Design and scaling of microscale Tesla turbines
Transcript of Design and scaling of microscale Tesla turbines
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: [email protected]
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.