Elsevier

Chemical Engineering Science

Volume 65, Issue 6, 15 March 2010, Pages 2199-2212
Chemical Engineering Science

Enhancing radial temperature uniformity and boundary layer development in viscous Newtonian and non-Newtonian flow by transverse oscillations: A CFD study

https://doi.org/10.1016/j.ces.2009.12.022Get rights and content

Abstract

Radial heat transfer in laminar pipe flow is limited to slow thermal conduction which results in a wide temperature distribution over the pipe cross-section. This is undesirable in many industrial processes as it leads to an uneven distribution of fluid heat treatment. Often the fluids involved are relatively viscous and processing them under turbulent conditions is impractical and/or uneconomical. On the other hand, the use of static in-line mixers to promote radial mixing may be prohibited in hygienic processes because they are difficult to keep clean. In this paper, we use a validated Computational Fluid Dynamics (CFD) model to show that the imposition of a transverse vibration motion on a steady laminar flow generates sufficient chaotic fluid motion which leads to considerable radial mixing. This results in a large enhancement in wall heat transfer as well as a near-uniform radial temperature field accompanied by a substantial heating of the inner region of the flow. Vibration also causes the temperature profile to develop very rapidly in the axial direction reducing the thermal entrance length by a large factor, so that much shorter pipes could in principle be used to achieve a desired temperature at the outlet. The effects are quantitatively demonstrated for Newtonian and non-Newtonian pseudoplastic fluids of different viscosities, for a wide range of vibration amplitudes and frequencies. For processes where vibrational motion can be implemented the benefits can be very significant.

Introduction

Laminar flow conditions of Newtonian and non-Newtonian fluids with heat transfer prevail in a number of industrial operations such as the processing of polymer melts, pharmaceutical formulations, and foodstuffs where the fluids to be heated (or cooled) are often viscous and temperature dependent. In continuous food sterilisation, for example, laminar flow of liquid foods of relatively high viscosities leads to significant variations in velocity over the pipe cross-section, thus resulting in a considerably wide residence time distribution. The fluid near the pipe centre flows at velocities significantly higher than the mean flow velocity, while the fluid flowing near the pipe wall travels at much lower velocities. This, in turn, results in a significant radial temperature distribution which translates into a wide variation of product sterility and nutritional quality across the pipe. The optimisation of such thermal processes poses a challenging manufacturing problem. The overriding importance of safety often results in the food being exposed to a more severe process than is desirable from a quality aspect, resulting in poor sensory and nutritional attributes, especially with sensitive products (Barigou et al., 1998). To ensure product safety, the velocity distribution must be taken into account when choosing the length of the holding tube so that all of the fluid is subjected to the required temperature for a sufficient length of time. It is common practice in the food processing industry to assume laminar Newtonian flow and design the length of the holding tube based on twice the mean velocity (Lareo et al., 1997). Although this ensures sterility of the whole product, it causes overexposure of the slow-moving parts resulting in a significant deterioration in product quality. However, without confidence in the design data, processes will always be overdesigned for safety.

These issues have been demonstrated in a limited number of studies (e.g. Jung and Fryer, 1999; Liao et al., 2000). Jung and Fryer (1999) used a finite-element Computational Fluid Dynamics (CFD) model to simulate the steady-state laminar flow of Newtonian and non-Newtonian power-law food fluids in circular pipes with a uniform wall temperature. Liao et al. (2000) also used a CFD model to investigate the thermo-rheological behaviour of a starch dispersion in a continuous sterilisation process at a uniform wall temperature. These studies demonstrated that validated CFD models are useful tools for investigating such systems and confirmed the wide radial variation in temperature. Since the rheological properties of most food liquids are dependent on temperature, the radial temperature gradients in the pipe may lead to significant variations in the rheological properties of the fluid leading to a distorted velocity profile, which in turn alters the temperature distribution. A more uniform temperature profile, however, would reduce such variations in the rheological properties, thus making the flow behaviour of the fluid more predictable. In addition, to minimise the loss in product quality caused by the presence of a wide temperature distribution, it is necessary to enhance the transfer of heat to the inner parts of the fluid so that all parts of the fluid receive equal thermal treatment. Therefore, methods of increasing radial mixing must be sought in order to improve the uniformity of the temperature distribution (Jung and Fryer, 1999).

