While raising turbine inlet temperature improves the efficiency of the gas-turbine cycle, the increasing turbine-cooling losses become a limiting factor. Detailed prediction of those losses is a complex process, thought to be possible only for specific designs and operating conditions. A general, albeit approximate, model is presented to quantify those cooling losses for different types of cooling technologies. It is based upon representing the turbine as an expansion path with continuous, rather than discrete, work extraction. This enables closed-form solutions to be found for the states along the expansion path as well as turbine work output. The formulation shows the key factor in determining the cooling losses is the parameter scaling the ratio of heat to work fluxes loading the machine surfaces. Solutions are given for three cases: internal air-cooling, transpiration air cooling, and internal liquid cooling. The first and second cases represent lower and upper bounds respectively for the performance of film-cooled machines. Irreversibilities arising from flow-path friction, heat transfer, cooling air throttling, and mixing of coolant and mainstream are quantified and compared. Sample calculations for the performance of open and combined cycles with cooled turbines are presented. The dependence and sensitivity of the results to the various loss mechanisms and assumptions is shown. Results in this paper pertain to Brayton-cycle gas turbines with the three types of cooling mentioned. Reheat gas turbines are more sensitive to cooling losses due to the larger number of high-temperature stages. Those are considered in Part 3.