The standard linearized model of a controlled heat exchanger in the form of a system of two first-order PDEs is interpreted as an abstract semigroup system on an appropriately selected Hilbert state space. We analyse some its basis properties: exponential stability, admissibility of an observation functional and regularity of the system impulse response. Next we give a proof of a version of the circle criterion which bases on the use of the Leray-Schauder fixed point theorem jointly with a uniqueness result. The circle criterion is then implemented to a feedback control system of the heat exchanger with a proportional controller and a non-linear valve in the feedback loop. All considerations are illustrated by a numerical example of the heat exchanger installed in central heating system at the AGH University of Science and Technology, Cracow, Poland, for which we estimate feasible settlements of the proportional controller.