On adiabatic-isothermal mixed boundary conditions in heat transfer

Mixed boundary conditions of the adiabatic-isothermal type are often encountered in heat transfer problems. In some cases, singular behaviour may lead to erroneous results in determining the temperature and flow fields and evaluating the heat transfer. In this study, steady heat conduction in a square plate with mixed boundary condition on a straight boundary has been studied using finite difference and control volume methods. A method has been proposed to overcome the difficulties encountered in singular cases. Convergence criteria used in the numerical treatment of these problems have been discussed.

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Über Mischrandbedingungen adiabat/isotherm in der Wärmeübertragung

Zusammenfassung Mischrandbedingungen vom Typ adiabat/isotherm treten häufig bei Wärmeübertragungsproblemen auf. In manchen Fällen führt das singuläre Verhalten an gewissen Stellen zu fehlerhaften Resultaten für die Temperatur- und Strömungsfelder und damit für das Wärmeübertragungsverhalten. In dieser Untersuchung wird die stationäre Wärmeleitung in einer quadratischen Platte unter einer Mischrandbedingung an einem geraden Randstück betrachtet, und zwar mittels der Methoden der Finiten Differenzen und des Kontrollvolumenprinzips. Zur Überwindung der an singulären Stellen auftretenden Schwierigkeiten empfiehlt sich ein spezielles Verfahren. Die bei der numerischen Behandlung dieses Problems eingesetzten Konvergenzkriterien werden diskutiert.

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List of symbols [A] coefficient matrix - [b] right hand side vector - h step size, corner width - l computation domain length, height - Nu Nusselt number - q heat flux - Ra Rayleigh number - Q _L heat flux through the lower boundary - Q _U heat flux through the upper boundary - T temperature - U, V velocity components - [x] solution vector - x, y coordinate system Greek letters error defined in Eq. (4) - error defined in Eq. (5) - stream function Financial support by Natural Sciences and Engineering Research Council of Canada is acknowledged.     

Analysis of viscous dissipation effect on thermal entrance heat transfer in laminar pipe flows with convective boundary conditions

Consideration is given to the influence of viscous dissipation on the thermal entrance region laminar pipe flow heat transfer with convective boundary condition. The Eigenfunction series expansion technique is employed to solve the governing energy equation. The results for axial distributions of dimensionless bulk and wall temperatures, local Nusselt number as well as modified local Nusselt number are presented graphically forNu ₀ =0.1, 2, and 100. The complicated variations of conventional local Nusselt number is due to the inappropriate definition of conventional heat transfer coefficient in this problem. A modified local heat transfer coefficient, based on the difference of bulk fluid temperature and wall temperature, is introduced. Its value can clearly indicate the extent and the direction of heat exchange between the fluid in the pipe and the ambient. The effects of outside Nusselt number are also investigated. Significant viscous dissipation effects have been observed for large Br.     

Exact solutions for variable-thickness inhomogeneous elastic plates under various boundary conditions

In this paper, an exact solution to the governing equations of the bending of a variable-thickness inhomogeneous rectangular plate is presented. The procedure is applicable to variable-thickness inhomogeneous rectangular plates with two opposite edges simply supported. The remaining ones subjected to a combination of clamped, simply supported, and free boundary conditions and between these two edges the plate may have varying thickness. The procedure is valuable in view of the fact that tables of deflections and stresses cannot be presented for variable-thickness inhomogeneous orthotropic plates as for uniform-thickness homogeneous isotropic plates even for commonly encountered loads because the results depend on the inhomogeneity coefficient and the orthotropic material properties instead of a single flexural rigidity. Numerical results, useful for the validation or otherwise of approximate solutions, are tabulated. The influences of the degree of the inhomogeneity, aspect ratio, thickness parameter and degree of non-uniformity on the deflections and stresses are investigated. This paper is partially supported by the Deanship of Scientific Research at King AbdulAziz University (Grant no. 172/427).     

Flow and heat transfer of nanofluids over stretching sheet taking into account partial slip and thermal convective boundary conditions

The effect of flow slip on the nanofluid boundary layer over a stretching surface is studied. The present results provide a basic understanding on the effects of the slip boundary condition on heat and mass transfer of nanofluids past stretching sheets subject to a convective boundary condition from below. The results show that an increase of thermophoresis parameter or slip factor would decrease the reduced Nusselt number in some cases.     

