Laminar dispersion in capillaries: Part I. Mathematical analysis

Abstract

Abstract Since 1953, when G. I. Taylor first considered the problem, numerous studies of the miscible displacement of fluids in capillaries have produced several approximate mathematical solutions which are purported to be valid under different conditions. Their form and ranges of applicability have been in conflict to some extent, since no exact solution is available to check these expressions. This study has resulted in exact numerical solutions to this problem with both axial and radial molecular diffusion accounted for. The range of parameters investigated is wide enough for comparison with all known analytical and empirical results and covers τ from 0.01 to 30 and N Pe from 1 to 23,000. It is shown that for sufficiently large values of τ the Taylor‐Aris theory is valid and thus results for all τ and N Pe of any practical interest are now known. Axial molecular diffusion is significant at lower values of the Peclet number but the magnitude of N Pe at which this occurs depends on the value of τ. In general, axial molecular diffusion is important for Peclet numbers less than about 100. Present results show that there is no justification for Bailey and Gogarty's empirical modification which yields an exponent of 0.541 rather than 0.50 for τ in Equations (35) and (36). Also, for the system studied here, no justification was found for the conjecture of Bournia et al. that Aris' low N Pe modification may not account for axial diffusion properly. Simple expressions given by Equations (48) and (50) were developed empirically and they give with good accuracy the average concentration distribution over wider ranges of N Pe and τ than previously reported expressions.