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calculation models and applications is presented in Section 2.16 and an exam-ple calculation of a turbulent Bunsen flame, based on the equations derived in Sections 2.8–2.15, is presented in Section 2.17. Section 2.18 concludes this chapter.

2.2 Laminar and Turbulent Burning Velocities

The most important quantity in premixed combustion is the velocity at which the flame front propagates normal to itself and relative to the flow into the unburnt mixture. This velocity is called the laminar burning velocity sL. It is a thermo-chemical transport property that depends primarily on the fuel-to-air equivalence ratioφ, the temperature in the unburnt mixture, and the pressure. It has been measured for various fuels over a wide range of these parameters (cf.

Law, 1993). It also can be calculated numerically using elementary or reduced reaction mechanisms and molecular transport properties. For that purpose one considers a planar steady state flame configuration normal to the x direction with the unburnt mixture at x → −∞, and the burnt gas at x → +∞. The one-dimensional balance equations for continuity, mass fractions of the chemical species, and energy following from (1.30), (1.62), and (1.76) are

∂(ρu) The continuity equation may be integrated to show that the mass flow rate through the flame is constant. This defines the burning velocity s_L⁰:

(ρu)−∞= ρs⁰L



u. (2.4)

Here the suffix 0 indicates that the flame is planar and the flow is one dimen-sional. The momentum equation has been used in the limit of small Mach num-bers to obtain the condition p= constant throughout the flame (cf. Williams, 1985a, p. 143). Solving (2.2) and (2.3) with prescribed values for Yi,uand Tuand zero gradient or equilibrium boundary conditions downstream yields the burn-ing velocity as an eigenvalue of the problem. Results of numerical calculations of laminar burning velocities for hydrogen, methanol, and hydrocarbon fuels up to propane may be found in Peters and Rogg (1993). The influence of heat losses due to radiation have been analyzed, for instance, by Kennel et al. (1990).

Numerical and asymptotic analyses based on reduced chemical mechanisms are

Figure 2.2. Burning velocities calculated by Mauss and Peters (1993) with a detailed mechanism, containing up to C₂-hydrocarbons, compared with data compiled by Gu et al. (2000) for atmospheric methane–air-flames.

reviewed by Seshadri and Williams (1994) and Seshadri (1996). As an ex-ample, burning velocities of methane–air-flames calculated by Mauss and Peters (1993), based on an elementary mechanism containing species up to C2-hydrocarbons, are compared with a recent compilation of experimental data by Gu et al. (2000) and are shown as a function of the equivalence ratio φ in Figure 2.2.

The very existence of a property with dimension [m/s] introduces new scaling laws. The analysis in this chapter will emphasize the consequences that result from the laminar burning velocity as an additional property in a fluid-dynamical system. An immediate consequence of the existence of a velocity scale in a diffusive medium is the existence of a length scale. Length scales in laminar flames have been discussed by Peters (1991). A suitable definition of a flame thickness of a premixed flame is

F = (λ/cp)₀

ρs⁰L



u

(2.5) (cf. G¨ottgens et al., 1992). Here the heat conductivityλ and the heat capacity cp

are evaluated at the inner layer temperature T₀while, as a consequence of (2.4), the product of the densityρ and the laminar burning velocity s⁰L is evaluated in the unburnt gas. If the definitionF = D/s⁰Lrelating the flame thicknessF

to the diffusivity D and the burning velocity is used, as in (2.23) below, and s⁰_L is taken as s⁰_L,u, (2.5) defines the diffusivity D as

D= sL⁰,uF = (λ/cp)₀/ρu. (2.6)

2.2 Laminar and turbulent burning velocities 71

orifice adjustment screw

fuel from supply needle

fuel orifice

air air

premixing region Bunsen tube mixing chamber

streamline (particle track) Bunsen

flame cone

α

Figure 2.3. The design of a classical Bunsen burner showing the premixing in the Bunsen tube and the Bunsen cone at the exit of the burner.

To address the various issues and physical phenomena associated with flame propagation, we first consider a classical experimental device, the Bunsen burner shown in Figure 2.3. Gaseous fuel from the fuel supply enters through an orifice into the mixing chamber, into which air is entrained from the out-side through adjustable openings. The opening area of the fuel orifice may be adjusted by moving the needle through an adjustment screw into the orifice, thereby allowing the velocity of the jet entering into the mixing chamber to be varied and the entrainment of the air and the mixing to be optimized. The mixing chamber must be long enough to generate a fully premixed gas.

