General solution of one-dimensional,time dependent coupled heat flow systems

A new, powerful method of analysis, involving the combined use of finite integral transform and finite element techniques, is presented for the solution of time dependent heat flow systems composed of many one-dimensional elements connected through the nodes. This method leads to an eigenvalue problem which is not of the conventional Sturm-Liouville type. A procedure for the determination of the eigenvalues is described. The solution obtained is in the form of an infinite series and contains quasi-steady and transient terms. The general solution obtained can be applied in the mathematical modelling of many engineering applications such as the determination of the penetration of the daily temperature cycle into buildings, the analysis of heat transfer in array of extended surfaces in compact heat exchangers, and many others.     

Approximate analytical solution for one-dimensional tissue freezing around cylindrical cryoprobes

An approximate analytical solution for the temperature distribution and interface motion is determined for the freezing of blood-perfused tissue around a cylindrical cryoprobe. The solution is based on an improved quasi-steady model in which assumed temperature profiles in the frozen and unfrozen tissue are used to determine the interface motion. The approximate solution satisfies all temperature boundary conditions as well as the transient heat equations at the interface. Due to blood perfusion in the unfrozen tissue, a steady state is reached where the interface becomes stationary. The solution converges to the exact steady state interface location. Improvement over the quasi-steady solution and the accuracy of the present theory are verified by comparison with numerical solutions for the limiting case of zero blood perfusion and metabolic heat production. Results show that a typical quasi-steady error of 73% is reduced to 8% using the present theory. Parametric charts are presented to evaluate the effect of the governing parameters on interface location.     

Numerical solution of three-dimensional backward heat conduction problems by the time evolution method of fundamental solutions

The time evolution method of fundamental solutions (MFS) is proposed to solve three-dimensional backward heat conduction problems (BHCPs). The time evolution MFS is obtained through the linear superposition of diffusion fundamental solutions. Through a correct treatment of temporal evolution, the MFS can be implemented to solve strongly ill-posed problems. The numerical results demonstrate the accuracy and stability of the MFS for three-dimensional BHCPs with high levels of noise. This represents the first implementation of MFS to solve three-dimensional BHCPs, and demonstrates that time evolution MFS is a stable and powerful numerical scheme which has the potential to significantly improve the solution of three-dimensional backward heat conduction problems.     

A general solution for one-dimensional multistream heat exchangers and their networks

A mathematical model for predicting the steady-state thermal performance of one-dimensional (cocurrent and countercurrent) multistream heat exchangers and their networks is developed and is solved analytically for constant physical properties of streams. By introducing three matching matrices, the general solution can be applied to various types of one-dimensional multistream heat exchangers such as shell-and-tube heat exchangers, plate heat exchangers and plate-fin heat exchangers as well as their networks. The general solution is applied to the calculation and design of multistream heat exchangers. Examples are given to illustrate the procedures in detail. Based on this solution the superstructure model is developed for synthesis of heat exchanger networks.     

An effective conduction length model in the enthalpy formulation for the stefan problem

In the fixed-grid finite-volume formulation, so called the enthalpy formulation, for the Stefan problem, the temperature and the front movement show step-like history, which is a well-known characteristic of the enthalpy method. This paper presents an effective conduction length model to mitigate such an oscillatory behavior as well as to support the physical reasoning. The proposed model is based on the simple fact that the heat flux across the boundary of phase-change cells should be estimated with the distance between the phase front and the center of neighboring cell. The model is applied to one-dimensional Stefan problems with various Stefan numbers. The numerical results show that the proposed model can smooth the spurious oscillation of the history of temperature and the evolution of front movement.     

A non-integral technique for the approximate solution of transport problems

An approximate method for solving transport problems, called the ‘non-integral’ method, has been presented. The method is based on the criterion that the total error over the domain of interest which results from introducing an approximate solution into the governing equation of a transport problem must be kept to a minimum. The method was demonstrated in three examples. Comparisons of the results with corresponding existing solutions were given.     

The UNIT algorithm for solving one-dimensional convection-diffusion problems via integral transforms

A unified approach for solving convection-diffusion problems using the Generalized Integral Transform Technique (GITT) was advanced and coined as the UNIT (UNified Integral Transforms) algorithm, as implied by the acronym. The unified manner through which problems are tackled in the UNIT framework allows users that are less familiar with the GITT to employ the technique for solving a variety of partial-differential problems. This paper consolidates this approach in solving general transient one-dimensional problems. Different integration alternatives for calculating coefficients arising from integral transformation are discussed. Besides presenting the proposed algorithm, aspects related to computational implementation are also explored. Finally, benchmark results of different types of problems are calculated with a UNIT-based implementation and compared with previously obtained results.     

