14- tesla turbomachinery

14

Tesla Turbomachinery

Warren Rice

Arizona State University, Tempe, Arizona, U.S.A.

Tesla-type turbomachinery is distinguished by the fact that the rotor iscomposed of at parallel corotating disks, spaced along a shaft. Athroughow of uid between the disks results in momentum exchangebetween the uid and the disks and hence shaft torque and power. Tesladesigned, built, and tested such machines but was not able to achieveindustrial applications of them.

In the subsequent years, many investigations have been carried out todetermine the performance and efciency of this type of turbomachinery.These have been both analytical and experimental. Most of the investiga-tions have had a certain limited application as the objective, with regard tosize and speed as well as the nature of the operating uid. However, someof the investigations have tried to establish the generalized performance ofTesla-type turbomachines. In general, it has been found that the efciency ofthe rotor can be very high, at least equal to that achieved by conventionalrotors. But it has proved very difcult to achieve efcient nozzles in the caseof turbines. For pumps and compressors, efcient diffusion after the rotorhas proven difcult to achieve. As a result, only modest machine efciencieshave been demonstrated. Principally for these reasons the Tesla-type

Copyright 2003 Marcel Dekker, Inc.

turbomachinery has had little utilization. There is, however, a widespreadbelief that it will nd applications in the future, at least in situations in whichconventional turbomachinery is not adequate. This includes the use withvery viscous uids, uids containing abrasive particles, and two-phaseuids.

The ow in the rotor may be either laminar or turbulent, with certainadvantages and disadvantages in each of the regimes. It is necessary tocarefully distinguish which type of ow is considered in reports by thevarious investigators.

Herein, the principles of the Tesla-type turbomachinery are reviewedand the problems with nozzles and diffusers (which limit the machineefciency) are discussed. The analytical methods that have been founduseful in modeling and calculating the ow in the rotor are reviewed, and theexperimental results obtained by some investigators are described. Anextensive list of references is presented, and summaries are made of the mainresults and conclusions that have been reported.

Nomenclature

h Spacing between disksr General radial coordinate in space between disksri Inner radius of diskro Outer radius of diskQ Volume ow rate between two disksuu Average (disk-to-disk) radical component of velocityvvo Average (disk-to-disk) tangential component of velocity, at

outer radius of diskn Kinematic viscosity of uidm Viscosity of uidO Angular velocity of pair of disksr Density of uid

INTRODUCTION

Tesla is most widely recognized for his outstanding contributions in theelds of generation, transmission, and utilization of electrical power.However, he was also the inventor of a unique type of turbomachinery thatcan be applied as liquid pumps, liquid or vapor or gas turbines, and gascompressors [1]. The Tesla turbine was widely discussed in the semitechnical


press at the time of the invention [27]. The Tesla turbomachinery isdistinguished by the fact that the rotor is composed of at parallelcorotating disks spaced along a shaft. A throughow of uid between thedisks results in momentum exchange between the uid and the disks andhence shaft torque and power. Tesla designed, built, and tested suchmachines but was not able to achieve signicant industrial application ofthem. Figures 1 and 2 are schematic diagrams of a Tesla-type pump andturbine.

As a turbine, the multiple-disk rotor is contained in a housingprovided with nozzles to supply high-speed uid approximately tangentialto the rotor. The uid ows spirally inward and nally exhausts from therotor through holes or slots in the disks near the shaft. As a pump orcompressor, uid enters the rotor through holes near the shaft, ows spirallyoutward, and exhausts from the rotor into a diffuser such as a volute scroll.Shaft power is delivered by the turbine, and shaft power must be supplied tothe pump or compressor. The performance and efciency of the rotor ofsuch machines are dependent on the combination of shaft speed, disk innerand outer diameters, spacing between the disks and the uid properties, andow rate. The performance of the overall turbine is strongly dependent alsoon the efciency of the nozzles and the nozzlerotor interaction. Theperformance of the pump is also strongly dependent on the interactionof uid leaving the rotor with that in the volute and on the efciency ofdiffusion in the volute. In general, ow in the rotor may be laminar orturbulent or mixed.

