Forced convection with viscous dissipation in the thermal entrance region of a circular duct with prescribed wall heat flux

The thermal entrance forced convection in a circular duct with a prescribed wall heat flux distribution is studied under the assumptions of a fully developed laminar flow and of a negligible axial heat conduction in the fluid, by taking into account the effect of viscous dissipation. The solution of the local energy balance equation is obtained analytically by employing the Laplace transform method. The effect of viscous dissipation is taken into account also in the region upstream of the entrance cross-section, by assuming an adiabatic preparation of the fluid. The latter hypothesis implies that the initial condition in the entrance cross-section is a non-uniform radial temperature distribution. Two special cases are investigated in detail: an axially uniform wall heat flux, a wall heat flux varying linearly in the axial direction.     

MHD convective flow adjacent to a vertical surface with prescribed wall heat flux

The steady magnetohydrodynamic (MHD) mixed convection flow adjacent to a vertical surface with prescribed heat flux is investigated. The governing partial differential equations are transformed into a system of ordinary differential equations, which is then solved numerically by a finite-difference method. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. Both assisting and opposing flows are considered. It is found that dual solutions exist for the assisting flow, besides that usually reported in the literature for the opposing flow.     

Falkner-Skan problem for a static and moving wedge with prescribed surface heat flux in a nanofluid

An analysis is carried out to study the problem of the steady flow and heat transfer over a static or moving wedge with a prescribed surface heat flux in a nanofluid. The governing partial differential equations are transformed into a set of nonlinear ordinary differential equations using similarity transformation, before being solved numerically by the Keller box method and the Runge-Kutta-Fehlberg method with shooting technique. The features of the flow and heat transfer characteristics are analyzed and discussed. Three different types of nanoparticles are considered, namely copper Cu, alumina Al₂O₃ and titania TiO₂ with water as the base fluid. It is found that the skin friction coefficient and the heat transfer rate at the surface are highest for copper-water nanofluid compared to the alumina-water and titania-water nanofluids. Moreover, the heat transfer rate at the surface increases with the Falkner-Skan power law parameter m.     

Axial heat conduction in a moving semi-infinite fluid

This paper considers the steady state conduction of heat from a wall to a fluid moving at a uniform velocity. The wall is heated by a step change in temperature. Although this appears to be a classical heat conduction problem, its application to various convective heat transfer problems is new. The mathematical procedure leads to the computation of the temperature field and the heat transfer coefficient. In the presence of a step change in the wall temperature, it is shown that the Stanton number is a function of the Peclet number alone. The acquired analytical results show that, near the thermal entrance location, heat conduction dominates and the local heat flux becomes independent of velocity. This phenomenon applies to classical convection problems in various-shaped ducts.     

Temperature field in flow over a stretching sheet with uniform heat flux

An analysis is made of the temperature distribution in the flow of a viscous incompressible fluid caused by the stretching of a sheet which issues from a slit into the fluid. The velocity of the sheet is proportional to the distance from the slit and the sheet is subject to uniform heat flux. It is shown that temperature at a point decreases with increase in the Prandtl number P. For a given surface heat flux, the temperature of the stretching sheet is also determined for several values of P.     

Analytical solution to the problem of heat transfer in an MHD flow inside a channel with prescribed sinusoidal wall heat flux

An MHD laminar flow through a two dimensional channel subjected to a uniform magnetic field and heated at the walls of the conduit over the whole length with a sinusoidal heat flux of vanishing mean value or not, is studied analytically. General expressions of the temperature distribution and of the local and mean Nusselt numbers are obtained by using the technique of linear operators in the case of negligible Joule and viscous dissipation and by taking into account the axial conduction effect. The principal results show that an increase of the local Nusselt number with Hartmann number is observed, and, far from the inlet section, the average heat transfer between the fluid and the walls shows a significant improvement at all values of Hartmann number used when the frequency of the prescribed sinusoidal wall heat flux is increasing in the case of vanishing mean value of the heat flux and this is true especially at low Peclet numbers.     

A generalized relation between the local values of temperature and the corresponding heat flux in a one-dimensional semi-infinite domain with the moving boundary

The paper presents generalized relation between the local values of temperature and the corresponding heat flux in a one-dimensional semi-infinite domain with the moving boundary. The generalized relation between the local values of temperature and the corresponding heat flux has been achieved by the use of a novel technique that involves generalized derivatives (in particular, derivatives of non-integer orders). Confluent hyper-geometric functions, known as Whittaker’s functions, appear in the course of the solution procedure, upon applying the Laplace transform to the original transport equation. The relation is written in the integral form and provides a relationship between the local values of the temperature and heat flux.     

Dual free-convective flows in a horizontal annulus with a constant heat flux wall

Natural convection in a horizontal annulus with a constant heat flux wall is investigated for the fluids of 0.2?Pr?1. The outer cylinder is kept at a constant temperature, and the inner cylinder is heated with a constant heat flux. By using a numerical approach in solving the unsteady governing equations of flow and temperature fields, it is shown that dual steady solutions exist above a critical Rayleigh number.     

