Planar solidification of a warm flowing-liquid under different boundary conditions

Planar solidification of a warm flowing liquid with the convective heat transfer to the growing solid layer, has been analysed for the boundary conditions of constant temperature, constant heat flux and convective heat flux at the surface respectively. The mathematical formulation of the problem resulted in a coupled set of two differential equations in temperature and solid thickness as function of position, time and the problem parameters. Analytical expressions for the temperature distribution within the growing solid layer, the rate of solidification and the solidification time are obtained. The perturbation techniques employed here is simple and straight forward in contrast with the earlier techniques. Good agreement between the experimental results and the present solutions is obtained for the convective heat flux boundary condition. The results of this analysis are useful in the design and analysis of experiments dealing with freezing/melting in one dimension. The role of the parameter Stefan number which is small for phase change materials, is discussed in context with the storage of thermal energy.     

Approximate solutions to the Stefan problem with internal heat generation

Using a quasi-static approach valid for Stefan numbers less than one, we derive approximate equations governing the movement of a phase change front for materials which generate internal heat. These models are applied for both constant surface temperature and constant surface heat flux boundary conditions, in cylindrical, spherical, plane wall and semi-infinite geometries. Exact solutions with the constant surface temperature condition are obtained for the steady-state solidification thickness using the cylinder, sphere, and plane wall geometries which show that the thickness depends on the inverse square root of the internal heat generation. Under constant surface heat flux conditions, closed form equations can be obtained for the three geometries. In the case of the semi-infinite wall, we show that for constant temperature and constant heat flux out of the wall conditions, the solidification layer grows then remelts.     

Effects of the number and distribution of fins on the storage characteristics of a cylindrical latent heat energy storage system: a numerical study

A numerical study of the effects of the number and distribution of fins on the storage characteristics of a cylindrical latent heat energy storage system (LHESS) was conducted. Due to the low thermal conductivity of phase change materials (PCMs) used in LHESS, fins were added to the system to increase the rate of heat transfer and charging. Finite elements were used to implement the developed numerical method needed to study and solve for the phase change heat transfer (melting of PCM) encountered in a LHESS during charging. The effective heat capacity method was applied in order to account for the large amount of latent energy stored during melting of the PCM and the moving interface between the solid and liquid phases. The effects of increasing the number and distribution of fins on the melting rate of the PCM were studied for configurations having between 0 and 27 fins for heat transfer fluid (HTF) velocities of 0.05 and 0.5?m/s. Results show that the overall heat transfer rate to the PCM increases with an increase in the number of fins irrespective of the HTF velocity. It was also observed that the total amount of energy stored after 12?h increases nearly linearly with the addition of fins up to 12 fins; further addition of fins increasing the total energy stored by ever smaller amounts.     

Modeling thermal insulation of firefighting protective clothing embedded with phase change material

Experiments and research on heat transport through firefighting protective clothing when exposed to high temperature or intensive radiation are significant. Phase change material (PCM) takes energy when changes from solid to liquid thus reducing heat transmission. A numerical simulation of heat protection of the firefighting protective clothing embedded with PCM was studied. We focused on the temperature variation by comparing different thicknesses and position conditions of PCM combined in the clothing, as well as the melting state of PCM and human irreversible burns through a simplified one-dimensional model. The results showed it was superior to place PCM between water and proof layer and inner layer, in addition, greater thickness increased protection time while might adding extra burden to the firefighter.     

Analytical approximation for solidification processes in PCM storage with internal fins: imposed heat flux

Uses of thermal energy storage systems have expanded notably in recent decades. In thermal energy systems, internal heat transfer enhancement techniques such as fins are often used because of the low thermal conductivity of the phase change materials (PCMs). In this paper, solidification of a PCM is studied in a rectangular storage with horizontal internal plate fins and an imposed constant heat flux on the vertical walls. A simplified analytical solution is presented and its results are compared to those for a numerical approach based on an enthalpy method. The fraction of solidified PCM in storage is calculated with the derived analytical model which determines how much of the storage is solidified after a certain time. The results show that the analytical model satisfactorily estimates the solid–liquid interface and the temperature distribution for the fin, which are useful in the design of PCM-based thermal energy storage or cooling systems.     

