AN ANALYSIS OF SELECTED TECHNEQUES
m MICROCALORIMETRY

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNiVERSiTY
DAVlD RUSSELL THOMPSON
1970

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AN ANALYSIS OF SELECTED TECHNIQUES
IN MICROCALORIMETRY

presented by
David Russell Thompson

has been accepted towards fulfillment
of the requirements for

Ph.D. Agr. Eng.

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ABSTRACT

AN ANALYSIS OF SELECTED TECHNIQUES
IN MICROCALORIMETRY

BY

David Russell Thompson

The objective of the research conducted was to ana-
lyze three aspects of microcalorimetry. Many processes
with small heats of reaction are being studied with micro-
calorimeters. The magnitude and the variations with time
of the reaction heats may be of considerable importance.
As the scientific understanding of food processes in-
creases, the measurement of the heat of immersion and
other small thermal effects will become more important.
Because the required accuracy of these measurements has
increased, a more detailed analysis of microcalorimeters
has become necessary.

The calibration of certain microcalorimeters was
investigated by studying the sensitivity coefficients.

It was shown that the unknown parameters of the instruments
could be determined simultaneously if the calibration
experiment was appropriately designed. The theory of

sensitivity coefficients was extended to include the

 

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David Russell Thompson

determination of unconstrained variables. The parameters
of the microcalorimeter are expected to change slightly
during the experiment. However, it was shown that unless
the reaction heat is known to follow a given functional
form, the changes cannot be measured during the experiment.
The sensitivity coefficients also indicate that accurate
compensation heating is impossible.

The effectiveness of metallic sinks for maintaining
a uniform temperature environment around a set of twin re-
action vessels was studied. A numerical technique was
devised to model a cross section of a solid cylindrical
thermal sink with separate cavities for the reaction and
reference cells. The influence of eleven parameters re-
lated to metallic sinks on the relative error was investi-
gated. The simulation indicated that the relative measure-
ment error can be maintained below 0.1% if the metallic
sink is properly designed.

A family of techniques for determining the reaction
heat as a function of time is presented. These techniques
are based on Duhamel's theorem with least squares smooth-
ing. The new methods are shown to be more accurate than
techniques reported earlier for some functional forms of

the reaction heat.

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AN ANALYSIS OF SELECTED TECHNIQUES

IN MICROCALORIMETRY

BY

David Russell Thompson

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Agricultural Engineering

1970

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The aut‘

AC KNOWLEDGMENTS

The guidance of Dr. F. W. Bakker-Arkema is acknowl-
edged with gratitude by the author. The indispendable help
of Dr. J. V. Beck is also gratefully acknowledged. Appreci-
ation is extended to Dr. L. Dugan and Dr. D. R. Heldman for
serving on the author's guidance committee.

Financial assistance from the National Aeronautics
and Space Administration and from the United States Army
Natick Laboratory is thankfully acknowledged.

The author is also appreciative of the understanding
and support of his wife, Janet, during this study. His
thanks are also extended to her for the help she gave in

preparing the manuscript.

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III.

IV.

TABLE OF CONTENTS

Page

INTRODUCTION 0 O O O O O O O O O O O l
1.1 General Remarks . . . . . . . . . l
1.2 Hygroscopic Bonds. . . . . . . . . 2
1.3 Thermodynamics of Immersional Wetting . . 5
REVIEW OF MICROCALORIMETRY . . . . . . . 7
2.1 Introduction . . . . . . . . . . 7
2.2 microcalorimeter Jackets . . . . . . 8
2.2.1 Adiabatic Microcalorimeters . . . 9
2.2.2 Nonadiabatic Microcalorimeters . . 12
2.2.3 Conduction Microcalorimeters. . . 14

2.3 Temperature Measurement. . . . . . . 19
2.3.1 Background. . . . . . . . . 19
2.3.2 Mercury-in-Glass Thermometers . . 23
2.3.3 Resistance Thermometers . . . . 24
2.3.4 Resistance Thermometry Circuits. . 28
2.3.5 Thermocouples. . . . . . . . 32
2.3.6 Quartz Crystal Thermometers . . . 36

2.4 Temperature Control . . . . . . . . 36
2.5 Calibration and Compensation . . . 40
2.6 The Sample Holder. . . . . . . 44
2.7 Stirring. . . . . . . . . . 47
2.8 Sensitivities . . . . . . . . 51
2.9 Some Sources of Error . . . . . 52
OBJECTIVES O O O I O O O O O O O 5 6
SENSITIVITY COEFFICIENTS . . . . . 57
4.1 Background . . . . . . . . . 59
4.2 Results . . . . . . . . . . 64

iii

VI.

VII.

4.2.1 Calibration of Instrument
Parameters with Only a Known
Heat Input . . . . . . . .

4.2.2 Calibration of Instrument
Parameters with Heat from Cali—
bration and Work . . . . . .

4.2.3 Sensitivity of Temperature to
Reaction Heat . . . . . . .

THE EFFECTIVENESS OF METALLIC SINKS. . . .

5.1

5.2

The Computer Model . . . . . . .

5 1.1 Selection of a Numerical Method .
5.1.2 The Grid System . . . . . .
5 l 3 Tests on the Model . . . . .

Results . . . . . . . . . . .

DETERMINING THE REACTION HEAT AS A FUNCTION
OF TIME I O O O C O O I O O O O O

6.1 Background . . . . . . . . . .
6.2 Improved Techniques for Finding the Heat
of Reaction . . . . . . . . . .
6.2.1 An Improved Inverse Technique. .
6.3 Computer Simulation . . . . . . .
6.4 Results . . . . . . . . . . .
SUMMARY AND CONCLUSIONS. . . . . . . .
BIBLIOGRAPHY O O O O O O I O O 0 O
APPENDICES
Appendix
A. The Heat Transfer Equations for Cells
on the Boundary of the Sink . . . .
B. The Computer Program of the Heat Sink.

iv

Page

64

67
72
75
78
78
80
86

88

94
94
97
101

104
108

112

114

122

135

LIST OF TABLES

Table

Page
1. The Maximum Relative Errors Introduced by
the Non-ideal Sink . . . . . . . . . 91
2. The RMS value of the Errors of Several Inverse
Techniques. . . . . . . . . . . . 109

Figure
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LIST OF FIGURES

Figure Page

1. The Sensitivities of Temperature with
Respect to the Constants cl and c2 for
the Initial Condition of T- = 0 in a
Microcalorimeter Without Any Heat Input
Due to Work. A Constant Calibration Heat
of 0.017 ca1./sec. was Added from t=0 to
t=100 sec. The Maximum Values of the
Dimensionless Sensitivity Coefficients
Were 1.0 and 0.069 for c1 and c2,
Respectively. The Maximum Temperature
was 0.129° C . . . . . . . . . . . 66

2. The Sensitivities of Temperature with
Respect to the Constants c and c2 for the
Initial Condition of Ti=0 in a Microcalori-
meter Without any Heat Input Due to Work.
The Calibration Heat Function was an In-
creasing Ramp from 0 at t=0 to 0.04
cal./sec. at t=200 sec. The Maximum Value
of the Dimensionless Sensitivity Coef-
ficients were 1.0 and 0.041 for c1 and c2,
Respectively. The Maximum Temperature was
.228° C. . . . . . . . . . . . . 68

3. The Sensitivities of Temperature with Respect
to the Constants c and c2 for the initial
Condition of Ti=0 in a Microcalorimeter
Without Any Heat Input Due to Work. The
Calibration Heat Function was a Decreasing
Ramp from 0.04 ca1./sec. at t=0 to 0 at.
t=200 sec. The Maximum Value of the
Dimensionless Sensitivity Coefficients were
1.0 and 0.081 for c1 and c2, Respectively.
The Maximum Temperature was 0.245° C . . . 69

vi

Figure

4.

A-S.

A-6.

A-7.

A-8.

The Sensitivity Coefficients of Temperature

with Respect to c1, c2, and q for the

Initial Condition of Ti=-clq ycz in a Micro-
calorimeter with Heat Input Uue to Work. A
Constant Calibration Heat of 0.016 ca1./sec.

was Added from t=0 to t=1000 sec. The
Maximum Value of the Dimensionless sensi-

tivity Coefficients was 1.0, 0.74, and 0.69

for cl, c2, and qw, Respectively. The
Maximum Temperature was .288° C . . . .

The Grid System for the Model of the Sink .

The Functional Form of the First Assumed
Reaction Heat . . . . . . . . . .

The Second Functional Form Selected to Test
the Stability of the Inverse Techniques .

The Third Functional Form Selected to
Represent a Rapid Reaction. . . . . .

Graphical Computation of the Dot Product. .
The Variables Needed to Compute cl. . . .
A Type One Cell . . . . . . . . . .

The Similar Triangles Used to Find (:15 for a
Type One Cell . . . . . . . . . .

A Type Two Cell . . . . . . . . . .

The Similar Triangles Utilized to Find d

for a Type Two Cell . . . . . . 5

A Type Three Cell . . . . . . . . .

A Type Four Cell. . . . . . . . . .

vii

Page

71

82

90

106

107
126
126

126

126

130

130
130

130

I . INTRODUCTION

1.1 General Remarks

 

Thermal effects accompany many chemical and bio-
logical phenomena. The magnitude of these effects may be
of considerable theoretical and practical importance. The
stability and structure of molecules, the conditions of
chemical equilibria, and the heat exchange in industrial
processes can all be calculated from a knowledge of these
thermal effects.

Many processes have been studied by calorimetric
methods. Heats of combustion and other processes that
evolve large quantities of thermal energy have been exten—
sively investigated. The advent of microcalorimetric
measurements has extended the range of calorimetric studies.
Various examples of these microthermal processes are re-
viewed by Swietoslawski (1946).

Microcalorimetric techniques have been used to mea-
sure the energy released by radioactive materials, the
heat produced by growing plants, by bacteria, and by the
germination of seeds. The gradual change in structure of
metallic alloys have been studied using microcalorimetors.

The thermal phenomena which take place when different kinds

of cement react with water give a direct indication of how
they can best be used in both cold and warm seasons. The
evolution of heat by the adsorption or the absorption of

gases and vapors has also been investigated.

1.2 Hygroscopic Bonds

 

Drying is one of the most widely used methods of
food preservation. Drying or dehydration of agronomic
products reduces the transportation and packaging costs
and can increase shelf life. The importance of drying has
provided the stimulus for a large body of scientific
literature. Much of this work has been primarily experi-
mental in nature. Consequently vast quantities of data
exist but only a small portion of it relates to the funda-
mental aspects of the drying process.

Microorganisms, metabolic processes, oxidation,
desiccation, abosrption of foreign odors, water damage,
and faulty preparation can all reduce the storage life of
biological materials. The moisture content of dried pro-
ducts must be low enough to retard the action of micro-
organisms and metabolic processes. Oxidation may cause
rancid flavors if the moisture content is too low. Thus,
the bond energy between the water molecule and the product
must be sufficiently high to make the water unavailable to
microorganisms although some water molecules should be left
to reduce the number of sites available for oxidation.

The optimum moisture content has been related to the

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monomolecular layer discussed by Brunauer, Emmett, and
Teller (1938). Microcalorimetric measurements of this
bond energy are expected to refine this concept.

Ngoddy (1969) explained the importance of the theo-
retical development of the isothermal curves for biological
materials. Equations representing the adsorption isotherms
are not only the most convenient form in which to plot
experimental data, but also they represent an important
step in the theoretical treatment of the drying process.
Adamson (1967) pointed out that although a theoretically
derived isotherm must be able to fit the experimental data,
this is not a sufficient test. He noted that relationships
derived from different models have been found to yield
equations which are graphically and even algebraically
identical. The distinguishing characteristic is the heats
and possibly the entropies of adsorption. Jura and Hill
(1951) showed that theoretically the entropies can be ob-
tained from the combination of heats of emersion data and
a single isotherm with considerably more accuracy than
from calculations dependent on the isotherms at two tem-
peratures.

The water in biological materials may exist as free,
chemically bound, and adsorbed moisture (Kuprianoff, 1958).
The hygroscopic point as defined by Lewis (1921) divides
the free moisture in food from the moisture that is
subject to surface sorption phenomena. The free water

is subject to the forces of osmosis, liquid diffusion,

capillary action, and gravity and is removed during the

so called constant rate drying period. Lykov (1955) pre-
sents a unifying theory of transport for free water.
Transport of moisture below the hygroscopic point is pri-
marily related to the surface forces. For this reason

this type of moisture is called adsorbed moisture. The
falling rate drying period is related to the removal of
this type of moisture. The transport of moisture below

the hygroscopic point is still the object of many studies.
It is generally agreed that this moisture is held by forces
of the van der Waal's type. The variety of sites and
energy levels available for water molecules on a particular
surface is reflected in the adsorption isotherm.

Several problems prevent the direct measurement of
the various site energy levels. Frequently the reactions
at surfaces are very slow and the energies involved are
very small. Water molecules released near a surface may
not select the highest energy site for adsorption. Thus,
the apparent energy release by a few molecules of water
added to the product, is not representative of the maximum
site energy available. Chessick and Zettlemoyer (1959)
indicated that most of these problems are surmounted by
the heat-of-immersion technique. The heat-of-immersion
or heat-of-wetting (also called the "Pouillet effect") is
usually a small exothermic quantity approximately equal to

the heat of adsorption. In order to avoid carrying the

negative sign, Boyd and Harkins (1942) introduced the term
heat of emersion and defined it as the negative of the

heat of immersion.

1.3 Thermodynamics of Immersional Wetting

 

Chessick and Zettlemoyer (1959) have reviewed the
thermodynamics of adsorption and immersional wetting. All
of their equations are on a per unit area basis; however,
the porous nature of biological materials may necessitate
utilizing a per unit of mass basis. Either way, the heat
of immersion is approximately equal to the difference in
the energies of the solid surface and the solid-liquid
interface because volume changes during the immersion
process are usually negligible.

The energy of adhesion of a liquid to a solid is
defined to be the sum of the surface energies of the liquid
and solid less the energy of the solid-liquid interface.
Therefore, the adhesion energy is approximately equal to
the heat of immersion plus the surface energy of the
liquid.

The adsorption and immersional processes are also
related through the heat effects. The integral heat of
adsorption of a given number of moles of adsorbate in the
vapor state at equilibrium pressure and temperature is
equal to the difference in the heats of immersion before
and after the moles are adsorbed plus the molar heat of

liquefaction times the number of adsorbed moles. The net

heat of adsorption is defined to be the difference in the
heats of immersion. The net heat of adsorption is also
equal to the difference in the molar energies of the ad-
sorbed film and the liquid times the number of moles.

Isosteric heats of adsorption can be determined from
heat of immersion data also. The integral of the isosteric
heat of adsorption over the number of moles adsorbed is
defined to be equal to the heat of adsorption. Therefore,
the difference in the heats of immersions before and after
the adsorption of N moles is equal to the integral of the
isosteric heat from zero to N less N times the molar heat
of liquefaction.

Entropy changes have also been related to heats of
immersion and the isothermal curves. If the entire entropy
change can be attributed to the adsorbate, Jura and Hill
(1951) showed that the difference in entropy between the
adsorbed film and the bulk liquid is equal to the differ-
ence in the heats of immersion per mole plus the film pres-
sure per mole all divided by the absolute temperature less
the Boltzman constant times the natural log of the rela-

tive equilibrium pressure.

II. REVIEW OF MICROCALORIMETRY

2.1 Introduction

 

In principle the design and operation of a micro-
calorimeter is simple. The basic parts of the calorimeter
include the product under study, water, one or more ther-
mometers, perhaps an electric heater, and devices for mix-
ing and stirring. A calorimeter is surrounded by an
insulative jacket in which thermometers may be located;
it has a provision for controlling its temperature. Heat
may be transferred between the calorimeter and the jacket ,
by conduction, by convection, or by thermal radiation.
Energy in the form of work may also be transferred to the
calorimeter by a mechanical stirrer or by compression or
expansion against an external pressure. Energy in the form
of heat may be supplied by an electric current used to
operate a heater or by the flow of electricity through a
resistance thermometer.

At least two microcalorimeters are commercially
available. Sunner and Walso (1966) reported on a micro-
calorimeter made in Sweden that is capable of measuring
the heat of immersion. A French built microcalorimeter

is being imported and marked by Imass Corporation. This

 

   

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instrument has sufficient sensitivity to measure the heat
of immersion but physically the instrument is not designed
for this application.

Although two microcalorimeters are commercially
available, Wilhoit (1967) reported that often it will still
be necessary for the experimenter to modify existing instru-
ments or construct them entirely to his own specification.
This review is not intended to discuss all the instruments
which have been used. However, an attempt will be made to
discuss the variations in the parts of a typical calori-
meter. A very large number of new calorimeters are being
built each year, but most of these involve only slight

modifications of the basic parts.

2.2 Microcalorimeter Jackets

 

Most microcalorimeters now in use can be classified
according to the type of thermal relationship between the
reaction vessel and the calorimeter jacket. The thermal
relationship is indicated by the thermal time constant of
the calorimeter--the rate of heat transfer between the re—
action vessel and the calorimeter jacket divided by the
heat capacity of the reaction vessel. The thermal time
constant of an adiabatic microcalorimeter is infinite
since no heat loss occurs. Microcalorimeters classified
as nonadiabatic have time constants between 1,000 and
100,000 seconds. Most nonadiabatic microcalorimeters have

time constants of about 20,000 seconds. The thermal time

constant of conduction microcalorimeters is less than
1,000 seconds.

