Kinetic modelling of regioselectivity in alkenes hydroformylation over rhodium

Article abstract of DOI:10.1016/j.molcata.2009.06.023

Propene hydroformylation was performed with rhodium triphenylphosphine catalyst with hydrogen and CO partial pressure varied in the range of 8-15 bar. The regioselectivity was independent on the concentration of reactants, displaying dependence only on the ligand concentration. A kinetic model was proposed based on the mechanism of alkene hydroformylation and compared with experimental observations. Numerical data fitting was performed showing good correspondence of reaction rates and regioselectivity with experimental data.

Full text of DOI:10.1016/j.molcata.2009.06.023

Journal of Molecular Catalysis A: Chemical 315 (2010) 148–154
Contents lists available at ScienceDirect
Journal of Molecular Catalysis A: Chemical
journal homepage: www.elsevier.com/locate/molcata
Kinetic modelling of regioselectivity in alkenes hydroformylation
over rhodium
∗
Dmitry Yu. Murzin , Andreas Bernas, Tapio Salmi
Åbo Akademi University, Biskopsgatan 8, 20500 Turku/Åbo, Finland
a r t i c l e i n f o
a b s t r a c t
Article history:
Available online 30 June 2009
Propene hydroformylation was performed with rhodium triphenylphosphine catalyst with hydrogen and
CO partial pressure varied in the range of 8–15 bar. The regioselectivity was independent on the concentra-
tion of reactants, displaying dependence only on the ligand concentration. A kinetic model was proposed
based on the mechanism of alkene hydroformylation and compared with experimental observations.
Numerical data fitting was performed showing good correspondence of reaction rates and regioselectivity
with experimental data.
Dedicated to 100th anniversary of Prof. M.I.
Temkin.
Keywords:
Hydroformylation
Alkene
©
2009 Elsevier B.V. All rights reserved.
Rhodium
Regioselectivity
Modelling
1
. Introduction
ity, our recent experimental data [5] obtained in the pressure range
of synthesis gas below 15 bar showed independence of selectivity
from CO. However, the regioselectivity of HRh(CO)(PPh ) was seen
to be dependent on the concentration of PPh3 as the following rate
equations were found to be adequate to explain experimental data
Homogeneous catalysis by metal complexes, widely utilized in
academic research, also made its way into industry. One of the most
prominent examples of applied homogeneous catalysis is hydro-
formylation, or oxo-synthesis, which is an important industrial
process for the production of aldehydes from alkenes (Fig. 1) [1,2].
One of the commonly studied catalysts for this reaction is
HRh(CO)(PPh ) . The currently accepted mechanism for Rh/PPh
3
2
−
0.54
r = k C
CH CCOCRh
C
(1)
(2)
i
i
alkene
2
PPh3
−
0.30
3
rn = knCalkeneCH CCOCRh
C
2
PPh
3
3
3
hydroformylation is shown in Fig. 2 [3]. Usually only one cycle is
discussed, and mechanistic aspects of regioselectivity are rather
seldom addressed [4].
The regioselectivity (e.g. the selectivity to linear over branched
aldehydes) can be very high reaching routinely 70–92% and even
a linear to branch ratio of 500 was attained depending on tem-
perature, ligand concentration and hydrogen to CO ratio [1,2]. In
industrial conditions excess of phosphine ligand is used, although
activity is deteriorated, but selectivity towards linear aldehyde is
improved.
and subsequently
rn
ri
kn
=
(3)
0.16
kiC
PPh
3
where rn and r correspond to formation of linear and branched
i
aldehydes, respectively.
Despite the industrial importance of hydroformylation reaction
and substantial amount of research devoted to understanding the
mechanism of this reaction, there are only a few kinetic studies
where the kinetic equations are derived from mechanistic con-
siderations and compared with experimental observations. In our
previous contribution [5] only a simple kinetic model of power
law type has been applied. In this paper, a mechanistic and more
extended kinetic model is evaluated.
The addition of excess phosphine ligand shifts the phos-
phine dissociation equilibrium towards the more selective
HRh(CO)(PPh3)2 catalyst. Although it is reported sometimes in the
literature that higher CO partial pressures lower the product linear-
The aim of this contribution is to discuss the mechanism
of hydroformylation and to compare the kinetics, which cor-
responds to the mechanism, with experimental observations,
focusing mainly on regioselectivity. The theory of complex reac-
∗ Corresponding author. Tel.: +358 2 215 4985.
