Heat conduction from a cylindrical source with increasing radius
Introduction. The theory of heat flow due to conduction from a moving heat source is of interest in a number of applications; for example, welding [8]**, heat exchangers [2, 3] and progressive freezing of a liquid [2, 3]. In this paper we consider heat conduction in an infinite homogeneous medium from the surface of a cylinder of finite length whose radius is increasing with time. This problem arises in connection with secondary oil recovery by an underground combustion process, e.g., [4; 7], The restriction of the cylindrical source to a finite length corresponds to considering an oil reservoir of finite thickness and including vertical heat losses to media bounding the reservoir. The differential equation describing this problem is written and the Greens function method is applied to obtain a solution in the form of an integral. A limiting value of this integral is then obtained for the case of the source moving at a constant velocity with no vertical losses. The problem is to evaluate a limit of the form lim(_,» j'0 f(t, r)dr when jit, t) has an asymptotic representation, for {t — t)/t sufficiently large, which can be integrated explicitly. For the integrand considered in Sec. 2, it is shown that the integral can be divided into two parts, namely