Design and scaling of microscale Tesla turbines

Vedavalli G Krishnan1 Vince Romanin

2 Van P Carey

2 and Michel M Maharbiz

1

1 EECS, University of California Berkeley, CA., US

2 ME, University of California Berkeley, CA., US

E-mail: vedavalli@berkeley.edu

Abstract. We report on the scaling properties and loss mechanisms of Tesla turbines and provide design

recommendations for scaling such turbines to the millimeter scale. Specifically, we provide design, fabrication and

experimental data for a low pressure head hydro Tesla micro-turbine. We derive the analytical turbine performance for

incompressible flow and then develop a more detailed model that predicts experimental performance by including a variety

of loss mechanisms. We report the correlation between them and the experimental results. Turbines with 1 cm rotors, 36%

peak efficiency (at 2 cm3/sec flow), and 45 mW unloaded peak power (at 12 cm

3/s flow) are demonstrated. We analyze the

causes for head loss and shaft power loss and derive constraints on turbine design. We then analyze the effect of scaling

down on turbine efficiency, power density and RPM. Based on the analysis, we make recommendations for the design of

~1 mm microscale Tesla turbines.

Keywords

Tesla turbine, viscous turbine, hydro turbine, power MEMS, microscale turbine.

Nomenclature

b spacing between disks (m)

ro rotor radius (m)

ri exhaust radius (m)

ξr r/ro ; ξi = ri / ro

ε aspect ratio = b / ro

t disk thickness (m)

Ndisk number of disks in rotor

J rotor moment of inertia (kg.m2)

c clearance: rotor tip and enclosure (m)

s gap: end disk and enclosure (m)

Wnoz nozzle width (m)

Hnoz nozzle height (m)

Lnoz nozzle length (m)

Dnoz hydraulic dia. of the nozzle

ρ density of the fluid (kg /m3)

μ dynamic viscosity (kg s /m5)

ν kinematic viscosity = ρ / μ (m2/s)

z axial coordinate

r radial coordinate

θ angular coordinate

RPM rotor revolutions/min ( /min)

ω rotor angular velocity = 2π RPM /60 (/s)

vtip rotor tip speed; normalizing factor = ω ro (m/s)

subscripts o (outer- at rotor entry)

i (inner –at rotor exit)

r (at rotor radius “r”)

vtan(r), vrad(r) tangential and radial velocity of flow (m/s)

φ(z) fluid velocity profile in axial (z) direction

vθ(r), vr(r) axially averaged vtan(r), vrad(r) (m/s)

Ur dimensionless average radial velocity =vr (r)/ vtip

Vr dimensionless average tangential velocity = vθ (r)/ vtip

Wr dimensionless relative tangential velocity = Vr –ξr

NRE rotational Reynolds number = ω b2 / ν

Redisk disk Reynolds number = ω ro2 /ν = NRE / ε

2

Renoz nozzle Reynolds number = 2π Uo Redisk

n fluid profile; n=2 parabolic ; n=6 uniform

α Nendl visco-geometric number = NRE Uo / ξ2

Rem* modified Rotor Reynolds number = 4 NRE Uo

q flow rate = 2π Uo b ω ro2 Ndisk (m

3/s)

qdisk flow rate / disk pair = q / Ndisk (m3/s)

mass flow rate between a disk pair = ρ qdisk (kg/s)

p pressure (Pascal)

P dimensionless pressure = p / ρ (vtip )2

τ rotor torque (N-m)

T dimensionless torque = τ / (ro2 b ρ (vtip )

2 Ndisk)

Work done = torque * angular velocity = τ ω (Watt)

Input power = flow rate * head = q p (Watt)

η efficiency = T / 2π Uo P = τ ω / q p = ⁄

1. Introduction

Motivation

The goal of this work is to provide design guidelines for small Tesla turbines and a scaling methodology for optimum

design of microscale turbines. At scales at which inertial forces dominate –which include conventional power generation

turbomachinery- inertial turbines are preferred over Tesla ‘friction’ turbines. However, as is well known, inertial

turbines suffer heavy losses as they are scaled down. At scales approaching a few cm3 of turbine volume, surface

area-to-volume ratio increases, surface tension, adhesion, and cohesion forces begin to dominate inertial forces. In

contrast, Tesla rotors use kinematic viscosity and surface effects (rather than inertia) to convert flow energy into

rotational motion and are thus interesting candidates for miniaturized micro-scale power generation machinery. As such,

such turbines may find use both in ultra-small-profile heat engines and in the scavenging of energy from low pressure

head flow. To date, no comprehensive work exists that details the scaling constraints and performance trade-offs when

attempting to engineer very small (~ 2 cm3) Tesla friction turbines. The 2 cm

3 rotor turbine presented here is, to our

knowledge, the smallest hydro Tesla turbine reported, with an unloaded peak power of 45 mW at 12 cm3/s flow and peak

efficiency of 36% at 2 cm3/sec flow. Moreover, the entire turbine is built using a variety of modern commercial rapid

prototyping methods, making its construction accessible to almost anyone. This work discusses the design of

miniaturized turbines (1 – 60 mm diameter) that are capable of producing 8 mW – 150 W rotational power output. These

turbines operate at low rotational Reynolds numbers (NRE ~ 1–40) corresponding to laminar flow.

Basic Operation

Tesla turbines were first proposed more than 100 years ago by Nikola Tesla [1]. In this turbine (figure 1), the adhesion

and viscosity of a moving medium are used to propel closely spaced disks into rotation. The fluid enters the inner space

between the disks from the periphery and exits through central holes near the axle (dotted lines). There are no constraints

or obstacles intended to couple inertial forces (i.e. vanes) as in traditional turbines. The fluid enters tangentially at the

periphery and makes several revolutions while spiraling towards the central exhaust (dotted lines). During this process,

it transfers momentum to the disks.

