The Design of a Micro-turbogenerator by Andrew Phillip Camacho Department of Mechanical Engineering and Materials Science Duke University Date: Approved: Jonathan Protz, Supervisor Devendra Garg Rhett George Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mechanical Engineering and Materials Science in the Graduate School of Duke University 2011
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The Design of a Micro-turbogenerator
by
Andrew Phillip Camacho
Department of Mechanical Engineering and Materials ScienceDuke University
Date:
Approved:
Jonathan Protz, Supervisor
Devendra Garg
Rhett George
Thesis submitted in partial fulfillment of the requirements for the degree ofMaster of Science in the Department of Mechanical Engineering and Materials
Sciencein the Graduate School of Duke University
2011
Abstract(micro-turbogenerator design)
The Design of a Micro-turbogenerator
by
Andrew Phillip Camacho
Department of Mechanical Engineering and Materials ScienceDuke University
Date:
Approved:
Jonathan Protz, Supervisor
Devendra Garg
Rhett George
An abstract of a thesis submitted in partial fulfillment of the requirements forthe degree of Master of Science in the Department of Mechanical Engineering and
Materials Sciencein the Graduate School of Duke University
4.6 Turbine shaft power and load power as a function of rotor speed . . . 120
xiv
Acknowledgements
I would first like to thank my advisor, Dr. Jonathan Protz, for all of his help and
guidance over these past few years. With the exception of my parents, no one has
ever done as much for me. I would also like to thank my parents and sisters for their
support over this time. Most of this work is a direct result of the work of others
before me at MIT and GaTech, and I wish to strongly acknowledge that and thank
all of them for contributing to the scientific literature on this subject. I want to
thank Will Gardner and Ivan Wang for not only greatly assisting me academically,
but for also being good friends throughout this period. I also wish to thank Yuxuan
Hu for her emotional support over the years. You’ve always been there for me, both
during the highs and the lows. And while I will forever deny it, everyone claims that
you motivated me to become a hard worker and a better person. Things did not work
out between us, but I wish you my best. I know you will have a happy, fulfilling,
and successful life. I also want to thank Gwen Gettliffe for being by my side during
much of this time and giving me a reason to keep on going. You are a wonderful
person in so many ways, and I wish things had worked out differently between us,
but I take comfort in knowing that at least we will continue to be good friends. A
strong degree of gratitude goes to Kathy Parrish for simplifying all of the graduate
school requirements, I would have been lost without you. I also want to thank all of
my professors over the years for being patient and allowing me to build a stronger
fundamental understanding of some very abstract concepts, in particular Dr. George
xv
for his E.E. insights and Dr. Shaughnessy for his compressible flow insights. I also
want to thank Dr. Garg for allowing me to pursue my interest in robotics during this
time. Kip Coonley and Justin Miles were especially kind in allowing me to use their
electronic equipment, sensors, and SMT soldering equipment. I would not have been
able to do my experiments without that support, thank you. Lastly, I would like to
thank Logos Technologies, DARPA, and Duke University for financial support over
this period.
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1
Introduction
1.1 Background and Motivation
1.1.1 Concept
In the mid 1990’s, the idea of using micro-sized devices to produce electrical power
from combustible fuels was presented by Dr. Epstein et al. (1997). The intent was
to use micro-sized heat engines in lieu of chemical batteries or fuel cells in specific
applications due to certain advantages that micro-heat engines would posses. To
demonstrate this, a quick synopsis of the strengths and weaknesses of these different
power sources must be given.
Chemical batteries are typically associated with high power densities but low
energy densities. As an example, conventional lithium-ion batteries are capable of
high power densities in the 1000 W/kg range, but also typically exhibit low energy
densities in the 150 W-hr/kg range [Panasonic Corporation (2009)]. In contrast,
fuel cells typically demonstrate high energy density due to the large specific energy
of their fuels and a good conversion efficiency, but also low power densities. A
compilation of commercial fuel cells by Narayan and Valdez (2008) showed energy
densities as high as 805 W-hr/kg, but with an accompanying specific power of only
1
2 W/kg. Other fuel cells were shown that demonstrated power densities as high as
18 W/kg, but this occurred at the expense of a much lower conversion efficiency as
demonstrated by a specific energy of 121 W-hr/kg, less than a lithium ion battery.
