Constant Heat Flux Case Research Articles

A general analytical solution for fluid flow and heat convection through arbitrary-shaped triangular ducts: A variational analysis

This paper presents an analytical solution for fluid flow and heat transfer inside arbitrarily-shaped triangular ducts for the first time. The former analytical solutions are limited to the special case of isosceles triangular ducts. The literature has no report about the analytical solution for the general case of arbitrarily-shaped triangular ducts. Due to the significant role of fluid flow through non-circular channels in industry and the large number of triangular shapes, a method for solving the heat transfer problem for all triangular shapes is needed. The heat transfer of a fluid flow through a channel with an arbitrary triangular cross-section for the case of constant heat flux at the walls is solved in this work for the first time, considering viscous dissipation. Here, the functionals of flow and heat transfer equations are derived, and the resulting Euler–Lagrange equations are solved using the Ritz method. The effect of the duct geometry on the velocity profile and friction coefficient is studied in detail. The effect of the Brinkman number on the temperature distribution and Nusselt number is investigated for both cooling and heating cases. The results reveal that the critical Brinkman Number distinguishes between the cooling and heating cases and represents the critical point at which the Nusselt number approaches infinity. The value of the Nusselt number decreases with the increase of the Brinkman number in both the wall cooling and heating modes. It is also found that the equilateral triangle exhibits the minimum friction coefficient and the maximum value of the Poiseuille number.

Utilization of the Crank-Nicolson technique to investigate thermal enhancement in 3D convective Walter-B fluid by inserting tiny nanoparticles on a circular cylinder

<abstract> <p>The current study is based on the mechanism of mixed convection and solar thermal radiation in Walters'-B fluid considering tera-hybrid nano-structures using convective boundary constraints (CBC) and (CHF) constant heat flux. The heat transmission phenomenon of the current study is taken into account under the influence of triple-suspended nanoparticles. The current problem has several potential applications, including improvements in solar thermal energy systems, nanofluids, aerospace, cooling processes, automotive engineering, and numerical modeling methods. A numerical approach, namely Crank-Nicolson, is utilized in the modeling of 3D Walter's B fluid past over a 3D circular cylinder whose radius varies sinusoidally for evaluation of velocity and temperature distributions. For mathematical modeling, the Cartesian coordinate system was used for the current study. Comparative analysis between constant heat flux (CHF) and convective boundary constraints (CBC) was demonstrated graphically against multifarious parameters towards the temperature profile and velocity profiles along the x-axis and in the y-axis. Moreover, comparative analysis for dissimilar parameters was manifested for Nusselt number through tables, and graphically for skin friction co-efficient and Nusselt number and has shown excellent accuracy. It was estimated that by enhancing values of <italic>Q<sub>sr</sub></italic>, <italic>C</italic>, <italic>H<sub>s</sub></italic> and <italic>E<sub>c</sub></italic>, it was addressed that temperature curve increases for CHF and CBC cases.</p> </abstract>

Effect of temperature and heat flux boundary conditions on hydrogen production in membrane-integrated steam-methane reformer

In order to operate reforming plants at maximum efficiency it is important to investigate their operation under various conditions. This study aims to perform a comprehensive computational analysis to examine the effect of thermal boundary conditions on SMR and hydrogen recovery and, consequently, enriching the literature on hydrogen generation through steam-methane reforming. The studied thermal boundary conditions include constant and variable inlet temperature, outer wall temperature, and heat flux on outer wall of the reactor. A constant inlet flow rate of 0.001 kg/s (methane mass flow rate = 0.00018 kg/s), steam-to-methane ratio (or steam-to-carbon ratio, i.e., S/C) of 4 is considered in the reformer side for all the cases. Furthermore, the difference of pressure between both faces of the membrane is kept constant. Sweeping flow is incorporated to further raise the hydrogen partial pressure difference. Steam with flow rate of 0.001 kg/s is used to induce the sweeping effect throughout the study. Constant and variable temperatures/heat fluxes along with the effect of radiation in the reformer and transient inlet temperatures are examined in this study in terms of CH4 conversion ( %), H2 recovery ( %), H2 mass flow rates (kg/s), hydrogen mass flux through the membrane (kg.m−2. s−1), and other important parameters. About 1.7 times increase in methane conversion is noted as the reformer temperature is increased from 500 to 700 °C. The hydrogen recovery is found to decrease with the increase in temperature. Around 57 % and 56 % decrease in methane conversion and 42 % and 11.5 % rise in hydrogen recovery is observed with segmented case as compared to the constant heat flux and constant temperature cases, respectively. As compared to the reference 600 °C constant wall temperature, the increasing temperature profile results in 10.6 % rise in methane reforming and 11.2 % drop in the recovery of hydrogen. On the other hand, the decreasing temperature profile demonstrates 9.4 % decrease in conversion and 10.2 % elevation in recovery rate in contrast with the constant temperature case.

