We study the evolution of the solid-liquid interface during melting and solidification of a material with constant internal heat generation and prescribed heat flux at the boundary for a plane wall and a cylinder. The equations are solved by splitting them into transient and steady-state components and then using separation of variables. This results in an ordinary differential equation for the interface that involves infinite series. The initial value problem is solved numerically, and solutions are compared to the previously published quasi-static solutions. We show that when the internal heat generation and the heat flux at the boundary are close in value to each other, the motion of the phase change front takes longer to reach steady-state than when the values are farther apart. As the difference between the internal heat generation and the heat flux increases, the transient solutions become more dominant and the numerical solution of the phase change front does not reach steady-state before the outer boundary or centerline is reached. The difference between the internal heat generation and the heat flux at the boundary can be used to control the motion and speed of the interface. The problem has applications for a nuclear fuel rod during meltdown.

In this study, the analogy between transient heat conduction and mass transfer is applied to investigate the dissolution behavior of solid particles in liquids, particularly, for the transport phenomenon associated with the controlled drug release process. Mathematical modeling is established assuming the shrinking core is solely caused by the diffusion mechanism. The transport governing equations for the dissolution process of controlled drug release are compared with the transient heat conduction differential equations. Analogous quantities, certain analytical solutions and numerical solutions for complex geometry are obtained to demonstrate the dissolution behavior of this specific type of solid particles in liquids based on the proposed shrinking core model. It is found that the shape of the drug capsule plays an important role for effective and timely release of drug content after intake. Among the three shapes investigated herein, sphere, cube and cuboid, we conclude that the drug concentration in a cuboid shaped drug head depletes the quickest whereas the spherical shaped head dissolves the slowest.

In advanced heat transfer courses, a technique exists for reducing a partial differential equation where the dependent variable is a function of two independent variables, to an ordinary differential equation where that same dependent variable becomes a function of only one independent variable. The key to this technique is finding out what the similarity variable to make this transformation is. The difficulty is that the form of the similarity variable is not intuitive, and many heat transfer textbooks do not reveal how this variable is found in classical problems such as viscous and thermal boundary layer theory. It turns out that one way to find this variable is by utilizing the integral technique. By employing the integral technique to boundary layer theory, it will be shown that when the approximate functional relationship for the dependent variable (temperature, velocity, etc) can be represented by an nth order polynomial, the similarity variable can be found very simply. This is seen to be a good tool especially in heat transfer education, but has applications in research as well.

In the Compressed Air Energy Storage (CAES) approach, air is compressed to high pressure, stored, and expanded to output work when needed. The temperature of air tends to rise during compression, and the rise in the air internal energy is wasted during the later storage period as the compressed air cools back to ambient temperature. The present study focuses on designing an interrupted-plate heat exchanger used in a liquid-piston compression chamber for CAES. The exchanger features layers of thin plates stacked in an interrupted pattern. Twenty-seven exchangers featuring different combinations of shape parameters are analyzed. The exchangers are modeled as porous media. As such, for each exchanger shape, a Representative Elementary Volume (REV), which represents a unit cell of the exchanger, is developed. The flow through the REV is simulated with periodic velocity and thermal boundary conditions, using the commercial CFD software ANSYS FLUENT. Simulations of the REVs for the various exchangers characterize the various shape parameter effects on values of pressure drop and heat transfer coefficient between solid surfaces and fluid. For an experimental validation of the numerical solution, two different exchanger models made by rapid prototyping, are tested for pressure drop and heat transfer. Good agreement is found between numerical and experimental results. Nusselt number vs. Reynolds number relations are developed on the basis of pore size and on hydraulic diameter. To analyze performance of exchangers with different shapes, a simplified zero-dimensional thermodynamic model for the compression chamber with the inserted heat exchange elements is developed. This model, valuable for system optimization and control simulations, is a set of ordinary differential equations. They are solved numerically for each exchanger insert shape to determine the geometries of best compression efficiency.