Thus fuel cells and batteries exist at opposite ends of a trade-off spectrum, with
batteries demonstrating low energy density but high power density, and fuel cells
demonstrating high energy density but low power density. In between these two
extremes are hydrocarbon combustion based heat engines. As an example, a 1000 W
Honda generator using 2 gallons of gasoline has a specific energy of 506 W-hr/kg and
a specific power of 40 W/kg [Honda Corporation (2001)]. Reasonably high energy
densities can be achieved despite a low conversion efficiency due to the large specific
energy of hydrocarbon fuels (12,500 W-hr/kg for gasoline).
As a result of these trade-offs, these device classes are each well suited to different
applications. In particular, in cases where reasonably high power and energy densities
must be obtained such as in automobiles and aircraft, heat engines are most often
used. This trend continues down to the size where relatively cheap and conventional
machining practices can be utilized. As an example, small sized piston and gas
turbine engines are used for long endurance RC hobby aircraft. Below such sizes
however, relatively cheap manufacturing processes with the necessary precision no
longer exist. For this reason, batteries and fuel cells are currently used exclusively
at these smaller sizes for applications that would be better suited for a heat engine.
This work discusses the design of a heat engine, a micro-turbogenerator, that
could be manufactured to fill this void. Its operation is analogous to a convention-
ally sized natural gas power plant, where a gas turbine utilizing a Brayton cycle is
driven by the combustion of fuel, and the resulting shaft power is outputted to an
electric generator to create electricity. A figure of the engine core without the elec-
trical components can be seen in figure 1.1. At the macro-size level, gas turbines are
excellent devices for converting heat from combustion into shaft power due to their
2
Figure 1.1: A micro-turbine engine operating a Brayton cycle, Spadaccini et al.(2003)
simple operation, high power density, and ability to use a wide range of fuels. And at
the micro-size level, a turbogenerator is an ideal choice also due to its theoretically
simple operation and low complexity; there are minimal moving parts, the compres-
sion, combustion, and expansion processes are steady state, and all the components
can be designed with planar geometry, allowing the utilization of micro-fabrication
techniques. In addition, when compared to its conventionally sized counterpart a
micro-heat engine has the advantage of benefiting from the cube-squared law, where
as the characteristic length of the engine decreases, its power density increases. This
can be shown as follows.
For a control volume, the output shaft power of a turbine is
9Wt ηt 9mcppTt4 Tt5q (1.1)
where ηt is the turbine efficiency, 9m is the mass flow, cp is the specific heat capacity at
constant pressure, Tt4 is the total temperature pre-turbine, and Tt5 is the total tem-
perature post-turbine. Factoring out Tt4 and using an isentropic expansion process,
this can be shown to be
9Wt 9mcpTt4p1 rPt5Pt4sγ1γ q (1.2)
where Pt represents the total pressure of the gas. As can be seen, for fixed conditions
such as efficiency, pressure, and temperature, the power output will scale linearly
3
with the mass flow. And again for fixed conditions, the mass flow will scale linearly
with the area, or with the square of a characteristic length, L2. As a result, the
power output will scale with the square of the characteristic length.
9W 9 9m 9 A 9 L2 (1.3)
The volume of the device however will simply be proportional to characteristic length
of the system cubed, L3. Therefore for fixed conditions, the ratio of power to volume,
or the power density of the device will be
9W
V9 L2
L39 1
L(1.4)
Such a simple analysis shows the advantage of micro-sized gas turbines. If one were
to simply scale a gas turbine with an outer diameter of 1 meter to the size of a micro
gas turbine with an outer diameter of 1 cm and maintain its operating conditions
and efficiency, the power density of the device would increase 100 fold.
In order to extract electrical energy from a gas turbine however, an electrical
generator must be attached to its output shaft. It follows then that a similar analysis
must be presented to show the scaling laws that govern electrical generators. For
this work, a permanent magnet micro-generator was selected for reasons to be later
explained.
As shown by Faraday’s law of induction, the induced voltage in a loop of coil is
equal to the time rate of change of the magnetic flux in the area circumscribed by
the coil.
ε dΦ
dt(1.5)
where the flux is determined by
Φ » »
~B d ~A (1.6)
4
As such, the induced voltage should be proportional to the area, or a characteristic
length squared, and the rotational speed of the magnetic field over the generator
coils.
