Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1982 An algebraic propeller model using Eulerian turbomachine theory. Wilken, Dennis Ray. Massachusetts Institute of Technology http://hdl.handle.net/10945/20043 LiE.sARY DUDLEY NAVAL POSTGRADUATE SCHOOL MONTERcY, CALIF. 93940 . An Algebraic Propeller Model Using Eulerian Turbomachine Theory by Dennis Ray Milken B.S.E.E. , Purdue University < 1974) SUBMITTED TO THE DEPARTMEI^ OF OCEAN ENGINEERING IN PARTIAL FULFILLMENT OF THE DEGREES OF OCEAN ENGINEER AND MASTER OF SCIENCE IN MECHANICAL ENGINEERING at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY MAY 1982 © Dennis Ray Nil ken 1982 The author grants to M.I.T. permission to reproduce and to distribute copies o-f this thesis document in whole or in part. / '.Y DUDLEV NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIF. 93940 , : . . An Algebraic Propeller Model Using Eulerian Turbc»nachine Theory by N Dennis Ray Milken o-f Ocean Engineering on May 7, 1982 partial -fulfillment o-f the requirements -for the degrees o-f Ocean Engineer and Master o-f Science in Mechanical Engineering. Submitted to the Department in ABSTRACT Utilizing -four -quadrant experimental data determined by Miniovich converted to the modified advance and angl e-of-advance coefficient form, complete Karman-Knapp Circle Diagrams are plotted for a number of 3 bl aded propellers. Reducing the family of curves to a rational basis, algebraic equations are derived to model the performance of a three bladed propeller through all four quadrants using Eulerian turbomachine theory. The result is a set of equations uihich can be used to generate complete propeller thrust and torque characteristics given only incomplete performance data suitable for use in steady state design or dynamic simulation. Thesis Supervisor: Title: Dr. Henry M, Paynter Professor of Mechanical Engineering To Professor Paynter uiithout whose support and assistance could not have completed this project. I To my uii-fe and children without whose love and encouragement I wouldn't have had any reason to. "Inawa/f science might be described as paranoid thinicing applied to nature: we are looking -for natural conspiracies! -for connections among apparently disparate data." -CARL SA6«i^ Using Eulerian Turbomachine Theory An Aloebraic Propeller Model CONTENTS Title Page Abstract Dedication CHAPTER 1 1-1 1 2 3 INTRODUCTION Primary Motivation for this Type Method o-f Modeling 6 1-2 Some Current Propeller Modeling Methods 7 the Modeling Equations 9 1-3 The Objective CHAPTER 2 o-f DERIV^TICBM OF THE ALGEBRAIC MODEL 2-1 Fundamentals o-f Eulerian Turbomachines 10 2-2 Ideal, Canonical and Perfect Model Equations for Ship Propellers 13 2-3 Determination of the Algebraic Modeling Equations 18 CHAPTER 3 INTRODUCTION OF ADDITIONAL TERMS AND OTHER DETAILS IN PROPELLER CHARACTERISTICS 3-1 Pitch as a Primary Determinant 40 3-2 Number of Blades and Blade Area 41 3-3 Effects of Cavitation 42 3-4 Hul /Propel er Interactions 44 3-5 Other Considerations 45 1 CHAPTER 4 1 PREDICTION OF PROPELLER PERFORMANCE USING THE MODELING EQUATIONS 4-1 Comparison with Kfery Similar Propellers 47 4-2 Extrapolation to Other Propellers 53 4-3 Extrapolation to Missing Quadrants 54 CHAPTER 5 SUMMARY 5-1 An Overview of the Method 5-2 The Potential APPENDIX A -for Further Applications ESSEhfTIALS OF EULERIAN TURBOMACHINES 55 57 59 The Per-fect Eulerian Turbomachine The Practical Eulerian Turbomachine The Dissipation Function Per Unit Variables APPENDIX B SELECTED EXPERIMENTALLY DETERMINED PROPELLER DATA CONVERTED TO '4-QUADRANT' FORM 64 APPENDIX C THE BOND GRAPH REPRESENTATION 71 LIST OF REFERENCES 74 CHAPTER 1 . 1 INTRODUCTION 1 Primary motivation for this kind of modelino method architect or anyone interested in the modeling of a naval For propellers as a part of the design procedure leading to integrated ship design, the problem long persisted where one to estimate the performance of a required is an propeller for ship modeling purposes before the design has progressed to the where one is reasonably confident as to what the point desired propeller should look like. the final Even more frustrating is problem of having somehow determined the geometry of the propeller but not having sufficient experimental data to allow a comprehensive computer simulation. Even where it has existed, considerable effort has been expen- ded to convert the data into useable form for ccwnputer simu- lations since the practice of tabulating the results as func- tions of an advance coefficient has difficulty in the vicinity of zero ship or shaft speed. The problems propeller allowances associated formidable since a complete are as a must make for complicated fluid motions which are only just beginning to be understood. difficult with modeling a machine such to In model particular, it is still quite pass completely from propeller geometry to detailed lift and drag forces required for specific the propeller characteristics. However, tation there now exists a considerable amount of documen- for modeling machines y^ery much like a propeller. For attention turbines and pumps have had the of modelers and those who perform simulations -for industrial example, axial decades. Taken together, propellers, turbines and pumps are in fact, members of a general class of machines known as machines. Models for characteristics and design. esis machines many of the large industrial been quite successful have in predicting machine turbo- performance and are widely used in control system A general model for propellers which synthcould incorporate that data base would have a firmer basis in experience and first principles. 1.2 Some current propeller model no methods present there are two common methods used by designers At propeller obtain involves a i data for computer simulations. The to first experimental determination of the characteristics of physical model of the full size propeller followed by con- version of the data into the familar thrust and torque coefficient form. Such coefficients are then used in a table look-up scheme within the simulation program to obtain the necessary thrust and torque values. There are advantages to this kind of method. First, the determination of the propeller characteristics ensure through experimentation (scaling considerations that the data used in the simulation will aside) be as accu- rate as possible. Secondly, the use of a model of the intended propeller acts as a test of the geometry and reveals at an early date any serious conceptual problems. The disadvantages of such a method are fairly obvious. To test a model have a propeller at an early stage in a design, o-F defined idea as to the range of operation of a well propeller and its desired geometry. simulation, sive all you must the To allow for a comprehen- you must test and record the model data in four quadrants of operation. Since cost is always a signi- consideration, ficant the the machining of a propeller model and operational costs of the set-up sizeable investment in the design. data problem of As alluded to earlier, raw due The experimentally be converted to a form that avoids such must a the to "singularities' of the advance coefficient at zero ship speed and zero shaft rpm. data represent tests suitable for computer simulations not is and determined difficul- ties. The second method of modeling propeller characteristics involves complex and relatively recent work in propeller namics. The Cummings C23 results. As undoubtedly computer models of Professors are quite thorough and provide a "complete" model, at Kerwin some hydrodyand [13 revealing these computer programs are the vangard in propeller theory. With a firm basis in hydrodynamic theory, the continual evolution of these programs will ultimately achieve very accurate results. Unfortunately, these programs are imnensely long and require consi- derable amounts of CPU time. Because of the attendant expense, they are not yet practical for incorporation directly into ship simulation program. 