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Research Papers: Fundamental Issues and Canonical Flows

# Mass Transfer in a Rigid Tube With Pulsatile Flow and Constant Wall Concentration

[+] Author and Article Information
T. E. Moschandreou1

Department of Medical Biophysics, University of Western Ontario, London, ON, N6A 5C1, Canadatmoschan@uwo.ca

C. G. Ellis

Department of Medical Biophysics, University of Western Ontario, London, ON, N6A 5C1, Canadacgellis@uwo.ca

D. Goldman

Department of Medical Biophysics, University of Western Ontario, London, ON, N6A 5C1, Canadadgoldma2@uwo.ca

1

Corresponding author.

J. Fluids Eng 132(8), 081202 (Aug 18, 2010) (11 pages) doi:10.1115/1.4002213 History: Received April 07, 2010; Revised June 16, 2010; Published August 18, 2010; Online August 18, 2010

## Abstract

An approximate-analytical solution method is presented for the problem of mass transfer in a rigid tube with pulsatile flow. For the case of constant wall concentration, it is shown that the generalized integral transform (GIT) method can be used to obtain a solution in terms of a perturbation expansion, where the coefficients of each term are given by a system of coupled ordinary differential equations. Truncating the system at some large value of the parameter $N$, an approximate solution for the system is obtained for the first term in the perturbation expansion, and the GIT-based solution is verified by comparison to a numerical solution. The GIT approximate-analytical solution indicates that for small to moderate nondimensional frequencies for any distance from the inlet of the tube, there is a positive peak in the bulk concentration $C1b$ due to pulsation, thereby, producing a higher mass transfer mixing efficiency in the tube. As we further increase the frequency, the positive peak is followed by a negative peak in the time-averaged bulk concentration and then the bulk concentration $C1b$ oscillates and dampens to zero. Initially, for small frequencies the relative Sherwood number is negative indicating that the effect of pulsation tends to reduce mass transfer. There is a band of frequencies, where the relative Sherwood number is positive indicating that the effect of pulsation tends to increase mass transfer. The positive peak in bulk concentration corresponds to a matching of the phase of the pulsatile velocity and the concentration, respectively, where the unique maximum of both occur for certain time in the cycle. The oscillatory component of concentration is also determined radially in the tube where the concentration develops first near the wall of the tube, and the lobes of the concentration curves increase with increasing distance downstream until the concentration becomes fully developed. The GIT method proves to be a working approach to solve the first two perturbation terms in the governing equations involved.

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## Figures

Figure 6

Concentration β Re(CFl) for A1/A0=1 and ωt=2πk at (a) x=0.12, (b) x=0.2, and (c) x=0.31

Figure 1

Geometry of problem considered

Figure 2

Eigenfunctions ψ associated with Graetz problem

Figure 3

(a)–(c) First nine solutions Re(C¯1k), k=1,…,9, of Eq. 10 for 0≤x≤0.31 and ω=2, 0≤x≤1 and ω=5, and 0≤x≤0.1 and ω=5, respectively. (d)–(e) Numerical and analytical plots of C1k for k=1,2 and k=3,4, respectively.

Figure 4

Numerical solution of Eq. 7 compared with the analytical solution for x=0.07 and ω=2

Figure 5

Concentration β Re(CFl) for A1/A0=1 and ωt=2πk at (a) x=0.005, (b) x=0.01, and (c) x=0.07

Figure 7

Radial variation of C1(x,r): (a) Im(C1) at x=0.12. (b)–(d) Variation over an oscillation period of β Re(C1(r,x)eiωt) for A1/A0=1 at x=0.01 and ω=2, x=0.12 and ω=2, and x=0.12 and ω=5, respectively.

Figure 8

Bulk concentration C1b versus frequency ω at (a) x=0.07, (b) x=0.12, (c) x=0.24, (d) x=0.48, and (e) x=0.01

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