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International Compressor Engineering Conference
School of Mechanical Engineering
1984
Optimisation of Compressor Valve Designs Using
the 'Nelder-Mead' Simplex Method
P. C. Shu
A. B. Tramschek
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Shu, P. C. and Tramschek, A. B., "Optimisation of Compressor Valve Designs Using the 'Nelder-Mead' Simplex Method " (1984).
International Compressor Engineering Conference. Paper 457.
http://docs.lib.purdue.edu/icec/457
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OPTIMISATION
USING
THE
OF
COMPRESSOR
"NELDER-MEAD"
VALVE
SIMPLEX
DESIGNS
METHOD
P.C. Shu, Lecturer, Compressor Division,
Jiaotong University, Xian, China.
A.B. Tramschek, Senior Lecturer, Department of Thermodynamics,
University of Strathclyde, Glasgow, Scotland.
ABSTRACT
cylinder some 12 independent variables have
to be considered in a valve design optimisation
In turn each variable may have
procedure.
values lying within certain ranges - the so
called explicit constraints - according to considerations of structure, strength and other
The present investigpractical requirements.
ation placed explicit constraints upon port diameters, cover plate thicknesses, permitted valve
lifts, valve spring stiffnesses, valve spring
Implicit constraints were put on the
preloads.
variables by the specification of maximum values
for the velocity with which the moving cover
(An impact
plate struck a motion limiting stop.
velocity constraint).
A combination of a compressor simulation model
and an optimisation technique was used to predict
enhanced compressor performance through an
The Nelder-Mead
improvement in valve design.
Simplex optimisation technique was used to determine optimum combinations of various compressor
valve design ~arameters in a multi-variable,
In this investigation
multi-constraint problem.
10 independent variables, 15 explicit and 2 implicit constraints, were manipulated to achieve a
minimum value of the chosen optimisation function - namely energy consumption per unit of flow
An
delivered at specified operating conditions.
external barrier function comprising the sum· of
the objective function and a penalty term ·was used
The paper ilto prevent constraint violation.
lustrates the progress to the attainment of optimum values of the independent variables as well
as the corresponding performance parameters (indicated .power, delivery capacity, valve power
The paper illosses, objective function).
lustrates a technique which is of great potential
value in research and development work associated
with compressors.
The present investigation was centred on the
need to obtain an optimum valve designfrcm a thermodynamic viewpoint and thus· the objective function
was selected to be energy consumption per unit of
It was the task of the optimiflow delivered.
sation procedure to minimise the value of the
objective function for specific compressor design
conditions (compressor speed, suction pressure,
suction temperature, discharge pressure).
Values of the objective function were calculated
using a digital computer and a mathematical
An
model of the compressor and its valves.
optimum valve design was achieved by combining
the mathematical model and the "Nelder-Mead"
Simplex optim1sation method.
INTRODUCTION
The process of achieving the best combination of
a set of independent variables under certain constraints in order to obtain an optimum value of
an objective function may be called design optAccording to one's definition, the
imisation.
optimum value of the objective function may represent a maximum or a minimum value of that funcDesign optimisation may utilise many
tion.
methods and techniques but the choice of a particular method is influenced by the nature of the
variables and the constraints as well as the relationships between variables, constraints and
objective function.
Previous publications [ 3 ], [ 4 ], [ 5 ]
concerning work relating to the optimisation of
compressor valves performed at the University of
Strathclyde have reported the use of various
optimisation methods and discuss the advantages
and disadvantages of these methods vis-a-vis
search speed, constraint handling and the identification of local optima as opposed to a global
The work reported in this
optimum condition.
paper utilised a Simplex Method [ 2 ], a method
generally considered to be both powerful and effective for many optimisation problems.