E-mail address: dmurzin@abo.fi (D.Yu. Murzin).
1381-1169/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.molcata.2009.06.023

D.Yu. Murzin et al. / Journal of Molecular Catalysis A: Chemical 315 (2010) 148–154
149
◦
stirring speed. The temperature could be maintained within ± 1 C.
The reactor had facilities for sampling of the liquid phase as well
as the gaseous content. The reactor was connected to reservoirs
of alkene and nitrogen. Synthesis gas containing H /CO mixtures
2
could be fed at a constant pressure to the reactor by means of
a pressure regulator (Brooks 5866, Brooks 0154) and the reactor
was equipped with transducers for on-line measurements of pres-
sure (Keller Type PA21 SR/80520.3-1) and temperature which was
followed up on a PC for continuous data logging.
Fig. 1. Hydroformylation of alkenes.
The catalyst precursor and the ligand were dissolved in 2,2,4-
trimethyl-1,3-pentanediol monoisobutyrate. The solution was
charged into the steel reactor. The reactor was sealed tight and the
heating jacket was attached. The gases fed into the reactor were dis-
persed in the liquid phase by the aid of a sinter. Propene was flushed
though the liquid phase at the flow rate of 22 l/h for 7 min at 1.2 bar
and room temperature. The reactor was kept closed for 25 min to
await gas–liquid equilibrium. The reactor was heated to the reac-
◦
tion temperature of 70–115 C while the total pressure increased to
∼
6 bar. After stabilization of the temperature, a zero sample was
taken from the liquid phase. The stirring was kept on during this
entire procedure. The reaction time was initialized to zero as soon
as the reactor was pressurized with hydrogen–carbon monoxide
syngas. Samples of the volume 1 ml were taken from the liquid
phase.
3. Reaction mechanism
It is generally accepted that the active catalyst form is a four-
coordinated intermediate like HRh(CO)(PPh ) . According to [1] the
3
2
catalytic cycle with this intermediate gives only predominantly lin-
ear aldehyde. Such an intermediate can undergo transformations,
leaving one ligand and accepting one CO molecule. Monophosphine
complexes are not selective giving an equimolar mixture of linear
and branched aldehydes. The mechanism, presented in Fig. 3 should
in principle give dependence of regioselectivity on the CO pressure
and triphenylphosphine concentration.
Experimental data, generated in [5], show no dependence of
regioselectivity on the CO concentration within the studied range
of conditions, which imposes some restrictions on the transfor-
Fig. 2. Hydroformylation mechanism [3].
tions with the notions of reaction routes, developed by Temkin [6]
will serve as a tool.
2
. Experiment procedure
mations between RhHLCO and RhHL CO. It is supposed that the
2
cycle where RhHL CO is involved leads to a higher amount of lin-
2
The experiments were carried out in a stirred and pressurized
00-ml reactor made of stainless steel (Parr 4561). It had an internal
cooling loop and was equipped with an automatic temperature con-
trol system consisting of an external electric heating jacket coupled
to a steering unit (Parr 4843), which was also used to control the
ear aldehyde, while the cycle with RhHLCO is not regioselective, in
the extreme case giving an equimolar ratio of linear and branched
aldehyde.
The reaction mechanism, given in Fig. 3, will be utilized for
derivation of the rate equations. It can be presented in the
3
Fig. 3. Hydroformylation mechanism with two cycles.

150
D.Yu. Murzin et al. / Journal of Molecular Catalysis A: Chemical 315 (2010) 148–154
following form:
3.1. Derivation of rate equations
Equilibria for steps 1, 2 and 7 give the following equations:
CRhHL
= K C
PCO
1
RhHL
3
(5)
(6)
CO
3
K K C
PCO
1
2
RhHL
3
CRhHL
=
CO
2
CL
K K K7C
PCO
1
2
RhHL
3
CRhHLCO
=
(7)
2
C
L
In principle rigorous analysis of reaction kinetics requires appli-
cation of steady-state approximation to steps 3–6 and 8–11 in
the reaction mechanism (4). Since the reaction rate displayed first
order dependence in the concentrations of alkene, CO and hydro-
gen [5] according to Eqs. (1) and (2), it is possible to simplify the
kinetic analysis and assume that there is only one rate-limiting
step in each catalytic route, while the other steps are at quasi-
equilibria. Although the rate-determining step in hydroformylation
with Rh/PPh3 systems is not fully understood, it can be assumed
that the rate-determining step is H2 addition in both routes.