ε 9 A Ω 9 L2 Ω (1.7)
As shown in the maximum power transfer theorem, for any given voltage, the maxi-
mum output power is related to the resistance of the generator coils by
9Wmax ε2
4 Rcoil
(1.8)
with the resistance of the coils being
R ρ LA
9 1
L(1.9)
Therefore, plugging in our voltage and resistance relationships, the power of the
generator system should scale as
9Wmax 9 L5 Ω2 (1.10)
However, with regard to turbomachinery, the tip speed of the turbine blades must
always be commensurate with the speed of the gases, which do not scale with the
size of the device. As such
Ω 9 1
L(1.11)
Inserting this conclusion into the power relationship shows that
9W 9 L3 (1.12)
As such, the power density of the magnetic generator should scale as
9W
V9 L3
L39 1 (1.13)
5
and therefore no benefit would be seen with miniaturization. However, such a benefit
is observed in real world scenarios as a result of scaling laws that affect heat dissipa-
tion. As shown in figure 1.2, for a given current density, the heat creation will scale as
L3. However, conductive and convective heat dissipation will scale with the surface
area or L2, making the ratio of heat dissipation to heat creation scale as 1L
. As a
Figure 1.2: Scaling laws for heat dissipation in a conductor, Cugat et al. (2003)
result, larger current densities are acceptable in micro-sized generators than in their
conventionally sized counterparts. This allows generator power density to an extent
also scale with the inverse of the characteristic length of the system at the expense
of efficiency. This has been confirmed in permanent magnet micro-generator tests
by Arnold et al. (2006b), where a power density of 59MW m3 was reported, whereas
typical macro-sized generators typically achieve around 20MW m3. Alternatively,
with size reduction, volume that was previously dedicated to cooling elements can
instead be dedicated to power producing elements, allowing the power density to
increase while maintaining efficiency.
As demonstrated, given equal non-scaling operating conditions such as tip speed,
efficiency, pressure ratio, etc., the power density of both the gas turbine and perma-
nent magnet generator scale indirectly proportional to the characteristic length of
the turbogenerator system. However, as shown in this thesis, the efficiencies of these
components typically do not scale favorably with size, and thus the power density
does not scale quite as well as this simple analysis would suggest. Regardless, the
power density does scale well enough for micro-turbogenerators to provide sufficient
amounts of power for many applications. Likewise, given sufficient components effi-
6
ciencies, the utilization of hydrocarbon fuels enables these devices to achieve energy
densities many times greater than conventional batteries. As shown in figure 1.3,
using ethanol as fuel, a thermal efficiency of roughly 13% and a generator efficiency
of 80% results in a device whose energy density can reach five times that of current
lithium-ion batteries. Thus there are many potential advantages associated with
Figure 1.3: Maximum energy density of a micro-turbogenerator relative to lithium-ion batteries as a function of thermal efficiency, generator efficiency, and fuel source
micro heat engines, particularly turbogenerators. They exhibit high energy density
relative to batteries due to their fuel source. They also possess sufficiently high
power densities due to scaling laws to power mobile devices and should theoretically
exhibit higher power densities than commercial fuel cells. For these reasons, micro-
turbogenerators are ideal power sources to run certain classes of devices, in particular
mobile devices that require endurant power sources and have relatively large power
requirements such as robotics, uav’s, and high power portable electronics. There
exists one other potential advantage associated with micro-turbogenerators. Due
to the power density scaling laws, if the devices are fabricated cheaply using large
volume manufacturing techniques as in the semiconductor industry, multiple micro-
7
turbogenerators could be used in parallel to provide a large power output at a fraction
of the size of an equivalent macro sized machine.
1.1.2 Challenges
Despite all the advantages that could be attributed to micro-turbogenerators, a self
sustaining engine has never been designed and run due to the many difficulties and
challenges associated with their design, fabrication, and operation.
Gas turbines often utilize a Brayton cycle, where ideally gases are isentropi-
cally compressed, then heated reversibly, and then isentropically expanded through
a turbine. As can be seen in figure 1.4, due to heat addition, the change in total
temperature across a compressor and turbine with the same pressure ratio diverges.
