NBSIR 73-184

A Cell Model for Isoperibol Calorimeters

K. L. Churney*. E. D. West**, and G. T. Armstrong*

Institute for Materials Research*

Institute for Basic Standards**

National Bureau of Standards

Washington, D. C. 20234

Churney, :k.. L., West, E. D., Armstrong, G. T.,

A cell model for isoperibol calorimeters,

Proc. 1st Natl. Conf. on Calorimetry,

Zakopane, Poland, Sept, 8-I6, 1973, PP.

1-41 (Polish Academy of Scionc©, Institute

of Chemical Physics, Warszawa, Poland,

1973)

.

Prepared for Physical Chemistry Division

institute for IVIateriais Research

National Bureau of Standards

Washington, D. C. 20234

NBSIR 73-184

A CELL MODEL FOR ISOPERIBOL CALORIMETERS

K. L. Churney*, E. D. West**, and G. T. Armstrong*

Institute for Materials Research^

Institute for Basic Standards**

National Bureau of Standards

Washington, D. C. 20234

April 1973

Prepared for Physical Chemistry Division

National Bureau of Standards

Washington, D. C. 20234

U. S. DEPARTMENT OF COMMERCE. Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts. Director

A Cell Model for Isoperibol Calorimeters

K. L. Churney, E. D. West, and G, T. Armstrong

National Bureau of Standards

Washington, D.C. 20234

Abstract

A Calorimeter can be modeled as a large number of volume elements

or cells in each of which the temperature may be considered uniform,

and each of which can store heat and exchange heat with other cells.

Application of the first law of thermodynamics to this set of cells

leads to representations of the usual calorimetric equations for

the energy change expressed in terms of measurable or estimatable

heat capacities, heat transfer coefficients, temperatures, and work

terms for the individual cells. Analysis of the results yields a framework within which most of the design and measurement problems

of isoperibol calorimeters can be treated.

i
t
Table of Contents

Page

1. Introduction and summary 1

2. Description of the cell model 11

2.1 Aneroid calorimeters . . 15

2.2 Calorimeters with stirred liquids 22

3. Non- linear heat transfer during the main period .... 24

3.1 Development of problem 24

3.2 Changes in state of substances in the calorimeter . 32

4. Details

4.1 Evaluation of parameters in equation (1) 40

Aneroid calorimeters 40

Stirred liquid calorimeters 44

4.2 Solution of equation (1) for rating periods .... 49

Derivation of equations (2) - (5) 49

Specific properties of rating periods 53

4.3 Solution of equation (1) for the main periods;

linear heat transfer 56

Derivation of equations (6) and (7) 56

Interpreation of equation (6) 59

Comments on surface S 62

4.4 Non-linear heat transfer during the main period . . 65

Derivation of equation (9) 65

Derivation of equations (10) - (14) 67

Equivalence of applying first lavz to

region enclosed by surfaces A and S 68
Page

Comment on conduction calorimeters 69

Heater lead problem 70

r- Radiation shield calorimeter 73

4.5 Some consequences of the theory concerning the

energy equivalent 77

Basic results for A.^^ = 0 77

Consequences concerning heat capacity

contribution to energy equivalent; inferences

concerning weighting factors 79

Consequences concerning dependence of energy

equivalent on location of sources 81

Consequences for convergence rating period . . 81

Separation of magnitudes of X-^ and A.2

comments on heat conduction calorimeter .... 82

Table of Symbols 84

5. References 88

6. Acknowledgements 91

Illustrations

1 Sketch of surfaces A and S for a calorimeter 92

2 Sketch of a cell located in a gas-solid boundary ... 93

3 Sketch of a heater current lead 94

4 Sketch of a radiation shield calorimeter . 95
1. Intrcxiuction and summary

A calorimeter is an instrument for measuring the energy of a

phencmenon evidencing itself as an observable thermal effect. In

current practice, this energy is considered to have been measured

when it has been related to the metre, kilogram, and second; i.e.

the International System of Units (S.I.).

In each of the experiments required for the measurement, the

process of interest which may be an internal energy change, a heat

transfer, or an amount of work done causes a thermal effect in the

calorimeter which is evidenced by a change in temperature or, for

example, a phase change in the calorimeter or some part of the

calorimeter. Typical experiments consist of observing the thermal

effects caused by a measured amount of electrical work in an "empty"

or "full" calorimeter or of comparing the thermal effects of a

chemical reaction and a measured amount of electrical work.

In each case, the observed thermal effect is related to the

internal energy change, heat transfer, or work done by applying the

first law of thermodynamics to the system identified as the calorimeter.

This is done by equating the increase in internal energy of the system

to the work done on the calorimeter ' plus the heat transferred to

the calorimeter from its surroundings.

Isoperibol calorimeters''', calorimeters having a constant temperature

environment, are in common use for measuring many classes of thermo-

^To be distinguished frcxn phase change calorimeters as the Bunsen

ice calorimeter.

1
chemical and thermodynamic properties (i.e. enthalpies of solution

and combustion^ enthalpies of reaction^ and enthalpy changes between

high temperatures and standard temperatures)^ energies of biological

processes^ the energy of laser or nuclear radiation, and the energies

of other processes.

The accuracy of results calculated using accepted

2 procedures [1] for this type of calorimeter has been subject to

question [2]. The details of this particular controversy were resolved

[3,4], The question of the accuracy of the results obtained with this

3

type of calorimeter , however, remained unanswered [5] because the

theory of operation of isoperibol calorimeters was inadequate. At

that time, the theory could be said to be based on either the analysis

of an unrea listically sirnple model such as the two body model [4] or

else on White's [6] apparently more comprehensive but ambiguous

treatment. White does not describe clearly the model to which his

theory applies so that it is hard or impossible to judge whether a

calorimeter satisfies the conditions required for his conclusions to

be valid. This point has been discussed earlier ([4], p. 4206).

For this reason, the limitations of accepted practices and concepts

that have been adopted to reduce systematic error in the measurements

using isoperibol calorimeters, as well as other types of calorimeters,

were still ambiguous.

2Numbers in brackets refer to references at the end of the paper.

3

To be fair, the same can be said to apply to a greater or lesser

extent to other types of calorimeters.

2
It should be emphasized that without any theory of operation of

a calorimeter there can be no independent means of judging experimental practice. A useful measurement theory must be based on the

analysis of a model of a calorimeter for it is in only this way,

for example, that general principles used in experimental practice

acquire any useful meaning. For a measurement theory, based on an

appropriate model, to be useful it must be stated in sufficiently

precise terms that all of its assumptions and consequences can be

subjected to the tests of experiment. The degree of precision of

the theory in this respect determines to what extent a consideration

of systematic error in a calorimetric measurement can be carried out

in a rational way.

If the theory fails to meet all the tests of experiment and

experience, it must be discarded and a better theory developed.

For example, the simplest derivation of the method for calculating

results obtained with an isoperibol calorimeter is based on the

assumption that the temperature of the calorimeter is uniform.

This assumption leads to the consequence that an experiment can be

ended immediately after a power input to the calorimeter is completed.

Since this is not confirmed by experiment, this model of an

isoperibol calorimeter must be rejected.

By the same token, interpretation of an experimental test in a

way not justified by theory is tantamount to making a systematic

error. An example of this kind of situation is summarized later in

3
this section in connection with the interpretation of calorimetric

results obtained for different levels of power input to a calorimeter.

Thus, analyses of more realistic models of an isoperibol calorimeter

are of decisive importance not only in devising a measurement procedure

for the real instrument but also in establishing the validity of

the measurements made with it. To the extent that the sophistication

of the model is not commensurate with the accuracy desired for the

measurements, the results of experimental tests will have less

significance.

A quantitative judgment of the adequacy of a theory for a given

accuracy requirement is not a simple matter. Perhaps an unambiguous

decision can never be made. Suffice it to say, however, that if

the experimentalist ignores the consequences of measurement theory,

he precludes the possibility of making any rational independent

judgment as to the nature or magnitude of the systematic error inherent

in his calorimetric measurements.

The presentation of the analysis of the model in this paper has

been divided into two parts--the main text in sections 2. and 3. and

the details in section 4. In the main text, we have selected and

presented those results that we consider to be of central importance.

In the details, the supporting mathematical development for the main

text is given as well as some further conclusions and insight into

the behavior of an isoperibol calorimeter predicted by the model.

The presentation of results in the main text proceeds from the less

to the more complex measurement problems, in terms of the analysis of

4
this model. We believe that analysis of this model offers a general

framework within which most of the problems of isoperibol calorimeters

can be examined. The analysis is far from complete; this will require,

for example, numerical calculations. A feature of the model which

should prove extremely fruitful is that it is amenable to such

numerical calculations. A brief introduction to the model and a

summary of the main conclusions are presented in the remaining part

of this section.

This paper represents the second in a series of analyses [7,8]

of more sophisticated models of an isoperibol calorimeter based on

the application of the first law of thermodynamics to the calorimeter

or some part of the calorimeter as a thermodynamic system. The

first law is applied with symbolism and meanings as follows:

the increase in internal energy of a system, is equal to the sum

of Q, the heat transferred to the system from the surroundings, and

W, the work done on the system. Heat is the energy transfer caused

by a temperature difference between the system and its surroundings

and work is the change in energy of the system caused by processes

that are ultimately reducible to raising or lowering a weight in the

surroundings. Thus, the common expressions, "electrical heat" and

"heat of stirring" represent quantities which are treated as work

in this paper.

The expression "storage of or conversion to heat", as used in

this paper, means an increase in the internal energy of a system that

5
can be completely described as a sum of terms each of which is the

product of the heat capacity of some part of the calorimeter and an

actual increase in temperature for this part of the calorimeter.

Thus^ in order to produce an observable thermal effect in an isoperibol

calorimeter, energy or work is converted to heat inside the calorimeter.

The positions in space inside the calorimeter where this occurs

are referred to as the "location of or geometric distribution of

the sources of heat or energy".

The model for an isoperibol calorimeter analyzed in this paper

is a natural extension of the "two body" model [8]. In the "two body"

model, the calorimeter is considered to consist of two cells enclosed

by the surroundings, and having the following characteristics. The

temperature of the surroundings is constant and uniform. The

temperatures of the two cells are uniform but generally different from

each other and the temperature of the surroundings. The heat capacities

of the cells are constant--independent of temperature and time--but

are not the same in general. The two cells exchange heat with each

other at a rate equal to the product of a heat transfer coefficient

and the difference in temperature of the cells. Each cell exchanges

heat with the surroundings at a rate equal to the product of a heattransfer coefficient and the difference in temperature of the cell

and its surroundings. The heat transfer coefficients are constant

but may be different. When heat transfer coefficients are constant,

this type of heat transfer is referred to as "linear".

6
In this paper, we consider the calorimeter to consist of an

arbitrarily large number of cells which again exchange heat in

proportion to their temperature differences; an approach that

parallels the work of Margas, Tabaka, and Zielenkiewicz [9a] and Davids and

Berger [9b]. Temperature of the surroundings is now generalized to be non-uniform

but constant. In section 4.2, we give the connection between this

model and the linear partial differential equations and boundary

conditions for heat flow in the continuum model for an aneroid isoperibol calorimeter [7]. In the following summary of results, we

give the main conclusions; specific conditions for the validity

of these conclusions are to be found in the main text.

In section 2, 4.2, and 4.3, we consider the case in which no

chemical reaction or phase transition takes place in the calorimeter,

the heat capacities of the cells are constant, and heat transfer is

linear. Also , we consider only those processes of interest where work

is done on the cells which is, in general, different for each cell.

During the time intervals in an experiment when the process of

interest is not taking place, we establish the existence of rating

periods (sections 2.1 and 4.2). In a rating period, the rate of

change of temperature in each cell is proportional to the difference

between the convergence temperature (i.e. steady state) and the actual

temperature at any particular time. The constant of proportionality,

the cooling constant which we designate as -''^j^^ is the same for all

cells. These results have been derived previously for the continuum

and two body models [7,8]. These results are extended to include

7
stirred fluid calorimeters (section 2.2) to the extent that our

equations for heat flow by convection (section 4.1) are valid.

During the time interval in which the process of interest is

taking place^ the main period^ we again establish (section 2.1^ 2.2

and 4.3) that the results may be expressed in the usual [7] form:

work done on the calorimeter equals an energy equivalent times a

corrected temperature rise.

These results are a complete analogue of those derived for a

continuum model [7] and the details show that the energy equivalent

depends on the location of the sources of energy in the calorimeter.

We call this the problem of equivalent sources. (We show th^t the

energy equivalent depends upon the geometrical distribution of the

work input to the calorimeter. This is a less restrictive assumption

than that made previously [7], which was that the power input to the cells

of the calorimeter during the main period is a product of a function of

time and a function of cell location.)

It should be pointed out that this form of expressing the results

again verifies (see [10]) that experiments carried out with different

heating rates reveal nothing about the problem of equivalent sources.

Also, the problem of equivalent sources is not necessarily eliminated

by making the temperature -time curves the same in "calibration" and

"unknown" experiments. This is due to the fact that a difference

in the locations of the sources can cause a difference in the heat

exchange with the surroundings even if the temperature time curves

are the same (section 4.3). The difference in heat exchange shows

8
up purely as a difference in the energy equivalents for the two

experiments. The heat exchange is properly accounted for regardless

of the shape of the temperature- time curve provided the sources are

in the same location in a practical sense. It may be desirable to

reproduce the temperature -time curves in calorimeters having nonlinear thermal properties. However, Swietoslawski ' s [11] principle

of substitution with its emphasis on the temperature-time curve

should be modified to emphasize the gecmetrical equivalence of

sources

As in the continuum model [7] and two body model [4,8] we show

(section 4.3) that energy equivalent is not simply the sum of heat

capacities of the cells of the calorimeter. New results are that,

by inspection of the relative temperature distribution in a rating

period and at the convergence temperature (see section 4.2, eq. (4-29)

and section 4.5) one can infer what these weighting factors are, and

that the heat capacity of each cell makes a positive contribution to

the energy equivalent (section 4.3). All these results apply to

stirred fluid calorimeters within the limitations mentioned previously.

In the second part of the paper, section 3, we show what general

constraints must be placed on the design of the calorimeter in order

that heat transfer can be non-linear (i.e. no assumptions of

linearity are made) during the main period in the region of the

calorimeter where the process of interest is taking place. These

constraints are in accord with those inferred from analysis of the

two body model [4,8].