Figure 1. Tesla turbine

Previous Work

Tesla turbine performance has been characterized by many researchers. Rice's [2] analysis was among the first and

claims turbines can be made up to 90% efficient, and designs on paper by Ho-Yan and Lawn et al claim over 70%

efficiency [3] [4]. Deam et al [5] argued that at small scales (sub-cm diameters) viscous turbines outperform

conventional bladed turbines and can provide ~40% efficiency. Hoya et al and Guha et al [6], [7] analyzed these

devices (with computational models, experimentation, and analysis for medium to large sized turbines) and claimed

25% efficiencies and demonstrated nozzle designs that could improve upon this efficiency. Though derived for meso-

and macro-scale turbines, all the works above form an excellent basis for verification of micro turbine design. A large

body of literature does exist on microscale inertial turbines and similar power-generating microelectromechanical

systems (MEMS); these systems usually operate between 100k and 1 M rpm and at least one order higher power density

[8] [9] [10].

The initial design for our 2 cm3 turbines was derived from design graphs presented by Lawn for macroscale turbines [4].

The fabrication and experimental results were disclosed in an earlier paper by Krishnan et al [11]. We predicted the test

turbine performance using an analytical solution posed by Carey [12] and verified it with an ANSYS simulation of the

micro turbines. The correlation of the experimental results with the analytical prediction and ANSYS simulation was

reported by Romanin and Krishnan et al [13].

There have been many attempts to employ various motive media in Tesla turbines. Designs with power densities ranging

from 5 mW/cm3 to 30 W/cm

3 are reported by various researchers and manufacturers [3] [4] [11] [14]. In general, the

reasons behind the wide variation in the power density of the designs are not well explained and the efficiency

discrepancy between the theory and practice is not adequately quantified.

Present Work and Methodological Overview

In this paper, we first derive an analytical model for incompressible flow and then add loss models to it. We present the

effect of turbine physical and operating parameters on its performance. Using the 300 mm turbine design by Lawn et al

[4] as reference, we present a design methodology for scaling down from 400 mm to 1 mm diameter rotors maintaining

better than 40% efficiency.

Our turbine model is based on the analytical solution (integral perturbation model) for the rotor momentum and pressure

drop posed by Romanin et al [15]. The ideal rotor momentum transfer and head drop are first derived for

non-dimensional flows. The actual turbine performance is then calculated by adding the losses incurred across the

turbine. Losses due to the nozzle path friction and the disk friction dominate the performance loss in the low laminar

flow regions. The volume loss, exit kinetic energy loss and bearing loss increase in the high flow, high rotor speed

regions. There is also impact loss in the slot nozzles at the nozzle-rotor interface. These losses are functions of the

turbine parameters and the performance goes down as the system scales down to the millimeter scale, resulting in

different optimum operating regions for the macro and the micro turbines. As a case study, this analysis is applied to our

2 cm3 turbine and theoretical and predictions are compared with experiment. We present a scaling report, wherein the

effect of main turbine parameters (rotor radius, interdisk space, rotor thickness, number of disks, tip clearance,

rotor-enclosure gap, nozzle width, nozzle height and exhaust to entry radius ratio) on the turbine performance is

detailed. Tesla rotor behavior is very sensitive to the rotor and nozzle dimensions and stable, reliable performance

demands high accuracy and precision in fabrication which gets harder to meet as the turbine scales down. Keeping this

in focus, different scaling techniques are investigated and recommendations are made for the micro turbine design.

In Materials and methods we provide fabrication, assembly and testing details for the turbines. In Theory and Modeling,

we discuss the various models of the turbine namely: analytical (integral perturbation), predicted and experimental (test

system) models. In Loss models subsection we summarize all significant losses. In Experimental Results we report

experimental data, trends, correlation to prediction, and mapping of the test system results over the predicted and

analytical results. In Design approach we recommend the design parameters and constraints for the turbine and

methods for minimizing various losses. In Scaling approach, we present the scaling effects of turbine parameters on

efficiency, power output, power density, rotor speed, head and flow and make recommendations for picking optimum

operating points as rotors scale down. We conclude with a recapture of our findings and future directions.

2. Materials and Method

Turbine fabrication and assembly

In this section we briefly discuss the fabrication, experiment and the data analysis for the micro turbine. Additional

details can be found in [11].

Disks of 1 cm and 2 cm diameter with three different center exhaust hole patterns were fabricated using commercial

photoetching (Microphoto, Inc., Roseville, MI) on 125 µm thick, 300 series full hard stainless steel sheets (figure 2a-2c).

A square axle with rounded ends was used to enable automatic alignment of the disks. Four rotors are fabricated to fit

into same enclosure (table 1). Rotors varied in interdisk spacing from 125 μm to 500 μm and in exhaust to entry ratio

from 0.47 to 0.51

Figure 2.. Rotors: (a) Assembled three 1cm and 2 cm diameter rotors (b) White light microscopy (20x) showing 125

µm disk and post-assembly gap uniformity of rotor stack; (c) Photo-etched stainless steel disks, bronze square axle

Figure 3. Exploded view of Turbine enclosure with 8 Nozzles N1-N8

Nozzle design plays a critical role in turbine performance [7]. [16]. To explore the nozzle parameter space, we used 3D

plastic rapid prototyping (ProtoTherm 12120 polymer, 50 μm layer thickness, High-Resolution Stereolithography 3,

FineLine Prototyping, Inc., Raleigh, NC) which allowed us to build designs which would otherwise be un-machinable.