The aim of this paper is to present a simple model, based on a time and space integral method, for prediction of preheating and conversion time of a charring solid particle exposed to a non-oxidative hot environment. The main assumptions are 1) thermo-physical properties remain constant throughout the process; 2) temperature profile within the particle is assumed to obey a quadratic function with respect to the space coordinate; 3) pyrolysis initiates when the surface temperature reaches a characteristic pyrolysis temperature; 4) decomposition of virgin material occurs at an infinitesimal thin layer dividing the particle into char and virgin material regions; 5) the volume of the particle remains unaltered; 6) volatiles escape through the pores immediately after formation. Employing assumption (2) allows one to convert the energy conservation equation of the particle, which is basically described in the form of a partial differential equation (PDE), into an ordinary differential equation (ODE) by performing space integration. Next, by applying approximate time integration the ODE is transformed into an algebraic equation. Applying this approach to the preheating and pyrolysis stages of a thermally thin charring solid particle leads to a set of algebraic equations which provides reactor designers with a convenient means for computation of the heating up time, mass loss history and total conversion of particle. The accuracy of the simple model is assessed by comparing its prediction with that of a one-dimensional detailed pyrolysis model. Overall, good agreement is achieved indicating that this new model can be used for engineering and design purposes.

Automotive industry frequently needs to test new products, according to different production parameters, in order to determine the actual thermal behavior of bodies before mass production is implemented. Numerical simulation of these processes can reduce the very expensive and time consuming experimental procedures. For the drying and hardening process of the top paint applied in the coating process, the body temperature must be raised according to the paint manufacturer regulations. Consequently, prediction of temperature distribution of the car body during various zones of ovens is very vital in the design and performance analysis of the paint dryers. In this research, a novel semi-analytical approach has been used to predict the body temperature variation during the curing process. Considering the energy balance for the body, a set of differential equation has been extracted, depending on the oven zone. These equations can be solved numerically to find the transient temperature profile of the car body. Some parameters in these equations have been achieved by experimental procedure. The results show that the present model predictions are in a good agreement with the experimental data. Therefore, the developed model has a reasonable accuracy and can be used as an efficient robust approach to distinguish overall thermal behavior of the body. These techniques can be used to optimize the design of curing paint oven.

An analytical approach for solving both transient and steady state conduction and surface radiation problems is presented. The method is based on the use of a Green’s function, and the temperature field is obtained by solving an integral equation. This is in contrast to the approach presented in radiative heat transfer texts in which temperature profiles are obtained from the simultaneous solution of coupled integral and differential equations. The analysis presented in this paper provides insight into the solution of this important class of problems. The method is illustrated by solving two representative problems. The first problem considered is the steady state analysis of radiating fins, which are frequently incorporated in the design of spacecraft. The second problem considered is the transient analysis of a radiating target, which is used to determine the temporal response of radiation thermometers.

Analytical solution to the hyperbolic damped wave conduction and relaxation equation is developed by a novel method called the relativistic transformation method. The hyperbolic PDE is decomposed into a time decaying damping component and a Klein-Gardon type equation for the wave temperature. The PDE that describes the wave temperature is transformed to a Bessel differential equation by using the relativistic transformation. The relativistic transformation, η = τ 2 − X 2 is symmetric in space and time. The solution obtained for the transient temperature to a semi-infinite medium was compared with success to a Laplace transform solution reported by other investigators. A approximate analytical solution is obtained for the transient temperature by realizing that the integration constants from the solution of Bessel differential equation in the transformation variable is with respect to the transformation variable which is a function of space and time. So the boundary conditions can be used to solve for integration “constants” that are functions of one variable. The solution consists of three regimes. There is no discontinuity at the wave front. A inertial zero transfer regime, a second rising regime characterized by Bessel composite function of the zeroth order and first kind and a third falling regime characterized by modified Bessel composite function of the zeroth order and first kind.

This paper presents a computational tool for the evaluation of engine performance and exhaust emissions for four stroke multi-cylinder spark ignition engine which uses gasoline as fuel. Gas dynamics flow in multi-cylinder intake and exhaust systems are modeled by using one dimensional unsteady compressible flow equations. The hyperbolic partial differential equations are transferred into a set of ordinary differential equations by using method of characteristics and solved by finite difference method. Compatibility relationships between local fluid velocity (U) and sonic velocity (a) are expressed in terms of Riemann variables, which are constant along the position characteristics. The equations are solved numerically by using rectangular grid in the flow direction (x) and time (t). In this model Nitric Oxide concentration is predicted by using the rate kinetic model in the power cycle and along the exhaust pipes. Carbon Monoxide is computed under chemical equilibrium condition and then empirical adjustment is made for kinetic behaviors based upon experimental results. A good agreement is obtained in the comparison of computed and experimental results of instantaneous cylinder pressure, manifold pressure and temperature, and NO, CO emissions level.

