Nonisothermal calorimetry for fast thermokinetic reaction analysis: Solvent-free esterification of n -butanol by acetic anhydride

Article abstract of DOI:10.1021/op900298x

An enhanced small-scale reaction calorimeter has been built for nonisothermal applications. Its unique design, combining compensation heater and heat flow sensors together with a solid intermediate thermostat is particularly suited for data oriented process development (determination of chemical reaction parameters, i.e. rate constants, reaction enthalpies, and reaction monitoring with optional in situ devices) in a wide range of applications in chemical and life-science oriented industries. The performance of the calorimeter is successfully demonstrated for kinetic investigation under nonisothermal conditions. Three different methods for determining the time-resolved reaction heat have been tested. The first is based on the traditional heat balance, the second on the twin principle, while the third is a novel method based on a rigourous heat flow modelling using mathematical finite methods. As a case study, we investigated the esterification of n-butanol with acetic anhydride catalysed by tetramethylguanidine using a temperature ramp from 30 to 80 °C. Each method accounts differently for the dynamics and the heat accumulation in the system. However, all three methods show minor differences in the resulting kinetic parameters and reaction enthalpies. In this temperature range, kinetic and mechanistic analysis resolved two competitive parallel catalytic and noncatalytic steps.

Full text of DOI:10.1021/op900298x

Organic Process Research & Development 2010, 14, 524–536
Nonisothermal Calorimetry for Fast Thermokinetic Reaction Analysis: Solvent-Free
Esterification of n-Butanol by Acetic Anhydride
Gilles Richner, Yorck-Michael Neuhold,* and Konrad Hungerb u¨ hler
Institute for Chemical and Bioengineering, Department of Chemistry and Applied Biosciences, ETH Zurich, H o¨ nggerberg HCI,
8093 Zurich, Switzerland
Abstract:
Several publications are dedicated to reaction calorimeters,
reviewing their different principles (power compensation, heat
flow, and heat balance), and their operational modes (adiabatic,
An enhanced small-scale reaction calorimeter has been built for
nonisothermal applications. Its unique design, combining com-
pensation heater and heat flow sensors together with a solid
intermediate thermostat is particularly suited for data oriented
process development (determination of chemical reaction param-
eters, i.e. rate constants, reaction enthalpies, and reaction monitor-
ing with optional in situ devices) in a wide range of applications
in chemical and life-science oriented industries.The performance
of the calorimeter is successfully demonstrated for kinetic inves-
tigation under nonisothermal conditions. Three different methods
for determining the time-resolved reaction heat have been tested.
The first is based on the traditional heat balance, the second on
the twin principle, while the third is a novel method based on a
rigourous heat flow modelling using mathematical finite methods.
As a case study, we investigated the esterification of n-butanol with
acetic anhydride catalysed by tetramethylguanidine using a tem-
perature ramp from 30 to 80 °C. Each method accounts differently
for the dynamics and the heat accumulation in the system.
However, all three methods show minor differences in the resulting
kinetic parameters and reaction enthalpies. In this temperature
range, kinetic and mechanistic analysis resolved two competitive
parallel catalytic and noncatalytic steps.
2,6-8
isoperibolic,isothermal,temperaturemodulations,andPeltier).
Isothermal mode is often preferred to reduce the amount of
information to be interpreted (e.g., temperature dependency
8
of the chemical system). However, temperature modulations
(mostly temperature ramps) are widely used for crystallisa-
9
10
tion and polymerisation. Moreover, for simple reactions,
the activation energy, E , can be determined simultaneously
to the reaction enthalpy, ∆ H, and the pre-exponential factor
of the Arrhenius equation in one single nonisothermal
a
r
1
1
experiment, while the usual approach requires to perform
the reaction at several different temperatures to draw an
1
2
Arrhenius plot. That way, a considerable gain of time and
of test substance is expected. However, it is well-known that
the determination of the reaction heat during a temperature ramp
is a challenging task as the temperature ramp often shifts the
baseline of the heat signals due to temperature dependencies
of the overall heat transfer coefficient, the heat loss, and/or
11,13
because heat may undesirably accumulate into the system
and may require several time-consuming calibrations.
To compensate for baseline shift due to the system itself,
twin-system calorimeters, such as a Differential Scanning
14
Calorimeter (DSC) or a Differential Reaction Calorimeter
Introduction
15
(
DRC), have been designed for the differential measurement
During early stages of process development chemical and
pharmaceutical industry needs flexible and versatile tools to
assess information about chemical reaction systems. For several
decades, calorimetry has become a standard analytical technique
in laboratories for monitoring the heat liberated or absorbed by
chemical and biological reactions. Reaction calorimetry (RC)
allows simulating industrial plants at a litre-scale, including
dosing, mixing, and controlling of the reactor temperature and/
or pressure, and using simultaneously multiple in situ analytical
devices. RC has been successfully employed during process
development for investigation of thermal safety but also to
elucidate reaction mechanisms and associated activation ener-
gies, rate constants, and heats of reaction, and to optimise
chemical processes in a data oriented way.
principle such that a sample vessel and a reference vessel run
in parallel. Naturally, for a good consideration of the dynamics
of the system, the physical properties of the reference should
be close to those of the sample. In a single vessel, a similar
approach consists of repeating the measurement with a nonre-
16
active reference.
(
(
6) Karlsen, L. G.; Villadsen, J. Chem. Eng. Sci. 1987, 42, 1153–1164.
7) Landau, R. Thermochim. Acta 1996, 289, 101–126.
1
(8) Zogg, A.; Stoessel, F.; Fischer, U.; Hungerb u¨ hler, K. Thermochim.
Acta 2004, 419, 1–17.
2,3
(
9) Monnier, O.; Fevotte, G.; Hoff, C.; Klein, J. P. Chem. Eng. Sci. 1997,
52, 1125–1139.
4,5
(10) Grolier, J.-P. E.; Dan, F. Thermochim. Acta 2006, 450, 47–55.
(
11) Hoffmann, W.; Kang, Y.; Mitchell, J. C.; Snowden, M. J. Org. Process
Res. DeV. 2007, 11, 25–29.
(
12) Pastr e´ , J.; Zogg, A.; Fischer, U.; Hungerb u¨ hler, K. Org. Process Res.
DeV. 2001, 5, 158–166.
*
Author to whom correspondence may be sent. E-mail: bobby.neuhold@
chem.ethz.ch.
(13) Weisenburger, G. A.; Barnhart, R. W.; Clark, J. D.; Dale, D. J.;
Hawksworth, M.; Higginson, P. D.; Kang, Y.; Knoechel, D. J.; Moon,
B. S.; Shaw, S. M.; Taber, G. P.; Ticknert, D. L. Org. Process Res.
DeV. 2007, 11, 1112–1125.
(
1) Ferguson, H. F.; Frurip, D. J.; Pastor, A. J.; Peerey, L. M.; Whiting,
L. F. Thermochim. Acta 2000, 363, 1–21.
(
(
(
(
2) Regenass, W. J. Therm. Anal. 1997, 49, 1661–1675.
3) Stoessel, F. J. Therm. Anal. 1997, 49, 1677–1688.
4) Regenass, W. Thermochim. Acta 1977, 20, 65–79.
5) Fischer, U.; Hungerb u¨ hler, K. In The InVestigation of Organic
Reactions and Their Mechanisms; Maskill, H., Ed.; Blackwell: Oxford,
UK, 2006; pp 198-226.
(14) Lacey, A. A.; Price, D. M.; Reading, M. In Modulated-Temperature
Differential Scanning Calorimetry; Reading, M., Hourston, D. J., Eds.;
Springer: Netherlands, 2006; pp 1-81.
(15) Andr e´ , R.; Bou-Diab, L.; Lerena, P.; Stoessel, F.; Giordano, M.;
Mathonat, C. Org. Process Res. DeV. 2002, 6, 915–921.
5
24
•
Vol. 14, No. 3, 2010 / Organic Process Research & Development
10.1021/op900298x  2010 American Chemical Society
Published on Web 05/03/2010

