Non-isothermal condition

Most chemical reactions are greatly affected by temperature. The previous chapters discussed reactions at isothermal condition, however, industrial reactors often operate under non-isothermal condition. This is because chemical reactions strongly depend on temperature, either absorbing (i.e., endothermic) or generating (i.e., exothermic) a large amount of heat. 

Example 5.14 A power law fluid with constants i]q= 1.2 x lO Ns/m and n = 0.35 is injected through a centre gate into a disc cavity which has a depth of 2 mm and a diameter of 200 mm. If the injection rate is constant at 6 X 10 m /s, estimate the time taken to fill the cavity and the minimum injecdon pressure necessary at the gate for (a) Isothermal and (b) Non-isothermal conditions. 

Decomposition of BeS04 (either reagent grade or freeze-dried) yielded BeO [771] and kinetic analyses based on the contracting area equation [eqn. (7), n = 2] for 875—990 K gave values of E from 213 to 226 kJ mole-1. Under non-isothermal conditions, the values of E varied between... 

In order to maintain the gas flow, the pump must raise the pressure at the upstream end of the pipe to some value P3, which is greater than P2. The required value of P3 will depend somewhat on the conditions of flow in the pipe. Thus for isothermal conditions, P3 may be calculated from equation 4.55, since the downstream pressure P2 and the mass rate of flow G are known. For non-isothermal conditions, the appropriate equation, such as 4.66 or 4.77, must be used. 

The inlet concentration of monomer and initiator were each separately varied in a very slow sinusoidal manner. The Dj, was again predicted to increase in comparison with the non-perturbed case, but they concluded that different results might be observed with regard to the magnitude and direction of the change in the polydispersity under non-isothermal conditions. 

The physical reason for the velocity slip is the fact that close to the wall the gas is not in thermal equilibrium. For the same reason, a temperature jump is induced, and a more detailed investigation based on the kinetic theory of gases shows that heat transfer and momentum transfer are coupled. Expressions for velocity slip and temperature jump valid in the case of non-isothermal conditions are given by... 

Empirical grey models based on non-isothermal experiments and tendency modelling will be discussed in more detail below. Identification of gross kinetics from non-isothermal data started in the 1940-ties and was mainly applied to fast gas-phase catalytic reactions with large heat effects. Reactor models for such reactions are mathematically isomorphical with those for batch reactors commonly used in fine chemicals manufacture. Hopefully, this technique can be successfully applied for fine chemistry processes. Tendency modelling is a modern technique developed at the end of 1980-ties. It has been designed for processing the data from (semi)batch reactors, also those run under non-isothermal conditions. 

Table 4 Rate constants calculated by non-linear least square method under non-isothermal condition...

COMPARISON OF CHEMILUMINESCENCE FOR ISOTHERMAL AND NON-ISOTHERMAL CONDITIONS WITH THOSE OBTAINED BY DSC... 

As discussed in Section 5.1, the extension to non-isothermal conditions is straightforward under the assumption that the thermodynamic properties are constant. 

In this case also the slope of the temperature coefficient dE° /dT, under non-isothermic conditions, as a function of 1 /T affords AH°C. 

A comprehensive review of electrothermal atomization devices has been published (94). The review includes a discussion of commonly encountered problems such as atom loss through non-pyrolytic graphite, non-isothermal conditions, differences in peak height and peak area measurement, etc. 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

The affect of diffusion on catalyst selectivity in porous catalysts operating under non-isothermal conditions has been examined by a number of workers. The mathematical problem has been comprehensively stated in a paper [21] which also takes into account the affect of surface diffusion on selectivity. For consecutive first-order exothermic reactions, the selectivity increases with an increase in Thiele modulus when the parameter A (the difference between the activation energy for reaction... 

Differential Scanning Calorimetry (DSC) This is by far the widest utilized technique to obtain the degree and reaction rate of cure as well as the specific heat of thermosetting resins. It is based on the measurement of the differential voltage (converted into heat flow) necessary to obtain the thermal equilibrium between a sample (resin) and an inert reference, both placed into a calorimeter [143,144], As a result, a thermogram, as shown in Figure 2.7, is obtained [145]. In this curve, the area under the whole curve represents the total heat of reaction, AHR, and the shadowed area represents the enthalpy at a specific time. From Equations 2.5 and 2.6, the degree and rate of cure can be calculated. The DSC can operate under isothermal or non-isothermal conditions [146]. In the former mode, two different methods can be used [1] ... 

Many apparent discrepancies can be found in the experimental results reported in literature for NSRC operation. They are usually caused by inconsistent experimental conditions, which have to be taken into account carefully (cf. Burch, 2004). Actual temperature, non-isothermal conditions in the test reactor, the composition of the gas mixture (presence of C02 and H20, ratio of NO/N02 at the inlet, the used reducing components), transport limitations and dynamics of the measurements are the most important ones. 

