Review of Combustion - Semantic Scholar · for engines and furnaces. Most combustion flows, particularly those in fires, are very complicated to study. Concept on applied physics

International Journal on Architectural Science, Volume 6, Number 2, p.38-69, 2005

38

A BRIEF REVIEW ON COMBUSTION MODELING Y. Gao Department of Building Engineering, Harbin Engineering University, Harbin, Heilongjiang, China W.K. Chow Areas of Strength: Fire Safety Engineering, Research Centre for Fire Engineering The Hong Kong Polytechnic University, Hong Kong, China (Received 2 March 2005; Accepted 10 June 2005) ABSTRACT There is a strong demand in applying combustion modeling for building fires, burning materials and of course for engines and furnaces. Most combustion flows, particularly those in fires, are very complicated to study. Concept on applied physics and chemistry would be applied. A review on combustion modeling will be presented in this paper. Effects of turbulence on flame propagation are focused. Some fundamental aspects of combustion modeling for numerical modeling of premixed and diffusion turbulent combustion are discussed. Eddy break-up model and its expansion, Eddy dissipation model, simplified PDF model with different reaction rate and flame surface, and flamelet model are included. 1. LAMINAR AND TURBULENT

PREMIXED FLAMES Combustion process of premixed gases depends on the flow condition. The flow velocity of the combustible premixed gases is not high for laminar flames, and there are no fluctuations. The flame surface appears smooth, with combustion rather steady. The flame surface will supply heat and activated radicals to the nearby thin layer of combustible premixed gases which are not yet burnt, through heat transfer and molecular diffusion. The flame will then spread along the surface, called laminar flames. When the flow velocity of the combustible premixed gases is relatively high or the flow area is relatively large, a large quantity of large and small eddies with random rotations and movement are produced in the fluid flow. These eddies will fluctuate back and forth, up and down through the flow streamlines. The flame surface will be folded and deformed to be thicker and shorter, kept on rolling and making noises. Such flames are called turbulent flames. In contrast to laminar flames, the transport of heat and activated radicals from the turbulent flame surface to the not yet burnt premixed gases are governed by the eddy movement of the fluid to excite and intensify, which depends on the fluid flow. The spreading rate of turbulent flames is much faster than that for laminar combustion.

1.1 Spreading Rate of Laminar Flames The spreading rate of laminar flames is a basic characteristic parameter of the combustion process. To effectively control the combustion in practical processes, the mechanism of flame spreading and the influences of various physical and chemical factors on it should be clearly understood. That is to understand the specific transfer process of chemical reactions of combustion from one layer of reacted premixed gases to another layer of freshly premixed gases. At present, there are two schools of thought on the mechanism of flame spreading: thermal theory and diffusion theory. For thermal theory, chemical reactions in the flame are induced by the introduction of heat which thermally activate the molecules. The movement of the chemical reaction zone (called the flame front) in space depends on the heat transfer rate from the reaction zone to the freshly premixed gases. This theory does not deny that there are activated radicals and diffusion in the flame, but it is generally believed that the activated radicals in the combustion process do not have a dominant effect on the chemical reactions. On the contrary, the diffusion theory assumes that chemical reactions in the flame are mainly due to the diffusion of activated radicals (such as H and OH) to the freshly premixed gases, which enhances the chain reactions. Among these two theories, the thermal theory more resembles the practical situations. The thermal theory of laminar flame spreading proposed by the Russian scientist Zeldovich (1938) and his coworkers Frank-Kamenetkii (1940), Semenov et al. (1951),

International Journal on Architectural Science

39

and based on the literature results of others as well, is considered the most widely accepted flame spreading theory at present. Zeldovich’s theory of flame propagation is described in the following.

1.1.1 Zeldovich’s analysis of flame propagation Based on the analysis of the flame front structure, the thermal theory of flame spreading was proposed by Zeldovich and coworkers with six assumptions: 1. Chemical reactions in the flame are the result

of thermal activation of molecules. The movement of the flame front in space depends on the heat transfer between the burning gases and the freshly mixed gases. Although there will be some effects due to diffusion of activated radicals, the effect is not significant and can be neglected.

2. The combustion reactions studied are simple reactions and not chain reactions. The reactions have very high activation energies and take place at high temperature close to the combustion temperature.

3. The reactants and combustion products are both ideal gases, their physical parameters including heat transfer coefficient λ, heat diffusion coefficient a, diffusion coefficient D and caloric value are all constants in the process, taken as the average value in the temperature zone studied. The heat diffusion coefficient is equal to the diffusion coefficient, i.e. a = D, meaning that temperature field and concentration field are similar.

4. The reactions are of equal pressure and thermally insulated, i.e. pressure diffusion and heat loss are not considered.

5. Air flow is one-dimensional thermally insulated flow. The gas parameter is only a one-dimensional function of space with negligibly small variation of kinetic energy.

6. The flame front is a calm plane front. Under the above assumptions, the equation for the spreading rate of laminar flames can be derived based on the energy equation of gases. The heat transfer equations in the flame are derived in the following. The energy equations for one-dimensional steady flow with chemical reactions are listed. Consider a small control volume in the flame front, with volume dv of dxdydz. When premixed combustible gases flow past the micro-volume from left to right, its temperature T increases gradually along the flow direction. Heat

dxxQ + is transferred from the high temperature

zone on the right side to the left side of the control volume dv with xQ . Since there is air flow through the control volume, the heat enthalpy brought in is dxxH + . The changes in energy in the control volume depend on these three heat flows. The heat flow introduced would increase the enthalpy of the gas in the control volume, i.e.

xx dx x x x

QQ Q Q dx Qx+

∂ − = + − ∂

xQ Tdx dydz dxx x x

λ∂ ∂ ∂ = = ∂ ∂ ∂

2

2

T dvx

λ ∂=

∂

The enthalpy brought away by the gas would reduce the enthalpy of gas in the control volume, i.e.

( ) dvxTcudxTdydzCu

xdx

xH

HH ppx

xdxx ∂∂

=∂∂

=∂∂

=−+ ρρ

The heat released per unit time in the control volume from the chemical reaction with rate w (mol/m3·s) and heat of reaction q (KJ/mol) would increase the enthalpy of gas.

wqdvwqdxdydzQch == For stable flame spreading, the enthalpy of gas in the above control volume will change with time. The sum of the increase in the above three values of enthalpy should be zero, i.e.

02

2

=+∂∂

−∂∂ wq

xTCu

xT

pρλ (1-1)

This is the energy equation for one-dimensional steady flows with chemical reactions. It gives the distribution of gas temperature in the flame front with x. Equation (1-1) means that in the control volume, the sum of the net heat gain from heat transfer and the heat released from the chemical reaction is equal to the energy flowing out from the gas flow. The balance between energy flowing in and out is the energy conservation within the control volume. Continuity equations for one-dimensional constant laminar flows with chemical reactions are:

constmuuu ==== ∞∞ 00ρρρ (1-2)


40

Under certain boundary conditions of ( )T x , the concentration of the reactant is: When −∞=x , 0TT = , 0CC = ; When +∞=x , fTT = , 0=C . The spreading rate of laminar flames LS can be calculated from equations (1-1) and (1-2). However, it is very difficult to obtain highly accurate results. Different approximation methods had been proposed. Among which, the “zoning approximation method” proposed by Zeldovich et al. is comparatively simpler. Two zones are proposed in the laminar flame front, a pre-heating zone and a chemical reaction zone. The influence of chemical reactions can be neglected in the pre-heating zone. The first-order derivative of temperature can be neglected in the reaction zone. Based on the different characteristics of these two zones, the value of

)( dxdT for each zone can be calculated by equations (1-1) and (1-2). The value of LS can then be obtained. Zeldovich’s approximation method is described as follows: (1) In the pre-heating zone, since the temperature

is low, the rate of chemical reactions is very slow and can be neglected. Equation (1-1) can be written as:

02

2

=−dxdTuc

dxTd

pρλ (1-3)

The boundary conditions are:

When −∞→x , 0TT = , 0=dxdT ;

When px δ= , iTT = , idx

dTdxdT )(=

Where Ti is the ignition temperature of the premixed gases in the pre-heating zone. When iTT < , the premixed gases are still at the pre-heating condition, the increasing heat flux to the reaction zone is larger than the ongoing heat flux from the premixed gases, i.e.

02

2

>dx

Tdλ

When iTT > , the premixed gases have already been ignited and entered the chemical reaction zone. At this time, the heat flux out from the gases is larger than the heat flux in, i.e.

02

2

<dx

Tdλ

When iTT = , 02

2

=dx

Tdλ , the premixed

gases have started to be ignited, the situation of heat flux in has started to change to heat flux out. Therefore, Ti can be defined as the minimum ignition temperature. Ti is a turning point on the curve )(xfT = . Integrating equation (1-3) gives:

)(10TTuc

dxdT

ipip

−=

ρ

λ (1-4)

(2) In the chemical reaction zone, since the

temperature is close to the burning gas temperature fT , the temperature gradient decreases rapidly to zero. Therefore, the second term (first-order derivative of temperature) in equation (1-1) can be neglected to give:

02

2

=+ qwdx

Tdλ (1-5)

The boundary conditions are:

When px δ= , iTT = ，idx

dTdxdT )(= ;

When ∞=→x , fTT ≈ ， 0≈dxdT .

Integrating equation (1-5) gives:

2 f

i

T

Tic

dT q wdTdx λ

= ∫ (1-6)

In the stable process, at the interface of

iTT = , the heat flux entering from the reaction zone should be equal to the heat flux transferred to the pre-heating zone, so

icip dxdT

dxdT

=

λλ

i.e.


41

( )01 2 f

i

T

p i T

quc T T wdTρλ λ

− = ∫ (1-7)

From the above equation,

( )0 22

0

2 f

i

T

L Tp f

q wdTm u Sc T T

λρ ρ= = =−

∫

Here, uρ is the mass burning rate, and the flame spreading rate LS can be obtained from the following equation:

( )20 0 0

2( )

f

i

T

L Tp i

m q wdTSC T T

λρ ρ

= =−∫ (1-8)

Ignition temperature Ti in equation (1-8) is an unknown. At present, a suitable simple method is not yet available for calculating the value of Ti. The following approximation is commonly used. Since the chemical reaction rate is very slow in the low-temperature pre-heating zone,

0

0fT

TwdT =∫ ; the integral equation f

i

T

TwdT∫

in equation (1-6) can be substituted by the integral equation

0

iT

TwdT∫ , i.e.

0

2 fT

Tic

dT q wdTdx λ

= ∫ (1-9)

In the high-temperature chemical reaction zone, since i fT T≈ , 0( )p iuc T Tρ − can be substituted by 0( )p fuc T Tρ − ,

( )01

p fip

dT uc T Tdx

ρλ

= −

(1-10)

Equating (1-10) to (1-9), the following expression for LS can be obtained,

( ) ( )0

2 2

0 0

2fT

TL

p f

wdTqSc T T

λ

ρ=

−

∫ (1-11)

Assuming that

0fC is the initial mole

concentration of the premixed gases,

( )0 0 0f p fC q C T Tρ= −

Therefore, equation (1-11) can be written as:

( )0

0 0

12

20 00

2 2fT

TL

f p f pf

wdT wSC c C cT T

λ λρ ρ

= = −

∫

(1-12)

where ( )

0

0

fT

T

f

wdTw

T T=

−

∫ is the average reaction

rate.

Since 0

0p

acλρ

= is the heat diffusion rate of

the premixed gases, 0fCw

τ = is the average

reaction time in the flame, then equation (1-12) can be written as:

( )1

2 10 20

2L m

m

aS a ττ

= ≈

(1-13)

Equation (1-13) shows that the spreading rate of laminar flame is directly proportional to the square root of the heat diffusion coefficient of the premixed gases, and inversely proportional to the square root of the average chemical reaction time. It can be seen that LS is a physical parameter which depends only on the physical and chemical nature of premixed gases. With known chemical reaction principles and other physical parameters, equation (1-12) can be used to estimate the spreading rate of laminar flame. The key is how to obtain the integral of ( )w T . This is a very crude model to solve for LS by this approximation. Therefore, modifications were proposed by some researchers later. It should be pointed out that the significance of the theory of laminar flame spreading is not on calculating the value of LS , but on revealing the effects of major physical and chemical parameters of various combustible mixed gases on the combustion process.

