Well Stirred Reactor Stabilization of flames Well Stirred Reactor (see books on Combustion ) Stabilization of flames in high speed flows (see books on Combustion )
Stabilization of flames Although the diffusion flames are the most used (for reasons already discussed), they often present major disadvantages, in particular relating to losses by radiation, soot, etc. Premix flames eliminate these disadvantages but have narrow ranges of stability, particularly if high flow rates are used, then occurring blowoff (note that the stabilization is based only on a balance between flow velocity and rate of flame propagation, and this is only changed by heat and radicals losses to the solid walls). A Atmosphere Burner s wall Good stabilization can be achieved by locating a flame in a recirculation zone, with strong mixing of the gases, and high residence time of reactants in the reaction zone. And if the fuel and oxidant can be mixed in that zone of extensive mixing, the better the typical problems of premixed flames are eliminated.
Flames stabilized in recirculation zones The recirculation zones and flame stability in these zones will be discussed later. Now let us examine the stability criteria (i.e., the residence time of reactants in the recirculation zone)
Well Stirred Reactor model Let us consider the combustion in a reaction zone (with both reactants and products) with very high mixing, and hence uniform composition That zone is fed with the reactants. The same mass flow rate of an homogeneous mixture of reactants and products leaves it (note that reactants leave that zone, and thus they must be consumed immediately downstream) The (physical and mathematical) model of such a very high mixing zone is called Well Stirred Reactor There is a limit to the amount of reactants that can be introduced in the mixing zone, above which combustion is extinguished Important is also the possibility of re-ignition if the reaction is extinguished. Note that the extinction is related to the chemical kinetics at high temperature, and the re-ignition to the chemical kinetics at low temperature A good physical and experimental model for the Well Stirred Reactor is the Longwell s bomb (Longwell, Frost & Weiss 1953)
Longwell s bomb Premixed reactants Exit of gases Refractory wall This reactor was tested with various fuels, pressures and temperatures. The results vary only slightly for most hydrocarbons and are relatively insensitive to the presence of additives N molar flow rate (kmol air s -1 ) flame impossible V volume of the bomb (m 3 ) P pressure in the bomb (atm) flame possible flame impossible
Mathematical model of the WSR Assumptions: bi-molecular, one step chemical reaction fuel + oxidant products adiabatic reaction steady regime all molecular masses are equal; all specific heats are equal Let us consider the following conserved scalars: y fu T T0 = y fu,0 c and Q y ox = y ox,0 s c T T Q 0 = y + s ( y y ) ox, 0 fu fu,0 (6-1) (6-) 3 The reaction rate = E R K p y y T fu fu ox exp (6-3) R0 T can be written as (cf (-1)) R fu = K p y fu,0 c T T Q 0 y ox,0 s c T T Q 0 T 3 exp E R T 0 (6-4)
Let us represent the variation of the reaction rate as a function of temperature (using normalized scales for R fu and for T) Rfu * 1.0 * fu R = R R fu fu,max τ = T T T0 T ad 0 (6-5) (6-6) 0.8 0.6 0.4 Note that R fu,max and T ad are defined for the particular conditions of p, T 0, y fu,0 and y ox,0 of each case 0. 0.0 0.0 0.5 1.0 τ The air / fuel mixture enters the chamber of volume V. Its mass flow rate is m&. The degree of turbulence and mixing are so high that the gas in the reactor has an homogeneous temperature and concentrations, uniformly distributed throughout the volume V. (Da = 0) Mass balance to the fuel m& Thermal balance m& Q = m& ( y y )Q = m& τ c fu, consum fu fu, 0 ( y y ) R V fu fu, 0 = fu ( T ) ad T 0 (6-7) (6-8)
The two balances (6-7) and (6-8) yield m& τ c T ad T 0 Q = R fu V = R fu,max R * fu V (6-9) Let us define the non-dimensional entity L, called chemical loading of the reactor m c Tad T0 L = & Q R V ( ) fu,max (6-10) L represents the ratio between 1) the energy that could be released if the entire mixture would reacted, and ) the maximum energy that the chemical kinetics could release * The equation to solve is then L τ = R fu (6-11) Let us solve this equation graphically representing the curves of R fu and of in the graph of the previous slide * L τ
The two curves intersect in one or three points. The intersection at the low value of τ corresponds to the mixture entering the reactor being too much for what the kinetics allows to react. There is no combustion and the gases leave the reactor without reacting. For high values of L (L 3 in the figure) the two curves intersect E only at the point where there is no combustion. The other two solutions do not exist because the mixture flow rate entering the reactor is too great for what the kinetics allows to react. The chemical loading L is the highest that allows chemical reaction to occur. It is called critical chemical loading For values lower than L there are two possible solutions (A and B). However, only one of them is stable. E
Let us consider L 1 and the solution A. If a disturbance momentarily decrease the temperature (lowering τ ) the rate of reaction) drops, leading to further lowering of τ, and consequently lowering R fu. The reaction is extinguished in E. If a disturbance momentarily increase the temperature, the reaction rate increases and the system evolves to the point B. E Let us consider now L 1 and the solution B. To a momentary increase of τ, the system responds by lowering R fu, and hence lowering τ ; and to a momentary decrease of, the system responds by increasing R fu, and hence increasing τ. Hence, solution A is instable, and B is stable.