Eight nozzles (N1-N8) were designed using three different shapes, three different areas, and four different angles of

entry (figure 3, table 2)). Spring loaded Ruby Vee bearings (1.25 mm OD, Bird Precision, Waltham, MA) connect the

rotor shaft to the housing These perform well at <10000 RPM. Adjusting the bearings’ position, the rotors are located

with respect to the nozzles.

Table 1. Rotors – 1cm diameter

Disks Gap(μ m) ri / ro

R1 20 125 0.47

R2 20 125 0.51

R3 13 250 0.47

R4 8 500 0.47

Table 2. Nozzle Specifications

Type Area

(mm2)

Length

mm

Width

mm

Width

arc o

Angle to

Tangent o

N1 Slit 3.28 3.,5 1 19 15

N2 Slit 3.28 3.5 1 16 25

N3 Slit 2.28 2.5 1 37 0

N4, N8 Slit 3.28 3.5 1 37 0

N5 5Array 0.69 0.4 0.4 8 15

N6 Slit 3.28 3.5 1 14 35

N7 Slit 7,14 4.0 2 56 15

Turbine operation and experiment

The experiment setup: A gear pump (EW-74014-40, Cole-Parmer) was used to produce flow while the pressure at the

nozzle inlet was measured (DPG8000-100, Omega Engineering). During operation, the rotation of the turbine was

recorded using a high speed video camera (FASTCAM-X 1024PCI, Photron). Thermocouples at the top and bottom of

the enclosure (5SC-TT-K-40-36, Omega Engineering) monitored turbine temperature (figure 4). Eight systems with

different nozzles and rotors were tested. Pressure pexpt vs. flow rate qexpt measurements were recorded for all the systems.

The rotational Reynolds number NRE = ω b2 / ν was found to be in the desired region of < 15 for the 20 disk stacks at

flow rates from 2 cm3/s – 20 cm

3/s, where ν is fluid kinematic viscosity and ω is rotor angular velocity.

Figure 4. Experimental system

Data collection and analysis

Data collection began when the turbine was at rest. Flow was then initiated, and once the rotor speed stabilized, flow was

halted, and data collection continued until the turbine returned to rest. Angular accelerations and decelerations were

computed from video data by performing polynomial curve fit on the frequency vs. time data and extracting the fitted

curve’s slopes at given frequencies. At any RPM, the acceleration of the turbine multiplied by J, the moment of inertia of

the rotor represents the torque being exerted by the fluid on the rotor, minus the resistive torque caused by the bearing

friction of the rotor mechanism, and the deceleration of the rotor multiplied by J, represents the resistive torque of the

rotor. The sum of the magnitudes of torques, τ represents the total torque exerted by the fluid on the rotor, and is used to

calculate the unloaded torque. The work done is derived by multiplying torque with angular velocity ω of the rotor as in

(1). The experimental efficiency is calculated using this, as the bearing loss can be recovered with suitable bearings. A

similar method was used by Hoya to calculate the unloaded torque and work done [6].

⁄ (1)

Experimental Uncertainty

Turbine design, fabrication, and test set-up were designed for rapid iteration and simplicity, for the sake of identifying

problems in micro-turbine design and for deriving optimum design parameters. The broad array of turbine parameters

allowed exploration of performance trends and the experimental uncertainty is estimated as follows. Fabrication, test

procedure and test data analysis each contributed an uncertainty of 4%, 5% and 10% respectively. In here all are treated

as independent random processes and the overall uncertainty is estimated as 12%.

3. Theory and Modeling

Below we first discuss the basic analytical model for a microscale turbine. The analytical model inputs the rotor

dimensions, flow profile, normalized flow parameters and Reynolds number and computes the rotor flow and pressure

drop characteristics. The flow momentum in the analytical model is verified using ANSYS simulation and the turbine

model is verified using published articles [4]. The head loss and shaft power loss models are derived from both

experimental measurements and the analytical model.

Analytical Turbine model

A basic analytical treatment of flow and pressure drop between adjacent rotating disks in a Tesla rotor was presented by

Romanin and Carey [15] and is used here to generate the model. The following assumptions are made to simplify the

equations.

Flow is incompressible, steady, laminar and two-dimensional: flow axial velocity = vz = 0

The flow field is radially symmetric, so all angular derivatives of the flow field are zero including at the outer

periphery of the rotor. Though this assumption is not true for a single nozzle entry, our ANYSY flow

simulations showed [13] that flow is symmetric within 10% of the entry.

Entrance and exit effects are not considered in this model. Only flow between adjacent rotating disks is

modeled.

The ratio of interdisk spacing to disk radius (aspect ratio), b/ro, is less than 0.05

In this model, the fluid profile φ(z) in the rotor interdisk space is given in terms of a profile number n ( equation (2),

figure 9). Axially averaged tangential vθ and radial vr velocities of the flow are calculated from the fluid profile and the

fluid tangential and radial velocities.. We apply n=2 for parabolic profile flow with individual nozzles for each disk pair

and n = 6 for uniform profile flow with slit nozzle scanning across all the rotor disks.

[ (

) ] (2)

The analysis henceforth uses dimensionless parameters. Size, velocity and pressure parameters are normalized by ro, the

rotor radius, vtip, , the rotor tip velocity and ρ v2

tip respectively where ρ is fluid density. At rotor normalized radius ξr =

ri / ro, Pr`, the rotor pressure gradient and Wr`, the fluid relative tangential gradient are derived based on the fluid profile

n, radial velocity at the rotor entry Uo, Reynolds number Rem* = 4 NRE Uo, and RPM as in (3). The rotor drop P and the

relative tangential exit velocity Wi are derived at the rotor exhaust by integrating iteratively for ξr = [ 1 ξi ].

