Laminar Flame Speed and Sensitivity Analysis, Slide di Fluidodinamica

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Combustione e formazione di inquinanti

Laminar flame speed and sensitivity analysis

Fabio Cecchetto

Exam ofCombustione e formazione di inquinanti – Prof. A. Frassoldati

Introduction

• The propagation speed of the premixed flame with respect to the unburned gases is called burning velocity or flame speed , 𝑺𝑺 (^) 𝑳𝑳.
• Engineers differentiate between: Laminar flame Laminar flame speed is a property of the mixture (fuel structure, stoichiometry) and thermodynamic conditions upon mixture ignition (pressure, temperature)

Turbulent flame Turbulent flame speed is a function of the same parameters, but also heavily depends on the flow field. As flow velocity increases and turbulence is introduced, a flame will begin to wrinkle, then corrugate and transport properties will be enhanced by turbulent eddies in the flame zone.

Detailed theory: Premixed 1D Flames

• The study of the structure of premixed flame will be presented under the simplifying assumptions of:  One-dimensional flame;  Stationary conditions;  A flow reactor with axial dispersion , both thermal and material.
• Some equations and properties have to be known:

 Conservation equations (mass flow, species and energy);  Detailed chemical kinetic;  Thermodynamics properties;  Transport properties (diffusion coefficient, thermal conductivity);  Computing grid.

𝑷𝑷𝑷𝑷 (^) 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂 =

𝝉𝝉 (^) 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂 𝒅𝒅𝒂𝒂𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒂𝒂𝒅𝒅𝒅𝒅 𝝉𝝉 (^) 𝒄𝒄𝒅𝒅𝒅𝒅𝒄𝒄𝑷𝑷𝒄𝒄𝒄𝒄𝒂𝒂𝒅𝒅𝒅𝒅^ =^

𝑳𝑳𝟐𝟐�𝔇𝔇 𝑳𝑳� (^) 𝒅𝒅 =

𝒅𝒅 � 𝑳𝑳 𝔇𝔇

Detailed theory: Premixed 1D Flames

• The corresponding steady-state governing conservation equations are: 1. Continuity equation (Mass Flow equation) 𝑚𝑚̇ = 𝜌𝜌𝜌𝜌 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 2. Species conservation equations

𝑚𝑚̇

𝜌𝜌𝜌𝜌𝜔𝜔𝑘𝑘 𝑉𝑉𝑘𝑘 + 𝜌𝜌 Ω𝐾𝐾 𝑀𝑀𝑊𝑊̇̇𝑘𝑘 = 0 k = 1, … , N

3. Energy conservation equation

λ

𝑘𝑘=

𝑁𝑁 𝐶𝐶𝑃𝑃𝑘𝑘 𝜔𝜔𝑘𝑘 𝑉𝑉𝑘𝑘

𝑘𝑘=

𝑁𝑁̇ Ω𝐾𝐾 𝑀𝑀𝑊𝑊𝑘𝑘 𝐻𝐻𝑘𝑘 = 0

Formation due to chemical reactions

Diffusion velocity: molecular and thermal diffusion

Detailed theory: Premixed 1D Flames

• For this kind of propagating flame, the mass flow rate, or, equivalently, laminar flame speed, is an eigenvalue of the system ; so it must be determined as a part of the solution.
• Another constraint is required:

fixed fixed

dm

dx

T x x T

• It is possible to fix the temperature at one point: this is sufficient to allow for the solution of the flame speed eigenvalue, because we release 1 DOF.

Detailed theory: Premixed 1D Flames

• It is important to fix this point in such a way that the gradients tend to vanish at the inlet , otherwise, the calculated flame velocity will be affected by heat loss at the boundary.

fixed fixed

dm

dx

T x x T

Thermal Theory: Mallard - Le Chatelier

• This thermal theory is based on some assumptions; the flame is considered:
  1. Homogeneous, laminar and premixed;
  2. Monodimensional and stationary;
  3. Deflagrating and adiabatic.
• The fundamental hypothesis is that:
• For temperature up to 𝑇𝑇𝑖𝑖𝑖𝑖(in the Zone [1] ), it is possible to neglect reaction terms, because flame temperature is too low to express reactivity.

The heat conducted from the Reaction Zone [2] is equal to that necessary to raise the premixed mixture to the “ignition temperature”, 𝑇𝑇𝑖𝑖𝑖𝑖

HEAT

This thermal theory does NOT consider material back-diffusion of hydrogen radicals in the preheating zone

Thermal Theory: Mallard - Le Chatelier

• So, approximating the slope of the profiles linearly ( order of magnitude approach ), the energy balance becomes:

p^ (^ ig 0 )^ (^ f ig )

L

mc T T T T

m S

λ δ ρ

f ig^1

L p (^) ig

T T

S

c T T

𝝀𝝀 is thermal conductivity; 𝒎𝒎̇ is the mass flow rate per unit area (flux); 𝜹𝜹 is the reaction zone thickness.