Exergy transfer characteristics of forced convective heat transfer through a duct with constant wall heat flux

The examination of exergy transfer characteristics caused by forced convective heat transfer through a duct with constant wall heat flux for thermally and hydrodynamic fully developed laminar and turbulent flows has been presented. The exergy transfer Nusselt number is put forward and the dependence relationships of the exergy transfer Nusselt number on the heat transfer Nusselt number, Reynolds number and Prandtl number are obtained. Expressions involving relevant variables for the local and mean convective exergy transfer coefficient, non-dimensional exergy flux and exergy transfer rate, etc. have been derived. By reference to a smooth duct, the numerical results of exergy transfer characteristics for fluids with different Prandtl number are obtained and the effect of the Reynolds number and non-dimensional cross-sectional position on exergy transfer characteristics is analyzed. In addition, the results corresponding to the exergy transfer and energy transfer are compared.     

Solutions for modelling moving heat sources in a semi-infinite medium and applications to laser material processing

This study describes the analytical and numerical solution of the heat conduction equation for a localised moving heat source of any type for use in laser material processing, as welding, layered manufacturing and laser alloying. In this paper, the analytical solution for a uniform heat source is derived from the solution of an instantaneous point heat source. The result is evaluated numerically and is compared to existing solutions for the moving point source and a semi-ellipsoidal source. Next, the result is used to demonstrate how such model can be used to study the effect of the heat source geometry. Besides, this solution reveals that a melting efficiency higher than 0.37 (= 1/e, a maximum value stated by Rykalin [N. Rykalin, A. Uglov, A. Kokora, O. Glebov, Laser Machining and Welding, Mir Publishers, Moscow, 1978]) can be obtained. To investigate the effect of the temperature dependence of the material parameters, in particular the latent heat of fusion, a finite difference model is implemented. It is shown that the enthalpy method is most suited to implement the latent heat of fusion. A numerical evaluation for Ti–6Al–4V, reveals that the effect of the latent heat is rather small, except when the conductivity is very low, e.g. when scanning in a loose powder bed. The results demonstrate that analytical and numerical solutions can be effectively used to calculate the temperature distribution in a semi-infinite medium for finite 3D heat sources. In this way, a tool to investigate the importance of different processing parameters in laser manufacturing is obtained.     

Temperature distribution in a rectangular plate heated by a moving heat source

In this paper, a solution to the problem of heat conduction in a rectangular plate subjected to the activity of a moving heat source is presented. The temperature of the plate changes because a limited area on the plate surface is heated by a heat source. The heat source moves along an elliptical trajectory which always remains within the plate area. An exact solution to the problem in an analytical form is obtained by applying the Green’s function method. Exemplary results of numerical calculations to determine the temperature distribution in the plate are presented.     

Radiant and convective heat transfer for flow of a transparent gas in a short tube with sinusoidal wall heat flux

The present analysis accounts for combined convective and radiant heat transfer to a fluid flowing in a short tube with prescribed wall heat flux. The heat flux distribution used was of sine shape with maximum at the middle of the tube. This solution is known to represent the axial power variation in a nuclear reactor core. The tube wall and gas bulk temperatures were obtained by successive substitutions for the wall and gas energy balance equations. The integrals were approximated by Sympson's rule and initial guesses for the iterative process were based upon limiting cases for pure radiation and pure convection. The results of the combined solution compared with the pure radiation approach show a decrease of 30 percent for the maximum wall temperature using black surface ($\text{ε=1}$). For this same situation, the increase in the gas temperature along the tube shows a reduction of 58 percent when compared to the pure convection solution.     

Thermodynamic optimization of a coiled tube heat exchanger under constant wall heat flux condition

In this paper the second law analysis of thermodynamic irreversibilities in a coiled tube heat exchanger has been carried out for both laminar and turbulent flow conditions. The expression for the scaled non-dimensional entropy generation rate for such a system is derived in terms of four dimensionless parameters: Prandtl number, heat exchanger duty parameter, Dean number and coil to tube diameter ratio. It has been observed that for a particular value of Prandtl number, Dean number and duty parameter, there exists an optimum diameter ratio where the entropy generation rate is minimum. It is also found that with increase in Dean number or Reynolds number, the optimum value of the diameter ratio decreases for a particular value of Prandtl number and heat exchanger duty parameter.     

Time and space resolved wall temperature and heat flux measurements during nucleate boiling with constant heat flux boundary conditions

The lack of time and space resolved measurements under nucleating bubbles has complicated efforts to fully explain pool-boiling phenomena. In this work, time and space resolved temperature and heat flux distributions under nucleating bubbles on a constant heat flux surface were obtained using a 10 × 10 microheater array with 100 μm resolution along with high-speed images. A numerical simulation was used to compute the substrate conduction, which was then subtracted from the heater power to obtain the wall-to-liquid heat transfer. The data indicated that most of the energy required for bubble growth came from the superheated layer around the bubble. Microlayer evaporation and contact line heat transfer accounted for not more than 23% of the total heat transferred from the surface. The dominant heat transfer mechanism was transient conduction into the liquid during bubble departure. Bubble coalescence was not observed to transfer a significant amount of heat.     

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.