Effect of axial dispersion

FIGURE 8,22. Combined effects of axial dispersion and mass transfer resistance for a favorable Langmuir equilibrium system ( ==0.33). a) Constant pattern breakthrough curves for various values of (fluid film resistances + axial dispersion), (h) The same curves plotted on a modified time scale with 7 = t/(1 + B). [Reprinted with permission from Chem. Eng. Sci 30. Garg and Ruthvcn (ref. 54). Copyright 1975, Pcrgamon Press, Ltd.) 

FIGURE 8.23, Theoretical constant-pattern breakthrough curves for irreversible adsorption showing combined effects of axial dispersion and (external) mass transfer resistance. 1.0 corresponds essentially to plug flow with external film resistance while oo corresponds to axial dispersion with negligible mass transfer resistance. Curves are calculated from expression given by Acrivos.  

An equivalent analysis of the combined effects of axial dispersion and mass transfer resistance has been presented by Rhee and Amundson, based on shock layer theory. From the mass balance over the shock layer it may be shown that the propagation velocity [Eq. (8.13)] is not affected by mass transfer resistance or axial dispersion. For an equilibrium system with axial dispersion the differential mass balance [Eq. (8.1)] becomes, under constant pattern conditions. 

It is evident that the dispersion and mass transfer resistance play essentially equivalent roles in Eq. (8.55). Except when the isotherm approaches the irreversible limit the seeond-order term is much smaller than the first-order term and it shown that to a good approximation the profile depends only on the value of the combined parameter 1 /Pe + w (l - w )/St. Since Pe/St = 8 this is equivalent to the previous result. 

In an adiabatic adsorption column the temperature front generally travels at a velocity which is different from the velocity of the primary mass transfer front and, since adsorption equilibrium is temperature dependent, a secondary mass transfer zone is established coincident with the thermal front. In a system with finite heat loss from the column wall one may approach either the isothermal situation with a single mass transfer zone or the adiabatic situation with two mass transfer zones, depending on the relative rates of heat generation and dissipation from the column wall. In the former case the effect of finite heat transfer resistance is to widen the mass transfer zone relative to an isothermal system. 

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Effect of Axial Dispersion on Column Performance. Another assumption underlying standard design methods is that the gas and the Hquid phases move in plug-flow fashion through the column. In reaHty, considerable departure from this ideal flow assumption exists (4) and different fluid... 

Fig. 17. Effect of axial dispersion in both phases on solute distribution through countercurrent mass transfer equipment. A, piston or plug flow B, axial...

Rapid Approximate Design Procedure. Several simplified approximations to the rigorous solutions have been developed over the years (57—60), but they aU. remain too compHcated for practical use. A simple method proposed in 1989 (61,62) uses a correction factor accounting for the effect of axial dispersion, which is defined as (57)... 

Axial Dispersion Effects In adsorption bed calculations, axial dispersion effects are typically accounted for by the axial diffusionhke term in the bed conservation equations [Eqs. (16-51) and (16-52)]. For nearly linear isotherms (0.5 < R < 1.5), the combined effects of axial dispersion and mass-transfer resistances on the adsorption behavior of packed beds can be expressed approximately in terms of an apparent rate coefficient for use with a fluid-phase driving force (column 1, Table 16-12) ... 

Adiabatic Reactors. Like isothermal reactors, adiabatic reactors with a flat velocity profile will have no radial gradients in temperature or composition. There are axial gradients, and the axial dispersion model, including its extension to temperature in Section 9.4, can account for axial mixing. As a practical matter, it is difficult to build a small adiabatic reactor. Wall temperatures must be controlled to simulate the adiabatic temperature profile in the reactor, and guard heaters may be needed at the inlet and outlet to avoid losses by radiation. Even so, it is hkely that uncertainties in the temperature profile will mask the relatively small effects of axial dispersion. 

To attempt clarification of this situation the effect of axial dispersion on the chromatogram was examined in two ways ... 

In this section, we apply the axial dispersion flow model (or DPF model) of Section 19.4.2 to design or assess the performance of a reactor with nonideal flow. We consider, for example, the effect of axial dispersion on the concentration profile of a species, or its fractional conversion at the reactor outlet. For simplicity, we assume steady-state, isothermal operation for a simple system of constant density reacting according to A - products. 

Note that setting one of the terms on the left side of the equation equal to zero yields either the batch reactor equation or the steady-state PFTR equation. However, in general we must solve the partial differential equation because the concentration is a function of both position and time in the reactor. We will consider transients in tubular reactors in more detail in Chapter 8 in connection with the effects of axial dispersion in altering the perfect plug-flow approximation. 

