Heat transfer in the magnetohydrodynamic flow of a power-law fluid past a porous flat plate with suction or blowing

An analysis is made of the steady magnetohydrodynamic flow of a power-law fluid past an infinite porous flat plate subjected to suction or blowing. A uniform transverse magnetic field is applied normal to the plate. It is shown that for small magnetic field parameter M, the steady solutions for velocity distribution exist for a pseudoplastic (shear-thinning) fluid for which the power-law index n satisfies 1/2 < n ≤ 1 provided that there is suction at the plate. For blowing at the plate the steady solutions for velocity distribution exist only when n is of the form p/q, where p is an odd positive integer and q is an even positive integer provided 1/2 < n < 1. Velocity at a point is found to increase with increase in M. The solution of the energy equation governing temperature distribution in the flow of a pseudoplastic fluid past an infinite porous plate subjected to uniform suction reveals that the temperature at a given point increases with increase in M.     

Numerical investigation of natural convection phenomena in a uniformly heated circular cylinder immersed in square enclosure filled with air at different vertical locations

In this model, a numerical study of two dimensional steady natural convection is performed for a uniform heat source applied on the inner circular cylinder in a square air (Pr = 0.7) filled enclosure in which all boundaries are assumed to be isothermal (at a constant low temperature). The developed mathematical model is governed by the coupled equations of continuity, momentum and energy and is solved by finite volume method. The effects of vertical cylinder locations and Rayleigh numbers on fluid flow and heat transfer performance are investigated. Rayleigh number is varied from 10³ to 10⁶ and the location of the inner cylinder is changed vertically along the centerline of the enclosure from − 0.25 L to 0.25 L upward and downward, respectively. It is found that at small Rayleigh numbers does not have much influence on the flow field while at high Rayleigh numbers have considerable effect on the flow pattern. In addition, the numerical solutions yield a two cellular flow field between the inner cylinder and the enclosure. Also, the total average Nusselt number behaves nonlinearly as a function of locations. Results are presented in terms of the streamlines, isotherms, local and average Nusselt numbers. Detailed results of the numerical has been compared with literature ones, and it gives a reliable agreement.     

Transient laminar compressible boundary layers over a permeable circular cone near a plane of symmetry

The transient laminar compressible boundary layer over a circular cone at an angle of attack near a plane of symmetry in hypersonic flow has been investigated. The case of the boundary layer near the windward and leeward planes has been considered. The effect of suction is included in the analysis which plays an important role in obtaining unique solution. We have examined the situation where the flow is steady at time t = 0 and at time t > 0, the total enthalpy at the wall is suddenly increased and subsequently maintained at that value. This imports unsteadiness in the flow field. The effects of the variable fluid properties, non-unity Prandtl number and viscous dissipation are considered. By suitable transformations, the coupled nonlinear parabolic partial differential equations with three independent variables governing the flow have been reduced to partial differential equations with two independent variables. The resulting partial differential equations have been solved by using an implicit finite-difference scheme in combination with the quasilinearization technique. Computations have been carried out from the initial steady state to the final steady state. It is found that in a small time interval immediately after the start of the impulsive motion, the direction of the heat transfer changes. The surface shear stresses in the streamwise and cross-wise directions and the surface heat transfer, in general, increase with time and attain final steady state values rather quickly (i.e., spin-up time is small). The total enthalpy at the wall strongly affects the surface shear stresses in the streamwise and cross-flow directions and the surface heat transfer, the suction strongly affects the surface shear stress in the streamwise direction and the surface heat transfer, and the cross-flow parameter strongly affects only the cross-flow surface shear stress.     

