Note: Descriptions are shown in the official language in which they were submitted.
method. ~n~--in~all~tion enable on ho One Hal thy
Knowledge in adva~çe at a given moment of the result to
which a oven develoP'nq chemical eddy necessarily leads
and, on the other Honeywell the req~l~tiQn this tedium Jo
arrive it_ Particular exult fixed in advance to be
ensured .
It is an object of the invention to provide a me-
trod enabling, on the one hand, the knowledge in advance
at a given moment of the result to which a given develop-
in chemical medium necessarily leads and, on the other
hand, the regulation of this medium to arrive at a part-
cuter result fixed in advance, to be ensured.
In industry numerous chemical media are changing
through the variation of the parameters which characterize
them, namely concentration of the various constituents,
phi temperature and the like.
By way of examples of such media, may be mentioned
- acid chemical conversion baths,
- alkaline or acid baths for decreasing metals,
- water contained in boilers and heating installations,
and the like.
The use of these media leads to desired results in
certain cases and undesired in other cases and Jay be
us manifested in the above said examples respectively:
- by more or less favorable properties of coatings
obtained by chemical conversion,
- by a more or less thorough state of cleanliness
of the decreased surfaces,
- by variable scale deposition states of boilers or
heating installations.
Now, it is important for the industrialist to Know
at any moment if the result obtained at the moment con-
corned by the use of the corresponding medium is indeed
that which is expected.
However the response is often long and difficult to
obtain. Thus, for example in the case of a chemical con-
version bath,
- 5 or 6 weeks must be available to know whether
the resistance to salt fog of the coating obtained is
5 higher than 1000 hours or not,
- expensive and bulky equipment must be available
(electron scanning microscope) to know the crystalline
structure of the coating obtained.
And it is indeed certain that once these tests have
lo been carried out, that is to say, for example six weeks
later, it is possible for the bath to have changed to an
important extent, and its composition can then be very
different from that which it had at the moment when the
coating under test was obtained. It follows that a modify-
lo cation which should possibly have been made in the bath at
the moment concerned to arrange so that the result obtain-
Ed is that desired, has in fact been overshot at the mow
mint when the result is known as a result of the tests
concerned.
methods for regulating changing chemical media at-
ready exist. However, in these methods, it is customary to
measure repetitively at least one of the parameters of
such a medium, to compare each time the value measured
with a reference value and to act on the parameter con-
corned so as to modify the measured value according to the
reference value, for example, by bringing, if the measured
parameter is a constituent of the changing medium, the
concentration of this constituent to the reference value
my adding suitable regenerating substances.
And it is indeed certain that in a changing comma-
eel medium of the type concerned, these operations cannot
guarantee the constancy of the desired results.
The problem which Applicants therefore propose to
resolve is that of perfecting a method suitable for enable
in the user:
- on the one hand, to know, from the measurement of
394L~
various changing parameters of a developing chemical
medium, the characteristics of the result to which the
medium concerned leads at the selected moment,
- on the other hand, in the case where the value
found by this measurement shows that the result concerned
does not correspond to that desired, to determine first a
group ox modifications of the variables or parameters of
the medium suitable to arrange that the result to which
the so-modified medium will lead, corresponds to that
desired, then to introduce this group of modifications
into the developing chemical media concerned by action on
the parameters concerned.
Applicant has had the merit of perfecting a method
enabling this object to be achieved, this method being
characterized by the fact
- that, in the case of a given developing chemical me-
drum and for at least one of the results to which this
medium leads, a so-called provisional equation is eta-
blushed of the type
0 1 1 b2X2 bkXk box by OX I
k-1,kX~_1Xk I---+ bkkXk +- Jo bnnxn2 +---
in which
- Y represents the result to which the medium leads,
X1~ X2 --- Ok On represent variables or pane-
meters of the developing chemical medium prom which the
result Y depends and
- boy by by bun are coefficients with respect
to which the linearity of the equation is expressed, the
value of these coefficients being determined by calculi-
lions based on regression methods known in themselves,
- that each of said variables or parameters is measured
repetitively by as many probes or detectors and
- that, on the one hand, at a given moment t, what is
to say at the end of one of the repetitive measurements,
the value of each of the equations Y is calculated by
using the measured values for thy various variables at the
moment t,
- that, on the other hand, as the case may require and
in the case where the value found for one or other of the
s equations Y does not correspond to the desired result,
there is determined by calculation a group of variations
or modifications to be imparted to the variables or pane-
meters, this group of variations giving far Y the de trod
value and constituting among all the possible solutions of
the provisional equation Y, that which is most economical
and easiest to realize, and
- that, on the other hand again, as many regulating
members as independent available parameters are acted upon
individually, providing to each of these regulating mom-
biers a signal adapted to bring the variable or parameter associated therewith to the value corresponding to the
above said solution of the provisional equation Y which is
most economical and easiest to obtain.
