Heat transfer resistance, effect

Keywords polymerization kinetics, polymerization reactors, mathematical modelling, molecular weight distribution (MWD), chemical composition distribution (CCD), Ziegler-Natta catalysts, metallocenes, microstructure, isotacticity distribution, mass transfer resistances, heat transfer resistances, effects of multiple site types. 

Additionally, the surfactant properties of filmers reduce the potential for stagnant, heat-transfer-resisting films, which typically develop in a filmwise condensation process, by promoting the formation of condensate drops (dropwise condensation process) that reach critical mass and fall away to leave a bare metal surface (see Figure 11.2). This function, together with the well-known scouring effect on unwanted deposits keeps internal surfaces clean and thus improves heat-transfer efficiencies (often by 5-10%). 

A perfect temperature-controlled heat-transfer surface is difficult to achieve, but it is closely simulated in practice by using a control fluid on one side of, for example, a metal tube. The tube wall should be thin and, ideally, the heat-transfer resistance comparatively large for the other fluid on the working side of the tube the latter surface is then effectively temperature-controlled and responds only to changes in the control fluid. 

The ratio of the observed reaction rate to the rate in the absence of intraparticle mass and heat transfer resistance is defined as the elFectiveness factor. When the effectiveness factor is ignored, simulation results for catalytic reactors can be inaccurate. Since it is used extensively for simulation of large reaction systems, its fast computation is required to accelerate the simulation time and enhance the simulation accuracy. This problem is to solve the dimensionless equation describing the mass transport of the key component in a porous catalyst[l,2]... 

On the other hand, it has been argued that the resistance to heat transfer is effectively within a thin gas film enveloping the catalyst particle [10]. Thus, for the whole practical range of heat transfer coefficients and thermal conductivities, the catalyst particle may be considered to be at a uniform temperature. Any temperature increases arising from the exothermic nature of a reaction would therefore be across the fluid film rather than in the pellet interior. 

The effects of non-uniform distribution of the catalytic material within the support in the performance of catalyst pellets started receiving attention in the late 60 s (cf 1-4). These, as well as later studies, both theoretical and experimental, demonstrated that non-uniformly distributed catalysts can offer superior conversion, selectivity, durability, and thermal sensitivity characteristics over those wherein the activity is uniform. Work in this area has been reviewed by Gavriilidis et al. (5). Recently, Wu et al. (6) showed that for any catalyst performance index (i.e. conversion, selectivity or yield) and for the most general case of an arbitrary number of reactions, following arbitrary kinetics, occurring in a non-isothermal pellet, with finite external mass and heat transfer resistances, the optimal catalyst distribution remains a Dirac-delta function. 

It is difficult to solve the system of Eqs. (39)—(41) for these boundary conditions. However, certain simplifying assumptions can be made, if the Prandtl number approaches large values. In this case, the thermal boundary layer becomes very thin and, therefore, only the fluid layer near the plate contributes significantly to the heat transfer resistance. The velocity components in Eq. (41) can then be approximated by the first term of their Taylor series expansions in terms of y. In addition, because the nonlinear inertial terms are negligible near the wall, one can further assume that the combined forced and free convection velocity is approximately equal to the sum of the velocities that would exist when these effects act independently. Therefore, for assisting flows at large Prandtl numbers (theoretically for Pr -> oo), Eq. (41) can be rewritten in the form ... 

In this section we have presented and solved the BVPs associated with the diffusion and reaction that take place in the pores of a porous catalyst pellet. The results were expressed graphically in terms of the effectiveness factor rj versus the Thiele modulus d> for two cases One with negligible external mass and heat transfer resistances, i.e., when Sh and Nu —> oo, and another with finite Sh and Nu values. This problem is very important in the design of fixed-bed catalytic reactors. The sample results presented here have shown that for exothermal reactions multiple steady states may occur over a range of Thiele moduli d>. Efficient numerical techniques have been presented as MATLAB programs that solve singular two-point boundary value problems. 

Here rB is the intrinsic rate of reaction, neglecting the mass and heat transfer resistances, r is the actual rate of reaction that takes the mass- and heat-transfer resistances into account, and rj is the effectiveness factor that accounts for the effect of mass- and heat-transfer resistances between the bulk fluid and the catalyst pellet. By using r] we are able to express the rate of reaction in terms of the bulk concentration and the temperature while the reaction is in fact taking place inside the pellet. 

