A New Idea of Fractal-Fractional Derivative with Power Law Kernel for Free Convection Heat Transfer in a Channel Flow between Two Static Upright Parallel Plates

Nowadays some new ideas of fractional derivatives have been used successfully in the present research community to study different types of mathematical models.Amongst them,the significant models of fluids and heat or mass transfer are on priority.Most recently a new idea of fractal-fractional derivative is introduced;however,it is not used for heat transfer in channel flow.In this article,we have studied this new idea of fractal fractional operators with power-law kernel for heat transfer in a fluid flow problem.More exactly,we have considered the free convection heat transfer for a Newtonian fluid.The flow is bounded between two parallel static plates.One of the plates is heated constantly.The proposed problem is modeled with a fractal fractional derivative operator with a power-law kernel and solved via the Laplace transform method to find out the exact solution.The results are graphically analyzed via MathCad-15 software to study the behavior of fractal parameters and fractional parameter.For the influence of temperature and velocity profile,it is observed that the fractional parameter raised the velocity and temperature as compared to the fractal operator.Therefore,a combined approach of fractal fractional explains the memory of the function better than fractional only.

Mixed convection heat transfer in horizontal channel filled with nanofluids

The laminar fully developed nanofluid flow and heat transfer in a horizonal channel are investigated. Highly accurate solutions for the temperature and nanopavticle concentration distributions are obtained. The effects of the Brownian motion parameter Nb, the thermophoresis parameter Nt, and the Lewis number Le on the temperature and nanoparticle concentration distributions are discussed. The current analysis shows that the nanoparticles can improve the heat transfer characteristics significantly for this flow problem.

Nanofluids Transport Model Based on Fokker-Planck Equation and the Convection Heat Transfer Calculation

In current research about nanofluid convection heat transfer, random motion of nanoparticles in the liquid distribution problem mostly was not considered. In order to study on the distribution of nanoparticles in liquid, nanofluid transport model in pipe is established by using the continuity equation, momentum equation and Fokker-Planck equation. The velocity distribution and the nanoparticles distribution in liquid are obtained by numerical calculation, and the effect of particle size and particle volume fraction on convection heat transfer coefficient of nanofluids is analyzed. The result shows that in high volume fraction （ 0 _-- 0.8% ）, the velocity distribution of nanofluids characterizes as a ＂cork-shaped＂ structure, which is significantly different from viscous fluid with a parabolic distribution. The convection heat transfer coefficient increases while the particle size of nanoparticle in nanofluids decreases. And the convection heat transfer coefficient of nanofluids is in good agreement with the experimental result both in low （0 ~〈 0.1% ） and high （ q = 0.6% ） volume fractions. In presented model, Brown motion, the effect of interactions between nanoparticles and fluid coupling, is also considered, but any phenomenological parameter is not introduced. Nanoparticles in liquid transport distribution can be quantitatively calculated by this model.

Convection Heat Transfer from Heated Thin Cylinders Inside a Ventilated Enclosure

Experimental study was conducted to determine the effect of velocity of axial fan,outlet vent height,position,area,and aspect ratio(h/w)of ventilated enclosure on convection heat transfer.Rectangular wooden ventilated enclosure having top and front transparent wall was made up of Perspex for visualization,and internal physical dimensions of box were 200 mm×200 mm×400 mm.Inlet vent was at bottom while outlet vents were at the side and top wall.Electrically heated cylindrical heat source having 6.1 slenderness ratio was fabricated and hanged at the centre of the enclosure.To calculate heat transfer rates,thermocouples were attached to the inner surface of heat source with silica gel.Heat source was operated at constant heat flux in order to quantify the effect of velocity of air on heat transfer.It was observed that average Nusselt number was increased from 68 to 216 by changing velocity from 0 to 3.34 m/s at constant modified Grashof number i.e.5.67E+09.While variation in outlet height at the front wall did not affect heat transfer in forced convection region.However,Nusselt number decreased to 5%by changing the outlet position from top to the front wall or by 50%reduction in outlet area during forced convection.Mean rise in temperature of enclosure increased from 8.19 K to 9.40 K by increasing aspect ratio of enclosure from 1.5 to 2 by operating heat source at constant heat flux i.e.541.20 w/m^(2).

