Low peclet number heat transfer in the thermal entrance region of parallel-plate channels with unequal wall temperatures

The problem of low-Peclet-number thermal entry heat transfer for plane Poiseuille flow in parallel-plate channels with uniform but unequal wall temperatures is approached by the eigenfunction expansion method utilizing the Gram-Schmidt orthonormalization procedure. The formulation considers axial heat conduction and allows upstream heat penetration through the thermal entrance. Numerical results are obtained for the case with entrance condition parameter θ₀ = 1 and Peclet number Pe = 1, 5, 10 and 50. The effect of Peclet number on temperature distributions in both upstream and downstream regions is studied. At Pe = 50, the concept of thermal boundary layer is applicable and the present series solution does not yield physically reasonable temperature distribution locally near the upper plate at the thermal entrance. The difficulty may be attributed to the nature of thermal boundary conditions at the thermal entrance and the transition from elliptic problem to parabolic problem with the increase of Peclet number.     

Low peclet number heat transfer for power law non-newtonian fluid with heat generation

Heat transfer for power law non-Newtonian fluid with heat generation and axial conduction is analyzed. Radial and axial temperature distribution and the Nusselt number inside a tube are obtained in terms of nonorthogonal series expansion. Eigenvalues and eigenfunctions are given for different values of various parameters. The effects of Peclet number, power law model index, viscous dissipation, and heat generation on the temperature distribution and Nusselt number are discussed. Comparison of the present results for extreme cases with those obtained by previous workers shows good agreement.     

Low peclet number particle-to-fluid heat and mass transfer in packed beds

For low Peclet numbers most of the experimentally obtained particle-to-fluid heat and mass transfer coefficients in packed beds were found to be some orders of magnitude below the values predicted for a single sphere in cross flow.From theoretical considerations one should expect the transfer coefficients in packed beds to exceed the single sphere predictions as they actually do for higher Peclet numbers.The obvious discrepancy between theory and experiment can be cxplained by a simple model accounting for a nonuniform distribution of the void fraction. The model consists of a packed bed of uniformly sized particles with an average void fraction ψ, where a small part ? of the total cross-sectional area f is assumed to have a larger void fraction ψ₂. Since the same pressure drop applies to both parts of the bed, the superficial velocity will be much larger in the section with the larger void fraction, especially in the range of low Reynolds numbers.Even though in both parts of the packed bed the individual transfer coefficients are taken from a correlation which is based on the single sphere predictions (as it is valid in the range of high Peclet numbers), the apparent overall transfer coefficients for the nonuniform system become much lower, and show the same characteristic variation with Peclet number and the ratio of particle diameter to bed height as the majority of the experimental data.     

Laminar heat transfer in the thermal entrance region

Series solutions for heat transfer in the thermal entrance region are presented for laminar flow in conduits (plane ducts, tubes, annular passages). The present solution of the equation of energy has been obtained for either a step change in heat flux or wall temperature and is valid for all time-independent non-Newtonian fluids, providing that the flow-field is such that v_θ = 0, v_r = 0, v_g = v_g(r) From its general nature with regard to the flow-field, such a solution is a convenient alternative to the eigenvalue solutions when applied in the thermal entrance region where a large number of eigenvalues and eigencoefficients are required.     

Laminar mixed convection in the thermal entrance region of horizontal rectangular channels with uniform heat input axially and uniform wall temperature circumferentially

This paper presents a numerical study on laminar mixed convection in the thermal entrance region of horizontal rectangular channels with uniform heat input axially and uniform wall temperature circumferentially. A relatively novel numerical method of solution is developed to obtain the developing velocity and temperature fields. The values of Prandtl number are 0.7 and 7.2, corresponding to air and water, respectively. The values of Rayleigh number are 0, 10⁴, 3 × 10⁴ and 10⁵. The channel aspect ratios considered are 0.2, 0.5, 1, 2 and 5. Variations in local friction factor ratios and local Nusselt numbers are presented. It is found that the circumferential boundary condition of uniform wall temperature significantly increases the value of local Nusselt number as compared to that found in earlier works under the boundary condition of uniform wall heat flux. But the boundary condition effect on the friction factor is shown to be comparatively minor. The asymptotic solutions at z → are compared to the existing numerical data with good agreement.     

