Conservation Law of Centrifugal Force and Mechanism of Energy Transfer Caused in Turbomachinery

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1 Poceedings of the 4th WSEAS Intenational Confeence on luid Mechanics and Aeodynamics, Elounda, Geece, August 1-3, 006 (pp337-34) Consevation Law of Centifugal oce and Mechanism of Enegy Tansfe Caused in Tubomachiney TAKAHAR TANAKA Depatment of Mechanical Engineeing Kobe nivesity Rokko, Nada, Kobe JAPAN Abstact: - Consevation low of Centifugal foce indicates that the fluid paticle, which is foced to otate in the otating flow passage, causes two diffeent kind otational motions. ne is the appeaent otational motion aound the axis of otation caused togethe with impelle blades. And the othe is the substantial otational votex motion aound itself. Head ise is caused by the infinitely small but infinitely lage numbe of fluid paticles substantial otational votex motion aound themselves caused in the otating flow passage by the impelling action. Key-Wods: - Theoetical pump head, Enegy change, Consevation law on Centifugal foce. 1 Intoduction Theoetical head is one of the most fundamental and impotant subject to be established in the study of tubomachiney. Stepanoff, A. J. [1] consideed that mechanical enegy is supplied to fluid paticles contained between two adjacent impelle blades at time t=0, theefoe, enegy change is caused on those fluid paticles. While, Tanaka, T. [] consideed that mechanical enegy is supplied to the system. Then, hydaulic enegy change has to be caused not only on the fluid paticles contained between two adjacent impelle blades at time t=0, but also caused on the fluid paticles enteing the impelle blades fom the leading edge of impelle inlet in time inteval dt. They applied the pinciple of consevation of angula moment of momentum to those masses of liquids in the otating flow passage of impelle blades and got theoetical equations epesent the pump head. Howeve, they do not explain any inteelation with the mass weight flow ate at all. In this viewpoint, theoetical pump head is investigated. In this pape, theoetical consevation low is intoduced on Centifugal foce and it is applied to the fluid flow between otating impelle blades and a easonable inteelation, which epesents the pump head and flow ate, is obtained. The souce of pump head and mechanism of head ise ae also discussed. Intoduction of Consevation Law on Centifugal oce Mechanical enegy is supplied to impelle blade and tansfeed to fluid paticle as hydaulic enegy. Tansfeed hydaulic enegy is constucted fom mass weight flow ate gq and pump head H [3]. Then, maximum pump head is obtained at zeo mass weight flow ate in the pactical opeation of pump. Now, let us conside two dimensional fluid flow in the otating flow passage of Centifugal pump at zeo flow ate and image two abital cicula lines A and B aound cente. luid paticles otate togethe with impelle blades at constant angula velocity. If thei otational adiuses have the elation <, cicula line B locates outside of cicula line A. Centifugal foce at outside cicula line B is given by = ρ ( ) = ρω (1) Hee, adius coesponds to dischage adius of a pump. If the leading edge of impelle blade stats at cente, the magnitude of Centifugal foce at adius, given by equation (1), which coesponds to pump head H Th, may become the maximum. If the Centifugal pump is set at the depth H measued vetical downwad fom the wate suface W-W, the pessue head at the location of impelle dischage is given by = ρ g () H 1

2 Poceedings of the 4th WSEAS Intenational Confeence on luid Mechanics and Aeodynamics, Elounda, Geece, August 1-3, 006 (pp337-34) Pessue foce of ested wate at vetical depth H due to gavitational acceleation g is balanced with the Centifugal foce of the fluids due to impelle blades otational motion. While thei acting diections ae opposed. Then = (3) Theoetical pump head H is given by H = H 1 = ω (4) g ig.1 Illustation to show the consevation low on Centifugal foce caused at otating flow passage aound cente at constant angula velocity. Then, let us ty to establish a new theoy, which epesents the case that impelle blade does not stat its leading edge at cente, but stats at some distances away fom the cente. Now, conside a cicula line A, which