Improving Heat Transfer in Parabolic Trough Solar Collectors by Magnetic Nanofluids

A parabolic trough solar collector(PTSC)converts solar radiation into thermal energy.However,low thermal efficiency of PTSC poses a hindrance to the deployment of solar thermal power plants.Thermal performance of PTSC is enhanced in this study by incorporating magnetic nanoparticles into the working fluid.The circular receiver pipe,with dimensions of 66 mm diameter,2 mm thickness,and 24 m length,is exposed to uniform temperature and velocity conditions.The working fluid,Therminol-66,is supplemented with Fe3O4 magnetic nanoparticles at concentrations ranging from 1%to 4%.The findings demonstrate that the inclusion of nanoparticles increases the convective heat transfer coefficient(HTC)of the PTSC,with higher nanoparticle volume fractions leading to greater heat transfer but increased pressure drop.The thermal enhancement factor(TEF)of the PTSC is positively affected by the volume fraction of nanoparticles,both with and without a magnetic field.Notably,the scenario with a 4%nanoparticle volume fraction and a magnetic field strength of 250 G exhibits the highest TEF,indicating superior thermal performance.These findings offer potential avenues for improving the efficiency of PTSCs in solar thermal plants by introducing magnetic nanoparticles into the working fluid.

Relationship between the Intensity of Secondary Flow and Convection Heat Transfer in a Helically Coiled Circular Tube with Uniform Wall Temperature

Numerical method is used to investigate fully developed laminar flow in helically coiled circular tube in this paper.The non-dimensional parameter(secondary flow Reynolds number Se)based on absolute vorticity flux along the mainstream is used to indicate the intensity of secondary flow caused by the centrifugal effect in helically coiled circular tube.The relationship between the intensity of secondary flow and the intensity of laminar convective heat transfer is studied.The effects of curvature and torsion on the enhancement of heat transfer are also considered.The results reveal that the absolute vorticity flux along the mainstream can be used to indicate the local or averaged intensity of secondary flow;the non-dimensional parameter of the absolute vortex along the main flow determines the convective heat transfer and friction factor.The relationships of Nusselt number and friction factor with the Se are obtained.The effect of curvature on Nusselt number is obvious,but the effect of torsion on Nusselt number is less obvious.

A Coupled Discrete Unified Gas-Kinetic Scheme for Convection Heat Transfer in Porous Media

In this paper,the discrete unified gas-kinetic scheme(DUGKS)is extended to the convection heat transfer in porous media at representative elementary volume(REV)scale,where the changes of velocity and temperature fields are described by two kinetic equations.The effects from the porous medium are incorporated into the method by including the porosity into the equilibrium distribution function,and adding a resistance force in the kinetic equation for the velocity field.The proposed method is systematically validated by several canonical cases,including the mixed convection in porous channel,the natural convection in porous cavity,and the natural convection in a cavity partially filled with porous media.The numerical results are in good agreement with the benchmark solutions and the available experimental data.It is also shown that the coupled DUGKS yields a second-order accuracy in both temporal and spatial spaces.

Numerical and theoretical investigations of heat transfer characteristics in helium-xenon cooled microreactor core

Helium-xenon cooled microreactors are a vital technological solution for portable nuclear reactor power sources.To exam-ine the convective heat transfer behavior of helium-xenon gas mixtures in a core environment,numerical simulations are conducted on a cylindrical coolant channel and its surrounding solid regions.Validated numerical methods are used to determine the effect and mechanisms of power and its distribution,inlet temperature and velocity,and outlet pressure on the distribution and change trend of the axial Nusselt number.Furthermore,a theoretical framework that can describe the effect of power variation on the evolution of the thermal boundary layer is employed to formulate an axial distribution cor-relation for the Nusselt number of the coolant channel,under the assumption of a cosine distribution for the axial power.Based on the simulation results,the correlation coefficients are determined,and a semi-empirical relationship is identified under the corresponding operating conditions.The correlation derived in this study is consistent with the simulations,with an average relative error of 5.3%under the operating conditions.Finally,to improve the accuracy of the predictions near the entrance,a segmented correlation is developed by combining the Kays correlation with the aforementioned correlation.The new correlation reduces the average relative error to 2.9%and maintains satisfactory accuracy throughout the entire axial range of the channel,thereby demonstrating its applicability to turbulent heat transfer calculations for helium-xenon gas mixtures within the core environment.These findings provide valuable insights into the convective heat transfer behavior of a helium-xenon gas mixture in a core environment.

