Nonisothermal conditions

The general problem of diffusion-reaction for the overall effectiveness factor D is rather complicated. However, the physical and chemical rate processes prevailing under practical conditions promote isothermal particles and negligible external mass transfer limitations. In other words, the key transport limitations are external heat transfer and internal mass transfer. External temperature gradients can be significant even when external mass transfer resistances are negligibly small. 

The analogous Biot number for heat is defined as the ratio of internal to external heat transport resistances and is used together with (BOm to assess the relative importance of all heat and mass transport limitations  

In order to express the parameters appearing in 2 in terms of measurable physical properties of the catalyst particles and the catalyst bed, the external surface area is equated to 

The porosity and density of the catalyst bed, Gb and ps, respectively, and the particle diameter, dp, are the physical properties used in estimating am- The global rate can be expressed per unit volume, per unit weight, or per unit surface area, according 

The Damkohler number Da, which is a measure of the ratio of the surface chemical reaction rate to the external mass transfer rate, can also be defined as an observable quantity in terms of the global rate when multiplied by the internal effectiveness factor ri [13]  

Heat effects caused by chemical reactions inside the catalyst particle are accounted for by setting up an energy balance for the particle. Let us consider the same spherical volume element as in the case of mass balances. Qualitatively, the energy balance in the steady state can be obtained by the following reasoning  

Heat conduction is described by the law of Fourier, and several simultaneous chemical reactions are assumed to proceed in the particle. Quantitatively, expression 5.127 implies that 

After dividing Equation 5.129 by r Ar and allowing Ar transformed to the following differential equation  

This expression is valid for a spherical geometry only. It is trivial to show that for an arbitrary geometry, the energy balance can be written using the form factor (s)  

The energy balance Equation 5.131 has the following boundary conditions  

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We might try to measure the temperature coefficient of the Galvani potential for an individual electrode under nonisothermal conditions then only the temperature of the test electrode would be varied, while the reference electrode remains at a constant temperature and retains a constant value of Galvani potential (Fig. 3.2). 

In the kinetic analysis of the experimental data with an autoclave, the non-linear least square method was used to estimate the rate constants under nonisothermal conditions. The simulation of liquefaction calculated by substituing the estimated values into the rate equations showed good agreement with experimental values. 

ILLUSTRATION 10.5 DETERMINATION OF THE VOLUME REQUIREMENTS FOR OPERATION OF A TUBULAR REACTOR UNDER NONISOTHERMAL CONDITIONS WITH HEAT EXCHANGE... 

Grant and coworkers [8] studied the dehydration kinetics of piroxicam monohydrate using both model-free and model-fitting approaches in an effort to understand the effects of lattice energy and crystal structure. The dehydration kinetics was found to differ when determined under isothermal and nonisothermal conditions. Ultimately, the dehydration behavior of piroxicam monohydrate was determined by details of the crystal structure, which was characterized by an absence of channels and a complicated hydrogen-bonding network, and ab initio calculations proved useful in understanding the structural ramifications of the dehydration process. 

It is obvious that nonisothermal conditions induced by microwave heating lead to very different results from those obtained under conventional heating conditions. In summary microwave effects like superheating, selective heating and hot spots, can all be characterized by temperature gradients ranging from macroscopic to molecular scale dimensions. 

Data on gas systems, again using a point source, have been obtained by Bernard and Wilhelm (B6), Dorweiler and Fahien (D20), Fahien and Smith (F2), and Plautz and Johnstone (P6). Plautz and Johnstone measured dispersion coefficients under isothermal and nonisothermal conditions and found that there was a difference between the two only for low Reynolds numbers. 

Most industrially relevant transformation processes are not isothermal and even in a controlled laboratory environment, it is difficult to perform experiments that are completely isothermal. The kinetics of nonisothermal phase transformations are more complex, of course, but there are some useful relationships that have been developed that allow for the evaluation of kinetic parameters under nonisothermal conditions. One such equation takes into account the heating rate, (p usually in K/min, used in the experiment [4] ... 

Curing of Epoxy-amine Systems at Nonisothermal Conditions. 139... 

In building mathematical models of product formation in a mold it is possible to treat a polymeric material as motionless (or quasi-solid), because the viscosity grows very rapidly with the formation of a linear or network polymer thus, hydrodynamic phenomena can be neglected. In this situation, the polymerization process itself becomes the most important factor, and it is worth noting that the process occurs in nonisothermal conditions. 

As stated in the previous section, the use of a phenomenological kinetic equation derived from Eq. (5.1), for a system that does not verify the required restrictions for its use, may lead to different kinetic expressions when trying to fit experimental results obtained under isothermal and nonisothermal conditions. In particular, it may be observed that different kinetic parameters result by varying the heating rates in nonisothermal experiments. 

An important corollary of this analysis is related to the inverse problem. We may state that a constitutive rate equation for a particular system will be adequate if and only if it may be used to fit experimental results obtained under isothermal and nonisothermal conditions. 

