Natural convection on horizontal,inclined, and vertical plates with variable surface temperature or heat flux

An analysis is performed to study the flow and heat transfer characteristics of laminar free convection in boundary layer flows from horizontal, inclined, and vertical flat plates in which the wall temperature T_w(x) or the surface heat flux q_w(x) varies as the power of the axial coordinate in the form T_w(x) = T_∞ + axⁿorq_w = bx^m. The governing equations are first cast into a dimensionless form by a nonsimilar transformation and the resulting equations are then solved by a finite-difference scheme. Numerical results for fluids with Prandtl numbers of 0.7 and 7 are presented for three representative exponent values under each of the nonuniform surface heating conditions. It has been found that both the local wall shear stress and the local surface heat transfer rate increase as the angle of inclination from the horizontal γ increases or as the local Grashof number increases. An increase in the value of the exponent n or m enhances the surface heat transfer rate, but it causes a decrease in the wall shear stress. Correlation equations for the local and average Nusselt numbers are obtained for the special cases of uniform wall temperature (UWT) and uniform surface heat flux (UHF). Comparisons are also made of the local Nusselt numbers between the present results and available experimental data for the UHF case, and a good agreement is found to exist between the two.     

The analysis of conjugate problems of conduction and convection for non-Newtonian fluids past a flat plate

In this paper, the conjugate problems of laminar forced convection in non-Newtonian fluid flow and heat conduction inside a heated flat plate is studied. A conjugate parameter ζ is proposed to reflect the characteristics of the conjugate problems. The value of the conjugate parameter lies among 0 and 1 and the two limiting values correspond to the ordinary convection problem with boundary condition of constant wall heat flux (ζ = 0) and constant wall temperature (ζ = 1), respectively. In addition, the power-law model is used for non-Newtonian fluids with exponent n < 1 for pseudoplastics, n = 1 for Newtonian fluids and n > 1 for dilatant fluids. Furthermore, the coordinates and dependent variables are transformed to yield computationally efficient numerical solutions that are valid over the entire range of conjugate problems and the whole regime of the non-Newtonian fluids. The effects of the conjugate parameter, the power-law viscosity index and the generalized Prandtl number on the temperature profiles, as well as on the local heat transfer rate are clearly illustrated.     

Analysis of laminar dissipative flow and heat transfer in a porous saturated circular tube with constant wall heat flux

A theoretical study of the thermal development of forced convection was performed using a circular tube filled with a saturated porous medium, with constant wall heat flux, and with the effect of viscous dissipation. The solution was obtained using the method of separation of variables. The Sturm–Liouville system was solved for the eigenvalues. Ordinary differential equations for the eigenfunctions were solved numerically by the fourth-order Runge–Kutta method. Results show that, in the presence of the viscous dissipation, both the level and distribution of temperature are altered remarkably, even for small values of the Brinkman number, Br, which is the ratio of heat generation caused by viscous dissipation to the value of heat flux at the wall. The value of the local Nusselt number, Nu, is demonstrably independent of Br, unlike the situation in which the wall temperature remains constant.     

Unsteady forced convection heat transfer on a flat plate embedded in the fluid-saturated porous medium with inertia effect and thermal dispersion

For an unsteady forced convection on a flat plate embedded in the fluid-saturated porous medium with inertia effect and thermal dispersion, this paper presents a precise and rigorous method to obtain the entire solutions from one-dimensional transient conduction (ξ=0) to steady forced convection in porous medium (ξ=1) under conditions of uniform wall temperature and uniform heat flux, respectively. It is worth noted that the rate of unsteady heat transfer can be accelerated by the thermal dispersion, which may be regarded as the effect of mixing or agitating, to enhance the heat transfer in porous medium. Additionally, it is found that the time response, from the transient heat conduction to a steady forced convection in Darcy's flow, is τ=1, and is independent of wall heating condition and thermal dispersion strength (φ).     

