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5. KINETICS AND SCALE UP – SAFETY ANALYSIS (STEP 3)

Calculation of adiabatic thermal transformation and heat accumulation from non-isothermal DSC measurements

5.1 Introduction

The precise prediction of reaction progresses in adiabatic conditions is necessary for the safety analysis of many technological processes. Calculations of an adiabatic temperature-time curve for the reaction progress can also be used to determine the decrease of the thermal stability of materials during storage at temperatures near the threshold temperature for triggering the reaction. Due to insufficient thermal convection and limited thermal conductivity, a progressive temperature increase in the sample can easily take place, resulting in a thermal runaway.

Several methods have been presented for predicting reaction progress of exothermic reactions under heat accumulation conditions [14-32]. However, because decomposition reactions usually have a multi-step nature, the accurate determination of the kinetic characteristics strongly influences the ability to correctly describe the progress of the reaction. The use of simplified kinetic models for the assessment of runaway reactions can, on one hand, lead to economic drawbacks, since they result in exaggerated safety margins. On another hand, it can cause dangerous situations when the heat accumulation is underestimated. For self-heating reactions occurring adiabatically, incorrect kinetic description of the process is usually the main source of prediction errors.

5.2 The kinetic based approach for the determination of Time to Maximum Rate under adiabatic conditions (TMRad)

5.2.1 Concept

Kinetic parameters calculated from DSC measurements are used for describing thermal behavior of larger amount of substance because the kinetics of any reaction is the same for ten milligrams of substance and for one ton. However, during scale-up two important factors have to be considered:

(i) the application of advanced kinetics, which properly describes the complicated, multistage course of the decomposition process,

(ii) the effect of heat balance in the energetic system, as the sample mass is increased by a few order of magnitude compared to the thermoanalytical experiments.

Because decomposition reactions usually have a multi-step nature, the isoconversional analysis enables a more accurate determination of the kinetic characteristics compared to simplified kinetic based approach that are essentially based on simplified kinetics assumption like ‘let’s assume that the reaction is zero order etc…’

The application of the DSC data for the simulation of the adiabatic measurements seems to be, for the first glance, not obvious. However, the closer look on the differences and similarities of the processes occurring under different conditions of the heat exchange allows to understand that the adiabatic properties such as self-heating rate or time to maximum rate can be correctly determined from the DSC results obtained or isothermally, or better, at different heating rates. The main difference between DSC run and fully adiabatic process lays in the conditions of the heat exchange. Due to the fact that in the thermoanalytical experiments the heat evolved during reaction is fully exchanged with the environments, the heating rates under which the experiment are carried out can be arbitrarily chosen. In turn, after determination of the kinetic parameters from the results of few experiments done with different heating rates, one can predict the reaction rate under any temperature ramp.

Due to the totally different heat exchange conditions in adiabatic conditions the heating rate of the process (called now self-heating rate) cannot be controlled anymore, being dependent only on the kinetics, adiabatic temperature rise and φ-factor. However, even in this case we can simulate the reaction rate. The scheme of the general idea of this procedure is presented in the next figure.

Fig. 5.1 : Scheme of the presentation of the adiabatic temperature rise in the form of four sequences of the time periods Δt in which the heating rates are constant and amount in turn to β1, β2, β3 and β4.

Let us assume that the temperature increase under adiabatic conditions in the coordinates time - temperature can be presented schematically as in Fig. 1 in the form of four sequences of the short periods of time Δt in which the heating rate is constant. In time periods 0-t1, t1-t2, t2-t3 and, finally, t3-t4 the heating rates are β1, β2, β3 and β4, respectively. As shown previously, one can easily predict the reaction rate under any bvalue. During the temperature change schematically presented in Fig. 1, the reaction course can therefore be calculated as a sequence of processes occurring at known heating rates β1-β4. Using the infinitesimal Δt values for the description of the self-heating rate one can predict the reaction course under the adiabatic temperature change. The process occurring under adiabatic temperature rise is expressed as a succession of an infinite number of processes occurring at constant heating rates which can now be easily calculated by described method.

