Purdue University
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International Compressor Engineering Conference
School of Mechanical Engineering
1992
A Review on Fixed-Percentage Tolerances for
Compressor Performance Parameters
K. W. Yun
United Technologies Carrier Corporation
Follow this and additional works at: http://docs.lib.purdue.edu/icec
Yun, K. W., "A Review on Fixed-Percentage Tolerances for Compressor Performance Parameters" (1992). International Compressor
Engineering Conference. Paper 925.
http://docs.lib.purdue.edu/icec/925
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A REVIE W ON FIXED -PERC ENT AGE TOLE
RANC ES FOR
COMP RESSO R PERFO RMAN CE PARA METE
RS
K.W. Yun
United Technologies Carrier Corp.
Syracuse, New York 13221 U.S.A.
ABSTRACT
Applying a fued-pe rcenmg e tolerance band to compre
ssor perform ance parameters is a
commo n practic e in the industry. While fmding the
practice reasona ble, its pitfalls are review ed
from the custom er's as weU as the compre ssor DJlllluf
acturer's standpoint. Enefgy efficiency
ratio needs to be controlled separately from capacit
y and input power, although it is derived from
the two. Depending on the purpose, differe nt
levels of fixed pen;eotages are suggested.
Statistically, a bivariate normal distribution can be
assumed for typical perform ance pamme terS.
A set of I02 paired bivariate data for capacity and
input power from an actual production audit
has been statistically analyu :d to verify that actual data general
ly fit the suggested dhtribu tion.
Correlation analysis shows a slight positive correla
tion between capacity and input power against
the assumed theoretical randomness.
INTRO DUCI 10N
Besides cost, delivery, quality, and reliability factors, accepta
nce of a compre ssor at any
given time is judged by its performance. Criteria or
parame ters for compre ssor perfOTDIBilCe are
usually capacit y (BTU's or calorie s per hour), input
watts, energy efficiency ratio (EER), sound
pressure level (SPL) and vibration. Among tbese,
EER is not an independent parnme ter in the
sense that it is derived indirectly as the ratio of capacit
y to input power. Among these criteria ,
the lim three are the most commo nly used ones.
PRAC TICES IN THE INDUSTRY
Produc t evaluation, production audit and quality
control deal with these parameters for
control purpos es. Original equipment manufacturers
use them in accepting produc t performance.
In the compressor manufacturing industr y, it is ComJIIO
il practic e to aUow a fixed-p ercenta ge
toler.mce band to the published or nominal value of
a given perfonn ance parame ter. Certainly,
nothing is particularly wrong with the practice and
it has served the industry fairly weU.
Accept ing the practice as a sound one, we should
establish its theoretical basis and
statistical significance verified with data generat ed
in an actual produc tion environ ment.
CUSTO MERS • AND MANU FACT URER S• PERSP
ECTIV ES
Let us examin e the fixed-percentage tolerance band not
just fmm the manufa cturer's point
of view, but also fmm the custom er's, First, from
the manufa cturer's point of view, the fixedpercentage tolerance band is a practical control tool
in product 3SSUl11IlCC. The rule is simple
enough to be popularly applied. Where produc t model
proliferation is the case it is quite a
convenient control scheme. On the other band, there
are some pitfalls. One of the pitfaUs is
the "as long as within ±. x percent" syndro me.
As actual mean value deviates from the
published nominal value, the risk of performance fallout
increases. Anothe r pitfall is that the
fixed-percentage rule is nOl very flexible as nomina
l perform ance of mass produc ed compre ssors
could have a cyclic trend. A oonnal production
environ ment sees both short-and-long-term
shifting trends. Application of the rule witbout respect
to time ~ve is not a rntional
approach.
1307
ed as long as
it is difficult to reject product shipp
Next, from customer's point of view,
based on
case
a
make
could
mers
the tole.ance band . Custo
the product perfo rman ce falls into
rmance deficiency is
perfo
prove
to
but
ed,
shipp
ct
produ
of
statistical analysis of performance
resources or lack the
custo mers have limited time and
realistically a diffic ult task. First ,
ent based on poor
argum
All
-acy.
accw
to the required
capability to determine the deficiency
perforJllance does
Poor
tion.
m bas an inherent limita
performance in an air conditioning syste
ce.
rman
perfo
r
resso
comp
not nece;sarily come from poor
S
SCO PE OF CONTROL PARAMETER
power.
manufacturers are usually capacity, input
are
Perionnance par.uneten controlled by
three
or
two
first
the
,
these
Of
level, and vibration.
energy efficiency ratio, sound pr~
other band, as ooise gets
the
On
ns.
