Radial Temperature Differences

GP 2] [R 2] The radial temperature distribution was determined by modeling, using a worst-case scenario (5 Nl h stoichiometric mixture without inert 100% conversion 80% selectivity) [102], The maximum radial temperature difference amounts to approximately 0.5 K. Thus, isothermal behavior in the radial direction can be diagnosed. 

Figure 13B shows the calculated temperature differences for the same cases as considered before, but with catalyst beds diluted with silicon carbide to one third of the original catalyst concentration. It can be seen that the temperature differences are appreciably smaller than in the undiluted case (note the differences in temperature scale between Figures 13A and 13B). The dilution with good thermally conducting material is particularly effective at the low velocities in short beds because the convective contribution to the effective heat conductivity is then relatively small. It can be inferred that in microflow reactors (D = 1 cm L = 10 cm) and in bench-scale reactors (D = 2 cm L = 1 m) with diluted beds radial temperature differences are less than 1-2 °C for the considered cases, which is quite acceptable. 

Below are listed the radial profiles for two axial positions within the reactor. Notice that at Z = 0.2, there are significant radial gradients while by Z = 0.6 these gradients are all but eliminated. If the inlet and coolant temperatures are around 360°C, then the centerline and inner wall temperatures at Z = 0.2 are 417°C and 385°C, respectively, giving a radial temperature difference of 32°C. 

The radial temperature difference between Tr-cvimder and To, i.e., A Trad, is, therefore, expressed as... 

A radial temperature difference measuring apparatus made by applying the cnnstant-hcating-ratc method... 

In Fig. 70, Rate of increase in temperature and Radial temperature difference denote the variation of the ihermoelectroniotive force of the thermocouple 2, /.e, the variation of the r,.q,/,w , with time and the variation of the value of Trad with lime, respectively. 

The individual values of and (A Tcaj)wm of the sawdusts of fifteen wood species tested herein were measured by means of a radial temperature difference measuring apparatus at two temperatures, i.e., a temperature near 100 °C and a temperature near room temperature, respectively. The arithmetic mean of the value of CTc of the sawdust of a wood species calculated by substituting the values of 0 and (A T,-acd im measured each at a temperature near 100 °C into Eq. (85) and that calculated by substituting the values measured each at a temperature near room temperature into the same equation is presented as the value of O c of the sawdust of the wood species in Table 21 in (1) of Subsection 8.4.4, although the two values of O c calculated each by substituting the individual values of 0 and (A Trud) m measured each at the two temperatures into Eq. (85) were nearly equivalent to each other for the sawdust of every wood species. Some values of de thus calculated each for the sawdusts of a few wood species are presented in Table 20. 

Temperature difference, in the adiabatic self-heating test, or in the adiabatic oxidatively-heating test, between the r /and the Tatn, or between the temperature of a chemical and the K. Radial temperature difference effected in the non-steady state between the periphery and the axial center of the specimen of a powdery chemical of the TD type, including every gas-permeable oxidatively-heating substance, charged in a cylindrical cell heated at a very slow rate, 0, [K]. 

All these flow types appear more or less in a series one after the other during the evaporation of a liquid in a vertical tube, as Fig. 4.30 illustrates. The structure of a non-adiabatic vapour-liquid flow normally differs from that of an adiabatic two-phase flow, even when the local flow parameters, like the mass flux, quality, etc. agree with each other. The cause of this are the deviations from thermodynamic equilibrium created by the radial temperature differences, as well as the deviations from hydrodynamic equilibrium. Processes that lead to a change in the flow pattern, such as bubbles coalescing, the dragging of liquid drops in fast flowing vapour, the collapse of drops, and the like, all take time. Therefore, the quicker the evaporation takes place, the further the flow is away from hydrodynamic equilibrium. This means that certain flow patterns are more pronounced in heated than in unheated tubes, and in contrast to this some may possibly not appear at all. 

Grushka et al. [4] have provided a useful rule of thumb that the radial temperature difference in the capillary (ATRadiab see Figure 18.2) should not be allowed to exceed 1.5 °C otherwise there is an unacceptable loss of efficiency due to peak broadening. 

It has been shown that the radial temperature difference is directly proportional to the power per unit length and is independent of the cooling efficiency of the instrument. The following equation applies for aqueous electrolytes ... 

Radial temperature difference across electrolyte (see Figure 18.2)... 

A special complication occurs when scaling up a reactive extrusion process. Where in small-scale extruders the surface-to-volume ratio is quite large, this ratio diminishes proportionally to the screw diameter. As a consequence, the relative amount of heat transferred in large production machines can be much lower than in similar laboratory size equipment. Moreover, in large-scale equipment the heat that is released in the middle of the channel has to be transported over a much larger distance than in small extruders. This leads to increasing radial temperature differences in larger extruders. Therefore, an analysis of the heat transfer characteristics of the extruder and its dependence on scale is essential for the reactive extrusion process. 

In high-temperature chromatography using conventional columns, preheating the mobile phase before entering the column is recommended in order to avoid radial temperature differences. 

The axial temperature profile in an experimental fixed bed reactor and at least one value of the wall temperature should be measured. The value of the radial temperature difference can also be estimated by Eq. (4.10.168) ... 

Note that radial temperature profiles in the thin reactor are not considered, and integration is only needed in the axial direction. [Detailed measurements show that radial temperature differences are less than 20 K (Kern, 1998).] The integral in Eq. (4.11.30) is determined graphically by the area under the respective function, as shown in Figure 4.11.14 for an activation energy of 300 kj mol and a maximum temperature of 1200 °C. 

The values given in Table 4.11.2 show that the influence of radial dispersion of heat, axial dispersion of mass and heat, and the influence of radial variations in the bed structure (wall effects) are neghgible. However, radial dispersion of heat may have an influence, although the maximum radial temperature difference of 3 K still... 

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