A bifurcation study of convective heat transfer in a hele-shaw cell

Steady-state multiplicity characteristics of convective heat transfer within a Hele-Shaw cell are investigated. The Navier-Stokes equations and the energy equation are averaged across the narrow gap, d, of the cell. The resulting two-dimensional, stationary equations depend on the following parameters: (i) the length to height aspect ratio γ, (ii) the tilt anle ? (iii) the Prandtl number Pr, (iv) an inertia parameter ξ = d²/ 12a², and (v) the Grashof number. Gr = Q_gβga⁵/kv². Here a is the height of the cell and Q, is the heat generation rate per unit volume. The complete structure of symmetric and asymmetric stationary solutions are traced using recent algorithms from bifurcation theory. In the double limit of ξ → 0 and Gr → ∞ such that Ra = 4GrPrξ remains finite (where Ra is the Rayleigh number for the Darcy model) the Hele-Shaw model reduces to that of the Darcy model.     

Stagnation Region Heat Transfer of Axisymmetric Impinging Jets on Solid Surfaces

A theoretical model is proposed to evaluate the heat transfer characteristics of axisymmetric impinging fluid jets on the horizontal hot surface in the stagnation region using the energy integral method. A generalized expression involving various modeling parameters such as Nusselt number, nozzle‐to‐plate distance, Prandtl number, Reynolds number, and the modeling parameter k is obtained from the analysis. Present predictions are found to agree well with the test data involving a wide range of coolant type, Reynolds number, and nozzle‐to‐plate distance. In addition, a mechanistic correlation is suggested between the modeling parameter k and flow parameter, i.e., Reynolds number.     

RADIATION EFFECT ON FREE CONVECTION LAMINAR FLOW FROM AN ISOTHERMAL SPHERE

Natural convection laminar flow from an isothermal sphere immersed in a viscous incompressible optically dense fluid in the presence of radiation effects has been investigated. The governing boundary layer equations are transformed into a non dimensional form and the resulting nonlinear systems of partial differential equations are reduced to local non-similarity boundary layer equations, which are solved numerically by two distinct, efficient methods, namely, (i) implicit finite difference method together with the Keller box scheme and (ii) local non-similarity method. Numerical results of the velocity and temperature profiles of the fluid are presented. The results of the shearing stress and the heat transfer rate in terms of skin-friction coefficient and Nusselt number respectively are also presented for a wide range of the radiation-conduction parameter or Planck number R _d (=0.0, 1.0, 2.0, 3.0), the surface heating parameter θ _w (=1.1, 1.2, 1.4), and Prandtl number Pr (=7.0, 10.0, 15.0, 20.0).     

UNSTEADY FLUID AND HEAT FLOW INDUCED BY A STRETCHING SHEET WITH MASS TRANSFER AND CHEMICAL REACTION

The effect of chemical reaction on the flow, heat, and mass transfer within a viscous fluid on an unsteady stretching sheet is examined. The stretching rate, temperature and concentration of the sheet, and the chemical reaction rate are assumed to vary with time. The time-dependent boundary layer equations governing the flow are reduced through a convenient similarity transformation to a set of ordinary differential equations, which are numerically solved by applying the fourth-order Runge-Kutta-Fehlberg scheme with the shooting technique. Results for the velocity, temperature, and concentration distributions as well as the wall temperature and concentration gradients are presented graphically for various values of the unsteadiness parameter A, Prandtl number Pr, Schmidt number Sc, and chemical reaction parameter γ.     

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.     

Laminar Flow and Heat Transfer Characteristics in Jackets of Triangular Flow Channels

Laminar flow and heat transfer characteristics of jacketed vessel with triangular flow channels were numerically studied under hydrodynamically and thermally fully developed conditions. Constant heat flux at theheated wall was assumed. The numerical program code interms of vorticity, stream function, axial velocity com ponent and energy equations was written based on a finite volume method. Based on the numerical results, the flow and temperature field were given, and the effects of Dean and Prandtl numbers on flow and heat transfer were ex amined, and the correlations of flow resistance and mean Nusselt number were developed for the jacket. The results show that the structure of secondary flow is steady two vortices in the investigated range of dimensionless curvatureratio and Reynolds number. Two peaks of local Nusselt number increase significantly with Prandtl and Dean num ber increasing, but the local Nusselt numbers near two ends and at the center of the heated wall increase only slightly. The center and two ends of heated wall are the poor positions for heat transfer in the jacket. Compared with the outer half coil jacket at the same area of heated wall, curvature radius, Reynolds number and Prandtl number, e jacket of triangular flow chmnel has lower flow resistance and less mean Nusselt number.     