Convective heat transfer in channels with high wall ribs

The features of the convective heat transfer intensified by the turbulization caused by high ribs are experimentally studied using laboratory tube-in-tube heat exchangers.     

Low-Reynolds number heat transfer enhancement in sinusoidal channels

Fully developed laminar flow and heat transfer in three-dimensional, streamwise-periodic sinusoidal channels with circular and semi-circular cross-sections are considered. Computational fluid dynamics (CFD) is used to investigate the effect of Reynolds number (5?Re?200) and amplitude to half wavelength ratio (0.222?A/L?0.667) on heat transfer enhancement and pressure drop for steady, incompressible, constant property, water (Pr=6.13) flows in geometries with L/d=4.5 for the constant wall heat flux (H2) and constant wall temperature (T) boundary conditions.The flow field in the sinusoidal geometries is increasingly dominated by secondary flow structures (Dean vortices) with increasing Reynolds number and A/L. These vortices act to promote convective heat transfer enhancement, resulting in high rates of heat transfer and low pressure loss relative to fully developed flow in a straight pipe. Heat transfer enhancement exceeds the relative pressure-drop penalty by factors as large as 1.5 and 1.8 for the circular and semi-circular cross-sections, respectively.     

Laminar forced convection in the thermal entrance region of curved pipes with uniform wall temperature

The thermal entrance region heat transfer problem for fully developed laminar flow in curved pipes with uniform wall temperature is approached by an alternating direction implicit method for the parabolic energy equation for a flow regime with Dean number ranging from 0 to an order of 100. This work represents an extension of the classical Graetz problem in straight tubes to curved pipes. The graphical results for temperature developments in the form of temperature profiles through the horizontal and vertical planes, isothermals and local Nusselt number variations in the thermal entrance region are presented in such a way as to illustrate clearly the interaction between the secondary flow and the developing temperature field for Prandtl numbers of 0.1, 0.7, 10 and 500. For a given Dean number, the effect of Prandtl number is to shorten the thermal entrance length (I/Gz) and the temperature field develops rather rapidly with large Prandtl number. The effect of Dean number is similar to that of Prandtl number with Dean number effect becoming much more appreciable at high Prantdl numbers than at low Prandtl number.     

Laminar mixed convection in the thermal entrance region of horizontal isothermal rectangular channels

Laminar mixed convection in the thermal entrance region of horizontal isothermal rectangular channels for moderate and small Prandtl number (air) is investigated using the vorticity-velocity formulation of the Navier-Stokes equation. The numerical results, including the cross-stream velocity vectors, local Nusselt numbers and local friction factor ratios are presented for the aspect (width/height) ratios 0.5, 1.0 and 2.0, Rayleigh numbers 0 ～ 2 × 10⁵ and Prandtl numbers 0.7, 5.0 and 100. The strength and pattern of the secondary flow induced by buoyancy effects is found to depend on the magnitude of Rayleigh number and aspect ratio, and the secondary flow leads to a significant enhancement of heat transfer in the entrance region. The classical Graetz problem is shown to be a limiting case which applies only when Ra ≤ 10.³ The behavior of the local Nusselt number for Pr = 100 compares well with the existing results for Pr → ∞.     

