Three Fairly New British Language References

Three Fairly New British Language References Three Fairly New British Language References Three Fairly New British Language References By Maeve Maddox Motivated by the lively debates about where to put commas, and the controversy over â€Å"gone missing,† I’ve added some up-to-date British references to my print reference library. The three newcomers to my shelves are: Penguin Dictionary of English Grammar by R. L. Trask, 2000. As the title implies, this guide arranges topics and terms in alphabetical order. It includes every permutation of terminology from the traditional ones I grew up with to the innovations born of transformational grammar and Quirk Grammar. Here one can find definitions of subject raising, subjuncts, adjuncts and conjuncts, along with more immediately useful terms as double negative, paradigm, relative pronoun and usage. A lot of the terms are, however, a bit esoteric. While it’s a great resource for me in my line of work, there’s probably nothing here you can’t find online at OWL or any of the other free references mentioned in Online Style Guides. Penguin Guide to Punctuation by R. L. Trask, 1997. Trask does more than present rules and made-up textbook examples. His personality comes through as he discusses badly punctuated passages, often speculating as to why certain errors are made. It’s extremely readable, whatever page you open to. Of the ten chapters, seven deal with specific punctuation marks: 2: The Full Stop, the Question Mark and the Exclamation Mark 3: The Comma 4: The Colon and the Semicolon 5: The Apostrophe 6: The Hyphen and the Dash 7 Capital Letters and Abbreviations 8 Quotation Marks Chapter 1 explains the practical importance of punctuation. Chapter 7 gives rules for capitalizing and abbreviating. Chapter 9 deals with typographical considerations and Chapter 10 discusses the punctuation of essays and letters. I’m still in the process of getting acquainted with it, but this punctuation guide promises to be a treasure. Having British usage all in one place will be a great help as I write future posts. Penguin Writer’s Manual by Martin H. Manser and Stephen Curtis, 2002. As might be expected, there’s some overlap with the other two books. This one book has everything a writer needs in a basic reference. Part One deals with the mechanics of writing: 1 Grammar 2 Usage 3 Vocabulary 4 Spelling 5 Punctuation 6 Abbreviations. Part Two gets into the specifics of style, revision, and types of writing. There’s also a generous glossary of grammatical terms. In case youre wondering: Quirk grammars: A series of grammars of English written by Randolph Quirk and his colleagues. Though rather traditional in orientation, these grammars are informed by contemporary linguistic research. They introduce a certain amount of novel terminology. Want to improve your English in five minutes a day? Get a subscription and start receiving our writing tips and exercises daily! Keep learning! Browse the Book Reviews category, check our popular posts, or choose a related post below:"Based in" and "based out of"75 Synonyms for â€Å"Talk†How to Style Titles of Print and Online Publications

Postmodernism in animation Research Paper Example | Topics and Well Written Essays - 1000 words

Postmodernism in animation - Research Paper Example Baudrillard and Frederic Jameson to popular animation series such as Beavis and Butt-head, Drawn Together and Harvey Birdman that aimed to target adult audiences with their content and themes. The first part of the paper intends to explore the specific theories presented by commentators on postmodernism, in order to apply the elements of the concepts with regards to the animated television series under discussion thereby concluding that aspects of postmodernism and its cultural components are visible in the animated representations of popular culture. Lyotard’s description and assessment of the world, in his essay titled â€Å"The Postmodern Condition† puts forward the limitations of believing in the positivity associated with a social and cultural agreement, thereby, postulating that postmodernism should in fact strive to work against the foundations of this widespread consensus which is an unattainable or utopian state due to the aspects related to the emergence and progress of popular culture which preaches plurality rather than singularity, this assertion led to the development of the concept of metanarratives and recognized the need to promote knowledge that is held by the general population rather than what is advocated by dominant cultural and political forces through the means of Grand narratives (til B.A-prà ³fs and Bjà ¶rnsson 2006). However, for Baudrillard (1994) the notions of postmodernism are associated with the representation of what he terms as hyperreality, such that in the society of today symbols or signs are no longer associable to their existence in reality but due to the emergence of a simulacrum can be traced to the subsistence of other elements. Perhaps, the most viable link of Baudrillard’s theory of hyperreality can be traced to the projections of media such as film and television which create the hyperreality and then propogate it (til B.A-prà ³fs and Bjà ¶rnsson 2006). With its comprehensions of society and culture, postmodernism has

Transformation & Tessellation Essay Example | Topics and Well Written Essays - 250 words

Transformation & Tessellation - Essay Example Reflection involves flipping the object on the plane, creating a mirror image of the object on the same plane. Translation involves moving, or "sliding" the object within the plane, with the points moving on the same direction or with the same distance. Transformation can be applied in the real world in creating 3D images and models of objects, like houses, buildings, or even human beings. In order to capture the real essence of the object being imitated (or transformed), the same dimensions should be used, however, on a smaller scale. Tessellation, on the other hand, occurs when objects in a plane create a pattern without overlapping with each other, and without leaving gaps in between the object (Seymour and Britton, 1989). These patterns are often a subject of artworks, and one can see a lot of these kinds of patterns on books and on the Internet, in pages where one is asked to count the number of polygons in a page. One of the most common examples of this kind of patterns can be found on floor tiles, which follow the same principle: each tile should fit a given space, without overlapping with another tile, and without any gaps in between. The objects can be regularly shaped polygons, like squares, triangles, octagons, etc., with each side touching another objects side (Seymour and Britton, 1989). Irregularly shaped objects can also be used, which proves to be more complicated than using regular polygons. Common objects such as soccer balls, jigsaw puzzles and honeycombs also have a tessellated

Concert Orchestra experience Essay Example for Free

Concert Orchestra experience Essay I went to the UNT Concert Orchestra on Wednesday, October 3rd, 2012. It was held in Winspear Hall at the Murchison Performing Arts Center at 8:00 pm. The Concert was led by Conductor Clay Couturiaux and featured soloist Christopher Deane, who played the Marimba. The first piece was Variations on a Theme of Tchaikovsky, Op. 35a (1894) by Anton Arensky (1861-1906). The piece was written in 1894, in tribute to Pyotr Il’yich Tchaikovsky (1840-1893). It was based on the theme from the poem Legend†, written by Richard Henry Stoddard (1825-1903). This poem portrays the crucifixion of Christ. Arensky admired Tchaikovsky so much that he used the theme of â€Å"Legend† for a set of variations in the second movement of his Second String Quartet. This piece’s style is a themes and variations. Its instrumentation includes Cello solo, 2 Flutes, 2 Oboes, 2 Clarinets (A), 2 Bassoons + 2 Horns (F) + Violins I, Violins II, Violas, Cellos, and Double Basses. The second piece was Concerto for Marimba and Orchestra, Op. 34 (1957) by Robert Kurka (1921-1957). This piece introduced the marimba, which proved to the musical world that it could contend with instruments that had been used in orchestras and also provide a unique sound to the traditional orchestras played in regular concerts. This piece’s style is solo concerto. Its instrumentation includes