Whilst such studies have served to highlight the importance of the problem posed by a non-uniform temperature profile in viscous flow, effective technological solutions to this problem are still missing. Radial mixing can be achieved by turbulent flow conditions. However, the fluids in question are usually of such high viscosities that processing them under turbulent conditions is impractical and may not be very economical. Alternatively, radial mixing can be achieved by inserting static mixers but their intricate geometries make them difficult to clean, and where hygiene is of the essence as in food and pharmaceutical production, the risk of contamination prohibits the use of such devices.

Pulsating flow and mechanical vibration have been shown in many reported works to enhance the heat flux and Nusselt number in Newtonian pipe flows (e.g. Klaczak, 1997; Gündoğdu and Çarpinlioğlu, 1999; Lee and Chang, 2003). However, the effects on the radial temperature distribution and the development of the thermal boundary layer in viscous fluids have not been studied. In this paper, we use CFD modelling techniques to investigate these aspects in the laminar flow of Newtonian and inelastic non-Newtonian fluids when subjected to forced transverse mechanical vibration. The effects are demonstrated for a range of vibration conditions and rheological properties taking into consideration the temperature-dependence of the viscosity of the fluids involved.

Section snippets

Fluid rheology and velocity profiles

The fluids considered in this study were Newtonian or inelastic non-Newtonian pseudoplastic (or shear-thinning) of the power law type, i.e.τ=kγ˙nwhere k is the consistency index, γ˙ the shear rate, and n the flow behaviour index. The apparent viscosity function, η, is then given byη=kγ˙n1The fully developed laminar velocity profile is given by the expression (Chhabra and Richardson, 1999)u(r)=u¯(3n+1n+1)[1(rR)n+1/n]where r is the radial position, R the pipe radius, and u¯ the mean velocity of

Governing equations

The equations that form the basis of the description of flow and heat transfer for a generalised Newtonian fluid in a pipe are the three transport equations written in their general form (Bird et al., 1987), thusContinuity·U=0where U is the velocity vector; MotionρDUDt=p+[·ηγ˙]+ρgwhere t is the time and g the gravitational acceleration. The term on the left-hand side represents mass per unit volume times acceleration, while the forces on the right-hand side are, successively, the pressure

CFD simulations

CFD simulations were conducted of the steady-state and vibrational flow of Newtonian and inelastic shear-thinning power law fluids with a temperature-dependent viscosity as described by Eqs. (2), (4). The values of the activation energy Ea and the factor k0 used here are typical of a range of food fluids (Steffe, 1996). The flow and vibration parameters, in addition to the rheological and thermal properties of the fluids used are given in Table 1.

Three-dimensional simulations were set up and

Validation of CFD model

Though CFX is a generally well validated code, the computational work was validated either by comparing results with theoretical solutions or experimental data from the literature where possible. The intention here was to try and validate the CFD model as much as possible so as to maximise confidence in the numerical results. The various stages of the validation process are described below.

Results and discussion

The CFD simulations conducted showed that transverse vibration can have a substantial effect on the radial temperature distribution of the fluid. There was also a large impact on the mean heat transfer coefficient. The extent of these effects was found to depend on the vibration amplitude and frequency, fluid rheology, and pipe length.

Conclusions

A validated CFD model was used to show that a transverse mechanical vibration imposed at the wall of a pipe conveying a non-isothermal laminar flow generates a vigorous swirling fluid motion represented by a strong vorticity field and complex spiralling fluid streamlines and trajectories. For a pipe with an isothermal wall, vibration leads to large enhancement ratios in wall heat transfer coefficient as well as a near-uniform radial temperature distribution accompanied by a substantial heating

Notation

Avibration amplitude, m
Cpspecific heat capacity, J kg−1 K−1
Dpipe diameter, m
Eenhancement ratio, dimensionless
Eaactivation energy, J g−1 mol−1
fvibration frequency, Hz
GrGraetz number, dimensionless
hsteady-state heat transfer coefficient, W m−2 K−1
hvvibrational heat transfer coefficient, W m−2 K−1
kfluid consistency index, Pa sn
k0pre-exponential factor, Pa sn
Lpipe length, m
LHhydrodynamic entrance length, m
Lththermal entrance length, m
m˙mass flowrate, kg s−1
NuNusselt number, dimensionless
nflow behaviour index,

References (18)

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