locates some distances away fom the otational cente, which coesponds to the leading edge of impelle inlet. Its adius is theefoe smalle than that of cicula line B. Centifugal foce at otational adius is given by = ρ ( ) = ρω (5) Then, diffeence of Centifugal foces and is given by equations (1) and (5) as follow, ρ B = ρ ( ) ρ( ) = ρω ( ) (6) ρ ρ Radiuses and have the elation = (7) Then, equation (6) can be ewitten as follow: In othe wods, Whee, = ρ ( ) ρ( ) = ρω (8) = ρ ( ) ρ( ) = ρ( ) (9) = ω (10) om the physical paametes that oganaze the equation (10), it could be ecognized that may coespond to the peipheal velocity of otational motion along the cicula line, and the cicula line may coespond to the locus of otational motion at adius at angula velocity. Cicula line touches intenally with the oute cicula line B and locates its cente on the cicle line A. Radius of cicula line is abitay between and. It is due to the adius of inside cicula line A egadless whethe the adius of outside cicula line B is fixed at constant o not if we conside unit adius =1. If the adius becomes lage towad adial outwad, adius becomes small. Cicula line foms vaious kinds cicula lines fo the change in otational adius between and. Howeve, all otational motions caused along those cicula lines ae caused unde constant angula velocity equivalent to those of cicula lines A and B at cente. This is clea fom equation (10). Now, equation (9) can be epesented as follow, = + (11) ρ ( ) = ρ( ) + ρ( ) (1) In othe wods,

3 Poceedings of the 4th WSEAS Intenational Confeence on luid Mechanics and Aeodynamics, Elounda, Geece, August 1-3, 006 (pp337-34) Whee ρω = ρω [ + ] (13) = ρ ( ) = ρω (14) These state that the summation of magnitude of Centifugal foce due to otational motion along the cicula line A and that due to otational motion along the cicula line is always equivalent to that due to otational motion along the cicula line B. Rotational adius is abitay between and. And above inteelation is always satisfied egadless whethe is fixed at constant o not if unut adius =1. Then, these equations could be called the Consevation Law of Centifugal foce due to otational motion at constant angula velocity [4]. In othe viewpoints, magnitude of Centifugal foce due to otational motion along the outside cicula line B is constucted fom two diffeent kinds otational motions. ne is that of Centifugal foce due to otational motion along the inside cicula line A whose otational axis locates at cente, which is equivalent to that of outside cicula line B. And the othe is that of Centifugal foce due to otational motion along the ciculaline whose line touches intenally with the oute cicula line B and locates its otational cente on the inside cicula line A. In othe expession, adius A of inside cicula line A and adius of imaginay cicula line constuct adius B of outside cicula line B and they all otate at same angula velocity. 3. If Cicula Line X locates utside of Cicula lines A and B Let us conside a cicula line X in addition to cicula lines A and B in the otating flow passage. If thei otational adiuses have the elation < < X, cicula line X locates outside of cicula lines A and B. Then, Centifugal foce X of outside cicula line X can be witten as follow = ( + ) + (15) X X ρ ( ) = ρ [( + ) + ] (16) X In othe wods, ρω = ρω [( + ) + ] (17) X In moe geneal, let us conside infinitely lage numbe of cicula lines C, D, V, W, X in addition to A and B. If thei adiuses have the elation < < C < D < < V < W < X, they also locate outside of cicula lines A and B. Then Centifugal foce X of outside cicula line X can be witten as follow = ( + ) (18) X X ρ ( ) = ρ[( + ) ] (19) X In othe wods, ρω = ρω [( + ) ] (0) X These indicate that Centifugal foce X of outside cicula line X is a function only of summation of inceased Centifugal foces. If otational adius of inside cicula line A is fixed at constant, and if otational adiuses of additionally added cicula lines B, C, D, V, W, X ae lage than that of cicula line A, total otational adius inceases towad outside. Then magnitude of total Centifugal foce also inceases as much. Total magnitude of Centifugal foces X is explained as the summation of inceased Centifugal foces in addition to that of oiginal Centifugal foce of cicula line A. If oiginal otational cicula line A does not exist, its magnitude could be explained simply as the summation of inceased Centifugal foces due to inceased otational adiuses. See equations (18) and (19). In othe wods, if angula velocity is constant it might be a function only of summation of inceased otational adiuses. See equation (0). 4. If Cicula Line Y locates between Cicula lines A and B Let us conside a cicula line Y in addition to A and B again. If thei otational adiuses have the elation < Y <, cicula line Y locates between cicula lines A and B. Then, Centifugal foce of outside cicula line B can be witten as follow, 3

