surface, traverses into the wick. Near the tip of the wick, the vapour movement is almost negligible. This liquid-vapour counter-flow and vapour re-condensation phenomena inside the wick has also been found previously in the one-dimensional analysis in [16]. Figure 6 shows the net heat flux supplied by the flame to the wick for evaporation. The heat flux distribution is nearly constant along the cylindrical surface of the wick except near the base of the wick and at the top corner of the wick. The rise in heat flux near the top of the wick might be due to the two-dimensional effect of the wick geometry. The top corner receives heat both from the top and the side. Figure 7 shows the saturation and non-dimensional temperature profiles along the cylindrical surface of the wick which is exposed to the candle flame. The figure shows that the saturation reaches zero at the tip of the wick. Near the base of the wick, there is small region which is liquid. Here no evaporation takes place at the surface and all the heat supplied at this region of the wick surface is simply conducted into the wick. The temperature in this region is well below the boiling temperature of the wax. There is also a temperature gradient at the base of the wick which indicates some heat lost to the wax pool through conduction inside the wick. Figure 8(a) shows the non-dimensional isotherms in the gas phase (non-dimensionalised by T∞ = 300 K). Figure 8(b) shows the reaction rate contour equal to 5 × 10−5 g cm−3 s−1 . This contour is defined as the boundary of the visible flame [22]. The flame length H is defined as the difference between the locations of the flame tip and flame base. The flame diameter D is defined as twice of the largest perpendicular distance from the line of symmetry to the reaction rate contour of the visible flame. Figure 8(b) shows a flame length of 2.8 cm and a flame diameter of 0.74 cm. In this candle flame, the location of the maximum temperature occurs at x = 2.7 cm and y = 0, i.e. Tmax =2135K. In normal gravity, the maximum temperature location is close to the flame tip. In addition, there is a small quenching distance(about 1 mm), which detaches the flame from the candle wax. The detailed flow field in the gas phase is qualitatively similar to the results obtained by Alsairafi et al. [4, 5] and hence is not presented in this paper.

M.P. Raju and J.S. T’ien T

s

1

1

0.8

0.9

s

saturation

0.7

0.85

T

0.6

0.8

0.5

0.75

0.4

0.7

0.3

0.65

0.2

0.6

0.1

0.55

0

Non-dimensional temperature

0.95

0.9

0.5 0

1

2

3

4

X/Dw Figure 7. Plot of saturation and temperature variation along the cylindrical surface of the wick for a self-trimmed candle flame at normal gravity.

2.5 1ge

r (cm)

2 1.5

Tmax = 7.117

1

1.1

0.5 0-1

0

2

4

1

6

7 2

3

4

5

X (cm) (a) 1.5 1

r (cm)


380

1ge

0.5

5E-05

0 -0.5 -1 -1.5-1

0

1

2

3

4

X (cm)

(b) Figure 8. Gas temperature contours for a self-trimmed candle flame (non-dimensionalised by T∞ = 300 K); (b) fuel reaction rate contour (5 × 10−5 g cm−3 s−1 ) for a self-trimmed candle flame.


Combustion Theory and Modelling 5.2.