(3)

⁄ ⁄

⁄

Efficiency estimate

The mechanical efficiency of the rotor ηrm is derived from the utilized fluid momentum. The ideal (simple analytical)

turbine head Pideal is calculated by adding the reversible kinetic energy KEin at the rotor entry to the head drop P in the

rotor and the ideal turbine efficiency ηideal is calculated using this turbine head. The estimated turbine efficiency ηpred is

calculated using the experiment head Pexpt as in (4).

⁄ (4)

⁄

⁄

Loss model

Central to the concerns in this paper is a thorough understanding of turbine loss mechanisms at the scales of interest.

There is efficiency loss in the turbine due to fluid frictional loss in the nozzle, disk friction loss in the clearance between

disk and the housing, mechanical loss in the bearing, unused head loss from volume leakage caused by inadequate

sealing, unused kinetic energy loss at the exhaust and impact loss due to geometry mismatch between the nozzle exit and

rotor entry. Disk friction is described and quantified by Daily et al [17]; nozzles losses are given by the Darcy-Weisbach

[18]. Nendl discusses flow turbulence in the rotor [19]. The losses have been measured, derived, simulated and reported

in the literature [7] [16] [20] Zeng et al provide an overview of losses in hydroturbines used in power generation [21].

The loss models are discussed in Appendix A.

Based on the investigation we categorized the losses into two types and modeled them as functions of flow rate and shaft

power. There is head loss due to friction in the nozzles, pressure drop in the rotor, unused kinetic energy at the exhaust,

volume leakage due to poor sealing and friction in the bearing. This is accounted for as an equivalent head loss modeled

as a second order polynomial in flow rate [21] . The tip frictional loss due to the trapped fluid between the rotor tip and

the cylindrical enclosure and the disk frictional loss due to the fluid in the gap between the rotor end disk and the

corresponding enclosure wall are modeled as a fraction of shaft power [17].

Test system model

The test system model is derived in two steps. First, the head loss ploss is modeled as a polynomial in flow rate and the

coefficients are derived from ideal heads and corresponding test heads at different flow rates. Next the shaft power loss

Tloss is modeled as the average difference between the predicted and test efficiencies. For this system, a0 =0, a1 = 1.81,

a2 = 0.017, Tloss = 0.586 , q is in cm3 / min and ploss is in Pascal. These estimates are used to map the ideal turbine

efficiency to predicted efficiency first (ηid2pr ) and to experimental efficiency next (ηid2ex ) (5).

( ) (5)

⟨{ ⁄ }⟩

(

)

( )

4. Results

Experimental results

Below we present experimental results from fabricated turbines along with observed trends in the turbine performance.

We discuss the mapping of the test results into the predicted and analytical result.

Figure 5. For R1, R3, R4 (rotor- disk space) with Nozzle4

Figure 6. For N3, N4, N7 (nozzle- length, width) with Rotor1

Performance trends of turbines

Decreasing interdisk space (for a given mass flow rate) increased efficiency in experimental data, and the predicted data

(figure 5). This is consistent with theory.

Increasing the velocity at the inlet to the rotor by decreasing the nozzle area (preserving mass flow rate) increased

efficiency up to a limit. Though higher velocity increases the kinetic energy and thus the efficiency, lower nozzle area

Table 3. Rotor1 with Nozzle 4 (R1-N4) had highest power output and R3-N3 highest efficiency

Rotor#-

Nozzle#

Flow

( cm3/s)

P

(bar) Rotation (rpm) NRE

Power

(mW)

eff

(%)

R1-N3 8 0.15 5590 9.3 20.3 18.4

R2-N3 8 0.13 5264 8.6 19.8 19.7

R3-N3 10 0.19 6522 43 16.9 9.3

R3-N3 2 0.01 1243 8.1 0.4 36.6

R1-N4 12 0.23 7247 12 45.0 17.3

R1-N5 6 0.29 4639 7.6 13.0 8.1

R1-N7 12 0.17 5807 9.5 23.2 11.9

increases the nozzle loss and lowers efficiency (figure 6, table 3).

Increasing interdisk space (R1 to R3) or increasing inner to outer radius ratio (R1 to R2) moved the efficiency peak to

lower flow rates. The higher aspect ratio of R3 and the lower active area of R2 both require slower flow to ensure similar

momentum transfer efficiency as R1

Maximum efficiency was achieved at low flow rates. The 13 disk rotor stack (R3) realized 36% efficiency for 2 cm3/s

flow rate at 0.4 mW shaft power (table 3).

Analytical to experimental mapping

For rotor1-nozzle3 tests are conducted at flow rates ranging from 2 cm3/s to 15 cm

3/s. The experimental, the predicted

and the ideal efficiencies are derived using equations (1) to (4). Then equation (5) is used to map the ideal efficiency to

the predicted and the predicted efficiency to the experimental (figure 7).

Figure 7. Rotor1 Nozzle3 test system efficiencies. Ideal turbine efficiency maps to the prediction first (ηid2pr ) ; then to

experimental efficiency (ηid2ex )

Figure 8. Rotor1 performance - projection of experimental and predicted efficiencies for Nozzles 3,4 and 7 ( N3, N4,

N7) on ideal efficiency surface of rotor-1 with uniform flow profile n = 6.

For rotor1 an ideal performance surface is generated with Uo = 0.1. The relative tangential velocity Wo and the Reynolds

number Rem* are varied over the operating range of -0.5 to +0.5 and 0.1 to 2 respectively. The predicted and the

experimental performance with the three nozzles R1-N3, R1-N4 and R1-N7 at a medium and high flow rate with 0.08 <

Uo<0.11 is picked and mapped onto the ideal rotor1 surface (figure 8).