4 Poceedings of the 4th WSEAS Intenational Confeence on luid Mechanics and Aeodynamics, Elounda, Geece, August 1-3, 006 (pp337-34) = + + (1) ρ ( ) = ρ( + + ) () In othe wods, ρω = ρω ( + + ) (3) In moe geneal, let us conside infinitely lage numbe of cicula lines C, D, X, and Y. If thei otational adiuses have the elation < C < D < < X < Y <, they locate between cicula lines A and B. Then, Centifugal foce of outside cicula line B can be witten as follow, = (4) ρ ( ) = ρ ( ) (5) In othe wods, ρω = ρω ( ) (6) These indicate that if outside otational adius is fixed at constant o if we conside the unit adius =1, total magnitude of Centifugal foce at outside adius does not changes at all by the numbe of cicula lines inside of it. 5. Physical Popety of Cicula line ne of pevious esults of theoetical discussion is that if outside otational adius is fixed at constant o if we conside the unit adius =1, otational adius does not change by the numbe of cicula lines inside of it. This is vey significant and meaningful. If we conside infinitely lage numbe of cicula lines between cicula lines A and B, fluid paticles geometical otational adiuses, which coesponds to adius in ig.1, become infinitely small. Whilst, thei summation of imaginay otational motions, which coespond to Centifugal foce due to cicula line, is emained constant. In othe wods, infinitely lage numbe fluid paticles otate at infinitely small otational adiuses in the otating flow passage between and and esultsd to the Centifugal foce due to the imaginay otational motion along the cicula line at otational adius, which is theefoe, equivalent to summation of thei Centifugal foces. This might be clea fom equations (4), (5) and (6). In opposed viewpoints, the infinitely small but infinitely lage numbe fluid paticles votex motions ae caused at angula velocity by the impelling action of impelle blades and distibute in the otating flow passage between and. And they constuct the Centifugal foce. This indicates that infinitely small but infinitely lage numbe fluid paticles votex motions cause substantial magnitude of Centifugal foce equivalent to that explained by imaginay cicula line. In othe wods, the summation of magnitudes of Centifugal foces caused by the infinitely small but infinitely lage numbe fluid paticles votex motions caused in the impelling egion between and is equivalent to the magnitude of Centifugal foce constucted by imaginay cicula line at otational adius. om these viewpoints it could be said that fluid paticles, which is foced to otate togethe with impelle blades aound the axis of otation, may not only otate aound the cente, but also otate aound itself. In addition to these, both of those angula velocities ae equivalent to those of cicula lines A and B at cente of the flow passage. 6. Physical Meaning of Cicula line Hee the most significant is the magnitude of Centifugal foce of otational motion along the imaginay cicula line at adius in ig.1. It is obvious that fluid paticle does not cause this kind eal otational movement in the pactical otational motion of fluid paticles in the flow passage of impelle blades. This indicates that geometical cicula line due to imaginay otational motion at adius could be ecognized as that it might be the geometical hypothetical desciption o the locus line appeaed accidentally o following the fundamental enegy tansfe mechanism visible way to show the total amount of fluid paticles substantial otational votex motion at angula velocity. If the otational cente of cicula line on the cicula line A changes its location along the adius line fom the location of adius of outside cicula line towad the otational cente, Centifugal foce changes its magnitude. Magnitude of Centifugal 4

