A multidimensional method has been devised to solve the conjugate conduction-convection heat tmnsfer process at the sudace of a moving valve of finite thickness within the flow field of an opemh'ng four-stmke internal combustion (IC) engine. Heat exchange processes between the valve and the gases aajacent to these boundaries were also computed during the portions of the engine cycle when the valve was closed. Boundaries of the solution scheme were extended fired distances into the piston and cylinder liner. The valve was simuhted as having a small but measumble thickness for the purpose of heat transfer calculations and as being immeasumbly thin for the purpose of other flow-field calculnrions. The effects of fluid enrminment caused by valve motion were also considered and modeled. The implicit finite-aifference solution of the governing equations for the primiiive variables in the flow field was conducted i n three regions: one jixed in space and time, one using a stretching and compressing compulotional mesh, and one that moved with lime withoul stretching or compressing. Valve, cylinder liner, and piston head energy equation solutions were obtained i n a fourih region (nonflow-field region), which was matched to the three adjacentflow-field regions. This work reporls use of the model to simulate a portion of an exhaust stroke for an axisymmerrie fourstmke engine piston. Forfinal nms, a 20 x 26 mesh was used lo solve the conjugate heat transfer problem i n thefluidjlow field, valve, piston top, and cylinder liner. A 17 x 20 mesh was used for otherflow-field calculnrions.