Scheme 1. Esterification of n-butanol (BuOH) by acetic anhydride (AA), catalysed by 1,1,3,3-tetramethylguanidine (TMG)
To study reaction mechanisms, in situ IR-spectroscopy is
often preferred to calorimetry. Multivariate spectra provide more
information on the chemical system, and optimisation of the
kinetic parameters is more robust than the monovariate calo-
rimetric data. However, under nonisothermal conditions, the
validity of Lambert-Beer’s law is often limited due to
the boiling point of the reaction mixture may be reached
19
increasing dangerously the pressure of the reactor. To our
knowledge, only few investigations studied the base-catalysed
esterification, and the researchers reported mainly a pseudo-
20
21,22
second-order reaction mechanism, with or without
fast
equilibrium of an activated catalyst complex. All these studies
were performed below 50 °C under multiple isothermal condi-
tions. In the present study, we investigated the reaction from
30 to 80 °C to cover also the temperature range reported in
17
temperature dependent spectral shifts in the absorbance spectra.
The goal of this study is to demonstrate that nonisothermal
calorimetric investigations without time-consuming calibration
steps may provide quick kinetic and mechanistic information
without performing an extensive kinetic study. To do so, we
first test the performance of our reaction calorimeter to control
the reaction temperature under nonisothermal conditions, since
until now, our prototype calorimeter has been used under strictly
isothermal conditions only. As a case study, the kinetics of the
solvent-free base-catalysed esterification of n-butanol by acetic
anhydride is reinvestigated. We applied three different methods
to perform the heat balance. The first two methods have been
already commonly used and employ either a global heat balance
or a measured reference. In the third method, we present a novel
approach based on a heat flow model of our reactor and a
simulated blank. Kinetic modelling with only few nonisothermal
measurements indicate a mechanism of two competitive parallel
reactions.
18
literature for the esterification of acetic anhydride by methanol.
The experiments were carried out in a semibatch mode using
the following procedure. The reactor was initially flushed with
dry N and then loaded with 22 mL of acetic anhydride (Acros
2
Organics, 99+%) and 1.25 mL of 1,1,3,3-tetramethylguanidine
(ABCR-Chemicals, 99%). After the steady-state has been
reached (T ) 30 °C), 21.4 mL of n-butanol (Acros Organics,
r
99%, extra pure) was fed at 3 mL/min with a syringe pump
(kdS 200, kdScientic) and a gastight Hamilton syringe (series
1000, 25 mL). The temperature ramp was started after the end
of the dosing to avoid perturbance by heat of mixing during
the nonisothermal phase. The ramp was applied simultaneously
r j
to the temperatures of the reactor, T , the jacket, T and the cover,
T
coVer (see Figure 1). A heating rate of 1 °C/min was chosen in
11,16
line with typical heating rates reported in literature
and also
due to limitation in the temperature control of the cover, TcoVer
.
An overview of the reaction conditions is given in Table 1.
Instrumentation. All experiments were performed in a high
performance small-scale reaction calorimeter, which has been
under ongoing development in our laboratories for the last 10
years. As illustrated in Figure 1, the system is composed of
one stirred vessel (in Hastelloy) and a hexagonal metal jacket
Experimental Section
Reaction. Ecological impact from chemical and pharma-
ceutical industry has been gradually decreasing during the past
2
few years, encouraged by political decisions for CO reduction
factors. Solvent-free reaction environment is a promising
solution to produce chemicals under green conditions as pre-
and post- treatment of solvent (purification, separation and
recycling) can be avoided. However, as no thermal buffer and
high chemical concentrations are present, the reaction has to
be particularly well-characterised. As an example, we investi-
gated the solvent-free esterification of n-butanol (BuOH) by
acetic anhydride (AA), catalysed by 1,1,3,3-tetramethylguani-
dine (TMG) to form butyl acetate (BuOA) and acetic acid (AH),
Scheme 1. This reaction is of particular relevance in industry
for solvent, pharmaceutical, perfume and explosive manufacturing.
Few studies showed that an autocatalytic behaviour of
(
in copper) designed as an intermediate thermostat. The jacket
temperature is controlled by six Peltier elements (Melcor, HT
-12-40-T2) placed on the lateral sides of the jacket. The heat
6
pumped through the Peltiers is carried away by coolers
connected to a cryostat (Haake P1-C50P). The temperature of
the bulk is controlled by an electric compensation heater, whose
power varies as the reaction liberates or consumes energy.
Sample volumes of 20-50 mL bridge the gap between
commercially available reaction calorimeters typically working
in the one-litre scale, and microvolume thermo-analytical
instruments. Thus, the device is particularly useful for research
18
esterification between anhydrides and alcohols may lead to
several safety issues. For example, in case of thermal runaway,
(
19) Gibson, N.; Roger, R.; Wright, T. Inst. Chem. Eng. Symp. Ser. 1987,
02, 61–84.
1
(
16) Bickerton, J.; Timms, A. W. In Hazards XVI: Analysing the Past,
Planning the Future; Pantony, M. F., Ed.; Institution of Chemical
Engineers (IChemE): Manchester, UK, 2001; pp 109-123.
(20) Ehly, M.; Gemperline, P.; Nordon, A.; Littlejohn, D.; Basford, J. K.;
De Cecco, M. Anal. Chim. Acta 2007, 595, 80–88.
(21) Gemperline, P.; Puxty, G.; Maeder, M.; Walker, D.; Tarczynski, F.;
Bosserman, M. Anal. Chem. 2004, 76, 2575–2582.
(22) Puxty, G.; Fischer, U.; Jecklin, M.; Hungerb u¨ hler, K. Chimia 2006,
60, 605–610.
(
17) Zogg, A.; Fischer, U.; Hungerb u¨ hler, K. Chem. Eng. Sci. 2004, 59,
5
795–5806.
(
18) Widell, R.; Karlsson, H. T. Thermochim. Acta 2006, 447, 57–63.
Vol. 14, No. 3, 2010 / Organic Process Research & Development
•
525