Swietoslawski (1946) reported that most calorimeters
provide thermal insulation without regulating the jacket
temperature. To make the heat exchange between a calori-
meter and the jacket more uniform, the calorimeter was
placed inside one or two empty cylindrical containers.
Usually the calorimeter held a large amount of water so
that its temperature change was very small. In this way
reactions that produced a large amount of heat could be
studied without considering heat loss to the surroundings.
With the advent of microcalorimetry this approximation
could no longer be made. Thus the type of jacket employed
became a distinguishing characteristic of microcalori-

meters.

2.2.l Adiabatic Microcalorimeters

 

If the temperature of a calorimeter's jacket is
maintained at the temperature of its reaction vessel, no
heat loss will occur and the jacket becomes an adiabatic
shield. Maintaining the temperature equality is an ideal
situation that cannot be exactly obtained in actual prac-
tice. Sturtevant (1946) listed four advantages to be
realized by using the adiabatic procedure. First, the
overall heat transfer coefficient is more nearly constant
if the temperature differences are minimized. As a temper-

ature difference increases, the value of the convective

10

and radiative transfer coefficients change. Thus, heat
loss due to imperfect adiabatic control can be calculated
more accurately than heat losses resulting from larger
temperature variations. Secondly, the maintenance of very
small temperature differences reduces the convective heat
transfer coefficient. It is thus permissible to use wider
air spaces for insulation. Third, errors due to evapor-
ation from an incompletely sealed calorimeter are less
serious if the adiabatic method is used. Fourth, the
adiabatic procedure gives more reliable results in pro-
longed calorimetric experiments. In such cases the heat
loss in other type calorimeters in contrast with the
adiabatic type may be as large as the total quantity of
heat to be measured. Morrison §E_al, (1951) employed an
adiabatic microcalorimeter to investigate the relatively
slow process of gas adsorption on solid surfaces.

To increase the accuracy of the adiabatic control,
it is usually necessary to enclose the adiabatic shield
in an isothermal or a second adiabatic shield. Stubble-
field (1965) submerged his adiabatic calorimeter in a
water bath controlled to within one hundredth of a degree
centigrade. Other investigators have reported using the
outer shield for advantages other than just thermal pro-
tection. Morrison and L05 (1950) used liquid hydrogen to
provide an isothermal shield around their adiabatic
calorimeter and also to provide the low temperature neces-

sary for their experiment. Howard and Culbertson (1950)

11

used two concentric adiabatic shields or jackets. The
outer shield was a heavy steel vacuum chamber so that con-
vective heat transfer could be nearly eliminated if small
temperature differences did occur. The inner and outer
shields were carefully insulated. Rather than measuring
the temperature rise of the reaction chamber, these
authors determined the heat of the reaction by measuring
the energy necessary to maintain the inner shield at
equilibrium with the reaction chamber.

At least two variations of the adiabatic micro-
calorimeter have been reported. Belousov and Ponner (1968)
utilized a kerosene bath to provide the jacket for their
microcalorimeter. The temperature of the bath was main—
tained below the temperature of the calorimeter by an
amount sufficient to exactly balance the heating effects
of the resistence thermometers used in the calorimeter.

In other words, the heat input from the electrical energy
was balanced by the heat loss to the jacket. Abrosimov

and Krestov (1967) used an isothermal steel chamber as a
jacket for their microcalorimeter. They reduced the air
pressure inside the steel jacket to 10.5 torr. The radi-
ation heat transfer was reduced by chrome plating the
reaction vessel and the steel chamber. The sensitivity of
the calorimeter was reported to be plus or minus 0.1 milli-
calories. Corrections for heat loss were reported to

change the experimental results by less than 2%. Although

12

the calorimeter was nonadiabatic, the heat loss compared

very well with many adiabatic calorimeters.

2.2.2 Nonadiabatic Microcalorimeters

 

Most nonadiabatic calorimeters employ constant tem-
perature, isothermal, jackets. This type of calorimeter
is distinguished from the conduction calorimeter by a
relatively low heat flow between the reaction vessel and
the calorimeter jacket. The equipment required to maintain
an isothermal shield is substantially less expensive than
that needed for an adiabatic shield or jacket. The primary
disadvantage of the nonadiabatic microcalorimeter is the
increased heat loss from the reaction vessel to the shield.
Long term reactions cannot be studied in nonadiabatic
microcalorimeters because of the inadequate knowledge of
the heat loss. The accuracy of the heat transfer calcu-
lations is limited in nonadiabatic calorimeters because
the overall heat transfer coefficient becomes increasingly
dependent on the temperature difference as this difference
increases. Reversely, the temperature difference can be
known with greater accuracy when an isothermal jacket is
used. It is difficult to prevent temperature gradients
from existing within an adiabatic shield. However, an
isothermal shield can be sufficiently massive or else
consist of a well stirred fluid so that temperature
gradients are minimized. In order to minimize temperature

gradients within the isothermal jacket, Zverev and Krestov

13

(1968) used a heavy aluminum block inside a well stirred
bath of water which in turn was inside an environmentally
controlled chamber.

Twin calorimeter vessels can be utilized with any
type of jacket. However, they are most frequently employed
inside isothermal shields. Bartell and Suggitt (1954)
studied the heat of wetting of several inorganic substances
in twin calorimeter vessels in a nonadiabatic microcalori-
meter. Both vessesl were treated identically except for
the release of a small quantity of the inorganic solid in
the reaction vessel. In this way, the electrical and
mechanical heat inputs could be neglected. Ostrovskii
gt_§l. (1968) used four vessels in one isothermal enclosure.
This instrument is capable of testing two reactions simul-
taneously. In order to reduce the inaccuracy of heat loss
calculations, electrical energy was added to either the
reaction vessel or the comparison vessel to maintain their
temperatures as near equal as possible. The authors
recognized that the temperature equality was not maintained
and used heat loss calculations to correct for this.

Berger et a1. (1968) used twin vessels in an isothermal
enclosure and employed a numerical technique to approxi—
mate the heat loss from the reaction vessel. The tempera-
ture difference between the twin vessels was measured
rather than measuring the difference between each vessel

and the sink.

 

 

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2.2.3 Conduction Microcalorimeters

 

If heat is removed from the reaction vessel at
approximately the same rate as it is produced by the re-
action, the temperature of the reaction vessel remains con-
stant. For this reason this type of calorimeter is some-
times labeled isothermal; however, more frequently it is
called a conduction microcalorimeter. The heat of the
reaction being studied is determined by measuring the heat
loss rather than the temperature rise of the reaction
vessel. Conduction calorimeters are frequently used to
study reactions that require long periods of time to run
to completion. Generally the only stirring used in a con-
duction calorimeter is that required to initiate the re-
action.

The original conduction microcalorimeters were de-
signed by Tian and modified by Calvet (Calvet and Prat,
1963). The Calvet conduction microcalorimeter resembles
a twin vessel nonadiabatic calorimeter. It consists of a
reaction vessel and a reference vessel, each inside a
metal block which in turn is enclosed by an isothermal
shield. The reaction and reference vessels are tall,
slender containers. No mechanical stirring is provided;
thus the temperature of the vessels is not uniform and
the heat transfer to the heavy metal block is a function
of location. A battery of thermocouples is judiciously
placed between the vessel and the heat sink. It can be

shown that the sum of the emf's produced by the

15

thermocouples is directly proportional to the heat loss if
one pair of thermocouples is located in each heat flux
tube. This of course assumes that the temperature differ-
ence remains small. If the thermal effects of the reaction
under study are larger than can be readily removed by con-
duction through the bank of thermocouples, Joule heating

or Peltier cooling is supplied. In order to eliminate

some of the methodical errors, the difference in potential
between the bank of thermocouples surrounding the reaction
cell and that surrounding the reference cell is recorded.
The integral of the potential recorded over time is pro-
portional to the heat of the reaction under study plus or
minus the heating or cooling provided. For an analysis of
the reaction kinetics, Tian and Calvet have devised tech-
niques which reconstruct the heat input as a function of
time. One of the principle difficulties with the Calvet
microcalorimeter is the amount of work required to assemble
and mount the banks of thermocouples.

The heat of reaction in a heat burst microcalori-
meter is all conducted through thousands of thermocouple
wires to the sink. Benzinger and Kitzinger (1963) explain
the principal of heat burst microcalorimetry in the follow-
ing way. In the classical calorimeter, that is a calori-
meter with either an adiabatic or nonadiabatic jacket, the
heat generated in the reaction vessel escapes very slowly.
If an instantaneous reaction proceeds in the reaction

vessel, the temperature of this vessel will rise slightly.

16

Since the heat escapes very slowly, the temperature would
remain almost constant for a very long period of time. The
temperature change could be measured with the thermometric
device almost immediately. Thus the remaining "potential
thermal energy" would be wasted.

Now instead of one thermometric device or perhaps a
few thermocouples, imagine a reaction vessel connected to
the sink with an almost infinite number of thermocouples.
Each couple would measure the temperature of an almost
infinitesimally small segment of the reaction vessel. A
jump in potential would be registered in each of the indi—
vidual couples in response to the instantaneous chemical
reaction. Each of these potential jumps would be of the
same magnitude as the single jump registered in the classi-
cal calorimeter. The heat produced by the reaction would
be transferred very rapidly from the reaction vessel to
the sink through the thermocouple wires. Thus the response
of the thermocouples would decay very rapidly. However,
all of the single electrical potentials could be made
additive by wiring the thermocouples in series. The re-
sult would be one very high though rapidly decaying spike
of potential. In effect the duration of the potential in
the classical calorimeter has been traded for the height
of the potential. It should be noted that many of the
classical microcalorimeters have used banks of thermo-
couples for measuring the temperature change of the re-

action vessel. Generally, these thermopiles included less

17

than one hundred fifty pairs of junctions. The thermal
response time of these instruments remained very long. In
contrast, heat burst microcalorimeters employ thousands of
thermocouples and the thermal response time is made as
short as is practical.

In a phase change conduction microcalorimeter, the
quantity of heat produced is measured by the amount of an
isothermal phase change which it produces in the calori-
meter material. The ice calorimeter is the most common
and the most convenient type of phase change calorimeter.
Basically an ice calorimeter consists of the reaction
vessel sealed in an outer vessel which is filled with air
free water--ice, and mercury. The bottom of the outer
vessel is connected by means of a small glass tube to a
device for measuring the change in volume of the water ice
mixture. The concentric vessels are completely immersed
in a mixture of ice and water in a Dewar flask. Thermal
effects occurring in the reaction vessel transfer heat to
the water ice mixture in the outer vessel and hence cause
part of the ice or melt or water to freeze. The amount of
heat transferred is determined by measuring the change in
volume. Valee (1962) has reported a very simple design
for an ice calorimeter. His instrument is constructed
of concentric glass containers. Because of the isothermal
characteristic of the ice calorimeter, it is possible to

keep the heat exchange with the ambient very low. The

18

apparatus is therefore particularly useful for determining
small or very slow heat effects.

Some disadvantages of ice calorimeters have been re-
ported. Swietoslawski (1946) reported that some secondary
phenomena apparently occur inside the ice microcalorimeter.
He used twin ice calorimeters of conventional size and
equal volume to determine whether or not the rate of volume
changes would be the same when both calorimeters were
placed in the same large ice bath. The results of this
experiment were negative. The same author theorized that
the erratic behavior was caused by the hydraulic head
within the calorimeter which prevented the existence of a
thermodynamic equilibrium. He did succeed in building an
ice calorimeter which could measure heating rates as low
as 0.03 calories per hour. However, this required con-
siderable skill on the part of the experimenter and still
occasional erratic results were obtained.

A second disadvantage of the ice calorimeter is that
it can be employed only at 0 degrees C. This disadvantage
is partially removed by employing substances other than
ice. Sturtevant (1945) reported that diphenylmethane,
diphenyl ether, and acetic acid have all been used in
place of ice. Phase changes other than the freezing-
melting one can also be used. Chessick, Young, and
Zettlemoyer (1954) reported using a phase change micro-

calorimeter with liquid nitrogen being vaporized.

l9

Statistically the overall error for measuring heat of
immersion in their calorimeter was less than 5 per cent.
If both of the reactants to be studied can be pumped
through small tubes, a flow type microcalorimeter may be
employed. Flow calorimeters represent a special type of
conduction calorimeter. Stoesser and Gill (1967) described
a precision flow microcalorimeter. The reactants are con-
tinuously pumped into the mixing chamber and the product
is continuously removed. The thermal effect of the re-
action is determined by measuring the equilibrium rate of
heat removal. Obviously it is not possible to measure the
heat of immersion of a solid using the flow calorimeter.
However the stopped flow calorimeter such as was suggested

by Berger and Stoddart (1965) suggests some possibilities.

2.3 Temperature Measurement

 

2.3.1 Background

 

Wilhoit (1967a) reported that the calorimetrist is
likely to spend more of his time in measuring and control-
ling temperature than any other single activity. Progress
in microcalorimetry is dependent on the availability of
suitable thermometers, sufficient sensitivity, and sta-
bility in the temperature range of interest. To be suit-
able for use in a microcalorimeter, a thermometer must not
introduce appreciable disturbing influences on the measure-
ment. In other words, the sensor must be small in size

and in heat capacity and not conduct excessive heat to or

20

from the calorimeter. Usually remote recording and rapid
response to changes of temperature are important charac-
teristics.

The various calorimeter designs have differing
requirements for the temperature measuring equipment.
Frequently even one calorimeter will require several kinds
of temperature measurement. Therefore it is not possible
to generally recommend a particular thermometer for micro-
calorimeter applications. Comparison calorimeters use
thermometric devices to measure the temperature difference
between the reaction vessel and a reference vessel.
Makrides and Hackerman (1959) reported using twin Dewar
flasks in a constant temperature bath for reaction and
reference vessels. They treated each of the flasks
identically except for the absence of a reaction in the
reference cell. Identical equipment was installed in the
two flasks, similar sample holders were broken in each
flask, and the stirrers were connected by a positive drive
system, so that energy inputs to each flask was as near
identical as possible.

Other authors, for example Young and Bogel (1932)
have used references without any equipment in them in an
effort to maintain a stable reference temperature. Some
correction for the work input to the reaction vessel is
then necessary unless this energy input can be kept

sufficiently low (Privalov and Monaselidge, 1966).

 

“a"

 

U)

'6

(2

(u

21

The difference between the temperatures of two sur-
faces must be measured in a comparison calorimeter. A
similar measurement is necessary also in a nonadiabatic
calorimeter if the heat exchange between the calorimeter
and the jacket is to be evaluated. Frequently these
temperature sensors are placed directly on the surfaces.
When this is done, it should be established that the
thermometer does not introduce undesirable disturbances
in the temperature distribution. The primary thermometer
requirements for this type of measurement are a high
sensitivity (since the differences may be small) and a
short response time (because temperature changes may be
rapid).

Compensation calorimeters are very similar to the
comparison units except that the reference vessel is main-
tained at the temperature of the reaction vessel. Berg-
hausen (1954) reported controlling the reference vessel
temperature to within one micro-degree C of the tempera-
ture of the reaction vessel. If the two vessels are
identical and are treated identically except for the re-
action occurring in the reaction vessel and the energy
added to the reference vessel, the heat of reaction can
be determined by measuring the heat added to the reference
vessel. A similar temperature sensor is required in
calorimeters using adiabatic jackets. The requirements
for these temperature sensors are very similar to the

requirements for the sensors in the comparison

Yl'u. '. ‘I V..."

 

'1

.4‘

‘1:

ti

22

calorimeters. The primary difference is that the accuracy
of the sensors in the compensation calorimeter is unimpor-
tant since it serves only as a component in the control
circuit as opposed to the sensor in the comparison calori-
meter which is being used to obtain a measurement.

A few microcalorimeters with one vessel have been
designed. In this case the temperature sensor must be
capable of measuring the absolute temperature instead of
a relative temperature between two bodies. Frisch and
Mackle (1965) reported using this type of calorimeter.

The accuracy of this type of instrument is usually limited
for rather obvious reasons. If a temperature sensor must
measure a temperature change from 100 degrees to 101 de-
grees with an accuracy of 1%, the accuracy of the sensor
must be 0.01%. On the other hand if a sensor must sense
the difference between two bodies and the difference varies
from zero to one degree, the measurement can be made with
an accuracy of 1% with a sensor that is accurate to only
plus or minus 1%. High accuracy temperature sensors
usually either add thermal energy to the bodies which they
are sensing or are thermal conductors to the ambient.

Kington and Smith (1964) have combined some of the
advantages of the absolute temperature calorimeter with
those of the comparison calorimeter. They use a single
reaction vessel but compare its temperature to a metal
block whose temperature in turn is followed by an absolute

temperature measuring device. In this way they are able

 

,-
VV

 

on.

 

 

v‘A1

”Iv.

.

v“.

V'-

“A
'»

 

 

“a
I:

~
u

 

,.

("J
I

23

to measure the absolute temperature changes of the reaction
vessel but have isolated the undesirable characteristics of
the absolute temperature measuring equipment.

Absolute temperature sensors are also necessary to
measure the temperature at which the measurement is made.
Accuracy for these measurements usually is much less im-
portant than for the measurement of the change in tempera-
ture in the reaction vessel. Usually accuracies of plus

or minus 0.1 degree or even 1 degree C are sufficient.

2.3.2 Mercury-in-Glass Thermometers

Mercury-in-glass thermometers have played an im-
portant role in calorimeters designed for measuring the
heats of combustion. Although they are not often used in
microcalorimeters, they should not be entirely overlooked.
These thermometers allow the measurement of temperatures
with moderate precision in the range of -35 degrees to 360
degrees, or even higher if the mercury is under pressure.
The short interval mercury thermometer is the most impor-
tant in calorimetric work. The Beckmann type can be read
with a lense to 0.001 degrees or with a microscope to 0.5
millidegrees C (Keffler, 1930). Barry (1920) has described
a thermometer with a range of 15 to 21 degrees C which
can be read to 0.1 millidegree. He considers this to be
about the limit of precision obtainable with mercury

thermometers.