Reciprocating compressor valves may be described
by the followihg variables :- port diameter (or
area), covering plate diameter (or area), covering plate thickness, permitted lift, valve spring
stiffness, valve spring preload. i.e. 6 variables
For a compressor having a single
per valve.
suction valve and a single discharge valve per
218
VARIABLES
5. Finite positive valve spring preloadin gs
If a compressor valving system is deemed to comprise a simple spring loaded disc covering a port
then for a single cylinder having one suction
valve and one discharg e valve the 12 independ ent
variables at the disposal of the designer may be
represen ted in matrix notation as :X = (x 1 , x , x , - - - - 2
3
where in the present study
x1, x6
x2, x7
X )
n
n
= 1,
12
discharg e
suction,
discharg e
x3, XB =
x4, x9 = suction, discharg e
x5, x10 = suction, discharg e
x11'x12 = suction, discharg e
valve port diameter
7. Impact velocity constrai nts
Maximum and minimum values for disc thickness
Hs1 $ x1 $ Hs2
Hd2
x6 '
Finite positive values for valve port diameters
o
x7:?
0
3. Finite positive values for permitted valve lifts
0
lying wholly within the compress or cylinder .
This condition was recognise d by placing a
limit on the sum of the valve port diameter s
valve spring preload
valve disc diameter
directly by specifyin g upper and lower bounds to
their values.
x2 ~
6. Non overlapp ing suction and discharg e valves
IMPLICIT CONSTRAINTS
A number of independ ent variable s were constrain ed
2.
0
valve permitted lift
valve spring stiffness
EXPLICIT CONSTRAINTS
Hd1 ~
x 10 Q
+X '.3_0
2
7~ 3
where D is the compress or cylinder diameter
The variables x , x
11
12 can be eliminate d from the
study if fixed values
are introduce d for the ratio
valve disc diameter
.
,1
1
valve port diameter ' l.e. x11 x2 or x12 x7.
By utilising this result the total number of independen t variable s used in the present study was
reduced to 10.
1.
0
X
= suction, discharg e valve disc thickness
= suction,
x5 ~
Limits were placed upon the maximum values for
the velocitie s with which the m3ving element in a
valve strikes a displarem ent limiting stop.
Practica l experien ce has determine d that the
stresses associate d with valve breakage s are related to the impact velocity and the types of
material being brought into contact.
The values
used are empirica l and the constrai nts are implicit since the impact velocity depends in a complicated way upon the valve geometry and the compressor operating condition s.
The impact velocities are calculate d by the mathemat 1cal model
and are clearly dependen t variable s which cannot
be constrain ed explicit ly.
Consider ation of the foregoing statemen ts reveals
that some 17 limiting condition s (constra ints)
are applicab le in the present study.
In any
given· study the number of constrai nts will vary
accordin g to the number of independ ent variable s
employed and the restricti ons placed upon them by
the designer in the light of his previous practical
experien ce.
GENERAL TREATMENT OF CONSTRAINTS
It is convenie nt to represen t the constrai nt conditions in a unified manner which may then be
used when penalty functions are introduc ed.
~ x8 /:. y d
Thus in general G. (X)~ 0
j = 1,2, ••••17
J
or for the 17 constrai nts applied in the present
study
4. Finite positive values for valve spring stiffness.
G2
=
=
G3
= c3
x6
Hd1)
G4
= .c
Hd2
x6)
G1
219
c1
x - H )
1
s1
cz
Hs2
x1)
4
=
G5
C5x2
Thus
where
G6
=
C6x7
G7
=
C7x3
G9
=
C9x8
G10
=
C10(Yd - xB)
G11
= c11x4
G12
=
c12x9
G13
=
c13x5
G14
=
c14x10
'2JXl (,(X)
indicated adiabatic efficiency
The relationship between the variables X and the
objective function FCXJ is a highly non linear
function and is evaluated numerically using the
compressor simulation model programmed for a
digital computer.
PENALTY FUNCTIONS
An external penalty function is introduced into
the optimisation routine to prevent any variable
taking on values which cause a constraint to be
The penalty function is obtained by
violated.
adding an extra penalty term to the objective
function
.J•I7
+
C ( -2 D - x2
15 3
G15
= - - -1.:..___ _
volumetric efficiency
=
CB(Ys- x3)
GB
F (X)
G16
=
c16( Vs
- v
G17
=
c17( vd
- vd
7
k: .
Joj
)
- X
¥[Min [G. (X), o]
J
The meaning of the term Min [G· (X), 0] is that
the penalty term is zero and tKe penalty function
is equal to the objective function when all the
const~aint conditions are satisfactory
s
[G. (X)q 0].
