Internal Temperature Gradients

Integrating from r = 0 to r = r, using the boundary conditions given by Eqns. (9-4a) and (9-Sa), and simplifying 

If the reaction is exothermic, the temperature at the center of the catalyst particle is higher than at the surface, i.e., r(0) 7 s. If the reaction is endothermic, the opposite is true. 

The largest absolute value of T (O) — 7 occurs when Ca(0) 0, i.e., when the effectiveness factor is very low. This value is given by 

The subscript max denotes the largest absolute value of the temperature difference. If the extranal mass-transfer resistance is negligible (Ca,s Ca,b) 

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First, we will consider thin objects - more specifically, those that can be approximated as having no spatial, internal temperature gradients. This class of problem is called thermally thin. Its domain can be estimated from Equations (7.11) to (7.12), in which we say the physical thickness, d, must be less than the thermal penetration depth. This is illustrated in Figure 7.7. For the temperature gradient to be small over region d, we require... 

For example, Narsimhan30 derived an equation for the thermal decomposition rate based on the idea that the entire particle rapidly reaches the decomposition temperature, so there are no internal temperature gradients. All the heat reaching the reaction interface is used up in the decomposition, so the reaction rate depends on the rate of heat transport. For both spherical and cylindrical particles,... 

The derivation of the internal temperature gradient can only be performed numerically, even for simale kinetics. Again the 5% criterion is used for the internal effectiveness factor. Assuming that the rate can be simplified into a temperature and concentration dependent part, this yields... 

Under the assumption that inside a particle the reactant concentration becomes zero it can easily be shown that the ratio of the external and internal temperature gradients is... 

Whether the criteria for intra- or extraparticle heat transport limitations are more severe than the corresponding ones for mass transport depends on the absolute value of the products ys x ft, and yb x Pe, respectively. In some cases the former product exceeds unity, but more often not [8], so internal temperature gradients do not occur frequently. The latter product is a factor Bim/Bih larger, generally >10 (gases), which means that in this case the extraparticle temperature gradient occurs earlier than an extraparticle concentration gradient. 

Single stage Multistage Polymerization reactions Convenient to operate Ease of variation or parameters Internal temperature gradients can be high... 

Internal temperature gradient has slow time response for high-pressure operation... 

Concerning internal temperature gradients, a criterion which guarantees that the deviation of the observed volumetric rate from the isothermal rate is less than 5% is given in an analogous way by [11] ... 

For all practical purposes internal temperature gradients can be neglected in laboratory reactors. 

This result shows close agreement with the experimental value of 2K and the low internal temperature gradient, as the value of [i is small. 

Equations (8.85) and (8.86) hold for any particle geometry. Substitution of typical values for the quantities appearing in the left-hand side of Eqn. (8.86), for exothermal reactions resulting in a typical value for p, ranging from 0 to 0.3, leads to the conclusion that, for most practical purposes, the internal temperature gradient can be neglected. 

To account for possible internal-temperature gradients, Eq. (11-42) is used instead of Eq. (11-43). Since the temperature effect on the intrinsic rate is expressed by the Arrhenius function, Eq. (11-42) may be written... 

Within a catalyst pellet the internal temperature gradient is rarely >1 to 2°C between the surface and the center. Assume temperature at the center of the catalyst = surface temperature. 

The hot end of a thin solid rod is made flush with the target surface. Thermocouples, located along the probe axis, are used to determine the axial internal temperature gradient that is assumed to be linear. The heat flux is calculated using a one-dimensional conduction equation. In most cases, the probe has shields to minimize heat flux from the sides. Figure 6.9 shows an example of a portable, water-cooled total heat flux probe used for inserting into high temperature furnace environments. 

Tc — T ), according to Eq. (5.50). However, once the effectiveness factor is low enough to make the center concentration almost zero, further increases in the rate constant k shift the reaction zone closer to the external surface, so heat does not have to be transferred as far. The internal temperature gradient becomes steeper as k increases, as shown in Figure 5.10, and (Tc — T ) remains constant. 

The limitations on heat transfer rates in conventionally conducted freeze drying operations have led early to the attempt to provide internal heat generation with the use of microwave power [21,22]. Theoretically, the use of microwaves should result in a very accelerated rate of drying because the heat transfer does not require internal temperature gradients and the temperature of ice could be maintained close to the maximum permissible temperature for the frozen layer without the need for excessive surface temperatures. 

A large number of catalytic reactions are exothermic and are accompanied by thermal effects. For relatively fast intrinsic kinetics as compared to the mass and heat transfer phenomena, the development of internal temperature gradients can be expected. Heat and mass transfer balances have to be solved simultaneously to estimate concentration and temperature profiles under steady-state conditions. 

In Equation 2.214 the temperature difference between bulk and pellet surface is compared with the maximum internal temperature gradient. The ratio between... 

As an approximation suitable for small units that can be considered ideal (that is, without internal temperature gradients and with a constant transport coefficient) and in equilibrium situation, AT can be calculated from (Ebert, 1971)... 

Carberry JJ. On the relative importance of external-internal temperature gradients in heterogeneous catalysis. Industrial and Engineering Chemistry Fundamentals 1975 14 129-131. 

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