An eppmxime~e simple analytic theory of criticality for thermal hot spot!, in given which leads to esti_ matm of tha ui~ical explosion parameter 6,. a dimensionless rate of heat production. within I M of the values computad by the sphere. by Merahanov, Baraykin and Contkovskaya, and the slab. The raJult may ba axpmssed as follows. If an inert symmetrical hot spar of unit dimension!, and unit diff@MCy is initially a( unit dimansionless temperature above its sunoundinga, the temperature of its ctntm falls with lima, and if this drop in temperature is denoted by &a( 8.) at dimensionless (ime A lf%wia~ number). where cl&) can be evaluated by conventional fhermal conduction theory, then for a hot spot generating heat accurding to the Arrhenius law with high activation energy and having an initial dimemioalaas temperature. artiers 6 is the aCtivati0n energy, R is the universal gas constant, Ta is the initial temperature of the hoC spot, and TI is that of the surroundings. the relation between 6, and fJa is approximately 8, !!JI 0 dA A+s, = a,, nod it links 6, and 9a db'actly thmugh the ~ooiing characteristic of the equivalent inert hot spot. The effeetr of shape, initial distribution of temperature, and boundary conditions are embodied in this onawrms. Foorasphetethisresultgives A low and a h&b vahm of 6, can @ve the same value of B., but only Ihe hi&% value is relevant here. The rma1meat b cxpactad to ap@y to shspas where the position of the madmum temperature can ba aawamad and to initial tampentw ditMenc9s high enough for the reactivity of the material outdde tbe bnl spot to ba aa&ctcd. Tha uitical condition has been used for evalusting + as a function of surface Bar uamfw fw piacaa 01 pilm of mlf-haating materials placed in cooliig surroundinga. There W d&am ldewrt lo Ibe d9fWti0a of safe conditions for the industrial iHOCeSing and storage d such matmiata whaa tba llnallye material iF hoi. The form of the relation betwaen 6'. and 0, rppen NJ ba s+@lcaady &epandant on the 'YnitiaI" temperatam distribution, i.e. on the pro-cx59aahvdvadhcbof~tionofthahotspot. sequently loses heat to its surroundings, accelerat-lhephtaae%otspot'intheconhxtofthemwd ing heat generation in heem and being reactive upkl&nShcett~~u §adto~riba itself, c73ntinuing to generate heat because of its SltBU~laak@tIlaSQf~t&lWhiChhas high temperature. The criticality prol+m b to beaaq&&ly&edtoa&&tempemtlJre.
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Bnd under what conditions the hittiy high tern--