• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.13 no.1
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• pp.31-40
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• 2009

In the present work transformation of dimensionless heat diffusion equation for the solution of moving boundary problems have been formulated. The formulation is based on 1-D, 2-D and 3-D, unsteady heat diffusion equations. These equations are rst turned int dimensionless form by using dimensionless quantities and their transformation was formulated in liquid and solid phases. The salient feature of this work is that during the transformation of dimensionless heat diffusion equation there arises a convective term $\tilde{v}$ which is responsible for the motion of interface in liquid as well as solid phase. In the transformed heat equation, a correction factor $\beta$ also arises naturally which gives the correct transformed flux at interface.

• East Asian mathematical journal
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• v.35 no.1
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• pp.99-108
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• 2019

In this paper, we introduce a numerical approach to solve heat-diffusion equation with discontinuous diffusion coefficients in the three dimensional rectangular domain. First, we study the support operator method and suggest a new method, the continuous velocity method. Further, we apply both methods to a diffusion process for neurotransmitter release in an individual synapse and compare their results.

• Water Engineering Research
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• v.2 no.3
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• pp.187-196
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• 2001

An approximate analytical solution of a nonlinear hydro-thermo coupled diffusion equation is derived using the dimensionless form of the equation and transformation method. To derive an analytical solution, it is drastically assumed that the product of first order derivatives in the non-dimensionalized governing equation has little influence on the solution of heat and moisture behavior problem. The validity of this drastic assumption is demonstrated. Some numerical simulation is performed to investigate the applicability of a derived approximate analytical solution. The results show a good agreement between analytical and numerical solutions. The proposed solution may provide a useful tool in the verification process of the numerical models. Also, the solution can be used for the analysis of one-dimensional coupled heat and moisture movements in unsaturated porous media.

• Bulletin of the Society of Naval Architects of Korea
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• v.19 no.4
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• pp.19-29
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• 1982

Galerkin's finite element method is applied to a two-dimensional heat convection-diffusion problem arising in the hydrodynamic lubrication of thrust bearings used in naval vessels. A parabolized thermal energy equation for the lubricant, and thermal diffusion equations for both bearing pad and the collar are treated together, with proper juncture conditions on the interface boundaries. it has been known that a numerical instability arises when the classical Galerkin's method, which is equivalent to a centered difference approximation, is applied to a parabolic-type partial differential equation. Probably the simplest remedy for this instability is to use a one-sided finite difference formula for the first derivative term in the finite difference method. However, in the present coupled heat convection-diffusion problem in which the governing equation is parabolized in a subdomain(Lubricant), uniformly stable numerical solutions for a wide range of the Peclet number are obtained in the numerical test based on Galerkin's classical finite element method. In the present numerical convergence errors in several error norms are presented in the first model problem. Additional numerical results for a more realistic bearing lubrication problem are presented for a second numerical model.

• Journal for History of Mathematics
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• v.15 no.3
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• pp.59-64
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• 2002

This paper investigates a history of Fourier Series for the heat equation and how deeply it is related to modern famous three equations, Navier-Stokes equations in fluid dynamics, drift-diffusion equations in semiconductor, and Black-Scholes equation in finance. We also propose improved models for the heat equation with finite propagation speeds.

• Journal of Korean Society for Atmospheric Environment
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• v.16 no.4
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• pp.339-350
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• 2000

The three-dimensional new corrdinate system over a single hill double hills and complex terrain with a single hill and a rectangular obstacle was generated using a body-fitted coordinate system. Control of the coordinate line distribution in the field was executed by generalizing the elliptic generating system to Poisson equation. ▽2ξ=P. The new coordinate system was well fitted to the surface boundary of single hill and double hills. But in the case of complex terrain with hill and rectangular obstacle there was smoothing tendency around the rectangular obstacle. In order to show the validity of the body-fitted coordinate system the heat diffusion equation was transformed and the temperature distribution was calculated over the various terrain. The results showed the temperature distribution was very symmetrical and stable around hills and obstacle. As a result the couple of a body-fitted coordinate system and the heat diffusion equation were executed successfully. Wind field over complex terrain with hill and rectangular obstacle which represent urban area was simulated stably in body-fitted coordinate system. The qualitative result show the enhancement of wind speed at the upwind direction of a hill and a rectangular obstacle and the recirculation zone at the downwind direction.

• Transactions of the Korean Society of Mechanical Engineers
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• v.10 no.3
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• pp.349-356
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• 1986

For the extension of application in flash method measuring the thermophysical properties of materials, the heat diffusion equation with the heat transfer loss from front, rear, and circumferential surfaces of two layer cylinderical sample is mathematically analyzed by means of Green's function for axially symmetric pulse heating on the front of samples. The solutions are applied to determine the unknown thermal diffusivity of the two materials and analyzed the measurement error due to heat loss and finite pulse time effects.

• Fire Science and Engineering
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• v.1 no.1
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• pp.11-18
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• 1987

This study is for calculating the equation of the inner temperature in the concrete, mainly by the theory or heat conduction in the solid. The results are as follow ; 1. The equation of the Fourier's heat diffusion is used formally to get the distribution of inner temperature or the concrete members, and this is programed by using the computer. 2. As study in the past, heat constant of concrete is calculated for function of temperature described recommendation heat constant value in comparison with the existing inner heating experimental result.

• Proceedings of the KSME Conference
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• 2008.11b
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• pp.2334-2339
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• 2008

This paper proposes an improved mathematical model for predicting the frosting behavior on a two-dimensional fin considering the heat conduction of heat exchanger fins under frosting conditions. The model consists of laminar flow equation in airflow, diffusion equation of water vapor for frost layer, and heat conduction equation in fin, and these are coupled together. In this model, the change in three-dimensional airside airflow caused by frost growth is accounted for. The fin surface temperature increased toward the fin tip due to the fin heat conduction. On the contrary, the temperature gradient in the airflow direction(x-dir.) is small throughout the entire fin. The frost thickness in the direction perpendicular to airflow, i.e. z-dir., decreases exponentially toward the fin tip due to non-uniform temperature distribution. The rate of decrease of heat transfer in the airflow direction is high compared to that in the z-direction due to more decrease in the sensible and latent heat rate in x-direction.

• Transactions of the Korean Society of Mechanical Engineers
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• v.16 no.6
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• pp.1171-1178
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• 1992

An experiment was performed to describe the heat and mass transfer occurring between hot waste gas and cold water through direct contact in a direct contact heat exchanger. This model was then used to obtain an equation of overall heat transfer coefficent based on heat exchanger volume. The diffusion heat transfer rate is 2-3 times larger than the convection heat transfer rate as results of condensation of the water vapor contained in the waste gas. The boiler efficiency increases over 10%.