In an advantageous embodiment, the calculations
above mentioned are effected by means of a digital computer
which is provided with appropriate data bases.
The invention also relates to other features which
are preferably used at the same tire and which will be
more explicitly considered below.
us It will, in any case, be well understood by means
of the additional description, the drawing relating there-
to and the examples, the said additional description and
examples relating to advantageous embodiments.
Figure 1 of these drawings is a diagram relating
30 to an equation Y ; Figure 2 is a diagrammatic represent-
lion of an installation suitable for practicing the method
according to the invention.
Figures pa and 3b taken together, on the one hand,
and pa, 4b and 4c taken together, on the other hand, each
35 represents a logigram relating to the practicing of the
method according to the invention.
~Z239~s3
Figure 5 is a block diagram of an example of an
installation enabling the invention to be put into pray-
lice.
Figure 6 is an example of a projection of a hype-
surface.
Figure 7 which is similar to Figure 5 shows an
other example of the said installation.
In order, consequently, to provide the means
enabling, on the one hand, to know in advance, at a given
moment, the result to which an evolving chemical medium
necessarily leads and, on the other hand, to ensure the
regulation of this medium to arrive at a predetermined
result fixed in advance, procedure is as follows or in
equivalent manner.
For a given developing chemical medium, it may be
necessary to know at any moment the result or property Ye
to which it leads this result or property being the lung-
lion ox a certain number of variables X1, X2, ..... Xi, of
the medium and of which the result depends, Ye and the Ye-
Z rubles X1, X2, ....... Xi being connected by the provision-
at equation
y = b -I b X + Buicks bit ~iXi it i
in which boy by -- by -- are coefficients, unknown at
the start and determined in the manner which will be India
acted below, the effective value Ye translating at a given
moment the result to which leads the medium taken in its
composition corresponding to this same moment.
By assuming that the result concerned Ye must be,
for example, equal or greater than the value N, it is no-
Cicero that, for the medium at the moment envisaged to be
considered as suitable, that at this moment the measure-
mint of the variables X1, X2, ....... Xi should provide for
I a value
Ye N
the measurement of the variables X1, X2, ........ Xi being
realizable by as many detectors or probes dipping into the
medium and assigned respectively to each of the variables
X1 l X2' ' Xi'
If Ye so measured is 3 N, the medium leads to the desired result ; if not modifications must be introduced
5 into the medium.
However before anything else, it is necessary of
course to establish the equation Ye and consequently,
- to select the variables Xi that are considered
as being relevant of the medium (these may be the
concentration of certain constituents, the temperature,
the pi and the like),
- to determine the values of the coefficients by
-- by
The choice of the variables is determining for
obtaining a provisional equation of good quality, the
omission of an influerltial parameter resulting in the
establishment of an erroneous equation. For each problem
set, the choice of the parameters results from the prior
knowledge of the technicians skilled in the art.
The determination of the coefficients by nieces i-
tales recourse to calculations based on conventional
regression methods (which will not be dwelt on here but
whose principle is to be found in the work of L. LIBRA,
A. MYRON, JO FENELON, "Possessing of statistical
I data, methods and program, DUNNED) using the data
provided by a certain number of experiments. From the
practical point of view, this procedure is as follows:
- the one or more properties Y to be measured are
selected,
- the experimental parameters Xi or variables
which will describe the equation are selected,
- for each parameter Xi a range of variations is
selected which is a function of the problem set and ox the
prior knowledge of the experimentation,
- a simple equation model is selected, it being
noticed that the first model selected can be of the linear
3~3
type:
1 0 by X1 + - + by Xi +- --+ b X
or linear with interactions
1 0 1 1 Buicks + bjXj +--~ bnXn + + by X X +
or non linear with square terms:
n n n
y by + Buicks bijXiXi I Buicks -
- the number of experiments defined by the choice
lo of the model is carried out and for each of them the value
of the response Y obtained is collected,
- the coefficients byway... of the model are eel-
quilted.
The model is then tested. This model being chosen
"a priori", it is in fact indispensable to test it to
accept it and to use it or, on the contrary, to reject it.
With this object, some additional experiments are carried
out for which the values of each of the parameters are
fixed simultaneously at an equal distance from the ends of
their range of variations and the response obtained is
compared with the response calculated by the model. If the
two results are neighboring, the model is accepted. In the
contrary case, the model is made more elaborate by the
addition of supplementary terms, for example, quadratic
terms.
The provisional equation representing the new
model is then written:
1 2 b1X1 I Buicks Jo bj~j +--+ bijXiXj+
+ bit Xi + by; Xj + --
The additional experiments necessary for the eel-
culation of the coefficients of the quadratic terms are
carried out and the model is tested again.
