Low temperature behavior 63

Part II: Parametric Studies

4.3 Variation of Equivalence Ratio 51

4.4.2 Low temperature behavior 63

63

Two other steps with similar activation energies are reactions 2b and 11 b in table 3.4a.t These possibilities are ruled out because their reactants only include species whose concen- trations are zero initially and their temperature exponents have a positive sign.

the higher temperature cases. Among the two is H02 which has been cited earlier for its possible role in the start-up regime. The other one is H₂0₂ which has shown little significance. In the scale of figure 4.4.3b, their dimensionless mole fractions based on the definition of equation 4.4.4b are

{

XHo2(To=850J()~5 and X H2 0 2 (To

=

850 J() ~ 623

while all other radicals have X(To = 850 [() less than 0(10-²)} Despite this finding, a fit of equation 4.4.5 on the three lowest temperature data fails to yield any parameters* relevant to the reactions of H0₂ or H₂0_{2 .} Therefore, those two species are not the determining factors for the start-up time. The excessive build up of H0₂ and H202 merely means that the reactions responsible for their destruction are not very active at low temperature while the counter part of these reactions remain relatively active. Instead, a graduate transfer of dominance must be occurring near 900 J(. It is for this reason that the correlation equation cannot be applied to determine the rate controlling reaction. The change in start- up time following the transition is again expected to satisfy the fit in equation 4.4.5 and the best fit should resemble the more or less linear portion in figure 4.4.7. Unfortunately, to determine the extent of the transition and to obtain enough low temperature data for parametric fit through numerical calculations would become excessively time consuming.

Linearly extrapolated behavior would estimate a start-up time of 467 and 3 ^X 10³¹ seconds if the transition extends to To equals 800 and 500 J( respectively. Calculation of the first case shows that start-up is not yet encountered at 500 seconds, implying that the transition extends below 800 J(; the second case is not attempted.

Theoretically, the start-up time is finite unless the temperature goes to zero. Nev- ertheless, an exponentially fast increase in start-up time is observed even at 850 J(. The trend suggests that the time may approach infinity at a non-zero temperature. It may be useful to extrapolate from the low temperature data what the limiting temperature is. In a combustion problem for aircraft engines, the time scale is the combustor residence time which is of

o

(millisecond) in the ^SCRAMJET engine at interest. In this respect, the start-up time is nearly infinite for the 850 J( case and is definitely infinite in the 800 J( case. The addition of a fourth parameter in equation 4.4.5 may model the existence of a limiting temperature. The modified formula is

loge(ts)

~

^10ge(Po)

⁺

^{PI loge (To}

~

^Tm)

⁺

_{P2 To}

~

^{Tm ' (4.4.7)}

U C behaves the same way except with larger CH 0 2 (To

=

^{850 K)} and _{CH 20 2} (TO

=

^{850 K).}

* Exact solutions are obtained since there are only three parameters and three data points.

where T m is introduced such that

lim is -+ 00 . ToLTm

For extrapolation purpose, four data points (To

=

850,900,950,1000 K) are used to solve for the parameters exactly. The result is

T m

=

805.92I(

Po

=

1.98265 x 10-11 PI

=

^-2.56665

P2

=

641.693

Use of the inflexion point data in equation 4.4.6 leads to a remarkably close T m of 806.08 J(.

Segments of the resulting curves are plotted in figure 4.4.7. The other three parameters, unfortunately, do not bear any meaning in this formulation.

4.4.3 Explosive rise behavior

Following the slow start-up, chain branching sets in. The result is an explosive com- bustion process signified by rapid heat release and temperature rise. Termination of the explosive stage is signified by the inflexion in heat release where chain breaking has just become dominant. Some effort is dedicated in this section to explore the behavior from the start-up to the inflexion.

In contrast with the effect of equivalence ratio, the rise following start-up occurs at a comparable rate for all initial temperatures. Evidences are shown pictorially in the mag- nified insert in figure 4.4.1b and quantitatively in table 4.4.1. According to the table, the slopes at the inflexion points change within 55% except for the highest and lowest tem- perature cases. The impact of low temperature on start-up behavior has been mentioned.

Different chain branching characteristic is therefore expected during fast-rise. At the higher end, the expanded curve for To equals 2500 J( in figure 4.4.2b helps to explain the departure.

The 'bump' which occurs at ,...., 6 microsecond suggests that the kinetic in the initial stages is different from the other cases where no similar structure is observed. The departure of C_H02 at To equals 2500 K from the trend of the other initial temperatures shown in figures 4.4.3a and 4.4.3b provides some indication of what might be involved. However, no attempt has been made to analyze the particular kinetic since no extended effect is noticeable. The data in figures 4.4.4a and 4.4.4b for the inflexion points"* show that the unusual trend diminishes rapidly with time.

** Also see figure 4.4.5a and 4.4.5b which will be described later.

Recall from the results of figure 4.4.4b that the mole fractions of 0, OH, H and H₂0 are nearly constant at the inflexion point independent of initial temperature. Combining this with the results of figure 4.4.3b, the change in mole fractions from start-up to inflexion must also be constant. Then in the spirit of equations 4.4.5, the dominant kinetic in the explosive stage may also be extracted from parametric fit. The relevant time duration in this case is the rise time given by

and the relevant composition is

A dimensionless change in mole fraction ~x can be defined referring to the conditions at To equals 1500 J(:

This is similar to equation 4.4.4b, but the mole fractions at ts are not zero and cannot be removed. ~x of four species at six initial temperatures are plotted in figure 4.4.5b.

The lines are generally close to unity. Similar dimensionless change in concentrations are plotted in figure 4.4.5a for comparison. The lines in this case are not as close to unity. A previously selected radical, H02 , is not included in the plots since its composition varies much wider over initial temperatures. On the other hand, H202 appears to be of rising importance in this stage. Their dimensionless changes in mole fraction are shown in figure 4.4.6.

A parametric fit of rise time verse initial temperature over the range To 2500 J( produces the following result:

{

Po

=

2.23496 X 10-¹ P1

=

1.65088 P2 = 1976.16

{

f3 = 0.65088

Ea = 3924.65 cal/mol

The deviation is 0.0243. According to

jj

and

E

_{a ,} the dominant reactions include

(_ 3800 cal/mOl) kJ=(1.59X1012)To.oe RuT

950 to

and

which corresponds to reactions 2, 17 and the reverse step of reaction 19

t

in table 3.6.3a.

The first reaction is in agreement with Mikolatis[loJ's branched chain cycle of reactions 2, 3 and 4 in the fast-rise stage. The importance of the last two reactions is not exactly certain.

The low temperature data again do not fit into the above result. The presence of a transition as in the start-up period may best explain the departure. However, the possible role of H0 2 and H2 O2 cannot be overlooked either. In the lowest temperature case, the change in mole fraction for all radicals is approximately one percent of that at To = 1500 K except for the above two. Some indication is given in figure 4.4.6. For example, LlX for H2 O 2 is -9 for the lowest temperature case, implying a depletion. The previous analysis has shown that an excessive amount of H20 2 is created during the start-up stage at low tem- perature (c.f. section 4.4.2). This means that a large amount of H202 is rapidly destroyed during fast-rise which may in part contribute to the departure in behavior.