We are thus, in many instances, more interested in the transient behaviour early in a reaction than we are in the more easily studied final or equilibrium state. With this in mind, we shall be concerned in our early chapters with simple models of chemical reaction that can satisfy all thermodynamic requirements and yet still show oscillatory behaviour of the kind described above in a well-stirred closed system under isothermal or non-isothermal conditions. 

Maximum Release. The analytical model described above assumes that the liquid phase is completely stagnant. While this may be true in an ideal laboratory experiment where a small sample can be kept isothermal at a specified temperature, in large scale systems where non-isothermal conditions exist, both natural convection and molecular diffusion will contribute to mass transfer. This combined effect, which is often very difficult to assess quantitatively, will result in an increase in fission-product release rate. Therefore, in making reactor safety analyses, it is desirable to be able to estimate the maximum release under all possible conditions. 

Fig. 3.6. Effectiveness factor as a function of the Thiele modulus for non-isothermal conditions for a sphere121 ...

Let s proceed with the second case as the best studied area. In this case the analysis should be based on the general properties of flow of abnormally viscous liquids in tubing and channels under isothermal and non-isothermal conditions. Generally, the process of moulding may be considered as being isothermal when the following condition is satisfied ... 

C. This figure is appropriate for adiabatic polymerization, which approximates reality in reactive processing of large articles with high volume-to-surface ratios. In this case, it is impossible to remove the heat effectively and to avoid intense local temperature jumps. Therefore, it is essential to know how to calculate temperature increase for reactions proceeding in non-isothermal conditions. The time dependence of viscosity in this situation can be written as... 

In current industrial practice, reactive processing carried out in non-isothermal conditions, for both inherent and other reasons, such as changes in temperature at the surface of an article during the process cycle. Inherent reasons are the existence of inner heat sources, which can be of chemical origin (enthalpy of reaction), heat of phase transition from crystallization of a newly formed polymer or heat dissipated due to the flow of a reactive mass. 

The key to modelling the crystallization process is the derivation a kinetic equation for a(t,T). It is possible to find different versions of this equation, including the classical Avrami equation, which allows adequate fitting of the experimental data. However, this equation is not convenient for solving processing problems. This is explained by the need to use a kinetic equation for non-isothermal conditions, which leads to a cumbersome system of interrelated differential and integral equations. The problem with the Avrami equation is that it was derived for isothermal conditions and... 

Polymer synthesis is usually accompanied by a strong exothermal effect and, as a general rule, real process proceeds in non-isothermal conditions. This problem was considered in Section 2.6 for polyester synthesis, and the results discussed above are of generally applicable to all non-isothermal processes. 

The situation becomes more complicated if experiments are carried out in non-isothermal conditions. First of all,many non-isothermal measurement procedures are possible. The selection of a particular method depends on the process characteristics and methods of interpretation. Scanning calorimeters, which measure the quantity of heat released as the ambient temperature is varied linearly. The rate of temperature change can be varied by the experimenter. 

The theoretical foundation for this kind of analysis was, as mentioned, originally laid by Taylor and Aris with their dispersion theory in circular tubes. Recent contributions in this area have transferred their approach to micro-reaction technology. Gobby et al. [94] studied, in 1999, a reaction in a catalytic wall micro-reactor, applying the eigenvalue method for a vertically averaged one-dimensional solution under isothermal and non-isothermal conditions. Dispersion in etched microchannels has been examined [95], and a comparison of electro-osmotic flow to pressure-driven flow in micro-channels given by Locascio et al. in 2001 [96]. 

If batch reactions are occasionally at constant temperature (isothermal), most reactions are started at a lower initial temperature and the temperature is increased to its desired value, sometimes by using the heat of reaction the reaction is performed under non-isothermal conditions. Different strategies of temperature control are technically practiced ... 

Conditions (66) and (67) ensure the existence of Lyapunov s convex function for eqns. (17) GGjdNi = fit. With a known type of the potentials /i, for which condition (1) is fulfilled, one can obtain Lyapunov s thermodynamic functions for various (including non-isothermal) conditions. Thus, for an ideal gas and the law of mass action [16]... 

Abstract A poromechanics formulation for transversely isotropic chemically active poroelastic media under non-isothermal conditions is presented. The formation pore fluid is modeled as a two-species constituent comprising of the solute and the solvent. The model is applied to study the thermo-chemical effects on the stress and pore pressure distributions in the vicinity of an inclined borehole drilled in a chemically active transversely isotropic formation under non-isothermal conditions. 

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