Lecture 8 Laminar Diffusion Flames: Diffusion Flamelet Theory 8.-­‐1 •  Systems, where fuel and oxidizer enter separately into the combus7on chamber •  Mixing takes place by convec7on and diffusion •  Only where fuel and oxidizer are mixed on the molecular level, chemical reac7ons can occur •  The 7me scale of reac7on is much shorter than the 7me scale for diffusion →  diffusion is rate determining •  This is why flames in non-­‐premixed combus7on are called diffusion flames Candle flame: A classical example of a diffusion flame 8.-­‐2 Example: Candle Flame •  The flow entraining the air into the flame is driven by buoyancy •  The paraffin of the candle first melts due to radia7ve heat from the flame to the candle •  It mounts by capillary forces into the wick and evaporates to become paraffin vapor, a gaseous fuel •  The combus7on zones in a diffusion flame are best described by an asympto7c expansion for very fast chemistry star7ng from the limit of complete combus7on 8.-­‐3 •  To leading order one obtains the adiaba7c flame temperature, which is a func7on of mixture frac7on only •  The asympto7c expansion around this limit will then describe the influence of finite rate chemistry •  If the expansion takes the temperature sensi7vity of the chemistry into account diffusion flame quenching can also be described •  By introducing the mixture frac7on as an independent coordinate for all reac7ng scalars, a universal coordinate transforma7on leads in the limit of sufficiently fast chemistry to a one-­‐dimensional problem for the reac7on zone •  This is the basis of the flamelet formula7on for non-­‐premixed combus7on 8.-­‐4 Flamelet Structure of a Diffusion Flame •  Assump7ons: Equal diffusivi7es of chemical species and temperature •  The balance equa7on for mixture frac7on, temperature and species read •  Here the low Mach number limit that leads to zero spa7al pressure gradients has been employed, but the temporal pressure change has been retained 8.-­‐5 Flamelet Structure of a Diffusion Flame •  The equa7on for the mixture frac7on does not contain a chemical source term, since elements are conserved in chemical reac7ons •  We assume the mixture frac7on Z to be given in the flow field as a func7on of space and 7me: Z=Z(xα ,t) 8.-­‐6 •  Then the surface of the stoichiometric mixture can be determined as •  Combus7on occurs in a thin layer in the vicinity of this surface if the local mixture frac7on gradient is sufficiently high •  Let us locally introduce an orthogonal coordinate system x1, x2, x3 aZached to the surface stoichiometric mixture •  x1 points normal to the surface Zst , x2 and x3 lie within the surface •  We replace the coordinate x1 by the mixture frac7on Z and x2, x3 and t by Z2 = x2, Z3 = x3 and t = τ •  This is a coordinate transforma7on of the Crocco type 8.-­‐7 •  Here the temperature T, and similarly the mass frac7ons Yi, will be expressed as a func7on of the mixture frac7on Z •  By defini7on, the new coordinate Z is locally normal to the surface of stoichiometric mixture •  With the transforma7on rules we obtain the temperature equa7on in the form •  The transforma7on of the equa7on for the mass frac7on is similar 8.-­‐9 •  If the flamelet is thin in the Z direc7on, an order-­‐of-­‐magnitude analysis similar to that for a boundary layer shows that is the domina7ng term of the spa7al deriva7ves •  This term must balance the terms on the right-­‐hand side •  All other terms containing spa7al deriva7ves can be neglected to leading order •  This is equivalent to the assump7on that the temperature deriva7ves normal to the flame surface are much larger than those in tangen7al direc7on 8.-­‐9 •  The term containing the 7me deriva7ve is important if very rapid changes, such as ex7nc7on, occur •  Formally, this can be shown by introducing the stretched coordinate ξ and the fast 7me scale σ
•  ε is a small parameter, the inverse of a large Damköhler number or a large ac7va7on energy, for example, represen7ng the width of the reac7on zone 8.-10
•  If the 7me deriva7ve term is retained, the flamelet structure is to leading order described by the one-­‐dimensional 7me-­‐dependent equa7ons •  Here is the instantaneous scalar dissipa7on rate at stoichiometric condi7ons •  It has the dimension 1/s and may be interpreted as the inverse of a characteris7c diffusion 7me •  It may depend on t and Z and acts as a prescribed parameter, represen7ng the flow and the mixture field 8.-­‐11 •  As a result of the transforma7on, the scalar dissipa7on rate implicitly incorporates the influence of convec7on and diffusion normal to the surface of the stoichiometric mixture •  In the limit χst → 0, equa7ons for the homogeneous reactor are obtained
