Tube-wall reactor assumptions

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)]... 

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 ... 

A simple estimation of the temperature profile inside a tube reactor starts from an energy balance for the system [7]. Since we are particularly interested in the performances of a tube reactor and a catalytic wall reactor the following simplifying assumptions are made ... 

For viscous solutions, the assumptions of plug flow are not strictly valid. If the velocity profile is not flat, polymer solution near the tube wall will move more slowly than that near the center of the tube. Since the slow-moving polymer near the wall remains in the reactor longer, it will polymerize to a higher conversion (or extent of reaction) than the bulk material. This higher conversion will then compound the viscosity problem. Studies on the effect of this deviation from plug flow in tubular polymerization have to be carried out by Hamer and Ray [4,5]. 

The second term in the heat balance accounts for heat removal by a circulating liquid at temperature TV through a heat transfer area A, on the assumption of an overall heat transfer coefficient U across the tube wall. Note that represents the rate per unit volume of the reactor. Hence,... 

Here in Chapter 1 we make the additional assumptions that the fluid has constant density, that the cross-sectional area of the tube is constant, and that the walls of the tube are impenetrable (i.e., no transpiration through the walls), but these assumptions are not required in the general definition of piston flow. In the general case, it is possible for u, temperature, and pressure to vary as a function of z. The axis of the tube need not be straight. Helically coiled tubes sometimes approximate piston flow more closely than straight tubes. Reactors with square or triangular cross sections are occasionally used. However, in most of this book, we will assume that PFRs are circular tubes of length L and constant radius R. 

Tubular reactors are particularly subject to plugging. In addition, for viscous reactants (polymerization), the assumption of plug flow may not be valid, since the material near the center of the tube will move with a higher velocity than material near the walls. 

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