Experimental and simulation studies of primary vacuum freeze-drying process of random solids at microwave heating

This paper concerns experimental and theoretical studies of freeze-drying process at microwave heating. Two kinds of random solids were dried: material which are assumed to have no internal porosity (ground glass), as well as one containing internal porosity (Sorbonorit 4 activated carbon). Formulated one-dimensional two-region model of freeze-drying process at microwave heating takes into account unknown a priori sublimation temperature T_s(t) and mass concentration of water vapor C_s(t) at moving ice front. Steady capacity of internal heat source is correlated with electric field strength E and dissipation coefficient K(T) in both regions of the material to be dried. Linear temperature dependency of dissipation coefficient is assumed and described by two regression parameters: μ_1i and μ_2i for dry (i = I) and frozen (i = II) bed, respectively. A correlation between both measured and calculated temperatures of the sample and actual electric field strength was observed. Fairly good agreement between experimental and simulated results was stated.     

Three-dimensional thermocapillary-buoyancy flow in a shallow molten silicon pool with Cz configuration

In order to understand the characteristics of surface patterns on silicon melt in Czochralski furnaces, we conducted a series of unsteady three-dimensional numerical simulations of thermocapillary-buoyancy flow of a shallow molten silicon pool with Czochralski configuration (depth d = 3 mm). The crucible sidewall is maintained at constant temperature. Bottom and free surfaces are adiabatic or allow heat transfer in the vertical direction. The simulation results indicate that two flow transitions occur with increasing the radial temperature difference along the free surface. At first, the steady two-dimensional flow becomes steady three-dimensional flow and then oscillatory three-dimensional flow. The critical conditions for the onset of the instability were determined. Characteristics of the steady and the oscillatory three-dimensional flows were discussed.     

Flow and heat transfer of an electrically conducting fluid of second grade over a stretching sheet subject to suction and to a transverse magnetic field

An analysis is performed for flow and heat transfer of a steady laminar boundary-layer flow of an electrically conducting fluid of second grade subject to suction and to a transverse uniform magnetic field past a semi-infinite stretching sheet. The governing partial differential equations are converted into ordinary differential equations by a similarity transformation and an analytical solution for this flow is utilized. The effects of viscous dissipation and work due to deformation are considered in the energy equation and the variations of dimensionless surface temperature and dimensionless surface temperature gradient with various parameters are graphed and tabulated. Two cases are studied, namely, (i) the sheet with constant surface temperature (CST case) and (ii) the sheet with prescribed surface temperature (PST case).     

Numerical simulation of steady natural convection heat transfer in a 3-dimensional single-ended tube subjected to a nanofluid

In this paper, 3-dimensional numerical simulation of steady natural convective flow and heat transfer are studied in a single-ended tube with non-uniform heat input. Apart from some other applications, it serves as a simplified model of the single-ended evacuated solar tube of a water-in-glass evacuated tube solar water heater. It is assumed that the sealed end of tube to be adiabatic and also the tube opening to be subjected to copper–water nanofluid. Governing equations are derived based on the conceptual model in the cylindrical coordinate system. The governing equations have been then approximated by means of a fully implicit finite volume control method (FVM), using SIMPLE algorithm on the collocated arrangement. The study has been carried out for solid volume fraction 0 ≤ φ ≤ 0.05 and maximum heat flux 100 ≤ q_m ≤ 700. Considering that the driven flow in the tube is influenced by the dimensions and the inclination angle of the solar tube, the flow patterns and temperature distributions are presented on different cross sectional planes and longitudinal sections, when the tube is positioned at different orientations.     

MHD natural convection in a laterally and volumetrically heated square cavity

A numerical study is presented of unsteady two-dimensional natural convection of an electrically conducting fluid in a laterally and volumetrically heated square cavity under the influence of a magnetic field. The flow is characterized by the external Rayleigh number, Ra_E, determined from the temperature difference of the side walls, the internal Rayleigh number, Ra_I, determined from the volumetric heat rate, and the Hartmann number, Ha, determined from the strength of the imposed magnetic field. Starting from given values of Ra_E and Ha, for which the flow has a steady unicellular pattern, and gradually increasing the ratio S = Ra_I/Ra_E, oscillatory convective flow may occur. The initial steady unicellular flow for S = 0 may undergo transition to steady or unsteady multicellular flow up to a threshold value, Ra_I,cr, of the internal Rayleigh number depending on Ha. Oscillatory multicellular flow fields were observed for S values up to 100 for the range 10⁵-10⁶ of Ra_E studied. The increase of the ratio S results usually in a transition from steady to unsteady flow but there have also been cases where the increase of S results in an inverse transition from unsteady to steady flow. Moreover, the usual damping effect of increasing Hartmann number is not found to be straightforward connected with the resulting flow patterns in the present flow configuration.     

