Note : Les descriptions sont présentées dans la langue officielle dans laquelle elles ont été soumises.
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Description
Isothermal P sitive Displacement ~achinery
Background of the Invention
INTRODUCT I ON
There are in general two types of machinery used
either to do work on or to have work done by the com-
pression or expansion of gases. These two generic
types of machinery are positive displacement and
turbine. The positive displacement type includes
various mechanically driven or driving pistons or
vane type rotors. A volume of gas is carried at rela-
tively low velocity from one volume to a different
one, either larger or smaller depending upon the func-
tion of compressor or engine. In the othee type of
machinery, turbines, the gas flow through blades
occurs at a velocity of roughly the speed of sound of
the gas. It is well known to those designing such
machinery that the turbines can be made more efficient
than positive displacement machinery. The reason for
this difference in efficiency has frequently been
obscure. A knowledge of the source of this ineffi-
ciency will allow positive displacement machinery to
be designed in a fashion such that the inefficiency
or loss is reduced by a significant factor to a mini-
mal value. There is, of course, the well-recognized,
additional loss of energy in positive disp'Lacement
machinery due to the friction between whatever is the
displacer, piston or vanes, and the walls of the
chamber. The turbine in turn avoids this inefficiency
but has others such as the friction of aerodynamic
flow at velocities near the sound speed.
HEAT EXCHANGE AND TOTAL ENE~:Y LOSS
Frictional loss between sliding parts is impor-
tant, but not usually the principal energy loss in
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the system. However, I will focus on one property of
positive displacement machinery that does cause a
major inefficiency and that is not well understood.
This is the heat exchange between the gases being
compressed or expanded and the walls of the positive
displacement volume. This heat exchange is usually
accepted as fundamental. Instead, I claim it can be
significantly reduced or enhanced as best suits the
purpose of the machinery, reduced in the case of adia-
batic cycles, and enhanced in the case of isothermalcycles.
HEAT EXCHANGE WITH THE W~LLS
Let us consider first compressors, although these
comments can be equally applied to expansion engines
with an inversion of terms. If a gas is adiabatically
compressed, it becomes both hotter as a function of
compression as well as increased in pressure. The
increase in temperature and pressure follow the well
known relations of the adiabatic law. ~n some cases,
as in a gas compressor, the additional temperature
created in the gas is later rejected as a nuisance,
although a significant fraction, even a malor frac-
tion, of the useful work may be wasted in the rejec-
tion of this heat. In any gas compressor, where this
heat is rejected, it is more efficient to reject this
heat as early in the cycle as possible so that less
work is done achieving a desired volume of cooler
compressed gas. In other cases where a compressor is
used, as in a Rankine cycle heat pump or in a compres-
sion cycle of various internal combustion engines,e.g., supercharging, this departure from an adiabatic
compression due to heat exchange of the working fluid,
i.e. gas, with the walls of the compressor is a major
disadvantage and inefficiency of the system. A point
of my related invention, the subject of Ca~adian
Patent Application Serial No. 411,2~6, filed
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September 13, 1982, is that by proper design of the
input and output ports of adiabatic positive displace-
ment machinery this heat exchange can ~e reduced to a
small value.
The mechanism for this heat loss is turbulent
motion of the working fluid making contact with the
walls during compression or expansion. There are two
parts to this heat exchange: (l! the heat exchange
between the gas and the wall if the wall were held
isothermal, and (2) the heat impedance of the wall
itself. It turns out that the heat impedance of the
wall is such that the wall acts as a time lag averag-
ing reservoir coming to a temperature equa:l to the
mean temperature of the gas at a delayed phase of the
stroke. The t;me phase lag as well as the magnitude
of heat exchange are both detrimental to adiabatic
efficiency.
T~ER MAL ~ K IN DEPTH
One can calculate the heat mass of the wall dur-
~0 ing the transient contact with the gas by calculating
the thermal skin depth within the time of heat contact.
The thermal skin depth, d, of penetration of heat (or
cold) within the given time t is expressed mathemati-
cally as
d = ~(K/CV) t]l/2
where Cv is the specific heat of the wall material, K
is thermal conductivity, and t is the time. (K/CV)
is often called the diffusion coefficient. For typical
materials where Cv is 1 calorie cm~3 deg~l, and the
time = 10-2 sec (for a stroke at 3000 rpm) or longer,
the skin depth will vary between 3 x 10-3 cm for a
plastic with K = 10-3 cal cm~3 deg~l at the highest
speed to 3 x 10-2 cm for a metal and a large slow
piston. Even the smallest skin depth corresponds to
a heat mass equivalent to several centimeters of air
or freon at atmospheric pressure. Therefore the heat
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mass of the skin depth of tlle wall in contact with
the gas will be comparable to or larger than the heat
mass of the gasO It is usual in engineering practice
to neglect this skin depth factor and assume that the
wall takes on a temperature which is the time average
of the heat flow from the gas. In this case the pri-
mary factor in determining heat loss is the theoreti-
cal heat exchange of the gas with an assumed isothermal
wall almost independent of wall properties. Later I
will show the importance of the time dependent phase
lag of the heat flow. First I will demonstrate the
skin depth effect. We assume that the walls of the
chamber will be smooth and then the heat loss will be
governed by the turbulent flow exchange wi~h a smooth
wall.
Description of the Drawings
Figure 1 is a diagram of transient heat transfer
into a uniform material;
Figure 2 is a PV diagram of various heat cycles;
Figures 3 to 6 are graphs of work during a cycle
for Stirling cycle machines with different phase angles
and showing the affect of losses on performance;
Figure 7 is a side cross-sectional schematic
drawing of a typical Stirling cycle heat pump;
Figure 8 is a side cross sectional view in gener-
ally schematic form of a bellows Stirling cycle heat
pump embodying the present invention;
Figure 9 is a side cross-sectional detail view
of the regenerator and part of a bellows;
Figure 10 is a detailed side cross-sectional
view of a portion of a rippled bellows with baffles;
Figure 11 is a side cross-sectional view in gener-
ally schematic form of a free piston heat pump embody-
ing the present invention;
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Figure 12 is a side cross-.sectional view in gener-
ally schematic form of a heat driven heat pump embody-
ing the present invention;
Figure 13 is a side cross-sectional view in gener-
5 ally schematic form of a heat driven engine embodyingthe present invention; and
Figure 14 is a side cross-sectional view in gener-
ally schematic form of an isothermal air compressor
embodying the present invention.
EXPLANATION OF DIFFUS IVE HEAT FLOW
In Figure 1 I show the classic solution of the
diffusion of heat from one reservoir 1 into a second
reservoir 2. Let us assume that 1 is hotter at Tl
and is a turbulent gas with essentially infinite abi-
lity to transport heat up to a barrier 3. The heatdiffuses into, or out of, region 2 with a diffusivity
K/CV. Then the distribution of heat or temperature,
T, as a function of depth, x, follows a sequence of
"error function" solutions in which
~ = T2 + (Tl - T2) exp(-x2/d2)
or T = T2 + (Tl - T2) e(-x2/d2)
where as before
d = ~(K/CV) t]l/2
The distance d is the centroid of the depth of penetra-
tion of the thermal wave. The three curves labeleddl, d~, d3 are the temperature profiles of times tl,
t2, t3, where tl is less than t2, is less than t3,
with characteristic skin depths dl being less than d2
being less than d3. If Tl is time dependent as it
would be in a cylinder with alternatively hot or cold
gases, then the actual distribution of temperature
should be a simple addition of such solutions. In
this sense "cold", i.e. Tl is less than T~, can pene-
trate into the wall just as well as hot, Tl is greater
than T2. The skin depth is just the characteristic
averaging depth of each temperature variation in a
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time t. The heat mass described by each curve is H =
d(Tl - T2)Cv and hence the longer the time the heat
has to "soak" in, the greater the heat transferred.
Typical diffusivities and skin depth heat masses are
shown in Table 1 for various materials. A frequency
of 3000 rpm is chosen as an example and the skin depth
heat mass is compared to 8~1 compressed combustion
gases typical of an Otto cycle engine. The diffusive
properties of air without turbulence are added for
comparison. One can see that the purely diffusive
heat flow in air leads to a skin depth heat mass that
is very small, 10-3 of that of the wall. Therefore
for the wall heat mass to be important requires aug-
mentation of the gas heat flow by turbulence.
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TABLE 1
Diffusivity, skin depth, heat mass of various materials
assume 3000 rpm, t = 1/~2f) = 0.01 sec.