5.2.2 Heat transfer mechanisms

Heat transfer deals with systems that lack thermal equilibrium, and thus it is a non equilibrium phenomenon. The basic requirement for heat transfer is the presence of a temperature difference. Heat is transferred in the direction of decreasing temperature. The temperature difference is the driving force for heat transfer. The rate of heat transfer in a certain direction depends on the magnitude of the temperature gradient in that direction. The larger the temperature gradient, the higher the rate of heat transfer.

Heat can be transferred in three basic modes:
- conduction,
- convection,
- radiation.
All modes of heat transfer require the existence of a temperature difference. All modes are from the high-temperature medium to a lower-temperature one.

Conduction

Conduction is the transfer of energy from the more energetic particles of a substance to the adjacent less energetic ones as a result of interactions between the particles. Conduction can take place in solids, liquids, or gases. In gases and liquids conduction is due to the collisions and diffusion of the molecules during their random motion. In solids conduction is due to the combination of vibrations of the molecules in a lattice and the energy transport by free electrons.

Fig. 5.2 : Heat transfer by conduction in solids, liquids or gases.

The range of thermal conductivities is enormous. As we see from Fig. 5.2 [30], λ varies by a factor of about 105 between gases (poor heat conductor or insulator) and diamond (good heat conductor) at room temperature. One should study and remember the order of magnitude of the thermal conductivities of different types of materials. This will be a help in avoiding in future computations, and it will be a help in making assumptions during problem solving.

Fig. 5.3 : Thermal conductivities of materials. Pure crystals and metals have the highest thermal conductivities, and gases and insulating materials the lowest.

Convection

Convection is the mode of energy transfer between a solid surface and the adjacent liquid or gas that is in motion. Convection is commonly classified into three sub-modes:
- Forced convection,
- Natural (or free) convection,
- Change of phase (liquid/vapor, solid/liquid, etc.)

Radiation

Radiation is the energy emitted by matter in the form of electromagnetic waves (or photons) as a result of the changes in the electronic configurations of the atoms or molecules. Heat transfer by radiation does not require the presence of an intervening medium. Radiation is emitted by bodies because of their temperature.

5.2.3 Kinetic based approach - Determination of the TMRad

In heat transfer problems it is convenient to write a heat balance and to treat the conversion of chemical energy into thermal energy as heat generation. The energy balance in that case can be expressed as

The solution of the problem leads to a thermal stability diagram, i.e. for each starting temperature the Time to Maximum Rate under adiabatic conditions (TMRad) can be obtained easily.

5.2.3.1 Batch reactor in case of cooling failure

The energy balance of an exothermic reaction taking place in a batch reactor can read as follow

with M: mass, Cp: specific heat, T: temperature, U: overall heat transfer coefficient, A: contact surface between the sample and the container, ΔHr : heat of reaction, indices c, s and env: container, sample and environment, respectively. In case of cooling failure the overall heat transfer coefficient U=0 for achieving the adiabatic conditions. In a fully operationally adiabatic environment all the heat release goes to heat the sample and the container. If there is thermal equilibrium within the solution and the wall

then the whole system will have the same temperature rise rate and we can simplify the above equation to:

that can be rewritten as

with:

- the adiabatic temperature rise:

- the phi factor: φ =

- the reaction rate

For batch reactor with large sizes (>1 m3), it can be assumed that  Ms>>Mc(jacket)  so that we obtain

(with φ  1)

In case of isoconversional analysis we have :

with

Note, for comparison, in case of simplified zero order kinetic assumption we would have :

It comes from the above equations that one can use the kinetic based approach for predicting the reaction progress a(t) and rate da/dt as well as the development of the temperatures Ts(t) and dT/dt and adiabatic induction times at any selected starting temperatures.

5.2.3.2 Adiabatic calorimeter

There are several issues that need to be taken into consideration for the examination of adiabatic conditions or reconstructing an Accelerating Rate Calorimeter experiment starting from the kinetic based approach. The energy balance in case of an adiabatic calorimeter can be expressed as before

Because the presence of a temperature difference leads to heat transfer, an adiabatic calorimeter constantly attempts to achieve equilibrium states by keeping Tenv = Ts. As a consequence there is no driving force for heat transfer and the chemical reactions run adiabatically. We obtain as before

However, one has to take into account the thermal inertia (or φ-factor) to consider the effect of the vessel’s inertia i.e. the reaction heat that is transferred in part into heating of the bomb. It can be observed that the φ factor influences the simulation of an ARC experiment. The φ factor influences:

- the because it comes from above description that

- and the TMRad in different levels depending on the type of decomposition kinetics.

