Numerical Analysis of a New Mixed Formulation for Eigenvalue Convection-Diffusion Problems

A mixed formulation is proposed and analyzed mathematically for coupled convection-diffusion in heterogeneous medias. Transfer in solid parts driven by pure diffusion is coupled with convection-diffusion transfer in fluid parts. This study is carried out for translation-invariant geometries (general infinite cylinders) and unidirectional flows. This formulation brings to the fore a new convection-diffusion operator, the properties of which are mathematically studied: its symmetry is first shown using a suitable scalar product. It is proved to be self-adjoint with compact resolvent on a simple Hilbert space. Its spectrum is characterized as being composed of a double set of eigenvalues: one converging towards −∞ and the other towards +∞, thus resulting in a nonsectorial operator. The decomposition of the convection-diffusion problem into a generalized eigenvalue problem permits the reduction of the original three-dimensional problem into a two-dimensional one. Despite the operator being nonsectorial, a complete solution on the infinite cylinder, associated to a step change of the wall temperature at the origin, is exhibited with the help of the operator’s two sets of eigenvalues/eigenfunctions. On the computational point of view, a mixed variational formulation is naturally associated to the eigenvalue problem. Numerical illustrations are provided for axisymmetrical situations, the convergence of which is found to be consistent with the numerical discretization

Effect of hydrocarbon adsorption on the wettability of rare earth oxide ceramics

Vapor condensation is routinely used as an effective means of transferring heat, with dropwise condensation exhibiting a 5 − 7x heat transfer improvement compared to filmwise condensation. However, state-of-the-art techniques to promote dropwise condensation rely on functional hydrophobic coatings, which are often not robust and therefore undesirable for industrial implementation. Natural surface contamination due to hydrocarbon adsorption, particularly on noble metals, has been explored as an alternative approach to realize stable dropwise condensing surfaces. While noble metals are prohibitively expensive, the recent discovery of robust rare earth oxide (REO) hydrophobicity has generated interest for dropwise condensation applications due to material costs approaching 1% of gold; however, the underlying mechanism of REO hydrophobicity remains under debate. In this work, we show through careful experiments and modeling that REO hydrophobicity occurs due to the same hydrocarbon adsorption mechanism seen previously on noble metals. To investigate adsorption dynamics, we studied holmia and ceria REOs, along with control samples of gold and silica, via X-Ray photoelectron spectroscopy (XPS) and dynamic time-resolved contact angle measurements. The contact angle and surface carbon percent started at ≈0 on in-situ argon-plasma-cleaned samples and increased asymptotically over time after exposure to laboratory air, with the rare earth oxides displaying hydrophobic (>90°) advancing contact angle behavior at long times (>4 days). The results indicate that REOs are in fact hydrophilic when clean and become hydrophobic due to hydrocarbon adsorption. Furthermore, this study provides insight into how REOs can be used to promote stable dropwise condensation, which is important for the development of enhanced phase change surfaces.United States. Office of Naval ResearchUnited States. Dept. of Energy (MIT S3TEC Energy Research Frontier Center, Award No. DE- FG02-09ER46577)National Science Foundation (U.S.) (Graduate research fellowship)National Science Foundation (U.S.) (Graduate Research Fellowship Program, Grant No. 1122374)Irish Research Council for Science, Engineering, and Technology (Marie Curie Actions under FP7

Stabilising falling liquid film flows using feedback control

Falling liquid films become unstable due to inertial effects when the fluid layer is sufficiently thick or the slope sufficiently steep. This free surface flow of a single fluid layer has industrial applications including coating and heat transfer, which benefit from smooth and wavy interfaces, respectively. Here, we discuss how the dynamics of the system are altered by feedback controls based on observations of the interface height, and supplied to the system via the perpendicular injection and suction of fluid through the wall. In this study, we model the system using both Benney and weighted-residual models that account for the fluid injection through the wall. We find that feedback using injection and suction is a remarkably effective control mechanism: the controls can be used to drive the system towards arbitrary steady states and travelling waves, and the qualitative effects are independent of the details of the flow modelling. Furthermore, we show that the system can still be successfully controlled when the feedback is applied via a set of localised actuators and only a small number of system observations are available, and that this is possible using both static (where the controls are based on only the most recent set of observations) and dynamic (where the controls are based on an approximation of the system which evolves over time) control schemes. This study thus provides a solid theoretical foundation for future experimental realisations of the active feedback control of falling liquid films

Predictions of Heat Transfer and Flow Circulations in Differentially Heated Liquid Columns With Applications to Low-Pressure Evaporators

Numerical computations are presented for the temperature and velocity distributions of two differentially heated liquid columns with liquor depths of 0.1 m and 2.215 m, respectively. The temperatures in the liquid columns vary considerably with respect to position for pure conduction, free convection, and nucleate boiling cases using one-dimensional (1D) thermal resistance networks. In the thermal resistance networks the solutions are not sensitive to the type of condensing and boiling heat transfer coefficients used. However, these networks are limited and give no indication of velocity distributions occurring within the liquor. To alleviate this issue, two-dimensional (2D) axisymmetric and three-dimensional (3D) computational fluid dynamics (CFD) simulations of the test rigs have been performed. The axisymmetric conditions of the 2D simulations produce unphysical solutions; however, the full 3D simulations do not exhibit these behaviors. There is reasonable agreement for the predicted temperatures, heat fluxes, and heat transfer coefficients when comparing the boiling case of the 1D thermal resistance networks and the CFD simulations