External mass transfer Isothermal conditions

A fluid passing over a solid surface develops a boundary layer in which the velocity parallel to the surface varies rapidly over a very short distance normal to the direction of flow. The velocity is zero at the solid surface but approaches the bulk-stream velocity at a distance less than a millimeter from the surface. Mixing occurs in the main fluid stream where reactants and products are transported at rates that depend on the nature of flow. The fluid velocity near the surface is low with little mixing therefore, mass transport perpendicular to the surface is by molecular diffusion. Although mass transport in the main stream is essentially independent of the molecular diffusion coefficient, it is proportional to DA-mix very 

Consider the unrmolecular, irreversible solid-catalyzed gas-phase reaction, A—yB, carried out in a fixed-bed reactor packed with completely nonporous particles. Assume that the chemical steps of adsorption-reaction-desorption are represented by first-order kinetics and that the bulk temperature, Tin and surface temperature, Ts, are the same around a particle located at any point along the length of the reactor. 

Under steady-state conditions, the rate of reaction will be balanced by the rates of mass transfer of reactants and products. If -Ra)p is defined as the experimentally measured global rate of disappearance of A per unit mass of catalyst (mol/g-s), the rates of steps 1, 3-4-5, and 7 in the 5-step sequence will all be equal to -Ra)p. 

Step 1 Mass transfer of reactant A from the bulk fluid to the catalyst surface 

Step 7 Mass transfer of B from the catalyst surface to the bulk fluid 

---

The linear driving force (LDF) approximation is obtained when the driving force is expressed as a concentration difference. It was originally developed to describe packed-bed dynamics under linear eqm-librium conditions [Glueckauf, Trans. Far Soc., 51, 1540 (1955)]. This form is exact for a nonlinear isotherm only when external mass transfer is controlling. However, it can also be used for nonlinear sys-... 

Isocratic Elution In the simplest case, feed with concentration cf is apphed to the column for a time tp followed by the pure carrier fluid. Under trace conditions, for a hnear isotherm with external mass-transfer control, the linear driving force approximation or reaction kinetics (see Table 16-12), solution of Eq. (16-146) gives the following expression for the dimensionless solute concentration at the column outlet ... 

If recirculation rates are 10 to 15 times the feed rate, the reactor would tend to operate nearly isothermally. High velocities past the bed of particles could eliminate almost completely any external mass-transfer influence on the reactor performance. By varying the circulation rates, the reaction condition for which the mass transfer effect is negligible can be established. Except for the rapidly-decaying catalyst system, steady state can be achieved effectively. Sampling and product analysis can be obtained as effectively as in the fixed-bed reactor. Residence-time distributions for the fluid phases can be measured easily. High fluid velocities would cause less flow-maldistribution problems. 

Isothermal and isobaric conditions. Pick law describes diffusion in the membrane. External mass transfer limitations are negligible. The model is solved for given reactant concentrations at the opposite membrane sides. 

Hydrazine has been studied extensively for use in monopropellant thrusters for space flights of long duration. Thrusters are used or altitude control of communication satellites. Here the decomposition of hydrazine over a packed bed of alumina-supported iridium catalyst is of interest. " In a proposed study, a 2% hydrazine in 98% helium mixture is to be passed over a packed bed of cylindrical particles 0.25 cm in diameter and 0.5 cm in length at a gas-phase velocity of 15m/s and a temperature of 750 K. The kinematic viscosity of helium at this temperature is 4.5 X 10 nF/s. The hydrazine decomposition reaction is believed to be externally mass transfer-limited under these conditions. If the packed bed is 0.05 m in length, what conversion can be expected Assume isothermal operation. 

In order to check the validity of the model described in Section 2, integration of Eqs. (6, 7) was carried out, considering the experimental conditions of the three dynamic runs in the boundary conditions (10). As already indicated, once the boundary conditions and the equilibrium isotherms are assigned, the only parameters of the model are the mass transfer coefficients kg and A simple best fit procedure led to a very satisfactory agreement between model and experimental results. It is important to observe that all three breakthrough curves were obtained using the same value for the external mass transfer coefficient (namely, e=5.M0 m/s) this is in agreement with the physical nature of kg, which only depends on fluid dynamic conditions. On the other hand, the internal mass transfer coefficient ki varied with the total normality of the solution used. In particular, for A =0.89 eqW it was, =1.4-10 eq/m for A7=22.6eq/m it was eq/m for A =44.4eq/m it was... 

The general problem of diffusion-reaction for the overall effectiveness factor D is rather complicated. However, the physical and chemical rate processes prevailing under practical conditions promote isothermal particles and negligible external mass transfer limitations. In other words, the key transport limitations are external heat transfer and internal mass transfer. External temperature gradients can be significant even when external mass transfer resistances are negligibly small. 

Figure 4.5.15 External effectiveness factor as a function ofthe ratio ofthe (measurable) effective reaction rate to the maximum rate (complete control by external mass transfer) fora constant Arrhenius number of 20. For a Prater number jSex < 0, the reaction is endothermic, for /Sex > 0 exothermic, and for /Sex = 0 we have isothermal conditions. Arrows and dashed line indicate ignition, as explained in the text.