Thus, by degrees, in sequential steps, the model is
refined until the prediction is of good quality.
The equation so established is represented by a
hypersurface in a space of no dimensions for n factors.
To materialize the incidence of the various lag-
ions, the hypersurface is projected successively on Jo
sub spaces of three dimensions of which two are constituted
by two variables thaw it is desired to study and the third
S by the response Y, the remaining variables being fixed at
constant values ; this is translated by contollr lines in
the plane of the two variables, the response being mate-
realized by the values of the contour lines; thus it is
possible to contemplate projecting the hypersurface repro-
setting the equation Ye on the plane of the parameters Xanadu X4; by way of example, it may be imagined that Ye
represents the resistance to salt fog of a metal surface
which has been previously subjected to a chemical convert
soon treatment and to a paint coating, X3 representing,
for example, the content of the developing medium, that is
to say of the conversion bath, of Zen ions and X4 the
content of the same bath of P04 ions, the remaining Ye-
rubles X1 and X2 being fixed at constant values.
There will be as many sets of lines Y = f(X3, X4)
as selected pairs X1, X2.
In this case, the hypersurface is translated in the
plane defined by X3, X4 by a certain number of lines eel-
led "isoresponse lines" or contour lines and denoted by
C1, C2 Con each of these lines corresponding to a given
value of the resistance to salt fog and each point of each
of the lines representing a pair of values of X3 and X4
which results in a metal surface having the resistance to
salt fog corresponding to the line.
In Figure 1 is shown a diagram projection of the
hypersurface on the plane X3, I showing the following
isoresponse lines:
line C1 : resistance to salt fog: 80 h
line C2 : resistance to salt fog: 100 h
line C3 : resistance to salt fog: 400 h
line C4 : resistance to salt fog: 500 h
line C5 : resistance to salt fog: 800 h
line I : resistance to salt fog: 900 h
line C7 : resistance to salt fog: 9S0 h
line C8 : resistance to salt fog: 1000 h.
If it is assumed that, to be sati~factcry, the
5 resistance to salt fog of the treated surfaces must be
from 950 to 1000 h, all the pairs of values of X3 and I
corresponding to points situated on one of the lines C7
and C8 or in the surface comprised between these lines
lead to the desired result.
lo In other words if, at a given moment, the measure-
mint of X3 and X4 results in a point Pi corresponding Jo
this condition, the resistance to salt fog of the treated
part will be satisfactory; if, on the contrary, it results
in a point Pi not corresponding to this condition, that is
lo to say located outside of the area defined by the lines C7
and I , the result will not be satisfactory an it will
be necessary to introduce into the developing tedium, that
is to say into the conversion bath, a modification of the
concentrations X3 and X4 which brings back within said
area the point situated outside of the latter.
And the modification taken among all the possible
modifications leading to this result must be that which is
easiest and Yost economical to realize.
This determination is carried out at each moment by
us effecting the necessary calculations to determine, for
each of the different variables concerned and no only X3
and X4, of each of the provisional equations Ye, Ye -Y
... the simplest and most economical modification among
all the possible modifications leading to a satisfactory
result.
For example, calculations may be carried out by
means of a calculator provided with necessary data bases,
particularly data relating to the provisional equations
Ye' 2 '' n
To enable all of the calculations to be carried
out, there should be available at the given moment for
3~3
each provisional equation all of the values of the cores-
pounding parameters, which can be produced by way of toe
repetitive measurements considered above by detectors or
probes dipping into the changing chemical odium con-
corned.
The modification is then produced by the dispatch to one or several devices of a group of devices adapted to
act on the various parameters, of signals suitable for
triggering a group of responses on the part of these
0 devices, which responses are translated at the level of
each parameter by a specific change, this set of changes
resulting in the above said modification, selected and
determined by the calculations.
For convenience of description, it is assured in
lo the following and in particular in the examples that the
developing chemical medium taken into consideration is a
sequence of phosphatation treatments of a chemical convert
soon bath with zinc, it being understood that the prince-
pies which will be developed with respect to this part-
cuter example of a developing chemical medium are easily transposed to any other developing chemical medium taking
into account the preceding more general considerations.
It is known that the conversion baths concerned
comprise nickel, zinc, phosptlate ions and are accelerated
us by nitrates and/or chlorates.
In addition, the sequence of treatments can come
prose conventionally, for example, a prior conditioning
treatment of the surface and a subsequent treatment con-
sitting of a passivating rinse; it goes without saying
that the sequence can comprise other treatments, but the
latter are not taken into consideration here for convey
niece of the description.
Figure 2 shows diagrammatically an installation
suitable for practicing the method according to the
invention.
