This paper investigates the size minimization of a high-temperature shell-and-tube heat exchanger for an externally fired Brayton cycle. On the cold side, the working fluid is an air-water mixture, while on the hot side, the exchanger is heated by flue gas with a high excess-air ratio and elevated water vapor content. Since the high-temperature heat exchanger is one of the most costly components of the system, its downsizing may substantially improve both economic viability and operating performance. The proposed concept involves the addition of water to the compressed air stream. This increases heat absorption in the evaporation region and enhances the effective specific heat of the air-steam mixture, thereby reducing the required heat transfer surface area and, potentially, the overall exchanger size. To carry out parametric calculations and design optimization, an in-house computational code was developed for a counter-current shell-and-tube heat exchanger. The model is based on a one-dimensional formulation that combines steady-state mass and momentum conservation equations for both working fluids with transient energy equations for the fluids and the heat transfer wall. This mixed steady-state/transient approach enables iterative determination of the final steady-state solution at a substantially lower computational cost than fully transient multidimensional models. The governing equations are discretized using the upwind method, ensuring numerical stability and robust prediction of temperature and pressure distributions along the exchanger. The thermophysical properties of air and water are evaluated using the REFPROP package. The results indicate that water addition modifies the thermodynamic properties of the working mixture, enhances heat transfer, and affects exchanger geometry, offering a promising approach to reducing the size and cost of high-temperature heat exchangers in externally fired Brayton cycle systems.