It should be noted that an assumption of steady-state for all
routes in the cycle does not give a kinetic equation consistent with
experimental kinetic regularities.
It holds, therefore, for the intermediate species in the first route
that
(4)
K K K C
PCOPA
1
2
3
RhHL
3
CRhHL
=
(8)
(9)
COA
2
CL
The elementary steps above can be described by four reaction
routes, i.e. sets of stoichiometric numbers of steps. Elementary
reactions are grouped in steps, and chemical equations of steps
contain reactants and intermediate products. A set of stoichiomet-
ric numbers of steps is defined as a reaction route. Routes must
be essentially different, and it is impossible to obtain one route
through multiplication of another route by a number, although
their respective overall equations can be identical. The number
of basic routes, P, is determined by P = S + W − I, where S is the
number of steps, W is the number of balance equations, and
I is the number of intermediates. Balance equations determine
the relationship between adsorbed intermediates. Such equations
can correspond to the total coverage equal to unity. In mech-
anism (4) A stands for alkene, N and I denote, respectively,
linear and branched aldehydes. On the right hand side of equa-
tions for the steps stoichiometric numbers along the four routes
2
CO
K K K K C
P
PA
1
2
3
4
RhHL
3
CRhL2COAH =
CL
and for the second
2
K K K7K C
P
1
2
8
RhHL
3
CO
CRhHL(CO)2
=
(10)
(11)
2
C
L
2
K K K7K K C
P
PA
1
2
8
9
RhHL
3
CO
C
=
RhHL(CO)2A
2
L
C
The rates along the first and the third routes are expressed by
2
k5K K K K C
P
PAPH2
1
2
3
4
RhHL
r(I) = r5 = k5PH2 CRhL
=
3
CO
(12)
(13)
2(CO)
2
AH
CL
2
k K K K K C
P
P P
H2
6
1
2
3
4
RhHL
A
r(III) = r6 = k6PH2
C
RhL2(CO)2AH
=
3
CO
(
1)
(4)
N
–N
that the overall chemical equations do not contain intermediates
6,7].
According to the rule of Horiuti–Temkin, the number of inde-
are given. These numbers are selected in a way so
CL
while the rate equations for the second and the fourth routes take
a form
[
2
CO
pendent routes can be determined by subtracting from the number
of steps, equal to 11 in Eq. (4), number of intermediates, equal
k10K K K K K C
P
P P
1
2
7
8
9
RhHL
A
H2
r(II) = r10 = k10PH C
3
=
2
RhHL(CO)2
A
2
L
C
to 8 {RhHL , RhHL CO, RhHL CO, RhHL COA, RhHL (CO) A (note
3
3
2
2
2
2
(14)
that this is five-coordinated Rh species, similar to the one pre-
sented in Fig. 2), RhHLCO, RhHL(CO) , RhHL(CO) A} and adding
2
2
the number of balance equations (here there is one balance
equation which is related to the concentration of all rhodium
containing intermediates). Following this rule, it can be con-
cluded that there are four independent routes. The first two steps
in the mechanism represent so-called hanging vertex, their sto-
ichiometric numbers being equal to zero, and such steps are
at steady-state can be considered as quasi-equilibrium ones.
Equilibrium step 7 links two catalytic cycles (Fig. 3) together.
Mechanism, presented in Eq. (4) is simplified, as it does not
distinguish between different geometrical positions (axial or equa-
torial) of phosphines; however, considerations in the present
manuscript are sufficient to describe the kinetic regularities
regarding rate and regioselectivity dependencies on the ligand con-
centrations.