the marimba and the orchestra. The third piece was Pictures at an Exhibition (1874) by Modest Mussorgsky (1839-1881). This piece was inspired by the paintings of the artist Viktor Hartmann (1834-1873). This piece’s style is an orchestral suite. Its instrumentation includes 3 Flutes (2nd and 3rd doubling Piccolos), 3 Oboes (3rd doubling Cor Anglais), 2 Clarinets in A and Bb, Bass Clarinet in A and Bb, Alto Saxophone, 2 Bassoons, Double Bassoon, 4 Horns in F, 3 Trumpet in C, 3 Trombones, Tuba, Timpani, Percussion (xylophone, triangle, rattle, whip, side drum, bass drum, cymbals, suspended cymbal), 2 Harps, Celesta, and Strings. I picked the pieces was Variations on a Theme of Tchaikovsky, Op. 35a (1894) by Anton Arensky and Concerto for Marimba and Orchestra, Op. 34 (1957) by Robert Kurka. Both of these pieces were distinctly different than one another. The piece by Arensky depicts a sense of deep sadness and despair as a whole. It starts out containing elements of intimacy and moves towards a slow moving harmony. The structure of the music matched the structure of the original poem. The variations of sounds expressed many shifting moods such as a dialogue between instruments. Mood changed quickly throughout the piece and showed different parts of the melody, from increments of joy, to sadness, to a deep sorrow. The rhythm seamlessly continued throughout the piece acting towards each of the different themes described in its construction. The piece by Kurka produced a new and different type of classical music that is unique to the orchestra. The use of the marimba stood out from the traditional orchestral instruments. The first movement begins with an alternation between the marimba and the orchestra. Its upbeat sound resonates in a catchy chiming sound whose rhythm is clear yet unexpected. It provides a playful side to a usually stern and focused orchestra. As the second movement begins, it as if the marimba is communicating to the orchestra itself. As if it is trying to fit in with these classic types of instruments through its unique dynamics and resounding tone. It seems to clash with its orchestral counterparts. By the third movement, it seems as if all the instruments reach an agreement on the legitimacy of the marimba through its colorful and exciting solo. Although both pieces are completely different than one another, they both exhibit emotion. Arensky exhibits cruel sounding music that discusses the importance of religion and a series of events that affects a wide variety of people. It evokes a sense of despair that expresses a deep sounding melody. Kurka exhibits a different type of music that discusses the marimba’s rise to becoming a part of classical orchestra. Its colorful timbre expresses a joyful and unique melody that pleases the human ear. Anton Arensky (12 July 1861 -25 February 1906), was a Russian composer of Romantic classical music, a pianist and a professor of music. Pyotr Tchaikovsky was the greatest influence on Arenskys musical compositions. Indeed, Rimsky-Korsakov said, In his youth Arensky did not escape some influence from me; later the influence came from Tchaikovsky. He will quickly be forgotten. The perception that he lacked a distinctive personal style contributed to long-term neglect of his music, though in recent years a large number of his compositions have been recorded. Therefore, his values are seemingly non-existent because of the major influence of Tchaikovsky and absence of his own personal work. Throughout the performance I did perceive a strong sense of historical value and defines not who Arensky was, but his role model Tchaikovsky and how his music conveyed a strong sense of religious value. Kurka’s Concerto for Marimba and Orchestra was the first marimba work to enjoy both widespread public appeal and widespread recognition of having a high level of musical sophistication fit for the concert hall. It debuted during the modern style period. It provided important historical value by Kurka finally representing everything that early marimba composers set out to do in one piece: create a sophisticated and serious musical work that is both challenging to the performer and which has widespread public appeal. I perceived an ongoing struggle throughout the piece, but as the performance continued it conveyed the struggle the instrument had to do in order to become a prominent part of the classical orchestra. Citatation Keunning, G. (1999). Symphony of the canyons. Retrieved from http://lasr. cs. ucla. edu/geoff/prognotes/mussorgsky/pictures. html Strain, James. Vida Chenoweth. Percussive Notes 32. 6 (1994): 8-9. Print. Stevens, Leigh Howard. An Interview with Vida Chenoweth. Percussive Notes 15. 3 (2002): 22-25. PAS Online Archive . Weir, Martin. Catching up with Vida Chenoweth. Percussive Notes 32. 3 (1994): 53-55. Print.

Herbert Hoover Essay -- History

Herbert Hoover Herbert Hoover called it a "noble experiment." Organized crime found it to be the opportunity of a lifetime. Millions of Americans denounced it as an infringement of their rights. For nearly 14 years—from Jan. 29, 1920, until Dec. 5, 1933--the manufacture, transportation, and sale of alcoholic beverages was illegal in the United States. The 18th, or Prohibition, Amendment to the Constitution was passed by Congress and submitted to the states in 1917. By Jan. 29, 1919, it had been ratified. Enforcement legislation entitled the National Prohibition Act (or more popularly, the Volstead act, after Representative Andrew J. Volstead of Minnesota) was passed on Oct. 28, 1919, over President Woodrow Wilson’s veto. The 18th Amendment to the Constitution of the United States not only prohibited the "manufacture, sale, or transportation of intoxicating liquors," but their importation and exportation also. It was adopted after a nationwide crusade by temperance groups, notably the Women’s Christian Temperance Union, or WCTU. The amendment was enforced and defined by Congress in the Volstead Act. One result of the amendment was that the production and sale of alcoholic beverages became the province of organized crime. Americans did not stop drinking, and their demands for liquor were met by wide-scale smuggling and bootlegging, much of which was controlled by such gangs as that led by Al Capone in Chicago. The era of prohibition ended in 1933 when the 18th Amendment was repealed by the twenty-first Amendment. The stage was set for more than a decade of combat between the "wets" and the "drys"—those determined to keep drinking and those determined to enforce the law. In retrospect, the period has been called the Roaring Twenties and the Jazz Age. New music appeared along with new dances, a new feminism, and a general relaxation of standards after the rigorous years of World War I. The new mood was in complete contrast to the moral earnestness of many Americans who were determined to remain the ideal "Victorians." Organized efforts to limit the use of alcoholic beverages began in the United States during the 1820s. A by-product of the religious revivalism sweeping the nation, Prohibition soon became part of the whole social reform movement that preceded the Civil War. The earliest reformers called for moderation, not total abstinence, but as ... ...bition did not achieve its goals. Instead, it added many problems to those that it intended to solve. It came along in a social period where it was just simply unrealistic to have any success. The only beneficiaries to that of Prohibition were bootleggers, crime bosses, and the forces of big government in all of its corrupt forms. Though it failed to improve health, welfare, or America as a whole, the experiment with prohibition affords some valuable lessons. With this learning experience as part of the past, America should be able to confront its modern remnants in all of their assorted varieties. Bibliography Coffey, Thomas M. The Long Thirst: Prohibition in America, 1920-1933 New York: W. W. Norton & Co., 1975. Krout, J. A. The Origins of Prohibition, New York City: Russell & Russell, 1996. Lee, Henry. How Dry We Were, Enlewood Cliffs, NJ: Prentice Hall, 1963. Rorabaugh, W. J. The Alcoholic Republic- An American Tradition, New York Oxford University Press, 1979. Turner, George Kibbe. "The City of Chicago, A Study of Great Immoralities," McClure’s Magazine, April 1927 (vol. 28). Warburton, Clark. "The Results of Prohibition," Auburn Press, 1996.