Exact solutions of multi-term fractional difusion-wave equations with Robin type boundary conditions

General exact solutions in terms of wavelet expansion are obtained for multiterm time-fractional difusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial diferential equations are converted into time-fractional ordinary diferential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-difusion problems are given to validate the proposed analytical method.     

Exact solutions of multi-term fractional diffusion-wave equations with Robin type boundary conditions

General exact solutions in terms of wavelet expansion are obtained for multi- term time-fractional diffusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial differential equations are converted into time-fractional ordinary differ- ential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-diffusion problems are given to validate the proposed analytical method.     

Effects of outflow boundary condition on convective heat transfer with strong recirculating flow

Three practices of treating outflow boundary condition were adopted in computations for convective heat transfer of a two-dimensional jet impinging in a rectangular cavity. The three practices were local mass conservation method, local one-way method and fully developed assumption. The numerical solutions of the three methods were compared with test data obtained via, naphthalene sublimation technique. It was found that the fully developed assumption was inappropriate, and the local one-way method could provide reasonably good results for the cavity bottom, while for the lateral wall the results with this method qualitatively differed from the test data. The solution with the local mass conservation method was the best. It thus suggested that for a problem expected with a strong recirculating flow at the exit of the computation domain, the local mass conservation method be adopted to treat the outflow boundary condition.     

Effects of Lewis number on coupled heat and mass transfer in a circular tube subjected to external convective heating

Heat and mass transfer in a circular tube subject to the boundary condition of the third kind is investigated. The closed form of temperature and concentration distributions, the local Nusselt number based on the total external heat transfer and convective heat transfer inside the tube, as well as the Sherwood number were obtained. The effects of Lewis number and Biot number on heat and mass transfer were investigated.     

Numerical experiments on convective heat transfer in water-saturated porous media at near-critical conditions

Fluid and heat flow at temperatures approaching or exceeding that at the critical point (374 °C for pure water, higher for saline fluids) may be encountered in deep zones of geothermal systems and above cooling intrusives. In the vicinity of the critical point the density and internal energy of fluids show very strong variations for small temperature and pressure changes. This suggests that convective heat transfer from thermal buoyancy flow would be strongly enhanced at near-critical conditions. This has been confirmed in laboratory experiments. We have developed special numerical techniques for modeling porous flow at near-critical conditions, which can handle the extreme nonlinearities in water properties near the critical point. Our numerical simulations show strong enhancements of convective heat transfer at near-critical conditions; however, the heat transfer rates obtained in the simulations are considerably smaller than data reported from laboratory experiments by Dunn and Hardee. We discuss possible reasons for this discrepancy and develop suggestions for additional laboratory experiments.     

Exact solutions for extended boundary value problems

Summary Extended definition of a stress tensor for a non-Newtonian fluid brings in higher degree derivatives with coefficients as powers of non-Newtonian parameter in the differential equations of motion. Yet, these differential equations need to be solved subject to the same boundary conditions as in the corresponding Newtonian flow problem. A technique is developed to obtain exact solutions for such an extended boundary value problem. Some flow problems forWalters liquidB are considered.     

Experimental characterization of convective heat transfer with MWCNT based nanofluids under laminar flow conditions

This study describes an investigation on the convective heat transfer performance of aqueous suspensions of multiwalled carbon nanotubes. The results suggested an increase on heat transfer coefficient of 47 % for 0.5 % volume fraction. Moreover, the enhancement observed during thermal conductivity assessment, cannot fully explain the heat transfer intensification. This could be associated to the random movements among the particles through a fluid, caused by the impact of the base fluid molecules.     

An iterative method to solve the heat transfer problem under the non-linear boundary conditions

The aim of the paper is to determine the approximation of the tangential matrix for solving the non-linear heat transfer problem. Numerical model of the strongly non-linear heat transfer problem based on the theory of the finite element method is presented. The tangential matrix of the Newton method is formulated. A method to solve the heat transfer with the non-linear boundary conditions, based on the secant slope of a reference function, is developed. The contraction mapping principle is introduced to verify the convergence of this method. The application of the method is shown by two examples. Numerical results of these examples are comparable to the ones solved with the Newton method and the commercial software COMSOL for the heat transfer problem under the radiative boundary conditions.     