8 a 1 .3 The Objective The objective o-f the Modeling Equations this thesis will be to derive a set rela- o-f algebraic equations that will serve as a simple tively o-f tool for naval architects in the ship design process. By being able to predict accurately propeller per-formance in all rants operation based on Knowledge obtained from o-f experiments the naval and in a form suitable for computer quad- -four limited simulation, architect can save considerable time and expense in the simulation portion of the design. Experimental determina- tion of the points of minimum dissipation would be sufficient to allow for a comprehensive simulation for forward or crash- back maneuvers. Such a set of equations may not eliminate the need experiment in the final stages of the design, for actual however they would greatly simplify the initial Additionally, where experimental data exists for propellers in only the first quadrant, used to accuracy. requirements. particular the equations could be reconstruct the remaining three quadrants with good DERIVATION CHAPTER 2 2. 1 o-f the ALGEBRAIC MODEL Fundamentals of Eulerian turbcwnachines The practice data o-f presenting open water propeller nondintensional as cients versus simply advance thrust <Kt) and torque an absolute advance coefficient <J) coefficient C3],C4] Eulerian turbcxnachines. That is, (Kq) coe-f-fi- (lambda) incorporates which inherently restrict the machines of tions per-formance or assump- interest by considering only to high Reynolds numbers the performance of the machine is governed by Dimensional analysis on the inertia and pressure forces. pression ex- : (Thrust) F=f<'v»p,D,N,p) yiel ds Kt= F/PD4N2=f (^v'p/ND) =f<J) <2.1) Similarly for torque, Kq= T/pD5N2=g<v^p/ND) =g<J) <2.2) where Vp is the speed of advance of the propeller through the water and D any characteristic propeller diameter). performance data This dimension (usually single parameter representation of propeller yields the powerful result that the of geometrically similar propellers will, performance when plotted together, generate to good approximation a single curve. The same assumptions would , from a fluid mechanics stand- point, give Euler's equation of inviscid motion. That is, pv3eGRAD(v) = -6RAD(P) 10 (2.3) Or, more general in a to incorporate unsteady -flows, -form PD<v)/Dt = -6RAD<P> <2.4) where D/Dt is the substantive derivative. In conjunction with equation (2.4), we express mass conserva- tion by: D<P)/Dt where P is independent PGRADJ^v + o-f =0 position (2.5) and v is the vector -fluid velocity. Re-ference C5] chapter 3 and Re-ference 161 are use-ful for a more detailed analysis. A mach ine satis-fying equat i ons experimentally, -fied ed 2. 4) demonstrate Eulerian similitude". That will < is, and (2.5) will, what is as ver i- re-ferred to as geometrically similar machines produce similar per-formance characteristics when presentin properly chosen nondimensi onal a Casanady -form. C73 re-fers to this as a 'per-formance law'. Based on the discussion above, formance done characteristics with derived small -from o-f the determination a scale models. the model o-f large turbcwnachine is The -full scale the per- usually results are results by recognizing that at corres- ponding points on a characteristic curve, all nondimensi onal parameter ccmbinations are also equal. However, there are problems with scale e-ffects not evident in 11 . such single parameter representation. a In addition geo- to metric dissimilarities, such as roughness and clearances, differences are also present. drodynamic hy- These include Rey- nold's and Froude number variations as well as cavitation compressibility number differences. some Attempting to incorporate would of these variations into a performance law or all and require that the law be multidimensional. As an example, one could attempt to formulate a nondimensional representation for thrust and torque characteristics by: Kt=f<J,Re) Kq=f<J,Re) . equivalence of the advance coefficient in Here, the ction with the Reynolds's number would could one Similarly, lead also form yet another to conjun- similitude. expression for thrust and torque Kt=f<J,P) Kq=f<J,P> where P represents a nondimensional ratio for propeller pitch. Such a formulation could be applicable to Kaplan controllable turbines or pitch propellers where the blade pitch is varied mechanical y 1 Multi-dimensional performance laws unfortunately lead to families As an example, a a problem which single parameter laws avoid. of curves, Bovet C8] presents efficiency versus output at constant head for a Kaplan turbine for various tings. An formance entail analytical blade set- treatment for a multi-dimensional per- terms of generating data would a matrix of possible states with dimension equal to the law in 12 numerical number o-f nondimensional characteristic parameters. Obviously, analysis techniques used to generate characteris- dimensional tic nondimensional performance parameters are limited in their applicability in performance simulation. tion To allow incorpora- Mould fluid or geometrical affects into the analysis o-f require another approach. That approach should produce a for- mulation which retains the advantages of similitude yet allows for some fluid or geometrical effects. 2.2 Ideal, canonical and perfect ship propeller equations Figure a propeller blade cross shows 2.1 distance from the propeller hub arbitrary radial condition of zero nominal dicates speed of advance. some at under the Figure 2.1a in- the tangential velocity of the blade element (Vr) fluid velocity (Vf ) <D) section . Figure 2.1b shows the lift (L) and drag and forces with respect to an effective angle of attack a. The magnitude of the effective angle of attack is a function of the circulation which causes an induced velocity [33. If we define a torque exerted by the element as Fr then, Fr= LSin<oc) Further, defining + DCos<a) Ff as a force perpendicular to Fr <2.6) in the direction of fluid motion (fluid pressure term), Ff= LCosCo) - DSinCo) 13 (2.7) 1 Kfr <a) Fi Qure 2 14 . Since L and D are expressed as: L= Cl<i4pAvXv) <2.6) ; D= Cd<J4pAvXv) , and <2.7) can be rewritten: CdJfVr) (2.8) i4PAv(CdXVr - ClXV-f) <2.9) Fr= A^pAvCClX^f F-f= In per unit (see Appendix A), m= v(ClXq + (2.8) and (2.9) become: CdJfn) (2.10) p= v(ClXn - CdJCq) (2.11) Or using the canonical m= + notation, RgJ^q + RdSCn (2. 12) p= Rg3«n - RdJfq (2. 13) where Rg=rq3ev rq=rq(P) v=SQRT(nXn + q3Cq) Rd=rdJev rd=rdO) ^=ARCTAN(q/n) The correspondence with the bond graph representation shown in Appendix B is now readily apparent. The mechanical power can be expressed as Fr^Kfr while the -fluid power can be expressed as F-fX'v'-f. 15 Thus, dissipation losses can . be -Found to be (2.14) Pd= Fr3?Vr - FfXV-f In per unit, (2.14) becomes Pd= m^n - <2. 15) p3Sq Substituting (2.12) and <2.13) yields Pd= RdMvS^v Hence, as would be expected, <2. 16) an no power lost in dissipation have without drag. The equations ideal it i-f would ship propeller could reduce to the generate rather lift trivial expressions m = Rg3?q <2. 17a) p = RgXn <2. 17b) with Rg representing an energy trans-formation modulus. be considered as a general ation o-f separation o-f a number losses. o-f Writing Rd in a more general such losses such that at the point loss, propel Rd 1 combin- loss term that includes a loss mechanisms such as -form is identically zero [93, er 16 o-f minimum we can de-fine Rd can drag -form and typical dissipation a PERFECT Rd = W<P)3f<<n-q)5«<n-q)/v) Using -for <2.18) expression general the Kutta condition to -formulate a the modulus Rg would suggest Rg = rg di-f-fering -from -»^ XO) 3«( <nXn) -<qXq) )/v <2.19) the -form proposed by Paynter by the and (2.19) arbitrariness in into (2.12> and (2.13) and recognizing WO) and XO) rg Substituting which could represent some linear -function in n. <2.18) term the we can arrive at the -following , expressions: m = rgSSq * p = rgXn -^ Kl<P)5Cv<n-q) <2.20) K20) 5^v<n-q) (2.21) where Kl and K2 now incorporate the individual character given The equations above propeller. -for a PRACTICAL re-flect Rl propeller by including an a PERFECT propeller (zero minimum dissipation loss) can be au^nented to a o-f additional represent term to the actual non-zero minimum dissipation loss. m = rg^q + KKP) 3fv<n-q) * R13^n (2.22) p = rgXn -»• K2 ( P) 3C v ( n -q) - R23^q (2.23) and R2 would be -functions o-f 17 the advance coe-f icient