Empirical Colebrook equation implicit in unknown flow friction factor (λ) is an accepted standard for calculation of hydraulic resistance in hydraulically smooth and rough pipes. The Colebrook equation gives friction factor (λ) implicitly as a function of the Reynolds number (Re) and relative roughness (e/D) of inner pipe surface; i.e. λ0=f(λ0, Re, e/D). The paper presents a problem that requires iterative methods for the solution. In particular, the implicit method used for calculating the friction factor λ0 is an application of fixed-point iterations. The type of problem discussed in this "in the classroom paper" is commonly encountered in fluid dynamics, and this paper provides readers with the tools necessary to solve similar problems. Students’ task is to solve the equation using Excel where the procedure for that is explained in this “in the classroom” paper. Also, up to date numerous explicit approximations of the Colebrook equation are available where as an additional task for students can be evaluation of the error introduced by these explicit approximations λ≈f(Re, e/D) compared with the iterative solution of implicit equation which can be treated as accurate.
隐含未知流动摩擦系数(λ)的经验Colebrook方程是计算光滑和粗糙管道水力阻力的公认标准。Colebrook方程将摩擦系数λ隐式地表示为管内表面雷诺数Re和相对粗糙度e/D的函数;即λ0=f(λ0, Re, e/D)。本文提出了一个需要用迭代方法求解的问题。特别地，用于计算摩擦系数λ0的隐式方法是不动点迭代的一种应用。在这篇“课堂论文”中讨论的问题类型是流体动力学中经常遇到的问题，本文为读者提供了解决类似问题所需的工具。学生的任务是用Excel来解方程，在这篇“课堂上”的论文中解释了这个过程。此外，到目前为止，Colebrook方程的许多显式近似是可用的，作为学生的额外任务，可以评估这些显式近似λ≈f(Re, e/D)与隐式方程的迭代解相比所引入的误差，隐式方程可以被视为准确。
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An example of hydraulic design project for teaching purpose is presented. Students’ task is to develop a looped distribution network for water (i.e. to determinate node consumptions, disposal of pipes, and finally to calculate flow rates in the network’s pipes and their optimal diameters). This can be accomplished by using the original Hardy Cross method, the improved Hardy Cross method, the node-loop method, etc. For the improved Hardy Cross method and the node-loop method, use of matrix calculation is mandatory. Because the analysis of water distribution networks is an essential component of civil engineering water resources curricula, the adequate technique better than the hand-oriented one is desired in order to increase students’ understanding of this kind of engineering systems and of relevant design issues in more concise and effective way. The described use of spreadsheet solvers is more than suitable for the purpose, especially knowing that spreadsheet solvers are much more matrix friendly compared with the hand-orientated calculation. Although matrix calculation is not mandatory for the original Hardy Cross method, even in that case it is preferred for better understanding of the problem. The application of commonly available spreadsheet software (Microsoft Excel) including two real classroom tasks is presented.
{"title":"Spreadsheet-Based Pipe Networks Analysis for Teaching and Learning Purpose","authors":"D. Brkić","doi":"10.31219/osf.io/7ynvw","DOIUrl":"https://doi.org/10.31219/osf.io/7ynvw","url":null,"abstract":"An example of hydraulic design project for teaching purpose is presented. Students’ task is to develop a looped distribution network for water (i.e. to determinate node consumptions, disposal of pipes, and finally to calculate flow rates in the network’s pipes and their optimal diameters). This can be accomplished by using the original Hardy Cross method, the improved Hardy Cross method, the node-loop method, etc. For the improved Hardy Cross method and the node-loop method, use of matrix calculation is mandatory. Because the analysis of water distribution networks is an essential component of civil engineering water resources curricula, the adequate technique better than the hand-oriented one is desired in order to increase students’ understanding of this kind of engineering systems and of relevant design issues in more concise and effective way. The described use of spreadsheet solvers is more than suitable for the purpose, especially knowing that spreadsheet solvers are much more matrix friendly compared with the hand-orientated calculation. Although matrix calculation is not mandatory for the original Hardy Cross method, even in that case it is preferred for better understanding of the problem. The application of commonly available spreadsheet software (Microsoft Excel) including two real classroom tasks is presented.","PeriodicalId":41809,"journal":{"name":"Spreadsheets in Education","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2016-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69636444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is prompted by a recent call by the International Commission on Mathematical Instruction (ICMI) for the study of mathematical modeling as technology-enhanced didactic inquiry into relations between mathematics and the real world. It reflects on activities designed for a teacher education course that focuses on the computer spreadsheet as a tool for concept development through situated mathematical problem solving. Modeling activities described in this paper support the epistemological position regarding the interplay that exists between the development of mathematical concepts and available methods of calculation. The spreadsheet used is Microsoftr °Excel 2001.
{"title":"Spreadsheet-Enhanced Problem Solving in Context as Modeling","authors":"S. Abramovich","doi":"10.1155/2014/345146","DOIUrl":"https://doi.org/10.1155/2014/345146","url":null,"abstract":"This paper is prompted by a recent call by the International Commission on Mathematical Instruction (ICMI) for the study of mathematical modeling as technology-enhanced didactic inquiry into relations between mathematics and the real world. It reflects on activities designed for a teacher education course that focuses on the computer spreadsheet as a tool for concept development through situated mathematical problem solving. Modeling activities described in this paper support the epistemological position regarding the interplay that exists between the development of mathematical concepts and available methods of calculation. The spreadsheet used is Microsoftr °Excel 2001.","PeriodicalId":41809,"journal":{"name":"Spreadsheets in Education","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2014-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2014/345146","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64447354","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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