456
I n d E n g C‘hem Res. 1990, 29, 456-463
Effect of Laminarizing Flow on Postflame Reactions in a Thermally Stabilized Burner Lance R. Collins’ and Stuart W. Churchill* Department of Chemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104
A thermally stabilized burner consists of a ceramic tube through which premixed fuel and air are fed. Owing to the minimal backmixing, the changes in composition and temperature across the channel, this temperature flamefront are virtual step functions. For ethane and air in a 9.5” rise results in a correspondingly abrupt decrease in Reynolds number from 3000-6000 to 650-1550. The turbulent velocity profile upstream from the flamefront thereupon undergoes a gradual transition to the parabolic form characteristic of laminar flow. Results are presented herein for the postflame reactions in this region of decaying turbulence using an extended free-radical model for the kinetics and a previously computed field for the velocity. The computed concentrations of NO differ only slightly from those based on the postulate of plug flow, but the computed concentrations of CO are as much as 25% higher. T h e assertion of Aris and others that the reduction in conversion due to laminar flow is negligible, although correct per se, is thus midleading if the concentration of the reactant itself (here CO) is of primary interest. The thermally stabilized burner ITSB) is unique in that the thermal feedback required to stabilize a flame from premixed fuel and air in flow through a ceramic tube is accomplished by wall-to-wall radiation and in-wall conduction as contrasted, for example, with molecular diffusion for a Bunsen-type burner, recirculation for a bluffbody burner, and eddy mixing for a turbulent-jet burner, all of which produce backmixing of burned gas as well as energy. The avoidance of backmixing in the TSB results in a very thin zone of combustion, consisting of a virtual step in composition and temperature. Additionally, because heat losses from the TSB are ordinarily negligible, the temperature of the gas stream is nearly constant longitudinally and radially but has different values upstream and downstream from the flamefront. In a TSB with a channel diameter greater than about 7 mm, the narrow range of stable combustion for all hydrocarbons other than methane falls in the turbulent regime of flow upstream of the flamefront (see Pfefferle and Churchill (1984)). However, the sharp increase in temperature across the flamefront results in a corresponding increase in viscosity, and thereby a decrease in the Reynolds number, Dumpll, to a value that corresponds to the laminar regime. The TSB is thus only a partial exception to the assertion of Churchill and Pfefferle (1985) that turbulent flow can scarcely be achieved in a tubular homogeneous reactor. The conditions chosen for the calculations of this investigation of the combustion of premixed ethane and air in a refractory channel 9.5 mm in diameter, namely Re,, the Reynolds number at the inlet to the TSB, and ‘P, the equivalence ratio (fuel-to-air ratio/stoichiometric fuel-toair ratio), are listed in Table I, tugether with the corresponding values of Re,, the Reynolds number downstream from the flamefront, and Ya, the adiabatic flame temperature for complete combustion to COz and H,O only. The thermal effect of the slight amount of NO formed in the TSB is negligible, although the equilibrium value would change T , significantly. Re, is based on T, and the corresponding composition. The values of 3000 and 6000 for Re1 encompass the narrow range of flow for stable combustion of ethane and air in the TSB. whose range, to *Author to whom inquiries should be sent. ‘Current address: Department of Chemical Engineering, Pennsylvania State University, 158 Fenske Laboratory. University Park, P A 16802
0888-5885/90/2629-0456$02.50/0
Table I. Conditions for Calculation of Postflame Kinetics (Ethane-Air and a 9.5-mm Channel) T..” K Re, (uustream) Resn (downstream) 0.70 1730 3000 774 1162 4500 1549 6000 0.85 1957 3000 704 4500 1056 6000 1409 1.00 2157 3000 653 4500 950 6000 1307 Based on complete adiabatic reaction of fuel to COz and H,O(,, and neglecting formation of NO.