Example 9.3 Nonisothermal Drag Flow of a Power Law Model Fluid Insight into the effect of nonisothermal conditions, on the velocity profile and drag flow rate, can he obtained by analyzing a relatively simple case of parallel-plate nonisothermal drag flow with the two plates at different temperatures. The nonisothermicity originates from viscous dissipation and nonuniform plate temperatures. In this example we focus on the latter. 

The mathematical formulation of the fiber-spinning process is meant to simulate and predict the hydrodynamics of the process and the relationship between spinning conditions and fiber structure. It involves rapid extensional deformation, heat transfer to the surrounding quenching environment, air drag on the filament surface, crystallization under rapid axial-orientation, and nonisothermal conditions. 

When heated under nonisothermal conditions, the maximum volatile product evolution temperature was 425°C for isotactic PP, yielding volatile products comprising dienes, alkanes, and alkenes. Furthermore, the hydrogen content of pyrolysis products obtained by flash pyrolysis at 520°C indicates the magnitude of the flammability problem in term of its fuel-forming potential.23 The flammability of volatiles is further enhanced by the abundance of unsaturated less-volatile fuel fragments that behave as secondary fuel sources and which decompose further.24... 

So far it has been assumed that both reactions are first order and the pellet can be treated as isothermal. It may be obvious to note that under nonisothermal conditions the ratio of the intrinsic activation energies and, if necessary, the ratio of the external heat transfer coefficients will also affect the apparent selectivity of the catalyst. In addition, if the kinetic orders of the two reactions are different, this will also influence selectivity. 

So far, only isothermal operation has been considered. Under nonisothermal conditions, in Type II reactions a change of the apparent selectivity, caused by temperature variations across the pellet and/or the interphase boundary layer, may also be expected once the two reactions exhibit different activation energies. This holds irrespective of whether the kinetic order of the two reactions is equal or not. Nonisothermal Type II problems have been analyzed, for example, by 0stergaard [81]. 

The effect of using too large a volume of solution in the dilatometer is to extend the acceleration period, resulting in too high a rate. This is symptomatic of nonisothermal conditions. 

In practice the heat effects associated with chemical reactions result in nonisothermal conditions. In the case of a batch reactor the temperature changes as a function of time, whereas an axial temperature profile is established in a plug flow reactor. The application of the law of conservation of energy, in a similar... 

Duff et al. [27] reported a study made by means of DSC and WAXD on SPS/ PPE blends of various compositions, precipitated from ethylbenzene solutions, compression molded at 330 °C for 2 min and then slowly cooled to room temperature. In particular, the WAXD patterns show that in sPS-rich blends (>50 50 wt%) sPS is in a 0 or (3 form, while small amounts of a are present in the 50 50 wt% blend. The kinetics of crystallization and the mechanism of nucleation of sPS were investigated under isothermal and nonisothermal conditions as a function of blend composition and molecular weights of the components. The experimental curves show that the half-time to crystallization, t j2, increases with increasing content and molecular weight of PPE, but is not influenced by the molecular weight of sPS. The crystallization kinetics were... 

Theonly important current application of tubular reactors in polymer syntheses is in the production of high pressure, low density polyethylene. In tubular processes, the newer reactors typically have inside diameters about 2.5 cm and lengths of the order of I km. Ethylene, a free-radical initiator, and a chain transfer agent are injected at the tube inlet and sometimes downstream as well. The high heat of polymerization causes nonisothermal conditions with the temperature increasing towards the tube center and away from the inlet. A typical axial temperature profile peaks some distance down the tube where the bulk of the initiator has been consumed. The reactors are operated at 200-300°C and 2000-3000 atm pressure. 

The TGA data presented in Figure 1 were obtained under nonisothermal conditions in an atmosphere of oxygen. As shown, the sample was first rapidly heated to 200°C. held at that temperature for 2 min, and then heated to 560°C at 6K,/min, follow ed by an isothermal period at the final temperature. I hc bulk of the sample burnt out at about 52 min, prior to attaining the final temperature. From this point on, the mass signal appears to be due to slow burnout of residual, presumably more unreactive, material. 

If in the rare case that the reactor is accomplishing a constant pressure gas-phase reaction at nonisothermal conditions ... 

Consider a PFR operating at nonisothermal conditions (refer to Figure 9.4.1). To describe the reactor performance, the material balance. Equation (9.1.1), must be solved simultaneously with the energy balance. Equation (9.2.7). Assuming that the PFR is a tubular reactor of constant cross-sectional area and that T and C, do not vary over the radial direction of the tube, the heat transfer rate Q can be written for a differential section of reactor volume as (see Figure 9.4.1) ... 

To provide an example of how the sensitivity may be elucidated, consider a tubular reactor accomplishing an exothermic reaction and operating at nonisothermal conditions. As described in Example 9.4.3, hot spots in the reactor temperature pro-... 

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