Thermal Heat Heat mass of
5Conductivity Capacity Diffusivity skin depth
Watts/cm2/ Cal cm~~/ D/Cv Cv(Dt~l/2
deg C/cm deg C/cm cm2 sec~l cal cm~~
Carbon 0.5 0.81 0.13 0.0164
Steel
Stainless 0.14 0.81 0.036 0.0087
Steel
Nickle- 0.11 0.81 0.028 0.0076
Chrome
Phosphorus 2.2 0.84 0~55 0.035
Bronze
Berylilium 0.8 0.84 0.20 0.021
Copper
Aluminum 1.6 0.58 0.57 0.025
Alloy
Carbon 0.28 0~3 0.2 0.0075
Coke
Aluminum 0.30 0.8 0.08 0.013
Oxide
Ceramic
Silicon 0.016 0.8 0.004 0.003
Dioxide
Fused
Air 2.3 x 10-4 2.86 x 10-4 0.19 1~,25 x 10-5
atmospheric
Air 2.3 x 10-4 2.3 x 10-3 0.023 3.5 x 10-5
8 x atmospheric
The heat capacity of air plus fuel, eight-fold
compressed - 5 x 10-3 cal cm~3 or roughly twice that
of compressed air alone. A volume defined by a
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cylinder and piston at maximum compression with aver-
age dimension of 1 cm will have a heat mass of the
charge that is 30% of the heat mass of the thermal
skin depth of the wall of carbon steel. If part of
the wall is covered by an oil film or carbon black of
lower diffusivity, this fraction will be larger. Thus
a small part of the heat of either compression or
combustion delayed to the next period of compression
or combusion can be significant. It will increase
the energy of compression, and reduce the efficiency
of the cycle.
TURBULENT HEAT E~CHANGE WITH A SMOOTH SURFACE
If a gas flows in a smooth-wall pipe, then the
properties of turbulent fluid heat exchange are such
that the gas will reach thermal equilibrium with the
wall after moving roughly 50 pipe diameters (American
~andbook of Physics, 1963). This is also the viscous
slowing down length, or the length in which kinetic
energy is dissipated. The quantity "50 pipe diameters"
is determined by the peculiar properties of the laminar
sub-layer. This is the boundary layer between turbu-
lent fluid flow and smooth ~ipe wall. In the case of
the cylinder or other compression volume the appropri-
ate consideration is the distance the fluid (or gas)
travels in contact with the wall during the time of a
stroke. If the gas enters from a valve with a high
velocity relative to the chamber, then the gas will
circulate many times within the compression chamber
during the time of a compression or expansion stroke.
The number of cycles of circulation can be roughly
estimated by the ratio of the velocity of the gases
entering through the input valve to the velocity of
the piston. The average ratio of the valve area to
piston area is frequently about 20 to 1 (Taylor, 1~66),
so that gases entering the cylinder have velocities
between 10 to ~0 times that of the piston velocity.
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In general the gases enter the chamber non-
symetrically with respect to the compression volume
so that the turbulence genera~ed by the flow will be
greater than that induced in a normal pipe flow of a
fluid moving through a pipe. Therefore the heat
exchange with the wall will be greater when the tur-
bulence is greater. We expect roughly e-Eold of heat
exchange within roughly 10 circulation times because
the gas flowing by corners will be more turbulent
than straight pipe flow. Therefore the typical piston
with restricted inlet valves will allow heat exchange
of the gas with the wall of roughly half the differen-
tial heat of the gas during the time of compression
or expansion stroke. Since the differential tempera-
ture of the wall relative to the gas is roughly 1/2the total temperature difference, then roughly 1/4 of
the heat is lost to the wall. It is this large heat
exchange which accounts for the primary inefficiency
of such gas handling machines. The only way to avoid
this heat loss is to allow the gases to enter the
compression volume with low velocities. Then the
distance the gas moves during a stroke is small
(measured in diameters) and the heat exchange will be
small. If the flow velocity of the entering gas care-
fully matches the velocity of the piston or othercompression members, then we expect a weakly turbulent
boundary layer, i.e. not perfect laminar flow but
instead a low turbulence. The near absence of turbu-
lence I call near-laminar flow and hence the crucial
design is to create near-laminar flow of the input
gas to the compression or expansion cycle. If the
flow is to be near-laminar, at the piston velocity,
then the inlet port area must be close to the full
piston area. Or similarly, in an expansion engine,
the inlet ports must again be equal to the piston
area. This also applies to rotating vane machinery.
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THE I~EFFICIENCY DUE TO THE EXCHANGE OF ~EAT OF
THE GAS WITH THE WALL IN AN ADIABATIC CYCL~
Suppose a gas initially at temperature Tl is
compressed such that its final temperature would be
T3 if it were a perfect adiabatic compression but
instead is held isothermally at an intermediate tem-
perature T2 during the latter part of compression.
Then Tl is less than T2 is less than T3, and then the
heat energy in the gas after it leaves the piston
will be less than it would be by the ratio T2/T3.
(The mass of the gas is conserved). Therefore the
inefficiency factor of an adiabatic cycle or the heat
loss is just the difference (T3 - T2) divided by the
heat that would have been in the gas (T3 - Tl).
Depending upon the cooling of the cylinder walls and
other factors T2 might be only half way between Tl
and T3, and therefore compression machinery would be
50% efficient in following an adiabatic compression.
The temperature T2 that the wall reaches will be a
~0 complicated function of the heat exchange process and
the cooling of the walls. In general the gas will
not come into equilibrium at every point in the stro~e~
and so only an approximation to this heat loss will
actually occur. However, the fact that a simple cal-
culation indicates that up to 50% of the theoreticalmaximum heat can be exchanged is sufficient reason to
try to design machinery where one avoids this heat
short ci~cuit and its attendant loss in efficiency.
If the wall remained isothermal at temperature
T2, then this heat loss to the walls would be an
actual advantage in a compressor as, for example, a
refrigeration cycle or normal air compressor. However,
the heat e~change of the gas to the wall is more com-
plicated than this. If the gas can lose heat to the
wall in part of the cvcle it can also gain heat from
the wall in another part of the cycle if the wall is
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hotter than the gas. The wall will be hotter than
the gas for a transient time due to the skin depth
effect. This latter effect of heating the gas from
the wall is particularly harmful to the efficiency of
the compressor because the heating of the gas occurs
at its induction when the wall is hotter than the
inlet gas. The gas is then compressed with higher
heat than the ideal adiabatic cycle, and hence more
work is required than would be required for the
idealized cycle. Thus the heat is exchanged with a
harmful phase lag. Let us illustrate these ideal
cycles with and without heat exchange with the wall,
Figure 2.
The gas is drawn into the cylinder during the
induction stroke starting at temperature To along the
constant pressure PO to the volume, VO. In the ideal
cycle it starts compression at volume, VO along the
pure adiabatic curve 1, reaching the final reservoir
pressure Pl at volume Vl and temperature Tl. Several
possibilities due to heating the gas by the wall exist:
(1) If the gas is heated by + Tdiff only
during induction, then the pressure-volume relation
will remain the same. That is, since the gas is only
heated by the walls during induction and not during
compressionl by assumption, the compression will be
adiabatic (curve 1) and therefore will arrive at the
same state Vl, Pl, but at a higher temperature T =
(Tdiff + To)~To x Tl. The excess heat will be later
rejected, therefore requiring more work to deliver
the same mass of gas.
(2~ Heat can be added after the start of
compression and the gas will follow the curve 2,
steeper than the pure adiabatic one. The gas tempera-
ture is then likely to exceed the wall temperature,
transferring heat from the gas back to the wa~l and
the curve will bend over, curve 3, less steep than
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the adiabatic curve 1. The work required will be
greater. Curve 4 is more realistic in that wall cool-
ing of the compressed gas at the end of the cycle may
actually reduce the final gas temperature, T4 at V~,
below Tl at Vl of the adiabatic case, but the net
work still exceeds the adiabatic case.
(3) The wall can be cooled perfectly and
retained at the temperature To~ the gas can exchange
heat with the wall perfectly and then the compression
is isothermal along curve 5. This is the minimum
work cycle to obtain cold gas at the ~inal temperature
Ts = To~ It usually cannot be achieved in practice,
again because (1) the skin depth argument that isolates
the interior from the exterior on a transient basis,
and (2) turbulent heat exchange is only partially
effective in a normal cylinder and piston.
SUMMA~Y OF HEAT ~OSS AND ADI~BATIC CYCLE
The heat exchange occurs because of turbulent
flow in the induction gas. The maximum gas mass or
minimum temperature To is maintained during induction
only if either the walls are retained at temperature
To or induction is near-laminar flow. During compres-
sion the same argument applies. However the thermal
skin depth argument says that if the wall is thick
compared to the skin depth t it will average the heat
flow on the outside, but inside it will alternately
be hot and then cold in a thin layer. If the gas is
turbulent, this alternately hot and cold heat reservoir
will cause heating of the induction air at the worst
time, causing the compressed gas to reach a hotter
temperature T3 that in turn heats the gas still further
and requires still more work, and so forthr until the
higher average temperature of the walls allows the
heat to be carried away. This is an inefficient com-
pressor. It is better to reduce the heat exchangebetween the gas and walls by decreasing the turbulence
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and having near-laminar ~low induction as well as
compression.
ISOTHERMAL COMPRESSION
The opposite extreme of an adiabatic compression
(or expansion) is an isothermal one where the heat is
taken out (or introduced) continuously throughout the
stroke. An engine cycle based upon this continuous
heat exchange during both compression and expansion
is called a Stirling cycle. The usual machinery,
which employs a piston and a cylinder, designed for
such a cycle has a difficulty similar to that of an
adiabatic cycle, namely, heat dif~usion in and out of
the wall to a partial degree. In other words, only
part of the gas heat is exchanged. In the isothermal
case we want all the heat to be exchanged many times
during the stroke. The two independent effects of
the skin depth argument as well as the decay during
the stroke of the inlet turbulence ensures a half-way
result.
