Course Overviewmarchese/combustion11/... · Web viewWhere DTk is the thermal diffusion coefficient, which arises from an effect called thermophoresis. 11. Stability Limits of Premixed

MECH 558 Combustion Class Notes - Page: 1notes09 Premixed Laminar Flames Text: Ch.7, Ch. 8

Technical Objectives: Show that the laminar flame speed is proportional to the square root of the reaction rate and

thermal diffusivity.

Describe qualitatively, the structure of a premixed laminar flame.

Describe how flame speed varies with pressure, initial temperature, equivalence ratio, fuel type and type of inert.

Perform detailed freely propagating premixed laminar flame calculations using CHEMKIN.

Describe the definition of a flammability limit, why they exist and why CHEMKIN cannot calculate them.

Describe the concept of a minimum quenching distance and derive a simple expression for the quenching distance.

1. MotivationSo far we have learned about chemical thermodynamics (what will the products be if CH4 is reacted with O2?) and chemical kinetics (what chemical reactions must take place in the gas phase at the molecular level to oxidize CH4?). Next, we modeled some 0-dimensional chemically reacting systems with detailed chemistry and then we discussed the importance of mass transfer in combustion. In this section, we will begin to examine actual flames. Flames include not only chemistry, but also convection (bulk flow of a gas) and molecular diffusion (transport of mass and energy in the presence of a gradient).

2. The Governing Equations for 1-Dimensional Chemically Reacting Flow SystemsNow that we understand something about chemical kinetics and have had an introduction to mass transfer, it is possible to write down the governing equations for 1-D chemically reacting flow systems. For derivations of these equations, see Chapter 7 of Turns:

Conservation of Mass

(7.3)

Species Conservation

(7.16)

Energy Conservation

(7.67)

The above equations can be used to solve a variety of chemical reacting flow problems such as premixed laminar flames, diffusion flames, ignition problems, etc. In this lecture, we will be applying a simplified version of the equations to solve for the premixed laminar flame speed. Then, we will use CHEMKIN to solve the equations in full glory with detailed chemical kinetics.


3. The Overall Structure of a Premixed Laminar FlameRecall that flames can be classified in two general categories: premixed and diffusion flames. A candle is an example of a diffusion flame. A Bunsen burner is an example of a premixed flame. Consider the following Bunsen burner flame:

Definition of the Laminar Flame Speed (or Laminar Burning Velocity), sL

The laminar flame speed, sL, is defined as the velocity of the unburned gases entering the combustion zone in the direction normal to the combustion zone as shown below:

General Structure of the Premixed Flame

If you were to traverse a thermocouple and a gas sampling probe through the flame, you would observe the following structure:


There are three main regions in a premixed flame:

1.

2.

3.

4. Simplified Analysis of the Laminar Flame SpeedIn 1883, Mallard and Le Chatlier proposed a simple theory which predicted that the laminar flame speed, sL is related to the overall reaction rate and the thermal diffusivity as follows:

(n11.1)

The results were based on a simple derivation. Here we will provide a slightly more detailed derivation that can be used to solve for the laminar flame speed, sL, incorporating the following assumptions:

1. One-D, steady flow

2. Constant pressure (momentum equation is trivially satisfied)

3. Lewis Number = 1 for all species (k/cp = D)

4. All species have identical and constant specific heat

5. One step overall chemical reaction (1 kg fuel + kg oxidizer → +1 kg products)

6. Fuel is completely consumed at the flame ( <1)

Consider the following control volume with a thickness, x, located somewhere within a premixed laminar flame:

Conservation of Mass:

For a steady, one-dimensional flow, the conservation of mass requires:

(7.4b)


Conservation of Species:

For each of the species (F, Ox), the following species equation applies:

(7.8)

For the fuel, equation (7.8) becomes:

(8.6a)

And for the oxidizer, the equation becomes:

(8.6b)

Where, since there is a single 1-step reaction, the fuel consumption rate and oxidizer consumption rate are directly related by the constant :

(8.4)

Conservation of Energy

Invoking the unity Lewis number assumption, the energy equation becomes:

(8.7b)

where hc is the heat of combustion of the overall reaction.

