Tube flow radial conduction

An experimental evaluation of homogeneous continum models of steady state heat transfer in packed beds of low tube/particle diameter ratio has been carried out. It was found that both axial and radial conduction effects were important in such beds for N j 500, which covers the flow range in many industrial reactors. Heat transfer resistance at the wall was significant, but of secondary importance. 

In the common case of cylindrical vessels with radial symmetry, the coordinates are the radius of the vessel and the axial position. Major pertinent physical properties are thermal conductivity and mass diffusivity or dispersivity. Certain approximations for simplifying the PDEs may be justifiable. When the steady state is of primary interest, time is ruled out. In the axial direction, transfer by conduction and diffusion may be negligible in comparison with that by bulk flow. In tubes of only a few centimeters in diameter, radial variations may be small. Such a reactor may consist of an assembly of tubes surrounded by a heat transfer fluid in a shell. Conditions then will change only axially (and with time if unsteady). The dispersion model of Section P5.8 is of this type. 

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 ... 

The flow patterns, composition profiles, and temperature profiles in a real tubular reactor can often be quite complex. Temperature and composition gradients can exist in both the axial and radial dimensions. Flow can be laminar or turbulent. Axial diffusion and conduction can occur. All of these potential complexities are eliminated when the plug flow assumption is made. A plug flow tubular reactor (PFR) assumes that the process fluid moves with a uniform velocity profile over the entire cross-sectional area of the reactor and no radial gradients exist. This assumption is fairly reasonable for adiabatic reactors. But for nonadiabatic reactors, radial temperature gradients are inherent features. If tube diameters are kept small, the plug flow assumption in more correct. Nevertheless the PFR can be used for many systems, and this idealized tubular reactor will be assumed in the examples considered in this book. We also assume that there is no axial conduction or diffusion. 

The preceding discussion of experimental and theoretical studies pertaining to lateral migration applies only to the case of a rigid spherical particle suspended in a fully-developed, laminar, Newtonian flow within a circular tube. As discussed in subsequent paragraphs, related experimental and theoretical work on radial migration has also been conducted for a variety of situations in which one or more of these restrictions is relaxed. 

Two-dimensional (2-D) models allow for the change in temperature and reaction rate constant with tube radius. Most 2-D homogeneous models still assume plug flow of the gas and uniform radial concentration. (Calculations show that radial mixing is rapid enough to minimize the concentration differences.) Several radial increments are used for the computations, and the heat flux is set proportional to the radial temperature gradient and the local conductivity, kg. The conductivity can be taken as a constant or as a function of radial position. 

The length and diameter of the tube and the particle size (hydratdic diameter) also affect flow distrihution within the packed tube. If the ratio of the tube diameter to that of the particle diameter is above 30, radial variations in velocity can be n lected, and plug (piston) flow behavior can be assumed. The ratio of the tube length to particle diameter is also important if this ratio exceeds 50, axial dispersion and axial heat conduction effects can be ignored. These efiects bring notable simplifications into the modeling of PBRs, which are discussed in Chapter 3. 

Dispersion of solute bands in cells of different dimensions have been experimentally measured with concentric inlet and outlet connections (13), but this data although pertinent to some detectors (e.g. conductivity detectors) is not generally useful for optical detectors where inlet and outlet tubes normally have to be radially oriented. Radial entry and exit flow from the detector cell introduce significant radial mixing and thus reduces the extent of the dispersion that would be expected to result from low... 

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