Laminar or turbulent steady premixed flames can be established on a Bunsen burner. If the velocity of the mixture is sufficiently large, the flow inside the Bunsen tube becomes turbulent. Turbulence may also be generated by a turbu-lence grid at the upper end of the mixing chamber, for example. If the velocity of the flow issuing from the tube is larger than the laminar or the turbulent

vt,b= vt,u

vn,b

u b

vt,u

vn,u

s_L,u

vb

Oblique flame front vu

vu α

Figure 2.4. Kinematic balance for a steady oblique flame.

burning velocity sLor sT, respectively, a flame cone is established. The angle α of the cone is a measure for the laminar or turbulent burning velocity, as will be shown in the following.

The kinematic balance between the flow velocity and the burning velocity is illustrated for a steady oblique flame in Figure 2.4. The oncoming flow velocity vectorvuof the unburnt mixture is split into a componentvt,uthat is tangential to the flame front and a componentvn,u normal to the flame front. Owing to gas expansion within the flame front the normal velocity componentvn,bon the burnt gas side is larger thanvn,u, since, because of continuity, the mass flowρvn

in normal direction through the flame must be the same in the unburnt mixture and in the burnt gas:

(ρvn)u = (ρvn)b, (2.7)

while the density decreases. Therefore vn,b= vn,uρu

ρb. (2.8)

The tangential velocity componentvtis not affected by the gas expansion and remains the same:

vt,b= vt,u. (2.9)

2.2 Laminar and turbulent burning velocities 73 Vector addition of the velocity components in the burnt gas in Figure 2.4 then leads tovb, which points into a direction that is deflected from the direction of vuin the unburnt mixture.

Finally, since the flame front is stationary in this example, the burning ve-locity with respect to the unburnt mixture must be equal to the flow veve-locity of the unburnt mixture normal to the front. For a laminar flow the laminar burning velocity is obtained from the kinematic balance

s⁰_L,u= vn,u. (2.10)

Similarly, for a turbulent flow the turbulent burning velocity sT,uis equal to the mean normal velocity,

s_T⁰_,u= ¯vn,u, (2.11)

indicating that the turbulent burning velocity is an averaged quantity. With the angle of the Bunsen flame cone in Figure 2.4 denoted byα, the normal velocity is

vn,u= |vu| sin α, (2.12)

for both laminar and turbulent flow. Therefore the laminar and the turbulent burning velocities with respect to the unburnt gas are

s⁰_L,u= |vu| sin α,

s_T⁰_,u= | ¯vu| sin α. (2.13) This allows one to determine the burning velocity experimentally by measuring the flow velocity and the cone angle α. It will be shown below that for a given fuel–air mixture the turbulent burning velocity depends on the turbulence intensityv^and the integral length scale. In general, not only the mean flow velocity ¯vu but also v^ and  depend on the radial direction in a turbulent Bunsen flow. Therefore the flame angle and also the flame brush thickness, to be introduced below, varies with radial distance.

A particular phenomenon occurs at the flame tip of a Bunsen flame. If the tip is closed, which is in general the case for turbulent flames in the mean (but not necessarily for laminar flames), the burning velocity at the tip, being opposite and therefore equal to the flow velocity, is a factor 1/sin α larger than the burning velocity through the oblique part of the cone. This local increase of the burning velocity is caused by the merging of the flame fronts leading to an enhanced burnout. Finally, it is shown in Figure 2.3 that the flame is detached from the rim of the burner. This is caused by heat loss to the burner, which,

optical access

Figure 2.5. Spherical flame propagation in a combustion vessel.

in the region very close to the rim, leads to temperatures at which combustion cannot be sustained.

Another example of an experimental device for measuring burning velocities is the combustion vessel (Figure 2.5) within which a flame is initiated by a central spark. The spherical flame propagation that follows may optically be detected through quartz windows and the flame propagation velocity drf/dt may be recorded. This set-up is designed to generate a nonstationary motion of the flame front. It can be used to measure laminar and turbulent burning velocities. In the latter case a turbulent flow field must be generated before spark ignition. Using four mutually opposed high speed fans, a nearly homogeneous isotropic turbulence field has been generated by Andrews et al. (1975), Abdel-Gayed and Bradley (1977), Abdel-Abdel-Gayed and Bradley (1981), and in many subsequent studies by the Leeds group (cf. Bradley, 1992). One or two flame kernels were initiated by electrical sparks. The flame kernel development and the turbulent burning velocity are measured by high speed cinematography. This configuration is very suitable for studying the unsteady flame development at early times.

To investigate the structure of steady turbulent flames and their turbulent burning velocities, one would like to generate a steady planar turbulent flame in a sufficiently isotropic turbulent flow field. Three different configurations, namely the conical Bunsen flame, the rod-stabilized V-flame, and the turbulent flame stabilized in the stagnation flow in front of a disk, have been compared by Cheng and Shepherd (1991). They find that, while the measured turbulent burning velocities are consistent in these configurations, the turbulent transport processes are geometry dependent.

