Analytical Study of Natural Convection in a Cavity With Volumetric Heat Generation

Department of Mechanical Engineering (MAE), Maharashtra Institute of Technology, Poona, Maharashtra, India
Journal of Heat Transfer (Impact Factor: 1.45). 02/2006; 128(2). DOI: 10.1115/1.2137761


A semi-analytical method for natural convection in a two dimensional rectangular enclosure, with uniform volumetric heat generation, having insulated horizontal boundaries, and isothermal vertical boundaries, has been studied here. In this method, the governing equations for natural convection, have been solved under the assumption that for a cavity with small aspect ratio, the flow in the central region of the cavity is only in the vertical direction. It is found that for the cavities with small aspect ratio, the temperature in central region of the cavity is nearly constant along the horizontal direction. However, there is a uniform temperature gradient in the vertical direction, which can be related to the maximum temperature in conduction. The velocity profiles and temperature profiles obtained in the present work, are compared with the numerical simulations by Fluent and a fair agreement is found between these results.

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    • "Therefore, the role of internal heat generation becomes very important in several applications including storage of radioactive materials, combustion and fire studies, geophysics, reactor safety analysis and metal waste form development for spent nuclear fuel. However, there are only few studies available in which the effect of internal heating on convective flow in a fluid layer has been investigated [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44]. Viscosity is a physical property of fluids. "
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    ABSTRACT: In this paper, a theoretical investigation has been carried out to study the combined effect of rotation speed modulation and internal heating on thermal instability in a temperature dependent viscous horizontal fluid layer. Rayleigh–Bénard momentum equation with Coriolis term has been considered to describe the convective flow. The system is rotating about it is own axis with non-uniform rotational speed. In particular, a time-periodic and sinusoidally varying rotational speed has been considered. A weak nonlinear stability analysis is performed to find the effect of modulation on heat transport. Nusselt number is obtained in terms of amplitude of convection and internal Rayleigh number, and depicted graphically for showing the effects of various parameters of the system. The effect of modulated rotation speed is found to have a stabilizing effect for different values of modulation frequency. Further, internal heating and thermo-rheological parameters are found to destabilize the system.
    Full-text · Article · Jul 2014 · Ain Shams Engineering Journal
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    • "the governing equations in terms of streamfunction, vorticity and temperature formulation, Joshi et al. (2006) conducted a semi-analytical study of natural convection in cavities of different aspect ratios with uniform volumetric heat generation, considering two different boundary conditions, viz., all isothermal walls and only adiabatic horizontal walls. Daniels and Jones (1998) used matched asymptotic expansions method to analyze natural convection in a shallow rectangular cavity due to internal heat generation , where the nonlinear convective effects were studied. "
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    ABSTRACT: The generalized integral transform technique (GITT) is employed to obtain a hybrid numerical-analytical solution of natural convection in a cavity with volumetric heat generation. The hybrid nature of this approach allows for the establishment of benchmark results in the solution of non-linear partial differential equation systems, including the coupled set of heat and fluid flow equations that govern the steady natural convection problem under consideration. Through performing the GITT, the resulting transformed ODE system is then numerically solved by making use of the subroutine DBVPFD from the IMSL Library. Therefore, numerical results under user prescribed accuracy are obtained for different values of Rayleigh numbers, and the convergence behavior of the proposed eigenfunction expansions is illustrated. Critical comparisons against solutions produced by ANSYS CFX 12.0 are then conducted, which demonstrate excellent agreement. Several sets of reference results for natural convection with volumetric heat generation in a bi-dimensional square cavity are also provided for future verification of numerical results obtained by other researchers.
    Full-text · Article · Dec 2013 · Brazilian Journal of Chemical Engineering
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    • "However, there are relatively very few studies available in which the effect of internal heating on convective flow has been investigated. Some of these studies are Roberts [14], Tveitereid and Palm [15], Tveitereid [16], Yu and Shih [17], Bhattacharya and Jena [18], Takashima [19], Tasaka and Takeda [20], and Joshi et al. [21]. The non-linear stability analysis becomes inevitable if one wants to consider the heat transport in the system as linear stability analysis is inadequate to provide any information about the heat transport and details of pattern formation. "
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    ABSTRACT: In this paper, we study the combined effect of internal-heating and time-periodic gravity modulation on thermal instability in a viscous fluid layer, heated from below. The time-periodic gravity modulation, considered in this problem can be realized by vertically oscillating the fluid layer. A weak non-linear stability analysis has been performed by using power series expansion in terms of the amplitude of gravity modulation, which is assumed to be small. The Nusselt number has been obtained in terms of the amplitude of convection which is governed by the non-autonomous Ginzburg-Landau equation derived for the stationary mode of convection. Effects of various parameters such as internal Rayleigh number, Prandtl number, and amplitude and frequency of gravity modulation have been analysed on heat transport. It is found that the response of the convective system to the internal Rayleigh number is destabilizing. Further, it is found that the heat transport can be controlled by suitably adjusting the external parameters of the system.
    Full-text · Article · Sep 2013 · International Journal of Non-Linear Mechanics
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