Heat Transfer Analysis of Sisko Fluid Flow over a Stretching Sheet in a Conducting Field with Newtonian Heating and Constant Heat Flux

The present study investigates the steady three-dimensional flow of a Sisko fluid over a bidirectional stretching sheet under the influence of Lorentz force. Heat transfer effects have been carried out for constant heat flux and Newtonian heating systems. The transformed governing equations of the flow model are solved by using the spectral relaxation method (SRM), taking into account similarity transformations. The effects of controlling parameters on flow and derived quantities have been presented in the form of graphs and tables. Numerical benchmarks are used to characterise the effects of skin friction and heat transfer rates. It is noticed that in the case of Newtonian heating, the rate of heat transfer is higher than that in the constant heat flux case. As the stretching parameter increases, the fluid temperature decreases in both Newtonian heating and constant heat flux. It was discovered that successive over (under) relaxation (SOR) approaches will considerably boost the convergence speed and stability of the SRM system. The current findings strongly agree with earlier studies in the case of Newtonian fluid when the magnetic field is absent.

Combined influence of cross‐diffusion and variable fluid properties on free convection flow past a vertical cone

AbstractThe free convective flow of an incompressible viscous fluid over an isothermal vertical cone with variable viscosity and variable thermal conductivity is examined in the presence of the Soret and Dufour effects. As thermal and solutal boundary conditions at the cone's surface, the constant temperature and concentration (WTC) and constant heat and mass flux (HMF) cases are taken into account. The successive linearization method is applied to linearize a system of nonlinear differential equations that describes the flow under investigation. The numerical solution for the resulting linear equations is attained by means of the Chebyshev spectral method. The obtained numerical results are compared and found to be in good agreement with previously published results. The impact of significant parameters on the heat and mass transfer rates is evaluated and presented graphically for the WTC and HMF situations. In both cases, the Soret number increases the skin friction coefficient and rate of heat transfer while decreasing the Sherwood number. With an increase in the Dufour parameter, the coefficient of skin friction and Sherwood numbers increase while the heat transmission rate decreases.

Fluid flow and convective heat transfer in a rotating rectangular microchannel with various aspect ratios

The features of flow fluid and convective heat transfer have been investigated numerically in a rotating rectangular U-shaped microchannel for various aspect ratios (AR). Pure water is used as working fluid and 3D steady simulations are performed for Reynolds number of 400. The effects of aspect ratio (AR = 0.25–4), slip/no-slip conditions, rotational speeds in the range of 0–300 rad/s on the velocity profiles/contours, heat transfer, pressure drop, Nusselt number, and thermal performance coefficient are studied. The results show that an increase in AR (for AR>1) or a decrease in AR (for AR<1) provided an increase in pressure drop and heat transfer. Besides, contrary to hydrophilic microchannel, the heat transfer and pressure drop decrease in microchannel with the hydrophobic surfaces for all ARs considered. In addition, the thermal performance coefficient (E) is used as a balance between heat transfer augmentation and the power consumed. It is found that the E is higher for microchannel with hydrophobic surface than that of with hydrophilic one for all ARs examined. Moreover, the rotating microchannel is more efficient in respect to heat transfer enhancement than that of the stationary case, especially at AR = 1. Moreover, the results for two thermal boundary conditions of constant heat transfer and constant wall temperature conditions are compared at AR = 1 and it is found that the total Nusselt number is higher for constant wall temperature case than that of constant heat flux case in rotating microchannel while it is contrary in stationary one.