k11 K K K K K C
P2 P P
RhHL
CO
A
H2
3
1
2
7
8
9
r(IV) = r11 = k11 PH C
=
2
RhHL(CO)2A
2
L
C
(
15)
The balance equation relates all the catalytic intermediate
species
CRhHL3 + CRhHL
+ CRhHL
+ CRhHLCO + CRhHL
+ CRhL
CO
CO
A
COA
COAH
2
3
2
2
+ CRhHL(CO)2 + CRhHL(CO)2
=
CRhHL3 (1 + K1PCO + K K P /CL + K K K7PCO/CL2
1
2
CO
1 2
2
CO
2
+
K K K P P /CL + K K K K P P /CL + K K K7K P /CL2
1
2
3
CO
A
1
2
3
4
A
1
2
8
CO
2
CO
2
L
0
+ K K K7K K P P /C ) = C
(16)
1
2
8
9
A
Rh

D.Yu. Murzin et al. / Journal of Molecular Catalysis A: Chemical 315 (2010) 148–154
151
Combining Eqs. (12), (13) and (16) as well as (14)–(16) the
explicit expressions for the reaction rates along the routes are
2
k5K K K K P P PH /CL
1
2
3
4
A
2
r(I)
CO
0
=
C
2
2
2
CO
2
2
Rh
1
+ K1PCO + K K P /CL + K K K7PCO/CL + K K K P P /CL + K K K K P P /CL + K K K7K P /C L2 + K K K7K K P PA/CL
1
2
CO
1
2
1
2
3
CO
A
1
2
3
4
A
1
2
8
1
2
8 9
CO
CO
(
17)
2
k K K K K P P PH /CL
6
1
2
3
4
A
2
r(III)
CO
0
=
C
2
2
2
CO
2
CO
2
Rh
1
+ K1PCO + K K P /CL + K K K7PCO/CL + K K K P P /CL + K K K K P P /CL + K K K7K P /C L2 + K K K7K K P PA/CL
1
2
CO
1
2
1
2
3
CO
A
1
2
3
4
A
1
2
8
1
2
8
9
CO
(
18)
2
2
k10K K K7K K P P PH /C
1
2
8
9
A
2
r(II)
CO
L
0
=
C
Rh
2
2
2
CO
2
2
1
+ K1PCO + K K P /CL + K K K7PCO/CL + K K K P P /CL + K K K K P P /CL + K K K7K P /C L2 + K K K7K K P PA/CL
1
2
CO
1
2
1
2
3
CO
A
1
2
3
4
A
1
2
8
1
2
8 9
CO
CO
(
19)
2
2
k11 K K K7K K P P PH /C
1
2
8
9
A
2
r(IV)
CO
L
0
=
C
Rh
2
2
2
CO
2
CO
2
1
+ K1PCO + K K P /CL + K K K7PCO/CL + K K K P P /CL + K K K K P P /CL + K K K7K P /C L2 + K K K7K K P PA/CL
1
2
CO
1
2
1
2
3
CO
A
1
2
3
4
A
1
2
8
1
2
8
9
CO
(
20)
The first order in CO and alkene indicates that in denominator
Analogously the generation rate for the branched aldehyde is
obtained
unity as well as terms containing P2 and P can be neglected, which
CO
A
gives
k K K /K7P P PH CL(1 + (k K7K K /k K K ))
6
3
4
CO
A
2
11
2
8
9
6 3 4
0
Rh
ri =
C
k5K K K P P PH /CL
2
3
4
CO
A
2
1 + aC + bC
r(I)
=
C
C
0
,
L
L
2
L
Rh
1
+ K /CL + K K7/C
2
2
ꢀꢀ
ꢀꢀꢀ
kPpPH P (k + k CL)
2
CO
0
=
C
(27)
(28)
k K K K P P PH /CL
2
L
Rh
6
2
3
4
CO
A
2
1 + aC + bC
r(III)
=
0
Rh
(21)
L
2
L
1
+ K /CL + K K7/C
2
2
where
k6
k5
k11
k10
8
k11 K7K K9
ꢀ
ꢀ
ꢀꢀꢀ
= kꢀ
2
k
=
and
k
=
k10K K7K K P P PH /C
2
8
9
CO
A
2
5
3 4
k K K
r(II)
L
0
=
=
C
C
,
2
L
Rh
1
+ K /CL + K K7/C
2
2
Analysis of Eqs. (25) and (27) shows that the reaction rate is first
2
L
order in CO, alkene and hydrogen in agreement with experimen-
tal data and that an order in the ligand concentration is negative
in agreement with experimental observations [5]. Regioselectivity
k11 K K7K K P P PH /C
2
8
9
CO
A
2
r(IV)
0
Rh
(22)
2
L
1
+ K /CL + K K7/C
2
2
(
the ratio between linear and branched aldehyde) has the following
The generation rates for the components are written as follows:
dependence on the ligand concentration:
(
I)
(II)
ꢀ
rN = r + r
rn
rs = r
1 + k CL
=
(29)
ꢀ
ꢀ
ꢀꢀꢀ
k + k C
L
2
i
k5K K K P P PH /CL + k K K7K K P P PH /C
2
3
4
CO
A
2
10
2
8
2
L
9
CO
A
2
L
0
Rh
=
=
C
When it is compared with the empirical power law dependence
1
+ K /CL + K K7/C
2
2
0.16
(Eq. (3)) rs∼C
, it is clear that the term in denominator con-
PPh
3
k5K K /K7P P PH CL + k K K P PAPH
ꢀꢀꢀ
3
4
CO
A
2
10
8
9
CO
2
taining k C could be neglected. Additionally graphical comparison
between the power law dependence and Eq. (29) given in Fig. 4 sug-
0
Rh
L
C
(23)
2
L
1
+ CL/K7 + C /K K7
2
ꢀ
ꢀ
gests that k = k /k5 ≈ 1 at non-negligible ligand concentration,
6
which means that the cycle involving RhHLCO is not regioselective,
providing an equimolar ratio of linear and branched aldehyde.