The Systematic Phonics Case Education Essay

It is apparent from the epoch of 80s and 90s where rational bookmans and instructors presented new influential thoughts sing instruction of English. The field of instruction of reading is a topic that is immensely researched and still under farther research. Harmonizing to Pearson ( 2002 ) , â€Å" Man-made phonics was the chief method of learning since the beginning of twentieth century, this type of learning comprises exercising of larning missive names, sounds of letters and after so intermixing of these † ( pg, 1 ) . As the twentieth century progressed, it brought more extremist alterations of attack. Smith ( 1971 ) focused on the country sing the development of the capableness to read. Harmonizing to him, the reading has something that an person learned to make instead than something an person was taught. Man-made phonics is non a new of learning reading, in fact, its function of being outstanding and popular instruction has been an unintended result of the whole linguist ic communication acceptance attacks in British schools. Harmonizing to Rutter ( 2006 ) , every bit far as man-made phonics is concerned the rating of research has to be set long term and it needs to be made certain that kids with larning troubles should be assisted with different ways in order to do difference. Stuart ( 2006 ) emphasized that new options has to be seek so that recommendations on national degree could be made, Stuart made this suggestion to Rise â€Å" the current research grounds is non sufficient for leting reliable judgements of the effectivity of implementing different attacks to systematic structured phonics learning † ( Stuart, 2006 ; p11 ) .Systematic phonics CaseAmerican National Reading Panel ( NRP ) , in footings of research grounds and instruction of reading related inquiries, was amongst the critical subscribers who reported in learning kids in English reading ( NICHD, 2000 ) . The study consisted of inquiries sing the early literacy such as ‘Does systematic phonics direction aid kids learn to read more efficaciously than non systematic phonics direction or direction learning no phonics? ‘ ( P92 ) . ‘Are some specific phonics programmes more effectual than others? ( P93 ) . The decision from these inquiries was that ‘specific systematic phonics programmes are all significantly more effectual than non-phonics programmes ; nevertheless, they do non look to differ significantly from each other in their effectivity although more grounds is needed to verify the dependability of consequence sizes for each programme ‘ ( NICHD, 2000, pg93 ) . In another case, a comprehensive instruction of reading attacks research was commissioned by England ‘s Department for Education and Skills ( DfES ) in order to polish the NRP methodological analysis by bring forthing a randomised controlled tests ( RCT ) tests. Research workers in their work concluded that grounds has been seen in RCT surveies which could turn out the effectivity of one signifier of systematic phonics compared with that of other ( Torgerson et al, 2006 ) . Rutter ( 2006 ) besides commented that the of import facet of determination is that RCTs are one signifier of optimum research conditions. The existent message that is apparent from the meta-analysis, carried out by NRP and its limitation to RCTs, is the thought of importance of literacy acquisition every bit far as systematic instruction of phonics is concerned ( Torgerson et al, 2006 ) . Rose ( 2006 ) besides agreed with this decision by saying that â€Å" the importance of systematic phonic work is huge and could be more effectual if incorporated with man-made attack, after looking at its grounds which is wide-ranging † ( Rose, 2006, p20 ) . However he emphasizes that regardless of commonalty amidst systematic phonics and man-made phonics, it is therefore the man-made phonics which can offer much better class to going skilled readers for early scholars ( p19 ) .Man-made phonics instanceTwo surveies were reported by Johnston and Watson ( 2004 ) , in which the 2nd experiment was carried out before the first one and is of greater importance since it is related to intercession. This intercession varied from normal to extra schoolroom tuition get downing six hebdomads after school entry. The excess preparation kept on traveling for 10 hebdomads holding 2 categories per hebdomad wholly consisted of 114 printed words. The one group was taught 2 letters per hebdomad by agencies of assorted games played where kids matched images and words by merely pulling their attending to initial w ord sounds and letters of those sounds while other group was taught in all places of the words such as enhanced acquisition and blending of the missive sounds in all places, while being taught 2 letters per hebdomad every bit good ( Johnston and Watson, 2004, p347 ) . What writers concluded was that the group with man-made phonics were far much better in footings of reading and spelling every bit compared to analytic phonics group and therefore, the man-made phonics proved to be effectual attack to learning spelling, reading phonemic consciousness as comparison to analytics phonics ( Johnston and Watson, 2004 ) . Rose ( 2006 ) besides stresses that man-made phonics gives more indispensable accomplishments that allows the bulk of students to read and compose in front of their chronological age. Harmonizing to her, the 20 per centum pupils who have jobs with literacy still have better foundation of the reading rudimentss and merely necessitate excess clip and engagement.Pearson ‘s FindingssPearson ( 2003 ) became portion of fact-finding programme to derive some penetrations of the kids sing their reading position, their advancement every bit good as things that were their facets of success. During the 6 month period, she met the student twice, questioning the kids for about 30 proceedingss. Children were assured of their confidentiality in order to garner honorable responses so that school can improvize on its judicial admission in the coming academic old ages. She conducted semi-structured interviews from the students, and to assist them she used a ocular prompt based on Kelly ‘s attack ( Leadbetter et al, 1999 ) . The thought she got from both interviews was related to the public presentation e.g. both the gender expressed that they stumble when they read out loud and that they do n't wish reading in the schoolroom and maintain on spellings make them bury what to read. When the students ‘ position was asked sing the good readers, they commented that the good reader read louder and faster and that the difference between good reader and bad reader is that, the good reader makes the hapless book sound good while bad readers makes a good book sound drilling. Hence the result was that good readers have much better frequence every bit good as have much better reading. These positions of students were seen apparent even after nine months despite of the hint that their thoughts of literacy in the secondary school were developing. Few students continued to the thought that there reading is affected when they get prep and that reading is largely non the portion of their prep. Therefore the basic f eeling of kids sing reading can be confusing and if concepts of kids ‘s reading are to develop helpfully so there should be a strategic program for this. Assurance is another factor act uponing the public presentation degree Fahrenheit reading, less assurance can demo apparent diminution in their reading public presentation. Persistent and on-going encouragement of students in their reading can promote their public presentation and there are staff members holding peculiar accomplishments, they can portion the same accomplishment with other staff members to assist pupil improvise. Harmonizing to Pearson ( 2003 ) , the influential function of household is besides imperative in the procedure of reading. In her interviews with kids, she found out that students appreciated transporting out reading with person they knew although this chance was non gettable all the clip. Families promoting the privation of student to hold reading spouse might assist them in come oning and therefore will get down to hold more acute involvement in reading and may purchase books of their ain involvement for their reading calling. In footings of feedback, due to the deficiency of instruments used there were no specific standards for students ‘ thought of advancement as compared to prove consequences. However, at primary instruction phase, kids are really competent to track the advancement they made by agencies of utilizing information like coloring material they are on, the groups they are working in and or the degree with which they are asked to read. Though, this sort of system was recognized to be least available at the phase of secondary school instruction. Introducing wide stairss at the secondary degree may let the kids to track and place the advancement they have made in reading efficaciously. Hence, the critical phase for schools to ease or detain students ‘ accommodation is the period of passage that includes the Year seven. Supporting the reading procedure, hiking the assurance degrees, influential function of household and the students ‘ feedback are the most effectual ingredients in developing literacy accomplishments for both primary and secondary stage of the school.DecisionThe scope to which instruction of reading should do the stuff appropriate to be taught has been still in the Centre of statements sing reading teaching method. There is disagreement traveling on sing the all right ways to poising work on whole texts with sub-word-level work. One manner to attach man-made phonics learning firmly in an redolent context is to straight associate it to pupil ‘s books and other complete texts. The Rose Report has by now started to hold a consecutive influence on national educational policy in the United Kingdom since harmonizing to the study, the inst ructors and trainee instructors should be required to learn reading through man-made phonics foremost and fast. The interviews conducted by Sue Pearson gave two factors that encouraged the schools in order to give serious consideration to the findings, foremost was the honest responses from the students since they gave positive free of vacillation response and did n't felt forced sing their reading advancement. Second the students took engagement really earnestly. The interviews conducted gave wide scope of future considerations to schools since students shared their likes and disfavors in the reading advancement. Majority of them seem to hold better advancement in reading when they were motivated and acquiring aid from household or person they wished to read with. The information provided by the interview may play a cardinal function in the secondary school in footings of be aftering a proper focal point on the literacy and heightening student ‘s accomplishments. Hence the research of Sue Pearson consequently, discovered student ‘s aspirations, both in footings of short-run and lo ng-run every bit far as instruction of reading is concerned and will be an on-going aid for instructors in the chance.

