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.     

Freezing of liquids in turbulent flow inside tubes

The problem of steady-state freezing of liquids in turbulent flow inside a tube with its walls kept at a uniform temperature lower than the freezing temperature of the liquid is solved for a wide range of Reynolds and Prandtl numbers, 10⁴ ≤ Re ≤ 10⁴ and 0 ≤ Pr ≤ 10³. The effects of Prandtl and Reynolds number on the location of the solid-liquid interface and on the heat transfer rate as a function of position along the tube are established.     

Mixed convection heat transfer in porous media in the non-darcy regime

In this study mixed convection heat transfer in a homogeneous porous duct of square cross section in a horizontal orientation is examined. Results from a generalized Forchheimer model are compared with that from the Darcy model. The heat transfer rate and the flow behavior depend on the following parameters: Grashof number, Gr = Q'gβKa/kv², an axial flow pressure drop parameter, ζ = (aK/vμ)dp'/dz', an inertial parameter ξ = mK/a, appearing in the Forccheimer model and the Prandtl number, Pr = C_pμ/k. In the Darcy limit, ξ → 0, the role of the axial flow parameter, λ is reduced to a mere scale factor and the flow behavior is determined by a single parameter, λ = Gr · Pr. Both the Darcy and the Forchheimer models exhibit dual solutions and a hysteresis behavior over a certain range of Gr. Such parametric dependence can be used as an additional tool along with carefully designed experiments to determine the importance of inertial and Prandtl number effects on convective heat transfer in porous media.     

Natural convection in a porous medium at high rayleigh numbers — application of prandtl's analogy

The problem of natural convection heat transfer from an isothermal vertical cylinder to a saturated porous medium is solved by the integral method making use of one-seventh profiles for velocity and temperature variation in the boundary layer. Prandtl's analogy is used to obtain the wall heat transfer coefficients. The results obtained for Rayleigh numbers greater than 10⁹ and Prandtl numbers 4 and 0.12 are compared with experimental data from literature show satisfactory agreement. Also an approximate closed form solution is presented for the case of a flat wall to predict local Nusselt numbers.     

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 → ∞.     

Natural convection and flow simulation in differentially heated isosceles triangular enclosures filled with porous medium

A finite element analysis is performed to investigate the effects of uniform and non-uniform heating of bottom wall on natural convection flows within isosceles triangular enclosures filled with porous medium. The detailed analysis is carried out in two cases depending on various thermal boundary conditions:

(I)

two inclined walls are maintained at constant cold temperature while the bottom wall is uniformly heated;

(II)

two inclined walls are maintained at constant cold temperature while the bottom wall is non-uniformly heated.

The present numerical procedure adopted in this investigation yields consistent performance over a wide range of parameters of Darcy number, Da (10^-5?Da?10^-3), Rayleigh number, Ra (10³?Ra?10⁶) and Prandtl number, Pr (0.026?Pr?1000) in all the cases mentioned above. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. It is observed that at small Darcy numbers, the heat transfer is primarily due to conduction irrespective of Pr. As the Darcy number increases, there is a change from conduction dominant regime to convection dominant regime. Flow circulations are also found to be strong functions of Pr at large Da (Da=10^-3) and multiple circulation cells occur at small Pr with Ra=10⁶. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case, but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. As average Nusselt number is same on both the inclined walls, the average Nusselt number for bottom wall is times that of the inclined wall which is well matched in two cases considered for verifying the thermal equilibrium of the system. The correlations are proposed for average Nusselt number as functions of Ra for various Darcy and Prandtl numbers.     

THE EFFECT OF A MAGNETIC FIELD ON NATURAL CONVECTION IN A SHALLOW CAVITY HEATED FROM BELOW

A numerical study is presented for magnetohydrodynamic free convection of an electrically conducting fluid in a shallow cavity heated from below and cooled from above. The side walls are maintained adiabatic. A uniform magnetic field, inclined with an angle 0 with respect to the horizontal plane, is externally imposed. The investigation covers the range of the Rayleigh number, Ra, from 1.8 × 10₃ to 3 x 10₄the Hartmann number, Ha, from 0 to 35, the Prandtl number, Pr, from 0.005 to 1 and aspect ratio of the cavity, A = 6. The effect of the magnetic field on the flow structure is presented. For supercritical convection it is found that, upon increasing Ha, the number of roll cells in the cavity increases when it is perpendicular to it. The imposition of an inclined magnetic field gives rise to new flow patterns with tilted lateral cell walls. The effect of both strength and orientation of the magnetic field on the overall heat transfer is found to be significant.     