8.-12
Steady Solu7ons of the Flamelet Equa7on: The S-­‐Shaped Curve •  Burning flamelet correspond to the upper branch of the S-­‐shaped curve •  If χst is increased, the curve is traversed to the lec un7l χq is reached, beyond which value only the lower, nonreac7ng branch exists
•  Thus at χst = χq the quenching of Quenching
the diffusion flamelet occurs •  The transi7on from the point Q to the lower state corresponds to the unsteady transi7on 8.-­‐12a •  The neglect of all spa7al deriva7ves tangen7al to the flame front is formally only valid in the thin reac7on zone around Z = Zst •  There are, however, a number of typical flow configura7ons, where is valid in the en7re Z-­‐space •  As example, the analysis of a planar counterflow diffusion flame is included in the lecture notes 8.-­‐13 The Planar Counterflow Diffusion Flame •  Counterflow diffusion flames are very ocen used experimentally because they represent an essen7ally one-­‐dimension diffusion flame structures •  If one assumes that the flow veloci7es of both streams are sufficiently large and sufficiently removed from the stagna7on plane, the flame is embedded between two poten7al flows, one coming from the oxidizer and one from the fuel side 8.-­‐14 Flow equa7ons and boundary condi7ons •  Prescribing the poten7al flow velocity gradient in the oxidizer stream the veloci7es and the mixture frac7on are there •  Equal stagna7on point pressure for both streams requires that the veloci7es in the fuel stream are 8.-­‐15 •  The equa7ons for con7nuity, momentum and mixture frac7on are given by 8.-­‐16 Numerical Simula7on of Counterflow Diffusion Flame 1
0.00014
0.00012
CH4
Atmospheric Methane Counterflow Diffusion Flame at a = 100 s-­‐1 0.0001
0.6
8 10-5
H
6 10-5
0.4
Mass Fraction
1.2 10-5
2500
4 10-5
O2
0.2
2 10-5
1 10-5
2000
-5
• 
-4
-3
-2
-1
0
y [mm]
1
2
Oxygen leakage at high strain rate 3
0
8 10-6
1500
6 10-6
1000
4 10-6
500
0
2 10-6
-5
-4
-3
-2
-1
0
y [mm]
1
2
3
0
CH Mass Fraction
0
Temperature [K]
Mass Fraction
0.8
Example: Analysis of the Counterflow Diffusion Flame •  Introducing the similarity transforma7on one obtains the system of ordinary differen7al equa7ons in terms of the non-­‐dimensional stream func7on and the normalized tangen7al velocity 8.-­‐17 •  Furthermore the Chapman-­‐Rubesin parameter C and the Schmidt number Sc are defined •  The boundary equa7ons are •  An integral of the Z-­‐equa7on is obtained aswhere the integral I(η) is defined as 8.-­‐18 •  For constant proper7es ρ = ρ∞, C = 1 f = η sa7sfies and •  The instantaneous scalar dissipa7on rate is here where and have been used 8.-­‐19 •  When the scalar dissipa7on rate is evaluated with the assump7ons that led to one obtains •  For small Z one obtains with l‘ Hospital's rule •  Therefore, in terms of the velocity gradient a the scalar dissipa7on rate becomes showing that χ increases as Z2 for small Z 8.-20
Diffusion Flame Structure of Methane-­‐Air Flames •  Classical Linan one-­‐step model with a large ac7va7on energy is able to predict important features such as ex7nc7on, but for small values of Zst, it predicts the leakage of fuel through the reac7on zone •  However, experiments of methane flames, on the contrary, show leakage of oxygen rather than of fuel through the reac7on zone 8.-­‐48 •  A numerical calcula7on with the four-­‐step reduced mechanism has been performed for the counter-­‐flow diffusion flame in the stagna7on region of a porous cylinder •  This flow configura7on, ini7ally used by Tsuji and Yamaoka, will be presented in the next lecture 8.-­‐49 •  Temperature profiles for methane-­‐air flames •  The second value of the strain rate corresponds to a condi7on close to ex7nc7on •  It is seen that the temperature in the reac7on zone decreases 8.-­‐50 •  Fuel and oxygen mass frac7on profiles for methane-­‐air flames •  The oxygen leakage increases as ex7nc7on is approached 8.-­‐51 •  An asympto7c analysis by Seshadri (1988) based on the four-­‐step model shows a close correspondence between the different layers iden7fied in the premixed methane flame and those in the diffusion flame 8.