3.3.2.1 Reaction Heat–Adiabatic Temperature Rise– MTSR

The integration of the heat signal versus time gives us the total reaction heat, usually expressed in kJ or kcal. From this reaction heat, the adiabatic temperature rise of the synthesis reaction can be calculated according to Eq. (3.3):

(3.3)

where

• ΔH_R: reaction enthalpy (kJ/kg)
• c_p: specific heat capacity of the reaction mixture (kJ/kgK)

The MTSR can be calculated merely by adding the adiabatic temperature rise to the reaction temperature (Eq. 3.4):

(3.4)

Whereas the molar reaction enthalpy is an intrinsic property of a specific reaction, the adiabatic temperature rise is dependent on the reaction conditions. In the (hypothetical) example in Table 3.2, this difference is demonstrated.

This example clearly shows the importance of reaction conditions, with the adiabatic temperature rise being more than 10 times higher in case 1. This dramatic difference can be fully attributed to the effect of the solvent acting as a heat sink. Working at higher dilution in a solvent with a higher heat capacity can drastically reduce the possible consequences of reaction that runs out of control. Unfortunately, running a process at higher dilution has an impact on the overall economy, so both aspects should be considered.

The adiabatic temperature rise is often used as a measure for the severity of a runaway reaction. A process with an adiabatic temperature rise of less than 50 K is usually considered to pose no serious safety concerns, at least when there is no pressure increase associated with the reaction. When a process has an adiabatic temperature rise of more than 200 K, a runaway reaction would most probably result in a true thermal explosion, and hence such processes require a very thorough safety study.

3.3.2.2 Thermal Accumulation

According to Eq. (3.2), the thermal accumulation can be calculated by the partial integration of the heat signal. Thermal accumulation is an important parameter in the assessment of the safety of a process. If a problem occurs during a process (cooling failure, stirrer failure, etc.), it is common practice to stop the addition of chemicals immediately. In case there is no thermal accumulation, the reaction will also stop immediately, and there will be no further heat generation that can lead to a temperature increase in the reactor. A reaction with 0% thermal accumulation is therefore called dosing controlled. If there is thermal accumulation however, part of the reaction heat will still be set free after the dosing has been stopped, and hence the temperature in the reactor can increase.

Because of the importance of the thermal accumulation, the MTSR is often specified as either being MTSR_batch or MTSR_semi‐batch. For the calculation of the former, the total adiabatic temperature rise is added to the reaction temperature, whereas, for the latter, the adiabatic temperature rise is first multiplied by the percentage of thermal accumulation. An example is given in Table 3.3.

This example shows the big difference in intrinsic safety of the process between the two cases. Should a cooling failure occur in case 1, the temperature would never be able to rise significantly above the process temperature, provided of course that the dosing is stopped as soon as the failure occurs. In case 2, on the other hand, the temperature would increase to 120 °C without the possibility to cool, even when dosing is stopped immediately.

So, obviously, low thermal accumulation is to be preferred for any semi‐batch process. A high degree of thermal accumulation is a sign that the reaction rate is low relatively to the dosing rate. Two possible measures can be taken to decrease the thermal accumulation of a given process:

1. Increase the reaction rate. This can be done by increasing the reaction temperature. Increasing the reaction rate means also increasing the heat rate of the reaction, so a calorimetry experiment at this new (higher) process temperature is required to make sure that the cooling of the reactor can cope with the heat generation under normal process conditions. Obviously, the increased temperature will lead to a smaller safety margin between reaction temperature and possible decomposition temperature, and this should be dealt with appropriately.
2. Decrease the dosing rate.

3.3.2.3 Heat Rate

Whereas a correct determination of the total reaction heat is important, the rate at which this heat is being liberated is at least equally important for a proper safety study. Where a process is run under identical conditions both at large scale and in the calorimeter, the heat rate (in W/kg) is scale independent. It should be kept in mind, however, that the cooling capacity of a reaction calorimeter is in most cases several orders of magnitude higher than that of a large‐scale production vessel. A reaction calorimeter might still be able to keep a constant heat rate of 200 W/kg under control, running this process at production scale will most certainly lead to a runaway reaction. In such a case, the process should be redesigned as to decrease the heat rate, and ideally the calorimetry experiment should be repeated under the new process conditions to make sure that no unwanted side effects occur (higher thermal accumulation, sudden crystallization, formation of extra impurities, etc.).

Not only the absolute value of the heat rate has to be considered, the duration of the heat evolution is also important. A peak in the heat evolution that surpasses the available cooling capacity but which only lasts for a short period of time is not necessarily problematic. If such a peak is observed, one should calculate the corresponding adiabatic temperature rise and evaluate its consequences. E.g. a heat rate of 200 W/kg for two minutes would give rise to a temperature increase of 12 °C under adiabatic conditions, assuming a specific heat of 2 J/gK. If a cooling capacity of 50 W/kg is available, this will only be 9 °C. The issues one might encounter when scaling up a reaction to meet the heat removal capacities of the production vessel will be discussed in more detail.

3.4.1.1 Film Theory

When designing an exothermic reaction for scale‐up, it is important to know what the heat removal capacity of the reactor at production scale is. Unfortunately this is easier said than done. The heat transfer between the heat transfer medium in the jacket and the reaction mixture is usually described in terms of a series of resistances, the so‐called film theory. It considers three main factors governing the heat transfer in a stirred tank reactor: the resistance of the inner film (boundary reaction mixture–vessel wall), the resistance of the vessel wall and the resistance of the outer film (boundary vessel wall–heat transfer fluid). This can be expressed numerically:

(3.5)

where

• U: overall heat transfer coefficient (W/m²K)
• h_r: inner film transfer coefficient (W/m²K)
• d: thickness of the vessel wall (m)
• λ: thermal conductivity of the vessel wall (W/mK)
• h_c: inner film transfer coefficient (W/m²K)
• U_max: maximum heat transfer coefficient (W/m²K)

From this equation, it can be understood that there are two main contributions to the overall heat transfer coefficient: one that is solely dependent on the characteristics of the reaction mixture and one that is solely dependent on the characteristics of the reactor. Indeed, the inner film transfer coefficient h_r is a measure for the resistance to heat transfer between the reaction mixture and the vessel wall and is strongly correlated to the physicochemical properties of the reaction mixture (viscosity, density, heat capacity, etc.) and the stirring speed. U_max on the other hand can be interpreted as the maximum obtainable heat transfer coefficient in a certain reactor in the hypothetical case when the inner film resistance would approach zero. This term comprises of two contributing parts: one that is due to the thermal conductivity of the vessel wall and one that is due to the outer film coefficient. These two terms are solely dependent on the characteristics of the reactor.

