The stability of thermal oscillations in a reactive slab: reduction to Hill’s equation

1982 ◽  
Vol 384 (1787) ◽  
pp. 455-462
Keyword(s):  
Surface Temperature ◽  
Hill’S Equation ◽  
Periodic Surface ◽  
Temperature Oscillations ◽  
Hill's Equation ◽  
Chemically Reacting ◽  
The Stability ◽  
Thermal Oscillations ◽  
Surface Temperature Variation ◽  
Time Periodic

The thermal stability of an exothermic chemically reacting slab with time-periodic surface temperature variation is examined. It is shown, on the basis of a good approximation due to Boddington, Gray and Walker, that the behaviour depends on the solutions of an ordinary differential equation of first order. The equation contains a modified amplitude, for small values of which it can be reduced to a particular form of Hill’s equation. Critical values of the Frank-Kamenetskii parameter, as a function of the amplitude ϵ and frequency ω of the surface temperature oscillations, are derived from the latter equation. For ω = 2π and 0 ≼ ϵ ≼ 2 the values are in good agreement with previously calculated ones.

Thermal explosion theory for a slab with time-periodic surface temperature variation

1980 ◽  
Vol 370 (1740) ◽  
pp. 73-88 ◽  
Keyword(s):  
Surface Temperature ◽  
Temperature Variation ◽  
Thermal Explosion ◽  
Temperature Oscillation ◽  
Conservation Equation ◽  
Half Width ◽  
Periodic Surface ◽  
Rectangular Mesh ◽  
Surface Temperature Variation ◽  
Time Periodic

The problem of the onset of thermal explosion in a slab subject to time-periodic surface temperature variation has been investigated where the slab is symmetrically heated by an exothermic zero-order chemical reaction. The main purpose of the investigation was to obtain the critical Frank-Kamenetskii parameter #c(e, w) as a function of amplitude e and frequency w/2n of the surface temperature oscillation. We have shown that with values of this parameter, such that 0 < e, w) steady temperature oscillations can be maintained within the slab, but that 8 > 8c{e, w) must lead to thermal explosion. For a period of oscillation of 24 h and suitable values for the thermal diffusivity and half-width of the slab, o) 27i, and the parameter range 0 < e ^ 4 covers ambient temperature fluctuations likely to be encountered by hazardous materials, such as in the hold of a ship in tropical seas. The problem has been examined in three different ways, (i) A comparison theorem for partial differential equations has been used to determine an analytical bound on 8,the result shows th at stable oscillations exist for all oj, provided 8^ 8C e~e (8C = 0.878), this represents a lower bound to the stability surface in (8, e, oj) space, (ii) Perturbation theory, for small amplitude, has been used to determine the critical parameter in the form oj) — 8C + e1 281(oj) + ... with £x(w) a function of frequency. Comparison with the exact numerical solution shows that this gives values which differ by less than about 0-2 % for oj^ 2k and 0 ^ e^ 1. (iii) The energy conservation equation has been solved numerically over a rectangular mesh representing the half-width of the slab and one period of steady oscillation. For given e, oj such solution could be found for 8 sufficiently small; it is shown th at the breakdown of the numerical process is associated with criticality, allowing the limiting parameter to be determined. This method has been used to obtain curves of £c(e, oj) versus e for oj = 2 4n, 8tt and the range 0 ^ ^ 4.

Stability of General Hill’s Equation With Three Independent Parameters

1972 ◽  
Vol 39 (1) ◽  
pp. 276-278 ◽  
Author(s):  
K. Hamer ◽  
M. R. Smith
Keyword(s):  
Perturbation Method ◽  
New Method ◽  
Order Of Approximation ◽  
Hill’S Equation ◽  
Stability Boundaries ◽  
First Order ◽  
Hill's Equation ◽  
The Stability ◽  
Independent Parameters

The stability of Hill’s equation with three independent parameters, two of which are small, is analyzed using a perturbation method. It is shown that, except for periodic terms of a special type, existing methods of determining stability boundaries fail. A new method, which works successfully to the first order of approximation, is described.

On the Stability of the Damped Hill’s Equation With Arbitrary, Bounded Parametric Excitation

1993 ◽  
Vol 60 (2) ◽  
pp. 366-370 ◽  
Author(s):  
C. D. Rahn ◽  
C. D. Mote
Keyword(s):  
Asymptotic Stability ◽  
Parametric Excitation ◽  
Optimal Control Theory ◽  
Damping Ratio ◽  
Viscous Damping ◽  
Conservative Estimate ◽  
Hill’S Equation ◽  
Hill's Equation ◽  
Time Optimal ◽  
The Stability

The minimum damping for asymptotic stability is predicted for Hill’s equation with any bounded parametric excitation. It is shown that the response of Hill’s equation with bounded parametric excitation is exponentially bounded. The parametric excitation maximizing the bounding exponent is identified by time optimal control theory. This maximal bounding exponent is balanced by viscous damping to ensure asymptotic stability. The minimum damping ratio is calculated as a function of the excitation bound. A closed form, more conservative estimate of the minimum damping ratio is also predicted. Thus, if the general (e.g., unknown, aperiodic, or random) parametric excitation of Hill’s equation is bounded, a simple, conservative estimate of the damping required for asymptotic stability is given in this paper.

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