Figure 1. (Top) Schematic of the combined reaction calorimeter, CRC.v5, with independent PID control of T
r j
, T , and Tcover. (Bottom) Images
of (a) the calorimeter CRC.v5, (b) the electrically heated cover allowing for eight inserts, and (c) the copper jacket with the cooling system.
and development in fine and pharmaceutical chemical industry
where only small amounts of test substances are available.
Despite the small volume of the reaction vessel, an attenuated
total reflection Fourier transform infrared (ATR-FTIR) DiComp
probe (Mettler-Toledo) is mounted directly at its bottom. A
magnetically coupled pitched blade turbine with speeds up to
temperature are up to 60 bar and about 200 °C, respectively.
Instrumental control and data acquisition are operated with the
Labview software, version 8.5, at a sampling rate of 10 Hz
running on a 3.40 GHz Pentium 4 personal computer. The
working principle of the calorimeter combines the compensation
heater and heat balance principles. For details, we refer to Zogg
23
2000 rpm is used for stirring. The maximum pressure and
et al.
5
26
•
Vol. 14, No. 3, 2010 / Organic Process Research & Development

3
Table 1. Experimental conditions for the semi-batch
esterification of n-butanol (BuOH) by acetic anhydride (AA),
catalysed by 1,1,3,3-tetramethylguanidine (TMG) under
non-isothermal conditions
is given in eq 2, density, Fdos [kg/m ], and heat capacity, cp,dos
[
27
J/(kg ·K)], are taken from literature.
q˙ dos ) fdos · Fdos · cp,dos
·
(
Tdos - Tr
)
(2)
initial reactor temperature, Tr,0
initial jacket temperature, Tj,0
initial cover temperature, Tcover,0
30 °C
28 °C
The power, q˙ stirr, dissipated by the stirrer is negligible (<0.05
W). In the CRC.v5, the power due to cooling, q˙ cool is measured
at real time by the six heat flow sensors covering each lateral
side of the jacket.
The heat flow sensors do not cover the entire lateral surface
of the jacket. Thus, a convective heat flow bypasses the heat
flow sensors due to the temperature gradient between the jacket
28 °C
50 °C
∆
T
r
, ∆T
j
, ∆Tcover
start of the ramp
10 min after starting the dosing
1 °C/min
heating rate
cryostat temperature, Tcryo
initial reactor content
45 °C
22 mL (0.233 mol) AA + 1.25
mL (10 mmol) TMG
21.4 mL (0.233 mol) BuOH
3 mL/min
dosing
dosing rate, fdos
stirrer speed
pressure
500 rpm
atmospheric
j
(T) and the coolers (Tcryo), see also Figure 1. When the
temperature of the cover and the jacket are equal and the
environmental temperature is constant ((0.5 °C), the heat loss,
The calorimeter used in this study is our latest version, the
CRC.v5. Novelties of CRC.v5 compared to the previous version
q˙ loss, is expressed as a function of T
j
and Tcryo, only:
24
CRC.v4, are (a) an electrical heater mounted directly into the
cover, aiming at the diminution of heat loss and reducing the
thermal inertia of the cover (a new system for thermostatting
the cover, allowing also cooling, is under development and has
q˙ loss ) a ·
(
Tj - Tcryo
)
+ b
(3)
where the two temperature-independent parameters a [W/K]
and b [W] are determined at steady-state directly before and
after the reaction.
The heat accumulation is given by the sum of the individual
heat capacities and corresponding temperature gradients at the
various locations of the reactor, where the thermocouples are
located. In our case, we consider only the temperature of the
25
been patented ), and (b) six heat flow sensors (Captec, France)
located between the jacket and the Peltiers for measuring the
heat flow due to cooling, q˙ cool. (The expressions heat flow sensor
and heat flux sensor are both present in literature. As we are
2
interested in the flow [W] and not the flux [W/m ], heat flow is
preferred here.) The advantages of measuring q˙ cool with heat
r j
reactor, T , and of the jacket, T:
flow sensors instead of doing a heat balance over the Peltier
23,24
elements, as performed previously,
the temperature drop across the sensors is extremely reduced,
b) the real heat flow, q˙ cool, is proportional to measured voltage
are the following: (a)
dTr
dt
dTj
dt
q˙ acc ) Cp,r
+ Cp,j
(4)
(
and can be measured at real time, and (c) the sensitivity of the
sensor, e.g. the proportionality constant between the heat flow
and the measured voltage, is independent of the temperature.
Data Treatment. The elucidation of the kinetic parameters
and the reaction enthalpies requires first the determination of
the power profile due to the reaction, q˙ react. Three different
methods for calculating q˙ react are used: the global heat balance,
the blank experiment, and the simulated blank. Finally, the
technique for optimising the kinetic parameters and the reaction
enthalpies is explained.
Cp, r is the total heat capacity of the bulk [J/K], and Cp, j is the
total heat capacity of the calorimeter [J/K], including the jacket
and the vessel, commonly called calorimeter constant. Cp, r is
calculated experimentally with temperature oscillation calorim-
26
28
etry, and Cp, j is calibrated from an experiment without
chemical reaction, applying a temperature ramp. The derivatives
of the temperatures were calculated employing a Savitzky-Golay
29
filter, with the following parameters: 121 data points, corre-
sponding to 12.1 s of experiment, and a sixth-order polynomial
fit.
Method 1: EValuation with the Global Heat Balance. In reaction
calorimetry, the classic method to determine the heat of reaction
is to perform a global heat balance of the calorimeter, as given in
Method 2: EValuation with the Blank Experiment. With this
method, the evaluation of the power due to the reaction, q˙ react
,
7
requires to repeat the measurement with a reference. The
assumption is that systematic errors are common to both reactive
eq 1. For the notation, we refer to the list of symbols.
16
and reference experiments and therefore cancel out. Here, as
a reference, we dosed 21.4 mL of AA into 23.25 mL of AA.
Assuming q˙ dos, q˙ loss, and q˙ acc cancel out in both experiments,
q˙ react is determined by the difference between the measurement
and the reference:
q˙ acc ) q˙ react + q˙ comp + q˙ mix + q˙ stirr + q˙ dos - q˙ cool
-
q˙ loss (1)
The power due to dosing, q˙ dos corresponding to the power
required to bring the dosed material at the reactor temperature
meas
cool
blank
meas
comp
blank
comp
q˙ react
)
(
q˙
- q˙ cool
)
-
(
q˙
- q˙
)
(5)
(
(
(
(
23) Zogg, A.; Fischer, U.; Hungerb u¨ hler, K. Ind. Eng. Chem. Res. 2003,
4
2, 767–776.
24) Visentin, F.; Gianoli, S. I.; Zogg, A.; Kut, O. M.; Hungerb u¨ hler, K.
Org. Process Res. DeV. 2004, 8, 725–737.
where meas and blank are the reactive and blank (reference)
experiments, respectively. Naturally, as the physical properties
25) Richner, G.; Wohlwend, M.; Neuhold, Y.-M.; Godany, T.; Hunger-
b u¨ hler, K. German Patent DE 102008020989 (A1), 2008.
26) Richner, G. Dynamic study of a new small scale reaction calorimeter
and its application to fast online heat capacity determination. Doctoral
and Habilitation Thesis, Nr. 17972; Department of Chemistry, Swiss
Federal Institute of Technology (ETH): H o¨ nggerberg, Switzerland,
(27) Yaws, C. Chemical Properties Handbook; McGraw-Hill: New York,
1999.
(28) Richner, G.; Neuhold, Y.-M.; Papadokonstantakis, S.; Hungerb u¨ hler,
K. Chem. Eng. Sci. 2008, 63, 3755–3765.
2008; DOI: 10.3929/ethz-a-005686831; http://dx.doi.org/10.3929/ethz-
(29) Maeder, M.; Neuhold, Y.-M. Practical Data Analysis in Chemistry;
Elsevier: Amsterdam, 2007.
a-005686831.
Vol. 14, No. 3, 2010 / Organic Process Research & Development
•
527

