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The Generalized Graetz Problem in Finite Domains
2012
SIAM Journal on Applied Mathematics
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We consider the generalized Graetz problem associated with stationary convectiondiffusion inside a domain having any regular three-dimensional translationally invariant section and finite or semi-infinite ...
In the case of finite domains a similar basis can be exhibited, and the mode's amplitudes can be obtained from the inversion of newly defined finite domain operator. ...
The aim of the present contribution is to provide the mathematical analysis and numerical methods for solving the generalized Graetz problem in semi-infinite and finite domains, as well as effective numerical ...
doi:10.1137/11082542x
fatcat:iuwkosnczna7xh3zzzysdlcmwm
Numerical computation of 3D heat transfer in complex parallel heat exchangers using generalized Graetz modes
2014
Journal of Computational Physics
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It is based on a spectral method that allows to re-cast three-dimensional heat exchangers into a two-dimensional eigenvalue problem, named the generalized Graetz problem. ...
Several examples of heat exchangers are analyzed, their numerical convergence is tested and the numerical efficiency of the approach is illustrated in the case of Poiseuille flow in tubes. ...
The mathematical properties of the generalized Graetz modes have been studied in [21] [22] [23] . Generalized Graetz modes have been first used to solve problem (1) on infinite domains in [21] . ...
doi:10.1016/j.jcp.2014.02.037
fatcat:74ifvskp6jab3feihzdtdy5uki
Numerical modeling of heat and mass transfer in cylindrical ducts
2010
Archives of Thermodynamics
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The problems under consideration are often denoted as extended Graetz problems. ...
Calculations were carried out gradually decreasing the mesh size in order to examine the convergence of numerical method to analytical solution. ...
analytically extended Graetz problem. 2 Cylindrical duct ANSYS ICEM CFD software was used to generate the geometry of a cylindrical duct and finite volume grid. ...
doi:10.2478/v10173-010-0002-0
fatcat:jh3rt7j7zfbwtkbsondz4sqe2u
Conjugated Graetz problems—I
1981
Chemical Engineering Science
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These solid-fluid problems may be viewed as extensions of the extended, one-phase, Graetz problem with prescribed wall flux [2] . ...
The aforementioned operator is shown to be symmetric in its domain, possessing a denumerable set of eigenvalues and a complete set of eigenvectors. ...
Acknowledgemen&-The computer facilities for this work were generously provided by Purdue University. ...
doi:10.1016/0009-2509(81)80172-8
fatcat:wo26a53ekvex5mrta4qmj67ci4
Numerical Solution of Extended Graetz Problem by Orthogonal Collocation and Crank-Nicholson Method
2015
American Journal of Heat and Mass Transfer
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A numerical method was developed in this work, for visualization of the temperature profile, whose strategy of calculation is based on the orthogonal collocation method followed by the finite difference ...
The present set of themes related to the investigations of heat transfer by convection and the transport phenomenon in a cylindrical pipe in laminar flow, which is to explore the evolution of the temperature ...
Braga et al. (2014) considered an extended version of a Graetz problem with axial diffusion in an infinite domain. ...
doi:10.7726/ajhmt.2015.1011
fatcat:m6mv7sbhbvdzbp3q6nfrwpmihi
Page 789 of Mathematical Reviews Vol. , Issue 84b
[page]
1984
Mathematical Reviews
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Authors’ summary: “A finite domain D, is enveloped by a finite domain D,. The domain D,, in turn, is in contact with two thermal baths with time-dependent periodic temperatures 7,(¢) and 7,(1). ...
In the two articles cited in the heading of the first paper under review,
a “complicated Hilbert space approach” was used to solve the
extended Graetz problem. ...
Numerical study of heat convective mass transfer in a fully developed laminar flow with constant wall temperature
2015
Case Studies in Thermal Engineering
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boundary condition" is commonly called the Graetz Problem that our goal is to get the steady temperature distribution in the fluid. ...
The simplified energy equation is solved numerically by the orthogonal collocation method followed by the finite difference method (Crank-Nicholson method).The calculations were effected through a FORTRAN ...
[2] considered an extended version of a Graetz problem with axial diffusion in an infinite domain. ...
doi:10.1016/j.csite.2015.08.003
fatcat:mmd5uo2z5featij7kz3mfkz5x4
An unconditionally stable finite-difference method for the solution of multi-dimensional transport equation
2020
Ain Shams Engineering Journal
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The visibility of using this method for the solution of the Multi-dimensional Transport equations is demonstrated through the solution of each of the Burgers' equation and the Graetz-Nusselt problem for ...
