External and Internal Temperature Gradient

If the experimental results of Kehoe and Butt [114] in Figs. 3.7.a-2 and 3.7.a-3 are studied, note that the external heat transfer resistance can be appreciable, and, especially for the isothermal pellet, must be considered. The same mass and heat balance equations (3.7.a-l, 2) [or (3.7.a-8,9)] are used, but surface boundary conditions expressed in terms of the finite external heat and mass transfer resistances are used. 

The determination of the maximum temperature differences between bulk fluid, catalyst pellet surface, and catalyst pellet interior in terms of directly observable quantities is a veiy useful tool in the study of catalytic reactions. Only if these temperature differences are significant need one be concerned with further extensive analysis of the transport phenomena. 

Lee and Luss [122] provided such results in terms of the observable (Weisz) modulus and the external effective Sherwood and Nusselt numbers. The steady-state mass and heat balances for an arbitrary reaction, using slab geometry, are 

The right-hand side of Eq. 3.7.b-7 is the sum of the external and internal temperature differences, as pointed out by Hlavacek and Marek [123]. The maximum temperatuie difference is for complete reaction, when C, = 0  

The final step is to obtain C//C in terms of an observable rate, which is the vohune-averag rate in the peUet  

External temperature gradients are much more likely. In transient situations, temperatures exceeding the steady-state maximum temperature of (3.13.1-5) could exist [Wei, 1966 Georgakis and Aris, 1974]. 

Integrating once from the pellet center to the surface, and utilizing (3.13.2-3) and (3.13.2-4), leads to  

A second integration and rearrangement gives the overall temperature difference 

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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... 

Figure 5.10 External and internal temperature and concentration gradients (a)...

Thus, it is found that the size ratio hjh in this case is directly proportional to the strength ratio, whereas in Example 6.1, Equation 6.3, it was inversely proportional to the square root of the strength ratio. Why is this Examine Equations 6.15 and 6.16 O is proportional to AT but AT is proportional to h. However, in the beam bending problem O was inversely proportional to IP in Equation 6.1. The reader should keep in mind this difference in stresses caused by external loads and internal temperature gradients. (As a mental challenge, consider combined external loads with internal temperature gradients.)... 

This example also shows the effects of mass- and enei y-transfer resistances within the catalyst pellet. The temperature increases toward the center of the pellet and increases the rate, but the oxygen concentration goes down, tending to reduce the rate. The global value of 49.8 x 10" is the resultant balance of both factors. Hence the net error in using the bulk conditions to evaluate the rate would be [(49.8 — 43.6)/49.8] (100), 12.5%. In this case the rate increase due to external and internal thermal effects more than balances the adverse effect of internal mass-transfer resistance. The procedure for calculating the effects of internal gradients on the rate is presented in Chap. 11. 

A temperature gradient would also be expected. For an isothermal case, with rj set equal to 1, multiple steady-state solutions may be found (see Figure 10), and the concentration gradient is very significant at temperatures above 427°C (800°F). The non-isothermal catalytic effectiveness factors for positive order kinetics under external and internal diffusion effects were studied by Carberry and Kulkarni (8) they also considered negative order kinetics. 

Diffusion of reactants to the external surface is the first step in a solid-catalyzed reaction, and this is followed by simultaneous diffusion and reaction in the pores, as discussed in Chapter 4. In developing the solutions for pore diffusion plus reaction, the surface concentrations of reactants and products are assumed to be known, and in many cases these concentrations are essentially the same as in the bulk fluid. However, for fast reactions, the concentration driving force for external mass transfer may become an appreciable fraction of the bulk concentration, and both external and internal diffusion must be allowed for. There may also be temperature differences to consider these will be discussed later. Typical concentration profiles near and in a catalyst particle are depicted in Figure 5.6. As a simplification, a linear concentration gradient is shown in the boundary layer, though the actual concentration profile is generally curved. 

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. 