There was little activity concerning the Tesla turbomachinery until arevival of interest began in the 1950s [817]. Research was widespread afterthat, particularly in the area of analytical modeling of the ow in the spacebetween two disks in a rotor, and continues at present. References [1879]are representative. Many attempts have been made to commercialize Tesla-

Figure 1 Schematic diagram of a Tesla-type turbine.


type turbomachines, especially as pumps, but no widespread applicationsare apparent. While rotor efciencies can be very high in this type ofturbomachine, there are inherent losses in the uid ows entering andexiting the rotor. As a turbine, the nozzles are necessarily long andinefcient. As a pump or compressor, the diffuser or volute must handleow with a very small entering angle, which causes the volute to be veryinefcient. For these and other reasons, actual Tesla turbomachines haveefciencies much less than might be expected from consideration of the owin the rotor. There is little or no literature devoted to the ows that cause themain losses in Tesla-type turbomachinery. Tesla-type turbomachinery isvariously referred to in the literature as multiple-disk or friction or shearforce (or, nonsensically, boundary-layer) turbomachinery.

MULTIPLE-DISK ROTORS HAVING LAMINAR FLOW

A way of analytically modeling the ow in a multiple-disk rotor known asbulk-parameter analysis was used by early investigators [7, 911, 16, 17, 21,24, 49]. It is usable for both laminar and turbulent ow. In this method ofanalysis, the frictional interaction between the uid and the disks isrepresented in terms of an empirical uid friction factor. This results inordinary differential equations, which, together with simple boundary andentrance conditions, constitute a problem easily solved using computer-implemented stepwise calculations. The method is limited in usefulness bythe inadequacy of the friction factor concept.

In the case of laminar ow, much better analytical modeling can beachieved. Partial differential equations can be established rather simplytogether with appropriate boundary and inlet conditions. Because the owpath is long compared with the disk spacing, order-of-magnitude arguments

Figure 2 Schematic diagram of a Tesla-type pump.


can be made to simplify the equations somewhat and reduce them topseudo-parabolic form. The problem can then be solved by any one ofseveral available means.

Some investigators have used truncated series substitution methodol-ogy to make solutions [23, 25, 29, 32], but the method has accuracylimitations. A much more widely used method is to solve the problemusing computer-implemented nite-difference methods; Refs. [19] and [33]are typical examples. Integral methods have also been used, mainly toshorten the computer execution time required for solution [38]. Most of theliterature has considered the uid to be incompressible, although someinvestigators have calculated laminar compressible ows [48, 53, 65, 71].

Wu [78], in 1986, made detailed comparisons of results from all of theavailable literature. It was found that when the results of investigationsusing similar solution means were compared there was close agreement,within accuracy limitations. It was also concluded that the nite-differencesolutions all obtained good accuracy and produced essentially the sameresults, and such solutions are recommended to be used.

It is also clear from the literature that the calculated results for laminarows between corotating disks are in excellent agreement with experimentalresults for such ows [35, 44]. While future efforts may lead to increasedaccuracy or shorter computer execution time, present published methodol-ogy and results are sufcient for use in the design of laminar-ow multiple-disk rotors for pumps, compressors, and turbines. Some generalizedinformation for such rotors has been presented [45, 46].

The results of a solution of a ow consist of detailed knowledge of thedistribution of the velocity and pressure throughout the ow between thedisks. From this, the relationship between the mass ow rate and thepressure change from inlet to exit become known and allow determinationof design facts such as the number of disks required for an application ofinterest. The solutions are parameterized using the following or equivalentparameters: for outward ows (pumps), Reynolds number rOh2=m, ow-rate number Q=2pr2iOh, and radius ratio ro=ri; for inward ows(turbines) the additional parameter vvo=Oro is required. The results canbe arranged to provide values of dimensionless torque, dimensionlesspressure, dimensionless power, rotor efciency, etc.

With proper use of the analytical results, the rotor efciency usinglaminar ow can be very high, even above 95%. However, in order to attainhigh rotor efciency, the ow-rate number must be made small, whichmeans high rotor efciency is achieved at the expense of using a largenumber of disks and hence a physically large rotor. For each value of ow-rate number there is an optimum value of Reynolds number for maximumefciency. With common uids, the required disk spacing is dismayingly


small, causing laminar-ow rotors to tend to be large and heavy for aprescribed throughow rate.