reaso
good
for
ol
contr
ones looked at as product assurnnce
, acts as an important
and vibration, primarily sound level
customers· increasing attention, sound
contr ol par.u neter .
as long as
whether we should also control EER
A relevant issue which arises here i.s
itioning
cond
air
as
First,
yes.
ite
defin
a
is
er
. The answ
EER.
ol
capacity and input power are controlled
contr
to
rer
factu
manu
the
for
nt, it i.s only fair
system EER is scrutinized as a requireme
capacity and input power
that
eter,
param
t
enden
indep
an
not
One migh t argue that EER i.s
n a fued ± 5
capacity and power are controlled withi
determine the value. However, if ooly
could result
EER,
nal
nomi
the
of
nt
J>ef'e
-10
cts with
percent limit, otherwise unaccepcable produ
e 1).
in being ~epted (Poin t ·A· in Figur
Fig .l Con trol
....,
]
l. 10
Par ame ters
Lim its for Per form anc e
-----------~
-r------------------------------.:i>'"·{•
0!:
'.
.,
1.05
/
...
:J
~~~
,. ....--
/:,~.. .)§:
/
a:"
0
I
.95
~It:~
D
'
,
Q
'A'
Wear
Loos " Fit
.90
. 90
• 95
Wat t Rat to
I. 00
I. 05
1. 10
(Ac tual /Ra ted)
olled as a
y situation. EER should also be contr
Cenainly, this cannot be a satisfactor
same ±
The
.
ntage
perce
r
prope
the
ion is concerned with
sepat'llte param eter. The next quest
tical analysis of the prO!iuction
statis
the
as
,
nable
reaso
is
nt,
x percent limit, such as ± 5 perce
data will sbow in later discussion.
ee solid
should be discussed further. F~, ~thr
There are, however, two points which
. These are
limits
ol
contr
ite
defin
the
e
defin
Ow•
ol winO
lines of bottom boundaries of the •contr
lines of the
maximum watts. The other three dotted
the minimum capacity and EER and the
1308
upper boundaries may seem somewhat questionable limit<;. One might :u-gue:
why limit better
performance? The limits should still be enforced as performa
nce beyond tbese limits are
practically not feasible even with a well-designed compres
sor and controlled manufacturing.
Fwtherm ore, the limits are also helpful as an alert w possible
testing errors. If the mean
performa nce shifts from the published oomiual value as shown
in Figure l, then the window may
have to be shifted. The second point of concern is with the
shift of data points or the control
window. The three arrows point to the general directions of shift for
effects of run-in (or breakin), loose assembly fits, and mechanical wear of intemiil pans.
SOUND AND VIBRATION AS CONTR OL PARAM ETERS
As pointed out earlier, these panuneters are getting ever-inc
reasing attention. Applying
a control limit to these paramete rs is a more complex matter.
For example , setting a limit in
overall SOWld pressure level (SPL) does not necessarily provide
a satisfactory control of sound
quality. Frequen cy content is often more importan t tbao the
overall SPL value. It is tempting
to use a fixed-percentage upper limit for these paramet ers too,
but one must recognize that SPL
represents a proportional number and a fixed-percentage cannot
represent a true control limit.
It is more appropri ate w use a spc;ific upper SPL limit,
not a nominal SPL plus a fixedpe.-ceotage. In addition to this limit, specifk controls on frequenc
y content may be set.
"GOOD " FIXED- PERCE NTAGE AS A CONTR OL LIMIT
Accepting the fixed-percentage control limit as a popular,
reasonable, and useful
approach, the uext question is what should the control limit
be. Without talcing a survey in the
industry, ± 5 percent is known as a "good" limit and widely practiced
. Despite the possible lack
of a theoretical basis for the control limit, we regard the
limit as a reasonable compromise
between custome r's and manufac turer's needs. The control
range eDCOmpasses reasonable
variations in manufacturing factors (vMiability in parts, machinin
g, assembly, and testing errors). ·
The actual production audit dam in this paper proves the presump
tion.
The next question is if the same control limit is appropri ate for
both rating a compres sor
and providin g samples for the custome r's unit development.
For these special ptli"J)OSt::S, one can
make a good case for a tighter tolerance requirem ent. For
compres
performa nce rating
purpose, a good practice is to select samples with perform ance whichsorapproach
es the mean
nominal values as closely as possible. Note that the publishe
d value may not be the same as the
actual nominal value. There is no sense to select intention
ally a biased sanJple, as production
is controUed within a fixed range. For a custome r sample, it is
importan t not to seud a "good
looking" biased sample. This will only hurt the custome r
and manufac turer later. Since the
manufacturer must live up to the perfonna nce represented
by the sample shipped, the sample
should represent tbe nominal producti on perform ance. Thus,
a tighter nnge, such as ± 2.5 or
3 percent is reasonable for custome r·sampti ng purposes.