J
The factors C. are used to ensure that the same
The external penalty function is the sum of the
objective function and the penalty term which is
made much larger than the objective function when
one of the constraints is violated. [G j (X) ( 0].
¥kis the so called penalty factor whi~h ensures
that the penalty term is made much greater than
The Simplex method has
the objective function.
to discard a design point which would entail a
constraint violation and the penalty function
technique provides a convenient method of ensuring that all variables stay within their permitted bounds.
J
order of magnitude prevails for all the constraint
functions G. thereby making it possible to introJ
duce penalty functions which will prevent any
variable from exceeding prescribed bounds.
In the present case
1.0
c1,cz--C9, c1o' c15' c16' c17
c11' c1z
c13, c14
= 0.01
= 10.0
THE SIMPLEX METHOD
OBJECTIVE FUNCTION
In the SIMPLEX method an N variable optimisation
problem requires the formation of an N + 1 sided
polygon with one of the vertices corresponding to
an initial choice of variables (possibly an
Each vertex reexisting design configuration).
presents a given combination of the independent
variables and is associated with a corresponding
In the
value of the objective function .F (X).
present study the SIMPLEX method requires that
the vertex associated with the maximum value of
.F(X) be discarded and a new vertex (having a
lower value of f(X)) is introduced to create a
The process is repeated and the
new polygon.
shape and position of the polygon change as the
technique seeks to establish a combination of
variables which yields a minimum value of :F (XJ.
Unsatisfactory vertices are replaced by procedures involving reflection, expansion, contraction,
reduction.
For a given compressor operating condition a compressor design might be deemed to be an optimum
if it requires a minimum power input per unit
Thus considering a
volume of fluid delivered.
single compressor cycle
.
1
(Q/Q
0
)
(W /W.)"
0
1
= volume of fluid induced at suction
conditions
Q = cylinder swept volume
0
w. = indicated cycle work
where Q
1
w0 = theoretical
adiabatic cycle work
220
Figure 1 illustrat es in a 2 dimensio nal manner
the basic steps of the SIMPLEX method.
The
figure represen ts an N + 1 sided polygon for an
N dimensio nal optimisa tion problem.
Xhrepres ents
the vertex having the highest value of F(X).
X represen ts the vertex having the lowest value
1
of F(X) and Xb represen ts the centroid of the
polygon formed when the point xh is excluded .
In figures 1 (a), (b), (c) X is the point obtainr
ed by reflectin g the point ~h through the centroid
3)
Calculat e the values of the variable s corresponding to the centroid of the polygon that
excludes valu.e.s correspon ding to the point Xh.
=~
i.e. xb
4)
if
..,
xi
i
1h
Test whether the current polygon is small
enough to establish that an optimum
condition has been achie.;.v.:..ed;;,.·:..__...,.__
t'(~cxi)
~
xb.
The dista~ce (Xh- Xb)_is equal to the
distance (Xb - Xr).
Pain~ Xe correspon ds to the
situation where the point Xh is reflected to a
point twice as far from the centroid as ~h was
N : 1/
- F(Xb)) 2
E
,,,
If this condition is satisfied then the optimisation has been completed and the optimum
values of the variables correspon d to those
of the point ~ 1 , the point having the lowest
value of F(X).
In the present problem
from the centroid .
was set as 0.001.
i.e.
Xb - Xe
= 2(Xb
i.e.
e
5)
In figure
(c) the point Xc represen ts a contraction of the reflected point xr t~wards the
centroid Xb.
As illustrat ed point Xcis midway
between points Xr, Xb.
ship xr
6)
t~roug~
= 2Xb
xb to xr using the relation -
- xh.
Recalcul ate F(X~).
moving in the direction of the reflectio n
has yielded an improvement and this process
may be taken a stage further by reflectin g
xb through xr_to ~§using the relations hip
Xe
must be remembered that whilst the illustrat ion
in figure 1 is a two dimensio nal represen tation
= 2Xr
- Xb.
If F(Xe){Fc Xl) then xe is considere d to be
an improved point and the point Xe is substituted in the simplex in place of ~h(expan­
sion as in ~jgure 1 a).
The procedur e then
returns to step 2.
of the SIMPLEX method, the terms X, Xh, x are
1
N dimensio nal arrays represen ting combinat ions of
N variable s.