5 Poceedings of the 4th WSEAS Intenational Confeence on luid Mechanics and Aeodynamics, Elounda, Geece, August 1-3, 006 (pp337-34) foce becomes lage if the location of otational cente moves adial inwad fom the suface of cicula line B towad the cente. If it locates on the suface of cicula line B, both the otational adius and the magnitude of Centifugal foce become the minimum (zeo). If it becomes lage and eaches the cente, both the otational adius and the magnitude of Centifugal foce become the maximum, which ae equivalent to those of otational adius and Centifugal foce. 7. Inteelation between Pump Head and Mass Weight low Rate om the above discussion, it is clea that Centifugal foce coesponds to pump head H. Cicula line, which poduces the Centifugal foce, is fomed at adius and its otational cente locates on the cicula line A. This indicates that magnitude of Centifugal foce is the lagest at cicula line B because its cicula line touches intenally with the cicula line B and the minimum (zeo) at the inside cicula line A because it is the location of otational cente. In othe wods, pump head H is the lagest at the suface of cicula line B at the tailing edge of impelle outlet and the minimum (zeo) at the cicula line A at the leading edge of impelle inlet. Pump head H is zeo at adius means that flow ate Q is the maximum, theefoe ectangula velocity tiangle is fomed at the impelle inlet. Because tansfeed hydaulic enegy E H is constucted fom mass weight flow ate gq and pump head H. This indicates that adius coesponds to pump head and adius coesponds to flow ate Q, and thei summation coesponds to tansfeed hydaulic enegy. If lage adius is selected in the pactical design, flow ate Q becomes lage but pump head H becomes small. If small adius is selected, flow ate Q becomes small, but head H becomes lage. 8. veview of Theoetical Results In the pevious discussion, it became clea that two diffeent kinds otational motions ae caused in case of fluid paticles otational motion. ne is the otational motion aound the axis of otation and the othe is the fluid paticles own otational votex motion aound itself. om these viewpoints, it could be said that fluid paticles may not only cause the otational motion aound the axis of otation, but also causes the fluid paticles own otational votex motion aound itself in the otating flow passage of impelle blades. This theoetical esult could be expanded to the idea that if the fluid paticle tuns its flow diection at a cuved flow passage and causes a Centifugal foce, fluid paticle may not only cause its flow diections change along the cuved flow passage, but also causes a otational votex motion aound itself. These indicate that pessue head aise due to the fluid paticles otational motion and the flow condition at the cuved flow passage ae not only detemined by the fluid paticles otational motion aound the axis of otation o the flow diections change along the cuved flow passage, but also detemined by the fluid paticles own otational votex motion aound themselves. This esult of discussion indicates that fluid paticles otational motion aound the axis of otation and the flow diections change along the cuved flow passage cannot be discussed without consideing the infinitely small fluid paticles otational votex motion aound itself and without the application of votex theoy to the fluid paticles own otational votex motion aound itself. 9. Application to the luid low in Centifugal Pump In case of fluid paticles otational motion, otational motion is not same to that of mateials. It is vey obvious that otational motion does not always stat at cente, but stats vey often at an abitay adius point lage than zeo. luid paticle stats its otational motion at some distances away fom cente. This kind otational motion is seen fo example in the otating flow passage fomed by impelle blades in hydaulic machines, such as pumps, blowes, fans, compessos, wate tubines, gas tubines, and so on. Let us conside pactical application of consevation low on Centifugal foce to the fluid flow in the otating flow passage between impelle blades in Centifugal pump, next. See ig.. In this case, otational adius coesponds to the tailing edge of impelle outlet. Rotational adius coesponds to the leading edge of impelle inlet. If the Centifugal pump is set at the depth H measued vetical downwad fom the wate suface W-W, the pessue head is given at the location of impelle dischage by equation (). And this pessue foce that diects vetical downwad is balanced with the Centifugal foce that diects adial outwad given by equation (14). Then, theoetical pump head H is given by 5