24

Mercury thermometers have several disadvantages.
Sensitive mercury thermometers necessarily have a rela-
tively high lag and therefore are not suitable for the
observation of rapidly changing temperatures. The volume
of the bulb of a good mercury-in-glass thermometer will
change when heated. It may not regain its original
volume for several days. Errors of as much as 0.01 de-
grees for each 10 degrees the bulb has been heated above
the normal temperature are common. The difficulty in
adapting them to remote indicating, recording, and control-
ling is also a disadvantage. Busse (1941) had discussed
the various errors associated with mercury-in-glass ther-
mometers. These thermometers are too large, their lag
times are too long, and the thermal conduction along their
stems is too large for them to be employed inside the re-
action vessel of microcalorimeters. However they are

being used to measure the temperature of the jackets.

2.3.3 Resistance Thermometers

 

The measurement of temperature as a function of the
electrical resistance of a sensitive element is a familiar
method. Resistance thermometers may be produced from a
coil of metallic wire such as copper, nickel, or platinum;
a semi-conducting element such as carbon or germanium or
from a semi-conducting metallic oxide. Because of the
considerable versatility, reliability, and sensitivity,
resistance thermometers have a wide application to calori-

metry.

25

Daneman and Mergner (1967) indicated that platinum
is the most popular resistance thermometer sensor because
it can be obtained in a highly reproducible and pure con-
dition, is relatively unaffected by aging or by gases and
fumes over a wide temperature range, can be drawn into
fine wires, and has a high melting point. Many other
metals and alloys have temperature coefficients of re-
sistivity somewhat larger than platinum's 0.4 per cent
per degree. However, these other materials are not as
widely employed as platinum for precise measurements be-
cause of their greater susceptibility to external physical
and chemical effects.

Watson (1966a) indicated that because mechanical
strain alters the electrical resistance of metals, plati-
num resistance thermometers must be wound of annealed wire
on mica supports in such a way that the metal is subjected
to as slight a strain as possible when the thermometer is
heated or cooled. For most applications the coil is en—
closed in a sealed glass or silica tube. However, to
minimize the lag of the thermometer for calorimetric
applications, the coil may be enclosed in a thin metal
case. In this way it is possible to achieve a time re-
sponse as short as one micro second.

The principal disadvantages of platinum resistance
thermometers are their relatively high cost and the self
heating caused by the energy dissapation in the sensors.

The self heating disadvantage is of course a characteristic

26

of all resistance thermometers. Other sources of errors
with these thermometric devices include the resistance of
the leads and thermoelectric emf's in the circuits.

Loeffler (1967) indicated that the cost of resistance
thermometers can be decreased if nickel sensors are substi-
tuted for the platinum units. The thermal coefficient of
resistivity of nickel is larger than that for platinum.
However the nickel sensors are more non-linear. Rall and
Hornbaker (1967a) reported that Balco, an iron nickel alloy
has similar characteristics to those of nickel. Westrum
and Eyring (1952) have utilized copper wire as a resistance
thermometric sensor in an automatic control circuit in a
compensation type calorimeter. The copper sensor is much
less expensive than any of those previously mentioned.
However, its thermal coefficient of resistivity is not as
well defined nor as large as that of nickel or Balco. In
spite of these problems, Quigg (1966) was able to measure
small temperature differences with a copper sensor.

The resistance of certain semiconductors made from
inpurity doped metallic oxides is strongly dependent on
temperature. A thermistor is made by connecting two lead
wires to a small head of this material. Even when covered
with glass, thermistors are usually one millimeter or less
in diameter. The temperature coefficient of resistance for
thermistors is large and negative. Richards (1967) re-
ported a typical coefficient of minus 4.5 per cent per

degree. The magnitude of this coefficient is about eleven

27

times that of platinum. Usually thermistors are manu-
factured to have a high resistance in the temperature
range in which they are to be employed so that the re-
sistance of the leads can be ignored and the self heating
will be minimized. The functional relationship between
resistance and temperature for thermistors is highly non-
linear and is difficult to represent mathematically. How-
ever, for use over a limited temperature range, it is
possible to use an exponential function to fit the re-
lationship reasonably well.

The stability and reproducability of the resistance
of thermistors is generally lower than for metallic re-
sistance thermometers. Therefore, they are not generally
practical for making accurate measurements of absolute
temperature. Nevertheless, Hutchinson (1947) and Muira
et_§l. (1956) both were able to use thermistors to measure
the absolute temperature change of a reaction vessel em-
ployed for studying the heats of immersion. Over short
time periods, sensitivities of plus or minus 10 micro
degrees have been reported by Sharifov et_al. (1966).
Manufacturing tolerances for thermistors tend to be very
large. However, matched thermistors are now available
for comparison measurements in microcalorimetry (Nancollas
and Hardy, 1967). These authors also indicate that the
stability problem is reduced by using very low voltages.
The heat capacity of most thermistors is very low. How-

ever, the surface area for heat exchange is also usually

28

low; therefore, the time response may not be any better,
if as good as, metallic resistance thermometers.

Some of the disadvantages of thermistors are being
overcome by replacing them with germanium temperature sen-
sors. Maker (1968) reported measuring temperatures with
a germanium transitor with the base connector left open
circuited. This sensor is shown to have a sensitivity at
least as high as that of a thermistor, to have a linear
resistance as a function of temperature, and to be stable
over long time periods. The disadvantage of this tempera-
ture sensor is the relatively long time constant. Pure
crystals of germanium have been shown to have a slightly
lower sensitivity than the germanium transitor. However
the crystals can be manufactured to have large surface
areas and thus very small time constants. Rall and Horn-
baker (1967) indicated that germanium crystals may have an
accuracy and sensitivity as high as those of any resistance

thermal sensor available.

2.3.4 Resistance Thermometry Circuits

Several methods are available for accurately deter-
mining the change in resistance of the resistive ther-
mometers. Daneman and Mergner (1967) have discussed
direct current methods in precision resistance thermometry.
Each of the direct current bridges mentioned below is
described by them in more detail. The Mueller bridge is

the most commonly used method for accurately measuring the

29

absolute temperature with platinum resistance thermometers
in the Western Hemisphere. It utilizes elements of equal
resistance in the ratio arms of the bridge circuit to pro-
vide the following characteristics. If the resistance of
the leads to the temperature sensor is constant, this
bridge can compensate for the lead resistance with very
high accuracy. The circuit employed reduces the effect

of switch contact resistances in the measuring arm. How-
ever, the contact resistance of the internal switches of
the bridge is very critical. This bridge is available in
several models with accuracies ranging from one part in

one million to one part in one hundred million. The latter
accuracy represents a temperature measurement of plus or
minus 0.1 millidegree. With the present technology, errors
in thermometric measurements prevent accuracies any greater
than can be achieved with the Mueller bridge.

In the Eastern Hemisphere the Smith type III D.C.
bridge is the most frequently used detector in precision
resistance thermometry. The following characteristics of
this bridge are a result of the ratio of the resistance
of the calibrated variable arm and the thermometer resis-
tance being about one hundred. This bridge is easier to
use and the contact resistance of the internal switches
is not as critical as with the Mueller bridge. The dis-
advantages of the Smith bridge include less accurate com-
pensation for the resistance of the leads to the tempera-

ture sensor, and the effect of switch contact resistances

30

in the measuring arm are greater than in the previously
mentioned bridge.

Several automatic or direct reading bridges are
available. However, their accuracy is less than that of
the Mueller or the Smith type III bridges. The potentio-
metric method of measuring resistance offers some ad-
vantages in thermometry. A constant current is passed
through the thermometer and an external standard resistance
which is connected in series. The resistance of the ther-
mometer may be calculated from the measured ratio of the
potential drops across the thermometer and the standard
resistance. If a potentiometer is used for this measure-
ment, the procedure may be simplified by connecting the
standard resistance in place of the standard cell. A
potentiometer is somewhat more economical than a bridge
of the same accuracy, although the potentiometric method
requires a more sensitive null detector and a source of
more accurately controlled current. It has a very definite
advantage for the measurement of low resistances. Since
no current flows in the potential leads, their resistance
has very little effect on the measurement. Hall (1966)
indicated that the most serious disadvantage of potentio-
meters is that their maximum accuracy does not equal that
of the best bridges.

Alternating current methods of resistance thermometry
are employed more frequently in microcalorimetry than are

the D.C. methods. By using narrow band filters, it is

31

possible to eliminate nearly all stray voltages in alter-
nating current equipment. High gain alternating current
amplifiers are less expensive than their D.C. counterparts.
Inductors and capacitors can sometimes be used advan-
tageously in alternating current equipment. There are also
several disadvantages associated with A.C. equipment. The
leads from the temperature sensor to the detector and from
the detector to the amplifier or any other equipment must
be kept short, and their positions relative to each other
and to other equipment in the room must remain constant.
Any relative motion of the leads will change the inductance
and capacitance characteristics of the leads. The A.C.
properties of the resistance temperature sensor must also
be considered. Morrison and Hanlan (1957) reported using
a Wheatstone bridge with thermistor type temperature sen-
sors in their calorimetry work. This bridge or simple
modifications of it has been frequently used in micro-
calorimetry. Daneman and Mergner (1957a) indicated that
there are at least three bridges that are more accurate
than the Wheatstone design. The Hill and Miller bridge
is accurate to one micro degree. The Frisch and Mackle
bridge is more frequently used in calorimetry even though
its accuracy is somewhat less. For comparative tempera-
ture measurements the Foord, Langlind, and Binnie bridge
is frequently used.

For microcalorimetric measurements, it is usually

necessary to have continuous records of the temperatures

32

measured. Over very small temperature ranges, the in-
balance in any of the bridges discussed is approximately
proportional to the change in a resistance thermometer.
Very sensitive galvanometers or electronic null detectors
can be connected across the terminals of these bridges to
give a continuous read out of the temperature change. For
continuous recording of the temperature change, a high
gain, low noise amplifier can be connected across the
bridge terminals. If an A.C. bridge is being used, a
narrow band amplifier should be selected. Similar tech-
niques are used for automatic recording when the potentio-
metric method is used. However, Wilhoit (1967) indicated
that the instrumentation for automatic recording with
potentiometers is somewhat less expensive than that for

bridges.

2.3.5 Thermocouples

 

In microcalorimetry temperatures are frequently
measured with thermocouples. The electric potential
generated by a thermocouple is directly proportional to
the temperature difference between the hot and cold junc-
tions. The thermoelectric power and thus the electrical
potential per degree of temperature difference is of the
order of 50 microvolts per degree, depending on the metals
and temperature. Thermocouples are well suited for measur-
ing temperature differences. They may also be used for

absolute measurements by keeping one junction, the reference

33

junction, at a constant temperature. Holmes (1968), how-
ever, indicated that thermocouples generally cannot be
relied upon to be better than about one part in one thou-
sand. Therefore, to have an accuracy of 0.1 millidegree,
the hot and cold junction must be within 0.1 degree C.
Thermocouples can be made to have low heat capacities and
small lags if constructed of fine wires, and thus are
capable of following rapid temperature changes. Another
important advantage of thermocouples is that many junctions
may be connected in series and carefully mounted on a sur-
face or throughout a volume to average out slight tempera-
ture irregularities in the body under observation. Kit-
zinger and Benzinger (1967) reported covering the reaction
chamber with 10,000 thermocouple junctions.

A large variety of materials are being used in the
manufacture of thermocouples for calorimetry. Pirro (1967)
reported that generally platinum-platinum rhodium thermo-
couples are preferable for high precision measurements.
However, Chromel-alumel is attractive particularly because
of its very small diameter and relatively high thermo-
electric power. King (1968) reported that gold-nickel
thermocouples, if properly constructed, are desirable for
measuring temperature differences and for sensing surface
temperatures. Chromel-constantan junctions are also em-
ployed in microcalorimeters.

One difficulty of junctions in which neither lead is

copper is that normally a junction between the thermocouples

34

and the lead wires will be necessary. Evans and Richards
(1952) indicated that chromel-constantan and copper-
constantan junctions could be combined so as to avoid
heterogeneous junctions between the thermocouples or ther-
mopile and the leads.

The measurement of low D.C. voltages is required when
thermocouples are employed. The reduction of noise and
extraneous sources of potential below the level of a few
microvolts requires the exercise of some care. Potentials
of this magnitude are produced by electrostatic effects,
changes in the magnetic field, insulation leakage, thermo-
electric effects, inhomogeneities in the wires in the
thermocouple, and instability in the ground. The junctions
are usually pressed together or welded. An electroplating
process for producing large numbers of thermocouples for
microcalorimetry has also been reported by Evans §t_al.
(1968).

The electric potential generated by a thermocouple
is usually measured with a potentiometer. Since the
potentials developed by thermocouples are small, potentio-
meters of high precision are required for calorimetric
work. Interpolation between the steps of the last decade
of the potentiometer is accomplished with galvanometers
or ammeters (Lindeck method) rather than with slide wires.
Frequently the White double potentiometer is combined
with thermocouple sensors in microcalorimetry. If a

relatively small number of thermocouples are used, a

35

sensitive galvanometer or indicating electronic D.C. null
meter may be required. Gucker §t_§l, (1939) and others
have reported also utilizing extremely sensitive recorders
to monitor the potentiometer output. When very large
numbers of thermocouples are connected in series, less
sensitive detectors can be selected. The nanovoltmeter

is an alternate way of measuring the thermocouple poten-
tial (Berger gt_al., 1968).

Some potential errors in thermocouple measurements
are accented when surface temperatures are being measured.
Heat conduction along the leads to the sensor will cause
the sensor temperature to vary slightly from the tempera-
ture of the surface. Watson (1966b) indicated that this
problem can be reduced by mounting the sensors below the
surface or placing them into grooves cut into the surface.
If this is not feasible, then the leads should parallel
the wall for 50 to 100 lead diameters.

Maintaining good thermal contact between the sensor
and the surface may be difficult. If a thermel is measur-
ing the temperature of a metallic surface, good electrical
insulation must be provided between the sensor and the
surface. This further aggravates the thermal contact
problem. Watson (1966) suggested flattening the tip of
the sensor or using thermocouples that have been especially
prepared for the measurement of surface temperatures to

overcome the thermo-contact problem.

36

2.3.6 Quartz Crystal Thermometers

 

Gardner et a1. (1967) discussed a microcalorimeter
that replaced the usual thermister with a quartz crystal
thermometer. They indicated increased stability and
greater accuracy were obtained with the quartz sensor.
This thermometer is based upon a principal which is rela-
tively new to precision thermometry. It utilizes the
temperature dependence of the resonant vibrational fre-
quency of a quartz crystal. These crystals have served
for some time as a standard of frequency. However, it is
possible by cutting the crystal in a different way, to
produce a sensor with a resonant frequency that is nearly
a linear function of temperature. A commercial version
with a digital readout is available with a resolution of

0.1 millidegree (Wilhoit, 1967a).

2.4 Temperature Control

 

The problem of temperature control arising in micro-
calorimetry generally belong to one of two types--the
maintenance of constant temperature or the maintenance of
a constant temperature difference. The temperature control
apparatus consists of three essential parts. The thermo-
regulator which senses the temperature or temperature
difference, the relay or amplifier, and the heater. The
relay or amplifier may either be automatic or manual. The

heater can be an electrical resistance element, a cooling

37

coil, or it may be the addition of warm or cold water to
a well stirred bath.

The most convenient means of constant temperature
control is afforded by a mercury regulator in conjunction
with a sensitive electromagnetic relay. Swietoslawski
(1946) discussed some of the important design criteria for
this type of temperature controller in order to achieve an
accuracy of plus or minus one millidegree. In order to
decrease the lag of the controller, the surface area of
the container should be as large as possible in relation-
ship to its volume. To increase the sensitivity of the
regulator the volume should be fairly large, perhaps one
or two liters, and filled with a liquid such as toluene
which has a higher coefficient of expansion than mercury.
Of course, mercury is used in the part of the regulator
at which the circuit is made and broken.

The current carried by the mercury contact should be
as low as possible. Currents of the order of magnitude of
1 milliamp may cause sparking when the relay current is
broken. The result is a fouling of the mercury surface
in the capillary with an accompanying excessive sticking
of the meniscus. Zettlemoyer et al. (1953), used a thyra-
tron relay in connection with a mercury thermoregulator
so that the current at the mercury contact was of the
order microamperes. Very low amplitude vibrations have
also been shown to reduce the sticking of the mercury

meniscus. Zverev and Crestov (1968) substituted visual

38

observation and manual on-off control for the automatic
control and electric current flow. This of course permits
the use of a much smaller capillary as well as eliminating
the arching of the electric current.

A mercury thermoregulator provides only on-off con-
trol. Therefore the bath temperature will necessarily
oscillate about an equilibrium point. To minimize this
oscillation, either the room in which the bath is located
should be maintained at very nearly but slightly below the
temperature of the bath, or a small steady heater should
be provided that will supply most of the required heat
energy. The control heater should be adjusted so that it
is on approximately one-half of the time. The oscillation
of the control temperature can be substantially reduced by
giving very small but rather high frequency vertical
oscillations to the contacting wire. As the wire passes
through the lower part of its oscillation, electric con-
tact will be made. The period of the contact will vary
with the height of the mercury. If the period of the
oscillation of the wire is short as compared with the
regulator action, this has the effect of partially chang-
ing the control to proportional or continuous.