Laminar mixed convection boundary layers induced by a linearly stretching permeable surface

The boundary layer flow on a linearly moving permeable vertical surface is studied when the buoyancy force assists or opposes the flow. Similarity and local similarity solutions are obtained for the boundary layer equations subject to power law temperature and velocity variation. The effect of various governing parameters, such as Prandtl number Pr, injection parameter d, and the mixed convection parameter λ=Gr_x/Re_x², which determine the velocity and temperature distributions, the heat transfer coefficient, and the shear stress at the surface are studied. The heat transfer coefficient increases as λ assisting the flow for all d for uniformly or linearly heated surface and as Pr increases it becomes almost independent of λ. However, as the temperature inversely proportional to the distance up the surface, the buoyancy has no effects on the heat transfer coefficient. Critical buoyancy parameter values are obtained for vanished shear stress and for predominate natural convection. Critical values are also presented for predominate buoyancy shear stress at the surface for assisting or opposing flow. A closed form analytical solution is also presented as a special case of the energy equation.     

Models for transient conduction in a flat plate subjected to a variable heat flux

This work presents analytical models allowing to identify the transient temperature distribution in a flat plate. The plate is exposed to a convective heat transfer on a face and to a heat flux on the other one. The heating flux is Heaviside (crenel type) and is maintained during a t₁ time. The heating phase is followed by a relaxation one. The theoretical method is original because it uses Green’s functions method to determine the analytical solutions of the heat propagation equation in the plate during the heating and relaxation phases. These analytical solutions allow to identify the temperature distribution as well as wall heat flux versus time. The results of our work can be useful at different levels: during the identification of parameters (such as the thermal conductivity or the thermal diffusivity of a plate), during the identification of the boundary conditions (like the heating flux or the convection coefficient) in industrial processes using this kind of systems, or even with educational intents for teaching transient conduction.     

On temperature prediction at low Re turbulent flows using the Churchill turbulent heat flux correlation

The present work investigates the prediction of mean temperature profiles in turbulent channel flow using the fraction of the heat flux due to turbulence. According to this new model, suggested by Churchill and co-workers, fully developed flow and convection can be expressed as local fractions of the shear stress and the heat flux density due to turbulent fluctuations, respectively. The fully developed temperature profile can be predicted if the velocity field and the turbulent Prandtl number are known. Temperature profiles for Pr between 0.01 and 50,000 have been obtained theoretically and with simulations through the use of Lagrangian methods for both plane Poiseuille flow and plane Couette flow. The half channel height for all simulations was h = 150 in wall units. The theoretical predictions have been found to agree with the data quite well for a range of Pr, but there are deviations at very high Pr.     

Integral transform solution of transient forced convection in external flow

The Generalized Integral Transform Technique (GITT), under its partial transformation mode, is employed in the hybrid numerical–analytical solution of transient laminar forced convection over flat plates, subjected to arbitrary time variations of wall heat flux applied from above. From the available Blasius velocity distributions and employing a coordinate transformation to account only for the thermally affected region along the main flow direction, the transient temperature distribution is expanded in terms of eigenfunctions obtained from the diffusion operator in the transversal direction. The resulting coupled system of partial differential equations for the transformed potentials is numerically solved in terms of the dimensionless time variable and longitudinal coordinate, by making use of the method of lines implemented in the Mathematica routine NDSolve. Numerical solutions for the wall temperature and heat transfer coefficient are then readily computed, yielding the time evolution and the longitudinal distribution of these parameters, for any specified wall heat flux time function.     

Natural convection from a vertical cone in a porous medium due to the combined effects of heat and mass diffusion with non-uniform wall temperature/concentration or heat/mass flux and suction/injection

An analysis has been carried out to study the non-Darcy natural convention flow of Newtonian fluids on a vertical cone embedded in a saturated porous medium with power-law variation of the wall temperature/concentration or heat/mass flux and suction/injection with the streamwise distance x. Both non-similar and self-similar solutions have been obtained. The effects of non-Darcy parameter, ratio of the buoyancy forces due to mass and heat diffusion, variation of wall temperature/concentration or heat/mass flux and suction/injection on the Nusselt and Sherwood numbers have been studied.     

