We describe a mathematical model for iron particle formation and growth within a chemical reactor, which we use to study carbon nanotube (CNT) formation. This model was used to understand the effect of reactor conditions (e.g. temperature and flow rate) on the CNT diameters, which are known to be coupled to the iron particle dynamics, since the CNT size has a major effect on material properties of the final product. The CNTs were made at 1144-1265 degrees C in a vertical tubular reactor, 7.0 cm diameter x 1 m high, fed with ethanol, ferrocene and thiophene carried into the top by hydrogen gas. The ferrocene decomposed to form iron particles, which acted as a catalyst to form CNTs from the decomposed ethanol. The CNTs then agglomerated, giving a 'sock' concentric with the reactor, but of smaller diameter; the sock converged near the bottom of the reactor, emerging as a vertical thread wound onto a spindle. This paper gives analyses to predict: (i) velocity and temperature profiles for the gas; (ii) iron particle number, balancing chemical rate of ferrocene decomposition against diffusion and convection; (iii) iron particle volume fraction, governed by ferrocene decomposition, coagulation, and diffusion. The results show that at the top of the reactor, flow from the 1 cm diameter injection tube generates an annular vortex of hydrogen; however below about 10 cm from the top, the hydrogen flow is well described by a parabolic velocity profile (Poiseuille). The gas temperature is uniform (1265 C) below about 40 cm from the top. The iron particle size in the reactor is controlled by the non-dimensional groups (k) over cap (c) and (beta) over cap, which relate ferrocene dissociation and coagulation to axial gas convection through the reactor. For increasing values of these groups the iron particle size is smaller at a given distance below the injector. We then compare this model to experimental results for CNT diameter within the fibre, which show good qualitative agreement with the results for catalyst size. Two hypotheses may account for sock formation: (1) The model predicts that the iron particles grow bigger near the wall, because of their longer residence times. These big particles are unsuitable for CNT growth and moreover likely to be poisoned by coking. Near the reactor centre-line, there is insufficient shear to promote CNT agglomeration. These two effects imply an optimum radius for agglomeration, and hence sock formation. (2) Several investigators have observed that, in Poiseuille flow, spherical particles accumulate at about 0.185 x tube radius measured from the wall. If this applies to CNTs, it might explain sock formation. Furthermore, use of the groups (beta) over cap and (k) over cap (c), implies that for scale-up, the ratio of the radius to the average vertical velocity in the reactor remains constant, so the reactor output is proportional to the cube of the reactor radius. Thus, an industrial-scale reactor should be practicable. (C) 2010 Elsevier Ltd. All rights reserved.