Simulation of Full and Part-Load Performance of a Free-Piston Gas Generator by Electronic Analog Methods
>*
UNITED STATES
NAVAL POSTGRADUATE SCHOOL
THESIS
SIMULATION OF FULL AND PART- LOAD PERFORMANCE
OF A FREE- PISTON GAS GENERATOR
BY ELECTRONIC ANALOG METHODS
Alan F. Barnes
mm
12ND P2238
(1-59)
S^
YKN0XLIB RAW
MQNT8CY CA 93943-S101
SIMULATION OF FULL AND PART- LOAD PERFORMANCE
OF A FREE-PISTON GAS GENERATOR
BY ELECTRONIC ANALOG METHODS
*****
Alan
F.
Barnes
SIMULATION OF FULL AND PART- LOAD PERFORMANCE
OF A FREE -PISTON GAS GENERATOR
BY ELECTRONIC ANALOG METHODS
by
Alan
Barnes
F.
//
Lieutenant Commander, United States Navy
Submitted in partial fulfillment of
the requirements for the degree of
MASTER OF SCIENCE
IN
MECHANICAL ENGINEERING
United States Naval Postgraduate School
Monterey, California
1960
?«wi,, /^
DUDLEY KNOX LIBRARY
M POSTGRADUATE SCHOOt
101
MOMTFREY CA
SIMULATION OF FULL AND PART- LOAD PERFORMANCE
OF A FREE-PISTON GAS GENERATOR
BY ELECTRONIC ANALOG METHODS
by
Alan
F.
Barnes
This work is accepted as fulfilling
the thesis requirements for the degree of
MASTER OF SCIENCE
IN
MECHANICAL ENGINEERING
from the
United States Naval Postgraduate School
ABSTRACT
In this investigation, use was made of an electronic analog
for simulating the operation of a free-piston gas generator.
The
basic concept of the analog was developed in a previous investiga-
tion and involved the creating of the various free-piston forces as
functions of displacement and integrating these combined forces in
accordance with Newton's second law of motion to obtain a solution
representing piston motion
-
all by electronic analog means.
Modifications were made to the basic analog circuit of the previous investigation to overcome an unstable operating condition.
The modified circuit was then employed for determining performance
characteristics of an actual free-piston gas generator under varied
load conditions.
The results of these studies showed that the operation of a
free-piston gas generator could be simulated to a reasonable degree
of accuracy and stability by use of an electronic analog and that
this method might be a valuable means for design and performance
predictions of free-piston engine systems.
ii
ACKNOWLEDGMENT
The writer wishes to express his appreciation for the assistance
and encouragement given him by Professor Paul
F.
Pucci of the U,
Naval Postgraduate School during this investigation.
He is also
indebted to the Department of Mathematics and Mechanics of the U.
Naval Postgraduate School for use of the Boeing Electronic Analog
Computer.
iii
S.
S.
TABLE OF CONTENTS
Section
1.
Title
Introduction
Page
1
2.
Characteristics of the Free-Piston Engine System
3
3.
Dynamics of Piston Motion
6
4.
General Electronic Analog of the Free-Piston
Gas Generator
9
5.
System Inputs, Parameters and Performance Relations
12
6.
Electronic Analog of the Two-Stroke Standard Diesel
Cycle
19
7.
Electronic Analog of the Reciprocating Compressor
25
8.
Electronic Analog of the Bounce Cylinder "Gas Spring"
27
9.
Electronic Analog of the Friction Force
29
10.
Electronic Analog of the Free-Piston Gas Generator
30
11.
Operating Characteristics of the Free-Piston Gas
Generator
32
12.
Simulation of Full and Part Load Performance of the
SIGMA GS-34 Free-Piston Gas Generator on the Analog
Computer
35
13.
Conclusions
56
14.
Bibliography
58
Time and Magnitude Scaling
60
Appendix II
Programming Analog Computer for SIGMA GS— 34
^ree-Piston Gas Generator - Full and Part
Load
62
Appendix III
Full and Part Load Performance Calculations
from Computer Solution Results
66
Appendix IV
Analog Computer Symbols
74
Appendix V
Description of Equipment
76
Appendix
I
iv
LIST OF ILLUSTRATIONS
Figure
Page
1.
Basis Applications of the ree-Piston System
4
2.
Spring-Mass Analog for Free-Piston Gas-Generator System
6
3.
Forces Acting on the Free Piston
7
4.
Basic Electronic Analog Computer Arrangement for FreePiston Gas Generator
n
5.
Illustration of Displacement and Clearance Parameters
for Pistons in Contact at Midpoint
13
6.
Electronic Analog of the Two-Stroke Standard Diesel
Cycle
22
7.
Electronic Analdg of the Reciprocating Compressor
26
8.
Electronic Analog of the Bounce Cylinder "Gas Spring"
28
9.
Electronic Analog of the Friction Force
29
10.
Electronic Analog of the Free-Piston Gas Generator
31
11.
Pressure versus Displacement Diagram for Engine
Cylinder at Full Load
39
12.