5. Discussion- design

Design approach

With control of flow profile (figure 9a) and operating Reynolds number, the non-dimensional rotor behavior can be

maintained the same across the scaling. Given a subset of specifications (from available head, flow, power input,

desired RPM, power density, power output, and size) we can derive a range of turbines using the non-dimensional

operating points. Based on the fabrication restrictions and other specifications, the turbine design can be narrowed down

(figure 10). A list is developed with all of the parameters, constraints and their effect on the turbine performance. The

aim of the design is to maximize rotor performance and to minimize losses.

The optimal performing rotor

The five dimensionless parameters n, Vo, Uo, NRE, ξi that affect the rotor performance are studied to pick an operating

range for lossless turbines. These parameters also control the number of revolutions fluid makes before exiting the rotor

(figure 9c).

Profile of the flow n

Uniform flow with n=6 results in less rotor drop (figure 9b) and the efficiency curves broaden allowing for higher

rotational speed and higher power output compared to parabolic flow n=2.

The non-dimensional fluid tangential entry velocity, Vo

For a normalized average tangential velocity, Vo, less than 1, the rotor imparts a portion of its torque to the fluid,

resulting in a sharp drop in shaft power and efficiency; when Vo is near 1, the fluid makes many turns inside the rotor

before it reaches the exhaust transferring a large portion of its momentum to the rotor, but at low power. As Vo increases

above 1, the power transfer increases, but the efficiency drops due to increase in kinetic energy loss at the exhaust. Tesla

suggested a normalized velocity of 2.0 [14] and Lawn et al [4] used values between 0.8 and 1.3. The optimum range for

Vo is between 1.1 and 1.3, where power density gain of 20% can be achieved for an efficiency loss under 5%.

(a) (b) ( c)

Figure 9. Interdisk flow characteristics at Uo= 0.08, Vo= 1.3, NRE= 12, ξi= 0.2. (a) Flow profiles in the interdisk space,

for n =2,4,6,8 (b) Non-dimensional pressure drop in the rotor ,for flow profiles with n = 2, 4, 6 ,8. (c). Rotor fluid

streamlines wih a micro 2 mm, a macro 200 mm and our test rotor 1 of 1cm diameter.

The non-dimensional fluid radial entry velocity (flow rate indicator), Uo

As the normalized radial velocity, Uo decreases, the efficiency increases and the power density decreases. When radial

velocity is high, the efficiency drops but the power density increases. The optimum range is between 0.01 and 0.06.

The modified Reynolds number, Rem*, and rotational Reynolds number, NRE

Rem* is the rotor flow Reynolds number and is equal to 4 Uo NRE. For convenience, we discuss NRE which is independent

of the flow parameter. NRE varies between 1 -15 for the water turbines presented here with the optimum value of ~4 for

300 mm turbines; similar rotor performance is achieved at NRE ~8 for the mini 10 mm turbines, and at NRE ~12 for the

micro 2 mm turbines

The exhaust (inner) to rotor (outer) radius ratio, ξi

When exhaust radius is large (>0.6 – does not make many revolutions), fluid exits the rotor without transferring all its

momentum to the rotor. When this is small (< 0.2 – may exceed Nendl limit of 10) [19], the fluid at the exhaust turns

turbulent. Optimum range for this parameter is between 0.3 and 0.4.

Minimization of losses

In an ideal turbine, the efficiency would be determined by the rotor drop P and kinetic energy at the rotor input KEin. In

a real system, there are many sources of loss and, importantly, these are scale dependent (Appendix A). Figure 10 shows

the performance of a 2 mm rotor with no loss, with nozzle loss, with both nozzle loss and diskfriction loss.

Head loss minimization

Nozzle loss is the major contributor to head loss. All other head loss contributors can be minimized by good design

practices. Nozzle loss depends on the turbine dimensions and operating flow rates. As turbines scale down Renoz, the

nozzle Reynolds number drops, increasing the loss incurred. The following observations are relevant to scaling.

We can minimize nozzle loss by designing nozzles such that Renoz ~ 2100 for nozzles with relative roughness

roughnoz > 0.02 and Renoz as high as possible for smoother nozzles (equation (A.1)).

The position and orientation of the nozzles should be adjusted for maximum arc-width coverage for a given

nozzle width (Wnoz) while minimizing the volume loss into the clearance. The length of the nozzles (Lnoz)

should be minimized using techniques such as plenum chambers [7].

In a slit nozzle the hydraulic diameter of the nozzle is given by Dnoz = 2 Wnoz Hnoz / ( Wnoz + Hnoz) , where Hnoz

is the height of the nozzle which scales with the number of disks. The Reynolds number Renoz, can be

modified by changing the number of disks. In an individual nozzle per disk case, optimum Dnoz is equal to

interdisk space b resulting in higher nozzle loss, though the individual nozzles can reduce the leading edge

losses, gap losses and may reduce overall loss.

The height of the slit nozzle should span the entire length of the active rotor disks for maximum efficiency.

The end disks and the turbine enclosure at the end disks can be made larger to contain all the fluid volume into

the rotor space.

Shaft power loss minimization

Gap loss can be reduced by increasing the gap, reducing the fluid entrapment with better sealing and drainage.

As tip friction depends on t/c, by decreasing the disk thickness or by increasing clearance performance can be

improved. When increasing rotor tip clearance, proper sealing should be provided to prevent fluid from

escaping through the clearance into the gap at the ends of the rotor [17].

Higher rotor speed increases disk Reynolds number thus reducing the tip loss, though it increases the bearing

loss.

Minimization of other loses

Impact loss at the leading edge can be minimized by reducing the disk thickness.

Shaft-less rotors accommodate higher power transfer while maintaining desired exhaust area. Roughening the

rotor surface reduces the centripetal loss while maintaining the momentum transfer.

