We consider the problem of predicting the rate of mass transfer to a fluid flowing parallel to the axes of randomly placed aligned tubes, a model of hollow-fibre contactors. The analysis is carried out for the limiting cases of short contactors, for which the concentration boundary layers remain thin compared with the radius of the tubes, and for the fully developed case corresponding to very long tubes. Numerical simulations for random arrays are carried out for N randomly placed tubes within a unit cell of a periodic array. It is shown that the mass transfer coefficient for the fully developed case is vanishingly small in the limit N to infinity. This suggests that the mass transfer coefficient for a random array of tubes of radius a enclosed in a shell of radius S will vanish logarithmically as the ratio S/a is increased. This behaviour arises due to the logarithmically divergent nature of concentration disturbances caused by each tube in the plane normal to its axis. A theory is developed for determining conditionally averaged velocity and concentration fields and its predictions are shown to compare very well with the results of rigorous numerical computations. The predictions of the theory are also shown to compare well with the measurements of the mass transfer coefficients in hollow-fibre contactors reported in the literature.