1.1.2 Premixed turbulent flame propagation In practice, most of the combustible mixed gases are burnt under turbulent conditions. The fluid flow is fluctuating with many rolling eddies. The spreading rate of turbulent flames is much faster than that of laminar flames. Turbulent flames have different shapes. Combustible mixed gases in internal combustion engines also undergo


42

turbulent combustion. The average flame spreading rate in the gas engines is 15 m/s to 30 m/s, despite the fact that the pressure and temperature inside the gas tank are relatively high, that will not be the spreading rate of laminar flames. In other burning facilities, combustible mixed gases flowing at high velocity or through large ducts with a sudden increase in the flow area, or flowing past fixed objects are all turbulent combustion. In all kinds of fire environment, the flame is also a typical turbulent combustion process. The propagation of flame in laminar uniformly mixed gases has been discussed above. In general, laminar flame spreading is only found in mixed gases of fuel and oxygen or air where the airflow velocity is relatively small. In ordinary confined circular pipes, with Reynolds number for the mixed gases larger than 2300, the flow will change from laminar to turbulent. The flame front surface will become very complicated. Under such circumstances, the mass burning rate will increase significantly. This indicates that the spreading rate of turbulent flames is much faster than that of laminar flames. The reason for the increase in the spreading rate of turbulent flames and the patterns of increase are preliminarily discussed in the following. It should be pointed out that the spreading rate of turbulent flames TS means the movement velocity of the turbulent flame front on the hypothesized uniform surface along the normal direction relative to the freshly mixed gases. To understand the combustion of turbulent mixed gases, the special characteristics of turbulent flow should be understood first. Air flow in which gases (or eddies) are moving irregularly is called turbulent flow. Turbulent flow and laminar flow are different in which there are irregular molecular movement and irregular eddy movement in turbulent flow. Experimental results indicated that the stronger the turbulence, the faster the rate of flame spreading. A typical method to describe turbulent flow is to use the two characteristics of uniform eddies - turbulence scale and turbulence intensity. Turbulence scale 'l is defined as the movement distance of an eddy before it “disappears”. It has the same order of magnitude with the size of the eddy itself. Therefore, the larger the turbulence scale 'l , the larger the fluctuations of the air flow.

'l can be understood as similar to the molecule average free range, a parameter which describes the thermal movement of molecules. Turbulence intensity is described by the root mean square of

the average eddy pulsation velocity, i.e.

2' iu u= cm/s (1-14)

where 'u is the average eddy pulsation velocity. In principle, the concept of turbulence intensity is similar to the average velocity of molecular diffusion. The faster the pulsation velocity, the stronger the turbulence. According to the simple theory of molecular motion, the equation for

diffusion coefficient is 13

D v l= . By analogy,

2''' iulul ≈≈ε (1-15)

where ε is the turbulent diffusion coefficient. Both turbulence scale and turbulence intensity can be measured by experimental methods. These two parameters represent the basic characteristics of turbulent flow, which describe the fluctuation intensity and affected area in turbulent gas movement. When the gas flow is in turbulent state, its transport property is very strong. In other words, the heat transfer coefficient Tλ , viscosity coefficient Tµ and diffusion coefficient ε of turbulent flow are much larger (a few hundred times) than the transport coefficients in laminar flow. The transport medium of physical quantities (heat, mass and momentum) in turbulent flow is not through single molecules, but eddies. The size of an eddy is much larger than that of a molecule to transport much larger amount of physical quantities. Because of the increase in the transport ability of turbulent flow, the spreading rate of turbulent flames is much faster than that of laminar flames. Air flow in a fire or in combustion chambers is usually turbulent. For example, laminar duct flow would become turbulent flow when Re > 2300; and air flow around a cube would be changed into turbulent flow when Re > 105. In the combustion chambers of piston engines, air-jet engines and rocket engines, mixed gases are all at turbulent state. Therefore, it is very important to study flame spreading for turbulent mixed gases. The basic principle of flame spreading of uniform turbulent flow is similar to that of laminar flow. It is because flame spreading is the movement of the chemical reaction zone due to heat and mass exchange between the combustion products and freshly mixed gases. However, the turbulent properties of air flow have a great influence on the combustion process. Experimental results indicated that the flame spreading rate of turbulent


43

flow is much faster than that of laminar flow. It is important to study the reason for the faster flame spreading rate of turbulent flow and the relationship between the different airflow parameters. Besides, the flame front surface of turbulent flow is always pulsating and folded. Noise is generated during the combustion process. Such phenomenon is absent in the laminar combustion process. 1.1.3 Turbulent flame propagation When combustible gas mixtures are at turbulent state, a large quantity of eddy fluctuations are produced inside the fluid. That leads to rapid spreading and diffusion of the momentum and energy of the fuel and oxygen, and increases the flame spreading rate. Different turbulent states have different turbulence intensity and turbulence scale, and hence different effects on combustion. The flame shapes and internal structures are also different. Studies had been carried out by many researchers in the past years. Several kinds of models of turbulent flame structure were proposed with two being most popular. The first group is the flame surface model with foldings on the flame surface. The second group is the control volume model where turbulent combustion takes place in the assigned control volumes. a. Flame surface folding model

It was proposed by Damköhler and Schelkin that turbulent flame is the result of foldings occurred on the laminar flame surface induced by turbulent flow. The basic structure of turbulent flame surface is suggested to be laminar. Because of turbulent pulsations, the burning flame surface is deformed, curved, folded, and being torn into closed pieces of different sizes. The flame surface is expanded to give more combustible mixed gases burnt out per unit time. Therefore, the flame spreading rate of turbulent flow is much faster than that of laminar flow. This kind of flame surface folding model is widely adopted since they are easy and convenient to use. Experiments on the burning flame of premixed gases of propane-oxygen were carried out by Damköhler with a gas burner. The variations of flame spreading rate of turbulent flow under different Reynolds number were studied. It was found that: i) when Re < 2300, the ratio of flame spreading rate of turbulent flow to that of laminar flow ST/SL = 1, i.e. in laminar state; ii) when 2300 ≤ Re ≤ 6000, / ReT LS S ∝ ; iii) when

Re > 6000, / ReT LS S ∝ . The two latter cases indicated that the flame spreading rate of turbulent flow is faster than that of laminar flow. With different turbulence intensity and scale, the variation patterns of the spreading rate of turbulent flame are different.

(1) Small-scale turbulent flame (2300 ≤ Re ≤

6000)

When the turbulence intensity is low, the fluctuations and velocity pulsations are small. The size of the fluctuating eddy inside the fluid is smaller than the thickness of the laminar flame combustion zone. Large deformations of the flame surface would not be induced, but only some corrugations. Small-scale eddies would speed up the heat transfer and mass diffusion rate inside the combustion zone (molecular transport and also eddy transport). The flame surface would increase slightly, and the thickness of the flame region would also increase.

(2) Large-scale weak turbulent flame

When the fluid fluctuations increase and give rise to large-scale eddies, its turbulence scale is larger than the thickness of the combustion zone. Under turbulent pulsations, the flame surface would become folded and deformed. But the turbulent pulsation velocity is not very large. The eddy movement velocity is smaller than the flame surface normal movement velocity. Therefore, the fluid eddies cannot break through the flame surface, and result in partial fluctuation and deformation. On each of the micro-flame surface, the flame is still propagating at the spreading rate of laminar flame along the normal direction. Due to the large size of the eddies and irregular movement, the flame surface changes continuously to become an irregular curved surface with a lot of uneven foldings. Obviously, if the flame surface foldings are expanded, the flame surface area F will increase greatly. If the flame is still propagating at the spreading rate of laminar flame SL, then the amount of combustible mixed gases burnt out per unit time is:

LT SFm ρ= (1-16) If the surface area FL of the not yet folded laminar flame is the mean position of the flame surface, i.e. the basic surface, then the increase in the spreading rate of turbulent flame is only because of the folding of the laminar flame surface. Therefore, the ratio of the spreading rate of turbulent flame to


44

that of laminar flow is the ratio of variations in the flame surface area,

LTLT FFSS // = (1-17)

( ) LLTT SFFS /= (1-18) In order to determine the turbulent flame surface area FT, some simplifications are needed. According to Schelkin, the turbulent flame folded surface could be assumed to be made up of continuous identical cone surfaces. The surface area of each cone is:

22 hrrFc += π (1-19) The bottom surface area of each cone is:

2rFb π= (1-20) In equation (1-19), h is the average height of the cones, and r is the average radius of the bottom surface of the cone. Thus,

2222 )/(1/// rhrhrrFFFF bcLT +=+== ππ (1-21) Substituting (1-18) gives:

( )21 rhSS LT += (1-22)

The size of the foldings on the flame surface depends on the turbulent condition. The size of the folded flame cone and the size of the turbulent eddies are both directly proportional to the turbulence scale 'l , i.e.

'r l∝ . The folded flame cone height is directly related to the impingement on the flame surface by the turbulent pulsation velocity 'u ,

'h u t∝ (1-23) The effect time t is related to the eddy size and the flame spreading rate,

'/ Lt l S∝ (1-24) Substituting the above equation gives:

' '/ Lh u l S∝ (1-25) i.e.

( ) ( )22( / ) ' '/ / ' '/L Lh r u l S l u S∝ ∝ (1-26)

Substituting equation (1-22) gives:

( )2/'1 LLT SukSS += (1-27) Usually, k is taken as 1.

(3) Large-scale strong turbulent flame (Re > 6000)

When the turbulence intensity is sufficiently large, the turbulent pulsation velocity exceeds the spreading rate of laminar flame. The fluid eddies would break through the flame surface and tear it into dispersed small pieces. The surface of each piece has a thin layer of closed laminar combustion zone. The burning of those small pieces could be in the unburnt mixed gas zone, and also in the burning combustion products zone. The spreading rate of turbulent flame of this kind of burning mainly depends on the ratio of pulsation velocity to the spreading rate of laminar flame. The following equation was given by Tanford after modification,

)/'(1/ LLT SuASS += (1-28) where A is taken from 1 to 3.5.

b. Control-volume model

Under certain situations with high turbulence intensity, the combustion reactions are not concentrated in the thin combustion zone, but penetrated into the deeper zone. The thickness of the flame zone is about several ten times of the laminar flame surface, the concentration and temperature distribution in the flame are very different from laminar flow. Because of this, a micro-volume combustion model was proposed by Summerfield et al. When the turbulence intensity is very high, the fluid would be fluctuated by eddies and divided into control volumes. Among the control volumes of different sizes, the part of turbulent flame penetrating into the unburnt mixed gases is not formed by simple laminar flame surfaces, but a series of mixed gas “islands” of different degrees of reaction penetrated into a large reaction zone in the combustible mixed gases. Therefore, combustion takes place inside the volume. Within the existing time of each “island”, the internal temperature and concentration are partially balanced; and each “island” is different from one another. This combustion model had been used by


45

Shelinkow to estimate the spreading rate of turbulent flame. Results were also compared with the experiments. The results showed that micro-volume burning plays an important role in the spreading of turbulent flame. The structure of turbulent flame is very complicated. The above flame structure models are helpful in understanding the mechanism of turbulent flame spreading, but further verifications, research and development are still needed. Experiments were conducted by Williams and Bollinger on turbulent flame produced by propane-air mixture, ethane-air mixture and ethyne-air mixture with a Bunsen burner. The results were compiled into the following empirical formula:

0.26 0.240.18 ReT LS S d= cm/s (1-29)

1.1.4 The concept of self-turbulent flame It has been found that the spreading rate of turbulent flame TS determined by experiments is much larger than that calculated by using the turbulence intensity ( )'u in approach air flow. In order to explain the difference between the experimental value and the calculated value, the theory of self-turbulent flame is one of those commonly used. That theory states that the reason for the difference between the two values is that the flame itself has fluctuations, giving turbulent flow that affects the combustion process and increases the value of TS . The theory of self-turbulent flame was firstly proposed by Landau, explaining turbulent flame is due to instability of the plane laminar flame front. In studying the effect of small fluctuations on a plane flame front, it was found that if the small fluctuations caused the flame front to curve (i.e. deviated from the balanced plane position), this kind of curving would continue to develop with time, and finally leading to the breaking of the front and making the air flow turbulent. When air flows through the undisturbed plane flame front, since there is no variation in the tangential velocity (zero) but only variation in the normal velocity, the streamlines would not become curved. If there are occasional fluctuations (in fact often happens) which make the flame front surface become curved into “wave form”, the streamlines passing through the front surface would be

deflected due to the refraction of streamlines. When the streamlines pass through the protruding zone, the streamlines would disperse. The airflow velocity would decrease and pressure would increase. When passing through the sunken zone, the streamlines would gather together and accelerate the air flow. At the same time, the pressure would decrease. This kind of pressure field at the flame front surface would make the protruding part of the front surface expand and the sunken part contract continuously. Finally, the front surface would break and the flow would become turbulent. In this kind of turbulent flow, the pre-heating of freshly mixed gases would not be simply by molecular heat transfer, but through strong turbulent mixing. Since the front surface has been broken through, its width would increase significantly. Therefore, the nature of self-turbulent plane laminar flame front is that: the ever-increasing small fluctuations of flame and tangential part would induce the breaking of the flame front itself. However, this kind of laminar flame is only unstable in large space; whereas in finite small space, it is stable. It is because the volume and duct wall have confined the curving of the large wavelength, the flame has a tendency to maintain its plane. At the sunken part, freshly mixed gases are heated more vigorously, and so the spreading rate of flame is faster to keep the plane front. This phenomenon is more obvious when the curved radius is relatively small. Although the theory of self-turbulent flame has not yet been investigated thoroughly, it has important significance. With this theory, many problems regarding turbulent combustion have been cleared up. To verify this theory, experiments have to be carried out to directly obtain the turbulence intensity before and during combustion. Nevertheless, there are difficulties in conducting such experiments. 2. DIFFUSION COMBUSTION In many of the combustion facilities, premixed gases cannot be formed easily due to the nature of the fuel. Fuel and oxygen (or air) are usually supplied into the combustion chamber separately. The mixing of fuel with oxygen and chemical reactions happen at the same time. Under the high-temperature environment inside the combustion chamber, chemical reactions take place rapidly, but the mixing of fuel and oxygen is much slower. Therefore, it is the rate of mixing that controls the combustion rate. This is the basic characteristic of diffusion combustion. Depending on the state of the fuel, there are


