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chemical engineering research and design 89 (2011) 347–351
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Chemical Engineering Research and Design
journal homepage: www.elsevier.com/locate/cherd
Short communication
Numerical analysis of residence time distribution in
microchannels
A. Vikhansky
School of Engineering and Material Science, Queen Mary, University of London, United Kingdom
abstract
This article describes a numerical approach, which allows for the analysis of the residence time distribution (RTD) in
microchannels. While the traditional methods provide the RTD at the outlet of the reactor, we consider the distribu-
tion of the tracer’s age across the entire flowfield. The equation for the tracer’s age distribution is solved by a modified
method of moments and the distribution function is calculated by a reconstruction procedure. As an example we
consider a Dean vortex-based micromixer.
© 2010 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: RTD; Dean flow; Microchannels
1. Introduction
The residence time distribution (RTD) is one of the key char-
acteristics that determines the performance of a chemical
reactor (Danckwerts, 1953; Levenspiel, 1972; Cozewith and
Squire, 2000). Although the concept of RTD has been well-
established during the last 50 years and has been recently
extended to unsteady-state flows (Rawatlal and Starzak, 2003),
RTDs remain a focus of interest in the chemical engineer-
ing community. Recent progress in microfluidic technology
has turned researchs’ attention to the RTDs in microcdevices,
usually operated under laminar regimes (Vikhansky, 2008a;
Adeosun and Lawal, 2009; Cantu-Perez et al., 2010; Vikhansky
and MacInnes, in press)
The problem is formulated as following. Consider the flow
of a viscous incompressible liquid through a microchannel,
e.g., similar to that shown in Fig. 1. The flowfield is described
by the Navier–Stokes equations, namely
∇ · u = 0,(∂
t
+ u · ∇) u =− ∇ p + ∇
2
u, (1)
where u, p, and are velocity, pressure, density and the
kinematic viscosity. Due to the non-uniform velocity, different
particles spend different time in the channel and the par-
E-mail address: a.vikhansky@qmul.ac.uk.
Received 30 November 2009; Received in revised form 13 May 2010; Accepted 18 June 2010
ticles leaving the channel at time t have been injected into
the channel at different times in the past. The flux-averaged
concentration of an admixture at the outlet C
out
(t) is related
through the RTD to the flux-averaged concentration at the
entrance C
in
(t
′
) (where t
′
≤ t) as
C
out
(t) =
∞
0
E(t, )C
in
(t − )d. (2)
This formula constitutes the mathematical definition of
E(·,·)(Levenspiel, 1972).
The spread in the residence times implies that different
portions of the fluid have different chemical composition.
Knowledge of the RTD allows for exact calculation of the yield
of a first-order reaction and provides a crude estimate for the
yield of higher-order reactions. Surprisingly, in spite of the
widely recognized importance of the RTD, very little has been
done to develop the corresponding numerical algorithms. The
most commonly used numerical methods to calculate the RTD
is based on particle tracking (Cantu-Perez et al., 2010), which
works as follows. Consider a massless particle advected by the
flowfield: when a particle enters the channel its age s is 0 and
then it increases with time according to the differential equa-
tion ds/dt = 1. The position and age of a particle is governed
0263-8762/$ – see front matter © 2010 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.cherd.2010.06.010