Heat transfer in a fluid-to-fluid screen heat exchanger is analyzed from first principles. The screens are treated as an ensemble of pin fins and an empirical heat transfer coefficient accounts for convection heat transfer at the fin surface. Pressure drop and simultaneous axial conduction in the screen matrix and the wall separating the fluid streams are modeled. Expressions are obtained that relate dimensionless length ratios to exchanger effectiveness and pressure drop. The “mesh ratio,” defined as the ratio of fin diameter (d) to spacing (s), prevails throughout the results. The key findings are: (1) the existence of an optimal ratio of fin length (a) to fin diameter that maximizes thermal performance (arising from the competition between the fin-length dependent heat transfer coefficient and fin surface area), (2) increasing a/d greater than optimal increases exchanger length and reduces pressure drop; for a/d less than optimal heat transfer is depressed and pressure drop increased, and (3) the pressure drop is linear with overall Ntu and varies as d−2, (1 + d/s)6, and approximately the square of the mass flow rate per width of exchanger. An exact solution for axial conduction is presented that is valid in the limit of large Ntu and equal fluid capacity rates. Axial conduction is seen to decrease with increasing Ntu and mass flow rates and reduced fin a/d ratio. Predictions from the model are validated by comparing with published effectiveness and pressure-drop data.