DOCSLIB.ORG

Similitude and Dimensional Analysis III Analysis of Turbomachines

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Similitude and Dimensional Analysis III Analysis of Turbomachines Similitude and Dimensional Analysis III Hydromechanics VVR090 Analysis of Turbomachines • pumps (centrifugal, axial-flow) • turbines (impulse, reaction) Dimensional analysis useful to make generalizations about similar turbomachines or distinguish between them. Relevant variables with reference to power (P): • impeller diameter (D) • rotational speed (N) • flow (Q) • energy added or subtracted (H) [H] = Nm/kg = m2/s2 • fluid properties such as viscosity (m), density (ρ), elasticity (E) 1 Archimedean Screw Pump Rotodynamic Pumps Radial flow pump (centrifugal) Axial flow pump (propeller) 2 Turbines Pelton Kaplan Dimensional Analysis for Turbomachines Assume the following relationship among the variables: fPDNQH{ ,,,,,,,μρ E} = 0 Buckingham’s P-theorem: 3 fundamental dimensions (M, L, T) and 8 variables imply that 8-3=5 P-terms can be formed. Select ρ, D, and N as variables containing the 3 fundamental dimensions to be combined with the remaining 5 variables (P, Q, H, m, and E). Possible to use other variable combinations that contain the fundamental dimensions. 3 Buckinghams’ P-Theorem ρ, D, N combined with m yields: ab c d Π=μρ1 DN Solving the dimensional equations gives: ρND2 Π= =Re 1 μ Derive other P-terms in the same manner: 22 22 ρND ND 2 ρ, D, N combined with E Æ Π= = =M 2 E a2 P ρ, D, N combined with P Æ Π ==C 3 ρND35 P Q ρ, D, N combined with Q Æ Π ==C 4 ND3 Q H ρ, D, N combined with H Æ Π= =C 5 ND22 H 4 Summarizing the results: P 35= fCC',,Re,M{}QH ρND Or: Q = fCC''{} , , Re, M ND3 PH H 22= fCC'''{}PQ , , Re, M ND Previous analysis: PQH∼ ρ Form a new P-term: P CP IV Π=',,Re,M3 = fCC{}QH ρQH CQH C Incompressible flow with CQ and CH held constant: P ==ηf V {}Re ρQH H hH = hydraulic efficiency 5 Alternative Approach Assume that the relationship between P and ρ, Q, and H is known, and that h includes both Re and mechanical effects. Assume the following relationship (incompressible flow): fDNQH{ ,,,,η=} 0 An alternative dimensional analysis gives: HQ⎧⎫ =ηf ',⎨⎬ ND22⎩⎭ ND 3 Typical Plot of Experimental Data Spread represents variation with h (effects of Re) HD∼ 2 QD∼ 3 6 Alternative Dimensionless Terms Specific speed (pumps): NQ N = s H 3/4 (represents actual speed when machine operates under unit head and unit flow) • common to relate h to Ns • characterize classes of pumps etc Specific speed (turbines): NP N = s ρH 5/4 7 Application of Dimensional Analysis to Pipe Friction Assume the wall shear stress (τo) depends on: • mean velocity (V) • diameter (d) • mean height of roughness projections (e) • fluid density (ρ) • fluid viscosity (m) The following relationship should hold: fVde{τρμ=o ,,,,,} 0 Buckinghams’ P-Theorem V, d, ρ combined with τo yields: abcd Π=τ1 oVd ρ Solving the dimensional equations gives: τ Π= o 1 ρV 2 Further analysis gives: e Vdρ Π= Π= 2 d 3 μ 8 The following relationship may be derived: τ⎧⎫o Vdρ e⎧ e ⎫ 2 ==ff',⎨⎬⎨⎬ 'Re, ρμVdd⎩⎭⎩⎭ e = relative roughness d Hydraulically smooth and rough flow Darcy-Weisbach Friction Formula Frictional losses in a pipe: LV2 hf= L dg2 Energy equation: τo L hL = ρgRh ⎧ e ⎫ ff= ''⎨ Re, ⎬ ⎩⎭d 9 Moody Diagram 10.