This explains why a correct scale‐up of the heat transfer characteristics is so difficult: since U is dependent on both the process and the reactor, it ideally should be determined or calculated separately for each vessel–process combination. This is a rather time‐consuming process, as can be seen from the list actions that should be undertaken:

1. Determine the U_max of the calorimeter at the process temperature.
2. Determine U for the reaction mixture under process conditions in the calorimeter.
3. From 1 and 2, calculate h_r for the reaction mass in the calorimeter.
4. Calculate h_r for the reaction mass in the production vessel (literature scale‐up rules).
5. Determine the U_max of the production vessel at the intended jacket temperature.
6. From 4 and 5, calculate the according U‐value for the reaction mixture in the production vessel.

For a thorough description of the theory behind this approach, the reader should refer to the literature [12, 13]. In the following part, we will briefly discuss some major issues related to the determination of the heat transfer coefficient at production scale.

3.4.1.2 Determination of U

The most widely used approach to determine the heat transfer characteristics of a reactor is by means of the Wilson plot [14]. The heat transfer coefficient is determined experimentally (by means of a cooling curve) at several different stirring speeds. When plotting the reciprocal heat transfer coefficient at a certain jacket temperature versus the stirring speed to the power −2/3 (Eq. 3.6), a straight line is obtained, with the intercept being equal to .

(3.6)

where

• n = stirrer speed (in rpm)
• c^te = constant.

An example of such a Wilson plot is shown in Figure 3.4.

This is intrinsically the most reliable way to determine the U_max for any reactor, since the result is independent of the solvent being used for the cooling experiment. When repeating the same experiments with a different solvent, the experimental points will differ, and the slope of the line will differ, but U_max will remain the same.

This method can be used for the characterization of a reaction calorimeter where a large number of either cooling curves or U determinations via the calibration heater can be run in an automated way. Since U_max is dependent on the filling degree and jacket temperature, a vast range of U determinations are necessary for a proper description of the heat transfer properties of the reaction calorimeter. It is our experience that isopropanol is the most suitable solvent for obtaining good Wilson plots in the reaction calorimeter.

This method can be applied to the reaction calorimeter in programmed mode, and it becomes rather cumbersome however for use at plant scale. Since each measuring point in the curve has to be obtained from 1 cooling curve at one stirring speed, it becomes very time‐consuming to gather the data needed for this plot. Therefore, another approach can be used for the estimation of U_max at production scale. As mentioned above, constructing the Wilson plot with different solvents will alter the slope but not the intercept. If water is used for the construction of the Wilson plot, the slope turns out to be very low, i.e. the contribution of 1/h_r is low. This implies that determining the U‐value with a reactor filled with water at the highest possible stirring speed will yield a value that is close to U_max. This way, a good approximation of the maximum heat transfer capacity of the vessel can be obtained from only one experiment. This approximation will obviously be less accurate, but the error made is always on the safe side: the U_max will be underestimated, and hence in reality there will be more cooling power available than anticipated. An example of this approach is being shown in Figure 3.5, where the heat transfer coefficients for a typical stainless steel reactor and a glass‐lined reactor of 6000 L are shown as a function of jacket temperature for a filling degree of 50 and 85%. This shows clearly the large range of U‐values that can be encountered in practice.

It is important to note that the U_max value is a function of the jacket temperature rather than the reactor temperature because it is linked to the properties of the jacket and vessel wall, which is always at approximately the same temperature as the jacket. This implies that cooling curves should be determined with a constant temperature difference between the jacket and reactor. If a cooling curve is recorded with the jacket constantly at its lowest temperature, the temperature dependence of the U_max is lost. Moreover, when calculating the final overall heat transfer coefficient of the reactor from its respective U_max value, the intended temperature offset between the reactor and jacket should be kept in mind. This is less of an issue in the reaction calorimeter, since the observed temperature differences between the jacket and reactor are usually a lot smaller than in a large‐scale reactor. Obviously, the values from Figure 3.5 apply only to the specific reactors in this specific plant, as different layouts in the cooling system, different materials of construction, different heat transfer media, and different temperature control strategies can all have a large influence on the heat transfer characteristics of a vessel.

3.4.1.3 Influence of the Inner Film Coefficient

Now that the U_max has been determined at both lab scale and production scale, let us turn our attention to the other term in Eq. (3.5), i.e. the inner film coefficient h_r. This film coefficient is a measure for the resistance toward heat transfer between the reaction mass and the vessel wall. It is mainly governed by the stirring speed and the physical properties of the reaction mixture: a highly viscous reaction mixture will have more difficulties in dissipating reaction heat to the reactor wall than, for instance, pure water. Unfortunately enough, the influence of the inner film coefficient can be quite pronounced: if it were relatively small in comparison with the U_max term, it could be neglected, and one single heat transfer coefficient could be used for each vessel, irrespective of the reaction mixture.

When the U_max of the reaction calorimeter is known at the (jacket) temperature and fill degree used, the inner film coefficient can be calculated from the overall heat transfer coefficient as determined in the calibration procedure according to Eq. (3.5).

As the inner film coefficient is dependent on the mixing characteristics of the vessel used, it is scale dependent and should be scaled up accordingly. This is usually done according to Eq. (3.7):

(3.7)

where

• : the inner film coefficient at large scale
• : the inner film coefficient at calorimeter scale
• D: vessel diameter at large scale
• d: vessel diameter at calorimeter scale
• N: stirring speed at large scale
• n: stirring speed at calorimeter scale
• = viscosity number at large scale

The viscosity number is the ratio of the viscosity of the reaction mixture at the reaction temperature and its viscosity at the jacket temperature. When considering standard organic reactions in solution, this ratio is quite close to unity, so this factor is usually neglected. When studying polymerization reactions, however, this effect should be taken into account.