J -f -for a given propeller, but in a simpler model could be reduced to constants. As will be seen later, the values a effect on the shape direct o-f the 4-quadrant curves the slope given machine and in particular, and thrust radial s. KIO) and K20) The degree would be identical. on experimental based o-f the model o-f -for o-f the zero torque agreement between Ki and K2 data would be a measure o-f -for advance coe-f-f methodology C 12] . o-f the K<P) data. The by Miniovich CIO] and converted to a modi-fied icient -form in order Cll] was used in an to -formulate a general model be expanded upon in a more detailed analysis. ence determining R2 and rg based on experimental as Rl, produced the validity the algebraic model ina equations o-f propeller characteristics requires a determination data a an idealized propeller equations <2.22) and <2,23) as a model term as well o-f a given rg. -for 2.3 Determination Using Rl and R2 have o-f abbreviated could that See also Refer- Such an analysis would include an investigation of those factors which collectively yield the individual charac- ter of a given machine yet have defineable specific ces, the details of which are beyond the scope of this thesis. In Reference Cll], influen- experimental results for thrust and torque are tabulated for eighteen three bladed propellers with vary- ing expanded area ratios and propeller Table 2.1 pitch-to-diameter ratios. The designations are those used by Baker and Patterson. is provided as a quick reference for EAR values for each propeller. 18 and P/D B3 and B4 in Appendix B are plots Figures series propellers. o-f To be useful, must these curves To do so, rationalized to a consistent basis. data for two rauj o-f the points be o-f minimum dissipation Mere -found for each of the propellers (See Appendix A> and region !> in was fixed B3 the at (quadrant turbine operation and in the region of normal The mapping of the to those points of minimum dissipation is done over the (-1,-1) at entire q vs That is, third the curves. the q vs n plane. Hence pattern image'' of the point of minimum dissipation in the for positive torque and thrust quadrant first in plane recognizing the 'mirror n the curves. into dissipation (quadrant pump operation normal of III) data that the point of minimum such B4 (l,i> Those points were then used to scale figures . reflected is quadrant on the negative torque and thrust scaling is reflected through the origin the rather than simply across the zero torque and thrust radial s. Figures 2.2 and 2.3 are plots of the thrust and torque respectively for three propellers selected for their curves varying expanded area and pitch-to-diameter ratios and are representative the of form of the 4-quadrant curves for all of the propellers. With the data plotted in this form, the similarity of the torque and thrust curves versus the advance angle is striking. In addition, the general form of the curves is quite consistent in the regularity of the specific perturbations which influence all of the propellers in the same manner. 19 Figure 2.2 20 Figure 2.3 21 i 1 12 Prop No. 3 4 5 6 EAR 0.5 0.5 0.5 0.5 0.5 0.5 P/D 0.6 0.8 1.0 1.2 1.4 1.6 7 8 9 10 11 12 EAR 0.8 0.8 0.8 0.8 0.8 0.8 P/D 0.6 0.8 1.0 1.2 1.4 1.6 13 14 15 16 17 18 EAR 1.1 1.1 1.1 1.1 1.1 1.1 P/D 0.6 0.8 1.0 1.2 1.4 1.6 Prop No. Prop No. Table 2. continuing, Be-fore point must I out there that is an approximation which was used which requires an explanation. As Figure B2 thrust (head) not Appendix 8 demonstrates, in in degrees 180 measured clockwise and angle generally computed -for quite -for lines apart. -from The di-f-ference high small. between spec -fie I call speed the skew axial See re-ference [133. o-f the curves o-f machine. were o-f The skew quite 1 is angle were all degree. The o-f losses small, approximated to a PERFECT propeller by the skew angle between the two radial s -found -from the 22 The pumps -flow the magnitude Since the skew angle is radial angle. the propellers used in this analysis skew angle is a -function angle the the first quadrant zero thrust less than 3 degrees and were generally less than amount zero o-f the -first and second quadrants are obviously the second minus 180 degrees skew the the dividing minimum dissipation calculations. The scaling values for q and As recalculated using these adjusted angles. then the and -first The apart. lated third quadrant asymptotes approximation was were n were a result, 180 degrees Justified since the calcu- -felt minimum dissipation angles were obtained with a confi- dence of plus or minus 2 degrees due to the required extrapol- ations in the corresponding portion of the experimental Such approximation also simplifies the derivation of an modeling equations considerably. 2.2 lists and the approximate minimum angles both quadrants for six of the eighteen along thrust in angles. The the propellers with the calculated and approximated magnitudes of and torque radial s at those the Table calculated dissipation angles data. the approximate values are shown in parenthesis. Prop. No. Min. Dissipation angles Radial Length thrust torque 35.2 < 34.6) -146.0 (-145.4) 11.52(8.97) 3.92(3.80) 15.39(14.88) 25.83(22.44) 2 39.5 ( 38.5) -142.5 (-141.5) 5.64(5.02) 4.98(4.59) 13.54(11.32) 25.65(23.88) 3 44.5 ( 44.3) -136.0 (-135.8) 4.95(4.82) 5.02(4.90) 10.54( 9.96) 5 53.5 ( 53.0) -127.5 (-127.0) 5.06(4.78) 6.88(6.31) 9.27( 8.85) 26.99(20.92) 15 45.6 ( 45.6) -134.5 (-134.5) 6.99(6.90) 4.82(4.79) 12.10(12.04) 20.36(19.61) 6.99(6.59) 7.48(7.07) 11.96(11.47) 20.99(18.76) 1 18 58.0 ( 57.8) -122.5 (-122.3) Table 2.2 23 19.99(18.68) . equations Using setting and <2.20) rg equal required to constant (1) -fit and (2.21) for the PERFECT to n, we can plot the experimental and solving the -functions K<P) data by setting m and p to a data Kl and K2 at each -for propeller Since Kl and K2 must be positive at all point. we account for times, sign change in m and p in the different quadrants as the calculations proceed. By plotting Kl and -K2 as functions of we can gain a quick qualitative the normalized advance angle, feel for the similarity of Kl and K2. 2.6 are plots the Figures 2.4, of Kl and -K2 for propellers 9, and 2.5, and 15 18 respective! y As is readily apparent, even with <n) , and Kl functions in a K2 are y/ery similar. response simplified function for rg The behavior of the to changes in K<F) pitch-to-diameter and area ratios is also apparent and are manifest in the expanded amplitude variations and relative smoothness of the plots. the basis of that similarity, On KO) we form a new pair of functions by the straightforward transformation: Ka(P) = <K1 + K2)/2 Kb<P) = (Kl - K2)/2 such that Kl equals Ka+Kb and K2 represents an measure their difference. of average of equals Ka-Kb. the two functions while Hence, Kb is Figure 2.7 is a plot of the Ka a Ka and Kb functions for propeller number 15. A different approach may be to account for the general ampli- tude differences in Kl and K2 on eitherside of the zero thrust and torque radials and themselves. 24 Four gain parameters could u •H 25 IT) (U u en •H 26 O u 3 D^ •H Em 27 <a) 15 225 <b) Figure 2.7 28 be obtained each propeller such that when Ka and -for maximum amplitude of KbO) will be plotted, the reduced. This was done are Kb considerably propeller number 15 by gain scaling -for the average amplitudes as outlined above. Figure 2.8 shcMus the KbO) now approaches what may be Ka and Kb. result for con- sidered as individual propeller nuance characteristics. Since Ka<P) we is essentially doubly periodic over the range of ^ approximate form yet another function that can be used to the entire range of F Kl and K2 valid over KaO) plots of -135 for P < < 45 and By overlaying the . 