Load more

Recommended publications
  • Chapter 5 Dimensional Analysis and Similarity
    Chapter 5 Dimensional Analysis and Similarity Motivation. In this chapter we discuss the planning, presentation, and interpretation of experimental data. We shall try to convince you that such data are best presented in dimensionless form. Experiments which might result in tables of output, or even mul- tiple volumes of tables, might be reduced to a single set of curves—or even a single curve—when suitably nondimensionalized. The technique for doing this is dimensional analysis. Chapter 3 presented gross control-volume balances of mass, momentum, and en- ergy which led to estimates of global parameters: mass flow, force, torque, total heat transfer. Chapter 4 presented infinitesimal balances which led to the basic partial dif- ferential equations of fluid flow and some particular solutions. These two chapters cov- ered analytical techniques, which are limited to fairly simple geometries and well- defined boundary conditions. Probably one-third of fluid-flow problems can be attacked in this analytical or theoretical manner. The other two-thirds of all fluid problems are too complex, both geometrically and physically, to be solved analytically. They must be tested by experiment. Their behav- ior is reported as experimental data. Such data are much more useful if they are ex- pressed in compact, economic form. Graphs are especially useful, since tabulated data cannot be absorbed, nor can the trends and rates of change be observed, by most en- gineering eyes. These are the motivations for dimensional analysis. The technique is traditional in fluid mechanics and is useful in all engineering and physical sciences, with notable uses also seen in the biological and social sciences.
    [Show full text]
  • RHEOLOGY #2: Anelasicity
    RHEOLOGY #2: Anelas2city (aenuaon and modulus dispersion) of rocks, an organic, and maybe some ice Chris2ne McCarthy Lamont-Doherty Earth Observatory …but first, cheese Team Havar2 Team Gouda Team Jack Stress and Strain Stress σ(MPa)=F(N)/A(m2) 1 kg = 9.8N 1 Pa= N/m2 or kg/(m s2) Strain ε = Δl/l0 = (l0-l)/l0 l0 Cheese results vs. idealized curve. Not that far off! Cheese results σ n ⎛ −E + PV ⎞ ε = A exp A d p ⎝⎜ RT ⎠⎟ n=1 Newtonian! σ Pa viscosity η = ε s-1 Havarti,Jack η=3*107 Pa s Gouda η=2*108 Pa s Muenster η=6*108 Pa s How do we compare with previous studies? Havarti,Jack η=3*107 Pa s Gouda η=2*108 Pa s Muenster η=6*108 Pa s Despite significant error, not far off published results Viscoelas2city: Deformaon at a range of 2me scales Viscoelas2city: Deformaon at a range of 2me scales Viscoelas2city Elas2c behavior is Viscous behavior; strain rate is instantaneous elas2city and propor2onal to stress: instantaneous recovery. σ = ηε Follows Hooke’s Law: σ = E ε Steady-state viscosity Elas1c Modulus k or E ηSS Simplest form of viscoelas2city is the Maxwell model: t 1 J(t) = + ηSS kE SS kE Viscoelas2city How do we measure viscosity and elascity in the lab? Steady-state viscosity Elas1c Modulus k or EU ηSS σ σ η = η = effective [Fujisawa & Takei, 2009] ε ε1 Viscoelas2city: in between the two extremes? Viscoelas2city: in between the two extremes? Icy satellites velocity (at grounding line) tidal signal glaciers velocity (m per day) (m per velocity Vertical position (m) Vertical Day of year 2000 Anelas2c behavior in Earth and Planetary science
    [Show full text]
  • Similitude and Theory of Models - Washington Braga
    EXPERIMENTAL MECHANICS - Similitude And Theory Of Models - Washington Braga SIMILITUDE AND THEORY OF MODELS Washington Braga Mechanical Engineering Department, Pontifical Catholic University, Rio de Janeiro, RJ, Brazil Keywords: similarity, dimensional analysis, similarity variables, scaling laws. Contents 1. Introduction 2. Dimensional Analysis 2.1. Application 2.2 Typical Dimensionless Numbers 3. Models 4. Similarity – a formal definition 4.1 Similarity Variables 5. Scaling Analysis 6. Conclusion Glossary Bibliography Biographical Sketch Summary The concepts of Similitude, Dimensional Analysis and Theory of Models are presented and used in this chapter. They constitute important theoretical tools that allow scientists from many different areas to go further on their studies prior to actual experiments or using small scale models. The applications discussed herein are focused on thermal sciences (Heat Transfer and Fluid Mechanics). Using a formal approach based on Buckingham’s π -theorem, the paper offers an overview of the use of Dimensional Analysis to help plan experiments and consolidate data. Furthermore, it discusses dimensionless numbers and the Theory of Models, and presents a brief introduction to Scaling Laws. UNESCO – EOLSS 1. Introduction Generally speaking, similitude is recognized through some sort of comparison: observing someSAMPLE relationship (called similarity CHAPTERS) among persons (for instance, relatives), things (for instance, large commercial jets and small executive ones) or the physical phenomena we are interested.