Usually, the standard method of correcting the time for the reaction under adiabatic conditions is

This is however an approximation.

5.2.3.3 Dewar

In dewar tests we encounter walls that consist of layers of different materials. The energy balance in that case can be handled just like exothermic reaction in the adiabatic calorimeter or batch reactor in case of cooling failure.

The net heat rate through a three-layered wall (glass-vacuum-glass) of thicknesses L1, L2 and L3 with convection on both sides is presented in the next figure.

Fig. 5.4 : Schematic description of a Dewar. The net heat rate through a three-layered wall (glass-vacuum-glass) of lengths L1, L2 and L3 with convection on both sides.

The rate of the heat transfer between air and the wall at the outer surface is by convection. The heat transfer through the three-layered wall is conduction. If the reaction mixture is a liquid, heat transfer between the liquid and the wall at the inner surface is by convection. Rate of heat convection into the wall, rate of heat conduction through the wall and rate of heat convection from the wall can be expressed as:

In an analogous manner to Newton’s law of cooling it is sometimes convenient to express heat transfer through a medium UA(Tenv-Ts) using the thermal resistance concept:

Where

As h1<< h2, λ1<<λ2 and λ3<<λ2 (because λvacuum<<λglass), it can be assumed that

In addition, Ms>>Mc(glass) so that after insertion into the energy balance

and simplification we can write

Note that

In case of large adiabatic temperature rise ΔTad and because the total resistance Rtotal is important because of the vacuum we can assume that   and we obtain

which leads basically to the same results as before for exothermic reaction in the adiabatic calorimeter or batch reactor in case of cooling failure.

5.3 Construction of a thermal safety diagram: runaway time as a function of process temperature under adiabatic conditions (TMRad = f(T)) for the examined substance

The decomposition of the examined substance follows an exothermal process. Using the heat of reaction (ΔHr = -554.8±17.4 J/g) and assuming a heat capacity Cp = 1.5 J/(g·K), one can calculate the reaction progress due to self-heating for any ΔTad = (-ΔHr)/Cp/Φ. Notes:

(i) Cp can be easily determined with Calisto Software
http://www.akts.com/calisto-overview.html
http://www.akts.com/live-video-calisto-processing-video-cp-regression.html

(ii) Cp value has been assumed = 1.5 J/(g·K) for illustration (correct value has to be introduced by end user for final report !).

As long as thermal safety analysis is concerned, adiabatic induction time is defined as the time which is needed for self-heating from the start temperature to the time of maximum rate (TMRad) under adiabatic conditions. Depending on the decomposition kinetics and ΔTad, the choice of the starting temperatures strongly influences the time to runaway and the rate of the temperature evolution under adiabatic conditions. Figure 5.5 presents the starting temperature and corresponding adiabatic induction time TMRad relationship.

Figure 5.5 Thermal safety diagram: Starting temperature and corresponding adiabatic induction time TMRad relationship of the examined material. The choice of the starting temperatures strongly influences the adiabatic induction time. ΔHr = -554.8±17.4 J/g and ΔTad=(-ΔHr)/Cp/Φ=369.9±11.6°C for Φ = 1 and Cp = 1.5 J/g/°C.

