Time constants, calorimeter

The reciprocal value of the constant p/fi has a dimension of [<] and is called the calorimeter time constant r. This is another important characteristic of a heat-flow calorimeter ... 

This simplified equation is equivalent to Tian s equation [Eq. (16)], and it appears that n is indeed the time constant r of the calorimeter. Thence, the successive coefficients n in Eq. (29) may be called the calorimeter time constants of 1st, 2nd,. .., ith order. When the Tian equation applies correctly, all time constants r, except the first r may be neglected. Since the value of the coefficients n of successive order decreases sharply [the following values, for instance, have been reported (40) n = 144 sec, r2 = 38.5 sec, r3 = 8.6 sec, ri 1 sec], this approximation is often valid, and the linear transformation of many thermal phenomena produced by the thermal lag in the calorimeter may actually be represented correctly by Eqs. (16) or (30). It has already been shown (Section IV.A) that the total heat produced in the calorimeter cell is then proportional to the area limited by the thermogram. 

From Tian s equation [Eq. (30)3, it appears that in order to transform the calorimeter response g(t) into a curve proportional to the thermal input f(t), it is sufficient to add, algebraically, to the ordinate of each point on the thermogram g(t), a correction term which is the product of the calorimeter time constant n, by the slope of the tangent to the thermogram at this particular point. This may be achieved manually by the geometrical construction presented on Fig. 10. 

The value of the calorimeter time constant r (= n), may be determined from the cooling curve which is recorded, for instance, when a Joule heating, which produced a constant deviation A0 of the base line (Fig. 11), is suddenly stopped (16). The comparison of Eqs. (14) and (15) shows that the cooling curve is represented by... 

Backman, P., Bastos, M., Hall6n, D., Lonnbro, P., Wadso, I. (1994). Heat conduction calorimeters Time constants, sensitivity and fast titration experiments. J. Biochem. Biophys. Meth. 28,85-100. 

The value of the time constant depends upon the calorimeter itself p and upon the heat capacity of the calorimeter cell and of its contents p. Typical, but necessarily approximate, values of the time constant for some heat-flow microcalorimeters are given in Table II. 

As already indicated, Tian s equation supposes (1) that the temperature of the external boundary of the thermoelectric element 8e, and consequently of the heat sink, remains constant and (2) that the temperature Oi of the inner cell is uniform at all times. The first condition is reasonably well satisfied when the heat capacity of the heat sink is large and when the rate of the heat flux is small enough to avoid the accumulation of heat at the external boundary. The second condition, however, is physically impossible to satisfy since any heat evolution necessarily produces heat flows and temperature gradients. It is only in the case of slow thermal phenomena that the second condition underlying Tian s equation is approximately valid, i.e., that temperature gradients within the inner cell are low enough to be neglected. The evolution of many thermal phenomena is indeed slow with respect to the time constant of heat-flow calorimeters (Table II) and, in numerous cases, it has been shown that the Tian equation is valid (16). 

The time constant t is therefore very simply related to <1/2. Moreover, it can be shown also (IS) that the time constant represents the time necessary for attaining thermal equilibrium—and a stable base line—if the calorimeter cooling was linear and followed the tangent in the beginning of the cooling (Fig. 11). 

It should be recalled, at this point, that the value of the time constant is related to the heat capacity of the calorimeter cell and of its contents [Eq. (15)]. For meaningful results, it is therefore essential that the arrangement of the inner cell remain identical for both the Joule heating and the thermal phenomenon under investigation. Strictly speaking, the time constant would be unchanged if it were possible to keep the thermal paths completely identical in both cases. This condition is, of course, very difficult to meet. 

The efficiency of this method has been demonstrated for several types of heat-flow calorimeters. The rather long time constant of a Calvet-type calorimeter (200 sec), for instance, is decreased to 10 sec, when exact Peltier cooling is used (61). Similarly, the time constant of calorimeters... 

It must be noted that although the calibration cell is very different from the adsorption cell [Fig. 18, cells (1) and (2)3, the heat capacity of both cells is not very different, as the similar values of the time constant of the calorimeter containing one cell or the other indeed show (350 sec in the case of the calibration cell and 400 sec in the case of the adsorption cell) (55). This is explained by the fact that in both eases, the calorimeter cell is almost completely filled with a metal. However, the glass tube which is immersed in the calorimeter cell and the pressure changes which occur in the course of the adsorption experiments may be the sources of variable thermal leaks. The importance of these leaks was appreciated by means of the following control experiments. 

The duration of each dosing experiment is about 15-50 minutes (depending on the sample and of the time constant of the calorimeter), which was long enough to yield well-resolved heat-flow peaks and a stable horizontal baseline of the microcalorimeter. For all catalysts presented here, adsorption always reached thermodynamic equilibrium. Prior to adsorption measurements, the samples were pretreated in the calorimetric cell by heating overnight under vacuum. 