9
In addition^ we derive a general design criterion for the energy

equivalent of the calorimeter to be independent of source location

with which one may classify (section 3.1), as well as analyze (section

4.4) designs. This criterion formalizes And generalizes the concept

given previously [7]. It also emphasizes the importance of another

aspect of the problem of equivalent sources connected with, for example,

the heater current lead problem of an electrical heater (section 3.2

and 4.3). It is our belief that this design criterion and the

experimental test for equivalence of sources given here and previously

[7] should now be incorporated in the principle of substitutjion.

In the final section of the main text (section 3.2), we analyze what

general constraints must be placed on the design and operation of

the calorimeter in order that we can properly electrically calibrate

a calorimeter in which a chemical reaction or transition from one

phase to another takes place. These contraints are in accord with

those given previously [8]. Indications of how the analysis can be

extended to drop calorimetry and flow calorimetry are outlined. Some

comments on heat capacity measurements are given in section 4.5.

These constraints or criteria are general in the sense that they require

a minimum of assumptions. They are not meant to imply that specific

different arrangement could not solve, for example, the problem of

equivalent sources.

10
2. Description of the cell model.

In this section and in section 3.1, we shall consider a calorimeter

in which no chemical reaction or physical change in state (i.e. phase

trans it i on) occur s

The basic problem is to describe the temperature distribution

which changes with time in a calorimeter which is surrounded by scane

opaque (to radiation) constant-temperature region (the surroundings)

in which the temperature does not change with time but may vary from

point to point. The change in energy stored in the calorimeter and

the heat exchanged with the surroundings during an experiment can

then be deduced from the temperature distribution in the calorimeter.

An experiment is divided into three parts in time. In the first

part, the initial rating period, the observed calorimeter temperature

is either constant or varies exponentially with time. In the second

part, the main period, the temperature is raised by some means such

as supplying electrical power for a limited time interval to a resistance heater. In the third part, the final rating period, the observed

calorimeter temperature is again a constant or varies exponentially

with time.

In order to describe the temperature distribution in the calorimeter

we treat the region inside the constant-temperature surroundings as

an aggregate of an arbitrary but sufficiently large number n of volume

elements or cells such that each is small enough that its temperature

can be considered uniform. The location of the cells will be regarded

as fixed in space but the boundaries shall be considered to be flexible

to the extent that the mass in each volume element is the same at all

temperatures it assumes during an experiment. Each cell may store

11
heat and exchange heat with other cells. Heat exchange may be by

conduction, by thermal radiation, or by steady forced convection^

(in fluids) although some problems associated with convection have

not been solved, as will become clear later. Heat exchange by free

convection is excluded.

The first step in the analysis is to apply the first law of

thermodynamics to a single cell inside the surroundings. We equate

the rate of increase in the internal energy of a cell to the sum of

the rates of absorption of heat by the cell and of doing work on the

cell. For the i^th cell, the rate of increase in energy is the product

of its heat capacity, C^, and the rate of change of its temperature,

dT^(t)/dt. The rate at which heat is transferred to the ^th cell

from the j_th cell is proportional to the difference in temperature

between the two cells. The constant of proportionality is a heat

transfer coefficient, h.., which applies to conduction, radiation,

or forced convection or to combinations of these. Thus the rate

of heat transfer is h..[T.(t) - T.(t)]. In the case of conduction

h^j represents the thermal conductivity multiplied by the cross

sectional area between cells and divided by a characteristic distance

between them. In the case of radiation, h. . represents the coefficient

found for the approximation derived from the Stefan-Boltzmann law

and valid only for small temperature differences--that the radiant

energy transfer at a given average temperature of two bodies is

proportional to the temperature difference between them. In the

case of forced convection, h. . is proportional to the rate of

increase of internal energy caused by flow of liquid from cell j to

cell i. More precise definitions of h. . are given in section 4.1.

We use the term heat as well as energy exchange by steady forced

convection because we assume a constant velocity distribution in the

fluid. Thus, energy transfer occurs only when the temperatures of

the cells are different.

12
In addition to heat exchange among the n cells inside the

constant -temperature surroundings heat exchange may occur between

the i.th cell and the surroundings. To include this^ we consider the

surrounding also to be divided into cells, again small enough so

that the temperature in each cell is uniform. Then some closed surface,

S, that completely encloses the n cells of the calorimeter is drawn

such that the contiguous cells, whose inner boundary constitute this

surface, are in the surroundings. That is, the temperatures, T^, of

each of these contiguous cells, though different, is independent of time

throughout an experiment. Where the boundarj'^ suface S is located is

not so important as that it exist. (Some consequences of this assumption

are given at the end of section 4.3). Then, the rate of heat transfer

from the sth cell in the surface S to the ith cell in the calorimeter

is h. [T - T,(t)]. This term is significant because it yields the

is s 1 ^ ^

total rate of heat transfer to the calorimeter from the surroundings

when summed over all cells i(l < i < n) and cells s.

The rate of doing work on the ith cell is represented by the sum of

a constant power, y due for example, to the rate of doing work against

the internal stresses of the ith cell by its immediate neighbors, to

electrical power supplied to a resistance thermometer, or to power due

to stirring, and a time-varying power P^(t), for example, electrical

power supplied to a resistance heater during the main period. P^(t)

is zero except during the main period.

Summing over all cells £ in the surroundings and the other interior

cells inside the surroundings that exchange heat with a particular

cell i_, we can write, for the application of the first law to the ^th eel

13
C.^(t) = S^h. JT^ - T.(t)] + E.h. .[T.(t) - T.(t)] + P.° + P.(t) (1)

j i (1 < i f

^ny of the h. . and h. are zero, because the ith cell cannot

ij IS ' -

exchange heat with all other cells. Except for radiant heat

exchange (see section 4.1), h, . or h. will be zero except for direct

IJ IS

heat exchange between cell i and cells £ and J_,

Eq. (1) can be written for each of the n cells inside the constanttemperature boundary, giving a set of n equations interrelating

the temperature of the cells. Accordingly, we have appended t!he

notation 1 < i < n to eq . (1). Details of the physical basis for

eq, (1) in possibly more familiar terms, the corresponding linear

partial differential equations and boundary conditions for heat

transfer assumed for the model, are given in section 4.1.

In sections 4.2 and 4.3, eqs. (1) are put into matrix form, for

ease of manipulation, and details of their solution to yield the

results discussed in sections 2.1 and 2.2 are summarized.

14
2.1 Aneroid calorimeters

For calorimeters in which there is no stirred fluid, h. . and h.

' ij xs are positive and h. , is symmetric (i.e. h. . = h..).

In rating periods^ all P^(t) are zero. Assuming all C^,

'^is' '^ij'

and to be independent of temperature and time^ solution of eqs. (1)

gives the following form of the temperature as a function of time in

the i^th cell

T (t) = T . + 2 a (A )d (X )exp[\ t], 1 < i < n (2)

1 oolj^KiKlK K ~~

The terms in eq. (2) are as follows: (1) T , is the temperature

ool of the i.th cell in the steady state due to all the constant power

inputs, P.° for 1 < i < n, and the constant temperatures, T for all s.

(2) The are all real, negative numbers and are the eigenvalues of a

matrix that is generated by writing eqs. (1) in matrix notation (see

section 4.2). The sum over k is over all n eigenvalues X^, 1 ^ k < n

of this matrix (i.e. some of the may be identical). (3) The d^(X^)

are constants which depend on the thermal properties of the materials

of the calorimeter and their placement in the calorimeter. The ^^0^-^

also depend upon the temperature distribution at time t = 0

(T.(0) for 1 < i < n) as well as in the steady state (T. for 1 < i < n)

1 — — Ico — —

Details are given in section 4.2.

Some cells of which temperatures are given by equation (2) belong

to the thermometer and in particular to the sensing element. If more

than one cell is involved, the observed temperature, that is, the

6

The d^(\), ^ ^ i ^ proportional to the entries of the eigenvector

for the eigenvalue of the matrix mentioned above.

15
instantaneous temperature sensed by the thermometer, T*(t), can be

represented by a weighted average^

n n

T*(t) =S.ou.T.(t); Sou. = 1 (3)

The weighing factors O)^ are defined to be zero for cells outside

the sensitive part of the thermometer so the sum in eq. (6) may be

taken over all n cells of the calorimeter.

Eq. (6) implies a linear relationship between the observed temperature

and the temperatures of cells making up the sensitive part of the

thermometer. The value of T*(t) must be obtained by use of any

calibration information, such as the temperature-resistance relationship

of a resistance thermometer. In a simple case of a uniform resistance

wire considered to be made up to 20 cells, the uj. are all equal to 1/20.

16
When the exponential terms for for k > 1 in eq. (2) have become

negligible at some time t^, the calorimeter is in a rating period and

the equation reduces to one term

T.(t) - T^. - a.apd.(Xpexp[X^(t - t^], 1 < i < n (4)

since it can be shown that the eigenvalue of smallest magnitude is

distinct (i.e. unrepeated^ see section 4.2). The general conditions for the

absolute value of A.^ being much larger than the absolute value of A.^^

have not been derived but some indication that this may be true in

general is given by consideration of simple models of calorimeters

(see the last part of section 4.5).

Equation (4) can easily be put in the customary form [1] for

determining the cooling constant, multiply eq. (4) by cju^ and sum

over all i; substitute from equation (3) to obtain

T*(t) - T * = exp[\, (t - t )] Suj.a. (X.)d. (X,).

Differentiation of this equation and substitution of T*(t) - T^* gives

dT*/dt = A, [T*(t) - T *] (5)

As can be seen from eq. (4) or eq. (5), the cooling constant is the same

for any location of the thermometer inside the surroundings. Other

important properties of rating periods associated with the fact that

the ^^C^]^) °f ^1* (4) all positive are derived and summarized in the

last paragraph of section 4.2.

During the main period, not all the P^(t) are zero. Assuming as

in the rating periods that all C, h. , h. , are constant, the solution ^ x' IS-* ij '

of eq. (1) yields a more complex equation for the temperature of the i^th

17
cell containing the terms in eq . (2) plus terms involving integrals of

the form Jp. (T)e'^k^'' ^"^dT where t is the variable of integration,

o (See section 4.3.) Using this equation and eq. (2)^ which^ respectively^

describe the temperature distribution in the calorimeter as functions

of time in the main period and in the rating periods^ one can deduce

the increase in internal energy and the total heat (transferred to or

absorbed by) the calorimeter from the surroundings during the main

period to derive the results to be summarized below. This procedure

is the same as was used in the continuum model presented previously

[7]. An equivalent but simpler procedure^ summarized in section 4.3^

is to integrate^ directly, the matrix form of eqs. (1) from a time t^^

in the initial rating period to a time t^ in the final rating period

to obtain

t W - E {T.(tJ - T.(t.) + X J^[T . - T.(t)]dt} (6)

1 1 z 1 i t

°^

^ (1 < i < n)

r2 In eq . (6) W is equal to Z.J P. (t)dt or the total work done on the

1 # calorimeter by sources of energy in the process of interest.

The total work rather than just the work on i.th cell appears in eq. (6)

because the temperature of the ith cell is altered by all the time

varying power inputs P.(t), 1 < i < n to the calorimeter. E is an

1 i

energy equivalent of the calorimeter that will be obtained if the

temperature is measured at the i.th cell. is proportional to but is

not exactly equal to the total increase in the internal energy of the

entire calorimeter from the beginning to the end of the main period

(see section 4.3, eq. (4-42)). In particular, E^ is proportional to

a weighted rather than a direct sum of the heat capacities of all the

g Strictly, W includes the net work done on the calorimeter that produces

a change in strain at the boundary surface S. We assume this part of W is proportional to T^(t2) - T^Ct^) (see section 4.1) which requires, for example, that the pressure external to the surface S be constant

throughout the duration of an experiment. In any event, this part of W will be relatively small and tend to cancel between "calibration" and unknown" experiments.
cells of the calorimeter (see last paragraph of section 4.3). The

weighting factors are all positive but are not equal^ in general^

unless is identically zero (see section 4.5) which is not possible

for an isoperibol calorimeter. In general^ will be different for

each cell and^ for a given cell, will be different, in general, for

different relative magnitudes of the set of values of J^P^(t)dt, 1 <

i < n unless is identically zero (see section 4.3 an^ 4.5).

Eq. (6) states that work associated with the process under study, W,

is equal to an energy equivalent nultiplied by the corrected temperature

rise of convential isoperibol calorimetry [1] as measured in the ith

cell.

If it is desirable to represent the thermometer as consisting of

more than one cell, division of both sides of eq. (5) by E . , multiplication by cu^, summation over all n cells, and use of equation (3)

gives t W = E* {T*(tJ - T*(tJ + Xj [T * - T*(t)]}, (7)

Z 1 L" CO 1 the energy equivalent E* is now a weighted average of the E^ related

to the various cells making up the thermometer and is determined by a n known work input W. Formally, E* is the reciprocal of E i.

i=l E.

1

The ienergy equivalent E*, as E^, also depends in general on the

geometric distribution of the work associated with the process under

study. It can be shown (see eq. 4-45, section 4.3), as was shown

previously [7], that this is due to the fact that in any real isoperibol

calorimeter, the heat transferred to the calorimeter from the surroundings

t. during the main period is proportional to both f [T . - T,(t)]dt a nd •J coi 1

to W of eq. (6). The proportionality constant for W is a function of

19
the geometric distribution of the work input to the calorimeter and

the heat transfer coefficients associated with the various

cells of the calorimeter. (This proportionality constant is zero^

h in general^ only if is zero). Thus even if

J [T^^ - T^(t)]dt

*-l is the same for two experiments having different geometric distribution

of the work input to the cells^ the heat transferred to the calorimeter

during the main period will be different and this shows up as a

difference in energy equivalents for the two experiments.

Eqs. (2) and (7)^ for example^ are analogous to the equations

(6) and (2) developed previously for the continuum model of an aneroid

calorimeter [7] and therefore this analysis reproduces the results

given in that paper which may be summarized as follows: (I) The

observed change in temperature is proportional to the change in internal

energy of the calorimeter. (II) Conventional methods of determining

the corrected temperature rise are valid for these more sophisticated

models. The corrected temperature rise completely accounts for the

heat transfer to the calorimeter from its surroundings only if the

calorimeter is designed so that this heat exchange is the same for the

different source locations of a "calibration" and "unknown" experiment.