For an isothermal cycle we want (1) the thermal
impedance of the wall to be small in order to conduct
heat back and forth easily, and (2) we wanl: the heat
mass of the wall thermal skin depth to be large com-
pared to the gas heat mass. In this fashion the inside
wall temperature will remain isothermal, i~e., will
average the temperature fluctuations and remain at
the outside temperature.
Then the wall can be cooled or heated continuously
and maintained both inside and outside at constant
temperature. Then if the gas is maintained in close
thermal contact with the wall that bounds the compres-
sion or expansion volume during the stroke an iso-
thermal compression or expansion process can be
achieved.
The transient heat exchange due to partial tur-
bulence and thermal skin depth is deleterious to all
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positive displacement heat machinery. As a useful
measure Taylor (1966) ascribes about 30~ efficiency
loss to heat loss in a gasoline engine and Up to 50~
heat loss in a diesel engine. In other words a gaso-
line engine could be 45% efficient insteacl of 30% an~a diesel could be 7~ efficient rather than 35~ to
40%. These are large potential gains and therefore
warrant the following complexity to achieve these
results. Conversely, Stirling cycles are particularly
useful for heat pumps, and here the lack of effective
heat transfer can make up to factors x2 difference in
the performance of such machinery because both the
compressor and the expander are affected by heat
transfer.
General Vescription of the Invention
ISOTHERMAL CYCLE MACHINES GENERALLY
In an isothermal cycle we want to exchange heat
continously between the contained gas and a thermal
reservoir. I have described above how the skin depth
in the cylinder walls prevents the heat from penetrat-
ing to an external reservoir during part of a single
stroke, and how the skin depth reservoir exchanges
heat with the gas in the worst fashion for efficiency.
Therefore if we wish to exchange the gas heat frequently
during the cycle, both the thermal impedance of the
heat of the gas to the wall as well as the thermal
impedance of the heat through the wall must be made
small. To achieve this result the surface to volume
ratio must be made as large as possible, and in some
3~ cases the turbulence must be maximized.
The surface to volume ratio of a given geometric
volume is minîmum for a sphere or right circular
cylinder whose length equals its diameter. This ratio
is 3/radius for both geometries. In a sphere or right
circular cylinder, the interior volume is a maximum
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distance from a wall. This is the ideal ~eometry for
adiabatic cycles. In contrast, for isothermal cycles,
I desire all 1uid to be close to a wall so that the
surface area is greatly increased comparecl to a right
c;rcular cylinder of equivalent volume. ~ factor of
10 increase in area is a minimum value ancl signifi-
cantly larger ratios are possible ancl desirable for
e~ficient isothermal machinery.
BELLOWS COMPRESSOR AND EXPAND~R
To fulfill the above-described objective5, I pro-
vide, in accordance with my invention, a compressor-
expander comprising a variable volume chamber defined
by flexible, thin metal bellows side walls that are
so configured as to ensure that all of the gas in the
lS chamber is close to a metal wall and no mass of gas
is remote from the wall and hence no volume of gas is
quasi-adiabatic, but, instead, all the gas is isother-
mal in thermal contact with a metal wall. This is
achieved by one of three bellows configurations. In
one case the bellows are designed with a small inside
radius, say from approximately 1/5 to 1/10 o~ the
outside radius, so that the area of the hole and hence
the inside volume is small (e.g. 1/25 = 4~ for a 1/5
ratio of inside to outside radius~, of the outside
area or convolution volume. In the second case a
pair of nested bellows, one inside the other, is used
as the compression-expansion volume. Here the annular
space between the bellows is made small again to ensure
that all gas is close to a metal heat transfer surface.
In the third version, a single peripheral bellows
with an inside radius of about 1/3 to 2/3 of the out-
side radius is fitted with baffles, one at each con-
volution, fastened into the seam between the discs.
The baffles divide up the central space and provide
close thermal contact with the central gas that is
remote from the bellows and prevent the central gas
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from becoming adiabatic. Holes suitably arranged in
the centers, the perimeters, or both, of each baffle
in staggered positions baffle-to-baffle provide radial
and circumferential flow patterns that bet:ter dis-
tribute the gas and promote heat transfer. The holes
in the baffles are, of course, designed with a view
to avoiding excessive, harmful gas flow friction.
The objective of the bellows design is to give a
large surface area for heat exchange, create turbu-
lence, have a thin wall for heat conduction and pro-
vide sufficient radial thermal conductivity to carry
heat from within the chamber to the outside. ~refer-
ably, the ratio of bellows wall area to the area of a
right circular cylinder of equivalent volume should
not be less than lO:l. In addition, there are to be
no large trapped volumes of gas that are near adiaba-
tic, but instead all the gas is to remain isothermal
in close contact with the walls. A cup type displacer
inside the bellows that displaces the gas at the end
of stroke is not sufficient; the extended stroke volume
is large and not in contact with the walls and there-
fore is a major efficiency loss.
The heat must be transferred from the inside gas
through the wall to the outside gas. The criterion
of successful heat exchange is that the inside gas
must remain isothermal during the time of compression
or expansion and this temperature should be the same
as that of the external reservoir of gas. Therefore
thermal lag is the inverse of successful heat trans-
port. There are several thermal lags:
(l~ The transfer of heat from the internal
gas to the metal bellows walls.
(2) The temperature drop through the metal
wall.
(3) The external heat transfer to the
reservoir from the metal walls.
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The second thermal lag, (2) above, is small and
therefore will ~e discussed and e~iminated first.
For a bellows compressor or expander the surface area
of the many convolutions of the walls is 50 to 10Q
times greater than for the same internal volume of
gas in a normal cylinder. In a normal cylinder the
ratio of the heat mass of the thermal skin depth to
that of the internal gas is less than 10. The heat
mass of the thermal skin depth of the bellows is very
large, 100 to 1000 times, compared to the heat mass
of the internal gas~ Hence the small heat of the
internal gas in a cycle does not significantly change
the wall temperature, and the wall remains very nearly
isothermal during a cycle. For the same reason the
thermal lag of the metal bellows becomes negligible.
The temperature difference of the two walls, inside
and outside, can be calculated to be extremely small,
less than 1C or useful size machinery. Then the
major thermal lags are the heat transfer to the inter-
nal and external bellows surfaces.
The external heat transfer can be made large andhence the thermal lag small by inducing a high velo-
city flow of a fluid (usually air) around the external
bellows surfaceO The external fluid can be exchanged
many times in a convolution within a cycle time. This
external surface naturally induces turbulence and
high heat transfer. On the other hand the internal
gas may not be as turbulent and hence will not exchange
heat within a cycle as many times. However, the pro-
cess of induction (and exhaust) of the gas introducesturbulence. The width to length ratio of the gas
space between bellows convolutions is small and
enhances heat transfer. Oscillations of the bellows
can be introduced (they will occur naturally~ during
a stroke; this shuttles the internal gas from one end
to the other during a stroke and induces a large
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turbulence and hence heat transfer. A combination of
th~se effects results in a large heat transfer inter-
nally and hence small temperature lag and an efficient
isothermal compressor (or expander).
The hea~ mass of the wall acts as an averaging
thermal reservoir so that the external heat transfer
can take place during the full cycle. To keep the
temperature lag small, the ratio of the effective
heat mass of the wall to the heat mass of the gas
should be very large. The effective heat mass of the
wall is the smallest of elther the thickness or the
thermal skin depth. Therefore, if the skin depth is
larger than the wall thickness, the wall thickness
becomes the wall heat mass. Mechanical consi~erations
on the other hand like oscillating mass, spring con-
stant and fatigue life indicate that a small wall
thickness is desirable but limited by the stress
induced by the gas pressure.
Stirling cycle heat pumps and motors use a pair
of compressor-expanders interconnected via a regenera-
tor. As described in more detail below, it is just
as important in the case of the regenerator to mini-
mize losses as it is in the compressor-expanders.
Gas flow friction losses, the most important of the
possible energy losses in the regenerator, should not
exceed 3~. The regenerator should be designed to
provide about 5 to 10 heat exchange lengths. The
dead space of the regenerator should not e~ceed about
one-fifth of the compressed volume of the working gas
or about 10~ of the displacement volume in order to
minimize the reduction in the specific power.
Summary of the Invention
The present invention is characterizecl in that
the ratio of the surface area of the bellows-like
walls of the ~ariable volume chamber to the volume of
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the chamber and the configurations of the convGlutions
of the bellows-like walls are such as to ensure during
each stroke numerous heat exchanges be~ween the work-
ing gas in the chamber and the bellows-like walls by
both laminar and turbulent heat transfer, thereby to
ensure that heat is conducted to and through the
bellows-like wall and thence to and from a thermal
reservoir external to the bellows-like wall and to
produce a substantially constant temperature cycle.