6 Poceedings of the 4th WSEAS Intenational Confeence on luid Mechanics and Aeodynamics, Elounda, Geece, August 1-3, 006 (pp337-34) H = H 1 = ω (7) g Rotational adius of cicula line vaies between and, which coesponds to the distance between the leading edge of impelle inlet and the tailing edge of impelle outlet of the impelle blade. Then, equation (7) indicates that pump head H, that is, the magnitude of Centifugal foce is the minimum (zeo) at adius at the leading edge of impelle inlet and the maximum at adius at the tailing edge of impelle outlet. B A ig.4 Illustation to show the example of application of consevation low of Centifugal foce caused in the flow field aound the cente. Hee attention has to be paid to the facts that the fluid paticle that locates on the inside cicula line A otates at peipheal velocity at otational adius and that locates at the outside cicula line B otates at peipheal velocity at otational adius. Howeve, substantial pactical magnitude of fluid paticles Centifugal foce is not the value given by equation (5) but zeo and that of Centifugal foce is not the value given by equation (1) but that given by equation (14). This indicates that substantial pactical Centifugal foce indicated by cicula line is supplied by the fluid paticles otational votex motion. These indicate that essentially two diffeent kinds fluid paticles otational motions: appeaance and substantial otational motions have to be consideed in the otating flow passage. o example, magnitude of Centifugal foce obtained as a esult of theoetical discussion is explained as the diffeence between that of Centifugal foce and that of Contifugal foce by applying the consevation law on Centifugal foce intoduced in this investigation. Resultant Centifugal foce coesponds to theoetical head H Th. Magnitude of Centifugal foce coesponds to dischage pump head H obtained at the tailing edge of impelle outlet. Magnitude of Centifugal foce coesponds to suction pump head H 1 obtained at the leading edge of impelle inlet. om these it could be said that theoetical head H Th is explained as the diffeence between dischage and suction heads, H and H 1, by applying the consevation law on Centifugal foce. In othe wods, substantial pump head H P is given by H H 1 in the pactical pump technology. This indicates that the dischage pump head H is not the substantial pessue head but the appeaance pessue head at the tailing edge of impelle outlet and the suction pump head H 1 is not the substantial pessue head but the appeaance pessue head at the leading edge of impelle inlet. 10. Conclusions - Consevation low of Centifugal foce indicates that the fluid paticle, which is foced to otate in the otating flow passage of Centifugal pump, causes two diffeent kind otational motion. ne is the otational motion aound the axis of otation togethe with impelle blades. And the othe is the otational votex motion aound itself. Head ise is caused by the infinitely small but infinitely lage numbe of fluid paticles otational votex motion aound themselves caused by the impelling action in the otating flow passage of impelle blades. 11. Refeences 1. Stepanoff, A. J., Centifugal and Axial low Pumps, Theoy, Design, and Application, nd Edition, John Wiley & Sons, Inc., 1967, p.76.. Tanaka, T., An Investigation on Definition of Theoetical Head Poduced by Impelle Blade in Centifugal Pump, Poceedings of the 3d Intenational Confeence on Heat Tansfe, luid Mechanics, and Themodynamics (HEAT 004), Cape Town, South Afica, June Tanaka, T., An Investigation on Enegy Tansfe Mechanism Caused in Rotating low Passage of Tubomachiney: New Concept of Physical Paametes in Rectangula Coodinate System ASME Pape No.EDSM , June Tanaka, T. Consevation Low of Centifugal oce and Its Application to luid low in Tubomachiney, ASME Pape No.EDSM , July