(Ill) The energy equivalent of the calorimeter depends on the location

of the thermometer and the relative locations of energy sources to be

coirpared.

The dependence of the energy equivalent on the location of sources

constitutes a major obstacle to accurate calorimetry and its elimination^

in so far as possible^ is a major if not the paramount design problem

20
of calorimetry. This problem, called the equivalence of sources,

is examined further in section 3.1.

21
2.2 Calorimeters with stirred liquids

Because of the large number of calorimeters which contain stirred

water^ it is desirable to extend the analysis as far as we can to

include forced convection. That part of h. , that is due to forced

convection is proportional to the component of the velocity in the

direction of cell j to cell i and the sign of this velocity may be

either positive or negative. From this^ one can see that if heat is

transferred entirely by convection, h. . = -h,. and the sign of h,

,

may be either positive or negative.

The case of steady laminar fluid flow, in which fluid velocities

are constant in speed and direction, can be treated as given in

section 4.1. If the cells are small enough, it can be shown that

the net h^^ (i.e. due to conduction, radiation, and convection) is always positive

but h. . is asymmetric (h, , ^ h,.). Also, A, is still real, negative,

and distinct and the real part of A.^ for k > 1 are also negative and

have a magnitude greater than A.^. We have not shown that the for

k > 1 cannot be complex and, in general, for an asymmetric h^^ they

will be complex and occur in complex conjugate pairs. Consequently

eq. (2) will be modified to the extent that for any distinct pair

of complex conjugate roots exp(A^t) must be replaced by exp(b^t)cos (w^t +

9

V, ) where w, , v, , and b, are real and b, is negative. However, all

k k k^ k k ° ^

the other conclusions given for aneroid calorimeters still apply since

the remainder of the analysis (see section 4.3) holds for the case

when h^j is asymmetric, but is positive.

In eq. (2), d.(X^) and 3l-(X-^^) are complex if is c d*(X )a'KA, ) = d . (X*)a . (a*) , where * = complex conju ikik itciiC

omplex and

gate.
Turbulent flow^ which is undoubtedly the most common condition

in stirred-f luid calorimetry is very difficult to handle with confidence

because heat transfer in turbulent convection has not been worked out as

well as for heat transfer by radiation^ conduction^ and steady laminar

convection. For the simplest empirical formula given by mixing length

theory for heat transfer by turbulence for liquids stirred at a constant

rate^ h^^ is as3mimetric and positive (see section 4.1). In this restricted

case the results of section 2.1 again apply.

23
3. Non-linear heat transfer during the main period

3.1 Development of the problem

The analysis and^ hence, the results of sections 2.1 and 2.2

are based on essentially three constraints which we summarize as

follows. We assumed (1) that eq. (1) is valid everywhere in the calorimeter and all C, h.,, and h. are constant (independent of temperature

and time) during the main period and (2) that these conditions apply

during rating periods. We assumed that (3a) the temperatures of surface

S of the surroundings are constant and, also, that (3b) there is no

transfer of radiation across this surface and (3c) the total net

transfer of mass across the surface is zero. Thus, the results of

those sections strictly apply to adequately constrained experiments

for example an experiment when electric power is supplied to a resistance heater during the main period (see the end of section 4.3

for further comments).

All of these constraints may be relaxed to a considerable extent

within the framework of a linear theory of heat transfer provided

certain restrictions on either the design or operation of the

calorimeter (or both) are made. For example, to permit analysis of

devices used in absolute radiometric measurements, the assumption

that the constant- temperature surroundings surface is opaque to

radiation (3b) must be removed. As has been discussed in detail

previously [7] (see section 4.1), this 'can be done provided two

conditions are met. First, the total rate of transfer of radiant

energy to the calorimeter during rating periods is constant (so its

effect can be included in the P.° terms of eq . (1)). Second, radiant

24
energy transfer occurs only between solid surfaces; that Is^

absorption of radiant energy in the gases inside the surface S can

be neglected.

There is a significant number of types of experiments in which

non-linear heat transfer inside the calorimeter occurs. In addition^

some of the C, h, . and h. permanently change during the experiment;

1 IJ IS

they are different in the initial and final rating periods. An import

ant class of experiments of this type are those in which a chemical

reaction or phase transition takes place in the calorimeter. The

conditions for removing constraint (1), so that the case of nonlinear heat transfer during the main period is permitted but all the

C , h and h. are the same in both rating periods are given in this

i' xj xs section. This affords some further insight into the problem of equiva

lence of sources. Using the results of this analysis, we consider

the case of a chemical reaction occurring in the main period in

section 3.3.

25
In order to treat the case of non-linear heat transfer during

the main period^ we restrict the calorimeter to one in which there is

some closed surface A that is inside the constant -temperature surrounding

surface S and that meets assumptions (3b) and (3c) of the previous

section. In the region between A and S there are m cells, and to

each cell eq. (1), with all C., h. and h. . constant, applies throughout •* 1^ IS ij )

'^'^

the duration of an experiment. In addition, we require that the heat

transfer coefficients of the a_th cell of the envelope of contiguous

cells whose outer boundaries constitute the surface A be constant,

first, with respect to the cells between A and S, and second, with

respect to the ceils s making up surface S. These heat transfer

coefficients are, respectively, h, for 1 < i < m and h for all s.

The number of cells enclosed by the surface A will be i, chosen so that

+ m = n, and they will be numbered from 1 to ^. No confusion

results with the numbering from 1 to m of the cells between A and S.

We may apply conservation of energy at each moment in time to

either the entire calorimeter or just the cells inside the surface A.

Leaving to section 4.4 a summary of the proof that the final equations

will be the same in either case, we choose here to apply it to the

cells inside the surface A. This gives

^ Jl ^ 2.C.dT.(t) = dQ(A) + E.P.° + SP. (t) (8)

1 TT^ dt 1^ 1 ^ dt

In eq. (8), E.C.dT. (t) /dt is the total ra.te of increase of the

1^ ^ ^ Jl internal energy of the cells in A. The work terms, 2.P.° and

SP.(t), have the same meaning as before (ie. P. (t) = 0 except during

1 ^

^ the main period). dQ(A)/dt is the rate of heat transfer to the cells

inside A from the constant- temperature surroundings surface S and the
cells between the surfaces S and A in the inward direction as shown

in fig. 1.

dQ(A)/dt^ see section 4.4^ is given by

dQ(A) = S Z h- [T - T (t)] - 2 h'T (t) (9) -— saass a a a a at

m

i;3.(A)[P.° + p. (t) - C.dT.(t)]

•1 X 1 1 1 X dt

In eq. (9)^ h' is the total heat transfer coefficient by all paths^

a s —^

' —

direct and indirect, between the ath cell of A_, whose temperature

is T (t), and the sth cell of the surface whose temperature is T . Si s h' is zero for aneroid calorimeters and, in general, H h' =0.

a ° a a S^(A) is a dimensionless positive constant that has a value bounded by

one and zero, is a function of only the various heat transfer coefficients

of the cells between A and S, and depends upon the location of

surface A. Physically, eq. (9) states that dQ(A)/dt is the sum of three

parts. The first and second part, S S h' [T - T (t) ] and -2 h'T (t), ^ ^ ^ a s as s a a a a respectively, are the rate of heat transfer across the surface A assuming

the power input to the cells between A and S and the rate of change of

internal energy of the cells between A and S is zero. The first part

is caused by non-zero values of T - T (t) and the second part is due to

s a heat transfer across the surface A by convection caused by deviations of

T (t) from strict uniformity at each moment in time. (In eq. (9) each

3. T (t) is considered to be independent). The third part may be regarded as the

asum of the fractions of the total power input, V^" + P_j^(t), to each

i^th cell between the surfaces A and S which is not stored as internal

energy and appears as heat transferred across the surface A in the inward

direction. Simple descriptions (for example, see [12]) neglect this

second term. This is neither necessary nor is it desirable since

the (imprecise) locations of both surfaces A and S preclude the

assumption of zero mass betvjeen these two surfaces.

27
Eqs. (8) and (9) apply at all times during an experiment. During

the main period each of the for the cells inside A may depend upon

temperature and/or time but during rating periods we assume they

each have the same value. Combining eqs. (8) and (9) and integrating

from time t^ in an initial rating period to time t^ in the final

rating period gives_, without any approximations, (see section 4.4),

j2 t m t t Sj ^P.(t)dt + i:.3.(A); 2p.(t)dt = E^lT^Ctp - T^(t^) + X{ 2[T^ - T^(t)]dt

It- 1 t. t^

^ ^ (10)

The left hand side of eq. (10) is of the general form of the work

associated with the process under study. This equals an energy

equivalent, E^, times a corrected temperature rise as measured by

a temperature T (t) defined by eq . (11) where S h' ' = 2 h' + h'. A s as s as a

T,(t) = W^2V!l (11) ^ Z (Z h" ) a s as

Eq. (10) differs from the analogous equations in section 2,

eqs. (6) and (7), in three important respects. First, no assumption

about the nature of the heat transfer inside the surface A during the

main period is made so eq. (10) is valid for heat transfer of any

type inside A during the main period. Second, the work terms in

eq. (10) are in two parts. The first part is the total work

done on the cells inside the surface A, E,J P.(t)dt. The second

part, frcrni our previous discussion in connection with eq. (9), is

equal to that fraction of the work done on the cells between surfaces

A and S during the main period that appears as heat crossing the

surface A and cannot be accounted for by the measurement of T^(t)

28
throughout the time-course of the experiment. For an electrical

calibration, it can be shown (see section 4.4) that this second work

term is exactly analagous to the heater lead correction described by

Ginnings and West [13], Third, the energy equivalent, E^, is independent

of the location of the sources inside the surface A,

The invar iance of E. with the location of a source inside the

A

surface A really requires that two conditions be fulfilled. First,

we must be able to "correct" the work associated with the process under

study for any associated work effects in the region between A and S.

This requires either certain design restrictions or else auxiliary

measurements, since these effects are not accounted for by the usual

temperature measurements (ie. in this case).

To the extent that the locations of the surfaces A and S are

uncertain, these work effects should be made as small as possible.

In particular, for an electrical calibration experim.ent, this requires

that the total work supplied to the resistance heater current leads

be small, by both design and choice of operating conditions (see

section 4.4), as well as meeting other design criteria [13]. A

sufficient test for the particular design and operating conditions would

be to determine if the energy equivalent varies with different

heater element resistances [14]. The same pair of current leads must

be used in all experiments. Since the heater elements are the

sources, their location must be the same unless the second or following

condition is fulfilled.

The second condition is that we must be able to measure T^(t).

In term.s of eq. (3), the sensitive element of the thermometer must

measure the temperature of all the cells forming the surface A with

29
weighting factors cu. = h" ) /i; (D h '

' IS ) for i = all a, and u), = 0 as a s as ' i

for i elsewhere. As White has pointed out [15], direct measurement

of the required is not possible with any accuracy and the u)^ are

not proportional to the areas of the various parts of the surface A.

Thus, a surface thermometer cannot, in general, be constructed to

exactly meet the second condition for equivalence of sources although

it may well prove superior to a thermometer placed in a single location

on the surface A.

Since we cannot measure directly for any calorimeter as

specified up to this point, we must constrain the design of the calorimeter so that the measurement of the temperature in a single location

is equivalent to determining T (t) of eq. (11). One such design choice

is to attempt to make the temperature of some closed surface in the

calorimeter isothermal. This condition is met to a greater or lesser

extent in all calorimeters which contain a well-stirred fluid and

also, for example, by the laminated wall design developed for an

aneroid calorimeter by Prosen and Johnson [16].

A second design choice is to construct the calorimeter so

2 h'' is zero except for a small region on A. This condition is

s as very nearly met by the attachment of radiation shields to the calorimeter "vessel" with a good thermal shunt (see [17], [18]).

It can be shown (see section 4.4) that for this type of design

the temperature measured anywhere on the surface of the outermost

(i.e. closest to the surface S) radiation shield, as well as at the

30
thermal shunt, is equivalent to measuring T^(t), Because of this

fact, the design of the stirred-water calorimeter developed by Coops

(see [19] for details) is analogous to this type of calorimeter (the

centrifugal stirrer, because of its mixing action, is the thermal

shunt) . -

A third design choice is to place a thermometer in the region

between A and S that measures 2 Z h''[T - T (t) ] directly, essentially asass a

•' ' '

the design used in seme conduction calorimeters (i.e. when the

thermometer is a thermopile placed between the surface A and the constant

temperature "block"). Ideally, the approach should be to attempt to

force all the heat transferred to the surroundings to pass via the

thermometer [20] since otherwise the ambiguity associated with the

surface temperature measurements mentioned previously may arise.

This means the heat transfer coefficients between the calorimeter

and surroundings are necessarily large and the magnitude of \^ is

large. We have inferred in section 2.1 that as (or these heat

transfer coefficients) increases, the variation in energy equivalent

with different locations of sources may increase. The possible

competition between these factors suggests that in cases where there

are no other limitations, other designs may be preferrable. (For

other comments on conduction calorimeters, see sections 4.4 and 4.5).

A suitable experimental test for designs intended to meet the second

test for equivalence of sources has been described previously [7,8].

This test is the determination of the variation in the energy equivalent

for different locations of an electric heater inside the surface A.

Both the planning, execution, and interpretation of such tests must be

done with care (i.e. see [21] or [22,23] for earlier worlO.

31
3.2 Changes in state of substances in the calorimeter

In this section, we shall restrict the calorimeter to be one

which by design and experimental test is such that the energy

equivalent is independent of the location of the sources for a

fixed thermometer location. This means the measured thermometer

temperature T>^(t) will be treated as equivalent to a measurement of

T. (t) of eq. (11). We will again assume as in section 3.1 the C,

h, h. , h. , and P.° for all the cells between A and S are constant

ij IS-' la-' 1

throughout the duration of an experiment. Maintaining the above

limitations, we shall now consider the case where the heat capacities

of at least some of the cells inside the surface A and their associated

heat transfer coefficients, are different in the initial and final rating

periods. (All the for the cells in A are assumed to be the same

for both rating periods.)

For purposes of illustration, it will be useful to consider the

case where a chemical reaction occurs in a combustion bomb located

inside the surface A, though the results will be applicable to other types of

changes in state. In order to handle the experiment where a chemical

reaction occurs inside this bomb, we must first reformulate the left hand

side of eq. (8). Reformulation is necessary because eq. (8) implies

the rate of change of the internal energy of the cells bounded by A

i

is expressible in the form S^C^dT^(t) /dt throughout an experiment.