Pressure versus Displacement Diagram for Compressor
Cylinder at Full Load
40
13.
Pressure versus Displacement Diagram for Bounce
Cylinder at Full Load
40
14.
Pressure versus Displacement Diagram for Engine
Cylinder at Three-Quarters Load
43
15.
Pressure versus Displacement Diagram for Compressor
Cylinder at Three-Quarters Load
44
16.
Pressure versus Displacement Diagram for Bounce
Cylinder at Three-Quarters Load
44
17.
Pressure versus Displacement Diagram for Engine
Cylinder at One-Half Load
45
18.
Pressure versus Displacement Diagram for Compressor
Cylinder at One-Half Load
46
Figure
Page
19.
Pressure versus Displacement Diagram for Bounce
Cylinder at One-Half Load
46
20.
Pressure versus Displacement Diagram for Engine
Cylinder at One-Quarter Load
47
21.
Pressure versus Displacement Diagram for Compressor
Cylinder at One-Quarter Load
48
22.
Pressure versus Displacement Diagram for Bounce
Cylinder at One-Quarter Load
48
23.
Thermal Efficiency on LHV Basis of Combined Gasifier49
Turbine System versus Load from Computer Solution Results
24.
Fuel Consumption versus Load from Computer Solution
Results
50
25.
Turbine Inlet Pressure and Temperature versus Load from
Computer Solution Results
51
26.
Frequency and Gas Flow Rate versus Load from Computer
Solution Results
52
27.
Operating Pressures versus Load Employed in Computer
Simulation of SIGMA GS-34 Gas Generator
53
28.
Inner and Outer Dead Points as Measured from Midpoint
of Engine Cylinder versus Load from Computer Solution
Results
54
29.
Ratio of Air/Fuel and Engine/Compressor Air versus
Load from Computer Solution Results
55
30.
Actuation of Relays
77
31.
Equipment Components Employed in Investigation
79
vi
NOMENCLATURE
English Letter Symbols:
a
-
Potentiometer constant, dimensicnless
A
-
Cross-sectional area,
b
-
Bounce cylinder clearance with engine pistons in contact, ft
BSFC
-
Brake specific fuel consumption,
c
-
Compressor cylinder clearance with engine pistons in
contact,
2
ft
Ibm fuel/shphr
ft
C
-
Coulomb friction force,
Ibf
C
-
Electrical capacitance,
x\£d
-
Frequency, cyctes/min
-
Frictional force, Ibf
h
-
Enthalpy, BTU/lbm
IDP
-
Inner dead point piston position,
k
-
Ratio of specific heats, dimensionless
LHV
-
Lower heating value of fuel, BTU/lbm
m,
-
Mass of air delivered by the compressor,
m
-
Mass of engine intake air,
-
Mass of fuel,
M
-
Mass of piston, Ibm
n.
-
Polytropic exponent for bounce cylinder compression and
expansion processes, dimensionless
n
-
Polytropic exponent for compression cylinder compression
and expansion processes, dimensionless
n
-
Polytropic exponent for engine cylinder compression
process, dimensionless
-
Polytropic exponent for engine cylinder expansion
process, dimensionless
f
r
F
rn
n
f
f
8
ft
Ibm
Ibm
Ibm
vxi
English Letter Symbols (continued)
ODP
-
Outer dead point piston position, ft
P
-
Ambient air pressure, lbf/ft
P
-
Compressor intake pressure,
P
-
Compressor discharge pressure, lbf/ft
-
Scavenge air receiver pressure,
-
Engine cylinder pressure during scavenging, lbf/ft'
-
Gas delivery pressure,
P.
-
Maximum or initial bounce cylinder pressure^
Q
-
Energy of fuel < BTU
R
-
Gas constant for air, ft-lbf/lbm-°R
R
-
Electrical resistance, ohms
s
-
Stroke,
shp
-
Shaft horsepower, hp
t
-
Time, sec
T
-
Ambient air temperature,
T,
-
Compressor discharge temperature, °R
T
-
Engine exhaust gas temperature, °R
T
-
Engine intake air temperature, °R
T
-
Gas delivery temperature,
V
-
Engine cylinder volume at point of port closure, ft
Wk
-
Work,
x
-
Engine piston displacement from midpoint or point of
contact of pistons,, ft
x
-
Engine piston displacement at port closure, ft
2
lbf/ft
2
.
P
2
lbf/ft
2
B
P
P
P
lbf/ft
2
t
©
b
lbf/ft
ft
°R
°R
3
ft-lbf
vxlx
English Letter Symbols (continued)
x,y s z
-
Piston velocity, ft/sec
x
-
Piston acceleration, ft/sec
y
-
Compressor piston displacement from compressor cylinder
head,
z
2
ft
-
Length of air intake portion of compressor stroke, ft
-
Bounce piston displacement from bounce cylinder he*d, ft
Subscripts:
b
-
Refers to bounce cylinder
c
-
Refers to compressor cylinder
e
-
Refers to engine cylinder
f
-
Refers to operational amplifier feedback
s
-
Refers to isentropic process
Greek Letter Symbols:
°^ i
-
Magnitude scaling factor, units of i/volt
°^ t
-
Time scaling factor, computer time/real time
A
-
Difference, dimensionless
£
-
Relay bias voltage, volts
W
-
Efficiency, percent
Miscellaneous:
i
-
Scaled computer voltage quantity - related to real
physical quantity, i, by scale factor, o< - thus
i
= ©<„
i
1
ix
Introduction
1.