Extensive investigations have been made of Tesla-type liquid pumpsusing laminar-ow rotors [64]. It was found that overall pump efciency waslow even when rotor efciency was high because of the losses occurring atthe rotor entrance and exit mentioned earlier.

CRITERIA FOR THE OCCURRENCE OF LAMINAR AND OFTURBULENT FLOW IN MULTIPLE-DISK ROTORS

It is established that a wide range of types (or regimes) of ow can occurbetween corotating disks including wholly laminar ow, laminar ow withregions of recirculation, wholly turbulent ow, laminar ow proceedingthrough transition to turbulent ow, and turbulent ow proceeding throughreverse transition or relaminarization to laminar ow. There isexperimental evidence concerning these ows [26, 27, 31, 35, 42, 44], andthe hydrodynamic stability of laminar ows between corotating disks hasbeen investigated [31, 66]. Wu [78], considering all available experimentaland analytical evidence, concluded that the viscogeometric number aproposed by Nendl [42, 55] most adequately characterizes the ow regime,where a uuh2=vr. While the data of the various investigators are somewhatcontradictory, a reliable guide appears to be that ow is laminar for a < 10,in transition for 104a < 20, and turbulent for a520. The variousinvestigators introduce rather pragmatic and differing denitions oflaminar and turbulent, which further complicates the matter. It isdesirable that it be explored further both experimentally and by additionalhydrodynamic stability analysis.

MULTIPLE-DISK ROTORS HAVING TURBULENT FLOW

The bulk-parameter (friction factor) type of analysis of ow in amultiple-disk type of rotor was referenced in the foregoing discussion oflaminar ows. It has been used extensively to calculate turbulent owsbut with mostly unknown degrees of adequacy. The transition criteriapresented are not applicable for bulk-parameter use, the hydraulicdiameter concept is required, and the assumption must be made thatpipe-ow frictional correlations are valid for ows between corotatingdisks. Nevertheless, such analyses have been the only analytical predictionmethod available until relatively recently, and methods now becomingavailable are not yet well established. Wu [78] gives an extended


comparison of the results of various bulk-parameter methods of analysiswith several sources of experimental data for turbulent ows betweencorotating disks.

Momentum integral equation methods have been and continue to beused to calculate the subject turbulent ows [26, 34, 36]. In these methods,ordinary differential equations describing the ow are written based onassumed distribution forms for the velocity and on empirical expressions forthe stresses in the ow. Some of these rely on experience and/or specializedexperimental results and are found not to be universal. Within carefullynoted bounds, however, these methods can be very successfully used by anexperienced investigator and have yielded strong insights into and designinformation for the turbulent ows of interest.

In principle, the set of partial differential equations applicable forturbulent ow can be written, provided with appropriate boundary andentrance conditions, and then solved in some manner using computerimplemented calculations. In practice, there are several aspects of theprocess that cannot be executed with condence that the calculated resultswill agree with reality. Principal among these is that a model or mechanismdescribing the turbulent transport of linear momentum must be prescribed.Furthermore, the partial differential equations are much more complicatedfor turbulent than for laminar ow. In the entrance region, the equationsmust be solved in elliptical form before changing over to simplied pseudo-parabolic equations to continue the calculations.

Truman [73] carried out calculations for turbulent ow betweencorotating disks using eddy-viscosity/mixing-length turbulence (stress)modeling, and using a nite-difference method for solving the partialdifferential equations. Wu [78] used a slightly modied version of thecomputer-implemented calculations provided by Truman, to comparecalculated results with experimental data for turbulent ow due to Bakke[26] and Kohler [34]. In general, agreement was good but with somequalications and reservations discussed in detail by Wu [78]. It was notedthat at high rotor angular speeds the calculations indicated recirculation inthe turbulent ow that could not be detailed since the pseudo-parabolicequations are not applicable for recirculating ows.

Truman et al. [76] have also calculated similar ows using ananisotropic two-equation turbulence model. More analytical study andmuch more experimental investigation of the turbulent ow betweencorotating disks are needed to establish the most applicable methods forcalculating the ows with condence and for the results to be useful in thedesign of turbomachines. Wu [78] has outlined an experimental apparatusand procedure that might provide the next needed insights for progress inthis area.