Forced convection heat transfer from a heated rotating sphere in laminar flow revisited

In a paper published recently in Journal of Heat Transfer (Feng [1] ) the forced convective heat transfer from a heated rotating sphere in laminar flow was investigated numerically using a direct numerical simulation. There were significant discrepancies between [1] and previous studies. In the present paper we solved the same problem with another numerical method. The present fluid dynamics results are compared with the results of two previous numerical works and another experimental work and the comparison was very good. Afterwards the heat transfer problem was solved and we found large discrepancies between our results and those of Feng [1] which reach up to 38% for the isothermal case. New correlations for the mean Nusselt number for air have been developed for isothermal and constant heat flux case. The results for the heat flux case are new in the literature.

A LOOP THERMOSYPHON FOR LIQUID COOLED MINICHANNELS HEAT SINK WITH PULSATE SURFACE HEAT FLUX

The period operation of power electronic acts as switching element, where the power dissipated consists of pulses at certain duty cycle, the semiconductor temperature oscillates and varies as a waveform. In the present study, an experimental investigation was carried out for a loop thermosyphon order to evaluate the effect of pulsate surface heat flux on the single-phase buoyancy driven convection of ethylene glycol flow through a minichannels heat sink with hydraulic diameter 1.5 mm. An electric heater block is used to supply the heat flux to minichannels heat sink in a rectangle waveform. The study is done at different heat flux frequencies of 2.777×10-3 Hz, 8.333×10-4 Hz, 5.555×10-4 Hz and 4.166×10-4 Hz, while the heat flux amplitude (2 watt), Rayleigh number (1864) and duty cycle (50 %) are kept constant. The results revealed that for a range of the measured frequency for the complete power cycle and due to unsteady state operation conditions, the pulse heat flux pattern is close to a rectangle-wave, this generates the fluid outlet temperature pattern close to a triangle-wave. The fluid outlet temperature increases with the decreases of heat flux frequency and tends to reach to the fluid outlet temperature for a constant and continuous heat flux case. Due to closed-loop of thermosyphon, the fluid inlet temperature is changed in pattern like that the fluid outlet temperature change.

Natural convection of viscoplastic fluids in an enclosure with partially heated bottom wall

In this paper, heat transfer characteristics and flow patterns of viscoplastic fluid due to natural convection in a two-dimensional heated enclosure have been investigated both computationally and experimentally. The enclosure is heated partially at the bottom wall at its centre. The side walls are isothermally cooled. The rheology of the enclosed fluid is represented by Bingham model. In addition, an experimental investigation on the viscoplastic test fluids, prepared from Carbopol Ultrez 20 polymer, has also been carried out. The proposed computational model has been validated with our own experimental results. The results in terms of flow and thermal characteristics are presented for different values of Bingham number (Bn) varying between 0 and 30, Rayleigh number (Ra) ranging 104 to 106, Prandtl number (Pr) from 1 to 100 and heated lengths of the bottom wall (ε). The critical value of yield stress of enclosed fluid for which convection circulation diminishes and conduction becomes the sole mode of heat transfer in the enclosure has also been evaluated. The effect of constant heat flux boundary condition at the heated portion has been compared with that of constant temperature (isothermal) boundary conditions. It has been found that convection is weak in constant heat flux case compared to the isothermal condition case.