This is not true during the main period of an experiment when the

chemical reaction occurs. This can be seen by noting that while the

integral of the above expression over the main period may be large,

the total change in the internal energy of the cells bounded by A

may be very small. For example, if the cells bounded by A were "completely

isolated", (i.e. the region bounded by A has no interaction with its

surroundings in terms of bath heat or mass transfer and work).

32
the change in their internal energy would be zero. Another reason

the left hand side of eq. (8) must be reformulated is to separate

the reactants and products of the chemical reaction from the inert

calorimeter parts^ so that the internal energy change associated

with the chemical reaction can be expressed in terms of measurable

quantities.

For our purposes it will suffice to return to the basic concept

that the left hand side of eq. (8) is the rate of increase of the

internal energy of the cells inside A^ which we denote by dU. (t)/dt.

During the rating periods, dU. (t)/dt is expressible as IC.dT, (t) /dt,

as was done previously. During the main period, however, du^(t)/dt

is due to both a rate of temperature change and the rate of change

in internal energy caused by the chemical reaction. If no heat exchange

occurs with either the cells between the surfaces A and S or the constant

temperature surface S and no time varying power is supplied to the

cells bounded by the surface A, these two contributions to dU^(t)/dt

will be equal and opposite in sign, ie . dU (t)/dt =0.

In order to express the energy of the chemical process in terms

of measurable quantities, we divide the cells inside A into two groups.

The first group will be called the reaction zone and are those cells

whose chemical form and associated heat capacities have changed by

virtue of a chemical reaction. The internal energy of these cells will

be called U (t^) at the time t , the subscript r denoting reactants,

r i 1 and U (t_) at the time t , the subscript p denoting products. The

P ^ second group, the remainder of the cells inside the surface A, will be

called the empty system, referring to the fact that by design and

construction the products or reactants can be removed from the

33
calorimeter. The internal energy of the empty system at times t^^ and

t2 will be denoted by U^(t^) and U^(t2) respectively. Then ve may

write for the change in internal energy of the cells bounded by A

Now suppose no reaction occurred in the reaction zone (ie. the

reactant are present in the experiment in the final rating period) but

the relative temperature distribution in the final rating period is

identical to that which would have prevailed as if the reaction had occurred.

We will call the internal energy in the reaction zone in this hypothetical

state ^^(^2)^ the subscript being underlined to emphasize it is a

hypothetical state, in general.

Adding and subtracting ^^(^2^ right hand side of eq. (12)

gives

+ [U^(t,) - u (tJl

p Z ^ Z

Up to this point, neither of the quantities in brackets are

interpre table in terms of measurable quantities. Suppose now that

the final rating period is the convergence rating period when the

products are present. Since the convergence temperature is a steady

state, the temperature distribution is independent of the heat

capacities of any of the cells in the calorimeter, but will certainly

depend in general on heat transfer coefficients.

34
It is shown, see section 4.2, that if the constant temperature

of the surroundings surface S is uniform and the constant power inputs

to the calorimeter approach zero, the convergence temperature

distribution also tends to become independent of the heat transfer

coefficients of the cells in the calorimeter. That is, it becomes

uniform and equal to the temperature of the surroundings.

In this case, U (t„) can be associated with a real state. If

} r 2 the final rating period is taken as a convergence rating period and the

above constraints are met, it can be shown that in place of eq. (10)

we may write

In eq. (14) the first bracketed term on the left is the negative

of the internal energy change of the reaction as if it had occurred

at the convergence temperature, T^^. The next two terms on the left

hand side are the work terms that were discussed in the previous

section. On the right hand side of the equation, E^^ is the energy

equivalent of the calorimeter with the reactants present, ^^(^2) is

equal to T. and \^ is a cooling constant determined in the initial ^ Aoo 1 °

rating period.

By a minor change in the analysis, it can be shown that if the

initial rating period is at the convergence temperature and if the

same constraints on the calorimeter design are met, then eq. (14) again

1

1

1

35
holds except that in this case E is replaced by the energy

equivalent when the products are present,

^A^*"!^^

rather than ^^(^2);

is equal to and A.^ is the cooling constant of the calorimeter

determined during the final rating period.

It is to be noted that the temperature to which the chemical

reaction or change in state is to be referred is ccsnpletely unambiguous

We believe that the possible uncertainty on this point as evidenced

by differences in accepted procedures used in calorimetry (ie. compare

[24] and [25]) is now resolved. . ^

The constraints on the calorimeter operation and design are the

1 most general ones that fulfill the condition that either the final

or initial rating period temperature distribution be independent of

changes in heat capacities and heat transfer coefficients of the

cells of the reaction zone. Other conditions may serve to achieve

the same aim.

For example, as A.^ decreases in magnitude it can be shown (see section

4,5), that because the effect of a gradient on the constant temperature

surroundings on the convergence temperature distribution is reduced,

the effect of a change of a heat transfer coefficient inside the

surface A will be correspondingly reduced. Also it is more important

to make the constant power inputs negligible to the cells inside A

and in particular those near the reaction zone. In cases, where this

is not possible as, for example, in solution calorimeters, it suffices

to note that if ratio of the power input divided by the heat transfer

coefficients are small, the effect of small changes in heat transfer

coefficients is correspondingly reduced (see section 4.5).

36
It should be pointed out that an additional constraint is required

when the energy equivalent is determined by measuring the corrected

temperature rise caused by a known /3J for a standard chemical reaction.

This is because it is assumed that energy equivalent of the calorimeter

for the conditions applicable to the reaction under study, whose is

unknown, can be calculated by adding or subtracting, as required, the

appropriate heat capacities from the energy equivalent measured in the

standard chemical reaction. It is shown (see section 4.5) that as

\^ approaches zero, the energy equivalent, E^, becomes more nearly an

exact sum of the heat capacities of the cells inside the surface A or

conversely, if is not zero, it is never, in general, an exact sum

although the weighting factors in the sum are always positive. Thus,

for this situation it is desirable to design the calorimeter so

is small. If the energy equivalent is determined by electrical heating,

it is not necessary to make these corrections. However, in this case

the problem of equivalence of sources is more important.

To conclude this discussion of using a calorimeter for determining

an energy of reaction, a few words about initiation and termination of

the reaction are in order. To start the reaction, some work, though

it may negligible in many cases, must be done on the calorimeter,

(ie . in our example, it might be by doing electrical work on a fuse

which ignites a sample). The measured amount of work constitutes the

work terms on the left hand side of eq. (14).

Practically, the reaction will always stop of its own accord in a

finite time: when either one of the reactants is exhausted or the net

rate of formation of products is zero for either chemical equilibrium or

kinetic considerations. Whether or not it has stopped within the time scale we

37
have selected for our experiment can be established by whether or not

a final rating period is reached [ 4 ] (ie. eq. (5) applies with T*(t)

replaced by T^(t), and and are independent of the time selected

for the beginning of the final rating period).

The above analysis applies equally well for a measurement of a

physical change in state (ie. heat of fusion or transition). For

example, in this type of experiment electrical work is necessary to

heat a substance from a temperature below a transition point to a point

above the transition point. Since part of the change in energy will

not in general be associated with the actual heat of transition,

appropriate heat capacity corrections as well as a separate determination

of the transition temperature itself will be required. The analysis

is quite specific about how these corrections should be made.

If, for example, we assume that the energy equivalent of the

calorimeter has been determined when the reactants are present and we

measure an energy of fusion in which the final temperature is above

the melting point (and near the convergence temperature), then

^p^^2^ ' ^r^V = S^^A^^2^ " ^f^ ^fCTf) + C^[Tf - \(^2^^ ^^^^

In eq. (15 ) is the heat capacity of the products and is the

heat capacity of the reactants at or near the fusion temperature,

T^ and ^(T^) is the required energy of fusion.

Equation (l4) can be applied equally well to drop calorimetry

provided we make the following assumption:

38
(1) The cells of the calorimeter and constant temperature surfaces are

arranged so that all heat radiated in the direction of the calorimeter is

absorbed by elements inside the calorimeter and none by cells between the

surface A and surface S. (2) The temperature of the surroundings before

and after the sample has been dropped are the same and independent of

time

That is^ eq. (14) is equally valid if the closed surface S is

allowed to be open for part of the main period provided we are able

to properly account for the increase in internal energy^ the work

term, and the heat transfer term by measurements of a temperature,

T^(t), associated with the surface A. (Otherwise additional measurements would be required.) It is clear that the boundary S can

similarly open during rating periods provided the preceding statement

is true and the rate of increase of internal energy, of doing work,

and absorbing heat, are constant. Hence the analysis can be generalized

to flow systems.

The other restrictions concerning the validity of eq. (14) still

apply. The remainder of the analysis will not be presented since it

closely follows the arguments presented in detail for the "two body"

model of a drop calorimetric experiment given previously [4].

The problem of measuring heat capacities with isoperibol

calorimeters due to the fact that the energy equivalent is not, for

example, an exact sum of the heat capacities of the cells inside

A has been mentioned before [7]. Some comments on this problem are

given in section 4.5.

39
4. Details

4.1. Evaluation of the parameters in equation (1)

parameters C., h, , h..,

is' J.2 and P^** of eq. (1) or determination of numerical values for a

specific model is based on the connection between eq. (1) and the

description of the calorimeter heat flw problem in terms of partial

differential equations and boundary conditions. We consider a

calorimetric system for which heat transfer inside the surface S

can be described by equations that are linear with respect to

temperature and time. This excludes free convection which from a

design standpoint poses no special problem [26 ]• For aneroid

calorimeters^ heat transfer occurs by conduction in solids and by

both radiation and conduction in gases. When the absorption of

radiant energy by the gas is negligible^ the heat transfer problem

can be formulated [ 7 ] in terms of the equation for heat conduction

in solids and gases, eq. (4-1), the flux and temperature boundary

conditions at gas-solid interfaces, eqs. (4-2) and (4-3), respectively,

and the boundary condition at the surface S of the calorimetric system

eq. (4-4).

IH-I)

CH-z)

( H-l)

(V-V)

40
In eq. (4-1), c is the heat capacity per unit volume. It is

defined by ^~ = (^)^ where e is the internal energy per unit

volume. (de/dT) is to be evaluated along the actual sequence of

states occurring as a function of time at each position. For solids

which are required to be elastic, c is approximately that at constant

volume [27] f while, for gases, it will usually be that at constant

pressure or volume depending on the specific circumstances. T is

the temperature as a function of time at a particular position, k

is the thermal conductivity. p° and p(t), are, respectively, the

constant and time varying components of the power generated per unit

volume by sources both internal and external to the unit volume under

consideration. P" and P(t) will include the rate of doing work against

the internal stresses at a particular position.

In eqs. (4-2) and (4-3), the subscripts a and g denote the

solid and the gas phases, respectively. (~r~) ^i^d (*—. are the

^ ^ ^n'a ^n)g

derivatives of the temperature with respect to position in the solid

and gas, respectively, at the boundary A in the direction of the outward

normal to A viewed from solid and gas, respectively. h(A',A) is the

radiant heat transfer coefficient by all paths, direct and indirect,

between a unit area on surface A and solid surfaces A'' enclosing A [ 7].

The range of temperatures to which eqs. (4-1) - (4-4) apply is assumed

to be small enough that c, k, p°, and h(A' ,A) are independent of

temperature (and time) though they will vary with position.

Consider new the interior of S to be subdivided into cells fixed

in space and of such size such that the temperature of

41
each cell can be assumed to a first approximation as uniform. In addition,

we shall assume that the boundaries of each shall be flexible to the

extent that the mass in each cell is constant (so that the increase in

internal energy due to a change in strain or volume is not caused by

a change in mass). We will, for example, use the procedure described

by MacNeal [29], see [30] Chp. 6) in which the cells can be constructed

as polygons such that the interfaces are normal to a single interior

point of each cell to which we assign values of temperature to cell i

and Tj to each of the cells j_ contiguous with cell 1,

Consider first the case where the cell i is located in the

interior of a homogenious solid (or gas). Integrating eq. (4-1)^

using Green's Theorem, over the volume, V^, enclosed by cell 1, that has

a total surface A^y and calling n the outward normal to the surface

of cell i

^

c £EelVr ^in-vTdA t |)p'dV + ^

pa)dV (H-5)

Assume c, k, p°, and p(t) are independent of position inside the cell,

and are equal to c., k., p.°, and p.(t), respectively. Define A..

1^ 1^ 1

' X7

IT y ij

to be the area of the surface between cell i and i, 1. , to be the

distance between the interior points of cells i and j, k^^ to be the

thermal conductivity at the interface A.,, and V. to be the volume of

3-3 1

cell i. Then using central differences to approximate grad T, eq. (4-5)

can be approximated by

.Y^im I kdii ivt) -Tiit)] , pi\-^ pxtw, (1-6)
To illustrate how the boundary conditions of eq. (4-2) and

eq. (4-3) are treated^ consider a simple two dimensional case where

a solid-gas boundary, approximated as a plane, passes through the

interior point of cell 1 constructed as shown in fig. (2). Assume

that in the volume, V , of cell 1 in the solid phase c, k, p°, and

J.a p(t) are independent of position and have the values c , k , p

°

, a a la

and pj^ (t) and in the volume of cell 1 in the gas, V , are again

independent of position but have the values C„, k , p

°

, and p. (t)

.

8 g g Ig

Integrating eq. (4-1) first over the volume of cell 1 in the solid

and then over the volume in the gas, and using eq . (4-2) (and the procedure used to derive eq. (4-6) one obtains the two equations

A^^ is the area of the boundary enclosed in cell 1. Integrating eq.

(4-3) over the area A,, one obtains

Id

where A , is the area of the ith cell exchanging radiant energy with

jb ^

area A,, of cell 1. Adding the preceding three equations gives

lb

c.V, d^it), |, tI^(A,bAjKbAijT-(t)-T;(t)]
where CtV- = c V. + c V- and similarly for p °V- and p, (t)V,

.

1 1 a la g Ig ^ ^11 ^11

Comparing eqs, (4-6) and (4-7) with eq . (4-1)^ one obtains the following

set of relations for the case of aneroid calorimeters.