This installation comprises
- three vessels shown diagrammatically at 1, 2 and 3
and containing respectively the conditioning, conversion
proper and passivating rinse baths,
- six probes or detectors shown diagrammatically at
5 4 to 9, constituted by pH-meters, selective electrodes,
thermometers and the like and dip respectively, as regards
probe 4, into the vessel 1, as regards probes 5, 6 and 7,
into the vessel 2 and, as regards the probes 8 and 9, into
the vessel 3,
- a computer ox processor RAM PROM 10 into which
have been introduced on the one hand in the form of a pro-
gram the algorithms for resolving the various contemplated
equations Yin, on the other hand stored therein the per ma-
next data and coefficients, this processor being connected
to the probes 4 to 9 from which it can consequently no-
chive the measured values through interfaces comprising
converters A/N,
- a control channel or device 11 connected to the
processor 10 and adapted to control, under the influence
of signals emitted by the processor and through devices
shown diagrammatically by arrows 12, 13 and 14, the supply
of substances particularly suitable chemical products to
the various vessels 1, 2 and 3 and
- as the case may require, a printer 15 providing
us as a print-out the variations of the various equations Yin
on which it is intended to operate and of which the data
have been provided to the processor.
I\ In respect of the above said processor, there are
indicated in Figures awoke and awoke logigrams or flow-
sheets showing one of the possible approaches to, firstly charge into the memory PROM of the processor isoresponse
lines as digital samples, on the other hand, to search for
the optimum of the response Y my the method of the gray
dint.
Plotting and memorizing of the isores~onse lines
The case illustrated by the logigram of Figures pa-
12
3c corresponds to that of a polynomial equation of which certain terms are linear functions of one of the parade-
lens an other functions of two parameters.
The start is by indicating, to the scratch pad or
RAM of the processor, the number N of experimental factors
Xi which take part in thy determination of Y. I and J are
then introduced, which are the two values of i each cons-
tituting pairs for which isoresponse curves will be plot-
ted. In the course of the progress of the program, I and J
will obviously take all the pairs of values possible bet-
wren 1 and N, with the exception of the pairs for which
IT and for each pair (ZOO), K) there will be applied,
to the N-2 remaining variables, all the sets of constant
values within a determined finished list.
Once this loading of the program has been carried
out, there is introduced (frame 20) the equation of the
response function which, for simplification, has been
assumed to be of linear form with interactions:
ion N
20 Y = by E b. K + b...... K X(j~
it 1 i-l
joy
Before tracing the curves, the value of Y for all
points of a rectangular matrix corresponding to values of
us K and K distributed regularly, will be determined.
Preferably, so that the matrixes corresponding to
all the pairs of values may have the same number of
points, it is necessary to carry out a standardization and
a centering taking into account the range of possible
variations for each of the parameters or factors X. This
amounts to varying the number representing each parameter
X between -1 and I which involves the introduction of a
scale factor. For this, there are introduced into the
Mom limiting values of the parameters (frame 22) and
then there is determined, for each pair of parameters
corresponding to an isoresponse curve, the scale factor.
The frame 24 corresponds to a matrix with P points for
K and Q points for XtJ), corresponding to a first set
of constant values of all the Us other than K and K.
The tracing of the curves can then be done from
the minimal values X(I)MIN and X(J)MI~ provided for K
and X(J~ (for example ambient temperature, on the one
hand, concentration of one constituent nil, on the other
hand). After initialization of these values at 26, the
calculation of all the points in sequence is done, by
employing two loops one of which is included in the other.
The frame shown at 28 corresponds to the determination of
all the points of the matrix corresponding to a given
value of one of the parameters before passing to a new
value of this parameter to recommence the calculation.
Thus the values taken by Y for all the points of
the matrix are determined. The isoresponse curves are then
traced by an interpolation whose parameters are introduced
at 30. The spacing pitch or step of the isoresponses (Ye-
lutes of Y for which it is desired to have curves avail-
blew will be chosen as a function of the fineness of
adjustment desired.
The operation will be carried out from the points
of the matrix, using an interpolation program which may be
classical.
At the end of the operation, diagrammatically shown
at 32, all the isoresponse values for each couple of stank
dardized values of K and ZOO will be displayed and
memorized.
The same operation will be repeated for all the
pairs or couples of parameters XtI), K with all the N
sets of constant values applied to the fixed parameters (N
being a predetermined number).
Once the data are thus stored in the processor,
the latter can be used to optimize the function Y by the
mathematical gradient method, used for finding a maximum
or minimum value of a function.
3~3
14
OPtimiz~tion,of yo-yo the q~adient,_~et,,hod
The logigram shown in Figures pa to 4c takes up
again a fraction of the preceding logigram. However it
does not put into operation this common portion for the
s totality of the field of variation of the parameters X,
but only around the experimental point, instead of recomb
mincing from the minimal values.