a first approximation, is independent of the diameter of the channel. The TSB has two other unique operational characteristics that are relevant to this work: (1) as many as seven stationary states have been predicted (Chen and Churchill, 1972) and confirmed experimentally (Bernstein and Churchill, 1977), and (2) the large heat capacity of the wall as compared to that of the enclosed gas results in great thermal inertia and stability with respect to changes in the operating conditions. The first of these characteristics allows operation with different times of residence in the postflame zone while all other conditions are maintained unchanged, and the second allows operation at pseudostationary states as the flamefront drifts slowly along the channel in response to a minor perturbation, for example, in the rate of flow of fuel and/or air. Prior modeling of the reactions occurring in a TSB has indicated (Pfefferle, 1984; Pfefferle and Churchill, 1986a) that fuel-lean mixtures of hydrocarbons are oxidized so rapidly and completely to CO and H 2 0 at the flamefront that “prompt” NO, is not formed. Modeling of the reactions before and in the flamefront can therefore be avoided by considering stoichiometric and fuel-lean mixtures only and postulating the indicated composition out of the flamefront. An extensive set of free-radical mechanisms is, of course, necessary to model the postflame reactions that result in the formation of “thermal” NO, and the oxidation of CO to C02. Prior modeling of this behavior by Tang and Churchill (1980a,b), Tang et al. (19811, Pfefferle (1984), Churchill and Pfefferle, (1985), and Pfefferle and Churchill (1986a,b) has all been based on the postulate of plug flow. This modeling, as well as the as@ 1990 American
Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 3, 1990 457 sociated experimental work, has shown that the TSB produces exceptionally low concentrations of NO and negligible concentrations of NOz at the expense of modest residual concentrations of CO, owing to the short residence times, high temperatures, and minimal backmixing. The primary objective of the current investigation has been to take into account the actual field of flow in the TSB for representative conditions and thereby evaluate the effect of the postulate of plug flow on the composition of the gas leaving the burner. The velocity field ahead of the flamefront is essentially that for fully developed turbulent flow. The velocity of the gas is accelerated in passage through the flamefront owing to the decrease in density accompanying the increase in temperature, and the pressure decreases correspondingly. At the same time, owing to the increase in viscosity, the Reynolds number decreases below the critical value for transition to laminar flow (nominally 2100). Thereupon the turbulence begins to decay, and a nonfluctuating velocity field begins to develop, eventually approaching the parabolic distribution for fully developed laminar flow. A theoretical solution for this developing velocity field has been derived and tested experimentally in collateral work (Collins and Churchill, 1990). That solution is utilized in the kinetic calculations herein. Aris (1965) demonstrated theoretically that the maximum loss of conversion due to laminar flow vis-84s plug flow occurs for a first-order reaction with a mean residence time equal to ( k ’ ) - l and is limited to 11%. Cutler et al. (1988) recently reviewed the multitude of studies on the effect of postulating plug flow for a tubular reactor operating in the laminar regime and concluded that the associated error is negligible with respect to experimental uncertainties and other common idealizations. Since the velocity field downstream from the flamefront in a TSB undergoes a gradual transition from nearly plug flow to nearly laminar flow, the error due to the postulate of plug flow might be expected to be less than that found by Aris and Cutler et al. However, because of the radically changing flow, the complex kinetics, and the extreme conversions in the TSB, the generality of this premise appeared to be worthy of analysis.
Fluid-Mechanical Model and Solution Model. The field of flow in the TSB was modeled by Collins (1987) [also see Collins and Churchill (1990)l for the range of conditions summarized in Table I, which correspond to prior experimental work with mixtures of ethane and air in a 9.5-mm channel. The k-t model was used for the time-averaged transport of momentum, thereby representing the fundamentlly unsteady, three-dimensional motion by steady-state equations in two dimensions. The modeling was carried out separately for three regions. In region 1, upstream of the zone of combustion, the inlet temperature and composition were postulated to prevail, and the distance from the inlet was postulated to be sufficient to achieve fully developed turbulent flow at the flamefront. In region 2, which corresponds to the thin but finite zone of combustion, the fuel was postulated to burn to CO and H 2 0 at a rate given by a global, Arrhenius-type expression. However, the temperature was postulated to rise in proportion to the heat of combustion to COPand H20k,rather than to CO and H,O( ) In region 3, downstream from the zone of combustion, txe temperature was postulated to be uniform at the adiabatic value corresponding to complete conversion to C02and H,O. The density was also assumed to be constant in region 3. This latter slight idealization, which can readily be justified, has the great merit of de-
80 .