If F(Xe)~F(Xl) replace point xh by point
Xr and return to step 2.
OPTIMISATION PROCEDURE
Set up the initial N + 1 sided polygon in
which one of the vertices correspon ds to a
7)
feasible combinat ion of the variables X •
0
2)
If converge nce has not been achieved then the
point with the highest v~lue of F(X), Xh is
refle:ted
In figure 1 (d) the ~oint Xc represen ts a contraction of the point Xh ~awards the centroid .
In
figure 1 (e) the point xre represen ts a reduction
in the size of the polygon so that the polygon
has been reduced to half its initial size.
It
1)
is
Compare the values of the objective function
F(X) for each of the vertices of the current
polygon and establish the vertices having the
If F(X )}, F(X ), judge whether
_
r 1
F(Xr)»F( X_) where (X.) covers all points
l
1
in the simplex excluding the point (Xh).
If this condition is not fulfilled then point
Xr is a better
p~int than Xh.
Point Xr is
substitu ted for Xh' reflectio n as in Figure 1b
and the routine returns to step 2.
minimum and maximum values of F(X), i.e.
FcX) l ' FcX)h.
8)
<
If however F(Xr) FcXh) some improvem ent
has been achieved by a move in the reflectio n
direction but the reflectio n has been excessive.
Contract Xr to Xc when using
as in Figure 1c
221
Figu~es 3 and 4 show the progress of the optiml~atlon process as it seeks to establish an opFigtlmum ~o~b1nation of the design variables.
Substitute point Xc for point Xh. and return to
step Z.
ure 3 1llustrates the variation of the independent variables whilst Figure 4 shows the effect
Both figures are
upon the derived parameters.
drawn to a base of iteration number (the number
of times the compressor simulation routine was
called following the formation of the initial
Figures 3 and 4 show that the optim~implex).
lSed values of both the independent variables and
the objective function may differ significant ly
For the present
from their initial values.
study some 200+ iterations were needed to attain
an optimum combination and significant improvements are predicted for the compressor performance parameters as revealed by the marked reduction in the value of the objective function.
The delivered volume remained approximate ly constant at 26 litre/min but the indicated power redThis reduction in indicuced from 136W to 65W.
ated power represents a reduction in the energy
required to drive the compressor and reveals the
potential of design optimisation techniques.
If F(Xr)itF(Xh) the current polygon is contracted using Xc = (Xb + Xh)/Z.
9)
If F(Xc)<F(Xh ), replace Xh by Xc as in
If this condition
Figure 1d and return to 2.
is too
polygon
current
the
satisfied
is not
large and includes a point where
The polygon is reduced in size
F(X) ~ F(Xh).
by moving each vertex towards the vertex
having the lowest value of F(X) i.e. F(X ).
1
This move is achieved by writing
Xi = (Xi + X1 )/2.
and returning to step Z.
as in Figure 1e
The computer program embodying the optimisation
algorithms is relatively simple and is linked
directly to a number of routines which evaluate
the objective function, check the constraint funcFigure Z
tions and set up the penalty functions.
Figures similar to figures 3 and 4 are valuable
in illustrating the manner in which the independent var1ables and the dependent quantities change
Such
as the optimisation procedure progresses.
fig~r~s are needed when determining whether the
opt1m1sat1on procedure should be stopped. Figure 4
clearly shows that this particular optimisation
had to all intents and purposes converged after
Marginal gains were predicted
ZOO iteratio~s.
from iterations ZOO - 250 and virtually no changes
emerge from iterations beyond Z50.
illustrates the way in which a control function
subroutine links the various parts of the necessary computer program.
When the optimisation process has converged
values of the independent variables for the centroid of the simplex are printed out together with
the correspondin g values of the objective function,
valve opening and closing angles, compressor capa-
An optimisation can show which par·ameters are
truly significant and which constraints exercise
the greatest influence on the value of the objectAlteration of the value of a conive function.
straint may render that constraint inactive as
far as the optimisation procedure is concerned.