STATIS TICAL ANALY SIS OF PERFO RMANC E PARAM
ETERS
As we are dealing with statistical perfonw mce data, it is
appropri ate
to consider a
statistical analysis of such data. Starting with capatity , input
power and EER as control
variables, we shall first c::haracterize the variable, then
suggest an appropri ate statistical
distribution function. The assumed distribution flmctloo
will then be verified with actual
production data.
Data to be examiD ed- To examine an actual producti on situation
, we shall analyze the
production audit data of a rolling piston-type rQtary compres
sor model. The 102 sets of rawscore data cover 28 weeks of producti on by a certain compres
sor manufac turer.
1309
d capa city and input
:s- The data is in the form of paire
Mea sure d Perf~ Vari ablc
re 2 depicts the data
Figu
.
data
d 102 pairs constitutes a bivariate
pow er data. Tbe set of measure
in a scatt er diagram.
d1t Da ta
Fig .2 Per for ma nce Au
940 0
.s::
"
Rat ing Poi nt
920 0 I-·
;J
+'
••
IIl
c sm:m
"u
+'
-
880 0
-.
860 0
~
•
Ill
....
u"'
0
0
nt- e.3 5
Co rre lati on Co eff lcie
.L
840 e,,90
820
810
800
830
840
Inp ut Pow er in Wa tt
tly shifted from tbe rated
city and inpu t pow er are sligh
In this particular case, mean capa
rmance as one must
perfo
rated
the
in
ly justify a change
mean values. -This does not necessarilong-term time pecspective.
from
look at prod uctio n perf orma nce
the functional
diagram prov ides an insig ht into
The locus of points in the scatter
from the othe r
ber
num
ltant
resu
a
is
the variables. As the EER
data that
sents
repre
relationsbip tbat ell.ists between
data
biva riate
have bivariate data sets. The
distr ibute d (whi ch is
ally
two para mete rs, basically we
norm
are
bles
varia
two
the
we assume that
occu n; in orde red pairs . If
deal ing with a biva riate norm al
of the analysis to follow), we are
a reasonable assu mpti on in light
distribution.
that mor e capacity requ ires ·
interdepeudent on e~K:h otbe r in
Capa city and inpu t pow er are
hip betw een two parameterS.
here is not a functional relations
more pow er. How ever , the issue
ucts are inde pend ent or not.
prod
in
rs
mete
tions of tbe two para
Ralher, we are intere:sU:d if varia
retical stan dpoi nt, it wou ld
theo
a
from
e it ollC way or the othe r
·
Whi le it may be possible to prov
.
data Whi le the resu lt obtained
thesis statistically with actual
of com pres sors or
s
be mor e direct to test out a hypo
type
r
othe
to
ied
appl
ly
ot be universal
from this set of data cano
for statistical interf~.
11131lufacturets, it give s a clue
to vary. Valu es of
random variables and mus t be free
selec ted in the
Both varia bles are assu med to be
omly
rand
unit
n
uctio
prod
the
on
depending
riate norm al
biva
the
ooe varia bles occu r at rand om,
of
think
to
nt
enie
equa l status. It is conv
sample. Both variables have
3.
surface, sucb as shown in Fig
distribution as a three-dimensional
1310
>.
...,-\
+'
....
~
!:
<ll
0
'&v
0'&
>.
+'
.Q
"'
.0
0
....
c...
Input Power
Fig.3 Bivariate Normal Distribution
In the three-dimensional diagram, the exact shape of this figure, which looks like a
fireman's hat, will depend on how closely the variables are related. Here, we can see that:
(i)
The frequencies are concentrated in a elliptical area with the major axis inclined upward
to the right. There are no very low capacity with high power nor very high capacity with
very low power.
(ii)
The frequencies pile up along the major axis, reaching a peak near the center of the
distribution. They thin out around the edges vanishing entirely beyond the borders of the
ellipse.
Assumed Distribution Function - Capacity and input power vary together in a joint
distribution. If the form of the joint distribution of two variables, X andY, is normal, the joint
distribution is a bivariate normal distribution. A special joint distribution of X and Y is assumed
in making statistical inferences in simple correlation analysis - a bivariate normal distribution.
If we slice the surface at any level of Y or X, the shape of the resulting cross sections are
normal.
To start with, it is a reasonable assumption !hat the paired variables will a bivariate
normal distribution function. A bivariate normal distribution means that each of two variables
is distributed about the olher normally. Because of the nature of the bivariate normal
distribution, the values of either of the variables are distributed normally for a: fued value of the
other variable. This distribution has five parameters, mean and standard deviation for each
variable and !he correlation coefficient, p, of which r is an estimator. The parameter, p,
measures the closeness of the population relation between X and Y; it determines the narrowness
of the ellipse containing !he major portion of the observations.