-­‐52 •  The outer structure of the diffusion flame is the classical Burke-­‐Schumann structure governed by the overall one-­‐step reac7on with the flame sheet posi7oned at Z = Zst •  The inner structure consists of a thin H2 -­‐ CO oxida7on layer of thickness of order ε toward the lean side and a thin inner layer of thickness of order δ slightly toward the rich side of Z = Zst •  Beyond this layer, the rich side is chemically inert, because all radicals are consumed by the fuel 8.-­‐53 •  The comparison of the diffusion flame structure with that of a premixed flame shows that •  Rich part of the diffusion flame corresponds to the upstream preheat zone of the premixed flame •  Lean part corresponds to the downstream oxida7on layer •  The maximum temperature corresponds to the inner layer temperature of the asympto7c structure 8.-­‐54 •  The plot of the maximum temperature also corresponds to the upper branch of the S-­‐shaped curve •  The calcula7ons agree well with numerical and experimental data and they also show the ver7cal slope of T0 versus χ-1st which corresponds to ex7nc7on 8.-­‐54 Appendix Steady State CombusEon and Quenching of Diffusion Flames with One-­‐Step Chemistry •  If the unsteady term is neglected we obtain an ordinary differen7al equa7on that describes the structure of a steady state flamelet normal to the surface of stoichiometric mixture •  It can be solved for general reac7on rates either numerically or by asympto7c analysis 8.-­‐21 •  In the following we will express the chemistry by a one-­‐step reac7on with a large ac7va7on energy, assume constant pressure and the radia7ve heat loss to be negligible •  We will analyze the upper branch of the S-­‐shaped curve •  We will introduce an asympto7c analysis for large Damköhler numbers and large ac7va7on energies •  In the limit of large Damköhler numbers which corresponds to complete combus7on the chemical reac7on is confined to an infinitely thin sheet around Z = Zst 8.-­‐22 •  Assuming constant cp the temperature and the fuel, oxidizer, and product mass frac7on profiles are piecewise linear func7ons of Z •  This is called the Burke-­‐Schumann solu7on 8.-­‐23 8.-­‐24 •  We define the reac7on rate as to show that is able to describe diffusion flame quenching •  For simplicity we will assume T1 = T2 = Tu. •  Then, for one reac7on with 8.-­‐25 •  The temperature and the fuel and oxygen mass frac7on are expanded around Zst as where ε is a small parameter to be defined during the analyses •  The exponen7al term in the reac7on rate may be expanded as where the Zeldovich number is defined as 8.-­‐27 •  If all other quan77es in are expanded around their value at the stoichiometric flame temperature one obtains where is the Damköhler number 8.-­‐28 •  The differen7al equa7on is cast into the same form as the one that governs Liñán's diffusion flame regime by using the further transforma7on to yield 8.-­‐29 •  There are evidently two ways to define the expansion parameter ε •  Either by seong or by seong •  The first one would be called a large ac7va7on energy expansion and the second one a large Damköhler number expansion •  Both formula7ons are interrelated if we introduce the dis7nguished limit, where the rescaled Damköhler number is assumed to be of order one 8.-­‐30 •  Thus a definite rela7on between the Damköhler number and the ac7va7on energy is assumed as ε → 0 •  We set to obtain Liñán's equa7on for the diffusion flame regime •  The boundary condi7ons are obtained by matching to the outer flow solu7on 8.-­‐31 •  The essen7al property of this equa7on, as compared to the large Damköhler number limit δ → ∞ is that the exponen7al term remains, since δ was assumed to be finite •  This allows ex7nc7on to occur if the parameter δ decreases below a cri7cal value δq
•  Liñán gives an approxima7on of δq in terms of | γ |
•  For small values of Zst ex7nc7on occurs at the transi7on to the premixed-­‐
flame regime. He obtains 8.-­‐32 •  Characteris7c profiles for the temperature over Z with δ as a parameter
•  There is a limi7ng profile Tq(Z) corresponding to δq
•  Any solu7on below this profile is unstable, and the flamelet would be ex7nguished
8.-­‐33 •  The ex7nc7on condi7on δ = δq defines with and a maximum dissipa7on rate χq at the surface of stoichiometric mixture for a flamelet to be burning, namely 8.-­‐34 •  We may interpret χst as the inverse of a characteris7c diffusion 7me •  If χst is large, heat will be conducted to both sides of the flamelet at a rate that is not balanced by the heat produc7on due to chemical reac7on •  Thus the maximum temperature will decrease un7l the flamelet is quenched at a value of χst = χq
