Chapter 8. Chemical Reactivity

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Chapter 8. Chemical Reactivity

The learning objectives for this chapter are to:

Identify processes where a chemical reaction hazard may be present.
Understand a chemical compatibility matrix.
Understand the important questions for any chemical reaction.
Be aware of the various calorimeters and procedures used to characterize chemical reactions.
Apply the theoretical model used to analyze calorimeter data.

Reactive chemical hazards have resulted in many accidents in industrial operations and laboratories. Preventing reactive chemical accidents requires the following steps, which are discussed in this chapter:

Background understanding. This includes case histories and important definitions. Case histories provide an understanding of the consequences, frequency, and breadth of reactive chemical accidents. Definitions provide a common fundamental basis for understanding. This topic is presented in Section 8-1, and the discussion there is supplemented by Chapter 14, “Case Histories and Lessons Learned.”
Commitment, awareness, and identification of reactive chemical hazards. This is achieved through proper management and the application of several methods to identify reactive chemical hazards. This topic is discussed in Section 8-2.
Characterization of reactive chemical hazards. A calorimeter is typically used to acquire reaction data, and a fundamental model is used to estimate important parameters to characterize the reaction. This characterization is described in Section 8-3.
Control of reactive chemical hazards. This includes application of inherent, passive, active, and procedural design principles. This topic is discussed in Section 8-4, and the discussion there is supplemented by additional material in Chapter 9: “Introduction to Reliefs”; Chapter 10: “Relief Sizing”; and Chapter 13: “Safety Strategies, Procedures, and Designs.”

8-1 Background Understanding

In October 2002, the U.S. Chemical Safety and Hazard Investigation Board (CSB) issued a report on reactive chemical hazards.¹ The CSB analyzed 167 serious accidents in the United States involving reactive chemicals from January 1980 through June 2001. Forty-eight of these accidents resulted in a total of 108 fatalities, or an average of 5 fatalities per year. The report authors concluded that reactive chemical incidents are a significant safety problem. They recommended that awareness of chemical reactive hazards be improved for both chemical companies and other companies that use chemicals. They also suggested that additional resources be provided so that these hazards can be identified and controlled.

¹Improving Reactive Hazard Management (Washington, DC: U.S. Chemical Safety and Hazard Investigation Board, October 2002).

On December 19, 2007, an explosion occurred at T2 Laboratories in Jacksonville, Florida, killing 4 people and injuring 32. The facility was producing a chemical product to be used as a gasoline octane additive. The explosion was caused by the bursting of a large reactor vessel due to a runaway reaction. A runaway reaction occurs when the process is unable to remove adequate energy from the reactor to control the temperature. The reactor temperature subsequently increases, resulting in a higher reaction rate and an even faster rate of heat generation. Large commercial reactors can achieve heating rates of several hundred degrees Celsius per minute during a runaway.

The CSB investigated the T2 Laboratories accident and found that the company engineers did not recognize the runaway hazards associated with this chemistry and process and were unable to provide adequate controls and safeguards to prevent the accident. Even though the engineers were degreed chemical engineers, they did not have any instruction on reactive hazards. To remedy this omission, the CSB recommended adding reactive hazard awareness to chemical engineering curriculum requirements.

A chemical reactivity hazard is “a situation with the potential for an uncontrolled chemical reaction that can result directly or indirectly in serious harm to people, property or the environment.”² The resulting reaction may be very violent, releasing large quantities of energy and possibly large quantities of toxic, corrosive, or flammable gases, liquids, or solids. If this reaction is confined in a container, the pressure within the container may increase very quickly, eventually exceeding the pressure capability of the container, resulting in an explosion. The reaction may occur with a single chemical, called a self-reacting chemical (e.g., monomer), or with another chemical, called either a chemical interaction or incompatibility.

²R. W. Johnson, S. W. Rudy, and S. D. Unwin. Essential Practices for Managing Chemical Reactivity Hazards (New York, NY: AICHE Center for Chemical Process Safety, 2003).

Note that the hazard is due to the potential for a chemical reaction. Something else, such as a process upset, must occur for this hazard to actually result in an accident. However, as long as reactive chemicals are stored and used in a facility, the reactive chemical hazard is always present.

One difficulty associated with reactive chemicals hazards is that they are difficult to predict and identify. Common materials that we use routinely by themselves with apparent negligible hazard may react violently when mixed with other common materials or react violently when the temperature or pressure is changed.

Chemical reactivity hazards are likely to be found with the following operations:

Intentional chemical operations, where chemicals react by design, including chemical plants.
Mixing and physical processing, which may include the following operations: combine, formulate, crush, blend, screen, dry, distill, absorb, heat, dilute, and so on.
Storage, handling, and repackaging, which does not include mixing, but may include warehousing, tank storage, dividing, and other processes.

Note that many of these operations do not occur in a traditional chemical plant.

According to the U.S. Chemical Safety Board, only 25% of reactive chemicals incidents occur in chemical reactors, while 75% occur in piping where reactions were not expected. An estimated 90% of reactive chemical incidents could have been prevented with the application of information already available to the public.

Raw materials, process streams, products, and wastes in any process must be reviewed and evaluated to determine if any potential reactive chemical hazards are possible. If sufficient data are not available, the chemicals involved should be subjected to screening evaluations or laboratory tests. This must include the intended chemistry, unintentional reactions (for example, polymerizations and decompositions), and reactions that may occur from inadvertent mixing of incompatible materials (including waste streams).

Table 8-1 shows the key steps in reactivity hazard evaluation. The first step is to compile available reactivity information, from both corporate and outside sources. The second step is to complete an initial screening of hazards, as detailed in Section 8-3. The third step is to assess other potential hazards, as shown in Table 8-2. The fourth step is to evaluate process upsets and other operating conditions and use already developed kinetic models and other information to estimate the consequences of these upsets. Finally, you must conduct a comprehensive and rigorous review of the hazards and safeguards to ensure that the reactive hazards have been addressed properly.

Table 8-1 Key Steps in Reactivity Hazard Evaluation

Compile available reactivity information:
1. Existing reactive chemicals databases within company
2. Open literature (see Table 8-6)
3. Estimation from heats of formation (see Section 8-2)
4. Calorimetry data (see Section 8-3)
5. Development of complete kinetic models (see Section 8-3)
Complete initial screening of hazards (see Section 8-3):
1. Heat of reaction per mass
2. Maximum temperature and pressure rates
3. Maximum reaction temperature and pressure
4. Detected onset temperature
5. Temperature of no return
6. Time to maximum rate
Assessment of other potential hazards to determine if additional testing is required (see Table 8-2).
Evaluation of process upsets and safe operating conditions using rate information and kinetic models, as needed1
Review of hazards and safeguards by a multidisciplinary group as part of the overall risk management program

Source: Dow Chemical Company. AICHE Faculty Workshop, Freeport, TX, June 2017.

Table 8-2 Reactive Hazards: Other Considerations

Accidental mixing, which is a very common issue
Process upsets, including increased temperature, which may result in an unknown reaction
Endothermic reactions, which may be a hazard due to gaseous product generation
Any change in process or operating conditions
Different stoichiometries from the intentional chemistry1
Contamination by another chemical or substance
Change in scale, particularly with increasing size or quantities1
Change in mixing, either increased or decreased
Autocatalytic reactions, which may appear stable but may become reactive after an induction time; this includes inhibited polymers
Reactions with maximum pressures that may exceed the design pressure limit of equipment
Many others….

8-2 Commitment, Awareness, and Identification of Reactive Chemical Hazards

The first step in this process is to commit to manage reactive chemical hazards properly. This requires commitment from all employees, especially management, to properly identify and manage these hazards throughout the entire life cycle of a process. Necessary elements of this process include laboratory research and development; pilot plant studies; and plant design, construction, operation, maintenance, expansion, and decommissioning.

Figure 8-1 is a flowchart useful for preliminary screening for reactive chemicals hazards. This figure contains seven questions to determine if reactive chemical hazards are present.

A flowchart with seven questions reveals the presence of reactive chemical hazards. — **Figure 8-1** Screening flowchart for reactive chemical hazards. An answer of “yes” at any decision point moves more toward reactive chemistry. See Section 8-2 for more details. (Source: R. W. Johnson, S. W. Rudy, and S. D. Unwin. *Essential Practices for Managing Chemical Reactivity Hazards* (New York, NY: AICHE Center for Chemical Process Safety, 2003).)

Preliminary screening of reactive chemical hazards is done using a flowchart with seven questions. The questions are as follows: Is intentional chemistry done at your facility? Any other chemistry besides combustion with air? Any mixing or combining of substances? Any other physical processing, Heat generated during mixing or processing? Any hazardous substances stored or handled? and Any specific reactive hazards? If the answers to the questions are ''Yes,'' the flowchart leads to the conclusion, Chemical reactivity hazards expected, else if the answers are ''No'' the flowchart leads to the statement, Chemical reactivity hazards not likely.

Is intentional chemistry performed at your facility? In most cases this is easy to determine. The bottom line is: Are the products that come out of your facility in a different molecular configuration from the raw materials? A precise answer to this question is required prior to moving forward in the flowchart.
Is there any mixing or combining of different substances? If substances are mixed or combined, or even dissolved in a liquid or water, then it is possible that a reaction, either intended or unintended, may occur.
Does any other physical processing of substances occur in your facility? This could include size reduction, heating/drying, absorption, distillation, screening, storage, warehousing, repackaging, and shipping and receiving.
Are there any hazardous substances stored or handled at your facility? The Safety Data Sheet (SDS) is a good source of information here.
Is combustion with air the only chemistry intended at your facility? This includes combustion of common fuels such as natural gas, propane, and fuel oil. Combustion is a special reactive hazard that is handled by separate codes and standards and is not addressed here.
Is any heat generated during the mixing, phase separation, or physical processing of substances? Heat generation when chemicals are mixed is a prime indication that a reaction is taking place. Note that many chemicals do not release much heat during the reaction; thus, even if limited heat is released, a chemical reaction may still be occurring. Some physical heat effects, such as absorption or mechanical mixing, can also cause heat generation. This heat release, even though not caused by chemical reaction, may increase the temperature and cause a chemical reaction to occur.
Are there any specific reaction hazards that occur? Specific reaction hazards are shown in Table 8-3 with detailed lists of chemical categories and chemicals provided in Appendix D. Functional groups that are typically associated with reactive chemistry are shown in Table 8-4.

Table 8-3 Specific Reactive Chemical Hazards

Pyrophoric and spontaneously combustible: Readily react with the oxygen in the atmosphere, igniting and burning even without an ignition source. Ignition may be immediate or delayed.

Identification: SDS or labeling identifies this as “spontaneously combustible or pyrophoric.”

NFPA Flammability rating of 4.

DOT/UN Hazard Class 4.2 (spontaneously combustible solids).

Examples: Aluminum alkyl, Grignard reagent, finely divided metals, iron sulfide, triethyl aluminum.

See Table D-1 in Appendix D.

Peroxide-forming: React with oxygen in the atmosphere to form unstable peroxides.

Identification: Not easily identified as a peroxide former from SDS or other resources.

Examples: 1,3-Butadiene, 1,1-dichloro-ethylene, isopropyl and other ethers, alkali metals.

See Table D-2.

Water-reactive chemicals: Chemically react with water, particularly at normal ambient conditions.

Identification: Usually identified as water-reactive on SDS.

May be identified as DOT/UN Hazard Class 4.3 (dangerous when wet).

May be labeled as “dangerous when wet.”

NFPA Special Rating with symbol.

Examples: Sodium, titanium tetrachloride, boron trifluoride, acetic anhydride.

See Tables D-3 and D-4.

Oxidizers: Readily yield oxygen or other oxidizing gas, or readily react to promote or initiate combustion of combustible materials.

Identification: Identified as an oxidizer on the SDS.

DOT/UN Hazard Class 5.1 (oxidizing agent) or other rating groups.

NFPA Special Rating with symbol OX.

Examples: Chlorine, hydrogen peroxide, nitric acid, ammonium nitrate, ozone, hypochlorites,

benzyl peroxide.

See Table D-5.

Self-Reactive: Self-react, often with accelerating or explosive rapidity.

Identification: Generally identified on SDS or labeling as “self-reactive.”

NFPA stability or reactivity rating of 1 or higher.

Polymerizing: Monomers combining together to form very large, chain-like or cross-linked polymer molecules.