Using this equation, the inner film coefficient at production scale can be calculated, and hence the overall heat transfer coefficient is now known, according to Eq. (3.5). In order to get a more quantitative feeling for the influence of the different parameters on the overall heat transfer coefficient, let us take a look at a realistic numerical example.

3.4.1.4 Shortcuts to U‐Value Determinations

When the procedure for the determination of correct heat transfer data as described above is out of reach, there are other possibilities to make a rough estimation of the U‐values. If one chooses to go for these simplified estimation methods, care is needed to include a wide enough safety window when a batch is run for the first time in a certain reaction vessel.

A first possible estimation method is to simply use the heat transfer coefficient for a reactor filled with the neat solvent in which the reaction is to be performed. Cooling curves for some solvents are often readily available from cleaning campaigns. Since these cleaning cycles are usually repeated quite regularly, an indication of the evolution of the heat transfer characteristics of the reactor over time can also be obtained. It excludes the need for the separate determination of U_max with water, which is not often used as a cleaning solvent in a temperature cycle, and of the entire characterization of the U_max behavior of the reaction calorimeter. So if a reaction is to be run in a methanol solution at 30 °C, one could simply calculate the U‐value at that temperature from a cooling curve with neat methanol. This approach will yield acceptable results, as long as the reaction mixture is not strongly heterogeneous or highly viscous.

Another possible approach is to use very conservative general heat transfer coefficients in the scale‐up calculations. One could, for instance, record a cooling curve for methanol, calculate the U‐values from this curve, and then base all calculations on 50% of the heat transfer coefficient found. Although this method does not consider any specific effect of the physical properties of the reaction mixture on the heat transfer coefficient and should therefore only be used with great care and large safety margins, it can be an easy tool to give some guidance in the scale‐up calculations.

Again, it should be stressed that when the correct U‐values are not known and can only be roughly estimated, broad safety margins should be incorporated in the process, and the reactor data from the first batch should be checked carefully for any inconsistencies.

3.4.1.5 Practical Use of U‐Values

So now the U‐value of the reaction mixture at production scale has been determined, but what can we do with it? The main use of heat transfer coefficients is to allow for a correct calculation of dosing times, making sure that all heat that is being generated during the reaction can be safely removed. This is illustrated in the example below for a dosing controlled reaction. When dealing with a non‐dosing controlled reaction (i.e. with significant thermal accumulation), one should make sure that the available cooling capacity matches the heat release rate as observed in the calorimetry experiment at any time.

3.4.2.1 Gas Speed

In the scale‐up of a process in which gas is being liberated, it is of utmost importance to make sure that all the gas that is set free can be evacuated from the reactor safely, without causing any pressure buildup. When a chemical plant is being designed, a certain layout for the scrubber lines is worked out. This specific layout implies that there is a maximum gas speed in the piping in order to avoid pressure buildup, entrainment of powders, or unintended changes in flow pattern or even flow direction. A maximum gas speed that is often used is 5 m/s. If the maximum design gas speed and the minimum diameter through which the gas has to pass are known, the resulting maximum gas flow can be calculated. Some typical piping sizes and the corresponding maximum gas flow to meet the 5 m/s criterion are given in Table 3.5.

These figures clearly demonstrate the pronounced effect of the diameter of the narrowest piping the gas has to pass on its maximum flow rate: doubling the diameter allows a fourfold increase in gas flow.

The part of Table 3.5 shows some interesting scale‐down data. In these columns, we have calculated what the corresponding gas flow at lab scale is. For instance, if the narrowest piping in the scrubber line is a DN50, a maximum flow rate of 71 m³/h can be allowed at production scale. Using this maximum gas flow and assuming it concerns a 6000 L reactor, we can calculate the gas flow at lab scale. In this case, the corresponding gas flow in a 2 L reaction calorimeter would be 392 ml/min. This flow can be detected easily, but if the reaction is run in a 100 ml calorimeter, the corresponding gas flow is only 20 ml/min. One can imagine that such a low gas flow rate can be overseen easily during the process development work. This illustrates the importance of an accurate gas flow measurement combined with each reaction calorimetry experiment. Especially when small‐scale reactors are used (<1 L), care should be taken in choosing the appropriate gas flow measuring device.

These values for the maximum allowed gas flow apply mainly to the desired synthesis reaction. Exceeding this flow to a limited extent might result in operational problems such as slight pressure buildup, process gases entering a neighboring reactor that is connected to the same scrubber line, or suboptimal condenser and scrubber performance but will not necessarily lead to pronounced safety issues. When gas flow rates are considered an order of magnitude higher, vent sizing calculations come into play. This will be briefly discussed at the end of this chapter.

3.4.2.2 Reactive Gases

In the previous discussion, the only parameter of concern was the gas flow rate. In many cases, however, the gas being emitted is reactive by itself, and this can cause particular safety problems. One example of having a very high yielding but unfortunately enough undesired synthetic reaction is depicted in Scheme 3.1.

These two processes were both run in the same plant. By coincidence, they were being run at exactly the same time in two neighboring reactors. The reaction depicted at the top resulted in the emission of hydrogen chloride, while reaction at the bottom was releasing ammonia. Both reactors were connected to the same scrubber lines, and they inevitably reacted with each other, forming ammonium chloride in large amounts.

This solid material blocked the scrubber lines, and the reaction heat being evolved was large enough to partly melt the plastic scrubber lines. Fortunately enough there were no serious consequences, but this demonstrates the need for a broad safety overview in any chemical plant.

3.4.2.3 Environmental Issues

Although this factor is not related to process safety in the strict sense, the importance of the gas flow rate for environmental compliance should be mentioned here as well. It is important to know the layout of the gas treatment facility of the plant where the process is going to be run. If a carbon absorption bed is used, it is important to keep an overview of what the capacity of this bed is for the process gas being emitted: some gasses are retained better than others, and some gasses might even lead to dangerous hot spot formation in the bed. If a catalytic oxidation installation is used, it is important to know that some compounds (like hydrogen and alkenes) will lead to overheating in the installation if the flow is too high. And when no air treatment facility is installed at all, one should always make sure that the gas streams being emitted are within all environmental requirements. This assessment needs to be made for each production plant separately.

3.5.2.1 Reaction Enthalpy

Some typical decomposition energies for the most common thermally unstable groups are given in Table 3.7.

