Abstract: Abstract – Water or steam at supercritical pressure (p > 22,1 MPa) exhibits complex variations of its properties, density, heat capacity, viscosity and heat conductivity, depending on the fluid temperature in the range 280°C < T < 500°C. As a consequence of these property variations, the heat transfer in channel or pipe flows with heated walls behaves unusual and is strongly nonlinear, depending on heating rate, enthalpy, mass flux and pipe diameter. Either, heat transfer ‘deterioration’ with associated high temperatures up to 750°C of the channel walls, or heat transfer ‘enhancement’ are observed and well documented in experiments. Various methods to predict the wall temperature of a heated channel or pipe exist, e.g. empirical correlations, a lookup table, or CFD codes, but often large errors and deviations to experiments occur due to a lack of understanding of the relevant flow physics and the absence of reliable turbulence models for these flows. Here, the analytical or semi-analytical method offers the advantage, that the influence of parameters can be identified and the importance of thermo-physical mechanisms on flow and heat transfer may be understood better. Among such mechanisms are, for example, the insulating effect of a low-conductivity near-wall layer, the heat transport of a high-heat capacity fluid by turbulent-eddy turnover, upward or downward buoyancy, re-laminarization and thermal inlet effects. In the present work, an attempt is made to extend existing analytical models for the constant-property heat transfer in channel or pipe flows to variable-property fluids and identify relevant mechanisms of heat transfer deterioration or enhancement by comparison to experiments. Most important, but not fully understood, is the effect of a local, very high maximum of the heat capacity near the pseudo-critical temperature (e.g. 382°C at 25 MPa) on the heat transfer.
The classical assumptions of quasi fully-developed flow, Prandtl’s mixing-length eddy viscosity, as well as the two layer concept of a logarithmic wall-layer and a laminar sub-layer, are extended to a fluid with variable properties. An additional turbulence effect is taken into account by modelling the properties of the turbulent flow by weighted averaging with a probability density function of estimated temperature fluctuations. For given pipe radius, wall heat flux, mass flux and bulk enthalpy the local heat-transfer coefficient and the axial wall temperature distribution can be determined from the numerical integration of a simple ordinary differential equation. Mechanisms of heat transfer deterioration or enhancement can be identified.
Results are compared to various experiments of heated pipe flows at supercritical pressure. Good agreement is obtained in some cases with different ‘types’ of heat transfer deterioration and enhancement, which are classified systematically. In some parameter regions a strong deviation from a fully developed state exists, meaning that for these experimental cases neither the present theory nor correlations or a look-up table can be applied, unless an additional ‘thermal inlet model’ is used. The theory contributes to an understanding of heat transfer of supercritical flows and provides equations for its prediction. These can also be used for fast prediction tools or the development of numerical wall functions, which are necessary for an efficient numerical simulation of flows at supercritical pressure using Computational Fluid Mechanics.
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