Examples: Acrolein, ethylene and propylene oxide, styrene, vinyl acetate. See Table 8-6.

Shock-sensitive: React on impact.

Example: Picric acid.

Thermally decomposing: Large molecules breaking into smaller, more stable molecules.

Rearranging: Atoms in the molecule rearranging into a different molecular structure, such as a different isomer.

Incompatible materials: Incompatible materials contacting each other.

Examples: Ammonia–methacrylic acid; caustic soda- epichlorohydrin; acids–bases.

Note: See R. W. Johnson, S. W. Rudy, and S. D. Unwin. Essential Practices for Managing Chemical Reactivity Hazards (New York, NY: AICHE Center for Chemical Process Safety, 2003), for additional detail on these classifications, as well as Appendix D for more detailed lists of these materials.

Table 8-4 Reactive Functional Groups

Azide	N₃
Diazo	−N=N−
Diazonium	−N⁺X^-
Nitro	−NO₂
Nitroso	−NO
Nitrite	−ONO
Nitrate	−ONO₂
Fulminate	−ONC
Peroxide	−O−O−
Peracid	−CO₃H
Hydroperoxide	−O−O−H
Ozonide	O₃
N-haloamine
Amine oxide
Hypohalites	−OX
Chlorates	CIO₃
Acetylides of heavy metals

Source: Conrad Schuerch. “Safe Practice in the Chemistry Laboratory: A Safety Manual.” In Safety in the Chemical Laboratory, vol. 3, ed. Norman V. Steere (Easton, PA: Division of Chemical Education, American Chemical Society).

Among the most difficult reactive chemical hazards to characterize are incompatible chemicals, shown at the bottom of Table 8-3. Common materials that we use routinely and safely by themselves may become highly reactive when mixed. These materials may react very quickly, possibly producing large amounts of heat and gas. The gas may be toxic or flammable.

The easiest way to show graphically the various interactions between chemicals is a chemical compatibility matrix, like that shown in Table 8-5. The chemicals are listed on the left-hand column of the table and also across the top. The chemicals selected may be all of the chemicals in a facility, or just the chemicals that may come in contact with each other during routine or emergency situations. Clearly, listing all the chemicals provides a conservative result, but may result in a large and unwieldy matrix.

Each entry in the chemical compatibility matrix shows the interaction between two chemicals in the table. Thus, the entry just to the right of hydrochloric acid in Table 8-5 represents the binary interaction between acetic anhydride and hydrochloric acid solution.

Table 8-5 Chemical Compatibility Matrix and Hazards, as Predicted by CRW4.02

Chemical	Acetic anhydride	Hydrochloric acid solution	Methanol	Sodium hydroxide, solid
Acetic anhydride
Hydrochloric acid solution	N
Methanol	C	C
Sodium hydroxide, solid	N	N	N
solid
Water	N	C	Y	C

Key: N: Not compatible

C: Caution

Y: Compatible

The chemical compatibility matrix considers only binary interactions between two chemicals. Binary interactions would be expected during routine operations, while combinations of several chemicals may occur during emergency situations. However, once the hazards are identified using the binary interactions of all chemicals, additional hazards due to combinations of more than two chemicals are unlikely.

Individual Chemical Hazards and Functional Groups

Chemical	Reactive hazard	Functional group
Acetic anhydride	Water-reactive	Anhydride
Caustic soda, beads	Water-reactive	Base
Hydrochloric acid solution	Mildly air-reactive	Acid, inorganic, non-oxidizing
Methanol	Highly flammable	Alcohol

Once the chemicals are identified, the binary interactions are filled in. Information for these interactions can be obtained from a variety of sources, as shown in Table 8-6. Perhaps the easiest source to use is the Chemical Reactivity Worksheet (CRW).³ This worksheet is provided free of charge by the Office of Emergency Management of the U.S. Environmental Protection Agency (EPA), the Emergency Response Division of the National Oceanic and Atmospheric Administration (NOAA), and the AICHE Center for Chemical Process Safety. The software contains a library of more than 5000 common chemicals and mixtures and considers 43 different organic and inorganic reactive groups. The CRW also provides information on the hazards associated with specific chemicals as well as the reactive group(s) associated with those chemicals, as shown at the bottom of Table 8-5. Because the CRW tends to be conservative in its predictions of binary interactions, the results must be carefully interpreted. Also, the CRW program provides chemical compatibility information only at room temperature and pressure; thus, care must be taken in applying CRW results at different temperatures and pressures.

³Lewis E. Johnson and James K. Farr. “CRW 2.0: A Representative-Compound Approach to Functionality-Based Prediction of Chemical Hazards.” Process Safety Progress 27, no. 3 (2008): 212–218.

Table 8-6 Sources of Information on Chemical Reactivity Hazards

Source	Location
Your company database	Unique to company
Safety Data Sheet (SDS)	Provided by chemical manufacturer or on the Web.
Chemical Reactivity Worksheet (CRW)	National Oceanic and Atmospheric Administration (NOAA) http://response.restoration.noaa.gov/
Brethericks Handbook of Reactive	Elsevier Publishers
Chemical Hazards, 8th ed. P. Urben, ed. (2017)	www.elsevier.com
Sax’s Dangerous Properties of Industrial	John Wiley & Sons
Materials, 12th ed. R. J. Lewis, ed. (2012)	www.wiley.com
Sigma Aldrich Library of Chemical	Sigma-Aldrich
Safety Data, R. E. Lenga, ed. (1988)	www.sigmaaldrich.com
Fire Protection Guide to Hazardous	National Fire Protection Association (NFPA)
Materials (2010)	www.nfpa.org
CHETAH: Computer Program for	American Society for Testing and Materials (ASTM)
Chemical Thermodynamics and	www.astm.org
Energy Release Evaluation

Another source of information on reactive chemicals is a program called CHETAH (Chemical Thermodynamics and Energy Release Evaluation). This program is able to predict the reactive chemical hazards using heats of formation, Gibbs energy, and functional groups. CHETAH was originally developed by Dow Chemical and is very useful as an initial screening tool for reactive chemical hazards.

Example 8-1

A laboratory contains the following chemicals: hydrochloric acid solution, acetic anhydride, methanol, and caustic soda (NaOH) beads. Draw a chemical compatibility matrix for these chemicals. What are the major hazards associated with the chemicals?

Solution

The chemicals are entered into the CRW software; the chemical compatibility matrix is shown in Table 8-5. Table 8-5 also lists the major hazard and reactive groups associated with each chemical.

Note that water must be added to the mixture since the hydrochloric acid solution contains water.

From Table 8-5, it is clear that the mixing of any of these chemicals will result in a reactive hazard situation! All of these chemicals must be stored separately, and management systems must be enforced to ensure that accidental mixing does not occur.

Three combinations (hydrochloric acid + caustic soda; acetic anhydride + methanol; methanol + caustic soda) liberate a gaseous product, at least one of which is flammable. Two combinations (hydrochloric acid + acetic anhydride; hydrochloric acid + caustic soda) result in an intense or violent reaction. One combination (acetic anhydride + caustic soda) results in a reaction with explosive violence and/or forms explosive products. From the individual hazards table at the bottom of Table 8-5, two of the chemicals (acetic anhydride and caustic soda) are water-reactive, one chemical (hydrochloric acid solution) is mildly air-reactive, and methanol is flammable. The prediction of air reactivity with the hydrochloric acid solution is probably a bit conservative.

All personnel using these chemicals in the laboratory must be aware of both the individual chemical hazards and the reactive hazards that result when these chemicals are mixed.

8-3 Characterization of Reactive Chemical Hazards Using Calorimeters

Chemical plants produce products using a variety of complex reactive chemistries. It is essential that the behavior of these reactions be well characterized prior to using these chemicals in large commercial reactors. Calorimeter analysis is important to understand both the desired reactions and any unintended reactions.

Table 8-7 lists some of the important questions that must be asked to characterize reactive chemicals. Answering these questions is necessary to design control systems to remove heat from the reaction to prevent a runaway; to design safety systems, such as a reactor relief, to protect the reactor from the effects of high pressure (see Chapters 9 and 10); and to understand the rate at which these processes occur. The answers must be provided at conditions as close as possible to actual process conditions.

Table 8-7 Important Questions for the Characterization of Reactive Chemicals

At what temperature does the reaction rate become large enough for adequate energy to be produced for heating of the reaction mixture to be detected?
What is the maximum temperature increase due to adiabatic self-heating of the reactants?
What is the maximum self-heat rate? At what time and temperature does this occur?
What is the maximum pressure during the reaction? Is this pressure due to the vapor pressure of the liquid, or due to the generation of gaseous reaction products?
What is the maximum pressure rate? At what time and temperature does this occur?
Are there any other side reactions that occur, particularly at temperatures higher than the normal reaction temperature? If so, can questions 1 through 5 be answered for these side reactions?
Can the heat generation from the desired chemistry heat the reaction mass under adiabatic conditions to a temperature at which another reaction occurs?
Can the heat generated by chemical reaction (desired or undesired) exceed the capability of the vessel/process to remove heat? At what temperature does this occur?

For exothermic reactions, heat is lost through the walls of the reactor vessel to the surroundings. The higher these heat losses, the lower the temperature inside the reactor. Conversely, the lower the heat losses, the higher the temperature within the reactor. We can thus conclude that we will reach the highest reaction temperatures and highest self-heating rates when the reactor has no heat losses—that is, when the process is adiabatic.

The heat losses through the walls of the reactor are proportional to the surface area of the reactor vessel. As the vessel becomes larger, the surface-to-volume ratio becomes smaller and the heat losses through the walls have a smaller effect. Thus, as the vessel becomes larger, the behavior of the vessel approaches adiabatic behavior.

Many chemical plant personnel believe that a large reactor will self-heat at a rate that is much slower than that for a much smaller vessel. In reality, the reverse is true: A larger reactor vessel will approach adiabatic conditions and the self-heat rates will be significantly faster.

Heat removal rates do not scale linearly with increased reactor volume. This scaling problem has been the cause of many incidents. Reactions tested in the laboratory or pilot plant frequently showed slow self-heat rates that were controlled easily using ice baths or small cooling coils. However, when these reactions are scaled up to large commercial reactors, sometimes having volumes of 20,000 gallons or more, the self-heat rates are sometimes orders of magnitude higher, resulting in an uncontrollable temperature increase and explosion of the reaction vessel. Large commercial reactors with energetic chemicals such as acrylic acid, ethylene or propylene oxide, and many others can achieve self-heat rates as high as hundreds of degrees Celsius per minute!

Introduction to Reactive Hazards Calorimetry

The idea behind the calorimeter is to safely use small quantities of material in the laboratory to answer the questions listed in Table 8-7. Most of the calorimeters discussed in this chapter have test volumes from a few milliliters to as large as 130 mL. Larger test volumes more closely match industrial reactors, but also increase the hazards associated with the laboratory test.

The calorimeter technology presented here was developed mostly in the 1970s. Much of the early development was done by Dow Chemical,⁴,⁵ although numerous researchers have made significant contributions since then.

⁴D. L. Townsend and J. C. Tou. “Thermal Hazard Evaluation by an Accelerating Rate Calorimeter.” Thermochimica Acta 37 (1980): 1–30.

⁵U.S. Patent 4,439,048, March 27, 1984.

Table 8-8 summarizes the commonly used calorimeters that are available for reactive chemicals testing. All of the calorimeters hold the sample in a small sample cell. These devices have a means to heat the test sample and measure the temperature of the sample as a function of time. Most also have the capability to measure the pressure inside the closed sample cell.

Table 8-8 Types of Calorimeters Most Commonly Used to Study Reactive Chemicals

Calorimeter	Supplier	Type	Typical test vessel volume (mL)	Nominal phi factor (ϕ)	Adiabatic reaction tracking limits (k/min)	Operation time	Comments
Differential Scanning Calorimeter (DSC)	Various	Open	<1	Not applicable	Not applicable	1 hour	Used mostly for initial screening.
							Closed sample cells can be used.
Advanced Reactive System Screening Tool (ARSST)	Fauske and Associates www.fauske.com	Open	10	1.05	0.1–200	Hours	Used mostly for initial screening due to short operation time.
Accelerating Rate Calorimeter (ARC)	Various	Closed	10	1.5	0.04–20	1+ day	Most effective for reactions with low self-heat rates.
							Need to adjust temperatures for Φ factor.
Vent Sizing Package 2 (VSP2)	Fauske and Associates www.fauske.com	Closed	100	1.05	0.05–600	1+ day	Useful for reactions with very high self-heat rates.
							Can also be used to identify if two-phase flow occurs during relief discharge.
Automatic Pressure Tracking Adiabatic Calorimeter (APTAC)	NETZSCH www.netzsch.com	Closed	130	1.10	0.04–400	1+ day	Useful for reactions with high self-heat rates. Also has capability to automatically inject reactants and collect products.