ΔHr = -554.8±17.4 J/g
Cp = 1.5 J/g/K
ΔTad=(-ΔHr)/Cp= 369.9±11.6°C

Temperature (°C) (TProcess)	Time (h) (TMRad for TProcess and ΔTad=369.9°C)	Time (h) (TMRad for TProcess+1°C and ΔTad=381.5°C) (pessimistic)	Time (h) (TMRad for TProcess-1°C and ΔTad=358.3°C) (optimistic)
89	0.95	0.87	1.05
88	1.03	0.94	1.14
87	1.12	1.02	1.24
86	1.21	1.1	1.34
85	1.32	1.19	1.46
84	1.43	1.29	1.58
83	1.55	1.4	1.72
82	1.69	1.52	1.87
81	1.83	1.66	2.03
80	1.99	1.8	2.21
79	2.17	1.96	2.4
78	2.36	2.13	2.62
77	2.57	2.32	2.85
76	2.8	2.52	3.11
75	3.05	2.75	3.39
74	3.33	3	3.71
73	3.63	3.27	4.05
72	3.97	3.57	4.42
71	4.34	3.89	4.83
70	4.74	4.25	5.29
69	5.18	4.65	5.79
68	5.67	5.08	6.34
67	6.21	5.56	6.94
66	6.8	6.09	7.61
65	7.46	6.67	8.35
(*)64	(*)8.18	(*)7.31	9.16
(*) Means that if Φ=1 and Cp=1.5 J/g/°C the determined TMRad is about 8 hours (8.18 h) for an initial temperature of about 64°C (for that temperature a more conservative value for TMRad is about 7.31 h)
63	8.97	8.02	10.06
62	9.85	8.8	11.05
61	10.83	9.66	12.15
60	11.9	10.61	13.37
59	13.09	11.66	14.72
58	14.4	12.82	16.21
57	15.86	14.11	17.86
56	17.48	15.53	19.7
55	19.27	17.11	21.73
54	21.26	18.87	23.99
(**)53	(**)23.47	(**)20.81	26.51
(**) Means that if Φ=1 and Cp=1.5 J/g/°C the determined TMRad is about 24 hours (23.47 h) for an initial temperature of about 53°C (for that temperature a more conservative value for TMRad is about 20.81 h)
52	25.92	22.97	29.3
51	28.65	25.37	32.42
50	31.69	28.03	35.88

Table 5.1 Thermal safety table: Starting temperatures and corresponding TMRad for the examined material under adiabatic heat accumulation conditions. ΔHr = -554.8±17.4 J/g and ΔTad=(-ΔHr)/Cp/F=369.9±11.6°C for Φ=1 and Cp=1.5 J/g/°C.

Figure 5.6 illustrates the applications of the above equations for the simulation of the thermal behaviour under adiabatic conditions. The simulated T-time relationships are presented with a starting temperature of about 52.78°C leading to TMRad of 24 h.

Note that isothermal conditions can be numerically retrieved by setting an exceptionally large value of φ to make the adiabatic temperature rise ΔTad insignificant. If the φ is very high all heat released by the reaction is dissipated to the surrounding. As a consequence, the sample temperature remains constant i.e. isothermal:

for very large values of φ we have

and

The self-heat rate curves under adiabatic conditions can be calculated similarly (see figure 5.8). It can be observed that a self heat rate of 0.02 K/min (with Φ = 1), which corresponds to the typical detection limit of adiabatic calorimeters, occurs after about 17 hours, it means 7 hours before TMRad of 24 h for an initial process temperature of 52.78°C. It can be also seen from the simulation that at that time the reaction progress amounts already to about 0.022 (about 2.2%) (Fig. 5.8).

Similar calculations can be performed for different factors (see figures 5.9) for an initial temperature of about 52.78°C.

6. SELF ACCELERATING DECOMPOSITION TEMPERATURE (SADT) - (STEP 3)

6.1 Introduction

Additionally to the prediction of the reaction progress at any temperature profile in mg-scale and under adiabatic conditions, the simulations of thermal properties in kg-scale such as Self Accelerating Decomposition Temperature (SADT) can be carried out. The DSC signals of a material’s decomposition are processed using AKTS Thermokinetics software’s unique numerical techniques to create an accurate kinetic model. Subsequently, this kinetic model is used by the AKTS-Thermal Safety Software to predict the possible thermal runaway for a specific container type and size under any global temperature environment.

The method for predicting the thermal behaviour of the energetic materials such as the determination of SADT strongly depends on the sample mass due to the significant influence of the heat generated during the reaction course. At the mg-scale, all the evolved heat dissipates to the surroundings and does not affect the temperature of the heated material. Whereas at the ton-scale, the system can be considered adiabatic, because almost all generated heat remains in the sample and there is potential for a thermal runaway decomposition. From a practical perspective for the kg-scale, the temperature change of the test material results from two different processes that together determine the heat balance, which is defined by the heat generated during the thermal decomposition and heat loss to the environment. The rate of heat generated during an exothermic decomposition increases exponentially as the temperature rises but the rate of the heat loss remains in a linear manner. Therefore, a clear understanding of the heat transfer mechanism is necessary for an accurate determination of SADT.