Here we consider a spherical catalyst pellet with negligible intraparticle mass- and negligible heat-transfer resistances. Such a pellet is nonporous with a high thermal conductivity and with external mass and heat transfer resistances only between the surface of the pellet and the bulk fluid. Thus only the external heat- and mass-transfer resistances are considered in developing the pellet equations that calculate the effectiveness factor rj at every point along the length of the reactor. 

Surface reaction with diffusion and heat transfer resistance In fast exothermic reactions, in addition to grad c, also grad T (TG Ts) is present in the boundary layer between the gas bulk phase and the catalyst surface. For the outer effectiveness factor qext this means that... 

Although multiplicities of the effectiveness factor have also been detected experimentally, these are of minor importance practically, since for industrial processes and catalysts, Prater numbers above 0.1 are less common. On the contrary, effectiveness factors above unity in real systems are frequently encountered, although the dominating part of the overall heat transfer resistance normally lies in the external boundary layer rather than inside the catalyst pellet. For mass transfer the opposite holds the dominating diffusional resistance is normally located within the pellet, whereas the interphase mass transfer most frequently plays a minor role (high space velocity). 

From the analysis of Equation 18> it follows that the main variables that affect the error in the reaction rate are E and P due to their effect on T and TC Thus, very good responses are obtained from a one-dimensional model when reaction conditions are mild (moderate values of E and P). It can also be seen that for these conditions, the influence of the distribution of the radial heat transfer resistances between the bed and the wall, given by the Biot number, is small. E.g., for T = 673°K, Tw = 643°K and E = 12.5 kcal/mol, the maximum er, found for Big -> < is 2.8%. 

In fluid-solid systems the interparticle gradients - between the external surface of the particle and the adjacent bulk fluid phase - may be more serious, because the effective thermal conductivity of the fluid may be much lower than that of the particle. For the interparticle situation the heat transfer resistances, in general, are more serious than the interparticle mass transfer effects they may become important if reaction rates and reaction heats are high and flow rates are low. Hie usual experimental test for interparticle effects is to check the influence of the flow rate on the conversion while maintaining constant the space velocity or residence time in the reactor. This should be done over a wide range of flow rates and the conversion should be measured very accurately. 

The methods outlined by Satterfield94 for taking into account the effects of intraparticle mass- and heat-transfer resistances on the effective reaction rate are applicable to three-phase reactors and, therefore, they will not be repeated here. The importance of these resistances depends upon the nature of the reaction and... 

Experimental Measurements of Reaction Kinetics. The reaction expressions discussed in the following model the intrinsic reaction on the catalyst surface, free of mass-transfer restrictions. Experimental measurements, usually made with very fine particles, are described by theoretically deduced formulas, the validity of which is tested experimentally by their possibility for extrapolation to other reaction conditions. Commonly the isothermal integral reactor is used with catalyst crushed to a size of 0.5-1.5 mm to avoid pore diffusion restriction and heat-transfer resistance in the catalyst particles. To exclude maldistribution effects and back mixing, a high ratio of... 

Note that in the limit of external diffusion control, the activation energy 0b,—>0, as can be shown when substituting Eq. (7-110) inEq. (7-108). For more details on how to represent the combined effect of external and intraparticle diffusion on effectiveness factor for more complex systems, see Luss, "Diffusion-Rection Interactions in Catalyst Pellets. Heat-Transfer Resistances A similar analysis regarding external and intraparticle heat-transfer limitations leads to temperature... 

For endothermic reactions, do mass- and heat-transfer resistances have complementary or counterbalancing effects on the global rate ... 

In the foregoing illustration the temperature rise at the catalyst surface had a beneficial effect on selectivity. This is because the activation energy for the desired reaction was greater than that for the reaction producing by-product C. If were less than E2, external heat-transfer resistance would have reduced the selectivity for exothermic reactions. 

For an endothermic reaction there is a decrease in temperature and rate into the pellet. Hence 17 is always less than unity. Since the rate decreases with drop in temperature, the effect of heat-transfer resistance is diminished. Therefore the curves for various are closer together for the endothermic case. In fact, the decrease in rate going into the pellet for endothermic reactions means that mass transfer is of little importance. It has been shown that in many endothermic cases it is satisfactory to use a thermal effectiveness factor. Such thermal 17 neglects intrapellet mass transport that is, ri is obtained by solution of Eq. (11-72), taking C = Q. 

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