    [Show full text]
  • Rheology Bulletin 2010, 79(2)
    The News and Information Publication of The Society of Rheology Volume 79 Number 2 July 2010 A Two-fer for Durham University UK: Bingham Medalist Tom McLeish Metzner Awardee Suzanne Fielding Rheology Bulletin Inside: Society Awards to McLeish, Fielding 82nd SOR Meeting, Santa Fe 2010 Joe Starita, Father of Modern Rheometry Weissenberg and Deborah Numbers Executive Committee Table of Contents (2009-2011) President Bingham Medalist for 2010 is 4 Faith A. Morrison Tom McLeish Vice President A. Jeffrey Giacomin Metzner Award to be Presented 7 Secretary in 2010 to Suzanne Fielding Albert Co 82nd Annual Meeting of the 8 Treasurer Montgomery T. Shaw SOR: Santa Fe 2010 Editor Joe Starita, Father of Modern 11 John F. Brady Rheometry Past-President by Chris Macosko Robert K. Prud’homme Members-at-Large Short Courses in Santa Fe: 12 Ole Hassager Colloidal Dispersion Rheology Norman J. Wagner Hiroshi Watanabe and Microrheology Weissenberg and Deborah 14 Numbers - Their Definition On the Cover: and Use by John M. Dealy Photo of the Durham University World Heritage Site of Durham Notable Passings 19 Castle (University College) and Edward B. Bagley Durham Cathedral. Former built Tai-Hun Kwon by William the Conqueror, latter completed in 1130. Society News/Business 20 News, ExCom minutes, Treasurer’s Report Calendar of Events 28 2 Rheology Bulletin, 79(2) July 2010 Standing Committees Membership Committee (2009-2011) Metzner Award Committee Shelley L. Anna, chair Lynn Walker (2008-2010), chair Saad Khan Peter Fischer (2009-2012) Jason Maxey Charles P. Lusignan (2008-2010) Lisa Mondy Gareth McKinley (2009-2012) Chris White Michael J.
    [Show full text]
  • Chapter 8 Dimensional Analysis and Similitude
    Chapter 8 Dimensional Analysis and Similitude Ahmad Sana Department of Civil and Architectural Engineering Sultan Qaboos University Sultanate of Oman Email: [email protected] Webpage: http://ahmadsana.tripod.com Significant learning outcomes Conceptual Knowledge State the Buckingham Π theorem. Identify and explain the significance of the common π-groups. Distinguish between model and prototype. Explain the concepts of dynamic and geometric similitude. Procedural Knowledge Apply the Buckingham Π theorem to determine number of dimensionless variables. Apply the step-by-step procedure to determine the dimensionless π- groups. Apply the exponent method to determine the dimensionless π-groups. Distinguish the significant π-groups for a given a flow problem. Applications (typical) Drag force on a blimp from model testing. Ship model tests to evaluate wave and friction drag. Pressure drop in a prototype nozzle from model measurements. CIVL 4046 Fluid Mechanics 2 8.1 Need for dimensional analysis • Experimental studies in fluid problems • Model and prototype • Example: Flow through inverted nozzle CIVL 4046 Fluid Mechanics 3 Pressure drop through the nozzle can shown as: p p d V d 1 2 f 0 , 1 0 2 V / 2 d1 p p d For higher Reynolds numbers 1 2 f 0 V 2 / 2 d 1 CIVL 4046 Fluid Mechanics 4 8.2 Buckingham pi theorem In 1915 Buckingham showed that the number of independent dimensionless groups of variables (dimensionless parameters) needed to correlate the variables in a given process is equal to n - m, where n is the number of variables involved and m is the number of basic dimensions included in the variables.