Exact solutions for fractional diffusion equation in a bounded domain with different boundary conditions

We give an analytical treatment of a time fractional diffusion equation with Caputo time-fractional derivative in a bounded domain with different boundary conditions. We use the Fourier method of separation of variables and Laplace transform method. The solution is obtained in terms of the Mittag-Leffler-type functions and complete set of eigenfunctions of the Sturm–Liouville problem. Such problems can be used in the context of anomalous diffusion in complex media, as well as for modeling voltammetric experiment in limiting diffusion space.     

Hydromagnetic convective heat transfer in vertical pipes

In the present analysis, we consider the effect of radial magnetic field on the steady flow produced by the combined free and forced convection in an annulus between two coaxial vertical cylinders. A numerical solution of the problem is obtained by using Runge-Kutta-Merson method. For Rayleigh number Ra<0, that is, when the temperature of the pipes decreases as their height increases, the velocity increases with |Ra|. However, it reduces as the Hartmann number M increases. On the other hand, when Ra>0, there occurs back flow controlled by the effect of the magnetic field. Further, the influence of Rayleigh number and Hartmann number on the temperature is also discussed.Nomenclature c _p specific heat at constant pressure - g acceleration due to gravity - H _r applied magnetic field - H _z induced magnetic field - p pressure - T temperature of the fluid - T ₁, T ₂ temperatures of the inner and outer cylinders at z=0 - U _z velocity - coefficient of volume expansion - density - _w reference density - coefficient of viscosity - _e magnetic permeability - _e electrical conductivity - thermal conductivity - _m magnetic diffusivity     

Exact bending solutions of orthotropic rectangular cantilever thin plates subjected to arbitrary loads

Exact bending solutions of orthotropic rectangular cantilever thin plates subjected to arbitrary loads are derived by using a novel double finite integral transform method. Since only the basic elasticity equations for orthotropic thin plates are used, the method presented in this paper eliminates the need to predetermine the deformation function and is hence completely rational thus more accurate than conventional semi-inverse methods, which presents a breakthrough in solving plate bending problems as they have long been bottlenecks in the history of elasticity. Numerical results are presented to demonstrate the validity and accuracy of the approach as compared with those previously reported in the literature     

Similarity conditions for heat transfer in incompressible two-dimensional laminar boundary layer flow

Summary Similarity conditions are presented for the solution of some problems of heat transfer in incompressible two-dimensional boundary layer flow. The treatment holds for forced convection as well as for free convection. For free convection no a priori restriction is made with respect to geometry or temperature distribution of the solid surface. For forced convection the treatment is restricted to uniform bulk flow parallel to a flat surface of non-uniform temperature or heat flux. The results are summarized in some tables that facilitate comparison with older work.     

Radiative heat transfer in a hydromagnetic nanofluid past a non-linear stretching surface with convective boundary condition

Heat transfer characteristics of a two-dimensional steady hydromagnetic natural convection flow of nanofluids over a non-linear stretching sheet taking into account the effects of radiation and convective boundary condition has been investigated numerically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The local similarity solutions are obtained by using very robust computer algebra software Maple 13. The results corresponding to the dimensionless temperature profiles and the reduced Nusselt number, Sherwood number and skin friction coefficient are displayed graphically for various pertinent parameters. The results show that temperature within the boundary layer is enhanced with the increase of the Biot number, buoyancy due to nanoparticle concentration, strength of the applied magnetic field, Brownian motion parameter, and thermophoresis parameter. An opposite trend is observed for the increase of the buoyancy due to temperature, stretching index, and the radiation parameter. The results also show that the local rate of heat transfer strongly depends on the nonlinear stretching index, radiation parameter, Biot number, Brownian motion parameter, and thermophoresis parameter.     

Numerical simulation of heat transfer in a pipe with non-homogeneous thermal boundary conditions

Direct numerical simulations of heat transfer in a fully-developed turbulent pipe flow with circumferentially-varying thermal boundary conditions are reported. Three cases have been considered for friction Reynolds number in the range 180–360 and Prandtl number in the range 0.7–4. The temperature statistics under these heating conditions are characterized. Eddy diffusivities and turbulent Prandtl numbers for radial and circumferential directions are evaluated and compared to the values predicted by simple models. It is found that the usual assumptions made in these models provide reasonable predictions far from the wall and that corrections to the models are needed near the wall.