Laminar Conjugate Natural Convection And Surface Radiation In Horizontal Annuli

Abstract

Numerical studies of two-dimensional laminar conjugate natural convection flow and heat transfer in horizontal annuli formed between inner heat generating solid cylinders and outer isothermal circular boundary are performed with and without the effect of surface radiation. The two configurations of the concentrically placed inner cylinder are a circular cylinder (CC configuration) and a square cylinder (SOS, i.e., Square-On-Side, configuration). The mathematical formulation consists of the continuity equation, momentum equations with Boussinesq approximation and the solid and fluid energy equations. Numerical solutions are obtained by discretising the governing equations on a collocated mesh (non-staggered mesh) and the pressure-velocity coupling is taken into account via the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm. A cylindrical polar coordinate system is employed for CC configuration and a Cartesian coordinate system is used for the SOS configuration. The convective terms are discretised with donor-cell differencing scheme and the diffusion terms, with central differencing scheme. The algebraic equations resulting from the discretisation of the governing equations are solved using the line by line TDMA (Tri-Diagonal Matrix Algorithm). A global iteration scheme over each time step is used for better coupling of temperature and the flow variables and steady-state solutions are obtained by time-marching. Steady-state results of conjugate pure natural convection are obtained for the volumetric heat generation and outer radius based Grashof number ranging from 104 to 1010, for solid-to-fluid thermal conductivity ratios of 1, 5, 10, 50 and 100, and for the aspect ratios of 0.2 and 0.4, with air as the working medium (Pr=0.708) for the CC and SOS configurations. The flow and temperature distributions are presented in terms of isotherms and streamline maps. Results are presented for several quantities of interest such as local and average Nusselt numbers on the inner and outer boundaries, dimensionless local temperatures on the inner boundary and dimensionless maximum and average solid cylinder temperatures. The results show that the flow in the annulus is characterized by double or quadruple vortex patterns. Of the dimensionless maximum solid temperature, average solid temperature and average inner boundary temperature, the first two are much sensitive to solid-to-fluid thermal conductivity ratio. Surface radiation effects are studied numerically in conjugation with natural convection. The coupling with surface radiation arises through the solid-fluid interface thermal condition. To account for the radiation effects, configuration factors among the subsurfaces of the inner and outer boundaries formed by the computational mesh are determined. Results are obtained for CC and SOS configurations for emissivities ranging from 0.2-0.8, with the other parameters as in pure natural convection case. It is found that even at low surface emissivity, radiation plays a significant role in bringing down the convective component and enhancing the total Nusselt numbers across the annulus. The presence of radiation is found to reduce the dimensionless temperatures inside the solid and homogenise the temperature distribution in the fluid. The radiative Nusselt number is about 50-70 % of the total Nusselt number depending on the radiative parameters chosen. This factor emphasizes the need for taking into account the coupling of radiation and natural convection for the accurate prediction of the flow and heat transfer characteristics in the annulus. The solution of the conjugate problem facilitates the determination of the solid temperature distribution, which is important in connection with the safety aspects of various thermal energy systems. Correlations as functions of Grashof number and thermal conductivity ratio are constructed for the estimation of various quantities of interest for the two configurations and aspect ratios for pure natural convection and for combined natural convection and radiation cases. The results are expected to be useful in the design of thermal systems such as spent nuclear fuel casks during transportation and storage, underground transmission cables and cooling of electrical and electronic components.