    [Show full text]
  • Dimensional Analysis and Similitude Lecture 39: Geomteric and Dynamic Similarities, Examples
    Objectives_template Module 11: Dimensional analysis and similitude Lecture 39: Geomteric and dynamic similarities, examples Dimensional analysis and similitude–continued Similitude: file:///D|/Web%20Course/Dr.%20Nishith%20Verma/local%20server/fluid_mechanics/lecture39/39_1.htm[5/9/2012 3:44:14 PM] Objectives_template Module 11: Dimensional analysis and similitude Lecture 39: Geomteric and dynamic similarities, examples Dimensional analysis and similitude–continued Example 2: pressure–drop in pipe–flow depends on length, inside diameter, velocity, density and viscosity of the fluid. If the roughness-effects are ignored, determine a symbolic expression for the pressure–drop using dimensional analysis. Answer: We will apply Buckingham Pi-theorem Variables: Primary dimensions: No of dimensionless (independent) group: file:///D|/Web%20Course/Dr.%20Nishith%20Verma/local%20server/fluid_mechanics/lecture39/39_2.htm[5/9/2012 3:44:14 PM] Objectives_template Module 11: Dimensional analysis and similitude Lecture 39: Geomteric and dynamic similarities, examples Similitude: To scale–up or down a model to the prototype, two types of similarities are required from the perspective of fluid dynamics: (1) geometrical similarity (2) dynamic similarity 1. Geometric similarity: The model and the prototype must be similar in shape. (Fig. 39a) This is essential because one can use a constant scale factor to relate the dimensions of model and prototype. 2. Dynamic similarity: The flow conditions in two cases are such that all forces (pressure viscous, surface tension, etc) must be parallel and may also be scaled by a constant scaled factor at all corresponding points. Such requirement is restrictive and may be difficult to implement under certain experiential conditions. Dimensional analysis can be used to identify the dimensional groups to achieve dynamic similarity between geometrically similar flows.
    [Show full text]
  • Rheology and Mixing of Suspension and Pastes
    RHEOLOGY AND MIXING OF SUSPENSION AND PASTES Pr Ange NZIHOU, EMAC France USACH, March 2006 PLAN 1- Rheology and Reactors Reactor performance problems caused by rheological behaviors of suspensions et pastes 2- Rheology of complex fluids Definition Classification of mixtures Non-Newtonian behaviors Behavior laws of viscoplastic fluids Thixotropy Viscosity equations Rheological measurements 3-Factors influencing the rheological behavior of fluids 4- Mixing of pastes in agitated vessels Agitator and utilization Geometric parameters Dimensional numbers Dimensionless numbers 1- Rheology and Reactor DESIGN OF REACTOR FOR SCALE UP Flux Production Flux input output Accumulation Mass balance: ⎛⎞Aj ⎛⎞AAj,,in ⎛j out ⎞⎛Aj ⎞ ⎜⎟+=⎜⎟⎜ ⎟+⎜ ⎟ Flux ⎜⎟Flux Accumulation ⎝⎠⎝⎠Production ⎝ ⎠⎝ ⎠ 1 DIMENSIONS OF REACTOR IN VIEW OF SCALE CHANGE PERFORMANCE OF REACTOR: Thermodynamic and kinetic Hydrodynamic of the reaction Operating parameters: Composition Nature of reagents Conversion rate Pressure, temperature RTD Concentrations R Output Flow In Out Residence time Mass and heat transfer Geometric of reactor SIMILARITY PRINCIPLE: Geometric similitude Energetic similitude Kinematic similitude Thermal similitude 2 ENCOUNTERED PROBLEMS WITH REACTOR Existence of dead matter and recirculation: Stagnant fluid R Recirculation Presence of preferred passages R OBJECTIVE: Correct the flows or take it into consideration while designing the reactor 3 Ribbon impellers (agitators) for mixing Complex fluids 4 anchor Helicoidal ribbon Archemedian ribbon impeller 5 6 PLAN
    [Show full text]
  • An Introduction to Dimensional Analysis David Dureisseix
    An introduction to dimensional analysis David Dureisseix To cite this version: David Dureisseix. An introduction to dimensional analysis. Engineering school. Lyon, France. 2016, pp.20. cel-01380149v3 HAL Id: cel-01380149 https://cel.archives-ouvertes.fr/cel-01380149v3 Submitted on 12 Apr 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NoDerivatives| 4.0 International License An introduction to dimensional analysis David Dureisseix D´epartement G´enieM´ecanique, INSA de Lyon April 12, 2019 This document is a short (and hopefully concise) introduction to dimensional analysis and is not expected to be printed. Indeed, it relies on URL links (in colored text) to refer to information sources and complementary studies, so it does not provide a large bibliography, nor many pictures. It has been realized with the kind help of Ton Lubrecht and Marie-Pierre Noutary. Photography by KoS, 2008, distributed under a CC BY-SA 3.0 license 1 Contents 1 Goals of dimensional analysis3 2 Physical quantities and
