Tian equation

In the case of a Tian-Calvet calorimeter, we note that the presence of a temperature difference between the internal and external enclosure will lead to the creation of an electromotive force that is proportional to this difference (Seebeck effect). 

Consequently, a current will flow in the thermocouples between the internal and external enclosure. There is then a thermal exchange that is proportional to i (Peltier effect). 

the two effects add up and the expression of the flux W2 exchanged between the internal and external enclosure is obtained  

0 and 0j respectively denote the temperature of the external and internal enclosure. 

We then only have to calibrate the current i generated by the thermocouples using a known power to obtain the value of the flux ). 

---

This fundamental equation of heat-flow calorimeters is called the Tian equation. 

The proportionality constant g includes such parameters as the number of thermoelectric couples in the pile, their thermoelectric power, and the gain of the amplification device. It is supposed, moreover, that the response of the recording line is considerably faster than the thermal lag in the calorimeter. The Tian equation may also be written therefore ... 

The quantity of heat which is developed by the process under investigation is determined by integrating the Tian equation [Eq. (12), for instance],... 

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). 

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. 

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 calorimetric technique used in the titration experiment illustrated in Figure 9 allows short time intervals between the injections due to a comparatively low time constant for the instrument in combination with the electrical compensation technique. Rather, slow heat conduction microcalorimeters can be used in fast titration experiments if a dynamic correction, based on the Tian equation (equation (17)), is employed (Bastos et al., 1991 Backman et al., 1994). 

The conduction of heat through the thermopiles produces a power output that changes with time, shown by the Tian equation ... 

If we assume that the signal X provided by the calorimeter is proportional to the electromotive force E, and therefore to A0, then the Tian equation becomes ... 

The commercially available Calvet microcalorimeters with batch vessels of 100 mL active volume have rather high time constants of the thermal signal so that a smearing of the power-time curves occurs. If one is not only interested in the mean heat output of an animal during prolonged penods, but also in the amplitudes of locomotor activities desmearing techniques have to be applied to the registered slope. The classical approach is performed by the Tian equation... 

Figure 1. Recorded power-time curve (solid line) of a Milos wall lizard (Podarcis milensis) and the desmeared original signal (thin line) calculated with the Tian equation [38],...

The third calibration parameter, which is needed when rapidly changing processes are studied, is the time constant. This is a measure of the thermal inertia of the sample that blurs details in rapid events. The time constant is used in the Tian equation - named after a pioneer in isothermal calorimetry - to correct for this. Typical time constants in isothermal calorimeters are 100-1000 s. As the main hydration has timescales much longer than this, the Tian equation is not needed in cement calorimetry when the main hydration is studied, but it is needed when early reactions are studied. Further information on the Tian equation is given by Wadso (2005), and other similar methods are discussed by Evju (2003). 

It is not trivial to state how good a certain type of calorimeter or a certain type of calorimetric measurement is in terms of precision (or accuracy), as many factors (room temperature stability, reference balance, etc.) influence this. A problem is also that different errors influence different types of measurements in different ways. For example, an error in the calibration coefficient contributes the same error to 1 and 7 days measured heats but an error in the baseline is much worse for the 7 days heat as the thermal powers are low in the end of measurement. For a short (30 min) measurement of initial reactions, the baseline error is normally of no importance as very high thermal powers are measured more important is the time constant and how well the Tian equation can deconvolute the rapidly changing signal. 

---