where Qd is the gas flow rate in the downcomer, QL is the liquid circulation flow rate, Pi is pressure at point i of the reactor (1 is top of the riser, 2 is top of the downcomer, 3 is bottom of the downcomer, 4 is bottom of the riser), Cd is the hydraulic resistance coefficient, Ad is the downcomer cross-sectional area, UL is the linear liquid velocity, g is the gravitational acceleration, qL is the liquid density, and ud is the downcomer gas holdup.

The calculation of Qd thus requires knowledge of the liquid flow rate, the pressures, and the geometry of the reactor. This equation represents the maximum recycling of gas in the downcomer, which will take place only if all the energy dissipated in the downcomer is invested in gas compression.

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