Heat transfer in lid driven channels with power law fluids in a hydrodynamic fully developed flow field

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Abstract

We present a finite element numerical study of heat transfer in lid driven channels with fully developed axial flow for non-Newtonian power law fluids. The effect of channel aspect ratio and material properties on temperature distribution and wall heat transfer are studied. The results show that in comparison with Newtonian fluids the shear thinning property of the fluids acts to reduce the local viscous dissipative heating and as a result the axial local fluid temperature is reduced. Applications of the results to scraped-surface heat exchanger design and operation are recommended.

Introduction

This study concerns flow and heat transfer in lid driven channels. As well as a purely theoretical importance (Shankar & Deshpande, 2000), such flows are directly relevant to many complex industrial applications, such as screw extruders (Kokini, Ho, & Karwe, 1992) and scraped-surface heat exchangers (SSHEs) (Harrod, 1986). Though there are major differences between all of these industrial processes, they may be classified together in the sense that the flows involved can almost invariably be characterised as low Reynolds number/high Prandtl number flows of highly viscous non-Newtonian fluids. Many previous studies have concerned such flows. Numerous 2D studies examined the flow of Newtonian fluids in lid driven cavities (see, for example, Burggraf, 1966; Nallasamy & Prasad, 1977; Pan & Acrivos, 1967). A range of non-Newtonian materials, such as viscoelastic fluids (Grillet, Yang, Khomani, & Shaqfeh, 1999), Bingham fluids (Mitsoulis & Zisis, 2001) and power law fluids (Martin, 1969) have also been analysed in a two-dimensional setting. Though such studies can contribute much understanding, for non-Newtonian fluids the tangential and the axial flows in such devices are coupled. It is therefore important to know the consequences of axial flow and, in particular, how the tangential (cross) flow interacts with the axial flow. Asymptotic results for a three-dimensional isothermal flow in an SSHE were derived by Fitt & Please (2001) and Karwe & Jaluria (1990) carried out numerical studies of heat transfer within a single screw extruder. In both Fitt & Please (2001) and Karwe & Jaluria (1990), simplifications were made by assuming creeping flow of power law fluids with small annular-gap/perimeter ratios so that the sidewall effects could be neglected.

The role of viscous dissipation in steady-state 2D forced convection heat transfer was investigated numerically for lid driven cavities (Sun et al., submitted for publication). A further study (Sun et al., 2004) extended the work of Sun et al., submitted for publication to include the effects of shear thinning and realistic SSHE geometry. A key conclusion from Sun et al., submitted for publication was that, at the singularity corners of the flow region, the apparent viscosity, fluid temperature and viscous dissipation were all smaller for shear thinning fluids compared to the Newtonian case. Quasi-three-dimensional effects (i.e. assuming an axial hydrodynamic fully developed flow) were included in a numerical study of an isothermal lid driven channel in Sun et al. (2006). The results reported in Sun et al. (2006) showed that for hydrodynamic fully developed flow, both the pressure drop and flow field were dependent on the channel aspect ratio, power law index, the tangential and axial Reynolds numbers and their ratios. Similar to previously reported results in ducts (see, for example, Hartnett & Kostic, 1989), though this study concerned turbulent rather than laminar flow), the results of Sun et al. (2006) also confirmed that for (laminar) flow in a lid driven channel a notable “drag reduction effect” (resulting in a reduced axial pressure gradient) was found with shear thinning power law fluids. However, at high tangential Reynolds numbers it was also found that the tangential flow causes strong distortion in the axial flow so that the drag reduction effect was reduced or even reversed.

The current study further generalises the work discussed above by considering quasi-3D heat transfer for power law fluids in lid driven channels with hydrodynamic fully developed laminar flow. The general modus operandi is as follows: under the assumption of a temperature independent viscosity, the fully developed velocity field may be determined first (decoupled with temperature) as in Sun et al. (2006). Neglecting the axial conduction term in the energy equation, the temperature field is then calculated using a marching scheme.

Consistency checks are carried out by comparing the numerical results with available published results. The computations cover a range of physical and non-dimensional parameters, including power law index, channel aspect ratio, tangential velocity, Brinkman number, axial Peclet number and thermal boundary (heating/cooling) conditions. In each case, the temperature distributions, axial local average temperature and axial local heat flux are given. The main aim of the work is to understand the characteristics of heat transfer in a lid driven device such as an SSHE when the effects of wall heating, (quasi-3D), fully developed axial flow and the presence of shear-thinning fluids must all be taken into account.

Section snippets

Governing equations

Typical fluids processed in SSHEs and extruders are characterised by relatively large viscosities and high Prandtl numbers. An important consequence of this is that the thermal boundary layer is much thinner than the velocity boundary layer. For this reason, the assumption that the velocity profile is fully developed before the fluid enters the heating zone is justified. For materials with temperature independent viscosity, this further implies that the velocity field does not change with

Results and discussion

For power law fluids, the non-dimensional parameters are interdependent. For instance, if the lid velocity is altered then the Reynolds number, Peclet number and Brinkman number will all be different. Parametric studies were therefore carried out in two different ways. One was to change individual non-dimensional quantities. The other was to change individual physical parameters for scale-up purposes.

Conclusions

Finite element methods have been successfully used to study heat transfer for power law materials in lid driven channels with fully developed axial flow. Where comparison is possible, very good agreement is found between the numerical results and previously published data. Close to the singularity corners the velocity gradients are very large. For constant viscosity Newtonian fluids the viscous dissipation is therefore also very large. However, for shear thinning fluids the viscosity is reduced

Acknowledgements

This research is supported by The University of Reading and Chemtech International, Reading. The authors are grateful for helpful insights provided by Dr. N. Hall-Taylor of Chemtech International and Professor M. Baines of Reading University.

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