0.0
0.2
0.4
0.6
0.8
1.o
ria
Figure 1. Axial development of the radial distribution of the time-mean velocity across and downstream from the flamefront for Rel = 3000 and 9 = 1.0.
coupling the fluid-mechanical behavior from the postflame chemical kinetics. The temperatures, pressures, velocities, etc., were related at the common boundaries of regions 1 and 2 and regions 2 and 3. (A small mismatch in temperature, composition, etc., between regions 2 and 3 was specified in order to yield a finite zone of combustion.) The principal idealization in this model is not the postulate of constant density in region 3 but rather the neglect of the energy fed back from region 3 to region 1 for stabilization. (Such thermal feedback can actually produce temperatures within the channel that are above the adiabatic flame temperature.) However, experimental measurements have demonstrated that this idealization is closely compensated for by the use of the full heat of combustion to C 0 2 and H20e, in region 2. Method of Solution. The above partial-differential model was solved by finite-difference calculations for a range of representative conditions. The details of the fluid-mechanical modeling and the process of solution as well as the results and their interpretation are given in full by Collins (1987) and are summarized by Collins and Churchill (1990). Only those results directly relevant to the kinetic calculations are described herein. Numerical Results. Figure 1illustrates the computed values of the radial profiles of the time-mean velocity in the axial direction: (1)upstream from the flamefront ( z / D = 0-); (2) immediately downstream from the flamefront ( z / D = O+); (3) a t z / D = 5; and (4) a t z / D = 25. These velocities are for a stoichiometric mixture of ethane and air and a preflame Reynolds number of 3000, resulting in an adiabatic flame temperature of 2157 K and a postflame Reynolds number of 653. As the gas stream passes through the flamefront, the velocity is seen to increase several-fold due to the decrease in density accompanying the increase in temperature. As the gas proceeds downstream, the velocity profile becomes more peaked owing to the gradual laminarization and development of the flow. For these conditions, the process of laminarization and development is essentially complete in 25 diameters (240 mm). Radial profiles of k1/2/u,, which is a measure of the relative magnitude of the turbulent fluctuations in velocity, are illustrated in Figure 2, and radial profiles of I ’ / D = k 3 l 2 / ( e D )which , is a measure of the scale length of the
458
-
h d . Eng. Chem. Res., Vol. 29, No. 3, 1990
0.8
-\
0.4
-
0.01
\ I ,
03--
0.M)
0.0
0.4
0.2
0.6
1 .o
0.8
0.0 -i
rla Figure 2. Decay of the local turbulent kinetic energy, k , downstream from the flamefront for Re, = 3000 and @ = 1.0.
I
0.5
\
00
O?
3
06
0 4
fl=:o
OX
1 0
ria
Figure 3. Decay of the scale length of the turbulence downstream from the flamefront for Re, = 3000 and @ = 1.0.
turbulent fluctuations, are illustrated in Figure 3, both for the same conditions as in Figure 1. These characteristics of the turbulence decrease with distance from the flamefront as expected. Their maximum values move toward the centerline because the rate of decay of turbulence is greater near the wall. Illustrative time-mean particle paths are shown in Figure 4 for the same conditions as for Figures 1-3. These curves were computed by numerical integration of the expressions iir(r,zj = dr/dt
(1 1
ii,{r,z) = d z / d t
(2)
and The required local values of ur and U , were obtained by interpolation of the values obtained at the grid points used in the numerical solution of the fluid-mechanical model. The results can be expressed functionally as r =f h o l
(3) (4)
where ro is the initial radius (at z = 0) for a particular particle path. Then r can be plotted vs z for a chosen ro
.
