Analysis of a Stirling-Cycle Power Convertor
for Domestic Combined Heat and Power
D. J. Buckmaster and W. S. Newman, Senior Member, IEEE
Abstract—Use of a Stirling-cycle convertor is analyzed with
respect to domestic application to micro Combined Heat and
Power. An engine that can convert heat into electric power has
the potential to extract more availability from fuel consumption
than combustion for space heating alone. A specific conceptual
design is presented, illustrating that the technical aspects are
manageable. A dynamic analysis shows that such a system could
be integrated with line power without expensive tie-in electronics.
A cost analysis indicates an installed-system target price for
practical consumer adoption.
Index Terms—Stirling engine; linear alternator; combined
heat and power; co-generation
I. BACKGROUND AND MOTIVATION
Typically, homes are heated through combustion of
hydrocarbon fuels (with a majority of U.S. homes using
natural gas). Thermodynamically, this process is wasteful,
since the fuel is combusted at a relatively high temperature
and the heat produced is mixed with air at room temperature,
thus generating significant entropy and corresponding loss of
thermodynamic availability. A more efficient use of the heat
would be to extract as much electric energy as possible and
use the balance for space heating—a concept known as
“Combined Heat and Power” (CHP). Assuming a typical
heat-exchanger temperature of
TH =400C (673K) in a gas-
fired furnace and a room temperature of
energy
TH =20C (239K), for
QH of heat released by combustion, a theoretically
perfect Carnot engine would be able to extract an amount of
work W =
QH (TH − TL ) / TH ≈ 0.56QH .
As an example, the home of one of the authors consumed
64MCF of natural gas in January, 2011 at a cost of
approximately $700. Combustion of this gas corresponds to
67GJ of heat at a cost of $10.54/GJ. For the same region, the
cost of electricity to the consumer (fully loaded) was
$0.1385/kWh. As an upper bound, if 67GJ of waste heat came
from a theoretically perfect heat engine operating between
400C and 20C, gas consumption equivalent to 120GJ of heat
would produce 53GJ of electric energy and still provide the
desired 67GJ of waste heat. At the fully-loaded consumer
price of $10.54/GJ of heat energy from natural gas, 120GJ
would have cost $1,265. However, if the electricity produced
could be sold back at the consumer’s rate of $0.1385/kWh
($38.47/GJ), the value of the electricity produced would be
$2,039. Thus, for this month, instead of paying $700 for heat,
978-1-4577-0776-6/11/$26.00 ©2011 IEEE
the consumer would have received space heating for free plus
a payment of $774 for electric energy sold.
The above limiting analysis assumes an unrealistic
efficiency, as well as ignores a barrier in selling net electric
power at the consumer’s rate. While all but four states
currently have some form of net-metering regulation [1],
details vary as to how consumers are credited for net
production. Importantly, in many states consumers are
reimbursed for excess production at a “wholesale” rate that is
significantly lower than the retail cost [2]. Thus, more
pragmatically, the optimal power to produce over a month
would be no more than the homeowner consumes, resulting in
zero net electric energy consumption. Continuing from the
same anecdotal example, a representative electric consumption
was 850kWh/mo (3.06GJ). Producing this much electric
energy per month would have a consumer value of $118/mo,
resulting in zero electric bills. If the system were run
continuously to produce the desired 3GJ of energy per month,
the generator should be sized to approximately 1kW.
To further improve the realism, assume the energy
convertor operates at 50% of Carnot efficiency (28%
conversion of heat to electric energy for temperature
reservoirs of 400C and 20C). To produce 850kWh of electric
power at this efficiency would require approximately 10.9GJ
of gas energy consumption and would exhaust 7.9GJ of waste
heat. Correspondingly, an extra 3 GJ of heat energy,
purchased for $31.62, would be required to produce the
desired 3GJ of electric energy, worth $118, for a savings of
approximately $86. This savings could be realized for each
month during which at least 7.9GJ of space heating was
needed (which was 9 out of 12 months for this anecdotal
example). For this example, the annual value of a CHP
system operating at 50% of Carnot efficiency would be
approximately $775/yr. Such savings would fluctuate with the
cost of gas and the cost of electricity. For a simple payback of
5 years, the installed system cost would have to be less than
$4,000 to be justified. This rough analysis establishes an
estimate for the size and installed cost of a practical domestic
CHP system.