An electronic analog of the operation of a free-piston gas generator or gasifier based on the "spring-mass" nature of the free-piston
engine system was undertaken by LT Arthur E. PLOW, USN.
QJ
In that
investigation, the first law of thermodynamics was solved on an analog
computer three concurrent times representing electronic analogs of an
internal combustion engine, a reciprocating compressor, and a bounce
cylinder "grs spring".
Friction forces were also introduced.
The re-
sults of these four analogs were combined and integrated in accordance
with Newton's second law of motion to yield a solution representing
piston motion.
In addition a comparison of analog computer solution
results and actual performance data at approximate rated output conditions was also made for the SIGMA
2
organization of France model GS-34
free-piston gasifier.
The results of LT PLOW's investigation showed that the various
nonlinear "gas-spring" forces of the gasifier (engine, compressor and
bounce cylinder forces) as well as a friction force could be simulated
by electronic analog methods.
The operation of the bounce cylinder
analog, however, was non-stable or drifting and tended to disrupt the
problem.
The innaccuracy of this analog resulted from errors in a
two-quadrant electronic multiplication performed in this circuit,
these errors being due to the inherent inability of an electronic multi-
plier to give uniform results in multiplications involving more than
one quadrant (i.e. both inputs having the same sign).
Numerals in brackets refer to bibliography.
2
9
»
Societe Industrielle Generale de Mecanique Appliquee (SIGM/),
V^nissieuXs, France.
Relatively few techniques for predicting the design and performance
of free-piston engine systems may be found in the available literature.
These, for the most part s involve rather complex numerical processes
as illustrated by London's
[_2j
thermodynamic -dynamic analysis of a
free-piston gas generator system in which a calculated "basic design"
may be extrapolated by certain affinity relations to different plant
capacities and dimensions.
In this investigation the electronic analog method as developed by
PLOW £l[] but modified for stability of operation was employed for deter-
mining performance characteristics of a free piston gas generator under
*
varied load conditions.
To this end full and partial load studies were
made of the SIGMA GS-34 gasifier by introducing appropriate fixed geometric and variable operating parameters to the analog problem.
This
procedure could be adapted as well to performance predictions of modified gas generator designs and thus would be of considerable benefit in
the development of different sized machines.
In the available literature there is little information on predic-
tion of part load behavior of a free-piston gas generator.
The analog
method might well be a useful means in this type of analysis.
2.
Characteristics of the Free-Piston Engine System.
The free-piston engine system consists of an opposed piston
s
super-
charged two-stroke Diesel engine driving a reciprocating air-compressor
and may be combined with a turbine which utilizes the expansion of the
exhaust gases down to an atmospheric condition.
The engine reciprocat-
ing work is directly used to provide the compressor work requirement.
The free piston system is characterized by (a) constructional simplicity
due to the absence of cranks and bearings;
from vibration provided by th
(b) almost complete freedom
crankless opposed piston design; and (c)
the absence of piston-cylinder side thrust and therefore reduction of
cylinder wear as introduced by crank- connecting rod systems.
As shown in Fig.
piston system.
Fig.
1,
1
there are two basic applications of the free-
(a)
represents an internal-combustion engine air
compressor combination (free-piston air compressor) in which the useful
output is compressed air fot pneumatic purposes.
Fig.
1
(b)
shows an
air compressor internal combustion-engine combination (free-piston gas
generator or gasifier) for the production of hot gases under pressure.
The gasifier illustrated has an inboard compression,, outboard bounce,
and common central combustion chamber.
Useful shaft work is derived
from the hot pressurized gas by expansion to atmospheric pressure
through a turbine.
i.e.,
This investigation involved the latter configuration,
the free piston gas generator or gasifier.
AIR
INTAKE
VALVES
SCAVENGE
SCAVENGE
CYLINDER
CYLINDER
COMPRESSOR^
COMPRESSOR
CYLINDER
-CYLINDER
BOUNCE
CYLINDER
COMPRESSED
AIR
FUEL INJECTOR
(a).
FREE -PISTON
COMPRESSOR
AIR INTAKE
SYSTEM.
VALVES
COMPRESSOR
COMPRESSOR
CYLINDER
CYLINDER
BOUNCE
BOUNCE
CYLINDER
CYLINDER
FUEL
RECEIVER
INJECTOR
(P-IOO PSIA)
(T-IOOO°F)
—
*•-)
-CZfl
TURBINE
NET PLANT
WORK OUTPUT
EXHAUST
TO ATMOSPHERE
(b).
FREE- PISTON
Figure
1.
