Gradients of temperature

In an ideal CSTR, there are no gradients of temperature or composition, only the overall changes. 

Dapremont, O., Cox, G.B., Martin, M., Hilaireau, P., and Colin, H., Effect of radial gradient of temperature on the performance of large-diameter high-performance liquid chromatography columns I. Analytical conditions,. Chromatogr. A, 796, 81, 1998. 

In such studies one may also eliminate intraparticle gradients of temperature and composition by using very fine catalyst particles or by confining the catalytic species to the exterior surface of a nonporous or impervious pellet. Unfortunately, the conditions that are optimum for the elucidation of the intrinsic chemical kinetics are often inappropriate for use in... 

The Earth formed as a set of elements and compounds, ultimately distributed in zones, with a large gradient of temperature and consequently providing an enormous store of energy. 

The part of the sample along which the gradient of temperature is produced has a cylindrical form of length L = 5.75 0.03 mm, and radius r = 4.00 0.01 mm, giving a geometrical factor g = A/L = 8.74 0.09 mm at room temperature. The thermal contacts at the end of the sample have been realized by means of two gold-plated copper screws (Scj, Sc2) 4 mm in diameter. The threading in the sample had a depth of 5 mm. Since... 

Next, consider the gradients of temperature. If the reaction is exothermic, the center of the particle tends to be hotter, and conversely for an endothermic reaction. Two sets of gradients are thus indicated in Figure 8.9. Heat transfer through the particle is primarily by conduction, and between exterior particle surface (Ts) and bulk gas (Tg) by combined convection-conduction across a thermal boundary layer, shown for convenience in Figure 8.9 to coincide with the gas film for mass transfer. (The quantities T0, ATp, A7y, and AT, are used in Section 8.5.5.)... 

Pyrolysis of propane was accomplished by bubbling the gas through a shallow bed of molten lead at 1400 F and 4 psig (Fair et al, Chem Eng Prog 53 433, 1953). The vessel was assumed to have no gradients of temperature or... 

Porous solids are not good conductors of heat, so reactions with appreciable heats of reaction can develop significant gradients of temperature and reaction rate along the pore, as well as over an external film. 

Porous catalyst particles are complex devices with appreciable internal gradients of temperature and composition, but these factors can be taken account of by the concept of catalyst effectiveness which is sometimes calculable. 

Another issue that needs attention is that in large-scale beds, phenomena absent in the laboratory reactor may develop. For example, in commercial beds, axial gradients of temperature may appear, which are absent in bench beds due to the small diameters usually used in them. In the worst scenario, the controlling mechanism, and thus the whole behavior of the system, could change from the small to the large scale. 

The flame is not, however, a discontinuity. There are definite gradients of temperature and composition because of conduction of heat and diffusion of reaction products into the fresh gas. The temperature gradients have been studied by three techniques direct measurement with very fine thermocouples (24, 25, 43) refraction of a narrow slit of light (11, 16) and tracing the path of a stroboscopically lighted dust particle and computing temperature from its direction and velocity (l, 27). 

Porous solids are not good conductors of heat, so reactions with appreciable heats of reaction can develop significant gradients of temperature and reaction rate along the pore, as well as over an external film. Enhancement of a rate by temperature can counteract the effect of falling concentration. Exothermic reaction rates in pores, as a consequence, can be much greater than at the surface condition. Another peculiarity that can arise with adiabatic reactions is multiple steady states. 

As noted earlier, however, numerous variables have been identified as significant in SFC, including temperature the type of stationary phase the polarity, density (or pressure), and modifier content of the mobile phase and the corresponding gradients of temperature, density (pressure), and composition. Moreover, from chemometric principles it is clear that any procedure which does not consider all the significant variables simultaneously will seldom, if ever, locate the true set of optimum conditions. This point is illustrated in the section below. ... 

They represent heat and mass fluxes with unit gradients of temperature and concentration, respectively, and so can be considered as specific heat and mass transfer rates. 

The second law of thermodynamics not only gives us a direction for time but also gives us a macroscopic explanation for the direction for irreversible processes in steady-state systems. For heat flow and diffusion, give a microscopic reason why the flows are in the direction opposite to the gradient of temperature and concentration, respectively. 

We thus see that the affinity always has the same sign as the rate of the process. If the affinity is positive A>0, the rate must be positive v>0 indicating that the irreversible process proceeds in the forward direction whereas, if the affinity is negative A < 0, the rate must be negative v < 0 meaning that the process proceeds in the backward direction. When the affinity decreases to zero A - 0, the rate of process also decreases to zero and the process is in equilibrium. This property of affinity is characteristic of all kinds of irreversible processes such as the transfer of heat under a gradient of temperature and chemical reactions under a gradient of thermodynamic potentials. 

The principal features of the vertical T,S structure of the Black Sea waters are shown in Fig. 3. The upper mixed layer (UML) of the Black Sea in the warm period of the year has a thickness less than 10 m (see Fig. 3a). At this time, it is underlain by the layer of the seasonal pycnocline (ther-mocline) this layer is also thin (10-20 m) but features high vertical gradients of temperature (0.2-0.3 °Cm 1) and, correspondingly, of water density... 

Therefore, since qw = 0 if the wall is adiabatic, the gradient of temperature at the wall must be zero if the wall is adiabatic. 

Therefore, the gradient of temperature at all radii is the same. Hence, if Tm is the mean temperature at any section of the pipe it follows that ... 

Simultaneous measurements of the rate of change, temperature and composition of the reacting fluid can be reliably carried out only in a reactor where gradients of temperature and/or composition of the fluid phase are absent or vanish in the limit of suitable operating conditions. The determination of specific quantities such as catalytic activity from observations on a reactor system where composition and temperature depend on position in the reactor requires that the distribution of reaction rate, temperature and compositions in the reactor are measured or obtained from a mathematical model, representing the interaction of chemical reaction, mass-transfer and heat-transfer in the reactor. The model and its underlying assumptions should be specified when specific rate parameters are obtained in this way. 

Where A, is the effective heat conductivity of the catalyst particle and (-AHt) the heat effect of reaction i. The effective conductivity is the quantity which (when multiplied by the gradient of temperature) yields the true rate of energy transport in the porous structure of the solid. 

For a more complete discussion of the one-dimensional flame see J. O. Hirschfelder and C. F. Curtiss, Third Symposium on Combustion and Flame and Explosion Phenomena, The Williams Wilkins Company, Baltimore, 1949, and also J. O. Ilirsch-fekier et al., Fourth Symposium (International) on Combustion, The Williams Wilkins Company, Baltimore, 1953. The local velocity Vto will also depend on gradients of temperature (Sec. VIII. 10). 

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