Functional Group	∆H (kJ/mol)	Functional Group	∆H (kJ/mol)
Diazo ─N=N─	−100 to −180	Nitro ─NO2	−310 to −360
Diazonium salt ─N≡N+	−160 to −180	N‐hydroxide ─N─OH	−180 to −240
Epoxide	−70 to −100	Nitrate ─O─NO2	−400 to −480
Isocyanate ─N=C=O	−50 to −75	Peroxide ─C─O─O─C	−350

This table clearly shows that merely looking at a molecular structure can already give an indication about the decomposition potential of a reagent. Note however that the data in this table are given in kJ/mol, whereas DSC data are usually reported in J/g. The latter gives in fact a better indication of the intrinsic energy potential of the product, since the influence of an instable group will obviously be much larger in a small molecule than in a very large one. It is therefore no surprise that many of the most dangerous reagents (in terms of thermal stability) are indeed small molecules: hydroxylamine, nitromethane, cyanamide, methylisocyanate, hydrogen peroxide, diazomethane, ammonium nitrate, etc.

According to the observed decomposition enthalpy, a first assessment of the energy potential can be made. A decomposition energy of only 50 J/g is very unlikely to pose serious problems, even when the product would decompose entirely. On the other hand, a decomposition energy in the order of magnitude of 1000 J/g should be considered as problematic, and the first choice should always be to avoid the use of such chemicals as much as possible.

As already previously mentioned (Eq. 3.3), the reaction heat is directly proportional to the adiabatic temperature rise. For ∆T_ad to be known, the c_p of the compound or reaction mixture is needed. The heat capacity can be determined separately in the DSC, but in order to get a first indication, an estimated value can be used as well. Usually a c_p of 2 J/gK can be used for organic solvents or dilute reaction mixtures, 3 J/gK for alcoholic reaction mixtures, 4 J/gK for aqueous solutions, and 1 J/gK for solids (conservative guess). A rough classification of severity of a decomposition reaction based on its reaction enthalpy and corresponding adiabatic temperature rise (in this case for c_p = 2 J/gK) is given in Table 3.8.

One final remark should be made here about endothermic decompositions. Some compounds decompose endothermically, and whereas one might consider them therefore to be harmless, special attention for these compounds is sometimes needed. Decomposition reactions in which gaseous products are formed (such as elimination reactions) are often endothermic. The intrinsic risk of this type of decompositions lies therefore not in the thermal consequences but in a possible pressure buildup. For this type of compounds, an evaluation of the possible pressure buildup associated with the decomposition is recommended.

3.5.2.2 Onset Temperature

Even when a compound has a very high decomposition energy, it can still be possible to use it safely in a process, provided that the difference between the process temperature and the decomposition temperature is sufficiently high. The term “onset temperature” is used quite frequently to denote the temperature at which a reaction or decomposition starts. In reality, however, there is no such thing as an onset temperature, since the temperature at which a reaction starts is strongly dependent on the experimental conditions. The point from which on a deviation from the baseline signal can be observed is determined by the sensitivity of the instrument, the sample size, and the heating rate of the experiment. This implies that great care is needed when comparing “onset temperatures” obtained with different methods, but it does not completely rule out the use of this parameter in early safety assessment.

It should be mentioned here that the description above holds for the way in which the term “onset temperature” is usually interpreted in the process safety field. In other fields of research, the term onset is defined as the point where the tangent to the rising curve at the inclination point crosses the baseline. This definition is, e.g. used in the calibration of a DSC by measuring the melting point of a suitable metal (mostly indium). In this text, however, the term “onset” will always refer to the temperature at which a first deviation from the baseline signal is observed.

A rule of thumb that has been used quite extensively in the past is the so‐called 100 °C rule. This rule states that a process can be run safely when the operating process temperature is at least 100 °C below the observed onset temperature. This rule has been shown to be invalid in certain cases, so it should certainly not be used as a basis of safety. This does not mean however that it is completely useless. If a process is to be run on a relatively small scale (50 L or less) in a well‐stirred reaction vessel, natural heat losses to the environment are usually sufficiently large for the 100 °C rule to be valid. If the process is to be run at a larger scale, a proper and more detailed study of the decomposition kinetics (and especially of the TMR_ad) should be made, as will be discussed further on in this chapter. Here as well, it should be stressed that these remarks only apply to the thermal stability of the products, and extra care should be taken when gas evolution comes into play.

3.5.2.3 Reaction Type

When dealing with DSC data of compounds or reaction mixtures, merely looking at the shape of the peak can give some clues about the type of reaction taking place. The DSC run shown in Figure 3.7 consists of several different overlapping peaks, indicating a rather complex reaction type. Very sharp peaks are indicative for autocatalytic reactions, and extra care with this type of decompositions is needed [16].

Autocatalytic reactions are reactions in which the reaction product acts as a catalyst for the primary reaction. This implies that the reaction might run slowly at a certain temperature for a while, but as time passes more catalyst for the reaction is formed, and hence the reaction rate increases over time. Such reactions are therefore also called self‐accelerating reactions. This is in contrast with the more classical behavior of reactions following Arrhenius kinetics where the reaction rate stays constant (zero order) or decreases (first order or higher) with time at a certain temperature. Having determined the decomposition reaction in such a case on a fresh sample will therefore always give the worst‐case decomposition scenario, the initial rate measured will be the maximum rate for that sample at that particular temperature. This is not the case for autocatalytic reactions, where the reaction rate is strongly dependent on the thermal history of the sample. Measuring the reaction rate for a pristine sample might lead to an underestimation of the risk associated to the decomposition of this sample when it has been subject to a certain thermal history (e.g. prolonged residence time at higher temperature due to a process deviation). An example of an isothermal and a scanning DSC run for both an autocatalytic and a first‐order reaction are shown in Figure 3.8.

As pointed out in the above, an autocatalytic reaction behavior is easily recognized by isothermal DSC experiments. If the temperature in the sample remains constant, the heat release over time will decrease in case of an nth‐order reaction but will show a distinct maximum in case of an autocatalytic reaction. Although not always as easy as in an isothermal DSC run, autocatalytic reactions can also be recognized in scanning DSC experiments where the peak shape is notably sharper than for nth‐order reactions. This is shown in Figure 3.8.