of the reference (AA dosed in AA) and those of the sample
BuOH dosed in AA:TMG) are different (e.g., at 30 °C, FAA
071 g/L, FBuOH ) 802 g/L, cp, AA ) 1.869 kJ/g, cp, BuOH ) 2.170
kJ/g ), this induces an error in the determination of q˙ react. This
is an inevitable issue also known from investigations with twin
reactor systems such as DSC and DRC.
during the ramp the deviations are within the noise level (0.03
°C for T , 0.01 °C for T and 0.04 °C for Tcover). The
(
1
)
r
j
compensation heater provides a fast control of the reactor
temperature, and the powerful cooling system allows an ideal
control of the jacket temperature. In comparison, under iso-
thermal conditions at 40 °C in commercially available 75 mL
and 5 L reactors, the reactor temperature deviates above the
27
Method 3: EValuation with the Simulated Blank. The
evaluation of the power due to the reaction, q˙ react, by heat flow
simulation of the blank is similar to method 2, but the reference
is simulated by a dynamic heat flow model rather than being
measured. The heat flow and temperature profiles through the
reactor content, the vessel and the jacket are simulated for a
preselected reactor content. More details are given in Appendix
20
set point up to 13 and 40 °C, respectively.
This demonstrates that the CRC.v5 provides a well controlled
environment also when working under nonisothermal condi-
tions. The corresponding powers of the compensation heater
and of the cooling are shown in Figure 2b.
Heat of Reaction. In the next sections, we assess the three
different methods for the determination of q˙ react, and the issues
regarding possible change of the global heat transfer, the heat
loss, and the heat accumulation during the reaction.
26
A and by Richner. q˙ react is determined by the difference
between the measurement and the simulated reference:
meas
cool
sim
cool
meas
comp
sim
comp
q˙ react
)
(
q˙
- q˙
)
-
(
q˙
- q˙
)
(6)
Global Heat Transfer. In the CRC.v5, the power due to
cooling, q˙ cool, is measured online via heat flow sensors, without
calibration of the global heat transfer, UA. In contrast, in their
where meas and sim are the reactive (measured) and the
simulated (reference) experiments. Partial volumes of the
different components are assumed additive, i.e. a possible excess
volume is not considered.
11
nonisothermal kinetic study with RC1, Hoffmann et al.
calibrated UA before and after the reaction, interpolated the
values in between, and performed about 20 additional calibra-
tions of UA to validate the interpolation. In the CRC.v5, due to
the calibration-free measurement of the cooling power, the
possible changes of UA would not be an issue for method 1
8
Kinetic Analysis. Zogg et al. reviewed different evaluation
techniques for the identification of reaction enthalpy and reaction
model parameters under isothermal conditions from calorimetric
measurements. With nonisothermal conditions, the similar
techniques can be applied. For the simultaneous identification
of the reaction enthalpy and the model parameters (rate constant
and activation energy), we used a model-based evaluation, for
which a reaction mechanism must be postulated. The reaction
(global heat balance). In method 2 (blank experiment), a change
of UA during the reaction would deviate from the reference
measurement. For such a case, a second reference with the
16
products should be measured. The drawback is indubitably
the increased number of experiments per investigation, which
may not even be feasible if, e.g., the phase of the product
changes within the temperature range. Finally, with method 3,
the possible change of UA during the course of the reaction is
already accounted for by the heat flow model. The internal heat
transfer coefficient is a model parameter that is reoptimised
during the course of the experiment. Further details are given
in Appendix A.
rate r
differential equations (ODE) corresponding to the rate law, see
also eq 11. The temperature dependence of the rate constant k
is given by the Arrhenius equation, eq 7, where kref,i is the rate
constant at the reference temperature Tref
i i
(k, t) for a reaction i is given by the set of ordinary
i
.
e-(Ea,i/R)(1/T-1/Tref)
(7)
r
, the reaction
ki ) k
ref,i
Heat Loss. Solvent vaporisation and condensation in the
vessel are known to contribute to the total heat loss and
consequently shift the baseline of the heat signal. In this case,
heat loss is often evaluated by trial and error adjustment of the
For the respective number of reactions N
r
enthalpies, ∆ H1...Nr, the activation energies Ea,1...Nr and the
reference rate constant, kref,1...Nr, are determined for the experi-
mental calorimetric data using Matlab’s nonlinear least-squares
curve-fitting tool, lsqcurVefit, with eq 8 as the objective function.
30
baseline. According to Henry’s law, the mass fraction of
solvent in the gas phase should decrease with increasing reactor
pressure. In the CRC.v5, 25 mL of water at 75 °C have been
pressurised from 1 to 45 bar resulting in a decrease of heat
loss from 2.99 to 2.98 W, only. Consequently, we neglect
vaporisation/condensation effects and heat loss can be calibrated
as a function of the apparatus only, independently of the reactor
content. Due to the fast temperature control of the cover, the
heat loss can be calculated as a function of the jacket and
cryostat temperatures only (assuming a constant temperature
of the environment during the course of the reaction, in our
case, (0.5 °C), and thus be determined by a proportional
correlation.
Nr
2
min
q˙ react - Vr r (-∆ H )
(8)
∑
i
r
i
{
[
]
}
kref,1...N ,∆rH1...N ,Ea,1...N
r
r
r
i)1
Additionally, we determined the total process enthalpy, ∆H,
that includes reaction enthalpy and heat of mixing, by a simple
model-free evaluation technique that consists of directly inte-
grating q˙ react
.
Results and discussion
Temperature Control. Figure 2a shows the temperatures
of the reactor, the jacket, and the cover during the course of
the esterification under nonisothermal conditions. The temper-
Heat Accumulation. In small-scale reactors, the heat ac-
cumulated by embedded probes may represent a large contribution
ature deviations from the set points are below 0.2 °C for T
and T and 0.4 °C for Tcover. In all cases, the maximum deviation
appears directly after the start and the end of the ramp while
r
j
(
30) Ubrich, O.; Srinivasan, B.; Lerena, P.; Bonvin, D.; Stoessel, F. Chem.
Eng. Sci. 2001, 56, 5147–5156.
5
28
•
Vol. 14, No. 3, 2010 / Organic Process Research & Development