A numerical investigation of laminar free convective heat transfer in a vertical channel with asymmetrical heating has been presented. Uniform wall temperatures are prescribed as thermal boundary conditions. The governing differential equations were solved by a finite volume method. The SIMPLER algorithm for pressure velocity coupling was adopted. A new iterative scheme based on mass balance at inlet and outlet has been used. By solving the flow as an elliptic problem, the effect of vertical diffusion of thermal energy, which was neglected in previous numerical studies, was taken into consideration. Variation of the mean velocity and average Nusselt number for Rayleigh number range of 10 to 10 3 , channel aspect ratio range of 10 to 10 3 and Prandtl numbers of 0.72 and 5 are determined. For uniform wall temperature the average Nusselt number based on wall to ambient temperature difference and heat flux at different points are compared to the experimental results. The results revealed that the average Nusselt number from the thermally active surface in an asymmetric channel to be higher than from a comparable surface in a symmetric configuration, for fixed channel aspect ratio, at low values of Rayleigh number. There was no appreciable change in Nusselt number when Prandtl number was changed from 0.72 to 5. The minimum axial pressure for free convective flow of air is the highest for the symmetric heating condition.

Jet flows have been observed flowing from the tops of vapor bubbles during nucleate boiling in a variety of situations. This paper analyzes the physical mechanisms that cause jet flows to emanate from the tops of bubbles moving along microwires. The flows were analyzed by numerically solving the governing differential equations for the 3-D velocity and temperature distributions around the bubble and the heated wire as the bubble moves along the wire. The results show that the flow is most likely driven by the temperature difference from the front to the back of the bubble resulting from the bubble motion which would cause Marangoni flow. The Marangoni flow provides thrust to push the bubble forward. Comparisons with experimental observations suggests that the condensation heat transfer at the bubble interface must be restricted by noncondensable gases would increase the surface temperature gradient which would in turn increase the resulting Marangoni flow.

We present solutions for solid-liquid phase change in materials that generate internal heat. This problem is solved for both cylindrical and semi-infinite geometries. The analysis assumes a temperature profile in the solid phase and constant temperature boundary conditions on the exposed surfaces. We derive differential equations governing the solidification thickness for both geometries as functions of the Stefan number and the internal heat generation (IHG). For the cylindrical geometry, the solidification layer obtains a steady-state value which is related to the inverse of the square root of the IHG. The solutions to the semi-infinite geometry problem show that when the surface is cooled to below the freezing point, a solidification layer forms along the edge and begins to grow until it reaches a maximum, then begins remelt.

In this paper, both nucleus formation and bubble growth during boiling in microchannels were investigated. A series of visualized experiments were conducted to observe the boiling nucleation and bubble dynamics restricted within parallel microchannels on a silicon wafer. The channels were rectangular and had selected length scale ranging from 50 to 100 microns. A high-speed CCD camera was employed together with a microscope to dynamically record the boiling images. The rates of bubble growth were measured in the channels. The phase change nucleus formation theory was used to determine the initial position of the bubble. The bubble growth rate was described by two ordinary differential equations deduced from the microlayer evaporation theory. The calculation and experimental results were reasonably coincided.

A finite element method is presented for the solution of a free boundary problem which arises during planar melting of a semi-infinite medium initially at a temperature which is slightly below the melting temperature of the solid. The surface temperature is assumed to vary with time. Two different situations are considered (I) when thermal diffusivity is independent of temperature and (II) when thermal diffusivity varies linearly with temperature. The differential equation governing the process is converted to initial value problem of vector matrix form. The time function is approximated by Chebyshev series and the operational matrix of integration is applied, a linear differential equation can be represented by a set of linear algebraic equations and a nonlinear differential equation can be represented by a set of nonlinear algebraic equations. The solution of the problem is then found in terms of Chebyshev polynomial of second kind. The solution of this initial value problem is utilized iteratively in the interface heat flux equation to determine interface location as well as the temperature in two regions. The method appears to be accurate in cases for which closed form solutions are available, it agrees well with them. The effect of several parameters on the melting are analysed and discussed.