The viscous dissipation effect on heat transfer and fluid flow in micro-tubes

The numerical modeling of the conjugate heat transfer and fluid flow through the micro-tube was presented in the paper, considering the viscous dissipation effect. Three different fluids with temperature dependent fluid properties are considered: water and two dielectric fluids, HFE-7600 and FC-70. The diameter ratio of the micro-tube was D_i/D_o = 0.1/0.3 mm with a tube length L = 100 mm. The laminar fluid flow regime is analyzed. Two different heat transfer conditions are considered: heating and cooling and three different Br = 0.01, 0.1 and 0.5. The influence of the viscous heating on Nu and Po is analyzed and compared with Br = 0.     

Measurements of current and water distribution for a micro-PEM fuel cell with different flow fields

Time-dependent measurements for the mapping of current distribution via a segmented flow field plate approach in a micro-proton exchange membrane (PEM) fuel cell were conducted and the effects of flow field configuration were studied and discussed under fixed operating conditions. The results show that, among four flow fields studied herein, the interdigitated flow channel has the most uniform transient current distribution with a much higher water content at an early phase (say t < 0.5 h) than those of the other three channels: serpentine, mesh and parallel, indicating an adequate oxygen concentration of the airflow on the cathode. In addition, the effect of water content on current distribution was also examined and discussed. It was found that the volume of water in flow channels could reach a steady value of 45% for all four flow fields after a 3-h operation.     

A numerical study on entropy generation induced by turbulent forced convection in curved rectangular ducts with various aspect ratios

The present paper analyzes the entropy generation induced by turbulent forced convection in a curved rectangular duct with external heating by numerical methods. The problem is assumed as steady, three-dimensional and turbulent. The flow features, including the secondary flow motions, the distribution of local entropy generation as well as the overall entropy generation in the whole flow fields, are analyzed. For a baseline case with Re = 20,000, external heat flux q? = 0.112 and aspect ratio γ = 1, the results show the entropy generation induced by the frictional irreversibility concentrates within the regions adjacent to the duct walls, whereas the entropy generation resulted from the heat transfer irreversibility only significantly occurs near the outer wall of the duct where the external heat flux imposed. Except the baseline case, two additional cases with aspect ratio equal to 0.25 and 4 are calculated. Through the comparison of the three aspect-ratio cases, it is seen that the resultant entropy generations in the flow fields for the three cases are all dominated by the frictional irreversibilities. Among the three aspect-ratio cases, the resultant entropy generation is minimal in the γ = 1 case. Accordingly, the case with γ = 1 is concluded to be the optimal aspect ratio under the current flow condition based on the minimal entropy generation principle.     

Local analysis of heat transfer inside corrugated channel

Experiments are performed to study effects of hydrodynamic conditions on the enhancement of heat transfer for single phase flow. These experiments have been conducted for a wide range of Reynolds numbers, (0 < Re < 7500) in order to obtain the different regimes from steady laminar to turbulent. A two-dimensional corrugated test section which has been instrumented with thermocouples can be heated by electrical cartridges. The local temperature measurements are used to evaluate the local and global heat transfer coefficient of the wavy heat exchanger. As expected, the heat transfer is always higher than those in rectangular channel; it is essentially due to the mixing induced by the recirculation in the wake of the corrugations.     