The calorimeter should be as close as experimentally possible to adiabatic behavior to ensure that the results are representative of a large reactor. The adiabatic conditions will also ensure the “worst-case” results—that is, the highest temperature and pressure and the highest self-heat rates and pressure rates.

All of the calorimeters listed in Table 8-8 have two modes of operation. The most common mode, called the thermal scan mode, heats the sample at a constant temperature rate (for example, 2°C per minute) until the reaction rate becomes large enough for adequate energy to be produced for the calorimeter to detect the reaction heating. The Differential Scanning Calorimeter (DSC) and the Advanced Reactive System Screening Tool (ARSST) continue to heat beyond this temperature. All of the other calorimeters stop the heating once the reaction self-heating is detected, and the calorimeter then switches to an adiabatic mode by matching the sample temperature to the temperature external to the sample.

The other mode of heating is to heat the sample to a fixed temperature and then wait for a specified time to see if any reaction self-heating is detected. If self-heating is detected, the calorimeter is placed immediately in adiabatic mode. If no self-heating is detected after a specific time, the temperature is again incremented, and the process is repeated. This is called the heat–wait–search mode.

The thermal scan mode is typically used for reactive chemicals studies. The heat–wait–search mode is used for chemicals that have a long induction time, meaning they take a long time to react or may be autocatalytic. Several calorimeters—the Accelerating Rate Calorimeter (ARC), Vent Sizing Package (VSP2), and Automatic Pressure Tracking Adiabatic Calorimeter (APTAC)—have the capability to use both heating modes during the same run.

In adiabatic mode, the calorimeter attempts to adjust the heaters outside the sample cell to match the temperature inside the sample cell. This ensures that there is no heat flow from the sample cell to the surroundings, resulting in adiabatic behavior.

Calorimeters are classified as either open or closed. In an open calorimeter, the sample cell is either open to the atmosphere or open to a larger containment vessel. Two calorimeters in Table 8-8 (DSC and ARSST) are classified as open. For the traditional DSC, the sample cell is completely open to the atmosphere, and no pressure data are collected. Some DSCs have been modified to use sealed capillary tubes or sealed high-pressure metal holders. For the ARSST, the small sample cell (10 mL) is open to a much larger (350 mL) containment vessel. A pressure gauge is attached to the containment vessel, but only qualitative pressure data are collected from this gauge. The ARSST containment vessel is also pressurized with nitrogen during most runs to prevent the liquid sample from boiling so that higher reaction temperatures can be reached.

Since the ARC is a closed system, this calorimeter can also be used to determine the vapor pressure of the liquid mixture. This is done by first evacuating the ARC and then adding the test sample. The pressure measured by the ARC is then the vapor pressure, which can be measured as a function of temperature.

It is fairly easy to insulate a small sample cell to approach near-adiabatic conditions. However, the sample cell must be capable of withstanding reaction pressures that may be as high as hundreds of atmospheres. The easiest approach to achieve high-pressure capability is to use a thick-walled vessel capable of withstanding this pressure. The problem with this strategy is that the thick walls of the sample cell will absorb heat from the test sample, resulting in less than adiabatic conditions. The presence of a vessel that absorbs heat increases the thermal inertia of the test device and reduces both the maximum temperature and the maximum temperature rate.

The thermal inertia of the apparatus is represented by a phi factor, defined as

ϕ=Combined heat capacity of sample and containerHeat capacity of sample=1+Heat capacity of containerHeat capacity of sample(8-1) $\begin{matrix} \begin{matrix} ϕ & = \frac{Combined heat capacity of sample and container}{Heat capacity of sample} \\ = 1+ \frac{Heat capacity of container}{Heat capacity of sample} \end{matrix} & (8-1) \end{matrix}$

Clearly, a phi factor as close as possible to 1 represents less thermal inertia. Most large commercial reactors have a phi factor of approximately 1.1.

Table 8-8 lists the phi factors for the most common calorimeters. The ARC uses a thick-walled vessel to contain the reaction pressure; it has a high phi factor as a result. The ARSST uses a thin-walled glass sample cell that is open to the containment vessel. Since this glass vessel contains only the test sample and does not need to withstand the pressure developed by the reaction, it has a low phi factor.

The VSP2 and the APTAC use a unique control method to reduce the phi factor. Both of these calorimeters use thin-walled containers to reduce the heat capacity of the sample cell. Figure 8-2 shows a schematic of the VSP2 calorimeter. The sample is contained in a closed, thin-walled test cell, which is held in a containment vessel. The control system measures the pressure inside the test cell as well as the pressure inside the containment vessel. The control system is able to rapidly adjust the pressure inside the containment vessel to match the pressure inside the test cell. The pressure difference between the inside of the test cell and the containment device is kept as low as possible, usually less than 20 psi. As a result, the thin-walled sample cell does not rupture.

Schematic diagram of Vent Sizing Package (VSP2) Calorimeter. — **Figure 8-2** Vent Sizing Package (VSP2) showing the control system to equalize the pressure between the sample cell and the containment vessel.

Vent Sizing Package (VSP2) Calorimeter with its various components marked is shown. The sample is placed inside a thin walled closed test cell or sample container, which is insulated from the outer containment vessel. The capacity of the containment vessel is about 4000 cubic centimeters. A magnetic stirrer is placed inside the sample container and the container is heated using an adiabatic heater. Nitrogen is constantly supplied to the containment vessel as well as to the sample container. Two other valves are fixed, in which one is an exhaust valve and through the other valve, nitrogen is supplied to the containment vessel.

The adiabatic reaction tracking limits shown in Table 8-8 relate to the lower rate at which the self-heating is detected and the upper reaction rate that the calorimeter can follow. The operation time in Table 8-8 is an approximate idea of how long it takes to do a single, standard run on the apparatus. Some calorimeters (DSC and ARSST) have short run times; these devices are useful for completing numerous screening studies to get an initial idea of the reactive nature of the material before moving to a more capable calorimeter that has a longer operation time.

The DSC consists of two small sample cells, one containing the unknown sample and the other containing a reference material. The two samples are heated, and the DSC measures the difference in heat required to maintain the two samples at the same temperature during this heating. This apparatus can be used to determine the heat capacity of the unknown sample, the temperature at which a phase change occurs, the heat required for the phase change, and the heat changes due to a reaction. With a traditional DSC, the trays are open to the atmosphere, so this type of calorimeter is classified as an open type. The DSC can also be modified to use sealed glass capillary tubes or small metal containers. The DSC is mostly used for quick screening studies to identify if a reactive hazard is present and to get an initial idea of the temperatures at which the reaction occurs. The traditional DSC does not provide any pressure information since it is open. If sealed glass capillary tubes are used, limited pressure information can be obtained from the pressure at which the tube ruptures.

Figures 8-3 and 8-4 show typical temperature and pressure scans for an APTAC device. In Figure 8-3, the sample is heated at a constant temperature rate until an exotherm is detected. The calorimeter then switches to adiabatic mode, in which it measures the temperature of the sample and attempts to match the external temperature to the sample temperature. This ensures near-adiabatic behavior. During adiabatic mode, the sample self-heats by the energy released by the reaction. Eventually, the reactants are all consumed and the reaction terminates. Self-heating stops at this point and the temperature remains constant. At the end of the run, the calorimeter heating is turned off and the sample cools.

Two graphs for the APTAC data reaction are shown. — **Figure 8-3** APTAC data for the reaction of methanol and acetic anhydride.

The first graph shows an increasing curve. The horizontal axis represents "Time (in minutes)," ranging from 0 to 300, in increments of 50 and the vertical axis represents "Temperature (in degrees Celsius)" ranging from 0 to 200, in increments of 20. The curve rises from (0, 18), extends, and remains constant between (180, 170) to (200, 180), and slides down to (280, 60). The starting region of the curve is "Calorimeter heating," the increasing point is marked "Exotherm detected," the rising curve region is marked "Adiabatic mode," the starting point of the constant value is marked, "reaction finished," the end point of the constant value is marked "End of run," the sliding region is marked, "Cooling." The second graph shows a rising curve. The horizontal axis "Time (minutes)" ranges from 0 to 300, in increments of 50 and the vertical axis "Pressure (psi)" ranges from 0 to 500, in increments of 50. The curve tends to be constant from (0, 40) to (170, 40), rises, slides, and in turn, it again rises and slides down. The constant region of the curve is marked "Exotherm detected," the rising region is marked "Adiabatic mode," the peak value is marked "Reaction finished," the second peak region is marked "End of run," the decreasing curve is marked "Cooling."

The plot at the bottom of Figure 8-3 shows the pressure scan for an APTAC run. The pressure increase is due to the vapor pressure of the liquid sample. This pressure increases exponentially with temperature. The initial decrease and then subsequent increase in pressure after the reaction is finished would merit additional study: It might be due to the decomposition of one of the reaction products.

Figure 8-4 shows a calorimeter run using the heat–wait–search mode. The sample is incrementally heated using a user-defined temperature step, and then the calorimeter waits for a specified time, watching for any reaction self-heating. Several heat–wait–search cycles are required before an exotherm is detected and the calorimeter goes into adiabatic mode. At the completion of the reaction, the calorimeter goes back into the heat–wait–search mode, looking for additional reactions at higher temperatures.

Two graphs for APTAC data for thermal decomposition of di-tert-butyl peroxide are shown. — **Figure 8-4** APTAC data for the thermal decomposition of di-tert-butyl peroxide.

The first graph shows a curve in the form of a step with peak values. The horizontal axis "Time (in minutes)" ranges from 0 to 700, in increments of 100 and the vertical axis "Temperature (in degrees Celsius)" ranges from 0 to 300, in increments of 50. The graph starts at (0, 25) and increases linearly. The calorimeter starts heating at the point (50, 25), "multiple heat-wait-search cycles" are marked between (120, 80) and (200, 100). "Begin adiabatic mode" is marked at (320, 120), the reaction gets finished at (420, 230). More Heat-wait-search cycles are marked between (450, 230) and (550, 250). The end of the run is marked at (580, 250). The cooling region ranges between (550, 250) and (650, 50). The second graph shows a curve with the horizontal axis representing "Time (minutes)" ranging from 0 to 700, in increments of 100 and the vertical axis representing "Pressure (psi)" ranging from 0 to 800, in increments of 100. The curve rises from (0, 20) to (650, 280) with peak values at (450, 600). The region from (0, 20) to (350, 50) is the "Begin Adiabatic mode." At (420, 620), the reaction gets finished. (550, 650) is the end of the run. The region from (550, 650) to (650, 280) is the cooling region.

Another mode of calorimeter operation is to inject a chemical at a specified temperature. It is also possible to heat the calorimeter with chemical A in the sample container, and then inject chemical B at a specified temperature. Both of these methods will result in an injection endotherm as the cold liquid is injected, but the calorimeter will quickly recover.

The calorimeter selected for a particular study depends on the nature of the reactive material. Many companies perform screening studies using the DSC or ARSST and then, depending on the screening results, move to a more capable calorimeter. If a very slow reaction is expected, then the ARC has the best ability to detect very low heating rates. If the reaction has a very high self-heat rate, then the VSP2 or APTAC is the calorimeter of choice.

Theoretical Analysis of Calorimeter Data

Assume that a reaction occurs in a closed, well-stirred reaction test cell. Assume a general reaction of the form

αA+βB+=Products(8-2) $\begin{matrix} α A + β B + = Products & (8-2) \end{matrix}$

We can then write a mole balance on reactant A as follows:

dCAdt=−k(T)CaACbB…(8-3) $\begin{matrix} \frac{d C_{A}}{d t} = - k (T) C_{A}^{a} C_{B}^{b} \dots & (8-3) \end{matrix}$

where

C is the concentration (moles/volume),
k(T ) is the temperature-dependent rate coefficient (concentration^1-n/time), and
a, b ... are the reaction orders with respect to each species.