    [Show full text]
  • Dimensional Analysis and Similitude
    Fluid Mechanics Chapter 8 Dimensional Analysis and Similitude Dr. Amer Khalil Ababneh Introduction Because of the complexity of fluid mechanics, the design of many fluid systems relies heavily on experimental results. Tests are typically carried out on a subscale model, and the results are extrapolated to the full-scale system (prototype). The principles underlying the correspondence between the model and the prototype are addressed in this chapter. Dimensional analysis is the process of grouping of variables into significant dimensionless groups, thus reducing problem complexity. Similitude (Similarity) is the process by which geometric and dynamic parameters are selected for the subscale model so that meaningful correspondence can be made to the full size prototype. 8.2 Buckingham Π Theorem In 1915 Buckingham showed that the number of independent dimensionless groups (dimensionless parameters) can be reduced from a set of variables in a given process is n - m, where n is the number of variables involved and m is the number of basic dimensions included in the variables. Buckingham referred to the dimensionless groups as Π, which is the reason the theorem is called the Π theorem. Henceforth dimensionless groups will be referred to as π-groups. If the equation describing a physical system has n dimensional variables and is expressed as then it can be rearranged and expressed in terms of (n - m) π- groups as 1 ( 2 , 3 ,..., nm ) Example If there are five variables (F, V, ρ, μ, and D) to describe the drag on a sphere and three basic dimensions (L, M, and T) are involved. By the Buckingham Π theorem there will be 5 - 3 = 2 π-groups that can be used to correlate experimental results in the form F= f(V, r, m, D) 8.3 Dimensional Analysis Dimensional analysis is the process used to obtain the π-groups.
    [Show full text]
  • Study of Dimensional Analysis and Hydraulic Similitude 1 2 Syed Ubair Mustaqeemp ,P Syed Uzair Mustaqeemp
    IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 8 Issue 5, May 2021 ISSN (Online) 2348 – 7968 | Impact Factor (2020) – 6.72 www.ijiset.com Study of Dimensional Analysis and Hydraulic Similitude 1 2 Syed Ubair MustaqeemP ,P Syed Uzair MustaqeemP 1 P BP E Student, Civil Engineering Department, SSM College of Engineering, Kashmir, India 2 P P B E Student, Civil Engineering Department, SSM College of Engineering, Kashmir, India Abstract: Dimensional analysis is a powerful tool in designing, ordering, and analysing the experiment results and also synthesizing them. One of the important theorems in dimensional analysis is known as the Buckingham Π theorem, so called since it involves non-dimensional groups of the products of the quantities. In this topic, Buckingham Π theorem and its uses are thoroughly discussed. We shall discuss the planning, presentation, and interpretation of experimental data and demonstrate that such data are best presented in dimensionless form. Experiments which might result in tables of output, or even multiple volumes of tables, might be reduced to a single set of curves—or even a single curve—when suitably non- dimensionalized. The technique for doing this is dimensional analysis. Also, Physical models for hydraulic structures or river courses are usually built to carry out experimental studies under controlled laboratory conditions. The main purposes of physical models are to replicate a small-scale hydraulic structure or flow phenomenon in a river and to investigate the model performance under different flow and sediment conditions. The concept of similitude is commonly used so that the measurements made in a laboratory model study can be used to describe the characteristics of similar systems in the practical field situations.