Hydraulic Machines

30 CHAPTER 5 TURBINES 5. 1 Introduction Hydraulic turbines are the machines which use the energy of water and convert it to mechanical energy. The mechanical energy developed by a turbine is used in running an electric generator which is directly coupled to the shaft of the turbine. The electric generator thus develops electric power, which is known as hydro-electric power. 5. 2 Elements of Hydraulic Power Plants Fig. 5. 1 General Layout of a Hydraulic Power Plant 31 Fig. 5. 1 shows a general layout of hydraulic power plant, in which an artificial storage reservoir formed by constructing a dam has been shown. 5. 3General Classification of Turbines Turbines are hydraulic machines that convert energy into rotating mechanical energy which in turn generators to produce electrical energy. Originally developed from the water wheels, hydraulic turbines are the prime mortars of importance in modern water power development. According to their hydraulic action, turbines are broadly divided int o two classes. (1) Impulse Turbine: Impulse turbines are more efficient for high heads. At the inlet to the turbine runner, pressure head can be completely converted into kinetic head in the form of a jet of water issuing from one or more nozzles.The free jet will be at atmospheric pressure before as well as after striking the vanes. The turbines are regulated by nozzles which may be a simple straight flow type or a deflector type. The impulse turbines are commonly represented by Pelton Wheels. Turgo turbine is also an impulse turbine but with different buckets, when compared with pelton. Turgo and cross flow turbines are relatively new developments in this class. The main advantages of these turbines are: †¢ They can be easily adopted to power variation with almost constant efficiency. †¢The penstock overpressure and the runner overspeed control are easier. †¢ The turbine enables an easier maintenance. †¢ Due to the jet the manufacturer of these turbines impose a better solid particle control, conducting, consequently, to a lower abrasion effect. (2) Reaction Turbine: A turbine can be made to rotate under the action of water flowing under pressure through the runner. In such turbines the penstocks, the inlet passage to the runner, passage between the runner vanes, all form a continuous passage for the flow under a pressure which continuously decreases from inlet to outlet.The turbine runner directly converts both kinetic energy as well as the pressure energy into mechanical energy. Reaction turbines are represented in modern practice by two principal types: the Francis turbine where the flow is directed radial to the runner axis and the Propeller type 32 where the flow is axial to the runner axis. Propeller turbines may be fixed blade or adjustable blade types. Kaplan turbine has adjustable blades. The main advantages of these turbines are: †¢ It needs lesser installation space. †¢ It provides a greater net head and a better prot ection against downstream high flood levels. It can have greater runner speed. †¢ It can attain greater efficiencies for high power values. In order to distinguish different turbines, the hydraulically salient features like pressure, head, flow direction and magnitude, speed and power etc. The general classification of hydraulic turbines is illustrated in Fig. 5. 2. Hydraulic Turbines Impulse Turbines Pelton Turgo Reaction Turbines Cross-Flow Fixed -Blade Propeller Kaplan Francis Tubular Deriaz Bulb Fig. 5. 2 General Classification of Turbines 5. 4 Number of Units It is normally cost effective to have a minimum number of units at a given small ydropower installation. Multiple units may, however, be necessary from the operational point of view so that even one unit breaks down or is in the routine maintenance, the power generation can be achieved to a certain extent. The efficiency curves of turbines show that the 33 efficiency of power generation from hydraulic turbines conside rably decreases at low flow ratios or power ratios. In multiple units, it is possible to maintain the higher efficiency even in low flows and the low loads by running a certain number of the units at a time depending upon the available discharge and the load demand.Multiple units thus, make the most effective use of water where the flow as well as the load variations are significant. 5. 5 Limits of Use of Turbine Types The selection of best turbines for any particular small hydropower site depends on the site characteristics, the dominant beings the head and the available flow. There are some limits on the range of these parameters in the selection of turbines. Each turbine type is best suited to a certain range of pressure head and the flow rate. For instance, Pelton wheels operate with low flows discharged under great pressures where as Propeller turbines are effective in high flows under low heads.Francis turbines fall in the medium category covering a wide range of different hea ds and discharges. The common practice of SHP systems is to develop standard unit sizes of equipment that will operate over a range of heads and flows. Either charts or nomographs are used to select appropriate units for site specific application. One such chart showing the head-flow range of normal SHP schemes applicable to each type of turbine is given in Fig5. 3. The graph also indicates the approximate power generation for each combination of the head and the discharge applicable to SHP schemes. Fig. . 3 Head-Flow Ranges for Different Turbines 34 5. 6 Pelton Wheel Pelton wheel is well suited for operating under high heads. A pelton turbine has one or more nozzles discharging jets of water which strike a series of buckets mounted on the periphery of a circular disc. The runner consists of a circular disc with a number of buckets evenly spaced round its periphery. The buckets have a shape of a double semi-ellipsoidal cups. The pelton bucket is designed to deflect the jet back thro ugh 165 ° which is the maximum angle possible without the return jet interfering with the next bucket.Fig. 5. 4 Pelton bucket General arrangement of a pelton wheel is shown in the Fig. 5. 5. For SHP schemes, Pelton wheels are easier to fabricate and are relatively cheaper. The turbines are in general, not subjected to the cavitation effect. The turbines have access to working parts so that the maintenance or repairs can be effected in a shorter time. Fig. 5. 5 General Arrangement of a Pelton Wheel 35 Traditionally, micro hydro pelton wheels were always single jet because of the complexity and the cost of flow control governing of more than one jet.Advantages of multi-jet: -Higher rotational speed -Smaller runner -Less chance of blockage Disadvantages of multi-jet: -Possibility of jet interference on incorrectly designed systems -Complexity of manifolds 5. 7 Francis Turbine Francis turbine is a mixed flow type, in which water enters the runner radially at its outer periphery and le aves axially at its center. Fig. 5. 6 illustrates the Francis turbine. The runner blades are profiled in a complex manner and the casing is scrolled to distribute water around the entire perimeter of the runner.The water from the penstock enters a scroll case which completely surrounds the runner. The purpose of the scroll case is to provide an even distribution of water around the circumference of the turbine runner, maintaining an approximately constant velocity for the water so distributed. The function of guide vane is to regulate the quantity of water supplied to the runner and to direct water on to the runner at an angle appropriate design. A draft tube is a pipe or passage of gradually increasing cross sectional area which connects the runner exit to the tail race. Fig. 5. 