This allows a gas turbine to not only create its own pressurized gas source, but also
deliver excess shaft power to a generator to create electric power. In real systems
however, all of these processes are done in an irreversible entropic manner. The
differences between the reversible and non-reversible cycle paths can also be seen in
figure 1.4. For both the compressor and turbine portions of the cycle, the amount of
Figure 1.4: Reversible and non-reversible TS diagrams for a Brayton cycle
8
energy transfer will be
9W 9m cp ∆Tt (1.14)
As can be seen in figure 1.4 however, for any given pressure ratio, ∆Tt is larger for
the compressor and smaller for the turbine in the non-isentropic case relative to the
isentropic case. The result of this is that the compressor requires more input power
to achieve its pressure ratio, and the turbine delivers less power output for the same
pressure ratio. And because the compressor receives its shaft power from the turbine,
there then exists some efficiency below which the Brayton cycle will no longer close
as a result of the compressor requiring more power than the turbine can provide at
a given pressure ratio. This can be shown as follows. As stated before, the power
output for a turbine is
9Wt ηt 9mcpTt4p1 rPt5Pt4sγ1γ q (1.15)
Likewise, the power input for a compressor is
9Wc ηc 9mcpTt2prPt3Pt2sγ1γ 1q (1.16)
Equating these two until the power requirements are equal results in
ηt ηc ¥ Tt2prPt3Pt2sγ1γ 1q
Tt4p1 rPt5Pt4sγ1γ q
(1.17)
and therefore, for any given pressure and temperature ratios, there are minimum
component efficiencies that are required for the turbine to supply sufficient shaft
power to the compressor. If the efficiencies are below this threshold, the turbine
cannot power the compressor, and therefore no high pressure gas will be available to
power the turbine, and the cycle will not close.
For micro-turbomachinery unfortunately, the typical sources of entropic losses are
much greater than in their conventional counterparts due to relatively higher viscous
9
effects and manufacturing constraints. Some of these turbomachinery loss sources
are tip clearance losses, trailing edge mixing losses, end wall losses, and viscous losses
in the boundary layer.
In addition, there are many forms of losses that are practically unique to micro-
turbine engines. For example, due to the planar geometry dictated by MEMS fabri-
cation constraints, there exist many right angle turns in the flow path of the working
fluid as shown in figure 1.5, which lower the total pressure of the fluid. Another issue
Figure 1.5: Gas flow path for a micro-turbine, Frechette et al. (2005)
that affects micro-turbomachinery especially is residual swirl leftover in the wake of
the turbine. As can be seen by analyzing the velocity triangles in figure 1.6, there ex-
ists some rotational speed (or tip speed) for the turbine blades where the tangential
velocity of the exit flow is zero, with the remaining flow having either only a radial
or axial component. This speed corresponds to the optimum speed of the turbine,
because all of the tangential kinetic energy of the gases has been transmitted to the
turbine shaft. However, as described in section 1.1.1, the turbine tip speed is inde-
pendent of size. As a result, extremely high values of RPM (often above 300,000)
are required for micro-turbomachinery, and this necessitates the use of high-speed
air bearings for the engine. Such bearing systems are extremely complex however
and have only recently shown sufficient progress to be reliably utilized as demon-
strated by Frechette et al. (2005). They also have yet to be correctly integrated
into a complete turbogenerator system. The highest observed rotational rate for a
fully integrated turbogenerator system utilizing gas bearings is currently only 40,000
10
Figure 1.6: Velocity triangles of a turbine stage
RPM as demonstrated by Yen et al. (2008b).
