Intrapellet heat transfer

A limiting case of intrapellet transport resistances is that of the thermal effectiveness factor In this situation of zero mass-transfer resistance, the resistance to intrapellet heat transfer alone establishes the effectiveness of the pellet. Assume that the temperature effect on the rate can be represented by the Arrhenius function, so that the rate at any location is given by r = A... 

In a porous catalyst, as reactants diffuse in the radial direction toward the center of the particle, reaction occurs on the pore walls, releasing or absorbing heat as required by the reaction. The interior surface of a porous catalyst is not as effective as its exterior because each point on the inner surface is exposed to a lower reactant concentration than that of the exterior (Cas). The net effect of intrapellet mass transfer resistance is to reduce the global rate beneath the rate evaluated at surface conditions. The net effect of intrapellet heat transfer resistance, on the other hand, depends on the exothermicity-endothermicity of the surface reaction and on the relative significance of... 

The starting point of a number of theoretical studies of packed catalytic reactors, where an exothermic reaction is carried out, is an analysis of heat and mass transfer in a single porous catalyst since such system is obviously more conductive to reasonable, analytical or numerical treatment. As can be expected the mutual interaction of transport effects and chemical kinetics may give rise to multiple steady states and oscillatory behavior as well. Research on multiplicity in catalysis has been strongly influenced by the classic paper by Weisz and Hicks (5) predicting occurrence of multiple steady states caused by intrapellet heat and mass intrusions alone. The literature abounds with theoretical analysis of various aspects of this phenomenon however, there is a dearth of reported experiments in this area. Later the possiblity of oscillatory activity has been reported (6). 

The effects of diffusional restrictions on the activity and selectivity of FT synthesis processes have been widely studied (32,52,56-60). Intrapellet diffusion limitations are common in packed-bed reactors because heat transfer and pressure-drop considerations require the use of relatively large particles. Bubble columns typically use much smaller pellets, and FT synthesis rates and selectivity are more likely to be influenced by the rate of mass transfer across the gas-liquid interface as a gas bubble traverses the reactor (59,61,62). 

Essentially all of the surface, of porous catalyst pellets is internal (see page 295). Reaction and mass and heat transfer occur simultaneously at any position within the pellet. The resulting intrapellet concentration and temperature gradients cause the rate to vary with position. At steady state the average rate for a whole pellet will be equal to the global rate at the location of the pellet in the reactor. The concentration and temperature of the bulk fluid at this location rhay not be equal to those properties at the outer surface of the pellet. The effect of such external resistances can be accounted for by the procedures outlined in Chap. 10. The objective in the present chapter is to account for internal resistances, that is, to evaluate average rates in terms of the temperature and concentration at the outer surface. Because reaction and transport occur simultaneously, differential... 

The effective thermal conductivities of catalyst pellets are surprisingly low. Therefore significant intrapellet temperature gradients can exist, and the global rate may be influenced by thermal effects. The effective conductivity is the energy transferred per unit of total area of pellet (perpendicular to the direction of heat transfer). The defining equation, analogous to Eq. (11-18) for mass transfer, may be written... 

For an endothermic reaction there is a decrease in temperature and rate into the pellet. Hence 17 is always less than unity. Since the rate decreases with drop in temperature, the effect of heat-transfer resistance is diminished. Therefore the curves for various are closer together for the endothermic case. In fact, the decrease in rate going into the pellet for endothermic reactions means that mass transfer is of little importance. It has been shown that in many endothermic cases it is satisfactory to use a thermal effectiveness factor. Such thermal 17 neglects intrapellet mass transport that is, ri is obtained by solution of Eq. (11-72), taking C = Q. 

In many catalytic systems multiple reactions occur, so that selectivity becomes important. In Sec. 2-10 point and overall selectivities were evaluated for homogeneous well-mixed systems of parallel and consecutive reactions. In Sec. 10-5 we saw that external diffusion and heat-transfer resistances affect the selectivity. Here we shall examiineHEieHnfiuence of intrapellet res ahces on selectivity. Systems with first-order kinetics at isothermal conditions are analyzed analytically in Sec. 11-12 for parallel and consecutive reactions. Results for other kinetics, or for nonisothermal conditions, can be developed in a similar way but require numerical solution. ... 

The boundary conditions at the external surface of the catalyst are T = Tsurface and Ca = Ca surface, and A effeciive is the effective thermal conductivity of the composite catalyst structure (i.e., 1.6 x 10 J/cm s K for alumina). Initially, the surface temperature and concentration of reactant A in Uie vicinity of a single isolated catalytic peUet are chosen to match the inlet values to the packed reactor. If external mass and heat transfer resistances are minimal, then bulk gas-phase temperature and reactant concentration at each axial position in the reactor represent the characteristic quantities that should be used to calculate the intrapellet Damkohler number for nth-order chemical kinetics ... 

It might be possible to neglect the external gas phase resistance to mass transfer relative to intrapellet diffusional resistance through a tortuous pathway. An increase in the gas stream flow rate reduces the external mass transfer resistance further. Remember that diffusion coefficients and mass transfer coefficients increase as one progresses from solids to liquids to gases. Hence, gas-phase mass transfer resistances are small, but the intrapellet gas-phase diffusional resistances should be significant, particularly when the intrapellet Damkohler number is quite large. In contrast, thermal conductivities and heat transfer coefficients increase... 

Analyze coupled mass and thermal energy balances with chemical reaction in an isolated pellet to estimate the intrapellet resistance to heat transfer. Results from step 1 are used to simplify the thermal energy balance. 

Manipulate the multicomponent thermal energy balance in the gas-phase boundary layer that surrounds each catalytic pellet. Estimate the external resistance to heat transfer by evaluating all fluxes at the gas/porous-solid interface, invoking continuity of the normal component of intrapellet mass flux for each component at the interface, and introducing mass and heat transfer coefficients to calculate interfacial fluxes. 

Answer Let the temperature-dependent ratio of effective intrapellet dif-fusivities eA( ) be a simple power function of dimensionless temperature 0. Hence, a(0) = 0 "- Now, integrate the thermal energy balance given by equations (27-27) and (27-35) that results from coupled heat and mass transfer in porous catalytic pellets ... 

Important results from earlier sections are summarized here to develop reactor design strategies when external resistances to heat and mass transfer cannot be neglected. Intrapellet resistances require information about... 

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