1
Ind. Eng. Chem. Res., Vol. 29, No. 3, 1990 459
Chemical Kinetic Model The kinetic model of Tang and Churchill (1980a), which includes 21 forward reactions, 12 species, and 21 equilibrium constants, was utilized in the current work. Pfefferle and Churchill (1986a) compared calculated compositions for this relatively simple model with those for a revised and extended one and concluded that, after a very short distance downstream from the flamefront, the differences were negligible for fuel-lean mixtures. To initiate the computation of the concentrations downstream from the flamefront, the ethane was postulated to have reacted completely to CO and H20. The mass fractions at the beginning of the post flame region were thus 56@ wco = 482.9 + 30@ wH20
54@ = 482.9 30@
+
duction of species i due to all of the mechanisms that involve that species. The integration of this coupled set of differential equations was carried our numerically for @=0.7, 0.85, and 1.0 (which result in different densities and temperatures), using the LSODE code of Hindmarsh (1980) and the CHEMKIN code of Kee et al. (1980). A PFR produces the greatest possible extent of reaction under isothermal conditions. Hence, these solutions constitute an upper bound for the production of NO, and a lower bound for the residual concentration of CO for each equivalence ratio. For a single, irreversible reaction of order equal to or greater than unity, carried out at constant temperature and density, the fractional conversion of the reactant can be shown to be a function of L = k’CRon-lz/u, only. For multiple, reversible reactions at constant temperature and density, the moles of the various species per mole of feed can be expressed as a function of L = k*z/u,, where k* is an arbitrary and generally indeterminate coefficient with the dimension of reciprocal time. As contrasted with k ’, k* depends complexly on the initial composition, which, for a single fuel, is defined by the equivalence ratio @, Thus, the computed results for the combustion of ethane in a PFR can be expressed as
k*(@)z
w,o =
WN02=
w,
5
0 (12)
This model is obviously in error in specifying zero concentrations for the various free radicals a t the exit from region 2. However, the pool of free radicals builds up very rapidly a t the adiabatic flame temperature, and the resulting accumulative error in the concentrations of CO and NO, as evaluated by comparison with the calculations of Pfefferle (1984), which accounts for the detailed reactions ahead of and in the zone of combustion, is negligible after a very short distance. Although reaction mechanisms for the formation of NO2 were included in the model, the computed concentrations were always negligible (Z/&\
(15)
The numerical results for the indicated kinetic model were tabulated in this latter form for subsequent use. Laminar-Flow Reactor. For the same idealized conditions as described above, the least possible extent of reaction occurs in fully developed laminar flow with negligible radial as well as longitudinal diffusion. A solution for such a laminar-flow reactor (LFR) can be derived from that for a PFR by applying eq 15 with u, replaced by the local velocity u = 2u,(l
- (r/a)2)
(16)
for each filament of the LFR. That is,
The integral eq 18 was evaluated numerically by using Simpson’s rule and the previously mentioned tabulation of values for a PFR. Thermally Stabilized Burner. If the same idealizations are made for the TSB as for the LFR (constant temperature and density and negligible radial and longitudinal diffusion), a numerical solution can be derived for the chemical conversion downstream from the flamefront by using the tabulated results corresponding to eq 3,4, and 15 as follows. The time required to reach an axial position z along any particular streamline, as defined by ro, is given by the inverse of eq 4. The radial location of that streamline at that time is given by eq 3. The composition at that point is given by eq 15 with z/u, for the PFR interpreted as the time of residence, t’, for the TSB. The axial component of the velocity at this location is provided
460 Ind. Eng, Chem. Res., Vol. 29, No. 3, 1990
1
P
w
w
8
8
B
B
I
1 .
V
,
I
0
20
I
2
.
1
4
6
,
8
:0
t
1
P-
Y
II
e
8
3
B
0
2
4
6
8
10
dum(ITS)
dum(lm)
Figure 5. Longitudinal variation of the mixed-mean concentrations of CO and NO, in the TSB for Q = 0.7.