Using air or magnetic bearings for small and micro turbines and ball bearings for bigger turbines minimizes the

bearing loss improving efficiency.

Figure 10. 2 mm micro rotor design curves as a function of rotor speed , flow parameter (RPM, Uo) at Vo = 1.2, b = 40

μm, uniform flow n = 6, ξi = 0.4 (a) Ideal Turbine efficiency (b) Efficiency with nozzle loss at nozzle roughness =

0.05 (c) Efficiency with nozzle loss and disk friction loss at gap = 160 μm , clearance = 16 μm, (d) Power density

W/cm3, (e) Nozzle Reynolds number, (f) Disk friction loss factor

6. Discussion- Scaling

Scaling approach

We take a practical approach and base our scaling on specifications of the turbine such as available head, available flow,

desired RPM and desired power density. A scaling function was derived and used for the consecutive evaluations of the

effect of other dimensional and operating parameters on the overall turbine efficiency and power output. A hydro

turbine of 300 mm rotor with 200 μm interdisk space described by Lawn et al [4] is used as the reference rotor for this

study.

Scaling rotor parameters

The turbine scales with the rotor diameter and all the nozzles and turbine dimensions can be related to the rotor

dimensions. A proportional scaling down of the whole turbine is not optimum, as in this case the power density varies

inversely with (scaling)4. Beans [14], Lawn et al [4] investigated the performance sensitivity to interdisk spacing and

showed about an order of magnitude difference in power output for the same size rotor with different disk spacing. To

study the effect of scaling, we scaled radius by r scale and the interdisk space by bscale = rkscale at k = 0.0, 0.15, 0.33, 0.5

and 1.0. Using k = 0.5, turbines can be designed to operate at given pressure head. At k = 0.33, the scaling preserves

power density. We also evaluated at k = 0.15, as this corresponds to our test turbine. Effect of k on power density and

interdisk spacing is shown (figure 11) for 1 mm to 400 mm rotor range.

Figure 11. Effect of scaling exponent ‘k’ on (a) Interdisk space (b) Power density. At the scaling exponent of k =

0.33, the power density is constant.

Scaling at constant power density: k = 0.33

Using k=0.33, design parameters for a mini 1cm and micro 2mm turbine were derived. Figure 10 shows the

performance results for both lossless and lossy 2 mm turbines. The trends show it is possible to design a ~50%

efficiency turbine with Watts/cm3 range power density in a 2 mm microscale turbine, if it operates at higher rotational

Reynolds number and flow rate parameters (with a concomitant increase in the RPM and the power output). It should

be noted though that these graphs do not include the volume loss, bearing loss and leading/trailing losses. Accounting

for these at an additional 10% loss, it thus appears feasible to fabricate a microscale rotor with ~40% efficiency.

Optimizing scaling practical turbines

Optimization is done using following method. Power density of about 2 W/cm3 is chosen as the target for design. To

standardize across practical rotors, we kept the rotor height to be equal to its radius, the disk thickness t to be half of

interdisk space b, the tip clearance to be the larger of 1% of the radius ro and 0.2*(t+b), the gap to be 2*(t+b), and the

nozzle roughness parameter ε to be inversely proportional to the radius. With this setup, the scaling effect is studied.

A three level approach is used to design and to specify operating regions for the turbines across scaling from 1 mm to

400 mm diameter range. First an operating parameter set is generated at k = 0.33, for the range to provide a better than

40% efficiency. Next the power scaling k for interdisk space is tuned to provide tighter power/cm3 across the range.

Last, the interdisk spacing is tuned linearly to adjust the mean power density to be 2 W/cm3.

Six values namely 1 mm, 4 mm, 10 mm, 20 mm, 40 mm, 200 mm diameter rotors are chosen and the maximum

efficiency operating points are derived for each within a range of power density. The resulting 5 sets of parameters Vo,

Uo, n, NRE, ξi (figure 12c) are used to derive the operating parameters for rotors from 1 mm to 400 mm diameter using

piecewise interpolation. The 1 mm rotor RPM is 130000. Power density varied 30:1, from 38 W/cm3 to 1.3 W/cm

3

(figure 12b) with efficiency variation from 0.54 to 0.71 (figure 12a).

We needed to bring down the relative power density of the smaller rotors, as well as the speed of the rotor. The interdisk

spacing for the small rotors is increased to accomplish this. A study is conducted for k from 0.29 to 0.33 along with

minor modifications to the optimized parameters (figure 13c). For our system, k = 0.3 minimized the power density

variation to 2:1, from 4.4 W/cm3 to 2.2 W/cm

3 (figure 13b), while keeping the efficiency in the range of 0.41 to 0.75

(figure 13a)

A percent change to interdisk space results in about -6% change to power density and -2% change to the rotor speed. The

interdisk space effect is studied at four steps varying it from -7% to 14% (figure 13). With minor changes to interdisk

space, the power/disk can be tuned almost 1:3 (figure 14b) without much change to efficiency (figure 14a) or RPM

(figure 14c). Using 1.4 W/cm3 specifications, a sample design for a 2 m head and 1 cm

3/s flow rate is derived (table 4).

Here the size is based on the head at 1.4 W/cm3 power density and the number of disks is based on the flow rate. The

actual power density achievable is related to the fabrication accuracy. In practice, a design optimization needs to be run

to maximize power output at the fabrication accuracy.

Figure 12. Level-1 design for 1 mm to 400 mm diameter rotors with k = 0.33, Vo = 1.3 , b(1 mm rotor) = 25.8 μm. (a)

System efficiency ( turbine with nozzle and disk friction loss) variation 0.54 to 0.73, ( b) Power density variation 38

W/cm3 to 1.3 W/cm

3, head in meter, (c) Flow control parameters.