Micro-turbine engines also typically suffer from issues associated with heat trans-
fer due to the small characteristic length of the devices. As an example, it is unfavor-
able for heat to transfer from the hot exhaust gases of the turbine to the gases within
or traveling to the compressor. This can be explained conceptually as follows. For a
turbomachine with an incompressible working fluid, the power transfer between the
fluid and the blades will be
9W 9m∆Ptρ
(1.18)
Conceptually then, one can think of a Brayton cycle as working across a fixed pres-
sure ratio, with the combustor existing solely to heat the temperature of the gas
and lower the density so that the turbine can extract more power during expansion
11
than the compressor requires for compression. For this reason, if the gas within the
compressor or the gas in route to the compressor is heated, then for a given shaft
RPM determined by material constraints, a lower pressure ratio will be achieved and
result in lower system efficiency. In conventional sized machines, the heat conduc-
tion from the turbine section to the compressor is often negligible relative to the
mass flow. However, for micro-sized machines with small characteristic lengths, the
thermal resistance is sufficiently low that strong measures must be taken to limit
heat conduction. As can be seen in figure 1.7, structural changes such as adding
Figure 1.7: Comparison of a first generation design relative to a more recent designshowing the thermal resistance structures between the turbine and compressor, Lang(2009)
thin and long thermal barriers between the turbine and compressor sections must be
made. This figure contrasts an early generation design versus a more contemporary
design. However, this small scale heat transfer effect can be used advantageously
as well. As an example, a recuperator with a high effectiveness transferring heat
12
from the exhaust gases to the pre-combustion gases can be made with a very small
form factor due to these scaling laws. For conventionally sized machines, the size
and weight penalties of these heat exchangers often outweigh their thermodynamic
benefits. Micro-sized heat engines however can use heat exchangers very effectively
to theoretically improve performance because heat exchangers are relatively much
smaller in micro-systems due to the scaling laws of heat transfer.
In addition to the efficiency problems associated with micro-turbomachinery,
there are many problems encountered by micro-generators due to their small size
as well. However, these problems are not as severe as those that affect micro-
turbomachinery components. The issues effecting permanent magnet micro-generators
will be discussed, because this type of generator is the one that is ultimately chosen.
One of the primary difficulties with micro-sized permanent magnet generators
is magnetizing the hard magnetic material. Ideally, there would be a square wave
magnetization pattern on the magnets alternating between north and south polarity
as shown in figure 1.10 (cm 0). However, due to leakage flux (figure 1.9) in the
magnetizing heads (figure 1.8) used to magnetize the material, trapezoidal magnetic
Figure 1.8: A magnetizing head used to permanently magnetize magnet mate-rial, Gilles et al. (2002)
patterns are typically encountered. This results in a decreased time rate of flux
resistance value for a single layer is negative, or alternatively that the ohmic losses
109
in the stator and power electronics are greater than the shaft power. This indicates
that generator system has no means, with only a single coil layer, with which to
dissipate or utilize sufficient shaft power to maintain its RPM.
3.2.6 Power Output, Efficiency, and Layer Count
With all of the loss mechanisms having been determined, we can now look at load
power and system efficiency as a function of layer count. With this information,
we can determine which generator design should be physically constructed based on
trade-offs between system performance, complexity, and cost as a result of increased
layer counts.
With generator efficiency defined as,
ηgen PLPshaft
(3.42)
the results are as shown in table 3.5. As before, notice that this generator cannot
Table 3.5: Load power and efficiency as a function of layer count
n PL pW q ηgen %
1 - -
2 2.82 28
3 4.73 47
4 5.62 56
5 6.12 61
6 6.43 64
7 6.63 66
8 6.78 68
9 6.89 69
10 6.97 70
function at this RPM with only a single coil layer.
In addition, as expected from figure 3.33, there are diminishing returns to in-
creasing the layer count. As such, a generator with 6 layers, a power output of 6.4
110
W, and an efficiency of 64% seems to be the best choice with regard to the system
trade offs.
This generator was designed within the physical parameter limits of micro-generators
in the literature. The performance can be improved by venturing outside of this de-
sign space (which may require fabrication innovations). As an example, assume
superior coil fabrication reduced the coil correction factor from 3 to 2, the magnet
thickness was increased from 0.5 mm to 1mm, the magnetization profile was improved
such that its correction factor went from 0.72 to 0.95, 3 laminations were placed in
the radial segments, and non-conductive low-hysteresis high-frequency ferrites were
used as the stator back iron in order to reduce the magnetic reluctance of the flux
path. With such a generator, the optimal efficiency would be observed with only two
coil layers and the generator would have an efficiency of 93%.
3.3 Conclusion
The concepts that govern electric generators were presented. Scaling laws were also
shown that demonstrated why permanent magnet generators offer superior perfor-
mance at small size when compared to electromagnetic induction machines.
The principles of permanent magnet performance were explained as were the
concepts of magnetic circuits. A 1 cm diameter permanent magnet generator was
designed and the performance values were determined. The device demonstrated an
output power of 6.4 W with an efficiency of 64% and a coil layer count of 6. With
specific improvements, the device demonstrated 9.3 W at an effeciency of 93% and
a coil layer count of only 2.
