TECHNOLOGY FOCUS - INORGANIC CHEMISTRY - Preferred Method: The diluent is provided by at least one of further portion of the separated reaction product of step (d), and at least a portion of the second-reaction-zone reaction product of step (f). The separated reaction product is substantially water-free. The water makes up at least 10 (preferably at least 20, especially at least 30) wt.% of the total reaction products. The non-petroleum feed is introduced in the first reaction zone under conditions so that substantially all the feed and hydrogen are in a continuous liquid phase within the first reaction zone. The removed reaction products of step (c) are cooled prior to separating water according to (d). The first and second reaction zones are reactors, where each reactor contains greater than 10% hydrogen gas by total volume of the reactor. In the method, no excess hydrogen gas is vented from the first and second reaction zones to facilitate controlling quantities of liquids within the reaction zones. The method further involves g) removing the reaction products produced in (f) from the second reaction zone; h) cooling the removed reaction products from (g); and (i) separating any gas phase and any liquid water phase from the cooled reaction products from (h) to form a separated reaction product that is free from any separated liquid water to provide a separated liquid reaction product. The method further involves combining at least one of further portion of the cooled reaction product of step (c), and at least a portion of the cooled reaction products of step (g) with at least one of the feeds of step (a) and step (d); and repeating steps (d) through (g) and optionally (h) for any additional reaction zones, as necessary, to provide a final reaction product. Preferred Components: The non-petroleum feed is comprised of pyrolysis oils. The pyrolysis oils are derived from at least one of cellulosic biomass materials and coal. The non-petroleum feed to be treated is comprised of oxygen (at least 20 wt.%). EXAMPLE - A pyrolysis oil derived from waste wood was hydrotreated in a system having two reaction zones, each containing a hydroprocessing catalyst. The pyrolysis oil had a specific gravity of 1.25, a sulfur content of 0.35% by weight, a nitrogen content of 0.28% by weight and an oxygen content of 35% by weight. Hydrogen gas was added to each the feeds prior to introduction into the reaction zones so that sufficient hydrogen necessary for the reaction was dissolved in the feeds introduced as a liquid phase. The reaction products were removed from each zone and cooled to 280 degrees F after each reaction zone. Water was removed from each of the cooled reaction products from each reaction zone. Approximately 40% of the water was removed after the first reaction zone and 60% was removed after the second reaction zone. All of the liquid reaction product from the first reaction zone with the water removed was used as the feed for the second reaction zone. Approximately 85% of the liquid reaction product from the second reaction zone with the water removed was used as a recycle ahead of the first reaction zone. Approximately 60% of the hydrogen was added to the first reaction zone and 40% of the hydrogen was added to the second reaction zone. In both cases the hydrogen addition was at a 10% excess above the solubility limit of the reaction zone feed mixture. The startup diluent used was a hydrocarbon similar to the water-free reaction product from the second reaction zone. After completion of the treatment, light hydrocarbons (15 wt.%), oil naptha (35 wt.%), diesel (15 wt.%), heavy ends (3 wt.%), and water (35 wt.%) were obtained.