Buckingham π theorem if there are k variables ݑଵ

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Blueprint Reading for Welders
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Chapter 4 / Exercise 1
Blueprint Reading for Welders
Bennett/Siy
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3.Buckingham πTheorem If there are k variables ݑ, ݑ, ݑ, ⋯ , ݑݑൌ ݂ሺݑ, ݑ, ⋯ , ݑwhere f is a function, then we can form (k r) πgroups ߨ, ߨ, ߨ, ⋯ , ߨ௞ି௥, so that ߨൌ ݃ሺߨ, ߨ, ⋯ , ߨ௞ି௥where r is the number of basic dimensionsand g is a different function. 4.Procedures for applying Buckingham πTheorem: a.Determine relevant variables for a given problem. b.Determine the dimensions of all variables and the number of basic dimensions. c.Determine the number of πgroups. d.Select repeating variables: e.Form πgroups. f.Check that the resulting πgroups are indeed non-dimensional. g.Express the final functional form using the πgroups. and
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Blueprint Reading for Welders
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Chapter 4 / Exercise 1
Blueprint Reading for Welders
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4  Experimental Platform and Procedures Overview A virtual laboratory platform is used in this experiment. The experimental system includes a ball suspended in a jet of water or air. You are asked to identify the dependence of the ball suspension height on various parameters such as the sizes of the ball and water jet, the fluid properties and the flow velocity etc. Specially, the relevant variables to be considered in this virtual experiment include: h: the suspension height of the ball (m); W: the weight of the ball (N); D: the diameter of the ball (m); d: the diameter of the jet (m); Q: the jet flow rate (m3/s); ρ: the density of the fluid (kg/m3); μ: the viscosity of the fluid (Ns/m2). Procedures: 1.Installation of Virtual Laboratory (VL) Use the links provided on Canvas to download the correct version of the VL. There are two versions, one for Mac users and the other for Windows users. 2.Data Collection Once you start the VL platform, you will be taken to the following login screen: Figure 1. The login window.
5  If you have not registered for the VL system previously, use your USYD email and follow the on-screen instructions to register. Otherwise log on to VL using your credentials. Once logging on to the system, you will be presented with the Welcome screen for Virtual Lab 3 – Modelling. If not, click one of the arrows at the lower-left end of the screen to select the correct experiment. Read carefully the information and instructions as shown in Figure 2. You can click <Learn More> to browse through the instruction pages or click the Red Cross button to skip the instructions. You will then enter the environment for Virtual Lab 3, as shown in Figure 3. Figure 2. The instruction page.Figure 3. The virtual laboratory environment. The on-screen information is self-explanatory. You will be able to change the Flow Rate, the Mass and Diameter of the Ball, and the Type of Fluid from the jet. The water
6  or air jet will evolve accordingly and the ball will be suspended at different heights, depending on the parameter settings. You can press the Save button () to record a data set, which also saves all relevant settings such as the fluid properties and jet diameter etc. Warning:Press the Save button only after the ball height is approximately stabilized.You can check your collected data in the data table displayed on the right of the screen.

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