The importance of dispersion and its influence on flow pattern and conversion in homogeneous reactors has already been studied in Chapter 2. The role of dispersion, both axial and radial, in packed bed reactors will now be considered. A general account of the nature of dispersion in packed beds, together with details of experimental results and their correlation, has already been given in Volume 2, Chapter 4. Those features which have a significant effect on the behaviour of packed bed reactors will now be summarised. The equation for the material balance in a reactor will then be obtained for the case where plug flow conditions are modified by the effects of axial dispersion. Following this, the effect of simultaneous axial and radial dispersion on the non-isothermal operation of a packed bed reactor will be discussed. 

Show that the effect of axial dispersion on the conversion obtained in a typical packed bed gas-solid catalytic reactor is small. As the starting point consider the following relationship (see Chapter 2, equation 2.30). 

If the model accounts for the effects of axial dispersion, then the system is described by an axial dispersion model in terms of two-point boundary ODEs, i.e., by... 

Predictions of the column height required for any given separation can be obtained by using either a staged approach or a transfer unit approach. The plug flow models for determining the height of a column are of limited value due to the effect of axial dispersion, which is caused by... 

Sylvester and Pitayagulsarn53,54 considered combined effects of axial dispersion, external diffusion (gas-liquid, liquid-solid), intraparticle diffusion, and the intrinsic kinetics (surface reaction) on the conversion for a first-order irreversible reaction in an isothermal, trickle-bed reactor. They used the procedure developed by Suzuki and Smith,51,52 where the zero, first, and second moments of the reactant concentration in the effluent from a reactor, in response to a pulse introduced, are taken. The equation for the zero moment can be related to the conversion X, in the form... 

Here, the parameter F = Uo]dJ2De( — t) considers the effect of intraparticle diffusion, Pe = V dJlEzi. takes into account the effect of axial dispersion, S = 3(1 — e)Kt/U0L considers the effect of total external mass-transfer resistance, and A0 = /j (l — )k dp/2UoL considers the effect of surface reaction on the conversion. In these reactions L/0l, s the superficial liquid velocity, dp is the particle... 

In a pulse reactor, the effect of axial dispersion on the peak width can be minimized by introducing a dispersion column ahead of the catalyst bed to broaden the Gaussian-shaped pulses. 

If the flow rate is sufficiently high to create turbulent flow, then Pe is a constant and the magnitude of the right-hand side of the equation is determined by the aspect ratio, L/d. By solving Equation, (8.4.12) and comparing the results to the solutions of the PER [Equation (8.4.3)], it can be shown that for open tubes, L/d, > 20 is sufficient to produce PER behavior. Likewise, for packed beds, L/d, > 50 (isothermal) and L d, >150 (nonisothermal) are typically sufficient to provide PER characteristics. Thus, the effects of axial dispersion are minimized by turbulent flow in long reactors. 

In Chapter 8, axial dispersion in tubular reactors was discussed. Typical industrial reactors have sufficiently high flow rates and reactor lengths so the effects of axial dispersion are minimal and can be neglected. A rule of thumb is that axial dispersion can be neglected if ... 

Although the model equation included the axial dispersion coefficient (Dl), plug flow was approximated by assigning a very large value to the Peclet number (uL/Dl). This is because the effect of axial dispersion is quite negligible in a small column and the model with the second derivatives can give more stable numerical results. 

The marked differences in flow profiles have a significant effect on the efficiency of separation in open tubes. This is discussed in many publications including Knox and Grant (41. In the case of an open tube, the flow variation across the tube in pressure drive means that as the solute moves along it is dispersed. This is counteracted by transverse molecular diffusion to give a resulting net dispersion that is described by the Taylor equation 16. The additional effects of axial dispersion leads to the HETP equation... 

The Flow Equation. Consider a differential cross-sectional slice, dx, at distance x from the feed end of the devolatilizer. A volatile component material balance across this slice will include net inputs due to mean axial flow and axial dispersion (the latter arising from the nip mixing action), and depletion through the regenerated surface films. In addition to the three assumptions made above, it is assumed that uniform conditions prevail throughout the length—i.e., constant Uy p, S, Wy D y etc.-and that the effect of axial dispersion may be characterized by a constant axial eddy diffusivity, E. The steady-state material balance for a volatile component across dx reduces to ... 

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