Radial temperature profile

The effect on the temperature development is that the front of freezing CO2 is moving faster through the bed close to the wall. These experimental results are compared with simulations with the one-dimensional model. The resulting breakthrough times match well with the experimental outcomes and are exactly positioned between the two measured temperatures. 

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Detailed modelling of the fixed bed production of phthalic anhydride from o-xylene is discussed by Froment and Bischoff (1990), involving both axial and radial temperature profile effects. 

Fig. 17. Least squares fits of the radial temperature profile in KATAPAK-M. Reprinted from Chemical Engineering Science, Vol. 54, von Scala et al., Heat Transfer Measurements and Simulation of KATAPAK-M Catalyst Supports, pp. 1375-1381, Copyright (1999), with permission from Elsevier.

From this illustration we can see that the added detail of the radial temperature profile near the wall that could be provided by CFD simulations does not help in obtaining better estimates for the standard heat transfer parameters. It also implies that experimental efforts to measure temperatures closer to the wall are, in fact, counter-productive. Finally, it is clear that the standard model with plug flow and constant effective transport parameters does not fit satisfactorily to temperature profiles in low-Abeds. These considerations have led us to look for improved approaches to near-wall heat transfer. 

Fig. 27b gives a more quantitative comparison between the radial temperature profiles for the full, 1-hole, and 1-bighole particles. There is a temperature... 

Fio. 27. (a) Near-wall temperature map for the 1-hole particles (b) radial temperature profiles for solid cylinders and cylinders with two different sizes of internal void. 

Fig. 34. Comparison of radial temperature profiles for WS packed with full cylinders, with and without heat sinks solid symbols are for temperatures averaged over fluid and solid, open symbols for temperatures averaged over fluid alone.

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

The first step in the solution procedure is discretization in the radial dimension, which involves writing the three-dimensional differential equations as an enlarged set of two-dimensional equations at the radial collocation points with the assumed profile identically satisfying the radial boundary conditions. An examination of experimental measurements (Valstar et al., 1975) and typical radial profiles in packed beds (Finlayson, 1971) indicates that radial temperature profiles can be represented adequately by a quadratic function of radial position. The quadratic representation is preferable to one of higher order since only one interior collocation point is then necessary,6 thus not increasing the dimensionality of the system. The assumed radial temperature profile for either the gas or solid is of the form... 

Fig. 3. Steady-state radial temperature profiles, type I conditions.

Fig. 11. Effect of thermal well on transient radial temperature profiles, type II conditions.

The heat flux at the wall at any z position can be determined from the radial temperature profiles using Fourier s law as... 

Bunnell, Irvin, Olson, and Smith (1949) Radial temperature profiles C Air Alumina Cylinders 3 ... 

Non-isothermal and non-adiabatic conditions. A useful approach to the preliminary design of a non-isothermal fixed bed reactor is to assume that all the resistance to heat transfer is in a thin layer near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the approximate design of reactors. Neglecting diffusion and conduction in the direction of flow, the mass and energy balances for a single component of the reacting mixture are ... 

Scale-up of structured reactors is usually easier than for packed-bed reactors. The major point is that the hydrodynamics are independent of the scale of the reactor (assuming a good inlet device). When the radial temperature profile is also independent of the scale, scale-up is straightforward. This is the case for millisecond reactors. In these reactors, rates are very high as a consequence, in exothermic reactions they operate adiabatically. So they scale easily. 

Figure 4. Radial temperature profiles in a laminar propane diffusion flame (24)...

Steady-State Behaviour The dashed line in Figure 2 shows a typical experimental axial temperature profile for conditions listed in Table I. The banded region in the vicinity of the hot spot includes those points (labelled a, b and c) in which radial temperature profiles were also measured using moving thermocouples. There, the upper and lower lines represent the highest measured temperature and the wall temperature, respectively, at those axial points. 

Heat transfer studies on fixed beds have almost invariably been made on tubes of large diameter by measuring radial temperature profiles (1). The correlations so obtained involve large extrapolations of tube diameter and are of questionable validity in the design of many industrial reactors, involving the use of narrow tubes. In such beds it is only possible to measure an axial temperature profile, usually that along the central axis (2), from which an overall heat transfer coefficient (U) can be determined. The overall heat transfer coefficient (U) can be then used in one--dimensional reactor models to obtain a preliminary impression of longitudinal product and temperature distributions. 

For an irreversible reaction, and assuming parabolic radial temperature profiles and that the radial change in conversion is small, it follows that ... 

Another assumption that will be made hardly needs stating, as it has come to be generally accepted. It will be assumed that there is a local resistance to heat transfer from the fluid to the wall of the reactor, giving rise to what is practically a discontinuity in the radial temperature profile at the inside surface of the wall. It has not been possible to determine the exact character of the wall effect that is observed, so that the description used here is not certainly the best one. On the basis of the experiments that have been made, however, this description is indistinguishable from an alternative one that assigns a reduced thermal conductivity to a layer adjoining the wall (see Y1 and Y3). 

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