Diffusion in a tubular reactor

The importance of diffusion in a tubular reactor is determined by a dimensionless parameter, 3>Ai/R2 = aL/(uR2), which is the molecular diffusivity of component A scaled by the tube size and flow rate. If SCff/- 2 is small, then the effects of diffusion will be small, although the definition of small will depend on the specific reaction mechanism. Merrill and Hamrin1 studied the effects of diffusion on first-order reactions and concluded that molecular diffusion can be ignored in reactor design calculations if... 

Example 3.2.11. Axial Conduction and Diffusion in a Tubular Reactor... 

The importance of diffusion in a tubular reactor is determined by a dimensionless parameter = <3j L/(uR ), which is the molecular diffusivity of component... 

The final idealized flow situation that we will consider is laminar flow in a tubular reactor in the absence of either radial or longitudinal diffusion. The velocity profile in such a reactor is given by... 

Example Consider the equation for convection, diffusion, and reaction in a tubular reactor. 

The PFR model assumes a flat velocity profile across the whole of the reactor cross-section in reality, this is impossible to achieve although in practice certain combinations of physical conditions are closely described by this assumption. If the Reynolds number, dupln, in a tubular reactor is less than about 2100, then the flow therein will be laminar and where the flow is fully developed, the velocity profile across the reactor will be parabolic in form. If one assumes that diffusion is negligible between adjacent radial layers of fluid, then it is relatively straightforward to derive the forms of E(t), E(0) and F(0) associated with this type of reactor [42]. These are given in the equations... 

Fick s diffusion law is used to describe dispersion. In a tubular reactor, either empty or packed, the depletion of the reactant and non-uniform flow velocity profiles result in concentration gradients, and thus dispersion in both axial and radial directions. Fick s law for molecular diffusion in the x-direction is defined by... 

Consider a steady flow of reactant A to products at constant density through an element of radius r, width 8r, and height 81 in a tubular reactor at isothermal condition. Suppose that radial and axial mass transfer is expressed by Fick s law, with (D and (De)r as effective diffusivities. The rate at which A reacts is (-rA), mol/m3 sec. A material balance on a tubular element of radii r and r + 8r and height 81 is carried out from... 

Exercise 9.9.4. Show that the distribution function of residence times for laminar flow in a tubular reactor has the form 2z /Zp, where tp is the time of passage of any fluid annulus and the minimum time of passage. Diffusion and entrance effects may be neglected. Hence show that the fractional conversion to be expected in a second order reaction with velocity constant k is 2B[1 + j lnu5/(5 + 1)] where B = akt n and a is the initial concentration of both reactants. (C.U.)... 

Deviations from ideal flow can be classified in two types. In one type of deviation elements of fluid may move through the reactor at different velocities, causing channeling and dead spots. For such behavior to occur, the elements of fluid must not completely mix locally, but remain at least partially segregated as they move through the reactor. The other deviation refers to the extent of the local or micromixing. For exam.ple, there may be some mixing or diffusion in the direction of flow in a tubular reactor. 

The reduction of FeS2 to FeS is carried out in a tubular reactor with upfiow of hydrogen and downflow of solids. The reactor will operate at 495°C and 1 atm with pure hydrogen. For these conditions gas-phase diffusion resistance is... 

The axial mixing in a tubular reactor can sometimes be described by a dispersion model. This model is based on the assumption that the RTD may be considered to result from piston flow on which is superimposed an axial dispersion. The latter is taken into account by means of a constant effective axial dispersion coefficient, Dax, which has the same dimensions as the molecular diffusion coefficient, Dm. Usually Dax is much larger than the molecular diffusion coefficient because it incorporates all effects that cause deviations from plug flow, such as variations in radial velocities, eddies, and vortices. 

Equations 14.2-3 and 14.2-4 bear a striking resemblance to the mass and energy balances for a batch reactor, Eqs. 14.1-13 and 14. There is, in fact, good physical reason why these equations should look very much alike. Our model of a plug-flow reactor, which neglects diffusion and does not allow for velocity gradients, assumes that each element of fluid travels through the reactor with no interaction with the fluid elements before or after it Therefore, if we could follow a small fluid element in a tubular reactor, we would find that it had precisely the same behavior in time as is found in a batch reactor. This similarity in the physical situation is mirrored in the similarity of the descriptive equations. 

To carry out an exothermic reaction in a tubular reactor under nearly isothermal conditions, a small diameter is needed to give a high ratio of surface area to volume. The reactor could be made from sections of jacketed pipe or from a long coil immersed in a cooling bath. The following analysis is for a constant jacket temperature, and the liquid is assumed to be in plug flow, with no radial gradients of temperature or concentration and no axial conduction or diffusion. 

Estimate the conversion obtainable in a tubular reactor under laminar flow conditions neglecting radial diffusion for the reaction presented in Example 3.3. The mean residence time is f = 10 min. 

Simple experiments performed both in a continuous-fed tubular reactor or a batch reactor can discriminate between external or internal regimes. For external diffusion, runs are made in a tubular reactor in order to evaluate the conversion (X) of the reagent at different ratios between the weight of the catalyst (W, kg) and the amount of feed (F, kg/h or L/h). If two runs at the same temperature give the same value of X at the same value of W/F, but with two different amounts of catalyst (i.e., Wi and W2), then external diffusion is absent. In fact, to have the same value of W/F with two different weights W, it is necessary to change F and... 

Satterfield CN, Resnick H, Wentworth RI Simultaneous heat and mass transfer in a diffusion-controlled chemical reaction part I studies in a tubular reactor, Chem Eng Prog 50 460-466, 1954. 

Figure 4.10.17 Conversion in an ideal plug flow reactor (C= 1) and in a tubular reactor with laminar flow (negligible molecular diffusion) for a first-order reaction (Do = kr) approximations for laminar flow as given by Eq. (4.10.31) are also shown.

Figure 4.10.58 First-order reaction in a tubular reactor according to the dispersion model [Eq. (4.10.115) dotted line laminar flow without molecular diffusion (Eq. (4.10.29)].

Uniqueness of the steady state solutions for chemical reaction occurring in a catalyst particle or in a tubular reactor with axial diffusion. Chem. Eng. Sci. 22, 253 (1967). 

The DVD-ROM contains PDF files of the last live chapters from the fourth edition of the Elements of Chemical Reaction Engineering, which is mostly graduate material. These chapters, which were omitted from this book but are included on the DVD-ROM are DVD Chapter 10. Catalyst Decay DVD Chapter 11, External Diffusion Effects on Heterogeneous Reactions DVD Chapter 12. Diffusion and Reaction DVD Chapter 13, Distribution of Residence Times for Reactors DVD Chapter 14. Models for Non Ideal Reactors and a new chapter. DVD Chapter 15, Radial and Axial Temperature Variations in a Tubular Reactor. 

Tubular reactors are used for some polycondensations. Para-blocked phenols can be reacted with formalin to form linear oligomers. When the same reactor is used with ordinary phenol, plugging will occur if the tube diameter is above a critical size, even though the reaction stoichiometry is outside the region that causes gelation in a batch reactor. Polymer chains at the wall continue to receive formaldehyde by diffusion from the center of the tube and can crosslink. Local stoichiometry is not preserved when the reactants have different diffusion coefficients. See Section 2.8. 

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