6 Francis Turbine 36 . 8 Kaplan Turbine It is an axial flow turbine which is suitable for relatively low heads. From Fig. 5. 7, it will be seen that the main components of Kaplan turbine such as scroll casi ng, guide vanes, and the draft tube are similar to those of a Francis turbine. Fig. 5. 7 5. 9 Kaplan Turbine Specific Speed The specific speed of any turbine is the speed in r. p. m of a turbine geometrically similar to the actual turbine but of such a size that under corresponding conditions it will develop 1 metric horsepower when working under unit head. Ns = NP H5/ 4 †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦(5. 1) where Ns = specific speedP = power in HP 5. 10 Characteristic Curves The turbines are generally designed to work at particular values of H,Q,P,N and ? o which are known as the designed conditions. It is essential to determine exact behaviour of the turbines under the varying conditions by carrying out tests either on the actual turbines or on their small scale models. The results of these tests are usually graphically represented and the resulting curves are known as characteristic curves. 37 -constant head characteristic curves -constant speed characteristic curves -constan t efficiency characteristic curvesIn order to obtain constant head characteristics curves the tests are performed on the turbine by maintaining a constant head and a constant gate opening and the speed is varied by changing the load on the turbine. A series of values of N are thus obtained and corresponding to each value of N, discharge Q and the output power P are measured. A series of such tests are performed by varying the gate opening, the head being maintained constant at the previous value. From the data of the tests the values of Qu, Pu, nu and ? o are computed for each gate opening. Then with Nu as abscissa the values ofQu, Pu and ? o for each gate opening are plotted. The curves thus obtained for pelton wheel and the reaction turbines for four different gate openings are shown in Fig. 5. 8. Fig. 5. 8 Constant head characteristics for Pelton wheel and reaction turbines 38 5. 11 Cavitation in turbines When the pressure in any part of the turbine reaches the vapour pressure of the flowing water, it boils and small bubbles of vapour form in large numbers. These bubbles are carried along by the flow, and on reaching the high pressure zones these bubbles suddenly collapse as the vapour condenses to liquid again.The alternate formation and collapse of vapour bubbles may cause severe damage to the surface which ultimately fails to fatigue and the surface becomes badly scored and pitted. This phenomenon is known as cavitation. In order to determine whether cavitation will occur in any portion of the turbine, D. Thomas has developed a dimensionless parameter called Thomas'cavitation factor ? which is expressed as ?= Ha ? Hv ? Hs H †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦(5. 2) where Ha = atmospheric pressure head Hv = vapour pressure head Hs = suction pressure head For Francis turbines: Critical cavitation factor ? c = 0. 625 (Ns/444)2 †¦Ã¢â‚¬ ¦.. (5. 3) For Propeller turbines: ? c = 0. 28 + [ Example 5. 1 1 Ns 3 ( )] 7. 5 444 †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. (5. 4) Est imate the maximum height of straight conical draft tube of a 18000 h. p. Francis turbine running at 150 r. p. m under a net head of 27 m. The turbine is installed at a station where the effective atmospheric pressure is 10. 6 m of water. The draft tube must sink at least 0. 77 m below the tail race. Ns = NP H5/ 4 = 327 39 ?c = 0. 625 (Ns/444)2 = 0. 339 Cavitation factor ? = Ha -Hv -Hs H Ha -Hv = 10. 6 m, H =27 m 0. 339 = 10. 6 -Hs 27 Hs = 1. 45 m Max length of the draft tube = 1. 45 + 0. 7 = 2. 22 m 5. 12 Governing of Turbines All the modern turbines are directly coupled to the electric generators. The generators are always required to run at constant speed irrespective of the variations in the load. This constant speed N of the generator is given by the expression N= 60 f p †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. (5. 5) where f = frequency (usually 50) p = numbers of pairs of poles 5. 13 Water Hammer A gate or valve at the end of the penstock pipe controls the discharge to the turbine. As s oon as this governor regulated gate opening is altered, the pipe flow has to be adjusted to the new magnitude of flow.In doing so, there are rapid pressure oscillations in the pipe, often accompanied by a hammering like sound. Hence this phenomenon is called as water hammer. 5. 14 Jet Speed The velocity of flow of the jet depends upon the total net head H at the base of the nozzle and is given by the nozzle equation: v = C v ? 2gH where the discharge coefficient velocity of the nozzle is taken as 0. 95. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦(5. 6) 40 5. 15 Bucket speed V= ? DN 60 †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦(5. 7) The bucket speed should be half of the jet speed. In practice, losses in the turbine cause the maximum efficiency to occur at slightly less than a half, typically 0. 46. V =0. 6 v 5. 16 Design of Pelton Wheel Runner diameter: Runner diameter can be found out from the rpm equation. D= 38 ? H N †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. (5. 8) where N = runner speed(rpm) H = net head Nozzle diameter: The nozzle diameter is given by the nozzle equation: d = 0. 54 ? Q 0. 5 1 ? 0. 25 H n jet †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. (5. 9) Jet ratio: Jet ratio D/d is a size parameter for the turbine. It has a value in a range of 10 to 24. For the high efficiency Pelton wheel design, the ratio of the runner diameter to the nozzle should be more than 9. Number of buckets: The number of buckets required for the efficient operation of the Pelton turbine is calculated as:N buc = 0. 5 ? D + 15 d †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. (5. 10) In practice, the selection and the detail design of the turbine units are carried out by the manufactures based on the model performances. 41 Example 5. 2 Powerhouse is equipped with a vertical shaft pelton turbine. The generator is provided with 6 pairs of poles. Design discharge is 1. 4 m3/s and net head is 425 m. The turbine will provide 6500 hp. Take coefficient of nozzle as 0. 95. Determine (a) the specific speed (b) velocity of jet (c) jet diameter (d) pitch circle diameter of the wheel (e) number of buckets (a) N= 60 f p = 60 x 50/6 = 500 rpmNs = = NP H5/ 4 500 6500 4255 / 4 = 20. 9 Use single jet pelton turbine (b) velocity of jet v = C v ? 2gH = 0. 95 ? 2Ãâ€"9. 81Ãâ€"425 = 86. 75 m/s (c) jet diameter d = 0. 54 ? = 0. 54 ? Q 0. 5 1 ? 0. 25 H n jet 1. 4 0. 5 425 0. 25 ? = 0. 14 m =14 cm 1 1 42 (d) pitch circle diameter D= 38 ? H N = 1. 57 m (e) Number of buckets N buc = 0. 5 ? N buc = 0. 5 ? D + 15 d 1. 57 + 15 0. 14 = 20. 6 = 21 5. 17 Work done of Pelton Wheel In turbines, the water flows on to the runner, which itself is rotating with a certain speed. The water flows over the runner and leaves the runner at its outlet point.We can speak of absolute velocity of water before it flows in the runner, the relative velocity of water w. r. t the runner and again the absolute velocity of water after it has left the runner. In order to ascertain the relationship between these velocities, the velocity vector diagram prove to be very useful. Fig. 5. 