Application of First-Order Differential Equation to Heat Convection in Fluid

Differential equation is very important in science and engineering, because it required the description of some measurable quantities (position, temperature, population, concentration, electrical current, etc.) in mathematical form of ordinary differential equations (ODEs). In this research, we determine heat transferred by convection in fluid problems by first-order ordinary differential equations. So in this research work first we discuss the solution of ordinary homogeneous and non-homogeneous differential equation and then apply the solution of first-order ODEs to heat transferring particularly in heat convection in fluid.

Optimization of fin geometry in heat convection with entransy theory

The entransy theory developed in recent years is used to optimize the aspect ratio of a plate fin in heat convection.Based on a two-dimensional model,the theoretical analysis shows that the minimum thermal resistance defined with the concept of entransy dissipation corresponds to the maximum heat transfer rate when the temperature of the heating surface is fixed.On the other hand,when the heat flux of the heating surface is fixed,the minimum thermal resistance corresponds to the minimum average temperature of the heating surface.The entropy optimization is also given for the heat transfer processes.It is observed that the minimum entropy generation,the minimum entropy generation number,and the minimum revised entropy generation number do not always correspond to the best heat transfer performance.In addition,the influence factors on the optimized aspect ratio of the plate fin are also discussed.The optimized ratio decreases with the enhancement of heat convection,while it increases with fin thermal conductivity increasing.

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.

Numerical Assessment of Nanofluid Natural Convection Using Local RBF Method Coupled with an Artificial Compressibility Model

In this paper,natural heat convection inside square and equilateral triangular cavities was studied using a meshless method based on collocation local radial basis function(RBF).The nanofluids used were Cu-water or Al_(2)O_(3)-water mixture with nanoparticle volume fractions range of 0≤φ≤0.2.A system of continuity,momentum,and energy partial differential equations was used in modeling the flow and temperature behavior of the fluids.Partial derivatives in the governing equations were approximated using the RBF method.The artificial compressibility model was implemented to overcome the pressure velocity coupling problem that occurs in such equations.Themain goal of this work was to present a simple and efficient method to deal with complex geometries for a variety of problem conditions.To assess the accuracy of the proposed method,several test cases of natural convection in square and triangular cavities were selected.For Rayleigh numbers ranging from 103 to 105,a validation test of natural convection of Cu-water in a square cavity was used.The numerical investigation was then extended to Rayleigh number 106,as well as Al_(2)O_(3)-water nanofluid with a volume fraction range of 0≤φ≤0.2.In a second investigation,the same nanofluids were used in a triangular cavitywith varying volume fractions to test the proposed meshless approach on non-rectangular geometries.The numerical results appear to be in agreement with those from earlier investigations.Furthermore,the suggested meshless method was found to be stable and accurate,demonstrating that it may be a viable alternative for solving natural heat transfer equations of nanofluids in enclosures with irregular geometries.

Heat Convection Between Two Confocal Elliptic Tubes Placed at Different Orientations

In this paper,transient and steady natural convection heat transfer in an elliptical annulus has been investigated.The annulus occupies the space between two horizontal concentric tubes of elliptic cross-section.The resulting velocity and thermal fields are predicted at different annulus orientations assuming isothermal surfaces.The full governing equations of mass,momentum and energy are solved numerically using the Fourier Spectral method.The heat convection process between the two tubes depends on Rayleigh number,Prandtl number,angle of inclination of tube axes and the geometry and dimensions of both tubes.The Prandtl number and inner tube axis ratio are fixed at 0.7 and 0.5,respectively.The problem is solved for the two Rayleigh numbers of 104 and 105 considering a ratio between the two major axes up to 3 while the angle of orientation of the minor axes varies from 0 to 90◦.The results for local and average Nusselt numbers are obtained and discussed together with the details of both flow and thermal fields.For isothermal heating conditions,the study has shown an optimum value for major axes ratio that minimizes the rate of heat transfer between the two tubes.Another important aspect of this paper is to prove the successful use of the Fourier Spectral Method in solving confined flow and heat convection problems.

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.

Two-Dimensional Simulation of the Navier-Stokes Equations for Laminar and Turbulent Flow around a Heated Square Cylinder with Forced Convection

Few studies jointly investigate thermal and turbulent effects. In general, these subjects are treated separately. The purpose of this paper is to use the Immersed Boundary Method (IBM) coupled with the Virtual Physical Model (VPM) to investigate incompressible two-dimensional Newtonian flow around a heated square cylinder at constant temperature on its surface with forced convection and turbulence. The VPM model dynamically evaluates the force that the fluid exerts on the immersed surface and the thermal exchange between both in the Reynolds numbers (Re) window 40 ≤ Re ≤ 5×103 . For simulations of turbulence the Smagorinsky and Spalart-Allmaras models are used. The first model uses the Large Eddy Simulation (LES) methodology and is based on the local equilibrium hypothesis for small scales associated with the Boussinesq hypothesis, such that the energy injected into the spectrum of the turbulence balances the energy dissipated by convective effects. The second model uses the concept Unsteady Reynolds Averaged Navier-Stokes Equations (URANS), with only one transport equation for turbulent viscosity, being calibrated in pressure gradient layers. The goal of this work is to analyse the combination of the heat-transfer phenomena with the turbulence for the thermo-fluid-structure interaction in a square cylinder. For this, it was developed a C/C++ code that requires low computational costs in regards to memory and computer facilities. It is observed that, with the increase of the Reynolds number, an increase of the drag coefficient occurs, as well as reinforces the influence of the pressure distribution downstream of the cylinder, which is strongly influenced by the formation and detachment of vortices on the upper and lower sides of the square cylinder.