Actual Distribution Function - To prove that we are indeed dealing with a bivariate
normal distribution, we must show that each of capacity, input power, and the resultant EER
variables has a normal distribution. This can be readily proved by l1rst converting the frequency
distribution into a cumulative frequency distribution, then plotting it on special normal paper.
1311
distribution. Thus, the bivaria te data meets
All three variables are found indeed to be nonnal
all requirements to be bivariate normal distribution.
study is to determine the strength
Correl ation analys is - The objective of a correlational
indicates the extent to which
tion
correla
The
of a relationship between paired observations.
e. Though the regression analysis
variabl
another
of
values
to
related
are
e
variabl
values of one
a correlation analysis does, we are not
can provide essentially the same information that
focuses solely on the strength of the
tion
Correla
interested in a regression analysis here.
that both X and Y values differ from sample
assume
we
model
tion
correla
the
In
ship.
relation
to sample.
of correlation here, which will show
We want to find a measure, namely, the coefficient
is an index of the degree of this
ient
coeffic
tion
correla
The
bility.
covaria
this
us the degree of
estimate r are intimately
sample
its
and
p
ient
coeffic
tion
correla
ion
covariability. The populat
tion.
connected in the bivariate normal distribu
es is 0.35. This lies between no
The computed covariahility between the two variabl
es varying in the same direction
variabl
both
1),
+
(p""
blity
covariance (p =0) and perfect covaria
the 95% confidence interval
gives
r=0.35
ient
(the plus sign). Then sample correlation coeffic
n two variables means that high
betwee
ship
relation
e
positiv
A
+0.51.
<
p
<
for p, +0.16
power, and low values of one with low
value of capacity are paired with high values of input
only; it is of no use in describing
relation
linear
of
e
values of the other. Correlation is a measur
nonlinear relations.
hypothesis that p=O with a two-sided
Test of Hypot hesis that p=O We wish to test the
distribution says that the region of
normal
test and significance level of 0.05. A table of the
6 for a sample batch of 102 cases,
r=0.34
sample
For
-1.96.
<
z
and
1.96
>
z
significance is
z~r;n=I~0.346x10.05"3.48
esis is not tenable and the sample
As z=3.48 is in the region of significance, the hypoth
p =0.
esis:
hypoth
the
reject
estimate is significant. Thus, we must
tion between capacity and input
What this means is that there is a slight positive correla
es. Howev er, this is not
variabl
random
perfect
not
are
ed
measur
es
variabl
power. Thus, the
ness.
random
perfect
affect
might
which
!actors
too surprising in view of various
DETERMINANTS OF PARAMETER VARIABILITY
e sources. Variations exist in part
Variability in performance results from multipl
We also recognize "break -in' run as
errors.
testing
and
fits,
ces,
clearan
ly
dimension, assemb
a variable.
in" and how long it is run are·
Time Variab ility - How compressor is 'broken
(i)
decreases with run time and
known determinants. The facts that input power
tion shifts horiwn tally in
distribu
the
that
t
sugges
slightly
only
capacity varies
Figure 2. This will increase EER.
(ii)
nent fit-up in assembly affects
Manuf acturtn g Assem bly - The tightness of compo
input power and capacity.
g
affectin
thus
,
leakage
gas
and
drag of mating pans
(iii)
n in the data may be due
Testin g Errors - It is not clear how much of the variatio
ance variables are
perform
ed
measur
that
ze
recogni
to testing errors. We should
observe values with testing
subject to errors in measurement. That is, we really
1312
error included. Testing errors are independently and normally distributed with
their own mean zero and respective variances. It is quite possible, however, that
different cases can bias the inherent random nature of testing errors.
CONCLUSIO N
Fb:ed-percen~~~ge IOkrance band to compressor performance pan~meters are commonly
used in
the industry, and the practice is reasonable. The "as long as within ±x%" syndrome without
considering statistical implication and manufacturing variability is not rational. A tighter
tolerance band for customer samples is justified. Along with capacity and input powcr, energy
efficiency ratio needs also be controlled, although it is derived from the two. Statisacally, a
bivariate-IIQ(mal distribution can be assumed for typical performance parameters. Char-""~cs
of such distribution describes ~tual production audit data. A set of 102 paired bivariate dala-,
capacity and input power, from an actual production audit has been statistically analy<IOO, Tile.
actual data generally fits well to the assumed bivariate normal distribution. Correlation ~~s,
however, shows a slight positive correlation between capacity and input power. This s~
that a strict randomness lacks in the production data.
1313