An experimental study of convective instability in the thermal entrance region of a horizontal parallel-plate channel heated from below

This paper presents results of an experimental investigation on the onset of longitudinal columnar vortices due to buoyant forces in the thermal entrance region of a horizontal parallel-plate channel heated from below. Channels with width 30 cm and heights h = 2, 3 and 4.5 cm were employed in the experiment. The instability of the convective motion is caused by the effects of temperature gradients in both longitudinal and vertical directions and was observed by a direct flow-visualization technique using smoke. Photographs of the front and top views of the vortex rolls are presented. Experimental results indicate that the wave number of vortex rolls remains constant along the flow direction and tends to approach the theoretical value as the channel width-height ratio increases. It was also observed that the critical Rayleigh number from flow visualization is 1.4 to 10 times higher than the critical value from linear theory.     

An experimental study of convective instability in the thermal entrance region of a horizontal parallel-plate channel heated from below

This paper presents results of an experimental investigation on the onset of longitudinal columnar vortices due to buoyant forces in the thermal entrance region of a horizontal parallel-plate channel heated from below. Channels with width 30 cm and heights h = 2, 3 and 4.5 cm were employed in the experiment. The instability of the convective motion is caused by the effects of temperature gradients in both longitudinal and vertical directions and was observed by a direct flow-visualization technique using smoke. Photographs of the front and top views of the vortex rolls are presented. Experimental results indicate that the wave number of vortex rolls remains constant along the flow direction and tends to approach the theoretical value as the channel width-height ratio increases. It was also observed that the critical Rayleigh number from flow visualization is 1.4 to 10 times higher than the critical value from linear theory.     

Analysis of laminar flow forced convection heat transfer with uniform heating in the entrance region of a circular tube

A new integral or boundary-layer solution for laminar flow heat transfer in the combined entrance region of a circular tube is presented for the case of constant wall heat flux. The solution is based on the hydrodynamic inlet-filled region concept originally proposed by Ishizawa (1966) and later adopted by Mohanty and Asthana (1978) to hydrodynamically developing flow through a circular tube. Unlike available boundary-layer solutions, the new analysis provides results which join smoothly and asymptotically to the fully developed values. Results for the Nusselt number were found to agree favorably with available numerical solutions.     

Combined buoyancy effects of thermal and mass diffusion on laminar forced convection in the thermal entrance region of horizontal square channels

Heat and mass transfer in laminar mixed convection in the thermal entrance region of horizontal square channels is investigated by using the vorticity-velocity formulation of the Naiver-Stokes equations. The numerical results, including the developments of temperature and concentration contours, Nu_z and friction coefficient ratios, fRe / (fRe)₀, are presented for an air-water system. The effects of bottom wall temperature and the relative humidity on the momentum, heat and mass transfer in the flow are examined in detail. Results show that the influences of the evaporation of the water vapor along the wetted wall on the heat and mass transfer and the ratio fRe / (fRe)₀ are rather substantial.     

Convective heat transfer of alumina nanofluids in laminar flows through a pipe at the thermal entrance regime

The convective heat transfer characteristics of aqueous alumina nanofluids were investigated experimentally under forced laminar tube flows. The particles had different shapes of cylinders, bricks and blades, and particle loading was between 0?C5 volume%. The nanofluids were characterized rheologically, and the heat transfer system was validated by using water without particles. In calculating Nusselt and Peclet numbers to assess heat transfer enhancement of nanofluids, physical properties of water were used so as not to exaggerate the amount of heat transfer. It was found that heat transfer coefficients of nanofluids are almost the same or a little smaller than that of water. The heat transfer coefficient can be reduced by the lowering the thermal conductivity of the nanofluid under shearing conditions and particle depletion by the cluster migration from the wall to the tube center. The reduction in thermophysical properties also contributes to the reduction in heat transfer coefficient. It has been concluded that nanofluids from metal particles with appropriate stabilizing agents can satisfy the requirements to be a practically usable nanofluid.     