The rate coefficient k(T ) is given by the Arrhenius equation Ae−Ea/RgT $A e^{- E_{a} / R_{g} T}$ , where A is the pre-exponential factor and Ea is the activation energy. Equation 8-3 can be written for each species for the reaction shown in Equation 8-2.

If the reactants are initially added to the reaction test cell in their stoichiometric ratios, then the concentrations will remain at those ratios throughout the reaction. Then

CBCA=βα(8-4) $\begin{matrix} \frac{C_{B}}{C_{A}} = \frac{β}{α} & (8-4) \end{matrix}$

and similarly, for all of the other species. Substituting into Equation 8-3,

dCAdt=−k(T)CaA(βαCA)b....=−k(T)(βα)b...Ca+b+A...(8-5) $\begin{matrix} \frac{d C_{A}}{d t} & = - k (T) C_{A}^{a} {(\frac{β}{α} C_{A})}^{b} ... . \\ = - k (T) {(\frac{β}{α})}^{b} ... C_{A}^{a + b +} ... \end{matrix} \begin{matrix} (8-5) \end{matrix}$

We can then write an equation for the concentration of species A in terms of an overall reaction order n = a+b+... and define a new rate coefficient. This results in

dCAdt=−k'(T)CnA(8-6) $\begin{matrix} \frac{d C_{A}}{d t} = - k' (T) C_{A}^{n} & (8-6) \end{matrix}$

This approach works only if all of the species are stoichiometric.

Another approach is to have all the reactants, except one, be in excess. In this case, the excess reactant concentrations can be assumed to be approximately constant during the reaction. Suppose that all the reactants, except species A, are in excess. Then,

dCAdt=−k(T)CaACbBoCcCo...=−k(T)(CbBoCcCo...)CaA=−k'(T)CaA(8-7) $\begin{matrix} \begin{matrix} \begin{matrix} \frac{d C_{A}}{d t} \end{matrix} & = - k (T) C_{A}^{a} C_{B o}^{b} C_{C o}^{c} ... \\ = - k (T) (C_{B o}^{b} C_{C o}^{c} ...) C_{A}^{a} \\ = - k' (T) C_{A}^{a} \end{matrix} & (\begin{matrix} 8-7 \end{matrix}) \end{matrix}$

where CBo is the initial concentration of species B. This approach is useful for determining the reaction order with respect to an individual species.

Consider now a situation where the reaction order is not known. Assume that the reaction can be represented by an overall nth-order reaction of the following form:

dCdt=−k(T)Cn(8-8) $\begin{matrix} \frac{d C}{d t} = - k (T) C^{n} & (8-8) \end{matrix}$

We can define a reaction conversion x in terms of the initial concentration C_o:

x=Co−CC0(8-9) $\begin{matrix} x = \frac{C_{o} - C}{C_{0}} & (8-9) \end{matrix}$

Note that when C =C_o, x = 0 and when C = 0, x = 1.

Substituting Equation 8-9 into Equation 8-8,

dxdt=k(T)Cn−1o(1−x)n(8-10) $\begin{matrix} \frac{d x}{d t} = k (T) C_{o}^{n - 1} {(1 - x)}^{n} & (8-10) \end{matrix}$

Dividing both sides of Equation 8-10 by k(To)Cn−1o, $k (T_{o}) C_{o}^{n - 1},$

1k(To)Cn−1odxdt=k(T)k(To)(1−x)n(8-11) $\begin{matrix} \frac{1}{k (T_{o}) C_{o}^{n - 1}} \frac{d x}{d t} = \frac{k (T)}{k (T_{o})} {(1 - x)}^{n} & (8-11) \end{matrix}$

Now define a dimensionless time as

τ=k(To)Cn−1otdτ=k(To)Cn−1odt(8-12) $\begin{matrix} \begin{matrix} τ = k (T_{o}) C_{o}^{n - 1} t \\ d τ = k (T_{o}) C_{o}^{n - 1} d t \end{matrix} & (8-12) \end{matrix}$

and Equation 8-11 simplifies to

dxdτ=k(T)k(To)(1−x)n(8-13) $\begin{matrix} \frac{d x}{d τ} = \frac{k (T)}{k (T_{o})} {(1 - x)}^{n} & (8-13) \end{matrix}$

Up to this point, all our equations have been cast with respect to the concentration within our reaction vessel. Unfortunately, the concentration is very difficult to measure, particularly when the reaction is very fast and the concentrations are changing very rapidly. Online instruments to measure the concentration directly are still very limited. A more direct approach is to withdraw a very small sample from the reactor and cool and quench the reaction instantaneously to get a representative result. Withdrawing the sample will also have an impact on the test sample since we are removing mass and energy from the vessel. Thus, measuring the concentration in real time is very difficult to do.

The easiest system parameter to measure is the temperature. This measurement can be done easily with a thermocouple and can be performed very rapidly in real time. We can relate the temperature to the conversion by assuming that the conversion is proportional to the entire temperature change during the reaction. This gives us the following equation:

x=T−ToΔTad=T−ToTF−To(8-14) $\begin{matrix} x = \frac{T - T_{o}}{Δ T_{a d}} = \frac{T - T_{o}}{T_{F} - T_{o}} & (8-14) \end{matrix}$

where

T_o is the initial reaction temperature, also called the reaction detected onset temperature,
T_F is the final reaction temperature when the reaction is completed, and
ΔTad $Δ T_{a d}$ is the adiabatic temperature change during the reaction.

Equation 8-14 contains two important assumptions:

The reaction is characterized by an initial temperature and a final temperature, and these temperatures can be determined experimentally with a fair amount of precision.
The heat capacity of the test sample remains constant during the reaction.

Assumption 1 is perhaps the more important. The initial or detected onset temperature must be the temperature at which the reaction rate becomes large enough for adequate energy to be produced for heating of the reaction mixture to be detected by a thermocouple. This temperature has been misinterpreted by many investigators in the past. Some investigators have incorrectly concluded that if a reactive material is stored below the initial or detected onset temperature, then that reactive mixture will not react and can be stored indefinitely. This is totally incorrect, since the reaction proceeds at a finite, albeit often undetectable rate, even below the detected onset temperature. Different calorimeters will also give different initial or detected onset temperatures since this is a function of the temperature measurement sensitivity of the calorimeter. The initial or detected onset temperature is used only to relate the concentration of the reactant to the temperature of the reactor—and nothing more.

Assumption 2 will fail if the heat capacity of the products is significantly different from the heat capacity of the reactants. Fortunately, the heat capacities for most liquid materials are about the same. If the reactants are mostly liquids and a significant fraction of the products are gases, then this assumption may fail. This assumption is approximately true if one or more of the reactants are in excess or the system contains a solvent in high concentration, resulting in a nearly constant liquid heat capacity.

The kinetic term in Equation 8-13 can be expanded as follows:

k(T)k(To)=Ae−Ea/RgTAe−Ea/RgTo=exp[EaRg(1To−1T)]=exp[EaRgTo(1−ToT)]=exp[EaRgTo(1−ToTo+ΔTadx)](8-15) $\begin{matrix} \frac{k (T)}{k (T_{o})} & = \frac{A e^{- E_{a} / R_{g} T}}{A e^{- E_{a} / R_{g} T_{o}}} = \exp [\frac{E_{a}}{R_{g}} (\frac{1}{T_{o}} - \frac{1}{T})] \\ = \exp [\frac{E_{a}}{R_{g} T_{o}} (1 - \frac{T_{o}}{T})] \\ = \exp [\frac{E_{a}}{R_{g} T_{o}} (1 - \frac{T_{o}}{T_{o} + Δ T_{a d} x})] \end{matrix} \begin{matrix} (8-15) \end{matrix}$

=exp⎡⎣⎢⎢⎢EaRgToΔTadTox1+ΔTadTox⎤⎦⎥⎥⎥(8-16) $\begin{matrix} = \exp [\frac{\frac{E_{a}}{R_{g} T_{o}} \frac{Δ T_{a d}}{T_{o}} x}{1+ \frac{Δ T_{a d}}{T_{o}} x}] & (8-16) \end{matrix}$

Now define B as the dimensionless adiabatic temperature rise and Γ as the dimensionless activation energy, given by the following equations:

B=ΔTadTo=TF−ToTo(8-17) $\begin{matrix} B = \frac{Δ T_{a d}}{T_{o}} = \frac{T_{F} - T_{o}}{T_{o}} & (8-17) \end{matrix}$

Γ=EaRgTo(8-18) $\begin{matrix} Γ = \frac{E_{a}}{R_{g} T_{o}} & (8-18) \end{matrix}$

Then Equation 8-13 reduces to the following dimensionless equation:

dxdτ=(1−x)nexp(ΓBx1+Bx)(8-19) $\begin{matrix} \begin{matrix} \frac{d x}{d τ} = {(1 - x)}^{n} \exp (\frac{Γ B x}{1+ B x}) \end{matrix} & (8-19) \end{matrix}$

The percent difference between ΓBx $Γ B x$ and ΓBx/(1+Bx) $Γ B x / (1 + B x)$ is given by 100Bx. Thus, for Bx = 0.05, the error is equal to 5%. For this case, we can simplify Equation 8-19 to

dxdτ≅(1−x)nexp(ΓBx)(8-20) $\begin{matrix} \frac{d x}{d τ} ≅ (1 - x)^{n} exp (Γ B x) & (8-20) \end{matrix}$

Equations 8-19 and 8-20 contain the following very important assumptions:

The reaction vessel is well mixed. This means that the temperature and concentration gradients in the liquid sample are small.
The physical properties—heat capacity and heat of reaction—of the sample remain constant.
The heat released by the reaction is proportional to the conversion.
The reaction conversion x is directly proportional to the temperature increase during the reaction.

By converting the dimensional equations into dimensionless form, we reduce the equations to their simplest form, thereby allowing for easy algebraic manipulation of the equations. We also identify the smallest number of dimensionless parameters required to describe our system. In this case, there are three parameters: reaction order n, dimensionless adiabatic temperature rise B, and dimensionless activation energy Γ. Finally, the equations are easier to solve numerically since all variables are scaled, typically between 0 and 1.

The problem with using a dimensionless approach is that the dimensionless parameters and variables are difficult to interpret physically. Also, the physical parameters (T, P, and so on) typically appear in more than one of the dimensionless parameters. Thus, while the dimensionless parameters can all be specified independently, this might result in a contradiction in the real physical parameters. There are also many different ways to define and select dimensionless parameters.

The reaction order, n, typically has a value between 0 and 2. A reaction order of 1 (first order) is most common, but higher, fractional orders greater than 1 may also occur. The dimensionless adiabatic temperature rise, B, typically has a value from 0 to about 2. The dimensionless activation energy can have a value ranging from about 5 to 50 or higher, depending on the activation energy.

The dimensionless Equations 8-19 and 8-20 depend on a number of dimensional parameters, including the detected onset temperature To, final temperature TF, activation energy Ea, and the value of the Arrhenius equation at the detected onset temperature k(To). Note that the detected onset and final temperatures are implicitly related to the Arrhenius reaction rate equation and cannot be specified independently.

Equations 8-19 and 8-20 can be easily integrated, using either a spreadsheet with the trapezoid rule or a mathematical package. A spreadsheet using the trapezoid rule might require a small step size; this should be checked to ensure that the results converge.

Figures 8-5 and 8-6 show the results for the integration of Equation 8-19. The conversion is plotted versus the dimensionless time. The time scale is logarithmic, so as to show a much larger range. Figure 8-5 shows how increasing the dimensionless adiabatic temperature rise B results in a reaction that occurs over a shorter time period with a steeper slope. Figure 8-6 shows how increasing the dimensionless activation energy Γ also results in a reaction that occurs over a shorter time period and a steeper slope. For the case with Γ = 20, the conversion responds very quickly.

A graph shows the increasing dimensionless adiabatic temperature rise in B. — **Figure 8-5** Equation 8-19 solved for increasing dimensionless adiabatic temperature rise B.

A graph shows four curves. The horizontal axis represents "Dimensions time, tau" that ranges from 0 to 1, in increments of 0.10 and the vertical axis represents "Reaction Conversion, x" that ranges from 0 to 1.2, in increments of 0.2. Four curves are labeled for the values of "B equals 3, 2, 1, and 0.5." The curve starts at (0, 0.5) and rises up to the right. The curve "B equals 3" extends to the right from (0.20, 0.9). The curve "B equals 2" extends to the right from (0.30, 0.9). The curve "B equals 3" extends to the right from (0.50, 0.9). The curve "B equals 0.5" extends to the right from (1, 0.9). All the curves overlap each other. Note: All the values are approximate.