A significant body of work going back decades exists with
regard to evaluating the utility of micro-CHP systems. In [8],
the merits of a variety of CHP systems (Stirling-engines, fuel
cells, etc.) are evaluated, and the authors concluded that up to
45% of installed central-heating boilers in the Netherlands
could be replaced by micro-CHP systems per year based on
the units’ average life-cycle. The authors of [9] focus on
larger, “mini-CHP” systems and conclude that there is a large
potential for CHP in high-load, non-industrial applications
such as office buildings and hospitals. Reference [10]
evaluates potential of different micro-CHP solutions for use in
a particular Belgian city, including an analysis of thencurrently (2009) available commercial products.
This
included three crank-driven and two “free-piston” Stirling
engines, for which the primary barrier to feasibility was a cost
2-5x higher than what the authors considered viable, despite
being the technologies nearest to commercial availability. In
[11], the authors went so far as to create mathematical models
based on appliance usage, home location, and insulation
properties to predict end-user cost savings from CHP systems.
Publications of this sort have been frequent over more than
two decades, especially because of the benefits pointed out in
our anecdotal analysis and in [12]: micro-CHP systems can
achieve upwards of 90% conversion from chemical energy to
“useful” energy in the form of electricity and heat, while
centralized electricity production and distribution can
squander up to 60% at the point of production, and another 5%
in transmission. In addition to cost-benefits provided to the
end user, micro-CHP quite simply allows for more efficient
use of scarce resources.
We next consider a specific technology to serve this
domain.
compact, result in higher alternator efficiencies and have
attractive start-up transient properties.
Another development promoting Stirling-cycle designs is the
emergence of commercial CAD packages for the design and
analysis of Stirling systems. A notable example is “Sage” [5],
a software system used in the present analysis for design of
Stirling engines. The Sage program assumes periodicity of
thermodynamic solutions, from which it can derive nonlinear
pressure waveforms, time-varying flow rates and heat transfer,
and power fluxes.
Figure 1 shows a schematic example of a Sage model for a
Stirling engine. The primary modules consist of: piston,
displacer, regenerator, temperatures of thermal reservoirs and
respective heat exchangers, expansion and compression
volumes, and specification of working fluid and charge
pressure. The schematic description allows specification of
myriad details for each of the modules, as well as operating
frequency, piston and displacer amplitude and phase, charge
pressure, piston and displacer geometry, dead volumes, heat
exchangers and regenerator design details. An illustrative
design example is detailed in Section III.
II. STIRLING-CYCLE CONVERTOR TECHNOLOGY
Among the variety of heat-engine alternatives, Stirling engines
promise attractive efficiency, simplicity and reliability.
Stirling convertors have been under development and testing
at NASA’s Glenn Research Lab since 1983 for intended use in
space power systems as described in [6]. An overview of
ongoing work is given in [7]. Using radioisotope-based
General Purpose Heat Sources, high-temperature heat can be
converted into AC electric power. A Stirling converter can
have only two moving parts—a displacer and a piston (where
an alternator’s armature is coupled to the piston). A relatively
modern variation is the thermo-acoustic Stirling convertor, in
which a moving displacer is eliminated, reducing the system
to a single moving part. An analysis of a high-efficiency, longlife (>100,000 hour) design for NASA space-missions is
described in detail in [3] and [4]. Another variation is to add a
second piston, which oscillates in opposition to the first piston,
sharing a common compression space and resulting in near net
cancellation of imbalance forces from the working piston
oscillations.