381

Effect of gravity

Gravity can affect processes both in the gas phase and in the porous wick. For the wick parameters chosen in this study, the self-trimmed wick is relatively short so gravity does not have a significant effect on the steady state heat and mass transport inside the wick directly [23] (in other word, gravity term can be neglected in the porous phase equations). Gravity does affect the gas phase, through the buoyancy term [4, 5]. And the heat feedback from the flame affects the saturation profiles in the porous phase and the self-trimming wick length. The wick length, in turn, influences the gas flame size. So through the coupling processes, gravity has a profound effect on candle burning. In this work, the gravitational acceleration has been varied as a parameter over the entire range (from zero to high gravity) within which self-trimming action occurs. Figure 9 shows the computed visible flames at different gravity levels for a self-trimmed candle flame in standard air conditions (1 atm and 21% O2 ). The reaction rate contour of wf = 5 × 10−5 g cm−3 s−1 is chosen to represent the boundary of the visible flame, which is a function of the local fuel, oxygen and temperature. In this series of study, the candle and wick dimensions are: wick diameter = 1 mm, candle body height = 2 cm, and candle body diameter = 5 mm. The wick length will be that of the self-trimming length which is determined as a part of the solution. Results reveal that at zero and micro gravities, the flames are wide and short as having been previously found in both experiments and computations [1–5, 24]. The flame becomes longer and narrower while the trimmed wick length becomes shorter as gravity increases. The shorter trim length is due to the decrease of flame standoff distance from the wick and the increased heat flux to the wick with gravity. The longer flames are the results of greater buoyant flow velocity and greater total burning rate (a small exception will be explained later). Note that there is no self-trimming observed for gravity levels greater than 5.1ge . The bottom figure in Figure 9 shows, for example, using the self-trimmed wick length corresponding to the 5.1ge case, that at 5.2 ge , the flame is blown off from the side of the wick and anchored at the top of the wick. Because the total burning rate is small in this configuration [4, 5, 23], plenty of liquid fuel is now able to reach the tip of the wick so it is no longer a self-trimmed situation. Furthermore, since the flame base is at a greater distance from the candle body, the total heat feedback (conduction plus radiation) may not be sufficient to sustain a liquid wax pool on top of the candle beneath the wick. In such situation, the candle flame will go out, although a liquid fuel lamp flame in the same condition may survive [4, 5]. Theoretically, for a liquid-fuelled lamp, a self-trimmed solution with very long wick (so that gravity weight of the liquid needs to be counted) can exist for the low-heat-feedback wake-stabilised flame. But this appears to be a not very realistic case and is not investigated here. Figure 10 shows the effect of gravity on the total burning rate and the self-trimming wick length of the candle. As mentioned before the trim length decreases monotonically with gravity. The burning rate first increases with gravity rapidly but reaches a maximum at about 2.5ge . It then decreases gradually until 5.1ge beyond which the side-stabilised flame blows off. The reason for this non-monotonic behaviour is due to the competition of heat flux and wick length. The total burning rate of the candle is the integration of the local burning rate (proportional to the local heat flux) over the entire wick surface. When gravity increases, the local heat flux increases but trim length decreases. The competition results in a maximum of the total burning rate. It should be noted, however, that the variation of total burning rate from 2.5 ge to 5.1 ge is relatively small. 5.3.

Effect of wick permeability

Wick permeability is a measure of the ability to transport fluid through the porous wick. Wicks with high permeability, offers less resistance to fluid motion. In general, wick permeability is


r (cm)

1

0ge

0.5

0 -1

0

1

2

3

4

5

x (cm)

r (cm)

1 -4

10 ge

0.5

0 -1

0

1

2

3

4

5

x (cm)

r (cm)

1 -2

10 ge

0.5

0 -1

0

1

2

3

4

5

x (cm)

r (cm)

1

1ge

0.5

0 -1

0

1

2

3

4

5

x (cm)

r (cm)

1

3ge

0.5

0 -1

0

1

2

3

4

5

x (cm)

r (cm)

1

5.1ge 0.5

0 -1

0

1

2

3

4

5

x (cm) 1

r (cm)


382

5.2ge

0.5

0 -1

0

1

2

3

4

5

x (cm)

Figure 9. Candle flames at various gravity levels for a self-trimmed candle.

a function of the wick porosity, the structure of the wick and the wick material. The wick permeability affects the saturation distribution inside the wick for a given heat feed back from the flame. This would affect the self-trimming length of the candle wick and hence the flame structure and the burning rate.

383

1.3 1.2 1.1

10 0.8

8

0.6

6

0.4

4

0.2

2

0

1

10

-1

g/ge

10

0

0

12 11 10 9 8

0.9 0.8

7

0.7

6

0.6

5

0.5

4

0.4

no self trimming observed beyond g = 5.1ge

0.3

2

0.2

1

0.1 0

3

Self trimming wick length (mm)

Total burning rate (mg/s)

1.4


12 1




0

5

10

0 15

g/ge Figure 10. Candle burning rate and self-trimmed wick length of the candle at various gravity levels for a self-trimmed candle flame.

