K.C. Cheng

Laminar forced convection heat transfer in curved rectangular channels

The purpose of this paper is to present flow and heat transfer results obtained by a point successive over-relaxation method for steady fully developed laminar flow in curved rectangular channels under the thermal boundary conditions of axially uniform wall heat flux and peripherally uniform wall temperature at any axial position. The numerical method yields solutions up to a reasonably high Dean number for the aspect ratios γ = 0.2, 0.5, 1, 2 and 5 considered in this study. It is noted that perturbation method is applicable only for relatively low Dean number region and boundary-layer technique is valid only for high Dean number regime. Graphical results for f Re(f Re)0 and sol Nu(Nu)0, respectively, vs. Dean number are presented for Pr = 0.73. Typical examples for axial velocity and temperature profiles, streamlines and velocity profiles for secondary flow and isotherms are also shown. For square channel, the effect of Prandtl number on heat transfer result is also investigated. Comparison of the result from this analysis and the result for high Dean number regime for the curved square channel available in the literature shows clearly that reasonable estimate can be made for the flow and heat transfer results for the Dean number ranging from 150 to 1000 where currently accurate solutions are not available.

Boundary vorticity method for laminar forced convection heat transfer in curved pipes

Abstract A finite-difference solution using a combination of line iterative method and boundary vorticity method is presented for the hydrodynamically and thermally fully developed laminar forced convection in curved pipes subjected to the thermal boundary conditions of axially uniform wall heat flux and peripherally uniform wall temperature at any axial position. The numerical solution converges up to a reasonably high Dean number where the asymptotic behavior for the flow and heat transfer results already appears. The Prandtl number effect on heat transfer result is clarified for the first time, and it is shown that all the heat transfer results for Pr ⩾ 1.0 can be correlated by a single curve using a new parameter ( K 2 Pr ) with reasonable accuracy. The numerical results from the present analysis are compared with the experimental and theoretical results available in the literature. The perturbation method is clearly shown to be invalid, and certain deficiency of the boundary-layer approximation method is pointed out.

Buoyancy effects on laminar heat transfer in the thermal entrance region of horizontal rectangular channels with uniform wall heat flux for large prandtl number fluid

The Graetz problem for fully developed laminar flow in horizontal rectangular channels with uniform wall heat flux is extended by including buoyancy effects in the analysis for the case of large Prandtl number fluid. A general formulation valid for all Prandtl numbers is presented and the limiting case of large Prandtl number is approached by a numerical method. The typical developments of temperature profile, wall temperature and secondary flow in the thermal entrance region are presented for the case of square channel γ = 1. Local Nusselt number variations are presented for the aspect ratios γ = 0.2, 0.5, 1, 2 and 5 with Rayleigh number as parameter. Due to entry and secondary flow effects, a minimum Nusselt number occurs at some distance from the entrance, depending on the magnitude of Rayleigh number. This behavior is similar to that observed in the thermal entrance region where the transition from laminar to turbulent flow occurs. The effect of Rayleigh number is seen to decrease the thermal entrance length, and the Graetz solution, neglecting buoyancy effects, is found to be applicable only when Rayleigh number is less than about 103. A study of the practical implications of large Prandtl number on heat transfer results for hydrodynamically and thermally fully developed case reveals that the present heat transfer results are valid for Prandtl number ranging from order 10 to infinity.

Natural convection effects on graetz problem in horizontal rectangular channels with uniform wall temperature for large Pr

Abstract The effect of buoyancy forces on laminar forced convective heat transfer in the thermal entrance region of horizontal rectangular channels with uniform wall temperature is studied by a numerical method for the case of large Prandtl number fluids. The numerical results are presented for the aspect ratios (width/height) 0·5, 1, 2 and Rayleigh numbers 0 ~ 5 × 10 5 . The correlation equations for the prediction on the onset of significant free convection effect are developed. The asymptotic behavior of local Nusselt number is compared against the known asymptote for the uniform wall temperature boundary condition. The entrance region where the free convection effect is significant is clearly established and the classical Graetz problem is shown to be a limiting case and is applicable only when Ra ⩽ 10 3 .

NATURAL CONVECTION EFFECTS ON GRAETZ PROBLEM IN HORIZONTAL ISOTHERMAL TUBES

Abstract The classical Graetz problem with natural convection effect in isothermally cooled or heated horizontal tubes is approached by a numerical method using large Prandtl number assumption. Numerical solutions are obtained for a range of Rayleigh numbers Ra = 0–10 6 . The developing secondary flow and temperature fields, bulk temperature, local and average Nusselt numbers are presented to study the natural convection effect. The Nusselt number results are compared against the experimental data and the agreement is found to be satisfactory.