Heat transfer enhancement by multiple swirling impinging jets with twisted-tape swirl generators

The flow and heat transfer characteristics of multiple swirling impinging jets (M-SIJs) with 3 × 3 in-line arrangement, on impinged surfaces are reported. The experiments were conducted with four different jet-to-jet distances (S/D = 2, 4, 6 and 8) at the constant nozzle-to-plate distance of L/D = 4. The swirling jets with the swirl numbers of 0.4 were associated with twisted tapes. The multiple conventional impinging jets (M-CIJs) were also tested, for comparison. The flow patterns on an impinged surface were visualized using oil film technique while the distributions of temperature field and Nusselt number on impinged surface were evaluated via a thermochromic liquid crystal (TLC) sheet coupled with image processing technique. The experimental results showed that the M-SIJs offered higher heat transfer rate on impinged surfaces than the M-CIJs of all jet-to-jet distances (S/D).     

Two-dimensional mixed convection heat transfer from confined tandem square cylinders in cross-flow at low Reynolds numbers

A two-dimensional numerical simulation is carried out to understand the effects of thermal buoyancy and Prandtl number on flow characteristics and mixed convection heat transfer over two equal isothermal square cylinders placed in a tandem arrangement within a channel at low Reynolds numbers. The spacing between the cylinders is fixed with four widths of the cylinder. The numerical results are presented for the range of conditions as: 1 ≤ Re ≤ 30, 0.7 ≤ Pr ≤ 100 (the maximum value of Peclet number being 3000) and 0 ≤ Ri ≤ 1 for a fixed blockage parameter B = 10%. The unsteady numerical simulations are performed with a finite volume code based on the PISO algorithm in a collocated grid system. The representative streamlines, vortex structures and isotherm patterns are presented and discussed. In addition, the overall drag and lift coefficients, recirculation length and average Nusselt numbers are determined to elucidate the role of Reynolds, Prandtl and Richardson numbers on flow and heat transfer. It is found that the flow is completely steady for the chosen ranges of the parameters.     

High-resolution simulations of magnetohydrodynamic free convection in an enclosure with a transverse magnetic field using a velocity–vorticity formulation

The differential quadrature method (DQ) is employed to simulate the effect of a transverse magnetic field on buoyancy-driven magnetohydrodynamic (MHD) flow in an enclosure. The DQ numerical procedure is adopted for solving the velocity–vorticity form of Navier–Stokes equations in two dimensions. These equations together with respective (appropriate) boundary conditions are solved numerically using a DQ method by a coupled algorithm for the velocity–vorticity–temperature coupled together with a bi-conjugate gradient iterative solver technique. The velocity–vorticity formation is properly utilized to obtain results in the range of Grashof numbers (10⁴ ≤ Gr ≤ 10⁵), Hartmann numbers (0 ≤ Ha ≤ 100), and Prandtl numbers (0.01 ≤ Pr ≤ 10), as well as aspect ratios A = L/H varying from 1 to 3 in a differentially heated cavity with a transverse magnetic field. The algorithm is then employed to compute the average Nusselt numbers and flow parameters for all the cases. With support from the present simulations of the heat transfer characteristics for a constant value of Gr within the cavity, the heat transfer rate is at its maximum for higher Pr and in the absence of MHD effects (Ha = 0), while it is lower with increase in external magnetic field strength in the lower region of the Prandtl number.     

Numerical analysis of entropy generation in mixed convection flow with viscous dissipation effects in vertical channel

A numerical investigation is performed into the entropy generated within a mixed convection flow with viscous dissipation effects in a parallel-plate vertical channel. In performing the analysis, it is assumed that the flow within the channel is steady, laminar and fully developed. The governing equations for the velocity and temperature fields in the channel are solved using the differential transformation method. The numerical results for the velocity and temperature fields are found to be in good agreement with the analytical solutions. The entropy generation number (N_s), irreversibility distribution ratio (Φ) and Bejan number (B_e) of the mixed convection flow are obtained by solving the entropy generation equation using the corresponding velocity and temperature data.     