Let us estimate now L crit At the point where L crit is tangent to * R fu = f ( τ ) R fu, it is only slightly less than unity, and τ is between 0.65 and 0.80 (corresponding, respectively, to low and high values of T E or, in other words, of T E /T ad ). Hence, L crit is usually in the range of 1. to 1.5 Let us analyze the factors that lead to extinction of combustion * L crit m crit c ( Tad T0 ) = & Q R fu,max V m& crit = Lcrit Q R V c ( T T ) fu,max ad 0 (6-1) Combustion extinguishes (i.e., L crit is exceeded) increasing above m& The value of m& crit is larger the larger Let us analize what R fu,max depends on V and the reaction rate R fu,max are Let τ crit be the τ corresponding to the temperature at the point of extinction the critical point. At that point the reaction rate R fu,max is : m& crit
R fu, crit = K p y fu,0 c τ crit ( T T ) τ ( T T ) ad Q 0 y [ T + τ ( T T )] ox,0 3 0 crit ad 0 exp s c crit T 0 ad Q 0 E R + τ crit 0 ( ) Tad T0 (6-13) R fu,crit increases sharply with T ad, due to the Arrhenius type of evolution. Hence, it depends strongly on φ R fu,max also increases with the square of the pressure (in fact, with p raised to the order of overall reaction - about 1.75 to 1.8 for hydrocarbons) Let us check how the critical velocity of introduction of reactants in the reactor varies with the pressure m& crit V p m& = ρ crit 0 u crit A m & V crit = ρ u A M A 0 crit = ucrit p V R0 T0 V p (6-14) u crit p (the speed of introduction of reactants may be increased when the pressure increases)
From the previous equation (6-14) it can be seen that u crit A V p V u crit A For a fixed pressure, and if the proportions are maintained, the larger the reactor is, the greater the introduction rate of the reactants can be large flames resist extinction better than small flames If the pressure drops, the reaction rate decreases, as well as the speed of feeding of the reactants, and the reactor is more susceptible to extinction. This is a problem for the combustion chambers of aircraft engines working at high altitude (lower pressure than at sea level) The behavior at altitude can be simulated in tests carried out at sea level (hence with a higher pressure than the real one) if a smaller volume is used m& crit V p The theoretical analysis above made it possible to establish a basis for variations in scale suitable for this type of test (but note that the dependence with p has the exponent 1.8)
It was seen that it was important to analyze the problem from the standpoint of residence time The residence can be defined as: t res = V V & t res = length of the reactor velocity at the reactor (6-15) (6-16) Definition (6-16) is not applicable in this case velocity inside the reactor) (it makes no sense to define a t res = V V& = V m& ρ 0 (6-17) For a given flow of reactants, the minimum residence time is the one that corresponds to the minimum volume that ensures stability in the reactor V m & crit max V = m & crit crit c = L crit ( T T ) ad Q R 0 fu,max c ( T T ) t res V crit L crit R ad se 0 fu, max Q (6-18)
Comparison model/experiments N molar flow rate (kmol air s -1 ) flame impossible V volume of the bomb (m 3 ) P pressure in the bomb (atm) flame possible flame impossible As the model predicts, the flame is more stable for: mixtures near the stoichiometry larger volumes (for the same molar flow rate of air) higher pressures (note that the exponent 1.8 - instead of - is the order of reaction is typical of hydrocarbons)
Flame stabilization in high speed flows The gas in the zones of recirculation has a residence time much higher (~30x) than in flames in typical flows, and an intensity of mixing that achieves nearly homogeneous conditions The question now is how to create recirculation zones that stabilize the flames The recirculation zones can be created on the wake of bluff bodies The figure shows the wakes of cylinders perpendicular to a flow For spheres there is no regime 3. Combustion also eliminates the regime 3, with regime extending up to Re 10 5
There are several methods to create a recirculation zone: bluff bodies sudden expansions impinging jets opposing jets swirl multiple obstacles The entire flow can receive the fuel...... or not
Ref.: F A Williams (1966)
There are numerous results in the literature for flames stabilized in recirculation zones Let us examine, in particular, cases of recirculation due to bluffs bodies and then let us develop a mathematical model for those situations Scurlock (1948) Cylinders Light HC u ext - ft s -1 p - lbf in - D - in
