Heat capacity measurement time constant

To make a heat capacity measurement, a constant electric current is passed through the heater cireuit for a known period of time. The system is the calorimeter and its contents. 

I lough and co-workers 11 have measured the liquid heat capacity of isopropyfarninc from 30"C to 70°C Data up to the boiling point are also available The data were extended by the equation, density times heat capacity equals a constant... 

The 0.38 h discontinuity corresponds to the hydration end point, above which the nonideality per mol of protein is constant. The heat capacity measurements define the hydration end point tightly. These measurements are sensitive to the interactions of water with both hydrogen-bonding and nonpolar protein surface groups and should reflect essentially all time-average chemistry associated with hydration. 

Since the heat capacity is measured using a differential scanning calorimeter, the dependence of Cp on time and temperatures is complicated because Cp is measured continuously while the sample is being heated or cooled at a constant rate. Therefore, the time for the heat capacity measurement is not well defined, and thermal history effects complicate the shape of the step at 7. To overcome this difficulty, one can approximate the real situation by changing the temperature in small discrete steps AT at time intervals At. Stephens and Moynihan et al. follow this procedure and calculate the change in the sample s enthalpy H and its heat capacity from Cp =LH/l T. However, we already have an explicit expression for the equilibrium enthalpy and heat capacity. Since the result for Cp, (8.2), depends partially on the rate of change of p with respect to T, we need only include the effect of p falling out of equilibrium (i.e., dp/dT tCS) for T< 7 in that result. ... 

At the time of the heat capacity measurements of Lounasmaa (1964c) (0.4. 0 K), the lattice and magnetic contributimis could not be separated but later Rosen (1968a) determined the limiting Debye temperature from elastic constants measurements to be 118 K, equivalent to a lattice contribution to the heat capacity of 1.18 mJ/(mol K ) so that the magnetic contribution could then be separately calculated from a reassessment of the measurements of Lounasmaa (1964c) over the temperature range of 0.50-1.25 K. When combined with the nuclear terms as determined in Part 11.11, results in a revised heat capacity equation valid to 1.25 K ... 

Even for an open system one can easily express the entropy flux as written in Eq. (2). Calorimetry can give information on the heat flux, dQ/dt, which increases the entropy of the system as written in Eq. (2). Since the temperature is constant inside the system during the time interval, dt, the heat flux into the system can be considered reversible. Any irreversibility caused by temperature jumps (needed to cause the heat flux) is pushed, as mentioned above, into the boundary and is of no interest to the description of the system. The changes caused by the flux of matter can be measured, for example, by thermogravimetry. The matter flux, dgnj/dr, of substance i across the system boundary in the time interval, dr must be multiplied with the molar change in entropy due to the flux, 5j. The molar entropy change must be known or measured separately — for example by heat capacity measurements from zero kelvin to T, as outlined by Eq. (12) of Fig. 2.4. 

Time constants. Where there is a capacity and a throughput, the measurement device will exhibit a time constant. For example, any temperature measurement device has a thermal capacity (mass times heat capacity) and a heat flow term (heat transfer coefficient and area). Both the temperature measurement device and its associated thermowell will exhibit behavior typical of time constants. 

Whereas heat capacity is a measure of energy, thermal diffusivity is a measure of the rate at which energy is transmitted through a given plastic. It relates directly to processability. In contrast, metals have values hundreds of times larger than those of plastics. Thermal diffusivity determines plastics rate of change with time. Although this function depends on thermal conductivity, specific heat at constant pressure, and density, all of which vary with temperature, thermal diffusivity is relatively constant. 

For a thermometer to react rapidly to changes in the surrounding temperature, the magnitude of the time constant should be small. This involves a high surface area to liquid mass ratio, a high heat transfer coefficient and a low specific heat capacity for the bulb liquid. With a large time constant, the instrument will respond slowly and may result in a dynamic measurement error. 

The temperature response of the measurement element shown in Fig. 2.13 is strictly determined by four time constants, describing a) the response of the bulk liquid, b) the response of the thermometer pocket, c) the response of the heat conducting liquid between the wall of the bulb and the wall of the pocket and d) the response of the wall material of the actual thermometer bulb. The time constants c) and d) are usually very small and can be neglected. A realistic model should, however, take into account the thermal capacity of the pocket, which can sometimes be significant. 