Boundary Conditions

The boundary conditions for the 1-D premixed laminar flame are as follows:

(8.9)


Solution

A simplified solution of the energy equation (8.7b) can be found by assuming the following simple linear temperature profile for the flame:

Integrating equation (8.7b) once with respect x from -∞ to ∞ yields:

(8.10)

Since dT/dx = 0 for both the upstream and downstream conditions, equation (8.10) becomes:

(8.11)

Since dT/dx = (Tb-Tu)/, it is possible to convert the integral on the RHS of (8.11) into an integral with respect to dT:

(8.12)

Substituting (8.12) into (8.11) results in:

(8.13)

Which can be integrated as follows:

(8.15)

where is the average reaction rate.

Equation (8.15) could be used to solve for the laminar flame speed, sL, however the flame thickness, , is still an unknown.

(8-15b)


To eliminate the flame thickness, from equation 8.15, we note that, since the chemical reaction rate varies exponentially with respect to the temperature, the reaction is confined to a narrow thickness, rxn:

As a first approximation, we can assume that rxn→0, which will allow us to integrate equation (8.10) once again from – ∞ to as follows:

(n11.2)

Which results in the simple relationship for the flame thickness:

(n11.3)

This relationship makes sense if you consider dimensional analysis.

Substituting (n11.3) into (8.15) results in the following relationship for the laminar flame speed:

(n11.4)

It can also be shown that heat of combustion for a simple one-step reaction is related to cp(Tb-Tu) as follows:

(n11.5)


Substituting (n11.5) into (n11.4) results in the following equation for the laminar flame speed:

(n11.6)

Equation (n11.6) shows that the laminar flame speed, sL, is related to the square root of the thermal diffusivity multiplied by the reaction rate, which is the result that was originally predicted by Mellard and Le Chatlier in 1883. (Note that our equation differs from equation (8.20) by a factor of ).

Example 9.1: Estimate the laminar flame speed for stoichiometric methane/air combustion, assuming the following global one-step reaction rate:



5. Flame ThicknessUsing typical values of measured flame speed:

sL = 40 cm/s

And, thermal diffusivity evaluated at 1300 K:

= 2 cm2/s

Shows that the typical flame thickness, , is approximately:

And a typical characteristic flame residence time, , is approximately:

6. Variation of Flame Speed with PressureAlthough the above analysis was very simple, equation (n11.6) is very useful to predict how flame speed will vary with varying pressure, temperature, inert, type of fuel, etc.

Recalling again equation (n11.6)

Equation (n11.6) shows that, for a given pressure, the laminar flame speed is related to the pressure as follows:

To find the total variation with pressure, we need to know how the reaction rate varies with pressure. Recall that the "order" of reaction, n, determines how the reaction varies with pressure based on law of mass action:

Therefore:

(n11.7)

Substituting in to the flame speed proportionality equation:

(n11.8)


So, for an overall order of reaction, n, the flame speed varies with pressure as follows:

(n11.9)

Note: This equation can actually be used to DETERMINE the overall order of reaction for a real fuel/oxidizer system, since the overall rate of reaction will not, in general, be the order specified by the law of mass action for the overall reaction. Recall, C7H16 + 11O2 would suggest that the order of reaction is 12. Flame speed would then have to vary as follows:

This is not observed. Most hydrocarbons have an overall order of reaction n=2. Thus, how does the flame speed for most hydrocarbons vary with pressure?

7. Variation of Flame Speed with TemperatureThe variation of flame speed with temperature can be derived similarly by substituting in the flame speed proportionality equation as follows:

Thermal diffusivity,

Where, it can be shown that thermal conductivity, k, is proportional to T1/2

Reaction Rate


Substituting into the flame speed proportionality:

The final result is thus:

(n11.10)

The importance of this relationship is that an overall activation energy of a fuel/oxidizer combination can be determined by plotting laminar flame speed vs. flame temperature on a ln(sL) vs. 1/T plot:

8. Variation in Flame Speed with Equivalence Ratio ()Equivalence Ratio.

Recall that the equivalence ratio is defined as the ratio of the actual fuel/air ratio to the stoichiometric fuel/air ratio:

Thus, a mixture with equivalence ratio of unity ( = 1) should theoretically result in complete combustion.


Variation in Flame Speed with Equivalence Ratio

The variation in flame speed with equivalence ratio is basically a function in the variation in adiabatic flame temperature with equivalence ratio:

Note: the adiabatic flame temperature is generally a maximum at slightly greater than ( = 1). Why?

