Lecture 4 Laminar Premixed Flame Configura6on 4.- 1
Bunsen Burner Classical device to generate a laminar premixed flame Gaseous fuel enters into the mixing chamber, into which air is entrained Velocity of the jet entering the mixing chamber may be varied Entrainment of air and mixing can be opcmized 4.- 2
Mixing chamber must be long enough to generate a premixed gas issuing from the Bunsen tube into the surroundings If the velocity of the issuing flow is larger than the laminar burning velocity to be defined below, a Bunsen flame cone establishes at the top of the tube 4.- 3
Kinema6c Balance for Steady Oblique Flame Incoming flow velocity v u of the unburnt mixture (subscript u) is split into a component v t,u tangencal to the flame and a component v n,u normal to the flame front Due to thermal expansion within the flame front, normal velocity component is increased, since the mass flow density ρv n through the flame must be the same in the unburnt mixture and in the burnt gas (subscript b) TangenCal velocity v t is not affected by gas expansion: 4.- 4
Vector addicon of the velocity components in the burnt gas leads to v b, which points into a direccon which is deflected from the flow direccon of the unburnt mixture Since flame front is staconary, the burning velocity s L,u with respect to the unburnt mixture must be equal to the flow velocity of the unburnt mixture normal to the front where α is the Bunsen flame cone angle This allows to experimentally determine an escmate of the burning velocity by measuring the cone angle α under the condicon that the flow velocity v u is uniform across the tube exit 4.- 5
A parccular phenomenon occurs at the flame Cp If the Cp is closed, the burning velocity at the Cp is equal to the flow velocity Therefore the burning velocity at the flame Cp is by a factor 1 / sin α larger than burning velocity through the oblique part of the cone ExplanaCon: The strong curvature of the flame front at the Cp leads to a preheacng by the lateral parts of the flame front and thereby to an increase in burning velocity A detailed analysis of this phenomenon also includes the effect of non- unity Lewis numbers by which, for instance, a difference between lean hydrogen and lean hydrocarbon flames can be explained 4.- 6
Finally, one observes that the flame is detached from the rim of the burner This is due to conduccve heat loss to the burner which leads in regions very close to the rim to temperatures, at which combuscon cannot be sustained 4.- 7
Spherical Constant Volume Combus6on Vessel Experimental device to measure laminar burning velocices Flame is inicated by a central spark Spherical propagacon of a flame then occurs May be detected opccally through quartz windows and the flame propagacon velocity dr f /dt may be recorded In this experiment neither flame front posicon nor displacement speed are staconary 4.- 8
Mass flow through flame front with velocity relacve to flame front posicon u At the flame front the kinemacc balance between displacement velocity, flow velocity and burning velocity with respect to the unburnt mixture is Similarly, the kinemacc balance with respect to the burnt gas is 4.- 9
In the present example, flow velocity in burnt gas behind flame is zero due to symmetry This leads to from which the velocity in the unburnt mixture is calculated as This velocity is induced by expansion of the gas behind the flame front 4.- 10
Furthermore it follows that the flame displacement speed dr f /dt is related to the burning velocity s L,u by Measuring the flame displacement speed then allows to determine s L,u Furthermore, from it follows with v b = 0: The comparison shows that the burning velocity with respect to the burnt gas is by a factor ρ u /ρ b larger than that with respect to the unburnt gas This is equivalent to the Bunsen burner case with: Convenience: We keep the notacon s L,b for the burning velocity with respect to the burnt gas 4.- 11
Governing Equa6ons for Steady Premixed Flames, Numerical Calcula6ons and Experimental Data Planar steady state flame configuracon normal to the x- direccon unburnt mixture at x - burnt gas at x + 4.- 12
Structure of an unstretched premixed methane/air flame at standard condicons from numerical simulacons 0.0003 0.14 2000 0.00025 2000 0.12 Temperature [K] 1500 1000 0.0002 0.00015 0.0001 Mass Fraction Temperature [K] 1500 1000 CO 2 0.1 0.08 0.06 0.04 Mass Fraction 500 5 10-5 500 CO 0.02 1 1.5 T 2 2.5 0 3 y [mm] H 1 1.5 2 2.5 3 3.5 4 4.5 0 5 y [mm] 4.- 12a
Flame structure for the case of a lean flame with a one- step reaccon The fuel and oxidizer are convected from upstream with the burning velocity s L Having the mass fraccons Y F,u and Y O 2,u at x - and diffuse into the reaccon zone The fuel is encrely depleted while the remaining oxygen is convected downstream where it has the mass fraccon Y O 2,u 4.- 13