Figure 2. Temperature and power during the esterification of BuOH and AA catalysed by TMG under nonisothermal conditions.
a) Measured temperatures of the reactor, T (black line), the jacket, T (red line), and the cover, Tcover (blue line). The dashed lines
(
r
j
represent the corresponding set-point temperatures. (b) Power of the compensation heater (black) and of the cooling (grey). The
vertical dashed grey and black lines represent the start and the end of the dosing and of the temperature ramp, respectively. The
experimental conditions are given in Table 1.
of the total heat accumulated by the system. We investigated the
effect of the bottom mounted IR-probe on the baseline by
measuring two references, with and without IR-probe. Figure 3
shows the resulting q˙ comp and q˙ cool. We can observe peaks at the
beginning and the end of the temperature ramp (at 800 s and
The broad peaks (width ∼400s) are due to heat accumulation
in thermally noncontrolled parts of the system. As shown in
Figure 3, these peaks are not observed on q˙ comp for the
measurement without the IR-probe while q˙ cool from both cases
perfectly superimposed. We can conclude that (a) the major
contribution to the broad peaks comes from the IR-probe and
3800 s). We may distinguish the narrow and the broad peaks with
a width of 50 s and ∼400 s, respectively.
(b) q˙ cool is not affected by the IR-probe. The difference of q˙ comp
The narrow peaks are due to the dynamics of the calorimeter,
that is of major importance for the deviations from the set point
is about 1.3 W that corresponds to a heat capacity of 80 W/K.
In method 1, few calibrations would be required to account for
this effect. With commercial calorimeters, this is usually
performed by the manufacturer and hidden in the control
software. Method 2 accounts perfectly for it, while for method
3, the shape of the IR-probe is too complex to be accurately
modelled within our heat flow model. Simulation with com-
mercial finite element software such as FemLab or Ansys could
provide more accurate results but the simulation of an entire
reaction would require extensive computational time and, thus,
cannot be used on an everyday basis. One could expect that
the heat accumulation in the cover would contribute to a
significant shift of the baseline. However, due to the well
8
temperatures. The dynamics should be fast to accurately control
the temperature of the reaction medium as discussed above.
The compensation heater and the cooling system have time
24
constants of 4 s and 15 s, respectively, which is more than 10
8
times faster than in standard reaction calorimeters. The
drawback is a strong peak of the power at the beginning and
the end of the temperature ramp. With methods 2 and 3, the
dynamics are taken into account by the blank measurement and
the heat flow modelling, respectively. Thus, both methods allow
for compensating the dynamics. Method 1 does not account
for the dynamics at all, and consequently shows the highest
peaks (see Figure 4).
Vol. 14, No. 3, 2010 / Organic Process Research & Development
•
529

Figure 3. Effect of the IR bottom-mounted probe on the calorimeter dynamics during a temperature ramp of 1 °C/min with 44.7
mL of AA as reactor content. Power of the compensation heater, q˙ comp, (full lines) and heat of cooling, q˙ cool, (dashed lines) with and
without IR-probe (black and grey lines, respectively).
Figure 4. Reaction power, q˙ react during the esterification of BuOH and AA catalysed by TMG under nonisothermal conditions, with
q˙ react determined by: method 1 (global heat balance, black), eq 1, method 2 (blank experiment, dark grey), eq 5, and method 3
(
simulated blank, light grey), eq 6. The vertical grey and black lines represent the periods of dosing and of the temperature ramp,
respectively. The experimental conditions are given in Table 1.
controlled temperature of the cover (see Figure 2), this cannot
be observed with the CRC.v5.
of TMG into AA liberates -0.85 ( 0.05 kJ/mol of TMG (not
shown here). The addition of BuOH into AA absorbed also
5.28 ( 0.10 kJ/mol of BuOH as shown later in Figure 7. Thus,
the dosing of BuOH into the TMG:AA mixture might follow
a complex equilibrium mechanism. The investigation of such
phenomena was not the scope of this study and not further
pursued. For the optimisation of the kinetic parameters, the
calorimetric measurements during the dosing period were not
considered.
The power of the reaction, q˙ react, determined from the three
different methods is shown in Figure 4. The deviations between
the three methods are observed only after the start and the end
of the temperature ramp for the reasons explained above. The
noise level that comes principally from the calculation of the
derivatives of the reactor and jacket temperatures is higher for
method 1 (0.28 W). The simulated reference is free of noise;
thus, the noise level with method 3 (0.15 W) is lower than that
with method 2 (0.22 W).
For the model-based evaluation we first postulated the
esterification to be a pseudo-second-order reaction mechanism
with a steady-state assumption on the TMG concentration after
Kinetics and Reaction Mechanism. During the dosing, the
profile of q˙ react shows an endothermic heat effect in addition to
the heat released by the esterification reaction. This endothermic
31
dosing, similarly to Puxty et al. The optimised kinetic
parameters obtained from the three different methods for the
heat balance are given in Table 2. Figure 5 shows q˙ react
20
effect was already observed and can, for example, be explained
by a shift of the equilibrium from the activated catalyst complex
towards its dissociated components during the dosing (i.e.,
volume change), or more complex solvation effects. The mixing
(
31) Puxty, G.; Neuhold, Y.-M.; Ehly, M.; Jecklin, M.; Gemperline, P.;
Nordon, A.; Littlejohn, D.; Basford, J. K.; De Cecco, M.; Hungerb u¨ -
hler, K. Chem. Eng. Sci. 2008, 63, 4800–4809.
5
30
•
Vol. 14, No. 3, 2010 / Organic Process Research & Development

