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Abstract A unified one-dimensional (1D), steady-state flow and heat transfer model is presented for the pipeline transport of fluids at high pressures, including the supercritical (SC) conditions. The model includes a generalized temperature equation, presented here for the first time, and accounts for all of the important effects, including the property variation, viscous dissipation, Joule-Thomson (J-T) cooling, and heat exchange with the surrounding. With appropriate approximations, this model can yield all isothermal and nonisothermal pipe flow solutions reported thus far. A generalized multizone integral method is developed which solves the two resulting algebraic equations for pressure and temperature in conjunction with a property database, such as the National Institute of Standard and Technology (NIST) reference fluid thermodynamic and transport properties (REFPROP). With appropriately selected number and size of the zones and using property values at the mean temperature and pressure within each zone, this integral method can accurately predict the complex effects of the governing parameters, such as the pipe diameter and length, inlet and exit pressures, mass flowrate, J-T cooling, and inlet and surrounding temperatures. Its accuracy for small-to-large diameter pipes has been ascertained by a comparison with the numerical solutions of the differential form of governing equations that requires a large number of small grids along the pipe and the values of mean properties within each grid. Indeed, this integral model can be used for the pipeline transport at both subcritical and supercritical pressures as long as the fluid does not encounter its anomalous states and the phase-change. 

All fluids exhibit large property-variations near the critical point in a region identified as the anomalous state. The anomaly starts in the liquid and extends well into the supercritical state, which can be identified thermodynamically using the Gibbs free energy (g). The specific heat, isobaric expansion, and isothermal compressibility parameters governing the transitions are: (cp/T), (vβ), and (vκ), rather cp, β, and κ. They are essentially the second-order derivatives of g and have two extrema (minimum, maximum); only maxima reported ever. When applied to the van der Waals fluid, these extrema exhibit closed loops on the phase-diagram to satisfy d3g = 0 and map the anomalous region. The predicted liquid-like to gas-like transitions are related to the ridges reported earlier, and the Widom delta falls between these loops. Evidently, in the anomalous region, both the liquid and the supercritical fluid need to be treated differently. Beyond the anomalous states, the supercritical fluids show monotonic, gradual changes in their properties. The analysis for argon, methane, nitrogen, carbon dioxide, and water validates the thermodynamic model, supports the stated observations, and identifies their delimiting pressures and temperatures for the anomalous states. It also demonstrates the applicability of the law of corresponding states. Notably, the critical point is a state where d3g = 0, the anomaly in the fluid's properties/behavior is maximal, and the governing parameters approach infinity. Also the following are presented: (a) the trajectory of the liquid–vapor line toward the melt-solid boundary and (b) a modified phase diagram (for water) exhibiting the anomalous region. 

Abstract Single-photon counters are single-pixel binary devices that click upon the absorption of a photon but obscure its spectral information, whereas resolving the color of detected photons has been in critical demand for frontier astronomical observation, spectroscopic imaging and wavelength division multiplexed quantum communications. Current implementations of single-photon spectrometers either consist of bulky wavelength-scanning components or have limited detection channels, preventing parallel detection of broadband single photons with high spectral resolutions. Here, we present the first broadband chip-scale single-photon spectrometer covering both visible and infrared wavebands spanning from 600 nm to 2000 nm. The spectrometer integrates an on-chip dispersive echelle grating with a single-element propagating superconducting nanowire detector of ultraslow-velocity for mapping the dispersed photons with high spatial resolutions. The demonstrated on-chip single-photon spectrometer features small device footprint, high robustness with no moving parts and meanwhile offers more than 200 equivalent wavelength detection channels with further scalability.