For example, in the present case, setting valve
impact velocity values of Vs = 2.5 m/s, Vd :5.0m/s
could be seen as establishing a boundary between
an active and an inactive impact velocity criterIf Vs)Z.5 m/s and Vd)5.0 m/s the same ulion.
timate optimum design was reached as was the case
with Vs = 2.5 m/s, Vd = 5.0 m/s. cf Table3,
By implication the calculated
columns 1 and 3.
values of the impact velocities are less than the
impact velocity constraint values and the constrThe optaints have effectively become inactive.
imum design was being constrained more vigorously
by one of the other explicit variable constraints .
city indicated power and volumetric efficiency obtained from the compressor simulation model.
RESULTS OF OPTIMISATION PROCEDURE
The previously described optimisation procedure
was applied to a reciprocatin g air compressor
having a bore of 38.1 mm, a stroke of 25.4 mm run
Optimisation was
at a speed of 1000 rev/min.
performed for a fixed pressure ratio of 3:1 when
the nominal suction conditions were, suction pressure 1 bar absolute, suction temperature Z90 K.
Calculations were performed for a fixed value of
the ratio (valve plate area/valve port area) of
Impact velocity constraints were set at
1.3 •
V = 2.5 m/s, Vd = 5.0 m/s for the suction and
8
Table tshows the
valves respectively .
d1scharge
values of the major design parameters for an existing compressor together with the parameter values
Table 2 also
used to commence the optimisation .
contains values for these parameters at the mid
point (after 100 iterations) and the endpoint
(after 240 iterations) of the optimisation process.
22Z
However wh~n Vs(2.5 m/s and Vd<5.0 m/s the impact veloc1ty constraints become active and either
Vs or Vd (or even both) determine the ultimate
A comparison of columns Z and 4
design point.
of Table 3 shows that whilst th9 final design
point is the same vd had been raised from Z.5 m/s
Clearly the suction valve impact
to 5.0 m/s.
velocity constraint is having a dominant effect
upon the outcome of the optimisation whilst the
discharge valve impact velocity constraint is
The present work reveals the importinactive.
ance of having correc t values for impact veloci ty
constr aints so that design proced ures are not constrain ed unnec essaril y.
TABLE 1
COMPRESSOR DATA
Two variab les were elimin ated rrom the optim isatby use of fixed values for the ratio (disc
diame ter/po rt diame ter).
Figure 5 shows that
this ratio has a profou nd effect upon the value
of the object ive functi on.
io~
PARAMETER
c.f. F(X)pp t fallin g from 1.54 to 1.3 and then to
1.15 as tne ratio (Av/AP) was reduce d from 1.5 to
1.3 and then to 1.1. See Table 4.
This result shows a clear need to minimise valve
seat,, widths , a featur e which must be tempered in
practi ce by the need for effect ive sealin g and acceptab le wear chara cteris tics.
UNIT
Cylind er Diame ter
mm
38.1
Stroke
mm
25.4
Conne cting Rod Length
mm
BB.9
rev/mi n
1000
Suctio n Pressu re
Bar
1 .0
Discha rge Pressu re
Bar
3.0
Suctio n Temperature
Ko
290
mm'
1450
Speed
CONCLUSIO\IS
The combin ation of optim isation and mathem atical
modell ing techni ques equips the design engine er
with a powerf ul tool with which to attack design
improvement problem s but does not absolv e him from
the exerci se of his profes sional judgement in
assess ing the suitab ility of any model used by him
or in the select ion of the constr aints used to
circum scribe a partic ular set of possib ilities .
Work report ed in this paper shows how the combination of a relativ ely simple mathem atical model of
a compre ssor [1] and the Simplex Optim isation
Technique was able to indica te the existe nce of an
optimum combin ation of some 10 valve design parameters subjec t to 15 explic it and 2 implic it constrain ts.
VALUE
Cleara nce Volume
(a)
(c)
( b)
REFERENCES
1.
2.
3.
4.
5.
MACLAREN J.F.T. , KERR S.V., "An analyt ical
and experi mental study of self-a cting valves
in a recipr ocatin g air com?re ssor". I.Mech .E.
Confer ence Indus trial Recipr ocatin g and
Rotary Compr essors, Design and Operat ing
Proble ms.
London, Oct. 1970.
x~Xf
NELDER J.A., MEAD R., "A Simplex Method for
functi on Minim isation ".
Computer Journa l,
Vol. 7, No. 4, Jan. 1965.