Calculations that are performed using finite difference or finite element models should have a sufficiently fine mesh or element distribution to properly represent internal conduction and external and internal boundary conditions. External features such as fins should be given special attention as temperature gradients can be severe, perhaps requiring separate detailed calculations in order to determine the heat flux to the main body. Consideration should be given to the choice of one, two or... 

The effect of any one of these factors is, however, in no way simple. For the temperature effect, for example, the following mechanisms at least may be postulated Fusion of micellae, desorption of solvent, dissociation or formation of aggregates resulting in change of particle size and shape, increase in external and internal Brownian motion and (jonsequeiit contraction of the fiber molecule, and change in adjustment to the flow gradient. All these processes may occur individually or in combination and can be entirely or partially compensated by each other. 

The absence of external and internal mass transfer limitations and of temperature gradient have been proved by preliminary experiments. The conversion of methane as a function of space-time at different total pressures at 360°C in the Berty reactor is depicted in the Figure... 

The concept of effectiveness developed separately for external or internal transport resistances can be extended to an overall effectiveness factor for treating the general diffusion-reaction problem where both external and internal concentration and temperature gradients exist The overall effectiveness factor, D, is defined for relating the actual global rate to the intrinsic rate, that is, -Ra)p to (-Ra)6- To stun up the definitions for y, 7], and D,... 

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

Figure4.5.19 Concentration and temperature around and in a porous catalyst plate (external and internal gradients, although in this section only internal transport phenomena are inspected).

The distinction between electrical and non-electrical processes has another aspect. An external electric field can induce an electric current, and the calculation of this response, as in Kubo s theory, leads to the formula for electric conductivity. In contrast a heat current, for example, is the response to an internal temperature gradient and depends only indirectly on the external forces which set up this temperature gradient. 

In condensation-corrosion by gases, there are three possible modes of interaction of a gas with the external and internal surfaces of a refractory. First, if the gas is condensable (referred to as a vapor), in the course of penetration of the porosity and moving down the temperature gradient in a wall, the gas may literally condense or liquefy progressively. Second, the gas may condense by virtue of dissolving in the refractory and third, it may condense... 

The smaller reactor approaches plug-flow behavior and exhibits a large temperature gradient. In this case, external recycle provides the same degree of back-mixing as is provided by internal circulation in the larger diameter reactor. 

For catalytic investigations, the rotating basket or fixed basket with internal recirciilation are the standard devices nowadays, usually more convenient and less expensive than equipment with external recirculation. In the fixed basket type, an internal recirculation rate of 10 to 15 or so times the feed rate effectively eliminates external diffusional resistance, and temperature gradients. A unit holding 50 cm (3.05 in ) of catalyst can operate up to 800 K (1440 R) and 50 bar (725 psi). 

Catalyst pellets often operate with internal temperatures that are substantially different from the bulk gas temperature. Large heats of reaction and the low thermal conductivities typical of catalyst supports make temperature gradients likely in all but the hnely ground powders used for intrinsic kinetic studies. There may also be a him resistance to heat transfer at the external surface of the catalyst. 

All of these steps are rate processes and are temperature dependent. It is important to realize that very large temperature gradients may exist between active sites and the bulk gas phase. Usually, one step is slower than the others, and it is this rate-controlling step. The effectiveness factor is the ratio of the observed rate to that which would be obtained if the whole of the internal surface of the pellet were available to the reagents at the same concentrations as they have at the external surface. Generally, the higher the effectiveness factor, the higher the rate of reaction. 

When the internal diffusion effects are considered explicitly, concentration variations in the catalytic washcoat layer are modeled both in the axial (z) and the transverse (radial, r) directions. Simple slab geometry is chosen for the washcoat layer, since the ratio of the washcoat thickness to the channel diameter is low. The layer is characterized by its external surface density a and the mean thickness <5. It can be assumed that there are no temperature gradients in the transverse direction within the washcoat layer and in the wall of the channel because of the sufficiently high heat conductivity, cf., e.g. Wanker et al. 

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