# Accelerating Rate Calorimetry (ARC) in Process Safety: Heat-Wait-Search, Adiabatic Kinetics, TMRad & Phi-Factor Correction

Accelerating Rate Calorimetry (ARC) is the premier experimental technique for evaluating the self-heating kinetics, thermal stability, and runaway potential of reactive chemicals under true adiabatic conditions.

While DSC provides fast screening at high heating rates ($5\text{ }^\circ\text{C/min}$) and RC1 evaluates desired synthesis reactions, the ARC quantifies slow secondary thermal decomposition reactions occurring over hours or days under zero heat loss conditions ($Q_{loss} = 0$). ARC data directly provides the Time to Maximum Rate under adiabatic conditions ($\text{TMR}_{ad}$) and the $TD_{24}$ safety limit, which are mandatory inputs for process safety management (PSM) and storage stability protocols.

# 1. Operating Principles: The Heat-Wait-Search (HWS) Mode

An Accelerating Rate Calorimeter places a sample ($1 - 10\text{ g}$) inside a spherical heavy-walled metal bomb (titanium, Hastelloy, or stainless steel) suspended inside a nickel-plated copper jacket enclosure equipped with radiant heaters.

+---------------------------------------------------------------------------------------+
|                      ARC HEAT-WAIT-SEARCH (HWS) OPERATING CYCLE                       |
+---------------------------------------------------------------------------------------+
| 1. HEAT STEP: Raise sample temperature by ΔThws (typically 5.0 °C or 10.0 °C).       |
| 2. WAIT STEP: Allow sample and spherical bomb temperatures to equalize (15-20 min).  |
| 3. SEARCH STEP: Monitor self-heating rate (dT/dt) for 10-15 minutes.                  |
|    --> IF (dT/dt) < 0.02 °C/min: Exotherm NOT detected. Return to HEAT STEP.           |
|    --> IF (dT/dt) ≥ 0.02 °C/min: EXOTHERM DETECTED! Lock into ADIABATIC TRACKING MODE. |
+---------------------------------------------------------------------------------------+

# True Adiabatic Tracking Mode

When the self-heating rate exceeds the sensitivity threshold (typically $(dT/dt)_{threshold} = 0.02\text{ }^\circ\text{C/min}$), the instrument locks into adiabatic mode. Top, side, and bottom jacket heaters dynamically match the sample temperature in real-time ($T_{jacket}(t) = T_{sample}(t)$), enforcing zero heat flux through the bomb walls:

Under adiabatic tracking, the sample heats itself purely from its own chemical reaction energy until the reaction is exhausted or maximum temperature/pressure limits are reached.

# 2. Governing Equations & Thermal Inertia ($\Phi$-Factor) Mathematical Correction

In an ideal adiabatic industrial reactor (e.g., a $20,000\text{ L}$ insulated storage tank), all heat released by the chemical reaction goes into heating the liquid reaction mass ($m_s C_{p,s}$). However, in a benchtop ARC experiment, a portion of the heat released is absorbed by the heavy metal sample vessel ($m_b C_{p,b}$).

# A. The Thermal Inertia / Phi-Factor ($\Phi$) Formula

The ratio of total system thermal capacity to sample thermal capacity is defined as the Phi-Factor ($\Phi$):

where:

• $m_b$ and $C_{p,b}$ are the mass and specific heat capacity of the metal bomb (e.g., Titanium: $C_{p,b} = 0.52\text{ J/(g}\cdot\text{K)}$).
• $m_s$ and $C_{p,s}$ are the mass and specific heat capacity of the chemical sample.

Standard ARC spherical bombs typically yield $\Phi \approx 1.5 - 3.0$.

# B. Mathematical Correction Equations for Measured ARC Data

Because the metal bomb absorbs energy, measured self-heating rates $(dT/dt)_{meas}$ and measured adiabatic temperature rises $\Delta T_{ad,meas}$ are dampened compared to a full-scale plant vessel ($\Phi_{plant} \approx 1.0$).

To correct raw experimental data to true adiabatic plant conditions ($\Phi = 1.0$), the following mathematical transformations must be applied:

# 1. True Adiabatic Temperature Rise ($\Delta T_{ad,true}$):

# 2. True Adiabatic Self-Heating Rate ($(dT/dt)_{ad,true}$):

# 3. True Adiabatic Pressure Rise Rate ($(dP/dt)_{ad,true}$):

# 4. True Adiabatic Temperature Trajectory ($T_{true}$):

# 3. Derivation & Calculation of Time to Maximum Rate ($\text{TMR}_{ad}$)

The Time to Maximum Rate under adiabatic conditions ($\text{TMR}_{ad}$) is the remaining time required for an adiabatic runaway reaction to accelerate from a given starting temperature $T_0$ to its point of maximum heat release rate $T_{max}$.