46

gaseous diffusion combustion and liquid spray combustion. Gaseous diffusion combustion can be further divided into laminar diffusion combustion and turbulent diffusion combustion, depending upon the flow condition of the fuel gas discharged into the combustion chamber. In most cases of fire incidents, there is a certain degree of pre-mixing before the fire actually starts. The fires then tend to undergo diffusion turbulent combustion. In general, combustions where the fuel and oxidizer are not premixed are called diffusion combustion or diffusion flames. In this kind of flames, the chemical reaction rate is much faster than the mass transfer rate induced by diffusion and the energy transfer rate due to heat transfer. Its obvious characteristics are that the chemical reaction rate is very fast on the flame surface. The combustion zone is a very thin surface and can be regarded as a mathematical surface. On one side of the combustion surface is the combustion gas, the other side is the oxidizer gas. Based on the nature of the mixed gas flow, diffusion flames can be divided into laminar diffusion flames and turbulent diffusion flames. Candles and matches burning in the air, and the burning of a single fuel liquid drop in an oxidized medium belong to laminar diffusion flames. Combustions in industrial furnaces (for heating up combustion gases or liquid fuel, e.g. furnaces in steel plants), turbo-machinery, some liquid-fuel rocket engines and internal combustion engines got turbulent diffusion flames. A classical example of laminar diffusion flames is the diffusion flames in coaxial circular ducts. When gaseous fuel and air are flowing separating in circular ducts of diameter d and 'd respectively, the velocities of the flows are equal. The flame shapes of this kind of burning can be divided into two types: if the amount of oxygen supplied exceeds the amount for complete combustion of the fuel, it will give rise to oxygen-rich diffusion flames. The flame surface will gradually contract to the axis of the circular duct and become a conical flame. On the contrary, if oxygen in the air is not sufficient, the flame will develop to the wall of the external duct and become a bell-shaped oxygen-deficient diffusion flame. The eddies inside the air flow make the flame shape more complicated. When the velocity of discharging fuel into the air is sufficiently high, laminar diffusion flames would become turbulent diffusion flames. In the laminar flame zone, the flame height would increase with the increase in airflow velocity, and reach a maximum value. For further increase in the jet velocity of the fuel, unstable turbulent flame would start to appear at

the flame top which would further increase the airflow velocity. The position where laminar flame is changing into turbulent flame (break-up point) would descend rapidly, and the whole flame height would also decrease. When the airflow velocity further increases, and the height of the break-up point reaches a certain value, the whole flame height would remain almost unchanged. The turbulent flame height (the flame should be quiescent relative to the environment) in air had been determined by Hawthorne et al. The observed flame height agreed well with the mixing length calculated by using the jet-flow mixing theory. This also proved that in turbulent diffusion flames, the chemical reaction rates are also similar to those in laminar diffusion flames, which have no control effect on the combustion process. 2.1 Laminar Diffusion Flames The first successful analysis of laminar diffusion flames was done by Back and Schumann. A numerical equation which can accurately calculate the flame shape and height was derived. The calculated flame shape and height agreed with the experiments.

2.1.1 Differential equations of diffusion flames Since the processes taken place inside the flame are very complicated, the following simplification assumptions are made:

On the flame surface, fuel and oxygen are mixed in proper ratio. Chemical reactions take place only on the flame surface and they are instantaneous.

The flows of fuel and oxygen are one-dimensional flows with uniform and same velocity. Diffusion of the components only takes place along the radial directions.

The mole number is not changed in the combustion reactions, i.e. the pressure is the same throughout the whole process.

The diffusion of fuel and oxygen in inert gases is regarded as the diffusion of two components, their diffusion coefficients are equal.

The product of density ρ and diffusion coefficient D of mixed gases is not related to temperature, it is constant in radial direction, i.e. ρD = constant (2-1)

At constant pressure, 1T

ρ ∝ , the diffusion


47

coefficient of two components is:

1.75D T∝ Therefore, the effect of temperature on the product ρD is relatively small. Thus, ρD can be approximated as a constant in radial direction. According to the above simplification assumptions, differential equations for diffusion flames can be set up by applying the mass conservation equation. The mass equation for component K with chemical reaction is:

( ),ln 1 DK jKk K

K K j

Y uD Y wDt Y Y x

ρ

ρ ρ

∂= −

∂

r

(2-2)

where ( ) ( ) ( ) ( )

zwY

yvY

xuY

xuY z,DKKy,DKKx,DKK

j

j,DKK

∂∂

+∂

∂+

∂∂

=∂

∂ ρρρρ r

(2-3) and

∂∂

+∂∂

+∂∂

==z

Ywy

Yvx

YuYDt

DYYDt

YD KKK

K

K

K

K 11In

(2-4) According to assumption (b),

=== w,vux 0 constant (2-5) Since only steady-state diffusion combustion process is considered,

0KYt

∂=

∂ (2-6)

Then,

∂∂

=z

YwYDt

YD K

K

K 1In (2-7)

According to assumption (a), chemical reactions only take place on the flame surface. If the process taken place on the flame surface is not considered, then

0Kw = According to assumption (d),

, ,I F I OD D D= = = constant (2-8)

Then the diffusion velocity DKvr can be calculated:

, lnD k Kv D Y= − ∇r

From this equation and assumption (e), the second term on the right side of equation (2-2) can be simplified,

( ) ( )k,DKKj

y,DKK

K

vYYx

vYY

rρρ

ρρ

∇−=∂

∂−

11

( )

2

1 ln

1

K KK

K K KK K K

Y D YY

D DY Y YY Y Y

ρρ

= − ∇ − ∇

= ∇ ∇ = ∇

As the diffusion flames under discussion are axisymmetric, it is more convenient to use cylindrical coordinate system. Vector analysis gives:

2 22

2 2 2

1 1K K KK

Y Y YY rr r r r zϕ

∂ ∂ ∂∂ ∇ = + + ∂ ∂ ∂ ∂ (2-9)

Since the distribution of KY is axisymmetric, KY is not related to ϕ . According to assumption (b) and neglecting the diffusion along the z direction,

0

0

K

K

Y

Yz

ϕ∂

=∂∂

≈∂

2

2

2

2

0

0

K

K

Y

Yz

ϕ∂

=∂

∂≈

∂

Therefore,

22

2

1 1K K KK

Y Y YY rr r r r r r

∂ ∂ ∂∂ ∇ = = + ∂ ∂ ∂ ∂

That gives:

( ) 2,

2

1 1K DK j K K

K j K

Y v Y YDY x Y r r r

ρρ

∂ ∂ ∂− = + ∂ ∂ ∂

(2-10)

Substituting equations (2-7) and (2-10) into equation (2-27), with 0Kw ≡ , the basic differential equation for diffusion flames is:

∂∂

+∂∂

=∂∂

rY

rrY

wD

zY KKK 1

2

2 (2-11)

where KY is the mass fraction of component K , D is the diffusion coefficient of two components, w is the flow rate of gaseous fuel (or oxidizer),


48

z is the distance of a certain plane of the flame from the coordinate of the discharge opening; and the subscript “ F” represents fuel, “ O” represents oxidizer.

In deriving equation (2-10), KYz

∂∂

is neglected.

But in equation (2-7), KYz

∂∂

is taken into account.

This is because 1) based on assumption (b), only radial diffusion is considered and axial diffusion is

neglected, i.e. r

Yz

Y KK

∂∂

<<∂∂ ; however in equation

(2-7), the value of z

Yw K

∂∂ is still quite large (as

the airflow velocity w is very large, up to 3300 cm/s). 2) as stated in assumption (b) that axial

diffusion can be neglected, but ln KD YDt

represents the variation of the mass fraction of component K induced by the air flow in the flow process which is not at all related to diffusion; therefore at steady-state,

zY

Yw

DtYD K

K

K

∂∂

=ln (2-12)

The concentration fields of fuel FY and oxygen

OY can be determined by the partial differential equation (2-11), only that the boundary conditions are different (not including the point on the flame surface, because at point 0Kw ≠ ). Thus, two differential equations have to be solved for the distribution of FY and OY . Adopting the mathematical method by Back and Schumann, a new variable is defined.

0F

F O

YY

s Y

= −

2

0

drr

rr

rr

f

f

f

′≤<

=

<≤ (2-13)

where fr is the radius of the flame surface, and

Fs is the ratio of fuel and oxidizer. According to assumption (a), Kw = ∞ on the flame surface, so the instantaneous reaction has finished when the fuel and oxygen have diffused to

fr r= . Therefore, it is defined that 0Y = (at

fr r= ). In this way, the diffusion problem becomes the diffusion of a single gas (fuel). The function ( ),Y Y r z= is the solution of the following partial differential equation.

∂∂

+∂∂

=∂∂

rY

rrY

wD

zY 1

2

2 '0

2dr≤ ≤ (2-14)

The boundary condition for equation (2-14) is:

(i) '0

2

0, 0dr r

Y Yr r= =

∂ ∂= =

∂ ∂ 0 z L≤ ≤ (2-15)

(ii) ,0

0,0

F

zF O

YY

s Y=

= −

02

'2 2

dr

d dr

≤ <

< ≤

(2-16)

where

0

1FF

O

YsY s

= = (2-17)

2.1.2 Solving the differential equations for

diffusion flames Equation (2-14) can be solved by using separation of variables and Bessell function.

( )2

,0 ,0 ,021

4 1,' 'P F O P

m m

d dY z r Y s Y Yd d ϕ

+∞

=

= − +

∑

( )1 0

22

0

2' exp

2

m m

m m

dJ J r

d DJ zu

ϕ ϕ

ϕ ϕ

×

(2-18)

where ,0 ,0 ,0P F F OY Y Yν= + ; mϕ is all the positive

root values of the equation 1

' 02dJ ϕ =

(rejecting the zero root 0 0ϕ = ); 1 0,J J are Bessell functions. a. Flame concentration field

The concentration distribution function of the fuel inside the flame and oxidizer outside the flame (on each plane of the flame) can be calculated by equation (2-18).

( )

( )

LzrrYY

Lzdrrr,zY

Lzrrr,zY

fOF

fO

fF

≤≤===

≤≤′

<<

≤≤<≤

00

02

00

where L is the flame height.


49

For example, given z1 = 10 mm and a series of 1 2 3, , ,r r r … nr , since some of the parameters in equation (2-18) are already known such as ,0 ,0, ', , ,F O Fd d Y Y s , etc., and some can be checked in the mathematics handbook such as 0 1, , mJ J ϕ , etc., based on equation (2-18), the value of 1( , )Y z r on a plane of the flame can be calculated. Given a series of 2 3, ,Z Z …, nZ values, the concentration distribution function

2 3, ,Y Y … nY on different planes of the flame can be calculated by using a similar method.

b. Flame shape

According to assumption (a), the chemical reactions on the flame surface are instantaneous. There would not be any fuel and oxidizer on the flame interface. When the two of them have diffused to the flame interface, they would react immediately to give combustion products. The experiment by Hottel and Hawthorne had proved this assumption. Thus, the mass fraction of these two components on the flame interface are both zero. Therefore, using 0Y = in equation (2-18), the corresponding fr r= , and that gives the flame boundary equation:

2

,0 ,0 ,02

4 0' 'P F O P

d dY s Y Y Sd d

− + =

(2-19)

where

( )∑∑∞+

=

∞+

=

′

==1 2

2

0

01

1 exp2

21m

mm

fmm

mmm

zwDdJ

rJdJSS

ϕϕ

ϕϕ

ϕ

(2-20) That gives:

2,0

,0

'4 4

F Of

P

s Yd dS Cd Y

= − = = constant (2-21)

Equation (2-21) is the boundary equation for diffusion flames. In the equation, only z and fr are variables, the others are constants. According to equation (2-21), the calculated ( )fz f r= is an implicit

function and cannot be transformed into an explicit function, which can only be solved by graphical method.

