Semenov model

As above except that the condition of the second kind assumed the Boundary for Bi -> 0, the Thomas model becomes the Semenov Model for Bi oo, the Thomas model becomes the Frank-Kamanetskii Model critical conditions for various geometries given. 

We follow the analysis of Frank-Kamenetskii [3] of a slab of half-thickness, rG, heated by convection with a constant convective heat transfer coefficient, h, from an ambient of Too. The initial temperature is 7j < 7 ,XJ however, we consider no solution over time. We only examine the steady state solution, and look for conditions where it is not valid. If we return to the analysis for autoignition, under a uniform temperature state (see the Semenov model in Section 4.3) we saw that a critical state exists that was just on the fringe of valid steady solutions. Physically, this means that as the self-heating proceeds, there is a state of relatively low temperature where a steady condition is sustained. This is like the warm bag of mulch where the interior is a slightly higher temperature than the ambient. The exothermiscity is exactly balanced by the heat conducted away from the interior. However, under some critical condition of size (rG) or ambient heating (h and Too), we might leave the content world of steady state and a dynamic condition will... 

This deals with purely convective heating, but defining 6 by Equation (5.12), we take h = k/r0 and r0 = V/S of the Semenov model. Equation (5.18) approximates the constant surface temperature case. Also, here the initial temperature is taken as Too- If we allow no convective cooling, the adiabatic equation becomes... 

Equation (5.23) is still based on the adiabatic system. A better, but still approximate, result can be produced from the nonadiabatic Semenov model where... 

FIGURE 3.3. Comparison of Critical Temperatures for Frank-Kamenetskii and Semenov Models (Right Cylinder Configuration). 

Two classical models have been described for runaway calculations in which the important difference between the two is in the degree of mixing. The first model, proposed by Semenov [165], applies to well stirred mixtures where the temperature is the same throughout the mixture. Heat removal occurs with a steep temperature gradient at the surface of the walls or coils, and is governed by the usual factors of area, temperature of coolant, and heat transfer coefficients. Case A in Figure 3.20 shows a temperature distribution by the Semenov model for self-heating. 

The Semenov model applies when the Biot number is close to zero, and the Frank-Kamenetskii model applies when the Biot number is large. The... 

FIGURE 3.20. Typical Temperature Distributions during Self-Heating in a Vessel. A = The Semenov Model B = The Frank-Kamenetskii Model C = The Thomas Model... 

The Semenov model, which applies to well stirred mixtures at a uniform temperature, is the basis of Equations (3-26) and (3-27) below ... 

Schisla, R. M., P. N. Lodal, and M. A. Paulonis (1996). "Use of the Semenov Model for the Analysis of Runaway Reactions Induced in Pipelines by Electrical Tracing." Proceedings of the 30th Annual Loss Prevention Symposium, February 26-28, 1996, New Orleans, LA, ed. R. P. Benedetti, J. J. Rooney, and K. Chatrathi, Paper 13d. New York American Institute of Chemical Engineers. 

Fig. 5.4. Thermal diagram for Semenov model of thermal explosion the rate of chemical heat release varies with the dimensionless temperature excess 0 according to/(0) = e the rate of heat transfer is given by the straight line with a gradient of l/ifi. For small ifi the loss line is steep and makes two intersections corresponding to two steady-states for large tli the loss line has a low gradient and does not allow steady-state intersection points the critical case corresponds to tangency of the heat release and heat loss lines.

Or, would it be supposed in Semenov s formulation that the walls of the container were of infinite thickness, of infinite heat content or of infinite thermal conductivity, so that To was kept constant over a long time Such a situation will not be realistic. Besides, it seems that such a supposition that the container wall has a large overall coefficient of heat transfer is inconsistent with the Semenov model that the rate of heat transfer from a self-heating fluid filled in a container and placed in the atmosphere under isothermal conditions, through the whole fluid surface, across the container walls, to the atmosphere is far less than the rate of thermal conduction in the fluid. 

Now, the characteristic of the Semenov model, which is illustrated by the diagram of the left-hand side in Fig. 3, is the spatially uniform distribution of internal temperature. It seems most natural to think that this mode of temperature distribution arises from the heat transfer mode that the heat generated by the exothermic decomposition reaction occurring in a very slowly self-heating fluid filled in a container and placed in the atmosphere under isothermal conditions is transmitted uniformly throughout the fluid on account of its own fluidity. 

The above condition is the formal definition of the Semenov model. The Semenov equation becomes effective under the condition above referred to and defines the Tf for an arbitrary volume of a self-healing fluid (a sclf-hcating liquid chemical, in general) of the TD type charged, or confined, in an arbitrary container and placed in the atmosphere under isothermal conditions. The Semenov equation is thus appropriate for the calculation of the TV for every liquid chemical of the TD type. ... 

In this connection, there arc cases where the very ultimate of the Semenov model, in which the condition, UH A - 0, holds, is referred to as the Semenov s boundary conditions or the Semenov extreme [12]. 

Besides, it is self-evident, as stated in the preceding section, that the spatial distribution of temperature, in particular, in the early stages of the self-heating process, or of the oxidatively-heating process, in a small-scale chemical of the TD type, including every small-scale gas-permeable oxidatively-heating substance, subjected to either of the two kinds of adiabatic tests, is the very ultimate of the Semenov model, because the condition, the Biot number = Ur A = 0, holds strictly in such a chemical. 

The Semenov model holds for a chemical of the TD type, irrespective of liquid or powder, charged in the Dewar flask used in the BAM test. For further details, refer to Section 6.8. 

As commented in a footnote in (2) of Subsection 6.4.3, only MNTS, in the ten powdery chemicals of the TD type tested herein, is powdery at room temperature but self-heats as a liquid at temperatures higher than its melting point. It is, therefore, required to calculate the Tc for MNTS by applying the Semenov equation. This calculation is performed in Section 6.8 for the ten powdery chemicals of the TD type, including MNTS, assuming that 400 cm each of these powdery chemicals of the TD type are each charged in the 500 cm Dewar flask, which is used in the BAM test, and to which the Semenov model applies, and are each placed in the atmosphere under isothermal conditions. 

If, however, the Semenov model holds for a powdery chemical of the TD type charged in one particular container, Eq. (72) is applicable to the calculation of the Tc for the powdery chemical as well. 

Consequently, a temperature profile develops within the mass which is mainly determined by the substance specific heat conductivity. The temperature profiles of those two limiting cases are presented schematically in Figure 4-7, As the Semenov model is of greater importance to chemical transitions performed in their respective reactors, the following elaboration shall focus on this part of the explosion theory. The other limiting case should be applied when assessing the storage of solid substances with dust explosive or self-reactive properties. 

All these approaches are ultimately inadequate because they do not propo ly recognize the distinction between criticality and sensitivity. Further, although Adler and Enig consider that their analysis corresponds to the Semenov model in the limit as B- oo, this is certainly not the case. "- ... 

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