To carry out measurements at a fixed temperature, the refrigerator temperature must be kept constant for a suitably long time. The problem of the temperature control depends not only on the refrigerator itself, but on the thermal characteristics of the experiment. Let us now consider an oversimplified case in which heat capacities are neglected the mixing chamber temperature of a dilute refrigerator (DR) is to be controlled by a resistive heater HR and a d.c. power supply. 

In the methods reported above, the temperature change AT used to measure the heat capacity C(T) was supposed to be so small that the time constant r = R C could considered constant in the AT interval. Let us consider, for example, the thermal discharge of a system with heat capacity C(T) a T and thermal conductance to the bath G(T) a T3 (e.g. a metal sample and a contact resistance to the bath at rB). A AT/TB = 10% gives a At/t = 20% over the interval AT, that is a time constant definitely not constant. 

To measure G(T), an electrical power PR was supplied to the heater (H), keeping the frame temperature constant. G(T) was obtained as the derivative of the PH(T) curve. The heat capacity of the sample was evaluated simply as C = r x G, where r was obtained by the fit of the thermal discharges around a set of fixed temperatures. A single time constant was always found. An example of thermal discharge at T 160mK is shown in Fig. 12.11. 

The experimentally determined time constant was Tq = 9.7 0.2 ms. The calculated time constant is approximately 11% higher than the measured one. This is a good agreement given the fact that an error of 15% was assumed as a consequence of the uncertainty of 0.2 MJ/°C m in the heat capacities. 

Calorimeters of Historical and Special Interest Around 1760 Black realized that heat applied to melting ice facilitates the transition from the solid to the liquid stale at a constant temperature. For the first time, the distinction between the concepts of temperature and heat was made. The mass of ice that melted, multiplied by the heal of fusion, gives the quantity of heal. Others, including Bunsen, Lavoisier, and Laplace, devised calorimeters based upon this principle involving a phase transition. The heat capacity of solids and liquids, as well as combustion heats and the production of heat by animals were measured with these caloritnelers. 

An exothermal reaction is to be performed in the semi-batch mode at 80 °C in a 16 m3 water cooled stainless steel reactor with heat transfer coefficient U = 300 Wm"2 K . The reaction is known to be a bimolecular reaction of second order and follows the scheme A + B —> P. The industrial process intends to initially charge 15 000 kg of A into the reactor, which is heated to 80 °C. Then 3000 kg of B are fed at constant rate during 2 hours. This represents a stoichiometric excess of 10%.The reaction was performed under these conditions in a reaction calorimeter. The maximum heat release rate of 30Wkg 1 was reached after 45 minutes, then the measured power depleted to reach asymptotically zero after 8 hours. The reaction is exothermal with an energy of 250 kj kg-1 of final reaction mass. The specific heat capacity is 1.7kJ kg 1 K 1. After 1.8 hours the conversion is 62% and 65% at end of the feed time. The thermal stability of the final reaction mass imposes a maximum allowed temperature of 125 °C The boiling point of the reaction mass (MTT) is 180 °C, its freezing point is 50 °C. 

Clearly one obtains the best performance for a given time constant with a detector that has the lowest possible heat capacity. The heat capacity of a crystal varies like C oc (T/0 )3, where On is the Debye temperature. Diamond has the highest Debye temperature of any crystal, so FIRAS used an 8 mm diameter, 25 fim thick disk of diamond as a bolometer (Mather et al., 1993). Diamond is transparent, so a very thin layer of gold was applied to give a surface resistance close to the 377 ohms/square impedance of free space. On the back side of the diamond layer an impedance of 267 ohms/square gives a broadband absorbtion. Chromium was alloyed with the gold to stabilize the layer. The temperature of the bolometer was measured with a small silicon resistance thermometer. Running at T = 1.6 K, the FIRAS bolometers achieved an optical NEP of about 10 14 W/y/IIz. 

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