To analyze these results and the theories that try to explain them, let us first see some data on the flow downstream of obstacles for high Re l = a 5 D D The drag coefficient C D is defined based on the drag force that the flow exerts on the body F C D = 1 ρ u A F force A transverse area of the body ρ density of the undisturbed fluid u upstream speed of the flow (6-19)
Values of C D vary widely with the shape of the body, but for those that are used in flame stabilization (and for the Re used) its values do not deviate much from unity The volume of this zone is D l per unit length two-dimensional geometry π D l 4 axi-symmetric geometry and the area is l π Dl per unit length two-dimensional geometry axi-symmetric geometry A balance for the momentum in the control volume defined by the line which defines the recirculation zone (and considering the uniform pressure) yields pressure in the body cross-sectional area F A D 1 = S l 1 area where S is exerted longitudinal area shear stress (6-0)
1 C D ρ u D = S l two-dimensional geometry (6-1) 1 C D ρ u D = 4 S l axi-symmetric geometry (6-) or C D 1 ρ u D = S l j with j = two-dimensional geometry (6-3) j = 4 axi-symmetric geometry S CD D = ρ u u j l (6-4) It is reasonable to assume as valid the Reynolds analogy in this turbulent shear layer, whereby the exchange of thermal energy and the momentum exchange rate follow the relation α c = S u = C D j D l ρ u α convection heat transfer coefficient ( h) c specific heat of the mixture (6-5)
Note that the units of equation (6-5) are units of a mass flux It can be considered then that the rate of mass exchange across the surface of the recirculation zone (i.e. the mass flow G) is given by G α S C D = = D u c u j l ρ = (6-6) The surface of the recirculation zone (control surface) was defined as the line between the fluid that enters the recirculation zone and the one that does not enter that zone. Note that this definition is valid only on average terms. There must be mass exchange (otherwise the combustion would cease once the reagents in the recirculation zone were consumed) It should be noted that the balance of exchange of mass fluxes is nil, but that the flux that enters (fresh gas) is different from the one that leaves (a mixture of reactants and products - within the recirculation zone the composition is homogeneous)
( ) The heat transfer per unit area is α T, with the temperature T r in the r T u recirculation zone and the temperature T u in the outer flow Note that the heat transfer to the obstacle is neglected Similarly, the transfer of fuel is ( y fu y ) G, u fu, r With the usual assumptions (bi-molecular one-step reaction,..., etc.) and for * the same definition of R fu, a balance of the fuel entering the combustion zone and that is burned is then G ( y y ) j l = fu, u fu, r R fu, r Dl (6-7) Q y fu, u fu, r Having in mind that ( ) ( ) ( )τ y = c T r T u = c T ad T u (6-8) it yields CD D * ( y y ) ρ u j l = R R Dl fu, u fu, r fu fu, max j l (6-9)
c Q C D l D * ( T T ) τ ρ u = R R D ad u fu fu, max (6-30) C D D l ρ u c Q R ( T T ) * ad fu, max D u τ = R fu (6-31) Note that all groups in (6-31) are non-dimensional Let us define for convenience the parameter charge of the bluff body L b (similarly to what was done with the chemical load of the reactor) L b u c = ρ Q R ( T T ) ad fu, max D u (6-3) L b represents the ratio between 1) the energy (per unit of the surface area of the recirculation zone) that could be released if the entire mixture reacted, and ) the maximum energy that the chemical kinetics can release per unit width of the obstacle Rewriting equation (-3) then yields
C D D l L b τ = R * fu (6-33) Note the similarity with the result in Well Stirred Reactor it can then be said that CD D L b varies between 1. and 1.5, crit extinction l Since D/l varies between 1/5 and 1/, and C D 1.0, L b,crit varies in the rnage of 5 to 15 Note that values referred for C D and D/l are valid for isothermal flows. In the reactive flow C D and D/l decrease, and therefore L b,crit reaches higher values (up to about 40) L b,crit increases when C D and/or D/l decrease (i.e., the higher l is). Note that it was assumed that the height of the recirculation zone is D, but this increase deviates from reality the larger C D is. The increase in C D, by increasing the height of the recirculation zone, can increase the stability of the flame The equation of the previous slide shows that u increases linearly with D and with R fu,max. This variation with R fu,max means that