GAS
GENERATOR-TURBINE
SYSTEM.
Basic Applications of the Free-Piston System.
Referring to Fig,
1
(b),
the gasifier consists of two opposed
pistons having equal and symmetric strokes.
The Diesel or power cylin-
der is in the center and operates as a two- stroke Diesel engine super-
charged to a pressure of several atmospheres (according to turbine load
condition).
The two single-acting compressor cylinders are located on
both ends of the central housing or scavenge air receiver.
The cushion
or bounce cylinders which store the energy for the return stroke are
located at the outboard ends of the gasifier.
Fresh air is taken in
at atmospheric pressure through the air intake valves and is discharged
into the scavenge air receiver surrounding the power cylinder.
This com-
pressed air is used for scavenging and charging of the power cylinder.
The hot combustion gases, mixed with the excess scavenging air, ex-
haust into the receiver and produce the useful power in the gas turbine.
3.
Dynamics of Piston Motion.
In the free-piston engine, the absence of a crank results in a
"spring-mass" system which operates at a natural frequency depending
on the mass of the piston assemblies and the nature of the nonlinear
"gas springs" within the cylinders.
London's Q23 spring-mass analog
for the free-piston gas generator configuration is shown diagrammatically
in Fig. 2 below:
Compressor Cylinder
Scavenge Space
Engine Cylinder
Bounce Cylinder
-r±
J-llTL
i/V^
k-JV*
*/VW^
AA
t
l/V*
r
fc/W
irr-r
-LT-
Inlet Port
Figure
2.
Exhaust Port
Spring-Mass Analog for Free-Piston Gas-Generator System.
Pressure or force versus volume or displacement diagrams for the
power, compressor and bounce cylinders corresponding to the nonlinear
spring forces of Fig.
2
are shown in Fig.
3.
The pressure volume dia-
gram of the power cylinder is representative of the Diesel cycle which
was employed in this investigation.
Also shown in this figure is a
constant friction force that is independent of piston speed and gas
pressures.
Justification for this assumption may be found in Bobrowsky
During the outward stroke of the pistons the energy produced in
the power or engine cylinder in addition to that of re-expansion of
4
3
0)
I
1
tu
j
\
i
\
o
w
o
\
\
fa
u
\
u
o
fa
u
o
u
3
\
\
\
\
o
01
^^ q
\^
x^
1
\
\
\x
\
\X
\
\\
fa
4
2
1
Volume or Displacement
Volume or Displacement
Power Cylinder
\\
\
01
(4
A
\J
^"x^
\
u
3
(0
W
\^
fa
(a)
\\
I
\
u
\\
l<
\
\
CO
(0
0>
\
I
\
at
2
3
Compressor Cylinder
(b)
1
<u
f
o
p
o
I
1
1
J
fa
/
/
^^^
y^
./^
2
Displacement
0)
o
u
o
fa
u
o
0)
u
3
(0
CO
0)
Id
fa
Volume or Displacemer t
(c)
Friction Force
(d)
Bounce Cylinder
.
Figure
3»
Forces Acting on the Free Piston,
air in the clearance space of the compressor cylinders less friction
losses is stored in the cushion or bounce cylinders s
Wk
engine
_
exp stroke
= Wk
Wk
„
,
bounce cyl
compr stroke
,
+ Wk £
.3.1
t
friction
per stroke
.
.
1
compressor
exp stroke
.
„
.
,
During the inward stroke the energy stored in the bounce cylinders is
returned to the system again discounting friction losses, compressing
and discharging the air in the compressor cylinders and compressing the
air used for combustion in the power cylinder,
Wk,
,
bounce cyl
exp stroke
= Wk
+ Wk
engine
compr stroke
compressor
compr stroke
+ Wk £
.3.2
fcJ
friction
per stroke
,
On a net work per cycle basis,
Wk
.
engine
cycle
=
Wk
compressor
cycle
+ Wk.
.
3.3
.
friction
per cycle
fc
.
For each half cycle the component work terms must balance since the
kinetic energy in the piston mass is zero at the inner and outer dead
points where piston velocity is zero.
Newton's Second Law relates piston resultant force
F
R
piston rate of change of momentum where the resultant force
and the
F
R
may be
evaluated as a function of piston position from
F_ (out stroke)
R
=
F„ (in stroke)
R
=
F
eng
+
F
compr
-
-
F.
bounce
F^
fr
3.4
-
F,
bounce
F
compr
-
F
eng
-
F-
fr
The work done by the resultant force acting on the piston is equal to
the gain of kinetic energy of the piston.
8
4„
General Electronic Analog of the Free-Piston Gas Generator.
In the previous section it was shown that the free-piston engine
may be considered as a "spring-mass" system with a single degree of
freedom wherein the nonlinear "gas spring" forces are of a thermodynamic
nature.
A constant static friction force opposing piston motion is
also present.