Heating and cooling conditions effects on the kinetic of phase change of PCM embedded in metal foam

Phase change materials (PCM) are attractive candidates for energy storage. They can store large quantities of energy in small volumes at nearly constant temperatures. Despite their advantage, their thermal conductivity is very low with a high-volume change during the melting and solidification process. One way to increase their poor thermal conductivity is to embed them into open cell metallic foams. In this paper, a numerical study is conducted on the effect of the heating and cooling conditions on phase change kinetics of paraffin embedded in a metal foam. Constant heating and sinusoidal heating are similarly investigated. For the constant heat flux, a step function ranging from +1800 W/m2 to −1800 W/m2 is considered, while for the variable heat flux, a sinusoidal function having a similar area as step function is considered at one wall of the container to provide heating and cooling of the PCM/Metal foam composite. A new mathematical model based on the Brinkmann-Forchheimer-extended Darcy equation and the local thermal non-equilibrium model (LTNE) is proposed by applying a two-energy equation. The paraffin phase change is modeled using the enthalpy-porosity method. The numerical results are validated by comparing them with the experimental data. The results showed that at the time of melting it has reduced with sinusoidal heating. The results also showed that the heat losses on the boundary have a greater effect in a sinusoidal heat flux case than in constant heat flux case and this effect is more important on the solidification than on the melting process of the paraffin.

Conjugate forced convective heat transfer in a sandwich panel with a Kagome truss core: The effects of strut length and diameter

Heat exchangers with a cellular metal core have become an alternative to those with louvered fins due to their high heat transfer performance and mechanical strength. Because of the multi-functionality, they are mostly used in systems such as rockets and space vehicles or in others in need of both high heat transfer performance and mechanical strength. The geometrical parameters of the unit cell of cellular metals have important effects on both fluid and heat transfer characteristics and these effects need to be studied. Accordingly, conjugate forced convective heat transfer in a sandwich panel with a Kagome truss core both with a constant heat flux (CHF) and a constant temperature (CT) boundary condition was studied numerically in this study for various values of the Reynolds number, strut length, and strut diameter. The 3D turbulent flow and heat transfer was modeled by the RNG k-epsilon turbulence model. The near-wall behavior of the flow was modelled by enhanced wall treatment. The channel was extended before and after the Kagome core to avoid entrance and exit effects. The computational results were obtained for the Reynolds numbers in the range of about 6000–17000, for the strut lengths of 3, 5 and 7 mm, and for the strut diameters of 1.3 and 1.6 mm. A nonlinear regression analysis was also performed for the pressure loss coefficient and average Nusselt number. The numerical results show that complex flow structures develop in the flow field due to the presence of unit cells. The results also show that the pressure drop along the Kagome core increases considerably with decreasing strut length and increasing strut diameter. The average Nusselt number increases significantly when the strut length is decreased and shows a small increase with an increase in the strut diameter. The effect of strut diameter on average Nusselt number becomes marginal for strut length of 3 mm. The average heat transfer rate for a fixed pumping power shows a considerable increase with a decrease in the strut length and a small increase with the strut diameter for high values of strut length. On the other hand, for the strut length of 3 mm, it shows a small decrease with an increase in the strut diameter for the CT case and remains almost constant for the CHF case.

Correction for semi-infinite assumption in the theories of natural convection and determination of average Nusselt number for finite inclined plates

It is established that large errors would be incurred if the average Nusselt number for natural convection above heated horizontal plates of finite size is determined by the similarity or integral theories which assume the plate is semi-infinite. Recognizing that boundary layers grow from both edges of a finite plate, correction factors are mathematically deduced for the isothermal case (Ft=22/5) and for the case of constant heat flux (Fq=21/3). Correction factors are also developed for near-horizontal orientations of the heated plate. The semi-infinite assumption does not alter the computation of the overall heat transfer rate for inclination angles at which the vertical mechanism of natural convection is dominant since then, even in the case of a finite plate, a single boundary layer covers the entire length of the plate. The theoretical rationalization developed in the paper brings the predictions of explicit algebraic expressions for average Nusselt number in line with those determined accurately by computational fluid dynamics (with extensive experimental verification), and provides physical explanation of the subtle thermo-fluid-dynamics of natural convection established by the present CFD simulations.

