Abstract: Abstract: A two-dimensional axisymmetric conjugate heat-transfer model was established for a 3 m counterflow cooling unit serving a liquid-ring vacuum pump. Nine cases combined cooling-water flow rates of 0.30-0.70 kg/s with inlet temperatures of 20-30 °C. The heat-duty difference between the medium and fine meshes was 0.368%. Under the nominal condition, the hot-side outlet temperature, heat duty, and cold-side pressure drop were 40.037 °C, 4.151 kW, and 599.9 Pa, respectively. At 25 °C, increasing the flow rate to 0.70 kg/s lowered the outlet temperature but raised the pressure drop to 1056.6 Pa, while the marginal heat-transfer gain declined. Considering heat-transfer benefit and hydraulic cost, 0.50 kg/s and 20-25 °C constitute a priority checking range within the model scope.