The values of C and h,. are necessarily positive, h,,

i ij

is symmetric (see [28]) and can be interpreted as a direct heat

transfer coefficient between nearest neighbors for conduction plus

(or) a net heat transfer coefficient (ie. via all paths) for surfaces

exchanging radiant energy. If ^ is the size of a linear dimension

2 of a cell h. ./C. varies as l/i- for conduction and as Jl for the

radiation contribution.

For calorimeters containing unstirred liquids we again exclude

free convection which imposes more severe design constraints (see [26]).

For liquids stirred at a constant rate or solids moving with a velocity

u^ the equation governing heat transfer [31] is

In eq. (4-9a) c, k, p(t) haves the same meaning as eq . (4-1)^ u is

the velocity vector at a fixed position, and p° now includes the rate

of dissipation of kinetic energy into heat per unit volume due to

viscous effects. Taking the liquid to be essentially incompressible

.44
c is the heat capacity at constant volume (we exclude free convection.

see [31] p. 188). Conservation of mass requires

div u = 0 (4-9b)

At solid-liquid boundaries, the tangential and normal components

of the velocities must be equal and zero, respectively. Thus if the

solid is stationary the flux and temperature boundary conditions

will have the same form as eq. (4-2) and eq. (4-3) with h (A,A' ) = 0.

Si For interior points of a stirrer shaft rotated at constant angular

velocity, eq. (4-1) will apply since u-grad T = 0 as well as eq. (4-2)

and (4-3).

For forced, steady laminar flow in the interior of a liquid, eqs.

(4-9) apply with u independent of time though varying with position

as for example in the laminar sublayer of the liquid. Transposing

and integrating the convective term of eq. (4-9a) over the volume

of cell i using eq . (4-9b) gives

Integrating eq. (4-9b) over the volume of cell i, one obtains

In both equations u. , is the velocity normal to the surface A. ,

in the direction from the interior point of cell j to cell i.

Combining the preceding two equations yields

a

45
so that the term to be added to the r.h.s. of eq. (4-8b) to account

for convection is of the form + c,A.,u../2 and is antisymmetric

(u, , = -u.,). Because of eq. (4-10), the sum 2,h, , where

_j is over all cells exchanging heat with cell _i will contain no

convective contributions. If the linear dimentions of a cell are

of size ^, then the part of h.,/C, due to convection will vary as

1/j^ and h^^ will be positive only when ^ is sufficiently small. For

water, this requires that ^ be less than 0.029/|ul, where \u] is the

magnitude of the velocity u.

When the flow is turbulent in a fluid stirred at a constant rate,

the velocity u is no longer independent of time. Both u and T can

be approximated [32,33] as the sums of fluctuating components,

u' and T', due to turbulence and mean components, u and T, (ie. u or

T averaged over times long with respect to the f lucuaticn s) . It can

be shown [32,33] that eqs. (9) still formally apply with u and T

3 ^ replaced by u and T provided we add the term -cE^^^ (u^T') to the r.h.s.

1 i

of eq. (4-9a), The uj are the components of u' with respect to the

bases of an orthogonal coordinate system. Since no entirely

adequate method for evaluating this term, the flux due to transport

of heat by turbulence, exists, we appeal to empirical mixing length

theory [33] and assume that

where e^ is positive and a function of position. (In general, e_j^

could be an asymmetric tensor of the second order [33]. However

the theory is at best approximate and the above assumption is essential

to later developments). Integrating, eq. (4-11), the heat flux due

46
to turbulence over a cell i in cartesian coordinates using eq. (4-10)

and the notation that e. . = e^ , e„- or e„ over the area of the interxj 1^ 2^ 3

face, A. . of cell i vith cell j, one obtains

Thus, this term has the same form as the heat transfer coefficient

for conduction and is to be added to the r.h.s. of eq. (4-8b) for

turbulent convective flow.

Thus, eq. (4-8b) will in general be given by

0.. e.-

^;AiJ U ii

2

hi , is associated with heat transfer by conduction and thermal radiation

and by the component of turbulent convection caused by the correlated

fluctuations in u , . , T., and T.. h, . is associated with the heat

transfer caused by convection calculated assuming ^j_y and T^

are uncorrelated and can be replaced by their values averaged over

a time long in comparison with their fluctuations (but short in

comparison to the time scale of a calorimetric experiment).

where

47
The method used to derive eqs. (1) and (4-8) from eqs. (4-1)

through (4-4) and eqs. (4-9a) and (4-9b) is the so called "integration

method) (see [30], chp. 6). Whatever method is used, we have adopted

the point of view that it be equivalent to breaking up the regions

into cells or regions in space having a uniform temperature

and that heat transfer between each of these cells retain the physical

meaning of the differential equations and boundary conditions previously

cited. The latter necessarily requires that ht^ be positive and

symmetric, and h^^ be of no fixed sign (but consistent with eq. (4-10))

and antisymmetric, and 2,h.. = S.h., = 0 where the sum over i is over

' J ij J all cells contiuous to cell i. The assumption that h. , is constant

requires differences in T^(t) - ^^(t) be small and the assumption that

h^j is positive requires that the cells be sufficiently small (beyond

the initial assumption about the existence of cells) that h'^^ > jh^^j.

Thus, eq. (1) and (4-8) are regarded as our basic assmuptions.

Though convergence of eqs. (1) and (4-8) to eqs. (4-1) through

(4-4) and eq. (4-9) seems plausible on physical grounds, we are

unaware of a suitable mathematical proof at this time. Because the

time variable is not broken into discrete intervals, the problem of

stability does not arise.

48
4.2 Solution ®f equation (1) for rating periods

Eq. (1) for rating periods (P^(t) =0, 1 ^ i ^ n^^Y t>6 written

in matrix form as

(CDtHjTlt) (4-12)

In, eq. (4-12)^ D is the ordinary differential operator with respect

to time, d/dt. C is the diagonal matrix diag (C^,

•'^i/^ ^

an nxn real asymmetric matrix having elements H. . = 2] h, + S.h,, , i 4 ° 11 S IS j ij '

and H. , = -h, , for 1 < i, 1 < n. T(t) and Y are the column matrices

[T.(t), T (t) —,T (t)]' and [2 h. T + P % — h T + P °]', 1''2 •'n siss I ' ^snss n

'

respectively. The superscript prime, used in the latter, denotes

the tranpose. By inspection eq. (4-12) can be rearranged to

where

X(f) ^nt)'Tco (4-14) - H y (4.15)

and H

' is the reciprocal of H.

Premu 1 tip lying both sides of eq. (4-13) by C gives

(l0-A)X(t)-O

where I is the identity matrix of order n and A is defined by

(4-16)

10

H and C can be assumed to have the same dimensions, with out loss in

generality, see end of section 4.3.

49
Taking the Laplace transform [34] of both sides of eq. (4-16), one

obtains

ai-A)xa)»Xfo) (4-18)

It can be shown (see [35], p. 176) that for a. general matrix A that

(4-19)

where

>l^j^>^*.

(4-20)

and

(4-21)

In eq. (4-21), |(^) is the characteristic polynomial (b6]^ ch. 19)

and , in eq. (4-19) - (4-21) , A.^ are the distinct roots, each having

a multiplicity m^(m^ > 1), of the characteristic equation of the

matrix A (i.e. eigenvalues of A):

(4-22)

where \ | indicates the determinant of the enclosed matrix. In

eq. (4-20), F(X) is the adjoint ([36], ch. 6) of Xl-A. Substituting

eq. (4-19) into eq. (4-18) and inverting the transform ([34 ] , p. 72) gives

50
the formal solution to eq. (1) in matrix form

xft) » Ijk 1,7 Kia<fe)X(o)f '•'"'ixp (^k^Wa-i )> (4-23)

Based only on the S3mimetry properties of H, the K,(A. ) and

have the following properties. For aneroid calorimeters, H is

symmetric and, hence, A is similar to a sjnranetric matrix since C is

-1 -1 -1

positive definite ([37], p. 67, prob. 27) (ie. A = -L (L HL )L

i i 1 -1-1

where L = diag (Cj^^, '^n^^

^"'^ L HL is symmetric, see [38],

p. 34). It follows ([36], ch. 21) that all the are real and A is

similar to a diagonal matrix. Thus, if has a multiplicity m^ =

r, the rank of '*^j^I~^ is n-r. The latter has two consequences. First,

it requires that all K.(A. ), j > 1, are all null (see [35], p. 61-62,

or [36], p. 170, prob. 26), Hence for aneroid calorimeters, no terms

in the r.h.s. of eq. (4-23) involving the product of exp(A.^t) and t appear,

Second, since K^(''^j^) is not null while K^(A.^), j > 1, are null ([35],

p. 17), K^('^^) has a rank r, satisfies the equation ([35], p. 61)

K,/;i(k)afel-A) =AfeI-A)K,(^fe) (4-24) = o

and may be factored ([35], p. 65) as

In. eq. (4-25), D^(A.^{^and ^^(A.^) are (rxn) and (nxr) matrices,

respectively, and (X ) and B- (X ) are proportional to the set of right and left

(4-25)

11

D with a subscript is a matrix, and without one is the differential

operator d/dt.

51
handed (use [35], p# 64-67 and see [38], p. 4) eigenvectors, respectively, for

a A.^ with a multiplicity r. (For the other properties of K^(^^)^ see

[35], chp. 3 or [36], p. 160, prob. 26).

For calorimeters containing stirred liquids, H is asymntetric and

we are able to say less about the (A.^) and A.^ from sjmimetry alone.

Since A is real, son^ of the may be ccxnplex and, if so, will occur

in complex conjugate pairs. Further, from the form of K. (A. )

(see [35], p. 75), it can be seen, for example, that sum of terms for

an unrepeated pair of complex conjugate A. will involve terms that

vary with time as exp(b^t) cos (w^t + v^) where b^, w^, and v^ are real.

Further, since we have not proven that A is similar to a diagonal

matrix in this case, it follows that C'^j^) *^ j 5

all be null. Thus, for repeated roots, we cannot exclude the possibility

of T(t) varying in an even more conplicated fashion. It will be seen

in subsequent analysis, hcwever, that proving whether or not A is

similar to diagonal matrix is not essential to our analysis.

The specific form of H determines the sign of the real parts

of A.^ and a number of properties of the eigenvalue and eigenvector

of smallest magnitude, X^. For arbitrary cell size for aneroid

calorimeters and for sufficiently small cell size for stirred-water

calorimeters, h. . is positive. In either case, at least one 2 h, is

' ij ^ ' S IS non-zero. Hence the matrices H and A are irreducibly ([30], p. 20)

diagonally dominant ([30], p. 23). Since all the diagonal entries of

A are negative, the real parts of the are negative ([30], p. 23 or

apply Th. 1.5 and 1.6). Since for some positive, real number s, the

12

matrix si + A > 0, A.^ the eigenvalue of smallest magnitude is real and

12

0 is the nxn null matrix.

52
distinct, and the corresponding eigenvector is positive ([30]^ pp. 30^

258), that is, -D^(X^) is a column vector and every entry, d^(X^), is positive,

Thus, after sufficient time has elapsed, the ith entry of eq. (4-23)

is given by eq. (4) with ^j^(^-j^) equal to

n

where b^C^t^) is the jth entry of the row vector B^(A.^), While we are

assured of the separation of the eigenvalue, and the eigenvalue

whose real part is closest to that of because A.^ is distinct,

a discussion of the probable magnitude of this separation is given in

section 4.5.

Certain specific properties of the rating periods are necessary

for all subsequent analyses. The first is that eq. (4) may be written

in the form used to derive eq. (5)

:

it (4-25)

The set of temperatures, Tooi^ ^ ^ ^ ^ ^} represent the temperature in the convergence rating period (when all dT^(t)/dt = 0) and

is identical to the steady state temperature distribution when the

time t goes to infinity. Fran eq. (4-15), one may write

IHI

where a. ./(h| is the _i, jth entry of H a. . is the ij jth entry of

the adjoint of H, (adj H) and jllj is the determinant of H. Since H is

irreducibily, diagonally dominant (see previous discussion) and by

definition H. . > 0 and H, , < 0 for all i 3^ j , it can be shown (see

11 xj - -"^

53
[30], p. 85) that c^^7|Hj > 0, all i, j. Since )h| is positive (H is

positive definite, see [30], p. 23) the a^^ are also positive. The

of the cells of some region within surface S are equal^ in general,

if and only if this region is isolated (see section 4.5).

By inspection of the form of H, one can see that

IHI = |j^I>jJ«<ji'I;^4'V4^'<ij (4-26)

In eq. (4-26), 'h. = h. or h . for a cell j not in contact with

js js SJ

the surface S and in contact with the surface S, respectively.

Thus, if we define the average temperature of the cells of surface

S as Tg, eq. (4-15) may be written as

Thus, eq. (4-27) says the convergence temperature of the ith cell

is equal to the sum of three terms : the average temperature of

the surroundings, a term which may be positive, zero, or negative

caused by the deviation of the temperature of the surroundings from strict

uniformity, and a positive term due to the constant power input to

the calorimeter. From this, we conclude the convergence temperature

distribution, in general, will be more nearly uniform, and hence,

independent of all h... h. , as well as all C, as the temperature

ij-^ IS-' 1

of the surroundings becomes uniform and all approach zero.

The second important property of rating periods is that by

differentiating eq. (4) one obtains eq. (4-28) and by comparing

eq. (4-28) with eq. (4-25) one obtains eq. (4-29).

54
Since the d^(Xj^) for all i have the same sign^ eq. (4-28) indicates

that if the rate of change of temperature is positive (or negative)

anywhere in the calorimeter, it is positive (or negative) everywhere

in the calorimeter. The ^^^C^^) cells of some region inside

the surface S are equal, in general, if and only if this region is isolated

(see section 4.5). Since all d^(^j^) are dependent only on the

thermal properties of the calorimeter eq. (4-29) indicates, first,

that the set of temperatures T^^, 1 < i < n, and the temperature of

any J.th cell is sufficient to specify the temperatures of the remaining

cells in the calorimeter during any rating period. Second, if

all the C^f ^±s

°^ calorimeter and the convergence temperature distirbution for two rating periods are the same, it follows

that the difference in temperature of an arbitrary i.th cell is

sufficient to specify the difference in temperatures of the remaining

cells between the two rating periods.

55
4.3 Solution of eq. (1) for the main period; linear heat transfer

Defining P(t) to be the column matrix [?^(t) , P^(t)]*; eqs.