SIMULATION OF STEEL COIL HEAT TRANSFER IN HPH FURNACE

The mathematical model has been estublished for the simulation of steel coil＇s heat transfer during annealing thermal process in HPH （high performance hydrogen） furnace. The equivalent radial thermal conductivity is adopted by statistical analysis regression approach through the combination of a large quantity of production data collected in practice and theoretical analyses. The effect of the number of coils on circulating flow gas is considered for calculating the convection heat transfer coefficient, The temperature within the coil is predicted with the developed model during the annealing cycle including heating process and cooling process. The good consistently between the predicted results and the experimental data has demonstrated that the mathematical model established and the parameters identified by this paper are scientifically feasible and the effective method of calculation for coil equivalent radial heat transfer coefficient and circulating gas flow has been identified successfully, which largely enhances the operability and feasibility of the mathematic- model. This model provides a theoretical basis and an effective means to conduct studies on the impact that foresaid factors may imposed on the steel coil＇s temperature field, to analyze the stress within coils, to realize online control and optimal production and to increase facilily output by increasing heating and cooling rates of coils without producing higher thermal stress.

An Investigation into the Influence of the Airflow Path on the Convective Heat Transfer for an Eddy Current Retarder Turntable

In order to improve the convective heat transfer relating to an eddy current retarder,the finite element model has been used to assess the performances of different possible designs.In particular,assuming the steady running state of retarder as the working condition,flow and temperature fields have been obtained for the rotor.The influence of airflow path on heat dissipation has been analysed,and the influence of the temperature field distribution on the performance of retarder has been discussed accordingly.The results show that when the steady running state of the turntable is considered,the maximum temperature is lower,the level of turbulence flow is mitigated,and the temperature distribution becomes more regular.These factors contribute to improve the heat dissipation ability of the retarder.

Nusselt number correlation for turbulent heat transfer of helium-xenon gas mixtures

A gas-cooled nuclear reactor combined with a Brayton cycle shows promise as a technology for highpower space nuclear power systems.Generally,a helium-xenon gas mixture with a molecular weight of14.5-40.0 g/mol is adopted as the working fluid to reduce the mass and volume of the turbomachinery.The Prandtl number for helium-xenon mixtures with this recommended mixing ratio may be as low as 0.2.As the convective heat transfer is closely related to the Prandtl number,different heat transfer correlations are often needed for fluids with various Prandtl numbers.Previous studies have established heat transfer correlations for fluids with medium-high Prandtl numbers(such as air and water)and extremely lowPrandtl fluids(such as liquid metals);however,these correlations cannot be directly recommended for such helium-xenon mixtures without verification.This study initially assessed the applicability of existing Nusselt number correlations,finding that the selected correlations are unsuitable for helium-xenon mixtures.To establish a more general heat transfer correlation,a theoretical derivation was conducted using the turbulent boundary layer theory.Numerical simulations of turbulent heat transfer for helium-xenon mixtures were carried out using Ansys Fluent.Based on simulated results,the parameters in the derived heat transfer correlation are determined.It is found that calculations using the new correlation were in good agreement with the experimental data,verifying its applicability to the turbulent heat transfer for helium-xenon mixtures.The effect of variable gas properties on turbulent heat transfer was also analyzed,and a modified heat transfer correlation with the temperature ratio was established.Based on the working conditions adopted in this study,the numerical error of the property-variable heat transfer correlation was almost within 10%.

Experimental Investigation on Flow and Heat Transfer of Jet Impingement inside a Semi-Confined Smooth Channel

Experimental investigation is conducted to investigate the flow and heat transfer performances of jet impingement cooling inside a semi-confined smooth channel.Effects of jet Reynolds number(varied from 10 000to 45000),orifice-to-target spacing(zn=1d—4d)and jet-to-jet pitches(xn=3d—5d,yn=3d—5d)on the convective heat transfer coefficient and discharge coefficient are revealed.For a single-row jets normal impingement,the impingement heat transfer is enhanced with the increase of impingement Reynolds number or the decrease of spanwise jet-to-jet pitch.The highest local heat transfer is achieved when zn/dis 2.For the double-row jets normal impingement,the laterally-averaged Nusselt number distributions in the vicinity of the first row jets impinging stagnation do not fit well with the single-row case.The highest local heat transfer is obtained when zn/dis 1.A smaller jetto-jet pitch generally results in a lower discharge coefficient.The discharge coefficient in the double-row case is decreased relative to the single-row case at the same impingement Reynolds number.