Figure 6. Longitudinal variation of the mixed-mean concentrations of CO and NO, in the TSB for = 0.85.
by the fluid-mechanical solution. Hence, the mixed-mean mole ratios can be computed from
seen to rise first at a decreasing rate and then almost linearly. The initially decreasing rate of production is presumably due to the relatively rapid rate of oxidation of CO and the associated decrease in the concentration of molecular oxygen. The subsequent linear rate of production, which suggests zero-order kinetics, is presumably due to the essentially constant concentrations of H20, C02, 02,and N2 and hence of the various free-radicals as well. For equivalence ratios of 0.85 and 1.0, the concentration of CO is seen to approach the equilibrium value in a short distance (or time). The mixed-mean concentration of CO actually falls below the global equilibrium value for these equivalence ratios at intermediate times. Such behavior is possible in complex reacting systems since the local concentrations of the free radicals which control the path of reaction depart significantly from their equilibrium or pseudosteady values (see, for example, Tang and Churchill, (1980a)1. The upper and lower bounds for the concentration of CO provided by the solutions for a LFR and PFR, respectively, were not included in Figures 5-7 because the differences, although significant, can barely be distinguished in the very condensed logarithmic scale of the ordinate. The reversed bounds for the concentration of NO, were also not included since the differences can barely be distinguished even in the arithmetic scale of that ordinate. The range of compositions encompassed by these bounding solutions is, however, shown quite clearly in Figure 8, in which the ratio of the mixed-mean concentration for a LFR to that for a PFR is plotted vs z / u m for @ = 0.7, 0.85, and 1.0. The concentration of CO is seen
The integral of eq 19 was evaluated numerically by using the tabulations corresponding to eq 3-5 and 15. In general, these numerical results can be expressed as (xi)TSB = fTsB(Re3,Z/(DReB),~,Z/Um) (20) However, insofar as eq 7 is a good approximation, ( X J T S B = fTssIz/(DRe,),~,t/UmJ 121) Furthermore, since for a specified value of both Re, and Re3 are presumed to vary only by virtue of u,, it is conjectured that ( 8 i ) T S B 31 fTSBI@,Z/Uml (22) The validity of eq 22 is tested below.
Results The computed mixed-mean concentrations of NO, and CO for the developing flow downstream from the flamefront are plotted vs z/u, in Figures 5 , 6 , and 7 for Re, = 3000 and equivalence ratios @ of 0.7, 0.85, and 1.0, respectively. In confirmation of eq 22, these curves closely approximate the computed values for Re, = 4500 and 6000 as well. Equilibrium concentrations for CO (which allow for equilibrium with respect to NO,) are included in Figures 5-7 for reference. Owing to the much slower kinetics, the equilibrium concentrations for NO, are completely out of the range of these plots. The concentration of NO, is
Ind. Eng. Chem. Res., Vol. 29, No. 3, 1990 461
1.6
1
I
10=0.7
2 0=0.85 3 0=1.0
0
2
a
6
4
10
dum(m) 1.o
I
10=0.7
2 Q=0.85 3 o=1.0
0.8 2
0 -
10
dum(m)
I
0
a
6
4
2
4
6
dum (ns) Figure 7. Longitudinal variation of the mixed-mean concentrations of CO and NO, in the TSB for @ = 1.0.
to be as much as 55% higher for the LFR than for the PFR and to depend critically upon the equivalence ratio. The strong dependence on stoichiometry arises primarily from its effect on the adiabatic flame temperature and thereby on both the rate of oxidation and the concentration at equilibrium. The ratio of concentrations of NO, is less than unity and departs but slightly from that value since it is being formed whereas CO is disappearing. Analogous curves for the ratio of the mixed-mean concentration for a TSB to that for a PFR are shown in Figure 9. The deviations for CO at equivalence ratios less than unity are seen to be very significant, approaching 26% for large z/u, at cf, = 0.7. It may be inferred from the corresponding curves in Figures 8 and 9 that neither bounding solution provides a satisfactory approximation over a complete range of conditions.
Figure 8. Longitudinal variation of the ratio of mixed-mean concentrations of CO and NO, for a LFR and PFR.
