Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion
Abstract
There are two types of analytical solutions of temperature/concentration in and heat/mass transfer through boundaries of regularly shaped 1-D, 2-D, and 3-D blocks. These infinite-series solutions with either error functions or exponentials exhibit highly irregular but complementary convergence at different dimensionless times, td. In this paper, approximate solutions were developed by combining the error-function-series solutions for early times and the exponential-series 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 normal-accuracy approximations (with less than 0.003 relative error) or the first two terms for high-accuracy approximations (with less than 10-7 relative error) for 1-D isotropic (spheres, cylinders, slabs) and 2-D/3-D 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 1-D slab solutions. The switchover dimensionless time was determined to minimize the maximum approximation errors. Furthermore, the analytical solutions of first-order heat/mass flux for 2-D/3-D rectangular blocks were derived for normal-accuracy approximations. These flux equations contain the early-time solution with a three-term polynomial in √td andmore »
- Publication Date:
- Research Org.:
- Montana State Univ., Bozeman, MT (United States); Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
- Sponsoring Org.:
- USDOE Office of Fossil Energy (FE)
- OSTI Identifier:
- 1430907
- Alternate Identifier(s):
- OSTI ID: 1409860; OSTI ID: 1476561
- Grant/Contract Number:
- FC26-05NT42587; DE‐AC02‐05CH11231; AC02-05CH11231
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Water Resources Research
- Additional Journal Information:
- Journal Volume: 53; Journal Issue: 11; Journal ID: ISSN 0043-1397
- Publisher:
- American Geophysical Union (AGU)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 42 ENGINEERING
Citation Formats
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., 2017.
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. https://doi.org/10.1002/2017WR021040
Zhou, Quanlin, Oldenburg, Curtis M., Rutqvist, Jonny, and Birkholzer, Jens T. Tue .
"Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion". United States. https://doi.org/10.1002/2017WR021040. https://www.osti.gov/servlets/purl/1430907.
@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 1-D, 2-D, and 3-D blocks. These infinite-series solutions with either error functions or exponentials exhibit highly irregular but complementary convergence at different dimensionless times, td. In this paper, approximate solutions were developed by combining the error-function-series solutions for early times and the exponential-series 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 normal-accuracy approximations (with less than 0.003 relative error) or the first two terms for high-accuracy approximations (with less than 10-7 relative error) for 1-D isotropic (spheres, cylinders, slabs) and 2-D/3-D 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 1-D slab solutions. The switchover dimensionless time was determined to minimize the maximum approximation errors. Furthermore, the analytical solutions of first-order heat/mass flux for 2-D/3-D rectangular blocks were derived for normal-accuracy approximations. These flux equations contain the early-time solution with a three-term polynomial in √td and the late-time solution with the limited-term 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 low-permeability blocks of varying shapes and sizes.},
doi = {10.1002/2017WR021040},
journal = {Water Resources Research},
number = 11,
volume = 53,
place = {United States},
year = {Tue Oct 24 00:00:00 EDT 2017},
month = {Tue Oct 24 00:00:00 EDT 2017}
}
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