Figure 11 shows the effect of wick permeability on the self-trimming length of the candle wick at two different gravity levels. It is observed that as the permeability is decreased, the level to which the liquid fuel rises above the candle shoulder decreases due to increased resistance to fluid motion. Therefore the self-trimming length of the wick is reduced. Figure 12 shows the variation of flame lengths with the wick permeability. The flame lengths are proportional to the exposed wick lengths and hence the trends are similar to that of the self-trimming length variations shown in Figure 11. Figure 12 also shows the variation of burning rate of the candle flame with permeability of the wick. The burning rate decreases with decrease in permeability. This is caused directly by the decrease in self-trimming length of the candle wick. The exposed length is reduced and hence the burning rate decreases. In brief, we can conclude that the wick permeability significantly affects the self-trimmed wick length, the flame structure and the burning rate of the candle flame. Therefore while conducting experiments on candle flames or wick stabilised flames; the wick permeability should also be taken into account.

5.4. Effect of wick diameter Increasing wick diameter increases the wick surface area per unit length of the wick. Thus for a wick of fixed length such as that specified in [4, 5], more liquid fuel is vaporised that yields a larger and longer flame. For the present self-trimmed candles, there is an additional factor that influences the flame appearance; a larger diameter wick also increases the self-trim length. Computed results in several low and normal gravity levels using three different wick diameters are summarised in Table 3(a). The computed results in [4, 5] for a specified 5 mm wick coated with liquid fuel are also shown in Table 3(b) for comparison.



8

7

-2

10 ge

6

5

1ge 4

3

0

2.5E-12

5E-12 2

Absolute Permeability (m ) Figure 11. Self-trimming length of the candle wick using different wick permeabilities.

Table 3(a), Figures 13 and 14 show that the self-trimmed wick length, the flame diameter, the flame height and the total candle burning rate increase with wick diameter. For a fixed wick length (5 mm), the flame diameter, the flame height and the total candle burning rate also increase with wick diameter as shown in Table 3(b) but to a less extent for the reason just mentioned. We also note that the maximum flame temperature decreases as the wick diameter is increased and as gravity is decreased. This is attributable to the flame radiative loss as discussed in [4, 5]. In the self-trimmed candle, the flame temperature is slightly lower because in most of the cases 1.2

4

1.1

3.8

1ge

3.6

0.9 0.8

3.4

10 -2ge

3.2

0.7

1ge

0.6

3 2.8

0.5 0.4

2.6

10 -2ge

0.3

2.4

0.2

2.2

0.1 0

2 2.5E-12

5E-12 2

Absolute Permeability (m ) Figure 12. Candle burning rates and flame lengths for different wick permeability.

Flame Length (cm)

1



384



385

Table 3(a). Effect of wick diameter and gravity on candle flame characteristics. Wick diameter Dw (mm) 1 2 3 1 2 3 1 2 3 1 2 3

Gravity level g/ge

Self-trimmed wick length (mm)

Flame diameter (cm)

Flame length (cm)


1 1 1 1 × 10−2 1 × 10−2 1 × 10−2 1 × 10−4 1 × 10−4 1 × 10−4 0 0 0

4.36 7.71 9.12 5.825 11.41 16.45 9.2 15.83 Ext. 10.1

0.7413 0.9388 0.9715 1.4931 2.0367 2.304 1.962 2.304 Ext. 1.2845 Ext. Ext.

2.8034 5.2727 6.8402 2.2623 3.604 4.24 1.1887 1.4576 Ext. 0.9212 Ext. Ext.

0.9222 2.147 4.0079 0.7027 1.592 2.308 0.34036 0.7040 Ext. 0.1348 Ext. Ext.