The chemical reaccon forms the product P and releases heat which leads to a temperature rise The mass fraccon Y P increases therefore in a similar way from zero to Y P,b as the temperature from T u to T b The products diffuses upstream, and mix with the fuel and the oxidizer Heat conduccon from the reaccon zone is also directed upstream leading to a preheacng of the fuel/air mixture. Therefore the region upstream of the reaccon zone is called the preheat zone. 4.- 14
The general case with mulc- step chemical kineccs The fundamental property of a premixed flame, the burning velocity s L may be calculated by solving the governing conservacon equacons for the overall mass, species and temperature. ConCnuity Species Energy 4.- 15
For flame propagacon with burning velocices much smaller than the speed of sound, the pressure is spacally constant and is determined from the thermal equacon of state Therefore spacal pressure gradients are neglected The concnuity equacon may be integrated once to yield The burning velocity is an eigenvalue, which must be determined as part of the solucon The system of equacons may be solved numerically with the appropriate upstream boundary condicons for the mass fraccons and the temperature and zero gradient boundary condicons downstream 4.- 16
Example: CalculaCons of the burning velocity of premixed methane- air flames using a mechanism that contains only C 1 - hydrocarbons and a mechanism that includes the C 2 - species as a funccon of the equivalence raco φ [Mauss 1993]. The two curves are compared with compilacons of various data from the literature It is seen that the calculacons with the C 2 - mechanism shows a beger agreement than only the C 1 - mechanism 4.- 17
Example: Burning velocices of propane flames taken from Kennel (1993) s L typically decreases with increasing pressure but increases with increasing preheat temperature 4.- 18
Burning Velocity The fundamental property of a premixed flame is its ability to propagate normal to itself with a burning velocity The burning velocity, to first approximacon, depends on thermo- chemical parameters of the premixed gas ahead of the flame only But: For Bunsen flame, the condicon of a constant burning velocity is violated at the top of the flame and curvature must be taken into account. In this chapter we will first calculate flame shapes We then will consider external influences that locally change the burning velocity and discuss the response of the flame to these disturbances 4.- 19
A Field Equa6on Describing the Flame Posi6on The kinemacc relacon between the propagacon velocity, the flow velocity, and the burning velocity that was derived for spherical flame propagacon may be generalized by introducing the vector n normal to the flame where x f is the vector describing the flame posicon, dx f /dt the flame propagacon velocity, and v the velocity vector 4.- 20
A Field Equa6on Describing the Flame Posi6on The normal vector points towards the unburnt mixture and is given by where G(x,t) can be idencfied as a scalar field whose level surfaces where G 0 is arbitrary, represent the flame surface The flame contour G(x,t) = G 0 divides the physical field into two regions, where G > G 0 is the region of burnt gas and G < G 0 that of the unburnt mixture 4.- 21
DifferenCaCng G(x,t) = G 0 with respect to t at G = G 0 gives Introducing leads to @G @t + v rg = s Ln rg The level set equacon for the propagacng flame follows using as 4.- 22
If the burning velocity s L is defined with respect to the unburnt mixture, then the flow velocity v is defined as the condiconed velocity field in the unburnt mixture ahead of the flame For a constant value of s L, the solucon of is non- unique, and cusps will form where different parts of the flame intersect Even an originally smooth undulated front in a quiescent flow will form cusps and eventually become flager with Cme This is called Huygens' principle 4.- 23
Exercise: Slot Burner A closed form solucon of the G- equacon can be obtained for the case of a slot burner with a constant exit velocity u for premixed combuscon, This is the two- dimensional planar version of the axisymmetric Bunsen burner. The G- equacon takes the form 4.- 24
With the ansatz and G 0 = 0 one obtains leading to As the flame is agached at x = 0, y = ± b/2, where G = 0, this leads to the solucon 4.- 25
The flame Cp lies with y=0, G = 0 at and the flame angle a is given by With it follows that which is equivalent to. This solucon shows a cusp at the flame Cp x = x F0, y = 0. In order to obtain a rounded flame Cp, one has to take modificacons of the burning velocity due to flame curvature into account. This leads to the concept of flame stretch. 4.- 26