An installation working according to the logigram
of Figures pa to 4c hence constitutes an autonomous unit.
o This logigram provides constraints, in order to reduce the
duration of the calculations and to avoid too frequent
adjustments of the parameters. The passages of the cowlick-
lotion loops is stopped when the improvement in Y obtained
by an additional traversal is less than a predetermined
value. In addition, the values of the responses must
remain within a field of validity defined by a radius R.
Figure pa shows the initialization operations of
the programs. N and the equation of the response function
Y, that is to say data identical with those used at the
beginning of the program of curve tracing are introduced.
To apply the gradient method, the partial derivatives
D~X(i)] = yucca of Y with respect to a first parameter Xi
frame 34) are then introduced and then the limiting
values and the algorithm of the gradient frame 36) with a
predetermined value, the same for all the Us. Then at 38
(Figure 4b) there is determined the direction of the in-
fluency on Y of a first variation of the set of parameters
before carrying out the search for the optimum, by the
application of the algorithm of the gradient. It is import
lent to note that the logigram reduces the multiplier coefficient applied to the derivative when the preceding
latter iteration lends to exceeding the validity radius R
and recommences the same calculation, but with the reduced
value. In the example shown, this reduction is done in a
ratio 2. If the new calculation leads again to exceeding
the radius R, a further reduction in the same ratio is
us
applied: the accuracy is thus very much increased.
The logigram limits, in addition, the number of
passages through the loop by stopping the iterations as
soon as the proximity of the optimum is reached, which is
manifested by the fact that the difference between two
successive results is less than a threshold DYE or that, at
the same time, the radius R has been exceeded and that the
coefficient L applied to the derivative in the algorithm
of the gradient has fallen below a predetermined value
(test 44, Figure 4c).
Figure 5 shows, by way of simple example, an ins-
tallation capable of performing the process which has just
been defined. In this Figure, the members corresponding to
Figure 2 are denoted by the same reference numeral.
In the embodiment shown in Figure 5, a periodic
scanning circuit 46 samples, for example, once hourly, the
parameters X, which in the case illustrated are two in
number and are provided respectively by the probes 5 and
6. The sampled value is applied to an analog-digital or
A/D-converter 48 and sent to a calculating unit 10, which
can be a microcomputer or a programmable controller. The
new set values of the parameters X completed by thy cowlick-
feting unit 10 are sent to circuits 50 and 52 (each on a
printed card) for regulating the parameters. In systems
with slow kinetics, lending themselves poorly to regular
lion, the circuit 50 could be provided to complete, from
data known on the system, of the set value which has just
been provided by the calculating unit 10 and of the actual
value provided by the probe 5, the amount of reagent to be
I added to the system to obtain the set value. The regulate
in operation can then be carried out by actuating the
opening of a valve 54 for a predetermined time. In the
case where the kinetics are faster, it is possible to use
a circuit 52 which also receives, for a servo regulation,
an output signal coming from the converter 48, the scan-
nine circuit 46 then providing constantly the value of the
16
corresponding parameter, in order to permit the change
thereof to be followed. The circuit 52 then stops the
opening of the corresponding valve 54 as soon as the
reference value is reached.
It is by employing the method and installation
according to the invention that the following examples
have been produced.
EXAMPLE 1 - Determination of provisional equations repro-
setting two properties of a metal surface
0 treated in a chemical conversion bath.
In this example, it has been chosen to form a model
and hence to provide two properties Ye and Ye of a convert
soon coating currently measured as a check in phosphate-
lion treatments, namely:
- the layer weight Ye expressed in g/m2,
- the crystalline structure density Ye expressed as
number of crystals per unit surface and measured by means
of a photograph taken with a scanning electron microscope
with a magnification of 1500 times.
Five variables X1 to X5 of the conversion bath have
been taken into consideration:
X1 : concentration of nickel ions
X2 : concentration of chlorate ions
X3 : concentration of zinc ions
X4 : concentration of phosphate ions
X5 : value of the free acidity or Awl
it being understood that, by Awl, is meant the value ox-
twined expressed in ml of N/10 Noah on a sample of 10 ml
of the bath determined by N/10 Noah on the color change of
methyl orange.
Samples of cold-rolled steel sheet of current
quality are submitted to treatments by immersion using
products marketed by Compagnie Frowns de Products
Industries and currently employed in the automobile in-
dusty for the preparation of car bodies before painting.
The steps and products employed comprise
I
17
- a hot alkaline decreasing with the use ox de-
greasing agents based on alkaline salts and on surface
active agents marketed by Compagnie Frowns de Products
Industries respectively under the trademarks REDLINE
s 1550 CF/23" and "RIDOSOL 550 OF",
- a cold rinse in running water,
- a surface conditioning with the use of a refining
agent based on titanium salt marketed by Compagnie Fran-
raise de Products Industries under the trademark "FIX-
I DINE 5",
- a chemical conversion treatment with the use of
an acid solution based on zinc phosphate and other convent
tonal ions marketed by Compagnie Frowns de Products
Industries under the trademark "GRENADINE 908",
lo - a cold rinse,
- a passivating rinse with the use of an agent
based on chromium ions marketed by Compagnie Frowns de
Products Industries under the trademark "DEOXYLYTE 41",
- a staving treatment.
