CONVECTIVE HEAT TRANSFER IN HYDRODYNAMICALLY-DEVELOPED LAMINAR FLOW IN ASYMMETRICALLY-HEATED ANNULI : A THREE-TEMPERATURE

Heat transfer in hydrodynamically-developed flow in asymmetrically-heated channels and annuli has been studied extensively. This study is an extension of earlier work where heat transfer in an asymmetrically-heated parallel-plate channel was examined in a resistor-network framework. It was shown that the formulation of the problem in terms of a delta thermalresistor network has several advantages. A delta network can also be used to represent heat transfer in asymmetrically-heated annuli. Nevertheless, the evaluation of the three paired convective resistances that characterize the network is not straightforward. In the present paper, a new technique based on solutions of the energy equation with perturbed boundary conditions is proposed. The proposed technique is first verified by comparison with the results previously obtained for the parallel-plate channel problem. A superposition solution to the energy equation is obtained for hydrodynamically-developed laminar flow in an asymmetrically-heated annulus. The developed technique is then applied to the annulus problem to obtain the corresponding resistances. Results are validated by examining limiting cases. INTRODUCTION The problem of calculating the forced-convective heat transfer in hydrodynamically developed flow in asymmetrically-heated channels and annuli has been studied extensively. Several analytical and numerical solutions have been published and many variations and extensions have been investigated. New developments in emerging areas such as fuel cells and micro flow devices have led to a renewed interest in heat transfer in asymmetrically-heated passages. Reviews of the classical treatments of the problem can be found in advanced heat transfer texts, e.g. [1,2]. In recent studies of the problem, the effects of viscous dissipation, porous media and nonNewtonian fluids have been examined. See [3-5], for example. Nusselt numbers based on the difference between the wall temperatures and the bulk fluid temperature are traditionally used to characterize internal-flow convective heat transfer. For asymmetrically-heated passages, a temperature ratio is used to specify the ordering of the boundary temperatures, particularly the placement of the inlet fluid temperature with respect to the wall temperatures. As discussed in a recent paper [6], this approach has shortcomings when applied to asymmetricallyNOMENCLATURE A [m] surface area Cn [-] series solution coefficient fn [-] eigenfunction H [m] channel width Q [W] heat transfer rate q [W/m] heat flux k [W/mK] thermal conductivity L [m] length Nu [-] Nusselt number Pr [-] Prandtl number Re [-] Reynolds number based on hydraulic diameter R [K/W] thermal resistance r [m] radius radial location r̅ [-] dimensionless radius: r̅ =r/r1 S [m] conduction shape factor T [K] temperature X [-] inverse Graetz number (Eq. 10) Greek letters θ dimensionless temperature: θi=(T-T0)/(Ti-T0) λ eigenvalue φ radius ratio: φ=r2/r1 Subscripts 0 (inlet) fluid 1 upper/outer wall 2 lower/inner wall Superscripts (2T) two-temperature solution * perturbed condition -heated channels with isothermal walls. Most notably: i) a nonphysical singularity occurs in the distribution of the wall Nusselt number at the point where the bulk fluid temperature reaches the temperature of either wall, and ii) the Nusselt numbers depend on temperature ratio, which is inconsistent with the physics of the problem. These features are apparent in results presented by Hatton and Turton [7] and Mitrovic et al. [8] for the parallel-plate channel, and by Mitrovic and Maletic [9] and Coelho and Pinho [5] for concentric annuli. It was shown [6] that the formulation of the parallel-plate channel problem in terms of a delta thermal-resistor network eliminates the non-physical singularities observed in the existing solutions. In the present paper, the earlier work on parallel-plate channels is extended to concentric annuli and the problem of forced convection in hydrodynamically-developed, laminar flow in an asymmetrically-heated annulus is examined in a 12th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics