451 Thedesignofatestrigandstudyoftheperformance ...aguha/research/...PartA_2009_Tesla_Turb… · 451 Thedesignofatestrigandstudyoftheperformance andefﬁciencyofaTesladiscturbine

451

The design of a test rig and study of the performanceand efficiency of a Tesla disc turbineG P Hoya and A Guha∗

Aerospace Engineering Department, University of Bristol, University Walk, Bristol, UK

The manuscript was received on 25 June 2008 and was accepted after revision for publication on 18 December 2008.

DOI: 10.1243/09576509JPE664

Abstract: A Tesla disc turbine and a flexible test rig have been designed and manufactured,and experimental results are presented. An analysis of the performance and efficiency of thedisc turbine is carried out. The design philosophy of the flexible test rig has been explained.Various complementary methods of measurement have been implemented and compared, andseveral operational experiences have been noted. A new simple method, the angular accelera-tion method, for measuring output torque and power in a Tesla turbine has been developed.This proved to be a successful method for overcoming the difficulties associated with thedetermination of very low torque at very high angular speed.

Keywords: disc, turbine, bladeless, Tesla, shear force, eddy-current brake, torque, power,efficiency

1 INTRODUCTION

In this article, we describe the design and manufactureof a disc turbine and of a flexible test rig for a system-atic study of its performance, and report some sampleresults. A disc turbine is a type of bladeless turboma-chinery that works by means of shear forces producedby the action of the fluid passing through the narrowgaps between a set of coaxial discs. It is also calleda Tesla turbine, named after its inventor Nikola Tesla(1856–1943), who is well known for his research andinventions related to electricity and induction motors.However, in 1913 he patented a revolutionary motor,consisting of a bladeless turbine [1]. This turbine wasformed of a series of flat, parallel, co-rotating discs,closely spaced and attached to a shaft, forming a rotor.The rotor rotated within a housing, with a small radialand axial clearance. The working fluid was injected in adirection nearly tangential to the rotor by means of aninlet nozzle, and exhausted through outlet ports in thediscs near the shaft. Tesla’s engine had the advantagethat it could also be used as a compressor, by power-ing the rotor with an external motor and modifying thehousing [2]. The direction of flow was thus inverted,

∗Corresponding author: Aerospace Engineering Department,

University of Bristol, University Walk, Bristol, BS8 1TR, UK.

email: [email protected], [email protected]

flowing outwards and exhausting tangentially into aspiral volute. The reported power output of the firstturbine developed by Tesla was 30 hp; this machinehad eight discs of 6 in diameter, and was powered bycompressed air [3]. Some years later, he built a newturbine with an 18 in diameter rotor, which reached200 hp using steam as the working fluid [3].

However, the lack of interest from contemporariesand the impossibility for Tesla to finance himself puta stop to his development of the bladeless turbine.Nevertheless, this engine was not forgotten and stud-ies have been carried out about its working from the1950s [4] onwards.

In 1952 Armstrong [5] built a ten-disc Tesla tur-bine that reached 1.11 hp and an efficiency of 14per cent. Later, in 1965, Rice [6] conducted researchon multiple-disc turbines and developed several tur-bomachines. Other authors, such as Beans [7], alsoinvestigated the bladeless turbine. However, the max-imum power reached by any of these new turbineswas much lower than those obtained by Tesla, gen-erally being below 2 hp, and the efficiency fluctuatedbetween 20 and 35 per cent.

According to Cairns [3], the working principle ofTesla turbomachinery can be explained by analogy tothe movement of a fluid within a pipe. When a fluidflows within a pipe, the friction with the wall of thepipe due to the viscosity of the fluid results in the wallbeing dragged along by the fluid; consider what would

JPE664 © IMechE 2009 Proc. IMechE Vol. 223 Part A: J. Power and Energy

452 G P Hoya and A Guha

happen if the pipe’s movement was not constrained. Ifinstead of a pipe, the fluid flows within two flat parallelplates, it would make them accelerate until they reachthe velocity of the fluid. Therefore, if these plates havethe shape of a disc, and these discs are mounted ona shaft, forming a rotor, the fluid passing between thegaps would make the rotor spin, accelerating to reachthe speed of the flow. At the same time that the rotoraccelerates, fluid velocity decreases from the maxi-mum at the periphery while it flows spirally inwardsup to the central exhaust ports. This effect is due to thefact that the kinetic energy of the fluid is transformedinto the work available at the shaft [3]. As explained byHasinger and Kehrt [8], the relative velocity betweenthe driving medium and the driven medium should besmall, in order to achieve an efficient Tesla turboma-chine. Most authors agree that the Tesla turbine is analmost pure impulse turbine, i.e. there is a large pres-sure drop in the nozzle and a small pressure drop inthe rotor.

To sum up, the Tesla turbine produces an exchangeof momentum by the use of the friction of the fluidpassing through the gaps between the discs [7]. Asthe angular velocity of the rotor increases, the viscousshear force decreases, thereby decreasing the outputtorque. The rotor reaches a steady angular speed whenthe output torque is balanced by the sum of exter-nally applied load and various losses in the system(e.g. friction at the bearings, leakage flow through theclearance gap between the rotor and housing, etc.).

The most important parameters that affect the per-formance and efficiency of disc turbomachinery, asoutlined by Cairns [3] and Rice [4], are as follows:

(a) spacing between the discs;(b) characteristics of the fluid and the flow, such as

velocity ratio;(c) conditions of the surfaces of the disc and radius

ratio;(d) radial and axial clearances between the rotor and

the housing.

After the success of Whittle and von Ohain, the gasturbine became the centrepoint of research and devel-opment and the understanding of its performance andoptimization has reached quite a mature stage [9–15]. The understanding of the performance of Teslaturbines is not nearly as thorough. Recently, severalreferences have been written [3, 16–20] on Tesla tur-bines; however, they usually describe the studies wehave mentioned above and do not include any newexperimental results.

An important advantage of the Tesla turbine is thatits construction is simple and it can be manufacturedin a local workshop. We therefore decided to build aTesla turbine, develop a flexible test rig and under-take a systematic study of the performance of theturbine (particularly in the context of non-availability

of modern experimental studies). The objectives of thecurrent study are as follows.

1. To design and manufacture a disc turbine.2. To design, manufacture and develop a flexible test

rig for the Tesla turbine, which, if possible, can alsobe used for testing Tesla compressors. This wouldentail the consideration and evaluation of variousmeasuring techniques overcoming the principaldifficulties that are associated with measurementsin a small Tesla turbine, viz. the determinationof very low torque at very high angular speed. Iffeasible, more than one method would be usedto measure a quantity so that consistency can bechecked.

3. To conduct a systematic study of the performanceof Tesla turbines.

2 CHARACTERISTICS OF THE TURBINE ANDITS HOUSING

A multiple-disc turbine has been initially designed asthe first representative turbine for detailed experimen-tal investigations, and it is planned to design and builda few more turbines in the future to study aspects suchas the effects of scaling. The overall features of thisfirst turbine were chosen according to the experiencenoted in the references [1, 3, 4, 7, 16, 21]. The discdiameter is 92 mm (3.6 in), the thickness of each discis 0.9 mm, and the rotor-to-housing diametrical clear-ance is 0.3 mm. An overall view of the turbine can beseen in Fig. 1(a). The discs have a single central outletport, since this configuration was found to be moreefficient by Davydov and Sherstyuk [21] and Rice [4].The diameter of this central hole is 25 mm. In orderto accommodate the outlet of the fluid, the shaft issupported as a cantilever by means of bearings insidethe base plate (Fig. 1(a)). The rotor is formed from theselected number of individual discs by keeping themtogether with the help of three locating pins built-in onto a thicker master disc attached to the shaft. Immedi-ately after the master disc, two discs are put togetherwith no spacing between them. The purpose is to usethem as the thicker end discs as used by Tesla [1]. Theydo not contribute to the momentum transfer betweenthe fluid and discs but prevent the air, injected by thenozzle, from moving out of the rotor instead of enter-ing in the gap between the discs [1, 16, 17]. The wholestack of discs is fixed to the master disc by using threebolts, as can be seen in Fig. 1(b). Hence, a total of sixpins go through the rotor, which is shrouded by thehousing. Then, the housing is bolted and sealed to thebase plate.

The shaft of the turbine is extended beyond thebase plate; this makes it possible to attach two discsto the shaft outside the housing. The first one is asmall disc, which, together with an optical sensor, is

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Design of a test rig and study of the performance and efficiency of a Tesla disc turbine 453

Fig. 1 (a) Overall view of the test rig and the turbine. Keys: (1) turbine housing, (2) base plate,(3) inlet pipe, (4) eddy-current brake, (5) optical sensor, (6) outlet Pitot tube, (7) pressuretappings of the housing, (8) data logger, (9) barometer, (10) power supply for the opticalsensor, (11) power supply for the electromagnet, and (12) strain gage amplifier.(b) Details of the turbine’s rotor. Keys: (1) discs, (2) bolt, (3) locating pin, and (4) exhaustport. (c) Details of the turbine’s housing. Keys: (1) end wall of the housing, (2) nozzle insert,and (3) outlet. (d) Mechanism to measure the reaction torque. Keys: (1) load cell and (2)reaction arm

used to measure the angular velocity of the rotor. Theother disc is larger and forms part of an eddy-currentbrake. Moreover, the extended shaft allows the use ofother methods for the measurement of output poweras well as the attachment of a driving device to powerthe rotor, so that just by designing and manufacturinga new housing the turbomachine could be used as acompressor, maintaining the same rotor and the baseunit. This enhances the flexibility of the test rig.

The turbine itself is also very flexible in the currentdesign since various parameters can be easily variedin order to study their effects on the performance andefficiency of the turbine. It is possible to change thenumber of discs, disc spacing, and geometry of thenozzle without changing the base plate, end plate, orthe housing. The disc spacing is varied by means of anumber of 0.2 mm thick washers placed on the three

locating pins (see the first paragraph of section 2). Thegeometry of the nozzle can be changed, thanks to theuse of an interchangeable nozzle insert. This nozzleinsert can be seen in its position in the housing inFig. 1(c) and its geometry is outlined with more detailsin Fig. 2(a). The width of the slot-shaped nozzle can beadapted to the different widths of the rotor by usinginserts with the shape of the nozzle channel, so thatthe unused area of the nozzle can be blocked and thejet can be directed more efficiently to the rotor andmajor leakages can be avoided.

The fluid passes from the central hole of the discsinto an exhaust pipe that runs through the end wall ofthe housing. The pipe can be moved axially inwards oroutwards so that its end can be adjusted with respectto the central hole of the discs. This variable posi-tion of the exhaust pipe was included in the design



Fig. 2 Scheme of the air injection system of the Teslaturbine. (a) CAD model of a nozzle insert. Keys:(1) direction of entrance of the flow, (2) directionof injection of the jet in the rotor, and (3) blockingsof the end of the nozzle slot. (b) Positions of sets ofpressure tappings with regard to the outlet of thenozzle (0◦)

since the end wall of the housing remains in a fixedposition, but the axial extent of the rotor assemblychanges according to the number of discs used (whichis variable).

3 TEST RIG AND MEASUREMENT TECHNIQUES

A test rig was developed to measure the variousparameters that are necessary to determine the per-formance and efficiency of the Tesla turbine. Most ofthe apparatus used is outlined in Fig. 1(a). Gas cylin-ders containing compressed air are used to providehigh-pressure air at the inlet of the nozzle. A pressureregulator, situated immediately after the gas cylinder,controls the inlet total pressure, which is usually inthe range of 2–4.5 bar. Static and total pressures atvarious locations are read and logged by a ZOC 22BScanivalve [22]. Temperatures, rotation per minute,and force measured in an RS 632–736 Load Cell arelogged by an Agilent 34970A Data Logger [23].

The fluid flows through a flexible pipe from the pres-sure regulator into the housing. Total pressure, staticpressure, and total temperature are measured at theinlet duct before the turbine. To do this, a type-K ther-mocouple is placed at the beginning of the inlet pipe,and a Pitot tube and a pressure tapping are placedin a fitting right before the entrance to the inlet noz-zle. Total pressure and total temperature are measuredagain at the outlet of the turbine.

Inside the housing, several pressure tappings are setin order to know the static pressure distribution in thechamber. Altogether nine tappings are used at threecircumferential positions (see Fig. 2(b)), each set atthree axial planes. The first two planes are covered bythe jet of the nozzle, whereas the third set is placedclose to the end wall inside the housing. The purposeof the two sets covered by the width of the jet is to

estimate the static pressure at the outlet of the noz-zle, since it was not possible to directly measure thepressure exactly at that position.

The flexibility of the turbine designed for thisresearch with regard to the number of discs anddisc spacing entails a problem: the sealing betweenthe end plates of the housing and the rotor is notassured. This problem was faced by blocking the endsof the nozzle slot, as explained before. However, itwas decided to check whether this was actually work-ing; to do this, a set of three tappings was placed atthe end of the housing. This allowed one to knowthe pressure distribution there, as well as to com-pare the pressures with those obtained in the areacovered by the jet of the nozzle by means of theother two sets of tappings. The results are discussedin section 6.

3.1 Determination of output power

3.1.1 Direct measurement of output power

In the test rig used in this research, the power out-put can be obtained directly by means of an electricgenerator connected to the shaft of the turbine. A suit-able motor that is sold in the market as a standardcarpentry tool was identified and some changes in theelectrical connections were made in-house so that itcould act as a generator. This generator could reachmaximum speeds up to 30 000 r/min and thus wasinitially considered suitable for the present purpose(the high revolutions per minute of the Tesla turbinemay restrict the use of other power and torque mea-surement devices). The kinetic energy of the shaft isconverted into electric energy, which can be measuredby determining the voltage and current. Hence, thepower can be calculated as P = v · i.

The use of the electrical generator has an additionaladvantage in the flexibility that it offers, because theelectrical machine can be run as a motor to drive aTesla compressor (with some suitable modificationsof the housing). However, the measured power wouldinclude any frictional losses in the bearings of the tur-bine shaft and the electrical generator itself. It wasfound that the frictional torque in the electrical gen-erator was high compared with the torque producedby the present Tesla turbine; hence this method was,in the end, not used for determining the final resultsof the current study.

3.1.2 Determination of power from torque andangular speed

Apart from the direct measurement of power, it canalso be calculated from equation (1) once the torqueand angular velocity are known

P = τω (1)



In the following, a method for determining angu-lar speed and several methods for determining theoutput torques that have been used in the presentinvestigation are described.

3.2 Determination of angular speed by anoptical sensor

The angular speed ω was obtained by means of anoptical sensor, which can be seen in Fig. 3. Themethodology for the use of this device is simple. Adisc with two through holes is attached to the shaft,so that as this wheel rotates within the gap of thesensor a frequency output is obtained, which is thenconverted to angular velocity in the data logger. Asthere are two holes, the software of the data logger wasprogrammed to divide the frequency readings by two,in order to obtain the actual angular speed. A hand-held tachometer was used once to double-check thereadings obtained by the optical sensor and the twomeasurements were consistent.

The optical sensor required a power supply, whichcan be seen in Fig. 1(a). This method was found tobe accurate and inexpensive. The angular velocityobtained was used in two different ways: First, ω isneeded to determine the power from equation (1), i.e.P = τω. Second, the time evolution of ω is also used inthe acceleration method, described in section 3.3, todetermine the torque τ as well.

3.3 Torque and power output measurement underunsteady state: angular acceleration method

According to Newton’s second law of motion, thetorque produced by the disc turbine can be calculatedas the product of moment of inertia of the rotating

Fig. 3 Apparatus used to measure the angular velocityof the rotor. Keys: (1) optical sensor, (2) disc withtwo through holes, (3) turbine shaft, and (4) baseplate

parts and the angular acceleration

τ = Iα (2)

The angular acceleration is obtained by numericallydifferentiating the angular velocity of the disc attachedto the shaft, measured by the optical sensor. The speedis measured during the acceleration phase (when com-pressed air is being injected into the turbine) as wellas during the deceleration phase, when the supply ofcompressed air is switched off after the rotor reachesits maximum speed and the turbine rotates by theeffect of the inertia. According to Newton’s first law ofmotion, the rotor should maintain a constant speedunder this situation unless there is a resistive, dis-sipative force (in the form of friction). The resistiveforce gives rise to a frictional torque, which tends todecelerate the rotor. It is assumed here that, at thesame rotational speed, the frictional torque opera-tive under the acceleration phase (when the supplyof compressed air is on) is the same as that opera-tive under the deceleration phase (when there is nosupply of compressed air and the friction is actingon its own). Therefore, if one measures the frictionaltorque during the deceleration phase and the netaccelerating torque during the acceleration phase,then one can write that, at any particular rotationalspeed, actual torque produced by the turbine = netaccelerating torque + frictional torque. Then, accord-ing to equation (1), the power can be calculated asP = τactualω = (τaccelerating + |τfrictional|)ω.

From equation (2) it can be deduced that it iscrucial to calculate the moment of inertia as accu-rately as possible for an accurate determination of thetorque. Since the geometry of the shaft and the rotorassembly is very complex, the moment of inertia wascalculated by means of the CAD files used for man-ufacturing the turbine, and the results were checkedlater by hand calculations on a simplified geometry.The numerical differentiation of angular velocity ver-sus time to obtain angular acceleration can be trickyif the actual experimental values at discrete levelsof time do not form a smoothly varying function.In this work, the angular velocity was therefore firstexpressed as a function of time by a high-order polyno-mial curve-fit through the data points. This algebraicpolynomial expression was then differentiated analyt-ically to generate the variation of angular accelerationwith time. Moreover, the angular acceleration wasalso determined by direct numerical differentiation infinite-difference form. The two results agreed closely,indicating that the velocity measured by the opticalsensor was smoothly varying and of good quality.

3.4 Tests under steady state: eddy-current braking

In order to make it possible to reach a steady state(i.e. to maintain a constant angular speed while



varying other parameters), an eddy-current brake wasdesigned, manufactured, and implemented to load theshaft of the turbine. When considering the differentpossibilities to achieve the steady state, three meth-ods were initially found: friction braking, regenerativebraking, and magnetic braking.

When testing machines such as internal combus-tion engines, the most usual method to load the shaftis to use a brake pad, in other words, friction brak-ing. In this way, by changing the braking force appliedto the brake, the angular speed is reduced until aconstant speed is reached. However, this method isnot suitable for the current disc turbine, since theangular speeds reached are extremely high and thebraking pads would wear out quite quickly. Moreover,the braking performance of this kind of brake is low ina high-speed region [24]. Therefore, this method wasruled out. The next method considered was the useof regenerative braking. This method consists of con-necting an electric generator to the shaft, so that bymeasuring the current and voltage created, the poweroutput can be determined (see section 3.1(a)). How-ever, the torque generated by the disc turbine was nothigh enough to overcome the resistance torque of theelectrical generator, so this method had to be ruledout too.

The best method for the present purpose was foundto be the use of a magnetic brake or eddy-current brake.This kind of brake consists of a non-ferromagneticdisc attached to the shaft and a magnet or elec-tromagnet between whose poles the disc rotates. Itpermits the loading of the shaft without any wear-ing, since there is no contact between the disc andthe poles, and the braking torque is controllable byvarying the input current of the electromagnet, soit can be adapted to the low torque produced bythe disc turbine. Figure 4 shows the eddy-currentbrake set used in the test rig, which consists of anelectromagnet, a disc, and a base for the electromag-net. This base was designed and manufactured sothat it can act as a flexible holding platform for theelectromagnet, which would work with various sizesof the Tesla turbine or various sizes of the electro-magnet itself. The critical functional requirement isthat the disc of the eddy-current brake is fixed tothe shaft of the Tesla turbine; the flexible design ofthe base allows changing the vertical and horizon-tal position of the electromagnet, in order to easilyadjust it and adapt it to possible future changes in thetest rig.

The design of the disc was a crucial part of the designof the brake, since the disc had to overcome the cen-trifugal forces acting on it as a result of the high angularvelocity, i.e. it must be safe to use it, but also had toprovide the appropriate braking torque. As far as theformer requirement is concerned, according to Fau-pel [25], the allowable speed of a rotating hollow disc

Fig. 4 Eddy-current brake. Keys: (1) electromagnet,(2) disc, (3) base for the electromagnet, and (4)base plate of the turbine

of constant thickness is

ωallowable

=√

8gσw

(3 + ν)ρro

(1

2 + (ri/ro)2[1 − (1 + 3ν)/(3 + ν)])

(3)

The angular speed given by equation (3) is the maxi-mum at which the disc can be rotated without the riskof plastic deformation or bursting. In equation (3),σw is the working stress, defined as σw = σy/Sf . Forthe current design, the material selected was an alu-minium alloy used in aerospace applications and asafety factor (Sf) of 1.5 was used.

According to Lee and Park’s theory [24], the brakingtorque produced by the eddy-current brake is given by

τb = τii2ω (4)

where

τi = βCσ�2Sd(

μ0 · Nlg

)2

(5)

β = 1 − 12π

[4 tan

(ba

)+ b

aln

(1 + a2

b2

)

− ab

ln(

1 + b2

a2

)](6)

C = 12

[1 − ab

π(1 + �/r)2(r − �)2

](7)

The braking torque was calculated by means ofequation (4) for several outer radii of the discs. By com-paring this torque with that obtained experimentally



by the turbine for several angular velocities (by theangular acceleration method described in section 3.3),a compromise was reached for the outer radius ofthe disc taking into account the allowable speedcalculated using equation (3).

This is one of the most used relations that can befound in the literature, but it has a limitation in that thecalculated braking torque matches accurately with theactual braking torque only at low angular speeds, sincethe relationship between braking torque and angu-lar speed is, in reality, non-linear at high speeds [26],whereas Lee and Park’s theory (equation (4)) givesa linear relationship. The reason for the differentbehaviour of the eddy-current brake in the low- andhigh-speed regions is that, according to Simeu andGeorges [27], at high speed the magnetic inductioncreated by the electromagnet is partially cancelled bythat created by the eddy currents, which makes thenet magnetic induction decrease and hence the brak-ing torque also decreases. According to Anwar [26], athigh speed the relationship between braking torqueand angular speed can be expressed as a second-orderpolynomial τb = f0 + f1i + f2i2, where each of f0, f1, andf2 are various second-order polynomials in ω. Althoughthis method gives a more accurate prediction for thebraking torque, nine constant coefficients are neededto specify f0, f1, and f2, and can only be determined byconducting experiments on an existing disc if anotherindependent method of torque measurement is avail-able (the coefficients need to be determined only once;thereafter the equation has predictive power). There-fore, for the preliminary design of a brake, one is forcedto use a more self-contained theory and, hence, Leeand Park’s theory was used here for the preliminarydesign even at high-speed conditions.

Another important parameter when designing theeddy-current brake was to find out the number of turnsN for the coil of the electromagnet. A suitable value ofN , for a desirable range in current i, may be estimatedfrom equation (5) for a braking torque higher than thatactually desired (since it was going to be lower actually,due to the inaccuracy of Lee and Park’s theory for ahigh-speed region). Moreover, if it was later found thatthe calculated number of turns was too low, more turnscould be added to the electromagnet.

3.5 Shaft power measurement: reaction torquemethod

An interesting characteristic of the design of the discturbine used in this research is that the whole machineis supported on bearings along the same axis as theshaft, so that it is free to rotate. This freedom can beresisted by attaching a reaction arm to the housing, itsmovement being stopped by a load cell. The reaction-arm–load-cell mechanism can be seen in Fig. 1(d). Ituses Newton’s third law of motion; the torque of the

turbine produces an equal and opposite torque in thehousing, necessary to hold the drive and the drivenequipment in place [28]. This reaction torque can beobtained by multiplying the force measured by theload cell and the length of the moment arm. The elec-tric signal from the load cell is conducted to a straingage amplifier, then to a low-pass filter, and finallyit is logged in the data logger. However, some prob-lems were found in the application of this method, asexplained in section 4.3.

4 OPERATIONAL EXPERIENCE

4.1 Inlet

During the research dealt with in this article, it wasfound that it is very important to analyse the validityof the pressure readings at the inlet of the turbine [29].Care is needed in designing the measurement posi-tions for the total pressure and temperature at theinlet. It was found that our flexibility in changing thenozzle quickly by the use of interchangeable insertscaused some restriction in the flow-path design atthe inlet. This operational experience has given us theguidance for an improved design in future. Second, itwas found (see section 6) that a good design of the noz-zle is important in improving the overall efficiency ofthe turbine. The mass flow at the inlet is estimated bythe laws of gas dynamics (section 5) by measuring thetotal pressure, total temperature, and static pressureat a location in the inlet pipe. Given the importanceof the mass flowrate in determining the performanceparameters of the disc turbine, an independent mea-surement of the mass flowrate would be desirable inthe future development of the project.

4.2 Outlet

After the nozzle, the fluid flows spirally inwardsthrough the disc spacings; shear forces cause anexchange of momentum between the fluid and thediscs and make the rotor rotate. The fluid bends by90◦ at the central holes of the discs and exhausts to theatmosphere through a pipe connecting the interior ofthe housing with the exterior, as explained in section 2.

Initially in this investigation, the conditions at theexhaust were measured by means of a Pitot-static tubeand a type-K thermocouple attached to it. This allowedthe determination of total pressure, static pressure,and total temperature at the outlet of the disc turbine.However, problems were found in both the pressureand temperature measurements. As far as the totaltemperature at the exhaust is concerned, a criticalproblem was found in its measurement: the temper-ature at the outlet was higher than that at the inlet.Obviously this is not realistic, since the total tempera-ture should not increase after a process in which work



is extracted from the fluid (the thermocouples havingbeen correctly calibrated).

In the test rig, the thermocouple that measured theoutlet total temperature was stuck to the Pitot-statictube, and the depth at which this probe could be intro-duced in the exhaust pipe was limited by an elbow.Therefore, the parameters were not being measuredright after the discs, but quite far away from them. Dueto the complexity of the flow at the exhaust and the sig-nificant temperature difference with the environment,a temperature recovery could be occurring. Hence, itwas decided to place the thermocouple closer to theactual exhaust of the rotor. In order to do this, the Pitot-static tube that was used initially was discarded and anew Pitot-static tube was employed. The advantageof this new probe is that it is straight, i.e. it does nothave bends or elbows, so that it could be introducedcompletely through the exhaust pipe up to the actualexhaust of the rotor. The thermocouple was then stuckto this new Pitot-static tube. With this new layout, theresults changed significantly. Now the outlet total tem-perature was lower than that at the inlet (however,the output power calculated from the difference ofthe inlet and outlet total temperatures is still signifi-cantly different from the output power measured bythe acceleration method (section 3.3)).

With regard to the pressure measurements, it wasfound in reference [28] that the measured total pres-sures seem to vary significantly at different points ofthe cross-section of the outlet pipe, and the measuredtotal pressure appeared to be lower than the staticpressure, which is not realistic. These difficulties hadnot been described in the literature consulted. It wasthought that this problem would be solved by meansof a longer Pitot-static tube described above, whichallowed the measurement of pressures directly afterthe rotor. However, this did not solve the problem.The flow at the exhaust is very complex and seemsto follow a spiral path through the exhaust pipe. Itis thought that a possible cause for this behaviour ofthe pressure readings could be the fact that the probeis misaligned relative to the direction of the flow.Then, the air is directed more directly to the statictaps rather than to the total pressure tap of thePitot-static tube.

In reference [30] it is seen that Pitot tubes with aconical nose are more sensitive to misalignment thanthose with a cylindrical external shape. Moreover, thesensitivity to misalignment of the latter can be reducedfurther if they have a 15◦ internal chamfer [30]. APitot tube with these characteristics was manufac-tured and the thermocouple was stuck outside thetube. The measured pressures with this modified Pitottube were higher than the static pressures measured inother tests, and for the first time were also higher thanthe atmospheric pressure. Unfortunately, it cannot besaid that this happened always; hence the effective-ness of this new Pitot is not completely demonstrated.

Moreover, although the measurement of the totalpressure at the outlet was improved, this probe didnot have static taps.

To sum up, it has been found that both temperatureand pressure readings at the outlet of the particu-lar turbine under consideration are difficult to obtainaccurately and the flow may not be suitable for aquasi-one-dimensional gas dynamics analysis. Fur-ther research is needed to find out the nature of thisexhaust flow; accurate values of the total pressure atexhaust are critical in defining the isentropic efficiencyof the Tesla turbine.

4.3 Reaction torque method

This method has been proposed by Drecq [31],Gindy [32], and Fleming [33], but none of the refer-ences give detailed descriptions for practical imple-mentation of the method. This method has beenimplemented in the present study (see section 3.4),but two problems were found. First, the flexible pipe(responsible for providing compressed air to the tur-bine) connected to the base plate applied a certainamount of reaction torque, which is not negligible ascompared to the low value of torque provided by theturbine (this was verified by measuring the force onthe load cell with and without the flexible pipe in posi-tion). Second, the transmission of the force betweenthe reaction arm and load cell was more complex thanexpected; it was found that the readings dependedstrongly on the nature of connection between the reac-tion arm and the load cell. Several attempts were madein order to obtain accurate readings from the load cell(which was calibrated properly) by changing the wayof connecting it to the arm. First, a pin was placed atthe lower side of the tip of the reacting arm, the pinresting freely on the load cell plate. So when the tur-bine produced power and the reaction arm wanted tomove downwards, the reaction force got transmittedto the load cell. However, the pin and the reaction armcould go up and detach from the load cell plate whena vibration was present. Moreover, the readings werenot repeatable, i.e. if a load was applied on the arm andafter that the load was removed, the measurement ofthe load cell was not the same; it did not return to zero.At first it was thought that the problem may be a resultof the friction between the tip of the pin (connectedto the arm) and the plate of the load cell. An attemptwas made to reduce this friction by attaching a wheelat the bottom of the pin, but this did not remove theproblem with repeatability.

When the reaction arm was attached to the plateof the load cell by screwing the pin into a threadedhole on the plate, it was found that this restrictedcompletely the movement of the mechanism, and noforce could be measured. A last attempt was made tosolve the problems with vibration and repeatability.



This involved the design and manufacture of a linkagebetween the reaction arm and the load cell. A pinwas screwed to the reaction arm and another pin wasscrewed into the plate of the load cell, and a third linkwas placed in the middle connected through pin jointsat both ends to the pins connected to the reaction armand load cell, respectively. Therefore, this mechanismallowed a horizontal movement of the reaction armwith regard to the load cell, whereas the vertical move-ment was transmitted to and measured by the loadcell. However, this linkage also did not solve the diffi-culties mentioned before, and inaccuracies still existin the reaction torque measurement.

4.4 Disc rubbing

According to Cairns [3], Tesla’s turbines were not per-sisted upon by the Allis Chalmers Company (the firstcompany that was interested in this turbomachineand manufactured some of the most powerful Teslaturbines) because of the finding that the discs hadstretched as a result of inertial forces. This problemhas not been described again by any of the authors inthe cited literature. However, in the current researchit was found that some discs had sometimes rubbedwith the housing. The problem seemed to occur onlyfor speeds >10 000 r/min and when more than fourdiscs were being used. We performed some prelimi-nary calculations for the allowable angular speed fromstress considerations with the help of equation (3)and for the disc stretching using the related theoreti-cal formulation. Such simple calculations show thatdisc stretching should not cause any rubbing in thepresent case. The test rig, however, showed noticeablevibration at very high angular speeds and when a largenumber of discs were used. It is thought that this waspossibly responsible for the rubbing marks. Only fourdiscs were used for producing the results given in thisarticle, for which vibration (or disc rubbing) was not aproblem.

5 CALCULATION PROCEDURES

A steady, adiabatic, compressible, quasi-one-dimen-sional flow of a perfect gas is assumed.

5.1 Inlet: mass flow and power input

The mass flow was estimated by means of the total andstatic pressures and total temperature readings at theinlet duct before the turbine. The Pitot tube at thatposition brings the fluid to rest, making it possibleto obtain the total pressure. This process is con-sidered adiabatic and reversible [34], i.e. isentropic.Then, knowing the static-to-stagnation pressure ratioand the stagnation temperature, the velocity of thefluid can be calculated. From the steady flow energy

equation it is obtained as

T0 = T(

1 + γ − 12

M 2

)(8)

Since the process is assumed to be isentropic, theMach number and the velocity of the fluid at the pointwhere pressures and temperatures are being measuredare respectively

M =√√√√ 2

γ − 1

[(p0

p

)(γ−1)/γ

− 1

]= f (p0, p) (9)

and

V =√√√√2cp T0

[1 −

(pp0

)(γ−1)/γ]

= f (p0, p, T0) (10)

Finally, the mass flow can be obtained by introducingthe equation of state of a perfect gas and equation (10)in the continuity equation for a one-dimensionalsteady flow

m = ρVA = pRT

VA = f (p0, p, T0) (11)

With regard to the power input provided by the fluid,it can be defined [35] as

Pinput = Q · p01 (12)

Q being the volume flowrate calculated with theparameters measured at the inlet duct and p01 the totalpressure obtained there.

5.2 Pressure losses before the nozzle

p01 was measured by a Pitot tube placed in a fittingright before the entrance to the nozzle inlet. Therewere some losses in the total pressure before reachingthis point due to a series of contractions and enlarge-ments in the fittings after the gas cylinder. These didnot affect the reliability of the measured p01, since itwas measured downstream of them; however, as thetotal pressure after the gas cylinder was known andthe dimensions of the changes in the diameter of theinlet duct were also known, a simple calculation forthe drop in total pressure was carried out by using theequation

�p0,loss =∑

Ki12ρV 2 (13)

The values of K were obtained from reference [35].The calculated pressure drop was within 5 per cent

of the measured difference in total pressure betweenthe gas cylinder and the nozzle inlet. This providesconfidence in the measured value of p01.



5.3 Nozzle

At the inlet of the nozzle, an estimated value of thestagnation pressure p01 and the stagnation tempera-ture T01 are known. The static pressure at the outletof the nozzle, p2, is estimated by means of the staticpressure measurements in a number of circumfer-ential positions inside the housing. As explained insection 3, two sets of pressure tappings were placedin the periphery of the housing, forming two paral-lel planes that were covered by the jet of the nozzle.Each set had three tappings, two of them being placedafter and before the jet, and as close to it as possible,and the third tapping placed opposite to the outlet ofthe nozzle (Fig. 2(b)). The estimated static pressure atthe outlet of the nozzle was then calculated by obtain-ing the tendency with the angle of measurement andextrapolating for that point.

An expression to deduce the total pressure in a com-pressible flow is outlined in references [30] and [36].The expression is

p02 = p2

⎡⎣1

2+

√14

+(

γ − 12γ

) (m

A2p2

)2

(RT02)

⎤⎦

γ /(γ−1)

= f (p, m, A, T02) (14)

This equation can be used to estimate the total pres-sure after the nozzle, since at that point all fourparameters that determine p02 are known.

In order to have an approximate idea of the effi-ciency of the nozzle, by knowing the estimated totalpressures for both inlet and outlet, the losses in theexpansion can be analysed. As explained by Mat-tingly [9], the main losses in a nozzle are usually relatedto the loss in the total pressure from inlet to outlet, aswell as to the over or under expansion

πN = p02

p01< 1 (15)

Another useful parameter to estimate losses in thenozzle is the loss coefficient, which, as outlined byCohen in reference [10], expresses the proportion ofenergy degraded by friction

YN = (p01/p02) − 11 − (p2/p02)

(16)

A typical value of the loss coefficient for gas turbinenozzles is YN = 0.05 [10].

As Shapiro explains in reference [37], the bound-ary layer is a critical factor for very small nozzles,since in this case its thickness can be as large as thechannel, and this may decrease the efficiency. There-fore, this is one more thing to take into account whenstudying the performance of the nozzle for the Teslaturbine. Higher velocities of the flow will lead to higher

Reynolds numbers and thus to smaller thicknesses ofthe boundary layer, making it possible to reach higherefficiencies in the nozzle.

5.4 The turbine as a whole: power and efficiency

The conditions at the inlet of the rotor are assumed tobe the same as those at the outlet of the nozzle; henceas explained before, properties at this point such asT02(T02 = T01), p2, and p02 can be determined. In addi-tion, at the outlet of the rotor (point 3) T03, p3, and p03

are measured. The mass flow can be recalculated atthe outlet, applying the continuity equation, in orderto see whether there has been any important leakagein the system.

The ideal power that should be developed by theturbine (isentropic power) is

Wisen = mcp(T01 − T ′03) (17)

where the ideal outlet temperature can be calculatedby analysing the isentropic expansion in the rotor

T ′03 = T02

(p02

p′03

)(1−γ )/γ

= T02

(p02

p03

)(1−γ )/γ

(18)

The output power due to the actual enthalpy drop is

Wen = mcp(T01 − T03) (19)

And then, the efficiency of the turbine defined as theratio of the output power due to the enthalpy drop andthe power involved in the corresponding isentropicprocess is

ηen,isen = Wen

Wisen

= T01 − T03

T01 − T ′03

(20)

Moreover, the efficiency can also be defined as the ratioof the actual power obtained by means of the angularacceleration method and the power involved in thecorresponding isentropic process

η ,isen = τω

mcp(T01 − T ′03)

(21)

A third way to define the efficiency of the Tesla discturbine is as the ratio of the actual power obtainedby means of the angular acceleration method and thepower of the input stream (equation (12))

η ,stream = τω

Qp01(22)

Previous researchers have used various ways to definethe efficiency of a Tesla disc turbine as outlined inTable 1.

It has been explained in section 4.2 that there areuncertainties in the measurement of pressure and



Table 1 The definition of efficiency as used by previous researchers

Armstrong [5] Rice [6] Davydov and Sherstyuk [21] Roddy et al. (for a pump) [35]

η = POutput

h1 − h′3

η = Pmechanical energy

PIsentropic expansion of the compressed airη = T0 − T3

T0(1 − p3/p0)η = Pdelivered to the fluid

Pshaft input= ρgH Q

τω

temperature at the outlet of the current Tesla turbinedue to the complexity of the exhaust flow. For this rea-son, the values of η ,stream defined by equation (22) areused to represent the efficiency in the results givenin this article; this definition is analogous to thatemployed by Roddy et al. [35].

6 RESULTS

Figure 5 shows the measured results of the currentdisc turbine with four discs of 92 mm diameter. Theangular acceleration method described in section 3.3was used to determine the accelerating and frictionaltorques, and the rotational speed is measured by anoptical sensor (section 3.2). Figure 5(a) shows the vari-ation of the actual torque (and its components) withrotational speed. Figure 5(b) shows the variation of theactual output power and the efficiency.

Fig. 5 Variation of output torque, power, and efficiencyof the turbine as a function of angular speed(using four discs, obtained at an input total pres-sure of 3.9 bar). (a) Torques obtained by apply-ing the angular acceleration method. (b) Actualpower and efficiency

6.1 Output torque, power, and efficiency

The output torque decreases with increasing angu-lar speed. The output power and efficiency initiallyincrease with increasing angular speed before theyreach a maxima. The maximum power reached by thepresent turbine was 140W (0.2 hp), at 25 000 r/min,at which the turbine reached a steady state on itsown, without using the eddy-current brake to load it.The efficiency of the turbine at this operating pointis approximately 25 per cent. These conditions wereobtained when a total pressure of 3.8 bar was appliedat the inlet of the nozzle.

A Tesla turbine has many variables, e.g. disc diame-ter, disc thickness, number of discs used, disc spacing,nozzle design (e.g. number and circumferential posi-tions of the nozzles, design of a plenum chamber,etc.), and design of the outlet. The various previousstudies are all different in these respects from oneanother or from the design used for the present study.Indeed, we did not find any publication in which anyresults have been compared with those from anotherstudy. Although a direct comparison is not possible,the trends of the various curves in Fig. 5 are simi-lar to the results given in the previous studies [5, 7].The physical reasoning behind these trends can beexplained in the following manner.

In the disc turbine, any relative velocity of the fluidacross the discs causes a viscous drag force that accel-erates the rotor. As the angular speed of the rotorincreases, the viscous drag force decreases. Therefore,for a fixed inlet pressure (that controls the fluid jetspeed at the inlet to the rotor), the torque decreaseswith increasing angular speed of the rotor, as seen inFig. 5. Theoretical calculations by Beans [7] for laminarand turbulent flows through the disc spaces showedthat the theoretically predicted torque decreases lin-early with increasing angular speed (the magnitudes ofthe theoretical and experimental values of the torquein Beans’ study were significantly different, but thesame linear trend was observed in both curves).

Since output power is equal to output torque mul-tiplied by angular speed, initially the output power ofa disc turbine increases (non-linearly) with increasingangular speed as seen in Fig. 5. After reaching the max-imum value, the output power of a disc turbine woulddecrease with any further increase in angular speed (asshown in reference [7]). Here a simple explanation forthis behaviour of the output power of a disc turbineis provided. In the last paragraph, it is shown, fromtheoretical considerations, that τ ≈ τ0 − cω, where τ0



and c are constants. Therefore, P = τω ≈ τ0ω − cω2.Hence, a maxima in the power curve is achieved. Inthe present study, when the output power was close toits maximum value, the angular speed became highfor the friction in the bearings, etc. to be of suchvalue that the rotor achieved a steady state (i.e. theangular speed could not increase beyond the point ofmaximum power).

The diameter of the discs used in the present study is92 mm; this is much smaller than the dimensions usedin most of the previous studies. For example, Teslahimself used a disc diameter of 152.4 mm [3], Arm-strong [5] used 177.8 mm, Beans [7] used 152.4 mm,and Rice [6] used 177.8 and 203.8 mm. The currentmachine (with a small disc diameter) therefore pre-sented greater challenges in terms of measurementtechniques since the torque is even smaller and themaximum angular speed is higher than that encoun-tered in previous studies. The method of acceleration,developed by us for small disc turbines and describedin section 3.3, needs to be mentioned in this respect.The attraction of the method lies in its simplicity andaccuracy: an (inexpensive) optical sensor is used toobtain all three important parameters: angular speed,output torque, and output power.

The efficiency values shown in Fig. 5(b) are calcu-lated by equation (22). The values of power shown inFig. 5(b) are obtained from P = τactualω = (τaccelerating +|τfrictional|)ω. The accelerating torque τaccelerating and thefrictional torque τfrictional are shown as functions ofangular speed in Fig. 5(a). It is seen that the fric-tional torque (due to friction in the shaft bearings,etc.) in the present assembly is high and it increasesrapidly with angular speed. At the maximum value ofthe angular speed shown in Fig. 5, the rotor assem-bly reached a steady state so that all power producedby the Tesla turbine was consumed at the shaft bear-ings. While evaluating the fluid dynamic performanceof the Tesla turbine itself, it did not seem fair tocharge the mechanical inefficiency of the bearings onthe turbines, and hence in Fig. 5(b) we have shownthe power that would have been available as outputif the bearings were perfect (the bearing performancecan be improved independently of the Tesla turbine,for example, using a higher grade bearing, better lubri-cation, a magnetic bearing, etc.). The correspondingefficiency shown in Fig. 5(b) can thus be considered tobe a sort of internal fluid dynamic efficiency of theTesla turbine. (In this respect, the efficiency shownhere is similar to ‘indicated efficiency’ in the contextof a reciprocating machine. Of course, it is easy to cal-culate the variation of τacceleratingω if one is interested inthe net shaft power available after the bearings.)

In section 4.2, the uncertainties in the measure-ment of pressure and temperature at the outlet dueto the complexity of the exhaust flow are described.These inaccuracies would have affected seriously theisentropic efficiency, and that is why those results are

not shown in this article. The performance resultsshown in Fig. 5 are not affected by uncertainties inoutlet measurements, since both the output torqueand power are obtained from the angular accelerationmethod (section 3.3) and the efficiency is calculatedfrom equation (22). According to equation (2), theaccuracy of the measured torque depends on theaccuracy of the moment of inertia I and the accu-racy of measured angular speed ω. I is calculatedby the Autodesk Inventor program from the origi-nal CAD files used for manufacturing the rotatingparts, and hence is determined reliably. The maxi-mum error in determining the revolutions per minutefrom the signal given by the optical sensor is specifiedin the manual of the data logger as ±0.01 per cent.An error analysis involving equations (2) and (1) sug-gests that dτ/τ ≈ dω/ω and dP/P ≈ 2 dω/ω. Hence,the torque and output power shown in Fig. 5 are quiteaccurate. The efficiency shown in Fig. 5 is calculatedfrom equation (22). The accuracy of the numerator inequation (22) has already been described. The accu-racy of the denominator depends on the accuracy ofinlet total pressure p01 and volume flowrate Q. Themanual of the Scanivalve specifies the error of the pres-sure measurement as ±0.1 per cent FS, i.e. ±0.05 psi(±0.00 345 bar). The main uncertainty is in the deter-mination of Q since it was not measured directly butwas estimated as the product of velocity and cross-sectional area of the inlet pipe. The average velocityis estimated by the one-dimensional gas dynamicapproximation, equation (10), which depends on mea-sured values of total pressure, static pressure, and totaltemperature (according to the manual of the datalogger and the specification IEC 584-2 for a type-Kthermocouple, the maximum error is ±(1 + 1.5) ◦C,i.e. ±2.5 ◦C). We have already mentioned that it isdesirable to have an independent measurement ofvolume or mass flowrate and we are currently explor-ing suitable methods for the future continuation ofthe project.

6.2 Variation of efficiency with differentparameters

Using the flexibility for changing parameters that theturbine used in this investigation permits, the effi-ciency was obtained by varying nozzle width, nozzleangle, disc spacing, and number of discs. It was foundthat the width of the nozzle did not influence the effi-ciency significantly. The optimum angle of injection ofthe fluid was found to be nearly tangential, althoughbetween 5◦ and 15◦ the efficiency was almost constant.As far as the disc spacing is concerned, the maximumefficiency was reached for a disc spacing of 0.6 mm,decreasing rapidly for spacings <0.4 mm and >1 mm.Finally, the efficiency increased when the number ofdiscs was increased.