In the case of chambers having two flexible thin
metal bellows-like walls, one nested within the other,
defining an annular chamber, the bellows-like walls
are closely spaced such that the chamber is substan-
tially free of trapped volumes that are not in close
diffusive turbulent thermal contact with one of the
walls. In the case of a single peripheral bellows-
like wall, the inside radius is from about one-third
to about one-tenth of the outside radius and the
inside central volume within the inside radius is
small and in close diffusive turbulent thermal contact
with the wall. Baffles connected to the bellows-like
walls within each convolution and having holes to
enhance the circulation of the working gas enhance
the heat transfer between the gas and the wall.
The bellows are designed so that no mass of gas
is ever more than a few millimeters (10 at most and
ordinarily in the range of 2-5) from a wal] surface.
The maximum spacing, moreover, is proportional to the
inverse of the square of the frequency [l/frequency2]
and the inverse of the initial pressure Pi. ~ence
the lower the operating frequency or the pressure,
the closer the maximum gas-wall spacing must be.
The bellows-like walls may comprise annular discs
joined and sealed at each inside and outside edge to
an adjacent disc, preferably by an elastomeric adhesive
1~4Z~l
-2~-
on the inside seams and by an elastomeric adhesive
and a crimped channel at the outside seam.
The following further characteristics of inven-
tion are preferable:
1. The movable end walls of the compression
and expansion chambers of heat pumps and motors
are driven harmonically at a phase angle of fxom
about 90 to about 120. Such a drive may be
imparted by a free piston positive displacement
engine operating on an open Otto or diesel cycle,
or by a linear electric motor.
2. A fluid is caused to flow through the
external thermal reservoir and over the surface
of the bellows-like walls externally of the
chamber for enhancement of the heat transfer
from the working gas to and through the bellows-
like walls.
~. The ratio of the area of the bellows-
like walls to the area of a right-circular
cylinder of equivalent volume should not be less
than about 10:1. In the case of bellows chambers
having baffles, the ratio of the tota:L area of
the bellows walls and the baffles should likewise
not be less than 10:1.
4. The heat mass of the thermal skin depth
of the bellows-like wall is not less than about
100 time~ the heat mass of the working gas in
the chamber.
5. The compression ratio of the machinery
~0 is of the order of 2:1 to 2.7:1, and preferably
at the higher end of the range.
In machines having regenerators (required in
heat pumps and motors) the dead volume of the regenera-
tor is less than about 10% of the displacement volume
of the heat pump or motor. The regenerator provides
for about 5 to about 10 heat exchange lengths, and
~Z~4~
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the heat mass of the metal in the regenerator is of
the order of 10 to 20 times the heat mass of the work-
ing gas. Most importantly, the gas flo~ friction
loss in the re~enerator must not exceed about 3~.
Heat pumps and motors comprise two (2) isothermal
units, each having a bellows compression chamber and
a bellows expansion chamber. In these machines, the
compression chamber and expansion chamber of the two
units are mec'nanically coupled to move conjointly~
A compressor embodying the invention is charac-
terized in that there is a single bellows compression-
expansion chamber having a suitably dri-~en end wall
and having valved supply and exhaust ports in the
other end wall.
An especially interesting machine is a low tem-
perature difference Stirling cycle heat pump driven
by a high temperature ~tirling cycle engine powered
by a hot gas, for example, exhaust from a burner,
solar heat, or some other waste heat. Such a combina-
tion is referred to as a Veullimier cycle.
Theory of the Invention
STIRLING CYCLE HEA~ PU~P THEORY
An isothermal heat pump or Stirling Engine has
been the subject of considerable r~search endeavor~
and yet the fact that such effort has resu:Lted in
only a small market penetration attests to the diffi-
culty of the subject. The current state of the art
is covered in the book Stirling CYcle-Machines/ 1973,
reprint 1976, G. Walker, Clarendon Press, Oxford.
There are, of course, some developments since then
(one of which is mentioned below), but the prior art
is best covered in the Walker book. (G. Walker's
latest book, "Free Piston Stirling Engines," 198~,
University of Calgary, Alberta, Canada, has been
~Z~Z9~
-22-
reviewed before publication and does not effect the
following discussion.)
The well known Stirling cycle is composed of two
isothermal functions, a compression and an expansion,
and a reversible transfer process (the regenerator).
The objective of this cycle is optimizing the energy
efficiency and the specific power. These will always
conflict. The practical use of such a cycle is as a
heat pump for transferring heat, or equally as an
engine for power. Generally heat pumps must work
efficiently between relatively modest temperature
differences, Delta T~T = 10%, compared to engines
that, depending upon materials and heat sources, will
utilize Delta T/T = 50~. Hence heat pumps are
emphasized for commercal reasons where Delta T/T is
small, i.e., where Delta T is characteristic of heat
pumps for domestic use, such as refrigeration, where
Delta T is 30C. and T is the absolute temperature,
typically 3~0 degrees. As a consequence, efficiency
becomes a major challenge. Just how serious small
losses are is shown in the accompanying graphs
(Figures 3 to 6 of the drawings).
Cycle Program
First the cycle program must be discussed. The
phase relationship between the components of the
~tirling cycle, compression; transfer; expansion;
transfer, can be idealized where each process takes
place separately. Indeed, the "rhombic drive" is a
logical development to achieve this nearly full separa-
tion of cy~le elements and presuma~ly higher efficiency.However, the added complexity, the added friction
loss, and most importantly the lack of time overlap-
ping of the cycle functions make the idealized cycle
drive less attractive than the simple "near" harmonic
motion of a circular crank and crank rod. The lack
of 'Itime overlapping cycle function" is a subtle point
2~
-23-
that will be developed in greater de~ail ]ater, but
briefly the specific power of a given machine is
limited in one part by the frequency. The efficiency
in turn is effected non-linearly by, and is highly
sensitive to~ the gas friction loss in the transfer
part of the cycle. To keep this to a minimum means
using the major fraction of the cycle for transfer.
This necessarily competes with the time required for
heat transfer duriny compression or expansion. Hence
in the harmonic cycle, these functions "o~erlap" in
time, and for a loss of volumetric ef~iciency we gain
in net specific power. On balance, the rhombic cycle
is probably not worth the complexity. Hence this
development will emphasize simple harmonic motion.
Before discussing efficiency the relative phase
angle between the volumes must be understood.
The specific power of a given machine (isothermal
cycle element) depends upon the work per cycle. This
work is
Work = Pi Vi (ln CR)
where Pi is the initial pressure, Vi is the initial
volume, (ln CR) is the natural logarithm of CR, and
C~ is the compression ratio.
If the maximum pressure is limited by the strength
of the materials, where PmaX ~= CR Pi] is a constant,
then the specific power will be proportional to (ln
CR)/CR. This function is maximum for CR ~ e = 2.7, a
rather simple result. Nevertheless, with a limit on
PmaX the dependence of specific power on C~ is very
weak, being for example 94~ of maximum when CR = 2.
However, bellows, the essential element of this inven-
tion, are unlike the more common machine e]ements in
that they are much stronger when compressed than when
they are expanded. As a consequence, the limiting
pressure becomes Pi, not PmaX~ In this case the
specific power is proportional to (ln CR) and is more
~Z()4Z~
-24-
sensitive to reductions in CR. For example, (ln CR) =
0.59 when CR = 2, a significant loss of speci~ic power
compared to 0.94 with the limit of PmaX. As a conse-
quence there is a significant motivation to maintain
the compressior. rat;o large like 2a 7 fold.
Harmonic Phase Angle
The optimum phase angle with the inclusion of
parameterized losses has been investi~ated~ The usual
investigation of phase angle (see Walker, supra.,
Figure 5.4) shows an insensitivity to phase angle
with no losses. When losses are included (see Figures
3 and 4 attached) for ~elta T/T = 0.54, CR = e = 2.72,
the relative useful work (intercept of the work curves
with the right ordinate) for the two phase angles
120 and 90 is reduced by 28~ for the case of no
losses. The useful work for 90 phase angle is only
63% of the useful work for 120 phase angle when there
is the small loss of 6~ e., a total cycle efficiency
of 94%. Hence, the phase angle is important, provided
one can obtain the small dead volume necessary for
keeping CR = 2.7. When the phase angle is 121 between
compression and expansion, the allowable dead volume
goes to zero if CR = 2.72 = e. Hence one finds a
strong motivation to use a large phase angle with
minimum dead volume.
Efficiency Losses
These same calculations of useful work are para-
meterized as a function of the overall individual
cycle efficiency N. N is an efficiency parameter
that expresses the fractional approach of the real
expansion or compression process to a reversible iso-
thermal process. The loss fraction (1 - N) is a
measure of the mechanical work lost due to non-ideal
processes in a full cycle. The extraordinary result
of these calculations is the revelation of the extreme
sensitivity of useful work of an isotherma] cycle to
~ZV4,~
-25-
such losses. We see in the case of Delta T/T = 0.54,
i.e., a hot engine, that the useful output: work of
such a cycle is reduced by a factor of 2 for a 120
phase angle and to 38% for a phase angle of 90 when
the efficiency is 94%. In the case of a low tempera-
ture difference this sensitivity is further exaggerated.
In Figures 5 and 6 the useful work for two temperature
differences of Delta T/T = 15% and ~elta T/T = 10~
are shown for the two phase angles of 120 and 90.
