Nonisothermal histories

Obtaining Kinetic Samples for Reactive Extrusion. To develop and test kinetic models, homogeneous samples with a well defined temperature-time history are required. Temperature history does not necessarily need to be isothermal. In fact, well defined nonisothermal histories can provide very good test data for models. However, isothermal data is very desirable at the initial stages of model building to simplify both model selection and parameter estimation problems. 

Except for radioactive decays, other reaction rate coefficients depend on temperature. Hence, for nonisothermal reaction with temperature history of T(t), the reaction rate coefficient is a function of time k(T(t)) = k(t). The concentration evolution as a function of time would differ from that of isothermal reactions. For unidirectional elementary reactions, it is not difficult to find how the concentration would evolve with time as long as the temperature history and hence the function of k(t) is known. To illustrate the method of treatment, use Reaction 2A C as an example. The reaction rate law is (Equation 1-51)... 

Still, sophisticated, exact, numerical, non-Newtonian and nonisothermal models are essential in order to reach the goal of accurately predicting final product properties from the total thermomechanical and deformation history of each fluid element passing through the extruder. A great deal more research remains to be done in order to accomplish this goal. 

The residence time of the polymer particles in the reactor and the temperature history experienced by these particles during their journey through the reactor is governed by the compressible nonisothermal Navier-Stokes equations. 

Somew hat similar results were obtained for saran char under near-isothermal conditions in oxygen. The sample temperature was increased at bOK/min to 435°C and then held there. The sample mass actually began to decrease prior to attaining isothermal conditions. This illustrates the difficulty of resolving the porosity development history under isothermal conditions where the burn-off behavior typically varies widely with the nature and preparation history of the sample. Nonisothermal temperature programs appear to be more efficient in this regard. 

Figure 1. Saran char sample mass history during nonisothermal activation in oxygen. I hc symbols correspond to the scattering curves in Figure 2.

To formalize the main hypothesis, the nonisothermal functional given by Eq. (16.1) is modified by a new isothermic functional with a modified time scale to account for the temperature history. Now, according to the basic hypothesis of linear theory, the specific form of the stress-strain relationship can be written as... 

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. 

Figure 14.3. Computed concentration and temperature histories for a nonisothermal batch reaction A - K - P with highly exothermic second step (from P. Gray and S. K. Scott [39]).

The basis of the method described by Hands and HorsfaU [1] consists of calculating the tanperature distribution as a function of lime first, thoi evaluating the dis-ttibution of the cure. Taking a temperature profile, a technique is described enabling the evaluation of the state of cure. When the form of the relationship between cure and temperature history is known, the basic cure parameters can be obtained from experiments in which temperature varies with time. When the relationship is not known, it is preferable to conduct experiments at various fixed temperatures in order to separate the dependence on time and temperature. An apparatus was built for measuring cure as a function of time at a constant temperature. The data obtained from this isothermal apparatus and a new mathematical model of cure was used to predict the cure distribution obtained in nonisothermal experiments. 

In equation 16, t is the time when the change in temperature occurs, is the exponent, and r is the characteristic relaxation time. Nonexponentiality memory effect) is reflected in the value of < 1. When treating nonisothermal situations arbitrary thermal history) the relaxation function can be represented by the superposition of responses to a series of temperature jumps constituting the actual thermal history. The Active temperature is defined as the actual... 

Even in systems which are of more than one dimension, such as a tube, one can envision the propagation of a planar front with the properties described in the preceding sections. Such waves may be solutions to the governing reaction-diffusion or reaction-conduction equations, but if they are to be realized and observed in practice they must also be stable to the inevitable small fluctuations in local concentration and temperature. There is a long history of stability analysis for nonisothermal flame propagation [30-32], although the absence of exact analytical solutions to even the 1-D flame front equation makes these rather difficult. The same questions about the stability of isothermal reaction-diffusion fronts seem not to have been addressed until only recently [12]. 

Specific volume of the polymer decreases with time and eventually levels ofiF as equilibrium is approached. As the aging temperature is lowered, longer times are required to reach equilibrium. This intrinsic isotherm behavior shows the temperature and time dependence of structural relaxation, but in practical applications, materials undergo complicated temperature histories instead of isothermal conditions therefore, nonisothermal conditions should be considered to fully characterize the relaxation behavior. 

Fig. 3.3 Long-term creep predictions by KAHR-Ote model, combined with the isothermal and nonisothermal effective time theory and original Kohlrausch function from short-term response. Thermal histories 97 °C - 21 °C—> 73 °C 27 °C 73 °C. KoMiausch function parameters Do = 0.460 GPa r = 1593 s, = 0.417. The inset is the nonisothtamal temperature history prior to long-term creep tests. Data reproduced with permission from Ref. [73]...

Guo, Y., Bradshaw, R.D. Long-term creep of polyphenylene sulfide (pps) subjected to complex thermal histories The effects of nonisothermal physical aging. Polymer 50, 4048 (2009)... 

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