Non-Fourier heat conduction in a hollow sphere with periodic surface heat flux

Presented is the analysis of non-Fourier effect in a hollow sphere exposed to a periodic boundary heat flux. The problem is studied by deriving an analytical solution of the hyperbolic heat conduction equation. Using the obtained analytical expression, the temperature profiles at outer and inner surfaces of the sphere are evaluated for various thermal relaxation times. By comparing the results of non-Fourier model with those obtained from Fourier heat conduction equation, the transition process from parabolic model to hyperbolic one is shown. The phase difference and amplitude ratio of boundary surfaces are calculated as functions of the thermal relaxation time and the results are depicted graphically.     

Cattaneo–Christov heat flux model for heat transfer of Marangoni boundary layer flow in a copper–water nanofluid

The Cattaneo–Christov heat flux is first utilized to explore the heat transfer characteristics of Marangoni boundary layer flow in a copper–water nanofluid. The Marangoni boundary layer flow is driven by exponential temperature. Five different types of nanoparticle shapes including sphere, hexahedron, tetrahedron, column and lamina are considered for the copper–water nanofluid. The nonlinear system of partial differential equations is reduced by similarity transformations and then solved numerically by the shooting method. It is found that sphere nanoparticle has better heat transfer enhancement than other nanoparticle shapes and both the temperature and the thickness of the thermal boundary layer are lower for the Cattaneo–Christov heat flux model than the classical Fourier's law of heat conduction.     

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.     

Bounds on natural convective heat transfer in a porous layer with fixed heat flux

Using the background field variational method, bounds on natural convective heat transfer in a porous layer heated from below with fixed heat flux are derived from the primitive equations. The enhancement of heat transfer beyond the minimal conduction value (the Nusselt number Nu) is bounded in terms of the non-dimensional forcing scale set by the ‘effective’ Rayleigh number (Rˆ) according to Nu ≤ 0.354Rˆ^1/2 and in terms of the conventional Rayleigh number (Ra) defined by the temperature drop across the layer according to Nu ≤ 0.125Ra. It is presented that fixing the heat flux at the boundaries does not change the linear dependence between Nusselt number and Rayleigh number at high Rayleigh number region.     

On two-dimensional heat conduction problem of a periodic stratified convective half-space with a moving heat source

The paper deals with the quasi-stationary two-dimensional heat conduction problem of a periodic two-layered convective half-space subjected to the action of moving heat source with constant velocity on its surface. The analysis is performed within the framework of the homogenized model with microlocal parameters [1]. By using the method of Fourier transforms the temperature distribution in the body is obtained. The numerical examples illustrating the effects connected with the laminated structure of the half-space are also presented.     

On heat conduction problem in a semi-infinite periodically laminated layer

The paper deals with the heat conduction problem of a semi-infinite periodically stratified layer heated by a constant heat flux directed according to the layering, normal to the boundary being a cross-section of the composite components. The free heat exchange with surroundings is considered on the remaining parts of the boundary. The body is assumed to be composed of n periodically repeated two-layered, perfect bonding lamine. The problem is solved on two ways: (1) directly as heat conduction problem or (2) by using the homogenized model with microlocal parameters [R. Kulchytsky-Zhyhailo, S.J. Matysiak, On some heat conduction problem in a periodically two-layered body. Comparative results, Int. Commun. Heat Mass Transf., in press]. The obtained results are compared and presented in the form of figures.     

Second law analysis of periodic heat conduction through a wall

Periodic heat conduction through a wall is a simple model for the behaviour of a building wall submitted to climatic temperature changes. Beyond the well-known definition of heat transfer, the present concern is thermodynamics, through two quantities. The first quantity is the total entropy generation (total over time period and wall thickness). Formal derivation shows that from this point of view also, the phenomenon is the superposition of stationary linear heat-conduction plus periodic heat-diffusion around a uniform temperature. The second quantity is defined by considering ideal Carnot cycles that would maintain the inner space at a prescribed temperature. Parameters that influence either quantities are explored, evidencing values of the wall thickness beyond which there does not seem to be any interest in increasing the wall thickness.     

Transient heat conduction in a composite slab by a time-varying incident heat flux combined with convective and radiative cooling

Numerical calculations based on finite difference approximations are carried out to assess the thermal response of a composite slab due to a time-varying incident heat flux. The slab, which consists of several layers of various material, is cooled by combined convection and radiation. The slab is rotating and the front and rear surfaces receive the incident heat flux in an alternating fashion. Temperature distributions versus time and space are presented.     

