Freezing-controlled penetration of a saturated liquid into a cold tube

The penetration of a saturated liquid (a liquid at its freezing temperature) into a tube that is initially empty and maintained at a temperature below the freezing temperature of the liquid is treated theoretically and experimentally. A convenient approximate method is introduced which involves postulating a reasonable functional form for the instantaneous shape of the frozen layer along the tube wall. Graphical velocity-time and penetration distance-time curves are presented displaying the principal effects of a single dimensionless parameter. In the limit of negligible liquid inertia, shown to be relevant to high Prandtl number materials, a closed-form expression for the liquid penetration length is obtained. The expression compares well with the experimental results.