Here a loss of about 2% (98% efficient) reduces the
useful work tc zero. If we are making a heat pump
rather than an engine, this sensitivity to cycle loss
means that the heating or cooling effect will require
more energy than the equivalent ideal Carnot cycle.
The conclusion is that irreversible cycle losses have
a major e~fect on the useful work or on the work
required to produce a given heat or cold as a heat
pump.
I next make a distinction between mechanical and
thermal losses. The cycle loss referred to above îs
a pressure loss and hence mechanical loss in the cycle.
Mechanical friction losses in the machinery as well
as gas friction loss in the regenerator transfer pro-
cess are similarly direct cycle losses. Temperature
loss, on the other hand, gives rise to cyc:Le losses
only in so far as the cycle pressure is effected. If
a regenerator accepts gas at a temperature Tl and
returns it, say, cooler at T2, then the heat corres-
ponding to Tl - T2 must be added to restore the gas
to the original isothermal value Tl. The process of
reheating the gas by the amount Tl - T2 can be accom-
plished by either of two processes: (1) by PdV work,
or (2) by additional heat flow from the reservoir at
Tl. The mechanical work requires expensive mechanical
energy, whereas the reheat from the reservoir is lower
"quality" energy by the ratio of the overall thermal
~z~z~
-26-
efficiency of the machine. For an isothermal cycle
the gas must be in thermal contact with the walls or
reservoir many times over, say 30 times, w~ithin a
given stroke (compression or expansion) in order that
the temperature and hence pressure not suffer a time
phase lag and hence direct cycle loss of say 1/30 or
3%. Therefore~ the thermal loss from the regenerator
should be restored in a time of 1/30 of the compres-
sion or expansion stroke. Therefore the thermal cycle
loss is less important than otherwise suspected.
This relative insensitivity of useful work to
regenerator temperature lag is noted in ~alker but
not understood. The conclusion is that cycle losses
in the compression expansion volume are more important
than thermal loss in the regenerator by the ratio of
(l/inefficiency~ of the machine.
The several phases in the cycle that lead to
direct cycle pressure loss are:
1. Mechanical friction of sliding parts.
2. Temperature lag due to lack of perfect
thermal contact between the walls and gas during com-
pression or expansion.
3. Pressure drop due to gas flow friction
in the regenerator.
The first loss of sliding friction is obvious,
and several Stirliny cycle machines using bellows as
the compression or expansion element just to reduce
the friction of sliding parts have been proposed in
the past. The second loss is the major loss in all
Stirling cycle machines. It is due to a significant
fraction of the gas behaving adiabatically during
compression or expansion so that the gas temperature
partially lags the reservoir temperature. If a volume
of gas were perfectly adiabatic then the temperature
variation during a stroke would be
Delta T = T (1- CR(Y~
iZ~4~
-27-
If CR = 2.72, then Delta T/T = 50~ for air and 95%
for helium. ~ence in order for the temperature phase
lag to be small the thermal contact with the walls
must be excellent. This extreme sensitivity to
trapped thermally isolated adiabatic volurne of a gas
is not generally recognized (Walker, 1973, 1976), and
is ignored in most all Stirling engine disclosures.
It is an object of this invention to reduce all ther-
mally isolated volumes of gas to a very minimum and
hence achieve high efficiency.
Finally the gas friction loss is more impGrtant
in the regenerator than the temperature lag by the
ratio (l/cycle efficiency), as has already been
explained. This sensitivity to flow friction is not
emphasized in the literature (Walker, 1973, 1976,
Chapter 7) and hence the design of regenerators is
uncertain and not complete. It is stated that "small
engines work better with the regenerator entirely
removed." A detailed analysis of why is not given.
It is an ob~ective of this invention to design the
regenerator as a rational optimization of all the
conflicting requirements.
DESIGN AND HEAT FLOW OF STIRLING CYCLE ENGINES
In the limit of large dimensions and high velo-
city, i.e., high Reynold's number, heat flow in a gastakes place generally by turbulent transport. However,
the distance of travel along and adjacent to a rough
wall must be considerable - like 5 to 10 channel widths
for one heat exchange length. For isothermal condi-
tions in a compression or expansion volume, as hasalready been pointed out, the gas must be in thermal
contact with the wall some 30 times during a stroke.
Therefore the gas must travel 150 to 300 channel
widths in turbulent flow to exchange heat. The fluid
friction loss must be small, less than 1% for cycle
efficiency. The friction loss will be the number of
~4Z~l
-2~-
thermal exchange lengths, 30, times the kinetic
pressure. The kinetic pressure is ~he pressure equi-
valent of the kinetic energy of gas flow. It is equal
to the density times the square of the velocity. This
implies that the kinetic pressure must be about 3 x
10-~ of Pi, or that the gas velocity must be less
than about t2% x sound speed). This maximum flow
velocity, 600 cm sec~l for air, or 1700 cm sec~l for
helium, is a practical upper limit in either the com-
pression element bellows or regenerator. At thesevelocities, and for typical bellows channel (convolu-
tions) widths, 1 to 2 millimeters, the Reyncld's number
of a tapered channel and half width in contact with
the walls turns out to be:
Rey = f (width/4) x velocity]/lkinematic viscosity]
approximately 1~000 for air
approximately 100 for helium
The higher sound speed of helium is compensated by
its much larger ~times 7) kinematic viscosity (i.e.,
viscosity/density) than that of air. These values of
Reynold's number are just where turbulent flow heat
exchange increases above laminar heat exchange, and
hence the heat flow will be partly laminar and partly
turbulent. It will be turbulent only if the maximum
velocities are induced. It will be laminar for most
pract~cal cases where helium is used as the working
gas. Since the advantage of helium (or hydrogen) for
Stirling cycle engines is so well recognized, a factor
of 8 to 10 improvement in performance, most practical
engines or heat pumps will use the light gas and then
the heat exchange will be primarily laminar and turbu-
lent heat exchange can be neglected. It is ironic
that heat flow in positive displacement adiabatic-
cycle engines is undesirable and primarily turbulent
and in isothermal machinery where we want heat flow
4'~9~l
~9
it is primarily laminar, but: this is the result of
channel size and gas properties.
LAMINAR HEAT FLOW IN STIRLING CYCLE BELLOWS MACHINES
Laminar heat flow can be characterized by a
diffusivity, D. For helium, D = pi-l cm2 sec~l, where
Pi is measured in atmospheres of absolute pressure.
In air it is 1/7 as much, or D = 1~7 Pi. Since Pi
for most practical machines using bellows and helium
at 1 to 2 atmospheres, D will be 1 to 1/2 cm2 sec~l.
The mean distance to a bellows convolution wall (2
walls per convolution) is 1/4 the convolution spacing.
The time constant for heat transfer is then
time = (width/4)2 Pi sec
A typical average spacing during the stroke is 1 mm
(2 mm extended spacing), so that the thermalization
time becomes:
time = 1/800 sec
If the thermalization must take place 30 times in a
stroke, then the minimum stroke time becomes 1/30
second, or a stroke frequency of 30 Hz to 15 Hz in
revolutions. Such a bellows of 50 convolutions would
have a stroke length of 10 cm. For a bellows of 20 cm
diameter, the displacernent volume would be 3,000 cm3
and~ for Pi = 2 atmospheres, the circulating work
would be 10 KW of which roughly 1/2, or 5 1~W, could
be used for heating, cooling, or power.
THE REGENERATOR DE5 IGN
The losses in a regenerator areo gas friction
pressure drop, limited gas wall heat exchange, dead
space volume, limited regenerator heat mass, and
regenerator mass conduction loss. The first is the
most important, as already pointed out~ The require-
ment cf heat exchange with the walls is roughly the
same as the compressor-expander heat exchange except
that heat exchange is not a direct mechanical energy
loss so that only 5 to 10 heat exchange lengths are
~2~9~
-30-
required. The dead space volume directly reduces the
specific power because it limits both the phase angle
as well as the compression ratio. The regenerator
dead volume should be no more than 1/10 o]E the com-
pressed volume or about 4~ of the displace~ent volume.
The limi-ting gas velocity has been calculated for
helium as 1705 cm seC-l for 30 exchange lengths, and
so twice this, 3 x 103 cm sec-l, can be used for the
regenerator, provided it is designed to be less than
7.5 thermal or friction exchange lengths long. Since
the displacement volume is (pi r3), and the displace-
ment or stroke occurs in a limiting time of 1/30 sec,
the effective regenerator cross-sectional area for
this example becomes:
Area = [displacement volume~/~(stroke time)~maximum
gas velocity)] = 30 cm2
Since the gas volume of the regenerator cannot be
more than 4% of the displacement volume, the effective
length of the regenerator must be:
length = (4% x displacement volume)/ Area
= 4% (stroke time) x (gas velocity) =
4 cm
= 0.4 x radius of bellows.
This very small length governs the geometry of the
machine. Before discussing this we must consider the
geometry of the gas cllannels and the heat exchange
medium of the regenerator.
The total length is 4 cm and roughly 8 exchange
lengths are desired. Then a heat exchange length
equal to a viscous exchange length of 0.5 cm is
desired.