Conjugate natural convection about a vertical cylindrical fin with lateral mass flux in a saturated porous medium

The problem of conjugate natural convection about a vertical cylindrical fin with uniform lateral mass flux in a fluid-saturated porous medium has been studied numerically. Solutions based on the third level of truncation are obtained by the local nonsimilarity method. The effects of the surface mass flux, the conjugate convection-conduction parameter, and the surface curvature on fin temperature distribution, local heat transfer coefficient, local heat flux, average heat transfer coefficient, and total heat transfer rate are presented. A comparison with finite-difference solutions for the case of constant wall temperature was made, and found in a good agreement.     

Unsteady forced convection with sinusoidal duct wall generation: the conjugate heat transfer problem

Transient heat transfer solutions are found for a fluid flowing within a parallel plate duct when there is sinusoidal generation with axial position in the duct wall. Solutions are found for wall temperature, surface heat flux and fluid bulk mean temperature as a function of position and time in this conjugated problem. To develop this solution, finite difference methods are used as well as the quasi-steady method and another method which employs a two integral representation for the surface heat flux. Accuracy limitations of the quasi-steady results are identified. Transient local Nusselt number predictions show its dependence upon time.     

Effects of viscous dissipation on the heat transfer in a forced pipe flow. Part 2: Thermally developing flow

In this part of the study, consideration is given to thermally developing laminar forced convection in a pipe including viscous dissipation. The axial heat conduction in the fluid is neglected. Two different thermal boundary conditions are considered: the constant heat flux (CHF) and the constant wall temperature (CWT). Both the wall heating (the fluid is heated) case and the wall cooling (the fluid is cooled) case are considered. The distributions for the developing temperature and local Nusselt number in the entrance region are obtained. Results show that the temperature profiles and local Nusselt number are influenced by the Brinkman number (Br) and the thermal boundary condition used for the wall. Significant viscous dissipation effects have been observed for large Br.     

Finite element analysis of couple stress micropolar nanofluid flow by non‐Fourier's law heat flux model past stretching surface

Numerical analysis has been done to investigate magnetohydrodynamics nonlinear convective flow of couple stress micropolar nanofluid with Catteneo‐Christov heat flux model past stretching surface with the effects of heat generation/absorption term, chemical reaction rate, first‐order slip, and convective boundary conditions. The coupled highly nonlinear differential equation governing the steady incompressible laminar flow has been solved by a powerful numerical technique called finite element method. The impacts of diverse parameters on linear velocity, angular velocity (microrotation), temperature, concentration profile, local skin friction coefficient, local wall couple stress, local Nusselt number, and Sherwood number are presented in graphical and tabular form. The result pointed out that the enhancement in material parameter β increases the velocity of the fluid while the couple stress parameter K has quite opposite effect. Heat and mass transfer rate of the fluid are enhanced by increasing material parameter while couple stress parameter shows the opposite influence. Moreover, heat and mass transfer rate are higher with the Catteneo‐Christov heat flux model than Fourier's law of heat conduction. The accuracy of the present method has been confirmed by comparing with previously published works.     

Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate

The combined free convection boundary layer flow with thermal radiation and mass transfer past a permeable vertical plate is studied when the plate moves in its own plane. The plate is maintained at a uniform temperature with uniform species concentration and the fluid is considered to be gray, absorbing–emitting. The coupled unsteady non-linear momentum, energy and concentration equations governing the problem is obtained and made similar by introducing a time-dependent length scale. The similarity equations are solved numerically using superposition method. The resulting velocity, temperature and concentration distributions are shown graphically for different values of parameters entering into the problem. The numerical values of the local wall shear stress, local surface heat and mass flux are shown in tabular form.     

A general analytical solution to the equation of transient forced convection with fully developed flow

A general solution to the energy equation under zero wall temperature or zero heat flux boundary condition for the decay of an inlet and initial temperature distribution of an incompressible transient turbulent flow heat transfer between two parallel plates is given. It is shown that these solutions may then be used to obtain solutions due to unit steps in wall temperature or wall heat flux which is sufficient to sort out prescribed wall temperature and prescribed wall heat flux boundary condition. The results are confirmed experimentally by the frequency method. An experimental apparatus has been designed, built and used for this purpose.     

