A reduced-order model is derived for electroosmotic flow in a microchannel of nonuniform cross section using the Karhunen-Loève Galerkin (KLG) procedure. The resulting reduced-order model is shown to predict electroosmotic flows accurately with minimal consumption of computer time for a wide range of zeta potential zeta and dielectric constant epsilon. Using the reduced-order model, a practical method is devised to estimate zeta from the velocity measurements of the electroosmotic flow in the microchannel. The proposed method is found to estimate zeta with reasonable accuracy even with noisy velocity measurements.