MACLAREN J.F.T. , KERR S.V., HOARE R.G.,
"Optim isation of compre ssors and valve design an initia l study using a direct search
techni que". Proc. 3rd Purdue Compressor
Technology Confer ence.
July 1976.
/p.,~~e
X~XI
Xb
Xb
(d)
(e)
FIG.l. PROCEDURE OF SIMPLEX METHOD
MACLAREN J.F.T. , KERR S.V., HOARE R.G.,
"Optim isation of compre ssor valve design
using a Complex Method".
Proc. 4th Purdue
Compressor Technology Confer ence, July 1978.
DEMPSTER W., TRAMSCHEK A.B., MACLAREN J.F.T. ,
"Optim isation of a refrig eratio n compre ssor
valve design using a modifi ed Complex Method".
Proc. XVl th Int. Cong. of Refrig . Commission
82 Paris. Sept. 1983.
Xh
Xh
Kc P..,
Control
ObjE!'ctiv@
Function
F(i)
Funct1on
Subroutm e
Constrai nt Function
G(X)
Penalty Function
p(i)
Fig. 2. THE COMPOSITION SCHEME OF COMPUTER
PROGRAM
223
TABLE
PROGRESS
TO
UNITS
ITEM
2
OPTIMUM
DESIGN
EXISTING
COMPRESSOR
VALUES
INITIAL
VALUE
I.N=:1
VALUE
AT
I. N==1 DO
VALUE
AT
I.N==240
Suction Valve
Plate thickness
Port diameter
Valve plate lift
Valve spring stiffness
Valve spring preload
mm
mm
mm
N/m
N
3.3
8.4
0.6
200.0
0.2
1.0
6.0
0.4
50.0
0.05
1. 57
10.03
1. 07
89.9
0.14
1.23
12.13
1.00
69.8
0.18
Discharge Valve
Plate thickness
Port diameter
Valve plate lift
Valve spring stiffness
Valve spring preload
mm
mm
mm
N/m
N
3.7
8.4
0.6
420.0
0.5
1 .o
3.0
0.2
250.0
0.25
2.6
8.1
0.56
539.0
0.42
3.25
12.45
0.56
470.0
0.23
litre/min
25.2
6.07
16:73
78.72
87.2
77.6
1.47B
25.9
6.64
70.5
136.83
89.6
44.7
2.498
26.2
2.56
17.89
76.89
90.5
79.5
1.391
25.0
2.32
12.0
66.8
86.3
91 .5
1. 267
Performance Parameter
Volume flowrate
Suction power loss
Discharge power loss
Indicated power
Volumetric efficiency
Indicated efficiency
Objective function
Limiting conditions
w
·w
w
"'
'"
"''"
Av/Ap = 1.3
TABLE
EFFECT
OF
IMPACT
Impact velocity at
valve stop
Suction Valve
Plate thickness
Port diameter
Valve plate lift
Valve spring stiffness
Valve spring preload
Discharge Valve
Plate thickness
Port diameter
Valve plate lift
Valve spring stiffness
Valve spring preload
Performance Parameter
Volume flowrate
Suction power loss
Discharge power loss
Indicated power
Volumetric efficiency
Indicated efficiency
Objective function
Limiting conditions
VELOCITY
2.5 m/s ;
vs
CONSTRAINTS UPON
vs )/ 2.5
m/s
vd~s.o
vd
= 5.0 m/s
3
OPTIMUM
VALUES
v =1.25
s
Vd=5.0
OF
VARIOUS
v =2.5
s
Vd=2.5
PARAMETERS
vs =1.25
Vd=2.5
mm
mm
mm
N/m
N
1.23
12.13
1.00
69.8
0.18
1.01
16.32
0.39
64.0
0.08
1.17
14.09
1.18
114.7
0,13
1. 01
16.32
0,39
64.0
0.08
mm
mm
mm
N/m
N
3.25
12.45
0.56
470.0
0.23
3.61
8.99
0.59
1376.0
0.67
3.31
11 .16
0.59
774.0
0.78
3.61
8.99
0.59
1377.0
0.67
25.0
2.32
12.0
66.8
86.3
91 .5
'1.'1.67
26.4
4.01
16.65
78.2
91.2
78.1
1. 398
27.0
1. 74
13.25
72.5
93.1
84.3
1.274
26.4
4.01
16.65
78.2
91.2
78.1
1.398
litre/min
w
w
w
%
"'
'"
Av/Ap = 1 .3
Iteration Number•240
224
DISCHARGE
VALVE
THICKNESS
600
3-0
Q\SCHARGE VALVE
,.--..
z
E
sutrT citrvA LvE" P"o ff tliAM -ETE R-IS
VALVE SPRING STIFFNESS
'-../
0
500
PORT DIAMETER
0-3
0
--'
OISCHARGE
w
0-2
ll.