Table 2. Overview of the kinetic parameters and reaction enthalpy for the esterification of n-butanol with and without TMG
a
for different reaction mechanisms and three evaluation methods of q˙ react
b
c
ref,1 [M-2 s-1]
ref,2 [M-2 s-1]
model
experiment ∆H [kJ/mol]
∆
r
H [kJ/mol]
k
Ea,1 [kJ/mol]
k
Ea,2 [kJ/mol]
With TMG
-
-
-
4
4
4
catalytic
parallel
eq 9
method 1
method 2
method 3
-54.4 ( 2.6
-48.8 ( 3.6
-48.3 ( 1.6
-64.0 ( 2.8
-59.4 ( 3.3
-59.5 ( 2.1
3.2 ( 0.2 × 10
2.8 ( 0.2 × 10
3.4 ( 0.2 × 10
27.4 ( 0.7
34.3 ( 1.6
31.0 ( 2.4
-
-
-
-
-
-
With TMG
eqs 9 and 10 method 1
method 2
-
-
-
4
4
4
-5
-5
-5
-54.4 ( 2.6
-48.8 ( 3.6
-48.3 ( 1.6
-57.1 ( 3.7
-55.4 ( 3.4
-54.4 ( 1.4
3.2 ( 0.6 × 10
2.7 ( 0.4 × 10
3.4 ( 0.1 × 10
15.7 ( 5.8
18.2 ( 4.9
13.6 ( 4.7
1.4 ( 0.4 × 10
1.0 ( 0.4 × 10
1.7 ( 0.6 × 10
84 ( 16
87 ( 12
88 ( 8
method 3
Without TMG
method 1
method 2
method 3
-
-
6
6
noncatalytic eq 10
-47.7 ( 2.0
-48.6
-47.7 ( 1.1
-67.3 ( 1.0
-69.2
-67.0 ( 1.4
-
-
-
-
-
-
3.2 ( 0.1 × 10
84 ( 0.4
91.2
83.5 ( 2.0
d
-6
2.6 × 10
3.6 ( 0.4 × 10
With TMG
Puxty et al.
Puxty et al.
Puxty et al.
2
3
3
2
1
1
+3
-4
literature
eq 9
-
-
-
- 43-8
4.0 ( 1 × 10
37.3 ( 0.01
-
-
-
-
-
-
-4
e
-
-
5 × 10
30
-4
e
6 × 10
31
a
Method 1: global heat balance. Method 2: blank experiment. Method 3: simulated blank. Errors for this study are given in one standard deviation based on three
b
c
replicates; see text and given references for more details on errors. Tref ) 40°C. ∆H is given by the integral of q˙ react, and thus includes also the heat of mixing.
r
∆ H is the
heat of reaction obtained from optimisation excluding the dosing period, and thus does not include the heat of mixing. One single experiment. e Measurement based on
d
near-IR spectroscopy.
Figure 5. Modelled and measured power of the reaction, q˙ react, during the esterification of BuOH and AA catalysed by TMG under nonisothermal
conditions. The measured data (full grey line) are determined with methods 2 (a) and 3 (b), the modelled data are given for a pseudo-second-
22
order catalytic mechanism with the optimised parameters determined in this study (full black line), and from literature (black dotted line,
31
black dashed line, with ∆ H ) -43 kJ/mol, see Table 2). The experimental conditions are given in Table 1.
r
Vol. 14, No. 3, 2010 / Organic Process Research & Development
•
531

determined with methods 2 and 3, and the resulting simulation.
For comparison, simulations with published kinetic parameters
are also shown. Apparently, none of the simulations fits the
measurement reasonably. This is particularly the case for the
relatively flat power profile between 600 and 3000 s corre-
sponding to the temperature ramp.
dCBuOH
f
dos
)
)
)
)
)
)
-r (k ,t) - r (k ,t) +
· (Cdos,BuOH - CBuOH(t))
31
1
1
2
2
dt
dCAA
Vr(t)
fdos
-r (k ,t) - r (k ,t) -
· CAA(t)
1
1
2
2
dt
dCBuOA
V (t)
r
fdos
r (k ,t) + r (k ,t) -
· CBuOA(t)
· CAH(t)
1
1
2
2
dt
dCAH
Vr(t)
f
Autocatalytic behaviour is frequently observed during es-
dos
r (k ,t) + r (k ,t) -
1
1
2
2
18
Vr(t)
terification between anhydrides and alcohols, due to the
formation of organic acid. However, as TMG is a strong base,
the acid activity is assumed to be low, and no autocatalysis is
expected. Here, we propose that a base-catalysed and an
uncatalysed reaction path take place simultaneously, following
a parallel mechanism of pseudo-second-order (with constant
dt
dCTMG
f
dos
-
· CTMG(t)
dt
dVr
V (t)
r
fdos
k1 · CBuOH(t) · CAA(t) · CTMG(t)
dt
r (k ,t)
)
)
)
1
1
r (k ,t)
k2 · C
(t) · CAA(t)
2
2
BuOH
CTMG after dosing) and second-order rate laws, respectively.
(
- E
/R
)
(1/T-1/Tref
)
a,1
k1
k2
kref,1 · e
kref,2 · e
-
(
Ea,2/R
)
(1/T-1/Tref)
k1
)
BuOH + AA + TMG f BuOA + AH + TMG (9)
(
11)
k2
To show that within this temperature range the uncatalysed
reaction already is competing, we investigated the reaction under
the conditions given in Table 1 but without the catalyst. Figure 7
shows the resulting fits for q˙ react calculated with methods 2 and 3.
BuOH + AA f BuOA + AH
(10)
The resulting ordinary differential equations (ODEs) are
given in eq 11.
Figure 6. Fitted (black line) and measured (grey line) reaction power, q˙ react, during the esterification of BuOH and AA catalysed
by TMG under nonisothermal conditions. The kinetic model includes both a catalytic and noncatalytic reaction step. The contributions
to q˙ react of both reaction steps are given by the dashed and dotted lines, respectively. q˙ react is determined with methods 2 (a) and 3
(
b). The reaction conditions are given in Table 1.
5
32
•
Vol. 14, No. 3, 2010 / Organic Process Research & Development