2.2 Fundamental Aspects of Laminar Diffusion Flames with Fast Chemistry (One-Step Reaction)

Non-premixed diffusion flames are observed in most of the fire accidents and in many industrial applications such as spray burning and other combustion engines. Thermal decomposition and phase change of combustible solid materials, combustion in surrounding air are also non-premixed combustion. The simplest combustion reaction of carbon-hydrogen fuel is a one-step reaction:

( 1)F sO s P+ → + (2-22) Fuel (F) and oxygen (O) react to give combustion product (P) under stoichiometric number s. The mixture fraction f is defined as:

2

1 2

f φ φφ φ−

=−

where

0F

YYs

φ = −

The subscripts 1 and 2 refer to fuel and oxygen respectively. The reaction between fuel and oxygen is described by fast reacting chemistry. Under practical situations, various fuel and oxygen will undergo chemical reactions immediately after mixing. This assumption implies that the reaction time is much shorter than the mixing time. The simplest fast reaction is a one-step irreversible reaction. The instantaneous fuel, oxidizer and combustion product are related as:

; 0

; 0

0, 0; ( ); ( 1) (1 )

0, ( ); 0; ( 1) (1 )S F B S pr B S

S F B S pr B S

f f Y Y sY f f Y s Y f f

f f Y Y f f Y Y s Y f f

φ

φ

≤ ≤ = = − = + −

≥ ≥ = − = = + − (2-23) where

(1 )B F SY Y f= − Corresponding to the situation of φ = 0, the stoichiometric number of the mixture fraction sf is given as:

0

0S

F

YfsY Y

=+

(2-24)

It implies that there is no oxidizer on the fuel side, and there is no fuel on the side with oxidizer. The


50

mean Favre-averaged concentrations can be expressed as:

( ) ( ) ( )dfx,fP~fYxY~ iiirr

∫=1

0 (2-25)

Favre Probability Density Function (PDF) is often used to integrate the spatial distribution of mixture fraction f. This spatial distribution also includes the effect of turbulent fluctuations.

If the reaction is reversible, there will be both fuel and oxidizer at stoichiometric numbers close to the mixture fraction. Under the assumption of fast reaction, both the forward and backward reactions are faster than the mixing process, so that the mixed chemical components can attain dynamic equilibrium. To close the set of equations, the equation constant can be used. In solving the PDF transfer equation, the dependence on the flame position, turbulent conditions and other parameters can be considered. For the more general form of PDF commonly used, there is a simpler variable relationship. Therefore, the following variable of mixture fraction is usually added.

The most commonly used PDF functions are Beta function and Clipped-Gaussian distribution. Beta function is:

( ) ( )( )∫ −−

−−

−

−= 1

0

11

11

1

1

dfff

fffP~βα

βα (2-26)

where the parameters α and β are determined by the following equations:

( )[ ] ( ) f~f~,f~f~f~f~ αβα −=−−= 111 2 Clipped-Gaussian distribution is: ( )ff~

[ ]21 2

1 1exp ( ) ( ) ( 1)(2 ) 2

f H f H fµσ π σ

− = − − −

( )( )

−−+ ∫ ∞−

0 221 )(

21exp

21 dfff

σµ

πσδ

( )( )

−−−+ ∫

∞

1

221 )(

21exp

211 dfff

σµ

πσδ

(2-27)

where δ and H are deltahe and Heaviside function respectively. Distribution parameters µ and σ can be checked from the chart. 2.3 Simple Chemical Reacting System

(SCRS) and Mixture Fraction Fuel and oxidizer will undergo a series of reactions to give combustion products. For example, the combustion of the simplest hydrocarbon methane CH4 involves more than 40 basic dynamic reactions. The component equation of the mass fraction for a particular component jY can be written as:

jjjjj S)gradYdiv(Γ)uYdiv(ρ)Y(ρt

+=+∂∂ r (2-28)

jS is the source term of the chemical reaction in

the component transport equation, the production and dissipation rate of the component. The sum of the mass fraction of fuel, oxidizer and other components should be equal to 1, i.e.

1

1N

jj

Y=

=∑

where N is the total number of components. The heat energy released from combustion can be represented by the following transport equation:

hh Sgradh)div(Γ)uhdiv(ρh)(ρt

+=+∂∂ r (2-29)

The source term of the above transport equation of enthalpy includes radiation loss and increase, pressure work and chemical energy. Energy consumption due to viscosity under low consumption number can be neglected. The corresponding temperature can be calculated from the calorific value,

f f

p

h Y HT

c−

=

where fH is the calorific value of the fuel.

1( ) ref

T

p pTref

c c dTT T

=− ∫

1

N

p j jj

c m c=

=∑

where jc is the specific heat of species j.


51

The local density of the mixture depends on the concentrations of the combustion reactants and products and the mixing temperature, which can be calculated by:

1

Nj

jj

PYRT M

ρ

=

=

∑ (2-30)

where jM is the molecular weight of species j.

The variations of temperature and density would influence the flow field. By solving the above component equation (2-28) and enthalpy equation (2-29) by coupling with the laminar flow and turbulent flow basic equation set, the combustion flow field can be solved numerically. The simplest combustion process given by Spalding et al. is the simple chemical reacting system (SCRS). The reaction between the fuel and a certain proportion of oxygen to give combustion products is believed to be a one-step infinitely fast chemical reaction. 1 kg of fuel＋s kg of oxidant ＝(1＋s) kg of products The combustion of CH4 can be written as: CH4 ＋ 2O2 → CO2 ＋ 2H2O That means 1 mole of CH4 reacts with 2 moles of O2 would give 1 mole of CO2 and 2 moles of H2O. This can be written as a reaction equation:

1 kg of CH4 ＋ 6416

kg of O2 → 64116

+

kg

of products The stoichiometric number is:

64 416oxygens fuel= = =

In SCRS, infinitely fast chemical reaction implies that all intermediate products are neglected. The transport equations for the mass fraction of fuel and oxygen are:

( ) ( ) ( )f f f f fY div Y u div gradY Stρ ρ∂

+ = Γ +∂

r

(2-31)

( ) ( ) ( )o o o o oY div Y u div gradY Stρ ρ∂

+ = Γ +∂

r

(2-32)

Defining the following variable,

f osY Yφ = − Usually, it is assumed that the mass exchange coefficients jΓ in each component transport

equation are equal constants, φΓΓΓ of == . Equations (2-31) and (2-32) can be written as the transport equation of φ :

( ) ( ) ( ) ( )f odiv u div grad s S Sx φρφ ρφ φ∂

+ = Γ + ⋅ −∂

r

For the assumption of one-step reaction, ( ) 0f os S S⋅ − = , the equation is simplified as:

( ) ( ) ( )div u div gradt φρφ ρφ φ∂

+ = Γ∂

r (2-33)

Defining the non-dimensional variable f as the mixture fraction:

0

1 0

f φ φφ φ−

=−

(2-34)

The subscript O represents the side with oxygen, and the subscript 1 represents the side with fuel. f = 0 means the value of the mixing point φ when there is only oxygen, f = 1 means the value of φ when there is only fuel. The above equation can be expanded as:

0

1 0

f o f o

f o f o

sY Y sY Yf

sY Y sY Y

− − − = − − −

(2-35)

On the fuel side, [ ]11

1 , 0f om m = =

On the oxygen side, [ ]00

0 , 1f om m = =

Equation (2-35) can be simplified as:

[ ][ ]

,00

,1 ,001

f o o f o o

f of o

sY Y Y sY Y Yf

sY YsY Y

− − − − + = =+ − −

(2-36)

In the stoichiometric mixing ratio, the stoichiometric mixture fraction stf when fuel and oxygen cannot exist together in the combustion products is defined as:


52

,0

,1 ,0

ost

f o

Yf

sY Y=

+ (2-37)

Fast chemical reactions imply that no excessive oxygen and fuel would coexist in the combustion products.

If 0oY > , then 0fY = , and if ,stf f<

,0

,1 ,0

o o

f o

Y Yf

sY Y− +

=+

(2-38)

If 0fm > , then 0oY = , and if ,stf f>

,0

,1 ,0

f o

f o

sY Yf

sY Y+

=+

(2-39)

These two equations indicated that the linearity of

fY and oY depends on the mixture fraction f.

The mixture fraction f is related to φ linearly and described by the transport equation.

( ) ( ) ( )ff div f u div gradftρ ρ∂

+ = Γ∂

r (2-40)

In solving the above equation under suitable boundary conditions, the mixture fractions f of oxygen and fuel at the solid wall are both 0. By using equations (2-38) and (2-39), the mass fractions of oxygen and fuel on the burning surface are:

,10 ;1

sto f f

st

f fY Y Yf

−= =

− for 1stf f≤ < (2-41)

,00 ; stf o o

st

f fm Y Yf−

= = for 0 stf f< < (2-42)

Through Favre averaging, the above transport equation in turbulent combustion can also be solved to describe the density variations in the combustion flow. In using the PDF method in turbulent models, the variables ,f oY Y , etc. can be obtained by weighing the instantaneous values of PDF of the mixture fraction. The mean average value ϕ of the property parameter ϕ can be given as:

( ) ( )1

0f p f dfϕ ϕ= ∫ (2-43)

Here, ϕ is a certain variable of function f, and

( )p f is the probability density function. Different probability distribution functions can be used. Better results can be obtained by using clipped Gaussian and beta function. 2.4 An Example of Numerical Modeling of

Laminar Diffusion Flames Calculations on combustion are very important for increasing the efficiency of engines and reducing the production of pollutants. However, numerical modeling of combustion is difficult to carry out. First of all, the control equations are very complicated. There is not only a set of fluid dynamics equations, but also a set of chemical reaction equations, so that the set of control equations is very large. Also, a lot of components are involved in the chemical reactions which undergo several basic reactions. Generally, finite rate chemical reaction model involves tens of components which undergo several hundreds of intermediate basic reactions. Since the chemical reaction rate is very fast, the time-scale is much smaller than that of the fluid transport. As each basic reaction has its own time-scale, there will be serious time-scale problems; or called the rigidity problems of the system. In addition, the combustion process also includes heat transfer, radiation, convection and diffusion of the chemical components, and also anisotropy due to buoyancy. Two-phase turbulent combustion makes the problem even more complicated. Since the physical gradient at the flame edge is very large, finer grids are required at the flame edge in order to obtain more accurate numerical solution. Numerical modeling of combustion, though difficult as listed in above, is more attainable with the rapid development of computer technology. Here, a simplified two-dimensional laminar diffusion flame is considered. A one-step irreversible infinitely fast reaction of five species is considered:

C3H8 + 5O2 + 18.8N2 → 3CO2 + 4H2O + 18.8N2

Here, propane is the fuel, air is the oxidizer (nitrogen acts only as a diluent without involving in the reaction), carbon dioxide and water are the products. The first example focused on a chemical reaction model and its simplified version. Here, only the basic equations and boundary conditions in the numerical process are listed, but not the calculated results and the finite difference method used.


53

2.4.1 Numerical modeling of laminar diffusion flame with infinitely fast reaction rate

1. Control equations

For a two-dimensional laminar diffusion flame, neglecting radiation and adopting a one-step irreversible infinitely fast chemical reaction model, its control equations are:

(a) Continuous equation

0u vt x yρ ρ ρ∂ ∂ ∂+ + =

∂ ∂ ∂ (2-44)

(b) Momentum equation

2 23

u uu uv P u vt x y x x x yρ ρ ρ µ

∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + − − ∂ ∂ ∂ ∂ ∂ ∂ ∂

0u vy y x

µ ∂ ∂ ∂

− + = ∂ ∂ ∂ (2-45)

2 2 03

v uv vv P gt x y y

v u v ux x y y y x

ρ ρ ρ ρ

µ µ

∂ ∂ ∂ ∂+ + + +

∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂− + − − = ∂ ∂ ∂ ∂ ∂ ∂

(2-46)

(c) Components equation

F F F F

F

Y uY vY Yt x y x Sc x

ρ ρ ρ µ ∂ ∂ ∂ ∂∂+ + − ∂ ∂ ∂ ∂ ∂

0FF

F

Y wy Sc y

µ ∂∂− + = ∂ ∂

(2-47)

O O O O

O


ρ ρ ρ µ ∂ ∂ ∂ ∂∂+ + − ∂ ∂ ∂ ∂ ∂

0OO

O

Y wy Sc y

µ ∂∂− + = ∂ ∂

(2-48)

2 2 2 2

2

N N N N

N


ρ ρ ρ µ ∂ ∂ ∂ ∂∂+ + −

∂ ∂ ∂ ∂ ∂

2

2

0N

N

Yy Sc y

µ ∂∂− = ∂ ∂

(2-49)

( )2 2 11CO F O NY Y Y Y β= − − − (2-50)

( )( )

2 2 11 1H O F O NY Y Y Y β= − − − − (2-51)

(d) Energy equation

Pr Prh uh vh P h ht x y t x x y yρ ρ ρ µ µ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + − − − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

1 1

Pr Pr1 1 0Pr Pr

N Nj j

j jj j

Y Yh h

x Sc x y Sc yµ µ

= =

∂ ∂ ∂ ∂ − − − − = ∂ ∂ ∂ ∂ ∑ ∑

(2-52)

(e) State equation

PMRT

ρ = (2-53)

(f) Enthalpy equation

( , )jh h Y T= (2-54)

Here, u and v are velocity components, P is pressure, ρ is density, T is temperature, jY is the mass component of the j species (here, j can be C3H8, O2, CO2 or H2O), h is enthalpy, µ is viscosity coefficient, Pr is Prandtl number, Sc is Schmidt number (Prandtl number and Schmidt number are taken as 0.7 in this paper), jw is the production rate of

the j species, 1β is the ratio of the products

CO2 and H2O, and M is the average molecular mass.