As previously developed in Plow £lj
the differential
9
equation of motion of the piston in a gas generator is a mathematical
expression of Newton's Second Law (Force =
mass x
acceleration) and
is given by
F
e
(x,x)
t* F c (*»*)
"
F
b
<*>*)
"
F
fr
<x)
^»
*.l
M is the piston mass
where
F
e
(x,x) is the engine cylinder force on the piston
F
(x,x)
is the compressor cylinder force on the piston
F,
(x 9 x)
is the bounce cylinder force on the piston
F-
(x)
fr
is the
friction force
x
is the piston displacement
x
is the piston velocity
x
is the piston acceleration
It will be seen in later sections that the differential equations from
which the engine,, compressor and bounce cylinder forces or pressures
are obtained are functions of piston displacement and velocity and that
the friction force is dependent on piston velocity,,
Diagrams illustrat-
ing the 'gas spring" and friction forces acting on the piston are shown
in Fig,
3o
The electronic analog computing arrangement (from Plow C^G) f° r tne
solution of equation 4,1 is shown in Fig,
4,
The circuitry of the
blocks representing the engine, compressor, bounce and friction analogs will be described in detail in following sections.
Useful re-
sults from the computing arrangement of Fig. 4 for determination of
gas generator performance data will consist of piston displacement as
a
function of time and pressure-displacement diagrams for the engine,
compressor, and bounce cylinders.
appear in terms of scaled voltages.
10
Pressures and displacements will
+ X
Engine
-Pi
Analog
"V./
Compressor
-Pc
Analog
-X
+ X
Bounce
Analog
+p b
Friction
Analog
+ F^ r
- X
Figure 4.
Basic Electronic Analog Computer Arrangement for
Free-Piston Gas Generator.
11
„
5.
System Inputs, Pararaeters 9 and Performance Relations,,
Inputs to the system will consist of;
5.1
!o
Fixed geometric parameters
2.
Initial piston displacement position (at outer dead point)
3.
Various cylinder pressure conditions:, which are:
a.
Compressor intake and delivery pressures,,
bo
Engine pressure at start of compression,,
c„
Initial bounce cylinder pressure.
4
Fuel quantity.
5o
Thermodynamic' parameters such as polytropic exponents.
Geometric parameters.
Variables and fixed quantities depending upon the geometry of the
free-piston system are as follows;
A
1.
Engine piston area,
2
Compressor piston area,
3.
Bounce piston area
4.
Engine piston displacement from midpoint or point of
.
e
A,
8
A
„
.
contact of pistons s x.
5.
Compressor piston displacement from compressor cylinder
head s y.
6.
Bounce piston displacement from bounce cylinder head,
7.
Engine piston displacement at port closure^ x
8.
Piston mass 9
.
M
The following relations also hold for the above parameters;
12
z.
e
and
y
a:
X
+
z
=
b
-
=
dx
dt
The quantities
c
dy_
b
5.1,2
C9
5,1.3
x9
=
dt
and
5.1.1
c
- dz
5.1.4
dt
in the above equations represent the clear-
ances of the compressor and bounce cylinders respectively when the
pistons are in contact at the midpoint or center of the machine.
5
Fig.
below shows these relations for one-half of the gas generator configura-
tion.
H-h h-
b
H
\~\
J
I
1
I
t
Inlet Port
x
Figure
5.
Illustration of Displacement and Clearance
Parameters for Pistons in Contact at Midpoint.
5.2
Pressures.
In the free-piston gas generator ambient air of about one at-
mosphere is compressed in the compressor to several atmospheres absolute pressure depending on turbine load condition and is delivered to
13
the scavenge air receiver during the inward stroke of the pistons
During the latter part of the power or outward stroke and commencement
of inward stroke of the pistons, when both the exhaust ports and the intake or scavenging ports are open 8 the scavenging air flows through the
scavenge ports and through the full length of the power cylinder,
forc-
ing the exhaust gases through the exhaust ports and into the turbine inlet receiver (See Fig.
1).
Some of the scavenging air follows the ex-
haust gases into the receiver for good scavenging, but a portion of
this supercharge air remains within the power cylinder and is compressed for the next combustion cycle during the inward stroke of the piston.
Intake of fresh air into the compressor cylinders occurs during the out-
ward piston stroke.
The significant pressures in the free piston system are thus;
-
ambient air pressure.
P,
-
compressor intake pressure.
3.
P,
-
compressor discharge pressure.
4.
P
-
scavenge air receiver pressure.
-
engine cylinder pressure during scavenging.
1.
P
2.
o
s
5.
P
6.
P
operating pressure at turbine inlet.
Valve pressure drop allowances of five percent of upstream absolute pressure were arbitrarily chosen for each port passage of the gas
generator.
These include the compressor intake and discharge valves and
the engine scavenge and exhaust ports.
The ratio of downstream to up-
stream pressure (absolute) for the compressor valves and engine ports is
therefore equal to 0.95.
14
—
5.3
.
Temperatures
As will be seen in succeeding subsections,
the significant tempera-
tures in the free-piston problem for the determination of the masses of
and compressor air,
m
,
and for the determina-
tion of generator gas delivery temperature,
T
,
are as follows:
engine intake air,
,
1.