9 shows the velocity triangles at the tips of the bucket of a pelton wheel. At the outlet tip velocity triangles are different depending upon the magnitude of u corresponding to which it is slow, medium or fast runner. Inlet velocity diagram VVri Vai =Vwi 43 Outlet velocity diagram V Vwo ? ? Vfo Vro Vao Fig. 5. 9 Velocity triangles V = bucket velocity Vai = absolute velocity of jet at inlet tip Vai = Cv 2 gH1 Vao = absolute velocity of jet at outlet tip Vri = relative velocity of jet at inlet = Vai-V Vro = relative velocity of jet at outlet = k. Vri Vwi = velocity of whirl at inlet =Vai Vwo = velocity of whirl at outlet = V-VroCos ? Vfo = velocity of flow at outlet Mass/sec m =? Q =? a Vai=? ?/4 d2 Vai †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦(5. 11) Workdone on the bucket/sec (power developed by turbine) P = m (Vwi -Vwo) V †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦(5. 12)Maximum hydraulic efficiency ? h max = 1 (1 + kCos? ) 2 †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦(5. 13) The hydraulic efficiency is maximum when the bucket speed is equal to half of the velocity of jet. 44 Example 5. 3 The head available at entrance to the nozzle supplying a pelton wheel is 300 m and the coefficient of velocity for the nozzle is 0. 98. The wheel diameter is 1. 8 m and the nozzle diameter is 125 mm. The buckets deflect the jet through 165 °. Assuming the relative velocity of the jet is reduced by 16%, calculate the theoretical speed in rev per min for the maximum hydraulic efficiency.What is the hydraulic efficiency when running at this speed, and what is the power developed? Deflection angle =165 ° = (180- ? ) ? = 15 ° k = 0. 84 hf hn H'=Vai2/2g H For max hydraulic efficiency V/Vai =0. 5 Vai = Cvv2gH1 = 75 m/s V =Vai/2 = 37. 5 m/s V =? DN/60 N =60V/? D = 398 rpm ?hmax = 1/2 (1+ k Cos ? ) = 90. 55 % mass/sec = m = ? Q = ? ?/4 d2 Vai = 920 kg/sec H1 45 Inlet diagram: V Vri From velocity diagram Vwi = Vai =75 m/s Vri = Vai- V =37. 5 m/s Vwi = Vai Outlet diagram: Vwo = V -Vro Cos ? V = V – k Vri Cos ? Vwo Vro Power = m V ( Vwi -Vwo) = 234600 Watts Vao =7 m/s 46 CHAPTER 6CENTRIFUGAL PUMP 6. 1 Introduction Centrifugal pumps are classified as rotodynamic type of pumps in which dynamic pressure is developed which enables the lifting of liquids from a lower to a higher level. The basic principle on which a centrifugal works is that when a certain mass of liquid is made to rotate by an external force, it is thrown away from the central axis of rotation and a centrifugal head is impressed which enable it to rise to a higher level. Now, if more liquid is constantly made available at the centre of rotation, a continuous supply of liquid at a higher level may be ensured.Since in these pumps the lifting of the liquid is due to centrifugal action, these pumps are called ‘centrifugal pumps'. 6. 2 Advantages of centrifugal pumps over reciprocating pumps The main advantage of a centrifugal pump is that its discharging capacity is very much greater than a reciprocating pump which can handle relatively small quantity of liquid only. A centrifugal pump can be operated at very high speeds without any danger of separation and cavitation . The maintenance cost of a centrifugal pump is low and only periodical check up is sufficient .But for a reciprocating pump the maintenance cost is high because the parts such as valves etc. , may need frequent replacement. 6. 3 Component Parts of a Centrifugal Pump The main component parts of a centrifugal pump are: -impeller -casing -suction pipe -delivery pipe 47 Fig. 6. 1 Component part of a centrifugal pump 6. 4 Workdone by the Impeller The expression of the workdone by the impeller of a centrifugal pump on the liquid flowing through it may be derived in the same way as for a turbine. The liquid enters the impeller at its centre and leaves at its periphery. Fig. 6. shows a portion of the impeller of a centrifugal pump with one vane and the velocity triangles at the inlet and outlet tips of the vane. V is absolute velocity of liquid, u is tangential velocity of th e impeller, Vr is relative velocity of liquid, Vf is velocity of flow of liquid, and Vw is velocity of whirl of the liquid at the entrance to the impeller. Similarly V1,u1,Vr1,Vf1 and Vw1 represent their counterparts at the exit point of the impeller. 48 Fig. 6. 2 Velocity triangles for an impeller vane ? = the impeller vane angle at the entrance ? = the impeller vane angle at the outlet = the angle between the directions of the absolute velocity of entering liquid and the peripheral velocity of the impeller at the entrance ? = the angle between the absolute velocity of leaving liquid and the peripheral velocity of the impeller at the exit point Work done per second by the impeller on the liquid may be written as Work done = W ( Vw1 u1 – Vw u) ————-(6. 1) g where W kg of liquid per second passes through the impeller. Since the liquid enters the impeller radially ? = 90 and hence Vw = 0. Thus equation (6. 1) becomes Work done = W (Vw1u1) —â⠂¬â€Ã¢â‚¬â€Ã¢â‚¬â€-(6. ) g 6. 5 Head of a Pump The head of a centrifugal pump may be expressed in the following two ways: (a) Static head (b) Manometric head (or total head or gross head) 49 Fig. 6. 3 Head on a centrifugal pump (a) Static Head Static head is the vertical distance between the liquid surfaces in the pump and the tank to which the liquid is delivered by the pump. Static head (or lift) Hs = hs + hd where hs = static suction lift hd = static delivery lift †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. (6. 3) 50 (b) Manometric Head Manometric head is the total head that must be produced by the pump to satisfy external requirements.If there are no energy losses in the impeller and the casing of the pump, then the manometric head Hm will be equal to the energy given to the liquid by the V ? 1u1 g impeller, i. e Hm = . But if losses occur in the pump then V ? 1u1 ? losses of head in the pump g †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. (6. 4) Applying Bernoulli’s equation between the points, O at the liqui d surface in the pump and 1 in the suction pipe just at the inlet to the pump (i. e. , at the centre line of the pump), the following expression is obtained if the liquid surface in the sump is taken as datum. 0= p s Vs2 + + hs + h f s ? 2g ps Vs2 ? [ + hs + h fs ] 2g ? †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. (6. 5) where ps is the pressure at point l ; Vs is the velocity of flow in the suction pipe ; hs is the suction lift and hfs is the head loss in the suction pipe which includes the head loss due to friction and the other minor losses. It may however be pointed out that if the pump is situated below the level of the liquid surface in the sump, hs will be negative. Equation (6. 5) indicates that at the inlet to the pump there is always a suction or vacuum pressure developed which will be recorded by the vacuum gauge provided at this point as shown in Fig. . 3. The head expressed by equation (6. 5) is called the suction head of the pump. Also, applying Bernoulli’s equation between points 1 and 2, which is just at the outlet of the impeller and is assumed to be at the same level as point 1, then since the impeller imparts a head equal to (Vw1u1/g) to the liquid the following expression is obtained: p s V s2 V? 1u1 p 2 V12 + + = + + hLi ? 