ACTUAL TESLA-TYPE TURBOMACHINERY

Many individuals and groups attempting to commercialize Tesla-typeturbomachines have designed, constructed, and operated them. Pumps havereceived the most interest, but compressors and turbines have also been builtand operated. While much useful test data have no doubt been recorded,very little has been published or otherwise made known because of aperceived need for keeping information proprietary. Thus, this large body ofinformation is not available to most investigators.

Most Tesla-type turbines and pumps have been designed usingintuition and simple calculations or empirical experience and rules ofthumb. This has almost always led to the use of large spaces between thedisks corresponding to turbulent ow between the disks.

There is published information concerning actual Tesla-type pumps[17, 18, 20, 21, 28, 34, 37, 47, 49, 50, 54, 59, 64, 67]. All the information pre-sented shows lower efciency than was desired. In general, the pumpsdesigned with little prior calculated rotor design information showedefciencies below 40%. Adequate calculated rotor design informationresulted in higher efciency but still in the range of 4060%. All thingsconsidered, it seems probable that Tesla-type pumps of reasonable size,operating with common uids, will not exceed an efciency of 65% at best,with careful attention to volute/rotor matching. Many Tesla-type pumpsintended to be commercially competitive have not been provided with avolute or any other type of diffuser.

A number of Tesla-type turbines have been constructed for use withsteam, gas, and water. Very few data have been published concerning them.There are references with published information concerning experimentalturbines [26, 811, 13, 16, 24, 51, 63]. As with pumps, the turbine efciencieshave been low and the machine sizes large per unit of power delivered. All ofthe turbines reported probably operated with turbulent ow in the rotor.

Most recent pumps and turbines of the Tesla type have used rotorscomposed of disks with a central hole, rather than the spoke constructionused by Tesla. The disks are carried on throughbolts extending from athicker master disk fastened to the shaft; thus, the rotor is overhung.Although most machines have used thick disks, there seems to be littlereason to do so since the disks have only small forces acting on them and arenot required to be perfectly at.

Many design questions remain unanswered at this time. Whether thedisks should be rough or smooth, for best performance, is controversial, asis the question concerning appropriate rotor-to-housing seals. There aresome ideas concerning ways to improve volute or nozzle performances thatremain undemonstrated. An early idea of improving performance of rotors


by composing them of nested cones rather than at disks has been shown toproduce no performance advantages and to introduce structural handicaps[20, 79].

CONCLUSION

Tesla-type turbomachinery probably cannot prove competitive in anapplication in which more conventional machines have adequate efciencyand performance. Thus, it cannot be expected to displace conventionalwater pumps or conventional water turbines or gas turbines. Tesla-typeturbomachinery should be considered in applications in which conventionalmachines are inadequate. This includes applications for small shaft power,or the use of very viscous uid or non-Newtonian uids. There is somereason to believe that multiple-disk turbomachines can operate withabrasive two-phase ow mixtures with less erosion of material from therotor. For that reason, they should be further investigated for applicationsto produce power from geothermal steam and particle-laden industrial gasows. There may also be unique applications possible using ceramic disks.There is considerable evidence that multiple-disk turbomachinery can bequieter in operation than is conventional machinery and that the noiseproduced is more nearly white noise without a prevailing sound signature.Multiple-disk pumps are well known to resist cavitation [20, 49]. It is theonly type of turbomachinery that can be easily constructed in a relativelyprimitive machine shop.

ACKNOWLEDGMENT

This review was presented earlier, in Rice [80].

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Handbook of Turbomachinery, Second EditionTable of ContentsChapter 14: Tesla TurbomachineryNomenclatureINTRODUCTIONMULTIPLE-DISK ROTORS HAVING LAMINAR FLOWCRITERIA FOR THE OCCURRENCE OF LAMINAR AND OF TURBULENT FLOW IN MULTIPLE-DISK ROTORSMULTIPLE-DISK ROTORS HAVING TURBULENT FLOWACTUAL TESLA-TYPE TURBOMACHINERYCONCLUSIONACKNOWLEDGMENTREFERENCES

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