Thermal resistance constant

Figure 1.5 Indium heated at 5°C/min showing an almost triangular melting profile typical of single crystal melt at lower scan rates. The slope of the leading edge of the melt of a pure material such as indium gives a value for the thermal resistance constant Rq.

In practice, instrument companies provide software to perform purity calculations, but there is still a lot of care needed in the method. First of aU it has to be noted that thermal gradients across a sample will affect the rate of melting and the resulting peak shape. For this reason sample sizes must be small, typically about 1 mg, and the scan rates slow, typically l°C/min. Even so the rate of transfer of heat to the sample, expressed as the thermal resistance constant Ro, will influence the rate of melt, and this varies from instrument to instrument and with pan type used. Rq must therefore be determined under the conditions of the test and used in the calculations. See Section 1.5.2 for a description of how this is measured. It is very important that this is done correctly whilst in one sense the method is not based on the... 

Thermal resistance constant (Ro) In a DSC this is a measure of the resistance to the flow of energy input into a sample. Usually measured from the leading edge of the melt of indium it gives a measure of the rate of flow of energy into a sample, taking into account that the instrumental pathway energy must flow before it reaches the sample. 

An alternative method known as slicing and scaling has been developed (23,24). In this, the rate of diffusion is determined on a thin specimen (6—10 mm thick) and a scaling factor S used to relate the results to a thick specimen. For a material satisfying the requirements of a constant diffusion and constant initial pressure,, the same ratio of time thickness provides the same values of p and %. Thus the thermal resistance of a specimen of thickness at time can be obtained by conditioning a specimen of thickness over a time given by... 

A guarded hot-plate method, ASTM D1518, is used to measure the rate of heat transfer over time from a warm metal plate. The fabric is placed on the constant temperature plate and covered by a second metal plate. After the temperature of the second plate has been allowed to equiUbrate, the thermal transmittance is calculated based on the temperature difference between the two plates and the energy required to maintain the temperature of the bottom plate. The units for thermal transmittance are W/m -K. Thermal resistance is the reciprocal of thermal conductivity (or transmittance). Thermal resistance is often reported as a do value, defined as the insulation required to keep a resting person comfortable at 21°C with air movement of 0.1 m/s. Thermal resistance in m -K/W can be converted to do by multiplying by 0.1548 (121). 

In the case of a temperature probe, the capacity is a heat capacity C == me, where m is the mass and c the material heat capacity, and the resistance is a thermal resistance R = l/(hA), where h is the heat transfer coefficient and A is the sensor surface area. Thus the time constant of a temperature probe is T = mc/ hA). Note that the time constant depends not only on the probe, but also on the environment in which the probe is located. According to the same principle, the time constant, for example, of the flow cell of a gas analyzer is r = Vwhere V is the volume of the cell and the sample flow rate. 

Conducted heat is that going in through cold store surfaces, tank sides, pipe insulation, etc. It is normally assumed to be constant and the outside temperature an average summer temperature, probably 25-2/°C for the UK, unless some other figure is known. Coldroom surfaces are measured on the outside dimensions and it is usual to calculate on the heat flow through the insulation only, ignoring other construction materials, since their thermal resistance is small. 

Thermal resistance, gas constant, auto-correlation function, radius... 

The lattice may be distorted because of several reasons as vacancies, interstitials, dislocations and impurities. These lattice defects cause the so-called impurity scattering which produces the term i ei. At low temperatures, i ei is the constant electronic thermal resistance typical of metals. 

Fig. 4.6. Scheme for the calculation of the thermal time constant of a sample connected to a thermal bath by a thermal resistance of negligible value (see text). 

Such a program should be used in small signal circuits where the values of the components are constant. For this reason, we supplied a very small test energy AE in order to produce very small change AT and considered the thermal resistance and heat capacities constant in the calculations for each heat sink temperature. 

The third block in Fig. 2.1 shows the various possible sensing modes. The basic operation mode of a micromachined metal-oxide sensor is the measurement of the resistance or impedance [69] of the sensitive layer at constant temperature. A well-known problem of metal-oxide-based sensors is their lack of selectivity. Additional information on the interaction of analyte and sensitive layer may lead to better gas discrimination. Micromachined sensors exhibit a low thermal time constant, which can be used to advantage by applying temperature-modulation techniques. The gas/oxide interaction characteristics and dynamics are observable in the measured sensor resistance. Various temperature modulation methods have been explored. The first method relies on a train of rectangular temperature pulses at variable temperature step heights [70-72]. This method was further developed to find optimized modulation curves [73]. Sinusoidal temperature modulation also has been applied, and the data were evaluated by Fourier transformation [75]. Another idea included the simultaneous measurement of the resistive and calorimetric microhotplate response by additionally monitoring the change in the heater resistance upon gas exposure [74-76]. 