Another feature explored at NASA has been the use of gas
bearings with carefully tapered features on the piston and
displacer. This technology has been demonstrated and is
subject to ongoing testing for space-mission applications with
life-requirements in excess of 60,000 hours [15]. The analysis
of [15] further concludes that all of the technologies necessary
for long-lived Stirling conversion-systems exist today,
including bearings, materials resistant to temperature-variation
induced fatigue, and linear alternators.
By carefully choosing stiffnesses of mechanical springs on the
piston(s) and displacer, the moving parts can be tuned to
oscillate near a natural resonance. Such systems can be more
Fig 1: Sage model of Stirling engine. Upper fig: top-level view,
consisting of pressure source, piston, displacer and lumped
thermodynamics of the heat engine. Lower fig: icons that represent
and contain details within the heat engine.
III. ILLUSTRATIVE DESIGN
A specific Stirling-engine design for residential combined
heat and power is described here. Due to net-metering
regulations, the optimal electric-power production capability
of such an engine would be approximately 1kW (although this
value that may vary as consequence of numerous factors). A
reasonable assumption for the temperature of the heat source
is approximately 400C.
Integration with a home heating system requires a relatively
high-temperature for the convertor’s “cold” reservoir in order
to appropriately heat water for storage or space-heating.
Correspondingly, the present design assumes an exhaust
temperature of 66C.
Tables 1-4 summarize design details for a Stirling engine
capable of producing 1kW of electric power from 3.4kW of
heat input and providing 2.4kW of waste heat at 66C. Per
Sage analysis, the system efficiency is 59.3% of the Carnot
efficiency, resulting in converting 29.4% of the heat energy
into electrical energy.
This design assumes use of dual, opposed pistons that
cyclically expand and compress a working fluid of Helium
with a system charge of 0.7MPa (100 psi). Together, these
pistons displace a volume of +/- 161cm^3. Accounting for gas
volume within the system (including compression and
expansion spaces, regenerator porosity, heat-exchanger
passages and dead volume), the system pressure varies
(nearly) sinusoidally from 0.61MPa to 0.79MPa at 60Hz.
Length
Heater
Rejecter
25.000
cm
Temperature
672.0
K
Tube Inner Diameter
0.333
cm
Number of Tubes
156
Length
10.0
cm
Temperature
338.7
K
Channel Width
0.106
cm
Channel Height
0.500
cm
Number of Channels
396
Fin Thickness
0.050 cm
Table 4: Heat exchanger parameters
Fig 2 shows the resulting predicted internal temperature
variations at the two heat exchangers.
Working fluid
He
Charge pressure
0.7 MPa (100 psi)
Operating frequency
60Hz
Compression space mean vol 265 cm3
Pmax/Pmin pressure ratio
1.288
Expansion space mean vol
274 cm3
Heat input at 400C (750F)
3.4kW
Heat rejected at 65C (150F)
2.4kW
Mechanical work extracted
1.0kW
Table 1: System operating conditions
Amplitude
1.000 cm
Phase
0.000 rad
Piston
Face Area (each)
80.4 cm2
Spring Constant (each)
10.0 kN/m
Amplitude
1.215 cm
Phase
1.071 rad
Outer Diameter
13.4 cm
Displacer Drive Rod Diameter
2.77 cm
Hot-facing Area
141.0 cm2
Cold-facing Area
135.0 cm2
Spring Constant
42.6 kN/m
Table 2: Piston and displacer parameters
Length
4.141
cm
Inner Flow Diameter
13.500
cm
Outer Flow Diameter
20.786
cm
196.213
cm2
Cross-Sectional Flow Area
Porosity
Fig2: Sage-predicted temperature variations in the Stirling-engine
variable volume spaces.
0.900
Mesh Fiber Diameter
58.708 μm
Table 3: Regenerator parameters
Fig 3: Positions and velocities of displacer and pistons and
mechanical power output from pistons at nominal operating
condition.