45 F < < and 225 again averaging we obtain an approximate \«F> function applic- able to both the thrust and torque curves and valid for the that we are two ranges of P stated above. To able retain all of the information available such reconstruct to necessary, we functions are the original Kl and K2 must define four additional formed intervals of -135 < P functions < 45 degrees and 45 < Those functions. by first separating Ka and Kb P < 225 as on the degrees. Labled: We Kal = Ka -135 < P < 45 Ka2 = Ka 45 < P < 225 degrees Kbl = Kb -135 < P < 45 Kb2 = Kb 45 < P < 225 degrees. degrees degrees then define the four functions that represent the average and differences of these functions on the two intervals of P. 29 = <Kal+Ka2)/2 K 1 = <Kal-Ka2)/2 K 2 K =: <Kbl + Kb2)/2 3 = (Kbl-Kb2)/2 K 4 Algebraic manipulation will shcx«i that: -135 < ^ < 45 deg Kl = K +K +K +K 3 4 45 < P < 225 deg Kl = K -K +K -K 3 4 -135 < P < 45 deg K2 = K +K -K -K 3 4 45 < P < 225 deg. K2 = K -K -K +K 3 4 Our approximate KO) -function is now -found from: 12 12 12 12 KO) = <K +K )/2 1 3 valid on the intervals o-f -135<F<45 and 45<^225 degrees. the two functions Kl and K2 were the same and the identical I-f underlying function was responsible for the thrust and torque characteristics on each side of the 4-quadrant asymptote, then K , K , would be zero. In and K 3 1 surronary, K<P) is an average of 4 Kl and K2 which is then symmetrized for the -^ and - torque and thrust curves. Figure 2.9 shcxus the result for number 9 on the interval -PI To the of . < PI. square K(F) characteris- function we can formulate an expansion of Two of the methods which are available involve the Walsh functions and the Fourier series. waves are superimposed and approximate simple sinusoids. ful P determine some of the underlying fundamental tics K<P) < propeller With the Walsh in functions, the limit can be used to They could be particularly use- here since the K<F) function does have ^ery sharp minimums at the 45 and -135 degree angles. 30 The Fourier series is some- -135 <a) 15 Kb -135 45 <b) Figure 2.8 31 225 CM 0) u Cn •rH Cl4 32 . more familar and what -functions, Walsh to the the Fouries series can be used the limit, in As a dual used here. ujas to approximate square waves. By obtaining -finally, Table potential a lists 2.3 propellers 9, the first 15 and this -family -for Fourier -five K<P) convenient point approaches have propellers. o-f coe-f-f zero to begin the expansion. equation we icien ts for The Fourier expansion is taken over 18. function angles the by model KO) -for between the minimum dissipation angles. the interval given Fourier series expansion the where <2.24> The provides expansion defined is c and At those a is by equation basic modeling <2.25) By substituting the K(P) expression into the equations (2.20) and (2.21), we can determined cctfnpare the experimentally thrust and torque curves to the model by the equations over the range of -1 < n < -•I . In plotting doing so, we are assuming that the differences in Kl and K2 are sufficiently to permit small their cxnission. Justification of this assumption lies in the accuracy of the curves generated by the model equations Figures 2.10 when compared to actual and 2.11 shcHfi experimental the results for propeller data. number 18. KO) = c aO -»• = SQRT< SUM< a<k)Cos<kx) a3fa -» b3«b + b(k)Sin<kx> ) <2.24) (2.25) ) 33 Figure 34 2. 10 Figure 2.11 35 Propel er Number 1 abc abc ab 9 15 5.20 6.27 -.26 .64 .64 k«2 -2.02 -1.06 2.28 k=s3 -1.25 .27 1.28 k=4 -.59 -.87 1.06 k=0 5.20 k=l .59 18 • 6.27 10.31 .59 .87 -.23 -2.68 -1.15 2.92 -.57 -.19 .78 -.53 c 10.31 2.98 2.99 -3.27 -1.29 3.52 .60 -1.25 .12 1.26 .54 -.78 .01 .78 Table 2.3 To demonstrate the power o-f the curve resulting -from the model utilizing only the zeroth term the Fourier series expansion o-f is plotted along with the curve plotted using the -first Doing terms. so reveals that the di-f-ficulty -fitting the in proper K -function lies essentially in the dissipation o-f the 4-quadrant curves. the K(F> In a sense, both indicate the torque Additionally, it ple, the a-f-fects are lumped is obvious that quite accurate results KO) assumed -for the For a loss o-f -fit As more terms can be improved in power simulation -for most accurate results for thrust and torque 36 -for turbine operation can to be a constant. expansion are added, dissipation regions. in term. ahead propeller operation and the mechanism and thrust characteristics and KO) be achieved with of is expected since What is interesting is that the that a conwnon dissipation together in the normal regions -function represents a modulating dissipation term the basic modeling equations. plots this three the exam- values would require a large number number o-f o-f terms. The limitation on terms that can properly be added to the lies in the residue o-f the expansion the Kl and K2 functions. Recalling that KCP> was -formed -from an averaging process involving Kl and K2, the -functions which represent their di-f-ferences were -formed to allow retrieval the original Kl and K2 data. o-f that these di-fferences degrade the K<P> To the extent approximation, their impact should properly be less than the last term added to the expansion. Again terms areas required o-f the Walsh -functions may reduce the number , particularly in the normal pump the 4-quadrant diagram. be used to construct the is -f ami 1 n3Cn-^q3?q turbine and The model equations can also ar Kt and Kq versus J curves. Kt given by equation <2.26) and Kq by equation <2.27) simply q/n where R= o-f J . is and THETA=Arctan J. 2 Kt = l./R3{cos (THETA) (2.26) 2 Kq = l./R3ecos <THETA) Here, the R and n re-fer <2.27) to the appropriate values -for the thrust and torque curves. By selecting an n and solving the modeling equations and <2.21) for q with a given KO) , <2.20) quickly Kt and Kq can be determined for any value of J. Figure 2.12 is a plot of the Kt and Kq curves for propeller number 18 using the zeroth term of the K<^) expansion and the first three terms for J Note that the scale factors used 37 in constructing the > zero. rationa- plots lized lating Kt o-f thrust and torque must be considered in calcu- and Kq so that their ranges o-f values are those normally encountered in the literature. The scale -factors used -for propeller number 18 in constructing Firgure 2.12 can deduced from Table 2.2. 38 be Kt .75 >^^^ "^^ ^%^ 1 Term >\ .45 3 Terms Actual \\ .15 1.6 ,8 J <a) lOKq 1.2 1 Term 3 Terms .9 — .3 1.6 .8 J <b) Figure 39 2. 12 CHAPTER 3. 1 As INTRODUCTION of ADDITIONAL TERMS and DETAILS in PROPELLER CHARACTERISTICS 3 OTHER Pitch as a primary determinant the variation be discussed in Chapter 4, will pitch in expressed as a pitch-to-diameter ratio <P/D) has a significant From ler. propel- the performance characteristics of a given on e-f-fect designer's standpoint, a addresses Saunders C33 obtain- pitch-to-diameter selection primarily on the basis of ing maximum propeller efficiency, particularly when utilizing chart data in the initial stages of the series the perspective of the modeling equations however, is more it convenient to see the relationship between the relative to diameter ratio and specific speed. descriptions present meter on From design. References C 143 pitch and the use of the specific speed C 15] para- . Specific speed variations have a significant effect on the quadrant diagram not easily seen in the propellers 4- presented here. Since propellers are in general '^ery high specific speed the impact of variations of that parameter is some- machines, what obscure. ions on The principle effect of specific speed variat- the 4-quadrant diagram was only touched on in the plots of the thrust and torque characteristics discussed ear- lier. It is seen primarily in the location of and torque radials of Table 2.2. speed increases, As the the radials tend to rotate the zero thrust relative specific counterclockwise while the skew angle addressed in Chapter 2 tends to decrease. 40 How variations in pitch a a-f-fect the specific characteristics of propeller is best seen in an analysis of the performance of controllable, <CRP) Reference . is quite thorough in examining quantitatively the 1161 of pitch propellers reversible impact pitch on the efficiency and thrust and torque coefficients of a CRP propeller. is seen in the modeling equations as a factor in the determination of the amplitude of the function. An increase in the pitch-to-diameter ratio has dominant K<P) It two noticeable effects. tive amplitudes results which the various harmonic components of in First, it tends to increase the rela- a somewhat wider fluctuation KO) in the of 4- quadrant curve in the principle dissipation regions. Secondly, it has a direct affect on the zeroth term which represents the average value of the K<P) fun- expansion The net effect is ction. (corresponding Obviously, an to a larger shrinking of the 4-quadrant curves torque and thrust coefficients). the 'dc' value of the expansion is dominant since examination of Figures 2.2 and 2.3 would indicate that the harmonic content of K<^) cally all Fourier the of in quadrant I'v' in the dissipation regions, is apparently proportionally specifiequal for three propellers. 