The five variables of the chemical conversion bath
which are taken into account are measured by any convent
tonal means currently used: chemical determination, pi
measurement, potentiometryl ionic conductivity and the
like.
16 experiments are carried out to determine the
constants of the corresponding provisional equation which
is written for Ye:
1 0 1 1 b2X2 + b3X3 = box + b5X5 + b~2X1X2
13 1 3 b14X1~4 + b15X1X5 b23X2X3 + b24X2X4
25X2X5 + b34X3X4 + b35X3X5 + b45X4X5
and or each experiment there are measured
- the layer weight of each sample treated, by disk
solving the coating in a hot chronic acid solution ( 10% r
60~C, 15 minutes),
- the number of crystals present within a window of
7.5 cm diameter of each specimen treated, these crystals
18
being counted on a photograph taken with the scanning
electron microscope with a magnification of 1500 times.
To determine the characteristics of these export-
mints so as to be able to determine the above said cons-
s tents, the procedure is as follows:
For each of the five factors taken into account, a
field of variation is defined as follows:
0.8 g/l Zen 1.4 g/l
g/l P04 23 g/l
lo 0.3 g/l Coo - 0.9 g/l
+.?~
0.5 g/l No 0.8 g/l
0.9 ml Noah N/10 Awl 1.3 ml Noah N/10.
Taking into account the preceding equation, it is
necessary to determine p Pi coefficients connected with
lo the terms of the sty degree and with the rectangular terms
(p representing the number of parameters with, in the pro-
sent case p = 5), to which has been added the constant
term. This leads to carrying out the 16 experiments shown
in the hollowing table for which each of the parameters
takes only the extreme values of its range of variation.
lo 3
19
E:xp. No C103- Zen Pickle Ye p do Ye no
(s/m2 ) __
O.Sg/l 0.3g/1 Ogle 15g/1 1.3ml 1.74147
2 0.5g/1 O.9g/1 0.8g/1 15g/1 0.8ml1.14 160
3 0.8g/1 O.9g/1 1:).8g/115g/1 1.3ml1.19 168
_ . .. . . _
4 0.5g/1 0.3g/1 1.4g/1 15g/1 0.8ml 1.61 117
. _ _ . .
0.5g/1 O.9g/1 1.4g/1 15g/1 1.3ml 1.78 91
1 o . . . _ _ .
6 0.8g/1 0.3g/1 0.8g/1 23g/1 1.3ml 1.39 128
. _ . _ .
7 0.5g/1 0.3g/1 1.4g/1 23g/1 1.3ml 1.86 101
_ _ . .. _ _
8 0.8g/1 O.9g/1 1.4g/1 15g/1 0.8ml 1.29 I
. _ _ _ _ . _ . . _ _ . _. . _ _
9 0.8g/1 O.9g/1 0.8g/1 23g/1 0.8ml 1.1g 205
. _ _ , . . .. . _
0.8g/1 0.9g/1 1.4g/1 23g/1 1.3~1 1.41 66
11 0.5g/1 0.9g/1 1.4g/1 23g/1 0.8ml 1.61 I
_ . . . _ . . _ _ .
12 0.5g/1 0.9g/1 0.8g/1 23g/1 1.3ml 1.36 157
. _ _ _ . _ _ . _
20 13 0.5g/1 0.3g/1 0.8g/1 23g/1 0.8ml 1.34 126
.. .... .. _ ..
14 0.8g/1 0.3g/1 O.Bg/l 15g/1 `0.8ml 1.39 181
. . _ _
0.8g/1 0.3g/1 1.4g/1 23g/1 0.8ml 1.81 97
. = . . _ _ , . _ . . _ . . _
16 0.8g~1 0.3g/1 0.8g/1 1~/1 0.8ml 1.91 90
.. ---- _ __
To be independent of the factor units, to simplify
the calculations and the interpretations end to compare
the effects of the factors with one another, the values of
each factor are centered and reduced.
The calculation of the coefficients b. is then
done by the method of least squares described in detail in
the previously cited book. Only the most influential goof-
fishnets are taken into account in the equation.
After processiTIg of the data, the equations of the
two properties of the coating, namely respectively the
layer weight pdc and the number of crystals no per unit
surface area, are:
Y pdc = 1,~0 0,16 X3- 0,13 X2+ 0,079 X5- 0,070 X~X5 (I)
3 2 X3 14 X1X3~10 X2 X4 (II)
EXAMPLE 2 - Determination of the provisional equation no-
preventative of a property of metal surface
treated in a chemical conversion bath.