Figure 13. Leve-2 design for 1 mm to 400 mm diameter rotors with k = 0.3, NRE = 5, b(1mm rotor) = 32.5 μm. (a)

System efficiency (turbine with nozzle and diskfriction loss) variation 0.41 to 0.75, (b) Power density variation 2.2

W/cm3 to 4.4 W/cm

3, (c) Flow control parameters.

Figure 14. Level-3 design graphs for rotors from 1 mm to 400mm in diameter - all parameters as in figure 13; k = 0.3,

NRE = 5, b(1mm rotor) = 32.5 μm. Further tuning effect of interdisk scaling at -7% , 0, +7% and +14% (a) ηsystem ,

Efficiency curves for the turbine with nozzle and disk friction loss (b) Power in Watts/disk, (c) RPM.

Table 4. design specification for head = 2m flow = 1cm3/s and power density 1.4 w/cm

3

Power Eff. RPM Dia.

thick b Ndisk ξi Gap Clearance Hnoz Wnoz roughmax

9 mW 0.48 29900 2 mm 40 μm 40 μm 15 0.4 120 μm 24 μm 910 μm 310 μm 15 μm

Practical limits: the 1 mm3 engine

Scaling down below 1 mm rotors may not be practical. Tesla turbine performance is very sensitive to fabrication

accuracy and material stability. Though Tesla rotors do not have obstructing vanes, the particulate size in the fluid

dictates the lower limit of the interdisk space. The interdisk space to disk radius ratio needs to be smaller than 0.05 for

operation efficiency and that indirectly limits the minimum radius of the rotor discs. Additionally, smaller rotors operate

at lower flow rates resulting in higher frictional losses, thus bringing efficiency down. (Arguably, this problem can be

mitigated by increasing the rotor speed and tangential velocity for a combined optimization of power output and

efficiency but these numbers may become unfeasible for < 1 mm turbines).

7. Conclusions

We have shown here that it is possible to fabricate sub–cm Tesla turbines with commercially available technology and

with careful design it is possible to achieve close to 40% efficiency even when scaling to mm-diameter rotors. We

caution that the benefit of higher efficiency and usefulness of this hydro turbine may be limited to the 1 mm to 60 mm

range in an open loop system (although the range could be larger with other fluids and with incompressible flow). When

considered in conjunction with fabrication capabilities, the work provides a guide to what is achievable as down-scaling

of these systems is considered. The analysis here is focused on incompressible flow with water as the medium.

Extension of the analysis to other fluids and further analysis when dealing with compressible flow are future steps.

Acknowledgment

I like to thank Ms. Zohoro Iqbal, Mr. Frederick Dopfel for their help in the experiments, members of Maharbiz group

for various discussions and advice on fabrication, Joseph Gavazza of Electrical machine shop for help in fabrication.

8. Bibliography

[1] N. Tesla.United States of America Patent 1,061,206, 1913.

[2] W. Rice, "An Analytical and Experimental Investigation of Multiple Disk Turbines," Journal of Engineering for

Power, pp. 29-36, 1965.

[3] B. P. Ho-Yan, "Tesla Turbine for Pico Hydro Applications," Guelph Engineering Journal, 2011.

[4] R. W. Lawn M. J, "Calculated Design Data for the Multiple-Disk Turbine using Incompressible Fluid," Journal of

Fluids Engineering, Transactions of the ASME, pp. 272-258, 1974.

[5] T. R. Deam, E. Lemma, B. Mace and R. Collins, "On Scaling Down Turbines to Millimeter Size," Journal of

Engineering for Gas Turbines and Power, pp. 052301--9, 2008.

[6] G. P. Hoya and A. Guha, "The design of a test rig and study of the performance and efficiency of a Tesla disc

turbine," Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, pp.

451-465, 2009.

[7] A. Guha and B. Smiley, "Experiment and analysis for an improved design of the inlet and nozzle in Tesla disc

turbines," Journal Power and Energy, pp. 261-277, 2009.

[8] S. A. Jacobson and A. H. Epstein, "AN INFORMAL SURVEY OF POWER MEMS," in The International

Symposium on Micro-Mechanical Engineering, 2003.

[9] D. R. F. V. Jan Peirs, "A microturbine for electric power generation," Sensors and Actuators A, no. 113, pp. 86-93,

2004.

[10] F. Herrault, B. C. Yen, C.-H. Ji, Z. Spakovszky, J. H. Lang and M. G. Allen, "Fabrication and Performance of

Silicon-Embedded Permanent-Magnet Microgenerators," Journal of MicroMechanical Systems, vol. 10, no. 1, pp.

4-13, 2010.

[11] V. G. Krishnan, Z. Iqbal and M. M. Maharbiz, "A micro Tesla Turbine for power generation from low pressure

heads and evaporation driven flows," in Solid-State Sensors, Actuators and Microsystems Conference

(TRANSDUCERS), 2011 16th International, Beijing, 2011.

[12] V. P. Carey, "Assessment of Tesla Turbine Performance for Small Scale Rankine Combined Heat and Power

Systems," Journal of Engineering for Gas Turbines and Power, vol. 132, pp. 122301-1 122301-8, 2010.

[13] V. Romanin and V. G. Krishnan, "Experimental and Analytical study of sub-watt scale Tesla turbine performance,"

in Proceedings of the ASME 2012 IMECE, Houston, 2012.

[14] E. Beans, "Performance Characteristics of a Friction Turbines," Mechanical Engineering, Pennsylvania State

University, 1961.

[15] V. D. Romanin and V. P. Carey, "An integral perturbation model of flow and momentum transport in rotating

microchannels with smooth or microstructured wall surfaces," Physics of Fluids, p. 082003, 2011.