11

Nj

j

YM

M=

= ∑

There are altogether 11 equations in the above, giving a set of differential equations for solving 11 unknowns: u, v, P, ρ, T, h,

2 2, , ,F O CO H OY Y Y Y and

2NY . 2. Simplified mathematical model

In the above control equations, those chemical reaction production rates are more difficult to handle. The equation can be simplified by using the assumption of infinitely fast reaction. Assume Fw is proportional to Ow ,

OF

wws

= (2-55)

and

2F O NSc Sc Sc Sc= = = (2-56)


54

Combining equations (2-47) and (2-48),

0

O OF F

Y YY Ys s

x Sc x y Sc xµ µ

∂ − ∂ − ∂ ∂ − − =∂ ∂ ∂ ∂

(2-57)

Making OF

YYs

φ = − , equation (2-54) can be

written as:

0u vt x y x Sc x y Sc yρφ ρ φ ρ φ µ φ µ φ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + − − = ∂ ∂ ∂ ∂ ∂ ∂ ∂

(2-58) By further simplification,

Oin

Fin Oin

f φ φφ φ

−=

− (2-59)

Here, Finφ and Oinφ represent the values of

Fφ and Oφ at the opening respectively, equation (2-58) can be written as:

0f uf vf f ft x y x Sc x y Sc yρ ρ ρ µ µ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + − − = ∂ ∂ ∂ ∂ ∂ ∂ ∂

(2-60) Assuming the Lewis (Pr/Sc) number is 1, i.e. Pr 1Sc

= (2-61)

Then the energy equation can be written as:

Prh uh vh P ht x y t x xρ ρ ρ µ∂ ∂ ∂ ∂ ∂ ∂ + + − − − ∂ ∂ ∂ ∂ ∂ ∂

0Pr

hy y

µ ∂ ∂− = ∂ ∂

(2-62)

Finally, a set of simplified equations is obtained:

0u vt x yρ ρ ρ∂ ∂ ∂+ + =

∂ ∂ ∂ (2-63)

2 23

u uu uv P u vt x y x x x yρ ρ ρ µ

∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + − − ∂ ∂ ∂ ∂ ∂ ∂ ∂

0u vy y x

µ ∂ ∂ ∂

− + = ∂ ∂ ∂ (2-64)

v uv vv v ut x y x x yρ ρ ρ µ

∂ ∂ ∂ ∂ ∂ ∂+ + − + ∂ ∂ ∂ ∂ ∂ ∂

2 2 03

v u P gy y x y

µ ρ ∂ ∂ ∂ ∂

− − + + = ∂ ∂ ∂ ∂

(2-65)

0Pr Pr

h uh vh P h ht x y t x x y yρ ρ ρ µ µ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + − − − = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

(2-66)

0f uf vf f ft x y x Sc x y Sc yρ ρ ρ µ µ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + − − = ∂ ∂ ∂ ∂ ∂ ∂ ∂

(2-67)

2 2 2 2 2 0N N N N NY uY vY Y Yt x y x Sc x y Sc y

ρ ρ ρ µ µ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + − − = ∂ ∂ ∂ ∂ ∂ ∂ ∂

(2-68)

0P MRT

ρ = (2-69)

( , )jh h T Y= (2-70)

There are eight equations on the eight unknowns

2, , , , , , , Nu v P T f h Yρ to be

solved.

3. Boundary conditions and the related parameters

Taking a two-dimensional diffusion flame with two sides closed as an example, its boundary condition can be expressed as:

(a) Fuel supply condition at x = 0, in iny y y− < < :

,0,1

F

vf

ρ ρ===

F

F

F

u uT Th h

===

20NY = (2-71)

(b) Air supply condition at x = 0,

out iny y y− < < − or in outy y y< < :

2

,0,

0.767

air

N

vY

ρ ρ=

==

0

air

air

u uT Tf

=

=

=

airh h= (2-72)


55

(c) Fixed wall (thermally insulated) condition at outy y= or outy y= − :

0u v= =

2 0NYT f hy y y y yρ ∂∂ ∂ ∂ ∂= = = = =

∂ ∂ ∂ ∂ ∂ (2-73)

(d) Outflow condition at Lx x= :

2

2

2 0NYv T f hx x x x x x

ρ ∂∂ ∂ ∂ ∂ ∂= = = = = =

∂ ∂ ∂ ∂ ∂ ∂

( ) 0uρ∇⋅ =r (2-74)

If the set of partial differential equations are solved by discretizing with staggered grids, pressure condition is not needed. Other relevant parameters are:

8.31434R = J/mole·K

0 101.3P = kPa 298F airT T= = K

Pr = 0.7 0.7Sc =

44FM = g/mole

32OM = g/mole

228NM = g/mole

244COM = g/mole

218H OM = g/mole

55 104261100025320 −− ×+×≅+= .T.baTµ3.63636s =

1 0.647β = 2

1 2 3h a T a T a= + +

Here,

2 2 2 2 2 2

1 1 1 1 1 1 1 11 F F O O N N CO CO HO HOa Y a Y a Y a Y a Y a= + + + +

2 2 2 2 2 2


2 2 2 2 2 2


1 3 2 30.27467 10 , 0.3395, 689.74F F Fa a a−= × = = −

for C3H8

1 4 2 30.1987 10 , 0.2148, 65.8O O Fa a a−= × = = − for O2

2 2

1 4 2 30.394 10 , 0.203, 2201.5CO CO Fa a a−= × = = − for CO2

2 2 2

1 4 2 30.7652 10 , 0.3932, 3335.5H O H O H Oa a a−= × = = − for H2O

2 2 2

1 4 2 30.208 10 , 0.2345, 71.77N N Na a a−= × = = − for N2

4. Procedure

The above simplifications have combined FY and OY and introduced f. As there is

not a chemical reaction production term in the f equation, the set of equations can be solved rather easily. The specific solving steps can be described by the flowchart in Fig. 1.

Fig. 1: Solution Chart

,µ ρ are known, solving momentum equation and continuity equation

Solving energy equation and component equation

New fluid parameters

,µ ρ

Present time step ,µ ρ

, ,u v P,h f

,jY T

Next time step


56

In each time step, assuming that µ and ρ are already known, the continuous equation and momentum equation can be solved to obtain the new , ,u v P ; and then the energy equation and component equation are solved to obtain ,h f and

2NY , and then get FY ,

OY and T. The new fluid parameters µ and ρ can be obtained from the new jY and T. The steps are repeated until convergence. Practically, it is not necessary to obtain accurate results in one time step. Approximately convergent results can be obtained by just one multi-grid finite cycle as seen in later sections. How FY , OY and the product components can be obtained after getting f as shown here. The chemical reaction is assumed to be irreversible and infinitely fast, there can only be three situations: a) rich fuel and lean air: only fuel and products left after reaction; b) rich air and lean fuel: only air and products left after reaction; c) fuel and air in stoichiometric mixture proportion: no fuel and no air left after reaction, but only the products. In summary, this kind of infinitely fast chemical reaction does not allow the coexistence of gas and fuel after the reaction. If there is fuel left, there would be no air; and vice versa. In following the above definitions,

Oin

Fin Oin

f φ φφ φ

−=

−， Fin Oinφ φ≠

O

FYYs

φ = − (2-75)

Defining sf as the critical value which

0φ = :

Oins

Oin Fin

f φφ φ

=−

(2-76)

Obviously, there will be three situations:

(a) If sf f= , fuel and air are in right proportion, then

( )2 21

00

1

F

O

CO N

YY

Y Yβ

==

= −

( )2 21(1 ) 1H O NY Yβ= − − (2-77)

(b) If , 0, Os F

Yf f Ys

φ< < < , there is lean fuel

and rich air, then

( )2 21

0

1

F

O

CO N

YY s

Y s Y

φ

β φ

== −

= + −

( )2 21(1 ) 1H O NY s Yβ φ= − + − (2-78)

(c) If , 0, Os F

Yf f Ys

φ> > > , there is rich fuel

and lean air, then

( )2 21

0

1

F

O

CO N

YY

Y Y

φ

β φ

==

= − −

( )2 21(1 ) 1H O NY Yβ φ= − − − (2-79)

Here,

( )Fin Oin Oinfφ φ φ φ= − + In this way, the reactant and product components can be solved. The new ρ and µ can be obtained:

0P MRT

ρ =

aT bµ = +

Here, M is the average molecular weight. Considering the actual flow, at least one more transport equation has to be solved for the product

2COY or 2H OY . They are

assumed to have proper proportion to simplify the numerical solution process. The following parameters (without including gravity) are selected as examples for solving the laminar diffusion flames:

0.318iny = cm

2.540outy = cm 30.00L = cm 3.000Fu = cm/s


57

9.880Ou = cm/s 298.0FT = K

298.0OT = K The Reynolds number is:

Re 150O O out

O

u yρµ

= ≈

2.4.2 Numerical modeling of combustion with

finite reaction rate For simplicity, in the following discussion on combustion problems with finite reaction rate, the chemical reaction is assumed as a one-step irreversible reaction: C3H8 + 5O2 + 18.8N2 → 3CO2 + 4H2O + 18.8N2

(2-80) The control equations and boundary conditions are similar as in the previous section. They are listed again below:

0u vt x yρ ρ ρ∂ ∂ ∂+ + =

∂ ∂ ∂ (2-81)

u uu uv Pt x y xρ ρ ρ∂ ∂ ∂ ∂

+ + +∂ ∂ ∂ ∂

2 2 03

u v u vx x y y y x

µ µ ∂ ∂ ∂ ∂ ∂ ∂

− − − + = ∂ ∂ ∂ ∂ ∂ ∂

(2-82)

v uv vv v ut x y x x yρ ρ ρ µ

∂ ∂ ∂ ∂ ∂ ∂+ + − + ∂ ∂ ∂ ∂ ∂ ∂

2 2 03

v u P gy y x y

µ ρ ∂ ∂ ∂ ∂

− − + + = ∂ ∂ ∂ ∂ (2-83)

F F F F

F


ρ ρ ρ µ ∂ ∂ ∂ ∂∂+ + − ∂ ∂ ∂ ∂ ∂

0F FF

F F

Y Y wx Sc x y Sc y

µ µ ∂ ∂∂ ∂− − + = ∂ ∂ ∂ ∂

(2-84)

O O O O

O


ρ ρ ρ µ ∂ ∂ ∂ ∂∂+ + − ∂ ∂ ∂ ∂ ∂

0O OO

O O

Y Y wx Sc x y Sc y

µ µ ∂ ∂∂ ∂− − + = ∂ ∂ ∂ ∂

(2-85)

2 2 2N N NY uY vYt x y

ρ ρ ρ∂ ∂ ∂+ +

∂ ∂ ∂

2 2

2 2

0N N

N N

Y Yx Sc x y Sc y

µ µ ∂ ∂∂ ∂− − = ∂ ∂ ∂ ∂

(2-86)

( )2 2 11CO F O NY Y Y Y β= − − − (2-87)

( )( )

2 2 11 1H O F O NY Y Y Y β= − − − − (2-88)

0Pr Pr

h uh vh P h ht x y t x x y yρ ρ ρ µ µ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + − − − = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

(2-89)

PMRT

ρ = (2-90)

( , )jh h Y T= (2-91)

There are 11 equations on 11 unknowns

2 2, , , , , , , , ,F O CO H Ou v P T h Y Y Y Yρ and

2NY . The boundary conditions can be described as follows: (1) 0, in inx y y y= − < <

, 0,

1, 0

F

F

F F

F O

u u vT T h hY Y

ρ ρ== == == =

2

0NY = (2-92) (2) 0, out inx y y y= − < < − or in outy y y< < ,

, 0,

0, 0.233

air

air

air air

F O

u u vT T h hY Y

ρ ρ== =

= == =

2

0.767NY = (2-93) (3) outy y= (fixed wall)

0u v= =

2 0NYT f hy y y y yρ ∂∂ ∂ ∂ ∂= = = = =

∂ ∂ ∂ ∂ ∂ (2-94)


58

(4) Lx x= (outflow)

2

2

2 0NYv T f hx x x x x x

ρ ∂∂ ∂ ∂ ∂ ∂= = = = = =

∂ ∂ ∂ ∂ ∂ ∂

( ) 0uρ∇⋅ =r (2-95)

The parameters in the above equations and boundary conditions are the same as in the previous section. For the finite reaction rate model, the fuel equation and oxygen equation cannot be combined to cancel out the reaction term. The transport equation containing the reaction term has to be solved:

F F F FY uY vY Yt x y x Sc x

ρ ρ ρ µ∂ ∂ ∂ ∂∂ − − − − ∂ ∂ ∂ ∂ ∂

0FF

Y wy Sc y

µ ∂∂− + = ∂ ∂

(2-96)

Taking

118.6 10 , 300.1, 1.65

Ea b a b RT

F F Ow a Y Y e

A Ea b

ρ−+=

= × == =

The finite rate combustion model has a stiffness problem on the time scale. Since the chemical reaction rate is generally very fast, the time interval must be very short in using the explicit difference method. A much longer spatial increment can be used for the flow process. The explicit method cannot be adopted generally. The implicit approach has to be used. But this would make the diffusion of fuel difficult, and in turn greatly delay the combustion process. To overcome the above difficulties, the three-step method is adopted in this example calculation:

1 2 3t t t t∆ = ∆ + ∆ + ∆ (2-97) (1) In step 1t∆ , using 0F Ow w= = , the flow at

this time only has convection and diffusion, and no combustion.