These simplifications finally lead to the following equations:
ꢀ
k K K /K7P P PH CL + k K K PCOPAPH2
6
3
4
CO
A
2
11 8 9
(
III)
(IV)
0
rI = r
+ r
=
C
2
L
Rh
1
+ CL/K7 + C /K K7
2
(24)
kPpPH P
kPpPH P (1 + k CL)
2
CO
0
Rh
2
CO
0
Rh
ri =
C
,
rn =
C
(30)
1
+ aCL + bC2
which were used for data fitting.
1 + aCL + bCL2
L
Eq. (23) could be modified
k5K K /K7P P PH CL(1 + (k K7K K /k5K K ))
3
4
CO
A
2
10
8
9
3 4
0
Rh
rN =
C
2
L
1
+ CL/K7 + C /K K7
3.2. Data fitting
2
ꢀ
kPpPH P (1 + k CL)
2
CO
0
In order to elucidate the applicability of the kinetic model,
described above, the system of Eq. (30) was solved numerically in
the parameter estimations with the backward difference method
by minimization of the sum of residual squares with non-linear
regression using the simplex and Levenberg–Marquardt optimiza-
tion algorithms implemented in the software Modest [8].
=
C
(25)
(26)
2
Rh
1
+ aCL + bCL
with
k5K K
k10K7K K9
1
K7
1
3
4
kꢀ =
8
k =
,
a =
, b =
K K7
2
K7
5
k K K
3
4

152
D.Yu. Murzin et al. / Journal of Molecular Catalysis A: Chemical 315 (2010) 148–154
Table 1
Values of the estimated parameters.
Parameter
Estimated value
Est. relative standard error (%)
2
A1
A2
a
0.21 × 10
12.1
23.0
4.0
3
0.11 × 10
75.8
b
Ea1
2650
0.76 × 10
9.7
25.9
5
Note: Dimensions: [Ea] = J mol 1, [A1] = dm8 mol min , and [A2] = dimensionless.
−
−2
−1
Eq. (30) was in fact modified to include concentrations of alkene,
hydrogen and CO:
ꢀ
kCpCH C
+ aCL + bC2
kCpCH C (1 + k CL)
2
CO
0
Rh
2
CO
0
Rh
ri =
C
,
rn =
C
(31)
1
1 + aCL + bCL2
L
3
3
3
In Eq. (31) CP (mol/dm ), CH (mol/dm ), C (mol/dm ), [Rh]
2
CO
(
mass fraction), and [L] (mass fraction) denote concentration of
Fig. 4. Comparison between Eqs. (3) and (29) for kꢀꢀꢀCL ≈ 0.
propene, concentration of hydrogen and carbon monoxide, mass
fraction of rhodium, and mass fraction of ligand, respectively. The
solubility of hydrogen and carbon monoxide was studied previously
[
9]:
Fig. 5. Comparison between experimental and calculated data at some selected. Solid line: predicted value, ꢀ: propene, ꢁ: isomeric aldehyde, and ꢂ: normal aldehyde.