The performance of micro-turbogenerators using these two generators can be
compared against batteries in terms of energy density. Mattingly et al. (2006) gives
111
the performance of an ideally recuperated Brayton cycle as
ηtherm 1 PRk1k
T4T2
Assuming a compressor efficiency of 50%, a core turbine efficiency of 50%, and a
power turbine efficiency of 50% (all obtained using high speed radial turbomachin-
ery), a pressure ratio of 1.5, and a turbine inlet temperature of 1000K, results in
an thermal efficiency of 8.28%. As shown in figure 3.41, with generator efficiencies
of 64% and 93% from our low-end and high-end designs, the energy density of these
engines would be 3x and 4.5x greater respectively than lithium-ion batteries
Figure 3.41: Energy density of our designed micro-turbogenerators relative to Li-ion batteries
112
4
Experimental Results of an Ejector DrivenMicro-turbogenerator
4.1 Introduction
This section demonstrates an alternative thermodynamic cycle that was used to
overcome the challenges of the Brayton cycle at the micro scale. In this cycle, the
engine is designed around static pumping devices, an injector and an ejector [Gardner
et al. (2010a)]. Both are based on the same fundamental principle: a high momentum
motive fluid is mixed with a low momentum suction fluid, resulting in a discharge
fluid with less overall momentum and thus a higher pressure. The ejector is used
primarily to create a pressure gradient across the turbine, while the injector pumps
liquid into a high pressure boiler. The benefit of these components lies in their static
and turbo-machine independent operation. Unlike a turbo-compressor, the ejector is
uncoupled from the turbine and thus provides a pressure gradient across the turbine
regardless of the turbine’s rotor speed so long as heat is applied to the boiler.
This eliminates many problems associated with the startup phase and guarantees
that the cycle will close at any operating efficiency. In addition, because there are no
113
Figure 4.1: Control volume of the ejector mixing region
moving parts associated with these components, they can be more easily fabricated
as their manufacturing tolerances are not as critical as those associated with rotating
micro-turbomachinery components. Lastly, the suction provided by the ejector can
be utilized to prime the hydrostatic gas bearings, removing the need for an external
pressure source.
This section will discuss both the thermodynamic model concerning this type
of cycle and the preliminary experimental results of an ejector-driven micro-turbo-
generator.
4.2 Thermodynamic Cycle
The proposed cycle is derived from a steam locomotive cycle and an after-burning
Brayton cycle (Fig. 2). Combustion takes place downstream of the turbine, and
the generated heat is split between preheating the turbine inlet air with the use of
a recuperator and vaporizing ethanol in the boiler. Note that for this cycle ethanol
114
Figure 4.2: Schematic of an after-burning thermodynamic cycle driven by an in-jector
is initially used both as the fuel and as the motive vapor for both the injector and
ejector. The power from the turbine can be estimated from the incompressible flow
assumption, the ideal gas law, and by recalling that the total pressure at the turbine
inlet is roughly equal to ambient pressure.
9Wt
9ms
ηt∆Ptρ
ηtPamb Pt,s
PambRairTt4 (4.1)
The pressure difference between the ambient and total suction pressures can be
determined by the conservation of mass and momentum for the control volume shown
in Fig. 1, and assuming isentropic expansion in the diffuser such that the exit
dynamic head is approximately zero. The result is
Pamb Pt,s12ρu2
m
σ
2 σ α2σ3
p1 σq2 2ασ2
p1 σq 2ασ
p1 σq
(4.2)
115
where σ is the area ratio and α is the ejector entrainment ratio.
σ AmAd
α 9ms
9mm
The system rejects heat in the ejector discharge flow and by the cooling and conden-
sation of the ethanol vapor. However, the heat rejected in condensation typically far
exceeds the heat rejected from the cooling of the exhaust gases. The overall thermal
cycle efficiency can be approximated as
η 9Wt
9Wt 9mmhfg(4.3)
In addition, by further assuming the turbine power is significantly less than the
rejected heat we can re-approximate the cycle efficiency as
η 9Wt
9mmhfg(4.4)
Combining equations (4.1) - (4.4) we obtain
η ηtRairTt4ασ
2 σ α2σ3
p1σq2 2ασ2
p1σq 2ασ
p1σq
12ρu2
m
hfgPamb(4.5)
where σ, um, and Tt4 are design parameters (um is a function of boiler pressure).
