Laminar natural convection of yield stress fluids in annular spaces between concentric cylinders

https://doi.org/10.1016/j.ijheatmasstransfer.2019.04.092Get rights and content

Highlights

  • Natural convection of Bingham fluids was studied numerically in horizontal annuli.

  • The yield number has a significant impact on the heat transfer performance.

  • At the same yield number, the plug regions can increase with increasing Ra.

  • The unyielded plugs tend to accumulate at the bottom of the annulus.

  • Increasing in the annular gap width causes a faster growing of plug region.

Abstract

This paper aims to investigate the laminar natural convection of Bingham fluids between horizontal concentric cylinders. The governing equations are solved numerically by using the Galerkin’s weighted residuals scheme of finite element method with a non-uniform unstructured triangular grid. The effects of various parameters such as the Rayleigh number (103Ra106), yield number (0YYmax), ratio of gap width to inner cylinder radius (0.5L/Ri5) and fixed Prandtl number (Pr=100 since results are not sensitive to Pr in the tested range) on the flow and temperature profiles as well as on the heat transfer and morphology of yielded/unyielded regions are presented and discussed. The heat transfer rate shows a positive dependence on the Rayleigh number and L/Ri, however, there is an optimum value for L/Ri which maximizes the heat transfer rate. The results also show that the yield number has a significant impact on the flow and temperature fields and the heat transfer performance. In fact, the heat transfer rate decreases with the rise of the yield number and above a certain value of Y, the convection does not occur, and heat transfer is only due to conduction. It is found that while the enhancement of the Rayleigh number augments the heat transfer, it is not necessarily associated with a decrease in the size of the unyielded zones.

Introduction

Natural convective heat transfer in horizontal annular fluid layers is an important phenomenon in engineering systems due to its wide applications in heat exchangers, cooling of electronic equipment, practical heat transfer equipment, etc. As a result, extensive experimental and theoretical works have been reported in the literature dealing with issues of modeling and simulations of the natural convection in such configuration. In the following, some of the most important studies in this field are briefly discussed. One of the first numerical and experimental works about the natural convection of Newtonian fluids in annular spaces between concentric cylinders has been done by Kuehn and Goldstein [1] for several Rayleigh (102Ra105) and Prandtl (0.5Pr10) numbers. This study shows good agreement between the numerical results and experimental measurements of heat transfer and temperature distribution. An experimental study has also been done by Kuehn and Goldstein [2] to determine the influence of eccentricity on natural convection heat transfer through a fluid bounded by two horizontal isothermal cylinders. After that, transient natural convection heat transfer within a concentric cylindrical annulus has been studied by Tsui and Tremblay [3]. Caltagirone [4] and Burns and Tien [5] analyzed this problem for the porous medium and showed that natural convection may develop in the insulating material and contribute significantly to the heat transfer process. In recent years, Abu-Nada et al. [6] considered the problem related to natural convection within a horizontal annulus in nanofluids and Habibi Matin and Ahmed Khan [7] studied this problem for non-Newtonian power-law fluids.

In our knowledge, there is no research dealing with the natural convection of a yield stress fluid in annular spaces between concentric cylinders. However, analysis of this problem in other geometries has attracted considerable attention in recent years. The existence of a “yield stress” is traditionally recognized to be responsible for the complicated transition between classical solid-like and liquid-like behavior. If the material is not sufficiently stressed, it will not deform and behaves like a solid, but once the yield stress is exceeded, it will deform and flow according to different constitutive relations. To analyze the stress-deformation behavior of viscoplastic materials, three ideal models have been proposed, which are the Bingham plastic model, the Herschel- Bulkley model, and the Casson model [8].

The first numerical studies concerning natural convection of viscoplastic fluids seem to be those of Zhang et al. [9] and Vikhansky [10]. They analyzed the effects of fluid yield stress on the classical Rayleigh–Bénard instability in the cavity and show that the Rayleigh-Bénard convection of a Bingham fluid, when submitted to small perturbations, is linearly stable. Analysis of natural convection in an enclosure filled with a Bingham fluid has been done by Turan et al. [11], [12] and Yigit et al. [13] using Fluent simulation. They showed that the value of heat transfer in the case of Bingham fluids decreases with increasing Bingham number, and, for large values of Bingham number (Bn), heat transfer takes place principally due to thermal conduction. Study of natural convection of viscoplastic material extended to the Herschel-Bulkley fluids by Li et al. [14], Hassan et al. [15] and Aghighi and Ammar [16]. They analyzed the effects of power-law index (n) on viscoplastic fluid behavior and indicated that at the same nominal value of Bn, the plug regions increase for shear-thinning fluids n < 1. Recently, Rayleigh–Bénard convection of viscoplastic fluids obeying the Casson model has been investigated by Aghighi et al. [17]. Based on the results obtained in this work, it can be concluded that the minimum heat transfer of viscoplastic fluids can be obtained by choosing the Casson model.

Study of the convective heat transfer of viscoplastic fluids has been extended to more complex configurations by other researchers. In this field, Nirmalkar et al. [18] have been analyzed the laminar free convection heat transfer in a Bingham fluid from a heated horizontal cylinder. They showed that the maximum rate of heat transfer does not occur at the front stagnation point due to the formation of the polar caps of the unyielded material. On the other hand, natural convection from a circular cylinder in a square cavity has been investigated by Sairamu et al. [19] for Bingham fluids and, Recently, natural convection in a square cavity with an inner cold circular/elliptical cylinder filled with viscoplastic fluids has been simulated by Kefayati and Tang [20] using Lattice Boltzmann Method.

In the light of the above literature review, it appears that the analysis of natural convection in annular spaces between concentric cylinders was not reported previously for a yield stress fluid. Therefore, the aim of the present paper is to undertake a comprehensive numerical investigation on this topic, with the main scope to investigate the fluid flow and heat transfer characteristics of a viscoplastic fluid in horizontal concentric annuli for different operating conditions.

Section snippets

Mathematical formulation

The schematic diagram of the physical model and coordinate system is given in Fig. 1. The concentric annular region is formed by two horizontal cylinders of radii Ri and Ro. The annulus between the two cylinders is filled with a yield stress fluid obeying the Bingham model. The inner cylinder surface is maintained at a constant high-temperature Th and the outer cylinder at a constant low-temperature Tc. Thermo-physical properties of the fluid are assumed to be constant except for the density

Method of solution

The non-dimensional coupled conservation equations of mass, momentum, and energy related to the two-dimensional laminar natural convection of yield stress fluids in annular spaces between concentric cylinders are discretized by developing a numerical code based on the Galerkin weighted residual finite element method. The nonuniform unstructured grid (Fig. 2) is constructed by means of first-order triangular elements. The advantages of unstructured meshing are manifold, the main being that it

Effects of yield stress

In this section, numerical results have been presented for various values of the yield number (0YYmax) and Rayleigh number (103<Ra<106) at constant values of the Prandtl number (Pr=100) and ratio of gap width to inner cylinder radius (L/Ri=2). The effects of these parameters on heat and momentum transport have been discussed in detail.

Conclusions

In this study, a finite element numerical code based on the unstructured triangular grid has been developed for analyzing the two-dimensional steady-state natural convection of viscoplastic fluids obeying the Bingham model in annular spaces between concentric cylinders. The inner cylinder surface is maintained at a constant high-temperature and the outer cylinder at a constant low-temperature.

The effects of yield number and ratio of gap width to inner cylinder radius on heat and momentum

Conflict of interest

The authors declared that there is no conflict of interest.

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