Study of non-isothermal incompressible flow and heat flux of nano-ferrofluid with induced magnetic induction

Presently we have studied the nonlinear magnetohydrodynamic convective boundary-layer flow of water- based nano-ferrofluid adjacent to a non-isothermal stretching sheet. Three different ferrofluid suspensions are considered, namely water containing magnetite (Fe3O4), (Mn − Zn)and cobalt ferrite (CoFe2O4) ferriteferro-particles. Magnetic field is sufficiently strong to induce induction effects. Applying similarity transformations, the boundary region equations for mass, momentum, magnetic induction and energy are reduced to coupled, multi-degree, nonlinear differential equations. The non-dimensional boundary value problem is then solved computationally under specified wall and free stream conditions, with Maple quadrature and shooting techniques. Non-isothermal as well as constant heat flux cases are examined. The key parameters emerging in the model are the reciprocal of magnetic Prandtl number, Prandtl number, the reciprocal of magnetic body force parameter, heat source/sink parameter, nano-particle volume fraction and power-law index. The influence of these parameters on flow fields like velocity, temperature, and gradient of magnetic stream function for different ferro-particle cases, is presented graphically. Detailed elucidation of the results is also included. Validation is conducted with a finite element method (FEM). Significantly greater temperatures are observed for magnetite ferro-particles as compared with cobalt or Manganese-Zinc ferroparticles. Stream function (Magnetic) is found to be suppressed with increasing power-law index. With increasing magnetic force number there is a substantial elevation in induced magnetic field. Increasing reciprocal of magnetic Prandtl number i.e. decreasing magnetic Prandtl number leads to a depression in velocity i.e. flow deceleration, for the Fe3O4 − H2O magnetite ferrofluid. The study proposed here is of concern to the processing of nano-ferro-materials.

NUMERICAL STUDY ON THE IMPORTANCE OF THE MIXED CONVECTION AND THE IMPACT OF THE COOLING RATE ON THE SOLIDIFICATION OF A PURE MATERIAL

The present paper investigates numerically the solidification process of a pure metal, injected inside a channel, aiming to overview theoretically this complex phase change process, using COMSOL Multiphysics. The study is based on the Voller and Prakash model with fixed grid technique [1]. Results interpretation have been steered toward guessing the effect on the whole process of a varying entrance melt velocity; having proven that melt flow has a negative effect on the solidification rate. Also, the decreasing mid values of temperature transition (MVTT) would give fine numerical solution because it offers less thick mushy zone; this allows a good understanding to the solidification rate, mushy zone spreading and flow structure nearby solid region. Then, two different variable cooling rates functions were suggested comparing them to constant heat flux case. Results indicate that linear function cooling heat flux is the best function in terms of economy of cooling costs.

Exact Solutions and Stability Analysis of a Nonlinear Model of Open-Ocean Deep Convection that Allows Multiple Steady States

We present analytical solutions of the two-basin model of open-ocean deep convection. Originally suggested by Whitehead [Whitehead, J.A. (2000), Stratified convection with multiple states, Ocean Modelling, 2(3-4), 109-121], the model allows regimes with multiple steady states (multiple equilibria). We provide the full analytical description of the steady states for the particular case of constant surface heat flux from the ocean to the atmosphere, and explore analytically stability of the equilibria within the Lyapunov theory. The results show that, for this particular case, the steady state is unique and stable for all dynamic flow regimes. We also present analytical expressions for dependence of critical values of sea-surface heat flux, at which transitions between the dynamic regimes occur, on the model parameters.

Laminar Forced Convection in Transversely Corrugated Microtubes

Forced convection heat transfer in fully developed laminar flow in transversely corrugated tubes is investigated for nonuniform but constant wall heat flux as well as for constant wall temperature. Epitrochoid conformal mapping is used to map the flow domain onto the unit circle in the computational domain. The governing equations are solved in the computational domain analytically. An exact analytical solution for the temperature field is derived together with closed form expressions for bulk temperature and Nusselt number for the case of the constant heat flux at the wall. A variable coefficient Helmholtz eigenvalue problem governs the case of the constant wall temperature. A novel semi-analytical solution based on the spectral Galerkin method is introduced to solve the Helmholtz equation. The solution in both constant wall heat flux and constant wall temperature case is shown to collapse onto the well-known results for the circular straight tube for zero waviness.