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.

Analysis and calculating for preheat temperature and average convection heat change coefficient in the continue annealing furnace

The aim of the thesis is to utilize essential theory of heat transfer,to use correlative expressions to calculate average convection heat change coefficient and heating temperature of strip in Jet Preheat Furnace (JPF),make the calculating results accordant with production data,and make the calculation to be used the process of production.The method is to collect:entry temperature and speed of strip,temperature and speed of N_2 - H_2,to analyse heat transfer according to length and thickness of strip,jet hole and mutual position of jet piping in JPF,to analyse heat transfer and built the physical model.In mathematic model, Martin correlative expressions are tried to calculate using the data from production,and are modified in part properly.At the same time,heat boundary condition is analysed with theory of impact jet and production data.The conclusion is obtained that boundary condition is rarely average numerical value of temperature of strip and N_2 - H_2 with impact jet condition,instead of a relation of function of temperature of strip, temperature and speed of N_2- H_2,array of jet holes,diameter of hole,distance between hole and strip,and acquired a calculating expression.In calculation of examples,the thesis collected and calculated 15 kinds of strips.The thickness of strips are 2=0.51~1.41 mm,material DQ - IF、DDQ,EDDQ,SEDDQ and 340DDQ.Main assess numerical value is temperature value after strip is heated with certain speed and within section of time.Maximum error in 9 groups of numerical value in the thesis is 3.36%comparing with production data.The correlative expressions can be used in production to adjust temperature of strip through changing speed and temperature of N_2 - H_2 and speed of strip.The correlative expressions are compiled computer process.The process can be applied in on line control of production by rapid calculating speed.

Mixed convection effects on heat and mass transfer in a non Newtonian fluid with chemical reaction over a vertical plate

This paper studies mixed convection,double dispersion and chemical reaction effects on heat and mass transfer in a non-Darcy non-Newtonian fluid over a vertical surface in a porous medium under the constant temperature and concentration.The governing boundary layer equations,namely,momentum,energy and concentration,are converted to ordinary differential equations by introducing similarity variables and then are solved numerically by means of fourth-order Runge-Kutta method coupled with double-shooting technique.The velocity,temperature concentration,heat and mass transfer profiles are presented graphically for various values of the parameters,and the influence of viscosity index n,thermal and solute dispersion,chemical reaction parameter χ are observed.

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.

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).

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.

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.

Contrasts of Atmospheric Circulation and Associated Tropical Convection between Huaihe River Valley and Yangtze River Valley Mei-yu Flooding

The significant differences of atmospheric circulation between flooding in the Huaihe and Yangtze River valleys during early mei-yu （i.e., the East Asian rainy season in June） and the related tropical convection were investigated. During the both flooding cases, although the geopotential height anomalies always exhibit equivalent barotropic structures in middle to high latitudes at middle and upper troposphere, the phase of the Rossby wave train is different over Eurasian continent. During flooding in the Huaihe River valley, only one single blocking anticyclone is located over Baikal Lake. In contrast, during flooding in the Yangtze River valley, there are two blocking anticyclones. One is over the Ural Mountains and the other is over Northeast Asia. In the lower troposphere a positive geopotential height anomaly is located at the western ridge of subtropical anticyclone over Western Pacific （SAWP） in both flooding cases, but the location of the height anomaly is much farther north and west during the Huaihe River mei-yu flooding. Fhrthermore, abnormal rainfall in the Huaihe River valley and the regions north of it in China is closely linked with the latent heating anomaly over the Arabian Sea and Indian peninsula. However, the rainfall in the Yangtze River valley and the regions to its south in China is strongly related to the convection over the western tropical Pacific. Numerical experiments demonstrated that the enhanced latent heating over the Arabian Sea and Indian peninsula causes water vapor convergence in the region south of Tibetan Plateau and in the Huaihe River valley extending to Japan Sea with enhanced precipitation; and vapor divergence over the Yangtze River valley and the regions to its south with deficient precipitation. While the weakened convection in the tropical West Pacific results in moisture converging over the Yangtze River and the region to its south, along with abundant rainfall.

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.

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.

An Improvement of the Mass Flux Convection Parameterization Scheme and its Sensitivity Tests for Seasonal Prediction over China

A modified cumulus parameterization scheme, suitable for use in a seasonal forecast model, is presented. This parameterization scheme is an improvement of the mass flux convection scheme developed by Gregory and Rowntree (1989; 1990). This convection scheme uses a 'bulk' cloud model to present an ensemble of convective clouds, and aims to represent shallow, deep, and mid-level convection. At present, this convection scheme is employed in the NCC T63L20 model (National Climate Center, China Meteorological Administration). Simulation results with this scheme have revealed some deficiencies in the scheme, although to some extent, it improves the accuracy of the simulation. In order to alleviate the deficiencies and reflect the effect of cumulus convection in the actual atmosphere, the scheme is modified and improved. The improvements include (i) the full estimation of the effects of the large-scale convergence in the lower layer upon cumulus convection, (ii) the revision of the initial convective mass flux, and (iii) the regulation of convective-scale downdrafts. A comparison of the results obtained by using the original model and the modified one shows that the improvement and modification of the original convection scheme is successful in simulating the precipitation and general circulation field, because the modified scheme provides a good simulation of the main features of seasonal precipitation in China, and an analysis of the anomaly correlation coefficient between the simulation and the observations confirms the improved results.