Fig 3 shows the nominal displacements and velocities of
the pistons and displacer, as well as the net power required to
maintain these profiles. The displacer requires no control
force to produce its desired amplitude and phase. By choosing
an appropriate mechanical spring constant and displacer
dimensions (with respect to computed fluid-flow pressure
drops through the regenerator), the displacer resonates at the
desired amplitude, phase and frequency due to the timevarying internal pressures acting on opposite faces of the
displacer. The pistons also have tuned mechanical springs, but
substantial external forces are required to achieve the desired
amplitudes and phases at 60Hz. These external forces are
braking forces that extract power from the heat engine. These
forces are to be produced by rectilinear, oscillating alternators
integrated with the pistons (thus moving as a solid body). The
alternators are to be designed to provide nominally 110VAC at
60Hz at the design operating conditions of the heat engine.
Fig. 3 illustrates that the power extraction profile has the
waveform of a sinusoid squared, corresponding to a sinusoidal
braking force 180-deg out of phase with a sinusoidal velocity.
The power extracted peaks at 1kW in each of the 2 alternators,
which nets 1kW average power out, as desired.
Details of the linear alternator design are not provided here.
Ongoing work at the NASA GRC includes magnetic analysis
[16], characterization of magnet deterioration and alternator
efficiency [17], reliability [18], and evaluation of constructed
systems with efficiencies of 85% [19]. Assuming a similar
design for the present context, a linear alternator would exert a
force proportional to current and a back-EMF proportional to
velocity, where the constant of proportionality,
K m , is
identical for both effects.
A prospective barrier for a practical, low-cost system is that
it may be complex and expensive to construct the power
electronics required to achieve the design operating conditions
and to tie in the power production with line power. However,
an advantage of designing a Stirling engine to operate at line
frequency and voltage is that the system might require no tiein or control electronics at all (other than a relay to engage or
disengage the system). This will be analyzed in the next
section.
IV. DYNAMIC ANALYSIS
One of the challenges of home power generation is how to
tie in to the grid. Rectifying power from AC to DC then
inverting the DC to AC and synchronizing with line power
requires expensive components. A prospect for Stirling
convertors, however, is that the alternator can be designed to
provide AC power at the nominal line voltage and frequency.
This alone, however, would be inadequate for grid tie-in. The
convertor’s alternator must be synchronized precisely with the
line phasing and the alternator’s voltage must be only very
slightly higher than the line voltage, which can vary by 10%.
Surprisingly, the Stirling convertor is remarkably simple to
interface to line voltage. By connecting the alternator to line
voltage, the Stirling convertor will pull in to the proper
amplitude and phase, and the desired power production will be
delivered to the line. The line voltage will enforce that the
alternator back-EMF be close to line voltage, since a
significant difference between these voltages would induce
large currents in the alternator, thus producing corrective
forces that would cause the pistons to oscillate to nearly match
the back-EMFs to line voltage. If no heat were provided to the
engine, the alternators would behave as motors, consuming
power and cooling the tip of the unit (thus acting as a cooler
instead of a heater). Even under these dramatically offnominal operating conditions, the piston back-EMFs would be
close to line voltage.
Under nominal operating conditions, the pistons would tend
to overstroke due to power input from the heat source, unless
active braking were provided by the alternators. However,
any tendency to overstroke (at the design frequency of 60Hz)
would result in back-EMF waveforms with voltages in excess
of line voltage. As a result, potentially significant current
would flow in the alternators, providing power to the line and
providing substantial braking forces to the pistons. The
system would equilibrate with the back-EMF voltages being
near to line voltage and in synchrony. At this operating point,
the pistons would necessarily be oscillating at their design
amplitude.
Proving these assertions requires constructing a dynamic
model from the steady-state thermodynamic analysis. To
perform this conversion, an intermediate step is to find the
sensitivity coefficients of piston and displacer positions and
velocities on expansion-space and compression-space
pressures. This can be done by analyzing steady-state
operations at conditions that are perturbations from nominal,
from which sensitivity coefficients for a pressure-waveform
model can be derived.
