Contents lists available at ScienceDirect International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt Role of finite element based grids and simulations on evaluation of Nusselt numbers for heatfunctions within square and triangular cavities involving multiple discrete heaters ☆ Debayan Das, Leo Lukose, Tanmay Basak * Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600036, India ARTICLE INFO Keywords: Natural convection Multiple heaters Finite element Fractional error Nusselt number Heatfunction ABSTRACT Nusselt number is an important non-dimensional parameter which quantifies the heat transfer rate. Local Nusselt number is useful in predicting the heat transfer rate along the various hot and cold sections of the side walls in a discretely heated enclosed cavity. In addition, the overall heat balance in an enclosed cavity (total heat delivered by the hot isothermal walls should be equal to the total heat gained by the cold isothermal walls) can be validated via the average Nusselt numbers. Current finite element based simulations and post-processing have been carried out in order to analyze the influence of the multiple heaters on the Nusselt number along various sections (hot and cold) of the side walls in discretely heated square and triangular (design 1 and design 2) cavities. The working fluid is considered to be air (Pr = 0.7) and the numerical studies have been carried out for a large range of Rayleigh number (Ra =10 3 –10 5 ) for four different biquadratic elements (24 × 24, 28 × 28, 32 × 32 and 34 × 34). The current work also estimates the fractional error in the heat balance (ϵ) and it is clearly inferred that ϵ is comparatively lower for 34 × 34 biquadratic elements. Current work also reveals that the fractional error (ϵ) is mainly induced due to the sharp variations in the Nusselt number at the cold-hot junctions along the side walls. The present study also involves the detailed evaluation of the heatfunction (Π) expressions along the cold-hot junctions of the side walls. The computations of the heatfunctions are intrinsically related to the Nusselt numbers of the hot-cold junctions. 1. Introduction Natural convection with distributing heating methodology has gained wide attention in the recent times as this type of heating strategy leads to the increase in the thermal efficiency of a system compared to the conventional heating processes such as differential heating, and bottom wall heating [1–3]. Most of the earlier works with distributed/ discrete heating have been carried out in the square, rectangular or cylindrical enclosures [4–9]. The shape of the enclosure (such as tri- angle, and trapezoidal) varies for various industrial process [10–12]. The triangular configurations with heaters located on the side walls, occur abundantly in many electronic devices and hence, natural con- vection within discretely heated triangular cavities have been studied by various authors in the recent past [13–15]. Most of the earlier works of distributed heating involve the presence of the single heater along the side wall(s) of the triangular enclosures [13–15]. However, nu- merical studies based on the presence of the multiple heaters along each side wall of the triangular enclosures are yet to appear in the literature. Heatlines have been found to be an effective mathematical tool for visualizing the heat flow within the enclosures [1]. The mathematical representation of heatline is given by heatfunction (Π) and the di- mensionless form of Π is associated with the average Nusselt number [2,4]. The computation of Nusselt number is sensitive along the hot/cold junctions and this strongly influences the heatfunction or the heat flow distribution within the cavity. The average Nusselt number is an indicator of the heat balance for CFD simulations such that the total heat delivered by the hot isothermal walls should be equal to the total heat gained by the cold isothermal walls at steady state. In continuation with an earlier work [16], the heat balance has to be estimated based on the Nusselt number evaluation within the discretely heated square and triangular (design 1 and design 2) cavities involving multiple heaters along each side wall. Three different cases (case 1: larger heater near the lower portion and smaller heater near the top portion, case 2: larger heater near the lower portion and smaller heater near the central portion, case 3: larger heater near the central portion and smaller heater near the lower portion) are considered based on the locations of http://dx.doi.org/10.1016/j.icheatmasstransfer.2017.09.008 ☆ Communicated by A.R. Balakrishnan. * Corresponding author. E-mail addresses: debayan8221@gmail.com (D. Das), leolukose1993@gmail.com (L. Lukose), tanmay@iitm.ac.in (T. Basak). International Communications in Heat and Mass Transfer 89 (2017) 39–46 0735-1933/ © 2017 Elsevier Ltd. All rights reserved. MARK