Resistance thermometers are frequently applied in
precise isothermal temperature control. Makrides and
Hackerman (1959) used a thermister in a bridge circuit,
the output of which was amplified by a thyratron circuit

for controlling the heater. For control purposes carefully

39

made temperature sensors are not necessary. Therefore,
homemade sensors such as the copper wire resistance sensor
are frequently constructed (Westrum and Eyring, 1952). The
Wheatstone bridge circuit is usually satisfactory for con-
trol circuits. If a direct current bridge is selected,
the output is usually indicated by a galvanometer which
either controls the heater by an on-off photocell or by
means of a phase-shifting thyratron circuit in conjunction
with a photocell so that proportional control is obtained.
If the bridge input is A.C., the output is usually then
amplified directly by a thyratron circuit as indicated
above.

Although any of the thermometric devices discussed
in the previous section are satisfactory for maintaining
a constant temperature difference between two bodies, the
differential characteristic of thermocouples makes them a
more frequent choice. Thermocouple control is frequently
selected in adiabatic calorimetry. Morrison and Hanlan
(1957) observed the deflection of a sensitive galvanometer
connected to a thermopile and manually maintained adiabatic
conditions by adding hot or cold water to the bath as
necessary. Sturtevant (1937) reported maintaining the
jacket to calorimeter vessel temperature difference to
less than 0.2 millidegrees by manually controlling the
jacket heating current. He also used a sensitive galvano-
meter to observe the potential of a thermel. Sauer (1968)

attempted to provide automatic control of adiabatic

40

conditions with thermocouple sensors. However, the
accuracy of his control was much less than that of the
previously mentioned systems. The compensation type
calorimeter also requires a means of maintaining a minimum
temperature difference. Berghausen (1954) used a D.C.
breaker amplifier with 124 pairs of thermocouple junctions
to maintain a reference vessel to within one microdegree
of the reaction vessel in his microcalorimeter. For very
accurate control work, proportional controls are always
used and sometimes some degree of automatic compensation
for changes in the temperature in the surroundings is also

provided (Sturtevant, 1938).

2.5 Calibration and Compensation

 

Prior to the twentieth century the heat capacity of
a calorimeter was frequently estimated from a knowledge of
masses and the specific heats of the materials used in its
construction. This method is rarely used in any calori-
meters today and never used in microcalorimeters. Experi-
mental calibration is essential for several reasons. First
a calorimeter is not a perfectly isolated body. Even in an
adiabatic calorimeter there are connections between the
jacket and the calorimetric vessel. Part of the heat
required to raise the temperature of these connections
comes from the calorimeter vessel and part of it comes
from the jacket. The proportion that comes from each

source depends on the control system used with the jacket.

41

The only way to determine this proportion is to experi-
mentally calibrate the instrument under conditions as
nearly as possible like those during the measurements.
White (1928) has shown that even constant lags in the
adiabatic control have a negligible effect if the experi-
mental calibration heat exactly duplicates that of the
measured reaction. The magnitude of the heat loss and
heat inputs other than the reaction heat must also be
determined through an experimental calibration.

Most compensation energy used in microcalorimeters
is also some form of electrical energy. The accuracy,
availability, and relatively low cost of instruments for
measuring electrical energy make them very desirable for
calibration and compensation.

The most frequent application of electrical energy
in calibration or compensation is in the form of Joule
heating. For this purpose resistive heating elements are
built into most calorimeter vessels. Heating elements are
usually homemade from any one of several materials. Small
gage manganin wire of sufficient length to make 25 to 250
ohms is frequently noninductively wound onto a Lucite or
mica frame. The entire assembly is then insulated by
applying several thin layers of Glyptal varnish or Parylene
C. Manganin is frequently selected because of its constant
resistive properties. Gucker §£_§l, (1939) found that the
resistance of a manganin heater changed less than 0.01 per

cent for a current change from 0.005 to 0.015 amperes.

42

However, other materials such as constantan are also used
in the heater elements. The properties of constantan are
nearly constant with changes in current and temperature
and it has the additional advantage of being more readily
available than manganin. Belousov and Ponner (1968) re-
ported using a constantan heater that was mounted in
grooves into the calorimeter vessel. Nichrome heaters
have also been reported (Stubblefield, 1965).

Several different electrical circuits have been used
with these resistive heaters. Until the last year or two
most of the heater circuits have either been a parallel or
series combination of the heater element and a standard
resistor. Four leads were provided both from the heater
element and the standard resistor so that the potential
drops across these two resistive elements could be com-
pared with a potentiometer. A more accurate method of
determining the energy dissipated in the series type heater
circuit has been proposed by Vorob'ev and Klyuchnikov
(1967).

More recently Firth and VanMal (1968) suggested that
heater circuits should be compared on their ability to
meet four conditions. First, the power dissipated in the
heater element should not be altered significantly by
variations in the element's resistance. Second, it should
be possible to vary the energy dissipation from the heater
over a wide range without violating the first condition.

Third, it should be possible to substitute other heater

43

elements in the circuit without altering the character-
istics of the circuit with respect to the first two require-
ments. Fourth, because mercury cells and other batteries
are frequently used as energy sources, the current drain
should be minimized. Firth and VanMal went on to show that
a combination series and parallel network would be much
more satisfactory than the more traditional series or
parallel networks. Picker (1968) suggested employing
Zener diodes as heating elements in a series circuit rather
than using more elaborate circuitry with the traditional
resistive elements. Since the potential drop across the
diode is constant, the current is directly proportional to
the energy dissipated in the diode. Therefore, the sta-
bility of the power supply is not as critical as with other
resistive elements in similar circuits. For compensation
purposes it is sometimes desirable to remove heat energy
rather than adding it. Calvet and Prat (1963) have re—
ported using Peltier cooling with apparently good success.
Other authors have questioned the accuracy of this method.
Non-electrical means of calibration have also re-
ceived some attention in the last few years. The heating
value of some chemical reactions is well established.
Vanderzee and Swanson (1963) and Hale et a1. (1963) calcu-
lated independently the heat of neutralization of HCl with
NaOH. Their values and the value published by Parker of
the National Bureau of Standards in 1965 are considered

to be accurate within one part in 500. One advantage of

44

using a chemical reaction of this type is that the reaction
proceeds (and the heat is liberated) instantaneously. A
disadvantage of this method is that although it may cali-
brate the equipment, it does not calibrate the equipment
with the experimental reaction materials in it. Therefore,
an electrical calibration is still necessary. Calvet and
Prat (1963) reported calibrating a microcalorimeter with
radioactive materials. Radioactive materials are a very
stable heating source. However, the walls of the calori-
meter vessel must be sufficiently thick to absorb the
radiation from the materials. Also it is a steady heating
which can be easily supplied by the electric calibration

heater.

2.6 The Sample Holder

 

Almost all reports of heat of immersion or heat of
adsorption studies indicate the sample of material to be
immersed is sealed in a glass or metallic bulb. The bulb
is broken under the surface of the liquid in the reaction
vessel to initiate the reaction. A bulb serves two func-
tions: first, as a container for the sample to separate
it from the solution used in the immersion study and
secondly, as a container which can be evacuated or out-
gassed prior to the experiment. Typical outgassing con-
ditions for heat of immersion samples of constant thermal
conditions between 110 and 400 degrees until a residual

pressure of one namo-meter of mercury was attained have

45

been reported by Whalen (1961, 1962). This treatment has
been demonstrated to be necessary in order to yield re-
producible heat of immersion values. Unfortunately the
bulb is the most important source of uncertainty in immer-
sional calorimetry. When the bulb is broken, a small
amount of heat is given off. Millard et_al. (1955) re-
ported that the heat of breaking a ten milliliter glass
sample bulb is 0.15 plus or minus 0.02 calories. Other
reports indicated higher or lower values; however, fre-
quently very erratic results have been reported.

Guderjahn §t_al. (1958) were the first to systemati-
cally study the heat of bulb breaking. They evacuated and
sealed three sets of bulbs. The first series of bulbs
were small five cubic centimeter glass containers blown
in their own lab. The volume was varied by partially
filling some of the bulbs with glass beads or water. The
results of the tests on these bulbs were somewhat erratic.
The authors credited this to varying degrees of residual
strain.

A second series of metal bulbs of various volumes
having openings at the ends covered with thin brassfoil
were tested. The foil withstood the vacuum but could
easily be penetrated by a plunger to admit water with a
very small expenditure of mechanical energy. The heat
evolved when a bulb was broken was approximately equal to

the product of the atmospheric pressure and the volume of

46

the empty bulb. A rather small amount of scatter was ob-
served with the metal bulbs.

The third series consisted of glass bulbs commercially
produced with a volume of about ten cubic centimeters and
then annealed. The heat of breaking these bulbs exhibited
slightly more scatter than did the heat for the metal bulbs;
however, all of the points fell close to the PV energy line.
The authors concluded that the major contribution to the
heat of breaking the bulbs is the work done by the pressure
of the gas above the liquid. Heat is released in the turbu-
lent flow as the liquid rushes into the bulb.

Since a tightly sealed reaction vessel was not used,
some evaporation may have occurred from the surface of the
water in their reaction vessel to raise the humidity of the
air that entered from the ambient to replace the volume of
the bulb. This could amount to as much as 0.1 calories for
a 10 ml bulb. However, the air in the cabinet surrounding
the calorimeter vessel was about five degrees below the
temperature of the ambient. Thus, if the ambient humidity
was 50 per cent or greater, the air that entered the re-
action cell was probably nearly saturated. Guderjahn §E_al.
concluded that under the conditions of the experiment,
adsorption of heat due to evaporation of water was not
significant. If these authors are correct in attributing
the heat of bulb breaking to the PV energy, then it is not
correct to adjust the heat of immersion data by the energy

adsorbed when breaking an empty bulb.

47

Young and Bursh (1960) found that they could remove
some of the erratic behavior of the bulb breaking by using
a spring loaded breaking-rod coupled with a solenoid. They
attempted to determine the heat of bulb breaking by varying
the amount of sample material put into similar bulbs. The
apparent change in heat of immersion of these samples was
attributed to the heat of bulb breaking. They fitted a
curve to the data and extrapolated it to zero sample weight
and called the intercept the heat of breaking. The range
of their data was not adequate and the scatter was too

large to provide a further check on the PV work theory.

2.7 Stirring

 

Most microcalorimeters include some means of stirring
or mixing the contents of the reaction vessel. In general,
the temperature of a body can be observed only at one or at
best at a finite number of points. Therefore, not only the
jacket, but also the calorimeter vessel should be at uni-
form and measurable temperatures if the heat losses are to
be accurately evaluated and corrected. This temperature
uniformity can be obtained in two ways--by taking advantage
of the high thermal conductivity of some metals or by
thorough stirring of the liquid.

The second reason stirrers are frequently employed
is to mix the reactants. If a powder is being mixed with
a liquid, an upward or downward stirring motion is selected,

depending on whether the powder is heavier or lighter than

48

the liquid. In most cases the stirring of a liquid is
accomplished by a propeller. Greater efficiency is
achieved by mounting the propeller in a tube through which
the liquid streams. The propeller type stirrer may serve
other special purposes. Hutchinson (1947) blew sample
holder glass bulbs on the ends of glass tubes and then
slipped the stirrer propeller over the tube down to the
bulb. In the vessel the sample holder was rotated to turn
the propeller. Bartell and Suggitt (1954) used their
stirring rods also as the breaking rods for their glass
sample bulbs. Sturtevant (1937) used a reciprocating
manually operated stirrer for mixing the reactants in his
microcalorimeter. The stireer consisted of five thin
platinum discs on a platinum wire. The stireer was oper-
ated for 30-45 seconds at the outset of each experiment to
mix the reactants. Berger and Bowman (1964) reported good
mixing action by using a single stroke of a ball mixer.
Several techniques have been devised to avoid the
direct connection between the calorimeter and its surround-
ings by a stirring shaft and to permit tight sealing of
the calorimeter. Callot and Fouvry (1966) used a magnetic
arrangement to bring the reactants together efficiently.
Winterhalter and VanNess (1966) used continuous magnetic
stirring to mix their reactants and to maintain a uniform
temperature. Evans _e_t__ai_l. (1968) and others have rotated,
tipped, or swung their reaction vessels to cause mixing

of the reactants and in some cases limited temperature

49

leveling. Several authors have reported using no stirring
except for the action caused by breaking the sample bulb.
Belousov and Ponner (1968) placed their reaction vessel in
a bath of mercury. The heat conduction of the mercury was
considered to be high enough to cause temperature uni-
formity at the surface of its container. Calvet and others
have used long thin calorimeter vessels so that rapid
temperature equalization is obtained even in the absence
of stirring. The heat loss to the surroundings is in-
creased because of the large surface to volume ratio.
Calvet distributed a large number of thermocouples over
the surface of the calorimeter in order to more accurately
evaluate this heat loss. These thermocouples further in-
crease the rate of heat loss to the surroundings and thus
the uniformity of the temperature within the reaction
vessel.

The efficiency of the stirring is important. If the
reaction kinetics are being studied, the temperature uni-
formity and mixing of the reactants must be accomplished
as rapidly as possible. This requires that all materials
beyond the immediate range of the stirring, or other
temperature-equalization mechanisms, should have as low
a heat capacity and as high a thermal conductivity as is
possible. In other words, the calorimeter vessel should
probably be metallic and its supports should be as light

as possible. Berger and Stoddart (1965) determined the

50

mixing efficiency of their microcalorimeter with an in-
stantaneous reaction, the neutralization of HCl with NaOH.

If a stirrer is used, the measured heat of the re-
action must be corrected for the heat of stirring. Several
authors indicated difficulty in obtaining an accurate cor-
rection factor for the energy due to the stirring process.
Boyd and Harkins (1942) found that introducing more than
10 grams of powder in 650 cc of liquid caused a viscosity
change that significantly changed the heating rate of the
stirrer. Millard gt_gl. (1955) reported that it was neces-
sary to use a voltage controller on the stirring motor in
order to obtain reproducible heats of stirring. Evans and
Richards (1952) found that even matched stirrers driven
at the same rate of speed in identical calorimeter vessels
caused a drift in temperature between the vessels of ten
to twenty-five microdegrees per minute.

Similar considerations to those above also apply in
the design of the jacket or sink of most microcalorimeters.
Either a well stirred bath or a heavy metallic shield is
employed. In a well stirred bath a much higher rate of
stirring is employed than in the reaction vessel since the
heat of stirring is of little significance. The size and
shape of the metallic solid shield or jacket is the subject
of some debate among calorimeterists. However, no studies

on this subject have been reported.

51

2.8 Sensitivities

 

The heat of immersion of dry powders in liquids
varies with the surface area available for adsorption and
the energy level of the sites available for adsorption.
Energy levels from 0.1 to 27.0 calories per gram have been
reported for the heat of immersion of various dried powders.
The energy available per site decreases with increasing
rehydration. Therefore, greater sensitivity is required
for studying the differential heats of immersion than for
determining the heat of immersion of a dry powder.

The sensitivities that have been reported for the
various microcalorimeters vary considerably. The volume of
the reaction vessels also varies over a wide range. How-
ever, most of the basic designs have been reported to have
sensitivities approaching one millicalory per 100 m1 of
solution. The sensitivities of the calorimeters have been
determined by either statistically analyzing the data
gathered from them or by determining the minimum amount of
calibration energy necessary to give a deflection on the
indicating equipment. This provides no check against
systematic or methodical errors. Very few authors have
reported any analysis of design or any computations of
expected sensitivities based on their design. A few
calorimeterists have attempted to optimize the wire
diameter and the number of thermocouples in thermopiles
in order to minimize heat loss while maximizing the

potential output of the thermel. A few authors have also

52

looked at the degree of similarity and symmetry required
when using twin calorimeter vessels. Berger et a1 (1968)
used a computer model to evaluate the conductive heat loss
from their reaction vessel. They reported using this com-
puter program to assist in the design of the instrument.

Most of the calorimeters discussed provide an easy
method for determining the integral heats of the reaction.
However most calorimetric studies today are also interested
in the reaction kinetics, or in otherwords the heat pro-
duction as a function of time. Several schemes have been
devised for reconstructing the heat input as a function of
time for the various types of calorimeters. However for
the most part these schemes have either been insensitive
to rapid changes in the heat input or over sensitive to

noise in the measured values.

2.9 Some Sources of Error

 

Errors in microcalorimetry can arise from many
sources. No attempt will be made here to list all possible
sources of error. However, it does seem worthwhile men-
tioning some of the more important sources that have been
reported by other authors. Some of these have already been
mentioned earlier in this paper.

McGlashan (1964) lists four errors and means of pre-
venting them. The heat of evaporation is large compared
with the heat of immersion. Therefore, this potential

source of error must be eliminated. One way to prevent

53

evaporation is to eliminate any contact between the liquid
and any vapor spaces in or around the calorimeter. As was
mentioned earlier, the mixing of the reactants must be com-
plete. The complete process that takes place within the
reaction vessel must be known. Particular care must be
exercised to either prevent or account for any changes in
pressure or volume. The fourth source of error indicated
by this author is contamination of the reactants. The sur-
face of the powder to be immersed must not be contaminated
with adsorbed molecules. In otherwords once the sample

has been prepared, it should not be brought into contact
with air or particularly damp air.

Vendeborgh and Spall (1968) suggested that particular
care be exercised with respect to the self heating of
resistance thermometers. It may not be sufficient to
merely maintain the energy of the self heating at a negli-
gible level with respect to the heat of the reaction being
studied. The self heating of the temperature sensor causes
its temperature to be slightly above the temperature it is
to measure. If the current flow through the resistive
temperature sensor changes due to an unbalance of the
measuring circuit, the temperature difference between the
sensor and the temperature to be measured will change.

