The heat exchanger is important in practical thermal processes, especially those of (i) the molten-salt storage schemes, (ii) compressed air energy storage schemes, and (iii) other load-shifting thermal storage presumed to undergird a Smart Grid. Such devices, although central to the utilization of energy from sustainable (but intermittent) renewable sources, will be unfamiliar to many scientists, who nevertheless need a working knowledge of them. This tutorial paper provides a largely self-contained conceptual introduction for such persons. It begins by modelling a novel quantized exchanger , impractical as a device, but useful for comprehending the underlying thermophysics. It then reviews the one-dimensional steady-state idealization which demonstrates that effectiveness of heat transfer increases monotonically with (device length)/(device throughput). Next it presents a two-dimensional steady-state idealization for plug flow and from it derives a novel formula for effectiveness of transfer; this formula is then shown to agree well with a finite-difference time-domain (FDTD) solution of the two-dimensional idealization under Hagen-Poiseuille flow. These results are consistent with a conclusion that effectiveness of heat exchange can approach unity, but may involve unwelcome tradeoffs amongst device cost, size, and throughput.