Unsteady flow and heat transfer on an accelerating surface with blowing or suction in the absence and presence of a heat source or sink

The problem of unsteady flow and heat transfer in the laminar boundary layer on a linearly accelerating surface with suction or blowing in the absence and presence of a heat source or sink is considered. The governing partial differential equations for this investigation are transformed into the non-dimensional equations by using pseudo-similarity time and pseudo-similarity coordinate. The resulting two points boundary-value problem is solved numerically by the central finite difference method associated with Newton's iteration from the initial stage (ξ=0) to a steady state (ξ=1) completely. A parametric study is performed to illustrate the effects of Prandtl number, power-law surface temperature (PLST) or power-law heat flux (PLHF), heat sink or heat source, and suction or blowing parameter on the dynamic velocity and temperature fields as well as the transient development of the skin-friction coefficients and the Nusselt number. These results are depicted graphically to display special aspects of unsteady flow and heat transfer characteristics in all time.     

NATURAL CONVECTION FLOW OF SECOND-GRADE FLUID ALONG A VERTICAL HEATED SURFACE WITH POWER-LAW TEMPERATURE

In this study we investigated the effects of free convection flow of a viscoelastic second-grade fluid along a vertical flat surface with power-law temperature distribution. The boundary layer equations for the momentum and the energy transport have been reduced to local similarity equations using appropriate transformations. Solutions of the reduced equations are obtained employing the local non-similarity method as well as the implicit finite difference method against ξ (the local Deborah number) in the range [0, 10] for fluids having Prandtl numbers of 10, 50, and 100. The regular perturbation solutions are also been obtained for smaller values of ξ together with the Padé approximation. Results thus obtained are discussed in terms of the local skin friction and local rate of heat transfer for different values of the physical parameters, like n and Pr. Effect of the Deborah number, De, on the velocity and temperature profiles has also been shown graphically. It is observed that both the local skin friction and heat transfer coefficients decrease with increase in the value of De for given values of n and Pr.     

螺旋回热器传热性能的研究

本文在相似理论指导下研究了螺旋回热器稳定流动状态下的传热性能.分别用液体和空气作为传热介质,在比较大的Dn和Pr变化范围内测量了传热系数.用线性回归分析法关联出了Nu与Dn和Pr之间的准数方程.同时用修正的Wilson标绘法确定传热系数,获得了与测量结果比较满意的一致的结果.     

On the Prandtl or Schmidt number dependence of the turbulent heat or mass transfer coefficient

Numerical experiments using a direct numerical simulation (DNS) of turbulent flow between two parallel plates in conjunction with Lagrangian scalar tracking (LST) of trajectories of thermal markers in the flow field are conducted for Prandtl or Schmidt numbers between 0.01 and 50,000. The LST methodology is used to generate mean temperature profiles as a function of the entry distance in the case of a step change in heat or mass flux at the walls of the channel. The heat transfer coefficient and the Nusselt number ratio, Nu(x)/Nu(x→∞), downstream from the step change in the wall flux are determined for the range of Pr or Sc fluids examined. Relations between the heat or mass transfer coefficient at the fully developed part of the channel and Pr or Sc are proposed for low and high Pr or Sc cases. Finally, unified correlations, which provide the heat or mass transfer coefficient for all Pr or Sc, in the Reynolds number range examined, are proposed. Also, the exponent of the asymptotic dependence of the eddy diffusivity close to the wall is obtained.     