When a decomposition reaction is known to be autocatalytic, extra care is needed in order to avoid its triggering. For such compounds, any unnecessary residence time at elevated temperatures should be avoided. Extra testing might be appropriate to reflect the thermal history the product will experience at full production scale, since, e.g. heating and cooling phases may take considerably more time than in small‐scale experiments. It should also be kept in mind that temperature alarms are not always an efficient basis of safety for this type of decomposition reactions: since the temperature rise can be very sudden, this type of alarm might simply be too slow to ensure sufficient time is available to take corrective actions.

3.5.4.1 Adiabatic Temperature Profile

Let us consider the situation as being depicted in Figure 3.9. The reaction mixture is at a constant temperature of 120 °C when a cooling failure takes place. At this temperature, the reaction starts relatively slowly, and hence the temperature increases, albeit at a slow pace. Since most chemical reactions proceed faster at higher temperatures, the reaction rate (and thus the temperature rise rate) will increase as the reaction continues. This acceleration continues until finally the depletion of the reagents slows the reaction down again and a stable final temperature is achieved. The S‐shaped temperature curve seen in the graph is very characteristic for an adiabatic runaway reaction. As indicated on the graph, there are three main parameters that describe the process of a runaway reaction, i.e. the adiabatic temperature rise (ΔT_ad), the TMR, and the maximum self‐heat rate (max SHR). The first two have been discussed earlier in this chapter, and the max SHR is a measure for the maximum speed with which the reaction occurs and can be directly correlated to the power output of the reaction: a SHR of 1 °C/min corresponds to 33 W/kg reaction mixture for an organic medium with a c_p of 2 and to 70 W/kg in aqueous medium

This is important for vent sizing calculations and the assessment of using the boiling point as a safety barrier, as will be discussed further on in this chapter.

3.5.4.2 Heat–Wait–Search Procedure

The most commonly applied method for adiabatic testing is the so‐called heat–wait–search procedure, as depicted in Figure 3.10. In this procedure, the sample is introduced in the instrument at room temperature and then heated (heating step) to the starting temperature of the test. The sample is then allowed to equilibrate at this temperature (waiting step), followed by the so‐called search step. During this step (which usually takes between 5 and 30 minutes), the sample temperature is monitored to see if there is any sign of an exothermic reaction taking place. If the temperature rise rate under adiabatic conditions during this period is below the chosen detection threshold (typically 0.02 or 0.03 °C/min), the temperature is increased with a couple of degrees and the cycle starts all over again, until either an exotherm is detected or the preset final temperature has been reached. When an exotherm has been detected, the instrument will track the sample temperature adiabatically until the temperature rise rate drops below the threshold value (end of the reaction), after which the heat–wait–search cycle starts again. Alternatively, the run is aborted during an exotherm if the upper temperature limit of the experiment has been surpassed.

Although adiabatic calorimeters can usually operate in other thermal modes as well, the heat–wait–search procedure is still the most widely used because it allows determining the onset temperature of the exotherm with great accuracy while keeping the experimental time acceptably short.

3.5.4.3 Thermal Inertia: φ‐Factor

As mentioned above, one of the main reasons why it is hard to extrapolate adiabatic behavior at large scale from small‐scale lab work is the dramatic difference in heat losses between these two working environments. There is another reason as well, which plays a very important role in the interpretation of adiabatic calorimetry, i.e. the thermal inertia or φ‐factor:

(3.8)

where

• m_c: mass of the container (vessel or sample cell) (g)
• m_s: mass of the reaction mass (g)
• c_pc: heat capacity of the container (J/gK)
• c_ps: heat capacity of the reaction mass (J/gK)

When heat is generated in the reaction mixture, this heat will be used to increase the temperature of not only the reaction mixture itself but also the container, being the vessel at large scale or the test cell at small scale. The φ‐factor is therefore a measure of which fraction of the thermal mass of the entire system is due to the thermal mass of the reaction mixture and which part is due to the container.

In large‐scale equipment, the φ‐factor of a vessel during a runaway will be close to unity: i.e. the thermal mass of the vessel itself (mainly the jacket) will be low compared with the thermal mass of the reaction mixture (i.e. φ = 1). In small‐scale laboratory equipment, the φ‐factor is usually significantly higher than 1. In order to perform a lab scale experiment at a φ‐factor that is close to unity, one would need to use a very light test cell that can accommodate a large amount of sample. The influence of the φ‐factor on the runaway behavior of a system is very pronounced [17, 18], as can be seen in Figure 3.11. In this figure, the same adiabatic runaway profile is given for a sample being tested in two different test cells, one with a (hypothetical) φ‐factor of 1, the other one with a φ‐factor of 2 (simulations). As can be seen, the curves differ drastically. In every aspect, the curve obtained with φ = 1 is by far more severe than the one obtained with φ = 2. The total adiabatic temperature rise scales linearly with φ, i.e. the observed ΔT_ad in a φ = 2 experiment will be exactly half of the ΔT_ad in a φ = 1 experiment:

(3.9)

The TMR scales almost linearly with φ in most cases, but the max SHR scales far from linear with φ. The max SHR in an experiment with φ = 1 can easily be 10 times higher than in the φ = 2 experiment! The numerical data for the curves as depicted in Figure 3.11 are given in Table 3.10.

Figure 3.11 also gives the runaway behavior of one reaction in a test cell with a (hypothetical) φ‐factor of 1 as compared with the same run in test cell with a φ‐factor of 2. In this case, however, the difference between the two runs is even more pronounced. The reaction consists of two consecutive reactions, and running this reaction at φ = 1 will result in a temperature profile where the first exotherm continues into the second one, leading to a very rapid temperature rise. In the run with φ = 2, the temperature rise from the first exotherm will be far less pronounced, and this will lead to a significant time interval between the two exotherms. Hence the severity of this run will be significantly lower than that of the run with φ = 1.

These two examples show the importance of the φ‐factor on the experimental results. Ideally one would try to perform the adiabatic measurement in a low φ test cell. In many cases this is difficult to obtain experimentally, and a proper extrapolation to low‐φ conditions is needed.