(1) during the main period may be written in matrix form (see eq. (4-12)

as r- (rD-A)X(f)= C"p(i) (4-30)

To determine the form of T(t) during the main period, we consider

the case for aneroid calorimeters where A is similar to a diagonal

matrix. Applying the analysis given in section 4.2, one obtains

m) (4-31) = mhr« = Z^Ki«fc)[xfo)f c"J P(T)c *dT3e

As mentioned in section 2.1, eq. (4-31) may be used to obtain eq. (6)

of that section. However a simpler method can be used that does not

require that A be similar to a diagonal matrix and is as follows,

Premultiplying both sides of eq. (4-30) by the adjoint, F(D),

of the matrix operator, ID-A, yields

4>^D)xrt)= F(D)c"'p(f) (4.32)

where |)(D) is the determinant operator

<|)(D)= |Dr-A|= a.D\a,D''t...+a„.,Ofa„ (4.33)

The coefficients a^, 0 ^ i ^ are the corresponding coefficients

of X in the characteristic equation of A, eq. (4-22). Using eq . (4-33),

we now integrate eq. (4-32) from an initial rating period ending at

time to a final rating period beginning at time t^. Assuming that

D P(t) =0, 1 ^ k < n-2, in both rating periods, one obtains

56
*l (4-34)

where F(0) is the adjoint of -A and

(4-35a)

From eq. (4-25) written in matrix form, DX(t) = X^X(t) , one concludes

that during rating periods

Substituting eqs. (4-35 into equation (4-34) one obtains

= F(o)c''Cpmt

Using the fact that substituted in place of D in eq. (4-33) yields

j)(Xp = 0 and that (see [36], p. 151)

Flo) ^ adjC-A) . ^dAjA « -A"^ ^ H"'G

One obtains from eq. (4-36)

or the set of equations

57
where

(4-38b)

The expression for can be put into a form better suited for physical

interpretation by noting that from section 4.2^

^l^^l^

satisfies the equation

(4-40)

Premultiplying both sides of eq. (4-40) by C and looking at the

ith entry of the resulting equations gives

Summing both sides over i from 1 to n and using the definition of

and using eq. (4-26) gives:

(4-41)

58
Dividing the numerator and denominator of eq. (4-41) by d^(Xj^) and

inserting into eq. (4-38c) gives

Eq. (4-38a) can be expressed in terms of the instantaneous temperature

of the thermometer^ eq. (3) of section 2.0, if T^(t) is replaced by

T*(t), T^^ by T^, and is replaced by E* where in place of eqs.

(4-38c) or (4-38e) we have

(4-38e)

The interpretation of the quantities in eqs. (4-38a), (4-38b), and

(4-38d) are given in the text. In eq. (4-38c), A.^ is the cooling

constant. |h|/2^

*^ij^j steady-state heat transfer coefficient,

between the calorimeter and surroundings as measured at the ith cell

if constant power is supplied from the source of time -varying power

with a geometrical distribution that is identical to the set of 'Ij's

of eq. (4-38d). (That is suppose Pj (t) = ' and Pj'/P' = q^ where

P' = 2.P,'. From eq. (4-27), the difference in the convergence

1^ J temperature measured at the ith cell when P(t) =0, T and P(t) = P',

ooX T^^, is given by

IHI IrL-Toci) . P'

59
In eq. (4-38e) the numerator is equal to the increase in internal

energy of the calorimeter, ALT, divided by the change in the temperature as measured at the ith cell, T. (t-) - T,(t-), in the final and

initial rating periods. This follows from eq. (4-29) since

tV-.l]tini{^,)-li(^,)] .-[£jCjd;t»,)][Ti(0-T.<t,)J/,i.ft,)

n The term )d . (X,) /d . (X) = h, ' in the denominator of eq. (4-38e)

•j^J s JS J i 1 1

is -XALJ/[T^(t2 - T.(tj^))] from eq. (4-4<) . It can be shown to be

equal to the heat transfer coefficient between the calorimeter and

surroundings, as measured at the ith cell, if the denominator of

eq. (4-38e) is one. This follows from eq. (4-37), the ith entry of

which is

(4-43)

(4-44)

Premultiplying by 2 h. j summing over all i, and using eqs. (4-26)

S X s and (4-29) gives, after interchanging i with j

Premultiplying eq. (4-43) by h^', and subtracting frcan eq. (4-44)

' ' IHI (4-45a)

The left hand side of eq. (4-45a) is the heat absorbed by the calorimeter

plus the work done by constant power sources during the main period

60
This may be established as follows. Summing eq. (1) overall i, 1 <

i < n^ we have:

at ^'^j

The second double sum on the r.h.s. of eq. (4-45b) can be broken into

a sum over the set of i and j corresponding to interior cells (cells

not adjacent to the surface S) and the set of i or j corresponding to

boundary cells (cells contiguous to surface S) . The first sum involving

the exchange of heat between interior cells vanishes because S.h. , =

J ij

S.h,, =0 but the second sum will not vanish, in general, because

for the ith boundary cell 2,h^ . = -S h. if the surface S is located

J ij s IS

in a Stirred fluid. In can be shown that (4-45b) reduces to

dt

where 'h^^ has been defined in connection with eq . (4-26). It should

be noted that^ so long as the exchange of heat between the cells

forming surface S and the cells enclosed by S is linear^ eq. (4-45c)

is valic whether or not the heat exchange between interior cells is

13 linear or not. The terms involving T^ and P° may be eliminated by

noting that in a convergence rating period dT^(t)/dt = 0 when T^(t) =

T 1 < i < n. Substituting these conditions into eq. (4-54c)^

ool — —

subtracting the resulting equation from eq. (4-54c), and integrating

from time t^ to t2 ccmpletes the proof.

13

the l.h.s. of eq. (4-45c) is the rate of increase of internal energy

of the system bounded by the surface S; the r.h.s. is the total rate of

doing work on the system and the rate of absorption of heat by the

system. Heat exchange between interior cells cannot contribute to

the latter.

61
If the denominator of eq. (4-38e) is one, the term involving

W vanishes. It can be shown (see section 4,5) that this

occurs, in general, if and only if the region in which the various > 0 is isolated (i.e. = 0, see section 4.5).

From the previous discussion in section 4.2 and the ,

.

' s and

d^(X^)'s, we can make the following statements:

(1) The energy equivalent, E^, as measured by the corrected

temperature rise of the ith cell, will, in general, be different

for each ith cell and will also be different for different

geometrical distributions of the work input, a different set of

q^, (since Q'-j/jlll are not equal for all i, j inside the region

where q^ > 0 unless the region is isolated).

(2) The energy equivalent is always positive, if W is positive

(allcy../|H| > 0, d.(A )d.(A. ) > 0).

(3) The energy equivalent is not equal to the sum of the heat

capacities of cells of the calorimeter (since all d^(\^) are

not equal) but the contribution of the heat capacity of every

cell to the energy equivalent is positive (d^(A.^) /d^ (X^) > 0).

In sections 4.2 and 4.3 we assume that C and H are non-singular

matrices having the same dimensions or, equivalently, that the

temperature of every cell inside the surface S changes with time (even

if only infinites imally) while every cell forming the surface S is

constant. This assumption involves no loss in generality provided

the temperature of the surroundings is constant since eq. (1) for

the s^th cell making up the surface S is

0 = E.h^.[T.(t) - T ] + 2'h ,(T - T' ) + P° + P (t) iSli^' s sss s s"^ s s

62
The sum over s' involves all cells in the region comprising the

surroundings (including those making up the surface S) and the sum over

i involves all cells inside the surface S that exchange heat with

the sth cell. In order for T and all T' to be constant, S.h .T. (t) = — s s 'isii

-P (t) . From the properties of the matrix form of the resulting

s equations for all the cells in the region comprising surroundings , it can be seen that the temperatures of another envelope of cells

that completely encloses the surface S are uniquely related to the

set of temperatures ( and vice versa). Thus, all locations of

the surface S may be reduced to a single boundary surface S such

that all temperature inside this surface change with time.

It should be noted that since, by definition, ^-j^^^j^) j ^^j^) zero outside the surface S and greater than zero inside this surface,

and since d^(A.^) /d^ (?^^) is conceptually measurable from the temperature

distribution in the convergence rating period and any other rating

period (see eq. (4-29)), the location of the boundary surface, S,

is in principle (i.e. not practically) determinable from experiment.

The implication of the assumption that each of the temperatures,

T , is constant necessarily requires that the regulator of the

s constant -temperature jacket (see [1]) have "reset" as well as

"proportional" control [39], and that all leads, stirrer shafts, etc.

be well-tempered (thermally) [17] in their passage through the

jacket. Also, the resistance of the current leads of an electrical

heater must be sufficiently small that the change in temperature due

to passage of current through the leads is negligible (see section 4.4).

The importance of good temperature regulation of the jacket and thermal

tempering is further emphasized by the requirement that the location

63
of the boundary surface S must be the same in both "calibration"

and "unknown" experiments.

64
4.4 Non linear heat transfer during the main period

Eq. (9) is derived as follows. By analogy with eq. (4-45c),

dQ(A)/dt is given by

The terms involving T^(t) are eliminated by writing eq. (1) for all

the m cells between A and S in the matrix forms

H r = y

A) Al

Hj^ is a (pom) matrix whose entries are defined by

column vectors having, respectively, the entries

1 < i < m. Letting G = (G exists since H^^ is irreducibily

diagonally dominant) we have

^

(4-48)

Substituting the ith entry of T of eq. (4-48) into eq. (4-46) and

rearranging gives

65
where

w . rn

By adding all the rows or columns of to the i^th row or column and

expanding H^^ by cofactors of this row or column one obtains

Inserting eq. (4-50) into the expression for R gives eq. (9) with

The terms involving T and T in eq . (9) can also be written in

S 3, the form EEh'T - E2h"T where

s a as s s a as a

By inspection^ if H^^ is symmetric then h^ = 0 for each a and if T^(t)

is uniform at each moment in time 2h'T =TSh'=0. IflL. is

a a a a a a lA

as3rmmetric, then G is as3rmmetric and vice versa so that for stirred

fluids G.. 4 G.., l<i<m.

Since every entry of G is greater than 0 (see section 4.2)^

6^ (A) is greater than zero and from eq. (4-50) is less than one.

66
Eqs. (10) and (14) may derived as follows. Substituting eq . (9)

into eq. (8) and rearranging using T^(t) as defined in eq. (11)^ gives

Ml M " • (4-53)

m

where h = S 2 h", T = S E h' T /h, , P° = S.3.(A)P.% P^(t) - ^ A s a as'^ s s a as s M i M

E.3.(A)P.(t), P° = 2.P?, and P (t) = 2.P_(t). The terms involving

j^l 1 i A -^1 i A ^1 i

Tg, P^^ P^ may be eliminated by noting that in a convergence rating

period, dU^(t)/dt = 0 and dT^(t)/dt =0, 1 < i < m, P^(t) = P^(t) = 0,

and T. (t) = T. . Substituting these conditions into eq . (4-53) and

A Aco substracting the resulting equation from eq. (4-53) gives

h^ can be expressed in terms of by noting that eq. (4-45) holds

for rating periods when it can be shown that eq. (4-25), eq. (4-28), and

eq. (4-29) are valid with T^(t) replacing T.(t), with d^(X^) =

(T, Z h" d (/V,))/h. replacing d. (X.). and T , replacing T . Substituting

asasa l A xi' oa o^x

the equations analogous to eqs. (4-45) and (4-48) into eq. (4-54) after

replacing dU(A) dt with Z.C.dT. (t) /dt gives

1^ ^

^

^ _ -

' " if^-aL(A,) ti;"p-(A)CcaL(x.) (4.55)

Integrating eq. (4-54) between the times t^ and t^ gives

(4-54)

t

(4-56)

67
To derive eq. (10), we replace U (t ) - U(t.) with 2.C.[T.(t^) - A z i ^1 1 i 2

T^(tj^)] in eq. (4-56), use the equation analogous to eq. (4-29)

and eq. (4-55), and rearrange. is then given by

_ A il LJ _ (H-Slh)

To derive eq. (14) one must reinterpret the steps in the previous

procedure. In deriving eq. (4-54) we assume the convergence rating

period and, in particular T^, is that appropriate to the initial

rating period and also assume that the parameters in eq. (4-55) applyto the initial rating period only. Then, substituting eq. (13)

for ^^(^2^ ~ ^A^*"!^ following the same procedure as before

eq. (14) where E^^ replaces E^ in eq. (4-57) and the parameters on

the r.h.s. of eq. (4-57) are those for the initial rating period only.

Identical results are obtained if instead of eq. (8), one applied

conservation of energy at each moment in time to the entire calorimeter.

Calling dUg(t)/dt the rate of increase in internal energy of the

calorimeter and dQ(S)/dt the rate of transfer of heat to the calorimeter

in the direction from S to A one may write

(4-58)

Applying conservation of energy to the m cells between the surfaces

A and S

"

5t at dt

'

Solving the eq. (4-59) for dQ(S)/dt and substituting into the eq. (4-58)

68
along with the definition of dU_(t)/dt^

MH)^ fcidTili)

7i di ' at

<*-^°>

gives, upon rearrangement, eq. (4-53).

By inspection, of eq . (4-55) is independent of the location of

the sources. The numerator of the r.h.s. of eq. (4-57) is not

proportional to the increase in the internal energy of the cells inside

A (since some 3^ (A) > 0), the denominator is not equal to one (since

all d (^, ) are not equal), and E is not exactly additive with respect

a J. A. to the heat capacities of the cells inside the surface A (since the

d^(^j^) are not equal for 1 ^ i ^^)« These statements are, however

true (section 4.5) if and only if the surface A is located in some region

that is isolated (from either the remainder of the cells inside

surface S or surface S itself).