[16] W. Rice, "Tesla Turbomachinery," in Handbook of Turbomachinery, CRC Press, 1994.

[17] J. W. Daily and R. E. Nece, "Chamber Dimension Effects on Induced Flow and Frictional Resistance of Enclosed

Rotating Disks," Journal of Basic Engineering, Transactions of ASME, pp. 217-230, 1960.

[18] L. Moody, "Friction Factors for Pipe Flow," Transactions of the A.S.M.E., pp. 671-684, 1944.

[19] D. Nendl, "Eine Theoretische Betractung der Tesla-Reibungspumpe," VDI-Forsh.Heft 527, pp. 29-36, 1973.

[20] A. F. R. Ladino, "Numerical simulations of the flow field in a Friction-type Turbine ( Tesla Turbine)," Institute of

Thermal Powerplants, Vienna University of Technology, Vienna, 2004.

[21] Z. Yun, G. Yakun, Z. LiXiang, X. TianMao and D. Hongkui, "Torque Model of hydro turbine with inner energy

loss characteristics," Sci China Tech, pp. 2826-2832, 2010.

9. Appendix-A : Turbine loss models

Head loss contributors

Nozzle loss

Nozzles loss is calculated using Darcy-Weisbach (A.1) where Lnoz, Dnoz, g are nozzle length, hydraulic diameter and the

acceleration due to gravity. The friction factor is a complex function of velocity of flow Vnoz and the pipe relative

roughness . Research on friction factors has been consolidated to a usage form by Rouse and Moody [18]. Moody

presented the friction factor as a function of Reynolds number and pipe roughness ratio in a set of diagrams. For our

nozzle, we chose the applicable range of graphs from the Moody diagram and used a piece wise approximation to derive

the friction factor.

⁄ ⁄ (A.1)

⁄

⁄

The frictional loss is estimated for the nozzles N3, N4, and N7 over the tested flow rates of 2 cm3/s to12 cm

3/s. Reynolds

number varied from 700 to 8000 in the nozzles resulting in laminar to turbulent flow. For the turbulent flow, the

roughness factor of the nozzles is applied to derive the drop. As the nozzles are fabricated using 3D rapid plastic

prototyping with 50 μm resolution, a roughness factor of 0.05 is applied for the head calculations in the turbulent flow

regions resulting in head loss ranging from 5 to 3000 Pascal. This corresponds to a range of 0.1% to 10% of the

measured turbine head. The nozzles were also modeled in COMSOL and the head drop was verified for a number of

flow rates. It is notable that as the turbine scales down, the Reynolds number drops, increasing the loss incurred.

Rotor loss

Fluid flow characteristics such as flow profiles, the rotor Reynolds number, and rotor roughness affect the pressure drop

in the rotor. The uniform profile created by the slit geometry of the nozzle and maintained by rotor roughness causes less

drop compared to the parabolic profiles from individual disk spaced nozzles for the same average velocity due to the

reduction in centripetal force, (equation (3), figure 9b) . There is also loss due to turbulence near the exhaust. Nendl [19]

developed a visco-geometric constant αN which defines flow in between the corotating disks at any radial position r, as

laminar for αN < 10, transitional for 10 < αN < 20, and turbulent for αN > 20 (A.2).

⁄

(A.2)

Kinetic Energy loss at the exhaust

Higher tangential or radial fluid velocities relative to rotor speed result in inefficient transfer of the fluid energy to the

rotor. The fluid exits the rotor with unspent kinetic energy.

Leakage

Leakage due to escaping water between the periphery of the rotor and the enclosure contributes to loss in efficiency.

Previously it has been reported the loss due to this is less than 5% [21].

Leading and Trailing flow losses

The fluid exiting the slit nozzles encounters a rotor disk edge or a rotor disk gap resulting in impact loss. At the

exhaust, the fluid makes a 90o and suffers losses depending on the position of the disk in the rotor assembly [20]. This

loss can be modeled as a second order function of flow rate [21]. It is included in our loss model and is estimated as

having much lower effect on the loss compared to the first order flow rate effect for our test systems

Shaft power loss contributors

Disk friction loss

The water trapped in the gap s between the enclosure and the end disk of the rotor rotates at about half the speed of the

rotor; this inflicts a friction loss. An additional frictional loss occurs in the clearance c between the cylindrical enclosure

walls and the rotor tips (thickness t) for each disk. Both of these losses were analyzed using a single disk in a closed

enclosure and reported by Daily et al. [17] . The power loss due to disk friction is proportional to RPM 3 ro

5, similar to

the shaft power for a given rotor configuration. The disk friction loss can thus be defined as a fraction of shaft power.

⁄ (A.3)

The frictional torque loss due to the gap depends on whether the disk Reynolds number is laminar or turbulent and

whether the flow in the gap is merged or separate. A merged flow assumption is valid for small turbines due to the small

gap size (and would be a conservative estimation of losses even for larger turbines). The turbulent-laminar boundary

depends on the gap-to-radius ratio and the multiplicative constant is derived from the graphs presented by Daily. This

gap friction loss is shared by all the disks (A.3). The loss due to tip friction occurs for every disk (A.4). The

non-dimensional torque loss per disk is given by the addition of gap and tip coefficients (A.5).

(A.4)

( ) (A.5)

The performance loss due to trapped fluid in the gap can be reduced by increasing the number of disks. As tip friction

depends on t/c, by decreasing the disk thickness performance can be improved. The effect of tip loss increases as the

turbine scales down.

Bearing loss

The bearing loss is a function of rotor speed and, in our test, the bearing loss is accounted in the deceleration of the rotor.