1、核能電廠核能電廠 CFD安全分析研究室安全分析研究室Institute of Nuclear Engineering and ScienceNPP CFD Modeling and Safety Analysis LabChapter 7 Dimensional AnalysisModeling, and SimilitudeInstitute of Nuclear Engineering and ScienceNPP CFD Modeling and Safety Analysis Lab核能電廠核能電廠 CFD安全分析研究室安全分析研究室2MAIN TOPICS7.1 Dimensiona
2、l Analysis7.2 Buckingham Pi Theorem7.3 Determination of Pi Terms7.4 Comments about Dimensional Analysis7.5 Determination of Pi Terms by Inspection7.6 Common Dimensionless Groups in Fluid Mechanics7.7 Correlation of Experimental Data7.8 Modeling and Similitude7.9 Typical Model Studies7.10 Similitude
3、Based on Governing Differential EquationInstitute of Nuclear Engineering and ScienceNPP CFD Modeling and Safety Analysis Lab核能電廠核能電廠 CFD安全分析研究室安全分析研究室37.1 Dimensional Analysis_1l A typical fluid mechanics problem in which experimentation is required, consider the steady flow of an incompressible New
4、tonian fluid through a long, smooth-walled, horizontal, circular pipe. l An important characteristic of this system, which would be interest to an engineer designing a pipeline, is the pressure drop per unit length that develops along the pipe as a result of friction.Institute of Nuclear Engineering
5、 and ScienceNPP CFD Modeling and Safety Analysis Lab核能電廠核能電廠 CFD安全分析研究室安全分析研究室4Dimensional Analysis_2v The first step in the planning of an experiment to study this problem would be to decide on the factors, or variables, that will have an effect on the pressure drop.v Pressure drop per unit length
6、Pressure drop per unit length depends on FOUR variables:piping diameter (D); velocity (V); fluid density (); fluid viscosity (m)Institute of Nuclear Engineering and ScienceNPP CFD Modeling and Safety Analysis Lab核能電廠核能電廠 CFD安全分析研究室安全分析研究室5Dimensional Analysis_3l To perform the experiments in a meani
7、ngful and systematic manner, it would be necessary to change on of the variable, such as the velocity, which holding all other constant, and measure the corresponding pressure drop.l Difficulty to determine the functional relationship between the pressure drop and the various factors that influence
8、it.Institute of Nuclear Engineering and ScienceNPP CFD Modeling and Safety Analysis Lab核能電廠核能電廠 CFD安全分析研究室安全分析研究室6Series of TestsInstitute of Nuclear Engineering and ScienceNPP CFD Modeling and Safety Analysis Lab核能電廠核能電廠 CFD安全分析研究室安全分析研究室7Dimensional Analysis_4v Fortunately, there is a much simpler
9、 approach to the problem that will eliminate the difficulties described above.aCollecting these variables into two nondimensional combinations of the variables (called dimensionless product or dimensionless groups) Only one dependent and one independent variable Easy to set up experiments to determi
10、ne dependency Easy to present results (one graph)Institute of Nuclear Engineering and ScienceNPP CFD Modeling and Safety Analysis Lab核能電廠核能電廠 CFD安全分析研究室安全分析研究室8Plot of PressureDrop Data Using dimensionless product or dimensionless groupsInstitute of Nuclear Engineering and ScienceNPP CFD Modeling an
11、d Safety Analysis Lab核能電廠核能電廠 CFD安全分析研究室安全分析研究室Pumping PowerThe h-s diagram is usually used to calculate the net work and efficiency of RC. This process can be described as follows. Institute of Nuclear Engineering and ScienceNPP CFD Modeling and Safety Analysis Lab核能電廠核能電廠 CFD安全分析研究室安全分析研究室107.2 Bu
12、ckingham Pi Theorem_1v A fundamental question we must answer is how many dimensionless products are required to replace the original list of variables ?v The answer to this question is supplied by the basic theorem of dimensional analysis that statesv If an equation involving k variables is dimensionally homogeneous, it can be reduced to a relationship among k-r independent dimensionless products, where r is the minimum number of reference dimensions required to describe the variables.Buckingham Pi Theorem Pi terms