In the previous paragraph, the basic considerations of FEM modelling have been laid out. The outcome of a static thermal simulation based on this model is a 3-d temperature field T x,y,z). In this section it is discussed, how the characteristic figures, such as thermal resistance and thermal time constant of the membrane, can be deduced. 

The time constant will show a temperature dependence owing to thermal resistance variation and temperature-dependent heat capacitance. The differential - and more general - form of Eq. (3.33) is ... 

In case of a homogeneous temperature distribution in the heated area, h corresponds to the temperature coefficient of the heater material, otherwise h includes the effects of temperature gradients on the hotplate. As a consequence of the aheady mentioned self-heating, the applied power is not constant over time, and the hotplate cannot be simply modelled using a thermal resistance and capacitance. Replacing the right-hand term in Eq. (3.28) by Eq. (3.35) leads to a new dynamic equation ... 

For thermal characterization and temperature sensor calibration a microhotplate was fabricated, which is identical to that on the monoHthic sensor chips, but does not include any electronics. The functional elements of this microhotplate are connected to bonding pads and not wired up to any circuitry, so that the direct access to the hotplate components without electronics interference is ensured. The assessment of characteristic microhotplate properties, such as the thermal resistance of the microhotplate and its thermal time constant, were carried out with these discrete microhotplates. 

In order to determine the thermal time constant of the microhotplate in dynamic measurements, a square-shape voltage pulse was applied to the heater. The pulse frequency was 5 Hz for uncoated and 2.5 Hz for coated membranes. The amplitude of the pulse was adjusted to produce a temperature rise of 50 °C. The temperature sensor was fed from a constant-current source, and the voltage drop across the temperature sensor was amplified with an operational amplifier. The dynamic response of the temperature sensor was recorded by an oscilloscope. The thermal time constant was calculated from these data with a curve fit using Eq. (3.29). As already mentioned in the context of Eq. (3.37), self-heating occurs with a resistive heater, so that the thermal time constant has to be determined during the cooHng cycle. 

Table4.6. Experimentally determined values of the thermal resistance, the thermal time constant, and the temperatm-e homogeneity for uncoated and coated, transistor-heated microhotplates...

We take the heat transfer coefficient a to be independent of the jet velocity and of the residence time in the vessel. Physically, this assumption together with the assumption of complete mixing of the substance in the reaction vessel and of a constant mean temperature throughout the vessel corresponds to the idea that for heat transfer the governing factor is the thermal resistance from the internal wall of the vessel to the outside space in which the temperature is kept at T0. In other words, our assumption corresponds to the concept of a vessel which is thermally insulated from outside. 

In the emulsion phase/packet model, it is perceived that the resistance to heat transfer lies in a relatively thick emulsion layer adjacent to the heating surface. This approach employs an analogy between a fluidized bed and a liquid medium, which considers the emulsion phase/packets to be the continuous phase. Differences in the various emulsion phase models primarily depend on the way the packet is defined. The presence of the maxima in the h-U curve is attributed to the simultaneous effect of an increase in the frequency of packet replacement and an increase in the fraction of time for which the heat transfer surface is covered by bubbles/voids. This unsteady-state model reaches its limit when the particle thermal time constant is smaller than the particle contact time determined by the replacement rate for small particles. In this case, the heat transfer process can be approximated by a steady-state process. Mickley and Fairbanks (1955) treated the packet as a continuum phase and first recognized the significant role of particle heat transfer since the volumetric heat capacity of the particle is 1,000-fold that of the gas at atmospheric conditions. The transient heat conduction equations are solved for a packet of emulsion swept up to the wall by bubble-induced circulation. The model of Mickley and Fairbanks (1955) is introduced in the following discussion. 

The constant k takes into account the heat flow from the warm vapour to a coolant flowing behind a metal wall (tube). Its reciprocal represents the sum of thermal resistances on the gas side, at the wall and on the coolant side. 

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