Picking an a reasonable area ratio of σ = 0.5, the highest cycle efficiencies will
occur for an entrainment ratio of 0.53. Using these values and assuming a turbine
efficiency of 50%, we can approximate the overall efficiencies for different parameters
as shown in Table 4.1.
4.3 Experiment and Results
An experiment was conducted to demonstrate that an ejector powered by ethanol
vapor could create a pressure gradient across a turbine and drive its attached micro-
116
Table 4.1: Efficiency approximations
Case 1 Case 2
PboilerPs
3 30
Tt4 500 K 1600 K
η 1.13% 5.36%
generator to deliver electrical power. The turbine was originally designed and micro-
machined as a radial flow impulse turbine with an NGV outer-diameter of 10 mm
and blade heights of 250 µm (Fig. 4.3). However, due to the configuration and choice
Figure 4.3: The original micro-turbine design bonded to the rotor of a permanentmagnet generator with protruding leads
of materials, we believe eddy current losses in the surrounding material prevented
the turbine from operating on design. A new turbine with a rotor outer-diameter
of 11 mm and blade heights of 750 µm was 3D printed out of ABS plastic for the
final experiments. This allowed the turbine to reach a speed closer to its design
RPM. The ejector was micro-machined with a throat diameter of 719 µm, an area
ratio of 1 : 8, and was driven by ethanol vapor from a conventionally-sized boiler.
The turbine rotor was bonded to a 3-phase Faulhaber 1202-H-006-BH permanent
117
magnet DC motor with the outputs being rectified to DC with the use of 1N5817
Schottky diodes (Fig. 4.4 ). The rectified DC source was then connected across a
Figure 4.4: 3D printed turbo-generator connected to power electronics (boiler andejector not shown)
variable resistor that was adjusted until the maximum power output was obtained.
The conditions at the optimal operating point are shown in Table 4.2. Power from
Table 4.2: Experimental results
Property Units Value
Rotor Speed RPM 27,360
Turbine Pressure Ratio - 1.05
Boiler Pressure atm 15
VDC V 1.49
Turbine Inlet Temp. K 293
Power mW 7.5
the engine was also used to light a row of LEDs (Fig. 4.5)
118
Figure 4.5: LEDs powered by the micro-turbine generator
Note that maximum power does not take place by setting the load resistance
to the equivalent stator resistance as prescribed in the maximum power transfer
theorem. The reason for this is that the maximum power transfer theorem assumes
a fixed voltage source, where as for a turbo-generator the voltage is coupled to the
rotor speed. Therefore, the load resistance must be set such that the turbine can
reach its design speed (Fig. 4.6).
4.4 Conclusion
This section has demonstrated that, with no moving parts, an ejector can produce
a pressure gradient to drive a micro-turbine and generate power. An advantage of
this operating mode is that, unlike a standard Brayton cycle, there is no minimum
required efficiency for the cycle to close and the engine to function. In addition, the
ejector provides a means of creating the pressure gradient required by hydrostatic
gas journal bearings. This will allow the bearings to operate hydrostatically at low
speeds until the RPM increases, thereby allowing hydrodynamic bearing operation.
The manufacturing tolerances of these static pumping devices can also be much less
stringent than those of micro-turbomachinery.
119
Figure 4.6: Turbine shaft power and load power as a function of rotor speed
The thermodynamic cycle was analyzed with both compressible and incompress-
ible flow assumptions, and a basic method of estimating the cycle thermal efficiency
was presented. Experiments were conducted to demonstrate the viability of a power
cycle designed around an ejector-driven micro-turbine.
120
5
Conclusion and Future Work
5.1 Summary and Conclusions
The basic operating principles of micro heat engines were presented. They were
shown to represent a good balance between power density and energy density, and
this characteristic, in comparison to fuel cells and batteries, was shown to make them
good candidates for mobile applications that require both decent power densities and
energy densities.
The basic scaling laws that govern both turbomachinery and permanent magnet
generator power density were presented. It was shown that for turbomachinery,
the power density scales indirectly proportional with the characteristic length of the
system. For this reason, their power densities are observed to increase as they are
reduced in size. For permanent magnet generators, their power density was at first
determined to be scale independent. However, the heat dissipation capability of
the generator windings was shown to increase with reduced size as a result of the
increased ratio of surface area to heat generation. This should allow for more current
density and therefore power density at small sizes at the cost of efficiency or allow for
121
the replacement of cooling elements with power producing elements at small sizes,
thereby increasing power density with a constant efficiency.