Figure 7. Fitted (black line) and measured (grey line) reaction power, q˙ react, for the esterification of BuOH and AA without catalyst
under nonisothermal conditions. The kinetic model is comprised by the noncatalytic reaction step only. q˙ react is determined with
methods 2 (a) and 3 (b). The reaction conditions are given in Table 1.
Obviously, the reaction “starts” after 1600 s at about 45 °C, which
may explain that the uncatalysed reaction was not observed in the
published studies only covering a temperature range of 30-50 °C.
The endothermic peak before 500 s is due to the heat of mixing of
BuOH into AA. The second peak at 900 s (start of the temperature
ramp) is due to the dynamics of the calorimeter as already discussed
above. Activation energy and reaction enthalpy (see Table 2)
to the fit with the catalysed reaction step only. The activation energy
of the uncatalysed reaction, optimised for the data with and without
TMG, are equal within the error limits. E
reaction are 25% and 50% below the values given in literature.
However, Figure 4 in Puxty’s publication (3D contour plots of
the response surfaces) does show a high ambiguity (valley shape)
regarding the minimum of the objective function for the determi-
nation of the kinetic parameters.
a
and kref of the catalysed
31
31
18
correspond well to the data published by Widell and Karlsson,
who used methanol instead of n-butanol (E ) 68.1 kJ/mol and
H ) -67.1 kJ/mol). In this particular case, as TMG is not
a
Theoretically, from one single nonisothermal measurement, the
simultaneous evaluation of the model parameters of two parallel
reactions results in two different sets of optimal parameters (e.g.,
∆
r
present, autocatalytic behaviour may also occur. However, con-
sidering the good quality of the fit, including an additional step
would not be reasonable.
two local minima of the objective function eq 8, with N ) 2).
r
Complementary information such as an additional measurement
under different conditions or analysis with chromatographic
techniques is necessary. Here, we assumed that the catalysed
reaction is faster than the uncatalysed one, and thus, we set the
parameter constraints in the objective function accordingly. With
Finally, the kinetic parameters for the two parallel reactions,
eqs 9 and 10 have been simultaneously optimised for the
calorimetric measurements with TMG used as catalyst. As the
reaction enthalpy, ∆
r
H, is a state function, i.e. depending only on
H is assumed to be the
the initial and final states of the system, ∆
r
r
all three methods, the constraint lower and upper bounds were ∆H:
-
6
-2 -1
same for both reaction steps. The resulting parameters are given
in Table 2 and illustrated in Figure 6 for methods 2 and 3.
Obviously, the additional contribution of the uncatalysed reaction,
shown as the dotted line in Figure 6, allows to explain the profile
of q˙ react. The quality of the fit, given by the sum of the residuals,
is improved by about 25% with the parallel reaction steps compared
-100 to -1 kJ/mol, kref,1: 10 to 0.1 m s , Ea,1: 1-100 kJ/
8
mol, kref,2: 10 to 3 × 10 m-2 s , and Ea,2: 50-100 kJ/mol.
Any initial estimates within these boundaries converged to the same
optimised parameters.
-
-5
-1
The errors in Table 2 are calculated by the standard deviation
of the optimised values, based on three replicates of the
Vol. 14, No. 3, 2010 / Organic Process Research & Development
•
533

experiment. Note that the errors given for the corresponding
easily be adapted and implemented into commercial calorimeters
capable of measuring the heat of cooling at real time, e.g. via
heat flow sensors.
22
literature values based on a second-order global analysis tend
to be unreasonably small due to possible nonrandom contribu-
tions between the individual experiments and too large a degree
Model-based kinetic analysis of a single nonisothermal
experiment should be done with caution to avoid erroneous or
rather naive interpretations. Reasons for this can be due to the
experimental setup (e.g., inefficient stirring) but also due to the
mechanistic model that may lead to inevitable ambiguities in
the kinetic parameters. For this study, the well-stirred vessel
allowed us to exclude mass transfer effects. A reasonable
assumption in the form of a simple and validated constraint
broke the mathematical ambiguity.
32
of freedom. Thus, the errors from both studies are not
straightforwardly comparable. Surprisingly, despite the simul-
taneous optimisation of five parameters to a monovariate
29
measurement, the variance/covariance matrix did not show
strong correlations between the five parameters. We can observe
that the activation energy of the catalysed reaction, Ea,1 and the
rate constant of the uncatalysed reaction, kref,2 have a high error
(
about 30%). However, the difference between the simulated
reaction powers is below the noise level of a measurement. For
comparison, in his publication on nonisothermal calorimetry,
Hoffmann et al. also gave an error of 50% on the pre-
exponential factor of a second order reaction.
11
List of Symbols and Definitions
Table 3. List of symbols and definitions
Conclusions
symbol
A
a
b
unit
description
heat exchange area
The reaction calorimeter, CRC.v5, combines the principles
of power compensation and heat balance. For the first time,
the performance of the CRC.v5 was demonstrated under
nonisothermal conditions. Usually, it is a challenging task to
account for possible baseline shift due to temperature modula-
tion without extensive time-consuming calibration. The CRC.v5
allows for an independent control of the reaction, the jacket,
and the cover temperatures. The heat of cooling is measured at
real time with heat flow sensors, and thus does not require
calibration of the global heat transfer coefficient.
m2
W/K
W
calibration parameter of the heat loss
calibration parameter of the heat loss
concentration of acetic anhydride
concentration of butyl acetate
concentration of acetic acid
concentration of n-butanol
concentration of 1,1,3,3-tetramethylguanidine
total heat capacity of the reactor content
total heat capacity of the jacket
heat capacity of the dosing
error between actual and set temperature values
activation energy of the ith reaction
rate constant of the ith reaction of nth order
rate constant of the ith reaction of nth order at Tref
gain of the controller
C
AA
M
M
M
M
CBuOA
C
C
C
C
C
AH
BuOH
TMG
p, r
M
J/K
J/K
J/(kg·K)
K
p, j
c
e
p,dos
E
a, i
kJ/mol
n-1
k
i
(L/mol) /s
We applied three different methods for the determination
of the reaction power employing either a global heat balance,
a measured reference or, as a novelty, a simulated reference. A
rigourous dynamic heat flow model of the calorimeter allows
the simulation of a reference measurement rather than measuring
it, thus saving a considerable amount of experimental time. The
shifts due to the reactor dynamics and the heat accumulation
of the system are not equally accounted for in all three methods
of heat balancing. The choice of one method depends on the
reaction and operating conditions. Reaction time, heating rate,
and reactor dynamics are three important factors to consider.
For example, if the reaction time is faster than the dynamics,
method 2 should be preferred. If large changes of physical
properties of the reaction mixture are expected, method 3 should
be used. For other cases, method 1 could provide enough
accuracy. In our case study, only minor differences in the kinetic
parameters and reaction enthalpies have been observed. Obvi-
ously, the good thermal dynamics of our reactor does not favor
one of the methods for heat balancing. The results of the
experiments indicate that the solvent-free base-catalysed es-
terification of n-butanol by acetic anhydride follows a mecha-
nism of two competitive parallel catalytic and noncatalytic steps.
The reaction enthalpy and the model parameters (rate constant
and activation energy) of both reactions could be simultaneously
evaluated from one single nonisothermal calorimetric measure-
ment using a model-based evaluation method, under the
assumption that the activation energy is lower for the catalysed
reaction than for the uncatalysed reaction.
kref,i
K
(L/mol)n-1/s
W/K 2
h
bottom
W/(m ·K)
heat transfer coefficient of the heat loss at the
bottom of the vessel
internal heat transfer coefficient (reactor side)
heat transfer coefficient of the heat loss from the
jacket to the coolers
2
h
h
r
W/(m ·K)
2
side
W/(m ·K)
2
h
top
W/(m ·K)
heat transfer coefficient of the heat loss from the
jacket to the cover
∆
∆
H
J/mol
J/mol
mL/min
-
process enthalpy
reaction enthalpy of the ith reaction
dosing rate
number of reactions
heat accumulation
r
H
i
f
dos
N
r
q˙ acc
W
q˙ comp
q˙ cool
q˙ dos
W
W
W
W
W
W
W
W
power of the compensation heater
heat flow of the cooling
heat flow of dosing
q˙ mix
q˙ loss
q˙ PID
q˙ phase
q˙ react
q˙ stirr
q˙ V,ext
r
heat due to mixing
heat flow lost to the environment
simulated power calculated by the PID controller
heat flow due to phase change
heat produced by the reaction
power due to the stirrer
density of heat from an external source
radial coordinate
W
W/m3
m
r
t
i
mol/(L·s)
reaction rate of the ith reaction
time
temperature
temperature of the cryostat and of the coolers
temperature of the dosing
temperature of the environment
temperature of the jacket
temperature of the reactor content
set-point temperature of the reactor content
heat transfer coefficient
volume of the reactor content
Cartesian coordinate
s
T
T
T
T
T
T
T
U
V
z
K or °C
K or °C
K or °C
K or °C
K or °C
K or °C
K or °C
cryo
dos
env
j
r
r,set
2
W/(m ·K)
L or m3
r
m
λ
F
τ
W/(m·K)
thermal conductivity
fluid density
derivative time of the PID
integral time of the PID
time constant of the compensation heater
angular coordinate
dead time of the reactor temperature measurement
blank measurement
dosing
measurement
simulation
kg/m3
D
s
s
s
rad
s
τ
τ
I
Method 3 requires preceding experimental work to determine
the parameters for the heat flow model. In comparison, with
commercial reaction calorimeters, the dynamics is empirically
corrected by numerous calibrations (performed generally by the
manufacturer). From the scientific point of view, we consider
a physically meaningful model superior to an empirical calibra-
tion due to an intrinsic understanding. This technique could
q
comp
φ
φ
Tr
blank
dos
meas
sim
0
initial value
5
34
•
Vol. 14, No. 3, 2010 / Organic Process Research & Development