3.5.4.4 Interpretation of Adiabatic Experiments

Adiabatic experiments can be performed for different reasons, but usually the main goal is to get a representative idea of what the temperature and pressure profile could be in a full‐scale reactor in case of a runaway reaction. If the adiabatic experiment is performed at a φ‐factor close to unity, the match between the two will indeed be close. Let us take a look at the most important parameters to analyze.

1. Adiabatic temperature rise. This value can be directly extracted from the thermal profile. When dealing with very violent reactions, it is probably not possible to obtain the total adiabatic temperature rise as the maximum safety temperature or pressure of the equipment will be surpassed and the experiment will be automatically stopped. This is not necessarily a problem since ΔT_ad can be obtained from other experiments as well (e.g. from DSC), and for this type of violent reactions, the temperature profile close to the onset temperature is far more important. Whether the final temperature would be 1000 or 700 °C is relatively unimportant, it will be a full‐blown thermal explosion anyhow. It should be kept in mind that the observed ΔT_ad should be multiplied with the φ‐factor in order to obtain the correct adiabatic temperature rise in case of a large‐scale runaway (Eq. 3.9).
2. Onset of the exotherm. Here as well, the term onset refers to the point where deviation from the baseline can be observed and is hence instrument dependent. In adiabatic calorimetry a detection threshold of 0.02 °C/min is often used, which corresponds to a 1.4 W/kg in case of an aqueous reaction medium. Referring to Table 3.9, we know that the natural heat loss of a 5000 L reactor at 80 °C and an ambient temperature of 20 °C is 1.6 W/kg. These two figures match quite closely, so the temperature at which the exotherm is detected in the adiabatic calorimeter with this sensitivity is most likely to be the temperature at which exothermicity will be first noticed under adiabatic conditions at large scale (at least for temperatures higher than 80 °C). If the onset temperature is well above the MTSR, the decomposition is unlikely to be triggered, even in case of a cooling failure during the synthesis reaction. If the onset temperature is close to the MTSR, a calculation of the TMR_ad should be made, as will be discussed further on.
3. Pressure profile. A careful analysis of the pressure profile should be made after each experiment. Very often, the pressure signal will be more sensitive to detect the start of a decomposition reaction than the temperature signal [19]. If a slow pressure increase during any of the search periods is observed, a decomposition reaction with gas evolution should be suspected. Some software packages allow for a direct overlay of the vapor pressure of any chosen solvent related to the sample temperature. This can be very indicative to discern between permanent gas formation and vapor pressure. If this is not possible, looking at the pressure profile during the wait and search period should yield the same information: if the pressure remains constant during this stage (when the sample is at a constant temperature), vapor pressure is the most important contribution to the overall pressure. If the pressure rises during this stage, formation of a permanent gas is most likely to happen. If the pressure drops at a certain point, a leak of the test cell has most probably occurred.
4. Self‐heat rate (temperature rise rate). The temperature rise rate of a runaway reaction can be calculated by taking the first derivative of the temperature versus time plot. Two things should be kept in mind when analyzing the SHR: firstly the dramatic influence of the φ‐factor on the SHR, as discussed above, and secondly the fact that the associated pressure rise rate can have far more serious consequences. So in case the SHR is low at any time (e.g. below 2°C/min) in a low‐φ experiment (φ < 1.2) and there is no strong pressure increase, the consequences of the runaway reaction are unlikely to be severe.
5. Pressure rise rate. The analysis of the pressure rise rate data is far from trivial. First thing to keep in mind is the influence the free headspace volume in the adiabatic calorimeter has on the finally observed pressure profile. If a permanent gas is being formed during the runaway reaction, the observed pressure increase will be considerably larger when the test has been performed with a small free headspace (e.g. a filling degree of 90% of the test cell) than in the case of a large free headspace (e.g. 50% filling degree). Therefore, it is advisable to use a ratio reaction mixture versus free headspace that is comparable with the situation at a large scale. As mentioned before, this is only relevant when dealing with the formation of a permanent gas, not when considering vapor pressure data. Moreover, as the pressure rise rate is in any case directly correlated to the temperature rise rate, the remarks made in the point above about the influence of the φ‐factor hold here as well. The pressure rise rate can be calculated back to a gas evolution rate (see also the Example Problem 3.3), and if the thus obtained gas flow is below the design limits of the installation under normal process conditions, no problems are to be expected. If this limit is exceeded to a limited extent, some operational issues can be suspected (limited condenser capacity, disturbed flow patterns in the venting line, etc.) without serious safety consequences. If the gas flow rate is considerably above this design limit, proper vent sizing calculations are needed to make sure that the emergency relief system is sufficient to cope with a runaway reaction. This point will be briefly discussed in a later paragraph.

Let us now consider a real‐life example of such an adiabatic experiment. In Figure 3.12, a heat–wait–search experiment of a relatively concentrated solution of dibenzoyl peroxide in chlorobenzene is shown. The experiment was conducted in glass test cell with a φ‐factor of 1.5.

A close inspection of the temperature profile indicates that the thermal activity starts already at 45 °C, but the detection threshold of 0.02 °C/min is only reached at 58 °C. At this temperature the instrument goes into tracking mode, and a max SHR of more than 40 °C/min is reached after 280 minutes. The observed adiabatic temperature rise is 140 °C, but in reality it will be higher since the run was aborted at 200 °C in order to prevent leakage of the silicone septa used to seal the glass test cell. An overlay of the vapor pressure curve with the pressure profile shows that a large part of the observed pressure is due to gas evolution of the decomposition reaction. The maximum pressure rise rate is very high, more than 200 bar/min. Keeping in mind that this experiment was conducted at a relatively high φ‐factor, it is clear that the severity of this runaway is totally unacceptable for introduction at a large scale. An obvious safety advice would be to investigate the use of a more dilute solution of this compound, or to turn to other, more stable reagents.

3.5.4.5 Using the Boiling Point as a Safety Barrier

When a runaway reaction takes place in a reactor, the temperature of the reaction mixture can reach the boiling point. This can either be an extra risk that needs to be taken into account or it can act as an efficient safety barrier [20].

If the heat rate under adiabatic conditions at the boiling point is low, part of the solvent will be evaporated, but the temperature will remain constant, and this will temper the runaway reaction. When the detected onset temperature in the adiabatic calorimeter is close to the boiling point, this can be a very effective safety barrier.

