Transient freezing in laminar tube-flow




Martinez, Eugene Paul

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A new model is presented for treating transient freezing in laminar tube-flow. The equation of liquid motion is constrained by requiring its head-loss versus flow-rate relationship to match that of a linear pump characteristic curve. This feature enables the model to describe, in a practical manner, freezing conditions which may lead to occlusion of the tube with solid-phase material. Density differences between phases are neglected and paraboloidal velocity profiles are utilized in conjunction with an integral equation of motion. The energy equations omit only axial conduction and viscous dissipation. The resulting boundary-value problems are coupled through a nonlinear energy-balance condition at the moving phase-change interface. The location of this moving interface is the primary dependent variable to be determined by the solution. Immobilization of the moving boundary is achieved by applying distinct transformations in the solid region and the liquid region. Although the transformed description of the problem remains nonlinear, it is very well posed for solution by numerical-analysis methods. The energy equations are decomposed by the method of lines; and the resulting sets of ordinary differential equations, which describe temperature along lines of constant axial position and time, are linear in their transformed radial variables. They are solved by superposition of particular solutions in an iterative process at fixed time until their solutions are compatible with those of the interface-energy balance and the equation of motion. Solutions presented demonstrate the sensitivity of tube occlusion to both temperature conditions and liquid pumping capability. Steady state results are compared with the results found in existing literature. Agreement is excellent with error being less than one percent over a major portion of the tube length. Near the entrance the maximum error is within fifteen percent. Error in the transient results is estimated to be of the same order after effects of the starting discontinuity subside.