EVERYDAY EXAMPLES OF ENGINEERING CONCEPTS

Transcription

1 EVERYDAY EXAMPLES OF ENGINEERING CONCEPTS F7: Simiitude & dimensiona anaysis Copyright 04 This work is icensed under the Creative Commons Attribution-NonCommercia-NoDerivs.0 Unported License. To view a copy of this icense visit or send a etter to Creative Commons 444 Castro Street Suite 900 Mountain View Caifornia 9404 USA. Everyday Exampes from of 6

2 This is an extract from 'Rea Life Exampes in Fuid Mechanics: Lesson pans and soutions' edited by Eann A. Patterson first pubished in 0 (ISBN: ) which can be obtained on-ine at and contains suggested exempars within esson pans for Sophomore Fuids Courses. They were prepared as part of the NSF-supported project (#04756) entited: Enhancing Diversity in the Undergraduate Mechanica Engineering Popuation through Curricuum Change". INTRODUCTION (from 'Rea Life Exampes in Fuid Mechanics: Lesson pans and soutions') These notes are designed to enhance the teaching of a sophomore eve course in fuid mechanics increase the accessibiity of the principes and raise the appea of the subject to students from diverse backgrounds. The notes have been prepared as skeeta esson pans using the principe of the 5Es: Engage Expore Expain Eaborate and Evauate. The 5E outine is not origina and was deveoped by the Bioogica Sciences Curricuum Study in the 980s from work by Atkin & Karpus in 96. Today this approach is considered to form part of the constructivist earning theory. These notes are intended to be used by instructors and are written in a stye that addresses the instructor however this is not intended to excude students who shoud find the notes and exampes interesting stimuating and hopefuy iuminating particuary when their instructor is not utiizing them. In the interest of brevity and carity of presentation standard derivations common tabes/charts and definitions are not incuded since these are readiy avaiabe in textbooks which these notes are not intended to repace but rather to suppement and enhance. Simiary it is anticipated that these esson pans can be used to generate ectures/essons that suppement those covering the fundamentas of each topic. It is assumed that students have acquired a knowedge and understanding the foowing topics: first and second aw of thermodynamics Newton s aws free-body diagrams and stresses in pressure vesses. This is the fourth in a series of such notes. The others are entited Rea Life Exampes in Mechanics of Soids Rea Life Exampes in Dynamics and Rea Life Exampes in Thermodynamics. They are avaiabe on-ine at Eann A. Patterson A.A. Griffith Chair of Structura Materias and Mechanics Schoo of Engineering University of Liverpoo Liverpoo UK & Roya Society Wofson Research Merit Award Recipient Engeman Laura (ed.) The BSCS Story: A History of the Bioogica Sciences Curricuum Study. Coorado Springs: BSCS 00. Atkin J. M. and Karpus R. (96). Discovery or invention? Science Teacher 9(5): 45. e.g. Trowbridge L.W. and Bybee R.W. Becoming a secondary schoo science teacher. Merri Pub. Co. Inc Everyday Exampes from of 6

3 MODELLING 7. Topic: Simiitude and dimensiona anaysis Engage: Take your toy boats from the into cass. If you don t have any then you coud either borrow or buy some; or at the end of the previous cass invite students to bring in their own toys. Show them a video from YouTube of a ship in rough seas (search using Abeie Fandre and show the cip of this name 4. Expore: We have a payed with boats in the. Discuss whether the behavior of toy boats wi provide a good mode for predicting the behavior of fu-scae ships at sea. We coud get out of the and conduct the experiment in the controed environment of the aboratory; but woud the behavior in the aboratory be a good prediction of the performance on the high seas? Ask students to construct a ist of factors that coud differ between the ab and the and then construct a master ist on the board. Expain: You shoud have ended up with a ong ist of factors that coud vary between the ab and the fuscae performance at sea. Expain that it is advantageous to group variabes such as pressure density ength viscosity veocity into non-dimensiona groups and then to conduct experiments to estabish the functiona reationship between the groups rather than the variabes. This consideraby reduces the amount of experimentation required and can aso hep in ensuring simiitude between experiments and the prototype or fu-scae version. Eaborate: An American physicist Edgar Buckingham ( ) showed that the number of nondimensiona groups required to correate the variabes in a certain process is given by n-m where n is the number of variabes to be grouped and m is the number of basic dimensions incuded amongst the variabes. So we might expect the force F acting on our ship and toy boat woud be a function of fuid density ρ; dynamic viscosity μ; gravity g; the speed of the ship v; and a characteristic dimension of the ship. So F f g v 4 Everyday Exampes from of 6