The first-order derivative of the Transport equation is represented by a one-sided multi-level finite-difference. The resulting scheme is an explicit marching type iterative solution. ...
not exist in the solution in Graetz problem where it has not a rapid change. ...
doi:10.1016/j.asej.2020.05.008
fatcat:rsd35qrvrzg4vl2ahtbjipkltq
Analytical properties of Graetz modes in parallel and concentric configurations
[article]
2020
arXiv
pre-print
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The generalized Graetz problem refers to stationary convection-diffusion in uni-directional flows. ...
In this contribution we demonstrate the ana-lyticity of generalized Graetz solutions associated with layered domains: either cylindrical (possibly concentric) or parallel. ...
effective algorithmic framework for the computation of generalized Graetz mode decomposition in cylindrical or parallel configurations. ...
arXiv:1812.05351v2
fatcat:uxkixlet7fdwpl3nmsxuseb7ey
Shape optimization for the generalized Graetz problem
2014
Structural And Multidisciplinary Optimization
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The problem boils down to the optimization of generalized eigenvalues on a two phases domain. ...
This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 10965 Abstract We apply shape optimization tools to the generalized Graetz problem which is a convection-diffusion ...
The mathematical analysis and numerical methods designed to solve the generalized Graetz problem in semi-infinite and finite domains are presented in Fehrenbach et al. (2012) . ...
doi:10.1007/s00158-013-1032-4
fatcat:6pxk2rtnlzg2bc4g2dl7syuauq
A characteristic/finite element algorithm for time-dependent 3-D advection-dominated transport using unstructured grids
2003
Computer Methods in Applied Mechanics and Engineering
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The practical utility and effectiveness of the proposed numerical scheme was further demonstrated by solving the Graetz-Nusselt problem, i.e. high Peclet number mass/heat transport in a fully developed ...
The algorithm showed the expected error behaviour and good performance in modeling advection-dominated transport problems. ...
Fig. 10 . 10 A schematic of the mapping used to create the mesh in test case 3 (Graetz-Nusselt problem). Solid lines represent the box in which the structured mesh was generated. ...
doi:10.1016/s0045-7825(02)00627-8
fatcat:kpxmpfp235d2npa5w77ypncbhi
The Case of Neumann, Robin, and Periodic Lateral Conditions for the Semi-infinite Generalized Graetz Problem and Applications
2018
SIAM Journal on Applied Mathematics
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This analysis is the cornerstone of numerical methods to solve Graetz problem 11 in finite pipes, semi infinite pipes and exchangers of arbitrary cross section. ...
The authors of [15] coined this problem as the "generalized Graetz problem". 39 Numerical methods for the approximation of this operator and error estimates where 40 provided in [4]. 41 The objective of ...
Resolution of the problem in a finite domain. ...
doi:10.1137/17m1157507
fatcat:hxkigpks7jdnhhuuilw6zfblpm
An assessment of the accuracy of numerical solutions to the graetz problem
1985
International Communications in Heat and Mass Transfer
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It has been well documented in prior studies that numerical solutions to the classical Graetz problem in the entrance region of pipes produce estimates of Nusselt number that are greatly in error. ...
In this report, we examine this problem and attempt to provide plausible explanation for the discrepancy. 209 210 N. Conley, A. Lawal and A.S. Mujumdar Vol. 12, No. 2 * -6 X M with X =I0 . ...
Acknowledgements The authors are indebted to Professor W. Thorpe for providing extensive computer time on the Amdahl V7A of the McGill Computing Centre. Nomenclature ...
doi:10.1016/0735-1933(85)90069-7
fatcat:4yk4bmf3hreotjodxg6xoyehwq
Exact Analytical Solution of Boundary Value Problem in a Form of an Infinite Hypergeometric Series
2017
International Journal of Mathematical Sciences and Computing
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This paper proposes an exact solution of the classical Graetz problem in terms of an infinite series represented by a nonlinear partial differential equation considering two space variables, two boundary ...
The mathematical derivation is based on the method of separation of variables whose several stages were illustrated to reach the solution of the Graetz problem. ...
[5] presented an exact solution for a Graetz problem with axial diffusion and flow heating effects in a semi-infinite domain with a given inlet condition. ...
doi:10.5815/ijmsc.2017.01.03
fatcat:qv7x5vxrkrdsrksuct2u3cnpci
Exact Graetz problem solution by using hypergeometric function
2017
International Journal of Heat and Technology
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This paper proposes an exact solution of the classical Graetz problem in terms of an infinite series represented by a nonlinear partial differential equation considering two space variables, two boundary ...
In addition, the analytical solution was compared to the numerical values obtained previously by Shah and London. ...
[6] presented an exact solution for a Graetz problem with axial diffusion and flow heating effects in a semi-infinite domain with a given inlet condition. ...
doi:10.18280/ijht.350216
fatcat:5lg5xtrdjjfr3a6sohstfwm7xu
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