Criticality in a nearly circular cylinder

Abstract

The problem of determining the critical conditions in a non-circular cylinder of infinite length is examined, in which the reaction is of zero order and where the Frank-Kamenetskii approximation to the reaction rate has been made. The generators are assumed parallel and the boundary of a cross section is given by r = 1 + ε cos θ ( ε ≪ 1), where ( r , θ ) are plane polar coordinates. The surface conditions assumed are (i) uniform temperature and (ii) newtonian cooling. By suitable expansions of the interior temperature and the Frank-Kamenetskii parameter δ , it is shown that the critical value of δ for (ii) is δ crit ( ε, B ) = 8 σ c /(1 + σ c ) 2 exp[ – (4 σ c )/ B (1 + σ c )] x [1 - ε 2 { σ 2 c /1 + σ c ) + (1/ B ) σ c (1 + 3 σ c )/(1 + σ c ) 2 } + O ( ε 4 )], Where σ c = -2/ B + (1 + 4/ B 2 ) ½ and where B is a Biot number propor­tional to the surface-heat transfer coefficient. For (i), the simple result δ crit ( ε , ∞) = 2 - ε 2 + O ( ε 4 ), is obtained.