This change may be significant even though the energy
given off by the self heating is negligible.

White (1928) cautions against two additional sources

of error. Convective heat losses are not necessarily

54

directly proportional to temperature differences. In fact
for small temperature differences the convective heat loss
is not even a smooth function. Therefore unless the heat
loss is through a vacuum, the magnitude of this loss must
be determined when the temperature is similar to that that
will occur during the actual experiment. White also cau-
tions the calorimeterist about the lag of temperature
sensors. Modern sensors have greatly reduced this lag;
however, in some installations it is still important.

Calvet and Prat (1963) listed some other causes of
errors. If heat loss is being evaluated between two sur-
faces, either the surfaces must both be at uniform temper-
atures or else thermocouples or other temperature measuring
devices must be distributed over the surfaces properly. If
the distribution of the thermocouples on the surfaces is
irregular, the emf in the detector will not be proportional
to the heat flow between the surfaces. Parasitic thermo-
electric emf's are also a source of error. These can be
suppressed by having all electrical circuits and equipment
enclosed within the calorimeter jacket. The calorimeter
should be isolated from any sources of vibration. The
changes in mechanical strain caused by vibration or shock
produce small quantities of heat. The ambient humidity
must be low. Numerous thermal and electrical effects are
caused by a humid atmosphere.

Numerous authors have indicated additional sources

of errors. If comparison or compensation measurements are

55

being used, the similarity of the two vessels is extremely
important. Evans and Richards (1952) found that seemingly
identical stirrers produced slightly different heats of
stirring when driven at the same rate of speed. Picker
(1968) found that changes in the dimensions of his twin
chambers as small as three micrometers or differences in

the mass of the vessels as low as one milligram caused
measurable errors. Several authors have found that it is
difficult to provide true adiabatic conditions or to pro-
vide accurate compensation. Howard and Culbertson (1950)
found it necessary to use some corrections in their
adiabatic calorimeter because their temperature tended to
drift. Ostrovskii g£_al. (1968) found it was experimentally
impossible to reduce the error due to imperfect compensation

in their reference vessel to below 3 per cent.

III . OBJECTIVES

As a result of the review of literature in micro-
calorimetry, the following three objectives were selected
for the research:

1. The sensitivity coefficients and the amount of
information that can be obtained from micro-
calorimetry measurements.

2. A numerical technique to study the effects of
several parameters associated with metallic
sinks on the relative temperature measurement
error.

3. A new method for determining the rate of heat
production in a reaction as a function of time

from the temperature data.

56

IV. SENSITIVITY COEFFICIENTS

The sensitivity coefficients for experiments designed
to measure parameters or coefficients should be studied
prior to the construction of equipment. Beck (1964) indi-
cated some of the information that can be obtained from the
sensitivity coefficients. If the effect of the measured
characteristic with respect to a parameter of interest is
zero, this parameter cannot be determined regardless of
the accuracy of the measurement. If more than one param-
eter is to be determined simultaneously, the sensitivity
coefficients with respect to these parameters must be
linearly independent.

Unlike in previous sensitivity studies, one of the
parameters to be determined in a microcalorimeter may be
an unconstrained variable. In some instances the heat of
the reaction being studied may be known to follow a
particular functional form from the reaction kinetics.

Then the problem of determining the reaction heat reduces
to one of establishing the constant parameters in the
function. This case is no different from other sensi-

tivity studies. However, frequently the functional form

57

58

of the reaction heat is unknown or in other words the re-
action heat is an unconstrained variable. The experiment
must then supply sufficient information to determine a
constant reaction heat value for each time increment, At.

The value of an unconstrained variable can be deter-
mined as a function of the independent variables if the
sensitivity coefficient is non-zero. Furthermore, if the
sensitivity coefficient is not zero, the measured variable
can be varied indefinitely by varying the variable of
interest. Therefore, no more information is available
from the measured variable than the values of the uncon—
strained variable if the sensitivity coefficient is not
zero. In the microcalorimeter, if the sensitivity of
temperature with respect to reaction heat is non-zero at
time, t, a unique value for the reaction heat may be
determined during the preceding time increment. But no
information on any other parameter, regardless of the
value of its sensitivity coefficient is available from the
temperature measurement simultaneously. Conversely, if the
sensitivity of temperature with respect to reaction heat
is zero at time t, the value of the reaction heat during
the previous time increment cannot be determined, but
information is available for determining other parameters
with non-zero independent sensitivity coefficients.

The required sensitivity of the measurement instru-
ments can be predicted from the sensitivity coefficients.

After the experimental work has been started, the

59

sensitivity coefficients should be compared with a sta-
tistical accuracy. If the statistical accuracy does not
compare closely with the accuracy predicted from the
sensitivity coefficients, the system should be checked for
systematic errors. The sensitivity coefficients also pro-
vide a means of determining the optimal experimental

technique.

4.1 Background

 

There are two methods for determining the sensitivity
coefficients from a mathematical model of the experimental
system. The sensitivity coefficients should be determined
analytically when this is possible. Generally an analytic
solution provides a better understanding of the system. If
it is not possible to determine the sensitivity coefficients
analytically, numerical techniques must be employed. Beck
(1966) suggested a method for determining the sensitivity
of a measured variable, T, with respect to a constant
parameter, c:

T =

c 3c cs

3T = T[c(l+€)] - T(c) (1)
Co

where e is a small number.

If the sensitivity of the measured variable is to be
determined with respect to a second variable, y, the
method must be changed to the more traditional derivative

formula:

60

 

.._. .33: = Tme) - Hy)
Ty ay/yo e (2)

Both analytical and numerical techniques have been
employed to determine the sensitivity coefficients in this
study. In order to determine the sensitivity coefficients
numerically, the appropriate differential equation was
solved using the modified Runge-Kutta technique presented
by Gill (1951). Equation (1) or (2) was used depending on
whether the parameter was a constant or another function.
The sensitivity coefficients were nondimensionalized by
multiplying them by either the value of the constant
divided by the maximum temperature or the maximum value
of the parameter divided by the temperature maximum if the
parameter was a function. In order to more easily detect
linear dependence, unitized sensitivity coefficients--the
sensitivity coefficients divided by the maximum value which
it assumes through the range of interest--were graphed on
the Calcomp plotter.

The calorimeter to be studied can be described by

the following equation:
C E? = qw(t'T) + qr(t) + qb(t) + qe(t)
- H[T(t) - TS(t)] (3)

where

C is the heat capacity of the reaction vessel and
its contents.

61

T is the temperature of the reaction vessel.

t is the time.

is the energy input to the reaction vessel from
resistance thermometric devices and/or stirring.
It is called the heat input due to work.

q is the heat of reaction.

qb is the heat generated by breaking the sample
container.

is the calibration heat input.
H is the overall heat transfer coefficient. It
includes radiation, conduction and convection

from the total surface area of the reaction vessel.

T is the temperature of the cavity surrounding the
reaction vessel.

The values of each of the variables in (3) were estimated
from values given in the literature and from basic heat
transfer considerations. The heat capacity of the reaction
vessel and its contents, C, was considered to be constant.
The numerical value of the heat capacity was assumed to be
15 calories per degree C if no heat input due to work was
included. If stirring or resistance thermometers are em-
ployed in the calorimeter vessel, the physical size of the
vessel must be increased; therefore, the heat capacity was
increased to 200 calories per degree C. The overall heat
transfer coefficient, H, was assumed to have a numerical
value of 0.05 calories per degree C-second, if a heat in-
put due to work was included. If energy due to work was
neglected, the value for H was 0.01 calories per degree

C-second.

62

The heat input due to work, qw, is a function of
temperature because the self heating of resistance ther-
mometers depends on the inbalance of the bridge or potentio-
meter. It is also a function of time because the viscosity
of the solution may vary or the speed of the stirrer may
change slightly. Either of these variations will cause a
change in the mechanical input of energy. Because these
variations can be limited to small values and the functional
form is a function of the equipment and the experimental
conditions, the mechanical heat input was assumed to be a
constant. If this term was included in the equation, its
numerical value was set at 0.01 calories per second.

It was assumed throughout this study that the calori-
meter being considered should be capable of measuring the
reaction heat, qr, as a function of time. Since all micro-
calorimeters remain at essentially the same temperature
throughout the experiment (even adiabatic ones), any
dependence of the heat of reaction on temperature should
be negligible. Several functional forms were assumed for
the reaction heat during this study. The total amount of
heat evolved due to the reaction was also varied.

The heat caused by breaking the sample bulb, qb, was
assumed negligible throughout all of the calculations. It
can be shown that the heat of reaction and the heat due to
bulb breaking cannot be simultaneously determined as a
function of time. Therefore the heat due to sample bulb

breaking must be determined in a separate experiment in

63

which this term will be the heat of reaction. It was shown
that neglecting this term does not change the remaining
sensitivity coefficients significantly.

Several heating rates and functional forms were tried
for the calibration heat, qe. The sensitivity coefficients
are expected to be functions of the calibration heat.
Therefore, it is anticipated that the amount of information
that can be obtained about the parameters also depends on
the form and magnitude of the calibration heat. The over-
all heat transfer coefficient is a function of temperature;
therefore the integral over time of the calibration heat
must be similar to the anticipated integral heat of re-
action.

The functional form was restricted to those that can
be accurately generated without elaborate equipment. Con-
stant value functions, sine waves, and square waves were
all tried because they are easily generated electrically.
Ramp (linear increasing or decreasing) functions may be
slightly more difficult to generate electrically, but they
were employed in this study also. If the sensitivity co-
efficients are known for the ramp and constant calibration
heat functions, it should be possible to predict the
approximate shape of the coefficients for any heating
function. Impulse calibration heating functions were
also simulated because these can be accurately generated

by a chemical reaction.

64

The temperature of the surroundings, Ts' is usually
independent of the other variables. Therefore if it does
not remain constant or in a constant relationship to the
temperature of the reaction vessel, it must be measured.
The effect of small fluctuations in the ambient tempera-
ture on the sensitivity coefficients was tested numerically
and found to be negligible. Therefore, for most of the
calculations T8 was considered to be a constant, the zero

of the temperature scale.

4.2 Results

 

4.2.1 Calibration of Instrument
Parameters with Only a Known
Heat Input

 

 

 

The experiment, which represents the calibration of

the Calvet microcalorimeter, can be mathematically simu-

 

lated by
dT _
d—t ‘ clqe + °2T ‘4)
where
1 °c 1
cl = C [cal] and c2 = -H/C [sec 1

Although there are several possible initial conditions,
only two are very likely. If a constant heat has been
applied for a sufficient length of time so that the heat

loss matches the heat input, the initial temperature is

65

Ti = -clqe/c2. For this initial condition the dimension-

less sensitivity coefficients with respect to c1 and c2 are
positive and negative unity, respectively. Therefore, only
one of the constants can be determined from this experiment.

If the no heat equilibrium condition, Ti = 0, is

selected, the temperature of the reaction vessel is given

c
T = EL q (l - ec2t) (5)

2 e
From this equation the maximum temperature, Tm’ and the
dimensionless sensitivity coefficients for a constant

calibration heat input can be found directly:

C1
Tm = — E_ qe (6)
2
c
—l T = l - ec2t (7)
T C
m l
c
—g T - (1 - c t)ec2t - 1 (8)
T c 2
m 2

The sensitivities have also been numerically computed and
plotted in Figure 1. These two sensitivity coefficients
are clearly linearly independent even for short calibration
experiments; therefore, it is possible to determine c1 and

c simultaneously in one calibration experiment.

2

h‘z I.\U. {I § JIA BAH 5: h Ilil~9tumuz in,“ A! liiAl‘ h IZ‘H

 

UNITIZED SENSIBILITY COEFFICIENT

66

 

 

LO“- ,
/
_ /
/
.ai- /
/
/
1..
/
TC: /
.6 - /
/
_ /
/ T°2
/
.4._ /
/
_ /
/
/
/
2 ._ /
/
./
./
.L. /
/
/
//

c), ’“’1 g g; 4.1 1 1 1 !

O 25 50 75 I00 l25 I50 I75 200

TIME (SECONDS)

Figure l. The sensitivities of temperature with respect
to the constants c1 and oz for the initial condition of Ti = 0
in a microcalorimeter without any heat input due to work. A
constant calibration heat of 0.016 cal./sec. was added from
t = 0 to t = 100 sec. The maximum values of the dimensionless
sensitivity coefficients were 1.0 and 0.069 for cl and c2,
respectively. The maximum temperature was 0.129° C.

67

Sensitivities for non-constant forms of the cali-
bration heat input were tested numerically. For heating
functions that increase with time, the sensitivities tend
to become more nearly linearly dependent, Figure 2. De-
creasing heat functions emphasize the non-dependence and

increase the sensitivity with respect to c Figure 3.

2!

4.2.2 Calibration of Instrument
Parameters with Heat from
Calibration and Work

 

 

 

This instrument is mathematically modeled by

dT _
dt’— C1(qw + qe) + CZT (9)
If the initial condition Ti = 0 is selected, the tempera-
ture of the reaction vessel is given by
c
_ _ _l _ c t

T - c2 (q8 + qw)(l e 2 ) (10)
and the sensitivity coefficients with respect to c1 and qw
are clearly linearly dependent. The sensitivity coef-
ficient with respect to c2 and either c1 or qw can be
determined simultaneously from this experiment.

A second initial condition that could be used with
this instrument is that of an initial temperature such that
the heat loss was matched by the heat input from the work
energy and the calibration heater. However, this leads to

a reaction vessel temperature and to sensitivity coef-

ficients that are all constant with respect to time. This

I z .Ifl'. .. . .PIVWV i)! I I IOIIIQI..vZII'AI ‘. ~\“ h i)“'

 

68

O
4
I

in
f

UNITIZED SENSIBILIT Y COEFFICIENT

ID
I

 

 

1 1 I 41 .L
00 25 so 7'5 IOO 1'25 :30 I75 200
TIME (SECONDS)

Figure 2. The sensitivities of temperature with respect
to the constants c1 and c2 for the initial condition of Ti = 0
in a microcalorimeter without any heat input due to work. The
calibration heat function was an increasing ramp from 0 at
t = 0 to 0.04 cal./sec. at t = 200 sec. The maximum value of
the dimensionless sensitivity coefficients were 1.0 and 0.041
for Cl and c2,respectively. The maximum temperature was
.228° C.

lb z‘,-I1VI:I :h 4‘30 *1 1- ..HIIHIANZIJ‘A” A! It‘lr‘ § 5 \Z‘ib

 

 

LG
I— -3‘
z
I!

9
u.
u.
‘8
c;.6‘
).
I:
='
9
“z’
m .4‘
a)
o
m
E
t
g .2.
o ’1 I i i I 1 i I
o 25 50 75 IOO :25 1‘50 I75 200

TIME (SECONDS)

Figure 3. The sensitivities of temperature with respect
to the constants c1 and oz for the initial condition of Ti = 0
in a microcalorimeter without any heat input due to work. The
calibration heat function was a decreasing ramp from 0.04
cal./sec. at t = 0 to 0 at t = 200 sec. The maximum value of
the dimensionless sensitivity coefficients were 1.0 and 0.081
for c1 and c2, respectively. The maximum temperature was
0.245° C.

70

initial condition would only provide information for one of
the unknowns.

A third initial condition is an initial temperature
such that the heat loss is matched by the heat input due
to work only. The resulting dimensionless sensitivity co-

efficients are:

l l c t
—— T = + ——————— [q + q (l - e 2 )] (ll)
Tm cl qe + qw w e
C q
2 e c t
a—— T = (1 - c t)e 2 - l (12
Tm c2 qe + qw 2
q q
T! T = _____:’ (13)
m qw qw qe

The linear independence of these coefficients can be seen
in Figure 4. It is therefore possible to simultaneously
determine all three unknown constants when a constant cali-

bration heat is added and the initial temperature is

The sensitivity coefficients for this experiment
were also studied numerically. Other functional forms of
the calibration heat, qe, produce sensitivity curves
similar to those in Figure 4. The important variable in
the calibration experiment is time. For the first three
minutes of the experiment the sensitivity coefficient with

respect to c varies less than 1 per cent from a constant

2

value. Only in calibration experiments running ten to

HEAT PRODUCTION (CALORIES/SECOND)

71

 

0.61-

1 I l l l l

l L f . . -
460 600 ' 800 I000 200
TIME (SECONDS)

 

1;, 1
0.5 O ZOO

Figure 4. The sensitivity coefficients of tempera-
ture with reSpect to cl, c2, and qw for the initial con-
dition of Ti = -clqw/c2 in a microcalorimeter with heat
input due to work. A constant calibration heat of 0.016
cal./sec. was added from t = 0 to t = 1,000 sec. The
maximum value of the dimensionless sensitivity coefficients
was 1.0, 0.74, and 0.69 for c1, c2, and qw, respectively.
The maximum temperature was .288° C.

72

fifteen minutes can qw and c be determined simultaneously.