◦
Conditions: reactant, 1.2 bar overpressure of propene at 25 C (approx. 0.15 mol); solvent, 150 ml of 2,2,4-trimethyl-1,3-pentanediol monoisobutyrate; metal precursor,
(
acetylacetonato)dicarbonylrhodium(I); rhodium concentration, 100 ppm; H2-to-CO ratio, 1:1; reaction time, 180 min. The reaction temperature, pressure of CO and H2,
◦
◦
◦
and ligand concentration was varied as follows. (I) T = 100 C, PH = 5.5 bar, PCO = 5.5 bar, CL = 0.005 (mass); (II) T = 100 C, PH = 5.5 bar, PCO = 5.5 bar, CL = 0.02; (III) T = 100 C,
2
2
◦
◦
◦
PH2 = 5.5 bar, PCO = 5.5 bar, CL = 0.04; (IV) T = 100 C, PH = 5.5 bar, PCO = 5.5 bar, CL = 0.08; (V) T = 85 C, PH = 5.5 bar, PCO = 5.5 bar, CL = 0.005; (VI) T = 100 C, PH = 5.5 bar,
2
2
2
PCO = 5.5 bar, CL = 0.005;.

D.Yu. Murzin et al. / Journal of Molecular Catalysis A: Chemical 315 (2010) 148–154
153
Fig. 6. Objective function vs. kinetic parameters.
The lumped kinetic constant k was assumed to follow the Arrhe-
nius dependence:
Fig. 7. Contour plots: A2 vs. A1, b vs. a, a vs. A1, b vs. A1, EA1 vs. A1, a vs. A2, b vs. A2,
EA1 vs. A2, a vs. EA , and b vs. EA .
1
1
ꢀ
ꢀ
ꢁꢁ
Ea
R
1
T
1
k = A exp
−
−
(32)
Tmean
ꢀ
where A, Ea, T and Tmean denote frequency factor, activation energy,
the gas constant, reaction temperature and mean temperature of
the experiments, respectively.
therefore the lumped constant k was set to be temperature inde-
pendent. The model is still over-parametrized and could be further
simplified by assuming negligible temperature dependences of
equilibrium constants, which means in other words independence
of parameters a and b on temperature.
It was observed experimentally that regioselectivity did not
show any dependence on T [5] in the studied temperature range,

154
D.Yu. Murzin et al. / Journal of Molecular Catalysis A: Chemical 315 (2010) 148–154
The sum of squares in the parameter estimation was minimized
ucts. Kinetic hypothesis allowed some simplifications, in particular
−6
using a step size of 0.1 and 1 × 10
for the absolute and rela-
that one cycle is selective to linear aldehydes, while the second
one gives stoichiometric amounts of linear and branched products.
A complicated mechanism involving different intermediates was
translated into tractable rate expressions. Numerical data fitting
was performed and it was concluded that the advanced kinetic
model, based on mechanistic considerations, describes well exper-
imental data, in particular regioselectivity. At the same time it
should be noted that the general kinetic scheme was somewhat
simplified while deriving kinetic equations by assuming one rate-
determining step in each catalytic route, which was justified by the
obtained kinetic regularities. For more complicated kinetics, these
simplifications could be relaxed.
tive tolerances of the simplex and Levenberg–Marquardt optimizer,
starting with the former method and changing to the latter one
while approaching the minimum.
Comparison between estimated and calculated values is given
in Fig. 5, showing very good description of the data. The degree of
explanation was 94% and sum of residuals was 1.65. The values of
the estimated parameters, the estimated relative standard errors
in %) are shown in Table 1.
The value of activation energy for hydroformylation for Rh com-
plexes was reported to be 71 kJ/mol for octene [10], and 116 kJ/mol
for hexene [11], while for propene the value of activation energies
for hydroformylation with supported ionic liquid phase catalyst was
(
6
3 kJ/mol [12], which is very close to the theoretical values [4] as
Acknowledgements
well as those experimentally obtained in the present study.
The table demonstrates that the parameters are well
This work is a part of activities with Process Chemistry Centre
(2000–2011) by Academy of Finland. The authors are grateful to Pro-
fessor Dieter Vogt (Eindhoven University of Technology) for many
useful comments and suggestions.
ꢂ
ꢂ
ꢂ
2
ꢂ
identified. The objective function SRS = l(ꢀ) = y − yp
=
w
nsetsnobs(k)nydata(j,k)
ꢃ ꢃ
ꢃ
2
(
yijk − ypijk) w dependence for the parameters
ijk
k=1 j=1
i=1
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there is practically no correlation between parameters.
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