If the initial and final rating periods are the convergence

temperature, as is frequently the case in conduction calorimeters,

in section of eq. (4-57b) into eq. (14) yields

Thus, h. is the steady state heat 'A

cells inside the surface A and its

analyzing the data in this form is

other types of isoperibol calorime

transfer coefficient between all the

surroundings and the method of

analogous to the method used in

ters where the magnitude of A.^ is

69
not large. The same deductions concerning equivalence of sources apply

in either case since these are associated with an analysis of the teme

m t. S.J p.(A)P.(t)dt and T (t) of eq. (10),

1^ '^l J J ^ m

The evaluation of the terms Z! .8 . (A) P . (t) = P,,(t) for the

case of an experiment where electrical power is supplied to a resistance

heater can be performed as follows. We consider only the

case given by West and Ginnings [13] which is illustrated by

fig. 3. This involves the assumption that the heat transfer

coefficients of each of the m' cells of the current lead between

surfaces A and S with the other cells between A and S is negligible

in comparison to the heat transfer coefficients between the cells

of the lead. Following the analysis given by West and Ginnings^

we assume that the pth cell of the current lead is located

14

in the surface S and changes with time. This is permissible since

it can be seen from its derivation that eq. (9) is also valid

when T is a function of tine . It is assumed that none of the other

s cells forming the surface S change with time. The temperature of the

elements of the surface A including the ^ + m' + Jth element are

all assumed to change with time and are different. The number of

elements of the heater inside p^ and inside, A, ^'-m', are assumed

to be sufficiently large that the heater lead can be assumed to be

in contact with the surfaces A and S for an infinite distance,

thermally speaking.

Consider those terms, [ (dQ(A) /dt] ^, in eq. (9) which apply to

the heater. They may be written in the form (note P^° =0^ ^ ^ ^

^^A)f ^Jh' , tVH^^f'^ ^P(^^lB'(A)'l^C'Jv(i)m^^-

^ If some of the Tg change with time these Tg and the associated ^^Ks

must be known (or measured). In this case, neither Tp or Tp+m'+i

need .^e known provided we know all other T and T (see [13] or eq. (4-62)).

70
(4-62)

In eq. (4-61) we assume the power input per unit length is uniform

and equal to P(t). The variation in time of [T (t) - T , ..^(t)]

p prm +i

can be expressed in terms of the temperatures of the elements in S

and A adjacent to the heater using eq. (4-48).

In eq. (4-62), h and h are assumed to be the uniform heat

S 3

transfer coefficients per unit length with the surfaces S and A

respectively. The sums over s and a are over only those elements of

S and A, respectively, just adjacent to the heater lead.

7y l<j<P+-^', is the difference between the entries lying in

the £th and (£ + m' + l.)th rows but in the same jj^h column of the reciprocal

of the (p + X') square triadiagonal matrix t^^jl which is defined by:

In this matrix, h is the heat transfer coefficient between adjacents

cells of the current lead.

By sjmimetry it can be shown that if h = h , then =0.. Thus,

^ s -^3 3 the power term will vanish in eq. (4-62) under these circumstances.

71
The third term on the r.h.s. of eq. (4-61) can be evaluated from

eq. (4-49) to be

Flt)Z; P(t)J> Ij 0^.'

where G , . is the entry lying in the m'th row and jth column of the

reciprocal of the m' -square tridiagonal matrix t^^j] whose entries

are defined by

In this matrix, h is again the uniform heat transfer coefficient

between adjacent elements of the heater lead. If one substitutes

eqs. (4-63) and (4-62) into eq. (4-61) the net power term corresponding to Pjjj(t) of eq. (4-54) when h^ 4 h^ is

(4-63)

The reason for the initial boundary selection was simplicity of

calculation.

This net power term is exactly analagous that derived before [13],

but has been shown to be equally valid when the temperatures of the

elements of the surfaces S are different and the temperatures of the

elements of A are not only different but change with time.

Frcrni an examination of the manner in which this new power term is

derived here or as before [13], it can be seen the original

72
assumptions amount to the condition that the matrices tl^jl

[q|j] approximate the heat exchange problem for the increment in

temperature in the lead due to passage of current. (The heat exchange

when P(t) = 0 is completely accounted for by the measurement of T (t)

or its equivalent.) The operating condition mention in section 3.1 is

that the current in the lead must be sufficiently small that heat

transfer by thermal radiation can be neglected.

Since it has been assumed that the locations of the surfaces A

and S are known and h' , . ,

, is an unknown parameter, p+1, p + m' + ljj

one may regard the location of potential taps on the leads as

tantamount to specifying where the surfaces A and S are located in

the vicinity of the current leads assuming this unknown parameter

is zero. The experimental test cited in section 3,1 is a method for

verifying this location of potential taps.

The comment made in section 3.1 concerning the type of radiation

shield calorimeter described by Ginnings and West [17] or West and

Westrum [18] is established as follows. Consider, as a practical example,

a radiation shield calorimeter as sketched in figure 4 with two radiation

shields, I and II. If these shields are designed properly, the overall

heat transfer coefficient h^^ between arbitrary elements a on surface

A and b on surface B is effectively zero unless A^ = T^, that so called

"lip" or "shunt". If follows that the temperatures, T , on the

shield I should depend only upon the temperature T'.

73
Now apply eq. (4-48) to each of the elements of making up

surface A.

In eq, (4-64), m' is the number of elements between B

and S and each G . is derived from a matrix involving the various heat

transfer coefficients of all these m' elements. CXir assumption requires

Selecting some arbitrary cell, a = k, of the surface A, one obtains

from eq. (4-64) (ie. replace a by k in eq. (4-64) and subtract from

eq. (4-64).

74
(4-65)

The last sum of eq. (4-65) cannot vanish because otherwise the surface

A would behave as if it had zero heat capacity.

Because S and A are closed surfaces^ eq. (9) applies. Substituting

eq. (4-65) into eq. (9) gives

(4-66)

In eq. (4-66), m is the total number of elements between A and S and

h' and 3 . (A) are to be derived from a matrix containing the various

as J heat transfer coefficients of only these elements. Carrying out

an analysis analagous to that given previously in this section will

show that the energy equivalent derived from a measurement of T^(t)

will be independent of the spatial distribution of the power input to cells

75
inside the surface B but not to elements between the surfaces B and

S since the last two terms in eq. (4-66) are not zero.

I'

^

76
4.5 Some consequences of the theory concerning the energy equivalent.

In this section^ we establish the validity and examine some of

the consequences of the statements made in both the main text and

details that^ in general^ the energy equivalent is an exact sum of

heat capacities and is independent of the location of the sources

when = 0. The case when A.^ = 0 is called the "limiting case".

To start with^ if A.^ = 0, then some region inside the surface S

is isolated (i.e. exchanges no heat or mass with its environment) and

vice versa. For convenience, we shall call this region that bounded

by the surface A and allow the position of this surface to be arbitrary

to the extent it may be either inside or coincide with the surface S.

Suppose first that the region bounded by the surface A is isolated.

(We shall always use the expression "surface A" to denote the surface

A and A alone shall denote the matrix A defined in section 4.2.)

Then the matrices H and A are factorable:

to be associated with the cells between surfaces A and S and those

inside surface A, respectively. Both H and A are singular since

and A^ are singular; = 0 since A is singular. If the surface A

coincides with the surface S, the row sums of the entries of the rows of

matrices A and H are zero (see section 4.2 for definition of entries

of A and H) and again A is singular. The converse case is required

because X = 0 means A is singular (see [36], p. 151) and, since C is

where we have ordered the matrices H and A so

77
non singular, H is singular. In general, this implies the property

that both A and H are factorable or the row sum of their entries are

zero. (Thus, = 0 is not possible for real isoperibol calorimeters.)

Using this result, we show that the a. . of eq. (4-38e) (or

eq. (4-45) (or eq. (4-45) and eq. (4-26)) which are the entries

of the adjoint of H, adj H, are, in general, equal and differ from

zero inside the surface A and are zero outside the surface A only

when = 0 and vice versa. We also show the same result for the

entries of Dj^(Xj^) of eq. (4-25). Suppose first the region bounded by

the surface A is isolated.

Partitioning the adjoint of H, adj H, and Dj^(A.j^) to be conformable

with H and A, respectively, we have the result

Since and A^^ are non singular, adj H^^ = 0 and t^^j^C^]^) 1 = ^

(see [36], p. 76). Since and A^ are of rank n - 1 (i.e. the first

principal minors are irreducibly diagonally dominant^, (adj H) and

A

[D^(^^)]. have a rank of one and are not null (see [31], p. 50). Since

J. J. ir\ the sums of the entries of the rows of H. , A. are zero, all the entries of

A A

^

(adj H)^ or [^^^^''^'p^^ °^

'^'^i^^I^-'a

equal and positive. Suppose, on the

hand, that the entries of (adj H)^ or equal. From the

definition of the entries of H and A this requires X^ = 0 and h. , = 0

for all cells a' making up a surface A' that encloses A that is inside

surface S and i refers to all cells between the surfaces S and A', the

region M' , Also adj H^, = 0, and ~ 0* later purposes,

the surfaces A and A' can be taken as identical.

78
om

As indicated in sections 4.3 and 3.2 the important consequence

of the results for Dj^(Xj^) is that, in general, the energy equivalent

is not an exact sum of the heat capacities of cells inside A since

= 0 ±s not possible for a real isoperibol calorimeter. (i.e. fr

eq. (4-57) one can see that if the cells inside A are isolated,

Jl X, = 0 and E, = 2,C.). To obtain an estimate of the deviation of

1 A j^i X d^(Xj^) /d^ (/V^) from unity, where i. is a cell bounded by surface A,

assume all are equal, all are zero, let a ' be a cell in those

forming the surface A, and define e_j^^, by

Then from eq. (4-29) one can see that during a rating period

Thus, if the calorimeter is spherical (ie. if the surfaces A and S

are concentrically located spheres, and the thermal properties of the

cells between A and S are spherically symmetric), »

1 for a cell, 1 ^ i ^

-^'^ inside the envelope of cells forming surface

Aj e = I for a cell a forming surface A: and e. , < 1 for a cell

aa ° la'

1 < i < m between A and S.

The expression for E., eq. (4-55) can be written in terms of e. ,

= J _ —

79

as
where is the number of cells inside but not including those maki

up the surface A. For a spherical calorimeter^ the difference E^(full)

E^(empty) will be larger than the true heat capacity of a sample.

From this expression for E^, it can be seen that for the region inside

the surface A containing cells i, 1 < i < , where one locates a

sample whose heat capacity is to be measured, ®. , and e , must be

3.3. as close to one and zero, respectively, as possible. Calorimeter

vessels containing vanes welded to the walls of the vessel are an

example of a design to make ^j^g' close to one.

A sufficient experimental test to determine whether or not e. , la

for the cells in the sample region deviate appreciably from one is

» not, as proposed previously [8 to determine whether or not E^

increases linearly as sample as added to the calorimeter (ie . suppose

e. , is constant for each cell i that contains the sample). Rather

la

the procedure should be to extrapolate E^ to = 0 constant for

both full and empty calorimeter, which is the usual practice (see

[40]). (As \, -- 0, B,(A) -- 0 and h , -- 0) . Just how this

1 X as extrapolation should be carried out has not been worked out and will

depend upon the construction of the calorimeter. For example, it is

evident from eq. (4-57) that in the event that h^ can be approximated

2 as a power series in and terms higher than \^ can be neglected,

then a plot of E^ versus A.^ would be the proper method of extrapolation.

This assumption for h^ has not been verified, however, except for the

case where is extremely small (see [38], p. 67).

80
The consequence of the result for a-, . when the cells bounded by

A are isolated is that the energy equivalent is also independent of the

source location regardless of whether or not the thermometer measures

a temperature equivalent to so long as the thermometer

is located inside the surface A. (The denominator of eq. (4-38e)

becomes one as h. , 1 < i < m and all a, go to zero because

„ la' — — ' ° n m n ^ijqj/ H > 1 ^i^^h^^ and from eq . (4-67) ^.E^h.^d . (\^)/d. (\^) —

;

m

SjE^h . In the case of the equations of section 4.3, this follows

^1 a xa ^ since then dQ(A)dt = 0 at all times).

A third less important consequence of the limiting case is that

the non uniformity of the convergence temperature distribution produced

by a npn uniform temperature distribution on the surface S is reduced

as the cells inside A beccsne more isolated. This is perhaps best

illustrated by a simple model in which the matrix H is given by

10

O

13

2i

where h = + h^^, and h^^ = h^^ + h^^ , and s and s' refer to the

two cells of the surroundings surface S that have different constant

temperatures^ T^ and T^ , . If h^^ and h^^ are small^ and one expands

I h| to the first order of small quantities, one obtains for the difference

in the convergence temperature distribution if T ^ T , s s

J_ [h24,'fe,A'^4.'^ta] LT4,-r4!i

81
The effect of constant power input P^*'^ ^ ^ ^ ^ -^f these differences

in convergence temperatures is

3 o

o

where = ^lO^^^lO ^20^ ^2 " ^^20^^ ^^10 ^20^* ^^"^ effect

of increasing the isolation of the calorimeter reduces the already

small gradient in the convergence temperature distribution due to a

non uniform temperature distribution in the surroundings. However, this

does not reduce the gradient in the convergence temperature distribution

due to the constant power inputs. The effect of constant power

input is reduced if either h^^ 1 < i, J ^ ^ large relative to the

values of (thus producing a small change in the distribution for

a given percent change in a h^^), all P^°'s are zero, or if the power

input to the region where the process of interest takes place is reduced

(ie. let this region be cells 1 and 2, let h„_ = 0, and look at T„ ZU ZCD T, ).

As indicated in section 4,2, we have not established the general conjditions under which and the real part of are well separated. Some

suggestion that this is plausible, however, comes from simple numerical

calculations. For example consider a three cell model where A

is given by

82
O 41 -(

Letting the cell number equal the number of the diagonal entry of

this matrix, this corresponds to the situations where cell 3 exchanges

heat with only cell 2, cell 2 exchanges heat with cells 3 and 1,

and cell 1 exchanges heat with only cell 2 and the surroundings

which has a uniform temperature. (h^^^ = 1, h^^ = h^^^ = e, h^^ = = !•)

The calculated results for various values of e are as follows

e (-\) (-^3) d2(\)

.198 1.55 3.25 .328 .591 .737

.5 .145 1.40 2.45 .233 .632 .739

.1 .044 1.10 2.06 .065 .689 .721

.01 .005 1.01 2.01 .007 .705 .709

0 1 2 0 .707 .707

It can be seen that as e goes to zero cells one and two become

isolated from cell 1 and have equal d^(Xj^) for cells two and three.

As e increases d^(A.^) and d^C-^^) become progressively different but

the separation of roots and is maintained; d^CA.^) >

^2^^l^ ^

dj(^^) as we have shown on an apriori basis using physical arguments.

These results suggest that the separation of and X^ apply not only

in the case where the magnitude of A.^ is small^ conventional isoperibol

calorimeters, but also when it is relatively large, as in the case of

conduction calorimeters.