2

Varying the calibration heat has the greatest impact on the

sensitivity with respect to 01'

4.2.3 Sensitivity of Temperature
to Reaction Heat

 

 

The purpose of microcalorimetry studies is to deter-
mine the reaction heat. For this study the reaction heat
was considered to be an unconstrained variable; therefore
it is the only parameter than can be determined during the
reaction if its sensitivity coefficient is not zero. The
dimensional sensitivity coefficient of temperature with
respect to reaction heat, given by equation (14), is de-
rived directly from either equation (4) or equation (9)
with any initial condition that is independent of the re-
action heat. Thus equation (14) is correct either with or

without heat due to work or bulb breaking:

T = - — (1 - eczt) (14)

Three important conclusions can be drawn from the
general character of the sensitivity with respect to the
heat of reaction. First, the sensitivity is non-zero for
all time greater than zero. Therefore, a value for the
heat of reaction can be determined uniquely for each time
increment during the reaction. Thus, after the reaction
is initiated, no other information on any other parameters

is available from the temperature measurements. Second,

73

several authors have reported difficulty achieving accurate
adiabatic control or accurately providing compensation heat
to the reference vessel. Since the sensitivity coefficient
is zero at the start of the heating cycle, it is obvious
that to maintain the equality of the two temperatures is
not only experimentally difficult but theoretically im-
possible. For in effect every time increment begins a new
reaction of an unknown heating rate, and until some time
has elapsed the sensitivity of the temperature to the re-
action heat is zero. Third, stirring and other inputs of
heat due to work may decrease the accuracy of the micro-
calorimeter, but it does not influence the sensitivity of
the temperature measurements to the reaction heat.

The necessary sensitivity of the temperature measure-
ment apparatus for a microcalorimeter may also be deter-
mined from equation (14). If a constant rate of heat pro-
duction is to be determined during a time increment, At,
with an accuracy of : Aqr, the temperature measurement
equipment must be sensitive to temperature changes, AT,

at least as small as:

AT=-—r-(l-e2) (15)

For example, to determine the reaction heat to within one
millicalory for each one second interval in a calorimeter
with a heat capacity of 15 calories per degree C and an

overall heat transfer coefficient of 0.01 calories per

74

degree C-second, it would be necessary to measure the

temperature to within 6.3 x 10-5

° C. This assumes that
there are no other errors to reduce the accuracy. In
general the parameters of the calorimeter and the time

increment will not be known exactly and may impose a more

stringent requirement than is given by equation (15).

V. THE EFFECTIVENESS OF METALLIC SINKS

Several techniques have been used for providing uni-
form ambient conditions around the reaction vessel in
microcalorimeters. When twin vessels are employed the
uniformity requirement becomes more stringent than when a
single vessel is employed. One of the principle advantages
of twin vessels is that the temperature of the sink may be
permitted to vary with time without introducing an error.
Well stirred baths of water or other liquids have fre-
quently been utilized to maintain the temperature uniform-
ity condition. A number of the microcalorimeters reported
in the last decade have substituted a large metallic block
for the well stirred bath. A simpler design and a more
stable ambient thermal condition are achieved by substi-
tuting the solid metallic sink for the well stirred bath.

Various designs of the metallic sinks have been re-
ported. Berger (1968) reported using a cylindrical alumi-
num block. One oval shaped cavity contained both the
reaction and comparison vessels. The aluminum block was
set in styrofoam insulation. When operated in a closed

room this sink was considered to provide a uniform thermal

75

76

boundary for the two vessels. In contrast to this
cylindrical block, the sinks reported by Calvet and Prat
(1963) are much more complicated. Although Calvet also
uses a cylindrical metallic heat sink, his calorimeter
vessel is placed in an individual cavity cut into the
block. The block is made of aluminum or sometimes silver.
The block sets between the bases of two truncated solid
metallic cones centrally placed within an outer thick
metal cylinder. This cylinder is in turn enclosed in a
series of concentric thermally controlled metal canisters.
The temperature measurement technique is also a distin-
guishing characteristic between the Berger and Calvet
microcalorimeters. Calvet measured the potential differ-
ence between two banks of thermocouples. One bank measured
the temperature difference between the reaction vessel and
its cavity in the sink and the second bank measured the
temperature difference between the comparison cell and its
cavity. Berger employed a single thermopile to directly
measure the temperature difference between his twin vessels.
The thermal sensitivities of these two instruments are
similar in spite of the large variations in design.

It is very difficult to evaluate the effectiveness
of the metallic sinks experimentally. Microcalorimeters
require temperature measurement equipment with the highest
sensitivities available for measuring the temperature
difference between the reaction and comparison vessel.

However, this temperature difference is much larger than

77

the temperature variation in a good sink. Some indication
of the effectiveness of the metallic sink can be obtained
by increasing the reaction heat to a level that causes
measurable thermal variation within the sink. However, it
is not easy to relate the latter measurements to the
effectiveness of the sink for reactions with small heat
outputs. Both radiative and convective heat transfer
characteristics are functions of the temperature differ-
ences, and therefore the sink's ability to maintain a
uniform temperature with reactions of low heating value

is not the same as its effectiveness with reactions of
higher heating value. Furthermore because of the exact-
ing requirements or similarity between the reaction and
comparison vessels and their cavities, the cost of testing
several physical models is extremely high.

This study attempted to model a cross section of a
cylindrical metal sink. Individual cylindrical cavities
for the reaction and comparison cells were included. The
effectiveness of the sink was determined by computing the
relative error in the temperature measurements. That is,
the temperature measurements from the simulated real system
were compared with the measurements from a reaction vessel

enclosed in an absolutely constant environment.

Q

78

5.1 The Computer Model

 

5.1.1 Selection of a
Numerical Method

 

 

A number of numerical methods have been suggested for
the solution of the transient heat conduction equation.
These methods are classified into two broad categories, as
explicit or implicit methods. The forward difference
technique is the most commonly used explicit method. The
second order spacial derivatives utilize the temperature
values at the present time. The first order time deriva-
tives are evaluated from the present and future tempera-
tures. The stability of the forward difference technique
is dependent on the size of the time and space increments.
This technique could not be employed in the model of the
heat sink because the space increments must be varied to
account for the unusual configuration. This would cause
instability for any reasonable time step.

The implicit method suggested by Crank and Nicolson
(1947) is stable for all possible time and space incre-
ments. However, this method requires the solution of a
large number of simultaneous, linear, algebraic equations
at each time step. This means that the computer time re-
quired for this technique is very large. Furthermore it
would be difficult to generalize this technique to three
dimensions. This particular study was only concerned with

the two dimensional model. However, it was desirable to

79

use techniques that could be easily generalized to three
dimensions at some future time.

The backward difference implicit method contains
fewer terms to be evaluated than the Crank-Nicolson method.
This method still requires a large amount of computer time,
particularly for large grid systems or in three dimensions.

A third implicit method known as the "alternating-
direction-implicit-procedure," ADIP, has been suggested
by Peaceman and Rachford (1955). The amount of calcu-
lations required by this method per time step is much
less than other implicit methods, but the authors indi-
cated that the necessary calculations are somewhat larger
than that required by explicit methods.

In 1957 Saul'ev suggested a new explicit technique.
Allada and Quon (1966) compared an extension of the tech-
nique suggested by Saul'ev and some of the other numerical
methods. Their method, the alternating-direction-explicit-
procedure, or ADEP, was much faster than any of the implicit
techniques tested. They concluded that the ADEP appears to
combine the computational ease of explicit with the sta-
bility of implicit methods.

Barakat and Clark (1966) compared another extension
of the Saul'ev method with various numerical techniques.
The extension reported by these authors utilizes the
same molecule equation that was developed by Saul'ev but
solves the equation twice at each node for each time step.

First, the equation is solved for each node beginning at

80

one corner of the object being thermally modeled and pro-
gressing to the opposite corner. Second, the equation is
resolved proceeding through the array of nodes in the
opposite direction. Barakat and Clark claimed that a more
accurate model was obtained by averaging these two solu-
tions for each time step period. Their technique required
twice the computer time of the technique proposed by
Allada and Quon. Barakat and Clark also employed the
von Neumann method for stability analysis which is de-
scribed in Richtmyer (1957) to show that either of the
extensions of the Saul'ev method are unconditionally
stable.

The Saul'ev technique was selected for the numerical
model of the heat sink. The model was written so that
either the Allada-Quon or the Barakat-Clark extensions

could be applied.

5.1.2 The Grid System

The irregular boundary of the heat sink made the
selection of a coordinate system difficult. Two consider-
ations were paramount in the selection of the grid system.
Since the primary interest is in the temperatures around
the cavities, the system must accurately model the heat
sink in these areas. Furthermore, if a number of differ-
ent sizes and configurations were to be modeled, the

boundary conditions must be determined at each node by the

81

computer. Therefore, ease of programming the boundary
conditions in a general way was also important.

A third consideration was also taken into account in
the final selection of a grid system. It is very difficult
to guarantee that a computer program actually performs the
operations intended. The probability of errors existing
in the program can be greatly reduced if the final program
is developed from an initial crude but simple model. If
the original model is sufficiently simple, the errors in
it can be removed with reasonable certainty. If the modi-
fications are of modest magnitude, it is possible to main-
tain a low probability of errors occurring in the program.

A Cartesian network was constructed on the cross
section of a metallic sink for the first crude model,
Figure 5. A cell, the smallest area enclosed by the lines
in the network, was considered to either lie entirely in
the sink or entirely outside of the sink, depending on
whether more of its area lay inside or outside of the
boundary of the sink. The resulting simulated boundary is
illustrated by the dashed line in Figure 5. The tempera-
ture nodes used in the heat transfer calculations were
considered to be located at the centroid of each cell.
Since none of the nodes were located on the boundary, the

following convective boundary equation was written:

.. - _ E _ 2
g— h(Tm Tw) — d (Tw Ti’j) [cal-/cm 1 (l6)

.xcflm mnu mo HmUOE ocu How Empmmm peso mne .m musaflm

.50 3553?, a: 29.543/

 

 

 

 

¥ : mm
$6 1 I
Em..— r. N

 

 

 

 

1"

 

 

 

 

82

 

 

 

 

 

 

 

J \\\\ ..\N\\\ n fil

m . . M
\ . A \\
$mn XZ—m nufl\M\.

 

 

 

 

 

. \\e _ 3.1 .

 

 

 

 

 

s.
\ w.
\s

 

 

 

 

 

 

 

ex

pzu.m2<

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

83

where
q is the rate of heat transfer, cal./sec.
A is the boundary area, cmz.
T is the ambient temperature, °C.

T is the wall temperature, °C.

T. . is the temperature of the node in the cell of
interest, °C.

h is the convective heat transfer coefficient,
cal./°C-cm2-sec.

k is the conductivity of the sink, cal./°C-cm-sec.
and

d is the distance from the node to the surface, cm.

The wall temperature was eliminated in this equation leav-
ing a multiplicative constant of heat transfer between the

node and the ambient given by:

_ hA
C5 ‘ h'd——+ k [cm] (17)
such that
q = kcsvroo - Ti’j)[ca1.] (18)

This constant was appropriately employed in the Saul'ev
model.

This model was modified to improve the approximation
at the boundary. In the improved model the size and shape
of each cell lying on the boundary of the sink was modi-
fied. For programming purposes each cell was recognized

as having one of six possible relationships with the

84

boundary of the sink. In order to visualize these relation-

ships, only the second quandrant of the external boundary
was considered. The sides of each cell were numbered one
through four, proceeding clockwise from the 9:00 o'clock
position, Figure 5. If the cell was cut by the boundary
of the sink, the new surface was numbered five. Once the
relationships for the cells in the second quadrant near
the external boundary were found, the relationships for
any other quadrant of the external boundary or any of the
quadrants for the internal boundaries could be found by
judiciously changing the numbering of the sides of the
cell. If a cell laid entirely external to the mass of the
sink, it was assigned a type of negative one. If the cell
was located entirely within the sink, it was assigned a
type of zero.

All cells that laid on the boundary of the sink were
assigned a type of l, 2, 3, or 4. If the boundary of the
sink intersected sides three and four of a cell, that cell
was assigned a type of one. If sides two and four were
cut by the boundary, the cell was assigned a type two
designation. In type three cells, the first and third
sides were cut by the boundary of the sink. The first and
second sides of a type four cell were cut by the sink's
border.

For the mathematical model, the boundary of the sink
was replaced in each cell by a straight line. The straight

line intersected the border of the cell at the same

85

locations as did the boundary of the sink. The numerical
value for the cell's volume, the areas for heat transfer
to other cells, and to the ambient, and the distance from
the temperature node in this cell to the adjacent tempera-
ture nodes and to the boundary were all adjusted. The
temperature node for each cell that was intersected by the
boundary was assumed to lie at the centroid of the re-
mainder of the cell. The necessary mathematical relation-
ships for these adjustments are derived in Appendix A.

After the numerical values for the thermal coefficient
of the cells intersected by the boundary of the sink had
been adjusted, the necessary adjustments were made to the
adjacent cells. Finally, adjustments were made to any
cells that were directly influenced by the external and
internal boundaries of the sink. In final form the pro-
gram would accept information on the thermal character-
istics, the heat transfer values, the radii of the sink
and chambers, the location of the chambers, and the function
describing the heat produced by the reaction. The computer
model would then determine the necessary boundary condi-
tions, compute the temperatures of the reaction and compari-
son cells, and find the temperature distribution in the
sink for each time step. The model also computed the
temperature of a reaction vessel in an ideal sink and

printed out the relative error at given time intervals.

86

5.1.3 Tests on the Model

The accuracy of the model was tested with several
techniques. The computation of boundary conditions by the
model was checked by hand calculations on the various cell
types. The application of the Saul'ev method and the
stability of the technique was checked by printing out
the entire array of temperatures for each time step. It
was thus possible to see if the technique was being cor-
rectly applied to every node.

The Saul'ev technique has been shown by Barakat and
Clark (1966) to be unconditionally stable. However, the
coefficient in front of the temperature of the node of
interest at the previous time may under some circumstances
become negative. Physically, this is a violation of the
second law of thermodynamics, and may cause the calculated
values to oscillate about the true solution. The fact that
the method is stable means that the oscillations should be
damped. Because of their reduced volume, cells inter-
sected by the boundary of the sink are more likely to
cause the coefficient to become negative. In order to
partially offset this condition, the heat transfer from
the ambient was always employed at the future time. Even
with this precaution, temperature oscillation was ob-
served for very small boundary cells when large time steps
were employed.

The ability of the model to accurately predict

transient thermal behavior in the sink was also observed.

87

The size of the cavities and the reaction heat were both
reduced to zero. The initial temperature of the sink was
then set to a numerical value of l and the temperature of
the ambient was established at 0. The transient tempera-
ture as predicted by the model at various locations agreed
with the curves published by Boelter gt_al. (1942).

The maximum size of the time and space increments
that could be employed without introducing significant
errors was determined experimentally. The relative error
predictions did not vary more than a few percentages for
increments in length varying up to one-fourth of a chamber
diameter. However, increasing the step size from 20 per
cent to 40 per cent of the chamber diameter caused the
predicted relative error to decrease 15 per cent. The
maximum time step allowable depended on the thermal
properties of the sink, the square of the space increment,
and especially on the size and shape of the boundary cells.
The maximum time step that would not produce oscillations

was found experimentally to be:

(19)

 

where
At is the time step, sec.
Ax is the space increment, cm.

a is the thermal diffusivity, cmZ/sec.

88

The Allada-Quon technique was compared with the Barakat-
Clark method. The same results were obtained from each;
therefore, the Allada-Quon procedure was adopted to reduce

the necessary computer time.

5.2 Results

 

The numerical model of the sink was utilized to
study the effect of elevenparameterscuithe relative error
in the temperature measurement. The parameters investi-
gated include: the kind of metal in the sink, the heat
transfer coefficient from the ambient to the sink, the
external temperature control, the diameter of the sink,
the distance between the centers of the chambers and the
center of the sink, the diameter of the chambers, the heat
capacity of the twin cups, the amount of heat generated by
the reaction, the length of the reaction, the overall heat
transfer coefficient between the reaction vessel and the
sink, and the Calvet measurement configuration as opposed
to the Berger measurement configuration. The model was
solved for all the parameters in an initial condition.
Then the parameters were individually varied and the model
was then resolved. Because of the large number of param-
eters, very few of the interrelationships were investi-
gated. The study was restricted to the two dimensional
model of the sink. In effect the model simulated a sink
and twin vessels of infinite length. In a finite sink

that totally encloses the twin vessels, the relative error

89

should be slightly less than was predicted by the model.
The magnitude of this error was not investigated.

The model was initially solved for an aluminum sink
twenty centimeters in diameter. The cavities were five
centimeters in diameter and located four centimeters off
center, Figure 5. The heat transfer coefficients to the
ambient and cups were 0.000136 and 0.000271 calories per
degree centigrade-centimeter squared-second, respectively.
Both represent heat transfer through still air, however
the internal coefficient includes the conduction along the
thermocouple wires. The heat capacity of each vessel was
fifteen calories per degree centigrade. The ambient
temperature was assumed to be a uniform zero degrees. The
Calvet measurement technique was assumed. Five calories
of reaction heat were added in the functional form shown in
Figure 6.

The results of the investigation are summarized in
Table l. The significance of the relative errors depends
on the measurements to be made with the instrument. For
very precise measurements an error of 0.1 per cent could
be significant. However, for heat of immersion studies
and for any measurements on biological materials, the same
error can usually be neglected. Therefore, for these
:measurements the standard aluminum sink is adequate.

Calvet reported employing a silver sink. The results
in Table 1 show that a copper sink would provide approxi-

inately the same accuracy. The radiation heat transfer

HEAT PRODUCTION

.254—

.20 r

(CALORIES / SECOND)

.OS‘r-

90

.lOv—

 

 

o I I l I ! .— J. i
0 IO 20 30 4O 50 60 TO 80
TIME (SECONDS)

Figure 6. The functional form of the first
assumed reaction heat.

91.

TABLE 1. The maximum relative errors introduced by the non-ideal sink.

