• Title/Summary/Keyword: Peclet condition

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Extended Graetz Problem Including Axial Conduction and Viscous Dissipation in Microtube

  • Jeong Ho-Eyoul;Jeong Jae-Tack
    • Journal of Mechanical Science and Technology
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    • v.20 no.1
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    • pp.158-166
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    • 2006
  • Extended Graetz problem in microtube is analyzed by using eigenfunction expansion to solve the energy equation. For the eigenvalue problem we applied the shooting method and Galerkin method. The hydrodynamically isothermal developed flow is assumed to enter the microtube with uniform temperature or uniform heat flux boundary condition. The effects of velocity and temperature jump boundary condition on the microtube wall, axial conduction and viscous dissipation are included. From the temperature field obtained, the local Nusselt number distributions on the tube wall are obtained as the dimensionless parameters (Peclet number, Knudsen number, Brinkman number) vary. The fully developed Nusselt number for each boundary condition is obtained also in terms of these parameters.

A Study on the Transport of Soil Contaminant (A Development of FDM Model for 3-D Advection-Diffusion Equation with Decay Term) (토양 오염원의 이동에 관한 연구 (감쇠항이 있는 3차원 이송-확산 방정식의 수치모형 개발))

  • Kim, Sang-Jun
    • Journal of Korea Water Resources Association
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    • v.45 no.2
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    • pp.179-189
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    • 2012
  • To simulate the transport of pollutant, a numeric model for the advection-diffusion equation with the decay term is developed. This is finite-difference model using the implicit method (with the weight factor ${\alpha}$) and Gauss-Seidel SOR(successive over-relaxation). This model is compared to the analytical solutions (of simpler dimensional or boundary conditions), and in the condition of Peclet number < 5~20, the result shows stable condition, and Crank-Nicolson method (${\alpha}$=0.5) shows the more accurate results than fully-implicit method (${\alpha}$=1). The mass of advection, diffusion and decay is calculated and the error of mass balance is less than 3%. This model can evaluate the 3-D concentrations of the advection-diffusion and decay problems, but this model uses only the finite-difference method with the fixd grid system, so it can be effectively used in the problems with small Peclet numbers like the pollutant transport in groundwater.

AN ADAPTIVE FINITE DIFFERENCE METHOD USING FAR-FIELD BOUNDARY CONDITIONS FOR THE BLACK-SCHOLES EQUATION

  • Jeong, Darae;Ha, Taeyoung;Kim, Myoungnyoun;Shin, Jaemin;Yoon, In-Han;Kim, Junseok
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.4
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    • pp.1087-1100
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    • 2014
  • We present an accurate and efficient numerical method for solving the Black-Scholes equation. The method uses an adaptive grid technique which is based on a far-field boundary position and the Peclet condition. We present the algorithm for the automatic adaptive grid generation: First, we determine a priori suitable far-field boundary location using the mathematical model parameters. Second, generate the uniform fine grid around the non-smooth point of the payoff and a non-uniform grid in the remaining regions. Numerical tests are presented to demonstrate the accuracy and efficiency of the proposed method. The results show that the computational time is reduced substantially with the accuracy being maintained.

The Modified Eulerian-Lagrangian Formulation for Cauchy Boundary Condition Under Dispersion Dominated Flow Regimes: A Novel Numerical Approach and its Implication on Radioactive Nuclide Migration or Solute Transport in the Subsurface Environment

  • Sruthi, K.V.;Suk, Heejun;Lakshmanan, Elango;Chae, Byung-Gon;Kim, Hyun-su
    • Journal of Soil and Groundwater Environment
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    • v.20 no.2
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    • pp.10-21
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    • 2015
  • The present study introduces a novel numerical approach for solving dispersion dominated problems with Cauchy boundary condition in an Eulerian-Lagrangian scheme. The study reveals the incapability of traditional Neuman approach to address the dispersion dominated problems with Cauchy boundary condition, even though it can produce reliable solution in the advection dominated regime. Also, the proposed numerical approach is applied to a real field problem of radioactive contaminant migration from radioactive waste repository which is a major current waste management issue. The performance of the proposed numerical approach is evaluated by comparing the results with numerical solutions of traditional FDM (Finite Difference Method), Neuman approach, and the analytical solution. The results show that the proposed numerical approach yields better and reliable solution for dispersion dominated regime, specifically for Peclet Numbers of less than 0.1. The proposed numerical approach is validated by applying to a real field problem of radioactive contaminant migration from radioactive waste repository of varying Peclet Number from 0.003 to 34.5. The numerical results of Neuman approach overestimates the concentration value with an order of 100 than the proposed approach during the assessment of radioactive contaminant transport from nuclear waste repository. The overestimation of concentration value could be due to the assumption that dispersion is negligible. Also our application problem confirms the existence of real field situation with advection dominated condition and dispersion dominated condition simultaneously as well as the significance or advantage of the proposed approach in the real field problem.