Thermodynamic optimization of contact melting of phase change material inside a horizontal cylindrical capsule

Finite time thermodynamic analysis is applied to the contact melting process of phase change material inside a horizontal cylindrical capsule. With the minimum entropy generation in given time as the objective function the quadratic nonlinear ordinary differential equation that the optimal melting process should be satisfied is derived. The dimensionless liquid height, melt liquid film thickness, Nusselt number, melting rate, optimal wall temperature and entropy generation are obtained by the numerical method. The optimal results are discussed and compared with the unoptimizable analytical results under the condition of constant wall temperature. It is found that the heat transfer and thermodynamics performance of the optimal melting process is better than that of the melting with constant wall temperature.     

Weak 2D Convective Plumes in a Sloping Permeable Layer

This is a study of thermal plumes in a permeable fluid-saturated slab of a porous medium (of finite uniform thickness but otherwise infinite extent) that is heated by either an instantaneous or a steady line source embedded in the medium. The slab, which may be horizontal or sloping, is initially at ambient conditions; the impervious upper surface remains at the ambient temperature, while the impervious base is thermally insulated. The transient temperature distribution, the surface heat flux and the convective flow velocity are calculated for small instantaneous line heat energy sources. They show how the flow develops, spreads and slows as time progresses, and an estimate of the time to decay is given. Steady-state temperature and velocity profiles are calculated for embedded line sources that provide heat energy at a small constant rate, and the surface heat flux distribution is calculated.     

Transient conjugate heat transfer from a hemispherical plate during free liquid jet impingement on the convex surface

This paper considers the analysis of transient heating of a hemispherical solid plate of finite thickness during impingement of a free liquid jet. A constant heat flux was imposed at the inner surface of the hemispherical plate at t = 0 and heat transfer was monitored for the entire duration of the transient until a steady state condition was reached. Calculations were done for Reynolds number (Re) ranging from 500 to 1,500 and dimensionless plate thicknesses to nozzle diameter ratio (b/d _n) from 0.083 to 1.5. Results are presented for local and average Nusselt number using water as the coolant and various solid materials such as silicon, constantan, and copper. It was detected that increasing the Reynolds number decreases the time for the plate to achieve the steady-state condition. Also, a higher Reynolds number increases the Nusselt number. Hemispherical plate materials with higher thermal conductivity maintain lower temperature non-uniformity at the solid–fluid interface. Increasing the plate thickness decreases the maximum temperature in the solid and increases the time to reach the steady-state condition.     

Performance analysis and optimization of radiating fins with a step change in thickness and variable thermal conductivity by homotopy perturbation method

Although tapered fins transfer more rate of heat per unit volume, they are not found in every practical application because of the difficulty in manufacturing and fabrications. Therefore, there is a scope to modify the geometry of a constant thickness fin in view of the less difficulty in manufacturing and fabrication as well as betterment of heat transfer rate per unit volume of the fin material. For the better utilization of fin material, it is proposed a modified geometry of new fin with a step change in thickness (SF) in the literature. In the present paper, the homotopy perturbation method has been used to evaluate the temperature distribution within the straight radiating fins with a step change in thickness and variable thermal conductivity. The temperature profile has an abrupt change in the temperature gradient where the step change in thickness occurs and thermal conductivity parameter describing the variation of thermal conductivity has an important role on the temperature profile and the heat transfer rate. The optimum geometry which maximizes the heat transfer rate for a given fin volume has been found. The derived condition of optimality gives an open choice to the designer.     