(2) In step 2t∆ , putting Fw and Ow into the right term of the differential equation, as in using the explicit method.

(3) In step 3t∆ , using

{ }

{ }11NE

Na b a b RTF F O Fw A Y Y e Yρ

− ++ − =

{ }{ }11

NENa b a b RT

O F O Ow B Y Y e Yρ− ++ −

=

(2-98)

The terms Fw and Ow are linearized and then combined to the left term. It should be pointed out that in here,

2313 tt,tt ∆∆∆∆ >>>> . This technique has effectively overcome the time scale problem to allow the smooth diffusion of fuel and oxygen. This term in turn increases the rate of convergence significantly. Specific solution steps have been described previously and represented by the flowchart in Fig. 1. The finite rate multi-step reversible reaction model is very complicated and will not be described in detail here. 3. PREMIXED TURBULENT

COMBUSTION 3.1 Eddy Break-Up (EBU) Model and the

Time-Averaged Reaction Rate of Turbulent Flow

The eddy break-up (EBU) model was proposed by Spalding and later modified by Magnusen and Hiertager. There are many successful examples in using the model. It is based directly on a physical assumption on the turbulent reaction rate. As the reaction rate in laminar flow depends on the horizontal mixing, whereas the reaction rate in turbulent flow depends on the mixing of the turbulent eddies. Experimental results indicated that there were strong interactions between the turbulent flow and reactions. The reaction would affect the turbulent flow through the heat released to induce variations in density. On the other hand, turbulent flow might enhance the mixing and heat transfer of components through concentration and temperature pulsations, and in turn affect the reaction rate. Based on one-step reaction dynamics, the instantaneous reaction rate in laminar or turbulent flows can be represented in the form of Arrhenius equation:

)exp(Π1 RT

EYBw sms

Z

s

mS −=

=ρ (3-1)

where ws is the reaction rate of the s component, i.e. s component amounts (Kg/m3·s) produced or consumed per unit time per unit volume; E is the


59

activation energy, R is the common gas constant, s sY ρ ρ= is the mass fraction of the s component,

B is the pre-exponential factor, and sm m=∑ is the order of reaction. When the reaction consists of two components, fuel F and oxidizer O, and the reaction is assumed as a second-order reaction where the concentrations of the two components have an effect, the reaction rate is:

2 2exp( )S F O F OEw B Y Y k Y Y

RTρ ρ= − = (3-2)

When conducting Reynolds expansion of the above instantaneous reaction rate, i.e. taking

, , , ,F F F O O OY Y Y Y Y Y T T T k k k′ ′ ′ ′= + = + = + = + That gives:

2 2(1 )s F O F Ow kY Y kY Y Fρ ρ= = + (3-3)

where

0F O O F OF

F O F O F O

Y Y k Y k Y Yk YFY Y kY kY kY Y′ ′ ′ ′ ′ ′ ′′ ′

= + + + >

Therefore,

2

s F Ow kY Yρ≠ For turbulent flow, the reaction rate of the average value is not equal to the average value of the reaction rate. As the value of F is greater than 0, and

exp( ) exp( )E Ek B BRT RT

= − > −

Putting in k gives:

2exp( )s F O

Ew B Y YRT

ρ≠ −

Note that exp( ) exp( )E RT E RT− > − is the relationship between the concentration pulsations of different substances induced by turbulence. The relationship between concentration and temperature pulsations, and the temperature pulsations itself have enhanced and increased the time-averaged reaction rate. This is consistent with the early assumptions made on turbulent gas combustion. Folding occurred on the surface of laminar flames had been proposed by Damköhler et

al. 50 years ago. The burning flame surface was deformed by the turbulent pulsations. Not only folding occurred, the surface was also broken into pieces of different sizes to expand the flame surface. Therefore, more combustible mixed gases were burnt per unit time. As a result, the spreading rate of turbulent flames is much faster than that of laminar flames. If the spreading rate of laminar flames is LS , then for turbulent flow,

' 21 ( )T L T LS S k u S= + Usually,

Tk is taken as 1, and 'u is the turbulent pulsation velocity. Turbulent reactions are difficult to model because of the exponent functions in terms ' '

Fk Y , ' 'Ok Y

and ' ' 'F Ok Y Y , etc. in equation (3-3), i.e.

exp( )Ek BRT

= − . Physical knowledge for

modeling the above relations directly is still not clear. The EBU model proposed by Spalding has solved the problems on the turbulent reaction rate. If the chemical reaction rate is determined by the rates of production and dissipation of turbulent eddies and the mixing rate of reactant molecules, then the turbulent mixing or turbulent pulsation-controlled fuel reaction rate by Spalding is:

2)( FT,F Yk

~w ′ερ (3-4)

The instantaneous fuel concentration has to be calculated in this equation. This instantaneous quantity is related to the average value of the fuel component to close the set of equations by Magnusen and Hiertager. The reaction rate of fuel can be expressed as )( k/Y~w FT,F ε . However, on the flame side near to the fuel, the supply of oxygen is not constant. The eddies of the oxidizer would confine the combustion rate in the above equation. Therefore,

)s(/ k/Y~w OF ε and the reaction rate can be expressed as:

{ }min , /F R F Ow C Y Y skερ= − (3-5)

For turbulent premixed flames, a term on the concentration of combustion product has to be added,


60

min , / , exp( )F R F O F O PEw C Y Y s Y Y A

k RTερ = − −

(3-6)

RC is the empirical constant, usually taken as 0.35 to 0.4. It is believed in some works that RC might be related to the form of PDF of the mixture fraction. In the EBU model, the components of the oxidizer and combustion product can be expressed as:

1 FO F

S

YY Y ξξ−

= − − , 1pr F OY Y Y= − −

Although EBU model has already been widely applied, its chemical reaction dynamic process is still not very satisfactory. For example, some problems concerning the CO species cannot be represented by fast chemistry. Essentially, the results calculated by EBU are the same as those PDF results assumed by fast chemistry.

3.2 Eddy Dissipation Model (EDC) EBU model is the earliest model used in turbulent premixed combustion, then for non-premixed flames. Later, the concept was extended to eddy dissipation model (EDC). The fuel burning rate given by equation (3-6) can be rewritten as:

min( , , )(1 )

O PF F

Y Yw Yk s sερ α ρ β=

+ (3-7)

α and β are adjustable parameters of closed equations. Practically, for larger values of α , the adjustment and closing of the equations are more difficult. The above equation indicated that the reaction rate is also affected by the fuel and other components. The burning gases and the energy carried would ignite other reaction substances. In the distribution space of the mixture, there is not just diffusion combustion, but also situations where the mass fraction is lower than the average fuel mass fraction. In the EBU model, there are only the average concentrations of fuel and oxidizer FY and OY . These might be the minimum value of the turbulent dissipation rate of fuel or oxidizer that is dominating the reaction rate of turbulent diffusion flames. In the EBU model proposed by Spalding, the reaction rate (as equation (3-4)) is only related to the turbulent dissipation rate of the premixed gas. Whereas in the EDC model, the reaction rate is

related to the minimum value of turbulent dissipation rate of the premixed gas, oxidizer and combustion products with higher temperature. 3.3 Expansion of the EBU Model Considering the EBU-Arrhenius model used in partly premixed reactions, when F = 0, equation (3-3) is the reaction rate of the time-averaged value based on Arrhenius mechanism. Each term related F in equation (3-3) is related to the concentration pulsations 2Y ′ of each species. In analyzing equation (3-3) qualitatively, it can be seen that the turbulent reaction rates sw and ,s Aw

are related to 2Y ′ . Practically, the mixing ability and chemical reaction ability of the fuel and oxidizer have an effect on the reaction rate sw . To express the effect of the turbulent mixing ability and chemical reaction ability, two characteristic times are introduced, reaction time:

,c F F AY wτ = diffusion (turbulent mixing) time:

2

1 2TT

l l kD k

τε

= = =

where l is the turbulent mixing length, TD is the turbulent diffusion coefficient, k is the turbulent energy and ε is the turbulent dissipation rate. When T Cτ τ >> 1, it is a diffusion-controlled reaction flow, and non-premixed diffusion combustion. Cτ is much smaller than Tτ , and the reaction rate is very fast. Another extreme case is that T Cτ τ << 1, which is a reaction dynamics-controlled chemical reaction flow. Under general conditions, the whole flow field is at a state 1T Cτ τ ≈ , after a certain degree of mixing of different reaction components. Both the reaction dynamics and diffusion would be affected. Based on these physical foundations, it is assumed in the expanded EBU model that, in premixed reaction flow where both Arrhenius reaction dynamic mechanism and turbulent mixing mechanism have an effect,

1( , )s T Cw f τ τ= or 2 .. ..( , )s s A sTw f w w=

Although it is known that .. ..

1 1,s A sTC T

w wτ τ

∝ ∝ ,

because of the lack of understanding of the reaction mechanism, the following approximation is made in the EBU model. In the flow field,

C Tτ τ< or .. ..sT s Aw w< , taking . ..s sTw w≈


61

T Cτ τ< or . .s A s Tw w< , taking . . .s s Aw w≈ i.e. [ ]. .min ,s s T s Aw w w= (3-8)

2 2

exp( )s F O F OEw B Y Y k Y Y

RTρ ρ= − =

and .s Tw (based on the design of the EBU model as in equation (3-4)) is directly proportional to the mean-square of concentration pulsations, i.e. pulsation frequency 2Y ′ , or inversely proportional to pulsation time Tτ . Eddy break-up frequency

2ST,s Y

k~w ′ε (3-9)

For the reaction rate of fuel F,

.F T R Fw C Ykερ= (3-10)

Results predicted by using the simplified equation (3-10) did not deviate from those by equation (3-9). Although EBU model is relatively simple, the effect of pulsation time on the reaction rate can be described. The predicted results agree with the experimental results in many cases.

3.4 Simplified PDF Model of Fast Reaction

Flow T Cτ τ >> 1 (Diffusion-Controlled)

When T Cτ τ >> 1, the reaction rate is much faster than turbulent mixing. The reaction flow is controlled by the comparatively slower diffusion process. That is the situation for diffusion flames. For the condition T Cτ τ >> 1, solutions with certain simplifications could only give some rough qualitative estimation. Numerical method has to be used to achieve more accurate results. Experiments have proved that under general circumstances, an infinitely thin flame surface does not exist. In turbulent diffusion flames, there would be a region where time-averaged values of fuel and oxygen exist together in the same space. But the two of them would not exist at the same time, i.e. the instantaneous values of the two do not coexist. Through introducing the concept of conservation scalar or mixing fraction and fast reaction, and using the assumed probability density distribution function, the problem can be solved. For turbulent flow, there also exists one-step chemical reactions. Zeldovich’s transformation can yield a conservation scalar and mixing fraction.