3.2 Number of blades and blade area The effect of variations in numbers of blades was not examined in detail. tion of However, as pointed out in Chapter 4, the formula- the modeling equations was quite general 41 and utilizes what is recognized as the actuator disk theory to obtain their initial relationships. A comparison o-f 4-quadrant diagrams -for three bladed and seven bladed propellers has not indicated any potential problems. the amount blade area expressed here o-f Yet Figures B4(a> ratio. to Mhat is really the primary determinant is , B4(b> discount any major influence , area expanded as 2.2 and 2.3 would seem even o-f EAR on torque and thrust characteristics in general. cavitation 3.3 E-f-fects o-f Cavitation is probably the most dif-ficult aspect Rightly so, operation with which to deal. a lot o-f propeller o-f time being expended to describe the mechanics of cavita- energy is Referen- tion and its influence on propeller characteristics. ces C4], C 16] , £17], quantitatively the ance. Reference C ler to and 16] CIS], and C 19] explore qualitatively and perform- role of cavitation in propeller relates the speed of advance of a propel- the cavitation number. By defining the cavitation number (sigma) with equation <3.1>, sigma=2g<h-i-hA-hW/<v'a3^*^a <3. 1) where g is the acceleration due to gravity, h propeller hA the atmospheric head axis beneath the water surface, pressure, propeller hV the depth of the the vapor pressure of water speed of advance, it 42 is shown and that (using Ua the typical . . values) Sigma is related to Va by the relationship of equation <3.2) si^a= 1070/VaXVa How the cavitation number with Va in Knots. is seen in Reference C4], Chapter of (3.2) 16, a-f-fects Kt Figure 108 as a fall off Kt and efficiency as the advance ratio decreases. appear and Kq It that the effect of changes in sigma for a higher J smaller, also seen in Reference C does is 16] Yet nowhere is the role of cavitation included directly in the modeling equations. Since the intention of the modeling equa- tions is to provide a reasonable method of determining propeller characteristics at an early stage of a ship design say for the purposes of simulation, perhaps an extension to cavitation affects is unwarranted. include However, we can anticipate where such affects may be present. Equations (2.22) and (2.23) can be modified to indicate eral In gen- relationship to cavitation number shown below. m=rg(si^a)Xq + Kl(P,sigma) 5^vX(n-q) + Rl(sigma)3(n (3.3) p=rg(sigma)36n + K20,sigma) - R2(sigma)3«q (3.4) some the presence of cavitation is handled by cases, 3^vX(n-q) an effective reduction of the propeller blade area. This could be incorporated dissipation into the K(P) functions. However, the greater would alter the 4-quadrant zero thrust radial s to 43 hence, some extent, that Rl is also a function of sigma Notice case. the Rl and R2 terms are included in this since the losses will be reflected from the hydraulic to mechanial side. Although the potential affects, such of cavitation assumed a step may be beyond the scope of the equations. model is there for inclusion Again, intent of the equations is the a relatively simple means to simulate propeller provide formance for first-cut design purposes. that goal Mill Any extension to per- beyond increase the complexity of the equations rapid- ly. 3.4 Hul /propel er interactions 1 Two 1 of the familar parameters which reflect hull/propeller < t) interactions are the thrust and the wake fraction <w) consequence velocity the influence deduction of factor The wake fraction appears as . a of the disparity in ship forward velocity and the as seen by the propeller. The wake fraction is de- fined as: w = where Kf <»v'-Va)/V is the ship velocity and Va the velocity that the propeller sees. The thrust deduction factor the propeller through results from the requirement to produce enough thrust to a resistance of Rt propel the of ship while also reducing the high pres44 sure area over the ship's stern. De-fined as: t = <T-Rt)/T T is the net required thrust to propel where the ship at a given speed. Values for both -factors are usually obtained from over little modeling vary of the the approach was one of considering the C 16] would indicate that wide range of ship speeds. In in an open water condition and actuator disk approach essentially taken here, and t terms hence, w were not included or necessary. and t a equations, propeller for ui Reference tests. model corrections Owing to the allowances for complicated inflow fluid geometries due to disturbances ahead of the propeller have been ignored. Reference C 16] also addresses the pitch dependency deduction factor and gives general ranges of thrust ratios of -^ poration of the thrust deduction and wake fractions detai 1 1 to -1. of the for P/D t References C4] and C20] address the incorin more . 3.5 Other considerations As formulated, many of the modeling equatins simply do the fine details of propeller design. however, including blade rake, skew-back, edges, details hub fairing which not Most of the requirements 45 these the shaping of blade and blade profiles are in large go beyond address for the measure initial stages isics o-f Once the 'macroscopic' character- propeller design. o-f type is determined, a generic propeller cavitation and blade loading requirements speci-fic examined but only in the detail or can o-f be tests by actual model allcMAied in propeller -frcwn the begin- a complex computer program specializing by the impact hydrodynamics. A tacit assumption which has been incorporated ning is that the nondimensional tainable advance -from per unit quantity q is , the dimensional values Va and in turn, There is, hcwuever , a term called the axial directly accounted speed o-f the -flow rate through the propeller. which is included in the momentum theory not propeller o-f ob- -for in-flow -factor a C4] -for propeller the calculations. in It action has been assumed to be a quanti-f iabl e constant. The e-f-ficiency the behavior calculation o-f o-f o-f a given the K<P) propeller and its relationship function has not been the e-f-ficiency o-f a given explored. machine is to The straight- forward, but the qualitative relationship of the efficiency to pitch-to-diameter or expanded area ratio equations is yet to be determined. dissipation fran is lish or in a modeling That the points of minimum points of best efficiency can experimental data or inferred from other accepted, the be determined considerations but the use of the modeling equations to estab- 'best' efficiency machine given specific requirements is a topic that begs further work. 46 operating CHAPTER 4. PREDICTION o-f PROPELLER PERFORMANCE MODELING EQUATIONS 4 THE Comparison with very similar propellers 1 only a few Ni th the eighteen propellers available o-f igated in detail regarding the speci-fic theless o-f USING the -family o-f propellers. peller numbers 9, As seen by a comparison and 18, 15, o-f o-f overall. The plots o-f e-f-fect Appendix B. o-f to 1.0 1.6 the Kl and K2 -functions in the dissipa- regions while slightly increasing the average tion EAR -function retains the P/D ratios o-f pro- o-f Intuitively at least, the higher EAR 0.8 to 1.1. to smooth out KO) the same general shape over the range appears -functions, never- is possible to extend the results to the remainder it and EAR'S KO) invest- magnitude EAR is more easily seen in the raw data In Figure B4(b) has its most pronounced e-f-fect -for example, at ^ equals -i-PO the higher and -90 degrees. This is also shown in Figures 2.2, 2.3, 2.4, and 2.5. A point by point comparison will also indicate that the variation EAR has little in e-f-fect on the thrust other than at those two particular points o-f curve. Seen more clearly in Figure B4<a), it Figures 2.2 through 2.5. what obvious turbine region the 4-quadrant is rein-forced in The expanded area ratio has as some- larger e-ffect on the Kl curve (torque), the normal characteristics o-f particularly in the 4-quadrant curve. This is not Figure B4<b) where the plot the raw data would indicate that EAR has essentially no e-ffect outside the prin- in ciple dissipation regions. 