No abstract

Reported experimental and computational results confirm that both the flow features and heat-transfer rates inside a condenser depend on the specification of inlet, wall, and exit conditions. The results show that the commonly occurring condensing flows’ special sensitivity to changes in exit conditions (i.e., changes in exit pressure) arises from the ease with which these changes alter the vapor flow field in the interior. When, at a fixed steady mass flow rate, the exit pressure is changed from one steady value to another, the changes required of the interior vapor flow toward achieving a new steady duct flow are such that they do not demand a removal of the new exit pressure imposition back to the original steady value—as is the case for incompressible single phase duct flows with an original and “required” exit pressure. Instead, new steady flows may be achieved through appropriate changes in the vapor/liquid interfacial configurations and associated changes in interfacial mass, heat-transfer rates (both local and overall), and other flow variables. This special feature of these flows has been investigated here for the commonly occurring large heat sink situations, for which the condensing surface temperature (not heat flux) remains approximately the same for any given set of inlet conditions while the exit-condition changes. In this paper’s context of flows of a pure vapor that experience film condensation on the inside walls of a vertical tube, the reported results provide an important quantitative and qualitative understanding and support an exit-condition-based categorization of the flows. Experimental results and selected relevant computational results that are presented here reinforce the fact that there exist multiple steady solutions (with different heat-transfer rates) for multiple steady prescriptions of the exit condition—even though the other boundary conditions do not change. However, for some situations that do not fix any specific value for the exit condition (say, exit pressure) but allow the flow the freedom to choose any exit pressure value within a certain range, experiments confirm the computational results that, given enough time, there typically exists, under normal gravity conditions, a self-selected “natural” steady flow with a natural exit condition. This happens if the vapor flow is seeking (or is attracted to) a specific exit condition and the conditions downstream of the condenser allow the vapor flow a range of exit conditions that includes the specific natural exit condition of choice. However, for some unspecified exit-condition cases involving partial condensation, even if computations predict that a natural exit-condition choice exists, the experimental arrangement employed here does not allow the flow to approach its steady natural exit-condition value. Instead, it only allows oscillatory exit conditions leading to an oscillatory flow. For the reported experiments, these oscillatory pressures are induced and imposed by the instabilities in the system components downstream of the condenser.

Accurate steady and unsteady numerical solutions of the full 2-D governing equations -that model the forced film condensation flow of saturated vapor over a semi-infinite horizontal plate (the problem of Cess [1] and Koh [2]) -are obtained over a range of flow parameters. The results presented here are used to better understand the limitations of the well-known similarity solutions given by Koh [2]. It is found that steady/quasi-steady film wise solution exists only if the inlet speed is above a certain threshold value. Above this threshold speed, steady/quasi-steady film condensation solutions exist and their film thickness variations are approximately the same as the similarity solution given by Koh [2]. However these steady solutions differ from the Koh solution [2]regarding pressure variations and associated effects in the leading part of the plate. Besides results based on the solutions of the full steady governing equations, this paper also presents unsteady solutions that characterize the steady solutions' attainability, stability (response to initial disturbances), and their response to ever-present minuscule noise on the condensing surface. For this shear driven flow, the paper finds that if the uniform vapor speed is above a threshold value, an unsteady solution that begins with any reasonable initial guess, is attracted in time, to a steady solution. This long time limiting solution is the same -within computational errors -as the solution of the steady problem. The reported unsteady solutions that yield the steady solution in the long time limit also yield "attraction rates" for non-linear stability analysis of the steady solutions. The attraction rates are found to diminish gradually with increasing distance from the leading edge and with decreasing inlet vapor speed. These steady solutions are generally found to be stable to initial disturbances on the interface as well as in any flow variable in the interior of the flow domain.The results for low vapor speeds below the threshold value indicate that the unsteady solutions exhibit nonexistence of any steady limit of film wise flow in the aft portion of the solution. Even when a steady solution exists, the flow attainability is also shown to be difficult (because of waviness and other sensitivities) at large downstream distances.

Accurate steady and unsteady numerical solutions of the full 2-D governing equations – that model the film condensation of saturated vapor flowing over a horizontal plate (the problem of Cess [1] and Koh [2]) – are obtained and new results on the solutions’ unsteady response to disturbances are presented. The computations reveal important features of this classical condensing flow problem. The results highlight the scope and limitations of the well-known similarity solution given by Koh [2]. For the steady problem formulation, the paper discusses the similarities and differences between the solution obtained by solving the full 2-D governing equations and the one obtained semi-analytically by the similarity solution approach of Koh [2]. It is shown that the pressure variations in the vapor domain near the leading edge, though small, are important in deciding condensation dynamics (steady and unsteady) and cannot, in general, be neglected, as is the case with the similarity solution. For this shear driven flow, by considering the unsteady solutions, the paper finds that any initial guess leads to an unsteady solution which is attracted to a long-term steady solution (which is same as the solution as the steady problem). However, the attraction rates gradually diminish with increasing distance from the leading edge and decreasing inlet speed. The steady solutions for this external flow problem are generally found to be stable to initial disturbances at the interface or in the interior of the flow domain. However, since these flows can only be physically realized on suitable finite length portions of the plate, the issue of their stability and sensitivity to exit pressure disturbances and ever-present noise (through exit pressure or bottom plate) is also considered. For example, for the finite domain realization of this problem, it is found that the flows are stable to small initial disturbances to the nearly uniform value of exit pressure. These finite domain realizations of the flow are unique in the sense that they allow different non-uniform steady pressure prescriptions leading to different steady solutions – particularly near the exit zone. As a result, near the exit of a long plate, large unsteadiness is expected due to sensitivity to small exit pressure noise/fluctuations. The exit pressure noise for finite domain realization of these flows is expected because of practical difficulties in maintaining constant uniform pressures at downstream locations of the top and exit boundaries. It is shown that the transverse component of gravity does not affect the solution or its dynamic response except for the expected changes in the nature of hydrostatic pressure variations.

This paper presents computational simulations for internal condensing flows over a range of tube/channel geometries — ranging from one micro-meter to several millimeters in hydraulic diameters. Over the mm-scale, three sets of condensing flow results are presented that are obtained from: (i) full computational fluid dynamics (CFD) based steady simulations, (ii) quasi-1D steady simulations that employ solutions of singular non-linear ordinary differential equations, and (iii) experiments involving partially and fully condensing gravity driven flows of FC-72 vapor. These results are shown to be self-consistent and in agreement with one another. The paper demonstrates the existence of a unique solution for the strictly steady equations for gravity and shear driven flows. This paper also develops useful correlations for shear driven and gravity driven annular stratified internal condensing flows (covering some refrigerants and common operating conditions of interest). A useful map that marks various transitions between gravity and shear dominated annular stratified flows is also presented. For the micro-meter scale condensers, computations indentify a critical diameter condition (in non-dimensional terms), below which the flows are insensitive to the orientation of the gravity vector as the condensate is always shear driven. Large pressure drop, importance of surface tension, and vapor compressibility for μm-scale flows are also discussed. With the help of comparisons with 0g flows, the paper also discusses effects of transverse gravity on the solutions for horizontal channel flows.