Enhancing radial temperature uniformity and boundary layer development in viscous Newtonian and non-Newtonian flow by transverse oscillations: A CFD study
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.where k is the consistency index, the shear rate, and n the flow behaviour index. The apparent viscosity function, η, is then given byThe fully developed laminar velocity profile is given by the expression (Chhabra and Richardson, 1999)where r is the radial position, R the pipe radius, and 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), thuswhere U is the velocity vector; where 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
A vibration amplitude, m Cp specific heat capacity, J kg−1 K−1 D pipe diameter, m E enhancement ratio, dimensionless Ea activation energy, J g−1 mol−1 f vibration frequency, Hz Gr Graetz number, dimensionless h steady-state heat transfer coefficient, W m−2 K−1 hv vibrational heat transfer coefficient, W m−2 K−1 k fluid consistency index, Pa sn k0 pre-exponential factor, Pa sn L pipe length, m LH hydrodynamic entrance length, m Lth thermal entrance length, m mass flowrate, kg s−1 Nu Nusselt number, dimensionless n flow behaviour index,
References (18)
- et al.
Heat transfer in two-phase solid-liquid food flows
Food and Bioproducts Processing
(1998) - et al.
Vibrational flow of non-Newtonian fluids
Chemical Engineering Science
(2001) - et al.
CFD analysis of viscous non-Newtonian flow under the influence of a superimposed rotational vibration
Computers & Fluids
(2008) - et al.
Optimising the quality of safe food: computational modelling of a continuous sterilisation process
Chemical Engineering Science
(1999) - et al.
Non-isothermal laminar pipe flow – II
Experimental. Chemical Engineering Science
(1973) - et al.
The fluid mechanics of two-phase solid-liquid food flows
Food and Bioproducts Processing
(1997) - et al.
Role of thermo-rheological behaviour in simulation of continuous sterilization of a starch dispersion
Trans IChemE
(2000) - et al.
The Graetz–Nusselt problem for a power-law non-Newtonian fluid
Chemical Engineering Science
(1956) - Barth, T.J., Jesperson, D.C., 1989. The design and application of upwind schemes on unstructured meshes. AIAA, Paper...