8.-­‐35 •  Burning of the flamelet corresponds to the upper branch of the S-­‐shaped curve •  If χst is increased, the curve is traversed to the lec un7l χq is reached, beyond which value only the lower, nonreac7ng branch exists
Quenching
•  Thus at χst = χq the quenching of the diffusion flamelet occurs •  The transi7on from the point Q to the lower state corresponds to the unsteady transi7on 8.-­‐36 •  Auto-­‐igni7on, which would correspond to an unsteady transi7on from the point I to the upper curve, is unlikely to occur in open diffusion flames, since the required very large residence 7mes (very small values of χst) are not reached Auto-ignition
•  An example for auto-­‐igni7on in non-­‐premixed systems is the combus7on in a Diesel engine •  Here interdiffusion of the fuel from the Diesel spray with the surrounding hot air leads to con7nuously decreasing mixture frac7on gradients and therefore to decreasing scalar dissipa7on rates •  This corresponds to a shic on the lower branch of the S-­‐shaped curve up to the point I where igni7on occurs 8.-­‐37 Time and Length Scales in Diffusion Flames •  We will define the chemical 7me scale at ex7nc7on as •  This defini7on is mo7vated by expression •  By comparing this with the 7me scale of a premixed flame with the same chemical source term one obtains 8.-­‐38 •  Here ρu s_L has been calculated from and for a stoichiometric premixed flame
8.-39 •  This indicates that there is a fundamental rela7on between a premixed flame and a diffusion flame at ex7nc7on: •  In a diffusion flame at ex7nc7on the heat conduc7on out of the reac7on zone towards the lean and the rich side just balances the heat genera7on by the reac7on •  In a premixed flame the heat conduc7on towards the unburnt mixture is such that it balances the heat genera7on by the reac7on for a par7cular burning velocity •  These two processes are equivalent 8.-­‐40 •  A diffusion flame, however, can exist at lower scalar dissipa7on rates and therefore at lower characteris7c flow 7mes •  The flow 7me in a premixed flow is fixed by the burning velocity, which is an eigenvalue of the problem •  Therefore combus7on in diffusion flame offers an addi7onal degree of freedom: that of choosing the ra7o of the convec7ve to the reac7ve 7me, represented by the Damköhler number defined in as long as χst < χq
8.-­‐41 •  This makes non-­‐premixed combus7on to be beZer controllable and diffusion flames more stable •  It is also one of the reasons why combus7on in Diesel engines which operate in the non-­‐premixed regime is more robust and less fuel quality dependent than that in spark igni7on engines where fuel and air are premixed before igni7on 8.-­‐42 •  Equa7ons and may now be used to calculate chemical 7me scales for diffusion flames: •  Ex7nc7on of the hydrogen-­‐air diffusion flame occurs at a strain rate aq =14260/
s and that of the methane-­‐air flame at 420/s 8.-­‐43 •  The laZer es7mate is of the same order of magnitude as tc for stoichiometric premixed methane flames •  In diffusion flames, in contrast to premixed flames, there is no velocity scale, such as the burning velocity, by which a characteris7c length scale such as the premixed flame thickness could be defined •  There is, however, the velocity gradient a, the inverse of which may be interpreted as a flow 7me •  Based on this flow 7me one may define an appropriate diffusive length scale 8.-­‐44 •  Based on this flow 7me 1/a one may define an appropriate diffusive length scale •  Dimensional analysis leads to a diffusive flame thickness •  Here the diffusion coeffcient D should be evaluated at a suitable reference condi7on, conveniently chosen at stoichiometric mixture •  Assuming a one-­‐dimensional mixture frac7on profile in y-­‐direc7on as for the insteady mixing layer the flame thickness in mixture frac7on space may be defined 8.-­‐45 •  Here is the mixture frac7on gradient normal to the flamelet •  This flamelet thickness contains the reac7on zone and the surrounding diffusive layers •  The equa7on leads with and to where χref represents the scalar dissipa7on rate at the reference condi7on 8.-­‐46 •  If χref is evaluated at Zst and is used, it is seen that ΔZF is of the order of Zst, if Zst is small •  With an es7mate ΔZF = 2 Zst the flame thickness would cover the reac7on zone and the surrounding diffusive layers in a plot of the flamelet structure in mixture frac7on space 8.-­‐47