环形通道内层流入口段换热及流体变物性的影响

采用SIMPLER算法对环形通道内二维定常轴对称入口段流动与换热进行了数值计算, 研究了两种边界条件下的层流流动与换热规律, 给出了不同Prandtl数以及半径比率下沿程Nusselt数的变化曲线,同时还给出了流体物性随温度变化对流动与换热的影响, 最后还拟合出了不同半径比率下平均Nusselt数的关联式.计算结果还表明,环形通道能强化传热,强化程度随半径比率减小而增大,且入口段的换热强化与其较高的径向速度有关.     

Enhancing thermal conductivity of C/SiC composites containing heat transfer channels

In this study, CNTs/SiC micro-pillars at controlled content ratios were introduced into C/SiC composites as heat transfer channels to improve the thermal conductivity in the thickness direction. The thermal conductivities and bending strengths before and after heat treatment at 1650 °C were investigated and the results were discussed. The theoretical calculations and finite element analyses confirmed that CNTs/SiC micro-pillars successfully worked as heat transfer channels. The theoretical thermal conductivity calculated by effective medium theory (EMT) model was 19.25 W/m⋅k and agreed-well with the experimental value. The measured thermal conductivity was estimated to 20.69 W/m⋅k and improved to 22.36 W/m⋅k after heat treatment. The latter was 3.56-fold higher than that of traditional C/SiC and attributed to increased grain growth during heat treatment. The optimal bending strength before heat treatment was recorded as 324.5 ± 23.74 MPa due to microstructure evolution caused by CNTs. After heat treatment, the bending strength improved by 138 % with ductile fracture mode attributed to ordered layer structure of PyC interphase and complex phase composition of the composites. These features benefited the abundant propagation of cracks and energy consumption. In sum, introduction of heat transfer channels into C/SiC composites provided a new way to improve the thermal conductivity in thickness direction of ceramic matrix composites.     

Prandtl number effect on combined free and forced laminar convection in the thermal entrance region of a horizontal tube

This note presents the Prandtl number effect on combined free and forced laminar convection in the thermal entrance region of a horizontal tube by a numerical vorticity-velocity method. The tube wall is heated with a uniform wall heat flux. Vorticity-velocity governing equations in cylindrical coordinates and an evaluation of boundary vorticity on the tube wall are presented along with a numerical method of solution. Local Nusselt number variations are shown for Pr = 100 and 10 and compared with the existing data for Pr → ∞. It is observed that the large Prandtl number assumption is valid for Pr = 10 within a difference of 4% at Ra = 1 × 10⁵. The numerical results are compared against the available experimental data with good agreement.     

Thermal instability of plane poiseuille flow in the thermal entrance region

The onset of thermal convection in plane Poiseuille flow is investigated theoretically. New stability equations are derived by using the propagation theory considering the variations of disturbance amplitudes in the main flow direction. In the thermal entrance region an analytical procedure to predict the critical conditions for extremely small Prandtl-number fluids is described, based on the local similarity. For x_c≤0.01 the critical Rayleigh numbers are well represented in the whole domain of the Prandtl number by Ra_c = 200(1 + 0.123Pr^-1)Ra _C =200(1+0.123Pr⁻¹)x _C ⁻¹ under the conventional boundary layer theory. It is of much interest that the time-independent, three dimensional disturbances become more stable with a decrease in the Prandil number.     

End wall effects on thermal stratification and heat transfer in a vertical enclosure with offset partitions

End wall effects on thermal stratification and heat transfer in a vertical enclosure with offset partitions has been studied by numerically solving the governing differential equations. Two limiting end wall conditions are investigated — adiabatic end walls and perfectly conducting end walls. Two offset partition positions and three partition heights are considered. It is observed that with adiabatic end walls, the heat transfer from the vertical hot and cold walls is always greater. The effect of end wall conditions is most significant when the top partition is offset toward the cold wall and the bottom partition toward the hot wall. In this position strong thermal stratification in the core is observed. When the direction of offset is reversed, i.e., top partition is moved closer to the hot wall and the bottom partition closer to the cold wall, strong stratification effects are noted in the partition-near side wall region. Adiabatic end wall conditions promote these stratification effects.