83
Table 1. Summary of sjnnabols

Symbol Definition

Matrix operations for a matrix H:

adj H adjoint of H (transpose of a matrix formed from

H by replacing its entries with the co-factors

of the entries)

|h|| determinant of H

H reciprocal of H

diag( ) diagonal matrix

Notation (location: equation number—section)

A nxn matrix, eq. (4- 17) -4.

2

a^(A.^) eq. (4), text before eq. (4-25) -4.2

C nxn matrix, text after eq. (4-12) -4.

2

C. i^th entry of C, heat capacity of j.th cell, text

^ before eq. (l)-2., eq. (4-8a)-4.1

C text after eq. (15) -3.

3

p

C text after eq. (15) -3.

3

-r D ordinary differential operator with respect to

time, d/dt

T>^(\) eq. (4-25) -4.2

d,(X ) i^th entry of eigenvector of A for the eigenvalue

^ (1 < h < n)

d,(\j^) i^th entry of eigenvector of A for the eigenvector

(corresponds to the tth cell inside the surface S)

d^(A.j^) entry of eigenvector of A for the eigenvector

corresponding to a cell a forming the surface A

84
Definition

energy equivalent measured at ^th cell, text after

eq. (6) -2.1, eq. (4-38e)-4.3

energy equivalent determined with a thermometer,

eq. (7)-2.1, text after eq. (4-38e)-4.3

energy equivalent determined with a thermometer

that measures T^(t), eqs. (10), (11) -3.1

energy equivalent determined with a thermometer

that measures T. (t) with the reactants present

in the calorimelcer, text after eq. (14) -3.

2

adj(Al-A), text after eq. (4-22) -4.

2

adj(-A), text after eq. (4-34) -4.

3

nxn matrix, text after eq. (4-12) -4,

2

raxm submatrix of H whose diagonal entries are

for the m cells in the region between the surfaces

A and S, text after eq. (4-47) -4.4

ith diagonal entry of matrix H, text after eq.

(4-12)-4.2

±, jth entry of matrix H, text after eq, (4-12) -4.

2

heat transfer coefficient between ^th cell inside

surface S and the sth cell forming surface S,

text before eq. (17-2., (eq. (4-8b)-4.1, text

after eq. (4-10) -4.1)

heat transfer coefficient between _ith cell and j^th

cell; both cells inside the surface S, text before

eq. (l)-2., eq. (4-8b)-4.1, text after eq. (4- 10) -4.1

total (i.e. all paths) heat transfer coefficient

between the ath cell of surface A and sth cell of

surface S, text after eq. (9)-3.1, eq. (4-51)-4.4

subscript, any cell inside surface S

nxn matrix, diag(l, ,1)

subscript, any cell inside surface S

85
Symbol Definition

IC|| (Xj^) nxn matrix, eq. (4-20) -4.

2

i number of cells enclosed by surface A

m number of cells between surfaces A and S

n number of cells enclosed by surface S

P? constant power input to i^th cell, text before

^ eq. (l)-2., eq. (4-8c)-4.1

P.(t) time varying power input to i.th cell, text

^ before eq. (l)-2., eq. (4-8d)-4.1

s subscript, any cell forming the surface S, text

before eq. (l)-2.

t time

t^ a time after which exp(A.^t), k > 1, can be

neglected, text before eq. (4) -2.1

beginning of main period or end of initial rating

period

end of main period or beginning of final rating

period

T(t) nxl column matrix, text before eq. (4-12) -4.

2

T nxl column matrix, eq. (4-15) -4.

2

00 T(0) T(t) when t = 0

T^(t) i.th entry of T(t), temperature of i.th cell at time t

T^^ i.th entry of T^, convergence temperature of i^th cell

T^(0) ith entry of T(0), temperature of ith cell at t = 0

T constant temperature of sth cell forming surface S, see text before eq. (1)-2.1, text at end of 4.3

T (t) temperature of ath cell forming surface A, see text

^ before eq. (8), (9)-3.1

T*(t) temperature of thermometer at time t, see eq. (3) -2.1

86
Definition

T*(t) at t =

°°

eq. (11)-3.1

text after eq. (15) -3.

2

text after eq. (15)-3.2

internal energy of region enclosed by surface A

at time t, text before eq. (12) -3.

2

text before eq. (12) -3.

2

text before eq. (12)-3.2

text before eq. (12)-3.2

text before eq. (13)-3.2

total work done on all cells enclosed by the

surface S, text after eq. (6) -2.1

nxl column matrix, T(t) - T , eq. (4-16)-4.2

00 X(t) when t = 0

i, 2th entry of adj text after eq. (4-25) -4.2,

eq. (4-27)-4.2, eq. (4-38c)-4.2, text 4.5

text after eq. (9)-3.1, eq. (4-49)-4.4

kth eigenvalue of the matrix A

eigenvalue of matrix A having the smallest magnitude

weighting factors for thermcmeter, eq. (3) -2.1

87
5. References

[I] Rossini, F. D,, Ed., Experimental Thermochemistry, Measurement

of Heats of Reaction, (Interscience Publishers, Inc., New York,

NY, 1959).

[2] Macleod, A, C, Trans. Faraday Soc. 63, 289 (1967).

[3] Jessup, R, S., J, Appl. Phys. 13, 128 (1942).

[4] West, E. D., and Churney, K. L. , J, Appl. Phys. 39, 4206 (1968).

[5] Hubbard, W. N. , O'Hare, P. A. G., and Feder, H. M, , Experimental

Inorganic Thermochemistry, p. 115, Book, Annual Review of Physical

Chemistry, Ed. Eyring, H. , Christensen, C. J., Johnston, H. S. (Annual Reviews, Inc., Palo Alto, California, 1968).

[6] White, W. P., The Modern Calorimeter (The Chemical Catalog

Co., New York, 1928).

[7] West, E. D., and Churney, K. L. , J. Appl. Phys, 41, 2705 (1970).

[8 ] Churney, K. L, Armstrong, G, T., and West, E. D., Macrocalorime tryHow Accurate?, Status of Thermal Analysis, Ed,, 0. Menis, NBS

Special Publication 338, Oct. (1970) (U.S. Govt. Printing Office,

Washington, D.C., SD Catalog No. C13. 10:338).

[9a] Margas, E., Tabaka, A., and Zielenkiewicz, W. Bull. Acad. Pol. Ser.

Sci. Chim. 20, 323 (1972); 329 (1972).

L9b] Davids, N. and Berger, R. L. , Currents in Modern Biology 3,

169 (1969).

[10a] West, E. D. , J. Res. Natl. Bur. Stand. 67A, 331 (1963);

[10b] West, E. D., Trans. Faraday Soc. 59_, 2200 (1963).

[II] Swietoslawski, W. , J. Am. Chem. Soc. 39, 2295 (1917); Book,

Microcalorimetry (Reinhold Publishing Corp., New York, NY, 1946),

pp. 28 ff.

[12] Wilhoit, R. C, J. Chem. Education, 44, No. 7, A571 (1967),

see eq. (6), p. A588; eq. (5) is equivalent to our eq. (9) only

if the iacket coincides with the surface S and P.° = 0 for

all cells between A and S.

[13] Ginnings, D. C, and West, E. D., Rev. Sci. Instr. 35, 965 (1964).

[14] West, E. D., and Ishihara, S., Rev. Sci. Instr. 40, 1356 (1969).

[15] White, W. P., J. Am. Chem. Soc. 40, 1858 (1918).

[16] Prosen, E. J., and Johnson, W., to be published.

[17 ] Ginnings, D. C, and West, E. D. , Chapter 4, Book, Experimental

Thermodynamics Vol. I, Calorimetry of Non-Reacting Systems,

Ed., McCullough, J. P., and Scott, D. W. (Plenum Press,

New York, NY, 1968).

88
[18] West, E. D., and Westrum^ E. F.^ Jr., Chapter 9, Book, Experimental

Thermodynamics, Vol. I, Calorintetry of Non- Reacting Systems,

Ed., McCullough, J. P., and Scott, D. W. (Plenum Press, New York,

NY, 1968).

[19] Mosselman, C, and Dekker, H. , Rec. Trav. Chim. 88, 162 (1969);

Coops, J., Adrianse, N., and Van Nes, K. , Rec. Trav. Chim. 75,

237 (1956).

[20] Benzinger, T. H., and Kitzinger, C, Chapter 5, Book, TemperatureIts Measurement and Control in Science and Industry, Vol. 3,

Part 3 (Reinhold Publishing Corp., New York, NY, 1963).

[21] West, E, D., Case, W. E., Rasmussen, A. L, and Schmidt, L. B.,

J. Res. Natl. Bur. Stand. 76A , 13 (1972), see section 10.

[22] Prosen, E. J., Rossini, F. D., J. Res. Natl. Bur. Stand. 33,

255 (1944).

[23] Dickinson, H. C, Bull. Bur. Std. U, 189 (1914).

[24] Hubbard, W. N., Scott, D. W., and Waddington, G., Chapter 5,

Ref. [1], pp. 99, eq. (86).

[25] Prosen, E. J., Chapter 6, Ref. [1], pp. 130.

[26] Ref. [16], pp. 104.

[27] Landau, L. D., and Lifshitz, E. M, , Theory of Elasticity

(Pergamon Press, New York, NY, 1959), Chapter 1, and pp. 119-121.

[28] Ref. [9a], footnote 2.

[29] MacNeal, R. H. , Quart. Appl. Math. 11, 295 (1953).

[30] Varga, R. S., Matrix Iterative Analysis (Prentice Hall,

Englewood Cliffs, New Jersey, 1962).

[31] Landau, L. D. , and Lifshitz, E. M. , Fluid Mechanics (Pergamon

Press, New York, NY, 1959), pp. 184ff .

; from the discussion following

eq. (49.2), s in eq. (49.4) may be replaced by e for an incompressible fluid.

[32] Kay, J. M. , An Introduction to Fluid Mechanics and Heat Transfer

(Cambridge University Press, Cambridge, England, 1963), Chapter 6.

[33] Hinze, J. 0., Turbulence (McGraw Hill, New York, NY, 1959), pp. 24,

Chapter 5.

[34] Goldman, S., Transformation Calculus and Electrical Transients

(Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1949).

89
[35] Frazer, R. A., Duncan, W. J., and Collar, A. R., Elementary

Matrices and Some Applications to Dynamics and Differential

Equations (The l^cMillan Company, New York, NY, 1947).

[36] Ayres, F., Jr., Theory and Problems of Matrices (Schaum

Publishing Co., New York, NY, 1962).

[37] Bellman, R., Introduction to Matrix Analysis (McGraw-Hill Book

Company, New York, NY, 1970).

[38] Wilkinson, J. H. , The Algebraic Eigenvalue Problem (Clarendon

Press, Oxford, 1965).

[39] Westrum, E, F., Jr., Furukawa, G. T., and McCullough, J. P.,

Chapter 5, Book, Experimental Thermodynamics Vol. I, Calorimetry

of Non-Reacting Systems, Ed. McCullough, J. P., and Scott, D. W.

(Plenum Press, New York, NY, 1968).

[40] Cole, A. G., Hutchens, J. 0., Robie, R. A., and Stout, J. W.,

J. Am. Chem. Soc. 82, 4807 (1960).

90
6 . Acknowledgements

The authors are particularly indebted to R. N. Goldberg and

T, B. Douglas for their comments on the manuscript, and to

Miss Debra Miro for typing the manuscript.

One of us (K. L.C.) particularly wishes to acknowledge the

numerous helpful discussions of the contents of this paper with

R. L. Nuttall and for mathematical cOTments by Morris Newman.

91

Fig. 2. Sketch of a cell located in a gas-solid boundary

('), interior point of a cell; B, gas-solid boundary;

area of gas solid boundary in cell 1.

93
y 1*1 1

1

///// /

8;.

t—l <D + 5

o

•\

I-) CO

t-l <u

y-K U

•U <fl X

a •p-l (U (U CO • M O

!-t CO <U CO IH P 0) O H CO r-\

?-l CO - •U

(U M 0)

to <: + !^U

CO o

O

CO

CO 0)o

CO + U-J

<4-J O

CJ 3 e m U CO i-J <U +1-4

CO CO a, O

0)

•rl

Pt4

94
Fig. 4. Sketch of a radiation shield calorimeter

S, surface S; surface A; surface B; T, temperature

of "shunt" at surface B, I, shield I; II, shield II.

95

Jk

FORM SBS-114A (1-7!) U.S. DEPT. OF COMM. | 1. PUBLICATION OR REPORT NO. BIBLIOGRAPHIC DATA mtjctd tq i q/ SHEET 1 NBSIR 73-184

2, Gov't Accession No.3. Recipient's Accession No.

4. TITLE AND SUBTITLE

A Cell Model for Isoperibol Calorimeters

5. Publication Date

6. Performing Organization Ctnie

7. AUTHOR(S)

K. L. Churney. E. Do West, and C T. Arnistrona

8. Performing Organization

NBSIR 73-184

9. PERFORMING ORGANIZATION NA^iE AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/ Work Unit No. 316-0117

1 1. v_ontract/ Grant No,

12. Sponsoring Organization Name and Address 13. Type of Report & Period

Covered

14. Sponsoring Agency Code

IS. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summaty of most significant information. If document includes a significant

bibliography or literature survey, mentior.it here,)

A calorimeter can be modeled as a large number of volume elements or cells

in each of which the temperature may be considered uniform, and each of which

can store heat and exchange heat with other cells. Application of the first law

of thermodynamics to this set of cells leads to representations of the usual

calorimetric equations for internal energy change expressed in terms of

measurable or estimatable heat capacities, heat transfer coefficients, temperatures, and work terms for the individual cells. Analysis of the results

yields a framework within which most of the design and measurement problems

of isoperj.bol calorimeters can be treated.

17. KEY WORDS (Alphabetical order, separated by semicolons) calorimetry; energy equivalents; energy

measurement; heat transfer; interr^l energy measurement ; isoperibol calorimeters;

measurement theory.

18. AVAILABILITY STATEMENT 19. SECURITY CLASS

(THIS REPORT)

21. NO. OF PAGES

unlimited.

UNCLASSIFIED

1 1 FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

20. SECURITY CLASS

(THIS PAGE)

22. Price

UNCLASSIFIED

USCOMM-DC 66244.P71

)