This paper discusses a mathematical model for airflow through a cross-flow plate heat exchanger. The exhaust air is used to heat the supply air. Three kinds of plates are considered: made of aluminium, copper, and steel. The purpose of this research was to verify which material used to build the plate heat exchangers uses the exhaust air heat more efficiently. The method of the Trefftz function was used to determine approximate solutions to the analysed problem. The results obtained for 1.2 mm-thick plates and for external winter, summer, and spring–autumn temperatures are discussed. The results indicate that if the efficiency and price of the metals are considered, then steel is the best material for the plate heat exchanger. Thanks to the use of thin steel plates and the reduction in air exchange time to a few minutes, a cheap and efficient cross-flow heat exchanger can be obtained.