A Local Thermal Non-Equilibrium Analysis of Fully Developed Forced Convective Flow in a Tube Filled with a Porous Medium

A local thermal non-equilibrium model has been considered for the case of thermally fully developed flow within a constant heat flux tube filled with a porous medium. Exact temperature profiles for the fluid and solid phases are found after combining the two individual energy equations and then transforming them into a single ordinary differential equation with respect to the temperature difference between the solid phase and the wall subject to constant heat flux. The exact solutions for the case of metal-foam and air combination reveal that the local thermal equilibrium assumption may fail for the case of constant heat flux wall. The Nusselt number is presented as a function of the Peclet number, which shows a significant increase due to both high stagnant thermal conductivity and thermal dispersion resulting from the presence of the metal-foam.     

Effects of the heat transfer fluid velocity on the storage characteristics of a cylindrical latent heat energy storage system: a numerical study

A numerical study of the effects of the thermal fluid velocity on the storage characteristics of a cylindrical latent heat energy storage system (LHESS) was conducted. Due to the low thermal conductivity of phase change materials (PCMs) used in LHESS, fins were added to the system to increase the rate of heat transfer and charging. Finite elements were used to implement the developed numerical method needed to study and solve for the phase change heat transfer (melting of PCM) encountered in a LHESS during charging. The effective heat capacity method was applied in order to account for the large amount of latent energy stored during melting of the PCM and the moving interface between the solid and liquid phases. The effects of the heat transfer fluid (HTF) velocity on the melting rate of the PCM were studied for configurations having between 0 and 18 fins. Results show that the overall heat transfer rate to the PCM increases with an increase in the HTF velocity. However, the effect of the HTF velocity was observed to be small in configurations having very few fins, owing to the large residual thermal resistance offered by the PCM. However, the effect of the HTF velocity becomes more pronounced with addition of fins; since the thermal resistance on the PCM side of the LHESS is significantly reduce by the large number of fins in the system.     

Heat transfer to an underlying fluid due to the axisymmetric spreading of material on the surface

Similarity boundary layer solutions are found for the fluid underlying an axisymmetric spreading material layer. Three thermal boundary conditions for the fluid-material interface are considered, corresponding to constant temperature interface, adiabatic interface with heat source at origin, and constant heat flux interface. The boundary layer thickness is proportional to the distance from origin. Physical significance is discussed.     

Transient hyperbolic heat conduction in thick-walled FGM cylinders and spheres with exponentially-varying properties

This paper focuses on non-Fourier hyperbolic heat conduction analysis for heterogeneous hollow cylinders and spheres made of functionally graded material (FGM). All the material properties vary exponentially across the thickness, except for the thermal relaxation parameter which is taken to be constant. The cylinder and sphere are considered to be cylindrically and spherically symmetric, respectively, leading to one-dimensional heat conduction problems. The problems are solved analytically in the Laplace domain, and the results obtained are transformed to the real-time space using the modified Durbin’s numerical inversion method. The transient responses of temperature and heat flux are investigated for different inhomogeneity parameters and relative temperature change values. The comparisons of temperature distribution and heat flux between various time and material properties are presented in the form of graphs.     

Experimental study of enhanced heat transfer by addition of CuO nanoparticle

An energy storage system has been designed to study the thermal characteristics of paraffin wax with an embedded nano size copper oxide (CuO) particle. This paper presents studies conducted on phase transition times, heat fraction as well as heat transfer characteristics of paraffin wax as phase change material (PCM) embedded with CuO nanoparticles. 40?nm mean size CuO particles of 2, 5 and 10% by weight were dispersed in PCM for this study. Experiments were performed on a heat exchanger with 1.5–10?l/min of heat transfer fluid (HTF) flow. Time-based variations of the temperature distributions are revealed from the results of observations of melting and solidification curves. The results strongly suggested that the thermal conductivity enhances 6, 6.7 and 7.8% in liquid state and in dynamic viscosity it enhances by 5, 14 and 30% with increasing mass fraction of the CNEPs. The thermal conductivity ratio of the composites can be augmented by a factor up to 1.3. The heat transfer coefficient during solidification increased about 78% for the maximum flow rate. The analysis of experimental results reveals that the addition of copper oxide nanoparticles to the paraffin wax enhances both the conduction and natural convection very effectively in composites and in paraffin wax. The paraffin wax-based composites have great potential for energy storage applications like industrial waste heat recovery, solar thermal applications and solar based dynamic space power generation with optimal fraction of copper oxide nanoparticles.     