47 o-f The e-f-fect o-f Beginning with the raw data obvious. see the e-f-fect ination o-f An exam- contrary to intuition, the closer to the origin corresponding the higher the relative magnitudes Ct and Cq 2.5 and 2.6 reveal nearly we can Figure B3(a), the analysis. of P/D ratio throughout that a given point lies, Figures o-f Figures 2.2 and 2.3 show this in-fluence quite well. Note however, the more changes in pitch-to-diameter ratio is much coe-f-f icients. comparison A o-f o-f that the in-fluence of P/D ratio is the same for both the thrust and torque characteris- tics. Table 2.3 points out that the various principle frequen- cies of the K<P) average 'dc' function change in amplitude as well as The substantial variations in term. the peak the amplitudes of the K<P) function are also evidenced in the of Appendix B. data are intriguing. B3(a) in The apparent critical points raw Figure in The increase in P/D ratio is reflected the relative specific speeds of the various propellers is the demonstrated in the gradual and counterclockwise rotation zero thrust and torque radial s. Those apparent of critical points must lie along the pivot axis of that rotation but the reason for the correspondence isn't clear. By examining Tables 2.2 and 2.3 and Figures 2.2 through 2.6 as the figures of Appendix B, well as with \fery rating the is possible to reasonable accuracy the appropriate values points and the fundamental values of propellers of the data base. thrust it With that KO) infer of for all knowledge, and torque plots could be generated and compared Figures 2.2 and 2.3. 48 the of the with By plotting the higher harmonic amplitudes versus the a ccwnparison can be harmonic amplitude, zeroth made between three o-f propellers examined in detail regarding the K<P) -function, the Figure the amplitudes of shows 4.1 propellers number 9, would ratios area suggest particular a for By extrapolating along the and 18. 15, for a propeller whose pitch-to-diameter curves second and -first against the zeroth harmonic amplitude plotted harmonics the and expanded zeroth harmonic amplitude, we can approximate the values of the Fourier series expansion terms for the first two harmonics. As propeller number example, an has 1 a pitch-to-diameter ratio of 0.6 and an expanded area ratio of 0.5. zeroth amplitude is equal harmonic to four, Assuming the first two the harmonic amplitudes can be inferred by extending the curves to the value appropriate modeling series against the curves. Figure zeroth harmonic. with those amplitudes of equations Fourier the of expansion terms, experimentally a Using respective the comparison can determined thrust the made be and 4.2 shcxus the result for the positive torque thrust curve. Owing to the influence of zero, some curve. values of is expected in that region of error However, the the higher harmonics near J as K<^) Figure 4.2 indicates, expansion terms yield the equals thrust even extrapolated quite reasonable results for the of the curves. The value of the zeroth harmonic (first term of the Fourier series expansion) has a substantial normal pump and 49 turbine portions influence in the 1 8 5 Zeroth Harmonic Amplitude <a) Zeroth Harmonic Amplitude <b) Fi gure 4 50 . Actual Model Figure 4.2 51 neighborhood higher order equals zero equals zero and acts with J o-f neglected the terms to produce part of the error near values Figure 4.2. o-f With more the study J the o-f trends in the harmonic amplitudes with pitch-to-diameter ratio better approximations expanded area ratio, and and torque curves experimental the o-f data. error in Figure 4.2 is in the plot The scaling which was produce the rationalized data depends on the values at the point A o-f to and q n o-f those are points, icient y so to cause errors much like those of Figure 4.2. 1 comparison weakness o-f the curves Figure 4.2 also points out o-f the sharper peaks in the curve number o-f reproduce terms o-f a Fourier series expansion the experimental simulation quadrant large is required to data in the -fourth quadrant dissi- pation region. A piecewise continuous approximation -fourth to gener- by the modeling equations indicates that a -fairly ated a expansion methods in -forming the K(F) approxi- The absence mation. o-f the Fourier series representation as opposed o-f other possible the done Those scale -factors to errors in the location sensitive quite su-f -f minimum dissipation. o-f the thrust similar propellers will be possible. o-f Another important source o-f o-f would provide improved o-f K<F) in results -for a requiring better correlation to experimental during say, transient analysis. 52 data 4.2 Extrapolation to other propellers the modeling equations are quite As developed in this thesis, and although they general cular set o-f three they *r# equally Appli' aded propellers, b1 cable to other types sents 2-quadrant data propellers as well. o-f -for parti- their establishment to a otAie a -few seven Hecker [21] prebl propellers aded including a hybrid. Plotting the experimental data in the same as with the three bladed machines, manner the similarity o-f -form o-f to . o-f the experimental that generated by the modeling equations using only the various parameters was quite suc- approximate cess-ful earl- the curves to those presented ier. Although not shcxun here, a comparison data one is struck with values -for the principle areas Although dissipation regions as expected, o-f error were the the comparison did not indi- cate any serious problems with the applicabilty ling equations. A in more detailed comparison the o-f the o-f moderesults obtained from the modeling equations to much di-f-ferent propellers is yet to be done. 4-quadrant Considering experimental data -for the scarcity a large variety o-f types and the limitations inherent in this -form tion, such a ccMnparison wasn't possible. However, o-f the equations to date would tend to en-force -for their wide ranging applicability. 53 o-f complete propeller presenta- o-f the success the argument 4.3 Extrapolation to missino quadrants of the advantages One o-f the modeling equations lies in their ability to reconstruct essentially the entire 4-quadrant diagram using only partial information. Particularly when suffic- mini- ient information is available to determine the point of mum dissipation, a "universal" K<^) function can be scaled to fit the rationalized data and the appropriate values of and ^v'p ND at the point of minimum dissipation substituted to generate the nondimensional either Ct and Cq values or values of thrust and torque directly. If only the first quad- rant point of minimum dissipation is known, can be approximated with little loss of particularly model the dimensional the third quadrant accuracy. This is useful where first quadrant data exists but the from which the data was taken is no longer available. Extending the concept slightly, a comparison of available data on the rating points of various propellers could form a base from which the approximate rating point of machine could be inferred. mental with specific Thus, if only ^ery limited experi- results are available, constructed a data the 4-quadrant curve could some confidence by comparing the modeling equations results with whatever information is available. 54 be SUMMARY CHAPTER 5 An overview o-f the method The application o-f Eulerian turbcwnachine theory to the 5. 