Flame stretch Flame stretch consists of two contribucons: Flame curvature Flow divergence or strain For a one- step large accvacon energy reaccon and with the assumpcon of constant properces, the burning velocity s L is modified by these two effects as s 0 L is the burning velocity for an unstretched flame and length. is the Markstein 4.- 27
The flame curvature κ is defined as which may be transformed as The Markstein length appearing in is of the same order of magnitude and proporconal to the laminar flame thickness The raco is called the Markstein number 4.- 28
For the case of a one- step reaccon with a large accvacon energy, constant transport properces and a constant heat capacity c p, the Markstein length with respect to the unburnt mixture reads This expression was derived by Clavin and Williams (1982) and Matalon and Matkowsky (1982) Here is the Zeldovich number, where E is the accvacon energy and the universal gas constant, and Le is the Lewis number of the deficient reactant The expression is valid if s L is defined with respect to the unburnt mixture Different expression can be derived, if both, s L and are defined with respect to the burnt gas [cf. Clavin, 1985] 4.- 29
Effect of Flame Curvature We want to explore the influence of curvature on the burning velocity for the case of a spherical propagacng flame Since the flow velocity is zero in the burnt gas, it is advantageous to formulate the G- equacon with respect to the burnt gas: where r f (t) is the radial flame posicon The burning velocity is then s 0 L,b and the Markstein length is that with respect to the burnt gas. Here we assume to avoid complicacons associated with thermo- diffusive instabilices 4.- 30
In a spherical coordinate system, the G- equacon reads where the encre term in round brackets represents the curvature in spherical coordinates We introduce the ansatz to obtain at the flame front r=r f This equacon may also be found in Clavin (1985) 4.- 31
This equacon reduces to for It may be integrated to obtain where the inical radius at t=0 is denoted by r f,0 This expression has no meaningful solucons for indicacng that there needs to be a minimum inical flame kernel for flame propagacon to take off It should be recalled that is only valid if the product For curvature correccons are important at early Cmes only 4.- 32
Effects of curvature and strain on laminar burning velocity Curvature Effect on Laminar Burning Velocity from Experiments and Theory Strain Effect on Laminar Burning Velocity from Numerical SimulaCons Laminar premixed stoichiometric methane/air counterflow flames Laminar premixed stoichiometric methane/air spherically expanding flames φ = 0.8 Note: s L,u s L,b /7 φ = 1 4.- 32a
Flame Front Instability IllustraCon of the hydro- dynamic instability of a slightly undulated flame Gas expansion in the flame front leads to a defleccon of a stream line that enters the front at an angle A stream tube with cross- secconal area A 0 and upstream flow velocity u - widens due to flow divergence ahead of the flame 4.- 33
Expansion at the front induces a flow component normal to the flame contour As the stream lines cross the front they are deflected At large distances from the front the stream lines are parallel again, but the downstream velocity is At a cross seccon A 1, where the density is scll equal to ρ u the flow velocity due to concnuity and the widening of the stream tube is 4.- 34
The unperturbed flame propagates with normal to itself The burning velocity is larger than u 1 and the flame propagates upstream and thereby enhances the inical perturbacon Analysis performed with following simplificacons Viscosity, gravity and compressibility in the burnt and unburnt gas are neglected Density is disconcnuous at the flame front The influence of the flame curvature on the burning velocity is retained, flame stretch due to flow divergence is neglected 4.- 35
Analysis results in dispersion relacon where s is the non- dimensional growth rate of the perturbacon = 1 f df dt = d ln f dt L =0 r is density raco and k the wave number PerturbaCon grows exponencally in Cme only for a certain wavenumber range 0 < k < k* with L 4.- 52
Without the influence of curvature ( L =0), flame is uncondiconally unstable For perturbacons at wave numbers k > k* a plane flame of infinicvely small thickness, described as a disconcnuity in density, velocity and pressure is uncondiconally stable This is due to the influence of the front curvature on the burning velocity L =0 As one would expect on the basis of simple thermal theories of flame propagacon, the burning velocity increases when the flame front is concave and decreases when it is convex towards the unburnt gas, so that inical perturbacons become smoother L 4.- 53
