Nonisothermal conditions effects

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. 

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 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... 

Effects of space time under nonisothermal conditions. The above discussions around the effects of space time on a membrane reactor performance are limited to isothermal conditions. The behavior of the reaction conversion in response to space time can be further complicated under nonisothermal conditions. 

In general it should be noted that isothermal models are not strictly isothermal, but would typically display oscillations under nonisothermal conditions as well. In the case of supported catalysts at atmospheric pressure, oscillations are probably never purely isothermal. Usually, however, thermal effects tend to amplify instabilities, so that a model that predicts oscillations for the isothermal case will most probably predict oscillations under nonisothermal conditions. 

The above discussion of effectiveness factors is valid only for isothermal conditions. When a reaction is exothermic and nonisothermal, the effectiveness factor can be significantly greater than 1 as shown in Figiue 12-7. Values of t greater than 1 occur because the external smface temperature of the pellet is less than the temperature inside the pellet where the exothermic reaction is taking place. Therefore, the rate of reaction inside the pellet is greater than the rate at the surface. Thus, because the effectiveness factor is the ratio of the actual reaction rate to the rate at smface conditions, the effectiveness factor... 

Nonuniform temperatures, or a temperature level different from that of the surroundings, are common in operating reactors. The temperature may be varied deliberately to achieve optimum rates of reaction, or high heats of reaction and limited heat-transfer rates may cause unintended nonisothermal conditions. Reactor design is usually sensitive to small temperature changes because of the exponential effect of temperature on the rate (the Arrhenius equation). The temperature profile, or history, in a reactor is established by an energy balance such as those presented in Chap. 3 for ideal batch and flow reactors. 

Such an approach is of practical value in analysing the operation of existing industrial reactors. However, when new technological processes are designed, it is necessary to investigate various nonisothermal conditions of synthesis, as well as the effect of the rheology of the system on the peculiarities of its kinetic behavior. 

Figure 9.4. Effectiveness factor as function of modified Thiele modulus under nonisothermal conditions (from Weisz and Hicks [27]).

One should not mistake an acceleration of a polymerization reaction due to a rise in the temperature under nonisothermal conditions with a true gel effect from a rise in viscosity. The gel effect can occur when the temperature of the reaction is kept constant. 

On the other hand, the effectiveness factor varies significantly with temperature. Therefore, for nonisothermal conditions, especially in the exothermic reactions, it will vary since the temperature varies within the pores or particles due to the temperature gradient caused by the chemical reaction. It is then necessary to construct an... 

Isothermal conditions will be dealt with in this section to study the diffusion effect and will consider the nonisothermal conditions in the next section to investigate the coupled effect of diffusion interaction among species and the heat release. 

Effect of pressure For isothermic conditions the operation pressure does not influence conversion, of course, and the space-time-yield increases linearly with P. If the reactor operates under nonisothermic con-ditons the temperature rises and this causes the conversion to increase as is shown in Fig. 22. The space-time-yield is higher than under isothermic conditions but increases yet almost linearly with pressure. The rise in temperature from 258 to 288 C under nonisothermic conditions and a pressure of 3 MPa brings a gain in overall conversion from 0.78 to 0.96. The linear increase of the space-time-yield with synthesis pressure is in full agreement with the findings of Hall et al. 

By using a one-dimensional heterogeneous model, in nonisothermal conditions (by solving the energy balances above mentioned), the effect of heat profiles can be studied in the reactor. 

Weisz and Hicks [39] solved these differential equations numerically to determine the concentration profile in the particle, using Equation 2.68 for eliminating the exponential temperature dependence. The internal effectiveness factor t] was then calculated by using Equation 2.64, which is applicable to nonisothermal conditions if ky is evaluated at the surface temperature, Ts ... 

The same argument can be used for the lack of nonisofhermal models in literature. As it has been made clear in this chapter, the temperature plays a vital role in determining the overall magnitude of poisoning. In realify, PEM fuel cells run at a nonisothermal condition. The temperature, especially, increases in the catalyst layer due to the heat generahon during the electrooxidation reaction. Consequently, more work is needed to understand the effects of the nonuniformity of femperature on confaminafion of PEM fuel cells. 

Carberry, J.J. (1961) The catalytic effectiveness factor under nonisothermal conditions. AIChE J., 7, 350-351. 

The interpretation of Equations 3.95 and 3.96, as well as Equations 3.97 and 3.98, is simple if the number of molecules in the chemical reaction increases, the expression attains a positive value (since > 0 and > 0). Simultaneously, the terms in parentheses, in Equations 3.95 through 3.98, become larger than unity (>1) inside the reactor. Under isothermal conditions (T = To), the volumetric flow rate inside the reactor is thus increased. Under nonisothermal conditions—with strongly exothermic or endothermic reactions— the temperature effect, that is, the term T/Tq, results in a considerable change in the volumetric flow rate. 

It was shown in Chapter 4 that the rate of reaction is a function of temperature and concentration. The application of the subsequent equations developed were simplest for isothermal conditions since is then generally solely a function of concentration. If nonisothermal conditions exist, another equation must be developed to describe any temperature variations with position and time in a reactor. For example, in adiabatic operation, the enthalpy (heat) effect accompanying the reaction can be completely absorbed by the system and result in temperature changes in the reactor. As noted earlier, in an exothermic reaction, the temperature increases, which in turn increases the rate of reaction, which in turn increases the conversion for a given interval of time. The conversion, therefore, would be higher than that obtained under isothermal conditions. When the reaction is endothermic, the decrease in temperature of the system results in a lower conversion than that associated with the isothermal case. If the endothermic enthalpy of reaction is large, the reaction may essentially stop due to the sharp decrease in temperature. 

It is clear that we have neglected all details of the flow near the free surface, including any curvature. The flow near the three-phase moving contact line, where the melt, the mold wall, and the air all meet, is very difficult to analyze with the noslip boundary condition and is not completely understood. The time to fill the mold isothermally is insensitive to the details of this flow, but the flow near the contact line has a significant effect on the orientation of polymer chains and on the temperature profile under nonisothermal conditions. 

For the design and analysis of fixed-bed catalytic reactors as well as the determination of catalyst efficiency under nonisothermal conditions, the effective thermal conductivity of the porous pellet must be known. A collection of thermal conductivity data of solids published by the Thermophysical Properties Research Centre at Purdue University [ ] shows "a disparity in data probably greater than that of any other physical property ". Some of these differences naturally can be explained, as no two samples of solids, especially porous catalysts, can be made completely identical. However, the main reason is that the assumed boundary conditions for the Fourier heat conduction equation... 

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