Ambient air temperature,
2.
Engine intake air temperature,
T
pressor discharge temperature,
T,
3.
5.4
m
T
.
,
Engine exhaust gas temperature, T
assumed equal to com-
.
Masses of Engine Intake Air and Compressor Discharge Air.
Gas generator performance characteristics require the determination
of engine intake and compressor discharge air masses.
Employing the equation of state of a perfect gas, the mass of
engine intake air
where
and
V
R
m , will be
e
is the engine cylinder volume at the point of port closure
the gas constant for air
(R
=
Assuming engine intake air temperature,
charge temperature,
T
,
53.3 ft-lbf/lbm
T
,
-
°R).
equal to compressor dis-
and considering compression and expansion pro-
cesses in the various cylinders to be polytropic then
m<* =
—Pp Vp7-—
—
L'
C
m
*t (P^nc-iff
15
)
5.4.2
where
n
is the polytropic compression exponent
in the compressor.
Again employing the equation of state of a perfect gas
of air delivered by the compressor per cylinder per cycle 9
s
the mass
m.,
d
is given
by
m,
d
A
where
-
P.
i
A
c
Tto
«...
Y.
5.4.4
i
'
is the area of the compressor cylinder and
Y
is the length
of the air intake portion of the compressor stroke.
The above expressions for engine intake and compressor discharge
air masses involve only one 'half of the free-piston gas generator and
would have to be doubled to obtain total masses for both cylinders or
sides of the system.
5.5
Power Output of the System.
In the actual free-piston gasifier-turbine combination hot combus-
tion gases mixed with excess scavenging air exhaust from the gas generator into the turbine receiver and then flow through the gas turbine which
alone produces useful power.
The thermodynamic state of the gases at the
turbine inlet is dependent on the pressure and temperature of the ex) aust
gases.
The product of the isentropic available energy and the gas flow
rate results in the rate at which work can be obtained from the exhaust
gases.
By multiplying this quantity by the turbine isentropic efficiency
the effective power output at the turbine shaft finally can be determined.
The exhaust gas temperature of a reciprocating internal combus-
tion engine system is substantially less than the working substance tem-
perature at the end of the power stroke.
16
X-ondon £2 J considers the engine
exhausr
iture equal to the mass ave
irature of the
"blow- down" period and has dei
gas temperature equation for a constant
t
TV
r
/
xhaust,
v
_^
T..-^[lW*-Ufej.
Ps J
.
.
5.5.1
'
Subscripts
5
and 6 represent the state points at beginning and end of
exhaust or "blow-down" period (See Fig. 3).
Temperature at state
5
is
determined from the equation of state of a perfect gas,
where
m
is mass of engine intake air as obtained from equation
e
5.4.1 and
R
is the gas constant
for air.
The generator gas delivery temperature,
T
is then equal to
,
the mass average between engine exhaust gas temperature,
scavenge gas or engine intake air temperature s
T
,
T
,
and
and is given by,
Tt = 2&- Te + (1 - %*) TP
5.5.3
,
where m /m.
is the ratio of engine air to compressed air delivered,
m
and
m,
being obtained from equations 5 4.1 and 5.4.4 respectively,
and
T
assumed equal to compressor discharge temperature,
ed from
T,,
is obtain-
„
TP = Td = To
~Wl)
with quantities as defined in preceding subsections
.«.?
)
5.5.4
With the working substance considered a perfect gas
?
due to re-
latively low delivery pressures of the free piston gasifier 3 enthalpy
of exhaust gases may be considered a function of temperature alone
and isentropic available energy determined
The power out of the gas
turbine is the product of the isentropic available energy^ total exhaust
gas flow rate considering
the two sides of the gas-generator, and tur-
bine isentropic efficiency.
Including appropriate conversion factors,
the equation representing power output in horsepower is given by,
(shp)
where^h
m.
M
turbine
=
i^slCn&UXndXfrXKi,)
33,00
and
h
5.5.5
is the isentropic available energy (Btu/lbm),
is the mass
(Ibm) of air delivered by the compressor per
cylinder per cycle,
f
,
?
is the piston frequency in cycles per minute,
is the turbine isentropic efficiency.
18
6.
Electronic Analog of the Two-Stroke Standard Diesel Cycle,
Figo 3(a) shows a representative pressure versus volume indicator
diagram for the two-stroke air standard Diesel
invest igat ion o
I
;
in this
With the absence of a crank mechanism in the free-
piston gas generator^ the stroke of the system and hence the engine com-
pression ratio and compressor and bounce cylinder clearance volumes are
all variable quantities dependent on the load conditions.
The only fix-
ed geometrical point for the piston in the engine cylinder is that of
port closure.
The thermodynamic state of the engine intake air mass at
this point can be determined for a given compressor discharge pressure
which is also dependent on load conditions,,
As previously considered^ the working substances in all the cylinders
of the free-piston system are assumed to be perfect gases and compression
and expansion processes polytropic.
Again referring to the indicator dia-
gram of Figo 3(a) s the two-stroke standard Diesel cycle can be seen to
consist of the following processes?