Numerical investigation on convective heat transfer over two heated wall-mounted cubes in tandem and staggered arrangement

In this study, laminar convective heat transfer over two heated wall-mounted cubes is investigated.Two cubes, which are under constant heat flux, are placed in different tandem and staggeredarrangements on a base plate. This problem is studied for different streamwise and spanwisedistances between two cubes in different Renolds number （Re）, by using finite-volume method.Effects of these parameters are considered on flow and heat transfer characteristics. The resultsshow that the temperature distribution is strongly dependent on flow structure and varies with anychange of flow pattern in different arrangements of cubes. In addition, it is observed that the dragcoefficient, which is influenced more by pressure forces, in staggered arrangement, is greater thantandem arrangement. Results show that by increasing the spanwise distance the amount of meanNusselt number （Nu） of Cube 2 becomes the same as Cube 1.

Simulation of Heat Transfer Characteristics in Channel with Square Rib Under Uniform and Turbulent Inlet Conditions

The detailed flow structures and closely-related heat transfer characteristics are investigated along the wall of a cooling channel with rib tabulator by computation.Three typical Reynolds numbers defined by the rib height are set at 200,500,1300,and the Mach numbers is 0.2,respectively.Two inlet boundary conditions,including the uniform and the fully-developed turbulent conditions,are used to study the turbulence effects on the characteristics of heat transfer in the vicinity of rib and wall.Results show that the local Nusselt number increases when the Reynolds number rises from 200 to 1300.At lower Reynolds number,the turbulent inlet condition generates more tangible heat transfer enhancement.At higher Reynolds number,however,the uniform inlet condition contributes more to the convective heat transfer effects.The paper discovers that the high Nusselt number has a consistent correlation with the positive and negative sign alteration of the shear layer on the wall,which satisfactorily explains the mechanisms of heat transfer enhancement due to the flow.

THE PREDICTIONS OF CONVECTIVE HEAT TRANSFER ON TURBINE BLADE AIRFOIL BY USING LOW－REYNOLDS NUMBER TURBULENCE MODEL

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Heat Transfer Process of Bubble during the Occurrence of Cavitation in Hydraulic System

Thermal characteristic of cavitation has great influence on the process of occurrence,development and collapse of bubble in hydraulic system. By choosing the stage of bubble growth as the research object,combining with the characteristic of the process of bubble occurrence and development in hydraulic system, and ignoring the impact of thermal radiation,the heat transfer situation of bubble growth was analyzed under appropriate assumptions of thermodynamic conditions in the bubble generation and development process. The mathematical expression of the temperature change of bubble was deduced using thermodynamic principle. Through combining the expression with classic Rayleigh-Plesset Equation,numerical calculation was carried out and the temperature variation over time( or bubble radius) was obtained. The influences of convective heat transfer coefficient of bubble and polytropic exponent on the thermodynamic process of bubble were analyzed. Finally,the thermal characteristic of bubble growth after cavitation occurrence was summarized.

Natural Convection Melting in a Rectangular Heat Storage Tank of Carbon Nanotube Dispersed Latent Heat Storage Material