The property taken into consideration is again the
yen weight.
For the determination of the corresponding equal
lion, one may take into account not only the chemical
constituents but also physical factors such as tempera-
lure, stirring speed and the like, or discrete parameters,
i.e. digital samples such as the nature or surface state
of a substrate.
In the present case, the equation Ypdc was deter-
mined from seven variables.
Six of these variables, respectively X1, I X3,
X4, X5 and X7, relate to the chemical conversion bath,
namely,
X : concentration of zinc ions
X2 : concentration of chlorate ions
X3 : concentration of phosphate ions
X4 : accelerator proportion sodium nitrite)
X5 : temperature of the bath
X7 : value of the free acidity
the seventh, X6, representing the concentration of the
surface conditioning bath relative to the step which pro-
cedes the chemical conversion treatment proper.
my operating a in Example 1 as to the operational
method, the equation Ypdc obtained is written:
Y pdc = 2,15 - 0,19 X6 + 0,16 X1 - 0,13 X2 + 0,13 X5
0,11 X3 - 0,10 X4 + 0,07 X7 (III).
EXAMPLE_ - Practical examples of use.
A chemical conversion coating such as that
obtained with the conversion event marketed by Compagnie
Frenzies de Products Industries under the trademark
"GRENADINE 908" must meet the following requirements:
12~23~3
21
- its thickness or layer weight must be suffix Jo
ciently small, in the region of 2 to 2.5 g/m2, to permit
good adherence of the paint,
- its crystalline structure must be the densest
S and most regular possible to minimize porosity and to
improve the protection that it confers against corrosion.
It is this requirement "paint adherence-density of
crystalline structure" which it is proposed to resolve by
means of the equation indicated in Example 2.
lo In general, the parameters of such a bath vary
according to the ranges below:
0.8 g/l Zen 1.4 g/l
15 g/l P043 23 g/l
0.3 g/l Coo - 0.9 g/l
0.5 g/l I 0.8 g/l
0.9 ml Noah N/10 S Awl 1.3 ml Noah N/10
50-C < temperature of the conversion bath < 60-C
1.0 ml accelerator proportion 2.0 ml
1.0 g/l < concentration of the "FIXODINE 5"
surface conditioning bath < 3.0 g/l.
Each of the parameters X1-X7 are measured at
successive moments, for instance at intervals of half an
hour For instance:
a) Measurement at the moment to:
Concentration of the surface conditioning bath: 2 g/l
Parameters of the chemical conversion bath:
Zen++ : 0.8 g/l
Clue- : 0.6 g/l
pox : 19 g/l
accelerator proportion: 1.5 ml
temperature: 55-C
free acidity: 0.9 ml.
This set of values is introduced into the equation
(III) Y pdc, once the said values have been centered and
reduced:
Y pdc = 2.15 - 0.19 X6 + 0.16 X1 0 13 I + 0.13 I +
0.11 X3 - 0.10 X4 + 0.07 X7.
:.
I
Consequently, at the moment to:
Y pdc = 2.06 g/m2.
This value of the coating weight is situated bet-
wren the limits of the range, i.e. 2 and 2.5 g/m2, which
has been recommended for the obtention of a good adherence
of the paint.
Consequently, it is not necessary to modify the
parameters.
b) Measurement at the moment I
0 Concentration of the surface conditioning bath: 1 g/l
Parameters of the chemical conversion bath:
Zen : 1.4 g/l (X1)
Clue- : 0.3 g/l (x2)
P04 : 19 g/l (X3)
accelerator proportion: 1.0 ml
temperature: 55'C
free acidity: 1.0 ml.
The result thus obtained by introducing this set
of values into the equation (IV), is Y pdc = 2.77 g/m .
This calculated layer weight value is outside of
the acceptable range. This is the result of the drifting
of one or of several parameters, which it is necessary to
correct by way of the most economical adjustment.
To this purpose, one uses the projections of the
hypersurface having eight dimensions and representing the
equation (III) on the planes of which each one corresponds
to two parameters. For example, the Figure 6 shows, for
constant values of X2, X3, X4, X5 and X7 corresponding to
the measurement at the moment to, the isoresponse curves
Jo or level curves in the plane X1, X6, wherein:
X1 is the content of the conversion bath in Zen ions,
X6 is the content of the surface conditioning bath in
FIXODINE 5.
Each level curve corresponds to a same value of
the layer weight Y (in g/m2). The values of the other
parameters are:
23
I : 0 3 g/l of Cloy-
X3 : 19 g/l of POX
X4 : 1 ml of accelerator
X5 : 55-C
S X7 : 1.0 ml of free acidity.