2g g ? 2g †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.. (6. 6) where p2 is the pressure and V1 is the absolute velocity of the liquid leaving the impeller and hLi is the loss of head in the impeller. 51 6. Specific Speed of Centrifugal Pumps In order to compare the performance of different pumps, it is necessary to have some term which will be common to all centrifugal pumps. The term used for this purpose is the specific speed. The specific speed of a centrifugal pump is the speed at which the specific pump must run to deliver unit quantity against unit head, the efficiency being the same as the actual pump. Ns = NQ H 3/ 4 †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦(6. 7) where Ns= specific speed N = rotational speed(rpm) H = total head 6. 7 Performance of Pumps- Charact eristic CurvesA pump is usually designed for one speed, flow rate and head in actual practice, the operation may be at some other condition of head on flow rate, and for the changed conditions, the behaviour of the pump may be quite different. Therefore, in order to predict the behaviour and performance of a pump under varying conditions, tests are performed and the results of the tests are plotted. The curves thus obtained are known as the characteristic curves of the pump. The following three types of characteristic curves are usually prepared for the centrifugal pumps : a) Main and operating characteristics. (b) Constant efficiency or Muschel curves . (c) Constant head and constant discharge curves. Main and Operating Characteristics In order to obtain the main characteristic curves of a pump it is operated at different speeds. For each speed the rate of flow Q is varied by means of a delivery valve and for the different values of Q the corresponding values of manometric head Hm, shaft H. P. , P , and overall efficiency ? are measured or calculated. The same operation is repeated for different speeds of the pump. Then Q v/s Hm ; Q v/s P and Q v/s ? urves for different speeds are plotted, so that three sets of curves, as shown in Fig. 6. 4 are obtained, which represent the 52 main characteristics of a pump. The main characteristics are useful in indicating the performance of a pump at different speeds. During operation a pump is normally required to run at a constant speed, which is its designed speed, (same as the speed of the driving motor). As such that particular set of main characteristics which corresponds to the designed speed is mostly used in the operations of a pump and is, therefore, known as the operating characteristics.A typical set of such characteristics of a pump is shown in Fig. 6. 5 Fig. 6. 4 Main characteristics of a centrifugal pump Fig. 6. 5 Operating characteristic curves of a centrifugal pump 53 6. 8 Parallel or Series Operation of Pu mps Pumps in series Centrifugal pumps generate a relatively low head delivering a fairly high rate of discharge. Normally a pump with a single impeller can be used to deliver the required discharge against a maximum head of about 100 m. But if the liquid is required to be delivered against a still larger head then it can be done by using two or more pumps in series. Fig. 6. 6 Three stage centrifugal pumpIf the required head is more than that can be provided by one pump, the pumps are connected in series. The same discharge passes through both pumps but the head developed by one pump add the other. The total head developed is obtained by adding together the value of the head of each pump corresponding to the relevant discharge. Pumps in Parallel The multi-stage pumps or the pumps in series as described earlier are employed for delivering a relatively small quantity of liquid against very high heads. However, when a large quantity of liquid is required to be pumped against a relativel y small head, then it may 4 not be possible for a single pump to deliver the required discharge. In such cases two or more pumps are used which are so arranged that each of these pumps working separately lift the liquid from a common sump and deliver it to a common collecting pipe through which it is carried to the required height Fig. 6. 7. Since in this case each of the pumps deliver the liquid against the same head, the arrangement is known as pumps in parallel. If Q1, Q2, Q3†¦.. , Qn are the discharging capacities of n pumps arranged in parallel then the total discharge delivered by these pumps will beQt = (Q1+Q2+Q3+†¦Ã¢â‚¬ ¦+Qn) †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. (6. 8) If the discharging capacity of all the n pumps is same, equal to Q , then the total discharge delivered by these pumps will be Qt = nQ Fig. 6. 7 Two centrifugal pumps arranged in parallel 55 A centrifugal pump, having four stages in parallel, delivers 11m3/min of Example 6. 1 liquid against a head of 24. 7m , the diameter of the impeller being 225mm and the speed 1700 rpm. A pump is to be made up with a number of identical stages in series, of similar constriction to those in the first pump, to run at 1250 rpm, and to deliver 14. 5 m3/min, against a head of 248m.Find the number of stages required for the second pump. 1st Pump 2nd Pump Q = 11m3/min Q = 14. 5m3/min H = 24. 7 m H = 248 m N = 1700 rpm N =1250 rpm D = 225 mm Specific speed N s = NQ H 3/ 4 Q for one pump = 11/4 = 2. 75 m3/min Ns = 1700 2. 75 24. 7 3 / 4 = 254 For 2nd pump, with identical stages in series i. e multi-stage pump, if each stage is similar to those of each stage is similar to those of the first pump. The specific speed of each stage Ns = 254 Ns = 254 = NQ H 3/ 4 1250 14. 5 H 3/ 4 H = 49. 64 m Total head required = 248 m No of stages required = 248/49. 64 = 5 stages 56 CHAPTER 7DIMENSIONAL ANALYSIS, HYDRAULIC SIMILITUDE AND MODEL INVESTIGATION 7. 1 Dimensional Analysis Dimensional analysis is a mathematical method of obtaining the equations, changing units, determining a convenient arrangement of variable of a physical relation. In an equation expressing a physical relationship between quantities, absolute numerical and dimensional equality must exit. In general, all such physical relationships can be reduced to the fundamental quantities of mass M, length L and time T. It is a based on the assumption that the phenomenon can be expressed by a dimensionally homogeneous equation, with certain variable.The dimensional analysis is widely used in research work for developing design criteria and also for conducting model tests. 7. 2 Dimensions and Units All physical quantities are measured by comparison. This comparison is always made with respect to some arbitrarily fixed value for each independent quantity, called dimension(e. g. , length, mass, time, etc. ). Since there is no direct relationship between these dimensions, they are called fundamental dimensions. Some other quantities such as area , volume, velocity, force etc. can not be expressed in terms of fundamental dimensions and thus may be alled derived dimensions. There are two systems for fundamental dimensions namely FLT (i. e force, length, time) and MLT (i. e. , mass, length, time). One common system employed in dimensional analysis is the M,L,T system. Table is a listing of some of the quantities used in fluid flow, together with their symbols and dimensions. 57 Quantity Symbol Dimensional Form Length l L Time t T Mass m M Velocity v L T-1 Acceleration a L T-2 Force F M L T-2 Pressure P M L-1 T-2 Discharge Q L3 T-1 Power P M L2 T-3 W,E M L2 T-2 Density ? M L-3 Dynamic viscosity  µ M L-1 T-1 Kinematic viscosity ? L2 T-1 Surface tension M T-2 Work,energy 7. 