A small variation of the circular shape of the hodograph theorem states that for every elliptical solution of the twobody problem, it is possible to find an appropriate inertial frame such that the speed of the bodies is constant. We use this result and data from the NASA JPL Horizon Web Interface to find the best fitting ellipse for the trajectory of Mercury, Venus, Earth, Mars, and Jupiter. The process requires us to find procedures to obtain the plane and ellipse that best fit a collection of points in space. We show that if we aim for the plane that minimizes the sum of the square distances from the given points to the unknown plane, we obtain three planes that appear to divide the set of points equally into octants, one of these being our desired plane of best fit. We provide a detailed proof of the hodograph theorem.
Pilot-scale hydrothermal liquefaction (HTL) of biowaste is a critical step toward commercialization of the HTL technology. Despite many HTL studies conducted with wet biomass, including food waste, few were performed with a pilot-scale continuous plug-flow reactor (PFR), with the biocrude yield and quality analysis based on dewatering (ASTM D2892 Annex X1). This paper describes the development and performance evaluation of a mobile pilot-scale HTL continuous PFR, with a processing capacity of 60 L/ h of wet feedstock and 6 L/h of biocrude production. The reactor system was designed for reaction conditions of up to 325 °C and 17.25 MPa. The reactor has a volume of 28.88 L with an additional counterflow heat exchanger volume of 18.07 L. Two types of food wastes, from a food processing plant and grocery store, were processed at 280 °C for 30 min, producing biocrude oil yields of 52.19 and 47.06 wt %, energy recoveries of 68.17 and 70.77%, and carbon recoveries of 66.91 and 64.78%, respectively. Due to its high feedstock capacity and reaction volume, large amounts of biocrude oil and post-HTL wastewater (PHW) were obtained from this pilot-scale reactor to allow downstream research on upgrading biocrude oil for transportation fuel as well as PHW treatment and nutrient recovery.
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