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10THE IMPLEMENTATION OF THE CONVECTIVE MODE OF COMBUSTION FOR GRANULAR MIXTURES OF Ti + xC
B. S. Seplyarskii*", R. A. Kochetkov", T. G. Lisina", and M. I. Alymov"
aMerzhanov Institute of Structural Macrokinetics and Materials Science, Russian Academy of
Sciences, Chernogolovka, Moscow, 142432 Russia
*e-mail: [email protected]
DOI: 10.24411/9999-0014A-2019-10153
The occurrence of convection-driven coflow combustion of Ti + xC (0.5 < x < 1) granules at a nitrogen pressure drop of 1-2 atm was established in [1, 2]. By saying convection-driven combustion, we mean a mode of fast burning in a coflow of reactive gas. Hereafter, the term conduction-driven combustion will refer to a mode with burning velocity described by the theory of filtration combustion [3]. To date, the existing numerical models [4, 5] give no answer to the question which conditions are needed for the onset of transition between these modes. In this work, we attempted to fill this gap using coflow combustion of granulated Ti + xC mixtures (x = 0.5, 0.75, 1.0) as an example.
Burning velocity u is found as u = d/t, where d is mean granule size (1 mm) and t is the time required for ignition of a granule. A magnitude of t can be determined using well-known expressions for warmup of a semi-infinite body [6] at kind III boundary conditions:
T - T Tg_TL
T - T
Tg T
= 1 -^(ro)
where Tg is combustion temperature of granules, T0 is the initial temperature, Tig is the temperature of granule surface at the moment of its ignition, 9(0) = exp(ro2)(1 -erfro) is
additional error function, and ro(Tg, T0, Tig) = a*(Q)yfat /X. Here a*(Q) is the coefficient of interphase heat exchange at volume rate Q of gas flow. For Ti + C granule material thermal conductivity X is equal 1 W/(mK), and thermal diffusivity a is 10-2 cm2/s. We obtain:
Uc(Q) = d[ah *(Q)/ro(Tg,T0, Tig) X]2/a (1)
We assume that Tig = 1155 K (a ^ P transition in Ti) in case of nitrogen flow and 1933 K (Ti melting point) in case of argon flow [1, 2]. As follows from (2), in order to calculate the u(Q) function, we have to know the dependence of a* on Q.
As is known, in case of porous media, the expression for the interphase heat exchange coefficient a can be represented in the form [9]:
a(Q) = pgQcgPr-2/3(1 - s)-1¥(s)/4s (2)
where ¥(s) = 0.508 - 0.56(1 - s) for s < 0.4 and ¥(s) = 1 - 1.164(1 - s)2/3 for s > 0.4; Pr stands for Prandtl number, 0 < s < 1 indicates the extent of open porosity, pg stands for gas density, Cg is heat capacity of gas, 5 is sectional area of sample (2 cm2). It can be assumed [2] that Pr = 0.8 for nitrogen and 0.6 for Ar, s = 0.5.
The results obtained for Q = 800 l/h nitrogen flow a =262 W/( m2K). It is smaller than semi-
XV International Symposium on Self-Propagating High-Temperature Synthesis
empirical a* = 2711 W/(m2K) taken from [2] by a factor of 10.4, which can be associated with a rough surface of reactive Ti-C granules and irregular packing of reactive and inert mixtures [10, 11]. Assuming that the ratio a*/a = 10.4 holds true for other Q, the a* values for others Q were obtained simply by scaling-up on a factor of 10.4.
Given that a convection-driven combustion mode can also be realized in Ar flow, we have to use appropriate values of aAr*. For this to be done, the aAr(Q) magnitudes derived from (2) were multiplied by 10.4.
Since expression (2) contains no thermophysical parameters, the results obtained for mixtures with x = 1.0 can be expected applicable to those with x = 0.5 and 0.75. The values of Tg, T0, and Tig used in calculating ro(Tg, T0, Tig) are given in Table 1, along with resultant values of ro and u.
Table 1. Calculated values of ro and u for combustion of Ti + xC mixtures in a CvC mode.
To, K Tig, K Tg Ti + C =3300K Ti + 0.75C Tg = 3100 K Ti + 0.5C Tg = 2600 K
œ u , mm/s œ u , mm/s œ u, mm/s
N2 300 1155 0.35 9 105Q2 0.38 8 105Q2 0.51 4 105Q2
Ar 300 1933 0.90 8 10-6Q2 1.03 6 10-6Q2 1.71 2-10-6Q2
Figure 1 shows burning velocity u vs. Q (nitrogen coflow, convective-driven mode) for Tig = 1155 K and different Tg. Straight lines show burning velocities uf predicted by filtration combustion theory [3] (nitrogen coflow, conduction-driven mode) for mixtures with x = 1 (2), 0.75 (4), and 0.5 (6): data points at Q = 0 correspond the those measured in [2]. The points of intersection in Fig. 1 define a theoretical boundary at which the mode transition may take place. In case of Ar flow, the factors at Q2 (see Table 1) are one order of magnitude smaller compared to those in nitrogen flow, whereas the calculated uf values are close to those observed for combustion in nitrogen. Thus, a point of possible intersection is far above Q =1100 l/h. Therefore, realization of a convective-driven mode in Ar in conditions [2] is impossible.
150 u, mm/s 1
100
/ 3
2 701 l/h 756 l/h \ W , c
50 ■ 4
6
^ 539 l/h
e > 200 400 6< 10 800 1000 1200 Q, №
Fig. 1. Burning velocity u vs. Q (nitrogen coflow, convective-driven mode): T0 = 300 K, Tig = 1155 K, and Jg = 3300 (1), 3100 (3), and 2600 K (5). Straight lines show burning velocities uf predicted by filtration combustion theory for mixtures with x = 1 (2), 0.75 (4), and 0.5 (6): the magnitudes at Q = 0 correspond to those measured in [2].
Figure 2 presents the calculated u values as a function of z = Tg - Tig for convective-driven modes at several Q. All dependences are well approximated by exponentials whose pre-exponential factor grows with increasing Q.
Experimental verification of the above calculation was done by using the experimental setup described in [3]. We estimated an experimental boundary value of 732 l/h, while the calculated intersection point is at 779 l/h.
1000 1200 1400 WOO 1800 2000 2200 2400
Z=T -T., K
s ig
Fig. 2. Calculated u values as a function of z = Tg - Tig at Q = 420 (1), 800 (2), and 1100 l/h.
Key factors responsible for the onset of transition from conduction-driven to convection-driven coflow combustion in granulated Ti + xC (0.5 < x < 1) mixtures are (a) volume rate Q of gas flow, (b) difference between the ignition and combustion temperatures for granules, and (c) burning velocity at Q = 0. The measured Q values at the points of transition from conduction-driven to convection-driven combustion well agree with theoretical predictions for Ti + 0.5C mixture.
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