2.2 Laminar and turbulent burning velocities 75

turbulence grid

stagnation point

y x

burnt gas

unburnt gas v

v

Figure 2.6. Two turbulent counterflow premixed flames.

A modification of the stagnation flow geometry is the counterflow configu-ration shown in Figure 2.6, where two planar flames are stabilized in a divergent axially symmetric flow between two opposed ducts. There are a number of ex-perimental and theoretical papers using this configuration to study the structure of turbulent premixed flames (cf. Kostiuk et al., 1993a,b, Bray et al., 1991, and Wu and Bray, 1997). In this configuration, which we will consider for the laminar case in an example in Section 2.5, the component of the velocity in the y direction decreases from the values at the exits of the ducts to zero at the stagnation point but is approximately independent of the x direction. Therefore the two flame fronts are approximately normal to the y direction. Disadvantages of this configuration are the existence of a considerable mean strain and the fact that the two flame fronts may interact with each other.

To avoid these influences on the turbulent flame structure, while maintaining a turbulent flow normal to the main flow direction, a new type of burner, called the “weak swirl burner” has been designed by B´edat and Cheng (1995). This device is shown schematically in Figure 2.7. It consists of a flame tube, which, in addition to the main premixed flow, has four tangential air inlets that generate a circumferential velocity component. This velocity component, however, is restricted only to a small annular region of the outlet flow at the perimeter close to the burner rim but leaves the center core flow unchanged. The weak swirl generates a slightly diverging flow close to the burner rim that stabilizes the downward propagating flame at the vertical position where the mean flow velocity equals the turbulent burning velocity. Turbulent quantities such as v^ and are nearly independent of the radial direction. This setup produces stable premixed turbulent flames for a wide range of premixtures and turbulence

freely stabilized flame

supply of fuel-air mixture turbulence

generator

tangential air jet

Figure 2.7. The weak swirl burner by B´edat and Cheng (1995) with a freely stabilized premixed flame normal to the mean flow.

intensities. Detailed measurements of the velocity field were reported by Cheng (1995). Turbulent burning velocities and scalar fields were recently measured by Plessing et al. (2000).

To illustrate the kinematic balance among flame propagation velocity, flow velocity, and burning velocity the case of an unsteady radially propagating flame will be discussed next, using the laminar notation, for simplicity. We will ignore the initial flame development, where influences from the spark still prevail, and consider a well-established flame at times, where the increase of pressure (and thereby of the temperature due to adiabatic compression) within the vessel is still negligible. Effects of flame curvature will not be considered here but in the second example in Section 2.5 below. For the unsteady case, the propagation velocity drf/dt of the flame front results from an imbalance of the flow velocity and the burning velocity, here written with respect to the unburnt mixture:

drf

dt = vu+ s⁰L,u. (2.14)

This can also be written with respect to the burnt gas as drf

dt = vb+ s⁰L,b. (2.15)

2.2 Laminar and turbulent burning velocities 77 In a moving frame of reference attached to the flame the balance of the mass flow rate through the front is

ρu

In the present example of a laminar spherical flame the flow velocityvb in the burnt gas behind the flame is zero because of symmetry. With the gas expansion parameter defined as

Using (2.14) the velocity of the unburnt mixture at the front is calculated as vu = γ

1− γs⁰_L,u. (2.19)

This velocity is induced by gas expansion within the flame front. Introducing (2.19) into (2.18) we see that the propagation drf/dt velocity is related to the burning velocity s⁰_L,uby

(1− γ )drf

dt = sL⁰,u. (2.20)

Measuring the propagation velocity drf/dt then allows us to determine sL,u. Furthermore, from (2.15) it follows withvb = 0 that

drf

dt = s⁰L,b. (2.21)

Comparing (2.20) and (2.21) shows that the burning velocity with respect to the burnt gas is a factorρu/ρb = 1/(1 − γ ) larger than that with respect to the unburnt gas,

s⁰_L,b= s⁰_L,u

1− γ. (2.22)

In the following we will use the notation (ρs⁰L)= ρus_L⁰_,u= ρbs⁰_L,bfor the mass flow rate through a laminar flame front, where the brackets indicate that the mass flow rate is a constant. The notation ( ¯ρs⁰T) has the equivalent meaning for a turbulent flame front. The burning velocities s_L⁰ or s⁰_T appearing in the text indicate that the location, at which s⁰_Lor s_T⁰are to be evaluated, is not specified.

For general considerations, such as regime diagrams and scaling laws it is implicitly understood that s_L⁰ or s_T⁰ should be evaluated with respect to the unburnt mixture. For simplicity of notation, we will also remove the suffix 0.