The use of a filter determined by Eq. (3.15) allows the true form of the signal to be restored (Fig. 3.8, curve 1). In this case, both the magnitudes of the quantities measured and the qualitative shape of the experimental curve change. The use of this filtration method has enabled us to broaden the frequency range of the instrument by about an order of magnitude and to reduce the effective time constant of the calorimeter from 4 min to 30 sec. 

Figure 2. Temperature-time curves for adiabatic type calorimeters (with a low time constant). Curves A and C show curves following the release of a short heat pulse in an ideal adiabatic calorimeter and a semiadiabatic calorimeter, respectively. Curves B and D show the curves from experiments where a constant thermal power was released between t, and t2 for an ideal adiabatic calorimeter and a semiadiabatic calorimeter, respectively. For the ideal adiabatic instrument the slope of the curve during the heating period is proportional to the thermal power, P.

Equation (17) is usually called the Tian equation. In cases where significant temperature gradients are present within the reaction vessel, two or more time constants must be used. When the change in rate of a process is small, the value for X(dU/dt) will often be insignificant compared to the value for U (equation (17)). With heat conduction calorimeters used in work on cellular systems, this is typically the case and the heat production rate is then, with a good approximation, given by the simple expression... 

The thermal inertia of a heat conduction calorimeter is described by its time constant x (equation (18)). In practice, X is given by... 

B) Ordinary calorimeters no fixed relationship between Tq and Ts. These are inertial systems, characterized by a time-constant, and mainly include isoperibol calorimeters. 

For example, when NH3 is used to probe the acidity of zeolites at a given temperature, the time needed to establish thermal equilibrium after each dose at first increases with increasing adsorbed amount, passes through a maximum, then decreases rapidly and, finally, reaches a value close to the time constant of the calorimeter. 

The response of the calorimeter depends on the instrumental time constant, as given in (13). In general, useful kinetic information can be gained only when the rate of a reaction is significantly slower than the time constant. However, a mathematical correction can be made for reactions that are slightly faster than the instrument time constant. ... 

The use of the corrections B in Fig. A.l 1.2 needs two calibration runs of the DSC of Fig. 4.54. The heat capacities of the calorimeter platforms, C pi and C pi, and the resistances to the constantan body, R pi and R pi, must be evaluated as a function of temperature. First, the DSC is ran without the calorimeters, next a run is done with sapphire disks on the sample and reference platforms without calorimeter pans. From the empty run one sets a zero heat-flow rate for and This allows to calculate the temperature-dependent time constants of the DSC, written as = C piRspi and Tr = CrpiRrpi, and calculated from the equations in the lower part of Fig. A. 11.2. For the second run, the heat-flow rates are those into the sapphire disks, known to be mCpQ, as suggested in Figs. 4.54 and 4.70. The heat-capacity-correction terms are zero in this second calibration because no pans were used. From these four equations, all four platform constants can be evaluated and the DSC calibrated. 

By analogy to the OTR measurement (see Equ. 3.43), the dynamic method can also be used for heat transfer measurements. The bioreactor is then treated as an adiabatic calorimeter with constant internal heat production (Falch, 1968). A typical temperature curve of the dynamic calorimetry is shown in Fig. 3.20b in analogy to Fig. 3.15b. At time Iq, cooling is interrupted, and the rate of heat production may be directly calculated from the linear... 

If a maker of a calorimetric system decides to apply the method of determining the heat effect resulting from Eq. (1.99), it would be most convenient to establish a set of parameters (e.g. a, X, p, Cp) such that the calorimeter in which the heat K is generated fulfils the conditions needed to apply this equation. A probe to determine such a set of parameters was undertaken by Utzig and Zielenkiewicz [18] for a simple physical model as an approximation of a real calorimetric system. The relation between the dimensionless parameter x the physical parameters of the system and qy was elaborated. In the parameter x the t value corresponds to the time interval after which the body temperature changes can be described by one exponential term, while r is a time constant. 

The form of the transmittance H(s) indicates that the time constant r is a decisive parameter for characterizing the inertial properties of the object (calorimeter). This also means that the value of the time constant determines the course of the output function, the character of which is approached more closely for either proportional or integrating objects. Simply, the values of r control the inertial, damping properties of the object. Different values of the function x(t), depending on the values of r, are responses to the same heat forcing (Fig. 2.13). 

Different methods are presented for determining the time constant of a calorimeter treated as an inertial object of first order. The methods used to determine the dynamic parameters of calorimeters that are inertial objects of order higher than one are also discussed. All these numerical methods, algorithms and listing programs are described in detail in [67]. 

The dynamic properties of a calorimeter treated as an inertial object of first order are characterized unambiguously by the time constant r. 

To evaluate the time constant on the basis of the heat balance equation of a simple body, different input functions are used. Consider the determination of rby applying the input step function [Eq. (2.45)] under conditions where the initial temperatures of the calorimeter and isothermal shield are the same. Equation (2.19) can then be written in the form... 

The changes in temperature T(t) in time, caused only by the initial temperature difference between the calorimeter and shield, are also used to determine the time constant t. In this case, it is assumed that y(r) = 0, T(0) == To = 0 and the cooling process is expressed by equation... 

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