4 The fundamenta dimensions of these quantities are F N ( = kg m s-) MLT - kg m - ML - N s m - ML - T - g m s - LT - v m s - LT - m L So we have six variabes and three basic dimensions (M: mass L: ength and T: time) and thus we wi need three ( 6 ) non-dimensiona or Pi groups (hence the name Buckingham-Pi approach). This means we can reduce the number of variabes from six to three. To find the first group we can seect the dependent variabe F and form a non-dimensiona group by introducing variabes with the appropriate dimensions in such a way as to create the non-dimensiona group. If a dimension exists on its own as the ony dimension of a variabe then it shoud be considered ast. So in this case starting with [M] we can introduce and then v to have [T - ] but then we have [L - ][L ] = [L - ] and we need [L] so we must introduce. So F v This is effectivey the ratio of the shear force on the hu to the inertia forces. The process can be repeated choosing to form the second group v Note that the same repeating variabes are used to achieve the non-dimensiona group. This group is the Reynods number and describes the ratio of the inertia to shear forces in the fuid. And finay for third group using g we can obtain g v This is the Froude number and describes the ratio of inertia to gravitationa forces for a fuid with a free surface. So the functiona reationship for the behavior of the ship either as a mode (toy boat) or fu-scae in the is F g v v v In order to achieve simiitude between a mode and a prototype Pi groups are made equivaent. So for instance if a tug boat in your is 6cm ong and an -going tug is m ong then above woud impy that g v g v or v v And the speed of your mode needs to be / (= 5 ) of the speed of the -going tug which is just as we since a typica speed for the version is knots or 5.6 m/s. The Everyday Exampes from 4 of 6

5 mode woud have to move at 0.4 m/s (=5.6/) or about 9 inches/sec which is quite fast for the. Evauate: Ask the students to attempt the foowing exampes: Exampe 7. The suction of a vacuum ceaner can be equated with the pressure drop across its fan p which is in turn reated the fan diameter D; its axia ength ; the rotationa speed ; the inet and outet diameters d and d and the air density. Find the functiona reationship between these groups. Soution: p f D d d We can express the dimensions of the variabes as foows: p Pa (N m - ) ML - T - D m L m L rad s - T - d m L d m L kg m - ML - There are seven variabes and three basic dimensions so there wi be four non-dimensiona groups. The repeating variabes are D and. The first group can be formed around p and we obtain p D For the second group use the next non-repeating variabe and so on to give and D D Exampe 7. d d D and 4 D p d D D d D In order to study the interaction of a micro-surgery device and the fow in an artery a five times scae mode of an artery is to be constructed. The voume fow rate Q in the artery is beieved to be a function of frequency of the heart beat ; artery diameter D; the fuid density ; Everyday Exampes from 5 of 6

6 viscosity ; and the pressure gradient p/. Identify the dimensiona groups and estimate the voume fow rate required if saine is used as the work fuid instead of bood. Soution: p Q D We can express the dimensions of the variabes as foows: Q m s - L T - s - T - D m L kg m - ML - N s m - ML - T - p/ Pa/m (N m - ) ML - T - There are six variabes and three basic dimensions so there wi be three non-dimensiona groups. The repeating variabes are f D and. The first group can be formed around p and we obtain Q D Then taking each of the non-repeating variabes in turn p and D D So if the working fuid is changed from bood to saine equivaence must be maintained so D s b b s b b 0. b sds assuming the saine is formuated to give approximatey the same viscosity as bood the mode heart beat needs to be about /9 th of the natura heart rate. Now for fowrate equivaence of the first Pi group is required so D s s Qs Qb Qb 7 Dbb Q b Hence the voume fowrate woud need to be about 70 times the natura vaue in the body. b Everyday Exampes from 6 of 6