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In-sItu coal liquefaction processing is a new technology. Considerable research and development efforts are needed in this field before field testing can be undertaken. The important variables in the process are coal and the properties of coal, solvent, flow regime, surface treatment, geology, heat transfer, and permeability enhancement. In the present work, the heat transfer problem was studied extensively to estimate the heat loss through thermal conduction. This study treated a general heat transfer problem of a solvent flowing in a cylindrical channel embedded in a medium with a moving boundary. Three nonlinear conjugated differential equations describe the process. The boundary velocity and coefficients of the system equations may be temperature-dependent. New numerical methods were devised to solve these three differential equations simultaneously. The numerical methods consist of a combination of two stable methods, namely the ADI iteration method for the parabolic equation and the alternating variable method for the hyperbolic equations. These two methods give very fast concergenee and are well suited to nonlinear equations. Usually, two iterations give satisfactory results. Each method can be applied independently.