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.     

Modeling of forced convective heat transfer of a non-Newtonian nanofluid in the horizontal tube under constant heat flux with computational fluid dynamics

In this paper, convective heat transfer effect on the non-Newtonian nanofluid flow in the horizontal tube with constant heat flux was investigated using computational fluid dynamics (CFD). For this purpose, non-Newtonian nanofluid containing Al₂O₃ and Xanthan aqueous solution as a liquid single phase with two average particle sizes of 45 and 150 nm and four particle concentrations of 1, 2, 4 and 6 wt.% and two concentrations of Xanthan aqueous solutions (0.6,1.0 wt.%) were used. Effect of particle size and concentration of Xanthan solution on convective heat transfer coefficient was investigated in different Reynolds numbers (500 < Re < 2500) for various axial locations of tube. The results showed that heat transfer coefficient and Nu number of non-Newtonian nanofluid increased with increasing concentration of Xanthan solution. By applying the modeling results, an equation was obtained for Nusselt number prediction using the dimensionless numbers. The results showed that the correlated data were in very good agreement with predicted data. The maximum error was around 5%.     

Non-Fourier heat conduction in a finite medium subjected to arbitrary periodic surface disturbance

The non-Fourier transient heat conduction in a finite medium under arbitrary periodic surface thermal disturbance is investigated analytically. In order to obtain the desired temperature field from the known solution for non-Fourier heat conduction under a harmonic disturbance, the principle of superposition along with the Fourier series representation of an arbitrary periodic function is employed. The developed method can be applied for more realistic periodic boundary conditions occurred in nature and technology.     

Surface effects of ribbon heaters on critical heat flux in nanofluid pool boiling

This paper deals with a study of enhanced critical heat flux (CHF) and burnout heat flux (BHF) in pool boiling of water with suspended silica nanoparticles using Nichrome wires and ribbons. Previously the current authors and other researchers have reported three-digit percentage increase in critical heat flux in silica nanofluids. This study investigates the effect of various heater surface dimensions, cross-sectional shapes as well as surface modifications on pool boiling heat transfer characteristics of water and water-based nanofluids. Our data suggest that the CHF and BHF decrease as heater surface area increases. For concentrations from 0.1 vol% to 2 vol%, the deposition of the particles on the wire allows high heat transfer through inter-agglomerate pores, resulting in a nearly 3-fold increase in burnout heat flux at very low concentrations. The nanoparticle deposition plays a major role through variation in porosity. The CHF enhancement is non-monotonic with respect to concentration. As the concentration is increased, the CHF and BHF decrease prior to increasing again at higher concentrations. Results show a maximum of 270% CHF enhancement for ribbon-type heaters. The surface morphology of the heater was investigated using SEM and EDS analyses, and it was inferred that the 2 vol% concentration deposition coating had higher porosity and rate of deposition compared with 0.2 vol% case.     

Periodic B-spline basis for quasi-steady periodic inverse heat conduction

A method based on the use of periodic B-splines and the integral transform technique is proposed for the solution of quasi-steady periodic linear inverse heat conduction problems. Previous approaches based on a finite Fourier series representation of the unknown surface condition are best suited to smooth time variations of the surface condition. Now, using a B-spline representation, problems with discontinuities or abrupt variations in the surface condition can be handled readily. The versatility of the B-spline basis allows prior information concerning the general functional behavior of the surface condition to be better incorporated into the model.     

Axially symmetric problems of heat conduction in a periodically laminated layer with vertical cylindrical hole

The distributions of temperature and heat fluxes in a periodically laminated layer with a vertically located cylindrical hole are obtained approximately within the homogenized model with microlocal parameters. The method of solution is based on the Weber-Orr integral transforms. The effects of geometrical and mechanical parameters of the composite structure on the thermal fields are discussed and presented in graphical form.     