6.2 Equivalent spherical radius

SADT can be calculated for simple geometries such as spheres or infinite cylinders. These sample holders applied for calculations do not exactly correspond to the real containers geometries. However, the use of an equivalent radius representing all possible container geometries with equivalent spherical volumes allows in the first reasonable approximation for the determination of the SADT:

where
rs,eq = the equivalent spherical radius corresponding to the sample volume in the container
Vs = sample volume 
Ms = sample mass 
rs = sample density

6.3 The energy equation

When heat is transferred to surrounding environment the temperature profile within a solid body or a liquid depends upon the rate of heat generation, its capacity to store the part of this heat, and the rate of heat transfer to its boundaries. The energy balance over a volume element can be expressed as

and from Fourier's law

As the chemicals stored in containers can be solids or liquids, we must extend the heat conduction equation to allow for the motion of the fluid. After some restrictive assumptions [30], we obtain the energy equation:

where λ, ρ, cP, T, , qr mean: thermal conductivity, density, specific heat, temperature, fluid with a velocity field s(x,y,z) and heat generated per unit volume by the decomposition reaction

with dα/dt the rate of the decomposition reaction expressed by the Arrhenius type equation as those applied in Friedman analysis andDTad the adiabatic temperature rise expressed by the heat of reaction ΔHr and the specific heat cp,s : ΔTad = (-ΔHr)/cP,s. The energy equation is the same as for a solid body, except for the enthalpy transport, or convection term .

To perform the exact heat balance the numerical techniques like finite element analysis, or finite differences or finite volumes can be applied to solve the energy equation. The sample is virtually divided into the set of adjoining elements (see Fig. 1). These elements are organized in a virtual mesh and described by the advanced thermokinetics based on the Friedman analysis in each node of time and space.

Figure 6.1. Generalized heat balance over a container and a volume element.
(A) Kinetic parameters calculated from DSC measurements, independent of the sample mass, enable the determination of the reaction rate required for the heat balance.
(B) Heat balance depends on the sample mass and has to be calculated by numerical techniques.

6.3.1 For SOLIDS in containers:

If the examined substance in a sample holder is a SOLID, the main resistance to the heat transfer lies in the bulk of the substance. In SOLIDS, heat transfer occurs mainly by heat conduction and not by convection:

because = 0

Therefore, the property which quantifies the ease with which a material transfers heat is the thermal conductivity. As previously mentioned, thermal conductivity depends mainly on the medium’s phase, temperature, density and molecular bonding within a solid structure. It means that in case of solids (and only in that case) there is a relevant temperature gradient ‘within’ the solids from the center to the container wall depending on the surrounding temperature. This behaviour is illustrated by the calculation of the temperature changes in a sphere of polystyrene (PS) stored in a sample holder of polyethylene (PE) (Fig.6.2) in a surrounding temperature of 1°C.

After considering cylindrical coordinates and some simplifications for the heat conduction,

one can write

where: 
J is a geometry factor dependent on the type of the container: J=0 for the infinite plate, J=1 for the infinite cylinder and J=2 for the sphere and dα/dt = 0 (no decomposition reaction).

6.3.2 For LIQUIDS in containers

6.3.2.1 For LIQUIDS with high viscosity

In case of LIQUIDS, heat is transferred not only by conduction but mainly by convection. In convection, heat is transferred from one point to another through a moving fluid, as a result of the mixing of different portions of the fluid. The concept of the Nusselt number (Nu) is generally used to determine the heat transfer coefficient h. The Nusselt number is a dimensionless number which measures the enhancement of heat transfer from a surface, compared to the heat transfer that would be measured if only conduction could occur. It is calculated by:

where x is the position along the interface in the direction of fluid flow. The Reynolds number (Re), Prandtl number (Pr) and Grashof number (Gr) are dimensionless terms which depend on the velocity and on the physical properties such density, viscosity and expansion volume of the fluid. There are two sub-segments of convection, forced and natural. In forced convection, the motion of the fluid is the result of some mechanical work, such as a pump moving the fluid. For liquids in containers, convection is caused by buoyancy forces due to density differences caused by temperature variations in the fluid. At heating the density change in the boundary layer will cause the fluid to rise and be replaced by cooler fluid that also will heat and rise. This continues phenomenon is called free or natural convection and the Grashof number is generally used to approximate the ratio of the buoyancy force to the viscous force acting on a fluid. However, for chemical reaction mixtures, physical properties such as volume coefficient of expansion or viscosity are usually unknown. Therefore, because heat is transferred mainly by convection the following simplifications can be made:

where the size of N increases with lower viscosity and the boundary conditions read