A generalized approach to heat transfer in pipe flow with internal heat generation

In the last few years, there has been a renewed interest in the molten salt reactor (MSR), one of the “Generation IV International Forum” concepts, which adopts a circulating molten salt mixture as both heat generator (fuel) and coolant. The heat transfer of a fluid with internal heat generation depends on the strength of the source whose influence on the heat exchange process is significant enough to demand consideration. At present, few studies have been performed on the subject from either an experimental or a numerical point of view.This study considers fluids with a wide range of Reynolds numbers, flowing through smooth and straight circular tubes within which the flow is hydrodynamically developed but thermally developing (conditions of interest for MSR core channels). The study aims at an assessment of the heat transfer modelling for a large variety of fluids (with Prandtl numbers in the range 0≤Pr≤10⁴), in particular taking into account the influence of the internal heat generation on the temperature distribution, which plays an important role in the case of molten salts for nuclear reactors. To this purpose, the general and unified solution of the heat transfer equation is applied to the turbulent Graetz problem with boundary conditions of the third kind and arbitrary heat source distribution, incorporating recent formulations for turbulent flow and convection.Computed results are shown to be in a good agreement with experimental data concerning heat transfer evaluations for both fully developed and thermally developing flow conditions, over a large range of Prandtl numbers (10^?2<Pr<10⁴). Finally, a preliminary correlation, which includes the Prandtl number range of interest for molten salts, is proposed for the Nusselt number predictions in the case of simultaneous uniform wall heat flux and internal heat generation.     

Finite element analysis of natural convection in a triangular enclosure: Effects of various thermal boundary conditions

The phenomena of natural convection in a right-angled triangular enclosure is studied numerically. A penalty finite element analysis with bi-quadratic elements is used for solving the Navier-Stokes and energy balance equations. The detailed study is carried out in two cases depending on various thermal boundary conditions:

(a)

Vertical wall is uniformly or linearly heated while inclined wall is cold isothermal.

(b)

Inclined wall is uniformly or linearly heated while vertical wall is cold isothermal.

In all the cases horizontal bottom wall is adiabatic and the aspect ratio of the lengths of base and height is considered to be one. The present numerical procedure adopted in this investigation yields consistent performance over a wide range of parameters of Rayleigh number Ra(10³?Ra?10⁵) and Prandtl number Pr(0.07?Pr?1000) in all the cases mentioned above. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. It has been found that at low Rayleigh numbers (Ra?10⁴), the isotherms are almost parallel near the bottom portion of the triangular enclosure while at Ra=10⁵, the isotherms are more distorted. This is because the heat transfer is primarily due to conduction for lower values of Rayleigh number. As Rayleigh number increases, there is a change from conduction dominant region to convection dominant region, and the critical Rayleigh number corresponding to on-set of convection is obtained. It has been shown that the average Nusselt number for vertical wall is times that of the inclined wall as the bottom wall is adiabatic and that verifies the thermal equilibrium of the system for all case studies.     

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.     

Joule-Thomson effects on turbulent graetz problem for gas flows in pipes with uniform wall temperature

The Joule-Thomson effect is known to be important in arctic gas pipelines. The Joule-Thomson effects on forced convective heat transfer in the thermal entrance region of pipes with uniform wall temperature are studied for steady fully developed turbulent gas flows by the Graetz method. Thermal entrance heat transfer results are presented for Prandtl number 0.72, Reynolds number 10⁵ and Brinkman number ± 0.1, ± 1.0 with Joule-Thomson parameter Jμ ranging from 0 to 1.0 to cover the possible range in practical applications. Bulk temperatures and Nusselt numbers are also presented for fully developed flow with Reynolds numbers from 5 × 10³ to 10⁶. For given Prandtl and Reynolds numbers, the asymptotic Nusselt number is found to be dependent on the Joule-Thomson parameter only and is independent of Brinkman number. The fully developed bulk temperature is a linear function of Brinkman number and a linear relationship exists between the bulk temperature parameter (-θ_bf/Br) and the Joule-Thomson parameter Jμ for given Prandtl and Reynolds numbers.     

A turbulent transport model with emphasis on heat transfer to liquid metals

An analysis of the energy transport to eddies in transit from the turbulent core to the wall region coupled with the elementary surface renewal and penetration model is presented for the turbulent tube flow of liquid metals. Previous formulations of the various forms of the surface renewal and penetration model have been based on the assumption that eddies arrive at the surface with initial temperature T₁ equal to the bulk stream temperature T_b. However, due to the significance of molecular transport during the flight of eddies from the turbulent core to the wall region, the employment of this assumption immediately restricts the model to the analysis of the mean heat flux for fluids other than liquid metals. In contrast, the present analysis gives rise to an expression for the mean Nusselt number for turbulent tube flow which is shown to be applicable to fluids with moderate values of the Prandtl number (Pr = 0.5 - 5.0) as well as to the flow of liquid metals at high values of the Reynolds number.     