A dispersion system fluid can convect even if the dispersoid is a solid phase.Therefore,heat exchange performance can be improved while maintaining fluidity using a material with high thermal conductivity as the dispersoid.This study presents the melting performance evaluation results of a latent heat storage material with a carbon nanotube(CNT)dispersion system with high thermal conductivity,which enhances the thermal conductivity of the latent heat storage material and does not limit free convection.Increasing the thermal conductivity and enhancing the melting convection of the heat storage material result in increased latent heat storage speed.In this study,the thermal conductivity of the latent heat storage material was successfully increased by dispersing CNTs in the material.When 0.1%(in mass)of multi-wall CNT(MWCNT)was dispersed in a paraffin-based latent heat storage material,the shear stress increased by 1.5 times at a shear rate of 500 s^(-1),while taking into account the potential effects of convective inhibition.Therefore,a latent heat storage experiment was conducted in a rectangular heat storage tank using the CNT dispersion composition ratio as a parameter.A rectangular vessel with a heated vertical surface was used for the latent heat storage experiment.The melting speed was determined by comparing the amount of latent heat stored in a CNT-dispersed latent heat storage material and a single-phase latent heat storage material sample.The experimental results show that the time required for the latent heat storage material to completely melt in the heat storage tank was the shortest for the single-phase latent heat storage material sample.However,the fastest melting progress was observed for the sample with 0.02%(in mass)MWCNT content in the melting rate range of up to approximately 40%in the tank.The results indicate that this phenomenon is caused by the difference in the melting rates in the upper part of the tank.The generated data are useful for determining the shape and heat transfer surface arrangement of the latent heat storage tank.

Numerical study of convective heat transfer in static arrangements of particles with arbitrary shapes:A monolithic hybrid lattice Boltzmann-finite difference-phase field solver

A compressible lattice Boltzmann-finite difference method is extended by the phase-field approach into a monolithic scheme to study fluid flow and heat transfer through regular arrangements of solid bodies of circular,elliptical and irregular shapes.The advantage of using the phase-field method is demon-strated both in its simplicity of accounting for flow and thermal boundary conditions at solid surfaces with irregular shapes and in the capability of generating such complex-shaped objects.For an array of discs,numerical results for the overall solid-to-gas heat transfer rate are validated via experiments on flow through arrays of hot cylinders.The thus validated compressible LB-FD-PF hybrid scheme is used to study the dependence of heat transfer on flow and thermal boundary conditions(Reynolds number,temperature difference between the hot solid bodies and the inlet gas),porosity as well as on the shape of solid objects.Results are rationalized in terms of the residence time of the gas close to the solid body and downstream variations of gas velocity and temperature.Perspective for further applications of the proposed methodology are also discussed.

Comparative Experimental Study on Heat Transfer Characteristics of Building Exterior Surface at High and Low Altitudes

The external surface heat transfer coefficient of building envelope is one of the important parameters necessary for building energy saving design,but the basic data in high-altitude area are scarce.Therefore,the authors propose a modified measurement method based on the heat balance of a model building,and use the same model building to measure its external surface heat transfer coefficient under outdoor conditions in Chengdu city,China at an altitude of 520 m and Daocheng city at an altitude of 3750 m respectively.The results show that the total heat transfer coefficient(h_(t))of building surface in high-altitude area is reduced by 34.48%.The influence of outdoor wind speed on the convective heat transfer coefficient(h_(c))in high-altitude area is not as significant as that in low-altitude area.The fitting relation between convection heat transfer coefficient and outdoor wind speed is also obtained.Under the same heating power,the average temperature rise of indoor and outdoor air at highaltitude is 41.9%higher than that at low altitude,and the average temperature rise of inner wall is 25.8%higher than that at low altitude.It shows that high-altitude area can create a more comfortable indoor thermal environment than low-altitude area under the same energy consumption condition.It is not appropriate to use the heat transfer characteristics of the exterior surface of buildings in low-altitude area for building energy saving design and related heating equipment selection and system terminal matching design in high-altitude area.

Experimental Study and Thermal Modelling of Cocoa Shell Convective Drying in an Indirect Solar Dryer

The concern of the present work is the convective drying of empty cocoa shells in an indirect solar dryer. Some drying experiments, using one sample, were carried out. During the experiments, the sample is introduced in the drying chamber. Then at steady time intervals, the sample is withdrawn from the drying chamber, for a rapid weighing. After each weighing, the sample is reintroduced in the dryer. At each time interval, the ambient temperature of the drying chamber and its relative humidity γ are measured by a thermo-hygrometer. From the experimental data, a theoretical determination of the moisture evaporated from the product was performed and a good agreement was found between the theoretical and experimental values, confirmed by the value of the RMSE. Those calculations used the constants in the Nusselt number found in literature. Then those constants were evaluated again, to get new values more suitable with the experimental data. The dimensionless numbers of Nusselt, Grashof and Prandtl were calculated. That allowed the calculation of the average value of the Nusselt number. The average convective heat transfer coefficient was determined.