It comes for above that for liquids with high viscosity the energy equation leads to

Finally considering cylindrical coordinates and some simplifications for the heat conduction, one can write

The boundary conditions at the interface between the wall and the liquid can be approximated by

6.3.2.2 For LIQUIDS with low viscosity

For liquids with low viscosity we have

where the boundary conditions read

As a consequence for liquids with low viscosity, one can proceed with the following approximation:

Where Lc, Tc,1 and Tc,2 mean: thickness of the container wall, the temperature at the outer and inner surface respectively.

The net heat rate is

and finally using the resistance concept one can estimate the resistance of the container wall as follow:

Considering the resistance of the Dewar wall

Tt results that a container filled with a liquid chemical has similar thermal behavior just like a Dewar if their heat transfer is similar

where Bid and Bic are the Biot numbers of the Dewar and the container respectively. The Biot number expresses the ratio of the resistance within the wall by the convection resistance at the surface of the wall.

The above condition has to be fulfilled for describing the thermal behavior for liquids in large containers based on an experiment performed at Dewar scale. It results that a 500 ml Dewar vessel can correctly predicts thermal behaviour of a 500 L package filled with a liquid.

Illustration:
The liquid in Dewar is initially at a temperature of 50°C and the surrounding temperature is 20°C. Taking the heat transfer coefficient from air to Dewar hd,1 = 30.25 W/(m2·K), one can determine which package size filled with the same liquid has the same thermal behaviour as the Dewar. The Dewar of Vd=500 ml can be modelled as a 2.5 cm radius, 12.73 cm–long cylinder. The thermal properties of Dewar, container and the heat transfer coefficient are constant. The properties of the Dewar and container are Ld = 2.5 mm, λd = 0.01 W(m·K) and Lc = 3 mm, λc = 1 W(m·K), respectively.

From the above we have

Assuming a spherical container we can write

And finally

This example is in accordance with other observation reported elsewhere [28]. A 500 ml Dewar vessel can be used to correctly predict thermal behaviour of a 500 L package filled with a liquid, but not filled with a solid where conductive resistance occurs in the bulk! For liquids stored in the containers, there is no relevant temperature gradient within the liquids from the center to the wall (for reasonable container sizes). In such a situation, it can be assumed that the main contribution to the resistance to the heat transfer is given by the container wall and the surrounding convective resistance. Depending on the thickness and type of container walls one can distinguish two cases: 

• If the wall thickness of the container is significantly large compared to the equivalent radius of the containers, then the thermal properties of the container wall (such as e.g. PE) play a relevant role in terms of resistance to heat transfer.
• If the thickness of the wall is narrow compared to the equivalent radius of the container, then the influence of the type of the wall (as long as it is not a ‘good insulator’) in terms of resistance to heat transfer is much less significant. For example, a 3 mm wall can be considered as small compared to 22.85 cm being the equivalent radius of a 50 L sample holder.

Results of the determination of temperature profile in a 50L container filled by water with temperature of 20°C and a surrounding temperature of 1°C is presented in Fig. 4. It can be observed that there is no longer relevant ‘conductive’ resistance arising from the liquid with e.g. λs,eq =100 W/(m·K). It means that one can still use the heat conduction equation to approximate the heat transfer process for liquids. However, instead of the ‘true’ value of the thermal conductivity (such as e.g. 0.6 W/(m·K) for water) an equivalent value such as e.g. 100 W/(m·K) should be used. It means also that the numerical applications of large equivalent thermal conductivity values will no longer have influence on SADT in case of liquid reactive chemicals where dα/dt ≠ 0 and ΔTad ≠ 0 (see next chapter).

