Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion
There are two types of analytical solutions of temperature/concentration in and heat/mass transfer through boundaries of regularly shaped 1D, 2D, and 3D blocks. These infiniteseries solutions with either error functions or exponentials exhibit highly irregular but complementary convergence at different dimensionless times, t _{d0}. In this paper, approximate solutions were developed by combining the errorfunctionseries solutions for early times and the exponentialseries solutions for late times and by using time partitioning at the switchover time, t _{d0}. The combined solutions contain either the leading term of both series for normalaccuracy approximations (with less than 0.003 relative error) or the first two terms for highaccuracy approximations (with less than 107 relative error) for 1D isotropic (spheres, cylinders, slabs) and 2D/3D rectangular blocks (squares, cubes, rectangles, and rectangular parallelepipeds). This rapid and uniform convergence for rectangular blocks was achieved by employing the same time partitioning with individual dimensionless times for different directions and the product of their combined 1D slab solutions. The switchover dimensionless time was determined to minimize the maximum approximation errors. Furthermore, the analytical solutions of firstorder heat/mass flux for 2D/3D rectangular blocks were derived for normalaccuracy approximations. These flux equations contain the earlytime solution with a threeterm polynomial inmore »
 Authors:

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 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
 Publication Date:
 Grant/Contract Number:
 FC2605NT42587; AC0205CH11231; FE0023323
 Type:
 Accepted Manuscript
 Journal Name:
 Water Resources Research
 Additional Journal Information:
 Journal Volume: 53; Journal Issue: 11; Journal ID: ISSN 00431397
 Publisher:
 American Geophysical Union (AGU)
 Research Org:
 Montana State Univ., Bozeman, MT (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
 Sponsoring Org:
 USDOE Office of Fossil Energy (FE)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 42 ENGINEERING
 OSTI Identifier:
 1430907
 Alternate Identifier(s):
 OSTI ID: 1409860; OSTI ID: 1476561
Zhou, Quanlin, Oldenburg, Curtis M., Rutqvist, Jonny, and Birkholzer, Jens T.. Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion. United States: N. p.,
Web. doi:10.1002/2017WR021040.
Zhou, Quanlin, Oldenburg, Curtis M., Rutqvist, Jonny, & Birkholzer, Jens T.. Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion. United States. doi:10.1002/2017WR021040.
Zhou, Quanlin, Oldenburg, Curtis M., Rutqvist, Jonny, and Birkholzer, Jens T.. 2017.
"Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion". United States.
doi:10.1002/2017WR021040.
@article{osti_1430907,
title = {Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion},
author = {Zhou, Quanlin and Oldenburg, Curtis M. and Rutqvist, Jonny and Birkholzer, Jens T.},
abstractNote = {There are two types of analytical solutions of temperature/concentration in and heat/mass transfer through boundaries of regularly shaped 1D, 2D, and 3D blocks. These infiniteseries solutions with either error functions or exponentials exhibit highly irregular but complementary convergence at different dimensionless times, td0. In this paper, approximate solutions were developed by combining the errorfunctionseries solutions for early times and the exponentialseries solutions for late times and by using time partitioning at the switchover time, td0. The combined solutions contain either the leading term of both series for normalaccuracy approximations (with less than 0.003 relative error) or the first two terms for highaccuracy approximations (with less than 107 relative error) for 1D isotropic (spheres, cylinders, slabs) and 2D/3D rectangular blocks (squares, cubes, rectangles, and rectangular parallelepipeds). This rapid and uniform convergence for rectangular blocks was achieved by employing the same time partitioning with individual dimensionless times for different directions and the product of their combined 1D slab solutions. The switchover dimensionless time was determined to minimize the maximum approximation errors. Furthermore, the analytical solutions of firstorder heat/mass flux for 2D/3D rectangular blocks were derived for normalaccuracy approximations. These flux equations contain the earlytime solution with a threeterm polynomial in √td and the latetime solution with the limitedterm exponentials for rectangular blocks. The heat/mass flux equations and the combined temperature/concentration solutions form the ultimate kernel for fast simulations of multirate and multidimensional heat/mass transfer in porous/fractured media with millions of lowpermeability blocks of varying shapes and sizes.},
doi = {10.1002/2017WR021040},
journal = {Water Resources Research},
number = 11,
volume = 53,
place = {United States},
year = {2017},
month = {10}
}