Analysis of laminar flow forced convection heat transfer in the entrance region of a circular pipe

A numerical solution, for incompressible, steady-state, laminar flow heat transfer in the combined entrance region of a circular tube is presented for the case of constant wall heat flux and constant wall temperature. The development of velocity profile is obtained from Sparrow's entrance region solution. This velocity distribution is used in solving the energy equation numerically to obtain temperature profiles. Variation of the heat transfer coefficient for these two different boundary conditions for the early stages of boundary layer formation on the pipe wall is obtained. Local Nusselt numbers are calculated and the results are compared with those given byUlrichson andSchmitz. The effect of the thermal boundary conditions is studied by comparing the uniform wall heat flux results with uniform wall temperature.     

压电智能结构有限变形下热-机耦合的有限元分析

采用有限元方法研究了结构在热载荷作用下变形与热传导之间的耦合特性．分析表明，结构变形较小，非线性效应很弱时，变形对材料的热传导系数影响很小，对结构的温度分布几乎没有影响；当变形增大，非线性效应增强时，变形对材料的热传导特性影响显著，热载荷作用下结构的温度变化和变形与现行不考虑热-机耦合效应所得结果产生明显差异．因此，为实现压电智能结构形状(振动)的精确控制，分析及实施控制时须考虑热-机耦合及变形对热传导系数的影响．     

Analytical model for melting in a semi-infinite PCM storage with an internal fin

The most PCMs with high energy storage density have an unacceptably low heat conductivity and hence internal heat transfer enhancement techniques such as fins or other metal structures are required in latent heat thermal storage (LHTS) applications. Previous work has concentrated on numerical and experimental examination in determining the influence of the fins in melting phase change material. This paper presents a simplified analytical model based on a quasi-linear, transient, thin-fin equation which predicts the solid-liquid interface location and temperature distribution of the fin in the melting process with a constant imposed end-wall temperature. The analytical results are compared to the numerical results and they show good agreement. Due to the assumptions made in the model, the speed of the solid-liquid interface during the melting process is slightly too slow.     

Axial heat conduction effects in the entrance region of circular ducts

This paper describes the transport of thermal energy within a small distance after an abrupt wall temperature change in a circular duct. In general, the axial conduction becomes significant when the Peclet number is small. The results indicate that the inclusion of axial conduction in the fluid substantially increases the wall heat flux at near the thermal inlet location. The exact series solution leads to a modified Graetz type problem. This exact solution is augmented by an asymptotic solution describing the wall heat flux near the thermal entrance location.     

Laminar mixing and heat transfer for constant heat flux boundary condition

In a recent paper we have investigated mixing and heat transfer enhancement in a mixer composed of two circular rods maintained vertically in a cylindrical tank. The rods and tank can rotate around their revolution axes while their surfaces were maintained at a constant temperature. In the present study we investigate the differences in the thermal mixing process arising from the utilization of a constant heat flux as a boundary condition. The study concerns a highly viscous fluid with a high Prandtl number for which this chaotic mixer is suitable. By solving numerically the flow and energy equations, and using different statistical tools we characterize the evolution of the fluid temperature and its homogenization. Fundamental differences are reported between these two modes of heating or cooling: while the mixing with an imposed temperature results in a homogeneous temperature field, with a fixed heat flux we observe a constant difference between the maximal and minimal temperatures that establish in the fluid; the extent of this difference is governed by the efficiency of the mixing protocol.