3 Methods of Dimensional Analysis The methods of dimensional analysis are: -Buckingham's ? theorem -Ralyeigh's method Buckingham's ? Theorem If there are ‘n' variables in a dimensionally homogeneous equation, and if these variables contain ‘m' fu ndamental dimensions such as (M,L,T) , they may be grouped into (nm) non-dimensional independent ? terms. Mathematically, if a variable x1 depends upon independent variables x2, x3,x4, †¦. ,xn, the functional equation may be written as x1 = f (x2, x3, x4, †¦. , xn) The equation may be written in its general form as 58 f1 ( x1, x2, x3,†¦Ã¢â‚¬ ¦. xn) = C In this equation there are ‘n' variables. If there are ‘m' fundimental dimensions, the according to ? theorem f2 ( ? 1,? 2,? 3,†¦.. , ? n-m ) = C1 e. g Q = f (d,H, µ,? ,g) f1 (Q, d, H,  µ, ? , g) = C n = 6; m = 3; (n-m) = 3 f2 ( ? 1, ? 2, ? 3) = C1 Procedure 1. First of all, write the functional relationship with the given data. 2. Then write the equation in its general form. 3. Choose ‘m' repeating variables and write separate expressions for each term. Every ? term will contain the repeating variables and one of the remaining variables. The repeating variables are written in exponential form . 4.With the help of the principle of dimensional homogeneity, find out the values of the exponents by obtaining simultaneous equations. 5. Substitute the value of these exponents in the ? term. 6. After the ? terms are obtained, write the functional relation in the required form. e. g Q = f ( d, H,  µ, ? , g) f1 (Q, d, H,  µ, ? , g) =C n = 6 , m = 3 , (n-m) = 3 f2 (? 1,? 2,? 3)=C1 Choose ? , g, d as repeating variable, ?1 = ? a1 gb1 dc1 Q ? 2 = ? a2 gb2 d c2 H ? 3 = ? a3 gb3 dc3  µ 59 Selection of Repeating Variable 1. The variables should be such that none of them is dimensionless. 2. No two variables should have the same dimensions. . Independent variables should be as far as possible, be selected as repeating variable. ? > fluid property ? > flow characteristics l > geometric characteristics Example 7. 1 A V-notch weir is a vertical plate with a notch angle ? cut into the top of it and placed across an open channel. The liquid in the channel is backed up and forced to flow through the notch. The discharge Q is some function of the elevation H of upstream liquid surface above the bottom of the notch. In addition it depends upon gravity and upon the velocity of approach Vo to the weir. Determine the form of discharge equation: ?V ? Q = gH 5/2 f ? ,? ? ? gH ? ? ? Q = f (H, g , Vo , ? ) f1 ( Q, H ,g ,Vo, ? ) = C Choose g and H as repeating variables n = 5; n-m =3 ; m=2 ?1 = Ha1 gb1 Q = (L)a1 (LT-2)b1 L3 T-1 ?2 =Ha2 gb2 Vo = (L)a2 (LT-2)b2 LT-1 ?3 = ? (M)o (L)o (T)o = (L)a1 (LT-2)b1 L3 T-1 a1+b1+3 =0 ? a1 = -5/2 -2b1-1 =0 ? b1 = -1/2 ?1 = H-5/2 g-1/2 Q = Q vg H 5/2 (M)o (L)o (T)o = (L)a2 (LT-2)b2 L T-1 a2 = -1/2 60 b2 = -1/2 ?2 =H-1/2 g-1/2 Vo= Vo vgH Q f2 ( gH 5/ 2 Q gH 5/ 2 , Vo ,? gH ) = C1 Vo ,? gH ) Vo ,? gH ) =f( Q = gH 5/2 f ( Example 7. 2 Q = VD 2 f [ Prove that the discharge over a spillway is given by the relation gD H ,] V D where V= velocity of flowD = depth of throat H = Head of water g = Acceleration due to gravity Q= f (V,D,H,G) f1 (Q,V,D,H ,G) = C Choose V and D as repeating variables n = 5, m =2, n-m =3 ?1= Va1 Db1 Q = (LT-1)a1 (L)b1 (L3T-1) ?2 = Va2 Db2 H = (LT-1)a2 (L)b2 (L) ?3 = Va3 Db3 g = (LT-1)a3 (L)b3 (LT-2) M0L0T 0 = (LT-1)a1 (L)b1 (L3T-1) 0 = -a1-1 ; a1 = -1 0 = a1+b1+3 ; b1= -2 ?1 = V -1 D -2 Q 61 ?1 = Q VD2 M0L0T 0 =(LT-1)a2 (L)b2 (L) 0 = -a2 a2+b2+1 =0 ; b2 = -1 ?2 = V0 D-1 H = H/D M0L0T 0 = (LT-1)a3 (L)b3 (LT-2) 0 = -a3-2 ; a3 = -2 0 = a3+b3+1 ; b3 =1 ?3 = V-2 D g gD V = f2 ( gD H , )=0 VD 2 D V Q VD 2 Q , = f( gD H ,) V D Q = VD 2 f ( 7. 4 D H ,) V D HYDRAULIC MODELS Hydraulic models, in general, may be either true models or distorted models. True models have all the significant characteristics of the prototype reproduced to scale (geometrically similar) and satisfy design restrictions (kinematic and dynamic similitude). Model-prototype comparisons have clearly shown that the correspondence of behaviour is often well beyond expected limitations, as has been attested by the successful operation of many s tructures designed from model tests. 7. 5 Hydraulic Similitude To know the complete working and behaviour of the prototype, from its model, there hould be a complete similarity between the prototype and its scale model. This similarity is 62 known as hydraulic similitude. From the subject point of view, the following three types of hydraulic similitude are important. (1) Geometric similitude (2) Kinematic similitude (3) Dynamic similitude GEOMETRIC SIMILITUDE The model and the prototype are identical in shape, but differ only in size. (The ratios of all the corresponding linear dimensions are equal) . Let L = Length of the prototype B = Breadth of the prototype D = Depth of the prototype l,b,d = corresponding values of the model L Linear ratio Lr =Area ratio Ar = ( Volume ratio Vr = ( l = BD = bd L B D )2 = ( )2 = ( )2 l b d L B D ) 3 = ( ) 3 = ( )3 l b d KINEMATIC SIMILITUDE The model and the prototype have identical motions. ( The ratios of the velocities at corresponding points a re equal) Let V1 = velocity of liquid in prototype at point 1 V 2 = velocity of liquid in prototype at point 2 v1,v2 = corresponding values of the model Velocity ratio Vr = V1 V2 = = †¦.. v1 v 2 DYNAMIC SIMILITUDE The model and prototype have identical forces. (The ratios of the corresponding forces acting at corresponding points are equal). F Force ratio Fr = 1 = f1 F2 †¦.. f2 63 7. 6 CLASSIFICATION OF MODELS (1) Undistorted model (2) Distorted model Undistorted model A model which is geometrically similar to the prototype is known as undistorted model. Distorted model Model does not have complete geometric similarity with the prototype, is known as distorted model. 7. 7 Comparison of an Undistorted Model and the Prototype If the model is to be overall similar to the prototype, then all the three similarities (i. e, geometric, kinematic, dynamic ) should exist. But this is not possible in actual practice, as it is difficult to exist two types of similarities simultaneousl y.In general, and undistorted model of a prototype is made keeping in view the geometric similarity only and the remaining similarities are then compared for the scale ratio. 7. 8 Velocity of Water in Prototype for the Given Velocity of an Undistorted Model Consider an undistorted model geometrically similar to a proposed prototype like a weir, dam, spillway etc. Let h = head of water over the model v = velocity of water at a point in the model H,V = corresponding values for the prototype 1/s = scale ratio of the model to the prototype Velocity of water in the model v = Cv v2ghVelocity of water on the corresponding point in the prototype V = Cv v2gH C v 2 gh v = = V C v 2 gH h 1 = H s V = v vs Example 7. 3 The velocity at a point on a spillway model of a dam is 1. 3m/sec for a prototype of model ratio 1:10. What is the velocity at the corresponding point in the prototype? 64 Velocity in the model v = 1. 3 m/s Model ratio, 1/s = 1/10 s = 10 Velocity in the prototype V = v vs V =1. 3 x v10 = 4. 11 m/sec 65 References: 1. Hydraulics and Fluid Mechanics P. N. Modi and S. M. Seth 2. Hydraulics, Fluid Mechanics and Hydraulic Machines R. S. Khurmi 3. Fluid Mechanics Victor L. Streeter