Note that in the presence of a decomposition reaction such as the reduction of an ester with a Lithium aluminium hydride solution (Fig. 3.3) the more complicated and precise finite element analysis can also be used to retrieve the adiabatic behavior (Fig. 5.5.). The adiabatic behavior can be easily achieved with a very low value of thermal conductivity for the evacuated layer of container wall such as Dewar for simulating the behavior of a good insulator. For illustration, a sample holder with a wall thickness of 3 mm having extremely low values of λc = 10-10W/(m·K), ρc = 10-10 g/cm3, cp,c = 10-10 J/(g·K) insures numerically a perfect insulation for the examined substance (Fig. 6.4). As a result the adiabatic conditions are numerically retrieved and lead basically to the same thermal runway curve as presented in Fig. 5.5.

6.4 Method for the calculation of Self Accelerating Decomposition Temperature (SADT)

The Self-Accelerating Decomposition Temperature (SADT) is an important parameter that characterizes thermal hazard possibility under transport conditions of self-reactive substances. SADT is used in international transportation regulations and is referenced in the United Nations presented in “Recom mendations on the Transport of Dangerous Goods, Manual of Tests and Criteria” (TDG) [29]. Globally Harmonized System (GHS) [30] has inherited SADT as a classification criterion for self-reactive substances. According to the Recommendations on TDG, SADT is defined as “the lowest temperature at which self-accelerating decomposition may occur with a substance in the packaging as used in transport”. An important feature of SADT is that it is not an intrinsic property of a substance but “…a measure of combined effect of ambient temperature, decomposition kinetics, packaging size and heat transfer properties of the substance and its packaging” [28-32].

The Manual of Tests and Criteria of the United Nations regarding transport of dangerous goods and on the globally harmonized system of classification and labeling of chemicals indicates that the characterization of materials is based on heat accumulation storage tests.

The regulatory compliance definitions are:

(i) SADT is the lowest environment temperature at which overheat in the middle of the specific commercial packaging exceeds 6 °C (ΔT6) after a lapse of a seven day period (168 hours) or less. This period is measured from the time when the packaging center temperature reaches 2°C below the surrounding temperature.

(ii) SADT is the critical ambient temperature rounded to the next higher multiple of 5 °C.

The first definition is based on two essential parameters – maximal permissible overheating temperature and minimal acceptable induction period. The second definition considers only one parameter: the critical ambient temperature of thermal runaway rounded to the next higher multiple of 5°C without introducing any fixed transportation time into the definition.

The first definition is based on two essential parameters – maximal permissible overheating temperature and minimal acceptable induction period. The second definition considers only one parameter: the critical ambient temperature of thermal runaway rounded to the next higher multiple of 5°C without introducing any fixed transportation time into the definition.

Bearing in mind the abovementioned SADT definitions, several additional factors have to be considered for the proper determination of SADT:

• the intrinsic properties of the test material such as, kinetic parameters of decomposition (activation energy, pre-exponential factor in the Arrhenius equation) and physico-chemical properties such as thermal conductivity, specific heat and density.
• the external properties of the sample that can be changed arbitrarily, e.g. sample mass, geometry and physical properties of the sample holder/container.

From the above factors, especially two namely thermal conductivity and sample mass may influence the calculation of SADT in a significant way. As thermal conductivities of chemicals are not always exactly known and as both, container volume and sample mass, can be arbitrarily changed, their influence on SADT was examined in broad range of their changes. Simulations have been done for:

• sample holders shaped as polyethylene (PE) spheres with internal volumes of: 5, 10, 25, 50, 100, 250 and 500 L.
• following equivalent thermal conductivities λs,eq: 0.1, 1, 10 W/(m·K).

Illustration of choice of possible parameters regarding container materials and chemical substance for SADT calculations can be found in Table 6.1.

Density ρ	1 g/cm3 (assumed value)
Specific heat Cp	1.5 J/g/K (assumed value)
thermal conductivity  λ	0.1 W/(m·K) < λeq  < 10 W/(m·K) (through numerical variations)
Equivalent spherical radius	10.6 cm < spherical equivalent radius < 49.2 cm (5 L < sample holder volume < 500 L)
heat of reaction ΔHr (J/g)	-554.8 J/g (measured from DSC)
Heat transfer coefficient h	5 W/(m2·K)

Table 6.1. Sample holder illustration and physico-chemical properties of examined substance.

