Theory of Heatburst Calorimetry F. H. Verhoff Department of Chemical Engineering, Uniuersity of Notre Dame, Notre Dame, Ind. 46556
A mathematical theory based on the thermal conduction in a heat sink has been developed to explain the principle of heatburst calorimetry. In particular, this theory says that the time integral of the temperature deviation from steady state at any point in the heat sink is either approximately or exactly proportional to the integral of the total heat flux from the reaction cavity. Two simple examples are used to illustrate the theory. The theory and examples suggest various considerations to be taken into account for the design and operation of heatburst calorimeters.
IN THE RECENT PAST, heatburst calorimetry has been a usable analytical technique for the measurement of various thermodynamic properties of reactions and solutions (1,2). Among the advantages of this method of calorimetry are the approximate isothermal conditions, the absence of the need for heat capacity information, and the ability to use small samples. These characteristics apply to the calorimeters of different designs (e.g., 1, 3, 4 ) . All of the particular designs for these calorimeters are based upon the principle of heatburst calorimetry which has been shown to be true by numerous experiments. This principle states that if the heat of reaction or solution is liberated in a confined space and all of the heat is transferred through a thermopile to a heat sink, then the integral of the output of the thermopile will be proportional to the heat transferred and, hence, proportional to the heat liberated. The goal of this paper is to develop a theory which gives a precise definition to the above principle and which may suggest other heatburst thermocalorimetric designs and operating procedures. Basic Theory. To develop the theory we will consider the heat to be liberated in a confined space and we will be interested in the temperature distributions which result in the material surrounding this confined space. These materials would usually include the reaction vessel, the thermopile, and the large heat block; collectively these materials will be called the heat sink. The equation in vector notation which describes the temperature distribution in the heat sink (assuming the thermal properties to be independent of time and temperature) is the following:
C,(;)
=
r
I
= =
V
=
thermal conductivity of the material
T(7,r) = temperature of material +
position vector time vector differential operator
The density, thermal conductivity, and heat capacity are dependent upon the position vector 7 , because different materials exist a t different values of the position vector. The boundary conditions associated with this equation for the case of a heatburst calorimeter are listed below. T T
=
=
To
--k(r,)VT.;
To (& t
= =
=
0 for all 3
(2)
;.
(3)
far away from the heat source =
qn(rn,t)@
=
surface of cavity
(4)
Here Tois the approximate temperature of the heat sink and is the static temperature before the heat of reaction or solution is liberated. The heat flux normal to the surface qn(rn,r)can depend upon position and o n time, although we will be primarily interested in the case where qn can be written as a function of time multiplied by a function of position. This means that the last boundary condition (Le., Equation 4) can be written as -k(;.)VT.Z
=
qn(Tn,t)= g(t)h(;.)
?n
=
surface of cavity ( 5 ) g(t) = magnitude of the heat flux Btu/hr.ft* h(3,) = surface distribution of heat flux
I n actual calorimeters, this situation should be closely approximated since distribution of heat flux is probably independent of the magnitude of the heat flux as long as the physical properties can be considered independent of temperature. If, in addition, the same container and approximately the same amount of liquid are used, then the function h ( 7 ) should be the same for two successive experiments. Thus, the calorimeter can be calibrated in one experiment and this calibration used in other experiments. Equations 1, 2 , 3, and 5 can be made dimensionless by defining the following variables (T - To)k*
0 =
where
p(7)
k(7)
q* R* 3
density of the material heat capacity of material
+
e = -
(1) C. Kitzinger, and T. H. Benzinger, “Principle and Method of Heatburst Microcalorimetry and the Determination of Free Energy, Enthalpy and Entropy Changes” in “Methods of Bio-
chemical Analysis,” D. Glick, Ed., Interscience Publishers, Inc., New York, 1960, Vol. VIII, Chap. 8. (2) T. H. Benzinger and C. Kitzinger, “Microcalorimetry, New Methods and Objectives” in “Temperature, Its Measurement and Control in Science and Industry,” J. D. Hardy, Ed., Reinhold Publishing Corp., New York, 1963, Vol. 111, Chap. 5 . (3) I. Wadso, Acta Chem. SOC.,22,927 (1968). (4) S. N. Pennington and H. D. Brown, Chem. Instrum., 2, 167 (1969).
r =
r R*
(7)
k* t R* 2p* C,*
where the quantities with the asterisk represent characteristic quantities. For example, R* might be the radius of the spherical cavity in which the heat is liberated and k* might be the thermal conductivity of the heat sink. I n dimensionless quantities, Equations 1 , 2 , 3, and 5 become
ANALYTICAL CHEMISTRY, VOL. 43, NO. 2, FEBRUARY 1971
183
By a change of variables this equation can be changed to the
and
7 far from heat source
8 = 0@
.n’
-K(~,)vO
=
y(T)P(7,) @
7,
=
(1 1)
surface of cavity (12)
I n these equations, the following functions are used.
a(7) = K(7)
(28)
If we let 7, approach infinity, these integrations can be written as
c,*
k(;R*) k*
= -
g(R* 2p*C,*T/k*)
(1 5 )
4*
P(7d
y(Z).f(;,x)dxdZ
P* P(~R*)c,(~R*)
=
Y(T)
Jo J 0
F(;,TO) =
=
h(fnR*)
(16)
This last expression (Equation 29) then states mathematically the general principle of heatburst calorimetry. I n words this principle says that the time integral of the dimensionless temperature at any point in the heatsink is proportional to the total heat given off in the cavity, Le., the time integral of the heat flux from the cavity. This statement uses the fact that the following integral is proportional to heat flux.
In order to find a useful relationship between heat flux and temperature in the material, we transform the equations with respect to time using the Laplace transform. S%(-;,S) = cu(?)V. [K(7)V0(7,S)l
(17)
(7,s) = 0 7 far from heat source -K(7,)VO(7,,S).n= y(S)P(7,) @7, = %
surface of cavity
(18)
K(