A graph shows four curves for increasing dimensionless activation energy, Tau. — **Figure 8-6** Equation 8-19 solved for increasing dimensionless activation energy, Γ.

In the graph, the horizontal axis ranges from 0 to 1, in increments of 0.10 and the vertical axis represents "Reaction Conversion, X" that ranges from 0 to 1, in increments of 0.2. The curve for "tau equals 20" rises from (0, 0.5) to (0.001, 1). The curve for "tau equals 10" rises from (0, 0.03) to (0.4, 1), the curve for "tau equals 5" rises from (0, 0.30) to (0.30, 1), the curve for "tau equals 1" rises to the right from (0, 0.30) to (1.5, 1). A line connects all the curves. Note: All the values are approximate.

One of the important questions in Table 8-7 is the maximum self-heat rate and the time at which this rate is achieved. The maximum self-heat rate is important for designing heat transfer equipment to remove the heat of reaction: Larger self-heat rates require larger heat transfer equipment. Some chemicals, such as ethylene oxide and acrylic acid, will self-heat at rates of several hundred degrees Celsius per minute. The time at which the maximum self-heat rate occurs is also important for designing heat transfer equipment as well as for emergency response procedures.

The maximum self-heat rate can be found by differentiating Equation 8-19 with respect to the dimensionless time and then setting the result to zero to find the maximum. This equation can be solved for the conversion xm at which the maximum self-heat rate occurs. After a lot of algebra, the following equation is obtained:

xm=12nB[−(Γ+2n)+Γ[Γ+4n(1+B)]−−−−−−−−−−−−−−√](8-21) $\begin{matrix} \begin{matrix} x_{m} = \frac{1}{2 n B} [- (Γ + 2 n) + \sqrt{Γ [Γ + 4 n (1 + B)]}] \end{matrix} & (\begin{matrix} 8-21 \end{matrix}) \end{matrix}$

where xm is the conversion at the maximum rate. The maximum rate is then found by substituting xm into Equation 8-19.

Figure 8-7 is a plot of Equation 8-21 for a first-order reaction. As Γ increases, an asymptotic function is approached. This function is close to the Γ = 64 curve shown. Also, there is a maximum xm for each curve shown by the vertical tic mark on each curve; this maximum xm increases as Γ increases and also occurs at lower B values. Each curve intersects the x axis.

A graph portrays conversion at maximum self-heat rate for a first-order reaction. — **Figure 8-7** Conversion at maximum self-heat rate for a first-order reaction. The vertical line on each curve represents the maximum value.

A graph depicts conversion at maximum self-heat rate, X subscript m (vertical axis) versus dimensionless adiabatic temperature rise, B (horizontal axis) for a first-order reaction is shown. The horizontal axis ranges from 0 to 5 in increments of 0.5 and the vertical axis ranges from 0 to 1 in increments of 0.2. Seven curves are drawn for different values of dimensionless activation energy, gamma. The first curve representing gamma equals 1, originates from (1, 0), increases gradually through (2, 0.1), and ends at (5, 0.2). The maximum value of the curve lies at (5, 0.2), that is represented using a small vertical line. The second curve representing gamma equals 2, originates from (0.5, 0), increases gradually through (1, 0.3), and ends at (5, 0.3). The maximum value of the curve is marked at (3, 0.3). Similarly, five other curves representing gamma equals 4, 8, 16, 32, and 64 are drawn one above the other, that originate from the range between (0.3, 0) and (0.1, 0), increases gradually and ends in the range between (5, 0.5) and (5, 0.9). The maximum values of the 3rd, 4th, 5th, 6th, and 7th curves are marked at (2, 0.5), (1.5, 0.7), (1.2, 0.8), (1.1, 0.9), and (1, 0.95) respectively. Note: the above-mentioned values are approximate.

Equation 8-21 can be differentiated one more time with respect to B to find the maximum value of xm. After a fair amount of algebra, the maximum value is found to occur at

B=4nΓ+1xm=Γ4n+Γ(8-22) $\begin{matrix} \begin{matrix} B = \frac{4 n}{Γ} + 1 \\ x_{m} = \frac{Γ}{4 n + Γ} \end{matrix} & (8-22) \end{matrix}$

The intersection of the curves on Figure 8-7 with the x axis means that the maximum self-heat rate occurs at the detected onset temperature—that is, when τ = 0. The value of B at this intersection can be found from Equation 8-21 by setting xm = 0 and solving for B. The result is

B=nΓ(8-23) $\begin{matrix} B = \frac{n}{Γ} & (8-23) \end{matrix}$

The time at which the maximum rate occurs can be found by two methods. The first method is to integrate Equation 8-19 to determine the value of the dimensionless time τ_m at the conversion, xm, and then converting this to the actual time. Another approach is to separate variables in Equation 8-19 and integrate. The result is

∫0xmdx(1−x)nexp(ΓBx1+Bx)=∫0τmdτ=τm(8-24) $\begin{matrix} \int_{0}^{x_{m}} \frac{d x}{{(1 - x)}^{n} \exp (\frac{Γ B x}{1 + B x})} = \int_{0}^{τ_{m}} d τ = τ_{m} & (\begin{matrix} 8-24 \end{matrix}) \end{matrix}$

where τ_m is the dimensionless time at which the maximum rate occurs. Note that this time is relative to the detected onset temperature.

Equation 8-24 can be solved numerically for τ_m. The results are shown in Figures 8-8 and 8-9 for two different ranges of Γ. Each curve has a maximum, shown by the vertical tick mark. This maximum occurs at lower Γ values as the dimensionless adiabatic temperature rise B increases. This maximum must be solved for numerically—an algebraic solution is not possible.

A graph portrays time to maximum self-heat rate for a first-order reaction. — **Figure 8-8** Time to maximum self-heat rate for a first-order reaction: low Γ range. The vertical lines on each curve represent the maximum value.

A graph of dimensionless time to maximum rate, tau subscript m (vertical axis) versus dimensionless activation energy, gamma (horizontal axis) for a first-order reaction, n equals 1, is shown. The horizontal axis ranges from 0 to 10 in increments of 1 and the vertical axis ranges from 0 to 0.4 in increments of 0.05. Seven curves are drawn for different values of adiabatic temperature B. The first curve representing B equals 0.5, originates from (2, 0), increases gradually through (5, 0.35), and ends at (10, 0.3). The maximum value of the curve is marked using a small vertical line at (6, 0.35). The second curve representing B equals 0.75, originates from (1.3, 0), increases gradually through (3, 0.3), and ends at (10, 0.2). The maximum value of the curve is marked at (4.5, 0.32). Similarly, five other curves representing B equals 1, 1.5, 2, 3, and 4 are drawn one below the other, that originate from the range between (1, 0) and (0.2, 0), increases gradually and ends in the range between (10, 0.18) and (10, 0.05). The maximum values of the 3rd, 4th, 5th, 6th, and 7th curves are marked at (3.5, 0.3), (2.8, 0.28), (2.5, 0.25), (1.5, 0.21), and (1.2, 0.18) respectively.

A graph compares time to maximum self-heat rate and activation energy with different values of adiabatic temperature for a first-order reaction. — **Figure 8-9** Time to maximum self-heat rate for a first-order reaction: high Γ range.

A graph of dimensionless time to maximum rate, tau subscript m (vertical axis) versus dimensionless activation energy, gamma (horizontal axis) for a first-order reaction, n equals 1, is shown. The horizontal axis ranges from 10 to 40 in increments of 5 and the vertical axis ranges from 0 to 0.4 in increments of 0.05. Seven decreasing curves are drawn for different values of adiabatic temperature B. The first curve representing B equals 0.5, originates from (0, 0.28), decreases gradually through (20, 0.15), and ends at (40, 0.08). The second curve representing B equals 0.75, originates from (0, 0.2), decreases gradually through (20, 0.1), and ends at (40, 0.06). Similarly, five other curves representing B equals 1, 1.5, 2, 3, and 5 are drawn one below the other, that originate from the range between (0, 0.15) and (0, 0.03), decrease gradually and end in the range between (40, 0.05) and (40, 0.01).

Example 8-2

A chemical reactor currently has a temperature of 400 K. Given the calorimetry data provided below, calculate:

The current conversion
The current self-heat rate
The time since the detected onset temperature
The time to the maximum self-heat rate
The maximum self-heat rate

Calorimeter data:

Reaction order, n:	1 (first-order)
Detected onset temperature, To:	377 K
Final temperature, TF:	483 K
k(To) :	5.62 × 10^–5 s^–1
Activation energy, Ea:	15,000 cal/mol K

Solution

From the calorimetry data provided:

BΓ=TF−ToTo=483 K−377 K377 K=0.282=EaRgTo=15,000 ca1 / mol(1.987 ca1/ mol K)(377 K)=20.0 $\begin{matrix} \begin{matrix} B \end{matrix} & = \frac{T_{F} - T_{o}}{T_{o}} = \frac{483 K - 377 K}{377 K} = 0.282 \\ Γ & = \frac{E_{a}}{R_{g} T_{o}} = \frac{15, 000 ca1 / mol}{(1 . 987 ca1/ mol K) (377 K)} = 20.0 \end{matrix}$

The current conversion is found from Equation 8-14:

x=T−ToTF−To=400 K−377 K483 K−377 K=0.217 $x = \frac{T - T_{o}}{T_{F} - T_{o}} = \frac{400 K - 377 K}{483 K - 377 K} = 0.217$
The current self-heat rate is calculated using Equation 8-19:

dxdτ=(1−x)nexp(ΓBx1+Bx)=(1−0.217)×exp[(20.0)(0.282)(0.217)1+(0.282)(0.217)]=2.48 $\frac{d x}{d τ} = {(1 - x)}^{n} \exp (\frac{Γ B x}{1+ B x}) = (1 - 0.217) \times \exp [\frac{(20.0) (0.282) (0.217)}{1 + (0.282) (0.217)}] = 2.48$

This must be converted to dimensional time. From Equation 8-12,

t=τk(To)Cn−1o(8-25) $\begin{matrix} t = \frac{τ}{k (T_{o}) C_{o}^{n - 1}} & (8-25) \end{matrix}$

and using the definition of the conversion given by Equation 8-14:

dTdt=(TF−To)k(To)Cn−10(dxdτ)(8-26) $\begin{matrix} \frac{d T}{d t} = (T_{F} - T_{o}) k (T_{o}) C_{0}^{n - 1} (\frac{d x}{d τ}) & (\begin{matrix} 8-26 \end{matrix}) \end{matrix}$

And it follows for n = 1,

dTdt=(TF−To)k(To)dxdτ=(483 K−377 K)(5.62×10−5 s−1)(2.48)=0.0148K /s=0.886 K /min $\begin{matrix} \begin{matrix} \frac{d T}{d t} \end{matrix} & = (T_{F} - T_{o}) k (T_{o}) \frac{d x}{d τ} = (483 K - 377 K) (5.62 \times 10^{- 5} s^{- 1}) (2.48) \\ = 0 . 0148K /s = 0 . 886 K / \min \end{matrix}$
The dimensionless time since the detected onset temperature is found by integrating Equation 8-19. This can be done easily using either a spreadsheet or a numerical package. The results are shown in Figure 8-10. From Figure 8-10, at a conversion of 0.217, the dimensionless time is 0.141. The dimensionless time can be converted to actual time using Equation 8-25. For a first-order reaction, n = 1 and

Figure 8-10 Conversion plot for Example 8-2.