Unsteady Free Convection Boundary Layer Flows of a Bingham Fluid in Cylindrical Porous Cavities

We consider two unsteady free convection flows of a Bingham fluid when it saturates a porous medium contained within a vertical circular cylinder. The cylinder is initially at a uniform temperature, and such flows are then induced by suddenly applying either a new constant temperature or a nonzero heat flux to the exterior surface. As time progresses, heat conducts inwards and this may or may not overcome the yield threshold for flow. For the constant temperature case, flow begins immediately should the parameter, Rb, which is a nondimensional yield parameter, be sufficiently large. The ultimate fate, though, is full immobility as the cylinder eventually tends towards a new constant temperature. For the constant heat flux case, the fluid remains immobile but will begin to flow eventually should Rb be sufficiently large. The two cases have different critical values for Rb.

Solution of the Phase Change Stefan Problem With Time-Dependent Heat Flux Using Perturbation Method

Theoretical understanding of phase change heat transfer problems is of much interest for multiple engineering applications. Exact solutions for phase change heat transfer problems are often not available, and approximate analytical methods are needed to be used. This paper presents a solution for a one-dimensional (1D) phase change problem with time-dependent heat flux boundary condition using the perturbation method. Two different expressions for propagation of the phase change front are derived. For the special case of constant heat flux, the present solution is shown to offer key advantages over past papers. Specifically, the present solution results in greater accuracy and does not diverge at large times unlike past results. The theoretical result is used for understanding the nature of phase change propagation for linear and periodic heat flux boundary conditions. In addition to improving the theoretical understanding of phase change heat transfer problems, these results may contribute toward design of phase change based thermal management for a variety of engineering applications, such as cooling of Li-ion batteries.

Numerical study of natural convection in a cavity with discrete heat sources

Natural convection heat transfer in a cavity has always attracted researchers interest because of its numerous applications in engineering, e.g. , cooling of electronic parts, thermal insulators, buildings ventilation systems, solar collectors, and nuclear reactors. In general, due to favorable factors, such as process simplicity, cost-effectiveness, low noise, and the possibility of recovery, the process of natural convection has many uses in various industrial applications. In this study, a square cavity with air inside is considered. Three heaters are situated at the bottom wall, the top wall is maintained at a constant cold temperature, and the two side walls are insulated. The lattice Boltzmann method is used for simulation, and the overall goals are to optimize the installation location and heater length and also investigating the effects of the amplitude and oscillation period of heat flux fluctuation. The results indicate that increasing the difference between amplitudes and oscillation periods of heat flux in heaters causes the flow within the cavity to stabilize more quickly and also increases temperature oscillation due to larger amplitudes and periods. Also, in the cavity with three heaters, the average temperature of the middle heater remains unchanged relative to the case of constant heat flux. This research can be used in the design of an appropriate cooling system for electronic components to ensure effective and safe operational conditions.

Heat transfer of laminar pulsating flow in a rectangular channel

Pulsating flow has been found to both enhance and reduce heat transfer, although the explicit conditions that deliver heat transfer enhancement have yet to be identified. The current work builds on experimental and theoretical hydrodynamic analyses of earlier studies to investigate the effect of flow rate pulsations on the driving temperature difference for the case of constant heat flux, using a novel analytical solution to the energy equation in a rectangular channel. It is found that the oscillating temperature profiles are formed primarily as a result of fluid displacement against the temperature gradient, although diffusion at low frequencies and low Prandtl numbers obscures the mechanism. The temperature gradient at the wall is fixed for the case of constant heat flux, despite enhanced velocity gradients at the wall. The overall time-averaged change in Nusselt number with respect to steady flow is universally negative due to an enhanced axial heat flow towards the channel entrance, in agreement with similar studies in pipes. However, analysis of the time-dependent bulk temperature predicts that heat transfer is enhanced over the second half-cycle, where the flow rate is below it’s mean value. Hence, future work should investigate heat transfer enhancement using truncated sinusoidal flow rates.