Thermal instability of blasius flow along horizontal plates

Abstract The thermal instability of laminar forced convection flow along a horizontal semi-infinite flat plate heated isothermally from below or cooled isothermally from above is investigated for disturbances in the form of stationary longitudinal vortices which are periodic in the spanwise direction. The analysis uses non-parallel flow model considering the variation of the basic flow and temperature fields with the streamwise coordinate as well as the transverse velocity component in the disturbance equations. The critical values of the Grashof number Gr∗ L = Gr∗ x Re 3 2 X are obtained for Prandtl numbers ranging from 10−2 to 104. The Prandtl and Reynolds numbers effects on vortex-type instability for Blasius flow along horizontal plates are clarified.

Graetz problem in curved pipes with uniform wall heat flux

The thermal entrance region heat transfer problem (Graetz problem) for fully developed laminar flow in curved pipes with uniform wall heat flux is approached by an alternating direction implicit method for Dean numbers ranging from 0 to an order of 100. The effects of using several different finite-difference approximations for convective terms due to secondary flow in the energy equation on heat transfer result are studied. The effect of secondary flow on developing temperature field in the thermal entrance region is studied by considering the temperature profiles, isothermals and axial distributions of average wall temperature and bulk temperature. Heat transfer results are presented for Pr = 0.1, 0.7, 10 and 500.

The steady state ice layer profile on a constant temperature plate in a forced convection flow—II. The transition and turbulent regimes

Abstract The process of transition from laminar to turbulent flow on an ice surface was found to be very significantly different from that on a flat plate. The influence of the ice surface occurs because the shape of ice layer responds to the changes in heat-transfer coefficient that occur in the transition regime. As a result of this interaction two modes of transition were observed. Each mode was associated with a distinctive ice profile shape and Reynolds number for the onset of turbulent heat transfer. For both modes the onset Reynolds number was substantially lower than that for a flat plate. For some experimental parameters an interesting hysteresis phenomenon occurred in which the steady state ice layer could take either one of the two characteristic shapes depending on how equilibrium was approached. The decrease in the ice layer thickness that occurs in the transition region was also observed to have a major effect on the heat-transfer rates in the turbulent regime for some distance downstream of transition.

Flow transitions and combined free and forced convective heat transfer in a rotating curved circular tube

Abstract The simultaneous effects of curvature, rotation and heating/cooling of the tube complicate the flow and heat transfer characteristics beyond those observed in the tubes with simple curvature, rotation or heating/cooling. The phenomena encountered are investigated for steady, hydrodynamically and thermally fully developed laminar flow in circular tubes. A full second-order perturbation solution is obtained under the condition that the wall heat flux is uniform with peripherally uniform wall temperature. The results cover both the nature of flow transitions and the effect of these transitions on temperature distribution, friction factor and Nusselt number. When the rotation is in the same direction as the main flow imposed by a pressure gradient and the fluid is heated, the flow and heat transfer remain similar to those observed in stationary curved tubes, radially rotating straight tubes or mixed convection in stationary straight tubes. There are, however, quantitative changes due to the combined effects of centrifugal, Coriolis and buoyancy forces. A more complex behaviour is possible when the rotation is opposite to the flow due to the pressure gradient or when the fluid is cooled. In particular, the inward Coriolis force and/or buoyancy force may cause the direction of the secondary flow to reverse. The flow reversal occurs by passing through a four-cell vortex flow region where overall, the centrifugal, Coriolis and buoyancy forces just neutralize each other.

Viscous dissipation effects on convective instability and heat transfer in plane poiseuille flow heated from below

The effects of viscous dissipation on the onset of instability for longitudinal vortices in the thermal entrance region of a horizontal parallel-plate channel are studied by a numerical method for the case when the lower plate is heated isothermally and the upper one is cooled isothermally. Numerical results are obtained for Pr=0.1, 0.7, 10 and 100. It is found that viscous heating has a destabilizing influence. The effect is significant for large Prandtl number fluid (Pr≳10) but is insignificant for small Prandtl number fluid (Pr≲0.7). The viscous dissipation effects on thermal entrance region heat transfer for basic flow (without secondary flow) are also studied.