A graph of reaction conversion, X (vertical axis) versus dimensionless time, tau (horizontal axis) is shown. The horizontal axis ranges from 0 to 0.35 in increments of 0.05 and the vertical axis ranges from 0 to 1.2 in increments of 0.2. An increasing curve, originating from the origin is drawn, that represents at a conversion of 0.217, dimensionless time is 0.141 and at a conversion of 0.741, dimensionless time is 0.249. The increasing curve ends at (0.35, 1).

t=τk(To)=0.1415.62×10−5 s−1=2510 s=41.8 min. $t = \frac{τ}{k (T_{o})} = \frac{0.141}{5.62 \times 10^{- 5} s^{- 1}} = 2510 s = 41.8 \min .$

This is the time since the detected onset temperature.
The conversion at the maximum rate is given by Equation 8-21. Substituting the known values:

xm=12nB[−(Γ+2n)+Γ[Γ+4n(1+B)]−−−−−−−−−−−−−−√]xm=1(2)(1)(0.282)[−(20+2)+(20)[20+(4)(1+0.282)]−−−−−−−−−−−−−−−−−−−−√]xm=0.741 $\begin{array}{l} x_{m} = \frac{1}{2 n B} [- (Γ + 2 n) + \sqrt{Γ [Γ + 4 n (1 + B)]}] \\ x_{m} = \frac{1}{(2) (1) (0.282)} [- (20 + 2) + \sqrt{(20) [20 + (4) (1 + 0.282)]}] \\ x_{m} = 0.741 \end{array}$

From the numerical solution, Figure 8-10, the dimensionless time at this conversion is 0.249. The actual time is

t=τk(To)=0.2495.62×10−5 s−1=4432 s=73.9 min. $t = \frac{τ}{k (T_{o})} = \frac{0.249}{5.62 \times 10^{- 5} s^{- 1}} = 4432 s = 73.9 \min .$

This is with respect to the time the detected onset temperature is reached. The time from the current reaction state (x = 0.217) to the time at the maximum rate is 73.9 min – 41.8 min = 32.1 min.

Thus, the reaction will reach the maximum rate in 32.1 min.

Figure 8-8 or 8-9 could be used to solve this problem directly. However, it is not as precise.
The maximum self-heat rate is found using Equation 8-19 with x_m = 0.741:

dxdτ=(1−x)nexp(ΓBx1+Bx)=(1−0.741)×exp[(20.0)(0.282)(0.741)1+(0.282)(0.741)]=8.21 $\frac{d x}{d τ} = {(1 - x)}^{n} \exp (\frac{Γ B x}{1+ B x}) = (1 - 0.741) \times \exp [\frac{(20.0) (0.282) (0.741)}{1+ (0.282) (0.741)}] = 8.21$

It follows that

dTdt=(TF−To)k(To)dxdτ=(483 K−377 K)(5.62×10−5s−1)(8.21)=0.0489 K/s=2.93 K/min $\begin{matrix} \begin{matrix} \frac{d T}{d t} \end{matrix} & = (T_{F} - T_{o}) k (T_{o}) \frac{d x}{d τ} = (483 K - 377 K) (5.62 \times 10^{- 5} s^{- 1}) (8.21) \\ = 0 . 0489 K/s = 2 . 93 K/min \end{matrix}$

The maximum self-heat rate, along with the heat capacity of the reacting liquid, could be used to estimate the minimum cooling requirements for this reactor. Also, the time to maximum rate could be used to estimate the residence time, operating temperature, and conversion for the reactor.

Estimation of Parameters from Calorimeter Data

Prior to using the theoretical model, we need to estimate from calorimeter data the parameters required for the model. These include the detected onset temperature To, final temperature TF, reaction order n, activation energy Ea, and the value of the Arrhenius equation at the detected onset temperature k(To). The calorimeter typically provides temperature versus time data, so a procedure is required to estimate these parameters from these data.

The detected onset and final temperatures are perhaps the most important because the conversion, as well as the theoretical model, depends on knowing these parameters reasonably precisely. If we plot the rate of temperature change dT/dt versus time, we find that the temperature rate is very low at the beginning and end of the reaction. Thus, this procedure will not work.

A better procedure is to plot the logarithm of the rate of temperature change dT/dt versus –1000/T. This is best illustrated by an example.

Example 8-3

Figure 8-3 shows temperature-time and pressure-time data for the reaction of methanol and acetic anhydride in a 2:1 molar ratio. Using these data, determine the detected onset and final temperatures and the dimensionless adiabatic temperature rise B for this system.

Solution

A plot of the logarithm of the temperature rate versus –1000/T is shown in Figure 8-11. The temperature rates at early times are shown on the left-hand side of the plot; the temperature rates at the end of the experiment are shown on the right. When the calorimeter detects exothermic behavior, it stops the heating. This is shown as a drop in the plotted temperature rate on the far left side of Figure 8-11. The detected onset temperature is found as the temperature at the beginning of the nearly straight-line section of the curve. The detected onset temperature from Figure 8-10 is found to be

Onset and final temperatures are estimated using a graph of temperature rate versus temperature. — **Figure 8-11** Graphical procedure to estimate the detected onset and final temperatures.

A graph of temperature rate, dT over dt (vertical axis) versus negative 1000 over T (horizontal axis) is shown. The horizontal axis is represented in terms of Kelvin inverse and ranges from negative 3.6 to negative 2 in increments of 0.2. The vertical axis is represented in terms of Kelvin per minute and ranges from 0.01 to 1000 in increments of multiples of 10. A curve is drawn that originates from (negative 3.5, 0.15), slightly increases and drops down toward (negative 3.4, 0.015). The curve then rises to (negative 3.4, 0.1), that is marked as Onset temperature. From this point, the curve increases gradually till (negative 2.1, 104) that is marked as the maximum rate. The curve, after few fluctuations, drops down linearly toward (negative 2.2, 0.1), that is marked as the final temperature.

−1000To=−3.35To=298 K $\begin{array}{l} - \frac{1000}{T_{o}} = - 3.35 \\ T_{o} = 298 K \end{array}$

The final temperature is found from the right-hand side of the plot, where the temperature rate drops off suddenly and returns to almost the same temperature rate as the detected onset temperature. The final temperature is determined from Figure 8-11 to be

−1000TF=−2.25TF=404 K $\begin{array}{l} - \frac{1000}{T_{F}} = - 2.25 \\ T_{F} = 404 K \end{array}$

Figure 8-11 can also be used to identify the maximum temperature rate as well as the temperature at which this occurs. This is found from the peak value in Figure 8-11. From the data, the maximum temperature rate is 104 K/min (1.73 K/s), which occurs at a temperature of 429 K.

The dimensionless adiabatic temperature rise is given by Equation 8-17. Thus,

B=TF−ToTo=447 K−298 K298 K=0.500 $B = \frac{T_{F} - T_{o}}{T_{o}} = \frac{447 K - 298 K}{298 K} = 0.500$

At this point in the procedure, we have the detected onset and final temperatures and the dimensionless adiabatic temperature rise B. The next parameter to estimate is the reaction order. The reaction order can be estimated by rearranging Equation 8-19 as follows:

1(1−x)ndxdτ=exp(ΓBx1+Bx)(8-27) $\begin{matrix} \frac{1}{{(1 - x)}^{n}} \frac{d x}{d τ} = \exp (\frac{Γ B x}{1+ B x}) & (8-27) \end{matrix}$

From Equation 8-12, dτ=k(To)Cn−1odt $d τ = k (T_{o}) C_{o}^{n - 1} d t$ , and it follows that

1(1−x)ndxdτ=1(1−x)nk(To)Cn−1odxdt=exp(ΓBx1+Bx)(dx/ dt)(1−x)n=k(To)Cn−1oexp(ΓBx1+Bx) ln[(dx/ dt)(1−x)n]=ln[k(To)Cn−1o]+ Γ(Bx1+Bx)(8-28) $\begin{matrix} \begin{array}{l} \frac{1}{{(1 - x)}^{n}} \frac{d x}{d τ} = \frac{1}{{(1 - x)}^{n} k (T_{o}) C_{o}^{n - 1}} \frac{d x}{d t} = \exp (\frac{Γ B x}{1+ B x}) \\ \frac{(d x / d t)}{{(1 - x)}^{n}} = k (T_{o}) C_{o}^{n - 1} \exp (\frac{Γ B x}{1+ B x}) \\ \ln [\frac{(d x / d t)}{{(1 - x)}^{n}}] = \ln [k (T_{o}) C_{o}^{n - 1}] + Γ (\frac{B x}{1+ B x}) \end{array} & (\begin{matrix} 8-28 \end{matrix}) \end{matrix}$

Equation 8-28 can be modified in terms of the actual temperature. Since x=(T−To)/(TF−To) $x = (T - T_{o}) / (T_{F} - T_{o})$ and Bx/ (1+Bx)=(T−To)/ T $B x / (1 + B x) = (T - T_{o}) / T$ , then

ln[(dT/dt)(TF−To)1−n(TF−T)n]=ln[k(To)Cn−1o]+Γ(T−To)/T(8-29) $\begin{matrix} \ln [\frac{(d T / d t)}{{(T_{F} - T_{o})}^{1 - n} {(T_{F} - T)}^{n}}] = \ln [k (T_{o}) C_{o}^{n - 1}] + Γ (T - T_{o}) / T & (8-29) \end{matrix}$

For a first-order reaction, n = 1, and Equation 8-29 becomes

ln[dT/dtTF−T]=ln[k(To)]+Γ(T−To)/T(8-30) $\begin{matrix} \ln [\frac{d T / d t}{T_{F} - T}] = \ln [k (T_{o})] + Γ (T - T_{o}) / T & (8-30) \end{matrix}$

The procedure for determining the overall reaction order n, using either Equation 8-28, 8-29, or 8-30, is as follows. The detected onset and final temperatures and the dimensionless adiabatic temperature rise B are already known from Example 8-3. First, the overall reaction order is guessed. First-order (n = 1) is a good place to start. Then the left-hand side of either Equation 8-28, 8-29, or 8-30 is plotted versus Bx/(1+Bx) or (T−T_o)/T. If the guessed reaction order is correct, a straight line is obtained. It is possible to have fractional overall reaction orders, so some effort might be required to estimate the reaction order that produces the best fit to the data. For any reaction order not equal to unity, the initial concentration of reactant must also be known. Finally, as a bonus, once the reaction order is identified, the intercept of the straight line with the y axis provides an estimate of k(To)Cn−1o $k (T_{o}) C_{o}^{n - 1}$ or k(T_o) if the reaction is first order.

Equation 8-30 can be reduced even further to a more traditional form:

ln[dT/ dtTF−T]=lnA−EaRgT(8-31) $\begin{matrix} \ln [\frac{d T / d t}{T_{F} - T}] = \ln A - \frac{E_{a}}{R_{g} T} & (8-31) \end{matrix}$

Equation 8-31 states that if a system is first-order, then a plot of the left-hand side versus –1/RgT should produce a straight line with slope of Ea and intercept ln (A). Equation 8-31 is commonly seen in the literature to determine the kinetic parameters. It is not as general as Equation 8-28.

Example 8-4

Using the data in Figure 8-3 and the results of Examples 8-2 and 8-3, estimate the reaction order and the value of k(T_o) for the methanol + acetic anhydride reaction (2:1 molar ratio). Use these parameters to estimate the pre-exponential factor A and the activation energy Ea. Use the theoretical model to calculate the maximum self-heat rate and corresponding temperature. Compare to the experimental values.

Solution

The first step in this problem is to convert the temperature data into conversion data, x. This is done using the detected onset and final temperatures from Example 8-3 and Equation 8-14. Then (dx/dt) is calculated using the trapezoid rule or any other suitable derivative method. Equation 8-29 is then applied with an initial assumption that n = 1. We then plot ln(dT/dtTF−T) $\ln (\frac{d T / d t}{T_{F} - T})$ versus Bx/(1+Bx)=(T−To)/T $B x / (1 + B x) = (T - T_{o}) / T$ . These results are shown in Figure 8-12. If we eliminate some of the early and later points in the data that diverge from the straight line—these points have less precision—and fit a straight line to the remaining data, we get a very straight line with an R² value of 0.9997. This confirms that the reaction is first-order.

The kinetic parameters n and k(T subscript o) are determined using a graph. — **Figure 8-12** Determination of the kinetic parameters n and k(To).

A graph of natural logarithm of ((dT over dt) over (T subscript F, minus T subscript 0)) (vertical axis) versus Bx over (1 plus Bx) (horizontal axis) is shown. The horizontal axis ranges from negative 0.05 to 0.35 in increments of 0.05 and the vertical axis ranges from negative 14 to 0 in increments of 2. The graph is plotted with several experimental data points. The data points increase linearly in the range between (0.05, negative 11) and (0.3, negative 2). These linear points are connected using a straight line. The line represents the equation, y= 29.454x minus 11.289 and R squared equals 0.9997.

From the straight-line fit, the slope is equal to our dimensionless parameter Γ and the intercept is equal to ln [k(T_o)]. Thus, Γ = 29.4 and k(To)=−11.3 $k (T_{o}) = - 11.3$ . It follows that k(To)=1.24×10−5s−1. $k (T_{o}) = 1.24 \times 10^{- 5} s^{- 1} .$

From the definition for Γ given by Equation 8-18, the activation energy E_a is computed as 72.8 kJ/mol. Since k(To)=Aexp(−EalRgTo) $k (T_{o}) = A \exp (- E_{a} l R_{g} T_{o})$ , we can calculate the pre-exponential factor as A = 7.27 × 10⁷ s⁻¹.