If the heat rate at the boiling point is relatively high, other effects might come into play:

1. Evaporation of the solvent. If the boiling point is reached, part of the solvent will start to evaporate. This in itself will have a cooling effect, but when the vapor is no longer condensed and returned to the reactor, the reaction mixture will become more concentrated. This in turn will lead to an increased reaction rate and also to an increased boiling temperature. This should be taken into account when relying on the reflux barrier as a basis of safety.
2. Swelling of the reaction mass. If the reaction mixture starts to boil vigorously, a kind of “champagne effect” may take place, leading to an increase of the volume of the reaction mass. In such a case the reactor content might even be forced out of the reactor into the condenser and scrubber lines.
3. Flooding of the vapor line. This effect will occur particularly when countercurrent condensers are used (i.e. the vapor and the condensate flow in opposite direction through the condenser). If the vapor flow through the condenser is too high, this flow will prevent the condensed liquid to flow back into the reactor, and this liquid will be carried along with the vapor flow. Here as well, solvent will enter the scrubber lines with all possible problems associated with it.

Provided that all of the abovementioned factors are taken into account, the reflux barrier can be used as a very effective safety barrier. For a more quantitative description, the reader is referred to the literature [3].

3.5.5.1 Determination from 1 DSC Run

A first approximation of the temperature at which the TMR_ad is 24 hours (we will call this temperature TMR_ad,24h) can be made from one single DSC run. A full discussion of the theory behind this approach is out of scope here, so the reader is referred to the original publication by Keller et al. [18]. We will however point out the basic concept with an example.

The idea behind this approach is quite simple: measure the heat release at one temperature, and assume that the reaction follows zero‐order reaction kinetics with a low activation energy of 50 kJ/mol. From this, the heat release at any temperature and hence the TMR_ad,24h can be calculated. The assumptions on which this approach is based and the practical calculations are discussed below.

1. Zero‐order assumption. A “classical” behavior for a reaction is that it follows nth‐order Arrhenius kinetics. This means that the reaction rate increases with concentration and temperature. This implies that the reaction rate will decrease over time when the reaction temperature is constant (see Figure 3.8) as the concentration of the reagents drops with increasing conversion. In a reaction that follows zero‐order kinetics, however, the reaction rate is independent of the concentration. This means that the reaction rate remains constant at a given temperature from the start (0% conversion) until the end (full conversion) of the reaction. Assuming this type of reaction kinetics leaves out the concentration dependence and makes the calculations a lot easier. It should be kept in mind however that this is an assumption; in reality the reaction is most likely not going to follow these kinetics. It is however a “safe” assumption, since the reaction rate will be overestimated as the decrease in reaction rate with decreasing reagent concentration is neglected. The fact that this is indeed a “worst‐case” assumption is discussed thoroughly in the original publication.
2. Low activation energy. The dependence of the reaction rate on the temperature is dictated by the activation energy: the decrease in reaction rate when lowering the reaction temperature is more pronounced for a reaction with high activation energy than for a reaction with a low activation energy. Since we will try to extrapolate the reaction rate at lower temperatures from one single measurement at one temperature, the activation energy needs to be known or estimated. A correct determination of the activation energy is possible by means of DSC, but not always straightforward. Therefore, the activation energy is assumed to be 50 kJ/mol, which is very low for organic reactions and decompositions. Choosing a low activation energy will again be on the safe side, since it will tend to overestimate the reaction rate at lower temperatures.
3. Determination of the reaction rate (heat rate) at one temperature. A correct determination of the heat rate at one temperature is needed, preferably in a relatively early stage of the reaction since this will minimize the error introduced by the zero‐order assumption. The heat signal should be well separated from the baseline, however, in order to obtain an accurate signal. Keller et al. suggest to search for the temperature at which the heat rate is 20 W/kg. This is a sensitivity that is well within reach of any decent DSC apparatus. In the example shown in Figure 3.13, this heat rate is observed at 111 °C.
4. The heat rate at other temperatures can now be calculated according to Eq. (3.10):

(3.10)

where

• q₀: heat rate at the new temperature
• q_onset: heat rate at the onset temperature (in this case 20 W/kg)
• E_a: activation energy (50 kJ/mol)
• R: universal gas constant (8.31 J/mol K)
• T_onset: onset temperature (in this case 111 °C or 384 K)
• T₀: new temperature at which the heat rate is to be calculated

1. From this, the TMR_ad can be calculated for any temperature according to

Eq. (3.11):

(3.11)

If this calculation is performed for a number of temperatures, TMR_ad,24h can be determined, as shown in Table 3.11.

In this example (using a c_p of 2 kJ/kgK), the TMR_ad,24h is estimated to be 31 °C. Another interesting point we can learn from this table is that a heat release of only 10 W/kg corresponds to a TRM_ad of less than one hour!

This example shows how the TMR_ad,24h can be extrapolated from only one DSC experiment. Because of the assumptions made, this will only be a rough estimate that can differ considerably from the true TMR_ad,24h. The merit of this method however lies in the fact that all assumptions are on the safe (conservative) side. If the thus obtained TMR_ad at the MTSR is longer than 24 hours, no further testing is needed. If it is shorter than 24 hours, a more accurate determination of the TMR_ad,24h might be needed, as will be discussed below.

One final remark is needed about autocatalytic reactions. Since the thermal history of a sample is so important in the characterization of autocatalytic (decomposition) reactions, their TMR_ad is much harder to determine. The method described here should therefore not be used for this type of reactions, and more elaborate adiabatic testing will be needed.

3.5.5.2 TMR_ad from 1 Adiabatic Experiment

Probably the best way to determine the TMR_ad,24h accurately is to perform a number of adiabatic experiments in a low‐φ test cell, each at a different starting temperature, and then determine the TMR for each of these experiments until the temperature at which this TMR is 24 hours is found. Obviously, this will be a very time‐consuming procedure, and better alternatives are to be sought for. It would be beneficial if we could extract a reliable TMR_ad,24h from one single adiabatic heat–wait–search experiment at a somewhat higher φ‐factor. This way, we would run an experiment under adiabatic conditions, which is closer to the real situation in a vessel during a runaway reaction than a scanning DSC experiment. Also, if we can run in HWS mode, the experimental time will be reduced significantly, and running at a higher φ‐factor (e.g. 1.5) is experimentally easier than at a φ‐factor close to unity. The question is how to extrapolate the data from this single adiabatic experiment to other temperatures and other φ‐factors.