One-step chemical reaction is:

/ /(1 )F O prw w s w s+ → + Introducing Zeldovich’s transformation,

/F OY Y sφ = − . In turbulent flows, , FYφ and OY are all instantaneous quantities. The function φ is a kind of integrated concentration, also called conservation scalar. This is because there is no source term in the conservation equation of φ . The equations for the instantaneous component diffusion of fuel and oxygen are:

( ) ( ) ( )FF j F F

j j j

YY u Y D wt x x xρ ρ ∂∂ ∂ ∂

+ = −∂ ∂ ∂ ∂

(3-11)

( ) ( ) ( )OO j O O

j j j

YY u Y D wt x x xρ ρ ∂∂ ∂ ∂

+ = −∂ ∂ ∂ ∂

(3-12)

Multiplying equation (3-12) by 1/s, and subtracting it from equation (3-11), the φ equation without a source term can be obtained,

( ) ( ) ( )jj j j

u Dt x x x

φρφ ρ φ∂ ∂ ∂ ∂+ =

∂ ∂ ∂ ∂ (3-13)

The above is valid for both non-premixed diffusion-controlled reaction flow and premixed reaction flow under all situations. Introducing the mixing fraction for non-premixed diffusion-controlled situations,

2

1 2

f φ φφ φ−

=−

where 1φ and 2φ are the values of φ on the fuel side and on the oxygen side respectively,

1 1 2 ,21, / 1/F OY s sφ φ φ= = = − = − . Therefore f1 = 1, f2 = 0, the mixing fraction f expresses the degree of mixing of the two components. F is also a conservation scalar, the instantaneous conservation equation for f under diffusion-controlled turbulent flow is,

( ) ( ) ( )jj j j

ff u f Dt x x xρ ρ∂ ∂ ∂ ∂

+ =∂ ∂ ∂ ∂

(3-14)

This equation is similar to equation (2-40) under laminar flow conditions. Assume that the reaction is infinitely fast when


62

compared with turbulent mixing (diffusion), i.e. T Cτ τ >> 1. The basic assumption in diffusion

combustion is that fuel and oxygen would not exist at the same time or in the same space. There exists a reaction surface or flame surface that both concentrations are zero. The whole flow field is divided into two regions. On the oxygen side, there are both oxygen and combustion products. However, fuel and oxygen cannot exist together. Defining

∫→=

T

maxTYdt

TY

0

1lim ， 'ϕ ϕ ϕ= −

Obviously, on the flame surface

0F OY Y= =

2

1 2

1 11 (1 ) 1F

sf fs s

φφ φ

= = = =− + +

In the region 0 Ff f< < , which is the side near to

the oxygen, there is no fuel, i.e. 0FY = . Therefore,

2 2

1 2 1 2

1( / ) ( )( / ) 11 11 ( )

OO O

Y sY s Ysfs

s

φ φ φφ φ φ φ

+− − − −= = = =

− − ++

where 1

2

s φφ

=

or

1 (1 ) 1O

F

fY f sf

= − + = + (3-15)

On the side near to the fuel, there is no oxygen in the region 1Ff f< < , i.e. 0OY = . Then

2

1 2

(1 ) 11 (1 ) 1

F F FY Y s sYfs s

φφ φ

− + += = =

− + +

or

(1 ) 11

FF

F

f fs fYs f

−+ −= =

− (3-16)

The time-averaged values FY and OY of fuel and oxygen can both be represented as the function

of the time-averaged value of the mixing fraction f . Similarly, the correlation between the time-

averaged temperature T and the time-averaged mixing fraction f can be obtained.

3.5 k fε− − Flame Surface Model

By using equations (3-15) and (3-16) on OY and FY , and the instantaneous value of f in equation (3-

14), through Reynolds expansion and taking the time-averaged value, the equation of time-averaged value of f can be obtained,

( ) ( ) ( )efj

j j f j

ff u ft x x x

µρ ρ

µ∂ ∂ ∂ ∂

+ =∂ ∂ ∂ ∂

(3-17)

Considering the , ,k fε equation on turbulent convection and diffusion and forming a set of equations,

( ) 0jj

ut xρ ρ∂ ∂+ =

∂ ∂ (3-18)

( ) ( ) ( )i

ii i j eff u

j j j

uu u u St x x xρ ρ µ ∂∂ ∂ ∂

+ = +∂ ∂ ∂ ∂

(3-19)

( ) ( ) ( )effj k

j j k j

kk u k Gt x x x

µρ ρ ρε

σ∂ ∂ ∂ ∂

+ = + −∂ ∂ ∂ ∂

(3-20)

1 2( ) ( ) ( ) ( )effj k

j j j

u C G Ct x x x kε

µ ε ερε ρ ε ρεσ

∂ ∂ ∂ ∂+ = + −

∂ ∂ ∂ ∂ (3-21)

( ) ( ) ( )effj

j j f j

ff u ft x x x

µρ ρ

σ∂ ∂ ∂ ∂

+ =∂ ∂ ∂ ∂

(3-22)

This is called the k fε− − model. By solving these equations, and according to the function relation of ( ), ( )F OY f Y f and ( )T f ,

FY , OY and T in the average concentration and temperature fields can be obtained. These can also be found on the flame surface 0O FY Y= = ,

i.e. 11

fs

=+

isolines. However, the results

obtained are still too simplified and not very satisfactory. The reason is that an infinitely thin flame surface with zero concentration of fuel and oxygen does not exist in real turbulent diffusion flames. But on the flame surface with a finite thickness, the time-averaged values of fuel and


63

oxygen concentration coexist in the same space. Therefore, there are problems in the above expressions on ( )F FY Y f= and ( )O OY Y f= .

3.6 gfk −−− ε Model of Partial

Instantaneous Non-Mixing Since the reaction is very fast, fuel and oxygen cannot exist at the same time. However, their time-averaged values can exist at the same place. For the time-averaged condition of the flow field, a reaction surface does not exist. But the flame surface model can be extended to introduce the concept of partial instantaneous non-mixing in the turbulent eddies. The spatial distribution of time-averaged concentration (and temperature) are expressed in equations (3-15) and (3-16). Such distribution might occur randomly between the instantaneous values prY and FY , or between

prY and OY , but each value of f would appear according to a certain PDF. When 1 Ff f> > , there is no oxygen, the instantaneous relationship between FY and f is:

( ) /(1 )F F FY f f f= − − (3-23) And when 0 Ff f< < , there is no fuel, the instantaneous relationship between OY and f is:

1 /O FY f f= − (3-24) The above two equations on the relation between the instantaneous values are not always valid. They are valid only for certain f values, and each value of f would appear only at a certain PDF. The above two equations showed that all instantaneous variables such as FY , OY , prY and

T can be represented as the function of the conservation scalar quantity f; but the relationship between the time-averaged values of FY , OY and T with the time-averaged value f is still unknown. In real case, the time-averaged flame surface is not infinitely thin, instead it is a region with finite thickness. The above concept of partial instantaneous non-mixing is used in the submodel for establishing the internal state of turbulent eddies. For the whole flow field, all time-averaged equations are still equations (3-18) to (3-22). In addition, mean-square equation 2g f ′= of the mixture fraction pulsations has to be solved. As derived by Spalding et al.,

( ) ( )jj

g u gt xρ ρ∂ ∂

+∂ ∂

1 2

2( ) ( )effg eff g

j g j j

g fC C gx x x k

µ εµ ρσ

∂ ∂ ∂= + −∂ ∂ ∂

(3-25)

The set of equations consisting of equations (3-18) to (3-22), together with equation (3-15) is called the gfk −−− ε model. Since the ultimate numerical solution is to find the time-averaged concentration and temperature FY , oxY and T

from f and g, the PDF method has to be used.

If the PDF is known, then FY , oxY and T can

be obtained from f and g. If any instantaneous scalar quantity ϕ is the function of the instantaneous value of f, ϕ = ( )fϕ . Its probability of occurrence is:

( , ) ( , )p x d p f x dfϕ ϕ = The time-averaged value of ϕ is:

1

0( ) ( ) ( , )x f p f x dfϕ ϕ= ∫

The instantaneous relationship given by equations (3-23) and (3-24):

1 1

0 0( ) ( ) ( )

1F

F FF

f fY Y f p f df p f dff

−= = − ∫ ∫ (3-26)

1 1

0 0( ) ( ) 1 ( )o o

F

fY Y f p f df p f dff

= = −

∫ ∫ (3-27)

The problem now is how to find p(f). Generally, p(f) cannot be known in advance. One of the methods is to solve the transport equation of p(f), or called the probability density distribution function model. But certain terms in the equation cannot be modeled easily. Another method is to assume a p(f) in advance, referred to as the simplified PDF model. For example, assuming that the pulsations are step functions. It can also be assumed that PDF has a Gauss distribution with cut-off intermittent pulsations, PDF here is represented as:

( ) (0) (1)p f A Bδ δ= +

[ ]21 1exp ( ) ( 1)

22f D f D fµσσ π

− + − − −

(3-28)


64

where δ is the Dirac-delta function (at 0f = and 1f = ), and ( )D f is the Heaviside step function.

( )0, ( ) 00, 1

DD

ξ ξξ ξ< =

> =

0

0 ( )A p f df−∞

= ∫

01( )B p f df

∞= ∫

2

01 1( ) exp

22fp f µσσ π

− = −

where µ is the f value that corresponds to the maximum value of p(f), and σ is the mean square of deviation. At this time, the time-averaged values of f and mean-square pulsation are:

21

0

1exp22

f ff B dfµσσ π

− = + −

∫ (3-29)

221 2

0

1exp22

f fg B df fµσσ π

− = + − −

∫

(3-30) f and g at different positions of the flow are

obtained by solving the gfk −−− ε equations. µ and σ are unknown to be formed from f and g first. Different values of σ and µ are assumed and substituted into the above two equations to see whether it gives f and g consistent with their known values. More work is required in using PDF with cut-off Gauss distribution than using PDF with step functions pulsations. Past results indicated that the gfk −−− ε equation sets combined with the simplified PDF agreed better with the experimental data, when compared with the gfk −−− ε flame surface model. The cut-off Gauss distribution PDF model is slightly better than step functions pulsations PDF model, and the differences are very little. The partial instantaneous non-mixing model described previously is only applicable to simple diffusion flames where fuel and oxygen are provided separately; and to one-step fast reactions. In order to extend the fast reaction assumption to more complicated diffusion flames encountered in engineering and taking into account that real

reactions are not one-step reactions, fast reaction model of instantaneous balance inside the turbulent eddies is proposed. A different definition of mixture fraction is introduced here. In combustion chambers and in rooms where fires occur, part of the fuel has already been premixed with part of the air prior to the combustion. Air would also be entrained later on. Mixture fraction should be divided into the mass of the primary flow and the mass of the secondary flow for calculating the instantaneous spatial distribution of mixing ratio. Defining f by:

p

p s

mf

m m=

+ (3-31)

where pm is the primary flow mass, and sm is the secondary flow mass. f defined in here represents the instantaneous “degree of mixing” at any one point, or the partial instantaneous mixing ratio. Based on this definition of f, any other conservation scalar quantity ψ can be represented as:

(1 )p sf fψ ψ ψ= + − (3-32) where pψ , sψ are the values of ψ in the primary and secondary mass flow respectively. For an adiabatic system, the gas enthalpy h can also be taken as a conservation scalar quantity. When the heat diffusion coefficient and the diffusion coefficients of other elements are equal,

(1 )p sh fh f h= + − (3-33) Since f is a conservation scalar quantity, the conservation equation of its time-averaged value f and mean-square pulsation 2'g f= can be

determined by the equation given previously. Using the determined f and g and the assumed simplified PDF form p(f), the PDF form at different positions in the space can be obtained. If the relationship ϕ(f) between any scalar ϕ and f is known, then the time-averaged value of this scalar can be obtained.

1

0( )p f dfϕ ϕ= ∫

or

1

0( )p p s s p f dfϕ α ϕ α ϕ ϕ

−

+= + + ∫


65

where ,p sα α are the corresponding ( )p f

values for 1f = and 0f = ; and pϕ and sϕ are the values of ϕ in the primary and secondary mass flow respectively. The partial dynamic equilibrium concept based on the assumption of fast reaction is used to establish the relationships between the enthalpy, pressure, element components and temperature, as well as gas components mass fraction. Pressure variations of the turbulent reaction flow are not large. For adiabatic systems, ϕ ＝ ϕ(f), so

( ), ( ), ( )s sT T f f Y Y fρ ρ= = = ; and the time-averaged temperature, density and concentration fields ( ), ( ), ( )sT f f Y fρ can be directly obtained by using p(f). For non-adiabatic systems, the conservation equation of energy can be solved as:

( ) ( ) ( )ej h

j j h j

hh u h St x x x

µρ ρσ

∂ ∂ ∂ ∂+ = −

∂ ∂ ∂ ∂ (3-34)

In the equations,

( , )( , )( , )s s

T T f hf h

Y Y f hρ ρ===

The time-averaged values of T , ρ , sY have to be solved from

1

0( , )p p s s p f h df dhϕ α ϕ α ϕ ϕ

−

+= + + ∫ (3-35)

Since it is difficult to find the probability density function ( , )p f h related to h, using

( , ) ( )p f h p f= and obtain

∫−

+++=

1

0df)f(psspp ϕϕαϕαϕ (3-36)

The diffusion flame mechanism of this model is similar to the dynamic equilibrium model. However, there are limitations in applying this kind of partial dynamic equilibrium model. The reaction might be too fast in practice to get a partial dynamic equilibrium. 3.7 Simplified PDF Assumptions of Finite

Reaction Rate For more general situations, instantaneous laminar reaction is assumed within the turbulent eddies. But the reaction rate is finitely fast and not under

partial dynamic equilibrium state. It is because most of the combustions are inequilibrium diffusion flames. The simplified PDF model has the following set of time-averaged equations: Species equation:

( ) ( ) ( )e ss j s s

j j Y j

YY u Y wt x x x

µρ ρσ

∂∂ ∂ ∂+ = −

∂ ∂ ∂ ∂ (3-37)

Mixture fraction equation:

( ) ( ) ( )ej

j j f j

ff u ft x x x

µρ ρσ

∂ ∂ ∂ ∂+ =

∂ ∂ ∂ ∂ (3-38)