1.
Polytropic compress ion* at exponent
n
e
9
of engine charge
after port closure.
2.
Injection and burning of fuel to give addition of energy
at constant pressure during first portion of power stroke.
3.
Polytropic expansion 9 at exponent
n\
of engine charge and
e
products of combustion.
4.
Engine "blow-down" of exhaust gases at constant volume.
5.
Scavenging at constant pressure.
For a polytropic process involving a perfect gas
PV
-
constants,
19
6.1
where
n
is Che polytropie exponent.
dif ters.nt.iat ion and con-
By
sidering time as the Independent variable, we obtain
d-b
v
6.2
With the volume given by
V = Ae X >
6.3
and
6.4
then equation 6.2 becomes
d£>
n Px
^t
X
Following the procedure in Plow £lj
.5
,
the above two- stroke
standard Diesel cycle processes can be defined in terms of equation
6.5 as follows:
1.
Compression process.
(AM
- _ "^ex
^t /n
e
V
2.
.
x
Combustion process.
v di-b 'P
!.
Expansion process.
fdP&
P
ne Pe. X
\
5-8
in the above equation is engine cylinder pressure.
s
electronic analog of the compression or expansion processes
as represented by equations 6.6 and 6.8 is as follows
20
1.
M r&
Integrate
- P
to obtain
initial conditions,
P
with an appropriate
.
P
2.
Multiply -P
-n'P x
3.
e
by n x
or n'x
e
e
to give
-
or
n P x
e e
as appropriate.
Divide by
x
resulting in the formation of equation 6.6
or 6.8 and thus returning to the starting point of the
analog circuit loop.
The analog of the Diesel cycle applying the above processes and
procedure is shown schematically in Fig.
6.
Also shown is the modifi-
cation of the circuit of P»low [_lj employing single quadrant electronic
multiplication for more uniform and stable operation.
In this investi-
gation single quadrant multiplication was employed in each of the circuits simulating engine, compressor and bounce cylinder pressures.
This
was accomplished through the use of additional switching relays and
sign changing amplifiers in order that the electronic multipliers (BEM
1
involved in each of these circuits would function only in one
and DEM)
quadrant.
A detailed explanation of the circuit of Fig.
1.
follows:
6
The relay switch in the feedback of Amplifier
closed whenever
is greater
than
°
x
the initial condition,
P
1
is
establishing
x
p
.
P
2.
The electronic multiplier and its associated amplifier,
Amplifier
2,
together give a product of one- fiftieth of
the variable inputs to the multiplier.
1
Circuit component symbols defined in Appendix IV,
21
O
m
Oi
^
•o
(6
"0
c
CO
o
u
U
en
i
I
0)
4J
o
60
O
r-i
u
c
o
u
o
o
00
En
22
g
3,
Relay
1
is actuated by a change in sign of velocity,
and applies the exponent
n /2
when the velocity
is negative and the exponent
n"/2
when the velocity
x,
is positive,
4,
Division of a variable by another variable quantity is
accomplished by connecting an electronic multiplier in
the feedback of an amplifier as in the case of Amplifier 4.
The divisor
s
-x s must always be negative for stable opera-
tion, while the variable input to Amplifier 4 may have
either polarity.
The coefficient potentiometer,
a
in
,
conjunction with the dividing network 9 is adjusted to give
ten times the quotient of the inputs,
5,
Relay
2
is actuated by a change in sign of velocity,
and in conjunction with the
Q„
f
relay makes
fl
x,
=
'
dt
simulating injection of fuel at the commencement of the
power stroke.
6,
Relay
3,
connected in the feedback of Amplifier
actuated by a change in sign of velocity
8
x,
in Amplifier 6 performing the integration of
is also
6,
and results
-
n'Px
1
10
whenever
x is positive.
e
The integration of this quantity
produces a result which is proportional to the total instantaneous fuel energy which in turn is compared with the
desired value of
Qf
in Amplifier 7,
When the fuel energy
becomes equal to the desired value as represented by a
negative voltage corresponding to
is actuated^
reconnecting
23
Pt
ra
LHV„
the
Q
relay
HS back to Amplifier 1
through Relay
This
uel
!
ct ion
5
i
use
i
multi]
I
th actuated by a ch
Ln
to
multiplier Is always negative In sign.
5
are there fo
\
til
Lectronic
Relay
24
5 ar, ^
.
quired and so connected to ob-
tain the proper polarity feedback to Amplifier
stroke.
Relay
sign of velocity, x.
i is so actuated that the Input, x 9
plifier
input and
the polyttop
3
7
2„
1
for each
7.
Electronic Analog of the Reciprocating Compressor.
A typical pressure versus volume indicator diagram for a recipro-
cating compressor
is
The cycle is considered
represented by Fig. 3(b).
to consist of polytropic compression and expansion of air at exponent
n
and constant pressure air intake and discharge.