    [Show full text]
  • SIMILITUDE MODELING in RDC ACTIVITIES Rosli Darmawan1, Nor Mariah Adam2, Nuraini Abdul Aziz2 and M
    SIMILITUDE MODELING IN RDC ACTIVITIES Rosli Darmawan1, Nor Mariah Adam2, Nuraini Abdul Aziz2 and M. Khairol Anuar M. Ariffin2 1Malaysian Nuclear Agency, 2Universiti Putra Malaysia ABSTRACT Research and development activities would involve the scaled modelling activities in order to investigate theory, facts, thesis or concepts. In commercialisation activities, scaling-up proses is necessary for the development of pilot plants or prototypes. The issue with scaled modelling is the similarity between the small scaled model and the full scaled prototype in all aspects of the system such as physical appearance, dimension and the system behaviour. Similarly, for scaling-up process, physical parameters and behaviour of a smaller model need to be developed into a bigger prototype with similar system. Either way, the modelling process must be able to produce a reliable representation of the system or process so that the objectives or functions of the system can be achieved. This paper discusses a modelling method which may be able to produce similar representation of any system or process either in scaled-model testing or scaling-up processes. INTRODUCTION Throughout the research, development and commercialisation (RDC) activities, more often than not there will be a process where the suggested idea or concept needs to be fully tested. For example, in chemical research, a new formulation study would be conducted in a laboratory with laboratory size apparatus. The study may produce the desired formulation under laboratory environment. However, if the same process would be transferred to a production plant, the same product may not be able to be produced similarly in all aspects as per laboratory product.
    [Show full text]
  • The British Society of Rheology
    THE BRITISH SOCIETY OF RHEOLOGY To cite: R J Poole (2012), The Deborah and Weissenberg numbers. The British Society of Rheology, Rheology Bulletin . 53(2) pp 32-39 1 The Deborah and Weissenberg numbers Rob Poole School of Engineering University of Liverpool, United Kingdom Since their introduction to the literature nearly fifty years ago the Deborah and Weissenberg numbers have proved invaluable for rheologists in quantifying viscoelastic effects. Despite their different origins, and the fact that they quantify different effects, they are frequently used as synonyms. This situation is perhaps partly explainable given that in a wide range of flows of complex fluids characterised by a relaxation time ( λ) the use of simple single characteristic length ( L) and velocity scales ( U) does indeed result in identical definitions! In this short article we return to the original papers to highlight the different effects that the Deborah and Weissenberg numbers measure. The Deborah number As every good rheologist knows the Deborah number owes its name to the Prophetess Deborah who, in the Book of Judges, proclaimed “The mountains flowed before the lord” A. The definition is due to Reiner and we can do no better than to quote his original paper B “Deborah knew two things. First, that the mountains flow, as everything flows. But, secondly, that they flowed before the Lord, and not before man, for the simple reason that man in his short lifetime cannot see them flowing, while the time of observation of God is infinite . We may therefore well define a nondimensional number the Deborah number D = time of relaxation/time of observation.” AAs Huilgol notes “Reiner quoted the Prophetess Deborah as saying “The mountains flowed before the Lord” while the King James version says “The mountains melted from before the Lord”, and the New English Bible has “Mountains shook in fear before the Lord,” (See Judges 5:5) [R R Huigol (1975) “On the concept of the Deborah number”.
    [Show full text]