Mixture fraction mean-square pulsation equation:

( ) ( )jj

g u gt xρ ρ∂ ∂

+∂ ∂

2

1 2( )eg e g

j g j j

g fc c gx x x

µ εµ ρσ κ

∂ ∂ ∂= + − ∂ ∂ ∂

(3-39) where f is defined by equation (3-31). The time-averaged reaction rate is:

( ) ( )dffpfwRTEBYYw ss ∫=

−=

1

0212 expρ

(3-40) The corresponding instantaneous values in laminar combustion based on theory and data from laminar diffusion are:

( )( )

s sY Y fT T f=

=

And assuming that the PDF takes the form:

( ) ( , , )p f p f f g= (3-41) where both f and g are the function of spatial location. According to the above model, the solving steps are: 1. Solve for the continuity equation,

momentum equation and the , , ,f gκ ε equations for the turbulent time-averaged flow to get the preliminary spatial distribution of f and g;


66

2. Calculate sw from the values of f and g,

and the given values of ( , , )f f gρ and ( )sY f , ( )T f ;

3. Put sw into the conservation equation (3-37)

of species s to solve for sY ;

4. Solve the conservation equation of enthalpy h;

5. Calculate T from h and sY ;

6. Solve again by including the effect of the estimated T on density variation, on the momentum equation and the , , ,f gκ ε equations;

7. Repeat the above steps until convergence. 3.8 Simplified PDF Model of Premixed

Reaction Flow (Reaction Flow Controlled by Both Diffusion and Chemical Reaction)

For premixed reaction flow, i.e. the reaction flow is controlled by both diffusion and chemical reaction, 1T

c

ττ ≈ . Though the predicted results

are closer to practical situations, the EBU-Arrhenius model described previously is still unsatisfactory. The simplified PDF model was developed for studying premixed reaction flow. Assuming that the reaction is a one-step second-order reaction, the instantaneous reaction rate is:

2 exp( )s F OEw B Y Y

RTρ= − (3-42)

If the joint PDF is known,

( , , , , )F Op T Y Y xρ Then the time-averaged reaction rate is:

OFs pRTEYYBw ρ )exp(

0 0

1

0

1

0

2 −= ∫ ∫ ∫ ∫∞ ∞

×OFOF dYdYdTdx,Y,Y,T,p ρρ )( (3-43)

The joint PDF is easier to find if the relation between the instantaneous variables , , ,F OT Y Yρ , etc. with a certain single instantaneous variable can be obtained through experiments or other concepts. (1) Level of reaction and instantaneous laminar

adiabatic flame model

Assuming that inside each turbulent eddy is an instantaneous adiabatic laminar flamelet, and Pr 1Sc= = , then the relationship between the instantaneous , , ,F OT Y Yρ and reaction level “ε ” can be established. ε is defined as:

( )1

1 1 pOF

F O m F

c T TYY T TY Y T T Y Q

ε ∞∞

∞ ∞ ∞ ∞

−−= − = − = =

− (3-44) Therefore, the following relationships of instantaneous values can be obtained,

(1 )F FY Yε ∞= − (3-45)

(1 )O OY Yε ∞= − (3-46)

(1 )T T ε∞

= + Λ (3-47)

1(1 )ρ ρ ε −∞= + Λ (3-48)

11m F

p

T Y QT c T

∞

∞ ∞

Λ = − = (3-49)

The subscripts ∞ and m represent the inflowing values and the maximum value respectively. It can be seen that once the PDF ( )( , , , )F Op T Y Y pρ ε= is known, the time-averaged reaction rate can be obtained.

1

0( ) ( )s sw w p dε ε ε= ∫ (3-50)

(2) Relationship between the time-averaged

reaction level and other time-averaged quantities From the energy equation and conservation equation of either instantaneous component

FY or OY , the equation of the instantaneous reaction level can be obtained. For example, the conservation equation for FY can be written as:

( ) ( ) ( )jj j j

u D Wt x x x

ερε ρ ε ρ∂ ∂ ∂ ∂+ = +

∂ ∂ ∂ ∂

(3-51) The time-averaged reaction level ε can also be obtained from the following equation:


67

( ) ( ) ej

j j j

u Wt x x xε

µ ερε ρ εσ ∂ ∂ ∂ ∂

+ = + ∂ ∂ ∂ ∂

(3-52) where 1

0( ) ( )W W p dε ε ε= ∫ .

Similar to the previous method in deriving the g equation, the conservation equation of reaction level mean-square pulsation

2'η ε= (neglecting the reaction term) can also be derived:

( ) ( )jj

ut xρη ρ η∂ ∂

+ =∂ ∂

2

1 2( )ee

j j j

c cx x xη η

η

µ η ε εµ ρησ κ

∂ ∂ ∂= − + ∂ ∂ ∂

(3-53) It is different from solving f and g equations in the way that, before finding W , ε and η cannot be obtained in advance. Any one scalar quantity ϕ and its mean-square pulsation including ε and η can be represented as:

εεεϕϕ dx,p∫=1

0)()( r (3-54)

212

0' ( ) ( , )p x dϕ ϕ ε ϕ ε ε = − ∫

r (3-55)

εεεε dx,p∫=1

0)( r (3-56)

212

0' ( , )p x dη ε ε ε ε ε = = − ∫

r (3-57)

(3) Simplified PDF model

A PDF ( )p ε can be assumed first and if there are intermittent pulsations at ε = 0 and ε = 1, PDF takes the following form:

( ) ( )( , ) (0) (1)p x x xε α δ β δ= +r r r

[ ] ( ) ( )( ) ( 1) ,D D x f xε ε γ ε+ − −r r

(3-58)

where ( )0δ and ( )1δ are the Dirac-delta

function at ε = 0 and ε = 1 respectively, and D is the Heaviside function.

( )

1

0

2

1

( 1

( ) , '

f d

f f ,

α β γ

ε ε

ε ε ε ε

+ + =

=

=

∫ ） (3-59)

Therefore,

εεεγβεεεε dfdp ∫∫ +==1

0

1

0)()( (3-60)

Mdp γεεεεεεε −−=−=′ ∫ )(1)()(1

0

22

(3-61) where

εεεε dfM ∫ −=1

0)()(1

When ε = 0, 0ε = , and so α has no contribution to ε ,

IWdpWW γεεε max1

0)()( == ∫ (3-62)

where

εεε dfWW

∫=1

0max

)()(I

When ε = 0 and ε = 1, ( ) 0W ε = , W would

reach its maximum value maxW at a certain value of ε greater than zero and smaller than 1. In solving the equations, apart from solving the continuity equation, momentum equation, and the k and ε equations, equations (3-53), (3-54), (3-59), (3-60) and (3-63) have to be solved jointly. To solve for these five equations, the method of trial and error has to be used. This kind of model has only been used for calculating one-dimensional turbulent combustion, and not in the prediction of real flames. In recent years, more complicated PDF models have been developed which include the combined model of the given PDF and laminar flamelet (folded flame surface or volumetric combustion), multi-dimensional PDF model and the model for solving the transport equation of PDF, etc. Statistical fluid dynamics and probability concept are applied to derive turbulent combustion. The calculations involved are too complicated, for engineering application.


68

3.9 Brief Introduction of Laminar Flamelet Model

Laminar flamelet combustion model is different from SCRS in the way that it does not require a linear relationship between the mixture fraction, mass fraction and temperature. More complicated experimental correlation relationships between these variables are considered. From the laminar flamelet relation, the species mass fraction can be derived. The mixture fraction of the transport equation can be solved. This is an improvement for the infinitely fast reaction model by including additional parameters and relations with finitely fast chemical reaction rate. When the Dämkohler number is very large,

1a

c

Dα

τ∝ , cτ would be very small and that goes

back to the SCRS situation. For non-premixed turbulent flames, some of the premixed combustion systems have already got many good starting work. For example, by using premixed flamelet model to substitute the diffusion flamelet model and taking assumption of flamelet, the flame surface density model was developed. When the mixture fraction changes in passing through the flamelet, the concentrations and temperatures of all chemical species also change. By using a probe to pass through different positions of the flamelet, the concentration, temperature, soot concentration, viscosity, enthalpy and pressure of all species are all functions of the mixture fraction in the flamelet. A simple variation diagram of combustion products, oxygen and fuel concentration can be derived as functions of the mixture fraction. In the laminar flamelet model, it is assumed that chemical reactions only take place within the basic element ε in the laminar flame, that is, ε exists near the fire source. If turbulent flame is believed to be composed of many ‘tiny parts’, then the concept of laminar flamelet can be applied to turbulent flames such as fires. Each ‘tiny part’ is composed of undisturbed laminar diffusion flames. PDF can be used to describe the average state of turbulent diffusion flames. Suppose chemical reactions only take place in the thin flamelet, the chemical reaction time is shorter than other turbulent and transport processes. Flames under normal atmospheric pressure and buoyancy would behave like this. The uncertainties of the patterns of turbulent flow field can be coupled with the complicated multi-species chemical reactions. This assumption represents the practical situations which the concentration,

temperature and other thermal parameters of the instantaneous components are the functions in conserving total quantity. Therefore, the uncertainties of turbulent flow field and chemical reactions are treated separately. The assumed flamelet state relations can be saved as library files first. Detailed chemical dynamic processes can be combined into the Computational Fluid Dynamics CFD model without increasing the computing time. These flamelet relations can be used for many times in flow field calculations. In this way, it is not necessary to calculate the chemical reaction process in each computing in all node time intervals. In using the mixture fraction through the conservation equations, the state, enthalpy, viscosity, temperature and soot concentration of all chemical species can be obtained from experiments on laminar flames or from computation. Usually, gas probe sampling would be conducted at different positions in the thin flamelet. For example, starting from the oxygen side, the mixture fraction is zero. While moving towards the center of the flamelet with increasing temperature, a certain quantity of combustion products including the components is taken from the fuel to determine the concentration of carbon and hydrogen atoms, or other related molecules such as O, CO2 and free radicals H and HO2 at that point. REFERENCES 1. R.A. Strehlow, Combustion fundamentals, New

York: McGraw-Hill (1985).

2. J. Buckmaster, “The effects of radiation on stretched flames”, Combustion Theory and Modelling, Vol. 1, No. 1, pp. 1-11 (1997).

3. Y.B. Zel’dovich, G.I. Barenblatt, V.B. Librovich and G.M. Makhviladze, The mathematical theory of combustion and explosion, Moscow: Nauka, (1980).

4. N. Peters and B. Rogg, Reduced kinetic mechanisms for applications in combustion systems, Berlin: Springer (1993).

5. N. Peters, “Numerical and asymptotic analysis of systematically reduced reaction schemes for hydrocarbon flames”, In: Numerical simulation of combustion phenomena, Berlin: Springer, pp. 90-109 (1985).

6. F.A. Williams, Combustion theory, 2nd edition, Redwood City: Addison-Wesley (1985).

7. N. Peters, “Partially premixed diffusion flamelets in nonpremixed turbulent combustion”, In: 20th Symposium (International) on Combustion. Pittsburgh: The Combustion Institute, pp. 353-360 (1984).


69

8. I. Glassman, Combustion, 3rd edition, New York: Academic, pp. 205-215 (1996).

9. I. S. Wichman, N. Lakkaraju and B. Ramadan, “The structure of quenched triple flames near cold walls in convective flows”, Combustion Science and Technology, Vol. 127, pp. 141-165 (1997).

10. W. M. Pitts, “Large-scale turbulent structures and the stabilization of lifted turbulent jet diffusion flames”, In: 23rd Symposium (International) on Combustion, Pittsburgh: The Combustion Institute, pp. 661-668 (1990).

11. F. Takahashi and V.R. Katta, “A numerical analysis of a methane diffusion flame over a flat plate”, In: Proceedings of the Central States Section of the Combustion Institute, 27-29 April 1997, pp. 164-169 (1997).

12. R.W. Bilger, “The structure of turbulent non premixed flames”, In: 22nd Symposium (International) on Combustion, Pittsburgh: The Combustion Institute (1988).

13. R. Borghi, “Turbulent combustion modelling”, Progress in Energy and Combustion Science, Vol. 14, pp. 245-292 (1988).

14. C.M. Coats, “Coherent structures in combustion”, Progress in Energy and Combustion Science, Vol. 22, pp. 427-509 (1996).

15. N. Peters, “Laminar flamelet concepts in turbulent combustion”, In: 21st Symposium (International) on Combustion. Pittsburgh: The Combustion Institute, pp. 1231-1250 (1986).

16. A.G. Gaydon and H.G. Wolfhard, Flames, their structure, radiation, and temperature, 4th edition, London: Chapman & Hall (1979).

17. J. Warnatz, U. Maas and R.W. Dibble, Combustion: physical and chemical fundamentals, modelling and simulation, experiments, pollutant formation, Berlin; New York: Springer (1996).

Documents

Review of Combustion - Semantic Scholar · for engines and furnaces. Most combustion flows, particularly those in fires, are very complicated to study. Concept on applied physics