The computing arrangement for solution of the polytropic equation
for the compressor^
dPc
»
nc P&
-
dt
y
7.1
,
y
is similar to that of the engine
cycle
However, in this case diode
limiters are employed on the amplifier generating
P
to establish
limiting voltages corresponding to the intake and discharge pressures.
This is in conformance with the circuit developed in Plow [l^ with the
exception that single quadrant electronic multiplication was again employed involving the necessary two extra switching relays and additional
sign changing amplifier.
A schematic diagram of the compressor analog with the single quadrant multiplication feature is shown in Fig.
7.
The various functions
of the circuit components in performing the polytropic compression and
expansion processes are essentially the same as those described for the
engine analog in the preceding section.
25
J-l
o
CO
10
0)
u
a
E
o
o
60
c
•ft
u
«
o
O
u
a
1-1
CJ
&
a)
O
60
O
i—i
CO
3
c
o
o
o
w
u
60
26
8.
Electronic Analog of the Bounce Cylinder "Gas Spring".
For the bounce cylinder analog, air is alternately expanded and
compressed by reversible polytropic processes at exponenet
ing in a pressure-volume diagram as shown in Fig.
3(d).
n
result-
The differential
equation involved for the computing loop is
dPt = - *uP*>i
dt
,
8.1
2
and the schematic diagram of the analog circuit is shown in Fig. 8.
The employment of single quadrant multiplication in this circuit had
its most advantageous effect.
In the corresponding circuit of Plow £l^
the absence of this feature resulted in errors which became magnified
with time by the lack of an initial condition imposed each cycle.
This
modification resulted in stability of the bounce cylinder circuit or the
ability of the pressure-displacement curve to reproduce or retrace itself over a number of cycles of operation,
27
i
60
c
a
en
CO
u
GJ
OJ
>
c
o
a
u
c
O
PQ
a;
•u
M
p
O
i—
c
<
u
«f-I
c
o
u
U
u
<y
f-f
w
oo
a
3
60
L
28
9.
Electronic Analog of the Friction Force.
In this investigation a coulomb friction force,
considered.
"C", only was
This force was assumed to be independent of piston speed
and gas pressures Q?J arid therefore constant in magnitude.
An illustra-
tion of the coulomb friction force is given in Fig. 3(c).
The friction analog of Plow £lj involving both coulomb and viscous
friction was modified to produce a voltage corresponding to coulomb
friction alone and is shown below:
i.e.
X
+ Ffr
1
—"VWWV
Figure 9.
Electronic Analog of the Friction Force.
The limiters in the feedback of the friction analog amplifier
serve the purpose of establishing the equal magnitude but opposite
polarity voltages,
"C",
corresponding to a constant friction force,
at the output of the amplifier.
With this circuit, employing the
negative of piston velocity voltage as the input, the appropriate output polarity voltage for a complete stroke is obtained at the instant
of commencing the instroke and outstroke of the piston.
Those voltages
are of the proper polarity for the complete analog circuit to be shown
in the
following section.
29
10.
Electronic Analog of the Free-Piston Gas Generator.
The preceding electronic analogs for simulation of the engine, com-
pressor and bounce cylinder and friction forces were individually assembled and tested using the artificial stroke input circuit of Plow flj
.
The artificial stroke consisted of displaced sinusoid voltages simulating piston motion at about one cycle in ten seconds based on real time.
These voltages included the variable
"x".
"y", and "z" displacement
voltages employed in the engine, compressor, and bounce cylinder analogs
respectively.
On tying
together these component analog circuits with additional
amplifiers to sum and integrate the various piston forces to obtain its
motion, the complete free-piston analog was obtained and the artifical
stroke input no longer required.
The complete circuit is shown in Fig. 10 and corresponds to the
basic computer arrangement of Fig.
30
4.
PJ
•X
I
i?
>4
i->
CO
c
o
/ts_
CO
CO
o
o
Q.
c
o
I
+
I
+
OJ
0)
1-1
a>
X
4J
60
O
r-t
UJ
CO
i
c
o
o
OJ
M
60
In
$t^ffi
31
Operating Characteristics of the Free-Piston Gas Generator.
11.
In the free-piston gas generator,
the omission of a crank mechan-
sim permits a free stroke and therefore variable operating conditions
of piston movement.
Since there is no fixed limits for the stroke, the
motion of pistons should respect certain ranges of inner and outer dead
point positions, IDP and ODP, as measured from the midpoint of the engine
cylinder.
The limits of ODP should fall within ODP
becomes insufficient and ODP
max
°
°
mm where scavenging
.
corresponding to mechanical contact at
IDP must lie between the point of
the outer limit of piston travel.
contact of the two rpistons at the
midpoint (IDP
r
mm
.
0) and IDP
=
max
where ignition temperature is no longer reached by compression (200 psi).
IDP
mm or clearance also should not be so small that excessive peak com.
bustion pressures are reached in the power cylinder as compression pressure is dependent on the super-charge pressure and IDP
.
.
Operating sta-
bility of a gasifier is thus the ability to keep the piston motion within
the limits just defined.
For the SIGMA GS-34 gas generator the approxi-
mate operating limits f