Natural convection in a heat generating fluid bounded by two horizontal concentric cylinders

Convective flow and heat transfer of a Boussinesq fluid contained between two horizontal concentric cylinders is investigated under the effects of two driving mechanisms – an externally-imposed temperature gradient across the annulus, and a uniform internal heat generation. Numerical results for flow field and temperature distribution are obtained in terms of four dimensionless parameters, namely the radius ratio, R, the Prandtl number, Pr, the Rayleigh number, Ra*, and the ratio, S, between the characteristic temperature induced by internal heating and the applied temperature difference between the boundaries. Depending on the value of S, the flow pattern is made up of either one or two vortices in each half cavity, and heat is transferred into or out of the cavity through the hot wall. In particular, for a certain value of the applied temperature difference, the hot wall apparently acts as a thermally-insulated boundary, the internal heat is completely lost through the cold wall, and the fluid undergoes a transition from a bicellular to a unicellular flow regime.     

BOUNDARY LAYER FLOW AND DOUBLE DIFFUSION OVER AN UNSTEADY STRETCHING SURFACE WITH HALL EFFECT

The present investigation is concerned with the effect of Hall currents on boundary layer flow, and heat and mass transfer of an electrically conducting fluid over an unsteady stretching sheet in the presence of a strong magnetic field. The electron-atom collision frequency is assumed to be relatively high, so that the Hall effect is assumed to exist, while the induced magnetic field is neglected. The governing time-dependent boundary layer equations for momentum, thermal energy, and concentration are reduced using a similarity transformation to a set of coupled ordinary differential equations. The similarity ordinary differential equations are then solved numerically by the successive linearization method together with the Chebyshev pseudo-spectral collocation method. Effects of the Prandtl number, Pr, Schmidt number, Sc, magnetic field, M, Hall parameter, m, and the unsteadiness parameter, A, on the velocity, temperature, and concentration profiles as well as the local skin friction coefficient and the heat and mass transfer rates are depicted graphically and/or in tabular form. Favorable comparisons with previously published work on various special cases of the problem are also obtained.     

Impingement heat transfer under a confined slot jet. Part II: Effects of surface motion and throughflow

Local and average heat transfer coefficients were measured for a confined turbulent slot jet impinging on a moving surface at which there may be throughflow. Profiles of the local convective coefficient at the impingement surface were obtained using a fast responding, highly sensitive porous heat flux sensor. The decrease in average heat transfer with surface motion is not negligible, ～20%, at values of the surface motion parameter, M_vs, comparable to those used in industry. The enhancement of heat transfer by throughflow at a moving impingement surface is linearly additive and, when expressed as δSt, is proportional to only the throughflow parameter, M_us, with a proportionality constant of 0.17 which is independent of Re, M_vs or extent of the heat transfer surface.     

纳米流体在芯片微通道中的流动与换热特性

对去离子水及体积分数分别为0.15%和0.26%的水基γ-Al₂O₃纳米流体在当量直径为194.5 μm的硅基梯形芯片微通道内的层流流动和换热特性进行了实验研究。考察了Reynolds数、Prandtl数以及体积分数对流动换热的影响。结果发现,使用纳米流体后,压降无明显增加,纳米流体的流动阻力特性与去离子水基本相同;对流换热Nusselt数较去离子水有明显提高,且随着体积分数的增加而增加;相同泵功下换热热阻显著下降。实验还发现纳米流体的强化传热效果在较高温度时更加明显。根据实验数据得到了梯形硅微通道内低浓度纳米流体的层流对流换热关联式。研究结果对于集成高效芯片散热系统设计具有重要意义。     

Effects of filler on heat transmission behavior of flowing melt polymer composites

A probe consisting of a cartridge heater was inserted into melted polypropylene composites (PP/mica = 5, 10, and 20 wt%) flowing in a die having a diameter of 10 mm, and the effects of mica on the heat transfer coefficient and the heat transfer behavior from the surface of the probe to the flow compound material were examined experimentally. As a result, the heat transfer coefficient of the PP/mica composites increases 15–20% depending on the flow velocity and filler content. In general, the Prandtl number (Pr) for high‐viscosity (η) melted polymer is greater than one (Pr > 1), and the Prandtl number for polypropylene composites increases as the viscosity increases and decreases depending on the flow velocity and the temperature. The relationship between Pr and η is directly proportional, and the β value, which is given by the slope of the Pr ? η diagram, is a constant that shows the viscosity dependency of the Prandtl number. The β value can be determined by the ratio of the specific heat to the thermal conductivity. The Nusselt number depends on the mica content of the filler and increases gradually. POLYM. ENG. SCI. 46:1387–1393, 2006. © 2006 Society of Plastics Engineers     