The conversion at the maximum self-heat rate is given by Equation 8-21. Substituting the known dimensionless values,

xm=1(2)(0.5)[−(29.4+2)+29.4[29.4+4(1+0.5)]−−−−−−−−−−−−−−−−√]=0.861 $x_{m} = \frac{1}{(2) (0.5)} [- (29 . 4+2) + \sqrt{29.4 [29.4+4 (1+0.5)]}] = 0.861$

The temperature at this conversion is

T−ToTF−To=xm=0.861T=426 K $\begin{matrix} \frac{T - T_{o}}{T_{F} - T_{o}} = x_{m} = 0.861 \\ T = 426 K \end{matrix}$

This compares very well to the experimental maximum rate temperature of 429 K.

The maximum self-heat rate is estimated from Equation 8-19, with n = 1:

dxdτ=(1−x)exp(ΓBx1+Bx)=(1−0.861)exp[(29.4)(0.500)(0.861)1+(0.5)(0.861)]=967 $\frac{d x}{d τ} = (1 - x) \exp (\frac{Γ B x}{1+ B x}) = (1 - 0.861) \exp [\frac{(29.4) (0.500) (0.861)}{1+ (0.5) (0.861)}] = 967$

The maximum self-heat rate in dimensional units is calculated from Equation 8-26:

dTdt=(TF−To)k(To)dxdτ=(447 K−298 K)(1.24×10−5s−1)(967)=1.78 K /s=107 K / min $\frac{d T}{d t} = (T_{F} - T_{o}) k (T_{o}) \frac{d x}{d τ} = (447 K - 298 K) (1.24 \times 10^{- 5} s^{- 1}) (967) = 1 . 78 K / s = 107 K / min$

This compares to the experimental value of 1.73 K/s (104 K/min). The experimental maximum self-heat rate is usually lower than the theoretical value due to the dampening effect of the sample container heat capacity. Figure 8-13 is a plot of the experimental and theoretical temperature–time plots. The agreement is very good.

A graph of temperature versus time compares experimental data with theoretical model. — **Figure 8-13** Comparison of experimental data with theoretical model prediction.

A graph of temperature, in terms of degrees Celsius (vertical axis), versus time, in terms of minutes (horizontal axis), is shown. The horizontal axis ranges from 0 to 200 in increments of 50 and the vertical axis ranges from 0 to 200 in increments of 20. The graph is plotted with several experimental data points such as (0, 20), (50, 30), (100, 60), and (150, 100). These data points portrays, as time increases, temperature also increases till (100, 180), which then remains constant at 180 degree Celsius for the next few minutes and then starts decreasing. The theoretical model is represented using a curve that follows the experimental value till (120, 180), which then deviates from the experimental data and remains constant at 180 degrees Celsius.

Summary of Calorimeter Analysis: Methanol 1 Acetic Anhydride

Reaction order, n:	1
Detected onset temperature, T_o:	298 K
Final temperature, T_F:	447 K
Dimensionless adiabatic temperature rise, B:	0.500
Dimensionless activation energy Γ	29.4
Activation energy, E_a:	72.8 kJ/mol
Pre-exponential factor, A:	7.27 × 10⁷ s^–1
Maximum self-heat rate, experimental:	104 K/min
Maximum self-heat rate, theoretical:	107 K/min
Temperature at maximum self-heat rate, experimental:	429 K
Temperature at maximum self-heat rate, theoretical:	426 K
Time to maximum rate:	32.1 min

Another parameter for characterizing the hazard associated with a reaction is the temperature of no return. The temperature of no return is the temperature at which the reaction heat generation exceeds the heat losses through the reactor vessel walls to the surroundings. A key to operating a chemical reactor safely is to maintain the normal operating temperature below the temperature of no return.

If we equate the heat generation within a reactor vessel to the heat losses through the walls,

mCp(dTdt)max=UA(TNR−Ts)(8-32) $\begin{matrix} m C_{p} {(\frac{d T}{d t})}_{\max} = U A (T_{N R} - T_{s}) & (\begin{matrix} 8-32 \end{matrix}) \end{matrix}$

where

m is the mass of reactant in the reactor vessel (mass),
C_P is the heat capacity or the reacting liquid (energy/mass),
(dTdt)max ${(\frac{d T}{d t})}_{\max}$ is the maximum self-heat rate of the reaction liquid (temperature/time),
U is the overall heat transfer coefficient between the reactor contents and the surroundings (energy/area × time),
A is the external heat transfer area of the reactor vessel (area),
T_NR is the temperature of no return (degrees), and
T_Sis the temperature of the surroundings (degrees).

Equation 8-32 can be solved for the temperature of no return, TNR:

TNR=Ts+(mCpUA)(dTdt)max(8-33) $\begin{matrix} T_{N R} = T_{s} + (\frac{m C_{p}}{U A}) {(\frac{d T}{d t})}_{\max} & (8-33) \end{matrix}$

Adjusting the Data for the Heat Capacity of the Sample Vessel

Up to now, the calorimeter data analysis has assumed that the sample vessel container has a negligible heat capacity. This situation is not realistic: The sample container absorbs heat during the reaction and reduces the total temperature change during the reaction. Without the container, the final reaction temperature would be higher.

The effect of the sample container on the temperature can be estimated from a heat balance. The goal is to estimate the change in temperature of the isolated sample, without the container. The heat generated by the isolated reacting sample is distributed between the sample and the sample container. In equation form,

[mCPΔTad]Isolated Sample[mSCsPΔTsad]IsolatedΔTsadΔTsad=[mCPΔTad]Sample + Container=[(mSCsP+mCCcP)ΔTs+cad]Sample+Container=[mSCsP+mCCcPmSCsP]ΔTs+cad=ΦΔTs+cad(8-34) $\begin{matrix} \begin{matrix} {[m C_{P} Δ T_{a d}]}_{Isolated Sample} & = {[m C_{P} Δ T_{a d}]}_{Sample + Container} \\ {[m_{S} C_{P}^{s} Δ T_{a d}^{s}]}_{Isolated} & = {[(m_{S} C_{P}^{s} + m_{C} C_{P}^{c}) Δ T_{a d}^{s + c}]}_{Sample+Container} \\ Δ T_{a d}^{s} & = [\frac{m_{S} C_{P}^{s} + m_{C} C_{P}^{c}}{m_{S} C_{P}^{s}}] Δ T_{a d}^{s + c} \\ Δ T_{a d}^{s} & = Φ Δ T_{a d}^{s + c} \end{matrix} & (8-34) \end{matrix}$

where

Φ is the phi factor presented earlier in Equation 8-1 (unitless),
m is the mass,
C_P is the heat capacity (energy/mass),
T is the temperature (degrees),
s denotes the sample,
c denotes the container, and
ad denotes adiabatic.

Equation 8-34 shows that, for example, if the reaction occurs in a reaction container with a phi factor of 2, then the container absorbs half of the heat generated by the reaction and the measured adiabatic temperature increase is half of the temperature increase that would be measured by an isolated sample.

The dimensionless theoretical model can be adjusted by replacing the dimensionless adiabatic temperature rise B by ΦB to account for the heat capacity of the container.

The initial reaction temperature is not affected much by the container presence. However, the final reaction temperature is estimated from the following equation:

TF=To+ΦΔTs+cad(8-35) $\begin{matrix} T_{F} = T_{o} + Φ Δ T_{a d}^{s + c} & (8-35) \end{matrix}$

Heat of Reaction Data from Calorimeter Data

Calorimeter data can be used to estimate the heat of reaction of the reacting sample. This is done using the following equation:

ΔHrx=ΦCVΔTadmLR/mT(8-36) $\begin{matrix} Δ H_{r x} = \frac{Φ C_{V} Δ T_{a d}}{m_{L R} / m_{T}} & (8-36) \end{matrix}$

where

ΔHrx $Δ H_{r x}$ is the heat of reaction, based on the limiting reactant (energy/mass),
Φ $Φ$ is the phi factor for the vessel container, given by Equations 8-1 and 8-33 (unitless),
CV $C_{V}$ is the heat capacity at constant volume for the reacting liquid (energy/mass-degrees),
ΔTad $Δ T_{a d}$ is the adiabatic temperature rise during the reaction (degrees),
mLR is the initial mass of limiting reactant (mass), and
mT is the total mass of reacting mixture (mass).

Using Pressure Data from the Calorimeter

The pressure information provided by the calorimeter is very important, especially for relief system and pressure vessel design. If the calorimeter is the closed vessel type, the vessel can be evacuated prior to addition of the sample. The pressure data are then representative of the vapor pressure of the reacting liquid—important data required for relief sizing.

The pressure data are used to classify reaction systems into four different types.⁶ These classifications are important for relief system design, discussed in Chapter 10. The classifications are based on the dominant energy term in the liquid reaction mass as material is discharged through the relief system.

⁶H. G. Fisher et al. Emergency Relief System Design Using DIERS Technology (New York, NY: AICHE Design Institute for Emergency Relief Systems, 1992).

Volatile/tempered reaction: Also called a vapor system. In this case, the heat of vaporization of the liquid cools, or tempers, the reaction mass during the relief discharge. Tempered reactions are inherently safer since the cooling mechanism is part of the reaction mass.
Hybrid/tempered reaction: Noncondensable gases are produced as a result of the reaction. However, the heat of vaporization of the liquid dominates to cool the reaction mass during the entire relief discharge.
Hybrid/nontempered reaction: Noncondensable gases are produced but the heat of vaporization of the liquid does not always dominate during the entire relief discharge.
Gassy/nontempered reaction: The reaction produces noncondensable gases and the liquid is not volatile enough for the heat of vaporization of the liquid to have much effect during the entire relief discharge.

The pressure data from the calorimeter are used for classifying the type of system.

The initial and final pressure in the closed vessel calorimeter can be used to determine if the system produces a gaseous product. In this case, the calorimeter must be operated so that the starting and ending temperatures are the same. If the initial pressure is equal to the final pressure, then no gaseous products are produced and the pressure is due to the constant vapor pressure of the liquid—this is a volatile/tempered reaction system. If the final pressure is higher than the starting pressure, then vapor products are produced and the reaction system is either hybrid or gassy.
If the system is classified as hybrid or gassy from step 1, then the time and temperature at which the peaks in the temperature rate and the pressure rate occur is used. If the peaks occur at the same time and temperature, then the system is hybrid. If the peaks do not occur at the same temperature, then the system is gassy.

Application of Calorimeter Data

The calorimeter data are used to design and operate processes so that reactive chemical incidents do not occur. This is, perhaps, the most difficult part of this procedure since each process has its unique problems and challenges. Experience is essential in this procedure. Expert advice is recommended for interpreting calorimetry data and for designing controls and safeguards.

The calorimeter data are most useful for designing relief systems to prevent high pressures due to runaway reactors. This is done using the procedures discussed in Chapter 10.

Calorimeter data are also useful for determining the following design elements of the process:

Reactor vessel size
Reactor vessel pressure rating
Type of reactor: batch, semi-batch, or tubular
Reactor temperature control and sequencing
Semi-batch reactor reactant feed rates
Heat exchanger duty to achieve required reactor cooling
Cooling water requirements and cooling water pump size
Condenser size in a reactor reflux system
Maximum concentrations of reactants to prevent overpressure in the reactor
Catalyst concentrations
Alarm/shutdown setpoints
Maximum fill factor for batch and semi-batch reactors
Solvent concentrations required to control reactor temperature
Operating procedures
Emergency procedures
Reactant storage vessel design and storage temperature
Relief effluent treatment systems

8-4 Controlling Reactive Hazards

If the flowchart shown in Figure 8-1 and discussed in Section 8-1 shows that reactive hazards are present in your plant or operation, then the methods shown in Table 8-9 are useful to control these hazards and prevent reactive chemicals incidents. Table 8-9 is only a partial list of methods—many more methods are available. The methods are classified as inherent, passive, active, or procedural. The methods at the top of the list are preferred because inherent and passive methods are always preferred over active and procedural methods. The procedural methods, even though they are low on the hierarchy, are essential for any effective reactive chemicals management system. These are discussed in detail in Johnson et al.⁷

⁷R. W. Johnson, S. W. Rudy, and S. D. Unwin. Essential Practices for Managing Chemical Reactivity Hazards (New York, NY: AICHE Center for Chemical Process Safety, 2003).