"-I1
I 0 =0.7 2 Q =o.a5 3 Q=1.0
1.3
l
l
0.98
Interpretation Previous analysis of the error due to the postulate of plug flow for a reactor operating in the laminar regime, of which a recent review is provided by Cutler et al. (1988), have invariably used the fractional conversion of a reactant as a criterion. Cutler et al. are themselves concerned primarily with the associated error in rate constants derived from experimental data. Aris (1965) has shown that, for an irreversible first-order reaction, the ratio of the mixed-mean fractional conversion of the reactant in a LFR without diffusion to that in a PFR approaches unity for both small and large values of L = k'z/u, and passes through a mininum of 0.88 at L = 0.94. Orders of reaction greater than unity produce even lesser deviations. This result, although definitive in its own terms, is misleading
0.94
-1
0.924 0
2 Q-0.85 3 Q=1.0
.
, 2
.
, 4
.
, 6
.
,
.
8
dum( w )
Figure 9. Longitudinal variation of mixed-mean concentrations of CO and NO, in a TSB as normalized by those for a PFR.
if the residual concentration rather than the fractional conversion is of interest, as is the case for a pollutant.
462 Ind. Eng. Chem. Res., Vol. 29, No. 3, 1990
n
R
O
1
1.0
1.1 170.6
1.5 2.0
2.0 1.333
3.0 1.1313
For a first-order reaction, the ratio of moles of the residual reactant is
R =
(XR)LFR =
2EdL/21
(23) (XK)PFR P-L which increases monotonically from unity for L = 0 and approaches an unbounded asymptotic value given by 4e4L/i6 + L ) for large L. For a second-order reaction, H = (1 + 2L)[l - 2L + 2L2 In 11 + 1/13 (24) I _ _ -
which increases monotonically from unity to 4 / 3 as L = CR,k't/um increases. In general, for a single, irreversible reaction of nth order greater than unity, the asymptotic value for large L = CROn-lk'z/umis
Numerical values of R { m )for various orders of reaction are listed in Table 11. Clearly, the residual concentration for laminar flow may be much greater than that which can be achieved in plug flow (or in a batch reactor). The dependence of (uJCO)LFR/(U;CO)PFR on z/um,as illutrated in Figure 8, is more complex than the relationship above for first-order and second-order reactions, as might be expected owing to the multiple and reversible mechanisms of reaction. However, the results for single reactions provide a quantitative rationale for the large deviations that are computed for CO for developing flow in the TSB as well as for fully developed laminar flow. (For the small changes in composition involved in the current work, ( w ~ ~(w,&)FR ~ ~ does ~ /not differ significantly from R as defined in eq 23.) The small range of ( w , ~ ) L F R / ( w N O ) ~ F R , as illustrated in Figure 8, is not only a consequence of the limitations of the conversion of a reactant per Aris but also of the dependence on essentially zero-order reactions for which the velocity distribution has no effect.
Effectiveness of a TSB The results of this investigation also illustrate the capability of the TSB for clean combustion. Figure 5 indicates that, despite the reduced performance due to developing laminar flow, remarkably low concentrations of both NO (less than 1.2 ppm) and CO (less than 0.08 mol % ) can be achieved for @J = 0.7 and say z/(DRe,) = 0.01. For a 9.5" tube and the values of Re, in Table I, this implies a flame stabilized about 7 5 mm from the exit of the TSB. If the equivalence ratio is increased, the concentration of CO rises in accordance with equilibrium; the NO increases somewhat but remains rate-controlled. For a stoichiometric mixture, the concentration of NO, can easily be kept below 50 ppm, but equilibrium prevents the reduction of CO below 170. This superior performance of the TSB with respect to the production of NO, arises from the avoidance of backmixing, which is much more deleterious than the deviations from plug flow indicated by the current work. Conclusions The computed concentrations of CO and NO for a thermally stabilized burner fall between those for plug flow and those for fully developed laminar flow, approaching
the latter with distance owing to the rapid decay of turbulence and the gradual redevelopment of the flow following the drop in Reynolds number across the flamefront. Homogeneous reactors invariably operate in the laminar regime of flow but are customarily modeled in terms of plug flow. This simplification is ordinarily rationalized on the basis of theoretical analyses which indicate that the maximum possible deviation in the conversion of the reactant is negligible (