From the pressure-waveform
dependencies, along with parameters of masses, piston face
areas, mechanical spring constants, and alternator parameters
(back-EMF constant, resistance and inductance), one can
construct a dynamic model that relates piston and displacer
accelerations to velocities, positions, line voltage and
alternator currents. The process to derive such a model is not
detailed here, but techniques to do so are presented in [13] and
[14].
For a dynamic analysis, a state vector consists of: the
positions of both pistons and the displacer, the velocities of all
three of these components, and the currents in the two
alternators. This 8-D state vector evolves according to a
linearized state-space model of the system dynamics (related
through the pressure-wave influence coefficients) as excited
by a sinusoidal (line) voltage. In the present analysis, the two
alternators are excited in parallel, and each alternator is
designed for 110V at 60Hz under nominal operating
conditions of the convertor.
Fig. 4 shows a start-up transient for the dynamic model of
the example Stirling-convertor design. Initially, all three
moving parts (dual opposed pistons and displacer) are at their
rest positions (center of stroke, enforced by mechanical
springs). Sinusoidal line voltage is suddenly applied to the
two alternators (emulating enabling the system with a relay).
The two pistons pull into synchronism, reaching their design
amplitudes and phasing almost immediately. The displacer
takes several cycles (about 70ms) to reach its design
amplitude. During the transient, none of the components
overstroke or behave chaotically.
If the line voltage varies, the pistons will vary their
amplitudes proportionally. Thus a 10% variation in line
voltage will have no disproportionate influence on the system
operating conditions.
integrated with line power without the need for tie-in
electronics. This feature contributes to the economic viability
of a domestic CHP system.
To make Stirling convertors practical for residential use,
the installed system cost must be low. Variables such as
energy tax credits, availability of low-interest home-equity
loans and electricity and natural gas rates all affect an
economic decision, but it is nonetheless clear that a viable unit
must be manufacturable at relatively low cost. A prospect to
be explored is a thermo-acoustic design. This technology
allows for elimination of one of the moving parts (the
displacer), replacing it with a tuned cavity and inertance tubes.
While the analysis and design of such systems can be
complex, the result can be potentially cheap to manufacture.
In continuing work, we plan to fabricate and test a Stirlingconvertor design based on the analysis presented here.
VI. REFERENCES
[1]
[2]
[3]
Fig 4: Simulation of start-up transient. Pistons and displacer start
from rest at zero displacement. Line power is applied, and all three
components pull into synchronism with the desired phasing and
amplitudes.
[4]
It is also irrelevant what the line-voltage phase is upon
start-up (engagement of the alternator coils). The Stirling
convertor is surprisingly robust. It would behave like the
equivalent of a negative appliance—plugging it in would
consume negative power.
To halt the convertor, it is only necessary to short the
alternator coils. This would result in dynamic braking,
bringing the pistons (and, consequently) the displacer to rest.
[5]
V. CONCLUSIONS AND FUTURE WORK
[9]
Integrating a heat engine with a residential furnace can
accomplish more efficient utilization of finite fuel resources.
This is apparent from cost savings of electricity that can be
produced from a combined heat and power system. Due to the
nonlinear constraint of net-metering regulations, however, it is
not always economically advantageous for the consumer to
produce any more electric energy than they would consume
each month. A consequence of this constraint is that the
efficiency of a heat engine that produces electricity is almost
irrelevant, provided the waste heat is desired for space heating.
In testing at NASA, it has been demonstrated that Stirling
converters can be built to be extremely reliable, requiring no
maintenance for decades of operation. Another attractive
feature, evaluated in simulations presented here, is that a
Stirling convertor designed to operate at line voltage and
frequency requires no automatic control system and can be
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VII. BIOGRAPHIES
Wyatt Newman, Ph.D., P.E., is a professor in the EECS Dept
at CWRU with research in mechatronic systems. He has
degrees in engineering science (Harvard College), electrical
engineering (Columbia U.) and mechanical engineering
(M.I.T.).
David Buckmaster is a graduate student in the EMAE Dept at
CWRU. He has degrees in Computer, Systems/Control, and
Electrical Engineering.