THE CHARACTERISTICS OF HEAT TRANSFER AND CHEMICAL REACTION FOR THERMAL CRACKING OF ETHANE IN TUBULAR REACTOR (에탄 열분해 반응이 동반된 관형 반응기에서의 열전달 및 화학반응 특성 연구)

  • Shin, C.Y.;Ahn, J.
    • Journal of computational fluids engineering
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    • v.21 no.1
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    • pp.43-49
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    • 2016
  • Thermal cracking is commonly modeled as plug flow reaction, neglecting the lateral gradients present. In this paper, 2-dimensional computational fluid dynamics including turbulence model and molecular reaction scheme are carried out. This simulation is solved by means of coupled implicit scheme for stable convergence of solution. The reactor is modeled as an isothermal tube, whose length is 1.2 m and radius is 0.01 m, respectively. At first, The radial profile of velocity and temperature at each point are predicted in its condition. Then the bulk temperature and conversion curve along the axial direction are compared with other published data to identify the reason why discussed variations of properties are important to product yield. Finally, defining a new non-dimensional number, Effect of interaction with turbulence, heat transfer and chemical reaction are discussed for design of thermal cracking furnace.

Numerical Model of One-Dimensional Advection-Diffusion Equation Applying Split-Operator Method (연산자 분리기법에 의한 1차원 이송-확산방정식의 수치모형)

  • Lee, Jeong-Gyu;Gang, Chang-Gu;Lee, Jong-In
    • Journal of Korea Water Resources Association
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    • v.30 no.2
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    • pp.143-154
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    • 1997
  • A numerical model for solving advection-diffusion equation is presented by splitoperator method combining the Holly-Preissmann scheme with a fifth-degree interpolating polynomial for advection operator and the explicit scheme porposed by Hobson et al. for diffusion operator. To examine the developed model, the obtained numerical solutions are compared with both the analytic solution and those from the existing models for the instantaneous source (Gaussian hill) and the continuous source (advanced front) at upstream boundary with constant velocity and diffusivity condition. For the various cases having different Courant and Peclet numbers, it is shown that the present study provides stable solutions even for Courant numbers exceeding one. The result obtained by the present study also agree well with existing analytical solutions for both cases. The proposed explicit scheme somewhat releases the conventional restriction of explicit schemes for determining the time step size and provides satisfactory results for relatively large time step size.

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The Study of Finite Element Method for Analyses of Travelling Magnetic Field Problem (운동자계 문제의 해석을 위한 유한요소법에 관한 연구)

  • Chang Ho-Sung
    • Journal of the Korean Institute of Illuminating and Electrical Installation Engineers
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    • v.19 no.4
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    • pp.108-116
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    • 2005
  • This paper presents finite element analyses solution in the travelling magnetic field problem. The travelling magnetic field problem is subject to convective-diffusion equation. Therefore, the solution derived from Galerkin-FEM with linear interpolation function may oscillate between the adjacent nodes. A simple model with Dirichlet, Neumann and Periodic boundary condition respectively, have been analyzed to investigate stabilities of solutions. It is concluded that the solution of Galerkin-FEM may oscillate according to boundary condition and element type, but that of Upwind-FFM is stable regardless boundary condition.

An Analytical Study on The Structure and Boundary Conditions of The Premixed Flame Stabilized in Conductive Cylindrical Tubes (전도성 원형관 내에 안정화된 예혼합 화염의 구조와 경계 조건에 관한 이론해석)

  • Kim, Nam-Il
    • Journal of the Korean Society of Combustion
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    • v.11 no.3
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    • pp.8-17
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    • 2006
  • When a flame is stabilized in a tube of a finite thickness, a conductive heat transfer through the tube significantly changes the wall temperature and affects the flame characteristics. Thus the tube length and thermal boundary conditions affect on the structure of the flame in a conductive tube. A one-dimensional analytical study was conducted by employing two energy equations for tubes and mixtures and a species equation for the mixture. Variation of the maximum temperatures and indicating displacements were observed. A parametric study on the effects of inner Peclet numbers, normalized wall conductivities, and heat transfer conditions of the tube was conducted. This study provides essential data for a more efficient computational simulation of the flame stabilized in conductive tubes.

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Enhancement of Heat Transfer by Chaotic Stirring (혼돈적 교반에 의한 열전달 향상)

  • Suh, Y.K.
    • Korean Journal of Air-Conditioning and Refrigeration Engineering
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    • v.6 no.1
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    • pp.20-28
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    • 1994
  • A numerical study has been performed on enhancement of heat transfer in a forced convection of the modified driven-cavity flow which was previously found by the author to give a regular or chaotic stirring depending on the parameter value. It is found that for the present case wherein heat is transmitted between fluid and the surrounding walls, the chaotic stirring enhances the heat transfer at high Peclet numbers. The optimal condition of the flow modulation for the best heat transfer can be predicted by purely investigating the hydrodynamic facet, i.e. the stirring effect.

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Numerical Simulation of Dendritic Growth of the Multiple Seeds with Fluid Flow (유체 유동을 동반한 다핵 수치상결정의 미세구조성장에 대한 수치해석적 연구)

  • Yoon, Ik-Roh;Shin, Seung-Won
    • Transactions of the Korean Society of Mechanical Engineers B
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    • v.33 no.7
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    • pp.469-476
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    • 2009
  • Most material of engineering interest undergoes solidification process from liquid state. Identifying the underlying mechanism during solidification process is essential to determine the microstructure of material thus the physical properties of final product. In this paper, effect of fluid convection on the dendrite solidification morphology is studied using Level Contour Reconstruction Method. Sharp interface technique is used to implement correct boundary condition for moving solid interface. The results showed good agreement with exact boundary integral solution and compared well with other numerical techniques. Effects of Peclet number and undercooling on growth of dendrite tip of both single and multiple seeds have been also investigated.