In one of the classical papers about adiabatic calorimetry, Townsend and Tou [21] evaluated in great detail the analysis of experimental adiabatic data. In this article, they present a method to extrapolate the experimental TMR to other temperatures and other φ‐factors. A full description of the kinetic evaluation made in the original paper is out of scope here, but the general concepts and the practical use of this approach are discussed below.

We start from an adiabatic experiment, where we can determine the TMR at the onset temperature directly from the temperature versus time plot. The first extrapolation to be made is from the experimental φ‐factor (i.e. >1) to the “ideal” case of φ = 1. Townsend and Tou state that for most relevant decomposition reaction with a high activation energy, the TMR scales linearly with φ (Eq. 3.12):

(3.12)

Moreover, a method is described to extrapolate TMR data to lower temperatures as well. Assuming the reaction follows zero‐order kinetics, it can be shown that there is a linear correlation between the logarithm of the TMR and the inverse temperature according to Eq. 3.13:

(3.13)

Hence plotting the logarithm of the TMR versus the inverse temperature will yield a straight line with a slope proportional to the activation energy and the intercept being equal to the logarithm of the frequency factor.

Using these two equations, the TMR_ad,24h can be determined if the TMR_ad is known at a number of different temperatures. Since the approach is partly based on zero‐order assumptions, it is important to focus on the early part of the exotherm, since the influence of a decrease in concentration due to conversion can be neglected there. This way, the experimentally obtained TMR_ad at the onset temperature and at a couple of temperatures that are slightly higher are determined. These values are then corrected for the experimental φ‐factor according to Eq. (3.12). Finally a straight line is fitted through the plot of ln(TMR_ad) versus 1/T, and the point at which the TMR_ad is equal to 24 hours can be read from the graph. This approach is illustrated in the example below.

Let us consider the adiabatic experiment as represented in Figure 3.12. The φ‐factor used in this experiment was 1.5. The onset of the exothermicity is detected at 58 °C, 146 minutes after the start of the experiment. The maximum rate is reached after 429 minutes; hence the TMR_ad at this temperature is 283 minutes. We will consider the TMR at five different points in the early part of the exotherm, i.e. at 58, 60, 62, 64, and 66 °C. The corresponding TMR_ad at the experimental φ‐factor and at φ = 1 are calculated, as shown in Table 3.12.

The according plot based on these data is shown in Figure 3.14. It can be seen that the correlation is indeed linear for both experimental and φ‐corrected data points. From the graph, the TMR_ad,24h can be obtained directly. In this case, the TMR will be 24 hours at 38.3 °C.

The advantage of this method over the above method based on one DSC measurement is the increased accuracy. The reason for this can be found in different aspects of this approach:

1. This method is based on an adiabatic experiment that will be by definition more representative for the situation in a large‐scale reactor during a runaway reaction than a DSC experiment.
2. The temperature range over which the extrapolation takes place is fairly limited: in our example the difference between the experimental onset temperature and the finally obtained TMR_ad,24h is only 20 °C (as compared with 80 °C in the DSC example).
3. The extrapolation for the φ‐correction is also limited (from φ = 1.5 to φ = 1).
4. By using only data points in the early part of the reaction, the error introduced by assuming zero‐order kinetics is limited. Indeed, in this early stage with low conversion, the concentration of the reagents can be assumed to be constant.

Also for this method of determining TMR_ad,24h, a word of caution is needed. Only the data points in the early part of the exotherm are used; consequently we are dealing with very low heat rates at that moment. This poses high demands on the quality of the experimental data: a small amount of drift in the temperature stability of the instrument (either positive or negative drift) can have a profound effect on the final result. It is our experience that a well‐operated adiabatic calorimeter should be able to deliver reliable results, provided that regular drift checks on empty test cells are performed to confirm the stability of the instrument.

Here as well, extra attention is needed when dealing with strongly autocatalytic reactions, since this method might overestimate the TMR_ad for that kind of reactions, leading to unsafe extrapolations. When a very sudden and sharp temperature increase is noticed in an adiabatic experiment, autocatalysis should be suspected, and more testing will be appropriate. When in doubt, an iso‐aging experiment at a temperature close to the calculated TMR_ad,24h should be conducted to check the validity of the calculations.

3.5.5.3 Kinetic Modeling

Both of the abovementioned methods make it possible to extract the TMR_ad,24h from one single experiment but with a limited accuracy due to the assumptions made. There are more advanced methods available as well, which ask for a larger number of experiments and more advanced mathematical models, but which lead to more accurate description of the reaction and allow for a broader range of simulations.

One possible approach to kinetic modeling is the fully mechanistic description of the reaction, which implies a complete understanding of the (decomposition) reaction at a molecular level. The reaction is therefore split up into its elementary reactions, and for each of those the frequency factor, reaction order and activation energy are determined. It goes without saying that not only is this method quite elaborate from an experimental and computational point of view, but it also enables the widest range of process conditions that can be simulated (different concentrations, temperatures, φ‐factor, etc.) [22, 23]. This type of modeling is usually applied to the desired chemical reaction and less often to the undesired decomposition reactions.

Another possibility is the so‐called nonparametric kinetic modeling. In this approach, a general kinetic model of the (decomposition) reaction is constructed based on a number (usually 5) of DSC experiments with different heating rates [24, 25]. This kinetic description is said to be model free, meaning that there are no explicit assumptions being made about the reaction type. This type of modeling can lead to an accurate description of the reaction and enables the simulation of any temperature profile for the sample studied. With advancements both in readily available computing power and in fundamental understanding of the underlying theoretical kinetic models, this type of modeling is becoming more widespread and has even reached a point where some large companies tend to prefer modeling over adiabatic testing. It is worth pointing out however that, in order to guarantee a safe process development, the combination of data from different sources (modeling and experimental isothermal, non‐isothermal, and adiabatic data) is the preferred way to get to a fundamental understanding of a process during intended, non‐intended, and storage conditions.

The reader is referred to the literature for a more detailed discussion of the different possibilities of kinetic modeling for both safety studies and process development [23, 24, 26–28].