Values of j/jnt, and of selected reactions are listed in Table 4.5.6, which show that only for the dissociation of N2O may the effectiveness factor Jjpore.max exceed unity (up to Jjpore.max = 100 for < = 1, Figure 4.5.23). Thus, we conclude that 7pore is rarely influenced by heat transfer, and, only for control by external mass transfer, the overheating of a catalyst may enhance the effective rate constant compared to isothermal conditions (Topic 4.5.5). 

Assuming isothermal conditions and neglecting bulk flow, radial concentration gradients in the pore and external mass transfer resistance, the following dimensionless form of the pore plugging model is derived ( ). 

Most of the previously used expressions to account for incomplete catalyst wetting in trickle-beds are summarized in Table I. All of these, with the exception of the last one, are based on the assumptions of a) plug flow of liquid, b) no external mass transfer limitations, c) isothermal conditions, d) first order irreversible reaction with respect to the liquid reactant, e) nonvolatile liquid reactant, f) no noncatalytic homogeneous liquid phase reaction. 

It is clear from the foregoing discussion that the general problem of diffusion-reaction for the overall effectiveness factor is quite involved. Fortunately, however, the physical and chemical processes at work under realistic conditions favor isothermal pellets and negligible external mass transfer resistances. A more detailed examination of this is in order. Combining Eqs. 4.32 and 4.33 results in ... 

The basic approximations made in arriving at the reactor point effectiveness are (1) isothermal pellet, (2) negligible external mass transfer resistance, and (3) estimation of the pellet center concentration by a simple relationship when the reaction is not severely diffusion-limited. The first two approximations are quite adequate in view of the fact that the mass Biot number is of the order of hundreds under realistic reaction conditions. Both theoretical and experimental justifications for these approximations have been given in Chapter 4. The first approximation will be relaxed when reactions affected by pore-mouth poisoning are considered since a definite temperature gradient then exists within the pellet. An additional approximation is the representation of the difference between the Arrhenius exponentials evaluated at the pellet surface and the bulk-fluid temperatures by a linear rela-... 

Under the conditions of negligible external mass transfer resistance and an isothermal pellet, Eqs. 10.32, 10.37, and 10.38 can be removed from consideration for the case of uniform deactivation. However, Eq. 10.32 still needs to be retained in a limited form to account for the temperature drop in the deactivated outer shell in the case of shell-progressive deactivation. 

It has been shown that under realistic reaction conditions, the major transport resistances are external heat transport and internal diffusion. The pellet can be treated as isothermal and the external mass transfer resistance can be neglected. While these conclusions are valid for gas-solid systems, they are not valid for liquid-solid systems as evident from the range of the Biot number ratio (10 -10 ) given in Table 4.2 for the liquid-solid systems. 

The importance of internal diffusion can also be appreciated from a different point of view the fact that the internal diffusion plays a pivotal role in internal and external transport processes. For negligible concentration gradient in the pellet, Eq. 4.57 still holds. However, the value of r Da will be larger than that for diffusion-limited case for the same intrinsic rate since 17 is larger and therefore the pellet will be more isothermal as Figure 4.7 reveals. Further, a relativdy large Biot number for mass under realistic conditions still ensures negligible external mass transfer resistance. It is seen then that in the absence of diffusional resistance, the pellet tends to be more isothermal and the only major resistance is likely to be external heat transfer. 

As discussed fully in Chapter 4, the Biot number is large enough under typical reaction conditions to allow isothermal treatment of catalyst pellets. Furthermore, the external mass transfer resistance (interphase mass transfer) can also be... 

The most appropriate choice of the force X for the molecular diffusion through the membrane under isothermal conditions without external forces being applied to the mass transfer of the th component is the chemical potential gradient ... 

As the mass transfer resistance at the external surface of the silicalite crystal particle has negligible effects on the overall mass transfer in the present LC system (13), local equilibrium at the external surface thus can be assumed to simplify the mathematics involved in the numerical solution. Since the adsorption isotherm data of alcohols from dilute aqueous solution in silicalite, as reported by Milestone and Bibby (2), can be described by a Langmuir type equation, the following Langmuir type equation is used as the boundary condition at the external surface of the silicalite crystal particle ... 

Air at 294 K and 1 atm enters a fixed-bed adsorber at a flow rate of 0.146 m3/s with a benzene vapor concentration of 29 g/m3. The cylindrical adsorber is 0.61 m in inside diameter and is packed to a height of 1.83 m with 331 kg of silica gel particles having an effective diameter of 2.6 mm and an external porosity of 50%. The adsorption isotherm for benzene has been determined experimentally and found to be linear over the concentration range of interest, given by q = kc, where q is in kg benzene/kg gel, c is in kg benzene/m3 of gas, and k = 4.127 m3 of gas/kg of gel. It has been estimated that the overall volumetric mass-transfer coefficient for the conditions prevailing in the bed is Kc.a = 8.79 s-1. Assuming isothermal and isobaric operation, calculate ... 

---