# A. Mathematical Integral Derivation

By definition, $\text{TMR}_{ad}$ is calculated by integrating the reciprocal of the adiabatic self-heating rate $q(T) = (dT/dt)_{ad}$ from $T_0$ to $T_{max}$:

# B. Zero-Order / Low-Conversion Kinetic Approximation

Assuming zero-order kinetics ($1 - \alpha \approx 1$) at the early onset stage of decomposition, the Arrhenius self-heating rate is:

Substituting this rate expression into the $\text{TMR}_{ad}$ integral yields the classical Townsend-Tou formula:

where:

• $T_0$ is the absolute starting temperature ($\text{K}$).
• $(dT/dt)_{ad,0}$ is the true adiabatic self-heating rate at $T_0$ ($\text{K/s}$ or $\text{K/min}$).
• $E_a$ is the activation energy derived from the slope of $\ln\left[\left(\frac{dT}{dt}\right)_{ad}\right]$ versus $\frac{1}{T}$.

# 4. Pressure Trajectory & $(dP/dt)$ Hazard Characterization

In addition to temperature monitoring, pressure transducers connected to the ARC bomb log real-time pressure generation $P(t)$ and pressure rise rates $(dP/dt)$.

# Identifying Pressure Generation Sources

1. Vapor Pressure Acceleration: In tempered systems containing volatile solvents, pressure follows the exponential vapor pressure curve $P_{sat}(T)$. Pressure drops back to initial values upon cooling.
2. Non-Condensable Gas Generation: Decomposition of azides, diazonium salts, peroxides, or nitro compounds generates non-condensable gases ($\text{N}_2, \text{CO}_2, \text{O}_2$). Pressure increases rapidly and remains elevated after the vessel cools back to ambient.

# 5. Sample Analysis Illustration & Step-by-Step Result Derivation

To demonstrate the full numerical workflow of an ARC analysis, consider an experimental thermal stability test on an Organic Peroxide Intermediate Solution.

# A. Experimental Setup & Raw Readings

• Bomb Spec: Titanium spherical bomb ($m_b = 8.10\text{ g}$, $C_{pb} = 0.52\text{ J/(g}\cdot\text{K)}$).
• Sample Mass ($m_s$): $4.50\text{ g}$ solution ($C_{ps} = 2.10\text{ J/(g}\cdot\text{K)}$).
• Detected Onset Temperature ($T_{0,meas}$): $105.0\text{ }^\circ\text{C}$ ($378.15\text{ K}$) at $(dT/dt)_{meas} = 0.025\text{ }^\circ\text{C/min}$.
• Final Decomposition Temp ($T_{f,meas}$): $245.0\text{ }^\circ\text{C}$.

+---------------------------------------------------------------------------------------+
|                    ARC RAW TO PHI-CORRECTED DATA TRANSFORMATION                       |
+---------------------------------------------------------------------------------------+
| 1. Compute Phi Factor:  Φ = 1 + (8.10 g × 0.52) / (4.50 g × 2.10) = 1.446             |
| 2. True ΔTad:           ΔTad,true = 1.446 × (245.0 °C - 105.0 °C) = 202.4 K           |
| 3. True Onset Rate:     (dT/dt)true,onset = 1.446 × 0.025 °C/min = 0.03615 °C/min      |
| 4. Kinetic Fitting:     Slope d[ln(dT/dt)]/d(1/T) yields Ea = 115.4 kJ/mol           |
+---------------------------------------------------------------------------------------+

# B. Final Result Derivation & Safety Limits Table

# 6. Hazard Identification Parameters & Safety Thresholds

From the $\text{TMR}_{ad}$ vs. Temperature curve (often called the TD24 Plot), process safety engineers extract critical operational thresholds:

# 7. Where to Use ARC in Chemical Process Safety

1. Storage & Shipping Stability Assessment: Determine maximum ambient storage temperature ($SADT$ - Self-Accelerating Decomposition Temperature) for bulk storage tanks and intermediate bulk containers (IBCs).
2. Process Hold-Time Validation: Evaluate safety margins during planned or unplanned batch interruptions (e.g., overnight holds of active reaction masses).
3. Kinetic Model Parameterization: Supply activation energies ($E_a$), pre-exponential factors ($A$), and reaction orders ($n$) for advanced finite element thermal runaway simulations (e.g., AKTS, Netzsch Thermokinetics).