The processor will then calculate, starting from
the value of X, the optimum variations to be imparted to
the chemical conversion bath and/or the conditioning bath
to bring the value of Y within the acceptable range, for
lo instance to a value close to 2 g/m2.
To this purpose, at each investigation, all the
values of X are introduced into the working memory, either
automatically, or manually at the end of an analysis of a
bath sample. The processor then determines by way of the
gradient method, the optimum variations of each of the
parameters leading to the obtention of the desired value
of the layer weight.
When using the gradient method, it is possible to
make movements over the hypersurface at each point per pen-
dicularly to the isoresponse curves, the gradient being the direction which permits the most rapid shifting
towards an optimum.
One calls the gradient of a function u = f(x,y,z),
the vector whose projections on the coordinate axes are
the corresponding partial derivatives of the given lung-
lion.
grad u = dud i + duo j + duo k
i, j, k representing unit values of the reference system
chosen.
The gradient of a function of three variables is
directed, at each point, along the normal to the surface
passing through this point. The direction of the gradient
of a function at a given point is the direction along
which the function has the greatest growth speed at this
point, that it to say for 1 = grad u the derivative duo
do
24
reaches its greatest value which is equal to:
Dow + Dow + duo 2
do dye do
In the same way the gradient of a function with n
variables u = f(x,y,z, no is defined as
grad u = duo i + duo j + duo k + ... + duo n
do dye do dun
The latter is at each point directed along the normal to
the hypersurface passing through this point.
After the treatment, the processor provides the
optimum values:
X1 1.09 g/l of Zen+
X2 : 0.56 g/l ox Cloy
X3 : 16 g/l of POX
I : 1.33 ml
X5 : 51 C
X6 : 2.25 g/l
X7 : 0.91 ml
corresponding to Y - 1.94 g/m (selected lower than the
optimum due to the fact that the developing continues).
It is then assumed that the adjustment should be
made by
- providing FIXODINE 5 into the conditioning bath, in the
form of an aqueous premix containing 3 g/l of active
components,
- providing of phosphatizing product starting from two
concentrated aqueous solutions:
one of these solutions, A, giving the possibility
of regeneration of -the conversion bath during nor-
I met working, containing the ions Zen , P043 ,
No , for instance in the following formulation (OWE
by weight)
3.0 < Zen++ < 6.5
< H3PO4 < 45
0.1 < No < 1.9
the free acidity of which is comprised between 3
I
and 11 ml of N Noah (test sample to 1 ml),
the total acidity being comprised between 9.8 and
13.4 of N Noah (test sample: 1 ml),
. the other solution, B, providing the possibility
of correcting the increases in the free acidity
and of the zinc concentration of the bath, and
containing the ions P04 , No , for instance in
the following formulation:
2~+4
lo 0.1 < No <
4.2 < pi < 3.8
3 < Act < 5 ml Noah N to 1 ml)
- providing accelerators in the chemical conversion bath
obtained separately starting from concentrated aqueous
lo solutions:
one of these solutions, C, containing the nitrite
ions in a formulation comprised for instance bet-
wren 16 and 20%1
. the other solution, D, containing chlorate ions in
a formulation comprised for instance between 20
and 30%.
The adjustment of the preparation bath may be car-
fled out in a very simple way. The processor 10 sends to
the corresponding control card a value representative of
us the increase in contents to be obtained, i.e. 2.25 - 1 =
1.25 g/l. The circuit comprises a microprocessor or merely
a coding ROME. tread Only Memory) which determines,
starting from that value and from the volume of the bath,
the volume of premix to be added and opens an electrovalve
with constant flow-rate proper to introduce the premix
during the necessary time.
The adjustment of the surface conditioning bath is
somewhat more complicated. The processor determines the
amounts of solutions A, I, C and D introduced and nieces-
spry to bring the bath to the required composition and
sends the corresponding information to the operating
control cards 56. Each one of these cards comprises a
circuit for the determination of the time during which the
introduction is necessary from a corresponding tank 58 as
well as a power circuit which opens the appropriate
electrovalve 60 during the necessary time.
Another card 62 r capable to being addressed alike
cards 56, receives the temperature information and con-
trots a switching circuit 64 which regulates the electric
eel power applied by a source 66 on a heating resistor 58,
in the direction of a decrease in the case under consider
ration.
* *
It is also possible to regulate a bath of the type
concerned by effecting a compromise between several pro-
parties.
Thus, it is possible to use simultaneously the
provisional equations Y pdc (I) and Y no (II) of example
1. The system then permits the parameters to be developed
simultaneously to obtain the best compromise between the
layer weight and the structure density.
As is self-evident and as emerges already from the
foregoing, the invention is in no way limited to those of
its types of application and embodiments which have been
more especially envisaged; it encompasses thereof, on the
contrary, all modifications.