Transient natural convection in 3D tilted enclosure heated from two opposite sides

In the present work, we investigate numerically the natural convection flow in 3D cubic enclosure tilted at an angle (γ) with respect to the vertical position. The enclosure is heated and cooled from the two opposite walls while the remaining walls are adiabatic. The numerical procedure adopted in this analysis yield consistent performance over a wide range of parameters. Simulations have been carried out for Rayleigh numbers Ra ranging from 10³ to 1.3 × 10⁵, Prandtl number, Pr, (0.71 ≤ Pr ≤ 75) and inclination angle γ (0° ≤ γ ≤ 90°). Particular attention is focused on the three-dimensional steady effects that can arise in such configuration that seem to be unknown in the literature, even for relatively small values of the Rayleigh number. The 3D flow characteristics and thermal fields are analyzed in terms of streamlines, isotherms and Nusselt numbers. A periodic behavior of the 3D flow has been observed at Ra = 8.5 × 10⁴ with a fundamental frequency of 8.27. The Hopf bifurcation is localized. In addition, time-dependent solutions reveal that the flow characteristics depend on the inclination angle γ. The effects of Prandtl number on heat transfer and fluid flow is significant for Pr ≥ 6.     

Numerical analysis of natural convection for a porous rectangular enclosure with sinusoidally varying temperature profile on the bottom wall

Numerical investigations of steady natural convection flow through a fluid-saturated porous medium in a rectangular enclosure with a sinusoidal varying temperature profile on the bottom wall were conducted. All the walls of the enclosure are insulated except the bottom wall which is partially heated and cooled. The governing equations were written under the assumption of Darcy-law and then solved numerically using finite difference method. The problem is analyzed for different values of the Rayleigh number Ra in the range 10 ≤ Ra ≤ 1000, aspect ratio parameter AR in the range 0.25 ≤ AR ≤1.0 and amplitude λ of the sinusoidal temperature function in the range 0.25 ≤ λ ≤ 1.0. It was found that heat transfer increases with increasing of amplitude λ and decreases with increasing aspect ratio AR. Multiple cells were observed in the cavity for all values of the parameters considered.     

Numerical investigation on developing laminar forced convection and entropy generation in a wavy channel

The present paper investigates the developing laminar forced convection and entropy generation in a wavy channel with numerical methods. The effects of aspect ratio (W/H) and Reynolds number (Re) on entropy generation are the major concerns. The studied cases cover W/H = 1, 2 and 4, and Re range from 100 to 400. The flow features, including secondary flow motion and temperature distribution as well as the detailed distributions of local entropy generation due to frictional and heat transfer irreversibilities are reported. Through the evaluations of entropy generation in the whole flow field, the case of W/H = 1 is found to have the minimal entropy generation among all of the analyzed cases. Besides, the higher Re is found to be beneficial for obtaining the lower values of the total resultant entropy generation in the flow field. Accordingly, the case with W/H = 1 and higher Re is suggested to be used under the current flow conditions, so that the irreversibility resulted from the developing laminar forced convection in the wavy channel could be least and the best exergy utilization could be achieved.     

Predictions of buoyancy-induced flow in various across-shape concave enclosures

The present study was conducted to numerically investigate the steady laminar buoyancy-driven and convection heat transfer characteristics within three different across-shape concave enclosures for the Prandtl number of 0.71 and 4, the Grashof number range 10⁴ ≤ Gr ≤ 2 × 10⁵_, and the gap range 0 ≤ H₁/H₂ ≤ 0.25. The steady Navier-Stokes equations, governing the flow under Boussinesq approximation, are solved with the dimensionless stream function-vorticity formulation in terms of curvilinear coordinates using the finite difference method. The results show that the effects of various shapes, the strength of the vortex is relatively bigger in the rectangular-rectangular concave enclosure than in the rectangular-circular concave enclosure at the same Grashof number. Heat transfer from the different across-shape concave enclosures is evaluated, and flow and heat transfer characteristics are discussed.