An engineering foundation for controlling heat transfer in one-dimensional transient heat conduction problems

This paper develops an engineering foundation for controlling heat transfer in one dimensional transient heat conduction problems based upon concepts borrowed from vibration control problems. The foundation distinguishes between modal control, distributed control, discrete control and direct feedback control and then singles out direct feedback control because its simplicity. An example demonstrates modal control and direct feedback control of the variation of the transient temperature field in a one dimensional slab geometry.     

A new LRBFCM-GBEM modeling algorithm for general solution of time fractional-order dual phase lag bioheat transfer problems in functionally graded tissues

The main objective of this article is to propose a new hybrid modeling algorithm based on combining local radial basis function collocation method (LRBFCM) and general boundary element method (GBEM) for solving time fractional-order dual phase lag bioheat transfer problems in functionally graded tissues. The LRBFCM was developed and implemented using an implicit time-stepping technique and Caputo time fractional derivative for solving the fractional-order governing equation without dual phase lags. Due to suitability of the GBEM for modeling of bioheat transfer in functionally graded tissues. Therefore, GBEM is applied for solving the dual phase lags governing equation without fractional-order derivative. The numerical results are depicted graphical forms to show the effects of functionally graded parameter, fractional-order parameter and anisotropy on the nonlinear temperature distribution. Also, these numerical results demonstrate the validity and accuracy of the proposed algorithm and technique.     

On estimating thermal diffusivity using analytical inverse solution for unsteady one-dimensional heat conduction

A modified procedure for calculating the thermal diffusivity of solids based on temperature measurements at two points and the semi-infinite boundary condition is presented. The method makes use of a solution to the unsteady one-dimensional inverse heat conduction problem for the semi-infinite solid. The procedure gives accurate results based on temperature changes produced by an arbitrary fluctuating heat flux source at the boundary.     

Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions

A finite difference method is used to solve the one-dimensional Stefan problem with periodic Dirichlet boundary condition. The temperature distribution, the position of the moving boundary and its velocity are evaluated. It is shown that, for given oscillation frequency, both the size of the domain and the oscillation amplitude of the periodically oscillating surface temperature, strongly influence the temperature distribution and the boundary movement. Furthermore, good agreement between the present finite difference results and numerical results obtained previously using the nodal integral method is seen.     

Simple measurement of thermal diffusivity and thermal conductivity using inverse solution for one-dimensional heat conduction

A new method is proposed for measuring thermal diffusivity and thermal conductivity simultaneously using the inverse solution for one-dimensional unsteady heat conduction. Unlike previous method proposed by authors, the new procedure does not require the temperature measurement for a long time duration after the temperature starts changing at a sensor position; and then a selection of time duration can be chosen such that the measured temperature change becomes large enough to ensure a required accuracy for the estimated values of thermal diffusivity and thermal conductivity. The measurement is usually completed within 3 min until the temperature rise at the thermocouple position reaches a certain temperature level, for example 1% of an error level. This method has the additional advantage of being independent of the surface condition, except for the requirement of two or three sensing positions in the material. The accuracy of the estimated values is also similar to the error level of the sensor at these positions.     

Dynamic simulation of the vertical zone-melting crystal growth

Dynamic behavior of heat transfer, fluid flow, and interfaces in the vertical zone-melting (VZM) crystal growth is studied numerically. The model, which is governed by axisymmetric unsteady-state momentum and heat transfer and interface balance in the system, is solved by a robust finite-volume method. Single crystal growth of NaNO₃ in a computer-controlled transparent multizone furnace is simulated as examples. The effects of gravity levels and heater temperature are considered. Multiple steady states obtained at stationary cases are used as initial conditions to illustrate the transient response and the stability of the VZM crystal growth to the pulse and step changes in thermal environments. For unstable cases, periodically oscillatory flow and growth rate occurring at intermediate values of the Rayleigh number are observed. The upper flow cells beneath the feed front seems to be responsible to the instability, and this is consistent with the observation during crystal growth experiments. For stable cases, a steady state can be achieved smoothly, and the calculated results are in good agreement with the ones from a pseudo steady-state model.     

New derivations of the fundamental solution for heat conduction problems in three-dimensional general anisotropic media

This work presents two new methods to derive the fundamental solution for three-dimensional heat transfer problems in the general anisotropic media. Initially, the basic integral equations used in the definition of the general anisotropic fundamental solution are revisited. We show the relationship between three, two and one-dimensional integral definitions, either by purely algebraic manipulation as well as through Fourier and Radon transforms. Two of these forms are used to derive the fundamental solutions for the general anisotropic media. The first method gives the solution analytically for which the solution for the orthotropic case agrees with the well known result obtained by the domain mapping, while the fundamental solution for the general anisotropic media is new. The second method expresses the solution by a line integral over a semi-circle. The advantages and disadvantages of the two methods are discussed with numerical examples.