6.3 Losses in the nozzle

The values of two loss parameters (equations (15)and (16)) for the nozzle were determined in order toassess the influence of this loss on the overall efficiencyof the disc turbine. The value of the total pressure atthe outlet of the nozzle, p02, is estimated by meansof equation (14), as explained before in section 5.2.The results obtained for the loss parameters are givenin Table 2.

These results show that there is a large loss of totalpressure, giving rise to a high value of the loss coef-ficient YN (compare this with a typical value of theloss coefficient for gas turbine nozzles, which is usu-ally about YN = 0.05 [10]). We are exploring the nozzledesign and our current team members are planningto conduct specific tests to study nozzle performanceand possible ways of improving it. However, our find-ings are in line with a general comment made byRice [4] about the low efficiencies of the nozzles inTesla turbines, even though Rice does not quote anynumerical value for the loss coefficient. In reference [4]Rice writes:

In general, it has been found that the efficiency of rotorcan be very high, at least equal to that achieved byconventional rotors. But it has proved very difficult toachieve efficient nozzles in the case of turbines. […] Asa result, only modest machine efficiencies have beendemonstrated.

The results given in Table 2 show that the losses in thenozzle decrease when the velocity increases; a contrib-utory factor for this effect has been explained beforein section 5.

In order to double-check the validity of these results,a different test was carried out; the rotor and the baseplate were taken out, so that the housing was openand it was possible to access the outlet of the noz-zle directly, as can be seen in Fig. 1(c). Then, the inletpipe was connected directly to the inlet of the noz-zle and a Pitot tube was placed at the outlet. Thismade it possible to measure the total pressure atthe outlet of the nozzle directly; this experimentalresult for p02 matched well with the estimated valuefrom equation (14) (with which Table 2 has been con-structed). It can therefore be confirmed that the highloss in the nozzle contributes strongly to the rather lowefficiency of the turbine.

Even though the efficiency of a Tesla turbine is gen-erally low, it may still find use in special areas becauseof its several attractive features. The first advantageis the simplicity of its design and manufacture. It is

Table 2 Loss parameters for the nozzle

Inlet total pressure πN YN V2 (m/s)

3.0 bar 0.803 0.488 322.653.5 bar 0.820 0.407 346.07

also inexpensive. The turbine can be useful for situa-tions involving small shaft power, very viscous fluidsor non-Newtonian fluids, or non-conventional fuels(e.g. reference [38] gives a feasibility report on the useof Tesla turbines with biomass as the fuel). It is believedthat Tesla turbines can cope better with particle-ladentwo-phase flows [4]. (References [39] to [43] describethe general aspects of two-phase flows).

6.4 Pressure distribution inside the housing

In order to check the efficacy of the sealing betweenthe rotor and the end wall of the housing and whetherthis was causing part of the fluid injected by the nozzleto bypass the rotor, a set of three tappings was placedcircumferentially in a plane parallel to the end wall ofthe housing, and close to it. The pressures measuredthere were compared with the readings obtained fromthe other two sets of tappings, outlined in section 3,which are situated within the axial bounds of the rotor.The results show that the pressures at the end wall ofthe housing were higher than the atmospheric pres-sure, and in the same range as those obtained in thetappings under the influence of the jet from the nozzle.Beans [7] suggested that part of the jet, in his turbine,went into the gap between the rotor and the housing.While we tried to minimize this unwanted flow of thejet into the gap, more detailed experiments are neces-sary to establish whether there may be a recirculatingflow within the gap.

7 CONCLUSIONS

The design and manufacture of a Tesla disc turbineand of a flexible test rig to study systematically theperformance of the turbine are described. Experimen-tal results for a 92 mm diameter turbine are presented,which shows the variation of torque, output power,and efficiency as a function of angular speed. Mea-surements of total pressure, temperature, and staticpressure are also taken at many locations at the inlet,outlet, and inside the housing for an understandingof the thermo-fluid dynamics of the machine. Manydesign considerations and operational experiences arediscussed.

Several procedures for measuring torque and powerare considered and implemented for cross-checkingthe measured values. Difficulties with many methods,which arise as a result of the high revolutions perminute and low torque output of the Tesla disc tur-bine, still remain. We have developed a new method,the angular acceleration method, for measuring thetorque and power in a Tesla turbine (section 3.3). Thisproved to be a successful method; it is also the easi-est and the most inexpensive of the available methodssince the angular speed, output torque, and outputpower are all measured by an optical sensor.



Present experiments showed that the losses occur-ring in the nozzle are large and hence this needs to betackled for improving the overall efficiency of the Tesladisc turbine. The output flow was found to be complexand more experiments (and probably an improveddesign) are necessary to assess whether a meaning-ful quasi-one-dimensional approach is feasible for thisflow and to resolve the difficulties of measurement oftotal pressure and temperature at the outlet.

ACKNOWLEDGEMENTS

The authors are grateful to, among others, KevenChappell, Sam Beale (Rolls-Royce), Lindsay Clare,Steve Macqueen, Sandy Mitchell, Steve Burrow, SamMcGarey, Peter Monson, and the technicians of theengineering faculty of the University of Bristol.

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43 Guha, A. Computation, analysis and theory of two-phaseflows. Aeronaut. J., 1998, 102(1012), 71–82.

APPENDIX

Notation

a length of the pole of the electromagnet(perpendicular to radius of disc)

A areab width of the pole of the electromagnet

(parallel to radius of disc)cp specific heat capacity at constant

pressureC compensating factor for braking torqued thickness of the disc of the eddy-current

brakeg acceleration due to gravityi applied currentI moment of inertiaK head loss coefficientlg distance of air gapm mass flowM Mach numberN number of turnsp pressureP powerQ volume flowrate

r radius of the discR gas constant� distance between the centre of the disc

and the centre of the poleS surface area of the pole of the

electromagnetSf safety factorT temperaturev voltageV absolute velocity

α angular accelerationβ compensating factor for braking torqueγ ratio of specific heatsη efficiencyμ0 permeability of airν Poisson’s ratioρ densityσ electrical conductivityσw working stress = σy/Sfσy yield stressτ torqueτb braking torqueτi braking torque constantω rotor angular velocity

Subscripts

0 stagnation conditions1 conditions at the inlet of the nozzle2 conditions outlet of the nozzle/inlet of

the rotor3 conditions at the exhaust from the

turbineen power due to enthalpy dropisen power of the isentropic processstream power due to the input stream of fluid output from the angular acceleration

methodi innero outer

Superscript

′ ideal (isentropic) conditions


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