Predicted Maximum

Parameter Varied per Cent Error

 

Standard conditions* 0.104
Silver heat sink 0.054
Copper heat sink 0.058
External heat transfer coefficient increased to

0.0136 ca1./°c-cm2-sec. 0.092
External heat transfer coefficient increased to

0.1 cal./°C-cm2—sec. 0.067
Silver heat sink with external heat transfer

coefficient of 0.0136 cal./°C-cm2-sec. 0.050
Copper heat sink with external heat transfer

coefficient of 0.0136 cal./°C—cm2-sec. 0.054
Vary external temperature 0.1° C across the sink 0.126 or 0.082

Copper sink with external temperature varied 0.1° C
and external heat transfer coefficient of
0.0136 cal./°C-cm -sec. 0.938

Copper sink with temperature variation and
external heat ransfer coefficient of

0.1 cal./°C-cm -sec. 3.56
Diameter of sink reduced to 16 cm. 0.131
Distance to cavity center reduced to 3 cm. 0.077
Diameter of cavity increased to 6 cm. 0.136

Size of reaction vessel reduced such that

heat capacity is 6 ca1./°C. 0.103
Heat of reaction reduced by a factor of 10 0.105
Reaction rate reduced by a factor of 10 0.099

Internal heat transfer coefficient increased
to 0.001 cal./°C-cm -sec. 0.417

Berger Measurement technique 10

 

*The standard conditions were: an aluminum sink 20 cm in diameter with
cavities 5 cm in diameter and located 4 cm off center. The heat transfer
coefficients to the ambient and cups were 1.36 x 10"4 and 2.71 x 10‘4
cal./°C-cm2-sec., respectively. The heat capacity of each vessel was
15 cal./°C. Five cal. of heat was added during a period of 90 sec. The
ambient was a uniform 0°. The Calvet measurement technique was simulated.

92

from the copper sink would be higher than from the silver,
but if this was a serious disadvantage the copper could be
plated with chrome or gold.

The heat transfer coefficients can have an important
influence on the relative error. The effect of increasing
the external heat transfer coefficient by a factor of 800
was only about one-tenth as large as the effect of increas-
ing the internal heat transfer coefficient by a factor of
four. However, if the external temperature was not uni-
form, then the magnitude of the external heat transfer
coefficient was very important.

Changes in the physical size caused only small vari-
ations in the relative error. Increasing the diameters of
the cavities by one centimeter, decreasing the distance
between the cavities by two centimeters, or decreasing the
diameter of the sink by four centimeters would change the
relative error only 0.03 per cent.

Since variations in the physical size had only a
small effect on the relative error in the two dimensional
model, changes in the third dimension would not be expected
to be important either. This provides some justification
for studying only the two dimensional problem. Also, it
indicates that the elaborate sinks reported by Calvet are
very little better than a simple cylindrical block like

Berger's.

93

Changes in the heat capacity of the reaction vessel,
the amount of heat produced, or the length of the reaction
did not alter the relative error significantly.

The large relative error shown in Table l for the
Berger measurement technique does not reflect on the
accuracy of his instrument. It does mean that the data
collected from this instrument must be corrected for heat
transfer. The thermocouple arrangement in the Berger
instrument is easier to construct than the Calvet arrange-
ment. In the balance, it is necessary to correct the data
from the Berger instrument for heat transfer. The Calvet
instrument can be designed so that no data correction is

necessary.

VI. DETERMINING THE REACTION HEAT

AS A FUNCTION OF TIME

6.1 Background

 

In order to study the reaction kinetics in a micro-
calorimeter, it is necessary to determine the heat of
reaction as a function of time. The recent literature on
microcalorimeters indicates an interest in reproducing the
reaction heat curve from the temperature data. This
problem represents the inverse of the usual problem. In
the traditional heat transfer problem, the boundary and
initial conditions along with the internal heat generations
are known and the interior temperature distribution is
sought. The problem studied here attempts to determine
the heat generation from measurements of the temperature
distribution within the body.

Several approximate procedures exist for solving such
an inverse problem. If the general functional form of the
reaction heat is known, the problem becomes one of deter-
‘mining the parameters in the function. A nonlinear least
squares technique can be utilized to find the parameter
values that minimize the sum of the square of the differ-

ences between the experimental and predicted temperature

94

95

values. The same technique can be used for an unknown
functional form by approximating the reaction heat by a
polynominal or some other general function. However, this
technique tends to be insensitive to erratic changes in
the heating function.

Calvet and Prat (1963) have discussed two techniques
for determining the reaction heat as a function of time.
The first technique was developed by Tian (Calvet and
Prat, 1963). He inverted the differential equation that
describes the Calvet calorimeter so that the reaction heat
was given as a function of temperature:

2 .1.

q = —— T + C1 (20)

0.104
d“ I-El

c
C1
This procedure has several weaknesses. It implicitly
assumes that the temperature indicated by the thermopile
in the Calvet microcalorimeter is the same as the temper-
ature of the reaction solution. Because of the size of the
reaction vessel and the lack of stirring, this assumption
may not be correct unless the reaction proceeds slowly.
For fast reactions Calvet included two exponential lag

terms to give adjusted temperatures, T1' to be used in

equation (20):

(J)
2 1. dT

96
where

ml and wz are time constants in the expoential lag
terms.

Calvet emphasizes that his calorimeter does not have any
temperature uniformity requirements for the reaction
vessel. However, the inverse technique without a lag
factor, equation (20), assumes that the temperature of
the vessel is uniform. Equation (21) relaxes this require-
ment, but it still requires the temperature distribution
to be the same during the calibration and reaction experi-
ments. The evaluation of the reaction heat with the Tian
equation requires that the derivative of the temperature
measurements be taken with respect to time. Derivatives
of data inherently increase the magnitude of any noise in
the measurements. Thus, if the Tian or Calvet equations
are employed, small errors in the temperature measurements
are expected to cause the predicted heat of reaction values
to oscillate. Calvet reported compensating for 90 to 95
per cent of the reaction heat in his microcalorimeter in
order to increase the measurement accuracy. It may be
possible to eliminate the need for the compensation energy
if the technique for determining the reaction heat as a
function of time can be improved.

The inverse problem is not unique with calorimeters.
Determining a surface heat flux from internal temperature

measurements is also an inverse problem. Beck (1967) has

97

proposed a family of methods for solving this inverse
problem based on the numerical inversion of the Duhamel
integral equation. He found that a simple inversion
similar to that proposed by Stolz (1960) was unstable for
small time steps but the stability could be increased by
using future temperature measurements in a least squares
approximation of the future heat fluxes. The technique
proposed in the next section for determining the heat of
reaction closely parallels the method developed by Beck.

6.2 Improved Techniques for Finding
the Heat of Reaction

 

 

If the temperature of the reaction vessel is uniform,

Duhamel's theorem in the form:

8¢(t-1)dl
3t

 

t
T(t) = Ti +I qm (22)

0
may be used to calculate the heat of reaction. In this
equation 0 is the temperature response at t seconds after
a unit step increase in the reaction heat. The function,
¢, can be obtained from an exact or numerical solution if
the thermal characteristics of the calorimeter are known.
The exact solution was used throughout this study. The
uniformity of the temperature in the reaction vessel de-
pends on the dimensions of the vessel and the rate of the
reaction. If there is no internal heat generation and if
the reaction vessel is approximated by a solid cylinder,

the error introduced by the assumption of uniform

98

temperature will at no time exceed one-half the value of
the Biot number (Kreith, 1965, p. 128). Thus, to maintain
the error below 1 per cent, the following restriction

should be observed:

_ 11};
Bi .. ks < 0.02 (23) l

where

Bi is the Biot number. r

 

h is the convective heat transfer coefficient,
cal./°C-cm2-sec.

k is the conductive heat transfer coefficient of
the solid, cal./°C-cm-sec.

L is the characteristic length obtained by dividing
the volume by the surface area, cm.
Approximating the reaction vessel and its contents by a
solid is not expected to introduce a significant error
because the extremely small temperature gradients will not
establish convective currents in the solution.

For a reaction vessel with a height that is five
times its radius, the restriction in equation (23) requires
the radius of the reaction vessel to be less than 0.5 cm
if it is filled with water (k = 0.00149 cal./°C-cm2-sec.).

The restriction may not be sufficiently stringent
if the heat from a rapid reaction is being measured. The
temperature distribution in an infinite cylinder with uni-
form heat generation (Kreith, 1965, p. 46) can be compared

with the convective heat loss from the same cylinder to

99

show that equation (23) is valid for a constant heat
generation. However, Calvet (1963) reported that rapid
reactions caused temperature gradients to exist in a re-
action vessel that maintained adequate uniformity of
temperature for slower reactions. The uniformity require-
ment for rapid reactions was not investigated during this 3
study. I
A more general form of Duahmel's theorem would per-

mit the temperature of the reaction vessel to be deter- 1

 

mined as a function of location as well as time. However,
this was not considered for two reasons.

First, the only justification for applying Duhamel's
theorem at several locations would be that the temperature
uniformity requirement could be relaxed. However, if the
uniformity of temperature is not maintained, natural con-
vective currents will be established in the reaction
vessel. The numerical solution of the differential
equations describing natural convection in a finite cy-
linder represents a major study by itself. Forced con-
vection is also present in the reaction vessel when the
reactants are mixed. Finally, if 0 were known as a function
of location and time, the set of integral equations that
results from applying Duhamel's equation at several loca-
tions in the vessel would have to be solved simultaneously.

The difficulty of measuring temperatures inside the
reaction vessel was the second reason for not applying

Duhamel's theorem as a function of location. The accuracy

100

of the temperature sensors is reduced if they are subjected
to mechanical strain. Therefore each sensor would have to
be supported and protected from the currents in the re-
action solution. These supports would increase the diffi-
culty of determining 0.

If the integral in equation (22) is solved numerically

by the trapezoidal rule, the heat generation, qM, at time,

t = MAt, can be found directly:

 

2 M-1
_ __ _ . :
qM — C TM 2 E qn¢ M-nAt (qO — 0) (24)
l n—l
where
c1 = At/C, sec.°C/cal.
c2 = -HAt/C.
M = t/At.
n = A/At.
qM = the heat generation at time, MAt, cal.
q = the heat generation at time, nAt, cal.
n
TM = the measured temperature at time, MAt, °C.
' = fl — ° _
¢ M-n dt(t A), C/cal. sec.

For the Calvet calorimeter described by equation (4),
¢'M_nAt can be replaced by the derivative of the analytical
solution, ec2(M-n). It was assumed in the derivation of
equation (24) that the temperature measurements were

started before any heat was added and that the initial

temperature was zero. As would be expected from the study

101

of sensitivity coefficients, this equation tends to
oscillate about the true solution and is unstable for
small At's. The reason is that only temperature infor-
mation at the present time when the sensitivities are at

a minimum is employed.

6.2.1 An Improved Inverse Technique

 

The improved procedures given below are similar to

the technique discussed by Beck (1968) for determining a

‘-e“-_-we-‘

surface heat flux. In order to improve the stability and
to reduce the oscillation expected from equation (24),
more temperature measurements and a least squares smooth-
ing were utilized to find qM. Temperatures at times less
than MAt are independent of the heat generation at time
MAt; therefore, since the earlier heats are already
established, the earlier temperature measurements cannot
be employed. On the other hand, temperature measurements
at a future time do contain information on the present
rate of heat generation. In general Duhamel's theorem can

be applied at future times to give:

M-l I j-l ' qn+.
TM+j - C1 nilqn¢ M-n+j niqu+n¢ j-n + 2
(qo E 0) (25)

Where

¢'a_b = exp[cz(a-b)]

102

for the Calvet microcalorimeter. The heat generation term

for future times may be expressed by

= A0 + A i + A 12 + . . . + Anin (26)

qM+j 1 2

in which the coefficient A0 is equal to qM' Assuming

that r, the number of future temperature measurements E
"ml

utilized is greater than n, the coefficients in this

equation can be determined by minimizing the sum of the

 

squares of the differences between the measured and pre- 1
dicted future temperatures.

The general temperature T in equation (25) can be

M+j

combined with the polynominal for the future heat gener-

ation:
-1 n
_ I :
TM+j — Cl :lqn¢ M-n+j + p: Apij (qo _ 0) (27)
where
j-1 p
c . = z £P¢'. £ + l— (28)
p3 .Q,=O j- . 2

Differentiating the sum of squares with respect to the

coefficients, AP, gives the set of linear equations:

= 0 (29)

"04H

0[TM+j - Te,M+j)ij

103

for

p=0, l, 2, . . . n

where the subscript e denotes future measured temperatures
as opposed to the future temperatures predicted by the
polynominal expression for future heat generation terms.
With some algebraic manipulation, this set of equations

can be rewritten as:

7')
E Br,spAs = Srp p=0, 1, 2, . . . n (30)
s—O
where
r 0 i s i n
B = B = 2 C .C .
rISP rips j=0 5] P3 0 < p i n

and
r M-l r
S = 2 C . T . - Z 2 C .q 0' _ .
rp j=0 pj e,M+j/{:l n=l j=0 p3 n M n+3
or
r C . M-l
S = Z —21 T . Z q 0 _
rp j=0 c1 e,M+j n=l n rp,M n
where
r
¢rp,M-n = jio ij¢M-n+j

“m h 1%.-f 61W
9

104

This system of equations can be readily solved for A0 = qM

provided n is not too large.

6.3 Computer Simulation

 

The stability and accuracy of the Tian method,
equation (20), the inverse Duhamel equation given in (24),

and six of the low order members of the family of tech-

:5.

niques suggested by (30) were compared in a computer simu-

lation. Calvet and Prat (1963) did not report what method 1
was employed to determine the derivative in equation (20).

For this study the derivative at time MAt was evaluated

by the following two techniques:

 

 

SE = TM - TM'1 (31)
dt MAt At

and
d3 = TM+l - TM-l (32)
dt MAt 2At

The six techniques suggested by equation (30) included
constant, linear, and quadratic (n = 0,1,2) polynomials
for the future heats with one and two extra data points
for each (for n = 0, r = l and 2; for n = l, r = 2 and 3;
and for n = 2, r = 3 and 4). Artificial temperature data
was generated numerically for several heat generation
functions. The inverse techniques were then employed to
estimate the heat generation as a function of time. The

root mean square, RMS, value of the differences between

105

the heat generation function and the predicted values was
then computed.

Three heat generation functions were selected. The
first function, Figure 6, was intended to represent a
moderately slow, smooth reaction heat. This function pro-
vided an opportunity to compare the accuracy of the various E
techniques under conditions that might be similar to an
actual heat of immersion study. The effect of noise in

the temperature data was also studied with this function

 

by adding noise with a peak value of less than 1 per cent
of the maximum temperature to the numerically computed
temperature values. The random number function, RANF(-l),
of the CDC 3600 computer was utilized to generate the

noise, y, by the function:
y = 0.004[RANF(-1) - 0.5] (33)

The second function, Figure 7, was a step input of
heat. The behavior of the various techniques near the
abrupt change in heating rate provided some insight into
the stability of the methods. Although it is not a I
rigorous definition, if the predicted heating values tended
to oscillate with increasing magnitude after the step
change, the technique was considered unstable for that
size time step.

The third function, Figure 8, was selected to repre-

sent the reaction heat of a moderately fast reaction.

(CALORIES / SECOND)

HEAT PRODUCTION

.025“-

.020*

.OI5'

.OIO

106

 

 

 

I in l I I I I

 

0 IO 20 3'0 40 50 6'0 70 00

TIME (SECONDS)

Figure 7. The second functional form selected to
test the stability of the inverse techniques.

HEAT PRODUCTION (CALORIES/ SECOND)

107

l.2-

O.8<- I

 

0.6‘-

 

 

 

o I l J L I I I I
O 5 IO I5 20 25 3O 35 40

TIME (SECONDS)

 

Figure 8. The third functional form selected
to represent a rapid reaction.

108

6.4 Results

 

The results of the computer simulation are summarized
in Table 2. The significance of the numbers in this table
becomes more evident when they are compared with the average
heating rate for each of the functions. The average heat—

ing rates were 0.056, 0.014, and 0.075 calories per second

 

for functions one, two, and three, respectively. The RMS

value of the error for all of the techniques using a 0.1

 

second time step on the first function was below 1 per cent I,
of the average heating rate. The Tian equation with the ‘
derivative evaluated over 2At and several of the Duhamel
methods with least squares smoothing maintained the RMS
value of the errors below 1 per cent of the average heating
rate with a time step of one second.
The accuracy of all of the methods was reduced when
noise was added to the temperature data. Although the peak
level of the noise was less than 1 per cent of the maximum
temperature measured, the smallest RMS value for the errors
was just below 10 per cent of the average heating rate. If
the RMS value of the artificial noise is used for AT in
equation (15), the minimum RMS value expected in the errors
of the reconstructed heating rates can be predicted from
the sensitivity coefficients. For the smallest through
largest time steps employed, the approximate minimum ex-

pected error was 10_1, 10-2, and 10-3 calories per second,
respectively. This means that the smallest RMS value of

the errors to be expected with a 0.1 second time step is

TABLE 2. The

g3» 2-

1(39

RMS value of the errors of several inverse techniques.

Root-Mean-Square of the Errors

 

 

 

 

 

 

 

 

 

 

Time
Technique Step . Heating Func- ‘ . .
(Sec) :eatlvg tion No. 1 Heating Heating
unction . . Function Function
No 1 With NOISe No 2 No 3
° in the Data ' '
Tian with _4 _1 _4 _3
derivative 0.1 4.08 x 10 2.39 x 10 4.63 x 10 6.46 x 10
given by: 1 4 05 10'3 2 53 10‘2 2 91 x 10" 6 38 x 10’2
TI‘ - TF4- 1 O x I x O .