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.     

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。     

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.     

Free convection heat transfer over a horizontal plate at low prandtl number

Boundary layer equations for free convection heat transfer along a semi-infinite horizontal plate are derived by giving more importance to the energy equation. The equations are obtained for low Prandtl number and two separate polynomials are used to approximate the temperature and velocity profiles in these regions. The rate of heat transfer is compared with the available analytical and numerical results based on conventional boundary layer equations.     

NATURAL CONVECTION OF A BINARY MIXTURE IN A VERTICAL CLOSED ANNULUS

This article reports an analytical and numerical study of natural convection of a binary mixture within a vertical closed annulus. Neumann boundary conditions for temperature are applied to the vertical walls of the enclosure, while the short walls are insulated. The solutal buoyancy forces are assumed to be induced either by the imposition of constant fluxes of mass on the vertical walls (double-diffusive convection, a = 0) or by temperature gradients (Soret effect, a = 1). The governing parameters for the problem are the thermal Rayleigh number R_T, Prandtl number Pr, Lewis number Le, buoyancy ratio ?, aspect ratio A, constant a, and curvature parameter η. An analytical solution, based on the assumption of parallel flow over a large portion of enclosure, is derived. Numerical confirmation of the analytical results is also presented.     

Computational Modelling on Heat Transfer Study of Molten Silicon During Multi-crystalline Silicon Growth Process for PV Applications

Multi-crystalline silicon is an important material with advantages of low-production cost and high conversion efficiency for photovoltaic solar cells. Directional solidifi cation has become the main technique for producing mc-Si ingots for solar cell applications. The study is performed in the framework of the incompressible Navier-Stokes equation in the Boussinesq approximation with convection-conduction equations. The computations are carried out in a two dimensional (2D) axisymmetric model by the finite- element method. The influence of the Reynolds numbers, total heat flux and velocity streamline pattern on the silicon melt was simulated and analyzed for various Rayleigh numbers between 10 to 10 ⁶ with the help of a numerical technique. The following key findings are presented in this paper. The velocity field value is increased above 0.02(m/s), heat flux value is increased to 10 ⁴(W/m ²), when the Rayleigh number is increased above 1000. Reynolds numbers are also studied in five parallel horizontal cross-sections of the melt silicon region for various Prandtl numbers at a critical Rayleigh number of 1000. Reynolds numbers are varied between 100 and 10 ⁵ for the Rayleigh numbers between 10 to 10 ⁶. Meanwhile, the melt has high fluctuation when the Prandtl number is increased above 0.01. The flow is converted from laminar to turbulence at a critical Rayleigh number 1000 and Prandtl number 0.01. These results provide important information for controlling the melt fluctuations during the solidification process which are used to increase the average grain size in growing silicon multicrystals and reduce the dislocation density.     

TURBULENT HEAT TRANSFER IN A CIRCULAR TUBE WITH VISCOUS DISSIPATION

Abstract

The problem of developed turbulent conveclive heat transfer with viscous dissipation is considered for flow in a circular tube. Two cases are considered, Prandtl number equal to unity and large Prandtl number. For Prandtl number equal to unity, an exact expression for the recovery factor is presented which accounts for the radial variation in velocity profile. This exact expression is evaluated using the turbulent logarithmic velocity distribution. For large Prandtl numbers, a general asymptotic expansion for the recovery factor is determined which includes both wall and core contributions. A comparison of these results suggests that the large Prandtl number formula found for the recovery factor should be accurate for all Prandtl numbers greater than about unity.     

NATURAL CONVECTION OF A BINARY MIXTURE IN A VERTICAL CLOSED ANNULUS

This article reports an analytical and numerical study of natural convection of a binary mixture within a vertical closed annulus. Neumann boundary conditions for temperature are applied to the vertical walls of the enclosure, while the short walls are insulated. The solutal buoyancy forces are assumed to be induced either by the imposition of constant fluxes of mass on the vertical walls (double-diffusive convection, a = 0) or by temperature gradients (Soret effect, a = 1). The governing parameters for the problem are the thermal Rayleigh number R_T, Prandtl number Pr, Lewis number Le, buoyancy ratio ϕ, aspect ratio A, constant a, and curvature parameter η. An analytical solution, based on the assumption of parallel flow over a large portion of enclosure, is derived. Numerical confirmation of the analytical results is also presented.