Hydromagnetic mixed convection stagnation flow with suction and blowing

This paper deals with steady, two-dimensional, mixed convection flow of an electrically-conducting and heat-absorbing fluid near a stagnation point on a semi-infinite vertical permeable surface at arbitrary surface heat flux variations in the presence of a magnetic field. Similarity equations are derived and solved numerically by an implicit and accurate finite-difference method. Graphical solutions for the local skin-friction coefficient and the local Nusselt number are presented and discussed for various parametric conditions. These results are presented to illustrate the influence of the Hartmann number, wall mass transfer coefficient, heat absorption coefficient, Prandtl number and the mixed convection or buoyancy parameter.     

Viscous dissipation effects on the asymptotic behaviour of laminar forced convection for Bingham plastics in circular ducts

The present study concentrates on the effects of viscous dissipation and the yield shear stress on the asymptotic behaviour of the laminar forced convection in a circular duct for a Bingham fluid. It is supposed that the physical properties are constant and the axial conduction is negligible. The asymptotic temperature profile and the asymptotic Nusselt number are determined for various axial distributions of wall heat flux which yield a thermally developed region. It is shown that if the asymptotic value of wall heat flux distribution is vanishes, the asymptotic value of the Nusselt number is zero. The case of the asymptotic wall heat flux distribution non-vanishing giving a value of the Nusselt number dependent on the Brinkman number and on the dimensionless radius of the plug flow region was also analysed. For an infinite asymptotic value of wall heat flux distributions, the asymptotic value of the Nusselt number depends on the dimensionless radius of the plug flow region and on the dimensionless parameter which depends on the asymptotic behaviour of the wall heat flux. The condition of uniform wall temperature and convection with an external isothermal fluid were also considered. The comparison with other existing solutions in the literature in the Newtonian case is analysed.     

Second law analysis of fully developed laminar flow for rectangular ducts with semicircular ends

Second law analysis is performed analytically for rectangular ducts with semicircular ends in laminar flow. Two different situations are considered for the analysis. In the first case, boundaries of duct are considered in constant wall temperature. In the second case, constant wall heat flux boundary conditions applied. Entropy generation is obtained for various cross sectional areas, various wall heat flux and Reynolds numbers. It is found that with the increasing aspect ratio (β) values for both constant wall temperature (CWT) and constant heat flux (CHF) total entropy generation increases, however, required pumping power also increases.     

Slip MHD liquid flow and heat transfer over non-linear permeable stretching surface with chemical reaction

In the present study the magnetohydrodynamic (MHD) liquid flow and heat transfer over non-linear permeable stretching surface has been presented in the presence of chemical reactions and partial slip. By means of proper similarity variables, the fundamental equations of the boundary layer are transformed to ordinary differential equations which for the fixed values of the x-coordinate along the plate local similarity solution would be valid appropriately. The ordinary differential equations are solved numerically using an explicit Runge–Kutta (4, 5) formula, the Dormand–Prince pair and shooting method. As a result, the velocity profiles, the concentration profiles, temperature profiles, the wall shear stress, the local Sherwood number and the local Nusselt number for the various values of the involved parameters of the problem are presented and discussed in details.     

Transient two-phase flow and heat transfer with localized heating in porous media

The transient behavior of two-phase flow and heat transfer in a channel filled with porous media was numerically studied in this paper. Based on the two-phase mixture model, numerical solutions were obtained using the Finite-Volume Method (FVM). Two methods to treat the discontinuous diffusion coefficient in the energy equation, i.e. the harmonic mean method and the “modified” Kirchhoff method were compared. It was found that the “modified” Kirchhoff method was better in dealing with the rapid change in the diffusion coefficient. Three different cases, with discrete heat flux applied at (1) the upper wall, (2) lower wall and (3) both the upper and lower walls were studied. The velocity and temperature fields for these cases were discussed. The results show that the liquid and vapor flow fields, as well as the temperature and liquid saturation fields have distinctly different features with the change in heating location. An analysis of the vapor volume fraction indicates that the largest amount of vapor with the highest vapor generation rate was for the case in which the heat flux is applied from the lower wall.