Maximum flame temperature (K) 2138 2130 2120.4 1769 1731.6 1703.2 1186.0 1163.5 Ext. 1110.5 Ext. Ext.

computed, the wick length is greater than 5 mm (used in Table 3(b)) and the bigger flame induces more radiative loss. Consequently it is more prone to extinction as can be seen in the case of Dw = 3 mm, g/ge = 10−4 in Table 3(a) (extinction) and in Table 3(b) (no extinction). It should be noted that although candle is used in the title and in all the discussions, the method used in this paper is equally applicable to other wick stabilised liquid-fueled flames such as in oil lamps. In the latter case, however, wick self-trimming will not occur if the initial wick length is shorter than the theoretical value since, unlike candle, the exposed wick length will not increase during the burning process. Finally, we like to make a remark concerning experiments and the comparison with the present model results. The present model shows that the wick properties such as permeability affect the self-trim wick length and hence the flame dimensions. But there is no such data reported on candle wicks that we can find. So although all the predictions of the model appear to be qualitatively correct and quantitatively in the right ballpark, a detailed comparison with experiments is not yet Table 3(b). Effect of wick diameter and gravity on candle flame characteristics with a specified wick length at 5 mm wick length (from [4, 5]) Wick diameter Dw (mm) 1 2 3 1 2 3 1 2 3 1 2 3

Gravity level g/ge

Specified wick length (mm)

Flame diameter (cm)

Flame length (cm)


1 1 1 1 × 10−2 1 × 10−2 1 × 10−2 1 × 10−4 1 × 10−4 1 × 10−4 0 0 0

5 5 5 5 5 5 5 5 5 5 5 5

0.77794 0.8565 0.9715 1.3833 1.5953 1.8154 1.751 1.8554 1.9603 1.4931 Ext. Ext.

3.2627 4.8356 5.6702 2.1577 2.9164 3.6891 1.2424 1.3136 1.5635 0.9450 Ext. Ext.

1.09347 1.6131 2.0073 0.61362 0.846 1.133 0.24 0.264 0.374 0.15855 Ext. Ext.

Maximum flame temperature (K) 2142 2130 2120.4 1777 1760 1743 1216 1166 1176 1130.5 Ext. Ext.

M.P. Raju and J.S. T’ien 20


18 16 14 -2

10 ge 12 10

1ge

8 6 4 2 0 0.5

1

1.5

2

2.5

3

Wick diameter (mm) Figure 13. Self-trimming length of the candle wick for different wick diameters. 5 4.5



386

4

1ge

3.5 3 2.5

-2

10 ge

2 1.5 1 0.5 0 0.5

1

1.5

2

2.5

3

Wick diameter (mm) Figure 14. Candle burning rates for different wick diameters.

possible at this moment. Furthermore, to achieve a self-trimmed steady state, the experiments may require sufficiently long period of time that can be a problem in studying the effect of gravity.

6.

Concluding remarks

A more complete model of candle burning is offered in this work by coupling of the transport processes inside the porous wick with the gas-phase flame. The porous media analysis includes capillarity-induced liquid flow, liquid vaporisation, vapour motion and re-condensation and



387

multi-phase heat transfer. The steady-state solution obtained not only yields the flame structure but also the detailed flow pattern and saturation distributions inside the wick One of the novel features of the present model is the capability to address self-trimming phenomena of candle burning. The self-trimming length of the candle wick is determined by the point in the wick where the liquid saturation reaches zero. Based on this criterion, the self-trimming wick length and the associated flame characteristics have been computed as a function of gravity level, wick permeability and wick diameter. It should be noted that saturation distribution in the wick depends on both the porous and gas-phase parameters. Consequently, the modelling and experiment of this type of interactive process require not only the specification of the gas-phase parameters but also the porous media properties which often are neglected.

Acknowledgments This work has been supported by NASA grant NNC04GB34G with Dr Daniel Dietrich as the grant monitor.