Tab. 6.2 presents the results of the SADT simulation for λeq values between 0.1 and 10 W/(m·K) and sample volumes between 5 and 500 L.

λs,eq W/(m·K)	SADT (°C) Sample volume (L) / Equivalent spherical radius (cm)
	5 / 10.608	10 / 13.365	25 / 18.139	50 / 22.854	100 / 28.794	250 / 39.08	500 / 49.237
0.1	44°C	43°C	41°C	39°C	36°C	31°C	30°C
1	45°C	45°C	44°C	44°C	43°C	42°C	40°C
10	45°C	45°C	45°C	44°C	44°C	43°C	42°C

Table 6.2. Dependence of SADT (°C) on equivalent thermal conductivity λs,eq and the amount of chemical substance expressed by the sample volume (L) or the equivalent spherical radius (cm).

The results in table 6.2 and figures 6.5 clearly show that both, thermal conductivity and sample volume influences SADT values. For a 50 L sample volume, SADT for solid substance with an assumed thermal conductivity λ of 0.1 W/(m·K) amounts to 39°C (see Tab. 6.2 and Fig. 6.5 A) whereas under assumption of larger heat transfer for an ‘apparent’ conductivity term λs,eq arbitrarily set to 1 and 10 W/(m·K), SADT amount to about 44°C respectively (see Tab. 6.2 and Fig. 6.5 B-C). A sensitivity analysis can easily show that two parameters mostly influence SADT:
(i) the thermal conductivity and sample volume that is mainly responsible for the heat transfer (accumulation & release)
(ii) the type of reactions (decelerating, autocatalytic,...) and heat of reactions responsible for the heat source.
Note: All simulations were performed with a density of 1 g/cm3 (assumption).

Not observing the different overall thermal transport behaviour between solid or liquid, the confusion solid-liquid for the SADT determination is still a common mistake. We think that this topic should be even better emphasized in the current UN-Regulations as stated in the conclusions of recent studies at the Swiss Institute for Safety and Security [28]:

1. “SADT-test using a 500 ml Dewar vessel correctly predicts thermal behaviour of a 500 l package filled with a liquid, but it will only be representative for an 8L package filled with a solid” (This is of course an exaggerated statement. When the author H. Fierz [28] calculated it, he probably took rather extreme parameters to emphasize the’ warning’ statement).

2. “The UN-test H.4 was obviously designed to make a direct scale-up to a predetermined package size possible. This does not work for solids. Extrapolated package sizes for solids are dramatically different from those for liquids.”

3. “For solids the UN-test H.4 results are on the unsafe side and should therefore not be used.”

4. “The concept of time constants and specific heat losses for cooling of solids are misleading and should be abandoned. For solids the cooling characteristics of the bulk in its packaging should be determined individually for each case or even better: the theory of Frank-Kamenetskii should be applied.” (or more generally the problem of the heat conduction equation combined with the correct expression for the decomposition kinetics should be solved).

7. CONCLUSION

This study focuses on prediction of the reactivity of a substance both in extended temperature ranges and under temperature conditions for which experimental data collection is difficult. Adequate predictive examination of investigated reactions requires about four DSC measurements carried out with different heating rates, generally in the range of 0.25 to 8 K/min. Moreover, applying the results obtained by Differential Scanning Calorimetry (DSC) advanced numerical techniques enable prediction of the reactivity in broad temperature range. In fact, numerical simulations can be used to replace experiments in situations, which are not directly accessible to measurement for example for timing reasons. The examples of such modelling analysis can be helpful for guiding the screening and development of new materials activities. If modelling proceeds in parallel with experimental studies, then it should result in lower costs in a project development phase.
The proposed method has therefore several advantages:

• It is fast: Reliable screening of candidate energetic materials by revelation of potentially unstable mixtures at prolonged times and temperature exposures can be done within few hours
• It is economical: The determination of the kinetics offers very significant time/expertise savings compared to real time analysis which can extend over prolonged periods, even months.
• It is convenient: DSC devices are widely available and only require a small amount of material.
• It is versatile: With one set of measurements different scenarios can be calculated. The method can be used to predict the rate of the reaction progress for any temperature profile.

‘Safety through design not by Accidents’

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