The Effects of Power Law Fluid Rheology on Coil Heat Transfer for Agitated Vessel in the Transitional Flow Regime

A CFD model of heat transfer from power‐law fluids to helical cooling coils in the transitional flow regime of a baffled tank mixed with a pitched blade turbine was developed with Fluent^TM. The model captured local temperature and velocity gradients. Simulations were run, varying Re, Pr, K and n. The results indicate that a Sieder‐Tate type correlation, with the exponent on and the coefficient in front of the Reynolds number being a function of n, is recommended for estimating ho. Also, a new two coil bank design was found to be more efficient when 450 < Re < 650.     

扭曲扁管管内流动与传热的三维数值研究

利用计算流体力学和数值传热学的方法,对扭曲扁管三维模型的流动与传热性能进行了数值模拟计算和理论分析,研究了管内流体的Re,Pr以及管子几何尺寸对其流动与传热性能的影响,结果表明,扭曲扁管具有较好的强化传热效果,在管内流体具有高Pr低Re数的条件下,其强化传热效果尤为明显。根据数值计算的结果数据,拟合出了努塞德数和阻力系数的准则公式,对扭曲扁管工程实际应用具有一定的参考价值。     

Numerical determination of fluid-to-particle mass and heat transfer coefficients in packed bed reactors

A method based on particle-resolved CFD is built and validated, to calculate the fluid-to-particle mass and heat transfer coefficients in packed beds of spheres with different tube-to-particle diameter ratios (N) and of various particle shapes with N = 5.23. This method is characterized by considering axial dispersion. The mass and heat transfer coefficients increase by 5%–57% and 9%–63% after considering axial dispersion, indicating axial dispersion should be included in the method. The mass and heat transfer coefficients are reduced as N decreases. The catalyst particles without inner holes show higher mass and heat transfer coefficients than the ones with inner holes, because of unfavorable fluid flow in inner holes. The bed of trilobes has the highest mass and heat transfer coefficients, being 85% and 95% higher than the one of spheres. This work provides a versatile method and some useful guidance for the design of packed bed reactors.     

Combined forced and free convection flow past a horizontal flat plate

The problem of simultaneous forced and free convection flow of a Newtonian fluid past a hot or cold horizontal flat plate is investigated by means of numerical solutions of the full equations of motion and thermal energy subject only to the Boussinesq approximation. These solutions span the parameter ranges 10 ≤ Re ≤ 100, 0.1 ≤ Pr ≤ 10, and –2.215 ≤ Gr/Re^5/2 ≤ 2.215 where Re, Pr, and Gr are based on the ambient free stream fluid properties and the overall plate length l. When Gr > 0, the boundary flow near the plate surface is accelerated relative to the corresponding forced convection flow, with a resulting increase in both the local skin friction and heat transfer coefficients. When Gr < 0, the boundary flow is decelerated, the local skin friction and heat transfer are decreased, and the flow actually separates for Gr/Re^5/2 < –0.8 when Pr = 0.7. In the latter circumstance, an increasing degree of upstream influence is observed as Gr/Re^5/2 is further decreased.     

固定床内错流传热——II.床层对壁给热系数的经验关联式

采用3种导热性能不同的固体颗粒为填充物,以空气为介质,在床层被加热的情况下,研究了固定床中内置圆管的错流传热.采用最小二乘法对实验数据进行拟合,得到以床层对壁的平均给热系数、气体的导热系数和床层中被加热圆管的管径计算的Nusselt准数经验关联式:Nuf=31(lb0/ls)1.4(Db/Dp)0.2Rep0.33Pr0.62,Rep=10~180,Db/Dp=28~116,lb0/ls=0.5~0.2.Reynolds数以固体颗粒的等外表面积当量直径进行计算.结果表明,在错流传热过程中,表征气体流动特性的参数Rep仍是错流传热的重要影响因素,Nusselt准数除与床层的结构参数Db和颗粒的当量直径Dp有关外,还与颗粒的导热系数ls和床层的导热系数lbo密切相关.