Multiple challenges that affect micro-turbogenerators were presented. Of prime
importance, were the efficiency of micro-turbomachinery and the power transfer ca-
pabilities of the generator. The efficiency of micro-turbomachinery was shown to be
important not only for reasons related to energy and power density, but also because
for a Brayton cycle, component efficiencies must meet a specific threshold in order
for the cycle to close and create net power. The power transfer capabilities of the
generator were determined to be important because for a generator converting its
input shaft power to electric power, if the power transfer capabilities do not far ex-
ceed the input shaft power, either the conversion efficiency will be low or the device
will accelerate beyond its design RPM and potentially fail.
The basic operating principles of turbomachinery were developed, with the dif-
ferent components of energy transfer being specifically delineated. The ratio of these
energy transfer mechanisms was shown to relate to turbine reaction and device ef-
ficiency. Loss models were developed to quantify entropy creation from tip leakage,
trailing edge mixing, and viscous boundary layers over the surface of the blades.
The total entropy creation was then related to lost work and turbine efficiency. The
results showed the efficiency and power density of various turbine configurations over
a large range of sizes. For the configurations analyzed, the high speed single stage
reaction turbine showed the best performance. The power density was also shown
to scale linearly as expected over a large range of diameters. However, at very small
scales, the effects of viscous losses superseded the benefits from the scaling laws,
resulting in a peak power density. The single stage reaction turbine was shown to
possess the highest peak power density. The practice of multi-staging was shown to
not be as beneficial at small scales as it is at large scales, because the gains associ-
ated with increased kinetic energy absorption are largely offset by the combination
122
of high viscous losses associated with small scale turbomachinery and the increased
wetted area. The conclusion was that for micro-turbomachinery, high speed, high
reaction, single stage radial designs are most effective due to favorable pressure gra-
dients, low wetted area, and large portion of energy transfer taking place through
lossless centrifugal pressure fields.
The operating principles of generators and power electronics were then presented.
The scaling laws for both permanent magnet generators and electro-magnetic induc-
tion machines were developed and showed that permanent magnet generators should
scale down more favorably than electro-magnetic machines. The basic concepts of
permanent magnet operation were explained, and this was tied together with mag-
netic circuit theory to show the flux generating capabilities of permanent magnets
in planar micro-generators. The flux generation of the magnets was related to volt-
age creation through Faraday’s law, which allowed us to model the generator as
an alternating current source with a fixed internal resistance in an electric circuit.
An analysis was done to show the relationship between efficiency, generator voltage,
internal resistance, and load power.
Models were presented for planar micro-generators to determine output voltage,
internal resistance, electrical losses, and electromagnetic losses based on geometry
and key design parameters. A 3 phase multi-layer permanent magnet generator
operating at 175,000 RPM with an outer diameter of 1 cm was designed and an
efficiency of 65% was shown. A similar device was designed with improved features
that would require fabrication innovations and demonstrated an efficiency of 93%.
Lastly, an ejector driven turbogenerator was designed, built, and tested. A basic
thermodynamic cycle was presented in order to estimate system efficiency as a func-
tion of design parameters. Experiments were conducted showing a power output of
7.5 mW at 27,360 RPM.
123
5.2 Future Work
For future work, geometrically similar turbines should be tested in thermodynami-
cally identical operating conditions to determine the validity of the models presented
here. Alternatively, high fidelity CFD computational studies could be conducted. A
similar loss study model should be done for micro-compressors taking into account
different behavior and flow separation as a result of their unfavorable pressure gra-
dients. The basic loss models however should be similar.
In addition, a multi-layer generator with the dimensions and parameters given
here should be fabricated and tested in order to validate or repudiate the performance
and loss models. Alternatively, 3D FEA studies could be conducted.
After these tasks are completed, an externally supported turbogenerator system
with an integrated compressor, combustion chamber, turbine, generator, and gas
bearing system should be designed, fabricated, and tested. With the lessons learned
from this process, a stand-alone self sustaining device that would not require external
support for bearings, fuel injection, etc., should be designed, fabricated, and run as
the worlds first fully functional micro-heat engine.
124
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