At 10 3.71 x 10'2 3.72 x 10'2 8.62 x 10"4 2.99 x 10'1
géigvziige 0.1 2.98 x 10’6 1.19 x 10"1 2.36 x 10'4 1.02 x 10'3
given by‘ 1 2.64 x 10‘4 1.29 x 10‘2 7.42 x 10'4 3.19 x 10'2
T - T
“flilizt‘fl—i 10 1.47 x 10'2 1.49 x 10'2 2.23 x 10 3 2.11 x 10 3

-5 -2 —7
Duhamel 0.1 1.00 x 10 7.96 1.02 x 10 2.04 x 10
with
no least 1 9.83 x 10'4 2 97 x 10 l 1.01 x 10 2 1.14 x 10 4
squares _2 _2 _3
10 4.13 x 10 5.57 x 10 9.65 x 10 1.02
-6 -1 -4 -3
Duhamel 0.1 5.14 x 10 1.20 x 10 2.04 x 10 1.25 x 10
with
least 1 4.02 x 10 4 1.23 x 10 2 9.05 x 10 4 3.88 x 10 2
squares _3 _3 _4 _2
n—O, r—l 10 8.58 x 10 8.05 x 10 7.14 x 10 9.74 x 10
-5 -2 -4 -3
Duhamel 0.1 2.11 x 10 5.59 x 10 3.02 x 10 2.79 x 10
with
least 1 1.63 x 10 3 5.69 x 10 3 1.14 x 10 3 7.80 x 10 2
squares _2 _2 _3 _2
n—o, r—2 10 1.81 x 10 1.81 x 10 1.03 x 10 9.43 x 10
-5 -1 -3 —4
Duhamel 0.1 7.83 x 10 1.75 x 10 5.87 x 10 4.34 x 10
with
least 1 7.34 x 10"“ 1.78 x 10 2 5.87 x 10 3 1.35 x 10 2
Sfiiaref -3 —3 -3 -1
- , r-2 10 7.33 x 10 6.69 x 10 5.56 x 10 5.89 x 10
-6 -1 —4 -3
Duhamel 0.1 1.35 x 10 1.18 x 10 2.21 x 10 1.21 x 10
with
least 1 1.33 x 10'4 1.17 x 10 2 1.05 x 10 3 3.86 x 10 2
squares _3 _3 _4 _2
n—l, r-3 10 8.62 x 10 8.99 x 10 5.77 x 10 9.54 x 10
-6 -1 -4 -4
Duhamel 0.1 1.59 x 10 3.03 x 10 1.74 x 10 5.33 x 10
with
least 1 1.88 x 10 4 3.25 x 10'2 1.03 x 10 3 1.80 x 10 2
squares _3 _3 -3 _1
n-2, r-3 10 4.47 x 10 5.03 x 10 1.44 x 10 2.53 x 10
-6 -1 -4 -4
Duhamel 0.1 1.57 x 10 1.78 x 10 2.05 x 10 9.16 x 10
with
least 1 2.06 x 10 4 1 87 x 10 2 1.03 x 10 3 2.54 x 10 2
squares _3 _2 _3 _1
n—Z, r-4 10 8.51 x 10 1.06 x 10 1 75 x 10 2.27 x 10

 

110

twice the average rate of heating. The values in Table 2
indicate that all of the methods are near the limit in
accuracy for the two shorter time steps. For the ten
second time step the RMS values of the errors were not
changed significantly by the noise.

The second heat function was selected to give some
indication of the stability of the inverse methods. The
Duhamel method without least squares smoothing was the
only technique that exhibited large oscillations. The
magnitude of the errors for the remaining methods was not
considered significant because a heat of reaction is not
expected to have a step change.

The third heating function was intended to approxi-
mate a rapid reaction. Because of the linear character of
the function, the Duhamel technique without least squares
smoothing could predict the heating function very accurately
for small time steps. A real heating function would not be
this linear and although it was not tested, the accuracy of
the Duhamel technique would be expected to decrease sharply
for a nonlinear function. The only other techniques that
exhibited RMS errors below 1 per cent of the average heat-
ing rate were two of the Duhamel methods with least square
smoothing. Each employed one extra data point in the
least squares smoothing.

This study indicated that none of the techniques
tested are unequivocally better than the remaining

techniques. For the slow heating function without noise

111

in the data, the Duhamel methods with least squares
smoothing were almost always better than the other tech-
niques. When noise was added to the data, the results
were mixed for the smaller time steps. However, the error
of the Duhamel method with least squares smoothing (n = 0,
r = 2) was about one-half of the next smallest error for
both time steps. The Duhamel methods with least squares
smoothing were slightly better than the remaining tech—
niques with the ten second time step. The Duhamel tech-
nique without least squares smoothing exhibited some
instability. The results with the rapid reaction were
mixed. The Tian equation with the derivative evaluated
over 2At was consistently better than the Tian technique
that utilized only adjacent data points. In general it
can be concluded that the selection of an inverse technique
should be based on the functional form of the heat of re—

action and the errors in the temperature measurements.

VII. SUMMARY AND CONCLUSIONS

A survey of literature indicated three facets of
Inicrocalorimetry needed further study. The technique
(employed by a calorimeterist is as much art as science.
(Phe metallic heat sinks being employed vary considerably
:in size and shape. And the techniques for determining the
areaction kinetics from temperature measurements are ex-
1:remely sensitive to errors in the measured data.

A study of the sensitivity coefficients of calori-
naeters with and without mechanical energy being added
1: evealed:

l. The parameters of the calorimeter cannot be
determined during the experiment unless the
functional form of the reaction heat is known.

2. The two parameters of a calorimeter without
mechanical energy being added can be determined
simultaneously by adding known heat provided the
temperature measurements are started before the
system reaches equilibrium.

3. If the rate of heat loss from the calorimeter
vessel is equal to the mechanical energy input,

the three parameters of the instrument may be

112

113

determined simultaneously by adding a known heat
over a period of 15 minutes or more.

A numerical model of a solid heat sink was utilized
1:0 determine the effectiveness of metallic sinks. This
nnodel indicates that the sinks generally introduce less
tzhan 0.5 per cent error in the measurements.

A family of methods for determining the reaction heat
£18 a function of time are introduced. The new methods are
sshown to be more accurate for some functional forms of the
1:eaction heat than the technique employed by Tian and

(Salvet.

BIBLIOGRAPHY

BIBLIOGRAPHY

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APPENDICES

APPENDIX A

THE HEAT TRANSFER EQUATIONS FOR CELLS

ON THE BOUNDARY OF THE SINK

THE HEAT TRANSFER EQUATIONS FOR CELLS

ON THE BOUNDARY OF THE SINK

Background

Although the following derivations are elementary,
they are considered essential for understanding the com-
puter model of the heat sink in Appendix B. Therefore,
these equations are presented to aid the reader. Only two
dimensional heat flow is considered, but the model is
assumed to have a unit depth so that the phrases "area
for heat transfer" and "cell volume" have meaning.

As was explained in Chapter V, a rectangular grid

was drawn over the cross section of a sink forming small

cells - A temperature node was assumed to be located at
the centroid of each cell. The problem is to describe in
general mathematical terms the necessary constants for a
trans ient heat transfer equation compatible with the
Saul ' ev technique for cells intersected by the border of

the Sink.

Imagine a five sided cell with a temperature at
time nAt of Tn. An equation is desired to give the
temperature of this cell at time (n+1)At. In general the

change in internal energy of the cell, AB: is equal to

the net heat flow into the cell:

122

123

Zqi = AE (A-l)

where qi is understood to be positive if the heat flow is
111v121rd4 and negative if the heat flow is outward. The heat
flow into the cell of interest from each of the adjacent

cells can be approximated by:

 

q. = d (T. — T) (A-2)

wTuezrea

is the thermal conductivity, ca1./°C-cm-sec.

is the area for heat transfer, cmz.

Q: 3’ W

is the distance between the temperature nodes, cm.

Ti is the temperature of node i, °C.

The: cihange in internal energy of the cell of interest can

be approximated by:

pC V
AB = j-E—(Tnfl - Tn) (A-3)
Where

0 is the density of the material in the cell,
gm/cm .

C3 is the specific heat of the material in the cell,
ca1./°C-gm.

\] is the volume of the cell, cm3.

At is a time increment, sec.

If the temperature of cells 1,

thIue

124

4,

and 5 is known at

(n+l)At, and the temperature of cells 2 and 3 is

knuc>vvn.at time nAt, equation (A-l) can be rewritten):

n+1

 

T

n+1 T

 

pC V
At

n+1 n

_ T )

(T

Each parameter group,
pC V
the constants kAt

n+1

n+1

can be replaced by c

k2A

d2

2 n

) + 2

(T

 

k5A5

d

n+1

) + 5

(T

k.A.
l
d.

i

 

6.

-Tn)+

n+1

k3A3
d3

 

(A-4)

, can be replaced by kci and

With these

cruarigyes and some algebraic manipulation, equation (A-4)

Cali t)e written in the following form:

n+1

T n+1

5

[CST

+

(c6

If the cell of
one through four may
relative to the cell
(i-l , j), cell two is

and Cell four is at (

written:

+ C

4
c - )TnJ/(c + c +
3 c2 6 1

interest is located at
be renamed to indicate
at (i,j). Cell one is
at (i,j+1), cell three

in-l).

(A-S)

(i,j), cells
their position
located at

is at (1+11j)r

Equation (A-5) can then be

125

n+1 _ n+1 n+1 n+1
Ti,j ‘ [C5T5 + C4Ti,j-l + ClTi-l,j
n n
+ C2T1,j+1 + (C6 C3 C2)Ti,j ]/
(c6 + C1 + c4 + cs) (A-6)
If there are only four sides to cell (i,j), (c5 = 0)

Eirlci if c1 = c2 = c3 = c4, equation (A-6) reduces to the
equation given by Allada and Quon (1966). Thus equation
(Ix-6) is compatible with the Saul'ev technique. The addi-
tlixonal cell, 5, has been included to account for heat
t:1:ansfer with the ambient. The problem is now reduced to
C>I1e of finding the constants cl, c2, c3, c4, c5, and c6.
Each of the first four constants represent an area
:Ec3r heat transfer divided by a distance. More specifically,

<2 is equal to the area of contact between cell (i,j) and

J.
(26311.(irl,j) divided by the distance between nodes (i,j)

Eirld.(i-l,j) dotted into the normal of the area. It can
13€3 seen in Figure A-l that the dot product of the distance
Eil’ldnormal is equal to the sum of the absolute distances
from each node to the plane of contact between the cells.

qurus the values of cl, c2, c3, and c4 are given by:

 

 

A1
c = (A-7)
- O + I I
l d1,3,1 dl-l,j,3
A
_ 2 _
c2 — (A 8)

di,j,2 + di,j+1,4

126

 

  

 

 

 

 

  
 
   

 

 

 

 

PLANE OF CONTACT /
“"1” ' [NORMALJF
' - T (i . i)
I DISTANCE,d o
| .
.L _ —— _ 0,1) 1 (PM); flu!"
. ‘3}

Figure A-l. Graphical Figure A-2. The variables
computation of the dot needed to compute 01'
product.

SINK
BOUNDRYVV/ ,
A la 3
/ 4 I
8 AX
d H31 3
'1'
“7 d. l D
i
84 AX _'J

Figure A-3. A type Figure A-4. The

one cell. similar triangles used to

find d5 for a type one cell.

 

 

127

A3
c = (A-9)
3 di’j’3 + di+l,j,l

arnfl

C = (A-lo)
di,j,4 + di,j-1,2

\nliere d. is the shortest distance from node (i,j) to

i,j,k
'tlne kth boundary of the same cell. Figure A-2 illustrates
tihe variables in equation (A-7).

The value of CS was derived in Chapter V and is

SJiven by:

c = 5 (A-ll)

 

Vvhere

h is the convective heat transfer coefficient,
ca1./°C-cm2-sec.

A iszthe area of the cell exposed to the ambient,
cm .

d. . is the shortest distance from the node (i,j) to
the ambient, cm.

k is the thermal conductivity of the material in
the sink, ca1./°C-cm-sec.

The constant c6 is defined by:

pC Vi .
Ce = —%AT"1 (21-12)

 

128
where

Vi j is the volume of cell (i,j).

As was shown in Chapter V, it is only necessary to
consider the cells intersected by the boundary of the sink
in the second quadrant. The four types of cells illus— Pi

trated in Figure 5 are considered individually below.

The equations for each type of cell assume that the points

 

of intersection of the boundary of the sink and the cell E I
walls is known. These points are easily found by assuming

the radius of the sink is the hypotenuse of a right tri-

angle whose sides are the coordinates of the point of

intersection. All equations assume that the boundary of

the sink is replaced in each cell by a straight line con-

necting the points of intersection. The node of the cell

is then moved to the centroid of the polygon. The notation

is also simplified from di . to d with the i,j under-

,j,k k
stood.

A Type One Cell

 

Figure A-3 illustrates the triangular form of a type
one cell. The distance from a side of a right triangle
to its centroid is one—third of the length of the remaining

side. Therefore,

 

129

 

 

63Ax
A3 = 63Ax ' (A-15)
A4 = 64Ax (A-16)
FT
_ 2 2 1’5 _ ’ "1
A5 — (64 + 63 ) Ax (A 17) -
and gj
6364(Ax)2 -
V = 2 (A-18)

 

The length of d5 can be found from the similar triangles
illustrated in Figure A-4. The length of side a is one-
third the length of side B and the small and large tri-

angles are equalangular. Therefore, the lengths:

d
D _ (A-l9)

IP'IDJ

or

636 Ax
2

 

(A-ZO)

4
212
3(63 + 64 )

A type one cell has no sides one or two; therefore,

A Type Two Cell

 

A cell of the second type is illustrated in Figure

A-5. The location of the centroid can be easily found if

130

 

tame-4
SINK |
BOUNDIW~y | I
l
// d I rda" AX
W
/ : ..
1 I .1

 

 

 

 

 

 

 

Figure A-5. A type Figure A-6. The similar
two cell. triangles utilized to find d5
for a type two cell.

 

 

rasz "’1
SINK BOUNDRY L SINK BOUNDRY?

/

./” — 7 // I

17 “:3.“ .3. T .5 .1... .x
'L 1‘4 l""d3 ""I‘ L

-I.— Ax

 

3.AX d4.

 

 

 

 

 

 

 

-I-*-— AX

 

Figure A-7. A type Figure A-8. A type
three cell. four cell.

131

the cell is visualized as the combination of a rectangle

and a triangle.

of the locations of the two centroids:

and

Similar triangles can be employed to find d
A-6).

rectangle, triangle, and cell.

2 2
Ax 2 +55 +°4

2 4
3 62 + 64

6

 

[1 + (04 - 62)2]%Ax

(a + 62)(Ax)2

4

 

2

are represented by the letters,

DB
3A

= D(B+E)

2A

The centroid of the cell is the average

(A-21)

(A-22)

(A-23)

(A-24)

(A-25)

(A-26)

(A-27)

(see Figure
The three points correspond to the centroids of the

If the lengths of the sides

(A-28)

(A-29)

 

132

 

 

 

 

and
DEd' + DE;
d = D (A-30)
D(E + -2-)
therefore '
2 2 II
5 4 2
There is no heat transfer with cell (i-1,j); therefore i (
k;
cl = 0. Notice that expressions for d are not
required since d1 = Ax - d3 and d2 Ax - d .

A Type Three Cell

 

The techniques employed to find the parameters of a

type two cell can be used to find the following parameters

of a type three cell (Figure A-7):

26 + 6
d = £5[ 1 3]

 

3 3 61 + 63
2 2
d AK 61 + 6163 + 63
4 3 61 + 63
2 2
6 + 6 6 + 6
d = 3 3 l 1 (Ax)2

 

5 3A5((S3 + 61)

A1 = Gle

(A-32)

(A—33)

(A-34)

(A—35)

133

 

 

A3 = 63Ax (A-36)
A = [1 + (6 - 6 )2]%Ax (A-38)
5 3 1
E“
and a
2 ' has
(63 + 61)(AX)
V = 2 (A-39)
The constant c2 is zero for a type three cell. E ‘

A Type Four Cell

 

A type four cell, Figure A-8, has heat exchange
through five surfaces. The location of the centroid and
4, and d5 are found by assuming that a

square consists of the cell and the triangle that has been

the distances d3, d

removed. The centroids of a square and a triangle are al-
ready known. The location of the centroid of the cell is
such that the average of it and the triangle's centroid is

the center of the square. The resulting parameters are:

 

 

 

2

a = 1 + 261 + (62 + 62 )(1 61) Ag. (A-40)

3 l + 61 + 62 — 6162 3

1 + 26 + (6 + 6 2)(l 6 I

a = 2 1 l 2 AI (A-41)

4 1 + 61 + 62 - 6162 3

2 2

a = l _ 6 6 + 61 (1 62) + 62 (1 61) (Ax)2

5 l 2 1 + 61 + 62 - 6162 3A5

(A-42)

134

 

 

A1 = 51Ax (A-43)
A2 = 626x (A-44)
A3 = A4 = AX (A-45)
A5 = [(1 — 6l>2(1 - 62>21% (A‘46’ F1
and
V = (1 + 61 + 6: - 6162)(Ax)2 (A_47) g L

The computer program in Appendix B finds the 61
values and stores them in an array called PRE(I,J,K). The
distances, di j k are computed in the function subroutine

I I

D. The ci's are computed by the subprogram CONST.

APPENDIX B

THE COMPUTER PROGRAM OF THE HEAT SINK

 

135

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138

 

 

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139

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140

 

 

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141

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142

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143

 

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144

 

 

44..o~.oo4 Op 00 won
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145

 

 

~044m Qucza manta Numzw guazm domuu mo.

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146

 

 

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