H 2 /air system

• We analyse a H 2 /air system in these

conditions:

• Atmospheric pressure, 𝑃𝑃 = 1𝑎𝑎𝑐𝑐𝑚𝑚;
• Initial temperature 𝑇𝑇 0 = 300𝐾𝐾;
• Within the flammability limits (4% - 75% vol).
• Some computer simulations were

performed by means of the software OpenSMOKE_Flame1D, each varying the equivalence ratio, 𝑷𝑷𝑷𝑷𝒂𝒂 (𝝓𝝓).

• The results, in terms of LFS, Laminar Flame

Speed, and Temperature are reported in the Table.

• LFS for H 2 is one order of magnitude

greater than other fuels: this is fully explained by kinetic aspects regarding the production of H· radicals.

Phi LFS [m/s] T [K] 0,3 0,0331 1186, 0,5 0,4830 1593, 0,7 1,2823 1981, 1,0 2,3670 2376, 1,5 3,1985 2255, 2,0 3,1796 2062, 2,5 2,8558 1902, 3,0 2,4694 1771, 3,5 2,0958 1664, 4,0 1,7531 1573, 4,5 1,4448 1493, 5,0 1,1749 1421, 5,5 0,9288 1351, 6,0 0,7230 1288, 6,5 0,5536 1234, 7,0 0,4183 1188,

H 2 /air system

• This data can be reported also in some charts:

0,

0,

1,

1,

2,

2,

3,

3,

0 1 2 3 4 5 6 7 8

LFS [m/s]

Equivalence Ratio [-]

1000

1200

1400

1600

1800

2000

2200

2400

2600

0 1 2 3 4 5 6 7 8

Temperature [K]

Equivalence Ratio [-]

 Certainly, there is a connection between the LFS and the flame temperature.  The AFT, Adiabatic Flame Temperature, is related to the flame temperature , 𝑇𝑇𝑓𝑓, in such a way: 𝑇𝑇𝑓𝑓 = 𝑇𝑇 0 + ∆𝑇𝑇𝑎𝑎𝑎𝑎 and ∆𝑇𝑇𝑎𝑎𝑎𝑎 =

−∆𝐻𝐻 0𝑅𝑅 𝑁𝑁𝑡𝑡𝑡𝑡𝑡𝑡 �𝐶𝐶𝑝𝑝  A modification in ∆𝑇𝑇𝑎𝑎𝑎𝑎 affects, firstly, 𝑇𝑇𝑓𝑓 and then the LFS, because acts directly on the reaction rate.  LFS has its maximum in the rich zone (KINETICS) , whilst 𝑇𝑇𝑓𝑓 reaches its peak in Φ ≅ 1.

KINETICS : In rich conditions, hydrogen radicals are present in large quantities.

Does only 𝑻𝑻𝒅𝒅 affect the LFS?

• We consider three different fuels, such as Acetylene, Ethylene and Ethane.
• Evaluating LFS of their atmospheric and stoichiometric flames, at 𝑇𝑇 = 2300𝐾𝐾 (kept constant by means of the addition/removal of the inert 𝑁𝑁 2 ), we obtain the following graph:

The differences in 𝑇𝑇𝑓𝑓 only partially explain the large variations in LFS , which also are of kinetic nature.

Sensitivity Analysis

C 2 -unsaturated fuels have greater LFS , due to a higher production of H· radicals. This does NOT apply to other unsaturated fuels , that generates resonantly stabilized radicals.

It generates radicals that are resonantly stabilized.

Sensitivity Analysis

• Sensitivity analysis is one of the most widely used tools in kinetic modeling.
• Typically, it is performed by perturbing the pre-exponential A-factors of the

individual reaction rate coefficients and monitoring the effect of these perturbations on the observables of interest, such as the overall reaction rate and, consequently, the LFS.

• The absolute value of the sensitivity

coefficients represents the degree of influence of that specific reaction on the LFS.

• The negative sensitivity coefficient means that any increase in the specific reaction rate constant could reduce the LFS, and vice versa.

Sensitivity Analysis

Methyl radicals inhibit LFS because they easily recombine. There is an initial growth up to a peak, followed by a rapid decrease.

Conversely, hydrogen radicals enhance LFS because of their action as branching agents. Fuels with higher ratio 𝑯𝑯 ⁄𝑪𝑪 have greater LFS.

Sensitivity Analysis

• Now, we understand why:
  • 𝐻𝐻 2 shows the highest flame speed;
  • 𝐶𝐶𝐻𝐻 4 is the less reactive, because methyl radicals tend to recombine;
  • 𝐶𝐶 2 𝐻𝐻 6 is the most reactive, because ethyl radicals decompose, 𝐶𝐶 2 𝐻𝐻 5 → 𝐶𝐶 2 𝐻𝐻 4 + 𝐻𝐻· (giving ethylene and H·, very reactive);
  • All of the 𝐶𝐶𝑛𝑛 𝑐𝑐 > 2 are very similar, included between methane and ethane and mainly depend on the 𝐶𝐶 1 /𝐶𝐶 4 sub-mechanism.