u should be maximum for stoichiometric (or slightly rich) mixtures u should be proportional to p (because ρ is proportional to p and R fu is proportional to p ) Let us compare the results obtained with experimental results Let us check the order of magnitude of u crit. Let us consider an example corresponding to a stoichiometric mixture L b,crit = 10 ρ = 1.000 kg m -3 c = 1.05 kj kg -1 K -1 Q = 4 10 3 kj kg -1 R fu,max = 115 kg m -3 s -1 T ad T u = 000 K The extinction speeds (u ext u crit ) obtained are the following: D u ext (model) u ext (Scurlock) 1.5 mm 87 m s -1 ( 160 m s -1 ) 1.6 mm 37 m s -1 10 m s -1 0.4 mm 9 m s -1 60 m s -1 Comparing these speeds with the results of Scurlock it is verified that the result may be acceptable for the cylinder of 1.7 mm, but not for the cylinders of 1.6 mm and especially of 0.4 mm
The reason lies in the extremely low Reynolds number for these cylinders - the flow is too different from the one used in the model The results from De Zubay show that u ext is proportional to p 0.95 and D 0.85. The result for the pressure is virtually the predicted one, as well as the one for D. As far as the speed is concerned, the result for u ext p -0.95 D -0.85 = 76 and for p = 15 psi and D = 0.5 is a speed u ext = 55 ft s -1 = 18 m s -1 The value of the model is higher than the experimental value of De Zubai (similarly to what was present in Scurlock s case), but then again the order of magnitude is correct The experimental values of Barrère and Mestre are lower than those of Scurlock and De Zubai. But note that Scurlock and De Zubai used to flow velocity around the obstacle, while Barrère and Mestre used the undisturbed flow velocity (and therefore lower values)
All experimental results show maximum stability for a slightly rich mixtures (due to the higher temperature reached by these mixtures). This shift to the rich side decreases with increasing Re. This shows that the deviation is due to the low diffusivity of the molecules of the heavy fuel to the recirculation zone, which has thus a leaner air/fuel ratio than the outer flow. An increase in turbulence increases the (turbulent) diffusion of heavier molecules, thus reducing that shift Considering the form of the obstacle on the results of Barrère and Mestre, the obstacle 3 has the lowest C D. Hence u ext should be the greatest. However it is the obstacle that leads to the smallest height and volume of the recirculation zone. These reductions turn out to be the predominant factor relative to the lower value of C D, leading to the lowest value of u ext for the three obstacles An important set of conclusions can now be drawn these results
since the extinction speed is proportional to D/ρ and R fu,max, this speed is thus proportional to pd. This allows the testing of models (where D is different) varying the pressure in inverse proportion to D. More common is the reverse: reduced scale model used at atmospheric pressure to simulate conditions at low pressure for tests at different scales the proportionality to R fu,max is also used, making it possible to vary R fu,max by the addition of inert (especially nitrogen and water vapour), thereby lowering the concentration of mass of y fu and y ox stability is enhanced reducing C D and D/l. Note that the reduction of C D is beneficial from the point of view of pressure drop. However, a reduction of D/l involves an increase of the combustion zone, which may be undesirable adding fuel as an additional means to stabilize the flame may be counterproductive: maximum stability occurs for (very) slightly rich mixtures, and decreases rapidly on both sides of that value
the use of power assistance can be beneficial in the event of extinction, but only if the conditions return to the non-extinction conditions. The electrical aid is not a means to stabilize a flame, but to (re)ignite it. Its use is irrelevant for the stabilization because the electrical energy fed into the system is negligible compared with that that is released in the combustion (it is the latter what maintains the continuous combustion of the reactants that reach the recirculation zone) such as in the Well Stirred Reactor, the use of additives is inconsequential in the stabilization of flame, since the stabilization of flame is controlled by chemical kinetics at elevated temperature (the molecules are already "broken" in smaller radicals). By contrast, in self-ignition (phenomenon of chemical kinetics at low temperature) the existence of additives is of immense importance in the process of breaking the molecules and formation of radicals