This paper describes a hybrid analytical-numerical solution of conjugate heat transfer of laminar flow in a circular duct with axial wall conduction. An improved lumped-differential formulation is used to treat heat conduction in the wall; the Generalised Integral Transform Technique (GITT) is used to solve the energy equation in laminar flow. The coupled partial differential equations in the solid and fluid regions are reduced to a set of coupled ordinary differential equations. Analytical solution is obtained for a truncated set of ordinary differential equations. Improvement is expected over classical lumped models which are only applicable to problems with Biot numbers less than 0.1.

The effects of thermal radiation upon the laminar free convection boundary layer of a vertical flat plate was studied for an absorbing, emitting and scattering gray fluid. The two-flux model was used to represent the radiation term in the energy equation. This method allows to obtain a simpler formulation than required for exact representation of radiation in the boundary layer. The two-flux model reduces the governing equations: continuity, momentum, energy and irradiation, to a set of a coupled partial differential equations. A finite difference scheme is used to transform the resulting equations into an ordinary differential equation system which simplifies the numerical solution. Results for the velocity and temperature profiles and heat fluxes present close agreement with the optical thin and thick limits. Comparison of two-flux results with data obtained using an exact representation of radiation show very small deviations. The method proposed proves to be useful to investigate the effect of the different radiation parameters on natural convection boundary layer, and to obtain numerical results for engineering applications.

Ablation is a thermal protection process with several applications in engineering, mainly in the field of airspace industry. The use of conventional materials must be quite restricted, because they would suffer catastrophic flaws due to thermal degradation of their structures. However, the same materials can be quite suitable once being protected by well-known ablative materials. The process that involves the ablative phenomena is complex, could involve the whole or partial loss of material that is sacrificed for absorption of energy. The analysis of the ablative process in a blunt body with revolution geometry will be made on the stagnation point area that can be simplified as a one-dimensional plane plate problem. In this work the Generalized Integral Transform Technique (GITT) is employed for the solution of the non-linear system of coupled partial differential equations that model the phenomena. The solution of the problem is obtained by transforming the non-linear partial differential equation system to a system of coupled first order ordinary differential equations and then solving it by using well-established numerical routines. The results of interest such as the temperature field, the depth and the rate of removal of the ablative material are presented and compared with those ones available in the open literature.

A relatively simple mechanistic model for the combustion of an aluminum particle in air is presented. The model assumes combustion to occur in two stages. In the first stage, phase transition and heterogeneous surface reactions take place until the melting temperature of the oxide is reached. In the second stage, a quasi-steady state diffusion flame is established allowing for the use of commonly employed flame sheet approximations. Modified Ranz-Marshall and standard drag correlations for a sphere are used to describe the unsteady heating, mass loss rate, and drag of the particle, with the surrounding gas. A system of non-linear ordinary differential equations are formulated and numerically integrated in time for predictions of particle mass, temperature and velocity with, and without, the effects of heterogeneous combustion. Results indicate that, within the assumptions of the current model, the effects of heterogeneous combustion have a significant impact on the overall particle burn time and temperature history for gas temperatures ranging from 1500 to 2500 K . At higher particle Reynolds number, and for temperatures greater than 2500 K, the effects of heterogeneous combustion are not as important and an ignition criterion based on the oxide melting temperature may be sufficient.

One of the most effective methods of treatment for cardiac arrhythmias is radio-frequency (RF) ablation. Many studies have shown that the tissue temperature distribution is the key factor influencing lesion shape and size, and that accurate prediction of this distribution is essential to the further improvement of the procedure. Temperature distributions can be obtained by solving the bioheat equation, which has been done in several studies using numerical techniques. This paper describes the development of an analytical solution that can be used as a bench mark for subsequent numerical solutions. Using integral transforms, the bioheat equation is reduced to an ordinary differential equation with time as the independent variable. The solution has the form of a surface integral within another surface integral. An integration routine that extends the trapezoidal method of integration in two dimensions to an analogous method in three dimensions has been developed in order to evaluate the analytical solution. A C program was written to implement this method, and the program was validated using a surface integral with a known analytical solution. The program was then used to generate temperature profiles at various time values and for different convection coefficients.