Heat exchange length = (width/4)2(velocity/D)
= 0.5 cm
velocity = 3,000; D = Pi = 2; then the channel
width = 0.07 cm
120~Z~
The channels must be 0.07 cm wide - hence corregated
metal is suitable. The thickness of the metal must
be determined by the heat mass, the lengthwise conduc-
tivity and the thermal skin depth of the metal. The
heat mass of the helium gas at 2 atmospheres is roughly
4 x 10-4 cal cm~3 deg~l so that if the heat mass of
the metal is to be x20 that of the gas, a total o
3~ cm3 of metal is required. Since the gas volume of
the regenerator is length x area - 120 cm3, this is
1~4 the volume of the regenerator. There,ore foil
1/4 of the channel spacing thick will supply the
required heat mass. The foil thickness becomes
0.02 cm. The thermal s~in depth in the metal in a
stroke time of 1/30 sec is skin depth = (D x time)l/2 =
0~08 cm for D = 10-2 cm2 sec~l for stainless steel.
Since the skin depth is very large compared to the
half fo;l thickness, thermal lag in the foil can be
neglected.
The longitudinal heat conduction through the
foil from hot to cold end is
heat loss = Delta T [larea x conductivity)/length]
= 0.063 x (Delta T) watts.
This is a negliyible heat loss for all rea<;onable
temperature differences limited by materia:L properties.
We therefore have designed a regenera~or that
meets all the design criteria. A regeneral:or of this
design has been built and tested, and in a'll respects
it agreed with this simple theory.
With this regenerator design and bellows
compressor-expander units we can define the full
thermal cycle machine.
REVIEW OF DESIGN CRITERIA
The compressor-expander units must be the heat
exchangers of the machine. Therefore the surface to
volume ratio must be as large as possibleO Metal
bellows uniquely satisfy the criteria. No gas volume
~Z~42~1
~32~
may remain isolated from a thermal reservoir for even
a small fraction of a cycle. Therefore no large
volume remote from the bellows walls may exist. The
smallest irreversible loss, e.g. on the order of 3%,
makes a significant difference to the performance of
the machine. Therefore the bellows compressor-expander
units must be either a nested pair of bel]ows with a
relatively small annular gap, a bellows design where
the inner diameter is very small compared to the out-
side radius, or a bellows with baffles at the convolu-
tions of the baffles that intersect the interior
volume. Since the area is proportional to the radius
s~uared, the inner hole size of such a single bellows
compressor expander unit should be of the order of
1/6 to 1/10 the outer radius.
In the regenerator gas friction pressure drop is
a more important design criteria than the many other
characteristics such as dead volume, heat exchange,
heat mass, conduction loss, and skin depth. All dead
volume should be minimized, even if the gas is iso-
thermal, so that a compression ratio approaching e-fold
(times 2.72) is maintained at the largest possible
phase angle approaching 120. Mechanical friction
losses should be maintained small.
RESUI.TING DESIGN
The two bellows compressor-expander units (one
hot and one cold) must be connected by the regenerator.
If they were separated, there would be no possibility
of transferring the working gas without prohibitive
3~ pressure loss or dead volume. This leads to the con-
figuration of a regenerator with compression-expander
units (either a nested bellows pair or a small inside
diameter bellows) at each end. The regenerator design
is described above. It is the mid-plane member that
supports one end of each bellows. The bellows are
then compressed or expanded against the regenerator
~Z04Z~
by a suitable mechanism. ~eating or cooling air,
gases or even a liquid will then be caused to flow
across each bellows to establish a hot and cold ends.
Since the heating or cooling fluid flow e~ternal to
the bellows need only make one heat exchange length
with the bellows wall, the velocity can be higher and
is nearly continuous. Therefore heat exchange can be
turbulent and significantly greater than inside the
bellows. The air or gas can blow across the surface
transverse or parallel to the bellows axis and with
or without swirl will give adequate heating or cooling.
If a liquid is used external to the bellows, it is
incompressable, and so two compression-expansion units
180 out of phase should be used in the heat exchange
volume.
A Stirling cycle heat pump having bellows com-
pressor and expander units at opposite ends of a
stationary regenerator has been proposed heretofore
(see Raetz U.S. Patent No. 4,010,621, issued March,
1977). The Raetz design uses heat exchangers separate
from the bellows walls, and the bellows leave large
trapped adiabatic volumes. The present invention
involves two critical differences from the Raetz
patent design that provide an efficiently working
machine - the use of the bellows as the heat exchange
elements and a configuration for the bellows working
chambers that ensures numerous heat exchanc~es between
the gas and the bellows in each cycle due t:o the
absence of large trapped adiabatic gas masses.
There are other prior patents and literature
disclosing bellows machines that look like the present
invention, but they do not describe or suggest all of
the requirements of this invention, which provide
remarkable increases in efficiency. Among such patents
and literature are the following: Frankl U.S. Patent
No. 1,808,g21, June 9, 1931; Kohler et al. U.S. Patent
~2~4~
-34-
No. 2,611,236, September 23, 1952; Schuman U.S. Patent
No. 3,827,675, August 6, 1974; the Walker book, supra.
DRIVING MECHANISM
The driving mechanism will be either a rstating
mechanical drive with cranks, crank arms, and cross
heads or it can be a free piston engine(s), either an
electrical linear motor or fuel driven engine. In
general, an Otto or diesel free piston engine will be
more efficient than a bellows heat pump engine because
of the limitations in hot side termperature imposed
by the highly and alternately stre.ssed bellows. How-
ever, a bellows heat engine and heat pump combination
can be made where a small, high temperature difference
engine unit drives a larger low temperature difference
unit as a heat pump with significank thermal gain.
The Otto or diesel free piston engines have the dis-
advantage of lubrication and wearing parts. A bellows
Stirling cycle engine will have longer life and pro-
vide more complete combustion.
BELLOWS DESIGN
Welded metal bellows are now a commercially avail-
able item from several manufacturers. In general
they are specialty items that are expensive and diffi-
cult to manufacture without flaws. In particular,
the fatigue life is limited by the metallurgy at the
stress concentration points adjacent to the welds
where the metallurgy is critical and partially degraded
from the original material.
A bellows in a Stirling cycle heat pump can always
contain a positive pressure, i.e. Pi greater than 1
atmosphere. In this case the inner diameter seam of
the convolution will be in compression and will not
flex as much and therefore not fatigue. This is
important for the bellows design where the inner
diameter is much smaller than the outer diameter
~g~:91
-35-
because, if the bellows were extended, the tensile
stress would be larger and rapidly fatigue the
bellows.
I recommend instead a hellows construction speci-
fically for use in small temperature difference, roomtemperature bellows heat pumps. Such a construction
is cheaper and conducive to a longer life because of
the lack of welds. The seams are glued with a modern
elastomer, such as silicon rubber, that has nearly
infinite flexure life. The inner seam of each convolu-
tion is glued with no support since it is always in
compression. The outer seam is glued but backed up
with a rolled crimped "U" shaped channel. The channel
and elastomers distribute the stresses better than a
welded joint.
I also prefer to provide a baffle plate in each
convolution of the single bellows so that the gas
cannot go as easily directly through the chambers
central hole of the bellows but instead must fGllow a
zig-zag path between convolutions. The hole size
must become progressively larger towards the regenera-
tor end of the bellows to keep the gas friction low
enough. These baffles also increase the strength of
the bellows against the squirming mode failure so
that a larger length to diameter can be used.
Bellows Springs
When a free piston drive is used for the displace-
ment of the Stirling cycle compressor-expander unit,
the problem frequently is the requirement of an energy
storage mechanism or spring. If a standard steel
spring is used, then the metal weight of the spring
for a given energy storage turns out to be large,
relati~e to the other components and the frequency is
reduced. This is particularly true where a linear
electric motor is used tied to a 60 cycle power line.
The electrical frequency can be reconstituted but
~zo~
-36-
such components are expensive and the efficiency is
reduced. Consequently, there is a requirement for a
high efficiency gas spring.
~ormal gas springs of a cylinder and a piston
suffer the usual partial adiabatic loss discussed so
often in the disclosure. In addition, sliding fric-
tion and leakage add to the losses. An isothermal
bel]ows gas spring on the other hand can be very much
more efficient, as already discussed. The ideal gas
spring is an opposed Stirling cycle isothermal
compressor-expander unit. Two opposed units thereore
supply the gas springs to the opposite member of the
pair. Consequently we disclose embodiments of free
piston machines as opposed units within a housing.
In this case, a mass is provided to cause a phase
delay from one spring to another. One spring is the
compressor and one the expander. A central mass is
driven between two units either as an electrical arma-
ture or if one ~air is a heat engine and the opposite
a heat pump, then the central mass between the two
units stores and transmits ~he energy from the engine
to the heat pump. A separate isothermal bellows spring
is disclosed that can be used in either application
where the weight of a metal spring is a disadvantage.
