In this paper heat transfer between gas and relatively fine particles, d(p) < 350 mu m, say, is firstly considered. In this system, particles are injected from the top into a gas stream flowing in a horizontal channel, so that acceleration of the particles to the horizontal velocity of the gas is not required, i.e. the inlet velocity can be resolved into the stream velocity in the horizontal direction and in the vertical direction at or above the terminal velocity. The particles are injected in such a way as to ensure the porosity is .99 or higher. Relatively high Numbers of Transfer Units arise. The calculation method applied can cope with parameter changes during the calculation, e.g. due to temperature changes. Values of N-O1 could be enhanced by multi-stage operation where particles from one curtain are lifted to become the feed for the up-stream curtain so that the system is overall counter-current. Heat transfer coefficients increase as particle diameter reduces, and surface area rises (for the same mass of particles). Particles employed are chosen at sizes where the particle resistance to heat transfer is very small. Secondly, two limiting examples of mass transfer and chemical reaction, without heat transfer, are briefly examined: chemical reaction between gas and catalyst particles, and gas absorption of carbon dioxide with chemical reaction. All these cases fall under the description of cross-flow. (c) 2012 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.