However, hydrodynamic and curvature effects are not the only influencing factors for flame front stability Flame stretch due to flow divergence, gravity (in a downward propagacng flame) and the thermo- diffusive effect with a Lewis number larger unity are stabilizing effects A more detailed discussion of these phenomena may be found in Clavin (1985) and Williams (1985) 4.- 54
Details of the Analysis for Hydrodynamic Instability The burning velocity is given by Reference values for length, Cme, density, pressure: Introduce the density rate: Dimensionless variables: 4.- 36
The non- dimensional governing equacons are then (with the asterisks removed) where ρ u = 1 and ρ= r in the unburnt and burnt mixture respeccvely. If G is a measure of the distance to the flame front, the G- field is described by: 4.- 37
With equacons the normal vector n and the normal propagacon velocity then are 4.- 38
Due to the disconcnuity in density at the flame front, the Euler equacons are only valid on either side of the front, but do not hold across it. Therefore jump condicons for mass and momentum conservacon across the disconcnuity are introduced [Williams85,p. 16]: The subscripts + and - refer to the burnt and the unburnt gas and denote the properces immediately downstream and upstream of the flame front. 4.- 39
In terms of the u and v components the jump condicons read Under the assumpcon of small perturbacons of the front, with e << 1 the unknowns are expanded as 4.- 40
Jump condicons to leading order and to first order where the leading order mass flux has been set equal to one: 4.- 41
With the coordinate transformacon we fix the disconcnuity at x = 0. To first order the equacons for the perturbed quancces on both sides of the flame front now read where ρ = 1 for ξ < 0 (unburnt gas) and ρ = r for ξ > 0 (burnt gas) is to be used. In case of instability perturbacons which are inically periodic in the h- direccon and vanish for x ± would increase with Cme. 4.- 42
Since the system is linear, the solucon may be wrigen as where σ is the non- dimensional growth rate, κ the non- dimensional wave number and i the imaginary unit. Introducing this into the first order equacons the linear system may be wrigen as The matrix A is given by 4.- 43
The eigenvalues of A are obtained by sesng det(a) = 0. This leads to the characteriscc equacon Here again U = 1/r, ρ = r for ξ > 0 and U = 1, ρ = 1 for ξ < 0. There are three solucons to the characteriscc equacon for the eigenvalues α j, j = 1,2,3. PosiCve values of a j sacsfy the upstream (ξ < 0) and negacve values the downstream (ξ > 0) boundary condicons of the Euler equacons. 4.- 44
Therefore Introducing the eigenvalues into again, the corresponding eigenvectors w 0,j, j = 1,2,3 are calculated to 4.- 45
In terms of the original unknowns u, v and the solucon is now For the perturbacon f (η, τ) the form will be introduced. 4.- 46
InserCng and into the non- dimensional G- equacon sacsfies to leading order with and x = 0 -, x = 0 + respeccvely. 4.- 47
This leads to first order to With the jump condicons can be wrigen as 4.- 48
The system then reads 4.- 49
Since equacon is linear dependent from equacons it is dropped and the equacons and remain for the determinacon of a, b, c and s(k). 4.- 50
Dividing all equacons by one obtains four equacons for The eliminacon of the first three unknown yields the equacon The solucon may be wrigen in terms of dimensional quancces as Here only the posicve root has been taken, since it refers to possible solucons with exponencal growing amplitudes. 4.- 51
The relacon is the dispersion relacon which shows that the perturbacon f grows exponencally in Cme only for a certain wavenumber range 0 < k < k*. Here k* is the wave number of which ϕ = 0 in which leads to 4.- 52
Exercise Under the assumpcon of a constant burning velocity s L = s L0 the linear stability analysis leads to the following dispersion relacon Validate this expression by insercng What is the physical meaning of this result? What effect has the front curvature on the flame front stability? 4.- 55
Solu6on The dispersion relacon for constant burning velocity s L = s L0, shows that the perturbacon F grows exponencally in Cme for all wave numbers. The growth s is proporconal to the wave number k and always posicve since the density rate r is less than unity. This means that a plane flame front with constant burning velocity is unstable to any perturbacon. 4.- 56
The front curvature has a stabilizing effect on the flame front stability. As it is shown in the last seccon, the linear stability analysis for a burning velocity with the curvature effect retained leads to instability of the front only for the wave number range whereas the front is stable to all perturbacons with k > k*. 4.- 57