Integral Transform Solution for Diffusion Equations Coupled Through Boundary Conditions F. SCOFANO NETO and R. O. C. GUEDES Department of Mechanical and Materials Engineering Instituto Militar de Engenharia Praça General Tibúrcio 80, Praia Vermelha 22290-270 - Rio de Janeiro - RJ BRAZIL Abstract: This contribution describes a hybrid analytical-numerical procedure for solving a class of heat and mass diffusion equations for two potentials which are coupled through boundary conditions. The ideas of the Generalized Integral Transform technique are employed and therefore an eigenfunction expansion based on a simple Sturm-Liouville problem is used. In order to discuss the merits of the procedure outlined in this work, this solution scheme is applied to a physical problem associated to the thermal development in parallel plate heat exchanger. Numerical results for the heat transfer quantities are presented and the overall performance of the proposed methodology is discussed. Key-Words: Eigenfunction Expansion, Diffusion Equations, Non-Classical Boundary Conditions. 1 Introduction Integral Transform methods have long been recognized as a powerful tool for the solution of linear diffusive equations and a review of such procedures can be found in Mikhailov and Ozisik [1] where several formulations of diffusion problems are presented together with their solutions. However, an inspection of this contribution reveals that, in most cases, a non- classical eigenvalue problem has to be addressed for which the reliable evaluation of the eigenquantities are only possible for particular cases. More recently, extensions of these classical ideas have been applied to different problems in linear and non-linear diffusion-convection problems. The so called Generalized Integral Transform Technique [2,3,4] is based on an eigenfunction expansion but, as it shall be demonstrated in the following section, the eigenproblem is chosen in such a way that it contains some relevant information about the physical problem and yet it is a simple one to solve. Therefore, in the first part of this paper a general linear diffusion problem for two potentials which are coupled along a boundary surface is formulated and its solution through the Generalized Integral Transform Technique is presented. As mentioned in Mikhailov and Ozisik [1], this class of problems has many applications including, for example, the extraction of sugar from sugar beet turning. In another section, a particular case of this general formulation, which is related to the thermal development of double plate heat exchanger, will be considered where the solution procedure outlined in this contribution is discussed. 2 Analysis This section starts by considering a sufficiently general diffusion problem for the potentials T and φ which are coupled through a boundary condition. For simplicity in the analysis, a two independent variable problem is chosen. Thus, the problem to be addressed is: ) , ( ) ( ) , ( ) ( ) , ( ) ( t x T x d x t x T x K x t t x T x w -         ∂ ∂ ∂ ∂ = ∂ ∂ 0 , in ), , ( 1 0 > ≤ ≤ + t x x x t x P (1) 1 0 in ), ( ) 0 , ( x x x x f x T ≤ ≤ = (2) 0 ) , ( 0 = ∂ ∂ x t x T , in t>0 (3) ) ( ) , ( ) ( ) , ( 1 1 1 t x t x T x K t x T φ β α = ∂ ∂ + , t>0 (4) ) ( ) , ( ) ( ) ( 1 1 t Q x t x T x K t d t d = ∂ ∂ + γ φ (5) 0 ) ( φ φ = o (6) Equation (1) is readily identified as a general linear diffusion, subjected to an initial condition, equation(2). This formulation assumes that the problem is symmetric at boundary x 0 while a third type boundary condition is prescribed at x 1 . As it can be inferred from equation (5), the potential φ(t)