The efficiency of such a spring is important which
means that the heat transfer from the gas to the walls
must be as high as possible. Since no transfer of gas
is required, the floating baffle bellows are optimum
and only small holes are needed to supply the allowed
initial equilibrium with a fill gas. Here, helium or
hydrogen is the preferred fill gas. The Q (inverse
dampiny coefficient) will not be as great as for metal
springs and the Q will be frequency dependent, but
the mass will be less, about one-tenth of t:hat for
equivalent energy storage of the metal spring. This
ratio of mass to energy is derived from the fact that
~Z~)~291
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the maximum energy density of steel stressed to a
conservative value of 30,000 psi is about 2 atmos-
pheres and the same as that used in the bellows. The
bellows on the other hand have a metal thickness that
is about one-tenth the spacing of the convolutions.
Therefore, the mass ratio is about one-tenth.
HOUSE SIZE HEAT PUMP
The thickness of the bellows wall depends upon
the working pressure and dimensions but with typical
available materials having good fati~ue life and
strength like steel, phosphorous bronze, or beryllium
copper, working at 2~5 atmospheres pressure, 600 cycles
per minute and 30 to 40 cm in both diameter and length,
the wall thickness becomes about 1/4 the thermal skin
depth. Then, the heat mass of the wall is the entire
wall thickness. For example, let the pressure ratio
of the cycle be 2.5:1; then the maximum pressure dif-
ference, Pdi~f, becomes:
Pdiff = 2.5 -1 = 1.5 atmospheres.
The span, s, is the difference in the inner and outer
radius of a bellows. In our example we choose a 7 inch
outer radius and a 6 inch inner radius. The span
becomes s = 1 inch. We choose a conservative metal
thickness of t = 0~004 inches. Then the wall stress5 due to the pressure alone becomes:
wall stress (pressure) = Pdiff s/t = 2600 psi.
This is a very small stress increment and bending
stresses of the bellows will be significantly larger.
For a fractional cycle time of 1/100 sec for
compression, the thermal skin depth, d in steel (D =
0.2 cm2sec~l) becomes:
d = 0.0~ cm = 0.016 inches or four times the
thickness.
Then the heat mass of the wall becomes the entire
wall thickness~ The fractional thermal lag of the
wall will be less than the ratio of the heat mass of
~z~ 9~
-38-
the gas to heat mass of the metal or 1% for lO con-
volutions of the bellows per inch. If the outside
wall is maintained at an adequately constant tempera-
ture by cooling or heating air flow, this small thermal
5 lag favors an isothermal cycle.
CO:EFFICIENT OF PERFORMANCE
All heat pumps have an efficiency called the
coefficient o~ performance (COP) which is the ratio
of the heat out to the mechanical work in. Typical
heat pumps for the home have a COP of 2 to 2.5. The
ideal efficiency is (T2)/(Tl - T2) where Tl and T2
are the respective temperatures of the hot and cold
reservoirs. Both cycle inefficiency as well as refri-
gerant properties lead to the small COP compared to a
typical ideal heat pump where (Tl - T2) or Tdiff =
30K, T2 = 300K and the ideal COP = lO for house
heating and cooling. The difference between ideal
and practical is the inefficiency of compressor and
expander and the necessity of Tdiff being larger for
typical refrigerants. For example: let Tdiff = 30K
leading to an ideal COP of lO. If the efficiency of
the compressor and expander is 80% each, then the net
COP is ~.9. (One unit of mechanical work is put in
producing 0.8 units of heat. The mechanical energy
recovered by the expander is 0.8 units of heat. The
mechanical energy recovered by the expander is 0.8 of
the Carnot efficiency of 0.9 for T = 3~C. Therefore
the net mechanical work is ~l-(0.9 x 0.8)] = 0.28.
The COP is useful heat/mechanical energy = 0O8~0~28 =
2.9~) The Carnot cycle can be used as wel] as an
isothermal one, but the isothermal one has the advan-
tage of higher working pressure for the same COP, and
in additi~n the COP improves for smaller Tdiff, inde-
pendent of constant stroke. The bellows machinery
also has the possibility of ha~ing smaller losses, no
additional heat exchangers and ease of construction.
~Z(~4
-39~
A typical design would be a volume ratio, CR =
2.41, ideal COP = ln, and a heat transfer o~ (G -1)
ln CR = 35~ of the heat of the gas (G is the ratio of
specific heats of the gas = 1~4 for air) or ln CR =
.88 of the pressure energy of the gas. For our example
of a volume displacement of 27,000 cm3, at 600 rpm,
the heat out would be 24 kw with an input of 2.4 kw +
24(1 - eff2) where eff is the efficiency of the com-
pressor and expander. If the efficiency is 95%, then
the input energy is 2 x 2.4 kw or the COP = 5. We
therefore see that the isothermal cycle offers a signi-
ficant advantage for heat pump machinery, provided
the efficiency of the compression and expansion
machinery is high. It should be recognized that the
isothermal compression can not easily be used for
normal refrigerant compression because the refrigerant
will condense to a liquid in the compression cycle
just as it normally would do in the condenser after
compression. The transfer of the liquid refrigerant
out of the compression volume before any expansion
takes place would be exceedingly difficult. Therefore
the isothermal cycles are practically limited to the
use of a gas during the entire cycle.
One can use a totally sealed system with a gas
of a higher value of G like helium or argon (G = 1.67)
and at a higher pressure and achieve greate, heat
output for a given cycle.
The crank-driven bellows isothermal machinery is
driven slowly (say 10 ~z) and so becomes bulky. It
is well suited to house heating and cooling. It is
also suited to air compressors because of l:he lower
mechanical work required in the isothermal cycle
needed to produce a given volume of "cold" compressed
air.
lZ~
-40-
Description of the Exemplary Embodiments
STIRLING C~C`LE BELLOWS HEAT PUMPS GENERALLY
Referring to Figure 7 a standard Stirling cycle
heat pump is shown schematically. The compressor
variable volume 1 and expander variable volume 2 are
connected for gas transfer through a heat exchange
regenerator 3. The compressor and expander are driven
by a drive 6 through crank arms 4 and 5 at: the rela-
tive angle 7 of 90. Cooling air (gas) is blown
across the compressor chamber 1 and similarly heating
air is blown across the expansion chamber 2.
In operation the compression of the gas in volume
1 heats the gas, but the high heat transfer through
the walls of volume 1 and to the cooling air keeps
the gas inside volume 1 isothermal at temperature Tl.
At the top of the stroke, the gas leaves the chamber
2 and passes through the regenerator 3. The regenera-
tor 3 is of the standard type and merely represents a
large heat mass, usually sponge metal, that transfers
the heat of the gas at Tl to a reservoir by cooling
the gas to temperature T2O This heat is returned
later during the reverse stroke. When volume 1 is
near constant at top dead center, the gas in volume 2
enters at temperature T2 and is expanded. As it cools
further by expansion it is reheated by the heat trans-
fer from the heating air through the walls of volume
2 maintaining the gas isothermal during expansion.
The return stroke of 2 returns the gas to 1 through
the regenerator to volume 1. The regenerat:or now
returns the heat Tl -T2 to the gas enterincl 1 and a
new compression cycle starts with gas at T~ and
remains isothermal. The energy transferrecl is propor-
tional to T2 ln Rc (Rc = compression ratio), and the
energy gained back in expansion is Tl ln Rc~ so that
the net work becomes (T2 -Tl) ln Rc The coefficient
of performance (COP) as a heat pump is then:
~Z~4~
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(heat delivered)/(energy used) = Tl/(Tl -
T2) or the ideal thermal efficiency.
The losses are the friction of the parts and the
inefficiency of heat transfer. The heat transfer is
why we use bellows for compression and expansion
volume.
BELLOWS HEAT PUMP
In the heat pump shown in Fig. 8, the compression
chamber 1 and expansion chamber 2 are of the single
bellows type having rippled baffles extending from
the outside diameter seams into the central volume.
The design parameters for such chambers are described
in more detail above. The chambers are connected for
gas transEer by a heat exchange regenerator 3. The
required characteristics for the design of the regenera-
tor have also been thoroughly described above. The
regenerator 3 is maintained in an insulating plate 9,
which may be made of a plastic in the case of heat
pumps, but for machines of similar design but for
high temperature use should be m~de of a ceramic.
The regenerator 3 is made from a strip of corrugated
or crinkled metal foil and a strip of flat foil rolled
up together in much the same manner as is shown in
Frankl U.S. Patent No. 1,808,921. The plate 9 is
affixed within a housin~ 10 that surrounds the entire
unit. Ports 11 in the housing provide for the supply
and discharge of a flow of cooling gas (usually air)
to and from the section of the housing cont:aining the
compressor and ports 12 admit and discharge a flow of
a heating gas (usually air) that has been blown through
the expansion chamber 2.
The ends of the bellows remote from the mounting
plate 9 are affixed to movable end walls 4 and 5 that
are driven through connecting rods by crank arms 6
and 7 on a motor driven shaft 8. The crank arms are
lZ~429~
-42-
at a relative angle of 60. Inasmuch as the compres-
sor and expander chambers are at opposite ends of the
machine, 180 apart, the desired phase angle of 120
is provided by setting the crank arms 60 apart in
order to obtain the phase angle, i.e., 180 - 60 =
120.