So, for any given fuel, a plot of flame speed vs. equivalence ratio follows the plot of flame speed vs. adiabatic flame temperature, since (n11.10) shows that:

Dominates in the flame speed proportionality equation. The flame speed also is maximum at slightly fuel rich (Fig. 8.15 in Turns):

9. Variation in Flame Speed with Fuel TypeThe following is a plot of flame speed vs. equivalence ratio for various fuel/air systems. The plot can be explained very well by considering the simple relationship of


Firstly, most alkanes and aromatics have maximum flame speeds of around 40 cm/s, with methane being slightly slower.

Why is the flame speed slightly lower for methane?

Ethene and Acetylene (T=2600 K) have higher flame speeds, because of their higher flame temperatures than most hydrocarbons (T=2300 K).

Hydrogen/Air has a much higher flame speed, even though it's adiabatic flame temperature is less then acetylene. Why?

10. Governing Equations for Detailed Laminar Flame CalculationsOver the past several decades, much work has been done to develop numerical models to study laminar flames with detailed chemical kinetics and multi-component molecular transport. Developed over the last several decades, the CHEMKIN software package is the leading software for these calculations.

For 1-D, premixed laminar flames, CHEMKIN solves the following system of equations:

Conservation of Mass

(7.4a)

Species Conservation

(8.22)

Energy Conservation

(8.23)

Equation of State

(2.2)


For multi-component mixtures with detailed chemistry and molecular transport, the diffusion velocity, vi,diff , for each species is calculated using either a mixture average approach or using the full blown multicomponent diffusion approach.

For the mixture average approach, binary diffusion coefficients are first calculated for every species with respect to each other using the following equation:

(n12.1)

where kB is the Boltzmann constant, mjk is the reduced mass, jk the collision cross section and (1,1)* is the “collision integral”.

The collision integral is based on what happens when two molecules come within close proximity of each other. As they first approach each other, they attract because of Van der waal’s forces. As they come closer in contact, they repel:

The value of the collision integral is based on the Lennard-Jones potential well depth, k for each molecule and its dipole moment, k.

For the mixture average formulation, the diffusion coefficient of species k, with respect to the mixture is calculated as follows:

(n12.2)

Where Djk is the binary diffusion coefficient for each species pair as calculated by (n12.1):

The diffusion velocity in equations (8.22) and (8.23) are then calculated from:

(n12.3)

Where DTk is the thermal diffusion coefficient, which arises from an effect called thermophoresis.

11. Stability Limits of Premixed Laminar FlamesWe have spent several weeks studying laminar flames and calculating the steady state laminar flame speed. However, there are several limiting phenomena that affect whether a laminar flame will exist or not. These limits are as follows:

1.

2.


3.

4.

5.

Each of these phenomena will be described in the following sections.

11.1 Flammability LimitsConsider the following plot of laminar flame speed vs. equivalence ratio for a hydrocarbon:

For all flammable gas mixtures, there exist mixture ratios above or below which a laminar flame will not propagate. These limits are called flammability limits.

Difference Between Flammability and Explosion Limits

Note that there is a big difference between a flammability limit and explosion limit (e.g. the H2/O2

explosion limits). Consider the following system:


In the region of the spark, the system is raised to a temperature within the H2/O2 explosion limits, resulting in chain branching, heat release…yet for sufficiently lean mixtures the flame will not propagate.

Lean Limit and Rich Limit

According to chemical thermodynamics, all mixtures of fuel and air should be flammable! However, experimentally, flammability limits are observed for all fuel-air mixtures.

Consider the following mixtures of CH4 and air, calculated using Chemkin and GRIMech 3.0:

6% CH4 in air:

5% CH4 in air:

A complete plot of laminar flame speed vs. equivalence ratio for CH4 and air shows that there are two limits, outside which a laminar flame will not propagate no matter how you try to ignite it.

The two limits are called the Lean Flammability Limit and the Rich Flammability Limit.

Flammability Limit Data

Fuel Lean Limit

Stoichiometric(F/A)mass

Rich Limit

Methane 0.46 0.058 1.64

n-Octane 0.51 0.062 4.25

Hydrogen 0.14 0.029 2.54

Acetylene 0.19 0.075 ∞

Why do Flammability Limits Exist?