1 simulation and ing history. success-ful o-f long and While the particular method used here more recent, somewhat pump/turbine systems has a model- this thesis represents an initial is at- tempt to apply the methodology to high specific speed extended Using machines. o-f the the model que, general. in equations were able mechanisms which contribute to tor- overall losses of the system can be seen to ''reflect' through the transformation of rotational mechanical energy. The plotting of the fluid to the i.e., energy and under- behavior out a common dissipation mechanism for thrust and point the importantly, area this procedure was able dissipation mechanisms of propellers more Perhaps ratios as a basis, a -family -for expanded some of the important fundamental reveal lying experimental data bladed propellers with varying three pitch-to-diameter to y/ery ccwnplete transl at ional to KO) function for different propellers showed their strong correlation in form, a correlation which could in a more thorough examination be seen in the relative magnitudes of the components of the expansions. Certainly, Fourier series the zeroth component of the expansion represents an important parameter since a quite reasonable fit can be made over much of the four quadrants utilizing it solely in the determination of the model 4-quadrant diagram. This method also introduces to the simulation 55 of propeller . characteristics a procedure values di-f-ficulty with singular reduces o-f avoids characteristics relatively lend they to only two, simple and basically th^nselves to computer the o-f the advance coefficient. both complete which of It are The model equations reasonably well behaved and quite smooth. are most number of required curves to represent the propeller which straightforward. manipulation Thus, little with dif f icul ty From the standpoint of the ship designer, ship dynamics initial ly which include a realistic the simulation propeller of the in stages of the design would be much easier and certain- more cost effective. A generic set of equations with the specific values of a given propeller rating point incorporated in them would be sufficient to provide reasonable results the initial steady design stages which are oriented more toward near state ahead and astern operation than a thorough simu- lation over all is in determined, quadrants. Once the geometry of the propeller the equations can be updated to provide in- creased accuracy. That accuracy depends on the number of terms of the expansion one is willing to incorporate into the simu- lation program. necessary test until Thus, a scale model of the propeller is the need arises for a self-propelled not model or a detailed investigation is required into the cavita- tion or vibration characteristics of the propeller. 56 5.2 The potential Since -further applications -for thesis represents an initial examination this applicability o-f propellers, Eulerian turbcwnachine theory to little has been addressed regarding a quantitative o-f various propeller types. It ccxnparison was demonstrated data bladed base -for this analysis could be applied propellers as well but only to the extent the that results obtained for the three bladed propellers which the the oi -formed to seven a quick o-f qualitative look for comparison purposes. There is a great deal that can be done to enhance the useful- ness of the modeling equations. K<P) particular, values for the In function could be tabulated based on the examination of available propeller data for various combinations of governing parameters. series For example, the first few terms of the Fourier expansions could be tabulated for given pitch-to-dia- meter and expanded area ratio combinations and used in a table look-up scheme such that a ship designer wishing to simulate a given propeller point and would need only select a combination of P/D and EAR rating particular The ratios. computer program could then select the necessary coefficients and ceed with the computations. Or, if a pro- comparative study of the rating points of various well designed propellers used in ship propulsion lished such were to be conducted, a data base could be estab- that a ship designer would need only generic type of propeller to begin a simulation. select a The computer would determine the appropriate rating point and corresponding 57 coe-f -f icients to initialize the program. The -form useful the modeling equations make o-f applications to -for propellers <CRPP) . system control use-ful reversible controllable, Not only are they convenient simulations, but they could -form synthesis. particularly them -for pitch conducting the necessary -framework system Such a control -for could a be automated propulsion systems particularly involving in CRP propel ers. 1 There are involving small other high harbor propellers craft K^ery applications including low speed screws like those used thrust, with uses potential tugs. or high The skew applications rates or propellers are also worthy of investigation. in to exotic perhaps hybrid Yet another use involves an examination of supercavi tat ing propellers and they modeling for simulation. With the promising results obtained here, follow-on efforts could be quite rewarding. 58 Essentials Appendix A o-f Eulerian Turbomachines a 'black-box' representation, In machine modeled as in Figure Al be can parameters are torque (t), velocity <'v') , propeller like any turbo- a rate o-f sha-ft and fluid flow rate <F) where , fluid an IDEAL rotation (N) Ne define . various the propeller as one that, in the process of power conversion from mechanical rotation to axial fluid flow, experiences no los- ses. Such a propeller may have little connection with reality, however the analysis of such a machine can provide qualitative insight into the mechanics of the energy transfer more comfortable process. model would include the effects of A various types of losses into a PRACTICAL propeller model diagrammed as in Figure A2. The losses can be calculated by equation Al, Pd = T^N - PMV where represents the power Pd <A1) dissipation. The amount of varies with the advance ratio such that two mini- dissipation mum will occur, one in the range of normal pump operation and the other approximately 180 degrees away in the four quadrant diagram. Using the positive semi-definite dissipation function described by Paynter C9], the equation for calculating power dissipation from the experimental al form) the data <in nondimension- is found to be simply: 6<^) = 2ltCqVp - CtJAp 59 <A2) "v* Figure Al LOSSES <> p u N Fiqure A2 60 dimensional same results will be obtained using either a The or a nondimensional By plotting angle the dissipation to the arctangent o-f J, at which function. the dissipation o-f the losses will angle o-f square root the equal -formulation versus the the quadratic nature result in a nearly linear graph such that o-f the the minimum dissipation occurs is easily dis- cernabl e. Figure A3 shows the behavior o-f the advance ingly enough, occurs sion scaling o-f -for the dissipation as a normal o-f (Interest- "del', the smaller turbine operation.) The exten- PERFECT Eulerian turbomachine a o-f comes to eliminate the appropriate amount -for -function 1. propeller number the two minimum values that the expression to icient the region in to coe-f-f o-f the general o-f proper -from dissipation so dissipation -function goes zero at the prescribed operating point (best e-f-ficiency or minimum dissipation). The PRACTICAL Eulerian turbomachine then supplements this dissipation function by an amount equal to the minimum loss. Per as Unit variables as used throughout this paper are the nondimensional predetermined ratio of a parameter base value. For a pump, value defined some to those base values are taken to be at the point of best efficency or point of minimum dissipation. obviously the Mhen presenting such data in a per reference operating point must reconstruct the original experiment. 