Figure A-1. Illustration of the boundary conditions for the heat flow model of the CRC.v5. (a) top view (b) side view. Note that
both the cover and the bottom mounted IR-probe (not shown here) are outside the boundaries of the model.
Appendix A
Modelling with Finite Difference Method. The temperature
and heat flow profiles through a solid body due to thermal
conduction are governed by Fourier’s law. In cylindrical
coordinates, it is given by:
[Haynes International] for Hastelloy (vessel). The heat capacity
of the bulk, Cp, r was determined experimentally by temperature
oscillations. The advantages are the following: (a) with tem-
perature oscillation, even during the course of a reaction, the
heat capacity can be determined, and (b) heat capacity of the
inserts (embedded probes), the stirrer, the optional baffles, and
the heater are accounted for. For details on temperature
δT(φ,r,z,t)
δt
λ
Fcp
δ
δT
δ
δT
δφ
δ δT
δz δz
q˙ V,ext
)
(r ) +
) +
) +
[
rδr δr
2
(
(
λ
]
r δφ
28
oscillation calorimetry, we refer to Richner et al. As the mass
(
A-1)
of reactants is known and the vessel content is assumed to be
perfectly mixed, λ and F of the bulk are not required.
The modelling of q˙ comp and q˙ cool includes the dynamics of
the calorimeter, characterised by the PID settings of the thermal
regulator, and the time constant and dead time of the thermo-
couples. The PID controller modelling for q˙ comp involves the
three separate parameters given by:
The finite difference model used in method 3 was based on
the discretized form of eq A-1. This equation was discretized
along the spatial variables r and z while we assumed a symmetry
along the radial axis (e.g., (δT)/(δφ) ) 0). An optimal
discretization considered 273 elements, leading to 273 ODEs
that were solved in Matlab.
In eq A-1, q˙ V, ext represents the external heat sources, such
as the heat loss, the power of cooling and the compensation
power. The boundaries of the model are illustrated in Figure
A-1. htop, hbottom and hside are three different heat loss coefficients
that were characterised by a set of experiments at steady-state
1
τI
t
δe(t)
δt
PID
comp
q˙
(t) ) K e(t) +
·
∫
e(t')dt' + τD ·
[
0
]
(A-3)
where T
r
, T
j
, Tcover, and Tcryo were alternately changed. The
is determined by an
with
internal heat transfer coefficient, h
r
e(t) ) Tr,set - T (t - φ )
(A-4)
optimisation procedure performed with Matlab’s single-variable
r
T
r
optimiser, fminbnd, over a fixed interval, given by eq A-2.
K, τ
I
D
, and τ are the gain [W/K], the integral time [s], and the
sim
comp
meas 2
comp
mhirn{[q˙
- q˙
] }
(A-2)
derivative time [s], respectively. φ is the dead time of the sensor.
The time constant characterises the dynamic response of a
sensor or, more generally, of a system. The theoretical response
A(t) to a signal step ∆A of a linear system of first order is A(t)
The material properties (thermal conductivity, λ, density, F,
33
and heat capacity, c
p
) were taken from literature for copper
(jacket) and air (gas phase of the reactor), and from the producer
) A + ∆A[1 - exp (-t/τ)]. Thus, combining the PID control,
0
Vol. 14, No. 3, 2010 / Organic Process Research & Development
•
535

eq A-3, the dead time, eq A-4, and the time constant, τ,
gives:
value at the previous time. Equations A-3-A-5 are applied
sim
analogously for q˙ as a function of T. The dynamic parameters
cool
j
were characterised by performing experimental temperature
steps on T and multiobjective optimisation. More details on
experimental conditions and optimisation procedures are given
sim
comp
sim
comp
PID
comp
sim
comp
q˙
(t) ) q˙
(t - 1) + [q˙
(t) - q˙
(t - 1)] ×
r
-
[
1 - e t/τqcomp] (A-5)
2
6
sim
elsewhere.
where q˙ comp is the simulated value of the compensation heater,
q˙ comp the target value given by eq A-3, and q˙ cs iomm p(t - 1) the
PID
(
(
32) Bugnon, P.; Chottard, J.; Jestin, J.; Jung, B.; Laurenczy, G.; Maeder, M.;
Merbach, A.; Zuberb u¨ hler, A. Anal. Chim. Acta 1994, 298, 193–201.
33) Perry, R., Green, D. W., Maloney, J. O. Perry’s Chemical Engineers’
Handbook, 7th ed.; McGraw-Hill: New York, 1997.
Received for review November 12, 2009.
OP900298X
5
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Vol. 14, No. 3, 2010 / Organic Process Research & Development