It is preferred, though not required, to make
the bellows with ripple~ walls, e.g. 20 as shown in
Figs. 9 and lOo The rippled bellows in each expansion
chamber 1,2 are attached to the plate 9 by a tubula-
tion 15, it being suitable for the end convolution 16
to be joined to the tubulation expansion portion 18
by an elastomeric adhesive 17, such as a silicon
adhesive. Fig. 9 also shows the regenerator as it
appears in cross section. ~very other line represents
the strip of flat foil and the remaining lines repre-
sent the strip of corrugated foil. This construction
provides a myriad of small heat transfer passages for
heat exchange between the gas and the regenerator
mass.
As can be observed in the more detailed illustra-
tion in Fig. 10 of the rippled bellows and baffles,
the convolutions are joined by welds or by an elasto-
meric adhesive at inside joints 22 and out<,ide joints
23. The strength of the outside joints 23 may be
augmented by a U-shaped crimped seal element 24. In
general, the adhesive joints are limited to relatively
low temperature uses. High temperature bellows for
heat engines will probably have to use welded construc-
tion. The baffles 25 have holes 26 near their peri-
meters that compel the gas to follow a tortuous path
27 in and out of the convolutions. Central holes 28,
which may be off center and staggered plate to plate
cause turbulent circulation 2g between the baffles,
thus ensuring close thermal contact of the internal
gas with the inside surfaces.
~2~
-~3-
The machine shown in Figs. ~ to 10 operates as
follows. The compression of the gas in volume 1 heats
the gas, but the high heat ~ransfer to the walls of
chamber 1 and to the cooling air keeps ~he gas inside
[chamber] 1 isothermal at temperature T1. During the
compression stroke of chamber 1, the gas enters the
chamber 2 from the regenerator 3. The regenerator
represents a large heat mass of small volume, small
impedance to gas flow, and small longitudinal thermal
conductivity that transfers the heat of the gas at Tl
to a reservoir by cooling the gas to temperature T2.
This heat i5 returned later during the reverse cycle.
The gas in volume 2 enters at temperature T2 and is
expanded. As it cools further by expansion, it is
reheated by the heat transfer from the heating air 10
through the walls of volume 2 maintaining the gas
isothermal during expansion. The return stroke of
volume 2 returns the gas to volume 1 through the
regenerator. The regenerator now returns the heat Tl
minus T2 to the gas entering 1, and a new compression
cycle starts with gas at T2 and remains isothermal.
The energy transfer is proportional to T2 :Ln CR, CR =
compression ratio, and the energy gained back in expan-
sion is T1 ln CR, so that the net work becomes (T2 ~
Tl) ln CR.
The coefficient of performance (COP) as a heat
pump is then: (heat delivered)/(energy used) = Tl/
(Tl - T2~ or the ideal thermal efficiency.
The losses are the friction of the parts, gas
transfer friction, and the inefficiency of heat
transfer. The heat transfer is why we use bellows
for compression and expansion volumes.
Referring again to Fig. 8, the compression
volumes are the inside variable volume chambers
defined by the rippled bellows with baffles. If
nested bellows are used, the annular space between
z~
-44-
the bellows is the variable compression volume. Each
set of bellows is driven by typical crank arms at a
phas2 angle difference of 60. The crank arms are
driven, in turn, by a motor (or driver qenerator).
Finally, the compression ratio and maximum pres-
sure is determined by the 60 crank angle or 120
phase angle between the compressor and the expander.
The minimum volume corresponds to when the bellows
are plus and minus 60 from top dead center. The
volume then is Volmin = 2 (1 - cos 60) = 1Ø The
maximum volume is then Volmax = 2 (1 + cos 60) = 3Ø
The compression ratio, CR = 3.0O When the regenerator
volume and dead volume of the bellows is added, about
0.3, the final compression ratio becomes 2.5. The
maximum pressure is then 5 atmospheres, for Pi = 2
atmospheres, or a pressure differential across the
bellows of 4.0 atmospheres, or 58 psi. ~his leads to
a reasonable stress in the bellows and hence long
fatigue life.
FREE PISTOh HEAT PUMP
Referring to Fig~ 11, electric coils 100 are
energized by an alternating current ~02 to alternately
oscillate a hollow laminated iron armature 103 that
resonates with two bellows compression chambers 104.
Regenerators 105 are fixed to the housing 7'. Bellows
expansion chambers 106 and heads 108 tend walls of
chambers 106~ are tied end to end by rods 109 so that
the heads oscillate as a unit. The volume external
to the chambers 104 and 106 and surrounded by the
housing 7' allow the circulation of cooling and heat-
ing gases-air to inlet ducts 110 in the center and
111 and 112 at the ends. The air exits through ducts
113, 114, and 115.
In operation, regenerator heat pump units 104,
105, and 106 act as gas springs to the resonant mass
of the armature 103. The oscillation of the mass of
~.Z~291
-45-
the armature 103 alternately compresses and expands
each heat pump unit. ~he phase lag in the harmonic
oscillation of each e~pander volume 106 relative to
its compressor volume 104 is determined by the mass
of the heads 108 and rods lO9r Since the effective
spring constant of the bellows can be adjusted by the
initial pressure Pi, the heat pump springs and oscil-
lating masses can be timed to give the appropriate
resonant frequency of the AC line 102. For the 60
cycle current, these units will be fairly small, about
a 2 cm stroke and 5 to 10 cm diameters, and Pi will
be 2 to 4 atmospheres. The bellows will be of the
baffle design to maximize the heat transfer at the
high frequency. To obtain the 120 phase lag, the
mass of the heads and rods will be such that its
natural frequency with the expander bellows 106 will
be slightly less than the 60 cycle ~urrent. The arma-
ture mass 103 will be such that its natural frequency
also will be slightly less than the ~ cycle current.
Phase stability occurs due to the required energy
input from the AC line. ~he ambient input air enter-
ing duct 110 comes out hotter at duct 114. The out-
put air exiting ducts 113 and 115 comes out cooler
than the input air at ducts 111 and 11~.
HEAT DRIVEN HEAT PUMP
Referring to Fig. 12, a free piston heat pump
can be driven by a free piston heat engine 3~ to aug-
ment the net heat output or give refrigerat:ion. The
configuration is the same as Fig. 11 (the electrically
3~ driven heat pump where two heat pumps are clriven by
one armature), but instead no electric coi]s are used
and one end becomes the heat engine. l'he expander
bellows 31 of the heat engine are smaller than the
compressor bellows 32 because of the high temperature.
The high temperature is derived from a source of hot
gas 33, such as combustion of a fuel like natural
291
-46-
gas. The high temperature bellows 31 are also of
welded construction to withstand high temperature
gases. The mass 34 serves to couple the energy from
the heat engine to the heat pump 35. The mass 34 is
such that its natural frequency is slightly less than
the natural frequency of the engine so it drives the
heat pump. In this fashion phase stable power will
flow from the engine to the heat pump.
BELLOWS STIRLING ~YCLE ~EAT ENGINE
The engine shown in Fig. 13 is very similar to
the heat pump of FigO 11, except that warm air
supplies energy to drive the Stirling cycle unit and
deliver output power through the connecting rods and
cranks to the shaft. The illustrated embodiment has
nested bellows chambers 36 and 37 and an annular
regenerator 38, each designed as described above.
ISOTHERMAL AIR COMPRESSOR
An air compressor is usually used where the adia-
batic heat of compression is rejected before the com-
pressed air is utilized. Under these circumstancesthe adiabatic heat is wasted. As I have already dis-
cussed, the average piston-cylinder combination is
part way between adiabatic and isothermal. Further-
more the thermal skin depth effect somewhal: enhances
the work per cycle above that expected fro~l partial
heat exchange alone. Hence there is an efficiency
advantage for air compressors of a purely isothermal
compression cycle. The nested bellows performs the
function for both compression and expansion. Moreover,
the friction of the bellows driving machinery can be
made small Gompared to a piston with rings inside a
cylinder. For these reasons a cooled, nested bellows
compressor will be significantly more efficient in
producing a given volume of cGld compressed air than
a partially adiabatic one. The ratio of work energy
-47-
between an ideal isothermal compression and an adia-
batic one is:
ratio of work = [(G -1) ln RC~/~G(Rc(G -l)/G - 1)]
where Rc is the compression ration and G is the ratio
of the specific heats of the air, G = 1.4. For a
typical compressor supplying 120 psi, Rc = 8.57, and
the ratio of work = 71%. Hence depending upon friction
and partial heat exchange in the cylinders, something
like 30~ reduction in waste heat can be obtained by
using an isothermal compressor.
Referring to Figure 14, the variable compression
volume is the annular chamber between rippled nested
bellows 41 and 42, with an optimal mid-plane separator
plate ~3. The bellows are driven by a drive ~4 through
a crank 45. A housing 47 surrounds the bellows for
directing cooling air from a fan 48 around the bellows
and through the hole within the inner bellows 42. A
head 49 with inlet and outlet valves 50 and 51 connect
to the suction and discharge plenums 52 and 53. In
operation the nested bellows are alternately compressed
and expanded by the action of the crank, and air is
alternately inducted into the annular space 54 between
the nested bellows, compressed and discharyed through
the duct and plenum 53 to a receiver, not shown. This
compressor provides a higher efficiency isothermal
cycle resulting from the high heat exchange of internal
and external gases.