61 unit be form, known to < u •H fa 62 . per unit variables has significant Using they allcxM free manipulation o-f easily o-f handled numbers such as the base values. the derivation dimensional 1 the per selection and -1 by proper -feel -for their behavior. propeller data which provided the basis -for the propeller equations was already in a o-f -form o-f Thus the per unit values give a qualita- tive as well as quantitative Since -^ o-f neighborhood can be made to range in the variables First, the applicable equations with- Secondly, the values out regard to dimensionality. unit advantages. such that the physical dimensions o-f the non- Minio- vich's propellers was no longer readily apparent, the analysis was done in a non-dimensional -form to retain its generality. This may prompt one to conclude that the computational results are somewhat arbitrary, and D to derive all the paper, however one need only select , N, the appropriate values. As utilized within the per unit values are -found by: p=P/Po 'v'p, t=T/To , n=N/No 63 , q=F/Fo , etc Selected Experimentally Determined Propeller Data Converted to ^4-Quadrant^ Form Appendix B data The by Miniovich CIO] and converted presented modified advance quadrant form coe-f-f coefficient) (torque shown in Figure Bl. constant in polar form coefficient) utilizing the the to 4- and Cq convention This is possible by recognizing that at a thrust or torque (which the Karman-Knapp Circle dia- utilizes), gram icien t -form Cll] was converted plotting Ct (thrust by to defining relations used the by Baker and Patterson shc^n below: Ct » T / pD2(^Jp2+N2D2) Cq = Q / PDSCv'pS+NaDZ) a = Arc tan »v»p/ND can be rewritten in per unit form as: R = 1 / SQRT( ABS(Ct) ) for the thrust curve R = 1 / SQRT( ABS(Cq) ) for the torque curve and a = Arctan q/n and As a result, -1.5<J<-0.667 the and overlap of .667< 1/J< 1 64 .5 for both. the data in the ranges of act as a partial check by \ plotting on top o-f each other. This kind of representation o-f the data yields in two closed curves the ccxnp-lete characterisof the propeller. tics method Wylie and Streeter [223 address this applied to solution of transient problems using as An additional digital computer. a advantage is the avoidance of the problems with singularities where ever propeller speed of advance or shaft rpm approaches zero. See also Reference [233. The curves are speed, fic axial describes also easily recognizible as typical for high speciflow pumps. See Reference C24] where Knapp the 4-quadrant representation pumps of and and outlines a procedure by which the representation turbines could be used to deduce the transient behavior. Figure B2 is a sketch of the flow vs speed plane indicating various the regions of operation. The zero head and torque radial s are the asymptotes to which the thrust and torque curves approach for a given machine. Two series are shown here. of Figures B3<a) and B3(b) are thrust and torque for propellers 1,3, and B4<b) 15. The plots and 6. Figures B4<a) are thrust and torque curves for propellers 3,9 and plots reveal the effect of variation diameter ratio and expanded area ratio. 65 in pitch-to- Figure Bl II <+) Discharge .Dissipation DISSIPATION NORMAL PUMP ( -> Speed <+) DISSIPATION DISSIPATIOr^'^ Zero Head Zero Torque NORMAL TURBINE III Speed <-)Discharge Figure 02 66 l\> Figure BSa 67 Figure B3b <68 Figure B4a 69 Figure B4b 70 Appendix C Paynter C9] The Bond Graph Representation presents the canonical bond graph representation for an Eulerian turbomachine as: GY RG Rq Rui It is also represented as a modulated qyrator series adjoined to a shunt resistor: Rs In the -first case, the system equations are easily -found to be: mssRgJfq-t-RwJ^n <DI) p=Rg3fn-Rq3Cq <D2> using the pump sign convention. 71 In the equations are -found to be the second case, m = Rg X q + Rs X <n-q) <D3) p — Rg 3C n - Rs <n-q) <D4) By converting <D3) 3f to canonical and <D4) it -form, is evident v where that Rs can be expressed as K<P) 3C V = SORT ) <D5) qXq)/v3{v <D6) ( nJfn + qXq and Rg = R6 A great deal Rs 3« <n3^n + simplification is a-f-forded by making the simple approximation quality KJ^v o-f + with little loss in o-f where K absorbs the individual character Rg = o-f a n and Rs = particular machine. As a more widely applicable model, be general dissipations could adjoined to the l-junctions to account for -fluid and mech- anical losses, i.e. Rdq Rdn 72 The equations in matrix form: m A Rs+Rdn Rg-Rs <D7) Rg-Rs Rs-Rdq <D8) more complete model Mould leave the dissipation elements as •functions individual of the advance coefficient to absorb part behavior of a machine and smooth out the of the require- ments in the Rs term to fit experimental data. The tion reader interested in pursuing the bond graph is representa- directed to Reference C25] for an 'entry level' proach to bond graphs and to Reference C26] for a more ced treatment. 73 ap- advan- LIST o-f REFERENCES "A Design Theory -for Superand Leopo1d,R., CI] Kerwin,J.E., 72, 1964 cavitating Propellers', Transactions SN(ii#1E, 'v'ol . "Numerical Prediction o-f Propeller Characteristics", Journal of Ship Research, SNAME Mar. 1973 [23 Cunwnings,D.E. , Vol. "Hydrodynamics in Ship Design C3] Saunders,H.E. Society o-f Naval Architects and Marine Engineers, 1957 , "Principles o-f Naval Architecture", Society Naval Architects and Marine Engineers, 1967 [43 ComstocK , J.P. o-f 1", , o-f "Mechanics C53 Duncan, W. J., Thorn, A. S., and Young, A. D. Fluids", i^erican Elsevier Publishing Company, New York 1970 , "Physical Fluid Dynamics", C63 Tri tton ,D. J. Reinhold Ccwnpany, New York 1977 'v'on , [73 Csanady ,6.T. "Theory o-f Turbomachines" Series in Mechanical Engineering, 1964 , [83 Bovet,G.A., TR^V^4S. , Nostrand McGraw-Hill ASME, 75:975<1953) [93 Paynter ,H.M. "The Dynamics and Control o-f Eulerian Turbomachines", TRANS. ASME, Sept. 1972, pp. 198-205 , "Investigation o-f Hydrodynamic Characteristics o-f Screw Propellers Under Conditions o-f Reversing and Calculation Methods for Backing o-f Ships", BuShips Translation Translated by Royer and Roger, Inc. International 697, Division, Mash. D.C. 1960 [103 Miniovich ,Y.A. , [113 Baker, D.N., and Patterson ,C.L. "Representation o-f Propeller Thrust and Torque Characteristics for Simulations", Naval Ship Research and Development Center, Annapolis Division Rpt MEL 202/67, Apr. 1968 , [123 Pubis, C. J., and Moken,A.J., "Four Quadrant Open Water Performance o-f Propellers in Terms o-f Modi-fied Advance Coe-f-ficients". Naval Ship Research and Development Center, Annapolis Division Rpt 6-175, May 1971 [133 Donsky,B., "Complete Pump Characteristics and the Effects of Specific Speeds on Hydraulic Transients", Journal of Basic Engineering, TRANS. ASME, 1961 [143 Kovats,A., "Design and Performance of Centrifugal and Axial -Flow Pumps and Compressors", MacMillan Comp., New York 1964 74 Propulsion Anal"Speci-fic Speed Method o-f Aug. 1954 Journal, 727, Engineer's Naval pg [15] SI ocum,S.E. ysis ", , "CRP Propeller Ship Propulsion Vol.1', NSRDC rpt 6-10 1, 3238, Annapolis, Maryland [16] Pubis, C. J., Dynamics Feb. 1971 "Per-formance Characteristics o-f Three Prop[17] Peck, J, G., ellers With Varying Pitch Distributions on an Inclined Sha-ft", NSRDC rpt SPD-497-02 Bethesda, Maryland Aug. 1974 O'Brien ,T. P. "The Design o-f Marine Screw [18] London, Huthinson and Co. Ltd. 1962 , [19] Wisl icenus,G.F. , Propellers", TRANS. ASME, 78:1707(1956) and Johnson ,R.D. , "Propulsion Dynamics Simu[20] Moken,A.J., S^sl 685 Submarine Using a Direct-Current Eleco-f the lation Ship Research and Development Center, Drive", Naval tric Annapolis Division, Rpt c27-823, Sept. 1974 "Two-Quadrant Per-formance o-f Various PropelBethesda, lers", Naval Ship Research and Development Center, M.D., Rpt 386-H-35 Sept. 1973 [21] Hecker,R., Streeter ,V.L. [22] Wylie,E.B., and McGraw-Hill International 1978 , , "Fluid Transients", [23] Marchal,M., Flesh, G., and Suter,P., "The Calculation o-f Waterhammer Problems by Means o-f the Digital Computer", Proceedings, Pumped International Symposium Waterhammer Storage Projects, A^E, Chicago, Nov. 1965 [24] KnappjR.T., "Ccwnplete Characteristics o-f Centri-fugal Pumps and Their Use in the Prediction o-f Transient Behavior", TRANS. ASME, Vol 59, Nov. 1937, pp683-689 . [25] Karnopp,D., and Rosenberg, R. "Analysis and Simulation o-f Multiport Systems, The Bond Graph Approach to Physical System Dynamics", M.I.T. Press, Cambridge, Mass. 1968 , [26] Paynter ,H.M. "Analysis and Design Systems", M.I.T. Press, Cambridge, Mass. 1961 , 75 o-f Engineering Thesis W58823 c.l 138-^87 Wllken An algebraic propeller model using Eulerian turboma chine theory. 199'-f67 Thesis W58823 c.l Wilken An algebraic propeller model using Eulerian turbomachine theory. thesW58823 ^1«™ '^'' ^'°^'^^^' "^°del using Euler 3 2768 001 95812 7 DUDLEY KNOX LIBRARY

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