References [1] Y. Shu et al., Modeling of candle flame and near-extinction oscillation in microgravity, J. Jpn. Soc. Microgravity Appl. 15 (1988), pp. 272–277. [2] Y. Shu, Modeling of candle flame and near-limit oscillation in microgravity, M.S. thesis, Case Western Reserve University, Cleveland, OH, 1998. [3] D.L. Dietrich, et al., Candle flames in non-buoyant atmospheres, Combust. Sci. Tech. 156 (2000), pp. 1–24. [4] A.S. Alsairafi, T. Lee and J.S. T’ien, Modelling gravity effect on diffusion flames stabilised around a cylindrical wick saturated with a liquid fuel, Combust. Sci. Tech. 176 (2004), pp. 2165–2191. [5] A. Alsairafi, A computational study on the gravity effect on wick-stabilized diffusion flames, Ph.D. Dissertation, Case Western Reserve University, Cleveland, OH, 2003. [6] M.C. Leverett, Capillary behavior in porous solids, AIME Trans 142 (1941), pp. 152–157. [7] S. Whitaker, Simultaneous heat, mass and momentum transfer in porous media: a theory of drying, Adv. in Heat Transfer 13 (1977), pp. 119–203. [8] J. Benard et al., Boiling in porous media: model and simulations, Trans. Porous Media 60 (2005), pp. 1–31. [9] T.S. Zhao and Q. Liao, On capillary driven flow and phase change heat transfer in a porous structure heated by a finned surface: measurements and modeling, Int. J. Heat Mass Transfer 43 (2000), pp. 1141–1155. [10] M. Kaviany and Y. Tao, A diffusion flame adjacent to a partially saturated porous slab: funicular state, J. Heat Transfer 110 (1988), pp. 431–436. [11] M.P. Raju and J.S. T’ien, Two phase flow inside an externally heated axisymmetric porous wick, J. Porous Med. Accepted for publication. [12] J.R. Howell, M.J. Hall and J.L. Ellzey, Combustion of hydrocarbon fuels within porous inert media, Prog. Energy Combust. Sci. 22 (1996), pp. 32–64. [13] Y. Huang, C.Y.H. Chao and P. Cheng, Effects of preheating and operation conditions on combustion in a porous medium. Int. J. Heat Mass Transfer 45 (2002), pp. 4315–4324. [14] K. Takeno and T. Hirano, Behavior of combustible liquid soaked in porous beds during flame spread. Proceedings of the 22nd Symposium (International) on Combustion, 1988, pp. 1223–1230. [15] H.R. Baum and A. Atreya, A model of transport of fuel gases in a charring solid and its application to opposed-flow flame spread, Proc. Combust. Inst. 31 (2007), pp. 2633–2641. [16] M.P. Raju and J.S. T’ien, Heat and mass transport in a one-dimensional porous wick driven by a gas phase diffusion flam, J. Porous Med. 10 (2007), pp. 327–342. [17] H.H. Bau and K.E. Torrence, Boiling in low-permeability porous media, Int. J. Heat Mass Transfer 25 (1982), pp. 45–55. [18] K.S. Udell and J.S. Flitch, Heat and mass transfer in capillary porous media considering evaporation, condensation and non-condensable gas effects, Heat transfer in porous media and particulate flows, ASME HTD 46 (1985), pp. 103–110.


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[19] T.A. Davis, Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method, ACM Trans. Math. Softw. 30 (2004), pp. 196–199. [20] T.A. Davis and I.S. Duff, 1997, An unsymmetric-pattern multifrontal method for sparse LU factorization, SIAM J. Matrix Anal. Applic. 18 (1997), pp. 140–158. [21] M.P. Raju. and J.S. T’ien, Development of multifrontal solvers for combustion problems, Numerical Heat Transfer. B 53 (2008), pp. 191–207. [22] G.D. Grayson, K.R. Sacksteder, P.V. Ferkul and J..S. T’ien, Flame spreading over a thin solid in lowspeed concurrent flow- drop tower experimental results and comparison with theory, Microgravity Sci. Tech. VII/2 (1994), pp. 187–195. [23] M. Raju, Heat and mass transport inside a candle wick, Ph.D. Dissertation, Case Western Reserve University, Cleveland, OH, 2007. [24] H.D. Ross, R.G. Sotos and J.S. T’ien, Observation of candle flames under various atmospheres in microgravity, Combust. Sci. Tech., Short Communication 75 (1991), pp. 155–160.

Combustion Theory and Modelling

Combustion Theory and Modelling

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