Convective stability of the horizontal reacting liquid layer in the presence of various complicating factors

The slowly reacting mode of combustion of gaseous mixtures in spherical vessels. Part 2: Buoyancy-induced motion and its effect on the explosion limits

“…While the initial work of Frank-Kamenetskii [2] considered constant density, so that heat transfer occurred only by conduction, it was soon acknowledged [3] that in gaseous reactive systems under normal gravity the density differences associated with the small changes in temperature, of the order of the F-K temperature for the slowly reacting mode of combustion, generate a convective motion having a non-negligible effect on the heat-transfer rate if the corresponding Peclet numbers are of order unity or larger. The special non-generic case of a slab configuration, with a reactive gas bounded by two isothermal infinite horizontal walls, has been considered in a number of previous theoretical analyses that address the effects of buoyancy-induced convection on thermal explosions [4][5][6][7]. As in the closely related case of the Rayleigh-Benard problem of convection produced by wall-temperature differences, a motionless quasi-steady combustion mode (similar to that described in Part 1) may exist for values of the Rayleigh (or Grashof) number below a critical value, at which a bifurcation occurs to a convective state, as described by linear stability analyses [4,5].…”

Thermal explosions in spherical vessels at large Rayleigh numbers

“…The asymptotic prediction (45) evaluated with K c ¼ 0:655 is compared in the middle plot of Fig. 1 with the critical Damköhler numbers computed numerically from integration of (5)- (7). As can be seen, the accuracy of the prediction is truly remarkable.…”

“…, indicates that for Da $ 1 the temperature increase associated with the chemical reaction is limited to small values of order Ra À1=4 , insufficient to produce a significant increase in the reaction rate from its near-wall value, and that temperature increments of order unity, needed to trigger the thermal explosion, require values of the Damköhler number of order Ra 1=4 . In this near-explosion regime, Da $ Ra 1=4 , the chemical reaction occurs mainly in the central region, while the boundary layer remains chemically frozen in the first approximation, because the transport rates there are larger than the reaction term by a factor of order Ra 1=4 , as can be seen from (7).…”

“…In this near-explosion regime, associated with small density differences of order q I =b, the contribution of the temporal density variation to mass conservation gives a higher-order correction, of order ðb RaÞ À1 , which has been neglected in (5). Also, leaving out terms of order 1=b, variations of transport coefficients and specific heat from their values evaluated for T ¼ T o and q ¼ q I have been omitted in (6) and (7). In the momentum Eq.…”

“…The problem for a slab configuration, with a reactive gas bounded by two isothermal infinite horizontal walls, has been addressed in numerous theoretical analyses [6][7][8][9][10]. As in the closely related case of the Rayleigh-Bénard problem of convection produced by wall-temperature differences, a motionless quasi-steady combustion mode may exist for values of the Rayleigh number below a critical value, at which a bifurcation occurs to a convective state, as described by linear stability analyses [6,7]. A noteworthy feature of the slab geometry is that, contrary to widespread expectations, convection does not always hinder the development of the thermal explosion [8].…”