Table 8-9 Hierarchy of Methods to Improve Reactive Chemicals Safety

Inherent

Use a reaction pathway that uses less hazardous chemicals.
Use a reaction pathway that is less energetic, slower, or easier to control.
Use smaller inventories of reactive chemicals both in the process and in storage. Reduce pipe length and size to reduce inventory.
Eliminate or reduce inventories of reactive intermediates.
Use reactive chemicals at lower concentrations or temper reactions with a solvent.
Control reactor stoichiometry and charge mass so that in the event of a runaway reaction, the pressure rating of the vessel will not be exceeded and the relief devices will not open.
Reduce shipping of reactive chemicals; produce on-site by demand if possible.
Design equipment and/or procedures to prevent an incident in the event of a human error.
Reduce pipe sizes to reduce leak rate. Provide orifice plates or flow restrictors.
Apply principles of human error to the design and operation of the process. Simplify both the process design and operation. Use simpler processes and chemistry.

Passive

Ensure that incompatible chemicals are always separated.
Provide adequate separation distances between storage vessels, reactors, and other process equipment using reactive chemicals.
Provide passive engineering controls, such as dikes and containment, to control reactive chemical spills.

Passive

Provide passive fire protection for chemical reactors, storage vessels, and process equipment. This includes insulation of reactors and storage vessels, and thermal coating of all mechanical supports.
Ensure adequate separation of the plant from adjacent plants and local communities.

Active

Screen all chemicals for reactive chemical hazards.
Provide or have access to experimental calorimeter data for all reactive chemicals.
Provide properly designed control systems to control reactive chemicals in the process.
Provide properly designed heat transfer equipment to remove energy released by reactive chemistry.
Identify and characterize all possible reactions, including reactions or decompositions at higher temperatures, reactions induced by fire exposure, and reactions due to contamination.
Use quench, stop, or dump systems to quickly stop out-of-control reactive chemistry.
Provide active fire protection and emergency response to reduce the size of reactive chemical incidents.
Provide reliable mixing systems on all chemical reactors and sensors capable of identifying mixing failures.
Provide double-block and bleed systems and other systems to prevent backflow of reactor contents to storage vessels.
Provide properly designed relief systems to prevent high pressures in the process due to reactive chemistry.
Ensure that all reactive chemicals requiring inhibitors in storage, particularly monomers, are tested regularly to ensure that inhibitor concentrations are adequate.
Provide reliable cooling water systems to exothermic reactors.
Provide an explicit measure of cooling water flow to a reactor, with proper alarms and control system interlocks to provide safe control in the event of cooling water failure.
Use semi-batch reactors rather than batch. The flow rate of reactant to the semi-batch reactor provides a means to control the heat release rate from the exothermic reaction.
Use a continuous reactor since the chemical inventory in the reactor is smaller and the feed can always be stopped.
Prevent a “sleeping reactor” in a semi-batch reactor by properly measuring and controlling temperatures and reactant concentrations.

Procedural

Provide reactive chemical reviews of existing processes and new processes.
Document chemical reactivity risks and management decisions.
Communicate and train on chemical reactivity hazards.
Manage process changes that may involve reactive chemicals.
Review and audit the reactive chemicals program to ensure that it is operating properly.
Investigate chemical reactivity incidents.
Provide proper resource allocation for reactive chemicals programs.
Ensure management line responsibility for reactive chemicals.
Provide quality control procedures to ensure that all reactive chemicals received are the correct chemicals at the correct concentrations, without hazardous impurities.

Note: This is only a partial list of possibilities.

A chemical reactive hazards program requires a considerable amount of technology, experience, and management to prevent incidents in any facility handling reactive chemicals. This requires awareness and commitment from all employees, from upper management on down, and the resources necessary to make it all work.

Problems

8-1. Download the Chemical Reactivity Worksheet for Windows and Mac for free from here: http://response.restoration.noaa.gov/reactivityworksheet.

A laboratory contains the following chemicals: acetic anhydride, cumene hydroperoxide, methanol, and sulfuric acid (aqueous). Note that water must also be included since the sulfuric acid is aqueous. Use the CRW program to produce a chemical compatibility matrix for these chemicals and the reaction products. Only consider combinations of the reactants, and not the products, although in a real situation this must be also be considered.

Your homework submission must include a copy of the chemical compatibility chart, the predicted hazards report, the intrinsic hazards report, and your summary of the major reactive hazards and how they should be handled.

8-2. A calorimetric analysis of a reactive system reveals the following results:

Detected onset temperature: 350 K

Final temperature: 450 K

Assuming a reaction order of 1, a plot of ln[dT/dtTF−T] $[\frac{d T / d t}{T_{F} - T}]$ versus (T−To)/T $(T - T_{o}) / T$ produces a straight line with slope of 30 and intercept of –10.0. The R² fit for this straight line is 0.99.

What is the value of the dimensionless adiabatic temperature rise, B?
What is the value of the dimensionless parameter Γ?
What is the value of the rate coefficient at the detected onset temperature, k(To) $k (T_{o})$ ?
What is the activation energy, in kJ/mol, and the pre-exponential factor A, in s^–1?
At what temperature, in K, does the maximum self-heat rate occur?

8-3. A batch reactor contains 5000 kg of reacting liquid. A calorimetric study has shown that the maximum self-heat rate for the reaction is 20°C per minute. If the reactor is adiabatic and all the heat generated by the reaction is used to vaporize the liquid, then calculate:

The maximum energy generation rate, in kJ/s, for the liquid reaction mass.
The maximum vaporization rate, in kg/s.

The answer from part b would be useful for sizing a relief device.

Additional data that may be useful:
Heat capacity of liquid: 2 kJ/kg K
Heat of vaporization of liquid: 250 kJ/kg
Molecular weight of vapor, M : 80
Vapor acts as an ideal tri-atomic gas.

8-4. Equation 8-21 was derived by taking the derivative of Equation 8-19, setting this to zero and solving for xm, the maximum rate. Perform the same procedure on Equation 8-20, which applies for small values of B. Example 8-2 determines a value of xm using the exact Equation 8-21. Compute xm using the equation derived for this problem and compare to the more detailed result. What is the percentage error?

8-5. A chemical reaction has been characterized via a calorimeter and the following results were obtained:

Reaction order, n: 1.00
Detected onset temperature, To: 350 K
Final temperature, TF : 612 K
Activation energy, Ea: 29.1 kJ/mol
Pre-exponential, A: 2×10^–2 s^–1

What are the values of the dimensionless parameters B and Γ
What is the maximum temperature rate (K/s) and at what temperature does this occur (K)?
How many seconds after the detected onset temperature does the maximum temperature rate occur? (Hint: Use Figure 8-8.)

8-6. The following data are raw data from the ARSST. Assume a first-order reaction for this system. The heating rate for the calorimeter is 0.3°C/min.

Plot the temperature rate versus –1000/T to estimate the detected onset and final temperatures.
Use Equation 8-31 to determine the kinetic parameters A and Ea for the original raw data.

Time (min)	Temperature (°C)
0	12.0
5.1	15.0
10.1	17.1
15.2	19.3
20.2	21.4
25.3	23.4
30.3	25.3
35.4	25.7
40.4	26.6
45.5	27.6
50.6	28.9
55.6	30.4
60.7	32.4
65.7	34.8
70.7	37.6
75.8	41.0
80.8	45.0
85.9	50.0
90.9	57.1
95.1	67.1
96.16	71.1
97.25	77.1
98.25	85.6
99.2	100.7
99.53	110.3
99.63	114.5
99.75	120.2
99.87	127.0
99.93	131.2
100.01	138.4
100.06	142.5
100.13	149.3
100.21	158.0
100.3	165.5
100.39	170.1
100.48	172.6
101.2	171.9

8-7. Starting from Equation 8-20, show that the maximum self-heat rate for small values of B is given by

xn=1−nΓB $x_{n} = 1 - \frac{n}{Γ B}$

What happens for large Γ? At x_m = 0? At ΓB = n ?

8-8. A commonly used equation in calorimetry analysis to determine the temperature at the maximum rate is given by

Tmax=Ea2nRg[1+4nRgTFEa−−−−−−−−−−√−1](8-37) $\begin{matrix} T_{\max} = \frac{E_{a}}{2 n R_{g}} [\sqrt{1+ \frac{4 n R_{g} T_{F}}{E_{a}}} - 1] & (8-37) \end{matrix}$

Derive this equation from Equation 8-21. Use this equation to calculate T_max for the ethanol acetic anhydride system of Examples 8-3 and 8-4. Compare to the experimental value.

8-9. Suppose 3.60 g of vacuum-distilled acrylic acid is placed in a titanium test vessel in an ARC. Estimate the phi factor for this experiment.

Heat capacity of liquid acrylic acid: 2.27 J/g-K
Heat capacity of titanium: 0.502 J/g-K
Mass of titanium test vessel: 8.773 g

8-10. Your company is designing a reactor to run the methanol + acetic anhydride reaction. You have been assigned the task of designing the cooling system within the reactor. The reactor is designed to handle 10,000 kg of reacting mass. Assume a heat capacity for the liquid of 2.3 J/kg-K.

Using the results of Examples 8-3 and 8-4, estimate the heat load, in J/s, on the cooling coils at the maximum reaction rate. What additional capacity and other considerations are important to ensure that the cooling coils are sized properly to prevent a runaway reaction?
Using the results of part (a), estimate the cooling water flow required, in kg/s. The cooling water is available at 30°C with a maximum temperature limit of 50°C.

8-11. Your company proposes to use a 10 m³ batch reactor to react methanol + acetic anhydride in a 2:1 molar ratio. The reactor has cooling coils with a maximum cooling rate of 30 MJ/min. Since the cooling is limiting, the decision was made to operate the reactor in semi-batch mode. In this mode, the methanol is first added to the reactor, then heated to the desired temperature, and the limiting reactant, acetic anhydride, is added at a constant rate to control the heat release.

The following data are available:

Reaction: CH₃OH + (CH₃CO)₂O ® CH₃COOCH₃ + CH₃COOH Methanol + acetic anhydride ® methyl acetate + acetic acid
Specific gravity of final liquid mixture: 0.97
Heat of reaction: 67.8 kJ/mol
Additional data are also available in Examples 8-3 and 8-4.

The company wants to produce this product as quickly as possible. To achieve this, at what temperature should it operate the reactor?
Assuming a maximum reactor fill factor of 80%, calculate the total amount of methanol, in kg, initially charged to the reactor. How much acetic anhydride, in kg, will be added?
What is the maximum addition rate of acetic anhydride, in kg/min, to result in a reaction heat release rate equal to the cooling capacity of the cooling coils?
How long, in hours, will it take to do the addition?

Additional homework problems are available in the Pearson Instructor Resource Center.

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Table of Contents for
Chapter 8. Chemical Reactivity

Chapter 8. Chemical Reactivity

8-1 Background Understanding

8-2 Commitment, Awareness, and Identification of Reactive Chemical Hazards

8-3 Characterization of Reactive Chemical Hazards Using Calorimeters

Introduction to Reactive Hazards Calorimetry

Theoretical Analysis of Calorimeter Data

Estimation of Parameters from Calorimeter Data

Adjusting the Data for the Heat Capacity of the Sample Vessel

Heat of Reaction Data from Calorimeter Data

Using Pressure Data from the Calorimeter

Application of Calorimeter Data

8-4 Controlling Reactive Hazards

Suggested Reading

Problems

Table of Contents for Chapter 8. Chemical Reactivity

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Chapter 8. Chemical Reactivity

8-1 Background Understanding

8-2 Commitment, Awareness, and Identification of Reactive Chemical Hazards

8-3 Characterization of Reactive Chemical Hazards Using Calorimeters

Introduction to Reactive Hazards Calorimetry

Theoretical Analysis of Calorimeter Data

Estimation of Parameters from Calorimeter Data

Adjusting the Data for the Heat Capacity of the Sample Vessel

Heat of Reaction Data from Calorimeter Data

Using Pressure Data from the Calorimeter

Application of Calorimeter Data

8-4 Controlling Reactive Hazards

Suggested Reading

Problems

Table of Contents for
Chapter 8. Chemical Reactivity