Heat transfer in cylinder spaces is important to the performance of many reciprocating energy conversion machines, such as gas compressors and Stirling machines. Work over the past 10 years has shown that heat transfer driven by oscillating pressure differs from steady state heat transfer, in magnitude arid phase. In-cylinder heat transfer under this oscillating condition can be out of phase with the temperature difference. For studies with closed piston-cylinder gas springs, this heat transfer phase shift has been successfully predicted with the use of a complex Nusselt number. Since a complex,number has both a magnitude and a phase, a complex Nusselt number can describe the phase shift between temperature difference and heat transfer. Quasi - steady heat transfer models, such as Newton's Law of Cooling, do not predict this phase shift. In this project, the problem of in-cylinder heat transfer with inflow and outflow was studied. The goal was to determine what the complex heat transfer coefficients were under these conditions. Because methods which measure the heat transfer directly, such as heat flux gauges, only give local results, past work has used pressure and volume measurements to calculate surface averaged values for the heat transfer. This becomes much more difficult to do with inflow and outflow because of the difficulty in accurately determining how much mass is in the cylinder at any given time. Two approaches were used to overcome this problem. They are the main substance of the work presented here. The actual experimental pressure and volume measurements were taken by Kafka (Virginia Tech Master's Thesis, 1994). / Master of Science