### Exemplification of Dimensional Analysis via MATLAB® Using Eigen Values

Received: 15 October 2016     Accepted: 2 November 2016     Published: 18 January 2017
Abstract

In this research article, a set of dimensional physical quantities is transformed into a dimensionless group (or ratio). For a given set of dimensional variables, the physical variables represent the rows and their dimensions represent the columns of a dimensions-matrix. The dimensions-matrix is rearranged both column- and row-wise. The columns are sorted in ascending order based on the column sum and then on the largest negative entry (i.e., cell value). On the other hand, the rows are sorted in descending order based on the number of non-zero entries found in each row and then on the higher first entry. With the aid of MATLAB®, it was found that the proposed method leads to a permutation matrix that has an Eigen vector whose elements represent the exponent for each physical dimensional quantity such that, at the end, a dimensionless group (or ratio) can be formulated, like Schmidt, Nusselt, Reynolds, and Peclet number. The method, however, was found to work well with a set of physical quantities where each is raised to an exponent of ±1.

 Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 3, Issue 1) DOI 10.11648/j.ijamtp.20170301.13 Page(s) 14-19 Creative Commons This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. Copyright Copyright © The Author(s), 2017. Published by Science Publishing Group
Keywords

Dimensional Analysis, Reynolds, Nusselt, Schmidt, Peclet, Froude, Exemplification, Chemical Engineering Education, MATLAB, Eigen Vector

References
 [1] Al-Malah, K. “Exemplification of the Concept of Diffusivity in Mass Transfer”, International Journal of Chemical Engineering and Applied Sciences, 2(4): 24-26 (2012). http://urpjournals.com/tocjnls/9_12v2i4_1.pdf. [2] Al-Malah, K. “Exemplification of a Semi-Infinite vs. Finite Medium in Heat/Mass Transfer”. Chemical Engineering & Process Techniques, 2(1): 1021 (2014). http://www.jscimedcentral.com/ChemicalEngineering/chemicalengineering-2-1021.pdf [3] Al-Malah, K. “Prediction of Normal Boiling Points of Hydrocarbons Using Simple Molecular Properties”, J. of Advanced Chemical Engineering, Vol. 3 (2013). doi:10.4303/jace/235654. http://www.ashdin.com/journals/JACE/235654.pdf. [4] Al-Malah, Kamal I. “RK-, SRK-, & SRK-PR-Type Equation of State for Hydrocarbons, Based on Simple Molecular Properties”. Journal of Applied Chemical Science International, 2(2): 65-74, (2015). http://www.ikpress.org/issue.php?iid=451&id=35. [5] Buckingham, E. "On Physically Similar Systems: Illustrations of the Use of Dimensional Equations." Phys. Rev. 4: 345-376, (1914). [6] Buckingham, E. "Model Experiments and the Form of Empirical Equations." Trans. ASME 37: 263, (1915). [7] Leona, A.S. and L. Zhub. “A dimensional analysis for determining optimal discharge and penstock diameter in impulse and reaction water turbines”, Renewable Energy, 71: 609–615, (2014). doi:10.1016/j.renene.2014.06.024. [8] Matuszak, A. “Dimensional Analysis can Improve Equations of the Model”. Procedia Engineering, 108: 526-535, (2015). doi:10.1016/j.proeng.2015.06.174. [9] Olmos, E., K. Loubiere, C. Martin, G. Delaplace, and Annie Marc. “Critical agitation for microcarrier suspension in orbital shaken bioreactors: Experimental study and dimensional analysis”, Chemical Engineering Science, 122: 545–554, (2015). doi:10.1016/j.ces.2014.08.063. [10] Sollund, H.A., K. Vedeld, and O. Fyrileiv. “Modal response of free spanning pipelines based on dimensional analysis”, Applied Ocean Research, 50: 13-29, (2015). doi:10.1016/j.apor.2014.12.001. [11] Al-Malah, K., “MATLAB®: Numerical Methods with Chemical Engineering Applications”, McGraw Hill, Inc., New York, USA. ISBN: 9780071831284 (2013). http://www.mhprofessional.com/product.php?isbn=0071831282
• APA Style

Kamal Isa Masoud Al-Malah. (2017). Exemplification of Dimensional Analysis via MATLAB® Using Eigen Values. International Journal of Applied Mathematics and Theoretical Physics, 3(1), 14-19. https://doi.org/10.11648/j.ijamtp.20170301.13

ACS Style

Kamal Isa Masoud Al-Malah. Exemplification of Dimensional Analysis via MATLAB® Using Eigen Values. Int. J. Appl. Math. Theor. Phys. 2017, 3(1), 14-19. doi: 10.11648/j.ijamtp.20170301.13

AMA Style

Kamal Isa Masoud Al-Malah. Exemplification of Dimensional Analysis via MATLAB® Using Eigen Values. Int J Appl Math Theor Phys. 2017;3(1):14-19. doi: 10.11648/j.ijamtp.20170301.13

• ```@article{10.11648/j.ijamtp.20170301.13,
author = {Kamal Isa Masoud Al-Malah},
title = {Exemplification of Dimensional Analysis via MATLAB® Using Eigen Values},
journal = {International Journal of Applied Mathematics and Theoretical Physics},
volume = {3},
number = {1},
pages = {14-19},
doi = {10.11648/j.ijamtp.20170301.13},
url = {https://doi.org/10.11648/j.ijamtp.20170301.13},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20170301.13},
abstract = {In this research article, a set of dimensional physical quantities is transformed into a dimensionless group (or ratio). For a given set of dimensional variables, the physical variables represent the rows and their dimensions represent the columns of a dimensions-matrix. The dimensions-matrix is rearranged both column- and row-wise. The columns are sorted in ascending order based on the column sum and then on the largest negative entry (i.e., cell value). On the other hand, the rows are sorted in descending order based on the number of non-zero entries found in each row and then on the higher first entry. With the aid of MATLAB®, it was found that the proposed method leads to a permutation matrix that has an Eigen vector whose elements represent the exponent for each physical dimensional quantity such that, at the end, a dimensionless group (or ratio) can be formulated, like Schmidt, Nusselt, Reynolds, and Peclet number. The method, however, was found to work well with a set of physical quantities where each is raised to an exponent of ±1.},
year = {2017}
}
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JO  - International Journal of Applied Mathematics and Theoretical Physics
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AB  - In this research article, a set of dimensional physical quantities is transformed into a dimensionless group (or ratio). For a given set of dimensional variables, the physical variables represent the rows and their dimensions represent the columns of a dimensions-matrix. The dimensions-matrix is rearranged both column- and row-wise. The columns are sorted in ascending order based on the column sum and then on the largest negative entry (i.e., cell value). On the other hand, the rows are sorted in descending order based on the number of non-zero entries found in each row and then on the higher first entry. With the aid of MATLAB®, it was found that the proposed method leads to a permutation matrix that has an Eigen vector whose elements represent the exponent for each physical dimensional quantity such that, at the end, a dimensionless group (or ratio) can be formulated, like Schmidt, Nusselt, Reynolds, and Peclet number. The method, however, was found to work well with a set of physical quantities where each is raised to an exponent of ±1.
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Author Information
• Department of Chemical Engineering, Higher Colleges of Technology, Ruwais, UAE

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