Natural convection boundary layer on a horizontal elliptical cylinder with constant heat flux and internal heat generation

In this paper the natural convection boundary layer on a horizontal elliptical cylinder with constant heat flux and temperature dependent internal heat generation is investigated. The mathematical problem is reduced to a pair of coupled partial differential equations for the temperature and the stream function, and the resulting nonlinear equations are solved numerically by cubic spline collocation method. Results for the local Nusselt number and the local skin-friction coefficient are presented as functions of eccentric angle for various values of heat generation parameters, Prandtl numbers and aspect ratios. An increase in the aspect ratio of the elliptical cylinder decreases the average surface temperature of the elliptical cylinder with blunt orientation, while it increases the average surface temperature of the elliptical cylinder with slender orientation. Moreover, an increase in the heat generation parameter for natural convection flow over a horizontal elliptic cylinder with constant heat flux leads to an increase in the average surface temperature of the elliptical cylinder.     

Estimation of time-dependent heat flux and measurement bias in two-dimensional inverse heat conduction problems

This paper presents the results from the adaptive estimator developed to estimate time-dependent boundary heat flux in two-dimensional heat conduction domain with heated and insulated walls. For the estimation, the algorithm requires only the temperatures measured at the insulated walls. In addition, the estimator also predicts the bias in the measurements. In modeling the system, it is assumed that the input flux and bias sequence dynamics can be modeled by a semi-Markov process. By incorporating the semi-Markovian concept into a Bayesian estimation technique, the estimator consists of a bank of parallel, adaptively weighted, Kalman filters. Computer simulation results reveal that the proposed adaptive estimator has improved estimation performance even for step changing heat flux and measurement bias.     

Correcting lateral heat conduction effect in image-based heat flux measurements as an inverse problem

The image deconvolution method is developed, which is coupled with the one-dimensional (1D) analytical inverse method to calculate more accurate heat flux fields by correcting the lateral heat conduction effect in image-based surface temperature measurements. The theoretical foundation is a convolution-type integral equation with a Gaussian filter (kernel) that relates a heat flux field obtained by using the 1D inverse method on a surface to the true heat flux field. The accuracy of this method is evaluated and the standard deviation in the Gaussian filter is determined for different materials through simulations. This method is used to calculate heat flux fields in temperature-sensitive-paint measurements on a 7°-half-angle circular cone at Mach 6 in a short-duration hypersonic wind tunnel. In addition, a simple method is proposed to solve a projection problem associated with image deconvolution for a highly curved developable surface.     

Wall coupling effect in channel forced convection with streamwise periodic boundary heat flux variation

The present paper deals with the laminar forced convection in a parallel-plane channel, and is aimed to investigate the effect of conducting walls. On the external boundaries of the duct walls a thermal boundary condition is prescribed, such that the wall heat flux longitudinally varies with sinusoidal law. The local energy balance equation is written separately for the fluid and the solid regions, with reference to the fully developed regime, and then solved both analytically and numerically. Moreover, the local and average Nusselt numbers in a longitudinal period are evaluated. The average Nusselt number, if regarded as a function of the dimensionless pulsation, displays an interesting feature. In fact, for all the considered cases, it has a minimum, so that there exists a value of the dimensionless pulsation such that the heat exchange between the fluid and the solid wall is considerably inhibited.     

Unsteady free convection of second‐grade nanofluid with a new time‐space fractional heat conduction

The Caputo and Caputo–Fabrizio derivative are applied to study a second‐grade nanofluid over a vertical plate. A comparative analysis is presented to study the unsteady free convection of a second‐grade nanofluid with a new time–space fractional heat conduction. The governing equations with mixed time–space fractional derivatives are non‐dimensionalized and solved numerically, and a comparison between the Caputo and the Caputo–Fabrizio models is made. It is found that the temperature is higher for the Caputo–Fabrizio fractional model than the Caputo model, but the higher velocity only exists near the vertical plate for the Caputo–Fabrizio model than the Caputo model. Moreover, the velocity for the Caputo model will exceed the Caputo–Fabrizio model as y evolves.     

Lack of oscillations in Dual-Phase-Lagging heat conduction for a porous slab subject to imposed heat flux and temperature

This study shows that the physical conditions necessary for thermal waves to materialize in Dual-Phase-Lagging porous media conduction are not attainable in a porous slab subject to a combination of constant heat flux and temperature (Neumann and Dirichlet) boundary conditions. It is demonstrated that the approximate equivalence between Dual-Phase-Lagging (DuPhlag) heat conduction model and the Fourier heat conduction in porous media subject to Lack of Local Thermal Equilibrium (La Lotheq) that suggested the possibility of thermal oscillations and resonance reveals a condition that cannot be fulfilled because of physical constraints.