Effectiveness from NTU (Counterflow)

Effectiveness is the fraction of the thermodynamically possible heat an exchanger actually moves, and for counterflow it depends on nothing but NTU and Cr. The shape of the curve is the practical lesson: at NTU = 1 with Cr = 0.5 you get ε ≈ 0.56; doubling the surface to NTU = 2 buys ε ≈ 0.75; doubling again to NTU = 4 buys only 0.88. Surface is bought at a rising price per point of effectiveness, which is why the argument in every exchanger review is about where on that curve the money stops making sense. Worked example: NTU = 1.5, Cr = 0.5 gives $x = e^{-0.75} = 0.4724$ and ε = 0.5276/0.7638 = 0.691.

Two limits deserve memorising. When Cr → 0, a boiling or condensing stream, the expression collapses to $\varepsilon = 1 - e^{-\mathrm{NTU}}$, the best any arrangement can do. When Cr = 1, the balanced exchanger, the algebra goes 0/0 and the true limit is ε = NTU/(1 + NTU); this page detects that case and uses the limit rather than returning a NaN. The trap is inverting in your head: given ε and Cr the NTU comes back cleanly, but given ε and NTU there is no closed form for Cr, because it sits in the exponent and the denominator at once. Leave Cr blank and the page finds it by bisection instead.