1.


2.

3.

12. Quenching DistanceIt has also been observed that premixed flames will not propagate through small passageways. This observation was the basis for the invention of the Coal Miner’s Safety Lamp in 1815 by Sir Humphrey Davis.

From a safety standpoint, quenching distance is good (flame arrestors, safety lamps), but from a micro power generation standpoint, quenching distance is a problem. What is the smallest microturbine that we can build?

Consider the premixed flame propagating at sL in a duct with a large diameter, D1. If the diameter of the duct, somewhere upstream is decreased to a smaller diameter, D2, will the flame continue to propagate?

We can define the quenching distance, d, as the diameter through which a flame will not propagate.

12.1 Measurement of Quenching Distance


The following apparatus is used to measure quenching distance:

Technique:

1)

2)

3)

4)

Note: the test section can be a circular tube of diameter, d, or a thin slot with thickness, t. Note that tube quenching distances are measured to be 20% to 50% larger than slot quenching distance (so, this is not a fundamental parameter, but depends on geometry!).

Why Do Laminar Flames Quench in Small Passages?Consider a laminar flame of thickness, , propagating through a small diameter tube of thickness d.

If the diameter is less than the quenching distance, the flame will quench. Why do flames quench in small passages?

1)

2)

So, when , the flame will quench. Why does this occur in narrow passages?


And, this is why it is very difficult to build a microcombustor!

12.2 Simple Derivation of the Quenching Distance, dConsider a laminar flame of thickness, , propagating through a narrow rectangular slot of thickness d, and width (into the page) of L:

The above flame will quench when the heat generation rate (from chemical reaction) is balanced by the heat loss rate (from conduction to the walls):

(8.34)

Where is the volumetric heat release in J/m3, which is related to the mass consumption rate of fuel as follows:

(8.35)

The conduction heat loss from the flame zone, to the slot walls can be expressed as follows:

(8.36)

The (dT/dy)wall term is a bit problematic, since we do not know the temperature profile. A lower bound for the temperature gradient would be:

In actuality, the temperature profile is likely much steeper, so we introduce an unknown constant, b, which has a value greater than 2:

(8.37)


Equating heat loss with heat generation results in the following:

(8.38a)

Solving for the quenching distance yields:

(8.38b)

(n14.1)

Recall equation (n11.4), which related the laminar flame speed to the heat of combustion and fuel consumption rate:

(n11.4)

If we assume the worst case scenario that the wall temperature is equal to the unburned gas temperature (Tw = Tu), then we can combine equations (n14.1) and (n11.4) to eliminate Tb, Tu, hc, and as follows:

(n14.2)


Recalling also simple relationship between flame speed, sL, and flame thickness, :

(n11.3)

we can combine (n11.3) and (n14.2), which shows that the quenching thickness is directly related to the flame thickness:

(n14.3)

Recalling that b > 2 from the arguments above, it is clear that quenching distance must be greater than the flame thickness, or:

(n14.4)

Equations (n14.2) and (n14.3) suggest the following:

- higher flame speed results in smaller quenching distance

- higher thermal diffusivity results in larger quenching distance

- quenching thickness is on the same order as flame thickness

The following plot shows a plot of flame thickness vs. equivalence ratio and quenching distance vs. equivalence ratio:


The following table lists some flammability limits and quenching distances for common fuels burning in air.


Example 9.2 Design of a flashback-safe combustor.

Design a combustor to avoid flashback. A mixture of methane and air will be introduced into a tube at 5 atm and 300 K. A flame will be established at the exit of the tube. Find the maximum tube diameter that will prevent flashback.


Example 9.3 Design a microscale combustor

A microscale combustor is being designed to operate on CH4/air with a constraint that the entire combustor must be no larger than 100 microns (0.1 mm). Although this dimension seems to be significantly lower than any reasonable quenching distance, it may be possible to design it to operate at higher pressures.

Find: a) The required pressure necessary to avoid quenching at 0.1 mm. b) determine if this pressure is problematic from a structural standpoint.

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Course Overviewmarchese/combustion11/... · Web viewWhere DTk is the thermal diffusion coefficient, which arises from an effect called thermophoresis. 11. Stability Limits of Premixed