!,/')
VALVE SPRING PRELOAD
DISCHARGE
0-1
w
300
Vl
(!)
z
a: 200
ll.
!,/')
0
----
/-
SUCTION
:0::
u
a: 12
2.0
....
z
I.L.
I.L.
i=
VALVE ALLOWED LIFT
"'zw"'
2-5
I
Vl
-........._
SUCTIQ!::!__Y~.E_?.f>Rif'.!_G_~RELpAD_
--- ..: .•••,::::su~TIO~__y~LVE__l_H_I CK~ES~ ___
(!)
z
a:
400
U)
<(
g:
z
........
14
E
E
0-4
/:7--=::::~~----?UCTIO~ ~~LYE_ s~~~N-~ S~JFF~~~
100
....
60
120
160
ITERATION
Fig.3.
200
240
260
~
....I.L.
6
~
0-5
--'
<(
0
360
320
NUMBER
4
0
VARIATION OF VARIABLES DURING OPTIMISATIO
N
.Er:
- 29
=
~~~..;;;;;..;.~=:;.:_..,0·9 ~
-
120
z
0
;::: 1·6
u
z
=>
u.
UJ
80
INDICATED POWER
1·5
2:
tu
u
~1-4
m
IJ..
lb 0·5
0
SUCTION POWER LOSS
0
40
80
120
160
200
240
INTERATION NUMBER
280
320
FfG. 4. VARIATION OF PERFORMANCE PARAMET
ERS
225
to
w
> 1-5 0a:
--'
<(
ll.
>
w 6
a:
>
0
1-0 --'
>
40
~
<(
0
--'
VALVE ALLOWED LIFT
w
....
w
360
60
TABLE
EFFECT OF VALVE PLATE
Valve plate to valve
port area ratio
TO
4
PORT
VALVE
Av/Ap_
RATIO
AREA
UPON
OPTIMUM
VALUES
Av/Ap
1. 1
1.3
1. 5
Suction Valve
Plate thickness
Port diameter
Valve plate lift
Valve spring stiffness
Valve spring preload
mm
mm
mm
N/m
N
1. 38
13.12
0.8
193.6
0.13
1.23
12.13
1.0
69.8
0.18
1.93
13.38
1.02
126.7
0.02
Dischar5le Valve
Plate thickness
Port diameter
Valve plate lift
Valve spring stiffness
Valve spring preload
mm
mm
mm
N/m
N
3.94
12.02
0.54
706.0
0.43
3.25
12.45
0.56
470.0
0.23
3.92
11 .07
0.57
550.0
0.35
26.8
1. 71
6.18
65.2
92.6
93.7
1.153
25.0
2.32
12.0
66.8
86.3
91.5
1.267
I.N = 240
26.1
2.96
23.81
83.4
90.0
73.3
1. 516
Performance earameter
litre/min
Volume flowrate
Suction power loss
Discharge power loss
Indicated power
Volumetric efficiency
Indicated efficiency
Objective function
w
w
w
?~
""'
vs = 2.5
Limitin5J conditions
2·0
1·8
/
z
0
- 1·6
u
1-
z
::::>
lL.
1·4
w
>
1-
u
LJ.I
-.
1·2
vd
m/s
= 5.0 m/s
\\
~\ ........,_
\ \_
\
\
Av/Ap
~
I~ ...___
= 1· 5
~B =1· 3
~A_D
~1-1
aJ
0
1·0
0
I
40
120
80
ITERATION
FIG. 5
160
200
240
NUMBER
VARIATION OF OBJECTIVE
FUNCTION
226
WITH
RATIO Av/Ap